M'Tf^l^i """^ CO CO g NATIONAL ADVISORY COMMITTEE S FOR AERONAUTICS TECHNICAL MEMORANDUM 1364 THE PLANE PROBLEM OF THE FLAPPING WING By Walter Bimbaum Translation of *Das ebene Problem des schlagenden Flugels* Zeitschrift fiir augewandte Mathematik und Mechanik, Band 4, 1924. Washington January 19^4 ITYOF''' •'-'"'nA MTQ P' 'lENT E LIBRARY P.O. BOX 11701 1 GAINESVILLE, FL 32611-7011 USA 79 f ni ?^ ^nto^u^ NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL ME^DRANDUM I36I+ THE PLANE PROBLEM OF THE FLAPPING WING*^ By Walter Birnbaiom In connection with my report on the lifting vortex sheet^ which forms the essential basis of the following investigations, I shall show how the methods developed there are also suitable for dealing with the air forces for a wing with a circulation variable with time. I shall, in particular, develop the theory of a propulsive wing flapping up and down periodically in the manner of a bird's wing. I shall show how the lift and its moment result as a function of the flapping motion, what thrust is attainable, and how high is the degree of efficiency of this flapping propulsion unit if the air friction is disregarded. Finally, I shall treat an interesting case of dynamic instability for a spring- suspended wing; this phenomenon was confirmed by experiments at the Gottingen aerodynamic test laboratory. Professor Prandtl gave me his guidance concerning the present report, and I want to express here also my sincere gratitude for the abundant stimulation and energetic assist- ance he gave to me at all times . 1. General statements .- The calculations refer to the two-dimensional problem, that is, to the wing of infinite length, or bounded by plane side walls, or to a wing with so large an aspect ratio that the boundary effect is negligible. For the rest, the same assumptions as in the first report are valid concerning smallness of the air forces, slight camber of the wing, and so forth. The wing is ass-umed to extend from x = -l/2 to X = +3/2 so that the point x = becomes the center of pressiore of a plane wing with fixed angle of attack. Let v, in the direction of the positive X-axis, be the air velocity at a large distance. Let the posi- tive Y-axis point downward. By 7 = 7(x,t), I denote the density of the lifting vortices (now a function of the time) . At every variation with time of the density of circulation, free vortices of the density e(x,t) will separate from the lifting vortices and will drift away with the air flow. On the basis of the theorem about the conservation of circulation, or by integration of Euler's differential equation once along the pressure side and once along the suction side of the wing (compare the unabbreviated "Das ebene Problem des schlagenden Flugels" Zeitschrift fur augewandte Mathematik \md Mechanik, Band \, \S2}\, pp. 277-292. Abstract from the Gottinger dissertation of the same title. Avail- able in the University library at Gottingen and the state library at Berlin. Referent: Professor Prandtl. ^This periodical (Z.f .a.M.M.) , vol. 3, 1923, pp. 290-297. 2 MCA TM 136k report) , one finds easily that the vortices must satisfy at every point and at every time the following continuity eq.uation of the vortex density-^: ^ + M + V ^ = (1) St bt Sx Generally one will make suitable assimptions regarding 7(x,t) so that e may be found from the equation. If by/dt is designated by 7(x,t), its integral is e(x,t) = - ^ / 7(1, t +'i^)d| (2) ""^^Cx-vt) ^ ^ ' Where 9 is an arbitrary function to be determined by bovmdary condi- tions. After e has thus been fo\ind, there results the induced vertical velocity in first approximation as principal value of the integral w(x,t) = J- P 2_L1 d| (3) 2n J_ X - F Similarly to the former case, w now has to be put in relation with the moved wing contour. If the motion of the wing is indicated by the value of the ordinate at the point x, thus by y = y(x,t) (with this formu- lation one could also include a moved and simultaneously deforming wing) , the kinematic boundary condition reads ^ + V ^ = w(x,t) iS) dt dx ■^Details on the derivation of this equation may be found in a lecture of Prandtl "On the Formation of Vortices in the Ideal Fluid" at the hydro- aerodynamic conference at Innsbruck 1922. (Lectures concerning the field of hydro and aerodynamics, edited by Th. v. Karman and T. Levi-Civite, Berlin I92U) and in the original report quoted. MCA TM I36U This is the equation from which contour and motion of the wing are obtained if above said y , and therewith w, is prescribed. In analogy to the i|r(x-vt) wU,t + 1 > (9) MCA TM 136k J-l X - I J-1 X - I Ji ^^: — ^) from the three first fundamental functions^ ''0 = ^''Oa ^ ^^b ^ ^''Oc Where a, h, c now are complex nimibers. One has then also w = w^e^ w^ = aw. + hw., + cw^ Oa Ob Oc The integrals (9) and (lO) cannot be evaluated by elementary functions. The last integral (lO) in particular leads to an integral logarithm. The Euler constant C which appears as a consequence always occurs in combination with log 2 so that I shall introduce as transcendent number Z = log 2 - C = 0.11595 • • • especially for the present report. If one takes this fact into con- sideration one obtains the w in the form ^Oa "= ^0 ■*" °^l"^' "^ °''2'^' -'■°S ^' ■*" '^■i'^'^ + a^oo'^log cjd' + agoj'^ + a£ao'3log ui' (11) One obtains ^a =\ ^ - X ^ - Ji _ ^2 1 + X 'Ob = \|l - x2 7q^ = x^l - x2 NACA TM I36I+ and corresponding expressions with p^ and 7^^ are valid for Wq^ and Wq . The coefficients aj^, Pj^, y^ are polynomials in x the coefficients of which are linear and rational in Z (regarding their values compare the original report) . Besides^ the appearance of log co' shows that a treatment of the problem with a lifting line instead of the supporting surface must fail since to the transition to the lifting line there corresponds a transition to u)' = (v increases arbitrarily for decreasing wing chord) . For this reason, the theory of the flapping wing here presented is the simplest possible one. Bypassing the integral for y, we are now concerned with determining the wing motion in an appropriate manner and bringing it into accord with the expression for 7. Since the methods of the lifting vortex sheet are linear and since a deformation oscillation will not be taken into con- sideration, shape and mean angle of attack of the wing may be disregarded: It is then sufficient to calculate the oscillations of a plane wing with the mean angle of attack and to superimpose these oscillations as small fluctuations with time of the profile chord linearly on the arbi- trarily prescribed profile. For the amplitudes which are assumed to be small, one may disregard the fluctuation of the abscissa of a wing point as small of higher order, and has then as the most general motion still possible (compare fig. l) y = p(t) + x9(t) (12) p and cp are to be developed as Fourier series with respect to cdv; I retain only their first term y = (A + Bx)^'^ (15) Higher-harmonic oscillations would again have to be superimposed linearly. Fluctuations in the flight direction (which could be expressed by periodica fluctuations of v) also will be disregarded. I am designating the quantities A and B as complex stroke ampli- tude and amplitude of rotary oscillation, respectively, since equation (13) is formed by combination of the special translatory and rotary oscillations A = 1, B = (a) and A = 0, B = 1 (p) Thus 7q and all quantities linearly connected with it will be linear and homogeneous in A and B. For simplification of the calculation, the cases (a) and (p) may therefore be calculated separately and after- wards be combined linearly, for instance a = Aa^, + Ban etc. MCA TM 13614. 7 The complex circulation coefficients are found from equation (h) in the following manner: a, t , c are expressed in the form of the series 2 ? a = a + a o)' + a od' log oo' + a cu' + a.co'^log o)' + 2 2 ^ ^ ' log CD' + a/'cu'-' + aToa'-'log od' + ac-cjD ' log Yoo'-'log aoOD'-^log 0)' + aQ(JD'3log3oo' + . b = bo + c = Cq + . . (11^) Equation (k) has in our case the form w(x,t) = (Aid' + Bco'x + B)ve^'^ = fawQ^ +bwQ^ + cwQ^)e' GD'vt This is to be valid identically in x and t. If both sides are interpreted as series in co'^(log od')'^^ their coefficients must be identical, a, b, c therein are unknowns. Each coefficient yields a relation in x which at first cannot be expected to be identically f ulf illable by only three free values . It was intended to fulfill the condition at least at three suitably selected locations X]_, Xg, and Xo. Surprisingly, these conditions are now shown to be precisely of the second and not of a higher degree in x. At least this is the case up to the terms of third order, and there is no doubt that this will remain so for the higher terms as well. The coefficients of second degree in X determine, therefore, with their three subcoeff icients each one triplet of values, each of the a, b, c. In every new comparative coefficient there appears a new value triplet of this kind so that the thirty imknowns for which a formulation has been set up may be foimd 8 MCA TM 1364 uniquely from linear relations by successive evaluation. The calculation results in the following values: ao = 2Bv ai = 2vfA + ^ B(1 - 2Z)] ag = 2Bv [Taz - b(z - z2)] a^ = 2v l-AZ - B ai^ = 2v[a + B(1 - 2Zy| a^ = 2Bv ag = 2v AZ2 - Bfi + i Z - | Z^ + z3 (See footnote 5.) a^ = 2v p2AZ + B(i - Z + J^zA ag = 2v[a + B^l - 3Z)] an = 2Bv bo = b-j_ = Ij-Bv b2 = b, = 2v(a -h I b) 3 Cq = c-L = C2 = Co = Bv b^ = Ci = 0, i ^ Ij- The dimensions of all quantities are affected by the selection of the wing chord as unit length. A treatise of the author in the Zeitschrift flir Motorluftschiffahrt on the same subject has been arranged so that no objections are possible from the viewpoint of similarity mechanics . MCA TM I36I4. I want to point out here briefly that approximated values for the circulation coefficients are obtained also, if the effect of the free vortices is disregarded and the elementary calculation made in such a manner as if the momentary apparent angle of attack of each wing element were decisive for its circulation, namely the angle formed by the air velocity v relative to the moved wing element and the direction of the latter. Since in case of rotary oscillations (B j^ O) every wing element has another vertical velocity, the apparent angles of attack of the elements are all different, that is, the wing assumed to be plane behaves as if it had an apparent (dynamic) cuin/-ature which is periodical. The simple calculation (compare the original report, third part-, beginning of section II) yields accordingly circulation contributions of the two first fundamental functions, that is, only the following terms: aQ = 2Bv bQ = i^ = 2v(a + I b) b^ = 2Bv Thus except for higher terms and with consideration of the order of magni- tude of B (see below) a good approximation is obtained. All the rest follows readily from the circulation coefficients. I had introduced the quantity od' in order to enable an easier calculation and consideration also of complex vt = pv^rt^Am^^ + Bm3)e^'^^^ = pv^it^AmQ^ + Cm~)e = pv^jtcjD^Aii^ + CiSyj e .iojvt iODVt (15) 10 MCA TM 1364 Therein B = Coj; k^ = oikpj m^ = ojnp; k^^ = dk^', 1% = 'iUTa; kp = kyj mp = HLy I have introduced here the quantity C as new amplitude of rotary oscil- lation. This was done because in case of ordinary flapping motions B is of the order of magnitude oA as a simple consideration shows. The value of 00 is assumed to be small; thus A and C will be of the same order of magnitude so that calculation with C instead of B will be more convenient in practice. The third form of the air forces permits, for cases of equal stroke velocity Aioie-'-^^, comparison of these forces in a simple manner; the roughest approximation theory wovild yield constant coefficients k^, = 2i, k^ = 2. The coefficients k and m are complex numbers the constituents of which are given by the fol- lowing series developments . It is noteworthy that the series for m are finite. ^a = '^0' -^ i^a"» % = "V' + "\j"' <^ = '^'^'7 c^" a^' = k^' = -(1 - 2Z)a)^ - 2a) log od - ZnZay' + 2jta)3log cd + = ka_" = 2a) - na)^ + V^ " ^^J^^ + ^2a)3log o) > (16) 20)^ log 0) + MCA TM 136!^. 11 kg'=ky'=iky'=2-na)+ i^l + 2Z - 2Z^ - iW^ 2(1 - 2Z)ao2log CO - 2co^log^a3 - (l- - 2. + nZ - ^nZ^juy' + jt(l - 6z)aPlog 0) + 3i^<^ log ^ + • " _ V " _ 1 = k " = i ky" = (3 - 2Z)a) + 20) log 00 - rt(l - 2Z) ay 2n:cjo log cjd + lit. I \h 2 + Z - I jt^^ _ z^ + 2z3)co3 + (I jt^ - 1 + 2Z - ez^jcD^log CD - (1 - 6z)a)^log CJD - 2aPiog3cD + > (17) ma,' = aria,' = " 2 "^^ (18) mp ' = nty ' = j5 m^ ' = - ^ 3 ,,.2 CD nin ™ " 1 ™ ' = CD (19) It suggests itself to represent k^ and m^j by the initial posi- tion of "time vectors," visualized as rotating, in the complex number plane (as customary in alternating-current techniques) and to combine from them - with the parameters A and C (of which A may be assumed real without impairing the generality) - linearily, in the known manner, the ajnplitude coefficients k = Aka, + Cky (and correspondingly m) according to magnitude and phase. The diagram (figs. 2 and 3) shows the 12 MCA TM 136k curves of the end points of the vectors k and m as functions of the parameter cd. In the representation of the cvtrves for k and m which would yield the most accurate values for the graphical evalviation, the curves of the various approximations have been plotted side by side for comparison of the convergence of the series . Of particular interest is the case where the wing - without being affected by significant air forces - glides over an undulating streamline course, clinging to it as much as possible. To this corresponds the parameter C = -iA, which in fact yields small air forces of second order in o), namely i k' = 2^0^ + . . . i k" = [i - 2Zja)3 + . . . i m' = i cd2 i m" = a 0)3 A 2 A 8 3. The induced drag .