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Flutter of a buckled plate as an example of chaotic motion of a deterministic autonomous system
Journal of Sound of Vibration (1982) M(3), 333-344
FLUTTER
OF A BUCKLED
AN EXAMPLE
OF CHAOTIC
A DETERMINISTIC
PLATE
AS
MOTION
AUTONOMOUS
OF
SYSTEM?
E. H. DOWELL Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, U.S.A. (Received 28 November
1981, and in revised form 23 February 1982)
For aeroelasticity of plates and shells, the equations of motion are well established. Results obtained by numerical time integration have been compared to those obtained by topological theories of dynamics and also from experiment. chaotic self-excited oscillations may occur for this deterministic
All of these suggest that system.
1. INTRODUCTION
Non-linear aeroelasticity [l-4] is a rich source of static and dynamic instabilities and associated limit cycle motions. Fundamentally, aeroelasticity combines the classical fields of fluid and solid mechanics. Usually, as the name implies, it does so within the context of the sub-fields of aerodynamics combined with elasticity [5-81. The aerodynamic forces almost invariably are non-conservative. When acting upon a resonant elastic structure whose motion modifies these forces in a feedback sense, they lead to a complex and fascinating variety of dynamical behavior. It has been known for some time that a plate under a compressive in-plane load with a fluid flow over its (upper) surface may undergo complex motions [2] (see the sketch of plate geometry in Figure 1).
Figure 1. Sketch of plate geometry. w is the order of a plate thickness.
Plate deflection,
w, is shown to a greatly
exaggerated
For no fluid flow, but with a sufficiently large compressive load, -IV:, buckle into a statically deformed shape. By contrast, for no compressive
scale for clarity.
the plate will
load, but with a fluid flow of sufficiently large velocity, U,, the plate will flutter with a periodic, nearly harmonic motion. However, with both compressive load and a fluid flow, chaotic motion may occur which is thought to be a result of a strange attractor [g-12]. t An earlier version of the present work was published as Part I, E. H. Dowell, Nonlinear Aeroelasticity, in 1980 New Approaches to Non-linear Problems (edited by Philip J. Holmes). Philadelphia, Pennsylvania: SIAM.
333 0022-460X/82/230333+ 12 $03.00/O
@ 1982AcademicPressInc. (London)Limited
334
E.
H. DOWE1.I
The problem is considered here in its simplest formulation: i.e., a one-dimensional structural model (albeit non-linear) and (linear) piston theory aerodynamics (appropriate to high supersonic Mach numbers). The governing partial differential equation is reduced to a system of ordinary differential equations by using a modal expansion and Galerkin’s method. For this relatively simple model, Holmes [9, lo] has obtained a number of interesting results by using the methods of differential dynamics (for a two mode expansion). Here results are obtained (for two to six mode expansions) by using numerical integration in time. The resulting data are examined in the phase plane and also as Poincare plots, this examination being motivated by Holmes’ results. By systematically changing the compressive load and fluid flow parameters, the evolution of chaotic motion from simple, deterministic motion is studied. It is worthy of emphasis that at lower Mach numbers the aerodynamic model needed becomes more elaborate and leads to integro-differential equations. These have been successfully solved by numerical integration techniques [2]. However, it is unclear to what extent the methods of differential dynamics may be used for such equations. Hence, for the present class of problems, these latter methods appear primarily to be of value in explaining the qualitative character of the motion and in suggesting effective formats for presentation of the results of numerical solutions and their interpretation. 2. EQUATIONS
OF MOTION
Here only an abbreviated account is given as the underlying theory has been discussed thoroughly elsewhere [2,13,14]. The governing partial differential equation is
(1) where w is the plate transverse deflection, x is the streamwise spatial co-ordinate, t is time, D =Eh3/12(1 - v’) is the plate bending stiffness, N, = (Eh/2a) j,” (aw/ax)‘dx, which physically is the tension created by stretching of the plate due to bending, NF is the externally applied in-plane load, positive in tension, E is the modulus of elasticity and v is the Poisson ratio of the plate material, a is plate length, h is plate thickness, m is the mass/per unit length of the plate, pm is the fluid mass density, U, is the flow velocity, M is the flow Mach number, and Ap is the static pressure difference across the plate. A set of ordinary differential equations is obtained by using Galerkin’s method with the modal expansion, w = 1 a,(t) sin (n7rx/a). Equation (2) is compatible with so-called simply w = a2w/ax2 = 0 at x = (0, a). The result is (in non-dimensional notation)
+A( C[nm/(n”-m2)][1 m
-( - l)“+“]A,
supported
+(w/MA)“~A;)
(2) boundary
=P[l-(-
conditions:
i.e.,
l)“]/(nr),
n = 1,2, . . . , co,
(3)
where A,, =u,/h and later W = w/h, A =pooUoo3u3/MD, t_~=p,u/m, R, =NFu2/D, P =Apu4/Dh, T = t(D/mu4)“2, and a prime denotes a( )I&. These equations may be
CHAOTIC
FLUTTER
OF A BUCKLED
PLATE
335
numerically integrated to obtain time histories of motion. These are used to construct phase plane plots and Poincare plots [9-121. Systematic numerical studies are discussed in the following section. The choice of parameters is guided by the earlier results of Dowel1 [2,13,14] and Holmes [9, lo]. Among the parameters studied are A, a nondimensional flow velocity parameter, R,, a non-dimensional in-plane load parameter, the initial conditions, and P, a non-dimensional static pressure differential. All results were obtained by using a four mode expansion except for a few two and six mode calculations done for comparison. To make the nature of the mathematical model as transparent as possible, one can consider a two mode representation from equations (3), where various numerical constants are omitted for clarity. One has A~+A~(1+R,)-AA,+A”2&A;+(A:+A:)A~=P, A;I+A2(4+R,)+hA1+A1’2[2A;+(A:+A;)A2=0.
