UFDC Home myUFDC Home  |   Help
<%BANNER%>

# Robust Multicriteria Optimization of Surface Location Error and Material Removal Rate in High-Speed Milling under Uncertainty

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 E20101123_AAAACV INGEST_TIME 2010-11-23T15:15:07Z PACKAGE UFE0011626_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 30297 DFID F20101123_AABRKB ORIGIN DEPOSITOR PATH kurdi_m_Page_139.QC.jpg GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
80a476bc42938d90fec21d67752aff1b
SHA-1
b72994a5f38326e7b1712dcf702babf0f8313e2d
14855 F20101123_AABRJN kurdi_m_Page_132.QC.jpg
edea1a156423080f39c4a3177f01d3c3
19087 F20101123_AABRIZ kurdi_m_Page_125.QC.jpg
732abbd3208a2d3a7452a06ed8fdd58c
a73cd3d994bd0d173342475ed6e5e2c8a032e105
1053954 F20101123_AABQFX kurdi_m_Page_003.tif
e3c455b6f6ebec7eb8eb71c3724313bcda8f2f3b
F20101123_AABQGL kurdi_m_Page_018.tif
491c0d3ff8151f43cf0e3049d3c69cc8
05ac887f2d55021888c88e5022f06e6f2e671074
7555 F20101123_AABRKC kurdi_m_Page_139thm.jpg
39eb579835e017f7d83b6423cf3d7111
7186f239309bcf7720fdf5d4c2354bb55b987a5f
4296 F20101123_AABRJO kurdi_m_Page_132thm.jpg
ce9f7dcd5c88d3a7846f73e7c811efd2
4d53e7984edb41f418060f8d649197e16152af30
F20101123_AABQFY kurdi_m_Page_004.tif
66d601379ff2083c527acfe2d562b1bf
e8c852d254ce9626d3557a3413b6ebba2db88c8b
F20101123_AABQHA kurdi_m_Page_035.tif
19464131fdfa3e82fa154b1c44a37201
4d02ec775eb92a5b4aaf6f28222d1ef597f363e0
F20101123_AABQGM kurdi_m_Page_019.tif
0827105597e109f68fb91efc8823df9b
20a17aee1a122b8594077faf66a1af114ee4534c
7355 F20101123_AABRKD kurdi_m_Page_140thm.jpg
edd7c9d61259757367a6251c847fbe51
f3280db47f3ab213855b849fe2d1c75f0d252099
18955 F20101123_AABRJP kurdi_m_Page_133.QC.jpg
1993ca0e8e6570dbca4979f46e19487f
abf2838a8b9b86cfb2b979d21f82ac041827dc2d
25271604 F20101123_AABQFZ kurdi_m_Page_006.tif
9121437543b3b76e84b96c74da393a60
970e71a4efe4e1bee242e75199a838c937cbcbb7
F20101123_AABQHB kurdi_m_Page_036.tif
5d2f20103896238954846f66ebbe84c9
70b406832e9cabbc13c13be5ac02252759e0aa35
F20101123_AABQGN kurdi_m_Page_020.tif
6c98bca68a31ff50659da8689a9e7406
cd7c8b474705c1c5450f1bc78e04f5f2d86dba6e
29421 F20101123_AABRKE kurdi_m_Page_141.QC.jpg
0573fe87cd8bbac7c5d22a1daffd9de4
cb370608b5d3414e569ce203a541a4f805e45183
5221 F20101123_AABRJQ kurdi_m_Page_133thm.jpg
5bcf380ece4d82629cd90d648f55b93f
cba8aa41eaeea2ff47058c89b6af00e38f33aef2
F20101123_AABQHC kurdi_m_Page_037.tif
3b6575307df28b88ef835654d5c703a9
7e36c9fded9853930a1f2448ffd94bb338174f31
F20101123_AABQGO kurdi_m_Page_021.tif
cf2235b85a3b2e09bdb662469fde3cf4
434a58633781a2505151537d9e0db928a55e2a32
7460 F20101123_AABRKF kurdi_m_Page_141thm.jpg
7d101c361555f363e80e6a5f843594361906f56f
17605 F20101123_AABRJR kurdi_m_Page_134.QC.jpg
8ca0152a2916cb44dd04326edfcf1f2fdfa0c9c3
F20101123_AABQHD kurdi_m_Page_038.tif
2b27ce2e89f50c31ebab76ea202f5fbc
0ae33a2890e812f62375903e20814bb9e5192ac9
F20101123_AABQGP kurdi_m_Page_022.tif
553c2a929b21877d60bc3cd3a89a9131
29428 F20101123_AABRKG kurdi_m_Page_142.QC.jpg
df6a332cf57078da795a676ee54e06e1
38dc6194d3b5641674f51fc81e0393d8db346f9b
4893 F20101123_AABRJS kurdi_m_Page_134thm.jpg
12bf5147cfd80f6738ab59fd022f4f90
8311b85f9c708b86f549d9ab379eec538986dd94
F20101123_AABQHE kurdi_m_Page_039.tif
439f5a48145ae1617d513420e017d5a0
afa1bf41f368ede27b8d255a7a503f196ef14ce9
F20101123_AABQGQ kurdi_m_Page_023.tif
5717bf536f18716e762c88cc444bb641
97e6f8c204dfa1cdbee1108430afa3b6f4197300
7412 F20101123_AABRKH kurdi_m_Page_142thm.jpg
8b421be93543b29b58858753dd6b8b46
bc71565db52d1ab9c584c0557f8b29bdc2d8ef23
20036 F20101123_AABRJT kurdi_m_Page_135.QC.jpg
6020616acd1b881ceee0011f993bca5f248edbca
F20101123_AABQHF kurdi_m_Page_041.tif
c6b5e9ac5c9ab0dbebb28f2e1a12bfb6
F20101123_AABQGR kurdi_m_Page_024.tif
2fc347007c4178778fcf345924b37620
78322c14606789f2856e6589830075e9f1ff028d
25100 F20101123_AABRKI kurdi_m_Page_143.QC.jpg
7db2c0dd96cc79378196b0be835d88ae
bc757e25635bbf556da9f15f79e4c13523364c9c
5402 F20101123_AABRJU kurdi_m_Page_135thm.jpg
1d67e385b884ef5cd2ca29bc9ef2ae9f
77de34399a88bbabce5d8b8dd6390482bda5bccf
F20101123_AABQHG kurdi_m_Page_042.tif
1eebe57872833d52ec6096d311248d35
8690895f651665f32e0d50b62123f561a859acf8
F20101123_AABQGS kurdi_m_Page_025.tif
23425d9254744a066a3e4e3bd528b89e
b975a09b54f4662d69bd35750f88ac18451c646b
6222 F20101123_AABRKJ kurdi_m_Page_143thm.jpg
9829b3e5969d1a215371603a48241ac6
493a2d03fa4c4da50eac4c0f70ae84e5aa31705f
25758 F20101123_AABRJV kurdi_m_Page_136.QC.jpg
2ea424f6704538c6c42eafa19a4ab5ed
24c6f95e16b40002c05434bb002ee0f7ed939cb5
F20101123_AABQHH kurdi_m_Page_043.tif
6f22cb4598e938207d5bff621b74c2aa
6061791dbb2fa6bc83e151df28214e1a99c7fcaf
F20101123_AABQGT kurdi_m_Page_026.tif
1505fb080d589971e3aaa3f2510ea990
22064d9032834132432c4f06a29f789c3079c09b
17274 F20101123_AABRKK kurdi_m_Page_144.QC.jpg
a2cfa01b13e8e94e597c7d8b673a1d9a
598f54b493afa0fb9e3f5dac69ba556c0c0233e4
6646 F20101123_AABRJW kurdi_m_Page_136thm.jpg
ccf7f72fde93903d75f81fdb7e67c540
28290ab8eaf07edc1cabfbc1f81334af5e1ae886
F20101123_AABQHI kurdi_m_Page_044.tif
51a49edc41bec96001a3123455a6e159
F20101123_AABQGU kurdi_m_Page_028.tif
0c2f2897b38b3fbfca5e53516155a90c
5ca19f4bebd93f80f518a70e6c3a557d91fc7813
4800 F20101123_AABRKL kurdi_m_Page_144thm.jpg
5be381f386209fe196301777da4e4a25d77b9b39
28365 F20101123_AABRJX kurdi_m_Page_137.QC.jpg
ba6342b85b040257514f7b415dd9d2c4
de37dc1245169408cea8c42eab994e066d8ab52c
F20101123_AABQHJ kurdi_m_Page_046.tif
6a2de4d902af74dd297e23e13fa9b631
8abf7fb6d281a4087808e01af5708f4ea1c7d917
F20101123_AABQGV kurdi_m_Page_029.tif
5d69720ea045e7e87ac02bf3ac775bf8
27e1af50c69fd4ba264ec2edfd1250132a129fbf
7487 F20101123_AABRLA kurdi_m_Page_152thm.jpg
bb547136fe16d2dc4ab6645076fc0f00
301559232e2106259161d9638a9c4fc933a1a7f8
19704 F20101123_AABRKM kurdi_m_Page_145.QC.jpg
0743a8bba7fe313b47dfcf2569c8e59f
a781a3f254280c9c81b7a5cf0e39f81c3796bcf3
7398 F20101123_AABRJY kurdi_m_Page_137thm.jpg
0b58c959044cf9f7810b5070ebc5e333
09539f02dc8f6b6381ccdd5dc67a98e06538abf6
F20101123_AABQHK kurdi_m_Page_048.tif
1f44f4a3561d448b048bfefe0f2f228b
F20101123_AABQGW kurdi_m_Page_031.tif
42296b0cba7ffb56b642a52030db165e
136861f3acd27d6d68afdecf8517f073a300b91a
28634 F20101123_AABRLB kurdi_m_Page_153.QC.jpg
2d0ac2196949a353f86d9a3a61bd61d7
5416 F20101123_AABRKN kurdi_m_Page_145thm.jpg
f32bf9692089e3d213974d7b31eb0ab5
b4cbbace7e83e508b7072d7e20db42d0be1e1765
28974 F20101123_AABRJZ kurdi_m_Page_138.QC.jpg
3e9c13e648aa01903345983cf73e31a9
F20101123_AABQHL kurdi_m_Page_049.tif
0a0ed42638d499b7d8aea5c9a4fa47b2
1c27a6e9a9ab55258747ebf5e2fcf778a20a731d
F20101123_AABQGX kurdi_m_Page_032.tif
96de92fbea421e126ee516b396c965c8
aa5e62cd5bae481e0980288ef8417f9746539022
7210 F20101123_AABRLC kurdi_m_Page_153thm.jpg
9c3644184cf188a7ee60734c44457363
21300 F20101123_AABRKO kurdi_m_Page_146.QC.jpg
48801ce7eff773c83c3183ffec627907
F20101123_AABQIA kurdi_m_Page_066.tif
2d9f44b9e7d0b4611bd0ffb49fe16d17
F20101123_AABQHM kurdi_m_Page_050.tif
55c1830f769355b2151639dca29bd039
f3cf1f31384d5be418e0584076f692dde0b028ac
F20101123_AABQGY kurdi_m_Page_033.tif
b25f0ff7437e4a06148d0c143dc5e387
b4659fb1fa849f65e61b2d2b16d199d1d5aa0385
5845 F20101123_AABRLD kurdi_m_Page_154thm.jpg
8b4096885c3424d102f2c02ae271f500
50d94e1b55a5e8620210d34371610a664caa9f99
5872 F20101123_AABRKP kurdi_m_Page_146thm.jpg
d412c403d99cdfb27cd112477f380027
91a9d650b740f29251c7d5c31e8ac7e51e1cc213
F20101123_AABQIB kurdi_m_Page_067.tif
1ba018fe380e6c3f14bbc205af8a4e02
b78d58214da7571145a4efcd5b007c635afdf6a3
F20101123_AABQHN kurdi_m_Page_051.tif
7f2fd33475d80b9c939cbf32903f4b24
8f091732b5c0ed55fcc78853c0be74abf95466b4
F20101123_AABQGZ kurdi_m_Page_034.tif
efa3dc0923dbd455a6dacd9c6958e04b
11bd1e2f2db8115bb6cd18a5f0950682bb8e7e24
14919 F20101123_AABRLE kurdi_m_Page_155.QC.jpg
d180e3e717e156cba28ea450f55490b7
b7949c020a0d3db34ce81874ddbe0e6f5e960677
26847 F20101123_AABRKQ kurdi_m_Page_147.QC.jpg
72d8e3cedddbd588efa35ba18263c6ca
f1ee8d5b0273b6d27d7608541477f01b8a92be35
F20101123_AABQIC kurdi_m_Page_068.tif
2cea6fcbeaef7b2c499a11fb62685c03
F20101123_AABQHO kurdi_m_Page_052.tif
4b8a6fcd2548d919660e959a47ed9071
620369e96a625bccc0e87a6f0f7f60511770f498
4546 F20101123_AABRLF kurdi_m_Page_155thm.jpg
37deba72ff744e61716842b4a5ea2de4
c16e31a457f75a04fedfbcd4a688e716a8090d96
6844 F20101123_AABRKR kurdi_m_Page_147thm.jpg
0497fcc8cf92a389cd17a8215cb1b686
d0345cf5cf991e41ee91a8442a3710a3ca787ab5
F20101123_AABQID kurdi_m_Page_070.tif
9f5c6ddf53460be4ece8ee15b222214c
bd5ce287de24d566faa1cb44957782ed1f3f215d
F20101123_AABQHP kurdi_m_Page_053.tif
9afcf83548948520be0a0637858d1cc9
266714b04e00956d351a4ec534b94d1bf2432fbf
15049 F20101123_AABRLG kurdi_m_Page_156.QC.jpg
9f6b8650b1473473125746e54203d22a
7528 F20101123_AABRKS kurdi_m_Page_148thm.jpg
6644b79c41bbac42830094d64efc0a7a
5535ccd5eba5ea78ab6d6bf0bc4cda0f66fbc725
F20101123_AABQIE kurdi_m_Page_072.tif
8f82b1c9e1ce94bb137b066d6dc6378c
F20101123_AABQHQ kurdi_m_Page_054.tif
2e861c001bf714ab9d6a10d83a7f5585
c02449598e3081c8478c81d6968528725497799a
4206 F20101123_AABRLH kurdi_m_Page_156thm.jpg
8bc544082fb315fde56f284b5bc681fa
22ddb418dd4127cbe7eb8d626244f8bd46d69d9d
27714 F20101123_AABRKT kurdi_m_Page_149.QC.jpg
bd09c7d77569651656dfb199ef72b2cb
80900a623b415a0ebd269fc00dbaa887af47e195
F20101123_AABQIF kurdi_m_Page_074.tif
1d848f99b6a558e0b14494537b0e7469
82b9967e28d4dae753dee51e58452c5b66a49e8b
F20101123_AABQHR kurdi_m_Page_057.tif
bc03f17bb19b886cabaea88467dbb43973a6da59
21589 F20101123_AABRLI kurdi_m_Page_157.QC.jpg
4c710e409a73d8a3394894294d765f3c
2cfc545af4fee4fb385643dacf5e4097cef3c261
7102 F20101123_AABRKU kurdi_m_Page_149thm.jpg
583b01932f0ae6e16723f6ed14b0f20496b80b45
F20101123_AABQIG kurdi_m_Page_076.tif
45fdd5858bd03d636786f8ed46766062
90dd7c1bdef846e80989974721ceb67c80edd1e8
F20101123_AABQHS kurdi_m_Page_058.tif
58e75c5b5b06045d38d4b0fdbd741e1e
2dc11f97ec667c704891de898e920ab7ef78f47d
5838 F20101123_AABRLJ kurdi_m_Page_157thm.jpg
aeb008824b8cec85dd0f292fabe64020
afd9d0db8c5e63d93f1b9eeb695554d987122751
31055 F20101123_AABRKV kurdi_m_Page_150.QC.jpg
44fe683c1fdd9a573bb157c4a8efda92
1db79a8f861c610a01438a605bc27a9cf9a42e15
F20101123_AABQIH kurdi_m_Page_077.tif
96bae945ccfc0b457db5b4b17440f855
3f34eab38fa09cd67492fffa491b4fdf284d1a92
F20101123_AABQHT kurdi_m_Page_059.tif
70601849e3c7d1364139943d22102365
0f899267a137bb8d322c68b1d1f69d582b88ee90
20389 F20101123_AABRLK kurdi_m_Page_158.QC.jpg
0d9b1c3cc3fd8cb097a5497424e938d9
7b1bdd0946b4120f30d4cf7c1d0f501da5347b46
7748 F20101123_AABRKW kurdi_m_Page_150thm.jpg
9d163fe85efe8e194b6d55e9d9f398e6
f08ee2c0f9346d0653e0c747681f7e62b3b13a75
F20101123_AABQII kurdi_m_Page_079.tif
F20101123_AABQHU kurdi_m_Page_060.tif
bcf0b41345d40475f579efa53238ac32
fc752117f0c00f958046fce2858a088b3b0e5a5b
28663 F20101123_AABRMA kurdi_m_Page_168.QC.jpg
43f5a04d556dbf024d73307d23d9e2bf
667f41b0dda1080451194779e7e2f61289dd6a29
5499 F20101123_AABRLL kurdi_m_Page_158thm.jpg
87bab99e77e2e8855cf1eda7c663678e
844a44e23a2f79f1dbf2182a1bc3fbccc5349bba
29859 F20101123_AABRKX kurdi_m_Page_151.QC.jpg
abf3db8ac36a1e9532845e2e8985e769
ae12bd015957af5d06c67c99eda2626274529da0
F20101123_AABQIJ kurdi_m_Page_080.tif
11d32b1a33eb7ff359809f3a7c9210a3
b8998df572fec3055781319002a9d04a31ccdf13
F20101123_AABQHV kurdi_m_Page_061.tif
06ea3119ee86ff268c59ee4fb0222f2b
3192a414eb0e63e7f19442518c70ba6d21272d8e
15061 F20101123_AABRLM kurdi_m_Page_159.QC.jpg
cea5969929553da80130d2b1246c0c18
2974e8480604aea77e4defabdb228bac339ffe6d
7548 F20101123_AABRKY kurdi_m_Page_151thm.jpg
956169db06145f16f9c8a9a3ab83c1d4
ba1664375a136677d3d4642bb88b1b82279dd9e1
F20101123_AABQIK kurdi_m_Page_082.tif
6850f8a3d43183e817783983ecefa0ed
10c37efa4e178be7b898ba03226c9aeb2a3c3d1c
25265604 F20101123_AABQHW kurdi_m_Page_062.tif
d85a2d9d2bf428e197ec03f6ac20af98
70108da4c52fc397d17bf5ae9c604cdca01a79fc
7259 F20101123_AABRMB kurdi_m_Page_168thm.jpg
e6e9ca287af1b686c64ee88dc0357d40
4539 F20101123_AABRLN kurdi_m_Page_159thm.jpg
909bf22169ae8177142e2cfa03a42209
29553 F20101123_AABRKZ kurdi_m_Page_152.QC.jpg
45c454ee8e2a759d13b889568167b887
53f8b69cfdea935be794ba7f75425029e24cb8e3
F20101123_AABQIL kurdi_m_Page_083.tif
365fcd65f575e932c47d1a50d4960736
eee297ffd74f5c31d6edf2005ed610ffcceeee87
F20101123_AABQHX kurdi_m_Page_063.tif
57d22869e7eaf9e95b33837bcaaafd29
d50e5a24debe8157ebfbbb789d246c97b57da299
30469 F20101123_AABRMC kurdi_m_Page_169.QC.jpg
3a70bc6d498e5e2099c3cd1cbd704f14
c800ee8afd52b0db3b6611c1256a24f0ea84e753
4691 F20101123_AABRLO kurdi_m_Page_160thm.jpg
2700cee71a29a717b11d9d947422a516
F20101123_AABQIM kurdi_m_Page_084.tif
cf925b9113a184d907d7300b93e67585
780fb5ee2621533880a3d5dfbdf3af466934df52
F20101123_AABQHY kurdi_m_Page_064.tif
881b2d99c4a12050926c5bb4dfeace39
98fbcf80f7756201a57b5749355275a809385b35
F20101123_AABQJA kurdi_m_Page_098.tif
1572bb1b0983433dfeb14fb12de94da1
53ddb12826cc63fa670d4a448e073b04d4a2b31f
7666 F20101123_AABRMD kurdi_m_Page_169thm.jpg
be3f94f4a8111e8bb482f871500ecdda
0d00c01985649838ba9404ffa615e8eaa4ff4ec9
20394 F20101123_AABRLP kurdi_m_Page_161.QC.jpg
7c881c2f6ca9e7a2824ae23bb446ab0f
7d2aabc1780907bf797a9a627d40ab2b34d54ac0
F20101123_AABQIN kurdi_m_Page_085.tif
F20101123_AABQHZ kurdi_m_Page_065.tif
43b68843bf791d9bac8aa4fc85b961a1
85f49a5867b8441555e9b79e8f7d2b0bd370a85c
F20101123_AABQJB kurdi_m_Page_099.tif
295912ca563ed15e23030c874e8f7a462836c762
29011 F20101123_AABRME kurdi_m_Page_170.QC.jpg
318f8f1ca5c7c20b7b737892f2db76b6
9847fc83529e9b5eca67f0edb68285a72c94a339
5764 F20101123_AABRLQ kurdi_m_Page_161thm.jpg
841f4ddd1ae98c6d07541e39f27807af45dbce88
F20101123_AABQIO kurdi_m_Page_086.tif
963b7e9bea90fa27c069834b7cef5da0
8ca48d8799287c5045152e91aa97231ae3cbf5b9
F20101123_AABQJC kurdi_m_Page_100.tif
c2ecc97c6d2c046a545dd14260b08b57
fb6416e4ebe2465d7a11b5c895335c6523838f89
30203 F20101123_AABRMF kurdi_m_Page_171.QC.jpg
563e8b518902892428109ce4d9dd4182
44a53523625fc7c67c5e3db21ab8f310da9d72ba
18492 F20101123_AABRLR kurdi_m_Page_162.QC.jpg
833ca9e346a8c39107cf36cc29143d4a
F20101123_AABQIP kurdi_m_Page_087.tif
198289cebb5a3e2ca3e567e8dfb4dd3cf329644f
F20101123_AABQJD kurdi_m_Page_101.tif
cc15a56e9330dfefb9cdc7cc9c63e0ce
7562 F20101123_AABRMG kurdi_m_Page_171thm.jpg
03611a8204e0ce149695de15616bb299
80165de9c32bf805e010acbc2984c80e98b2bbfd
5231 F20101123_AABRLS kurdi_m_Page_162thm.jpg
a5767a402fa556f731b925db3ae29822
bcaaebc2553739c6004d8a0f4cb305874ac87fcb
F20101123_AABQIQ kurdi_m_Page_088.tif
c0c0a7676c5547b233269fc7b927ca2d
F20101123_AABQJE kurdi_m_Page_102.tif
4ab2e91a7bfda8a3bcfdec14085a7bf980846600
25668 F20101123_AABRMH kurdi_m_Page_172.QC.jpg
96264859e402a8f9fc1412b9c359aaf6
1ba55077c9da9c6f3d6772b8fb142c59e21e1f7f
17473 F20101123_AABRLT kurdi_m_Page_163.QC.jpg
F20101123_AABQIR kurdi_m_Page_089.tif
495afe8375bd45822838a60b751170e5
3c56f2643d6fba8418acbda3285cf3824ef05233
F20101123_AABQJF kurdi_m_Page_103.tif
62d012de6863364b4636601af2b869d2
9947519931e207edefbc15456f98ffafaefd5816
16447 F20101123_AABRMI kurdi_m_Page_173.QC.jpg
b2aa1b5800e490e1b05c054075d16209
62ae6b8c57dca281313312b39df65aa14883c177
4991 F20101123_AABRLU kurdi_m_Page_163thm.jpg
4d1e63208f67fffbaffb4596a17b304e
02013b6a212614b3af7a376e5936bd9906c8e777
F20101123_AABQIS kurdi_m_Page_090.tif
506b6b4767a5609457118c975f37d6a9e002db43
F20101123_AABQJG kurdi_m_Page_104.tif
bb63893606e02705a107a4fbea360ec0
7483d5f0748bdf9fdd0dddf4c95fbc330359ea93
15677 F20101123_AABRMJ kurdi_m_Page_174.QC.jpg
61b328b885c53542b66c2913170fd893
8a3695916400542dea42bc137d71d17c9561004d
20626 F20101123_AABRLV kurdi_m_Page_164.QC.jpg
e88ebe085da74846b4698f84e2cc766f
2ace67ab20ff01b5190b0f4ea897e77c47246ae6
F20101123_AABQIT kurdi_m_Page_091.tif
e0d70167e1f7058e267b9a3647cc75b4
45df7a91c5b936478064fda9475f00d815ca4ffa
F20101123_AABQJH kurdi_m_Page_105.tif
fab2b1314fde0a82bba0a876dac4dd21
0991bc1f08a174742fd7e32d0de131f679a50ea9
4596 F20101123_AABRMK kurdi_m_Page_174thm.jpg
3442bcc496d03ee03f86aa8d6c8c5d8b
509bcae78d01696f396319ec19b512d35d112aa6
6255 F20101123_AABRLW kurdi_m_Page_165thm.jpg
cc6e5e626296d28e17184640eb2dcc34
caa8a9e0dd27a4764782c9a228a6e3d1e3d38e5b
F20101123_AABQIU kurdi_m_Page_092.tif
5d195a527d580da16933fca3ed766f7b
9e8b608a149da3a586e7cb8acb583f62e56f4be1
F20101123_AABQJI kurdi_m_Page_106.tif
4f97cd640095e1930006fb3ac707121b
4613 F20101123_AABRNA kurdi_m_Page_183thm.jpg
d03952e813ddf6bbdcc5731a0a969e7a
e9562f068ff97d0c91646acf50d63456ea966696
17352 F20101123_AABRML kurdi_m_Page_175.QC.jpg
fe42a4a2545dec5da846b203d11c674a
327a8896b34dd30501c402f70a3f7044e4a8567e
28904 F20101123_AABRLX kurdi_m_Page_166.QC.jpg
538d0d719b5155c52b7c39e394c06a86
8f6ecb88d7a2c921cd288468189c5079c0a53721
F20101123_AABQIV kurdi_m_Page_093.tif
71f5229e35a3dfe921bcc1e5eba7f8b2
4c10d9489bbb450a3dde616ffc1d136290b18727
F20101123_AABQJJ kurdi_m_Page_107.tif
5cb755585d7b5b261506b81eb352989e
cd0c0e3a7db9909b3fd3a0fbf48ee671f02d1d4e
22382 F20101123_AABRNB kurdi_m_Page_184.QC.jpg
17803bcee33e8e4fde4530afab240280
6ebd8a65945dce8fb335b9018f3fed13edd2274f
4590 F20101123_AABRMM kurdi_m_Page_175thm.jpg
40fb71aeac4a3117e0a0383d71176aa7
2321575f2ab311b43d247f2cf4a9c322cc179f23
28092 F20101123_AABRLY kurdi_m_Page_167.QC.jpg
236f32d57cd5ab53a03f7ddb8055f8f0
F20101123_AABQIW kurdi_m_Page_094.tif
e89e5795e2972bb3be3d33fb0e980792
72bcfe18c316feab8cd44eff9d1ea6d70e50ab7d
F20101123_AABQJK kurdi_m_Page_108.tif
98f993b0cd440cb39e27600e5e959632
80e1f1de1a4b36ce90f4da7d76428eba4e789c65
21976 F20101123_AABRMN kurdi_m_Page_176.QC.jpg
247b15e30742a3d597a3491d57fb5f45
81b4a8437988fe49b0ea0c2e00c6f949a13f88d8
7104 F20101123_AABRLZ kurdi_m_Page_167thm.jpg
70882081fb21f5ee6be05e64204db886
426b8c78902cefe7891eb5bcc264bc066ed9a821
F20101123_AABQIX kurdi_m_Page_095.tif
ab972a1981bf9c6e31b8a9b43628d1cc
d0248cda962330f9e3fc876c35479657f65daa8b
F20101123_AABQJL kurdi_m_Page_109.tif
f2d8bfd878425a66704daec4a8c46a3a
0f7741cdfc0fb38d34db99d96a40f2311b320346
6010 F20101123_AABRNC kurdi_m_Page_184thm.jpg
9fc3a32d929eb50a1b5c0c55e437bb11e3bb977b
5626 F20101123_AABRMO kurdi_m_Page_176thm.jpg
e5f9ab25c493f9819b0c8815e30d3e2c
d360510661ced38180c8ef94ce4ddc0ce52059cd
F20101123_AABQIY kurdi_m_Page_096.tif
495d7797ca3fd8473603726522688df1
148226b1cde6ab9449ee227cf6d33e6e418d5421
F20101123_AABQKA kurdi_m_Page_127.tif
3fc7f6e816a0e33d9781d0972c5531eb
6ac09c715f47e8ef107dbe667f402e46fe832440
F20101123_AABQJM kurdi_m_Page_110.tif
cecec44545044e3870a753aa2ff7265c
25743 F20101123_AABRND kurdi_m_Page_185.QC.jpg
75ea675d262710fc709b007df1a63b30
20900 F20101123_AABRMP kurdi_m_Page_177.QC.jpg
ba1311ff72593ebf292a2c88bbdc7112
99950e7715e466eafa9989e653c36ba79203e824
F20101123_AABQIZ kurdi_m_Page_097.tif
F20101123_AABQKB kurdi_m_Page_128.tif
e1d3221fc6ea4d13b1873876127e454f
53b50916bc486e73f6296b514bf1e416aa2a5945
F20101123_AABQJN kurdi_m_Page_111.tif
b95734bd99523d6f3859a443a6854653
f183ce2eb6bc8b192e89e8b3c351cc5f7f1d63e3
6775 F20101123_AABRNE kurdi_m_Page_185thm.jpg
649955f77ec7808ff984fa41becf42ff
741cd3100a988c7a5e2039710ef7afbe8305b1f3
5840 F20101123_AABRMQ kurdi_m_Page_177thm.jpg
d8281acb250bd7a4f677ed737cae13a5
efdcdd475574269669a55b8311c29df46c61e864
F20101123_AABQKC kurdi_m_Page_129.tif
0c37733322e0063114af1c37500e65b3
F20101123_AABQJO kurdi_m_Page_112.tif
fa84c13bfa0b9618d0d03aa7258110d0
c308b2dd20f66ff846be2de1737d6cbc8251869d
29663 F20101123_AABRNF kurdi_m_Page_186.QC.jpg
29c75734cfd88745edf9689d3e9e6803
a72daacf9e90e7d957fe05aae627d898121114aa
20271 F20101123_AABRMR kurdi_m_Page_178.QC.jpg
26fafb5da50d1aa46cc015a33c89f248
F20101123_AABQKD kurdi_m_Page_130.tif
9e79f76e9eab314e2e6ec7c4edefef46
5c239b1e0f2f80f39c2ec1aec0b4b0956cae939f
F20101123_AABQJP kurdi_m_Page_113.tif
765fc430d3d531dc79c8e7c45eee219f
7589 F20101123_AABRNG kurdi_m_Page_186thm.jpg
d068862b05acb6e10a10ded204a3c51b
8ea015c9446aa391523564d9b212b288ecbd7078
5669 F20101123_AABRMS kurdi_m_Page_178thm.jpg
00c4d4a33f4088463295a18613713645
F20101123_AABQKE kurdi_m_Page_131.tif
716361dd4bc4a080c3e98c98355b806c
a0139b606b4d56fca337ff51bdefa46ced89e192
F20101123_AABQJQ kurdi_m_Page_114.tif
a6f53fe331707e6d152ddd099b8392b8
28100 F20101123_AABRNH kurdi_m_Page_187.QC.jpg
3a84f0b99abe2b2d98a8c705c0dec646
ed762c1b25907d3b0b9aee3a873f126c59ec34fc
17687 F20101123_AABRMT kurdi_m_Page_179.QC.jpg
6ee39569e3b1e68a2ecd01ae00ed9153
ee022b34d646812275a4af08a5c0aec4fa87c591
F20101123_AABQKF kurdi_m_Page_132.tif
22cc0e0f08a071f945e777ba7d1e47b5
dc649bea640d8546f86bd6ae2dfde015f59fe173
F20101123_AABQJR kurdi_m_Page_115.tif
b57fee89317213485770fbd4bb9ae158
eea621faaf9da1c321c8984eb5094db65aaf278e
7184 F20101123_AABRNI kurdi_m_Page_187thm.jpg
7a83f5f82f6c6a74e3038ed609efdf07
4879 F20101123_AABRMU kurdi_m_Page_179thm.jpg
07bf0b6be457c23ef38f73e554039b56
5426fe5cabe1cdb1604e3396cefdb53d262b8b68
F20101123_AABQKG kurdi_m_Page_133.tif
2675c609c6895d611a788f7fd5a80d2e27a1ffd5
F20101123_AABQJS kurdi_m_Page_116.tif
f9de2d62dc2403f16a8d5c6f5b8054530940bd23
30151 F20101123_AABRNJ kurdi_m_Page_188.QC.jpg
063de3ea26f3d2fa6f96ed90059dab73
297eb098437e1d5dfb269bf39a4ee20d19f93c51
4845 F20101123_AABRMV kurdi_m_Page_180thm.jpg
c2066fd9b515322b5db875df80650b99
f7c4b970eb7694244926b63768b72e2dfa4c58df
F20101123_AABQKH kurdi_m_Page_135.tif
6ebf198e7c0e4bdbc41a3cf9d0d30be3
496d7ffee02316803296c442311b8f1b9f61c2c1
F20101123_AABQJT kurdi_m_Page_117.tif
63d3241493b9d41364f25053bd65c32b
f966505945500903e1b3388b8f46e71a468c1e85
7649 F20101123_AABRNK kurdi_m_Page_188thm.jpg
3bc80dd6073c869fe164397e904faaf7
97d042585cefa05e420afbd6a0e4976cbb377693
18446 F20101123_AABRMW kurdi_m_Page_181.QC.jpg
afe6dcaa4c960ec07695d254c6c0432a
49dd257a993895e8afdcff4d10184ed44abcc36d
F20101123_AABQKI kurdi_m_Page_136.tif
b8346b609bec2943288136230f308398
5b03a2db4e6024f2ff6d69fa4f24de2efe470416
F20101123_AABQJU kurdi_m_Page_118.tif
bce4491185f62e783d0395edfcc37325
e2c74169476f5654710970ded4ce2ae9f6fb4abb
22112 F20101123_AABROA kurdi_m_Page_197.QC.jpg
77ea441815ee158206774499217ffe5d
93cc008df7bf68c719bb14f62cc7eb2c2032c1ea
30238 F20101123_AABRNL kurdi_m_Page_189.QC.jpg
0dda51a8b6e3c74208b846bfd5766762
a455fbbf23148d810f698763997d63594ec7e02f
5208 F20101123_AABRMX kurdi_m_Page_181thm.jpg
1cef83ce4ecbb07e471ce68d8ed7c4e8
2d3dedcafb4e15b47aa78ac432e4987571a87503
F20101123_AABQKJ kurdi_m_Page_137.tif
480b56618cbebb84f8390acbeb4f1a4e
481c9d523470b025a7231cd4c0d7eabf3fb35522
F20101123_AABQJV kurdi_m_Page_119.tif
b1b9e467437d7c660039be38e61bffd3aa8df29e
6356 F20101123_AABROB kurdi_m_Page_197thm.jpg
79d6f39669034a9797dc650fa38564c2
a03a45dbcfa2116e43fd498c459a47080a7d9a08
7512 F20101123_AABRNM kurdi_m_Page_189thm.jpg
546ae664f1946c063734dbe9bf0ac2f9
c09448097b5080b78e603bcc7d57cedd552be2b2
17763 F20101123_AABRMY kurdi_m_Page_182.QC.jpg
dbf5e9cd4604b15ffb18343237838c89
d71c64fca74ca4ed6e56c702aa3ca88ccea2ab6f
F20101123_AABQKK kurdi_m_Page_138.tif
0da61a2f9ec5ae9ba51ddcbb7da29a284db7b5f9
F20101123_AABQJW kurdi_m_Page_121.tif
cdfa4834e08b68f4ff7506da357d0998
23104 F20101123_AABROC kurdi_m_Page_198.QC.jpg
9cda23a6ff0b03eef900c9e9b9946360
29477 F20101123_AABRNN kurdi_m_Page_190.QC.jpg
4e883540e6839bd1f5e1382385f0d5c5fa555d7f
4614 F20101123_AABRMZ kurdi_m_Page_182thm.jpg
8ea7579030ab71b66bc94e2df8b22636
b6d84473aa456432cd522a932b594c9bee3cee44
F20101123_AABQLA kurdi_m_Page_159.tif
ee811fd9e12f8cbeba46889a01d48b1b
2cb500959b8469ece29db98735470cbde245c82c
F20101123_AABQKL kurdi_m_Page_139.tif
a7a65abcace56d54b0fcddac9e8f8066e9caf85a
F20101123_AABQJX kurdi_m_Page_124.tif
96877140ae66995fa439de0921b4f878
f2f192d4ba3e30178ffc5fbfcbedbff591678f63
7464 F20101123_AABRNO kurdi_m_Page_190thm.jpg
2d29140e469302d997af702a86113a1d
9633b75438c8c59099d05897d63ca662d398bbe2
F20101123_AABQKM kurdi_m_Page_140.tif
09d52a5f8894d887766d364136455e30
8392e5211b59f4f2ac0e71b21ac7933e9a1a58b4
F20101123_AABQJY kurdi_m_Page_125.tif
c5808888e158378f98afb8c55d43a5a1
6415 F20101123_AABROD kurdi_m_Page_199thm.jpg
49ce4950e4d340c271eb73fb08d814fc
30624 F20101123_AABRNP kurdi_m_Page_191.QC.jpg
83c9e8f1a9e5657efd4775f00b8b475a
2d76235c14c21fc5115a82771a49fdbb6e0061da
F20101123_AABQLB kurdi_m_Page_160.tif
89acee861dc93a5a4ccd3f29565ef464
7268f1cf8494fa62fb926bc6083b86fcf0bf235b
F20101123_AABQKN kurdi_m_Page_143.tif
d2ac24a3df232df3fb5c270a6db2c711
F20101123_AABQJZ kurdi_m_Page_126.tif
d98d563100e1cc74c71e9beb61c25a5f688f0356
23856 F20101123_AABROE kurdi_m_Page_200.QC.jpg
fde7e079478139a025c53e0dc453f61e
27ddc23e37922cf4930277359b77da5a46bcac95
7716 F20101123_AABRNQ kurdi_m_Page_191thm.jpg
27058dd6263696b16766bf81406e62ce
8e558ec164d149feb929ecb4a36cac712e396369
F20101123_AABQKO kurdi_m_Page_144.tif
bae53de74dd5cc99cd6f6b054a5e2ea5
08a9bac007b125c4bcc76bf61a41f747b66f51fa
F20101123_AABQLC kurdi_m_Page_161.tif
2c65233a1c56dc535b872c477aa59a85
70644daa56a89c1089d1131d9a2d2658f3a14b9c
19826 F20101123_AABROF kurdi_m_Page_201.QC.jpg
dd183d9072b550f6d54941303e43dc07dbdce391
23190 F20101123_AABRNR kurdi_m_Page_192.QC.jpg
8a60635d3aa283062cf9466c0a1cf522
351663866f4bf07cb43ca568957133ab904d2782
F20101123_AABQKP kurdi_m_Page_145.tif
81611808ae650225b8ac94f1716c3c5a
3c3a313e7a965ed10bac61042e88cd1d8a01782b
F20101123_AABQLD kurdi_m_Page_162.tif
2a55582f0a5a4a4b89c4b779a0da8a7d
7619ff4feea15e13c27bbee3ab1802f0da2a6ce6
5701 F20101123_AABROG kurdi_m_Page_201thm.jpg
dec4c379566742d2b1b26a269f829acf
df0f2ef5bcf36774db1bbc2f022f8a6234d248e8
6164 F20101123_AABRNS kurdi_m_Page_192thm.jpg
33b06d64e4cc4222192c35e2a716d23d332a61fc
F20101123_AABQKQ kurdi_m_Page_146.tif
4e7c6d11881c4b3dd920c994807dfb86
7fcf275f04ca89296ff7bb86191be1dd91db9fc7
F20101123_AABQLE kurdi_m_Page_163.tif
e9dde331bd40ab87312abf04183e1f16
0d96bf20cc678aa32db67df1dd731373173c46d9
12110 F20101123_AABROH kurdi_m_Page_202.QC.jpg
2ed740fd1a211ab90be986cfd6220ea9
6760 F20101123_AABRNT kurdi_m_Page_193.QC.jpg
6a77d9bf42da771db93beaba4024e493
112df73927cdea9e53355aaaa7289a0494f39eb9
F20101123_AABQKR kurdi_m_Page_147.tif
30a6878f3699372697c383816113f85c
5caf30b57c19263fbe55513cc25b1ef724914f65
F20101123_AABQLF kurdi_m_Page_165.tif
e3d842c69487b20c8d4b57be438dd909
fd9b1b61a3c2a6db8621ed15fd8e28c705af8b57
3599 F20101123_AABROI kurdi_m_Page_202thm.jpg
ed3e90a5ef68e7e75025f30b781cb0ab
1ae0c3b4fabcdd2856a08def56929f80a00ec9a5
2181 F20101123_AABRNU kurdi_m_Page_193thm.jpg
425469e5c7383eb97d7f8b027331e3f5a5c9db37
F20101123_AABQKS kurdi_m_Page_148.tif
3333fa162cb3d867fe62e802b1a33d6a
040aa72482ea7152cfd7d2d0dc912a3cae39fef8
F20101123_AABQLG kurdi_m_Page_166.tif
ec23a73dd3c88559e19bb9078eb5945c
1ce6c59b0e7ff1f4b68a4a70e5ab67d811daaab2
230969 F20101123_AABROJ UFE0011626_00001.mets FULL
4482cf503d26dd9d1c85035d3b5f6ea7
53229150a8611f5d1b53e02cc85c849a3fe6f483
20568 F20101123_AABRNV kurdi_m_Page_194.QC.jpg
0b1acab0aaf6e0149a150d78cacceb4b
365e5a26e4ec66bfce34133786f3caf8e44d5f42
F20101123_AABQKT kurdi_m_Page_150.tif
c55272c7b00715fffc4c833bd1dbd13e
3676e601291325533122819de1a1308a8d910d0a
F20101123_AABQLH kurdi_m_Page_167.tif
b778d1a12d25d8826d83e3fd12779284d84b995d
5797 F20101123_AABRNW kurdi_m_Page_194thm.jpg
e00525ea3e6e5be21fbb4fe428b6c6d0
7cf35fccfb1c50d74f7df6e1d2b0f0f80352e44f
F20101123_AABQKU kurdi_m_Page_151.tif
db2cbe048fc5fe8c05d50253da166366
bbb5c03262a7a84013f770532a7fff4ea9d602ef
F20101123_AABQLI kurdi_m_Page_168.tif
44d5f432fbc125b189d20e3ef364c8ab
295c609e6aa499d8afa17fda8e0f914e3423d6cb
23122 F20101123_AABRNX kurdi_m_Page_195.QC.jpg
1b1f782385bc1430de5832642d9977b2
3d464b867680999efffc71e3b114068be49c2924
F20101123_AABQKV kurdi_m_Page_152.tif
be97ab47e2db2fc89356488b3dc85b57
9c547be1c744864fb85f3ee92bb2cbdfe61038ce
F20101123_AABQLJ kurdi_m_Page_169.tif
2afc5649fe3d1eed828a9aba790aca22
61bac65fd1cb3f502722cb6d3e11e1df23b04456
6662 F20101123_AABRNY kurdi_m_Page_195thm.jpg
e0ecf87358449d3ecb8d28d989ca8377
8f11dce6e293fcb147f2a29c4540f164a7bcec6c
F20101123_AABQKW kurdi_m_Page_154.tif
727d1dbed2afef6e371ac64eb948399c
848a92e7cbdb535bb3152da07431c4d92ef58bdd
F20101123_AABQLK kurdi_m_Page_170.tif
66e6fab4f76db059e779a095ed9e302b
6764 F20101123_AABRNZ kurdi_m_Page_196thm.jpg
56096135869b7cd12062f273fd72883b
F20101123_AABQKX kurdi_m_Page_155.tif
33afa169cfaff69460473e2f9a9b1d5a
86e7e5ce1a60517e50b568a4c87ac1f826054765
F20101123_AABQMA kurdi_m_Page_187.tif
a1ee26fea315ba2acd3109073dc07db2
65ff0925fbcc5d0b378be571cbb4d9bb63c2c923
F20101123_AABQLL kurdi_m_Page_172.tif
346e9878f66b8723a8aa72eaec937b9c
25b43d1346209af2e33f06e80d527af0caccc832
F20101123_AABQKY kurdi_m_Page_157.tif
bba75ab9a4b3e8cece6161501a25de98
b55c68a219b0dc7f9e5d2dd6ea90e241534026d8
F20101123_AABQMB kurdi_m_Page_188.tif
0e35b0c30fb971e4704a6c92c077d288
4fdffa605acd2c43dae604d334304edc953e8132
F20101123_AABQLM kurdi_m_Page_173.tif
71296f1f88d7f5a4e58a08f153945908
0b40dfcefdf0b91f4bb1d5f473aa8928b9355dc5
F20101123_AABQKZ kurdi_m_Page_158.tif
919cba7cdd1b870be217d4cd3fc8809f
f3f143ea2d71f43f74c90218e94ef110665b3577
F20101123_AABQLN kurdi_m_Page_174.tif
21c5455d668e3ebde096cd8a9ff51249
9c4615a93c68367b5f310b48753879657a240e69
F20101123_AABQMC kurdi_m_Page_189.tif
27f22f2cdbb9f435366c6c5b897e3db5
F20101123_AABQLO kurdi_m_Page_175.tif
43e8f3b9baa9e350695dc40ce46ee06b
41e8e855a3f27b947d5e897953009df5f326f3c3
F20101123_AABQMD kurdi_m_Page_190.tif
a04d535e3dd925cbbcc4a967c829e528
722c5d29a6ab596c658d61b3eaaa63527932fb7b
F20101123_AABQLP kurdi_m_Page_176.tif
7ffa1801f3b5b38015e9bc45f84f4989
F20101123_AABQME kurdi_m_Page_191.tif
b41a88ab205f9ba8582989e1db432952
9ee54a5ae820b22efc89101358a86352d542fc5c
F20101123_AABQLQ kurdi_m_Page_177.tif
ef4d666b231f1d1deb04ed216cb1d0d0
40b7586688d57ea07f0b7a288e99cedf92e6807f
F20101123_AABQMF kurdi_m_Page_192.tif
ddb9650d9a24630d16518654e167947e
5f2b59a6cbd74bb89ee2bfdee9ab6a6edf46e27c
F20101123_AABQLR kurdi_m_Page_178.tif
a4ee34115e5c8d5802e151b0ff8d2923
d6e842eb78ca7c21a7b4be7a413b151bd51bb018
F20101123_AABQMG kurdi_m_Page_193.tif
5beee4740820d4a276fd40faa47b20be
ce1f7cecf4a66dfcbdcd36aaa1942c7db44b1284
F20101123_AABQLS kurdi_m_Page_179.tif
7b3f59beba98ded6bfe7c36f131c9ff7
8f58870c9dd84de210d983d0fa15005fb19bb312
F20101123_AABQMH kurdi_m_Page_195.tif
2cfa72a2dc2330af0810bb41da0985ce
a32239785d3008406a6713e44c3fa64bea63d73f
F20101123_AABQLT kurdi_m_Page_180.tif
e2871e72e42cab5d9ecc73172e398fb3449a6705
F20101123_AABQMI kurdi_m_Page_196.tif
2bbd47d99a5e2e7341966418b226974d
81c7b76bebfe0448afe9ae3edfe081cfcac001db
F20101123_AABQLU kurdi_m_Page_181.tif
F20101123_AABQMJ kurdi_m_Page_197.tif
8e747925d54be1885223d6478c9393e8
ff0519fbf3a900eb87810b34285d3d4454aa53ca
F20101123_AABQLV kurdi_m_Page_182.tif
e350dac8e0edc8210cacde0ca76e2462e6d18231
F20101123_AABQMK kurdi_m_Page_199.tif
8e1eb1438d651c597cc9586255778c6f
087752c32a7043b89bfddab100ab835fa4c0c382
F20101123_AABQLW kurdi_m_Page_183.tif
2859a565b1fc85c90a029fcaa4c0e6f2
8638b0832bb018feb0a1092823cc16d972db1e45
37666 F20101123_AABQNA kurdi_m_Page_016.pro
3ea448d5f5c1b2db7298f53bb8a933ac
b0e362ffde39b1da783199026dab1cf4b3774ca4
F20101123_AABQML kurdi_m_Page_200.tif
afecabd930ed9c960a9284803108b6e1
330a929715e0ed12506b347fa8367574fe99c0fb
F20101123_AABQLX kurdi_m_Page_184.tif
541c92490d93f73c25b26cc194e3056d
bd8787eaf458b0fb4f2eea827768af2f18dae037
46230 F20101123_AABQNB kurdi_m_Page_017.pro
7395e8eb226b5840742cb1556f6478ba
37c963f91306e0b82fec9e6b583127fa893359ec
F20101123_AABQMM kurdi_m_Page_202.tif
b7490ccb108de7cce1ce79473df83dd9
3590e93017736a240eaa74fdb91218e300948496
F20101123_AABQLY kurdi_m_Page_185.tif
35387744444936f7ce3d1e024b4a19ca
b4a7414804ce3442a0789d973efd14e0b53a5d9f
41230 F20101123_AABQNC kurdi_m_Page_018.pro
ae42b36356feba2934001898697ef81b
443143a4fee8547493969ebbec147b22b8d9fbd1
9429 F20101123_AABQMN kurdi_m_Page_001.pro
5ea805257507d5745f9fd88f2e3d40b3
ceebdd3013d3857bc0af3663bfb2529a1cfc6947
F20101123_AABQLZ kurdi_m_Page_186.tif
4f38487fb6797147872c09e87e8b28b8
b7aa16e23190015c85f6102c61958cf4c94d4e58
753 F20101123_AABQMO kurdi_m_Page_003.pro
abbb35ea33416768452b363facf894cb
44490 F20101123_AABQND kurdi_m_Page_019.pro
6295d9b5e44242c7b1a755ec4c0414a6
34449c06be81c7de7af0d8ee5a553d2ccb5dd8bf
28420 F20101123_AABQMP kurdi_m_Page_004.pro
cfa60e33eac2aa48e3453eb36e0a76d85b98b19f
48144 F20101123_AABQNE kurdi_m_Page_020.pro
be59fb7557a5e4f061850308b31b9349
bc7fea57deb4e0107eb557a90db0de8ef9d69763
77489 F20101123_AABQMQ kurdi_m_Page_005.pro
f33cbaae8641839a88ebe85956a060a0
e6965286d7c44943a558c6c36cc5e9622c1df0d4
51031 F20101123_AABQNF kurdi_m_Page_021.pro
90ec443e7fd1614411c641e2bcbb6fb7
112424 F20101123_AABQMR kurdi_m_Page_006.pro
613c071f2cb842a96c2c46238a30d426
1c5a0d76c84380367f55b84d94a61ea769c9a043
48233 F20101123_AABQNG kurdi_m_Page_022.pro
ba182453d534dd3c5bc3b6c20ebcdfc7
4d64e051264dd713b09be65d31ae9aac72df6939
75920 F20101123_AABQMS kurdi_m_Page_007.pro
839edf6a6f86cd3d9b5308eea68a014036231085
44589 F20101123_AABQNH kurdi_m_Page_023.pro
3b4009c582bcff0be8853c65f0147af6
602bf299e7b51352c659ec104bf81f76ddf0ea9e
52469 F20101123_AABQMT kurdi_m_Page_008.pro
7753e129c4abb5cf9ea52db4ef2fe01c
027c3ddca43453f22fd3e2fef88c46982ba9280b
15283 F20101123_AABQNI kurdi_m_Page_024.pro
4a5bf1e6d38552342a618fa034911762
8da767627489e8a37fc28e6febd3eaa40379acac
59166 F20101123_AABQMU kurdi_m_Page_009.pro
9fbcea26c1705d1b3db3c1093f304c7d
458b214c095b8e4cc8acaaf32e78d8c87d6e4126
38260 F20101123_AABQNJ kurdi_m_Page_025.pro
1bed003ce5ac1e39f50c640f71e2ae24
9ddac49a83168856f126a6a6725928a72c60f553
75352 F20101123_AABQMV kurdi_m_Page_010.pro
6e90052a1ba7b870ff2fbbd52df199a6
b93fbd8119f89b4c999bcf35dabd422e8d91cf4e
37615 F20101123_AABQNK kurdi_m_Page_027.pro
e431d2fbf1b00f682fff16bc2a79af59ce127d69
67645 F20101123_AABQMW kurdi_m_Page_011.pro
336c1ba30287d47037b7522ba5ca428a
e370939bea5c2b40dbd909dd171deb9535a2fe2d
42612 F20101123_AABQNL kurdi_m_Page_028.pro
dc977a93ebdefa5beeed573659d929e5287baf99
43514 F20101123_AABQMX kurdi_m_Page_012.pro
2e36cbdd5b03117aff98ef4b83a119d5
1279211675e66994b590e7b9afe6bf480efd588c
39253 F20101123_AABQOA kurdi_m_Page_044.pro
d0974bf19e982cf01f1472bfceee1cef
a326b5165ef8b3ffbddfc1a79884a598f7c80d34
51823 F20101123_AABQNM kurdi_m_Page_029.pro
a422d11f230908129c9a8d93dfc9ba96
29000 F20101123_AABQMY kurdi_m_Page_013.pro
5b5f1a33d6b90ba2db3dc257e33e9017
40229 F20101123_AABQOB kurdi_m_Page_046.pro
482aa309b5bb9d16510e19a7de09cb17
f9638da513e313f58fa0455f464becb328024a9f
27538 F20101123_AABQNN kurdi_m_Page_030.pro
a9c71ca3e3006caf7720cc146af67b87
818ba5a3b0e7d2f1fd4ea16ba5d5a68a2335e992
32724 F20101123_AABQMZ kurdi_m_Page_014.pro
f4d6bf8fb96e859949b36ab90cde5bb0
927523766739debeccf657b1b2b8c2e2969d95f4
43615 F20101123_AABQOC kurdi_m_Page_047.pro
efc63d1fce02a3d2de26f505a72f628c
9cf8be70ef6063f62fdba6ac6cf14c0d4032e818
37066 F20101123_AABQNO kurdi_m_Page_031.pro
37357e5043643450aa3ed8e879b3b616
26a936f39a0414e055900163cd80c496fa96066f
15136 F20101123_AABQOD kurdi_m_Page_048.pro
8d13ce984e4f09fa7b5352ac7107a3be
40976 F20101123_AABQNP kurdi_m_Page_032.pro
f0c2c3360c2bab396d099bfc01349f40
ca2433623888b21f7cba97a4fafca86806ca7e42
41079 F20101123_AABQNQ kurdi_m_Page_033.pro
7cd6a66efed7b0a45b110abc6334e1f7
37812 F20101123_AABQOE kurdi_m_Page_049.pro
5da6ce5ca9afd07c932f9df79d543942
06b8d05e1b4d0baf834d107a1f52bfc50a182579
42415 F20101123_AABQNR kurdi_m_Page_034.pro
d7a723b7d93d691306614bbe88931a53
814d3dd8b6d7f377bfaf4dacba8ce46966ae5ecd
39605 F20101123_AABQOF kurdi_m_Page_051.pro
93a25c032c47bbb44273579849a1f038
a664cabfbccc88c42bf42c566464b0b746f46fcb
46195 F20101123_AABQNS kurdi_m_Page_035.pro
eb04b8ec5e7bf5af57b2fb3a635e4d00206e8e44
30326 F20101123_AABQOG kurdi_m_Page_052.pro
b6f8e45c908689539e992e17c7ce8254
8996 F20101123_AABQNT kurdi_m_Page_036.pro
f3652052f5941a8d0ae1ee43205c7c58
34167 F20101123_AABQOH kurdi_m_Page_053.pro
ddcb9b5157c8640f94827b0ba51c5f15
ba0c6e3872e44fd3f84922d1eb45bf9896fb548d
43249 F20101123_AABQNU kurdi_m_Page_038.pro
41d74cc6970108f0d8c44cd847c75984
4542147d0f785beef2ce4544d2864d8dc87b052e
40184 F20101123_AABQOI kurdi_m_Page_054.pro
4600425994e98e3aefa79abd06beb445
c2750f5d726a847d01fe40a05da18e5d4c004429
40544 F20101123_AABQNV kurdi_m_Page_039.pro
07f674e9d1d0ff2e2ee700ba5c2922af
f2733156fd661fda962feb859518129144751c5d
34466 F20101123_AABQOJ kurdi_m_Page_055.pro
370fc7d1b71ebf5dd6945dd8317dfb86
a4fa474038803aff35da56bd6a3f3706b59e70b1
38592 F20101123_AABQNW kurdi_m_Page_040.pro
992339b3c224e89d24b01b5cc4087c56
56d15fae7c9c6c05af56b8d9b86e8b8055a89119
14426 F20101123_AABQOK kurdi_m_Page_056.pro
f4bb2080c8de597b56e4edcc6cdf1f79
7ec835c0d162e5128139fe22019fbab1d0e34e0d
21658 F20101123_AABQNX kurdi_m_Page_041.pro
59c1debdb203deae58ef9256792479f89b3ccc55
34737 F20101123_AABQPA kurdi_m_Page_076.pro
2f3a217eae48e0f5bb8fd5cdb7070bb1
81c5529a6a1c2eda0bd90006b3eda208c60d8644
8914 F20101123_AABQOL kurdi_m_Page_057.pro
ab152906abc2c0ba3cd5412cf7a76324
72043f304ec4b00027777d6b723347db5a6453af
38336 F20101123_AABQNY kurdi_m_Page_042.pro
dd6401ed8951ea44a049d763c940eb7c
c341d2e43b3b5571ec7c8d1476454196a32d2281
19246 F20101123_AABQPB kurdi_m_Page_077.pro
d96c89fa5a34eecba6cd1a3e0b78a75b
0b2013304f5fef197b181ca3688b2b61d12bdde4
50134 F20101123_AABQOM kurdi_m_Page_059.pro
4b79c66ce87e570421a138277f01cb78902e7500
41421 F20101123_AABQNZ kurdi_m_Page_043.pro
e352216f1b705d8809c58a09d7506f53
1225 F20101123_AABPMA kurdi_m_Page_002.pro
9d202a9d90bae73c840ff1b0f7dfb67a
9e792f1f8de769acc428ec2ec9a060e8c633e3eb
30178 F20101123_AABQPC kurdi_m_Page_078.pro
e4dd2d51c661af0fc94bfbca93049540
a06436a03ed5efce50dc5629e1e6294cbc4de395
19192 F20101123_AABQON kurdi_m_Page_060.pro
2bae6cfba498af699c67c3f64131e729
57931eced7abcd9a60f518a990fe9ea578b57cde
1988 F20101123_AABPMB kurdi_m_Page_118.txt
f33d095fd74f28cb39a14efe05316a9b
6769 F20101123_AABQPD kurdi_m_Page_079.pro
66e17c2e0de3c9b804a01da7995070d9
cc02b6c56cfe18402e87ce814fcdf6fd405670bf
10517 F20101123_AABQOO kurdi_m_Page_061.pro
51006f0830aebe32e790424168896bb8
15463 F20101123_AABPMC kurdi_m_Page_183.QC.jpg
4660866062c65e32f17e105f71db25c7
eb052ef50822aecbe31a50973ec6a51e2ab3ab74
43322 F20101123_AABQPE kurdi_m_Page_080.pro
326d2e0c12156c4732bfa6276eb61744
5b13b2510e5468deb3cde6169b83fe1daa5809cd
9454 F20101123_AABQOP kurdi_m_Page_062.pro
a11634a7943206d6abf01a8f3e596e23f7d14b4d
46524 F20101123_AABQOQ kurdi_m_Page_064.pro
95d2cedcf14d9b35460ef45c3b6f5423
181471cd1c8f54c62149ee8ef2edd5d5df549bca
24344 F20101123_AABPMD kurdi_m_Page_165.QC.jpg
9ee5d6c08ed046f058c3d7394d16f400
e4fcb0bc758a51f921bbed33bb796051a833e952
38568 F20101123_AABQPF kurdi_m_Page_081.pro
27abf5d2888a91f346c974f7fba9def4
86bf79050844243d6c6020c2c5f674d9d6182ca3
52633 F20101123_AABQOR kurdi_m_Page_065.pro
fc0ae926f6f23c0cdd6b2bfeea1fc7f0eb46b110
F20101123_AABPME kurdi_m_Page_069.tif
ebb50170f3d1b198d09bb99a6836e8b9
febf8370a2bf744609f01df0dd95215886c6b84a
19283 F20101123_AABQPG kurdi_m_Page_082.pro
a978cca68d859068436e40192e712d7c
bb0698d386734ae8c3112850c6473ebafb3d159e
29516 F20101123_AABQOS kurdi_m_Page_066.pro
f96c7d06dd898c579611ffb86906cfb6
6369feedfc020ac543befe207f9b3e14b0d897b6
4550 F20101123_AABPMF kurdi_m_Page_173thm.jpg
024cb0f4219fe766b3da1f078ffd3c6f5cf1c185
26513 F20101123_AABQPH kurdi_m_Page_083.pro
777f35c4aa29ba0e9df4a601ea361e7d
db743b3f3e1a8e054945ee072353b362f4c917f1
45513 F20101123_AABQOT kurdi_m_Page_067.pro
6b1554d26abe15a9e7a35079135ac54c
5ca4fdcb35090b4dfb67071efe6015c151d02114
16729 F20101123_AABPMG kurdi_m_Page_015.pro
00319d773b5e7b07ff659e867c3f29f7
8a31999c354231aa64942d1ba198b5e8629c0c9c
31271 F20101123_AABQPI kurdi_m_Page_084.pro
528a9ec0761a2119bab3019b775b13a2
61625cbfbe658fd99bd4a342a743643a3e091640
36398 F20101123_AABQOU kurdi_m_Page_068.pro
a527a8d84283fcddd9725ec25a51ea75f79294d7
70346 F20101123_AABPMH kurdi_m_Page_113.jpg
c4b3a8ea6476a5367176386bc72ab0c0
6b3065dcd0dd4d7c17347783e757fe152b2cc1fb
25619 F20101123_AABQPJ kurdi_m_Page_085.pro
63e38e350e2fab82da15bf9052c357ef
36d5b6525d9310dfdf4ea37e467fb0211acb0ca3
43136 F20101123_AABQOV kurdi_m_Page_069.pro
39cd89dfceff8f45c427cc30255cf288
504ae9522f1311deb6b3131562affdd87731a0cb
778867 F20101123_AABPMI kurdi_m_Page_085.jp2
0f407862ce5f351a8426349f5aa993e9
ae341d66fc2935f60a5ce8c37ee2ec30572168e6
35640 F20101123_AABQPK kurdi_m_Page_086.pro
c986050fc41c385127667c7dcdfe64c4
30875 F20101123_AABQOW kurdi_m_Page_070.pro
a5c9161c761bc2fceb4bf3b1f360d485
1d9c56d4e6bd92f314cf5bb024045fd6d366f521
55488 F20101123_AABPMJ kurdi_m_Page_027.jpg
3ef80c1c17a1b854d0c6d2d2bb90823f34e099a2
50755 F20101123_AABQPL kurdi_m_Page_087.pro
e33d5107519a39af43d2d9892e95871f
0fbbba99fe12c6fa29088ed1cc0bb0802abc3ff1
23618 F20101123_AABPLV kurdi_m_Page_199.QC.jpg
527029c0dd2d5affc1ee58fe5e2bde72
7104269193cc4d7e907b46bf62b1380eb00c3021
20866 F20101123_AABQOX kurdi_m_Page_071.pro
4e3dd1b16cffdd60233973a3bffe42ce
b56a2e277fec1729e6abc923794975898ecca144
32836 F20101123_AABQQA kurdi_m_Page_104.pro
85872b26e228e066de55bc1070bb6558
508ddf6418aef0296221740ce60f42238613ef44
37316 F20101123_AABQQB kurdi_m_Page_105.pro
d642a277258f1b1b460d2e5421289628
d161d82431521fedb80957ecff4fce12229c39bf
25802 F20101123_AABPMK kurdi_m_Page_119.jpg
e4fd6cdfcdddeabd6c56baff68ecc09f
60e36501ee28a3f835be9fabd38d6a32a8a263bc
4078 F20101123_AABQPM kurdi_m_Page_088.pro
44d770daa5225fbe5419f5a1d90eaa90
6f88b13cf539d2dd138a99ac7e629de56833208d
38656 F20101123_AABPLW kurdi_m_Page_026.pro
966718734eb0aa0e8d1e606f284020c6
64669e082d5ce96060615fde722bdcaa57cd0211
21014 F20101123_AABQOY kurdi_m_Page_073.pro
37cfacb170e6fe7508c8a34a95f9c5da
de04883f7ee5aa3f33df69a9b167a4ed2241a181
30607 F20101123_AABQQC kurdi_m_Page_107.pro
78686c8f14b5176e6a6c7f4b42cf5f5a
80632b77892aa57f531ff343454dcc48e4fe7974
F20101123_AABPML kurdi_m_Page_171.tif
bab76e10d142aec4b1b695a4af1b184c
dde4901bf378365c267b28d50dd94ae56552849e
32267 F20101123_AABQPN kurdi_m_Page_089.pro
7fd3a1a053916058963d906a38af082a05feb938
37198 F20101123_AABPLX kurdi_m_Page_130.jpg
a03921bc163d6ba5bab737cdc4880bf6
27787 F20101123_AABQOZ kurdi_m_Page_075.pro
b8dcb5337e8801163417aa6e3b01b03ca805b714
6074 F20101123_AABPNA kurdi_m_Page_002.jp2
2ebe45bbc1f159477670dfa4b6fd3ed1
dffa25fd8b8d2d8a3b0db03073bc007e0793ce98
27871 F20101123_AABQQD kurdi_m_Page_108.pro
86f82b8de9e7e72a9bf08c77b70f2836
6c08461e7e1874f75f6dc0d1da3a2f22a12567e9
F20101123_AABPMM kurdi_m_Page_040.tif
5e178c1429c7d84c06bbe3c1abf28a64
93294bb1d062b8d71eee39e4d0a3aa5204d81e33
31313 F20101123_AABQPO kurdi_m_Page_090.pro
7ba0a01d37aeeb70743465de022450d4
44a0e832eb3f689f94b4e71abd79fe0413689391
109447 F20101123_AABPLY kurdi_m_Page_140.jpg
a7e98f15e5014203f762541134d122cf
508ecbd8b98e7e3917f27a6db1007ca2012c6e8a
F20101123_AABPNB kurdi_m_Page_201.tif
38e7e707deccc5d402e6e9f646e8ef58
1e1fda12e93fd1e8d5014713bc599d75193cd39c
36826 F20101123_AABQQE kurdi_m_Page_109.pro
355624d6e5e519fe711f478cde1dd89a
1d928cc42a682683f62fc5714e3fbe2977a304db
F20101123_AABPMN kurdi_m_Page_141.tif
65af5e342b9fcc5497e46f3efdaa526c
17d62cbaea9ee96d9763cc7abefeab2ea1f51785
28865 F20101123_AABQPP kurdi_m_Page_091.pro
78076e466207c7a68c40ac015fecec9b
338347038aecf655cc75255a534bcc9301b5bae9
182538 F20101123_AABPLZ kurdi_m_Page_148.jp2
c2b385e30ee8d898ce2b3911df73f0aa
0d89ee425ee6042b9f3debb0bfb7425cf2f1d069
F20101123_AABPNC kurdi_m_Page_123.tif
01e5722958c8ec340599c1644bb6bba2
000cabbb520dbdfc51218b1568bb61aff85df7a2
50756 F20101123_AABQQF kurdi_m_Page_110.pro
d8ecbab2c9d6e112e09688f341150feb
78b324d4b2aa494613896efff0e0bb357d8538ed
22188 F20101123_AABPMO kurdi_m_Page_154.QC.jpg
3e23a9f7dbb28e6889dc880005beced5
454c4840494ff0a0e758444fc41f6fe4870d633e
30821 F20101123_AABQPQ kurdi_m_Page_092.pro
3bb938518ea17a1e78e61108ca5b186c
a1c6d47b2c080f6d830b1bd70450141edffea967
72565 F20101123_AABPND kurdi_m_Page_098.jp2
ab41f0e996fbe9406a1e2415580d7db7
d7abdd50fec60760b32f76ae234d136a6ba9554a
98590 F20101123_AABPMP kurdi_m_Page_134.jp2
86d0310e015728da5ac774d78390155a
772f75b3cfd2aa30d32be5c29b1acaa39735ddf8
24486 F20101123_AABQPR kurdi_m_Page_093.pro
4b853b6b5ae571477cf670883bc05186
470d806ae68176257e4ea89f6f6aa5fa88b36047
39950 F20101123_AABQQG kurdi_m_Page_111.pro
11347b68079b3be0f192c59187f6ee8d
d28837900324fc482e96ff4a7f08f1d012a032a7
103984 F20101123_AABPMQ kurdi_m_Page_158.jp2
c046b4d50f5c7935d9cbffc1cd6aa21f
4fb78d193cc6cf57d3896e990494cf70930a6f79
27661 F20101123_AABQPS kurdi_m_Page_094.pro
385509e0cd9d60c407d36c5b99451df7
807384f3e13d196eb49d1c7309538dbab8f41fa7
1051967 F20101123_AABPNE kurdi_m_Page_062.jp2
1be429a972073010fe2724f455915a74
eda34d2b0e358b2e6958a15109f87c475c4a6627
35289 F20101123_AABQQH kurdi_m_Page_112.pro
134a6a68781f785c6b4e7a063a870e0f
F20101123_AABPMR kurdi_m_Page_047.tif
e28a9c08d06a5f2b19018e97614608fa
c1780bc37ef6200aa3e9d63d59492699bee5dc72
15710 F20101123_AABQPT kurdi_m_Page_096.pro
53a13f8992ebba2722acd1a057a2d223
018dbc3128f37c7a3ccbf8929b763a4fd970eff2
F20101123_AABPNF kurdi_m_Page_081.tif
b8cfbdb4fc3302984ed8317f303f6daa
6201ddd59914042fd6a3e774d60c6bc16bbcc1bd
50460 F20101123_AABQQI kurdi_m_Page_113.pro
9381979339ff4cfa5d24537cc4e2eacd
6e56588e8b14d19c118f1fe422556024e521c281
16405 F20101123_AABPMS kurdi_m_Page_160.QC.jpg
0a12e60ffcba2919724af663f51800b4
8c094abe6aaf60ebbbdb19c3e5414ac51e62d42c
31822 F20101123_AABQPU kurdi_m_Page_097.pro
95309214967aa9c811313c63d10f42da
4eb8f62ddb152e742a0687e1cd46610949c9a322
65176 F20101123_AABPNG kurdi_m_Page_133.jpg
edc048c4b22606ab0ffb401915f95864
cd49d4d9d2fbf1c37128c97c91bac7c4a6fef677
21322 F20101123_AABQQJ kurdi_m_Page_114.pro
dae72d901484b9902c6a8c29a206d706
1cde03818b2ed2b926658ce1d24e148fc32a3ea2
22475 F20101123_AABPMT kurdi_m_Page_022.QC.jpg
3481f9ac7e7a9fc40b2d5ef71f1d0333
7e4d8ff3e4a22e20922b9c8c699758eba877748e
27169 F20101123_AABQPV kurdi_m_Page_099.pro
F20101123_AABPNH kurdi_m_Page_075.tif
20ca6907903af2fde0ddb05eab3547e8
65531698c1bb028ae175f0d105e53c4969bf7f63
17591 F20101123_AABQQK kurdi_m_Page_115.pro
e00c03813432de63ce5620f5514f8cd0
29202196af94b32b176d334481d33b95d308db90
109080 F20101123_AABPMU kurdi_m_Page_189.jpg
e76911cc3ff336dfdc670bfcc3ae1120
41526 F20101123_AABQPW kurdi_m_Page_100.pro
3b83b01c100cf31df08970b956a288c1
c2c5f395c6a45a5f85657be2622c5b96ec352559
846145 F20101123_AABPNI kurdi_m_Page_045.jp2
f2bd629c19e3da720ceb2d3bb057e98d
faf448f18f1edb86cd7e5593b291ac4decbcaea9
30621 F20101123_AABQRA kurdi_m_Page_132.pro
db0738993365ca91dda3a88b5ac89f38
6c7b39860ca8e32d2d419d8fee0a3471278895cd
21269 F20101123_AABQQL kurdi_m_Page_116.pro
517fb74f4f8ca11c75d7df141ea16baa
b49dd8f6df06b96ebd61b94448b1313b043a25b8
50847 F20101123_AABPMV kurdi_m_Page_121.pro
41025 F20101123_AABQPX kurdi_m_Page_101.pro
03f4d4c047962f42150117721b17e146
61c38de9ed9f2c36a89fb9f73444f967e839a336
6643 F20101123_AABPNJ kurdi_m_Page_200thm.jpg
2fe420ec8c346009714cc8a6cef159c3
c4a7e2e84ed61a3af101849950a17e8247b2ff49
43712 F20101123_AABQRB kurdi_m_Page_133.pro
25c86f3a54e55401f1bde09b81b2ee53
c70ac1d768199b69c1e4314ee2488dd01b6f8412
21435 F20101123_AABQQM kurdi_m_Page_117.pro
ffe15e02b608ffc4e55a0b7d39b1b623
4e246879d9d2d4818760a5b391b5de2a999f5f77
35008 F20101123_AABPMW kurdi_m_Page_098.pro
12cfcf7c6caf9147a4d85c8f14290635
48381 F20101123_AABQPY kurdi_m_Page_102.pro
425995bee23ebc4e02ea78587e9d0dc6
bbd64709a0daffc754e23e1c147faf71166309ef
F20101123_AABPNK kurdi_m_Page_122.tif
e2b04f67071c77fd70ef29837c18b476
39816 F20101123_AABQRC kurdi_m_Page_134.pro
c6e9700ddeb05de5051c5c99542382d0
a6341ea0169512b889a9e70e71e9fe0540968efc
50079 F20101123_AABQQN kurdi_m_Page_118.pro
680eb209e98871878bc81742468085ee
a533e87a9d9faf6f0d45fb36ba96160cf0b68fb9
24482 F20101123_AABPMX kurdi_m_Page_095.pro
79b971d9ee10bbe073cba76bf24e92d8
5601447388c7a2cb73a85417ae8e8d5a7185cf23
14789 F20101123_AABQPZ kurdi_m_Page_103.pro
17385def73b5193251255c8613b3d079
3b4564317b8fc60686c9957f9e4d779f96588461
81698 F20101123_AABPOA kurdi_m_Page_027.jp2
fc58fdae912699ceee3d814eff7ecbb8
853e10d99bae3d0579e368f5ea1346bc0cd7373b
5240 F20101123_AABPNL kurdi_m_Page_081thm.jpg
507b0cebf35a395136f8359e9907721860b5df01
41459 F20101123_AABQRD kurdi_m_Page_135.pro
bdbe5306d851ed1d8412bda8ea1f3dd7
b751bb074df609d51ec439675dcac317ae2e2614
14042 F20101123_AABQQO kurdi_m_Page_119.pro
125d9e117e81c05eff74d632a28e5092
c0d654a1a86ff854f222b99806ee8aea8f5a8461
36330 F20101123_AABPMY kurdi_m_Page_163.pro
5465141006272233e15df046ace9482def62771b
5771 F20101123_AABPOB kurdi_m_Page_006thm.jpg
f8cc82d7eebd84f4f4a0d566c1ce3822
9a21f371f610c106fbcd988f93ed49a0fbdc8873
1051940 F20101123_AABPNM kurdi_m_Page_005.jp2
612c06296efc913c04550ff6b939b285
3371713bec62a53ea2608f8f6034cef4f705f628
52567 F20101123_AABQRE kurdi_m_Page_136.pro
307395c3dcfa0f15746e3ec6c6db9259
b6bd8d8773fa26f4220fd09ceb9506291abdc2a7
58667 F20101123_AABQQP kurdi_m_Page_120.pro
0476a4cb99a0dfeea1c57ac8c5b538c5
510646b17ee6b1d8f337d58a741ab8a14d3e330a
55289 F20101123_AABPMZ kurdi_m_Page_173.jpg
e4aab1080304629d7ce5cc1887dd4c85
63c582287fa53c0e73e56cf7056fccffb5931d6d
1692 F20101123_AABPOC kurdi_m_Page_161.txt
6f69ebbc9972de410118841578d76292
6d0c19d272a2cf4d4368a04a567a454a67430d1b
69609 F20101123_AABPNN kurdi_m_Page_149.pro
9c6463f5e5c4f34f11e0ca8ae0a530bd45a4fa51
68476 F20101123_AABQRF kurdi_m_Page_137.pro
c813bf355e4f7e43c043400aab98b992
2f7b344ac66d2456536af5247e2e1dfa88a60f3f
5107 F20101123_AABQQQ kurdi_m_Page_122.pro
c4dd22b2a0046f540b57e00d87677441
7cea3f9b00122dd684c5689fb4e8f8ddfccc07e7
1267 F20101123_AABPOD kurdi_m_Page_107.txt
130aeab11e8cdf9f980408fcd8414b71
1790 F20101123_AABPNO kurdi_m_Page_031.txt
70558 F20101123_AABQRG kurdi_m_Page_138.pro
87ef9b8b40f235cdfe073db36b38ec73
52879167a4c675fdfe03b0241edc42b1130f5f33
11669 F20101123_AABQQR kurdi_m_Page_123.pro
f1cf0206759b9028b31328fa87d3f494
acd88050c0d74833fba8f2695ce617df19ac6cd6
19305 F20101123_AABPOE kurdi_m_Page_036.jpg
bc2a4bddbe31fb110ba3ba24b73d68a2
b37226dbc32d1e8135b037f16bfe9fb0aea2e7df
24306 F20101123_AABPNP kurdi_m_Page_196.QC.jpg
76375a71557472daf6443a4d33c0bc68c4558868
38345 F20101123_AABQQS kurdi_m_Page_124.pro
2d4146c3dfee405eed44c9ecc1fb21dd
bf86251904df4c86608d329e60c2da3fe9902bfd
16253 F20101123_AABPNQ kurdi_m_Page_094.QC.jpg
d28b1269009ca31d2d42b0c1154367b0
fa7dc761540eedcb05cf5bc30fe142dc3f903777
74186 F20101123_AABQRH kurdi_m_Page_139.pro
f7cdf58d3010809e67da4f945ff50bdc
80fe509e776559ac1ed860c4636ba4032a60e965
38256 F20101123_AABQQT kurdi_m_Page_125.pro
1767 F20101123_AABPOF kurdi_m_Page_012.txt
2e8f9f51a8fd80c48d555e29bc89f9b0
9b785568e3a5afe80221cce4cc5eb874b031a189
52103 F20101123_AABPNR kurdi_m_Page_159.jpg
ba63f50f04511ebfe4debf5b58567842
0dcf513e1772d49b2b6a62922d1b3cd2cb2f3cbd
70396 F20101123_AABQRI kurdi_m_Page_140.pro
ab4c6dfef6d661d580918b0ca58c5183
39721d6e9a664e051af72dfe648c1084b0b39c6e
34074 F20101123_AABQQU kurdi_m_Page_126.pro
9284f3136e75aef9dfaf7823c3363b6e
b467e53fa19ebed70c16ca0613184b579b19c2b3
F20101123_AABPOG kurdi_m_Page_056.tif
37913ee783435c81c8bcdeddd3681a0d
8a23241163f6d8295503e4a15f8f5d0ef8e7df1a
862652 F20101123_AABPNS kurdi_m_Page_112.jp2
2d9f3bb603bf5bb9805b8818f57daeab
fb4c48d2dd584ddc793be5bbc3dd8824e97fbe02
70119 F20101123_AABQRJ kurdi_m_Page_141.pro
10913ea3fc07d1cc6a7ce8e896a3276e
125cc2aeffb21f6410b91506af64426a8af55bf5
21426 F20101123_AABQQV kurdi_m_Page_127.pro
c224f5b8a1777d64ece39a184bec9d38
eebe3790bb42a148cf31a9007b38141a5d1be8bd
2115 F20101123_AABPOH kurdi_m_Page_165.txt
16a6f9530ec926b8ca40d443ac4b2bb3
5cf429f7e042e6ffe702fefdd38aaf4b4493eaa5
5734 F20101123_AABPNT kurdi_m_Page_092thm.jpg
1c1802d117b730785a5884031dca54a9
cabbb3e88dc047d814f76ec7bcaafbc7dca6cf9c
70804 F20101123_AABQRK kurdi_m_Page_142.pro
02db51d0733871dcb7c53f120c34eb82
00fe5fe401463fa73bc271069f4e2d0e38430f3e
28084 F20101123_AABQQW kurdi_m_Page_128.pro
064ea59c6e924577a66db5a5bf231215
c03feb05d8aefa5faffe3bba0f67a809d46ddf1c
70069 F20101123_AABPOI kurdi_m_Page_161.jpg
a2ab9e27a643e5968608ac4fec6cb5ed
fdcc2fdf3607cf45bd7a1cd70871a8da63420f49
1304 F20101123_AABPNU kurdi_m_Page_003thm.jpg
362df33ef6b76741a7e277ffaa626ce483a9942e
41894 F20101123_AABQSA kurdi_m_Page_161.pro
c2693954260884634878ab4bb8a1a2ce
c4020a70e2a040222f41a8a6e00c134e909e3a51
57217 F20101123_AABQRL kurdi_m_Page_143.pro
9b9f8c3c4112f31f7e23fe8f7bb96004
7f486db643be78d58f3a0223c64e2ddfef4d66e9
14077 F20101123_AABQQX kurdi_m_Page_129.pro
f5a8af2ce15c861d1492464c09192ba6
168df6187506326f1b1a13ff06aa59f1c55cd873
F20101123_AABPOJ kurdi_m_Page_045.tif
2cd000dce9a777086fe07df8d9d3a1a5
9d4b0f7e510b93395f9cc0ab04b4ca78815e5f08
18255 F20101123_AABPNV kurdi_m_Page_057.QC.jpg
3a0d00794d2cccd64dfde03808eca0dd
ae401791cd05158e102d8875b2e897af5376cebe
38476 F20101123_AABQSB kurdi_m_Page_162.pro
e26913704d69ed4b8516743842c3b95b
e490f0dd565370f2740802d7bc7d07b861de6923
36463 F20101123_AABQRM kurdi_m_Page_144.pro
a408db587254460a70fafcff4d4da576ca95e678
17604 F20101123_AABQQY kurdi_m_Page_130.pro
5a3a3a16565879937c70b91ca56c0daa
69ab6e581839d382ed76363f85955d1c4a33bc71
1427 F20101123_AABPOK kurdi_m_Page_144.txt
1e250dc645a6fbd8fece1a918690c5e8
b3f27f8142801620ae9e722eb7a4de79a740c632
86866 F20101123_AABPNW kurdi_m_Page_054.jp2
317f08cf918150407b34591c7e4b1496f5508383
42666 F20101123_AABQSC kurdi_m_Page_164.pro
d2ebfb16cea16154d4f1473cbfa971a1
8b1b84ee0a398474818e27467d99d3a033144961
39906 F20101123_AABQRN kurdi_m_Page_145.pro
627ccbf8ed11d3b6a3398cf249f021a2
eb7b323113f8945324e823c0ba3b473b1caf087d
33165 F20101123_AABQQZ kurdi_m_Page_131.pro
89aabae623c074fc20b95e6bc115c43f
ba584b621da083e4c8592787616dd17332d2f75a
2147 F20101123_AABPPA kurdi_m_Page_176.txt
0a488176033c6a487298506e63b5409e
4e377f6e32fcd31781c1c232574fc7df2b6c02eb
29405 F20101123_AABPOL kurdi_m_Page_173.pro
0a2b7ab7368373c90c1eb0f8c2415668
22959 F20101123_AABPNX kurdi_m_Page_087.QC.jpg
37151fbd817d9d88b8aed60cbe73de16
3a3c5abe268e226a8d2556e8157d1c2cf08708e9
69290 F20101123_AABQSD kurdi_m_Page_166.pro
9a3b8db1c3ca5326e765c982883a522d
00da3e1c13d6058716babecc9f876cd919067ab2
48788 F20101123_AABQRO kurdi_m_Page_146.pro
a170acb0bfc2abd75b002356861c055c
f5f69023030c07a0662e45e4ab03624417e5609a
756902 F20101123_AABPPB kurdi_m_Page_070.jp2
1b9b0e5feeb087f2af076b69dfa4b373
69891c9638d9685355670e3eb33de70292cdfc2c
F20101123_AABPOM kurdi_m_Page_078.tif
47c05879b583aa2db70e27cd62b587a8
d411b201c457f629b171c17cb635b45d7f4d1512
37261 F20101123_AABPNY kurdi_m_Page_045.pro
3c6cb64e8a38a3b31710cb77140d368a
4aeef32618f47c18e2ec5a56fd44fd0886fc99c5
68431 F20101123_AABQSE kurdi_m_Page_167.pro
11675966357779fc7f44e7711cd7726633c91a77
56973 F20101123_AABQRP kurdi_m_Page_147.pro
60c5b02ceb29df8d24524ae7d37c121a
80fed8838dac60de2110b8ccfd627347292b4167
F20101123_AABPPC kurdi_m_Page_153.tif
34711e57a959b4c46cae49087ac4d7c6
8abde77a6a94c27f7d92588467d69ff8f77765f9
2698 F20101123_AABPON kurdi_m_Page_011.txt
f1d67012e0284f93513b025fc7f40f34
4572d4ab4c3eeae38ef69ff97dbaf74187128ce2
20676 F20101123_AABPNZ kurdi_m_Page_055.QC.jpg
0ccd7bd14dbce24c99e14abe0008d4e6
69933 F20101123_AABQSF kurdi_m_Page_168.pro
1a51b63ab8379e5570e10a9b1857d55c
592d30b157b357b3e067571997548ca13565c447
72007 F20101123_AABQRQ kurdi_m_Page_148.pro
14627343aae58b23d1064f56721d1459
4622 F20101123_AABPPD kurdi_m_Page_006.txt
b1436057f227147067e4fe33747d2c34
b28b1d7592feb5805b2a4a5ab51668c8dcaf6130
50190 F20101123_AABPOO kurdi_m_Page_098.jpg
51c984ea423a4829489c0d72c490ff7a
58d93628c0bb4fd25bcc981119e8d5a32c46fd9c
74204 F20101123_AABQSG kurdi_m_Page_169.pro
2f288e727ddf9d486d6b56250f8db55c
76459 F20101123_AABQRR kurdi_m_Page_150.pro
1c34633bfa969b7d0f17a5367525efac
ccf91e65b6b8be40ecda168a442f7f7c5d706826
F20101123_AABPPE kurdi_m_Page_164.tif
295d141ea6d03d0a6e46558a2f3e9ee7
0b16d89981dbb0b47d08da7e8c2f34ae053bb823
52452 F20101123_AABPOP kurdi_m_Page_056.jpg
a45080b81d2e8acd96b165375efde8ec
69763 F20101123_AABQSH kurdi_m_Page_170.pro
7b5ed7bc624a2fe77011cb828ac30f17
9b8cff06c3eaa9b067a05afbe196a2f946b06958
69840 F20101123_AABQRS kurdi_m_Page_151.pro
591d37298b201607087f75013595733e
75a9edd4fcf4a913904d91aea4ec5bc4682d8d9d
F20101123_AABPPF kurdi_m_Page_027.tif
8160bc2de31a423e55764c8efa19d8a0
0cb1272ddcbf577d3f5f543477407ff9e47d6aab
4849 F20101123_AABPOQ kurdi_m_Page_050thm.jpg
135709f0a936c70a78f2e802034d110fb49f1cea
70513 F20101123_AABQRT kurdi_m_Page_152.pro
686b88a23e91f6e88de053aeef75bb01
ac0a3afab6a4fefa79f2447fbfc92441689d409a
F20101123_AABPOR kurdi_m_Page_005.tif
1ffa29242724057618a7f138ff5e20d1
dd95f2038930927833f065537602f0f3e0f874a4
70245 F20101123_AABQSI kurdi_m_Page_171.pro
f755c0d3f9af848e4ff0c912577193a0
db72033078a62f7dddaee72edefbd44b89ea8c9d
67895 F20101123_AABQRU kurdi_m_Page_153.pro
4c60c465818107612a1c63ddbe324bac
23069 F20101123_AABPPG kurdi_m_Page_020.QC.jpg
843ab996086025f3a974dc31610b41f0
1e69eb905c1f5fe6ac632ec43083a0feb1117724
3662 F20101123_AABPOS kurdi_m_Page_013thm.jpg
eff7a0a14468f591225f652489ff6d90
3cfabd4dae6e241fe97a11e5be61fd002b4d5145
61469 F20101123_AABQSJ kurdi_m_Page_172.pro
48d20e701cfe4c40f1b3d218a336d418
3bea4a18a241bd0ed5ced20aa3fa82be34d0dd36
47876 F20101123_AABQRV kurdi_m_Page_154.pro
772eac32deec8811618e4fa29a1bc924
0611be965313758127fc8d4febd0694c36c68b40
104618 F20101123_AABPPH kurdi_m_Page_167.jpg
7fbc204e5da8844c9dd7152b8e197695
8805e50573d85165b0d8daf61827025dd07d7ef9
3295 F20101123_AABPOT kurdi_m_Page_005.txt
918b06f120ef7b24f057a3329bd3bd70
9acfab7cb012f890aaeeb18a3e8e2f2040fdf976
37113 F20101123_AABQSK kurdi_m_Page_174.pro
ffe6df53d9b85d2ca153f4d8f30ff41d
0bcc4dc07b4a6e0ba8fc251eafb043148c59eeb6
34828 F20101123_AABQRW kurdi_m_Page_155.pro
c1ac815ca01c50291d03e0270a73e6f871d200fb
47099 F20101123_AABPPI kurdi_m_Page_165.pro
bcc1b765ebaf84208e686e430fbe2c37ab12ccee
21227 F20101123_AABPOU kurdi_m_Page_023.QC.jpg
ef45cca60b3037668ac3d78b7aae8688106a3fe0
71500 F20101123_AABQTA kurdi_m_Page_191.pro
35729ed2ea8f3fbb6724edaa96cd40ec
6a264063382fc1a6006b930cc3285c43b5542c9e
41321 F20101123_AABQSL kurdi_m_Page_175.pro
aa66777b2955420b0da44c521e459277
7102f2aa9020a60d245aaeeeef6fb91582c6504c
33988 F20101123_AABQRX kurdi_m_Page_156.pro
4827482d670c1ffe9f522628c826b7bfc45934de
20378 F20101123_AABPPJ kurdi_m_Page_202.pro
db2f98c982029afe017e66041b233599
268e48d66dcbbe411b3e850cba2de114526cbd7c
1909 F20101123_AABPOV kurdi_m_Page_020.txt
a5cf2b638f5e151baf6de7d4658f746e
0c00b8ca023106f338a4624992e23221676e49c3
51125 F20101123_AABQTB kurdi_m_Page_192.pro
ae97e7a38cba9821f223cb1cff413443
50230 F20101123_AABQSM kurdi_m_Page_177.pro
72c87e041a445f91dfc31564e8d3ab85
c942caf33bee12e2f1a8fec53d4cf463267d6f20
47248 F20101123_AABQRY kurdi_m_Page_158.pro
27c0212061f8e964ac8c3f5803f1a139
c9a8f9b1e3d5afb97c58786fb0aefbfc478bfe2f
93259 F20101123_AABPPK kurdi_m_Page_102.jp2
e4c56ae8c5f90b8e8b6039c9df07cb4c
0b5f6750ed34010fe8edcefb03255b37afebb6da
21692 F20101123_AABPOW kurdi_m_Page_051.QC.jpg
7dcefa74f3f66ae96861591d9695d579
9795 F20101123_AABQTC kurdi_m_Page_193.pro
f46c5d1265e9dac922c7fd0c6fb4165f169053c7
50381 F20101123_AABQSN kurdi_m_Page_178.pro
74c7587cf590891d8baa31babd81e225
c5136e33dc98abdae52870fb468facf102177690
36556 F20101123_AABQRZ kurdi_m_Page_160.pro
ab89f0d4bc0ec7a7141c1543092e33ee
107390 F20101123_AABPPL kurdi_m_Page_168.jpg
3ef36bcbae3e7fa9e0388f4f009fae3b
ea285fb8983535e6f1eeba17bb7b740540cbb6d8
17437 F20101123_AABPOX kurdi_m_Page_180.QC.jpg
511d7043ce7d1fd4603d349768b6c9b1
7bcf65e8694cf5febc171c2d9641af85ddb0e527
44248 F20101123_AABPQA kurdi_m_Page_058.pro
a495f4819f7a85201f856cd486044570
06acc0f8cf9554538c27757bba5a128a64c58724
49741 F20101123_AABQTD kurdi_m_Page_194.pro
04787c0be7ca6031ce5fb319929b50c499793b34
36965 F20101123_AABQSO kurdi_m_Page_179.pro
97d21b52c18f468a07c770f6513e6284
e4eb93b1f62fa30dfd500309e63b2ed726e66625
29040 F20101123_AABPPM kurdi_m_Page_050.pro
de76c5ccba788e993dd97606015058365276d522
32581 F20101123_AABPOY kurdi_m_Page_063.pro
d3c8d7824890a1e19701758d6e100210
259a6eaa1e8e71effd61a33238a9e6bbafdaa8d0
7391 F20101123_AABPQB kurdi_m_Page_170thm.jpg
6cd315d8cd144a2e31412ddf013f184c
55826 F20101123_AABQTE kurdi_m_Page_195.pro
72c5af932b4e9c6b6a8d461163b62254
dbcf8751b0ae57f1262084a894ebd7bbe5b09e34
35407 F20101123_AABQSP kurdi_m_Page_180.pro
08fcef766ef62d412037165b36d933a2
59933 F20101123_AABPOZ kurdi_m_Page_092.jpg
43323fdd853e7598c505c352176a9ed55a1fd33c
F20101123_AABPQC kurdi_m_Page_055.tif
214d52cb763dc03911a430cfdd351df4
F20101123_AABPPN kurdi_m_Page_194.tif
fc10ae3b6e8ebb6746661f55c40209e4
3a88d1c7ef3e5322f0dfb990b8aaaa9b4c474fb4
58285 F20101123_AABQTF kurdi_m_Page_196.pro
f9478b60c17b28d6101ec4efa6770642
e55c6ea4ff2eb8e56b88c147c454bb416617af0a
39578 F20101123_AABQSQ kurdi_m_Page_181.pro
ddac568b1483fc3bc23060f72bd736fc
d79aae2626f93641423b5e03b21f9d97a921b60d
45802 F20101123_AABPQD kurdi_m_Page_048.jpg
a9e210b4fa972deb3385a67573f42e87
f71e85a6e2951007ef4ef0e92216bdbdc0cc3f47
29304 F20101123_AABPPO kurdi_m_Page_140.QC.jpg
d439e31e835d6c0017d96b15bd6f4517
f125a17006d9cf7349437910f09da4691b78b86d
51024 F20101123_AABQTG kurdi_m_Page_197.pro
640b0ccd788693cc89c9a4f6a107eabd
34126 F20101123_AABQSR kurdi_m_Page_182.pro
03efda927bfe522b23b36a1bde64e5f1
6516 F20101123_AABPQE kurdi_m_Page_198thm.jpg
808830e40ccef6e9312a29e255115775
64012 F20101123_AABPPP kurdi_m_Page_058.jpg
52eef773eba9c104ccf9703e017d529f
f780a0cf668290692a85b2fa0e088bda8b50ae65
58057 F20101123_AABQTH kurdi_m_Page_198.pro
fa547b8d0cc23100c164cf66efbe1d57
d708b9b8da06df619e207314c64df5810bc9e742
33085 F20101123_AABQSS kurdi_m_Page_183.pro
ee87c9b432dcd45248e9815de3c379c5
9c139d4b61fdbcaec22cb225031265485b4f4d96
66403 F20101123_AABPQF kurdi_m_Page_067.jpg
aececdf677b6cb1ff52da085b4f8e234
a00ffba628104697cb40a6038cc18e76c8b2898e
61374 F20101123_AABPPQ kurdi_m_Page_043.jpg
6f402a233a0505050b02473bc5e063bc
de6ece5b7b88a82cdd15e2ab685cea2859d2594e
56228 F20101123_AABQTI kurdi_m_Page_199.pro
878cfe524b9a3ecf3fc22a5eda8fc838
44758 F20101123_AABQST kurdi_m_Page_184.pro
d7922a9ffbbfa9c5c6bc29e82a93df27
8dac05af216c03eb32a9e90bf4c2c9fb96fca293
25194 F20101123_AABPQG kurdi_m_Page_072.pro
3c384cb275e8f892dd75cf030a642854
fc52906247e6985731a81b2f6ae30c19d40107f2
181409 F20101123_AABPPR kurdi_m_Page_141.jp2
c9043d04fa7a159b18bb5d5ab6d287f4
d0d972c65746af4f46e5990ac65f04f5f5f8971e
53596 F20101123_AABQSU kurdi_m_Page_185.pro
1c64ecc238a525318f743b6918d2c9f3
023538a620d80002f377e041c79a640089354469
5708 F20101123_AABPPS kurdi_m_Page_084thm.jpg
f49a507cd7a091f1d2f245429986768e
5e3b9915da7e132ddb28955c20faca4039c555f1
59231 F20101123_AABQTJ kurdi_m_Page_200.pro
a698fa16beeaec11b88e88e4ed3baee9
686bef3f0cbf7efc03cf8286f85416d026fd1227
70954 F20101123_AABQSV kurdi_m_Page_186.pro
d1af304a2f076642d6c7371f8ac06526
e340f839a86190ed7b51885f0086cd8af3b506be
447 F20101123_AABPQH kurdi_m_Page_057.txt
af6126296e85d8bddf33aa645cf6ffcf
15115 F20101123_AABPPT kurdi_m_Page_004.QC.jpg
fb9058180efcb2c48839d76bb87e8128
225ea2f9fdb6fd9563898b59d7ebc03c3d139346
44742 F20101123_AABQTK kurdi_m_Page_201.pro
9789c38d0595b79a67769e162ed03598
03e25bef5342db0bdd524a33ba3a5e6201c9d0ce
68945 F20101123_AABQSW kurdi_m_Page_187.pro
d83ca99ee60822d48c6327aa39fabf4d
4c945673d8d992a023bec5dffb660fb01490f21b
68033 F20101123_AABPQI kurdi_m_Page_162.jpg
c57ee976ed98a1c2a874b7923e344cd2
ff53aed13e858c71f9c9e32c31ec671ebd42d94d
92759 F20101123_AABPPU kurdi_m_Page_172.jpg
80ddcde82c281965a3737850a6feed8c
519aa68711402c61ce35064731510851d9b7352f
1898 F20101123_AABQUA kurdi_m_Page_022.txt
bfb305ec0084b2cee743c6f886490f34
972158ecd866fc255627f4ee3485f3bf6b59cbd8
516 F20101123_AABQTL kurdi_m_Page_001.txt
86766d9a80680bed8ff3a260a1e1afb5
99d32f4358c7a3a6d15f91d406edd208b8e16cc4
75844 F20101123_AABQSX kurdi_m_Page_188.pro
6e2b5b868d201945eee23866ceff021a
23375 F20101123_AABPQJ kurdi_m_Page_021.QC.jpg
cb53a4c0f4d44c695d3649c5810e068f
1051983 F20101123_AABPPV kurdi_m_Page_009.jp2
6ba97fffcdf072b06ce42ab7097dcde5815a9efe
1803 F20101123_AABQUB kurdi_m_Page_023.txt
a0762d6ccdb08a52424ce83e8df11fdaf0e2defd
115 F20101123_AABQTM kurdi_m_Page_002.txt
b6468f3f9eba3ddf882bd4b381ef37cf
56856d6e0268df9d32579bd867c7430f8f04bfd4
70573 F20101123_AABQSY kurdi_m_Page_189.pro
79bbbb7a157272cde4ab1fa136a27309
2c5b4c0793632d229a0a21c327f4f088496f0cc7
73359 F20101123_AABPQK kurdi_m_Page_178.jpg
e684d390fff2385cb48817ed95d070d4
edac0e9926445e9525e1a079d1df8aae8be3a158
1434 F20101123_AABPPW kurdi_m_Page_160.txt
d9a1b759eba78e1ed3b204a2a70ae228
5b49e757f817a27e23fb8b096fbca6b7553070f1
657 F20101123_AABQUC kurdi_m_Page_024.txt
432cbaf4e4e06b8625f6016ac77365f6
7829c8bf6c118d2c3dca73ca6f19e2d8fb6cce0a
85 F20101123_AABQTN kurdi_m_Page_003.txt
ca7604355d329fd79829aaaac188b01a7cc45873
69996 F20101123_AABQSZ kurdi_m_Page_190.pro
f1e10277bcd8ccfd25cc68f1822d3986
21bba8f77b33c166aef0d036c3b79b9bca21cb90
61518 F20101123_AABPRA kurdi_m_Page_100.jpg
78705c7dac8ba9c343f158d2c63dfb70
c095eaf930d4394947e59be7ac21fa7f1debd0a4
F20101123_AABPQL kurdi_m_Page_071.tif
0f1402f351c2010a2fd0c60d58961d0e
83300713835ec8c249eff391f29be8b890ba6468
396923 F20101123_AABPPX kurdi_m_Page_123.jp2
d8131581e0462a845e7a18ee85cbeabc
1db8c4fa2bd62c635fb69f7143823dda680f6fe4
1603 F20101123_AABQUD kurdi_m_Page_025.txt
4f5ecb327ccaa74c2286ffe8242301d0
a5d91fc9e4f728671a8998c04f7e084e883cf4fb
1188 F20101123_AABQTO kurdi_m_Page_004.txt
68aa574e4b7c1c47d7e364f93ab3efe9
58a32358f3b7645a7bc9fbcefe12460573677f06
F20101123_AABPRB kurdi_m_Page_073.tif
4a7033824227d3df16b4cd7858fca5df
85da84b128df6e471d0fcbd4764327aab68d9d83
973690 F20101123_AABPQM kurdi_m_Page_084.jp2
03ccc4094b04c38081b7ccc3c3a8dd01
799aecaa8fb4b39fc66986c614a675266c0f598d
F20101123_AABPPY kurdi_m_Page_198.tif
f2f8e89ef04137ec6677fa59fd18ebde2e089359
1996 F20101123_AABQUE kurdi_m_Page_026.txt
1c9d9963c294123a947a5dc23aef71e4
5a3c03b5aa8be6256abbe61b6f73d46df19a28b2
3090 F20101123_AABQTP kurdi_m_Page_007.txt
bac3e62f4bc317f8402bd19965c877c0
8a26fa8be80aa931bc00cdb13dba7eee40957cb1
521802 F20101123_AABPRC kurdi_m_Page_037.jp2
e1528e4d480c40bfb29182642178b03b
1310 F20101123_AABPQN kurdi_m_Page_182.txt
e173f7286e38b6a201ed01360d844ffe
0f24a0bda5c954d316b0c40afb38c0c75f869bf2
5710 F20101123_AABPPZ kurdi_m_Page_164thm.jpg
95e7d4e6b3ee5d828cd857bf9879e821
e048e424e3fa48d80db66c09940866612c31004d
1580 F20101123_AABQUF kurdi_m_Page_027.txt
944660f456b25ca976b07470babb454b
2170 F20101123_AABQTQ kurdi_m_Page_008.txt
1137dfd2773e23f85f9ab05c833904d7
fd4528481fd3a57459baa683b956769ae0309cac
F20101123_AABPRD kurdi_m_Page_134.tif
e1c205bcb0db029bfc2f162a349f5a4caafe9714
62488 F20101123_AABPQO kurdi_m_Page_175.jpg
186eec0c9b2400f6b6e909967c659eec
cddae08edd379ba5c8560de0a11b83781db95ea3
2086 F20101123_AABRAA kurdi_m_Page_194.txt
f85e7d6c609648d45492906bfb98c150
2af6a7f1a6e7509d9c11fbd951c651a594951e2b
1707 F20101123_AABQUG kurdi_m_Page_028.txt
08f34370efd9da6efab7a9ca96c76ca8
2395 F20101123_AABQTR kurdi_m_Page_009.txt
b7ebd30cdc97c7192792da37e668e387
a64f41c1913c742fb58db3d49d72c27f7f663937
17724 F20101123_AABPRE kurdi_m_Page_046.QC.jpg
549cd31e943b871cddc68501235df4f8
b0271db559b8692c434c6f41f0eaaf7dc63e3931
F20101123_AABPQP kurdi_m_Page_095thm.jpg
6c4cb9e55b2d3bf9fda271305afc2ec7
1a79e5790b2699d072a7b20bc2f2e12815987fb9
2338 F20101123_AABRAB kurdi_m_Page_195.txt
0830d54b9ed4c8c80b123f78e3aac23b
5a475f01be406929ac2369f324f18a3df4f0aa81
2055 F20101123_AABQUH kurdi_m_Page_029.txt
ce83382f27c5d4b2c8f14bb54ee3833f
43c5a8c665d218c099a13b691b5c740096e962e4
1254 F20101123_AABQTS kurdi_m_Page_013.txt
617609a72f441b8e567747132610a11e
1eba92f2063d11310bdc1343e4aa8414fda744b1
16184 F20101123_AABPRF kurdi_m_Page_098.QC.jpg
a2bde213142b7e435d133af7493c664e
a8d46b538441cb00909f9f9418752d9da0d06976
1480 F20101123_AABPQQ kurdi_m_Page_105.txt
027f6f11c6f3e71ac44b1b810701a6d5
ec7f7777a63b6e0ee05809f7a575b3ec5c78f0bd
2411 F20101123_AABRAC kurdi_m_Page_196.txt
90dfd36c84018ca05006e962e144f23f
1260 F20101123_AABQUI kurdi_m_Page_030.txt
ec81248066da99c6aac8fafbbf888208
2b52eb828bbd7174a121308167901179b3b7e2ae
1404 F20101123_AABQTT kurdi_m_Page_014.txt
94409f579824af89060548141b640de9
4fd452ed91905f4f2e00207b3bb924ccb019fb4f
F20101123_AABPRG kurdi_m_Page_142.tif
e6451f4af240a23f1e193b5087204675
5163 F20101123_AABPQR kurdi_m_Page_076thm.jpg
31df6c777c67be25a58bcd4beff4eb79
b383086033f10e0d0e0d032cabf1540edfb83c88
9b9cce3f616fca4733bd6127260616c6
72679910173a8aa47d630c4de996d98d881e8f16
1843 F20101123_AABQUJ kurdi_m_Page_032.txt
5da37695250062dea31b90df74d00d55
dfcc10fb1f44deee2dc161cf27a446fda34f0d99
745 F20101123_AABQTU kurdi_m_Page_015.txt
eacf5d675e5e7fd320855743cf8d63a3
bf7a77aa27a6f2e81dac92c9fe43d2489aed42bb
18009 F20101123_AABPRH kurdi_m_Page_037.pro
22f86dfdddc1e1158b3a2f4c70caef96
e48261784c872d60860685f59c7ed1a4c2798d01
4483 F20101123_AABPQS kurdi_m_Page_030thm.jpg
d3bc2a3e764c0b96f47943e5ceccd4b1
a19eb97441b0b22099734734a2fd663bf3dc108c
2425 F20101123_AABRAE kurdi_m_Page_198.txt
f63210222ecb025b6205ddb9b31608ea
47c098ec44e051306c8b12551bd020b8d7bffef8
1681 F20101123_AABQTV kurdi_m_Page_016.txt
e90445b68362156373aaa0096be6d8bc
50acab60115fc7c0e5f6ef5a4866688337719504
F20101123_AABPQT kurdi_m_Page_030.tif
2021b13268dfc2d63f22f7a8c8711255
e49f5c6dd4f3c18d57f18d33f0328bef770f4f7d
2343 F20101123_AABRAF kurdi_m_Page_199.txt
1d43d7ae83d781b0c53fc94652f6d919
a43d3677b3f501cd77b11754b2fc8c3089baa21e
1731 F20101123_AABQUK kurdi_m_Page_033.txt
d2abf8abdbe2aebdf166119712669964
bb061eab21375e987c789fea7dfabeaf86e797e0
1832 F20101123_AABQTW kurdi_m_Page_017.txt
a0ecac90b39ea32e13c367e7c46d3f0b
c7229f06506be0c3aa52581a86809b8d001e8a1c
175085 F20101123_AABPRI kurdi_m_Page_137.jp2
d68584174fed4069862902be94c761af9a9bc02b
2247 F20101123_AABPQU kurdi_m_Page_121.txt
60a1661050079f5697f9d0d36bf9b446
27b4822ee6a814e39e7b21dd1ca60f0c1fd611c3
2466 F20101123_AABRAG kurdi_m_Page_200.txt
3968eaaac422cb680f970b096edab894
bf5d8f3ab86cf1936092213f450ece404221b549
F20101123_AABQVA kurdi_m_Page_049.txt
cb8fc2e27ee518e7b26d7a6b17ba5bbe
1690 F20101123_AABQUL kurdi_m_Page_034.txt
7f8b801fbf4af615112f67c2df3bcddf
6cb113c2437384e5dc322e0d04f54bb41b0fcab2
1743 F20101123_AABQTX kurdi_m_Page_018.txt
3cd4dcf6dbcfd7f503bda96cb958da35
F20101123_AABPRJ kurdi_m_Page_090thm.jpg
00b0eb75f61bdb19f800d7d863716087
32674425f33914f1bc27154bb5a84ee5824a76e2
1717 F20101123_AABPQV kurdi_m_Page_054.txt
5c3a334d255825df61f1ae9c4c0b328b
09b3cb4f251c5266946319fa90e723d54e406418
1879 F20101123_AABRAH kurdi_m_Page_201.txt
2270329228c214982b975f88a35cd6bf
65e17be729e276e19eb591b9771012f0fba54974
1592 F20101123_AABQVB kurdi_m_Page_050.txt
bde55d8f6ab9afdec12b2ea5029a1ff0
0ae7440e1711b13e3ed1555c78a244187219dd87
1827 F20101123_AABQUM kurdi_m_Page_035.txt
20c12441775e534168d5f8b0895d5a63
d5f79fbd1f5bcc7c9514e60096e2ca9458408e36
1799 F20101123_AABQTY kurdi_m_Page_019.txt
045210d2dc729cfccb65b4de90714669
988 F20101123_AABPRK kurdi_m_Page_115.txt
993240c0c8028e7dd8b9753c301198eb
c9520f9e4c72df70b9c5e24f824a7d83d766170c
5484 F20101123_AABPQW kurdi_m_Page_109thm.jpg
d18a7a26c1507896c661da079d1e6ed0
83ecd50266b16bf42ec19cb4a9177ed871f1b00c
858 F20101123_AABRAI kurdi_m_Page_202.txt
349b73e2a84693a1d48160d8e4fcaa31
dc666ac1f3f51a76e5367c5d98d25f9a6332d986
1884 F20101123_AABQVC kurdi_m_Page_051.txt
09c5ab5545c9829e78f71664a7e508e3637eabce
401 F20101123_AABQUN kurdi_m_Page_036.txt
5f96f9e09acb2564f7da7bef306c1cf1
8f04d4e1d215fba7219283f4b713b4cf2c9f2e26
2014 F20101123_AABQTZ kurdi_m_Page_021.txt
7f54da27cbd270c962efc7eb63d7fc26f28c8406
F20101123_AABPRL kurdi_m_Page_149.tif
570d05914c48867745bc32e4e19eee38
2bb2db53c73cc83de47e4b98b5aae55dacfc8490
3029 F20101123_AABPQX kurdi_m_Page_010.txt
22c67db06d5ee1161525efd1377d6b49
2013596 F20101123_AABRAJ kurdi_m.pdf
03b1e92a9fe82423d0638b7ffe7474b4
ea2515d6876324d702b25639a4ed763006baa061
1397 F20101123_AABQVD kurdi_m_Page_052.txt
dd93d93c5a2fb1a5c9392e3080cbca5d
e26f6c321492601a34b64846dcbbc8cfc395fb44
853 F20101123_AABQUO kurdi_m_Page_037.txt
844f85992a33fc6ffe4073d093694fe8
d22841e49194ee011a381253f744f6e223f753bc
25052 F20101123_AABPRM kurdi_m_Page_106.pro
37c10ecf9dacae54fcae5e456f7ed348
6602 F20101123_AABPQY kurdi_m_Page_172thm.jpg
a3cf693bf4304aab79f75af56d6bd66f
77399d99110823549d46c9395bff7207506e5685
7950 F20101123_AABRAK kurdi_m_Page_001.QC.jpg
bef4f94c0b5ed2c9751893428a4b97f8
43b0e6d9a51af81c81ce7e9ace66b8438f5f55a2
1708 F20101123_AABQVE kurdi_m_Page_053.txt
b73a799e9efb50b2fe8e06d62028a5f7
1762 F20101123_AABQUP kurdi_m_Page_038.txt
db855e705fa6b8e2aa2173bf882eaa84
F20101123_AABPRN kurdi_m_Page_156.tif
502a8e415053e5e7bf0098dcc39bab9e
7404 F20101123_AABPQZ kurdi_m_Page_166thm.jpg
0bc4dcd084beeacac148378df6f6b802
2b1a1d25257af1fb274387554717688baaa0ca65
26482 F20101123_AABPSC kurdi_m_Page_001.jpg
96c7dfaf27a7d8fe4f64e70dc37089b7
d41ca37764ff11799bb1d703eeb3084060dae21d
2606 F20101123_AABRAL kurdi_m_Page_001thm.jpg
7e6f85ac548a1fd34459589913647968
1643 F20101123_AABQVF kurdi_m_Page_055.txt
edd1bf351e977fa5f4f8af55e21dcb89
5e3dc653a4fb2e76638f96b465f7dbb88e6a074b
1759 F20101123_AABQUQ kurdi_m_Page_039.txt
7bf1d6cf9246c2f07645e081bb0e3b4e
7ab1b688473cb5e77e1175e20cf0399758780328
51497 F20101123_AABPRO kurdi_m_Page_157.pro
5acb58a6050caf8169ee24976264d52a
baae4efdc77870be40292c0395be46e17366c1ea
10440 F20101123_AABPSD kurdi_m_Page_002.jpg
f802f1279f26a1a71bc86a57f819d38c
60219fb46f67d4b4e36215085db6eb6866e3f180
7064 F20101123_AABRBA kurdi_m_Page_010thm.jpg
e330258bb8dc73baf3bfd82acedebe64
b51e5959fcc9c510c819a1c2a37cd5545fbaf8b0
3342 F20101123_AABRAM kurdi_m_Page_002.QC.jpg
77cf26af36df9730e75429db97da8999
d1c042fa02d6710f6e723fb4aeddac0b63e2ace4
931 F20101123_AABQVG kurdi_m_Page_056.txt
b67ec9fc31fe9385d4039c627f53cbba
1538 F20101123_AABQUR kurdi_m_Page_040.txt
85078 F20101123_AABPRP kurdi_m_Page_044.jp2
ee66095166bdcbbe37a966082b516f48
33cc84297de3b268aac2cf45d2350ace0f3582e0
9703 F20101123_AABPSE kurdi_m_Page_003.jpg
8a0201cedd6e5dbbf4f3aec920efd68a
83cccbca7e9f2ac46e68fde91059921a176c923f
24973 F20101123_AABRBB kurdi_m_Page_011.QC.jpg
825e94b71f659085b62082d065f7647e23bbaaf6
1438 F20101123_AABRAN kurdi_m_Page_002thm.jpg
24222cf86e38e114d45c699fdd37db9f
1914 F20101123_AABQVH kurdi_m_Page_058.txt
f2d1274960111e8421fef6a648788978
e96c89685f008fbb3a70285fbbd401fba61d8750
993 F20101123_AABQUS kurdi_m_Page_041.txt
1e73b688371c04c21d00be07112ae287
c1d5fe884dbe891a1e21c16a1f07e89b07298d07
836432 F20101123_AABPRQ kurdi_m_Page_104.jp2
9b88eea379656016b8bacf53e799c8c3
efe6703dc062f1167ea074334aeed2628c2885a4
45231 F20101123_AABPSF kurdi_m_Page_004.jpg
a50a052f74a8df886eba723caacb349c
73f7c2fc14c7d3ef5065a403cfa7de559638f3b9
6595 F20101123_AABRBC kurdi_m_Page_011thm.jpg
79666bcf20b4f88d55720cb0e486917a
3002 F20101123_AABRAO kurdi_m_Page_003.QC.jpg
05a90b9976aaff4b5783ffb56967ba66
e9dc08f9418f5ac71ebed4f0b0660720956908fa
1971 F20101123_AABQVI kurdi_m_Page_059.txt
821cc012b8607207561dea7137b7d5fa
1488cbbd66d3599ab6d95d9bfe20ab0869e64343
2001 F20101123_AABQUT kurdi_m_Page_042.txt
fe4f96c56857fa1b3dc861ce6244db7e
866416c643455008e3cb9ecaf83dabf5fa5126eb
13308 F20101123_AABPRR kurdi_m_Page_082.QC.jpg
114eee543276193ecc882088babd726d
1dd67bc90568c31de5b4cb9f2e8e9fb47f1c6829
71003 F20101123_AABPSG kurdi_m_Page_005.jpg
2b5d6ae3b14b997c8a6059df418ee7d0
8df8eb5e9ecdbb64f4406499752fddc9ff53c3ec
17919 F20101123_AABRBD kurdi_m_Page_012.QC.jpg
c72f80fd5016bfd53f09ee84501bf3f4
d0e4beb31d3e1f43d9eeaeacc0d8bffae6323137
4452 F20101123_AABRAP kurdi_m_Page_004thm.jpg
929 F20101123_AABQVJ kurdi_m_Page_060.txt
9ae84cda6930a03129dd5afe0cb9579a
4fdfeacced715cc4fed126fa292d86acd32c40f2
F20101123_AABQUU kurdi_m_Page_043.txt
d28e1b48dd535217363b7ce1f4e3b96e632e9f77
F20101123_AABPRS kurdi_m_Page_120.tif
7db18399ca2c08b03e7700ceda0eafdb
612fa90ff1ed0f130e943883fae2df6b5bdbd237
98135 F20101123_AABPSH kurdi_m_Page_006.jpg
61e7f583894036c2a704c290e613991d
174b8cfbea093d4643b2e1f19a705649585a1fe9
4697 F20101123_AABRBE kurdi_m_Page_012thm.jpg
01349e597139a30aa71223d9d9242ae55cf0225c
18445 F20101123_AABRAQ kurdi_m_Page_005.QC.jpg
bd5669c9a64fd3237110e507bb326101
894fa8b28f4183cf971d6772aa0b07c00ae3033a
448 F20101123_AABQVK kurdi_m_Page_061.txt
d1c7fccc0dba52019f327ced94eb2737
1e43f1276a58c15f86704d26edc04d4b6e289e36
1791 F20101123_AABQUV kurdi_m_Page_044.txt
32860 F20101123_AABPRT kurdi_m_Page_159.pro
37094d9b824723ee5eee3f6bdeef0fd8
1218899a539d915cb0e7dbc7ab8d1b4c4380f286
68542 F20101123_AABPSI kurdi_m_Page_007.jpg
d081667767b3f82a8e2eabab49b83119
589476283efaaf3857c78c24cb6b76b999273f75
13035 F20101123_AABRBF kurdi_m_Page_013.QC.jpg
9633d63e0c73d6768c803b014f46d9be
998c8d621fdb9aba71bb0047f7c12c655e514fef
4961 F20101123_AABRAR kurdi_m_Page_005thm.jpg
ba8e8c056dc1ce7676620b8d3d94748615c8d66e
1836 F20101123_AABQUW kurdi_m_Page_045.txt
3fbf00b15d08f1a612bec93257863912
469962c53e632937080cc462f262464ba3f8b8e2
1869 F20101123_AABPRU kurdi_m_Page_100.txt
032e32de43cdd5480e3d3dd2719f902fbc538bfc
13932 F20101123_AABRBG kurdi_m_Page_014.QC.jpg
152300e65171aee9fbce221d0bd4555e
6d5bc0b4757e246c125d69b22b6932d93c635507
24208 F20101123_AABRAS kurdi_m_Page_006.QC.jpg
29766e63770e2de70a49f33e5c3c1a5f
1195 F20101123_AABQWA kurdi_m_Page_077.txt
7e67a8bd2a6046f559e20b060a926567
a2aa53329dc86187b532ed6366740d9516a88467
670 F20101123_AABQVL kurdi_m_Page_062.txt
13e014e9c364aa4acfe9e6ee19c4806f
074f5235f6c8986dca88f27a185a46c27e16bede
1937 F20101123_AABQUX kurdi_m_Page_046.txt
09b772d535206e6cdb815d33f7537de4
6c05aae7cb8d9c89816a09d3347a698244ae87e0
53604 F20101123_AABPRV kurdi_m_Page_176.pro
310ba6a14d7bc7db93231f8b95504f23
da175c3081ea9e98f05f308c9b929ed76432efe7
63762 F20101123_AABPSJ kurdi_m_Page_008.jpg
91f2ac342c8f1e724d4ff12976a17083
6ce3f4ecfd61754a2bdecd45460b944cbe60d406
4203 F20101123_AABRBH kurdi_m_Page_014thm.jpg