- The induced drag is no longer linear in the circiilation so that the complex method could not be retained without new stipulations. It offers no longer any simplifications, and it is advisable to continue from here on the calculation with the imaginary constituent of all quantities in real form. If the calculation is carried out according to equation (7), W assumes the form W = Wq + W-L sin(ajvi: + q)j_) + W2 sin(2(jmrt + cpg) Here W]_ is different from zero only when the oscillation is superimposed on a constant angle of attack different from zero. W is in 00 small, of second order. The piirely periodical terms are, therefore, hardly sig- nificant; however, the temporal mean value Wq^ which is different from zero is important. I calculate only this value and write for it, for reasons of simplicity, again W. I equate A = A' + lA", B = B' + iB", C = C + iC" and may assiome, without restricting the generality. A" = 0. Then W becomes a quadratic form in A' = A, B', B" (or A, C, C"): a simple deliberation shows that the coefficients of B'2 and B"2 (or C'2 and C"2) are equal and that the coefficient of B'B" (or C'C") is zero. Thus W becomes W = pvSnJA^w^ + 2AB'w^p' + 2AB"w^p" + (B'2 + B"2)wpp| «|a2w^ + 2AC'w^y' + 2AC"w^y" + (C'2 + C"^)v^2. 2 (20) MCA TM 136k 13 The Wjj^ are again series in cjuP(cu log u))™ and have up to higher terms the values v^ = -CO' i2 + jta)3 _ /I jt2 - Z^joD^ - 2Z6D^log ao + OD^log^o) + . . , V, ap w. CD 0.7 i (3 - 2Z)cjo2 - i oi^log o) + 4 -^ 2 i (3 - 2Z)(xr' + i cD^iQg (j^ + !<. 2 ^ap - ^ ^a7 1 3 2 2 1^ fl «2 . 1 ,2 ^ 1 ,\ V8 2 2 / 3 i (1 - 2Z)co3log 00 = i a>3log^a) + 1 ^..2 >(21) '3P - :j ^77 - -^ '^ - ^ -' ■ ^'" '8 (n^ + 1)cd2 + 1 (3Tt2 + 2 - Uz2 - 4z)co3 + 00'- i (1 + 2Z)cD3log CJO - i (JO^log^OD + 2 / B 2 W may be positive or negative, according to the selection of parameters. Depending on the type of motion, one has, therefore, to expect drag or thrust. I postpone detailed discussion until after calculation of the power requirement and the efficiency. The case of gliding over an imdulating flow mentioned above, corresponding to C ' = 0, C" = -A, gives W = A^pv^jt -cD^log o) - (i - zU^ > Thus the selection of parameters made does not yet correspond exactly to the case W = 0; a small correction would have to be provided for this case. 11^ MCA TM 1361^ k. Work done at the ving .- In the free vortices hehlnd the wing, energy is contained which must be produced by mechanical work on the airplane. This may be done in two ways. The flapping motion may resvilt, as mentioned before, in positive or negative thrust. In the case of neg- ative thrust a propeller which overcomes this and all other resistances to flight is required for maintenance of equilibrium of motion. In the case of positive thrust a propeller is needed only until the thrust due to flapping exceeds the resistances to flight, whether the flight be uniform or accelerated. As to the work performed at the wing itself, the wing motion consumes, of couse, energy if thrust exists; the ratio of thrust power -L^. = -Wv and the total mechanical power L^ to be applied to the wing may then be denoted as aerodynamic efficiency of the flapping wing. In case of drag, two more possibilities exist. First, the flapping motion may require additional work . The efficiency defined above then becomes negative and arbitrarily large when the wing power L^ decreases more and more. Second, the case may occur that Lf becomes negative, that is, the wing then is supplied with energy from the air (indirectly by the propeller) and may vise that energy for surmounting the resistances in the oscillation mechanism, or may store it in the oscillation itself, that is, increase its amplitudes. Aside from this "incremental power" the propeller must, of course, in this case yield additionally the energy of the free vortices so that one may define the quotient of -Lf and the total propeller power as efficiency referred to the power absorption of the wing. This is then exactly the reciprocal value of the efficiency defined above which in this case, as the quotient of two negative numbers becomes positive but larger than one, thus loses its physical meaning. Lf is divided into two parts . One has -^ = p + 9X = ia3v(A + Bx)e^"^ ot Therewith L-^ = Kp becomes the "flapping power" and L^, = Mcp the power opposed to the rotary oscillation; thus the wing power is Lf = L^ + Lr* The power opposed to the drag is denoted by I^ = Wv. In the air there then remains in all L=Lf+Lyr = L^+Lj. + Ly. I indicate of all Lj^ again only the temporal mean values for which I obtain qioadratic forms of the same type as for W Li = pv^nJA^Zi^ + SAB'Zi^p, + 2AB"Zi^pM + (B'2 + B"2) lippj y (22) = pv^nh^l^^ + SAC'Zi^^. + 2AC"Zi^^, + (C'2 + C"2) Z^ 1 NACA TM 136k 15 L]- has finite series and is, in