Bending-Torsion Flutter of an Airfoil with Nonlinear Structural Characteristics Citation Peck, Jerry Clifford (1959) Bending-Torsion Flutter of an Airfoil with Nonlinear Structural Characteristics. Engineer's thesis, California Institute of Technology. doi:10.7907/P0DS-PX18. https://resolver.caltech.edu/CaltechETD:etd-02132006-152832 Abstract A theoretical analysis of the incompressible bending-torsion flutter of a two-dimensional airfoil with nonlinear structural characteristics is presented. The Method of Slowly Varying Parameters of Kryloff and Bogoliuboff is applied, and the steady-state oscillations are found. The stability of the steady-state oscillation is analyzed through the use of perturbation equations. A quasi-steady aerodynamic approximation is used, and closed-form solutions for the steady-state oscillations are found for the case of elastic torsional nonlinearities. A numerical example of a soft-hard stiffness characteristic is treated, and the steady-state and stability results given. Item Type: Thesis (Engineer's thesis) Subject Keywords: (Aeronautical Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): Sechler, Ernest Edwin (advisor) Caughey, Thomas Kirk (advisor) Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 1 January 1959 Record Number: CaltechETD:etd-02132006-152832 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-02132006-152832 DOI: 10.7907/P0DS-PX18 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 624 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 14 Feb 2006 Last Modified: 18 Oct 2023 23:25 Thesis Files Preview PDF (Peck_jc_1959.pdf) - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

BENDING.TORSION FLUTTER OF AN AIRFOIL WITH HONLIBEAR STRUCTURAL CHARACTERISTICS Thesia by derry C. Peck in Partial Fulfillment of the Recuiresents For the Degree of 4aronautical Engineer California Institute of Technology Pasadena, California 1959 ABSTRACT A theoretical analysis of the incompressible bending-torsion flutter of a two-dimensional airfoil with nonlinear structural characteristics is presented. The Method of Slowly Varying Parameters of Kryloff and Bogoliuboff is applied, and the steady~ state oscillations are found. The stability of the steady-state oscillation 41s analyzed through the use of perturbation equations. A quasi-steady aerodynamic approximation is used, and closed-form solutions for the steady-state oscillations are found for the case of elastic torsional nonlinearities. A numerical example of a softehard stiffness characteristic is treated, and the steady- state and stability results given. ACZNOWLEDGEMESTS The author wishes to express his sincere acpreciation to Professora BE. BE. Sechier, T. K. Caughey and W. W. Royea for their invaluable advice and encouragement in carrying out thie research. He is alao pleased to acknowledge mapy helpful discussions with fellow student B. R. Fleming. This research has been greatly aided by the Lockheed Leadership Fund, whose financial assistance is gratefully acknowledged. fo his wife the author expresses his appreciation for her skillful assistance in computation and in preparation of the marugeri pt. Ls Ii. iit. Ve VI. Vit. VIit. TABLE OF CONTENTS INTRODUCTION EQUATIONS CF MOTION 1. General Equations of Motion 2. Honlinear Structoral Forces 3. Linear Quasi-Steady Aerodynamic Forces 4%, Equations of Motion for Quasi-Steady Air Forces and Structural Nonlinesrities METHOD OF SOLUTICH 1, Introduction 2. Application of Method 3. Steady-State Oscillations 4, Stability of Steady-State Oscillations CLOSED-FORM SOLUTION FOR ELASTIC TORSIONAL NONLINEARITY RUMBRICAL EXAMPLE: SOFT}HARD QUIRTIC CHARACTERISTIC 1. Specification of Example 2. Steady-State Solution 3. Stability of Steady-State 4%, Discussion of Results GENERAL DISCUSSION AND CONCLUSIONS 1. Applicability and Advantages of S.V.P. Method . 2. General Features of Results REFERENCES APPENDIX: QUASI-STEADY AERODYNAMICS APPROXIMATION - ££ WwW w AR 35 37 1 I. IJHTRODUCTION Important aspects of the flutter phenomenon are basically none» linear in nature and this cannot be explained by the usual (linear) flutter theory. For example, linear theory predicts that once the oritical flutter speed is reached any small disturbance will initiate amplitude build-up without limit. However, limited amplitude flutter is a commonly encountered experimental fact. So also is the depen- denee of the stability upon the size of the initial disturbance, go that stability often persists beyond the linear critical speed antil a sufficiently large and sharp disturbance is encountered. These phenomena are typical of nonlinear systems in general, and indicate that the methods of nonlinear mechanics must be invoked to study such effects in detail. Previous work, both theoretical and experimental, has been done on nonlinear flutter problems. Analog computer studies of the effects