Effect of the lift coefficient on propeller flutter

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Title:
Effect of the lift coefficient on propeller flutter
Series Title:
NACA WR
Alternate Title:
NACA wartime reports
Physical Description:
16 p., 7 ℓ : ill. ; 28 cm.
Language:
English
Creator:
Theodorsen, Theodore
Regier, Arthur A
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:

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Subjects / Keywords:
Oscillating wings (Aerodynamics)   ( lcsh )
Propellers -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Summary: Flutter of propellers at high angles of attack is discussed, and flutter data obtained in connection with tests of models of large wind-tunnel propellers are analyzed and results presented. It is shown that in the high angle-of-attack range flutter of a propeller invariably occurs at a speed substantially below the classical flutter speed. The angle of attack at which flutter occurs appears to be nearly constant and independent of the initial blade setting. Thus, the blade simply twists to the critical position and flutter starts. Formulas have been developed which give an operating angle in terms of the design angle and other associated parameters, and these relations are presented in the form of graphs. It is seen that the flutter speed is lowered as the initial design lift coefficient is increased. It is further shown that by use of a proper camber of the propeller section the flutter speed may approach the classical value. A camber for which the blade will not twist is found to exist, and the corresponding lift coefficient is shown to be of special significance.
Bibliography:
Includes bibliographic references (p. 16).
Statement of Responsibility:
by Theodore Theodorsen and Arthur A. Regier.
General Note:
"Originally issued July 1945 as Advance Confidential Report L5F30."
General Note:
"NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were previously held under a security status but are now unclassified. Some of these reports were not technically edited. All have been reproduced without change in order to expedite general distribution."

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University of Florida
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Full Text
ACA L-l l


ACR No. L5F30


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS





WARTIME REPORT
OuGQNALLY ISSUED
July 1945 as
Advance Confidential Report L5F30

EFFECT OF TEE LIFT COEFFICIENT ON PROPELLER FLUTTER
By Theodore Theodorsen and Arthur A. Regier

Langley Memorial Aeronautical Laboratory
Langley Field, Va.



UNNERS1Ty OF FLOIDiA
OCU CP'- DEPI MrAENT
IS RAFTCNC LSRA(
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WASHINGTON


NACA WARTIME REPORTS are reprints of papers originally issued tu provide rapid distribution of
advance research results to an authorized group requiring them for the war effort. They were pre-
viously held under a security status but are now unclassified. Some of these reports were not tech-
nically edited. All have been reproduced without change in order to expedite general distribution.


L 161










































Digitized by Ihe Inerinei Archive
in 2011 with luniding Irom
Universal, ol Florida. C.eoige A. Smalhers Libra3ie, Wilh support from LY'RASIS and the Sloan Foundalion


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NACA ACR No. L5F30

NATIONAL ADVISORY COrfMTTBEE FOR AERONAUTICS


ArLVA:CL CO 3NriNTIrL REPORT

EFFECT OF TiHE LIFT COEFFICIE'NT ON PROPELLER FLUTTER

By Theodore Theodorsen a.nd Arthur A. Rsgier





Flutter of propellers at high angles of attack is
discussed, and flutter data obta~ ned in c..nneccion with
tests of models of lar.e wn-d-tunnel propellers are
anal,:zed and results presented. It is shown that in the
high angle-nf-attack range fluLter .f a propeller invaria-
bly occurs at a speed substantially below the classical
flutter speed. The angle of attack at which flutter
occurs appears to be nearly constant and independent of
the initial blade setting. Thus, the blade simply twists
to the critical position and flutter starts. Formulas
have been developed which give -n operating ansle in
terms of the design angle -nd other associated parameters,
and these relations are presented in the form of graphs.
It is seen that the flutter speed is lowered as the
initial design lift cocfficient Is increased. It is
further shown that by use of a proroer canber of the pro-
peller section the flutter 2peed ray approach the classi-
cal value. A camLber for which 'he olade will not twist
is found to exist, and the correa;.onding lift coefficient
is snown to be of special sl-niffcance.