4d27e485619e00cbdf86247dd5c3a07f
a916e2842b52eec93c9bd2a2f849594d358656ac
17840 F20101123_AABRAT kurdi_m_Page_007.QC.jpg
d86ab9e4e3ddcd07ea9a16ac2e285385
615500b7bd966a762f0aba09f5fdee0b0c8887b9
1509 F20101123_AABQWB kurdi_m_Page_078.txt
3c3b8b87c26c27c67a0b4fea1a5015d2
168da4d7b7c45335482fe30961941dbf4750395f
1350 F20101123_AABQVM kurdi_m_Page_063.txt
feec97b8e7128efd0c6aa7ec1f085bc0
28f9f16b1ae6ba30cf41a856d28fc8c30738b624
1829 F20101123_AABQUY kurdi_m_Page_047.txt
29456 F20101123_AABPRW kurdi_m_Page_148.QC.jpg
054e30d0072f46c37de1b69272e79d30
5663c27399b78cbbca3e848f5275703edde23e55
74581 F20101123_AABPSK kurdi_m_Page_009.jpg
c0be9d9d42850a1c23bc98f793754e48
8888 F20101123_AABRBI kurdi_m_Page_015.QC.jpg
622386ab7a1a53ff478b6c3e8d2c17ca
d8e4422b81c12173e0050a017cb986431208f058
4603 F20101123_AABRAU kurdi_m_Page_007thm.jpg
236dcfb7bbf01ae1431baac5fa48aacf
a1f4bc0612ae63ab37a14c084e062047ae6859ec
371 F20101123_AABQWC kurdi_m_Page_079.txt
f567ecfff180955dc6b34e013dc057c2
1906 F20101123_AABQVN kurdi_m_Page_064.txt
39dc1f14a89c6589db0c6ac6f918b0bc8bddce9b
837 F20101123_AABQUZ kurdi_m_Page_048.txt
9aae4309f1ff17a0f738c062ac6fba6b
30f46fc8519f6fb89e19bef9cea5a00a065eda91
514 F20101123_AABPRX kurdi_m_Page_193.txt
1d72f434c15ecf3e1bf30a9009813b5c35bce3a9
57882 F20101123_AABPTA kurdi_m_Page_025.jpg
439f194e600896273a791779808c4886
7cb4980e9921129cfd008515712da8422a5ea3ba
96583 F20101123_AABPSL kurdi_m_Page_010.jpg
283f79589e3313a171421887c0e5358fed8a8427
2794 F20101123_AABRBJ kurdi_m_Page_015thm.jpg
fa5bf7123d12f4d56d080d996d95a3fb
1726141ef7077e7baeb982cf16712a5a608ea1ea
1923 F20101123_AABQWD kurdi_m_Page_080.txt
9c3c2a3f876378896a49fdac984969fd
2096 F20101123_AABQVO kurdi_m_Page_065.txt
514167a4dd1da5c3a9779c5ff7ba3069
0a46bcfa5b07bab653539dcedda962904ff38735
16819 F20101123_AABPRY kurdi_m_Page_074.pro
fccd760d02fa3949d76a3edf61eba5ac
f0d569926c4dd26fd79e6735434f97742b973d2b
64110 F20101123_AABPTB kurdi_m_Page_026.jpg
27b44ac4113be92c6e15c8693cca34e7
0b2b0ef6430fecc1b4ca61ac8e1bf96b846b6cfa
86790 F20101123_AABPSM kurdi_m_Page_011.jpg
a355bd6f87309925e33fc50d17f188c5
c66fdd062b121f99bfff56b5ff34301fe52a233f
18318 F20101123_AABRBK kurdi_m_Page_016.QC.jpg
48b9b38b1f1f2281fc338927623e951f
0b15fb8e6f8794203349d586c6141ae743fb38a2
18433 F20101123_AABRAV kurdi_m_Page_008.QC.jpg
8e35ba1cb6bb1e2da44955d7e67259f6
fb3e8f56f8e8b154e53dfb849151efab8e67c2c4
1545 F20101123_AABQWE kurdi_m_Page_081.txt
6331381c64d9fcedb296438dec30a055
b9882b06e585f47faf894e992d1fe2f49cc6ea00
1337 F20101123_AABQVP kurdi_m_Page_066.txt
f705f16f68eb9b1d6fd85d51d44e7db0
73ccc5f8163160e6f41f7ddbea1cbb3cb9c46fcd
299329 F20101123_AABPRZ UFE0011626_00001.xml
76ed2d7f1db8a823775709729ea10fb3
ab172f512ab076795970117bf1526ffb9c4f73fb
60867 F20101123_AABPTC kurdi_m_Page_028.jpg
9a96427cb4cc30492fc16aafdef63eb9
849a3e6b5cacf0d900d4793af20bd6737b503909
62578 F20101123_AABPSN kurdi_m_Page_012.jpg
ca477259af65b07929e6a0df5e534ae7
3e875980fb0f9307e35262acd81e210a8d48c936
5226 F20101123_AABRBL kurdi_m_Page_016thm.jpg
3b92b6b73848af9cf014074267cdaefcb561150a
5071 F20101123_AABRAW kurdi_m_Page_008thm.jpg
c1938ceaeca71d55b732c1f753ef4e40
a556858d0eba063ea105f6b2e5ae5573433e8d51
944 F20101123_AABQWF kurdi_m_Page_082.txt
85085490a0aed7a9e08294891b678898
33b43b956f33857f071022583fcda0bd695e7409
1822 F20101123_AABQVQ kurdi_m_Page_067.txt
37dbb432d1c8bc95115037fcdd6b8f54
f1bef13d1f9abdd2a721e46a49229e17ce40a91d
72010 F20101123_AABPTD kurdi_m_Page_029.jpg
20c6e478d5067d609e5e144e72f6cfec
42703 F20101123_AABPSO kurdi_m_Page_013.jpg
144354642f2018b506933bf61a1e13f4
20356 F20101123_AABRCA kurdi_m_Page_026.QC.jpg
9cb799f2af80da1b36b71b2de0fd3f45
6b97c645f181200cf497e3871325a1682567b761
21530 F20101123_AABRBM kurdi_m_Page_017.QC.jpg
514bd7a07b1d07e66665b7c030b7a949
13ea790f35d29e6110b79969659d694d325ec573
21357 F20101123_AABRAX kurdi_m_Page_009.QC.jpg
a58964a090cdb7368970bed29b2a4c047a68f5ab
1209 F20101123_AABQWG kurdi_m_Page_083.txt
3d306ae1dc305e01e93fc3f64c50a598
f969e3ef80a0768956500828f766236419a3e3a8
1719 F20101123_AABQVR kurdi_m_Page_068.txt
9cdd0d25d31e1611153ce0c57183927b
45111 F20101123_AABPTE kurdi_m_Page_030.jpg
d5553be0726ae3d2ea08e79f22a88028
da7e0ac45aae2489c6620ff09fa261e74c176a6d
45986 F20101123_AABPSP kurdi_m_Page_014.jpg
40a2f550c4e5f90ae9411d6c3279aeff
5993 F20101123_AABRCB kurdi_m_Page_026thm.jpg
29e53e5431f0a228820ab5a33e1e66ce
179e2694945c66a9b8ecb8c6c1a510cc71312026
5969 F20101123_AABRBN kurdi_m_Page_017thm.jpg
29d0aeb4f4b720087817ddea32fd95ce
869b8573b10aaec173e0512f0d98d5b35d3de1e3
5801 F20101123_AABRAY kurdi_m_Page_009thm.jpg
b9b9416ba1847fe65fbf381c539a0d58
9df42a5522670bc682720315125ca3ff576973db
1379 F20101123_AABQWH kurdi_m_Page_084.txt
5b59be4ce6974f796736c1e746beebe3
f31118c67f56cfccdb33f8ef6e649fac24be9454
1896 F20101123_AABQVS kurdi_m_Page_069.txt
5a5f8d0259d7dc89223ba2a72b158937
de6292c52a9ca738df3978ca531f18bbbed277fd
54360 F20101123_AABPTF kurdi_m_Page_031.jpg
142ca79e33f15f72760a44d3565ea8a1
27665 F20101123_AABPSQ kurdi_m_Page_015.jpg
32d3b9a33ec0f2b5a9fca50d7ef01fc1
91fd13e0a1ea28677ea09b6b091fe5ca6f800796
18052 F20101123_AABRCC kurdi_m_Page_027.QC.jpg
e035a7c2e4627fb41145baa7616c5137
edbfddfa076c27098e10269836c4571bbc6660a7
19928 F20101123_AABRBO kurdi_m_Page_018.QC.jpg
2fca8f0981dd227ab89f172903d55dc0
583609a697f944592f5eb79ae3091ca995db9e87
26460 F20101123_AABRAZ kurdi_m_Page_010.QC.jpg
64378f2e07faa1962620a875364e91f4
0a01603a9d28bee176f794ecf7400a04ba23a713
1094 F20101123_AABQWI kurdi_m_Page_085.txt
735155e57847eab164b30a6c87e71ab9
de1ee2ed87c765d572c0d3743f81ef58ceaf6574
1446 F20101123_AABQVT kurdi_m_Page_070.txt
d9cea942b63ce29849ff4cc7dece0d39
a68cd4da51d88245aeab027ccab67a92ea88851b
58843 F20101123_AABPTG kurdi_m_Page_032.jpg
131847e6a43e5f1dc107a7cf390473ec
979bb7bc9c689013832eb556d7f8d12f6ebdd5b9
59537 F20101123_AABPSR kurdi_m_Page_016.jpg
d764fac0d285d1e0e999c552cb230bac
3a087d980547c3ed20b57e363833d095e7964d1a
5369 F20101123_AABRCD kurdi_m_Page_027thm.jpg
d00fa83b6b93febb84c2f013a5138506
870c53f81fbe2272ee904cf9a2a4cf5f1eaefbee
5487 F20101123_AABRBP kurdi_m_Page_018thm.jpg
9c266a1ac90eb596c38861fb10dca0c3
a1cb6aa8f8970c503d36cba3fba6d7b82e9868b0
1747 F20101123_AABQWJ kurdi_m_Page_086.txt
1c7287a226607beb4ceed280e93df317
737b8ed9c8e8daef23190840f4f29ff47a21929a
1308 F20101123_AABQVU kurdi_m_Page_071.txt
305c907300792011dbc78238fb8a0f63
5fe534c765be7d94a66e9724da28e504c93827e6
60805 F20101123_AABPTH kurdi_m_Page_033.jpg
7652c4748d581bb9d74d4093d63f6c47
65499 F20101123_AABPSS kurdi_m_Page_017.jpg
a730682c0743b6bd1e39034f92795d5b
3a5cecfc0c76a593fc73874cbc451215080b3bed
19504 F20101123_AABRCE kurdi_m_Page_028.QC.jpg
61aaa6f812e89755e16277bca4a31036
4982ed1de215e35f467a717acd550c198f0e7506
20847 F20101123_AABRBQ kurdi_m_Page_019.QC.jpg
cf50dbf6f2c42cca3f3aab31d976b5c4
d1dddcbc88fc5a52a1b52cccd3cfabddbb371934
2048 F20101123_AABQWK kurdi_m_Page_087.txt
b03bacc9d2c9c474916e0ee27c9e360d
eff8af34653c816af30164613b00c75592f64421
F20101123_AABQVV kurdi_m_Page_072.txt
19fac25e0f483e07fde8f41142a1f353
bdbe864b73aba334d7ed0de5c8e90a357a5acecf
62902 F20101123_AABPTI kurdi_m_Page_034.jpg
5a69638d511f1c6ce579befbfe8de134
2d3ea61dc1f325e730d9548444309c5d64399738
60973 F20101123_AABPST kurdi_m_Page_018.jpg
d1331a53bcb043e6077f9cc9a0077edb
8005dbb9e0fa5d1835d469511bd8225eb79c37bc
6108 F20101123_AABRCF kurdi_m_Page_028thm.jpg
dbe499542436c223eef9b147ecde0d51
ac49c892a46425bf5c7bb78a90e589800a13aa24
5891 F20101123_AABRBR kurdi_m_Page_019thm.jpg
586dbc901dab43801ca2bc4166b7841c
4a929ca5f4573ff5dfb3e50a527edd55c19aa59d
205 F20101123_AABQWL kurdi_m_Page_088.txt
b2bece5f7f882216f88af09e1e15d01c
edb41143d1123a2ec2c3c4a1daa710ce83795a6a
1013 F20101123_AABQVW kurdi_m_Page_073.txt
c9958d96b0541fe1f2856416d07f2636
e281cda03271f730e0634fa43c0cdc6d6a63fd46
68654 F20101123_AABPTJ kurdi_m_Page_035.jpg
6f7c106473aecc9c930e2d120201e33a
5cf84ca8b8993c9976d5c627329a71063f9e48c6
65517 F20101123_AABPSU kurdi_m_Page_019.jpg
cdcf1312830f652f0ee92796352bf506
0ea9d8177009b44dc2d4d590c0e33ca6960d61e4
23873 F20101123_AABRCG kurdi_m_Page_029.QC.jpg
f5104810bba3e509d3c7fefc6fee18d6
6409 F20101123_AABRBS kurdi_m_Page_020thm.jpg
888c2b6782691e12f342efa3e2e53ab2
bb5827bcc2f0bdc959c4aa0ecdd8cefbd7221ca6
1492 F20101123_AABQXA kurdi_m_Page_104.txt
622c1481e57ea47a7f1d9a9f3689eca1c6b8e401
791 F20101123_AABQVX kurdi_m_Page_074.txt
528f09c30fca51d147061963191587213742a22c
69634 F20101123_AABPSV kurdi_m_Page_020.jpg
f5136074912b683a9cd65a879f964ed1
18e32961b4c70f919a09bc6b9d36ce061a3a3676
6712 F20101123_AABRCH kurdi_m_Page_029thm.jpg
1f5044e7a84ddccd4b34d3bfd0493e8986bf7ee6
6489 F20101123_AABRBT kurdi_m_Page_021thm.jpg
ca7ee5351ae17c771360f0e22de237ea
3a94acab01f13d07843f9734c84f469bfb43dd6d
1222 F20101123_AABQXB kurdi_m_Page_106.txt
2abf90e08a74e459df13902e8b167da1
faf1a64e8c28b17d1b49ebf42939f3171d423405
1485 F20101123_AABQWM kurdi_m_Page_089.txt
a8510f05f17f2ece47c8923eb52c6cc7
4dcc1342149d2d54481f1aede77e33029ffabf24
1341 F20101123_AABQVY kurdi_m_Page_075.txt
d8bcdf02f5d735baa1330b8b3eae14c5
a7a14bf89a745179376cc365795745b51a10d67b
41428 F20101123_AABPTK kurdi_m_Page_037.jpg
43bfc64269ef1f43810bb800a05e9cbe
72354 F20101123_AABPSW kurdi_m_Page_021.jpg
a38023050964c834a75dcdeffe97493e
df2f2e86b77181dd6c0f5a3d40851f3bb233e34a
14782 F20101123_AABRCI kurdi_m_Page_030.QC.jpg
18f6ed7ea77c1d39c7216956ce850ef4
bbcd083a5f2f1abc9763e9381ec197c3b10007a7
6245 F20101123_AABRBU kurdi_m_Page_022thm.jpg
8fed84cddda427a217fa5cf05d61fe53
87307884c06cafa334f59e331e1dd7f722ecafa8
1115 F20101123_AABQXC kurdi_m_Page_108.txt
3b7647e0374da51c60ed208c3d603f82
1873 F20101123_AABQWN kurdi_m_Page_090.txt
a0d27a5f86da5c289f9f75eecf65f366
8e845fb4d304132609fcbba2afbcf4af72deca0a
1734 F20101123_AABQVZ kurdi_m_Page_076.txt
179c24043fcd37c2342bd8ae917a1108
8bbf0a0c541a2caf0488d16fe487ee2c8511e1d1
71654 F20101123_AABPUA kurdi_m_Page_055.jpg
3bd794641ef13a1c1035df1b21c2598c
713aae88ac8482062e420c236deaf006d4ab506c
64264 F20101123_AABPTL kurdi_m_Page_038.jpg
67c55a92b030ccc4262a6e83453193b0
aef3ff6858fb3de586105bf378eed2a03cd17efe
68757 F20101123_AABPSX kurdi_m_Page_022.jpg
c734b7efaefb456cdb4b070a5cbe7c72
d84957cfa53aae86e6e79360a1351faf4c7506cb
18316 F20101123_AABRCJ kurdi_m_Page_031.QC.jpg
cd2e9d37f266e2e770dc9e9caa5bbc6472511078
5851 F20101123_AABRBV kurdi_m_Page_023thm.jpg
1369dc1cd6e4e96d45c99dbe9aaec015
a071af0daa8cef00f6169c16a46cd0ecf9c0af55
F20101123_AABQXD kurdi_m_Page_109.txt
4441fa846fd153c8fbf57c7429aff911c52f2df7
1724 F20101123_AABQWO kurdi_m_Page_091.txt
43c124e83e6ea49a2c259968cd75a795
b554688769aff3c23c25e7b4e1dd51752a0c15f1
69636 F20101123_AABPUB kurdi_m_Page_057.jpg
ca8521bf8f6f863c42fd9780d1fc0cdf4ab5eacf
57455 F20101123_AABPTM kurdi_m_Page_039.jpg
ca31e2309b5504869ffc1fb73476a1962682f056
64626 F20101123_AABPSY kurdi_m_Page_023.jpg
1d218024361b9198b9f937c12de38e6d
9453226c71c6f2af773bde88eaebb76c53738d94
5631 F20101123_AABRCK kurdi_m_Page_031thm.jpg
8f6b5474c29d279118e705e9d01eac1f
2740 F20101123_AABQXE kurdi_m_Page_110.txt
47c9271ced10597aedba72227fbd1159
fb173112232fd76d1668ef7b7b498f32c1d391d8
1411 F20101123_AABQWP kurdi_m_Page_092.txt
8e3ae09bd8dd6a738baa1714f9d6c3364d4a6163
71531 F20101123_AABPUC kurdi_m_Page_059.jpg
c5719fba70b7d27a49d2a9fdff26c15a
db46585c0eccf54a8a734d0aa3992325879cdb38
57796 F20101123_AABPTN kurdi_m_Page_040.jpg
242de0dcdf11f4c2b5dc7a65ea55a617
9aa7f4da7a5d76c89812ab9e2c59b3a1b5a9bc67
26897 F20101123_AABPSZ kurdi_m_Page_024.jpg
75426f861ed3ed15b1b5ef132ede6193
0d196ee869c3f4cde5b499845bc6597858b7d66b
18790 F20101123_AABRCL kurdi_m_Page_032.QC.jpg
a392b240718e79d85a9ca455f789204f
8961 F20101123_AABRBW kurdi_m_Page_024.QC.jpg
23ac4e23c3b4575d7f631dc883659e07
b6cbf9a9d92a65e359646e479e7195e757269dd3
2028 F20101123_AABQXF kurdi_m_Page_111.txt
b6b436ee94f103b8812f3caa69c2e32d
da1a21258d40557afda7dc65dc59228e8ce6d118
977 F20101123_AABQWQ kurdi_m_Page_093.txt
95fe48f86baced9ab902b398d5a7f986
af5bd2f3a855cc0e730bba2b70d3b71669b575e0
41103 F20101123_AABPUD kurdi_m_Page_060.jpg
8b014800707242846d3365f46bb467eb
3041c5cd116160224654d61097d60c5947b71a14
43661 F20101123_AABPTO kurdi_m_Page_041.jpg
2becc8f1b069686fd5b5505dc1b87e3b
144f5a075d5360c7aee864d7263bd81109f5bfdf
F20101123_AABRDA kurdi_m_Page_039thm.jpg
69e30f31c476dd1f1df7c12065cb5f58
5712 F20101123_AABRCM kurdi_m_Page_032thm.jpg
10ac09ce91c89194c4a5a786120bc2ac
6650d216fed67cbd952937062ea0d5f0901a6ffb
2821 F20101123_AABRBX kurdi_m_Page_024thm.jpg
4012f241cbea233d8cd0aef8cff10250
26fb9338008aacbfe53d740ce8e8d52f45c97175
1501 F20101123_AABQXG kurdi_m_Page_112.txt
284bf859ce372459687ef4ab7657c99b
114522446b786aa846df7e3a412f48136528bd0f
1231 F20101123_AABQWR kurdi_m_Page_094.txt
d618f52ac016af5722d1c708eae28bb5d51c5a5d
28534 F20101123_AABPUE kurdi_m_Page_061.jpg
3e30f27859618e8ee3dbeb35797c975c
4fd79d0fd8ea1d81cfca04f39d51043347148ae7
64422 F20101123_AABPTP kurdi_m_Page_042.jpg
f4c787e67c5d5505bc20aef9313506d3
44cb15b9c24c5db0a2c3501b134594419ecf8d7a
19534 F20101123_AABRDB kurdi_m_Page_040.QC.jpg
d2a0f7638e359a14890675d6520ce02f
848897780d925f3f982e972e0fbc96377c25dbc1
19879 F20101123_AABRCN kurdi_m_Page_033.QC.jpg
55ed5f5fa1ee6fd591938642206637b0
0cd6ff43262e18e8a91b9e0649f8ab6488a06207
18495 F20101123_AABRBY kurdi_m_Page_025.QC.jpg
2726498e37c663dd1ef4ea392abe7991
47702e21a678c5b78034df0707081553c58595db
1985 F20101123_AABQXH kurdi_m_Page_113.txt
8f99a0c6f31161309078df76148f95f8
86c57c8eaab19e9cf8585a61725f752a9e32d777
1227 F20101123_AABQWS kurdi_m_Page_095.txt
47d1e02ea3da7b78dc9f873df58ba44c
5b35d6d3595f38a341fce5bfd63b5efbc178ba74
50502 F20101123_AABPUF kurdi_m_Page_062.jpg
d0ca9bf0fc9b46e7e0ee29fd96efa40f
e41036e1369c6f26e34956376924b199f4cfd01a
56802 F20101123_AABPTQ kurdi_m_Page_044.jpg
3977264227fc3609e7228afe4b81c856
5387 F20101123_AABRDC kurdi_m_Page_040thm.jpg
3f16c720d6c1b6a4bda31bebfcab4312
985099fb8204252d437b0e8469b55a9c9642a56b
5830 F20101123_AABRCO kurdi_m_Page_033thm.jpg
09485b337b1441e6fddf5c4f1daed0a8
52a375fdc127794b7b57ddb2be609f0124f8b723
5521 F20101123_AABRBZ kurdi_m_Page_025thm.jpg
2ea5bc2934f6c0b0e8fa3a7303a38a6799dfc9b3
1138 F20101123_AABQXI kurdi_m_Page_114.txt
ff37d68e3c755c3d9653583d51444ba6
0164f0cdea623316402503d571d0d7f9404ec5c4
816 F20101123_AABQWT kurdi_m_Page_096.txt
a659c6663b47a0e1d29bc2d3d0087e10
49442 F20101123_AABPUG kurdi_m_Page_063.jpg
dba9ba6a04dbde4d160e0bf732a9fd3c
dba04f72f6c6b1aa5920e15bc39206148dd72212
62892 F20101123_AABPTR kurdi_m_Page_045.jpg
eaf80ca5d3aca7af6691688338bd7dcc
91077 F20101123_AABQAA kurdi_m_Page_034.jp2
cf7446ffcee031f7760ea009610fb9a5
af5493ffc3529c4931e73edb8001929496046434
14347 F20101123_AABRDD kurdi_m_Page_041.QC.jpg
ba52d29191db50f900c6919e3eda270b
4a04d9e314040066e52be2735469c72785ddb1c0
19912 F20101123_AABRCP kurdi_m_Page_034.QC.jpg
9bf067bbd6cd485c1b79e0e5f939692f
dc077ddd49142e6eaf5316f014efbd66eb742a44
1247 F20101123_AABQXJ kurdi_m_Page_116.txt
d38c36d01071b64caf7d157519111b84
8c85e7b9bd099bdf8b405a5748c59e41d868c27f
1531 F20101123_AABQWU kurdi_m_Page_097.txt
f5ef8fa12f1d3f9cb44d3269b960a5d8
2f4fb1b3e83d6008ba3a457ac6c7b0164aa38c07
66240 F20101123_AABPUH kurdi_m_Page_064.jpg
e9ddb6b1180f09e3f033fa79276dc52f0db91f24
56790 F20101123_AABPTS kurdi_m_Page_046.jpg
eea2334d2e2bb60e5ac9c9f3644b9b08
54be1420af8fb4a04e31038f93fbe14f03568335
98771 F20101123_AABQAB kurdi_m_Page_035.jp2
39cb13649f7c6893bcfd2235fa9893f8
4158 F20101123_AABRDE kurdi_m_Page_041thm.jpg
c96847d3e55501c2b083fde77dc14615
5798a5c998a999776bf7aa62099ac1a752a6fcc5
5911 F20101123_AABRCQ kurdi_m_Page_034thm.jpg
029a50a8f4079fe7b405c459d10ba6a180ccf358
1183 F20101123_AABQXK kurdi_m_Page_117.txt
f1200134381aace8b26c159330a57e02
77386b89f0a01f4ced8a00302485506a180c5cff
F20101123_AABQWV kurdi_m_Page_098.txt
89698c26fdbfdd7e7404ed1f6dd05129
7b1b006cd4b852244efc8a0c090b27709e03583d
74033 F20101123_AABPUI kurdi_m_Page_065.jpg
1bbea9a30672f734575b31bda96e7f481f2b7a35
66232 F20101123_AABPTT kurdi_m_Page_047.jpg
ec1e56f6eff9480d7c149b325395ac0f
50ca71904f44d2df0fecef0dbfb4470e4805f46b
22284 F20101123_AABQAC kurdi_m_Page_036.jp2
15fea755b27edffa75b1eb5d617577ed
fe985c2ab6cf961d5f4262e30a2dd0d752220105
19576 F20101123_AABRDF kurdi_m_Page_042.QC.jpg
ac8b52a9826fe237de10d849b0b4abf3
e11b00986e1bdbe2d82b342c4898958ec396a141
22860 F20101123_AABRCR kurdi_m_Page_035.QC.jpg
5708aba39e6f327ed10000bd7d32b739
ef96f28479fc2a2c3fb134be939153f976dfd033
597 F20101123_AABQXL kurdi_m_Page_119.txt
1726 F20101123_AABQWW kurdi_m_Page_099.txt
afe186c70675c2ba1edb376903a6d9f2
a314dfd490b8e7f38116c9ef019f7a084123169c
55434 F20101123_AABPUJ kurdi_m_Page_066.jpg
72eedd239de0a629d0788f33ce04381b
c04663160aedd59192569d35524d2bde9cb228e6
53974 F20101123_AABPTU kurdi_m_Page_049.jpg
73a39f953b5699c4a7bda569ca772dd2
3f57ecf342406f42a9396d2e2473dca14868857e
3eee06b24886bbfd409a80662705ab4472abdba0
5619 F20101123_AABRDG kurdi_m_Page_042thm.jpg
f13e9a855a716eda3028e77ac14d9ff1
6210 F20101123_AABRCS kurdi_m_Page_035thm.jpg
437d17bb5303392554bcba421951d6c5
435e35f639e8b39ed51ee77151a76b26c6df10c5
1594 F20101123_AABQYA kurdi_m_Page_135.txt
58b456f3aaab143ee36a37a1144e7cb6
06397e63e4398cbf34760f80280a6802d00963e2
2493 F20101123_AABQXM kurdi_m_Page_120.txt
a04f7a7286e2a354026f92ce2a634db9
1744 F20101123_AABQWX kurdi_m_Page_101.txt
8385b2ab8a95d81619e176586ecd8b28
721eb6221bd8ce21df5033e9efd1bb6a15a37189
54022 F20101123_AABPUK kurdi_m_Page_068.jpg
e37d77645d42d634c778caa225632201
5f7579bb42b17f9aac81ee9bee0329b5b02341f5
51123 F20101123_AABPTV kurdi_m_Page_050.jpg
24b3c5de01dcdc4bbc706160593b23f8
79284474f1fa916a8598d050cae2f3e113a3f97a
85040 F20101123_AABQAE kurdi_m_Page_039.jp2
78991b73dab648dfe06ddfd820e91c35
2659ea1409f0bff4c6c41dc9bcab20dfa63d343d
19905 F20101123_AABRDH kurdi_m_Page_043.QC.jpg
35656ab6c4ce5c88d90e5746a8411b4a
547f997d888c1206222034e9a465ce7c17c142c8
6878 F20101123_AABRCT kurdi_m_Page_036.QC.jpg
7e0f976ccacd87d8d3d6c8c15cb16d6b
dca601dcc4215b8ea7023dc9aba480c9c84eeecc
2046 F20101123_AABQYB kurdi_m_Page_136.txt
c313e8f4694750635073acb1b00ec962
6e676f72638422b246a236834bd958a26423da37
2109 F20101123_AABQWY kurdi_m_Page_102.txt
4c8e7903548f727bbe1ca427e82f4110
002ac71435f795a153e057ea5cb3030cf472794a
71700 F20101123_AABPTW kurdi_m_Page_051.jpg
e0ea38f8e809508408f52d105fb2c32c
aefc7e0834edd9277e1bcec6554aede27ff9e72f
83480 F20101123_AABQAF kurdi_m_Page_040.jp2
b6ccfe6b7e921d906ec60cee5ae5d23c
5644 F20101123_AABRDI kurdi_m_Page_043thm.jpg
23d541a82c38caf4c0d05c4e6c7a6eed
ee68bca7bec479466e19e9239934292bd51104fb
2220 F20101123_AABRCU kurdi_m_Page_036thm.jpg
8bc3a13840ecb94aa303e98d707113ac
2643 F20101123_AABQYC kurdi_m_Page_137.txt
064439e702f24110caae167e22b83f9b
1f95ceda07d52ffd4ec71e99ee98c8132ecc107e
252 F20101123_AABQXN kurdi_m_Page_122.txt
0394e718bcd539cb19057d654d043736
cf5fcc3448afb655b11d17bf60554fa7873e376b
721 F20101123_AABQWZ kurdi_m_Page_103.txt
08f34f32050f76788b46ac238184284549623367
52477 F20101123_AABPTX kurdi_m_Page_052.jpg
503bf7c9d591ea38846e212f8771222c
198a4e3fe50152167a91b7ac2b42a775ccd2c5e8
630984 F20101123_AABQAG kurdi_m_Page_041.jp2
af9547788ac69f9bea46d24439479b1b90b5f9cd
64350 F20101123_AABPVA kurdi_m_Page_084.jpg
3b58fecd4493c482a35d5395336bdc90
b547388c1ffee18f29b1f0eb38e68290c9a6da20
60744 F20101123_AABPUL kurdi_m_Page_069.jpg
4999217c22401177555ba0dbbc531faa
90008384ba6628ed885d8a9426b839345be1f3c5
18623 F20101123_AABRDJ kurdi_m_Page_044.QC.jpg
49e3d8f92889d9c039a61e1482db995d
13459 F20101123_AABRCV kurdi_m_Page_037.QC.jpg
ed48e912a09f55d30904d89f8c427885
313c3d83c800de8fedd54c581fbd27998fb29365
2707 F20101123_AABQYD kurdi_m_Page_138.txt
cf10445fa26d68eab2c0d138546bd334
f5c82d155dba87178e1d5fa4257d36268d73343f
558 F20101123_AABQXO kurdi_m_Page_123.txt
82ca034222e9b6bf5f523695dd79a3e1
067031772081e1df9f68bac1bc06310a7b4278ff
59898 F20101123_AABPTY kurdi_m_Page_053.jpg
f794cb01e0932125e7a4e9b4cc318cc5
0698bf872d5cd808927f19350744408800abd77e
972696 F20101123_AABQAH kurdi_m_Page_042.jp2
ecd84c1c9d3ff28dc8744886c5803604
2301ed379767cce2a235f87a8122c95ddb2e5caa
50240 F20101123_AABPVB kurdi_m_Page_085.jpg
6a00407629168c402462f194e7c2ff63
38fa2970aa31c8709827797e1c0c49c38ecd44ce
55068 F20101123_AABPUM kurdi_m_Page_070.jpg
ca1cb6ab239dd1bfea62c1c35c86c1cf
8f3854675fd886c4307d9b888220b200a5e0b304
5528 F20101123_AABRDK kurdi_m_Page_044thm.jpg
a81e409dd02b003a38c7816a646f3320
ab0214e25b90274e247a8b20b1135a20f96476e4
4046 F20101123_AABRCW kurdi_m_Page_037thm.jpg
14ff1cca8156deaf5fc4c4f747039031
f513bb859164276382760a318278b545e4ceb929
2848 F20101123_AABQYE kurdi_m_Page_139.txt
e509eff8cb75532bbd68375a3a50acb3
a32a33177a3bb73c56fb75e03ec9e2194ab521c1
1704 F20101123_AABQXP kurdi_m_Page_124.txt
991cec92c74d97976968b9b51e33e8ba
62137 F20101123_AABPTZ kurdi_m_Page_054.jpg
0c33d5d72c1c32306402d79a174800a5
3473412ff70d84a2a39e1a7020c7a5e5ba9b8371
92894 F20101123_AABQAI kurdi_m_Page_043.jp2
029a3e9d49009ef5b280bf196e60182e
b8184cc627667ab1aa20851dcb10c45c46f45f1d
50629 F20101123_AABPVC kurdi_m_Page_086.jpg
6fc87b3dfdd3914efa7b96c1abdcb901
e653c13a11417d2286979c63d79c758254034214
39756 F20101123_AABPUN kurdi_m_Page_071.jpg
15e80d284e1df3a110e076d090e17a77
604d290da02a22758590aac573fee2679c56ea86
19821 F20101123_AABRDL kurdi_m_Page_045.QC.jpg
663434442ca7d7e8a33ab21fe53bf26eaf151fcc
2711 F20101123_AABQYF kurdi_m_Page_140.txt
e265a00655164a7a1c5ff1436995a953
607e84bca78541f1e1a5ff866ea5f0401bfc98ff
1883 F20101123_AABQXQ kurdi_m_Page_125.txt
9618b12c855a287ee3857c9011540d1e
113047e7b34e32659c845c07e1c4d3ab4b274090
85100 F20101123_AABQAJ kurdi_m_Page_046.jp2
e2c43d2885c1edc3627100144d0bf2dc
1c46a61a3a4866e18e481f502d19ec1a17d6686a
70744 F20101123_AABPVD kurdi_m_Page_087.jpg
dd0010975c84da15b0122763f2265579
d685ccc8f48ef5f4e2f61038cd18a9a27737784d
46712 F20101123_AABPUO kurdi_m_Page_072.jpg
18a168142f6190204e9e81fe0ace14f4476aa9d1
19309 F20101123_AABREA kurdi_m_Page_054.QC.jpg
ee4ed353ca598a8f67921b376c706bb7
77e042f03f65cecd926c689ac8c68919f694d1d4
5774 F20101123_AABRDM kurdi_m_Page_045thm.jpg
65331c9c484a6ca6a6225a60387dff1f
0a28c41de2a6e06c4dd569794377e154e9eb5d65
20717 F20101123_AABRCX kurdi_m_Page_038.QC.jpg
9d4a3f3ccdddf61e01e26716a97561a9
16c5a574627f25f039a1d74abbf345839af53901
2696 F20101123_AABQYG kurdi_m_Page_141.txt
a65d9aef63ebfdf8c6f3e1c836fdb861
1595 F20101123_AABQXR kurdi_m_Page_126.txt
1bd383ff7caa47b8533f3a3071b5d5f80a0ca29d
92695 F20101123_AABQAK kurdi_m_Page_047.jp2
518b6d77a15a2b32d796cd7722c0aefd
04cc1ee793e60b207980cb657051626a614819b1
13392 F20101123_AABPVE kurdi_m_Page_088.jpg
ec29ffffe003e713f435345ed8a180e8
2485287213ec7451968b7f4c3154c153f73f915c
54480 F20101123_AABPUP kurdi_m_Page_073.jpg
eb7ed32fd1a66a896debb25e9aceb9f6
7f1d4f2edf6c1fddff02d9abce1de2911410d157
5469 F20101123_AABREB kurdi_m_Page_054thm.jpg
237e263122f9b240ac0c5e92fe3a6099
e525ed305af1f3767f2a2546b151dcbe1734166f
5191 F20101123_AABRDN kurdi_m_Page_046thm.jpg
12732ab0abd457e16de30c821055ee635d176e18
6151 F20101123_AABRCY kurdi_m_Page_038thm.jpg
33427cc211bfb8e3d8801e091a2d701c
89b1a79db719e6060041c41c570fb192b29e4ac6
2720 F20101123_AABQYH kurdi_m_Page_142.txt
2b848cfd6c0ca51ac6bcb12d84fa9ea9
1031 F20101123_AABQXS kurdi_m_Page_127.txt
d20d33e91429e38fc55ce2cc597d1a1a
00284b79a35991ec741027f89ffeb2b4377b5367
608183 F20101123_AABQAL kurdi_m_Page_048.jp2
4f90588db358ec6dd1228439f322c72b
0871f7ca10dfa8b09f9bddcfc2ca7aec5a25b4d7
48518 F20101123_AABPVF kurdi_m_Page_089.jpg
99c9b16190b3385038506535206d4f5e
03621c3c85d0e16dacd6b0720b3e7da007beca47
40307 F20101123_AABPUQ kurdi_m_Page_074.jpg
a19e01f84e4db84effcbb53a63a9ac12656bb8b2
5828 F20101123_AABREC kurdi_m_Page_055thm.jpg
e0cf902162a1d7b22f996f573ea51ccd
9f410494b004bac4be93fa84619fc345be1e64b1
20495 F20101123_AABRDO kurdi_m_Page_047.QC.jpg
a874bb0ea0fe411f8fb09d2892872b2f
889cbe8d68f552c11b347b620320cd0310a6da9b
20098 F20101123_AABRCZ kurdi_m_Page_039.QC.jpg
342cd3755570b8b99a71853f687d1410
2251 F20101123_AABQYI kurdi_m_Page_143.txt
64b29a775a308553828313071ca6d38f
72a1a866ddc3f14b1babf2b6b6684d2ca5f051c8
F20101123_AABQXT kurdi_m_Page_128.txt
a98ae1d68966fd518449bcc108a28eaf
112e931a4b4e9bf77851f6f64b6c5a25577caf3a
112060 F20101123_AABQBA kurdi_m_Page_065.jp2
be9dbaee0d62eefce37af88089a721e7
79324 F20101123_AABQAM kurdi_m_Page_049.jp2
12485bfe9a1e93d264ce3ce2a67dd32b
94e875372d37a710910a0274e4470f66b017c0ef
45335 F20101123_AABPVG kurdi_m_Page_090.jpg
80fe40023cb2bbdb8b2169f149e980897497a6ac
52694 F20101123_AABPUR kurdi_m_Page_075.jpg
d6d41dca12e781a0ac5fb2ef26e7f2b3
c03b6fbc0ce85e38e95222963a5a4b7f53379068
15119 F20101123_AABRED kurdi_m_Page_056.QC.jpg
50f2b230f343f6dff9e06af9099b0fc5
c50aeae00e71f077c9045a21fda84f1a4bd5c612
6020 F20101123_AABRDP kurdi_m_Page_047thm.jpg
3fcf919472bd15521566dc3b64770ba4
1766 F20101123_AABQYJ kurdi_m_Page_145.txt
635 F20101123_AABQXU kurdi_m_Page_129.txt
e490923732d8f42235f7be7eb6417f03
223f1f78c115d0acd77c196ff9c86a6835e044c6
679044 F20101123_AABQBB kurdi_m_Page_066.jp2
566e80975ecbac3a84fc574e51a6d9c4
6a63d2c10cbdcd0c90150bfe816c3fb2e058a84f
745413 F20101123_AABQAN kurdi_m_Page_050.jp2
177a083e6864bfab35249ba31fcf3a3c
c791819b105cd75d3ffcf82eed3cfc49785a7e40
45676 F20101123_AABPVH kurdi_m_Page_091.jpg
338fc3a88740778c07efe171dc322126
bb2c02c655955f5a5fa5f7c2000ae284826f0fb7
58555 F20101123_AABPUS kurdi_m_Page_076.jpg
2e6957725e65115f9c2ecee3151cc9fc
6a646ae6e4635aa0ee44dfe671c5a9d3456280df
4527 F20101123_AABREE kurdi_m_Page_056thm.jpg
88620cb2fd31113a8ea3daaf57086ee29481c19c
13241 F20101123_AABRDQ kurdi_m_Page_048.QC.jpg
ffffa60e962ddfc4eb673870390c5fa3
b10eddbc2b731a55938e41b382a9180846524d94
1989 F20101123_AABQYK kurdi_m_Page_146.txt
eac1016ef000413aac9942ff9a349668
f9e8763f5366e6dda28bfc104c60be4dcff53340
790 F20101123_AABQXV kurdi_m_Page_130.txt
05eecd7741358684b7da5bf51ebf6a36
2770984b32c401ffa66dc5090e3fe7c3fe13dbe1
96948 F20101123_AABQBC kurdi_m_Page_067.jp2
8417e842fb2ec65464ab519b701b022f
2b121d05bf23072ae85fdcd2015f65ac83abdf61
1051804 F20101123_AABQAO kurdi_m_Page_051.jp2
64f9145fe3f05e1f98426730b08ef28a
dbf9f9357cabcccbf4ea1f67668f1fa156787ed5
56835 F20101123_AABPVI kurdi_m_Page_093.jpg
7afcf84dc8986dd27fc40634c8bdb699
759e60f9c5e952fff35d97157dbc8ea13a1752cb
43785 F20101123_AABPUT kurdi_m_Page_077.jpg
59250b113ec1fa2e8213766b1c10cef1
dedea8d6c824f3745d585e9e1627347a57db6895
5185 F20101123_AABREF kurdi_m_Page_057thm.jpg
9afb42f2af315de44b706fbbfb4ba1be
3907 F20101123_AABRDR kurdi_m_Page_048thm.jpg
8431c97b9c3960707986deb1b243fc28
2236 F20101123_AABQYL kurdi_m_Page_147.txt
bced2c4c1ee5b37a565eca0e619630f0
a11171fe6fc53ca513cc59103927118f9e878bb0
1455 F20101123_AABQXW kurdi_m_Page_131.txt
b8007852f143fc7b19ae10765b07dbf6
91695e8da876dc9ebf1ffdd33e2d1ccf81ef8cf0
77503 F20101123_AABQBD kurdi_m_Page_068.jp2
813492bfa249c81652f2f7477a7e59cc
c67902dc08f5db68aa497a3e294e1033e49c2be1
73825 F20101123_AABQAP kurdi_m_Page_052.jp2
3cc614c0f2c786ea38f41c06ccc5fc20
3d36a190d8707493854934b948b437e87176b51e
50029 F20101123_AABPVJ kurdi_m_Page_094.jpg
b4b1e14f6196e38a86fed89dafce58fe
493b9e03477cf65d81959f05327cc115fe0b8136
56413 F20101123_AABPUU kurdi_m_Page_078.jpg
6a7faea0a13dac37b0f2bda8737f5857
e6f32bc218e383073b89fa63ecf771a515854bea
19492 F20101123_AABREG kurdi_m_Page_058.QC.jpg
dfd1ec86f9d9696dfffdc2836145d0e00bbc3551
16982 F20101123_AABRDS kurdi_m_Page_049.QC.jpg
28817df112128631672b57df38240f04
1821 F20101123_AABQZA kurdi_m_Page_164.txt
600150214cca7ed9fff8a42e6aa6fda6819df71c
2764 F20101123_AABQYM kurdi_m_Page_148.txt
a4f1ff64fa1144496f52c81872df738b
464d50f49ff9c9309cb0de894b0a41c95e04c24a
1289 F20101123_AABQXX kurdi_m_Page_132.txt
8dcc35e0dbfdc66f4ca87627af43745f1a608779
91298 F20101123_AABQBE kurdi_m_Page_069.jp2
b31c9631e78d2b4456910f43fda7e3f5
4aa5e425d23e1f8ff4590203fea46d9c9fda97a6
806793 F20101123_AABQAQ kurdi_m_Page_053.jp2
066dd4d217b037f5487cf756524840c4
47336 F20101123_AABPVK kurdi_m_Page_095.jpg
a6193a0a8559ba84a98e9897282e770d
81f6a4a24c6a04c7fdbcc0a2bfd12fcf3dd6c15a
35375 F20101123_AABPUV kurdi_m_Page_079.jpg
fd1159977f2d0072a94758860e21f21a
b3b2209151e4be1a2c0b78df8d4c8fabf10680d7
5723 F20101123_AABREH kurdi_m_Page_058thm.jpg
c423f40fc96bc46efeb95ab9a2caf8bd
5037 F20101123_AABRDT kurdi_m_Page_049thm.jpg
a12912389fce8bb67cde411b66d2f8fd6eb579a0
2662 F20101123_AABQZB kurdi_m_Page_166.txt
18a162615d492bda813fdaf1e78e7670
f9b6128cd4d6e7e148847f0c005e34a9c2da942c
2670 F20101123_AABQYN kurdi_m_Page_149.txt
f85eb30e98cc19ee2397182636b8865c
0839e0ebfe0203fae448d4749478fc1b44ae9da1
1921 F20101123_AABQXY kurdi_m_Page_133.txt
695352113635b918cbe402cbb57c75db
105a5e551c297db10033b300f92b6af7c87fc59c
581662 F20101123_AABQBF kurdi_m_Page_071.jp2
d7cf7ba58b5f66446366a80d16618b4e
2af94907f44734ebe65304365304ddc33fb12fa5
1051944 F20101123_AABQAR kurdi_m_Page_055.jp2
401fab48ba7c8e8db6124ce718275433
3eb96404ff68be5da62ec53202bb3383dbaeb3ee
49991 F20101123_AABPVL kurdi_m_Page_096.jpg
f4a9e2415e4beba7d363bb1d56033665
fcabeb02761bc1c6547cc5004676bc25ede2b7a2
63247 F20101123_AABPUW kurdi_m_Page_080.jpg
457bd40dde5c3d576b7c18ce9f7208a5
e5dbbdea7cbe325a17a6942c996c04e700da223c
23100 F20101123_AABREI kurdi_m_Page_059.QC.jpg
5207764e51b9b2170d41c0bbee27e83f
806a998eefc519ea23735d91ab5d230864e81679
16193 F20101123_AABRDU kurdi_m_Page_050.QC.jpg
5fc3ff320fd5cdabf5a8862aeb4827e1
f8204d8f7a85af74e22ebeb4b5f967e28335f564
2625 F20101123_AABQZC kurdi_m_Page_167.txt
7be8735a87bd8a995f6a1d3000ee9cfc
f32ed3c7a4d9fb38376ec5870512ba452933d68f
1581 F20101123_AABQXZ kurdi_m_Page_134.txt
974e134911e3c5eb3dd6c4d4c5d190cf
0f6034b089b2d04fc2bf9719222f55bf5afc1871
712301 F20101123_AABQBG kurdi_m_Page_072.jp2
0f92015e300cca52ef1ac8c73c00d621
046c24a8214b4933269e183b1c212989340c3670
43794 F20101123_AABPWA kurdi_m_Page_114.jpg
6077b800d38ac241f0e24ffd5f62438a
95c0fb3a34a46693b3e53ed9708596c070787f81
1051941 F20101123_AABQAS kurdi_m_Page_056.jp2
7ddd7230d0a6484c0c3807e01e60146c075f17cb
57758 F20101123_AABPUX kurdi_m_Page_081.jpg
8bf581b89e116867fb351a9d3a53d65a
6474 F20101123_AABREJ kurdi_m_Page_059thm.jpg
6924e9deed219ac0ed9e58f05841b8df
3ccd0d3010fdcf948e53e1727f11458be938c60a
5964 F20101123_AABRDV kurdi_m_Page_051thm.jpg
fee8eee15d69a36275291d06ce9ef5ea
d4c65e6d3f2000f27a898e236865417ec591aff5
2683 F20101123_AABQZD kurdi_m_Page_168.txt
1fe12257a64ff5e9a62f2a7fbc9be97f
239f225d51c4a3873c5d5456a9cac8c0c56d9046
2928 F20101123_AABQYO kurdi_m_Page_150.txt
294cb7c9a19c6444f735b7cfe4dcc922
825568 F20101123_AABQBH kurdi_m_Page_073.jp2
5f2d75ef316306762899b0a879f709f8
0d3c964d58bb9c5f826eac4b0a05db44f2ce546d
37256 F20101123_AABPWB kurdi_m_Page_115.jpg
6d3e9dd2051ff5cde3a509dc896b84c6
c5e2534705efdda07d323ec98911f63561c3edd4
1051980 F20101123_AABQAT kurdi_m_Page_057.jp2
a3af781a5b39eaf862190c7dae0e5ef7de64d900
64981 F20101123_AABPVM kurdi_m_Page_097.jpg
a2be9b2c7a1f72c492520f3021cec1e4
e91eb0c61cda3ab51053d727cdfa13dd1015a371
43632 F20101123_AABPUY kurdi_m_Page_082.jpg
1b63652889f7510a768cd7d1700e0f39
c7a5d0642ac780c71d623e7b8cfc680e27ed7436
12103 F20101123_AABREK kurdi_m_Page_060.QC.jpg
c424b57cde2bed39ebcafb964aeea2ce
81629433b7f1b911fb83985fb54b88c04af2445a
16282 F20101123_AABRDW kurdi_m_Page_052.QC.jpg
23cdfe3f724d6fbf177277c69130b6a7
82334605440c14c4e1424bea005215010c746fa7
2851 F20101123_AABQZE kurdi_m_Page_169.txt
91c0f65c8a6ebd87a3769f8a167b7ea3
58af10e6cf25b377139eb7c2cb4fb051891b5903
2677 F20101123_AABQYP kurdi_m_Page_151.txt
18d98ba697e1c9fb237e5cd23b38b031
267b608c3b080c5b81c0d5cb0ebb36fc0bf3bf4f
42831 F20101123_AABPWC kurdi_m_Page_116.jpg
f422fdb55cc2c1e0bc84f597a8a5d5ee
92191 F20101123_AABQAU kurdi_m_Page_058.jp2
dc7b75076bd87b02c909230591a58130
75369af6c9ab9dc4d7e5d0c47c6924a33332e87a
39273 F20101123_AABPVN kurdi_m_Page_099.jpg
c1a394b543c1484d0bdee49508007c31
ee4c451a6838cc186ed79ea463cdded1a2266bf7
48572 F20101123_AABPUZ kurdi_m_Page_083.jpg
c204ae69bbb1cd41dbea7d7a10a13a76
8845b70e546fbceaf27a23bcc45a4b4957d73bb6
653177 F20101123_AABQBI kurdi_m_Page_074.jp2
28933f2cc47fdb83a90472220d04d0d59f54ec28
3965 F20101123_AABREL kurdi_m_Page_060thm.jpg
927a845d7717d4d1e0baa0e30f5d7d86f608245d
4830 F20101123_AABRDX kurdi_m_Page_052thm.jpg
2350abb72343304d2020ab144dbe3610
2679 F20101123_AABQZF kurdi_m_Page_170.txt
878a7cf73e92a9b72b4a16501cc55534
1e91de359a281b1a5ef6908f1799fbe735e5297a
2702 F20101123_AABQYQ kurdi_m_Page_152.txt
17e4d6e7f337c4ac567be4c47d1b06a7
14ceff2d7062126fa38364dfa93333c2ebf49bac
45421 F20101123_AABPWD kurdi_m_Page_117.jpg
97f235440e0b706935d179a41b18fdf0
045a9a7038ae4ff77d80e846340f11a37237356d
108683 F20101123_AABQAV kurdi_m_Page_059.jp2
8d6a1761cbb440caf2093fc1b39fdca2
208f1d9ffbb6e362059b00b312cea1fcb2976124
64780 F20101123_AABPVO kurdi_m_Page_101.jpg
78a1224ccc1efe07afdf902233786444
871334 F20101123_AABQBJ kurdi_m_Page_075.jp2
bce1ebe365b1d610e28dd2fe3f391f2c
3815ea7ae7e8aa2cea6fb01333e09dba5a3dc919
17048 F20101123_AABRFA kurdi_m_Page_068.QC.jpg
883e04a6f80dbf6fdc864c2c84db70de
9023 F20101123_AABREM kurdi_m_Page_061.QC.jpg
3eb016d86009824f0da3e0ce4d995b2d0247ec2d
2692 F20101123_AABQZG kurdi_m_Page_171.txt
cf59c82474d8f5e5747bde9f5422b768
c52b4d9e897e1d890fa3e13d67428dfab1302631
F20101123_AABQYR kurdi_m_Page_153.txt
ed7173e1a75ef9047b862fdfdaa5b745
71236 F20101123_AABPWE kurdi_m_Page_118.jpg
1bf60f15231ed2d4a73997a61c49749f
366b71b9faa67e32c2b03116dfabba60fe8cca85
72862 F20101123_AABPVP kurdi_m_Page_102.jpg
4b3b4d05b7d3f60ba66520cda6aa04e9
360fc81cd9ac7279465db25d2dc6be68bfae9ac5
836869 F20101123_AABQBK kurdi_m_Page_076.jp2
8b189f8a01761a0ac9d7b41ebbd22dca
5322 F20101123_AABRFB kurdi_m_Page_068thm.jpg
b83020d07f31690ab77ea29e69aba480
81f16fc5aef7f3e6e29bd57eb643f0f956fbbb70
2927 F20101123_AABREN kurdi_m_Page_061thm.jpg
0c63be9621f4f74528537935a3acc0c3
17967 F20101123_AABRDY kurdi_m_Page_053.QC.jpg
cd7c21bf7f688cc2c59742da17853d28
c3b645b59e72c365dbb0e6ace73cd0a777b9382b
2398 F20101123_AABQZH kurdi_m_Page_172.txt
3f146fd353c72627e701327cc8f21222
F20101123_AABQYS kurdi_m_Page_154.txt
e084d966df62ac5e392f187932dcf357
d852bc016647e1eeb77acd82c71753ee1b3cbb7d
80500 F20101123_AABPWF kurdi_m_Page_120.jpg
332962cc9ca39a17c83095340cbab446
546e03435995f57f093a0588509ffcddec6a8a83
54463 F20101123_AABQAW kurdi_m_Page_060.jp2
bd1bfb3daec64abaf8566ffd51c6d95e
37153 F20101123_AABPVQ kurdi_m_Page_103.jpg
838cb701c13a5c5aeb56fc689c738d69
58b36d6f51576e8abffbf7b8acc73fe998597a94
569116 F20101123_AABQBL kurdi_m_Page_077.jp2
66fa97b17508f48799b9347b1ba535966318cf15
20313 F20101123_AABRFC kurdi_m_Page_069.QC.jpg
c31038e6bff2791283d0006deeca778b
b1c2e4edfa0f36a748e6f2f853025c159161d406
13069 F20101123_AABREO kurdi_m_Page_062.QC.jpg
99049e0e1be846b3a1c7cc67e0e85f9a
5326 F20101123_AABRDZ kurdi_m_Page_053thm.jpg
f79471a811fb7e2a18fb4374062f6821
a897d1688d58ac6b92a1fdbf0e3cf23c10a65353
1343 F20101123_AABQZI kurdi_m_Page_173.txt
8ff5cefabefac62871138d7eb404f7ba
e3c74e1b90cfb164ecd573eb2d3441e9a7a23954
1626 F20101123_AABQYT kurdi_m_Page_155.txt
7647a2e3ac50973b0c6c820bc3eb9fec
de6fe542b6f7da16a76d3ba18e15ec253e7045b3
71402 F20101123_AABPWG kurdi_m_Page_121.jpg
e435bd15cb9a71923551ddea966d31fa
0c853e06f9b2d25928cab7e167b4c98367b1c8d9
34617 F20101123_AABQAX kurdi_m_Page_061.jp2
cdc12ca987fb6495299659e936541798
918b778bfb3e956f9b25b46b97d92c29c2246d6a
59777 F20101123_AABPVR kurdi_m_Page_104.jpg
ddc86a43a6c7b32084c9df7e2fe71257
af3aa2285d30c3213d2854d13a0b6a8381f312af
625384 F20101123_AABQCA kurdi_m_Page_094.jp2
a3155577e880cb9ecba4d3d70ff50809
e8393ac5e63b92ce99b7829917aaf8c57133a47b
881724 F20101123_AABQBM kurdi_m_Page_078.jp2
7a88042df752a6ec2d62dcb0a85619fc46d802ab
5997 F20101123_AABRFD kurdi_m_Page_069thm.jpg
0f80b0594d0ec4f6ae4cbc08592ff6a4
67666232d4b8531de6b4c9c5d46012f7a4f087ac
3671 F20101123_AABREP kurdi_m_Page_062thm.jpg
efd1ba026a3b7c10fdd11ba011c25612
bd7a3d52598ba6ddab100c12f91d46845604bf9c
1477 F20101123_AABQZJ kurdi_m_Page_174.txt
c0a7e696b98f91782cfe2b138cf47a4d
6277f14e85d0fcea3d72695be6e5e93ea96d540e
1322 F20101123_AABQYU kurdi_m_Page_156.txt
15007 F20101123_AABPWH kurdi_m_Page_122.jpg
089355572bffec26a547a420db5fb11e
9079103ca0c90fd481146966f1d44b415b776aef
72090 F20101123_AABQAY kurdi_m_Page_063.jp2
88e9ef1f21605819286aed40b3d353b1
156b08fd0c020e1627f00dcfb59978c6f46ccf82
55161 F20101123_AABPVS kurdi_m_Page_105.jpg
b93fdd649a10f80cc5e783e6bb49b6be
569289 F20101123_AABQCB kurdi_m_Page_095.jp2
fb22c706ae7c789f4fe01669e7a94321
6ba84a29bd09552cf2477bd832687fd1d9c4143c
600120 F20101123_AABQBN kurdi_m_Page_079.jp2
e09df653e4b7fd19038144f9e98eb8fc
901f768331628e04ef6ec0fc47d8569d1771f185
17046 F20101123_AABRFE kurdi_m_Page_070.QC.jpg
db723cbd45aacfe719779350a3e3dd7c
a6528e276d3bde321353ca6ba1f3068e855226b9
16078 F20101123_AABREQ kurdi_m_Page_063.QC.jpg
a061185ee15bd5fb0b9a0820dbe1572f
f38d9bb0c044f726f3c5e554f5a99c79160008f1
1622 F20101123_AABQZK kurdi_m_Page_175.txt
2074 F20101123_AABQYV kurdi_m_Page_157.txt
4c3b53010648066c36160f03b58d30e5
4d5531ee1e3122375231983873098ca4345ca2af
32908 F20101123_AABPWI kurdi_m_Page_123.jpg
92f8209c4b89712973ae1f0edb539060
99168 F20101123_AABQAZ kurdi_m_Page_064.jp2
68de391b932202b8737fbc5f91f346cee5caae9a
76628 F20101123_AABPVT kurdi_m_Page_106.jpg
7679929c41d8f1ede57d13d3deb1fa20
4228aed459fee500ff5501dc6d39115e6da6a974
990019 F20101123_AABQCC kurdi_m_Page_096.jp2
94869 F20101123_AABQBO kurdi_m_Page_080.jp2
6fe498bfab3e4b139be0bd575b88ff63
c1d98ddfc7e24d559e8bf88cd0920cb66de25476
4970 F20101123_AABRFF kurdi_m_Page_070thm.jpg
bed047d704a64a3b8d8598b1c5d39060
d58d47363603a4e133579da0e817a35a578d4e0e
4739 F20101123_AABRER kurdi_m_Page_063thm.jpg
cd25b2f9dd45ed1ff8cd50700bfab793
e14be9658a23626370b0d96a5e7f8177a5390367
2035 F20101123_AABQZL kurdi_m_Page_177.txt
12845e873738f4eb84e2ff151eb9e1fa
a3c4b15bfda0b7631aa38d822162d49f01eb8cc1
1908 F20101123_AABQYW kurdi_m_Page_158.txt
96fb6b77b53faee9b7e551284fee58c2
bff4c8b9f9d811236279bb383be89225145e12a5
53855 F20101123_AABPWJ kurdi_m_Page_124.jpg
2869570382dda1d129919258b28022e20bca74c9
65945 F20101123_AABPVU kurdi_m_Page_107.jpg
d5ac228bc52654cbd24635faffda96d5
e945f6cfd8bda1ef5f2dd823b712800b4b6ca9a7
1051969 F20101123_AABQCD kurdi_m_Page_097.jp2
36bc488d0d8a8850e80463789d0ca7eb
9e4f580b84bdbfa25598ce247291e70ffbc83370
84658 F20101123_AABQBP kurdi_m_Page_081.jp2
062da8271a1296bed04b1f4a12af4dd8
416d05daa1df1237bdc92b08b94fce78ec308894
12225 F20101123_AABRFG kurdi_m_Page_071.QC.jpg
76388e60847f729aa93cf9b918008507
25612d8f6e942150054fca8b4e53368f98c80827
21425 F20101123_AABRES kurdi_m_Page_064.QC.jpg
58e2c08cbf2bb242867e2366b6701d28
394df0b5068e4a5471b7f261cd6cf385602a620d
1957 F20101123_AABQZM kurdi_m_Page_178.txt
a05ab1da270cc1c25fcc599ac1e3045d
1271 F20101123_AABQYX kurdi_m_Page_159.txt
28b2edb726db420a4268edf12bfabec3
e81c86512b3d23df4b0557cd355de988550a9591
57290 F20101123_AABPWK kurdi_m_Page_125.jpg
ba2cb03a692579fe7d4567e776f44122
cbfc4f1420d759419fafcefaaf706e95edb9be0f
52648 F20101123_AABPVV kurdi_m_Page_108.jpg
3d39be2605ca28879732764c7756200af0c7992e
57703 F20101123_AABQCE kurdi_m_Page_099.jp2
b62dab1133560ae04f05a6b5274bb5df
372f185c519836283e4395d394e3420ede65c145
617477 F20101123_AABQBQ kurdi_m_Page_082.jp2
a5a865381e660859dc37a8b2f7863ea1
a47cbf464c616a903dc315c1f8c05e98a2324f63
3793 F20101123_AABRFH kurdi_m_Page_071thm.jpg
b98eb789045066430506d6f60c50f34e
5950 F20101123_AABRET kurdi_m_Page_064thm.jpg
6c5435f9133583099371a9355815d25b
b270ce4d85bea21478153f8af558cd9a40bdce1c
1422 F20101123_AABQZN kurdi_m_Page_179.txt
b47a150e7a397537504472930d6833d0
b1b383dc3a6a07dbb1a2e5a14e2f2883d5e3f7c7
F20101123_AABQYY kurdi_m_Page_162.txt
3c65e95a3c64c4f1b5589e09848aaa89
49196dba2015e68f8a6fa9440bfe87824debfded
49740 F20101123_AABPWL kurdi_m_Page_126.jpg
3958bdbac9577863902bc691d03052b589213c1f
61128 F20101123_AABPVW kurdi_m_Page_109.jpg
5c432239fbaf22bb3a986d4219f81b13
21216e258107552ee661ff0ebf0e38bbb89162dc
81521 F20101123_AABQCF kurdi_m_Page_100.jp2
7860e046153f24fcc8560c8451f82b44
586d5258e205f257ca6c2e82042d50e3ebdae499
704206 F20101123_AABQBR kurdi_m_Page_083.jp2
5314885a4af8e154fd6c3e9233bb0f90
2a994c2941e7d44ee4998b817cc891252b05055b
14215 F20101123_AABRFI kurdi_m_Page_072.QC.jpg
079f84c62945a4ede953ca10a6734ff8
fc7415f4f4d499a6996b8c3df145f1861260a0b9
24001 F20101123_AABREU kurdi_m_Page_065.QC.jpg
a2008e3802a25daf8d0d006007e0a01e
9d9afa3dee262205a48c165fa05ba6101e3116e9
1412 F20101123_AABQZO kurdi_m_Page_180.txt
35acc5c3797b0212ba6cbcf4456e2b99
d30fedaebc1540f4f37468ddd628bfda264fb1a4
1389 F20101123_AABQYZ kurdi_m_Page_163.txt
70bbf74532a43068b13a62c77a2db4ea
b2a82bd2962f014e0eab0f33daf48ac9a0b9a0c6
42559 F20101123_AABPWM kurdi_m_Page_127.jpg
cb078aec097402a8fa5cdc98b9558a0562362f25
65000 F20101123_AABPVX kurdi_m_Page_110.jpg
ae00e5dbae10dc08a67b46972cbfdc67
5455dda8ec771ae689b274d0c71755f3f0fbb33d
93658 F20101123_AABQCG kurdi_m_Page_101.jp2
06af9026188b391b0d6242ab5ea7b0f0
358e1cc3d7739a344301a6db69306ae95f885160
60714 F20101123_AABPXA kurdi_m_Page_144.jpg
d1ff1bc2edbe9f17cf424d4131508180
97b1847c7f7bef3089f3469bae9abfd5758a8894
75248 F20101123_AABQBS kurdi_m_Page_086.jp2
d5d67b81ea3bd693b05bb768b4ac5e79
cb54b53a1f90a193cb551351a32504e22764c0d8
4448 F20101123_AABRFJ kurdi_m_Page_072thm.jpg
b3f0b81dcfb00650c1591a6def7031ec
a2412d9e3afa865b79c3e649c49a76809db9ba07
6729 F20101123_AABREV kurdi_m_Page_065thm.jpg
b946abdb787e30da988bd000cbb11638
e67ca60ab52df7f4224ca23b44b03774275e07b3
68202 F20101123_AABPVY kurdi_m_Page_111.jpg
e8f1e34b75db1be33a55fe6e70d494cc
b4c598bda670d77e5af7a5d6e4e4de010970a0f0
528025 F20101123_AABQCH kurdi_m_Page_103.jp2
49314595b2dbb57e7a37216c27aa0463
fbd9e3837a8d3e8eeaf48d23225f9a8044f78339
71090 F20101123_AABPXB kurdi_m_Page_145.jpg
d46afbf60ed73d371c6fed0f37c4937c
d9a96d91be6234883703354eebbbc7ca67d4f5fc
106775 F20101123_AABQBT kurdi_m_Page_087.jp2
7ba74c8564711c5c1d03b7cc72e9d3820c009f75
17184 F20101123_AABRFK kurdi_m_Page_073.QC.jpg
f460f13182e0840459a390d2f7371c09
01394d788417be71b52b4b6d8117790a5bdf7b35
16980 F20101123_AABREW kurdi_m_Page_066.QC.jpg
15265eaa7f34d1e2b5a3f1f82d843ed0
6240fb789ab5e7f9e42d5976b2971637ce585d74
1649 F20101123_AABQZP kurdi_m_Page_181.txt
69e0fdc65690c7fef8081769d7fa7a5897e078ca
42881 F20101123_AABPWN kurdi_m_Page_128.jpg
d9db31f33dfd707663eb2a5aee8ed723
19fc9ef9009263f2f705e12a567f019bd465e8ed
61095 F20101123_AABPVZ kurdi_m_Page_112.jpg
72279b7e6b3ba8ac76d2fb6fbf5af7d5
82655 F20101123_AABQCI kurdi_m_Page_105.jp2
4b34145858e6238eb269e7ebdbf12dbf
c65ac23ea9c0f822b59da5882d24f1229d268283
79937 F20101123_AABPXC kurdi_m_Page_146.jpg
8cee598a3c06e3e30407b07a6c8445cb
ac4a71c3315a93559c6a010f312751d6b6675491
11679 F20101123_AABQBU kurdi_m_Page_088.jp2
0727b1dc41c59445e6d458af14f4fb82a1548e58
4878 F20101123_AABRFL kurdi_m_Page_073thm.jpg
4dd8056dff4241a7a352eb9437b09e1808e25e2e
4924 F20101123_AABREX kurdi_m_Page_066thm.jpg
04c538337d03d75b14f1e7973530d8f5
1315 F20101123_AABQZQ kurdi_m_Page_183.txt
a4b4550199e1cf8656263db55a8b6162
54a17de869d61343e3c4e1e1b83900f0b50377d1
43255 F20101123_AABPWO kurdi_m_Page_129.jpg
998d4af7d05306dd348d1d454cc73095
5a2c8ee5f0a41894dbd0831290816a602b7dedcb
F20101123_AABQCJ kurdi_m_Page_106.jp2
7ee32ddedfe7c84956964d0379817622
005445eb2a7f85a47ebc11e8a58c75da3b734987
100295 F20101123_AABPXD kurdi_m_Page_147.jpg
e13b2fa60df95edde8bc636cae4901ea
73139 F20101123_AABQBV kurdi_m_Page_089.jp2
d63414c07f8530e454d719aa86e71104
d38c05807473e273374dc418138e5e2695a39f36
4151 F20101123_AABRGA kurdi_m_Page_082thm.jpg
0c409a6c6c79a430b59bcac3b164ccb9
1a32e4419e20d278f6ff61e2452804cf691423d6
11867 F20101123_AABRFM kurdi_m_Page_074.QC.jpg
8037d494cd682fcea483ebe64de681a029cbfbb5
20417 F20101123_AABREY kurdi_m_Page_067.QC.jpg
dd007f4280b7a2a322a4e5e3627f69af
0a5a0814fcf03efeb55c5e55e6eeda969d98cdb6
2070 F20101123_AABQZR kurdi_m_Page_184.txt
72244935d2aa5e78c7b299275c36a5dc
02f96c4e3a2e41f2f466e6f235eb89101e33f79d
49564 F20101123_AABPWP kurdi_m_Page_131.jpg
eb8de1cceab6381cf7e2311ec4c59026
097a5a9186d1624bcc5c138a3a72a3aed8eb4653
934050 F20101123_AABQCK kurdi_m_Page_107.jp2
ddfb74130c908ab5afc6f256b6409ac5
109163 F20101123_AABPXE kurdi_m_Page_148.jpg
c9b6b8d1989c28aba8d450840625345f
0d57fbd595fd075cc1383a443fda253d4105dc91
66347 F20101123_AABQBW kurdi_m_Page_090.jp2
c8f5e51979e56153a038aa92bd147ebd
2d8356c793be5f144d3aa0db49e2fe03257b6301
15315 F20101123_AABRGB kurdi_m_Page_083.QC.jpg
a0bb82d2a91aa2bb80a09a0221f0baa7
ef75f8e35c6421e40fd0272390e73e3367173470
3762 F20101123_AABRFN kurdi_m_Page_074thm.jpg
9151b1a4335ee67cd5a1e9dc4c76e5c5
3a1c6660aefdaab5a3126d9f34fbd95724eecfd7
2195 F20101123_AABQZS kurdi_m_Page_185.txt
a1f0008443cf102ba4413d8e80931581384edd65
52030 F20101123_AABPWQ kurdi_m_Page_132.jpg
4ee8cf6abfcce4e2e764a614c2254130
de7e01568d35775b00ac586207459156303f7821
687819 F20101123_AABQCL kurdi_m_Page_108.jp2
9bf5d7a4f109735a056490664064557841e92041
104389 F20101123_AABPXF kurdi_m_Page_149.jpg
bd1510c0e8d59af1cf2bcd81b2ac1bdb
4548 F20101123_AABRGC kurdi_m_Page_083thm.jpg
33d7c02c924b46b03094a2cd1433b818
cb3cb0ffddb7b892799e2dcd1c0c25046c5aff2e
16113 F20101123_AABRFO kurdi_m_Page_075.QC.jpg
0b9984d9c1ed617c8aeb6fa9266608ca
98c39ffee39ff35d3611298ca6635b5d168960f8
6132 F20101123_AABREZ kurdi_m_Page_067thm.jpg
daea37888e6c5872884e220e184b84f7
a885b91c38159da1707a9114a9f48a9341d51201
2727 F20101123_AABQZT kurdi_m_Page_186.txt
61b58529f0322e4e511b2ab67be194cb
a7cf652780401d6c74a95274092c7acaa07a1fa5
62658 F20101123_AABPWR kurdi_m_Page_134.jpg
bb9cca92460d7fa940a338e576986165
33811ea49a496d79ddc6ca047c91b597f886f08d
83802 F20101123_AABQDA kurdi_m_Page_125.jp2
e15cff7096c751913220b3be33df7e93
f6a6954e089e939320eabdbf8fc415bc2a9043df
984674 F20101123_AABQCM kurdi_m_Page_109.jp2
dbaeb591f00660d25685e52debefd6bc
116140 F20101123_AABPXG kurdi_m_Page_150.jpg
94b5ba4b6597f5f0eda86cf022aa9d8d
7ab9235be30c0dc27951d6a4cdb68af867f99d8b
66069 F20101123_AABQBX kurdi_m_Page_091.jp2
61727b58092baa2751ae4dcd80d4ae82
7b75542df65e414477fb4445e03e2a444d5d6f1f
19740 F20101123_AABRGD kurdi_m_Page_084.QC.jpg
b2f1547795c3e3a7bb39c7c8fe903e3f
d15b7bc30683f9819134a3c39048ef1819af526e
4772 F20101123_AABRFP kurdi_m_Page_075thm.jpg
7b16ef51570443f6212b728e49df393d
1169d964d10ab42dc0b484775d637efe2d0c3441
2645 F20101123_AABQZU kurdi_m_Page_187.txt
d5a6863dc97d18e92620f2eb55588f0d
73104 F20101123_AABPWS kurdi_m_Page_135.jpg
68f98963bc68e285be06a573c2aabb52
47fb1313f86c36ddfc9531a3eb9e4bd3bb5c7b12
75060 F20101123_AABQDB kurdi_m_Page_126.jp2
ee84a02e9758e91fd836ded0434ed8b7
38c3ee475aeb38800bbe8bb0cbaf067062801308
1034140 F20101123_AABQCN kurdi_m_Page_110.jp2
0ff464ce336e4f736ce3dd9e07433b0e
a8fc5c75dd5e62a61b425c48ca17fe535ab92e0c
109456 F20101123_AABPXH kurdi_m_Page_151.jpg
856558 F20101123_AABQBY kurdi_m_Page_092.jp2
aed94571abdf6c57becf6f6335815271
15356 F20101123_AABRGE kurdi_m_Page_085.QC.jpg
17e3029cd15fde8dfa7d4b4b329c1367
31b895ed4f103566ee9b997acd7328990aed2fe3
17817 F20101123_AABRFQ kurdi_m_Page_076.QC.jpg
21c70c37969f02a528aaa45da0db2606
2913 F20101123_AABQZV kurdi_m_Page_188.txt
209f6f81c977eceac9623063038bb319f8eb6de8
97153 F20101123_AABPWT kurdi_m_Page_136.jpg
8b78b3234633b4b2020b8dd64a0b68a0
900c26c41cf9ecdf3ba1eee4a7e014e22d4caf37
56566 F20101123_AABQDC kurdi_m_Page_127.jp2
7b30da7a0253e599285e9b9c8b8bac73
cdea60e9cd35406232ba0025b3a5488dac679b14
1051955 F20101123_AABQCO kurdi_m_Page_111.jp2
5cd74eebd572aa1a472fd7c2463b7f2a
de814ccfcdb55cb41cfc3165f94449273bfabb74
108628 F20101123_AABPXI kurdi_m_Page_152.jpg
3676e7105b6b0a26d8b79b0eeb34a6ac
fdc7f533c4da74fb1c8cbc184d0f8b58467860a1
941384 F20101123_AABQBZ kurdi_m_Page_093.jp2
f04eb68141c694d292a28260abef0356
72a45ab1dcfafe3064e7a1305b9796c4047105c6
4771 F20101123_AABRGF kurdi_m_Page_085thm.jpg
c05398d89cced60a0c2ff30a567a41bc
964305e6b490920e4337cbb1c41eb7bbbfb96b86
12960 F20101123_AABRFR kurdi_m_Page_077.QC.jpg
3be23cb9cc1e5838f1dde06c6a418c9f
c1069718f819351b17917db214dba1524d9fc8dd
2706 F20101123_AABQZW kurdi_m_Page_189.txt
567efaf628d94bf19652d7e1c3c0cdce
ea37edf6471b534c87fd4351cede07af603d9a71
104337 F20101123_AABPWU kurdi_m_Page_137.jpg
fbdb1fd298847a9463234a3b6b992946
9b30761c7b239d0d0ee16488f7273e9ee87d61dc
60429 F20101123_AABQDD kurdi_m_Page_128.jp2
5f5c1b1d34df0c5f6bdd2b1047ff9249
bd81de9e93ba8899a9405cda3ba17009b0e17e3d
109661 F20101123_AABQCP kurdi_m_Page_113.jp2
a9ce7ef340b5a57712a1a8ef9c7ba006bf52446a
105096 F20101123_AABPXJ kurdi_m_Page_153.jpg
9419b3522da2b4687f2711626476a41b
e181847d649f2d85acb80e82ea2a17c7b2142a7e
17269 F20101123_AABRGG kurdi_m_Page_086.QC.jpg
210d29049dbdecc43b13a6e3e241effc
3c648cbb61ef52d6fdc951ff0e1c15125d75ef47
4499 F20101123_AABRFS kurdi_m_Page_077thm.jpg
ae60300f6294cffedb9f11f9efe783c0
2687 F20101123_AABQZX kurdi_m_Page_190.txt
3a8b1b748b0dc226f1b954155b8ed06bdf46a6aa
107264 F20101123_AABPWV kurdi_m_Page_138.jpg
3312c6dc3d450335b58a0a0aefe740fd
517323 F20101123_AABQDE kurdi_m_Page_129.jp2
73b3344b17968336d72c05d4c3cfe47a
548200 F20101123_AABQCQ kurdi_m_Page_114.jp2
670e2f152c81f01b22f2d4ee8b021b9c
0d4f2e0eb868530fc46964ae4dffb1c9d03a7dd2
78362 F20101123_AABPXK kurdi_m_Page_154.jpg
330a3cd66bb764b87a27d71a836688bd
f8e3ed6da21b7d4c14603aac9e19cba41cdbf8f8
5150 F20101123_AABRGH kurdi_m_Page_086thm.jpg
c12b8d3d15881f962af6242c879fcf81
17355 F20101123_AABRFT kurdi_m_Page_078.QC.jpg
b22b271a3618dea4e00a02e69e0d415c
b7345d040a76401993e1324e6db0f69e4115c822
2739 F20101123_AABQZY kurdi_m_Page_191.txt
6fb24df3eaf2aa119a14e7526df6421846d1bd44
113123 F20101123_AABPWW kurdi_m_Page_139.jpg
173f64d45ebe5f9facbbf9566c475c18
e146624a3ccafcb1fd87e926ed130d94d9e99d04
47414 F20101123_AABQDF kurdi_m_Page_130.jp2
4b24ce3c63358ec8f37101ce7f2a1641
586dacf57551c9beefb79e1b48296b767ab9ab5c
556918 F20101123_AABQCR kurdi_m_Page_115.jp2
b7ab98d4499fb518c03e8f09aeaa0e42
0a266a196d6050394122e2ee3201c5b09b47ef99
51430 F20101123_AABPXL kurdi_m_Page_155.jpg
47c4bc25b6542884b3ee508105184481
95111e433136f054c26bfc24ba80ff97f4f51464
6261 F20101123_AABRGI kurdi_m_Page_087thm.jpg
4c03db056e3a3fd8d63a71fb1218ff8b
a74c176db42a12092271ac844f7c7b1e4c4f6441
4898 F20101123_AABRFU kurdi_m_Page_078thm.jpg
078a610321c0e227010459fe16fe2268e2236d2b
2033 F20101123_AABQZZ kurdi_m_Page_192.txt
c6701307f1c18c042064b40735d1106f
041921fb2746691d140abf31e592ec8369ac76a2
107851 F20101123_AABPWX kurdi_m_Page_141.jpg
c3dd767f7977208c1fa760ca8079072a
c17730ca40694668012525274e09fff24393f9cc
73083 F20101123_AABQDG kurdi_m_Page_131.jp2
072cfc21f8992be010c310d32e6cd65c
e2cda591820bb99eeff7dab5e9324c7beaf31dc4
61738 F20101123_AABPYA kurdi_m_Page_179.jpg
f1e119e4eba635a4415f170cd22a4bef
51e9310be413398184111c8b4705ca2b89080a17
642833 F20101123_AABQCS kurdi_m_Page_116.jp2
b87c55f2bac5a858e8ee03b3d60360a0
2de80d2dd5fd05818d91c4d1dc08b91d009e58fd
53522 F20101123_AABPXM kurdi_m_Page_156.jpg
53ff912e7718d890ddac86bf3030b628
3da682c60d9b210b98fe727d89cf717e4fff14c4
4547 F20101123_AABRGJ kurdi_m_Page_088.QC.jpg
2e94d57b3d6304c411683d0d385e9d42
10688 F20101123_AABRFV kurdi_m_Page_079.QC.jpg
303c147b8757c2aa55d9eab26e9024cf51835c57
105184 F20101123_AABPWY kurdi_m_Page_142.jpg
9365b854759bdd2e804a4818bedf9ffd
b399c76e916a0aff063340f8b5fbb859c9dd1444
76012 F20101123_AABQDH kurdi_m_Page_132.jp2
604be6af88d51cb9b79dbd3ee055f33c
0e34e48f3dacf5d960caa70403401109e0b29728
61020 F20101123_AABPYB kurdi_m_Page_180.jpg
4a8d8ea363bcd15bfcb95c97177a4651
c19f6e8c43acf99de325e7759841d0f51f299098
510381 F20101123_AABQCT kurdi_m_Page_117.jp2
c6bd375a6cc19f3ba08b038540c8ca24
5ba0f716443312c3afc34bbf40aece5a2b80f6d5
77332 F20101123_AABPXN kurdi_m_Page_157.jpg
8df22347e220dee0318be48dbcff3ba4
e4de801d5584bbb4cbc583c01709839ed8ef5e23
1691 F20101123_AABRGK kurdi_m_Page_088thm.jpg
dff40f0ff041726238c4278431070831
95a51ae58feb83fac63ac8d3e690092a09c6b46d
3588 F20101123_AABRFW kurdi_m_Page_079thm.jpg
73aa178cc266579c63c68500ee5e60ba
2c1400a5b6483c0fa745c0ce9ac022f201095584
89099 F20101123_AABPWZ kurdi_m_Page_143.jpg
fef9cebdff780f0943632355a7e06b0a
e39c83d92baab152b2c81a22b3e4faeddaa71a37
96495 F20101123_AABQDI kurdi_m_Page_133.jp2
7a2ba7f58bf529c33539539ab6fcc54a
e2895aa21ba2dbfe15dc8a9260b0f10bb36efd36
63808 F20101123_AABPYC kurdi_m_Page_181.jpg
ee2e2060a5031d2d554fc6331cf07411
107154 F20101123_AABQCU kurdi_m_Page_118.jp2
97d150a04dbc7605fc2eb7f05f8660a3
ca18a779cba7b8925c9d8b494378bb583d225c21
16400 F20101123_AABRGL kurdi_m_Page_089.QC.jpg
f4e5075017d6604fd8de33c49f39c915
6d7f4b21fd93c1b570019cbd918d167d62edb9ec
20868 F20101123_AABRFX kurdi_m_Page_080.QC.jpg
287de78f2108443ecd52b18de7898f54
121512 F20101123_AABQDJ kurdi_m_Page_135.jp2
2102f926857b3708db9c9037c3f13bc4
dba004e00d5606a19cb62f400988091c93768ff0
63027 F20101123_AABPYD kurdi_m_Page_182.jpg
45d82806228e1e089b7cc371370a1ebc
f2fd013f863c4b78c57d23e1424347f0c6f5d6ef
32783 F20101123_AABQCV kurdi_m_Page_119.jp2
5aae9fbf6b1f5e9fc55de608415b19a6
397aa3eef095ed28d88d3da6d31428c1b0415555
73183 F20101123_AABPXO kurdi_m_Page_158.jpg
e0db872edf4f91430b8cfa9005abbafb78b70f38
F20101123_AABRHA kurdi_m_Page_099.QC.jpg
414a30809d54c53e0d4de37c261cfe7a
d730fcca5978de7315c7ecf3b8d3f1da9db9e7e2
4973 F20101123_AABRGM kurdi_m_Page_089thm.jpg
a9044906b050e2755ac33a52582df3e7
f2be81a0dbbe59b34af2752bd0e074f37a2f4ba6
5841 F20101123_AABRFY kurdi_m_Page_080thm.jpg
9503ba4d58f47ac8a9202ee203bfc9f0
c58a56e15a936d64d4d08428d0636d3197566b17
176561 F20101123_AABQDK kurdi_m_Page_136.jp2
eeb5f6e3397d309506a33652dd40be2a
529c10cba92f79538976ef25786b39925796decc
52899 F20101123_AABPYE kurdi_m_Page_183.jpg
38a5a2bc847e8641c5a18c6f3ec0df80
119500 F20101123_AABQCW kurdi_m_Page_120.jp2
33d363fcbfd0006c29032fa7a4fa72d7
7a12eea9279a7142b4a117bb655a39f29ba5f2be
61916 F20101123_AABPXP kurdi_m_Page_160.jpg
c4e6e3b8fa3750a16234f3993a25796b
60e8c81e8deb5c0e56ae452c599277d3dd19bbfb
4467 F20101123_AABRHB kurdi_m_Page_099thm.jpg
05432b059d3c26a0b9e7291b997f0f8483abf5a6
15734 F20101123_AABRGN kurdi_m_Page_090.QC.jpg
eea68e82ef065c8e4f84e0a4c8f15248
18528 F20101123_AABRFZ kurdi_m_Page_081.QC.jpg
808e7c65fcba86e3f7d83be96f00fb89f2ca3927
179025 F20101123_AABQDL kurdi_m_Page_138.jp2
96f3bdf21ebe5393cb965b18916c535f
4a21e2aa29d04b4a9426416109ab3b47972486e2
83608 F20101123_AABPYF kurdi_m_Page_184.jpg
5508c2f2b3f9afab312d9f3533222c00
1e5a9640f2719e03295e77ab4852dbc88421c96e
103735 F20101123_AABQCX kurdi_m_Page_121.jp2
23581bc7ac3d3bf1b5728b36fff17152
60d5e5f245597f30faf819ba6e1d8c3830ba1ddc
63418 F20101123_AABPXQ kurdi_m_Page_163.jpg
0bd19fe9cc0ee9792b7a505e246cdaa1
bc46026170ce04d968f7ceb081ac9299e6f3a9e3
19545 F20101123_AABRHC kurdi_m_Page_100.QC.jpg
670851a35ae330c50cc284645cc932a7
ba570563b60d334f5f94b703cb759720dbfc3cec
15365 F20101123_AABRGO kurdi_m_Page_091.QC.jpg
a95594e4349d3758fa931288b63ab626
d011272e4e2b6808402e6d0d6fc26b37d0c163bb
188939 F20101123_AABQDM kurdi_m_Page_139.jp2
7da73573aee9fab8a502cefa9a5785e3
8781d62474058c9289d59102cdb237cfdf645404
96188 F20101123_AABPYG kurdi_m_Page_185.jpg
5f46864d2a99e8ed38b6ffca4aa98a3b
a59b41232f16b6df535d12d59503fc2d5acd1c80
F20101123_AABPXR kurdi_m_Page_164.jpg
30525bab63e4eddd5686813902b08a43
93ce6b1c6e26ec0c2402e91ebfe3114de2e4f48c
75319 F20101123_AABQEA kurdi_m_Page_155.jp2
742c1e67c5ab95720d3d85a6ceb3d314
5a9a66277aff2bf4aa4d2484aa12e06d2ff38c09
5539 F20101123_AABRHD kurdi_m_Page_100thm.jpg
e167cafe81d20483cbcdd5be0d92d11b
29dee2be6682a1c0cff97282f1cec20f71676f51
4764 F20101123_AABRGP kurdi_m_Page_091thm.jpg
ec400892a7a424db8a62e8bcffe15119
b7277712945a979c18ef3cca185717ee5008d592
182232 F20101123_AABQDN kurdi_m_Page_140.jp2
6b23fa2206c6606958590c3334f96c42
7540b59ec5d0f09000201381fab346d6fccc74aa
109672 F20101123_AABPYH kurdi_m_Page_186.jpg
5ab78eb30ab46fb47e2f5779cfc76e89893ba8c8
13662 F20101123_AABQCY kurdi_m_Page_122.jp2
ed311ce9cd4328e20765cf589540ea14
c9df5ee93480d91f4cd3e426acf4d15929410371
92342 F20101123_AABPXS kurdi_m_Page_165.jpg
37abb07476d65f182fb5c5927f857749
76132 F20101123_AABQEB kurdi_m_Page_156.jp2
56fc9a2d636d88932458792ec6f6b360
55d15718d6d560f1d70a0d9833e6a57e3a06b6f6
20884 F20101123_AABRHE kurdi_m_Page_101.QC.jpg
1fbd5b91cd275e8949d9ba6df81e3489
871dfceccc6fc236991633e36ca0f922d9f92b36
19407 F20101123_AABRGQ kurdi_m_Page_092.QC.jpg
669edd7740cb4d17d42e86222e780b59
f19336f5fe971cd4db551e95c1963a036db02836
177217 F20101123_AABQDO kurdi_m_Page_142.jp2
4f2ec5f42e43ff9dfda09d0ab5ba47a0
a6e603f1ae338ba57d30d4cdfce66bde2af5004b
105188 F20101123_AABPYI kurdi_m_Page_187.jpg
a0febb01dcae394c0ef3894b44889068
4f3270cc21ca4dcfb7146a622e8b64f9ddde837e
82815 F20101123_AABQCZ kurdi_m_Page_124.jp2
9e1fa45d44ae0eae9a0828f36a2e392e
106840 F20101123_AABPXT kurdi_m_Page_166.jpg
06e09ed085761fc9a9ef76c18ea5c25c
b34f7970801ef326b7c1599a101f93d10ee855da
110854 F20101123_AABQEC kurdi_m_Page_157.jp2
ed2e745b77db23d0f4cf327dc491ba53
4ff9f9c4ea6e14fa90de9b20147baf14b6f8aa04
5789 F20101123_AABRHF kurdi_m_Page_101thm.jpg
627ea28abc7007b9ce1bdb7237c8b885
18195 F20101123_AABRGR kurdi_m_Page_093.QC.jpg
c237216d41e2d8f5c8cd9839d7e6aff9
216f9ac9061f38312b1cf36bbf095bf5430ae935
139345 F20101123_AABQDP kurdi_m_Page_143.jp2
0f0a082bd2cf39c4aeb4aff9514449b2
71ab33ce482bcc84211c59a4d44ab3ff4dcb1426
114571 F20101123_AABPYJ kurdi_m_Page_188.jpg
a5f97e1edf3f3eb56672a2e4dce371e1
50963a3aa4875a016a8ab4d54fb49e6e0692a46b
114421 F20101123_AABPXU kurdi_m_Page_169.jpg
4089ac69aa0d6e13d263f119a3abd791
9d826c7ba85b6ecb4879f2b171e2c2a0d1ee401e
77554 F20101123_AABQED kurdi_m_Page_159.jp2
bb71d7940d1d17b6376754d030cee96a
22550 F20101123_AABRHG kurdi_m_Page_102.QC.jpg
a8d4b341b3d1c4fe81b826ce5386762b
abc443696da9bb881ff3b96d2d3315a073dd9163
5595 F20101123_AABRGS kurdi_m_Page_093thm.jpg
b153cfb334d3177e5d01d4b20934c325
89218 F20101123_AABQDQ kurdi_m_Page_144.jp2
ae646a42ed28d57837129363f792ae48
2a21356d27166e2bebe991630abbf302af49b19a
108549 F20101123_AABPYK kurdi_m_Page_190.jpg
fc964893ab18eb5df307e56351ea063d
856df8c664a7fcc22db8890939d5542a9d0e637f
107967 F20101123_AABPXV kurdi_m_Page_170.jpg
6e6f3f5542cdfd9589c5138c4171484d
2ac568a23cdc1dfa76736f6af0548db74b831f52
96928 F20101123_AABQEE kurdi_m_Page_160.jp2
cd1c030097a177f5a523130844c47352
94b158c9a2b81d3e53d364b6d1efb95ba420a6e1
6321 F20101123_AABRHH kurdi_m_Page_102thm.jpg
66c77df106f45750478cb7ae41246512
8448591531bec9d3887f514a5190f04f81dbce2c
4902 F20101123_AABRGT kurdi_m_Page_094thm.jpg
b5b73b81a9aa80909dc009d366ecf385
3179b8fa9908edb4c90ff698912eefe3c299d134
119856 F20101123_AABQDR kurdi_m_Page_145.jp2
4d17d50ea1f1ae24878b65292117d540
4267d8260658734c98a4e8b47a2a582c1329072a
109808 F20101123_AABPYL kurdi_m_Page_191.jpg
c11d83f588c138f77fb6a3d28cd6d9d8
f3364c90c62dc0cfc7a931792e0b2bc9f9fcf94f
107522 F20101123_AABPXW kurdi_m_Page_171.jpg
2f5a43572256d8c66460f4c0e600461e
59b3f54005341fca47d4a6dd95de41ca5ff83093
112210 F20101123_AABQEF kurdi_m_Page_161.jp2
6b311aebefbbe542e06a15c1b5476cb5
e230f8dc0f2f7d694f9dda4d78f3e8e0f35b9201
11921 F20101123_AABRHI kurdi_m_Page_103.QC.jpg
4890b59692c648da3a2899d342c38575
507c1731aa7e325f9f77aa4f5a50f4fd8d684b5f
13889 F20101123_AABRGU kurdi_m_Page_095.QC.jpg
80f00e079932f1ec2398c1c443be7dc5
a58a053bd962dd9298d476714b9b16af88b8c2bc
1051981 F20101123_AABPZA kurdi_m_Page_006.jp2
460e63cd6d90af74b31a944f5ddc0cc5
705424ac231d7490f2b4e12af815a4c42c2bf951
132733 F20101123_AABQDS kurdi_m_Page_146.jp2
3131593de11eb4f906e2f1829c5b9314
94be4ef2249f8bb8ce94ca361e55e1deac7d1751
83338 F20101123_AABPYM kurdi_m_Page_192.jpg
04ff56846f9f0e788052f4e128f67a2d
59e83b21f95918ca1f7dee6a67101935a9a41672
56016 F20101123_AABPXX kurdi_m_Page_174.jpg
6dcdcdaffded41f80a9b9b372fdbb926
0aafd54708b219bf8d679e62a7b54769d88dcea0
112140 F20101123_AABQEG kurdi_m_Page_162.jp2
da28ca24e5e0bdc84fb7a1e12ebafcb45e822829
3917 F20101123_AABRHJ kurdi_m_Page_103thm.jpg
e2e2c7f0eda2d431f4821b4d20636bc6
fa4b2fe2724eab0080ed502fe89c558fcac77a80
15561 F20101123_AABRGV kurdi_m_Page_096.QC.jpg
277a63639f0d56423db0dc42bdf628e9
64102fa200d5f81cf7417a10abea31fd6a4145e7
1051963 F20101123_AABPZB kurdi_m_Page_007.jp2
bc54262fc1ec30aa3a77bc2e1717523a
ede8c8751a28da4d8821e701f95c06e8569cfbcc
172704 F20101123_AABQDT kurdi_m_Page_147.jp2
d02a0daf6ab07be8de6382518f0b7982
1aa0d632d2308e5162d5e45c54682be4a6330bf7
22379 F20101123_AABPYN kurdi_m_Page_193.jpg
acd7c351d0899856068b66dfaa5a6728
78643 F20101123_AABPXY kurdi_m_Page_176.jpg
3a2035da567baca97cb6e263f02b09b7
509db4815340591241d3a06b51e3f19b60661f2a
106101 F20101123_AABQEH kurdi_m_Page_163.jp2
6538fccddfb3c597a5c43b4045b587b8
0a80ee71aab22006899cabe0c1dafb6c1879d481
19222 F20101123_AABRHK kurdi_m_Page_104.QC.jpg
88ee4ac537241a8c8c8ea2e6f01ca6ba
d040ec04b173dfaced2dea582500811869618791
4833 F20101123_AABRGW kurdi_m_Page_096thm.jpg
a092231040e3e8c93fa06b1606fdc48f
1b555d8e06e93a4f5774f0635c95184b58a78b16
1051973 F20101123_AABPZC kurdi_m_Page_008.jp2
402e18c59e3375272b15cb467d1364f0
175807 F20101123_AABQDU kurdi_m_Page_149.jp2
41e47c8836d1a2585e75dc746ca1c95e
4c06238af8aace7af6e0aa0ce222e7661c6477e8
71342 F20101123_AABPYO kurdi_m_Page_194.jpg
25e51a73e48a1bda49102fa0e8642774
5e3c15d66b4df6119e6935da8b5ff282dcd02f30
76144 F20101123_AABPXZ kurdi_m_Page_177.jpg
b076cf8cb9fc53336fdb5d16192f626f
dd9116a6620a1793dbffb8fbb7cb233999bf7f75
121923 F20101123_AABQEI kurdi_m_Page_164.jp2
3124c9f13330512a5e75926bea411556
bc22170fb1aba6f4b988fdf847043f1d9996eed9
5272 F20101123_AABRHL kurdi_m_Page_104thm.jpg
18138513a4432d7268f491d569bb0c9db2e82e19
19231 F20101123_AABRGX kurdi_m_Page_097.QC.jpg
b045b73f126285f99f2c2661f1887d42
c175f42eac7b4733f513bce8beb091cb76d3f2cd
1051979 F20101123_AABPZD kurdi_m_Page_010.jp2
9e67932117e871f775652f10de8525135ab247d1
195019 F20101123_AABQDV kurdi_m_Page_150.jp2
7a153074f99c964b2e0442f47478ab8b
24bca04fd66d726f647164576ec1bf6109a9621c
162014 F20101123_AABQEJ kurdi_m_Page_165.jp2
8235db2017cfcb1c358ab8a681640fd2
1c8b75fe608b33275b3798a353c1f95d304984f6
5585 F20101123_AABRIA kurdi_m_Page_112thm.jpg
ecbce94d75e560ea1f9522b32af0674b
66ab6aecb96df5770eec540113bbdfc2019425d7
18211 F20101123_AABRHM kurdi_m_Page_105.QC.jpg
b3471550250b12a07ce0066051b307c5
5633 F20101123_AABRGY kurdi_m_Page_097thm.jpg
aed33156b241dc6fe9fc96803569e1e0
def99782b75970b1b6b8cbbb4910b3c884e570fa
F20101123_AABPZE kurdi_m_Page_011.jp2
1dd7b42e2af0e1a71f8d38d0af371e2d
e0e1f89fd3d0164c79c3fbd88b48992bb19bb00f
182055 F20101123_AABQDW kurdi_m_Page_151.jp2
f7668dea6ae2208736dd3160800ea894
0b74ffe9a67080a68a748d1d83492da99e8a2d49
80856 F20101123_AABPYP kurdi_m_Page_195.jpg
dbf923b7fb9325363eaa5fffa9e88533
3cbd5b387c4f558cc348c6a2fe57faf3a356cc60
177361 F20101123_AABQEK kurdi_m_Page_166.jp2
fc02d15149a388a599d4e3bcfa2c6d20
4d594f5757515cb36e0e97df849f7b91ee11d6fc
23881 F20101123_AABRIB kurdi_m_Page_113.QC.jpg
e81f112b4805bf1a7dc32426afd1501d
43005157d7cc32b14eed9ee03c329f7e1230a69a
5213 F20101123_AABRHN kurdi_m_Page_105thm.jpg
5e5c1e75818603957ffa9939aef9b59b
673857971f73d9aa645483b2efc15958b696b3e5
5118 F20101123_AABRGZ kurdi_m_Page_098thm.jpg
891197a03df760a0929f262f46b0a0e2
fdd404e24c27b1a735080528e766436fd877b601
F20101123_AABPZF kurdi_m_Page_012.jp2
c9d2a812f0d6c2e4bcd4713b2151151d
16fe39cef7aab6d3433e46385ab78c6a44672a6a
182264 F20101123_AABQDX kurdi_m_Page_152.jp2
c2bbf1c0db3bafebf7c2194433774b20
7dc01e495162a48395a7e0cd6374baac2a54a04f
82947 F20101123_AABPYQ kurdi_m_Page_196.jpg
994eec80774a280804666d5552458933
d38c5ce241feb0702d1f55fff6234f1e9e658f96
174177 F20101123_AABQEL kurdi_m_Page_167.jp2
82311d59642817f779f87283a2089b86
dd9c7192c9f595730fd4907e46cb2c69005ae8fe
6706 F20101123_AABRIC kurdi_m_Page_113thm.jpg
ce79ba9fa1c6710e71fd7dc45e37e60c
dd4cd5382038b5aec083dd4cb1e2228c214429cf
23188 F20101123_AABRHO kurdi_m_Page_106.QC.jpg
f6e550244b4aeae6b0471a41b2ff4c9f9cae4005
59271 F20101123_AABPZG kurdi_m_Page_013.jp2
f238950bf38afdb93edae7048126e03a
a648fd3bcd2b3537937bbeae146e7f7ec4de004c
175838 F20101123_AABQDY kurdi_m_Page_153.jp2
c17bed659e921aec9dfdcfca6edf07bb
726a105410b8b4883f209f5f9bb219521bd239e8
74215 F20101123_AABPYR kurdi_m_Page_197.jpg
777877153dcf2078b1788ba75bd93401
103390 F20101123_AABQFA kurdi_m_Page_182.jp2
900c720f816ed330d7712822fc987753
3ca377219209e6a70762de02009ce7fbf951aaa0
178195 F20101123_AABQEM kurdi_m_Page_168.jp2
6c87a0ac06870d2ce2c5ea09278b728186e7e718
14016 F20101123_AABRID kurdi_m_Page_114.QC.jpg
e2999633256667179853629542479fea
30db5bcb1bacb171920beb600d1e50c8eaf3084f
6508 F20101123_AABRHP kurdi_m_Page_106thm.jpg
e4ca93ba73b5b22e26e5b4af29607a42
cfd29c73efa84ab06cd03c5083eb26263b7c3234
65865 F20101123_AABPZH kurdi_m_Page_014.jp2
e23d8cc07eba944aceac1f7eabd96093
e40c067a2c99831abebb9b489562ccdbe4a82ee3
80537 F20101123_AABPYS kurdi_m_Page_198.jpg
c1a0d1573f4524564d6aedef96a1dd16
e4db8919b5d02b87881084f02d09c4965e3862ce
82780 F20101123_AABQFB kurdi_m_Page_183.jp2
14af8477dcdc0731c993080064e71898
e119defe9b1e0f5a5569e7da47552b1a3ae7e65a
190838 F20101123_AABQEN kurdi_m_Page_169.jp2
93a55f74c55c02d4bf2f69652e82fef7a0f9bc12
4146 F20101123_AABRIE kurdi_m_Page_114thm.jpg
e3a2b27d461279c0e15c052920fd1061
6962b30350c109a5616d4fa87d7214db3ba0d2f1
20587 F20101123_AABRHQ kurdi_m_Page_107.QC.jpg
4a8abef17e34eb913f8250b71a652a73
06d281e5b3d778ec162a025931d6b346d43495e8
116460 F20101123_AABQDZ kurdi_m_Page_154.jp2
7623025e9513a9109be8728e53e5c18c
3471d5ab37226221344ac7b2274947deb36723c9
81044 F20101123_AABPYT kurdi_m_Page_199.jpg
0034b08cc3061d0373d788a9ebdebe9f
caec376ef3158bfb1d44364dc62eb8f30352355a
144534 F20101123_AABQFC kurdi_m_Page_184.jp2
e1c658f1789b53aec53ee4c2578354c5
059de1dc5df527dda09a296a4de5dfd93c9dc80f
181688 F20101123_AABQEO kurdi_m_Page_170.jp2
1e61099913ac2f3ce603ac945aa30ba1
861664a8517a778d6d3471883c94875552e90297
36178 F20101123_AABPZI kurdi_m_Page_015.jp2
d9ae6fca250da2d9c0258c440f3f7b9d
4f80d4d475934efb8ba3ac2504bc19d5d1fc9986
12014 F20101123_AABRIF kurdi_m_Page_115.QC.jpg
7a6835d887ddb3f3e25184f082d415ee
0bcd2ec81ff8ea5811738ac69a907a27c045640d
5968 F20101123_AABRHR kurdi_m_Page_107thm.jpg
f65cd16df61ac2517698b270d0e7d4da
74947d46679ceba0f35145bef8b5cce8eebefbb8
83962 F20101123_AABPYU kurdi_m_Page_200.jpg
5d91e50a47c47e0d95db4bb90b50ca67
297ec9f1d475bda1017558963dcd4e1a06edc9e9
169768 F20101123_AABQFD kurdi_m_Page_185.jp2
956b48d22df747c45e5856383705037b
e15478b786e1226482776906e13537b930b8d1e1
181620 F20101123_AABQEP kurdi_m_Page_171.jp2
dacb13051128474bf7d12cc1fcb30173
91398c477283d4fab4c74bfbe1883eafa1471fd5
83757 F20101123_AABPZJ kurdi_m_Page_016.jp2
3874 F20101123_AABRIG kurdi_m_Page_115thm.jpg
03102e622f4e1361a7e73e78bd21d5e4
41bfa3c7d1e9cdc7d98cbef88217ba7ee570b253
16517 F20101123_AABRHS kurdi_m_Page_108.QC.jpg
ca0a732a071906e19a0390bbe15c74bcc1979a34
67232 F20101123_AABPYV kurdi_m_Page_201.jpg
bdf22fec0956b18278e13925c77c8026
d5d77d424df8106f0f1123761e7c6b01ac94cfbd
183333 F20101123_AABQFE kurdi_m_Page_186.jp2
a9c7c7e201a25dda62ef86240e4528d5
0c23ef490894441a9509eb7d21d554c7d6c07a1c
147608 F20101123_AABQEQ kurdi_m_Page_172.jp2
6bcaee154a9da2de8fecfb780ec5aec9
3e8c2c9c0ef6c353e0cf8382fa791cc06e17a304
98490 F20101123_AABPZK kurdi_m_Page_017.jp2
bd4428ae332e6dca10175868741921e0
3cd4bf4daa805dc585e2e56ea0fe9802f4b900f8
12854 F20101123_AABRIH kurdi_m_Page_116.QC.jpg
433162d847921448f4093be2c101173b
05f57790b13085b5fbecaf328b8219ba7aafd51b
5146 F20101123_AABRHT kurdi_m_Page_108thm.jpg
cd8a67b20f9bf30ec5c8116c9b43328f
67064fe791cf299e624971cf7d67c3db70c85326
35699 F20101123_AABPYW kurdi_m_Page_202.jpg
a3e5223741e085c930cbdde5ba438e14
29cc68a5ce97b0f857873b72c388af50afdc0e39
174469 F20101123_AABQFF kurdi_m_Page_187.jp2
8cc1e3186a3313e96c0de4bdc7f84911
b17d94962cf5d63903c4087d4e3bb171f8584889
79313 F20101123_AABQER kurdi_m_Page_173.jp2
d58940ce5008100ee657cf5804e215d4
3c69f01146102139f877dff0106da9b178a02e00
90263 F20101123_AABPZL kurdi_m_Page_018.jp2
132b6992a60b2f45bca0b8168b696595
f7ba75145d59ab802571e42be958652ed56d0377
3894 F20101123_AABRII kurdi_m_Page_116thm.jpg
57171035c164e70a946132206deab18b
db5de45af1e6f4d07b2d9b9073b77ed647f1c25b
17617 F20101123_AABRHU kurdi_m_Page_109.QC.jpg
5a317255eac9aa2271e9c63c91ef75775f09e622
27108 F20101123_AABPYX kurdi_m_Page_001.jp2
ac5fcaf42062f6ed2f8a37021252a49c4ed3e8b9
193061 F20101123_AABQFG kurdi_m_Page_188.jp2
af338edf20423336b77544d6caef34e4
4538847433bcc9132477ce5210918981c226781b
80847 F20101123_AABQES kurdi_m_Page_174.jp2
3a42bb7042ea32158f3fb972c3f942c3
d74429c62d82e4a15179a3b8403d8db5bed3af04
97264 F20101123_AABPZM kurdi_m_Page_019.jp2
715fc5023bf840bd005937563bcc9722
abf1a71c7127d1d5de73b1d1e37dd86af06581e9
13458 F20101123_AABRIJ kurdi_m_Page_117.QC.jpg
8b489557cc4c172e82184e109ea401dc
d8a1cc5ce56837ef60df80834005dcb9ea135310
19659 F20101123_AABRHV kurdi_m_Page_110.QC.jpg
a7f047f15febc98ba68b8d0910798f00aaf61b9d
4607 F20101123_AABPYY kurdi_m_Page_003.jp2
32d487dcaf3bf035a4927b4b214117bb
e166f2da47930dfcea400af41e3e15e309b27011
183406 F20101123_AABQFH kurdi_m_Page_189.jp2
11ca39a91745deab0c86dd2707f582a80ac35a1d
90234 F20101123_AABQET kurdi_m_Page_175.jp2
86099878c527d137443d119bf403f521
e7d081c657d6c042de61b37aa5d34703fa76ebae
104912 F20101123_AABPZN kurdi_m_Page_020.jp2
90fe792285885e5d2b57fc3cf167ff1fcc2f72c9
4311 F20101123_AABRIK kurdi_m_Page_117thm.jpg
ed7bef207abc4a3a4d80e6af98926ec2
F20101123_AABRHW kurdi_m_Page_110thm.jpg
91f195ec4d9248d4919c0e2acdeeac22
737cbd4cb14a17cb8572f4c25b45d26ed3553c28
62947 F20101123_AABPYZ kurdi_m_Page_004.jp2
6dc93a3fae1e2ca7f22a146e3e130937
22855c2fe3bf6346a1893906f04fdb8ed7671ca1
181961 F20101123_AABQFI kurdi_m_Page_190.jp2
964ce5ff26c0772d4b2973e1fb038d6c
aa982f12538c28b57cb98b5de56a2baed6bde589
114210 F20101123_AABQEU kurdi_m_Page_176.jp2
0f049094b0d1ee0e7fe110751fce4e70
065a0c3f1b0bf9e6649aa1689059b37b536c3832
109123 F20101123_AABPZO kurdi_m_Page_021.jp2
9988eecf591511df2cf9eb4431f48ec0
7b64432666a3719748a8cfb836f1a604e466c3b0
22856 F20101123_AABRIL kurdi_m_Page_118.QC.jpg
3fb77179112c585da1c6784f1bf3862e47777062
20945 F20101123_AABRHX kurdi_m_Page_111.QC.jpg
da799b065593599a8edf12522466bec2
2810e0968976f81b3d1b81bd1bab2d212fd48dce
183959 F20101123_AABQFJ kurdi_m_Page_191.jp2
087380d365ec9c25e03e9f2c47ba4aee
7e5ff9238102c47e61c5e50dd5425cda8d40995f
108745 F20101123_AABQEV kurdi_m_Page_177.jp2
a20de58959187e62e528b2b3702e170c
c73127f3f09e94f0dba6cf2b446694777031d5a1
103409 F20101123_AABPZP kurdi_m_Page_022.jp2
0e9cb61c896389219e42362ee429d82e
5642 F20101123_AABRJA kurdi_m_Page_125thm.jpg
121fa6b83782c759b6fd10f1442a92b3
7def90b7c54c6a5f99b606ac3a04685d8ebaa577
6286 F20101123_AABRIM kurdi_m_Page_118thm.jpg
982d54bc529f1dc4b49825b27cf223b3
3f0c12313bafb7792a75c44d2da7891b5bb810a1
5653 F20101123_AABRHY kurdi_m_Page_111thm.jpg
2181bd8161bf2e58418b13796c2807cdcc4ed1a6
125775 F20101123_AABQFK kurdi_m_Page_192.jp2
8f7cc5fe0341c40282092e28233fc73f
75384a9faccc8592a1592fdd260b38c12bfb1179
111399 F20101123_AABQEW kurdi_m_Page_178.jp2
b32f99a7abc3736007346ce38c1b41e7
b22e4b1eb48c7a6fa47810f6a89a21df7c636fb0
17292 F20101123_AABRJB kurdi_m_Page_126.QC.jpg
f6ffc2f734f826a07839bb4a4e194a0b
bb3016997a63dc01f06b3d651bea78e9f6656d03
8608 F20101123_AABRIN kurdi_m_Page_119.QC.jpg
f626b7c650a00df3866a1478bdbbb64b
3a05a6c51f41fc475b84b426ee47bd9d96e63362
18530 F20101123_AABRHZ kurdi_m_Page_112.QC.jpg
662182364c7a9feb644f5dc52b604f1a
68207210025b95cfd3d2023520d638a5cef06109
24731 F20101123_AABQFL kurdi_m_Page_193.jp2
e7a6eac10602dc4a0d52b4de55eb6466
7f035966786ebbc18a8fa066c6789303990b42b5
96309 F20101123_AABQEX kurdi_m_Page_179.jp2
545a0f646b6df55316c39de2859b62d6
d9086f32435f995400823cec823324669a7d33a2
97653 F20101123_AABPZQ kurdi_m_Page_023.jp2
f63de8604c8be2bd4da5d48f0a9c234d
ef40c2d4e20113ae1188a031dcf4dc5b6b950bc7
5382 F20101123_AABRJC kurdi_m_Page_126thm.jpg
282ea4b5baec1bfca80b13861bbdf93b0a8f7dd0
2642 F20101123_AABRIO kurdi_m_Page_119thm.jpg
5fd33b0b9025dc3d040ec64d26bf1c1ffedbb05e
F20101123_AABQGA kurdi_m_Page_007.tif
bac616e33d8569365685d74d669c7d7c
d2cd03af1ce0053578b7eafcb55247216308af95
104686 F20101123_AABQFM kurdi_m_Page_194.jp2
130c1cfd201e2df5afef1aa10ac4a763
94745 F20101123_AABQEY kurdi_m_Page_180.jp2
fa28dabc96b643bef44178bd044970b5
b6d92ecc26e7a2a6ed89b5b778d309476e3085ba
36367 F20101123_AABPZR kurdi_m_Page_024.jp2
7188ae105f59981116c687a541b6992c
1423d4a01ef62a6466e87ceed766871186a3392a
13605 F20101123_AABRJD kurdi_m_Page_127.QC.jpg
0e33cd0ac21ef3d00236f7f282d7455a
a70bcc3cbffaf6c63a375d232d673dcfff316257
22516 F20101123_AABRIP kurdi_m_Page_120.QC.jpg
281631f92e86b1c0180861f075afd318
84525a8ce9400439020ff06e8839b5ccf5d1883d
F20101123_AABQGB kurdi_m_Page_008.tif
68135037c2c5d2a6a2a15c7c806e64eb
a7d5e14484bdb0b551f922b366d266a33ebb9404
116383 F20101123_AABQFN kurdi_m_Page_195.jp2
58560d1ab84d022f18d9c9079fb4e6b8
6b0212cdea26868d0d655fd0d9185a50e1c2f584
101442 F20101123_AABQEZ kurdi_m_Page_181.jp2
b2ab0b5cfdd084b6e0892c04f4346c03
e4eaa4834efc0465f94c1a21e105d13232beaef7
83599 F20101123_AABPZS kurdi_m_Page_025.jp2
7ebbf5d89da5f3fa8c09c287004f3c15
41b7221b8719f130b0c80dcbe3ff64e6af3ec7c3
4138 F20101123_AABRJE kurdi_m_Page_127thm.jpg
ca135bcf22b081835f89a4a4d95db3ca
91d71aa8f9dfd65036d17e0e2c42eae163fd09a2
F20101123_AABRIQ kurdi_m_Page_120thm.jpg
03ac3d401e9fcdf697be7a0493b55238c7192c86
F20101123_AABQGC kurdi_m_Page_009.tif
d2338c26172a60afb0cbea07d3500cef
3c66eda5f038c50c1d006fd593a297fc3036b892
124769 F20101123_AABQFO kurdi_m_Page_196.jp2
df0c322485c8078e98a4fe610136e28d
17f56d82f8730bc720686559f84416d641ba3fd3
871362 F20101123_AABPZT kurdi_m_Page_026.jp2
aba387cf76a6c98e020fea6f74439526
7f0354753a43cfd508651dff2c2e193ded088a1b
14382 F20101123_AABRJF kurdi_m_Page_128.QC.jpg
17663b4e4b4bf4b7102a7b3835df68b4363eda08
20005 F20101123_AABRIR kurdi_m_Page_121.QC.jpg
f44f07a7ca1c7c3b5ed94ac3c9e28845
b09741e5fe47807a81ac1d3863dc135f6ea7a192
F20101123_AABQGD kurdi_m_Page_010.tif
cdd80d01438794ba9b6e59b77431615a
107737 F20101123_AABQFP kurdi_m_Page_197.jp2
b882b11d338a90a7e60bcd6d21440dd2
94e3a49b3699c41de094d86a109d18b741b3355e
92699 F20101123_AABPZU kurdi_m_Page_028.jp2
4495 F20101123_AABRJG kurdi_m_Page_128thm.jpg
86a0809d7cfe7ae69aca0727a2d144e6
57d3c771c5d7d92a42ae6351bec0f3402031bae5
5860 F20101123_AABRIS kurdi_m_Page_121thm.jpg
7cfccb21bd44c20235b494e46404877019b716bc
F20101123_AABQGE kurdi_m_Page_011.tif
efb661a79d1e8f1f5075710af5db651e
85c890d3e0af70e3a480f2a31a7b075c340ecdeb
119137 F20101123_AABQFQ kurdi_m_Page_198.jp2
ac7f0096955977fb1b68c70a2c9bd72d
109615 F20101123_AABPZV kurdi_m_Page_029.jp2
d670ae8284080c26ae066de42c2bc0de
2495e707a7847ecbe7a3723092f18b72c0bf9ae3
13828 F20101123_AABRJH kurdi_m_Page_129.QC.jpg
6da8f93f9f33ccac7c70f4d2050bb646
4659 F20101123_AABRIT kurdi_m_Page_122.QC.jpg
d567796f99fc11d81c17f532ce66061e5094ef35
120707 F20101123_AABQFR kurdi_m_Page_199.jp2
303350a5be8ec66cca0e636f5f37b4fe
3f9532a2c9dd3bd0803f5db1b900c6f81f6dd5ed
601914 F20101123_AABPZW kurdi_m_Page_030.jp2
5a1423ab9e26d79419c9245a0993df0b63903676
F20101123_AABQGF kurdi_m_Page_012.tif
7403a18e10dafc39d080d1d7227556c1
9f5df00882cd239ec82720f0d4e09485ddd4076c
4067 F20101123_AABRJI kurdi_m_Page_129thm.jpg
92e892a286d83fed7a12bbaf7a7947f7
a36fa74c803c274d957a0c34e4bf574c2160dd55
1645 F20101123_AABRIU kurdi_m_Page_122thm.jpg
811304908f0bc45dfa934cd70ea47d6b
998fc797b3344d7b7f41ed689ef7be78a18ffd34
123549 F20101123_AABQFS kurdi_m_Page_200.jp2
9b2d48d900d795b51d74fa1c5ffb9789
711830c13502ec8d37a93d6cb37bfc81f1db17a4
80452 F20101123_AABPZX kurdi_m_Page_031.jp2
a12f19fbc5cdf8030bc91e6f5dd358bc
56c86082c650f3f2bca11ae90a28b1d4f99f537d
F20101123_AABQGG kurdi_m_Page_013.tif
22e18d652cf7f8267f5d13b84e6e3944
d33c6cb96555696c5ae678a899c1ede9ef7c5aba
12287 F20101123_AABRJJ kurdi_m_Page_130.QC.jpg
df95e70818d101d506f3fa288d32f7b2
d4c91061d25ee6120d0caae1735f994b617002c3
10665 F20101123_AABRIV kurdi_m_Page_123.QC.jpg
09d42fb8ec5711d9ac4856725fcdbc46
0761d859ac65c2f8716d578531dbae13354de3ec
98697 F20101123_AABQFT kurdi_m_Page_201.jp2
3ce99c56d7eb3be754c4021d022a996f
4f5e8c789605bf773653ee36f40794ee31cdb39a
86936 F20101123_AABPZY kurdi_m_Page_032.jp2
7a5a0bb17bfb1f981b58e8483b7f7daf
2e24bfa9115bfb55dbbd77ab501ddb3ddfe79782
F20101123_AABQGH kurdi_m_Page_014.tif
9eb8b3123fcd25e276c47163a9ba8900
c0282749729c1e706d642d3e17fe2f9a4efeaed3
3895 F20101123_AABRJK kurdi_m_Page_130thm.jpg
f36e46bae9798be70944f6dd4735ebf3
eb664159753f8e5e90a9a8416b5259fc0bc34afd
3480 F20101123_AABRIW kurdi_m_Page_123thm.jpg
48691 F20101123_AABQFU kurdi_m_Page_202.jp2
f0767f85641c143aa2af0e678874e25c
311b1fd08a2035fa0652915a39614c56ecd830ef
88737 F20101123_AABPZZ kurdi_m_Page_033.jp2
c0cc7136c0d0e8faba3210167c25df69
655ab385fcac21698bda7d137fbab4e0b5478ab4
F20101123_AABQGI kurdi_m_Page_015.tif
04ab1352439cc552ca11169bff69a3f4
15859 F20101123_AABRJL kurdi_m_Page_131.QC.jpg
483eb81735645d2b84e52902132dacf3
18280 F20101123_AABRIX kurdi_m_Page_124.QC.jpg
b86b935b7d2553ddc8ea383a62c08048
F20101123_AABQFV kurdi_m_Page_001.tif
edea97a44fa801433c5819cd08ba8de0
a5acff3e3e2e7e53a46c1ce5974ca9b3243bd0d4
F20101123_AABQGJ kurdi_m_Page_016.tif
d591744cc1fd42c86e76818d1b89cee7c9497981
7212 F20101123_AABRKA kurdi_m_Page_138thm.jpg
2b81a0e8bacaab4af93dec46e3ce3484
c918e6838af8038ed1ea257327515d6deaafd011
4984 F20101123_AABRJM kurdi_m_Page_131thm.jpg
ae5283e433105ccc3c156b5d4bf8b850
211f3c8fe8c60b08e5467f11b7a11a321d2b3dfa
5407 F20101123_AABRIY kurdi_m_Page_124thm.jpg
eb92fbb04a205218f67e4b4bcab0d94b
d56399587b21b69026e20568b59b6353f12b13fa
F20101123_AABQFW kurdi_m_Page_002.tif
2cbca5c62dbee3c76efb483bef288c90
F20101123_AABQGK kurdi_m_Page_017.tif
b23f6bccd933672dd7d25bf146ec0985
364ac774621945ca9d8754dbeef0822097f57009