general, small compared to L+. ike coefficients are individually ^wik ~ '"'ik (25) ^toa = "^^ " f ^^ "^ (tT " ^^1^^ ■*" 2Za)^log oj - cjo^log^cu + Ha3' = I Ha/' = J (3 - 2Z)a32 ^ | cD^log co i (1 - 2Z)od3 - ^ OO^log ^3 + 4 2 •, 1 ■, 1 Jt 2 /it'^ 1 „ 1 ,,2 1\ 3 tap ta7 2 U V8 2 2 8/ i (1 - 2Z)cL3log 0) - i CD^iogS^^ + Hpp - ^ H77 - ° > (2i|) ^roa = Wp' - ~ Va7' - ° 1 1 ^ ^v~^q" = — ira-y" = rr cjD-^ rap' CO ''ra/ 7 - 1 7 _ 1 2 ^rpp - ^ ir77 ~ 2 ^ )- (25) 16 MCA TM 13614. ^foa = 0)2 . I 0.3 + {i - ^^) 4 i4- k 9 o) + SZfOO log CD - o) log O) + O) ^fap- = ^ ^far' = ^ (3 - 2Z)a)2 + i a)2iog 21 (1 _ 2Z)ol3 - i a)3log OD + 4 2 IfaB" = - ifa-x" =icD-«a)2+(4 + -Z-i 22^5 lap CO far 2 k V8 2 2 / i (1 - 2Z)a)3log CO -i ooSlog^o) + 2 2 1 1 1 12 or > (26) 3t o n'' k cux 2 2 1 ■^ Z„o I = — Z_-y I = — oi-^ + 1 , « 2 n^ 3 ^ II = _ CD - 00-^ + ^ap" - - ^ar 2 ^PP = ;^ Vr = f - Hi cd2 W I «3 ^ i 2 l8 4 2L Z'^ 2 21 zjcD^ + - (1 + 2Z)£D3log CD - - Co3log2(D + (27) NACA TM I36U 17 L, as the energy of the vortex trail, can of course never become nega- tive and must therefore "be a positive q.\iadratic form - definitive in the amplitudes . The fact that L in the form here noted is capable also of small negative values is caused by the neglect of higher terms in the series development. What is obtained in this case, is therefore only the error, accidentally negative, of the almost vanishing vortex power. p Since, when C is used, all coefficients contain the factor o) , the latter has been cancelled in the graphical plotting which is in agree- ment with the presentation of the power for constant flapping velocity (fig. k) . The c\arves show that the essential terms always stem from the stroke amplitude A, possibly in combination of the latter with the amplitude of rotary oscillation, and that the corresponding coefficients increase somewhat more slowly than o)^. By rotary oscillation I meant above an oscillation about the z-axis. More generally, every oscillation where the ratio of A and C is real is a rotary oscillation about the fixed axis with the abscissa a where A then is A = -aCoj. It is shown that for not too large values of am, that is, for axes which do not lie at too great a distance, W and L are always positive; that is, it is not possible to obtain thrust or power absorption by rotary oscillations about fixed axes. Production of thrust always requires a stroke amplitude different from zero, power absorption req_uires addi- tionally a rotary oscillation lagging by about 90°- Pure stroke oscil- lation without rotation also produces thrust which in roughest approxi- 2 2 2 mation results as -W = A pv nco , similar to the so-called Knoller-Betz effect (if one calculates with the y-axis as the "polar" in the plane problem) . One obtains good insight into the variation of drag and power if one varies, for fixed absolute value of the amplitude ratio C/A = c, only the phase angle cp between the two oscillation components where one then has to put C = Ac' = a|c| cos cp, C" = Ac" = A[c| sin cp W is shown to become a minimum, the thrust thus a maximum, when the rotation leads by somewhat more than 90°, namely by cp = arc tan — —. It is plausible physically, too, that the thrust will assume large values precisely then when the phase is shifted by about l80° compared to the phase which is present for gliding free from air forces over the wave course. For every |c| there exists an 00 and vice versa for which the thrust maximum is absolute. One then has 18 MCA TM I36U ^77 W ti c (O)) = ^min = A^pv^rtWj^in ^77 = _ia.- J-a.2_ J_(l+ 3„2)^3 _ 2jt2 2rt3 L becomes a minimum and disappears for suitable co except for terms of the fifth order when the rotation is lagging by somewhat more than 90°. Finally, the wing power L|. becomes a minimum - thus, the absorbed power a maximum - when the rotation lags by somewhat less than 90°; namely for ^far" cp = arc tan - — '—. This maximum too becomes absolute when between c ^fa7' and CD the following condition is satisfied ^f77 c' (cc) = - ifyy ^fmin = ^^P^^-^l-fxaln Va7' ^ ^far"^ rt ifmin = ^foux = -1 + 2 CO + 'f77 Altogether, L becomes negative only when |c| > 1. MCA TM I36U 19 The efficiency , r J T)-]^ is valid for Lf > and T^, = — = - — = 1 - — where ^2 Lf Lf ^2 ^°^ ^f "^ °- > (28) is the quotient of two q_\iadratic forms and capable of a great many values . Generally, it "becomes negative for small or completely vanishing stroke amplitudes, and arbitrarily large with uj — ^0. The same is valid for rotations about a fixed not too remote axis. If, however, k f Q and also lim A j^ 0, t] approaches for o) — >• to the limit 1. (0=0 By a rotary component (c" = O) of equal or opposite phase tj is always deteriorated compared to c = Oj the same is true for c ' = 0, c" > 0. In contrast, the efficiency is improved for c ' = 0, -1 ^ c" ^ 0, as can be seen from the diagram (fig. 5). For c' = 0, c" = -1, T] becomes identically 1, and for c' =0, c" < -1 there results power absorption. The representation for fixed |c| in depend- ence on cp as a discontinuous single-wave-harmonic f\inction is very graphical for the efficiency as well. Figure 6 shows clearly at what phase angles the transition from power absorption to power production takes place. The most important ones among the coefficients found from the series developments have been compiled in the numerical table (table I) . 