The classic.-:l flutter speed and the divergence soeed
of a orooellcr are snovrn to be approximately the same
because of the centrifugal-force effects. It aonears
that a propeller will not flutter until the blade tvists
to a stall condition noar Lhe divergence speed. .Strooo-
scopic observation of several propellers confined this
theory. I wvas observed that, regardless of the initial
pitch setting of the propeller, the blades always twisted
to the stall condition before flutter commnenced. The
problem., of predicting propeller flutter is Lhus resolved
primarily into the calculation of the speed at which the
propeller will stall.









2 CIJDFIDEi7TIAL IAC". AA No. L573.


T 4m R DDUTIT ON


The present study of propeller flutter was conducted
in connection with the design of several large wind-tunnel
propellers for the Langley, Anaes, and Cleveland Labora-
tories of the NIA.CA. ,ind-turnel propellers are not
required to operate in a fully stalled condition and can
therefore be designed with small margin of safety against
flutter. Airplane propellers, on the other hand, must
have a considerable margin of safety since they are
required to operate in the stall or near-stall condition
in take-off. The results of the present tests are of
wide interest since they apply to the general problem of
the effect of high lift coefficients on the flutter
velocity of a pinpeller.

There are two principal types of flutter: (1) "clas-
sical" flutter and (2) "stall" flutter. Classical flutter
is an oscillatory instability of an airfoil operating in
a potentlsl flow. The problem of classical flutter was
solved tneoretlcally in reference 1. Stall flutter
involves separation of the flow and occurs on airfoils
ooerating near or in the stall condition of flow. Studer
(reference 2) studied this type of flutter experimentally
with an airfoil in two-di-.ensional "flow. H:e found that
the stall flutter sned was very much lo;er than the
classical flutter speed nd that, as the angle of attack
of the airfoil .:a3 increased, the change from classical
flutter to still flutter was rather abrant.

The problem of propeller flutter is somewhat dif-
ferent from the problem of ving flutter in that the
change between classical flutter and stall flutter appears
to be much more gradual. Ihis gradual change of flutter
speed with an3le of attack has not been clearly under-
stood, and attempts to calculate the flutter speed of
propellers operating under normal loads have not been
entirely satisfactory.

A third type of flutter, which rnay be referred to as
"wake" flutter because it occurs on propellers operating
at zero lift in their o.vn wake, has sometimes been
observed. Self-excited torsional oscillations of the
proDeller blade occur at frequencies which are integral
multiples of the rotational speod of the propeller. At
low speed the frequency of oscillation is equal to the
torsional frequency of the propeller blade in still air.


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NACA aCR No. L5F30


The first such oscillation appears when the propeller
rotational speed reaches approximately one-twentieth of
the blade torsional frequency. This oscillation dis-
appears as the rotational speed is increased but reap-
pears at each integral multiple of the propeller speed
until the classical flutter speed is reached. This type
of flutter is not important for normal propeller opera-
tion.


SYMB'OLS


L representative length of propeller blade

c chord of propeller section

b s1ir:!chcrd

t ti-ic-:ness of propeller section

R radius to tip of propeller

r radius to propeller section

K torsional stiffness of representative section

q eynanic pressure of relative air stream

p 6ensi uy

K ratio off J.r'.s of cylinder of ati' of diameter
sl, i ; .' I o- airfoil to rrass of airfoil

a angle o. i a .: k

Aa angle cf twist or deformation of blade at repre-
sentative section

amo angle of attack for which there is no twist

a0o angle of attack for zero lift


Cmc/4 moment coefficient about quarter-chord point

CL lift coefficient


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NACa ACE IN:'. L5350


CLu
CIx


Subscripts:

u untwisted or design

cr critical

I ideal

c compressible

i incompressible


CONFIDENTIAL


untwisted or design value of CL

lift coefficient for ideal no-twist condition

location of center of gravity as measured from
leading edge

location of center of gravity with reference to
elastic axis in terms of semichord

coordinate of torsional stiffness axis in terms
of semichord as measured from midahord posi-
tion

nonrc.inensl nal. radius of gyration of airfoil
section in terms of se:niichord referred to a

torsional frequency, radians per second

bending frequency, radians per second

Tjch number

divergence speed

flutter sneed

flutter speeo corrected for compressibility


Vf


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TACA ACR Jo. L5F50 CONFIDENTIAL 5


