<%BANNER%>

Aeroelastic Analysis and Optimization of Membrane Micro Air Vehicle Wings

University of Florida Institutional Repository
Permanent Link: http://ufdc.ufl.edu/UFE0021970/00001

Material Information

Title: Aeroelastic Analysis and Optimization of Membrane Micro Air Vehicle Wings
Physical Description: 1 online resource (179 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: aeroelasticity, membrane
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Fixed-wing micro air vehicles are difficult to fly, due to their low Reynolds number, low aspect ratio nature: flow separation erodes wing efficiency, the wings are susceptible to rolling instabilities, wind gusts can be the same size as the flight speed, the range of stable center of gravity locations is very small, etc. Membrane aeroelasticity has been identified has a tenable method to alleviate these issues. These flexible wing structures are divided into two categories: load-alleviating or load-augmenting. This depends on the wing?s topology, defined by a combination of stiff laminate composite members overlaid with a membrane sheet, similar to the venation patterns of insect wings. A series of well-validated variable-fidelity static aeroelastic models are developed to analyze the working mechanisms (cambering, washout) of membrane wing aerodynamics in terms of loads, wing deformation, and flow structures, for a small set of wing topologies. Two aeroelastic optimization schemes are then discussed. For a given wing topology, a series of numerical designed experiments utilize tailoring of laminate orientation and membrane pre-tension. Further generality can be obtained with aeroelastic topology optimization: locating an optimal distribution of laminate shells and membrane skin throughout the wing. Both optimization schemes consider several design metrics, optimal compromise designs, and experimental validation of superiority over baseline designs.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Ifju, Peter.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0021970:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021970/00001

Material Information

Title: Aeroelastic Analysis and Optimization of Membrane Micro Air Vehicle Wings
Physical Description: 1 online resource (179 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: aeroelasticity, membrane
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Fixed-wing micro air vehicles are difficult to fly, due to their low Reynolds number, low aspect ratio nature: flow separation erodes wing efficiency, the wings are susceptible to rolling instabilities, wind gusts can be the same size as the flight speed, the range of stable center of gravity locations is very small, etc. Membrane aeroelasticity has been identified has a tenable method to alleviate these issues. These flexible wing structures are divided into two categories: load-alleviating or load-augmenting. This depends on the wing?s topology, defined by a combination of stiff laminate composite members overlaid with a membrane sheet, similar to the venation patterns of insect wings. A series of well-validated variable-fidelity static aeroelastic models are developed to analyze the working mechanisms (cambering, washout) of membrane wing aerodynamics in terms of loads, wing deformation, and flow structures, for a small set of wing topologies. Two aeroelastic optimization schemes are then discussed. For a given wing topology, a series of numerical designed experiments utilize tailoring of laminate orientation and membrane pre-tension. Further generality can be obtained with aeroelastic topology optimization: locating an optimal distribution of laminate shells and membrane skin throughout the wing. Both optimization schemes consider several design metrics, optimal compromise designs, and experimental validation of superiority over baseline designs.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Ifju, Peter.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0021970:00001


This item has the following downloads:


Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E20101111_AAAABW INGEST_TIME 2010-11-11T18:27:49Z PACKAGE UFE0021970_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 53644 DFID F20101111_AABJOA ORIGIN DEPOSITOR PATH stanford_b_Page_058.pro GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
2c6e40c4f0211101bbd82221c413c1e5
SHA-1
be2c71a7ea67bf700840d0b96fed148d3b8f0c70
57171 F20101111_AABJNL stanford_b_Page_034.pro
49058ffe9bffd33bf0b6ef9ee2d469d7
c51c28f5f8bc2af8e7d86aff5ff5c110e5752bee
59965 F20101111_AABJMX stanford_b_Page_013.pro
c54ce144836a755c8c3c5dc7a79997a7
be4656395a8ad69fad184de83035680c6a71dfc5
35204 F20101111_AABIKJ stanford_b_Page_052.pro
e7468e418782a669fe9c2f52feddba75
d7364dd2ba3ba14a5cfdafcb18d9393d11d1e4bf
25271604 F20101111_AABIJV stanford_b_Page_065.tif
0c757dc189b47b822b0432133121a94d
8f2ed767d2d9b04beee233dad3670d14ec0a22ea
47308 F20101111_AABJOB stanford_b_Page_063.pro
a460679bc214ebb33a5e2e0f3f66769e
7103827417b57c0a06863b4c092082983a003f4f
54496 F20101111_AABJNM stanford_b_Page_035.pro
181bebe363f9dca995593e3ab0191057
3e80037e3a643dd39cf8aae568a0c6ae71ffb975
48316 F20101111_AABJMY stanford_b_Page_014.pro
53f2094367c9ee1033ef6ccfb14fce20
338e97f1b1664422376da6e7d7d8133ca004197a
F20101111_AABIKK stanford_b_Page_133.tif
7184aad0f463c9b8a3ce28162b5fbd5e
e02fcd92337429357ddba7a40ea9abb6b3bfb178
1051979 F20101111_AABIJW stanford_b_Page_010.jp2
5fe5ce41f9ddd12bdc911c9c558a04fe
8bfbaa56286f4629c0d7d6e7d100530499c313b7
47780 F20101111_AABJOC stanford_b_Page_065.pro
d530f042235f9f13fd04969b078d29d8
a3879895c84adc0d8807572c13c9360b8bdb3067
53590 F20101111_AABJNN stanford_b_Page_037.pro
ee64c877ebed55d83082f6b5f9743a31
8db17abb628f631002ff8e7a8a1b0fb35d4268fe
44202 F20101111_AABJMZ stanford_b_Page_015.pro
2eb04615b889b1b7bb6cb213f7d8f994
bf3cef4f43cca85c8971ed1c5a99cc23a3cd9597
26870 F20101111_AABILA stanford_b_Page_138.QC.jpg
bb7f91e264894cce6b23bbf9f1ba9eca
4f921629811bedcc06bf91805e21e44dd808bc30
53737 F20101111_AABIKL stanford_b_Page_042.pro
780b10135387c1eb1bd40fda27501479
171d73008f8390afd250230eea46502d0056b55f
96749 F20101111_AABIJX stanford_b_Page_092.jpg
f90a867fe04fd40d7253aa6396a027f6
e1810b788af7dfbcd9429c0decd18d855aeb5286
42045 F20101111_AABJOD stanford_b_Page_066.pro
aae1fdfd1df682bcd5edfc87ffd3994a
90049b6a4fc372ae55cf8efb02c19796babd3ca2
28524 F20101111_AABJNO stanford_b_Page_038.pro
79e61229c973aee55cb6bfed5666d985
0cf1fb85851355129ea0ec5eb9410379dd6db0d7
83786 F20101111_AABILB stanford_b_Page_125.jpg
49a89835b91837eb43138dace1d38c51
c08fda4476b4423408bd08a522065ce818bda405
F20101111_AABIKM stanford_b_Page_001.tif
22664a0d40be1a519edcac791b4f3ff7
73d75dd0ec389084b04b92cdf2356ee2c2ef91b8
7229 F20101111_AABIJY stanford_b_Page_037thm.jpg
277f496d7293160bfeb50a9b621db395
7ae4ed20703a88b8ce4c9a3490985a8b47dd9294
41637 F20101111_AABJOE stanford_b_Page_068.pro
0ce9954ae3630f38122b85e51c4241a7
4ee60ed5862d363e0fc3ce1db8086bdbf05c5ce7
31018 F20101111_AABJNP stanford_b_Page_039.pro
ab1325f57645bfc465fa944ee9feeff5
49f031b02c68cbcd1ee10a6c75bcbcda126198fb
1557 F20101111_AABILC stanford_b_Page_052.txt
4514616b0595ae6f8b72b775853a03c0
aed615cca5d455d64f11d34b6b9decbba0572f10
41984 F20101111_AABIKN stanford_b_Page_070.pro
2897f71ce9a81fedda9b27d431d55313
cf949d86f8ea9af62d711d61dd5272ecb3e1918c
2190 F20101111_AABIJZ stanford_b_Page_163.txt
ebf2e503999683a30e521bce56215a5f
b0c6fc412302a827ee02dc32bca205663f1a71f0
32301 F20101111_AABJOF stanford_b_Page_069.pro
7fe66678a3bda20f704d471efb5ba60f
9f4a963acf8b4c3f92d896e523ba83820ba6ff77
55834 F20101111_AABJNQ stanford_b_Page_040.pro
671e8ab5ebaca9e8bb06cfa870e10225
94eefeb18a6e5328b6b71921ccc760f1bd737e9e
F20101111_AABILD stanford_b_Page_151.tif
c32fd826979f33f65a8f8d3d41d6d9ec
705d01203930779c0d15ca77538fe650c22ede09
1051985 F20101111_AABIKO stanford_b_Page_092.jp2
836c227cfde8b39d76bf76d82852fb34
3e5030c401234a0695196cf2a005b87c3dce2d57
53291 F20101111_AABJOG stanford_b_Page_072.pro
762172d93a8c521609bd4fc9dfe58742
a0773db82c472cdfe2800edd9276851b9d0469f6
54470 F20101111_AABJNR stanford_b_Page_041.pro
cfe9896410fa1dc71e61d8f34d2d4aae
4c7aac99a7fa0f530e98646ea202ff517034f070
2374 F20101111_AABILE stanford_b_Page_107.txt
c4b223be36e3f7a1023fa24e75b84acf
6987264a50d62891cf869714ea7a649a25fc8561
F20101111_AABIKP stanford_b_Page_134.tif
a2a8b5a2343f83ac194edf3700f0a495
a74a1e0e12a1ad9e23f9da15e05c4ed6a05f79d0
48879 F20101111_AABJOH stanford_b_Page_073.pro
173c0447a3950253ec1c761982b1bbd9
a219d61573e5cc2ef08b7160b61d0b67abc73a29
54879 F20101111_AABJNS stanford_b_Page_043.pro
2bc83a4af6db586837d1645b63e6f160
ace883c89b16165fda980ed13c1e17dfe0d529e5
86916 F20101111_AABILF stanford_b_Page_089.jpg
a02a30acfc112cf3f1d533154f86bd95
023874a13dc5b7b5509f1df030e86f0d11e3e670
85315 F20101111_AABIKQ stanford_b_Page_043.jpg
1f8e264a77bf6dbcd2db1f5a7be69c22
a4cc9d1cfb962dfb5cfc0f27ded40fba3d259983
51778 F20101111_AABJOI stanford_b_Page_075.pro
b4f001b7b22b2cb5b1841533264c6c51
cdedfab55498bb1259d134b5288f06afec5b3535
55125 F20101111_AABJNT stanford_b_Page_046.pro
172f85a4d59082f936dc84c2ce8008ba
27ffc2b74d3c5da28996d615884ec691e8f03655
71377 F20101111_AABILG stanford_b_Page_056.jpg
abb67a630e6ed5e9005b7d934f3876f1
935330249c5afb54ba025bec42eee3374e0183aa
56861 F20101111_AABIKR stanford_b_Page_136.pro
6870e148f5987fd0192bce303578ecef
48ab9b86d9f1a7d1be1564925757029e200f2bf6
49585 F20101111_AABJOJ stanford_b_Page_078.pro
8e82318ae9c44334735327801a9092ff
d0cb0637c0d30b5e999e75c5f05f655e93351ee9
53462 F20101111_AABJNU stanford_b_Page_051.pro
56c355c150bdb8a90e31ecc3b6cc50b7
c9452f63f30279d2d942aec91f77f11273718027
F20101111_AABILH stanford_b_Page_031.tif
72221f2b491fa23ac884dca443b627e3
4e310b47dd7fabb41d987c3839c91d45d4710895
7377 F20101111_AABIKS stanford_b_Page_060thm.jpg
2d5a4e0e14a65a8cbbec823ebbeb95ba
5815f27d18c44dcf29dac753860201da4849d20a
48724 F20101111_AABJNV stanford_b_Page_053.pro
9b676a7a0329cf7498e199390f7e6f62
a9569a5434dc82cc6b7360e93a0c5d29f29de540
F20101111_AABIKT stanford_b_Page_084.tif
3e333012f0a20e54e3c686acb29f64a2
ae7e2848de8810b350cd1cefe9994892f2615fa4
42803 F20101111_AABJOK stanford_b_Page_079.pro
a39a2153f05f7cec4aad48e721f9333c
05e90032c8dfbdb2eba6b1cd882b45b3b120786c
39812 F20101111_AABJNW stanford_b_Page_054.pro
43a635d948a1a5e1f70f983ad8c3d8d6
57021c2aabbc2042be48fe213327fe11f3f69158
7309 F20101111_AABILI stanford_b_Page_016thm.jpg
7bcc5d7ae242e9f1d314703707854fd2
1cb8c79d44ff00c88fe5cfe69676164c04dd9578
6792 F20101111_AABIKU stanford_b_Page_068thm.jpg
11d3df571df5f4d20880e9de735d6d0a
3ac53df2b255563b6506d7e6c85b280d8d2f7743
55706 F20101111_AABJPA stanford_b_Page_108.pro
bf45b11cafd873d1be73484b1785a080
f5fcee1318b781e1965114e496ec1078ac60b7ee
32678 F20101111_AABJOL stanford_b_Page_080.pro
43b82fd026f244f10b7da5f42e89b7ed
828a8acbec682567a43fa20f57f4096f7a7099d8
56631 F20101111_AABJNX stanford_b_Page_055.pro
e103ebdf30f28a5699945a9e0fc251ef
8c5b8de455feceafce542e3f60c4039cdc0a50bf
2187 F20101111_AABILJ stanford_b_Page_029.txt
db2887ab0b7ff95765969b443344eeed
a64a41a3d9423fd0611d2607bed46a29da4b7978
85059 F20101111_AABIKV stanford_b_Page_018.jpg
43cc52244a879d9f80bf9bc2a20608a0
828061b2d045e9246646c86b633dfea750b2bee1
42501 F20101111_AABJPB stanford_b_Page_111.pro
f076da896541444ee06b7c24c2c19bf4
7c3c1cd120f9ba1254839c9c5483c97f5008665f
54881 F20101111_AABJOM stanford_b_Page_081.pro
edff30e9a61c9f4a5991de240c49d3aa
bf1507b7b1cce7c4f138f55f680e0dc8ebeea27c
37240 F20101111_AABJNY stanford_b_Page_056.pro
cd8960b0fdec5e7710a53652e780da0b
30e00db3a1526d1196766498a091b0b903f12b65
2200 F20101111_AABILK stanford_b_Page_103.txt
52f8ea411709c45d54a38241af445508
7725c8ba8e2183e2d73227748a99c4866cb7faca
74801 F20101111_AABIKW stanford_b_Page_179.jpg
f59d8fccfd3a581d58eabf60fd3797c0
8f56ff523c6c41272e9fb066d1730dc5ec2b8092
53931 F20101111_AABJPC stanford_b_Page_112.pro
d45b7df88d6b7678114cb40074743e3a
b44a52ed809ebea3babfff5a09ec1fb12b721a5c
56505 F20101111_AABJON stanford_b_Page_083.pro
f6f7be1e4038bcbe8b52c42f5751ffce
b07ef63cea8ba1baad9261c45cfdc408f18679fc
53345 F20101111_AABJNZ stanford_b_Page_057.pro
31011c8995207f3339d1ab43d00e7976
166b9219d17f625f70c2aad76d5a27d76e841451
28216 F20101111_AABILL stanford_b_Page_033.QC.jpg
596f857bfa08d929045e04df56b523e4
b6233f95085d24df2fc9f107476055637bd4f726
F20101111_AABIKX stanford_b_Page_096.jp2
6a9d95cb5b52d29212adfd29cc8966c4
b31d456fc30e73c07724bcdd92be4e08d7608ca0
81110 F20101111_AABIMA stanford_b_Page_138.jpg
2db0de62881ace44fe1c4d254e275e4e
8ff08f9623c41279c8b25a6998415e165f18393e
52008 F20101111_AABJPD stanford_b_Page_113.pro
356f8fe7bd223296280dac8b6f7eb490
0b9aa38fd9cffdfc5bb53a38e62461fe13b5a7a2
54074 F20101111_AABJOO stanford_b_Page_085.pro
102c725ba27736f7c7d141ca32d2977b
790c00ac28ffb8df7e916b61f107af351fac9749
88913 F20101111_AABILM stanford_b_Page_019.jpg
d49756817416305d03da133a1e44d54e
f61726ec51e7c46ff7501424b856fa343cc465c9
787486 F20101111_AABIKY stanford_b_Page_131.jp2
19085c55585d2ab61d2a9819ad5953ed
c2de109f3566e16cae84af7e7bad162185fc74e7
2173 F20101111_AABIMB stanford_b_Page_165.txt
e555a86262f353dce52b07a6aaaf4e4a
94bfd5c178d284daff4f54eb91bface587664c4c
70666 F20101111_AABJPE stanford_b_Page_115.pro
19ea140756d6ae1ccceb69c03f8c620f
030529112415dd72efc66939cc723b94d641bf98
40055 F20101111_AABJOP stanford_b_Page_087.pro
cdfc9367d64b7ff9241d60e4156aafb3
35d11a42f0240d0b4d5d8b8f56e2e4bf451fc180
F20101111_AABILN stanford_b_Page_089.tif
92741064fb6fdba4b44d690174728fe8
01ebd5f0f89bd5672f6f2004f5be766eea2ea410
55531 F20101111_AABIKZ stanford_b_Page_134.pro
8fae97e57a22bace74ebf2155ca1a2b4
370bf51118bd46ce9451dcb4a97ce00098d8b78d
7490 F20101111_AABIMC stanford_b_Page_036thm.jpg
1b1ff3a31f1ed3f196bd59cd979285d9
004a098112f0af89592936367e96feb0a54fedc3
56151 F20101111_AABJPF stanford_b_Page_116.pro
ba26914170da44fc45246d9c95a49612
b1636896c418e2981f9843b7b9e740e219250efc
55683 F20101111_AABJOQ stanford_b_Page_089.pro
cb3609edb47ae5291687f1ab64cc7290
97203d5c066beec247a153cd4bcd903be82bb26a
27583 F20101111_AABILO stanford_b_Page_134.QC.jpg
8019692ae7125d124e1f7935cfb66d65
36bbccf94f771268801cbacd4e4e41b5b7d14050
1949 F20101111_AABIMD stanford_b_Page_091.txt
a333d560a1de0884404acedf06281220
20f8e317357069b68e95e8aa5a2e57c104a9d08f
43139 F20101111_AABJPG stanford_b_Page_118.pro
2db07fd7f01b1c3275ddc0927375ca4c
07da74ab4994efb6bff765981c813941e9a7f4d3
45502 F20101111_AABJOR stanford_b_Page_092.pro
84903becffe65b6f65e89fd51d0e9c6c
8e2a8c6f220d87bb097ef27467b6d211eb91f5f8
5935 F20101111_AABILP stanford_b_Page_003.pro
a29d8de6f9d68c944fbdba2eb5a14322
8f64b908a5d12ea0214dbc35f3a418a17e7cc19f
6832 F20101111_AABIME stanford_b_Page_103thm.jpg
d9bfb63acb603deae33c1c910012defd
e5a0787c69d5609c64d7bf3a069f847dc2b81452
44786 F20101111_AABJPH stanford_b_Page_124.pro
b6c4f6233ec19e3b6c9c2e181827696e
22bc95fdc047a1a412c4c300049303bbf30aad96
56069 F20101111_AABJOS stanford_b_Page_093.pro
0dcea40fb69fc07cdad413dd28081225
eb7fdbf597392752df4d9ab832b085a02966bc1b
1292 F20101111_AABILQ stanford_b_Page_074.txt
e4a66a0a5c5a7e2516740161691cc398
36ffb60cf7170887e2e5c07e119252eee12850ca
2142 F20101111_AABIMF stanford_b_Page_075.txt
cb6d8b6eb4d6cee494eb3e7fa39f2a3d
8e41d55808cc59a26909edfcfe47264f6f85a03b
53446 F20101111_AABJPI stanford_b_Page_125.pro
98a078cd90ff31509f3b34d62f7bed7e
c4e30499f0e623d92126265c631121d21e947e6d
54076 F20101111_AABJOT stanford_b_Page_095.pro
3db27840114d1abad85ec6772edc052c
4cdd2c3230d6a532c2228581c36d1446f9b9decc
8639 F20101111_AABILR stanford_b_Page_154thm.jpg
817ef5c4f3af8950d6be0a9d5bd86041
9b0b4d6563ea83dbd3565eab7726a340942344e6
F20101111_AABIMG stanford_b_Page_130.tif
18316ce7e64856c747b09fa3168a5cbb
9465d126d173a7d78344e9862eedd726cf9b52b8
52899 F20101111_AABJPJ stanford_b_Page_126.pro
a441490e0d80f0dc55d6da2b508756bc
ae0d5544d2b148b34f007588a4f316df8dc15ef0
31828 F20101111_AABJOU stanford_b_Page_096.pro
63c96bc4eda069a6e136949811c183c4
2d8de6c2818f757f6b2265ebbf1bdf5427e83ad1
41713 F20101111_AABILS stanford_b_Page_135.pro
7b44ef495fcb9dff0be551b8db956a8c
f88d3d166a57ed3fce4c4d22848efd7f2dd4b8ee
24192 F20101111_AABIMH stanford_b_Page_052.QC.jpg
d54dfa559b8ef263932541c4df66c0c3
eb7afff05b67131ac76a7d611d6e453c751df3f0
42753 F20101111_AABJPK stanford_b_Page_127.pro
51cd2587a25bf943384d915f22663910
f38a17932c2d3fa4266e69face1001ca76f5c6d7
52366 F20101111_AABJOV stanford_b_Page_098.pro
7b2626fbc739ad2ef8bc1b9421c28bb2
784b9d21fccf407090ab37ff3a8eb63ee6539bc7
7256 F20101111_AABILT stanford_b_Page_088thm.jpg
c88e27439125f6be79dcccffde213c27
ac37ac7724c22f7f5eaaa40d23ac5e9cb903f0fb
F20101111_AABIMI stanford_b_Page_061.tif
472d78e9c5e1797c9ba3b309464edda5
673b3d019e892097223deafbdfd3e5492bff9619
37677 F20101111_AABJOW stanford_b_Page_100.pro
926848a75f23c2a38ea0601c8863b82f
7b6c7df692bf96e3f1a8ee97311d50fe6518d1d7
7423 F20101111_AABILU stanford_b_Page_126thm.jpg
b562a9f45c84d5eb56b39ad782e0f54c
8808ead15b0cde7d6bcb02f373572608881e1d3d
67034 F20101111_AABJQA stanford_b_Page_167.pro
1b9e4c165b022c41898a46cf39840b01
2537c705b8b65b8b2c7eeb2172725b152afebee1
49457 F20101111_AABJPL stanford_b_Page_129.pro
7c36597ccfcf2cfedf724ac7c2f8eda7
3b29c9421cd911c26ae90b965aa88b8253278a14
50605 F20101111_AABJOX stanford_b_Page_101.pro
640f238d5879e0c6d4ac5822190a79fc
3bccc527cebe244c0704519bdb6af8032084cdb0
F20101111_AABILV stanford_b_Page_120.tif
04404a2246426434f0e24d7029d0a94e
dd08071fe94b7bc9787ce643dd81ed4ecfbf975c
F20101111_AABIMJ stanford_b_Page_026.tif
3bf478461eeb73cc19d2e2788d510768
b41d607f0d83ed7a10e2bf705aec38c521c02197
59395 F20101111_AABJQB stanford_b_Page_168.pro
431ceb24aa4a42207170986140c1eafc
d0a59585be2f7e38867e37d40786e5a455e76be3
39783 F20101111_AABJPM stanford_b_Page_130.pro
fefe3fde9bcd34f7789df675cf634360
5236bd036265beba7ac98222dc04e93b9ace5e73
35149 F20101111_AABJOY stanford_b_Page_102.pro
5b02324b2be2717e996531339f5d226d
9c3222b0601a5a2c058c16d81b2a2ce2bdac3187
F20101111_AABILW stanford_b_Page_109.tif
438cc2a9a603a7e370df3e1c4d2d0c73
f085a323686dc88f8facd33a1eaf8770839c0655
45943 F20101111_AABIMK stanford_b_Page_179.pro
1e5c923010ca5b3f2fae84cec793d0fa
4bcdf6d1452e08c9b596755a31f555c98eb3eca3
67099 F20101111_AABJQC stanford_b_Page_170.pro
1a0d8a5fdb4b6c97de111b79f6cef836
816c4f087112bc4b9989ae5d1f2398e1be43a558
37751 F20101111_AABJPN stanford_b_Page_132.pro
2dfc8c0980194c579fb845b0b09dff3a
7132b4331e20abec089bae145867a62e37df3e63
53975 F20101111_AABJOZ stanford_b_Page_105.pro
a67dd534b84b752068101affe761f551
e9c5323381931c26300a4ac390c56358a7190274
46658 F20101111_AABILX stanford_b_Page_137.pro
e24df0f10c14a4bb1d0d51e93f3f316c
0715552075000a91dd124d2647ee50714cc98a62
F20101111_AABINA stanford_b_Page_067.tif
fd200e6b66c97b4a5e3634e3b2f2563d
b9d1625784f9cc4458aa6c7b8654632890074e24
23690 F20101111_AABIML stanford_b_Page_068.QC.jpg
6ff158a8ad547a2bce6b36154b7a8122
558cbc1bfff74908af4d0281af9105d1c7c5341b
66152 F20101111_AABJQD stanford_b_Page_171.pro
a01aa04fc6eb73f8a42a9a62856bf078
df0e7e41c3aa77e27aee9ca76acb7d072b0f8754
50638 F20101111_AABJPO stanford_b_Page_140.pro
a90055e7918aeb4ec6bdfc631b425d3e
c2a13c3a19f185ce7a922fd8e2447c705dbb2d0a
888102 F20101111_AABILY stanford_b_Page_087.jp2
b484b0750c793d21556d5ad798a8fc14
97edbbf76db0f47aba4fe376cd582a5b1074620d
F20101111_AABINB stanford_b_Page_038.tif
ac9bcf42d6f1995526cff763c5e73aa1
49a350ee700e1431c5caf060b19df8b11bb6d0a1
26546 F20101111_AABIMM stanford_b_Page_123.QC.jpg
245bbd70ccf7ae735fffd772904662b1
3f38afef422879ca3cb13b3d1dd83c420d0f3b4d
68840 F20101111_AABJQE stanford_b_Page_172.pro
3378b75a2c19b485bc5da80b9fd7d451
5c81a341f5fba2467cbb8bdfe0472c2ad9f6a112
40503 F20101111_AABJPP stanford_b_Page_141.pro
2c1e15c9bf7250f39911a71c73b4b51c
a3e0a9de266fc9fd18cbd6ef45b12066b1faf279
45655 F20101111_AABILZ stanford_b_Page_059.pro
9c32f16fb488e243755b1c84c9339f00
7151d016952f715f36cc6237ea26b7d3317af708
5425 F20101111_AABINC stanford_b_Page_005thm.jpg
cbb77e62fa86f9942866bdb7f2c954b8
66edb91fd702c112592dd3975b0bed8fa6aeea97
45872 F20101111_AABIMN stanford_b_Page_094.pro
27d01fd2e746953c0a45dcfd88303a5a
b2d0d83af00ad01d98364d08d2c67673147ac2dc
66473 F20101111_AABJQF stanford_b_Page_173.pro
a8c3724207136f40cd2882bf5a770a5a
a93cb54cd41cc00e6fb68495120059c793e96e91
42406 F20101111_AABJPQ stanford_b_Page_142.pro
5643144e0fa2e225e391336183195e78
b6079cdb831edb37b849f8d64d49d5b460f9e696
95 F20101111_AABIND stanford_b_Page_002.txt
6cdebe28a68e4cf7737e8427a1bef8fe
bdc9bc866a4b914bc8348bba1fd6897f919cebf3
1051977 F20101111_AABIMO stanford_b_Page_049.jp2
01b9366c0708da2070e9ae99b533de9b
c8b0855f79891ee10c4f3dc3e08a1078977bb0fd
66180 F20101111_AABJQG stanford_b_Page_174.pro
544e49d85be2d06f7856c1fa5367c2fa
241914dfd7e2c018c7d7b959fae2c490070c8d64
38271 F20101111_AABJPR stanford_b_Page_145.pro
88f2914b48a33e3d5092f69a11e11b1a
a76cc4599a3b18eb081ecdc015dc568478f9ba5c
1850 F20101111_AABINE stanford_b_Page_154.txt
204332d0357efc90f9003fecc69cfce3
aaf300eda096c39828ec465eb42912bff365534d
2655 F20101111_AABIMP stanford_b_Page_169.txt
ec2f99624af87a3f7ea067991a18b549
1ab502f0eaf1318dc37692a626a468841444cbc7
63579 F20101111_AABJQH stanford_b_Page_177.pro
a47b9626e832b9d68764821c35a67b7f
06beec03a79792377da4eca4cf6e37cfaaa33856
40004 F20101111_AABJPS stanford_b_Page_146.pro
adbeb55a0c6781f1f5e25406082f4faf
74489f8a6f5e000e4e768a98c894648e896d1997
30259 F20101111_AABINF stanford_b_Page_175.QC.jpg
7501a4b74e04e26fbcc6255920eefad6
fdf82f8ab08845919cd6bfc29d779a50473006b2
98284 F20101111_AABIMQ stanford_b_Page_006.pro
b3c5b3b30a12f6adbff5e689bebb0653
d9cb2f24a6b24510ae1cfa8c0fb7b76b6de5ee95
27347 F20101111_AABJQI stanford_b_Page_178.pro
cfbebd968a24b76bba1934677095becb
9bbe3cdd2f83781393f80be9a0439fdc20d463b7
53538 F20101111_AABJPT stanford_b_Page_147.pro
ffba518bc5026cf1e01f0675d888bbe6
9aa15e0c77aa46c59b5d9ece202e9a95025604d0
F20101111_AABING stanford_b_Page_177.tif
c94cf030136613e6b9dcb0c7f7be2244
9f0a3348e9facf1a3ffd898b9430c78f823b3d51
7653 F20101111_AABIMR stanford_b_Page_043thm.jpg
d98bf9d2c87b57ec7ff8f67fa4f6770a
7fba3d3d381a32d46b437fa2e614e92bf6a9e9d0
479 F20101111_AABJQJ stanford_b_Page_001.txt
19a93ba903020b8a2e3d2caa6d676eb0
54fe52f4aaa54cc19d3d30184936bfa343c5e676
39555 F20101111_AABJPU stanford_b_Page_149.pro
ff81d8fb33087c8f6f75e3a18077ba21
c8c5126735a3424a0e8a65141529b341b9448541
1008757 F20101111_AABINH stanford_b_Page_071.jp2
34cd2c23ac29d98fd088d016a4d19f8d
54139b31923333e8d8fbcb2686f94b616b506aa3
1454 F20101111_AABIMS stanford_b_Page_123.txt
9f2ff89f0dd27db988fb09ac266a0cc8
fb598e62e2b1849aed831d3c7aee8d274482dbdd
294 F20101111_AABJQK stanford_b_Page_003.txt
0b2e67e2bb46a2b8f3977a1132ae1b2b
2031f15e93e756d98cf7a1a7f5e28e75e6aeaaa3
55967 F20101111_AABJPV stanford_b_Page_150.pro
c6e8fd3db05923cb29e13d16b48b31ef
1e91ce4fc86b3fe97068e4282c4691e4d74bfe14
19790 F20101111_AABINI stanford_b_Page_069.QC.jpg
0114a636e9e3fd35a48cd21c2965c827
a729b8b282bac7478f42f078a3d78d2cbf046370
8446 F20101111_AABIMT stanford_b_Page_159thm.jpg
92d047fb37a7ead2b676dc48b0a320f9
a36aab330cea5076d51f74615cb88b779c2c0b17
4034 F20101111_AABJQL stanford_b_Page_006.txt
a555a79a8068848710213618f4253d71
4caeffa03c109128e194946ed1b17d8b1209a001
42564 F20101111_AABJPW stanford_b_Page_154.pro
efe178130bc9dedd879b39c5788c71e5
61da43900fc0372e2b409157d13cfd6aec78ff4c
6463 F20101111_AABINJ stanford_b_Page_156thm.jpg
0af39729ee183fffbb41a37fdd1e604a
f804e3c1f1b837cf02d4e86ff4ef0b01e1f8b9c6
1051978 F20101111_AABIMU stanford_b_Page_094.jp2
4a79b79d38469ecf80724aa93e81a102
d3f46b68fbc8e9e66367c301ac3154e64e7c77c7
2177 F20101111_AABJRA stanford_b_Page_027.txt
6f97ff44183fc689eba30423f14195bc
a70bb21ce1e64d8fc228335ad675373c07e9f3a9
40975 F20101111_AABJPX stanford_b_Page_155.pro
be2046411d986d6e497b792ee2dcc1e5
a31747b9f0e69af3c57027b351e78cfb363ed895
F20101111_AABIMV stanford_b_Page_118.tif
30ef27bc9a3eeb8f5c52a4d9948a5e00
1cadb81ab18f9721b89c292cf0cd7d639b78221b
2260 F20101111_AABJRB stanford_b_Page_028.txt
d50b133a95a0e4374b296738973a1bc5
9d846ec4003af745d1471de5defdf1465fe51bed
843 F20101111_AABJQM stanford_b_Page_007.txt
f142d621e31714e639e2e99a108c9100
c9d0fdef06e9bed5f9fd8f82e121fb0f1a2d32e3
55070 F20101111_AABJPY stanford_b_Page_162.pro
f8b298fd72ba547e1cefa3906eb21056
bf501e8de6dc95eeafd2ce7a995aa34f95c6e5ec
16965 F20101111_AABINK stanford_b_Page_003.jpg
edac9dc73f6971bfe2134cc2977c856e
971098bf6e8bc25e5210845cafa4fa44332ddd65
29660 F20101111_AABIMW stanford_b_Page_120.pro
dd629c9f72723db3ac76405926bfe1e4
bb42f8838dacba986e222cf7f8f1f1b460c815b1
2202 F20101111_AABJRC stanford_b_Page_030.txt
658e888eb7c669edf1ae1251a0db0857
48a43cedc361b7348d6bc7ad396ca09d6bae9902
2990 F20101111_AABJQN stanford_b_Page_008.txt
ae4e5edacbb3891da4448431d967edec
db5543d8aade245a02e75d8c382cf54becd93a06
55195 F20101111_AABJPZ stanford_b_Page_164.pro
e3e6116f2e5273d23d4976dbe40664d9
cd6e7ba712b3138ae33087a8caadd6db4d49c8fc
82963 F20101111_AABIOA stanford_b_Page_147.jpg
745f2a1fab421d565a4a3cba83a88081
3fb8b96c999b0d7110c75c2a92416382be3440b0
73293 F20101111_AABINL stanford_b_Page_119.jpg
493ac59e16c7436507b469816fcf7888
15eeb2a4b2234509ea64aa377e58abdbc6d81c51
F20101111_AABIMX stanford_b_Page_175.jp2
fd10d1f490a2d27ce2b9d3b7066638e7
79f1bbc19973ab126bff7ff389f49a93967d3aad
2130 F20101111_AABJRD stanford_b_Page_032.txt
656fa31d229eaec45f42c6e583307ece
33b972100a5cf2c17364cfe3ee45b48f2d937b8b
2860 F20101111_AABJQO stanford_b_Page_009.txt
af5d1696c3fdcc8e9ef1e839195d9ce8
8bd2ee00b437316b66db6c07e6702d35e7888ca9
2240 F20101111_AABIOB stanford_b_Page_161.txt
546e67da87b3827fd07ab48f5df4d343
1ccee39a8844f9c98ff174ab73d9392845e24594
23769 F20101111_AABINM stanford_b_Page_118.QC.jpg
b80efc9ec997479d4179051557deb564
923d26f97de6ef9fd802dc8204c78c821896e816
2861 F20101111_AABIMY stanford_b_Page_007thm.jpg
8c819e470c58e025316516209690f9aa
780f1ad5a3480c6adc3117ca2e2177c67ecc5780
2272 F20101111_AABJRE stanford_b_Page_034.txt
48e318725661e438ce8b9550f5316da6
6a72e4a05c98ea7791b42dd8b617d97d6a0f2f0e
3085 F20101111_AABJQP stanford_b_Page_010.txt
8333751311d0599dfe7e97071ed6df5c
8cfe4ff966e371e90bd004708b3d99af322ed7ec
2166 F20101111_AABIOC stanford_b_Page_164.txt
39311a97caf74e0440b4e3f0f021171b
08b30343cec9430747fabc490980a81d221fe2ee
1051970 F20101111_AABINN stanford_b_Page_169.jp2
d8b33c6f17bcaaed3c98654630ba4930
06e10d58fa80fe49db352e307a7168532671772d
30224 F20101111_AABIMZ stanford_b_Page_176.QC.jpg
ec0ab8352167257f01bf4700925914b8
fb323e6351343b70ba14778a6633d7afb5bd086a
2110 F20101111_AABJRF stanford_b_Page_037.txt
954df18f15c7ebc0163741b7531c5b6a
7ad23ffb2c372ae005cb8cf33fc61f12ea3fab73
2102 F20101111_AABJQQ stanford_b_Page_012.txt
b85d1f59310d8771325dd2121f428042
80134b46c6a8c35ffaf565d88eb57987d88ca59c
107529 F20101111_AABIOD stanford_b_Page_115.jpg
7a288b68633a2a8a424f0eb796756508
335d4903251d0c432935e8905a0151e4256b5852
1483 F20101111_AABINO stanford_b_Page_121.txt
9dc81b2e2677d70eb997082c2fe90943
49d77c3a2ffa9bde5f51a80b9cc2df4043d6628c
1583 F20101111_AABJRG stanford_b_Page_039.txt
cefff89cb93d268fb57977cff5c8f7f3
71a0adc14c8a3b223e5376fcebdd03bee23af6a9
2470 F20101111_AABJQR stanford_b_Page_013.txt
927abb242e34afeefe014b158aef19d5
f70e76576515b572d710c26750f3d8824afb54c8
F20101111_AABIOE stanford_b_Page_122.tif
494c131ed5590a598c97cd44518ebf38
930e1d086ad13821e8174bf766c6676ea89203f0
7906 F20101111_AABINP stanford_b_Page_177thm.jpg
9a6fcc4ef3b5039e1c644cbc1b8f3df8
971e82133910a41fa040134747b8e1f2fa9e0cd3
2161 F20101111_AABJRH stanford_b_Page_043.txt
939e3ac20293897c261ac0194041a226
1695ca72ec6a3941d4c834105b960fbe07728b8a
1920 F20101111_AABJQS stanford_b_Page_014.txt
c9675781a2220b2dc5b07530cec782eb
7666b0a10bbed9510d63f80c06930e3ba6ca1bad
2094 F20101111_AABIOF stanford_b_Page_098.txt
89321d9ff4cac5987fdddddd9275b915
1a67b53a5393c18f771a29203597a6f718e319c7
F20101111_AABINQ stanford_b_Page_135.tif
2fe7826fd834e1a55b58426166f4c193
87d415acd4ae0b5f0fdcb151ef38d27d50a504b5
2163 F20101111_AABJRI stanford_b_Page_046.txt
a05e20a7022436c69e775ca211eaa0e2
1b0725ebae9d9320445ee3490223d8fbcd8f6d34
1770 F20101111_AABJQT stanford_b_Page_015.txt
18b028ab54a75b314bacd8ff56e4d00c
514c04969e726f9a39f9bd94a9278c5e2db359c1
F20101111_AABIOG stanford_b_Page_127.tif
ee1fe2471e9e90b946ddc39ebf5a8341
c01c0b961e2dcccce5f339d5c0fdc52739cfbe5c
2268 F20101111_AABINR stanford_b_Page_062.txt
bf0d35f07e02947f9559a80f8a7af75d
f047345779fd9dd4c2a8fc736279e8bc824fe006
1829 F20101111_AABJRJ stanford_b_Page_048.txt
05f93400326ace50c5fe3189c756683d
618ef645a0e49751dea3905790746cb4ec8e8610
2143 F20101111_AABJQU stanford_b_Page_016.txt
5700f8cf112bed12f9180f269eda1089
011dc86fa5680b2501e906bf03c664ce650b6ee2
1562 F20101111_AABIOH stanford_b_Page_151.txt
2096723b1f9f3ec05cf1ab8e5b7f5335
384ea3726652ebec7ea25e69c6a40438c38a3487
2227 F20101111_AABINS stanford_b_Page_131.txt
27dd6d30fde267f771cfe0bf699d0bed
935b826c548e8dcb61afd332de1720e30bef2b71
2160 F20101111_AABJRK stanford_b_Page_058.txt
9573f178a6ac40caaeaad7d57663b1c8
a640effac805a78a1d920b631a65d82e45fcde13
2231 F20101111_AABJQV stanford_b_Page_019.txt
1d9dd29f9341c416c71149d9760a6b73
2422f7a0361d63eecadcfade19ba1dfcf587b181
F20101111_AABIOI stanford_b_Page_170.tif
34fee07b4f30d1159094d9cd930f3b80
aab4ae9097b09feb46c6b6b7ba9c63c442f91db9
7261 F20101111_AABINT stanford_b_Page_112thm.jpg
2c69d2c3541dc24743143765bae38b25
be982359a981f41f8c3c30af9a6c18490dfdb754
2073 F20101111_AABJRL stanford_b_Page_059.txt
563316314f2a584f505de31d1bea5617
9d85449b039ad6661cbda572513f13fbe1e3afdd
1043 F20101111_AABJQW stanford_b_Page_021.txt
439c6604c38cd054c9d907b3ebb97b00
0e73fa5a76012a9694b774dc15959ae0cd52cea5
25525 F20101111_AABIOJ stanford_b_Page_054.QC.jpg
2113faab1f3feaaf4b19e48ed20f67d0
a84fe14e72f02997e11acce07f53fc735135415f
29065 F20101111_AABINU stanford_b_Page_139.QC.jpg
de41ad3b0e5d53249f0414af7a4fea9c
3992da2e828055aa4c9bae987b8d9cc3773ec7be
2487 F20101111_AABJSA stanford_b_Page_086.txt
d6366033959b33cd15910043b17235b0
f41409ec52047b5444d678b439f099a56965053c
1875 F20101111_AABJRM stanford_b_Page_061.txt
8f8f5ae4dd1d3001cbbd520d155d2783
f613652f326c5e96b4c4665729d372567553c359
2267 F20101111_AABJQX stanford_b_Page_024.txt
a320e78ec57dab7271842283ac8a8011
6c208d7ef0a86d5727132b309f85f8982336fb44
74430 F20101111_AABIOK stanford_b_Page_148.jpg
fb5c83e23cd9a872b62806f098832dd3
8365179efb352961211644f26065711f2cdced6f
29792 F20101111_AABINV stanford_b_Page_091.QC.jpg
eeefe41b9c2dc640a870336798a46e42
bfb9cc42c4c83c2a0ede8eab290da31d38815d1d
2737 F20101111_AABJSB stanford_b_Page_088.txt
31bcb608542d7baa5e1ecf5bee38261c
4783a7e46b8ebd46f3c2604b38f8e2012a9dd313
2117 F20101111_AABJQY stanford_b_Page_025.txt
210dff0a61161f85412cd193c692caf6
d76cecfa25a6f71fc2b2230bf8247a70060cc1ba
2357 F20101111_AABINW stanford_b_Page_001thm.jpg
d9f73122a837bea3392d023f14b3ee33
1c3140e2def68ca79bd3f120e19305dc5139cf7d
1936 F20101111_AABJSC stanford_b_Page_090.txt
d354abf86532d1d325f66ee0f6056eb5
2981e278184c3700e9f5eacd3d799c3023af26bf
2164 F20101111_AABJRN stanford_b_Page_065.txt
2a5c8d3ad76974cc0027c1687307e7be
532367ac74bb2b51bcad4da9a46a9efb9873d524
2022 F20101111_AABJQZ stanford_b_Page_026.txt
37ddfb359e94b6d0082c3538c1ffc264
b6a2469edfd35b7fd2a5f85e4ec1c5fb84755f1a
27080 F20101111_AABINX stanford_b_Page_051.QC.jpg
8443785d5354fb7eaacaef4b8f4074dc
0357367fdaab4ad0d48e7f3fef8b561a3bd5fd14
28921 F20101111_AABIPA stanford_b_Page_090.QC.jpg
7a70830f1015e22d08650092c14ee5e2
d63f3cddbb6162e3d061e0f0e864c5e142e82d28
6309 F20101111_AABIOL stanford_b_Page_039thm.jpg
588659b721ed42c2905f0b796998ef92
76097d67f2e86aeaa44d839c0e53bff7977af26f
1839 F20101111_AABJSD stanford_b_Page_092.txt
ba7251148d49fd57b1c95a57e9d49cdb
08ac50f99d3a1c44a53ba085b12cfaa462c042e3
F20101111_AABJRO stanford_b_Page_066.txt
9e411e0b3fdc6e5d834a87066ffc5095
997f42602eff8c21af695921879295255cb9ffcf
1051944 F20101111_AABINY stanford_b_Page_068.jp2
0aae3e8866b8f050b17b0c5716af0db1
1326c5b14246c2a7cb646bae02ce5980701bb6a1
73254 F20101111_AABIPB stanford_b_Page_100.jpg
2a640b99bbdac0d2addac69b774b0243
d2b4cde3cdb5dac055c614df1fdddc6cc6a49456
F20101111_AABIOM stanford_b_Page_046.tif
5bd73dfb421add3cfd8802daaf52b7eb
51703c2029eb275bf82c407740514804246f0c50
1852 F20101111_AABJSE stanford_b_Page_094.txt
d850fe9e9d9393ad6753c1fd1ecc39ee
7fb7f751a4f3cf4059d1f4429c3a9e172ba7b85d
2037 F20101111_AABJRP stanford_b_Page_067.txt
b5e8adfea12b8e92a6b555b50c32ec60
167c51cf56cc1cbd415abcd9d741a68cd73b2189
78728 F20101111_AABINZ stanford_b_Page_015.jpg
d5f116be725ee899e959b62d3521639f
676ade8eea886859af260e6a80b0d50535cc4fb6
7327 F20101111_AABIPC stanford_b_Page_013thm.jpg
5ed436d6fb9ab8436438316fd7c91336
4448ed70eddede901942fc3ad152ee144d0bbcef
53485 F20101111_AABION stanford_b_Page_018.pro
ed402e72b288371cd611aca5c44ffb8b
92703ed1916371c15fce70c476bab424a080af91
1451 F20101111_AABJSF stanford_b_Page_096.txt
d368dd3912aa13761f5718fac1504fdd
2b0b9a6ae3f02360fe0589d79419dede0f772019
1541 F20101111_AABJRQ stanford_b_Page_069.txt
2e4ec172baa752fdf7c307663a36fcbc
298b195c7511444781144499df48ee9d0a37bb04
2024 F20101111_AABIPD stanford_b_Page_109.txt
2dbc752495c622638b9262bfe341b242
2d689875405855cf7a56101747e3fcb6ce022924
2167 F20101111_AABIOO stanford_b_Page_158.txt
c16e01f5f1e28bc8f98ade6299818dde
9b2b74198421f5a6bc9b8a5e8b52b40d82c703e3
2263 F20101111_AABJSG stanford_b_Page_097.txt
993aecca8b9b055d6cd76baf4080c11e
481677ad0ebd421a6c7d2056d6dfa1773a44251b
2135 F20101111_AABJRR stanford_b_Page_072.txt
d1a9311a0cd2c1f9495a302db3a5821d
21625b6662b3305347cbf4e15cef109fea5ea490
46677 F20101111_AABIPE stanford_b_Page_103.pro
2b72b7a880fb9d77fa88b889d954eae0
57f76187db1e5f8b28e25c441e7a68894212f546
1723 F20101111_AABIOP stanford_b_Page_004.txt
c4df5f49a2a3c9845962223dafd4694c
dbeda95355d0c6059b0be276f2983f9c6a84c4c1
F20101111_AABJSH stanford_b_Page_099.txt
1a62e171c56ef26d98ac571f1e49ef88
b26e7c69bdfa68b3623a622765ccb60d442d599b
2041 F20101111_AABJRS stanford_b_Page_073.txt
b08ce53623aca3f324467cd2d13c1a9a
f23b9cd743e0e08b8cd0b1025d9378c81397fa31
6869 F20101111_AABIPF stanford_b_Page_012thm.jpg
c7dcd7999d6cb2168a641dee07b1c3d5
455686b646bf5d3bb176acc9a4a51940dba02833
2210 F20101111_AABIOQ stanford_b_Page_036.txt
ada26dc7abcf2eda5384b39cd6d09a1c
a2a4fda224e9348b56079c00af329802e578e8f1
2279 F20101111_AABJSI stanford_b_Page_101.txt
92af7e5f84a67d77a2498e868c6d4183
eaa3d37eed4605640b4b7bd72936fc92ff75b937
2229 F20101111_AABJRT stanford_b_Page_076.txt
218e72137cd7aa488260812528a93164
8b401d5600fcf3313c82db57e2cc3bd59e122e1e
86698 F20101111_AABIPG stanford_b_Page_029.jpg
4296a582230c6ea6a5660b0c6020108f
a51f5295d0ff012fcab8b332d5c5e8269b51953d
2170 F20101111_AABIOR stanford_b_Page_049.txt
cb3d3fa845552ac317a2dde8692647ee
6857c2f4551f3686469fe87fd40be83c38cec2b1
1502 F20101111_AABJSJ stanford_b_Page_102.txt
948d287558c7a0cfe03dcff3a84fbc3c
a0cec91178938f0c4f992e76ba96470708857638
2235 F20101111_AABJRU stanford_b_Page_078.txt
04d8a0c0f872347fe9e70923dacd35bf
9ca1dfce7dd10ee912cad0eb53cbb3505fe5270d
38714 F20101111_AABIPH stanford_b_Page_131.pro
a1d51272def118c91ebd82847e3602cd
07277d07d786d00067f6c602242d26b246fa0f0f
F20101111_AABIOS stanford_b_Page_159.tif
dac137abcd128f402bd1b55bd968d5ea
d4941f20d8cbb474fb9a54df707d32be8d2590b9
2518 F20101111_AABJSK stanford_b_Page_105.txt
8f408dbad2fbf9f1f0451d1293ada7fc
1dfce355c4b51c8426754b2ee42e2b4d00b9531a
2199 F20101111_AABJRV stanford_b_Page_079.txt
fcae88c9df73e2c08cf8792020afbcf7
edadb383aca037e935a107bb14345ede3d230769
3615222 F20101111_AABIPI stanford_b.pdf
64cd8c44c03d22f6b6b1340532f66dec
dfef70218c117f482454e9c80ed39a855cf9d940
27753 F20101111_AABIOT stanford_b_Page_124.QC.jpg
7de4a48c5e20afee1b6d2d691e1d4b4c
fbe5ae455d6d44e0e72f07a3ace664ee270d53a0
2228 F20101111_AABJSL stanford_b_Page_106.txt
e694d30418a51613d8f2823e9d93a998
0ec5728544b151a7501fd8822acc54e6c6203e59
1801 F20101111_AABJRW stanford_b_Page_080.txt
a72c02051d51d8d94e90bdec92a31bb5
6989ad76dce764cd6695789fec9c6747cf4d5478
1051935 F20101111_AABIPJ stanford_b_Page_013.jp2
aa80b3661debaa4ff01ee1fc394c0d8e
adeaaf69c0281278a9cb42b79d515fa464621cfb
7241 F20101111_AABIOU stanford_b_Page_155thm.jpg
09e4ee0dd74484c5a8b523ca8e470e38
524b71ee56c9be2ddccce48bcffd9fcee96364d8
2127 F20101111_AABJTA stanford_b_Page_143.txt
c433c8983ecd865e9f4989c6e4eb2ad5
92466787a4eccf6e10179e4b00377a28bf01cba7
2186 F20101111_AABJSM stanford_b_Page_108.txt
a07092b8e9f9ef099e6873dccf2b3e80
9265afcc628c70095a0e13e86302c9b4fed43d54
2473 F20101111_AABJRX stanford_b_Page_082.txt
ff6fdd9bfbd427696775b07e0ae6f7dd
da7d1cc8dc63fef1ed72b750957f1583fd88a3ee
1051946 F20101111_AABIPK stanford_b_Page_117.jp2
d08071ecf53c0fef0bc048fae3c3853d
39f7bc268609bf2001c14cc62b98150b89c4e49f
77438 F20101111_AABIOV stanford_b_Page_079.jpg
27bbe73c0b724447564dc5af32823d14
cc8eed6a290d3b8b45803b081ee52992b7cfe9e9
1659 F20101111_AABJTB stanford_b_Page_145.txt
6a2974d02e998c3c1fd41f7d78a4242a
a21144ec88d10776aa01d6b803cec7da82937dde
2093 F20101111_AABJSN stanford_b_Page_111.txt
844b67058a558aa89af072573b002313
c8120ad8aead06ad5209cb1a5505738e5ddecffa
2776 F20101111_AABJRY stanford_b_Page_083.txt
1c4cb51bca751025790e711de04df107
053141f6c66bfd781fa811b51059cba6ad469ce2
23142 F20101111_AABIPL stanford_b_Page_133.QC.jpg
b8bf1acab0e11e7c330e008bcdf958ff
437e01fdd819f3091148d1260b414f2935a69dac
87851 F20101111_AABIOW stanford_b_Page_114.jpg
4696cb9ec30882f16e2e770461e02549
13fcf690b28ea822ae78c51e0f368978e2152f4b
1866 F20101111_AABJTC stanford_b_Page_146.txt
a8171863c768dffa807402ca623e25d8
7e4e05f2f953240934e189dd473c855d689e7ad7
2171 F20101111_AABJRZ stanford_b_Page_084.txt
bdb2bbe0058d509f5d021d5c2007061a
00385f3c4823eb24ebb2bf6276fcefea2508e249
25194 F20101111_AABIQA stanford_b_Page_155.QC.jpg
404706b39d00de0e9fa63a23307ec933
65b6b5f0d33e6fd100a04a92abff0e6df8fbfc3f
36487 F20101111_AABIOX stanford_b_Page_152.pro
2fe8523a9b9cf1c013205e18ecdaf342
96cdbceebe7b5c17d2d7e0af72160850a6c6d696
2109 F20101111_AABJTD stanford_b_Page_147.txt
775afd603735f6fbd498bf156db77d83
522a79ed7c051f7e3f08ca137643d4f472dde13e
2182 F20101111_AABJSO stanford_b_Page_114.txt
65837a9b5059966142d063a73ede0d39
57a98775cffd6391e13493406f5e9bd7d0dba91e
F20101111_AABIQB stanford_b_Page_087.tif
b329d7b5b266447818807b928edaa336
d82f94e547fda69bb757191c2143d968b9467ac1
88143 F20101111_AABIPM stanford_b_Page_024.jpg
b8be7c46405811bb4e5f7f41d4e32dd9
cc4d990362fb438f21332526ca0ad5206588ed4b
F20101111_AABIOY stanford_b_Page_004.tif
550be938ed4245ed44be6fc25fa40112
1a950a505fe57f0e5b3c3696193c89b6930b97df
1867 F20101111_AABJTE stanford_b_Page_149.txt
95a28e40e4567f58699b38de3699dfdf
66868cc2f9f4b7014be51b295d990050411b5db7
2224 F20101111_AABJSP stanford_b_Page_116.txt
a817b15a0c2e9e02c1fe3d0ae164c099
c1be5ccb0d4330ed3dddf1deccc7c1daf397f634
F20101111_AABIQC stanford_b_Page_092.tif
27b979c60de4a6e2d1462294fb4ac810
9ec617951ee7f64d109318469895a2d1c6bd66d4
2455 F20101111_AABIPN stanford_b_Page_081.txt
46232e3958ab6f5d4be4520f1287682d
145165e76a7d3738c328c0a44d4db9e0e92cf9eb
39431 F20101111_AABIOZ stanford_b_Page_048.pro
105be9a3d8f9fbbeb4c615f0637063b7
89d68e59db7687b4e7a889ff8160ecd27d380d78
2321 F20101111_AABJTF stanford_b_Page_150.txt
662e97b764e27cfd012b34f2128d5ca5
6cfb1aaee140f1cf2f0d6a98c743d7e75ac09739
F20101111_AABJSQ stanford_b_Page_125.txt
94996107dbf6bbe05408e67ae5f8b55a
418f11f17e8e2cedad2cf2d6b17d6a64f4bf1a10
2153 F20101111_AABIQD stanford_b_Page_017.txt
fb7d024be6d5ee9f65897d15c78dc045
7762972ea99e865dde61898e00a9cc6e6163bc39
2769 F20101111_AABIPO stanford_b_Page_175.txt
7e6d4fc863bc5ca69885ff051da92185
1f21a162dac330a6c6c49fb4bb0357dcfb95597c
1719 F20101111_AABJTG stanford_b_Page_152.txt
051ddc9d5d7874e51f4eabec530eb3f5
8a1a4ce782d3da3d58ed0f619df2c6196416a273
F20101111_AABJSR stanford_b_Page_128.txt
641888c856bde5d5d5eb49c482049953
b42ee24ae35b056f463e8e10dcf90dd344104a8a
2196 F20101111_AABIQE stanford_b_Page_051.txt
48b63129b9f28b0108a3634b36dc9fc5
c259598c6904c37aff52d32b5a1f03611f59907e
27902 F20101111_AABIPP stanford_b_Page_097.QC.jpg
83f370a51809589867679f72f1b8e8d4
8c97fd3e6a2ac821860522d548c53883ff1277cf
1592 F20101111_AABJTH stanford_b_Page_156.txt
96ebe16133b67e5f840399aa6e7c2012
9357c28c32955ea6467e548aac4ad729dbe2f740
2136 F20101111_AABJSS stanford_b_Page_129.txt
af15eaf8945ad0b4027629c03534b25a
7b86a50870d902e95bd8e445cb0f356b0724d47d
26103 F20101111_AABIQF stanford_b_Page_018.QC.jpg
d93a72aaf2b69ad353f118e5420b6252
c7010be66523006dc71ff24f203dbd3275fe9686
1051932 F20101111_AABIPQ stanford_b_Page_075.jp2
2a812aedcf60f32d8a2ccaffa1bba5cc
fb988873c0dd749b94f1ca04e3869c1b8ecde605
1775 F20101111_AABJTI stanford_b_Page_157.txt
ee191924bbe725f8199a09a402d13d4d
6a18a951fd27006167c5eccd14bd573eb3967c67
2152 F20101111_AABJST stanford_b_Page_132.txt
304f9ab83ed9039814f7c86cfd642027
946fae6983a4839169572473374d5a3017933019
54005 F20101111_AABIQG stanford_b_Page_122.pro
6e5b6230d0e485a2b55e33d94fe1693a
56127741a22cefc912c22df0e39b17ffe678e534
2095 F20101111_AABIPR stanford_b_Page_022.txt
898faa7ba538c590ca7da98d571db757
02f7190d89b26fcf5fe11769435c02ea1fa71aca
F20101111_AABJTJ stanford_b_Page_159.txt
538ead37efa499740b0056d78b3145c3
9882126e4c689d629dc3ab6ba0cbebb1f124956d
2176 F20101111_AABJSU stanford_b_Page_133.txt
137477b5e4b26180d1641d2226ef4c51
a16e30b3a2208da5c73de0502e78195dec778833
2030 F20101111_AABIQH stanford_b_Page_118.txt
c62778ad24387010a85297dd134e541b
b275fda151639d0266f2596499f1b962c129b03f
85218 F20101111_AABIPS stanford_b_Page_112.jpg
9ee6843fb0c0bdc106ca109d25c807a6
afac8cce4f2215e525a44a565a8354fed4ceaddc
1236 F20101111_AABJTK stanford_b_Page_160.txt
2542b354c15fa390a9e52f1837b8818d
0e5c75d0d48f0cbcaabc7440b0b06dded5b5ad1f
2212 F20101111_AABJSV stanford_b_Page_134.txt
5a574c7326263420d229f350804c16aa
eeca99f34b17e00fd2e43c2829a43b7c6467b1d2
7356 F20101111_AABIQI stanford_b_Page_047thm.jpg
5af9d3101e05531086eed4aa1922df49
b230361e829966a221fea676755b22d698fe98b5
5704 F20101111_AABIPT stanford_b_Page_131thm.jpg
58a00ddc8f1092de8247578a03d7f429
85a46d33d66e9b9fe454d8cf2b41fd1905af1f09
2168 F20101111_AABJTL stanford_b_Page_162.txt
ad1660dd90d1ea619a28d1635cbac955
a7d630dcaecf77d3e64d30758e754fb3aa756f5f
2053 F20101111_AABJSW stanford_b_Page_139.txt
9c6054c66198dbb50ede20391b178ab3
8d228d2522eab8b93d57a7076ec45f6478aedc79
8133 F20101111_AABIQJ stanford_b_Page_139thm.jpg
f2cc71c29b2a0ae258c64facee901214
9c62ec745b8a1a7284c4751ea0a6215833551afd
79637 F20101111_AABIPU stanford_b_Page_012.jpg
6b0d84ecf4449f9105cec4bbdcfb9198
e54f7b274bb36abda49dcfed01e838b7e18d27f9
9910 F20101111_AABJUA stanford_b_Page_007.QC.jpg
fbd4264dcb010a559479a3a2ab8d21c6
221ad05402d922e51db7e0acc2c9f649b7b4d6a2
2678 F20101111_AABJTM stanford_b_Page_167.txt
37836c1f270a0d364d2799a3c0cb6ee5
8ad3f3e3883263b460dd6a1e255cbdb206f6f963
2075 F20101111_AABJSX stanford_b_Page_140.txt
495ec3e681c2f1ee1340c31963aacff8
367736392318381d1e58fea259886233c98dc781
1051926 F20101111_AABIQK stanford_b_Page_077.jp2
04771873d4a7ab6a3f3fba45f4f027a0
a2d17d2591c8cbe6dd3e065e3587e807a992816c
25105 F20101111_AABIPV stanford_b_Page_086.QC.jpg
628c88e5c4a1e9ef11349817c2862877
380169f88c6e5452f2f0fbc15026980f365df382
26007 F20101111_AABJUB stanford_b_Page_008.QC.jpg
aeecbdaf015e873314fa09acbe6a2a3c
af5f69fed4f63f738d604360bd3557475c460ad5
2632 F20101111_AABJTN stanford_b_Page_171.txt
27cc63d164713116eb69b058e3cd54c8
7154b6c16444ba77e0379cdc16c97e4e91d7e57f
1712 F20101111_AABJSY stanford_b_Page_141.txt
6544bb7362f0068e1525c63a6ebc6b95
71e403e6d4135ddd646e446a8c86d099c2028ac0
F20101111_AABIQL stanford_b_Page_090.tif
c17cb71fbeafdae5c806800c7846c251
4fd8bc8a1d5bb1f5b089b566f9a55213bdcc6d5f
F20101111_AABIPW stanford_b_Page_042.tif
f3ca0038675d50d07f06ad375aed73ba
494486edce2f9a2c6db9580a1b040ae60f3be4a7
6535 F20101111_AABJUC stanford_b_Page_008thm.jpg
b99f27a2ad8d2ceebe8109d242b809dd
da2af8e314ea2c3a61aef4cc11b16dc941a25882
2745 F20101111_AABJTO stanford_b_Page_172.txt
b56f873ea270c5e259f506265726bfee
a87351fa8f5a2ef516408a84bcbc31eaf878abc9
1991 F20101111_AABJSZ stanford_b_Page_142.txt
527bc8427c3a103d375381b2278c1a51
2161bd762aa0739d9d0430cd8b000bab0d203b0d
1051974 F20101111_AABIQM stanford_b_Page_171.jp2
afff0692564b0027147a07225c99d6f8
dcadfe31db1d69f46651ab488fba78e75d646d06
7968 F20101111_AABIPX stanford_b_Page_167thm.jpg
da6f6ca55f780d29248f929317ea7919
406865c873302b3f325d52465f59639e1a8c3567
2038 F20101111_AABIRA stanford_b_Page_119.txt
29a1df2bbbae025aa3b98d85869f89fc
c9de7fa7560d6aefa58d90e68c9916391e89e003
6924 F20101111_AABJUD stanford_b_Page_009thm.jpg
9af02885519a50d6c331e5549929e2c8
c88b23fc266cb1fc32dd6ce45b794a003fd7ae42
1051967 F20101111_AABIPY stanford_b_Page_042.jp2
a9ea893fa3c5bcd0a6dd97327fa89283
1806823d3260323bcdd532a8ad73df15c4ddc16c
78170 F20101111_AABIRB stanford_b_Page_109.jpg
0ef2ad17343cc11d3efb63fe8e0d0d1b
b778354d02d135d5f78ec620dceb58d311f70a45
28452 F20101111_AABJUE stanford_b_Page_010.QC.jpg
9e50a2d869c767b8236fe2e0825361c8
0e5d3b76c43b9dd6015744a95a1c8ab5327c9e7f
2697 F20101111_AABJTP stanford_b_Page_173.txt
85880cc3b576084e9a2a48c79d20cf7c
a34795ea32ae69c4eaa89a4186b0ce686671cdea
1051712 F20101111_AABIQN stanford_b_Page_052.jp2
a58fb4e95d53e0065ff6288f16ff0350
4e94e17be07ea278d87c148bfd91305e81fd96e5
69251 F20101111_AABIPZ stanford_b_Page_152.jpg
904590116cf3a52d00b083741d582f98
e24148208e7a907fbec6cf69253a98f709a85454
22331 F20101111_AABIRC stanford_b_Page_151.QC.jpg
42e59d291e858c97c5f34d84a89e1b2d
f26140d158ee0d2fd5ffc280504b538c8a6d0a0f
7228 F20101111_AABJUF stanford_b_Page_010thm.jpg
2e511c24a2189bdba8b06cfc7f49b1f2
f3ac490170dbdb27784cb886caad3c23cb5e318c
2725 F20101111_AABJTQ stanford_b_Page_176.txt
f9da0dab57d4f507f878361cac2c0c63
4d61b15be40b54bcc8b45fb829449e01a8a4efb9
1914 F20101111_AABIQO stanford_b_Page_120.txt
736dfd0a7eeb1a360e9aa5bb0dd3d672
7fb1eeff9f0af34cf3253cf056b9ff7f74e27ef4
F20101111_AABIRD stanford_b_Page_068.tif
0e790eed4c2066cce01c5a7d96996362
da2e30e65cefe28db55b02478d5ba3636b973567
208087 F20101111_AABKAA UFE0021970_00001.mets FULL
b2c6f3c427f6098ec303291dce6c49ef
ab3295b0f6759aaad91d6859517eef0dc5e4ed29
6433 F20101111_AABJUG stanford_b_Page_011thm.jpg
8a70800523fe12840a60f3d970b7a37c
a6ae8aeb62a0ca6d89bc5e1d719fc5df2610566c
1120 F20101111_AABJTR stanford_b_Page_178.txt
2dabdf7600dbf2a04e3bee004132ba33
ceed9133cdbb144f7ec2d46bcb6656667c4cef5f
2132 F20101111_AABIQP stanford_b_Page_085.txt
07bf0cb21ec5fc02422c2b67a22ec1fc
73a853d8b100725db767ad2d4a35272a3d9c9090
7616 F20101111_AABIRE stanford_b_Page_033thm.jpg
19ad2a7c1da83d10669d5a884dd12415
6f56350532c1ffb015025fe62a68f3c51ae5e8d2
23791 F20101111_AABJUH stanford_b_Page_012.QC.jpg
f643bcf632097298278dde605ffce14e
3652ae4abe4b74a87b092645050a342ff5af890d
1858 F20101111_AABJTS stanford_b_Page_179.txt
314ee440f5a47f42da799274875faba9
93a5d9e60f8ca9552224598700e426a2a720df20
26312 F20101111_AABIQQ stanford_b_Page_025.QC.jpg
e5c845b3daa3ff43ebc41b1b11638d6f
d577448a5d2879ffa43722bfc69b6781c6e0fb22
5624 F20101111_AABIRF stanford_b_Page_132thm.jpg
85e214fc7f551f611fed2b375b6e218e
8c266352db3473e4cad4c2772a441c2f48e5bcf8
26887 F20101111_AABJUI stanford_b_Page_013.QC.jpg
38fe580b4421478f72a1e1e6ed2ecbe3
34db6f8c600564ad21dab67525569bcaae2c6a21
7635 F20101111_AABJTT stanford_b_Page_001.QC.jpg
2ad3a155566f440ae4e830743beff617
9a343f519ddc2093374771899fe175ec5feba2cd
2183 F20101111_AABIQR stanford_b_Page_035.txt
7d43f16e9224f243024219505ae0fd2a
34e44852ccbbc6ca5756eaa014702bd1739deba6
981259 F20101111_AABIRG stanford_b_Page_056.jp2
a2adc0d75b14fe292925279a72374df1
5540f934e3262a1167d210f8bd5b6fcb5d99100b
7088 F20101111_AABJUJ stanford_b_Page_015thm.jpg
7e1573f4c8b420a97fb534f5bbf47ce0
817f2e37abdb4a4f4d1b75f3c6bd35cd3e7a99b8
3157 F20101111_AABJTU stanford_b_Page_002.QC.jpg
4ec47ea02f88e15b19b291bbbf35b299
77c5ca3cc9151adc8e743937104f0944d09a70d2
1472 F20101111_AABIQS stanford_b_Page_077.txt
90d23a9079bacd5885c6d369da69c378
66d9c22805de727a6c5ae6246acc7c87695a2bd7
84485 F20101111_AABIRH stanford_b_Page_105.jpg
37f5495e66804c37694daf44d761f49c
1e47ec2ff8744e121b4dba5a01de69a41148f64a
27737 F20101111_AABJUK stanford_b_Page_017.QC.jpg
a797b3b804869593d619e6fbb4904186
e91f1b7be9a7c418a3de3139ae1b86e1bfe37a06
1351 F20101111_AABJTV stanford_b_Page_002thm.jpg
96fcbe1f5cad17df79382afbc1a5d97b
39521478947d5e834ca37ecf20648690cacd4dff
F20101111_AABIQT stanford_b_Page_005.jp2
530476e1d99bb3f419c6c8a76e267e5e
9b60c54eae82d247cb3a387ff985eba9c9a587d3
7997 F20101111_AABIRI stanford_b_Page_174thm.jpg
0df36d1681e50c81c2d7ad5bf481f208
699ec0c8e1c3fe9ba3766c6bf8dc2ae9e0bf2005
7269 F20101111_AABJUL stanford_b_Page_018thm.jpg
67c2cdda816c1ed8b3fbf5aa54ac0c9a
29caaff334d2b4355366704f0706a444346bf5c2
5061 F20101111_AABJTW stanford_b_Page_003.QC.jpg
ffea4fe63215bf9b74a8b48825756092
650f9dee3063a990cc5740e1614861a02e2c321c
7192 F20101111_AABIQU stanford_b_Page_142thm.jpg
a1e70ed7ea3455012746b95c8731a14b
80ccc8d3022408dc3b1b62fc3661de13fb1cdd04
1935 F20101111_AABIRJ stanford_b_Page_020.txt
e64029fa125e0f00fbef2fe62f5db87d
c79684375ab94c3b1dd4da6775e7ea1c6f7158aa
28339 F20101111_AABJVA stanford_b_Page_036.QC.jpg
52b83e9d1e27c9e6814239fa9b3df68d
a35f2775660535d20cce3b5f074780839dd8eec2
7312 F20101111_AABJUM stanford_b_Page_019thm.jpg
148273fd89a9b65da922fc9155bcc5e8
1c52345d8b5078e931eaa81cdaafad580ea1d86e
F20101111_AABJTX stanford_b_Page_003thm.jpg
a13edf2f2e5915df75c8d9345928e5ef
3c2e05045bb61f409cb85305261ffac31b2c09fb
F20101111_AABIQV stanford_b_Page_012.jp2
800e61a4d8b5c961ee5c9487ade50d4c
4a1364cab5f1c99d6e6739d732d9557261dfac4d
75430 F20101111_AABIRK stanford_b_Page_053.jpg
f7f4615e2b3451800e47347628ec6606
8ee3cd45f98593ce9e1bcf56dfefbb19f3f06605
26478 F20101111_AABJVB stanford_b_Page_037.QC.jpg
70ca248f5dd29f6bacb5a24877b46355
5a64a313ec74962e0ff621cb8a58b8f09623a837
24579 F20101111_AABJUN stanford_b_Page_020.QC.jpg
55f13da0067a0da2ec29b0207e9c48e1
4974142a04a33f212ded1faba577eb07dd5160e8
21697 F20101111_AABJTY stanford_b_Page_004.QC.jpg
cbe7dd62c7fdb3073a1047dab46962bd
b2fb12d6225fff87cee4429f6c2387cf6a563b49
27462 F20101111_AABIQW stanford_b_Page_046.QC.jpg
abc03679cecef05512352087c8f03e9d
4ee2fd2a0670b7de53fd6ae5787d46a8db63c16c
F20101111_AABIRL stanford_b_Page_141.tif
977f1b874f9b305312331682c5e9b769
ac7dd78e49109f45f334f580d286a3214fb916b7
16094 F20101111_AABJVC stanford_b_Page_038.QC.jpg
eeff9c05f79b94b2e6c5d005bed054c1
7147057d00fe9d4c60d0453169877ab8f0d30ffe
7558 F20101111_AABJUO stanford_b_Page_023thm.jpg
c66b4e9e8331d933b273af6d2a8a2398
7a5a5fd3df0a9c91f93fa656d11f168db2964351
20475 F20101111_AABJTZ stanford_b_Page_006.QC.jpg
0276342ba401074366134faca13c0a7d
712fd85201e10f5696f1748fc5f0a9f93a760129
35313 F20101111_AABIQX stanford_b_Page_123.pro
2cfb22ca121b3ffaba26daa422e5e19e
c5cd0cf4771dbbc7e841a6ad362aea7cbd180407
46667 F20101111_AABISA stanford_b_Page_090.pro
a424c0df4bde8a2d274fe5111f1aa87b
53ae7788773bb7315aa47d9b8f3e4a35eb9b4607
80527 F20101111_AABIRM stanford_b_Page_082.jpg
d14bd3696e2b2fa44caa81b4ba128168
51bdb7732f859cdfdfacabb1edb194c2abd28de6
21485 F20101111_AABJVD stanford_b_Page_039.QC.jpg
73aeab67c7e606bae5e181c14322c925
ec4b7dab13c2a6bba26c562c930c59fa281275b7
7572 F20101111_AABJUP stanford_b_Page_024thm.jpg
b6c40ca722680bed6fa5a844d7326583
5d2d4b5138a0ec10c007171b85a19fd2e601326c
85550 F20101111_AABIQY stanford_b_Page_134.jpg
a7f58b710601a88429c71ecde6cda849
ef163d32b1c2afbd4e5a8fed3c8f932aafbf4853
7374 F20101111_AABISB stanford_b_Page_014thm.jpg
e9e7ec5c0ad599220edb1341e529c9e9
efb5b8f934c8a392eee931c597991a64e4470df7
1051973 F20101111_AABIRN stanford_b_Page_144.jp2
53a42dac31f4c6697f98b97a0f211d20
13e528f4f75099f05123f4cf3b7ff3fc280d2b64
7436 F20101111_AABJVE stanford_b_Page_041thm.jpg
d0b01110d219202247fd0446415a7e20
15547c4127973b1d5e4d9d30b0a7854140f78b0b
43069 F20101111_AABIQZ stanford_b_Page_148.pro
a5310f2fee4386e6fea7fa5beba51963
6dfdd68076f34835c7dc89f95c7c2aecf8cdc9f1
933173 F20101111_AABISC stanford_b_Page_157.jp2
e4d8887b75ea5bf68cdce757ba8713a9
c3e3f85bce5dcca9a627bfd815cea2b292a3e283
26792 F20101111_AABJVF stanford_b_Page_042.QC.jpg
3158dbafeca3bdbfbd3508919cc65521
723f30b2e17c33be03354f16fef4c6f380fb87bf
25450 F20101111_AABJUQ stanford_b_Page_026.QC.jpg
11fdd173073d9b862eb9f497ae5a4de1
6e960d2f6a8810839d8570160c0106556d034b93
7268 F20101111_AABISD stanford_b_Page_042thm.jpg
d4185e6605a1d96f53cb54d3c1c5fcbe
1004b6333dadb1288af9e84c47ba1dfdc439b5c3
27630 F20101111_AABIRO stanford_b_Page_110.QC.jpg
406fd01ba47feadf8f43b5cbab918749
cad42ce957975db123dd54ce7884eac4f1480969
7548 F20101111_AABJVG stanford_b_Page_044thm.jpg
9eb9aa980520c22301c0600883600a92
579f57a7be578c178b1640d0e84fbbe71003a046
7029 F20101111_AABJUR stanford_b_Page_026thm.jpg
0d62b206d3536053c7ab901b24bc06dc
669a57cda98a07b709ba57c33d5abf4b5fa4a59d
81223 F20101111_AABISE stanford_b_Page_083.jpg
8b7ea5bd5858f0f384c408c813cf3fdd
82643d7b30cbce357d0dfa9aa980e37e3fd90640
F20101111_AABIRP stanford_b_Page_072.jp2
2096c662df759538ebdc0281539d8429
c61eca65d68ab0a38dfd0bb8206a0d531004d09b
27214 F20101111_AABJVH stanford_b_Page_045.QC.jpg
c54d186cf594b0a24e1caca01ecd2744
3091c99e4dc08b9db3ae7c97967a4ab6db03a93d
7596 F20101111_AABJUS stanford_b_Page_028thm.jpg
c101a267bfc452cd57a64647dc329c4b
679865133fdc7999e5761beb4915cd3234142ec2
26174 F20101111_AABISF stanford_b_Page_126.QC.jpg
953eb7e53c43901159da5d04fcc0a9c0
681994d664a47d05f28afa5ff75d9d7034d14f30
109645 F20101111_AABIRQ stanford_b_Page_169.jpg
e02f44949842df0a6f478fa339a5b678
3aa5ac1c4b4f85830ce78e08e0beaac1a2ecdea1
7387 F20101111_AABJVI stanford_b_Page_045thm.jpg
d7320196e19e5deb2b760df7718e4cbb
09a9aa719a50f7bac7bbc378ef6a39a4f2ac1b69
28001 F20101111_AABJUT stanford_b_Page_030.QC.jpg
8daedee2554e647be751f89b88a2f548
d1162b7bcaf9f7b6f0dd66d57c0200796fb4523b
28012 F20101111_AABISG stanford_b_Page_114.QC.jpg
b1a159dfd45619acf9ee63e62cabbb26
8c1ef14fbf609b927100cba1a1f35a54bda99ca7
931859 F20101111_AABIRR stanford_b_Page_152.jp2
d95b8b69b4cf26e4f0cf9d535d97277c
e7b724bc384b6c02c594306271146207f4487974
F20101111_AABJVJ stanford_b_Page_046thm.jpg
408004c9d4d2bf670633e54775c72f25
0f1fa2e1f124fb5f656b12948ac93cbfe08d2dc9
7753 F20101111_AABJUU stanford_b_Page_031thm.jpg
a09489e4c9451dc600b29a1dabaf0160
25766d5e92e34a5043702a0a359b7d33d6c5383a
55148 F20101111_AABISH stanford_b_Page_088.pro
78bcc8435127d00c58982096129c6fac
77212182fa7dba3204b02c26daca1e6e20f2b88e
103819 F20101111_AABIRS stanford_b_Page_170.jpg
0bc5b204b458a6daa77155075f4f7276
7d8478215036d7561354e57e820da85bfd7c1158
27732 F20101111_AABJVK stanford_b_Page_047.QC.jpg
f574a9b5af777d9e4c328c4529357eca
f80101389dce86e44d8c98dd2010001cd09ee7a9
27192 F20101111_AABJUV stanford_b_Page_032.QC.jpg
78a77e30dcae095346a96100798cc1ce
093c3a4def701f7fc1878c04fdde579ad6f12a72
2016 F20101111_AABISI stanford_b_Page_135.txt
54eef2fe7839a6235307965e5a557dc0
b59308ffa0f16513d53b7d3154ac1aebd8fa10eb
F20101111_AABIRT stanford_b_Page_097.tif
7d01f7a019c5efcb16d58c2db2f5d40b
dc3014dc3520aed21141f078d0c3ce2683d2a130
12873 F20101111_AABJVL stanford_b_Page_050.QC.jpg
d78dbe391113424fb93273547e2979df
01bba6c0858bf0ffd46a19d3e14d4e910e80f99d
28113 F20101111_AABJUW stanford_b_Page_034.QC.jpg
cdd31379cbfc1287a3d95b3a3e3f6ad9
52f2dbabc86c56061115919b6f4da3f52f1ebeb9
54712 F20101111_AABISJ stanford_b_Page_045.pro
9585a9e5de58e9bf6d6c6f2d613fe6c0
e175cd179dfb952852de81a6e912169b5bb44ec2
2501 F20101111_AABIRU stanford_b_Page_113.txt
ba85b58340a39d647a62daac8fafdd4b
4a4ba8704c57193b0d346eb5e31bad75c082832d
5492 F20101111_AABJWA stanford_b_Page_069thm.jpg
26b1570836002ad43a20813cc99fe55c
b4d249e3712f90d04e8530655fa4b84df694ce89
7242 F20101111_AABJVM stanford_b_Page_051thm.jpg
e252b9d3a649463753ba4458cdd7f34f
90daa5b322fd9cbc56db39581046e94fc9ab8090
7595 F20101111_AABJUX stanford_b_Page_034thm.jpg
0cdd1cdb5249a7b6c44ed094ae5bef6e
a9b2258457e591f15b3b13832b8a165bdf5319f2
F20101111_AABISK stanford_b_Page_167.tif
ae5ddb6274fee629198713ca330ea887
1c3edc0c531b584e2ae2f9cdacaee2cae4b513f5
7071 F20101111_AABIRV stanford_b_Page_128thm.jpg
3d5f28c553ea3c5e0e1fb303c4479770
961ee7f7786f9cbebb8dd485f53278fe4b4823f5
6192 F20101111_AABJWB stanford_b_Page_070thm.jpg
d1d918d3e1ab1ecdb1fee8b0a4a437e1
7b3c8305a9164b47a9e26287011c200470b21b0f
6865 F20101111_AABJVN stanford_b_Page_052thm.jpg
06b9a77e4cb1c7efdc562523da9e7d28
76b9dd80e88a033e6fbc3c0db7bf2dd33df1e374
27504 F20101111_AABJUY stanford_b_Page_035.QC.jpg
4b9d2c789f22c11f02cb31f48564b43d
43044801ad08828129005cac169bf4958d00eea3
7216 F20101111_AABISL stanford_b_Page_082thm.jpg
bbacd0482c4f0581d3aabdef11327a16
253ad450833844ad3612277d0de9d164533de7a5
66572 F20101111_AABIRW stanford_b_Page_169.pro
428cb160211ee522c2b02054716f3c6a
ad85830134317cbb674ef146b3834865540744ea
6396 F20101111_AABJWC stanford_b_Page_071thm.jpg
5db8848b9960f33db91a444a05d148c2
85babc13f6ca242db51707e19904b77004e644a5
5657 F20101111_AABJVO stanford_b_Page_056thm.jpg
f5bdc808056a117ecfed1b98518f0683
0432bbe5c70afafbf3a6db7e03c2c0605a588ae6
7621 F20101111_AABJUZ stanford_b_Page_035thm.jpg
d50a0ead770ef8dcf7c632b8de268f43
b32450b7939c4b89fc9f2d6d050984fe7067dd4c
7221 F20101111_AABITA stanford_b_Page_083thm.jpg
a210c95183eb1e1968790a79482a5644
9b898a500bcf3155ab81158d939ee943927fcc77
48754 F20101111_AABISM stanford_b_Page_104.pro
454501290c37541c16daf460d25c0782
7ecfbf3873b3aa6616c562d7a9993aef377ac5a3
1051941 F20101111_AABIRX stanford_b_Page_153.jp2
afd81a4f3bfcff0e452058f5206fcbe3
4a11316730e52c59d860f9311dac03b41651dcf3
26904 F20101111_AABJWD stanford_b_Page_072.QC.jpg
157b3a7c4c3fd52a045797d039ca0650
1b1ccf845783f1dc060f4852464be9449e7279c8
7190 F20101111_AABJVP stanford_b_Page_057thm.jpg
40dd2c7a80210574724f8a008291b6b2
522992cab5a8023e9ea3daabe9b7ee2f30da02a3
F20101111_AABITB stanford_b_Page_157.tif
25be40fda3ca81f611d5855911b7c5e3
fae6d57cbc3958fc672fb7e4104e4a40637bcabd
8155 F20101111_AABISN stanford_b_Page_171thm.jpg
24ad56b78f383a4e3ad77146b348b0fc
0e1c9f0a503433c2f9c3db5582f297892cee8855
55750 F20101111_AABIRY stanford_b_Page_163.pro
24a621352a2f192f2fefd6f0a68cb253
aca7bfeee2dea1665b75d93c70cce6f6ce0b55be
7424 F20101111_AABJWE stanford_b_Page_072thm.jpg
8bd5158cc8660e1f5622dd6d1a613d10
2e380931a125e996f7d4c7876914256f489a21aa
25965 F20101111_AABJVQ stanford_b_Page_058.QC.jpg
4a8cb8229b6f0a9ef37b8ce214cea102
e17991a6a24bc4ba9a48c887a31ea9ae2c0feaf1
F20101111_AABITC stanford_b_Page_033.txt
9c1720cbb33126a9fc7037633a54903b
f21fd5c13c03645f50b5a4b4e7e911e02c6b5207
81382 F20101111_AABISO stanford_b_Page_022.jpg
5ccbe3ff4cd22b48b2be2c859b4f5fdf
d11485c853e86e602eefce1e904a7bc5ddb90ca7
980059 F20101111_AABIRZ stanford_b_Page_127.jp2
b9b666ae9d302edebf32b45ae96c8586
49262c8ee2d64f3e9d9a29f3ae8a5ea7156e1c7e
23320 F20101111_AABJWF stanford_b_Page_073.QC.jpg
de6d0cea1a09463a7e246bcc8edcc04b
06f0f712a8ba3d3fa365326ec0a729b9d7c8fb71
80033 F20101111_AABITD stanford_b_Page_166.jpg
66b1b6612746259b874e23266e5de669
bfd5ac823e3b791fd895c57c3fcd494ce0ccc76e
26538 F20101111_AABJWG stanford_b_Page_075.QC.jpg
5e7549e99ce049a48083087a9ae20029
7348fbe8b9c7cad97d3ef93a181ac220216ef776
18718 F20101111_AABJVR stanford_b_Page_061.QC.jpg
a2d5469df5c60d8a449c44863a446c14
7e7b9a5ff91c54d8ba43488efeb3ad03d60b516f
86122 F20101111_AABITE stanford_b_Page_045.jpg
b87a6527ce888f8210753a9f5cc0ea18
f028b8d06a7c439be84c34e6731cf633594c708c
43571 F20101111_AABISP stanford_b_Page_138.pro
3f0244233695f41687d4a9d7f10feb12
86940f13b0c97bb20cfb2b3dd2f0d7e0e32d98e3
27388 F20101111_AABJWH stanford_b_Page_076.QC.jpg
b8a3e6a554247157ddab7ed0ba6ff5dc
452cf40a61bda1554b5120d84eb683bd2ebd7f29
5557 F20101111_AABJVS stanford_b_Page_061thm.jpg
2dcb3a123c8204d2e282b3ad80ad4aa5
2b0ec2063f1ff3bbab75a3d70af9a9f605645677
65980 F20101111_AABITF stanford_b_Page_069.jpg
dd723da13fad2acabd717649e43419d9
667e3ffaf6d2b0bbd963c071faf3e30df74bc99d
58692 F20101111_AABISQ stanford_b_Page_131.jpg
387e4d58fd8b5c0963593f781042738d
af00717a43bb21cd8285d579850f3b489b8878e7
7701 F20101111_AABJWI stanford_b_Page_076thm.jpg
141b15927717f0143a197d148bb70cd7
92d202f59af2fab1d4eb76eb89be4fc48b6f8022
28546 F20101111_AABJVT stanford_b_Page_062.QC.jpg
07974225e1fea4d74d212e986a9045ee
00e7693b7104ffd110b41f5c18841607bbfc6666
F20101111_AABITG stanford_b_Page_082.pro
71eeb606476bd23742d62a2f56132cfc
ee7fc6c22579320f869c1a22958e45951e4bd93e
8310 F20101111_AABISR stanford_b_Page_175thm.jpg
74ed34903fe7bb86a33ea923ee1f73b7
e7f7fbb1f04b65dcf7974af96d896d727c09bce9
23397 F20101111_AABJWJ stanford_b_Page_077.QC.jpg
7f20bbce4c33806304e753c47d8bf78a
46eac59b33cd9d01a451acb75623d0dbdcb0a551
7469 F20101111_AABJVU stanford_b_Page_063thm.jpg
4bd216d9dcde7245285eb6f29ebb7f60
2cb7f15fd3a7dd79ad46c301e97a752607421919
7317 F20101111_AABITH stanford_b_Page_030thm.jpg
41b3267464ca46cfb70650864834e18c
1cb49c4697839b2abeff97bde4c0c4287bc794f1
92977 F20101111_AABISS stanford_b_Page_013.jpg
8978370f9b854620d833a6d42d0fb3a7
c046b840637e6b32c1f5ff010d39fac2cf770891
7603 F20101111_AABJWK stanford_b_Page_078thm.jpg
694dfead20d6d623343ef617f5b2e5ae
89b4a51a943fdbed19c226e37026f3729d8e3178
28307 F20101111_AABJVV stanford_b_Page_064.QC.jpg
4556650276a014147bf926b1dd0942ff
c5a1bccdf52e68fc9e862353240db17ab216e0e4
F20101111_AABIST stanford_b_Page_064.txt
aa979b2e4c14c08b8286bb5caf3875a1
c886b7ba3fdf895865cdd9e501bbf6ef11741026
6428 F20101111_AABITI stanford_b_Page_119thm.jpg
732d1c2e38ac36f4ddf3e63e76f21e18
34cf1736c2705e4079a1cb752fef0bc3f452f3a0
25864 F20101111_AABJWL stanford_b_Page_079.QC.jpg
c5977a59acfd243515a43697afb5d0e3
7a00c5b03ae27283b60b7a126797c71ab0bcacf6
7573 F20101111_AABJVW stanford_b_Page_064thm.jpg
fd3bb33a70518413c23d9b38fe93af07
a4933ff6edc93e570ceb9db5155ae11c2e47041c
F20101111_AABISU stanford_b_Page_106.jp2
59f50871c9dff019a4ae256ef65190eb
f39000027bdf5a991cfee6346c1dd4e6fdf099ee
85645 F20101111_AABITJ stanford_b_Page_159.jpg
144d12ba2af0f032afbcbbfa865ed086
730703425a262fc0a2c5f691714a0d93d1780d60
29306 F20101111_AABJXA stanford_b_Page_094.QC.jpg
f1bf32604401c333434b8fcd430bf0e9
3bf5731d6f5bcb648b642a34b60107b34dac8d88
23604 F20101111_AABJWM stanford_b_Page_080.QC.jpg
f781213997597d093d0bc255cb88a7b2
c324dd18d0ba3120718faf0dc4119eba7a1fd178
6303 F20101111_AABJVX stanford_b_Page_065thm.jpg
ead7eef8378230ebc4e03130cec82c10
fc8f7f15ffb72816e28c0698480951a83bd1801b
F20101111_AABISV stanford_b_Page_063.tif
8b6cb736a1d7b9213885e4eb5da11b56
47f66279d7031eb6708a15ac51383695df7196ba
26780 F20101111_AABITK stanford_b_Page_009.QC.jpg
429fda90c3686f8284bb200d3a8484cc
ec9a2abb0d73eeb8a91506b324d8db0b7b10c045
27048 F20101111_AABJXB stanford_b_Page_095.QC.jpg
359357c4e7391225d4f8bde6c0c1a5f7
8d1445e4f525475594b9e3f092bbd919fa4bd70d
7133 F20101111_AABJWN stanford_b_Page_080thm.jpg
fd25392459170e2ef4fb020674ba0769
79313081b192cca3cfea3b7eeaf14dc67a3fdc36
6745 F20101111_AABJVY stanford_b_Page_066thm.jpg
9782d714891b051f7ba2ce2d1ffa2bc0
3a94cc452c8e05d4f94e84073ac7e5d74324d3dc
F20101111_AABISW stanford_b_Page_110.tif
435ae98e5efa50ecb21811d34123d7fb
4668b7ea2e927c0ff08813cb5a2c14bc7e70857f
72297 F20101111_AABITL stanford_b_Page_066.jpg
53b86692ad3dd1dfefe063d0ea15fe12
a01d4df6910809b01eb804047e1d508c765485b1
22983 F20101111_AABJXC stanford_b_Page_096.QC.jpg
7aa6db4613f67c159518171836136fb9
f9265253c513c9e21febcd2bfd16b1ec9d889634
27792 F20101111_AABJWO stanford_b_Page_081.QC.jpg
a83404a05550a49aa6615b8fd251698b
298bd67b26a8239eb2a19be18fc0b59267ee4e18
25123 F20101111_AABJVZ stanford_b_Page_067.QC.jpg
16fdbb47208438505748f45534c9f140
4eadefb5442105629814267609580e4e7e83be36
2205 F20101111_AABISX stanford_b_Page_093.txt
d2b595e91682d865b83538e83f6d5ee9
062ccae0c89c232b332a1dc2f9e6900b120e70a4
F20101111_AABIUA stanford_b_Page_164.tif
925f3a87061c946bc54a3b11933c8f6f
b1610ccc36fcae9157904e19691749ac96411ddb
82632 F20101111_AABITM stanford_b_Page_141.jpg
d89b9d8d0b44db7f91bb020768482059
5769f476f2eb0267263fabf9bffc47705fc9cd64
7416 F20101111_AABJXD stanford_b_Page_097thm.jpg
79ab8a269468f70a9c6e8b76ccdcd288
ccdd86265f2767b3bc3c9472830a89eef6b08518
26452 F20101111_AABJWP stanford_b_Page_082.QC.jpg
784605c4913f0a52ef067f507bf5e8ab
e28283f26770599631c6dc689ccbad83dcd8ea77
24444 F20101111_AABISY stanford_b_Page_053.QC.jpg
fef6e62f0766e207b72e878fa79ce55f
b0604a833394f1ccde861725d92f548529771dbd
F20101111_AABIUB stanford_b_Page_113.jp2
8920ce465835c891ac9b547e5284c288
0af9b9d3c0bf6a316c231c4a51f91a172f6f4da6
F20101111_AABITN stanford_b_Page_040.tif
741bbaecb01ae4c1db56d300466d82e2
4c05e279493a40640ab9b0db268da2beb251589f
7123 F20101111_AABJXE stanford_b_Page_098thm.jpg
b6579e4d0d8b77464f5453743203540d
d4e7e0ac1c2644d921a2f3e11e545a4c468f8192
25494 F20101111_AABJWQ stanford_b_Page_083.QC.jpg
ec117b02b42d2fcfcbbc1c912cb5ad63
8bf7dccd32289a8d8d219e46ccecbdb3a43cc012
7124 F20101111_AABISZ stanford_b_Page_109thm.jpg
f87545d8b24fe3ba79227f65143156eb
29d13dfc8ef7aca3eedcd428a969d27a05360210
1051986 F20101111_AABIUC stanford_b_Page_080.jp2
040647f751fcd0b510454be66038c01e
447549022064e61edb9dc91b71838b17c0eec982
1924 F20101111_AABITO stanford_b_Page_127.txt
7cf12bbcef111d6bb67bb8ed0026e478
4c09b7352b4012a6c5c17cf13ec1c5cbf00bf81b
25420 F20101111_AABJXF stanford_b_Page_099.QC.jpg
6d33965d8062af3693c71a34a237585c
cc481107daf7872516c95c6f9bf280a303e837d8
24266 F20101111_AABJWR stanford_b_Page_084.QC.jpg
3f10a178cbe2e0e9081d2e9b8f11ea84
8872f74a22c394084de211ecbf25a4052fc275d3
42059 F20101111_AABIUD stanford_b_Page_159.pro
4d92eacafebfe1722c2f2eb93382ee91
b8f9c158b4db75f41794218b407999e7e88c6d8f
90324 F20101111_AABITP stanford_b_Page_154.jpg
dfb7220b4b5342f4a6475a5115387912
3d7698e655f20bd8161078178f3d7b7602255f88
7215 F20101111_AABJXG stanford_b_Page_100thm.jpg
30df4ba4edf56168d715d512d2a45a47
2e92b4ced7f82b7d4fecc118da94049e385c3006
86959 F20101111_AABIUE stanford_b_Page_108.jpg
9c85c7b79a45c04e2f86259b4e7e5c80
b5c6a9e99e0f5a1df48db61ea48f6b8f1013ce01
26183 F20101111_AABJXH stanford_b_Page_101.QC.jpg
a3febb13168db9c2e748fa5e9c1a51d0
9e13d52bcfd07997d9a89893f55af013f5d6c47e
6615 F20101111_AABJWS stanford_b_Page_084thm.jpg
e062a4781b7f15455754a051adfa1689
ee76183c6358b6a272dc7f91645a73649bc98b92
7983 F20101111_AABIUF stanford_b_Page_091thm.jpg
254619c2da40b7e459457191485b3b49
19dc50207d70befa5ce6315f57be73443c38456c
4349 F20101111_AABITQ stanford_b_Page_074thm.jpg
b4a85d031230001e1d8c70a9b86763ae
b9e54d60772e68b14164eaf5272cf3b93993dc8f
24178 F20101111_AABJXI stanford_b_Page_103.QC.jpg
22c6fb0bcc2878c0d18a8364944196eb
8e47e103ca7d59b3796b0c2d960a9a5c28d2072f
6904 F20101111_AABJWT stanford_b_Page_086thm.jpg
590163448723d1de968ec712db33a54c
4f61397e127c00e8c37221109cd596c978d539ce
F20101111_AABJAA stanford_b_Page_062.tif
0977ddb89df554fd6f64e49928e3fb6a
64cfda1969142dd7a2a52a2e55389a70c3324b5e
82535 F20101111_AABIUG stanford_b_Page_099.jpg
2cd46fc7be101a6691bde0f969630c9c
7c9058a7b52e8f83a6e725ea5479a51e388986f5
1699 F20101111_AABITR stanford_b_Page_056.txt
bde912cdbe82ae9f4fce2ffbbc0f513c
6ef991c0d92135259f8d3f915a94fb6fc982ae2b
27168 F20101111_AABJXJ stanford_b_Page_104.QC.jpg
77107e6799b0e29dc296b09be6591e42
5cc6b5334f08dd48eaacedad7453d81ef87c73f9
F20101111_AABJWU stanford_b_Page_088.QC.jpg
9615037204bb173cc6a7a382c64cc63e
83120640be374a428ae262e8edbaa66e6ca0be70
F20101111_AABJAB stanford_b_Page_172.jp2
b969c72476268fd607eea6ee2b248c3c
4c1d2975ae0cdcc0ef17e076acd4ceee27ea4471
23138 F20101111_AABIUH stanford_b_Page_066.QC.jpg
9df4b13cd5f5d114d6df1220c508028d
15219c01bffd075b564c01c75cc57c2664a4dab7
F20101111_AABITS stanford_b_Page_014.tif
1f9d3a60768077dd24e45dc5d94e302a
65f66618cd6d9d66225c6bebeef2cabbba41d8ad
7619 F20101111_AABJXK stanford_b_Page_104thm.jpg
3171335b5a20ad1dd3c721777a748641
efc2b9a3ff328ed3b00d16b3707410c7138a6f0d
27652 F20101111_AABJWV stanford_b_Page_089.QC.jpg
76ab80a384bac8214370bd7c0ab3be83
8f9272fe119ceaeaa14c98482096f3dc7a42a3c7
77895 F20101111_AABJAC stanford_b_Page_129.jpg
38b0eec2dd95dcc264680e2d80a061d5
9163e206709978625421ba0508a32125e1ad1bd4
50670 F20101111_AABIUI stanford_b_Page_038.jpg
ebc7286309cff0d758758d79645a43bb
efff773e3b4d0d22eb035ba7b8334176efcaaf4d
F20101111_AABITT stanford_b_Page_010.tif
7c5abab6035af57b4f1ec475b3a26af9
bfdd0f149a483f5478a72eee154d7bfd9a19df63
7058 F20101111_AABJXL stanford_b_Page_106thm.jpg
e6a8d922bf6716f9b82d0c8de3123e93
6444e0232c26d47cbda752a3a18aaa192b3044d9
8020 F20101111_AABJWW stanford_b_Page_090thm.jpg
2303df1dfa0a5e3575074b5541ad98ce
ce921f065278f5ddf681e1303e868255d1b130a5
21869 F20101111_AABJAD stanford_b_Page_048.QC.jpg
4d49de7f78ea496a32a05a41bfc75d5e
acc31f1d80102c4bc296c27fcb4d17ed9c1f1015
74510 F20101111_AABIUJ stanford_b_Page_073.jpg
cad86c76d058e60db2fa68075f4deaa7
95703a96d0494571880b7c68103b7e6581f8eff8
27294 F20101111_AABITU stanford_b_Page_060.QC.jpg
546630999d412f23e8f82e89b0bfd2ed
8ea446cee7e6dc630e507d0599c63b1e04781d6a
19657 F20101111_AABJYA stanford_b_Page_121.QC.jpg
49f030caa22f3a51c20ceede7c3c2379
8b5a66ef897f62bed43e0fa36f76df07f139629f
24795 F20101111_AABJXM stanford_b_Page_107.QC.jpg
c27c8aa29dedeeae752eb916f2d4ec8e
9d076863cd3f656696662f9ba55f5e13f389482b
8071 F20101111_AABJWX stanford_b_Page_092thm.jpg
e2bf65847ef0cf8fcdf29849f874bef9
21c2ccb1c945d01bc56dcca0a23ac82309db2b97
67748 F20101111_AABJAE stanford_b_Page_176.pro
08b6946db418b7d62c5d056dbf11328f
08539402caa8e512817f5af9d1d758fb2f985df2
7284 F20101111_AABIUK stanford_b_Page_101thm.jpg
802c96cf2c1a4d1dce9e25e5b69e88d1
09348683b29a3a8cfc935449a2d944dd310b0255
27439 F20101111_AABITV stanford_b_Page_055.QC.jpg
08d45db5e9cbfd9bdb6f35dddb37f492
d1ff672ffb55fb3ae4ad202082083bdc0934d64e
27449 F20101111_AABJYB stanford_b_Page_125.QC.jpg
8c861708727942f0b702f6627c013487
98cffe3e962851e7058a02566ab4ef2e3cfcc9a5
6768 F20101111_AABJXN stanford_b_Page_107thm.jpg
0577e9c2977a16d2d04b67161647bd97
897fa6b26a8b9b7ed77195429e1b98b0e817fe1c
27465 F20101111_AABJWY stanford_b_Page_093.QC.jpg
002e842d76469f3714abe60cece55ac3
0d998e42389f8a28c1aa9615ba1474b08b8705d5
7203 F20101111_AABIUL stanford_b_Page_162thm.jpg
fcd267e452bc13be6640c12a947abb52
9a5311e65c728dbdbde6fcb934656059ba825729
13487 F20101111_AABITW stanford_b_Page_021.QC.jpg
a23e73db67b5597375de244c181ad361
857e9a0de4fe6a2fdad0fedcd5d234dba1a61b6a
73993 F20101111_AABJAF stanford_b_Page_135.jpg
3eac0003aade7b4b68827bbf8120b20d
81b7c12ca03e95daf89a5c4770881ebfcfad63c8
7402 F20101111_AABJYC stanford_b_Page_125thm.jpg
a6bd1fb3e46cfdee8a4501f23119cea6
c2dc7d2c34042c622cc83eec102fd402e26badeb
7608 F20101111_AABJXO stanford_b_Page_108thm.jpg
849b8da0b1d91f59ddcbfc55e297b7ac
bc781af0ce3da0f8d178002b1a02e0c672616987
7510 F20101111_AABJWZ stanford_b_Page_093thm.jpg
005444c0663e76eea3797d4cee380e2e
7da337b0cc5b75976ae718fcbe6b99aacf21876e
6738 F20101111_AABIVA stanford_b_Page_133thm.jpg
8568013b1956f92421c80cb880d23374
21b26aafa075e3c991efddd4909feaad507f7a06
1051984 F20101111_AABIUM stanford_b_Page_020.jp2
1249c17fb15cc7f125952a0751d2c846
0038e399884d28af19b86b4618be72253ec9bd49
7065 F20101111_AABITX stanford_b_Page_135thm.jpg
61cdb77273a469d26ce5f0b9d7cf5dc2
6b2bdad246823a7704182d235f05bc616353f3d1
F20101111_AABJAG stanford_b_Page_043.tif
e2c5b85ce5561eb7b9dd309c1ac323d0
bf28ef8e3fdaccf6ffd2032524c9f44bba24babb
6502 F20101111_AABJYD stanford_b_Page_127thm.jpg
18f530ab225423cfa7e63d5aefdadf93
f238150d6bdcb091b45cce7f28707ed7c571f9ea
24487 F20101111_AABJXP stanford_b_Page_109.QC.jpg
7a35e7168d550ccd805c0022a45af875
91c9fdb9f8ac1b7cbdf8baef8071334a37d6ddc5
17765 F20101111_AABIVB stanford_b_Page_120.QC.jpg
55e975f9aadfe04384011ae8157e33e2
c2800fa8b635af5829b96a07284d391191c23c8a
F20101111_AABIUN stanford_b_Page_066.jp2
021dd2261ecda4db8234745d69f96435
c9278b75b6aa7d814123ff7c81019cd1188fbfd4
F20101111_AABITY stanford_b_Page_146.jp2
1a72688f85603ba9bc8711c1bb995597
ebac78efc601eb6d9d66679a32267edb9f45ddf0
649429 F20101111_AABJAH stanford_b_Page_038.jp2
1a0352b1b2699bdb126e40b6489017f4
de1f94834db9674f606d444717ffdbc6a349f150
25951 F20101111_AABJYE stanford_b_Page_128.QC.jpg
7f49f1459fbd5aabf7079ef702156a82
ad1c33519ed72b392e8633e95518cc9097438d74
F20101111_AABJXQ stanford_b_Page_110thm.jpg
012f1da0915abcf9bd4cf7234367f2cd
d7dd476e995077446d94d78ca6b3b31a1b7a4e6a
F20101111_AABIVC stanford_b_Page_009.tif
0c1d1fc90c2d082380b44579065a4341
00bc288c1e3b27afac78ce3d3d55543782cb0ee3
F20101111_AABIUO stanford_b_Page_169.tif
73fd29650ca6f614b3e59a4bfb67e33a
66074b36956751d54e5592fcd8bce7b49ab37759
102899 F20101111_AABITZ stanford_b_Page_177.jpg
920a456784f1a1bd3e50088eba13ad17
579276964a12b90780f2beac9c7d631247eb600b
F20101111_AABJAI stanford_b_Page_083.tif
ffcbf5bfb8873f06a1d5e2e329a3e269
9d75e9370ef29f721978b5433cdeb0a92d6c1623
25759 F20101111_AABJYF stanford_b_Page_129.QC.jpg
7bfbc13c379835094b3c99aafee6f537
d1fc72c287364162c663eb2aa70aeb2e3b5ca56d
24417 F20101111_AABJXR stanford_b_Page_111.QC.jpg
20ed998ea766df185392ab1d8b68273c
165fd866a0d9f6b6a8b2f08ca557f1bb5108fec0
7497 F20101111_AABIVD stanford_b_Page_054thm.jpg
1c8c21bc07dd59c78b40cbe8cf30085c
561da9f6885e65e85e3156b02651748c41dc1948
25346 F20101111_AABIUP stanford_b_Page_106.QC.jpg
d7e118b3954277e1843049b57936b587
1bf5e6d72f17c9b70e5089f19c7f1f2e6e66e55f
1051971 F20101111_AABJAJ stanford_b_Page_124.jp2
89dc8b0013b4106c9bbf5521794ed13f
bf41c1eba5cba04afd313040654e2a7f0c13c9ec
6886 F20101111_AABJYG stanford_b_Page_129thm.jpg
d26c3317c18f99dc52cf463f01acc696
35adfef60f493e456efeed5a93455cc85eb7f499
6979 F20101111_AABJXS stanford_b_Page_111thm.jpg
21637e29f6c734be2fd39fca7b184bfc
d1b5e0156370ccc200e3053be0f735b9789a87fe
1890 F20101111_AABIVE stanford_b_Page_070.txt
6093897994a9c941a7086dd2ef88238f
57a3306037c9a92d72aa83dee0c2da5ada80d89f
28099 F20101111_AABIUQ stanford_b_Page_040.QC.jpg
b7df5daacc7c4204342734a482d3b7c9
3f502d185978d1b7bfe45959fca205c3544bfb7e
1035721 F20101111_AABJAK stanford_b_Page_179.jp2
8abf8d37523354a67165e820373f364d
31a056bd17e844f05593cc7f1ca6ea9069de6c88
6723 F20101111_AABJYH stanford_b_Page_130thm.jpg
cb67b5c0654ef723dd21710a6c004e65
a1a3f6a9253514718989e917c4cb32041ef3e65a
55021 F20101111_AABIVF stanford_b_Page_110.pro
d5285306cf9c46e684686b3ac454808d
6816cf5155f93701c960388531872e8eec680b12
54477 F20101111_AABJAL stanford_b_Page_049.pro
bcc0f3591c92274614f888c81a6aa215
681bf5952d5266673155c2e9dbdd1d4641ed0536
19281 F20101111_AABJYI stanford_b_Page_131.QC.jpg
38ce03a046a1ee926cd77b28018d855f
5d5b67eb33fe9c9b1c9aaae5b910fe29713e6f2f
26856 F20101111_AABJXT stanford_b_Page_112.QC.jpg
a0c797a9b0b8b9edc77364706823f618
c60e2b33261b9740256d158eb2aeb4268c6f99e9
25092 F20101111_AABIVG stanford_b_Page_021.pro
6e78c50f15045702004a4a7c7450a296
7a42a9adb32986ee82055b416d1203742679058a
1051961 F20101111_AABIUR stanford_b_Page_089.jp2
9af26274f6ecb773b4101d7fe57a579d
691d591bc53cd4f7286eac91d34285745c2587d4
21789 F20101111_AABJBA stanford_b_Page_070.QC.jpg
41508a527e9f62fd1f0891d2376fbade
4d04ecfc726279d39ed037a4f9ad6885ccb772f2
21714 F20101111_AABJAM stanford_b_Page_005.QC.jpg
c23fbf1ea87220684442f2cabfa21ba9
c5694a1011ea8cb88a5737575bdcfe5bdd0599e8
20144 F20101111_AABJYJ stanford_b_Page_132.QC.jpg
6b92ea9878f925257d60cebcba21368e
50127fbe183aae14d6fb26bf0e5bf6815efb3c53
7122 F20101111_AABJXU stanford_b_Page_113thm.jpg
f894fb54ec68489633e7a9da7ffc56b4
7c74c27cb1978a010d327de13be19a9aeacecc21
23389 F20101111_AABIVH stanford_b_Page_119.QC.jpg
87f6642cfb61595d5955673a45cdd194
96b4ec733d2cd842150289ad50a2dbc72deeb9f9
28053 F20101111_AABIUS stanford_b_Page_043.QC.jpg
ef40c01a0424dbd900dcf90f814e3e1b
787697539b7c3a3e27ab0648edcbf967c747f15f
49979 F20101111_AABJBB stanford_b_Page_166.pro
df472502f9072b6d50923fd83dccd290
59e36a0397e800be7e94b4c94bfa88572e50e61c
4398 F20101111_AABJAN stanford_b_Page_038thm.jpg
a03b13ad470f6a5e83ec2db62a9ded3b
5a3ca4422425e1ac017d7d9e5b472d330323d998
24219 F20101111_AABJYK stanford_b_Page_135.QC.jpg
e311d54563271af50555281379b4c65a
f24b858dd253c9f3ad668e28640ae1fe8fc65306
7553 F20101111_AABJXV stanford_b_Page_114thm.jpg
63516cddcc1113262b14ce3e2597a765
296df58aa6526eb9d80c9c187c801dded65818ef
F20101111_AABIVI stanford_b_Page_081thm.jpg
0e435472539c4d73aa5d7b6affa08db9
ef0668c66667b62404a10acc3e0ecf1b021d6cf1
20552 F20101111_AABIUT stanford_b_Page_087.QC.jpg
97178c3049468e386d50fa2169edbb60
79503241fb5a683c31fc420f1dd00667378e363e
27264 F20101111_AABJBC stanford_b_Page_143.QC.jpg
e6ac0ec3d422f1860f5981f5f5a92b47
b89cf96f027d8e47d7a5cfffee02122ed03a89c5
F20101111_AABJAO stanford_b_Page_128.tif
6498c662ad1ab93b0fdebad3c26cfd50
46419eae08559338b6ec89b29056e698ab67fe99
28169 F20101111_AABJYL stanford_b_Page_136.QC.jpg
9187b38d78207d5c422a11ec2820bc09
8f2404b788742be6264eccabfe016146079468f4
31777 F20101111_AABJXW stanford_b_Page_115.QC.jpg
29def224ad814d483392ec5dcb4e6964
5d6c6c668513c35a73820795682e33183c9c0e91
35716 F20101111_AABIVJ stanford_b_Page_156.pro
b8bab8bea7907e559ac3c7b0141c9377
88bdf99afad3a1a62c561d74c0866994f5f7795d
F20101111_AABIUU stanford_b_Page_077thm.jpg
bbe9f6134ec6aa5158b001129140d7e7
95aeb3e90265f143764b7cf1304d3bcaaf607924
55227 F20101111_AABJBD stanford_b_Page_117.pro
e87e141bf3f28a2b0a69cb4415761656
cf77b429d79707af6ca72263fa47c9192d424129
26940 F20101111_AABJAP stanford_b_Page_122.QC.jpg
fe346dbd04bb78f692f4a1df8daafe8d
6b3f9d2e9678b01bf5e881e14a3642393f52b3ac
7170 F20101111_AABJZA stanford_b_Page_153thm.jpg
55a6660f03e81c81154b05c6cabcc4aa
d4a8362cb6a9eaf7a243e747b289a176a8465ef4
7639 F20101111_AABJYM stanford_b_Page_138thm.jpg
b9826005fd309ca9cb5aa213e76c784e
5141219b1580440e393fb31a17af976681c21220
28272 F20101111_AABJXX stanford_b_Page_116.QC.jpg
8576ff3383f13e4d73d1713c8b99ed94
1cc08891a4df09be6261d06350365d29affc8b1b
50534 F20101111_AABIVK stanford_b_Page_076.pro
815d3b8a36b58d39fdbb32d32ea7e8c9
b23c89840d9275967dbc648fcd777a328d3c0099
F20101111_AABIUV stanford_b_Page_018.jp2
b8b7e3448f365c6a6e060a29a2edb9ad
bab6453eec15225fd08e4239c9239d3c26f88518
2688 F20101111_AABJBE stanford_b_Page_170.txt
175e2cb73a8e64fd51c4508bf09d6bb1
d0dd05bab2c56c864d4a152a821ec476aa3ac90c
29195 F20101111_AABJAQ stanford_b_Page_154.QC.jpg
9f5abfbcfeb10046ad48dd048d2dafcd
826a7331ff8bbf46e2e8b9c0546695619a36a032
22280 F20101111_AABJZB stanford_b_Page_156.QC.jpg
7c1e70b824ded40e9b4f26d167036bea
0048310187c302cb9876edd444eb04c2ce8a918b
28792 F20101111_AABJYN stanford_b_Page_140.QC.jpg
cab0fe21f5609974abb1f0da896ad9fa
fd9357ee705c71fdbedd6b8c90a4bd897d97b055
7339 F20101111_AABJXY stanford_b_Page_117thm.jpg
64a54fee2073311dc1ad39e97c47feae
a8a48127b0f862949e945fc7efbfc24b9dadaed6
6850 F20101111_AABIVL stanford_b_Page_053thm.jpg
c20f99b68b332b2ef36486840d720122
12bede5c721ba62e959bc5e33ca5754108dc78e0
F20101111_AABIUW stanford_b_Page_106.tif
072535efcba3dbf93b714cb81257bd60
07879c0ed8517d50441cae5f0d9628d73b413a0d
F20101111_AABJBF stanford_b_Page_161.tif
73930982952aaf6d4d6c796c74c493a7
e27d2a52257782acff3e89fd82f3e2d5909a5462
2206 F20101111_AABJAR stanford_b_Page_031.txt
f452e275537651978d8e0caeee4c6ed1
eebbebba3cebd6b6eaa791a6c9a04de1d12e5361
22779 F20101111_AABJZC stanford_b_Page_157.QC.jpg
398620b5468939694e9211eea90e1298
7ac5e57c44502868f658efdd572c916e69cfe209
7851 F20101111_AABJYO stanford_b_Page_140thm.jpg
3e9d2129da6eaf2a21478c3e3686b6d6
9ee29fb9170a7f872c6718d07ea6027d3c4e6c0c
5121 F20101111_AABJXZ stanford_b_Page_120thm.jpg
e41093bb6911889262776a7b6106bec5
dcc7e14b39277ec793812a7562a2c7f7dd0b1a08
F20101111_AABIVM stanford_b_Page_033.tif
d873b81e44aaa30f1af5fbdc8a584713
bed08270289409091f9acbcf4f0d2da82ce180c7
F20101111_AABIUX stanford_b_Page_144.tif
6779a73bc5d10865fc0134190511fbc2
cfceb5fb1ebbca047a265d7b62025577d8f9c570
27490 F20101111_AABJBG stanford_b_Page_105.QC.jpg
94d1f5bed5328f6655db209a5d564201
d1fd2ca8baecf715e60bbe9f85a85d75863538bd
F20101111_AABIWA stanford_b_Page_045.txt
06927334813cbfa4f6f2177d2bc73996
3b7c2b96a73fb6197058f8f5c506887302413400
F20101111_AABJAS stanford_b_Page_074.tif
d7100568f242a1b66265c857c15788ee
1d8f93b0785c28bb6bf9b4147ff21eaccf5643f5
6382 F20101111_AABJZD stanford_b_Page_157thm.jpg
f6c480a7dc7e4d63c88e4f692a11f9b8
593382b2b482096e9b3d8c385e6ec9bcda9e34d7
8464 F20101111_AABJYP stanford_b_Page_141thm.jpg
ed775c45b84a4aef362e10e4556d47ee
46373e5a0e2e703fd5b692f1d66bd71ac579c54d
2421 F20101111_AABIVN stanford_b_Page_055.txt
5f8b1da8283ffa3de39e3ed853d58268
e178abebb9b484ddaa06b1a8e5cb34292605b3c4
F20101111_AABIUY stanford_b_Page_137.jp2
07aaddba4118a6db118aa1e77310dd53
25529afc54e57670a19c26fd547b4d2b662b20af
27870 F20101111_AABJBH stanford_b_Page_027.QC.jpg
205f5ff181bfb89df5d315609efee8f0
5b18bdfa6f9b39dd6eade5e1be8d192438637d33
15224 F20101111_AABIWB stanford_b_Page_074.QC.jpg
6033683a045efb3f01975b182be55333
a4dfcd64e6c7623fb756c11cd9411f9836652bf5
22876 F20101111_AABJAT stanford_b_Page_127.QC.jpg
35293b67304fafa6e864449237bdc181
69e02cbda1c08866bd822985eb6e9ed8b1d21c48
7440 F20101111_AABJZE stanford_b_Page_158thm.jpg
546f151fd017fcedc874e14cec6c64a4
1844a6d8829458df2513c8f55a61c31c8bbdd39e
7526 F20101111_AABJYQ stanford_b_Page_144thm.jpg
94b36f422082410c598b977afd131ab6
1d54b75f4204d8fdea1d55b73c7d3ecc8ccfde01
27767 F20101111_AABIVO stanford_b_Page_137.QC.jpg
56d6d08f3cc532ac3966ae4e78771192
3027e73ca96d6e441955417c7b3d2cc5c823be0a
55590 F20101111_AABIUZ stanford_b_Page_114.pro
3c1ac9dc8f0449364e8198eaed6bce91
c76897c865c9e87e52b5b1dd010f906d449ca826
87876 F20101111_AABJBI stanford_b_Page_144.jpg
83023a1e26bbc7a5b51c4dbe08e7c10c
c53e59ab714604eeb10f83f83b432d884c15657e
68166 F20101111_AABIWC stanford_b_Page_004.jpg
869e7ba6a1e4c3b04ab9f114bbb783c0
eadaadda6889e712709b39a52c073ad9e55a4efb
77488 F20101111_AABJAU stanford_b_Page_086.jpg
9a9cfc1bf3c0a94afb68d4590721d1eb
e829a24b288bf63af3da763ed6a45b4c00a9e5a8
27972 F20101111_AABJZF stanford_b_Page_159.QC.jpg
0b8219f22487f17842c63350a889b6ff
3e58a6435dad3895159db140157f9fb861c6b128
27025 F20101111_AABJYR stanford_b_Page_145.QC.jpg
5a2b9abcccb98f3d8865e20359120df1
bade7811b4e3c9b8ceea36f98d6448becca76198
F20101111_AABIVP stanford_b_Page_060.jp2
3852249120b57858e6dfb7d283082530
5afd5fbf20872743efb6d1aa19065c00a70abfb4
F20101111_AABJBJ stanford_b_Page_131.tif
6e23d84afceadd66758bec2b806a08da
6df5ae20a85a562704307d5ca82cf0ca11d9ce3e
1999 F20101111_AABIWD stanford_b_Page_063.txt
86726b6815ad4096186588c4c5d9d16b
25964b8d19b2bece24043cfe40457b7f03d5a0be
20507 F20101111_AABJAV stanford_b_Page_056.QC.jpg
329aefe3bf5788de0dae0c6b76d23b31
8edec5e994ba53f4e389e064e047f75ca55003b1
16235 F20101111_AABJZG stanford_b_Page_160.QC.jpg
0b6e763e505baf984a7c1e0da9d4e666
299216f586d687adb0da56f092afbecf8d593c24
8239 F20101111_AABJYS stanford_b_Page_145thm.jpg
7c72009236d0eb9460e16d9cb508088b
5335d39a14067f723eec3adbedc7b28de8a4a8e4
7350 F20101111_AABIVQ stanford_b_Page_089thm.jpg
ca99507579c4712257442f7c0d26aa76
aadc6ad17875938e03cddc2167f47c1b92040e54
F20101111_AABJBK stanford_b_Page_122.txt
5b11d59b7c179033b2d55c162375a52e
f535c5137e5a5e9ab48bc3b37cefc3f1503c8199
2661 F20101111_AABIWE stanford_b_Page_174.txt
f155c5e1565d64baebc349c62807c532
a853baa50b96c1bff3bbbd24bafe74e60a2bb8ef
F20101111_AABJAW stanford_b_Page_086.tif
0d468fe7043adcc0ca55360dbc77a078
4ee8d7b8c2d7880209fc8e240ec4c7ab32fc6f01
4943 F20101111_AABJZH stanford_b_Page_160thm.jpg
a90ae9169c3bafc2318383653b29762d
54e0768a731b38579f68be39b478a439d9254a6f
24945 F20101111_AABJYT stanford_b_Page_146.QC.jpg
1cafc49733bd8b9e5e3416647732be8d
8fdc7fe67740d608fd08381bc387f5445e76dca2
54198 F20101111_AABIVR stanford_b_Page_143.pro
3a114b640c15249e0ad4c8576e1738e1
af1eb58c7a00e3667e7a96eafe965126c846e09a
1051939 F20101111_AABJBL stanford_b_Page_041.jp2
b4bcb175cdd4a57668716227e9b382af
1a0bfe1faf118b7b388d1a27bb7d4f8fd00da4e3
49380 F20101111_AABIWF stanford_b_Page_107.pro
5c7a027f0c3215ba9160c21508968d22
9149fcf2f94084a572359df304ec0d5dc35ea3e5
F20101111_AABJAX stanford_b_Page_153.tif
609ea2f2e36d1330eee17378aec45f8d
204453d35e543f093b3033fe5f86e26f72534a80
27708 F20101111_AABJZI stanford_b_Page_161.QC.jpg
dadda4863de3ceb5d64f272f75006b5b
f05735f8ac72a23b4cf8dc2203b73d3b2dc9932c
48322 F20101111_AABJCA stanford_b_Page_084.pro
a738bf0ea12ce22e2a39308591b86e3b
f0675ecaef432942d37c6379ad5ec76d79971577
27622 F20101111_AABJBM stanford_b_Page_041.QC.jpg
0f6d6313612d3fa363a1e7d5c9baf51c
6f172ff377e3c80e51671a52ddeffb0240446bba
F20101111_AABIWG stanford_b_Page_028.jp2
53eae7bd3cbbef1befc9ce2e198a2ffe
fff52b5df44f3041cfd3df01db669e0aa03326c0
83455 F20101111_AABJAY stanford_b_Page_058.jpg
add034956019e8e70c2bf8492501bf2f
d0c41a825079c3d3fc9d0506ce48bdde518d3aa4
7334 F20101111_AABJZJ stanford_b_Page_161thm.jpg
044dd1a6c8c5db726456026384a7a287
6afa53d75fa6bc845c824b436890027faadbbe00
7054 F20101111_AABJYU stanford_b_Page_146thm.jpg
e67844889266df106c654246838844d5
5ed792bd1a08726eab78d4b4a97ab0e08b73e274
F20101111_AABIVS stanford_b_Page_149.tif
15095bbef4b00ccc7f3eccd50cd6ba0a
fe1e05beecb57c939118147f8c0d6d921882f425
6996 F20101111_AABJCB stanford_b_Page_166thm.jpg
b14bae107f99041830f7c7d05dff65f7
1d21ef6610bcadc5d56edd35fda0d2cdafd6f34e
7981 F20101111_AABJBN stanford_b_Page_115thm.jpg
cd9e66eab7d1689eb3216776a513f7f2
734bbe332110963e019123682d18752b3865ae43
F20101111_AABIWH stanford_b_Page_155.txt
13e43d087ca6ed5505489660b22dadc7
526fdb828c7c30d4e5ac4e27c24887b0581e6aae
1051937 F20101111_AABJAZ stanford_b_Page_016.jp2
7584442db77c0b04c888d6cc122f887b
0dd13f6b7b999a356e363b9485e71e62745abec1
27428 F20101111_AABJZK stanford_b_Page_162.QC.jpg
1f9ba9b9a3bec2eaa066706e0c52c578
0290150704201614fdd826888139fa9ca58b81a9
7070 F20101111_AABJYV stanford_b_Page_147thm.jpg
a6b2417cf6bbe63f20079d47e4ebe23d
aeb94080f822b9915eea4ccbc87d1bf186d367bd
1051934 F20101111_AABIVT stanford_b_Page_023.jp2
06373a3f31a8485c6fb02532f8ca727b
def33e8c892ea4e30f6eb2e3a0cd1d446f0055f5
6569 F20101111_AABJCC stanford_b_Page_149thm.jpg
93c7633646a4cef8aea36f741d165b6c
4746f5e6ada9ead6df96b2133c014635411d9005
24825 F20101111_AABJBO stanford_b_Page_113.QC.jpg
e435aa4d7551016a8137faefdeca31af
3b703c0b497a993c1d03a58befa67d4bbdcbb94a
1797 F20101111_AABIWI stanford_b_Page_138.txt
8998acdd773d59f1071137eb2dc9c35a
9afe41b976882d8c42724d63b1ed13992dc87072
27629 F20101111_AABJZL stanford_b_Page_163.QC.jpg
71c137787bb9cab3bbac387ea5f2091e
c9016a442cf5177a478c38afd00c9835e5db932c
6834 F20101111_AABJYW stanford_b_Page_148thm.jpg
7d922ceb53492785fbc4ae4ae820fdb5
e2e5b26c1c0e032460221172aba2388ab4c22b2d
2126 F20101111_AABIVU stanford_b_Page_126.txt
c52110e2fa43c9d9d1e0f2223befcae7
b8c5b7fc781c2d8fb9bec524ffb9fcd02d67a4f4
6575 F20101111_AABJCD stanford_b_Page_179thm.jpg
caf6eb9a4732b9f31111f02e8e1e78e2
7699443cc75cd9418043124617318a3126327c93
27982 F20101111_AABJBP stanford_b_Page_144.QC.jpg
df8b8b7b594afc101036be6085132508
8138955a940116a30031af67cd403cd2d14099c0
F20101111_AABIWJ stanford_b_Page_076.tif
03214fdba3254063f30c02f02d175ee2
bfeefafc3c4bc387636c9112db98558db8cdb449
7305 F20101111_AABJZM stanford_b_Page_164thm.jpg
f13894a45e34712547e396f39c82b2a8
eab64d2ecfcf7e5fcfe6d7b18b94c85838809fe1
27997 F20101111_AABJYX stanford_b_Page_150.QC.jpg
fb5325d34abfd42e952cffdfe88b46a2
eba019ea547db0b86e5e4c191183541dc8ffed32
1051980 F20101111_AABIVV stanford_b_Page_078.jp2
7637b8781affa867b721ef4615cf1504
17feede8820cf9c1de9a8dab9fa5d166f81e9a38
70813 F20101111_AABJCE stanford_b_Page_059.jpg
08f24ba88838ead34841acabc0f82499
a9f125c284083f949c34128253012f2577265f31
72128 F20101111_AABJBQ stanford_b_Page_077.jpg
ad4d9805e87431f12ee8763790229108
594e168ab284227c51d98e841e06ac0d30cf0b0e
62604 F20101111_AABIWK stanford_b_Page_087.jpg
506dc5042e9913c6f033c0b509506ab8
222f036c193ed02fbb826a8308e4bfb769e80b1b
27016 F20101111_AABJZN stanford_b_Page_165.QC.jpg
069c9320d5236d56eb63103700db4ac4
c40c974032c83a536b0bfaa10109b45b1f13dac2
7411 F20101111_AABJYY stanford_b_Page_150thm.jpg
bdfc35e705fe32084b37ebfd9b568b0e
c34c67d5377cfc6fd64bb60663dcb4f42850f3c9
772492 F20101111_AABIVW stanford_b_Page_120.jp2
2066db7d3bfd8c810f3d8dfc3bda52c6
eec85338ca4c713a38554afe0012e3649edf432d
27228 F20101111_AABJCF stanford_b_Page_085.QC.jpg
47cd5cbbe7eb15480a5a81353206614e
9435586fbb368889e5fe2a137c981a0336a3f0f6
132622 F20101111_AABJBR stanford_b_Page_003.jp2
6929bdd4f1eba1b131be72f2394c60c7
96f81abe5f03ad7e32761fe583e2b00ea9013167
656925 F20101111_AABIWL stanford_b_Page_074.jp2
ccca101edef807e0a24ef58f098a8bab
07332f1bdd5ac6e2212bf9bd01173b8d5912329d
7341 F20101111_AABJZO stanford_b_Page_165thm.jpg
cc15695fc7e845a1c82bf81d1ff8d653
4e86a2840c2698306236d047e42cf8ff1238f2be
6636 F20101111_AABJYZ stanford_b_Page_151thm.jpg
345717cba8ab993c0d4f73ac3dd2aab5
b3c6067dd692c4294c2647c069d5c1c8c03e4b34
72528 F20101111_AABIVX stanford_b_Page_130.jpg
475eb05a4297bbdce3f1aebd683c4102
a857e41602c0978784cf7d663346795d38f1588f
269653 F20101111_AABJCG UFE0021970_00001.xml
86ee728474a0e91168fade4bf2dde2dc
81311f043bacd1c1df4ac44e90a32aaf7fffc026
7270 F20101111_AABIXA stanford_b_Page_143thm.jpg
e5b33926b01bdf362868879c3cea3ca4
0945f3eb188b6315a1249724b29823d59d13458b
86016 F20101111_AABJBS stanford_b_Page_150.jpg
4f84b0781fcdd8401b0d38beb8c4b69a
ef14688676c123702e3b2c4a286207c3a9a6cbd6
26193 F20101111_AABIWM stanford_b_Page_098.QC.jpg
faccd8e30dd417b65b27be59e7a88a4d
4e0068c296cce4053874ffb7f23eea3e34292019
25204 F20101111_AABJZP stanford_b_Page_166.QC.jpg
6cf2280ce9c74a19fe995316bada57e5
ce142cb397b2aefe23121586966dc46980f94495
22951 F20101111_AABIVY stanford_b_Page_152.QC.jpg
36e15792309f35c5f99111c001a529c8
5d40d2259977ffc315087d093745df930e5f4d08
27712 F20101111_AABIXB stanford_b_Page_164.QC.jpg
0c37a39d83ca1402d5315288ed8b4c82
51d8937824cc8ed0acc7b72832d54a9b384ae8c2
24871 F20101111_AABJBT stanford_b_Page_142.QC.jpg
7cfb046f1193f4ab7d8f069c3e2296f7
588a32fdae4f3dd877efe8bef4d147930bb732e4
F20101111_AABIWN stanford_b_Page_029.tif
8bdbd937ac90b2717e66fbf4f691eddf
5f79d826e3a5b84af23178767e78076d601bce59
7579 F20101111_AABJZQ stanford_b_Page_168thm.jpg
2a4ccfeb8be74cf161c901bbb6b90a41
8fabdf665252ad8939430b663bcbdf6395786d05
7319 F20101111_AABIVZ stanford_b_Page_134thm.jpg
ac1fd099a8204d0407d97f948ce6e7b3
5eacb854fd7f50e79a0c9fa4cf499e56dbda37a0
89316 F20101111_AABIXC stanford_b_Page_055.jpg
8d0b4d3bd15c6b7f462c94036b9e9cf3
41193070bd402ab573a5254496dcbeda6dadb7ce
27442 F20101111_AABJBU stanford_b_Page_141.QC.jpg
1edf96a06aa2fca0c79c3fba68a98578
cbd3b72ca545274e2eb960bf7d2a39de6035060c
46782 F20101111_AABIWO stanford_b_Page_074.jpg
2c8d708628f1dcdb794e8843a901a932
bfb7623b81becc6994574ecdef5549b959538a48
30386 F20101111_AABJZR stanford_b_Page_169.QC.jpg
a8094b58c11c996be217a2ffb598859a
d9c5c2bdd6f080d19c73675bb93a47c390542e5c
26705 F20101111_AABJCJ stanford_b_Page_001.jpg
5cd74793b3431372a41b56be6edb68e4
ce9206ac19b7eca41144587c31ace1c269603f42
7789 F20101111_AABIXD stanford_b_Page_079thm.jpg
6b46b9d9c6c9ba2da797865e17821603
41d107daff18b2b36f73e635efe1623bed6ac94d
28070 F20101111_AABJBV stanford_b_Page_019.QC.jpg
77ba45d0020fd0404c859ce36006ca5b
c72c3d029856b4cc5a0fdd0010ca22aa22ca54ae
6902 F20101111_AABIWP stanford_b_Page_059thm.jpg
5a65407b8d46cffb869d260c9a515e09
dcefef63ce5956f6c92baa2c64a761bb763d5e26
30031 F20101111_AABJZS stanford_b_Page_170.QC.jpg
65ca8bf7c1df1f8a5eba4e9c689087fc
a93cc6f9b44487c007d654f87d113717fb4f49c4
10340 F20101111_AABJCK stanford_b_Page_002.jpg
8a2cb5cab3f2f7b62074d026ffe58c39
0abfaff645c45b36796a7073e86d2afac3fd27d7
1022322 F20101111_AABIXE stanford_b_Page_048.jp2
e3e9cc0629755c4da9e50c4127ff9b57
7cdf54d46bb89a33b4dfa9b54278deddb2caaa17
5642 F20101111_AABJBW stanford_b_Page_121thm.jpg
54145a8d59f610826afa295290d8bd02
6dd98e54aa9ea9dd0a2965bee9f5cac9ab1cb459
27662 F20101111_AABIWQ stanford_b_Page_078.QC.jpg
7dc93aad07bf3b2170d54d15b309d14d
2aaa8fe1380a07eab5ec92bc62aba376ad9da06b
8227 F20101111_AABJZT stanford_b_Page_172thm.jpg
429d4af5dd93a2872697652ae7baaa2d
9f33151b914da9aaa1c9b8d1461399caef55cf12
78344 F20101111_AABJCL stanford_b_Page_005.jpg
4ef0aef7e467a8b90805cb1eab472ac9
52311d09234abe0d08a2929481851a4b193af3c9
F20101111_AABIXF stanford_b_Page_088.tif
b167710ef65e4abf9a88984540d5114f
aa6c51163d81a7b16ae22f7b8c32788fd7476fb5
1144 F20101111_AABJBX stanford_b_Page_038.txt
39a0a567d2cdf842a45e13dd91b6145b
2507f70d517875af85caec2c3549109190139ca7
1051964 F20101111_AABIWR stanford_b_Page_098.jp2
94e8e347019c5814d37e30742fa4957f
bfc41bb3ecda76cddad40c7e9279522b14293c83
30058 F20101111_AABJZU stanford_b_Page_173.QC.jpg
8605953d5c92d632179f0dcaea15538c
af529089d101f898c0a2ac9f1c641dc2abd55c69
91563 F20101111_AABJCM stanford_b_Page_008.jpg
d11db14fa914c17210375aa791e5042c
dead50e13d1c243b5dbf0a3f62f06092988f996f
2169 F20101111_AABIXG stanford_b_Page_117.txt
b40dc94e687392cf52010a84b4597129
0e1a41bdecb8dbf69746cd89d681687ffc3cd0db
F20101111_AABJBY stanford_b_Page_058.jp2
789225864a17461f9840b2b14f58a7c0
830e6947740e1fec8797d3bcf6c881dd4125da9c
28447 F20101111_AABIWS stanford_b_Page_031.QC.jpg
b88745095ed8720e1ed69ad43d6e0dd6
c4b4cc0364de79dba5a502183c233083d8f5f26c
84979 F20101111_AABJDA stanford_b_Page_032.jpg
c1cd57f5a1894657d0c58da741c06415
6dd1e21693e53ba0a7990f778fe01d7f93ab5900
92399 F20101111_AABJCN stanford_b_Page_009.jpg
2405ac3917af33948a57c4a5f2410239
4a473d52125361195d7383e4f09bf09333f619e4
7385 F20101111_AABIXH stanford_b_Page_102thm.jpg
610e8df0a1e8acd0308bfb62d464f386
11bdfa682b5d15369131147071214fffe92e19a0
48276 F20101111_AABJBZ stanford_b_Page_106.pro
538feea5b25f4cd3bae9822db1d26e02
ce4155074ebce4150d7478e07eac320466d215a8
88915 F20101111_AABJDB stanford_b_Page_036.jpg
b6981f7633ffe4b7d5257c5131f8af64
3e206f637af8a1ada615aaf6ea7319a81211ecbf
7940 F20101111_AABJZV stanford_b_Page_173thm.jpg
2ef17c519271be126689e8f39d42d132
beacfd6593361bd2d38f6a415779c91bf48735bf
97541 F20101111_AABJCO stanford_b_Page_010.jpg
b24d4ac2274c60e8cc3a2a6c168586be
4c5b2a893fac39854e6dc6d69df1bb633c50c48b
7277 F20101111_AABIXI stanford_b_Page_032thm.jpg
635383cc97b1b88d28a1a0b6d4fc8e1d
b6e5ea9f51b889842ac5c0e713075ae83297ff99
89337 F20101111_AABIWT stanford_b_Page_034.jpg
ad5cbbf1930ab7ffc970c83c8c5ea7e1
14331f8daeccba317cb5fbc296f80b901b52afc5
84776 F20101111_AABJDC stanford_b_Page_037.jpg
c6a6756aeb05a3fe8672686a05e096c0
159c39b941d5259fa8ffe3afd06805a82069d4cd
29781 F20101111_AABJZW stanford_b_Page_174.QC.jpg
21a8ca548980ec08937584500785d390
00335e6cba2a01e990b412eb9b18454b90b2068e
85822 F20101111_AABJCP stanford_b_Page_011.jpg
bd1e949cd32e886d02abe123747806f0
33a4a1a488daab3f17ca47dbb18f59ae61f9ad23
F20101111_AABIXJ stanford_b_Page_178.tif
5c2d0be00751c3bc62b6e379a92270b6
c820f5f99315dab1f25a11388fe46a3795f3f4e1
27559 F20101111_AABIWU stanford_b_Page_153.QC.jpg
3ee55ac546afcb8d133052b2cb8c97bd
0ca8817dbbe933463145b50621fe995bfb19002d
87301 F20101111_AABJDD stanford_b_Page_040.jpg
fe1981ed7050d3b4dc3027917a6c318d
33dbfdd94501a8ec93de3f2c29365dee2ee8cb39
8242 F20101111_AABJZX stanford_b_Page_176thm.jpg
ff2d4a1410a16b2f4206457a0fd6c344
f3e76e4f6a367c313fe7e3baab1bc619c4bc8544
83128 F20101111_AABJCQ stanford_b_Page_014.jpg
7178b99d9d4b94d08cb2d9edfc14faab
5b9b9dcec70b4879ee9b765d6e90c7b077fda85f
82956 F20101111_AABIXK stanford_b_Page_095.jpg
bde19ace586ec7c5d664f7f079345d2d
646fe3b039b09aff40d34e15710762f3b986d17b
F20101111_AABIWV stanford_b_Page_158.tif
1862ae615c5d9b4774f5e731db11a1a0
6c6f9f4128ac09b8756bd0518d98ae5194f79d2a
85501 F20101111_AABJDE stanford_b_Page_041.jpg
ad28fc91ec1b1c0c469710036a2c613d
a685f23ca1c741539bc41ccc044dbc87501142cc
28514 F20101111_AABJZY stanford_b_Page_177.QC.jpg
05686db15785fdae9b82eecce83dd593
fa92a2e63d4901c3d8fcf499fa7e3f27dba04874
85568 F20101111_AABJCR stanford_b_Page_016.jpg
133a8a7efedfc9a8f7d394edf36b9989
726c98370e1c5038757b6abedb7aca403c997506
57707 F20101111_AABIXL stanford_b_Page_024.pro
13c03a54cf5c6263787976ce0dd52c6e
82d4a2b2aa96bed2f88ec14b118caaaafc2a78a6
F20101111_AABIWW stanford_b_Page_107.tif
701e2421045e5c3b4c58a3630e2ce45a
788635312f6c3deff78cd3f6e3e4c3dda0e7a3d5
83674 F20101111_AABJDF stanford_b_Page_042.jpg
5ca13e533e01d3e1a22448b659d576d1
6e88b724df07f59aa732c12753b9a867ca263f0c
23705 F20101111_AABJZZ stanford_b_Page_179.QC.jpg
1fd1d56cbc56ebbdaf62c5a8ad99a39a
b014a106589ba4f0b33974fcdcde1102bd01c8d8
23698 F20101111_AABIYA stanford_b_Page_065.QC.jpg
da1feb2215a498f168bc3744078b948a
4f3e7e97df383c8d1d636f31c50a06c438996ae4
78125 F20101111_AABJCS stanford_b_Page_020.jpg
09bfa054a6c59e7bc418c15c1169edc5
20299979d2cf1eac1c87cf52efb178642ef8848c
2219 F20101111_AABIXM stanford_b_Page_047.txt
6e9735ca7fa00f89c3c02e1920a7bfb6
73e8a4a41656c9a6d676238ea15f46eebdd77d41
50746 F20101111_AABIWX stanford_b_Page_022.pro
65d839d881eb0fa1ddc04e5131cf8291
88070ff2535d1bfba8986b5343dcc7535cc40c91
86872 F20101111_AABJDG stanford_b_Page_044.jpg
730f381d4fcc7eb266238f6894b8b312
0ccc0fef607bcc82b30d0838b3e12afaa5758337
F20101111_AABIYB stanford_b_Page_130.txt
e9ad4fe57e05643a70e5868ae99b88e6
739d2afaff91b237d4398486cfb2c0ee8ee52c14
43708 F20101111_AABJCT stanford_b_Page_021.jpg
8fe66f73ff7fa0748cf8dfca77eed4bd
720a84f321c72cdcfadbef0a49959b5071f0df27
F20101111_AABIXN stanford_b_Page_048thm.jpg
2aec64666492cb8d5c98583d3e622f17
2c7ab45caa8fc3cf3fd3ca24e6e66441a36526d1
F20101111_AABIWY stanford_b_Page_049thm.jpg
bf1e5b4f931fd973b1400fd3fc67081a
3d236dec88ab086ff3dc66775afad2e8a57cdbfa
85900 F20101111_AABJDH stanford_b_Page_046.jpg
9d2d7cf5923447e22f1850a3d386128a
a0661b2670c2a2acaa04999bb222651310806c40
F20101111_AABIYC stanford_b_Page_053.tif
0afdfb5c58a1ac389ce383a742d0a659
fa1f07b2ae8cc9f7339e85709065001a8f8ec986
85193 F20101111_AABJCU stanford_b_Page_023.jpg
02b952a8c9a181bdc4c26715c1194053
8b37de061825aa3700a586636e4d5b5a7e050876
7688 F20101111_AABIXO stanford_b_Page_062thm.jpg
a7982db837ab2d973c98fb03f925f052
e794cdf4b125dea77b592854ec408ba31f166ad8
57742 F20101111_AABIWZ stanford_b_Page_064.pro
f6f26d46e0787a0111ac062807c14683
c22bcd484e3d4092b060c944ec3a4bf9339c39c7
86936 F20101111_AABJDI stanford_b_Page_047.jpg
e163d51697707a24c42fc98ab836ccb9
28fc6b66607ffba0f459350cd3a6681b98deef21
54393 F20101111_AABIYD stanford_b_Page_060.pro
00692e2987c67d6ebb2b1bf53a41b515
4989bef1ff83b7cc8225472df57d64db2a96525c
83639 F20101111_AABJCV stanford_b_Page_025.jpg
49146ca506edc5d45a29fb2fdca28128
4e5a26088f6af2fb98c2069c05d590c2437889e3
F20101111_AABIXP stanford_b_Page_058.tif
fc3e49b560715d89bf676cc027712313
2d2823300f4aed5df5a08554a41053491a8a2534
68673 F20101111_AABJDJ stanford_b_Page_048.jpg
8acd5ad4074eb83ff5e9fc2be608b813
d5f88e051603f0304748010c130b1b59183875de
1051969 F20101111_AABIYE stanford_b_Page_139.jp2
4206b5f3c2fdef5a090778a5b55035b8
0e0397415bb75a6d7f191a4bf2dee6c97b03871c
79845 F20101111_AABJCW stanford_b_Page_026.jpg
ce514b62441d7a366c587b221a2b1bcf
72ab03b6504e8da621bd1a0c1f029ec5f1c02058
F20101111_AABIXQ stanford_b_Page_081.tif
6c4582bb353619d60fea10a8001c9ae8
f2371832b42884847d7b626078d7fd521583f10a
85255 F20101111_AABJDK stanford_b_Page_049.jpg
e25b7b58973677c8d8718a9e7bce4df7
cbb7fe7e6901f539841b93bc79b9bee707e7af5c
88828 F20101111_AABJCX stanford_b_Page_028.jpg
bfdeb961aa7162f59f29bb6e33220bac
d9e061e6ed49a0b3c67f67c202387dd168d9d494
1051929 F20101111_AABIXR stanford_b_Page_163.jp2
b7dbebc64a19d64ded8ac0f3e1662fcc
0945ac5d934cdd6a5ea45186c0f3054578c5a8af
39365 F20101111_AABJDL stanford_b_Page_050.jpg
6be44f3a69f464f27a31224695eeb8e4
b1f0b0aafb845c154eea86c29e697d46b76e0bd7
1051940 F20101111_AABIYF stanford_b_Page_176.jp2
881e2a8a8c145b6acedb73d6966dab8f
5c6b3bbd5ca209155e5dcb4b95311a8523a5bb54
87207 F20101111_AABJCY stanford_b_Page_030.jpg
49b11fb61a6ae52dedfc126e1ce80671
2668ae261463ef421a3b50fbc73e01794fa47105
F20101111_AABIXS stanford_b_Page_050.tif
7f5a5e6667ced9b5fafb76b0304b794d
5fb1e76cffc2f664b567953473e67601c39e32f2
83266 F20101111_AABJEA stanford_b_Page_076.jpg
47bdb3675796742fe1e49b4b6fab7976
922c8b1ad6b29f90b06bc9e31d94ae38eb9e503e
84697 F20101111_AABJDM stanford_b_Page_051.jpg
cf66f185f6855c56e3a4baf336a4ae74
a730eba16537b2769ac101df7c29c84a5d262460
7196 F20101111_AABIYG stanford_b_Page_085thm.jpg
ddba43c1721c9a06eebe69155678e88a
29cd58d71beab5829a13fb54e8dd32edfbfbe1e4
88250 F20101111_AABJCZ stanford_b_Page_031.jpg
7424a91faeccf1377ef6a7477109cbfa
c8d38852ec4e037a366599bdccad0d47bfb23073
F20101111_AABIXT stanford_b_Page_078.tif
17107501a90e363ea9d4f4ac140b8269
e9054f6d9039fb124a0796dd011a9f7250a99e35
84192 F20101111_AABJEB stanford_b_Page_078.jpg
3585d56b76e09676d198fc7ee430387a
26917e34d940e78c8b1f384ee153c2a3f72e330a
79153 F20101111_AABJDN stanford_b_Page_052.jpg
d0360f7cd54f799f83a9d3419a15b562
af8cfb7341db8b6b614225e3f608b2d5c0bd2fbc
F20101111_AABIYH stanford_b_Page_108.jp2
8af491678a40f5182ac776852db5a52e
d62c1e9af2e68f7d004a76fc88cf52d2fa370aa1
70316 F20101111_AABJEC stanford_b_Page_080.jpg
3bb59f8004a6dcc0cf8e120d755dab72
2b4b92b83b20f0f0a13f5e1c4ab5b1c4a898654c
78730 F20101111_AABJDO stanford_b_Page_054.jpg
8f0c76dc24205523b938b26b3dabd7a4
f7ee0b36ce030f343b1ff4a77fa80998cf184138
22965 F20101111_AABIYI stanford_b_Page_071.QC.jpg
e2dea6278e4e0ab20891b09066ce43d4
a60202b2f5acdf4bf342b14adbbd76b00a2fe899
6078 F20101111_AABIXU stanford_b_Page_004thm.jpg
73a2f2b74db5d4766a408472c95e8ac9
68456c6445661cf1331de3d0806a3010a0b243b9
88353 F20101111_AABJED stanford_b_Page_081.jpg
f5b5723987c4eb24821850d2a19914df
e52e9d00b4cd51c41630caca3667516c5fceb8fb
83509 F20101111_AABJDP stanford_b_Page_057.jpg
259195d0f1363c002773b288ccc840ca
4673c72c3a73ad8a84aecb81e7c33681f7510a7e
55896 F20101111_AABIYJ stanford_b_Page_019.pro
ad4496c1294c9e63c11949e59685b0d6
3422edc98f2d6ab2c83c6ea99196b79c96aef345
55619 F20101111_AABIXV stanford_b_Page_153.pro
0c0191b081f44b310d67b8a18390cb31
110ecbcf30ea2b0ce9697ebff71d64684d88ca0e
75354 F20101111_AABJEE stanford_b_Page_084.jpg
c7a712c410c15c2741b6edc4c33d3212
4b00d2706231c67654b2d889cca84249bcc99dfe
85434 F20101111_AABJDQ stanford_b_Page_060.jpg
8d822c24f5f4e5d073dcbf541750bbe8
c5fc75557c6a4a6d5c2f62d4295fd4690e3fe6c9
75045 F20101111_AABIYK stanford_b_Page_123.jpg
9fccc9526619d58cd8e8e1128ce7d84d
13bfa2c7ca8148e37b2b3c06eb400dde92c5ba5e
F20101111_AABIXW stanford_b_Page_025.jp2
882c60700e619bf82694dd1e77b90b63
a3c204db93796c5dcf61e2ab3e77df5c89578bce
93411 F20101111_AABJEF stanford_b_Page_090.jpg
b132fd00df0a9320adcdafe486cfa4d4
91d9dc9f0f9be364ded9f8410ce1dd16c43110c5
59940 F20101111_AABJDR stanford_b_Page_061.jpg
e1905eb371d522f4130c544e8a393ece
27416e028db1345db7652ccc4c9c18d2d6f1a11e
14254 F20101111_AABIYL stanford_b_Page_178.QC.jpg
513dee3e24b181fc2ebbfbc6c911f4b3
8272c4d649e73d3eda162efcc9fb18b1141dfddc
1696 F20101111_AABIXX stanford_b_Page_054.txt
ebd2ef974616f5a89401ef01258528ec
7819d34097178e1d9cbf2578be515571823d1126
95487 F20101111_AABJEG stanford_b_Page_091.jpg
793536c7ac136afcdfd29029fda5c6c6
35f55f74dca741d2781981cedb05712d21eb1d5a
76844 F20101111_AABIZA stanford_b_Page_103.jpg
652fafa1f7ac640e5e60e2c9e03d2208
3f2505a05e3e2bc9b4b5bb87635cd0c721999845
89332 F20101111_AABJDS stanford_b_Page_062.jpg
bd10773157eac1a67ad9f38d0a6f9b96
68658438b568090da4450d5a3a2012cc7ac8eeb4
67270 F20101111_AABIYM stanford_b_Page_156.jpg
6943e4aa128c71cf081b78d781d04c74
aafe6b298eaac89b86d21084be7a308cb208c3a6
F20101111_AABIXY stanford_b_Page_018.txt
f11ec276772a15595309895ba9e48f66
fdedd7dc99867c3a136938a96076d4edf96f7740
F20101111_AABJEH stanford_b_Page_093.jpg
0a9f37e8b3641bed53d07833bbe4c9e5
a813228efcba3b1eb79063f710c8429570120cb9
1051983 F20101111_AABIZB stanford_b_Page_082.jp2
be506ed9d6451db2694f99b20d8a7565
e6f8d9005361b7ba631c8d114d066fa7ac8f63f7
79659 F20101111_AABJDT stanford_b_Page_063.jpg
da11507fd5d29550a3abd6808236046d
10c9eb324bfd8aa7a9634f9d160f5ebbfac3a0fc
F20101111_AABIYN stanford_b_Page_153.txt
c426364c48d866c803a32535760374f4
52f21e4e1d87ebebd8099074a9eaec652512216c
97450 F20101111_AABIXZ stanford_b_Page_005.pro
dd131d7ab3f7e8ff44dfa2e2cd3b701b
1945afc6b9b5c15e15a3d8ee3b291b06aa724503
72692 F20101111_AABJEI stanford_b_Page_096.jpg
a727ee179c15b70dc6662a49f147255c
0fd139e6ed920c6405215035aef12762239d993a
1688 F20101111_AABIZC stanford_b_Page_100.txt
67c22608920e296adc979c11c51a3333
77db8fdc56751919531e369220a1e55ccffa7c7c
77250 F20101111_AABJDU stanford_b_Page_065.jpg
399d1e3f0d6cf575ca5fc4b00d105cda
ad33545e7586eb3be9994c4ea81bbd1bd6a8590d
51573 F20101111_AABIYO stanford_b_Page_128.pro
cc09f64e7fceafe004b1aaf83e6ee28f
bd56d46fe87d40a8daeb2c43310a7e962000468d
89017 F20101111_AABJEJ stanford_b_Page_097.jpg
8ee775c3a9ed148e563cea98308bcbe9
f264835b949e1b5e5008d78ab4fd504d086c9b3c
87056 F20101111_AABIZD stanford_b_Page_163.jpg
8d1286ba2d1ef25e555885674c7d1ab4
821b85733504ad7a35b562566a539a5fd0223b14
79786 F20101111_AABJDV stanford_b_Page_067.jpg
b3469472090696189fcf50f70958ad20
c66a1d003ded74f9a72748115d025702a72b5fb2
F20101111_AABIYP stanford_b_Page_066.tif
b90a09e26d843495fdd1b4b4c8dd5bde
608900230973c6e0c8f2669722ea1c557b5bf0fd
83280 F20101111_AABJEK stanford_b_Page_101.jpg
c9825003972051da568e1aca107aa3fc
ce72bf65b405d9944e1446c52a2250968fb6248e
F20101111_AABIZE stanford_b_Page_091.tif
372ef0319cbaf50c4d189d844d09ac5d
4f2b7edfe48858ed4aa216ee467ae42626c76566
76012 F20101111_AABJDW stanford_b_Page_068.jpg
969e9d93fc5f57c376a69397486fbccb
6ecc416156d45e16259b38f92b862ce85c1124c6
F20101111_AABIYQ stanford_b_Page_171.tif
55cd7fa456d90e690d586a1b74d665a9
c4c019b04570c2e2306d18baa50138c7c080d5d7
88948 F20101111_AABJFA stanford_b_Page_136.jpg
19f77fedd6d5ae141306367dce410538
1ba149daa36e4e81d5fc95e3da4688b88ab9fb86
71020 F20101111_AABJEL stanford_b_Page_102.jpg
8b4f40b57762161df6c3b291b8b39f87
7563a0f84560834250c6ed77a6f1c60ae2f85ef1
66265 F20101111_AABIZF stanford_b_Page_039.jpg
d43f6eeaf0bc686a019d239a04e6fa86
4c79d28e1b3cab138d4dd1fbd23229baba7c680a
68210 F20101111_AABJDX stanford_b_Page_070.jpg
09606c4162868fb23474236fd965dcf8
8db8c7acdcd30177a4b550c78a6ca6e457c21f78
6445 F20101111_AABIYR stanford_b_Page_096thm.jpg
8be5e5e4293394ead49eec2e68cb0b31
b7f4182e07a4aae1d7ddbe7734381a4d1faf0a28
82583 F20101111_AABJEM stanford_b_Page_106.jpg
052b5b2bb297727525015673d21cbd16
d8bb21763ebb65d8f9daa5226578711da53b6308
37149 F20101111_AABIZG stanford_b_Page_121.pro
435146a43f2819d247741cf1019817a7
10750be41a6601710898d095cc24d21278a978ba
71644 F20101111_AABJDY stanford_b_Page_071.jpg
a8473295b414256153bfafb3702fb3cb
c3223a207459364951c7399d5dfea3f357cf8f5a
7897 F20101111_AABIYS stanford_b_Page_094thm.jpg
49fae63143f405eb24dd51f295126f9e
84141d1083e1494d47883d5af2c264bad2df0407
90503 F20101111_AABJFB stanford_b_Page_139.jpg
bc3a49b1b6b535a1c74a48b18b10ea7a
a949dc17117f69add36ebe1bb6be540e0cf79e97
78842 F20101111_AABJEN stanford_b_Page_113.jpg
c5bd58c86b43909e0062ad971a423a44
c7509ed0760b46cdc1518b436d177023fdc6c27c
53785 F20101111_AABIZH stanford_b_Page_086.pro
7f058011fe3b5011f7eff71a42f8ae38
c06848e6b05d446fd3022255eee734a97da18b81
83520 F20101111_AABJDZ stanford_b_Page_072.jpg
646d34ec8febd2cabce6d6544c1f6220
c4cafed4692a77588c4e4e11dc3d58d183c24b9c
F20101111_AABIYT stanford_b_Page_102.jp2
1aecc269230864ea86e94eb658813e2e
480f70649d5c6f848feca8f9ecf1137db2c34d33
88382 F20101111_AABJEO stanford_b_Page_116.jpg
ddb3d2abee2ba715d0ad68ee4dd447de
b76a1ab64c4d770eb0d9cd1a913af44792d761dd
F20101111_AABIZI stanford_b_Page_101.tif
bcd5c7b7010b10462af81ed7bf0196c6
da4888d4e3ad5b441dadfa69ad2fe1c529e1d492
F20101111_AABIYU stanford_b_Page_022.tif
823cd5dd4d4be77b199eab59eb2bc0f4
184cb1908d3df8f371061c6b09db30f1fc46592c
84517 F20101111_AABJFC stanford_b_Page_143.jpg
b09ba4c4a3145b9bd194a39e6db43bcf
7e898f3872c80ce25edf51ab14d4f0e99df6d3a6
87152 F20101111_AABJEP stanford_b_Page_117.jpg
ccb5ba557d87b36020cfd35328120d9d
9d198d8d9b2c177f673b3f1ffd54a1678e8d48e6
64124 F20101111_AABIZJ stanford_b_Page_011.pro
b5c6367d7f01148567fae70b0fca49cf
d4ba2a9b9c0cbb489361e5e127df5ac0471e27be
80647 F20101111_AABJFD stanford_b_Page_145.jpg
d421633601a4a81685505f708b1c8561
9e364a30faad764aff71fc2c50cba688385f5e12
74333 F20101111_AABJEQ stanford_b_Page_118.jpg
ee4d9dec7581190ae7a8719f37404782
a21314f0fba4241ad25497f2fc490010f66e86b4
89522 F20101111_AABIZK stanford_b_Page_140.jpg
c25f802acd3408081e7ff15ae02e7d05
ae481d110552f6369ae01003524f670a0fe834ed
1714 F20101111_AABIYV stanford_b_Page_148.txt
d86e2157bb168ee63d7baf4279c364c7
5b4275d6cf703ddd008af2d6c1843e13289795d2
78352 F20101111_AABJFE stanford_b_Page_146.jpg
c2ed4223f7799ae0517b1e349bb77e4d
ab21cd5ef073806b3dabd7ccc6231533ee4f6408
54902 F20101111_AABJER stanford_b_Page_120.jpg
49063ba13512431f8f01727188e890ad
e54ee1814e6b0efce01669c1740d4c650f7e4ac7
76233 F20101111_AABIZL stanford_b_Page_111.jpg
113cb125f260ddeeb156074ec937a5b3
f62a420c1e8f31f001fbdf717420960ea899f1ad
24086 F20101111_AABIYW stanford_b_Page_148.QC.jpg
135ecbefa9e323fab854b76439c3754f
debfbadc4b738f46462e3f629a6760eaf72863af
73481 F20101111_AABJFF stanford_b_Page_149.jpg
856dca816623821234707c56f151f953
bdf99c97d6e907e7c706193b43d12533aeb22651
61400 F20101111_AABJES stanford_b_Page_121.jpg
0a9b0a6004c2528505a28ab56320f1aa
7bf837f669fe6cbdd48cf02a25547f16bec3b89f
27778 F20101111_AABIZM stanford_b_Page_117.QC.jpg
eae4c206332135e463ae27e1d8bd97e1
8f65f8d468f35f2f3589fbefb5ed0ec8f9066d84
27537 F20101111_AABIYX stanford_b_Page_158.QC.jpg
243fc0c6a3d855ed96d9a542c2a5ce18
f90abecedfa91a6fd7253ac4ec0a6eee3326bd07
67762 F20101111_AABJFG stanford_b_Page_151.jpg
c7471b51844c1c632be32f98979b227e
e9332255fba99c6fafb6a30b7be51b6acd666c7d
85882 F20101111_AABJET stanford_b_Page_122.jpg
b40a1874e29a6fef71e2d7ad4697a860
f12998daf7b6a2a36370d0354cf2caca95770bd8
7445 F20101111_AABIZN stanford_b_Page_017thm.jpg
f31150341b19d3ecfb56df0cba829583
34b09dca4146c69fe4f8267212f4f7694d14e7b2
1051951 F20101111_AABIYY stanford_b_Page_149.jp2
80875a7e12ee408e63c40d9f8fac263c
feb17c51ae00314fe2ac400bf955e943b6502ab2
87306 F20101111_AABJFH stanford_b_Page_153.jpg
2702cd6217f3490aba55bfdfd1aaac44
dbc8181c7c9c6695589400492c8db14344a6dd02
88208 F20101111_AABJEU stanford_b_Page_124.jpg
ed20bb6cd93aa2df4a9a321ea2606564
3ef7a8095962b16cd1bd8b8055caa0ead051b9f3
4078 F20101111_AABIZO stanford_b_Page_178thm.jpg
1da9899bf4bb85da58dc64b99439c4db
54c58690a59575717a63568203209fb8662a8d6b
815 F20101111_AABIYZ stanford_b_Page_050.txt
e3ae98e7b8f3f927a8c4da03495aed29
cb8cad819e3dd98f52d41cd2a19879389ed5df1f
69231 F20101111_AABJFI stanford_b_Page_157.jpg
65aee1f0c9de5b62ce946cf2c59648ae
6f78422ae3b1c40f8a5c266b007b4cd3a9c9f8fb
83552 F20101111_AABJEV stanford_b_Page_126.jpg
0e926b181c078cfc35ca5955538b0717
be39c32beb1f54827d3ee99940c152a33222f0d2
1900 F20101111_AABIZP stanford_b_Page_137.txt
584cf42e9b2c492fcc7879c646c6d390
6a62d1bdf21720b639438a4848ab5aea52fb689c
52466 F20101111_AABJFJ stanford_b_Page_160.jpg
59d3348b8555046115cff680eaca735d
9dc6f066106cf00b8f281d0c2fa560c801867d78
71905 F20101111_AABJEW stanford_b_Page_127.jpg
4eb0de46b1aedd7d357b6a6fb0b3ef3d
43f2592340906903dded6a87b28c669ba6172a49
58001 F20101111_AABIZQ stanford_b_Page_062.pro
85fe3185a26e199380fa5588eb26a426
1b37a1f399c3cee13ef7082e28eea78f67b9571a
88110 F20101111_AABJFK stanford_b_Page_161.jpg
0542accbd11313e382a2fba859748201
2d7fb0d226398e6b7236f50350fca2c66354786f
80852 F20101111_AABJEX stanford_b_Page_128.jpg
a444995a801c77e471839a5c40262f3b
72260913c13b4900fcd55e5a973b6602a82b12a3
3840 F20101111_AABIZR stanford_b_Page_021thm.jpg
433e90b571680455916b68000a27f719
8ce97f03f685678e8182aff6c11b165d7c2719e5
F20101111_AABJGA stanford_b_Page_008.jp2
3f8f091cbd0bbf29279c7df0ae6e6fbd
d09325fffd59d5068a14b5b4e2dcd7e9ba3ab21c
86647 F20101111_AABJFL stanford_b_Page_162.jpg
95ebbb911bbea3ba1f18a46faf50fb64
0c81001ff90d1404f43017eecd7f3f0118669757
59967 F20101111_AABJEY stanford_b_Page_132.jpg
063d9a1ec9e055d01dc4cc929c3a65ec
201884a8ce96c1aa56b999a8435542b92b5490bb
1022956 F20101111_AABIZS stanford_b_Page_130.jp2
fc0fff4503d5abc19968f2b0391bec3f
8446d12601eebcbdf6b11a84e0858d453f64ef6e
F20101111_AABJGB stanford_b_Page_011.jp2
28abae19ed557d1f67a04f02384b5124
c1133e5252c8904e18258b879b8ae6a467f631ac
86640 F20101111_AABJFM stanford_b_Page_164.jpg
f4c3aa8cea639c8b1b21eeb6fbf054fc
c26ba3f886917e62473bc2a0e8c0c6875460df5f
74458 F20101111_AABJEZ stanford_b_Page_133.jpg
77ea93a95844736787ee7175e3656260
7572343b0211da5f5e15b81a86ea704aa34c6b4a
F20101111_AABIZT stanford_b_Page_110.txt
bbb0b4d06cee0ae97cfb583600487781
63730366662adf8150064a2767672245fb0c120a
85502 F20101111_AABJFN stanford_b_Page_165.jpg
8f29e9e65d58739ea08f6650fda0bd1d
cafb185ad1e9ead6cdaaf631e5fdab08c8c3bfed
88534 F20101111_AABIZU stanford_b_Page_064.jpg
49b189899bcd4b17fa16e668b7e64b73
158dc00f0071eebed6221cb015f8ba4d067eef97
1051966 F20101111_AABJGC stanford_b_Page_014.jp2
760c5f8ab624d8b35aaaef1487292228
002de26c546ada4e7ec70488d0d6051c62d86977
109256 F20101111_AABJFO stanford_b_Page_167.jpg
a446e8e0e24a500ffde493eb94e7046d
e505df938e2244a9f608d3ae13c6863baab37626
7452 F20101111_AABIZV stanford_b_Page_105thm.jpg
f923033609b7d87d707a0abe7dc47bca
0210580755561dc3711a3d619984120c701779e2
1051839 F20101111_AABJGD stanford_b_Page_015.jp2
435cbbafc9fa293f7c5751b38852865f
9ef1d573e88173f03c285cd5298d3d36954d3682
107389 F20101111_AABJFP stanford_b_Page_171.jpg
ec06d530494a5a667b76f6510f2ab4cf
20daeec9285c4d35fb51730c48d0bd6bca765315
F20101111_AABJGE stanford_b_Page_017.jp2
bf189fc31a1b876442ca652d361fd9ab
114007ff92b9a89c9aa94172146e741ab643b89b
112541 F20101111_AABJFQ stanford_b_Page_172.jpg
00fc7a3a31a9b827845b730371a16e0f
2c734e78a4d4d4d27d2d3533e2fb9c26be8efbc4
F20101111_AABIZW stanford_b_Page_097.jp2
1ceed24aac59290f91c5df100a943add
218bfff2be41cc87ba1010a4b95a8e6cc2de656a
1051954 F20101111_AABJGF stanford_b_Page_019.jp2
d6f9c27a05a804d5ec6b2a31211d199a
7a62e1b1b1728a660b6a6c985bafb34b2708f81b
105151 F20101111_AABJFR stanford_b_Page_173.jpg
2e472d27686798b65faca184952f04ec
b7852460c48139242adcc16e12d716ebd2c11575
19946 F20101111_AABIZX stanford_b_Page_050.pro
1fa951e4eb1dbfb4ca2118d483610cb2
9f7f3c7ef5e21e387edeb139beab3844c420b365
563642 F20101111_AABJGG stanford_b_Page_021.jp2
79437542d2bec7eacdd3d788c8900cf7
93aac6d60e34e493d857b2ffb5ffca9f82231541
106836 F20101111_AABJFS stanford_b_Page_174.jpg
b17b936cc4be9edd42518ff5eb0119d8
a840651d988f11533ee202ded78d28a4e0c7b4e0
25816 F20101111_AABIZY stanford_b_Page_014.QC.jpg
2a2cd7792c433276bae9c234bea521aa
0fc614bac2bc1d5a60d4a697c9c7f772ef8a533f
F20101111_AABJGH stanford_b_Page_022.jp2
321546f63baac906b897751c2bbaabc3
94b9daa5bf23fcc3eadd59cb4e58faa1c9fede9c
112163 F20101111_AABJFT stanford_b_Page_175.jpg
181897a2f88c866137d07e5df40fa808
8ec1e71425a5d17709f3253c39d93b9b4b3919de
1051917 F20101111_AABIZZ stanford_b_Page_110.jp2
e823c1ef38a0923e6ea7195ad0c487cd
5101163f157afa6166565aef434b01ba2bee6e0a
F20101111_AABJGI stanford_b_Page_024.jp2
eb4c3c82d74c8436e94adbcf7fbd0248
ffae015e4069b05ab98ae98560111fa2d1dd58e7
108750 F20101111_AABJFU stanford_b_Page_176.jpg
2555a5cf89d3d0bd91a9230e923affb3
a65e569370a945978829448d3d965899d0988f03
F20101111_AABJGJ stanford_b_Page_026.jp2
6019a41fbdd7d1fb4daf7b3291949640
106f9b0ce3c1c6b134f58c9e2acc5d3d5daeb4e0
49654 F20101111_AABJFV stanford_b_Page_178.jpg
6b9285a8b8ec66d1baeb478586be3279
82ccc4a3dea0757ed5004a4d081dcb10a5233741
F20101111_AABJGK stanford_b_Page_029.jp2
1bbe396a45b43ea510ad0350304bded2
703889c7df240fc961036a3b36f9f51354889694
247803 F20101111_AABJFW stanford_b_Page_001.jp2
39d1c461b7591296658c65eb860cd58a
5a96d847ab522873d7c8661030856f911a9e9e39
1051972 F20101111_AABJHA stanford_b_Page_062.jp2
213800805e7b29a75f494360f057b352
513b9c6bccd66aa2fc7199cfaa30959fa8417431
1051976 F20101111_AABJGL stanford_b_Page_030.jp2
e38d1eb70c3941fb5531d0ab1bd58c0e
8bf267780e0beb336f9ff87fbde5546fe2ab85a1
30327 F20101111_AABJFX stanford_b_Page_002.jp2
3a1881c5447ae1b5309cb0569acfc970
41f9e7498a3ffea2c228f30876c56dc8e70fc44b
F20101111_AABJHB stanford_b_Page_063.jp2
63e85d6381aed9afc2f2918bba3b3eb7
5d29a95793b9bd33a99f2754ab2dc7e63dddc767
F20101111_AABJGM stanford_b_Page_033.jp2
8b1ccf78974ad9c19bb5864b40cec6a6
e0eec799e789636c2d6c3fc43d4d6ec2169de235
F20101111_AABJFY stanford_b_Page_006.jp2
09eed12dda777ae84c8d1aa4934f6b59
676ec467185f5a836b2e21a48a3543d1ef6cd080
F20101111_AABIEA stanford_b_Page_035.tif
7778af384caaaca6c0651630dcb5bacd
292807cd50d745e6327043ebf97967bfb683c375
F20101111_AABJHC stanford_b_Page_064.jp2
4c13c7c49482b8ba5206e8adb5eac473
5dd41abded26230c153d8dafc35e043a349cf6da
F20101111_AABJGN stanford_b_Page_035.jp2
4bfd85ddeb2e557eec6deb21244a8f89
d0b17b627bc087a4b0153535af5fd6e682ff4610
786033 F20101111_AABJFZ stanford_b_Page_007.jp2
292e81b00f8ef3fcf4a7470656934eaf
1593e9ef7f7a1584eb9ef65d1d70a18bcb709444
F20101111_AABJGO stanford_b_Page_037.jp2
89009d716a3191bdb78f5fa7479c234c
0861850577cf187fbbcd3c3f8b23339550d68a8a
1978 F20101111_AABIEB stanford_b_Page_166.txt
302e0a61279b699c1cda0c0659726928
4ca20dd366fdd22ebf9b7d9327aaa2281d34f18c
F20101111_AABJHD stanford_b_Page_065.jp2
69917fc7467bee27909b6eb8ba83ee8b
d579d7a333823d44523333e0a7f538bd1a5264bd
100212 F20101111_AABJGP stanford_b_Page_039.jp2
bd8ff1d64422a7c76c914334205b665c
d7e323f87923dfa6b673433e9526d6d4e91fdf2b
77299 F20101111_AABIEC stanford_b_Page_142.jpg
a4e18edd990892df06ab2d6ab0b87d07
835fa84cbea6cb16b9bebc7963684ca257d4c8b5
F20101111_AABJHE stanford_b_Page_067.jp2
38ece0ca26501e39a34ddaab14f0651b
07084c0303f372e405d4eb662619af01b5c0336c
1051920 F20101111_AABJGQ stanford_b_Page_040.jp2
d4c05bbe239804c817721462bd1f5eb1
ecbc508dc278c5afcd851c75b1783c4009623fef
1051962 F20101111_AABIED stanford_b_Page_043.jp2
bb656346abb171281211070101ebd6d5
fa5d739bc38258c975532765d57e32346a02cf34
F20101111_AABJHF stanford_b_Page_069.jp2
de5f4ea6e267b86e1150b6c9ee34b365
4615e7cd28f4df3bf6229a909e942f7f6f8ef40e
F20101111_AABJGR stanford_b_Page_045.jp2
e8f3021a15e581f8b06394b34573a08c
91c0dc2e6ec5469ec4a7e2b7802295df3dbb8cdb
86906 F20101111_AABIEE stanford_b_Page_110.jpg
ac18ad9b3dad9c5617a5569a0514f9a8
f8a2065aad367e8602cbdcb4fa461a66fef897d1
1051945 F20101111_AABJHG stanford_b_Page_073.jp2
6a24fe50576824df7751c08181fa8a76
ce3f9fa741cb795648c066cd1c9c70ed2a0df334
F20101111_AABJGS stanford_b_Page_047.jp2
54ca82a89e3e5bdad63b0391234a26d2
7937094c1bd5629c361c1f580c227d34e3142c16
F20101111_AABIEF stanford_b_Page_095.txt
5b2711c7898cc8171ab2673453eebe74
891d696ddacbec09b31ac8d221a5a7b2913ce0d7
F20101111_AABJHH stanford_b_Page_076.jp2
927d5387376126a76aa1239287c4c343
9bf9a40d87a0086901e93a1785b1e6c85b4f93ce
94662 F20101111_AABIDR stanford_b_Page_094.jpg
284e8e4d61c16115e1fa7fcb2449483e
7b8487b6f1af367f0bbe1e850feb509cb4a6550f
598746 F20101111_AABJGT stanford_b_Page_050.jp2
ae17e021a3709b5bda32663d3e1ab125
a296b82b8903b5102041f666d2acf71c09bebd19
F20101111_AABIEG stanford_b_Page_173.jp2
0b93b97c2e267a3e578d4758e2901d17
e52f7a09788d005729d22d14d678bea80724ff15
F20101111_AABJHI stanford_b_Page_079.jp2
30a56805e1eaf8d4163763cd77d499e0
31f4d2262f20a2038f2e421805159bfd7c32643b
31211 F20101111_AABIDS stanford_b_Page_171.QC.jpg
5b604a98f0531daf6659eedbddd3d38d
65c74c0ec54fb574c6d28347bb2bbf01ad2c1bd7
F20101111_AABJGU stanford_b_Page_051.jp2
855622c6bc70fc73fe0cf2f180c4f3d1
6e41d5f2b48006ec177b281d6475627273bf622e
1051975 F20101111_AABIEH stanford_b_Page_147.jp2
92244eeb5e3cb0134d7f9a5d8604edf6
a07f52d718a7be8fadf5783d18a1207b732353fa
1051922 F20101111_AABJHJ stanford_b_Page_081.jp2
1e08bfe674d7a64b70f87e95b20409c1
c61482b3973a048cedbec363fec73bf51cb72e4d
86220 F20101111_AABIDT stanford_b_Page_137.jpg
5965ee594d6f03ac7acfba0a74d13135
f062e538d8fbb339951c07340dbc8e9ad28fa626
F20101111_AABJGV stanford_b_Page_053.jp2
e82d12aed8dd1d59338278950b63d376
40cb7d6f3ee46d1259d69a19b4798d614ebee0ea
F20101111_AABIEI stanford_b_Page_132.tif
a005964b33a673ef12cca2ce566e4b1c
c2d6c943bcaa1431f3ab3eb44ceb38a0edb6a8a1
F20101111_AABJHK stanford_b_Page_083.jp2
ab1cc090acbbac00b1b85ace15e20c1e
cf8d6f8f145d3ab1b06d647dcb94ed586dccc908
4082 F20101111_AABIDU stanford_b_Page_050thm.jpg
5e6f4c9b8649dc6aff40de1419dcdb33
df260920d9b7d2922d836852de0266d380b0b8e9
F20101111_AABJGW stanford_b_Page_054.jp2
1b0d6b12e3dbfbd48b2a78b9d6687042
b4ab7465f677270e812d9fb63990b3d1a3e1c3ed
1051921 F20101111_AABJIA stanford_b_Page_116.jp2
cbd3e8337ed4dc342ab6a8507316aa8e
4ea5040dfa1006cf09b4e0a8998642271264965e
7496 F20101111_AABIEJ stanford_b_Page_123thm.jpg
1371d84df52a1a422ba0b02d911f73be
bfbb3168484c8f06cec1d453296fd118f5930b83
F20101111_AABJHL stanford_b_Page_084.jp2
056e097e3236df64a428e03c93c30f67
a40636a944466749e3aa62d1c0037ca0614acdbd
27426 F20101111_AABIDV stanford_b_Page_029.QC.jpg
788139dd3c07041b342337a85dfdaef7
d8ea0a60ecd4d200200eac5ae0cae487c8e8f41f
1051942 F20101111_AABJGX stanford_b_Page_055.jp2
b6427e06a0dd758cf434b19be9a97d55
3dc971ca7ac255c9addaf12f12912851b5427216
1049941 F20101111_AABJIB stanford_b_Page_118.jp2
f9003754e30534bd1007b67b7d7c8a08
471381eefbf11f30575092402fec2ad1c2bc528f
98321 F20101111_AABIEK stanford_b_Page_168.jpg
66009cb2bc2ad4c76642cbe0975b1ccf
34616f51830dd02a847d7cdaec8930bf895ff5d7
F20101111_AABJHM stanford_b_Page_086.jp2
b78957a3152040260959e9a2a762ed8d
923edfda5c7624d54519a837b6fd0209a8fb25df
54378 F20101111_AABIDW stanford_b_Page_023.pro
c22e69282a6ea3d66975e9c2e3ed56b9
a70a725297e413103802dbcb27589e248ad3bdf1
1051894 F20101111_AABJGY stanford_b_Page_057.jp2
1c098a3fecf306686398e464935e112d
68a0a4c14b4325a3bbbd778dba4194de00b849bf
1037340 F20101111_AABJIC stanford_b_Page_119.jp2
99162c8cb00d3e17f3e62ac19b59958c
888313342d54cc601fbeb50b1ddc87f742a6708a
F20101111_AABIEL stanford_b_Page_156.tif
22a4c3e4cba065a4a7d76954d3ce86e2
45c263974d8a919335f43ac353a53b74629fedf9
F20101111_AABJHN stanford_b_Page_088.jp2
197b5877a57e2fa62095c4a8a1389bd7
a8e85a50fe60eb5cf9a225c070502d67fb91fb41
2454 F20101111_AABIDX stanford_b_Page_071.txt
e55bf9fb294fac6c791ae125d637a5d7
64c2a53d127ee61be658672437b13c98d29b31d4
848748 F20101111_AABJGZ stanford_b_Page_061.jp2
39cbf244279009e15295ea1d2e33f77f
018a777bebd385c75e282a256c6684000cb1b598
90527 F20101111_AABIFA stanford_b_Page_088.jpg
0dac2a2f35203cc7250641901a8f0f9d
274cdeb574303e4da71ff4da9f309ab081718f10
1051982 F20101111_AABJID stanford_b_Page_122.jp2
dd307e96ff491b83fd2d18c610b4a78d
c093bc6c557cdd3a27df21713c185bc225514eeb
1051914 F20101111_AABIEM stanford_b_Page_031.jp2
0a3a2befaeae99f75a36c08581d875a8
ee25860c06865e60b8ea54befcde0629db0baeb3
F20101111_AABJHO stanford_b_Page_090.jp2
c9c1d06acb1c799498dce2508baf29b5
f009754221ac1a8c5a7c9613327b7fa431495cbe
F20101111_AABIDY stanford_b_Page_016.tif
4f5b5415dbf916e251fdd37bc0f65bf8
8832825c4fdd090e2762bf2592cd86fb25b0e628
55895 F20101111_AABIFB stanford_b_Page_097.pro
05fd4457d2f2f9da689f6b7d0b593afb
6e47993c62bafa69cb155587a7b90e50ea0a2d94
7493 F20101111_AABIEN stanford_b_Page_029thm.jpg
0aa25c97164f52475647c4fcae828edb
a1f327ac9abb0d3b78987721a7f4ff1587519b68
1051943 F20101111_AABJHP stanford_b_Page_093.jp2
9b78aa26048d89dbc38f775b7a44b4b1
2c3a2a54f22fc4a7e1241dabbfec2a5b7f3cdda0
85897 F20101111_AABIDZ stanford_b_Page_035.jpg
f60bd236f0148c3241902c0629f8f466
75f313e79ca4e993e2e600046138e61aa9705652
97976 F20101111_AABJIE stanford_b_Page_123.jp2
50cd3f15aa870a40ba65004002d4cf12
2c997f6c40b155ad8f3bb5669ce3fc55ffaa0407
27225 F20101111_AABIEO stanford_b_Page_016.QC.jpg
dbdf311af5e000b87aaf14a50baca2e9
3ac17bb2aec0bbda7f05147f8e20157d610e7045
1051981 F20101111_AABJHQ stanford_b_Page_095.jp2
54af9ca889b50979efdc79c16126fc36
285a8fd3a75e4abc4a61dc1de07c6eb58ccb46f1
7127 F20101111_AABIFC stanford_b_Page_099thm.jpg
1d1d87933b368db2e97a32be1c03b3c3
7d59b48ebbc70834d980048f5535cabb5e169258
F20101111_AABJIF stanford_b_Page_125.jp2
5e8d1ba684e153b977ca8d45c75e4c87
e626022e23244e542bf2f395fe22001fb964d0d5
F20101111_AABIEP stanford_b_Page_030.tif
d407c433626eac2338a4db5701e61577
7ac8dd8273ffb82097aa48e331677f1f50df1da7
1051957 F20101111_AABJHR stanford_b_Page_099.jp2
f5a2fd303803defb6ba46d13113ae840
3fef3a9a7793910c0c9984841c3827c8169a0642
F20101111_AABIFD stanford_b_Page_161.jp2
3c667064ae86ddadde3952da747f4266
f774ce73d1e63b1e581b6e4cacd6fb4513943325
F20101111_AABJIG stanford_b_Page_126.jp2
f26a997e5f8af9ab570e58eabdf379f9
30bc34bc1da7dee52f4bd3997cfbfa8155b04f95
F20101111_AABJHS stanford_b_Page_100.jp2
a714cf8b962e7b2538334575e3a16695
495846ee69b8f10e2e399160c48c380302747d5c
86500 F20101111_AABIFE stanford_b_Page_158.jpg
97f0976b5448243dc5fbe83c60fb37e1
d0c3eae760c5260dcebffb7d5002ba13da176abb
F20101111_AABIEQ stanford_b_Page_011.tif
646cbd9f27bc017e4a91803d2e7ead36
1c466d63fd9352f63ca229eeab119163d6ed4eef
791092 F20101111_AABJIH stanford_b_Page_132.jp2
83c52f9d293e82031a6449e061ec9751
2908181306a55f85a108198b001f0bc948b3cd09
F20101111_AABJHT stanford_b_Page_103.jp2
2889f9b5b87a39cef38caa6cc6fde748
36b37f2278c2ee4bf5fe132241e02f6c6c98b2a0
F20101111_AABIFF stanford_b_Page_044.jp2
47df085ca61273e6a97421bfa1a7d174
6af6ca2a3cdf01b7423f5ac0f735cfa5c87d817a
23815 F20101111_AABIER stanford_b_Page_100.QC.jpg
b3b6c04a01673faf073c481636e6f7c9
9b6ec7a6489748e92a7a8f9e79797c9a9404a30a
1035537 F20101111_AABJII stanford_b_Page_133.jp2
f0758e2d6cebb0e3b5cbc91d0582561b
18656ebf1b6c27b898a87af715c593f2c89d3bbb
1051949 F20101111_AABJHU stanford_b_Page_104.jp2
ab9b261d38073ec3c24b8b2da6ade551
4c3a8d8cc66da8fa896ae3e4a63a0cff82601081
47797 F20101111_AABIFG stanford_b_Page_091.pro
0ba6728e246e3e2bfb5fffb437da9e4b
76b8fd3fc531bb6fbd8b6872f782ae2f657429b6
22668 F20101111_AABIES stanford_b_Page_130.QC.jpg
4876702ab95a0d8e90639aeaf51adaf3
480b04599040852ab787711322943f9441cd5670
F20101111_AABJIJ stanford_b_Page_134.jp2
f255f45bba3c81b4481799295fdf345a
d39eb4ef7a271b7e07050d26bf8af860ef7e7822
F20101111_AABJHV stanford_b_Page_107.jp2
b9fb76c61124ec063e7e59cb9264ace4
be85bedb720a10a7311dcb98de22ee28c2eb25e9
F20101111_AABIFH stanford_b_Page_172.tif
6a20e870ecc5875cb94ce902661b07ad
7b0f237740de3ec5dbce5290a9f8f1d96aa572e3
7164 F20101111_AABIET stanford_b_Page_022thm.jpg
15213c08b1d047b9038c18c94faa0997
1dd525ac8114305e474bf3da579c9c5a644d4320
982815 F20101111_AABJIK stanford_b_Page_135.jp2
e241fc4004064df8e97f07c03bd922f3
b7e93ba02dd7cfabbd8c4f4d250be14c2d6ea5de
F20101111_AABJHW stanford_b_Page_109.jp2
e2717e7786fb482adb7e1071828f20f5
2d1f753104575de18403beae2506d84c5cddf872
F20101111_AABIFI stanford_b_Page_027.jp2
c5c4ffcb49918a16644d51b92eb47af1
59ed0de9d8a929f94bbe611836865325f50710e8
28108 F20101111_AABIEU stanford_b_Page_044.QC.jpg
8e740d9e6de25d03218ffce63cfb44c8
6df3605cb4f9828c9ab48eb31ca4faac3c5a9e47
1051852 F20101111_AABJJA stanford_b_Page_164.jp2
dfcf691f7a06a64a9cd083a799fb21b8
ecb9e6018e9c8693b18c27dd5330e10b04224b7e
F20101111_AABJIL stanford_b_Page_136.jp2
90b204015a10cebb4ce548b7d0a2b40c
89896ba3c80e9b51edb7f1a3733a5478c98659c6
F20101111_AABJHX stanford_b_Page_111.jp2
6999e098858acde76de05e347b4692bd
639aa97e01837e58a610542f277c11389c0e98cd
F20101111_AABIFJ stanford_b_Page_103.tif
ea2f958296aa3fc69640465c384676c0
fd8222539bc7c48eade226b0a1c0bb2088dce4d1
F20101111_AABIEV stanford_b_Page_144.txt
bea453cb64b9fe9ede2fb14005d20ca7
e8d248e795699188f5e260a94ec133e13973fa3e
1051956 F20101111_AABJJB stanford_b_Page_165.jp2
1429cf1c35f06830a674b41c4733487c
ee8cdf0d7c70468c94b81091f87c5fee1ed20988
F20101111_AABJIM stanford_b_Page_138.jp2
36fbe0eab9e47870da9c1f5bb0e6f2d5
3ec7093721dbbe05d2840f5fe8d7e13c35227521
1051857 F20101111_AABJHY stanford_b_Page_114.jp2
8dc384be68eef3bf503e93160a31dd1d
4b8ea96788d94da98819910fb702ddb3c9b3b5ab
86497 F20101111_AABIFK stanford_b_Page_104.jpg
3cd6ac37c82d129d32a86f9b614b49c7
21c2d38693e9fb347ed1a27293c8ae585d5ca4ab
2140 F20101111_AABIEW stanford_b_Page_057.txt
539489c7123fcaf3eaa400b1d125f8fe
8a30a22400a3ff21f343bd7ee6529e19b538211a
F20101111_AABJJC stanford_b_Page_166.jp2
81d741c3d116131df1244b730cdbd500
24644d72ce422b17f296c38412b7fd9f3abb009d
F20101111_AABJIN stanford_b_Page_140.jp2
f55ddd17c6ec266878e534d70843dd1a
fe5811f4308e0748bc727c37680f4ca16fd339f1
F20101111_AABJHZ stanford_b_Page_115.jp2
b2f483f778c833e8dca048f9445e5348
5ce2ca926582558280d43fc0dd7b95ee24c824e0
7702 F20101111_AABIGA stanford_b_Page_116thm.jpg
56ee7c61b34dd1b1356eb0482a96a7da
62bffccfcdd1ac5bcbd15de1239b8164ad7bc50f
F20101111_AABIFL stanford_b_Page_042.txt
6cc8f14bea7a98159913897450a319b4
df177778d77376875da35dd653c7727ff0f0407f
1781 F20101111_AABIEX stanford_b_Page_068.txt
65305976583430305445d860379060af
d22a56934ad8172d552161af0b486eb91489899d
F20101111_AABJJD stanford_b_Page_168.jp2
e79ff8b57e3fc2eb202b0c5a9282667d
e0eebedf936f3a92b808cd190dcc21ca27046655
1051959 F20101111_AABJIO stanford_b_Page_141.jp2
4fca829bb1af90e2bf6c1beba73cd0e5
01fbe33942c0465353e9059d3c080587c4088d67
7722 F20101111_AABIGB stanford_b_Page_124thm.jpg
f671653b40e3130385f8021ac2233684
e970e9ee8ccf431e727fd85f6a8aab8f45eb50c6
F20101111_AABIFM stanford_b_Page_046.jp2
44e3028bde00a90fa39740b2500001bc
4d2a03b57766b52b0522e05cddcd1b60253da530
49727 F20101111_AABIEY stanford_b_Page_067.pro
a2c2cf4357c83a61f74261af7d501b15
d2f2f9aec0154fa3167522bec98a1240f81d9fce
1051955 F20101111_AABJJE stanford_b_Page_170.jp2
386bbc368045f9b4aaf5c6d67cbd9fbc
80fef28b09253a657d4bdf995d5446ee5bfed879
F20101111_AABJIP stanford_b_Page_142.jp2
2aaf86845a80bdce06c59695bb7b48f8
0b421e97fb19926dc03ff68a777c1322b7f6bb0b
39140 F20101111_AABIGC stanford_b_Page_157.pro
d5931010f30ec1a1f56068fa9d554f41
669ab3fa5cd0ed0b4adbcfe017b2ad8573719b2a
F20101111_AABIFN stanford_b_Page_104.tif
fea3c462d4f19c681bf3d2d934dad344
e1f48119c48c36e6880215124518abb833958352
7449 F20101111_AABIEZ stanford_b_Page_136thm.jpg
76f1f7a47c874cfb71527223d58f264c
46170743fc72d075b5d3c93c38592ea07ccf0841
1051963 F20101111_AABJIQ stanford_b_Page_143.jp2
d5a3b53c61e5ab24449a43c9dd8e8ca1
4a28697688863a66e3df91ee28ac684f0480d38a
2103 F20101111_AABIFO stanford_b_Page_053.txt
393fb5abdd468acbebaf5047046f9d61
8bd662c4a051d4baa62d44e973fa5fa812601564
F20101111_AABJJF stanford_b_Page_174.jp2
44451ae1278d76a2774d4dd7432f637c
cc7b772798852d0e5db5d830316cd1d5c4f73b19
F20101111_AABJIR stanford_b_Page_145.jp2
8c19872c1a30fd1dae03db66d3fbafb5
6ced3f63d67a7dd2cc4e9bff8a0649d7305a9ba2
81668 F20101111_AABIGD stanford_b_Page_006.jpg
eda41a7ecc86154928bf7b23dfdad055
d9a363b5becbf156261a0978b8988f6bca9a8fac
F20101111_AABIFP stanford_b_Page_023.tif
ee40052f391c61822bbc2f1b8a7146c8
147eb8b65738b33706cc4b0600f952527cee3d0d
F20101111_AABJJG stanford_b_Page_177.jp2
701f67a2b6715b7a13e3327da19a0e20
435c19275928791e8672447fa33a5bbfc9852c08
F20101111_AABJIS stanford_b_Page_148.jp2
ce417938135e17447843bfb8eca05b25
f933b956e37ee2872ac0ffc157219f455cb58244
78688 F20101111_AABIGE stanford_b_Page_155.jpg
493948c164524a908d57a5a112d98820
3883b6fa5b7059aa84c8cb75223960d31f3403ad
7666 F20101111_AABIFQ stanford_b_Page_040thm.jpg
979e296e08f8295a68fbda926da7945c
12f8cb93b0b54e16d6bc6f593649ac1fd585e484
638071 F20101111_AABJJH stanford_b_Page_178.jp2
d7de99e1bc4d524cde5a09e36fd47dde
c6040ba11909ce12a3435844bf28823278db0c05
F20101111_AABJIT stanford_b_Page_150.jp2
31fbefdd25c169968cc150f5bf53ed6f
bb6ec0f4fca47353dd643b861f71aa5cdf2ef8ec
24452 F20101111_AABIGF stanford_b_Page_011.QC.jpg
3491d33f49d3ed29c729f7b5e8619db4
92356e09ba782c74fc2a93dd2d0817d9fe897e73
5390 F20101111_AABIFR stanford_b_Page_006thm.jpg
a6f26fc47e50d119b5aa51e3232fe383
301b341970717d91a7feaef725e8ee749933470d
F20101111_AABJJI stanford_b_Page_002.tif
cf478114851292cee66ecaccc973ba32
f55c16666cbbbf5973f719883d2b483917a5cf0c
F20101111_AABJIU stanford_b_Page_154.jp2
11f50076bcb8d216adbe574185b58501
8fbdb52405e4938506788e0b2518ab5047c312cd
F20101111_AABIGG stanford_b_Page_009.jp2
5a73eb76f8aceaa036bccfb4329449f1
383d78cf2d71c4cb28e085a2da9010a48b93e58a
F20101111_AABIFS stanford_b_Page_096.tif
b1b23e8299b2b50c0f067cbbb4f4c68a
661ce317eae18ce48e1deeeb0550ac7d653d16a1
F20101111_AABJJJ stanford_b_Page_003.tif
e16d1772b2de1118e87dee95a40a7c58
db9ebfed3db57663746aacb32f2af9accf7c0b64
F20101111_AABJIV stanford_b_Page_155.jp2
353e679a78e5965d4d2bbfc5f1e0de94
5fc130d0c18c3e60336442b5623a6e84109420c6
55294 F20101111_AABIGH stanford_b_Page_161.pro
51bbb26e70b6f4aac03b2139806fee4f
5f92d41b9219684c4d66ca2d9a1af42a8098a6a3
24360 F20101111_AABIFT stanford_b_Page_059.QC.jpg
9b9cbf7ed07f93daabec5d8fa4113788
06a4f327cc57d33ce30d2a2a651f2bd390e41531
F20101111_AABJJK stanford_b_Page_005.tif
8c32a1cd03ff25c64e675a0eef9a2a00
f2ea109e94fc2f619b0414a595f347c1cdc83069
898162 F20101111_AABJIW stanford_b_Page_156.jp2
c6eaebe8c0ee7f5d2a1d85d563e004a1
5257082316fb1cdfb1f2af2515b3841bc78663ab
28645 F20101111_AABIGI stanford_b_Page_061.pro
143746f34dd7a13f365d547d15ad59f3
5b71cadf5f0ecfc72afc789429bb73aa0386fec3
3970 F20101111_AABIFU stanford_b_Page_005.txt
34c0b0b192b26cb8076fc8ab8619f8de
879a64a10ea645158a14127afd8d1e913b1b2e93
F20101111_AABJKA stanford_b_Page_036.tif
4b3c5e822a70d88d0ea9b491837068d9
8cd13f60295bf51923bfa6da100af0b7f2414c2f
F20101111_AABJJL stanford_b_Page_006.tif
79ab405876597c1b88336f4add230e4d
61dbf2d4c2e2e747eced55981ef5b5fa855eb647
1051931 F20101111_AABJIX stanford_b_Page_158.jp2
cab5e5d2fe480d96df274cf7474d79d4
197745494fea75f086951fbe1c792108b53cfd0f
7748 F20101111_AABIGJ stanford_b_Page_169thm.jpg
1c77a932af56eeef93aa128fd2b5b031
b778d187fc87bd96f7f9e5dc4fa27a6b00743c26
F20101111_AABIFV stanford_b_Page_167.jp2
452770344bdcb9a5f076765d44586e45
8146c831ec2b7d58f7da9c6ff9e37618506494c3
1053954 F20101111_AABJKB stanford_b_Page_039.tif
1c88869b5d4cf6438abe0a51568d0099
978649671256bc1a6b0585805c25094482c95ee5
F20101111_AABJJM stanford_b_Page_008.tif
1c49093c17286badbeb7994313d55083
7f602dfa48f4c5ecd0b9f482b6f59e0057117295
852825 F20101111_AABJIY stanford_b_Page_160.jp2
7730534cb42311f4a1760418cc90e646
2b43b0477e126c13008c54cb3e76b1aab4eb5795
7310 F20101111_AABIGK stanford_b_Page_058thm.jpg
fef3ce5cd60108d4b89320984ffcca4e
c09693285b032229bb8288f89d6aa5d3a28460bf
68897 F20101111_AABIFW stanford_b_Page_175.pro
830988713021b3aa675cc928ffa26979
493bcc7190cab322d9a055af6bb5103e228b0d2f
F20101111_AABJKC stanford_b_Page_041.tif
0aceda6881020035c91bcbf042d4b550
452f56dfe4c2a2c86f7d8fdd46a7c40f0d0d2141
F20101111_AABJJN stanford_b_Page_012.tif
7dead9b210bb7d8a96fc5a544d577e36
3511ddc138890cf4e51c663e2a97573a821385a5
F20101111_AABJIZ stanford_b_Page_162.jp2
f1e810685bdc8ca4107e9d576070444f
c749641f1f9d9be6d60a5af683b8b48c670d7b78
34234 F20101111_AABIGL stanford_b_Page_077.pro
c8f51c8224ac098ea45260ca9333b7b9
d6605070de13cc46880fd4668e21600b5d52f87d
2125 F20101111_AABIFX stanford_b_Page_104.txt
75683d94f07a6ac09e37976e7cb9e902
3ddab5873701652d2ccc569661d6e06d21eb33bc
F20101111_AABIHA stanford_b_Page_118thm.jpg
bbc4c4f08197fb846545ded5ae6f78d4
ab84d222bdde6130c7ee7033973093182f62cf7d
F20101111_AABJKD stanford_b_Page_044.tif
2ee8892ef748bbe6f014dcd7063f315c
59b8852269f03e72845cf42967a48c54fd2c1997
F20101111_AABJJO stanford_b_Page_013.tif
c280d90cc8fd4618cd930ec6124990bd
5a4f7b65da45499d40014ed369a10ed04d005ef1
833841 F20101111_AABIGM stanford_b_Page_121.jp2
d3f8c840788a6a889de174094417bb7e
b05fbe27c4181816420b06227258fd195c511207
49218 F20101111_AABIFY stanford_b_Page_099.pro
f2f10682514c4059eda8dce70fb29077
cf7f1223b9e8913249f355d4cf1335b387149aae
2595 F20101111_AABIHB stanford_b_Page_011.txt
8d9df6448efe38d2694d6a8bd786b7bc
9017c8c751211d294be0ebc6e741076bb70437a1
F20101111_AABJKE stanford_b_Page_045.tif
d074e7a9cb029f668fd72bfdff157e5e
28e1139bb6ef52a12031ab77b2709ce5b19ce898
F20101111_AABJJP stanford_b_Page_017.tif
646e375137045b49dc1ee3bc69bb4e33
a70249a8349b6848bf2b449a7f5d1855506bb616
7412 F20101111_AABIGN stanford_b_Page_122thm.jpg
9bc9f4395837f9363f116af91ba68aa5
3d380ffa345c936f36ae2936ce9ea47866bde54d
F20101111_AABIFZ stanford_b_Page_036.jp2
8e48e694834681cf88d8ce23bf79e799
cb76d90c339b5eb527eb91401a6448e73e19cad7
86207 F20101111_AABIHC stanford_b_Page_017.jpg
f781e7180fe043ab0393ac138ba29b77
93dfc1a557dd4ed2a39baae60711dd573f85cc94
F20101111_AABJKF stanford_b_Page_048.tif
4af0bca5b1e30e776a9a603197e57533
c21dba6dec20f4b5d5c570668e02f42026696321
F20101111_AABJJQ stanford_b_Page_018.tif
346e528b047feb7935f98e77937b4b19
ad75f37bc52857294aa8b387d5fdda8a547222d3
44176 F20101111_AABIGO stanford_b_Page_119.pro
0dffd4e640c199f0812e6e74ef12417f
443b68d8d5955582a9bd749a232f08049faded1c
2412 F20101111_AABIHD stanford_b_Page_087.txt
50fa06e7c1d41e9a54e29e54176a1a25
38c2661caadde44a3be31abb0e0de45f291052d5
F20101111_AABJJR stanford_b_Page_019.tif
d8c0541ea09ba7d051eb39e9371878fc
02145cd4ef0f37478a1e7108003c14f87b7a5836
1848 F20101111_AABIGP stanford_b_Page_124.txt
dc1bf0381ce14b87a015e09894384771
0939711d211e48f0e7af48943d39caa043d5bae9
F20101111_AABJKG stanford_b_Page_049.tif
79307e829ac5f71ddbc0a2893c636a99
d70ee18a64599b44b5f00089742518e285b7b8d6
F20101111_AABJJS stanford_b_Page_020.tif
80b78bb3ba7a8f9899018079e67493b8
f9783d2635aac9eee14a2de567caa40d0d5a9627
51154 F20101111_AABIGQ stanford_b_Page_139.pro
5a25fd4e013719a749bdf3637142a9d7
d644e068cd2c9d17bb01f62f2aedddd478ed396e
42699 F20101111_AABIHE stanford_b_Page_109.pro
bd47ac8c10aee2aca6321bcccdd2890e
800f1b73834131414b7ee6a3e7075b509c0d4d5a
F20101111_AABJKH stanford_b_Page_051.tif
f79e48e7bf0e27009a3d9f66ec3c8a28
805489fbb39207d61905a4e9b0624bed5b9e10c9
F20101111_AABJJT stanford_b_Page_021.tif
ded76c1ac4e47b2b6c8db41860b67b45
01dbaddd59f1cfbb426da3011c1954ddb064607e
F20101111_AABIGR stanford_b_Page_112.jp2
79b27160f2e8db50b9bcee2da794e18c
a83271c3decd3796dcccb6d4aebeae39d8e2fd59
48772 F20101111_AABIHF stanford_b_Page_020.pro
913a590944fe829c18d1f6418be92d37
5a1ace65961a743100df20da7f3deacd4df12397
F20101111_AABJKI stanford_b_Page_052.tif
46d0ace674f17f75c421428d5eef091e
dda4ca7a2ac03014240471f795a608cf66137c8f
F20101111_AABJJU stanford_b_Page_024.tif
170dc6c28c4b463f32ced9c8554d6eda
2e13c9175b7ee6de96d36494b68c075b21503f34
7386 F20101111_AABIGS stanford_b_Page_163thm.jpg
26e927f6e7da1c6e471071af9b98c592
3699a4f8404f2d7d336388c0dfbf1649c49abe2c
F20101111_AABIHG stanford_b_Page_024.QC.jpg
954a194f680bfa5c47aee717d5dd3521
cef86dd5bef7ccf39b00268242daa304d67579d3
F20101111_AABJKJ stanford_b_Page_054.tif
8f9e644d2ad92129ae2f823fc00415dc
d40464a0ce013cca327875b65531391db540e59d
F20101111_AABJJV stanford_b_Page_025.tif
a3c921569d12988dc97d53bc852a217d
7c756862b213fede9605c09e4149cb6ea4947d43
893809 F20101111_AABIGT stanford_b_Page_151.jp2
4789ebe75e488177b82fca74f212dec1
f91cce75b345b6d0021f00b3bc407723d9f71ab8
23580 F20101111_AABIHH stanford_b_Page_102.QC.jpg
a5d10a2c15869cba5ed2f24f345b681b
d95c7def6d02d7a2641b4faa4ab003154a4f13d3
F20101111_AABJKK stanford_b_Page_055.tif
0efcbbe638e8edf7a659477e31b68782
225216a4e63c9c7b72252d8c888b78f1f787937b
F20101111_AABJJW stanford_b_Page_027.tif
fb87aef24273ba4a9ea53dd94eca4a43
7aed21408b61e9fe6b57bbd2021eba7d28b21142
6665 F20101111_AABIGU stanford_b_Page_020thm.jpg
5be82621959d9bae70273b47f09b43d8
7c4a14382a39bfb4fd18faf5595dd236dfc69ac9
F20101111_AABIHI stanford_b_Page_105.jp2
3f978f438fb45541d4c9da60827e2fe5
1f4fb44e5f27a37638d36c2c7a25c7a1eccc9c02
F20101111_AABJLA stanford_b_Page_095.tif
a5ab49ea14e3a479ea13cd81f52caec0
a9d350eb32e13f3100a6729898f8a53cccee8597
F20101111_AABJKL stanford_b_Page_056.tif
c41c43fe1e4ada8d83eef6742ff12de4
cb84ccff41e2f86b37c54404abc69fe8796f5e08
F20101111_AABJJX stanford_b_Page_028.tif
1b834c9e23ade0d388ff460abfc2443c
b142fe74eb3bc075a27a47d8354e0e47f8fe43dc
F20101111_AABIGV stanford_b_Page_159.jp2
4e5d803f954683f4dd84f9fbc3f1f3ad
91975c340997b70083c0544b8b7e777e588a33e9
8118 F20101111_AABIHJ stanford_b_Page_170thm.jpg
a8e3fa6c096156f6314b38debc1e49c3
820f39b94bf1bf2d64ffafb03abfa61ca4dbabbc
F20101111_AABJLB stanford_b_Page_098.tif
d5560a7dc6ac59ffa6d50637bd568960
dac9efbc1cf2f8961b33a489459eb6ee4c20ce0c
F20101111_AABJKM stanford_b_Page_057.tif
d75cfe0bdd242347285dd61086976dd4
c4db01fba541692a4df39b50b155db0730ffa199
F20101111_AABJJY stanford_b_Page_032.tif
a7572ecc4858df597d489a334864c9f0
0f9c36925b220f7b85ff0aee403939358d483f84
26295 F20101111_AABIGW stanford_b_Page_057.QC.jpg
1d5d1d1a0ba86b408d23a47f7774f038
547254755828d52e4ba2c60490ba2019f9be86b8
55234 F20101111_AABIHK stanford_b_Page_165.pro
3060c4e0322c7b2443cf4e12ce93dcb1
01b17b00990a8213964e856a5bbfe75293fd4719
F20101111_AABJLC stanford_b_Page_099.tif
3700e7d1fd6d2b999a01c56f78d7214f
4fa1f0792fda263e39d65693c6e35ea0ef2ad4af
F20101111_AABJKN stanford_b_Page_059.tif
d0078727c2c470047f1e983f508bcc49
ff30a02aca8af74232bbad95c5c8dfdcac272cea
F20101111_AABJJZ stanford_b_Page_034.tif
4c75f91a8d3e6131792c0771f9127343
69c6f1e049c671af60f0a60977241e0c6bdb98ec
8048 F20101111_AABIGX stanford_b_Page_137thm.jpg
7bf816c44efa160c17fbd46d9e097d0b
514ec2d47da921e0140b66dce118ea7e5cee2046
26474 F20101111_AABIIA stanford_b_Page_147.QC.jpg
3f98e99c43dbeebc2b3cab1f9fddb605
604e39d49bfe50b78382e63c7f6b76e81851d7a4
2124 F20101111_AABIHL stanford_b_Page_112.txt
ba782970a09a1b6150f7361eb339255c
b3115f617c807b97e581ecbafed9743514af9b10
F20101111_AABJLD stanford_b_Page_100.tif
803e58a27bb39b45fbb91910623b3c72
5b3a56f5c6824bcd34e9e16c2cbb00b3add996bd
F20101111_AABJKO stanford_b_Page_060.tif
756a45a23e4fec5044ab0c1d6d00de59
d38cf28176dd2dd901086fbdff505b95e61e81b3
F20101111_AABIGY stanford_b_Page_032.jp2
ea74d2ca09420f1886ccdf114d957ec8
5f90d7360eacdfc43a13fec257f72da58fc9c4e4
30825 F20101111_AABIIB stanford_b_Page_151.pro
8693b6bbee7d955fc6618999755ee67c
220cdbc274f0b0c405fbd0e1fd44dfb1d72faf25
2371 F20101111_AABIHM stanford_b_Page_168.txt
350417ca963e463115dbf4433ce80dc6
18a1e69e9fa905a6a5310a5c48f38f378a92c8be
F20101111_AABJLE stanford_b_Page_102.tif
3ee6be4a107a657994569f8b9b98b1a0
d564d927ffc4d007ca780ceec02e79fc627c9d00
F20101111_AABJKP stanford_b_Page_064.tif
b5e7dc33681f04d8868372d555742da9
9313395d4572a9442f46dc9005ee76be23da4130
25623 F20101111_AABIGZ stanford_b_Page_022.QC.jpg
03de606eb029dcf9f238b26dc347d09e
6a22cdf6c194eda4e70b3b0dce2ceb29e9d47c95
56702 F20101111_AABIIC stanford_b_Page_047.pro
5dac7a31d633cea62acd2a24e7e9ef73
ac0f3cdcb5698e9b7620243bf8d9a57fe08cd1e8
82331 F20101111_AABIHN stanford_b_Page_098.jpg
6379a614f462dd26ed87dce692c985f8
9091c829c1414510b3b1f7a834f907b7929950da
F20101111_AABJLF stanford_b_Page_105.tif
c588a7d10eeea6102e22908438f53f43
a1f1138707983b595a99a8b7d2511d4536c92709
F20101111_AABJKQ stanford_b_Page_069.tif
a6a94db1dba8892ff657f6e48791e6eb
1ed9dd2eba54bd40de4ed5487fdd2a4a8839e53f
29991 F20101111_AABIID stanford_b_Page_167.QC.jpg
e10fafe526a5417015a2bdd94e5dfb98
e12c58169409b601b150d7115ac688ac7c81c267
F20101111_AABIHO stanford_b_Page_085.tif
f0b546ee2a60f16bba24775ab5bd45f3
b79b1c63627ecd293780dd18b569641f088273ab
F20101111_AABJLG stanford_b_Page_108.tif
4ea3e480814a255480290aa419cf4c44
5fe81ed58ae5b30c4237d9283a6fbaaf623aa8e4
F20101111_AABJKR stanford_b_Page_070.tif
11568419fdb5ae8b063cceec1cd774d7
640a2542036d7dd2d6735879dda6da0617ca7197
6018 F20101111_AABIIE stanford_b_Page_087thm.jpg
b5c2a79d38d4e7cb73ee55fe1ee49bf9
5d0b6d521e2d3250d0b9228784553176449b1b4e
25937 F20101111_AABIHP stanford_b_Page_063.QC.jpg
db1afbb8cf44916c1365e50ae332fbad
03aa38b4b9911442c20e5ffbe07068668405a166
F20101111_AABJKS stanford_b_Page_071.tif
b7feab5ad34e1b5d7c6f5cff45e4a62c
aa715bd67f72ad10e27ca43eda6d1424c848ce51
2989 F20101111_AABIHQ stanford_b_Page_115.txt
78667756f3ff5e4875c773e7c1cadc08
0369816e53901bc8f74d87859b126f2a2aeeee2f
F20101111_AABJLH stanford_b_Page_111.tif
b1b79de14555b79c1fb69d519824befd
4350b11629d9e31faed21281f9f1afa5b439ccfb
F20101111_AABJKT stanford_b_Page_072.tif
522f8512d90b7a48e2db4d164b4d72f2
9960441b0ce84846286fa94463aa4cc236832649
F20101111_AABIIF stanford_b_Page_128.jp2
c3b9db760104a80be79ae9dbe3405ef8
7f80f319d6b2bdbf720d13a7709777e840e86ad0
2144 F20101111_AABIHR stanford_b_Page_041.txt
4b189ee9a81e04a57de002321a24f5e4
7244b8d6740763ff3b58694c07b95adc984c8bf4
F20101111_AABJLI stanford_b_Page_112.tif
3c1aba1d6a121322f39aa87709c21a5e
f9200d0c4ea89cc8afd0df0db336301370baa97c
F20101111_AABJKU stanford_b_Page_073.tif
a265f7c1e3d7eba5569cd5f5a6a9d1c5
c9ea9f4ba31d4da4c4bc0e7265e91490c5095b4d
88935 F20101111_AABIIG stanford_b_Page_033.jpg
8a8d7757ef9e9cd7bcc0500629bcb675
4d0aa4eeaa6add31b9f1c40e0951de42063e191b
2562 F20101111_AABIHS stanford_b_Page_177.txt
6648c8ea2eb604c61988542cbfc2be28
d84301f2494e133559b289e069e0c2641e0cdf60
F20101111_AABJLJ stanford_b_Page_113.tif
85f7203d28e73c826ee00b9a70800d65
f5155567478296ed13aad068d5e93a790716f15c
F20101111_AABJKV stanford_b_Page_079.tif
bb8f6d0cf6d87cf406add12fab11597b
aa389d140768ffcbe0f18b7ae0c3258b8979e659
82101 F20101111_AABIIH stanford_b_Page_075.jpg
a8a08d9e8d1caf952fc8e238929a8f4f
1cc9efa445f6cadfd6103c6b36f54ad8facb46c1
28706 F20101111_AABIHT stanford_b_Page_168.QC.jpg
505897652c4f537c587c6896c02bf320
05a5ff98b2fa15a383846eaa46c5e8f83bdbcbf2
F20101111_AABJLK stanford_b_Page_115.tif
480255d22671f71b19ebfbdaf773b0b6
c5a2c9430dd11fe38036d908c2e5ade4d9c449a2
F20101111_AABJKW stanford_b_Page_080.tif
0d58e81ef5fd201fc68e0c84b98b3e1b
c96cbbb60929b33e93188b11c62e48baf60d65cc
F20101111_AABIII stanford_b_Page_139.tif
cace0f4ff3b644ea9a7f367b44f1a3ed
26231435b85991a5e1f6bfb06ce1e9adedda54f9
25543 F20101111_AABIHU stanford_b_Page_015.QC.jpg
cdd5297d8ca358740fdaeaafc8156027
0082def27d45653cf1f4ebb6a8de769dfeb99e58
F20101111_AABJMA stanford_b_Page_146.tif
c2558433aa6aac664fb6e2450f60ecec
8d788e2ea151eb897471bb219ee32c360ac8aa7f
F20101111_AABJLL stanford_b_Page_116.tif
b9020b69ae37bec4915fe65e36a35395
ce0ae75629098cb31275f112083527c1cb62fb35
F20101111_AABJKX stanford_b_Page_082.tif
f9054b8618bcb27816e223b08f3940f2
c96ac853489c609c512ea6ec7dbe8d71d53800f9
7342 F20101111_AABIIJ stanford_b_Page_027thm.jpg
a40b40b2adff5af5e81ee93681760240
82dc205f25b8d0a2feef96687d9d9fdcfb58dfb7
1051958 F20101111_AABIHV stanford_b_Page_129.jp2
da1d6759495009c5099b51df53c609ff
b4c94113136ac698c32f2af808a7623ad64e6219
F20101111_AABJMB stanford_b_Page_147.tif
03ca6f24edb5d07d537ac7822f95ee5e
26c6c6761f4ed87e79609105ff49054b8c02743e
F20101111_AABJLM stanford_b_Page_117.tif
14f24e9bcff7be571e2ffc975189051a
30db44b9cf6c3906625ba208d21a211ac2503798
F20101111_AABJKY stanford_b_Page_093.tif
fab8147f982bd5e1e54969005da50474
9115ce367c55748187dafb545d1bf23827ae3261
F20101111_AABIIK stanford_b_Page_114.tif
13e05afe3c3dd6c96f55e6e3c93cb38e
529add1e76d972201ed3afcbb791439121d187eb
968451 F20101111_AABIHW stanford_b_Page_059.jp2
991264c081994516d4c383da42150319
85dcf8a38af681a02e42465bcfab13b88c50da9d
F20101111_AABJMC stanford_b_Page_150.tif
417fc4e9e04fc39fb62e68c86255e969
aaad4f1dc4a5d0d4fe356cf9809ea85078ecbae1
F20101111_AABJLN stanford_b_Page_119.tif
28c20fd262942000d521c7689fe5db5e
3755ede1682e76df9dd7936daafe9072bbb2614b
F20101111_AABJKZ stanford_b_Page_094.tif
07c5d73ca436f2a528108de777dcb3c8
a60069b25fa5b318e48c14a6f72512567ef953a2
F20101111_AABIJA stanford_b_Page_060.txt
18d8d0dee990d3a711aec5a0317335b2
73dbb13a1d474845ab4287f7a8caed8de3964aab
F20101111_AABIIL stanford_b_Page_007.tif
8b5e19a49000faed896225a5f61f24a9
33e6fac40d77a29f9c36b6c5513ff66a93fff4d5
29860 F20101111_AABIHX stanford_b_Page_092.QC.jpg
34972a0a9df91d8157af06955dfac556
10bf356448569756a24e54f907b283e61054f261
F20101111_AABJMD stanford_b_Page_152.tif
f6f445b7ad0dff0381bbda5b3da1192b
67f5ec6465c2e544301541829d3e2952a80f2761
F20101111_AABJLO stanford_b_Page_121.tif
c7103d60b0572f108c9e619183759139
94b7d54e961a24f6c6586984885d4e8cbd337261
56309 F20101111_AABIJB stanford_b_Page_036.pro
87eaf3301c7044a8b086277373ec90f2
01d9acf13a961c6c83cb4229a9ae880a362fd854
910806 F20101111_AABIIM stanford_b_Page_070.jp2
e3265ce3d486e22d6cc76c17363c0e77
5584aa3ad9f24ac7ad01169663e5799dca23eaf4
937184 F20101111_AABIHY stanford_b_Page_004.jp2
e922a68f174a482b63a819501ac356c4
8f696057c0826262cbfcb88cab185940fa0ed6e9
F20101111_AABJME stanford_b_Page_154.tif
dc605f4861d22470288100acbaabf4ed
45af907a338c4f88115cc07bc2a075f71ce344b1
F20101111_AABJLP stanford_b_Page_123.tif
5449ec6f6acf3f4a4cee44d2b7f0f29a
66e3ae41ae7f8f1fa53e0897ebcd7bbf02d4297a
F20101111_AABIJC stanford_b_Page_044.txt
44ded5012dfba6e1b28aff96465d31a8
b87231e533260e9822bb8723edc6f32428728959
85205 F20101111_AABIIN stanford_b_Page_085.jpg
34302f2150456acea1f228b4892b13ea
337337fec4520d03772689850e687a760b5674b3
54920 F20101111_AABIHZ stanford_b_Page_158.pro
2bc237a2650b83e4db8c377c74ec099f
ba58e797f303c04fedeb9c7e97d7bcb3b1b9f807
F20101111_AABJMF stanford_b_Page_155.tif
5d0704e989d1e74fa712f4fa06547e19
98ed1bb978fd54f27c0b0b19901c88f82e4b592d
F20101111_AABJLQ stanford_b_Page_124.tif
fd4c4191595e87147d5843c92455e88f
c318c75aef1dc44218f92bd3ee8f87f7f0cad796
34789 F20101111_AABIJD stanford_b_Page_007.jpg
2b110824a4477031266d270489193b36
4dcddaefa7d79317e9477552bb256fb54a4dcfe5
86851 F20101111_AABIIO stanford_b_Page_027.jpg
d550e54f3f3698f303897ab9eab8e823
91629d5141222e518b785782417e26bf79aacd0c
F20101111_AABJMG stanford_b_Page_160.tif
6a99e4cd6fece9e62b82c9a874d6830a
aa35a785cf4e5bed3fa2adf68a9de19ff4cad3d9
F20101111_AABJLR stanford_b_Page_125.tif
4c4a8b256814637e12b6f9b1b50f2b15
ef776a8a2a89a45145cef68ce1340f1aeb08e1a6
27374 F20101111_AABIJE stanford_b_Page_049.QC.jpg
50c772bb981c73e76845d6ecccf308c2
908c2d1322d57a21bad77c37b4031f36982833b3
56612 F20101111_AABIIP stanford_b_Page_144.pro
1fc98cf6c66a59d0b55e8de5f68f05c3
6695dbedad6286bd5955bbcd17c956c76f119bd3
F20101111_AABJMH stanford_b_Page_163.tif
19ce126715f75c04ae2cc72965a346a5
24b7c5f69f468263a66c9b5b2d31cbb2fe81e766
F20101111_AABJLS stanford_b_Page_126.tif
ab0223ade7efa6714522d23c9ff6b412
b1d295e5aae685020ac677d2b34bf02f55285cf3
6653 F20101111_AABIJF stanford_b_Page_073thm.jpg
8cfa2a6bd9bf9d12c0d7064d2df51330
e38af9be4dcccdb582a1ac38f8dbeddc076ebb06
2185 F20101111_AABIIQ stanford_b_Page_089.txt
08f102a59cfb358eb39a1b8beb7a5bee
6ec42f9dd6d353791b031de096a26e7c2c2b6dfc
F20101111_AABJLT stanford_b_Page_129.tif
66f18f29e235f3431f75092e76785690
0bfbe28a44cc5bb0e1e0c78a66ddc9c0c2ed09b8
F20101111_AABIIR stanford_b_Page_047.tif
872446901ba7421d4c64520770adedfa
b97ba6652c9103bc40cdad4173a2c597c20a2617
F20101111_AABJMI stanford_b_Page_165.tif
d8b765787744968991b20373209a7951
b42d9af4033d8e9707a627ca4d19e8518db7fc98
F20101111_AABJLU stanford_b_Page_136.tif
e8075c08f817b93a50ab88c0230bec8e
ef6ba2c8ed83aaab8a84d1fa00362e59e2c00ef7
F20101111_AABIJG stanford_b_Page_091.jp2
64580cd568ec205e37363f836886aa44
97a6d4a7ed1138d9f8cb88086212baf2d7c84857
7186 F20101111_AABIIS stanford_b_Page_055thm.jpg
b316fdf1525c6861bdd6670324328767
9c124cfe6d3c3107685b6e60d8a1b5c2bae93074
F20101111_AABJMJ stanford_b_Page_166.tif
f7863e1e5913071a6dd8b369c0db2003
fb989b151d238633e97f79f59aa28fc0f8c43869
F20101111_AABJLV stanford_b_Page_137.tif
d4c5e145924e6478614cc90527138852
1d32f0a27a9242729e65acd7cba59a73e65b68de
F20101111_AABIJH stanford_b_Page_075.tif
8dfc5720f5425d9e0de094b88162812e
472b09b31bee28b22010d1b43ab07047d782ce21
45346 F20101111_AABIIT stanford_b_Page_071.pro
230eb81c5dc8ee10df0dda9c7f7263cc
0c0510186185844888f1cd2d804f9aea4632e18a
F20101111_AABJMK stanford_b_Page_168.tif
38adb537209fe9a2cdb236dfddde6cb1
221ef15b5a21016f65406f221aba3ce5feebcba2
F20101111_AABJLW stanford_b_Page_138.tif
5696747c683d44e677584052a28b4fe1
06ece83275b71e92f7da41af0eb4fc4099050a38
27555 F20101111_AABIJI stanford_b_Page_108.QC.jpg
ba03d9484e3cc1bd1d56e4e4799f83ab
4e046643d33cf3e847aeeaec9d8f2398c2941923
F20101111_AABIIU stanford_b_Page_162.tif
73f1d5b09b29a9351b31e1167b7a02e5
921343c8935b2913a5c56452259e29be13ba7383
54432 F20101111_AABJNA stanford_b_Page_016.pro
70d4c9e8e8f24485efe563cdcc79e527
e242c74722721927823354ba6b203d665fc353a8
F20101111_AABJML stanford_b_Page_173.tif
8af24cc114e9c7d654deb09c241a4168
6e95f870647e6b9010508344d9128e35aca08069
F20101111_AABJLX stanford_b_Page_140.tif
2dd8a323b193662196332a013bb80708
ca3b672215ea6b3ede9351e7456fe3b328cc7d9a
F20101111_AABIJJ stanford_b_Page_037.tif
20e4f5d68943e522ece37fd31fbb2dc6
1037182ff2ac3643dbf758063e574bcd33df2ec0
6894 F20101111_AABIIV stanford_b_Page_067thm.jpg
a6bc49450a6c8ca6dd74b450c719cfef
2e643fef976a53893700e0d59f3cc2e984d99ac1
54872 F20101111_AABJNB stanford_b_Page_017.pro
4b23f63ab4c1b376cee0387035ebd42a
461eaf39c4c9d63f95938fdf50fed9481174959d
F20101111_AABJMM stanford_b_Page_175.tif
616b446f77783a932a94b06880c36ca8
f6da42944449b7b9cb968026456a2005c4f34c79
F20101111_AABJLY stanford_b_Page_142.tif
7bf4449b4c4b22a3dbaf7c1161db8ccb
110d3d016a4502e29ba09dab091b889fd3a8a00e
F20101111_AABIJK stanford_b_Page_034.jp2
fe0ed8c80de1c4d54322e1e62bfa08f3
dc25a6c3c9e909d4c7947f45f0071bfcc953dba9
30065 F20101111_AABIIW stanford_b_Page_172.QC.jpg
e297d5627baaf7bea5220128e44e4f70
3da485dc1986dfabd74fd9fbef6988fe651ffe79
53716 F20101111_AABJNC stanford_b_Page_025.pro
e84a40f19b8bb21391c106c5fc369125
6baf6b15ab1ff2e3b3d7b983fb8c5deb6ccd8b72
F20101111_AABJMN stanford_b_Page_176.tif
0b4bccbbc17fe676c6cb6c2312e8cb0c
20a4c9ec90759460ce4a3c8ee932144e8410d078
F20101111_AABJLZ stanford_b_Page_145.tif
196194934be1fa43c8cddd843f7f3615
f2d1b6fd16ce4bf58a8771fe9c549649891604c2
49032 F20101111_AABIJL stanford_b_Page_133.pro
37e10125d96406399f79995b8fad961b
a9ec88711062cb40d15c39cfda437195a8f691da
6683 F20101111_AABIIX stanford_b_Page_152thm.jpg
a2c1682af5ed35ecfcdff2f38e80cbe0
dc16f732eb99276bdd8f44323ede35946d06820f
55622 F20101111_AABIKA stanford_b_Page_044.pro
8f4d6a0c2d07902bdb2b70e82cb30b24
263a61f82679b999ed738984e593aa5eb8c8b9c7
50296 F20101111_AABJND stanford_b_Page_026.pro
a61f1c67f7b96465ed4a6d10329d5f06
11dab28b5425c701eec81a108f77fa93e66d77aa
F20101111_AABJMO stanford_b_Page_179.tif
8640da4848ee6be8902c71f66073499f
d1cb893f824c5858b626c60d954d2e644d3ee0c8
2223 F20101111_AABIJM stanford_b_Page_040.txt
a1113f7342047561260d9a0c8bb3cb95
938478fa630e1bc5e7c6860ddc0c7377e05ab4b9
23106 F20101111_AABIIY stanford_b_Page_160.pro
6730190365451865d6a76c733dad01bd
71298a69b263ef2c9d2557d2d6a9d81c7f348258
26368 F20101111_AABIKB stanford_b_Page_074.pro
5ef562838b7bdfc645f454b8b9f71cbf
b7e1aee6e3eca2d860c8e57dd9e16f12902e32d6
55300 F20101111_AABJNE stanford_b_Page_027.pro
515ebabb71fd0a14641e014065a11bd7
cdf92b9fdd7b6b89aa3876e80d9d8374cfe2506d
7948 F20101111_AABJMP stanford_b_Page_001.pro
26f3f6dc15724a7866c1f5cae952772b
b3748cf11107b7c3f70b24d50f5283f94033cd09
2145 F20101111_AABIIZ stanford_b_Page_023.txt
b17df9f99047befbd8138d1d8f81715a
f09e2b8e24f6e9e7a3783a248bca0b0ee55a0c6e
F20101111_AABIKC stanford_b_Page_077.tif
8f53d23d3e1c5e52f3322fbcffd8ca77
a009c02cb59e1ad1d8c8c6c038b60c77629a6e33
7376 F20101111_AABIJN stanford_b_Page_025thm.jpg
3fcfe9d5249e876ec8d614fb46011a1b
bcdc1b04ca3ad9aa2b61cc7298e2aa8237419b67
56847 F20101111_AABJNF stanford_b_Page_028.pro
5023ce8f8a1d4e5d21d385a06bdff144
f18f4d47013db163d310860189fe16b17343d989
1019 F20101111_AABJMQ stanford_b_Page_002.pro
6d94621b09d6a3363b79f411be40dbd8
dae36a352747df3fe9daf7a82cdf83edb75df099
23403 F20101111_AABIKD stanford_b_Page_149.QC.jpg
ff51f1d2c7bbb739bded20367401949a
888437aef69936b46570636a53648885a47cf9de
F20101111_AABIJO stanford_b_Page_174.tif
464a690dfc4cc537a9410f6b896b57c8
b2d18a76a8b533faf8a32660df0b828624b983a8
55696 F20101111_AABJNG stanford_b_Page_029.pro
9d0f808fa26eb9b511dc7e1041c7ced7
9fb85c9c9a98ec503e5bb56d6a0020292e97feb5
41806 F20101111_AABJMR stanford_b_Page_004.pro
d2700abfbb74eda81aa827236b0ba15c
6de34cf5230c71147521efce793dfd86469f7b08
2236 F20101111_AABIKE stanford_b_Page_136.txt
5e677e7e32909b61419b8e88ea1e7b66
ed82a6742c1c43188e3f7061bb7e572a2820810f
F20101111_AABIJP stanford_b_Page_143.tif
aa61657b50fb188ce3248f05ebc54de3
d8b52ec79be2926b91c10a705292bc89e1235047
55926 F20101111_AABJNH stanford_b_Page_030.pro
77e1e0abfeada0603a463321c5ad933c
92abc2924c6c5dcdef86ce2353604d57ffb52265
19996 F20101111_AABJMS stanford_b_Page_007.pro
6cb217dbcd21cd67d6fa319748e01433
89e73870fcb90c4754e6c59f3b258db4237902e9
78003 F20101111_AABIKF stanford_b_Page_107.jpg
09164d86a542025f16fd034b304c6272
8f8722d877121273ee5182194d8fd0acc017a0d2
1051913 F20101111_AABIJQ stanford_b_Page_101.jp2
e3473c6172dd5ced3d537c36ade0cd8d
b93ffdf22c5ebb762fa46ee6c5e1ce81712923ae
56098 F20101111_AABJNI stanford_b_Page_031.pro
2c43f6bb610d70bf9e87099de2a7a0a8
e3ebd516773f059519b2a5adea0433c5867b1aba
75339 F20101111_AABJMT stanford_b_Page_008.pro
202dabfe37cfbda56fda7307034ecee3
ab176288b7916955d7fe548f5568ad950d6d5bc1
27863 F20101111_AABIKG stanford_b_Page_023.QC.jpg
701b3caa662e5ea36e2b07f717e9932b
e46cccf341ac7e9d6f7b6fbaee7ded50047cbccc
F20101111_AABIJR stanford_b_Page_015.tif
6c0f41634f09c2a7d293f6bd6dd56390
2a0e298a65835db96a8b693eca239ca1b54303c1
72488 F20101111_AABJMU stanford_b_Page_009.pro
ccd50993f49e52db927a6b683e37c115
b8b6f961dfedd521d441ba975a032246a1056850
F20101111_AABIJS stanford_b_Page_148.tif
8d23fbf2316b6d3da978989c8f8ec318
4d1f61996227a2a7ed602e15fa63c04748e89c36
54116 F20101111_AABJNJ stanford_b_Page_032.pro
68bbded949165fc0053baa96dc4a9d05
7f5b4358b466c2a260265b27ccc308362f6f80e7
78374 F20101111_AABJMV stanford_b_Page_010.pro
91705ebca515c739b2dc63e9716f0247
3e6cfe66f51e9b7f495c045cd9a6042bb9277555
7418 F20101111_AABIKH stanford_b_Page_095thm.jpg
71e37ea67bd1a7e009692a501302f377
09919ecce06078a755ce0aa7979b76bec65fee09
F20101111_AABIJT stanford_b_Page_085.jp2
8755a2f9c0e0c5b46e37105b4bf53b99
ef040b78f68d9790ab1b7ddb2de35fd3ee388437
56133 F20101111_AABJNK stanford_b_Page_033.pro
af4cf49bf7ab4865b786a410381aa8a8
d288d44dfacd5483c25d6582fa66897e6e5be705
48105 F20101111_AABJMW stanford_b_Page_012.pro
019143387fcecb097e9d74b291f7ac1d
8aaf86df7e4c26d210809a35923b93efe364f30c
7008 F20101111_AABIKI stanford_b_Page_075thm.jpg
1d8aefa55ef94fcb8d64a20050eeeddd
0bff045a38071fbb15c3aaf6423c3805849d425c
28926 F20101111_AABIJU stanford_b_Page_028.QC.jpg
62c6fd9d8f7f2a0189169eecd8834fd6
f9a8e5177a60e7907dff4d798c9ba864c6f3e58d







