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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( " .. 117011 ' :c. W FL 326117011 x2, , 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 illp: www.archiv\e.,rg details elleclollilcoelOOanq / .o' .? I  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 wndtunnel propellers are anal,:zed and results presented. It is shown that in the high anglenfattack 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 slniffcance. The classic.:l flutter speed and the divergence soeed of a orooellcr are snovrn to be approximately the same because of the centrifugalforce 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 windtunnel propellers for the Langley, Anaes, and Cleveland Labora tories of the NIA.CA. ,indturnel 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 nearstall condition in takeoff. 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 twodi.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. Selfexcited 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. CONfTFI DENTIA L NACA aCR No. L5F30 The first such oscillation appears when the propeller rotational speed reaches approximately onetwentieth 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 tiic: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 quarterchord point CL lift coefficient COtIFIDEIITIAL CONFIDENTIAL 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 notwist 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 CO;FIDE1TTIAL 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 CONFIDENTIAL 6 C1ON IDFNTIAL 1. 1CA At :) :". 0 The moment coefficient around the quarterchord 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 CONFIDENTIAL CONFIDENTIAL 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 rra be de..;ign:d to have zero t'"ist. A highly tapered pri:,eller tens to decrase its tip angle due to cen trifvLcal twisting ,ion.enr and a riorb 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 thb tlade. EXPRT"r'.1;AL STUDTIS OF F'LTTLR'T OP PROPELLERS AT TTTIH L i'I.7ITGS A number of pro.pellers of different designs, some representing ej.:isting "LC. windtunnel propellers for which data were available and others representing rro posed windtunnel 1iopellers, were tested as windtunnel fans in a small oon tunnel. A crosssectional 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 0.8radlus 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. Bladetip deflecion and twist and the pressure increase of the air passing through the fan were recorded. These data were used to give an ineDn'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.:' flatbittom Clark Y sections. Pr> peller A, for which data are given in figure LL, was a sixblade propeller 45 inches in diraeter. Propeller B was a singleblade propeller having the same diameter but CO ;FIDE TIA L CONFIDENTIAL NACA ACR Io. L5F50 twothirds 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 12percent thickness. The straight lines that converge at qq = 1.0 and CLu, = 0.57 in qcr CONFIDENTIAL CONFIDENTIAL 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 blae 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 uustioned, 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 secd 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 propllers Lhat operate in such a condition at timcs in tk':eoff. for example. C T T I AL 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 CONFIDENTIAL CONFIDENTIAL 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 centerofgravity'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 aerodynainiforce moment and the centrifugalforce 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 dynamicstiffness 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 0.8radius s.ation. CONFIDENTIAL CONFIDENTIAL 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 0.8radius'station: Type Clark Y flatbottom 12percentthick 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 CONFIDENTIAL CONFIDENTIAL ',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, HansLuzi: 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 edo '.ouinba.zj zl.+nt o ba C 4 %% I I + 4 Lai Z um SoT Vc x 10 0 3. 0 . E4 C L 0. 4 ,, 8 S0 m 0 S 4 00 . Q Q : 4 0 0 0 a a T I 9 OONm A 0 c _ Sop 'apnylTdW ae".4nlj uoTsjo, dTL * u 'apn4nldi ,a'lnTj Brpupuaq dT_. Fig. 5 S I. r. .1 5 I I II I 0 i UNIVERSITY OF FLORIDA *" :II I I 1 I11111 Ill l Iy 3 1 262081064783  " ~~~~ ~ ': 'i rl"'" "^N C , i ,I il ..!,:~ t ...:.;!i i ." ; I 4 *** . ii 1< 4 
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