of several typical structural nonlinearities on flutter were made by Woolston, et al. (1,2). A theoretical investigation, using the Method of Equivalent Linearization of Kryloft and Bogoliuboff, has been carried out by Shen and Hsu (3,4) and compares favorably with the analog computations, An aspect of the problem not previously considered is the stability of the various parts of the steadyastate amplitude- velocity carve. This is important because nonlinearities resulting in multiple-valued amplitudes are often encountered. A particularly convenient techique for the stability analysis is found in the _ Method of Slowly Varying Parameters, also by Kryloff and Bogoliuboff. In this method the problem is reduced to the solution of first order ordinary differential equations in the amplitudes and phases, so 2 that the perturbations about the steady state are governed by first order linear ordinary differential equations with constant coef. ficients. These equations are distinctly easier to study than the Mathieu-Hill equations resulting from the perturbation of the original second order equations for the displacements. In this study the application of the Method of Slowly Varying Parameters to the nonlinear flutter problem is treated, including both the steady-state and the stability analysis. Specifically, the incompressible bending-torsion flutter of @ twoudimensiional airfoil with structural nonlinearities is considered. A linear quasi-steady aerodynamic theory is used, allowing a closed form solution to be obtained. A closed~form solution for the case of uncoupled elastic torsional nonlinearities is presented, and a specific naswerical example is worled out. 3 II. EQUATIONS OF MOTION The equations of motion for the bending-torsion of a twoe dimensional airfoil are Fst mhimzybe+Ll =o (1) " Mc+La&+muzbh-mM= fo) (2) translational ("bending®) structural force, positive upward = 5 9 ae i BS & rotational ("torsicn”) structural moment, positive sounter~clockwise mase of the airfoil a a i. = mass moment of inertia about center of rotation io ] aerodynamic lift, positive upward YW = serodynanie moment, positive clockwise; the remaining symbols are shown in fig. 1. 2b b- ab ete xb | | Mean position Displaced | Elastie axis Center of gravity Fig. 1 Description of system y The aerodynamic and etructurel forces may be linear or nonlinear as eggs. i and 2 stand. In the present study structural nonlinearities only are considered, and a linear quasi-steady theory is used for the aerodynamic forces. Small nonlinearities in the structural forces are donsidered. These nonlinearities may be dependent on h,a, h and/or&, and thus may be elastic, damping, coupled, or a combination. Fe = weme [bh + EF (hx, h, &)| (3) Moz al [ 2 + €g(h,a,h, o)] (4) (4 The smallness of the nonlinearity is characterized by &. The precise statenent is hn of <-— E< ——__ UES 4(h,a,h,&) g(h, of, he, 6) 6) In the above equations w2m and «’I_ are the spring constants in bending and torsion, respectively. 3e The air forces used in this study are a quasi-steady approxi- mation to incompressible, unsteady air forces. The approximation is that of Dugundji's, in reference 5. Eriefly, the Theodorsen function ig set equal to unity and the density ratio, “, is assumed high enough to permit neglecting the apparent mass contribution to the bh and oxo terme. Full details are given in the appendix. 5 fhe use of the quasi-steady approximations allows a closed-form solution to be obtained. Of course, the presence of the Theodersen funetion precludes this for the complete unsteady formulation. The quasiesteady lift and moment are given by l= (eeu bU*) « + (ZpnbU)h + (en U)2(\-a)ex - (pmb?a) & (6) Ma (29 UN +.) a + (zenb*u) ($+a)h + (Zeb Ua)(S-a) & + (pub ¥a)h (7) where p is the air density, and U the air speed. 4, Using the above expressions for the forces the equations of motion become a wem{h + eF(h,oA.&)]+ mh + wxbo + ZoubUc + Zenbu h + prbUZ(\-ac - orb ac =O (8) 2 wy, Lla+ €q(h,o K&)| + Lo + meh 2pnbU (S+a)o — 2on U(d+a)h-Zenb Ua(s-a)x - en bah=o (9) The equations are nondimensionalized in the usual manner for flutter analysie, vig., define a nondimensional time, T= 4 t, denote derivatives with respect to tT by primes, and divide eq. 8 by empw* 6 and eq. 9 by puntw , Grouping terms, the equations become: Akt qa lpm ayo & 242 pall-alod’ +2Xuh + Za + EvX =O (10) (i) (2) (3) (4) (s) (é) (1) (8) i } , alii a) + php —3 ston - Zealand! + (4) Qo) (4) (12) (3) [XEu- Ble raja + E4gX =O (11) (\4) (\S) (16) fhe new symbole in eqs. 10 and 11 are standard flutter notation, and are defined below, Me ape » the wass ratio a k = an the reduced frequency, or Strowhal number. ry = | 2 > the nondimensional radius of gyration. = wn Rs LS. Ee re ot a ee § 8 re off oF g 2 g It is of interest to identify the origin of each of the terms in eqs. 19 and 11. The numbers below each term