AEROELASTIC ANALYSIS AND OPTIMIZATION OF MEMBRANE MICRO AIR
VEHICLE WINGS




















By

BRET KENNEDY STANFORD


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008

































2008 Bret Kennedy Stanford

































To my family, despite making fun of me for being in school for so long
To Angel, despite making fun of me for a bunch of other reasons
And to Fatty too, for only scratching me when I really deserve it, which is often









ACKNOWLEDGMENTS

Thank you to Dr. Peter Ifju, who offered his guidance on a countless number of research

projects, and still let me work on the ones he thought were dumb. Unquestionably the coolest,

smartest, most up-beat professor I've ever been around.

Thank you to Dr. Rick Lind, for consistently pointing out when I need to shave, or get a

haircut, or more frequently, both.

Thank you to Dr. Roberto Albertani, for sharing with me his passion for all things wind

tunnel related, and for sharing his equipment up at the REEF.

Thank you to Dr. Raphael Haftka and Dr. David Bloomquist for serving on my committee

and sitting through my long, scientifically-questionable presentations without complaining.

Thank you to Dr. Dragos Viieru, for imparting me with his vast knowledge of CFD.

Thank you to Dr. Wei Shyy, for all his help my first few semesters of grad school.

A final thank you to all the people who hung around the labs I worked in. Frank Boria,

who helped teach me the real names for various tools and hardware, which had previously been

known to me only as shiny metal things. A thanks-in-advance to Frank for taking all of my

future phone calls concerning mortgages, insurance, child rearing, etc, no matter how distraught

and hysterical they may be. Mujahid Abdulrahim, for discussing with me the ethics of returning

a rental car completely caked in mud, and going in reverse through a drive-thru. Wu Pin, for

relating countless unintentionally funny and creepy stories that I'll never forget, despite my best

efforts. I'll always wonder how you got into this country.









TABLE OF CONTENTS

page

A CK N O W LED G M EN T S ................................................................. ........... ............. .....

LIST OF TABLES .................. .....................................................7

LIST OF FIGURES .................................. .. ..... ..... ................. .8

A B S T R A C T ......... ....................... ............................................................ 12

CHAPTER

1 INTRODUCTION ............... ............................ .............................. 13

M motivation .......... ............................... ................................................13
Problem Statem ent ............................................................... .... ..... ........ 18
D issertation Outline .................................................................... .. ............ 19
C contribution s ................................................................................2 1

2 L ITE R A TU R E R E V IE W ........................................................................ .. .......................22

M icro A ir V vehicle A erodynam ics ................................................ .............................. 22
L ow R eynolds N um ber Flow s........................................................................... .... ... 22
Low A aspect Ratio W ings...................................... .............................................. 23
Low Reynolds Number Low Aspect Ratio Interactions ............................................24
R igid W ing O ptim ization ............................................................................. .................... 26
M icro A ir V vehicle A eroelasticity ........................................ ............................................28
Tw o-D im ensional A irfoils............................................ .................... ...............28
T hree-D im ensional W ings....................................................................... ..................3 1
A eroelastic Tailoring .............. ......... ............................. ............ 34
T op ology O ptim ization ............................................................................... ......................3 5

3 EXPERIMENTAL CHARACTERIZATION............................................. ............... 39

C losed L oop W ind T unnel........................................................................... .....................39
Strain G age Sting B balance .............................................................................. ............... 40
U uncertainty Q uantification ...................................................................... .................. 43
W ind T unnel C corrections ........................................................................ ..................43
V isual Im age C orrelation ......... ................................................................ ... ................ 44
D ata P ro c e ssio n ............................................................................................................... 4 7
U uncertainty Q uantification ...................................................................... .................. 48
M odel F abrication and Preparation ............................................................. .....................49

4 COMPUTATIONAL FRAMEWORK AND VALIDATION............................................51

Stru ctu ral S olv ers ........................................................ .............. 5 1









C om posite L am inated Shells......................................... ............................................52
M em brane M modeling ................. .................................. ................ .. ............. 56
Skin Pre-tension Considerations..................... ....... .............................. 62
F lu id S olv ers .................................................................................................................... 66
V ortex Lattice M methods ............................................................................. 66
Steady N avier-Stokes Solver ........................................... ....... ...................... 67
Fluid M odel Com prisons and V alidation ........................................... .....................70
A eroelastic Coupling .............. ......... ............................. ............ 72
M moving G rid T technique ......................................................................... ...................72
N um erical Procedure ..................................................... ........ .. ...... .... 73

5 BASELINE WING DESIGN ANALYSIS....................................................................... 75

W ing D eform ation .................................................................................... .. ................ .. 7 5
Aerodynamic Loads ....................................................... .. ....................... 83
F low Structures....................................................88

6 A ER O ELA STIC TA IL O R IN G .............................................................. ......... .................97

OFAT Simulations ............... .............................. ............................. 98
M em brane P re-T pension ......................................................................... ....................99
Single Ply L am inmates .................................................... .. ........................ 102
D double Ply L am inmates ......................... ............................................. 103
Batten Construction .......... ............... ..... ................ ...... .......... .. ........ .... 105
Full F actorial D designed E xperim ent......... ................. ...................................... ...............107
Experimental Validation of Optimal Design Performance ............................................116

7 AEROELASTIC TOPOLOGY OPTIMIZATION.................................... ............... 122

C om putational F ram ew ork ......... .. ................... ...................................... ........................125
M material Interpolation .......... ............................................. .... ......... 125
Aeroelastic Solver .................................... ... .. ......... ....... .... 128
A djoint Sensitivity A nalysis.................................................................................... 130
O ptim ization P rocedure............................................................................... ........... 133
Single-O objective O ptim ization ...................... .. .. ......... .. ........................ ......................134
M ulti-O objective O ptim ization ...................................................................... ..................149

CONCLUSIONS AND FUTURE WORK ................................. .................................161

R E F E R E N C E S .........................................................................167

B IO G R A PH IC A L SK E T C H ...................... .. .. ......... .. ............................... ......................... 179









6









LIST OF TABLES


Table page

4-1 Experimental influence matrix (mm/N) at points labeled in Figure 4-2 .............................55

4-2 Numerical influence matrix (mm/N) at points labeled in Figure 4-2.............. ..................56

5-1 Measured and computed aerodynamic characteristics, a = 6..................... ...............88

6-1 Optimal MAV design array with compromise designs on the off-diagonal, a = 12:
design description is (wing type, Nx, Ny, number of plain weave layers) .......................115

6-2 Optimal MAV design performance array, a = 12: off-diagonal compromise design
performance is predicated by column metrics, not rows ................................................115









LIST OF FIGURES


Figure page

1-1 Batten-reinforced m em brane wing design........................................ .......................... 14

1-2 Perimeter-reinforced membrane wing design.................................. .......................... 15

3-1 Schem atic of the wind tunnel test setup .............................................................................39

3-2 Quantification of the resolution error in the VIC system................................................. 48

3-3 Speckled batten-reinforced membrane wing with wind tunnel attachment .........................50

4-1 Unstructured triangular mesh used for finite element analysis, with different element
types used for PR and BR wings ..................................................................... 52

4-2 Computed deformations of a BR wing skeleton due to a point load at the wing tip (left)
and the leading edge (right) ....................................................................... ..................54

4-3 Compliance at various locations along the wing, due to a point load at those locations .......56

4-4 Uni-axial stretch test of a latex rubber membrane....................................... ............... 61

4-5 Circular membrane response to a uniform pressure.................... ..................61

4-6 Measured chordwise pre-strains in a BR wing before the tension is released from the
latex (left), and after (right) ....................................................................... ..................63

4-7 Monte Carlo simulations: error in the computed membrane deflection due to a spatially-
constant pre-strain distribution assum ption ............................................ ............... 65

4-8 Computed pre-stress resultants (N/m) in the chordwise (left), spanwise (center), and
shear (right) in a BR wing, corrected at the trailing edge for a uniform pre-stress
resultant of 10 N /m .............. ............................ ........... .... ...... ............66

4-9 C FD com putational dom ain.......................................................................... ....................68

4-10 Detail of structured CFD mesh near the wing surface....................... ................69

4-11 Computed and measured aerodynamic coefficients for a rigid MAV wing, Re = 85,000 ...71

4-12 Iterative aeroelastic convergence of membrane wings, a = 9........................................74

5-1 Baseline BR normalized out-of-plane displacement (w/c), a = 15.......................................76

5-2 Baseline BR chordwise strain (xx), a = 15 ............................................................ 77

5-3 Baseline BR spanwise strain (eyy), a = 15 ..................................................................77









5-4 Baseline BR shear strain (Exy), a = 15 ........................................................... ......... 78

5-5 Baseline PR normalized out-of-plane displacement (w/c), a = 15....................................79

5-6 Baseline PR chordwise strain (exx), a = 15.................................. .........................79

5-7 Baseline PR spanwise strain (Syy), a = 15 ..........................................................80

5-8 Baseline PR shear strain (xy), a = 15 .............................................. .......................... 80

5-9 Baseline BR aerodynamic and geometric twist distribution, a = 15..................................81

5-10 Baseline PR aerodynamic and geometric twist distribution, a = 15 ...................................82

5-11 Aerodynamic and geometric twist at 2y/b = 0.65.................................... ............... 83

5-12 Baseline lift coefficients: numerical (left), experimental (right)..........................................84

5-13 Baseline drag coefficients: numerical (left), experimental (right) .......................................86

5-14 Baseline pitching moment coefficients: numerical (left), experimental (right) ...................87

5-15 Baseline wing efficiency: numerical (left), experimental (right) ......................................87

5-16 Pressure distributions (Pa) and streamlines on the upper surface of a rigid (left), BR
(center), and PR wing (right), a = 0................................... ...............................90

5-17 Pressure distributions (Pa) and streamlines on the lower surface of a rigid (left), BR
(center), and PR wing (right), a = 0................................... ...............................91

5-18 Pressure distributions (Pa) and streamlines on the upper surface of a rigid (left), BR
(center), and PR wing (right), a = 15............................................................92

5-19 Pressure distributions (Pa) and streamlines on the lower surface of a rigid (left), BR
(center), and PR wing (right), a = 15............................................................94

5-20 Section normal force coefficients, and pressure coefficients (2y/b =0.5), a = 0 ...............96

5-21 Section normal force coefficients, and pressure coefficients (2y/b =0.5), a = 15 ..............96

6-1 Computed tailoring of pre-stress resultants (N/m) in a BR wing, a = 12 ...........................99

6-2 Computed BR wing deformation (w/c) with various pre-tensions, a = 12.........................100

6-3 Computed tailoring of pre-stress resultants (N/m) in a PR wing, a = 12............................101

6-4 Computed PR wing deformation (w/c) with various pre-tensions, a = 12..........................102









6-5 Computed tailoring of laminate orientation for single ply bi-directional carbon fiber, a =
12 ......... ....................................................................... ...... ................ 103

6-6 Computed tailoring of laminate orientations for two plies of bi-directional carbon fiber
in a B R w ing, a = 12 ........ ................................................................ ........ .. .... ........104

6-7 Computed tailoring of laminate orientations for two plies of bi-directional carbon fiber
in a P R w in g a = 12 ................. ..... .. ........................................ ..................... .......... 10 5

6-8 Computed tailoring of batten construction in a BR wing, a = 12 ......................................106

6-9 Computed normalized out-of-plane displacement (left) and differential pressure (right)
at x/c = 0.5, for various BR designs, a = 12 ...................................... ...............107

6-10 Computed full factorial design of a BR wing, a = 12 ..................................................109

6-11 Computed BR wing deformation (w/c) with one layer of plain weave (left), two layers
(center), and three layers (right), a = 12 ............................................ ......... .............. 109

6-12 Computed full factorial design of a PR wing, a = 12................................................ 111

6-13 Computed PR wing deformation (w/c) with one layer of plain weave (left), two layers
(center), and three layers (right), a = 12 ................. ......... .......... ............... .. 111

6-14 Computed design performance and Pareto optimality, a = 12........................................113

6-15 Experimentally measured design optimality over baseline lift ...................................118

6-16 Experimentally measured design optimality over baseline pitching moments ................19

6-17 Experimentally measured design optimality over baseline drag ................................. 120

6-18 Experimentally measured design optimality over baseline efficiency .............................120

7-1 W ing topologies flight tested by Ifju et al. [10] ....................................... ............... 123

7-2 Sample wing topology (left), aerodynamic mesh (center), and structural mesh (right).......124

7-3 Effect of linear and nonlinear material interpolation upon lift.............................................127

7-4 Measured loads of an inadequately reinforced membrane wing, U, = 13 m/s .................... 130

7-5 Convergence history for maximizing L/D, a = 30, reflex wing................ .................135

7-6 Affect of mesh density upon optimal L/D topology, a = 120, reflex wing......................137

7-7 Affect of initial design upon the optimal CD topology, a = 120, reflex wing....................138

7-8 Affect of angle of attack and airfoil upon the optimal CL topology ................................139









7-9 Affect of angle of attack and airfoil upon the optimal L/D topology............................. 140

7-10 Normalized out-of-plane displacements (top) and differential pressure coefficients
(bottom) for baseline and optimal topology designs, a = 12, reflex wing......................141

7-11 Deformations and pressures along 2y/b = 0.58 for baseline and optimal topology
designs, a = 12 reflex w ing....................................................................... 142

7-12 Normalized out-of-plane displacements (top) and differential pressure coefficients
(bottom) for baseline and optimal topology designs, a = 12, cambered wing .............145

7-13 Deformations and pressures along 2y/b = 0.58 for baseline and optimal topology
designs, a = 12 cam bered w ing ............................................. ............................ 146

7-14 Wing topology optimized for minimum CL, built and tested in the wind tunnel.............148

7-15 Experimentally measured forces and moments for baseline and optimal topology
designs, reflex w ing .................................................................. ....... .... 149

7-16 Convergence history for maximizing L/D and minimizing CL,, 6 = 0.5, a = 3, reflex
w ing .............. ...................... .............................................. ....... 15 1

7-17 Trade-off between efficiency and lift slope, a = 3, reflex wing..............................152

7-18 Normalized out-of-plane displacements (top) and differential pressure coefficients
(bottom) for designs that trade-off between L/D and CL,, a = 3, reflex wing..............154

7-19 Deformations and pressures along 2y/b = 0.58 for designs that trade-off between L/D
and C L a = 3 reflex w ing .............................................................................. .... 155

7-20 Trade-off between drag and pitching moment slope, a = 12, reflex wing ........................ 156

7-21 Trade-off between lift and lift slope, a = 12, cambered wing........................................157

7-22 Normalized out-of-plane displacements (top) and differential pressure coefficients
(bottom) for designs that trade-off between CL and CL,, a = 12, cambered wing..........159

7-23 Deformations and pressures along 2y/b = 0.58 for designs that trade-off between CL
and CL,, a = 120, cam bered wing ............................. ....................... ............... 160









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

AEROELASTIC ANALYSIS AND OPTIMIZATION OF MEMBRANE MICRO AIR
VEHICLE WINGS

By

Bret Kennedy Stanford

May 2008

Chair: Peter Ifju
Major: Aerospace Engineering

Fixed-wing micro air vehicles are difficult to fly, due to their low Reynolds number, low

aspect ratio nature: flow separation erodes wing efficiency, the wings are susceptible to rolling

instabilities, wind gusts can be the same size as the flight speed, the range of stable center of

gravity locations is very small, etc. Membrane aeroelasticity has been identified has a tenable

method to alleviate these issues. These flexible wing structures are divided into two categories:

load-alleviating or load-augmenting. This depends on the wing's topology, defined by a

combination of stiff laminate composite members overlaid with a membrane sheet, similar to the

venation patterns of insect wings. A series of well-validated variable-fidelity static aeroelastic

models are developed to analyze the working mechanisms camberingg, washout) of membrane

wing aerodynamics in terms of loads, wing deformation, and flow structures, for a small set of

wing topologies. Two aeroelastic optimization schemes are then discussed. For a given wing

topology, a series of numerical designed experiments utilize tailoring of laminate orientation and

membrane pre-tension. Further generality can be obtained with aeroelastic topology

optimization: locating an optimal distribution of laminate shells and membrane skin throughout

the wing. Both optimization schemes consider several design metrics, optimal compromise

designs, and experimental validation of superiority over baseline designs.









CHAPTER 1
INTRODUCTION

Motivation

The rapid convergence of unmanned aerial vehicles to continually smaller sizes and greater

agility represents successful efforts along a multidisciplinary front. Technological advances in

materials, fabrication, electronics, propulsion, actuators, sensors, modeling, and control have all

contributed towards the viable candidacy of micro air vehicles (MAVs) for a plethora of tasks.

MAVs are, by definition, a class of unmanned aircraft with a maximum size limited to 15 cm,

capable of operating speeds of 15 m/s or less. Ideally, a MAV should be both inexpensive and

expendable, used in situations where a larger vehicle would be impractical or impossible, to be

flown either autonomously or by a remote pilot. Military and defense opportunities are perhaps

easiest to envision (in the form of over-the-hill battlefield surveillance, bomb damage

assessment, chemical weapon detection, etc.), though MAVs could also play a significant role in

environmental, agriculture, wildlife, and traffic monitoring applications.

MAVs are notoriously difficult to fly; an expected consequence of a highly maneuverable

and agile vehicle that must be flown either remotely or by autopilot [1]. The aerodynamics are

beset by several unfavorable flight issues:

1. The operational Reynolds number for MAVs is typically between 104 and 105. Flow over the
upper wing surface can be characterized by massive flow separation, a possible turbulent
transition in the free shear layer, and then reattachment to the surface, leaving behind a
separation bubble [2]. Such flow structures typically result in a loss of lift, and an increase in
drag, and a drop in the overall efficiency [3].

2. The low aspect ratio wing (on the order of unity) promotes a large wing tip vortex swirling
system [4], which interferes with the longitudinal circulation of the wing [5]. Entrainment of
the aforementioned separated flow can lead to tip vortex destabilization [6]; the resulting
bilateral asymmetry may be the cause of the rolling instabilities known to plague MAV flight.

3. Sudden wind gusts may be of the same order of magnitude as the vehicle flight speed (10-15
m/s). Maintaining smooth controllable flight can be difficult [7] [8].









4. The range of flyable (statically stable) CG locations is generally only a few millimeters long,
which represents a strenuous weight management challenge [9].

These problems, along with a broad range of dynamics and control issues, can be alleviated

through the appropriate use of wing shape adaptation. Active morphing mechanisms have been

successfully used on a small class of unmanned air vehicles [1], but the limited energy budgets

and size constraints of micro air vehicles make such an option, at present writing, infeasible. As

such, the current work is restricted to passive shape adaptation.

Passive shape adaptation can be successfully built into a MAV wing through the use of a

flexible membrane skin [10]. The basic structure of these vehicles is built around a composite

laminate skeleton. Bi-directional graphite/epoxy plain weave or uni-directional plies are usually

the materials of choice, due to durability, low weight, high strength, and ease of fabrication: all

qualities well-exploited in the aviation industry [11]. The carbon fiber skeleton is affixed to an

extensible membrane skin, of which several choices are available: latex, silicone, plastic sheets,

or polyester [12]. The distribution of carbon fiber and membrane skin in the wing determines the

aeroelastic response, and is demonstrated by two distinct designs. The first utilizes thin strips of

uni-directional carbon fiber imbedded within the membrane skin, oriented in the chordwise

direction (Figure 1-1). The trailing edge of the batten-reinforced (BR) design is unconstrained,

and the resulting nose-down geometric twist of each flexible wing section should alleviate the

flight loads: decrease in CD, decrease in CL,, delayed stall (as compared to a rigid wing) [13].


b


Figure 1-1. Batten-reinforced membrane wing design.


Op- _74









A second design leaves the interior of the membrane skin unconstrained, while the perimeter

of the skin is sealed to a thin curved strip of carbon fiber (Figure 1-2). The perimeter-reinforced

(PR) wing deformation is closer in nature to an aerodynamic twist. Both the leading and the

trailing edges of each membrane section are constrained by the relatively-stiff carbon fiber. The

positive cambering (inflation) of the wing should lead to an increase in CL and a decrease (more

negative) in Cma [14].











Figure 1-2. Perimeter-reinforced membrane wing design.

While both of these wing structures can adequately perform their intended tasks (load

alleviation for the BR wing, load enhancement for the PR wing), several sizing/stiffness

variables exist within both designs, leading to an aeroelastic tailoring problem. Conventional

variables such as the laminate fiber orientation [11] can be considered, but the directional

stiffness induced by varying the pre-tension within the membrane skin may play a larger role

within the aeroelastic response [15]. Systematic optimization of a single design metric will

typically lead to a wing structure with poor performance in other important aspects. For

example, tailoring the PR wing structure for maximum static stability may provoke an

unacceptable drag penalty, and vice-versa. Of the aerodynamic performance metrics considered

here (lift, drag, efficiency, static stability, gust suppression, and mass) many are expected to

conflict, as with most engineering optimization applications. Formal multi-objective

optimization procedures can be used to tailor flexible MAV wing designs that strike an adequate









compromise between conflicting metrics, filling in the trade-off curves.

While thorough exploration of this aeroelastic tailoring design space can provide a

fundamental understanding of the relationship between spatial stiffness distribution and

aerodynamic performance in a flexible MAV wing, further steps towards generality can be

achieved by removing the constraint that the wing structures must utilize a BR or a PR design.

Topology optimization is typically used to find the location of holes within a homogenous

structure, by minimizing compliance under a constraint upon the volume fraction [16]. Here it is

used to find the location of membrane skin within a carbon fiber skeleton that will optimize a

given aerodynamic objective function.

This work will be able to highlight wing topologies with superior efficacy to those designs

considered above (for example, a wing with better gust suppression qualities than the BR wing),

as well as designs that strike a compromise between conflicting metrics (for example, a

topological combination of the BR and the PR wings). While the results may be more rewarding

than those obtained from tailoring, aeroelastic topology optimization is significantly more

complex. Tailoring requires 5-10 sizing and stiffness variables, but the topology optimization

may utilize thousands of variables: the wing is divided into a series of panels, each of which may

be membrane or carbon fiber. This necessitates a gradient-based algorithm, while evolutionary

algorithms or response surface approaches are feasible for the former problem.

Both aeroelastic tailoring and topology optimization are effective tools for exploiting the

passive shape adaptation of flexible MAV wings, but the computational cost is prohibitive. It is

not uncommon for aeroelastic optimization studies to require hundreds, or even thousands, of

function evaluations. Numerical modeling of flexible MAV wings is very challenging and

expensive: flow separation, transition, and reattachment [17], vortex shedding and pairing [18],









and wing tip vortex formation/destabilization [6] are all known to occur within the flow over low

aspect ratio wings at low Reynolds numbers. Structurally, the membrane skins used for MAV

construction are beset by both geometric and material nonlinearities [19]; the orthotropic nature

of the carbon fiber laminates must also be computed.

At present, no numerical model exists which can accurately predict all of the three

dimensional unsteady features of an elastic MAV wing (flow transition being a particular

challenge [8]). As such, an important step in aeroelastic optimization of MAV wings is careful

development of lower-fidelity numerical models. Both inviscid vortex lattice methods and

laminar Navier-Stokes solvers are investigated, along with linear and nonlinear membrane finite

elements (only static aeroelastic models are considered here). In light of the low-fidelity tools

that must be used (to maintain computational cost at a reasonable level), a second important step

in aeroelastic optimization is extensive experimental model validation. Three levels of model

validation are employed: validation of the structural response of individual components of the

membrane wing, validation of the aeroelastic behavior of various baseline membrane wings, and

validation of the superiority of the computed optimal wings (found either through tailoring or

topology optimization) over the baselines.

Further complications arise from complex objective functions. As discussed above, gust

response is an important performance metric for micro air vehicles [7], but systematic

optimization would require an unsteady model, non-homogenous incoming flow, and subsequent

time integration. Similarly, delaying the onset of wing stall would require several sub-iterations

to locate the stalling angle. Both objectives can be reasonably replaced by a minimization of the

lift slope, which is more amenable to a systematic optimization. Finally, the computational

complexity is further exacerbated by the multi-objective nature of the problem. Wing structures









that optimize a single objective function are of limited value; of greater importance is the array

of designs that lie along the Pareto optimal front.

This front is a trade-off curve comprised of non-dominated designs, one of which can be

selected based on additional considerations not included in the optimization: manufacturability,

flight specifications (duration, payload), etc. Computation of the Pareto front is costly when

gradient-based search routines are used for optimization, typically involving successive

optimization runs with various convex combinations of the objective functions. The success of

this technique depends on the convexity of the Pareto optimal front. More efficient methods for

computing the Pareto front are available if evolutionary algorithms or response surface methods

are employed.

Problem Statement

The static aeroelasticity of membrane micro air vehicle wings represents the intersection of

several rich aerodynamics and mechanics problems; numerical modeling can be very challenging

and expensive. Furthermore, MAVs are beset with many detrimental flight issues, and are very

difficult to fly: systematic numerical optimization schemes can be used to offset these problems,

improving flight duration, gust suppression, or static stability. Many optimization studies can be

considered for MAVs; the current work utilizes aeroelastic optimization, which will require

hundreds of costly function evaluations to adequately converge to an optimal design. As the

feasibility of such a scheme relies on a moderate computational cost, what is the lowest fidelity

aeroelastic model that can be appropriately used?

Model development requires extensive experimental validation, and several challenges

exist here as well. The forces generated by a MAV wing are very small, and highly-sensitive

instrumentation is needed. For deformation measurements, only vision-based non-contacting

methods are appropriate. What particular components are required to construct an adequate









experimental test-bed for flexible MAVs? What performance metrics should be compared

between numerical and experimental results for sufficient model validation?

Upon suitable validation of the aeroelastic model, two optimization studies are developed:

tailoring and topology optimization. Considering the former, with a given spatial distribution of

laminated carbon fiber and membrane skin throughout the wing, what is the optimal chordwise

and spanwise membrane pre-tension and carbon fiber laminate lay-up schedule? For the

aeroelastic topology optimization studies, with a given membrane pre-tension and laminate

orientation, what is the optimal distribution of carbon fiber and membrane skin throughout the

wing? What performance metrics should be optimized? As these metrics will surely conflict,

what multi-objective optimization schemes are appropriate for computation of the Pareto front?

Can the numerically-indicated optimal wing design structures be built and tested, and will the

experimental results also indicate superiority over similarly-tested baseline designs?

Dissertation Outline

This work begins with a detailed literature review of micro air vehicle aerodynamics (low

Reynolds number flows, low aspect ratio wings, unsteady flow phenomena), aeroelasticity

(membrane sailwings, flexible filaments), and optimization (rigid wing airfoil and planform

optimization, tailoring). I review the literature pertaining to topology optimization as well, with

a particular emphasis upon aeronautical and aeroelastic applications.

I then discuss the apparatus and procedures used for experimental characterization of the

membrane micro air vehicle wings. This includes a low-speed closed loop wind tunnel, a high

sensitivity sting balance, and a visual image correlation system. Information is also given

detailing wing fabrication and preparation. I summarize the computational framework, including

both linear and nonlinear structural finite element models. Three-dimensional viscous and

inviscid flow solvers are formulated, along with aeroelastic coupling and ad hoc techniques









devised to handle the membrane skin pre-tension. The estimated validity range of each model is

discussed.

I detail the deformation patterns, flow structures, and aerodynamic characterization of a

series of baseline flexible and rigid MAV wings, obtained both numerically and experimentally

for comparison purposes. Once the predictive capability of the aeroelastic model is well-

verified, these data sets are studied to uncover the working mechanisms behind the passive shape

adaptation and their associated aerodynamic advantages.

I then use a non-standard aeroelastic tailoring study to identify the optimal wing type and

structural composition for a given objective function, as well as various combinations thereof.

Wing types are limited to rigid, batten-reinforced, and perimeter-reinforced designs; structural

composition variables include anisotropic membrane pre-tension and laminate lay-up schedule.

Multi-objective optimization is conducted using a design of experiments approach, with a series

of aerodynamic coefficients and derivatives as metrics. The tailoring concludes with

experimental validation of the performance of selected optimal designs.

Finally, I formulate a computational framework for aeroelastic topology optimization of a

membrane micro air vehicle wing. A gradient-based search is used, with analytically computed

sensitivities of the same aerodynamic metrics as used above. The optimal wing topology is

discussed as a function of flight condition, grid density, initial guess, and design metric. I

optimize a convex combination of two conflicting objective functions to construct the Pareto

front, with a demonstrated superiority over the baseline wing structures employed in the tailoring

study. As before, the work concludes with experimental validation of the performance of

selected optimal designs.









Contributions

1. Develop a set of variable-fidelity aeroelastic models for low Reynolds number, low aspect
ratio membrane micro air vehicle wings.

2. Develop a highly-sensitive non-intrusive experimental test-bed for model deformation and
flight loads.

3. Optimization-based system identification of the wing structure's material properties.

4. Experimental aeroelastic model validation of flight loads and wing deformation.

5. Optimize multiple flight metrics by tailoring membrane pre-tension and laminate orientation.

6. Develop computational framework for topology optimization of membrane wings, with an
analytical sensitivity analysis of the coupled aeroelastic system.

7. Able to provide scientific insight into the relationship between optimal wing flexibility, flow
structures, and the resulting beneficial effects upon flight loads and efficiency.

8. Experimental validation of the superiority of selected optimal designs over baselines.









CHAPTER 2
LITERATURE REVIEW

Micro Air Vehicle Aerodynamics

A long history of flight testing, computational modeling, and wind tunnel work has

generally pushed the design methodology of successful fixed wing MAVs to a thin, cambered,

low aspect ratio "flying wing". Maximizing the wing area for a given size constraint obviously

leads to a low aspect ratio design. Further desire to minimize the size of a MAV negates the use

of horizontal stabilizers, replaced with a reflex airfoil for longitudinal static stability, wherein

negative camber present towards the trailing edge helps offset the longitudinal pitching moment

of the remainder of the wing. The superiority of thin wings for MAV applications can be shown

by both three dimensional inviscid simulations [20] and two-dimensional viscous simulations

[21] [22], where the drop in the adverse pressure gradients increases the lift and decreases the

drag towards stall. Similar tools, as well as wind tunnel testing, indicates the advantage of

cambered wings over flat plates [5]; beyond the obvious increase in lift, higher lift-to-drag ratios

are reported by Laitone [23]. Much work has also been done on locating suitable MAV

planform shapes. Torres identifies the inverse-Zimmerman as ideal, based upon size restrictions,

required angle of attack, and drag performance; the optimum shifts to an elliptical shape as the

aspect ratio is increased [3].

Low Reynolds Number Flows

Low Reynolds number laminar flow is likely to separate against an adverse pressure

gradient aft of the pressure recovery location (velocity peak) on the upper wing surface, even for

fairly low angles of attack. The formation of a turbulent boundary layer aft of a separation

bubble is a very mutable process: Reynolds number, pressure distributions, airfoil geometry,

surface roughness, turbulence intensity, acoustic noise, wall heating, and a-direction (whether









the angle of attack is being increased or decreased can lead to hysteresis [24]) are all cited by

Young and Horton [2] as highly influential on the formation of a bubble. Furthermore, the flow

will only reattach to the surface if there is enough energy to maintain circulating flow against

dissipation [25].

An extensive survey of low Reynolds number (3-104 5-105) airfoils is given by

Carmichael [17] (there are quite a few others, as reviewed by Shyy et al. [26]). The study finds

that, for the lower end of tested Reynolds numbers, the laminar separated flow does not have

time to reattach to the surface. Above 5-104, the flow will reattach, forming a long separation

bubble over the wing. At the upper end of the range of Reynolds numbers discussed by

Carmichael, the size of the bubble decreases, generally resulting in a decrease in form drag.

Increasing the angle of attack generally enhances the turbulence in the flow, which can also

prompt quicker reattachment and shorter bubbles [8]. The length of the separation bubble can

generally be inferred from the plateau-like behavior of the pressure distribution: the flow speeds

up before the bubble (dropping the pressure), and slows down after the bubble [27].

This description is a time-averaged scenario: in an unsteady sense, the inflectional velocity

profile across the separation bubble can develop inviscid Kelvin-Helmhotlz instabilities and

cause the shear layer to roll up. This leads to periodic vortex shedding and the required matching

downstream [18], and can cause the separation bubble to move back and forth [28]. Further

work detailing low Reynolds number flow over rigid airfoils can be found by Nagamatsu [29],

Masad and Malik [30], and Schroeder and Baeder [31].

Low Aspect Ratio Wings

Early work in low aspect ratio aerodynamics was sparked by an inability to fit

experimental data with linear aerodynamics theories, as reported by Winter [32] for aspect ratios

between 1.0 and 1.25. The measured lift is typically higher than predicted (similar to vortex lift









discrepancies seen on delta wings [27]), as the strong wing tip vortices interfere with the

longitudinal wing circulation. The most obvious indication of such an interaction is the high

stalling angles of low aspect ratio wings, where the downward momentum of the tip vortices can

keep the flow attached to the upper wing surface. Experimental work by Sathaye et al. [33]

using an array of pressure ports was able to confirm the deviation of the lift distribution from

elliptic wing theory.

Lian et al. [28] report a computed dip in the previously constant pressure coefficients over

the upper wing surface at 75% of the semi-span for high angles of attack, but only minor changes

on the bottom surface at the wing tip. These low pressure cells at the wing tip will grow in

intensity and spread inward towards the root as the angle of attack (and thus the strength of the

swirling system) is increased [34]. The cells are a nonlinear contribution to the wing's lift; their

growth with angle of attack increases CL, with angle of attack as well. Torres [3] gives a general

cutoff between a linear and a nonlinear CL-a relationship at an aspect ratio of 1.25. Low aspect

ratio corrections to the lift predicted by linear theory (among many) are given by Bartlett and

Vidal [35], while Polhamus [36] is able to collapse the measured profile drag data at various

aspect ratios to a single curve through the use of an effective two-dimensional lift coefficient.

Further experimental work is given by Kaplan et al. [37], who use measurements of the

trailing vortex structure off of low aspect ratio flat plates for adequate comparison with force

balance measurements. The authors indicate that the nonlinear lift curves may also be caused by

a loss of leading edge suction, and a rotation of the force vector into a flow-normal direction.

Viieru et al. [38] discusses the use of endplates to temper the induced drag from the tip vortices,

with reported improvements in the lift-to-drag ratio at small to moderate angles of attack.

Low Reynolds Number Low Aspect Ratio Interactions

Several interactions between the low aspect ratio and low Reynolds number aerodynamics









of MAVs are reported in the literature. Mueller and DeLaurier cite aspect ratio as the most

important design variable, followed by wing planform and Reynolds number. Free stream

turbulence intensity and trailing edge geometry are reported to be non-factors, and Reynolds

number is only important near stall [39]. Flow visualization experiments by Gursul et al. [40] on

swept, non-slender, low aspect ratio wings find the presence of primary and secondary vortices,

with stagnant flow regions outboard of the former. Vortex merging and other unsteady

interactions within the shear layers are found to be highly dependent on Reynolds number.

Kaplan et al. [37] report a fluctuation in the location of the vortex core off of a semi-

elliptical wing at 8,000 Reynolds number. Numerical simulations and flow visualization by

Tang and Zhu [6] of an accelerating elliptical wing show an unstable interaction between a

longitudinal secondary separated vortex and the tip vortices. This destabilization (for angles of

attack above 11) causes the tip vortex system to swing back and forth along the wing, leading to

bilateral asymmetry problems in roll. The authors also note a stationary separated vortex (rather

than the customary shedding) for angles above 33, possibly due to the vertical components of

the tip vortices.

Cosyn and Vierendeels [41], Brion et al. [42], and Stanford et al. [43], discuss numerical

wing modeling of lift and drag for comparison with wind tunnel experiments: the lack of a three-

dimensional turbulent-transition model is generally cited as the reason for poor correlation at

higher angles of attack. Results documenting the aerodynamics of a complete micro air vehicle

(wing with fuselage, stabilizers, propellers, etc) are scarce: wind tunnel experimentation by Zhan

et al. document longitudinal and lateral stability as a function of vertical stabilizer placement and

wing sweep [44]. Similar stability data is given by Ramamurti et al. [45] for a MAV wing with

counter-rotating propellers.









Gyllhem et al. [46] reports that the presence of a fuselage, motor, and stabilizers

surprisingly improves the computed maximum lift and stall angle (compared to simulations with

just the wing), but increases the drag as well. Experimental work by Albertani [47] finds just the

opposite: a decrease in lift of the entire vehicle, but less of a penalty when passive shape

adaptation is built into the wing. Waszak et al. [13] are able to show significant improvements in

efficiency if a streamlined MAV fuselage is used.

Rigid Wing Optimization

Though the main scope of the current work is to improve the aerodynamic qualities of

fixed micro air vehicle wings through the judicious use of aeroelastic membrane structures, much

successful work has been done with multidisciplinary optimization of the shape, size, and

components of a rigid MAV wing. These studies must often make use of low fidelity models

due to the large number of function evaluations required for a typical optimization run, and may

not be able to capture the complicated flow physics described above. Nonetheless, insight into

the relationship between sets of sizing/shape variables and a given objective function can still be

gained.

Early work is given by Morris [48], who finds the smallest vehicle that will satisfy given

constraints throughout a theoretical mission, using several empirical and analytical expressions

for the performance evaluation. Rais-Rohani and Hicks investigate a similar problem, using a

vortex lattice method (for computations of aerodynamic performance and stability, along with

propulsion and weight modules) and an extended interior penalty function method to reduce the

size of a biplane MAV [49]. Kajiwara and Haftka emphasize the unconventional need for

simultaneous design of the aerodynamic and the control systems at the micro air vehicle scale,

due to limited energy budgets [50].









Torres [3] uses a genetic algorithm to minimize a weighted combination of payload,

endurance, and agility metrics, with various discrete (wing and tail planform) and continuous

(aspect ratio, propeller location, angle of attack, etc) variables. Aerodynamic analysis is

provided by a combination of experimental data, analytical methods, and interpolation

techniques. The author cites convergence problems stemming from the discrete variables.

Genetic algorithms are also used in the work of Lundstrom and Krus [51] and Ng et al. [52]. The

latter indicates that these algorithms are more suited for the potentially disjointed design spaces

presented by MAV optimization efforts. A comparison between a genetic algorithm and

gradient-based sequential quadratic programming used to design winglets for a swept wing MAV

indicates the superiority of the former, with a vortex lattice method used for aerodynamic

analysis. However, a genetic algorithm may only be feasible for lower fidelity tools, due to the

large number of function evaluations required for convergence.

Higher fidelity aerodynamics tools (namely, thin-layer or full Navier-Stokes equation

solvers) are employed in recent studies. For example, a combined 2-D thin layer Navier-Stokes

model and a 3-D panel method is used by Sloan et al. [53], who use the outcome to construct a

response surface to optimize the wing geometry for minimum power consumption. As above,

the study reveals the superiority of thin wings, and finds that optimal airfoil shapes are

insensitive to aspect ratio. Lian et al. [54] use a full Navier-Stokes solver to maximize the lift-to-

drag ratio of a rigid MAV wing subject to various lift and wing convexity constraints, with

sequential quadratic programming search methods. Efficiency improvements are feasible by

decreasing the camber at the root and increasing at the tip, thereby decreasing the amount of flow

separation. Improvements are found to be more substantial at moderate angles of attack.

Given the computational complexities associated with MAV simulation, several research









efforts use wind tunnel hardware-in-the-loop for optimization. Load measurements from a sting

balance are fed into an optimizer as the objective function or constraints. Genetic or other types

of evolutionary algorithms are invariably used, as a sensitivity analysis requires finite difference

computations which are easily distorted by experimental error. Examples with MAVs are given

by Boria et al. [55] (optimize lift and efficiency with airfoil morphing), Hunt et al. [56] (optimize

the forward velocity and efficiency of an omithopter, with flapping rate and tail position as

variables), and Day [57] (planform optimization of a wing with variable feather lengths).

Micro Air Vehicle Aeroelasticity

The role of aeroelasticity in the study of membrane micro air vehicle wings differs greatly

from conventional aircraft. While certain aeroelastic instabilities do exist (typically involving

the lift slope approaching infinity [15], unstable flapping of a poorly constrained trailing edge

[58], or luffing [59]), classical problems like torsional divergence and flutter have little bearing

on MAV design, due to the low aspect ratio nature of the wings and the small operating dynamic

pressures [60]. Great savings are available in the form of load redistribution however, as

mentioned above: potential improvements in lift, drag, stall, and longitudinal static stability can

all be obtained. Lateral control improvements are also obtainable with membrane wings [1] [61].

Furthermore, chordwise bending of a wing section (aerodynamic twist) can often be ignored in

conventional aircraft (except, for example, when constructed from laminates with many off-axis

plies [11]), but such deformation is very prevalent in low aspect ratio membrane wings.

Two-Dimensional Airfoils

The aeroelastic membrane structure is dominated by three-dimensional structural and

aerodynamic effects, but much useful insight can be gained from two-dimensional simulations

and experiments. Such endeavors are obviously easier to undertake for PR-type membrane

wings, but three-dimensional reinforcement must be taken into account for a pure membrane (or









string in two dimensions) with geometric twist, as the structure alone cannot sustain a flight load

in a stable manner. A second option involves considering an elastic sheet with some

bending/flexural stiffness. A large variety of work can be found in the literature concerning two-

dimensional flexible beams in flow. For problems on a MAV scale, work tends to focus on flags

and organic structures such as leaves, seaweed, etc.

Fitt and Pope [58] derive an integro-differential flag equation for the shape of a thin

membrane with bending stiffness in unsteady inviscid flow, considering both a hinged and a

clamped leading edge boundary condition. Argentina and Mahadevan [62] solve a similar

problem, and are able to predict a critical speed that marks the onset of an unstable flapping

vibration, noting that the complex instability is similar to the resonance between a pivoting

airfoil in flow and a hinged-free beam vibration. Over-prediction of the unstable flapping speed

(when compared to experimental data) leads to the possibility of a stability mechanism wherein

skin friction induces tension in the membrane. Alben et al. [63] discuss the streamlining of a

two-dimensional flexible filament for drag reduction. In particular, they are able to show that the

drag on a filament at high angles of attack decreases from the rigid Uo2 scaling to Uo4/3.

Early work in the study of membrane wings without bending stiffness is given by Voelz

[64], who describes the classical two-dimensional sail equation: an inextensible membrane with

slack, fixed at the leading and trailing edges, immersed in incompressible, irrotational, inviscid

steady flow. Using thin airfoil theory, along with a small angle of attack assumption, Voelz is

able to derive a linear integro-differential equation for the shape of the sail as a function of

incidence, freestream velocity, and slack ratio. Various numerical solution methods are

available, including those by Thwaites [65] (eigenfunction methods) and Nielsen [66] (Fourier

series methods), to solve for lift, pitching moments, and membrane tension.









Multiple solutions are found to exist at small angles of attack with a finite slack ratio:

approaching 0 from negative angles provides a negatively-cambered sail, though the opposite is

true if this mark is approached from a positive value. The sail is uncertain as to which side of the

chord-line it should lie [65], a phenomenon which ultimately manifests itself in the form of a

hysteresis loop [15]. Variations on this problem are considered by Haselgrove and Tuck [67],

where the trailing edge of the membrane is attached to an inextensible rope, thereby introducing

a combination of adaptive aerodynamic and geometric twist. Increasing the length of the rope is

seen to improve static stability, but decrease lift.

Membrane elasticity is included in the work of Murai and Maruyama [68], Jackson [69],

and Sneyd [70], indicating a nonlinear CL-a relationship as strains develop within the membrane

at high incidence. Viscous flow models are employed in the work of Cyr and Newman [71] and

Smith and Shyy [72]. The latter cites viscous effects as having much more influence on the

aerodynamics of a sail wing than the effects of the assumptions made with linear thin airfoil

theory. Specifically, inviscid solutions tend to over-predict lift at higher angles of attack (or

large slack ratios), due to a loss of circulation caused by viscous effects about the trailing edge.

A comparison of lift and tension versus angle of attack with experimental data (provided by

Newman and Low [73], among others) yields mixed results; surprisingly, the lift is over-

predicted by the viscous flow model, yet the tension is under-predicted.

Smith and Shyy also note a substantial discrepancy in the available experimental data in

reported values of slack ratios, sail material properties, and Reynolds numbers, which may play a

role in the mixed comparisons [74]. Comparison of numerical and experimental data for two-

dimensional sails is also discussed by Lorillu et al. [75], who report satisfactory correlation for

the flow structures and deformed membrane shape. Unsteady laminar-turbulent transitional









flows over a membrane wing are studied by Lian and Shyy [8] (who correlate the frequency

spectrum of the vibrating membrane wing to the vortex shedding).

Three-Dimensional Wings

Three complicating factors can arise with the simulation of a three-dimensional membrane

wing, rather than the planar case [76]. First, the tension is not constant (in space or direction),

but is in a state of plane-stress. Secondly, the wing geometry can vary in the spanwise direction,

and must be specified. Finally, the membrane may possess a certain degree of orthotropy [59].

Most importantly, analytical solutions cannot generally be found. Simplifying assumptions to

this problem are given by Sneyd et al. [77] (triangular planform) and Ormiston [15] (rectangular

sailingg. Sneyd et al. reduce both the aerodynamics and the membrane deformation to two-

dimensional phenomena, where the third dimension is felt through a trailing edge cable.

Ormiston assumes both spanwise and chordwise deformation (but not aerodynamics), and

is able to effectively decouple the two modules by using only the first term of a Fourier series to

describe the inflated wing shape. Boudreault uses a higher-fidelity vortex lattice solver, but also

prescribes the wing shape, here using cubic polynomials [78]. Holla et al. [79] use an iterative

procedure to couple a double lattice method to a structural model, but assume admissible mode

shapes to describe the deformation of a rectangular membrane clamped along the perimeter. The

stress in the membrane is assumed to be always equal to the applied pre-stress (inextensibility,

which overwhelms the nonlinearities in the membrane mechanics). A similar framework is used

by Sugimoto in the study of circular membrane wings, where the wing shape is completely

determined by a linear finite element solver [80].

Jackson and Christie couple a vortex lattice method to a nonlinear structural model for the

simulation of a triangular membrane wing. Comparisons between a rigid wing, a membrane

wing fixed at the trailing edge, and one with a free trailing edge elucidate the tradeoffs in lift









between adaptive camber and adaptive washout [76]. Charvet et al. [81] study the effect of non-

homogenous incoming flow (vertical wind gradients and gusts) on a flexible sail. Schoop et al.

use a nonlinear membrane stress-strain relationship (hyperelasticity) with a vortex lattice solver

for simulation of a flat rectangular membrane wing [82].

Lian et al. [28] compute the unsteady aeroelasticity of a batten-reinforced membrane micro

air vehicle wing, with a nonlinear hyperelastic solver and a turbulent viscous flow solver, using

thin plate splines as an interfacing technique. Battens are simulated with a dense membrane.

The results indicate self-exciting membrane vibration on the order of 100 Hz, with a maximum

wing speed about 2% of the freestream, though overall aerodynamics are similar to that of a rigid

wing prior to stall. Stanford and Ifju [14] discuss steady laminar aeroelasticity of a perimeter-

reinforced membrane micro air vehicle wing, and are able to show the expected increase in lift

and stability. Significant drag penalties are seen to arise with increasing Reynolds numbers,

though the opposite is true for the rigid wing.

Complexities involving membrane wing models with both membranes and elastic shells

(such as batten reinforcement) can be found in the work of Stanford et al. [83] (linear mechanics)

and Ferguson et al. [84] (nonlinear). Higher-ordered membrane modeling with wrinkling (the

loss of one or more principle stresses) as pertaining to membrane wings is given in the work of

Smith and Shyy [85] and Heppel [86].

A large volume of work can be found dealing with experimental characterization of

membrane wings. Early wind tunnel work by Fink [87] details a full-scale investigation of an

11.5 aspect ratio sail wing with a rigid leading edge, wingtip, and root, and a cabled trailing

edge. The deformation is reported to be fairly smooth prior to stall, but visible rippling develops

along the membrane at the onset of stall. At low angles of attack, the slope of the lift curve is









unusually steep (an instability discussed by Ormiston [15], among others), as the strains in the

skin are low enough to allow for large changes in camber. Greenhalgh and Curtiss conduct wind

tunnel testing to study the effect of planform on a membrane wing; only a parabolic planform is

capable of sustaining flight loads without the aid of a trailing edge support member [88].

Galvao et al. [89] conduct tests on a membrane sheet stretched between two rigid posts, at

Reynolds numbers between 3-104 and 105. The results show a monotonic increase in membrane

camber with angle of attack and dynamic pressure, up to stall, as well as the aforementioned

steep lift slopes. De-cambering of the wing as the pressure on the upper surface increases due to

imminent flow separation is seen to ameliorate the stall behavior, as compared to a rigid plate.

Flow visualization of a batten-reinforced membrane MAV wing exhibits a weaker wing tip

vortex system than rigid wings [13], possibly due to energy conservation requirements [90].

Parks measures the vortex core of a BR wing 5% to 15% higher above the wing than for the rigid

case, though the flexible wing is seen to have a denser core-distribution of velocity for moderate

angles of attack [91]. Gamble and Reeder [92] measure the flow structures resulting from

interactions between propeller slipstream and a BR wing. The rigid wing spreads the axial

component of the propwash further along the wing (resulting in a higher measured drag),

whereas the membrane wing can absorb the downwash and upwash. A region of flow separation

is measured at the root of the rigid wing, significantly larger and stronger than that measured

from the membrane wing; the superiority decreases with larger Reynolds numbers.

Albertani et al. [9] detail loads measurements of both BR and PR wings, with dramatic

improvements in longitudinal static stability of both membrane wings over their rigid

counterpart. The BR wing has a noticeably smoother lift behavior in the stalled region, though

neither deforms into a particularly optimal aerodynamic shape: both incur a drag penalty.









Deformation measurements of a membrane wing under propwash indicate unsymmetrical (about

the root) wing shapes, a phenomenon which diminishes with higher angles of attack and dynamic

pressures [47]. Stults et al. use laser vibrometry to measure the modal parameters (shapes,

damping, frequency spectra) of a BR wing, which are then fed into a computational model for

simulation of static and dynamic deformations in both steady freestream and a gust [93].

Aeroelastic Tailoring

Although aeroelastic tailoring is generally defined as the addition of directional stiffness

into a wing structure so as to beneficially affect performance [11], this has traditionally meant

the use of unbalanced composite plates/shells. Despite the use of such laminated materials on

many MAV wing frames [94], there does not appear to be any tailoring studies on fixed micro air

vehicle wings. Some investigators have applied the concept to the design of flapping omithopter

wings [95] [96], where a bend/twist coupling can vary the twist-induced incidence of a wing to

improve thrust. Conventional tailoring studies can also be found applied to a larger class of

unmanned aerial vehicles [97] [98]. The latter study by Weisshaar et al. uses laminate tailoring

to improve the lateral control of a vehicle with an aspect ratio of 3. With ailerons, a wing

tailored with adaptive wash-in is shown to improve roll performance and roll-reversal speed,

though wash-out is preferred for a leading edge slat [98].

In addition to conventional laminate-based tailoring, drastic changes in the performance of

a membrane wing are attainable by altering the pre-tension distribution within the extensible

membrane. Holla et al. [79], Fink [87], Smith and Shyy [72], Murai and Maruyama [68], and

Ormiston [15] all note the enormous impact that membrane pre-tension has on aerodynamics: for

the two-dimensional case, higher pre-tension generally pushes flexible wing performance to that

of a rigid wing. For a three-dimensional wing, the response can be considerably more complex,

depending on the nature of the membrane reinforcement. Well-reported effects of increasing the









membrane pre-tension include: decrease in drag [89], decrease in CL, [15], linearized lift

behavior [72], increase in the zero-lift angle of attack [68], and more abrupt stalling patterns

[89]. Ormiston details aeroelastic instabilities in terms of the ratio of spanwise to chordwise pre-

tension [15].

Adequate control of membrane tension has long been known as a crucial concern to sailors

in order to efficiently exploit wind power [99]. Tension-control is similarly important to the

performance and agility of fighter kites: a wrinkled membrane surface will send the kite into an

unstable spin. When pointed in the desired direction, pulling the control line tenses and deforms

the kite, which thus attains forward velocity [100]. Biological inspiration for aerodynamic

tailoring of membrane tension can be seen in the wing structures of pterosaurs and bats. In

addition to membrane anisotropy (pterosaur wings have internal fibrous reinforcement to limit

chordwise stretching [101], while bat wings skins are measured to be 100 times stiffer in the

chordwise direction than the spanwise [102]), the tension can be controlled through a single digit

(pterosaurs) [59], or varied throughout the wing via multiple digits (bats) [103].

Work formally implementing membrane tension as a variable for optimizing aerodynamic

performance is very rare. Levin and Shyy [104] study a modified Clark-Y airfoil with a flexible

membrane upper surface, subjected to a varying freestream velocity. Response surface

techniques are used to maximize the power index averaged over a sinusoidal gust cycle, with

membrane thickness variation, elastic modulus, and pre-stress used as variables. The maximum

power index is found to coincide with the lower bound placed upon pre-stress, though lift and

efficiency are also seen to be superior to a rigid wing.

Topology Optimization

The basics of topology optimization are given by Bendsoe and Sigmund [16] and Zuo et al.

[105]: the design domain is discretized, and the relative density of each element can be 0 or 1.









Traditionally, this is done on a structure with static loads by minimizing the compliance under an

equality constraint upon the volume fraction, though recent work can be seen in the design of

compliant mechanisms [106] and channel flows [107] as well. Solving the problem with strictly

discrete variables is rare; Beckers [108] uses a dual method to solve the large-scale discrete

problem, while Deb and Goel [109] use a genetic algorithm. This latter option, though attractive,

requires a very large number of function evaluations even for a small number of variables.

The topology optimization problem is typically solved using the SIMP approach (solid

isotropic material with penalization): the density of each element is allowed to vary continuously

between 0 and 1. A nonlinear power-law interpolation provides an implicit penalty which

pushes the density to 0 or 1: intermediate densities are unfavorable, as their stiffness is small

compared to their volume [16]. An adjoint sensitivity analysis of the discrete system is required

to compute the sensitivity of the compliance (or other objective functions) with respect to each

element density, as the number of variables is much larger than the number of constraints [110].

A mesh-independent filter upon the gradients is also typically employed, in order to limit the

minimum size of the structure and eliminate checkerboards. Computation of the topological

Pareto trade-off curve can be done using a multi-objective genetic algorithm [109], or by

successively optimizing a weighted sum of conflicting objectives [111].

Aeronautical applications are given by Borrvall and Petersson [112], who divide a

computational domain into either fluid or solid walls to find the minimum drag profile of

submerged bodies in Stokes flows. Pingen et al. [107] solve a similar problem, using a lattice

Boltzmann method as an approximation to the Navier-Stokes equations. Drag is minimized by a

football shape (with front and back angles of 900) at low Reynolds numbers (where reducing

surface area is important), and a symmetric airfoil at higher Reynolds numbers (where









streamlining is more important).

Several examples can be found in the literature pertaining to compliance minimization of a

flexible aircraft structure. Flight loads are typically obtained from an aerodynamics model, but

the redistribution of these loads with wing deformation (aeroelasticity) is not included.

Balabanov and Haftka [113] optimize the internal structure of a transport wing, using a ground

structure approach (the domain is filled with interconnected trusses, and the cross-sectional area

of each is a design variable [16]) for compliance minimization. Eschenauer and Olhoff [114]

optimize the topology of an internal wing rib under both pull-up load maneuvers and internal

tank pressures, using a bubble method. Krog et al. [115] also optimize the topology of wing box

ribs, and discuss methods for interpretation of the results to form an engineering design, followed

by sizing and shape optimization. Luo et al. [116] compute the optimal topology of an entire

aerodynamic missile body, considering both static loads and natural frequencies.

Santer and Pellegrino [117] replace the leading edge of a wing section with a compliant

morphing mechanism, which is subjected to topology optimization. Rather than a compliance-

based objective function, the author's use airfoil efficiency, but as above, do not include

aeroelastic load redistribution. Such an aeroelastic topology optimization is an under-served area

in the literature. Maute and Allen [118] consider the topological layout of stiffeners within a

swept wing, using a three-dimensional Euler solver coupled to a linear finite element model.

Results from an adjoint sensitivity analysis of the three-field couple aeroelastic system [110]

[119] are fed into an augmented Lagrangian optimizer to minimize mass with constraints upon

the lift, drag, and wing displacement. The authors are able to demonstrate the superiority of

designs computed with aeroelastic topology optimization, rather than considering a constant

pressure distribution.









Gomes and Suleman [120] use a spectral level set method to maximize aileron reversal

speed by reinforcing the upper skin of a wing torsion box via topology optimization. Maute and

Reich [106] optimize the topology of a compliant morphing mechanism within an airfoil, by

considering both passive and active shape deformations. The authors are able to locate superior

optima with this aeroelastic topology optimization approach, as compared to a jig-shape

approach: optimizing the aerodynamic shape, and then locating the mechanism that leads to such

a shape.

At present, there is no research pertaining to aeroelastic topology optimization of

membrane wings, or micro air vehicle wings. Biological inspiration for this concept can be

found in the venation of insect wings however. For example, a pleated grid-like venation can be

seen in dragonfly wings, posteriorly curved veins in fly wings, and a fan-like distribution of

veins in the locust hindwing [121]. On the whole, the significance of this variation in wing

stiffness distribution between species is not well understood.









CHAPTER 3
EXPERIMENTAL CHARACTERIZATION

As will be extensively discussed below, numerical modeling of flexible MAV wings, while

conducive to optimization studies, is very challenging: at the present time, no model exists which

can accurately predict all of the unsteady flow phenomenon known to occur over a micro air

vehicle. As such, experimental model validation is required to instill confidence in the employed

models, highlight numerical shortcomings, and provide additional aeroelastic wing

characterization. All of the aerodynamic characterization experiments discussed in this work are

run in a closed-loop wind tunnel, a diagram of which can be seen in Figure 3-1. Only

longitudinal aerodynamics are of interest, and only a-sweep capability is built into the test setup.



Mounting
Bracket "


VC Cameras


Test Section



Incoming
Flow Sting
Speckled Balance
MAV Wing





Figure 3-1. Schematic of the wind tunnel test setup.

Closed Loop Wind Tunnel

The test facility used for this work is an Engineering Laboratory Design, Inc. (ELD) 407B

closed-loop wind tunnel, with the flow loop arranged in a horizontal configuration. The test









section has an inner dimension of 0.84 m on each side and is 2.44 m deep. The velocity range is

between 2 and 45 m/s, and the maximum Reynolds number is 2.7 million. The flow is driven via

a two-stage axial fan with an electric motor powered by three-phase 440 V at 60 Hz. The

controller is operated remotely with appropriately dedicated data acquisition software, wherein

the driving frequency is based upon a linear scaling of an analog voltage input. Suitable flow

conditions are achieved through hexagonal aluminum honeycomb cell, high-porosity stainless

steel screens, and turning vane cascades within the elbows of the closed loop. Centerline

turbulence levels are measured on the order of 0.2%. Optical glass window access is available

on the sidewalls and the ceiling.

A Heise model PM differential pressure transducer rated at 12.7 cm and 127 cm of water

(with a manufacturer-specified 0.002% sensitivity and a 0.01% repeatability) is used to

measure the pressure difference from a pitot-static tube mounted within the test section, whose

stagnation point is located at the center of the section's entrance. The Heise system is capable of

measuring wind speeds up to 45 m/s. A four-wire resistance temperature detector is mounted to

the wall of the test section for airflow temperature measurements.

Strain Gage Sting Balance

Several outstanding issues exist with measuring the aerodynamic loads from low Reynolds

number flyers. Several such airfoils are known to exhibit hysteresis loops at high angles of

attack. If the flow does not reattach to the wing surface (typically for lower Reynolds numbers

below 5-104 [17]) counterclockwise hysteresis loops in the lift data may be evident; the opposite

is true if a separation bubble exists via reattachment [24]. Adequate knowledge of such a loop is

obviously important as it effects vehicle control problems via stall and spin recovery. As

described by Marchman [122], the size of the hysteresis loop measured in a wind tunnel can be

incorrectly decreased by poor flow quality: large freestream turbulence intensity levels or









acoustic disturbances (noise emitted from the turbulent boundary layer along the tunnel walls,

the wind tunnel fan, etc. [24]). Mounting techniques are also presumed to cause an incorrect

relationship between Reynolds number and the zero-lift angle of attack among several sets of

published data [122].