5. Application to the flutter of elastically supported wings .- The derived laws could be practically applied in the investigation of a phenomenon our pilots observed in the last war. In the so-called sesqui- planes, the lower wing was fastened to one single spar only, thus was only slightly elastic against small deflections and rotations. In case of increased flight velocity, for instance in steep dives, there occurred sometimes vigorous flutter of the lower wing tips which underwent obviously unstable oscillations in the increased air flow. Of course, such unstable oscillations are possible only if energy is supplied to the oscillating system, and tMs occurs, according to ray investigations, only when the vector of the ajtiplitude of rotation lags by about 90° and when the ampli- tude of rotation itself is sufficiently large i\c\ must be > |A|j. Let us visualize again the gliding - almost free from air forces - of the wing over an undulating flow course where the airspeed (relative to the wing elements) has no vertical component. If the amplitiode of rotation is smaller than corresponds to this case, the motion is damped by the counteracting air force. If the amplitude of rotation is, on the con- trary, larger than in the case above and the wing therefore scoops more deeply into the air, the air force always acts in the direction of the motion, and the motion is excited. 20 MCA TM 1361^ I consider a wing supported on spars, elastic with respect to trans- lation in the y-direction and with respect to twist. In order to be able to go on from my formulas used so far, I introduce the directional forces per unit length in z-direction; I calculate therefore as if these forces were distributed continuously over the length of the wing. This assump- tion does not lead to any contradictions if the wing in itself, aside from its support, is sufficiently stiff. Since I disregard deflections in the flight direction, I can show that a wing supported on spars always has only three essential elasticity parameters, corresponding to the three constants of the work of deformation quadratic in p and cp . (Compare fig. 1 and the original report.) With respect to its elastic properties, this wing may therefore always be replaced by a wing which is supported only on one spar (the "elastic axis") with the abscissa a, elastic with respect to translation by means of the directional force c, and with respect to rotation by means of the directional moment 7, as schematically indicated in figure 1. The resultant of the elastic forces and its moment at the origin then are K = -c(p + acp) M = -cap - (ca^ + 7)

p,_^o _o\2j.^2j.p,2 r frequency - oc^. With 5 = (s - a)^ + r2 + q2 there i£ 03. .2 . ^^ ^.2 ^ IU_^g ^ \|52 .. I.q2r2) ,2 .'^ ^1.2 1.2 2r' A B 2ar' 1^6 ± ^6^ - i^q2r2j & ± \S^ - i4-q^r2 - 2r2 = -a 1.2 y^ = B^{x - ak)ei'%^j k = 1.2 (30) 22 NACA TM I36U The main oscillations therefore are rotary oscillations about fixed axes at the distances a^^ag. The quantity s - a forms a measure for the coupling. In general, there develop beats from both main oscilla- tions. If I write those in the form y = -a-]_B-]_e ievt agBge-i^^ + X |B,el ievt ^ B2e-"te^ 9} ,ic0\rt tD2_ + CDg = 2a) a>]_ - (02 = 26 « 0) The motion may be interpreted as an ordinary oscillation with an ampli- tude ratio slowly variable as to magnitude and phase. Therein there appears of course, periodically recurrent, the phase angle which corre- sponds in the air flow to the power absorption. If the air forces are to counteract the change of this phase angle, corresponding to con- tinuous supply of energy, it can be shown that the case of slight coupling (s - a small) for balanced or almost balanced frequencies of the uncoupled system is the best presupposition for this phenomenon. I am anticipating from the results of the following calculation with consideration of the air forces that without coupling (s = a) no unstable oscillation at all woiold be possible. The air-force coefficients