refer to the following list: ? (1) translational inertia due to bending acceleration (2) translational inertia due to torsional acceleration (3) apparent mase due to torsional acceleration (4) sft at the $ ~ ebord due to & 7 (5) combination of lift at the $~ chord due to 22% ang v léft at the t ~ chord due to apparent mass arising from Ud (6) linear structural restoring force in bending (7) sft at the ¢ - chord due to instantaneous angle of attack (8) nonlinear structural force in bending | (9) rotational inertia due to bending acceleration (10) apparent moment of inertia due to bending acceleration (41) rotational inertia due to torsional acceleration (12) aerodynamic moment arising from term (4) above (13) aerodynamic moment associated with term (5) above (143 linear structural restoring moment in torsion (15) sercdynamic moment arising from term (7) above (16) nonlinear structural force in torsion It is neceseary that eqs. 10 and 11 be "uncoupled" in the second derivatives for the application of the method of slowly prying parae Pa meters, Multiplying eq. 10 by ¥- x and eq. 11 by - & Za uo i and adding results in an equation in bh only. Multiply- 2 ing eq. 10 by « ey and @q. 11 by ra and adding results in an 4 equation in a vay. The two equations are: %, , defining (2 Bh + Zh + Bdge ah’ + ELEC-a) + %lg- aa] oe + KAZ RKA + {2 [4 Elgra] - PEAEK}a + EXUEI-%3) =O (az) (h2- %2)a" — lates yh - Zl4g-4) + H(\-a)ja’ - (7%,2Xh + [EX - 3G +arB& a + ePK(3 -~%F t) =o (13) This is the final form of the equations of motion te which the method of slowly varying parameters will be applied. 9 III. METHOD OF SOLUTION i. Introduction The Method of Slowly Varying Parageters is well imown in the theory of nonlinear oscillations (see references (6) and (7) for example). It is variously known ag the Method of the First Approxi- mation of Kryloff and Bogoliuboff, the Method of Slowly Varying Amplitude and Phase, and by the name used in this study. It should not be confused with the Method of Equivalent Linearization, also by Eryloff and Bogoliuboff, The 3S.V.P.* method requires that the equations to be studied be quasi~-linear, 1.6., that the nonlinearities be small. This makes the harmonic (first) approximation to the motion quite good and leadgsto the slow variation of the amplitude and phase. Approximations of higher order can be made but for practical purposes the first approxi. mation is almost always sufficient. While the S.V.P, method is tsually applied to single degree of | freedom systems, the methods of solution remain essentially the same for maltiple degree of freedom systema. First, the displacements corresponding to the p degrees of freedom are replaced by the ampli- tudes and phase angles. This changes the system of » second order equation to one of gn first order equations, the dependent variables being mn amplitudes, (n~- 1) phases (since one phase is arbitrary in the free vibration problem) and the frequency. Second, all quantities are averaged over a eycle, whieh leads to the harmonic motion approxi- mation to the more complex exact motion. This "first approximation” © * For convenience abbreviate Slowly Varying Parameters to 3.V¥.P. 10 contains all the significant nonlinear behsvior of quasi-linear systens, such ag limit cycles and amplitude dependence, The simplified first order equations resulting from the averaging process are used to study the steady-state oscillations by setting the t derivatives equal to zero. The regulting set of np linear simultaneous equations is solved for the steady-state amplitudes, phases, and frequency. fhe stability of the steady-state is studied oy perturbing the steady-state solutions. Sinee only small perturbations are considered the n first order equations become linear in amplitude and phase per~ turbations. Furthermore since the dependent variables are amplitude and phase, rather than displacement, the equations have conetant coefficients. Thus relatively simple linear theory can be used in the atabllity analysis. It will be noted that the oseillations considered are "single- frequency", i.e. only one frequency is associated with the multiple degree of freedom system. In an ordinary vibration problem this might be a considerable restriction, since one would expect to find as many characteristic frequencies as there are degrees of freedom. Fortunately, flutter oscillations are peculiarly suited to a single-frequency analy~ (Sia, It ie a well-documented fact (see references (8), (9), (10)) that dending-torsion flutter systems manifest “frequency coalescence” as the eritical flutter speed is approached. Foreed vibration studies, both experimental and theoretical, indicate that the two characteristic frequencies approach a common frequency at or near flutter. Thus the requirement of single-frequency cecillations is not restrictive in the present case. Li Change