Sensitivity is another concern, particularly in drag force measurements which may be as

low as 0.025 N (computed for a wing with a chord of 100 mm and a Reynolds number of 5-104).

An electrical resistance strain gage sting balance is typically used for force and moment

measurements. While strain gages typically provide the greatest sensitivity and simplicity, they

are also prone to temperature drift, electromagnetic interference, creep, and hysteresis.

An internal Aerolab 01-15 6-component strain gage sting balance is used for force/moment

measurements in the current work. Wind tunnel models are mounted to the sting balance by a

simple jaw mechanism. Each of the six channels is in a full Wheatstone-bridge configuration,

with five channels dedicated to forces, and one to a moment. Two forces are coincident with the

vertical plane of the model (traveled during an a-sweep), two are in the plane normal to the

previous (traveled during a P-sweep), one force is in the axial direction, and the moment is

dedicated to roll. Data acquisition is done with a NI SCXI 1520 8 channel programmable strain

gage module with full bridge configuration, 2.5 excitation volts, and a gain of 1000.

Other modules included in the system are a SCXI 1121 signal conditioner, 1180 feed-

through with 1302 breakout and 1124 D/A module. A NI 6052 DAQPad firewire provides A/D

conversion, multiplexing, and the PC connection. For a given flight condition, the output signals

from the six components are sampled at 1000 Hz for 2 seconds. The average of this data is sent

to one module for calculation of the relevant aerodynamic coefficients, and the standard

deviation of the data is stored for further uncertainty analysis. Signals from each channel are









recorded before and after a testing sequence, with no airflow through the tunnel, to provide an

estimation of the overall drift.

The sting balance is mounted to a custom-fabricated aluminum model arm within the wind

tunnel (seen in Figure 3-1). A U-shape is built into the arm, so that the structure curves well

behind the model and aerodynamic interactions are minimal. The arm extends through a hole in

the wall of the test section, and is then attached to a gearbox and a brushless servomotor for

pitching control. The motor is run by a single axis motion controller; a high precision US Digital

absolute encoder, connected to an SCXI 1121 module provides angle of attack feedback.

Pitching rates are on the order of 1 O/s. For a given flight condition, the aforementioned

instrumentation (the Heise and thermocouple connected through an RS232, and the sting

balance) is used to measure the pressure, temperature, and voltage signals. A set of tare voltages

(obtained prior to the test, with no flow through the tunnel) are subtracted from the sting balance

data, which is then filtered through the calibration matrix, and normalized by the subsequent

computations of flow velocity and air density. The numerous systems described above are

integrated to allow for completely automated wind tunnel testing for force/moment data, along

with a LABview GUI written for user inputs of the wing geometry, angle of attack array, and the

commanded wind speed.

Standard procedures [123] are used to calibrate the sting balance down to an adequate

sensitivity: 0.01N in drag (though still just 40% of the minimum given above). Such a resolution

is comparable to that found in the work of Pelletier and Mueller [34], but superior precision is

used by Kochersberger and Abe [124] and Moschetta and Thipyopas [125]. The calibration

matrix is determined through the use of known weights applied at control points in specified

directions. This calibration is able to predict the relationship between load and signal for a given









channel, as well as potential interactions (second-order interactions are not included) in both

single and multiple load configurations. Further information on the calibration of strain gage

sting balances for micro air vehicle measurements is given by Mueller [126] and Albertani [47].

Uncertainty Quantification

Two types of uncertainty are thought to contribute to the eventual error bounds of the sting

balance data. The resolution error is indicative of a measurement device's resolution limit: for

example, the inclinometer used to measure the pitch of a model can measure angles no finer than

0.1, an uncertainty that can be propagated through the equations to find its theoretical effect on

the aerodynamic coefficients using the Kline-McClintock technique [127]. The following

resolution errors are used: 3 Pa of dynamic pressure from the Heise, 1.2-10-7 V from the output

voltage of the strain gages (estimated from the quantization error of the 16-bit DAQ cards),

0.001 m2 from wing area measurements, and 0.002 m from chord length measurements. The

second source is the precision error, a measure of the repeatability of a measurement. This is

well quantified by the standard deviation of the voltage signals from 2000 samples at each angle

of attack, as described above. Uncertainty bounds are computed with a squared sum of the

resolution and precision errors (where the latter is magnified by Student's t at 95% confidence

and infinite degrees of freedom). The precision of the strain gage signals is found to contribute

the most error to the aerodynamic coefficients, particularly in the stalled regions. Typical

uncertainty percentages are 5% for CL, 7% for CD, 9% for L/D, and 20% for Cm. Theses values

can be expected to double during stall.

Wind Tunnel Corrections

Corrections are applied to the coefficients of lift, drag, and pitching moment based upon

wind tunnel blockage, and model flexibility effects. The solid blockage effect is due to the

presence of the model within the wind tunnel, thus decreasing the effective area of the test









section and increasing the flow velocity (and the coefficients) in the vicinity of the model [128].

Wake blockage occurs when the flow outside of the model's wake must increase, in order to

satisfy the flow continuity in a closed tunnel. In an open freestream, the velocity outside of the

wake would be equal to the freestream velocity. The effect of wake blockage is proportional to

the wake size, and therefore proportional to drag [3]. Streamline curvature blockages are the

effect of the tunnel walls on the streamlines around the model. The streamlines are compressed,

increasing the effective camber and lift [129]. Such corrections generally decrease both lift and

drag, while the pitching moment is made less negative, with percentage changes on the order of

2-3%.

Finally, flexibility effects within the wind tunnel setup must be accounted for. These

effects are primarily caused by the elasticity of the internal strain gage sting balance; under load

the wind tunnel model will pitch up via a rigid body rotation. Visual image correlation

(described below) is used to measure the displacement at points along the wing known to be

nominally rigid (specifically, the sting balance attachment points along the wing root). This data

then facilitates the necessary transformations and translations of the wing surface, and is used to

correct for the angle of attack. Aa is a positive monotonically increasing function of both lift and

dynamic pressure, and can be as large as 0.70 at high angles of attack [47].

Visual Image Correlation

Wind tunnel model deformation measurements are a crucial experimental tool towards

understanding the role of structural composition upon aerodynamic performance of a MAV

wing. The flexible membrane skin generally limits applications to non-contacting optical

methods, several of which have been reported in the literature. Galvao et al. [89] use stereo

photogrammetry for displacement measurements of a membrane wing, with a reported resolution

between 35 and 40 [tm. Data is available at discrete markers placed along the wing. Projection









moire interferometry requires no such marker placement (a fringe pattern is projected onto the

wing surface), and the resulting data set is full-field. However, displacement resolutions

reported by Fleming et al. [130] are relatively poor (250 itm), the dual-camera system must be

rotated during the a-sweep, and only out-of-plane data is available, making strain calculations (if

needed) impossible. Burner et al. [131] discuss the use of photogrammetry, projection moire

interferometry, and the commercially available OptotrakTM system. The authors find no single

technique suitable for all situations, and that a cost-benefit tradeoff study may be required.

Furthermore, the methods need not be mutually exclusive, as situations may arise wherein they

can be used in combination. For the current work, a visual image correlation system (VIC),

originally developed by researchers at the University of South Carolina [132], is used to measure

wing geometry, displacements, and plane strains.

The underlying principle of VIC is to calculate the displacement field by tracking the

deformation of a subset of a random speckle pattern applied to the specimen surface. The

random pattern is digitally acquired by two cameras before and after loading. The acquisition of

images is based on a stereo-triangulation technique, as well as the computing of the intersection

of two optical rays: the stereo-correlation matches the two 2-D frames taken simultaneously by

the two cameras to reconstruct the 3-D geometry. The calibration of the two cameras (to account

for lenses distortion and determine pixel spacing in the model coordinates) is the initial

fundamental step, which permits the determination of the corresponding image locations from

views in the different cameras. Calibration is done by taking images (with both cameras) of a

known fixed grid of black and white dots.

Temporal matching is then used: the VIC system tries to find the region (in the image of

the deformed specimen) that maximizes a normalized cross-correlation function corresponding to









a small subset of the reference image (taken when no load is applied to the structure) [132]. The

image space is iteratively swept by the parameters of the cross-correlation function, to transform

the coordinates from the original reference frame to coordinates within the deformed image. An

originally square subset in the un-deformed image can then be mapped to a subset in the

deformed image. As it is unlikely that the deformed coordinates will directly fall onto the

sampling grid of the image, accurate grey-value interpolation schemes [133] are implemented to

achieve optimal sub-pixel accuracy without bias. This procedure is repeated for a large number

of subsets to obtain full-field data.

In order to capture the three-dimensional features and deformation of a wind tunnel model,

twin synchronized cameras, each looking from a different viewing angle, are installed above the

wind tunnel ceiling, as can be seen in Figure 3-1. As the cameras must remain stationary

through the experiment (to preserve the information garnered from the calibration procedure), a

mounting bracket is constructed to straddle the tunnel, and prevent the transmission of vibration.

Optical access into the test section is through an optical glass ceiling. The results of conducting

visual image correlation tests with a glass interface between the cameras and the specimen have

been studied, with little benign effects reported [134]. Furthermore, the cameras are initially

calibrated through the window to ensure minimal distortion. Two 250 W lamps illuminate the

model, enabling the use of exposure times of 5 to 10 ms.

The twin cameras are connected with a PC via an IEEE 1394 firewire cable, and a

specialized unit is used to synchronize the camera triggers for instantaneous shots. A standard

acquisition board installed in the computer carries out the digitalization of the images, and the

image processing is carried out by custom software, provided by Correlated Solutions, Inc.

Typical data results that can be obtained from the VIC system consist of the geometry of the









surface in discrete x, y, and z coordinates (where the origin is located at the centroid of the

speckled area of interest, and the outward normal points towards the cameras, by default), and

the corresponding displacements along the wing (u, v, and w). The VIC system places a grid

point every N pixels, where N is user defined. A final post-processing option involves

calculating the in-plane strains (Exx, Ey, and exy). This is done my mapping the displacement

field onto an unstructured triangular mesh, and conducting the appropriate numerical

differentiation (the complete definition of finite strains is used).

Data Procession

The objective of most of the wind tunnel tests given in the remainder of this work is to

determine the deformation of the wings under steady aerodynamic loads, at different angles of

attack and free stream velocities, while simultaneously acquiring aerodynamic force data. Each

angle of attack requires a separate wind-off reference image: failure to do so will inject rigid

body motions (as the body moves sequentially from one angle of attack to the next) into the

displacement fields. If each reference image taken for VIC is of the fully assembled wing, the

amount of pre-strain in the wing is not included in the measured strain field, but only those

caused by the aerodynamic loads. This condition needs to be carefully considered in the

evaluation of the results, since the areas of relaxation of the pre-existing tension will generate

areas of "virtual" compression within the skin. The thin membrane cannot support a genuine

compressive stress (it will wrinkle), but negative Poisson strains are possible.

An alternative procedure uses the un-stretched sheet of latex rubber (prior to adhesion on

the wing) as a reference image. This provides the state of pre-strain in the membrane, as well as

the absolute strain field during wind tunnel testing, but makes the displacement fields very

difficult to interpret and is not used here. The pre-strain data is merely recorded (with a separate

set of reference and deformed images), but not used as a reference for further aerodynamic









testing. As mentioned above, the acquired displacement field will be composed of both elastic

wing deformation and rigid body motion/translations originating from the sting balance, the

latter of which must be filtered out. The computed strains are unaffected by these motions.

Uncertainty Quantification

In order to estimate the resolution error of the VIC system, a simple ad hoc experiment is

conducted. A known displacement field is applied to a structure, and then compared with the

field experimentally determined by way of image correlation. A thin latex membrane is

stretched and fixed to a rigid aluminum ring with a diameter of 100 mm. The center of the

membrane circle is then indented with a rigid steel bar with a spherical head of 8 mm diameter.

The bar is moved against the membrane by a micrometer with minimum increments of 0.25 mm.

Results, in terms of the error between commanded displacement (via the micrometer) and the

measured displacement at the apex of the membrane profile, directly beneath the axis of the

indentation bar, are given in Figure 3-2.

0.08


0.04




-0.04 -- .5 speckles/mm
-E- 3 speckles/mm
S --- 10 speckles/mm
-0.08
0 1 2 3 4 5
commanded displacement (mm)

Figure 3-2. Quantification of the resolution error in the VIC system.

Three different VIC setups are shown: 0.5, 3, and 10 speckles per millimeter of membrane,

the latter of which corresponds to 2 pixels per speckle (with half the membrane in view). As









expected, the error is smallest for the finest speckle pattern, whose readings randomly oscillate

about zero, with a peak error of 0.018 mm (0.6%). The coarser speckling patterns randomly

oscillate at an offset error of 0.04 mm, with a peak error of 0.077 mm (2.2%). This places the

resolution error for the VIC system between 10 and 20 [tm, about twice the resolution reported

for the photogrammetry system [89]. Though not explicitly discussed here, the strain resolution

is estimated to be between 500 and 1000 tE (a non-dimensional parameter independent of

speckle size), a high value (compared to strain gages, for example) owing to the differentiation

methods used.

Model Fabrication and Preparation

Only the wing (152 mm wingspan, 124 mm root chord, 1.25 aspect ratio) of the MAVs

seen in Figure 1-1 and Figure 1-2 is considered in this work. The camber at the root is 6.8% (at

x/c = 0.22), the reflex at the root is -1.4% (at x/c = 0.86), and 70 of positive geometric twist (nose

up) is built into the wingtip. The MAV wing has 7 of dihedral between 2y/b = 0.4 and the

wingtip. The fuselage, stabilizers, and propeller are omitted from both computations and

experiments. The leading edge, inboard portion of the wing, and perimeter (of the PR wings) are

constructed from a bi-directional plain weave carbon fiber laminate with 3000 fibers/tow, pre-

impregnated with thermoset epoxy. The battens (for the BR wings) are built from uni-directional

strips of carbon fiber. These materials are placed upon a tooling board (appropriately milled via

CNC) and cured in a convection oven at 2600 F for four hours. A wind tunnel attachment (to be

fastened to the aforementioned jaw mechanism) is bonded along the root of the wing between x/c

=0.25 and 0.8.

The latex rubber skin adhered to this wing surface is 0.12 mm thick, and approximately

isotropic. A random speckle pattern is applied to the latex sheet with flat black spray paint, and

then coated with a layer of dulling spray. Each paint speckle, while relatively brittle, has a small









average diameter (less than 0.5 mm) and is generally not connected to another speckle pattern;

the pattern should not provide significant reinforcement to the latex. If information concerning

the state of pre-strain in the skin is desired, a picture of the un-stretched latex sheet is taken for

future use as a reference in the VIC system. The latex is then appropriately stretched about a

frame (or not at all if a slack membrane is desired), and adhered to the upper carbon fiber wing

surface (which must be painted white) with spray glue. After the glue has dried, the excess latex

is trimmed away. A picture of the resultant wind tunnel model is given in Figure 3-3.











Figure 3-3. Speckled batten-reinforced membrane wing with wind tunnel attachment.









CHAPTER 4
COMPUTATIONAL FRAMEWORK AND VALIDATION

Several difficulties are associated with modeling the passive shape adaptation of a flexible

micro air vehicle wing. From a fluid dynamics standpoint, the low aspect ratio wing (1.25)

forces a highly three-dimensional flow field, and the low Reynolds number (105) implies strong

viscous effects such as flow separation, transition, and potential reattachment. Structurally, the

mechanics of the rubber membrane inflation are inherently nonlinear, and the orthotropy of the

thin laminated shells used for the wing skeleton is dependent on the plain weave fiber

orientation. Further difficulties arise with the inclusion of pre-tension within the membrane.

Only static aeroelasticity is considered here. Several computational membrane wing

studies have included unsteady effects [28] [81], and are thus able to study phenomena such as

vortex shedding [18], membrane vibration (unstable [62] or otherwise), unsteady interactions

between the separated flow and the tip vortices [6], and wind gusts [8]. Past wind tunnel work,

however, has indicated that MAV membrane inflation is essentially qausi-static for a large range

of angles of attack up to stall [135], and that adequate predictive capability still exists for those

flight conditions with obvious unsteady features [43].

Structural Solvers

The unstructured mesh used for finite element analysis can be seen in Figure 4-1. 2146

nodes are used to describe the surface of the semi-wing, connected by 4158 three-node triangle

elements. The same mesh is used for both batten and perimeter-reinforced computations, by

using different element-identification techniques, as seen in the figure. Greater local effects are

expected in the membrane areas of the wing, and the mesh density is altered accordingly. Nodes

that lie along the wing root between x/c = 0.25 and 0.8 are given zero displacement/rotation

boundary conditions, to emulate the restrictive effect of the wind tunnel attachment (Figure 3-3).









All nodes that lie on the wing root are constrained appropriately as necessitated by wing

symmetry.
















Figure 4-1. Unstructured triangular mesh used for finite element analysis, with different element
types used for PR and BR wings.

Composite Laminated Shells

Discrete Kirchhoff triangle plate elements [136] are use to model the bending/twisting

behavior of the carbon fiber areas of the wings: leading edge, root, perimeter, and battens. Due

to the comparative stiffness of these materials, linear behavior is assumed. The orthotropy of the

plates is introduced by the flexural stiffness matrix of the laminates, Dp, relating three moments

(two bending, one twisting) to three curvatures:

NL
D = Qk-(h/12 + hk Z) (4-1)
k=1

where NL is the number of layers in the laminate, hk is the thickness of the kth ply, Zk is the

normal distance from the mid-surface of the laminate to the mid-surface of the kth ply, and Qk is

the reduced constituitive matrix of each ply, expressed in global coordinates. Qk depends upon

the elastic moduli in the 1 and 2 directions (equal for the bi-directional laminate, but not so for

the uni-directional) El and E2, the Poisson's ratio v12, and the shear modulus G12. The finite

element stiffness matrix pertaining to bending/twisting is then found to be:










(4-2)


Ao

where T is a matrix which transforms each element from a local coordinate system to a global

system, Bp is the appropriate strain-displacement matrix [137], and Ao is the un-deformed area of

the triangular element. Kp is a 9x9 matrix whose components reflect the out-of-plane

displacement w and two in-plane rotations at the three nodes.

Similarly, in-plane stretching of the laminates (a secondary concern, but necessarily

included), is given by:

NL
Ap =ZQkhk (4-3)
k=1

where Ap is a laminate matrix relating three in-plane stress resultants to three strains.

Expressions similar to Eq. (4-2) are then formulated to compute Km, the 6x6 finite element

stiffness matrix governing in-plane displacements u and v at the three nodes. Km and Kp are then

combined to form the complete 15x15 shell stiffness matrix of each element, KI. Drilling

degrees of freedom are not included. Though some wing designs may use un-symmetric

laminates, coupling between in-plane and out-of-plane motions is not included.

Loads Model Validation/Estimation. The following method is used to both validate the

model presented above, and identify the relevant material properties of the laminates. A series of

weights are hung from a batten-reinforced wing (with 2 layers of bi-directional carbon fiber

oriented 450 to the chord line and 1 layer uni-directional battens, but no membrane skin) at nine

locations: the two wing tips, the trailing edges of the six battens, and the leading edge, as shown

in Figure 4-2. VIC is used to measure the resulting wing displacements. A linear curve is fit

through the load-displacement data of all nine points due to all nine loadings. The slopes of

these curves are used to populate the influence matrix in Table 4-1: the diagonal gives the motion


KP = T f B T DP.Bpn dA.T T









of a wing location due to a force at that location; the off-diagonals represent indirect

relationships.



S 4 2 8 6 4 2


79 5.7
7







Figure 4-2. Computed deformations of a BR wing skeleton due to a point load at the wing tip
(left) and the leading edge (right).

A genetic algorithm is then used for system identification. The six variables are the

material parameters: Ei, v12, and G12 of both the plain weave and the battens. E2 is assumed to be

equal to El for the plain weave, and equal to 10 MPa for the uni-directional battens. This latter

value has little bearing on the results, as the 1 direction corresponds with the axis of the batten.

The objective function is the sum of the squared error between the diagonals of the computed

and the measured influence matrix. The error terms are appropriately normalized before

summation and off-diagonal components are not considered in the optimization. For the genetic

algorithm, the population size is 20, the elitism count is 2, reproduction is via a two-point

crossover function with a 0.8 crossover fraction, and a uniform mutation function is used with a

0.01 mutation rate. Convergence is adequately achieved after 30 iterations, with each function

evaluation call requiring a single finite element analysis.

The resulting numerical influence matrix is given in Table 4-2. This matrix is symmetric,

whereas the experimental matrix is slightly un-symmetric, probably due to manufacturing errors.









For the plain weave, El = 34.8 GPa, v12 = 0.41, and G12 = 2.34 GPa. For the uni-directional

battens, El = 317.2 GPa, v12 = 0.31, and G12 = 1.05 GPa. The model correctly predicts the very

stiff leading edge (point 1), and the negative influence it has on the remainder of the wing (as

shown on the right of Figure 4-2). The rest of the points along the wing positively influence one

another. Errors between the two matrices are typically on the order of 5-10%; the numerical

wing is generally stiffer than the actual wing. As expected, the weakest battens are the longest,

found towards the root (points 8 and 9). The wingtips (points 2 and 3) generally have the

greatest indirect influence on the rest of the wing (as shown on the left of Figure 4-2).

Force-displacement trends at the nine locations along the wing, due to loads at those points

(the diagonal terms in the matrices) are given in Figure 4-3, showing a suitable match between

model and experiment. With the exception of the leading edge, two data points are given for

each load level, corresponding to the data from the left and right sides of the wing. Higher

fidelity methods for system identification of a carbon fiber MAV skeleton are given by Reaves et

al. [138], who utilize model update techniques with uncertainty quantification methods. This is

largely done due to the uncertainty in the laminate lay-up, predominately in ply overlapping

regions within the skeleton, which is not an issue for the current work.

Table 4-1. Experimental influence matrix (mm/N) at points labeled in Figure 4-2.
1 2 3 4 5 6 7 8 9
1 1.58 -2.90 -2.97 -3.00 -3.07 -3.47 -3.54 -3.05 -3.03
2 -2.93 104.65 3.94 50.85 5.88 36.47 7.65 26.06 9.99
3 -3.07 3.67 118.46 8.18 52.24 9.39 38.29 13.33 29.22
4 -3.68 49.05 5.69 329.11 8.71 44.14 10.51 33.05 13.23
5 -3.78 5.68 50.15 9.17 366.92 11.89 45.19 14.98 33.68
6 -4.33 36.41 7.92 44.63 10.68 547.50 13.41 38.46 15.23
7 -4.37 7.77 36.74 11.46 44.35 13.28 513.83 17.40 38.19
8 -4.75 24.30 9.02 30.15 11.24 34.68 12.63 757.00 15.81
9 -4.76 9.22 25.09 12.34 30.30 14.32 35.51 17.65 742.25









Table 4-2. Numerical influence matrix (mm/N) at points labeled in Figure 4-2.
1 2 3 4 5 6 7 8 9
1 1.49 -2.92 -2.92 -3.61 -3.61 -4.24 -4.24 -4.60 -4.60
2 -2.92 107.96 3.36 54.49 5.05 33.10 6.80 20.84 8.13
3 -2.92 3.36 107.96 5.05 54.49 6.80 33.10 8.13 20.84
4 -3.61 54.49 5.05 312.00 6.78 39.30 8.51 25.09 9.73
5 -3.61 5.05 54.49 6.78 312.00 8.51 39.30 9.73 25.09
6 -4.24 33.10 6.80 39.30 8.51 584.26 10.14 28.81 11.18
7 -4.24 6.80 33.10 8.51 39.30 10.14 584.26 11.18 28.81
8 -4.60 20.84 8.13 25.09 9.73 28.81 11.18 776.48 11.95
9 -4.60 8.13 20.84 9.73 25.09 11.18 28.81 11.95 776.48


0 5 10 15 20 25
w (mm)

Figure 4-3. Compliance at various locations along the wing, due to a point load at those
locations.

Membrane Modeling

In the modeling of thin, elastic membranes (with no resistance to a bending couple), three

basic options are available. If there exists a significant pre-strain field throughout the sheet,

linear modeling is possible by assuming inextensibility: the pre-strain overwhelms the strains

that develop as a result external loading. As these strains grow in magnitude (or if the membrane

is originally slack) a nonlinear model must be used, as the membrane's resistance depends upon

the loading (geometric nonlinearity). However, a linear constituitive relationship is still typically









valid up to a point, after which the membrane becomes hyperelastic (material nonlinearity), and

the stress-strain relationship changes with increasing load.

Linear Modeling. Geometric stress stiffening provides a relationship between in-plane

forces and transverse deflection [137], and is indicative of a structure's reluctance to change its

state of stress. For an initially flat membrane with a transverse pressure, the constitutive

equation is:

N, w,, +2 Ny w,xy +Ny w,y +p = 0 (4-4)

where w is the out-of-plane displacement (as above), Nx, Nxy, and Ny are the in-plane pre-stress

resultants, and p is the applied pressure field. For an isotropic stress field with no shear, this

equation reduces to the well known Poisson's equation [139]. This model assumes that the

displacement along the membrane is purely out-of-plane; thus the membrane is inextensible in

response to a pressure field (although extensibility is needed to apply the initial pre-stress field).

The resulting finite element model is fairly inexpensive, as each node has only one degree

of freedom, and standard direct linear solvers can be used. This model is thought to be accurate

for small pressures, small displacements, and large pre-stresses. Though it is not expected that

the MAV wing displacements will be particularly large (typically less than 10% of the root

chord), it is expected that a slack membrane skin may provide many aerodynamic advantages.

As the solution to Eq. (4-4) becomes unbounded as the pre-stress approaches zero, higher fidelity

models will also be pursued for the current work. MAV wing simulations with linear membrane

models can be found in the work of Stanford and Ifju [14], Thwaites [65], and Sugimoto [80].

Nonlinear Modeling. The nonlinear membrane modeling discussed in this section will

incorporate geometric nonlinearities, but Hooke's law is assumed to still be valid. For the

inflation of a circular membrane, Pujara and Lardner [140] show that linear and hyperelastic









constituitive relationships provide the same numerical solutions up to deformations on the order

of 30% of the radius, a figure well above the deflections expected on a membrane wing.

Geometric nonlinearity implies that the deformation is large enough to warrant finite strains, and

that the direction of the non-conservative pressure loads significantly changes with deformation.

Eq. (4-4) is still valid, only now the stress resultants depend upon the state of pre-stress, as well

as in-plane stretching, which in turn depends on the out-of-plane displacement. Furthermore, the

rotation of the membrane is no longer well-approximated by the derivative of w, rendering the

equilibrium equation nonlinear. Three displacement degrees of freedom are required per node

(u, v, w), rather than the single w used above. Finite element implementation of such a model is

described by Small and Nix [141] and Pauletti et al. [142].

The strain pseudo-vector within each element is given as:

Ss=s + L =Bo Xe + BL.Xe (4-5)

where Eo and EL represent the division of the linear (infinitesimal) and nonlinear contributions to

the Green-Lagrange strain, Xe is a vector of the degrees of freedom in the elements (three

displacements at the three nodes), and Bo and BL are the appropriate strain-displacement matrices

(the latter of which depends upon the nodal displacements) [143]. The pre-stress (if any) can be

included into the model in one of two methods. First, they can be simply added to the stresses

computed my multiplying the strain vector of Eq. (4-5) through the constituitive matrix. This

may cause problems if the imposed pre-stress distribution does not exactly satisfy equilibrium

conditions, or if there is excessive curvature in the membrane skin: the membrane will deform,

even in the absence of an external force.

A second option is to use the pre-stresses in a finite element implementation of Eq. (4-4),

then add the resulting stiffness matrix and force vectors to the nonlinear terms. For a flat









membrane with uniform pre-stress, the two methods are identical. The internal force in each

element Pe can be computed from the principle of virtual work:

Pe=T.J [BfB +(BL .Xe)/ e .Am .6.dV+T.K, X, (4-6)
Vo

where Am is the linear constitutive elastic matrix of the membrane, Am'- is the stress pseudo-

vector within each element, and Kw is the stiffness matrix representation of Eq. (4-4), containing

only terms related to the out-of-plane displacement w. The tangential stiffness matrix Ke is then

the sum of the geometric KI, constituitive KI, external Kxt and pre-stress stiffness matrix Kw:

K, =T- GT M-G-dV.T (4-7)
Vo

K, =T. [Bo +(BL .X,)/Xe ]T. Am .[Bo + (BL .Xe)/X,].dV. TT (4-8)
Vo

Kext = -8Fe/OXe (4-9)

where G is a matrix linking the nodal degrees of freedom to a displacement gradient vector

[144], M is a stress matrix whose elements can be found in [137], and Fe is the external force

vector. Computation of the skew-symmetric external stiffness matrix is given in [142]. Fe must

be written in the unknown deformed configuration:

F =T.(p.A/3).[I I I]T n (4-10)

where A is the deformed area of the triangle, p is the uniform pressure over the element, I is the

identity matrix, and n is the unit normal vector to the deformed triangular finite element. The

resulting non-linear set of equations is solved with Newton's recurrence formula [142].

The above method essentially separates the linear and nonlinear stiffness contributions. If

the pre-stress in the membrane is very large, Kw will overwhelm its nonlinear counterparts, and

membrane response will be essentially linear for small pressures and displacements. Continued









inflation will transition from linear to nonlinear response [145]. In the event of a slack

membrane, the membrane's initial response to a pressure will have an infinite slope until strains

develop and provide stiffness. Numerous membrane wing models use some variant of the

geometrically nonlinear model described above: Stanford et al. [43], Ormiston [15], Smith and

Shyy [72], Jackson and Christie [76], and Levin and Shyy [104].

Inflatable Diaphragm Validation. In order to validate the above membrane models, the

material properties of the latex are first identified with a uni-axial tension test. The test specimen

has a width of 20 mm, a length of 120 mm, and a thickness of 0.12 mm. The latex rubber sheets

are formed in a rolling process, implying an orthotropy, though specimens cut from different

orientations yield very similar results. VIC is used to monitor both the extensional and the

Poisson strains: data is sampled at 50 pixel locations within the membrane strip, and then

averaged. The resulting data can be seen in Figure 4-4, and is used to identity the linear elastic

modulus and the Poisson's ratio. A linear fit through the stress-strain curve results in a modulus

of 2 MPa; the nonlinear stress-softening behavior for higher strains is the hallmark of

hyperelasticity [146]. The Poisson's ratio for small strains is 0.5, a result of the material's

incompressibility.

Using these material parameters to construct the constitutive matrix Am, the finite element

model can be appropriately validated with the Hencky test [144]: a flat circular membrane (with

or without pre-tension), clamped along its boundary, and subjected to a uniform pressure [145].

A 57.15 mm radius is chosen in order to emulate the length scale of a micro air vehicle.

Although the problem is axisymmetric, a full circular mesh is used for numerical computations.

Experimentally, VIC is used to monitor the shape of the membrane, while a Heise pressure

transducer measures the pressure within a chamber, to the top of which the membrane sheet is










fixed. Results are given in Figure 4-5, in terms of the displacement of the membrane center

(normalized by the radius) versus pressure.

2 0.6 L


1.5




0.5
s


0 0.5 1 1.5 0 0.5 1 1.5
strain strain

Figure 4-4. Uni-axial stretch test of a latex rubber membrane.

1.5
0

o Data o
o o
SLinear o oo
I Nonlinear o
I0
0 00
00

taut membrane -C
0.5 =0.044 00



"'lak membrane

0 0.1 0.2 0.3 0.4 0.5
w/R (~=0)

Figure 4-5. Circular membrane response to a uniform pressure.

Two cases are considered: a slack membrane, and a taut membrane. Computational results

from both the linear and the nonlinear models formulated above are given. As expected, the

response of the slack membrane to an applied pressure is at first unbounded, but becomes finite

with the advent of the extensional strains. The linear model is useless for a slack membrane

(unbounded), but the nonlinear model can predict this behavior. The correlation between model









and experiment is adequate up to w/R = 0.22 (slightly lower than the value given by Pujara and

Lardner [140]), when the model begins to under-predict the inflated shape. Hyperelastic effects

appear after this point: Hooke's law over-predicts the stress for a given strain level (Figure 4-4),

and thus the membrane's resistance to a transverse pressure.

For the case with membrane pre-tension, VIC is used to measure the pre-strain in the

membrane skin (applied radially [147]), the average of which is then used for finite element

computations. The mean pre-strain is 0.044, with a coefficient of variation of 3.1%. For this

case, the linear model now has a small range of validity, up to w/R = 0.15. Prior to this

deformation level, linear and nonlinear models predict the same membrane inflation. The

response then becomes nonlinear, due to the advent of finite strains, but also because a relevant

portion of the uniform non-conservative pressure is now directly radially, rather than vertically.

The nonlinear model and experiment now diverge at w/R = 0.3: the addition of a pre-tension

field increases the range of validity of both the linear and the nonlinear membrane models.

Skin Pre-tension Considerations

A state of uniform membrane pre-tension, though numerically convenient [15] [80] [14], is

essentially impossible to actually fabricate on a MAV wing. One reason is that the latex sheets

used on the MAVs in this work are not much wider than the wingspan, subjecting the state of

pre-stress to end-effects. This may perhaps be remedied with larger sheets and a biaxial tension

machine, which hardly seems worth the effort for MAV construction. Another problem is the

fact that the wing is not a flat surface. Even if a state of uniform pre-tension were attainable, it

cannot be transferred to the wing without significant field distortions, particularly due to the

camber in the leading edge. A typical pre-strain field is given in Figure 4-6, as measured by the

VIC system off of a BR wing in the chordwise direction. The contour on the left is the pre-strain

field after the spray adhesive has dried, but before the latex surrounding the wing has been de-









pinned from the stretching frame (as discussed above). The contour on the right of Figure 4-6 is

the pre-strain after the excess latex has been trimmed away.


0.1





0

Figure 4-6. Measured chordwise pre-strains in a BR wing before the tension is released from the
latex (left), and after (right).

The pre-strains measured from the carbon fiber areas of the wing (leading edge, root,

battens) are meaningless, as the shell mechanics largely govern the response in these areas. The

large extensional strains (-12%) at the leading edge are indicative of the fact that the wing

skeleton is flattened against the membrane until the spray glue dries. At this point, the wing is

allowed to re-camber, causing the latex adhered to its top surface to stretch. The anisotropic

nature of the pre-tension field is very evident, with strains ranging from between 4% to 9% on

the left semi-wing and slightly higher on the right. Furthermore, when the surrounding latex is

de-pinned from its frame the membrane at the trailing edge contracts, leaving an area of almost

no tension (right side of Figure 4-6). This is a result of the BR wing's free trailing edge, and

would not be a problem with a perimeter-reinforced wing.

One numerical solution to such a problem is to interpolate the data of Figure 4-6 onto the

finite element grid, and compute the pre-stress within each element, as discussed by Stanford et

al. [43]. This method, though accurate, would require an experimental VIC analysis in

conjunction with every numerical analysis; not a cost-effective method for thorough exploration

of the design space. Eq. (4-4) however, is a natural smoothing operator [139]; simply averaging

the pre-strains for the computations, though crude, can in some cases be relatively accurate. The









match between measured and predicted membrane deformation for the taut case in Figure 4-5 is

very good, despite the fact that the numerical pre-strains were presumed uniform.

The error resulting from a uniform pre-stress assumption can be estimated with the

following method. The pre-strain distribution throughout a flat circular membrane is considered

a normally-distributed random variable: each finite element has a different pre-strain. The linear

membrane model of Eq. (4-4) is then used to compute the displacement at the center of the

membrane due to a hydrostatic pressure. The same membrane is then given a constant pre-strain

distribution (the average value of the randomly-distributed pre-strain), and the central deflection

is recomputed for comparison purposes. Monte Carlo simulations are then used to estimate the

average error at the membrane center, for a given coefficient of variation of the pre-strain.

The results of the Monte Carlo simulation are given in Figure 4-7. Each data point is the

percentage error between the central displacement computed with a non-homogenous random

pre-strain, and that with a constant pre-strain. Each error percentage is the average of 500 finite

element simulations. The radius of the circle is 57.15 mm, the thickness is 0.12 mm, the elastic

modulus is 2 MPa, the Poisson's ratio is 0.5, and the hydrostatic pressure is fixed at 200 Pa. The

mean pre-strain is 0.05, and the standard deviation is decided by the COV of each data point's

abscissa. Nonlinear membrane modeling is not used. The smoothing nature of the Laplacian

operator in Eq. (4-4) is very evident: even in the presence of 30% spatial pre-strain variability,

the error in assuming a constant pre-strain is still less than 5%. On one hand, the error in Figure

4-7 is probably under-predicted, as strain cannot truly be a spatially-random variable: on a local

scale measured strain may seem random, but on a global scale it must satisfy the compatibility

equations [146]. Both of these scale-trends are evident in Figure 4-6. On the other hand, Figure

4-7 represents the worst case scenario, as nonlinear membrane effects will dilute the importance









of the pre-tension [145], whatever it's distribution throughout the membrane skin.

50 ..........

40

30 -

I 20

10
0 -

0 0.2 0.4 0.6
spatial COV of pre-strain

Figure 4-7. Monte Carlo simulations: error in the computed membrane deflection due to a
spatially-constant pre-strain distribution assumption.

Though the above results indicate the appropriateness of using a constant membrane pre-

stress for MAV wing computations (despite an inability to reproduce this in the laboratory), the

tension relaxation at the free trailing edge of the BR wing (seen in Figure 4-6) should be

corrected for. Regardless of the amount of pre-tension placed in a batten-reinforced membrane,

the pre-stress traction normal to the free trailing edge will always be zero, producing a stress

gradient. This can be accounted for in the following manner:

1. Specify the pre-stress field within the membrane skin (uniform or otherwise).

2. Compute the traction due to this pre-stress along the outward normal, at each edge in a
membrane finite element that coincides with a free surface.

3. Apply a transverse pressure along each edge, equal and opposite to the computed traction.

4. Compute the resulting stress field (while holding the carbon fiber regions of the wing rigid),
add this field to the prescribed stress in step 1, and use the result as the new pre-stress
resultant field for aeroelastic computations.

The resulting pre-stress field will be very small along the free edge, and approach the

original specified value deeper into the wing towards the leading edge, as can be seen in Figure

4-8. For this example, about a fourth of the membrane area is affected by the free edge, while

the remainder retains a pre-stress close to the prescribed value (a result validated by Figure 4-6).









As mentioned above, this pre-stress correction only needs to be applied for simulations of a

batten-reinforced wing.

N N N
x 10 Y 10 xy 6

8 8



4 4-2

2 I2 -
-__ 4

0 0 -6

Figure 4-8. Computed pre-stress resultants (N/m) in the chordwise (left), spanwise (center), and
shear (right) in a BR wing, corrected at the trailing edge for a uniform pre-stress
resultant of 10 N/m.

Fluid Solvers

As discussed above, several viscous effects dominate the flow about a micro air vehicle

wing: laminar separation, turbulent transition and reattachment, periodic shedding and pairing,

and three-dimensional flow via wing tip vortex swirling. An inviscid flow solved such as the

vortex lattice method is unable to predict any of these effects (drag, in particular, will be severely

underestimated), but its small computational expense is attractive. Solving the steady Navier-

Stokes equations represents a substantial increase in cost, but an equally large step forward in

predictive capability. Some aspects of the flow (namely, turbulent transition over a separation

bubble, and subsequent shedding), still cannot be predicted with the methods presented here.

Vortex Lattice Methods

This section briefly describes a well-developed family of methods for predicting the steady

lifting flow and induced drag over a thin wing at small angles of attack. The continuous

distribution of bound vorticity over the wing is approximated by discretizing the wing into a









paneled grid, and placing a horseshoe vortex upon each panel. Each horseshoe vortex is

comprised of a bound vortex (which coincides with the quarter-chord line of each panel), and

two trailing vortices extending downstream. Each vortex filament creates a velocity whose

magnitude is assumed to be governed by the Biot-Savart law [27]. Furthermore, a control point

is placed at the three-quarter-chord point of each panel.

The velocity induced at the mth control point by the nth horseshoe vortex is:

m,n Cm,n n Vm,n Cm,n Wm,n Cm,n .F (4-11)

where u, v, and w are the flow velocities in Cartesian coordinates, F is the vortex filament

strength, and C' are influence coefficients that depend on the geometry of each horseshoe vortex

and control point combination. The complete induced velocity at each control point is the sum

of the contributions from each horseshoe vortex, resulting in a linear system of equations.

The strength of each vortex must be found so that the resulting flow is tangent to the

surface of the wing: the wing becomes a streamline of the flow. This requirement is enforced at

each control point by:

{U, cos(a)+um vm U, .sin(a)+wm)}VF(xm,ym,zm)= 0 (4-12)

where Uo, is the free-stream velocity, a is the angle of attack, and F(x,y,z) = 0 is the equation of

the surface of the wing. Inserting the relevant terms of Eq. (4-11) into Eq. (4-12) provides a

linear system of equations for the filament strength of each horseshoe vortex. Micro air vehicle

simulations that utilize a vortex lattice method are typically forced to do so by the computational

requirements of optimization (as is the case in the current work). Examples can be seen in the

work ofNg and Leng [52], Sloan et al. [53], and Stanford et al. [61].

Steady Navier-Stokes Solver

The three-dimensional incompressible Navier-Stokes equations, written in curvilinear









coordinates, are solved for the steady, laminar flow over a MAV wing. As before, the fuselage,

stabilizers, and propeller are not taken into account. The computational domain can be seen in

Figure 4-9, with the MAV wing enclosed within. Inlet and outlet boundaries are marked by the

flow vectors; velocity is specified at the inlet, and a zero-pressure boundary condition is enforced

at the outlet. The configuration shown in Figure 4-9 is for simulations at a model inclination of

0 angle of attack. For non-zero angles, the lower and upper surfaces will also see a mass flux,

rather than re-meshing the wing itself. The sidewalls are modeled as slip walls, and thus no

boundary layer forms. The MAV wing itself is modeled as a no-slip surface.







12c



x 8c

Figure 4-9. CFD computational domain.

Because no flow is expected to cross the root-chord of the wing (unsteady effects that may

lead to bilateral asymmetry [6] are not included; nor is propeller slipstream [45]), symmetry is

exploited by modeling only half of the computational domain (the plane of symmetry is also

modeled as a slip wall). A detailed view of the resulting structured mesh (the nodes that lie on

the plane of symmetry and the MAV wing) is given in Figure 4-10. 210,000 nodes fill half of

the computational domain, with 1300 nodes on the wing surface. This is a multi-block grid, with

four patches coinciding with the upper and lower wing surfaces. The wing itself has no

thickness. Such a flow model should be able to adequately predict the strong tip vortex swirling









system (and the accompanying nonlinear lift and moment curves [3]), as well as the laminar flow

separation against an adverse pressure gradient [2]. Similar laminar, steady flow computations

for low Reynolds number flyers can be found in the work of Smith and Shyy [72], Viieru et al.

[38], and Stanford et al. [43].

11 1111.111










Figure 4-10. Detail of structured CFD mesh near the wing surface.

In order to handle the arbitrarily shaped geometries of a micro air vehicle wing with

passive shape adaptation, the Navier-Stokes equations must be transformed into generalized

curvilinear coordinates: (x,y,z), rl(x,y,z), ((x,y,z). This transformation is achieved by [148]:

Ix Iy Lz f1i f2 f13
fx ]y z 21 f22 f23 (4-13)
x Cy z _f31 A32 A33

wherefj are metric terms, and J is the determinant of the transformation matrix:

J =(,) (4-14)


Using the above information, the steady Navier-Stokes equations can then be written in

three-dimensional curvilinear coordinates [149]. The continuity equation and u-momentum

equation are presented here in strong conservative form, with the implication that the v- and w-

momentum equations can be derived in a similar manner.









u, +V+W =


(4-15)


a(p.U.u) a(p.V.u) a(p.W.u)
+ k + --- (qii-u +q12 *u +q13 u
0 aq ac 8 J
+ -a -.(q21 u +q22 u +q23 .)J + -.(q31 .u+q32 u +q33 *u (4-16)

S(f "-p) a(f21 P) 1(f, p)'


where p is the density, p is the pressure, ti is the viscosity, qij are parameters dictated by the

transformation (expressions can be found in [149]), and U, V, and W are the contravariant

velocities, given by the flux through a control surface normal to the corresponding curvilinear

directions:

U= f -u+/2- V+ + 3-W
V = f21 + f22 v + ,23.w (4-17)
W = f31 +32 + 33.w

In order to numerically solve the above equations, a finite volume formulation is

employed, using both Cartesian and contravariant velocity components [148]. The latter can

evaluate the flux at the cell faces of the structured grid and enforce the conservation of mass. A

second order central difference operator is used for computations involving pressure and

diffusive terms, while a second order upwind scheme handles all convective terms [150].

Fluid Model Comparisons and Validation

Validation of both the linear vortex lattice method and the nonlinear CFD is given in

Figure 4-11, in terms of lift, drag, and longitudinal pitching moments (measured about the

leading edge) at 13 m/s. Pre-stall, the CFD model is able to accurately predict lift and drag

within the experimental error bars of the measured data. Drag is consistently over-predicted at

higher angles of attack; turbulent reattachment of separated flow is know to decrease the profile










drag [8], but is not included in the model. The magnitude of the pitching moment is slightly

over-predicted by the CFD, though the data still falls within the error bars, the slopes match well,

and the onset of nonlinear behavior (due to the low aspect ratio [3]) is well-predicted. The CFD

is also able to predict the onset of stall (via a loss of lift) at about 210, but loses its predictive

capability in the post-stall regime, as the flow is known to be highly unsteady.

1,4 1,4
1.2 1.2
1 1
0.8 0.8
0.6 06 ,
0.4 data 0.4 data
0.2 CFD 0.2 CFD
0 ------ VLM 0 VLM
-0.2 ... .......... .... ............-. 2 -0.2 ......................
-10 0 10 20 30 0 0.2 0.4 0.6
ac CD
0.1
6-
0.

-0.1 4

E -0.2 2
data 2
-0.3 C--
VLM 0 / CFD
-0.4 ------ VLM

-0.5 F -2 '
-0.2 0 0.2 0.4 0.6 0.8 1 12 1.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
CL CL


Figure 4-11. Computed and measured aerodynamic coefficients for a rigid MAV wing, Re =
85,000.

The vortex lattice method is accurate at low angles of attack, but the slope is under-

predicted (possibly due to an inability to compute the low-pressure cells at the wing tips, similar

to vortex lift discrepancies seen in delta wings [27]), and the wing never stalls. The drag

predicted by the vortex lattice method is necessarily augmented by a non-zero CDo (estimated









from the experimental data), and is moderately accurate up to 100 angle of attack. After this

point, the inviscid drag is under-predicted due to massive flow separation over the wing. No

significant differences can be seen between the pitching moments predicted by the CFD and the

vortex lattice method, until the aforementioned nonlinear behavior appears, which the inviscid

solver cannot predict.

Aeroelastic Coupling

Transfer of data between a structured CFD mesh and an unstructured FEA mesh is done

with simple polynomial interpolation. If information from grid A is to be interpolated to grid B,

the element from grid A (triangular for the FEA mesh, quadrilateral for the CFD mesh) is found

which is closest to each node from grid B. Except for nodes that lie on the wing border, these

elements will enclose their corresponding nodes. Polynomial shape functions for the desired

variable are formulated to describe its distribution within the element, and then the value at the

node is solved for. Such a method is found to adequately preserve the integrated forces and the

strain energies from one mesh to another, and is fairly inexpensive.

The un-deformed (due to aerodynamic loads) wing shape technically depends upon the

membrane pre-tension. This shape could be found with Eq. (4-4), setting the pressure source

terms to zero, and letting the wing shape at nodes upon the carbon fiber-latex boundary be

prescribed displacement boundary conditions. Such a scheme should result in slight concavities

along the membrane surface [151]. This effect is considered small, however, and is ignored for

the current work. Shear stress over the wing is also not included in the aeroelastic coupling.

Moving Grid Technique

For aeroelastic computations using the Navier-Stokes flow solver, a re-meshing algorithm

is needed to perturb the structured grid surrounding the flexible wing (no such module is

required when a vortex lattice method is used, as all of the nodes lie on the wing surface). For









the current work, a moving grid routine based upon the master-slave concept is used to maintain

a point-matched grid block interface, preserve grid quality, and prevent grid cross-over. Master

nodes are defined as grid points that lie on the moving surface (the wing surface of the micro air

vehicle, in this case), while the slave nodes constitute the remaining grid points. A slave node's

nearest surface point is defined as its master node, and its movement is given by:

Xs = Xs +-(Xm -xm) (4-18)

where Xs is the location of the slave node, Xm is the location of its master node, the tilde indicates

a new position, and 0 is a Gaussian distribution decay function:


0 = exp n min{500, (-X2 (Ys y)2 ( Z) (4-19)
xmI -m) + (Ym-ym) + (2m +Zm)2+

where E is small number to avoid division by zero, and 3 is a stiffness coefficient; larger values

of 3 promote a more rigid-body movement. Further information concerning this technique is

given by Kamakoti et al. [152].

Numerical Procedure

The steady fluid structure interaction of a MAV wing is computed as follows:

1. If computations involve a batten-reinforced wing, correct for the membrane pre-tensions at
the free trailing edge.

2. Solve for the aerodynamic pressures over the wing, using either the steady Navier-Stokes
equations or the tangency condition of the vortex lattice method.

3. Interpolate the computed pressures from the flow solver grid to the FEA grid.

4. Solve for the resulting wing displacement using either the linear or the nonlinear
membrane/carbon fiber model.

5. Interpolate the displacement onto the MAV wing of the flow solver grid.

6. If nonlinear CFD models are utilized, re-mesh the grid using the master/slave scheme.

7. Repeat steps 2-6 until suitable convergence is achieved: less than 0.1% change in the lift.










Less than ten iterations are usually adequate at modest angles of attack (3
Typical results are given in Figure 4-12 for the lift and efficiency of both a BR and a PR wing,

computed with a Navier-Stokes flow solver and a nonlinear membrane solver. The lift of the PR

wing monotonically converges (lift increases camber, which further increases lift), while the

history of a BR wing is staggered (lift decreases wing twist, decreasing lift). For the nonlinear

modules, step 2 requires between 150 and 250 sub-iterations, while step 4 can typically converge

within 20 sub-iterations. For the linear modules, the equations of state can be solved for directly.

0.65 5.8


0.6 5.6
S- BR BR
0.5---- PR --.4 -- PR


0.5 5.2 .... .
0 5 10 0 5 10
iteration iteration

Figure 4-12. Iterative aeroelastic convergence of membrane wings, a = 9.









CHAPTER 5
BASELINE WING DESIGN ANALYSIS

Three "baseline" micro air vehicle wing designs are considered in this section: first, a

completely rigid wing. Secondly, a batten-reinforced wing with no pre-tension in the membrane

skin, two layers of bi-directional carbon fiber at ply angles of 450 to the chord line (at the root

and leading edge), and one layer of uni-directional carbon fiber (fibers aligned in the chordwise

direction) for the battens. Third, a perimeter-reinforced wing with no pre-tension in the

membrane skin and two layers of bi-directional carbon fiber at ply angles of 450 to the chord line

(at the root, leading edge, and perimeter).

As a large number of function evaluations are not required for this strictly analysis section,

all numerical results are computed with the higher-fidelity methods discussed above: the steady

Navier-Stokes solver and the nonlinear membrane solver. Furthermore, all results are found at

U, = 15 m/s, a value towards the upper range of MAV flight. A higher velocity is chosen to

emphasize aeroelastic deformations. For a given flight condition, 10 VIC images are taken of

the deformed wing (at 1 Hz) and averaged together. Sting balance results are, as discussed,

sampled at 1000 Hz for 2 seconds, and then averaged.

Wing Deformation

Numerical and experimental out-of-plane displacements, normalized by the root chord, are

given in Figure 5-1 for a BR wing, along with a section of the data at x/c = 0.5, at 15 angle of

attack. As expected, the primary mode of wing deformation is a positive deflection of the

trailing edge, resulting in a nose-down twist of each flexible wing section. Deformations are

relatively small (-5%, or 6.2 mm), though still have a significant effect upon the aerodynamics.

The membrane inflates from between the battens (clearly seen in the section plot) towards the

leading edge, but at the trailing edge the wing shape is more homogenous and smooth, and no









distinction between batten and membrane can be made. This is presumably due to the pressure

gradient, with very high forces at the leading edge which dissipates down the wing. The carbon

fiber wing tips, though several orders of magnitude stiffer than the membrane, shows appreciable

twisting, indicative of the large suction forces from the tip vortex. Correlation between model

and experiment is acceptable, with the model slightly under-predicting the adaptive washout, and

over-predicting the local membrane inflation between the battens. Wing shapes and magnitudes

match well with time-averaged results reported by Lian et al. [28].

0.03
0.04 numerical experimental A
0.03aa '002

0.02 0.o I
0.01
0 -1 -0.5 0 0.5 1
2y/b

Figure 5-1. Baseline BR normalized out-of-plane displacement (w/c), a = 15.

Chordwise strains for the same case as above can be seen in Figure 5-2. The directional

stiffness of the battens generally prevents significant stretching in the chordwise direction. The

model predicts appreciable strain (1.4%) at the carbon fiber/membrane interface towards the

leading edge (due to inflation), almost no strain near the mid-chord region, and negative Poisson

strains at the trailing edge. Strains in the carbon fiber regions, while computed, are much smaller

than the membrane strains, and cannot be discerned in Figure 5-2. The measured chordwise

strain is very small and noisy, with no evident differences between the carbon fiber and

membrane regions. Much of the measured field lies below the system's strain resolution (-1000

te). Several noise spikes are also evident in this strain field, while the displacement field in

Figure 5-1 has none; the strain differentiation procedure is more sensitive to experimental noise

than the displacement temporal matching.








numerical experimental


S


x 103

10

5

0


Figure 5-2. Baseline BR chordwise strain (Exx), a = 15.
A better comparison between model and experiment is given in Figure 5-3, with the
spanwise strains. These extensions are essentially a product of the change in distance between
the battens as they deform. Both model and experiment indicate a peak in ey between the inner
batten and the carbon fiber root, though the model indicates this maximum towards the leading
edge (-1.2%), while the measurement places it farther aft. Strain concentrations at the trailing
edge are visible in both fields. Though still noisy, the VIC system's spanwise strain can
differentiate between battens and membrane. Suitable model validation is also seen in shear
(Figure 5-4). Both peaks and distributions of the anti-symmetric shear are well predicted. The
tips of the battens at the trailing edge cause a shear concentration, typically of opposite sign to
the strain in the rest of the membrane segments between each batten.

x 10-3 x 103
numerical experimental 10 .


S


-5 --... .-. .
-1 -0.5 0 0.5 1
2y/b


Figure 5-3. Baseline BR spanwise strain (Eyy), a = 15.


-2
-1 -0.5 0 0.5 1
2.%,b


15
10
5
0


a;.ai


w









0.01
I0.1 numerical experimental
0.005
0.050


-0.005 0 005
-0.01 -0.01
-1 -0.5 0 0.5 1
2y/b

Figure 5-4. Baseline BR shear strain (Exy), a = 15.

Normalized out-of-plane displacements for the perimeter-reinforced wing are given in

Figure 5-5. Deformations are slightly larger than with the BR wing (6%), and are dominated by

the membrane inflation between the carbon fiber leading and trailing edges. The membrane apex

occurs approximately in the middle of the membrane skin, despite the pressure gradient over the

wing. This location is a function of angle of attack, as the peak will move slightly forward with

increased incidence [74], [43]. The carbon fiber wing tip twists less than previously, thought to

be a result of the fact that the wingtip is not free in a PR configuration, but attached to the

trailing edge by the laminate perimeter. Some bending of the leading edge at the root can also be

seen, but not in the BR wing (Figure 5-1). Correspondence between model and experiment is

suitable, with the model again under-predicting wing deformation, but accurately locating the

apex. Slight asymmetries in the measured wing profile (also evident in the BR wing) are

probably a result of manufacturing errors (particularly in the application of the membrane skin

tension), and not due to flow problems in the wind tunnel.

As the amount of unconstrained membrane is greater in a PR wing than in a BR wing,

chordwise strains (Figure 5-6) are much larger as well: peak stretching (3%) is located at the

membrane/carbon fiber boundary towards the leading edge, as before. The magnitude and size

of this high-extension lobe is over-predicted by the model. Both model and experiment show a

region of compressive strain aft of this lobe, towards the trailing edge. This is a Possion strain









(and thus not a compressive stress), but the stress in this region does become slightly negative for

higher angles of attack. Erroneous computation of compressive membrane stresses indicates the

need for a wrinkling module. Though wrinkles in the membrane skin are not obviously visible in

the VIC measurements (possibly an unsteady process averaged out with multiple images),

wrinkling towards the onset of stall is a well-known membrane wing phenomena [87]. As

before, no appreciable strain is measured or computed in the carbon fiber areas of the wing.

0.06 0.06
06 numerical experimental 0.0

0.04
0,04
0.04 At 0.04

4 0,02


0 -1 -0.5 0 0.5 1
2./b

Figure 5-5. Baseline PR normalized out-of-plane displacement (w/c), a = 15.

0.03 numerical experimental 0.03

0.02 ( 0.02
00.0
0.01

0 0 05 1
x/c

Figure 5-6. Baseline PR chordwise strain (exx), a = 15.

Peak spanwise stretching (Figure 5-7) occurs at the membrane carbon fiber interface

towards the center of the wing root, and is well predicted by the model. The computed strain

field erroneously shows a patch of negative Poisson strain towards the leading edge, due to the

high chordwise strains in this area. One troubling aspect of the measured spanwise strains is the

areas of negative strains along the perimeter of the membrane skin: namely on the sidewalls

towards the root and the wingtip. Such strains have been measured in previous studies [9], but









their presence is peculiar. Basic membrane inflation mechanics indicates large extension at the

boundaries rather than compression [145] (as is computed by the model).

The compression may be membrane wrinkling (which, again, is not evident from Figure 5-

5, or may be an error in the VIC strain computations, potentially caused by the large

displacement gradients in this area of the wing. A third possibility is that the VIC is measuring a

bending strain at this point, where the radius of curvature is close to zero. The latex skin, though

modeled as a membrane, does have some (albeit very small) bending resistance due to its finite

thickness. The anti-symmetric shear strain field (Figure 5-8) shows good correspondence

between model and experiment, with accurate computations in-board, but slight under-

predictions of the high shear closer to the wingtip.

x 10-3
numerical experimental 0.02
20


0


^0 --0.01-
-1 -0.5 0 0.5 1
2 /b

Figure 5-7. Baseline PR spanwise strain (cyy), a = 15.

numerical experimental
0.01

0.01


-0.010
-1 -0.5 0 0.5 1
2y/b

Figure 5-8. Baseline PR shear strain (Exy), a = 15.

The aerodynamic twist (camber and camber location) and geometric twist angle









distributions for the baseline BR and rigid wings are given in Figure 5-9. The rigid wing is

characterized by positive (nose-up) twist and a progressive de-cambering toward the wingtip.

The carbon fiber inboard portion of the BR wing exhibits very similar wing twist to the rigid

wing. Past 2y/b = 0.3 however, both model and experiment show that the membrane wing has a

near-constant decrease in twist of 2-3: adaptive washout. Though this geometric twist

dominates the behavior of the BR wing, the membrane also exhibits some aerodynamic twist.

This occurs predominately in the latex between the battens, about 1% of the chord in magnitude.

The location of this camber has large variations: some portions of the wing are pushed back from

25% (rigid) to 75% (membrane), as shown by both model and experiment. Shifting the camber

aft-ward on low Reynolds number wings is one method to hinder flow separation through control

of the pressure gradient [27], and may play a role in the BR wing's delayed stall as well.

8f 0.08 A 0.8

6 exp. / exp.
--- 0.06 / .6 num.
S.........-----d- 0.04 ri g
\: nu o 0.6

2 \ num 0.4
S 0.02 --------- rigid
0
0 i.........g 0.2
-1 0 1 -1 0 1 -1 0 1
2y/b 2y/b 2y/b

Figure 5-9. Baseline BR aerodynamic and geometric twist distribution, a = 15.

The aerodynamic and geometric twist distributions for the baseline PR and rigid wings are

given in Figure 5-10. Membrane inflation adaptively increases the camber by as much as 4%,

though this figure is slightly under-predicted by the model. The location of this camber is shifted

aft-ward, though not as much as with the BR wing. The flexible laminate used for the wing

skeleton pushes the location of the camber at the root slightly forward. Like the BR wing

deformation, shape changes over the PR wing are a mixture of both aerodynamic and geometric









twist (though the former dominates). The laminated perimeter deflects upward farther than the

leading edge, resulting in a slight nose down twist. This is as much as 2 at the wingtips, slightly

under-predicted by the model.

8 0.1 0.5 exp.
1 exp. 0.08 num.
num. 0.06 0.4 -------- riid
S--------rd \ 0.
4.......... rigi Qi-. -
0.04 exp.
2 \\ S30.3 \ j
0.02
2 ^O / ----------rigid
0 -- 0 f 0.2 .
-1 0 1 -1 0 1 -1 0 1
2y/b 2y/b 2y/b

Figure 5-10. Baseline PR aerodynamic and geometric twist distribution, a = 15.

Wing twist and camber throughout the entire a-sweep are given in Figure 5-11, at a

flexible wing section at 2y/b = 0.65. The master slave moving grid algorithm [152] fails with

BR wings at angles of attack higher than 20: the steep displacement gradients between the

carbon fiber root and the membrane skin leads to excessive shearing within the CFD mesh

surrounding the wing.

The nose-down twist of both the BR and the PR wing increase monotonically with angle of

attack, thought the former is obviously much larger. Experimentally measured BR wing twist

has a linear trend (up to stall at about 22) with a, while the numerical curve is more nonlinear,

and under-predicts twist at moderate angles. Both model and experiment demonstrate a

moderate increase in camber of the BR wing, with a linear trend in a up to stall. After stall, the

camber of the BR wing increases substantially, from 5% to 8%.

The camber of the PR wing is much larger than the rigid wing, even at low and negative

angles of attack. This is due to the lack of pre-tension: even a moderate amount of force will

cause substantial deformations [147]. Both measurements and simulations of the PR wing are










difficult at lower angles than shown in Figure 5-11: the membrane is equally apt to lie on either

side of the chordline [65], and steady-state solutions don't exist. PR wing camber variations

with angle of attack are nonlinear (the development of finite strains cause a 1/3 power law

response to the applied load [72]), and are slightly under-predicted by the model.

The location of this camber in a PR wing moves somewhat forward for modest angles,

while the BR wing sees a significant aft-ward shift at the onset of stall. Both of these camber

location trends are well-predicted by the model. Experimental error bars for camber, though not

shown here, are on the order of 10% at low angles, less than 2% at moderate angles, and upwards

of 20% in the stalled region [43]. This stems not from uncertainty but from unsteady membrane

vibration, possibly due to vortex shedding as discussed by Lian and Shyy [8].

4 0.1 ------
rigid 0.8 000
-... PR t BR
2 0 4B R
co PR 0.08 0.6
o 0.6
0 BR PR
E PR
o0 0.06 o 0 o4--
-2\a BRO D.4 ,,,AA AAB
-2 rioid rigid
S0.04 0.2
-10 0 10 20 30 -10 0 10 20 30 -10 0 10 20 30
a a a

Figure 5-11. Aerodynamic and geometric twist at 2y/b = 0.65.

Aerodynamic Loads

Lift coefficients (both measured and predicted) throughout the a-sweep, with no model

yaw, are given in Figure 5-12, for the three baseline wing designs discussed above. For all six

data sets, lift slopes are very low (-0.05/0, about half of the value for two-dimensional airfoils

[27]) as expected from low aspect ratio wings. The downward momentum from the tip vortices

helps mitigate the flow separation, delaying stall to relatively high angles (18-22o). Focusing

first on the rigid wing, mild nonlinearities can be seen in the lift curve. Both model and










experiment indicate an increase in the slope by 25% between 0 and 15 angle of attack. This is

presumably due to a growth in the low pressure cells at the wing tips of the low aspect ratio wing

[3]. Such nonlinearities should become more prevalent for lower aspect ratios than considered

here (1.25). Model and experiment show good agreement for the lift over the rigid wing prior to

stall. At stall (where the static model's predictive capability is questionable due to unsteady flow

separation [18] and tip vortices [6]) the model slightly under-predicts the stalling angle and

CL,max; the loss of lift is more severe in the experimental data.

1.4 1.4
1.2 1.2 o -

/ 1- a





0.2 BR 02 A BR
0 y' ---- .PR 0 c o PR
-0.2 -0.82
-10 0 10 20 30 -10 0 10 20 30
a a

Figure 5-12. Baseline lift coefficients: numerical (left), experimental (right).

The adaptive inflation/cambering of the PR wing substantially increases the lift and the lift

slope as compared to the rigid wing. The lift curve of the PR wing is less nonlinear than the

rigid wing. This may be due to the nonlinear cambering seen in Figure 5-11, which is known to

decrease the lift slope [15] and can offset the growth of the tip vortices. Drastic changes in the

lift characteristics at low angles due to hysteresis effects [65], and a gradual onset of stall [89]

are not evident in either the numerical or the experimental data, perhaps because a relevant

portion of the wing is not composed of the flexible membrane. The model significantly under-

predicts CL,max of the PR wing, and erroneously computes that the wing stalls before the rigid









wing. Similar experimental work [9] at lower speeds also show early stall, again indicating the

sensitivity of Reynolds number to stall.

At angles of attack below 100, the BR wing has very similar lift characteristics to the rigid

wing, a fact also noted in the work of Lian et al. [28]. This is thought to be due to two offsetting

characteristics of a wing with both aerodynamic and geometric twist [67]: the inflation in

between each batten increases the lift, while the adaptive washout at the trailing edge decreases

the lift. Both of these deformations can be seen in Figure 5-1. At higher angles of attack, the

load alleviation from the washout dominates the deformation, and decreases both the lift and the

lift slope, as indicated by both model and experiment. Delayed stall is not present in the

measurements (though, as with the PR wing, has been measured at lower Reynolds numbers [9]),

and numerical BR wing modeling cannot be taken past 20 due to aforementioned problems with

the moving boundary.

Figure 5-13 shows drag coefficients through the a-sweep, with good experimental

validation of the model. As before, the drag of the rigid and the BR wings are very similar for

modest angles of attack. Above 100 the load alleviation at the trailing edge decreases the drag, a

streamlining effect [63]. It should be noted however that for a given value of lift, the BR wing

actually has slightly more drag than a rigid wing [9]. Regardless of whether the comparative

basis is lift or angle of attack, the PR wing has a drag penalty over the rigid wing. This is in part

due to the highly non-optimal airfoil shape of each membrane wing section: Figure 5-5 shows

the tangent discontinuity of the wing shape at the membrane/carbon fiber interface towards the

leading edge. Excessive inflation may also induce additional flow separation.

Longitudinal pitching moments (measured about the leading edge) are given as a function

of lift for the three baseline designs in Figure 5-14. Of the three, the PR wing is not statically










stable (based upon a negative Cm,AC), and the hinged trailing edge portion (seen in Figure 1-1 and

Figure 1-2) must be used for trimmed flight. Prior to stall, both the aeroelastic model and the

experiment indicate very similar behavior between the rigid and the BR wing, with mild

nonlinearities in the moment curves. This is ostensibly due to tip vortex growth, as before [3].

0.6 ..... 0.6 - -

0.5 0.5
rigid 0 o rigid a
0.4 ---------- BR 0.4 a BR
-.PR / 0 PR A
U 0.3 / 0.3 o

0.2 0.2 2 2
U03 0.2 0
0.1 0.1

0 ...... ........ 0 "
-10 0 10 20 30 -10 0 10 20 30
cc a

Figure 5-13. Baseline drag coefficients: numerical (left), experimental (right).

The PR wing has a 15% lower pitching moment slope than the rigid wing. This is a result

of the membrane inflation, which shifts the pressure recovery towards the trailing edge,

adaptively increasing the strength of the nose-down (restoring) pitching moment with increases

in lift and a. Steeper Cm slopes indicate larger static margins: stability concerns are a primary

target of design improvement from one generation of micro air vehicles to the next. The static

margin of a MAV is generally only a few millimeters long; properly fitting all the micro-

components on board can be difficult. Furthermore, the PR wing displays a greater range of

linear Cm behavior, possibly due to the fact that the adaptive membrane inflation quells the

strength of the low pressure cells [13].

Finally, L/D characteristics are given in Figure 5-15, as a function of lift. For low angles

of attack (and lift), the three wings perform similarly. At higher angles of attack (prior to stall),

the PR wing has the highest efficiency. The model incorrectly computes the BR wing to have










the best L/D for a small range of modest lift values. Correlation between model and experiment

is generally acceptable for the rigid and BR wings, though the L/D of the PR wing is

significantly under-predicted by the model, owing mostly to poor lift prediction at these angles

(Figure 5-12). The camber of the PR wing is subsequently under-predicted as well (Figure 5-

11), and may be a result of membrane vibration [89]. At no point does either model or

experiment indicate that the rigid wing has the best efficiency; perhaps surprising, given the fact

that neither wing deforms into a particularly optimal airfoil shape.

0.1 .... 0.1 -
rigid o rigid
0 *'-.. -..-- BR 0 a A BR

..PR -. & PR
-0.1 -0.1 a &
0


-0.3 -0.3
S0 -0.2



-0.4 -0.4

-0.5 -0.5
0 0.5 1 0 0.5 1
CL CL

Figure 5-14. Baseline pitching moment coefficients: numerical (left), experimental (right).


0 o
6 6 ,


4 4 0o

2 2 00
riid d o rigid
0 --------- BR 0 A BR
PR 0 PR

2 *. . . 2
0 0.5 1 0 0.5 1
CL CL

Figure 5-15. Baseline wing efficiency: numerical (left), experimental (right).









A quantitative summary of the last four figures is given in Table 5-1, for all three baseline

wings at 6 angle of attack. Experimental error bounds are computed as described above.

Aerodynamic sensitivities (as well as the pitching moment about the aerodynamic center) are

found with a linear fit through the pre-stall angles of attack. Error bounds in these slopes are

computed with Monte Carlo simulations. Computed lift, drag, and pitching moments

consistently fall within the measured error bars (the latter of which are exceptionally large),

though pitching moments are significantly under-predicted (10-30%). Sensitivities are also

under-predicted, though still fall within the large error bars associated with pitching moment

slopes. With the exception of L/D of a PR wing, trends between different wing structures are

well-predicted by the aeroelastic model.

Table 5-1. Measured and computed aerodynamic characteristics, a = 6.
S CL I CD
num. exp. error (%) num. exp. error (%)
rigid 0.396 0.384 0.024 3.10 0.070.0.069 + 0.007 1.15
BR 0.381 0.382 + 0.024 -0.16 0.067 0.069 + 0.007 -3.04
PR 0.465 0.495 + 0.031 -5.98 0.085 0.076 + 0.009 11.61
Cm I L/D A
num. exp. error (%) num. exp. error (%)
rigid -0.084 -0.063 0.033 -32.81 5.64 5.49 + 0.69 2.72
BR -0.087 -0.073 + 0.034 -19.39 5.70 5.49+ 0.68 3.77
PR -0.138 -0.131 0.042 -5.64 5.49 6.49+ 0.87 -15.36
I CL. Cm,AC
num. exp. error (%) num. exp. error (%)
rigid 0.049 0.051 0.003 -5.26 0.013 0.016 0.018
BR 0.044 0.048 0.004 -9.35 0.006 -0.001 + 0.020
PR 0.052 0.057 0.004 -9.21 -0.008 -0.015 + 0.026
Cma I dCm/dCL
num. exp. error (%) num. exp. error (%)
rigid -0.012 -0.010 + 0.004 -11.65 -0.246 -0.199 0.086 -23.07
BR -0.011 -0.009 0.004 -17.97 -0.244 -0.185 0.098 -31.88
PR -0.014 -0.013 0.006 6.01 -0.280 -0.229 0.105 -22.17


Flow Structures

Having established sufficient confidence in the static aeroelastic membrane wing model,









attention is now turned to the computed flow structures. No experimental validation is available

for this work, though whenever possible the results will be correlated to data in the previous two

sections, or results in the literature. Experimental flow visualization work for low aspect ratios

and low Reynolds number is given by Tang and Zhu [6] and Kaplan et al. [37]. Work done

explicitly on MAV wings is given by Gursul et al. [40], Parks [91], Gamble and Reeder [92], and

Systma [153].

The pressure distributions and flow structures are given in Figure 5-16 at 00 angle of attack

for the upper/suction wing surface of all three baseline wing designs. The plotted streamlines

reside close to the surface, typically within the boundary layer. For the rigid wing, a high

pressure region is located close to the leading edge, corresponding to flow stagnation. This is

followed by pressure recovery (minimum pressure), located approximately at the camber of each

rigid wing section. Pressure recovery is followed by a mild adverse pressure gradient, which is

not strong enough to cause the flow to separate. A further decrease in Reynolds number has

been shown to cause mild flow separation over the top surface for 00 however [14], [153]. A

small locus of downward forces are present over the negatively-cambered region (reflex) of the

airfoil, helping to offset the nose-down pitching moment of the remainder of the rigid MAV

wing, as discussed above. The reflex can also help improve the wing efficiency, compared to

positively-cambered wings [55]. There is positive lift of this wing at 00 (Figure 5-12), resulting

in a mild tip vortex swirling system. The low pressure cells at the wing tip are not yet evident.

Aeroelastic pressure redistributions of the upper surface of the BR wing are seen in the

form of three high-pressure lobes at the carbon fiber/membrane boundary interface towards the

leading edge. The membrane inflation in between each batten (Figure 5-1) results in a slight

tangent discontinuity in the wing surface. This forces the flow to slow down and redirect itself









over the inflated shape: such a deceleration results in a pressure spike. Aft of these spikes, the

pressure is slightly lower in the membrane skin than over each batten (due to the adaptive

camber), driving the flow into the membrane patches. This is a very small effect (mildly visible

in the streamlines) for the current case, but can be expected to play a large role with potential

flow separation, where the chordwise velocities are very small [154].




... .. ....... .. .





-80 -30 20 70 120 -80 -30 20 70 120 -80 -30 20 70 120

Figure 5-16. Pressure distributions (Pa) and streamlines on the upper surface of a rigid (left), BR
(center), and PR wing (right), a = 0.

For the PR wing (Figure 5-16), the pressure spike is stronger, and exists continuously

along the membrane interface. A significant percentage of this spike is directed axially,

increasing the drag (as seen in Figure 5-13). The adaptive inflation causes an aft-ward shift in

the pressure recovery location of each flexible wing section. The longer moment arm increases

the nose-down pitching moment about the leading edge (Figure 5-14), which is the working

mechanism behind the benevolent longitudinal static stability properties of the PR wing.

Furthermore, the aerodynamic twist increases the adverse pressure gradient over the membrane

portion of the wing: some flow now separates as it travels down the inflated shape, further

increasing the drag (as compared to the rigid wing).

Similar results are given for the lower/pressure side of the three wings at 0 angle of attack

in Figure 5-17. The flow beneath the rigid wing is dominated by an adverse pressure gradient

towards the leading edge, causing a large separation bubble underneath the wing camber. This









separated flow is largely confined to the in-board portions of the wing. Flow reattaches slightly

aft of the quarter-chord, after which the pressure gradient is favorable. The flow accelerates

beneath the negatively-cambered portion of the rigid wing: this decreases the local pressures,

further offsetting the nose-down pitching moment. The pressure distribution on the lower

surface is not greatly affected by the tip vortices, previously noted by Lian et al. [28].










S-130 -S2 -34 14 -130 -82 -34 14 -130 -82 -34 14

Figure 5-17. Pressure distributions (Pa) and streamlines on the lower surface of a rigid (left), BR
(center), and PR wing (right), a = 0.

For the BR wing (Figure 5-17), slight undulations in the pressure distribution are indicative

of the membrane inflation in between the battens. This causes the opposite of what is seen on

the upper wing: flow is slightly packed towards the battens [154], though the effect is minor, as

before. The adaptive aerodynamic twist of the PR membrane wing pushes the bulk of the

separated flow at the leading edge towards the root, and induces further separation beneath the

inflated membrane shape, as the air flows into the cavity against an adverse pressure gradient.

The location of maximum pressure is increased and pushed aft-ward to coincide with the apex of

the inflated membrane, increasing both the lift and the stability.

Flow structures over the upper surface at 150 angle of attack are given in Figure 5-18. At

this higher incidence, the adverse pressure gradient is too strong for the low Reynolds number

flow, and a large separation bubble is present at the three-quarter chord mark of the rigid wing.

Despite the nose-up geometric twist built into the wing (7 at the tip, Figure 5-9), flow separates









at the root first, and is confined (at this angle) to the in-board portion of the wing. This may be

due to the steeper pressure gradients at the root, or an interaction with the tip vortex system [5].

The reattached flow aft of the bubble (and the resulting pressure distribution) must be

viewed with a certain amount of suspicion. Such a reattachment is known to be turbulent

process [25], and no such module is included in the CFD (or even, to the author's knowledge,

exists for complex three-dimensional flows). Unsteady vortex shedding may accompany the

bubble as well [8], though time-averaging of vortex shedding is known to compare well with

steady measurements of a single stationary bubble [18]. The augmented incidence has

considerably increased the strength of the wing-tip vortex swirling system over the rigid wing.

The size of the vortex core is larger (indicative of the expected increase in induced drag [27]),

and the low pressure cells at the wing tip are very evident [3].













Figure 5-18. Pressure distributions (Pa) and streamlines on the upper surface of a rigid (left), BR
(center), and PR wing (right), a = 15.

As expected, the aeroelastic effects of the BR and PR wings are more predominate at 15

in Figure 5-18. For the BR wing, the three high-pressure lobes over the membrane/carbon fiber

interface are larger. Significant pressure-redistribution over the membrane stretched between the

outer batten and the wing tip can be seen as well. Adaptive washout slightly decreases the

intensity of the separation bubble, but has no noticeable effect on the pressure distribution at the

trailing edge of the upper BR wing surface. At 15, the aerodynamic twist of the PR wing is









considerably larger than before, as is the resulting pressure spike at the membrane-carbon fiber

interface. Despite the large adverse pressure gradient in this region, flow does not separate

(though it has been noted in other studies [153]).

The inflated membrane shape of the PR wing pushes the bulk of the flow separation closer

to the wing root. Some of this separated flow reattaches to the wing and travels into the wake,

while the rest travels spanwise. This flow is attracted either by the low pressures associated with

the adaptive cambering, or by the low pressures at the core of the tip vortex. Some of these

separated streamlines are entrained into the swirling system, an interaction that has been shown

to cause potential bilateral instabilities for high angles of attack [6]. This effect, not seen in the

rigid or BR wings, obviously cannot be further studied in this work, due to both the symmetry

and the steady assumptions made in the solver.

It can also be seen that the passive shape adaptation decreases the magnitude of the low

pressure cells at the wing-tip, by 9% for the BR wing and 13% for the PR wing, compared to the

rigid case. This indicates that the induced drag is decreased with flexibility, though this is only a

re-distribution of the total drag. Two possible explanations exist for the decrease in tip vortex

strength. The mechanical strain energy in the inflated membrane skin may be removing energy

from the vortex swirling system [90]. For the PR wing, the inflated membrane shape may act as

a barrier to the tip vortex formation, preventing the full swirling development at the wing-tip.

A similar effect is demonstrated in the work of Viieru et al. [38] by the use of endplates

installed on a rigid MAV wing. Whereas the endplates are able to decrease induced drag only at

moderate angles (afterwards the tip vortices increase in strength to overwhelm the geometrical

presence of the endplates), the phenomena demonstrated in Figure 5-18 is effective at all angles:

both the size of the membrane barrier and the strength of the vortex swirling grow in conjunction









with one another as the angle of attack increases. This decrease in tip vortex strength is also seen

in Figure 5-14: the nonlinear aerodynamics (from the low pressure cells at the tip) is evident in

the pitching moments of the rigid and BR wings, while the PR curve is very linear.

On the underside of the rigid wing at 15 angle of attack (Figure 5-19), the increased

incidence provides for completely attached flow behavior. The pressure gradient is largely

favorable, smoothly accelerating the flow from leading to trailing edge. From the previous four

figures it can be seen that separated flow over the bottom surface gradually attaches for

increasing angles of attack, while attached flow over the upper surface gradually separates

(eventually leading to wing stall). As time-averaged flow separation is likely to be unsteady

vortex shedding [18]: this explains the aforementioned membrane vibration amplitudes that

decrease to a quasi-static behavior, then increase through the a-sweep [135].













Figure 5-19. Pressure distributions (Pa) and streamlines on the lower surface of a rigid (left), BR
(center), and PR wing (right), a = 15.

Load alleviation on the lower surface of the BR wing is evidenced by a decrease in the

high-pressure regions associated with camber, and a growth of the suction region at the trailing

edge (the latter presumably due to a decrease in the local incidence). A high-pressure lobe also

develops at the trailing edge of the membrane panel between the carbon fiber root and the inner

batten. At higher angles, this region of the membrane does not locally inflate; it merely stretches

between the two laminates, acting as a hinge. The adaptive inflation of the PR wing causes a









significant redirection of the flow vectors beneath the wing, but does not induce flow separation.

Two sharp pressure drops are seen beneath the wing: one as the flow accelerates into the inflated

membrane shape, and the second as the flow accelerates out of from the membrane and

underneath the re-curved area of the wing.

Further delineation of the flow structures over the three baseline wings can be seen in

Figure 5-20 (a = 0) and Figure 5-21 (a = 150), with the sectional normal force coefficient and

the pressure coefficients over a flexible span station (2y/b = 0.5) of the wing. For the rigid wing

at 0, the sectional normal force peaks at 2y/b = 0.9 (due to the decreasing local chord length of

the Zimmerman planform, but also the low pressure cells left by the tip vortices) and then

experiences a sharp drop at the tip, as necessitated by the low thickness of the wing. Grid

resolution and errors from interpolating the pressures from the cell centers to the nodes [150]

prevent this curve from reaching the correct value of zero. No significant differences arise

between the computed cn of the BR and rigid wings at 0, as previously indicated by the similar

aerodynamic loads (Figure 5-12). The adaptive inflation of the PR wing increases the normal

force over most of the wing, including the stiff carbon fiber root.

Turning now to the pressure coefficients at 0 (Figure 5-20), both the BR and the PR wings

experience a pressure spike over the upper surface at 2y/b = 0.2, corresponding to the membrane

inflation. Outside of this location, pressure redistribution over the BR wing is negligible. The

PR wing shows an aft-ward shift in the high-lift forces over both the upper and lower surfaces.

Adaptive inflation is also seen to increase the severity of the adverse pressure gradient (leading

to the flow separation seen in Figure 5-16), and exacerbate the pressure gradient reversal over the

reflex portion of the wing.

At 15 angle of attack (Figure 5-21), the BR wing is more effective, able to alleviate the










load over the majority of the wing. An evaluation of the pressure coefficients at this angle

indicates that the majority of this reduction in lift occurs towards the trailing edge of the bottom

surface, where the suction forces are increased. Both flexible wing pressure spikes over the

upper surface are intensified at the higher angle, with the PR wing's approaching the strength of

the leading edge stagnation pressure. Sharp pressure drops are also visible on the underside of

the wing, as the flow accelerates into the membrane cavity. All three wings show a mild

pressure plateau associated with separation [27]; the plateaus of the flexible wings are shifted

towards the trailing edge.


1


0.5


- 0.


-0.5


0 0.2 0.4 0.6 0.8 1
2y/b

Figure 5-20. Section normal force coefficients,


0 0.2 0.4


0.6 0.8 1


x/c

and pressure coefficients (2y/b =0.5), a = 0.

1.5 .


1


0.5


0


-0-5


0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
2y/b x/c

Figure 5-21. Section normal force coefficients, and pressure coefficients (2y/b =0.5), a = 15.


S- Rigid
top -------- BR
,- -. ------------- PR


bottom









CHAPTER 6
AEROELASTIC TAILORING

The static aeroelastic modeling algorithm detailed above (using the Navier-Stokes flow

solver and the nonlinear membrane solver) can elucidate accurate quantitative dependencies of a

variety of parameters (CL, CD, Cm, L/D, CL,, Cma, mass) upon the wing structure. Having first

studied the general effect of wing topology (batten-reinforced and perimeter-reinforced

membranes, as well as rigid wings), attention is now turned to structural sizing/strength variables

within the BR and PR wings. Results from the previous section show that the membrane skin's

inflation/stretching dominates the aeroelastic behavior, indicating the importance of the pre-

tension in the membrane skin. Pre-stress resultants in the spanwise and chordwise directions are

both considered as variables. With the exception of the free trailing edge correction of the BR

wings detailed above, the pre-tension is constant throughout the wing. The laminate orientation

and number of plies used to construct the plain weave carbon fiber areas of the wing can be

varied as well. Finally, the number of layers in each batten of the BR wing can be altered,

though the orientation will be fixed so that the fibers run parallel to the chord line.

The sizing/strength parameters listed above leads to an optimization framework with 9

variables, if the number of layers in each of the three battens are permitted to differ, and the wing

type (BR, PR, rigid) is considered a variable as well. Some of the variables are discrete, others

continuous. The variables are not entirely independent either: the fiber orientation of the second

bi-directional plain weave ply is meaningless if only a single ply is used. Genetic algorithms are

well-suited to problems with a mixed integer-continuous formulation, can handle laminate

stacking sequence designs without a set number of layers (with the use of addition and deletion

modules [155]), are a cost-effective method of solving multi-objective problems [156], and can

navigate disjointed design spaces [52]. The computational cost of a genetic algorithm is









prohibitive however, typically requiring thousands of function evaluations for suitable

convergence; a single simulation using the static aeroelastic model described above takes 2-3

hours of processor time on a Compaq Alpha workstation.

A viable alternative is a designed experiment: the computational cost is lower, and

provides an effective investigation of the design space. For this work, one-factor-at-a-time

(OFAT) numerical tests are run to establish the effect of various structural parameters upon the

relevant aerodynamics. The three baseline wing designs used above will represent the nominal

wing designs (2 layers of plain weave at 450, one layer battens, slack membrane). Having

identified the structural variables that display the greatest sensitivity within the system, a full-

factorial designed experiment [157] will be run on a reduced set of variables. This data set can

then be used to identify the optimal wing type and structural composition for a given objective

function. Designs that strike a compromise between two objective functions are considered as

well. The work concludes with experimental wind tunnel validation of the performance of

selected optimal designs.

OFAT Simulations

The schedule of OFAT simulations is as follows: a 6-level full factorial design is

conducted for the chordwise and spanwise pre-stress resultants, 6 simulations for the orientation

of a single laminate of plain-weave, a 6-level full factorial design for the orientations of a two-

layer laminate plain weave, and a 3-level full factorial design for the number of layers used in the

three battens. Pre-stress resultants are bounded by 0 N/m (slack membrane) and 25 N/m (axial

batten buckling can be computed for a distributed axial force equivalent to 31 N/m of pre-stress

resultant in the membrane). The latter value corresponds to roughly 10% pre-strain. Plies of

plain weave carbon fiber are limited to two layers, while battens are limited to three.









Membrane Pre-Tension

Computed aerodynamic derivatives (CL, and Cm.) and efficiency (L/D) are given as a

function of the pre-stress resultants in the chordwise (Nx) and spanwise (Ny) directions for a BR

wing in Figure 6-1. The corresponding normalized wing displacement is given in Figure 6-2 for

a subset of the data matrix. All results are computed at 120 angle of attack, aerodynamic

derivatives are computed with a finite difference between 11 and 12. In a global sense,

increasing the pre-tension in the BR wing increases CL,, decreases Cma, and decreases L/D. The

increased membrane stiffness prevents effective adaptive washout (and the concomitant load

alleviation), and the wing performance tends towards that of a rigid wing. At 12 for a rigid

wing, CL, = 0.0507, Cm. = -.0143, and L/D = 4.908. Overall sensitivity of the aerodynamics to

the membrane pre-tension can be large for the derivatives (up to 20%), though less so for the

wing efficiency (less than 5%, presumably due to the conflictive nature of the ratio).

CL^x Cmx L/D



0.05
-0.013 T/ 5.2
0.045 20 -0.014 20 5.1 /20
10 -0.015 10 5 10
20 10 20 10 0 N 20 10 0 0 N
N N N

Figure 6-1. Computed tailoring of pre-stress resultants (N/m) in a BR wing, a = 12.

The BR wing is very sensitive to the pre-stress in the spanwise direction, but less so to

stiffness in the chordwise direction. This is seen in Figure 6-2: the slack membrane wing has a

trailing edge deflection of 2.5% of the root chord. Maximizing the spanwise pre-tension (with

the other direction slack) drops this value to 1%, while the opposite scenario drops the value to

only 1.9%. This is due to the directional stiffness of the battens (which depend on compliance









normal to their axis for movement), but also the trailing edge stress correction detailed above.

Despite the global trend towards a rigid wing with increased pre-tension, the changes are

not monotonic. A wing design with a minimum lift slope (for gust rejection, improved stall

performance, etc.) is found, not with a completely slack wing, but a wing with a mild amount of

stiffness (10 N/m) in the chord direction, and none in the span direction. Such a tactic removes

the aforementioned conflicting sources of aeroelastic lift in a BR wing. The pre-stress correction

eliminates most of the stiffness at the trailing edge (allowing for adaptive washout and load

alleviation), but retains the chordwise stiffness towards the leading edge, as seen in Figure 4-8.

The membrane inflation in this area is thus decreased, along with the corresponding increase in

lift due to camber.

N = 0 N/n N = 10 N/m N = 20 N/m



Nx= 0 N/m




N,=10N/m y


x

Nx = 20N/m




-0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025

Figure 6-2. Computed BR wing deformation (w/c) with various pre-tensions, a = 12.

Maximizing CL, (for efficient pull-up maneuvers, for example), is found by maximizing Ny

and setting Nx to zero; this eliminates the adaptive washout, but retains the inflation towards the

leading edge. Conversely, maximizing CL, with a constraint on the acceptable L/D might be









obtained by maximizing Nx and setting Ny to zero. Peak efficiency is found with a slack

membrane: this corresponds to minimum drag, which is not shown. It should be mentioned

however, that if a design goal is to maximize the lift slope (or minimize the pitching moment

slope for stability), a BR wing is most likely a poor choice.

Opposite trends are found for the PR wing (Figure 6-3 and Figure 6-4): increasing the pre-

tension decreases CL,, increases Cma, and increases L/D. Similar to before, added wing stiffness

decreases the adaptive inflation of the wing skin, and results tend towards that of a rigid wing.

Without the directional influence of the battens and the trailing edge stress correction needed for

the BR wing, the PR wing surfaces in Figure 6-3 are very smooth, and converge monotonically

for high pre-stress.

CLax Cma L/D




0.054015 4.9 /
20 -0.016 20 4.85 .1 20
0.052 0 4 0
10 -0.017 10 48
20 10 0 0 N 20 10 00 N 20 10 0 0N
N N N
Y y

Figure 6-3. Computed tailoring of pre-stress resultants (N/m) in a PR wing, a = 12.

As before, the PR wing is more sensitive to pre-tension in the spanwise direction than the

chordwise direction. The slack membrane wing inflates to 5% of the chord: maximizing tension

in the chord direction (with none in the span direction) drops this value to 3%, though the

opposite case drops the value to 1.5%. This is probably due to the fact that the chord of the

membrane skin is about twice as long as it's span. The sensitivity of a pressurized rectangular

membrane to a directional pre-stress is inversely-proportional to its length in the same direction,

as indicated by solutions to Eq. (4-4). Though the L/D of the PR wing is equally affected by pre-









stresses in both directions, the two aerodynamic derivatives in Figure 6-3 have a significantly

muted response to Nx. Such a result has noteworthy ramifications upon a multi-objective

optimization scenario. The longitudinal static stability is optimal for a slack membrane wing, but

the wing efficiency at this data point is poor. Maximizing Nx and setting Ny to zero greatly

improves the lift-to-drag ratio (only 0.2% less than the true optimum found on this surface), with

a negligible loss in static stability.

N = 0 N/m N = I t N/m Ny = 20 N/m



NO = 0 N/m




N, = 10 N/nmi




Nx = 20 N/mn




0 0.01 0.02 0.03 0.04 0.05

Figure 6-4. Computed PR wing deformation (w/c) with various pre-tensions, a = 12.

Single Ply Laminates

The same aerodynamic metrics are given in Figure 6-5 as a function of the ply angle (with

respect to the chord line) for a set of wings with a single layer of bi-directional carbon fiber at

the wing root, leading edge, and perimeter (for the PR wing only). The membrane wing is slack.

Due to the plain weave nature of the laminate, all trends are periodic every 900. Only fiber

orientations of 0, 45, and 900 automatically satisfy the balance constraint [155]. For the PR

wing, changing the fiber angle has a minor effect on the aeroelastic response, and optima are









mostly located at either 450 (where spanwise bending is largest) or 900 (where it is smallest).

This indicates that the PR wing, whose planform is dominated by membrane skin, can only take

advantage of different laminates inasmuch as the spanwise bending can increase or decrease the

aerodynamic membrane twist/cambering.

On the other hand, the BR wing relies mostly upon geometric twist (Figure 5-11), which

can be provided from unbalanced laminates via bend-twist coupling; the concept behind

traditional aeroelastic tailoring [11]. Of the 7 data points shown in Figure 6-5, orientations less

than 450 cause the wing to wash-in, while angles greater than 450 cause washout, the latter of

which minimizes CL, of a BR wing, as expected. Using laminate wash-in to counter the load

alleviation of the membrane washout (at 150) optimizes the wing efficiency. Aerodynamic

sensitivity of the BR wing to laminate orientation is also larger than that seen in the PR wing

because the carbon fiber skeleton is less constrained. The wing tip of the BR wing (where the

forces can be large, due to the tip vortices seen in Figure 5-18) is not connected to the trailing

edge via a perimeter strip.



PR 54 BR
rigid -0.014 rigid
0.05 5.2
-0.016
'0451. BR/ PR 5 rigid
-0.018 4.8 pR
0 30 60 90 0 30 60 90 0 30 60 90
0 0 0

Figure 6-5. Computed tailoring of laminate orientation for single ply bi-directional carbon fiber,
a =12.

Double Ply Laminates

Computed aerodynamic derivatives (CL, and Cma) and efficiency (L/D) are given as a

function of the ply orientations (61 and 02) of the two layers of bi-directional plain weave in a BR









wing (Figure 6-6) and a PR wing (Figure 6-7) at 120 angle of attack. As before, the membrane

skin is slack. Aeroelastic trends are expected to repeat every 900, and will be symmetric about

the line 01 = 02. This latter point is only true because bending-extension coupling in non-

symmetric laminates is ignored, though the effect of its inclusion would be very small as the

wing is subjected mostly to normal pressure forces.

CLax Cmm L/D



-0.0115 5.4


90 45 0 0 01 90 45 0 0 1 90 45 0 0 0
9 2 2 02


Figure 6-6. Computed tailoring of laminate orientations for two plies of bi-directional carbon
fiber in a BR wing, a = 12.

For the BR wing, efficiency is maximized and the lift slope is minimized when the fibers

make 450 angles with the chord and span directions. Static stability is improved when fibers

align with the chord. The response surface of the two stability derivatives are very noisy,

suggesting possible finite differencing errors, and all three surfaces in Figure 6-6 show little

variation (only Cma of the BR wing can be varied by more than 5%). Unlike any of the tailoring

studies discussed above, the PR wing shows the same overall trends and optima as the BR wing.

The surfaces for the PR wing, however, are much smoother but have less overall variation.

Of the sampled laminate designs, [15]2 and [75]2 will exhibit the greatest bend-twist

coupling, yet neither are utilized by the membrane wings. This fact, along with the similarity

between the PR and the BR surfaces, suggest that the orientation of a plain weave laminate with

two layers is too stiff to have much impact on the aerodynamics, which is dominated by

membrane inflation/stretching. The use of bi-directional plain weave is not the most effective









means of introducing bend-twist coupling in a laminate. The fact that the two fiber directions

within the weave are perpendicular automatically satisfies the balance constraint at angles such

as 45. This would not be the case if plies of uni-directional carbon fiber are utilized, but this is

prohibitive in MAV fabrication for the following reason. Curved, unbalanced, potentially non-

symmetric thin uni-directional laminates can experience severe thermal warpage when removed

from the tooling board, retaining little of the intended shape.

CLa Cma L/D



0.055 -0.0168 4.85
90 / 90 90
0.054 9 / 90 9
-0.0169 4.8 / 4
0.053 45 45 45
90 0 1,
90 45 0 01 90 45 0 1 90 45 0 0
2 02 02

Figure 6-7. Computed tailoring of laminate orientations for two plies of bi-directional carbon
fiber in a PR wing, a = 12.

Batten Construction

Computed lift slope and efficiency of a BR wing at 120 angle of attack is given in Figure 6-

8 as a function of the number of layers in each batten. The thickness of each batten can be varied

independently, though the number of layers is limited to three, resulting in 27 possible designs.

As before, the membrane skin is slack, and a two-layer plain weave at 450 makes up the

remainder of the wing. The normalized out-of-plane displacement and differential pressure

coefficients along the chord-station x/c = 0.5 for 4 selected designs is given in Figure 6-9.

As expected, the wing with three one-layer battens has the most adaptive washout, which

provides the shallowest lift slope, but also the best lift-to-drag ratio. Additional plies, regardless

of which batten they are added to, monotonically decreases the efficiency. The same technique

can be used to increase CL,, except for combinations of stiff battens towards the wing root and a









thin outer batten at the wing tip (331 and 332, for example, where the battens are numbered from

inner to outer and the integers indicate the number of layers), which can cause the lift slope to

decrease from these peaks. Design 223 shows the steepest lift slope of the wings in Figure 6-8.

0.048 5.26


5.24
0.047

5.22
0.046
5.2

0.045 5.18

11I 11111 1222222222333333333 111111111222222222333333333
111222333111222333111222333 111222333111222333111222333
123123123123123123123123123 123123123123123123123123123
# of batten layers # of batten layers

Figure 6-8. Computed tailoring of batten construction in a BR wing, a = 12.

The undulations in the differential pressure due to local membrane inflation from in

between the battens are clearly visible in Figure 6-9. Low pressure regions on the upper surface

of the membrane skin and high pressure on the lower surface (which slightly re-directs the flow

towards the battens [154]) results in the four high-lift lobes over the inflated membrane skin.