PAGE 1

ROBUST MULTICRITERIA OPTIMIZATI ON OF SURFACE LOCATION ERROR AND MATERIAL REMOVAL RATE IN HIGH-SPEED MILLING UNDER UNCERTAINTY By MOHAMMAD H. KURDI 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 2005

PAGE 2

PAGE 3

PAGE 4

iv ACKNOWLEDGMENTS I would like to thank my advisor Dr. T ony Schmitz for his advice and generous financial support of my resear ch. I would like to thank Dr. Haftka for his expert advice and inspiring questions. I woul d like to thank Dr. Mann for in troducing me to the field of time finite elements. I would like to thank the committee members Dr. Schmitz, Dr. Haftka, Dr. Mann, Dr. Schuller and Dr. Ak cali for their advice, time and effort. In completing my research I was lucky to be a member of the Machine Tool Research Center where I had th e opportunity to work with intelligent and hard working graduate students. I would like to thank all fellow members for their helpful suggestions and interactions. Also, I would like to thank Ms. Christine Schmitz for taking the time to edit the dissertation draft. I would like to thank my wife Caro lina for her continued support and encouragement. I would like to thank my da ughters Alanis and Alia for bringing laughter and joy to my life. Finally, I would like to thank my mom and dad for their endless encouragement and support.