occurring in the oscillation determinant are themselves functions of the frequency o)' which cannot be indicated by simple analj'-tical expressions so that the roots of the oscillation determinant cannot be obtained in a simple manner. However, since their existence is secured by the physical meaning of the problem, it is per- missible to introduce into the eqnoation instead of the coefficients k and m, the first terms of their series developments, all the more so since the occurring factor pQ generally is a small number; for it is sensible to break off the series for cos x in order to find from the polynomial obtained for instance approximately the first zero of this function whereas the same method is meaningless for e-^, since no zeros of this function exist in the finite domain. The oscillation determinant obtains the following form r^cD'^ + Scjuq^o)'^ + q^a:^3^ + PqCd'^ (r^ + s^)k^ - sko - smj^^ + mT] + Po'^O^il^^ ■*■ ^^^^a, - ^p - a^ + nip] + Po^(^"ip - ^p°t(,) = ° (31) MCA TM I36U 23 The roiighest approximation is the result of the calculation which bases the determination of the lift and moment coefficients - in the manner of the theory of the Knoller-Betz effect - on the momentary apparent angle of attack and the apparent (dynamic) curvature of the wing. Even with this procedure, there result in certain cases complex roots gd' with positive real parts which resiilt in an "increment" of the oscillations and thus correspond to dynamic instability. I shall not here discuss this approximation more closely. Also, I shall only briefly mention the case of the greatest instability since this case is trivial. It occurs if the elastic axis lies so far to the rear that at the slightest displacement of the wing from equilibrium position the air flow simply causes the apparatus to tip over aperiodically toward the rear. This always occurs as soon as a is positive and the air velocity sufficiently large, namely for a > or v^ > 2pQ 2pjta for a > 0. If I retain of k and m all terms up the second order in go', the period-equation of the wing becomes f(a)') = aQU)'^ + aQ'oj'^log co' + a^'ui'^log^cD' + bQOo'^ + bQ-co'Sipg cu' + CqU)'^ + CQ'CD'^lOg U)' + CQ"a)'^l0g^CD' + dQGD' + dQ'oo' log CD' + Cq = (32) With the coefficients ao = r^ + p^6 + PO^ -^ Po'(i + i Z - Z2) > bo = Pq^ + Pq^(^ - z) > Cq = CDQ^^g + p^v) - 2PqS + Pq^ dQ = cDq^PqX > 2k MCA TM 13614- - ^ 2 / 2^ 2 '0 V- en = ^n (l^n " SpQa ^0 = PQ 2(r2 + s2) - 2s(l - 2Z) - p^^i - 2z\ ^0" = "PQ^^^ ■" PO^ ^0' = -Po(2^ - Pq) Cq' = 2p(^2p + q^2 _ g^^ _ 2z)J Cq = do' = -2poao^' 5 = (s-a)2 + r2 + q^2>o ^ = 1 + 2(r2 + s2) _ s(3 - 2Z) > X = 1 + 2(a2 + q2) _ a(3 - 2Z) > e = (r2 + s2)(l - 2Z) - s(l - 2Z + 2z2) + | > 8 V = (q2 + a2)(l - 2Z) - a(l - 2Z + 2z2) + 1 > 8 z = 0.11593 MCA TM 136k 25 6. Numerical evaluation .- The equation can be solved only approxi- mately, of course. For this purpose, I first omitted the logarithmic terms and determined the roots c5' of the algebraic epilation (cjd') = agO)'^ + bg^'3 + CqCjo'^ + dQOj' + Cq = (33) g I regard this equation as an approximation, equate cd' = co' + X, and can now develop the logarithmic terms retaining the linear terms in X. Thus I obtain T.. R X = -od' log o)' I (3U) R = a'oi'^ + a "cD'^log 00' + b 'od'^ + c 'cu' + d '(l + od' log cd') S = imoCu'^+ aQ'a)'5(i + U log u)') + 2aQ"aD'3log a)'(l + 2 log o)') + 3bQa)'^ + bQ'aJ'2(l + 3 log cB) + 2cQa3' + Cq'od'(1 + 2 log co') + dg + dQ'(l + 2od' log d)')(l + log do') The procedure may be continued and yields the fiirther approximation R(a3')aL)' log o)' + g(cD') X' = S(a)') Finally, there follows the complex amplitude ratio B:A = b from one of the equations (29) . 26 MCA TM 1364 Finally one now has to learn the conditions for which the equation for cjd' has complex roots with positive real part, corresponding to excited oscillations or critical support of the wing. For the limiting case of dynamic indifference, I formulate the roots of the eqiiation (32) as p\arely imaginary and obtain, hy setting the real and the imaginary constituent of the equation equal to zero, the conditions aQ - — Slqjci:) + a^'cD log ao + a^"a)^loe^a) + b^ ' ^ o^^ 2 / Jt^ „1 2 ,2-, J I /« -, 2 ^ Icq - — c^ IcD - Cq (D log (D - d_'a3l— + (JO log 00 I a_' -^ (J^ + aQ"ncjD-^log 03 - bQ(ja - bQ'co log co Cp, ' T^ CD + d^' log cd( 1 - Jtoo) + dp, = + e^ = V (35) '0 2 From them ao would have to be eliminated in every special case whereby a conditional equation for any of the parameters s, a, r^ , 1 ) (i^o, pQ or for the flight velocity v results. As an approxi- mation, it is sufficient to investigate the equation (33) • Since the approximation (3^) shows in the case of instability, a small additional damping, the criteria for instability derived from the following equa- tions are necessary but not always sufficient; they are therefore valid only with the reservation of checking with the approximation (3^) • When they are satisfied, the wing may at any rate be regarded as critically supported. Before I set down the most general criterion for instability, I shall mention one which is very simple but suffices only for cases of great instability. This is the condition that the coefficient Cq becomes negative v2 > prt(2s - Pq) (36) MCA TM 136k 27 Written as a function of s , the same condition reads ? 