Journal of Sound and Vibration (1988) 123(1), 1-13

ANALYSIS OF LIMIT CYCLE FLUTTER OF AN AIRFOIL

INCOMPRESSIBLE FLOW

Z. C. YANG AND L. C. ZHAO

Department of Aircraft Engineering, Northwestern Polytechnical University, Xi" an, China

(Received 2 April 1987)

Both experimental and theoretical analyses are performed to investigate several types of self-excited oscillations of a two-dimensional wing model with non-linear pitching stiffness. Free play non-linearity receives especial attention. In the low speed wind tunnel test .of the wing model with free play in pitch, an interesting case of double limit cycle flutter is found. Subsequent harmonic balance analyses ascertain not only these two sustained oscillations, but also two other unstable limit cycles. A digital simulation method is also used for flutter analysis. The theoretical results agree well with the experimental data.

IN

1. INTRODUCTION

Dowell [1] has noted that non-linear aeroelasticity is a rich source of static and dynamic instabilities and associated limit cycle motions. On the other hand, in aircraft design practice, structural non-linearities are not rare; sometimes they are even very prominent. Along with large structural deformation problems, there are other important types of non-linearities. These are the so-called concentrated non-linear elements, such as those in all-moving tails, control surfaces and external stores. Hence from both theoretical and practical aspects, aeroelastic problems concerning non-linear effects are of interest and pose a challenge for research.

The first study of non-linear aeroelastic problems of an aircraft wing was perhaps the works of Woolston [2] and Shen [3]. They made an analysis by analog computer and harmonic balance methods. These two approaches, especially the latter, have been widely used thereafter. The contributions of Breitbach to this field are also well known, among which reference [4] contains an excellent discussion of structural non-linearities in aircraft flutter with both experimental and theoretical techniques employed. Later, in reference [5] the equivalent linearization concept was extended to the flutter analysis of an airplane with strong non-linearities in the control system. Mclntosh et al. [6] conducted a high quality wind tunnel test of a typical section with non-linear stiffness. Recently, Lee [7] developed an iterative algorithm based upon equivalent linearization concepts for a structure with non-linear elements existing simultaneously in more than one mode.

The aim of the present paper is to study the possible peculiarities of wing flutter due to structural non-linearities, based upon experimental observation of wind tunnel tests, and also to perform a thorough theoretical analysis of them. A uniform rectangular wind model is used in the low speed wind tunnel test for the reason that the formulae of two-dimensional, incompressible, unsteady aerodynamics have long been proved to be accurate, and hence experimental and theoretical results can be compared on a sound basis. By adjusting the spring devices, the stiffness characteristics of the model can be changed, and a total of 28 cases have been treated in the wincl tunnel test. Among these cases a hitherto unknown phenomenon, that at one wind speed there are two different

1 0022-460X/88/100001 + 13 $03.00/0 �9 1988 Academic Press Limited

Z. C. Y A N G A N D L. C. Z H A O

modes of limit cycle flutter, was observed. For the mechanism of this interesting case an explanation is found by means of harmonic balance analysis. Based upon the experimental observation of the limit cycle amplitudes, and by choosing appropriate initial displace- ments, the double-cycle phenomenon is reproduced in digital simulation records.

2. THEORETICAL ANALYSIS

The two-dimensional airfoil is shown in Figure 1. Governing equations of the airfoil with two degrees of freedom in pitch and plunge are

moh'+ S,~fft + Khh = Qh, Sah'+ l ~ + M ( a ) = Q,,, (1)

where h is the plunge displacement, a is the pitch displacement, the do t above these characters means d/d t and t is time, mo is the total mass per unit span in plunge, S,, is the mass static moment, I,, is the mass moment of inertia, and Kh is the plunging stiffness coefficient.

/ / / / Z

Figure 1. Sketch of two-dimensional airfoil.