(4)
The skew-symmetric terms involving A are responsible for the dynamic instability, flutter, while the R, terms (when R, ~0) can cause a static instability, buckling. The form of flutter modeled by equations (4) is called coupled mode flutter and is so named because a minimum of two modes is required to produce it. If one examines a root locus of equations (4) for infinitesimal perturbations about the trivial equilibrium, A 1= A2 = 0, then the two eigenvalues, which were the system natural frequencies at A = 0, approach each other as A increases and after near coincidence one of them passes into the unstable half plane of the root locus. Hence this form of flutter is also sometimes called merging frequency flutter [ 1,2]. Only this type of plate flutter has been treated by the qualitative theory of differential equations in the literature [9, lo]. By contrast another form of flutter is called single mode flutter [l, 21 and it arises because the aerodynamic forces create negative damping. It is discussed briefly here, before returning to equations (3) and (4). Analogous to equations (4), a single equation for A1 displays the essential features of this type of flutter: T A;+
I -m
K1[(v~)A1’*]A;((+)d~+Al(l+Rx)+A:=P.
(5)
The convolution integral is a mathematical representation of the physical fact that at time T the aerodynamic forces depend in general on the entire past history of the plate motion. This effect is important for Mach numbers, A4, near unity, but is unimportant at large M where equations (3) and (4) apply. A simple explanation of the non-linear limit cycle behavior of equation (5) for R, = 0 and P = 0 may be obtained by using the method of harmonic balance. Assuming AI=~Icosf2n7
(6)
one substitutes equation (6) into equation (5) and integrates over one period of motion (as the steady state is approached 7 + 00). Then one obtains KT [0/A 1’2]= 0,
R2=1+AT,
(738)
where non-essential constants have been dropped. Kf is the (real part of the) Fourier transform of K1 and physically is a damping coefficient. It is shown in Figure 2. The infinitesimal stability is obtained when A1 = 0. From equation (8) the frequency of the flutter oscillation is LQ = 1 (i.e., the single mode natural frequency) while from equation (7) the flutter velocity parameter, Af, is found from the condition that KT = 0 at A = Ar; see Figure 2.
336
I
E. H. DOWEL
Figure
2. Sketch of aerodynamic
damping
coefficient.
Now consider what happens when A >Af and A1 > 0. One still requires that K? = 0, and thus R2 = 1 +A:.
R/A iI2 = Rf/A;‘2,
(9, 10)
Solving equations (9) and (lo), one has ii, =
n2 = A/IQ,
[h/hf
- 1]“2.
(11912)
Equation (11) gives the limit cycle frequency and equation (12) the amplitude for A > hf. When R, ~0 one anticipates that chaotic motion may occur [2], but that issue is not pursued here.
3. NUMERICAL
STUDIES
For all results reported in what follows, p/M = 0.01, x/a = 0.75, v = O-3 and P = 0 unless otherwise noted. Equations (3) were solved by using a standard finite difference scheme on an IBM 3033 computer. From previous numerical simulation studies [2], it is known that the parameters A and R, govern the type of motion which may occur. Figure 3 displays a map in A, R, space which identifies the various types of motion which may occur. Only R, < 0 is considered, since this proves to be the interesting case. For small A and R,, the plate is flat and undeformed in the steady state. For small A, but moderate R,, the plate buckles. For small R,, but moderate A, the plate flutters with simple harmonic motion. For moderate A, R, a more complicated periodic limit cycle motion occurs and for sufficiently large R, and moderate to large A, chaotic motions ensue.
I (simple
x
200
-
Flat
I
I Llmlt
cycle
osc~//ot~on
hcrmonlc)
and
stable
100 Buckled,
R,
Figure
3. Sketch
of stability
regions.
- --,
but dynamically
stable
4 w/M =
0.1:
---,
F/M = 0.01; P = 0.
CHAOTIC
FLUTTER
OF A BUCKLED
337
PLATE
Results will be presented later in the phase plane for a representative point on the plate, x/a = 0.75. In anticipation of these, the distinctive types of motion which may occur are sketched in Figure 4 as follows. The buckled plate corresponds to two (non-linear static equilibria) points in the phase plane. Simple harmonic motion flutter is an ellipse. The more complicated periodic limit cyclic motion flutter comprises a smaller orbit about each buckled state and a larger orbit which evolves from the flutter motion. The chaos is beyond the author’s ability to sketch simply. Small
Moderate
A. R,
h, moderate
R,
W’
Small A, moderate
Rx
Moderate
A, large Rx
I
Figure
4. Sketch
of representative
phase plane orbit.