PAGE 5

v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES...........................................................................................................ix NOMENCLATURE........................................................................................................xiv ABSTRACT....................................................................................................................xvi i CHAPTER 1 INTRODUCTION........................................................................................................1 Justification of Work....................................................................................................1 Literature Review.........................................................................................................2 Optimization in Machining....................................................................................2 High-speed Milling Optimization..........................................................................3 Multi-objective Optimization................................................................................3 Stability and Surface Location Error.....................................................................4 Scope of Work..............................................................................................................6 2 MULTI-OBJECTIVE OPTIMIZATION.....................................................................8 Fundamental Concepts in Mu lti-Objective Optimization.............................................8 Single and Multi-objective Optimization..............................................................8 Definition of Multi-Objective Optimization Problem.........................................10 Definition of Terms.............................................................................................10 Pareto Optimality.................................................................................................11 Multi-objective Optimization Methods......................................................................12 Methods with a Priori Articulation of Preference s using a Utility Function......13 Weighted global criteria method..................................................................14 Weighted sum method..................................................................................14 Exponential weighted criterion....................................................................15 Weighted product method............................................................................15 Conjoint analysis..........................................................................................15 Methods with a Priori Articulation of Preference s without using a Utility Function...........................................................................................................16

PAGE 6

vi Lexicographic method..................................................................................16 Goal programming methods.........................................................................16 Methods for an a Posteriori Articulation of Preferences....................................16 Bounded objective function method............................................................17 Normal boundary intersection (NBI) method..............................................17 Normal constraint (NC) method...................................................................18 Homotopy method........................................................................................18 Choice of Optimization Method..........................................................................18 3 MILLING MULTI-OBJECTIVE OPTIMIZATION PROBLEM..............................20 Introduction.................................................................................................................20 Milling Problem..........................................................................................................20 Milling ModelEquation Chapter 3 Section 1.......................................................20 Solution Method..................................................................................................21 Problem Specifics................................................................................................22 Stability Boundary...............................................................................................22 Surface location error and stability boundary: C1 discontinuity..................23 TFEA convergence.......................................................................................24 Optimization Method..................................................................................................26 Particle Swarm Optimization Technique.............................................................26 Sequential Quadratic Programming ( SQP ).........................................................27 Problem Formulation..................................................................................................27 Problem Statement...............................................................................................27 Tradeoff Method..................................................................................................28 Robust Optimization............................................................................................29 Problem solution..........................................................................................29 Reformulation of problem............................................................................32 Bi-objective space........................................................................................37 Selection of spindle speed perturbation bandwidth......................................38 Case Studies.........................................................................................................41 Radial immersion ( a )....................................................................................41 Chip load ( c ).................................................................................................42 Discussion...................................................................................................................46 4 UNCERTAINTY ANALYSIS...................................................................................47 Milling Model.............................................................................................................48 Stability and Surface Location Error Analysis....................................................50 Bi-section Method Convergence Criterion..........................................................50 Number of Elements............................................................................................50 Numerical Sensitivity Analysis..................................................................................51 Truncation Error..................................................................................................51 Step Size..............................................................................................................52 Case Studies................................................................................................................52 Stability Sensitivity Analysis......................................................................................58 Surface Location Error Sensitivity Analysis..............................................................61

PAGE 7

vii Uncertainty of Stability Bounda ry and Surface Location Error.................................63 Input Parameters Correlation Effect....................................................................63 Monte Carlo Simulation......................................................................................64 Sensitivity Method...............................................................................................65 Latin Hyper-Cube Sampling Method..................................................................68 Robust Optimization under Uncertainty.....................................................................69 Discussion...................................................................................................................70 5 EXPERIMENTAL RESULTS...................................................................................72 Cutting Force Coefficients..........................................................................................72 Milling Forces.....................................................................................................72 Experimental Procedure......................................................................................74 Covariance Matrix (Linear Multi-Response Model)..................................................80 Compliant Tool M odal Parameters.............................................................................84 Stability Lobe Validation....................................................................................87 Stability Lobe Uncertainty..................................................................................87 Experimental Procedure......................................................................................90 Results.................................................................................................................91 Pareto Front Validation...............................................................................................93 Pareto Front Simulation Results..........................................................................93 Experimental Procedure and Results...................................................................95 Conclusions.......................................................................................................101 6 SUMMARY..............................................................................................................103 Robust Optimization Algorithm...............................................................................103 Limitations and Future Research..............................................................................104 APPENDIX A TIME FINITE ELEMENT ANALYSIS..................................................................106 B MATLAB CODE......................................................................................................115 LIST OF REFERENCES.................................................................................................177 BIOGRAPHICAL SKETCH...........................................................................................185

PAGE 8

viii LIST OF TABLES Table page 1 Classification of solutions........................................................................................13 2 Cutting conditions and modal parameters for the tool used in optimization simulations...............................................................................................................30 3 Cutting conditions, modal parameters and cu tting force coefficients used in biobjective space simulations......................................................................................37 4 Cutting conditions, modal parameters and cutting force coefficients used in radial immersion case study.....................................................................................41 5 Milling cutting conditions, modal paramete rs and cutting force coefficients used in chip load study case.............................................................................................43 6 Cutting force coefficients, modal parameters and cutting conditions of milling process......................................................................................................................49 7 Cutting coefficients for 1 insert endmill for slotting cutting tests............................77 8 Up milling cutting coefficients for 12% radial immersion......................................78 9 Estimated cutting force coefficients and their correlation matrix for 7475 aluminum and a 12.7 mm diameter solid carbide endmill with 4 teeth and 30 degree helix angle.....................................................................................................83 10 Tool modal parameters in x and y -directions...........................................................85 11 Correlation coefficient matrix for modal parameters...............................................85 12 Surface location error cutti ng conditions for two Pareto optimal designs with no uncertainty considered............................................................................................100

PAGE 9

ix LIST OF FIGURES Figure page 1 (a) Typical Pareto front in the criteria space (b) Design variables x1 and x2, and constraint in the design space.....................................................................................9 2 Pareto optimality and domination relation...............................................................13 3 Schematic of 2-DOF milling tool.............................................................................20 4 Surface location error and its absolute.....................................................................24 5 A typical stability boundary.....................................................................................24 6 Convergence of stability constraint for 5% radial immersion and different spindle speeds for an 18 mm axial depth.................................................................25 7 Schematic of milling cutting conditions and various types of milling operations...28 8 Stability, SLE f and M RRfcontours with optimum points overlaid............................31 9 A typical optimum point found; optimum point sensitivity with respect to spindle speed is apparent..........................................................................................31 10 Perturbed average of SLE f validation as optimizati on criterion that avoids spindle speed sensitive SLE f ...................................................................................33 11 Stability, SLE f and M RRf contours with optimum Pareto front points found using PSO and SQP (average perturbed spindle speed formulation).................................34 12 Pareto front showing optimum poi nts found using three optimization algorithms/formulations; the same trends are apparent............................................35 13 Variations in the eigenvalues, surfac e location error, and removal rate for PSO and SQP optima, where M RRf is the objective for both.............................................36 14 Average surface location error contour s for 300 rpm bandwidth perturbation, stability boundary and material removal rate (see Table 3).....................................38 15 Feasible domain........................................................................................................39

PAGE 10

x 16 Contour lines corresponding to constant spindle speed in feasible region of biobjective space.........................................................................................................39 17 Average surface location error contours for 100 and 300 rpm band width, stability boundary and materi al removal rate contours............................................40 18 Pareto front for spindle speed and axia l depth as design variables with radial immersion 0.508 mm, compared to the case where radial immersion is added as a third design variable..............................................................................................43 19 Pareto front using chip load as a third design variable compared to spindle speed and axial depth as design variables..........................................................................44 20 Stability, perturbed average SLE f and M RRf contours with optimum Pareto front points found using 100 rpm and 400 rpm bandwidth...............................................45 21 Schematic of 2-D milling model..............................................................................49 22 The effect of error limit in the bi section method on nume rical noise in the sensitivity calculation...............................................................................................53 23 Sensitivity of SLE with respect to Kx.......................................................................54 24 Comparison between 2nd and 4th order central difference formulas.........................55 25 The logarithmic derivative of axial depth with respect to inpu t parameters versus step size percentage..................................................................................................56 26 The variation of axial depth blim with respect to a 10% change in nominal input parameters................................................................................................................57 27 The variation of blim with respect to a 10% change in Kt and Kn. The sensitivity of blim with respect to each parameter is superimposed. Linearity of blim(Xi) can be observed (see Table 6).........................................................................................57 28 Sensitivity of axial depth blim to changes in modal mass M and modal stiffness K in the x and y -directions (see Table 6)..................................................................58 29 Sensitivity of axial depth blim to changes in modal damping C in the x and y directions..................................................................................................................59 30 Sensitivity of axial depth blim to changes in spindle speed. The spindle speed sensitivity is compared here to the modal mass and stiffness in y -direction............60 31 Sensitivity of axial depth blim to changes in force cutting coefficients in the tangential Kt and normal directions Kn.....................................................................60

PAGE 11

xi 32 Sensitivity of surface location error SLE to changes in modal parameters in y direction....................................................................................................................61 33 Sensitivity of SLE to cutting force coefficients........................................................62 34 Sensitivity of SLE to spindle speed and radial depth of cu......................................62 35 Confidence in stability b oundary due to input parameters uncertainties using Monte Carlo simulation............................................................................................65 36 Uncertainty boundary in axial depth limit using two standard deviation confidence interval...................................................................................................66 37 Uncertainty in axial depth using sensitivity and Monte Carlo methods..................67 38 Surface location error uncertainty with two standard deviation confidence interval on the nominal SLE .....................................................................................68 39 Example simulation of cutting forces to facilitate proper selection of dynamometer............................................................................................................75 40 Work-piece, dynamometer and tool setup................................................................76 41 Cutting coefficient in tangential direction ( Kt).........................................................77 42 Cutting coefficient in normal direction ( Kn).............................................................78 43 Simulated and measured forces fo r 0.12 mm/tooth chip load and 1000 rpm...........79 44 Simulated and measured cutting forces for 0.2mm/tooth chip load, b =0.4 mm and 5000 rpm............................................................................................................79 45 Simulated and measured forces at 20 krpm and b =0.4 mm for slotting...................80 46 Modal analysis test equipment typically used in machine tool structures...............86 47 Frequency response function measurement of tool in x -direction...........................86 48 Frequency response function measurement of tool in y -direction...........................87 49 Boxplot of stability lobes boundary uncertainty......................................................89 50 Histograms of axial depth limit distri butions for various spindle speeds................89 51 Probability plot of axial depth lim it distribution at 10000 rpm spindle speed.........90 52 Schematic of stability tests for partia l radial immersion cutting conditions............91

PAGE 12

xii 53 Stability lobe generated using mean values of input parameters with experimental results overlaid, also shown the boxplot co rresponding to each spindle speed used in the measurements..................................................................92 54 Fast Fourier Transform (FFT) of sound signals for selected stability tests.............93 55 Stability boundary using m ean values in the input pa rameters Pareto optimal designs are overlaid for two cases: mean va lues and uncertain input parameters...94 56 Pareto Front of perturbed average SLE and MRR The Pareto Front with uncertainty in axial depth is compar ed to the one with no uncertainty....................95 57 Surface location error experiment schematic...........................................................97 58 Measured surface location error of b= 2.12 mm and the reference dimension (A) error.......................................................................................................................... 98 59 Measured surface location error of b= 4.45 mm and the reference dimension (A) error.......................................................................................................................... 98 60 Boxplot of SLE uncertainty at spindle speed s for 4.45 mm axial depth case...........99 61 Measured surface location error of b=4.45 mm case...............................................99 62 Surface location error of preferred design conditi ons with no uncertainty considered in the optimization. Optimum spindle speeds are indicated in the figure......................................................................................................................100 63 Slotting cut with time in the cut divided into two elements...................................112

PAGE 13

xiv NOMENCLATURE slotting tranformation matrix modal damping in -direction modal damping in -direction number of elements () vector of objective x yA Cx Cy E Fx 0functions utopia point average cutting force average cutting force in x-direction average normal cutting force in x-direction x xc xeF F F F F average normal edge cutting force in x-direction average cutting force in y-direction average tangential cutting force in y-direction average tangential edy yc ycF F F ge cutting force in y-direction Identity matrix of size Q cutting force coefficients matrix defined in Appendix A tangential cutting force coefficient t nK KQIcK normal cutting force coefficient edge tangential cutting force coefficient edge normal cutting foce coefficient modal stiffness in -direction te ne x yK K Kx K modal stiffness in -direction sample size modal mass in -direction modal mass in -direction material removal rate numbex yy L Mx My MRR N r of teeth on the cutting tool particle swarm optimization PSO

PAGE 14

xv 2Q number of experimental runs cutter radius adjusted coefficient of determination surface location error sequential quadratic proadjR R SLE SQPthgramming used for utility function expanded uncertainty feasible design space two-element position vector milling model i input pe iU U X Xt X th iarameter vector of observations in i response feasible criterion space Z matrix of rank radial depth of cut axial depi iiY Z Qpp a b th of cut set of goals for objective functions maximum stable axial depth chip load deviation from the goals constraint limitj lim jb b c d eMRR f 0 cutting force coefficients vector defined in Appendix A surface location error objective function material removal rate objective set of inequality conSLE MRR jf f g straints absolute value of maximum characteristic multiplier set of equality constraints step size used to estimate numerical derivative parametelg h h pr in exponential weighted criterion rank of matrix number of response variables correlation coefficient between and combined standard ii xy cpZ r rxy u uncertainty vector of weights (preferences) -direction w x x

PAGE 15

xvi vector of design variables -direction parameter used in homotopy method vector of unknown constant parameters unbiased estimate x yy i of spindle speed perturbation(half of bandwidth) absolute error limit random error vector associated with response variance-covariancethi st ex matrix system characteristic multipliers radial depth angle radial depth angle at start of cut radial depth angle at end of cut spindle speed covariance matrix element of covariance matrix estimate of tooth passing period dampinij ijiji,j g factor

PAGE 16

xvii 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 ROBUST MULTICRITERIA OPTIMIZATI ON OF SURFACE LOCATION ERROR AND MATERIAL REMOVAL RATE IN HIGH-SPEED MILLING UNDER UNCERTAINTY By Mohammad H. Kurdi August 2005 Chair: Tony L. Schmitz Cochair: Raphael T. Haftka Major Department: Mechanic al and Aerospace Engineering High-speed milling ( HSM ) provides an efficient method for accurate discrete part fabrication. However, successful implementa tion requires the selection of appropriate operating parameters. Balancing the multiple process requirements, including high material removal rate, maximum part accurac y, chatter avoidance, and adequate surface finish, to arrive at an optimum solution is difficult without the ai d of an optimization framework. Despite the attractive gain in productivity that HSM offers, full realization of the benefits is dependent on the proper selection of cutting parameters. Parameters selected must achieve the required productivity while maintaining an acceptable accuracy. Milling models are used to aid in the proper selecti on of these cutting parameters. They provide information on whether a cutting condition is stable and/or predict the surface accuracy. However, this selection is rather tedious, costly and time consuming and might not even

PAGE 17

xviii provide an optimum solution. Parameters are se lected based on experi ence until a point is found that provide the productivity and su rface accuracy required. Difficulties encountered in this selection process incl ude sensitivity of surface accuracy to cutting parameters, uncertainties in several parameters in the milling model and the computational effort needed to account for stability and surface accuracy. Therefore, balancing the multiple requirements, including high material removal rate, minimum surface location error and chatter avoidance, to arrive at an optimum solution is difficult without the aid of op timization techniques. In this dissertation a robust optimizati on algorithm that accounts for the inherent process uncertainty and surface location erro r sensitivity is developed. Two optimization criteria are considered, namely, surface locati on error and material removal rate under the stability constraint. The trade off curve of surface location error versus material removal rate is calculated for the m ean values of input parameters, as well as for a confidence level in the stability boundary. An experime ntal validation of the robust optimization algorithm is also conducted, including an experi mental validation of the variation of the cutting forces as a function of spindle speed. The confidence level in the axial depth limit and surface location error prediction is found us ing two methods: 1) sensitivity analysis; and 2) sampling methods. The sensitivity st udy highlights the most significant factors affecting process stability a nd surface location error. The effect of input parameters correlation is included in the confidence le vel predictions using Monte Carlo and Latin Hyper-Cube sampling methods.

PAGE 18

1 CHAPTER 1 INTRODUCTION Justification of Work Intense competition in manufacturing places a continuous demand on developing cost-effective manufacturing processes w ith acceptable dimensional accuracy. Highspeed milling, HSM offers these benefits provided appropriate operating parameters are selected. Some typical applica tions include, but are not limited to, orthopedic surgery [1], end milling (pocketing) of airframe panels [2 ] and ball end milling of stamping dies [3, 4] in automotive manufacturing. Equation Chapter 1 Section 1 Despite the attractive gain in productivity that HSM offers, full realization of the benefits is dependent on the proper selection of cutting parameters. Parameters selected must achieve the required productivity while maintaining an acceptable accuracy. Milling models are used to aid in the proper selecti on of these cutting parameters. They give us information on whether a cutting condition is stable and/or they predict the surface accuracy. However, this selection is rather tedious, costly and time consuming and might not even provide an optimum solution. Parameters are selected based on experience until a point is found that provides the productivity and surface accuracy required. Difficulties encountered in this selection process incl ude sensitivity of surface accuracy to cutting parameters, uncertainties in several parameters in the milling model and the computational effort needed to account for stability and surface accuracy. Therefore, balancing the multiple requirements, including high material removal rate, M RR f

PAGE 19

2 minimum surface location error SLE f and chatter avoidance, to arrive at an optimum solution, is difficult without the aid of optimization techniques. Literature Review The literature review proceeds with a su mmary of previous implementations of optimization methods in machining, with par ticular attention to high-speed milling and multi-objective optimization. Also, a review of milling models for stability and surface location error is provided. Optimization in Machining Previous research in machining pr ocess optimization [5] has focused on mathematical modeling approaches to determin e optimal cutting parameters with regard to various objective functions. Three main obj ectives have been recognized: 1) maximum production rate or minimum cycle time [6-9]; 2) minimum cost [10-21]; and 3) maximum profit [12, 22], or a combined criterion ba sed on a weighted sum of these [23, 24]. The machining optimization problem can be formulated using deterministic and probabilistic approaches [11, 25] Several optimization techniques were used to handle both formulations. For the deterministic a pproach they include linear and nonlinear programming techniques [9, 15, 26, 27], while for the probabilistic approach chanceconstrained programming can be used [17, 28]. Other optimization techniques used in machining include graphical optimization [12, 22], polynomial geometric programming [6, 18-20, 29, 30], geometric programming [1 0] based on quadratic posylognomials (QPL) [31], goal programming with linear [32, 33] and nonlinear [34] goals, fuzzy optimization [35], and global search methods such as particle sw arm optimization [21] and simulated annealing [16].

PAGE 20

3 The machining optimization literature can al so be classified according to different constraints and design variab les handled. Several authors [7, 14] considered cost optimization for single-pass milling and turning [10, 17, 19, 20, 29]. The range of constraints considered are machine tool constr aints, such as cutting speed and feed rate, tool dynamics constraints such as cutti ng force, power and stability, and product constraints such as surface roughness. In reference, [17] some of the constraints considered are of probabilistic nature. Also, multi-pass peripheral and end milling to maximize production rate are considered [8] unde r a range of constraints with relevance to rough milling such as the machine tool limiting power, torque, f eed force and feedspeed boundaries while in anothe r work. In addition to the previous constraints, arbor rigidity and deflection are used [6]. High-speed Milling Optimization Few references are found on optimization of high-speed milling. The concept of adaptive learning (polynomial ne twork) [16] is used to construct a machining model. Simulated annealing was then used to mi nimize production cost for rough high-speed machining operations for three cutting condi tion parameters namely cutting speed, chip load and axial depth of cut. A similar study was done for low speed milling [21] where an artificial neural netw ork was used to build the machining model. However, particle swarm optimization was used to optimize production cost under machine, tool and product constraints. Multi-objective Optimization Multi-objective optimization (MOO) addre sses the issue of competing objectives using concepts first introduced by Edgewo rth [36] then expanded and developed by Pareto [37], the French-Italian economist who established an optimality concept in the

PAGE 21

4 field of economics based on multiple objectives A Pareto front [38] is generated that allows designers to trade off one objective against another. In the area of machining, Jha [24] st udied two objective f unction optimization based on cost and rate of production where example constraints were machine power, cutting speed limitations, depth of cut, and ta ble feed. The two objectives were combined using weights. Koulams [28] studied single-pass machining c onsidering the influence of tool chatter failure where a tool failure proba bility function effect was added as a penalty cost function to the objective function. Stability and Surface Location Error As explained earlier, th e full exploitation of HSM demands mathematical models to predict stability and surface location error. An unstable milling process is caused by a phenomenon called chatter. Among the first to describe chatter is Taylor, [39] who described chatter as the most obscure and de licate of all problems facing the machinist. Chatter [40] is a self-excited vibration that occurs if the chip width is too large with respect to the dynamic stiffness of the syst em. It causes undulations in the machined surface (poor surface finish) and could result in tool breakage. Extensive work has been done to generate stability bounda ries or lobes. The lobes define a region below which chatter is nonexistent. Two approaches are used to generate these lobes: 1) analytical [41] with a continuous cutting mode l or with an interrupted cu tting model [42]; and 2) time domain simulation [43, 44]. Surface location error is defined as the e rror in the placement of the milling cutter teeth when the surface is generated. This error depends on the interaction of workpiece/tool dynamic stiffness and the cutting fo rces. The correct prediction of this error depends on correct prediction of the cutting forces and resu lting deflections. Mechanistic

PAGE 22

5 models can be used to estimate these for ces. The cutting force is found by summing the forces acting on incremental sections of a he lical cutting edge [45, 46], then the surface location error is computed based on the static stiffness of the tool [47]. However, the effect of the deflection of the cutter on the cutting forces is not included. In an improvement of the previous model, the static deflection is fed back to correct the cutting forces [48, 49]. A more realistic regenerative force model [50] considered the effect of undulations in the surface gene rated by previous tooth passag e on the next tooth passage. In this model the dynamic deflection of th e tool imprints waviness on the generated surface. Using time domain simulation, surface lo cation error, cutting forces and stability lobes are predicted. An improvement on this model considered [42, 51-53] interrupted cutting as a factor influencing the stabili ty lobes and surface location error. A newly developed method uses time finite element an alysis (TFEA) to model the governing time delayed differential equation [ 54-58]. Regenerative cutting forces and dynamic deflection of the tool are all implicitly included in the governing differential equation. The advantage of this method is that it conc urrently provides surface location error and stability information on the milling process in a semi-analytical manner. In this method the governing differential equa tion is modeled by dividing th e time in the cut into a number of elements, where displacement and velocity continuity are enforced between each element. A discrete linear map is fo rmed by mapping the time in the cut to free vibration. The eigenvalues of the discrete map determine the stability boundaries, whereas fixed points of the dynamic ma p determine surface location error (SLE f ).

PAGE 23

6 Scope of Work The purpose of this dissertation is to us e optimization as a tool to efficiently determine preferred and r obust operating conditions in HSM considering multiple objectives. Although known optimization methods and machining models will be applied, there are a number of innovative aspects of this research. Firs t, proper formulation of the objective functions to account for practical application of the preferred conditions is necessary. The formulation should account for uncertainty in the milling model and sensitivity of objective(s) to process variab les. Uncertainty has not previously been considered. Second, two objectives are simulta neously optimized: su rface locat ion error SLE f and material removal rate, M RR f Stability and side bounds of design variables are considered as constraints. Prior research ha s focused only on the empi rical tool life, not the unavoidable milling dynamics and the inherent limitations they impose. The tradeoff curve (Pareto front) [38, 59] of M RR f and SLE f is generated based on nominal experimental model parameters. Experimental case studies are conducted to verify the validity of the Pareto front. The uncertainty in the milling model is addressed using Monte Carlo simulation and/or se nsitivity analysis, where a confidence interval is applied to the stability limit. The uncertainty of diffe rent input parameters such as cutting force coefficients, tool/work-piece dynamic parameters and milling process parameters are considered in the uncertainty prediction. This uncertainty is used in the selection of a robust design that would allow a venue for the practical application of the stability lobe theory at the shop floor.

PAGE 24

7 The dissertation organization proceeds as follows: Chapter 2 gives a general description of multi-objective optimization; Chapter 3 describes Pareto front generation formulation of the optimization problem, opt imization methods and case studies; Chapter 4 provides the uncertainty analysis of stab ility and surface location error; Chapter 5 describes the robust optimization algorithm and presents some practical case studies to verify stability lobes and selected design points on the Pareto front. Chapter 6 summarized the results and outlines future work in this area.

PAGE 25

8 CHAPTER 2 MULTI-OBJECTIVE OPTIMIZATION Fundamental Concepts in Mu lti-Objective Optimization Optimization is an engineering discipline wh ere extreme values of design criteria are sought. However, quite often there are multip le conflicting criteria that need to be handled. Satisfying one of thes e criteria comes at the expens e of another. Multi-objective optimization deals with such conflicting objectives. It provides a mathematical framework to arrive at an optimal design state which accommodates the various criteria demanded by the application. Equation Chapter 2 Section 1 This chapter begins with a comparison of singleand multiple-objective optimization. Next, the definition of the multi-objective optimization problem and terms are explained. Then, a summary of multi-obj ective optimization methods is presented. Finally, reasons are given for the choice of the multi-objective optimization method. Single and Multi-objective Optimization In single objective optimization one is faced with the problem of finding the optimum of the objective function. For ex ample considering the decision making involved in an investment (Fi gure 1). There are several possi ble designs in the feasible domain (A, B and 1-6). These designs are ma pped from the design space Figure 1 (b) into the criteria space Figure 1 (a). In the design space ther e are two design variables x1 (spindle speed) and x2 (axial depth) where the fe asible domain is limited by the

PAGE 26

9 constraint. If we are only con cerned about profit with no rega rd to risk (profit is our single objective), then poin t B would correspond to the maximum profit optimum design. A risk averse investor would choose risk as an objective func tion. The optimum design for the risk objective would correspond to point A. Depending on the objective function, constraints, and design variab les, different techniques are used to solve for the singleobjective optimum. However, in multi-objective optimizati on, a vector of objectives needs to be optimized. For the investment exam ple, two objectives are considered. In this two objective case there is no unique optimum, rather a set of optimum solutions is found. In Figure 1, for instance, points A, B and 5-6 are all candidate solutions. Depending on the decision makers risk aversi on, a single solution can be chosen from that set. ProfitRisk A B 1 2 3 4 $1000$5000 10% 90% Feasible domain 5 6 Feasible domain x 1 x 2 1 2 3 4 B 5 A 6 criteria space design space constraint (a) (b) Figure 1. (a) Typical Pareto front in the criteria space (b) Design variables x1 and x2, and constraint in the design space. The similarity between singleand multi -objective optimization makes it possible to use the same optimization algorithms as for the single-objective case. The only

PAGE 27

10 required modification is to transform the multiobjective problem into a single one. This may be accomplished in a number of ways, such as introducing a vector of preferences, w, to get a single objective as a weighted su m, or by solving one of the objectives for a different set of limits on the other objectives [6062]. In any case, a set of optimal solutions are found rather than a single one. It is worth noting, however, that when the objective functions are non-conf licting, the optimal set reduces to a single solution rather than a set. This can be related to the comm odity example. For instance, if we want to maximize both cost and quality, th en solution B is the only one. Definition of Multi-Objective Optimization Problem The mathematical representation of the multi-objective optimization problem is formulated as follows: 12 ,,..., 0, 1,2,..., 0, 1,2,..., T k j lMinimizeFxFxFxFx subjecttogxjm hxle (2.1) where subscript k denotes the number of objective functions F, m is the number of inequality constraints and e is the number of equa lity constraints; and n E x is the vector of design variables, where n is the number of independent design variables. Definition of Terms The feasible design space (inference space), X i s defined as the set of design variables that satisfy th e constraint set, or 0, 1,2,...,; and 0, 1,2,..., .jlxgxjmhxle (2.2)

PAGE 28

11 The feasible criterion space, Z (often called the cost space or attainable set) i s defined as the set of cost functions Fx such that x maps to a point in the feasible design space X or FxxX The preferences refer to the decision makers opi nion in terms of points in the criterion space. The preferences can be set a priori (before solution set is obtained) or a posteriori (after solution set is obtained). The preference function is an abstract function of points in the criterion space which perfectly satisfies the decision makers preferences. The utility function is an amalgamation of indivi dual utility functions of each objective that approxim ates the preference function, whic h typically cannot be expressed in mathematical form. The formation of a utility function requires insight into the physical aspects of each objective. This may require finding the Pareto front (explained next) in order to properly formulate the utility function. A utopia point is a point0kFZ that satisfies 0 ii F minimumFxxX for each 1,2...,. ik Pareto Optimality The multi-objective optimization problem has more than one global optimum. The predominant concept in defining an optimal point is that of Pareto optimality [37] which is defined as follows: a point, x X is Pareto optimal if there does not exist another point, x X such that FxFx and iiFxFx for at least one function. That is the set of Pareto optimal points dom inates any other optimal set. This can be defined by the domination relation [60], where a vector 1 x dominates a vector 2 x if: 1 x is

PAGE 29

12 at least as good as 2 x for all the objectives, and 1 x is strictly better than 2 x for at least one objective. To better understand the domin ation relation, or Pareto optimality, an example is provided [63] (Figure 2). A two-objective problem of maximizing f1 and minimizing f2 is addressed. Table 1 presents the set of solutions, classified with respect to each other. A solution P is designated as +,or = depending on whether it is better, worse or equal to a solution Q for the corres ponding objective. For example, comparing solutions A and B, we find that solution A is worse for f1 (maximizing f1) compared to B, therefore it is designate d as (-) for objective f1. Also, comparing objective f2 we find that solution A is worse than B (-). Now for a solution to belong to the non-dominated set it must be as good as the other solutions for bot h objectives and it must be strictly better for at least one objective. Considering solution A in Figure 2 we see it is worse than all other solutions (dominated); solution B is also wo rse than C for both objectives (dominated). Solution C is not dominated by point E (couple (+,-) at the intersection of the row E and the column C) and it does not dominate point E (couple (-,+) at the in tersection of the row C and the column E), therefore points C a nd E are non-dominated. Solution D is worse than C for both objectives theref ore solution D is dominated. Multi-objective Optimization Methods As explained earlier the solution to a multi-objective optimization problem is a Pareto optimal set that gives a tradeoff between the different objective functions considered. Depending on the decision makers pr eferences, a solution is selected from that set. Therefore multi-objective optimizati on methods can be categorized according to how the designer articulates his preferences (by order or by importance of objectives). This includes three cases: a priori a posteriori, and progressive articulation of

PAGE 30

13 preferences. A brief overview of the methods us ed is outlined. For a detailed description of the methods the reader is referred to reference 64. Table 1. Classification of solutions Solutions A B C D E A (-,-) (-,-) (-,-) (-,-) B (+,+) (-,-) (-,=) (-,=) C (+,+) (+,+) (+,+) (-,+) D (+,+) (+,=) (-,-) (-,=) E (+,+) (+,=) (+,-) (+,=) 2 4 6 8 10 12 14 16 1 2 3 4 5 6 f 1f 2 C E D B A Figure 2. Pareto optimality and domination relation. Methods with a Priori Articulation of Preferences using a Utility Function In these methods, the decision makers pref erences are incorporated as parameters in terms of a utility function a priori Typically these parameters can be coefficients, exponents, constraint limits, etc. These para meters determine the tradeoff of objectives before implementation of the optimization method. The optimum solution found would

PAGE 31

14 reflect the tradeoff made a priori Depending on whether the solu tion found turned out to satisfy the preferences or not, the decision ma ker can re-adjust the parameters to get a better solution. However the beau ty of these methods is that they do not require doing a multi-objective optimization problem since the a priori preferences and utility function reduce the optimization to a single one. Weighted global criteria method In this method, all objective functions are combined to form a single utility function. The weighted global criteri on is a type of utility function U in which parameters are used to model preferences. The simplest form of a general utility function can be defined as 1, 0, ork P iii iUwFxFxi (2.3) 1, 0,k P iii iUwFxFxi (2.4) where w is the vector of weights set by the decision maker such that 0 w and 11k i iw. The difference between the two above fo rmulations is rela ted to conditions required for Pareto optimality. Complete discussion can be found in reference [64]. Weighted sum method This is a special case of the weighted gl obal criteria method in which the exponent P is equal to one; that is, 1.k ii iUwFx (2.5) The method is easy to implement and guara ntees finding the Pareto optimal set, provided the objective function sp ace is convex. However, a uni formly distributed set of

PAGE 32

15 weights does not necessarily find a uniformly di stributed Pareto optimal set, which makes it difficult to obtain a Pareto solution in a desired region of the objective space. Exponential weighted criterion It is defined as follows: 11,i ik pFx pw iUee (2.6) where the argument of the summation repr esents an individual utility function for iFx. Weighted product method To avoid transforming objective functions with similar significance and different order of magnitude, one may consider the following formulation [65]: 1,iw k i iUFx (2.7) where iw are weights indicating the relative si gnificance of the obj ective functions. Conjoint analysis This method [66, 67] uses a concept borro wed from marketing, where a product is characterized by a set of attr ibutes, with each attribute having a set of levels. An aggregated utility function is developed by direct interaction with the customer/designer; the designer is asked to rate, rank orde r, or choose a set of product bundles. In engineering design studies, we can assume th at people will choose their most preferred product alternative. Conjoint anal ysis takes these sets of attrib utes and converts them into a utility function that specifies the preferen ces that the customer has for all of the products attributes and attri bute levels. The advantage of this method is that it automatically takes into account marginal dimi nishing utility (i.e., no cost is expended in a design that does not really have practical utility).

PAGE 33

16 Methods with a Priori Articulation of Preferences wi thout using a Utility Function Lexicographic method Here the objective functions are arranged in a descendi ng order of importance [68]. The highest preference objective is optimized with no regard to the other objectives, and then a single objective problem is solved consecutively (in order of preference of objectives) for a set of limits on the optimums of previously solved for objectives. This can be defined as 1,2,...,1,1, 1,2,...,i jjjMinimizeFx subject toFxFxjii ik (2.8) where i represents the functions position in the preferred sequence and j j F x represents the optimum of the jth objective function found in the jth iteration. Goal programming methods Here, goals j b are specified for each objective function jFx [69]. Then the total deviation from the goals, 1 k j jd, is optimized, where j d is the deviation from the goal j b for the jth objective. Methods for an a Posteriori Articulation of Preferences The inability of the decision maker to set preferences a priori in terms of a utility function makes it necessary to generate a Pareto optimal set after which an a posteriori articulation of preferences is made; such methods are sometimes referred to as cafeteria or generate-first-choose-later. These methods however requ ire the generation of the Pareto optimal set which may be prohibitively time consuming. It is worth noting that repeatedly solving the weighted sum approaches presented earl ier can be used to find the

PAGE 34

17 entire Pareto optimal solution for convex cr iteria space; however, these methods fail to provide an even distribution of points that can accurat ely represent the Pareto optimal set. Bounded objective function method In this method [70], the single mo st important objective function, sFx is minimized, while all other objective functions are added as constraints with lower and upper bounds such that 1,2,...,,iiilFxikis A variation of this method is the constraint [71] or trade-off method in which the lower bound il is excluded and the Pareto optimal set is obtained using a systematic variation of i This method is particularly useful in finding the Pareto optimal solution for convex or non-convex objective spaces alike. However, c hoice of the constraint vector must lie within the minimum and maximum of the objective functi on considered; otherwise, no feasible solution will be found. Also the distribution of the Pareto optimal solution will usually be non-uniform for the objective function(s) minimized. Normal boundary intersection (NBI) method This method provides a means for obtaining an even distribution of Pareto optimal points for a consistent variation in parameter vector of weights [72, 73], even with a nonconvex Pareto optimal set (a deficiency found in weighted sum method). For each parameter weight the NBI problem is solved to find an optimum point that intersects the criteria feasible space boundary, however for non-convex problems, some of the solutions found can be non Pareto optimal. Details of the method can be found in the references.

PAGE 35

18 Normal constraint (NC) method This method uses normalized objective functi ons with a Pareto filter to eliminate non-Pareto optimal solutions [74]. The indi vidual minima of th e normalized objective functions are used to construct the vertices of the utopia hyper-plane A sample of evenly distributed points on the ut opia hyper-plane is found from a linear combination of the vertices with consistently varied weights in criterion space. Each Pareto optimal point is found by solving a separate normalized si ngle-objective function with additional inequality constraints for the rema ining normalized objective functions. Homotopy method In this method the convex combin ation of bi-objective functions 121 f f is optimized for an initial value of the parameter Then homotopy curve tracking methods are used to generate the Pare to optimal solution curve for 0,1 whenever the curve is smooth [75, 76] or even non-sm ooth [62, 77] at points co rresponding to changes in the set of active constraints. Choice of Optimization Method The ease of implementation of the -constraint method [71] for a bi-objective problem makes it a good candidate method. In this method, one of the objectives is optimized for systematic variation of limits (12,...,i ) on the second objective. A uniform distribution of the Pareto optimal se t can be found for the constrained objective. There is no limitation on the convexity or nonconvexity of the objective space in finding the Pareto optimal set. However, choice of the constraint set of limits (12,...,i ) must lie within the minimum and maximum of the objective function consid ered; otherwise, no feasible solution would be found. In our case the material removal rate ( M RR f ) and

PAGE 36

19 surface location error (SLE f ) are the bi-objective criteria. The material removal rate objective would be a better choice for the c onstrained objective, since the set of limits (12,...,i ) of M RR f constraint can be more easily c onstructed according to designers preference, whereas that would be difficult for the SLE f objective.

PAGE 37

20 CHAPTER 3 MILLING MULTI-OBJECTIVE OPTIMIZATION PROBLEM Introduction In this chapter, a description of the milling problem and solution method used to solve the mathematical model is presented. Two optimization met hods of interest are briefly described. These methods are then applied to the multi-objective optimization problem and a discussion of results is provided. Equation Chapter 3 Section 1 Milling Problem Milling Model Equation Chapter 3 Section 1 The schematic for a two degree-of-freedom (2-DOF) milling process is shown in Figure 3 (repeated here). With the assumption of either a compliant tool or a structure, a summation of forces gives the following equation of motion: k y c y k x c x x y Figure 3. Schematic of 2-DOF milling tool

PAGE 38

21 ,00 0() ()()() + 00 0() ()()() mc kFt xtxtxt xx xx mc kFt ytytyt yy yy (3.1) where the terms mx, cx, kx and my, cy, ky are the modal mass, viscous damping, and stiffness terms, and Fx, and Fy are the cutting forces in the x and y directions, respectively. A compact form of the milling process can be found by considering the chip thickness variation and forces on each tooth (a detailed derivation is provided in references [54-58] and Appendix A): ()()()()()() X tXtXttbXtXt+ftb o MCKK c (3.2) where T X txtyt is the two-element position vector and M, C, and K are the 2x2 modal mass, damping, and stiffness matrices, Kc and 0 f (function of the cutting force coefficients) are defined in Appendix A, b is the axial depth of cut, = 60/( N ) is the tooth passing period in seconds, is the spindle speed given in rev/min (rpm), and N is the number of teeth on the cutting tool. As shown in Eq. (3.2), the milling model is dependent on modal parameters of the tool/w ork-piece combination and the cutting force coefficients. Solution Method As described in Chapter 1, a solution of Eq. (3.2) can be completed using numerical time-domain simulation [43, 44, 50] or the semi-analytical TFEA [54-58]. Compared to the first approach, TFEA can obtain rapid process performance calculations of surface location error,SLEf, and stability. The computational ef ficiency of TFEA compared to conventional time-domain simulation methods ma kes it the most attractive candidate for use in the optimization formulation. In this method a discrete linear map is generated that

PAGE 39

22 relates the vibration while the tool is in the cut to free vibratio n out of the cut. Stability of the milling process can be determined using the eigenvalues of the dynamic map, while surface location error (see Appendix A) is found from the fixed points of the dynamic map. Details can be found in references [54-58] An added advantage of TFEA is that it provides a clear and distinct definition of stability boundaries (i.e., eigenvalues of the milling equation with an absolute value greate r than one identify unstable conditions). Problem Specifics In this section, the calculation of the st ability boundary is analyzed, the continuity of surface location error and stability boundary is addressed, TFEA convergence is described, and sensitivity of the milling model to cutting force coefficients is defined. Stability Boundary In order to find the axial depth limit, blim, of neutral stability at corresponding input parameters, the bi-section me thod is used in the TFEA algorithm to solve for blim at which the maximum characteristic multiplie r is equal to one (stability limit) max1g (3.3) where is the eigenvalues of the dynamic map. An absolute error is used as a criterion for convergence 1 ii ibb b (3.4) where corresponds to the error tolerance and ibis the root corresponding to max1 at iteration i. The value of is set based on the numerical accuracy required in the calculation of blim. A value of 13 e can be adequate for the calculation of blim.

PAGE 40

23 Surface location error and stability boundary: C1 discontinuity Correct use of an optimization method de pends on its limitations. Gradient-based methods, for example, depend on C1 continuity (the first derivative of the function is continuous) of the objective functions ( M RRf and SLE f ) and stability constraint (Eq. (3.3)). The objective M RRf is defined analytically in Eq. (3 .6), where it is clear that it is C1 continuous. However, the SLE f and stability ( g ) functions are only found numerically using TFEA. A graphical descri ption of both functions provides some insight into the continuity of these functions. Figure 4 depicts the variation of SLEf and SLE f as a function of spindle speed for a typica l set of cutting parameters. Although SLEf is C1 continuous in the region where it is defined (stable region), SLE f is C1 discontinuous. This can be easily verified analy tically by considering the functions ()fxx and () 0 and 0 fxxxforxxforx The absolute function is clearly C1 discontinuous at0 x The same argument can be made for the near-zero SLE f range shown in Figure 4. In Figure 5 the variati on of stability function g versus spindle speed shows lobe peaks where C1 (slope) discontinuity of g is also observed. C1 discontinuity makes convergence of gradient-based optimization al gorithms near the di scontinuity rather difficult. This requires the use of multiple initial guesses in order to converge to even a local optimum.

PAGE 41

24 10 12 14 16 18 20 1 0 1 2 3 4 5 6 7 fSLE ( m) ( x 103 rpm ) fSLE |fSLE| Figure 4. Surface location error and its absolu te. Discontinuity of the absolute surface location error is apparent in the lower insert. 1 1.1 1.2 1.3 1.4 1.5 0.5 1 1.5 2 2.5 3 3.5 ( x 10 3 rpm )b (mm) Figure 5. A typical stabil ity boundary. The cusps where 1C discontinuity in the stability boundary are depicted. TFEA convergence The convergence of TFEA depends on the cu tting parameters. A higher number of elements must be used when convergence is not achieved. Either SLE f or the stability boundary g can be used to check for convergence. A typical procedure to test for the convergence of finite element meshes is to compare the change in the estimated value ( g or SLE f ) as the number of elements is increased (mesh refinement). In Figure 6, the

PAGE 42

25 dependence of convergence on the spindle speed is shown for a randomly selected cutting condition of 5% radial immersi on (percentage of radial depth of cut to tool diameter) and 18 mm axial depth. As seen in Figure 6, the flawed convergence for a small number of elements (=1) would give the impression of a sufficient number of elements. However, further increasing the number of elements (=12) shows poor convergence for the low speed. This can be due to the fact that as th e spindle speed decreases, the time in the cut increases, which requires a higher number of elements to achieve convergence. The fact that the optimization algorithm will pick milling parameters within the design space makes it necessary to choose a rather high number of elements to ensure convergence anywhere in the design space. However, a pena lty in computational time is incurred. 0 10 20 30 40 -20 0 20 40 No. of elementsg 1 2 4 6 0 10 20 30 40 50 No. of elementsg 1 20 25 30 35 40 0 2 4 6 8 10 No. of elementsg 1 10 15 20 0 1 2 3 x 104 No. of elementsg 1 500 (rpm) 1000 Figure 6. Convergence of stability constraint fo r 5% radial immersion and different spindle speeds for an 18 mm axial depth. We can see that convergence at lower speeds require substantially more number of elements.

PAGE 43

26 Optimization Method Optimization methods can be categorized according to the searching method used to find the optimum [78]. They are either direct where only the values of the objective function and constraints are used to guide the search strategy, or gradient-based, where first and/or second order de rivatives guide the search process. Particle swarm optimization (PSO) and sequential quadratic programming (SQP) will be used to test the feasibility of both methods, respec tively, for the problem at hand. Particle Swarm Optimization Technique Particle swarm optimization is an evolutionary comput ation technique developed by Kennedy and Eberhart [79, 80]. It can be used for solving si ngle or multi-objective optimization problems. To find the optimum so lution, a swarm of pa rticles explores the feasible design space. Each particle keeps tr ack of its own personal best (pbest) fitness and the global best (gbest) fitness achieved during design space exploration. The velocity of each particle is updated toward its pbest and the gbest positions. Acceleration is weighted by a random term, with separa te random numbers being generated for acceleration toward pbest and gbest. In order to accommodate constraints, Xiaohui et al. [79] presented a modified particle swarm optimi zation algorithm, where PSO is started with a group of feasible solutions and a feasibility function is used to check if the newly explored solutions satisfy all the constraints. All the particles keep onl y those feasible solutions in their memory while discarding infeasible ones.

PAGE 44

27 Sequential Quadratic Programming ( SQP ) The basic idea of this me thod is that it transforms the nonlinear optimization problem into a quadratic sub-problem around the initial guess. The nonlinear objective function and constraints are transformed into their quadratic and linear approximations. The quadratic problem is then solved iterati vely and the step size is found by minimizing a descent function along the search direction. Standard optimizati on algorithms may be used to solve the quadratic sub-problem. Usually SQP leads to identification of only lo cal optima. In order to better converge to the global optimum, a number of initial guesses is used to scan the design space and the optimum of th ese local optima is clos e to the global optimum. Problem Formulation In this section, the multi-objective optimization problem is defined and then a description of the tradeoff met hod is given. The problem soluti on is then presented in the order it has been addressed in the robust optimization section. Finally, discussion of the simulation results is provided. Problem Statement The problem of minimizing surface location error SLE f and maximizing material removal rate M RRfis stated as follows: ,,,,,,,,,, : ,,,,1SLEMRRminfabcNfabcN subjecttogbmaxabN (3.5) where g is the stability constraint obtain ed from the dynamic map eigenvalues,SLEf is found from the fixed points, and the mean M RRfis given as: MRRfabcN (3.6)

PAGE 45

28 where a, b, c, N and are radial depth of cut, axial de pth of cut, feed per tooth (chip load), number of teeth, and spindle speed, re spectively (Figure 7). From Eq. (3.5) it can be seen that only the stability constraint is not a function of the feed per tooth. In Eq.(3.5) SLEf and M RRf are explicitly stated as a function of cutting conditions (a, b, c, N and ). This reflects the relative ease by which th ese conditions can be adjusted to achieve optimality of the objectives. Chip load, c N=2 Axial depth, b Radial depth of cut, a x y Slotting x y Up milling x y Down milling ex ex = st =0 st =0 ex = st a R Figure 7. Schematic of milling cutting conditi ons and various types of milling operations. Tradeoff Method To address the multi-objective problem the constraint method is used, where the two-objective problem is tran sformed into a single objectiv e problem of minimizing one objective with a set of different limits on the second objective. E ach time the single objective problem is solved, the second objective is constrained to a specific value until a sufficient set of optimum points are found. Thes e are used to generate the Pareto front [38] of the two objectives. In the case that SLEf is chosen as the objective function to be minimized then Eq. (3.5) is transformed to:

PAGE 46

29 ,,,,, : ,,,,, 1 ... ,,,,,,1,SLE MRRiminfabcN s ubjecttofabcNeforik gabNmaxabN for a series of k selected limits (e) on MRRf. (3.7) Where the cutting conditions: a, b, c, N and are the design variables. On the other hand if M RRf is chosen as the objective function to be maximized, then Eq. (3.5) is transformed to: ,,,,, : ,,,,, 1 ... ,,,,,,1,MRR SLEiminfabcN s ubjecttofabcNeforik gabNmaxabN for a series of k selected limits (e) on SLEf. (3.8) It should be noted that a pplying Eq. (3.7) using the SQP method is more straightforward than Eq. (3.8). The reason is that in order to use a number of initial guesses along the SLE f contour in Eq. (3.8), the ax ial depth corresponding to that SLE f needs to be found, whereas in Eq. (3.7) the ax ial depth can be explic itly expressed found as shown in Eq. (3.6). Robust Optimization Problem solution In the first iteration of th e problem, only axial depth (b) and spindle speed ( ) are considered as design variables. Other cutting conditions are held fixed (Table 2) for a down milling cut. Modal parameters for a single degree-of-freedom tool with one dynamic mode in x and y directions are used (Table 2). The nominal values of the tangential (Kt) and normal (Kn) cutting force coefficients are 550 N/mm2 and 200 N/mm2, respectively. The SQP method is used to find the Pareto front using the formulation in Eq. (3.7). Here SLE f is minimized for a set of limits on M RRf. As mentioned earlier, the

PAGE 47

30 SQP method is a local search method that is highly dependent on C1 continuity of the objective function and constraints. To obtain a global optimum, a number of initial guesses are used along each M RRf constraint limit. A set of optimum points are obtained for these initial guesses. The minimum of thes e optimum points is nominated as a global optimum. The number of initial guesses is incr eased and another run of the optimization simulation is made to check the validity of that global optimum. Table 2. Cutting conditions and modal parameters for Tool used in optimization simulations M (kg) C (Ns/m) K (N/m) 0.0560 00.061 3.940 03.86 6 61.52100 01.6710 Tool diameter (mm) c (mm) a (mm) N 19.05 0.178 0.76 2 Kt (N/m2) Kn (N/m2) Kte (N/m)Kne (N/m) 550 x 106 200 x 106 0 0 In this formulation, the minimum SLE f points were found to favor spindle speeds where the tooth passing frequency is equal to an integer fraction of the systems natural frequency (Figure 8), which corresponds to the most flexible mode (these are the traditionally-selected best speeds which are lo cated near the lobe peaks in stability lobe diagrams). BecauseSLEfcan undergo large change s in value for small perturbations in at these optimum points, the formulation provid ed in Eqs. (3.7) and (3.8) leads to optima which are highly sensitive to spindle speed vari ation (Figure 8) To show the sensitivity of these optimum points, a typical optimum point is superimposed on a graph of SLE f vs. in Figure 9. It is seen that th e optimum point is located in a highSLEfslope region.

PAGE 48

31 5 10 15 20 25 30 0 2 4 6 8 10 12 (x 103 rpm)b (mm)51 01 52 02 53 04 05 0 m7 0 6 0 0 m m3/ s5 5 05 0 04 5 04 0 03 5 03 0 02 5 02 0 01 5 01 0 05 0 Stability Lobe Optimum points Figure 8. Stability, SLE f and M RRfcontours with optimum points overlaid. The figure shows that optimum points occur in regi ons sensitive to spindle speed variation (Table 2). 4 6 8 10 12 14 -5 0 5 10 fSLE (m) (x103 rpm) Surface location error Spindle speed sensitive optimum Figure 9. A typical optimum point found; op timum point sensitivity with respect to spindle speed is apparent (Table 2).

PAGE 49

32 Reformulation of problem The optimization problem was redefined in order to avoid convergence to spindle speed-sensitive optima. Two approaches were applied: 1) an addi tional constraint was added to the SLE f slope; and 2) the SLEf objective was redefined as the average of three perturbed spindle speeds. The la tter proved to be more robust than the former. This is due to the difficulty in setting the value of the SLE f slope constraint a priori. The spindle speed perturbed form of the problem transforms Eqs. (3.7) and (3.8) to ,,, 3 : ,, 1 ... ,,,1,MRRifbfbfb SLESLESLEmin subjecttofbeforik gbgbgb for a series of selected limi ,MRRts (e) on f (3.9) and ,,, 3 ,, : 1 ... ,,,1,MRR ifbfbfb SLESLESLEminfb s ubjecttoforik gbgbgb for a series of selected limits () on a ,SLEverage perturbedf (3.10) where is the spindle speed perturbation selected by the designer (a typical value for our analyses was 50 rpm). A study of spindle speed perturbation selecti on is provided in the next section. The validity of the perturbed SLE f average as a convergence criteria can be seen in Figure 10. In this figure the perturbed average SLE f is plotted with SLE f where points A and B correspond to highly and moderately spindle speed-sensitive SLE f respectively.

PAGE 50

33 The average perturbed SLE f at point A (high slope point) is shown to be higher than at point B. Therefore, using the perturbed average SLE f as an objective function criteria can avoid convergence to spindle speed sensitive SLE f (such as SLE f region near point A). 10.5 11 11.5 12 12.5 13 13.5 14 0.2 0.4 0.6 0.8 1 1.2 fSLE ( m) x x A B (x 103 rpm) Perturbed average fSLE fSLE around point A relatively more sensitive |f SLE | region Figure 10. Perturbed average of SLE f validation as optimizati on criterion that avoids spindle speed sensitive SLE f Shown in the figure are points A (close to steep slope region of SLE f ) and B (close to moderate slope region of SLE f ), the perturbed average of SLE f near A is higher than at B. Therefore, using the perturbed average as an optimum criterion is valid. The SQP method is used to solve Eqs. (3.9) and (3.10). In case Eq. (3.9) is implemented then initial guesses of and b (design variables) are made along the M RRf contour. In the other case (Eq. (3.10)) the initial guesses of and b are made along SLE f

PAGE 51

34 contour. The number of initial guesses along th e constraint is made such that convergence is towards a global optimum. The initial guesse s for the spindle speed are increased in 625 rpm increments for the corresponding sp indle speed range considered. Also, the PSO method is used to solve Eq. (3.8). When using PSO, the optimum points do not tend to converge to spindle speed sens itive optimums. Therefore, th ere is no need to solve the reformulated form of the problem in PSO. This leads to a fewer number of evaluations of SLE f and is a computati onally more efficient optimization method. A comparison of the three optimization sc hemes is shown in Figure 11 and Figure 12. Figure 11 shows the optima for each approach superimposed on the corresponding stability lobe diagram. In Fi gure 12, the Pareto fronts for the three methods are shown. The optimum points found using the two SQP formulations closely agree with the PSO method (Figure 12). Figure 11. Stability, SLE f and M RRf contours with optimum Pa reto front points found using PSO and SQP (average perturbed spindle speed formulation). The figure shows that optimum points are not in regions sensitive to spindle speed (Table 2).

PAGE 52

35 50 100 150 200 250 300 350 400 450 0 2 4 6 8 10 12 14 16 18 20 fMRR (mm3/s)perturbed average |fSLE| (m) SQP SLE objective SQP MRR objective PSO MRR objective Figure 12. Pareto front showing optimum points found using three optimization algorithms/formulations; the same trends are apparent. However, the SQP methods required additional computational time (Table 2). Although the PSO points show the same trend, some improvement in the fitness is still possible relative to the SQP results. Because the PSO search inherently avoided optimum points that are spindle speed inse nsitive, there is no need to use average perturbed fSLE as with SQP, which leads to a decreased number of fSLE evaluations in PSO. However, narrow optimum poin ts may go undetected when using PSO. As noted, when comparing the Pareto fr onts in Figure 12, it is seen that the PSO approach did not converge to the same fitness as SQP method. A check of the optimum points which correspon d to a value of SLE f = 4 m, for example, shows that PSO converged to 100 mm3/s, while SQP converged to 150 mm3/s. To better understand this

PAGE 53

36 result, the design space was divide d between the two design vectors, b and for SQP and PSO using a factor, a, that was normalized between 0 and 1. The PSO and SQP optimums were normalized to a = 0 and 1, respectively. Ne xt, the stability constraint ( g ), M RRf, and SLE f were plotted against that rati o. In Figure 13 it is seen that discontinuities exist in the SLE f constraint and the first de rivative of the eigenvalue constraint within this region. Although PSO is not significantly affected by a discontinuity in the derivative constraint, it can be affected by a discontinuity of the SLE f constraint, where the discontinuity tends to narrow the search region of the swarm. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.9 1 1.1 ag PSO optimum at 4 m SQP optimum 4 m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 afSLE m PSO optimum at 4 m SQP optimum 4 m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100 120 140 160 afMRR mm3/s PSO optimum at 4 m SQP optimum 4 m Figure 13. Variations in the eigenvalues, su rface location error, and removal rate for PSO and SQP optima, where M RRf is the objective for both. Th e discontinuities in the surface location error cause PSO to not converge on the SQP optimum.

PAGE 54

37 Bi-objective space In this section, the bi-objective domain (the feasible space of the objective functions) of average perturbed SLE f and M RRf for the set of input parameters listed in Table 3 for an up milling case is provided. Figure 14 shows the objective contours in the design space of spindle speed ( ) and axial depth (b). The respective bi-objective space is shown in Figure 15 and Figure 16. In Figure 15 the contours of constant axial depth are shown, while the contours of constant sp indle speeds are shown in Figure 16. These figures give an idea of the feasible design a nd bi-objective space. It can be seen that the bi-objective feasible sp ace can be non-convex (not all poi nts on a straight line connecting two points in the feasible domain belong to th at domain). This makes the choice of using the tradeoff method as a multi-objective optimi zation approach a suitable one, since this method can handle both convex and non-conve x problems. A good observation can be made from Figure 15, where it can be seen that for the high M RRf region with high b values, the relative sensitivity of SLE f increases compared to the lower M RRf region. Table 3. Cutting conditions, modal parameters and cutting force coefficients used in biobjective space simulations M (kg) C (Ns/m) K (N/m) 0.440 00.35 830 090 6 64.45100 03.5510 Tool diameter (mm) c (mm) a (mm) N 25.4 0.1 21.8 1 Kt (N/m2) Kn (N/m2) Kte (N/m)Kne (N/m) 700 x 106 20 x 106 46 x 103 33 x 103

PAGE 55

38 Selection of spindle speed perturbation bandwidth In Figure 10, it was shown that the average perturbation of SLE f provided an adequate optimization criteria. However, the choice of the spindle speed perturbation step size or bandwidth,2 depends on the designer pref erence. Any spindle speed perturbation in SLE f would avoid convergence to sensitive SLE f optima. Depending on the machining center spindle drive accuracy, the perturba tion bandwidth can be set accordingly. The average perturbed SLE f contours of 100 and 300 rpm bandwidth are shown in Figure 17 (use Table 3 paramete rs). The high slope region of average SLE f in the 100 rpm bandwidth case is repl aced by higher values of average SLE f making the optimization formulation favor insensitive spindle speed SLE f (x 103 rpm)b (mm)2 53 5 4 55 56 51 4 01 2 01 0 0701 051 0153 05 0 2 70 02 5 0 023 002 1 0 01 9 0 01 8 0 01 6 0 01 4 0 01 2 0 01 0 0 09 0 07 0 05 0 03 0 0 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Stability boundary fMRR mm3/s perturbed average |fSLE| m Zig zag line indicate the limit of the domain where SLE is calculated Figure 14. Average surface location error co ntours for 300 rpm bandwidth perturbation, stability boundary and material removal rate (see Table 3).

PAGE 56

39 perturbed average |f SLE | ( mm)4 7 m m 4 3 3 9 3 73 5 3 5 3 53 12 72 5 2 32 11 91 71 31 11 10 7 500 1000 1500 2000 2500 20 40 60 80 100 120 140 160 Feasible domain Pareto front f MRR (mm 3 /s) Figure 15. Feasible domain. C ontour lines corresponding to c onstant axial depth in the stable region in the bi-objective space (see Table 3). fMRR (mm3/s)perturbed average |fSLE| ( m)1515.114.6 x103 rpm15.415.610.116 13.9 14.6 12.2 11.219.6 18.314.917 500 1000 1500 2000 2500 20 40 60 80 100 120 140 160 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 Figure 16. Contour lines correspond ing to constant spindle speed in feasible region of biobjective space (see Table 3).

PAGE 57

40 (x 103 rpm)b (mm) 2 3 5 710142440598572534328 17 13 9 5 1700150013001100900700500300100 rpm bandwidth 10 12 14 16 18 20 1 2 3 4 5 6 (x 103 rpm)b (mm) 2 3579142030426181695746362719 368 1700150013001100900700500 300 300 rpm bandwidth 10 12 14 16 18 20 1 2 3 4 5 Stability boundary fSLE m fMRR mm3/s Figure 17. Average surface location error contours for 100 and 300 rpm band width, stability boundary and ma terial removal rate contours (see Table 3).

PAGE 58

41 Case Studies As opposed to the previous anal ysis of two design variables ( and b), two cases of an added third design variable were anal yzed. The first one was for radial immersion (a) and the second one was for chip load (c). These cases are compared to the two design variable case. Radial immersion ( a ) Previous simulations consid ered spindle speed and axia l depth of cut as design variables. Another simulation was completed using radial immersion as a third design variable for an up milling cut. It was compared to a two design variable case where radial immersion was held constant at 0.508 mm in a 25.4 mm to ol (Table 4). Figure 18 shows the Pareto front for these two cases. It is seen that adding radial immersion as a third design variable improved the value of perturbed average SLE f with respect to the constant radial immersion case. The optimum ra dial immersion found was 0.58 mm for all optimum points up to 500 mm3/s. In both simulations the same spindle speed perturbation (170 rpm) was used. As seen in Figure 18, a better calculation of the Pareto front (smoother than Figure 12) is found by using sm all increments in the spindle speed (each 100 rpm) initial guesses. However, the SLE f found in Figure 18 appear to be unrealistically small which ma y warrant further analysis. Table 4. Cutting conditions, modal parameters and cutting force coefficients used in radial immersion case study M (kg) C (Ns/m) K (N/m) 0.250 00.23 34.40 027.0 6 1.30100 6 01.2010 Tool diameter (mm) c (mm) a (mm) N 25.4 0.1 0.508 2 Kt (N/m2) Kn (N/m2) Kte (N/m) Kne (N/m) 700 x 106 210 x 106 0 0

PAGE 59

PAGE 60

43 0 100 200 300 400 500 600 0 0.005 0.01 0.015 0.02 0.025 0.03 fMRR (mm3/s)Perturbed average |fSLE| ( m) radial immersion as a third design variable radial immersion = 0.508 mm Figure 18. Pareto front for spindle speed and axial depth as design variables with radial immersion 0.508 mm, compared to the case where radial immersion is added as a third design variable. The optimum radial immersion for the latter case is 0.58 mm up to 500 mm3/s (see Table 4). Table 5. Milling cutting conditions, modal para meters and cutting force coefficients used in chip load study case M (kg) C (Ns/m) K (N/m) 0.0270 00.03 70 02 6 61.0100 01.610 Tool diameter (mm) c (mm) a (mm) N 12.7 0.1 0.635 2 Kt (N/m2) Kn (N/m2) Kte (N/m)Kne (N/m) 600 x 106 180 x 106 0 0

PAGE 61

44 0 200 400 600 800 1000 0 5 10 15 20 25 30 35 fMRR (mm3/s) Perturbed average |fSLE| (m) chip load constant chip load 3rd design c=0.16 to 0.2 mm/tooth Figure 19. Pareto front using chip load as a third design variable compared to spindle speed and axial depth as design variables. For the three design variable case, an improvement in the average surface loca tion error can be seen (see Table 5).

PAGE 62

45 (rpm)b (mm)264610163052122446468121620242832812182432425468844814 208122444 12001150 1100 1050950850 8007507006005505004504003502503006501000 90020015010050 10 15 20 25 30 35 40 2 4 6 8 10 12 14 16 18 Average fSLE ( m) for 400 rpm bandwidth fMRR (mm3/s) Stability boundary Optimum points 400 rpm bandwidth Optimum points 100 rpm bandwidth Figure 20. Stability, perturbed average SLEf, and MRRf contours with optimum Pareto front points found using 100 rpm and 400 rpm bandwidth. This case study shows the di fficulty in selecting op timum points based on experience (Table 5).

PAGE 63

46Discussion The formulations provided in Eqs. (3.9) and (3.10) proved adequate in finding the Pareto optimal set insensitive to spindle speed variation, provided an appropriate number of initial guesses is made. Also, the Eq. (3.9) formulation is easier to apply using the SQP method, where the initial guesses are made along the M RRf contour. The generation of the Pareto front for the multi-design variable case can be rather time-consuming. However, if the designer is given that freedom of choice, it might be a necessity. For example, the effect of adding chip load or radial immersion as a third design variable gave a substantial impr ovement in the surf ace location error in comparison to the two design variable case. This is counterintuit ive to using a lower value of c or a as means of reducing the surface location error. The effect of spindle speed perturbation bandwidth on the sensitivity of optimum points is rather complex. Qualitatively, in Figure 17 it is shown that increasing the bandwidth from 100 rpm to 300 rpm had the sa me effect of increasing the value of SLE f near the sensitive region. Further investiga tion is needed to establish a quantitative relation between bandwidth and sensitivity of optimum points.

PAGE 64

47 CHAPTER 4 UNCERTAINTY ANALYSIS In Chapter 3, optimization was used to find preferable designs for two objectives: material removal rate (MRR) and surface location error [48, 81, 82] (SLE), with a Pareto front, or tradeoff curve, found for the two competing objectives. Although the milling model used in the optimization algorithm is de terministic (time finite element analysis), uncertainties in the input parameters to the model limit the confidence in the optimum predictions. These input parameters include cutting force coefficients (materialand process-dependent), tool modal parameters and cutting conditions. By accounting for these uncertainties it is possible to arrive at a robust optimum operating condition. In previous studies [83-85], uncertainty in the milling process was handled from a control perspective. The uncertainty in the cutting force was accommodated using a control system. The force c ontroller was designed to comp ensate for known process effects and accounted for the force-feed nonlinea rity inherent in metal cutting operations. In this study, the uncertainties in the milling model are estimated using sensitivity analysis and Monte Carlo simulation. This en ables selection of a preferred design that takes into account the inhere nt uncertainty in the model a priori. This chapter begins with a description of the milling model and continues with a discussion of stability lobes and surface loca tion error analysis w ith regard to their numerical accuracy. Sensitivity analysis is discussed in the next section. Then, case studies for the numerical accuracy of the sensi tivities of the maximum stable axial depth, blim, and SLE are presented for a typical two degree-of -freedom tool. This enables us to

PAGE 65

48 carry out the stability lobe and surface locati on error sensitivity anal ysis in the next two sections. Sensitivity is used to determ ine the effect of input parameters on blim and SLE. This enables the determination of which pa rameter(s) is the highe st contributor to stability enhancement and SLE reduction. The uncertainties in blim and SLE predictions are then calculated using two methods 1) the Monte Carlo simulation; and 2) the use of numerical derivatives of the system characte ristic multipliers to determine sensitivities. The uncertainty in axial depth effects a reduction in the MRR, and the SLE uncertainty provides bounds on SLE mean expected value. This allo ws robust optimization that takes into consideration both performance and uncertainty. Equation Chapter 4 Section 1 Milling Model A schematic of a two degree-of-freedom milling tool is shown in Figure 21. The tool/work-piece dynamics and cutting forces are used to formulate the governing delay differential equation for the system. Solution of the delay differen tial equation is found using time finite element analysis (TFEA) [54-56]. This method provides the means for predicting the milling process stability and quality (SLE). However, the uncertainty in the input parameters to the solution method pl aces an uncertainty on the stability and SLE prediction. These parameters are divided into two groups; 1) uncerta inty from lack of knowledge of the tool modal matrices, K, C and M, and the cutting force coefficients (mechanistic force model); and 2) uncertaint y in other machining parameters, such as spindle speed, chip load and radial depth. To estimate the parameters in the former, modal testing is used to measure the dyna mic parameters while cutting tests are completed to estimate the cutting force coefficients. In the modal parameter estimation the peak amplitude method is used to fit th e measured frequency response function. In this method [86, 87], the peak of the ma gnitude of the frequency response function

PAGE 66

49 corresponds to the natura l frequency. From this the half power frequencies are used to estimate the damping ratio. Table 6 lists th e mean modal values for 25.4 mm diameter endmill having a 12 helix angle with 114 mm ove rhang length and the corresponding cutting force coefficients for 6061 aluminum (assuming a mechanistic force model, see Chapter 5). The cutting conditions are also list ed in the table. These parameters will be used in the simulations in this chapter for a down milling cut. K x x K y C x C y y Feed SLE Figure 21. Schematic of 2-D milling model. Surface location error (SLE) due to phasing between cutting force and tool displacement is also shown. Table 6. Cutting force coefficients, modal parameters and cutting conditions of milling process. M (kg)K (N/m x106)C (N.s/m)x 0.44 4.45 830.030 y 0.44 3.55 90.90.036Kt(N/m2 x106)Kn(N/m2 x106)Kne(N/m x103)6001806 Tool diameter (mm)radial dept h,a (mm)chip load, c (mm/tooth) 25.4 0.5080.1Kte(N/m x103)12 N 1

PAGE 67

50 Stability and Surface Location Error Analysis The stability lobes are used to repres ent the stable space of axial depth (b) and spindle speed of a milling process. In TFEA [54-57], a discrete map is used to match the tool-free vibration while out of the cut, w ith the tool vibration in the cut. The system characteristic multipliers ( ) of the map provide the st able cutting zone where max is less than one. TFEA provides a field of max in the design space of b and The limit of stability, blim can be found using root-finding numerical techniques. Here we use the bisection root-finding method. The convergence cr iterion of the bi-section method should account for the amplification of numerical noi se induced by sensitivity estimation. It should be noted that the number of elemen ts affects the accuracy of the estimation. For calculation of SLE in TFEA, the numerical noise is only due to the number of elements. In this section we will discuss th e effect of both the convergence criterion and the number of elements on th e sensitivity estimation of blim and SLE. Bi-section Method Convergence Criterion As described in Chapter 3 the axial depth limit, blim, was calculated using the bisection method (Eq. (3.4)). Although a relativel y large value of can be adequate for the calculation of the stability lobe s, a tighter limit is needed to calculate the sensitivities. This is attributed to amplification of nume rical noise in the deri vative calculation. This comparison is made in the Case Studies section. Number of Elements The accuracy of TFEA pred iction of stability and SLE is highly dependent on the number of elements used. The effect of the number of elements is even more apparent

PAGE 68

51 when calculating the sensitivity of the prediction, where a higher number of elements is needed to eliminate numerical noise from the sensitivity calculation. Numerical Sensitivity Analysis The sensitivity of axial de pth to input parameters / bX i is cumbersome to compute analytically using the TFEA method; therefore, a numerical derivative is used by implementing a small perturbation. Factors which affect accurate calculation of sensitivity to inputs include: 1) central difference truncation error; and 2) step size selection. Theref ore, a balance needs to be achieved in determining the sensitivity that provides a stable estimate of the sensitivity while maintaining computational efficiency. In the following, we describe these factors and their consideration in the calculati on of stability and SLE sensitivities. Truncation Error The central difference method is used in the sensitivity calculation. The formula for this method is 2 11, 2ibb b Oh Xh (4.1) where h denotes the step size in input parameter Xi, 1ibbXh 1ibbXh and O(h2) is the 2nd order truncation error. A higher order formula with 4th order truncation error O(h4) can also be used. However, as show n in Eq. (4.2), it is two times more computationally expensive than Eq. (4.1), 4 211288 12ibbbb b Oh Xh (4.2)

PAGE 69

52 In order to help decide whether the highe r truncation error formula need be applied (Eq. (4.2)), the sensitivity of blim with respect to modal stiffness Kx is calculated as a function of step size h This comparison is made in the Case Studies section. Step Size The step size, h in Eqs. ((4.1) and (4.2)) should be carefully chosen. This is especially important when there is numerical noise in the calculated blim due to the convergence criterion (Eq. (1)). The step si ze should be large enough to be out of the numerical noise range, however, not so larg e that the non-linear va riation in the output ( blim or SLE ) takes effect. The following section illustrates this idea. Case Studies In this section, numerical estimations of the sensitivity are made based on different variations of convergence criterion, number of elements, sensitivity analysis formula (Eq. (4.1) and Eq. (4.2)), and step size. The co mparisons are made for a 10 krpm spindle speed, 10 elements and 4310 x unless otherwise noted. The logarithmic derivative can be used in making these comparisons by evaluati ng the percentage of change in an output (axial depth, b ) due to a percentage change in the input, Xi. It is expressed as ln lni iib X b X bX (4.3) To illustrate the effect of convergence cr iterion, the logarithmic derivative of blim with respect to Mx (the X direction modal mass) is calculated for two error limits as a function of step size percentage %/100iihXX see Figure 22. It can be seen that a tighter error limit nearly eliminates the num erical noise in the derivative calculation.

PAGE 70

53 The effect of the number of elements on SLE sensitivity is illustrated in Figure 23, where the SLE sensitivity with respect to Kx is calculated. The /xSLEK is used to illustrate the effect of the number of elements because it is known that the SLE does not depend on the Kx stiffness (tool feeding direction being the xaxis). Therefore /0xSLEK which would amplify and illustrate more clearly the effect of the number of elements on the sensitivity estimation. The higher number of elements provides a larger stable region of sensitivity It should be noted that the 2nd order finite difference method is used in this sensitivity comparison and the bi-section convergence criterion is not applicable here since SLE is found from fixed points of the dynamic map (see Eq. (A.18) in Appendix A) when the cutting conditions provide a stable cut. 0 0.1 0.2 0.3 0.4 0.5 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 %hln(blim)/ln(Mx) = 3e-4 = 3e-10 2nd order central difference E=10 Figure 22. The effect of error limit in the bisection method on numerical noise in the sensitivity calculation (see Table 6).

PAGE 71

54 0 0.1 0.2 0.3 0.4 0.5 -6 -4 -2 0 2 4 6 8 10 12 14 x 10-23 %hSLE/Kx (m2/N) E = 10 E = 30 E = 50 Figure 23. Sensitivity of SLE with respect to Kx. The higher number of elements, E, provides more stable sensitivity estimation. The second order finite difference formula is used here (see Table 6). Figure 24 shows the effect of the central difference truncation error. A finite step size percentage is needed to reach a stable value of the derivative for both formulas. It can be seen that Eq. (4.2) gives a wider ra nge of step sizes at which the sensitivity calculation is stable. However, the improved stability range, or reduction in numerical noise, is not significant to sacrifice co mputational efficiency for its usage.

PAGE 72

55 0 0.1 0.2 0.3 0.4 0.5 0 0.5 1 1.5 2 2.5 x 10-9 %hblim/Kx (m2/N) 2nd Order central difference 4th order central difference Convergence limit = 3e-4 E=10 Figure 24. Comparison between 2nd and 4th order central difference formulas. The 4th order formula shows a wider stable regi on for step size, but higher computation time (see Table 6). The importance of step size selection can be illustrated by Figure 25, which shows the logarithmic derivative of ax ial depth with respect to input parameters versus step size percentage. It can be seen that the step size should be chosen high e nough to be out of the numerical noise range but not so high so that the non-linear variation is included (in this range of % h only is non-linear). The figure also i ndicates the relative sensitivity of axial depth to each input parameter, spindl e speed having the largest effect followed by modal mass and stiffness.

PAGE 73

56 0 0.5 1 1.5 2 -4 -3 -2 -1 0 1 2 h%ln(blim)/ln(Xi) Kx Cx Mx Kt Kn Convergence limit =3e-4 E=10 Figure 25. The logarithmic derivative of axia l depth with respect to input parameters versus step size percentage (see Table 6). From Figure 24 and Figure 25 it can be seen that h= 0.2% provides a stable sensitivity estimation. To verify that a typical step size of 0.2%, convergence limit 4310 E=10, and the 2nd order finite difference a pproximation give correct calculation of sensitivity, the variations of b to modal parameters and cutting coefficients are plotted in Figure 26 and Fi gure 27, respectively. Also, the slope predicted using Eq. (4.1) with h=0.2% is superimposed on the same plot. The suitable selection of h is indicated by the tangency of the predicted slope to the functional variation. On the other hand, it can be seen that wh en the variation is linear, th e linear approximation can be accurate for a large variation of the input parameter.

PAGE 74

57 90 95 100 105 110 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 Xi/Xiblim (mm) Ky Cy My Sensitivity Prediction Figure 26. The variation of axial depth blim with respect to a 10% change in nominal input parameters. The sensitivity of blim with respect to each pa rameter is superimposed. Linearity and non-linearity of blim(Xi) can be observed (see Table 6). 90 95 100 105 110 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Xi/Xi (x 100)blim (mm) Kt Kn Sensitivity Prediction Figure 27. The variation of blim with respect to a 10% change in Kt and Kn. The sensitivity of blim with respect to each parameter is superimposed. Linearity of blim(Xi) can be observed (see Table 6).

PAGE 75

58 Stability Sensitivity Analysis In this section, calculations of the sensitivity of blim to the input parameters are provided. The parameters used in the sensitiv ity calculations are provided in Table 7. In Figure 28 a comparison between the sensitivities of stiffness, K, and modal mass, M, are compared in the x (feed) and y-directions of the tool. As can be seen in the figure, the sensitivities in the x and y-directions are comparable in magnitude; however, the sensitivity in the y-direction is inaccurate near discontinuities in the system characteristic multipliers. This will be explained in the Un certainty section with a graphic depicting these discontinuities. Table 7. Parameters used in sensitivity analysis. h (%) E Central difference 0.2 10 2nd order 4310 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 (rpm x103)ln(blim)/ln(Xi) Kx Ky Mx My C1 discontinuities in blim Figure 28. Sensitivity of axial depth blim to changes in modal mass M and modal stiffness K in the x and y-directions (see Table 6).

PAGE 76

59 In Figure 29, the effect of damping on the stability is shown to be minimal compared to the modal stiffness and mass. This is a somewhat counter-intuitive result, but can be explained by regene ration (undulations in the cut surface experienced by the tooth in the current cut that are caused by the tooth vibration in the previous cut), which is a primary physical phenomenon that causes instability. The modal mass and stiffness have a great effect on the systems natural fr equency, which has a significant effect on regeneration. This also explai ns the result shown in Figure 30, where the sensitivity of axial depth blim to a change in spindle speed is significant and comparable to modal mass and stiffness. The effect of cutting force co efficients is shown in Figure 31, where the tangential cutting force coefficient, Kt, has more effect on the axial depth limit than the normal direction coefficient, Kn. 5 10 15 20 -30 -20 -10 0 10 20 30 (x 103)ln(blim)/ln(Xi) Kx Cx Cy C1 discontinuities in blim Figure 29. Sensitivity of axial depth blim to changes in modal damping C in the x and ydirections. The damping sensitivity is compared to modal stiffness sensitivity in the x-direction (see Table 6).

PAGE 77

60 5 10 15 20 -250 -200 -150 -100 -50 0 50 100 150 (rpm x103)ln(blim)/ln(Xi) Ky My rpm C1 discontinuities in blim Figure 30. Sensitivity of axial depth blim to changes in spindle speed. The spindle speed sensitivity is compared here to the modal mass and stiffness in y-direction (see Table 6). 5 10 15 20 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0 4 (rpm x103)ln(blim)/ln(Xi) Kt Kn Figure 31. Sensitivity of axial depth blim to changes in force cutting coefficients in the tangential Kt and normal directions Kn. Higher sensitivity can be seen for Kt (see Table 6).

PAGE 78

61 Surface Location Error Sensitivity Analysis The sensitivity of surface location error, SLE, to changes in input parameters is examined here. The parameters listed in Table 6 are used with b=1 mm and down milling case. In Figure 32, the sensitivity of SLE to changes in modal parameters in the ydirection is shown. Again, it can be seen that changes in Ky and My contribute more than Cy to a change in SLE. In Figure 33, the effect of cutt ing force coefficients is shown, where it is observed that the highest contributors to SLE sensitivity are Kt and Kte. Also, in Figure 34, SLE sensitivity to spindle speed and radial depth, rstep, is shown. Substantial sensitivity to spindle speed can be s een. This is due to the dependence of SLE on the relationship between the tool point frequency response and the selected spindle speed. As the spindle speed changes, it tracks different parts of the response. 5 10 15 20 0 50 100 150 200 250 (rpm x 103)Sensitivity Xi(SLE)/(Xi) (m) Ky My Cy Figure 32. Sensitivity of surface location error SLE to changes in modal parameters in ydirection (see Table 6).

PAGE 79

62 5 10 15 20 0 5 10 15 20 (rpm x103)Xi SLE/ Xi ( m) Kt Kn Kte Kne Figure 33. Sensitivity of SLE to cutting force coefficients (see Table 6). 5 10 15 20 0 100 200 300 400 500 (x 103)Xi (SLE)/ (Xi) ( m) rstep Figure 34. Sensitivity of SLE to spindle speed and radial depth of cut (see Table 6).

PAGE 80

63 Uncertainty of Stability Bounda ry and Surface Location Error Input Parameters Correlation Effect The correlation between the input parameters can have significant effect on the prediction of uncertainty. Neglecting the correl ation can result in e rroneous estimation of the uncertainty, especially when the input para meters are highly corr elated. Inclusion of the covariance matrix between pa rameters is necessary in this case. The input parameters can be classified into thr ee groups: dynamic modal paramete rs of the tool (work-piece assumed rigid), cutting force coeffici ents and machining parameters (e.g., radial step and spindle speed). In Chapter 5, estimation of th e correlation between parameters of the first two groups is explained and used in the uncertain ty prediction. The combined standard uncertainty uc can be found using sensitivities of output (blim or SLE) to input parameters. For the case of axial depth limit, uc is given as [88]: 2 1 2 limlimlim lim 1112,mmm ciij iii iijbbb ubuXuXX XXX (4.4) where u(Xi) refers to the standard unce rtainty in the input parameter Xi, u(Xi,Xj) is the estimated covariance between parameters Xi and Xj,. and m is the number of input parameters. The degree of the correlation between Xi and Xj is characterized by the correlation coefficient ,ij ij ijuXX rXX uXuX (4.5) In the Monte Carlo and Latin Hype-Cube sampling methods (described next), the multivariate normal distribution can be used to estimate the confidence level, in which case the covariance matrix between parameters controls the random sampling procedure.

PAGE 81

64 Monte Carlo Simulation The combined standard uncertainty, uc, of the stability boundary ( blim) and surface location error ( SLE ) can be predicted using Monte Carl o simulation. In this method, a random sample of size n is selected from the populati on of each input parameter. A normal distribution of the input parame ters is assumed. In the sample n the nominal value and standard deviation of each input pa rameter are used to generate the sample. The axial depth limit and surface location error are then calculated using TFEA for each point in the sample. The standa rd deviation of the predicted blim and SLE is then calculated from sample output for the range of spindle speeds of interest. It should be noted here that in doing so, no correlation between the input parameters is assumed, which is a common, and sometimes erroneous approach. To illustrate the effect of uncertainty in the input parameters on stability boundary uncertainty, standard uncertainties of 5% 0.5%, 0.001% and 0.5% are assigned to nominal values of the cutting force coeffici ents, modal parameters, radial step, and spindle speed, respectively. Th e values of the standard uncer tainties assigned correspond to practical variation in th e parameters involved. The parameters are assumed to be uncorrelated here. A sample size of 1000 is used. The stability boundary uncertainty is found, as shown in Figure 35, for one sta ndard deviation interval around the neutral stability boundary.

PAGE 82

65 5 10 15 20 2 4 6 8 10 12 14 (rpm x103)blim (mm) mean one std mean mean + one std Figure 35. Confidence in stability boundary due to input parameters uncertainties using Monte Carlo simulation (see Table 6). Sensitivity Method The combined standard uncertainty uc in axial depth limit while neglecting correlation between input parameters can be obtained from Eq. (4.4) as 2 lim lim 1 m ci i ib ubuX X (4.6) where u(Xi) refers to the standard unce rtainty in the input parameter Xi (same used for Monte Carlo method), and m is the number of input pa rameters. Although this relation assumes no correlation between input paramete rs it should be noted that cutting force

PAGE 83

66 coefficients ( Kt, Kn, Kte, Kne) and modal parameters ( K C M ) may be correlated in practice. The same standard uncertainty is assumed in the input parameters as in previous sections and the confidence level in axial depth limit is calculated for an interval of 2 uc( blim). Figure 36 shows the close agreement found using the two methods. However, it should be noted that the sensitivity method can be inaccurate near points where the function ( blim) is C1 discontinuous. Figure 37 shows the direct correspondence between the inaccurate sensitivity and C1 discontinuity in The C1 discontinuity in blim leads to inaccurate estimation of uc(blim) (see Eq. (4.6)). 5 10 15 20 0 2 4 6 8 10 12 14 16 18 20 (x 103)blim (mm) Sensitivity Monte Carlo Nominal Figure 36. Uncertainty boundary in axial depth limit using two standard deviation confidence interval. Uncertainty is calcul ated using sensitivity method and Monte Carlo method (see Table 6).

PAGE 84

67 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 1 2 3 uc (b) (mm) 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 -1 0 1 Real max( ) 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0.5 1 (x 103)Imag max( ) Derivative method Monte Carlo Simulation Figure 37. Uncertainty in axial depth us ing sensitivity and Monte Carlo methods. Inaccuracies in the sensitivity method can be seen near C1 discontinuity in the real and imaginary part of system characteristic multipliers (see Table 6). It should be noted here that predicting the uncertainty by Eq. (4.6) uses a linear approximation. The standard uncertainties assumed earlier are small where the linear approximation is still valid. However, if the uncertainties in the input parameters are large, then that linear approximation is no longer valid. In this case, simple random sampling methods (such as Monte Carl o simulation) are more appropriate. The surface location error uncertainty is found similarly using both methods. However, as shown earlier (see Figure 32 and Figure 34), the SLE sensitivities are accurate and do not depend on the characteristic multipliers continuity. Since the SLE is only defined for stable cutting conditions (see Eq. (A.18) in Appendix A) and explains

PAGE 85

68 the close prediction of uncertainty in SLE using sensitivity and Monte Carlo methods (Figure 38). 5 10 15 20 -15 -10 -5 0 5 10 15 20 25 30 (x 103)SLE 2uc(m) Monte Carlo method Sensitivity method Figure 38. Surface location error uncertainty with two standard deviation confidence interval on the nominal SLE Close agreement is observed (see Table 6). Latin Hyper-Cube Sampling Method This method was originally proposed as a variance reduction technique [89] in which the estimated variance is asymptotical ly lower than with simple random sampling (Monte Carlo method) [90, 91]. That is, for a sample size L this method gives a lower estimate of the output variance than is possi ble with the Monte Carlo method. The basic idea of this method is that each value (or range of values) of a variab le is represented in the sample, no matter which value turns out to be the most important. In this way, the

PAGE 86

69 sampling distribution is divided into a number of strata w ith a random selection inside each stratum. The Latin Hyper-Cube method will be used in Chapter 5 for predicting the standard combined uncertainty of the stabili ty and surface location e rror cutting tests in that chapter. Robust Optimization under Uncertainty In order to account for uncertainty in the ax ial depth stability limit, the safety factor design analogy is used here. The deterministic optimization algorithm implemented in Chapter 3 (Eq. (3.9)), repeated here, can be modified to account for the axial depth uncertainty. ,,, 3 : ,, 1 ... ,,,1,MRRifbfbfb SLESLESLEmin subjecttofbeforik gbgbgb for a series of selected li ,MRRmits (e) on f (4.7) Therefore, the axial depth b used in the stability constraint is set equal to an uncertainty inflated value. That is, b is replaced by b+ Ue, where ecUkub is the expanded uncertainty, k is a factor that estimates the uncertainty confidence interval and uc(b) is the combined standard uncertainty in th e axial depth. Therefore Eq. (4.7) becomes ,,, 3 : ,, 1 ... ,,,1,MRRi eeefbfbfb SLESLESLEmin subjecttofbeforik gbUgbUgbU for a series of ,MRRselected limits (e) on f (4.8)

PAGE 87

70 Discussion In this chapter, the sensitivities of axial depth limit and surface location error to model input uncertainties were studied. Numerical estimation of the sensitivities can be challenging, where several factors contribute to the accuracy of the estimation. Step size is one of the significant factors that affect the accuracy of the estimation. The sensitivity analysis aids in identify ing the relative contribution of the milling model input parameters to the sensitivity of either axial depth limit or surface location error. For the case of axial depth, the sp indle speed, followed by modal stiffness and mass, is the most significant contributor. In the case of cutting force coefficients, the tangential cutting force coefficient is found to contribute more to the sensitivity than the normal cutting force coefficient. As for the surface location error sensitivity, the same trend can be observed. However, for the cut ting force coefficients, the edge tangential cutting force coefficient signi ficantly contributes to the SLE The uncertainty in axial depth and surf ace location error was predicted using two methods: the sensitivity method and the Mont e Carlo simulation approach. Comparable agreement is shown. However, the sensitivit y method is more efficient computationally. For example, in the case of SLE uncertainty prediction, Monte Carlo simulation required 9.34 hours, while the sensitivity method needed only 0.26 hours (36 times more efficient). It is noted that for the uc(SLE) case, when the milling parameters are well into the stable region, the accuracy of the sensitivity method is not sacrificed at the cost of efficiency as is the case for uc(b) at discontinuities in the characteristic multipliers. Finally, the optimization algorithm introdu ced in Chapter 3 was modified to account for confidence levels in the axial dept h limit. This allows robust optimization to

PAGE 88

71 account for inherent uncertainty in the mean va lues of the input parameters. In Chapter 5 an implementation of this algorithm is demonstrated.