dependent variables h(T) and *( 7) in eqs. 10 and il as follows: ble) = a(t) sin[kt + & (ci\ (14) X(t) = A(t) Sin fer + @&(r)] (15) Since four new variables have been introduced to replace the original two, four relations aust be defined. The remaining two are: h(t) = Alt) ke cos [kr +4 (0)] (16) ae'(t) = dat) kcosfkr + &()] (17) Since a, and , ere functions of T eqs. 16 and 17 require a,/sin(kr+ &) + @/ a, cos(kTt+@) =O a3) ag! sin(ket + @2) + Qa cos(kt+ @)=0 (19) Differentiating eqs. 16 and 17 to obtsin the expressions for the second derivatives, and substituting into eqs. 12 and 13 yields two first order equations. Let 9, = kt +4, and 9, = kt+ %,. A -B[ak cos - aie(kaer)sin Ol + ZUEt Lg ta] a, kas6, + -,) ; rilNGee +%,( 77a] a,k cos 8, + a 2Xa,sinB, + {2148s (4 40)]- ZAK] aa sn + ELRUEF-§4) =O 20) iz | 2 _ (2-2)( 42 k coS@2 — Ark (k+ Uz) Sin ®| — wri (54+ a+ %)ak cosG - 2 ! x (\- . -~ Zz : gp ltla-4) + Hl! aarkcos Oz - KE LRKa, sin& + [eka -Z(b+ar¥y} az sn® + KX (§-%F) = © (21) The above equations and eqs. 18 and 19 form the set of four first order equations. Explicit expressions for ay ‘ as F 5 and ¢/ can be obtained by algebraic manipulation. The four explicit first order equations are (h- 8) a! = — Zn +%, ($+ al] 2, cosa, - J Weta) 4% a(4-al]a, cose, 50, KAZ RX- (AY -%2)] a, sin ®, C050, - {Zee Hl +2)] -kE ALK} ts C056, sin, eX PT-H 3) cosB, (22) (Ag 3 )ay = rae +art, wa ws, cos, + Fla (4-a)+ Hala, 0s'@, + kz RKa, sin 9, cos@, - {LAK ALK (A7-%3)] - Pe ratEy Az Wn Fz 0059,— ELK (G- % 5) cos 0, (23) (2 -22)¢/= = rats +Z Z(gra)|sin§, cos8, + pela Bas - al? 5n§, c05 8+ KI RX- (a2 2) sir 8, +a rE +a)] -kEAZK x}4 22 iD sinGe + (2h) EkX FZ AS-% 3) <in®, 13 (BG! = als t a+%)o 7. 58, sn, - GAs 2) +B (1-4) sinO, cos®, - LERKS sin8, sin Be +{k[EX- ULF E)-7 2 (rary zt sin 204 ELK g_%F) sinB, (25) Kote that these equations are exact. The approximation arises in the averaging which follows. Since the amplitudes and phases are slowly varying*, they may be regarded as constant during a cycle, and eqs, 22 to 25 may be averaged over a cycle. Thies eliminates explicit dependence on t, and simplifies the equations condiderably. Denote the average value of the nonlinear expressions as follows: Nic= Uzi-% %g) cos 9, 48, | Noc = ER (4-48) cos 2 58, ° . (26) Nig = eh eae %4) sin@, dO, (e] Nzs= eek - %$) sin8z dO, , | ° , The averaged equations are (Bai = - GA HBS 4 a)la, - a) +%A(5-@)] 2. c0se+ { [+e Gtall- Sb kk, i AX} a, SInY — Nic (27) * see p. CES regarding glow variation assumption i4 (NB )as = Z(bt a+ Beco’ + gle (g-a) +%, (1-2) ae + a, (A?-%2)@ = gh tea) + Ba -ad)a, Sin +-£ | PRX- (4-H), + {aul B(st4)]-gkenx} ar cose + Nis (29) On(Me UNG = G(tat+B) a, sine- LeERK A, cose + {KU aLEN]- dt gtetBljart Nes vhere Y= 4, -%, = 9, - 8, | _ The above equations are the basic results of the Method of Slowly Varying Parameters, and are used to study the steady-state oscilla tions and their stability. For future convenience, the equations may be written in simpler form with the definition of some new symbols. Let the equations be Qa, = Ba, + (-Csin@+ Veos¢) a2 — Nie (31) Maz = (EsmG+ Feos¢)a, + HA.-N,. (32) aQe’= -Aa,- (Dsin? + Coos¢\ar + Nis (33) a,0¢@ = (FsinQ-Ecose)a, - Ga, + Nos (34) i5 where A = -Epgex - (2) ) B= - 71+ &l5+a C= “1 et ES 44a) -LkE AY D = -Z[A(i-a) + %,a(z-a)] L 95) E= ZkXex Fag (gtat® G = -{EIRX- (a ED]- 2g +a4%y = 1 elg-a) + % ay] 5 QE AERP Hote that this form exhibits variables a,, a5 and ? wat not variables k and xX. This turns out to be convenient for later use. The steady-state oscillations, i.e., at constant amplitudes and phases, are the nonlinear equivalent of the flutter condition. Setting ale a, 4» and G, all equal to sero in eqs. 27 to 30 results in four sigsbrate simultaneous equations in the five variables B®» BV, k and KX. Hote that in the free vibration problem one phase is arbitrary and thus only the phase difference is significant. The equations can be solved to give velocity as a function of maplitude, i.6., a closed-form solution, for the case of elastic nonlinearities in one of the degrees of freedom, and such 4 solution is carried out in Perts IV and ¥. For more complicated cases, consi-e derable algebraic reduction can still be achieved, however, some iteration may be necessary to obtain numerical results. The procedure in the preceding paragraphs generates relations between amplitudes, phase, frequency, and velocity which may exist at steady-state, but makes no statement regarding the stability of gmall perturbations from the steady state. If, in fact, partes of the steady-state relations are unstable in the above sense they will not be encountered in practice, since they are perhaps more accurately described ae critical energy levels, which separate motions converging on different amplitudes, or limit cycles. Un. atable portions of the amplitude--velocity curve arise frequently due to soft~hard or similar types of stiffnesses, which result in multiple-valued amplitudes for certain ranges of velocity. Thus an investigation of the stability is seen to be an important part of the nonlinear flutter problem, Assume the steady-state equations have been golved and denote the steady-state values by 815° 829° Po Ky and x The perturbations are defined as follows: a,= a, + &, Az = Ay, + E2 , (36) =F * P= Ken be, Y=m-N2 Fe =@% + H2 J The perturbations are assumed to occur at constant velocity and the frequency perturbations are accounted for by nN . 