To investigate the possibility of chaotic motion, two trend studies were made. First and A was varied in the sequence 300, 250, 200, 175, 150, 130, 115, 100 (see Figures 5(a)-(d)). At A = 300 and 250 the phase plane plot, W vs. W, of the limit cycle shows an ellipse typical of pure flutter motion. For A = 200, the first deviations from the ellipse are evident, although the phase plane plot remains approximately a single closed curve. However, at A = 175, 150, 130 and 115 three loops are in evidence. The largest of these derives from the pure flutter motion, while the two smaller ones are associated with buckling or divergence. At A = 175 the three loops are particularly clearly defined and as A = 150,130 and 115 the motion becomes progressively more chaotic. At A = 100 the phase plane plot is simply a point, representative of a pure buckling or divergence instability. -37r*, Secondly, A was held fixed at 150 and R, varied in the sequence -2.5~*, -3*5?r*, -4r*, -5?r*, -6r2. At R, = -2.5~~ and -37r*, the three loops are particularly clearly defined and as R, = -3*57r*, -47~*, -59~~ and -6n*, the motion becomes progressively more chaotic. Nevertheless the basic three loop pattern remains albeit in a more obscure form. See Figures 5(c), 6(a) and (b). The chaotic motion which occurs for certain parameter combinations of A and R, might well be termed random. However, it is clear that such motion evolves continuously in parameter space from motion which is decidedly deterministic. Moreover even taken solely as motion at a point in parameter space (some fixed A and R,), it is manifestly bounded in the phase plane and can be characterized, for example, by minima and maxima of displacement and velocity, W and W’. Finally, it is noted, the chaos comes from the solution of deterministic equations. To further characterize the motion, autocorrelations and power spectra of the motion were calculated. Two representative power spectra are shown in Figures 7(a) and (b) for R, was held fixed at -4~~
338
E. H
DOWEL1
-6
-6 -15 -1.0
-0.75-0.50-0.25
W
Figure 5. Phase plane plot: effect of flow velocity. (d) A = 115.
0
0.25
0.50
0.75
1.0
I.25
W
R, = -4n*;
P = 0. (a) A = 300; (b) A = 200; (c) A = 150;
A = 150 and Rx/r2 = - 3 and -6 respectively. As can be seen the former results show two discrete frequencies dominating the power spectrum, which is the expected result for a periodic motion with two dominant motions in the phase plane (recall Figure 6(a)). However, in the latter case, Rx/r2 = - 6, the power spectrum, although still showing peaks, has a continuous distribution such as that conventionally expected of a random process. f is a non-dimensional frequency in cycles per unit time. Quite aside from its own intrinsic interest, the present results may serve as a paradigm for similar chaotic motions which result from deterministic partial differential equations. Perhaps the best known of these is the chaotic (turbulent) motion which is associated with the Navier-Stokes equations. There is at least one distinction, however, between the two. For the present problem there are two parameters, A and R,, each of which
CHAOTIC
FLU’lTER
OF A BUCKLED
24
50
I
339
PLATE
I
I
I
1
1 -1.0
/ -0.5
I 0
1 0.5
I
I
I
L I.0
I I.5
I 2.0
(b) 20
40
16
-
30 -
12
-12 -0.7
-40 -0.5
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
1 -1.5
-2.0
W
2.5
W
Figure 6. Phase plane plot: effect of in-plane load. A = 150; P= 0. (a) R, = -3~'; (b) R, = -6~'.
governs a distinctive instability, flutter and buckling (divergence). It is the interaction of these two essentially deterministic motions for certain combinations of A and R, that leads to chaotic motion. On the other hand only a single parameter, the Reynolds number = Re, appears in the Navier-Stokes equations (if one assumes incompressible flow). At this stage, one can only speculate that Re may play a dual role (as suggested by Lin’s interpretation of linear hydrodynamic stability theory [15]) and/or that more than one instability mode is governed by Re. 18.
1(a)
(b)
32 -
16 -
28 -
14-
24 -
12 -
20-
S 16-
12 -
a-
‘l-
O_ 0
6
Figure 7. Power spectra. A = 150; TMM = 288; DELT = 0.001; TPRINT= 20 x DELT. (a) Rx/p2 = -3.0; (b) RJlr2=40.
340
F:.
4. EFFECT
H.
DOWFI
1
OF INITIAL
CONDITIONS
For the base case of A = 150 and R, = --4~r’, various initial disturbance amplitudes were studied. In all cases the initial deflection of the plate was assumed to be a pure first natural mode: i.e., W(x, t= 0)=A,it = 0)sin (n-x/u I, with various A ,(f = 0)chosen. In addition to the base case value of A ~(t = 0)= 0.1,values of 0.2, 0.5. 1 .O and 2.0 were chosen. The long time steady state results were essentially the same, showing no sensitivity to the initial disturbance amplitude chosen. It should be emphasized that this result is not universal since Ventres and Dowel1 [16] have already shown that for plates of finite (large) length/width ratio initial conditions can play an important role. 5. EFFECT
OF STATIC
PRESSURE
DIFFERENTIAL
Another parameter of interest is a static pressure differential across the plate. This, of course, destroys the symmetry of the geometry by giving the plate deflection a preferred direction. Two trend studies were conducted. First of all, the simpler case of zero-in-plane P. At P = 0 the load, R, = 0,was studied at A = 400 for various pressure differentials, plate flutters in a limit cycle oscillation whose phase plane trajectory is an ellipse centered about the origin, W = W = 0.As P is increased from 0 to 100, 200 and 250 the elllipse decreases in size and its center moves to the right. At P = 300 the ellipse has collapsed completely to a point, the static pressure differential having stiffened the plate sufficiently so that flutter is completely suppressed. As expected the motion does not exhibit any tendency to chaos for R, = 0. Next the base case of A = 150 and R, = -47r’ was reconsidered for various amounts of static pressure differential P = 0, 12.5, 25, 37.5, 50, 56.25, 62.5, 68.75, 75, 100 and 200. For Pa68.75, the static loading was sufficient to suppress all flutter motion. For small P, say 0, 12.5, 25 and 37.5, the motion was qualitatively similar and exhibited a somewhat chaotic appearance. However, it will be recalled that for R = -37r’ the distinct three loops were present and, with that knowledge, these three loops may still be seen, 25
-20 -0.3
-0.1
0.1
0.3
0.7
0.5
0.9
I.1
I.3
I.5
W
Figure
8. Phase plane plot: effect of static pressure
differential.