PAGE 89

72 CHAPTER 5 EXPERIMENTAL RESULTS The milling model accuracy depends on reli able determination of cutting force coefficients and tool or work-piece modal parameters. These values are found experimentally and their uncertainties c ontribute to the uncertainty of the model prediction. In this chapter, the experime ntal procedure used to determine these parameters is described and then the op timization algorithm is executed using the experimentally determined input parameters to find the Pareto optimal points. Another set of experiments is completed to validate/i nvalidate these optimal points. Using the optimization algorithm, the strength and weakness of the mathematical model or solution method can be obtained. Equation Chapter 5 Section 1 Cutting Force Coefficients Milling Forces The average milling forces during one tooth period in the x and y -directions are [92, 93], cos22sin2sincos 82 2sin2cos2cossin 82ex s t ex stxtntene ytnteneNbcNb FKKKK NbcNb FKKKK (5.1) where teK and neK are the tangential and normal edge cutting force coefficients, respectively. In slotting tests (see Figure 7), the entry and exit angles of the cutter are

PAGE 90

73 0st and ex respectively. The average forces per tooth period for this case are found to be: 4 4xnne ytteNbNb FKcK NbNb FKcK (5.2) Equation (5.2) can be written as a function of chip load ( c ) as: ,,.qqcqeFFcFqxyz (5.3) The experimental procedure consists of completing multiple cutting tests at varying chip loads and recording the cutting forces. For each chip load increment, the average cutting forces in the x and y -directions are measured, and th en a linear regression of the data points is made to extract the cutting coefficients using Eqs. (5.2) and (5.3): 4 4 .ycye tte xcxe nneFF KK NbNb FF KK NbNb (5.4) For radial immersions less than the cutter diameter, the entry and exit angles differ from the slotting case. For up-milling (see Figure 7) the entry and exit angles of the cutter are 0st and 1cos1exa R where a is the radial depth of cut. Substituting in Eq. (5.1) gives: cos212sin2sincos1 82xtexnexexteexneexNbcNb FKKKK (5.5) Factoring Eq. (5.5) in terms of chip load c gives:

PAGE 91

74 xxcxeFFcF (5.6) cos212sin2 8 sincos1. 2xctexnexex xeteexneexNb FKK Nb FKK (5.7) Similarly, the following equations are obtained for the y-direction. 2sin2cos21cos1sin 82ytexexnexteexneexNbcNb FKKKK (5.8) yycyeFFcF (5.9) 2sin2cos21 8 cos1sin 2yctexexnex yeteexneexNb FKK Nb FKK (5.10) Writing Eqs. (5.7) and (5.10) in matrix form to solve for the cutting coefficients the final equation can be expressed as shown in Eq. (5.11). cos212sin2 00 2sin2cos21 00 8 sincos1 00 1cossin 00 2exexex xc t exexex yc n xe te exex ye ne exexNb F K F K F K Nb F K (5.11) The same procedure can be used to solve for the cutting coefficients in the downmilling case (Figure 7). Experimental Procedure Proper selection of a suitable dynamometer to measure the dynamic cutting forces is important. Some of the factors that need to be addressed are the calibration range of the

PAGE 92

75 dynamometer and its dynamic response. Simulation of the cutting forces helps in addressing the issue of cutting force magnit ude range. Using time-domain simulation of the cutting forces and approximate cutting coe fficient values, an estimate of the typical range of cutting forces can be found. Euler integration is used to solve for the tool displacement during the cut in the 2nd order differential equation (Eq. (3.2)) and find the corresponding cutting forces in the x and y-directions. An example is shown in Figure 39. It is seen that a dynamometer with the 0 kN to 5 kN range is acceptable, although the force levels are relatively small compared to the full scale value. A Kistler 9257A dynamometer with 5 kN range was available for these tests. One requirement for this dynamometer is that the cutting force is ap plied to the dynamometer not more than 25 mm above the top surface of the dynamometer. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -100 -50 0 50 Time (s)Fx (N) 0 0.1 0.2 0.3 0.4 0.5 -50 0 50 100 Time (s)Fy (N) Figure 39. Example simulation of cutting forc es to facilitate proper selection of dynamometer.

PAGE 93

76 A 25 mm thick 6061-T6 aluminum work-piece was sized to 100 mm x 85 mm, then faced and drilled to fit the dynamometer hole pattern as shown in Figure 40. Slotting cutting tests were made for a 25.4 mm diameter end mill with a 145 mm overhang and a single 12 helix insert for chip load range of 0.1-0.24 mm/tooth in 0.02 mm/tooth steps. The cutting forces in x and y-directions were measured for each chip load using an axial depth of 0.4 mm. Two sets of measurements were made for a 1000 rpm spindle speed. To address the influence of spindle speed on cutt ing coefficients, the above two sets were repeated for {5000, 10000, 15000 and 20000} rpm. The average value of the measured cutting forces was inserted into Eq. (5.4) to solve for the cutting coefficients. Average cutting coefficients of the two sets of meas urements at each spindle speed are listed in Table 8. Work-piece D y namomete r Figure 40. Work-piece, dynamometer and tool setup

PAGE 94

77 A regression analysis of the cutting force coefficients as a function of spindle speed was carried out. For Kt and Kn, a linear regression with logarithmic transformation of spindle speed indicates a stat istically significant relation with a P-Value of less than 0.007. Figure 41 and Figure 42 show the tr end line for this regression for both Kt and Kn, respectively. For the edge cutting force coefficients Kne and Kte the regression doesnt indicate a significant statistical relation between Kne or Kte and spindle speed. The PValue for the slope of the regressi on was 0.39 and 0.55, respectively. Table 8. Cutting coefficients for 1 insert endmill for slotting cutting tests (krpm) Kt (N/mm2 ) Kn (N/mm2) Kte (N/mm) Kne (N/mm) 1 1321 379 28 32 5 832 183 47 34 10 841 62 37 38 15 655 34 52 33 20 670 65 37 26 y = -504.17x + 1284.9 R2 adj = 0.91 0 200 400 600 800 1000 1200 1400 00.511.5 log10( (rpm) x10 3 )Kt (N/mm2) Figure 41. Cutting coefficient in tangential direction (Kt)

PAGE 95

78 y = -268.84x + 369.14 R2adj = 0.93 0 50 100 150 200 250 300 350 400 00.20.40.60.811.21.4 log10( rpm) x103)Kn (N/mm2) Figure 42. Cutting coefficient in normal direction (Kn) A similar set of measurements were ma de using partial radial immersion (up milling) for a 15000 rpm spindle speed. Equatio n (5.11) was used to find the cutting coefficients in this case. The results are provided in Table 9. Table 9. Up milling cutting coefficients for 12% radial immersion Kt( N/mm2) Kn( N/mm2) Kte( N/mm) Kne( N/mm) 833 431 6 8 To verify the fit, the cutting coeffici ents obtained were used in a time-domain simulation of the cutting forces. The measured forces were then ove rlaid on the simulated forces. Figure 43 shows a case for 0.12 mm/t ooth chip load and 1000 rpm. Also Figure 44 and Figure 45 show the fit for high er spindle speeds of 5000 and 20000 rpm, respectively.

PAGE 96

79 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 -40 -20 0 20 40 60 80 Time (secs)Fx (N) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 -50 0 50 Time (secs)Fy (N) Simulated force Measured force Simulated force Measured force Figure 43. Simulated and measured forces for 0.12 mm/tooth chip load and 1000 rpm. 0 0.02 0.04 0.06 0.08 0.1 0.12 0 50 100 Time (secs)Fx (N) 0 0.02 0.04 0.06 0.08 0.1 0.12 -80 -60 -40 -20 0 20 40 Time (secs)Fy (N) Simulated force Measured force Simulated force Measured force Figure 44. Simulated and measured cutti ng forces for 0.2mm/tooth chip load, b=0.4 mm and 5000 rpm.

PAGE 97

80 1.01 1.015 1.02 1.025 1.03 1.035 -20 0 20 40 60 80 100 Time (secs)Fx (N) 1.01 1.015 1.02 1.025 1.03 1.035 -50 0 50 Time (secs)Fy (N) Simulated force Measured force Simulated force Measured force Figure 45. Simulated and measur ed forces at 20 krpm and b=0.4 mm for slotting. Covariance Matrix (Linear Multi-Response Model) The regression analysis performed in th e previous section is a single response analysis. However, the measured re sponses are the forces in both the x and y-directions during a single measurement (dynamometer). Obviously this is a multi-response measurement. Therefore analysis of the da ta should take into consideration the multivariate nature of the data. The interrelationship existing between the variables could render univariate investigations meaningl ess. The development for a multi-response model follows the description in [94]. If we let Q be the number of experimental runs and r be the number of response variables measured for each setting (two in our case, i.e., Fx and Fy) of a group of variables (chi p load only in our case). The ith response model can be written in vector form as

PAGE 98

81 1,2,...,iiiiYZir (5.12) where Yi is an 1Qvector of observations in the ith response, Zi is an iQp matrix of rank pi (for the simple linear model pi = 2), i is a 1ip vector of unknown constant parameters, and i is an 1Q random error vector associated with ith response. The assumption of simple linear regr ession apply here, that is 0iE and iiiQVar I However, the covariance matrix be tween the responses is not zero, 1,2,..., ,1,2,..,;iiiQ ijijQVarir Covijrij I I (5.13) We denote the rr covariance matrix whose (i,j) th element is ,1,2,...,ijijr by For the case of two responses, Eq. (5.12) can be written in matrix form as: 01 111 121 11 02 222 121 12ZQQQ QQQYZ Y 0 0 (5.14) where 12 1 11Q Q Z Zc (5.15) where c represents the chip load vector (see Eq. (5.3)) and the left hand side vector of Eq. (5.14) represents the observed average cutting forces in the x and y-directions. From Eq. (5.13) it can be seen that has the following variance-covariance matrix, Var Q I (5.16) where is a symbol for the direct (or Kron ecker) product of matrices. The direct product of two matrices and Q I both of size rr gives an 22rr matrix which is

PAGE 99

82 partitioned as ij Q I where ij is the (i,j)the element of matrix The best linear unbiased estimate of is given by [95] 1 11 ZZZY (5.17) where Y is the left hand side of Eq. (5.14) The variance-covariance matrix of the estimated vector is 1 1 Var ZZ (5.18) Since is usually unknown, it is estimated using the following equation [95] 11 ,1,2,...,iNiiiiNjjjjj ijYY Q ijr IZZZZIZZZZ (5.19) It should be noted that ij is computed from the residual vectors which result from ordinary least-squares fit of the ith and jth single response models to their respective data sets. Using this estimate for in Eq.(5.19), an estimate of the variance of can be obtained. The cutting force coefficients ar e determined using a linear transformation KA (5.20) where the matrix A for slotting (see Eq. (5.4)) is 000 4 000 000 4 000 Nb Nb A Nb Nb (5.21)

PAGE 100

83 Therefore the variance-covari ance matrix of cutting force coefficients can be found as '. VarKAVarA (5.22) Using the procedure outlined above, the cutting force coefficients and their corresponding correlation matrices are calculat ed and listed in Table 10 for cutting tests carried out according to the same procedure de scribed earlier, noting that the correlation matrix is obtained directly from the covari ance matrix (see Eq. (4.5)) As indicated in Table 10, a high correlation between Kt and Kte and Kn and Kne is found. This high correlation is justified since both of th e corresponding cutting coefficients ( Kt and Kte or Kn and Kne) are obtained from the same regression fit and cutting force direction. However, a small correlation between the x and y -directions of the forces is found (between Kt and Kn or Kne) which may be due to experimental error. Table 10. Estimated cutting force coeffici ents and their correlation matrix for 7475 aluminum and a 12.7 mm diameter solid carbide endmill with 4 teeth and 30 degree helix angle. Kt (N/m2)Kn (N/m2)Kte (N/m)Kne (N/m) Mean 8.41E+082.53E+081.27E+041.01E+04 Standard deviation 2.19E+072.66E+071.70E+032.07E+03 Coefficient of variation 0.030.110.130.20 P Value 2.E-088.E-053.E-043.E-03 Correlation Coeff. MatrixKneKnKteKt Kne 1.00 Kn -0.931.00 Kte -0.130.121.00 Kt 0.12-0.13-0.931.00

PAGE 101

84 Compliant Tool Modal Parameters The cutting tests were conducted on a Makino V55 vertical milling machining center located at Techsolve, an Ohio-bas ed not-for-profit ma nufacturing research organization. The cutting tool was a 12 .7 mm diameter solid carbide end mill (CRHEC500S4R30-KC610M) with 100 mm overall length ( OAL ), 70 mm over-hang length, 30 helix angle, and 4 flutes. A relatively long tool over-hang length was used in order to obtain a compliant t ool that could reasonably be modeled as single degree of freedom. Four measurements of the frequenc y response function of the tool (Figure 46) were made after running the spindle for 30 s econds at a specific spindle speed then completing a tap test in the xdirection, then running the spindle for another 30 seconds and taking a tap test in the ydirection, then removing the ho lder from the machine, and replacing and repeating the above procedur e for a different speed. This measurement procedure enabled the estimation of the vari ation of the modal parameters due to the spindle thermal effect and holder replacemen t effect. Figure 47 and Figure 48 show the frequency response measurement of the tool in the x and ydirections respectively. Table 11 lists the fitted tool modal parameters obtained by the peak amplitude method (see Milling Model section in Chapter 4) with thei r average and standard deviation, and Table 12 lists the correlation coefficient matrix The correlation coefficient between Mx and Kx, for example, is calculated according to 4 1 44 22 11ii xx iixxxx i MK xxxx iiMMKK r MMKK (5.23)

PAGE 102

85 The values shown in Table 12 indicate strong correlation coefficients for xx M Kr and y y M Kr. This is expected since the natural frequency of the tool is constant so that Kx or Ky depend entirely on Mx or My, respectively. Also, since the tool is symmetric the correlations for y x M Mr and y x K Kr are also expected to be high, which is the case for 0.86yxMMr As for the correlation coefficient, 0.75yxCCr some correlation is expected since the tool-holder interface damping is ideally symmetric for the round tool. The minimal correlation indicated between the damping and other modal parameters can be justified since there is no direct relationship between damping and mass or stiffness. The mean, standard deviation and the corr elation matrix are used to generate the random sample of input modal paramete rs in order to estimate uncertainty. Table 11. Tool modal parameters in x and y -directions. measurement stateM (kg)C (N.s/m)K (N/m) M (kg)C (N.s/m)K (N/m) static 0.0324.344.8E+060.0329.094.3E+06 5 krpm and replacement 0.0322.054.4E+060.0237.252.6E+06 10 krpm and replacement 0.0322.664.3E+060.0229.542.9E+06 20 krpm and replacement 0.0224.183.9E+060.0229.853.4E+06 mean 0.0323.314.4E+060.0231.433.3E+06 standard deviation 0.0020.9763.16E+050.0043.3686.60E+05 coefficient of variation (CV) 0.070.040.070.200.110.20 xy Table 12. Correlation coefficient matrix for modal parameters. Correlation CoefficientMxCxKxMyCyKy Mx 1.00 Cx 0.231.00 Kx 0.990.131.00 My 0.860.690.801.00 Cy -0.09-0.75-0.05-0.501.00 Ky 0.660.880.580.95-0.641.00

PAGE 103

86 Machine Spindle Instrumented Hammer Accelerometer Sensor interface Tool Holder Figure 46. Modal analysis test equipment t ypically used in machine tool structures. 0 500 1000 1500 2000 2500 3000 3500 4000 -2 -1 0 1 2 x 10-6 Real (m/N) 0 500 1000 1500 2000 2500 3000 3500 4000 -4 -3 -2 -1 0 x 10-6 Imag (m/N) Frequency (Hz) static 5 krpm and replacement 10 krpm and replacement 20 krpm and replacement Mean Figure 47. Frequency response function measurement of tool in x -direction. Four sets of measurements are made to estimate spindle thermal and holder replacement effects.

PAGE 104

87 0 500 1000 1500 2000 2500 3000 3500 4000 -2 -1 0 1 2 x 10-6 Real (m/N) 0 500 1000 1500 2000 2500 3000 3500 4000 -3 -2 -1 0 x 10-6 Imag (m/N)Freq. (Hz) static 5 krpm and replacement 10 krpm and replacement 20 krpm and replacement Mean Figure 48. Frequency response function measurement of tool in y -direction. Four sets of measurements are made to estimate spindle thermal and holder replacement effects. Stability Lobe Validation In this section the stability lobe diag ram for the 70 mm over-hang length compliant tool is verified. First the stability limit un certainty is predicted using Latin Hyper-Cube and Monte Carlo sampling, then a description of the experimental procedure is provided and results are discussed. Stability Lobe Uncertainty Due to the relatively high uncertainty in input parameters (Table 11), the sensitivity method cannot be used because of the nonlin earity of the axial depth limit to the respective parameters. The alternative random sampling methods (Latin Hyper-Cube and Monte Carlo) are used instead. The stability lobes are generated using TFEA. A random sample of size L=1000 is generated from the normal di stribution for each input parameter

PAGE 105

88 group (cutting force coefficients and m odal parameters) using their corresponding standard deviation, mean values and cova riance matrix. Latin Hyper-Cube sampling is used to generate the samples for the cutti ng force coefficients and modal parameters groups. Also, a random sample of the same size is generated for the radial immersion and spindle speed using Monte Carlo Simulation with no correlation assumed. This random sample of input parameters is used to generate the stability lobe diagram uncertainty intervals. Figure 49 shows the boxplot (a plot used to show variation and measures of central tendency for a sample) of axial dept h as a function of spindle speed. The grey boxes indicate the range of minimum a nd maximum axial depths, the black boxes indicate the lower 2.5 percen tile and upper 97.5 percentile (95 % confidence interval), while the two lines indicate the median and mean of the sample. It is seen that the median and mean lines do not match, which indicates the distribution is skewed. Examination of the histogram at selected spindle speeds (see Figure 50) validates this conclusion. At 10000 rpm the distribution appears close to normal, however, in checking the normality of the distribution at this speed (see Figure 51) we find that it is in fact not normal with a P-value of less than 0.005. This is illustrat ed by the deviation of the observations from the normal probability line.

PAGE 106

89 Omega (rpm x 10^3)b (mm) 20000 19000 18000 17000 16000 15000 14000 13000 12000 11000 10000 9 8 7 6 5 4 3 2 1 0 Figure 49. Boxplot of stability lobes bounda ry uncertainty. The minimum and maximum values are shown for each spindle speed (grey boxes), the mean and median of axial depth limit are indicated by the line and circles respectivel y, also shown is the 2.5 and 97.5 percen tiles (black boxes). Axial Depth Limit (mm)Number of Neutral Stability Observations 5 6 4 8 4 0 3 2 2 4 1 6 0 8 80 60 40 20 0 4 2 3 6 3 0 2 4 1 8 1 2 0 6 0 0 160 120 80 40 0 6 0 5 4 4 8 4 2 3 6 3 0 2 4 1 8 80 60 40 20 0 4 8 4 2 3 6 3 0 2 4 1 8 1 2 0 6 100 75 50 25 0 3 5 3 0 2 5 2 0 1 5 1 0 0 5 0 0 150 100 50 0 4 8 4 0 3 2 2 4 1 6 0 8 0 0 240 180 120 60 0 10000 12000 14000 16000 18000 20000 Mean3.054 StDev1.009 N1000 10000 Mean1.341 StDev0.6144 N1000 12000 Mean3.807 StDev0.8778 N1000 14000 Mean2.222 StDev0.8063 N1000 16000 Mean1.321 StDev0.5377 N1000 18000 Mean1.384 StDev0.7592 N1000 20000Histogram of Axial Depth Limit Distribution for Various Spindle Speeds Figure 50. Histograms of axial depth limit di stributions for various spindle speeds.

PAGE 107

90 Axial Depth Limit (mm)Percent 7 6 5 4 3 2 1 0 -1 99.99 99 95 80 50 20 5 1 0.01 Mean3.054 StDev1.009 N1000 AD3.835 P-Value<0.005Probability Plot for 10000 rpm Figure 51. Probability plot of axial depth limit distribution at 10000 rpm spindle speed. Experimental Procedure The stability lobes were verified experime ntally for 25% partial radial immersion down milling and 0.1 mm/tooth chip load. The same tool with modal tool parameters listed in Table 11 was used. A 7475 aluminum work-piece was mount ed (see Figure 52) to a Makino V55 vertical machining center tabl e. Cutting tests with different axial depths were conducted at a range of spindle speeds from 10000 rpm to 20000 rpm in 1000 rpm steps. The stability of each cutting oper ation was determined by recording the sound signal of the cut. The Fast Fourier Tran sform was used to transform the sound timedomain sound signal into the frequency domain. An analysis of the signal frequencies identified the chatter frequency, if one ex isted (i.e., significant content was seen at frequencies other than the runout and tooth passing frequencies) It was observed that the

PAGE 108

91 chatter frequency when it existed was always slightly higher than the tool natural frequency, as expected. Tool Machine Spindle Data Acquisition Stability test Workpiece x z y Figure 52. Schematic of stability tests for partial radial immers ion cutting conditions. Results The cutting test conditions are shown in Figure 53 with the boxplot of axial depth limit uncertainty. Also, the stab ility lobe boundary is overlaid using mean values of input parameters. In order to identify the stability of each cut, as noted previously, the sound signal was analyzed. In Figure 54 we can see some of these signals in the frequency domain. As noted previously, the chatter fre quency occurs near the natural frequency of the tool. The natural frequency of the tool is approximately 2000 Hz ( 1 / 2 fKM nyy meanmean, see Table 11). For 13000 rpm and a 1.52 mm axial depth, and 14000 rpm and 3.05 mm, for example, the chatter frequencies occur near 2100 and 2200 Hz, respectively. It should be noted that the chatter frequencies were difficult to

PAGE 109

92 identify when the tooth passing frequency or one of its harmonics are near the tool natural frequency. This is evident from the cutting test at 10000 rpm and 16000 rpm where there is high amplitude near the tool na tural frequency. In that case, examinations of the cut surface of the wor kpiece help in identifying chatter (due to the corresponding rough surface finish). In Figure 53, the stability of the cutting conditions agreed well with the median of stability prediction almost ev erywhere along the spindle spee d. However, near fractions of the tool natural frequency (60/1530000 rpmnnfNf ), poor agreement between prediction and experimental result is observed. This may be attributed to confidence in the modal fitting near the natural frequency of the tool 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 b (mm) (rpm x 103) Stable Unstable Marginal Stability lobe, mean parameters Figure 53. Stability lobe generated using mean values of input parameters with experimental results overlaid, also shown the boxplot co rresponding to each spindle speed used in the measurements.

PAGE 110

93 1800 1900 2000 2100 2200 2300 0 10 20 30 Magnitude20000 rpm 1800 1900 2000 2100 2200 2300 0 5 10 15 16000 rpm 1500 2000 2500 0 1 2 3 4 15000 rpmMagnitude 1500 2000 2500 0 5 10 15 20 25 14000 rpm 1800 1900 2000 2100 2200 2300 0 10 20 30 40 13000 rpm Frequency (Hz)Magnitude 1500 2000 2500 0 1 2 3 4 10000 rpm Frequency (Hz) 4.06 mm, stable 3.56 mm, stable 5.08 mm, stable 1.02 mm, stable 1.52 mm, chatter 2.03 mm, chatter 2.03 mm, stable 3.05 mm, chatter 2.54 mm, marginal 6.1 mm, stable 7.11 mm, stable 3.56 mm, stable 4.06 mm, stable 1.02 mm, stable 1.52 mm, chatter Figure 54. Fast Fourier Transform (FFT) of so und signals for selected stability tests. Pareto Front Validation This section begins with the calculation of the Pareto Front for a specific single degree of freedom tool considering conf idence levels in the axial depth limit, blim, (see Robust Optimization section in Chapter 4), after which the ex perimental procedure of the tests is described, followed by the results. Pareto Front Simulation Results The Pareto front for SLE and MRR is generated for the same material (7475 aluminum) and tool used in the stability tests (Table 11 and Table 12) and the same cutting conditions of radial immersion and chip load. Two cases are considered: 1) no uncertainty in the input parameters; and 2) uncertainty in input parameters or axial depth limit, blim, where an uncertainty of limcUub is used. The robust optimization algorithm (Chapter 4) is used to generate the Pareto front for the input parameters

PAGE 111

94 uncertainty case. Figure 55 and Figure 56 illustrate the Pareto front for the aforementioned cases. It can be seen that the uncertainty designs predict higher SLE for the same MRR compared to the mean value one. Also, in Figure 56 the knee in both Pareto Fronts indicate the design beyond which the SLE increases at a higher rate, which makes that knee a preferred design poin t. In considering Figure 56, the SLE difference between uncertainty and without uncertainty case s is larger at higher MRR than at lower MRR This is attributed to the fact that as higher MRR is required, the axial depth, b, approaches blim. Here, the predicted uncertainty in blim changes the design variables ( b or ) substantially to account for th e uncertainty. This penalizes the SLE for the uncertainty case and makes it significantly larg er than the no uncertainty case at higher MRR. However, at lower MRR the axial depth b is far from blim and is therefore less affected by uc(blim) 10 11 12 13 14 15 16 17 18 19 20 0 1 2 3 4 5 6 (x 103)b (mm)2 0 04 0 06 0 08 0 0 1 0 0 01 2 0 0 1 4 0 0 MRR (mm3/s) No uncertainty Uncertainty uc(b) Stability using Mean Parameters 5.4 m 85.7 m 46 m 3.8 m 2.1 m Figure 55. Stability boundary using mean values in the input parameters Pareto optimal designs are overlaid for two cases: mean values and uncertain input parameters.

PAGE 112

95 0 200 400 600 800 1000 1200 1400 0 10 20 30 40 50 60 70 80 90 MRR (mm3/s)perturbed average SLE (m) Uncertainty uc(b) Uncertainty not considered Figure 56. Pareto Front of perturbed average SLE and MRR The Pareto Front with uncertainty in axial depth is compar ed to the one with no uncertainty. Experimental Procedure and Results As a first step in conducting the surface lo cation error tests, the work-piece was machined to a specified dimension (nominally 40 mm width) usi ng shallow axial depth slotting cuts (see Figure 57). Careful atten tion was paid to minimizing positioning errors of the machine by feeding from the same di rection prior to cutting (i.e., minimize the influence of reversal errors). The cutting conditions of tw o mean value Pareto optimal designs were selected from Figure 56 for th e case of no uncertainty. The first point corresponds to the knee design poi nt in the figure and the second point corresponds to the maximum MRR for the case that uncertainty was not considered. For these two design conditions of axial depth and spindle speed, f our additional cuts were made for each case by varying the spindle speed around the sel ected design (see Table 13). The purpose of these extra cuts was to check the sensitivity of the stability and surface location error to

PAGE 113

96 spindle speed. The work-piece webs, shown in Figure 57, were milled from both sides. A coordinate measuring machine (CMM) was used to measure the base of the web (dimension a ) and the top portion (dimension b ). The measured surface location error was then taken to be 3.175 mm, 2 ab SLE where the commanded radial depth was 3.175 mm. Each dimension ( a and b ) was measured 15 times in order to evaluate the CMM machine measurement repeatability. The 15 measurements had a maximum standard deviation of2 m (the machine accuracy is estimated at < 5 m based on recent calibration tests). The measured SLE for the set of cutting tests is shown in Figure 58 and Figure 59. It should be noted here that all cuts were stable. Therefore, all SLE results are shown in the figures. The error of the reference dimension ( a=40 mm) is also shown in these figures. This would identify if there is a trend in the measured SLE due to the errors in the reference dimension. The standard deviation of the reference dimension is 4 m and 8 m for 4.45 mm and 2.12 mm axia l depth cuts, respectively. To illustrate the effect of the helix angle of the tool (30) on the SLE the CMM measurement was repeated for distances of {1, 2, 3, and 3.4} mm from the top surface of the work-piece web along the tool axis. Figure 61 shows that the SLE varies along the axial depth of the cut. This va riation corresponds with previous SLE studies [40] where similar variation of SLE was observed. Although the measured SLE does not agree well with the mean predicted value (Figure 62), there is some agreem ent in the trend of median of SLE (Figure 60) and the measured SLE (Figure 59). Also the measured SLE is within the uncertainty bounds on SLE (Figure 60). The disagreement betw een the measured and predicted SLE can be attributed to: 1) the milling model used in th e prediction assumed straight cutter teeth (the

PAGE 114

97 actual tool had a 30 helix angle) whic h would yield higher SLE ; 2) the cutting force coefficients used in the prediction were measured for 8900 rpm, while the SLE cuts were made for around 15000 rpm. At higher spindle speeds the cutting fo rce coefficients (cutting forces) tend toward lower values. Th ese two factors explai n the high prediction of SLE for the 1400 mm3/s case relative to the meas ured one. However, for 700 mm3/s case they fail to explain the difference. This may highlight model w eaknesses at this level of axial depth (2.12 mm). Tool Machine Spindle Surface Location Error Test Workpiece x z y 40 mm 3.175 mm 3.175 mm A B 2.12 or 4.45 mm Aluminum 7475 Figure 57. Surface location e rror experiment schematic.

PAGE 115

98 15.5 15.55 15.6 15.65 15.7 15.75 15.8 35 40 45 50 55 60 b = 2.12 mmSLE (m) 15.5 15.55 15.6 15.65 15.7 15.75 15.8 10 20 30 40 mError in reference dimension A (rpm x103) Predicted 1.2 m Figure 58. Measured surface location error of b=2.12 mm and the reference dimension (A) error. 14.75 14.8 14.85 14.9 14.95 15 15.05 0 20 40 SLE (m) b = 4.45 mm 14.75 14.8 14.85 14.9 14.95 15 15.05 5 10 15 20 Error in reference dimension Am (rpm x103) 2 mm 1 mm 2 mm 3 mm 3.4 mm Predicted 93.4 m Figure 59. Measured surface location error of b=4.45 mm and the reference dimension (A) error.

PAGE 116

99 14.753 14.803 14.853 14.903 14.953 20 40 60 80 100 120 140 SLE (m) (rpm x103) Measured SLE Figure 60. Boxplot of SLE uncertainty at spindle speeds for 4.45 mm axial depth case. The upper tail of the SLE uncertainty is not present due to the undefined SLE in the unstable region. 1 1.5 2 2.5 3 3.5 5 10 15 20 25 30 35 40 45 50 55 CMM measurement location (mm)SLE (m) 14753 rpm 14803 rpm 14853 rpm 14903 rpm 14953 rpm Figure 61. Measured surface location erro r of b=4.45 mm case. The CMM probe measurement is repeated along the axial depth of the tool with {1, 2, 3, and 3.4} mm from the top surface of the work-piece web.

PAGE 117

100 Table 13. Surface location error cutting conditi ons for two Pareto optimal designs with no uncertainty considered. Tool # (100 mm OAL) Cut No.b (mm) (rpm) MRR (mm3/s) SLE (m) 12.1215617 22.1215667 32.1215767 42.1215567 52.1215517 64.4514853 74.4514803 84.4514753 94.4514903 104.4514953 CRHEC500S4R30-KC610M 7001.2 93.4 1400 10 11 12 13 14 15 16 17 18 19 20 -20 0 20 40 60 80 100 120 X: 15.6 Y: 1.761 SLE ( m) (x 103 rpm) X: 14.8 Y: 93.42 b = 2.12 mm b = 4.45 mm Figure 62. Surface location error of prefe rred design conditions with no uncertainty considered in the optimization. Optimu m spindle speeds are indicated in the figure.

PAGE 118

101 Conclusions The uncertainty in axial depth limit, blim, and SLE indicates a non-normal distribution, although for conve nience a normal distribution was used to estimate the confidence levels of blim and SLE This non-normality predicates the use of different measures of uncertainty bounds in order to a ccount for a specific confidence interval. There is good agreement between the predic tion of stability and the experimental results. It is shown that th ere is a distinct grey region of neutral stability boundary (marginal stability) rather th an a black and white step change between stable and unstable zones as suggested by the single stability boundary typi cally indicated in stability lobe diagrams. Also, the uncertainty identified in the blim boxplot indicates that the distribution is skewed to higher blim values near the tool system natural frequency. This was also confirmed by the experimental results where higher axial depths were generally feasible. The measured surface location error of th e Pareto design points didnt show high sensitivity to spindle speed variation. Th is shows the validity of the optimization algorithm selection of a design that mitigates the effect of spindle speed on SLE. In this worst case scenario of uncertainty in modal parameters (thermal effects and dynamic variations due to tool removal and replacement), there was substantial uncertainty seen in blim and SLE. Reduction of uncertainty in the input parameters may be necessary to fully realize the benefits of high-speed milling. This may be done by conducting more tests to lessen the uncertainty bounds and/or completing modal tests of the tool-holder assembly each time it is removed and replaced. Although no experimental verification of robust optimum designs was done, it is interesting to note that the predicted robust optimum design corresponding to 700 mm3/s

PAGE 119

102 had approximately the same value of measured SLE (Figure 55 and Figure 56). This is due to the fact that the robust design select ed a design with spindl e speed that was 1000 rpm lower than the measured one (with no uncertainty considered). Also, in considering uncertainty (see Figure 55), the 1400 mm3/s was not realizable. This may be due to an overestimation of the uncertainty in the input parameters. As mentioned previously, this requires more testing in order to better esti mate the uncertainty of input parameters.

PAGE 120

103 CHAPTER 6 SUMMARY This chapter provides a summary of the work completed in this dissertation with a detailed procedure on how to implement the robust optimization algorithm. Then the limitations of the milling model are addressed with suggestions for future research. Robust Optimization Algorithm In order to implement the robust optimi zation algorithm for a specific compliant tool/work-piece system, the followi ng steps should be taken: 1. Measure the tool/work-piece frequency response and complete a modal fit to the measured response. The confidence levels in the fitted modal parameters are estimated by repeating the measurement at different thermal states of the machining spindle. In case this measurement cannot be repeated each time the tool/holder assembly is removed from the spindle, then several measurements should be made wherein the tool/holder assembly is rem oved from the spindle and replaced. This will account for the dynamic non-repeatability due to tool/holder replacement. The mean, standard deviation, and correla tion between the modal parameters are calculated. Equation Chapter 6 Section 1 2. Measure the cutting force coefficients fo r the tool/workpiece material. Chapter 5 gives the procedure used in estimating the mean values of these coefficients and details the regression analys is needed to estimate the mean values, standard deviation, and correlation between these coefficients. 3. The confidence levels in the spindle speed and radial depth can be either estimated from experience or machine manufacturer data. 4. These steps enable estimation of the mean values, confidence levels (standard deviation), and correlation in the input parameters to the milling model. Depending on the confidence levels in these paramete rs, an uncertainty prediction method can be used to estimate the confidence le vels in the stability boundary and SLE Two methods can be used: 1) the sensitivit y method; 2) the sampling methods (Monte Carlo and Latin Hyper-Cube). If the coeffici ent of variation in the input parameters (especially K or M ) is larger than 1%, then the sensitivity method cannot be used due to the non-linearity of axial depth limit to these parameters. Chapter 5 gives an

PAGE 121

104 example on how to use the sampling methods to assign confidence levels on the stability boundary and SLE. 5. The stability boundary confidence level, U obtained in step 4 is used in the robust optimization algorithm. The algorithm formulation is repeated here ,,, 3 : ,, 1 ... ,,,1,MRRifbfbfb SLESLESLEmin subjecttofbeforik gbUgbUgbU for a series of s ,MRRelected limits (e) on f (6.1) The spindle speed perturbed SLE average is used to account for SLE sensitivity to spindle speed. A typica l value for the perturbation is 50 rpm to 100 rpm In order to calculate the trade-off curve between SLE and MRR the optimization algorithm is run for a series of limits on the MRR objective. 6. The trade-off curve is used to select optimum cutting conditions that match the designer preferences. Typically a knee in the curve would indi cate a preferential design where the highest MRR can be achieved for a moderate SLE This completes the description of the selection of robust cutting conditions. Limitations and Future Research In this section, the limitations of this research are discussed as well as the recommendations for future research. Th e limitations and recommendations are as follows: 1. Further efforts should account for the pot ential variation of the cutting force coefficients as a function of spindle sp eed. This entails a significant amount of experimental testing. 2. The peak amplitude method used to obtai n the fitted modal parameters of the tool/workpiece system does not perform we ll near the system natural frequency. This makes the model predictions poor at regions where more accuracy is actually needed. 3. A weakness in the solution method ( TFEA ) was observed at shallow axial depths of cut (2 mm in our tests). Further testing at this condition is needed to verify this discrepancy and account for it.

PAGE 122

105 4. The solution method ( TFEA ) assumes straight cutter teet h while most cutters have a helix angle. This makes the model pred ictions more conservative and can over predict the SLE

PAGE 123

106 APPENDIX A TIME FINITE ELEMENT ANALYSIS Mechanical Model A schematic diagram of two degree of freedom milling process is shown in Figure 1 (repeated here). With the assumption of either a compliant structure or tool, a summation of forces gives the following equation of motion: Equation Chapter 1 Section 1 k y c y k x c x x y Figure 1. Schematic of 2-DOF milling tool 00 0() ()()() +, 00 0() ()()() mc kFt xtxtxt xx xx mc kFt ytytyt yy yy (A.1)

PAGE 124

107 where the terms mx,y, cx,y, kx,y and Fx,y are the modal mass, damping, spring stiffness, and cutting forces in the flexible directions of the system. The x and y cutting force components on the pth tooth are given by: () cos()sin() () (), () ()sin()cos() Ft tt Ft xppp tp gt p Ft Fttt np pp yp (A.2) where gp(t) acts as a switching function It is equal to one if the pth tooth is active and zero if it is not cutting [54] The tangential and normal cutting force components, Ftp(t) and Fnp(t), respectively, are considered to be the product of linearized cutting coefficients Kt and Kn, the nominal depth of cut b and the instantaneous chip thickness wp(t): () (), ()tp tte p np nneFt KK bwtb Ft KK (A.3) where wp(t) depends upon the feed per tooth, h the cutter rotation angle p, and regeneration in the compliant structure directions: ()sin()()()sin()()()cos(). wthtxtxttytytt p ppp (A.4) Here = 60/N is the tooth passing period, is the spindle speed given in rpm, h is chip load (used instead of c to differentiate it from cosp defined later) and N is the total number of cutting teeth. The angular position of the pth tooth for a cutter with evenly spaced teeth is p(t)=(2 /60)t+ p 2 / N The total cutting force equations are found by summing the forces on each cutting tooth in Eq. (A.2) and substituting Eqs. (A.3) and (A.4) into Eq. (A.2):

PAGE 125

108 2 2 () () 22 () ()() ()() 22tene teneKscKs KcKs n t h KsKc Ft KsKsc x n t gtb p Ft KscKsKcKsc y xtxt nn tt ytyt KsKscKscKc nn tt 1 N p (A.5) where s = sin p(t) and c = cos p(t). A more compact form fo r the equation of motion is realized by making the following substitutions: 22 ()() 22 1 KscKsKcKsc N nn tt tgt c p p KsKscKscKc nn tt K (A.6) 2 ()() 2 1tene teneKscKs N KcKs n t ftgth o p KsKc p KsKsc n t (A.7) Using Eqs. (A.5), (A.6) and (A.7) Eq.(A.1) can be written as: ()()()()()() X tXtXttbXtXt+ftb o MCKK c (A.8) where T X txtyt is the two-element position vector and M, C, and K are the 22 x mass, damping, and stiffness matrices of Eq.(A.1). Time Finite Element Analysis (TFEA) The dynamic behavior of the milling proce ss is governed by Eq.(A.8). Since this equation does not have a closed form solu tion, an approximate solution is sought to understand the behavior of the system. On e such approximation technique used for dynamic systems is TFEA [54]. An approximate discrete linear map is constructed using time finite elements in the cut to exact ma pping of free vibration out of the cut, where mapping is performed on displacement and ve locity components of vibration [54-57]. The formulation of the dynamic map for the multiple degree of freedom systems closely

PAGE 126

109 follows the discretization procedure outlined in references [54], but has been presented here for completeness. Free Vibration When the tool is not in contact with th e work-piece, the system is governed by the equation for free vibration. The cutt ing forces then become zero: ()()() XtXtXt MCK0, (A.9) and the exact solution for the free vibration can be written with a state transition matrix ,fccttt where ct is that the time the tool leaves the material and ft is the duration of free vibration. Exact mapp ing of displacement and veloci ty components can be written in terms of state transition matrix as: Xtt c Xt f c ttt cc f Xt c Xtt c f (A.10) Vibration during Cutting When the tool is in the cut, its motion is governed by a time delayed-differential equation. Since this equation does not have a closed form solution, an approximate solution for the tool displacement is assumed for the jth element of the nth tooth passage as a linear combination of polynomials [54]: 4 () 1n X tat jiij i (A.11) Here 1 1j ttnt j k k is the local time within the jth element of the nth period, the length of the kth element is tk and the trial functions i( j(t)) are cubic Hermite polynomials defined in Eq. (A.12),

PAGE 127

110 23 132 1 23 2 2 23 32 3 23 2 4jj j tt jj jjj t jj ttt jjj jj j tt jj jj t jj tt jj (A.12) Substitution of the assumed solution of Eq. (A.11) into the equation of motion (Eq. (A.8)) leads to a non-zero erro r. The error from the assume d solution is weighted by multiplying by a set of test functions and setting the integral of the weighted error to zero. Two test functions are ch osen to be a constant 1( j)=1 and 2( j)= j/tj-1/2 (linear). The integral is taken over the time for each element, tj=tc/E, thereby dividing the time in the cut tc into E elements. The resulting two equations are 44 i=1i=1 4 i=1 0 0 4 1 1 nn aa pp jiijjjiijj n ba cp t jjiijj j d j n ba cp jjiijj i bf op jj M+C KK K p=1,2 (A.13)

PAGE 128

111 where Kc( j) and f o j have been used in place of previously defined Kc(t) and f ot to explicitly show the dependence on local time. In Eq. (A.13), the index j refers to the number of elements and the index i refers to the corresponding Hermite polynomial. The displacement and velocity at tool entry into the cut are specified by the coefficients of the first two basis functions on the first element: 11 n a and 12 n a The relationship between the ini tial and final conditions during free vibration can be mapped through th e state transiti on matrix as ,1 3 11 12 4n n a a E a a E (A.14) where E is the total number of finite elements in the cut and the last term in Eq. (A.14) is the displacement and velocity of the element as it leaves the cut. For the remainder of the elements in the cut, a continuity constraint is imposed to set the position and velocity at the end of one element (13 a and 14 a for the 1st element) equal to the position and velocity at the beginning of the next element (21 a and 23 a for the 2nd element), see Figure 63.

PAGE 129

112 a 11 a 12 a 13 = a 21 a 14 = a 22 cutting zon e a 23 a 24 element 1 element 2 free vibration zone a ji j refers to number of elements i refers to Hermite coefficient Figure 63. Slotting cut with time in the cut divided into two elements. Transition matrix maps the position and velocity exactly in free vibration zone, while elements map them in cutting zone. Equations (A.13) and (A.14) can be arranged into a global matrix mapping the position and velocity of each tooth passage in terms of the previous one. Equation (A.13) maps the cutting zone approximately, while Eq. (A.14) maps the free vibration zone exactly. The following expression is for the case when number of elements E = 3

PAGE 130

113 1111 12 000000 21 1111 0000 1212 22 2222 0000 31 1212 3333 32 0000 1212 33 34 n aa a II a NNPP a a NNPP a NNPP a a ,1 0 0 12 1 21 2 22 1 31 2 32 1 33 2 34 n a C a C a C a C a C a C a (A.15) where the sub-matrices and elements of the sub-matrices for the jth element are j j NN NN jj 1314 1112 N = N= 12 NNNN 21222324 j j jj 1314 1112 = = 12 21222324 PP PP PP PPPP (A.16) 0 -b c -b c 0 0 t ijij j j Nd p pijj jij t j j Pd p pijijjj t j j Cbfd pop jjj MC KK K (A.17) Equation (A.15) describes a discrete dynamical system, or map, that can be written as A=B+ -1 or =Q+ -1 aaC n n aaD n n (A.18)

PAGE 131

114 Stability Prediction The eigenvalues of the transition matrix Q=A-1B are called characteristic multipliers (CMs) and take on a discrete ma pping analogy to the characteristic exponents that govern stability for continuous systems. The condition for stability is that the magnitudes of the CMs must be in a modulus of less than one for a given spindle speed () and depth of cut (b) for the milling process to be asymptotically stable. Surface Location Error Surface location error is defined as the e rror in the placement of the milling cutter teeth when the surface is generated. When the milling process is stable, the surface location error can be obtained by extracting th e position of the tool when the surface is generated. At steady state, the displacement a nd velocity coefficients are constant and are found from fixed points (* a n ) of the dynamic map: 1 aaa nn n (A.19) Substitution of Eq. (A.19) into Eq. (A.18) gives the fixed point map solution or steady state coefficient vector: -1 =-D a nIQ (A.20) Since Q and D can be computed for the milling parameters, the fixed point displacement solution can be found and used to specify surface location error as a function of machining process parameters.

PAGE 132

115 APPENDIX B MATLAB CODE Robust Optimization Code Main program % M. Kurdi (12/1/2004) % surface location error and MRR robust optimization clc; clear all; close all; pack; global Min_speed Max_speed Min_ depth Max_depth MRR_c band; warning off all; band =500; Min_speed = 10e3; Max_speed = 20e3; Min_depth = 1e-6; Max_depth = 18e-3; nteeth = 1; tic % % %%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% % % Finding the initial guess % %%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% % % MRR_c MRR constraint in mm^3/s % Max_MRR = 400; hand = waitbar(0,'Please wait'); for MRR_c = 100:100:Max_MRR delete('MULTIPOINT OPTIMUMS.m') fid2 = fopen('MULTIPOINT OPTIMUMS.m','a'); speed_vec = 0:.1 :1.0; % spindle speed radial_vec = linspace(.03,1,11); % radial depth for i=1:length(speed_vec) % loop for spindle speed x0(2) = speed_vec(i); for j=1:length(radi al_vec) % loop for radial depth x0(3) = radial_vec(j); % figure % contour_plot; % obj_mrr; % hold on; a = x0(3) 25.4e-3; % radial depth of cut h = 0.1e-3; % feed per tooth % calculate initial depth, using MRR constraint and initial speed rpm = x0(2) (Max_speed Min_speed) + Min_speed;

PAGE 133

116 b = MRR_c / (a rpm h nteeth / 60 1e9); disp('axial depth(m) speed (rpm) radial immersion (m)'); design_point = [num2str(b),' ',num2str(rpm),' ',num2str(a)]; disp(design_point); x0(1) = (b Min_depth) / (Ma x_depth Min_depth); lb = [0 0 0.01]; ub = [1 1 1]; options = optimset('LargeSc ale','off','MaxFunEvals',50 );%,'Display','iter'); %fprintf(fid,'SLE_depth depth speed\n'); [x,fval,EXITFLAG] = fm incon(@obj,x0,[],[],[],[], lb,ub,@confun,options); [c,ceq] = confun(x); if EXITFLAG > 0 % solution found depth = x( 1) (Max_depth Mi n_depth) + Min_depth; rpm = x(2) (Max_speed Min_speed) + Min_speed; a = x(3); sle_exact = sle([rpm depth a]); % multipoint optimum file fprintf(fid2,'%e %e %e %e %e %e %e %e %e\n',depth,rpm,x(3),fva l,sle_exact,c(1),c(2),c(3),c(4)); end % end if loop end % end radial loop end % end spindle speed loop % finding minimu m of all solutions found fclose(fid2); fid2 = fopen('MULTIPOINT OPTIMUMS.m','r'); xx = fscanf(fid2,'%e %e %e %e %e %e %e %e %e\n',[9 inf]); xx = xx'; fclose(fid2); % find minimum value of sle_depth for the ra nge of speed initial % guesses and a particular MRR [minimum_sle,index]=min(xx(:,4)); fid3 = fopen('OPTIMUM POINTS.m','a'); % Optimum points file fprintf(fid3,'%e %e %e %e %e %e %e %e %e %e\n',MRR_c,xx(index,1),xx(index,2),xx( index,3),xx(index,4),xx(index,5),xx(ind ex,6),xx(index,7),xx(index,8),xx(index,9)); fclose(fid3); waitbar(MRR_c/Max_MRR,hand); end % end of MRR loop fclose(hand) Constraint Function % objective function for SLE / depth of cut function [c, ceq] = confun(x) global Min_speed Max_speed Min_ depth Max_depth MRR_c band;

PAGE 134

117 x1 = x(1) (Max_depth Min_depth) + Min_depth; x2 = x(2) (Max_speed Min_speed) + Min_speed; % MRR constraint on the first perturbed point c1 = confun1([x1 x2-band x(3)]); c2 = confun1([x1 x2 x(3)]); c3 = confun1([x1 x2+band x(3)]); % MRR constraint h = 0.1e-3; % m/tooth % b = x(1) ; % x(2) rpm % x(3) radial step a = x(3)*.0254; % radial depth in m nteeth = 1; MRR = a .* x1 h nteeth .* x2 / 60 1e9; c4 = MRR_c MRR; c = [c1 c2 c3 c4]; ceq = []; % objective function for SLE / depth of cut function c = confun1(x) % Input: % rpm speed (rpm) % E number of elements % Output: % CM eigen value for rpm and doc % b transition depth of cut (m) % b = x(1); rpm = x(2); E = 25; % number of elements Kt = 1295.9e6*(rpm/1000)^-0.2285; % N/m2 Kn = ((rpm/1000)^2*1.8485-54.604*(rpm/1000)+423.77)*1e6; Kte = ((rpm/1000)^2*-0.1335+3.2431*(rpm/1000)+27.216)*1e3; % N/m Kne = ((rpm/1000)^2*-0.0821+1.4447*(rpm/1000)+30.202)*1e3; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% % CUT PROCESS DESCRI PTION GEOMETRY/IMMERSION/PROCESS PARAMETERS %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% h = 0.1e-3; % feed per tooth nteeth = 1; % number of teeth Diam = 1; % inches rstep = x(3); % radial immersion (inches) TRAVang = acos(1-rste p/(Diam/2)); % angular travel during cutting LAGang = 2*pi/nteeth; % separation angle for teeth

PAGE 135

118 rho = acos(1-rstep/(Diam/2))/(2*pi ); % fraction of time in cut IMMERSION = rstep/Diam; opt = 'up'; if TRAVang>LAGang % MU LTIPLE TEETH ARE IN CONTACT teethNcontact = floor(TRAVang/LAGang) +1; else % SINGLE TOOTH IN CONTACT teethNcontact = 1; end %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% % LOAD SYSTEM IDENTIFICATION MATRICES %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% % load SYSTEM_ID_1mode Kx = 4.4528e+006; Ky = 3.5542e+006; Mx = 0.4362; My = 0.3471; Cx = 82.5955; Cy = 89.8606; % zeta_x = .02996; zeta_y = 0.02576; % freq_x = 362.75; freq_y = 362.71; % Kx = 1.308e6; Ky = 1.194e6; % Mx = Kx/(freq_x*2*pi)^2; % My = Ky/(freq_y*2*pi)^2; % Cx = zeta_x 2 Mx 2*pi freq_x; % Cy = zeta_y 2 My 2*pi freq_y; M =[Mx zeros(size(Mx)); zeros(size(Mx)) My]; C =[Cx zeros(size(Mx)); zeros(size(Mx)) Cy]; K =[Kx zeros(size(Mx)); zeros(size(Mx)) Ky]; lmx = length(Mx(1,:)); lmy = length(My(1,:)); lmx=1; lmy=1; DOF=2; Mx=M(1,1); My=M(2,2); DOF = lmx+lmy; V = [ones(1,lmx) zeros(1 ,lmy); zeros(1,lmx) ones(1,lmy)]; A = zeros((E+1)*2*DOF,(E+1)*2*DOF); B = A; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % BEGIN LOOP CALCULATIONS OVER RPM vs DOC FIELD %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% speed = rpm; omega = speed/60*(2*pi); % radians per second T = (2*pi)/omega/nteeth; % tooth pass period TC = rho*T*nteeth; % time a single tooth spends in the cut

PAGE 136

119 tf = T-TC; % time for free vibs tj = TC/E; % time for each element %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % SET CUTTER ROTA TION ANGLE FOR UP/DOWN-MILLING %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% switch opt case 'up' t0mat = [0 tj*(1:(E-1))]; % upmilling locat = 2*DOF+lmx+1:3*DOF; case 'down' tex = pi/omega; tent=tex-TC; % downmilling t0mat = [tent tent+tj*(1:(E-1))]; % downmilling locat = (E+1)*2*DOF-DOF-lmy+1:(E+1)*2*DOF-DOF; end %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% % STATE TRANSITION MATRIX %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% G1 = [zeros(size(M)) M; eye(size(M)) zeros(size(M))]; G2 = [K C; zeros(s ize(M)) -eye(size(M))]; G = -G1\G2; PHI = expm(G*tf); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% % N & P are used to create A & B which then become Q in..... a_n = Q a_n-1 + D %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% for e=1:E, t0 = t0mat(e); C1 = V'*[ -1/4*b*(-h*Kt*cos( 2*t0*omega+2*omega*tj)+2*h*Kn*omega*tjh*Kn*sin(2*t0*omega+2*omega*tj)+4*Kte*sin(t0*omega+omega*tj)4*Kne*cos(t0*omega+omega*tj)+h*Kt*co s(2*t0*omega)+h*Kn*sin(2*t0*omega)4*Kte*sin(t0*omega)+4*Kne*cos(t0*omega))/omega; 1/4*b*(2*h*Kt*omega*tjh*Kt*sin(2*t0*omega+2*omega*tj)+h*Kn*cos(2*t0*omega+2*omega*tj)4*Kte*sin(t0*omega+omega*tj)+4*Kne*cos (t0*omega+omega*tj)+h*Kt*sin(2*t0*ome ga)-h*Kn*cos(2*t0*omega)+4*Kte*sin(t0*om ega)-4*Kne*cos(t0*omega))/omega]; C2 = V'*[ 1/8*b*(h*Kt*sin(2*t0*omega+2*omega*tj)+h*Kt* cos(2*t0*omega+2*omega*tj)*omega*tj+h* Kn*cos(2*t0*omega+2*omega*tj)+h*Kn*s in(2*t0*omega+2*omega*tj)*omega*tj+8*K te*sin(t0*omega+omega*tj)-4*Kte *cos(t0*omega+omega*tj)*omega*tj-

PAGE 137

120 8*Kne*cos(t0*omega+omega*tj)4*Kne*sin(t0*omega+omega*tj)*omega*tj+h*Kt*sin(2*t0*omega)h*Kn*cos(2*t0*omega)8*Kte*sin(t0*omega)+8*Kne*cos(t0*omega )+h*Kt*tj*cos(2*t0*omega)*omega+h*Kn* tj*sin(2*t0*omega)*omega-4*K te*tj*cos(t0*omega)*omega4*Kne*tj*sin(t0*omega)*omega)/tj/omega^2; 1/8*b*(h*Kt*cos(2*t0*omega+2*omega*tj)+ h*Kt*sin(2*t0*omega+2*omega*tj)*omeg a*tj+h*Kn*sin(2*t0*omega+2*omega*tj)h*Kn*cos(2*t0*omega+2*omega*tj)*omega*tj8*Kte*sin(t0*omega+omega*tj)+4*Kte*cos (t0*omega+omega*tj)*omega*tj+8*Kne*co s(t0*omega+omega*tj)+4*Kne*sin (t0*omega+omega*tj)*omega*tjh*Kt*cos(2*t0*omega)-h*Kn*sin(2*t0*omega)+8*Kte*sin(t0*omega)8*Kne*cos(t0*omega)+h*Kt* tj*sin(2*t0*omega)*omegah*Kn*tj*cos(2*t0*omega)*omega+4*Kte*tj*co s(t0*omega)*omega+4*Kne*tj*sin(t0*o mega)*omega)/tj/omega^2]; P11 = [ 1/8*b*(3*Kt*sin(2*t0*omega+2*omega*tj)+3*K n*cos(2*t0*omega+2*omega*tj)+2*Kn*omeg a^4*tj^4+3*Kt*omega*tj*cos(2*t0*omega +2*omega*tj)+3*Kn*omega*tj*sin(2*t0*ome ga+2*omega*tj)+3*Kt*sin(2*t0*omega)3*Kn*cos(2*t0*omega)+2*Kt*cos(2*t0*omeg a)*omega^3*tj^3+2*Kn*sin(2*t0*omega) *omega^3*tj^3+3*Kt*omega*tj*cos(2*t0*omega)+3*Kn*omega*tj*sin(2*t0*omega))/o mega^4/tj^3, 1/8*b*(-3*Kt*cos(2*t0*omega+2*omega*tj)3*Kn*sin(2*t0*omega+2*omega*tj)+2*Kt*omega^4*tj^43*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+3*Kn*omega*tj*cos(2*t0*omega+2*om ega*tj)+3*Kt*cos(2*t0*omega )+3*Kn*sin(2*t0*omega)2*Kt*sin(2*t0*omega)*omega^3*tj^ 3+2*Kn*cos(2*t0*omega)*omega^3*tj^33*Kt*omega*tj*sin(2*t0*omega)+3*Kn*ome ga*tj*cos(2*t0*omega))/omega^4/tj^3; 1/8*b*(-3*Kt*cos(2*t0*omega+ 2*omega*tj)-3*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^4*tj^43*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+3*Kn*omega*tj*cos(2*t0*omega+2*om ega*tj)+3*Kt*cos(2*t0*omega )+3*Kn*sin(2*t0*omega)2*Kt*sin(2*t0*omega)*omega^3*tj^ 3+2*Kn*cos(2*t0*omega)*omega^3*tj^33*Kt*omega*tj*sin(2*t0*omega)+3*Kn*omega *tj*cos(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kn*omega^4*tj^4+3*Kt*omega*tj*cos(2* t0*omega+2*omega*tj)+3*Kn*omega*tj*s in(2*t0*omega+2*omega*tj)3*Kt*sin(2*t0*omega+2*omega*tj)+3*Kn*c os(2*t0*omega+2*omega*tj)+2*Kt*cos(2* t0*omega)*omega^3*tj^3+2*Kn*sin(2*t0*omega)*omega^3*tj^3+3*Kt*omega*tj*cos( 2*t0*omega)+3*Kn*omega*tj*sin(2*t0*omega)+3*Kt*sin(2*t0*omega)3*Kn*cos(2*t0*omega))/omega^4/tj^3]; P12 =[ 1/48*b*(6*Kn*omega*tj*sin(2*t0*omega+2*om ega*tj)+6*Kt*omega*tj*cos(2*t0*ome ga+2*omega*tj)+2*Kn*omega^4*tj^49*Kt*sin(2*t0*omega+2*omega*tj)+9*K n*cos(2*t0*omega+2*omega*tj)+12*Kn*ome ga*tj*sin(2*t0*omega)+12*Kt*omega*tj*cos(2*t0*omega)-

PAGE 138

121 6*Kt*omega^2*tj^2*sin(2*t0*omega)+6*Kn*om ega^2*tj^2*cos(2*t0*omega)+9*Kt*si n(2*t0*omega)-9*Kn*cos(2*t0*omega))/tj^2/omega^4, 1/48*b*(9*Kt*cos(2*t0*omega+2*omega* tj)+9*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^4*tj^4+6*Kt*omega*tj* sin(2*t0*omega+2*omega*tj)6*Kn*omega*tj*cos(2*t0*omega+2*omega* tj)+12*Kt*omega*tj*sin(2*t0*omega)+6* Kn*omega^2*tj^2*sin(2*t0*omega)+ 6*Kt*omega^2*tj^2*cos(2*t0*omega)9*Kt*cos(2*t0*omega)-12*Kn*omega*tj*cos(2*t0*omega)9*Kn*sin(2*t0*omega))/tj^2/omega^4; 1/48*b*(2*Kt*omega^4*tj^4+6*Kt*omega* tj*sin(2*t0*omega+2*omega*tj)+9*Kt*cos( 2*t0*omega+2*omega*tj)6*Kn*omega*tj*cos(2*t0*omega+2*omega *tj)+9*Kn*sin(2*t0*omega+2*omega*tj)+1 2*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga^2*tj^2*sin(2*t0*omega)+6*Kt*omega^ 2*tj^2*cos(2*t0*omega)-9*Kt*cos(2*t0*omega)-12*Kn*omega*tj*cos(2*t0*omega)9*Kn*sin(2*t0*omega))/tj^2/omega^4, 1/48*b*(6*Kt*omega*tj*cos(2*t0*o mega+2*omega*tj)6*Kn*omega*tj*sin(2*t0*omega+ 2*omega*tj)+2*Kn*omega^4*tj^49*Kn*cos(2*t0*omega+2*omega*tj)+9*K t*sin(2*t0*omega+2*omega*tj)12*Kt*omega*tj*cos(2*t0*omega)12*Kn*omega*tj*sin(2*t0*omega)+6*K t*omega^2*tj^2*sin(2*t0*omega)6*Kn*omega^2*tj^2*cos(2*t0*om ega)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega ))/tj^2/omega^4]; P13 =[ 1/8*b*(2*Kn*omega^4*tj^43*Kn*cos(2*t0*omega+2*omega*tj)+3*K t*sin(2*t0*omega+2*omega*tj)3*Kt*omega*tj*cos(2*t0*o mega+2*omega*tj)2*Kn*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)3*Kn*omega*tj*sin(2*t0*omega+2*omega*tj)2*Kt*omega^3*tj^3*cos(2*t0*omega+ 2*omega*tj)+3*Kn*cos(2*t0*omega)3*Kt*sin(2*t0*omega)-3*Kt*omega*tj*cos(2*t0*omega)3*Kn*omega*tj*sin(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kt*omega^4*tj^4+2*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)2*Kn*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+3*Kt*cos(2*t0*omega+2*omega*t j)+3*Kn*sin(2*t0*omega+2*omega*tj)+3*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)3*Kn*omega*tj*cos(2*t0*omega+2*ome ga*tj)-3*Kt*cos(2*t0*omega)3*Kn*sin(2*t0*omega)+3*Kt*omega*tj*sin(2*t0*omega)3*Kn*omega*tj*cos(2*t0*omega))/omega^4/tj^3; 1/8*b*(2*Kt*omega^4*tj^4+2*Kn*omega^3*tj^ 3*cos(2*t0*omega+2*omega*tj)+3*Kn *omega*tj*cos(2*t0*omega+2*omega*t j)-3*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)3*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)3*Kt*cos(2*t0*omega+2*omega*tj)+3*Kn*s in(2*t0*omega)+3*Kt*cos(2*t0*omega)3*Kt*omega*tj*sin(2*t0*omega)+3*Kn*omega *tj*cos(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kn*omega^4*tj^4+3*Kn*cos(2*t0*ome ga+2*omega*tj)+2*Kn*omega^3*tj^3 *sin(2*t0*omega+2*omega*tj)3*Kt*sin(2*t0*omega+2*omega*tj)+2*Kt *omega^3*tj^3*cos(2*t0*omega+2*omega*tj

PAGE 139

122 )+3*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)+3*Kn*omega*tj*sin(2*t0*omega+2*o mega*tj)3*Kn*cos(2*t0*omega)+3*Kt*sin(2*t0*omeg a)+3*Kt*omega*tj*cos(2*t0*omega)+3* Kn*omega*tj*sin(2*t0*omega))/omega^4/tj^3]; P14 =[ -1/48*b*(6*Kn*cos(2*t 0*omega+2*omega*tj)*omega^2*tj^212*Kn*omega*tj*sin(2*t0*omega+2*omega*tj)6*Kt*sin(2*t0*omega+2*ome ga*tj)*omega^2*tj^212*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)+2*Kn*omega^4*tj^49*Kn*cos(2*t0*omega+2*omega*tj)+9*K t*sin(2*t0*omega+2*omega*tj)6*Kn*omega*tj*sin(2*t0*omega)6*Kt*omega*tj*cos(2*t0*omega)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega))/tj^2/omeg a^4, 1/48*b*(-2*Kt*omega^4*tj^412*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+12*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+6*Kt*cos(2*t0*omega+2*omega* tj)*omega^2*tj^2+6*Kn*sin(2*t0*omega+2 *omega*tj)*omega^2*tj^2-9*K t*cos(2*t0*omega+2*omega*tj)9*Kn*sin(2*t0*omega+2*omega*tj)6*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga*tj*cos(2*t0*omega)+9*Kt*cos(2*t0*o mega)+9*Kn*sin(2*t0*omega))/tj^2/omega^4; 1/48*b*(2*Kt*omega^4*tj^412*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+12*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+6*Kt*cos(2*t0*omega+2*omega* tj)*omega^2*tj^2+6*Kn*sin(2*t0*omega+2 *omega*tj)*omega^2*tj^2-9*K t*cos(2*t0*omega+2*omega*tj)9*Kn*sin(2*t0*omega+2*omega*tj)6*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga*tj*cos(2*t0*omega)+9*Kt*cos(2*t0*o mega)+9*Kn*sin(2*t0*omega))/tj^2/omega^4, 1/48*b*(12*Kn*omega*tj*sin(2*t0*omega+2*omega* tj)-9*Kn*cos(2*t0*omega+2*omega*tj)2*Kn*omega^4*tj^4-6*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^212*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)+6*Kn*cos(2*t0*omega+2*omega*tj)*o mega^2*tj^2+9*Kt*sin(2*t0*omega+2*omega*tj)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega)-6*Kn*omega*tj*sin(2*t0*omega)6*Kt*omega*tj*cos(2*t0*omega))/tj^2/omega^4]; P21 =[ -1/80*b*(-15*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^215*Kn*sin(2*t0*omega+2*omega*tj)*omega^2*tj^2+60*Kt*cos(2*t0*omega+2*omega *tj)+60*Kn*sin(2*t0*omega+ 2*omega*tj)+4*Kn*omega^5*tj^5+60*Kt*omega*tj*sin(2 *t0*omega)10*Kn*omega^3*tj^3*cos(2*t0*omega)+15*Kn* omega^2*tj^2*sin(2*t0*omega)+15*K t*omega^2*tj^2*cos(2*t0*omega)+1 0*Kt*omega^3*tj^3*sin(2*t0*omega)60*Kn*omega*tj*cos(2*t0*omega)+10*Kn*sin(2*t0*omega)*omega^4*tj^4+10*Kt*co s(2*t0*omega)*omega^4*tj^4-60*Kn*sin(2*t0*omega)60*Kt*cos(2*t0*omega)+60*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)60*Kn*omega*tj*cos(2*t0*omega+2*ome ga*tj))/omega^5/tj^4, -1/80*b*(60*Kn*cos(2*t0*omega)+60*Kt*sin(2*t0*omega)15*Kn*cos(2*t0*omega+2*omega*tj)*omega^ 2*tj^2+15*Kt*sin(2*t0*omega+2*omega *tj)*omega^2*tj^2+60*Kn*omega* tj*sin(2*t0*omega+2*omega*tj)60*Kt*sin(2*t0*omega+2*omega*tj)+60*Kt *omega*tj*cos(2*t0*omega+2*omega*tj)+ 60*Kn*cos(2*t0*omega+2*omega*tj)+10*Kt*omega^3*tj^3*cos(2*t0*omega)-

PAGE 140

123 15*Kt*omega^2*tj^2*sin(2*t0*omega)10*Kt*sin(2*t0*omega)*omega^4*tj^4+10*K n*cos(2*t0*omega)*omega^4*tj^4+15*K n*omega^2*tj^2*cos(2*t0*omega)+60*Kt*om ega*tj*cos(2*t0*omega)+10*Kn*omega^ 3*tj^3*sin(2*t0*omega)+60*Kn*omega*tj*si n(2*t0*omega)+4*Kt*omega^5*tj^5)/ome ga^5/tj^4; 1/80*b*(60*Kn*cos(2*t0*omega)60*Kt*sin(2*t0*omega)+15*Kn*cos(2*t 0*omega+2*omega*tj)*omega^2*tj^215*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^260*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)-60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega)+15*Kt* omega^2*tj^2*sin(2*t0*omega)+10*Kt *sin(2*t0*omega)*omega^4*tj^4-10*Kn*cos(2*t0*omega)*omega^4*tj^415*Kn*omega^2*tj^2*cos(2*t0*omega )-60*Kt*omega*tj*cos(2*t0*omega)10*Kn*omega^3*tj^3*sin(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)+4*Kt*omega^5*tj^5)/omega^5/tj^4, -1/80*b*(60*Kt*omega*tj*sin(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)10*Kn*sin(2*t0*omega)*omega^4*tj^4-10*K t*cos(2*t0*omega)*omega^4*tj^415*Kt*omega^2*tj^2*cos(2*t0*omega)10*Kt*omega^3*tj^3*sin(2*t0*omega)+10*Kn*omega^3*tj^3*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega)+60*K n*sin(2*t0*omega)+60*Kt*cos(2*t0*ome ga)60*Kt*cos(2*t0*omega+2*omega*tj)+4*Kn*o mega^5*tj^5+60*Kn*omega*tj*cos(2*t0 *omega+2*omega*tj)+15*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+15*Kn*sin (2*t0*omega+2*omega*tj)*omega^2*tj^2-60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega*tj))/omega^5/tj^4]; P22 = [ 1/480*b*(-2*Kn*omega^5*tj^ 5-180*Kt*cos(2*t0*omega+2*omega*tj)180*Kn*sin(2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+135*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+30*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+30*Kn*sin(2*t0*ome ga+2*omega*tj)*omega^2*tj^2225*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+225*Kn*omega*tj*cos(2*t0*omega)120*Kt*omega^2*tj^2*cos(2*t0*omega)+ 30*Kt*omega^3*tj^3*sin(2*t0*omega)30*Kn*omega^3*tj^3*cos(2*t0*omega)120*Kn*omega^2*tj^2*sin(2*t0*om ega))/tj^3/omega^5, 1/480*b*(135*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+135*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*Kn*co s(2*t0*omega+2*omega*tj)+2*Kt*o mega^5*tj^5+30*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*cos(2*t0*omega)+ 120*Kn*omega^2*tj^2*cos(2*t0*omega)120*Kt*omega^2*tj^2*sin(2*t0*omega)30*Kn*omega^3*tj^3*sin(2*t0*omega)+ 225*Kt*omega*tj*cos(2*t0*omega)+225*Kn* omega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5;

PAGE 141

124 1/480*b*(135*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+135*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*K n*cos(2*t0*omega+2*omega*tj)2*Kt*omega^5*tj^5+30*Kt*sin(2*t0*o mega+2*omega*tj)*omega^2*tj^230*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*cos(2*t0*omega)+ 120*Kn*omega^2*tj^2*cos(2*t0*omega)120*Kt*omega^2*tj^2*sin(2*t0*omega)30*Kn*omega^3*tj^3*sin(2*t0*omega)+ 225*Kt*omega*tj*cos(2*t0*omega)+225*Kn* omega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5, -1/480*b*(2*Kn*omega^5*tj^5180*Kt*cos(2*t0*omega+2*omega*tj)-180*K n*sin(2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+135*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+30*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+30*Kn*sin(2*t0*ome ga+2*omega*tj)*omega^2*tj^2225*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+225*Kn*omega*tj*cos(2*t0*omega)120*Kt*omega^2*tj^2*cos(2*t0*omega)+ 30*Kt*omega^3*tj^3*sin(2*t0*omega)30*Kn*omega^3*tj^3*cos(2*t0*omega)120*Kn*omega^2*tj^2*sin(2*t0*om ega))/tj^3/omega^5]; P23 = [ -1/80*b*(-4*Kn*omega^5*tj^5-60*Kt*cos(2*t0*omega+2*omega*tj)60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+60*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+15*Kt*cos(2*t0*omega+2*omega *tj)*omega^2*tj^2+15*Kn*sin(2*t0*omega +2*omega*tj)*omega^2*tj^210*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*sin(2*t0*o mega+2*omega*tj)+10*Kt*omega^4*tj^ 4*cos(2*t0*omega+2*omega*tj)+10*Kn*omeg a^3*tj^3*cos(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega)+60*Kn*s in(2*t0*omega)+60*Kt*cos(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-15*K t*omega^2*tj^2*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega))/omega^5/tj^4, -1/80*b*(60*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)60*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^4*tj^4*sin(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*cos(2*t0*o mega+2*omega*tj)-10*Kn*omega^ 3*tj^3*sin(2*t0*omega+2*omega*tj)4*Kt*omega^5*tj^515*Kt*sin(2*t0*omega+2*omega*tj)*omega ^2*tj^2+15*Kn*cos(2*t0*omega+2*omega *tj)*omega^2*tj^215*Kn*omega^2*tj^2*cos(2*t0*omega)+15*K t*omega^2*tj^2*sin(2*t0*omega)60*Kt*omega*tj*cos(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)60*Kt*sin(2*t0*omega)+60*Kn*cos(2*t0*omega))/omega^5/tj^4; -1/80*b*(-60*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)60*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kn*cos(2*t0*omega+2*omega*tj)-

PAGE 142

125 10*Kt*omega^4*tj^4*sin(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*cos(2*t0*o mega+2*omega*tj)10*Kn*omega^3*tj^3*sin(2*t0*omega+ 2*omega*tj)+4*Kt*omega^5*tj^515*Kt*sin(2*t0*omega+2*omega*tj)*omega ^2*tj^2+15*Kn*cos(2*t0*omega+2*omega *tj)*omega^2*tj^215*Kn*omega^2*tj^2*cos(2*t0*omega)+15*K t*omega^2*tj^2*sin(2*t0*omega)60*Kt*omega*tj*cos(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)60*Kt*sin(2*t0*omega)+60*Kn*cos(2*t0*omega))/omega^5/tj^4, 1/80*b*(4*Kn*omega^5*tj^5-60*Kt*cos(2*t0*omega+2*omega*tj)60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+60*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+15*Kt*cos(2*t0*omega+2*omega *tj)*omega^2*tj^2+15*Kn*sin(2*t0*omega +2*omega*tj)*omega^2*tj^210*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*sin(2*t0*o mega+2*omega*tj)+10*Kt*omega^4*tj^ 4*cos(2*t0*omega+2*omega*tj)+10*Kn*omeg a^3*tj^3*cos(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega)+60*Kn*s in(2*t0*omega)+60*Kt*cos(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-15*K t*omega^2*tj^2*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega))/omega^5/tj^4]; P24 = [ 1/480*b*(2*Kn*omega^5*tj^5+180*Kt*cos( 2*t0*omega+2*omega*tj)+180*Kn*sin(2*t 0*omega+2*omega*tj)+225*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)225*Kn*omega*tj*cos(2*t0*omega+2*omega*tj)120*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2120*Kn*sin(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+30*Kn*omega^3*tj^3*cos(2*t0*o mega+2*omega*tj)+135*Kt*omega*tj*si n(2*t0*omega)-180*Kn*sin(2*t0*omega)180*Kt*cos(2*t0*omega)135*Kn*omega*tj*cos(2*t0*omega)+30*Kt *omega^2*tj^2*cos(2*t0*omega)+30*Kn*o mega^2*tj^2*sin(2*t0*omega))/tj^3/omega^5, 1/480*b*(225*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+225*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*K n*cos(2*t0*omega+2*omega*tj)30*Kt*omega^3*tj^3*cos(2*t0*omega+2*omega*tj)30*Kn*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+2*Kt*omega^5*tj^5+120*Kt*sin( 2*t0*omega+2*omega*tj)*omega^2*tj^2120*Kn*cos(2*t0*omega+2*omega*tj)*omega^2* tj^2+30*Kn*omega^2*tj^2*cos(2*t0* omega)30*Kt*omega^2*tj^2*sin(2*t0*omega)+ 135*Kt*omega*tj*cos(2*t0*omega)+135*Kn*o mega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5; 1/480*b*(-225*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)225*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+180*Kt*sin(2*t0*omega+2*omega*tj )180*Kn*cos(2*t0*omega+2*omega*tj)+30*Kt*omega^3*tj^3*cos(2*t0*omega+2*ome

PAGE 143

126 ga*tj)+30*Kn*omega^3*tj^3*sin(2*t0*om ega+2*omega*tj)+2*Kt*omega^5*tj^5120*Kt*sin(2*t0*omega+2*omega*tj)*omega^ 2*tj^2+120*Kn*cos(2*t0*omega+2*ome ga*tj)*omega^2*tj^230*Kn*omega^2*tj^2*cos(2*t0*omega)+30*K t*omega^2*tj^2*sin(2*t0*omega)135*Kt*omega*tj*cos(2*t0*omega)135*Kn*omega*tj*sin(2*t0*omega)180*Kt*sin(2*t0*omega)+180*Kn*cos(2*t0*omega))/tj^3/omega^5, 1/480*b*(2*Kn*omega^5*tj^5-180*Kt*c os(2*t0*omega+2*omega*tj)180*Kn*sin(2*t0*omega+2*omega*tj)225*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+225*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+120*Kt*cos(2*t0*omega+2*omeg a*tj)*omega^2*tj^2+120*Kn*sin(2*t0*o mega+2*omega*tj)*omega^2*tj^2+30*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj) -30*Kn*omega^3*tj^3*cos( 2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+135*Kn*omega*tj*cos(2*t0*omega)-30*Kt*omega^2*tj^2*cos(2*t0*omega)30*Kn*omega^2*tj^2*sin(2*t0*omega))/tj^3/omega^5]; P11 = [P11(1,1)*ones(lmx,1) P 11(1,2)*ones(lmx,1); P11(2,1)*ones(lmx,1) P11(2,2)*ones(lmx,1)]*V; P12 = [P12(1,1)*ones(lmx,1) P 12(1,2)*ones(lmx,1); P12(2,1)*ones(lmx,1) P12(2,2)*ones(lmx,1)]*V; P13 = [P13(1,1)*ones(lmx,1) P 13(1,2)*ones(lmx,1); P13(2,1)*ones(lmx,1) P13(2,2)*ones(lmx,1)]*V; P14 = [P14(1,1)*ones(lmx,1) P 14(1,2)*ones(lmx,1); P14(2,1)*ones(lmx,1) P14(2,2)*ones(lmx,1)]*V; P21 = [P21(1,1)*ones(lmx,1) P 21(1,2)*ones(lmx,1); P21(2,1)*ones(lmx,1) P21(2,2)*ones(lmx,1)]*V; P22 = [P22(1,1)*ones(lmx,1) P 22(1,2)*ones(lmx,1); P22(2,1)*ones(lmx,1) P22(2,2)*ones(lmx,1)]*V; P23 = [P23(1,1)*ones(lmx,1) P 23(1,2)*ones(lmx,1); P23(2,1)*ones(lmx,1) P23(2,2)*ones(lmx,1)]*V; P24 = [P24(1,1)*ones(lmx,1) P 24(1,2)*ones(lmx,1); P24(2,1)*ones(lmx,1) P24(2,2)*ones(lmx,1)]*V; N11 = -C+1/2*K*tj+P11; N12 = -M+1/12*K*tj^2+P12; N13 = C+1/2*K*tj+P13; N14 = M-1/12*K*tj^2+P14; N21 = M/tj-1/10*K*tj+P21; N22 = 1/2*M-1/12*C*tj-1/120*K*tj^2+P22; N23 = -M/tj+1/10*K*tj+P23; N24 = 1/2*M+1/12*C*tj-1/120*K*tj^2+P24; N1 = [N11 N12; N21 N22]; N2 = [N13 N14; N23 N24]; P1 = [P11 P12; P21 P22]; P2 = [P13 P14; P23 P24]; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%% % BUILD GLOBAL MATRICES

PAGE 144

127 %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%% A(1:2*DOF,1:2*DOF) = eye(2*DOF); A(2*DOF*e+1:2*DOF*e+2*DOF,2* DOF*(e-1)+1:2*DOF*(e-1)+2*DOF) = N1; A(2*DOF*e+1:2*DOF*e+2*DOF ,2*DOF*(e-1)+2*DOF+1:2*DOF*(e1)+2*DOF+2*DOF) = N2; B(2*DOF*e+1:2*DOF*e+2*DOF,2* DOF*(e-1)+1:2*DOF*(e-1)+2*DOF) = P1; B(2*DOF*e+1:2*DOF*e+2*DOF ,2*DOF*(e-1)+2*DOF+1:2*DOF*(e1)+2*DOF+2*DOF) = P2; B(1:2*(DOF),E*2*(DOF )+1:(E+1)*2*(DOF)) = PHI; Cvec(1:2*DOF,1) = zeros(2*DOF,1); Cvec(2*DOF*e+1:2*DOF*e+DOF,1) = C1; Cvec(2*DOF*e+DOF+1:2*DOF*e+2*DOF,1) = C2; end; % end # of elements loop Q = A\B; [vec,lam] = eig(Q); CM = max(abs(diag(lam))); D = A\Cvec; N1 = zeros(2*DOF,2*DOF); N2 = N1; P1 = N1; P2 = P1; c = CM 1; return %save TFEA_STABSLE_LOW ss zz CM ee IMMERSION SLE % NOTES % ss spindle speeds % zz depth of cut % CM charistic multipliers or eigenvalues % SLEsurface location error Objective Function % % average surface location error % % Input: % x(1) depth % x(2) speed % x(3) radial function SLE_AVER = obj(x) global Min_speed Max_speed Min_ depth Max_depth MRR_c band; x1 = x(1) (Max_depth Min_depth) + Min_depth; x2 = x(2) (Max_speed Min_speed) + Min_speed; % MRR constraint on the first perturbed point sle1 = obj1([x1, x2-band ,x(3)]); sle2 = obj1([x1, x2, x(3)]); sle3 = obj1([x1, x2+band ,x(3)]); SLE_AVER = (sle1+sle2+sle3)/3;

PAGE 145

128 % objective function for SLE function SLE = obj1(x) % Input: % x(1) depth % x(2) speed % x(3) radial b = x(1); rpm = x(2); rstep = x(3); % radial immersion (inches) E = 25; % number of elements % adjust cutting coefficients to spindle speed Kt = 1295.9e6*(rpm/1000)^-0.2285; % N/m2 Kn = ((rpm/1000)^2*1.8485-54.604*(rpm/1000)+423.77)*1e6; Kte = ((rpm/1000)^2*-0.1335+3.2431*(rpm/1000)+27.216)*1e3; % N/m Kne = ((rpm/1000)^2*-0.0821+1.4447*(rpm/1000)+30.202)*1e3; %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%% % CUT PROCESS DESC RIPTION GEOMETRY/IMMERSION/PROCESS PARAMETERS %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% h = 0.1e-3; % feed per tooth nteeth = 1; % number of teeth Diam = 1; % inches TRAVang = acos(1-rs tep/(Diam/2)); % angular travel during cutting LAGang = 2*pi/nteeth; % separation angle for teeth rho = acos(1-rstep/(Diam/2))/(2*pi); % fraction of time in cut IMMERSION = rstep/Diam; opt = 'up'; if TRAVang>LAGang % MU LTIPLE TEETH ARE IN CONTACT teethNcontact = floor(TRAVang/LAGang) +1; else % SINGLE TOOTH IN CONTACT teethNcontact = 1; end %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% % LOAD SYSTEM IDENTIFICATION MATRICES %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% % load SYSTEM_ID_1mode Kx = 4.4528e+006; Ky = 3.5542e+006; Mx = 0.4362; My = 0.3471; Cx = 82.5955;