The unsteady aerodynamic force and moment are expressed in terms of the Theodorsen function when the motion is simple harmonic such as in the linear critical flutter case:

Qh = -rrpb2( v& + [ ; - abe) -27rpVbC(k)[ "Ca + !; + (0"5 - a)bdt]'

Q,, = zrpb2[ ab( V& + / [ - abffz ) - 0 . 5 Vb& - b2 fft/8]

+ 2rrp Vb2(0"5 + a ) C ( k ) [ Va + !~ + (0-5 - a)bfi]. (2)

Here p is the air density, V is the air speed, b is the half-chord length of the airfoil, ab is the streamwise distance of the pitch axis from the mid-chord point, C(k) is the Theodorsen function for unsteady aerodynamics, k is the reduced frequency and equal to cob/V, and to is the vibration frequency (rad/s).

In equation (1), M ( a ) is the non-linear stiffness term. In this paper bilinear stiffness and free play are studied chiefly; a cubic power stiffness is used only for the comparison of different analytic methods. When M ( a ) = Kaa, where Ka is the pitching stiffness coefficient, the problem is reduced to a linear model. For different K,,, usually expressed in terms of to,~(to~ = K , / I , ) the linear flutter speed VF as a function of K~, can be obtained by the V-g method as shown in Figure 2. The experimental results are also plotted for comparison. Since in the wind tunnel test, the air flow about the airfoil cannot exactly be two-dimensional, a correction factor deduced from one typical test has been introduced in the aerodynamic formulae.

LIMIT CYCLE FLUTTER OF AN AIRFOIL 3

~ 0 I . . . . I . . . . I . . . .

40 50 60 70 Wa(rad/s)

Figure 2. Linear flutter speed as function of pitching stiffness.

2.1. HARMONIC BALANCE

For wing flutter problems, as Shen [3] pointed out, the system has a strong filtering property such that the fundamental harmonic is predominant in the flutter oscillation. Hence it is adequate to treat the critical flutter motion as simple harmonic. Thus in the pitch degree of freedom

a = A sin ( t o t ) , f (3a)

where A is the limit cycle vibration amplitude. Upon substituting this expression for a in M ( a ) , the first harmonic sine term of the

Fourier expansion of M(a) gives an equivalent linearized stiffness by which an equivalent linear aeroelastic system is established. For any such linear system, a critical flutter velocity VF can be found straightforwardly from a diagram such as Figure 2. According to this procedure, for every prescribed limit cycle amplitude A of pitching oscillation a corre- sponding flutter velocity Vr is determined, from which a VF US. A plot can be drawn. Figure 3 depicts schematically the procedure for the above analysis. Such diagrams are very useful to demonstrate the stability condition, although they are not used for quantita- tive prediction.

Vr

Figure 3. Harmonic balance analysis scheme for bilinear stiffness case.

4 z . c . Y A N G A N D L. C. Z H A O

From the above procedure, it !s evident that the analysis of non-linear flutter by harmonic balance is based on the synthesis of a series of consecutive linear flutter analyses, from which conclusions about the stability condition of the original non-linear aeroelastic system can be deduced.

2.2. D I G I T A L S I M U L A T I O N .

When the governing equation in the frequency domain of the non-linear airfoil aeroelas- tic system is extended to the Laplace domain, and the unsteady aerodynamics term is treated as a linear time lag element, the system may be treated as a automatic feedback control system which can be solved by modern control theory.

In control theory terminology, the input-output equations are

x = A x + B u , v = C x + D u , u = W y . (4)

Upon discretization of the equation~with a sampling period T, the state vector of the ( k + l)th sampling is found to be / '

x[(k + 1) T] = tb(T)x(kT) + G(T)a(kT) + B(T)v(kT),

v[ kT] = {u[(k + 1) T] - u(kT)}/T, (5)

y [ ( k + I ) T ] = C x [ ( k + I ) T ] + D u [ ( k + I ) T ] , u [ ( k + I ) T ] = W Y [ ( k + I ) T ] ,

where

q~(T) = exp (AT), G(T) = ~(r dr,

P(T) = rq~(r)B d r + r q~(r)B dr, (6)

v[kT] being the correction vector accounting for the variation of the input u during the samping period [kT, ( k + 1)T].

Once the digital simulation model of a non-linear system is formulated and initial conditions are given, the response at every sampling time represented by either a time history trace or phase plot can be obtained. Hence the flutter stability condition is determined.