A = 150; R, = -4~‘;
P = 50.
CHAOTIC
FLU-ITER
OF A BUCKLED
PLATE
341
somewhat dimly to be sure, at R, = -47r*. The most remarkable result occurs, however, at P = 50 and 56.25 (see Figure 8). Here the motion has a very distinct, non-chaotic character with two loops in evidence: that is, the static pressure differential has suppressed the chaotic character of the motion and one of the three loops which appear for smaller P. For P = 62.5 the motion returns to a chaotic state with no readily discernible pattern and for P 2 68.75, the limit cycle degenerates to a point and all motion ceases.
6. POINCARE
PLOTS
To gain further insight into the nature of the motion, an alternative format for presentation of the data was considered: i.e., a Poincare plot. For present purposes such a plot is one where the continuous phase plane diagram is sampled at discrete time intervals. Of course, since the time histories here were generated by a digital computer and are thus necessarily discrete, all of the previous phase plane diagrams were, strictly, Poincare plots. However, previously the time interval used was short compared to any time intervals characteristic of the motion, so that the diagrams were effectively continuous. The effect of systematically increasing this time interval was examined for two example cases, A = 150 and R, = -3~~ or -4~~. The former, R, = --37~~, is a periodic motion and as the sampling time interval becomes equal to the period of the motion, the Poincare plot reduces to a single point. The latter, R, = -41r’, is a chaotic motion where no such reduction is expected, but other interesting insights may be obtained. A unit time interval was selected to be AT = O-02. Results were obtained for R, = -3~~ and time intervals of 1, 2, 5, 10, 20, 34, and 40 units. For a small number of units the complete character of the phase plane diagram is evident; for 34 units essentially a single point is obtained indicating that this is the period of the periodic motion. Similar results were obtained for R, = -4~~. Because the motion is now chaotic the character of the motion becomes very difficuit to ascertain as the sampling time interval is increased. This is emphasized when the sampling points in the phase plane are unconnected. No period is found in which the phase plane diagram is reduced to a single point. An alternative Poincare plot may be constructed by specifying one of the dynamical variables and taking slices in variable space of the remaining co-ordinates. For example, one could specify A; = 0 and examine, at those times when this is true, the values of Ai, AZ, AL, As A;, . . . . However for even four modes, the remaining variable space has dimension 7 = (4 x 2) - 1. Hence it would be difficult to display and interpret such results. To reduce the dimension of this Poincare plot, one might proceed as follows (which is suggested by physical considerations). Consider a particular physical velocity, W’, and the corresponding displacement, W, for each such W’ which appears in the phase plane. (Of course, W is a linear combination of the A,, and W’ of the AA. Hence they are suitable lower dimensional representations of the individual co-ordinates, A,, and A;.) Furthermore the range of W, call it A W, (for each such W’) is a measure of the system chaos. A strictly periodic system would have a range of zero: i.e., for each such W’ there is a single W. Conversely if all W are possible between Wmin and W,,, the system motion might be termed completely chaotic. The larger R =A W/( W,, - Wmin), the more chaotic the motion. In Figures 9(a) and (b) Poincare plots are shown for W’= 0, A = 150 and R,.T’ = - 3 and - 6, respectively. The former results display the expected repetitive, periodic character for an oscillation with two dominant frequencies. For the latter a pattern appears to
342
.
.
.
.
.
.
.
. .
. .
.
. ..-*. .
Figure
9. PoincarC
.
.-*.
.
....*
plot. A = 1.50; W’=O.
be present, but it has a far more complicated suggestively, large scale turbulent motion.
7. EFFECT
OF NUMBER
.._-. _~----.....~ . -. .-.-...
.
.
.. . ..-.. ._ (a) Rx/r’=
character.
OF MODES
.
..
-3;
(b) R,/n2=
-6.
One might call this pattern,
RETAINED
Two mode calculations were made which gave qualitatively similar results to those with four modes. Based upon the present results and earlier studies [l, 41 with up to six modes, it is expected that four mode calculations will give quantitatively accurate results for the parameter combinations studied here.
8. CONCLUSIONS
Chaotic motions occur for certain combinations of in-plane compressive load and fluid flow velocity. These evolve continuously from simpler motions with changes in these two parameters. Phase plane plots effectively display the results; conventional Poincare plots are less useful, though extensions of these appear fruitful. Results from the qualitative theory of non-linear differential equations (differential dynamics) were helpful in motivating the present study and interpreting the results obtained. Finally there are several questions which frequently are asked about this work. These are dealt with here. How many modes are required to give the chaotic motion, how many are required to give essentially converged quantitative results, and what is the largest number that has been used? The answers are, respectively, two, four (for the range of non-dimensional parameters studied in this paper) and as many as 20 (the maximum number of modes employed being determined by computing cost). Four modes were normally used in the present calculations. Is the time step size used in the numerical simulation small enough to insure numerical stability of the results? The usual estimates of step size required for numerical stability for the corresponding linear system were made based upon the highest frequency mode and also numerical testing was done with various step sizes. These results were in good agreement and showed that for a sufficiently small time step, no numerical instability occurred and the results for the time simulation were closely repeatable.