PAGE 146

129 Cy = 89.8606; % zeta_x = .02996; zeta_y = 0.02576; % freq_x = 362.75; freq_y = 362.71; % Kx = 1.308e6; Ky = 1.194e6; % Mx = Kx/(freq_x*2*pi)^2; % My = Ky/(freq_y*2*pi)^2; % Cx = zeta_x 2 Mx 2*pi freq_x; % Cy = zeta_y 2 My 2*pi freq_y; M =[Mx zeros(size(Mx)); zeros(size(Mx)) My]; C =[Cx zeros(size(Mx)); zeros(size(Mx)) Cy]; K =[Kx zeros(size(Mx)); zeros(size(Mx)) Ky]; lmx = length(Mx(1,:)); lmy = length(My(1,:)); lmx=1; lmy=1; DOF=2; Mx=M(1,1); My=M(2,2); DOF = lmx+lmy; V = [ones(1,lmx) zeros(1 ,lmy); zeros(1,lmx) ones(1,lmy)]; A = zeros((E+1)*2*DOF,(E+1)*2*DOF); B = A; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % BEGIN LOOP CALCULATIONS OVER RPM vs DOC FIELD %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% speed = rpm; omega = speed/60*(2*pi); % radians per second T = (2*pi)/omega/nteeth; % tooth pass period TC = rho*T*nteeth; % time a single tooth spends in the cut tf = T-TC; % time for free vibs tj = TC/E; % time for each element %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % SET CUTTER ROTA TION ANGLE FOR UP/DOWN-MILLING %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%% switch opt case 'up' t0mat = [0 tj*(1:(E-1))]; % upmilling locat = 2*DOF+lmx+1:3*DOF; case 'down' tex = pi/omega; tent=tex-TC; % downmilling t0mat = [tent tent+tj*(1:(E-1))]; % downmilling locat = (E+1)*2*DOFDOF-lmy+1:(E+1)*2*DOF-DOF; end %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%

PAGE 147

130 % STATE TRANSITION MATRIX %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% G1 = [zeros(size(M)) M; eye(size(M)) zeros(size(M))]; G2 = [K C; zeros(s ize(M)) -eye(size(M))]; G = -G1\G2; PHI = expm(G*tf); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% % N & P are used to create A & B which then become Q in..... a_n = Q a_n-1 + D %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% for e=1:E, t0 = t0mat(e); C1 = V'*[ -1/4*b*(-h*Kt*cos(2*t0*o mega+2*omega*tj)+2*h*Kn*omega*tjh*Kn*sin(2*t0*omega+2*omega*tj)+4*Kte*sin(t0*omega+omega*tj)4*Kne*cos(t0*omega+omega*tj)+h*Kt*co s(2*t0*omega)+h*Kn*sin(2*t0*omega)4*Kte*sin(t0*omega)+4*Kne*cos(t0*omega))/omega; 1/4*b*(2*h*Kt*omega*tjh*Kt*sin(2*t0*omega+2*omega*tj)+h*Kn*cos(2*t0*omega+2*omega*tj)4*Kte*sin(t0*omega+omega*tj)+4*Kne*cos (t0*omega+omega*tj)+h*Kt*sin(2*t0*ome ga)-h*Kn*cos(2*t0*omega)+4*Kte*sin(t0*om ega)-4*Kne*cos(t0*omega))/omega]; C2 = V'*[ 1/8*b*(h*Kt*sin(2*t0*omega+2*omega*tj)+h*Kt* cos(2*t0*omega+2*omega*tj)*omega*tj+h* Kn*cos(2*t0*omega+2*omega*tj)+h*Kn*s in(2*t0*omega+2*omega*tj)*omega*tj+8*K te*sin(t0*omega+omega*tj)-4*Kte *cos(t0*omega+omega*tj)*omega*tj8*Kne*cos(t0*omega+omega*tj)4*Kne*sin(t0*omega+omega*tj)*omega*tj+h*Kt*sin(2*t0*omega)h*Kn*cos(2*t0*omega)8*Kte*sin(t0*omega)+8*Kne*cos(t0*omega )+h*Kt*tj*cos(2*t0*omega)*omega+h*Kn* tj*sin(2*t0*omega)*omega-4*K te*tj*cos(t0*omega)*omega4*Kne*tj*sin(t0*omega)*omega)/tj/omega^2; 1/8*b*(h*Kt*cos(2*t0*omega+2*omega*tj)+ h*Kt*sin(2*t0*omega+2*omega*tj)*omeg a*tj+h*Kn*sin(2*t0*omega+2*omega*tj)h*Kn*cos(2*t0*omega+2*omega*tj)*omega*tj8*Kte*sin(t0*omega+omega*tj)+4*Kte*cos (t0*omega+omega*tj)*omega*tj+8*Kne*co s(t0*omega+omega*tj)+4*Kne*sin (t0*omega+omega*tj)*omega*tjh*Kt*cos(2*t0*omega)-h*Kn*sin(2*t0*omega)+8*Kte*sin(t0*omega)8*Kne*cos(t0*omega)+h*Kt* tj*sin(2*t0*omega)*omegah*Kn*tj*cos(2*t0*omega)*omega+4*Kte*tj*co s(t0*omega)*omega+4*Kne*tj*sin(t0*o mega)*omega)/tj/omega^2];

PAGE 148

131 P11 = [ 1/8*b*(3*Kt*sin(2*t0*omega+2*omega*tj)+3*K n*cos(2*t0*omega+2*omega*tj)+2*Kn*omeg a^4*tj^4+3*Kt*omega*tj*cos(2*t0*omega +2*omega*tj)+3*Kn*omega*tj*sin(2*t0*ome ga+2*omega*tj)+3*Kt*sin(2*t0*omega)3*Kn*cos(2*t0*omega)+2*Kt*cos(2*t0*omeg a)*omega^3*tj^3+2*Kn*sin(2*t0*omega) *omega^3*tj^3+3*Kt*omega*tj*cos(2*t0*omega)+3*Kn*omega*tj*sin(2*t0*omega))/o mega^4/tj^3, 1/8*b*(-3*Kt*cos(2*t0*omega+2*omega*tj)3*Kn*sin(2*t0*omega+2*omega*tj)+2*Kt*omega^4*tj^43*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+3*Kn*omega*tj*cos(2*t0*omega+2*om ega*tj)+3*Kt*cos(2*t0*omega )+3*Kn*sin(2*t0*omega)2*Kt*sin(2*t0*omega)*omega^3*tj^ 3+2*Kn*cos(2*t0*omega)*omega^3*tj^33*Kt*omega*tj*sin(2*t0*omega)+3*Kn*ome ga*tj*cos(2*t0*omega))/omega^4/tj^3; 1/8*b*(-3*Kt*cos(2*t0*omega+2*omega* tj)-3*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^4*tj^43*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+3*Kn*omega*tj*cos(2*t0*omega+2*om ega*tj)+3*Kt*cos(2*t0*omega )+3*Kn*sin(2*t0*omega)2*Kt*sin(2*t0*omega)*omega^3*tj^ 3+2*Kn*cos(2*t0*omega)*omega^3*tj^33*Kt*omega*tj*sin(2*t0*omega)+3*Kn*omega *tj*cos(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kn*omega^4*tj^4+3*Kt*omega*tj*cos(2* t0*omega+2*omega*tj)+3*Kn*omega*tj*s in(2*t0*omega+2*omega*tj)3*Kt*sin(2*t0*omega+2*omega*tj)+3*Kn*c os(2*t0*omega+2*omega*tj)+2*Kt*cos(2* t0*omega)*omega^3*tj^3+2*Kn*sin(2*t0*omega)*omega^3*tj^3+3*Kt*omega*tj*cos( 2*t0*omega)+3*Kn*omega*tj*sin(2*t0*omega)+3*Kt*sin(2*t0*omega)3*Kn*cos(2*t0*omega))/omega^4/tj^3]; P12 =[ 1/48*b*(6*Kn*omega*tj*sin(2*t0*omega+2*om ega*tj)+6*Kt*omega*tj*cos(2*t0*ome ga+2*omega*tj)+2*Kn*omega^4*tj^49*Kt*sin(2*t0*omega+2*omega*tj)+9*K n*cos(2*t0*omega+2*omega*tj)+12*Kn*ome ga*tj*sin(2*t0*omega)+12*Kt*omega*tj*cos(2*t0*omega)6*Kt*omega^2*tj^2*sin(2*t0*omega)+6*Kn*om ega^2*tj^2*cos(2*t0*omega)+9*Kt*si n(2*t0*omega)-9*Kn*cos(2*t0*omega))/tj^2/omega^4, 1/48*b*(9*Kt*cos(2*t0*omega+2*omega* tj)+9*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^4*tj^4+6*Kt*omega*tj* sin(2*t0*omega+2*omega*tj)6*Kn*omega*tj*cos(2*t0*omega+2*omega* tj)+12*Kt*omega*tj*sin(2*t0*omega)+6* Kn*omega^2*tj^2*sin(2*t0*omega)+ 6*Kt*omega^2*tj^2*cos(2*t0*omega)9*Kt*cos(2*t0*omega)-12*Kn*omega*tj*cos(2*t0*omega)9*Kn*sin(2*t0*omega))/tj^2/omega^4; 1/48*b*(2*Kt*omega^4*tj^4+6*Kt*omega* tj*sin(2*t0*omega+2*omega*tj)+9*Kt*cos( 2*t0*omega+2*omega*tj)6*Kn*omega*tj*cos(2*t0*omega+2*omega *tj)+9*Kn*sin(2*t0*omega+2*omega*tj)+1 2*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga^2*tj^2*sin(2*t0*omega)+6*Kt*omega^ 2*tj^2*cos(2*t0*omega)-9*Kt*cos(2*t0*omega)-12*Kn*omega*tj*cos(2*t0*omega)9*Kn*sin(2*t0*omega))/tj^2/omega^4, 1/48*b*(6*Kt*omega*tj*cos(2*t0*o mega+2*omega*tj)-

PAGE 149

132 6*Kn*omega*tj*sin(2*t0*omega+ 2*omega*tj)+2*Kn*omega^4*tj^49*Kn*cos(2*t0*omega+2*omega*tj)+9*K t*sin(2*t0*omega+2*omega*tj)12*Kt*omega*tj*cos(2*t0*omega)12*Kn*omega*tj*sin(2*t0*omega)+6*K t*omega^2*tj^2*sin(2*t0*omega)6*Kn*omega^2*tj^2*cos(2*t0*om ega)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega ))/tj^2/omega^4]; P13 =[ 1/8*b*(2*Kn*omega^4*tj^43*Kn*cos(2*t0*omega+2*omega*tj)+3*K t*sin(2*t0*omega+2*omega*tj)3*Kt*omega*tj*cos(2*t0*o mega+2*omega*tj)2*Kn*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)3*Kn*omega*tj*sin(2*t0*omega+2*omega*tj)2*Kt*omega^3*tj^3*cos(2*t0*omega+ 2*omega*tj)+3*Kn*cos(2*t0*omega)3*Kt*sin(2*t0*omega)-3*Kt*omega*tj*cos(2*t0*omega)3*Kn*omega*tj*sin(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kt*omega^4*tj^4+2*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)2*Kn*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+3*Kt*cos(2*t0*omega+2*omega*t j)+3*Kn*sin(2*t0*omega+2*omega*tj)+3*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)3*Kn*omega*tj*cos(2*t0*omega+2*ome ga*tj)-3*Kt*cos(2*t0*omega)3*Kn*sin(2*t0*omega)+3*Kt*omega*tj*sin(2*t0*omega)3*Kn*omega*tj*cos(2*t0*omega))/omega^4/tj^3; 1/8*b*(2*Kt*omega^4*tj^4+2*Kn*omega^3*tj^ 3*cos(2*t0*omega+2*omega*tj)+3*Kn *omega*tj*cos(2*t0*omega+2*omega*t j)-3*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)3*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)3*Kt*cos(2*t0*omega+2*omega*tj)+3*Kn*s in(2*t0*omega)+3*Kt*cos(2*t0*omega)3*Kt*omega*tj*sin(2*t0*omega)+3*Kn*omega *tj*cos(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kn*omega^4*tj^4+3*Kn*cos(2*t0*ome ga+2*omega*tj)+2*Kn*omega^3*tj^3 *sin(2*t0*omega+2*omega*tj)3*Kt*sin(2*t0*omega+2*omega*tj)+2*Kt *omega^3*tj^3*cos(2*t0*omega+2*omega*tj )+3*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)+3*Kn*omega*tj*sin(2*t0*omega+2*o mega*tj)3*Kn*cos(2*t0*omega)+3*Kt*sin(2*t0*omeg a)+3*Kt*omega*tj*cos(2*t0*omega)+3* Kn*omega*tj*sin(2*t0*omega))/omega^4/tj^3]; P14 =[ -1/48*b*(6*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^212*Kn*omega*tj*sin(2*t0*omega+2*omega*tj)6*Kt*sin(2*t0*omega+2*ome ga*tj)*omega^2*tj^212*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)+2*Kn*omega^4*tj^49*Kn*cos(2*t0*omega+2*omega*tj)+9*K t*sin(2*t0*omega+2*omega*tj)6*Kn*omega*tj*sin(2*t0*omega)6*Kt*omega*tj*cos(2*t0*omega)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega))/tj^2/omeg a^4, 1/48*b*(-2*Kt*omega^4*tj^412*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+12*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+6*Kt*cos(2*t0*omega+2*omega* tj)*omega^2*tj^2+6*Kn*sin(2*t0*omega+2 *omega*tj)*omega^2*tj^2-9*K t*cos(2*t0*omega+2*omega*tj)9*Kn*sin(2*t0*omega+2*omega*tj)-

PAGE 150

133 6*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga*tj*cos(2*t0*omega)+9*Kt*cos(2*t0*o mega)+9*Kn*sin(2*t0*omega))/tj^2/omega^4; 1/48*b*(2*Kt*omega^4*tj^412*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+12*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+6*Kt*cos(2*t0*omega+2*omega* tj)*omega^2*tj^2+6*Kn*sin(2*t0*omega+2 *omega*tj)*omega^2*tj^2-9*K t*cos(2*t0*omega+2*omega*tj)9*Kn*sin(2*t0*omega+2*omega*tj)6*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga*tj*cos(2*t0*omega)+9*Kt*cos(2*t0*o mega)+9*Kn*sin(2*t0*omega))/tj^2/omega^4, 1/48*b*(12*Kn*omega*tj*sin(2*t0*omega+2*omega* tj)-9*Kn*cos(2*t0*omega+2*omega*tj)2*Kn*omega^4*tj^4-6*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^212*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)+6*Kn*cos(2*t0*omega+2*omega*tj)*o mega^2*tj^2+9*Kt*sin(2*t0*omega+2*omega*tj)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega)-6*Kn*omega*tj*sin(2*t0*omega)6*Kt*omega*tj*cos(2*t0*omega))/tj^2/omega^4]; P21 =[ -1/80*b*(-15*Kt*cos(2*t0*o mega+2*omega*tj)*omega^2*tj^215*Kn*sin(2*t0*omega+2*omega*tj)*omega^2*tj^2+60*Kt*cos(2*t0*omega+2*omega *tj)+60*Kn*sin(2*t0*omega+ 2*omega*tj)+4*Kn*omega^5*tj^5+60*Kt*omega*tj*sin(2 *t0*omega)10*Kn*omega^3*tj^3*cos(2*t0*omega)+15*Kn* omega^2*tj^2*sin(2*t0*omega)+15*K t*omega^2*tj^2*cos(2*t0*omega)+1 0*Kt*omega^3*tj^3*sin(2*t0*omega)60*Kn*omega*tj*cos(2*t0*omega)+10*Kn*sin(2*t0*omega)*omega^4*tj^4+10*Kt*co s(2*t0*omega)*omega^4*tj^4-60*Kn*sin(2*t0*omega)60*Kt*cos(2*t0*omega)+60*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)60*Kn*omega*tj*cos(2*t0*omega+2*ome ga*tj))/omega^5/tj^4, -1/80*b*(60*Kn*cos(2*t0*omega)+60*Kt*sin(2*t0*omega)15*Kn*cos(2*t0*omega+2*omega*tj)*omega^ 2*tj^2+15*Kt*sin(2*t0*omega+2*omega *tj)*omega^2*tj^2+60*Kn*omega* tj*sin(2*t0*omega+2*omega*tj)60*Kt*sin(2*t0*omega+2*omega*tj)+60*Kt *omega*tj*cos(2*t0*omega+2*omega*tj)+ 60*Kn*cos(2*t0*omega+2*omega*tj)+10*Kt*omega^3*tj^3*cos(2*t0*omega)15*Kt*omega^2*tj^2*sin(2*t0*omega)10*Kt*sin(2*t0*omega)*omega^4*tj^4+10*K n*cos(2*t0*omega)*omega^4*tj^4+15*K n*omega^2*tj^2*cos(2*t0*omega)+60*Kt*om ega*tj*cos(2*t0*omega)+10*Kn*omega^ 3*tj^3*sin(2*t0*omega)+60*Kn*omega*tj*si n(2*t0*omega)+4*Kt*omega^5*tj^5)/ome ga^5/tj^4; 1/80*b*(60*Kn*cos(2*t0*omega)60*Kt*sin(2*t0*omega)+15*Kn*cos(2*t 0*omega+2*omega*tj)*omega^2*tj^215*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^260*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)-60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega)+15*Kt* omega^2*tj^2*sin(2*t0*omega)+10*Kt *sin(2*t0*omega)*omega^4*tj^4-10*Kn*cos(2*t0*omega)*omega^4*tj^415*Kn*omega^2*tj^2*cos(2*t0*omega )-60*Kt*omega*tj*cos(2*t0*omega)10*Kn*omega^3*tj^3*sin(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)+4*Kt*omega^5*tj^5)/omega^5/tj^4, -1/80*b*(60*Kt*omega*tj*sin(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-

PAGE 151

134 10*Kn*sin(2*t0*omega)*omega^4*tj^4-10*K t*cos(2*t0*omega)*omega^4*tj^415*Kt*omega^2*tj^2*cos(2*t0*omega)10*Kt*omega^3*tj^3*sin(2*t0*omega)+10*Kn*omega^3*tj^3*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega)+60*K n*sin(2*t0*omega)+60*Kt*cos(2*t0*ome ga)60*Kt*cos(2*t0*omega+2*omega*tj)+4*Kn*o mega^5*tj^5+60*Kn*omega*tj*cos(2*t0 *omega+2*omega*tj)+15*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+15*Kn*sin (2*t0*omega+2*omega*tj)*omega^2*tj^2-60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega*tj))/omega^5/tj^4]; P22 = [ 1/480*b*(-2*Kn*omega^5*tj^5180*Kt*cos(2*t0*omega+2*omega*tj)180*Kn*sin(2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+135*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+30*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+30*Kn*sin(2*t0*ome ga+2*omega*tj)*omega^2*tj^2225*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+225*Kn*omega*tj*cos(2*t0*omega)120*Kt*omega^2*tj^2*cos(2*t0*omega)+ 30*Kt*omega^3*tj^3*sin(2*t0*omega)30*Kn*omega^3*tj^3*cos(2*t0*omega)120*Kn*omega^2*tj^2*sin(2*t0*om ega))/tj^3/omega^5, 1/480*b*(135*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+135*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*Kn*co s(2*t0*omega+2*omega*tj)+2*Kt*o mega^5*tj^5+30*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*cos(2*t0*omega)+ 120*Kn*omega^2*tj^2*cos(2*t0*omega)120*Kt*omega^2*tj^2*sin(2*t0*omega)30*Kn*omega^3*tj^3*sin(2*t0*omega)+ 225*Kt*omega*tj*cos(2*t0*omega)+225*Kn* omega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5; 1/480*b*(135*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+135*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*K n*cos(2*t0*omega+2*omega*tj)2*Kt*omega^5*tj^5+30*Kt*sin(2*t0*o mega+2*omega*tj)*omega^2*tj^230*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*cos(2*t0*omega)+ 120*Kn*omega^2*tj^2*cos(2*t0*omega)120*Kt*omega^2*tj^2*sin(2*t0*omega)30*Kn*omega^3*tj^3*sin(2*t0*omega)+ 225*Kt*omega*tj*cos(2*t0*omega)+225*Kn* omega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5, -1/480*b*(2*Kn*omega^5*tj^5180*Kt*cos(2*t0*omega+2*omega*tj)-180*K n*sin(2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+135*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+30*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+30*Kn*sin(2*t0*ome ga+2*omega*tj)*omega^2*tj^2225*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+225*Kn*omega*tj*cos(2*t0*omega)-

PAGE 152

135 120*Kt*omega^2*tj^2*cos(2*t0*omega)+ 30*Kt*omega^3*tj^3*sin(2*t0*omega)30*Kn*omega^3*tj^3*cos(2*t0*omega)120*Kn*omega^2*tj^2*sin(2*t0*om ega))/tj^3/omega^5]; P23 = [ -1/80*b*(-4*Kn*omega^5*tj^5-60*Kt*cos(2*t0*omega+2*omega*tj)60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+60*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+15*Kt*cos(2*t0*omega+2*omega *tj)*omega^2*tj^2+15*Kn*sin(2*t0*omega +2*omega*tj)*omega^2*tj^210*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*sin(2*t0*o mega+2*omega*tj)+10*Kt*omega^4*tj^ 4*cos(2*t0*omega+2*omega*tj)+10*Kn*omeg a^3*tj^3*cos(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega)+60*Kn*s in(2*t0*omega)+60*Kt*cos(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-15*K t*omega^2*tj^2*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega))/omega^5/tj^4, -1/80*b*(60*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)60*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^4*tj^4*sin(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*cos(2*t0*o mega+2*omega*tj)-10*Kn*omega^ 3*tj^3*sin(2*t0*omega+2*omega*tj)4*Kt*omega^5*tj^515*Kt*sin(2*t0*omega+2*omega*tj)*omega ^2*tj^2+15*Kn*cos(2*t0*omega+2*omega *tj)*omega^2*tj^215*Kn*omega^2*tj^2*cos(2*t0*omega)+15*K t*omega^2*tj^2*sin(2*t0*omega)60*Kt*omega*tj*cos(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)60*Kt*sin(2*t0*omega)+60*Kn*cos(2*t0*omega))/omega^5/tj^4; -1/80*b*(-60*Kt*omega*tj*co s(2*t0*omega+2*omega*tj)60*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^4*tj^4*sin(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*cos(2*t0*o mega+2*omega*tj)10*Kn*omega^3*tj^3*sin(2*t0*omega+ 2*omega*tj)+4*Kt*omega^5*tj^515*Kt*sin(2*t0*omega+2*omega*tj)*omega ^2*tj^2+15*Kn*cos(2*t0*omega+2*omega *tj)*omega^2*tj^215*Kn*omega^2*tj^2*cos(2*t0*omega)+15*K t*omega^2*tj^2*sin(2*t0*omega)60*Kt*omega*tj*cos(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)60*Kt*sin(2*t0*omega)+60*Kn*cos(2*t0*omega))/omega^5/tj^4, 1/80*b*(4*Kn*omega^5*tj^5-60*Kt*cos(2*t0*omega+2*omega*tj)60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+60*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+15*Kt*cos(2*t0*omega+2*omega *tj)*omega^2*tj^2+15*Kn*sin(2*t0*omega +2*omega*tj)*omega^2*tj^210*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*sin(2*t0*o mega+2*omega*tj)+10*Kt*omega^4*tj^ 4*cos(2*t0*omega+2*omega*tj)+10*Kn*omeg a^3*tj^3*cos(2*t0*omega+2*omega*tj)-

PAGE 153

136 60*Kt*omega*tj*sin(2*t0*omega)+60*Kn*s in(2*t0*omega)+60*Kt*cos(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-15*K t*omega^2*tj^2*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega))/omega^5/tj^4]; P24 = [ 1/480*b*(2*Kn*omega^5*tj^5+180*Kt*cos( 2*t0*omega+2*omega*tj)+180*Kn*sin(2*t 0*omega+2*omega*tj)+225*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)225*Kn*omega*tj*cos(2*t0*omega+2*omega*tj)120*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2120*Kn*sin(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+30*Kn*omega^3*tj^3*cos(2*t0*o mega+2*omega*tj)+135*Kt*omega*tj*si n(2*t0*omega)-180*Kn*sin(2*t0*omega)180*Kt*cos(2*t0*omega)135*Kn*omega*tj*cos(2*t0*omega)+30*Kt *omega^2*tj^2*cos(2*t0*omega)+30*Kn*o mega^2*tj^2*sin(2*t0*omega))/tj^3/omega^5, 1/480*b*(225*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+225*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*K n*cos(2*t0*omega+2*omega*tj)30*Kt*omega^3*tj^3*cos(2*t0*omega+2*omega*tj)30*Kn*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+2*Kt*omega^5*tj^5+120*Kt*sin( 2*t0*omega+2*omega*tj)*omega^2*tj^2120*Kn*cos(2*t0*omega+2*omega*tj)*omega^2* tj^2+30*Kn*omega^2*tj^2*cos(2*t0* omega)30*Kt*omega^2*tj^2*sin(2*t0*omega)+ 135*Kt*omega*tj*cos(2*t0*omega)+135*Kn*o mega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5; 1/480*b*(-225*Kt*omega*tj*cos( 2*t0*omega+2*omega*tj)225*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+180*Kt*sin(2*t0*omega+2*omega*tj )180*Kn*cos(2*t0*omega+2*omega*tj)+30*Kt*omega^3*tj^3*cos(2*t0*omega+2*ome ga*tj)+30*Kn*omega^3*tj^3*sin(2*t0*om ega+2*omega*tj)+2*Kt*omega^5*tj^5120*Kt*sin(2*t0*omega+2*omega*tj)*omega^ 2*tj^2+120*Kn*cos(2*t0*omega+2*ome ga*tj)*omega^2*tj^230*Kn*omega^2*tj^2*cos(2*t0*omega)+30*K t*omega^2*tj^2*sin(2*t0*omega)135*Kt*omega*tj*cos(2*t0*omega)135*Kn*omega*tj*sin(2*t0*omega)180*Kt*sin(2*t0*omega)+180*Kn*cos(2*t0*omega))/tj^3/omega^5, 1/480*b*(2*Kn*omega^5*tj^5-180*Kt*c os(2*t0*omega+2*omega*tj)180*Kn*sin(2*t0*omega+2*omega*tj)225*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+225*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+120*Kt*cos(2*t0*omega+2*omeg a*tj)*omega^2*tj^2+120*Kn*sin(2*t0*o mega+2*omega*tj)*omega^2*tj^2+30*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj) -30*Kn*omega^3*tj^3*cos( 2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+135*Kn*omega*tj*cos(2*t0*omega)-30*Kt*omega^2*tj^2*cos(2*t0*omega)30*Kn*omega^2*tj^2*sin(2*t0*omega))/tj^3/omega^5];

PAGE 154

137 P11 = [P11(1,1)*ones(lmx,1) P11(1,2) *ones(lmx,1); P11(2,1)*ones(lmx,1) P11(2,2)*ones(lmx,1)]*V; P12 = [P12(1,1)*ones(lmx,1) P12(1,2) *ones(lmx,1); P12(2,1)*ones(lmx,1) P12(2,2)*ones(lmx,1)]*V; P13 = [P13(1,1)*ones(lmx,1) P13(1,2) *ones(lmx,1); P13(2,1)*ones(lmx,1) P13(2,2)*ones(lmx,1)]*V; P14 = [P14(1,1)*ones(lmx,1) P14(1,2) *ones(lmx,1); P14(2,1)*ones(lmx,1) P14(2,2)*ones(lmx,1)]*V; P21 = [P21(1,1)*ones(lmx,1) P21(1,2) *ones(lmx,1); P21(2,1)*ones(lmx,1) P21(2,2)*ones(lmx,1)]*V; P22 = [P22(1,1)*ones(lmx,1) P22(1,2) *ones(lmx,1); P22(2,1)*ones(lmx,1) P22(2,2)*ones(lmx,1)]*V; P23 = [P23(1,1)*ones(lmx,1) P23(1,2) *ones(lmx,1); P23(2,1)*ones(lmx,1) P23(2,2)*ones(lmx,1)]*V; P24 = [P24(1,1)*ones(lmx,1) P24(1,2) *ones(lmx,1); P24(2,1)*ones(lmx,1) P24(2,2)*ones(lmx,1)]*V; N11 = -C+1/2*K*tj+P11; N12 = -M+1/12*K*tj^2+P12; N13 = C+1/2*K*tj+P13; N14 = M-1/12*K*tj^2+P14; N21 = M/tj-1/10*K*tj+P21; N22 = 1/2*M-1/12*C*tj-1/120*K*tj^2+P22; N23 = -M/tj+1/10*K*tj+P23; N24 = 1/2*M+1/12*C*tj-1/120*K*tj^2+P24; N1 = [N11 N12; N21 N22]; N2 = [N13 N14; N23 N24]; P1 = [P11 P12; P21 P22]; P2 = [P13 P14; P23 P24]; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%% % BUILD GLOBAL MATRICES %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%% A(1:2*DOF,1:2*DOF) = eye(2*DOF); A(2*DOF*e+1:2*DOF*e+2*DOF,2* DOF*(e-1)+1:2*DOF*(e-1)+2*DOF) = N1; A(2*DOF*e+1:2*DOF*e+2*DOF ,2*DOF*(e-1)+2*DOF+1:2*DOF*(e1)+2*DOF+2*DOF) = N2; B(2*DOF*e+1:2*DOF*e+2*DOF,2* DOF*(e-1)+1:2*DOF*(e-1)+2*DOF) = P1; B(2*DOF*e+1:2*DOF*e+2*DOF ,2*DOF*(e-1)+2*DOF+1:2*DOF*(e1)+2*DOF+2*DOF) = P2; B(1:2*(DOF),E*2*(DOF )+1:(E+1)*2*(DOF)) = PHI; Cvec(1:2*DOF,1) = zeros(2*DOF,1); Cvec(2*DOF*e+1:2*DOF*e+DOF,1) = C1; Cvec(2*DOF*e+DOF+1:2*DOF*e+2*DOF,1) = C2; end; % end # of elements loop Q = A\B;

PAGE 155

138 [vec,lam] = eig(Q); CM = max(abs(diag(lam))); D = A\Cvec; % Extract SLE coefficients if CM<1 SLE_vec = inv((eye(size(Q))-Q))*D; SLE = abs(sum(SLE_vec(locat))); else SLE = 100; end N1 = zeros(2*DOF,2*DOF); N2 = N1; P1 = N1; P2 = P1; return; Uncertainty Analysis Code Stability Uncertainty, Sensitivity Method % M. Kurdi (1/26/2005) % Function to find uncertainty in ax ial depth to change in cutting % coefficients, dynamic parameters and cutting process variables % Input: % b: depth of cut (m) % rpm: spindle speed % rstep: radial step (inches) % Kt % Kn % Kte % Kre % DELTA_Kt finite change in Kt % DELTA_b finite change in b % system_ID: Modal parameters % The derivative of Max eigen value is found for a miniscule perturbation % in input parameters, then its effect on the change of axial depth is % found. clear all; close all; clc;tic; % function uncer % percentage of uncertainty in cutting coefficients, dynamic parameters % and process parameters % tic; percent_Kcut = 0.05; % cutt ing coefficents uncertainty percent_Dyn = 0.005; % modal parameters uncertainty percent_rstep = 0.0001; % ra dial step uncertainty percent_rpm = 0.005; % spindle speed uncertainty % nominal values of process paramete rs and their calculated uncertainty rstep = 0.2;

PAGE 156

139 rpm_vec = 10000:200:30000; DELTA_rstep = percent_rstep*rstep; % cutting coefficient uncertainty Kt = 6e8; DELTA_Kt = percent_Kcut*Kt; Kn = .3*Kt; DELTA_Kn = percent_Kcut*Kn; Kte=0; DELTA_Kte = percent_Kcut*Kte; Kne=0; DELTA_Kne = percent_Kcut*Kne; % nominal values of dynamic parameters and their calcul ated uncertainty Kx = 4.4528e+006; Mx = 0.4362; Cx = 83; % Y direction parameters Ky = 3.5542e+006; My = 0.4362; Cy = 89.9; DELTA_Mx = Mx*percent_Dyn; DELTA_My = My*percent_Dyn; DELTA_Kx = Kx*percent_Dyn; DELTA_Ky = Ky*percent_Dyn; DELTA_Cx = Cx*percent_Dyn; DELTA_Cy = Cy*percent_Dyn; % to calculate the numerical deriva tive with respect to each input % variable set the miniscule change in each input % set miniscule change in input para meters to estimate the derivative step_percent = 0.002; dKt = step_percent*Kt; % N/m2 dKn = step_percent*Kn; % N/m2 dKte = step_percent*30; % N/m dKne = step_percent*30; % N/m drstep = step_percen t*rstep; % inch dKx = step_percent*Kx; % N/m dKy = step_percent*Ky; % N/m dCx = step_percent*Cx; % dCy = step_percent*Cy; dMx = step_percent*Mx; % Kg dMy = step_percent*My; % Kg h = waitbar(0,'Please wait...'); % computation here % for i=1:length(rpm_vec) waitbar(i/length(rpm_vec),h); rpm = rpm_vec(i); drpm = step_percent*rpm; % rpm

PAGE 157

140 DELTA_rpm = percent_rpm rpm; % Find depth of cut correspondi ng to stability bounda ry using nominal % settings of input parameters [b(i)] = bisection(rpm,rstep,Kt,K n,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % depth at boundary % Find numerical derivative of ma ximum eigenvalue with respect to input % parameters % perturb cutting coefficient Kt by dKt [b1] = bisection(rpm,rstep,Kt-dKt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); [b2] = bisection(rpm,rstep,Kt +dKt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen matr ix w.r.t cutting coefficient Kt d_b_Kt(i) = (b2-b1)/dKt/2; b1 =[]; b2 =[]; % perturb cutting coefficient Kn by dKn [b1] = bisection(rpm,rstep,Kt ,Kn-dKn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); [b2] = bisection(rpm,rstep,Kt ,Kn+dKn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % % % derivative of eigen matrix w.r.t cutting coefficient Kt d_b_Kn(i) = (b2-b1)/dKn/2; b1 =[]; b2 =[]; % perturb cutting coefficient Kte by dKte [b1] = bisection(rpm,rstep,Kt ,Kn,Kte-dKte,Kne,Mx,Kx,Cx,My,Ky,Cy); [b2] = bisection(rpm,rstep,Kt ,Kn,Kte+dKte,Kne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen matr ix w.r.t cutting coefficient Kt d_b_Kte(i) = (b2-b1)/dKte/2; b1 =[]; b2 =[]; % perturb cutting coefficient Kne by dKne [b1] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne-dKne,Mx,Kx,Cx,My,Ky,Cy); [b2] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne+dKne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen matrix w.r.t cutting coefficient Kne d_b_Kne(i) = (b2-b1)/dKne/2; b1 =[]; b2 =[]; % % perturb depth of cut rstep by drstep [b1] = bisection(rpm,rstep-drst ep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); [b2] = bisection(rpm,rstep+drst ep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen value w.r.t rstep of cut d_b_rstep(i) = (b2-b1)/drstep/2; b1 =[]; b2 =[]; % perturb spindle speed by drpm [b1] = bisection(rpm-drpm,rst ep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); [b2] = bisection(rpm+drpm,rst ep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen value w.r.t rpm d_b_rpm(i) = (b2-b1)/drpm/2; b1 =[]; b2 =[];

PAGE 158

141 % perturb Kx by dKx [b1] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx,Kx-dKx,Cx,My,Ky,Cy); [b2] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx,Kx+dKx,Cx,My,Ky,Cy); % derivative of eigen value w.r.t Kx d_b_Kx(i) = (b2-b1)/dKx/2; b1 =[]; b2 =[]; % perturb Ky by dKy [b1] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky-dKy,Cy); [b2] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky+dKy,Cy); % % derivative of eigen value w.r.t Ky d_b_Ky(i) = (b2-b1)/dKy/2; b1 =[]; b2 =[]; % perturb Cx by dCx [b1] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx,Kx,Cx-dCx,My,Ky,Cy); [b2] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx,Kx,Cx+dCx,My,Ky,Cy); % derivative of eigen value w.r.t Cx d_b_Cx(i) = (b2-b1)/dCx/2; b1 =[]; b2 =[]; % perturb Cy by dCy [b1] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy-dCy); [b2] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy+dCy); % derivative of eigen value w.r.t Cy d_b_Cy(i) = (b2-b1)/dCy/2; b1 =[]; b2 =[]; % perturb Mx by dMx [b1] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx-dMx,Kx,Cx,My,Ky,Cy); [b2] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx+dMx,Kx,Cx,My,Ky,Cy); % derivative of eigen value w.r.t Mx d_b_Mx(i) = (b2-b1)/dMx/2; b1 =[]; b2 =[]; % perturb My by dMy [b1] = bisection(rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My-dMy,Ky,Cy); [b2] = bisection(rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My+dMy,Ky,Cy); % % % derivative of eigen value w.r.t My d_b_My(i) = (b2-b1)/dMy/2; b1 =[]; b2 =[]; % DELTA_b(i) =( (DELTA_Kt d_b_Kt(i) )^2 + (DELTA_Kn d_b_Kn(i))^2 + ... (DELTA_Kne d_b_Kne (i))^2 + (DELTA_Kte d_b_Kte(i))^2 +... (DELTA_Kx d_b_K x(i))^2 + (DELTA_Mx d_b_Mx(i))^2 + ... (DELTA_Cx d_b_C x(i))^2 + (DELTA_Ky d_b_Ky(i))^2 +... (DELTA_My d_b_M y(i))^2 + (DELTA_Cy d_b_Cy(i))^2+... (DELTA_rstep d_b_rstep(i))^2 + (DELTA_rpm* d_b_rpm(i))^2)^0.5; end

PAGE 159

142 close(h); % d_b_rpm % % Find the uncertainty in depth of cut for a corresponding uncertainty in % % input paramters % % save uncer_march_10_smallconverror figure plot(rpm_vec*1/60/(sqrt( Ky/My)/2/pi),b*1000,'-g') % % set(gca,'fontname','times','fontsize',16); % xlabel('\Omega (x10^3 rpm)','fontsize',14) % ylabel('b (mm)','fontsize',14) % legend('Stability boundary nominal input','Stabili ty boundary uncertainty'); % axis([5 20 0 15]) Stability Uncertainty, Monte Carlo and Latin Hyper-Cube % % M. Kurdi (6/17/05) % 4 OAL TOOL % Program to complete LatinHyper and M onte simulation for TFEA stability lobes % clear all; close all; % function LatinHyper % tic; % chip_load=0.1e-3;% chip load % nteeth = 4; % Diam =0.5; % E=15; % N = 1000; % number of iterations % % % % AL 6061 % % percent_Kt = 7.13/100; % cutting coefficents uncertainty % % percent_Kn = 8.09/100; % % percent_Kte = 30.3/100; % % percent_Kne = 23.9/100; % % % 5 OAL TOOL UNCERTAINTIES % % percent_KX = 0.054; % modal parameters uncertainty % % percent_CX = .286; % % percent_MX =.045; % % % % percent_KY = 0.054; % modal parameters uncertainty % % percent_CY = .173; % % percent_MY =.055; % % 4 OAL TOOL UNCERTAINTIES due to thermal effect only

PAGE 160

143 % percent_MX = 0.074; % percent_CX = 0.042; % percent_KX = 0.073 ; % percent_MY = 0.2; % percent_CY = 0.107; % percent_KY = 0.2 ; % percent_rstep = 0.0005; % radial step uncertainty % percent_rpm = 0.005; % spindle speed uncertainty % % speed_min = str2num(input('Min_speed = ','s')); % % speed_max = str2num(input('Max_speed = ','s')); % % speed = speed_min:200:speed_max; % speed = 10000:100:20000; % h = waitbar(0,'Please wait...'); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%5 % % Cutting Coefficients %%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%% % % AL 6061 % % mean_Kt =7.06E+08; % N/m2 % % mean_Kn = 2.50E+08; % % mean_Kte = 1.29E+04; % N/m; % % mean_Kne = 6.57E+03; % % AL 7475 % mean_Kt = 690480868.527357; % mean_Kte = 12022.3004909002; % mean_Kn = 142535991.092323; % mean_Kne =11281.4601645315; % std_Kn=4009843*4.45; % N % std_Kne=310.909*4.45; % std_Kte=200.731*4.45; % std_Kt=2588583*4.45; % % std_Kt = percent_Kt*mean_Kt; % % std_Kn = percent_Kn*mean_Kn; % % std_Kte = percent_Kte*mean_Kte; % % std_Kne = percent_Kne*mean_Kne; % % Kne Kn Kte Kt % % AL 6061 % % SIGMA_K = [1.480E+07 -1.778E+11 -8.216E+06 9.871E+10; % % -1.778E+11 2.458E+15 9.871E+10 -1.365E+15; % % -8.216E+06 9.871E+10 9.163E+07 -1.101E+12; % % 9.871E+10 -1.365E+15 -1.101E+12 1.522E+16 % % ]; % % % AL 7475

PAGE 161

144 % SIGMA_K = [ 42157610.7365206 -506483170409.775 3598978.12573119 43238262783.6325; % -506483170409.775 7.00379474676691e+015 43238262783.6325 -597911116174549; % -3598978.12573128 43238262783.6335 17574719.1179838 211143357093.21; % 43238262783.6335 -597911116174562 -211143357093.21 2.91975098408051e+015]; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%% % % Modal Parameters % % X %%%%%%%%%%%%%%%% % % 5 OAL TOOL % % mean_Kx = 2.64E+06; % % mean_Mx = 0.049; % % mean_Cx = 8.972; % % dynamic parameters for 4 OAL tool % mean_Mx = 0.027 ; % mean_Cx= 23.309; % mean_Kx= 4359275.000 ; % std_Cx = percent_CX*mean_Cx; % std_Kx = percent_KX*mean_Kx; % std_Mx = percent_MX*mean_Mx; % % Mx Cx Kx My Cy Ky 5 OAL % % SIGMA = [3.85E-06 4.03E-03 2.48E+02 2.40E-06 -3.18E-03 1.31E+02; % % 4.03E-03 5.27E+00 2.69E+05 4.08E-03 -3.32E+00 2.19E+05; % % 2.48E+02 2.69E+05 1.61E+10 1.67E+02 -2.07E+05 9.15E+09; % % 2.40E-06 4.08E-03 1.67E+02 4.23E-06 -1.88E-03 2.24E+02; % % -3.18E-03 -3.32E+00 -2.07E+05 -1.88E-03 2.71E+00 1.04E+05; % % 1.31E+02 2.19E+05 9.15E+09 2.24E+02 -1.04E+05 1.19E+10 % % ]; % % Mx Cx Kx My Cy Ky 4 OAL % SIGMA = [4.04188E-06 0.000450265 631.110625 7.25563E-06 -0.000584252 878.998125; % 0.000450265 0.953490935 38828.325 0.00283473 -2.467636648 567721.5525; % 631.110625 38828.325 1.00042E+11 1068.011875 -51720.94813 1.21332E+11; % 7.25563E-06 0.00283473 1068.011875 1.76519E-05 -0.007067488 2638.261875;

PAGE 162

145 % -0.000584252 -2.467636648 -51720.94813 -0.007067488 11.34481701 1426512.396; % 878.998125 567721.5525 1.21332E+11 2638.261875 -1426512.396 4.36003E+11]; % % % Y %%%%%%% %%%%%%%%%%%% %%%%%%% % % 5 OAL TOOL % % mean_Ky = 2.26e+006; % % mean_Cy = 10.651; % % mean_My = 0.042; % % Y direction parameters 4 OAL TOOL % mean_Ky = 3301775.000; % mean_My = 0.021; % mean_Cy = 31.432; % std_My = percent_MY*mean_My; % std_Ky = percent_KY*mean_Ky; % std_Cy = percent_CY*mean_Cy; % % %%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% % % % Radial step inches % % %%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%% % mean_rstep = 0.25*.5; % std_rstep = percent_rstep*mean_rstep; % % %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% % randn('state',0) % Mode = lhsnorm([mean_Mx mean_Cx mean_Kx mean_My mean_Cy mean_Ky],SIGMA,N); % % Mode(:,1) is Mx random vector % % Mode(:,2) is Cx random vector % % Mode(:,3) is Kx random vector % % Mode(:,4) is My random vector % % Mode(:,5) is Cy random vector % % Mode(:,6) is Ky random vector % Cut_Coeff = lhsnorm([mean_Kne mean _Kn mean_Kte mea n_Kt],SIGMA_K,N); % % Cut_Coeff(:,1) Kne % % Cut_Coeff(:,2) Kn % % Cut_Coeff(:,3) Kte % % Cut_Coeff(:,4) Kt % sample = randn(N, 2); % for j=1:length(speed) % waitbar(j/length(speed),h) % for i=1:N % % Unless otherwise specifi ed, all dimensions in m % % Define input parameters %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% % % Cutting coefficients % Kt = Cut_Coeff(i,4);

PAGE 163

146 % Kn = Cut_Coeff(i,2); % Kte = Cut_Coeff(i,3); % Kne = Cut_Coeff(i,1); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% % % milling parameters % % Spindle speed % mean_rpm = speed(j); % std_rpm = percent_rpm*mean_rpm; % rpm = mean_rpm + std_rpm*sample(i,1); % % rstep % rstep = mean_rstep + std_rstep*sample(i,2); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% % % Dynamic parameters % % X direction is feed direction % Kx =Mode(i,3); % Mx = Mode(i,1); % Cx = Mode(i,2); % % Y direction parameters % Ky = Mode(i,6); % My = Mode(i,4); % Cy = Mode(i,5); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% % % Calculate axial depth corr esponding to input paramters % % that is on the stability boundaries % b(i,j) = bisection(rpm,rstep,Kt,Kn,Kte,Kne,Mx,Cx, Kx,My,Cy,Ky,chip_load,nteeth,Diam,E); % end % i end monte loop for one spindle speed % end % j end spindle speed range % std_dev = std(b) % b_mean = mean(b) % time=toc; % save Latin_AL7475 std_dev speed b_mean b time % close(h); % hold on; % h1 = plot(speed/1000,(mean (b)-2*std(b))*1000,'-r') % hold on; % h2 = plot(speed/1000,mean(b)*1000,'g-'); % hold on; % h3 = plot(speed/1000,(2*std (b)+mean(b)) *1000,'-r'); % legend([h1,h2,h3],'lower boundary','mean','upper boundary') figure plot(speed/1000,std(b)*2*1000) %

PAGE 164

147 % Function to stability lobe using bisection method. % Input: % rpm ; % rstep: radial immersion (inches) % Output: % b depth of cut (m) function [b] = bisection(rpm,rstep,Kt,Kn,Kte,Kne, Mx,Cx,Kx,My,Cy,Ky,h,nteeth,Diam,E) % E % h % feed per tooth % nteeth % number of teeth % Diam % inches TRAVang = acos(1-rs tep/(Diam/2)); % angular travel during cutting LAGang = 2*pi/nteeth; % separation angle for teeth rho = acos(1-rstep/(Diam/2))/(2*pi); % fraction of time in cut IMMERSION = rstep/Diam; opt = 'down'; if TRAVang>LAGang % MU LTIPLE TEETH ARE IN CONTACT teethNcontact = floor(TRAVang/LAGang) +1; else % SINGLE TOOTH IN CONTACT teethNcontact = 1; end %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% % LOAD SYSTEM IDENTIFICATION MATRICES %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%% M =[Mx zeros(size(Mx)); zeros(size(Mx)) My]; C =[Cx zeros(size(Mx)); zeros(size(Mx)) Cy]; K =[Kx zeros(size(Mx)); zeros(size(Mx)) Ky]; lmx = length(Mx(1,:)); lmy = length(My(1,:)); DOF = lmx+lmy; V = [ones(1,lmx) zeros(1 ,lmy); zeros(1,lmx) ones(1,lmy)]; A = zeros((E+1)*2*DOF,(E+1)*2*DOF); B = A; b_r = [1e-10 100e-2]; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % BEGIN LOOP CALCULATIONS OVER RPM vs DOC FIELD %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% while (abs(b_r(1) b_r(2)) / b_r(1) > 1e-6) warning off MATLAB:singularMatrix; warning off MATLAB:nearlySingularMatrix;

PAGE 165

148 b = (b_r(1) + b_r(2)) / 2; speed = rpm; omega = speed/60*(2*pi); % radians per second T = (2*pi)/omega/nteeth ; % tooth pass period TC = rho*T*nteeth; % time a single tooth spends in the cut tf = T-TC; % time for free vibs tj = TC/E; % time for each element %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % SET CUTTER RO TATION ANGLE FOR UP/DOWN-MILLING %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% switch opt case 'up' t0mat = [0 tj*(1:(E-1))]; % upmilling locat = 2*DOF+lmx+1:3*DOF; case 'down' tex = pi/omega; tent=tex-TC; % downmilling t0mat = [tent tent+tj*(1:(E-1))]; % downmilling locat = (E+1)*2*DOF -DOF-lmy+1:(E+1)*2*DOF-DOF; end %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% % STATE TRANSITION MATRIX %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% G1 = [zeros(size(M)) M; eye(size(M)) zeros(size(M))]; G2 = [K C; zeros (size(M)) -eye(size(M))]; G = -G1\G2; PHI = expm(G*tf); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% % N & P are used to create A & B which then become Q in..... a_n = Q a_n-1 + D %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% for e=1:E, t0 = t0mat(e); C1 = V'*[ -1/4*b*(-h*Kt*cos(2*t0*o mega+2*omega*tj)+2*h*Kn*omega*tjh*Kn*sin(2*t0*omega+2*omega*tj)+4*Kte*sin(t0*omega+omega*tj)4*Kne*cos(t0*omega+omega*tj)+h*Kt*co s(2*t0*omega)+h*Kn*sin(2*t0*omega)4*Kte*sin(t0*omega)+4*Kne*cos(t0*omega))/omega; 1/4*b*(2*h*Kt*omega*tjh*Kt*sin(2*t0*omega+2*omega*tj)+h*Kn*cos(2*t0*omega+2*omega*tj)-

PAGE 166

149 4*Kte*sin(t0*omega+omega*tj)+4*Kne*cos (t0*omega+omega*tj)+h*Kt*sin(2*t0*ome ga)-h*Kn*cos(2*t0*omega)+4*Kte*sin(t0*om ega)-4*Kne*cos(t0*omega))/omega]; C2 = V'*[ 1/8*b*(h*Kt*sin(2*t0*omega+2*omega*tj)+h*Kt* cos(2*t0*omega+2*omega*tj)*omega*tj+h* Kn*cos(2*t0*omega+2*omega*tj)+h*Kn*s in(2*t0*omega+2*omega*tj)*omega*tj+8*K te*sin(t0*omega+omega*tj)-4*Kte *cos(t0*omega+omega*tj)*omega*tj8*Kne*cos(t0*omega+omega*tj)4*Kne*sin(t0*omega+omega*tj)*omega*tj+h*Kt*sin(2*t0*omega)h*Kn*cos(2*t0*omega)8*Kte*sin(t0*omega)+8*Kne*cos(t0*omega )+h*Kt*tj*cos(2*t0*omega)*omega+h*Kn* tj*sin(2*t0*omega)*omega-4*K te*tj*cos(t0*omega)*omega4*Kne*tj*sin(t0*omega)*omega)/tj/omega^2; 1/8*b*(h*Kt*cos(2*t0*omega+2*omega*tj)+ h*Kt*sin(2*t0*omega+2*omega*tj)*omeg a*tj+h*Kn*sin(2*t0*omega+2*omega*tj)h*Kn*cos(2*t0*omega+2*omega*tj)*omega*tj8*Kte*sin(t0*omega+omega*tj)+4*Kte*cos (t0*omega+omega*tj)*omega*tj+8*Kne*co s(t0*omega+omega*tj)+4*Kne*sin (t0*omega+omega*tj)*omega*tjh*Kt*cos(2*t0*omega)-h*Kn*sin(2*t0*omega)+8*Kte*sin(t0*omega)8*Kne*cos(t0*omega)+h*Kt* tj*sin(2*t0*omega)*omegah*Kn*tj*cos(2*t0*omega)*omega+4*Kte*tj*co s(t0*omega)*omega+4*Kne*tj*sin(t0*o mega)*omega)/tj/omega^2]; P11 = [ 1/8*b*(3*Kt*sin(2*t0*omega+2*omega*tj)+3*K n*cos(2*t0*omega+2*omega*tj)+2*Kn*omeg a^4*tj^4+3*Kt*omega*tj*cos(2*t0*omega +2*omega*tj)+3*Kn*omega*tj*sin(2*t0*ome ga+2*omega*tj)+3*Kt*sin(2*t0*omega)3*Kn*cos(2*t0*omega)+2*Kt*cos(2*t0*omeg a)*omega^3*tj^3+2*Kn*sin(2*t0*omega) *omega^3*tj^3+3*Kt*omega*tj*cos(2*t0*omega)+3*Kn*omega*tj*sin(2*t0*omega))/o mega^4/tj^3, 1/8*b*(-3*Kt*cos(2*t0*omega+2*omega*tj)3*Kn*sin(2*t0*omega+2*omega*tj)+2*Kt*omega^4*tj^43*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+3*Kn*omega*tj*cos(2*t0*omega+2*om ega*tj)+3*Kt*cos(2*t0*omega )+3*Kn*sin(2*t0*omega)2*Kt*sin(2*t0*omega)*omega^3*tj^ 3+2*Kn*cos(2*t0*omega)*omega^3*tj^33*Kt*omega*tj*sin(2*t0*omega)+3*Kn*ome ga*tj*cos(2*t0*omega))/omega^4/tj^3; 1/8*b*(-3*Kt*cos(2*t0*omega+2*omega* tj)-3*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^4*tj^43*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+3*Kn*omega*tj*cos(2*t0*omega+2*om ega*tj)+3*Kt*cos(2*t0*omega )+3*Kn*sin(2*t0*omega)2*Kt*sin(2*t0*omega)*omega^3*tj^ 3+2*Kn*cos(2*t0*omega)*omega^3*tj^33*Kt*omega*tj*sin(2*t0*omega)+3*Kn*omega *tj*cos(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kn*omega^4*tj^4+3*Kt*omega*tj*cos(2* t0*omega+2*omega*tj)+3*Kn*omega*tj*s in(2*t0*omega+2*omega*tj)3*Kt*sin(2*t0*omega+2*omega*tj)+3*Kn*c os(2*t0*omega+2*omega*tj)+2*Kt*cos(2* t0*omega)*omega^3*tj^3+2*Kn*sin(2*t0*omega)*omega^3*tj^3+3*Kt*omega*tj*cos(

PAGE 167

150 2*t0*omega)+3*Kn*omega*tj*sin(2*t0*omega)+3*Kt*sin(2*t0*omega)3*Kn*cos(2*t0*omega))/omega^4/tj^3]; P12 =[ 1/48*b*(6*Kn*omega*tj*sin(2*t0*omega+2*om ega*tj)+6*Kt*omega*tj*cos(2*t0*ome ga+2*omega*tj)+2*Kn*omega^4*tj^49*Kt*sin(2*t0*omega+2*omega*tj)+9*K n*cos(2*t0*omega+2*omega*tj)+12*Kn*ome ga*tj*sin(2*t0*omega)+12*Kt*omega*tj*cos(2*t0*omega)6*Kt*omega^2*tj^2*sin(2*t0*omega)+6*Kn*om ega^2*tj^2*cos(2*t0*omega)+9*Kt*si n(2*t0*omega)-9*Kn*cos(2*t0*omega))/tj^2/omega^4, 1/48*b*(9*Kt*cos(2*t0*omega+2*omega* tj)+9*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^4*tj^4+6*Kt*omega*tj* sin(2*t0*omega+2*omega*tj)6*Kn*omega*tj*cos(2*t0*omega+2*omega* tj)+12*Kt*omega*tj*sin(2*t0*omega)+6* Kn*omega^2*tj^2*sin(2*t0*omega)+ 6*Kt*omega^2*tj^2*cos(2*t0*omega)9*Kt*cos(2*t0*omega)-12*Kn*omega*tj*cos(2*t0*omega)9*Kn*sin(2*t0*omega))/tj^2/omega^4; 1/48*b*(2*Kt*omega^4*tj^4+6*Kt*omega* tj*sin(2*t0*omega+2*omega*tj)+9*Kt*cos( 2*t0*omega+2*omega*tj)6*Kn*omega*tj*cos(2*t0*omega+2*omega *tj)+9*Kn*sin(2*t0*omega+2*omega*tj)+1 2*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga^2*tj^2*sin(2*t0*omega)+6*Kt*omega^ 2*tj^2*cos(2*t0*omega)-9*Kt*cos(2*t0*omega)-12*Kn*omega*tj*cos(2*t0*omega)9*Kn*sin(2*t0*omega))/tj^2/omega^4, 1/48*b*(6*Kt*omega*tj*cos(2*t0*o mega+2*omega*tj)6*Kn*omega*tj*sin(2*t0*omega+ 2*omega*tj)+2*Kn*omega^4*tj^49*Kn*cos(2*t0*omega+2*omega*tj)+9*K t*sin(2*t0*omega+2*omega*tj)12*Kt*omega*tj*cos(2*t0*omega)12*Kn*omega*tj*sin(2*t0*omega)+6*K t*omega^2*tj^2*sin(2*t0*omega)6*Kn*omega^2*tj^2*cos(2*t0*om ega)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega ))/tj^2/omega^4]; P13 =[ 1/8*b*(2*Kn*omega^4*tj^43*Kn*cos(2*t0*omega+2*omega*tj)+3*K t*sin(2*t0*omega+2*omega*tj)3*Kt*omega*tj*cos(2*t0*o mega+2*omega*tj)2*Kn*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)3*Kn*omega*tj*sin(2*t0*omega+2*omega*tj)2*Kt*omega^3*tj^3*cos(2*t0*omega+ 2*omega*tj)+3*Kn*cos(2*t0*omega)3*Kt*sin(2*t0*omega)-3*Kt*omega*tj*cos(2*t0*omega)3*Kn*omega*tj*sin(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kt*omega^4*tj^4+2*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)2*Kn*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+3*Kt*cos(2*t0*omega+2*omega*t j)+3*Kn*sin(2*t0*omega+2*omega*tj)+3*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)3*Kn*omega*tj*cos(2*t0*omega+2*ome ga*tj)-3*Kt*cos(2*t0*omega)3*Kn*sin(2*t0*omega)+3*Kt*omega*tj*sin(2*t0*omega)3*Kn*omega*tj*cos(2*t0*omega))/omega^4/tj^3; 1/8*b*(2*Kt*omega^4*tj^4+2*Kn*omega^3*tj^ 3*cos(2*t0*omega+2*omega*tj)+3*Kn *omega*tj*cos(2*t0*omega+2*omega*t j)-3*Kn*sin(2*t0*omega+2*omega*tj)-

PAGE 168

151 2*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)3*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)3*Kt*cos(2*t0*omega+2*omega*tj)+3*Kn*s in(2*t0*omega)+3*Kt*cos(2*t0*omega)3*Kt*omega*tj*sin(2*t0*omega)+3*Kn*omega *tj*cos(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kn*omega^4*tj^4+3*Kn*cos(2*t0*ome ga+2*omega*tj)+2*Kn*omega^3*tj^3 *sin(2*t0*omega+2*omega*tj)3*Kt*sin(2*t0*omega+2*omega*tj)+2*Kt *omega^3*tj^3*cos(2*t0*omega+2*omega*tj )+3*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)+3*Kn*omega*tj*sin(2*t0*omega+2*o mega*tj)3*Kn*cos(2*t0*omega)+3*Kt*sin(2*t0*omeg a)+3*Kt*omega*tj*cos(2*t0*omega)+3* Kn*omega*tj*sin(2*t0*omega))/omega^4/tj^3]; P14 =[ -1/48*b*(6*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^212*Kn*omega*tj*sin(2*t0*omega+2*omega*tj)6*Kt*sin(2*t0*omega+2*ome ga*tj)*omega^2*tj^212*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)+2*Kn*omega^4*tj^49*Kn*cos(2*t0*omega+2*omega*tj)+9*K t*sin(2*t0*omega+2*omega*tj)6*Kn*omega*tj*sin(2*t0*omega)6*Kt*omega*tj*cos(2*t0*omega)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega))/tj^2/omeg a^4, 1/48*b*(-2*Kt*omega^4*tj^412*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+12*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+6*Kt*cos(2*t0*omega+2*omega* tj)*omega^2*tj^2+6*Kn*sin(2*t0*omega+2 *omega*tj)*omega^2*tj^2-9*K t*cos(2*t0*omega+2*omega*tj)9*Kn*sin(2*t0*omega+2*omega*tj)6*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga*tj*cos(2*t0*omega)+9*Kt*cos(2*t0*o mega)+9*Kn*sin(2*t0*omega))/tj^2/omega^4; 1/48*b*(2*Kt*omega^4*tj^412*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+12*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+6*Kt*cos(2*t0*omega+2*omega* tj)*omega^2*tj^2+6*Kn*sin(2*t0*omega+2 *omega*tj)*omega^2*tj^2-9*K t*cos(2*t0*omega+2*omega*tj)9*Kn*sin(2*t0*omega+2*omega*tj)6*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga*tj*cos(2*t0*omega)+9*Kt*cos(2*t0*o mega)+9*Kn*sin(2*t0*omega))/tj^2/omega^4, 1/48*b*(12*Kn*omega*tj*sin(2*t0*omega+2*omega* tj)-9*Kn*cos(2*t0*omega+2*omega*tj)2*Kn*omega^4*tj^4-6*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^212*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)+6*Kn*cos(2*t0*omega+2*omega*tj)*o mega^2*tj^2+9*Kt*sin(2*t0*omega+2*omega*tj)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega)-6*Kn*omega*tj*sin(2*t0*omega)6*Kt*omega*tj*cos(2*t0*omega))/tj^2/omega^4]; P21 =[ -1/80*b*(-15*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^215*Kn*sin(2*t0*omega+2*omega*tj)*omega^2*tj^2+60*Kt*cos(2*t0*omega+2*omega *tj)+60*Kn*sin(2*t0*omega+ 2*omega*tj)+4*Kn*omega^5*tj^5+60*Kt*omega*tj*sin(2 *t0*omega)10*Kn*omega^3*tj^3*cos(2*t0*omega)+15*Kn* omega^2*tj^2*sin(2*t0*omega)+15*K t*omega^2*tj^2*cos(2*t0*omega)+1 0*Kt*omega^3*tj^3*sin(2*t0*omega)-

PAGE 169

152 60*Kn*omega*tj*cos(2*t0*omega)+10*Kn*sin(2*t0*omega)*omega^4*tj^4+10*Kt*co s(2*t0*omega)*omega^4*tj^4-60*Kn*sin(2*t0*omega)60*Kt*cos(2*t0*omega)+60*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)60*Kn*omega*tj*cos(2*t0*omega+2*ome ga*tj))/omega^5/tj^4, -1/80*b*(60*Kn*cos(2*t0*omega)+60*Kt*sin(2*t0*omega)15*Kn*cos(2*t0*omega+2*omega*tj)*omega^ 2*tj^2+15*Kt*sin(2*t0*omega+2*omega *tj)*omega^2*tj^2+60*Kn*omega* tj*sin(2*t0*omega+2*omega*tj)60*Kt*sin(2*t0*omega+2*omega*tj)+60*Kt *omega*tj*cos(2*t0*omega+2*omega*tj)+ 60*Kn*cos(2*t0*omega+2*omega*tj)+10*Kt*omega^3*tj^3*cos(2*t0*omega)15*Kt*omega^2*tj^2*sin(2*t0*omega)10*Kt*sin(2*t0*omega)*omega^4*tj^4+10*K n*cos(2*t0*omega)*omega^4*tj^4+15*K n*omega^2*tj^2*cos(2*t0*omega)+60*Kt*om ega*tj*cos(2*t0*omega)+10*Kn*omega^ 3*tj^3*sin(2*t0*omega)+60*Kn*omega*tj*si n(2*t0*omega)+4*Kt*omega^5*tj^5)/ome ga^5/tj^4; 1/80*b*(60*Kn*cos(2*t0*omega)60*Kt*sin(2*t0*omega)+15*Kn*cos(2*t 0*omega+2*omega*tj)*omega^2*tj^215*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^260*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)-60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega)+15*Kt* omega^2*tj^2*sin(2*t0*omega)+10*Kt *sin(2*t0*omega)*omega^4*tj^4-10*Kn*cos(2*t0*omega)*omega^4*tj^415*Kn*omega^2*tj^2*cos(2*t0*omega )-60*Kt*omega*tj*cos(2*t0*omega)10*Kn*omega^3*tj^3*sin(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)+4*Kt*omega^5*tj^5)/omega^5/tj^4, -1/80*b*(60*Kt*omega*tj*sin(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)10*Kn*sin(2*t0*omega)*omega^4*tj^4-10*K t*cos(2*t0*omega)*omega^4*tj^415*Kt*omega^2*tj^2*cos(2*t0*omega)10*Kt*omega^3*tj^3*sin(2*t0*omega)+10*Kn*omega^3*tj^3*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega)+60*K n*sin(2*t0*omega)+60*Kt*cos(2*t0*ome ga)60*Kt*cos(2*t0*omega+2*omega*tj)+4*Kn*o mega^5*tj^5+60*Kn*omega*tj*cos(2*t0 *omega+2*omega*tj)+15*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+15*Kn*sin (2*t0*omega+2*omega*tj)*omega^2*tj^2-60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega*tj))/omega^5/tj^4]; P22 = [ 1/480*b*(-2*Kn*omega^5*tj^5180*Kt*cos(2*t0*omega+2*omega*tj)180*Kn*sin(2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+135*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+30*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+30*Kn*sin(2*t0*ome ga+2*omega*tj)*omega^2*tj^2225*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+225*Kn*omega*tj*cos(2*t0*omega)120*Kt*omega^2*tj^2*cos(2*t0*omega)+ 30*Kt*omega^3*tj^3*sin(2*t0*omega)30*Kn*omega^3*tj^3*cos(2*t0*omega)120*Kn*omega^2*tj^2*sin(2*t0*om ega))/tj^3/omega^5, 1/480*b*(135*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+135*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)-

PAGE 170

153 180*Kt*sin(2*t0*omega+2*omega*tj)+180*Kn*co s(2*t0*omega+2*omega*tj)+2*Kt*o mega^5*tj^5+30*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*cos(2*t0*omega)+ 120*Kn*omega^2*tj^2*cos(2*t0*omega)120*Kt*omega^2*tj^2*sin(2*t0*omega)30*Kn*omega^3*tj^3*sin(2*t0*omega)+ 225*Kt*omega*tj*cos(2*t0*omega)+225*Kn* omega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5; 1/480*b*(135*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+135*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*K n*cos(2*t0*omega+2*omega*tj)2*Kt*omega^5*tj^5+30*Kt*sin(2*t0*o mega+2*omega*tj)*omega^2*tj^230*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*cos(2*t0*omega)+ 120*Kn*omega^2*tj^2*cos(2*t0*omega)120*Kt*omega^2*tj^2*sin(2*t0*omega)30*Kn*omega^3*tj^3*sin(2*t0*omega)+ 225*Kt*omega*tj*cos(2*t0*omega)+225*Kn* omega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5, -1/480*b*(2*Kn*omega^5*tj^5180*Kt*cos(2*t0*omega+2*omega*tj)-180*K n*sin(2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+135*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+30*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+30*Kn*sin(2*t0*ome ga+2*omega*tj)*omega^2*tj^2225*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+225*Kn*omega*tj*cos(2*t0*omega)120*Kt*omega^2*tj^2*cos(2*t0*omega)+ 30*Kt*omega^3*tj^3*sin(2*t0*omega)30*Kn*omega^3*tj^3*cos(2*t0*omega)120*Kn*omega^2*tj^2*sin(2*t0*om ega))/tj^3/omega^5]; P23 = [ -1/80*b*(-4*Kn*omega^5*tj^5-60*Kt*cos(2*t0*omega+2*omega*tj)60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+60*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+15*Kt*cos(2*t0*omega+2*omega *tj)*omega^2*tj^2+15*Kn*sin(2*t0*omega +2*omega*tj)*omega^2*tj^210*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*sin(2*t0*o mega+2*omega*tj)+10*Kt*omega^4*tj^ 4*cos(2*t0*omega+2*omega*tj)+10*Kn*omeg a^3*tj^3*cos(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega)+60*Kn*s in(2*t0*omega)+60*Kt*cos(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-15*K t*omega^2*tj^2*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega))/omega^5/tj^4, -1/80*b*(60*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)60*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^4*tj^4*sin(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*cos(2*t0*o mega+2*omega*tj)-10*Kn*omega^ 3*tj^3*sin(2*t0*omega+2*omega*tj)4*Kt*omega^5*tj^5-

PAGE 171

154 15*Kt*sin(2*t0*omega+2*omega*tj)*omega ^2*tj^2+15*Kn*cos(2*t0*omega+2*omega *tj)*omega^2*tj^215*Kn*omega^2*tj^2*cos(2*t0*omega)+15*K t*omega^2*tj^2*sin(2*t0*omega)60*Kt*omega*tj*cos(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)60*Kt*sin(2*t0*omega)+60*Kn*cos(2*t0*omega))/omega^5/tj^4; -1/80*b*(-60*Kt*omega*tj*co s(2*t0*omega+2*omega*tj)60*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^4*tj^4*sin(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*cos(2*t0*o mega+2*omega*tj)10*Kn*omega^3*tj^3*sin(2*t0*omega+ 2*omega*tj)+4*Kt*omega^5*tj^515*Kt*sin(2*t0*omega+2*omega*tj)*omega ^2*tj^2+15*Kn*cos(2*t0*omega+2*omega *tj)*omega^2*tj^215*Kn*omega^2*tj^2*cos(2*t0*omega)+15*K t*omega^2*tj^2*sin(2*t0*omega)60*Kt*omega*tj*cos(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)60*Kt*sin(2*t0*omega)+60*Kn*cos(2*t0*omega))/omega^5/tj^4, 1/80*b*(4*Kn*omega^5*tj^5-60*Kt*cos(2*t0*omega+2*omega*tj)60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+60*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+15*Kt*cos(2*t0*omega+2*omega *tj)*omega^2*tj^2+15*Kn*sin(2*t0*omega +2*omega*tj)*omega^2*tj^210*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*sin(2*t0*o mega+2*omega*tj)+10*Kt*omega^4*tj^ 4*cos(2*t0*omega+2*omega*tj)+10*Kn*omeg a^3*tj^3*cos(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega)+60*Kn*s in(2*t0*omega)+60*Kt*cos(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-15*K t*omega^2*tj^2*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega))/omega^5/tj^4]; P24 = [ 1/480*b*(2*Kn*omega^5*tj^5+180*Kt*cos( 2*t0*omega+2*omega*tj)+180*Kn*sin(2*t 0*omega+2*omega*tj)+225*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)225*Kn*omega*tj*cos(2*t0*omega+2*omega*tj)120*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2120*Kn*sin(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+30*Kn*omega^3*tj^3*cos(2*t0*o mega+2*omega*tj)+135*Kt*omega*tj*si n(2*t0*omega)-180*Kn*sin(2*t0*omega)180*Kt*cos(2*t0*omega)135*Kn*omega*tj*cos(2*t0*omega)+30*Kt *omega^2*tj^2*cos(2*t0*omega)+30*Kn*o mega^2*tj^2*sin(2*t0*omega))/tj^3/omega^5, 1/480*b*(225*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+225*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*K n*cos(2*t0*omega+2*omega*tj)30*Kt*omega^3*tj^3*cos(2*t0*omega+2*omega*tj)30*Kn*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+2*Kt*omega^5*tj^5+120*Kt*sin( 2*t0*omega+2*omega*tj)*omega^2*tj^2120*Kn*cos(2*t0*omega+2*omega*tj)*omega^2* tj^2+30*Kn*omega^2*tj^2*cos(2*t0*

PAGE 172

155 omega)30*Kt*omega^2*tj^2*sin(2*t0*omega)+ 135*Kt*omega*tj*cos(2*t0*omega)+135*Kn*o mega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5; 1/480*b*(-225*Kt*omega*tj*cos( 2*t0*omega+2*omega*tj)225*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+180*Kt*sin(2*t0*omega+2*omega*tj )180*Kn*cos(2*t0*omega+2*omega*tj)+30*Kt*omega^3*tj^3*cos(2*t0*omega+2*ome ga*tj)+30*Kn*omega^3*tj^3*sin(2*t0*om ega+2*omega*tj)+2*Kt*omega^5*tj^5120*Kt*sin(2*t0*omega+2*omega*tj)*omega^ 2*tj^2+120*Kn*cos(2*t0*omega+2*ome ga*tj)*omega^2*tj^230*Kn*omega^2*tj^2*cos(2*t0*omega)+30*K t*omega^2*tj^2*sin(2*t0*omega)135*Kt*omega*tj*cos(2*t0*omega)135*Kn*omega*tj*sin(2*t0*omega)180*Kt*sin(2*t0*omega)+180*Kn*cos(2*t0*omega))/tj^3/omega^5, 1/480*b*(2*Kn*omega^5*tj^5-180*Kt*c os(2*t0*omega+2*omega*tj)180*Kn*sin(2*t0*omega+2*omega*tj)225*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+225*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+120*Kt*cos(2*t0*omega+2*omeg a*tj)*omega^2*tj^2+120*Kn*sin(2*t0*o mega+2*omega*tj)*omega^2*tj^2+30*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj) -30*Kn*omega^3*tj^3*cos( 2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+135*Kn*omega*tj*cos(2*t0*omega)-30*Kt*omega^2*tj^2*cos(2*t0*omega)30*Kn*omega^2*tj^2*sin(2*t0*omega))/tj^3/omega^5]; P11 = [P11(1,1)*ones(lmx,1) P11(1, 2)*ones(lmx,1); P11(2,1)*ones(lmx,1) P11(2,2)*ones(lmx,1)]*V; P12 = [P12(1,1)*ones(lmx,1) P12(1,2) *ones(lmx,1); P12(2,1)*ones(lmx,1) P12(2,2)*ones(lmx,1)]*V; P13 = [P13(1,1)*ones(lmx,1) P13(1,2) *ones(lmx,1); P13(2,1)*ones(lmx,1) P13(2,2)*ones(lmx,1)]*V; P14 = [P14(1,1)*ones(lmx,1) P14(1,2) *ones(lmx,1); P14(2,1)*ones(lmx,1) P14(2,2)*ones(lmx,1)]*V; P21 = [P21(1,1)*ones(lmx,1) P21(1,2) *ones(lmx,1); P21(2,1)*ones(lmx,1) P21(2,2)*ones(lmx,1)]*V; P22 = [P22(1,1)*ones(lmx,1) P22(1,2) *ones(lmx,1); P22(2,1)*ones(lmx,1) P22(2,2)*ones(lmx,1)]*V; P23 = [P23(1,1)*ones(lmx,1) P23(1,2) *ones(lmx,1); P23(2,1)*ones(lmx,1) P23(2,2)*ones(lmx,1)]*V; P24 = [P24(1,1)*ones(lmx,1) P24(1,2) *ones(lmx,1); P24(2,1)*ones(lmx,1) P24(2,2)*ones(lmx,1)]*V; N11 = -C+1/2*K*tj+P11; N12 = -M+1/12*K*tj^2+P12; N13 = C+1/2*K*tj+P13; N14 = M-1/12*K*tj^2+P14; N21 = M/tj-1/10*K*tj+P21; N22 = 1/2*M-1/12*C*tj-1/120*K*tj^2+P22; N23 = -M/tj+1/10*K*tj+P23;

PAGE 173

156 N24 = 1/2*M+1/12*C*tj-1/120*K*tj^2+P24; N1 = [N11 N12; N21 N22]; N2 = [N13 N14; N23 N24]; P1 = [P11 P12; P21 P22]; P2 = [P13 P14; P23 P24]; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%% % BUILD GLOBAL MATRICES %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%% A(1:2*DOF,1:2*DOF) = eye(2*DOF); A(2*DOF*e+1:2*DOF*e+2*DOF,2* DOF*(e-1)+1:2*DOF*(e-1)+2*DOF) = N1; A(2*DOF*e+1:2*DOF*e+2*DOF ,2*DOF*(e-1)+2*DOF+1:2*DOF*(e1)+2*DOF+2*DOF) = N2; B(2*DOF*e+1:2*DOF*e+2*DOF,2* DOF*(e-1)+1:2*DOF*(e-1)+2*DOF) = P1; B(2*DOF*e+1:2*DOF*e+2*DOF ,2*DOF*(e-1)+2*DOF+1:2*DOF*(e1)+2*DOF+2*DOF) = P2; B(1:2*(DOF),E*2*(DOF )+1:(E+1)*2*(DOF)) = PHI; Cvec(1:2*DOF,1) = zeros(2*DOF,1); Cvec(2*DOF*e+1:2*DOF*e+DOF,1) = C1; Cvec(2*DOF*e+DOF+1:2*DOF*e+2*DOF,1) = C2; end; % end # of elements loop size(A) Q = A\B; [vec,lam] = eig(Q); CM = max(abs(diag(lam))); D = A\Cvec; % Extract SLE coefficients if CM<1 SLE_vec = inv((eye(size(Q))-Q))*D; SLE = abs(sum(SLE_vec(locat))); else SLE = nan; end N1 = zeros(2*DOF,2*DOF); N2 = N1; P1 = N1; P2 = P1; if CM < 1 b_r(1) = b; else b_r(2) = b; end

PAGE 174

157 end % while loop depth of cut Uncertainty SLE, Sensitivity Method % M. Kurdi (3/28/2005) % Function to find uncertainty in SLE to change in cutting % coefficients, dynamic parameters and cutting process variables % Input: % b: depth of cut (m) % rpm: spindle speed % rstep: radial step (inches) % Kt % Kn % Kte % Kre % DELTA_Kt finite change in Kt % DELTA_b finite change in b % system_ID: Modal parameters % The derivative of Max eigen value is found for a miniscule perturbation % in input parameters, then its effect on the change of axial depth is % found. clear all; close all; clc;tic; % function uncer % percentage of uncertainty in cutting coefficients, dynamic parameters % and process parameters % tic; percent_Kcut = 0.05; % cutt ing coefficents uncertainty percent_Dyn = 0.005; % modal parameters uncertainty percent_rstep = 0.0001; % ra dial step uncertainty percent_rpm = 0.005; % spindle speed uncertainty % nominal values of process paramete rs and their calculated uncertainty rstep = 0.2; b=1e-3; rpm_vec = 5500:50:5600; DELTA_rstep = percent_rstep*rstep; % cutting coefficient uncertainty Kt = 6e8; DELTA_Kt = percent_Kcut*Kt; Kn = .3*Kt; DELTA_Kn = percent_Kcut*Kn; Kte=0; DELTA_Kte = percent_Kcut*Kte; Kne=0; DELTA_Kne = percent_Kcut*Kne; % nominal values of dynamic parameters and their calcul ated uncertainty Kx = 4.4528e+006; Mx = 0.4362; Cx = 83;

PAGE 175

158 % Y direction parameters Ky = 3.5542e+006; My = 0.4362; Cy = 89.9; DELTA_Mx = Mx*percent_Dyn; DELTA_My = My*percent_Dyn; DELTA_Kx = Kx*percent_Dyn; DELTA_Ky = Ky*percent_Dyn; DELTA_Cx = Cx*percent_Dyn; DELTA_Cy = Cy*percent_Dyn; % to calculate the numerical deriva tive with respect to each input % variable set the miniscule change in each input % set miniscule change in input para meters to estimate the derivative step_percent = 0.002; dKt = step_percent*Kt; % N/m2 dKn = step_percent*Kn; % N/m2 dKte = step_percent*30; % N/m dKne = step_percent*30; % N/m drstep = step_percen t*rstep; % inch dKx = step_percent*Kx; % N/m dKy = step_percent*Ky; % N/m dCx = step_percent*Cx; % dCy = step_percent*Cy; dMx = step_percent*Mx; % Kg dMy = step_percent*My; % Kg h = waitbar(0,'Please wait...'); % computation here % for i=1:length(rpm_vec) waitbar(i/length(rpm_vec),h); rpm = rpm_vec(i); drpm = step_percent*rpm; % rpm DELTA_rpm = percent_rpm rpm; % Find depth of cut correspondi ng to stability bounda ry using nominal % settings of input parameters [sle(i)] = sle_f(b,rpm,rstep,Kt,K n,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % depth at boundary % Find numerical derivative of ma ximum eigenvalue with respect to input % parameters % perturb cutting coefficient Kt by dKt [sle1] = sle_f(b,rpm,rstep,Kt-dKt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); [sle2] = sle_f(b,rpm,rstep,Kt+dKt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen matr ix w.r.t cutting coefficient Kt d_sle_Kt(i) = (sle2-sle1)/dKt/2; dsleKt_log(i) = d_sle_Kt(i)*Kt/(sle(i))*2; sle1 =[]; sle2 =[];

PAGE 176

159 % perturb cutting coefficient Kn by dKn [sle1] = sle_f(b,rpm,rstep,Kt,Kn-dKn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); [sle2] = sle_f(b,rpm,rstep,Kt,Kn+dKn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % % % derivative of eigen matrix w.r.t cutting coefficient Kt d_sle_Kn(i) = (sle2-sle1)/dKn/2; dsleKn_log(i) = d_sle_Kn(i)*Kn/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb cutting coefficient Kte by dKte [sle1] = sle_f(b,rpm,rstep,Kt ,Kn,Kte-dKte,Kne,Mx,Kx,Cx,My,Ky,Cy); [sle2] = sle_f(b,rpm,rstep,Kt,Kn,Kte+dKte,Kne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen matr ix w.r.t cutting coefficient Kt d_sle_Kte(i) = (sle2-sle1)/dKte/2; dsleKte_log(i) = d_ sle_Kte(i)*Kte/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb cutting coefficient Kne by dKne [sle1] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne-dKne,Mx,Kx,Cx,My,Ky,Cy); [sle2] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne+dKne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen matr ix w.r.t cutting coefficient Kne d_sle_Kne(i) = (sle2-sle1)/dKne/2; dsleKne_log(i) = d_sle_Kne(i)*Kne/(sle(i))*2; sle1 =[]; sle2 =[]; % % perturb depth of cut rstep by drstep [sle1] = sle_f(b,rpm,rstep-drstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); [sle2] = sle_f(b,rpm,rstep+drstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen value w.r.t rstep of cut d_sle_rstep(i) = (sle2-sle1)/drstep/2; dslerstep_log(i) = d_s le_rstep(i)*rstep/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb spindle speed by drpm [sle1] = sle_f(b,rpm-drpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); [sle2] = sle_f(b,rpm+drpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen value w.r.t rpm d_sle_rpm(i) = (sle2-sle1)/drpm/2; dslerpm_log(i) = d_sle_rpm(i)*rpm/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb Kx by dKx [sle1] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx-dKx,Cx,My,Ky,Cy); [sle2] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx+dKx,Cx,My,Ky,Cy); % derivative of eigen value w.r.t Kx d_sle_Kx(i) = (sle2-sle1)/dKx/2; dsleKx_log(i) = d_sle_Kx(i)*Kx/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb Ky by dKy [sle1] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky-dKy,Cy);