80 that &k — 4? and XK do not require parturbation. Substituting eqs. 36 into eqs. 31 to 34 yields Qe! = B(4,+ &) — C(@, +82) sin(Y rm) + Plaz_+52) cos(@ +n) - [Ne (2t,, a é Qe= F(a, +6) an(Gsn) + F(a,+4 $1) <05(¢,4m) + H(@2,4 Bs) - [ Nec (a, ae, e) + w(t + Mesh é, + sent ; | (38) (+8, )Qy/ = -AC,,+8:)~D(%+E2) sin(@+n)— C (apt 62) Os(Gey) + [Ng 10% 2) \+ Ms |B + 2Ms| &, + + SB yt. . | (39) aa @ a¢ (a7,+€)@y, = F (a, +&))sin (Gen) - E(4,,+ +€,) cod +n) - (az + €,) + [No(@,,4, @) + IE + el Bat scl 4+: | (40) To first order in the perturbations, and with the subtraction of sume equal to zere by virtue of the fact that Biot Bag Por & and x are Solutions of the steady-state aquations, the perturbation equations become Qtr = (B- Ne )& + (-c sine, + Dost _ otc =|) Set (- Cae, cos, — Vag, sin® — atl 9 (1) 18 QE, = (Esng + Feat ~ |, ye, + (4 - Co \ be + (E 2, SE, ~ Fa, sm _ |.) n (dye ) = LANs \ oN 3 Yas co += gop Mal J a be Ca Fate EA “cine Nis] | aN (Percses a Cfo — ie cage, — ES Soin +g xl- é, sel yn. (43) Hote that, again, since only 7 = N4 -4o is significant there are only three equations. The above equations determine the stability of small perturbations about the steady state. Since they are linear differential equations with constant ooaefficlente they can be solved by standard techniques. Specifically, it can be assumed that - wre - AT _ At kaze, b2-Be , y=He"! Then eqs. Gi to 43 become N= 0B, + B,E. + UF NE, = a2 & + 62.E, +t %) AH = aE, + 63 E, + ¥sh where OTe Bs and % are defined in an obvious manner. For non» trivial solutions the determinant of the coefficients mist be zero. ig The determinant generates a cubic polynomial in A. ‘The stabil. ity, iee., real part of 2 greater than or less than sero, may then be atudied with the aid of Routh's criteria for cuble poly- nomials, or by actually aolving for the roots. In addition to stability studies, the perturbation equations provide an a posterior}, check on the assumption of slow variations. Since A characterizes the rate of change of the perturbations, ita magnitude is the critical factor, and the roots of the cubic in A give this information. It ie not alwaye necessary that all the roots be small, however, gince some may be associated with initial transients which do not affect the significant part of the perturbation motion. The example in Part V illustrates this point. Thera the three roots congist of one real, amall root which determines the stability, and two rather large, complex conjugate roots (negative real part) which correspond to a highly damped oscillatory mode which will ve in evidence only for a short time after the perturbation is initiated. Since the solutions obtained are approximate, the question arises as to whether some undesirable large roots may have been precluded by the averaging approximation itself. Such questions generally accompany g posterior), justifications of this kind, and are usually answered on physical grounds. For the present case of quasi-linear systems there appears to be little danger that such Values of A have been precluded. Further analyeis of the atability in these general terus is beyond the scope of this study. In any specific numerical example, however, the procedure is straightforward. in Part V such a cal- culation will be carried out. 20 IV. CLOSED-FORM SOLUTION FOR ELASTIC TORSIONAL NONLINEARITY Solution of the set of four simultaneous nonlinear algebraic equations for the steady-state oscillations (eqs. 27 to 30 or 31 to 34) appears to be a formidable task, however, there is a rela tively simple procedure which ylelds closed-form results in some cases, and almost closed-form results in other cases. The closed. form golutions are possible for the important case of nonlinearities in one degree of freedom, which are dependent only on the amplitude of that degree of freedom, i.