CHAOTIC
FLUTTER
OF A BUCKLED
PLATE
343
Is the result truly random or are the time histories repeatable from one run to the next for a given set of initial conditions? The results are repeatable from one computer run to the next for the same initial conditions. Hence calling the motion chaotic is perhaps a better choice of term than random. Do the results depend upon the initial conditions? Apparently not. Initial displacements in the first modal amplitude, Al, from 0.1 to 2 were chosen with no perceived difference in the steady state oscillations and their character. How do you know that the steady state oscillation has been reached and the transient has decayed? The time for the transient oscillation to decay can be computed for the corresponding linear system and also numerical experiments were performed for varying time simulation record lengths. Results from these are consistent and show that for a sufficiently long time simulation the transient has decayed. Typically, the maximum time of the simulation was T = 18 and the (conservative) time at which the steady results were taken to be reached was 9 (corresponding to, typically, 20 cycles of oscillation for the transient to decay and 20 cycles of steady state motion). Do similar results obtain when a plate of finite width which bends in two directions is considered? Yes. Perhaps the most profound and difficult question is why do chaotic motions occur? Here a partial, intuitive answer is given. The appearance of chaotic motion seems to arise as a consequence of the presence of two parameters, in this case flow velocity and mechanical in-plane load, which govern two distinct types of instability, in this case flutter (sometimes called Hopf bifurcation) and Euler buckling (sometimes called static bifurcation). In the corresponding linear model, flutter occurs as a result of the coalescence of two eigenfrequencies with increasing flow velocity and buckling occurs upon the vanishing of an eigenfrequency with increasing compressive mechanical in-plane load. Near the point in the parameter space of flow velocity and mechanical in-plane load where the flutter and buckling stability boundaries merge, chaos appears. As the mechanical in-plane load increases, the range of flow velocity for which chaotic motion occurs also increases. Also see the discussion by Holmes and Marsden in references [9] and [lo]. ACKNOWLEDGMENT
The author would like to thank the reviewers for several helpful suggestions.
REFERENCES 1. E.H. DOWELL,H.C.CURTISS,JR.,R.H.SCANLAN andF. SISTO 1979 AModern Course in Aeroelasticity. Leyden, The Netherlands: Sitjhoff-Noordhoff. 2. E. H. DOWELL 1975 Aeroelasticity of Plates and Shells. Leyden, The Netherlands: Noordhoff. 3. E. BREITBACH 1977 AGARD Report R-665. Effectsof structural nonlinearities on aircraft vibration and flutter. 4. E. SIMIU and R. H. SCANLAN 1978 Wind Effects on Structures. New York: John Wiley and Sons. 5. R. L. BISPLINGHOFF and H. ASHLEY 1962 Principles of Aeroelasticity. New York: John Wiley and Sons. 6. R. L. BISPLINGHOFF, H. ASHLEY and R. L. HALFMAN 1955 Aeroelasticity. Cambridge, Massachusetts: Addison-Wesley Publishing Co. 7. Y. C. FUNG 1955 An Introduction to the Theory of Aeroelasticity. New York: John Wiley and Sons. 8. R. H. SCANLAN and R. ROSENBAUM 1951 Aircraft Vibration and Flutter. New York: Macmillan Company.
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E. H. DOWEL1
9. P. J. HOLMES 1977 Journal of Sound and Vibration 53, 471-503.
10. 11. 12. 13. 14. 15. 16.
Bifurcations to, divergence and flutter in flow-induced oscillations: a finite dimensional analysis. P. J. HOLMES and J. MARSDEN 1978 Automatica 14, 367-384. Bifurcations to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis. J. MARSDEN and M. MCCRACKEN 1976 The Hopf Bifurcation and Its Application, Springer Applied Mathematics Series No. 19. Berlin: Springer-Verlag. S. SMALE 1967 Bulletin of the American Mathematical Society 73, 747-817. Differentiable dynamical systems. E. H. DOWELL 1966 American Institute of Aeronautics and Astronautics Journal 4, 12671275. Nonlinear oscillations of a fluttering plate, Part I. E. H. DOWELL 1967 American Institute of Aeronautics and Astronautics Journal 5, 18561862. Nonlinear oscillations of a fluttering plate, Part II. C. C. LIN 1955 The Theory of Hydrodynamic Stability. Cambridge University Press. C. S. VENTRES and E. H. DOWELL 1970 American Institute of Aeronautics and Astronautics Journal 8, 2022-2030. Comparison of theory and experiment for nonlinear flutter of loaded plates.
FLUTTER
OF A BUCKLED
AN EXAMPLE
OF CHAOTIC
A DETERMINISTIC
PLATE
AS
MOTION
AUTONOMOUS
OF
SYSTEM?
E. H. DOWELL Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, U.S.A. (Received 28 November
1981, and in revised form 23 February 1982)
For aeroelasticity of plates and shells, the equations of motion are well established. Results obtained by numerical time integration have been compared to those obtained by topological theories of dynamics and also from experiment. chaotic self-excited oscillations may occur for this deterministic
All of these suggest that system.