PAGE 177

160 [sle2] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky+dKy,Cy); % derivative of eigen value w.r.t Ky d_sle_Ky(i) = (sle2-sle1)/dKy/2; dsleKy_log(i) = d_sle_Ky(i)*Ky/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb Cx by dCx [sle1] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx-dCx,My,Ky,Cy); [sle2] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx+dCx,My,Ky,Cy); % derivative of eigen value w.r.t Cx d_sle_Cx(i) = (sle2-sle1)/dCx/2; dsleCx_log(i) = d_sle_Cx(i)*Cx/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb Cy by dCy [sle1] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy-dCy); [sle2] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy+dCy); % derivative of eigen value w.r.t Cy d_sle_Cy(i) = (sle2-sle1)/dCy/2; dsleCy_log(i) = d_sle_Cy(i)*Cy/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb Mx by dMx [sle1] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx-dMx,Kx,Cx,My,Ky,Cy); [sle2] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx+dMx,Kx,Cx,My,Ky,Cy); % derivative of eigen value w.r.t Mx d_sle_Mx(i) = (sle2-sle1)/dMx/2; dsleMx_log(i) = d_ sle_Mx(i)*Mx/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb My by dMy [sle1] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My-dMy,Ky,Cy); [sle2] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My+dMy,Ky,Cy); % % derivative of eigen value w.r.t My d_sle_My(i) = (sle2-sle1)/dMy/2; dsleMy_log(i) = d_ sle_My(i)*My/(sle(i))*2; sle1 =[]; sle2 =[]; DELTA_sle(i) =( (DELTA_Kt d_sle_Kt(i ))^2 + (DELTA_Kn d_s le_Kn(i))^2 + ... (DELTA_Kne d_sle_Kn e(i))^2 + (DELTA_Kte d_sle_Kte(i))^2 +... (DELTA_Kx d_sle_Kx(i))^2 + (DELTA_Mx d_sle_Mx(i))^2 + ... (DELTA_Cx d_sle_ Cx(i))^2 + (DELTA_Ky d_sle_Ky(i))^2 +... (DELTA_My d_sle _My(i))^2 + (DELTA_Cy d_sle_Cy(i))^2+... (DELTA_rstep d_sle_r step(i))^2 + (DELTA_rpm* d_sle_rpm(i))^2)^0.5 end close(h); % % Find the uncertainty in depth of cut for a corresponding uncertainty in % % input paramters time_total=toc; % save uncer_march_30_sle % subplot(2,1,1)

PAGE 178

161 % plot(rpm_vec/1000,sle*1e6,'-g',rpm _vec/1000,(sle+2*DELTA_sle)*1e6,'k',rpm_vec/1000,(sle-2*DELTA_sle)*1e6,'-k') % set(gca,'fontname','times','fontsize',16); % xlabel('\Omega (x10^3 rpm)','fontsize',14) % ylabel('SLE (\mum)','fontsize',14) % legend('Stability boundary, nom inal input','\pm2u_c(SLE)'); % axis([5 20 -12 28]) % subplot(2,1,2) % plot(rpm_vec/1000,d_sle_Ky*Ky*1e6,'<',rpm_vec/1000,d_sle_My*My*1e6,'>',... % rpm_vec/1000,d_sle_Cy*Cy*1e6,'o',r pm_vec/1000,d_sle_rpm.*rpm_vec*1e6,'+',... % rpm_vec/1000,d_sle_rstep*rstep*1e6,'^',rpm_vec/1000,d_sle_Kt*Kt*1e6,'s',... % rpm_vec/1000, d_sle_Kn*Kn*1e6,'*'); % legend('K_y','M_y' ,'C_y','\Omega','r_{step}','K_t','K_n'); % xlabel('\Omega (x10^3 rpm)','fontsize',14) % ylabel('x_i \partial(SLE)/\partial(x_i)'); % figure; % plot(rpm_vec/1000,dsleKy_log,'<',rpm_vec/1000,dsleMy_log,'>',... % rpm_vec/1000,dsleCy_log,'o',rpm_vec/1000,dslerpm_log,'+',... % rpm_vec/1000,dslerstep_log,'^',rpm_vec/1000,dsleKt_log,'s',... % rpm_vec/1000,dsleKn_log,'*'); % legend('K_y','M_y' ,'C_y','\Omega','r_{step}','K_t','K_n'); % xlabel('\Omega (x10^3 rpm)','fontsize',14) % ylabel('\partial(SLE)/\partial(x_i)x_i/SLE'); % figure; plot(rpm_vec/1000,abs(d_sle_Ky)*Ky*1e6,'.', rpm_vec/1000,abs(d_sle_My)*My*1e6,':',. .. % rpm_vec/1000,abs(d_sle_Cy)*Cy*1e6,'-',rpm_vec/1000,abs(d_sle_rpm).*rpm_vec*1e6,'-',... % rpm_vec/1000,abs(d_sle_rstep)*rstep*1e6,'^', rpm_vec/1000,abs(d_sle_Kt)*Kt*1e6,'s',... % rpm_vec/1000,abs(d_sle_Kn)*Kn*1e6); % legend('K_y','M_y' ,'C_y','\Omega','r_{step}','K_t','K_n'); % figure; % plot(rpm_vec/1000,abs(d_sle_Ky)*Ky./abs(d_sle_My)/My) % legend('K_y/M_y'); figure plot(rpm_vec/1000,DELTA_sle*1e6) xlabel('\Omega (x10^3 rpm)','fontsize',14) ylabel('u_c(SLE) (\mum)') figure plot(rpm_vec/1000,DELTA_sle*1e6,'-
PAGE 179

162 Uncertainty SLE, Monte Carlo and Latin Hype-Cube Sampling Methods % % M. Kurdi (6/17/05) % 4 OAL TOOL % Program to complete LatinHyper and Monte simulation for SLE clear all; % function LatinHyper tic; chip_load=0.1e-3;% chip load nteeth = 4; Diam =0.5; E=20; N = 1000; % number of iterations baxial=4.45e-3; % AL 6061 % percent_Kt = 7.13/100; % cutting coefficents uncertainty % percent_Kn = 8.09/100; % percent_Kte = 30.3/100; % percent_Kne = 23.9/100; % 5 OAL TOOL UNCERTAINTIES % percent_KX = 0.054; % modal parameters uncertainty % percent_CX = .286; % percent_MX =.045; % percent_KY = 0.054; % modal parameters uncertainty % percent_CY = .173; % percent_MY =.055; % 4 OAL TOOL UNCERTAINTIES due to thermal effect only percent_MX = 0.074; percent_CX = 0.042; percent_KX = 0.073 ; percent_MY = 0.2; percent_CY = 0.107; percent_KY = 0.2 ; percent_rstep = 0.0005; % radial step uncertainty percent_rpm = 0.005; % spindle speed uncertainty % speed_min = str2num(input('Min_speed = ','s')); % speed_max = str2num(input('Max_speed = ','s')); % speed = speed_min:200:speed_max; % speed = [ 14753 14803 14853 14903 14953]; 4.45 mm speed = [15517 15567 15617 15667 15767]; % 2.12 mm h = waitbar(0,'Please wait...'); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%5 % Cutting Coefficients %%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%

PAGE 180

163 % AL 6061 % mean_Kt =7.06E+08; % N/m2 % mean_Kn = 2.50E+08; % mean_Kte = 1.29E+04; % N/m; % mean_Kne = 6.57E+03; % AL 7475 mean_Kt = 690480868.527357; mean_Kte = 12022.3004909002; mean_Kn = 142535991.092323; mean_Kne =11281.4601645315; std_Kn=4009843*4.45; % N std_Kne=310.909*4.45; std_Kte=200.731*4.45; std_Kt=2588583*4.45; % std_Kt = percent_Kt*mean_Kt; % std_Kn = percent_Kn*mean_Kn; % std_Kte = percent_Kte*mean_Kte; % std_Kne = percent_Kne*mean_Kne; % Kne Kn Kte Kt % AL 6061 % SIGMA_K = [1.480E+07 -1.778E+11 -8.216E+06 9.871E+10; % -1.778E+11 2.458E+15 9.871E+10 -1.365E+15; % -8.216E+06 9.871E+10 9.163E+07 -1.101E+12; % 9.871E+10 -1.365E+15 -1.101E+12 1.522E+16 % ]; % AL 7475 SIGMA_K = [ 42157610.7365206 -506483170409.775 -3598978.12573119 43238262783.6325; -506483170409.775 7.00379474676691e+015 43238262783.6325 597911116174549; -3598978.12573128 43238262783.6335 17574719.1179838 211143357093.21; 43238262783.6335 -597911116174562 -211143357093.21 2.91975098408051e+015]; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% % Modal Parameters % X %%%%%%%%%%%%%%%% % 5 OAL TOOL % mean_Kx = 2.64E+06; % mean_Mx = 0.049; % mean_Cx = 8.972; % dynamic parameters for 4 OAL tool mean_Mx = 0.027 ; mean_Cx= 23.309; mean_Kx= 4359275.000 ;

PAGE 181

164 std_Cx = percent_CX*mean_Cx; std_Kx = percent_KX*mean_Kx; std_Mx = percent_MX*mean_Mx; % Mx Cx Kx My Cy Ky 5 OAL % SIGMA = [3.85E-06 4.03E-03 2.48E+02 2.40E-06 -3.18E-03 1.31E+02; % 4.03E-03 5.27E+00 2.69E+05 4.08E-03 -3.32E+00 2.19E+05; % 2.48E+02 2.69E+05 1.61E+10 1.67E+02 -2.07E+05 9.15E+09; % 2.40E-06 4.08E-03 1.67E+02 4.23E-06 -1.88E-03 2.24E+02; % -3.18E-03 -3.32E+00 -2.07E+05 -1.88E-03 2.71E+00 1.04E+05; % 1.31E+02 2.19E+05 9.15E+09 2.24E+02 -1.04E+05 1.19E+10 % % ]; % Mx Cx Kx My Cy Ky 4 OAL SIGMA = [4.04188E-06 0.000450265 631.110625 7.25563E-06 -0.000584252 878.998125; 0.000450265 0.953490935 38828.325 0.00283473 -2.467636648 567721.5525; 631.110625 38828.325 1.00042E+11 1068.011875 -51720.94813 1.21332E+11; 7.25563E-06 0.00283473 1068.011875 1.76519E-05 -0.007067488 2638.261875; -0.000584252 -2.467636648 -51720.94813 -0.007067488 11.34481701 -1426512.396; 878.998125 567721.5525 1.21332E+11 2638.261875 -1426512.396 4.36003E+11]; % % Y %%%%%%%%%%% %%%%%%%%%%%%%%% % 5 OAL TOOL % mean_Ky = 2.26e+006; % mean_Cy = 10.651; % mean_My = 0.042; % Y direction parameters 4 OAL TOOL mean_Ky = 3301775.000; mean_My = 0.021; mean_Cy = 31.432; std_My = percent_MY*mean_My; std_Ky = percent_KY*mean_Ky; std_Cy = percent_CY*mean_Cy; % %%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % % Radial step inches % %%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% mean_rstep = 0.25*.5; std_rstep = percent_rstep*mean_rstep;

PAGE 182

165 % %%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%% randn('state',0) Mode = lhsnorm([mean_Mx mea n_Cx mean_Kx mean_My mean_Cy mean_Ky],SIGMA,N); % Mode(:,1) is Mx random vector % Mode(:,2) is Cx random vector % Mode(:,3) is Kx random vector % Mode(:,4) is My random vector % Mode(:,5) is Cy random vector % Mode(:,6) is Ky random vector Cut_Coeff = lhsnorm([mean_Kne mean_K n mean_Kte mean_K t],SIGMA_K,N); % Cut_Coeff(:,1) Kne % Cut_Coeff(:,2) Kn % Cut_Coeff(:,3) Kte % Cut_Coeff(:,4) Kt sample = randn(N, 2); for j=1:length(speed) waitbar(j/length(speed),h) for i=1:N % Unless otherwise specified, all dimensions in m % Define input parameters %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% % Cutting coefficients Kt = Cut_Coeff(i,4); Kn = Cut_Coeff(i,2); Kte = Cut_Coeff(i,3); Kne = Cut_Coeff(i,1); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% % milling parameters % Spindle speed mean_rpm = speed(j); std_rpm = percent_rpm*mean_rpm; rpm = mean_rpm + std_rpm*sample(i,1); % rstep rstep = mean_rstep + std_rstep*sample(i,2); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% % Dynamic parameters % X direction is feed direction Kx =Mode(i,3); Mx = Mode(i,1); Cx = Mode(i,2); % Y direction parameters

PAGE 183

166 Ky = Mode(i,6); My = Mode(i,4); Cy = Mode(i,5); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% % Calculate axial depth corre sponding to input paramters % that is on the stability boundaries sle(i,j) = sle_f(baxial,rpm,rstep,Kt,Kn,Kte,Kne,Mx, Kx,Cx,My,Ky,Cy,chip_load,nteeth,Diam,E); end % i end monte loop for one spindle speed end % j end spindle speed range for i=1:length(speed) index = find(isnan(sle(:,i))==0); sle_mean(i) = mean(sle(index,i)); std_dev(i) = std(sle(index,i)); end time=toc; save Latin_AL7475SLE2p12 std_dev speed sle_mean sle time close(h); % hold on; % h1 = plot(speed/1000,(sle _mean-2*std_dev)*1e6,'-r') % hold on; % h2 = plot(speed/1000,sle_mean*1e6,'g-'); % hold on; % h3 = plot(speed/1000,(2*st d_dev+sle_mean)*1e6,'-r'); % legend([h1,h2,h3],'lower boundary','mean','upper boundary') % hold on; % for i=1:1000 % % plot(speed/1000,sle(i,:)*1e6,'.'); % end % % figure % % plot(speed/1000,std(sle)*2*1000) % % % % Input: % rpm ; % rstep: radial immersion (inches) % Output: % b depth of cut (m) function SLE = sle_f(b,rpm,rstep,Kt,Kn,Kt e,Kne,Mx,Kx,Cx,My,Ky,Cy,h,nteeth,Diam,E) % E=30; % % h = 0.1e-3; % feed per tooth % nteeth = 1; % number of teeth

PAGE 184

167 % Diam = 1; % inches TRAVang = acos(1-rs tep/(Diam/2)); % angular travel during cutting LAGang = 2*pi/nteeth; % separation angle for teeth rho = acos(1-rstep/(Diam/2))/(2*pi); % fraction of time in cut IMMERSION = rstep/Diam; opt = 'down'; if TRAVang>LAGang % MU LTIPLE TEETH ARE IN CONTACT teethNcontact = floor(TRAVang/LAGang) +1; else % SINGLE TOOTH IN CONTACT teethNcontact = 1; end %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% % SYSTEM IDENTIFICATION MATRICES %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% M =[Mx zeros(size(Mx)); zeros(size(Mx)) My]; C =[Cx zeros(size(Mx)); zeros(size(Mx)) Cy]; K =[Kx zeros(size(Mx)); zeros(size(Mx)) Ky]; lmx = length(Mx(1,:)); lmy = length(My(1,:)); DOF = lmx+lmy; V = [ones(1,lmx) zeros(1 ,lmy); zeros(1,lmx) ones(1,lmy)]; A = zeros((E+1)*2*DOF,(E+1)*2*DOF); B = A; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % BEGIN LOOP CALCULATIONS OVER RPM vs DOC FIELD %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% speed = rpm; omega = speed/60*(2*pi); % radians per second T = (2*pi)/omega/nteeth ; % tooth pass period TC = rho*T*nteeth; % time a single tooth spends in the cut tf = T-TC; % time for free vibs tj = TC/E; % time for each element %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % SET CUTTER RO TATION ANGLE FOR UP/DOWN-MILLING %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% switch opt case 'up' t0mat = [0 tj*(1:(E-1))]; % upmilling locat = 2*DOF+lmx+1:3*DOF; case 'down'

PAGE 185

168 tex = pi/omega; tent=tex-TC; % downmilling t0mat = [tent tent+tj*(1:(E-1))]; % downmilling locat = (E+1)*2*DOF -DOF-lmy+1:(E+1)*2*DOF-DOF; end %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% % STATE TRANSITION MATRIX %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% G1 = [zeros(size(M)) M; eye(size(M)) zeros(size(M))]; G2 = [K C; zeros (size(M)) -eye(size(M))]; G = -G1\G2; PHI = expm(G*tf); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% % N & P are used to create A & B which then become Q in..... a_n = Q a_n-1 + D %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% for e=1:E, t0 = t0mat(e); C1 = V'*[ -1/4*b*(-h*Kt*cos( 2*t0*omega+2*omega*tj)+2*h*Kn*omega*tjh*Kn*sin(2*t0*omega+2*omega*tj)+4*Kte*sin(t0*omega+omega*tj)4*Kne*cos(t0*omega+omega*tj)+h*Kt*co s(2*t0*omega)+h*Kn*sin(2*t0*omega)4*Kte*sin(t0*omega)+4*Kne*cos(t0*omega))/omega; 1/4*b*(2*h*Kt*omega*tjh*Kt*sin(2*t0*omega+2*omega*tj)+h*Kn*cos(2*t0*omega+2*omega*tj)4*Kte*sin(t0*omega+omega*tj)+4*Kne*cos (t0*omega+omega*tj)+h*Kt*sin(2*t0*ome ga)-h*Kn*cos(2*t0*omega)+4*Kte*sin(t0*om ega)-4*Kne*cos(t0*omega))/omega]; C2 = V'*[ 1/8*b*(h*Kt*sin(2*t0*omega+2*omega*tj)+h*Kt* cos(2*t0*omega+2*omega*tj)*omega*tj+h* Kn*cos(2*t0*omega+2*omega*tj)+h*Kn*s in(2*t0*omega+2*omega*tj)*omega*tj+8*K te*sin(t0*omega+omega*tj)-4*Kte *cos(t0*omega+omega*tj)*omega*tj8*Kne*cos(t0*omega+omega*tj)4*Kne*sin(t0*omega+omega*tj)*omega*tj+h*Kt*sin(2*t0*omega)h*Kn*cos(2*t0*omega)8*Kte*sin(t0*omega)+8*Kne*cos(t0*omega )+h*Kt*tj*cos(2*t0*omega)*omega+h*Kn* tj*sin(2*t0*omega)*omega-4*K te*tj*cos(t0*omega)*omega4*Kne*tj*sin(t0*omega)*omega)/tj/omega^2; 1/8*b*(h*Kt*cos(2*t0*omega+2*omega*tj)+ h*Kt*sin(2*t0*omega+2*omega*tj)*omeg a*tj+h*Kn*sin(2*t0*omega+2*omega*tj)h*Kn*cos(2*t0*omega+2*omega*tj)*omega*tj-

PAGE 186

169 8*Kte*sin(t0*omega+omega*tj)+4*Kte*cos (t0*omega+omega*tj)*omega*tj+8*Kne*co s(t0*omega+omega*tj)+4*Kne*sin (t0*omega+omega*tj)*omega*tjh*Kt*cos(2*t0*omega)-h*Kn*sin(2*t0*omega)+8*Kte*sin(t0*omega)8*Kne*cos(t0*omega)+h*Kt* tj*sin(2*t0*omega)*omegah*Kn*tj*cos(2*t0*omega)*omega+4*Kte*tj*co s(t0*omega)*omega+4*Kne*tj*sin(t0*o mega)*omega)/tj/omega^2]; P11 = [ 1/8*b*(3*Kt*sin(2*t0*omega+2*omega*tj)+3*K n*cos(2*t0*omega+2*omega*tj)+2*Kn*omeg a^4*tj^4+3*Kt*omega*tj*cos(2*t0*omega +2*omega*tj)+3*Kn*omega*tj*sin(2*t0*ome ga+2*omega*tj)+3*Kt*sin(2*t0*omega)3*Kn*cos(2*t0*omega)+2*Kt*cos(2*t0*omeg a)*omega^3*tj^3+2*Kn*sin(2*t0*omega) *omega^3*tj^3+3*Kt*omega*tj*cos(2*t0*omega)+3*Kn*omega*tj*sin(2*t0*omega))/o mega^4/tj^3, 1/8*b*(-3*Kt*cos(2*t0*omega+2*omega*tj)3*Kn*sin(2*t0*omega+2*omega*tj)+2*Kt*omega^4*tj^43*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+3*Kn*omega*tj*cos(2*t0*omega+2*om ega*tj)+3*Kt*cos(2*t0*omega )+3*Kn*sin(2*t0*omega)2*Kt*sin(2*t0*omega)*omega^3*tj^ 3+2*Kn*cos(2*t0*omega)*omega^3*tj^33*Kt*omega*tj*sin(2*t0*omega)+3*Kn*ome ga*tj*cos(2*t0*omega))/omega^4/tj^3; 1/8*b*(-3*Kt*cos(2*t0*omega+2*o mega*tj)-3*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^4*tj^43*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+3*Kn*omega*tj*cos(2*t0*omega+2*om ega*tj)+3*Kt*cos(2*t0*omega )+3*Kn*sin(2*t0*omega)2*Kt*sin(2*t0*omega)*omega^3*tj^ 3+2*Kn*cos(2*t0*omega)*omega^3*tj^33*Kt*omega*tj*sin(2*t0*omega)+3*Kn*omega *tj*cos(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kn*omega^4*tj^4+3*Kt*omega*tj*cos(2* t0*omega+2*omega*tj)+3*Kn*omega*tj*s in(2*t0*omega+2*omega*tj)3*Kt*sin(2*t0*omega+2*omega*tj)+3*Kn*c os(2*t0*omega+2*omega*tj)+2*Kt*cos(2* t0*omega)*omega^3*tj^3+2*Kn*sin(2*t0*omega)*omega^3*tj^3+3*Kt*omega*tj*cos( 2*t0*omega)+3*Kn*omega*tj*sin(2*t0*omega)+3*Kt*sin(2*t0*omega)3*Kn*cos(2*t0*omega))/omega^4/tj^3]; P12 =[ 1/48*b*(6*Kn*omega*tj*sin(2*t0*omega+2*om ega*tj)+6*Kt*omega*tj*cos(2*t0*ome ga+2*omega*tj)+2*Kn*omega^4*tj^49*Kt*sin(2*t0*omega+2*omega*tj)+9*K n*cos(2*t0*omega+2*omega*tj)+12*Kn*ome ga*tj*sin(2*t0*omega)+12*Kt*omega*tj*cos(2*t0*omega)6*Kt*omega^2*tj^2*sin(2*t0*omega)+6*Kn*om ega^2*tj^2*cos(2*t0*omega)+9*Kt*si n(2*t0*omega)-9*Kn*cos(2*t0*omega))/tj^2/omega^4, 1/48*b*(9*Kt*cos(2*t0*omega+2*omega* tj)+9*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^4*tj^4+6*Kt*omega*tj* sin(2*t0*omega+2*omega*tj)6*Kn*omega*tj*cos(2*t0*omega+2*omega* tj)+12*Kt*omega*tj*sin(2*t0*omega)+6* Kn*omega^2*tj^2*sin(2*t0*omega)+ 6*Kt*omega^2*tj^2*cos(2*t0*omega)9*Kt*cos(2*t0*omega)-12*Kn*omega*tj*cos(2*t0*omega)9*Kn*sin(2*t0*omega))/tj^2/omega^4; 1/48*b*(2*Kt*omega^4*tj^4+6*Kt*omega* tj*sin(2*t0*omega+2*omega*tj)+9*Kt*cos(

PAGE 187

170 2*t0*omega+2*omega*tj)6*Kn*omega*tj*cos(2*t0*omega+2*omega *tj)+9*Kn*sin(2*t0*omega+2*omega*tj)+1 2*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga^2*tj^2*sin(2*t0*omega)+6*Kt*omega^ 2*tj^2*cos(2*t0*omega)-9*Kt*cos(2*t0*omega)-12*Kn*omega*tj*cos(2*t0*omega)9*Kn*sin(2*t0*omega))/tj^2/omega^4, 1/48*b*(6*Kt*omega*tj*cos(2*t0*o mega+2*omega*tj)6*Kn*omega*tj*sin(2*t0*omega+ 2*omega*tj)+2*Kn*omega^4*tj^49*Kn*cos(2*t0*omega+2*omega*tj)+9*K t*sin(2*t0*omega+2*omega*tj)12*Kt*omega*tj*cos(2*t0*omega)12*Kn*omega*tj*sin(2*t0*omega)+6*K t*omega^2*tj^2*sin(2*t0*omega)6*Kn*omega^2*tj^2*cos(2*t0*om ega)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega ))/tj^2/omega^4]; P13 =[ 1/8*b*(2*Kn*omega^4*tj^43*Kn*cos(2*t0*omega+2*omega*tj)+3*K t*sin(2*t0*omega+2*omega*tj)3*Kt*omega*tj*cos(2*t0*o mega+2*omega*tj)2*Kn*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)3*Kn*omega*tj*sin(2*t0*omega+2*omega*tj)2*Kt*omega^3*tj^3*cos(2*t0*omega+ 2*omega*tj)+3*Kn*cos(2*t0*omega)3*Kt*sin(2*t0*omega)-3*Kt*omega*tj*cos(2*t0*omega)3*Kn*omega*tj*sin(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kt*omega^4*tj^4+2*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)2*Kn*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+3*Kt*cos(2*t0*omega+2*omega*t j)+3*Kn*sin(2*t0*omega+2*omega*tj)+3*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)3*Kn*omega*tj*cos(2*t0*omega+2*ome ga*tj)-3*Kt*cos(2*t0*omega)3*Kn*sin(2*t0*omega)+3*Kt*omega*tj*sin(2*t0*omega)3*Kn*omega*tj*cos(2*t0*omega))/omega^4/tj^3; 1/8*b*(2*Kt*omega^4*tj^4+2*Kn*omega^3*tj^ 3*cos(2*t0*omega+2*omega*tj)+3*Kn *omega*tj*cos(2*t0*omega+2*omega*t j)-3*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)3*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)3*Kt*cos(2*t0*omega+2*omega*tj)+3*Kn*s in(2*t0*omega)+3*Kt*cos(2*t0*omega)3*Kt*omega*tj*sin(2*t0*omega)+3*Kn*omega *tj*cos(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kn*omega^4*tj^4+3*Kn*cos(2*t0*ome ga+2*omega*tj)+2*Kn*omega^3*tj^3 *sin(2*t0*omega+2*omega*tj)3*Kt*sin(2*t0*omega+2*omega*tj)+2*Kt *omega^3*tj^3*cos(2*t0*omega+2*omega*tj )+3*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)+3*Kn*omega*tj*sin(2*t0*omega+2*o mega*tj)3*Kn*cos(2*t0*omega)+3*Kt*sin(2*t0*omeg a)+3*Kt*omega*tj*cos(2*t0*omega)+3* Kn*omega*tj*sin(2*t0*omega))/omega^4/tj^3]; P14 =[ -1/48*b*(6*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^212*Kn*omega*tj*sin(2*t0*omega+2*omega*tj)6*Kt*sin(2*t0*omega+2*ome ga*tj)*omega^2*tj^212*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)+2*Kn*omega^4*tj^49*Kn*cos(2*t0*omega+2*omega*tj)+9*K t*sin(2*t0*omega+2*omega*tj)6*Kn*omega*tj*sin(2*t0*omega)-

PAGE 188

171 6*Kt*omega*tj*cos(2*t0*omega)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega))/tj^2/omeg a^4, 1/48*b*(-2*Kt*omega^4*tj^412*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+12*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+6*Kt*cos(2*t0*omega+2*omega* tj)*omega^2*tj^2+6*Kn*sin(2*t0*omega+2 *omega*tj)*omega^2*tj^2-9*K t*cos(2*t0*omega+2*omega*tj)9*Kn*sin(2*t0*omega+2*omega*tj)6*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga*tj*cos(2*t0*omega)+9*Kt*cos(2*t0*o mega)+9*Kn*sin(2*t0*omega))/tj^2/omega^4; 1/48*b*(2*Kt*omega^4*tj^412*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+12*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+6*Kt*cos(2*t0*omega+2*omega* tj)*omega^2*tj^2+6*Kn*sin(2*t0*omega+2 *omega*tj)*omega^2*tj^2-9*K t*cos(2*t0*omega+2*omega*tj)9*Kn*sin(2*t0*omega+2*omega*tj)6*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga*tj*cos(2*t0*omega)+9*Kt*cos(2*t0*o mega)+9*Kn*sin(2*t0*omega))/tj^2/omega^4, 1/48*b*(12*Kn*omega*tj*sin(2*t0*omega+2*omega* tj)-9*Kn*cos(2*t0*omega+2*omega*tj)2*Kn*omega^4*tj^4-6*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^212*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)+6*Kn*cos(2*t0*omega+2*omega*tj)*o mega^2*tj^2+9*Kt*sin(2*t0*omega+2*omega*tj)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega)-6*Kn*omega*tj*sin(2*t0*omega)6*Kt*omega*tj*cos(2*t0*omega))/tj^2/omega^4]; P21 =[ -1/80*b*(-15*Kt*cos(2*t0*o mega+2*omega*tj)*omega^2*tj^215*Kn*sin(2*t0*omega+2*omega*tj)*omega^2*tj^2+60*Kt*cos(2*t0*omega+2*omega *tj)+60*Kn*sin(2*t0*omega+ 2*omega*tj)+4*Kn*omega^5*tj^5+60*Kt*omega*tj*sin(2 *t0*omega)10*Kn*omega^3*tj^3*cos(2*t0*omega)+15*Kn* omega^2*tj^2*sin(2*t0*omega)+15*K t*omega^2*tj^2*cos(2*t0*omega)+1 0*Kt*omega^3*tj^3*sin(2*t0*omega)60*Kn*omega*tj*cos(2*t0*omega)+10*Kn*sin(2*t0*omega)*omega^4*tj^4+10*Kt*co s(2*t0*omega)*omega^4*tj^4-60*Kn*sin(2*t0*omega)60*Kt*cos(2*t0*omega)+60*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)60*Kn*omega*tj*cos(2*t0*omega+2*ome ga*tj))/omega^5/tj^4, -1/80*b*(60*Kn*cos(2*t0*omega)+60*Kt*sin(2*t0*omega)15*Kn*cos(2*t0*omega+2*omega*tj)*omega^ 2*tj^2+15*Kt*sin(2*t0*omega+2*omega *tj)*omega^2*tj^2+60*Kn*omega* tj*sin(2*t0*omega+2*omega*tj)60*Kt*sin(2*t0*omega+2*omega*tj)+60*Kt *omega*tj*cos(2*t0*omega+2*omega*tj)+ 60*Kn*cos(2*t0*omega+2*omega*tj)+10*Kt*omega^3*tj^3*cos(2*t0*omega)15*Kt*omega^2*tj^2*sin(2*t0*omega)10*Kt*sin(2*t0*omega)*omega^4*tj^4+10*K n*cos(2*t0*omega)*omega^4*tj^4+15*K n*omega^2*tj^2*cos(2*t0*omega)+60*Kt*om ega*tj*cos(2*t0*omega)+10*Kn*omega^ 3*tj^3*sin(2*t0*omega)+60*Kn*omega*tj*si n(2*t0*omega)+4*Kt*omega^5*tj^5)/ome ga^5/tj^4; 1/80*b*(60*Kn*cos(2*t0*omega)60*Kt*sin(2*t0*omega)+15*Kn*cos(2*t 0*omega+2*omega*tj)*omega^2*tj^215*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^260*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)-60*Kn*cos(2*t0*omega+2*omega*tj)-

PAGE 189

172 10*Kt*omega^3*tj^3*cos(2*t0*omega)+15*Kt* omega^2*tj^2*sin(2*t0*omega)+10*Kt *sin(2*t0*omega)*omega^4*tj^4-10*Kn*cos(2*t0*omega)*omega^4*tj^415*Kn*omega^2*tj^2*cos(2*t0*omega )-60*Kt*omega*tj*cos(2*t0*omega)10*Kn*omega^3*tj^3*sin(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)+4*Kt*omega^5*tj^5)/omega^5/tj^4, -1/80*b*(60*Kt*omega*tj*sin(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)10*Kn*sin(2*t0*omega)*omega^4*tj^4-10*K t*cos(2*t0*omega)*omega^4*tj^415*Kt*omega^2*tj^2*cos(2*t0*omega)10*Kt*omega^3*tj^3*sin(2*t0*omega)+10*Kn*omega^3*tj^3*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega)+60*K n*sin(2*t0*omega)+60*Kt*cos(2*t0*ome ga)60*Kt*cos(2*t0*omega+2*omega*tj)+4*Kn*o mega^5*tj^5+60*Kn*omega*tj*cos(2*t0 *omega+2*omega*tj)+15*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+15*Kn*sin (2*t0*omega+2*omega*tj)*omega^2*tj^2-60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega*tj))/omega^5/tj^4]; P22 = [ 1/480*b*(-2*Kn*omega^5*tj^5180*Kt*cos(2*t0*omega+2*omega*tj)180*Kn*sin(2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+135*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+30*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+30*Kn*sin(2*t0*ome ga+2*omega*tj)*omega^2*tj^2225*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+225*Kn*omega*tj*cos(2*t0*omega)120*Kt*omega^2*tj^2*cos(2*t0*omega)+ 30*Kt*omega^3*tj^3*sin(2*t0*omega)30*Kn*omega^3*tj^3*cos(2*t0*omega)120*Kn*omega^2*tj^2*sin(2*t0*om ega))/tj^3/omega^5, 1/480*b*(135*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+135*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*Kn*co s(2*t0*omega+2*omega*tj)+2*Kt*o mega^5*tj^5+30*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*cos(2*t0*omega)+ 120*Kn*omega^2*tj^2*cos(2*t0*omega)120*Kt*omega^2*tj^2*sin(2*t0*omega)30*Kn*omega^3*tj^3*sin(2*t0*omega)+ 225*Kt*omega*tj*cos(2*t0*omega)+225*Kn* omega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5; 1/480*b*(135*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+135*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*K n*cos(2*t0*omega+2*omega*tj)2*Kt*omega^5*tj^5+30*Kt*sin(2*t0*o mega+2*omega*tj)*omega^2*tj^230*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*cos(2*t0*omega)+ 120*Kn*omega^2*tj^2*cos(2*t0*omega)120*Kt*omega^2*tj^2*sin(2*t0*omega)30*Kn*omega^3*tj^3*sin(2*t0*omega)+ 225*Kt*omega*tj*cos(2*t0*omega)+225*Kn* omega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5, -1/480*b*(2*Kn*omega^5*tj^5-

PAGE 190

173 180*Kt*cos(2*t0*omega+2*omega*tj)-180*K n*sin(2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+135*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+30*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+30*Kn*sin(2*t0*ome ga+2*omega*tj)*omega^2*tj^2225*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+225*Kn*omega*tj*cos(2*t0*omega)120*Kt*omega^2*tj^2*cos(2*t0*omega)+ 30*Kt*omega^3*tj^3*sin(2*t0*omega)30*Kn*omega^3*tj^3*cos(2*t0*omega)120*Kn*omega^2*tj^2*sin(2*t0*om ega))/tj^3/omega^5]; P23 = [ -1/80*b*(-4*Kn*omega^5*tj^5-60*Kt*cos(2*t0*omega+2*omega*tj)60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+60*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+15*Kt*cos(2*t0*omega+2*omega *tj)*omega^2*tj^2+15*Kn*sin(2*t0*omega +2*omega*tj)*omega^2*tj^210*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*sin(2*t0*o mega+2*omega*tj)+10*Kt*omega^4*tj^ 4*cos(2*t0*omega+2*omega*tj)+10*Kn*omeg a^3*tj^3*cos(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega)+60*Kn*s in(2*t0*omega)+60*Kt*cos(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-15*K t*omega^2*tj^2*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega))/omega^5/tj^4, -1/80*b*(60*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)60*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^4*tj^4*sin(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*cos(2*t0*o mega+2*omega*tj)-10*Kn*omega^ 3*tj^3*sin(2*t0*omega+2*omega*tj)4*Kt*omega^5*tj^515*Kt*sin(2*t0*omega+2*omega*tj)*omega ^2*tj^2+15*Kn*cos(2*t0*omega+2*omega *tj)*omega^2*tj^215*Kn*omega^2*tj^2*cos(2*t0*omega)+15*K t*omega^2*tj^2*sin(2*t0*omega)60*Kt*omega*tj*cos(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)60*Kt*sin(2*t0*omega)+60*Kn*cos(2*t0*omega))/omega^5/tj^4; -1/80*b*(-60*Kt*omega* tj*cos(2*t0*omega+2*omega*tj)60*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^4*tj^4*sin(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*cos(2*t0*o mega+2*omega*tj)10*Kn*omega^3*tj^3*sin(2*t0*omega+ 2*omega*tj)+4*Kt*omega^5*tj^515*Kt*sin(2*t0*omega+2*omega*tj)*omega ^2*tj^2+15*Kn*cos(2*t0*omega+2*omega *tj)*omega^2*tj^215*Kn*omega^2*tj^2*cos(2*t0*omega)+15*K t*omega^2*tj^2*sin(2*t0*omega)60*Kt*omega*tj*cos(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)60*Kt*sin(2*t0*omega)+60*Kn*cos(2*t0*omega))/omega^5/tj^4, 1/80*b*(4*Kn*omega^5*tj^5-60*Kt*cos(2*t0*omega+2*omega*tj)60*Kn*sin(2*t0*omega+2*omega*tj)-

PAGE 191

174 60*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+60*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+15*Kt*cos(2*t0*omega+2*omega *tj)*omega^2*tj^2+15*Kn*sin(2*t0*omega +2*omega*tj)*omega^2*tj^210*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*sin(2*t0*o mega+2*omega*tj)+10*Kt*omega^4*tj^ 4*cos(2*t0*omega+2*omega*tj)+10*Kn*omeg a^3*tj^3*cos(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega)+60*Kn*s in(2*t0*omega)+60*Kt*cos(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-15*K t*omega^2*tj^2*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega))/omega^5/tj^4]; P24 = [ 1/480*b*(2*Kn*omega^5*tj^5+180*Kt*cos( 2*t0*omega+2*omega*tj)+180*Kn*sin(2*t 0*omega+2*omega*tj)+225*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)225*Kn*omega*tj*cos(2*t0*omega+2*omega*tj)120*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2120*Kn*sin(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+30*Kn*omega^3*tj^3*cos(2*t0*o mega+2*omega*tj)+135*Kt*omega*tj*si n(2*t0*omega)-180*Kn*sin(2*t0*omega)180*Kt*cos(2*t0*omega)135*Kn*omega*tj*cos(2*t0*omega)+30*Kt *omega^2*tj^2*cos(2*t0*omega)+30*Kn*o mega^2*tj^2*sin(2*t0*omega))/tj^3/omega^5, 1/480*b*(225*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+225*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*K n*cos(2*t0*omega+2*omega*tj)30*Kt*omega^3*tj^3*cos(2*t0*omega+2*omega*tj)30*Kn*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+2*Kt*omega^5*tj^5+120*Kt*sin( 2*t0*omega+2*omega*tj)*omega^2*tj^2120*Kn*cos(2*t0*omega+2*omega*tj)*omega^2* tj^2+30*Kn*omega^2*tj^2*cos(2*t0* omega)30*Kt*omega^2*tj^2*sin(2*t0*omega)+ 135*Kt*omega*tj*cos(2*t0*omega)+135*Kn*o mega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5; 1/480*b*(-225*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)225*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+180*Kt*sin(2*t0*omega+2*omega*tj )180*Kn*cos(2*t0*omega+2*omega*tj)+30*Kt*omega^3*tj^3*cos(2*t0*omega+2*ome ga*tj)+30*Kn*omega^3*tj^3*sin(2*t0*om ega+2*omega*tj)+2*Kt*omega^5*tj^5120*Kt*sin(2*t0*omega+2*omega*tj)*omega^ 2*tj^2+120*Kn*cos(2*t0*omega+2*ome ga*tj)*omega^2*tj^230*Kn*omega^2*tj^2*cos(2*t0*omega)+30*K t*omega^2*tj^2*sin(2*t0*omega)135*Kt*omega*tj*cos(2*t0*omega)135*Kn*omega*tj*sin(2*t0*omega)180*Kt*sin(2*t0*omega)+180*Kn*cos(2*t0*omega))/tj^3/omega^5, 1/480*b*(2*Kn*omega^5*tj^5-180*Kt*c os(2*t0*omega+2*omega*tj)180*Kn*sin(2*t0*omega+2*omega*tj)225*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+225*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+120*Kt*cos(2*t0*omega+2*omeg a*tj)*omega^2*tj^2+120*Kn*sin(2*t0*o mega+2*omega*tj)*omega^2*tj^2+30*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)

PAGE 192

175 -30*Kn*omega^3*tj^3*cos( 2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+135*Kn*omega*tj*cos(2*t0*omega)-30*Kt*omega^2*tj^2*cos(2*t0*omega)30*Kn*omega^2*tj^2*sin(2*t0*omega))/tj^3/omega^5]; P11 = [P11(1,1)*ones(lmx,1) P11(1,2) *ones(lmx,1); P11(2,1)*ones(lmx,1) P11(2,2)*ones(lmx,1)]*V; P12 = [P12(1,1)*ones(lmx,1) P 12(1,2)*ones(lmx,1); P12(2,1)*ones(lmx,1) P12(2,2)*ones(lmx,1)]*V; P13 = [P13(1,1)*ones(lmx,1) P 13(1,2)*ones(lmx,1); P13(2,1)*ones(lmx,1) P13(2,2)*ones(lmx,1)]*V; P14 = [P14(1,1)*ones(lmx,1) P 14(1,2)*ones(lmx,1); P14(2,1)*ones(lmx,1) P14(2,2)*ones(lmx,1)]*V; P21 = [P21(1,1)*ones(lmx,1) P 21(1,2)*ones(lmx,1); P21(2,1)*ones(lmx,1) P21(2,2)*ones(lmx,1)]*V; P22 = [P22(1,1)*ones(lmx,1) P 22(1,2)*ones(lmx,1); P22(2,1)*ones(lmx,1) P22(2,2)*ones(lmx,1)]*V; P23 = [P23(1,1)*ones(lmx,1) P 23(1,2)*ones(lmx,1); P23(2,1)*ones(lmx,1) P23(2,2)*ones(lmx,1)]*V; P24 = [P24(1,1)*ones(lmx,1) P 24(1,2)*ones(lmx,1); P24(2,1)*ones(lmx,1) P24(2,2)*ones(lmx,1)]*V; N11 = -C+1/2*K*tj+P11; N12 = -M+1/12*K*tj^2+P12; N13 = C+1/2*K*tj+P13; N14 = M-1/12*K*tj^2+P14; N21 = M/tj-1/10*K*tj+P21; N22 = 1/2*M-1/12*C*tj-1/120*K*tj^2+P22; N23 = -M/tj+1/10*K*tj+P23; N24 = 1/2*M+1/12*C*tj-1/120*K*tj^2+P24; N1 = [N11 N12; N21 N22]; N2 = [N13 N14; N23 N24]; P1 = [P11 P12; P21 P22]; P2 = [P13 P14; P23 P24]; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%% % BUILD GLOBAL MATRICES %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%% A(1:2*DOF,1:2*DOF) = eye(2*DOF); A(2*DOF*e+1:2*DOF*e+2*DOF ,2*DOF*(e-1)+1:2*DOF*(e-1)+2*DOF) = N1; A(2*DOF*e+1:2*DOF*e+2 *DOF,2*DOF*(e-1)+2*DOF+1:2*DOF*(e1)+2*DOF+2*DOF) = N2; B(2*DOF*e+1:2*DOF*e+2*DOF ,2*DOF*(e-1)+1:2*DOF*(e-1)+2*DOF) = P1; B(2*DOF*e+1:2*DOF*e+2 *DOF,2*DOF*(e-1)+2*DOF+1:2*DOF*(e1)+2*DOF+2*DOF) = P2; B(1:2*(DOF),E*2*(DOF)+1:(E+1)*2*(DOF)) = PHI; Cvec(1:2*DOF,1) = zeros(2*DOF,1);

PAGE 193

176 Cvec(2*DOF*e+1:2*DOF*e+DOF,1) = C1; Cvec(2*DOF*e+DOF+1:2*DOF*e+2*DOF,1) = C2; end; % end # of elements loop  Q = A\B; [vec,lam] = eig(Q); CM = max(abs(diag(lam))); D = A\Cvec; % Extract SLE coefficients if CM<1 SLE_vec = inv((eye(size(Q))-Q))*D; SLE = (sum(SLE_vec(locat))); else SLE = nan; end N1 = zeros(2*DOF,2*DOF); N2 = N1; P1 = N1; P2 = P1;

PAGE 194

177 LIST OF REFERENCES 1. Plaskos, C., Hodgson, A.J., and Cinqui n, P., Modelling and Optimization of Bone-Cutting Forces in Orthopaedic Surgery. Medical Image Computing and Computer-Assisted Intervention Miccai 2003, Pt 1, 2003. 2878: p. 254. 2. De Lacalle, L.N.L., Lamikiz, A., Sanchez, J.A., and Cabanes, I., Cutting Conditions and Tool Optimization in the High-Speed Milling of Aluminium Alloys. Proceedings of the Inst itution of Mechanical Engi neers Part B-Journal of Engineering Manufacture, 2001. 215(9): p. 1257. 3. De Lacalle, L.N.L., Lamikiz, A., Salg ado, M.A., Herranz, S., and Rivero, A., Process Planning for Reliable High-Speed Machining of Moulds. International Journal of Production Research, 2002. 40(12): p. 2789. 4. Fallbohmer, P., Rodriguez, C.A., Ozel, T., and Altan, T., High-Speed Machining of Cast Iron and Alloy Steels for Die and Mold Manufacturing. Journal of Materials Processing Technology, 2000. 98(1): p. 104. 5. Ermer, D.S., A Century of Optimizing Machining Operations. Journal of Manufacturing Science and EngineeringTransactions of the ASME, 1997. 119(4B): p. 817. 6. Sonmez, A.I., Baykasoglu, A., Turkay, D., and Filiz, I.H., Dynamic Optimization of Multi-Pass Milling Operations Via Geometric Programming. International Journal of Machine To ols & Manufacture, 1999. 39(2): p. 297. 7. Wang, J. and Armarego, E.J.A., Computer-Aided Optimization of Multiple Constraint Single Pass Face Milling Operations. Machining Science and Technology, 2001. 5(1): p. 77. 8. Armarego, E.J.A., Smith, A.J.R., and Wang, J., Computer-Aided Constrained Optimization Analyses and Strategies for Multi-Pass Helical Tooth Milling Operations. CIRP Annals, 1994. 43(1): p. 437. 9. Ermer, D.S. and Patel, D.C. Ma ximization of the Production Rate with Constraints by Linear Programming and Sensitivity Analysis. In Proceedings of NAMRC. 1974.

PAGE 195

178 10. Ermer, D.S., Optimization of Cons trained Machining Economics Problem by Geometric Programming. ASME Journal of Engin eering for Industry, 1971. 93: p. 1067. 11. Hati, S.K. and Rao, S.S., Determination of Optimum Machining Conditions Deterministic and Probabilistic Approaches. ASME Journal of Engineering for Industry, 1975. 98: p. 354. 12. Hitomi, K., Analysis of Optimal Machining Speeds for Automatic Manufacturing. International Journal of Production Research, 1989. 27: p. 1685. 13. Shalaby, M.A. and Riad, M.S. A Li near Optimization Model for Single Pass Turning Operations. in Proc. 27th Int. MATADOR Conf. 1988. 14. Armarego, E.J.A., Smith, A.J.R., and Wang, J., Constrained Optimization Strategies and Cam So ftware for Single-Pass Peripheral Milling. International Journal of Production Research, 1993. 31(9): p. 2139. 15. Challa, K. and Berra, P.B., Automate d Planning and Optimization of Machining Processes: A Systems Approach. Computers and Industr ial Engineering, 1976. 1: p. 35. 16. Juan, H., Yu, S.F., and Lee, B.Y., The Optimal Cutting-Parameter Selection of Production Cost in HSM for Skd61 Tool Steels. International J ournal of Machine Tools & Manufacture, 2003. 43(7): p. 679. 17. Iwata, K., Murostu, Y., T., I., and Fu jii, S., A Probabilistic Approach to the Determination of the Optimum Cutting. ASME Journal of Engineering for Industry, 1972. 94: p. 1099. 18. Beightler, C.S. and Philips, D.T. Optimization in Tool Engineering Using Geometric Programming. In AIIE Trans. 1970. 19. Gopalakrishnan, B. and Al-Khayyal, F ., Machine Parameter Selection for Turning with Constraints: An Analytical Ap proach Based on Geometric Programming. International Journal of Production Research, 1991. 29: p. 1897. 20. Walvekar, A.G. and Lambert, B.K., An Application of Geometric Programming to Machining Variable Selection. International Journal of Production Research, 1970. 8: p. 241. 21. Tandon, V., El-Mounayri, H., and Kishawy, H., NC End Milling Optimization Using Evolutionary Computation. International Journal of Machine Tools and Manufacture, 2002. 42(5): p. 595.

PAGE 196

179 22. Boothroyd, G. and Rusek, P., Maximu m Rate of Profit Cr iteria in Machining. ASME Journal of Engin eering for Industry, 1976. 98: p. 217. 23. Agapiou, J.S., The Optimization of M achining Operations Based on a Combined Criterion Part I: The Use of Combined Objectives in Single-Pass Operations. ASME Journal of Engin eering for Industry, 1992. 114: p. 500. 24. Jha, N.K., A Discrete Data Base Multiple Objective Optimization of Milling Operation through Ge ometric Programming. ASME Journal of Engineering For Industry, 1990. 112: p. 368. 25. Abuelnaga, A.M. and Eldardiry, M. A., Optimization Methods for Metal-Cutting. International Journal of Mach ine Tools & Manufacture, 1984. 24(1): p. 11. 26. Milner, D.A., Use of Linear-Programming for Machinability Data Optimization. Mechanical Engineering, 1977. 99(7): p. 96. 27. Tae, J.K. and Dong, W.C., Adaptive Optimization of Face Milling Operations Using Neural Networks. J ournal of Manufacturing Sc ience and Engineering, 1998. 120: p. 443. 28. Koulams, C.P., Simultaneous Determin ation of Optimal Machining Conditions and Tool Replacement Policies in Constr ained Machining Economics Problem by Geometric Programming. International Journal of Production Research, 1991. 29(12): p. 2407. 29. Eskicioglu, H., Nisli, M.S., and K ilic, S.E. An Application of Geometric Programming to Single-Pass Turning Operations. in Proceedings of International MTDR Conference. 1985. Birmingham. 30. Petropoulos, P.G., Optimal Selection of Machining Rate Variables by Geometric Programming. International J ournal of Production Research, 1975. 13: p. 390. 31. Hough, C.L. and Goforth, R.E. Quadratic Posylognomials: An Extension of Posynomial Geometric Programming. In AIIE Transactions. 1981. 32. Philipson, R.H. and Ravindran, A., Application of Goal Programming to Machinability Data Optimization. ASME Journal of Mechanical Design, 1978. 100: p. 286. 33. Sundaram, R.M., An Application of Goal Programming Technique in Metal Cutting. International Journal of Production Research, 1978. 16: p. 375. 34. Subbarao, P.C. and Jacobs, C.H. App lication of Nonlinear Goal Programming to Machining Variable Optimization. In Proceedings of NAMRC. 1978.

PAGE 197

180 35. Fang, X.D. and Jawahir, I.S., Predic ting Total Machining Performance in Finish Turning Using Integrated Fuzzy-Set Models of the Machinability Parameters. International Journal of Production Research., 1994. 32: p. 833. 36. Edgeworth, F.Y., Mathematical Psychics, ed. P. Keagan. 1881, London; C.K. Paul. 37. Schwier, A.S., Manual of Political Economy, Transla tion of the French Edition (1927). 1971, London-Basingslohe: Macmillan Press Ltd. 38. Kalyanmoy, D., Multi-Objective Optimization Using Evolutionary Algorithms. 2001, West Sussex, UK: John Wiley. 39. Taylor, F.W., On the Art of Cutting Metals. Transactions of ASME, 1907. 28: p. 31. 40. Tlusty, J., Manufacturing Processes a nd Equipment. 2000, Upper Saddle River, NJ; Prentice Hall. 41. Tlusty, J. and Polacek, M. The Stability of the Machine Tool against Self Excited Vibration in Machining. In Production Engineering Research Conference. 1963. Pittsburgh. 42. Altintas, Y. and Budak, E., Analytical Prediction of Stability Lobes in Milling. CIRP Annals, 1995. 44(1): p. 357. 43. Tlusty, J., Zaton, W., and Ismail, F., Stability Lobes in Milling. CIRP Annals, 1983. 32(1): p. 309. 44. Tlusty, J., Basic Non-Lin earity in Machining Chatter. CIRP Annals, 1981. 30(1): p. 299. 45. DeVor, R.E., Kline, W.A., and Zdeblick, W.J. A Mechanistic Model for the Force System in End Milling with Application to Machining Airframe Structures. In North American Manufacturing Research Conference. 1980. 46. Tlusty, J. and MacNeil, P., Dynami cs of Cutting Forces in End Milling. CIRP Annals, 1975. 24. 47. Kline, W.A., DeVor, R.E., and Shareef J.R., The Prediction of Surface Accuracy in End Milling. ASME Journal of Engineering for Industry, 1982. 104: p. 272. 48. Kline, W.A. and DeVor, R.E., The Effect of Runout on Cutting Geometry and Forces in End Milling. International Journal of Machine Tool Design and Research, 1983. 23(2/3): p. 123.

PAGE 198

181 49. Sutherland, J.W. and Devor, R.E., An Improved Method for Cutting Force and Surface Error Prediction in Fl exible End Milling Systems. Journal of Engineering for Industry-Transactions of the ASME, 1986. 108(4): p. 269. 50. Tlusty, J. and Ismail, F., Special Aspects of Chatter in Milling. Journal of Vibration Acoustics Stress and Reliability in Design-Transactions of the ASME, 1983. 105(1): p. 24. 51. Merdol, S.D. and Altintas, Y., Multi Frequency Solution of Chatter Stability for Low Immersion Milling. Journal of Manufacturing Science and EngineeringTransactions of the ASME, 2004. 126(3): p. 459. 52. Budak, E. and Altintas, Y., Analytical Pr ediction of Chatter Stability in Milling Part 1: General Formulation. Journal of Dynamic Systems Measurement and Control-Transactions of the ASME, 1998. 120(1): p. 22. 53. Budak, E. and Altintas, Y., Analytical Pr ediction of Chatter Stability in Milling Part Ii: Application of the General Formulation to Common Milling Systems. Journal of Dynamic Systems Measurement and Control-Transactions of the ASME, 1998. 120(1): p. 31. 54. Mann, B.P., Dynamics of Milling Proce ss. 2003, Ph.D dissertation, Saint Louis, MO: Washington University. 55. Mann, B.P., Bayly, P.V., Davies, M.A., and Halley, J.E., Limit Cycles, Bifurcations, and Accuracy of the Milling Process. Journal of Sound and Vibration, 2004. 277(1-2): p. 31. 56. Mann, B.P., Insperger, T., Bayly, P.V., and Stepan, G., Stability of up-Milling and Down-Milling, Part 2: Experimental Verification. International Journal of Machine Tools and Manufacture, 2003. 43: p. 35. 57. Insperger, T., Mann, B.P., Stepan, G., a nd Bayly, P.V., Stability of up-Milling and Down-Milling, Part1: Altern ative Analytical Methods. International Journal of Machine Tools and Manufacture, 2003. 43: p. 25. 58. Halley, J.E., Stability of Low Radial Immersion Milling. 1999, M.Sc. thesis, St. Louis, Mo: Washington University. 59. Eschenauer, H.A., Koski, J., and Osyczk a, A., Multicriteria Design Optimization: Procedures and Applications. 1986, New York: Springer-Verlag. 60. Collette, Y. and Siarry, P., Multiobjective Optimization,.ed. D.R. Roy. 2003, Berlin: Springer-Verlag.

PAGE 199

182 61. Deb, K., Multi-Objective Optimizatio n Using Evolutionary Algorithms. 2001, West Sussex, UK: John Wiley. 62. Rakowska, J., Haftka, R.T., and Wats on, L.T., Tracing the Efficient Curve for Multi-Objective Control-Structure Optimization. Computing Systems in Engineering, 1991. 2(5/6): p. 461. 63. Deb, K., Multi-objective ge netic algorithms: Problem di fficulties and construction of test problems. Evolutionary Computational Journal, 1999. 7(3): p. 205. 64. Marler, R.T. and Arora, J.S., Survey of Multi-Objective Optimization Methods for Engineering. Structural Multidisciplinary Optimization, 2004. 26: p. 369. 65. Keeney, R.L. and Raiffa, H., Decisions with Multiple Objectives: Preferences and Value Tradeoff. 1993, New York: Cambri dge University Press Publishing. 66. Rangaswamy, A. and Shell, G.R., Usi ng Computers to Realize Joint Gains in Negotiations: Toward an '' Electronic Bargaining Table. Management Science, 1997. 43(8): p. 1147. 67. Rangaswamy, A. and Lilien, G.L., So ftware Tools for New Product Development. Journal of Marketing Research, 1997. 34(1): p. 177. 68. Coello Coello, C.A., An Updated Survey of G. A. Based Multi-Objective Optimization Techniques. In Technical Report Lania-RD-98-08. 1998, Xalapa, Veracruz, Mexico. 69. Miettinen, K.M., Nonlinear Multiobjective Optimization.1999, Boston: Kluwer Academic Publisher. 70. Hwang, C.-L., Masud, A.S., Paidy, S.R., and Yoon, K., Multiple Objective Decision Making, Methods and Applic ations: A State-of-the-Art Survey. Lecture Notes in Economics and Mathematical Systems, No. 164, ed. M. Beckmann and H.P. Kunzi. 1979, Berlin: Springer Verlag. 71. Haimes, Y.Y., Lasdon, L.S., and Wismer D.A. On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization. In IEEE Transactions Syst. Man Cybern. 1971. 72. Das, I. An Improved Technique for Choosing Parameters for Pareto Surface Generation Using Normal-Boundary Intersection. In ISSMO/UBCAD/AIASA, Third World Congress of Structural and Multidisciplinary Optimization. 1999. Buffalo: University of Buffalo, Center for Advanced Design.

PAGE 200

183 73. Das, I. and Dennis, J.E., Normal -Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems. SIAM Journal of Optimization, 1998. 8(631-657). 74. Messac, A., Ismail-Yahaya, A., and Mattson, C.A., The Normalized Normal Constraint Method for Gene rating the Pareto Frontier. Structural Multidisciplinary Optimization, 2003. 25: p. 86. 75. Salama, M. and Garba, J., Simultaneous Optimization of Controlled Structures. Computational Mechanics, 1988. 3: p. 275. 76. Shin, Y.S., Haftka, R.T., Watson, L. T., and Plaut, R.H., Tracing Structural Optima as a Function of Available Resources by a Homotopy Method. Computational Methods in Applie d Mechanical Engineering, 1988. 70: p. 151. 77. Rakowska, J., Haftka, R.T., and Wa tson, L.T., An Active Set Algorithm for Tracing Parametrized Optima. Structural Optimization, 1991. 3: p. 29. 78. Deb, K., Optimization for Engineeri ng Design: Algorithms and Examples. 1995, New Delhi: Prentice Hall. 79. Hu, X. and Eberhart, R. Solving Constrained Nonlinear Optimization Problems with Particle Swarm Optimization. In 6th World Multiconfer ence on Systemics, Cybernetics and Informatics. 2002. Orlando, USA. 80. Kennedy, J., Minds and Cultures: Partic le Swarm Implications for Beings in Sociocognitive Space. Adaptive Behavior, 1999. 7(3-4): p. 269. 81. Schmitz, T. and Ziegert, J., Examin ation of Surface Location Error Due to Phasing of Cutter Vibrations. Precision Engineering-Journal of the American Society for Precision Engineering, 1999. 23(1): p. 51. 82. Smith, S. and Tlusty, J., An Overview of Modeling and Simulation of the Milling Process. Journal of Engineering for Industr y-Transactions of the ASME, 1991. 113(2): p. 169. 83. Guerra, R.E.H., Schmitt-Braess, G., Habe r, R.H., Alique, A., and Alique, J.R., Using Circle Criteria for Verifying Asympt otic Stability in Pl-Like Fuzzy Control Systems: Application to the Milling Process. Iee Proceedings-Control Theory and Applications, 2003. 150(6): p. 619. 84. Kim, S.I., Landers, R.G., and Ulsoy, A.G., Robust Machining Force Control with Process Compensation. Journal of Manufacturing Science and EngineeringTransactions of the ASME, 2003. 125(3): p. 423.

PAGE 201

184 85. Rober, S.J., Shin, Y.C., and Nwokah, O.D.I., A Digital R obust Controller for Cutting Force Control in the End Milling Process. Journal of Dynamic Systems Measurement and Control-Transactions of the ASME, 1997. 119(2): p. 146. 86. Pandit, S.M., Modal and Spectrum Analysis: Data Dependent Systems in State Space. 1991: John Wiley and Sons Inc. 87. Ewins, D.J., Modal Testing: Theory and Practice. 1984, New York: John Wiley and Sons Inc. 88. Guide to the Expression of Uncertainty in Measurement, in ISO. 1995. 89. McKay, M.D. and Beckmann, R.J ., A Comparison of Three Methods for Selecting Values of Input Variables in th e Analysis of Output from a Computer Code. Technometrics, 1979. 21(2): p. 239. 90. Stein, M., Large Sample Properties of Simulation Using Latin Hypercube Sampling. Technometrics, 1987. 29(2): p. 143. 91. Cheng, J. and Druzdzel, M.J. Latin Hypercube Sampling in Bayesian Networks. In Thirteenth International Florida Artificial Intelligence Research Society. 2000. Menlo Park, CA: AAAI Press. 92. Budak, E., Altintas, Y., and Armare go, E.J.A., Prediction of Milling Force Coefficients from Orthogonal Cutting Data. Journal of Manufacturing Science and Engineering-Transacti ons of the ASME, 1996. 118(2): p. 216. 93. Altintas, Y., Manufacturing Automation : Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design. 2000, Cambridge: Cambridge University Press. 286. 94. Khuri, A.I. and Cornell, J.A., Re sponse Surfaces: Design and Analysis. 1996, New York: Marcel Dekker Inc. 95. Zellner, A., An Efficient Method of Es timating Seemingly Unrelated Regressions and Tests for Aggregation Bias. Journal of American St atistic Association, 1962. 57: p. 348.

PAGE 202

185 BIOGRAPHICAL SKETCH Mohammad Kurdi was born and raised in the suburbs of Amman, Jordan. He finished his B.Sc degree in mechanical e ngineering in 1995 from the University of Jordan. He worked in his dads family opt ometric practice while attending graduate school at the University of Jordan, where he obtained an M.Sc. in mechanical engineering in 1999. After graduate school he joined R oyal Jordanian Airlines as an aircraft maintenance engineer where he worked for one year then he moved to Jordan Petroleum Refinery Co. and worked for 2 years as a development engineer In August 2002 he enrolled in the mechanical engineering gra duate program at the University of Florida where he obtained a Master of Science in December, 2003, and a Ph.D degree in August, 2005.

## Material Information

Title: Robust Multicriteria Optimization of Surface Location Error and Material Removal Rate in High-Speed Milling under Uncertainty
Physical Description: Mixed Material

## Record Information

Source Institution: University of Florida
Holding Location: University of Florida
System ID: UFE0011626:00001

## Material Information

Title: Robust Multicriteria Optimization of Surface Location Error and Material Removal Rate in High-Speed Milling under Uncertainty
Physical Description: Mixed Material

## Record Information

Source Institution: University of Florida
Holding Location: University of Florida
System ID: UFE0011626:00001

Full Text

ROBUST MULTICRITERIA OPTIMIZATION OF SURFACE LOCATION ERROR
AND MATERIAL REMOVAL RATE IN HIGH-SPEED MILLING UNDER
UNCERTAINTY

By

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

2005

by

ACKNOWLEDGMENTS

I would like to thank my advisor Dr. Tony Schmitz for his advice and generous

financial support of my research. I would like to thank Dr. Haftka for his expert advice

and inspiring questions. I would like to thank Dr. Mann for introducing me to the field of

time finite elements. I would like to thank the committee members Dr. Schmitz, Dr.

Haftka, Dr. Mann, Dr. Schuller and Dr. Akcali for their advice, time and effort.

In completing my research I was lucky to be a member of the Machine Tool

Research Center where I had the opportunity to work with intelligent and hard working

graduate students. I would like to thank all fellow members for their helpful suggestions

and interactions. Also, I would like to thank Ms. Christine Schmitz for taking the time to

edit the dissertation draft.

I would like to thank my wife Carolina for her continued support and

encouragement. I would like to thank my daughters Alanis and Alia for bringing laughter

and joy to my life. Finally, I would like to thank my mom and dad for their endless

encouragement and support.

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES ................................ .......... .. .... .. .... .............. viii

LIST OF FIGURES ......... ......................... ...... ........ ............ ix

N O M E N C L A T U R E .................................................. ................................................ xiv

ABSTRACT ........................................................... xvii

CHAPTER

1 IN TR OD U CTION ............................................... .. ......................... ..

Justification of W ork ............................................... ..... ...... .............. .. 1
L literature R review ................. ..... .. .......................... ...... ........ .......... ...... .
O ptim ization in M achining......................................................... ............... 2
H igh-speed M killing O ptim ization...................................... ........................ 3
M ulti-objective O ptim ization ...................................................... ..... .......... 3
Stability and Surface Location Error ............... .............................................4
Scope of W ork ............................................................... ... .... ........ 6

2 MULTI-OBJECTIVE OPTIMIZATION .......................................... ............... 8

Fundamental Concepts in Multi-Objective Optimization ............................................8
Single and Multi-objective Optimization ............................................................8
Definition of Multi-Objective Optimization Problem.............. ...................10
Definition of Term s .............. ..................................................... 10
Pareto O ptim ality................................................................. ......... ......... 11
M ulti-objective Optimization M ethods ................................................................. 12
Methods with a Priori Articulation of Preferences using a Utility Function......13
Weighted global criteria method ............ .............................................14
W eighted sum m ethod............................................ ........... ............... 14
Exponential w weighted criterion ....................................... ............... 15
W eighted product m ethod ................................... ...................................... 15
C onjoint analysis ........ .................... ........................... ..........................15
Methods with a Priori Articulation of Preferences without using a Utility
F u n c tio n ..................................................................................................1 6

v

Lexicographic m ethod ...................................................................... 16
Goal program m ing m ethods......................................................................... 16
Methods for an a Posteriori Articulation of Preferences.............. .....................16
Bounded objective function method ............................ ...............17
Normal boundary intersection (NBI) method ...........................................17
N orm al constraint (N C) m ethod................................................................ 18
Hom otopy m ethod .................. .............. ..................... ..... 18
Choice of Optimization Method ............... .. ..... .........................18

3 MILLING MULTI-OBJECTIVE OPTIMIZATION PROBLEM ...........................20

In tro d u ctio n ........................................................................................................... 2 0
M killing Problem .................................20
Milling ModelEquation Chapter 3 Section 1.................. ............................ 20
Solution M ethod ....................................................... ................. 21
Problem Specifics ............. .................... ........ ... ...................... .. 22
Stability B boundary ............................................ .. ......... .. .... ..... ...... .. 22
Surface location error and stability boundary: C1 discontinuity ..................23
T F E A conv erg en ce ............................................................ .....................24
Optimization Method.................... .... ......... .. ................................. 26
Particle Swarm Optimization Technique.................................. ............... 26
Sequential Quadratic Programming (SQP) ............................... ................27
Problem Form ulation .................. ........................... .... ... ..... ............ 27
P problem Statem ent........... ............................................................ .. .... .. ... .. 27
Tradeoff M ethod .................. ....................................... .......... .... 28
R obu st O ptim ization ......... ........................................................ ...... .... ..... 29
Problem solution ...................... ................ .................. ........ 29
R eform ulation of problem .................................... ..................................... 32
B i-objective space ............ .......... .. .. .... .... ...................................... 37
Selection of spindle speed perturbation bandwidth.............................. 38
Case Studies....................... ..................41
Radial im m version (a)... ........ .................................... .... ........ .......... 41
C h ip lo a d (c ) ............................................................................. 4 2
D isc u ssio n ............................................................................................................. 4 6

4 UNCERTAINTY ANALYSIS ............................................................................47

M killing M odel ................................. .. ............. .............. ........... 48
Stability and Surface Location Error Analysis..................................................50
Bi-section Method Convergence Criterion........................................................ 50
Number of Elem ents .................. ............................. ......... ........ .... 50
N um erical Sensitivity Analysis ............................................................................ 51
Truncation Error ......... ...... .. .......... ................. ............. .... 51
Step S ize ................................................................... 52
C ase Studies................................................................................................................52
Stability Sensitivity A nalysis........................................................... ............... 58
Surface Location Error Sensitivity Analysis ................................... .................61

Uncertainty of Stability Boundary and Surface Location Error .............................63
Input Parameters Correlation Effect ................................ ................63
M onte C arlo Sim ulation ........................................................... .....................64
Sensitivity M ethod................................ ........ .......... .......... ...... ........ 65
Latin Hyper-Cube Sampling M ethod ...................................... ............... 68
Robust Optimization under Uncertainty ................ ..............................................69
D discussion ..................................... .................. ............... ........... 70

5 EXPERIMENTAL RESULTS ............................................................................72

C cutting F orce C oefficients............................................................... .....................72
M illin g F o rc e s ............................................................................................... 7 2
Experim ental Procedure ....................................... .............................. 74
Covariance Matrix (Linear Multi-Response Model)...............................................80
Com pliant Tool M odal Param eters....................................... .......................... 84
Stability Lobe V alidation ............................................................................. 87
Stability Lobe U uncertainty ............................................................................ 87
E xperim ental P procedure ........................................................... .....................90
R e su lts ......................................................................... 9 1
Pareto Front V alidation .................................................... .............93
Pareto Front Sim ulation R esults................................... .................................... 93
Experim ental Procedure and R esults................................................................ 95
C onclu sion s ......................................................................10 1

6 SUM M ARY ............... ............... ......... ................... .......... 103

Robust Optim ization A lgorithm ........................................ .......................... 103
Lim stations and Future Research ........................................................... ... .......... 104

APPENDIX

A TIME FINITE ELEMENT ANALYSIS ...................................... ............... 106

B MATLAB CODE.................. ....................... ......... 115

LIST OF REFEREN CES ........................................................... .. ............... 177

BIOGRAPHICAL SKETCH ............................................................. ............... 185

LIST OF TABLES

Table p

1 Classification of solutions ......................................................... ............... 13

2 Cutting conditions and modal parameters for the tool used in optimization
sim u latio n s ............................................................................................................... 3 0

3 Cutting conditions, modal parameters and cutting force coefficients used in bi-
objective space sim ulations ......................................................... .............. 37

4 Cutting conditions, modal parameters and cutting force coefficients used in
radial im m version case study .............................................................................. 41

5 Milling cutting conditions, modal parameters and cutting force coefficients used
in chip load study case ...................... .. ........................ .... .... ........... 43

6 Cutting force coefficients, modal parameters and cutting conditions of milling
p ro c e s s ........................................................................... 4 9

7 Cutting coefficients for 1 insert endmill for slotting cutting tests..........................77

8 Up milling cutting coefficients for 12% radial immersion ....................................78

9 Estimated cutting force coefficients and their correlation matrix for 7475
aluminum and a 12.7 mm diameter solid carbide endmill with 4 teeth and 30
degree h elix angle................................................ ................. 83

10 Tool modal parameters in x and y-directions. .............. ...... .................. 85

11 Correlation coefficient matrix for modal parameters........................ ...............85

12 Surface location error cutting conditions for two Pareto optimal designs with no
u n certainty con sidered ............................................. ......................................... 100

LIST OF FIGURES

Figure pge

1 (a) Typical Pareto front in the criteria space (b) Design variables xl and x2, and
constraint in the design space .................................. ............... ............... 9

2 Pareto optimality and domination relation. ....................................................... 13

3 Schem atic of 2-DOF milling tool................................ ........................ ......... 20

4 Surface location error and its absolute .......................................... ............... 24

5 A typical stability boundary. ............................................ ............................ 24

6 Convergence of stability constraint for 5% radial immersion and different
spindle speeds for an 18 mm axial depth. ..................................... ............... 25

7 Schematic of milling cutting conditions and various types of milling operations...28

8 Stability, IfSE and fJ contours with optimum points overlaid ............................ 31

9 A typical optimum point found; optimum point sensitivity with respect to
spindle speed is apparent...... ....................................................... ...... .........3 1

10 Perturbed average of fsLE validation as optimization criterion that avoids
spindle speed sensitive fsLE .........................................................33

11 Stability, IfS, and f, contours with optimum Pareto front points found using
PSO and SQP (average perturbed spindle speed formulation)............................ 34

12 Pareto front showing optimum points found using three optimization
algorithms/formulations; the same trends are apparent................ .............. ....35

13 Variations in the eigenvalues, surface location error, and removal rate for PSO
and SQP optima, where f, is the objective for both.............................................36

14 Average surface location error contours for 300 rpm bandwidth perturbation,
stability boundary and material removal rate (see Table 3)..................................38

15 F easib le dom ain .................................................. ................ 39

16 Contour lines corresponding to constant spindle speed in feasible region of bi-
objectiv e sp ace ..................................................... ................ 3 9

17 Average surface location error contours for 100 and 300 rpm band width,
stability boundary and material removal rate contours .........................................40

18 Pareto front for spindle speed and axial depth as design variables with radial
immersion 0.508 mm, compared to the case where radial immersion is added as
a third design variable. ........................................ ........................ 43

19 Pareto front using chip load as a third design variable compared to spindle speed
and axial depth as design variables. .............................................. ............... 44

20 Stability, perturbed average If,,E and f, contours with optimum Pareto front
points found using 100 rpm and 400 rpm bandwidth ...........................................45

21 Schem atic of 2-D m killing m odel. ................. ........................... ........................49

22 The effect of error limit in the bisection method on numerical noise in the
sensitivity calculation ................................. ............ ................. .. ..... 53

23 Sensitivity of SLE with respect to Kx. ..... ............ .. ......... ..................54

24 Comparison between 2nd and 4th order central difference formulas.........................55

25 The logarithmic derivative of axial depth with respect to input parameters versus
step size percentage ....................... ........ ............ ................... .. ......56

26 The variation of axial depth blum with respect to a 10% change in nominal input
p aram eters. ........................................................ ................. 57

27 The variation of b,,m with respect to a 10% change in Kt and K,. The sensitivity
of blum with respect to each parameter is superimposed. Linearity of blm,(Xi) can
be ob served (see T able 6) ................................................................. ............... 57

28 Sensitivity of axial depth blim to changes in modal mass M and modal stiffness
K in the x and y-directions (see Table 6) .... ........... ........ ......................... 58

29 Sensitivity of axial depth blum to changes in modal damping C in the x and y-
d irectio n s ............................................................................. 5 9

30 Sensitivity of axial depth blum to changes in spindle speed. The spindle speed
sensitivity is compared here to the modal mass and stiffness in y-direction...........60

31 Sensitivity of axial depth blum to changes in force cutting coefficients in the
tangential Kt and normal directions K........................................... ...............60

32 Sensitivity of surface location error SLE to changes in modal parameters in y-
d irectio n ............................................... ................. .........................6 1

33 Sensitivity of SLE to cutting force coefficients.................................................. 62

34 Sensitivity of SLE to spindle speed and radial depth of cu ................................62

35 Confidence in stability boundary due to input parameters uncertainties using
M onte C arlo sim ulation .......................... ..................... ................. ............... 65

36 Uncertainty boundary in axial depth limit using two standard deviation
confidence interval. ....................... ...................... ................... .. .....66

37 Uncertainty in axial depth using sensitivity and Monte Carlo methods. ...............67

38 Surface location error uncertainty with two standard deviation confidence
interval on the nominal SLE ........... .. ......... ............................ 68

39 Example simulation of cutting forces to facilitate proper selection of
dy n am o m eter ...................................................... ................ 7 5

40 W ork-piece, dynamometer and tool setup.............................................. .......... 76

41 Cutting coefficient in tangential direction (Kt) ....... ... ....................................... 77

42 Cutting coefficient in normal direction (K,) .......... .............. ........................78

43 Simulated and measured forces for 0.12 mm/tooth chip load and 1000 rpm..........79

44 Simulated and measured cutting forces for 0.2mm/tooth chip load, b=0.4 mm
and 5000 rpm .................................................... ................. 79

45 Simulated and measured forces at 20 krpm and b=0.4 mm for slotting ...................80

46 Modal analysis test equipment typically used in machine tool structures. ..............86

47 Frequency response function measurement of tool in x-direction ...........................86

48 Frequency response function measurement of tool in y-direction .........................87

49 Boxplot of stability lobes boundary uncertainty ............................................... 89

50 Histograms of axial depth limit distributions for various spindle speeds ...............89

51 Probability plot of axial depth limit distribution at 10000 rpm spindle speed.........90

52 Schematic of stability tests for partial radial immersion cutting conditions. ...........91

53 Stability lobe generated using mean values of input parameters with
experimental results overlaid, also shown the boxplot corresponding to each
spindle speed used in the measurements. ...................................... ...............92

54 Fast Fourier Transform (FFT) of sound signals for selected stability tests. ............93

55 Stability boundary using mean values in the input parameters Pareto optimal
designs are overlaid for two cases: mean values and uncertain input parameters. ..94

56 Pareto Front of perturbed average SLE and MRR. The Pareto Front with
uncertainty in axial depth is compared to the one with no uncertainty....................95

57 Surface location error experiment schematic. .................................. ............... 97

58 Measured surface location error of b=2.12 mm and the reference dimension (A)
error. .................................................................................9 8

59 Measured surface location error of b=4.45 mm and the reference dimension (A)
error. .................................................................................9 8

60 Boxplot of SLE uncertainty at spindle speeds for 4.45 mm axial depth case...........99

61 Measured surface location error of b=4.45 mm case .....................................99

62 Surface location error of preferred design conditions with no uncertainty
considered in the optimization. Optimum spindle speeds are indicated in the
figure. ............................................................................ 100

63 Slotting cut with time in the cut divided into two elements................................112

NOMENCLATURE

A slotting transformation matrix
Cx modal damping in x-direction
Cy modal damping in y-direction
E number of elements
F(x) vector of objective functions
F utopia point
F average cutting force
Fx average cutting force in x-direction
Fx average normal cutting force in x-direction
Fxe average normal edge cutting force in x-direction
FY average cutting force in y-direction
F average tangential cutting force in y-direction
Fy average tangential edge cutting force in y-direction
IQ Identity matrix of size Q
K, cutting force coefficients matrix defined in Appendix A
K, tangential cutting force coefficient
Kn normal cutting force coefficient
K, edge tangential cutting force coefficient
K edge normal cutting foce coefficient
Kx modal stiffness in x-direction
Ky modal stiffness in y-direction
L sample size
Mx modal mass in x-direction
My modal mass in y-direction
MRR material removal rate
N number of teeth on the cutting tool
PSO particle swarm optimization

Q number of experimental runs
SLE surface location error
U used for utility function
U, expanded uncertainty
X feasible design space
X(t) two-element position vector
X, milling model ith input parameter
IY vector of observations in ith response
Z feasible criterion space
Z, Q x p, matrix of rankp,
b axial depth of cut
bi set of goals for objective functions
b,, maximum stable axial depth
d deviation from the goals
e MRR constraint limit

QA cutting force coefficients vector defined in Appendix A
fSE surface location error objective function
f material removal rate objective
g, set of inequality constraints
g, absolute value of maximum characteristic multiplier
h, set of equality constraints
h step size used to estimate numerical derivative
p parameter in exponential weighted criterion
p, rank of Z, matrix
r number of response variables
1r correlation coefficient between x and y
uC combined standard uncertainty
Sector of weights (preferences)
x x-direction

x vector of design variables
y y-direction
a parameter used in homotopy method
, vector of unknown constant parameters
/ unbiased estimate of /
3 spindle speed perturbation (half of bandwidth)
E absolute error limit
E, random error vector associated with ith response
A variance-covariance matrix
A system characteristic multipliers
S radial depth angle at start of cut
ex radial depth angle at end of cut
Q spindle speed
Y covariance matrix
0o- i, j element of covariance matrix Y
Oyj estimate of o-
r tooth passing period
1 damping factor

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

ROBUST MULTICRITERIA OPTIMIZATION OF SURFACE LOCATION ERROR
AND MATERIAL REMOVAL RATE IN HIGH-SPEED MILLING UNDER
UNCERTAINTY

By

August 2005

Chair: Tony L. Schmitz
Cochair: Raphael T. Haftka
Major Department: Mechanical and Aerospace Engineering

High-speed milling (HSM) provides an efficient method for accurate discrete part

fabrication. However, successful implementation requires the selection of appropriate

operating parameters. Balancing the multiple process requirements, including high

material removal rate, maximum part accuracy, chatter avoidance, and adequate surface

finish, to arrive at an optimum solution is difficult without the aid of an optimization

framework.

Despite the attractive gain in productivity that HSM offers, full realization of the

benefits is dependent on the proper selection of cutting parameters. Parameters selected

must achieve the required productivity while maintaining an acceptable accuracy. Milling

models are used to aid in the proper selection of these cutting parameters. They provide

information on whether a cutting condition is stable and/or predict the surface accuracy.

However, this selection is rather tedious, costly and time consuming and might not even

provide an optimum solution. Parameters are selected based on experience until a point is

found that provide the productivity and surface accuracy required. Difficulties

encountered in this selection process include sensitivity of surface accuracy to cutting

parameters, uncertainties in several parameters in the milling model and the

computational effort needed to account for stability and surface accuracy. Therefore,

balancing the multiple requirements, including high material removal rate, minimum

surface location error and chatter avoidance, to arrive at an optimum solution is difficult

without the aid of optimization techniques.

In this dissertation a robust optimization algorithm that accounts for the inherent

process uncertainty and surface location error sensitivity is developed. Two optimization

criteria are considered, namely, surface location error and material removal rate under the

stability constraint. The trade off curve of surface location error versus material removal

rate is calculated for the mean values of input parameters, as well as for a confidence

level in the stability boundary. An experimental validation of the robust optimization

algorithm is also conducted, including an experimental validation of the variation of the

cutting forces as a function of spindle speed. The confidence level in the axial depth limit

and surface location error prediction is found using two methods: 1) sensitivity analysis;

and 2) sampling methods. The sensitivity study highlights the most significant factors

affecting process stability and surface location error. The effect of input parameters

correlation is included in the confidence level predictions using Monte Carlo and Latin

Hyper-Cube sampling methods.

xviii

CHAPTER 1
INTRODUCTION

Justification of Work

Intense competition in manufacturing places a continuous demand on developing

cost-effective manufacturing processes with acceptable dimensional accuracy. High-

speed milling, HSM, offers these benefits provided appropriate operating parameters are

selected. Some typical applications include, but are not limited to, orthopedic surgery [1],

end milling (pocketing) of airframe panels [2] and ball end milling of stamping dies [3, 4]

in automotive manufacturing.

Despite the attractive gain in productivity that HSM offers, full realization of the

benefits is dependent on the proper selection of cutting parameters. Parameters selected

must achieve the required productivity while maintaining an acceptable accuracy. Milling

models are used to aid in the proper selection of these cutting parameters. They give us

information on whether a cutting condition is stable and/or they predict the surface

accuracy. However, this selection is rather tedious, costly and time consuming and might

not even provide an optimum solution. Parameters are selected based on experience until

a point is found that provides the productivity and surface accuracy required. Difficulties

encountered in this selection process include sensitivity of surface accuracy to cutting

parameters, uncertainties in several parameters in the milling model and the

computational effort needed to account for stability and surface accuracy. Therefore,

balancing the multiple requirements, including high material removal rate, f ,

minimum surface location error fSLE and chatter avoidance, to arrive at an optimum

solution, is difficult without the aid of optimization techniques.

Literature Review

The literature review proceeds with a summary of previous implementations of

optimization methods in machining, with particular attention to high-speed milling and

multi-objective optimization. Also, a review of milling models for stability and surface

location error is provided.

Optimization in Machining

Previous research in machining process optimization [5] has focused on

mathematical modeling approaches to determine optimal cutting parameters with regard

to various objective functions. Three main objectives have been recognized: 1) maximum

production rate or minimum cycle time [6-9]; 2) minimum cost [10-21]; and 3) maximum

profit [12, 22], or a combined criterion based on a weighted sum of these [23, 24].