EFFECT OF LOADING OT7 THE TWIST OF A PROPELLER BLADE


The centrifugal force or. a propeller has a component
in the direction perpendicular to the relative flew, which
is very nearly equal to. the aerodynramic force. This
statel.lrit is exactly true if the propeller is designed to
avoid bending stresses and is approxi.rastely true in any
ce.se, since the bending forces are small compared wiuh
the aerodynamic forces. (See fig. 1.) The twist of the
proDeller at some representative section may be expressed
by the relation


,a =Lc2 dCL a- 4m (1)


The value of the critical velocity qcr for which
divergence occurs is obtained from equation (1). Diver-
gence evidently occurs for the condition

CL -- *3

or

'a
---l
au + (a am,)


For a->-)m then,


qcr = (2)




By substitution of this value of qcr for q in equa-
tion (1) the twist becomes

,4

a = (a CaT) qcr (5)

qcr


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6 C1ON IDFNTIAL 1. 1CA At- :) :". 0

The moment coefficient around the quarter-chord point
r.sa be written as

dCL /Q 1
c/ = C (o amo -


This equation may be rewritten as




o o 0 x 1\ dCL

\ daa

The relation between the untwisted or design value of
the lift coefficient CL and the actual or measured
value of CL resulting from the twist may be expressed
as

dCL
CL d (au a, + :.a)
da \o

or
dCL
CL CL a (5)


By substitution for A~ from equation (5) and by use of
the value of amo from equation (4) the following equa-
tion is obtained for CL:

q
1 1 4cr
CL=z L + c/. x 1 1-

qe q ,cr


COiTIDENTIAL








NACA ACR No. L5F50


or the following equivalent relation is obtained:



CLu = CL -q + Cc/4 (6)
qcr_ -


The increase in lift coefficient due to twist is evidently


"CL = -cr L + (7)
Scr 1'


There is no increase in CL, or no twist, if



CT/4 (8)
CsUI 1
x -


where CLuT indicates the value of CL at which no
twist occurs. This relation is plotted in figure 2.
This figure shows that, for the Clark Y airfoil with
center of gravity at 44 percent and Cm = -0.07, thr.
value of the lift coefficient at which no twist of the
blade is incurred is 0.37. In this case the angle of
zero twist is not very far from the ideal angle of attack
of the Clark Y airfoil, which is about 0.0. The following
discussion sh- ws that it is desirable to operate the pro-
peller at ths ideal angle of attack since operation at
this angle delays stall and thus obviously causes an
increase in the flutter speed. (See reference 5 for
discussion of ideal angle of attack.)

A propeller, if generated as a true helix, will not
be subjected to any centrifugal twisting moment; in fact,
if the blade width of the propeller is adjusted to achieve
the desired blide loading at an angle everywhere propor-
tional to the helix angle, there will be no twist. This
statement must be modified slightly, however, because the
radial generating lines through the leading and trailing


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NACA ACR ITo. L5F30


e..ges mna not be conlinuei to the center and, as a result,
the angle at the tip may be decreased. By proper plan
form and distribution of mass, therefore, a propeller
rr-a be de..;ign-:-d to have zero t'"ist. A highly tapered
pri:,eller tens to decr-ase its tip angle due to cen-
trifvLcal twisting ,ion.enr and a rior-b nearly rectangular
plan form induces an increase in the tip engle. These
effects are in reality small compared with the indirect
effect due to the aerodynamic forces resulting from the
binding of th-b tlade.


EXP-RT"r'.1;AL STUDTIS OF F'LTTLR'T OP PROPELLERS

AT TTTI--H L i-'I.7ITGS


A number of pro.pellers of different designs, some
representing ej.:isting "LC. wind-tunnel propellers for
which data were available and others representing rro-
posed wind-tunnel 1iopellers, were tested as wind-tunnel
fans in a small oo-n tunnel. A cross-sectional sicetch
of the test setup is shown in figure 3.