1. INTRODUCTION
Non-linear aeroelasticity [l-4] is a rich source of static and dynamic instabilities and associated limit cycle motions. Fundamentally, aeroelasticity combines the classical fields of fluid and solid mechanics. Usually, as the name implies, it does so within the context of the sub-fields of aerodynamics combined with elasticity [5-81. The aerodynamic forces almost invariably are non-conservative. When acting upon a resonant elastic structure whose motion modifies these forces in a feedback sense, they lead to a complex and fascinating variety of dynamical behavior. It has been known for some time that a plate under a compressive in-plane load with a fluid flow over its (upper) surface may undergo complex motions [2] (see the sketch of plate geometry in Figure 1).
Figure 1. Sketch of plate geometry. w is the order of a plate thickness.
Plate deflection,
w, is shown to a greatly
exaggerated
For no fluid flow, but with a sufficiently large compressive load, -IV:, buckle into a statically deformed shape. By contrast, for no compressive
scale for clarity.
the plate will
load, but with a fluid flow of sufficiently large velocity, U,, the plate will flutter with a periodic, nearly harmonic motion. However, with both compressive load and a fluid flow, chaotic motion may occur which is thought to be a result of a strange attractor [g-12]. t An earlier version of the present work was published as Part I, E. H. Dowell, Nonlinear Aeroelasticity, in 1980 New Approaches to Non-linear Problems (edited by Philip J. Holmes). Philadelphia, Pennsylvania: SIAM.
333 0022-460X/82/230333+ 12 $03.00/O
@ 1982AcademicPressInc. (London)Limited
334
E.
H. DOWE1.I
The problem is considered here in its simplest formulation: i.e., a one-dimensional structural model (albeit non-linear) and (linear) piston theory aerodynamics (appropriate to high supersonic Mach numbers). The governing partial differential equation is reduced to a system of ordinary differential equations by using a modal expansion and Galerkin’s method. For this relatively simple model, Holmes [9, lo] has obtained a number of interesting results by using the methods of differential dynamics (for a two mode expansion). Here results are obtained (for two to six mode expansions) by using numerical integration in time. The resulting data are examined in the phase plane and also as Poincare plots, this examination being motivated by Holmes’ results. By systematically changing the compressive load and fluid flow parameters, the evolution of chaotic motion from simple, deterministic motion is studied. It is worthy of emphasis that at lower Mach numbers the aerodynamic model needed becomes more elaborate and leads to integro-differential equations. These have been successfully solved by numerical integration techniques [2]. However, it is unclear to what extent the methods of differential dynamics may be used for such equations. Hence, for the present class of problems, these latter methods appear primarily to be of value in explaining the qualitative character of the motion and in suggesting effective formats for presentation of the results of numerical solutions and their interpretation. 2. EQUATIONS
OF MOTION
Here only an abbreviated account is given as the underlying theory has been discussed thoroughly elsewhere [2,13,14]. The governing partial differential equation is
(1) where w is the plate transverse deflection, x is the streamwise spatial co-ordinate, t is time, D =Eh3/12(1 - v’) is the plate bending stiffness, N, = (Eh/2a) j,” (aw/ax)‘dx, which physically is the tension created by stretching of the plate due to bending, NF is the externally applied in-plane load, positive in tension, E is the modulus of elasticity and v is the Poisson ratio of the plate material, a is plate length, h is plate thickness, m is the mass/per unit length of the plate, pm is the fluid mass density, U, is the flow velocity, M is the flow Mach number, and Ap is the static pressure difference across the plate. A set of ordinary differential equations is obtained by using Galerkin’s method with the modal expansion, w = 1 a,(t) sin (n7rx/a). Equation (2) is compatible with so-called simply w = a2w/ax2 = 0 at x = (0, a). The result is (in non-dimensional notation)
+A( C[nm/(n”-m2)][1 m
-( - l)“+“]A,
supported
+(w/MA)“~A;)
(2) boundary
=P[l-(-
conditions:
i.e.,
l)“]/(nr),
n = 1,2, . . . , co,
(3)
where A,, =u,/h and later W = w/h, A =pooUoo3u3/MD, t_~=p,u/m, R, =NFu2/D, P =Apu4/Dh, T = t(D/mu4)“2, and a prime denotes a( )I&. These equations may be
CHAOTIC
FLUTTER
OF A BUCKLED
PLATE
335
numerically integrated to obtain time histories of motion. These are used to construct phase plane plots and Poincare plots [9-121. Systematic numerical studies are discussed in the following section. The choice of parameters is guided by the earlier results of Dowel1 [2,13,14] and Holmes [9, lo]. Among the parameters studied are A, a nondimensional flow velocity parameter, R,, a non-dimensional in-plane load parameter, the initial conditions, and P, a non-dimensional static pressure differential. All results were obtained by using a four mode expansion except for a few two and six mode calculations done for comparison. To make the nature of the mathematical model as transparent as possible, one can consider a two mode representation from equations (3), where various numerical constants are omitted for clarity. One has A~+A~(1+R,)-AA,+A”2&A;+(A:+A:)A~=P, A;I+A2(4+R,)+hA1+A1’2[2A;+(A:+A;)A2=0.