The machining optimization problem can be formulated using deterministic and

probabilistic approaches [11, 25]. Several optimization techniques were used to handle

both formulations. For the deterministic approach they include linear and nonlinear

programming techniques [9, 15, 26, 27], while for the probabilistic approach chance-

constrained programming can be used [17, 28]. Other optimization techniques used in

machining include graphical optimization [12, 22], polynomial geometric programming

[6, 18-20, 29, 30], geometric programming [10] based on quadratic posylognomials

(QPL) [31], goal programming with linear [32, 33] and nonlinear [34] goals, fuzzy

optimization [35], and global search methods such as particle swarm optimization [21]

and simulated annealing [16].

The machining optimization literature can also be classified according to different

constraints and design variables handled. Several authors [7, 14] considered cost

optimization for single-pass milling and turning [10, 17, 19, 20, 29]. The range of

constraints considered are machine tool constraints, such as cutting speed and feed rate,

tool dynamics constraints such as cutting force, power and stability, and product

constraints such as surface roughness. In reference, [17] some of the constraints

considered are of probabilistic nature. Also, multi-pass peripheral and end milling to

maximize production rate are considered [8] under a range of constraints with relevance

to rough milling such as the machine tool limiting power, torque, feed force and feed-

speed boundaries while in another work. In addition to the previous constraints, arbor

rigidity and deflection are used [6].

High-speed Milling Optimization

Few references are found on optimization of high-speed milling. The concept of

adaptive learning (polynomial network) [16] is used to construct a machining model.

Simulated annealing was then used to minimize production cost for rough high-speed

machining operations for three cutting condition parameters namely cutting speed, chip

load and axial depth of cut. A similar study was done for low speed milling [21] where an

artificial neural network was used to build the machining model. However, particle

swarm optimization was used to optimize production cost under machine, tool and

product constraints.

Multi-objective Optimization

Multi-objective optimization (MOO) addresses the issue of competing objectives

using concepts first introduced by Edgeworth [36] then expanded and developed by

Pareto [37], the French-Italian economist who established an optimality concept in the

field of economics based on multiple objectives. A Pareto front [38] is generated that

allows designers to trade off one objective against another.

In the area of machining, Jha [24] studied two objective function optimization

based on cost and rate of production where example constraints were machine power,

cutting speed limitations, depth of cut, and table feed. The two objectives were combined

using weights. Koulams [28] studied single-pass machining considering the influence of

tool chatter failure where a tool failure probability function effect was added as a penalty

cost function to the objective function.

Stability and Surface Location Error

As explained earlier, the full exploitation of HSM demands mathematical models to

predict stability and surface location error. An unstable milling process is caused by a

phenomenon called chatter. Among the first to describe chatter is Taylor, [39] who

described chatter as "the most obscure and delicate of all problems facing the machinist."

Chatter [40] is a self-excited vibration that occurs if the chip width is too large with

respect to the dynamic stiffness of the system. It causes undulations in the machined

surface (poor surface finish) and could result in tool breakage. Extensive work has been

done to generate stability boundaries or lobes. The lobes define a region below which

chatter is nonexistent. Two approaches are used to generate these lobes: 1) analytical [41]

with a continuous cutting model or with an interrupted cutting model [42]; and 2) time

domain simulation [43, 44].

Surface location error is defined as the error in the placement of the milling cutter

teeth when the surface is generated. This error depends on the interaction of work-

piece/tool dynamic stiffness and the cutting forces. The correct prediction of this error

depends on correct prediction of the cutting forces and resulting deflections. Mechanistic

models can be used to estimate these forces. The cutting force is found by summing the

forces acting on incremental sections of a helical cutting edge [45, 46], then the surface

location error is computed based on the static stiffness of the tool [47]. However, the

effect of the deflection of the cutter on the cutting forces is not included. In an

improvement of the previous model, the static deflection is fed back to correct the cutting

forces [48, 49]. A more realistic regenerative force model [50] considered the effect of

undulations in the surface generated by previous tooth passage on the next tooth passage.

In this model the dynamic deflection of the tool imprints waviness on the generated

surface. Using time domain simulation, surface location error, cutting forces and stability

lobes are predicted. An improvement on this model considered [42, 51-53] interrupted

cutting as a factor influencing the stability lobes and surface location error. A newly

developed method uses time finite element analysis (TFEA) to model the governing time

delayed differential equation [54-58]. Regenerative cutting forces and dynamic deflection

of the tool are all implicitly included in the governing differential equation. The

advantage of this method is that it concurrently provides surface location error and

stability information on the milling process in a semi-analytical manner. In this method

the governing differential equation is modeled by dividing the time in the cut into a

number of elements, where displacement and velocity continuity are enforced between

each element. A discrete linear map is formed by mapping the time in the cut to free

vibration. The eigenvalues of the discrete map determine the stability boundaries,

whereas fixed points of the dynamic map determine surface location error (fsLE)

Scope of Work

The purpose of this dissertation is to use optimization as a tool to efficiently

determine preferred and robust operating conditions in HSM, considering multiple

objectives. Although known optimization methods and machining models will be applied,

there are a number of innovative aspects of this research. First, proper formulation of the

objective functions to account for practical application of the preferred conditions is

necessary. The formulation should account for uncertainty in the milling model and

sensitivity of objectives) to process variables. Uncertainty has not previously been

considered. Second, two objectives are simultaneously optimized: surface location error

fSLE and material removal rate, f, Stability and side bounds of design variables are

considered as constraints. Prior research has focused only on the empirical tool life, not

the unavoidable milling dynamics and the inherent limitations they impose. The tradeoff

curve (Pareto front) [38, 59] of fA, and fsLE is generated based on nominal

experimental model parameters. Experimental case studies are conducted to verify the

validity of the Pareto front. The uncertainty in the milling model is addressed using

Monte Carlo simulation and/or sensitivity analysis, where a confidence interval is applied

to the stability limit. The uncertainty of different input parameters such as cutting force

coefficients, tool/work-piece dynamic parameters and milling process parameters are

considered in the uncertainty prediction. This uncertainty is used in the selection of a

robust design that would allow a venue for the practical application of the stability lobe

theory at the shop floor.

7

The dissertation organization proceeds as follows: Chapter 2 gives a general

description of multi-objective optimization; Chapter 3 describes Pareto front generation

formulation of the optimization problem, optimization methods and case studies; Chapter

4 provides the uncertainty analysis of stability and surface location error; Chapter 5

describes the robust optimization algorithm and presents some practical case studies to

verify stability lobes and selected design points on the Pareto front. Chapter 6

summarized the results and outlines future work in this area.

CHAPTER 2
MULTI-OBJECTIVE OPTIMIZATION

Fundamental Concepts in Multi-Objective Optimization

Optimization is an engineering discipline where extreme values of design criteria

are sought. However, quite often there are multiple conflicting criteria that need to be

handled. Satisfying one of these criteria comes at the expense of another. Multi-objective

optimization deals with such conflicting objectives. It provides a mathematical

framework to arrive at an optimal design state which accommodates the various criteria

demanded by the application.

This chapter begins with a comparison of single- and multiple-objective

optimization. Next, the definition of the multi-objective optimization problem and terms

are explained. Then, a summary of multi-objective optimization methods is presented.

Finally, reasons are given for the choice of the multi-objective optimization method.

Single and Multi-objective Optimization

In single objective optimization one is faced with the problem of finding the

optimum of the objective function. For example considering the decision making

involved in an investment (Figure 1). There are several possible designs in the feasible

domain (A, B and 1-6). These designs are mapped from the design space Figure 1 (b) into

the criteria space Figure 1 (a). In the design space there are two design variables xl

(spindle speed) and x2 (axial depth) where the feasible domain is limited by the

constraint. If we are only concerned about profit with no regard to risk (profit is our

single objective), then point B would correspond to the maximum profit optimum design.

A risk averse investor would choose risk as an objective function. The optimum design

for the risk objective would correspond to point A. Depending on the objective function,

constraints, and design variables, different techniques are used to solve for the single-

objective optimum. However, in multi-objective optimization, a vector of objectives

needs to be optimized. For the investment example, two objectives are considered. In this

two objective case there is no unique optimum, rather a set of optimum solutions is

found. In Figure 1, for instance, points A, B and 5-6 are all candidate solutions.

Depending on the decision maker's risk aversion, a single solution can be chosen from

that set.

B
90% -. B constraint

design space
Feasible domain
3
*4 6 x2

2 t
S 5 B
4 0
criteria space

10%- A 1 A Feasible domain

$1000 Profit$5000 x

(a) (b)
Figure 1. (a) Typical Pareto front in the criteria space (b) Design variables x] and x2, and
constraint in the design space.

The similarity between single- and multi-objective optimization makes it possible

to use the same optimization algorithms as for the single-objective case. The only

required modification is to transform the multi-objective problem into a single one. This

may be accomplished in a number of ways, such as introducing a vector of preferences,

v, to get a single objective as a weighted sum, or by solving one of the objectives for a

different set of limits on the other objectives [60-62]. In any case, a set of optimal

solutions are found rather than a single one. It is worth noting, however, that when the

objective functions are non-conflicting, the optimal set reduces to a single solution rather

than a set. This can be related to the commodity example. For instance, if we want to

maximize both cost and quality, then solution B is the only one.

Definition of Multi-Objective Optimization Problem

The mathematical representation of the multi-objective optimization problem is

formulated as follows:

Minimize F(X) = (k), F (5),...,Fk()1]
subject to gJ (,)<0, j = 1,2,..., (2.1)
h, ()= 0, =1,2,...,e

where subscript k denotes the number of objective functions F, m is the number of

inequality constraints and e is the number of equality constraints; and x e E" is the

vector of design variables, where n is the number of independent design variables.

Definition of Terms

The feasible design space (inference space), X, is defined as the set of design

variables that satisfy the constraint set, or

{gx,()<0, j=1,2,...,m; andh(, )=0, =l1,2,...,e (2.2)

Thefeasible criterion space, Z, (often called the cost space or attainable set) is

defined as the set of cost functions F () such that V maps to a point in the feasible

design space X or (F ) Y G X .

The preferences refer to the decision maker's opinion in terms of points in the

criterion space. The preferences can be set apriori (before solution set is obtained) or a

posteriori (after solution set is obtained).

The preference function is an abstract function of points in the criterion space

which perfectly satisfies the decision maker's preferences.

The utilityfunction is an amalgamation of individual utility functions of each

objective that approximates the preference function, which typically cannot be expressed

in mathematical form. The formation of a utility function requires insight into the

physical aspects of each objective. This may require finding the Pareto front (explained

next) in order to properly formulate the utility function.

A utopia point is a point F0 e Zk that satisfies F = minimum{F () i e X} for

each i = 1,2..., k.

Pareto Optimality

The multi-objective optimization problem has more than one global optimum. The

predominant concept in defining an optimal point is that of Pareto optimality [37] which

is defined as follows: a point, x*" X, is Pareto optimal if there does not exist another

point, x e X, such that F(x)
That is the set of Pareto optimal points dominates any other optimal set. This can be

defined by the domination relation [60], where a vector x, dominates a vector x2 if: l is

at least as good as j2 for all the objectives, and 2, is strictly better than j2 for at least

one objective. To better understand the domination relation, or Pareto optimality, an

example is provided [63] (Figure 2). A two-objective problem of maximizingfi and

minimizing2 is addressed. Table 1 presents the set of solutions, classified with respect to

each other. A solution P is designated as +,- or = depending on whether it is better, worse

or equal to a solution Q for the corresponding objective. For example, comparing

solutions A and B, we find that solution A is worse forfi (maximizingfi) compared to B,

therefore it is designated as (-) for objectivefi. Also, comparing objective2 we find that

solution A is worse than B (-). Now for a solution to belong to the non-dominated set it

must be as good as the other solutions for both objectives and it must be strictly better for

at least one objective. Considering solution A in Figure 2 we see it is worse than all other

solutions (dominated); solution B is also worse than C for both objectives (dominated).

Solution C is not dominated by point E (couple (+,-) at the intersection of the row E and

the column C) and it does not dominate point E (couple (-,+) at the intersection of the row

C and the column E), therefore points C and E are non-dominated. Solution D is worse

than C for both objectives therefore solution D is dominated.

Multi-objective Optimization Methods

As explained earlier the solution to a multi-objective optimization problem is a

Pareto optimal set that gives a tradeoff between the different objective functions

considered. Depending on the decision maker's preferences, a solution is selected from

that set. Therefore multi-objective optimization methods can be categorized according to

how the designer articulates his preferences (by order or by importance of objectives).

This includes three cases: apriori, aposteriori, and progressive articulation of

13

preferences. A brief overview of the methods used is outlined. For a detailed description

of the methods the reader is referred to reference 64.

Table 1. Classification of solutions

Solutions A B C D E
A (-,-) (-,-) (-,-) (-, -)
B (+,+) (-,-) (-,=) (-,=)
c (+,+) (+,+) (+,+) (-,+)
D (++) (+,=) (-,=)
E (+,+) (+,=) (+,-) (+,=)

B D

10 12 14 16

Figure 2. Pareto optimality and domination relation.

Methods with a Priori Articulation of Preferences using a Utility Function

In these methods, the decision maker's preferences are incorporated as parameters

in terms of a utility function apriori. Typically these parameters can be coefficients,

exponents, constraint limits, etc. These parameters determine the tradeoff of objectives

before implementation of the optimization method. The optimum solution found would

reflect the tradeoff made a priori. Depending on whether the solution found turned out to

satisfy the preferences or not, the decision maker can re-adjust the parameters to get a

better solution. However the beauty of these methods is that they do not require doing a

multi-objective optimization problem since the a priori preferences and utility function

reduce the optimization to a single one.

Weighted global criteria method

In this method, all objective functions are combined to form a single utility

function. The weighted global criterion is a type of utility function U in which

parameters are used to model preferences. The simplest form of a general utility function

can be defined as

k
U= w,, (x))P, F, (x)> 0Vi, or (2.3)
i=1

U [w (x)]P, F (x)> Vi, (2.4)
i=1

where i5 is the vector of weights set by the decision maker such that 4i > 0 and

w = 1. The difference between the two above formulations is related to conditions

required for Pareto optimality. Complete discussion can be found in reference [64].

Weighted sum method

This is a special case of the weighted global criteria method in which the exponent

P is equal to one; that is,

k
U Zw, (x). (2.5)
i=1

The method is easy to implement and guarantees finding the Pareto optimal set,

provided the objective function space is convex. However, a uniformly distributed set of

weights does not necessarily find a uniformly distributed Pareto optimal set, which makes

it difficult to obtain a Pareto solution in a desired region of the objective space.

Exponential weighted criterion

It is defined as follows:

U (eP' -1)ePF(x), (2.6)
i=1

where the argument of the summation represents an individual utility function for F( (x).

Weighted product method

To avoid transforming objective functions with similar significance and different

order of magnitude, one may consider the following formulation [65]:

k w
U= J[F (x)] (2.7)
1=1

where w, are weights indicating the relative significance of the objective functions.

Conjoint analysis

This method [66, 67] uses a concept borrowed from marketing, where a product is

characterized by a set of attributes, with each attribute having a set of levels. An

aggregated utility function is developed by direct interaction with the customer/designer;

the designer is asked to rate, rank order, or choose a set of product bundles. In

engineering design studies, we can assume that people will choose their most preferred

product alternative. Conjoint analysis takes these sets of attributes and converts them into

a utility function that specifies the preferences that the customer has for all of the

product's attributes and attribute levels. The advantage of this method is that it

automatically takes into account marginal diminishing utility (i.e., no cost is expended in

a design that does not really have practical utility).

Methods with a Priori Articulation of Preferences without using a Utility Function

Lexicographic method

Here the objective functions are arranged in a descending order of importance [68].

The highest preference objective is optimized with no regard to the other objectives, and

then a single objective problem is solved consecutively (in order of preference of

objectives) for a set of limits on the optimums of previously solved for objectives. This

can be defined as

Minimize F (x)

subject to F (x) 1, (2.8)
i = 1,2,...,k

where i represents the function's position in the preferred sequence and FI (xi)

represents the optimum of the jth objective function found in the jth iteration.

Goal programming methods

Here, goals bi are specified for each objective function FJ (.) [69]. Then the total

k
deviation from the goals, \d is optimized, where d is the deviation from the goal
J=1

bJ for thejth objective.

Methods for an a Posteriori Articulation of Preferences

The inability of the decision maker to set preferences a priori in terms of a utility

function makes it necessary to generate a Pareto optimal set after which an aposteriori

articulation of preferences is made; such methods are sometimes referred to as cafeteria

or generate-first-choose-later. These methods however require the generation of the

Pareto optimal set which may be prohibitively time consuming. It is worth noting that

repeatedly solving the weighted sum approaches presented earlier can be used to find the

entire Pareto optimal solution for convex criteria space; however, these methods fail to

provide an even distribution of points that can accurately represent the Pareto optimal set.

Bounded objective function method

In this method [70], the single most important objective function, Fg (.), is

minimized, while all other objective functions are added as constraints with lower and

upper bounds such that / I, F (2) < c,, i = 1,2,...,k,i # s. A variation of this method is

thee constraint [71] or trade-off method in which the lower bound 1 is excluded and

the Pareto optimal set is obtained using a systematic variation of S,. This method is

particularly useful in finding the Pareto optimal solution for convex or non-convex

objective spaces alike. However, choice of the constraint vector s must lie within the

minimum and maximum of the objective function considered; otherwise, no feasible

solution will be found. Also the distribution of the Pareto optimal solution will usually be

non-uniform for the objective functions) minimized.

Normal boundary intersection (NBI) method

This method provides a means for obtaining an even distribution of Pareto optimal

points for a consistent variation in parameter vector of weights [72, 73], even with a non-

convex Pareto optimal set (a deficiency found in weighted sum method). For each

parameter weight the NBI problem is solved to find an optimum point that intersects the

criteria feasible space boundary, however, for non-convex problems, some of the

solutions found can be non Pareto optimal. Details of the method can be found in the

references.

Normal constraint (NC) method

This method uses normalized objective functions with a Pareto filter to eliminate

non-Pareto optimal solutions [74]. The individual minima of the normalized objective

functions are used to construct the vertices of the utopia hyper-plane. A sample of evenly

distributed points on the utopia hyper-plane is found from a linear combination of the

vertices with consistently varied weights in criterion space. Each Pareto optimal point is

found by solving a separate normalized single-objective function with additional

inequality constraints for the remaining normalized objective functions.

Homotopy method

In this method the convex combination of bi-objective functions (1- a)f + af, is

optimized for an initial value of the parameters Then homotopy curve tracking methods

are used to generate the Pareto optimal solution curve for a e [0,1] whenever the curve is

smooth [75, 76] or even non-smooth [62, 77] at points corresponding to changes in the

set of active constraints.

Choice of Optimization Method

The ease of implementation of the c -constraint method [71] for a bi-objective

problem makes it a good candidate method. In this method, one of the objectives is

optimized for systematic variation of limits (e, ,...,, ) on the second objective. A

uniform distribution of the Pareto optimal set can be found for the constrained objective.

There is no limitation on the convexity or non-convexity of the objective space in finding

the Pareto optimal set. However, choice of the constraint set of limits (e, e..., ) must

lie within the minimum and maximum of the objective function considered; otherwise, no

feasible solution would be found. In our case the material removal rate (fm) and

19

surface location error ( fLE) are the bi-objective criteria. The material removal rate

objective would be a better choice for the constrained objective, since the set of limits

(E, ..., ) of f, constraint can be more easily constructed according to designer's

preference, whereas that would be difficult for the fsLE objective.

CHAPTER 3
MILLING MULTI-OBJECTIVE OPTIMIZATION PROBLEM

Introduction

In this chapter, a description of the milling problem and solution method used to

solve the mathematical model is presented. Two optimization methods of interest are

briefly described. These methods are then applied to the multi-objective optimization

problem and a discussion of results is provided.

Milling Problem

Milling Model

The schematic for a two degree-of-freedom (2-DOF) milling process is shown in

Figure 3 (repeated here). With the assumption of either a compliant tool or a structure, a

summation of forces gives the following equation of motion:

Figure 3. Schematic of 2-DOF milling tool

mX 0 1x(t) c 0 xt) kX 0 x(t)= iXQ) (3.1)
0 my y(t) 0 Cy y(t) 0 ky y(t) Fy(t) I

where the terms mx, cx, kx and my, cy, k are the modal mass, viscous damping, and

stiffness terms, and F, and Fy are the cutting forces in the x and y directions, respectively.

A compact form of the milling process can be found by considering the chip thickness

variation and forces on each tooth (a detailed derivation is provided in references [54-58]

and Appendix A):

MX(t) + C(t) + KX(t) = Kc (t)b (t) (t r))+ (t) b (3.2)

where X(t)=[x(t) y(t)]T is the two-element position vector and M, C, and K are the

2x2 modal mass, damping, and stiffness matrices, Kc and f0 (function of the cutting

force coefficients) are defined in Appendix A, b is the axial depth of cut, r = 60/(NQ) is

the tooth passing period in seconds, Q is the spindle speed given in rev/min (rpm), and N

is the number of teeth on the cutting tool. As shown in Eq. (3.2), the milling model is

dependent on modal parameters of the tool/work-piece combination and the cutting force

coefficients.

Solution Method

As described in Chapter 1, a solution of Eq. (3.2) can be completed using numerical

time-domain simulation [43, 44, 50] or the semi-analytical TFEA [54-58]. Compared to

the first approach, TFEA can obtain rapid process performance calculations of surface

location error, fL,, and stability. The computational efficiency of TFEA compared to

conventional time-domain simulation methods makes it the most attractive candidate for

use in the optimization formulation. In this method a discrete linear map is generated that

relates the vibration while the tool is in the cut to free vibration out of the cut. Stability of

the milling process can be determined using the eigenvalues of the dynamic map, while

surface location error (see Appendix A) is found from the fixed points of the dynamic

map. Details can be found in references [54-58]. An added advantage of TFEA is that it

provides a clear and distinct definition of stability boundaries (i.e., eigenvalues of the

milling equation with an absolute value greater than one identify unstable conditions).

Problem Specifics

In this section, the calculation of the stability boundary is analyzed, the continuity

of surface location error and stability boundary is addressed, TFEA convergence is

described, and sensitivity of the milling model to cutting force coefficients is defined.

Stability Boundary

In order to find the axial depth limit, bl,,, of neutral stability at corresponding input

parameters, the bi-section method is used in the TFEA algorithm to solve for bm,, at

which the maximum characteristic multiplier is equal to one (stability limit)

gA =max < 1 (3.3)

where I is the eigenvalues of the dynamic map. An absolute error is used as a criterion

for convergence

b, < (3.4)
b

where E corresponds to the error tolerance and b is the root corresponding to

max = 1 at iteration i. The value of E is set based on the numerical accuracy required

in the calculation of b,,,. A value of E = le 3 can be adequate for the calculation of bl,,.

Surface location error and stability boundary: C1 discontinuity

Correct use of an optimization method depends on its limitations. Gradient-based

methods, for example, depend on C1 continuity (the first derivative of the function is

continuous) of the objective functions ( f and IfSL ) and stability constraint (Eq. (3.3)).

The objective f, is defined analytically in Eq. (3.6), where it is clear that it is C1

continuous. However, the IfSL and stability (g,) functions are only found numerically

using TFEA. A graphical description of both functions provides some insight into the

continuity of these functions. Figure 4 depicts the variation of fSL and IfS,, as a

function of spindle speed for a typical set of cutting parameters. Although fIS is C1

continuous in the region where it is defined (stable region), IfSLE is C1 discontinuous.

This can be easily verified analytically by considering the functions f(x) = x and

f (x) = Ix =x { for x > 0 and -x for x <0}. The absolute function is clearly C1 discontinuous

at x = 0. The same argument can be made for the near-zero IfSL range shown in Figure 4.

In Figure 5 the variation of stability function g, versus spindle speed shows lobe peaks

where C1 (slope) discontinuity of g, is also observed. C1 discontinuity makes

convergence of gradient-based optimization algorithms near the discontinuity rather

difficult. This requires the use of multiple initial guesses in order to converge to even a

local optimum.

2
1
0

10 12 14 16 18 20
S ( x31epr

Figure 4. Surface location error and its absolute. Discontinuity of the absolute surface
location error is apparent in the lower insert.

3.5

3

2.5

25

1.5

0.5
1 1.1 1.2 1.3 1.4 1.5
W(x 1 arpm)
Figure 5. A typical stability boundary. The cusps where C' discontinuity in the stability
boundary are depicted.

TFEA convergence

The convergence of TFEA depends on the cutting parameters. A higher number of

elements must be used when convergence is not achieved. Either If, or the stability

boundary g, can be used to check for convergence. A typical procedure to test for the

convergence of finite element meshes is to compare the change in the estimated value (g

or ISE I) as the number of elements is increased (mesh refinement). In Figure 6, the

dependence of convergence on the spindle speed is shown for a randomly selected cutting

condition of 5% radial immersion (percentage of radial depth of cut to tool diameter) and

18 mm axial depth. As seen in Figure 6, the flawed convergence for a small number of

elements (=1) would give the impression of a sufficient number of elements. However,

further increasing the number of elements (=12) shows poor convergence for the low

speed. This can be due to the fact that as the spindle speed decreases, the time in the cut

increases, which requires a higher number of elements to achieve convergence. The fact

that the optimization algorithm will pick milling parameters within the design space

makes it necessary to choose a rather high number of elements to ensure convergence

anywhere in the design space. However, a penalty in computational time is incurred.

40 50
500 (rpm)
1000 40
20 30
30

10

-20
0 10 20 30 40 2 4 6
No. of elements No. of elements
4
10 3 x10

8
2 /

S 4 \

2
0 0
20 25 30 35 40 10 15 20
No. of elements No. of elements
Figure 6. Convergence of stability constraint for 5% radial immersion and different spindle
speeds for an 18 mm axial depth. We can see that convergence at lower speeds
require substantially more number of elements.

Optimization Method

Optimization methods can be categorized according to the searching method used

to find the optimum [78]. They are either direct where only the values of the objective

function and constraints are used to guide the search strategy, or gradient-based, where

first and/or second order derivatives guide the search process. Particle swarm

optimization (PSO) and sequential quadratic programming (SQP) will be used to test the

feasibility of both methods, respectively, for the problem at hand.

Particle Swarm Optimization Technique

Particle swarm optimization is an evolutionary computation technique developed

by Kennedy and Eberhart [79, 80]. It can be used for solving single or multi-objective

optimization problems. To find the optimum solution, a swarm of particles explores the

feasible design space. Each particle keeps track of its own personal best (pbest) fitness

and the global best (gbest) fitness achieved during design space exploration. The velocity

of each particle is updated toward its pbest and the gbest positions. Acceleration is

weighted by a random term, with separate random numbers being generated for

acceleration toward pbest and gbest.

In order to accommodate constraints, Xiaohui et al. [79] presented a modified

particle swarm optimization algorithm, where PSO is started with a group of feasible

solutions and a feasibility function is used to check if the newly explored solutions satisfy

all the constraints. All the particles keep only those feasible solutions in their memory

The basic idea of this method is that it transforms the nonlinear optimization

problem into a quadratic sub-problem around the initial guess. The nonlinear objective

function and constraints are transformed into their quadratic and linear approximations.

The quadratic problem is then solved iteratively and the step size is found by minimizing

a descent function along the search direction. Standard optimization algorithms may be

used to solve the quadratic sub-problem.

Usually SQP leads to identification of only local optima. In order to better

converge to the global optimum, a number of initial guesses is used to scan the design

space and the optimum of these local optima is close to the global optimum.

Problem Formulation

In this section, the multi-objective optimization problem is defined and then a

description of the tradeoff method is given. The problem solution is then presented in the

order it has been addressed in the robust optimization section. Finally, discussion of the

simulation results is provided.

Problem Statement

The problem of minimizing surface location error If, I and maximizing material

removal rate f, is stated as follows:

min[ fsE (a, b, c, N, ),- fI f(a,b, c, N, ()],
(3.5)
subject to: g (b,Q) = max i(a,b,N,Q) <1

where g, is the stability constraint obtained from the dynamic map eigenvalues, fs, is

found from the fixed points, and the mean f, is given as:

fm = abcNQ, (3.6)

where a, b, c, N and 2 are radial depth of cut, axial depth of cut, feed per tooth (chip

load), number of teeth, and spindle speed, respectively (Figure 7). From Eq. (3.5) it can

be seen that only the stability constraint is not a function of the feed per tooth. In Eq.(3.5)

, fAE and f, are explicitly stated as a function of cutting conditions (a, b, c, N and ( ).

This reflects the relative ease by which these conditions can be adjusted to achieve

optimality of the objectives.

N=2 of cut, a Axi
Axial
depth,

Y Ost=O L y Pst=0 "y

x ex x
a R
I ex= 7 | ex I
Slotting Up milling Down milling
Figure 7. Schematic of milling cutting conditions and various types of milling operations.

To address the multi-objective problem the constraint method is used, where the

two-objective problem is transformed into a single objective problem of minimizing one

objective with a set of different limits on the second objective. Each time the single

objective problem is solved, the second objective is constrained to a specific value until a

sufficient set of optimum points are found. These are used to generate the Pareto front

[38] of the two objectives. In the case that f,, is chosen as the objective function to be

minimized then Eq. (3.5) is transformed to:

min fE{ (a, b, c, N,),
subject to: f, (a,b,c,N, ) ) (3.7)
gp (a,b,N,Q ) = maxl (a,b,N, Q) < 1,
for a series of k selected limits (e) on f,,.

Where the cutting conditions: a, b, c, N and n are the design variables. On the other

hand if f, is chosen as the objective function to be maximized, then Eq. (3.5) is

transformed to:

min f (a,b,c, N, ),
subject to : If, (a,b,c,N, Q) e,, for i = 1 ... k
(3.8)
gp (a,b,N, )= maxl(a,b,N, ) < 1,
for a series of k selected limits (e) on I f I.

It should be noted that applying Eq. (3.7) using the SQP method is more

straightforward than Eq. (3.8). The reason is that in order to use a number of initial

guesses along the Ifs, contour in Eq. (3.8), the axial depth corresponding to that IfL

needs to be found, whereas in Eq. (3.7) the axial depth can be explicitly expressed found

as shown in Eq. (3.6).

Robust Optimization

Problem solution

In the first iteration of the problem, only axial depth (b) and spindle speed (n2) are

considered as design variables. Other cutting conditions are held fixed (Table 2) for a

down milling cut. Modal parameters for a single degree-of-freedom tool with one

dynamic mode in x andy directions are used (Table 2). The nominal values of the

tangential (Kt) and normal (K,) cutting force coefficients are 550 N/mm2 and 200 N/mm2,

respectively. The SQP method is used to find the Pareto front using the formulation in

Eq. (3.7). Here Ifs I is minimized for a set of limits on f As mentioned earlier, the

SQP method is a local search method that is highly dependent on C1 continuity of the

objective function and constraints. To obtain a global optimum, a number of initial

guesses are used along each f, constraint limit. A set of optimum points are obtained

for these initial guesses. The minimum of these optimum points is nominated as a global

optimum. The number of initial guesses is increased and another run of the optimization

simulation is made to check the validity of that global optimum.

Table 2. Cutting conditions and modal parameters for Tool used in optimization
simulations

M (kg) C (Ns/m) K (N/m)
0.056 0 3.94 0 1.52x106 0
0 0.061 0 3.86 0 1.67x106
Tool diameter (mm) c (mm) a (mm) N
19.05 0.178 0.76 2
Kt (N/m2) Kn (N/m2) Kte (N/m) Kne (N/m)
550 x 106 200 x 106 0 0

In this formulation, the minimum f,, I points were found to favor spindle speeds

where the tooth passing frequency is equal to an integer fraction of the system's natural

frequency (Figure 8), which corresponds to the most flexible mode (these are the

traditionally-selected 'best' speeds which are located near the lobe peaks in stability lobe

diagrams). Because ,,L can undergo large changes in value for small perturbations in 2

at these optimum points, the formulation provided in Eqs. (3.7) and (3.8) leads to optima

which are highly sensitive to spindle speed variation (Figure 8) To show the sensitivity

of these optimum points, a typical optimum point is superimposed on a graph of fs,E Ivs.

2 in Figure 9. It is seen that the optimum point is located in a high ,,L slope region.

IF i

i*I
Ii

ii;

!I I

iii I
!i
!! i'
II i!
*uJ 1

I"i
i

i

!.F-
I
r
i I

I 3
I
i jj9

5 10 15 20 25 30
Q (x 103 rpm)

Figure 8. Stability, IfS, L and fJ contours with optimum points overlaid. The figure
shows that optimum points occur in regions sensitive to spindle speed variation
(Table 2).

Surface location error
-- Spindle speed sensitive optimum

4'

4 6 8 10 12 14
Q2 (x 13 rpm)
Figure 9. A typical optimum point found; optimum point sensitivity with respect to
spindle speed is apparent (Table 2).

Reformulation of problem

The optimization problem was redefined in order to avoid convergence to spindle

speed-sensitive optima. Two approaches were applied: 1) an additional constraint was

added to the IfS slope; and 2) the f,, objective was redefined as the average of three

perturbed spindle speeds. The latter proved to be more robust than the former. This is due

to the difficulty in setting the value of the If I slope constraint apriori. The spindle

speed perturbed form of the problem transforms Eqs. (3.7) and (3.8) to

in fLE (b, Q +3) + fsLE (b, Q) + fsLE (b, Q )|
3
subject to: f (b, ) < e,, for i = 1... k (3.9)
(g, (b, 6)n g (b,i)n g, (b,+ )} < 1,
for a series of selected limits (e) on f,

and

min f, (b, ),
S rSLE (b, Q + 3) + \fSLE (b, 4) + SLE ](b,Q1 S)
subject to: fSLE (bQ+-) Q) l fSLE' Q- 5 C, for i = 1...k
3
(g, (b, 6)n g, (b, )n g, (b, + )} < 1,
for a series of selected limits (s) on average perturbed Ifs ,

(3.10)

where 5is the spindle speed perturbation selected by the designer (a typical value for our

analyses was 50 rpm). A study of spindle speed perturbation selection is provided in the

next section.

The validity of the perturbed IfsL, average as a convergence criteria can be seen in

Figure 10. In this figure the perturbed average Ifs, is plotted with fsE where points A

and B correspond to highly and moderately spindle speed-sensitive fs,L respectively.

The average perturbed If, I at point A (high slope point) is shown to be higher than at

point B. Therefore, using the perturbed average |, I as an objective function criteria can

avoid convergence to spindle speed sensitive fE (such as IfsE region near point A).

Perturbed average fSLE
IfsLEI
---------------I-----
1.2

5" 0.8

around point A relatively
more sensitive fSLE| region
A
0.6

0.4

0.2
10.5 11 11.5 12 12.5 13 13.5 14
C (x 103 rpm)

Figure 10. Perturbed average of fSE validation as optimization criterion that avoids

spindle speed sensitive fSLE Shown in the figure are points A (close to steep

slope region of fSLE ) and B (close to moderate slope region of fSLE ), the

perturbed average of fsLE near A is higher than at B. Therefore, using the

perturbed average as an optimum criterion is valid.

The SQP method is used to solve Eqs. (3.9) and (3.10). In case Eq. (3.9) is

implemented then initial guesses of 2 and b (design variables) are made along the f,

contour. In the other case (Eq. (3.10)) the initial guesses of 2 and b are made along If,,

contour. The number of initial guesses along the constraint is made such that convergence

is towards a global optimum. The initial guesses for the spindle speed are increased in

625 rpm increments for the corresponding spindle speed range considered. Also, the PSO

method is used to solve Eq. (3.8). When using PSO, the optimum points do not tend to

converge to spindle speed sensitive optimums. Therefore, there is no need to solve the

reformulated form of the problem in PSO. This leads to a fewer number of evaluations of

IfsLE I and is a computationally more efficient optimization method.

A comparison of the three optimization schemes is shown in Figure 11 and Figure

12. Figure 11 shows the optima for each approach superimposed on the corresponding

stability lobe diagram. In Figure 12, the Pareto fronts for the three methods are shown.

The optimum points found using the two SQP formulations closely agree with the PSO

method (Figure 12).

5 eL bI\\\ b' iJr3ly" r
I':.. p \. \ SQP SLE objective
"" \ SQPMRRobjectve
4.5 0- PSO MRR objective

14 16 1 20 22 24 26 2 30
35
7

10

so -- 7. 71.-.

14 16 18 20 22 24 26 28 \$0
0 (x 1 rpm)

Figure 11. Stability, IfS| and f, contours with optimum Pareto front points found
using PSO and SQP (average perturbed spindle speed formulation). The figure
shows that optimum points are not in regions sensitive to spindle speed (Table 2).

20
S SQP SLE objective ? /
18 SQP MRR objective
1- PSO MRR objective
16

14,

12

10

f 6-

4-

2
0

50 100 150 200 250 300 350 400 450
fM,, mm /s)
Figure 12. Pareto front showing optimum points found using three optimization
algorithms/formulations; the same trends are apparent. However, the SQP
methods required additional computational time (Table 2).
Although the PSO points show the same trend, some improvement in the fitness is

still possible relative to the SQP results. Because the PSO search inherently avoided

optimum points that are spindle speed insensitive, there is no need to use average

perturbedfsLE as with SQP, which leads to a decreased number of fLE evaluations in

PSO. However, narrow optimum points may go undetected when using PSO.

As noted, when comparing the Pareto fronts in Figure 12, it is seen that the PSO

approach did not converge to the same fitness as SQP method. A check of the optimum

points which correspond to a value of If,,LE I = 4 rim, for example, shows that PSO

converged to 100 mm3/s, while SQP converged to 150 mm3/s. To better understand this

result, the design space was divided between the two design vectors, b and Q, for SQP

and PSO using a factor, a, that was normalized between 0 and 1. The PSO and SQP

optimums were normalized to a = 0 and 1, respectively. Next, the stability constraint

(g,), f., and IfS E were plotted against that ratio. In Figure 13 it is seen that

discontinuities exist in the fs, I constraint and the first derivative of the eigenvalue

constraint within this region. Although PSO is not significantly affected by a

discontinuity in the derivative constraint, it can be affected by a discontinuity of the fsA,

constraint, where the discontinuity tends to narrow the search region of the swarm.

1.1 PSO optimum at 4tm -
O SQP optimum 41tm
o

0.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
a
150
0 PSO optimum at 4pm
S100 O SQP optimum 41m

50 -

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
a
160
0 PSO optimum at 41tm
140 SQP optimum 41tm

I 120 -
1200
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
a

Figure 13. Variations in the eigenvalues, surface location error, and removal rate for PSO
and SQP optima, where f, is the objective for both. The discontinuities in the surface
location error cause PSO to not converge on the SQP optimum.

Bi-objective space

In this section, the bi-objective domain (the feasible space of the objective

functions) of average perturbed If, I and f, for the set of input parameters listed in

Table 3 for an up milling case is provided. Figure 14 shows the objective contours in the

design space of spindle speed (n2) and axial depth (b). The respective bi-objective space

is shown in Figure 15 and Figure 16. In Figure 15 the contours of constant axial depth are

shown, while the contours of constant spindle speeds are shown in Figure 16. These

figures give an idea of the feasible design and bi-objective space. It can be seen that the

bi-objective feasible space can be non-convex (not all points on a straight line connecting

two points in the feasible domain belong to that domain). This makes the choice of using

the tradeoff method as a multi-objective optimization approach a suitable one, since this

method can handle both convex and non-convex problems. A good observation can be

made from Figure 15, where it can be seen that for the high f, region with high b

values, the relative sensitivity of If, increases compared to the lower f, region.

Table 3. Cutting conditions, modal parameters and cutting force coefficients used in bi-
objective space simulations

M (kg) C (Ns/m) K (N/m)
0.44 0 83 0 4.45x106 0
0 0.35 0 90 0 3.55x106
Tool diameter (mm) c (mm) a (mm) N
25.4 0.1 21.8 1
Kt (N/m2) Kn (N/m2) Kte (N/m) Kne (N/m)
700 x 106 20 x 106 46 x 10 33 x 10

Selection of spindle speed perturbation bandwidth

In Figure 10, it was shown that the average perturbation of IfS provided an

adequate optimization criteria. However, the choice of the spindle speed perturbation step

size or bandwidth, 2,, depends on the designer preference. Any spindle speed

perturbation in If, would avoid convergence to sensitive If, optima. Depending on

the machining center spindle drive accuracy, the perturbation bandwidth can be set

accordingly. The average perturbed If,, contours of 100 and 300 rpm bandwidth are

shown in Figure 17 (use Table 3 parameters). The high slope region of average If, I in

the 100 rpm bandwidth case is replaced by higher values of average If, I, making the

optimization formulation favor insensitive spindle speed If

C.) Stability boundary
5.5 200 f mm /S
5 2C00 perturbed average |fsLE| m
4.5 00

%07 /h h :ei \ d,
S 3 100 \ n SLE :l: ul.

2.5 \ 200
\900 1000 4 C
2 \

1.5 700 0
1 z/ 35 "

10 12 14 16 18 20
S(x 10 rpm)
Figure 14. Average surface location error contours for 300 rpm bandwidth perturbation,
stability boundary and material removal rate (see Table 3).

160

,140

-120
Cl
S100

80 / Feasible domain

B 60

o 40

20
20 TPareto front

500 1000 1500 2000 2500
fMRR(mm3/s)
Figure 15. Feasible domain. Contour lines corresponding to constant axial depth in the
stable region in the bi-objective space (see Table 3).

120

100 -

80

60-

40-

20-

1-.2 ,, pK0'.....)
500 1000 1500 2000
fMR (mm )
Figure 16. Contour lines corresponding to constant spindle
objective space (see Table 3).

19

18

17

16

15

14

13

12

11

2500

speed in feasible region of bi-

100 rpm bandwidth

i
3

2

1
10

12 14 16 18
Q (x 103 rpm)

300 rpm bandwidth

12 14 16 18
Q (x 103 rpm)

Figure 17. Average surface location error contours for 100 and 300 rpm band width,
stability boundary and material removal rate contours (see Table 3).

5

4

E3

2

1

Case Studies

As opposed to the previous analysis of two design variables (2 and b), two cases

of an added third design variable were analyzed. The first one was for radial immersion

(a) and the second one was for chip load (c). These cases are compared to the two design

variable case.

Previous simulations considered spindle speed and axial depth of cut as design

variables. Another simulation was completed using radial immersion as a third design

variable for an up milling cut. It was compared to a two design variable case where radial

immersion was held constant at 0.508 mm in a 25.4 mm tool (Table 4). Figure 18 shows

the Pareto front for these two cases. It is seen that adding radial immersion as a third

design variable improved the value of perturbed average If,, with respect to the constant

radial immersion case. The optimum radial immersion found was 0.58 mm for all

optimum points up to 500 mm3/s. In both simulations the same spindle speed perturbation

(6 = 170 rpm) was used. As seen in Figure 18, a better calculation of the Pareto front

(smoother than Figure 12) is found by using small increments in the spindle speed (each

100 rpm) initial guesses. However, the If,, found in Figure 18 appear to be

unrealistically small which may warrant further analysis.

Table 4. Cutting conditions, modal parameters and cutting force coefficients used in
M (kg) C (Ns/m) K (N/m)
0.25 0 34.4 0 1.30 x106 0
0 0.23 0 27.0 0 1.20 x106
Tool diameter (mm) c (mm) a (mm) N
25.4 0.1 0.508 2
Kt(N/m) (N/m)) Kte (N/) K(Nm) Kne (N/m)
700 x 106 210 x 106 0 0

To study the effect of chip load on surface location error, it is added as a third

design variable in addition to spindle speed and axial depth. The parameters used in this

study are listed in Table 5 for a down milling case. For the two design variable case (0.1

mm/tooth chip load), the Pareto optimal points are found for two different bandwidths,

100 rpm and 400 rpm, respectively (Figure 20). It is noted that 2 of the optimum points

is almost constant up to 700 mm3/s ( 31,325 rpm) where it changes to another almost

constant 2 (29,500 rpm) for the higher f, range. Also the effect of bandwidth size

does not show significant effect on the optimum points found. Figure 19 shows the Pareto

front for a constant chip load of 0.1 mm/tooth compared to the three design variable case,

where the chip load (3rd design variable) side constraints are from 0.01 mm/tooth to 0.2

mm/tooth. An improvement in the average perturbed If,,E can be seen. It should be noted

here that for the latter case, 2 is also found to be constant (31,325 rpm same as two

design variable case) while the chip load increased from 0.16 to 0.2 mm/tooth. The effect

of adding the chip load is therefore seen as an improvement in the fitness of average

perturbed If,, objective, where further improvement is possible while eliminating the

need to switch to a lower speed (29,500 rpm) where the If, S error is much higher. This

explains the agreement between the two design variables case and three design variable

case in the f, range below 600 mm3/s. When higher f, is needed the two design

variable case fails to account for the f, constraint at the same spindle speed. However

the three design variable case (with chip load) can accommodate this by increasing the

chip load while keeping the spindle speed unchanged. This makes the I| in the three

design variable case substantially lower.

0.02

S0.015

S 0.01

0 100 200 300
fR (mm3s)

400 500 600

Figure 18. Pareto front for spindle speed and axial depth as design variables with radial
immersion 0.508 mm, compared to the case where radial immersion is added as a
third design variable. The optimum radial immersion for the latter case is 0.58
mm up to 500 mm3/s (see Table 4).

Table 5. Milling cutting conditions, modal parameters and cutting force coefficients used

M (kg) C (Ns/m) K (N/m)
0.027 0 7 0 1.0x106 0
0 0.03 0 2 0 1.6x106
Tool diameter (mm) c (mm) a (mm) N
12.7 0.1 0.635 2
Kt (N/m2) Kn (N/m2) Kte (N/m) Kne (N/m)
600 x 106 180 x 106 0 0

c=0.16 to 0.2 mm/tooth

200

400

Figure 19. Pareto front using chip load as a third design variable compared to spindle
speed and axial depth as design variables. For the three design variable case, an
improvement in the average surface location error can be seen (see Table 5).

600

f (mm 3s
MRR

800

1000

~c~s~

^-^-"

16 -0- Optimum points 100 rpm bandwidth

14

12

6
4lO

10 15 20 25 30 35 40
2 (rpm)

Figure 20. Stability, perturbed average If,,, and f, contours with optimum Pareto front points found using 100 rpm and 400 rpm

bandwidth. This case study shows the difficulty in selecting optimum points based on experience (Table 5).

Discussion

The formulations provided in Eqs. (3.9) and (3.10) proved adequate in finding the

Pareto optimal set insensitive to spindle speed variation, provided an appropriate number

of initial guesses is made. Also, the Eq. (3.9) formulation is easier to apply using the SQP

method, where the initial guesses are made along the f, contour.

The generation of the Pareto front for the multi-design variable case can be rather

time-consuming. However, if the designer is given that freedom of choice, it might be a

design variable gave a substantial improvement in the surface location error in

comparison to the two design variable case. This is counterintuitive to using a lower

value of c or a as means of reducing the surface location error.

The effect of spindle speed perturbation bandwidth on the sensitivity of optimum

points is rather complex. Qualitatively, in Figure 17 it is shown that increasing the

bandwidth from 100 rpm to 300 rpm had the same effect of increasing the value of IfS,,L

near the sensitive region. Further investigation is needed to establish a quantitative

relation between bandwidth and sensitivity of optimum points.

CHAPTER 4
UNCERTAINTY ANALYSIS

In Chapter 3, optimization was used to find preferable designs for two objectives:

material removal rate (MRR) and surface location error [48, 81, 82] (SLE), with a Pareto

front, or tradeoff curve, found for the two competing objectives. Although the milling

model used in the optimization algorithm is deterministic (time finite element analysis),

uncertainties in the input parameters to the model limit the confidence in the optimum

predictions. These input parameters include cutting force coefficients (material- and

process-dependent), tool modal parameters, and cutting conditions. By accounting for

these uncertainties it is possible to arrive at a robust optimum operating condition.

In previous studies [83-85], uncertainty in the milling process was handled from a

control perspective. The uncertainty in the cutting force was accommodated using a

control system. The force controller was designed to compensate for known process

effects and accounted for the force-feed nonlinearity inherent in metal cutting operations.

In this study, the uncertainties in the milling model are estimated using sensitivity

analysis and Monte Carlo simulation. This enables selection of a preferred design that

takes into account the inherent uncertainty in the model a priori.

This chapter begins with a description of the milling model and continues with a

discussion of stability lobes and surface location error analysis with regard to their

numerical accuracy. Sensitivity analysis is discussed in the next section. Then, case

studies for the numerical accuracy of the sensitivities of the maximum stable axial depth,

blm, and SLE are presented for a typical two degree-of-freedom tool. This enables us to

carry out the stability lobe and surface location error sensitivity analysis in the next two

sections. Sensitivity is used to determine the effect of input parameters on b,,m and SLE.

This enables the determination of which parameters) is the highest contributor to

stability enhancement and SLE reduction. The uncertainties in bm, and SLE predictions

are then calculated using two methods 1) the Monte Carlo simulation; and 2) the use of

numerical derivatives of the system characteristic multipliers to determine sensitivities.

The uncertainty in axial depth effects a reduction in the MRR, and the SLE uncertainty

provides bounds on SLE mean expected value. This allows robust optimization that takes

into consideration both performance and uncertainty.

Milling Model

A schematic of a two degree-of-freedom milling tool is shown in Figure 21. The

tool/work-piece dynamics and cutting forces are used to formulate the governing delay

differential equation for the system. Solution of the delay differential equation is found

using time finite element analysis (TFEA) [54-56]. This method provides the means for

predicting the milling process stability and quality (SLE). However, the uncertainty in the

input parameters to the solution method places an uncertainty on the stability and SLE

prediction. These parameters are divided into two groups; 1) uncertainty from lack of

knowledge of the tool modal matrices, K, C and M, and the cutting force coefficients

(mechanistic force model); and 2) uncertainty in other machining parameters, such as

spindle speed, chip load and radial depth. To estimate the parameters in the former,

modal testing is used to measure the dynamic parameters while cutting tests are

completed to estimate the cutting force coefficients. In the modal parameter estimation

the peak amplitude method is used to fit the measured frequency response function. In

this method [86, 87], the peak of the magnitude of the frequency response function

corresponds to the natural frequency. From this the half power frequencies are used to

estimate the damping ratio. Table 6 lists the mean modal values for 25.4 mm diameter

endmill having a 12 helix angle with 114 mm overhang length and the corresponding

cutting force coefficients for 6061 aluminum (assuming a mechanistic force model, see

Chapter 5). The cutting conditions are also listed in the table. These parameters will be

used in the simulations in this chapter for a down milling cut.

Feed
----------------

Figure 21. Schematic of 2-D milling model. Surface location error (SLE) due to phasing
between cutting force and tool displacement is also shown.

Table 6. Cutting force coefficients, modal parameters and cutting conditions of milling
process.

M 1. K(. I x10 6) C(N.s/m) i
x 0.44 4.45 83 0.030
Y 0.44 3.55 90.9 0.03
K,(. m2 x106) K ,(. 2 x106) K,,( mx103) K,(. mx103)
600 180 6 12
25.4 0.508 0.1 1

Stability and Surface Location Error Analysis

The stability lobes are used to represent the stable space of axial depth (b) and

spindle speed 12 of a milling process. In TFEA [54-57], a discrete map is used to match

the tool-free vibration while out of the cut, with the tool vibration in the cut. The system

characteristic multipliers (A ) of the map provide the stable cutting zone where max A

is less than one.

TFEA provides a field of max A in the design space of b and Q. The limit of

stability, blm can be found using root-finding numerical techniques. Here we use the bi-

section root-finding method. The convergence criterion of the bi-section method should

account for the amplification of numerical noise induced by sensitivity estimation. It

should be noted that the number of elements affects the accuracy of the estimation.

For calculation of SLE in TFEA, the numerical noise is only due to the number of

elements. In this section we will discuss the effect of both the convergence criterion and

the number of elements on the sensitivity estimation of bl.m and SLE.

Bi-section Method Convergence Criterion

As described in Chapter 3 the axial depth limit, blim, was calculated using the bi-

section method (Eq. (3.4)). Although a relatively large value of E can be adequate for the

calculation of the stability lobes, a tighter limit is needed to calculate the sensitivities.

This is attributed to amplification of numerical noise in the derivative calculation. This

comparison is made in the Case Studies section.

Number of Elements

The accuracy of TFEA prediction of stability and SLE is highly dependent on the

number of elements used. The effect of the number of elements is even more apparent

when calculating the sensitivity of the prediction, where a higher number of elements is

needed to eliminate numerical noise from the sensitivity calculation.

Numerical Sensitivity Analysis

The sensitivity of axial depth to input parameters (Sb / axi) is cumbersome to

compute analytically using the TFEA method; therefore, a numerical derivative is used

by implementing a small perturbation.

Factors which affect accurate calculation of sensitivity to inputs include: 1) central

difference truncation error; and 2) step size selection. Therefore, a balance needs to be

achieved in determining the sensitivity that provides a stable estimate of the sensitivity

while maintaining computational efficiency. In the following, we describe these factors

and their consideration in the calculation of stability and SLE sensitivities.

Truncation Error

The central difference method is used in the sensitivity calculation. The formula for

this method is

Bb b1- b
b +0(h2), (4.1)
OXA 2h

where h denotes the step size in input parameter X,, b b (X, + h) b = b(X, h) and

O(h2) is the 2nd order truncation error. A higher order formula with 4th order truncation

error O(h4) can also be used. However, as shown in Eq. (4.2), it is two times more

computationally expensive than Eq. (4.1),

b -b2 +8b, -8b +b
X, 12h ) (4.2)
8 X 12h

In order to help decide whether the higher truncation error formula need be applied

(Eq. (4.2)), the sensitivity of b,,m with respect to modal stiffness Kx is calculated as a

function of step size h. This comparison is made in the Case Studies section.

Step Size

The step size, h, in Eqs. ((4.1) and (4.2)) should be carefully chosen. This is

especially important when there is numerical noise in the calculated bum, due to the

convergence criterion (Eq. (1)). The step size should be large enough to be out of the

numerical noise range, however, not so large that the non-linear variation in the output

(b,,m or SLE) takes effect. The following section illustrates this idea.

Case Studies

In this section, numerical estimations of the sensitivity are made based on different

variations of convergence criterion, number of elements, sensitivity analysis formula (Eq.

(4.1) and Eq. (4.2)), and step size. The comparisons are made for a 10 krpm spindle

speed, 10 elements and e =3x10-4 unless otherwise noted. The logarithmic derivative can

be used in making these comparisons by evaluating the percentage of change in an output

(axial depth, b) due to a percentage change in the input, X,. It is expressed as

aln((b) _X Qb
(4.3)
aln(X) b c8X

To illustrate the effect of convergence criterion, the logarithmic derivative of blum

with respect to Mx (the X direction modal mass) is calculated for two error limits as a

function of step size percentage (/oh = AX, IX, x 100), see Figure 22. It can be seen that a

tighter error limit nearly eliminates the numerical noise in the derivative calculation.

The effect of the number of elements on SLE sensitivity is illustrated in Figure 23,

where the SLE sensitivity with respect to Kx is calculated. The OSLE / Kx is used to

illustrate the effect of the number of elements because it is known that the SLE does not

depend on the Kx stiffness (tool feeding direction being the x-axis). Therefore

8SLE / 8Kx = 0, which would amplify and illustrate more clearly the effect of the number

of elements on the sensitivity estimation. The higher number of elements provides a

larger stable region of sensitivity. It should be noted that the 2nd order finite difference

method is used in this sensitivity comparison and the bi-section convergence criterion is

not applicable here since SLE is found from fixed points of the dynamic map (see Eq.

(A.18) in Appendix A) when the cutting conditions provide a stable cut.

2nd order central difference = 3e-4
-0.2 E=10 = 3e-10

-0.4

-0.6

-0.8 -

-1.6

-1.8

2 i I I I
0 0.1 0.2 0.3 0.4 0.5
%h
Figure 22. The effect of error limit in the bisection method on numerical noise in the
sensitivity calculation (see Table 6).

54

-23
x10
14
1-- E=10
12 ---- E=30 -
--E=50
10

8

6

84-'i

%h
-2

-4-

-6 I I I I
0 0.1 012 03 0 .4 0.5

Figure 23. Sensitivity of SLE with respect to Kx. The higher number of elements, E,
provides more stable sensitivity estimation. The second order finite difference
formula is used here (see Table 6).

Figure 24 shows the effect of the central difference truncation error. A finite step

size percentage is needed to reach a stable value of the derivative for both formulas. It

can be seen that Eq. (4.2) gives a wider range of step sizes at which the sensitivity

calculation is stable. However, the improved stability range, or reduction in numerical

noise, is not significant to sacrifice computational efficiency for its usage.

-9
x 10
2.5,
2nd Order central difference
4th order central difference
2 Convergence limit e= 3e-4
E=10

1.5

O1 i i i

0.5

0 0.1 0.2 0.3 0.4 0.5
%h
Figure 24. Comparison between 2nd and 4th order central difference formulas. The 4th
order formula shows a wider stable region for step size, but higher computation
time (see Table 6).

The importance of step size selection can be illustrated by Figure 25, which shows

the logarithmic derivative of axial depth with respect to input parameters versus step size

percentage. It can be seen that the step size should be chosen high enough to be out of the

numerical noise range but not so high so that the non-linear variation is included (in this

range of %h only D is non-linear). The figure also indicates the relative sensitivity of

axial depth to each input parameter, spindle speed having the largest effect followed by

modal mass and stiffness.

1-

K2 C- -M K- Kn

-4 i\
^ ~~ ---

sensiiviesi i ver step size o see Table 6).

s = 3 x 10-4 E=10, and the 2nd order finite difference approximation give correct

calculation of sensitivity, the variations of b to modal parameters and cutting coefficients

are plotted in Figure 26 and Figure 27, respectively. Also, the slope predicted using Eq.
-3-

(4.1) with h 0.2% is superimposed on the same plot. The suitable selection ofh is

indicated by the tangency of the predicted slope to the functional variation. On the other

hand, it can be seen that when the variation is linear, the linear approximation can be

accurate for a large variation of the input parameter.
accurate for a large variation of the input parameter.

K
y
C
y
M
y
Sensitivity Prediction

5
-c

/7
^ -'-

//8r
^^*-Q^^

100
AX/X

Figure 26. The variation of axial depth b1lm with respect to a 10% change in nominal input
parameters. The sensitivity of blm with respect to each parameter is superimposed.
Linearity and non-linearity of bin(X,) can be observed (see Table 6).

4.8

Sensitivity Prediction

90 95 100 105 110
AXl/Xi (x 100)

Figure 27. The variation of blm with respect to a 10% change in Kt and K,. The sensitivity
of bl1; with respect to each parameter is superimposed. Linearity of bl,(Xi) can be
observed (see Table 6).

58

Stability Sensitivity Analysis

In this section, calculations of the sensitivity of bm,, to the input parameters are

provided. The parameters used in the sensitivity calculations are provided in Table 7. In

Figure 28 a comparison between the sensitivities of stiffness, K, and modal mass, M, are

compared in the x (feed) and y-directions of the tool. As can be seen in the figure, the

sensitivities in the x and y-directions are comparable in magnitude; however, the

sensitivity in they-direction is inaccurate near discontinuities in the system characteristic

multipliers. This will be explained in the Uncertainty section with a graphic depicting

these discontinuities.

Table 7. Parameters used in sensitivity analysis.

h (%) E C
0.2 10 2

20

15 -

10 -

X- 5'

-5

-10 -

-15

-20
5

central difference
nd order

Figure 28. Sensitivity of axial depth blim to changes in modal mass M and modal
stiffness K in the x and y-directions (see Table 6).

8

3x10 '

bh

S(rpm x103)
- Ky -

Mx My

I
:I n

'
iI

In Figure 29, the effect of damping on the stability is shown to be minimal

compared to the modal stiffness and mass. This is a somewhat counter-intuitive result,

but can be explained by regeneration (undulations in the cut surface experienced by the

tooth in the current cut that are caused by the tooth vibration in the previous cut), which

is a primary physical phenomenon that causes instability. The modal mass and stiffness

have a great effect on the system's natural frequency, which has a significant effect on

regeneration. This also explains the result shown in Figure 30, where the sensitivity of

axial depth bl,m to a change in spindle speed is significant and comparable to modal mass

and stiffness. The effect of cutting force coefficients is shown in Figure 31, where the

tangential cutting force coefficient, Kt, has more effect on the axial depth limit than the

normal direction coefficient, K,.

30
K
C
X
20 C

10 -

0_ ------

-10 \ t

-20- C1 discontinuities in bhmM

-30
5 10 15 20
0 (x 103)
Figure 29. Sensitivity of axial depth bl,m to changes in modal damping C in the x and y-
directions. The damping sensitivity is compared to modal stiffness sensitivity in
the x-direction (see Table 6).

-150

K ]C1 discontinuities in blm
-K
y

-200 _____ M
y

-250
5 10 15 20
Q (rpm x03)
Figure 30. Sensitivity of axial depth bl,m to changes in spindle speed. The spindle speed
sensitivity is compared here to the modal mass and stiffness in y-direction (see
Table 6).

U.4
0.4-----------------------------------------------
--... Kt
0.2 K

0-

-0.2

S-0.4

%
-0.6

S-0.8 Jk h I. ..

-1.2

-1.4
5 10 15 20

Q (rpm x03)
Figure 31. Sensitivity of axial depth bl,m to changes in force cutting coefficients in the
tangential Kt and normal directions K,. Higher sensitivity can be seen for Kt (see
Table 6).

Surface Location Error Sensitivity Analysis

The sensitivity of surface location error, SLE, to changes in input parameters is

examined here. The parameters listed in Table 6 are used with b=l mm and down milling

case. In Figure 32, the sensitivity of SLE to changes in modal parameters in they-

direction is shown. Again, it can be seen that changes in Ky and My contribute more than

Cy to a change in SLE. In Figure 33, the effect of cutting force coefficients is shown,

where it is observed that the highest contributors to SLE sensitivity are K, and Kte. Also,

in Figure 34, SLE sensitivity to spindle speed and radial depth, rtep, is shown. Substantial

sensitivity to spindle speed can be seen. This is due to the dependence of SLE on the

relationship between the tool point frequency response and the selected spindle speed. As

the spindle speed changes, it tracks different parts of the response.

250
K
Y
lM Y
200- C

150

100 l

50

5 10 15 20
Q (rpm x 103)
Figure 32. Sensitivity of surface location error SLE to changes in modal parameters in y-
direction (see Table 6).

15

K
ne

Q (rpm x103)
Figure 33. Sensitivity of SLE to cutting force coefficients (see Table 6).

500

400

step

300

200

100

Q (x 103)
Figure 34. Sensitivity of SLE to spindle speed and radial depth of cut (see Table 6).

Uncertainty of Stability Boundary and Surface Location Error

Input Parameters Correlation Effect

The correlation between the input parameters can have significant effect on the

prediction of uncertainty. Neglecting the correlation can result in erroneous estimation of

the uncertainty, especially when the input parameters are highly correlated. Inclusion of

the covariance matrix between parameters is necessary in this case. The input parameters

can be classified into three groups: dynamic modal parameters of the tool (work-piece

assumed rigid), cutting force coefficients and machining parameters (e.g., radial step and

spindle speed). In Chapter 5, estimation of the correlation between parameters of the first

two groups is explained and used in the uncertainty prediction.

The combined standard uncertainty uc can be found using sensitivities of output

(b,,m or SLE) to input parameters. For the case of axial depth limit, uc is given as [88]:

1m m 1m 1m1 OA
uc (bhm)2 = u (X) +2ZZ y '1imbu( XX), (4.4)

where u(X,) refers to the standard uncertainty in the input parameter X1, u(X,,Xj) is the

estimated covariance between parameters X, and X,,. and m is the number of input

parameters. The degree of the correlation between X, and X, is characterized by the

correlation coefficient

u X(,,X,)
r (X,, X)= u (4.5)
(x,) (x,)

In the Monte Carlo and Latin Hype-Cube sampling methods (described next), the

multivariate normal distribution can be used to estimate the confidence level, in which

case the covariance matrix between parameters controls the random sampling procedure.

Monte Carlo Simulation

The combined standard uncertainty, uc, of the stability boundary (blm) and surface

location error (SLE) can be predicted using Monte Carlo simulation. In this method, a

random sample of size n is selected from the population of each input parameter. A

normal distribution of the input parameters is assumed. In the sample n, the nominal

value and standard deviation of each input parameter are used to generate the sample.

The axial depth limit and surface location error are then calculated using TFEA for each

point in the sample. The standard deviation of the predicted bi,, and SLE is then

calculated from sample output for the range of spindle speeds of interest. It should be

noted here that in doing so, no correlation between the input parameters is assumed,

which is a common, and sometimes erroneous, approach.

To illustrate the effect of uncertainty in the input parameters on stability boundary

uncertainty, standard uncertainties of 5%, 0.5%, 0.001% and 0.5% are assigned to

nominal values of the cutting force coefficients, modal parameters, radial step, and

spindle speed, respectively. The values of the standard uncertainties assigned correspond

to practical variation in the parameters involved. The parameters are assumed to be

uncorrelated here. A sample size of 1000 is used. The stability boundary uncertainty is

found, as shown in Figure 35, for one standard deviation interval around the neutral

stability boundary.

mean one std
mean
mean + one std

5 10 15 20
2 (rpm x03)
Figure 35. Confidence in stability boundary due to input parameters uncertainties using
Monte Carlo simulation (see Table 6).

Sensitivity Method

The combined standard uncertainty uc in axial depth limit while neglecting

correlation between input parameters can be obtained from Eq. (4.4) as

I m b bh
uc (blhm)= lml(X),
vI \^ 8X, )

(4.6)

where u(X,) refers to the standard uncertainty in the input parameter X, (same used for

Monte Carlo method), and m is the number of input parameters. Although this relation

assumes no correlation between input parameters it should be noted that cutting force

coefficients (Kt, Kn, Kte, Kne) and modal parameters (K, C, M) may be correlated in

practice.

The same standard uncertainty is assumed in the input parameters as in previous

sections and the confidence level in axial depth limit is calculated for an interval of + 2

uc(bjzm). Figure 36 shows the close agreement found using the two methods. However, it

should be noted that the sensitivity method can be inaccurate near points where the

function (b,,m) is C1 discontinuous. Figure 37 shows the direct correspondence between

the inaccurate sensitivity and C1 discontinuity inA The C1 discontinuity in blum leads to

inaccurate estimation of uc(bizm) (see Eq. (4.6)).

20
Sensitivity
18 -- Monte Carlo
Nominal
16

14

12

10

5 10 15 20
Q (x 103)
Figure 36. Uncertainty boundary in axial depth limit using two standard deviation
confidence interval. Uncertainty is calculated using sensitivity method and Monte
Carlo method (see Table 6).
Carlo method (see Table 6).

Derivative method
S2 -- Monte Carlo Simulation

0

1
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0-

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1

Q (x 103)
Figure 37. Uncertainty in axial depth using sensitivity and Monte Carlo methods.
Inaccuracies in the sensitivity method can be seen near C1 discontinuity in the real
and imaginary part of system characteristic multipliers (see Table 6).

It should be noted here that predicting the uncertainty by Eq. (4.6) uses a linear

approximation. The standard uncertainties assumed earlier are small where the linear

approximation is still valid. However, if the uncertainties in the input parameters are

large, then that linear approximation is no longer valid. In this case, simple random

sampling methods (such as Monte Carlo simulation) are more appropriate.

The surface location error uncertainty is found similarly using both methods.

However, as shown earlier (see Figure 32 and Figure 34), the SLE sensitivities are

accurate and do not depend on the characteristic multipliers' continuity. Since the SLE is

only defined for stable cutting conditions (see Eq. (A. 18) in Appendix A) and explains

the close prediction of uncertainty in SLE using sensitivity and Monte Carlo methods

(Figure 38).

30

I \
o~ l\
5 "'

0 -'

5 \ i

0-

/

I\
!i \l
i ':

[I ,;

I~

Monte Carlo method
Sensitivity method

Q2 (x 103)
Figure 38. Surface location error uncertainty with two standard deviation confidence
interval on the nominal SLE. Close agreement is observed (see Table 6).

Latin Hyper-Cube Sampling Method
This method was originally proposed as a variance reduction technique [89] in

which the estimated variance is asymptotically lower than with simple random sampling

(Monte Carlo method) [90, 91]. That is, for a sample size L, this method gives a lower

estimate of the output variance than is possible with the Monte Carlo method. The basic

idea of this method is that each value (or range of values) of a variable is represented in

the sample, no matter which value turns out to be the most important. In this way, the

sampling distribution is divided into a number of strata with a random selection inside

each stratum. The Latin Hyper-Cube method will be used in Chapter 5 for predicting the

standard combined uncertainty of the stability and surface location error cutting tests in

that chapter.

Robust Optimization under Uncertainty

In order to account for uncertainty in the axial depth stability limit, the safety factor

design analogy is used here. The deterministic optimization algorithm implemented in

Chapter 3 (Eq. (3.9)), repeated here, can be modified to account for the axial depth

uncertainty.

min fSLE,, (b, +) + fSLE (b, ) + fSLE (b, 3)
3
subject to: f, (b, Q) < e,, for i = 1 ... k (4.7)
{g, (b, ) ng, (b, )n g, (b, + 6)} <1,
for a series of selected limits (e) on fe,

Therefore, the axial depth b used in the stability constraint is set equal to an uncertainty

inflated value. That is, b is replaced by b+ Ue, where Ue = kuc (b) is the expanded

uncertainty, k is a factor that estimates the uncertainty confidence interval and uc(b) is

the combined standard uncertainty in the axial depth. Therefore Eq. (4.7) becomes

min fsLE, (b, +) + fSLE (b, Q) + fSLE (b, )|
3
subject to: f, (b,Q) e,, for i = 1...k (4.8)
{g, (b+Ue, ) g,) (b+Ue,)n g, (b+Ue,,+6)}< 1,
for a series of selected limits (e) on f,,,

Discussion

In this chapter, the sensitivities of axial depth limit and surface location error to

model input uncertainties were studied. Numerical estimation of the sensitivities can be

challenging, where several factors contribute to the accuracy of the estimation. Step size

is one of the significant factors that affect the accuracy of the estimation.

The sensitivity analysis aids in identifying the relative contribution of the milling

model input parameters to the sensitivity of either axial depth limit or surface location

error. For the case of axial depth, the spindle speed, followed by modal stiffness and

mass, is the most significant contributor. In the case of cutting force coefficients, the

tangential cutting force coefficient is found to contribute more to the sensitivity than the

normal cutting force coefficient. As for the surface location error sensitivity, the same

trend can be observed. However, for the cutting force coefficients, the edge tangential

cutting force coefficient significantly contributes to the SLE.

The uncertainty in axial depth and surface location error was predicted using two

methods: the sensitivity method and the Monte Carlo simulation approach. Comparable

agreement is shown. However, the sensitivity method is more efficient computationally.

For example, in the case of SLE uncertainty prediction, Monte Carlo simulation required

9.34 hours, while the sensitivity method needed only 0.26 hours (36 times more

efficient). It is noted that for the uc(SLE) case, when the milling parameters are well into

the stable region, the accuracy of the sensitivity method is not sacrificed at the cost of

efficiency as is the case for uc(b) at discontinuities in the characteristic multipliers.

Finally, the optimization algorithm introduced in Chapter 3 was modified to

account for confidence levels in the axial depth limit. This allows robust optimization to

71

account for inherent uncertainty in the mean values of the input parameters. In Chapter 5

an implementation of this algorithm is demonstrated.

CHAPTER 5
EXPERIMENTAL RESULTS

The milling model accuracy depends on reliable determination of cutting force

coefficients and tool or work-piece modal parameters. These values are found

experimentally and their uncertainties contribute to the uncertainty of the model

prediction. In this chapter, the experimental procedure used to determine these

parameters is described and then the optimization algorithm is executed using the

experimentally determined input parameters to find the Pareto optimal points. Another set

of experiments is completed to validate/invalidate these optimal points. Using the

optimization algorithm, the strength and weakness of the mathematical model or solution

method can be obtained.

Cutting Force Coefficients

Milling Forces

The average milling forces during one tooth period in the x and y-directions are [92,

93],

=Nb [K,cos(2) -K [20-sin(20)]]+ N[-K, sin(0)+K, cos(0)]l
'(5.1)
S Nbc [K, [2-sin(2)] + Kcos(20)] Kcos() + Ksin )
< 8ffcos(O+ sin(

where K, and K are the tangential and normal edge cutting force coefficients,

respectively. In slotting tests (see Figure 7), the entry and exit angles of the cutter are

, = 0 and 0ex = .i, respectively. The average forces per tooth period for this case are

found to be:

Nb Nb
F= K c- Kne
4 KZ (5.2)
SNb Nb
F = Kc+- Ke
4 Z

Equation (5.2) can be written as a function of chip load (c) as:

FF = FcC + F, (q = x, y,z) (5.3)

The experimental procedure consists of completing multiple cutting tests at varying

chip loads and recording the cutting forces. For each chip load increment, the average

cutting forces in the x and y-directions are measured, and then a linear regression of the

data points is made to extract the cutting coefficients using Eqs. (5.2) and (5.3):

4F orF
Kt Kte y
Nb Nb (54)
4F(5.4)
K K ne
Nb Nb

For radial immersions less than the cutter diameter, the entry and exit angles differ

from the slotting case. For up-milling (see Figure 7) the entry and exit angles of the cutter

are b, =0 and x = cos 1- a- where a is the radial depth of cut. Substituting in Eq.

(5.1) gives:

F Nb= K, cos(2 )- 1]- K [2q -sin(2 )+-sin( (2J)] Nb sin )+ Ke[cos(ex)- 1]
S(5.5)
(5.5)

Factoring Eq. (5.5) in terms of chip load c gives:

Fx = Fxcc + Fxe (5.6)

F Nb [K, [cos(2x)-1] -K [2x sin (20)]]
(5.7)
F Nb _Kt, sin ( x) + Kn [cos(e)- ]].

Similarly, the following equations are obtained for the y-direction.

Nb[K, [2 sin(2qJe) +Kn [cos(2x)-l ]] [K cos(Oex)-1]+Ksin(ex)]

(5.8)

F= Fyc +F (5.9)

=c Nb [K, [2, -sin(2ex) + K [cos(25ex-1]]
(5.10)
Fy = [ K cos (ex) 1+ Ke sin ( )]

Writing Eqs. (5.7) and (5.10) in matrix form to solve for the cutting coefficients the

final equation can be expressed as shown in Eq. (5.11).

Fx; Nb[ cos(20e)-l -2 x + sin(2 ex) 0 0 K,
FC= 8-r 20 -sin(2 ex) cos(2,)-l 0 0 K,
&Fo 0 Nb[ -sin(0,) cos(ex)- KK,
SFye 0 0 2 1-cos(o) -sin( x) K,
(5.11)

The same procedure can be used to solve for the cutting coefficients in the down-

milling case (Figure 7).

Experimental Procedure
Proper selection of a suitable dynamometer to measure the dynamic cutting forces

is important. Some of the factors that need to be addressed are the calibration range of the

dynamometer and its dynamic response. Simulation of the cutting forces helps in

addressing the issue of cutting force magnitude range. Using time-domain simulation of

the cutting forces and approximate cutting coefficient values, an estimate of the typical

range of cutting forces can be found. Euler integration is used to solve for the tool

displacement during the cut in the 2nd order differential equation (Eq. (3.2)) and find the

corresponding cutting forces in the x and y-directions. An example is shown in Figure 39.

It is seen that a dynamometer with the 0 kN to 5 kN range is acceptable, although the

force levels are relatively small compared to the full scale value. A Kistler 9257A

dynamometer with 5 kN range was available for these tests. One requirement for this

dynamometer is that the cutting force is applied to the dynamometer not more than 25

mm above the top surface of the dynamometer.

50

0-

-50

-100
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Time (s)

100

50-

-50
0 0.1 0.2 0.3 0.4 0.5
Time (s)
Figure 39. Example simulation of cutting forces to facilitate proper selection of
dynamometer.

A 25 mm thick 6061-T6 aluminum work-piece was sized to 100 mm x 85 mm, then

faced and drilled to fit the dynamometer hole pattern as shown in Figure 40. Slotting

cutting tests were made for a 25.4 mm diameter end mill with a 145 mm overhang and a

single 12 helix insert for chip load range of 0.1-0.24 mm/tooth in 0.02 mm/tooth steps.

The cutting forces in x and y-directions were measured for each chip load using an axial

depth of 0.4 mm. Two sets of measurements were made for a 1000 rpm spindle speed. To

address the influence of spindle speed on cutting coefficients, the above two sets were

repeated for {5000, 10000, 15000 and 20000} rpm. The average value of the measured

cutting forces was inserted into Eq. (5.4) to solve for the cutting coefficients. Average

cutting coefficients of the two sets of measurements at each spindle speed are listed in

Table 8.

Figure 40. Work-piece, dynamometer and tool setup

A regression analysis of the cutting force coefficients as a function of spindle speed

was carried out. For Kt and K,, a linear regression with logarithmic transformation of

spindle speed indicates a statistically significant relation with a P-Value of less than

0.007. Figure 41 and Figure 42 show the trend line for this regression for both Kt and K,,

respectively. For the edge cutting force coefficients K,e and Kte the regression doesn't

indicate a significant statistical relation between K,e or Kte and spindle speed. The P-

Value for the slope of the regression was 0.39 and 0.55, respectively.

Table 8. Cutting coefficients for 1 insert endmill for slotting cutting tests

Q Kt Kn Kte Kne
(krpm) (N/mm2 (N/mm2) (N/mm) (N/mm)
1 1321 379 28 32
5 832 183 47 34
10 841 62 37 38
15 655 34 52 33
20 670 65 37 26

1400

1200

1000

800

600

400

200

0

0 0.5

1
logio(Q (rpm) x103)

Figure 41. Cutting coefficient in tangential direction (Kt)

400

350

300
y = -268.84x + 369.14

200

150

100

50

0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
3
logo( (rpm) x10)
Figure 42. Cutting coefficient in normal direction (K,)

A similar set of measurements were made using partial radial immersion (up

milling) for a 15000 rpm spindle speed. Equation (5.11) was used to find the cutting

coefficients in this case. The results are provided in Table 9.

Table 9. Up milling cutting coefficients for 12% radial immersion

Kt(N/mm2) Kn( N/mm2) Kte(N/mm) Kne(N/mm)
833 431 6 8

To verify the fit, the cutting coefficients obtained were used in a time-domain

simulation of the cutting forces. The measured forces were then overlaid on the simulated

forces. Figure 43 shows a case for 0.12 mm/tooth chip load and 1000 rpm. Also Figure

44 and Figure 45 show the fit for higher spindle speeds of 5000 and 20000 rpm,

respectively.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Time secss)

Simulated force
Measured force

0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Time secss)
Figure 43. Simulated and measured forces for 0.12 mm/tooth chip

-- Simulated force
Measured force

0 0.02 0.04 0.06 0.08
Time secss)

0.4

0.1 0.12

Simulated force
Measured force

0.02 0.04 0.06 0.08
Time secss)

0.1 0.12

Figure 44. Simulated and measured cutting forces for 0.2mm/tooth chip load, b=0.4 mm
and 5000 rpm.

40
20
0
-20
-40
-60
-80
0

I

100
80g o / Simulated force

20 '
0
-20
1.01 1.015 1.02 1.025 1.03 1.035
Time secss)

50- Simulated force
Measured force

-50

1.01 1.015 1.02 1.025 1.03 1.035
Time secss)
Figure 45. Simulated and measured forces at 20 krpm and b=0.4 mm for slotting.

Covariance Matrix (Linear Multi-Response Model)

The regression analysis performed in the previous section is a single response

analysis. However, the measured responses are the forces in both the x and y-directions

during a single measurement (dynamometer). Obviously this is a multi-response

measurement. Therefore analysis of the data should take into consideration the

multivariate nature of the data. The interrelationship existing between the variables could

render univariate investigations meaningless. The development for a multi-response

model follows the description in [94]. If we let Q be the number of experimental runs and

r be the number of response variables measured for each setting (two in our case, i.e., Fx

and Fy) of a group of variables (chip load only in our case). The ith response model can be

written in vector form as

Y =Z Z,+, i=1,2,...,r (5.12)

where Y, is an Q x vector of observations in the ith response, Z, is an Q x p, matrix of

rank p, (for the simple linear model p, = 2), f/ is a p, x 1 vector of unknown constant

parameters, and E, is an Q x 1 random error vector associated with ith response. The

assumption of simple linear regression apply here, that is E(E) = 0 and Var (es) = cIg.

However, the covariance matrix between the responses is not zero,

Var(s,)= ,, i = 1,2,...,r
(5.13)
Cov (E,,e = ,, i, j = 1,2,..,r;i i j

We denote the r xr covariance matrix whose (i,j) th element is o-j (i, j = 1,2,..., r)

by For the case of two responses, Eq. (5.12) can be written in matrix form as:

r Zo 0i 01 E
=] x2 8 + ]xl (5.14)
Y2 0 Z2 02 -2
Q xl Q x2_ 1 Q x 1

where
Z,=Z2 = 1 c (5.15)

where c represents the chip load vector (see Eq. (5.3)) and the left hand side vector of Eq.

(5.14) represents the observed average cutting forces in the x and y-directions. From Eq.

(5.13) it can be seen that E has the following variance-covariance matrix,

A =Var (s)= E o0 (5.16)

where 0 is a symbol for the direct (or Kronecker) product of matrices. The direct

product of two matrices E and I. both of size r xr gives an r2 xr2 matrix which is

partitioned as o-JI where -,, is the (i,j)he element of matrix E. The best linear unbiased

estimate of ,f is given by [95]

f =(Z'A 'Z) Z'A' 1 (5.17)

where Y is the left hand side of Eq. (5.14). The variance-covariance matrix of the

estimated vector / is

Var ()=(Z'A 'Z) (5.18)

Since E is usually unknown, it is estimated using the following equation [95]

Y' I,-Z, Z Z,) Z;Z I,-Z Z;'Z Z; ^Y
--j -- (5.19)
i,j = 1,2,...,r

It should be noted that j is computed from the residual vectors which result from

ordinary least-squares fit of the ith andjth single response models to their respective data

sets. Using this estimate for E in Eq.(5.19), an estimate of the variance of / can be

obtained. The cutting force coefficients are determined using a linear transformation

[K]= [A][] (5.20)

where the matrix A for slotting (see Eq. (5.4)) is
0 0 0
Nb
4 0
0 0 0
A Nb (5.21)
0 0 z 0
Nb
0 0 0 4
Nb

Therefore the variance-covariance matrix of cutting force coefficients can be found

as

Var(K})= [A]'Var(f)[A]. (5.22)

Using the procedure outlined above, the cutting force coefficients and their

corresponding correlation matrices are calculated and listed in Table 10 for cutting tests

carried out according to the same procedure described earlier, noting that the correlation

matrix is obtained directly from the covariance matrix (see Eq. (4.5)) As indicated in

Table 10, a high correlation between K, and Kte, and Kn and Kne is found. This high

correlation is justified since both of the corresponding cutting coefficients (K, and Ke or

K, and Kne) are obtained from the same regression fit and cutting force direction.

However, a small correlation between the x and y-directions of the forces is found

(between K, and Kn or Kne) which may be due to experimental error.

Table 10. Estimated cutting force coefficients and their correlation matrix for 7475
aluminum and a 12.7 mm diameter solid carbide endmill with 4 teeth and 30
degree helix angle.

Kt (N/m2) Kn (N/m2) Kte (N/m) Kne (N/m)
Mean 8.41E+08 2.53E+08 1.27E+04 1.01E+04
Standard deviation 2.19E+07 2.66E+07 1.70E+03 2.07E+03
Coefficient of variation 0.03 0.11 0.13 0.20
P Value 2.E-08 8.E-05 3.E-04 3.E-03
Correlation Coeff. Matrix Kne Kn Kte Kt
Kne 1.00
Kn -0.93 1.00
Kte -0.13 0.12 1.00
Kt 0.12 -0.13 -0.93 1.00