@., for single, uncoupled, elastic nonlinearities. The procedure will be shown for an elastic, un coupled nonlinearity in torsion. For an elastic torsional nonlinearity the functions Mi ee Woe N,., and W.,, defined in eqs. 26, take on a simple fora. is 28 on =< 42 720 % [- EKX%,AZ , Nex ELK ty g 6050,d0, = —-——-=*/ g(azsn&) cos ab, 2a ar O _ 7o a 2n-¢ Nie=r SKF / g(a; Sin &) [cos @, cos ¥— sin G sin] dGe -¢ Similarly, Ngoc =O (45) - BE ko ws? a, Zz a \t (46) where 5 tte 27 az ¢ Note that ¢ is a function of a,, torsional amplitude. With the definitions of eqs. 44 to 47 the steady-state equations corresponding to eqs. 31 to 34 become Ba, + [-(c+%ko) sine + Dees] a. =O (49) [Esing + Fos a, + Hae =O (50) Aa, + (Dain? 4 (C+ % ko) cose] az = (51) [Fsine- E costa, -(G-ka)az =O (52) Working with eqs. 49 and 51 leads to the system (-Bsin@+ Acse)a, + (C+ Rho) az zo (53) (Beos @ +Acin ¢@) a, + Pa,=o (54) (€ sin? + Fos $d, + Haz =O {55} (Eas? - Fsin@)a, + (G-ke)az=6 (56) 2a From eqs. 5% and 55 an expression for the amplitude ratio, in terms of Y only, is obtained. — @ = P __ H az B cosP+ Asin Esin@ + Foose (57) From eqn. 57 AH-DE gn@ = DF- BH J(de~ BH)? + (AK - DEY Cos Ox AH-DE > (38) /(DF-BY)*+ (AK- DE)? a _ ADE-BH)*+ (AK-DEY* az BE-AF J Substituting eqs. 58 into eqs. 53 and 56 yields ~B(DF- BH) + A(AH-De) — (AF- BEC + % ko) = 0 E(AH-DE) - F(DF-BH)-(AF-BE)(G-ka) =o (59) Working with eqs. 59 yields finally BH-DF + (c+& kr) E- (G-ks)A=o (60) DE-HA+ (c+ %ke) F -(G-ks) Boo (61) a3 When the definitions from eqs. 35 are substituted inte eqs. 60 and 61 only collection of teras is necessary to obtain relations between flutter frequency, flutter velocity, and torsional amplitude. Bquae tion 61 yields Lat _ Ag+ R(a-Z)a 42025 (62) = wD? = x a At42bara?- (Era) Equation 60 yields 1 AB BF - A R([l42R (RK) + 2(1-RHK) 4] 37 = (63) 2 ZlRK( 2 +4) -(grat%e~z)] In the above da 6 = SF. a sin %) Sin Qpd@, = {nc (2 on! (64) KA 20a, gG (425i Vz) Sin Ggdd2 = % 4) From eqs. 62 and 63 one obtains 2 (ZY ~ (E)e) = ER (1 y= de} KIA AED] - BAZ (1-H) + 2A (HRS bw, 2 x(ete +h-g)- R(a+s) (65) Since 4 4e a given function of a,, X, and hence 0, can be found directly from eqs. 62 and 65, thus forming a closed-form solution. In carrying out a caleulation the procedure would be ag follows. 1) Assume a value of &,, tia determining 5 . 2) Using &, a compute X. 3) Using X, compute U/bu, and ke 4) Phase diffe. a rence, ¢ » and amplitude ratio, ra can then be computed from the following formulas, derived from eqs. 58. tan Y = HER (Ha) + (I- RRS -a?)] (66) a Ae (I-A) + %e (ana) (67) az METAZ- Hf-RKAL] SING — [AZ + %(444)] cos” Equations 62, 65, 66, and 67 constitute a complete solution for the steady-state osaillations in the nonlinear flutter problem for an elastic toraional nonlinearity*. A numerical example is worked out in Part V. ity * Hote that when. = 0 (the linear case) eqs. 62 and 65 reduce to eqs. 3 and 4 of reference (5). 25 V. MUMBRICAL EXAMPLE: SOFT-HARD QUINTIC CHARACTERISTIC The exaaple chosen for this study is particularly interesting from the point of view of stability, since it manifests multiple- valued steady-state amplitudes over part of the velocity range. The perturbation analysis shows that the limit cycles corresponding to these amplitudes vary alternately in their stability, a result one might intuitively expect from single degree of freedom theory. Such multiple-valuedness arises from the fact that the relative size of the nonlinearity, fz » 248 not @ monotonia function of the amplitude. Examples of stiffness characteristics of this type are shown in fig. 2 , together with the relative size of the none linearity. The specific characteristic chosen is a soft-hard quintic, given by Ms = Wel, (a —-4a3 43205) (68) Bence 6g =—-Yu? + 324% (69) It is plotted in fig. 3. The system parameters are At = 9,25, of = 22) a=-0.4, B= OA, he lo, (70) 26 ‘sepnyydwp panjoa—sjdijjnw Ul JINSAd YIUM SaljlaDaUIjUOU JO SaZIS BAI{O|a4 PUD SOIjsi4aJODADYO sseuUjsiysS JO SajdwDxy “g‘bi4 yods- 4D} 4 4JOS - pad} pady-430S aN 5 te 27 0.5 | 0.4 / Wa Moment, , wl VA a Lf 0.2 0. Va O O 0.1 0.2 0.3 0.4 0.5 Torsional Displacement, & 0 0.1 02 o3 | | 05 -0.04 NN £9 § > ° -0.08 \ eg_.) i \ A -0.12 a Fig.3. Sott-hard quintic nonlinearity: Mg = [«- 407+ 320°|w,?1, 28 They are taken from reference (11), fig. IeA (m), where the linear ease is worked out for the full unsteady incompressible aerodynamics. The linear (a, = 0) case worked bere indicates that the quasi- steady approximation results in a reduction in the eritical velocity of about 25%. For the present purposes this error is reasonable, since it should have no direct bearing on the effect of the non Linearity. 2. Steady-State Solution Using the values given in eqs. 69 and 70 the governing for+ mules (eqs. 62 to 67) become as follows § = a5(-1.5 + loa?) (74) x = o6I7 +1175 (72) J ay _ S(t (2-y _ x (1:30 - 1.152 x ) 0.05 +2.5( xX o.0A)s (73) | buy o.l4 + ~ 0,004 Xx 2 1 /bw,\? [? ¥ (34) (7%) tan Y = ? kK (1.64 —0.144X) (75) a _ | 2.2996 (76) a2 k (1.152 - 0.05%) sin — 0.264 Cos@ The ealeulations for each point are begun by choosing a value 29 of Bos then r and all other quantities follow in auceesion. The results are plotted in fig. 4. 