(4)
The skew-symmetric terms involving A are responsible for the dynamic instability, flutter, while the R, terms (when R, ~0) can cause a static instability, buckling. The form of flutter modeled by equations (4) is called coupled mode flutter and is so named because a minimum of two modes is required to produce it. If one examines a root locus of equations (4) for infinitesimal perturbations about the trivial equilibrium, A 1= A2 = 0, then the two eigenvalues, which were the system natural frequencies at A = 0, approach each other as A increases and after near coincidence one of them passes into the unstable half plane of the root locus. Hence this form of flutter is also sometimes called merging frequency flutter [ 1,2]. Only this type of plate flutter has been treated by the qualitative theory of differential equations in the literature [9, lo]. By contrast another form of flutter is called single mode flutter [l, 21 and it arises because the aerodynamic forces create negative damping. It is discussed briefly here, before returning to equations (3) and (4). Analogous to equations (4), a single equation for A1 displays the essential features of this type of flutter: T A;+
I -m
K1[(v~)A1’*]A;((+)d~+Al(l+Rx)+A:=P.
(5)
The convolution integral is a mathematical representation of the physical fact that at time T the aerodynamic forces depend in general on the entire past history of the plate motion. This effect is important for Mach numbers, A4, near unity, but is unimportant at large M where equations (3) and (4) apply. A simple explanation of the non-linear limit cycle behavior of equation (5) for R, = 0 and P = 0 may be obtained by using the method of harmonic balance. Assuming AI=~Icosf2n7
(6)
one substitutes equation (6) into equation (5) and integrates over one period of motion (as the steady state is approached 7 + 00). Then one obtains KT [0/A 1’2]= 0,
R2=1+AT,
(738)
where non-essential constants have been dropped. Kf is the (real part of the) Fourier transform of K1 and physically is a damping coefficient. It is shown in Figure 2. The infinitesimal stability is obtained when A1 = 0. From equation (8) the frequency of the flutter oscillation is LQ = 1 (i.e., the single mode natural frequency) while from equation (7) the flutter velocity parameter, Af, is found from the condition that KT = 0 at A = Ar; see Figure 2.
336
I
E. H. DOWEL
Figure
2. Sketch of aerodynamic
damping
coefficient.
Now consider what happens when A >Af and A1 > 0. One still requires that K? = 0, and thus R2 = 1 +A:.
R/A iI2 = Rf/A;‘2,
(9, 10)
Solving equations (9) and (lo), one has ii, =
n2 = A/IQ,
[h/hf
- 1]“2.
(11912)
Equation (11) gives the limit cycle frequency and equation (12) the amplitude for A > hf. When R, ~0 one anticipates that chaotic motion may occur [2], but that issue is not pursued here.
3. NUMERICAL
STUDIES
For all results reported in what follows, p/M = 0.01, x/a = 0.75, v = O-3 and P = 0 unless otherwise noted. Equations (3) were solved by using a standard finite difference scheme on an IBM 3033 computer. From previous numerical simulation studies [2], it is known that the parameters A and R, govern the type of motion which may occur. Figure 3 displays a map in A, R, space which identifies the various types of motion which may occur. Only R, < 0 is considered, since this proves to be the interesting case. For small A and R,, the plate is flat and undeformed in the steady state. For small A, but moderate R,, the plate buckles. For small R,, but moderate A, the plate flutters with simple harmonic motion. For moderate A, R, a more complicated periodic limit cycle motion occurs and for sufficiently large R, and moderate to large A, chaotic motions ensue.
I (simple
x
200
-
Flat
I
I Llmlt
cycle
osc~//ot~on
hcrmonlc)
and
stable
100 Buckled,
R,
Figure
3. Sketch
of stability
regions.
- --,
but dynamically
stable
4 w/M =
0.1:
---,
F/M = 0.01; P = 0.
CHAOTIC
FLUTTER
OF A BUCKLED
337
PLATE
Results will be presented later in the phase plane for a representative point on the plate, x/a = 0.75. In anticipation of these, the distinctive types of motion which may occur are sketched in Figure 4 as follows. The buckled plate corresponds to two (non-linear static equilibria) points in the phase plane. Simple harmonic motion flutter is an ellipse. The more complicated periodic limit cyclic motion flutter comprises a smaller orbit about each buckled state and a larger orbit which evolves from the flutter motion. The chaos is beyond the author’s ability to sketch simply. Small
Moderate
A. R,
h, moderate
R,
W’
Small A, moderate
Rx
Moderate
A, large Rx
I
Figure
4. Sketch
of representative
phase plane orbit.