Transforming the governing differential equations (1) and (2) into the Laplace region and using a non-dimensional Laplace argument

= ( b / V ) s (7)

gives the non-dimensional governing equations in the Laplace region as

(p-o+ 1)~2H + (p-x~, - a)g:,~ + e~ + p-(bco,,/V)2/~ = -2C(/ ) [c~ + ~/~ + (0.5 - a).~c~ ],

(p.x,, - a ) /2H + (p-r~ + a2+0 �9 125)g2c~ + (0.5 - a)gff + M(~) / ( z rpb 2 V 2)

= (1 + 2a)C( / ) [ f f + g/4 + (0.5 - a)gff], (8)

where p-o = mo/(1rpb:), p- = m/(Trpb2), m is the mass per unit span in pitch, x,~ = S~,/(mb), 2 - L,/(mb2), H = h / b and /7, c~ are the Laplace transforms of H and a respectively. r a - -

C(g) is the approximate Laplace transform of C(k) ,

C(g) = {(1 + 1-774/)(1 + 10.61g)}/{(1 + 2.745g)(1 + 13.51g)}. (9)

After some manipulations the final governing equations for digital simulation are found to be

~2 FI = c , f l + c2ga + C, B4a + C3C( / ) [ a + ~ + B ~ a ] , (10)

~2 ~ = CsFI "1- C 6 ~ -'k C7B40t "1- C s C ( g ) [ t ~ q- .~/'~ q- B3.(c~ ] ,

where C~ ( i = 1 , 2 , . . . , 8 ) and B~( i=3,4) are composed of the model parameters and coefficients in C(g).

L I M I T C Y C L E F L U T T E R O F A N A I R F O I L

From equations (10) the block diagram of the simulation system can be constructed and drawn as shown in Figure 4. From this simulation block diagram, it can be seen that the non-linear aeroelastic system under investigation has four integral elements, two linear time lag elements, and one non-linear element which affects only the conjunction matrix W. All the matrices except W are diagonal, and hence the simulation calculation is rather rapid. The results for digital simulation of one case which was investigated, time history and phase plot, are shown in Figure 5.

3. EXPERIMENTAL INVESTIGATION 3.1. MODEL AND ITS MOUNTING

The model is a wooden solid rectangular wing segment with a NACA 0015 profile. Its inertial and geometrical parameters are a = - 0 . 1 5 2 , b = 0 . 1 2 5 ( m ) , x , ,=0.16, r ~ = 0.4711, tt = 108.4696, and /Zo = 145.3884.

' ( I Non-linear element I (

74S1~

Figure 41 Block diagram of digital simulation.

,Q

AAa AAA AAAAAAAAAAAAAAAAAAAA 'vvvvvvvvvvvvvv vvvlv vvlvvvl # t

Figure 5. Time history and phase plot of digital simulation results for a case with free-play stiffness.

6 z . c . YANG AND L. C. ZHAO

Two large fixed rectangular plates (one is transparent) adjacent to both tips of the wing model were used to keep the air stream as nearly two-dimensional as possible.

All flutter tests were performed in a closed circuit, low speed wind tunnel with an open test section of 1 meter diameter. The maximum air speed attainable is 60m/s . For protection of the model from strength failure due to high dynamic pressure, the model is designed to have its linear flutter speed below 30 m/s. The flutter regime for speeds above 30 m/s was not pursued in this investigation.

The fixtures and suspensions of the model and its installation in the wind tunnel are shown in Figures 6, 7 and 8. The wind model with the dummy weights is elastically supported and can oscillate in plunge and pitch modes.

The mounting systems on both port and starboard sides are identical. On each side a cross bracket suspended by four helical springs serves as the mainsuppor t of the model. The axle fixed on the wing tip side of the model is supported on a ball bearing fitted in

Figure 6. Sketch of the model mounting.

Figure 7. Spring system of the model.

