• ACR No . L5F3O NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT OMONALLy ISSUED July 19^5 as Advance Confidential Report L5F3O EFFECT OF THE LIFT COEFFICIENT ON PROPELLER FLUTTER By Theodore Theodorsen and Arthur A . Regier Langley Memorial Aeronautical Laboratory Langley Field, Va. NACA WASHINGTON 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 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 - l6l \ Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/effectofliftcoefOOIang / 1 ?> qoo kls NACA ACR No. L5F30 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ADVANCE CONFIDENTIAL REPORT EFFECT OF THE LIFT COEFFICIENT ON PROPELLER FLUTTER By Theodore Theodorsen and Arthur A. Regier SUMMARY Flutter of propellers at 'high angles of attack is discussed, and flutter data obtained in connection with tests of raooels 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 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 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 pro- peller section the flutter speed may approach the classi- cal 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. The classical flutter speed and the divergence speed of a propeller are shown to be approximately the same because of the centrifugal-force effects. It appears that a propeller will not flutter until the blade twists to a stall condition near the divergence speed. Strooo- scopic observation of several propellers confirmed this theory. It was observed that, regardless of the initial pitch setting of the propeller, the blades always twisted to the stall condition before flutter commenced. The problem of predicting propeller flutter is thus resolved primarily into the calculation of the speed at which the propeller will stall. CONFIDENTIAL NACA ACR No. L5P30 \ INTRODUCTION The present study of propeller flutter was conducted In connection with the design of several large wind-tunnel propellers for the Langley, Ames, and Cleveland Labora- tories of the NACA. l/vind- tunnel 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 propeller. 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 potential flow. The problem of classical flutter was solved theoretically in reference 1. Stall flutter involves separation of the flow and occurs on airfoils operating near or in the stall condition of flow. Studer (reference 2) studied this type of flutter experimentally with an airfoil in two-dimensional flow. He found that the stall flutter soeed was very much lower than the classical flutter speed and that, as the angle of attack of the airfoil was increased, the change from classical flutter to stall flutter was rather abrupt. The problem of propeller flutter is somewhat dif- ferent from the problem of wing flutter in that the change between classical flutter and stall flutter appears to be much more gradual. This gradual change of flutter speed with angle 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 may be referred to as "wake" flutter because it occurs on propellers operating at zero lift in their own wake, has sometimes been observed. Self -excited torsional oscillations of the proueller blade occur at frequencies which are integral multiples of the rotational speed of the propeller. At low speed the frequency of oscillation is equal to the torsional frequency of the propeller blade in still air. CONFIDENTIAL NACA ACR No. L5F30 CONFIDENTIAL 5 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. SUBOLS L representative length of propeller blade c chord of propeller section b semi chord t thickness of propeller section R radius to tip of propeller r radius to propeller section K torsional stiffness of representative section q dynamic pressure of relative air stream p density k ratio of mass of cylinder of air of diameter eqv.fi.3 co cho^d of airfoil to mass of airfoil a angle ol attack Aa angle of twist or deformation of blade at repre- sentative section a m angle of attack for which there is no twist aj angle of attack for zero lift Cm /i moment coefficient about quarter- chord point C L lift coefficient CONFIDENTIAL k CONFIDENTIAL Ni,CA ACE No. L5?30 Ct untwisted or design value of Cl C]^ lift coefficient for ideal no-twist condition x location oT center of gravity as measured from leading edge x a location of center of gravity with reference to elastic axis in terms of semi chord a coordinate of torsional stiffness axis in terms of semi chord as measured from mid chord posi- tion r a nondimensional. radius of gyration of airfoil section in terms of semi chord referred to a co Q torsional frequency, radians per second a>k tending frequency, radians per second M Mach number v-Q divergence speed Vf flutter speed Vf flutter speed corrected for compressibility Subscripts : u untwisted or design cr critical I ideal c compressible i incompressible CONFIDENTIAL NACA ACR No. L5F30 CONFIDENTIAL 5 EFFECT OF LOADING ON THE TWIST OF A PROPELLER BLADE T'Vs i The centrifugal force on a propeller has a component n the direction perpendicular to the relative flow, which is very nearly equal to the aerodynamic force. This statement is exactly true if the propeller Is designed to avoid tending stresses and is approximately true in any case, since the "bending forces are small compared with the aerodynamic forces. (See fig. 1.) The twist of the propeller at some representative section may be expressed by the relation _ / n \ dCr , Aa K = qLc^ (x - ± J^~ (a^ + Aa - c m ^ (1) The value of the critical velocity q cr for which divergence occurs is obtained from equation (1). Diver- gence evidently occurs for the condition A a — > °° or Aa a u + Aa - o ViQ -^1 For Aa— >ro then, (2) By substitution of this value of q cr for q in equa- tion (I) the twist becomes ^cr Aa = (a u - a mQ ) — (3) 1 - CONFIDENTIAL cr 6 confib] m cai : .