This inflation can be controlled in obvious ways: wing displacement is larger for design 111 than

design 333, throughout the entire length of the wing section in Figure 6-9. The wing

deformation of design 123 is comparable to design 111 towards the root of the wing, but tapers

off towards the wingtip, where it resembles design 333. In some cases, redistributing the batten

sizes causes a trade-off between local inflation and spanwise bending. The displacement of

design 123 is less than design 111 between 24% and 60% of the semispan, but the local inflation

between the stiffer battens is higher, causing greater redistribution of the flow and high

differential pressures. A similar comparison can be made between designs 321and 111 towards










the wingtip. Overall changes in the aerodynamics due to batten tailoring are relatively small

however, with 5% possible variability in CL, and 1.5% in L/D. Though not shown, the static

stability of the BR wing can be varied by 10% with batten tailoring.

0.025 ....


02 0.65
0.02 *y L
(16
0.015 0.55-
"0 I" i 0.5
S111 ---------..........- 123
/ ---------- 123 0.45 321
321 33 ^ ---333
0.005 321 .............. 333
0 ................
. . . . 0.35
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
2y/b 2y/b

Figure 6-9. Computed normalized out-of-plane displacement (left) and differential pressure
(right) at x/c = 0.5, for various BR designs, a = 12.

Full Factorial Designed Experiment

Of the structural sizing/strength parameters discussed above, spanwise membrane pre-

tension, chordwise pre-tension, the number of layers of bi-directional plain weave carbon fiber,

and the wing type (BR, PR, rigid) are considered in a designed experiment. As stated above, the

aeroelasticity of the MAV wing is dominated by the membrane inflation, and the laminate

stacking is a secondary effect (though Cma of a BR wing is moderately sensitive to fiber

orientation and batten thickness). The number of layers of plain weave carbon fiber, though not

explicitly discussed above, is included due to interesting discrepancies between laminate

tailoring with one layer (Figure 6-5) and tailoring with two layers (Figure 6-6 and Figure 6-7).

For this study, the number of layers in each batten is fixed at one, and all plain weaves are

oriented at 450 to the chordline.

A three-level, three-variable full factorial designed experiment is implemented for each









membrane wing. Only 1, 2, or 3 layers of carbon fiber are permitted, while pre-tension resultant

(chordwise or spanwise) is restricted to 0, 10 or 20 N/m. More than 3 layers is excessively stiff

and heavy; 1 layer may not be able to withstand flight loads or survive a crash. The upper cap on

pre-tension is, as discussed above, meant to prevent batten buckling. Each full factorial design

array requires 27 simulations for each membrane wing, a number which must be doubled to

obtain finite difference approximations of the lift and moment derivatives in angle of attack.

Including the two data points needed for the rigid wing, 110 computationally expensive

simulations are required. While a full factorial matrix is not the most economical choice for a

designed experiment (a central-composite design is an adequate fraction of the full factorial, for

example [157]), the uniform sampling will provide the best qualitative insight into the membrane

wing tailoring.

All 27 data points for the BR wing are given in Figure 6-10, in terms of CLa, Cma, and L/D.

The data points for a two-layer laminate are identical to those seen in Figure 6-1. The computed

normalized out-of-plane displacement of the BR wing with a slack membrane can be seen in

Figure 6-11, with one, two, and three layers of plain weave carbon fiber. These designs are the

three found on the z-axis of Figure 6-10. The variability in the aerodynamics with the three

design variables is substantial: 22% in the lift slope, 54% in the pitching moment slope, and 16%

in efficiency. As above, increasing the pre-tension in the BR wing increases CL,, decreases Cma,

and decreases L/D, though the trend is not monotonic. No prevalent trend exists for the number

of plain weave layers, demonstrating strong interactions with the membrane pre-tension.

For a slack BR membrane wing (Nx = Ny = 0), increasing the number of plain weave layers

significantly decreases the deformation of the wing tip and the adaptive washout at the trailing

edge. As seen in Figure 6-11, a three-layer BR wing is mostly characterized my local membrane








inflation among the battens. This demonstrates the degree to which the adaptive washout at the

trailing edge depends on the bending/twisting of the leading edge laminate (where the forces are

very high, seen in Figure 5-18), and also explains why tailoring the thickness of the battens,

discussed above, has only a minor effect upon the aerodynamics.

CLa ma L/D


0.052 -001 5.5
0.048
0.048410 -0.015 10 5
20 N 20 N 20 N
10 0 x 10 0 x 10 0 Nx
N N N
y y y
1 layer 2 layers 3 layers
Figure 6-10. Computed full factorial design of a BR wing, a = 12.


S


0 0.01 0.02 0.03 0.04 0.05
Figure 6-11. Computed BR wing deformation (w/c) with one layer of plain weave (left), two
layers (center), and three layers (right), a = 12.
This inability of the slack membrane wing to alleviate the flight loads decreases the

efficiency, but surprisingly, has little effect on the stability derivatives. One possible reason for

this is the negative deformations at the trailing edge of the three-layer MAV wing. The stiffer

wing adheres closely to the original, rigid wing shape, which contains reflex (negative camber) at

the trailing edge. The negative forces in this area push the membrane downward, increasing the

wing camber. Increasing the stiffness of the plain weave laminate may convert the BR wing

from a structure with adaptive washout to one with progressive de-cambering, leaving the


rTF









longitudinal stability derivatives relatively unchanged.

Greater variation with laminate thickness is seen for non-zero pre-tensions, particularly

when Nx = 20 N/m and Ny = 10 N/m. If a single layer of carbon fiber is used, this data point

represents the minimum lift slope. Like the double-layered laminates studied above, the BR

wing removes the camber due to membrane inflation (and thus the lift) at the leading edge with

high chordwise stiffness, and allows for adaptive washout with low spanwise stiffness

perpendicular to the battens. Such a design has biological inspiration: the bone-reinforced

membrane skins of pterosaurs [101] and bats [102] both have larger chordwise stiffness. For low

levels of pre-tension, decreasing the number of plain weave layers increases the efficiency; if the

membrane is highly-tensioned, the opposite is true. The L/D objective function is optimized

with a one-layer slack membrane BR wing. It can also be seen in Figure 6-10 that for high levels

of membrane pre-tension, there is little computed difference between 2 and 3 layer laminates.

Similar data is given in Figure 6-12 and Figure 6-13, for a PR wing. The single-layer PR

wing exhibits a substantial amount of adaptive washout, owning to deflection of the weak carbon

fiber perimeter. Two and three-layer laminates remove this feature completely, forcing the wing

into a pure aerodynamic twist. Regardless of the load alleviation along the trailing edge, the

steepest lift and moment curves are found with single-layer laminates, as the weak carbon fiber

reinforcement intensifies the cambering of the membrane wing. The two-dimensional equivalent

to this case is a sailing with the trailing edge attached to a flexible support. Well-known

solutions to this problem indicate that increasing the flexibility of the support improves the static

stability [67], a trend re-iterated in Figure 6-12.

None of the PR aerodynamic metrics or the displacement contours show a substantial

difference between two and three layer laminates. For the thicker laminates, increasing the










spanwise pre-tension provides steeper lift and pitching moment curves; the system has a low

sensitivity to chordwise pre-tension. This may be due to the membrane skin's shape: its chord is

much greater than its span, as discussed above.

CL Cna L/D



5
0.056 20 -0.016 20 20
0.054 4.8
0.052 10 -0.018 10 4.6 10
20 10 0 0 Nx 20 10 0 0 Nx 20 10 0 0 x
N N N
y y Y
S1 layer -A-- 2 layers 3 layers

Figure 6-12. Computed full factorial design of a PR wing, a = 12.








0 0.01 0.02 0.03 0.04 0.05 0.06

Figure 6-13. Computed PR wing deformation (w/c) with one layer of plain weave (left), two
layers (center), and three layers (right), a = 12.

For one-layer laminates, no clear trend between CL,, Cma, and pre-tension (chordwise or

spanwise) emerges. Whereas the thicker laminates prefer a slack membrane wing to optimize

longitudinal static stability, the one-layer wing optimizes this metric when 10 N/m is applied in

the span direction. The reflex in the airfoil shape may again be the reason for this. The mild

amount of spanwise pre-tension enforces the intended reflex in the membrane skin, and the

downward forces depress the membrane skin (seen in Figure 6-4). Slight increases in angle of

attack increases the inflation camber towards the leading edge, but decreases the reflex at the

trailing edge, resulting in a significant restoring moment. The efficiency of thick-laminate PR









wings is equally degraded by chordwise and spanwise pre-tensions. The opposite is true for

single-layers, where L/D can actually be improved with less tension.

The above data is recompiled in Figure 6-14, which plots the performance of the 27 BR

designs, the 27 PR designs, and the rigid wing, in terms of the lift slope and pitching moment

slope as a function of L/D. As seen many times in the above plots, the various objective

functions conflict: tailoring a wing structure for longitudinal static stability may induce a severe

drag penalty, for example. No wing design exists (typically) that will optimize all of the relevant

performance metrics, and compromise designs must be considered. The set of compromise

solutions fall on the design space's Pareto optimal front [156]. A Pareto optimal solution is non-

dominated: no solution exists within the data set that out performs the Pareto optimal solution in

all of the performance metrics.

Three Pareto fronts are given in Figure 6-14. The first details the tradeoff between

maximizing L/D while minimizing CL,. The second gives the tradeoff between maximizing L/D

while maximizing CL,, and the final front is a tradeoff between maximizing L/D while

minimizing Cma. It may be beneficial for a MAV wing to have a very steep lift slope (for

efficient pull-up maneuvers, for example) or very shallow (for gust rejection), so both are

included. All three of these objective functions could be used to compute a common Pareto

front, but visualization of the resulting hypersurface would be difficult. Furthermore,

maximizing CL, and minimizing Cma evolve from similar mechanisms, and seldom conflict.

The overlap between BR wings and PR wings in Figure 6-14 is minimal, with the latter

design typically having higher efficiency and shallow lift and moment slopes. The rigid wing

lies close to the interface between the two membrane wing types, but is not Pareto optimal. The

basic performance tradeoffs are readily visible: peak L/D is 5.49 (a single-layer BR wing with a










slack membrane), a design whose lift slope is 8% higher than the minimum possible lift slope,

18% lower than the maximum possible lift slope, and whose pitching moment slope is 34%

higher than the minimum possible moment slope.

Most of the dominated solutions do not lie far from the Pareto front, indicative of the fact

that all of the objective functions are obtained by integrating the pressure and shear distributions

over the wing. Substantial variations in the CFD state variables can be obtained on a local level

through the use of wing flexibility (Figure 5-21, for example), but integration averages out these

deviations. It can also be seen that two of the three Pareto fronts in Figure 6-14 are non-convex.

As such, techniques which successively optimize a weighted sum of the two objective functions

(convex combination) to fill in the Pareto front will not work; more advanced schemes, such as

elitist-based evolutionary algorithms [156], must be used.

........- ,.-0.01 .... .. ......
8 ritid
0.055 A BR -0.012- a
Do a PR A
A -0.014
0.05 r Pareto front
UA Pareto front 2 U A max L/D, min C
max L/D, max C -0.016- &
a E--0--0 : P0
Pareto front 1 n rigid
0.045 max lD. min C ,- -0.018 a A BR

-0.02
4.6 4.8 5 5.2 5.4 5.6 4.6 4.8 5 5.2 5.4 5.6
L/D L/D

Figure 6-14. Computed design performance and Pareto optimality, a = 12.

Having successfully implemented the designed experiment, the typical next step is to fit

the data with a response surface, a technique used by Sloan et al. [53] and Levin and Shyy [104]

for MAV work. Having verified the validity of the surrogate, it can then be used as a relatively

inexpensive objective function for optimization. Such a method is not used here for several









reasons. First, half of the design variables (wing type and laminate thickness) are discrete, which

as discussed by Torres [3], can cause convergence problems in conventional optimization

algorithms. Second, nonlinear curve fitting is likely required (membrane wing performance

asymptotically approaches that of a rigid wing for increased pre-tension), and the moderate

number of data points (only 9 for each wing type and laminate thickness) won't provide enough

information for an adequate fit. Finally, such a method may result in an optimal pre-stress

resultant of 5.23 N/m, for example. As discussed above, the actual application of pre-tension to

a membrane MAV wing is an inexact science, and such resolution could never be produced in

the laboratory (or more importantly, the field) with any measure of repeatability or accuracy.

A more practical approach is to simply treat the pre-tension as a discrete variable: taut (-20

N/m), moderate (-10 N/m), and slack (0 N/m). Figure 6-10 and Figure 6-12 now represent an

enumeration-type optimization, wherein every possible design is tested. Optimal wing designs in

terms of 7 objective functions (maximum L/D, minimum mass, maximum lift, minimum drag,

minimum pitching moment slope, maximum lift slope and minimum lift slope) are located

among the 55 available data points, and given along the diagonal of the design array in Table 6-

1. Results from the OFAT tests above are not included.

Satisfactory compromise designs are found by first normalizing design performance

between 0 and 1, and then locating a utopia point. This utopia point is a (typically) fictional

design point which would simultaneously optimize both objective functions. In the design trade-

of between L/D and Cma in Figure 6-14, the utopia point is (5.49, -0.0189). An adequate

compromise is the Pareto optimal design which lies closest to the utopia point; these are listed in

the off-diagonal cells in Table 6-1. This method is found to give a better compromise than

optimizing a convex combination of the two objective functions, presumably due to the non-









convexity of the Pareto fronts. Only compromises between 2 objective functions are considered

in this work. The corresponding performance of each design is given in Table 6-2. The value in

each cell is predicated upon the label at the top of each column; the performance of the second

objective function (row-labeled) is found in the cell appropriately located across the diagonal.

Table 6-1. Optimal MAV design array with compromise designs on the off-diagonal, a = 12:
design description is (wing type, Nx, Ny, number of plain weave layers).
max L/D min mass max CL min CD min Cma max CL. min CL.
max L/D BR,0,0,1L BR,0,0,1L PR,10,0,1L BR,0,0,1L BR,20,0,3L PR,20,0,2L BR,0,0,1L
min mass BR,0,0,1L PR,20,20,1L PR,0,0,1L BR,0,0,1L PR,0,10,1L PR,0,10,1L BR,20,10,1L
max CL PR,10,0,1L PR,0,0,1L PR,0,0,1L BR,0,10,1L PR,0,0,1L PR,0,0,1L BR,20,10,1L
min CD BR,0,0,1L BR,0,0,1L BR,0,10,1L BR,0,0,1L BR,20,0,3L BR,10,20,3L BR,10,0,1L
min Cma BR,20,0,3L PR,0,10,1L PR,0,0,1L BR,20,0,3L PR,0,10,1L PR,0,10,1L BR,20,0,3L
max CL. PR,20,0,2L PR,0,10,1L PR,0,0,1L BR,10,20,3L PR,0,10,1L PR,0,10,1L BR,0,20,3L
min CLa BR,0,0,1L BR,20,10,1L BR,20,10,1L BR,10,0,1L BR,20,0,3L BR,0,20,3L BR, 20,10,1L

Table 6-2. Optimal MAV design performance array, a = 12: off-diagonal compromise design
performance is predicated by column metrics, not rows.
max L/D min mass (g) max CL min CD min Cma max CL minCLl
max L/D 5.49 4.36 0.780 0.112 -0.015 0.054 0.047
min mass 5.49 4.10 0.817 0.112 -0.019 0.057 0.043
max CL 4.84 4.18 0.817 0.145 -0.018 0.056 0.043
min CD 5.49 4.36 0.716 0.112 -0.014 0.052 0.045
min Cma 5.05 4.16 0.817 0.134 -0.019 0.057 0.049
max CL. 4.90 4.16 0.817 0.141 -0.019 0.057 0.050
min CLa 5.49 4.31 0.673 0.119 -0.015 0.049 0.043

For reference purposes, the design performance of the rigid wing (at 120 angle of attack)

is: L/D = 4.908, mass = 6.36 grams, CL = 0.6947, CD = 0.1415, Cm, = -0.0147, and CL, = 0.0507.

As above, at no point does the rigid wing represent an optimum design (compromise or

otherwise). The compromise between minimizing the lift slope, and maximizing the lift slope is

identified by located the design closest to the normalized CL, of 0.5. This is found by a BR wing

design with peak pre-tension normal to the battens to limit adaptive washout, but no pre-tension

in the chordwise direction to allow for camber and lift via inflation. Both BR and PR wings are

equally-represented throughout the design array, with the exception of designs requiring load

alleviation: all compromises involving drag or lift slope minimization utilize a BR wing. The









majority of the optimal designs use a single layer of plain weave carbon fiber to take the most

advantage of wing flexibility. A single layer slack BR wing can minimize drag through

streamlining [63], for example, as a significant portion of the wing is deformed (Figure 6-11). A

few designs use 3 layers; only one design uses 2 layers.

A few compromise wing designs coincide with the utopia point: a one-layer BR wing with

a slack membrane maximizes L/D and minimizes the drag. A one-layer PR wing with no pre-

tension in the chordwise direction and 10 N/m in the spanwise direction provides the steepest lift

slope and pitching moment slope. Most compromise designs improve both objective functions,

compared to the rigid wing, but the system particularly struggles to maximize both L/D and lift

(above results indicate that efficiency improvements are driven by drag reduction), and to

maximize lift and minimize the lift slope.

The conflictive nature of the objective functions means that looking at designs that strike a

reasonable compromise between three or more aerodynamic metrics is of minor usefulness. It

should be noted however, that the design that lies closest to the utopia point of all 7 objectives

shown in Table 6-1 is a 2-layer BR wing with a slack membrane in the chordwise direction, and

10 N/m of pre-tension spanwise, similar to the design that lies closest to the normalized CL, of

0.5, as discussed above. Finally, mass minimization is obviously afforded with a single layer of

plain weave carbon fiber: membrane pre-tension then provides moderate and insignificant

deviations from this value, by changing the amount of latex used over the MAV wing.

Experimental Validation of Optimal Design Performance

The design results from the single-objective optimization studies (the diagonal of Table 6-

1, with the exception of the minimum mass design) are fabricated and tested in the wind tunnel.

Only loads are measured through the a-sweep, for comparison with the experimental data from

the three baseline wings designs in Chapter 5. As discussed above, each of these wing designs









utilize a single layer of plain weave carbon fiber. Two layers are typically used for MAVs of

this scale. Despite the extremely compliant nature of the wings (which is precisely why they

were located as optimal), all designs are able to withstand flight loads in the wind tunnel without

buckling. Whether they can withstand maneuver loads or strong gusts is still unknown however,

as is their ability to endure a flight crash without breaking.

Some wing designs display substantial leading edge vibration at very low and negative

angles of attack (presumably due to the vortex shedding from the separation bubble seen in

Figure 5-17), though deformation is observed to be quasi-static above 3 and prior to stall. The

required pre-stress resultants are converted into pre-strains using Hooke's law, and applied to a

square of latex rubber by uniformly stretching each side. VIC is used to confirm the pre-strain

levels, with spatial coefficients of variation between 10 and 20%, similar to data given in Figure

4-6 and by Stanford et al. [43].

Results for lift-related optima are given in Figure 6-15. The design that maximizes CL

(PR,0,0,1L) produces more lift than the baseline PR wing up to 100 angle of attack, though

within the error bars (not shown, but on the order of 5%). Above this angle the wing shows a

premature stall: CL, max is much lower than measured from the baseline PR wing, bearing closer

similarities to the rigid wing. The vibration and buffeting typically seen over MAV wings

towards stall is obviously magnified for these compliant designs; the coupling between the

shedding and the wing vibration may contribute to the loss of lift, as demonstrated in the work of

Lian and Shyy [8]. The upward deformation of the single layer trailing edge perimeter is

substantial (as seen in Figure 6-13), and the resulting adaptive washout may also play a role.

Similar results are seen for the wing design that maximizes CL, (PR,0,10,1L), though in this case

the lift slope is nearly identical to that measured from the baseline PR wing up to 100, after which









premature stall occurs. This benign stall behavior is not necessarily detrimental [49], though

unintended by the numerical model, caused by optimizing at a single angle of attack with a

steady aeroelastic solver.

1.5 .
1. max CL
--- min CLt

1 max CLa



0.5
0.5 -e--- rigid
-y BR
aPR

0 10 20 30


Figure 6-15. Experimentally measured design optimality over baseline lift.

The optimizer is considerably more successful when minimizing CL, with design

(BR,20,10,1L), as seen in Figure 6-15. The BR wings used in these tests are qualitatively

observed to have smaller vibration amplitudes, compared to the PR wings, at very low and very

high angles of attack. At low angles of attack, the lift of the optimal design is smaller than both

the baseline rigid and BR wings, though the lift slope is comparable. For moderate angles, no

significant differences are evident. After 100 however, the optimal design shows a clear drop in

lift slope, a very flat stalling region, and stalling angle delayed by 30 over the baseline designs.

CL, max is measured to be 9% less than that measured for the baseline BR wing.

Experimental validation results for the wing design minimizing the pitching moment slope

(PR,0,10,1L) is given in Figure 6-16. As before, performance of the baseline PR wing and the

optimal design are comparable up to 130. Above this angle, and through the stalling region, the

optimal design has a steeper slope than the baseline PR wing. At these angles, the nose-down

pitching moment is stronger than that seen in the baseline BR and rigid wings, but the slope is










similar. This is largely due to the linear pitching moment behavior previously noted on the PR

wings, possibly due to membrane inflation interference with the tip vortices [14]. Despite the

measured improvements over the baseline PR wing, the data indicates that longitudinal control

beyond stall (- 280) may not be possible [27]. Interestingly, the same wing design theoretically

minimizes the moment slope and maximizes the lift slope, but only the former metric is

considerably improved over the baseline.




-0.1

-0.2

-0.3 --- rigid
6-- BR
-0.4 PR
-*- min C
-0.5 ma
0 10 20 30
a

Figure 6-16. Experimentally measured design optimality over baseline pitching moments.

Similar validation results are given in Figure 6-17 and Figure 6-18, for the minimization of

drag and maximization of L/D. Both metrics are optimized by wing design (BR,0,0, 1L). The

drag is consistently lower than the three baseline designs up to 200. Accurate drag data for micro

air vehicles at low speeds is very difficult to measure, largely due to resolution issues in the sting

balance [34]. Questionable data typically manifests itself through atypically low drag.

Regardless, the veracity of the data from the optimal wing in Figure 6-17 may be confirmed by

the identical results at the bottom of the drag bucket with the rigid wing, where deformation is

very small. The data also compares very well with computed results. Unlike the baseline BR

wing, the optimal design has less drag at a given angle of attack and at a given value of lift (the

latter of which is visible in the drag polar, which is not shown). Past 200, the optimal design










shows more of a drag penalty than the baseline BR wing, which may also be attributed to larger

vibration amplitudes in the single-layer wing.

0.7

0.6 rigid
------- BR -
0.5 BR









0 10 20 30
-a-- PR





0.4
0.3

0.2

0.1


















0-- max --D--
0 10 20 30

Figure 6-18. Experimentally measured design optimality over baseline dragefficiency.
7

6

5

n 4

3 rigid

2 (j& BR

-*- max L/D

0 10 20 30


Figure 6-18. Experimentally measured design optimality over baseline efficiency.

The results for optimal efficiency (Figure 6-18) show substantial improvements over the

three baseline designs for a range of moderate angles: 8 18. The optimization is only

conducted at 120 angle of attack; whereas the previously considered optimal designs can be

reasonably considered ideal throughout most of the a-sweep (up to stall), the conflictive nature

of the lift-to-drag ratio is more complex. This can be seen in the numerical data of Figure 5-15,

where the baseline BR, PR, and rigid wings all have the highest L/D for different lift values. It is









expected that optimizing at different angles of attack will produce radically different optimal L/D

designs, but similar results may be retained for the remaining objectives.

Of the six aerodynamic objectives considered in this section, wind tunnel testing indicates

that two are unmistakably superior to the baselines over a large range of angles of attack

(minimum drag and maximum efficiency), and two have similar responses to one or more of the

baseline designs for small and moderate angles but are clearly superior for higher angles of

attack (minimum lift and pitching moment slopes). One objective (maximum lift) is slightly

better at moderate angles (though not beyond the measured uncertainty), but decidedly inferior

during stall, while another objective (maximum lift slope) is identical to the baseline for

moderate angles, and again inferior during stall.

With the exception of these latter two studies, this wind tunnel validation confirms the use

of numerical aeroelastic tailoring for realizable improvements to actual MAV wings. This is not

to indicate that the latter two studies have failed: the computed performance of the tailored wings

is not always significantly better than the baseline, and may be blurred by experimental errors.

The experimental data of these two designs is not significantly better than the baseline designs,

but not measurably worse either (for moderate angles).









CHAPTER 7
AEROELASTIC TOPOLOGY OPTIMIZATION

The conceptual design of a wing skeleton essentially represents an aeroelastic topology

optimization problem. Conventional topology optimization is typically concerned with locating

the holes within a loaded homogenous structure, by minimizing the compliance [16]. This work

details the location of holes within a carbon fiber wing shell, holes which will then be covered

with a thin, taut, rubber membrane skin. In other words, the wing will be discretized into a series

of panels, wherein each panel can be a carbon fiber laminated shell or an extensible latex rubber

skin. Rather than compliance, a series of aerodynamic objective functions can be considered,

including L/D, CL, CD, CL., Cma, etc.

While the two wing topologies discussed in the preceding section (batten- and perimeter-

reinforced wings) have been shown to be effective at load alleviation via streamlining and load

augmentation via cambering, respectively, both designs have deficiencies. The BR wing

experiences membrane inflation from in-between the battens towards the leading edge (Figure 5-

1), cambering the wing and contradicting the load alleviating effects of the adaptive washout at

the trailing edge. Furthermore, the unconstrained trailing edge is only moderately effective at

adaptive geometric twist, as the forces in this region are very small (Figure 5-20). If re-curve is

built into the wing section, the forces in this area may push the trailing edge downward, actually

increasing the incidence, and thus the loads.

The PR wing, being a simpler design, is more effective in its intended purpose (adaptive

cambering for increased lift and static stability), but the drastic changes in wing geometry at the

carbon fiber/membrane interfaces towards the leading and trailing edges of the membrane skin

are aerodynamically inefficient. Large membrane inflations are also seen to lead to potentially

unacceptable drag penalties as well. All of these deficiencies can be remedied via the tailoring









studies considered above, but the greater generality of an aeroelastic topology scheme (due to the

larger number of variables) would suggest better potential improvements in aerodynamic

performance. Furthermore, such an undertaking can potentially be followed by an aeroelastic

tailoring study of the optimal topology for further improvements, as discussed by Krog et al.

[115]: topology optimization to locate a good design, followed by sizing and shape optimization.

A flexible MAV wing topological optimization procedure has some precedence in early

micro air vehicle work by Ifju et al. [10], with an array of successfully flight tested designs

shown in Figure 7-1. Each of these designs consists of a laminated leading edge, wing tip, and

wing root; a series of thin strips of carbon fiber are imbedded within the concomitant membrane

skin. Both the BR and PR wings are present, along with slight variations upon those themes.

Ifju et al. [10] qualitatively ranks these wing structures based upon observations in the field and

pilot-reported handling qualities: a crude trial and error process led to the batten-reinforced

design as a viable candidate for MAV flight.




















Figure 7-1. Wing topologies flight tested by Ifju et al. [10].

Several challenges are associated with the optimization procedure considered here. First, a









fairly fine structural grid is needed to resolve topologies on the order of those seen in Figure 7-1.

The fine grid will, of course, increase the computational cost associated with solving the set of

FEA equations, as the number of variables in the optimization algorithm is proportional to the

number of finite elements. The wing is discretized into a set of quadrilaterals, which represent

the density variables: 0 or 1. These quadrilaterals are used as panels for the aerodynamic solver,

and broken into two triangles for the finite element solver, as shown in Figure 7-2. As in Figure

7-1, the wing topology at the root, leading edge, and wing tip is fixed as carbon fiber, to maintain

some semblance of an aerodynamic shape capable of sustaining lift. The wing topology in the

figure is randomly distributed.

/ desink domain










Figure 7-2. Sample wing topology (left), aerodynamic mesh (center), and structural mesh
(right).

Further complications are associated with the fact that these variables are binary integers: 1

if the element is a carbon fiber ply, 0 if the element is latex membrane. Several binary

optimization techniques (genetic algorithms [109], for example) are impractical for the current

problem, due to the large number of variables, but also due to the extremely large computational

cost associated with each aeroelastic function evaluation. A fairly standard technique for

topology optimization problems classifies the density of each element as continuous, rather than

binary [16]. Intermediate densities can then be penalized (implicitly or otherwise) to push the

design towards a pure carbon fiber/membrane distribution, with no "porous" material.









The sensitivity of each element's density variable upon the wing's aerodynamic

performance is required for this gradient-based optimization scheme. As before, the large

number of variables and the expensive function evaluations preclude the use of simple finite

difference schemes for computation of gradients. An adjoint sensitivity analysis of the coupled

aeroelastic system is thus required, as the number of design variables is much larger than the

number of objectives/constraints [110]. Further complications arise from the fact that second

derivatives are also required: important MAV aerodynamic performance metrics such as the

slope of the lift curve, for example, are sensitivity derivatives that depend upon the

characteristics of the aeroelastic system as well.

This chapter provides a computational framework for computing the adjoint aeroelastic

sensitivities of a coupled aeroelastic system, as well as interpolation schemes between carbon

fiber and membrane finite elements and methods for penalizing intermediate densities. The

dependency of the computed optimal topology upon mesh density, angle of attack, initial

topology, and objective function are given, as well as the resulting deformation and pressure

distributions. The wing designs created via aeroelastic topology optimization demonstrate a

clear superiority over the baseline BR and PR designs discussed above in terms of load

alleviation (former) and augmentation (latter), advantages which are further expounded through

wind tunnel testing. Multi-objective topology optimization is discussed as well, with the

evolution of the optimal wing topology as one travels along the Pareto optimal front.

Computational Framework

Material Interpolation

Topology optimization often minimizes the compliance of a structure under static loads,

with an equality constraint upon the volume. If the density of each element is allowed to vary

continuously, an implicit penalty upon intermediate densities (to push the final structure to a 0-1









material distribution) can be achieved through a nonlinear power law interpolation. This

technique is known as the solid isotropic material with penalization method, or SIMP [105].

For the two-material wing considered above (membrane or carbon fiber), the stiffness

matrix Ke of each finite element in Figure 7-2 can be computed as:

Ke= (K.(1-P)-Km).XP+Km+P.K (7-1)

where Kp and Km are the plate and membrane elements, respectively (the latter with zeros placed

within rows and columns corresponding to bending degrees of freedom). 3 is a small number

used to prevent singularity in the pure membrane element (due to the bending degrees of

freedom), and Xe is the density of the element, varying from 0 (membrane) to 1 (carbon fiber). p

is the nonlinear penalization power (typically greater than 3).

A common criticism of this power law approach is that intermediate densities do not

actually exist. This is a particular problem for the current application, where each element is

either carbon fiber or membrane rubber. The physics of these two elements is completely

different, as the carbon fiber is inextensible yet has resistance to bending and twisting, while the

opposite is true for the latex. An equal combination of these two (equivalent to stating that the

density within an element is 0.5), while computationally conceivable, is not physically possible.

The wing topology will not represent a real structure until the density of each element is pushed

to 1 (carbon fiber) or 0 (membrane).

The power law's effectiveness as an implicit penalty is predicated upon a volume

constraint: intermediate densities are unfavorable, as their stiffness is small compared to their

volume [16]. No such volume constraint is utilized here, due to an uncertainty upon what this

value should be. Furthermore, for aeronautical applications it is typically desired to minimize

the mass of the wing itself, as discussed by Maute et al. [118]. Regardless, the nonlinear power









law of SIMP is still useful for the current application, as demonstrated in Figure 7-3. Both linear

and nonlinear material interpolations are given for the lift computation, and the wing topology is

altered uniformly.

For the linear interpolation (i.e., without SIMP), the aeroelastic response is a weak function

of the density until X becomes very small (-0.001), when the system experiences a very sharp

change as X is further decreased to 0. This is a result of the large stiffness imbalance between

the carbon fiber laminates and the membrane skin, and the fact that lift is a direct function of the

wing's compliance (the inverse of the weighted sum of the two disparate stiffness matrices in Eq.

(7-1)). The inclusion of a nonlinear penalization power (p = 5), spreads the response evenly

between 0 and 1. Aeroelastic topology optimization with linear material interpolation

experiences convergence difficulties, as the gradient-based technique struggles with the nearly-

disjointed design space; a penalization power of 5 is utilized for the remainder of this work.


0.63

0.62 p = 5

0.61

0.6

0.59 T

0 0.2 0.4 0.6 0.8 1
X

Figure 7-3. Effect of linear and nonlinear material interpolation upon lift.

The results from Figure 7-3 suggest a number of other potential difficulties with an

aeroelastic topology optimization scheme. First, the sensitivity of the aeroelastic response to

element density is zero for a pure membrane wing (X = 0), as can be inferred from Eq. (7-1). As

such, using a pure membrane wing as an initial guess for optimization will not work, as the









design won't change. Secondly, two local optima exist in the design space of Figure 7-3, which

may prevent the gradient-based optimizer from converging to a 0-1 material distribution. To

counteract this problem, an explicit penalty on intermediate densities is added to the objective

function, as discussed by Chen and Wu [158]:

N,
R. sin(X, .7r) (7-2)
1=i1

where R is a penalty parameter appropriately sized so as not to overwhelm the aerodynamic

performance of the wing topology. This penalty is only added when and if the aeroelastic

optimizer has converged upon a design with intermediate densities, as will be discussed below.

Aeroelastic Solver

Due to the large number of expected function evaluations (- 200) needed to converge upon

an optimal wing topology, and the required aeroelastic sensitivities (computed with an adjoint

method), a lower-fidelity aeroelastic model (compared to that utilized in Chapters 5 and 6) must

be used for the current application. An inviscid vortex lattice method (Eq. (4-11)) is coupled to a

linear orthotropic plate model and a linear stress stiffening membrane model (Eq. (4-4)). The

latter module is perfectly valid in predicting membrane inflation as long as the state of pre-stress

is sufficiently large, as seen in Figure 4-5. Furthermore, in-plane stretching of the laminate is

ignored; only out-of-plane displacements (as well as in-plane rotations in the laminate) are

computed over the entire wing.

The vortex lattice method is reasonably accurate as well, despite the overwhelming

presence of viscous effects within the flow. As seen in Figure 4-11, the lift slope is consistently

under-predicted due to an inability to model the large tip vortices [3], and the drag is under-

predicted at low and high angles of attack due to separation of the laminar boundary layer [4].

Aeroelastic coupling is facilitated by considering the system as defined by a three field









response vector r:


r =(u z F ') (7-3)

where u is the solution to the system of finite element equations (composed of both

displacements and rotations) at each free node, z is the shape of the flexible wing, and r is the

vector of unknown horseshoe vortex circulations. The coupled system of equations G(r) is then:

K-u-Q-T
G(r)= z-zo-P.u =0 (7-4)
SC--L

The first row of G is the finite element analysis: K is the stiffness matrix assembled from

the elemental matrices in Eq. (7-1), and appropriately reduced based upon fixed boundary

conditions along the wing root. Q is an interpolation matrix that converts the circulation of each

horseshoe vortex into a pressure, and subsequently into the transverse force at each free node.

The second row of G is a simple grid regeneration analysis: Zo is the original (rigid) wing shape,

and P is a second interpolation matrix that converts the finite element state vector into

displacements at each free and fixed node along the wing. The third row of G is the vortex

lattice method. C is an influence matrix depending solely on the wing geometry (computed

through the combination of Eqs. (4-11) and (4-12)), and L is a source vector depending on the

wing's outward normal vectors, the angle of attack, and the free stream velocity. Convergence

of this system can typically be obtained within 25 iterations, and is defined when the logarithmic

error in the wing's lift coefficient is less than -5.

One potential shortcoming of this aeroelastic model can be seen in Figure 7-3, where the

computed lift of a wing with no carbon fiber in the design domain (X = 0) is larger than the lift

generated by the rigid wing (X = 1). This is due to a combination of membrane cambering

towards the leading edge, and a depression of the trailing edge reflex region. In reality, however,









the combination of a poorly-constrained trailing edge and unsteady vortex shedding will lead to a

large-amplitude flapping vibration, similar to that discussed by Argentina and Mahadevan [62].

Wind tunnel testing of this wing is given in Figure 7-4 at 13 m/s; the critical speed of flapping

vibration is approximately 3 m/s.

6 - - -
1.2
5-

0.8-
3

0.4

0.2OT
0

0 0.2 0.4 0.6 0 0.5 1
CD CL

Figure 7-4. Measured loads of an inadequately reinforced membrane wing, U. = 13 m/s.

As expected, the measured lift of the membrane wing is significantly less than that

measured from the rigid wing in the wind tunnel: the poorly-supported wing cannot sustain the

flight loads, while the large amplitude vibrations levy a substantial drag penalty. Even a mild

amount of trailing edge reinforcement (such as that seen in the upper left of Figure 7-1) will

prevent this behavior, but formulating a constraint that will push the aeroelastic topology

optimizer away from wing designs with a poorly-reinforced trailing edge is difficult, and is not

included. This section only serves to highlight one significant shortcoming of the aeroelastic

model used here, and to diminish the perceived optimality of certain wing topologies.

Adjoint Sensitivity Analysis

As the number of variables in the aeroelastic system (essentially the density of each

element) will always outnumber the number of constraints and objective functions, a sensitivity









analysis can be most effectively carried out with an adjoint analysis. The sought-after total

derivative of the objective function with respect to each density variable is given through the

chain rule:

dg ag +gT dr
+ (7-5)
dX X Or dX

where g is the objective function (a scalar for the single-objective optimization scheme

considered here; multi-objective optimization will be discussed below) and r is the aeroelastic

state vector discussed above. The term 8g/8X is the explicit portion of the derivative, while the

latter term is the implicit portion through dependence on the aeroelastic system [159]. Only

aerodynamic objective functions are considered in this work: the explicit portion is then zero,

unless the intermediate density penalty of Eq. (7-2) is included.

The derivative of the aeroelastic state vector with respect to the element densities is found

by differentiating the coupled system of Eq. (7-4):

dG(X, r) +G dr
0dGX -> -+A 0 (7-6)
dX AX dX

where A is the Jacobian of the aeroelastic system, defined by:

dG
A (7-7)
Or

Combining Eqs. 7-5 and 7-6 leaves:

dg dg dg _G
dg g g.A'.aG (7-8)
dX AX Or 8X

Using the adjoint, rather than the direct method to solve Eq. (7-8), the adjoint vector is:

a = ATg (7-9)
Or

The system of equations for the adjoint vector does not contain the density of each element









(X), and only needs to be solved once. For the aeroelastic system considered above, the terms

that make up the adjoint vector are:



A = -P I 0 (7-10)
0 dC/dz Fr- OL/az C

gr= 0 0 S} (7-11)
Or

where S is the derivative of the aerodynamic objective function with respect to the vector of

horseshoe vortex circulations. For metrics such as lift and pitching moment, g = ST-T, though

more complex expressions exist for drag. The sensitivities can then be computed as:

dg Bg_ dG
a (7-12)
dX 8X dX

Only the finite element analysis of the aeroelastic system contains the element densities, and so

this final term can be computed as:

OK
--u
8X
dG
dG 0 (7-13)
dX
0


Of all of the terms needed to undertake the adjoint sensitivity analysis, only the derivative

of the vortex lattice influence matrix C with respect to the wing shape z (a three-dimensional

tensor) is computationally intensive, and represents the majority of the cost associated with the

gradient calculations at each iteration. In order to solve the linear system of Eq. (7-9), a

staggered approach is adapted, rather than solving the entire system of (un-symmetric sparse)

equations as a whole, as discussed by Maute et al. [110]. Each sub-problem is solved with the

same algorithm used in the aeroelastic solver (direct sparse solver for the finite element









equations, and an iterative Gauss-Seidel solver for the vortex lattice equations), and as such, the

computational cost and number of iterations needed for convergence is approximately equal

between the aeroelastic solver (Eq. (7-4)) and the adjoint vector solver (Eq. (7-9)).

The second derivative of the objective function is required if aerodynamic derivative

metrics such as CL, and Cma are of interest. Two options are available for this computation. The

first involves a similar analytical approach to the one described above. This would eventually

necessitate the extremely difficult computation of 8A/8r, which is seldom done in practice [160].

Finite differences are used here:

ag2 1 (ag A g
S L- (a + ) A (c) (7-14)
9Xda Aa OX aX

The term ag/Oa can be computed using another finite difference, or with the adjoint method

described above, substituting the angle of attack for the element densities X.

Optimization Procedure

In order to ensure the existence of the optimal wing topologies, a mesh-independent filter

is employed along with the nonlinear power penalization. Such a filter acts as a moving average

of the gradients throughout the membrane wing, and limits the minimum size of the imbedded

carbon fiber structures. Such a tactic should also limit checkerboard patterns (carbon fiber

elements connected just at a corner node). The moving average filter modifies the element

sensitivity of node i based on the surrounding sensitivities within a circular region of radius rmin,

as discussed by Bendsoe and Sigmund [16]:

dg1 dg rmin -dist(i,j) ifdist(i,j)< rmn (7-15)
dg I I H X H = -n) (7 -15)
dX1 new- NxH l JdX 0 otherwise
j-1

As no constraints are included in the optimization (preferring instead the multi-objective









approach described below), an unconstrained Fletcher-Reeves conjugate gradient algorithm

[159] is employed. Step size is kept constant, at a reasonably small value to preserve the fidelity

of the sensitivity analysis. The upper and lower bounds of each design variable (1 and 0) are

preserved by restricting the step size such that no density variable can leave the design space,

forced to lie on the border instead. In order to increase the chances of locating a global optimum

(rather than a local optimum), each optimization is run with three distinct initial designs: Xo = 1

(carbon fiber wing), Xo = 0.5, and Xo = 0.1. A pure membrane wing (Xo = 0) cannot be

considered for the reasons discussed above.

Six objective functions are considered: maximum lift, minimum drag, maximum L/D,

maximum CL,, minimum CL,, and minimum Cma. Flight speed is kept constant at 13 m/s, but

both 3 and 12 angles of attack are considered, with a Aa of 1 for finite differences. Both the

reflex airfoil seen in Figure 5-16 and a singly-curved airfoil are used, though aspect ratio,

planform, and peak camber are unchanged. The stiffness of the carbon fiber laminates is as

computed by Figure 4-3, and the pre-stress of the membrane is fixed in both the chordwise and

spanwise directions at 7 N/m. No correction is applied to the free trailing edge, as such a

computation would render the pre-stress in this location very small, leading to unbounded

behavior of the linear membrane model. The circular radius rmin for the mesh-independent filter

is fixed at 4% of the chord.

Single-Objective Optimization

A typical convergence history of the aeroelastic topology optimizer can be seen in Figure

7-5, for a reflex wing at 3 angle of attack, with a maximum L/D objective function. The initial

guess is an intermediate density of 0.5. Within 4 iterations, the optimizer has removed all of the

carbon fiber adjacent to the root of the wing, with the exception of the region located at three-

quarters of the chord, which corresponds to the inflection point of the reflex airfoil. The material









towards the leading edge and at the wing tip is also removed. Further iterations see topological

changes characterized by intersecting threads of membrane material that grow across the surface,

leaving behind "islands" of carbon fiber. These structures aren't connected to the laminate wing,

but are imbedded within the membrane skin.

4.2

4.15

4.1

4.05
4.-5 / explicit
4 --- penalty
4 IJ -
"3 added

3.95 carbon fiber

3.9 initial design

3.85
*D nimembrane
3.8
0 50 100 150
iteration

Figure 7-5. Convergence history for maximizing L/D, a = 30, reflex wing.

These results indicate two fundamental differences between the designs in Figure 7-1 and

those computed via aeroelastic topology optimization. The first is the presence of"islands";

these designs can be built, but the process is significantly more complicated than with a

monolithic wing skeleton. Such structures could be avoided with a manufacturability

constraint/objective function (such as discussed by Lyu and Saitu [161]), but the logistics of such

a metric (as above, with the trailing edge reinforcement constraint) are difficult to formulate.

Furthermore, the aeroelastic advantages of free-floating laminate structures are significant, as

will be discussed below. A second difference is the fact that the designs of Figure 7-1 are

composed entirely from thin strips of carbon fiber embedded within the membrane, while the









topology optimization is apt to utilize two-dimensional laminate structures.

After 112 iterations in Figure 7-5, the optimization has largely converged (with only

minimal further improvements in L/D), but some material with intermediate densities remains

towards the leading edge of the wing. Many techniques exist for effectively interpreting gray

level topologies [162]; the explicit penalty of Eq. (7-2) is used here. Surprisingly, the L/D sees a

further increase with the addition of this penalty, contrary to the conflict between performance

and 0-1 convergence reported by Chen and Wu [158]. The explicit penalty does not significantly

alter the topology, but merely forces all of the design variables to their limits, as intended.

The final wing skeleton has three trailing edge battens (one of which is connected to a

triangular structure towards the center of the membrane skin), and a fourth batten oriented at 450

to the flow direction. The structure shows some similarities to a wing design in Figure 7-1 (third

row, first column), and appears to be a topological combination of a BR and a PR wing, with

both battens and membrane inflation towards the leading edge. The optimized topology

increases the L/D by 9.5% over the initial design, and (perhaps more relevant, as the initial

intermediate density design does not technically exist) by 10.2% over the rigid wing.

The affect of mesh density is given in Figure 7-6, for a reflex wing at 120 angle of attack,

with L/D maximization as the objective function. The 30x30 grid, for example, indicates that

900 vortex panels (and 1800 finite elements) cover each semi-wing. As the leading edge, root,

and wing tip of each wing are fixed as carbon fiber, 480 density design variables are left for the

topology optimization. One obvious sign of adequate convergence is the efficiency of the rigid

wing, with only a 0.44% difference between that computed on the two finer grids. The three

optimal wing topologies are similar, with three distinct carbon fiber structures imbedded within

the membrane skin: two extend to the trailing edge and the third resides towards the leading









edge. While the 20x20 grid is certainly too coarse to adequately resolve the geometries of

interest, the topology computed on the 30x30 grid is very similar to that computed on the 40x40

grid. The computational cost of each optimization iteration upon the coarser grid is 5 times less

than that seen for a 40x40 grid, and will be used for the remainder of this work.










20x20, L/D= 4.61 30x30, L/D= 4.58 40x40, L/D= 4.55
L/Digid = 4.54 L/Digid = 4.49 L/Dig = 4.47

Figure 7-6. Affect of mesh density upon optimal L/D topology, a = 12, reflex wing.

The affect of the initial starting design is given in Figure 7-7, for a reflex wing at 120 angle

of attack, with drag minimization as the objective function. As mentioned above, Xo = 1 (carbon

fiber wing), Xo = 0.5, and Xo = 0.1 are all considered. The three final optimal topologies are

very different, indicating a large dependency upon the initial guess and no guarantee that a global

optimum has been located. Nevertheless, the indicated improvements in drag are promising,

with a potential 6.7% decrease from the rigid wing. As expected, the denser the initial topology,

the denser the final optimized topology.

All three wing topologies utilize some form of adaptive washout for load and drag

alleviation. The structures must be flexible enough to generate sufficient nose-down rotation of

each wing section, but not so flexible that the membrane areas of the wing will inflate and

camber, increasing the forces. The wing structure in the center of Figure 7-7 (with Xo = 0.5)

strikes the best compromise between the two deformations, and provides the lowest drag. When

Xo = 1, the structure is too stiff, relying upon a membrane hinge between the carbon fiber wing









and root. When Xo = 0.1, the optimizer is unable to fill in enough space with laminates to

prevent membrane inflation. Of the three designs, this is the least tractable from a manufacturing

point of view as well.










Xo = 1, CD = 0.131 Xo = 0.5, CD = 0.125 Xo = 0.1, CD = 0.128
CDrigid = 0.134

Figure 7-7. Affect of initial design upon the optimal CD topology, a = 12, reflex wing.

The dependency of the optimal topology (maximum lift) upon both angle of attack and

airfoil shape are given in Figure 7-8, for both a reflex (left two plots) and a cambered wing (right

two plots). For the wing with trailing edge reflex, the optimal lift design looks similar to that

found in Figure 7-5: trailing edge battens that extend no farther up the wing than the half-chord,

a spanwise member that coincides with the inflection point of the airfoil, and unconstrained

membrane skin towards the leading edge, where the forces are largest. The optimizer has

realized that it can maximize lift by both cambering the wing through inflation at the leading

edge, and forcing the trailing edge battens downward for wash-in.

This latter deformation is only possible due to the reflex (negative camber) in this area,

included to offset the nose-down pitching moment of the remainder of the "flying wing", and

thus allow for removal of a horizontal stabilizer due to size restrictions. Increasing the angle of

attack from 3 to 120 shows no significant difference in the wing topology, slightly increasing the

length of the largest batten. At the lower angle of attack, up to 22% increase in lift is indicated

through topology optimization.
















a = 3, CL = 0.312 = 12, CL = 0.675 a = 3', CL = 0.561 a = 12', CL = 0.947
CL,rigid = 0.254 CL,rigid = 0.604 CL,rigid = 0.487 CL,rigid = 0.842

Figure 7-8. Affect of angle of attack and airfoil upon the optimal CL topology.

For the cambered wing (singly-curved airfoil, right two plots of Figure 7-8), the lift over

the rigid wing is, as expected, much larger than found in the reflex wings, but adequate stability

becomes critical. With the removal of the negatively-cambered portion of the airfoil, most of the

forces generated over this wing will be positive, and the topology optimizer can no longer gain

additional lift via wash-in. Imbedding batten structures in the trailing edge will now result in

washout, surely decreasing the lift. As such, the optimizer produces a trailing edge member that

outlines the planform and connects to the root (similar to the perimeter-reinforced wing designs),

restraining the motion of the trailing edge and inducing an aerodynamic twist.

Unlike the PR wing, this trailing edge reinforcement does not extend continuously from

the root to the tip, instead ending at 65% of the semi-span. This is then followed by a trailing

edge batten that extends into the membrane skin, similar to the designs seen for the reflex wing

in Figure 7-8. Why such a configuration should be preferred over the PR wing design for lift

enhancement will be discussed below. As before, increasing the angle of attack has little bearing

on the optimal topology, again increasing the size of the trailing edge batten. A potential

increase in lift by 15% over the rigid wing is indicated at the lower angle of attack.

Similar results are given in Figure 7-9, with L/D maximization as the topology design

metric. Presumably due to the conflictive nature of the ratio, the wing topology that maximizes

L/D is a strong function of angle of attack. For the reflex wing at lower angles, the optimal









design resembles topologies used above for lift enhancement (Figure 7-8), while at 120 the

design is closer in topology to one with minimum drag (Figure 7-7). Increasing lift is more

important to L/D at lower angles, while decreasing drag becomes key at larger angles. The drag

is very small at low angles of attack (technically zero for this inviscid formulation, if not for the

inclusion of a constant CDo), and insensitive to changes via aeroelasticity.

This concept is less true for the cambered wing (right two plots of Figure 7-9), where

designs at both 3 and 12 angle of attack utilize a structure with trailing edge adaptive washout.

At the lower angle, the topology optimizer leaves a large triangular structure at the trailing edge

(connected to neither the root nor the wing tip), and the leading edge is filled in with carbon

fiber. At the higher angle of attack, four batten-like structures are placed within the membrane

skin, oriented parallel to the flow, one of which connects to the wing tip. Potential

improvements are generally smaller than those seen above, though a 10% increase in L/D is

available for the cambered wing at 12.








a = 3, L/D = 4.18 a = 120, L/D = 4.58 a = 3, L/D = 4.78 a = 120, L/D = 4.41
L/Drigid = 3.78 L/Dri = 4.49 L/D ri = 4.70 L/Drig = 4.01
rigid 3.8L d rigid rigid

Figure 7-9. Affect of angle of attack and airfoil upon the optimal L/D topology.

Wing displacements and pressure distributions are given for select wing designs in Figure

7-10, for a reflex wing at 120 angle of attack. Corresponding data along the spanwise section

2y/b = 0.58 is given in Figure 7-11. As the wing is modeled with no thickness in the vortex

lattice method, distinct upper and lower pressure distributions are not available, only differential

terms. Five topologies are discussed, beginning with a pure carbon fiber wing. Lift-









augmentation designs are represented by a baseline PR wing and the topology optimized for

maximum lift. Lift-alleviation designs are represented by a baseline BR wing and the topology

optimized for minimum lift slope.

CL = 0.604 CL = 0.639 CL = 0.675 CL = 0.595 CL = 0.570


w/c
-0.01


0 0.01
0 0.01


0.02


0.03


ACp : :
-1 -0.5 0 0.5 1 1.5 2

Figure 7-10. Normalized out-of-plane displacements (top) and differential pressure coefficients
(bottom) for baseline and optimal topology designs, a = 12, reflex wing.

The differential pressure distribution over the rigid wing is largely similar to that computed

with the Navier-Stokes solver in Figure 5-18 and Figure 5-19: leading edge suction due to flow

stagnation, pressure recovery (and peak lift) over the camber, and negative forces over the reflex

portion of the wing. As expected, the inviscid solver misses the low-pressure cells at the wingtip

(from the vortex swirling system [3]), and the plateau in the pressure distribution, indicative of a

separation bubble [27]. This aerodynamic loading causes a moderate wash-in of the carbon fiber

wing (0.1), resulting in a computed lift coefficient of 0.604.

Computed deformation of the PR wing is likewise similar to that found above (Figure 5-5),

though the deformations are smaller, within the range of validity of the linear finite element

solver. The sudden changes in wing geometry at the membrane/carbon fiber interfaces lead to


P


~\









sharp downward forces at the leading and trailing edges, the latter of which exacerbates the

effect of the airfoil reflex. Despite this, the membrane inflation increases the camber of the wing

and thus the lift, by 6.5% over the rigid wing.

0.04 1.5

0.03

0.02 1 /
-0.5
0.0 1 0

0 o

-0.01
N_ .... -0.5
0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8
x/c X/C
p~




Figure 7-11. Deformations and pressures along 2y/b = 0.58 for baseline and optimal topology
designs, a = 12, reflex wing.

As discussed above, several disparate deformation mechanisms contribute to the high lift

of the MAV design located by the aeroelastic topology optimizer (middle column, Figure 7-10).

First, the membrane inflation towards the leading edge increases lift via cambering, similar to the

PR wing (the pressure distributions over the two wing structures are identical through x/c =

0.25). The main trailing edge batten structure is then depressed downward along the trailing

edge (due to the reflex) for wash-in, while the forward portion of this structure is pushed

upwards. This structure essentially swivels about the inflection point of the wing's airfoil, a

deformation which is able to further increase the size of the membrane cambering, and is only

possible because the laminate is free-floating within the membrane skin. It can also be seen

(from the left side of Figure 7-11 in particular) that the local bending/twisting of this batten









structure is minimal: the deformation along this structure is largely linear down the wing. The

intersection of this linear trend with the curved inflated membrane shape produces a cusp in the

airfoil. The small radius of curvature forces very large velocities, resulting in the lift spike at

46% of the chord.

This combination of wash-in and cambering leads to a design which out-performs the lift

of the PR wing by 5.6%, but the former effect is troubling. The wash-in essentially removes the

reflex from the airfoil (as does the aerodynamic twist of the PR wing), an attribute originally

added to mitigate the nose-down pitching moment. This fact leads to two important ideas. First,

thorough optimization of a single design metric is ill-advised for micro air vehicle design, as

other aspects of the flight performance will surely degrade. Its inclusion here is only meant to

emphasize the relationship between aeroelastic deformation and flight performance, and show

the capabilities of the topology optimization. A better approach is the multi-objective scheme

discussed below.

Secondly, if the design goal is a single-minded maximization of lift, a reflex airfoil is a

poor choice compared to a singly-curved airfoil, a shape which the topology optimizer strives to

emulate through aeroelastic deformations. Furthermore, if the design metric is an aerodynamic

force or moment, passive shape adaptations need not be used at all: simply compute the optimal

wing shape from the bottom row of Figure 7-10, and build a similar rigid wing. Mass

restrictions prevents such a strategy in traditional aircraft design (though a similar idea can be

seen in the jig-shape approach [163], where wing shape is optimized, followed by identification

of the internal structure which allows for deformation into this shape), but two layers of carbon

fiber can adequately hold the intended shape without a stringent weight penalty. However, if the

design metric is an aerodynamic derivative (gust rejection or longitudinal static stability, for









example), membrane structures must be used, as these metrics depend on passive shape

adaptation with sudden changes in freestream, angle of attack, or control surface deflection.

Referring now to the load-alleviating MAV wing structures of Figure 7-10 and Figure 7-

11, the deformation of the BR wing is relatively small, allowing for just 0.1 of adaptive

washout. As discussed above, the BR wing is very sensitive to pre-tensions in the span direction

(Figure 6-1); the structure is too stiff. Less than a 2% drop in lift from the rigid wing is obtained,

and the pressure distributions of the two wings in Figure 7-11 are very similar. What load

alleviation the BR wing does provide seems to be due to the membrane inflation from between

the leading edge of the battens, and the concomitant flow deceleration over the tangent

discontinuity, rather than the adaptive washout at the trailing edge.

The load alleviating design located by the topology optimizer (right column, Figure 7-10)

is significantly more successful. By filling the design space with patches of disconnected carbon

fiber structures (dominated by a long batten which extends the length of the membrane skin, but

is not connected to the wing's laminate leading edge), the MAV wing is very flexible, but none

of the membrane portions of the wing are large enough to camber the wing via inflation. Wing

deformation is the same magnitude as that seen in the PR-type wings, but the motion is located at

the trailing edge for adaptive washout, and lift is decreased by 5%. The local deformation within

the membrane between the leading edge and the long batten structure is substantial, and the flow

deceleration over this point sees a further loss in lift, as with the BR wing.

Similar results are given in Figure 7-12 and Figure 7-13, for a cambered wing at 120 angle

of attack. The three baseline wings are again shown (carbon fiber wing, PR, and BR), as well as

the designs located by the topology optimization to maximize lift and minimize lift slope. As the

forces are generally larger for the cambered airfoil, the deformations have increased to 5% of the









root chord. The negative forces at the trailing edge of the airfoil are likewise absent. As before,

the PR membrane wing effectively increases the lift over its carbon fiber counterpart through

adaptive cambering, along with aerodynamic penalties from the shape discontinuities at the

leading and trailing edge of the membrane skin.

CL = 0.842 CL = 0.915 CL = 0.947 C = 0.777 CL = 0.727






w/c
0 0.01 0.02 0.03 0.04 0.05








0 0.5 1 1.5 2

Figure 7-12. Normalized out-of-plane displacements (top) and differential pressure coefficients
(bottom) for baseline and optimal topology designs, a = 12, cambered wing.

There is an appreciable amount of upward deformation of the PR wing's trailing edge

carbon fiber strip, leading to washout of each flexible wing section, degrading the lift. As such,

the aeroelastic topology optimizer can maximize lift (Figure 7-12, middle column) by adding

more material to this strip and negating the motion of the trailing edge. As discussed above, this

strip does not continue unbroken to the wing tip, but ends at 65% of the semispan. The

remaining membrane trailing edge is filled with a free-floating carbon fiber batten. Such a

configuration can (theoretically) improve the lift in several ways, similar to the trailing edge

structure used for lift optimization in Figure 7-10.

Placing a flexible membrane skin between two rigid supports produces a trade-off: the









cambering via inflation increases lift, but this metric is degraded by the sharp discontinuities in

the airfoil shape. Towards the inner portion of the MAV wing, this trade-off is favorable for lift.

Towards the wingtip however (either due to the changes in chord or in pressure) this is no longer

true, and the topology optimizer has realized that overall lift can be increased by allowing this

portion of the trailing edge to washout, thereby avoiding the negative pressures seen elsewhere

along the trailing edge.

2.5
0.05
j 2


S0.03




0 -0.5

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8
X/C X/C
x/c x/c






Figure 7-13. Deformations and pressures along 2y/b = 0.58 for baseline and optimal topology
designs, a = 12, cambered wing.

The forward portion of this batten structure also produces a cusp in the wing geometry,

forcing a very strong low pressure spike over the upper portion of the airfoil, further increasing

the lift, as before. Due to the inviscid formulation, further grid resolution around this cusp will

cause the spike to grow larger, as the velocity around the small radius approaches infinity. The

presence of viscosity will attenuate the speed of the flow, and thus both the magnitude of the low

pressure spike and its beneficial effect upon lift. The aeroelastic topology optimizer predicts a

3.5% increase in lift over the PR wing, and 12.5% increase over the rigid wing, though the









veracity of the former beneficial comparison requires a viscous flow solver to ascertain the actual

height of the low-pressure spike at x/c = 0.68.

The batten-reinforced design of Figure 7-12 is substantially more effective with the

cambered wing, than with the reflex wing. As discussed above, reflex in the wing pushes the

trailing edge down, limiting the ability of the battens to washout for load reduction. This can

also be seen by comparing the airfoil shapes between Figure 7-13 and Figure 7-11: the cambered

wing shows a continuous increase in the deformation from leading to trailing edge, while most of

the deformation in the reflex wing is at the flexible membrane/carbon fiber interface. Aft of this

point, deformation is relatively constant to the trailing edge.

The 1.60 of washout in the cambered BR wing decreases the load throughout most of the

wing and decreases the lift by 8.5% (compared to the rigid wing), but, as before, the load-

alleviating design located by the topology optimizer (right column, Figure 7-12) is superior.

Similar to above, the design utilizes a series of disconnected carbon fiber structures, oriented

parallel to the flow, and extending to the trailing edge. The structures are spaced far enough

apart to allow for some local membrane inflation, but this cambering only increases the loads

towards the trailing edge. The discontinuous wing surface forces a number of high-pressure

spikes on the upper surface, notably at x/c = 0.2 and 0.6. This, in combination with the

substantial adaptive washout at the trailing edge, decreases the lift by 13.6% over the rigid wing

and by 5.6% over the BR wing.

Three of the wing topologies discussed above (minimum CL,, minimum drag, and

minimum pitching moment slope, all optimized for a reflex wing at 30 angle of attack) are built

and tested in the closed loop wing tunnel, as seen in Figure 7-14. Though the aeroelastic model

relies on a sizable state of pre-stress in the membrane skin to remain bounded, all three of the









wings are constructed with a slack membrane. This is to ensure similarity between the three

wings (pre-stress is very difficult to control), and also to compare the force and moment data to

the baseline membrane data acquired above (Figure 5-12 Figure 5-15).













Figure 7-14. Wing topology optimized for minimum CL, built and tested in the wind tunnel.

Results are given in Figure 7-15, for a longitudinal a-sweep between 0 and 300. All three

structures located by the topology optimizer show marked improvements over the baseline

experimental data, validating the use of a low fidelity aeroelastic model (vortex lattice model

coupled to a linear membrane solver) as a surrogate for computationally-intensive nonlinear

models. With the exception of very low (where deformations are small) and very high angles of

attack (where the wing has stalled), the optimized designs consistently out-perform the baselines.

As discussed above, this is not expected to be true for L/D, where design strategies vary strongly

with incidence (Figure 7-9).

It should also be noted that the three optimized designs in Figure 7-15 provide shallower

lift slopes, less drag, and steeper pitching moment slopes, respectively, than the experimental

data gathered from the designs utilizing aeroelastic tailoring (Figure 6-15 Figure 6-18). This

confirms the idea that topology optimization can out-perform tailoring of the baseline MAV

wings, as the former has a larger number of variables to work with. The two techniques need not

be mutually exclusive: having located suitable wing topologies, the designs can be subjected to a









tailoring study for further benefit to the flight performance.

1.5
0.6 0

S0.4 E -0.2
U C) U
0.5 0.2 -0.4


0 0
0 10 20 30 0 10 20 30 0 10 20 30
a a a






Figure 7-15. Experimentally measured forces and moments for baseline and optimal topology
designs, reflex wing.

Multi-Objective Optimization

The need to simultaneously consider more than one design metric for aeroelastic topology

optimization of MAV wings is demonstrated above: optimizing for lift prompts the algorithm to

remove the reflex, by depressing the flexible trailing edge. The downward forces provided by

the reflex offset the nose-down pitching moment of the remainder of wing, and are therefore

essential for stability. Design and optimization with multiple performance criteria can be done

my optimizing one variable with constraints upon the others (as discussed by Maute et al. [118]

for aeroelastic topology optimization). For MAV design however, the formulation and bounds

of these constraints are uncertain, and the method does not provide a clear picture of the inherent

trade-off between variables.

The current work minimizes a convex combination of two objective functions (as

discussed by Chen and Wu [158] for topology optimization). Successive optimizations with

different relative weighting between the two metrics can fill out the Pareto optimal front. The









computational cost of such an undertaking is large, and adequate location of the front is not

ensured for non-convex problems (such as seen in Figure 6-14). The objective function is now:


g =(l-). fl i1 mm +6. 2 .2m.. (7-16)
f -f f -f,
flmax l,mm ) f2,max 2,mm .

where 6 is a weighting parameter that varies between 0 and 1, and fi and f2 are the two objective

functions of interest. These functions are properly normalized, with the minimum and maximum

bounds computed from the single-objective optimizations (optimizing with 6 set as 0 or 1). Eq.

(7-16) is cast as a minimization problem, and the sign of f1 and f2 is set accordingly. As before,

the objective function can be augmented with the explicit penalty of Eq. (7-2) as needed.

Typical convergence history results are given in Figure 7-16, for simultaneous

maximization of L/D and minimization of the lift slope. The weighting parameter 6 is set to 0.5,

for an equal convex combination of the two variables. The values given for CL, (- 0.4) are

smaller than experimentally measured trends (- 0.5, from Table 5-1), as the inviscid solver is

unable to predict the vortex lift from the tip vortex swirling system [27]. Beginning with an

intermediate density (Xo = 0.5), the optimizer is able to decrease the convex combination (g)

from 0.7 to 0.3, using similar techniques seen above. All of the carbon fiber material adjacent to

the root, leading edge, and wingtip is removed. Intersecting streams of membrane material grow

across the wing, leaving behind disconnected carbon fiber structures.

The lift-to-drag ratio monotonically converges after 25 iterations, while the lift slope

requires 70 iterations to converge to a minimum value. An explicit penalty on intermediate

densities is employed at the 80 iteration mark, providing a moderate decrease in the combination

objective function. The lift-to-drag ratio is improved as well through the penalty, though the lift

slope suffers. As before, the penalty only serves to force the density variables to 0 or 1, and does

not significantly alter the wing topology.



























4.1 I I I I I 0.04


-0.039

3.9 C

3.8 .... .
Lj 0.038

0 20 40 60 80 100 120
iteration

Figure 7-16. Convergence history for maximizing L/D and minimizing CL,, 6 = 0.5, a = 3,
reflex wing.

The multi-objective results of Figure 7-16 can be directly compared to the single-objective

results of Figure 7-5, where only L/D must be improved. For the latter, L/D can be increased to

4.17, with the inclusion of trailing edge battens for adaptive wash-in, and an unconstrained

membrane skin towards the leading edge for cambering via inflation. This is a load-augmenting

design, and as such the lift slope is very high: 0.040. In order to strike an adequate compromise

between the two designs, the multi-objective optimizer leaves the trailing edge battens, but fills

the membrane skin at the leading edge with a disconnected carbon fiber structure. The L/D of

this design obviously degrades (4.05), but the lift slope is much shallower (0.038), as desired.

The Pareto front for this same trade-off (maximum L/D and minimum CLa) is given in









Figure 7-17, along with the performance of the 20 baseline MAV wing designs (Figure 7-1), and

the design located by the single-objective topology optimizer to maximize CL,. All results are

for a reflex wing at 3 angle of attack. Focusing first on the baseline wings, the BR and PR

wings represent the extremes of the group in terms of lift slope, as expected. The homogenous

carbon fiber wing has the lowest L/D (implying that for a reflex wing at this flight condition, any

aeroelastic deformation will improve efficiency, regardless of the type), while a MAV design

with 2 trailing edge battens as the largest L/D.

0.0421


0.041


0.04 \ a E


S 0.039 [
0 O



0.038



0.037 0* baseline
S' Pareto front
max CL deign
0.036
3.7 3.8 3.9 4 4.1 4.2
L/D

Figure 7-17. Trade-off between efficiency and lift slope, a = 3, reflex wing.

The aeroelastic topology optimization produces a set of designs that significantly out-

perform the baselines, in terms of individually-considered metrics (maximum and minimum lift

slope, maximum L/D), and multiple objectives: all of the baselines are removed from the

computed Pareto front. The optimized designs lay consistency closer to the fictional utopia point

as well, which for Figure 7-17 is at (4.18, 0.0366). The entirety of the Pareto front is not









convex, but the topology optimizer is still able to adequately compute it. The data points are not

evenly spaced either, with 6 = 0.4 and 0.2 both very close to the solution with optimal L/D (6 =

0). This would suggest that despite the normalizing efforts, maximizing L/D carries greater

weight than minimizing the lift slope, an imbalance which may be remedied through nonlinear

weighting [111]. The results of Figure 7-16 also indicate that using an explicit penalty to force

the design to a 0-1 density distribution favors L/D, but not CLa.

In terms of the two metrics in Figure 7-17, none of the designs along the Pareto optimal

front are technically superior: they are non-dominated, in that no other design exists within the

data set that out-performs another design in both metrics. Other performance indices, not

included in the optimization, can then be used to select an adequate design. For micro air vehicle

applications, payload, flight duration, or agility/control metrics can be used, as discussed by

Torres [3]. Realistic knowledge of the low-fidelity aeroelastic model's limitations (the perceived

superiority of an unconstrained membrane wing in Figure 7-3 is destroyed by large nonlinear

flapping vibrations [62], for example), or manufacturability [161] may also be used to select a

design. It should also be noted that at higher angles of attack, the trade-off between high

efficiency and low lift slopes does not exist. As discussed (Figure 7-9), increasing the incidence

promotes an aeroelastic structure with streamlining to improve L/D, a deformation that will also

decrease the lift slope.

Wing displacements and pressure distributions for selected wings along the Pareto front of

Figure 7-17 are given in Figure 7-18, for a reflex wing at 30 angle of attack. Corresponding data

along the spanwise section 2y/b = 0.58 is given in Figure 7-19. When 6 = 1 (single-objective

optimization to minimize the lift slope), the aeroelastic topology optimizer locates a design with

several disconnected structures imbedded within the membrane, including a long batten that









extends the length of the membrane skin. The wing is flexible enough to adaptively washout, but

the remaining patches of membrane skin are not large enough to inflate and camber.

S= 1.0 8 = 0.8 8 = 0.6 8 = 0.4 6 = 0.0
L/D =3.81 L/D 3.97 L/D 4.06 L/D 4.15 L/D 4.18






w/c
-0.01 0 0.01 0.02







ACP
-1.5 -1 -0.5 0 0.5 1 1.5

Figure 7-18. Normalized out-of-plane displacements (top) and differential pressure coefficients
(bottom) for designs that trade-off between L/D and CL,, a = 30, reflex wing.

Gradually adding weight to the L/D design metric removes the structures from the leading

edge of the membrane skin, leaving batten-like structures at the trailing edge of the wing. The

former transition allows the membrane to inflate and camber the wing, while the latter provides

wash-in through depression of the trailing edge. The cambering membrane inflation does not

grow monotonically with decreasing 6, but the trailing edge deformation does: from 0.250 of

washout to 0.750 of wash-in. The size of the depressed trailing edge portion also grows in size.

Decreasing 6 shifts the lift penalty (pressure spike on the upper surface) forward towards the

membrane/carbon fiber interface, and the lift spike (due to the surface geometry cusp at the

leading edge of the batten structures) aft-ward. However, the design that maximizes L/D (6 = 0)

has no spike, with a smooth pressure and displacement profile aft of the lift penalty towards the

leading edge. This may be indicative of the detrimental effect the airfoil cusp has on drag.









0.03


0.01 0.5

0
-0.5

-0.01

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8
x/c x/c
-~----8- ------~-E




Figure 7-19. Deformations and pressures along 2y/b = 0.58 for designs that trade-off between
L/D and CL,, a = 30, reflex wing.

The trade-off between the drag and longitudinal static stability of a membrane MAV wing

is very important: the latter is typically improved through large membrane inflations. The

resulting tangent discontinuities in the wing surface produce pressure spikes oriented axially, and

the exaggerated shape prompts the flow to separate above and below the membrane [14]. The

trade-off is given in Figure 7-20 for a reflex wing at 120, for both the 20 baseline designs and the

Pareto front located with topology optimization. Compared with the data seen in Figure 7-17,

the baseline designs at this higher angle of attack fail to adequately fill the design space; their

performance generally falls within a band. The streamlining of the BR wing provides the lowest

drag (of the baselines), but doesn't significantly out-perform the homogenous carbon fiber wing.

As expected, the PR wing has the largest static stability margin of the baselines, but the

drag penalty is large (and probably under-predicted by the inviscid flow solver). The topology

optimizer is able to locate a design with the same drag penalty, but a steeper pitching moment

slope: by 5.6% over the PR wing. The baselines designs, in general, lie closer to the Pareto front









than seen in Figure 7-17, but the optimized designs are still superior in terms of Pareto optimality

and individual metrics. The optimal drag design (3.8% less than the BR wing) begins with two

carbon fiber structures imbedded within the membrane skin, one of which is a long batten that

extends the length of the design domain.

-0.008

-0.0085



-0.009
-0.0095 0




-0.0 05

-0.011 o

-0.0115 o baseline
-e- Pareto front
-0.012
0.125 0.13 0.135 0.14 0.145 0.15
CD

Figure 7-20. Trade-off between drag and pitching moment slope, a = 12, reflex wing.

By adding weight to the static stability metric (Cma), this long batten breaks in two pieces;

the foreword section shrinks into a slender batten imbedded in the leading edge of the membrane

skin. The aft-ward section gradually accumulates along the trailing edge, merges with the root,

and forms the trailing edge support. As discussed above, this reinforcement does not connect

monolithically to the wingtip; this space is filled with a trailing edge batten. The superiority of

this design is confirmed by the wind tunnel data of Figure 7-15. The Pareto optimal front of

Figure 7-20 shows a more pronounced convexity than seen in Figure 7-17, though the data points

are still not evenly spaced with 6.









Similar data is given in Figure 7-21, for the trade-off between maximum lift and minimum

lift slope, for a cambered wing (no reflex) at 120 angle of attack. Such a trade-off is of interest

because minimizing the lift slope of a membrane MAV wing, while an effective method for

delaying the onset of stall or rejecting a sudden wind gust, typically decreases the pre-stall lift in

steady flight as well; a potentially unacceptable consequence. Certain aeroelastic deformations,

such as a passive wing de-cambering, would provide a wing with higher lift (than the baseline

carbon fiber wing, for example), but a shallower lift slope.

0. 042

0.041

0.04

0.039 0

0.038 'Ih

0.037

0.036 E 0

Sa moo i al or lo a rao baseline
ie -e---- Pareto front
0.034 \ max CLa design

0.033
0.7 0.75 0.8 0.85 0.9 0.95 1
CL

Figure 7-21. Trade-off between lift and lift slope, a = 12, cambered wing.

Such a motion is unusual for low aspect ratio membrane structures however: none of the

baseline designs have both larger lift and a smaller lift slope than the carbon fiber wing. The

correlation between CL and CL, within the set of baseline designs is very strong, and all the

designs fall very close to a single line, clustered in three groups. Any baseline design with

adaptive washout (free trailing edge) has lift slopes between 0.035 and 0.037, any overly-stiff









design with battens oriented perpendicular to the flow (or the carbon fiber wing) has a slope

between 0.038 and 0.039, and the PR wing has a lift slope of 0.041.

The strong data correlation is in sharp contrast to the results of Figure 7-17 for the reflex

wing, where the baseline structures are well-distributed through the design space. This

emphasizes the large role that the doubly-curved airfoil can play in producing many different

types of aeroelastic deformation, providing greater freedom to the designer and better

compromise designs. Despite this, the magnitude of the variability is higher for the cambered

wing, as the forces are generally larger: CL, can be varied by 14.5% for the reflex wing in Figure

7-17, but by 26.4% for the cambered wing in Figure 7-21. These numbers can be increased

further with the use of nonlinear membrane structures, but deformations must be kept at a

moderate level to preserve the fidelity of the linear finite element model in the current work.

As wing structures with high lift and shallow lift slopes are rare, the set of baseline designs

lies close to the Pareto front in Figure 7-21. None are superior however, in terms of individual

metrics or Pareto optimality. The designs located by the topology optimizer to maximize lift and

maximize lift slope are almost identical, though disparate designs can be obtained with a reflex

wing, as noted above. The PR wing is very effective for cambered wings at higher angles of

attack, and lies close to these two optimums. The slight convexity in the Pareto front produces

two designs with the sought-after higher lift and lower lift slope than the homogenous carbon

fiber wing. The topology highlighted in Figure 7-21 increases the lift coefficient from 0.842 to

0.876 and decreases the lift slope from 0.038 to 0.036, and is found from an equal weighting of

the two metrics (6 = 0.5).

Wing displacements and pressure distributions for selected wings along the Pareto front of

Figure 7-21 are given in Figure 7-22, for a cambered wing at 120 angle of attack. Corresponding









data along the spanwise section 2y/b = 0.58 is given in Figure 7-23. Shallow lift slopes are

provided with a series of disconnected batten structures oriented parallel to the flow. As a

weight for high lift is added to the objective function, a large carbon fiber region grows at the

trailing edge, but is connected to either the root or the wing tip. This allows for both washout

and membrane cambering, and produces the MAV design with higher lift and shallower lift

slopes than the carbon fiber wing (6 = 0.5). Further decrease in 6 flattens the chord of the

trailing edge structure and removes the disjointed battens at the leading edge, to maximize lift.

6 = 1.0 8 = 0.6 8 = 0.5 8 = 0.4 8 = 0.0
CL 0.727 CL = 0.776 C = 0.876 C = 0 927 C = 0.947





w/c
0 0.01 0.02 0.03 0.04 0.05 0.06
0 0.01 0.02 0.03 0.04 0.05 0.06


Ar


LP -
0 0.5 1 1.5 2 2.5

Figure 7-22. Normalized out-of-plane displacements (top) and differential pressure coefficients
(bottom) for designs that trade-off between CL and CL,, a = 120, cambered wing.

The locus of aeroelastic deformation clearly shifts from the trailing edge to the mid-chord

of the wing as the structures produce higher lift. Washout monotonically decreases with 6 (from

3 to 0.10 of wash-in). Membrane deformations are largest when 6 = 0.5, though the design that

maximizes lift shows the largest change in camber, owing to the significant adaptive washout of

the former, as discussed. Similarly, the aerodynamic penalty at the leading edge of the


Ivy


^r Ly










membrane/carbon fiber interface is largest with the compromise design. The severity of the

surface cusp (and the concomitant lift spike) increases with decreasing 6, emphasizing its

usefulness as a lift-augmentation device. As discussed above, the severity of this spike is

certainly over-predicted by the inviscid flow solver, though similar trends are seen using Navier-

Stokes solvers for wings with tangent discontinuities Figure 5-21.

0.06 2.5

0.05 2 -

0.04 ,A 1.5

0.03

0.02 0.5
0,
0.01 0


0 -0.5 .......
0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8
x/c x/c






Figure 7-23. Deformations and pressures along 2y/b = 0.58 for designs that trade-off between CL
and CL,, a = 12, cambered wing.









CHAPTER 8
CONCLUSIONS AND FUTURE WORK

The results given in this work detail a comprehensive research effort to understand and

exploit the static aeroelasticity of membrane micro air vehicle wings. The flow structures of

such wings are exceedingly complex, characterized by low Reynolds numbers (flow separation,

laminar-turbulent transition, reattachment, vortex shedding, vortex pairing), low aspect ratios

(strong tip vortex swirling, low pressure wing tip cells), and unstable interactions between the

two (vortex destabilization for bilateral asymmetry). The wing's structural mechanics are also

difficult to predict: a topologically-complex orthotropic wing shell is covered with a thin

extensible latex skin, a membrane with an inherently nonlinear response.

Aeroelastic fixed membrane wing topologies can be broadly divided into two categories:

load-alleviating, and load-augmenting. The former can use streamlining to reduce the drag, or

adaptive washout for gust rejection, delayed stall, or attenuated maneuver loads. The latter

increases the loads via adaptive cambering or wash-in, for improved lift and static stability; the

wing may also be more response to pull-up maneuvers, etc. Wing topology is given by a distinct

combination of stiff laminate composite members and a thin extensible rubber membrane sheet,

similar to the skeletal structure of a bird wing, or the venation patterns of insect wings.

This work discusses aeroelastic analysis and optimization in three phases. First, given a

set of wing topologies (a batten-reinforced design for adaptive geometric twist, a perimeter-

reinforced design for adaptive aerodynamic twist, and a homogenous laminate wing), how does

the membrane inflation affect the complex flow structures over the wing? Secondly, how can the

various sizing and strength variables incorporated within the wing structures be tuned to improve

flight performance, in terms of both individual metrics and compromise functions? Third, can

these baseline wing topologies be improved upon? How does the distribution of laminate shells









throughout a membrane skin affect the aeroelastic response?

No model currently exists that can accurately predict such aeroelasticity (the three-

dimensional transition is the biggest numerical hurdle), and so the current work utilizes a series

of low-fidelity aeroelastic models for efficient movement through the design space: vortex lattice

methods and laminar Navier-Stokes solvers are coupled to linear and nonlinear structural solvers,

respectively (detailed in Chapter 4). Due to the lower-fidelity nature of the models (despite

which, the computational cost of this coupled aeroelastic simulation is very large), experimental

model validation is required. Such characterization is conducted in a low speed closed loop wind

tunnel. Aerodynamic forces and moments are measured using a strain gage sting balance with an

estimated resolution of 0.01 N. Structural displacement and strain measurements are made with

a visual image correlation system; a calibrated camera system is mounted over the test section, as

discussed in Chapter 3.

Chapter 5 provides a detailed analysis of the flow structures, wing deformation, and

aerodynamic loads of a series of baseline membrane MAV wings. At small angles of attack, the

low Reynolds number flow beneath a MAV wing separates across the leading edge camber, the

flow over the upper surface is largely attached, and the tip-vortex swirling system is weak. The

opposite is true has the incidence is increased: the bubble on the upper surface grows, eventually

leading to stall. The lift curves of the low aspect ratio wings are typically shallow, with a large

stalling angle. Low pressure cells deposited on the upper surface of the wing tip by the vortex

swirling grow with angle of attack, adding nonlinearities to the lift and moment trends.

The structural deformation of a batten-reinforced wing has two main trends: the forces

towards the leading edge are very large, and induce membrane inflation in-between the battens.

This increases the camber over the wing, and thus the lift. A second trend comes from the free









trailing edge of the BR wing, which deflects upward for a nose-down twist, decreasing the wing

lift. These two effects tend to offset for lower angles of attack, and the aerodynamics follow the

rigid wing's very closely. At higher angles the adaptive washout dominates, decreasing the

incidence of a wing section by as much as 5 and decreasing the slope of the wing's lift curve.

Outside of the promise such a wing shows for gust rejection and benevolent stall, the data also

indicates that the streamlining decreases drag.

The deformation of a perimeter-reinforced wing is characterized by adaptive aerodynamic

twist: the membrane skin inflates, constrained at the leading and trailing edges by the stiff carbon

fiber perimeter. Lift, drag, and pitching moments are consistently stronger than measured from

the rigid and BR wings, as a result of the cambering motion. The slope of the pitching moment

curve is considerably steeper, providing much-needed longitudinal static stability to a wing with

severe space and weight constraints. The large drag penalty of the wing is partly due to a

pressure spike at the tangent discontinuity between the inflated membrane and the carbon fiber,

and partly due to the greater amount of separated flow over the PR wing. Interactions between

the separated longitudinal flow and the wing tip vortices are clearly visible in the PR wing,

possibly indicating a greater propensity for rolling instabilities. The stretching of the membrane

skin in the PR wing is more two-dimensional without the restrictive presence of battens.

It is shown in Chapter 6, both numerically and experimentally, that unconventional

aeroelastic tailoring can be used to improve MAV wing performance. The chordwise and

spanwise membrane pre-tension, number of plain weave carbon fiber layers, laminate

orientation, and batten thickness are all considered, with the first three variables identified as

critical through a series of one-factor-at-a-time tests. Increasing stiffness is seen to tend

aerodynamic behavior towards a rigid wing, thought many local optima exist and can be









exploited. A comprehensive numerical review of the design space is provided with a full

factorial designed experiment of the three aforementioned variables. This data is then used to

optimize six aerodynamic variables, as well as compromises between each. The six designs

resulting from the single-objective optimizations are built and tested in the wind tunnel: five

show improvements over the baseline designs, one has a similar response.

While the flexible wing structures have been shown to effectively alter the flow fields over

a MAV wing, aeroelastic topology optimization (Chapter 7) can be used to improve on the

shortcomings of the previously-considered baseline designs. Results are superior to those

computed via tailoring, as the number of variables is much larger: the wing is discretized into a

series of panels, each of which can be membrane or carbon fiber laminate. The computational

cost is severe: hundreds of iterations are expected for convergence, and a sensitivity analysis of

the coupled aeroelastic system must be conducted.

The optimization is able to identify a series of interesting designs, emphasizing the

relationships between flight condition, airfoil, design metric, and wing topology. For load

alleviation, the algorithm fills the membrane skin with a number of disconnected laminate

structures. The structure is flexible enough to washout at the trailing edge, but the patches of

exposed membrane skin are not large enough to inflate and camber the wing. Such a design has

less drag and a shallower lift curve than the batten-reinforced wing. For load augmentation, the

topology optimizer utilizes a combination of cambering, wash-in, and wing surface geometry

cusps to increase the lift over the perimeter-reinforced wing. As a wing design optimized for a

single metric is of minor usefulness, the topology optimizer is expanded to minimize a convex

combination of two metrics for computation of the Pareto front. Three such designs are built and

tested in the wind tunnel, confirming the computed superiority over the baseline wings.









Several future aeroelastic optimization studies are of interest. First, it is desired to upgrade

the model fidelity used in the topology optimization described above. In order to limit

computational cost, the work uses several linear modules: a vortex lattice solver and a linear

stress-stiffening membrane solver, computed on a relatively coarse topology grid. Such a model

is unable to capture several important nonlinearities, including flow separation and tip vortex

formation. This can be remedied by using an unsteady Navier-Stokes solver coupled to a

nonlinear membrane structural dynamics solver, increasing the computational cost by several

orders of magnitude. The large number of variables (-1000) requires the use of a gradient-based

optimizer; the higher-fidelity models will increase the complexities involved in the sensitivity

analysis of the coupled aeroelastic system as well.

Of particular interest is gust response: how the membrane wing responds to a sinusoidal

wind cycle, where it is desired to minimize the overall response for smoother flight. Objective

functions may be the change in lift, integrated over the gust cycle. A second interest is the wing

topology that delays the stall of the fixed wing. Conventional optimization formulations for this

problem are difficult, as the stall angle is not a direct output from the aeroelastic system, but the

angle at which the slope of the lift curve becomes negative. The optimizer will have to compute

the lift at a set number of (large) angles of attack, and interpolate between the data points to

estimate the stalling angle.

Secondly, these aeroelastic topology optimization techniques will be extended to flapping

micro air vehicle wings. The structure of these wings is very similar to the fixed wings discussed

above (thin membrane skins reinforced with laminate plies), and so the two-material model is

appropriate. As with the gust cycle, lift and thrust will have to be computed over an entire

flapping cycle, and then integrated to produce a scalar objective function. Furthermore, lift and









thrust will conflict: thrust relies on wing twist via deformation for thrust generation, while lift is

dependent upon the leading edge vortex, which can be disrupted by excessive deformations.

This requires successive optimizations of a convex combination of the two weighted

metrics to fill out the trade-off curve (assuming that this Pareto front is convex). The optimal

design can then be selected from this front based upon metrics not considered in the formal

optimization: trim requirements, manufacturability, etc. The flow structures that develop over

flapping wing systems are very complicated, unsteady vortex driven flows. Navier-Stokes

solvers can adequately handle these phenomena, but the computational cost may be prohibitive.

Topology optimization of flapping wings may require lower-fidelity aerodynamic methods for

effective navigation through the design space.

Finally, the aeroelastic topology optimization of both the fixed and flapping wings can be

followed by a tailoring study for additional improvements to the flight performance. This is a

standard optimization process: topology optimization, interpretation of the results to form an

engineering design, followed by sizing and shape optimization (or in this case, tailoring). Both

laminate thickness/orientation and membrane pre-tension can be used, as above. Membrane pre-

tension is difficult to control however, and will relax at the un-reinforced borders of the wing,

leading to a pre-tension gradient. Anisotropic membranes (through imbedded elastic fibers or

crinkled/pleated geometries) are an attractive alternative for directional wing skin stiffness. The

excess area of the skin may also be a useful variable. As the number of variables in a tailoring

study is relatively small (-10), gradient-free global optimizers such as evolutionary algorithms or

response surface techniques may become applicable.









REFERENCES


[1] Abdulrahim, M., Garcia, H., Lind, R., "Flight Characteristics of Shaping the Membrane Wing of
a Micro Air Vehicle," Journal ofAircraft, Vol. 42, No. 1, 2005, pp. 131-137.

[2] Young, A., Horton, H, "Some Results of Investigation of Separation Bubbles," AGARD
Conference Proceedings, No. 4, 1966, pp. 779-811.

[3] Torres, G., "Aerodynamics of Low Aspect Ratio Wings at Low Reynolds Numbers with
Applications to Micro Air Vehicle Design," Ph.D. Dissertation, Department of Aerospace and
Mechanical Engineering, University of Notre Dame, South Bend, IN, 2002.

[4] Shyy, W., Ifju, P., Viieru, D., "Membrane Wing-Based Micro Air Vehicles," Applied
Mechanics Reviews, Vol. 58, No. 4, 2005, pp. 283-301.

[5] Hoerner, S., Borst, H., Fluid-Dynamic Lift, Hoerner Fluid Dynamics, Brick Town, NJ, 1975.

[6] Tang, J., Zhu, K., "Numerical and Experimental Study of Flow Structure of Low-Aspect Ratio
Wing," Journal ofAircraft, Vol. 41, No. 5, 2004, pp. 1196-1201.

[7] Jenkins, D., Ifju, P., Abdulrahim, M., Olipra, S., "Assessment of the Controllability of Micro
Air Vehicles," Bristol International RPV/UAV Conference, Bristol, UK, April 2-4, 2001.

[8] Lian, Y., Shyy, W., "Laminar-Turbulent Transition of a Low Reynolds Number Rigid or
Flexible Airfoil," AIAA Journal, Vol. 45, No. 7, 2007, pp. 1501-1513.

[9] Albertani, R., Stanford, B., Hubner, J., Ifju, P., "Aerodynamic Coefficients and Deformation
Measurements on Flexible Micro Air Vehicle Wings," Experimental Mechanics, Vol. 47, No. 5,
2007, pp. 625-635.

[10] Ifju, P., Jenkins, D., Ettinger, S., Lian, Y., Shyy, W., Waszak, M., "Flexible-Wing-Based Micro
Air Vehicles," Confederation of European Aerospace Societies Aerodynamics Conference,
London, UK, June 10-12, 2003.

[11] Shirk, M., Hertz, T., Weisshaar, T., "Aeroelastic Tailoring-Theory, Practice and Promise,"
Journal ofAircraft, Vol. 23, No. 1, 1986, pp. 6-18.

[12] Griffin, C., "Pressure Deflection Behavior of Candidate Materials for a Morphing Wing,"
Masters Thesis, Department of Mechanical and Aerospace Engineering, West Virginia
University, Morgantown, WV, 2007.

[13] Waszak, M., Jenkins, L., Ifju, P., "Stability and Control Properties of an Aeroelastic Fixed Wing
Micro Air Vehicle," AIAA Atmospheric Flight Mechanics Conference and Exhibit, Montreal,
Canada, August 6-9, 2001.

[14] Stanford, B., Ifju, P., "Membrane Micro Air Vehicles with Adaptive Aerodynamic Twist:
Numerical Modeling," AIAA Structures, Structural Dynamics, and Materials Conference,
Honolulu, HI, April 23-26, 2007.









[15] Ormiston, R., "Theoretical and Experimental Aerodynamics of the Sail Wing," Journal of
Aircraft, Vol. 8, No. 2, 1971, pp. 77-84.

[16] Bendsoe, M., Sigmund, O., Topology Optimization, Springer-Verlag, Berlin, Germany, 2003.

[17] Carmichael, B., "Low Reynolds Number Airfoil Survey," NASA Contractor Report, CR
165803, 1981.

[18] Lin, J., Pauley, L., "Low-Reynolds-Number Separation on an Airfoil," AIAA Journal, Vol. 34,
No. 8, 1996, pp. 1570-1577.

[19] Mooney, M., "A Theory of Large Elastic Deformation," Journal ofApplied Physics, Vol. 11,
1940, pp. 582-592.

[20] Dodbele, S., Plotkin, A., "Loss of Lift Due to Thickness for Low-Aspect-Ratio Wings in
Incompressible Flow," NASA Technical Report, TR 54409, 1987.

[21] Gopalarathnam, A., Selig, M., "Low-Speed Natural-Laminar-Flow Airfoils: Case Study in
Inverse Airfoil Design," Journal ofAircraft, Vol. 38, No. 1, 2001, pp. 57-63.

[22] Kellogg, M., Bowman, J., "Parametric Design Study of the Thickness of Airfoils at Reynolds
Numbers from 60,000 to 150,000," AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV,
January 5-8, 2004.

[23] Laitone, E., "Wind Tunnel Tests of Wings at Reynolds Numbers Below 70,000," Experiments
in Fluids, Vol. 23, No. 5, 1997, pp. 405-409.

[24] Mueller, T., "The Influence of Laminar Separation and Transition on Low Reynolds Number
Airfoil Hysteresis," Journal ofAircraft, Vol. 22, No. 9, 1985, pp. 763-770.

[25] Gad-el-Hak, M., "Micro-Air-Vehicles: Can They be Controlled Better?" Journal ofAircraft,
Vol. 38, No. 3, 2001, pp. 419-429.

[26] Shyy, W., Lian, Y., Tang, J., Viieru, D., Liu, H., Aerodynamics of Low Reynolds Number
Flyers, Cambridge University Press, New York, NY, 2008.

[27] Katz, J., Plotkin, A., Low-SpeedAerodynamics, Cambridge University Press, Cambridge, UK,
2001.

[28] Lian, Y., Shyy, W., Viieru, D., Zhang, B., "Membrane Wing Mechanics for Micro Air
Vehicles," Progress in Aerospace Sciences, Vol. 39, No. 6, 2003, pp. 425-465.

[29] Nagamatsu, H., "Low Reynolds Number Aerodynamic Characteristics of Low Drag NACA 63-
208 Airfoil," Journal ofAircraft, Vol. 18, No. 10, 1981, pp. 833-837.

[30] Masad, J., Malik, M., "Link Between Flow Separation and Transition Onset," AIAA Journal,
Vol. 33, No. 5, 1995, pp. 882-887.









[31] Schroeder, E., Baeder, J., "Using Computational Fluid Dynamics for Micro Air Vehicle Airfoil
Validation and Prediction," AIAA Applied Aerodynamics Conference, Toronto, Canada, June 6-
9, 2005.

[32] Winter, H., "Flow Phenomena on Plates and Airfoils of Short Span," NACA Technical Report,
TR 539, 1935.

[33] Sathaye, S., Yuan, J., Olinger, D., "Lift Distributions on Low-Aspect-Ratio Wings at Low
Reynolds Numbers for Micro-Air-Vehicle Applications," AIAA Applied Aerodynamics
Conference and Exhibit, Providence, RI, Aug. 16-19, 2004.

[34] Pellettier, A., Mueller, T., "Low Reynolds Number Aerodynamics of Low Aspect Ratio
Thin/Flat/Cambered-Plate Wings," Journal ofAircraft, Vol. 37, No. 5, 2000, pp. 825-832.

[35] Bartlett, G., Vidal, R., "Experimental Investigation of Influence of Edge Shape on the
Aerodynamic Characteristics of Low Aspect Ratio Wings at Low Speeds," Journal of
Aeronautical Sciences, Vol. 22, No. 8, 1955, pp. 517-533.

[36] Polhamus, E., "A Note on the Drag Due to Lift of Rectangular Wings of Low Aspect Ratio,"
NACA Technical Report, TR 3324, 1955.

[37] Kaplan, S., Altman, A., 01, M., "Wake Vorticity Measurements for Low Aspect Ratio Wings at
Low Reynolds Numbers," Journal ofAircraft, Vol. 44, No. 1, 2007, pp. 241-251.

[38] Viieru, D., Albertani, R., Shyy, W., Ifju, P., "Effect of Tip Vortex on Wing Aerodynamics of
Micro Air Vehicles," Journal ofAircraft, Vol. 42, No. 6, 2005, pp. 1530-1536.

[39] Mueller, T., DeLaurier, J., "Aerodynamics of Small Vehicles," AnnualReview ofFluid
Mechanics, Vol. 35, No. 35, 2003, pp. 89-111.

[40] Gursul, I., Taylor, G., Wooding, C., "Vortex Flows Over Fixed-Wing Micro Air Vehicles,"
AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 14-17, 2002.

[41] Cosyn, P., Vierendeels, J., "Numerical Investigation of Low-Aspect-Ratio Wings at Low
Reynolds Numbers," Journal ofAircraft, Vol. 43, No. 3, 2006, pp. 713-722.

[42] Brion, V., Aki, M., Shkarayev, S., "Numerical Simulation of Low Reynolds Number Flows
Around Micro Air Vehicles and Comparison against Wind Tunnel Data," AIAA Applied
Aerodynamics Conference, San Francisco, CA, June 5-8, 2006.

[43] Stanford, B., Sytsma, M., Albertani, R., Viieru, D., Shyy, W., Ifju, P., "Static Aeroelastic Model
Validation of Membrane Micro Air Vehicle Wings," AIAA Journal, Vol. 45, No. 12, 2007, pp.
2828-2837.

[44] Zhan, J., Wang, W., Wu, Z., Wang, J., "Wind-Tunnel Experimental Investigation on a Fix-Wing
Micro Air Vehicle," Journal ofAircraft, Vol. 43, No. 1, 2006, pp. 279-283.









[45] Ramamurti, R., Sandberg, W., Lohner, R., "Simulation of the Dynamics of Micro Air Vehicles,"
AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 10-13, 2000.

[46] Gyllhem, D., Mohseni, K., Lawrence, D., "Numerical Simulation of Flow Around the Colorado
Micro Aerial Vehicle," AIAA FluidDynamics Conference and Exhibit, Toronto, Canada, June
6-9, 2005.

[47] Albertani, R., "Experimental Aerodynamics and Elastic Deformation Characterization of Low
Aspect Ratio Flexible Fixed Wings Applied to Micro Aerial Vehicles," Ph.D. Dissertation,
Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL,
2005.