L I M I T C Y C L E F L U T T E R O F A N A I R F O I L 7

Figure 8. Model installation in the wind tunnel.

the center of the cross bracket. The model can vibrate vertically, along with the cross bracket, as a whole body. Rotation of the model relative to the cross bracket is constrained by a leaf spring of which the root is clamped on the axle and the tip is held by two opposing screws driven through a slide block on the cross bracket. When the model rotates relative to the cross bracket, the leaf spring is deformed and provides a restoring moment. By adjusting the set of the slide block along the vertical arm of the cross bracket, the rotational stiffness can be changed at will. The leaf spring is dismountable, and a set of springs of different sizes also makes a change of stiffness possible. Both upper and lower portions of the axle have sockets for clamping the leaf spring. The tip of the upper leaf spring is free to swing with a small gap of which the length and center can be adjusted by turning the two screws on the slide block. When both upper and lower leaf springs are set, a bilinear stiffness is provided, while only one leaf spring produces a free play type non-linearity. Since both the root and tip end conditions are diMcult to formulate exactly, the rotational stiffness is obtained from a vibration test to determine the natural frequency of pure rotational when the plunging motion is restrained.

3.2. E X P E R I M E N T A L R E S U L T S

3.2.1. Model with bilinear stiffness

The bilinear stiffness curve is shown schematically in Figure 9. All specimens in this investigation have K~ > Kin, where K,, corresponds to the minimum linear flutter speed

S

td~

J

Figure 9. Sketch of bilinear stif[ness curve.

8 Z . C . Y A N G A N D L. C. Z H A O

Vm in Figure 2. Hence each test exhibits limit cycle flutter. For each specimen, the limit cycle amplitude grows with increasing air speed. All the test results agree with the theoretical ones quite well. In Figure 10 one of the test results is shown along with theoretical results from the harmonic balance analysis scheme shown in Figure 3.

3.2.2. Model with free play The free-play stiffness curve is shown in Figure 11. There is an unbalanced moment

due to the weight; hence a static deviation Ep exists and the stiffness curve is not symmetric. For each specimen the tip of the leaf spring is held first, and the wind tunnel is run

with increasing air speed to obtain a so-called linear flutter speed VL. Then the tip is released and with various gap widths and leaf spring lengths the non-linear wind tunnel test is performed. The test results can be classified into three different cases as follows.

(i) No non-linear flutter occurs: i.e., the system is always stable. The harmonic balance analysis scheme for this case is shown in Figure 12. The specimen in this category all have a softer leaf spring, K,, < K,,.

(ii) At a certain air speed below the associated VL, limit cycle flutter occurs and the amplitude grows with increasing speed until VL is reached when the flutter becomes divergent. The harmonic balance analysis scheme is shown in Figure 13, and two of the test results are shown in Figure 14 along with theoretical results.

(iii) At an air speed below VL, limit cycle flutter occurs with the unique feature that there exist two different flutter modes at the same air speed. One limit cycle with smaller

O.S

~ 0.8

0.7

! I

o-[-"

J �9 40

o +

o +

o -I-

o

2 3 A/or

I

Figure 10. Limit cycle flutter amplitude for bilinear stiffness case. to., =44.9 rad/s, to..,= 58 rad/s, toh = 43.9 rad/s, V L = 33.8 m/s, o- = 2.5 ~

J

t~

Figure 1 I. Sketch of free-play stiffness curve.

LIMIT CYCLE FLUTTER OF AN AIRFOIL

A

J I I I I I I I

l �9 1 e , ,

1 I

Figure 12. Harmonic balance analysis scheme for free-play case with no flutter, K~ < K,,,.

. . . . . . . . ~ - _ - _ _ _ v t _ . . . . . . . .

4 ! / , ~,K~K.. !

\

i I . . . . ~ - - - [

A

Figure 13. Harmonic balance analysis scheme for free-play case with single limit cycle flutter.

amplitude and lower frequency, hence appearing to be milder, is a pitch-dominant mode which occurs naturally from the wind tunnel turbulence excitation. On the contrary, the other limit cycle motion is much more violent with larger amplitude, higher frequency and a plunge-dominant mode. Only by a man-made, large initial displacement can the model attain this state.