-.. acs no. " The moment coefficient around the quarter- chord point may he written as dd This equation may be rewritten as a m ~ a l Cvv, /| ° ° / A dC T The relation between the untwisted or design value of the lift coefficient Ct and the actual or measured Ha value of Cl resulting from the twist may be expressed as dC L L da ^ L o ) or dC r da ^ + — ia (5) 3y substitution for Ac from equation (3) and by use of the value of a™ from equation (1+) the following equa- o tion is obtained for C^: c L - civ — + ^i c/k j - X - r- 1 c icr q cr/ h 'Icr C EVIDENTIAL NACA ACR No. L5FJ0 CONFIDENTIAL or the following equivalent relation is obtained: *-"*-± (*■ + *%) ■cr x -w The increase in lift coefficient due to twist is evidently ^c/k\ ac l = -- [c L + — ( 7 ) q cr There is no increase in Cj_,, or no twist, if Or C T =-■— ^L± (8) -u T i where Ct indicates the value of Ct 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 \\l\. percent and Qn r /I, = -0.07 > the value of the lift coefficient at which no twist of the blade is incurred is 0.37 • I n 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 O.L.0. The following discussion shows that it is desirable to operate the pro- peller at the ideal angle of attack since operation at this angle delays stall and thus obviously causes an increase in the flutter speed. (See reference 3 ? or 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 blade 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 CONFIDENTIAL CONFIDENTIAL NaCA ACR No. L5FJ0 edges may not be continued 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 may be designed to have zero twist. A highly tapered propeller tends to decrease its tip angle due to cen- trifugal twisting moment j and a more nearly rectangular plan form induces an increase in the tip angle. These effects are in reality small compared with the indirect effect due to the aerodynamic forces resulting from the bsnding of the blade. EXPERIMENTAL STUDIES OF FLUTTER OF PROPELLERS AT HIGH: LOADINGS A number cf propellers of different designs, some representing existing NACA wind-tunnel propellers for which data were available and others representing oro- posed wind-tunnel propellers- were tested as wind-tunnel fans in a small open tunnel. A cross-sectional sketch of the test setup is shown in figure J. ■u The lift coefficient of the blades was changed by changing the area of the tunnel exit. The value of C] was calculated from the relative wind direction at the 0.8-radius station and the angle of attack of the untwisted blade. The propeller tips were observed by stroboscope through a small window in the tunnel wall, centered in the plane of the propeller. Blade-tip deflection and twist and the pressure increase of the air passing through the fan were recorded. These data were used to give an independent check on the operating lift coeffi- cient. Owing to the nonuniform! ty of the blades, the variation of the lift coefficient with the radius, and other causes, the value of the operating lift coefficient could be. determined only within about 0.1 to 0.2. Although a number of propellers were tested, only results from the tests of two fairly representative pro- pellers are reported. The propellers were made of lami- nated spruce and had flat-bottom Clark Y sections. Pro- peller A, for which data are given in figure ]±, was a six-blade propeller J4.5 inches in diameter. Propeller 3 was a single-blade propeller having the same diameter but CONFIDENTIAL MCA ACR No. L5P30 JONPIDENTIAL two-thirds the chord and thickness of prooeller 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 the other propellers, location in the tunnel a booster fan was attached at the rear of the motor force the air through the tunnel during the tests peller B. but a to of pro- The vibration frequencies of the propellers are as follows • Mode First bencli no- Second bending Torsi on Vibration frequency (cps) Propeller A Propeller B Ik 355 172 3k0 Torsion and 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 may 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 I4.OO to p^G feet per second - the observed bending amplitude is small and the flutter frequency attains a higher value, 2.1+0 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, 3i_i_0 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 c l Ui = 0-37 the value of ,lj uj for a Clark Y flat- bottom airfoil of 12-percent thickness. The straight lines that converge at — S— = 1.0 and Cl =0.37 in i cr 30NPIDENTIAL 10 CONFIDENTIAL NACA ACR No. L5PJ0 \ figure 6 are lines representing constant values of Cj,. Figure 6 shows that, if the design lift coefficient Cy, u is 0.6, for instance, the blade will twist so that Cy_, is 1.0 at —•— = O.oJ, . Diagrams similar to figure 6 may ^cr be used bo obtain Cy, for any propeller in terms of Cr. LI Figure 6 also shows curves for data plotted with Cy uncorrected for compressibility and for the same data plotted with Cl^ corrected for compressibility by Glauert's formula = L U , "u c VI - M 2 The data corrected for compressibility show that, when the propeller was set with the blade at stall, the flutter speed q/q cr was only . 