[48] Morris, S., "Design and Flight Test Results for Micro-Sized Fixed Wing and VTOL Aircraft,"
International Conference on Emerging Technologies for Micro Air Vehicles, Atlanta, GA,
February 3-5, 1997.

[49] Rais-Rohani, M., Hicks, G., "Multidisciplinary Design and Prototype Development of a Micro
Air Vehicle," Journal ofAircraft, Vol. 36, No. 1, 1999, pp. 227-234.

[50] Kajiwara, I., Haftka, R., "Simultaneous Optimal Design of Shape and Control System for Micro
Air Vehicles," AIAA Structures, Structural Dynamics, and Materials Conference, St. Louis,
MO, April 12-15, 1999.

[51] Lundstrom, D., Krus, P., "Micro Aerial Vehicle Design Optimization using Mixed Discrete and
Continuous Variables," AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference,
Portsmouth VA, September 6-8, 2006.

[52] Ng, T., Leng, G., "Application of Genetic Algorithms to Conceptual Design of a Micro Air
Vehicle," Engineering Applications ofArtificial Intelligence, Vol. 15, No. 5, 2003, pp. 439-445.

[53] Sloan, J., Shyy, W., Haftka, R., "Airfoil and Wing Planform Optimization for Micro Air
Vehicles," Symposium ofRTO Applied Vehicle Technology Panel, Ottawa, Canada, October 19-
21, 1999.

[54] Lian, Y., Shyy, W., Haftka, R., "Shape Optimization of a Membrane Wing for Micro Air
Vehicles," AIAA Journal, Vol. 42, No. 2, 2004, pp. 424-426.

[55] Boria, F., Stanford, B., Bowman, W., Ifju, P., "Evolutionary Optimization of a Morphing Wing
with Wind Tunnel Hardware-in-the-Loop," Aerospace Science and Technology, submitted for
publication.

[56] Hunt, R., Hornby, G., Lohn, J., "Toward Evolved Flight," Genetic and Evolutionary
Computation Conference, Washington, DC, June 25-29, 2005.

[57] Day, A., "Optimization of a Micro Aerial Vehicle Planform Using Genetic Algorithms,"
Masters Thesis, Department of Mechanical Engineering, Worcester Polytechnic Institute,
Worcester, MA, 2007.









[58] Fitt, A., Pope, M., "The Unsteady Motion of Two-Dimensional Flags With Bending Stiffness,"
Journal ofEngineering Mechanics, Vol. 40, No. 3, 2001, pp. 227-248.

[59] Wilkinson, M, "Sailing the Skies: the Improbable Aeronautical Success of the Pterosaurs,"
Journal ofExperimental Biology, Vol. 210, No. 10, 2007, pp. 1663-1671.

[60] Bisplinghoff, R., Ashley, H., Halfman, R., Aeroelasticity, Dover, Mineola, NY, 1955.

[61] Stanford, B., Abdulrahim, M., Lind, R., Ifju, P., "Investigation of Membrane Actuation for Roll
Control of a Micro Air Vehicle," Journal ofAircraft, Vol. 44, No. 3, 2007, pp. 741-749.

[62] Argentina, M., Mahadevan, L., "Fluid-Flow-Induced Flutter of a Flag," Proceedings of the
National Academy of Science: Applied A///hei'/ntiL %, Vol. 102, No. 6, 2005, pp. 1829-1834.

[63] Alben, S., Shelley, M., Zhang, J., "How Flexibility Induces Streamlining in a Two-Dimensional
Flow," Physics ofFluids, Vol. 16, No. 5, 2004, pp. 1694-1713.

[64] Voelz, K., "Profil und Auftrieb Eines Segels," Zeitschriftf'ur Angewandte Mathematik und
Mechanik, Vol. 30, 1950, pp. 301-317.

[65] Thwaites, B., "The Aerodynamic Theory of Sails," Proceedings of the Royal Society ofLondon,
Vol. 261, No. 1306, 1961, pp. 402-422.

[66] Nielsen, J., "Theory of Flexible Aerodynamic Surfaces," Journal ofApplied Mechanics, Vol.
30, No. 3, 1963, pp. 435-442.

[67] Haselgrove, M., Tuck, E., "Stability Properties of the Two-Dimensional Sail Model," Society of
Naval Architects and Marine Engineers New England Sailing Yacht Symposium, New London,
CN, January 24, 1976.

[68] Murai, H., Murayama, S., "Theoretical Investigation of Sailwing Airfoils Taking Account of
Elasticities," Journal ofAircraft, Vol. 19, No. 5, 1982, pp.385-389.

[69] Jackson, P., "A Simple Model for Elastic Two-Dimensional Sails," AIAA Journal, Vol. 21, No.
1, 1983, pp.153-155.

[70] Sneyd, A., "Aerodynamic Coefficients and Longitudinal Stability of Sail Aerofoils," Journal of
Fluid Mechanics, Vol. 149, No. 7, 1984, pp.127-146.

[71] Cyr, S., Newman, B., "Flow Past Two-Dimensional Membrane Aerofoils with Rear
Separation," Journal of Wind Engineering andIndustrial Aerodynamics, Vol. 63, No. 1, 1996,
pp. 1-16.

[72] Smith, R., Shyy, W., "Computational Model of Flexible Membrane Wings in Steady Laminar
Flow," AIAA Journal, Vol. 33, No. 10, 1995, pp. 1769-1777.

[73] Newman, B., Low H., "Two-Dimensional Impervious Sails: Experimental Results Compared
with Theory," Journal ofFluidMechanics, Vol. 144, 1984, pp. 445-462.









[74] Smith, R., Shyy, W., "Computation of Aerodynamic Coefficients for a Flexible Membrane
Airfoil in Turbulent Flow: A Comparison with Classical Theory," Physics ofFluids, Vol. 8, No.
12, 1996, pp. 3346-3353.

[75] Lorillu, O., Weber, R., Hureau, J., "Numerical and Experimental Analysis of Two-Dimensional
Separated Flows over a Flexible Sail," Journal ofFluid Mechanics, Vol. 466, 2002, pp. 319-
341.

[76] Jackson, P., Christie, G., "Numerical Analysis of Three-Dimensional Elastic Membrane
Wings," AIAA Journal, Vol. 25, No. 5, 1987, pp. 676-682.

[77] Sneyd, A., Bundock, M., Reid, D., "Possible Effects of Wing Flexibility on the Aerodynamics
of Pteranodon," The American Naturalist, Vol. 120, No. 4, 1982, pp. 455-477.

[78] Boudreault, R., "3-D Program Predicting the Flexible Membrane Wings Aerodynamic
Properties," Journal of WindEngineering andIndustrial Aerodynamics, Vol. 19, No. 1, 1985,
pp. 277-283.

[79] Holla, V., Rao, K., Arokkiaswamy, A., Asthana, C., "Aerodynamic Characteristics of
Pretensioned Elastic Membrane Rectangular Sailwings," Computer Methods in Applied
Mechanics and Engineering, Vol. 44, No. 1, 1984, pp. 1-16.

[80] Sugimoto, T., "Analysis of Circular Elastic Membrane Wings," Transactions of the Japanese
Society ofAerodynamics andSpace Sciences, Vol. 34, No. 105, 1991, pp. 154-166.

[81] Charvet, T., Hauville, F., Huberson, S., "Numerical Simulation of the Flow Over Sails in Real
Sailing Conditions," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 63, No. 1,
1996, pp. 111-129.

[82] Schoop, H., Bessert, N., Taenzer, L., "On the Elastic Membrane in a Potential Flow,"
International Journal for Numerical Methods in Engineering, Vol. 41, No. 2, 1998, pp. 271-
291.

[83] Stanford, B., Albertani, R., Ifju, P., "Static Finite Element Validation of a Flexible Micro Air
Vehicle," Experimental Mechanics, Vol. 47, No. 2, 2007, pp. 283-294.

[84] Ferguson, L., Seshaiyer, P., Gordnier, R., Attar, P., "Computational Modeling of Coupled
Membrane-Beam Flexible Wings for Micro Air Vehicles," AIAA Structures, Structural
Dynamics, and Materials Conference, Honolulu, HI, April 23-26, 2007.

[85] Smith, R., Shyy, W., "Incremental Potential Flow Based Membrane Wing Element," AIAA
Journal, Vol. 35, No. 5, 1997, pp. 782-788.

[86] Heppel, P., "Accuracy in Sail Simulation: Wrinkling and Growing Fast Sails," High
Performance Yacht Design Conference, Auckland, New Zealand, December 4-6, 2002.

[87] Fink, M., "Full-Scale Investigation of the Aerodynamic Characteristics of a Model Employing a
Sailwing Concept," NASA Technical Report, TR 4062, 1967.









[88] Greenhalgh, S., Curtiss, H., "Aerodynamic Characteristics of a Flexible Membrane Wing,"
AIAA Journal, Vol. 24, No. 4, 1986, pp. 545-551.

[89] Galvao, R., Israeli, E., Song, A., Tian, X., Bishop, K., Swartz, S., Breuer, K., "The
Aerodynamics of Compliant Membrane Wings Modeled on Mammalian Flight Mechanics,"
AIAA Fluid Dynamics Conference and Exhibit, San Francisco, CA, June 5-8, 2006.

[90] Pennycuick, C., Lock, A., "Elastic Energy Storage in Primary Feather Shafts," Journal of
ExperimentalBiology, Vol. 64, No. 3, 1976, pp. 677-689.

[91] Parks, H., "Three-Component Velocity Measurements in the Tip Vortex of a Micro Air
Vehicle," Masters Thesis, School of Engineering and Management, Air Force Institute of
Technology, Wright Patterson Air Force Base, OH, 2006.

[92] Gamble, B., Reeder, M., "Experimental Analysis of Propeller Interactions with a Flexible Wing
Micro Air Vehicle," AIAA FluidDynamics Conference and Exhibit, San Francisco, CA, June 5-
8, 2006.

[93] Stults, J., Maple, R., Cobb, R., Parker, G., "Computational Aeroelastic Analysis of a Micro Air
Vehicle with Experimentally Determined Modes," AIAA AppliedAerodynamics Conference,
Toronto, Canada, June 6-9, 2005.

[94] Ifju, P., Ettinger, S., Jenkins, D., Martinez, L., "Composite Materials for Micro Air Vehicles,"
Society for the Advancement of Material and Process Engineering Annual Conference, Long
Beach, CA, May 6-10, 2001.

[95] Frampton, K., Goldfarb, M., Monopoly, D., Cveticanin, D., "Passive Aeroelastic Tailoring for
Optimal Flapping Wings," Proceedings of Conference on Fixed, Flapping, and Rotary Wing
Vehicles at Very Low Reynolds Numbers, South Bend, IN, June 5-7, 2000.

[96] Snyder, R., Beran, P., Parker, G., Blair, M., "A Design Optimization Strategy for Micro Air
Vehicles," AIAA Structures, Structural Dynamics, andMaterials Conference, Honolulu, HI,
April 23-26, 2007.

[97] Allen, M., Maute, K., "Probabilistic Structural Design of UAVs under Aeroelastic Loading,"
AIAA "Unmanned Unlimited" Conference, San Diego, CA, September 15-18, 2003.

[98] Weisshaar, T., Nam, C., Batista-Rodriguez, A., "Aeroelastic Tailoring for Improved UAV
Performance," AIAA Structures, Structural Dynamics, and Materials Conference, Long Beach,
CA, April 20-23, 1998.

[99] Garrett, R., The Symmetry of Sailing: The Physics of Sailingfor Yachtsmen, Adlard Coles,
Dobbs Ferry, NY, 1996.

[100] Eden, M., The Magnificent Book of Kites: Explorations in Design, Construction, Enjoyment,
andFlight, Sterling Publishing, New York, NY, 2002.









[101] Templin, R., Chatterjee, S., "Posture, Locomotion, and Paleoecology of Pterosaurs," Geological
Society of America Special Paper 376, 2004.

[102] Swartz, S., Groves, M., Kim, H., Walsh, W., "Mechanical Properties of Bat Wing Membrane
Skin," Journal ofZoology, Vol. 239, 1996, pp. 357-378.

[103] Norberg, U., "Bat Wing Structures Important for Aerodynamics and Rigidity," Zoomorphology,
Vol. 73, No. 1, 1972, pp. 45-61.

[104] Levin, O., Shyy, W., "Optimization of a Low Reynolds Number Airfoil with Flexible
Membrane," Computer Modeling in Engineering and Sciences, Vol. 2, No. 4, 2001, pp. 523-
536.

[105] Zuo, K., Chen, L., Zhang, Y., Yang, J., "Study of Key Algorithms in Topology Optimization,"
International Journal ofAdvanced Manufacturing Technology, Vol. 32, No. 7, 2007, pp. 787-
796.

[106] Maute, K., Reich, G., "Integrated Multidisciplinary Topology Optimization Approach to
Adaptive Wing Design," Journal ofAircraft, Vol. 43, No. 1, 2006, pp. 253-263.

[107] Pingen, G., Evgrafov, A., Maute, K., "Topology Optimization of Flow Domains Using the
Lattice Boltzmann Method," Structural and Multidisciplinary Optimization, Vol. 34, No. 6,
2007, pp. 507-524.

[108] Beckers, M., "Topology Optimization using a Dual Method with Discrete Variables," Structural
and Multidisciplinary Optimization, Vol. 17, No. 1, 1999, pp. 14-24.

[109] Deb, K., Goel, T., "A Hybrid Multi-Objective Evolutionary Approach to Engineering Shape
Design," International Conference on Evolutionary Multi-Criterion Optimization, March 7-9,
Zurich, Switzerland, 2001.

[110] Maute, K., Nikbay, M., Farhat, C., "Sensitivity Analysis and Design Optimization of Three-
Dimensional Non-Linear Aeroelastic Systems by the Adjoint Method," International Journal
for Numerical Methods in Engineering, Vol. 56, No. 6, 2002, pp. 911-933.

[111] Min, S., Nishiwaki, S., Kikuchi, N., "Unified Topology Design of Static and Vibrating
Structures Using Multiobjective Optimization," Computers and Structures, Vol. 75, No. 1,
2000, pp. 93-116.

[112] Borrvall, T., Petersson, J., "Topology Optimization of Fluids in Stokes Flow," International
Journal for Numerical Methods in Fluids, Vol. 41, No. 1, 2003, pp. 77-107.

[113] Balabanov, V., Haftka, R., "Topology Optimization of Transport Wing Internal Structure,"
Journal ofAircraft, Vol. 33, No. 1, 1996, pp. 232-233.

[114] Eschenauer, H., Olhoff, N., "Topology Optimization of Continuum Structures: A Review,"
Applied Mechanics Reviews, Vol. 54, No. 4, 2001, pp. 331-390.









[115] Krog, L., Tucker, A. Kemp, M., "Topology Optimization of Aircraft Wing Box Ribs,"
AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, NY, August 30-
September 1, 2004.

[116] Luo, Z., Yang, J., Chen, L., "A New Procedure for Aerodynamic Missile Designs Using
Topological Optimization Approach of Continuum Structures," Aerospace Science and
Technology, Vol. 10, No. 5, 2006, pp. 364-373.

[117] Santer, M., Pellegrino, S., "Topology Optimization of Adaptive Compliant Aircraft Leading
Edge," AIAA Structures, Structural Dynamics, and Materials Conference, Honolulu, HI, April
23-26, 2007.

[118] Maute, K., Allen, M., "Conceptual Design of Aeroelastic Structures by Topology
Optimization," Structural and Multidisciplinary Optimization, Vol. 27, No. 1, 2004, pp. 27-42.

[119] Martins, J., Alonso, J., Reuther, J., "Aero-Structural Wing Design Optimization Using High-
Fidelity Sensitivity Analysis," Confederation of European Aerospace Societies Conference on
Multidisciplinary Analysis and Optimization, Cologne, Germany, June 25-26, 2001.

[120] Gomes, A., Suleman, A., "Optimization of Aircraft Aeroelastic Response Using the Spectral
Level Set Method," AIAA Structures, Structural Dynamics, and Materials Conference, Austin,
TX, April 18-21, 2005.

[121] Combes, S., Daniel, T., "Flexural Stiffness in Insect Wings: Scaling and Influence of Wing
Venation," The Journal ofExperimental Biology, Vol. 206, No. 6, 2003, pp. 2979-2987.

[122] Marchman, J., "Aerodynamic Testing at Low Reynolds Numbers," Journal ofAircraft, Vol. 24,
No. 2, 1987, pp. 107-114.

[123] Recommended Practice R-091-2003, "Calibration and Use of Internal Strain-Gage Balances
with Application to Wind Tunnel Testing," AIAA, Reston, VA, 2003.

[124] Kochersberger, K., Abe, C., "A Novel, Low Reynolds Number Moment Balance Design for
Micro Air Vehicle Research," AIAA Fluid Dynamics Conference and Exhibit, Toronto, Canada,
June 6-9, 2005.

[125] Moschetta, J., Thipyopas, C., "Aerodynamic Performance of a Biplane Micro Air Vehicle,"
Journal ofAircraft, Vol. 44, No. 1, 2007, pp. 291-299.

[126] Mueller, T., "Aerodynamic Measurements at Low Reynolds Numbers for Fixed Wing Micro-
Air Vehicles," RTO Special Course on the Development and Operation of UA Vsfor Military
and Civil Applications, Von Karman Institute, Belgium, September 13-17, 1999.

[127] Kline, S., McClintock, F., "Describing Uncertainties in Single-Sample Experiments,"
Mechanical Engineering, Vol. 75, No. 1, 1953, pp. 3-8.

[128] Pankhurst, R., Holder, D., Wind Tunnel Technique, Sir Isaac Pitman and Sons, London, UK,
1952.









[129] Barlow, J., Rae, W., Pope, A., Low-Speed Wind Tunnel Testing, Wiley, New York, NY, 1999.

[130] Fleming, G., Bartram, S., Waszak, M., Jenkins, L., "Projection Moire Interferometry
Measurements of Micro Air Vehicle Wings," SPIE Paper 4448-16.

[131] Burner, A., Fleming, G., Hoppe, J., "Comparison of Three Optical Methods for Measuring
Model Deformation," AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 10-13,
2000.

[132] Sutton, M., Turner, J., Bruck, H., Chae, T., "Full Field Representation of the Discretely
Sampled Surface Deformation for Displacement and Strain Analysis," Experimental Mechanics,
Vol. 31, No. 2, 1991, pp. 168-177.

[133] Schreier, H., Braasch, J., Sutton, M., "Systematic Errors in Digital Image Correlation caused by
Intensity Interpolation," Optical Engineering, Vol. 39, No. 11, 2000, pp. 2915-2921.

[134] Sutton, M., McFadden, C., "Development of a Methodology for Non-Contacting Strain
Measurements in Fluid Environments Using Computer Vision," Optics andLasers in
Engineering, Vol. 32, No. 4, 2000, pp. 367-377.

[135] Albertani, R., Stanford, B., Sytsma, M., Ifju, P., "Unsteady Mechanical Aspects of Flexible
Wings: an Experimental Investigation Applied to Biologically Inspired MAVs," European
Micro Air Vehicle Conference and Flight Competition, Toulouse, France, September 17-21,
2007.

[136] Batoz, J., Bathe, K., Ho, L., "A Study of Three-Node Triangular Plate Bending Elements,"
International Journalfor Numerical Methods in Engineering, Vol. 15, No. 12, 1980, pp. 1771-
1812.

[137] Cook, R., Malkus, D., Plesha, M., Witt, R., Concepts andApplications ofFinite Element
Analysis, Wiley, New York, NY, 2002.

[138] Reaves, M., Horta, L., Waszak, M., Morgan, B., "Model Update of a Micro Air Vehicle (MAV)
Flexible Wing Frame with Uncertainty Quantification," NASA Technical Memorandum, TM
213232, 2004.

[139] Isenberg, C., The Science of Soap Films and Soap Bubbles, Dover, New York, NY, 1992.

[140] Pujara, P., Lardner, T., "Deformations of Elastic Membranes Effect of Different Constituitive
Relations," Zeitschrift fir Angewandte Mathematik und Physik, Vol. 29, No. 2, 1978, pp. 315-
327.

[141] Small, M., Nix, W., "Analysis of the Accuracy of the Bulge Test in Determining the Mechanical
Properties of Thin Films," Journal ofMaterials Research, Vol. 7, No. 6, 1992, pp. 1553-1563.

[142] Pauletti, R., Guirardi, D., Deifeld, T., "Argyris' Natural Finite Element Revisited," International
Conference on Textile Composites and Inflatable Structures, Stuttgart, Germany, October 2-5,
2005.









[143] Lian, Y., Shyy, W., Ifju, P., Verron, E., "A Computational Model for Coupled Membrane-Fluid
Dynamics," AIAA Fluid Dynamics Conference and Exhibit, St. Louis, MO, June 24-26, 2002.

[144] Wu, B., Du, X., Tan, H., "A Three-Dimensional FE Nonlinear Analysis of Membranes,"
Computers and Structures, Vol. 59, No. 4, 1996, pp. 601-605.

[145] Campbell, J., "On the Theory of Initially Tensioned Circular Membranes Subjected to Uniform
Pressure," Quarterly Journal of Mechanics andAppliedA it/h1eutii \, Vol. 9, No. 1, 1956, pp.
84-93.

[146] Mase, G., Mase, G., Continuum Mechanics for Engineers, CRC Press, Boca Raton, FL, 1999.

[147] Stanford, B., Boria, F., Ifju, P., "The Validity Range of Pressurized Membrane Models with
Varying Fidelity," Society for Experimental Mechanics, Springfield, MA, June 4-6, 2007.

[148] Tannehill, J., Anderson, D., Pletcher, R., Computational Fluid Mechanics andHeat Transfer,
Taylor and Francis, Philadelphia, PA, 1997.

[149] Shyy, W., Computational Modelingfor Fluid Flow and Interfacial Transport, Elsevier,
Amsterdam, The Netherlands, 1994.

[150] Thakur, S., Wright, J., Shyy, W., "STREAM: A Computational Fluid Dynamics and Heat
Transfer Navier-Stokes Solver: Theory and Applications," Streamline Numerics, Inc.,
Gainesville, FL, 2002.

[151] Lewis, W., Tension Structures: Form and Behavior, Thomas Telford Ltd, London, UK, 2003.

[152] Kamakoti, R., Lian, Y., Regisford, S., Kurdila, A., Shyy, W., "Computational Aeroelasticity
Using a Pressure-Based Solver," Computer Methods in Engineering and Sciences, Vol. 3, No. 6,
2002, pp. 773-790.

[153] Sytsma, M., "Aerodynamic Flow Characterization of Micro Air Vehicles Utilizing Flow
Visualization Methods," Masters Thesis, Department of Mechanical and Aerospace
Engineering, University of Florida, Gainesville, FL, 2006.

[154] Hepperle, M., "Aerodynamics of Spar and Rib Structures," MHAeroTools Online Database,
http://www.mh-aerotools.de/airfoils/ribs.htm, March 2007.

[155] Giirdal, Z., Haftka, R., Hajela, P., Design and Optimization of Laminated Composites Materials,
Wiley, New York, NY, 1999.

[156] Deb, K., Pratap, A., Agarwal, S., Meyarivan, T., "A Fast and Elitist Multiobjective Genetic
Algorithm: NSGA-II," IEEE Transactions on Evolutionary Computation, Vol. 6, No. 2, 2002,
pp. 182-197.

[157] Antony, J., Design of Experiments for Engineers and Scientists, Butterworth-Heinemann,
Boston, MA, 2003.









[158] Chen, T., Wu, S., "Multiobjective Optimal Topology Design of Structures," Computational
Mechanics, Vol. 21, No. 6, 2998, pp. 483-492.

[159] Haftka, R., Girdal, Z, Elements of Structural Optimization, Kluwer, Dordrecht, The
Netherlands, 1992.

[160] Maute, K., Nikbay, M., Farhat, C., "Coupled Analytical Sensitivity Analysis and Optimization
of Three-Dimensional Nonlinear Aeroelastic Systems," AIAA Journal, Vol. 39, No. 11, 2001,
pp. 2051-2061.

[161] Lyu, N., Saitou, K., "Topology Optimization of Multicomponent Beam Structure via
Decomposition-Based Assembly Synthesis," Journal of Mechanical Design, Vol. 127, No. 2,
2005, pp. 170-183.

[162] Hsu, M., Hsu, Y., "Interpreting Three-Dimensional Structural Topology Optimization Results,"
Computers and Structures, Vol. 83, No. 4, 2005, pp. 327-337.

[163] Rohl, P., Schrage, D., Mavris, D., "Combined Aerodynamic and Structural Optimization of a
High-Speed Civil Transport Wing," AIAA Structures, Structural Dynamics, andMaterials
Conference, New Orleans, LA, April 18-21, 1995.









BIOGRAPHICAL SKETCH

Bret Kennedy Stanford was born in Richmond, Virginia on September 30, 1981, though

his grandmother claims it was on September 29. School was never really an option for young

Bret, forced by his parents at an early age to join the circus instead. He was taught to read, write,

and juggle by a kindly group of clowns, despite his extreme terror of anything with big floppy

shoes, a phobia which continues unabated to this day. Bret was reunited with his parents two

years later, an act which was prompted by a recent increase in the Child Tax Credit. Several

brush-ins with the law led to the Stanford family's expulsion from Virginia, escorted to the North

Carolina border by a group of unsympathetic state troopers. The family subsequently relocated

to Tampa, Florida in the fall of 1988, though Bret's grandmother claims it was in the summer.

Bret's time in Tampa was mostly spent selling hand-carved limestone trinkets and jewelry

to tourists. At the age of 17, he was rejected from most of the universities along the eastern

seaboard, who were collectively unimpressed with his artesian and entertainment backgrounds.

A clerical error granted him acceptance to the University of Florida. He arrived in Gainesville in

the fall of 1999 (a date his grandmother generally agrees upon) with the intent of studying

French post-modem theatre. Nine convoluted years later he received his doctorate in aerospace

engineering. Upon graduation, he plans on throwing all of his newfound knowledge, books, and

lab journals into the River, in order to start a Beach Boys cover band. He hopes that his advisor

will handle all of the journal review replies as they come back from the editors, so the research

will not have gone to waste.





PAGE 1

1 AEROELASTIC ANALYSIS AND OPTIMI ZATION OF MEMBRANE MICRO AIR VEHICLE WINGS By BRET KENNEDY STANFORD A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008

PAGE 2

2 2008 Bret Kennedy Stanford

PAGE 3

3 To my family, despite making fun of me for being in school for so long To Angel, despite making fun of me for a bunch of other reasons And to Fatty too, for only scratching me wh en I really deserve it, which is often

PAGE 4

4 ACKNOWLEDGMENTS Thank you to Dr. Peter Ifju, who offered his guidance on a countless number of research projects, and still let m e work on the ones he thought were dumb. Unque stionably the coolest, smartest, most up-beat professor Ive ever been around. Thank you to Dr. Rick Lind, for consistently po inting out when I need to shave, or get a haircut, or more frequently, both. Thank you to Dr. Roberto Albertani, for shari ng with me his passion for all things wind tunnel related, and for sharing his equipment up at the REEF. Thank you to Dr. Raphael Haftka and Dr. David Bloomquist for serv ing on my committee and sitting through my long, scientifically-que stionable presentations without complaining. Thank you to Dr. Dragos Viieru, for impar ting me with his vast knowledge of CFD. Thank you to Dr. Wei Shyy, for all his help my first few semesters of grad school. A final thank you to all the people who hung aro und the labs I worked in. Frank Boria, who helped teach me the real names for various tools and hardware, which had previously been known to me only as shiny metal things. A thanks-in-advance to Frank for taking all of my future phone calls concerning mortgages, insuran ce, child rearing, etc, no matter how distraught and hysterical they may be. Mujahid Abdulrahim, for discussing with me the ethics of returning a rental car completely caked in mud, and going in reverse through a drive-thru. Wu Pin, for relating countless unintentionally funny and creepy stories that Ill never forget, despite my best efforts. Ill always wonder how you got into this country.

PAGE 5

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES.........................................................................................................................8 ABSTRACT...................................................................................................................................12 CHAP TER 1 INTRODUCTION..................................................................................................................13 Motivation...............................................................................................................................13 Problem Statement.............................................................................................................. ....18 Dissertation Outline........................................................................................................... .....19 Contributions..........................................................................................................................21 2 LITERATURE REVIEW.......................................................................................................22 Micro Air Vehicle Aerodynamics.......................................................................................... 22 Low Reynolds Number Flows.........................................................................................22 Low Aspect Ratio Wings.................................................................................................23 Low Reynolds Number Low Aspect Ratio Interactions............................................... 24 Rigid Wing Optimization....................................................................................................... 26 Micro Air Vehicle Aeroelasticity........................................................................................... 28 Two-Dimensional Airfoils...............................................................................................28 Three-Dimensional Wings............................................................................................... 31 Aeroelastic Tailoring.......................................................................................................... ....34 Topology Optimization.......................................................................................................... .35 3 EXPERIMENTAL CHARACTERIZATION........................................................................ 39 Closed Loop Wind Tunnel...................................................................................................... 39 Strain Gage Sting Balance...................................................................................................... 40 Uncertainty Quantification.............................................................................................. 43 Wind Tunnel Corrections................................................................................................43 Visual Image Correlation....................................................................................................... .44 Data Procession...............................................................................................................47 Uncertainty Quantification.............................................................................................. 48 Model Fabrication and Preparation........................................................................................ 49 4 COMPUTATIONAL FRAMEWORK AND VALIDATION............................................... 51 Structural Solvers............................................................................................................. .......51

PAGE 6

6 Composite Laminated Shells........................................................................................... 52 Membrane Modeling....................................................................................................... 56 Skin Pre-tension Considerations......................................................................................62 Fluid Solvers...........................................................................................................................66 Vortex Lattice Methods................................................................................................... 66 Steady Navier-Stokes Solver........................................................................................... 67 Fluid Model Comparisons and Validation......................................................................70 Aeroelastic Coupling........................................................................................................... ...72 Moving Grid Technique..................................................................................................72 Numerical Procedure....................................................................................................... 73 5 BASELINE WING DESIGN ANALYSIS............................................................................. 75 Wing Deformation..................................................................................................................75 Aerodynamic Loads................................................................................................................83 Flow Structures.......................................................................................................................88 6 AEROELASTIC TAILORING..............................................................................................97 OFAT Simulations..................................................................................................................98 Membrane Pre-Tension...................................................................................................99 Single Ply Laminates..................................................................................................... 102 Double Ply Laminates................................................................................................... 103 Batten Construction.......................................................................................................105 Full Factorial Designed Experiment..................................................................................... 107 Experimental Validation of Opti mal Design Performance................................................... 116 7 AEROELASTIC TOPOLOGY OPTIMIZATION............................................................... 122 Computational Framework................................................................................................... 125 Material Interpolation.................................................................................................... 125 Aeroelastic Solver......................................................................................................... 128 Adjoint Sensitivity Analysis.......................................................................................... 130 Optimization Procedure................................................................................................. 133 Single-Objective Optimization............................................................................................. 134 Multi-Objective Optimization.............................................................................................. 149 CONCLUSIONS AND FUTURE WORK .................................................................................. 161 REFERENCES............................................................................................................................167 BIOGRAPHICAL SKETCH.......................................................................................................179

PAGE 7

7 LIST OF TABLES Table page 4-1 Experimental influence matrix (m m /N) at points labeled in Figure 4-2................................55 4-2 Numerical influence matrix (mm/ N) at points labele d in Figure 4-2 ..................................... 56 5-1 Measured and computed aerodynamic characteristics, = 6 ................................................ 88 6-1 Optimal MAV design array with co m promise designs on the off-diagonal, = 12: design description is (wing type, Nx, Ny, number of plain weave layers)....................... 115 6-2 Optimal MAV design performance array, = 12: off-diagonal com promise design performance is predicated by column metrics, not rows................................................. 115

PAGE 8

8 LIST OF FIGURES Figure page 1-1 Batten-reinforced membrane wing design.............................................................................. 14 1-2 Perimeter-reinforce d m embrane wing design......................................................................... 15 3-1 Schematic of the wind tunnel test setup................................................................................. 39 3-2 Quantification of the reso lution error in the VIC system ....................................................... 48 3-3 Speckled batten-reinforced membra ne wing with wind tunnel attachm ent........................... 50 4-1 Unstructured triangular mesh used for fi nite elem ent analysis, with different element types used for PR and BR wings........................................................................................52 4-2 Computed deformations of a BR wing skeleton due to a poin t lo ad at the wing tip (left) and the leading edge (right)...............................................................................................54 4-3 Compliance at various locations along the wing, due to a point load at those locations ....... 56 4-4 Uni-axial stretch test of a latex rubber membrane.................................................................. 61 4-5 Circular membrane response to a uniform pressure............................................................... 61 4-6 Measured chordwise pre-strains in a BR wi ng before the tension is released from the latex (left), and after (right)............................................................................................... 63 4-7 Monte Carlo simulations: error in the com puted m embrane deflection due to a spatiallyconstant pre-strain distribution assumption.......................................................................65 4-8 Computed pre-stress resultants (N/m) in the chordwise (left), spanwise (center), and shear (right) in a BR wing, corrected at the trailing edge for a uniform pre-stress resultant of 10 N/m............................................................................................................66 4-9 CFD comput ational dom ain.................................................................................................. ..68 4-10 Detail of structured CF D m esh near the wing surface.......................................................... 69 4-11 Computed and measured aerodynamic co efficients for a rigid MAV wing, Re = 85,000 ... 71 4-12 Iterative aeroela stic convergence of m embrane wings, = 9.............................................. 74 5-1 Baseline BR normalized out-of-plane displacement (w/c), = 15 ....................................... 76 5-2 Baseline BR chordwise strain ( xx), = 15........................................................................... 77 5-3 Baseline BR spanwise strain ( yy), = 15.............................................................................77

PAGE 9

9 5-4 Baseline BR shear strain ( xy), = 15................................................................................... 78 5-5 Baseline PR normalized out-of-plane displacement (w/c), = 15 ........................................ 79 5-6 Baseline PR chordwise strain ( xx), = 15............................................................................ 79 5-7 Baseline PR spanwise strain ( yy), = 15.............................................................................80 5-8 Baseline PR shear strain (xy), = 15.................................................................................... 80 5-9 Baseline BR aerodynamic and geom etric twist distribution, = 15.....................................81 5-10 Baseline PR aerodynamic and geometric twist distribution, = 15 ...................................82 5-11 Aerodynamic and geometric twist at 2y/b = 0.65................................................................. 83 5-12 Baseline lift coefficients: numer ical (left), experim ental (right).......................................... 84 5-13 Baseline drag coefficients: nume rical (left), experim ental (right)....................................... 86 5-14 Baseline pitching moment coefficients: num erical (l eft), experimental (right)................... 87 5-15 Baseline wing efficiency: numeric al (left), experim ental (right)......................................... 87 5-16 Pressure distributions (Pa) and stream lines on the upper surface of a rigid (left), BR (center), and PR wing (right), = 0.................................................................................. 90 5-17 Pressure distributions (Pa) and streamlin es on the lower surface of a rigid (left), BR (center), and PR wing (right), = 0.................................................................................. 91 5-18 Pressure distributions (Pa) and stream lines on the upper surface of a rigid (left), BR (center), and PR wing (right), = 15................................................................................92 5-19 Pressure distributions (Pa) and streamlin es on the lower surface of a rigid (left), BR (center), and PR wing (right), = 15................................................................................94 5-20 Section normal force coefficients, and pressure coefficients (2y/b =0.5), = 0 ............... 96 5-21 Section normal force coefficients, and pressure coefficients (2y/b =0.5), = 15 .............. 96 6-1 Computed tailoring of pre-st ress resultants (N/ m) in a BR wing, = 12............................. 99 6-2 Computed BR wing deformation (w/c) with various pre-tensions, = 12 ......................... 100 6-3 Computed tailoring of pre-stress resultants (N/ m) in a PR wing, = 12............................ 101 6-4 Computed PR wing deformation (w/c) with various pre-tensions, = 12 .......................... 102

PAGE 10

10 6-5 Computed tailoring of lam inate orientation for single pl y bi-directional carbon fiber, = 12....................................................................................................................................103 6-6 Computed tailoring of lami nate orientations for two plies of bi-directional carbon fiber in a BR wing, = 12 .......................................................................................................104 6-7 Computed tailoring of lami nate orientations for two plies of bi-directional carbon fiber in a PR wing, = 12 .......................................................................................................105 6-8 Computed tailoring of batten construction in a BR wing, = 12 .......................................106 6-9 Computed normalized out-of-plane displaceme nt (left) and different ial pressure (right) at x/c = 0.5, for various BR designs, = 12 ...................................................................107 6-10 Computed full fact orial design of a BR wing, = 12 .......................................................109 6-11 Computed BR wing deformation (w/c) with one layer of plain weave (left), tw o layers (center), and three layers (right), = 12......................................................................... 109 6-12 Computed full factor ial design of a PR wing, = 12 ........................................................111 6-13 Computed PR wing deformation (w/c) with one layer of plain weave (left), tw o layers (center), and three layers (right), = 12......................................................................... 111 6-14 Computed design perfor m ance and Pareto optimality, = 12..........................................113 6-15 Experimentally measured design optimality over baseline lift.......................................... 118 6-16 Experimentally measured design optimality over baseline pitching moments.................. 119 6-17 Experimentally measured design optimality over baseline drag........................................ 120 6-18 Experimentally measured design optimality over baseline efficiency............................... 120 7-1 Wing topologies flight te sted by Ifju et al. [10] ................................................................... 123 7-2 Sample wing topology (left), aerodynamic me sh (center), and structural m esh (right).......124 7-3 Effect of linear and nonlinear material interpolation upon lift ............................................. 127 7-4 Measured loads of an inade quately reinforced m embrane wing, U = 13 m/s.................... 130 7-5 Convergence history for maximizing L/D, = 3, reflex wing ............................................ 135 7-6 Affect of mesh density upon optimal L/D topology, = 12, reflex wing ........................... 137 7-7 Affect of initial design upon the optimal CD topology, = 12, reflex wing....................... 138 7-8 Affect of angle of attack and airfoil upon the optimal CL topology..................................... 139

PAGE 11

11 7-9 Affect of angle of attack and airfoil upon the optim al L/D topology................................... 140 7-10 Normalized out-of-plane displacements (top) and differential pressure coefficients (bottom ) for baseline and optimal topology designs, = 12, reflex wing......................141 7-11 Deformations and pressures along 2y/b = 0.58 for baseline and optim al topology designs, = 12, reflex wing............................................................................................ 142 7-12 Normalized out-of-plane displacements (top) and differential pressure coefficients (bottom ) for baseline and optimal topology designs, = 12, cambered wing............... 145 7-13 Deformations and pressures along 2y/b = 0.58 for baseline and optim al topology designs, = 12, cambered wing..................................................................................... 146 7-14 Wing topology optim ized for minim um CL built and tested in the wind tunnel............... 148 7-15 Experimentally measured forces a nd m oments for baseline and optimal topology designs, reflex wing.........................................................................................................149 7-16 Convergence history for ma xi mizing L/D and minimizing CL = 0.5, = 3, reflex wing..................................................................................................................................151 7-17 Trade-off between e fficiency and lift slop e, = 3, reflex wing........................................ 152 7-18 Normalized out-of-plane displacements (top) and differential pressure coefficients (bottom ) for designs that trade-off between L/D and CL = 3, reflex wing................154 7-19 Deformations and pressures along 2y/b = 0.58 for designs that trade-off between L/D and CL = 3, reflex wing.............................................................................................155 7-20 Trade-off between drag and pitching moment slope, = 12, reflex wing ........................ 156 7-21 Trade-off between lift and lift slope, = 12, cam bered wing........................................... 157 7-22 Normalized out-of-plane displacements (top) and differential pressure coefficients (bottom ) for designs that trade-off between CL and CL = 12, cambered wing.......... 159 7-23 Deformations and pressures along 2y/b = 0.58 for designs that trade-off between CL and CL = 12, cambered wing..................................................................................... 160

PAGE 12

12 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AEROELASTIC ANALYSIS AND OPTIMI ZATION OF MEMBRANE MICRO AIR VEHICLE WINGS By Bret Kennedy Stanford May 2008 Chair: Peter Ifju Major: Aerospace Engineering Fixed-wing micro air vehicles are difficult to fly, due to their low Reynolds number, low aspect ratio nature: flow separa tion erodes wing efficiency, the wi ngs are susceptible to rolling instabilities, wind gusts can be the same size as th e flight speed, the range of stable center of gravity locations is very small, etc. Membrane aeroelasticity has been identified has a tenable method to alleviate these issues. These flexible wing structures are divide d into two categories: load-alleviating or load-augme nting. This depends on the wings topology, defined by a combination of stiff laminate composite members overlaid with a membrane sheet, similar to the venation patterns of insect wings. A series of well-validated variable-fidelity static aeroelastic models are developed to analyze the working mechanisms (cambering, washout) of membrane wing aerodynamics in terms of loads, wing deformation, and flow structures, for a small set of wing topologies. Two aeroelastic optimization schemes are then discussed. For a given wing topology, a series of numerical de signed experiments utilize tailoring of laminate orientation and membrane pre-tension. Further generality can be obtained with aeroelastic topology optimization: locating an optimal distribution of laminate shells and membrane skin throughout the wing. Both optimization schemes consider several design metrics, optimal compromise designs, and experimental validation of superiority over baseline designs.

PAGE 13

13 CHAPTER 1 INTRODUCTION Motivation The rapid co nvergence of unmanned aerial vehicles to continually smaller sizes and greater agility represents successful efforts along a mu ltidisciplinary front. Technological advances in materials, fabrication, electroni cs, propulsion, actuators, sensors, modeling, and control have all contributed towards the viable candidacy of micro air vehicles (MAVs) for a plethora of tasks. MAVs are, by definition, a class of unmanned ai rcraft with a maximum size limited to 15 cm, capable of operating speeds of 15 m/s or less Ideally, a MAV should be both inexpensive and expendable, used in situations wh ere a larger vehicle would be im practical or impossible, to be flown either autonomously or by a remote pilot. Military and defens e opportunities are perhaps easiest to envision (in the form of over-the-hill battlefield surveillance, bomb damage assessment, chemical weapon detection, etc.), tho ugh MAVs could also play a significant role in environmental, agriculture, wildlife, and traffic monitoring applications. MAVs are notoriously difficult to fly; an e xpected consequence of a highly maneuverable and agile vehicle that must be flow n either remotely or by autopilot [1]. The aerodynamics are beset by several unfavorable flight issues: 1. The operational Reynolds num ber for MAVs is typically between 104 and 105. Flow over the upper wing surface can be characterized by massi ve flow separation, a possible turbulent transition in the free shear layer, and then reattachment to the surface, leaving behind a separation bubble [2]. Such flow structures typically resu lt in a loss of lift, and an increase in drag, and a drop in the overall efficiency [3]. 2. The low aspect ratio wing (on the order of unity) prom otes a large wing tip vortex swirling system [4], which interferes with the l ongitudinal circulation of the wing [5]. Entrainment of the aforem entioned separated flow can lead to tip vortex destabilization [6]; the resulting bilateral asymmetry m ay be the cause of the rolling instabiliti es known to plague MAV flight. 3. Sudden wind gusts may be of the same order of magnitude as the vehicle flight speed (10-15 m/s). Maintaining smooth contro llable flight can be difficult [7] [8].

PAGE 14

14 4. The range of flyable (statically stable) CG lo cations is generally only a few millimeters long, which represents a strenuous weight management challenge [9]. These problem s, along with a broad range of dynamics and control issues, can be alleviated through the appropriate use of wi ng shape adaptation. Active morphing mechanisms have been successfully used on a small class of unmanned air vehicles [1], but the limited energy budgets and size constraints of micro air ve hicles m ake such an option, at present writing, infeasible. As such, the current work is restrict ed to passive shape adaptation. Passive shape adaptation can be successfully built into a MAV wing through the use of a flexible membrane skin [10]. The basic structure of these vehicles is built around a com posite laminate skeleton. Bi-directional graphite/epoxy pl ain weave or uni-directi onal plies are usually the materials of choice, due to durability, low we ight, high strength, and ease of fabrication: all qualities well-exploited in the aviation industry [11]. The carbon fiber skeleton is affixed to an extensible m embrane skin, of which several choices are available: latex, s ilicone, plastic sheets, or polyester [12]. The distribution of carbon fiber and me mbrane skin in the wing determ ines the aeroelastic response, and is demonstrated by two dis tinct designs. The first ut ilizes thin strips of uni-directional carbon fiber imbe dded within the membrane ski n, oriented in the chordwise direction ( Figure 1-1). The trailing edge of the batte n-reinforced (BR) design is unconstrained, and the resulting nose-down geom etric twist of each flexible wing section should alleviate the flight loads: decrease in CD, decrease in CL delayed stall (as compared to a rigid wing) [13]. Figure 1-1. Batten-reinf orced membrane wing design.

PAGE 15

15 A second design leaves the interior of the me mbrane skin unconstrained, while the perimeter of the skin is sealed to a thin curved strip of carbon fiber ( Figure 1-2). The perimeter-reinforced (PR) wing defor mation is closer in nature to an aerodynamic twis t. Both the leading and the trailing edges of each membrane section are cons trained by the relatively-stiff carbon fiber. The positive cambering (inflation) of the wi ng should lead to an increase in CL and a decrease (more negative) in Cm [14]. Figure 1-2. Perimeter-reinf orced membrane wing design. While both of these wing structures can ade quately perform their intended tasks (load alleviation for the BR wing, load enhancement for the PR wing), several sizing/stiffness variables exist within both desi gns, leading to an aeroelastic tailoring problem. Conventional variables such as the la minate fiber orientation [11] can be considered, but the directional stiffness induced by varying the pre-tension within the m embrane skin may play a larger role within the aeroelastic response [15]. Systematic optimization of a single design metric will typically lead to a wing structure with poor pe rform ance in other important aspects. For example, tailoring the PR wing structure for maximum static stability may provoke an unacceptable drag penalty, and vice-versa. Of the aerodynamic performance metrics considered here (lift, drag, efficiency, static stability, gust suppression, and mass) many are expected to conflict, as with most engineering optimi zation applications. Formal multi-objective optimization procedures can be used to tailor flex ible MAV wing designs that strike an adequate

PAGE 16

16 compromise between conflicting metrics, filling in the trade-off curves. While thorough exploration of this aeroel astic tailoring design space can provide a fundamental understanding of the relationship between spatial stiffn ess distribution and aerodynamic performance in a flexible MAV wi ng, further steps towards generality can be achieved by removing the constraint that the wing structures must utilize a BR or a PR design. Topology optimization is typically used to find the location of holes within a homogenous structure, by minimizing compliance unde r a constraint upon the volume fraction [16]. Here it is used to find the location of m embrane skin w ithin a carbon fiber skeleton that will optimize a given aerodynamic objective function. This work will be able to highlight wing topologi es with superior efficacy to those designs considered above (for example, a wing with be tter gust suppression quali ties than the BR wing), as well as designs that strike a compromise between conflicting metrics (for example, a topological combination of the BR and the PR wings ). While the results may be more rewarding than those obtained from tailoring, aeroelast ic topology optimization is significantly more complex. Tailoring requires 5-10 sizing and sti ffness variables, but the topology optimization may utilize thousands of variables: the wing is divi ded into a series of panels, each of which may be membrane or carbon fiber. This necessitates a gradient-based algorithm, while evolutionary algorithms or response surface approaches are feasible for the former problem. Both aeroelastic tailoring and topology optimi zation are effective tools for exploiting the passive shape adaptation of flexible MAV wings, but the computational cost is prohibitive. It is not uncommon for aeroelastic optimization studies to require hundreds, or even thousands, of function evaluations. Numerical modeling of fl exible MAV wings is very challenging and expensive: flow separation, tr ansition, and reattachment [17], vortex shedding and pairing [18],

PAGE 17

17 and wing tip vortex formation/destabilization [6] are all known to occur within the flow over low aspect ratio wings at low Reynol ds numbers. Structurally, the membrane skins used for MAV construction are beset by both geometric and material nonlinearities [19]; the orthot ropic nature of the carbon fiber lam inates must also be computed. At present, no numerical model exists whic h can accurately predict all of the three dimensional unsteady features of an elastic MAV wing (flow transition being a particular challenge [8]). As such, an important step in aeroe las tic optimization of MAV wings is careful development of lower-fidelity numerical models Both inviscid vortex lattice methods and laminar Navier-Stokes solvers are investigated, al ong with linear and nonlinear membrane finite elements (only static aeroelastic models are considered here). In light of the low-fidelity tools that must be used (to maintain computational cost at a reasonable level) a second important step in aeroelastic optimization is extensive experime ntal model validation. Three levels of model validation are employed: validation of the struct ural response of indi vidual components of the membrane wing, validation of the aeroelastic beha vior of various baseline membrane wings, and validation of the superiority of the computed optimal wings (found either through tailoring or topology optimization) over the baselines. Further complications arise from complex objec tive functions. As discussed above, gust response is an important performan ce metric for micro air vehicles [7], but systematic optim ization would require an unsteady model, non-homogenous incoming flow, and subsequent time integration. Similarly, delaying the onset of wing stall would require several sub-iterations to locate the stalling angle. Both objectives can be reasonably replaced by a minimization of the lift slope, which is more amenable to a syst ematic optimization. Finally, the computational complexity is further exacerbated by the multi-objective nature of the problem. Wing structures

PAGE 18

18 that optimize a single objective function are of limited value; of greater importance is the array of designs that lie along the Pareto optimal front. This front is a trade-off curve comprised of non-dominated designs, one of which can be selected based on additional considerations not included in the optimization: manufacturability, flight specifications (duration, payload), etc. Computation of the Pareto front is costly when gradient-based search routines are used for optimization, typically involving successive optimization runs with various convex combina tions of the objective functions. The success of this technique depends on the convexity of the Pa reto optimal front. More efficient methods for computing the Pareto front are available if evol utionary algorithms or response surface methods are employed. Problem Statement The static aeroelas ticity of membrane micro air vehicle wi ngs represents th e intersection of several rich aerodynamics and mechanics problems ; numerical modeling can be very challenging and expensive. Furthermore, MAVs are beset w ith many detrimental flight issues, and are very difficult to fly: systematic numerical optimization schemes can be used to offset these problems, improving flight duration, gust supp ression, or static stability. Ma ny optimization studies can be considered for MAVs; the curren t work utilizes aero elastic optimization, which will require hundreds of costly function evaluations to adequa tely converge to an optimal design. As the feasibility of such a scheme relies on a moderate computational cost, what is the lowest fidelity aeroelastic model that ca n be appropriately used? Model development requires extensive experimental valid ation, and several challenges exist here as well. The forces generated by a MAV wing are very small, and highly-sensitive instrumentation is needed. For deformation measurements, only visi on-based non-contacting methods are appropriate. What particular com ponents are required to construct an adequate

PAGE 19

19 experimental test-bed for flexible MAVs? What performance metrics should be compared between numerical and experimental re sults for sufficient model validation? Upon suitable validation of the aeroelastic m odel, two optimization studies are developed: tailoring and topology optimization. Considering the former, with a given spatial distribution of laminated carbon fiber and membrane skin thro ughout the wing, what is the optimal chordwise and spanwise membrane pre-tension and carbon fiber laminate lay-up schedule? For the aeroelastic topology optimization studies, with a given membrane pre-tension and laminate orientation, what is the optimal distribution of carbon fiber and membrane skin throughout the wing? What performance metrics should be optim ized? As these metrics will surely conflict, what multi-objective optimization schemes are appropr iate for computation of the Pareto front? Can the numerically-indicated optimal wing design structures be built and tested, and will the experimental results also indi cate superiority over similarl y-tested baselin e designs? Dissertation Outline This work begins with a detailed literature review of m icro air ve hicle aerodynamics (low Reynolds number flows, low aspect ratio wings unsteady flow phenomena), aeroelasticity (membrane sailwings, flexible filaments), and optimization (rigid wing airfoil and planform optimization, tailoring). I review the literatur e pertaining to topology optim ization as well, with a particular emphasis upon aeronautical and aeroelastic applications. I then discuss the apparatus and procedures us ed for experimental ch aracterization of the membrane micro air vehicle wings. This incl udes a low-speed closed loop wind tunnel, a high sensitivity sting balance, and a visual image correlation system. Information is also given detailing wing fabrication and pr eparation. I summarize the com putational framework, including both linear and nonlinear structural finite elem ent models. Three-dimensional viscous and inviscid flow solvers are formulated, along w ith aeroelastic coupling and ad hoc techniques

PAGE 20

20 devised to handle the membrane skin pre-tension. The estimated validity range of each model is discussed. I detail the deformation patterns, flow stru ctures, and aerodynamic characterization of a series of baseline flexible a nd rigid MAV wings, obtained both numerically and experimentally for comparison purposes. Once the predictive capa bility of the aeroelastic model is wellverified, these data sets are st udied to uncover the working mech anisms behind the passive shape adaptation and their associated aerodynamic advantages. I then use a non-standard aeroela stic tailoring study to iden tify the optimal wing type and structural composition for a given objective function, as well as va rious combinations thereof. Wing types are limited to rigid, batten-reinforced and perimeter-reinforced designs; structural composition variables incl ude anisotropic membrane pre-tensi on and laminate lay-up schedule. Multi-objective optimization is c onducted using a design of experiments approach, with a series of aerodynamic coefficients and derivatives as metrics. The tailoring concludes with experimental validation of the perfor mance of selected optimal designs. Finally, I formulate a computa tional framework for aeroelast ic topology optimization of a membrane micro air vehicle wing. A gradient-based search is used, with analytically computed sensitivities of the sa me aerodynamic metrics as used above The optimal wing topology is discussed as a function of flight condition, grid density, initial guess, and design metric. I optimize a convex combination of two conflicting objective functions to construct the Pareto front, with a demonstrated superiority over the base line wing structures em ployed in the tailoring study. As before, the work concludes with e xperimental validation of the performance of selected optimal designs.

PAGE 21

21 Contributions 1. Develop a set of variable-fidel ity aeroelastic m odels for lo w Reynolds number, low aspect ratio membrane micro air vehicle wings. 2. Develop a highly-sensitive non-intrusive experi mental test-bed for model deformation and flight loads. 3. Optimization-based system identification of the wing structures material properties. 4. Experimental aeroelastic model validation of flight loads and wing deformation. 5. Optimize multiple flight metrics by tailoring memb rane pre-tension and laminate orientation. 6. Develop computational framework for topology optimization of membrane wings, with an analytical sensitivity analysis of the coupled aeroelastic system. 7. Able to provide scientific insight into the re lationship between optimal wing flexibility, flow structures, and the resulting beneficial effects upon flight loads and efficiency. 8. Experimental validation of the superiority of selected optimal designs over baselines.

PAGE 22

22 CHAPTER 2 LITERATURE REVIEW Micro Air Vehicle Aerodynamics A long history of flight te sting, com putational modeli ng, and wind tunnel work has generally pushed the design methodology of succe ssful fixed wing MAVs to a thin, cambered, low aspect ratio flying wing. Maximizing the wing area for a gi ven size constraint obviously leads to a low aspect ratio design. Further desi re to minimize the size of a MAV negates the use of horizontal stabilizers, repla ced with a reflex airfoil for longi tudinal static stability, wherein negative camber present towards the trailing edge helps offset the longitudinal pitching moment of the remainder of the wing. The superiority of thin wings for MAV applications can be shown by both three dimensional inviscid simulations [20] and two-dimensional viscous simulations [21] [22], where the drop in the adverse pressure gradients increas es the lift and decreases the drag towards stall. Similar tools, as well as wind tunnel testing, indi cates the advantage of cambered wings over flat plates [5]; beyond the obvious increase in lift, higher lift-to-drag ratios are reported by Laitone [23]. Much work has also been done on locating suitable MAV planform shapes. Torres identifies the inverseZimmerman as ideal, based upon size restrictions, required angle of attack, and drag performance; th e optimum shifts to an elliptical shape as the aspect ratio is increased [3]. Low Reynolds Number Flows Low Reynolds num ber laminar flow is likely to separate against an adverse pressure gradient aft of the pressure re covery location (velocity peak) on the upper wing surface, even for fairly low angles of attack. The formation of a turbulent boundary la yer aft of a separation bubble is a very mutable process: Reynolds numbe r, pressure distributions, airfoil geometry, surface roughness, turbulence intensity, acoustic noise, wall heating, and -direction (whether

PAGE 23

23 the angle of attack is being increase d or decreased can lead to hysteresis [24]) are all cited by Young and Horton [2] as highly influential on the formati on of a bubble. Furtherm ore, the flow will only reattach to the surface if there is enough energy to maintain circulating flow against dissipation [25]. An extensive survey of low Reynolds num ber (34 55) airfoils is given by Carmichael [17] (there are quite a few othe rs, as reviewed by Shyy et al. [26]). The study finds that, for the lower end of tested Reynolds num be rs, the laminar separated flow does not have time to reattach to the surface. Above 54, the flow will reattach, forming a long separation bubble over the wing. At the upper end of the range of Reynolds numbers discussed by Carmichael, the size of the bubble decreases, genera lly resulting in a decrease in form drag. Increasing the angle of attack ge nerally enhances the turbulence in the flow, which can also prompt quicker reattach ment and shorter bubbles [8]. The length of the separation bubble can generally be inferred from the plateau-like behavior of the pressure distribution: the flow speeds up before the bubble (dropping the pressu re), and slows down after the bubble [27]. This description is a time-averaged scenario: in an unsteady se nse, the inf lectional velocity profile across the separation bubble can develop inviscid Kelvin-Helmho tlz instabilities and cause the shear layer to roll up. This leads to periodic vortex sh edding and the required matching downstream [18], and can cause the separation bubble to m ove back and forth [28]. Further work detailing low Reynolds num ber flow over rigid airfoils can be found by Nagamatsu [29], Masad and Malik [30], and Schroeder and Baeder [31]. Low Aspect Ratio Wings Early work in low aspect ratio aerody namics was sparked by an inability to fit experimental data with linear aerodyna mics theories, as reported by Winter [32] for aspect ratios between 1.0 and 1.25. The m easured lift is typically higher than predicte d (similar to vortex lift

PAGE 24

24 discrepancies seen on delta wings [27]), as the strong wing tip vortices interfere with the longitudinal wing circulation. Th e most obvious indication of such an interaction is the high stalling angles of low aspect ratio wings, wher e the downward m omentum of the tip vortices can keep the flow attached to the upper wing surf ace. Experimental work by Sathaye et al. [33] using an array of pressure ports was able to confirm the devia tion of the lift distribution from elliptic wing theory. Lian et al. [28] report a computed dip in the previous ly constant pressure coefficients over the upper wing surface at 75% of the sem i-span for high angles of attack, but only minor changes on the bottom surface at the wing tip. These low pr essure cells at the wing tip will grow in intensity and spread inward towards the root as th e angle of attack (and t hus the strength of the swirling system) is increased [34]. The cells are a nonlinear cont r ibution to the wings lift; their growth with angle of attack increases CL with angle of attack as well. Torres [3] gives a general cutoff between a lin ear and a nonlinear CLrelationship at an aspect ratio of 1.25. Low aspect ratio corrections to the lift pr edicted by linear theory (among many) are given by Bartlett and Vidal [35], while Polhamus [36] is able to collapse the meas ured profile drag data at various aspect ratios to a single curve th rough the use of an effective twodimensional lift coefficient. Further experimental work is given by Kaplan et al. [37], who use measurements of the trailing vortex structure off of low aspect ratio flat plates for adequate com parison with force balance measurements. The authors indicate that the nonlinear lift curves may also be caused by a loss of leading edge suction, a nd a rotation of the force vector into a flow-normal direction. Viieru et al. [38] discusses the use of endplates to temp er the induced drag fr om the tip vortices, with reported improvements in the lift-to-drag ra tio at small to moderate angles of attack. Low Reynolds Number Low As pect Ratio Interactions Several interactions betw een the low aspect ratio and low Reynolds number aerodynamics

PAGE 25

25 of MAVs are reported in the lite rature. Mueller and DeLaurier ci te aspect ratio as the most important design variable, followed by wing pl anform and Reynolds number. Free stream turbulence intensity and trailing edge geometry are reported to be non-factors, and Reynolds number is only important near stall [39]. Flow visualization e xperim ents by Gursul et al. [40] on swept, non-slender, low aspect ratio wings find the presence of prim ary and secondary vortices, with stagnant flow regions outboard of the former. Vortex merging and other unsteady interactions within the shear layers are found to be highly de pendent on Reynolds number. Kaplan et al. [37] report a fluctuation in the locati on of the vortex core off of a se mielliptical wing at 8,000 Reynolds number. Numerical simulations and flow visualization by Tang and Zhu [6] of an accelerating elliptical wing show an unstable interaction between a longitudinal secondary separated vortex and the tip vortices. This destabilization (for angles of attack above 11) causes the tip vor tex system to swing back and fo rth along the wing, leading to bilateral asymmetry problems in roll. The author s also note a stationary separated vortex (rather than the customary shedding) for angles above 33 possibly due to the vertical components of the tip vortices. Cosyn and Vierendeels [41], Brion et al. [42], and Stanford et al. [43], discuss numerical wing m odeling of lift and drag for comparison with wind tunnel experiments: the lack of a threedimensional turbulent-transition model is genera lly cited as the reason for poor correlation at higher angles of attack. Results documenting th e aerodynamics of a complete micro air vehicle (wing with fuselage, stabilizers, propellers, etc) are scarce: wi nd tunnel experimentation by Zhan et al. document longitudinal and la teral stability as a f unction of vertical stabilizer placement and wing sweep [44]. Similar stability data is given by Ramamurti et al. [45] for a MAV wing with counter-rotating propellers.

PAGE 26

26 Gyllhem et al. [46] reports that the presence of a fuselage, m otor, and stabilizers surprisingly improves the computed maximum lift a nd stall angle (compared to simulations with just the wing), but increases the drag as well. Experimental work by Albertani [47] finds just the opposite: a decrease in lif t of the entire vehicle, but less of a penalty when passive shape adaptation is built into th e wing. Waszak et al. [13] are able to show significant improvements in efficiency if a stream lined MAV fuselage is used. Rigid Wing Optimization Though the m ain scope of the current work is to improve the aerodynamic qualities of fixed micro air vehicle wings through the judicious use of aeroelastic membrane structures, much successful work has been done with multidiscip linary optimization of the shape, size, and components of a rigid MAV wing. These studies must often make use of low fidelity models due to the large number of function evaluations required for a typical optimization run, and may not be able to capture the complicated flow physics described above. Nonetheless, insight into the relationship between sets of sizing/shape va riables and a given objective function can still be gained. Early work is given by Morris [48], who finds the smallest ve hicle tha t will satisfy given constraints throughout a theoretical mission, using several empirica l and analytical expressions for the performance evaluation. Rais-Rohani and Hicks investigate a similar problem, using a vortex lattice method (for com putations of aerodynamic performance and stability, along with propulsion and weight modules) a nd an extended interior penalty function method to reduce the size of a biplane MAV [49]. Kajiwara and Haftka empha size the unconventional need for sim ultaneous design of the aerodynamic and the cont rol systems at the micro air vehicle scale, due to limited energy budgets [50].

PAGE 27

27 Torres [3] uses a genetic algorithm to minimi ze a weighted com bination of payload, endurance, and agility metrics, with various di screte (wing and tail planform) and continuous (aspect ratio, propeller location, angle of attack, etc) variable s. Aerodynamic analysis is provided by a combination of experimental data, analytical methods and interpolation techniques. The author cites convergence problems stemming fr om the discrete variables. Genetic algorithms are also used in the work of Lundstrm and Krus [51] and Ng et al. [52]. The latter indicates that thes e algorithms are more su ited for the potentially disjointed design spaces presented by MAV optimization efforts. A comparison between a genetic algorithm and gradient-based sequential quadratic programming used to design winglets for a swept wing MAV indicates the superiority of th e former, with a vortex lattice method used for aerodynamic analysis. However, a genetic algorithm may only be feasible for lo wer fidelity tools, due to the large number of function evalua tions required for convergence. Higher fidelity aerodynamics tools (namely, th in-layer or full Navier-Stokes equation solvers) are employed in recent studies. For ex ample, a combined 2-D thin layer Navier-Stokes model and a 3-D panel method is used by Sloan et al. [53], who use the outcome to construct a response su rface to optimize the wing geometry for minimum power consumption. As above, the study reveals the supe riority of thin wings, and finds that optimal airfoil shapes are insensitive to aspect ratio. Lian et al. [54] use a full Navier-Stokes so lv er to maximize the lift-todrag ratio of a rigid MAV wing subject to vari ous lift and wing convexity constraints, with sequential quadratic programming search methods Efficiency improvements are feasible by decreasing the camber at the root and increasing at the tip, thereby decreasi ng the amount of flow separation. Improvements are found to be more s ubstantial at moderate angles of attack. Given the computational complexities associated with MAV simulation, several research

PAGE 28

28 efforts use wind tunnel hardware-i n-the-loop for optimization. Load measurements from a sting balance are fed into an optimizer as the objective function or constraints. Genetic or other types of evolutionary algorithms are invariably used, as a sensitivity analysis requires finite difference computations which are easily dist orted by experimental error. Examples with MAVs are given by Boria et al. [55] (optimize lift and efficiency w ith airfoil morphing), Hunt et al. [56] (optimize the forward velocity and efficiency of an ornith opter, with flapping rate and tail position as variables), and Day [57] (planform optimiza tion of a wing with vari able f eather lengths). Micro Air Vehicle Aeroelasticity The role of aeroelasticity in the study of membrane micro ai r vehicle wings differs greatly from conventional aircraft. Wh ile certain aeroelastic instabi lities do exist (t ypically involving the lift slope approaching infinity [15], unstable flapping of a poorly constrained trailing edge [58], or luffing [59]), classical problems like torsional divergence and flutter have little bearing on MAV design, due to the low aspect ratio natu re of the wings and the sm all operating dynamic pressures [60]. Great savings are available in the for m of load redistribution however, as mentioned above: potential improvements in lift, dr ag, stall, and longitudina l static stability can all be obtained. Lateral cont rol improvements are also obta inable with membrane wings [1] [61]. Furtherm ore, chordwise bending of a wing section (aerodynamic twist) can often be ignored in conventional aircraft (except, for example, when constructed from laminates with many off-axis plies [11]), but such deformation is very prev alent in low aspect ratio m embrane wings. Two-Dimensional Airfoils The aeroelastic m embrane structure is domi nated by three-dimensional structural and aerodynamic effects, but much useful insight can be gained from two-dimensional simulations and experiments. Such endeavors are obviously easier to undertake for PR-type membrane wings, but three-dimensional reinforcement must be taken into account for a pure membrane (or

PAGE 29

29 string in two dimensions) with geometric twist, as the structure alone cannot sustain a flight load in a stable manner. A second option involve s considering an elastic sheet with some bending/flexural stiffness. A large variety of work can be found in the literature concerning twodimensional flexible beams in flow. For problem s on a MAV scale, work tends to focus on flags and organic structures such as leaves, seaweed, etc. Fitt and Pope [58] derive an integro-differential fl ag equation for the shap e of a thin membrane with bending stiffness in unsteady inviscid flow, consider ing both a hinged and a clamped leading edge boundary condition. Argentina and Mahadevan [62] solve a similar problem and are able to predict a critical speed that marks the onset of an unstable flapping vibration, noting that the complex instability is similar to the resonance between a pivoting airfoil in flow and a hinged-free beam vibration. Over-predicti on of the unstable flapping speed (when compared to experimental data) leads to the possibility of a stability mechanism wherein skin friction induces tension in the membrane. Alben et al. [63] discuss the streamlining of a two-dim ensional flexible filament fo r drag reduction. In particular, they are able to show that the drag on a filament at high angles of attack decreases from the rigid U 2 scaling to U 4/3. Early work in the study of membrane wings without bending stiffness is given by Voelz [64], who describes the classical twodimensional sail equation: an inextensible membrane with slack, fixed at the leading and trai ling edges, immersed in incompressible, irrotational, inviscid steady flow. Using thin airfoil theory, along with a small angle of attack assumption, Voelz is able to derive a linear integro-differential equati on for the shape of the sail as a function of incidence, freestream velocit y, and slack ratio. Various num erical solution methods are available, including those by Thwaites [65] (eigenfunction methods) and Nielsen [66] (Fourier series m ethods), to solve for lift, pitc hing moments, and membrane tension.

PAGE 30

30 Multiple solutions are found to exist at small a ngles of attack with a finite slack ratio: approaching 0 from negative angles provides a negatively-cambered sa il, though the opposite is true if this mark is approached from a positive valu e. The sail is uncertain as to which side of the chord-line it should lie [65], a phenomenon which ultimately manifests itself in the form of a hysteresis loop [15]. Variations on this problem are considered by Haselgrove and Tuck [67], where the trailing edge of the m embrane is attached to an inextensible rope, thereby introducing a combination of adaptive aerodynamic and geometric twist. Increasing the le ngth of the rope is seen to improve static st ability, but decrease lift. Membrane elasticity is included in the work of Murai and Maruyama [68], Jackson [69], and Sneyd [70], indicating a nonlinear CLrelationship as strains de velop within the membrane at high incidence. Viscous flow models ar e employed in the work of Cyr and Newman [71] and Sm ith and Shyy [72]. The latter cites vi scous effects as having m uch more influence on the aerodynamics of a sail wing than th e effects of the assumptions ma de with linear thin airfoil theory. Specifically, inviscid solutions tend to ov er-predict lift at higher angles of attack (or large slack ratios), due to a loss of circulation caused by viscous effects a bout the trailing edge. A comparison of lift and tension versus angle of attack with experimental data (provided by Newman and Low [73], among others) yields mixed result s; surprisingly, the lift is overpredicted by the viscous flow m odel, yet the tension is under-predicted. Smith and Shyy also note a substantial discrepa ncy in the available experimental data in reported values of slack ratios, sail material properties, and Re ynolds numbers, which may play a role in the mixed comparisons [74]. Comparison of numerical and experim ental data for twodimensional sails is also di scussed by Lorillu et al. [75], who report satisf actory correlation for the flow structures and defor med membrane shape. Unsteady laminar-turbulent transitional

PAGE 31

31 flows over a membrane wing are studied by Lian and Shyy [8] (who correlate the frequency spectrum of the vibrating membra ne wing to the vortex shedding). Three-Dimensional Wings Three com plicating factors can arise with the simulation of a three-dimensional membrane wing, rather than the planar case [76]. First, the tension is not constant (in space or direction), but is in a state of plane-stress Secondly, the wing geom etry can vary in the spanwise direction, and must be specified. Finally, the membra ne may possess a certain degree of orthotropy [59]. Most im portantly, analytical so lutions cannot generally be found. Simplifying assumptions to this problem are given by Sneyd et al. [77] (triangular planform) and Ormiston [15] (rectangular sailwing). Sneyd et al. reduce both the aerodyna m ics and the membrane deformation to twodimensional phenomena, where the third dimens ion is felt through a trailing edge cable. Ormiston assumes both spanwise and chordwis e deformation (but not aerodynamics), and is able to effectively decouple the two modules by using only the firs t term of a Fourier series to describe the inflated wing shape. Boudreault uses a higher-fidelity vortex lattice solver, but also prescribes the wing shape, he re using cubic polynomials [78]. Holla et al. [79] use an iterative procedure to couple a double latt ice m ethod to a structural mode l, but assume admissible mode shapes to describe the deformation of a recta ngular membrane clamped along the perimeter. The stress in the membrane is assumed to be always equal to the applied pre-stress (inextensibility, which overwhelms the nonlinearities in the membrane mechanics). A similar framework is used by Sugimoto in the study of circular membra ne wings, where the wing shape is completely determined by a linear finite element solver [80]. Jackson and Christie couple a vortex lattice m ethod to a nonlinear structural model for the simulation of a triangular membrane wing. Co mparisons between a rigid wing, a membrane wing fixed at the trailing edge, and one with a fr ee trailing edge elucidate the tradeoffs in lift

PAGE 32

32 between adaptive camber and adaptive washout [76]. Charvet et al. [81] study the effect of nonhom ogenous incoming flow (vertical wind gradients and gusts) on a flexible sail. Schoop et al. use a nonlinear membrane stress-strain relationship (hyperelasticity ) with a vortex lattice solver for simulation of a flat rectangular membrane wing [82]. Lian et al. [28] compute the unsteady aeroelasticity of a batten-reinforced m embrane micro air vehicle wing, with a nonlinear h yperelastic solver and a turbul ent viscous flow solver, using thin plate splines as an interf acing technique. Battens are simu lated with a dense membrane. The results indicate self-exciting membrane vi bration on the order of 100 Hz, with a maximum wing speed about 2% of the freestr eam, though overall aerodynamics ar e similar to that of a rigid wing prior to stall. Stanford and Ifju [14] discuss steady laminar ae roelasticity of a perim eterreinforced membrane micro air vehicle wing, and are able to show the exp ected increase in lift and stability. Significant drag penalties are seen to arise w ith increasing Reynolds numbers, though the opposite is true for the rigid wing. Complexities involving membrane wing models with both membranes and elastic shells (such as batten reinforcement) can be found in the work of Stanford et al. [83] (linear mechanics) and Ferguson et al. [84] (nonlinear). Higher-o rdered m embrane modeling with wrinkling (the loss of one or more principle stresses) as pertai ning to membrane wings is given in the work of Smith and Shyy [85] and Heppel [86]. A large volum e of work can be found dealing with experimental characterization of membrane wings. Early wind tunnel work by Fink [87] details a full-scal e investigation of an 11.5 aspect ratio sail wing with a rigid leading edge, wingtip, and root, and a cabled trailing edge. The defor mation is reported to be fairly sm ooth prior to stall, but visible rippling develops along the membrane at the onset of stall. At low angles of attack, the slope of the lift curve is

PAGE 33

33 unusually steep (an instability discussed by Ormiston [15], among others), as the strains in the skin are low enough to allow for large changes in cam ber. Greenhalgh and Curtiss conduct wind tunnel testing to study the effect of planform on a membrane wing; only a parabolic planform is capable of sustaining flight loads without the aid of a trailing edge support member [88]. Galvao et al. [89] conduct tests on a membrane sheet stretched between two rigid posts, at Reynolds numbers between 34 and 105. The results show a monotonic increase in membrane camber with angle of attack and dynamic pressu re, up to stall, as well as the aforementioned steep lift slopes. De-cambering of the wing as the pressure on the upper surface increases due to imminent flow separation is seen to ameliorate the stall behavior, as compared to a rigid plate. Flow visualization of a batten-reinforced membrane MAV wing exhibits a weaker wing tip vortex system than rigid wings [13], possibly due to energy conservation requirem ents [90]. Parks m easures the vortex core of a BR wing 5% to 15% higher above the wing than for the rigid case, though the flexible wing is s een to have a denser core-distrib ution of velocity for moderate angles of attack [91]. Gamble and Reeder [92] measure the flow st ructu res resulting from interactions between propeller slipstream and a BR wing. Th e rigid wing spreads the axial component of the propwash further along the wi ng (resulting in a higher measured drag), whereas the membrane wing can absorb the downwash and upwash. A region of flow separation is measured at the root of the rigid wing, signifi cantly larger and stronge r than that measured from the membrane wing; the superiority d ecreases with larger Reynolds numbers. Albertani et al. [9] detail loads measurements of both BR and PR wings, with dramatic im provements in longitudinal static stability of both membrane wings over their rigid counterpart. The BR wing has a noticeably smoothe r lift behavior in th e stalled region, though neither deforms into a particularly optimal aerodynamic shape: both incur a drag penalty.

PAGE 34

34 Deformation measurements of a membrane wing under propwash indicate unsymmetrical (about the root) wing shapes, a phenomenon which diminish es with higher angles of attack and dynamic pressures [47]. Stults et al. use la ser vibrom etry to measure th e modal parameters (shapes, damping, frequency spectra) of a BR wing, which are then fed into a computational model for simulation of static and dynamic deforma tions in both steady freestream and a gust [93]. Aeroelastic Tailoring Although aeroelastic tailoring is generally defi ned as the addition of directional stiffness into a wing structure so as to beneficially affect perform ance [11], this has traditionally meant the use of unbalanced com posite plates/shells. Despite the use of such laminated materials on many MAV wing frames [94], there does not appear to be an y tailoring studies on fixed m icro air vehicle wings. Some investigators have applied the concept to the design of flapping ornithopter wings [95] [96], where a bend/twist coup ling can vary the twist-indu ced incidence of a wing to im prove thrust. Conventional tailoring studies ca n also be found applied to a larger class of unmanned aerial vehicles [97] [98]. The latter study by Weisshaar et al. uses lam inate tailoring to improve the lateral control of a vehicle with an aspect ratio of 3. With ailerons, a wing tailored with adaptive wash-in is shown to im prove roll performance and roll-reversal speed, though wash-out is preferred for a leading edge slat [98]. In addition to conventional lam inate-based tail oring, drastic changes in the performance of a membrane wing are attainable by altering the pr e-tension distribution within the extensible membrane. Holla et al. [79], Fink [87], Smith and Shyy [72], Murai and Maruyama [68], and Or miston [15] all note the enormous im pact that m embrane pre-tension has on aerodynamics: for the two-dimensional case, higher pre-tension generally pushes flex ible wing performance to that of a rigid wing. For a three-dimensional wing, th e response can be considerably more complex, depending on the nature of the membrane reinfor cement. Well-reported e ffects of increasing the

PAGE 35

35 membrane pre-tension include: decrease in drag [89], decrease in CL [15], linearized lift behavior [72], increase in the zero -lift angle of attack [68], and more abrupt stalling patterns [89]. Ormiston details aeroelastic in stabilities in terms of the ratio of spanwise to chordwise pretension [15]. Adequate control of m embrane tension has long been known as a crucia l concern to sailors in order to efficiently exploit wind power [99]. Tension-control is sim ilarly important to the performance and agility of fighter kites: a wrinkled membrane su rface will send the kite into an unstable spin. When pointed in the desired dire ction, pulling the control line tenses and deforms the kite, which thus attains forward velocity [100]. Biological in spiration for aerodynam ic tailoring of membrane tension can be seen in the wing structures of pterosaurs and bats. In addition to membrane anisotropy (p terosaur wings have internal fibrous reinforcement to limit chordwise stretching [101], while bat wings skins are measured to be 100 times stiffer in the chordwise direction than the spanwise [102]), the tension can be c ontrolled through a single digit (ptero saurs) [59], or varied throughout the wing via m ultiple digits (bats) [103]. Work for mally implementing membrane tension as a variable for optimizing aerodynamic performance is very rare. Levin and Shyy [104] study a modified Clark-Y airfoil with a flexible m embrane upper surface, subjected to a vary ing freestream velocity. Response surface techniques are used to maximize the power index averaged over a sinusoidal gust cycle, with membrane thickness variation, elastic modulus, and pre-stress used as variables. The maximum power index is found to coincide with the lower bound placed upon pre-st ress, though lift and efficiency are also seen to be superior to a rigid wing. Topology Optimization The basics of topology optim ization are given by Bendse and Sigmund [16] and Zuo et al. [105]: the design domain is discretiz ed, and the relative density of each elem ent can be 0 or 1.

PAGE 36

36 Traditionally, this is done on a structure with static loads by minimizing the compliance under an equality constraint upon the volume fraction, thou gh recent work can be seen in the design of compliant mechanisms [106] and channel flows [107] as well. Solving the problem with strictly discre te variables is rare; Beckers [108] uses a dual method to solve the large-scale discrete problem while Deb and Goel [109] use a genetic algorithm. Th is latter option, though attractive, requires a very large number of function evaluations even for a sm all number of variables. The topology optimization problem is typically solved using the SIMP approach (solid isotropic material with penalization): the density of each element is allowed to vary continuously between 0 and 1. A nonlinear pow er-law interpolation provides an implicit penalty which pushes the density to 0 or 1: intermediate densit ies are unfavorable, as their stiffness is small compared to their volume [16]. An adjoint sensitivity analysis of the discrete system is required to compute the sensitivity of the compliance (or other objective functions ) with respect to each element density, as the number of variables is much larger than the number of constraints [110]. A m esh-independent filter upon the gr adients is also typically empl oyed, in order to limit the minimum size of the structure and eliminate chec kerboards. Computatio n of the topological Pareto trade-off curve can be done us ing a multi-objective genetic algorithm [109], or by success ively optimizing a weighted sum of conflicting objectives [111]. Aeronautical applications are given by Borrvall and Petersson [112], who divide a com putational domain into either fluid or solid walls to find the minimum drag profile of submerged bodies in Stokes flows. Pingen et al. [107] solve a similar problem, using a lattice Boltzm ann method as an approximation to the Navi er-Stokes equations. Drag is minimized by a football shape (with front and back angles of 90) at low Reynolds num bers (where reducing surface area is important), and a symmetric airfoil at higher Reynolds numbers (where

PAGE 37

37 streamlining is more important). Several examples can be found in the literature pertaining to compliance minimization of a flexible aircraft structure. F light loads are typically obtained from an aerodynamics model, but the redistribution of these loads with wing de formation (aeroelasticity) is not included. Balabanov and Haftka [113] optimize the internal structur e of a transport wing, using a ground structure approach (the dom ain is filled with interconnected trusse s, and the cross-sectional area of each is a design variable [16]) for compliance minimization. Eschenauer and Olhoff [114] optim ize the topology of an internal wing rib und er both pull-up load maneuvers and internal tank pressures, using a bubble method. Krog et al. [115] also optimize the topology of wing box ribs, and discuss m ethods for interpretation of th e results to form an engineering design, followed by sizing and shape optimization. Luo et al. [116] compute the optimal topology of an entire aerodynam ic missile body, considering both stat ic loads and natura l frequencies. Santer and Pellegrino [117] replace the leading edge of a wing section with a com pliant morphing mechanism, which is subjected to topology optimization. Rather than a compliancebased objective function, the authors use airf oil efficiency, but as above, do not include aeroelastic load redistribution. Such an aeroela stic topology optimization is an under-served area in the literature. Maute and Allen [118] consider the topological layout of stiffeners within a swept wing, using a three-dim ensional Euler solver coupled to a linear finite element model. Results from an adjoint sensitivity analysis of the three-field coupl e aeroelastic system [110] [119] are fed into an augmented Lagrangian optim izer to minimize mass with constraints upon the lift, drag, and wing displacement. The author s are able to demonstrate the superiority of designs computed with aeroelastic topology optim ization, rather than considering a constant pressure distribution.

PAGE 38

38 Gomes and Suleman [120] use a spectral level set method to maximize aileron reversal speed by reinforcing the upper skin of a wing torsion box via topology optim ization. Maute and Reich [106] optimize the topology of a compliant m orphing mechanism within an airfoil, by considering both passive and active shape deformati ons. The authors are ab le to locate superior optima with this aeroelastic topology optimiza tion approach, as compared to a jig-shape approach: optimizing the aerodynamic shape, and then locating the mechanism that leads to such a shape. At present, there is no research pertai ning to aeroelastic t opology optimization of membrane wings, or micro air vehicle wings. Bi ological inspiration for this concept can be found in the venation of insect wings however. Fo r example, a pleated grid-like venation can be seen in dragonfly wings, posterior ly curved veins in fly wings, and a fan-like distribution of veins in the locust hindwing [121]. On the whole, the signifi cance of this variation in wing stiffness distribution between sp ecies is not well understood.

PAGE 39

39 CHAPTER 3 EXPERIMENTAL CHARACTERIZATION As will be e xtensively discussed below, numerical modeling of flexible MAV wings, while conducive to optimization studies, is very challenging: at the present time, no model exists which can accurately predict all of the unsteady fl ow phenomenon known to occur over a micro air vehicle. As such, experimental model validation is required to instill confidence in the employed models, highlight numerical shortcomings and provide additional aeroelastic wing characterization. All of the aerodynamic characteriz ation experiments discussed in this work are run in a closed-loop wind tunnel, a diagram of which can be seen in Figure 3-1. Only longitudinal aerodynam ics are of interest, and only -sweep capability is built into the test setup. Figure 3-1. Schematic of the wind tunnel test setup. Closed Loop Wind Tunnel The test facility us ed for this work is an Engineering Laboratory Design, Inc. (ELD) 407B closed-loop wind tunnel, with the flow loop arranged in a horizont al configuration. The test Wind Tunnel Test Section Incoming Flow Mounting Bracket VIC Cameras Model Arm Sting Balance Speckled MAV Wing

PAGE 40

40 section has an inner dimension of 0.84 m on each side and is 2.44 m deep. The velocity range is between 2 and 45 m/s, and the maximum Reynolds number is 2.7 million. The flow is driven via a two-stage axial fan with an electric motor pow ered by three-phase 440 V at 60 Hz. The controller is operated remotely with appropriately dedicated da ta acquisition software, wherein the driving frequency is based upon a linear scaling of an analog voltage input. Suitable flow conditions are achieved through hexagonal alumin um honeycomb cell, high-porosity stainless steel screens, and turning vane cascades with in the elbows of the closed loop. Centerline turbulence levels are measured on the order of 0.2%. Optical glass wi ndow access is available on the sidewalls and the ceiling. A Heise model PM differential pressure trans ducer rated at 12.7 cm and 127 cm of water (with a manufacturer-specified 0.002% sensitiv ity and a 0.01% repeatability) is used to measure the pressure difference from a pitot-stat ic tube mounted within the test section, whose stagnation point is located at the center of the sec tions entrance. The Heise system is capable of measuring wind speeds up to 45 m/s. A four-wire re sistance temperature de tector is mounted to the wall of the test section for airflow temperature measurements. Strain Gage Sting Balance Several outstanding issues exis t with m easuring the aerodynami c loads from low Reynolds number flyers. Several such airfoils are known to exhibit hysteresis l oops at high angles of attack. If the flow does not reattach to the wing surface (typi cally for lower Reynolds numbers below 54 [17]) counterclockwise hysteresis loops in the lift data m ay be evident; the opposite is true if a separation bubble exists via reattachment [24]. Adequate knowledge of such a loop is obviously important as it effects vehicle control problem s via stall and spin recovery. As described by Marchman [122], the size of the hysteresis loop m easured in a wind tunnel can be incorrectly decreased by poor flow quality: large freestream turbulence intensity levels or

PAGE 41

41 acoustic disturbances (noise emitted from the tu rbulent boundary layer along the tunnel walls, the wind tunnel fan, etc. [24]). Mounting techniques are also presum ed to cause an incorrect relationship between Reynolds num ber and the zero-lift angle of attack among several sets of published data [122]. Sensitiv ity is another concern, particularly in drag force measurements which may be as low as 0.025 N (computed for a wing with a c hord of 100 mm and a Re ynolds number of 54). An electrical resistance strain gage sting bala nce is typically used for force and moment measurements. While strain gages typically provide the greatest sensitivity and simplicity, they are also prone to temperature drift, electroma gnetic interference, cree p, and hysteresis. An internal Aerolab 01-15 6-component strain gage sting balance is used for force/moment measurements in the current work. Wind tunnel models are mounted to the sting balance by a simple jaw mechanism. Each of the six channe ls is in a full Wheatst one-bridge configuration, with five channels dedicated to forces, and one to a moment. Two forces are coincident with the vertical plane of the mo del (traveled during an -sweep), two are in th e plane normal to the previous (traveled during a -sweep), one force is in the ax ial direction, and the moment is dedicated to roll. Data acquisition is done with a NI SCXI 15 20 8 channel programmable strain gage module with full bridge configurati on, 2.5 excitation volts, and a gain of 1000. Other modules included in the system ar e a SCXI 1121 signal conditioner, 1180 feedthrough with 1302 breakout and 1124 D/A module A NI 6052 DAQPad firewire provides A/D conversion, multiplexing, and the PC connection. For a given flight condition, the output signals from the six components are sampled at 1000 Hz for 2 seconds. The average of this data is sent to one module for calculation of the relevant aerodynamic coefficients, and the standard deviation of the data is stored for further uncer tainty analysis. Signals from each channel are

PAGE 42

42 recorded before and after a testing sequence, w ith no airflow through the tunnel, to provide an estimation of the overall drift. The sting balance is mounted to a custom-fab ricated aluminum model arm within the wind tunnel (seen in Figure 3-1). A U-shape is built into the arm so that the structure curves well behind the model and aerodynamic interactions are minimal. The arm extends through a hole in the wall of the test section, and is then attach ed to a gearbox and a brushless servomotor for pitching control. The motor is run by a single ax is motion controller; a high precision US Digital absolute encoder, connected to an SCXI 1121 module provides angle of attack feedback. Pitching rates are on the order of 1 /s. For a given flight condition, the aforementioned instrumentation (the Heise and thermocouple connected through an RS232, and the sting balance) is used to measure the pressure, temperat ure, and voltage signals. A set of tare voltages (obtained prior to the test, with no flow through th e tunnel) are subtracted from the sting balance data, which is then filtered through the calibra tion matrix, and normalized by the subsequent computations of flow velocity and air dens ity. The numerous systems described above are integrated to allow for completely automated wind tunnel testing for force/moment data, along with a LABview GUI written for user inputs of th e wing geometry, angle of attack array, and the commanded wind speed. Standard procedures [123] are used to calib rate the sting balance down to an adequate sensitivity: 0.01N in drag (though s till just 40% of the m inimum gi ven above). Such a resolution is comparable to that found in the work of Pelletier and Mueller [34], but superior precision is used by Kochersberg er and Abe [124] and Moschetta and Thipyopas [125]. The calibration m atrix is determined through the use of known we ights applied at contro l points in specified directions. This calibration is ab le to predict the relationship be tween load and signal for a given

PAGE 43

43 channel, as well as potential interactions (sec ond-order interactions ar e not included) in both single and multiple load configurations. Furthe r information on the calibration of strain gage sting balances for micro air vehicle measurements is given by Mueller [126] and Albertani [47]. Uncertainty Quantification Two types of uncertainty are t hought to contribute to the eventu al error bounds of the sting balance d ata. The resolution error is indicativ e of a measurement devi ces resolution limit: for example, the inclinometer used to measure the p itch of a model can measure angles no finer than 0.1, an uncertainty that can be propagated through the equations to find its theoretical effect on the aerodynamic coefficients usi ng the Kline-McClintock technique [127]. The following resolution errors are used: 3 Pa of dyna m ic pressure from the Heise, 1.2-7 V from the output voltage of the strain gages (estimated from the quantization error of the 16-bit DAQ cards), 0.001 m2 from wing area measurements, and 0.002 m from chord length measurements. The second source is the precision error, a measure of the repeatability of a me asurement. This is well quantified by the standard deviation of the voltage signals from 2000 samples at each angle of attack, as described above. Uncertainty bounds are computed with a squared sum of the resolution and precision errors (w here the latter is magnified by Students t at 95% confidence and infinite degrees of freedom). The precision of the strain gage signals is found to contribute the most error to the aerodynamic coefficients, pa rticularly in the stal led regions. Typical uncertainty percentages are 5% for CL, 7% for CD, 9% for L/D, and 20% for Cm. Theses values can be expected to double during stall. Wind Tunnel Corrections Corrections are app lied to the coefficients of lift, drag, and pitching moment based upon wind tunnel blockage, and model flex ibility effects. The solid bl ockage effect is due to the presence of the model within the wind tunnel, th us decreasing the effective area of the test

PAGE 44

44 section and increasing the flow velocity (and th e coefficients) in the vicinity of the model [128]. Wake blockage occurs when the flow outside of the m odels wake must increase, in order to satisfy the flow continuity in a closed tunnel. In an open freestream, the velocity outside of the wake would be equal to the freestream velocity. The effect of wa ke blockage is proportional to the wake size, and therefore proportional to drag [3]. Streamline curvature blockages are the effect of the tunnel walls on the stream lines around the model. The streamlines are compressed, increasing the effective camber and lift [129]. Such corrections gene rally decrease both lift and drag, while the pitching mom ent is made less negative, with per centage changes on the order of 2-3%. Finally, flexibility effects within the wind tunnel setup must be accounted for. These effects are primarily caused by the elasticity of the internal strain gage sting balance; under load the wind tunnel model will pitch up via a rigid body rotation. Visual image correlation (described below) is used to measure the displacement at points al ong the wing known to be nominally rigid (specifically, the sting balance attachment points al ong the wing root). This data then facilitates the necessary transformations and translations of the wing surface, and is used to correct for the angle of attack. is a positive monotonically incr easing function of both lift and dynamic pressure, and can be as large as 0.7 at high angles of attack [47]. Visual Image Correlation W ind tunnel model deformation measurements are a crucial experime ntal tool towards understanding the role of structural com position upon aerodynamic performance of a MAV wing. The flexible membrane skin generally limits applications to non-contacting optical methods, several of which have been repor ted in the literature. Galvao et al. [89] use stereo photogrammetry for disp lacement measurements of a membrane wing, with a reported resolution between 35 and 40 m. Data is available at discrete ma rkers placed along the wing. Projection

PAGE 45

45 moir interferometry requires no such marker placement (a fringe pattern is projected onto the wing surface), and the resulting data set is full -field. However, displacement resolutions reported by Fleming et al. [130] are relatively poor (250 m ), the dual-camera system must be rotated during the -sweep, and only out-of-plane data is av ailable, making strain calculations (if needed) impossible. Burner et al. [131] discuss the use of phot ogramm etry, projection moir interferometry, and the commercially available OptotrakTM system. The authors find no single technique suitable for all situa tions, and that a cost-benefit tradeoff study may be required. Furthermore, the methods need not be mutually ex clusive, as situations may arise wherein they can be used in combination. For the current work, a visual image correlation system (VIC), originally developed by researchers at the University of South Carolina [132], is used to measure wing geom etry, displacements, and plane strains. The underlying principle of VIC is to calcu late the displacement field by tracking the deformation of a subset of a random speckle pa ttern applied to the sp ecimen surface. The random pattern is digitally acquired by two cameras before and after loading. The acquisition of images is based on a stereo-triangulation techniqu e, as well as the computing of the intersection of two optical rays: the stereo -correlation matches the two 2-D frames taken simultaneously by the two cameras to reconstruct the 3-D geometry. The calibration of the two cameras (to account for lenses distortion and determine pixel spaci ng in the model coordinates) is the initial fundamental step, which permits the determina tion of the corresponding image locations from views in the different cameras. Calibration is done by taking images (with both cameras) of a known fixed grid of black and white dots. Temporal matching is then used: the VIC syst em tries to find the region (in the image of the deformed specimen) that maximizes a norma lized cross-correlation function corresponding to

PAGE 46

46 a small subset of the reference image (taken when no load is applied to the structure) [132]. The im age space is iteratively swept by the parameters of the cross-correlation function, to transform the coordinates from the original reference frame to coordinates within the deformed image. An originally square subset in the un-deformed image can then be mapped to a subset in the deformed image. As it is unlikely that the de formed coordinates will directly fall onto the sampling grid of the image, accurate grey-value interpolation schemes [133] are implemented to achieve op timal sub-pixel accuracy without bias. This procedure is repeated for a large number of subsets to obtain full-field data. In order to capture the three-dimensional feat ures and deformation of a wind tunnel model, twin synchronized cameras, each looking from a di fferent viewing angle, are installed above the wind tunnel ceiling, as can be seen in Figure 3-1. As the cameras must remain stationary through the experim ent (to preser ve the information garnered fr om the calibration procedure), a mounting bracket is constructed to straddle the tunnel, and prevent the transmission of vibration. Optical access into the test section is through an optical glass ceiling. The results of conducting visual image correlation tests with a glass inte rface between the cameras and the specimen have been studied, with litt le benign effects reported [134]. Furthermore, th e ca meras are initially calibrated through the window to ensure minima l distortion. Two 250 W lamps illuminate the model, enabling the us e of exposure times of 5 to 10 ms. The twin cameras are connected with a PC via an IEEE 1394 firewire cable, and a specialized unit is used to synchronize the camera triggers for instantaneous shots. A standard acquisition board installed in the computer carries out the digitalization of the images, and the image processing is carried out by custom soft ware, provided by Correlated Solutions, Inc. Typical data results that can be obtained from the VIC system consist of the geometry of the

PAGE 47

47 surface in discrete x, y, and z coordinates (where the origin is located at the centroid of the speckled area of interest, and the outward norma l points towards the cameras, by default), and the corresponding displacements along the wing ( u, v, and w). The VIC system places a grid point every N pixels, where N is user defined. A final post-processing option involves calculating the in-plane strains ( xx, yy, and xy). This is done my mapping the displacement field onto an unstructured triangular mesh, and conducting the appropriate numerical differentiation (the complete defini tion of finite strains is used). Data Procession The objectiv e of most of the wi nd tunnel tests given in the re mainder of this work is to determine the deformation of the wings under steady aerodynamic loads, at different angles of attack and free stream velocities, while simulta neously acquiring aerodynamic force data. Each angle of attack requires a separate wind-off reference image: failure to do so will inject rigid body motions (as the body moves seque ntially from one angle of attack to the next) into the displacement fields. If each reference image ta ken for VIC is of the fully assembled wing, the amount of pre-strain in the wing is not include d in the measured strain field, but only those caused by the aerodynamic loads. This condition needs to be carefully considered in the evaluation of the results, since the areas of rela xation of the pre-existing tension will generate areas of virtual compression w ithin the skin. The thin me mbrane cannot support a genuine compressive stress (it will wrinkle), but ne gative Poisson strains are possible. An alternative procedure uses the un-stretched sheet of latex rubber (prior to adhesion on the wing) as a reference image. This provides the state of pre-stra in in the membrane, as well as the absolute strain field duri ng wind tunnel testing, but makes the displacement fields very difficult to interpret and is not used here. The pre-strain data is merely recorded (with a separate set of reference and deformed images), but not used as a reference for further aerodynamic

PAGE 48

48 testing. As mentioned above, the acquired displ acement field will be composed of both elastic wing deformation and rigid body motion/translations originating from the sting balance, the latter of which must be filtered out. The computed strains are unaffected by these motions. Uncertainty Quantification In order to estim ate the resolution error of th e VIC system, a simple ad hoc experiment is conducted. A known displacement field is applied to a structure, and then compared with the field experimentally determined by way of imag e correlation. A thin latex membrane is stretched and fixed to a rigid aluminum ring w ith a diameter of 100 mm. The center of the membrane circle is then indented with a rigid steel bar with a s pherical head of 8 mm diameter. The bar is moved against the membrane by a micrometer with minimum increments of 0.25 mm. Results, in terms of the error between commanded displacement (via the micrometer) and the measured displacement at the apex of the membra ne profile, directly beneath the axis of the indentation bar, are given in Figure 3-2. Figure 3-2. Quantification of the re s olution error in the VIC system. Three different VIC setups are shown: 0.5, 3, and 10 speckles per millimeter of membrane, the latter of which corresponds to 2 pixels per speckle (with half the membrane in view). As

PAGE 49

49 expected, the error is smallest for the finest speckle pattern, whose readings randomly oscillate about zero, with a peak error of 0.018 mm (0.6%). The coarse r speckling patterns randomly oscillate at an offset error of 0.04 mm, with a peak error of 0.077 mm (2.2%). This places the resolution error for the VIC system between 10 and 20 m, about twice the resolution reported for the photogrammetry system [89]. Though not explicitly discussed here, the strain resolution is estim ated to be between 500 and 1000 (a non-dimensional parameter independent of speckle size), a high value (compared to strain gages, for example) owing to the differentiation methods used. Model Fabrication and Preparation Only the wing (152 mm wingspan, 124 mm root chord, 1.25 aspect ratio) of the MAVs seen in Figure 1-1 and Figure 1-2 is considered in this wor k. The cam ber at the root is 6.8% (at x/c = 0.22), the reflex at the root is -1.4% (at x/c = 0.86) and 7 of positive geometric twist (nose up) is built into the wingtip. The MAV wing has 7 of dihedral between 2y/b = 0.4 and the wingtip. The fuselage, stabilizers, and prope ller are omitted from both computations and experiments. The leading edge, inboard portion of the wing, and perimeter (of the PR wings) are constructed from a bi-directio nal plain weave carbon fiber laminate with 3000 fibers/tow, preimpregnated with thermoset epoxy. The battens (for the BR wings) are built from uni-directional strips of carbon fiber. These materials are plac ed upon a tooling board (appropriately milled via CNC) and cured in a convection oven at 260 F for four hours. A wind tunnel attachment (to be fastened to the aforementioned jaw mechanism) is bonded along the root of the wing between x/c = 0.25 and 0.8. The latex rubber skin adhered to this wi ng surface is 0.12 mm thick, and approximately isotropic. A random speckle pattern is applied to the la tex sheet with flat black spray paint, and then coated with a layer of du lling spray. Each paint speckle, while relatively brittle, has a small

PAGE 50

50 average diameter (less than 0.5 mm) and is genera lly not connected to an other speckle pattern; the pattern should not provide sign ificant reinforcement to the la tex. If information concerning the state of pre-strain in the skin is desired, a picture of the un-stretched latex sheet is taken for future use as a reference in the VIC system. Th e latex is then appropria tely stretched about a frame (or not at all if a slack membrane is desired), and adhered to the upper carbon fiber wing surface (which must be painted white) with spray glue. After the glue has dried, the excess latex is trimmed away. A picture of the resultant wind tunnel model is given in Figure 3-3. Figure 3-3. Speckled batten-re inforced membrane wing with wind tunnel attachment.