3e Onee the steady state solution has been obtained the coefficients of the perturbation equations, eqs. 41 to 43, are determined for any given amplitude. The asplitudes a,. = 0.1658 and a, * 0.3499 are chosen as representative; they both correspond to a value of u/U of 0.963, and their corresponding coefficients differ only linear where 5 ’ ay, ard ay, are involved. The two seta of perturbation equations, characteristic equations, and roote are as follows Perturbation equations. 0.2304! = —0.026HE, + 0,0332E, + 0.00930n ~~ 0.29048, = 0.0125 E,- 0.0164 E,- 9.004%0%% ; (77) 0.2304 y!' = - 0.9348, + 0.434 E,- 0.04?2 J Assuming &,= zt etc. and forming the determinant of the coefficients yields the characteristics equation »? + 0,245 )* + 0.1245) —0.000\2=0. (78) The roots are: A = C.cclo ; A23 = —-0.1\98 tT O2F%e. (79) 36 0.4 LL g 03 [ S o ao] rd ra 0.2 E Ss s Cc o ron ” rs} k ot, 90 95 100 105 Per cent linear flutter speed, v linear 105 vy % @ 100 °W linear 95 as 90 1.23 ai, aA — 1.22 < G25 (ea 1.21 61° y ~~ _ 60° Fig.4. Steady-state solution for soft-hard quintic nonlinearity. 31 Gp —2 0a a2 Perturbation equations: 0.2304 E/ = -0.0264%E, + 0,027\E2 + O.O1972) ) O.2304 EL = 0.0135E, - Ao164EZ - 0.00852 . (80) 0.2304 9 = -0.15BE, t C0565E,- 0.0482 N J Characteristic equations: 7+ 0.345 \* + ONOISA + 0.00178 =O (a1) Roote: A: = — O.O\75 5 Az25 = -OlU + 0.760 (82) The steady-state amplitude--velocity curve is double~valued in amplitude at speede 0.937 < (U/U,,....) < 1-¢ That this should be the case is clear from fig. 3, where it is seen that & is double- | Walued in <8 and é « Since Y is dependent on 3 only, it follows that & will be double-valued in J. The stability of the steady-state is détermined by the signe of the rests of the characteristic equations corresponding to the perturbation equations at each amplitude. For each amplitude studied a real root and two complex conjugate roots are found. For a = 0.3499 the real parte are all negative, and the perture 32 bations decay exponentially, so that the steady-state oscillation ig stable. For a, * 0.1658 the real root is positive hence the perturbations diverge and the steady-state oscillation is unstable. By computing and plotting the coefficients (or the roots) of the characteristic equation, the atability can be established for any range of aaplitudes of interest. osteriori, check on the slowness of the variation of the anplitudes and phase has received previous comment in Part III-4, There it was stated that, although in general the roots A must be small, large values will not disqualify the assumption of slow variation if they are associated with highly damped initial trans sients. The present example is a good illustration. The real root at both amplitudes is the critical stability reot, and is small, so that the number of cycles ( of basic motion ) required for double ( or half ) amplitude to be attained are a, = 0.1658: 44.5 cycles to double amplitude, a, = 0.3499: 3.56 aycles to half amplitude. Considering the least favorable of the above, a, 7 Oe3499, it is seen that the perturbation changes amplitude at a rate of about 19f-of ite initial amplitude per cycle (during the initial part of the motion). Assuming the perturbations are limited to 10% of the basie amplitude, then there will be a mwaxiaum variation of about 14% per cycle in the basic amplitude, which admits with little error the averaging approximation. Thus these perturbation modeg ere slowly varying. The complex conjugate roots are kather large, and thus - 33 correspond to a mode in which the perturbations vary rapidly. However, the real part is negative, so that after only one cycle thie mode has damped to 6% of its initial amplitude, and hence has essentially disappeared after a short time. The subsequent per« turbation motion consists of essentially only the slowly-varying mode, and it is this part of the motion which is of major concern. Thus the original assumption ia justified. 34 VI. GENERAL DISCUSSION AND CONCLUSIONS It hes been seen that the Method of Slowly Varying Paraueters yields rather concise closed-form solutions for the nonlinear bend- ing-torsion flutter problem using the simplified aerodynamics of the quasi-steady approximation. In addition, the steady-state equations are in a form that makes the perturbation equations excep. tionally easy to analyze, thus providing a straightforward stability analysis. The results of the stability analysis confirm what other non- linear flutter investigations have indicated, and what intuition leads one to expect. In references (3) and (4) bending-torsion flutter with free play in the torsional stiffness was analyzed by the Method of Equivalent Linearization. Multiple-valued amplitudes were found, but the question of stability of the limit cycles did not enter since essentially linear theory was used. In this theory the linear flutter velocity is found for the equivalent stiffnesses corresponding to the amplitude under consideration. Thus the region to the right of the steady-state curve is termed unstable, as shown in fig. 5. 