Furthermore, when the air speed is raised the amplitude of the milder limit cycle diminishes while that of the other one increases, until the milder one disappears and only the violent flutter mode exists similar to the second case just mentioned in (ii).

Case (iii), being unknown before, was studied further to understand better its mechanism by the aid of harmonic balance analysis. For this analysis the steady lift produced by the air stream must be taken into consideration. Because of this lift, the leaf spring is relaxed from the static deformation caused by the weight. This means that the preload decreases when the air speed increases. The equivalent linear stiffness of the free-play spring with preload is calculated by the following equations:

Keq= K,, for A < Ep,

Keq=K,,[~r/2+CS+(sin2ck)/2]/~r for Ep<A<(Ep+E,),

Keq=g~,[~r+Cb-tp+(sin2d~)/2-(sin2~)/2]/~ for A>(Ep+E,). (11)

l 0 Z. C. YANG AND L. C. ZHAO

0"9E

0"94

0-9~

0.90

0-88

:~ 0.86

0-84

0.82

0-80

0.78

I i i i I I I I , I -(a) -

I , , i,l T f I

5 10 20 50 40 50 2

- (b)

A l E t

I J i i i

I i l , , i l l f l 3 4 5 10 20 30 40

Figure 14. Single limit cycle flutter amplitude for free-play case. (a) to,, = 55.6 tad/s, toh = 47-5 rad/s, V L = 25 m/s, E, = 0.5*; (b) to,, = 59-2 rad/s, to n = 47.5 rad/s, Vt. = 28 m/s, E, = 0.6 ~ - - , digital simulation; O, harmonic balance; +, experiment.

Here (;b =arcsin (Ev/A) and tp =arcsin [(Ev+E,)/A]; the meanings of E v and /7, are shown in Figure 11.

By taking account all the situations actually occurring during the test, the harmonic balance analysis scheme can be constructed as shown in Figure 15. It is noted that in this figure, there is not solely a single equivalent stiffness curve as in Figure 12. For each air speed there must be one distinct equivalent stiffness curve for each associated preload state. The scheme of Figure 15 demonstrates clearly the occurrence of the two limit cycle at one air speed and their evolution with increasing air speed.

Actually, it may be said that there are four limit cycles with amplitudes A~, A_,, Aj, A4 as illustrated in Figure 16. Among these four amplitudes, Ai and A 3 correspond to unstable limit cycles, while A2 and A4 are stable ones. With increasing amplitude, the

"I~ -- -~" i- -- -- r -- -- -- -~ ~-# - - ' I - ~ i I I I -----6 .... ,'I r --_-- .... I---

I I I I I' I I I

A 4 i I ~

J , ', ', " - - 2 . . . . E i t \ " , , "", . " ' - - . _ . . . .

"Ep v ~ \ \ \ i i l "E^ Ep= .

[[ ......... I I

I

Figure 15 Harmonic balance analysis scheme for freeplay case wilh double limit cycle flutler,

L I M I T C Y C L E F L U T T E R O F A N A I R F O I L 11

/

A ' ' , ' ! ' u | ' \ \,, '--

\.~ A2"

Figure 16. Stable and unstable limit cycles for free-play case.

/

/~'a t. Keg

unstable and stable limit cycles occur alternatively as confirmed by the theory of oscilla- tions [8]. Furthermore, on the equivalent stiffness curve the small cross and circle refers to an unstable and stable limit cycle state respectively. These two states vary with the air speed V, and upon their coalescence bifurcation occurs. It can be seen that at air speeds V,, and Vh there are two bifurcation states of a different nature.

After having established the existence of two limit cycle states and the size of the two amplitudes, adequate initial conditions can be chosen to reproduce these states in a digital simulation. The results are comparable with test data as shown in Figure 17.