PAGE 51

51 CHAPTER 4 COMPUTATIONAL FRAMEWORK AND VALIDATION Several d ifficulties are associated with modeling the passive shape adaptation of a flexible micro air vehicle wing. From a fluid dynamics standpoint, the low aspect ratio wing (1.25) forces a highly three-dimensional flow field, and the low Reynolds number (105) implies strong viscous effects such as flow separation, transiti on, and potential re attachment. Structurally, the mechanics of the rubber membrane inflation are i nherently nonlinear, a nd the orthotropy of the thin laminated shells used for the wing sk eleton is dependent on the plain weave fiber orientation. Further difficulties arise with the inclusion of pretension within the membrane. Only static aeroelasticity is considered here. Several computational membrane wing studies have included unsteady effects [28] [81], and are thus able to study phenom ena such as vortex shedding [18], membrane vibration (unstable [62] or otherwise), unsteady interactions between the separated flow and the tip vortices [6], and wind gusts [8]. Past wind tunnel work, however, has indicated that MAV me m brane inflation is essentially qausi-static for a large range of angles of attack up to stall [135], and that adequate predictive capability still exists for those flight conditions with obvious unsteady features [43]. Structural Solvers The unstructured m esh used for finite element analysis can be seen in Figure 4-1. 2146 nodes are used to describe the surface of th e sem i-wing, connected by 4158 three-node triangle elements. The same mesh is used for both ba tten and perimeter-reinforced computations, by using different element-identificatio n techniques, as seen in the fi gure. Greater local effects are expected in the membrane areas of the wing, and the mesh density is altered accordingly. Nodes that lie along the wing root between x/c = 0.25 and 0.8 are given zero displacement/rotation boundary conditions, to emulate the restrictiv e effect of the wind tunnel attachment ( Figure 3-3).

PAGE 52

52 All nodes that lie on the wing root are constr ained appropriately as necessitated by wing symmetry. Figure 4-1. Unstructured triangular mesh used for finite element analysis, with different element types used for PR and BR wings. Composite Laminated Shells Discrete Kir chhoff triangle plate elements [136] are use to model the bending/twisting behavior of the carbon fiber areas of the wings: le ading edge, root, perim eter, and battens. Due to the comparative stiffness of th ese materials, linear behavior is assumed. The orthotropy of the plates is introduced by the flexural stiffness matrix of the laminates, Dp, relating three moments (two bending, one twisting) to three curvatures: NL 32 p kkkk k1h12hz DQ (4-1) where NL is the number of layers in the laminate, hk is the thickness of the kth ply, zk is the normal distance from the mid-surface of th e laminate to the mid-surface of the kth ply, and Qk is the reduced constituitive matrix of each ply, expressed in global coordinates. Qk depends upon the elastic moduli in the 1 and 2 directions (equal for the bi-directional laminate, but not so for the uni-directional) E1 and E2, the Poissons ratio 12, and the shear modulus G12. The finite element stiffness matrix pertaining to bending/twisting is then found to be:

PAGE 53

53 oTT pppp AA KTBDBT d (4-2) where T is a matrix which transforms each element from a local coordinate system to a global system, Bp is the appropriate strain-displacement matrix [137], and Ao is the un-deformed area of the triangular element. Kp is a 9x9 matrix whose compone nts reflect the out-of-plane displacement w and two in-plane rotations at the three nodes. Similarly, in-plane stretching of the lamina tes (a secondary con cern, but necessarily included), is given by: NL p kk k1h AQ (4-3) where Ap is a laminate matrix relating three in-pla ne stress resultants to three strains. Expressions similar to Eq. (4-2) are then formulated to compute Km, the 6x6 finite element stiffness matrix governing in-plane displacements u and v at the three nodes. Km and Kp are then combined to form the complete 15x15 shell stiffness matrix of each element, Ke. Drilling degrees of freedom are not included. T hough some wing designs may use un-symmetric laminates, coupling between in-plane a nd out-of-plane motions is not included. Loads Model Validation/Estimation. The following method is used to both validate the model presented above, and identify the relevant mate rial properties of the la minates. A series of weights are hung from a batten-reinforced wing (w ith 2 layers of bi-directional carbon fiber oriented 45 to the chord line a nd 1 layer uni-directional battens, but no membrane skin) at nine locations: the two wing tips, the tr ailing edges of the six battens, and the leading edge, as shown in Figure 4-2. VIC is used to measure the resu lting wing dis placements. A linear curve is fit through the load-displacement data of all nine poi nts due to all nine load ings. The slopes of these curves are used to popul ate the influence matrix in Table 4-1: the diagonal gives the motion

PAGE 54

54 of a wing location due to a force at that lo cation; the off-diagonals represent indirect relationships. Figure 4-2. Computed deforma tions of a BR wing skeleton due to a point load at the wing tip (left) and the leading edge (right). A genetic algorithm is then used for system identification. The six variables are the material parameters: E1, 12, and G12 of both the plain weave and the battens. E2 is assumed to be equal to E1 for the plain weave, and equal to 10 MPa fo r the uni-directional battens. This latter value has little bearing on the re sults, as the 1 direc tion corresponds with the axis of the batten. The objective function is the sum of the squared error between the diagonals of the computed and the measured influence matrix. The error terms are appropriately normalized before summation and off-diagonal components are not cons idered in the optimization. For the genetic algorithm, the population size is 20, the elitism count is 2, reproduction is via a two-point crossover function with a 0.8 cro ssover fraction, and a uniform mu tation function is used with a 0.01 mutation rate. Convergence is adequately achieved after 30 itera tions, with each function evaluation call requiring a single finite element analysis. The resulting numerical influence matrix is given in Table 4-2. This matrix is symmetric, whereas th e experimental matrix is slightly un -symmetric, probably due to manufacturing errors.

PAGE 55

55 For the plain weave, E1 = 34.8 GPa, 12 = 0.41, and G12 = 2.34 GPa. For the uni-directional battens, E1 = 317.2 GPa, 12 = 0.31, and G12 = 1.05 GPa. The model correctly predicts the very stiff leading edge (point 1), and the negative influence it has on the remainder of the wing (as shown on the right of Figure 4-2). The rest of the points alon g the wing positively influence one another. Errors between the two matrices are typically on the order of 5-10%; the numerical wing is generally stiffer than the actual wing. As expected, the weakest battens are the longest, found towards the root (points 8 and 9). The wingtips (points 2 and 3) generally have the greatest indirect influence on the rest of the wing (as shown on the left of Figure 4-2). Force-displacem ent trends at th e nine locations along the wing, due to loads at those points (the diagonal terms in the matrices) are given in Figure 4-3, showing a suitable match between model and experim ent. With the exception of th e leading edge, two data points are given for each load level, corresponding to the data from the left and right sides of the wing. Higher fidelity methods for system identification of a ca rbon fiber MAV skeleton are given by Reaves et al. [138], who utilize model update techniques with uncertainty quantification m ethods. This is largely done due to the uncertainty in the lami nate lay-up, predominat ely in ply overlapping regions within the skeleton, which is not an issue for the current work. Table 4-1. Experimental influence matrix (mm/N) at points labeled in Figure 4-2. 1 2 3 4 5 6 7 8 9 1 1.58 -2.90 -2.97 -3.00 -3.07 -3.47 -3.54 -3.05 -3.03 2 -2.93 104.65 3.94 50.85 5.88 36.47 7.65 26.06 9.99 3 -3.07 3.67 118.468.18 52.24 9.39 38.29 13.33 29.22 4 -3.68 49.05 5.69 329.118.71 44.14 10.51 33.05 13.23 5 -3.78 5.68 50.15 9.17 366.9211.89 45.19 14.98 33.68 6 -4.33 36.41 7.92 44.63 10.68 547.5013.41 38.46 15.23 7 -4.37 7.77 36.74 11.46 44.35 13.28 513.8317.40 38.19 8 -4.75 24.30 9.02 30.15 11.24 34.68 12.63 757.00 15.81 9 -4.76 9.22 25.09 12.34 30.30 14.32 35.51 17.65 742.25

PAGE 56

56 Table 4-2. Numerical influence ma trix (mm/N) at points labeled in Figure 4-2. 1 2 3 4 5 6 7 8 9 1 1.49 -2.92 -2.92 -3.61 -3.61 -4.24 -4.24 -4.60 -4.60 2 -2.92 107.96 3.36 54.49 5.05 33.10 6.80 20.84 8.13 3 -2.92 3.36 107.965.05 54.49 6.80 33.10 8.13 20.84 4 -3.61 54.49 5.05 312.006.78 39.30 8.51 25.09 9.73 5 -3.61 5.05 54.49 6.78 312.008.51 39.30 9.73 25.09 6 -4.24 33.10 6.80 39.30 8.51 584.2610.14 28.81 11.18 7 -4.24 6.80 33.10 8.51 39.30 10.14 584.2611.18 28.81 8 -4.60 20.84 8.13 25.09 9.73 28.81 11.18 776.48 11.95 9 -4.60 8.13 20.84 9.73 25.09 11.18 28.81 11.95 776.48 Figure 4-3. Compliance at vari ous locations along the wing, due to a point load at those locations. Membrane Modeling In the m odeling of thin, elastic membranes (w ith no resistance to a bending couple), three basic options are available. If there exists a significant pre-strain field throughout the sheet, linear modeling is possible by assuming inextensib ility: the pre-strain overwhelms the strains that develop as a result external loading. As th ese strains grow in magnit ude (or if the membrane is originally slack) a nonlinear model must be used, as the membranes resistance depends upon the loading (geometric nonlinearity). However, a linear constituitive relationship is still typically

PAGE 57

57 valid up to a point, after which the membrane b ecomes hyperelastic (mater ial nonlinearity), and the stress-strain relationship changes with increasing load. Linear Modeling. Geometric stress stiffening provide s a relationship be tween in-plane forces and transverse deflection [137], and is indicative of a stru ctures reluctance to change its state of stress. For an initially flat membrane with a transverse pressure, the constitutive equation is: xxxxyxyyyyNw,2Nw,Nw,p0 (4-4) where w is the out-of-plane displacement (as above), Nx, Nxy, and Ny are the in-plane pre-stress resultants, and p is the applied pressure field. For an isotropic stress field with no shear, this equation reduces to the we ll known Poissons equation [139]. This model assumes that the displacem ent along the membrane is purely out-of-plane; thus the membrane is inextensible in response to a pressure field (althoug h extensibility is needed to appl y the initial pre-stress field). The resulting finite element model is fairly inexpensive, as each node has only one degree of freedom, and standard direct linear solvers can be used. This model is thought to be accurate for small pressures, small displacements, and larg e pre-stresses. Though it is not expected that the MAV wing displacements will be particularly large (typically less than 10% of the root chord), it is expected that a slack membrane skin may provide many aerodynamic advantages. As the solution to Eq. (4-4) becomes unbounded as the pre-stress approaches zero, higher fidelity models will also be pursued for the current work MAV wing simulations with linear membrane models can be found in the work of Stanford and Ifju [14], Thwaites [65], and Sugimoto [80]. Nonlinear Modeling. The nonlinear membrane modeling di scussed in this section will incorporate geometric nonlinearities, but Hookes law is assumed to still be valid. For the inflation of a circular me mbrane, Pujara and Lardner [140] show that linear and hyperelastic

PAGE 58

58 constituitive relationships provide the same numerical solutions up to deformations on the order of 30% of the radius, a figur e well above the deflections expected on a membrane wing. Geometric nonlinearity im plies that the deformation is large enough to warrant finite strains, and that the direction of the non-cons ervative pressure loads significantly changes with deformation. Eq. (4-4) is still valid, only now the stress resultants depend upon the state of pre-stress, as well as in-plane stretching, which in turn depends on the out-of-plane displacement. Furthermore, the rotation of the membrane is no longer well-appr oximated by the derivative of w, rendering the equilibrium equation nonlinear. Three displacement degrees of freedom are required per node (u, v, w), rather than the single w used above. Finite element implementation of such a model is described by Small and Nix [141] and Pauletti et al. [142]. The strain pseudo-vector with in each elem ent is given as: oLoeLeBXBX (4-5) where o and L represent the division of the linear (inf initesimal) and nonlinea r contributions to the Green-Lagrange strain, Xe is a vector of the degrees of freedom in the elements (three displacements at th e three nodes), and Bo and BL are the appropriate stra in-displacement matrices (the latter of which depends upon the nodal displacements) [143]. The pre-stress (if any) can be included into the m odel in one of two methods. Fi rst, they can be simply added to the stresses computed my multiplying the strain vector of Eq (4-5) through the constituitive matrix. This may cause problems if the imposed pre-stress di stribution does not exactly satisfy equilibrium conditions, or if there is excessive curvature in the membrane skin: the membrane will deform, even in the absence of an external force. A second option is to use the pre-stresses in a finite element implementation of Eq. (4-4), then add the resulting stiffness matrix and for ce vectors to the nonlinear terms. For a flat

PAGE 59

59 membrane with uniform pre-stress, the two met hods are identical. The internal force in each element Pe can be computed from the principle of virtual work: oT eoLeemwe VV PTBBXXATKX d (4-6) where Am is the linear constitutive elastic matrix of the membrane, Am is the stress pseudovector within each element, and Kw is the stiffness matrix represen tation of Eq. (4-4), containing only terms related to the out-of-plane displacement w. The tangential stiffness matrix Ke is then the sum of the geometric K, constituitive Kc, external Kext and pre-stress stiffness matrix Kw: oTT VV KTGMGTd (4-7) oT T coLeemoLee VV KTBBXXABBXXTd (4-8) extee KFX (4-9) where G is a matrix linking the nodal degrees of fr eedom to a displacement gradient vector [144], M is a stress matrix whose elements can be found in [137], and Fe is the external force vector. Computation of the skew-symmetric external stiffness matrix is given in [142]. Fe must be written in the unknown de formed configuration: T epA/3 FTIIIn (4-10) where A is the deformed area of the triangle, p is the uniform pressure over the element, I is the identity matrix, and n is the unit normal vector to the defo rmed triangular finite element. The resulting non-linear set of equations is solved with Newtons recurrence formula [142]. The above m ethod essentially separates the line ar and nonlinear stiffness contributions. If the pre-stress in the membrane is very large, Kw will overwhelm its nonlinear counterparts, and membrane response will be essentially linear fo r small pressures and displacements. Continued

PAGE 60

60 inflation will transition from linear to nonlinear response [145]. In the event of a slack m embrane, the membranes initial response to a pre ssure will have an infinite slope until strains develop and provide stiffness. Numerous memb rane wing models use some variant of the geometrically nonlinear model de scribed above: Stanford et al. [43], Ormiston [15], Smith and Shyy [72], Jackson and Christie [76], and Levin and Shyy [104]. Inflatable Diaphragm Validation. In order to validate the above m embrane models, the material properties of the latex are first identified with a uni-axial tension test. The test specimen has a width of 20 mm, a length of 120 mm, and a thickness of 0.12 mm. The latex rubber sheets are formed in a rolling process, implying an orthotropy, though specimens cut from different orientations yield very similar results. VIC is used to monitor both the extensional and the Poisson strains: data is sample d at 50 pixel locations within the membrane strip, and then averaged. The resulting data can be seen in Figure 4-4, and is used to identity the linear elastic modulus and the Poissons ratio. A linear fit through the stress-strain curve results in a m odulus of 2 MPa; the nonlinear stress-softening beha vior for higher strains is the hallmark of hyperelasticity [146]. The Poissons ratio for small strains is 0.5, a result of the materials incom pressibility. Using these material parameters to construct the constitutive matrix Am, the finite element model can be appropriately validated with the Hencky test [144]: a flat circular membrane (with or without p re-tension), clamped along its boundary, and subjected to a uniform pressure [145]. A 57.15 mm radius is chosen in order to em ul ate the length scale of a micro air vehicle. Although the problem is axisymmetric, a full circul ar mesh is used for numerical computations. Experimentally, VIC is used to monitor the sh ape of the membrane, while a Heise pressure transducer measures the pressure within a chamber, to the top of which the membrane sheet is

PAGE 61

61 fixed. Results are given in Figure 4-5, in terms of the displacement of the membrane center (norm alized by the radius) versus pressure. Figure 4-4. Uni-axial stretch test of a latex rubber membrane. Figure 4-5. Circular membrane response to a uniform pressure. Two cases are considered: a slack membrane, a nd a taut membrane. Computational results from both the linear and the nonlinear models fo rmulated above are given. As expected, the response of the slack membrane to an applied pr essure is at first unbounded, but becomes finite with the advent of the extensional strains. The linear model is useless for a slack membrane (unbounded), but the nonlinear mode l can predict this behavior. The correlation between model

PAGE 62

62 and experiment is adequate up to w/R = 0.22 (slightly lower than the value given by Pujara and Lardner [140]), when the model begins to under-predic t the inflated shap e. Hyperelastic effects appear after this point: Hookes law over-predicts the stress for a given strain level ( Figure 4-4), and thus th e membranes resistance to a transverse pressure. For the case with membrane pre-tension, VIC is used to measure the pre-strain in the membrane skin (applied radially [147]), the average of which is then used for finite element com putations. The mean pre-strain is 0.044, with a coefficient of variation of 3.1%. For this case, the linear model now has a small range of validity, up to w/R = 0.15. Prior to this deformation level, linear and nonlinear models predict the same membrane inflation. The response then becomes nonlinear, due to the advent of finite stra ins, but also because a relevant portion of the uniform non-conservative pressure is now directly radially, rather than vertically. The nonlinear model and experiment now diverge at w/R = 0.3: the addition of a pre-tension field increases the range of validity of both the linear and the nonlinear membrane models. Skin Pre-tension Considerations A state of uniform membrane pre-tension, though numerically convenient [15] [80] [14], is essentially impossible to actually fabricate on a M AV wing. One reason is that the latex sheets used on the MAVs in this work are not much wider than the wingspan, subjecting the state of pre-stress to end-effects. This may perhaps be remedied with larger sheets and a biaxial tension machine, which hardly seems worth the effort for MAV construction. Another problem is the fact that the wing is not a flat surface. Even if a state of unifo rm pre-tension were attainable, it cannot be transferred to the wing without significant fi eld distortions, particularly due to the camber in the leading edge. A typical pre-strain field is given in Figure 4-6, as measured by the VIC system off of a BR wing in the chordwise direc tion. The contour on the left is the pre-strain field after the spray adhesive has dried, but before the late x surrounding the wing has been de-

PAGE 63

63 pinned from the stretching frame (as discu ssed above). The contour on the right of Figure 4-6 is the pre-strain after the excess la tex has been trimmed away. Figure 4-6. Measured chordwise pr e-strains in a BR wing before the tension is released from the latex (left), and after (right). The pre-strains measured from the carbon fiber areas of the wing (leading edge, root, battens) are meaningless, as the shell mechanics largely govern the response in these areas. The large extensional strains (~12%) at the leading ed ge are indicative of the fact that the wing skeleton is flattened against the membrane until the spray glue dries. At this point, the wing is allowed to re-camber, causing the latex adhered to its top surface to stre tch. The anisotropic nature of the pre-tension field is very evident, with strains ranging from between 4% to 9% on the left semi-wing and slightly higher on the righ t. Furthermore, when the surrounding latex is de-pinned from its frame the membrane at the trai ling edge contracts, leaving an area of almost no tension (right side of Figure 4-6). This is a result of th e B R wings free trailing edge, and would not be a problem with a perimeter-reinforced wing. One numerical solution to such a problem is to interpolate the data of Figure 4-6 onto the finite elem ent grid, and compute the pre-stress w ithin each element, as discussed by Stanford et al. [43]. This method, though accurate, would re quire an exp erimental VIC analysis in conjunction with every numerical analysis; not a cost -effective method for thorough exploration of the design space. Eq. (4-4) however, is a natural smoothing operator [139]; simply averaging the pre-strains for the co mputations, though crude, can in some cases be relatively accurate. The

PAGE 64

64 match between measured and predicted membrane deformation for the taut case in Figure 4-5 is very good, d espite the fact that the numerical pre-strains were presumed uniform. The error resulting from a uniform pre-stress assumption can be estimated with the following method. The pre-strain distribution throughout a flat ci rcular membrane is considered a normally-distributed random variable: each finite element has a different pre-strain. The linear membrane model of Eq. (4-4) is then used to compute the displacemen t at the center of the membrane due to a hydrostatic pressure. The same membrane is then given a constant pre-strain distribution (the average value of the randomly-distributed pre-st rain), and the central deflection is recomputed for comparison purposes. Monte Ca rlo simulations are then used to estimate the average error at the membrane center, for a give n coefficient of variation of the pre-strain. The results of the Monte Carlo simulation are given in Figure 4-7. Each data point is the percentage error betw een the central displa cem ent computed with a non-homogenous random pre-strain, and that with a constant pre-strain. Each error percentage is the average of 500 finite element simulations. The radius of the circle is 57.15 mm, the thickness is 0.12 mm, the elastic modulus is 2 MPa, the Poissons ratio is 0.5, and the hydrostatic pressure is fixed at 200 Pa. The mean pre-strain is 0.05, and the standard devia tion is decided by the COV of each data points abscissa. Nonlinear membrane modeling is not used. The smoothing nature of the Laplacian operator in Eq. (4-4) is very evident: even in th e presence of 30% spatial pre-strain variability, the error in assuming a constant pre-strain is still less than 5%. On one hand, the error in Figure 4-7 is probably under-predicted, as strain canno t truly b e a spatially-random variable: on a local scale measured strain may seem random, but on a global scale it must satisfy the compatibility equations [146]. Both of these scale-trends are evident in Figure 4-6. On the other hand, Figure 4-7 represen ts the worst case scenario, as nonlin ear membrane effects will dilute the importance

PAGE 65

65 of the pre-tension [145], whatever its distributio n throughout the m embrane skin. Figure 4-7. Monte Carlo simulations: error in the computed membrane deflection due to a spatially-constant pre-strain distribution assumption. Though the above results indicate the appropr iateness of using a constant membrane prestress for MAV wing computations (despite an inability to reproduce this in the laboratory), the tension relaxation at the free trailing edge of the BR wing (seen in Figure 4-6) should be corrected fo r. Regardless of the amount of pr e-tension placed in a batten-reinforced membrane, the pre-stress traction normal to the free trailin g edge will always be zero, producing a stress gradient. This can be accounted for in the following manner: 1. Specify the pre-stress field within the membrane skin (uniform or otherwise). 2. Compute the traction due to this pre-stress along the outward normal, at each edge in a membrane finite element that coincides with a free surface. 3. Apply a transverse pressure along each edge, equal and opposite to th e computed traction. 4. Compute the resulting stress fiel d (while holding the carbon fi ber regions of the wing rigid), add this field to the prescribed stress in st ep 1, and use the result as the new pre-stress resultant field for aeroelastic computations. The resulting pre-stress field will be very small along the free edge, and approach the original specified value deeper into the wing towards the leading edge, as can be seen in Figure 4-8. For this exam ple, about a fourth of the membrane area is affected by the free edge, while the remainder retains a pre-stress close to th e prescribed value (a result validated by Figure 4-6).

PAGE 66

66 As mentioned above, this pre-stress correction only needs to be applied for simulations of a batten-reinforced wing. Figure 4-8. Computed pre-stress resultants (N/m) in the chordwise (left), spanwise (center), and shear (right) in a BR wing, corrected at th e trailing edge for a uniform pre-stress resultant of 10 N/m. Fluid Solvers As discussed above, several viscous effects dom inate the flow about a micro air vehicle wing: laminar separation, turbulent transition an d reattachment, periodic shedding and pairing, and three-dimensional flow via wing tip vortex sw irling. An inviscid flow solved such as the vortex lattice method is unable to predict any of thes e effects (drag, in particular, will be severely underestimated), but its small co mputational expense is attractiv e. Solving the steady NavierStokes equations represents a substantial increase in cost, but an equally large step forward in predictive capability. Some asp ects of the flow (namely, turb ulent transition over a separation bubble, and subsequent shedding), still cannot be predicted with the methods presented here. Vortex Lattice Methods This sec tion briefly describes a well-developed family of methods for predicting the steady lifting flow and induced drag over a thin wing at small angles of attack. The continuous distribution of bound vorticity over the wing is approximated by discretizing the wing into a

PAGE 67

67 paneled grid, and placing a horseshoe vortex upon each panel. Each horseshoe vortex is comprised of a bound vortex (which coincides with the quarter-chord line of each panel), and two trailing vortices extending downstream. Each vortex filament creates a velocity whose magnitude is assumed to be governed by the Biot-Savart law [27]. Furthermore, a control point is placed at the three-quarter-cho rd point of each panel. The velocity induced at the mth control point by the nth horseshoe vortex is: xyz m,nm,nnm,nm,nnm,nm,nnuCvCwC (4-11) where u, v, and w are the flow velocities in Cartesian coordinates, is the vortex filament strength, and Ci are influence coefficients that depend on the geometry of each horseshoe vortex and control point combination. The complete induc ed velocity at each control point is the sum of the contributions from each horseshoe vortex resulting in a linear system of equations. The strength of each vortex must be found so that the resulting flow is tangent to the surface of the wing: the wing becomes a streamline of the flow. This requirement is enforced at each control point by: mm mmmmUcosuvUsinwFx,y,z0 (4-12) where U is the free-stream velocity, is the angle of attack, and F( x,y,z) = 0 is the equation of the surface of the wing. Inserting the relevant terms of Eq. (4-11) into Eq. (4-12) provides a linear system of equations for the filament streng th of each horseshoe vortex. Micro air vehicle simulations that utilize a vortex lattice method ar e typically forced to do so by the computational requirements of optimization (as is the case in the current work). Examples can be seen in the work of Ng and Leng [52], Sloan et al. [53], and Stanford et al. [61]. Steady Navier-Stokes Solver The three -dimensional incompressible Navier-S tokes equations, written in curvilinear

PAGE 68

68 coordinates, are solved for the steady, laminar flow over a MAV wing. As before, the fuselage, stabilizers, and propeller are not taken into account. The computational domain can be seen in Figure 4-9, with the MAV wing enclosed within. Inlet and outlet boundaries are m arked by the flow vectors; velocity is specifi ed at the inlet, and a zero-pressure boundary condition is enforced at the outlet. The configuration shown in Figure 4-9 is for simulations at a model inclination of 0 angle of attack. Fo r non-zer o angles, the lower and upper surf aces will also see a mass flux, rather than re-meshing the wing itself. The sidewalls are modeled as slip walls, and thus no boundary layer forms. The MAV wing its elf is modeled as a no-slip surface. Figure 4-9. CFD computational domain. Because no flow is expected to cross the root -chord of the wing (unsteady effects that may lead to bilateral asymmetry [6] are not included; nor is prop eller slipstream [45]), symmetry is exploited by modeling only half of the computational domain (the plane of symmetry is also modeled as a slip wall). A detailed view of the resulting structured mesh (the nodes that lie on the plane of symmetry and the MAV wing) is given in Figure 4-10. 210,000 nodes fill half of the com putational domain, with 1300 nodes on the wing surface. This is a multi-block grid, with four patches coinciding with the upper an d lower wing surfaces. The wing itself has no thickness. Such a flow model should be able to adequately predict the strong tip vortex swirling 8c z x y 11c 12c 6c

PAGE 69

69 system (and the accompanying nonlinear lift and moment curves [3]), as well as the laminar flow separation against an ad verse pressure gradient [2]. Similar laminar, steady flow computations for low Reynolds num ber flyers can be found in the work of Smith and Shyy [72], Viieru et al. [38], and Stanford et al. [43]. Figure 4-10. Detail of structured CFD mesh near the wing surface. In order to handle the arbitrarily shaped geometries of a micro air vehicle wing with passive shape adaptation, the Navier-Stokes equations must be transformed into generalized curvilinear coordinates: (x,y,z), (x,y,z), (x,y,z). This transformation is achieved by [148]: xyz111213 xyz212223 xyz3132331 J f ff f ff f ff (4-13) where fij are metric terms, and J is the determinant of the transformation matrix: x,y,z J ,, (4-14) Using the above information, the steady Navie r-Stokes equations can then be written in three-dimensional curvilinear coordinates [149]. The continuity equation and u-m omentum equation are presented here in st rong conservative form, with the implication that the vand wmomentum equations can be derived in a similar manner.

PAGE 70

70 UVW0 (4-15) 1112 13 21 22 23 31 32 33 11 21 31UuVuWu quququ J ququququququ JJ ppp fff (4-16) where is the density, p is the pressure, is the viscosity, qij are parameters dictated by the transformation (expressions can be found in [149]), and U, V, and W are the contravariant velocities, given by the flux through a contro l surface norm al to the corresponding curvilinear directions: 111213 212223 313233Uuvw Vuvw Wuvw fff fff fff (4-17) In order to numerically solve the above e quations, a finite volume formulation is employed, using both Cartesian and contravariant velo city components [148]. The latter can evaluate the flux at the cell faces of the structur ed grid and enforce the conservation of m ass. A second order central difference operator is used for computations involving pressure and diffusive terms, while a second order upw ind scheme handles all convective terms [150]. Fluid Model Comparisons and Validation Validation of both the linear vortex lattice method and the nonlinear CFD is given in Figure 4-11, in terms of lift, drag, and longitudinal pitching m oments (measured about the leading edge) at 13 m/s. Pre-stall, the CFD mode l is able to accurately predict lift and drag within the experimental error bars of the measured data. Drag is consistently over-predicted at higher angles of attack; turbulent reattachment of separated flow is know to decrease the profile

PAGE 71

71 drag [8], but is not included in the model. The magnitude of the pitching mom ent is slightly over-predicted by the CFD, though the data still falls within the error bars, the slopes match well, and the onset of nonlinear behavior (due to the low aspect ratio [3]) is well-predicted. The CFD is also able to predic t the onset of stall (via a loss of lift) at about 21 but loses its predictive capability in the post-stall regime, as th e flow is known to be highly unsteady. Figure 4-11. Computed and measured aerodynami c coefficients for a rigid MAV wing, Re = 85,000. The vortex lattice method is accurate at low angles of attack, but the slope is underpredicted (possibly due to an inability to comput e the low-pressure cells at the wing tips, similar to vortex lift discrepancies seen in delta wings [27]), and the wing never stalls. The drag predicted by the vortex lattice method is necessarily augmented by a non-zero CDo (estimated

PAGE 72

72 from the experimental data), and is moderately accurate up to 10 angle of attack. After this point, the inviscid drag is under-p redicted due to massive flow separation over the wing. No significant differences can be seen between th e pitching moments predicted by the CFD and the vortex lattice method, until the af orementioned nonlinear behavior appears, which the inviscid solver cannot predict. Aeroelastic Coupling Transfer of data between a structured CFD mesh and an unstructured FEA mesh is done with simple polynomial interpolation. If information from grid A is to be in terpolated to grid B, the element from grid A (triangular for the FE A mesh, quadrilateral for the CFD mesh) is found which is closest to each node from grid B. Ex cept for nodes that lie on the wing border, these elements will enclose their corresponding nodes. Polynomial shape functions for the desired variable are formulated to descri be its distribution within the element, and then the value at the node is solved for. Such a method is found to ad equately preserve the integrated forces and the strain energies from one mesh to a nother, and is fairly inexpensive. The un-deformed (due to aerodynamic loads) wing shape technically depends upon the membrane pre-tension. This shape could be f ound with Eq. (4-4), sett ing the pressure source terms to zero, and letting the wing shape at nodes upon the carbon fiber-latex boundary be prescribed displacement boundary conditions. Such a scheme should result in slight concavities along the membrane surface [151]. This effect is considered sm all, however, and is ignored for the current work. Shear stress over the wing is also not included in the aeroelastic coupling. Moving Grid Technique For aeroelastic computations using the Navier-Stokes flow solver, a re-meshing algorithm is needed to perturb the stru ctured grid surrounding the flex ible wing (no such module is required when a vortex lattice method is used, as all of the nodes lie on the wing surface). For

PAGE 73

73 the current work, a moving grid routine based upon the master-slave concept is used to maintain a point-matched grid block interface, preserve grid quality, and prevent grid cross-over. Master nodes are defined as grid points that lie on the moving surface (the wing surface of the micro air vehicle, in this case), while the slave nodes cons titute the remaining grid points. A slave nodes nearest surface point is defi ned as its master node, and its movement is given by: ssmmxxxx (4-18) where xs is the location of the slave node, xm is the location of its master node, the tilde indicates a new position, and is a Gaussian distribution decay function: 222 smsmsm 222 mmmmmmxxyyzz exp-min500, xxyyzz (4-19) where is small number to avoid division by zero, and is a stiffness coefficient; larger values of promote a more rigid-body movement. Further information concerning this technique is given by Kamakoti et al. [152]. Numerical Procedure The steady fluid structure interaction of a MAV wing is computed as follows: 1. If computations involve a batt en-reinforced wing, correct for the membrane pre-tensions at the free trailing edge. 2. Solve for the aerodynamic pressures over the wi ng, using either the steady Navier-Stokes equations or the tangency conditi on of the vortex lattice method. 3. Interpolate the computed pressures from the flow solver grid to the FEA grid. 4. Solve for the resulting wing displacement using either the linear or the nonlinear membrane/carbon fiber model. 5. Interpolate the displacement onto the MAV wing of the flow solver grid. 6. If nonlinear CFD models are ut ilized, re-mesh the grid using the master/slave scheme. 7. Repeat steps 2-6 until suitable convergence is achieved: less than 0.1% change in the lift.

PAGE 74

74 Less than ten iterations are usually adequa te at modest angles of attack (3< <18). Typical results are given in Figure 4-12 for the lift and efficiency of both a BR and a PR wing, com puted with a Navier-Stokes flow solver and a nonlinear membrane solver. The lift of the PR wing monotonically converges (lift increases camber, which furthe r increases lift), while the history of a BR wing is staggere d (lift decreases wing twist, d ecreasing lift). For the nonlinear modules, step 2 requires between 150 and 250 sub-it erations, while step 4 can typically converge within 20 sub-iterations. For the linear modules, the equations of state can be solved for directly. Figure 4-12. Iterative aeroelastic convergence of membrane wings, = 9.

PAGE 75

75 CHAPTER 5 BASELINE WING DESIGN ANALYSIS Three bas eline micro air vehicle wing designs are considered in this section: first, a completely rigid wing. Secondly, a batten-reinfo rced wing with no pre-tension in the membrane skin, two layers of bi-directiona l carbon fiber at ply angles of 45 to the chord line (at the root and leading edge), and one laye r of uni-directional carbon fiber (f ibers aligned in the chordwise direction) for the battens. Third, a perime ter-reinforced wing with no pre-tension in the membrane skin and two layers of bi-directional carbon fiber at ply an gles of 45 to the chord line (at the root, leading edge, and perimeter). As a large number of function evaluations are not required for this strictly analysis section, all numerical results are comput ed with the higher-fidelity met hods discussed above: the steady Navier-Stokes solver and the nonlinear membrane solver. Furthermore, all results are found at U = 15 m/s, a value towards the upper range of MAV flight. A higher velocity is chosen to emphasize aeroelastic deformations. For a given flight condition, 10 VIC images are taken of the deformed wing (at 1 Hz) and averaged together Sting balance results are, as discussed, sampled at 1000 Hz for 2 seconds, and then averaged. Wing Deformation Numerical and experimental outof-plane displacements, normalized by the root chord, are given in Figure 5-1 for a BR wing, along with a section of the data at x/c = 0.5, at 15 angle of attack. As expected, the prim ary mode of wing deformation is a positive deflection of the trailing edge, resulting in a nose-down twist of each flexible wing section. Deformations are relatively small (~5%, or 6.2 mm), though still ha ve a significant effect upon the aerodynamics. The membrane inflates from between the battens (clearly seen in the section plot) towards the leading edge, but at the traili ng edge the wing shape is more homogenous and smooth, and no

PAGE 76

76 distinction between batten and me mbrane can be made. This is presumably due to the pressure gradient, with very high forces at the leading edge which diss ipates down the wing. The carbon fiber wing tips, though several orders of magnitude stiffer than the membrane, shows appreciable twisting, indicative of the large suction forces from the tip vortex. Co rrelation between model and experiment is acceptable, with the model s lightly under-predicting the adaptive washout, and over-predicting the local membrane inflation betw een the battens. Wing shapes and magnitudes match well with time-averaged results reported by Lian et al. [28]. Figure 5-1. Baseline BR normalized out-of-plane displacement (w/c), = 15. Chordwise strains for the same case as above can be seen in Figure 5-2. The directional stif fness of the battens generally prevents signif icant stretching in the ch ordwise direction. The model predicts appreciable strain (1.4%) at the carbon fiber/membrane interface towards the leading edge (due to inflation), almost no strain near the mid-chord region, and negative Poisson strains at the trailing edge. St rains in the carbon fiber regions, while computed, are much smaller than the membrane strains, and cannot be discerned in Figure 5-2. The measured chordwise strain is very sm all and noisy, with no evid ent differences between the carbon fiber and membrane regions. Much of the measured fiel d lies below the systems strain resolution (~1000 ). Several noise spikes are also evident in th is strain field, while the displacement field in Figure 5-1 has none; the strain diffe rentiation procedure is m ore se nsitive to experimental noise than the displacement temporal matching.

PAGE 77

77 Figure 5-2. Baseline BR chordwise strain ( xx), = 15. A better comparison between model and experiment is given in Figure 5-3, with the spanwise strains. These extensions are essentia lly a product of the change in distance between the battens as they deform. Both m ode l and experiment indicate a peak in yy between the inner batten and the carbon fiber root, though the model indicates this maximum towards the leading edge (~1.2%), while the measuremen t places it farther aft. Strain concentrations at the trailing edge are visible in both fields. Though stil l noisy, the VIC systems spanwise strain can differentiate between battens and membrane. Suit able model validation is also seen in shear ( Figure 5-4). Both peaks and distributions of th e anti-sym metric shear are well predicted. The tips of the battens at the traili ng edge cause a shear concentration, typically of opposite sign to the strain in the rest of the memb rane segments between each batten. Figure 5-3. Baseline BR spanwise strain ( yy), = 15.

PAGE 78

78 Figure 5-4. Baseline BR shear strain ( xy), = 15. Normalized out-of-plane displacements for th e perimeter-reinforced wing are given in Figure 5-5. Deformations are s ligh tly larger than with the BR wing (6%), and are dominated by the membrane inflation between the carbon fiber l eading and trailing edges. The membrane apex occurs approximately in the middle of the membra ne skin, despite the pre ssure gradient over the wing. This location is a function of angle of att ack, as the peak will move slightly forward with increased incidence [74], [43]. The carbon fiber wing tip twists less than previously, thought to be a resu lt of the fact that the wingtip is not free in a PR configurati on, but attached to the trailing edge by the laminate perimeter. Some bendi ng of the leading edge at the root can also be seen, but not in the BR wing ( Figure 5-1). Correspondence betw een m odel and experiment is suitable, with the model again under-predicting wing deformation, but accurately locating the apex. Slight asymmetries in the measured wing profile (also evident in the BR wing) are probably a result of manufacturing errors (particularly in the app lication of the membrane skin tension), and not due to flow problems in the wind tunnel. As the amount of unconstrained membrane is greater in a PR wing than in a BR wing, chordwise strains ( Figure 5-6) are much larger as well: peak stretching (3%) is located at the m embrane/carbon fiber boundary towards the leading edge, as before. The magnitude and size of this high-extension lobe is over-predicted by the model. Both model and experiment show a region of compressive strain aft of this lobe, towards the trailing edge. This is a Possion strain

PAGE 79

79 (and thus not a compressive stress), but the stress in this region does become slightly negative for higher angles of attack. Errone ous computation of compressive membrane stresses indicates the need for a wrinkling module. T hough wrinkles in the membrane sk in are not obviously visible in the VIC measurements (possibly an unsteady pr ocess averaged out with multiple images), wrinkling towards the onset of stall is a well-known membrane wing phenomena [87]. As before, no appreciable strain is m easured or computed in the carbon fiber areas of the wing. Figure 5-5. Baseline PR normalized out-of-plane displacement (w/c), = 15. Figure 5-6. Baseline PR chordwise strain ( xx), = 15. Peak spanwise stretching ( Figure 5-7) occurs at the membrane carbon fiber interface towards the cente r of the wing root, and is well predicted by the model. The computed strain field erroneously shows a patch of negative Poisson strain towards the leading edge, due to the high chordwise strains in this area. One troubling aspect of the measured spanwise strains is the areas of negative strains along the perimeter of the membrane skin: namely on the sidewalls towards the root and the wingtip. Such stra ins have been measured in previous studies [9], but

PAGE 80

80 their presence is peculiar. Basic membrane infl ation mechanics indicates large extension at the boundaries rather than compression [145] (as is computed by the model). The com pression may be membrane wrinkli ng (which, again, is not evident from Figure 55, or m ay be an error in the VIC strain computations, potentially caused by the large displacement gradients in this area of the wing. A third possibility is that the VIC is measuring a bending strain at this point, wher e the radius of curvature is clos e to zero. The latex skin, though modeled as a membrane, does have some (albeit ve ry small) bending resistance due to its finite thickness. The anti-symmetric shear strain field ( Figure 5-8) shows good correspondence between m odel and experiment, with accurate computations in-board, but slight underpredictions of the high shear closer to the wingtip. Figure 5-7. Baseline PR spanwise strain ( yy), = 15. Figure 5-8. Baseline PR shear strain ( xy), = 15. The aerodynamic twist (camber and camber location) and geometric twist angle

PAGE 81

81 distributions for the baseline BR and rigid wings are given in Figure 5-9. The rigid wing is characterized by positive (nose-up ) twist and a progressive de-cam bering toward the wingtip. The carbon fiber inboard portion of the BR wing exhibits very sim ilar wing twist to the rigid wing. Past 2y/b = 0.3 however, both model and experiment show that the membrane wing has a near-constant decrease in twis t of 2-3: adaptive washout. Though this geometric twist dominates the behavior of the BR wing, the memb rane also exhibits some aerodynamic twist. This occurs predominately in the latex between th e battens, about 1% of the chord in magnitude. The location of this camber has large variations: some portions of the wing are pushed back from 25% (rigid) to 75% (membrane), as shown by bot h model and experiment. Shifting the camber aft-ward on low Reynolds number wings is one me thod to hinder flow separation through control of the pressure gradient [27], and may play a role in th e BR wing's delayed stall as well. Figure 5-9. Baseline BR aerodynamic and geometric twist distribution, = 15. The aerodynamic and geometric tw ist distributions fo r the baseline PR and rigid wings are given in Figure 5-10. Membrane inflation adaptively increases the camber by as much as 4%, though this figure is slightly under-p redicted by the m odel. The loca tion of this camber is shifted aft-ward, though not as much as with the BR wing. The flexible laminate used for the wing skeleton pushes the location of the camber at the root slightly forward. Like the BR wing deformation, shape changes over the PR wing ar e a mixture of both aerodynamic and geometric

PAGE 82

82 twist (though the former dominates). The lamina ted perimeter deflects upward farther than the leading edge, resulting in a slight nose down twist. This is as much as 2 at the wingtips, slightly under-predicted by the model. Figure 5-10. Baseline PR aerodynamic and geometric twist distribution, = 15. Wing twist and camber throughout the entire -sweep are given in Figure 5-11, at a flexible win g section at 2y/b = 0.65. The master slave moving grid algorithm [152] fails with BR wings at angles of attack higher than 20 : the steep d isplacement gradients between the carbon fiber root and the membrane skin leads to excessive shearing within the CFD mesh surrounding the wing. The nose-down twist of both the BR and the PR wing increase monotonically with angle of attack, thought the former is obviously much larger. Experimentally measured BR wing twist has a linear trend (up to stall at about 22) with while the numerical curve is more nonlinear, and under-predicts twist at moderate angles. Both model and experiment demonstrate a moderate increase in camber of th e BR wing, with a linear trend in up to stall. After stall, the camber of the BR wing increases subs tantially, from 5% to 8%. The camber of the PR wing is much larger than the rigid wing, even at low and negative angles of attack. This is due to the lack of pre-tension: even a moderate amount of force will cause substantial deformations [147]. Both measurements and simulations of the PR wing are

PAGE 83

83 difficult at lower angles than shown in Figure 5-11: the membrane is equally apt to lie on eithe r side of the chordline [65], and steady-state solutions dont exist. PR wing cam ber variations with angle of attack are nonlinear (the develo pment of finite strains cause a 1/3 power law response to the applied load [72]), and are slightly under-p redicted by the m odel. The location of this camber in a PR wing m oves somewhat forward for modest angles, while the BR wing sees a significant aft-ward shif t at the onset of stall. Both of these camber location trends are well-predicted by the model. Experimental error bars for camber, though not shown here, are on the order of 10% at low angles, less than 2% at moderate angles, and upwards of 20% in the stalled region [43]. This stems not from uncertainty but from unsteady membrane vibration, possibly due to vortex sh edding as discussed by Lian and Shyy [8]. Figure 5-11. Aerodynamic and ge ometric twist at 2y/b = 0.65. Aerodynamic Loads Lift coefficients (both measured and predicted) throughout the -sweep, with no model yaw, are given in Figure 5-12, for the three baseline wing designs discussed above. For all six data sets, lift slopes are very lo w (~0.05/ about half of the va lue for two-dimensional airfoils [27]) as expected from low aspect ratio wings T he downward momentum from the tip vortices helps mitigate the flow separation, delaying stall to relatively high angles (18-22). Focusing first on the rigid wing, mild nonl inearities can be seen in th e lift curve. Both model and

PAGE 84

84 experiment indicate an increase in the slope by 25% between 0 and 15 angle of attack. This is presumably due to a growth in the low pressure cells at the wing tips of the low aspect ratio wing [3]. Such nonlinearities should become more pr evalent for lower aspect ratios than considered here (1.25). Model and experim ent show good ag reement for the lift over the rigid wing prior to stall. At stall (where the static models predictive capability is questionable due to unsteady flow separation [18] and tip vortices [6]) the model slightly under-pr edicts the stalling angle and CL,max; the loss of lift is more severe in the experimental data. Figure 5-12. Baseline lift coefficients: numerical (left), experimental (right). The adaptive inflation/cambering of the PR wi ng substantially increases the lift and the lift slope as compared to the rigid wing. The lift curve of the PR wing is less nonlinear than the rigid wing. This may be due to the nonlinear cambering seen in Figure 5-11, which is known to decrea se the lift slope [15] and can offset the growth of th e tip vortices. Drastic chan ges in the lift characteristics at low angles due to hysteresis effects [65], and a gradua l onset of stall [89] are not evid ent in either the num erical or the experimental da ta, perhaps because a relevant portion of the wing is not composed of the flex ible membrane. The mo del significantly underpredicts CL,max of the PR wing, and erroneously computes that the wing stalls before the rigid

PAGE 85

85 wing. Similar experimental work [9] at lower speeds also show ear ly stall, again indicating the sensitivity of Reynolds num ber to stall. At angles of attack below 10, the BR wing has very similar lift charact eristics to the rigid wing, a fact also noted in the work of Lian et al. [28]. This is thought to be due to two offsetting characteristics of a wing with both aerodynam ic and geometric twist [67]: the inflation in between each batten increas es the lift, while the adaptive was hout at the trailing edge decreases the lift. Both of these defor mations can be seen in Figure 5-1. At higher angles of attack, the load alleviation from the washout dominates the deformation, and decreases both the lift and the lift slope, as indicate d by both model and experiment. De layed stall is not present in the measurements (though, as with the PR wing, has been measured at lower Reynolds numbers [9]), and num erical BR wing modeling cannot be taken past 20 due to aforementioned problems with the moving boundary. Figure 5-13 shows drag co efficients through the -sweep, with good experim ental validation of the model. As before, the drag of the rigid and the BR wings are very similar for modest angles of attack. Above 10 the load alleviation at the tr ailing edge decreases the drag, a streamlining effect [63]. It should be noted however that for a given value of lift, the BR wing actually has slightly m ore drag than a rigid wing [9]. Regardless of whether the comparative basis is lift or angle of attack, the PR wing has a dr ag penalty over the rigid wi ng. This is in pa rt due to the highly non-optimal airfoil sh ape of each membrane wing section: Figure 5-5 shows the tang ent discontinuity of the wing shape at the membrane/carbon fiber interface towards the leading edge. Excessive inflation may also induce additional fl ow separation. Longitudinal pitching moments (measured about the leading edge) are given as a function of lift for the three baseline designs in Figure 5-14. Of the three, the PR wing is not statically

PAGE 86

86 stable (based upon a negative Cm,AC), and the hinged trailing edge portion (seen in Figure 1-1 and Figure 1-2) must be used for trimmed flight. Prior to stall, both the aeroelastic model and the experim ent indicate very similar behavior be tween the rigid and the BR wing, with mild nonlinearities in the moment curves. This is os tensibly due to tip vortex growth, as before [3]. Figure 5-13. Baseline drag coefficients: numerical (left), experimental (right). The PR wing has a 15% lower pitching moment sl ope than the rigid wing. This is a result of the membrane inflation, which shifts the pressure recovery towards the trailing edge, adaptively increasing the strength of the nose-do wn (restoring) pitching moment with increases in lift and Steeper Cm slopes indicate larger static margins: stability concerns are a primary target of design improvement from one generation of micro air vehicles to the next. The static margin of a MAV is generally only a few m illimeters long; properly fitting all the microcomponents on board can be difficult. Furthermor e, the PR wing displays a greater range of linear Cm behavior, possibly due to the fact that the adaptive membrane inflation quells the strength of the low pressure cells [13]. Finally, L /D characteristics are given in Figure 5-15, as a function of lift. For low angles of attack (and lift), the three wings perform similarly. At higher angles of attack (prior to stall), the PR wing has the highest efficiency. The model incorrectly computes the BR wing to have

PAGE 87

87 the best L/D for a small range of modest lift values. Correlation between model and experiment is generally acceptable for the rigid and BR wings, though the L/D of the PR wing is significantly under-predicted by the model, owing mostly to poor lift prediction at these angles ( Figure 5-12). The camber of the PR wing is subsequently under-predicted as well ( Figure 511), and m ay be a result of membrane vibration [89]. At no point does either model or experim ent indicate that the rigid wing has the best efficiency; pe rhaps surprising, given the fact that neither wing deforms into a particularly optimal airfoil shape. Figure 5-14. Baseline pitching mo ment coefficients: numerical (l eft), experimental (right). Figure 5-15. Baseline wing efficiency: num erical (left), experimental (right).

PAGE 88

88 A quantitative summary of the last four figures is given in Table 5-1, for all three baseline wings at 6 angle of attack. Experimental error bounds are co mputed as described above. Aerodynamic sensitivities (as well as the pitc hing moment about the aerodynamic center) are found with a linear fit th rough the pre-stall angles of attack. Error bounds in these slopes are computed with Monte Carlo simulations. Co mputed lift, drag, and pitching moments consistently fall within the measured error bars (the latter of which are exceptionally large), though pitching moments are significantly under-pr edicted (10-30%). Sensitivities are also under-predicted, though still fall with in the large error bars asso ciated with pitching moment slopes. With the exception of L/D of a PR wing, trends between differe nt wing structures are well-predicted by the aeroelastic model. Table 5-1. Measured and computed aerodynamic characteristics, = 6. CL CD num. exp. error (%)num. exp. error (%) rigid 0.396 0.384 0.024 3.10 0.070 0.069 0.007 1.15 BR 0.381 0.382 0.024 -0.16 0.067 0.069 0.007 -3.04 PR 0.465 0.495 0.031 -5.98 0.085 0.076 0.009 11.61 Cm L/D num. exp. error (%)num. exp. error (%) rigid -0.084 -0.063 0.033-32.81 5.64 5.49 0.69 2.72 BR -0.087 -0.073 0.034-19.39 5.70 5.49 0.68 3.77 PR -0.138 -0.131 0.042-5.64 5.49 6.49 0.87 -15.36 CL Cm,AC num. exp. error (%)num. exp. error (%) rigid 0.049 0.051 0.003 -5.26 0.013 0.016 0.018 BR 0.044 0.048 0.004 -9.35 0.006 -0.001 0.020 PR 0.052 0.057 0.004 -9.21 -0.008 -0.015 0.026 Cm dCm/dCL num. exp. error (%)num. exp. error (%) rigid -0.012 -0.010 0.004-11.65 -0.246 -0.199 0.086 -23.07 BR -0.011 -0.009 0.004-17.97 -0.244 -0.185 0.098 -31.88 PR -0.014 -0.013 0.0066.01 -0.280 -0.229 0.105 -22.17 Flow Structures Having established sufficient confidence in th e static aeroelastic membrane wing model,

PAGE 89

89 attention is now turned to the computed flow stru ctures. No experimental validation is available for this work, though whenever possible the results wi ll be correlated to data in the previous two sections, or results in the literat ure. Experimental flow visuali zation work for low aspect ratios and low Reynolds number is given by Tang and Zhu [6] and Kaplan et al. [37]. Work done explicitly on MAV wings is given by Gursul et al. [40], Parks [91], Gamble and Reeder [92], and Systm a [153]. The pressure distributions and flow structures are given in Figure 5-16 at 0 angle of attack for the upper/suction wing surface o f all three baseline wing desi gns. The plotted streamlines reside close to the surf ace, typically within the boundary layer. For the rigid wing, a high pressure region is located close to the leading ed ge, corresponding to flow stagnation. This is followed by pressure recovery (minimum pressure), located approximately at the camber of each rigid wing section. Pressure recovery is followe d by a mild adverse pressure gradient, which is not strong enough to cause the flow to separate. A further decrease in Reynolds number has been shown to cause mild flow separation over the top surface for 0 however [14], [153]. A sm all locus of downward forces are present over the negatively-cambered region (reflex) of the airfoil, helping to offset the nose-down pitchi ng moment of the remainder of the rigid MAV wing, as discussed above. The reflex can also help improve the wing efficiency, compared to positively-cambered wings [55]. There is positive lif t of this wing at 0 (Figure 5-12), resulting in a m ild tip vortex swirling system. The low pre ssure cells at the wing tip are not yet evident. Aeroelastic pressure redistributions of th e upper surface of the BR wing are seen in the form of three high-pressure lobes at the ca rbon fiber/membrane boundary interface towards the leading edge. The membrane infl ation in between each batten ( Figure 5-1) result s in a s light tangent discontinuity in the wing surface. This fo rces the flow to slow down and redirect itself

PAGE 90

90 over the inflated shape: such a d eceleration results in a pressure spike. Aft of these spikes, the pressure is slightly lower in the membrane sk in than over each batten (due to the adaptive camber), driving the flow into the membrane patches. This is a very small effect (mildly visible in the streamlines) for the current case, but can be expected to play a la rge role with potential flow separation, where the chordw ise velocities are very small [154]. Figure 5-16. Pressure distributions (Pa) and streamlines on the upper surface of a rigid (left), BR (center), and PR wing (right), = 0. For the PR wing (Figure 5-16), the pressure spike is stronger, and exists continuously along the m embrane interface. A significant per centage of this spike is directed axially, increasing the drag (as seen in Figure 5-13). The adaptive inflation causes an aft-ward shift in the pressure recovery location of each flexible wing section. The longer mom ent arm increases the nose-down pitching moment about the leading edge ( Figure 5-14), which is the working m echanism behind the benevolent longitudinal sta tic stability properties of the PR wing. Furthermore, the aerodynamic twist increases th e adverse pressure gradient over the membrane portion of the wing: some flow now separates as it travels down the inflated shape, further increasing the drag (as comp ared to the rigid wing). Similar results are given for the lower/pressure si de of the three wings at 0 angle of attack in Figure 5-17. The flow beneath the rigid wing is dom inated by an adverse pressure gradient towards the leading edge, causing a large separa tion bubble underneath the wing camber. This

PAGE 91

91 separated flow is largely confined to the in-board portions of the wing. Flow reattaches slightly aft of the quarter-chord, after wh ich the pressure gradient is favorable. The flow accelerates beneath the negatively-cambered portion of the rigid wing: this decreases the local pressures, further offsetting the nose-down pitching moment The pressure distribution on the lower surface is not greatly affected by the tip vortices, previously noted by Lian et al. [28]. Figure 5-17. Pressure distributions (Pa) and streamlines on the lower surface of a rigid (left), BR (center), and PR wing (right), = 0. For the BR wing ( Figure 5-17), slight undulatio ns in the pressure distribution are indicative of the m embrane inflation in be tween the battens. This causes the opposite of what is seen on the upper wing: flow is slightly packed towards the battens [154], though the effect is minor, as before. The adaptive aerodynam ic twist of the PR membrane wing pushes the bulk of the separated flow at the leading edge towards the root, and induces furthe r separation beneath the inflated membrane shape, as the air flows into the cavity against an adverse pressure gradient. The location of maximum pressure is increased and pushed aft-ward to coincide with the apex of the inflated membrane, increasing both the lift and the stability. Flow structures over the upper surface at 15 angle of attack are given in Figure 5-18. At this higher incidence, the adve rse pressure gradient is too strong for the low Reynolds num ber flow, and a large separation bubble is present at the three-quarter c hord mark of the rigid wing. Despite the nose-up geometric twist bu ilt into the wing (7 at the tip, Figure 5-9), flow separates

PAGE 92

92 at the root first, and is confined (at this angle) to the in-board portion of the wing. This may be due to the steeper pressure gradients at the root or an interaction with the tip vortex system [5]. The reattach ed flow aft of the bubble (and th e resulting pressure distribution) must be viewed with a certain amount of suspicion. Such a reattachment is known to be turbulent process [25], and no such module is included in the CFD (or even, to the authors knowledge, exists for co mplex three-dimensional flows) Unsteady vortex shedding may accompany the bubble as well [8], though time-averaging of vortex sh edding is known to com pare well with steady measurements of a single stationary bubble [18]. The augmented incidence has consider ably increased the strength of the wingtip vortex swirling system over the rigid wing. The size of the vortex core is larger (indicativ e of the expected increase in induced drag [27]), and the low pressure cells at th e wing tip are very evident [3]. Figure 5-18. Pressure distributions (Pa) and streamlines on the upper surface of a rigid (left), BR (center), and PR wing (right), = 15. As expected, the aeroelastic effects of the BR and PR wings are more predominate at 15 in Figure 5-18. For the BR wing, the three highpressure lobes over the m embrane/carbon fiber interface are larger. Significant pressure-redistr ibution over the membrane stretched between the outer batten and the wing tip can be seen as well. Adaptive washout slightly decreases the intensity of the separation bubble, but has no noticeab le effect on the pressu re distribution at the trailing edge of the upper BR wing surface. At 15, the aerodynamic twist of the PR wing is

PAGE 93

93 considerably larger than before, as is the resu lting pressure spike at the membrane-carbon fiber interface. Despite the large adverse pressure gradient in this region, flow does not separate (though it has been noted in other studies [153]). The inflated m embrane shape of the PR wing pus hes the bulk of the fl ow separation closer to the wing root. Some of this separated flow reattaches to the wing and travels into the wake, while the rest travels spanwise. This flow is attr acted either by the low pressures associated with the adaptive cambering, or by the low pressures at the core of the tip vortex. Some of these separated streamlines are entrained into the swirling system, an interaction that has been shown to cause potential bilateral instab ilities for high angles of attack [6]. This effect, not seen in the rigid or BR wings, obviously cannot be further studi ed in this work, due to both the symm etry and the steady assumptions made in the solver. It can also be seen that the passive shape adaptation decreases the magnitude of the low pressure cells at the wing-tip, by 9% for the BR wing and 13% for the PR wing, compared to the rigid case. This indicates that the induced drag is decreased with flexibility, though this is only a re-distribution of the total drag. Two possible e xplanations exist for the decrease in tip vortex strength. The mechanical strain energy in the inflated membrane skin may be removing energy from the vortex swirling system [90]. For the PR wing, the inflated membrane shape may act as a barrier to the tip vo rtex formation, preventi ng the full swirling development at the wing-tip. A similar effect is demonstrated in the work of Viieru et al. [38] by the use of endplates installed on a rigid MAV wing. Wh ereas the endplates are able to decrease induced drag only at moderate an gles (afterwards the tip vortices incr ease in strength to overwhelm the geometrical presence of the endplates), the phenomena demonstrated in Figure 5-18 is effective at all angles: both the size of the m embrane barrier and the st rength of the vortex swir ling grow in conjunction

PAGE 94

94 with one another as the angle of at tack increases. This decrease in tip vortex strength is also seen in Figure 5-14: the nonlinear aerodyna m ics (from the low pressure ce lls at the tip) is evident in the pitching moments of the rigid and BR wings, while the PR curve is very linear. On the underside of the rigid wi ng at 15 angle of attack ( Figure 5-19), the increased incidence provides for com pletely attached flow behavior. The pressure gradient is largely favorable, smoothly accelerating the flow from leading to trailing edge. From the previous four figures it can be seen that separated flow over the bottom surface gradually attaches for increasing angles of attack, while attached flow over the upper surface gradually separates (eventually leading to wing stall). As time-aver aged flow separation is likely to be unsteady vortex shedding [18]: this explains the aforementione d m embrane vibration amplitudes that decrease to a quasi-static beha vior, then increase through the -sweep [135]. Figure 5-19. Pressure distributions (Pa) and streamlines on the lower surface of a rigid (left), BR (center), and PR wing (right), = 15. Load alleviation on the lower surface of th e BR wing is evidenced by a decrease in the high-pressure regions associated with camber, a nd a growth of the suction region at the trailing edge (the latter presumably due to a decrease in the local incidence). A high-pressure lobe also develops at the trailing edge of the membrane panel between the carbon fi ber root and the inner batten. At higher angles, this region of the memb rane does not locally inflate; it merely stretches between the two laminates, acting as a hinge. The adaptive inflation of the PR wing causes a

PAGE 95

95 significant redirection of the flow vectors beneat h the wing, but does not induce flow separation. Two sharp pressure drops are seen beneath the wing: one as the flow accelerates into the inflated membrane shape, and the second as the flow accelerates out of from the membrane and underneath the re-curved area of the wing. Further delineation of the flow structures over the three base line wings can be seen in Figure 5-20 ( = 0 ) and Figure 5-21 ( = 15 ), with the sectional normal force coefficient and the pressure coefficients over a flexible span sta tion (2y/b = 0.5) of the wing. For the rigid wing at 0, the sectional normal force peaks at 2y/b = 0.9 (due to the decreasi ng local chord length of the Zimmerman planform, but also the low pressu re cells left by the ti p vortices) and then experiences a sharp drop at the tip, as necessitated by the low thickness of the wing. Grid resolution and errors from interpolating the pressures from the cell centers to the nodes [150] prevent this curve from reaching the correct va lue of zero. No significant differences arise between the computed cn of the BR and rigid wings at 0, as previously indicated by the similar aerodynamic loads ( Figure 5-12). The adaptive inflati on of the PR wing increases the norm al force over most of the wing, incl uding the stiff carbon fiber root. Turning now to the pressure coefficients at 0 ( Figure 5-20), both the BR and the PR wings experience a pressure spike over the upper surface at 2y/b = 0.2, corresponding to the m embrane inflation. Outside of this location, pressure redistribution over the BR wing is negligible. The PR wing shows an aft-ward shif t in the high-lift forces over both the upper and lower surfaces. Adaptive inflation is also seen to increase the se verity of the adverse pr essure gradient (leading to the flow separation seen in Figure 5-16), and exacerbate the pre ssu re gradient reversal over the reflex portion of the wing. At 15 angle of attack ( Figure 5-21), the BR wing is more e ffective, able to allev iate the

PAGE 96

96 load over the majority of the wing. An evaluati on of the pressure coefficients at this angle indicates that the majority of this reduction in lift occurs towards the trailing edge of the bottom surface, where the suction forces are increased. Both flexible wing pressure spikes over the upper surface are intensifie d at the higher angle, with the PR wings approaching the strength of the leading edge stagnation pressu re. Sharp pressure drops are also visible on the underside of the wing, as the flow accelerat es into the membrane cavity. All three wings show a mild pressure plateau associated with separation [27]; the plateaus of the fl exib le wings are shifted towards the trailing edge. Figure 5-20. Section normal force coefficients, and pressure coeffi cients (2y/b =0.5), = 0. Figure 5-21. Section normal force coefficien ts, and pressure coeffi cients (2y/b =0.5), = 15.

PAGE 97

97 CHAPTER 6 AEROELASTIC TAILORING The static aeroelastic m odeling algorithm de tailed above (using the Navier-Stokes flow solver and the nonlinear membrane solver) can el ucidate accurate quantita tive dependencies of a variety of parameters (CL, CD, Cm, L/D, CL Cm mass) upon the wing structure. Having first studied the general effect of wing topology (b atten-reinforced and perimeter-reinforced membranes, as well as rigid wings), attention is now turned to structural sizing/str ength variables within the BR and PR wings. Results from the previous section show that the membrane skins inflation/stretching dominates the aeroelastic behavior, indicating the importance of the pretension in the membrane skin. Pre-stress resultan ts in the spanwise and chordwise directions are both considered as variables. With the excepti on of the free trailing e dge correction of the BR wings detailed above, the pre-tens ion is constant throughout the wi ng. The laminate orientation and number of plies used to construct the plai n weave carbon fiber areas of the wing can be varied as well. Finally, the num ber of layers in each batten of the BR wing can be altered, though the orientation will be fixed so that the fibers run parallel to the chord line. The sizing/strength parameters listed above leads to an optimization framework with 9 variables, if the number of layers in each of the three battens are permitted to differ, and the wing type (BR, PR, rigid) is considered a variable as well. Some of the variables are discrete, others continuous. The variables are not entirely independent either: the fiber orientation of the second bi-directional plain weave ply is m eaningless if only a single ply is used. Genetic algorithms are well-suited to problems with a mixed intege r-continuous formulation, can handle laminate stacking sequence designs without a set number of layers (with the use of addition and deletion modules [155]), are a cost-effec tive m ethod of solving multi-objective problems [156], and can navigate dis jointed design spaces [52]. The computational cost of a genetic algorithm is

PAGE 98

98 prohibitive however, typically requiring thousands of function evaluations for suitable convergence; a single simulation using the static aeroelastic model desc ribed above takes 2-3 hours of processor time on a Compaq Alpha workstation. A viable alternative is a designed experiment: the computational cost is lower, and provides an effective investigation of the desi gn space. For this work, one-factor-at-a-time (OFAT) numerical tests are run to establish the effect of various structural parameters upon the relevant aerodynamics. The three baseline wing designs used above will represent the nominal wing designs (2 layers of plain weave at 45, one layer batt ens, slack membrane). Having identified the structural variables that displa y the greatest sensitivity within the system, a fullfactorial designed experiment [157] will be run on a reduced set of variables. This data set can then be used to identify the optim al wing type and structural composition for a given objective function. Designs that strike a compromise between two objective functions are considered as well. The work concludes with experimental wind tunnel validation of the performance of selected optimal designs. OFAT Simulations The schedule of OFAT simulations is as fo llows: a 6-level full factorial design is conducted for the chordwise and spanwise pre-stre ss resultants, 6 simulations for the orientation of a single laminate of plain-weave, a 6-level fu ll factorial design for th e orientations of a twolayer laminate plain weave, and a 3-level full fact orial design for the number of layers used in the three battens. Pre-stress resultants are bounde d by 0 N/m (slack membrane) and 25 N/m (axial batten buckling can be computed for a distributed axial force equivalent to 31 N/m of pre-stress resultant in the membrane). The latter value corresponds to roughly 10% pre-strain. Plies of plain weave carbon fiber are limited to two la yers, while battens are limited to three.

PAGE 99

99 Membrane Pre-Tension Computed aerodynamic derivatives (CL and Cm ) and efficiency (L/D) are given as a function of the pre-stress resultants in the chordwise (Nx) and spanwise (Ny) directions for a BR wing in Figure 6-1. The corresponding norma lized wing disp lacement is given in Figure 6-2 for a subset of the data m atrix. All results ar e computed at 12 angl e of attack, aerodynamic derivatives are computed with a finite differe nce between 11 and 12. In a global sense, increasing the pre-tension in the BR wing increases CL decreases Cm and decreases L/D. The increased membrane stiffness prevents effec tive adaptive washout (and the concomitant load alleviation), and the wing perfor mance tends towards that of a ri gid wing. At 12 for a rigid wing, CL = 0.0507, Cm = -.0143, and L/D = 4.908. Overall sensitivity of the aerodynamics to the membrane pre-tension can be large for the derivatives (up to 20%), though less so for the wing efficiency (less than 5%, presumably due to the conflictive nature of the ratio). Figure 6-1. Computed tailoring of prestress resultants (N/m) in a BR wing, = 12. The BR wing is very sensitive to the pre-stress in the spanwise direction, but less so to stiffness in the chordwise direction. This is seen in Figure 6-2: the slack membrane wing has a trailing edge deflection of 2.5% of the root chord. Maxim izing the spanwise pre-tension (with the other direction slack) drops this value to 1% while the opposite scenario drops the value to only 1.9%. This is due to the directional sti ffness of the battens (which depend on compliance

PAGE 100

100 normal to their axis for movement), but also the trailing edge stre ss correction detailed above. Despite the global trend toward s a rigid wing with increased pre-tension, the changes are not monotonic. A wing design with a minimum lif t slope (for gust rejection, improved stall performance, etc.) is found, not with a complete ly slack wing, but a wing with a mild amount of stiffness (10 N/m) in the chord direction, and none in the span direction. Such a tactic removes the aforementioned conflicting sources of aeroelastic lift in a BR wing. Th e pre-stress correction eliminates most of the stiffne ss at the trailing edge (allowin g for adaptive washout and load alleviation), but retains the chordwise stiffn ess towards the leading edge, as seen in Figure 4-8. The m embrane inflation in this area is thus decreased, along with th e corresponding increase in lift due to camber. Figure 6-2. Computed BR wing deformation (w/c) with various pre-tensions, = 12. Maximizing CL (for efficient pull-up maneuvers, for example), is found by maximizing Ny and setting Nx to zero; this eliminates the adaptive wa shout, but retains the inflation towards the leading edge. Conversely, maximizing CL with a constraint on the acceptable L/D might be

PAGE 101

101 obtained by maximizing Nx and setting Ny to zero. Peak efficiency is found with a slack membrane: this corresponds to minimum drag, which is not shown. It should be mentioned however, that if a design goal is to maximize the lift slope (or minimize the pitching moment slope for stability), a BR wing is most likely a poor choice. Opposite trends are found for the PR wing ( Figure 6-3 and Figure 6-4): increasing the pretension d ecreases CL increases Cm and increases L/D. Similar to before, added wing stiffness decreases the adaptive inflation of the wing skin, and results tend towards that of a rigid wing. Without the directional influence of the battens a nd the trailing edge stress correction needed for the BR wing, the PR wing surfaces in Figure 6-3 are very smooth, and converge m onotonically for high pre-stress. Figure 6-3. Computed tailoring of prestress resultants (N/m) in a PR wing, = 12. As before, the PR wing is more sensitive to pr e-tension in the spanwise direction than the chordwise direction. The slack membrane wing in flates to 5% of the chord: maximizing tension in the chord direction (with none in the span direction) drops this value to 3%, though the opposite case drops the value to 1.5%. This is pr obably due to the fact that the chord of the membrane skin is about twice as long as its span. The sensitiv ity of a pressurized rectangular membrane to a directional pre-stress is inverselyproportional to its length in the same direction, as indicated by solutions to Eq. (4-4). Though the L/D of the PR wing is equally affected by pre-

PAGE 102

102 stresses in both directions, th e two aerodynamic derivatives in Figure 6-3 have a significantly muted respo nse to Nx. Such a result has noteworthy ramifications upon a multi-objective optimization scenario. The longitudinal static stab ility is optimal for a slack membrane wing, but the wing efficiency at this da ta point is poor. Maximizing Nx and setting Ny to zero greatly improves the lift-to-drag ratio (only 0.2% less than the true optimum found on this surface), with a negligible loss in static stability. Figure 6-4. Computed PR wing deformation (w/c) with various pre-tensions, = 12. Single Ply Laminates The same aerodynamic metrics are given in Figure 6-5 as a function of the ply angle (with respect to the chord line) for a set of wings with a single layer of bi-directional carbon fiber at the wing root, leading edge, and perim eter (for the PR wing only). The membrane wing is slack. Due to the plain weave nature of the laminate, all trends are periodic every 90. Only fiber orientations of 0, 45, a nd 90 automatically satisf y the balance constraint [155]. For the PR wing, changing the fiber angle has a m inor effect on the aeroel astic response, and optima are

PAGE 103

103 mostly located at either 45 (where spanwise bendi ng is largest) or 90 (whe re it is smallest). This indicates that the PR wing, whose planfo rm is dominated by membrane skin, can only take advantage of different laminates inasmuch as the spanwise bending can increase or decrease the aerodynamic membrane twist/cambering. On the other hand, the BR wing relies mostly upon geometric twist ( Figure 5-11), which can be provided from unbalanced laminate s via bend-twist coupli ng; the concept behind traditional aeroelastic tailoring [11]. Of the 7 data points shown in Figure 6-5, orientations less than 45 cause the wing to wash-in, while angles greater than 45 cause washout, the latter of which m inimizes CL of a BR wing, as expected. Using la minate wash-in to counter the load alleviation of the membrane washout (at 15 ) optimizes the wing efficiency. Aerodynamic sensitivity of the BR wing to laminate orientation is also larger than th at seen in the PR wing because the carbon fiber skeleton is less constrai ned. The wing tip of the BR wing (where the forces can be large, due to the tip vortices seen in Figure 5-18) is not connected to the trailing edge via a perim eter strip. Figure 6-5. Computed tailoring of laminate orientation for single ply bi-directional carbon fiber, = 12. Double Ply Laminates Computed aerodynamic derivatives (CL and Cm ) and efficiency (L/D) are given as a function of the ply orientations ( 1 and 2) of the two layers of bi-directional plain weave in a BR

PAGE 104

104 wing ( Figure 6-6) and a PR wing ( Figure 6-7) at 12 angle of atta ck. As before, the m embrane skin is slack. Aeroelastic trends are expected to repeat every 90, and will be symmetric about the line 1 = 2. This latter point is only true because bending-extension coupling in nonsymmetric laminates is ignored, though the effect of its inclusion would be very small as the wing is subjected mostly to normal pressure forces. Figure 6-6. Computed tailoring of laminate orientations for two plies of bi-directional carbon fiber in a BR wing, = 12. For the BR wing, efficiency is maximized and the lift slope is minimized when the fibers make 45 angles with the chord and span directio ns. Static stability is improved when fibers align with the chord. The response surface of the two stability derivatives are very noisy, suggesting possible finite differencin g errors, and all three surfaces in Figure 6-6 show little variation (only Cm of the BR wing can be varied by more th an 5%). Unlike any of the tailoring studies discussed above, the PR wing shows the sa me overall trends and optima as the BR wing. The surfaces for the PR wing, however, are much smoother but have less overall variation. Of the sampled laminate designs, [15]2 and [75]2 will exhibit the greatest bend-twist coupling, yet neither are utilized by the membrane wings. This fact, along with the similarity between the PR and the BR surfaces, suggest that th e orientation of a plain weave laminate with two layers is too stiff to have much im pact on the aerodynamics, which is dominated by membrane inflation/stretching. The use of bi-d irectional plain weave is not the most effective

PAGE 105

105 means of introducing bend-twist coup ling in a laminate. The fact that the two fiber directions within the weave are perpendicular automatically sa tisfies the balance constraint at angles such as 45. This would not be the ca se if plies of uni-dir ectional carbon fiber are utilized, but this is prohibitive in MAV fabrication for the followi ng reason. Curved, unbalanced, potentially nonsymmetric thin uni-directional laminates can experience severe thermal warpage when removed from the tooling board, retaining little of the intended shape. Figure 6-7. Computed tailoring of laminate orientations for two plies of bi-directional carbon fiber in a PR wing, = 12. Batten Construction Computed lift slope and effi ciency of a BR wing at 12 a ngle of attack is given in Figure 68 as a function of the number of la yers in each batten. The thickness of each batten can be varied independently, though the num ber of layers is limited to three, re sulting in 27 possible designs. As before, the membrane skin is slack, and a two-layer plain weav e at 45 makes up the remainder of the wing. The normalized out-of-plane displacement and differential pressure coefficients along the chordstation x/c = 0.5 for 4 sele cted designs is given in Figure 6-9. As expected, the wing with three one-layer battens has the m ost adaptive washout, which provides the shallowest lift slope, but also the best lift-to-drag rati o. Additional plies, regardless of which batten they are added to, monotonically decrea ses the efficiency. The same technique can be used to increase CL except for combinations of stiff battens towards the wing root and a

PAGE 106

106 thin outer batten at the wing tip (331 and 332, fo r example, where the battens are numbered from inner to outer and the integers indicate the numbe r of layers), which can cause the lift slope to decrease from these peaks. Design 223 show s the steepest lift slope of the wings in Figure 6-8. Figure 6-8. Computed tailoring of batten construction in a BR wing, = 12. The undulations in the differential pressure due to local membrane inflation from in between the battens are clearly visible in Figure 6-9. Low pressure regions on the upper surface of the m embrane skin and high pressure on the lo wer surface (which slightly re-directs the flow towards the battens [154]) results in the four high-lift lobes over the in fl ated membrane skin. This inflation can be controlled in obvious ways: wing displacement is larger for design 111 than design 333, throughout the entire le ngth of the wing section in Figure 6-9. The wing defor mation of design 123 is comparable to desi gn 111 towards the root of the wing, but tapers off towards the wingtip, where it resembles design 333. In some cases, redistributing the batten sizes causes a trade-off between local inflati on and spanwise bending. The displacement of design 123 is less than design 111 between 24% and 60% of the semispan, but the local inflation between the stiffer battens is higher, causi ng greater redistribution of the flow and high differential pressures. A similar comparison can be made between designs 321and 111 towards

PAGE 107

107 the wingtip. Overall changes in the aerodynamics due to batten tailoring are relatively small however, with 5% possible variability in CL and 1.5% in L/D. Though not shown, the static stability of the BR wing can be vari ed by 10% with batten tailoring. Figure 6-9. Computed normalized out-of-plane displacement (lef t) and differential pressure (right) at x/c = 0.5, for various BR designs, = 12. Full Factorial Designed Experiment Of the structural sizing/str ength parameters discussed above, spanwise membrane pretension, chordwise pre-tension, the number of layers of bi-dir ectional plain weave carbon fiber, and the wing type (BR, PR, rigid) are considered in a designed experiment. As stated above, the aeroelasticity of the MAV wing is dominated by the membrane inflation, and the laminate stacking is a seconda ry effect (though Cm of a BR wing is moderately sensitive to fiber orientation and batten thickness). The number of layers of plain weave carbon fiber, though not explicitly discussed above, is included due to interesting discrepancies between laminate tailoring with one layer ( Figure 6-5) and tailoring with two layers ( Figure 6-6 and Figure 6-7). For this s tudy, the number of layers in each ba tten is fixed at one, a nd all plain weaves are oriented at 45 to the chordline. A three-level, three-variable full factorial designed experiment is implemented for each

PAGE 108

108 membrane wing. Only 1, 2, or 3 layers of carbon fiber are permitted, while pre-tension resultant (chordwise or spanwise) is restricted to 0, 10 or 20 N/m. More than 3 layers is excessively stiff and heavy; 1 layer may not be able to withstand flight loads or survive a crash. The upper cap on pre-tension is, as discussed above meant to prevent batten buckli ng. Each full factorial design array requires 27 simulations for each membra ne wing, a number which must be doubled to obtain finite difference approximations of the lift and moment derivatives in angle of attack. Including the two data points needed for the rigid wing, 110 computationally expensive simulations are required. While a full factorial matrix is not the most economical choice for a designed experiment (a central-composite design is an adequate fraction of the full factorial, for example [157]), the uniform sampling will provide the best qualitative insight into the membrane wing tailoring. All 27 data points for the BR wing are given in Figure 6-10, in terms of CL Cm and L/D. The data points for a two-layer laminate are identical to those seen in Figure 6-1. The computed norm alized out-of-plane displacement of the BR wi ng with a slack membrane can be seen in Figure 6-11, with one, two, and th ree layers of plain weave carbon fiber. These designs are the three found on the z-axis of Figure 6-10. The variability in the aerodynam ics with the three design variables is substantial: 22% in the lift slope, 54% in the pitching moment slope, and 16% in efficiency. As above, increasing th e pre-tension in the BR wing increases CL decreases Cm and decreases L/D, though the tre nd is not monotonic. No prevalent trend exists for the number of plain weave layers, demonstr ating strong interacti ons with the membrane pre-tension. For a slack BR membrane wing (Nx = Ny = 0), increasing the number of plain weave layers significantly decreases the deformation of the wi ng tip and the adaptive washout at the trailing edge. As seen in Figure 6-11, a three-layer BR wing is most ly characterized m y local membrane

PAGE 109

109 inflation among the battens. This demonstrates the degree to which the adaptive washout at the trailing edge depends on the bending /twisting of the leading edge la minate (where the forces are very high, seen in Figure 5-18), and also explains why ta ilo ring the thickness of the battens, discussed above, has only a minor effect upon the aerodynamics. Figure 6-10. Computed full f actorial design of a BR wing, = 12. Figure 6-11. Computed BR wing deformation (w/c ) with one layer of plain weave (left), two layers (center), and three layers (right), = 12. This inability of the slack membrane wing to alleviate the flight loads decreases the efficiency, but surprisingly, has little effect on the stability derivatives. One possible reason for this is the negative deformations at the trailing edge of the three-layer MAV wing. The stiffer wing adheres closely to the origin al, rigid wing shape, which contai ns reflex (negative camber) at the trailing edge. The negative forces in this area push the membrane downward, increasing the wing camber. Increasing the stiffness of the plain weave laminate may convert the BR wing from a structure with adaptive washout to one with progressive decambering, leaving the

PAGE 110

110 longitudinal stability deriva tives relatively unchanged. Greater variation with laminate thickness is seen for non-zero pre-te nsions, particularly when Nx = 20 N/m and Ny = 10 N/m. If a single layer of carbon fiber is used, this data point represents the minimum lift slope. Like th e double-layered laminates studied above, the BR wing removes the camber due to membrane inflati on (and thus the lift) at the leading edge with high chordwise stiffness, and allows for ad aptive washout with low spanwise stiffness perpendicular to the battens. Such a design has biological inspiration: the bone-reinforced membrane skins of pterosaurs [101] and bats [102] both have larger chordwise stiffness. For low levels of pre-tension, decreasing the n umber of plain weave layers increases the efficiency; if the membrane is highly-tensioned, the opposite is true. The L/D objective function is optimized with a one-layer slack membrane BR wing. It can also be seen in Figure 6-10 that for high levels of membrane pre-tension, there is little com puted difference between 2 and 3 layer laminates. Similar data is given in Figure 6-12 and Figure 6-13, for a PR wing. The single-layer PR wing exhibits a substantial am ount of adaptive washout, owning to deflection of the weak carbon fiber perimeter. Two and three-layer laminates remove this feature completely, forcing the wing into a pure aerodynamic twist. Regardless of the load alleviation along the trailing edge, the steepest lift and moment curves are found with single-layer laminates, as the weak carbon fiber reinforcement intensifies the cambering of the me mbrane wing. The two-dimensional equivalent to this case is a sailwing with the trailing edge attached to a flexible support. Well-known solutions to this problem indicate that increasing the flexibility of the support improves the static stability [67], a trend re-iterated in Figure 6-12. None of the PR aerodynam ic metrics or the displacement contours show a substantial difference between two and three layer laminates. For the thicker laminates, increasing the

PAGE 111

111 spanwise pre-tension provides steeper lift and pitching moment curves; the system has a low sensitivity to chordwise pre-tension. This may be due to the membrane skins shape: its chord is much greater than its span, as discussed above. Figure 6-12. Computed full f actorial design of a PR wing, = 12. Figure 6-13. Computed PR wing deformation (w/c) with one layer of plain weave (left), two layers (center), and three layers (right), = 12. For one-layer laminates, no clear trend between CL Cm and pre-tension (chordwise or spanwise) emerges. Whereas the thicker lamina tes prefer a slack membrane wing to optimize longitudinal static stability, the one-layer wing optimizes this metr ic when 10 N/m is applied in the span direction. The reflex in the airfoil shape may again be the reason for this. The mild amount of spanwise pre-tension enforces the in tended reflex in the membrane skin, and the downward forces depress the membrane skin (seen in Figure 6-4). Slight in creases in angle of atta ck increases the inflation camber towards the leading edge, but decreases the reflex at the trailing edge, resulting in a significant restoring moment. The efficiency of thick-laminate PR

PAGE 112

112 wings is equally degraded by chordwise and span wise pre-tensions. The opposite is true for single-layers, where L/D can actually be improved with less tension. The above data is recompiled in Figure 6-14, which plots th e perform ance of the 27 BR designs, the 27 PR designs, and the rigid wing, in terms of the lift slope and pitching moment slope as a function of L/D. As seen many ti mes in the above plots, the various objective functions conflict: tailori ng a wing structure for longitudinal st atic stability may induce a severe drag penalty, for example. No wing design exists (typically) that will optimize all of the relevant performance metrics, and compromise designs mu st be considered. The set of compromise solutions fall on the design spaces Pareto optimal front [156]. A Pareto optimal solution is nondom inated: no solution exists within the data set that out performs the Pareto optimal solution in all of the performance metrics. Three Pareto fronts are given in Figure 6-14. The first details the tradeoff between m aximizing L/D while minimizing CL The second gives the tradeoff between maximizing L/D while maximizing CL and the final front is a trad eoff between maximizing L/D while minimizing Cm It may be beneficial for a MAV wi ng to have a very steep lift slope (for efficient pull-up maneuvers, for example) or very shallow (for gust rejection), so both are included. All three of these objective functions could be used to compute a common Pareto front, but visualization of the resulting hype rsurface would be difficult. Furthermore, maximizing CL and minimizing Cm evolve from similar mechanisms, and seldom conflict. The overlap between BR wings and PR wings in Figure 6-14 is minimal, with the latter design typically having higher effi ciency and sh allow lift and mo ment slopes. The rigid wing lies close to the interface between the two membrane wing types, but is not Pareto optimal. The basic performance tradeoffs are readily visible: peak L/D is 5.49 (a single-layer BR wing with a

PAGE 113

113 slack membrane), a design whose lift slope is 8% higher than the minimum possible lift slope, 18% lower than the maximum possible lift slope, and whose pitching moment slope is 34% higher than the minimum possible moment slope. Most of the dominated solutions do not lie far from the Pareto front, indicative of the fact that all of the objective functions are obtained by integrating the pr essure and shear distributions over the wing. Substantial variations in the CFD state variables can be obtained on a local level through the use of wing flexibility ( Figure 5-21, for example), but in tegration averages out thes e deviations. It can also be seen th at two of the three Pareto fronts in Figure 6-14 are non-convex. As such, techniques which success ively optimize a weighted sum of the two objective functions (convex combination) to fill in the Pareto front will not work; more advanced schemes, such as elitist-based evolut ionary algorithms [156], must be used. Figure 6-14. Computed design performance and Pareto optimality, = 12. Having successfully implemented the designed e xperiment, the typical ne xt step is to fit the data with a response surface, a technique used by Sloan et al. [53] and Levin and Shyy [104] for MAV work. Having verified the valid ity of the surrogate, it can then be used as a relatively inexpensive objective function for optimization. Such a method is not used here for several

PAGE 114

114 reasons. First, half of the desi gn variables (wing type and laminate thickness) are discrete, which as discussed by Torres [3], can cause convergence problem s in conventional optim ization algorithms. Second, nonlinear cu rve fitting is likely required (membrane wing performance asymptotically approaches that of a rigid wi ng for increased pre-tension), and the moderate number of data points (only 9 for each wing type and laminate thickness) wont provide enough information for an adequate fit. Finally, su ch a method may result in an optimal pre-stress resultant of 5.23 N/m, for example. As discusse d above, the actual applica tion of pre-tension to a membrane MAV wing is an inexact science, an d such resolution could never be produced in the laboratory (or more importantly, the field) with any measure of repeatability or accuracy. A more practical approach is to simply treat th e pre-tension as a discrete variable: taut (~20 N/m), moderate (~10 N/m), and slack (0 N/m). Figure 6-10 and Figure 6-12 now represent an enum eration-type optimization, wherein every possi ble design is tested. Optimal wing designs in terms of 7 objective functions (maximum L/D, minimum mass, maximum lift, minimum drag, minimum pitching moment slope, maximum lif t slope and minimum lift slope) are located among the 55 available data points, and given along the diagonal of the design array in Table 61. Results from the OFAT tests above are not included. Satisfactory compromise designs are f ound by first normalizing design performance between 0 and 1, and then locating a utopia point. This utopia point is a (typically) fictional design point which would simultaneously optimize both objective functions. In the design tradeof between L/D and Cm in Figure 6-14, the utopia point is (5.49, -0.0189). An adequate com promise is the Pareto optimal design which lies closest to the utopia point; these are listed in the off-diagonal cells in Table 6-1. This method is found to give a better compromise than optim izing a convex combination of the two obj ective functions, presumably due to the non-

PAGE 115

115 convexity of the Pareto fronts. Only compromi ses between 2 objective f unctions are considered in this work. The corresponding performance of each design is given in Table 6-2. The value in each cell is predicated u pon the label at the top of each column; the performance of the second objective function (row-labeled) is found in the cell appropriately located across the diagonal. Table 6-1. Optimal MAV design array with compromise designs on the off-diagonal, = 12: design description is (wing type, Nx, Ny, number of plain weave layers). max L/D min mass max CL min CD min Cm max CL min CL max L/D BR,0,0,1L BR,0,0,1L PR,10,0,1LBR,0,0,1L BR,20,0,3LPR,20,0,2L BR,0,0,1L min mass BR,0,0,1L PR,20,20,1L PR,0,0,1L BR,0,0,1L PR,0,10,1LPR,0,10,1L BR,20,10,1L max CL PR,10,0,1L PR,0,0,1L PR,0,0,1L BR,0,10,1LPR,0,0,1LPR,0,0,1L BR,20,10,1L min CD BR,0,0,1L BR,0,0,1L BR,0,10,1L BR,0,0,1L BR,20,0,3LBR,10,20,3L BR,10,0,1L min Cm BR,20,0,3L PR,0,10,1L PR,0,0,1L BR,20,0,3L PR,0,10,1L PR,0,10,1L BR,20,0,3L max CL PR,20,0,2L PR,0,10,1L PR,0,0,1L BR,10,20,3LPR,0,10,1L PR,0,10,1L BR,0,20,3L min CL BR,0,0,1L BR,20,10,1L BR,20,10,1LBR,10,0,1LBR,20,0,3LBR,0,20,3L BR, 20,10,1L Table 6-2. Optimal MAV design performance array, = 12: off-diagonal compromise design performance is predicated by column metrics, not rows. max L/D min mass (g)max CL min CD min Cm max CL min CL max L/D 5.49 4.36 0.780 0.112 -0.015 0.054 0.047 min mass 5.49 4.10 0.817 0.112 -0.019 0.057 0.043 max CL 4.84 4.18 0.817 0.145 -0.018 0.056 0.043 min CD 5.49 4.36 0.716 0.112 -0.014 0.052 0.045 min Cm 5.05 4.16 0.817 0.134 -0.019 0.057 0.049 max CL 4.90 4.16 0.817 0.141 -0.019 0.057 0.050 min CL 5.49 4.31 0.673 0.119 -0.015 0.049 0.043 For reference purposes, the design performance of the rigid wing (at 12 angle of attack) is: L/D = 4.908, mass = 6.36 grams, CL = 0.6947, CD = 0.1415, Cm = -0.0147, and CL = 0.0507. As above, at no point does the rigid wing re present an optimum design (compromise or otherwise). The compromise between minimizing th e lift slope, and maximizing the lift slope is identified by located the design closest to the normalized CL of 0.5. This is found by a BR wing design with peak pre-tension normal to the batte ns to limit adaptive washout, but no pre-tension in the chordwise direction to allow for camber a nd lift via inflation. Both BR and PR wings are equally-represented throughout the de sign array, with the excepti on of designs requiring load alleviation: all compromises involving drag or lift slope minimization utilize a BR wing. The

PAGE 116

116 majority of the optimal designs use a single laye r of plain weave carbon fiber to take the most advantage of wing flexibility. A single la yer slack BR wing can minimize drag through streamlining [63], for example, as a significant portion of the wing is deform ed ( Figure 6-11). A few designs use 3 layers; onl y one design uses 2 layers. A few comprom ise wing designs coincide with the utopia point: a one-layer BR wing with a slack membrane maximizes L/D and minimizes the drag. A one-layer PR wing with no pretension in the chordwise directi on and 10 N/m in the spanwise dire ction provides the steepest lift slope and pitching moment slope. Most compro mise designs improve both objective functions, compared to the rigid wing, but the system par ticularly struggles to maximize both L/D and lift (above results indicate that efficiency improvements are driven by drag reduction), and to maximize lift and minimize the lift slope. The conflictive nature of the objective functions means that l ooking at designs that strike a reasonable compromise between three or more aer odynamic metrics is of minor usefulness. It should be noted however, that the design that lies closest to the utopia poin t of all 7 objectives shown in Table 6-1 is a 2-layer BR wing with a slack m embrane in the chordwise direction, and 10 N/m of pre-tension spanwise, similar to the design that lies closest to the normalized CL of 0.5, as discussed above. Finally, mass minimizati on is obviously afforded with a single layer of plain weave carbon fiber: membrane pre-tensi on then provides moderate and insignificant deviations from this value, by changing the amount of latex used over the MAV wing. Experimental Validation of Optimal Design Performance The design results from the single-objectiv e optimization studies (the diagonal of Table 61, with the exception of the m inimum mass design) ar e fabricated and tested in the wind tunnel. Only loads are measured through the -sweep, for comparison with the experimental data from the three baseline wings designs in Chapter 5. As discussed above, each of these wing designs

PAGE 117

117 utilize a single layer of plain weave carbon fiber. Two layers are typically used for MAVs of this scale. Despite the extremely compliant natu re of the wings (which is precisely why they were located as optimal), all designs are able to withstand flight loads in the wind tunnel without buckling. Whether they can withstand maneuver loads or strong gusts is still unknown however, as is their ability to endure a flight crash without breaking. Some wing designs display subs tantial leading edge vibrati on at very low and negative angles of attack (presumably due to the vor tex shedding from the separation bubble seen in Figure 5-17), though deformation is observed to be quasi-static above 3 and prior to stall. The required pre-stress resultants are co nverted into pre-strains using Hookes law, and applied to a square of latex rubber by uniformly stretching each side. VIC is used to confirm the pre-strain levels, with spatial coefficients of variation between 10 and 20%, similar to data given in Figure 4-6 and by Stanford et al. [43]. Results f or lift-related optima are given in Figure 6-15. The design that maximizes CL (PR,0,0,1L) produces more lift than the base line PR wing up to 10 angle of attack, though within the error bars (not shown, but on the or der of 5%). Above this angle the wing shows a premature stall: CL, max is much lower than measured from the baseline PR wing, bearing closer similarities to the rigid wing. The vibrati on and buffeting typically seen over MAV wings towards stall is obviously magnified for these compliant designs; the coupling between the shedding and the wing vibration may contribute to the loss of lift, as demonstrated in the work of Lian and Shyy [8]. The upward deformation of the si ngle layer trailing edge perim eter is substantial (as seen in Figure 6-13), and the resulting adaptiv e washout may also play a role. Sim ilar results are seen for the wing design that maximizes CL (PR,0,10,1L), though in this case the lift slope is nearly identical to that measured from the baseline PR wing up to 10, after which

PAGE 118

118 premature stall occurs. This benign stall behavior is not necessarily detrimental [49], though unintended by the num erical model, caused by optim izing at a single angle of attack with a steady aeroelastic solver. Figure 6-15. Experimentally measured design optimality over baseline lift. The optimizer is considerably more successful when minimizing CL with design (BR,20,10,1L), as seen in Figure 6-15. The BR wings used in these tests are qualitatively observed to have sm aller vibrati on amplitudes, compared to the PR wings, at very low and very high angles of attack. At low a ngles of attack, the lift of the optimal design is smaller than both the baseline rigid and BR wings, though the lift slope is comparable. For moderate angles, no significant differences are evident. After 10 however, the optimal design shows a clear drop in lift slope, a very flat stalling region, and stalling angle delayed by 3 over the baseline designs. CL, max is measured to be 9% less than th at measured for the baseline BR wing. Experimental validation results for the wing de sign minimizing the pitching moment slope (PR,0,10,1L) is given in Figure 6-16. As before, performance of the baseline PR wing and the optim al design are comparable up to 13. Above this angle, and through the stalling region, the optimal design has a steeper slope than the base line PR wing. At these angles, the nose-down pitching moment is stronger than that seen in th e baseline BR and rigid wings, but the slope is

PAGE 119

119 similar. This is largely due to the linear pitc hing moment behavior prev iously noted on the PR wings, possibly due to membrane inflati on interference with the tip vortices [14]. Despite the m easured improvements over the baseline PR wing, the data indicates that longitudinal control beyond stall (~ 28) may not be possible [27]. Interestingly, the sa m e wing design theoretically minimizes the moment slope and maximizes the lift slope, but only the former metric is considerably improved over the baseline. Figure 6-16. Experimentally measured design optimality over baseline pitching moments. Similar validation results are given in Figure 6-17 and Figure 6-18, for the minimization of drag and m aximization of L/D. Both metrics are optimized by wing design (BR,0,0,1L). The drag is consistently lower than the three baseline designs up to 20. Accurate drag data for micro air vehicles at low speeds is very difficult to measure, largely due to resolution issues in the sting balance [34]. Questionable data typically manife sts itself through atypically low drag. Regardless, the ver acity of the data from the optimal wing in Figure 6-17 may be confirmed by the iden tical results at the bottom of the drag bu cket with the rigid wing, where deformation is very small. The data also compares very well with computed results. Unlike the baseline BR wing, the optimal design has less drag at a given a ngle of attack and at a given value of lift (the latter of which is visible in the drag polar, wh ich is not shown). Past 20, the optimal design

PAGE 120

120 shows more of a drag penalty than the baseline BR wing, which may also be attributed to larger vibration amplitudes in the single-layer wing. Figure 6-17. Experimentally measured design optimality over baseline drag. Figure 6-18. Experimentally measured design optimality over baseline efficiency. The results for optimal efficiency ( Figure 6-18) show substan tial im provements over the three baseline designs for a range of moderate angles: 8 18 The optimization is only conducted at 12 angle of attack; whereas the pr eviously considered optimal designs can be reasonably considered ideal throughout most of the -sweep (up to stall), the conflictive nature of the lift-to-drag ratio is more complex. This can be seen in the numerical data of Figure 5-15, where the baselin e BR, PR, and rigid wings all have the highest L/D for different lift values. It is

PAGE 121

121 expected that optimizing at different angles of attack will produce radically different optimal L/D designs, but similar results may be reta ined for the remaining objectives. Of the six aerodynamic objectives considered in this section, wind tunnel testing indicates that two are unmistakably superior to the base lines over a large range of angles of attack (minimum drag and maximum efficiency), and two have similar responses to one or more of the baseline designs for small and moderate angles but are clearly superior for higher angles of attack (minimum lift and pitching moment slopes) One objective (maximum lift) is slightly better at moderate angles (though not beyond the m easured uncertainty), but decidedly inferior during stall, while another objec tive (maximum lift slope) is identical to the baseline for moderate angles, and again inferior during stall. With the exception of these latter two studies, this wind tunnel validation confirms the use of numerical aeroelastic tailoring for realizable improvements to actual MAV wings. This is not to indicate that the latter two st udies have failed: the computed pe rformance of the tailored wings is not always significantly better than the baseline, and may be blurred by experimental errors. The experimental data of these two designs is not significantly better th an the baseline designs, but not measurably worse eith er (for moderate angles).