2 2s (pQ + ^) Pq2 + 0:^2 /g^2 ^ j,2 ^ ^^2 Pqv) < Fiirther limits are yielded "by the Routh condition ^ ^0 or ^0 < ^0 — + ^^ — — -r c-. ^0 ^'^ 0^2 (S + Pqv) + Pq^ _ 2PqS < |^r2 + p^e + >(5T) H + PQ e.-) In order to satisfy this equation, it will above all be required that V be sufficiently large, that is, (Xiq sirff iciently small. Fur- thermore, s must be positive; therewith s - a is positive, too, because, as mentioned before, a > would for siiff iciently large v always result in great instability. One can readily understand that this must be so. The centroidal axis of the wing as axis of inertia generally lags behind the elastic axis as line of application of the directional force. If the centroidal axis therefore lies, in flight direction, behind the elastic axis (s - a > O) , the wing has in its upward motion, on the average, a positive angle of attack; the opposite is true for the downward motion. Thus the air forces always take effect in the direction of the motion and amplify the oscillation; whereas in the case s - a <0 the opposite, that is, damping occurs. It can easily be confirmed that both degrees of freedom must act together for achievement of the oscillation. The calculation always yields roots with negative real parts if one of the degrees of freedom is suppressed (corresponding to ovn = 00 or q2 = 00) . This fact is confirmed by the failure of tests undertaken formerly in Gottinger with a wing with only one degree of freedom. 28 MCA TM I36I+ The power L produced or, respectively, absorbed by the wing is according to previous formulas L^ = Prpnv^lf (38a) ^f = ^foa ''■ 2^'^fa3' "^ If I put temporarily co' = 3 + io), the mean value of the entire wing energy for constant amplitude A is — 1 2 2 Q = 0^^ + o)^ + 2b ' iaiar? + su? ] + 1 ' Uia^ + su?j (b'2 + b"2) \L^ + ^ ' *T^^ l: \ ' Figure 1.- Elastic suspension of a Joukowsky profile with plane "spine." 32 NACA TM I56I+ 0J5 \i \P-2i 1 MOJ X \ A >0,10_ 0.08 0.1 i- ?0.06 *0.04 fO.05' I >0.02" 0.15 =f m" ^' DO* i mn ^ - U.r 0j08i - r> r )05i U.V. 0.06^ T\ <°b^ 2aj 0,10 0.08 0.04 ji m' m^ ;j - 0.005 •0.001 1 0.05 0.10 0.15 k' 0.08 k^ OJO, Figure 2.- Vectorial lift and moment coefficients for equal beat amplitudes. NACA TM 156^4- 53 \ K ) I 1 a .3.)V ^ 0.1 9\ ^0.12 y u 0.1 r 0.10 •<3 ?^°.o», 0.02, -1 r\c r\r\A^^S^ V O.Ob ^-^ ^^r^js:::^ V^0.02 0.06^ 0.2 2.)'^ ia -H ir r.'^r O.Ot "0.04 1 03 1 1.5 .. ,i._„. J 1.6 1.7 1.8 1 1.9 ^y 2.0 Figure 3.- Vectorial lift and moment coefficients for equal beat velocity. ^ NACA TM 1564 Figure 4.- Power and drag coefficients for equal beat velocity. (1) co-2zfaa; (2) a."2zfc.7'; (3) a)"^fa7"; (4) ao'Szf^r; (5) (Jo"2wafl,; (6) Go"2wa7'; (V) co'^Wa?"; (8) 03-2^77. MCA TM 1364 55 hi -r ^^5 S k Uc;=-' b.. = n 1 ^?cr- ^ L ^, ^ •>-^ ^0^^ ^ -t \ s>. ^-^ r~£ N Kv ^ k^ "*< - 1 ^^ 0.5 s \, ^ kc=-2 r f\ h. ^ K ^ c"= 0, N k^c'--2 >>. ^ N 3s V, N 3^ 0.02 0.04 0.06 1 0.08 0,10 aj0.l2 Figure 5.- Degrees of efficiency as a function of w for different amplitude ratios c. 56 NACA TM 1564 X 1 I i X . V- -< L-< w- < ;^-< M 1^ 1 ■0.9 -0.7 ■ae ' V, ' ^ r^ k. iJ = 0.02- > C ^__^ ^-< M r J I M ^ ii C=0.5 90° 180° 270° ■P 360° 1 1 ()— 0— — (^— — i — — s> Figures.- Degrees of efficiency as a function of the phase cp, NACA TM 1564 37 Sliding weights )ross- shaped metal sheet EL ^ "•" f ■V \ ^ Laminated springs e Spring joint WW Eo Figure 7.- Test arrangement. 58 NACA TM 1564 2) \l rt 9 M C OS S 2 - S I d a s ^ « 5 s ^ < w ii< 5 03 Z N «hH ■a > ^ t. c 0) •- c o 3 u >;< o 0) o i-H i-H W e - (1.2 lutte s; S fa a. a "S a a g o s t* boo 2 b iMa Px ^H> u g c» g ■^ V x: ■a 5 n ■a sd" CM d Q- P. iS •n » c ■«j« H J= a CJ Ol < CO ^S ^~*^ C CJ -IS m z N rt 3 < «aS 1-t ^ CM 5^ M n 0) a a o CM 0) ;^i tj fa n to t s S2 5 § as 3 H ?l£'13 S<: Wing Thee Vibr 1^ o z t« =3S — X '^ ^-hS .1 w M 1 rH CM a) I 0) .3 *« S 3 § w f. rt bJ) o ■ B i^a fa > H > a to &2 13 2 s^ H .SCO s s ^ « I s 5 < M ^ S - m z N rt 3 <; «• d B ■5 <" u <: z 1 < 2^ I uaS !" d •" tH rt (N a> a o u I CM ^1 Q* -^ I a s ^ « J s « Z N «• d S C O o (u o •35 S ^ ^ CM HH-aB