17 . As Cy was decreased from 0.85 to 0,65, q/q„ r increased from 0.37 to 0.62. It may be noted that in this range the flutter occurred at an approximately constant Cl of 1.1. A further increase in Tl/lcr caused a rather sharp drop in Cy, for flutter. The data uncorrected for compressibility show that, as Cy_ varies from O.78 to 0.55* Cy, for flutter varies from 1.1 to 0.82. The validity of the compressibility correction as applied to Cy, in figure 6 may be questioned, but results of the tests clearly indicate that the propeller twisted to a lift coefficient near unity before flutter occurred. The propeller can carry a lift coefficient exceeding unity without flutter if 1/ a i Cr . is low enough: however, as the classical flutter speed is approached, the flutter lift coefficient becomes less than unity. In other words, the amount of stall necessary to excite flutter is less as the classical flutter speed is approached. This conclusion is in agree- ment with the experiment of reference 2 and seems logical in view of the various flutter modes described by figure 5' The minimum flutter speed for a completely stalled propeller is of importance for propellers that operate in such a condition at times - in take-off, for example. CONFIDENTIAL NACA ACR No. L5F30 CONFIDENTIAL 11 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 Karman vortex street. In figure 6 it may be observed that this minimum appears at — S— a . 17 for ^cr 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/^cr as low as O.OI4.. 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 Ci^ for a constant propeller speed at —2— = 0.37* This twist was observed ^cr by means of telescope and stroboscope. The line on the figure is drawn through the point for an angle of twist of 0° and a value of Ci^ of O.37, as predicted by fig- ure 2. The slope of this line, which may be calculated by use of equation (3). is adjusted to fit the data. Equation (3) is based on a simplified propeller and gives the twist at the representative section. At Cl„ = O.78 and -iL. = O.37, equation (3) gives Aa = 2:l\° . The ^cr observed twist at the propeller tip for this condition was 5 • 1° • It w as observed by direct measurement that the torsional stiffness at the representative section was 2.3 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 J4. (p. 17 ) which is CONFIDENTIAL 12 CONFIDENTIAL NACA ACR No. I/jFJO (9) Reference I+ (p. 17 ) also gives an approximate flutter formula that appears to hold, very well for a heavy wing and for small values of coh/^a " tiie conditions for normal propellers. The approximate flutter formula is tow a (10) where x a 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 x a in formula (10). If the center-of -gravity ' location coincides with the location of the elastic axis," x a = 0, .the two formulas are iden- ti cal . It was shown in the discussion of figure 1 that there are two moments acting about the elastic axis, the aerodynamic-force moment and the centrifugal-force moment. If the aerodynamic 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 vp ~ 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 + x a , is expressed in terms of the semi- chord as measured from the midchord position. Equa- tion (10) may be written CONFIDENTIAL NACA ACR No. L5P30 CONFIDENTIAL 1J 1 v f _ . |rcT k 'a (11) D00„ HC 1 X - II- where x is the location of the center of gravity in fraction of chord as measured from the leading edge. Since vr> s vf , 1 2 ^cr p -l 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 compressibility correction for the subsonic range from reference L is M ^ ~ M . 2 In ~ Mi fc 1 n . !l 2 M • ^ M i i \ 2 8 where M c is the Mach number corresponding to flutter speed in compressible flow and M^ is the Mach 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« ^cr 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 semi span as the representative section for wings. Because of the velocity distribution on propellers, the representative section was taken at the O.G-radius station, CONFIDENTIAL Ik CONFIDENTIAL NACA ACR No. L^FJO \ CONCLUDING REMARKS 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 Committee for Aeronautics Langley Field, Va . CONFIDENTIAL NACA ACR No. L5F30 CONFIDENTIAL 15 APPENDIX SAMPLE CALCULATION OP FLUTTER SPEED CORRECTED FOR COMPRESSIBILITY FOR PROPELLER A Propeller section characteristics at .8-radius' station: Type - Clark Y flat-bottom 12-percent-thick airfoil x = O.lilj. r a 2 = 0.2^ Specific gravity = 0.5 _1_ ~ = 0.093 Propeller characteristics: R = 1.87 ft co a = 2rr (355) , b = ~^^( hzl ) ( . 098 ) = 0.092 2 2 R \ 2. J a v f s bu) a t J- ■ 2 f K 1 x - k (0.2li.)(i4.5)(0.25) S (0.0Q2)(2rr)(355) l/-— , , l ■ = 7?2 fps y 0.144 - .^5 CONFIDENTIAL 16 CONFIDENTIAL NACA ACR No. L5F30 \ Correction for compressibility: 772 ?,;. = J-J — =0.69 1120 c= i/»i a Ci-|»i**|«^ \ Vf = (0.6l2)(1120) = 685 fps Ar.V /685V V J = 0.79 y " V772/ This value represents the flutter speed corrected for compressibility. (See fig. 6.) REFERENCES 1. Theodorsen, Theodore: General Theory of Aerodynamic Instability and the Mechanism of Flutter. NACA Rep. No. li.96, 1935. 2. Studer, Hans-Luzi; Sxperimentelle Untersuchungen uber Flugelschwingungen. I.litteilung no. if., Inst. Aerod. Tech. H. S. Zurich, C-ebr. Leemann & Co. (Zurich), 1936. 3- Theodorsen, Theodore: On the Theory of Vving Sections with Particular Reference to the Lift Distribution. NACA Rep. No. 385, 1931. I4.. Theodorsen, Theodore, and Garrick, I. E. : Mechanism of Flutter - A Theoretical and Experimental Inves- tigation of the Flutter Problem. NACk Rep. 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