For quantitative analysis by the harmonic balance method, it is noticed first from the test and digital simulation results that the motion is biased. Hence instead of equation (3a) the mathematical expression of the limit cycle motion is changed to

a = Ao+A sin (cot). (3b)

Substituting this expression in M(a) gives rise to a constant term in the Fourier expansion of M(a) , before the first harmonic term. According to the concept that in one cycle of vibration, the elastic restoring force has to have zero mean, the constant term should vanish. This consideration provides a supplementary condition to determine Ao. For free play, the equations for Ao are

Ao=A[cos ~b~+~b t sin q~,-'z'(sin ~b,)/2]/Tr for ( E p - A o ) < A < ( E p + E , - A o ) ,

Ao= A[cos ~bl+ q~, sin ~bt--if, sin ifl + zr(sin if--sin ~b) /2-cos if]/~-

for A > ( E p + E , - A o ) , (12)

where ~bl = arcsin [(Ep-Ao)/A] and II = arcsin [(Ep + E , - A o ) / A ] . After having determined Ao, the equivalent stiffness can be calculated by referring to

the dynamic deviation (Ep-Ao) , where E e is the initial static deviation due to steady lift and weight. Only for a large enough amplitude (such as the larger one of the double limit cycle state), can the difference of the static and dynamic deviations be neglected. In contrast to the statement in reference [2], here the static deviation has an effect on the value Of equivalent stiffness for the double cycle flutter case.

12 z . c . YANG AND L. C. ZHAO

0 9 5

0"90

0 8 5

,,,..

0.95

0-9C

0"05

o-1

I I

(o)

Y O I I

I I i l ' I i l l l l

(b) i i i l l I

o +

, , , I , , , , !

0"5 1 i

A/Et

' I I i l l I I

/ , I , , , , I I

5 10 20 30

Figure 17. Double limit cycle flutter amplitudes for free-play case. (a) m,, =57.6 rad/s, ~h =47.5 rad/s, V L = 26 m/s, E, = 1-3~ (b) a~,, = 58"3 rad/s, a)l, = 47-5 rad/s, Vt. = 27 m/s, E, = l.l*. , Digital simulation; O, harmonic balance; +, experiment.

4. CONCLUSIONS

(1) For s tudy o f non-l inear flutter, experimental observat ion is impor tant as some unforeseen p h e n o m e n a may be revealed.

(2) Digital s imulat ion may be considered to provide an exact solution to the non-l inear flutter problem. However , since it is impractical to choose all possible initial condit ions leading to unforeseen phenomena , the numerical analysis may miss some impor tant feature o f this intriguing problem.

(3) By construct ing a heuristic analysis scheme, the harmonic balance method may be used not only to explain, but also predict, most o f the impor tant limit cycle flutter states with sufficient accuracy.

(4) The double limit cycle flutter possibly provides a meaningful model for more general studies o f analyt ic dynamics. It is in tended in future work to design a wing model with suitable parameters to manifest some general analytical characteristics for a non- linear dynamical system.

REFERENCES

1. E. H. DOWELL 1982 Journal o f Sound and Vibration 85, 333-344. Flutter of a buckled plate as an example of chaotic motion of a deterministic autonomous system.

2. D. S. WOLSTON, H. L. RUNYAN and R. E. ANDREWS 1957 Journal o f the Aeronautical Sciences 24, 57-63. An investigation of effects of certain types of structural nonlinearities on wing and control surface flutter.

3. S. F. SHEN 1959 Journal o f the Aerospace Sciences 28, 25-32, 45. An approximate analysis of nonlinear flutter problems.

L I M I T C Y C L E F L U T T E R O F A N A I R F O I L 13

4. E. BREITBACH 1978 Advisor), Group.for Aerospace Research and Development Report No. 665. Effects of structural nonlinearities on aircraft vibration and flutter.

5. E. BREITBACH 1980 NASA TN-1620. Flutter analysis of an airplane with multiple structural nonlinearities in the control system.

6. S. C. MCINTOSH, JR., R. E. READ, JR. and W. P. RODDEN 1980American lnstitute of Aeronautics and Astronautics Paper 80-0791. An experimental and theoretical study of nonlinear flutter.

7. C. L. LEE 1986 American Institute of Aeronautics and Astronautics Journal 24, 883-846. An iterative procedure for nonlinear flutter analysis.

8. N. MINORSKY 1962 Nonlinear Oscillations. New York: Van Nostrand.