PAGE 122

122 CHAPTER 7 AEROELASTIC TOPOLOGY OPTIMIZATION The conceptual design of a wing skeleton esse ntially represents an aeroelastic topology optim ization problem. Conventional topology optimization is typically concerned with locating the holes within a loaded homogenous structure, by minimizing the compliance [16]. This work details th e location of holes within a carbon fibe r wing shell, holes which will then be covered with a thin, taut, rubber membrane skin. In other words, the wing will be di scretized into a series of panels, wherein each panel can be a carbon fiber laminated shell or an extensible latex rubber skin. Rather than compliance, a series of aerodynamic objective functions can be considered, including L/D, CL, CD, CL Cm etc. While the two wing topologies discussed in th e preceding section (battenand perimeterreinforced wings) have been shown to be effectiv e at load alleviation via streamlining and load augmentation via cambering, respectively, both designs have defici encies. The BR wing experiences membrane inflation from in-betwe en the battens towards the leading edge ( Figure 51), cam bering the wing and contradicting the load a lleviating effects of th e adaptive washout at the trailing edge. Furthermore, the unconstrained trailing edge is only moderately effective at adaptive geometric twist, as the forces in this region are very small ( Figure 5-20). If re-curve is built into the wing section, the forces in this area may push the trailing edge downward, actually increasing the incidence, and thus the loads. The PR wing, being a simpler design, is more effective in its intended purpose (adaptive cambering for increased lift and static stability), but the drastic changes in wing geometry at the carbon fiber/membrane interfaces towards the lead ing and trailing edges of the membrane skin are aerodynamically inefficient. Large membrane in flations are also seen to lead to potentially unacceptable drag penalties as well. All of these deficiencies can be remedied via the tailoring

PAGE 123

123 studies considered above, but the greater generality of an aeroelastic topology scheme (due to the larger number of variables) would suggest better potential improve ments in aerodynamic performance. Furthermore, such an undertaki ng can potentially be followed by an aeroelastic tailoring study of the optimal topology for furthe r improvements, as discussed by Krog et al. [115]: topology optimization to lo cate a good design, followed by sizing and shape optim ization. A flexible MAV wing topological optimizati on procedure has some precedence in early micro air vehicle work by Ifju et al. [10], with an array of successfully flight tested designs shown in Figure 7-1. Each of these designs consists of a laminated leading edge, wing tip, and wing root; a series of thin stri ps of carbon fiber are im bedded within the concomitant membrane skin. Both the BR and PR wings are present, al ong with slight variati ons upon those themes. Ifju et al. [10] qualitatively ranks these wing structures b ased upon observations in the field and pilot-reported handling qualities: a crude trial and error process led to the batten-reinforced design as a viable candi date for MAV flight. Figure 7-1. Wing topologies flight tested by Ifju et al. [10]. Several challenges are as sociated with the optim ization procedure considered here. First, a

PAGE 124

124 fairly fine structural grid is needed to re solve topologies on the order of those seen in Figure 7-1. The f ine grid will, of course, increase the computational cost associated with solving the set of FEA equations, as the number of variables in the optimization algorithm is proportional to the number of finite elements. The wing is discretized into a set of quadrilaterals, which represent the density variables: 0 or 1. These quadrilate rals are used as panels for the aerodynamic solver, and broken into two triangles for the finite element solver, as shown in Figure 7-2. As in Figure 7-1, the wing topology at the root, leading edge, and wing tip is fixed as carbon fiber, to m aintain some semblance of an aerodynamic shape capable of sustaining lift. The wing topology in the figure is randomly distributed. Figure 7-2. Sample wing topol ogy (left), aerodynamic mesh ( center), and structural mesh (right). Further complications are associat ed with the fact th at these variables are binary integers: 1 if the element is a carbon fiber ply, 0 if th e element is latex membrane. Several binary optimization techniques (genetic algorithms [109], for example) are impractical for the current problem due to the large number of variables, but also due to th e extremely large computational cost associated with each aeroelastic function evaluation. A fairly standard technique for topology optimization problems classifies the density of each element as continuous, rather than binary [16]. Intermediate densities can then be penalized ( implicitly or otherwise) to push the design towards a pure carbon fiber/membrane distribution, with no porous material.

PAGE 125

125 The sensitivity of each elements dens ity variable upon the wings aerodynamic performance is required for this gradient-bas ed optimization scheme. As before, the large number of variables and the e xpensive function evaluations precl ude the use of simple finite difference schemes for computation of gradients. An adjoint sensitivity analysis of the coupled aeroelastic system is thus required, as the numbe r of design variables is much larger than the number of objectives/constraints [110]. Further complications ar ise from the fact that second derivatives are also required: im portant MAV aerodynamic performance metrics such as the slope of the lift curve, for example, are sensitivity derivatives that depend upon the characteristics of the aeroelastic system as well. This chapter provides a computational framew ork for computing the adjoint aeroelastic sensitivities of a coupl ed aeroelastic system, as well as interpolation schemes between carbon fiber and membrane finite elements and methods for penalizing intermed iate densities. The dependency of the computed optimal topology up on mesh density, angle of attack, initial topology, and objective function are given, as we ll as the resulting deformation and pressure distributions. The wing designs created via aeroelastic topo logy optimization demonstrate a clear superiority over the baseline BR and PR designs discussed above in terms of load alleviation (former) and augmentation (latter), advantages which are further expounded through wind tunnel testing. Multi-ob jective topology optimization is discussed as well, with the evolution of the optimal wing topology as one travels along the Pareto optimal front. Computational Framework Material Interpolation Topology optimization often minimizes the complia nce of a structure under static loads, with an equality constraint upon th e volume. If the density of each element is allowed to vary continuously, an implicit penalty upon intermediate densities (to push the final structure to a 0-1

PAGE 126

126 material distribution) can be achieved through a nonlinear power law in terpolation. This technique is known as the solid isotropic material with pe nalization method, or SIMP [105]. For the two-m aterial wing considered above (membrane or carbon fiber), the stiffness matrix Ke of each finite element in Figure 7-2 can be computed as: p epmemp1X KKKKK (7-1) where Kp and Km are the plate and membrane elements, re spectively (the latter with zeros placed within rows and columns corresponding to bending degrees of freedom). is a small number used to prevent singularity in the pure membrane element (due to the bending degrees of freedom), and Xe is the density of the element, varying from 0 (membrane) to 1 (carbon fiber). p is the nonlinear penaliz ation power (typically greater than 3). A common criticism of this power law approach is that intermediate densities do not actually exist. This is a particular problem fo r the current application, where each element is either carbon fiber or membrane rubber. The physics of these two elements is completely different, as the carbon fiber is inextensible yet has resistance to bending and twisting, while the opposite is true for the latex. An equal combination of these two (equivale nt to stating that the density within an element is 0.5), while computat ionally conceivable, is not physically possible. The wing topology will not represent a real structure until the density of each element is pushed to 1 (carbon fiber) or 0 (membrane). The power laws effectiveness as an im plicit penalty is predicated upon a volume constraint: intermediate densities are unfavorable, as their stiffness is small compared to their volume [16]. No such volume constraint is utilized here, due to an uncertainty upon what this value should be. Furthermore, for aeronautical a pplica tions it is typically desired to minimize the mass of the wing itself, as discussed by Maute et al. [118]. Regardless, the nonlinear power

PAGE 127

127 law of SIMP is still useful for the current application, as demonstrated in Figure 7-3. Both linear and nonlinear m aterial interpolations are given for the lift computation, and the wing topology is altered uniformly. For the linear interpolation (i.e., without SIMP ), the aeroelastic response is a weak function of the density until X becomes very small (~0.001), when the system experiences a very sharp change as X is further decreased to 0. This is a result of the large st iffness imbalance between the carbon fiber laminates and the membrane skin, a nd the fact that lift is a direct function of the wings compliance (the inverse of the weighted sum of the two disparate stiffness matrices in Eq. (7-1)). The inclusion of a non linear penalization power (p = 5) spreads the response evenly between 0 and 1. Aeroelastic topology optimi zation with linear material interpolation experiences convergence difficulties, as the gradie nt-based technique stru ggles with the nearlydisjointed design space; a penalization power of 5 is utilized for the remainder of this work. Figure 7-3. Effect of linear and nonli near material interpolation upon lift. The results from Figure 7-3 suggest a number of othe r potential difficulties with an aeroelastic topology optim ization scheme. First, th e sensitivity of the aer oelastic response to element density is zero for a pure membrane wing (X = 0), as can be inferred from Eq. (7-1). As such, using a pure membrane wing as an initial guess for optimization will not work, as the

PAGE 128

128 design wont change. Secondly, two local optima exist in the design space of Figure 7-3, which m ay prevent the gradient-based optimizer from c onverging to a 0-1 materi al distribution. To counteract this problem, an expl icit penalty on intermediate dens ities is added to the objective function, as discussed by Chen and Wu [158]: XN i i1RsinX (7-2) where R is a penalty parameter appropriately sized so as not to overwhelm the aerodynamic performance of the wing topology. This penalty is only added when and if the aeroelastic optimizer has converged upon a design with intermed iate densities, as will be discussed below. Aeroelastic Solver Due to the large number of expected functi on evaluations (~ 200) needed to converge upon an optimal wing topology, and the required aeroela stic sensitivities (computed with an adjoint method), a lower-fidelity aeroelastic model (compared to that utili zed in Chapters 5 and 6) must be used for the current application. An inviscid vortex lattice method (Eq. (4-11)) is coupled to a linear orthotropic plate model a nd a linear stress stiffening membrane model (Eq. (4-4)). The latter module is perfectly valid in predicting membra ne inflation as long as the state of pre-stress is sufficiently large, as seen in Figure 4-5. Furthermore, in-plane s tretching of the laminate is ignored; only out-of-plane displacements (as well as in-plane rotations in the laminate) are computed over the entire wing. The vortex lattice method is reasonably accu rate as well, despite the overwhelming presence of viscous effects within the flow. As seen in Figure 4-11, the lift sl ope is co nsistently under-predicted due to an inability to model the large tip vortices [3], and the drag is underpredicted at low and high angles of attack due to separation of th e lam inar boundary layer [4]. Aeroelas tic coupling is facilitated by consider ing the system as defined by a three field

PAGE 129

129 response vector r: T TTTruz (7-3) where u is the solution to the system of finite element equations (composed of both displacements and rotati ons) at each free node, z is the shape of the flexible wing, and is the vector of unknown horseshoe vortex circulat ions. The coupled system of equations G(r) is then: o() KuQ GrzzPu0 CL (7-4) The first row of G is the finite element analysis: K is the stiffness matrix assembled from the elemental matrices in Eq. (7-1), and appropriately reduced based upon fixed boundary conditions along the wing root. Q is an interpolation matrix that converts the circulation of each horseshoe vortex into a pressure, and subsequen tly into the transverse force at each free node. The second row of G is a simple grid regeneration analysis: zo is the original (rigid) wing shape, and P is a second interpolation matrix that conv erts the finite element state vector into displacements at each free and fixed node along the wing. The third row of G is the vortex lattice method. C is an influence matrix depending solely on the wing geometry (computed through the combination of Eqs. (4-11) and (4-12)), and L is a source vector depending on the wings outward normal vectors, the angle of att ack, and the free stream velocity. Convergence of this system can typically be obtained within 25 iterations, and is defi ned when the logarithmic error in the wings lift coefficient is less than -5. One potential shortcoming of this aeroelastic model can be seen in Figure 7-3, where the com puted lift of a wing with no carbon fiber in the design domain (X = 0) is larger than the lift generated by the rigid wing (X = 1). This is due to a combination of membrane cambering towards the leading edge, and a depr ession of the trailing e dge reflex region. In reality, however,

PAGE 130

130 the combination of a poorly-constrained trailing edge and unsteady vortex shedding will lead to a large-amplitude flapping vibration, similar to that discussed by Argentina and Mahadevan [62]. W ind tunnel testing of this wing is given in Figure 7-4 at 13 m/s; the cr itical speed of flapping vibration is approxim ately 3 m/s. Figure 7-4. Measured loads of an inad equately reinforced membrane wing, U = 13 m/s. As expected, the measured lift of the membrane wing is significantly less than that measured from the rigid wing in the wind tunnel: the poorly-supported wi ng cannot sustain the flight loads, while the large amplitude vibrations levy a substantial drag penalty. Even a mild amount of trailing edge reinforcement (suc h as that seen in the upper left of Figure 7-1) will prevent this behavior, bu t formulating a constr aint that will push the aeroelastic topology optimizer away from wing designs with a poorly-rein forced trailing edge is difficult, and is not included. This section only serves to highlight one significant shortcoming of the aeroelastic model used here, and to diminish the perc eived optimality of certain wing topologies. Adjoint Sensitivity Analysis As the number of variables in the aeroelas tic system (essentially the density of each element) will always outnumber the number of constraints and objective functions, a sensitivity

PAGE 131

131 analysis can be most effectivel y carried out with an adjoint an alysis. The sought-after total derivative of the objective function with respect to each density variable is given through the chain rule: Tdgggd dd r XXrX (7-5) where g is the objective function (a scalar for the single-objective optimization scheme considered here; multi-objective optimization will be discussed below) and r is the aeroelastic state vector discussed above. The term g/ X is the explicit portion of the derivative, while the latter term is the implicit portion th rough dependence on the aeroelastic system [159]. Only aerodynam ic objective functions are considered in th is work: the explicit portion is then zero, unless the intermediate density penalty of Eq. (7-2) is included. The derivative of the aeroelastic state vector with respect to the element densities is found by differentiating the coupled system of Eq. (7-4): d(,) d dd GXrGr 0A0 XXX (7-6) where A is the Jacobian of the aer oelastic system, defined by: G A= r (7-7) Combining Eqs. 7-5 and 7-6 leaves: T -1dggg d G A XXrX (7-8) Using the adjoint, rather than the direct method to solve Eq. (7-8), the adjoint vector is: -Tg aA r (7-9) The system of equations for the adjoint vector does not contain the density of each element

PAGE 132

132 (X), and only needs to be solved once. For th e aeroelastic system considered above, the terms that make up the adjoint vector are: dd K0-Q A-PI0 0CzLzC (7-10) T Tg 00S r (7-11) where S is the derivative of the aerodynamic objectiv e function with respect to the vector of horseshoe vortex circulations. For metr ics such as lift and pitching moment, g = ST though more complex expressions exist for drag. Th e sensitivities can then be computed as: Tdggd dd G a XXX (7-12) Only the finite element analysis of the aeroelas tic system contains the element densities, and so this final term can be computed as: d d K u X G 0 X 0 (7-13) Of all of the terms needed to undertake the ad joint sensitivity analysis, only the derivative of the vortex lattice influence matrix C with respect to the wing shape z (a three-dimensional tensor) is computationally intensive, and represents the majority of the cost associated with the gradient calculations at each iteration. In order to solve the linear system of Eq. (7-9), a staggered approach is adapted, rather than solv ing the entire system of (un-symmetric sparse) equations as a whole, as discussed by Maute et al. [110]. Each sub-problem is solved with the sam e algorithm used in the aeroelastic solver (direct sparse solver for the finite element

PAGE 133

133 equations, and an iterative Gauss-Seidel solver for the vortex lattice equations), and as such, the computational cost and number of iterations needed for convergence is approximately equal between the aeroelastic solver (Eq. (7-4)) and the adjoint vector solver (Eq. (7-9)). The second derivative of the objective func tion is required if aerodynamic derivative metrics such as CL and Cm are of interest. Two options are available for this computation. The first involves a similar analyti cal approach to the one described above. This would eventually necessitate the extremely difficult computation of A / r, which is seldom done in practice [160]. Finite d ifferences are used here: 2g1gg (( XXX (7-14) The term g/ can be computed using another finite difference, or with the adjoint method described above, substituting the angle of attack for the element densities X. Optimization Procedure In order to ensure the existence of the op tim al wing topologies, a mesh-independent filter is employed along with the nonlinear power penaliza tion. Such a filter acts as a moving average of the gradients throughout the membrane wing, and limits the minimum size of the imbedded carbon fiber structures. Such a tactic shoul d also limit checkerboard patterns (carbon fiber elements connected just at a corner node). The moving average filter modifies the element sensitivity of node i based on the surrounding sensitiv ities within a circular region of radius rmin, as discussed by Bendse and Sigmund [16]: X XN min min i,jj i,j N j1 ij new ii,j j1rdist(i,j)ifdist(i,j)r dg1 dg HXH 0otherwise dX dX XH (7-15) As no constraints are included in the optimi zation (preferring instead the multi-objective

PAGE 134

134 approach described below), an unconstrained Fl etcher-Reeves conjugate gradient algorithm [159] is employed. Step size is ke pt constant, at a reasonably sm a ll value to preserve the fidelity of the sensitivity analysis. The upper and lowe r bounds of each design variable (1 and 0) are preserved by restricting the step size such that no density variable can leave the design space, forced to lie on the border instead. In order to increase the chances of locating a global optimum (rather than a local optimum), each optimization is run with thr ee distinct initial designs: Xo = 1 (carbon fiber wing), Xo = 0.5, and Xo = 0.1. A pure membrane wing (Xo = 0) cannot be considered for the reasons discussed above. Six objective functions are considered: ma ximum lift, minimum drag, maximum L/D, maximum CL minimum CL and minimum Cm Flight speed is kept constant at 13 m/s, but both 3 and 12 angles of a ttack are considered, with a of 1 for finite differences. Both the reflex airfoil seen in Figure 5-16 and a singly-curved airf oil are used, though aspect ratio, planform and peak camber are unchanged. The stiffness of the carbon fiber laminates is as computed by Figure 4-3, and the pre-stress of the membra ne is fixed in both the chordwise and spanwise direction s at 7 N/m. No correction is applied to the free trailing edge, as such a computation would render the pre-stress in this location very small, leading to unbounded behavior of the linear membrane model. The circular radius rmin for the mesh-independent filter is fixed at 4% of the chord. Single-Objective Optimization A typical convergence history of the aeroelastic topology optimizer can be seen in Figure 7-5, for a reflex wing at 3 ang le of attack, with a maximum L/D objective function. The initial guess is an intermediate density of 0.5. Within 4 iterations, the optimizer has removed all of the carbon fiber adjacent to the root of the wing, w ith the exception of the region located at threequarters of the chord, which corresponds to the infl ection point of the reflex airfoil. The material

PAGE 135

135 towards the leading edge and at the wing tip is also removed. Further ite rations see topological changes characterized by intersec ting threads of membrane material that grow across the surface, leaving behind islands of carbon fiber. These stru ctures arent connected to the laminate wing, but are imbedded within the membrane skin. Figure 7-5. Convergence hi story for maximizing L/D, = 3, reflex wing. These results indicate two fundamental differences between the designs in Figure 7-1 and those com puted via aeroelastic t opology optimization. The first is the presence of islands; these designs can be built, but the process is significantly more complicated than with a monolithic wing skeleton. Such structures could be avoided with a manufacturability constraint/objective function (suc h as discussed by Lyu and Saitu [161]), but the l ogistics of such a m etric (as above, with the trai ling edge reinforcement constraint) are difficult to formulate. Furthermore, the aeroelastic advantages of freefloating laminate structures are significant, as will be discussed below. A second difference is the fact that the designs of Figure 7-1 are com posed entirely from thin strips of carbon fiber embedded within the membrane, while the

PAGE 136

136 topology optimization is apt to utilize tw o-dimensional laminate structures. After 112 iterations in Figure 7-5, the optimization has largely converged (with only m inimal further improvements in L/D), but some material with intermediate densities remains towards the leading edge of the wing. Many t echniques exist for effec tively interpreting gray level topologies [162]; the explicit penalty of Eq. (7-2) is used here. Su rprisingly, the L/D sees a f urther increase with the additi on of this penalty, contrary to the conflict between performance and 0-1 convergence reported by Chen and Wu [158]. The explicit penalty does not significantly alter the topology, but merely forces all of the design variables to their lim its, as intended. The final wing skeleton has thre e trailing edge battens (one of which is connected to a triangular structure towards the center of the membra ne skin), and a fourth batten oriented at 45 to the flow direction. The structure s hows some similarities to a wing design in Figure 7-1 (third row, first colum n), and appears to be a topolog ical combination of a BR and a PR wing, with both battens and membrane inflation towards the leading edge. The optimized topology increases the L/D by 9.5% over the initial design and (perhaps more relevant, as the initial intermediate density design does not techni cally exist) by 10.2% over the rigid wing. The affect of mesh density is given in Figure 7-6, for a reflex wi ng at 12 angle of attack, with L/D maximization as the objective function. The 30x30 grid, for example, indicates that 900 vortex panels (and 1800 finite elements) cover each semi-wing. As the leading edge, root, and wing tip of each wing are fixed as carbon fiber, 480 density design variables are left for the topology optimization. One obvious sign of adequate convergence is the efficiency of the rigid wing, with only a 0.44% difference between that computed on the two finer grids. The three optimal wing topologies are similar, with three distinct carbon fi ber structures imbedded within the membrane skin: two extend to the trailing ed ge and the third resides towards the leading

PAGE 137

137 edge. While the 20x20 grid is certainly too coar se to adequately resolve the geometries of interest, the topology computed on the 30x30 grid is very similar to that computed on the 40x40 grid. The computational cost of each optimization iteration upon th e coarser grid is 5 times less than that seen for a 40x40 grid, and will be used for the remainder of this work. Figure 7-6. Affect of mesh de nsity upon optimal L/D topology, = 12, reflex wing. The affect of the initial starting design is given in Figure 7-7, for a refl ex wing at 12 angle of attack, with drag minimization as the objective function. As mentioned above, Xo = 1 (carbon fiber wing), Xo = 0.5, and Xo = 0.1 are all considered. The three final optimal topologies are very different, indicating a larg e dependency upon the initial guess and no guarantee that a global optimum has been located. Nevertheless, the indicated improvements in drag are promising, with a potential 6.7% decr ease from the rigid wing. As expect ed, the denser the initial topology, the denser the final optimized topology. All three wing topologies utilize some form of adaptive washout for load and drag alleviation. The structures must be flexible enough to generate sufficient nose-down rotation of each wing section, but not so flexible that the membrane areas of the wing will inflate and camber, increasing the forces. The wing structure in the center of Figure 7-7 (with Xo = 0.5) strikes the best compromise between the two defo rmations, and provides th e lowest drag. When Xo = 1, the structure is too stiff, relying upon a membrane hinge between the carbon fiber wing

PAGE 138

138 and root. When Xo = 0.1, the optimizer is unable to fill in enough space with laminates to prevent membrane inflation. Of the three designs, this is the least tractable from a manufacturing point of view as well. Figure 7-7. Affect of ini tial design upon the optimal CD topology, = 12, reflex wing. The dependency of the optimal topology (max imum lift) upon both angle of attack and airfoil shape are given in Figure 7-8, for both a reflex (left tw o plots) and a cambered wing (right two plots). For the wing with tr ailing edge reflex, the optimal li ft design looks similar to that found in Figure 7-5: trailing edge batt ens that extend no farther up th e wing than the half-chord, a spanwise m ember that coincides with the in flection point of the airfoil, and unconstrained membrane skin towards the leading edge, where the forces are largest. The optimizer has realized that it can maximize lift by both cambering the wing through inflation at the leading edge, and forcing the trailing edge battens downward for wash-in. This latter deformation is only possible due to the reflex (negative camber) in this area, included to offset the nose-down pitching moment of the remainder of the flying wing, and thus allow for removal of a horizontal stabilizer due to size restrictions. Increasing the angle of attack from 3 to 12 shows no significant differe nce in the wing topology, slightly increasing the length of the largest batten. At the lower angle of attack, up to 22% increase in lift is indicated through topology optimization.

PAGE 139

139 Figure 7-8. Affect of angle of attack and airfoi l upon the optimal CL topology. For the cambered wing (singly-curved airfoil, right two plots of Figure 7-8), the lift over the rigid wing is, as expected, m uch larger than found in the reflex wings but adequate stability becomes critical. With the removal of the negati vely-cambered portion of the airfoil, most of the forces generated over this wing will be positive, and the topology optimizer can no longer gain additional lift via wash-in. Imbedding batten stru ctures in the trailing edge will now result in washout, surely decreasing the lift. As such, th e optimizer produces a trailing edge member that outlines the planform and connects to the root (sim ilar to the perimeter-reinforced wing designs), restraining the motion of the trailing edge and inducing an aerodynamic twist. Unlike the PR wing, this trailing edge rein forcement does not extend continuously from the root to the tip, instead ending at 65% of the semi-span. This is then followed by a trailing edge batten that extends into the membrane skin similar to the designs seen for the reflex wing in Figure 7-8. Why such a confi guration should be preferred ove r the PR wing design for lift enhancement will be discussed below. As before, increasing the angle of attack has little bearing on the optimal topology, again in creasing the size of the trailing edge batten. A potential increase in lift by 15% over the rigid wing is indicated at the lower angle of attack. Similar results are given in Figure 7-9, with L/D maximi zation as the topology design m etric. Presumably due to the conflictive nature of the ratio, the wing topology that maximizes L/D is a strong function of angle of attack. For the reflex wing at lower angles, the optimal

PAGE 140

140 design resembles topologies used above for lift enhancement ( Figure 7-8), while at 12 the design is closer in topology to one with m inimum drag ( Figure 7-7). Increas ing lif t is more important to L/D at lower angles, while decreasing drag becomes key at larger angles. The drag is very small at low angles of attack (technically zero for this inviscid form ulation, if not for the inclusion of a constant CDo), and insensitive to changes via aeroelasticity. This concept is less true for th e cambered wing (right two plots of Figure 7-9), where designs at both 3 and 12 angle of attack utiliz e a structure with trailing edge adaptive washout. At the lower angle, the topology optimizer leaves a large triangular structur e at the trailing edge (connected to neither the root nor the wing tip), and the leadi ng edge is filled in with carbon fiber. At the higher angle of attack, four batt en-like structures are placed within the membrane skin, oriented parallel to the flow, one of which connect s to the wing tip. Potential improvements are generally smaller than those seen above, though a 10% increase in L/D is available for the cambered wing at 12. Figure 7-9. Affect of angle of attack and airfoil upon the optimal L/D topology. Wing displacements and pressure distributions are given for select wing designs in Figure 7-10, for a reflex wing at 12 angle of attack. Corresponding data along the spanwise section 2y/b = 0.58 is given in Figure 7-11. As the wing is mode led with no thic kness in the vortex lattice method, distinct upper and lower pressure distributions are not available, only differential terms. Five topologies are discussed, be ginning with a pure carbon fiber wing. Lift-

PAGE 141

141 augmentation designs are represented by a base line PR wing and the topology optimized for maximum lift. Lift-alleviation designs are re presented by a baseline BR wing and the topology optimized for minimum lift slope. Figure 7-10. Normalized out-of-plane displacement s (top) and differential pressure coefficients (bottom) for baseline and optimal topology designs, = 12, reflex wing. The differential pressure distri bution over the rigid wing is larg ely similar to that computed with the Navier-Stokes solver in Figure 5-18 and Figure 5-19: leading edge suction due to flow stagnation, pressure recovery (and p eak lift) over the camber, and negative forces over the reflex portion of the wing. As expected, the inviscid so lver misses the low-pressure cells at the wingtip (from the vortex swirling system [3]), and the plateau in the pressu re d istribution, indicative of a separation bubble [27]. This aerodynamic loading causes a moderate w ash-in of the carbon fiber wing (0.1), resulting in a comput ed lift coefficient of 0.604. Computed deformation of the PR wing is likewise similar to that found above (Figure 5-5), though the defor mations are smaller, within the range of validity of the linear finite element solver. The sudden changes in wing geometry at the membrane/carbon fiber interfaces lead to

PAGE 142

142 sharp downward forces at the lead ing and trailing edges, the latter of which exacerbates the effect of the airfoil reflex. Despite this, the membrane inflation increases the camber of the wing and thus the lift, by 6.5% over the rigid wing. Figure 7-11. Deformations and pressures along 2y/b = 0.58 for baseline and optimal topology designs, = 12, reflex wing. As discussed above, several disparate deforma tion mechanisms contribute to the high lift of the MAV design located by the aeroelastic topology optimizer (middle column, Figure 7-10). First, the m embrane inflation towards the leading edge increases lift via cambering, similar to the PR wing (the pressure distributions over the two wing structures are identical through x/c = 0.25). The main trailing edge batten structur e is then depressed downward along the trailing edge (due to the reflex) for wash-in, while the forward portion of th is structure is pushed upwards. This structure essentially swivels about the inflection point of the wings airfoil, a deformation which is able to further increase the size of the membrane cambering, and is only possible because the laminate is free-floating within the membrane skin. It can also be seen (from the left side of Figure 7-11 in particular) that the local bending/twisting of this batten

PAGE 143

143 structure is minimal: the deformation along this structure is largely linear down the wing. The intersection of this linear trend with the curved inflated membrane shape produces a cusp in the airfoil. The small radius of cu rvature forces very large velociti es, resulting in the lift spike at 46% of the chord. This combination of wash-in and cambering leads to a design which out-performs the lift of the PR wing by 5.6%, but the former effect is troubling. The wash-in essentially removes the reflex from the airfoil (as does the aerodynamic tw ist of the PR wing), an attribute originally added to mitigate the nose-down pitching moment. This fact leads to two important ideas. First, thorough optimization of a single design metric is ill-advised for micro air vehicle design, as other aspects of the flight performance will surely degrade. Its inclusion here is only meant to emphasize the relationship between aeroelastic de formation and flight performance, and show the capabilities of the topology optimization. A better appro ach is the multi-objective scheme discussed below. Secondly, if the design goal is a single-minded maximization of lift, a reflex airfoil is a poor choice compared to a singly-curved airfoil, a shape which the topology optimizer strives to emulate through aeroelastic deformations. Furthe rmore, if the design metric is an aerodynamic force or moment, passive shape adaptations need not be used at all: simply compute the optimal wing shape from the bottom row of Figure 7-10, and build a similar rigid wing. Mass restrictions prevents such a st rategy in traditional aircraft design (though a sim ilar idea can be seen in the jig-shape approach [163], where wing shape is optim ized, followed by identification of the internal structur e which allows for deformation into this shape), but two layers of carbon fiber can adequately hold the inte nded shape without a stringent we ight penalty. However, if the design metric is an aerodynamic derivative (gust rejection or longitudinal static stability, for

PAGE 144

144 example), membrane structures must be us ed, as these metrics depend on passive shape adaptation with sudden changes in freestream, a ngle of attack, or control surface deflection. Referring now to the load-allev iating MAV wing structures of Figure 7-10 and Figure 711, the deform ation of the BR wing is relatively small, allowing for just 0.1 of adaptive washout. As discussed above, the BR wing is very sensitive to pre-tensions in the span direction ( Figure 6-1); the structure is too stiff. Less than a 2% drop in lift from th e rigid wing is obtained, and the pressure distributi ons of the two wings in Figure 7-11 are very similar. What load alleviation the BR wing does provide seem s to be due to the membrane inflation from between the leading edge of the battens, and the conc omitant flow deceleration over the tangent discontinuity, rather than the adaptive washout at the tr ailing edge. The load alleviating design located by the topology optimizer (right column, Figure 7-10) is sign ificantly more successful. By filling th e design space with patches of disconnected carbon fiber structures (dominated by a long batten whic h extends the length of the membrane skin, but is not connected to the wings laminate leading edge), the MAV wing is very flexible, but none of the membrane portions of the wing are large enough to camber the wing via inflation. Wing deformation is the same magnitude as that seen in the PR-type wings, but the motion is located at the trailing edge for adaptive was hout, and lift is decr eased by 5%. The local deformation within the membrane between the leading edge and the long batten structure is subs tantial, and the flow deceleration over this point sees a furthe r loss in lift, as with the BR wing. Similar results are given in Figure 7-12 and Figure 7-13, for a cambered wing at 12 angle of attack. T he three baseline wings are again sh own (carbon fiber wing, PR, and BR), as well as the designs located by the topology optimization to maximize lift and minimize lift slope. As the forces are generally larger for the cambered airfo il, the deformations have increased to 5% of the

PAGE 145

145 root chord. The negative forces at the trailing edge of the airfoil are likewise absent. As before, the PR membrane wing effectively increases th e lift over its carbon fi ber counterpart through adaptive cambering, along with aerodynamic penalties from the shape discontinuities at the leading and trailing edge of the membrane skin. Figure 7-12. Normalized out-of-plane displacement s (top) and differential pressure coefficients (bottom) for baseline and optimal topology designs, = 12, cambered wing. There is an appreciable amount of upward deformation of the PR wings trailing edge carbon fiber strip, leading to washout of each flexib le wing section, degrading the lift. As such, the aeroelastic topology op timizer can maximize lift ( Figure 7-12, middle column) by adding more m aterial to this strip and negating the moti on of the trailing edge. As discussed above, this strip does not continue unbroke n to the wing tip, but ends at 65% of the semispan. The remaining membrane trailing edge is filled with a free-floating carbon fi ber batten. Such a configuration can (theoretically) improve the lift in several ways, similar to the trailing edge structure used for lift optimization in Figure 7-10. Placing a flexible m embrane skin between tw o rigid supports produces a trade-off: the

PAGE 146

146 cambering via inflation increases lift, but this metr ic is degraded by the sh arp discontinuities in the airfoil shape. Towards the inner portion of the MAV wing, this trade-off is favorable for lift. Towards the wingtip however (either due to the cha nges in chord or in pressure) this is no longer true, and the topology optimizer has realized that overall lift ca n be increased by allowing this portion of the trailing edge to washout, there by avoiding the negative pr essures seen elsewhere along the trailing edge. Figure 7-13. Deformations and pressures along 2y/b = 0.58 for baseline and optimal topology designs, = 12, cambered wing. The forward portion of this batten structure also produces a cusp in the wing geometry, forcing a very strong low pressure spike over the upper portion of the airfoil, further increasing the lift, as before. Due to the inviscid formul ation, further grid resolution around this cusp will cause the spike to grow larger, as the velocity around the small ra dius approaches infinity. The presence of viscosity will attenuate the speed of the flow, and thus both the magnitude of the low pressure spike and its beneficial effect upon lift. The aeroelastic topology optimizer predicts a 3.5% increase in lift over the PR wing, a nd 12.5% increase over the rigid wing, though the

PAGE 147

147 veracity of the former beneficial comparison requires a viscous flow solver to ascertain the actual height of the low-pressure spike at x/c = 0.68. The batten-reinforced design of Figure 7-12 is substantially m ore effective with the cambered wing, than with the reflex wing. As discussed above, reflex in the wing pushes the trailing edge down, limiting the ability of the batt ens to washout for load reduction. This can also be seen by comparing the airfoil shapes between Figure 7-13 and Figure 7-11: the cambered wing shows a continuous increase in the deform ati on from leading to trailing edge, while most of the deformation in the reflex wing is at the flex ible membrane/carbon fiber interface. Aft of this point, deformation is relatively constant to the trailing edge. The 1.6 of washout in the cambered BR wing decreases the load throughout most of the wing and decreases the lift by 8.5% (compared to the rigid wing), but, as before, the loadalleviating design located by the to pology optimizer (right column, Figure 7-12) is superior. Sim ilar to above, the design utilizes a series of disconnected carbon fibe r structures, oriented parallel to the flow, and extending to the trai ling edge. The structures are spaced far enough apart to allow for some local membrane inflatio n, but this cambering only increases the loads towards the trailing edge. The discontinuous wing surface forces a number of high-pressure spikes on the upper surface, notably at x/c = 0. 2 and 0.6. This, in combination with the substantial adaptive washout at the trailing edge decreases the lift by 13. 6% over the rigid wing and by 5.6% over the BR wing. Three of the wing topologies discussed above (minimum CL minimum drag, and minimum pitching moment slope, all optimized for a reflex wing at 3 angle of attack) are built and tested in the closed loop wing tunnel, as seen in Figure 7-14. Though the aeroelastic model relies on a sizable state of pre-stress in the m emb rane skin to remain bounded, all three of the

PAGE 148

148 wings are constructed with a slack membrane. This is to ensure similarity between the three wings (pre-stress is very difficult to control), and also to compare the force and moment data to the baseline membrane data acquired above ( Figure 5-12 Figure 5-15). Figure 7-14. Wing topology optimized for minimum CL built and tested in the wind tunnel. Results are given in Figure 7-15, for a longitudinal -sweep b etween 0 and 30. All three structures located by the topol ogy optimizer show marked improvements over the baseline experimental data, validating the use of a low fidelity aeroelastic model (vortex lattice model coupled to a linear membrane solver) as a su rrogate for computationally-intensive nonlinear models. With the exception of very low (where de formations are small) and very high angles of attack (where the wing has stalled), the optimized designs consistently outperform the baselines. As discussed above, this is not expected to be tr ue for L/D, where design st rategies vary strongly with incidence ( Figure 7-9). It should also be noted that the three optim ized designs in Figure 7-15 provide shallower lift slopes, less drag, and steeper pitching m oment slopes, respectively, than the experimental data gathered from the designs utilizing aeroelastic tailoring ( Figure 6-15 Figure 6-18). This confirm s the idea that topology optimization ca n out-perform tailoring of the baseline MAV wings, as the former has a larger number of variab les to work with. The two techniques need not be mutually exclusive: having located suitable wi ng topologies, the designs can be subjected to a

PAGE 149

149 tailoring study for further bene fit to the flight performance. Figure 7-15. Experimentally measured forces and moments for baseline and optimal topology designs, reflex wing. Multi-Objective Optimization The need to simultaneously consider more th an one design metric for aeroelastic topology optimization of MAV wings is demonstrated above: optimizing for lift prompts the algorithm to remove the reflex, by depressing the flexible tr ailing edge. The downward forces provided by the reflex offset the nose-down pitching moment of the remainder of wing, and are therefore essential for stability. Design and optimization with multiple performance criteria can be done my optimizing one variable with constraints upon the others (as disc ussed by Maute et al. [118] for aeroelastic topology optim ization). For MAV design however, the formulation and bounds of these constraints are uncertain, and the method does not provide a clear pi cture of the inherent trade-off between variables. The current work minimizes a convex combination of two objective functions (as discussed by Chen and Wu [158] for topology optimization). Successiv e optimizations with different relative weighting betw een the two metrics can fill out the Pareto optimal front. The

PAGE 150

150 computational cost of such an undertaking is la rge, and adequate locati on of the front is not ensured for non-convex problems (such as seen in Figure 6-14). The objective function is now: 11,min 22,min 1,max1,min 2,max2,minffff g1 ffff (7-16) where is a weighting parameter that varies between 0 and 1, and f1 and f2 are the two objective functions of interest. These functions are pr operly normalized, with the minimum and maximum bounds computed from the single-object ive optimizations (optimizing with set as 0 or 1). Eq. (7-16) is cast as a minimizati on problem, and the sign of f1 and f2 is set accordingly. As before, the objective function can be augmented with th e explicit penalty of Eq. (7-2) as needed. Typical convergence history results are given in Figure 7-16, for simultaneous m aximization of L/D and minimization of the lift slope. The weighting parameter is set to 0.5, for an equal convex combination of the tw o variables. The values given for CL (~ 0.4) are smaller than experimentally measured trends (~ 0.5, from Table 5-1), as the inviscid solver is unable to p redict the vortex lift from the tip vortex swirling system [27]. Beginning with an inte rmediate density (Xo = 0.5), the optimizer is able to decrease the convex combination (g) from 0.7 to 0.3, using similar techniques seen ab ove. All of the carbon fiber material adjacent to the root, leading edge, and wingtip is removed. Intersecting streams of membrane material grow across the wing, leaving behind disc onnected carbon fiber structures. The lift-to-drag ratio monotoni cally converges after 25 itera tions, while the lift slope requires 70 iterations to converg e to a minimum value. An explicit penalty on intermediate densities is employed at the 80 iteration mark, pr oviding a moderate decrease in the combination objective function. The lift-to-d rag ratio is improved as well th rough the penalty, though the lift slope suffers. As before, the penalty only serves to force the density variables to 0 or 1, and does not significantly alter the wing topology.

PAGE 151

151 Figure 7-16. Convergence history fo r maximizing L/D and minimizing CL = 0.5, = 3, reflex wing. The multi-objective results of Figure 7-16 can be directly co mpared to th e single-objective results of Figure 7-5, where only L/D must be improved. For the latter, L /D can be increased to 4.17, with the inclusion of trailing edge batt ens for adaptive wash-in, and an unconstrained membrane skin towards the leading edge for cambe ring via inflation. This is a load-augmenting design, and as such the lift slope is very high: 0.040. In order to strike an adequate compromise between the two designs, the multi-objective optimizer leaves the trailing edge battens, but fills the membrane skin at the leading edge with a disconnected carbon fiber st ructure. The L/D of this design obviously degrades (4.05), but the lif t slope is much shallower (0.038), as desired. The Pareto front for this same trade-off (maximum L/D and minimum CL ) is given in

PAGE 152

152 Figure 7-17, along with the performance of the 20 baseline MAV wing designs ( Figure 7-1), and the design located by the single-objectiv e topology optim izer to maximize CL All results are for a reflex wing at 3 angle of attack. Focu sing first on the baseline wings, the BR and PR wings represent the extremes of the group in te rms of lift slope, as expected. The homogenous carbon fiber wing has the lowest L/D (implying that for a reflex wing at this flight condition, any aeroelastic deformation will improve efficiency regardless of the type), while a MAV design with 2 trailing edge battens as the largest L/D. Figure 7-17. Trade-off between efficiency and lift slope, = 3, reflex wing. The aeroelastic topology optimization produces a set of designs th at significantly outperform the baselines, in terms of individually -considered metrics (maximum and minimum lift slope, maximum L/D), and multiple objectives: all of the baselines are removed from the computed Pareto front. The optimized designs la y consistency closer to the fictional utopia point as well which for Figure 7-17 is at (4.18, 0.0366). The en tirety of the Pareto front is not

PAGE 153

153 convex, but the topology optimizer is still able to ade quately compute it. The data points are not evenly spaced either, with = 0.4 and 0.2 both very close to the solution with optimal L/D ( = 0). This would suggest that despite the normalizing efforts, maximizing L/D carries greater weight than minimizing the lift slope, an imba lance which may be remedied through nonlinear weighting [111]. The results of Figure 7-16 also indicate that usi ng an exp licit penalty to force the design to a 0-1 density dist ribution favors L/D, but not CL In terms of the two metrics in Figure 7-17, none of the designs along the Pareto optimal front are technically superior: th ey are non-dom inated, in that no other design exis ts within the data set that out-performs another design in bo th metrics. Other performance indices, not included in the optimization, can then be used to select an adequate design. For micro air vehicle applications, payload, flight duration, or agility/control metrics can be used, as discussed by Torres [3]. Realistic knowledge of the low-fidelity aeroelastic m odels limitations (the perceived superiority of an unconstrained membrane wing in Figure 7-3 is destroyed by large nonlinear flapping vibrations [62], for example), or manufacturability [161] may also be used to select a design. It should also be noted that at higher ang les of attack, the trade-off between high efficiency and low lift slopes doe s not exist. As discussed ( Figure 7-9), increasing the incidence prom otes an aeroelastic structure with streamlini ng to improve L/D, a deformation that will also decrease the lift slope. Wing displacements and pressure distributions for selected wings along the Pareto front of Figure 7-17 are given in Figure 7-18, for a reflex wing at 3 angle of attack. Corresponding data along the spanwise section 2y/b = 0.58 is given in Figure 7-19. When = 1 (sing le-objective optimization to minimize the lift slope), the aeroe lastic topology optimizer locates a design with several disconnected structures imbedded with in the membrane, including a long batten that

PAGE 154

154 extends the length of the membrane skin. The wing is flexible enough to adaptively washout, but the remaining patches of membrane skin ar e not large enough to inflate and camber. Figure 7-18. Normalized out-of-plane displacement s (top) and differential pressure coefficients (bottom) for designs that trade-off between L/D and CL = 3, reflex wing. Gradually adding weight to the L/D design metr ic removes the structures from the leading edge of the membrane skin, leavi ng batten-like structures at the trailing edge of the wing. The former transition allows the membrane to inflat e and camber the wing, wh ile the latter provides wash-in through depression of the trailing edge. The cambering membrane inflation does not grow monotonically with decreasing but the trailing edge deform ation does: from 0.25 of washout to 0.75 of wash-in. The size of the depressed trailing edge portion also grows in size. Decreasing shifts the lift penalty (pressure spike on the upper surface) forward towards the membrane/carbon fiber interface, and the lift spik e (due to the surface geometry cusp at the leading edge of the batten stru ctures) aft-ward. However, th e design that maximizes L/D ( = 0) has no spike, with a smooth pressure and displace ment profile aft of the lift penalty towards the leading edge. This may be indicative of the de trimental effect the airfoil cusp has on drag.

PAGE 155

155 Figure 7-19. Deformations and pressures along 2y /b = 0.58 for designs that trade-off between L/D and CL = 3, reflex wing. The trade-off between the drag and longitudina l static stability of a membrane MAV wing is very important: the latter is typically improved through large membrane inflations. The resulting tangent discontinuities in the wing surf ace produce pressure spikes oriented axially, and the exaggerated shape prompts the flow to separate above and below the membrane [14]. The trade-off is given in Figure 7-20 for a reflex wing at 12, for both the 20 baseline designs and the Pareto front located with t opology optim ization. Compared with the data seen in Figure 7-17, the base line designs at this higher angle of attack fail to adequately fill the design space; their performance generally falls with in a band. The streamlining of the BR wing provides the lowest drag (of the baselines), but doesnt significan tly out-perform the homogenous carbon fiber wing. As expected, the PR wing has the largest static stability margin of the baselines, but the drag penalty is large (and probably under-predict ed by the inviscid flow solver). The topology optimizer is able to locate a design with the same drag penalty, but a steeper pitching moment slope: by 5.6% over the PR wing. Th e baselines designs, in general, lie closer to the Pareto front

PAGE 156

156 than seen in Figure 7-17, but the optimized designs are still superior in terms of Pareto optimality and individual m etrics. The optimal drag desi gn (3.8% less than the BR wing) begins with two carbon fiber structures imbedded within the membra ne skin, one of which is a long batten that extends the length of the design domain. Figure 7-20. Trade-off between drag and pitching moment slope, = 12, reflex wing. By adding weight to the st atic stability metric (Cm ), this long batten breaks in two pieces; the foreword section shrinks into a slender batten imbedded in the leading edge of the membrane skin. The aft-ward section gradually accumulate s along the trailing edge, merges with the root, and forms the trailing edge support. As discu ssed above, this reinforcement does not connect monolithically to the wingtip; this space is filled with a trailing edge batten. The superiority of this design is confirmed by the wind tunnel data of Figure 7-15. The Pareto optimal front of Figure 7-20 shows a more pronounced convexity than seen in Figure 7-17, though the data points are still not evenly spa ced with

PAGE 157

157 Similar data is given in Figure 7-21, for the trade-off between maximum lift and minimum lift slope, for a cam bered wing (no reflex) at 12 angl e of attack. Such a trade-off is of interest because minimizing the lift slope of a membrane MAV wing, while an effective method for delaying the onset of stall or rejecting a sudden wi nd gust, typically decreases the pre-stall lift in steady flight as well; a potentially unacceptable c onsequence. Certain aeroelastic deformations, such as a passive wing de-cambering, would provi de a wing with higher li ft (than the baseline carbon fiber wing, for example), but a shallower lift slope. Figure 7-21. Trade-off betw een lift and lift slope, = 12, cambered wing. Such a motion is unusual for low aspect ratio membrane structures however: none of the baseline designs have both larg er lift and a smaller lift slope than the carbon fiber wing. The correlation between CL and CL within the set of baseline desi gns is very strong, and all the designs fall very close to a si ngle line, clustered in three gr oups. Any baseline design with adaptive washout (free trailing edge) has lift slopes between 0.035 and 0.037, any overly-stiff

PAGE 158

158 design with battens oriented pe rpendicular to the flow (or the carbon fiber wing) has a slope between 0.038 and 0.039, and the PR wing has a lift slope of 0.041. The strong data correlation is in sh arp contrast to the results of Figure 7-17 for the reflex wing, where the baseline structures are well-d is tributed through the design space. This emphasizes the large role that the doubly-curved airf oil can play in producing many different types of aeroelastic deformation, providing greater freedom to the designer and better compromise designs. Despite this, the magnitude of the variability is higher for the cambered wing, as the forces are generally larger: CL can be varied by 14.5% for the reflex wing in Figure 7-17, but by 26.4% for the cam bered wing in Figure 7-21. These numbers can be increased further with the use of nonlinear m embrane stru ctures, but deformations must be kept at a moderate level to preserve the fidelity of the linear finite element model in the current work. As wing structures with high lift and shallow lift slopes are rare the set of baseline designs lies close to the Pareto front in Figure 7-21. None are superior however, in terms of individual m etrics or Pareto optimality. The designs located by the topology optimizer to maximize lift and maximize lift slope are almost identical, though disp arate designs can be obtained with a reflex wing, as noted above. The PR wing is very effective for cambered wings at higher angles of attack, and lies close to these two optimums. Th e slight convexity in the Pareto front produces two designs with the sought-aft er higher lift and lower lift slope than the homogenous carbon fiber wing. The topology highlighted in Figure 7-21 increases the lift coefficient from 0.842 to 0.876 and decreases the lif t slope from 0.038 to 0.036, and is found from an equal weighting of the two metrics ( = 0.5). Wing displacements and pressure distributions for selected wings along the Pareto front of Figure 7-21 are given in Figure 7-22, for a cambered wing at 12 angle of attack. Corresponding

PAGE 159

159 data along the spanwise sect ion 2y/b = 0.58 is given in Figure 7-23. Shallow lift slopes are provided with a series of disconnected batten stru ctures oriented parallel to the flow. As a weight for high lift is added to the objective function, a large carbon fiber region grows at the trailing edge, but is connected to either the root or the wing tip. This allows for both washout and m embrane cambering, and produces the MAV design with higher lift and shallower lift slopes than the carbon fiber wing ( = 0.5). Further decrease in flattens the chord of the trailing edge structure and removes the disjointed battens at the leading edge, to maximize lift. Figure 7-22. Normalized out-of-plane displacement s (top) and differential pressure coefficients (bottom) for designs that trade-off between CL and CL = 12, cambered wing. The locus of aeroelastic deform ation clearly shifts from the trailing edge to the mid-chord of the wing as the structures produce higher lift. Washout monotonically decreases with (from 3 to 0.1 of wash-i n). Membrane deformations are largest when = 0.5, though the design that maximizes lift shows the largest change in camber, owing to the significant adaptive washout of the former, as discussed. Similarly, the ae rodynamic penalty at th e leading edge of the

PAGE 160

160 membrane/carbon fiber interface is largest with the compromise design. The severity of the surface cusp (and the concomitant lift spike) increases with decreasing emphasizing its usefulness as a lift-augmentation device. As discussed above, the severi ty of this spike is certainly over-predicted by the inviscid flow solver, though similar trends are seen using NavierStokes solvers for wings with tangent discontinuities Figure 5-21. Figure 7-23. Deformations and pressures along 2y /b = 0.58 for designs that trade-off between CL and CL = 12, cambered wing.

PAGE 161

161 CHAPTER 8 CONCLUSIONS AND FUTURE WORK The results given in this work detail a co m prehensive research effort to understand and exploit the static aeroelasticity of membrane mi cro air vehicle wings. The flow structures of such wings are exceedingly complex, characte rized by low Reynolds numbers (flow separation, laminar-turbulent transition, r eattachment, vortex shedding, vort ex pairing), low aspect ratios (strong tip vortex swirling, low pr essure wing tip cells), and unstable interactions between the two (vortex destabilization for b ilateral asymmetry). The wings structural mechanics are also difficult to predict: a topologically-complex orth otropic wing shell is covered with a thin extensible latex skin, a membrane with an inherently nonlinear response. Aeroelastic fixed membrane wi ng topologies can be broadly di vided into two categories: load-alleviating, and load-augmenting. The form er can use streamlining to reduce the drag, or adaptive washout for gust rejection, delayed sta ll, or attenuated maneuver loads. The latter increases the loads via adaptive cambering or wash-i n, for improved lift and static stability; the wing may also be more response to pull-up maneuvers, etc. Wing topology is given by a distinct combination of stiff laminate composite members and a thin extensible rubber membrane sheet, similar to the skeletal structure of a bird wing, or the venation patterns of insect wings. This work discusses aeroelastic analysis and optimization in three phases. First, given a set of wing topologies (a batten-reinforced design for adaptive geometric twist, a perimeterreinforced design for adaptive aerodynamic twis t, and a homogenous laminate wing), how does the membrane inflation affect the complex flow st ructures over the wing? Secondly, how can the various sizing and strength variab les incorporated within the wing structures be tuned to improve flight performance, in terms of both individual metrics and compromise functions? Third, can these baseline wing topologies be improved upon? How does the distribution of laminate shells

PAGE 162

162 throughout a membrane skin aff ect the aeroelastic response? No model currently exists that can accurate ly predict such aeroelasticity (the threedimensional transition is the biggest numerical hurdle), and so the current work utilizes a series of low-fidelity aeroelastic models for efficien t movement through the design space: vortex lattice methods and laminar Navier-Stokes solvers are coupled to linear and nonlinear structural solvers, respectively (detailed in Chapter 4). Due to th e lower-fidelity nature of the models (despite which, the computational cost of this coupled ae roelastic simulation is ve ry large), experimental model validation is required. Su ch characterization is conducted in a low speed closed loop wind tunnel. Aerodynamic forces and moments are measured using a strain gage sting balance with an estimated resolution of 0.01 N. Structural displa cement and strain measurements are made with a visual image correlation system; a calibrated camera system is m ounted over the test section, as discussed in Chapter 3. Chapter 5 provides a detailed analysis of th e flow structures, wing deformation, and aerodynamic loads of a series of baseline membrane MAV wings. At small angles of attack, the low Reynolds number flow beneath a MAV wing sepa rates across the leading edge camber, the flow over the upper surface is largel y attached, and the tip-vortex swirling system is weak. The opposite is true has the incidence is increased : the bubble on the upper surface grows, eventually leading to stall. The lift curves of the low aspe ct ratio wings are typically shallow, with a large stalling angle. Low pressure cells deposited on the upper surface of the wing tip by the vortex swirling grow with angle of attack, adding nonlinearities to the lift and moment trends. The structural deformation of a batten-reinfo rced wing has two main trends: the forces towards the leading edge are very large, and induce membrane infl ation in-between the battens. This increases the camber over the wing, and thus the lift. A second trend comes from the free

PAGE 163

163 trailing edge of the BR wing, which deflects upw ard for a nose-down twist, decreasing the wing lift. These two effects tend to offset for lower angles of attack, and the aerodynamics follow the rigid wings very closely. At higher angles the adaptive washout dominates, decreasing the incidence of a wing section by as much as 5 and decreasing the slope of the wings lift curve. Outside of the promise such a wing shows for gust rejection and benevolent stall, the data also indicates that the streamlining decreases drag. The deformation of a perimeter-reinforced wing is characterized by adaptive aerodynamic twist: the membrane skin inflates, constrained at the leading and traili ng edges by the stiff carbon fiber perimeter. Lift, drag, and pitching moment s are consistently stronger than measured from the rigid and BR wings, as a result of the camberi ng motion. The slope of the pitching moment curve is considerably steeper, providing much-need ed longitudinal static st ability to a wing with severe space and weight constrai nts. The large drag penalty of the wing is partly due to a pressure spike at the tangent discontinuity betw een the inflated membrane and the carbon fiber, and partly due to the greater amount of separate d flow over the PR wing. Interactions between the separated longitudinal flow and the wing tip vortices are clearly visible in the PR wing, possibly indicating a greater propensity for rolling instabilities. The stretching of the membrane skin in the PR wing is more two-dimensiona l without the restrictiv e presence of battens. It is shown in Chapter 6, both numerically and experimentally, that unconventional aeroelastic tailoring can be us ed to improve MAV wing performance. The chordwise and spanwise membrane pre-tension, number of plain weave carbon fiber layers, laminate orientation, and batten thickness are all considered, with the first three variables identified as critical through a series of one-factor-at-a-time tests. In creasing stiffness is seen to tend aerodynamic behavior towards a rigid wing, thought many local optima exist and can be

PAGE 164

164 exploited. A comprehensive numerical review of the design space is provided with a full factorial designed experiment of th e three aforementioned variables. This data is then used to optimize six aerodynamic variables, as well as compromises between each. The six designs resulting from the single-objective optimizations are built and tested in the wind tunnel: five show improvements over the baseline designs, one has a similar response. While the flexible wing structures have been shown to effectively alter the flow fields over a MAV wing, aeroelastic topology optimization (C hapter 7) can be used to improve on the shortcomings of the previously-considered base line designs. Results are superior to those computed via tailoring, as the number of variables is much larger: the wing is discretized into a series of panels, each of which can be membra ne or carbon fiber laminate. The computational cost is severe: hundreds of iterati ons are expected for convergence, and a sensitivity analysis of the coupled aeroelastic syst em must be conducted. The optimization is able to identify a seri es of interesting designs, emphasizing the relationships between flight c ondition, airfoil, design metric and wing topology. For load alleviation, the algorithm fills the membrane skin with a number of disconnected laminate structures. The structure is flexible enough to washout at the trailing edge, but the patches of exposed membrane skin are not large enough to in flate and camber the wing. Such a design has less drag and a shallower lift curve than the batte n-reinforced wing. For load augmentation, the topology optimizer utilizes a combination of cam bering, wash-in, and wing surface geometry cusps to increase the lift over th e perimeter-reinforced wing. As a wing design optimized for a single metric is of minor usefulness, the topo logy optimizer is expanded to minimize a convex combination of two metrics for computation of the Pareto front. Three such designs are built and tested in the wind tunnel, c onfirming the computed superior ity over the baseline wings.

PAGE 165

165 Several future aeroelastic optimi zation studies are of interest. First, it is desired to upgrade the model fidelity used in the topology optim ization described above. In order to limit computational cost, the work uses several linea r modules: a vortex lattice solver and a linear stress-stiffening membrane solver, computed on a relatively coarse topology grid. Such a model is unable to capture several important nonlinearit ies, including flow separation and tip vortex formation. This can be remedied by using an unsteady Navier-Stokes solver coupled to a nonlinear membrane structural dynamics solver, increasing the computati onal cost by several orders of magnitude. The large number of variables (~1000) require s the use of a gradient-based optimizer; the higher-fidelity models will increas e the complexities involved in the sensitivity analysis of the coupled aeroelastic system as well. Of particular interest is gust response: how the membrane wing responds to a sinusoidal wind cycle, where it is desired to minimize the overall response for smoother flight. Objective functions may be the change in lift, integrated ov er the gust cycle. A second interest is the wing topology that delays the stall of the fixed wing. Conventional optimization formulations for this problem are difficult, as the stall angle is not a direct output from the aeroelastic system, but the angle at which the slope of the lift curve become s negative. The optimizer will have to compute the lift at a set number of (large) angles of a ttack, and interpolate betw een the data points to estimate the stalling angle. Secondly, these aeroelastic topology optimiza tion techniques will be extended to flapping micro air vehicle wings. The structure of these wi ngs is very similar to the fixed wings discussed above (thin membrane skins reinforced with lami nate plies), and so the two-material model is appropriate. As with the gust cycle, lift and th rust will have to be computed over an entire flapping cycle, and then integrated to produce a s calar objective function. Furthermore, lift and

PAGE 166

166 thrust will conflict: thrust relies on wing twist via deformation for th rust generation, while lift is dependent upon the leading edge vortex, which can be disrupted by excessive deformations. This requires successive optimizations of a convex combination of the two weighted metrics to fill out the trade-off curve (assuming that this Pareto front is convex). The optimal design can then be selected from this front based upon metrics not considered in the formal optimization: trim requirements, manufacturability etc. The flow structures that develop over flapping wing systems are very complicated, uns teady vortex driven flows. Navier-Stokes solvers can adequately handle these phenomena, but the computational cost may be prohibitive. Topology optimization of flapping wings may requi re lower-fidelity aerodynamic methods for effective navigation through the design space. Finally, the aeroelastic topology optimization of both the fixed and flapping wings can be followed by a tailoring study for additional improvements to the flight performance. This is a standard optimization process: topology optimizati on, interpretation of th e results to form an engineering design, followed by sizi ng and shape optimization (or in this case, tailoring). Both laminate thickness/orientation and membrane pre-tension can be used, as above. Membrane pretension is difficult to control how ever, and will relax at the un-reinforced borders of the wing, leading to a pre-tension gradie nt. Anisotropic membranes (thr ough imbedded elastic fibers or crinkled/pleated geometries) are an attractive alte rnative for directional wing skin stiffness. The excess area of the skin may also be a useful variab le. As the number of variables in a tailoring study is relatively small (~10), gr adient-free global optimizers such as evolutionary algorithms or response surface techniques may become applicable.

PAGE 167

167 REFERENCES [1] Abdulrahim, M., Garcia, H., Lind, R., Flight Ch aracteristics of Shaping the Membrane Wing of a Micro Air Vehicle, Journal of Aircraft, Vol. 42, No. 1, 2005, pp. 131-137. [2] Young, A., Horton, H, Some Results of Investigation of Separation Bubbles, AGARD Conference Proceedings, No. 4, 1966, pp. 779-811. [3] Torres, G., Aerodynamics of Low Aspect Ra tio Wings at Low Re ynolds Numbers with Applications to Micro Air Vehicle Design, Ph .D. Dissertation, Department of Aerospace and Mechanical Engineering, University of Notre Dame, South Bend, IN, 2002. [4] Shyy, W., Ifju, P., Viieru, D., Membr ane Wing-Based Micro Air Vehicles, Applied Mechanics Reviews, Vol. 58, No. 4, 2005, pp. 283-301. [5] Hoerner, S., Borst, H., Fluid-Dynamic Lift, Hoerner Fluid Dynamics, Brick Town, NJ, 1975. [6] Tang, J., Zhu, K., Numerical and Experimental Study of Flow Structur e of Low-Aspect Ratio Wing, Journal of Aircraft, Vol. 41, No. 5, 2004, pp. 1196-1201. [7] Jenkins, D., Ifju, P., Abdulrahim, M., Olipra, S., Assessment of the C ontrollability of Micro Air Vehicles, Bristol International RPV/UAV Conference, Bristol, UK, April 2-4, 2001. [8] Lian, Y., Shyy, W., Laminar-Turbulent Tran sition of a Low Reynolds Number Rigid or Flexible Airfoil, AIAA Journal, Vol. 45, No. 7, 2007, pp. 1501-1513. [9] Albertani, R., Stanford, B., Hubner, J., Ifju P., Aerodynamic Coefficients and Deformation Measurements on Flexible Micro Air Vehicle Wings, Experimental Mechanics, Vol. 47, No. 5, 2007, pp. 625-635. [10] Ifju, P., Jenkins, D., Ettinger, S., Lian, Y., Shyy, W., Waszak, M., Fle xible-Wing-Based Micro Air Vehicles, Confederation of European Aerospace Societies Aerodynamics Conference, London, UK, June 10-12, 2003. [11] Shirk, M., Hertz, T., Weisshaar, T., Aeroelastic Tailoring-Theory, Practice and Promise, Journal of Aircraft, Vol. 23, No. 1, 1986, pp. 6-18. [12] Griffin, C., Pressure Deflec tion Behavior of Candidate Ma terials for a Morphing Wing, Masters Thesis, Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, WV, 2007. [13] Waszak, M., Jenkins, L., Ifju, P., Stability and Control Properties of an Aeroelastic Fixed Wing Micro Air Vehicle, AIAA Atmospheric Flight Mec hanics Conference and Exhibit, Montreal, Canada, August 6-9, 2001. [14] Stanford, B., Ifju, P., Membrane Micro Air Vehicles with Adaptive Aerodynamic Twist: Numerical Modeling, AIAA Structures, Structural Dynamics, and Materials Conference, Honolulu, HI, April 23-26, 2007.

PAGE 168

168 [15] Ormiston, R., Theoretical and Experimental Aerodynamics of the Sail Wing, Journal of Aircraft, Vol. 8, No. 2, 1971, pp. 77-84. [16] Bendse, M., Sigmund, O., Topology Optimization, Springer-Verlag, Berlin, Germany, 2003. [17] Carmichael, B., Low Reynolds Number Airfoil Survey, NASA Contractor Report, CR 165803, 1981. [18] Lin, J., Pauley, L., Low-Reynolds-Number Separation on an Airfoil, AIAA Journal, Vol. 34, No. 8, 1996, pp. 1570-1577. [19] Mooney, M., A Theory of Large Elastic Deformation, Journal of Applied Physics, Vol. 11, 1940, pp. 582-592. [20] Dodbele, S., Plotkin, A., Loss of Lift Due to Thickness for Low-Aspect-Ratio Wings in Incompressible Flow, NASA Technical Report, TR 54409, 1987. [21] Gopalarathnam, A., Selig, M., Low-Speed Na tural-Laminar-Flow Airfoils: Case Study in Inverse Airfoil Design, Journal of Aircraft, Vol. 38, No. 1, 2001, pp. 57-63. [22] Kellogg, M., Bowman, J., Parametric Design Stu dy of the Thickness of Airfoils at Reynolds Numbers from 60,000 to 150,000, AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 5-8, 2004. [23] Laitone, E., Wind Tunnel Tests of Wi ngs at Reynolds Numbers Below 70,000, Experiments in Fluids, Vol. 23, No. 5, 1997, pp. 405-409. [24] Mueller, T., The Influence of Laminar Se paration and Transition on Low Reynolds Number Airfoil Hysteresis, Journal of Aircraft, Vol. 22, No. 9, 1985, pp. 763-770. [25] Gad-el-Hak, M., Micro-Air-Vehicles: Can They be Controlled Better? Journal of Aircraft, Vol. 38, No. 3, 2001, pp. 419-429. [26] Shyy, W., Lian, Y., Tang, J., Viieru, D., Liu, H., Aerodynamics of Low Reynolds Number Flyers, Cambridge University Press, New York, NY, 2008. [27] Katz, J., Plotkin, A., Low-Speed Aerodynamics, Cambridge University Press, Cambridge, UK, 2001. [28] Lian, Y., Shyy, W., Viieru, D., Zhang, B., Membrane Wing Mechanics for Micro Air Vehicles, Progress in Aerospace Sciences, Vol. 39, No. 6, 2003, pp. 425-465. [29] Nagamatsu, H., Low Reynolds Number Aerodyn amic Characteristics of Low Drag NACA 63208 Airfoil, Journal of Aircraft, Vol. 18, No. 10, 1981, pp. 833-837. [30] Masad, J., Malik, M., Link Between Fl ow Separation and Transition Onset, AIAA Journal, Vol. 33, No. 5, 1995, pp. 882-887.

PAGE 169

169 [31] Schroeder, E., Baeder, J., Usi ng Computational Fluid Dynamics for Micro Air Vehicle Airfoil Validation and Prediction, AIAA Applied Aerodynamics Conference, Toronto, Canada, June 69, 2005. [32] Winter, H., Flow Phenomena on Plates and Airfoils of Short Span, NACA Technical Report, TR 539, 1935. [33] Sathaye, S., Yuan, J., Olinger, D., Lift Di stributions on Low-Aspect -Ratio Wings at Low Reynolds Numbers for Micro-Ai r-Vehicle Applications, AIAA Applied Aerodynamics Conference and Exhibit, Providence, RI, Aug. 16-19, 2004. [34] Pellettier, A., Mueller, T., L ow Reynolds Number Aerodyna mics of Low Aspect Ratio Thin/Flat/Cambered-Plate Wings, Journal of Aircraft, Vol. 37, No. 5, 2000, pp. 825-832. [35] Bartlett, G., Vidal, R., Experimental Inves tigation of Influence of Edge Shape on the Aerodynamic Characteristics of Low Aspect Ratio Wings at Low Speeds, Journal of Aeronautical Sciences, Vol. 22, No. 8, 1955, pp. 517-533. [36] Polhamus, E., A Note on the Drag Due to Lift of Rectangular Wings of Low Aspect Ratio, NACA Technical Report, TR 3324, 1955. [37] Kaplan, S., Altman, A., Ol, M., Wake Vorticity Measurements for Low Aspect Ratio Wings at Low Reynolds Numbers, Journal of Aircraft, Vol. 44, No. 1, 2007, pp. 241-251. [38] Viieru, D., Albertani, R., Shyy, W., Ifju, P., Effect of Tip Vortex on Wing Aerodynamics of Micro Air Vehicles, Journal of Aircraft, Vol. 42, No. 6, 2005, pp. 1530-1536. [39] Mueller, T., DeLaurier, J., A erodynamics of Small Vehicles, Annual Review of Fluid Mechanics, Vol. 35, No. 35, 2003, pp. 89-111. [40] Gursul, I., Taylor, G., Wooding, C., Vortex Flows Over Fixed-Wing Micro Air Vehicles, AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 14-17, 2002. [41] Cosyn, P., Vierendeels, J., Numerical Inve stigation of Low-Aspect-Ratio Wings at Low Reynolds Numbers, Journal of Aircraft, Vol. 43, No. 3, 2006, pp. 713-722. [42] Brion, V., Aki, M., Shkarayev, S., Numeri cal Simulation of Low Reynolds Number Flows Around Micro Air Vehicles and Comp arison against Wind Tunnel Data, AIAA Applied Aerodynamics Conference, San Francisco, CA, June 5-8, 2006. [43] Stanford, B., Sytsma, M., Albertani, R., Viieru, D., Shyy, W., Ifju, P., S tatic Aeroelastic Model Validation of Membrane Micro Air Vehicle Wings, AIAA Journal, Vol. 45, No. 12, 2007, pp. 2828-2837. [44] Zhan, J., Wang, W., Wu, Z., Wang, J., Wind-T unnel Experimental Investigation on a Fix-Wing Micro Air Vehicle, Journal of Aircraft, Vol. 43, No. 1, 2006, pp. 279-283.

PAGE 170

170 [45] Ramamurti, R., Sandberg, W., Lhner, R., Simulat ion of the Dynamics of Micro Air Vehicles, AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 10-13, 2000. [46] Gyllhem, D., Mohseni, K., Lawrence, D., Num erical Simulation of Flow Around the Colorado Micro Aerial Vehicle, AIAA Fluid Dynamics Conf erence and Exhibit, Toronto, Canada, June 6-9, 2005. [47] Albertani, R., Experimental Aerodynamics a nd Elastic Deformation Characterization of Low Aspect Ratio Flexible Fixed Wings Applied to Micro Aerial Vehicles, Ph.D. Dissertation, Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL, 2005. [48] Morris, S., Design and Flight Test Results for Micro-Sized Fixed Wing and VTOL Aircraft, International Conference on Emerging Technologies for Micro Air Vehicles, Atlanta, GA, February 3-5, 1997. [49] Rais-Rohani, M., Hicks, G., Multidisciplinary Design and Prototype Development of a Micro Air Vehicle, Journal of Aircraft, Vol. 36, No. 1, 1999, pp. 227-234. [50] Kajiwara, I., Haftka, R., Simultaneous Optimal Design of Shape and Control System for Micro Air Vehicles, AIAA Structures, Structural Dynam ics, and Materials Conference, St. Louis, MO, April 12-15, 1999. [51] Lundstrm, D., Krus, P., Micro Aerial Vehicle Design Optimization using Mixed Discrete and Continuous Variables, AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Portsmouth VA, September 6-8, 2006. [52] Ng, T., Leng, G., Application of Genetic Algor ithms to Conceptual Design of a Micro Air Vehicle, Engineering Applications of Artificial Intelligence, Vol. 15, No. 5, 2003, pp. 439-445. [53] Sloan, J., Shyy, W., Haftka, R., Airfoil and Wing Planform Optimization for Micro Air Vehicles, Symposium of RTO Applied Vehicle Technology Panel, Ottawa, Canada, October 1921, 1999. [54] Lian, Y., Shyy, W., Haftka, R., Shape Optim ization of a Membrane Wing for Micro Air Vehicles, AIAA Journal, Vol. 42, No. 2, 2004, pp. 424-426. [55] Boria, F., Stanford, B., Bowman, W., Ifju, P ., Evolutionary Optimization of a Morphing Wing with Wind Tunnel Hardware-in-the-Loop, Aerospace Science and Technology, submitted for publication. [56] Hunt, R., Hornby, G., Lohn, J., Toward Evolved Flight, Genetic and Evolutionary Computation Conference, Washington, DC, June 25-29, 2005. [57] Day, A., Optimization of a Micro Aerial Vehicle Planform Using Genetic Algorithms, Masters Thesis, Department of Mechanical E ngineering, Worcester Po lytechnic Institute, Worcester, MA, 2007.

PAGE 171

171 [58] Fitt, A., Pope, M., The Unsteady Motion of Tw o-Dimensional Flags With Bending Stiffness, Journal of Engineering Mechanics, Vol. 40, No. 3, 2001, pp. 227-248. [59] Wilkinson, M, Sailing the Skies: the Improba ble Aeronautical Success of the Pterosaurs, Journal of Expe rimental Biology, Vol. 210, No. 10, 2007, pp. 1663-1671. [60] Bisplinghoff, R., Ashley, H., Halfman, R., Aeroelasticity, Dover, Mineola, NY, 1955. [61] Stanford, B., Abdulrahim, M., Lind, R., Ifju, P., Investigation of Memb rane Actuation for Roll Control of a Micro Air Vehicle, Journal of Aircraft, Vol. 44, No. 3, 2007, pp. 741-749. [62] Argentina, M., Mahadevan, L., Fluid -Flow-Induced Flutter of a Flag, Proceedings of the National Academy of Science: Applied Mathematics, Vol. 102, No. 6, 2005, pp. 1829-1834. [63] Alben, S., Shelley, M., Zhang, J., How Flex ibility Induces Streamlining in a Two-Dimensional Flow, Physics of Fluids, Vol. 16, No. 5, 2004, pp. 1694-1713. [64] Voelz, K., Profil und Auftrieb Eines Segels, Zeitschrift fur Ange wandte Mathematik und Mechanik, Vol. 30, 1950, pp. 301-317. [65] Thwaites, B., The Aerodynamic Theory of Sails, Proceedings of the Royal Society of London, Vol. 261, No. 1306, 1961, pp. 402-422. [66] Nielsen, J., Theory of Flexible Aerodynamic Surfaces, Journal of Applied Mechanics, Vol. 30, No. 3, 1963, pp. 435-442. [67] Haselgrove, M., Tuck, E., Stability Propert ies of the Two-Dimensional Sail Model, Society of Naval Architects and Marine Engineers New England Sailing Yacht Symposium, New London, CN, January 24, 1976. [68] Murai, H., Murayama, S., Theor etical Investigation of Sailwing Airfoils Taking Account of Elasticities, Journal of Aircraft, Vol. 19, No. 5, 1982, pp.385-389. [69] Jackson, P., A Simple Model for Elastic Two-Dimensional Sails, AIAA Journal, Vol. 21, No. 1, 1983, pp.153-155. [70] Sneyd, A., Aerodynamic Coefficients and L ongitudinal Stability of Sail Aerofoils, Journal of Fluid Mechanics, Vol. 149, No. 7, 1984, pp.127-146. [71] Cyr, S., Newman, B., Flow Past Two-Dimensional Membrane Aerofoils with Rear Separation, Journal of Wind Engineeri ng and Industrial Aerodynamics, Vol. 63, No. 1, 1996, pp. 1-16. [72] Smith, R., Shyy, W., Computational Model of Flexible Membrane Wings in Steady Laminar Flow, AIAA Journal, Vol. 33, No. 10, 1995, pp. 1769-1777. [73] Newman, B., Low H., Two-Dimensional Impervi ous Sails: Experimental Results Compared with Theory, Journal of Fluid Mechanics, Vol. 144, 1984, pp. 445-462.

PAGE 172

172 [74] Smith, R., Shyy, W., Computation of Aerodynami c Coefficients for a Flexible Membrane Airfoil in Turbulent Flow: A Comparison with Classical Theory, Physics of Fluids, Vol. 8, No. 12, 1996, pp. 3346-3353. [75] Lorillu, O., Weber, R., Hureau, J., Numerical and Experimental Analysis of Two-Dimensional Separated Flows over a Flexible Sail, Journal of Fluid Mechanics, Vol. 466, 2002, pp. 319341. [76] Jackson, P., Christie, G., Numerical Analys is of Three-Dimensional Elastic Membrane Wings, AIAA Journal, Vol. 25, No. 5, 1987, pp. 676-682. [77] Sneyd, A., Bundock, M., Reid, D., Possible E ffects of Wing Flexibility on the Aerodynamics of Pteranodon, The American Naturalist, Vol. 120, No. 4, 1982, pp. 455-477. [78] Boudreault, R., -D Program Predicting th e Flexible Membrane Wings Aerodynamic Properties, Journal of Wind Engineeri ng and Industrial Aerodynamics, Vol. 19, No. 1, 1985, pp. 277-283. [79] Holla, V., Rao, K., Arokkiaswamy, A., Asth ana, C., Aerodynamic Characteristics of Pretensioned Elastic Membrane Rectangular Sailwings, Computer Methods in Applied Mechanics and Engineering, Vol. 44, No. 1, 1984, pp. 1-16. [80] Sugimoto, T., Analysis of Circular Elastic Membrane Wings, Transactions of the Japanese Society of Aerodynamics and Space Sciences, Vol. 34, No. 105, 1991, pp. 154-166. [81] Charvet, T., Hauville, F., Huberson, S., Numerical Simulation of the Flow Over Sails in Real Sailing Conditions, Journal of Wind Engineering and Industrial Aerodynamics, Vol. 63, No. 1, 1996, pp. 111-129. [82] Schoop, H., Bessert, N., Taenzer, L., On the Elastic Membrane in a Potential Flow, International Journal for Nume rical Methods in Engineering, Vol. 41, No. 2, 1998, pp. 271291. [83] Stanford, B., Albertani, R., Ifju, P., Static Fi nite Element Validation of a Flexible Micro Air Vehicle, Experimental Mechanics, Vol. 47, No. 2, 2007, pp. 283-294. [84] Ferguson, L., Seshaiyer, P., Gordnier, R., A ttar, P., Computationa l Modeling of Coupled Membrane-Beam Flexible Wings for Micro Air Vehicles, AIAA Structures, Structural Dynamics, and Materials Conference, Honolulu, HI, April 23-26, 2007. [85] Smith, R., Shyy, W., Incremental Potential Flow Based Membrane Wing Element, AIAA Journal, Vol. 35, No. 5, 1997, pp. 782-788. [86] Heppel, P., Accuracy in Sail Simulation: Wrinkling and Growing Fast Sails, High Performance Yacht Design Conference, Auckland, New Zealand, December 4-6, 2002. [87] Fink, M., Full-Scale In vestigation of the Aerodynamic Char acteristics of a Model Employing a Sailwing Concept, NASA Technical Report, TR 4062, 1967.

PAGE 173

173 [88] Greenhalgh, S., Curtiss, H., Aerodynamic Char acteristics of a Flexible Membrane Wing, AIAA Journal, Vol. 24, No. 4, 1986, pp. 545-551. [89] Galvao, R., Israeli, E., Song, A., Tian, X., Bishop, K., Swartz, S., Breuer, K., The Aerodynamics of Compliant Membrane Wings Modeled on Mammalian Flight Mechanics, AIAA Fluid Dynamics Conf erence and Exhibit, San Francisco, CA, June 5-8, 2006. [90] Pennycuick, C., Lock, A., Elastic Energy Storage in Primary Feather Shafts, Journal of Experimental Biology, Vol. 64, No. 3, 1976, pp. 677-689. [91] Parks, H., Three-Component Velocity Measur ements in the Tip Vortex of a Micro Air Vehicle, Masters Thesis, School of Engineer ing and Management, Air Force Institute of Technology, Wright Patters on Air Force Base, OH, 2006. [92] Gamble, B., Reeder, M., Experimental Analysis of Propeller Interactions with a Flexible Wing Micro Air Vehicle, AIAA Fluid Dynamics Conference and Exhibit, San Francisco, CA, June 58, 2006. [93] Stults, J., Maple, R., Cobb, R., Parker, G., Com putational Aeroelastic An alysis of a Micro Air Vehicle with Experimentally Determined Modes, AIAA Applied Aerodynamics Conference, Toronto, Canada, June 6-9, 2005. [94] Ifju, P., Ettinger, S., Jenkins, D., Martinez, L ., Composite Materials for Micro Air Vehicles, Society for the Advancement of Material and Process Engineeri ng Annual Conference, Long Beach, CA, May 6-10, 2001. [95] Frampton, K., Goldfarb, M., Monopoly, D., Cveti canin, D., Passive Aeroelastic Tailoring for Optimal Flapping Wings, Proceedings of Conference on Fixed, Flapping, and Rotary Wing Vehicles at Very Lo w Reynolds Numbers, South Bend, IN, June 5-7, 2000. [96] Snyder, R., Beran, P., Parker, G., Blair, M ., A Design Optimization Strategy for Micro Air Vehicles, AIAA Structures, Structural Dy namics, and Materials Conference, Honolulu, HI, April 23-26, 2007. [97] Allen, M., Maute, K., Probabi listic Structural Design of U AVs under Aeroelastic Loading, AIAA Unmanned Unlimited Conference, San Diego, CA, September 15-18, 2003. [98] Weisshaar, T., Nam, C., Batista-Rodriguez, A., Aeroelastic Tailoring for Improved UAV Performance, AIAA Structures, Structural Dynamics, and Materials Conference, Long Beach, CA, April 20-23, 1998. [99] Garrett, R., The Symmetry of Sailing: The Physics of Sailing for Yachtsmen, Adlard Coles, Dobbs Ferry, NY, 1996. [100] Eden, M., The Magnificent Book of Kites: Explorations in Design, Construction, Enjoyment, and Flight, Sterling Publishing, New York, NY, 2002.

PAGE 174

174 [101] Templin, R., Chatterjee, S., Posture, Locomotion, and Paleoecology of Pt erosaurs, Geological Society of America Special Paper 376, 2004. [102] Swartz, S., Groves, M., Kim, H., Walsh, W., Mechanical Properties of Bat Wing Membrane Skin, Journal of Zoology, Vol. 239, 1996, pp. 357-378. [103] Norberg, U., Bat Wing Structures Important for Aerodynamics and Rigidity, Zoomorphology, Vol. 73, No. 1, 1972, pp. 45-61. [104] Levin, O., Shyy, W., Optimization of a Lo w Reynolds Number Airfoil with Flexible Membrane, Computer Modeling in Engineering and Sciences, Vol. 2, No. 4, 2001, pp. 523536. [105] Zuo, K., Chen, L., Zhang, Y., Yang, J., St udy of Key Algorithms in Topology Optimization, International Journal of Adv anced Manufacturing Technology, Vol. 32, No. 7, 2007, pp. 787796. [106] Maute, K., Reich, G., Integrated Multidisci plinary Topology Optimization Approach to Adaptive Wing Design, Journal of Aircraft, Vol. 43, No. 1, 2006, pp. 253-263. [107] Pingen, G., Evgrafov, A., Maute, K., Topol ogy Optimization of Flow Domains Using the Lattice Boltzmann Method, Structural and Multidisciplinary Optimization, Vol. 34, No. 6, 2007, pp. 507-524. [108] Beckers, M., Topology Optimization using a Dual Method with Discrete Variables, Structural and Multidiscipli nary Optimization, Vol. 17, No. 1, 1999, pp. 14-24. [109] Deb, K., Goel, T., A Hybrid Multi-Objective Evolutionary Approach to Engineering Shape Design, International Conference on Evolutionary Multi-Criterion Optimization, March 7-9, Zurich, Switzerland, 2001. [110] Maute, K., Nikbay, M., Farhat, C., Sensitivity Analysis and Design Optimization of ThreeDimensional Non-Linear Aeroelastic Systems by the Adjoint Method, International Journal for Numerical Methods in Engineering, Vol. 56, No. 6, 2002, pp. 911-933. [111] Min, S., Nishiwaki, S., Kikuchi, N., Unifi ed Topology Design of Static and Vibrating Structures Using Multiobjective Optimization, Computers and Structures, Vol. 75, No. 1, 2000, pp. 93-116. [112] Borrvall, T., Petersson, J., Topology Op timization of Fluids in Stokes Flow, International Journal for Numerical Methods in Fluids, Vol. 41, No. 1, 2003, pp. 77-107. [113] Balabanov, V., Haftka, R., Topology Optimization of Transport Wing Internal Structure, Journal of Aircraft, Vol. 33, No. 1, 1996, pp. 232-233. [114] Eschenauer, H., Olhoff, N., Topology Optimiza tion of Continuum Structures: A Review, Applied Mechanics Reviews, Vol. 54, No. 4, 2001, pp. 331-390.

PAGE 175

175 [115] Krog, L., Tucker, A. Kemp, M., Topology Op timization of Aircraft Wing Box Ribs, AIAA/ISSMO Multidisciplinary Anal ysis and Optimization Conference, Albany, NY, August 30September 1, 2004. [116] Luo, Z., Yang, J., Chen, L., A New Pro cedure for Aerodynamic Missile Designs Using Topological Optimization Approach of Continuum Structures, Aerospace Science and Technology, Vol. 10, No. 5, 2006, pp. 364-373. [117] Santer, M., Pellegrino, S., Topology Optimizati on of Adaptive Compliant Aircraft Leading Edge, AIAA Structures, Structural Dy namics, and Materials Conference, Honolulu, HI, April 23-26, 2007. [118] Maute, K., Allen, M., Conceptual Desi gn of Aeroelastic Structures by Topology Optimization, Structural and Multidis ciplinary Optimization, Vol. 27, No. 1, 2004, pp. 27-42. [119] Martins, J., Alonso, J., Reuther, J., AeroStructural Wing Design Optimization Using HighFidelity Sensitivity Analysis, Confederation of European Aero space Societies Conference on Multidisciplinary Analysis and Optimization, Cologne, Germany, June 25-26, 2001. [120] Gomes, A., Suleman, A., Optimization of Ai rcraft Aeroelastic Response Using the Spectral Level Set Method, AIAA Structures, Structural Dy namics, and Materials Conference, Austin, TX, April 18-21, 2005. [121] Combes, S., Daniel, T., Flexural Stiffness in Insect Wings: Scaling and Influence of Wing Venation, The Journal of Ex perimental Biology, Vol. 206, No. 6, 2003, pp. 2979-2987. [122] Marchman, J., Aerodynamic Testing at Low Reynolds Numbers, Journal of Aircraft, Vol. 24, No. 2, 1987, pp. 107-114. [123] Recommended Practice R-091-2003, Calibration a nd Use of Internal Strain-Gage Balances with Application to Wind Tunnel Testing, AIAA, Reston, VA, 2003. [124] Kochersberger, K., Abe, C., A Novel, Low Reynolds Number Moment Balance Design for Micro Air Vehicle Research, AIAA Fluid Dynamics Conference and Exhibit, Toronto, Canada, June 6-9, 2005. [125] Moschetta, J., Thipyopas, C., Aerodynamic Pe rformance of a Biplane Micro Air Vehicle, Journal of Aircraft, Vol. 44, No. 1, 2007, pp. 291-299. [126] Mueller, T., Aerodynamic Measurements at Low Reynolds Numbers for Fixed Wing MicroAir Vehicles, RTO Special Course on the Development and Operation of UAVs for Military and Civil Applications, Von Karman Institute, Belgium, September 13-17, 1999. [127] Kline, S., McClintock, F., Describing Uncertainties in Single-Sample Experiments, Mechanical Engineering, Vol. 75, No. 1, 1953, pp. 3-8. [128] Pankhurst, R., Holder, D., Wind Tunnel Technique, Sir Isaac Pitman and Sons, London, UK, 1952.

PAGE 176

176 [129] Barlow, J., Rae, W., Pope, A., Low-Speed Wind Tunnel Testing, Wiley, New York, NY, 1999. [130] Fleming, G., Bartram, S., Waszak, M., Jenki ns, L., Projection Moir Interferometry Measurements of Micro Air Vehi cle Wings, SPIE Paper 4448-16. [131] Burner, A., Fleming, G., Hoppe, J., Compari son of Three Optical Methods for Measuring Model Deformation, AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 10-13, 2000. [132] Sutton, M., Turner, J., Bruck, H., Chae, T., Full Field Representation of the Discretely Sampled Surface Deformation for Displacement and Strain Analysis, Experimental Mechanics, Vol. 31, No. 2, 1991, pp. 168-177. [133] Schreier, H., Braasch, J., Sutton, M., Systematic Errors in Digital Image Correlation caused by Intensity Interpolation, Optical Engineering, Vol. 39, No. 11, 2000, pp. 2915-2921. [134] Sutton, M., McFadden, C., Development of a Methodology for Non-Contacting Strain Measurements in Fluid Environm ents Using Computer Vision, Optics and Lasers in Engineering, Vol. 32, No. 4, 2000, pp. 367-377. [135] Albertani, R., Stanford, B., Sytsma, M., Ifju, P., Unsteady Mechanical Aspects of Flexible Wings: an Experimental Investigation A pplied to Biologically Inspired MAVs, European Micro Air Vehicle Conference and Flight Competition, Toulouse, France, September 17-21, 2007. [136] Batoz, J., Bathe, K., Ho, L., A Study of Th ree-Node Triangular Plate Bending Elements, International Journal for Nume rical Methods in Engineering, Vol. 15, No. 12, 1980, pp. 17711812. [137] Cook, R., Malkus, D., Plesha, M., Witt, R., Concepts and Applications of Finite Element Analysis, Wiley, New York, NY, 2002. [138] Reaves, M., Horta, L., Waszak, M., Morgan, B ., Model Update of a Micro Air Vehicle (MAV) Flexible Wing Frame with Uncertainty Quantification, NASA Technical Memorandum, TM 213232, 2004. [139] Isenberg, C., The Science of Soap Films and Soap Bubbles, Dover, New York, NY, 1992. [140] Pujara, P., Lardner, T., Deformations of Elastic Membranes Effect of Different Constituitive Relations, Zeitschrift fr Angewandte Mathematik und Physik, Vol. 29, No. 2, 1978, pp. 315327. [141] Small, M., Nix, W., Analysis of the Accuracy of the Bulge Test in Determining the Mechanical Properties of Thin Films, Journal of Materials Research, Vol. 7, No. 6, 1992, pp. 1553-1563. [142] Pauletti, R., Guirardi, D., Deifeld, T., A rgyris Natural Finite Element Revisited, International Conference on Textile Composite s and Inflatable Structures, Stuttgart, Germany, October 2-5, 2005.

PAGE 177

177 [143] Lian, Y., Shyy, W., Ifju, P., Verron, E., A Co mputational Model for C oupled Membrane-Fluid Dynamics, AIAA Fluid Dynamics Conf erence and Exhibit, St. Louis, MO, June 24-26, 2002. [144] Wu, B., Du, X., Tan, H., A Three-Dimensi onal FE Nonlinear Analysis of Membranes, Computers and Structures, Vol. 59, No. 4, 1996, pp. 601-605. [145] Campbell, J., On the Theory of Initially Tens ioned Circular Membrane s Subjected to Uniform Pressure, Quarterly Journal of Mechanics and Applied Mathematics, Vol. 9, No. 1, 1956, pp. 84-93. [146] Mase, G., Mase, G., Continuum Mechanics for Engineers, CRC Press, Boca Raton, FL, 1999. [147] Stanford, B., Boria, F., Ifju, P., The Validit y Range of Pressurized Membrane Models with Varying Fidelity, Society for Experimental Mechanics, Springfield, MA, June 4-6, 2007. [148] Tannehill, J., Anderson, D., Pletcher, R., Computational Fluid Mechanics and Heat Transfer, Taylor and Francis, Philadelphia, PA, 1997. [149] Shyy, W., Computational Modeling for Fluid Flow and Interfacial Transport, Elsevier, Amsterdam, The Netherlands, 1994. [150] Thakur, S., Wright, J., Shyy, W., STREAM: A Computational Fluid Dynamics and Heat Transfer Navier-Stokes Solver: Theory and Applications, Streamline Numerics, Inc., Gainesville, FL, 2002. [151] Lewis, W., Tension Structures: Form and Behavior, Thomas Telford Ltd, London, UK, 2003. [152] Kamakoti, R., Lian, Y., Regisford, S., Kurdila A., Shyy, W., Computat ional Aeroelasticity Using a Pressure-Based Solver, Computer Methods in Engineering and Sciences, Vol. 3, No. 6, 2002, pp. 773-790. [153] Sytsma, M., Aerodynamic Flow Characterization of Micro Ai r Vehicles Utilizing Flow Visualization Methods, Masters Thesis, Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL, 2006. [154] Hepperle, M., Aerodynamics of Spar and Rib Structures, MH AeroTools Online Database, http://www.mh-aerotools.de/airfoils/ribs.htm March 2007. [155] Grdal, Z., Haftka, R., Hajela, P., Design and Optimization of Laminated Composites Materials, Wiley, New York, NY, 1999. [156] Deb, K., Pratap, A., Agarwal, S., Meyarivan, T., A Fast and Elitis t Multiobjective Genetic Algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, Vol. 6, No. 2, 2002, pp. 182-197. [157] Antony, J., Design of Experiments fo r Engineers and Scientists, Butterworth-Heinemann, Boston, MA, 2003.

PAGE 178

178 [158] Chen, T., Wu, S., Multiobjective Optim al Topology Design of Structures, Computational Mechanics, Vol. 21, No. 6, 2998, pp. 483-492. [159] Haftka, R., Grdal, Z, Elements of Structural Optimization, Kluwer, Dordrecht, The Netherlands, 1992. [160] Maute, K., Nikbay, M., Farhat, C., Coupled An alytical Sensitivity Analysis and Optimization of Three-Dimensional Nonlinear Aeroelastic Systems, AIAA Journal, Vol. 39, No. 11, 2001, pp. 2051-2061. [161] Lyu, N., Saitou, K., Topology Optimization of Multicomponent Beam Structure via Decomposition-Based Assembly Synthesis, Journal of Mechanical Design, Vol. 127, No. 2, 2005, pp. 170-183. [162] Hsu, M., Hsu, Y., Interpreting Three-Dimens ional Structural Topology Optimization Results, Computers and Structures, Vol. 83, No. 4, 2005, pp. 327-337. [163] Rohl, P., Schrage, D., Mavris, D., Combined Aerodynamic and Structural Optimization of a High-Speed Civil Transport Wing, AIAA Structures, Structural Dynamics, and Materials Conference, New Orleans, LA, April 18-21, 1995.

PAGE 179

179 BIOGRAPHICAL SKETCH Bret Kennedy Stanford was born in Rich m ond, Virginia on September 30, 1981, though his grandmother claims it was on September 29. School was never really an option for young Bret, forced by his parents at an ear ly age to join the circus instead He was taught to read, write, and juggle by a kindly group of clowns, despite his extreme terror of anything with big floppy shoes, a phobia which continues un abated to this day. Bret wa s reunited with his parents two years later, an act which was prompted by a re cent increase in the Child Tax Credit. Several brush-ins with the law led to the Stanford familys expulsion from Virginia, escorted to the North Carolina border by a group of uns ympathetic state troopers. The family subsequently relocated to Tampa, Florida in the fall of 1988, though Brets grandmother claims it was in the summer. Brets time in Tampa was mostly spent selli ng hand-carved limestone trinkets and jewelry to tourists. At the age of 17, he was rejected from most of the universities along the eastern seaboard, who were collectively unimpressed with his artesian and entertainment backgrounds. A clerical error granted him acceptance to the Univers ity of Florida. He arrived in Gainesville in the fall of 1999 (a date his grandmother genera lly agrees upon) with the intent of studying French post-modern theatre. Nine convoluted ye ars later he received hi s doctorate in aerospace engineering. Upon graduation, he plans on throwing all of his newfound knowledge, books, and lab journals into the River, in order to start a Beach Boys cover band. He hopes that his advisor will handle all of the journal review replies as th ey come back from the editors, so the research will not have gone to waste.