The lift coefficient of the blades was changed by
changing the area of the tunnel exit. The value of CLu
was calculated from the relative wind direction at the
0.8-radlus station and the angle of attack of the
untwisted bl&do.

The propeller tips were observed by stroboscope
th:irorgh a small window in the tunnel wall, centered in
the plane of the propeller. Blade-tip deflecion and
twist and the pressure increase of the air passing
through the fan were recorded. These data were used to
give an in-eD-n'ent check on the oo)rating lift coeffi-
clent. wingg to the nn!rniformity of the blades, the
variation of the lift coefficient with the rAdjus, and
other causes, the value of the operating lift coefficient
could be. determined only within about 0.1 to 0.2.

Although a num:ibEr of oroneller. were tester, only
results from the tests of two fairly representative pro-
pellers are reported. The pro'pell-,'rs were .rrade of lami-
nated spruce and ha.:' flat-bittom Clark Y sections. Pr>-
peller A, for which data are given in figure LL, was a
six-blade propeller 45 inches in diraeter. Propeller B
was a single-blade propeller having the same diameter but


CO ;FIDE TIA L


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NACA ACR Io. L5F50


two-thirds the chord and thickness of propeller A. The
propeller of smaller cross section, propeller B,.was used
to reduce the flutter speed and to make possible a study
of the flutter modes. Propeller B was tested in the same
location in the tunnel as the other propellers, but a
booster fan was attached at the rear of the motor to
force the air through the tunnel during the tests of pro-
peller B.

The vibration frequencies of the propellers are as
follows:

Vibration frequency
(cps)
.]ode_ )
Propeller A Propeller B

First bending 7h i 4

Second bending 26t 172

Tori on I, 5L0
SI ---

Torsion anc bending strain gages were attached to
propeller B, and the flutter amplitudes and frequencies
were recorded. The results of these tests are given in
figure 5. The flutter speed was changed by changing the
blade lift coefficient. It nma, be observed that the
bending amplitude is large and the flutter frequency is
low (170 cps) at the highest observed flutter speed. In
a lower range of flutter speed from 400 to 500 feet per
second the observed bending amplitude is small and the
flutter frequency attains a higher value, 20O cycles per
second instead of 170. This flutter evidently involves
the second bending mode, whereas the flutter at top speed
involves the first bending mode. At the lowest flutter
speed and the highest blade loading, the flutter reverts
to a condition of pure torsional oscillation at a fre-
quency corresponding to that of pure torsion measured in
free air, namely, 530 cycles per second.

The results of the flutter tests of propeller A are
shown in figure 6, which is a plot of equation (6) for
CLu = 0.37, the value of CLuI for a Clark Y flat-
bottom airfoil of 12-percent thickness. The straight
lines that converge at -qq = 1.0 and CLu, = 0.57 in
qcr


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10 CONFIDENTIAL NACA ACR No. L5F30


figure 6 are i.nes representing constant values of CL.
Figure 6 show; that, if the design lift coefficient CLu
is 0.6, f-'r instance, the blade will twist so that CL
is 1.0 at 0.7. 'Diagrar.s similar to figure 6 may
qc,
be used to obtain CL for any pro--.ller in ter::.s of CLu.

Figure 6 also shows curves for data plotted with CL,
uncorrected for compressibility and for the same data
plotted with CLu corrected for compressibility by
Glauertts formula


CLui
c Lu- M2

The data corrected for compre3s1bility show that, when
the propeller was set with the bla-e at stall, the flutter
speed q/ocr was only 0.17. As C xas decreased
from 0.85 to 0.65, q/qor increased from 0.57 to 0.62.
It may be note' that in this range the flutter occurred
at an approximately constant CL of 1.1. A further
incrtpse in q/q r cause% a rather zharo drop in C,
for flutter. The data uncorrected for compressibility
show that, as CLu varies from 0.7-:. to 0."5, CL for
flutter varies from 1.1 to 0..:2. The validity of the
compressibility correction as applied to CL in figure 6
may be uu-stioned, but results of the t.sts clearly
indicate that the propeller tested co a lift ccefficient
near unity before flutter occurred. The propeller can
carr'y a lift coefficient exceeding unity without flutter
if q/qcr is low enriough; however, as the classical
flutter soeed is approached, the flutter lift coefficient
becomes less than unity. In other .vords, the ar.ount of
stall necessary to excite flutter i; less as the classical
flutter s-ecd is a.pproachee. This crncluzacn is in agree-
ment with the -xper'lint of r&fCrencc 2 and se~rns logical
in view of the various flutter :nodes described by figure 5.