891 Stable Unstable U U linear Fig. 5 Equivalent linearization stability result 35 If such regions of “instability” are regarded as ranges of amplitudes, for a given velocity, where the oscillations are converging on a limit cycle of higher amplitude, then the results of both analyses are identical. However, besides the difference in point of view, the present analysis is the more rigorous. The rigor is not without sost, however. As the member of degrees of freedom considered ine creases, and if multiple nonlinearities are to be considered, and if nonsteady aerodynamics theory is used, then the solution of the steady-state equations becomes much more difficult. For this reason, it may well be that for complicated cases the method of equivalent linearization would be preferable, at least from the standpoint of the practicing flutter engineer. The applicability of the 5.V.P. method is of course not limited to elastic nonlinearities, or, for that matter, to structural none Linearities. A fruitful field for future research would be the investigation of the effects of various types of damping and @ero~ dynamic nonlinearities. Ze The steady-state amplitude-«velocity curve is of course directly dependent on the type of nonlinearity under consideration, and there will be as many type of curves as there are nonlinearities. However, it is seen from eqs. 62 and 65 that § can conveniently be regarded as a function of 3}, the averaged relative size of the nonlinearity, and for a given system J can be plotted against 5. Then from the amplitude-~ $ curves corresponding to the nonlinearity under consideration rapid cross-plotting will yield the amplitude-~veloc. ity curves. Since for moderate ranges of Py the Yoo dS relation 36 will be approximately linear, the shape of the amplitucde--velocity curves can be qualitatively or quantitatively determined once the amplitude-— 5 relation is given. Thus the steady-state results are relatively easily found and interpreted. The stability analysis does not yield so easily to rigorous generalization. However, intwition predicts that the limit cycles will alternate in their stability, and the specific example treated lends further support to such expectations. | justification of the assumption of slow variations introduces an interesting concept. The perturbation motion was seen to be composed of a slowly varying “stability” mode and a highly damped escillatory mode, which affected neither the stability nor the assumption of slow variations. Such secondary modes are net uncommon in multiple degree of freedom nonlinear systems, and their study may well lead to fuller understanding of auch systems. de Ze 5. be Vo 8. Ge 10. ii. 7 VII. REFERENCES ean le D. "eit Rae Be bes en soars: Re B. Ag Geet Scare 4 tere. a line aad Oo nivel Surf Dpe se Ligh ~ Woolston, 1 D. Se. reich He ae and Byrdsong, T. Ae Sone Readers Dat aS raul "De 1366 Dagnede Sey Biieck of Snani=Beeaty Air Forees on Cee » (1958), JOU. (1955), Jona Wiley ani Sons, Ne New York. Bollay, ¥. and Brown, 0. D., Some Experimental Results on Wing Flutter, (1941), Jour. Aero. Sciences, Vol. 8, PPe 313-318. Rocard, Y., Dynamic Instability, (1957), Grosby, Lockwood and Son, London. 38 VIII. APPENDIX: QUASI-STSADY AERCDYNAMICS APPROXIMATION The approximation to the aerodynamic forces used in this study . is due to Dugundji, and is described in reference (5). In this reference the details are not too clear, however, since approxima- tion is made after the air forces have been put inte the equations of motion. Therefore a derivation will be shown here. The approximation differs from the usual quasi-steady theory, as given in reference (8), in that one starts with the full un- steady expressions and reduces them to a much simpler, essentially quasi-steady, form. The unsteady expressions for harmonic motions are given by eqs. 9, Section 6.9 of reference 8, and are repeated below — _})2 oO . “pre (Fade) + tka + 2 Clea + t kh, + (4-a)ikew,] (Al) mie" i oo, pth? (La) 2 ck) [ay +Ekh, +(g-a)ik a] - er c . k? Ya (4?-aa,)- (4-2) ike, +a, (Ae) “YH (2) (3) In the above, C(k) is the Theodorsen function, and Ay and a, are complex amplitudes in bending and torsion, respectively. Essentially two approximations are made. First, the Theodorsen function is set equal to unity, its limiting value as k approaches 39 zero. Second, the contribution of bending acceleration to the apparent mass, term (1), amd the contribution of torsional accel- eration to the apparent moment of inertia, terms (2) and (3), are neglected. This approximation will be good if «i e With the above approximations, and reverting to general time derivatives where eqs. Al and £2 become Lit) rn h b, onbuz ue * get eet g tla ag a] aa a(hrallares(Z-a) Ta] + 2 h-(g-a ea, Collecting terms yields eqs. 8 and 9 of Part IL.