The miniLurim flutter speed for a completely stalled
propeller is of i'nportance for prop--llers Lhat operate
in such a condition at timcs in tk':e-off. for example.


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NACA ACR No. L5F30


There is no reliable method known for calculating this
minimum flutter speed for a completely stalled propeller,
and further investigation of the problem is required. As
the angle of attack is increased from a normal value, the
flutter speed will decrease until a minimum is reached.
There is evidence that this minimum is related to the
von Karman vortex street; the minimum probably corresponds
to a coincidence of the torsional frequency with that of
the von Karmin vortex street. In figure 6 it may be
observed that this minimum appears at -9.-- 0.17 for
qcr
propeller A. This value may be fair for wooden propellers
but cannot be taken as valid for metal propellers. Flut-
ter on metal propellers operating in the stall condition
has been observed to occur at values of q/qcr as low
as 0.04.

Figure 7 is of interest as a verification of the :
theoretical treatment in this paper. It gives the twist
of the propeller tip as a function of CLu for a constant
propeller speed at 0.37. This twist was observed
qcr
by means of telescope and stroboscope. The line on the
figure is drawn through the point for an angle of twist
of 00 and a value of CLu of 0.57, as predicted by fig-
ure 2. The slope of this line, which may be calculated
by use of equation (5), is adjusted to fit the data.
Equation (5) is based on a simplified propeller and gives
the twist at the representative section. At CLU = 0.78
and -3- = 0.37, equation (3) gives Aa = 2.4. The
qcr
observed twist at the propeller tip for this condition
was 5.10. It was observed by direct measurement that the.
torsional stiffness at the representative section was
2.5 times that at the tip. The observed tip twist is
therefore consistent with the expected value.


DETERMINATION OF DIVERGENCE SPEED AND

CLASSICAL FLUTTER SPEED


The divergence speed of a propeller can be calculated
from equation (2) if proper values are selected for L,
c, and K but may be more conveniently found from the
formula of reference 4 (p. 17) which is


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12 CONFIDENTIAL NACA ACR Ho. L5FS0


1
2 -
: v D r_ 2

a + a
2

Reference 1 (p. 17) also gives an approximate flutter
formula that appears to hold very well for a heavy wing
and for small values of w /wa the conditions for
normal propellers. The approximate flutter formula is

1
--_-_ (10)
"a + a + xa
2

where xa is the location of the center of gravity with
reference to the elastic axis in terms of the semichord.
It may be noted that formulas (9) and (10) are alike
except for an additional term xa in formula (10). If
the center-of-gravity'location coincides with the location
of the elastic axis,' xa = 0, -the two formulas are iden-
tical.

It was shown in the discussion of figure 1 that
there are two moments acting about the elastic axis, the
aerodynaini-force moment and the centrifugal-force moment.
If the aerodyna:nic forces are balanced by components of
the centrifugal force, the resulting moment is the same
as if the aerodynamic forces acted with a moment taken
about the center of gravity of the airfoil section. For
propellers, therefore, the dynamic-stiffness axis may be
taken at the center of gravity and the divergence speed
of the propeller will be given approximately by the flut-
ter formula, equation (10), or vD = vf. It may be men-
tioned that for normal propellers the location of the
elastic axis and the location of the center of gravity
are usually very close together.

The location of the center of gravity from equa-
tion (10), a + xa, is expressed in terms of the semi-
chord as measured from the midchord position. Equa-
tion (10) may be written


CONFIDENTIaL








NACA ACR No. L5F350


1
Vf ra 2 4
(11)
ob K 1
x -


where x is the location of the center of gravity in
fraction of chord as measured from the leading edge.
Since vD = vf,


q = pvf2
cr 2

Propellers usually operate near a Mach number of
one, and the compressibility correction therefore becomes
extremely important. As yet, there is no accurate knowl-
edge concerning the compressibility correction for the
flutter velocity near the velocity of sound. An approxi-
mate compressioility correction for the subsonic range
from reference I, is


2 Mi2 Mi
M.2 Mi 1 + .
\ 2 8

where Mc is the Mach number corresponding to flutter
speed in compressible flow and .ii is the V'ach number
corresponding to flutter speed in incompressible flow.
The flutter speed corrected for compressibility is tenta-
tively calculated in the appendix and is indicated in
figure 6 by the vertical line at -- 0.79.
qcr

The choice of the radius of the representative sec-
tion is open to some question. Since the velocity varies
approximately as the radius, this choice is rather impor-
tant. It has been customary to use the section at three-
fourths serispan as the representative section for wings.
because of the velocity distribution on propellers, the
representative section was taken at the 0.8-radius s.ation.


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CONFIDENTIAL NACA. aCR No. L5F50


CONCLUDING RE.aRBKS


It has been shown that the stall flutter speed of a
propeller is in general very much lower than the calculated
classical flutter speed.

The classical flutter speed may be attained only if
the propeller operates at the ideal angle of zero twist.
The ideal angle of zero twist depends on the moment coef-
ficient of the section, and the corresponding lift coeffi-
cient has been given by a simple relation.

It is desirable to have the design angle equal to
the ideal angle of attack in order that the speed at which
flutter occurs may be higher. The design angle should
therefore be equal both to the ideal angle and to the
ideal angle of zero twist.


Langley Memorial Aeronautical Laboratory
National Advisory Committe- for Aeronautics
Langley Field, Va.


CONFIDENT IAL








NACA ACR No. L5F50


APPENDIX

SAMPLE CALCULATION OF FLUTTER SPEED CORRECTED

FOR COMPRESSIBILITY FOR PROPELLER A

Propeller section characteristics at 0.8-radius'station:

Type Clark Y flat-bottom 12-percent-thick airfoil

x = 0.44

r2 0.21

Specific Igrvity = 0.5
1
=5

= 0.093


Propeller characteristics:

R = 1.87 ft

wa = 2T (355)


b R c _
b 2 2R


z (0.092)(2)r)(555)


(0.09S) = .092


(0.24)(45) (0.25)
o.4 0.25 = 772 fps
0o.U 0.25


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',2-


1
x -








CONFIDENTIAL IjCA kCR No. L5F50


Correction for compressibility:

772
Mi 2 = 0.69
1120


E= 2 1 -~ Mi +1Mi = 0.612

vf = (0.612)(1120) = 695 fps

vc 2 2
cr) V ^ (685\=0.79

This value represents the flutter speed corrected
for compressibility. (See fig. 6.)




REFER EIIC ES

1. Theodorsen, Theodore: General Theory of Aerodynamic
Instability and the Lechanism of Flutter. I'ACA Rep.
No. 496, 1955.
2. Studer, Hans-Luzi: Experimentelle Untersuchungen uber
FlUgelschwingungen. l.:tteilung no. L, Inst. Aerod.
Tech. H. S. Zurich, Gebr. Leemann & Co. (Zurich),
1956.

3. Theodorsen, Theodore: On the Theory of 'ing Sections
with Particular Reference to the Lift Distribution.
ILiCA Rep. Io. 583, 1931.
4. Theodorsen, Theodore, and Garrick, I. E.: Mechanism
of Flutter A Theoretical and Experimental Inves-
tigation of the Flutter Problem. NACA Rep. No. 685,
19.40.


COrF:IDENTIAL








NACA ACR No. L5F30 Fig. 1











u


04

oa W

O4

'-4
O D




S14

144
0






OS -- 4 O
o u .

< *









0 0
El0 0


0 00








1. 1 U







NACA ACR No. L5F30


0 .20


r/R

Figure 4.- Data for propeller A.

CONFIDENTIAL


Fig. 4








NACA ACR No. L5F30


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o

ba
C -4
%%


I I +


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Lai




Z um


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--V---c


x 10


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