To investigate the possibility of chaotic motion, two trend studies were made. First and A was varied in the sequence 300, 250, 200, 175, 150, 130, 115, 100 (see Figures 5(a)-(d)). At A = 300 and 250 the phase plane plot, W vs. W, of the limit cycle shows an ellipse typical of pure flutter motion. For A = 200, the first deviations from the ellipse are evident, although the phase plane plot remains approximately a single closed curve. However, at A = 175, 150, 130 and 115 three loops are in evidence. The largest of these derives from the pure flutter motion, while the two smaller ones are associated with buckling or divergence. At A = 175 the three loops are particularly clearly defined and as A = 150,130 and 115 the motion becomes progressively more chaotic. At A = 100 the phase plane plot is simply a point, representative of a pure buckling or divergence instability. -37r*, Secondly, A was held fixed at 150 and R, varied in the sequence -2.5~*, -3*5?r*, -4r*, -5?r*, -6r2. At R, = -2.5~~ and -37r*, the three loops are particularly clearly defined and as R, = -3*57r*, -47~*, -59~~ and -6n*, the motion becomes progressively more chaotic. Nevertheless the basic three loop pattern remains albeit in a more obscure form. See Figures 5(c), 6(a) and (b). The chaotic motion which occurs for certain parameter combinations of A and R, might well be termed random. However, it is clear that such motion evolves continuously in parameter space from motion which is decidedly deterministic. Moreover even taken solely as motion at a point in parameter space (some fixed A and R,), it is manifestly bounded in the phase plane and can be characterized, for example, by minima and maxima of displacement and velocity, W and W’. Finally, it is noted, the chaos comes from the solution of deterministic equations. To further characterize the motion, autocorrelations and power spectra of the motion were calculated. Two representative power spectra are shown in Figures 7(a) and (b) for R, was held fixed at -4~~
338
E. H
DOWEL1
-6
-6 -15 -1.0
-0.75-0.50-0.25
W
Figure 5. Phase plane plot: effect of flow velocity. (d) A = 115.
0
0.25
0.50
0.75
1.0
I.25
W
R, = -4n*;
P = 0. (a) A = 300; (b) A = 200; (c) A = 150;
A = 150 and Rx/r2 = - 3 and -6 respectively. As can be seen the former results show two discrete frequencies dominating the power spectrum, which is the expected result for a periodic motion with two dominant motions in the phase plane (recall Figure 6(a)). However, in the latter case, Rx/r2 = - 6, the power spectrum, although still showing peaks, has a continuous distribution such as that conventionally expected of a random process. f is a non-dimensional frequency in cycles per unit time. Quite aside from its own intrinsic interest, the present results may serve as a paradigm for similar chaotic motions which result from deterministic partial differential equations. Perhaps the best known of these is the chaotic (turbulent) motion which is associated with the Navier-Stokes equations. There is at least one distinction, however, between the two. For the present problem there are two parameters, A and R,, each of which
CHAOTIC
FLU’lTER
OF A BUCKLED
24
50
I
339
PLATE
I
I
I
1
1 -1.0
/ -0.5
I 0
1 0.5
I
I
I
L I.0
I I.5
I 2.0
(b) 20
40
16
-
30 -
12
-12 -0.7
-40 -0.5
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
1 -1.5
-2.0
W
2.5
W
Figure 6. Phase plane plot: effect of in-plane load. A = 150; P= 0. (a) R, = -3~'; (b) R, = -6~'.
governs a distinctive instability, flutter and buckling (divergence). It is the interaction of these two essentially deterministic motions for certain combinations of A and R, that leads to chaotic motion. On the other hand only a single parameter, the Reynolds number = Re, appears in the Navier-Stokes equations (if one assumes incompressible flow). At this stage, one can only speculate that Re may play a dual role (as suggested by Lin’s interpretation of linear hydrodynamic stability theory [15]) and/or that more than one instability mode is governed by Re. 18.
1(a)
(b)
32 -
16 -
28 -
14-
24 -
12 -
20-
S 16-
12 -
a-
‘l-
O_ 0
6
Figure 7. Power spectra. A = 150; TMM = 288; DELT = 0.001; TPRINT= 20 x DELT. (a) Rx/p2 = -3.0; (b) RJlr2=40.
340
F:.
4. EFFECT
H.
DOWFI
1
OF INITIAL
CONDITIONS
For the base case of A = 150 and R, = --4~r’, various initial disturbance amplitudes were studied. In all cases the initial deflection of the plate was assumed to be a pure first natural mode: i.e., W(x, t= 0)=A,it = 0)sin (n-x/u I, with various A ,(f = 0)chosen. In addition to the base case value of A ~(t = 0)= 0.1,values of 0.2, 0.5. 1 .O and 2.0 were chosen. The long time steady state results were essentially the same, showing no sensitivity to the initial disturbance amplitude chosen. It should be emphasized that this result is not universal since Ventres and Dowel1 [16] have already shown that for plates of finite (large) length/width ratio initial conditions can play an important role. 5. EFFECT
OF STATIC
PRESSURE
DIFFERENTIAL
Another parameter of interest is a static pressure differential across the plate. This, of course, destroys the symmetry of the geometry by giving the plate deflection a preferred direction. Two trend studies were conducted. First of all, the simpler case of zero-in-plane P. At P = 0 the load, R, = 0,was studied at A = 400 for various pressure differentials, plate flutters in a limit cycle oscillation whose phase plane trajectory is an ellipse centered about the origin, W = W = 0.As P is increased from 0 to 100, 200 and 250 the elllipse decreases in size and its center moves to the right. At P = 300 the ellipse has collapsed completely to a point, the static pressure differential having stiffened the plate sufficiently so that flutter is completely suppressed. As expected the motion does not exhibit any tendency to chaos for R, = 0. Next the base case of A = 150 and R, = -47r’ was reconsidered for various amounts of static pressure differential P = 0, 12.5, 25, 37.5, 50, 56.25, 62.5, 68.75, 75, 100 and 200. For Pa68.75, the static loading was sufficient to suppress all flutter motion. For small P, say 0, 12.5, 25 and 37.5, the motion was qualitatively similar and exhibited a somewhat chaotic appearance. However, it will be recalled that for R = -37r’ the distinct three loops were present and, with that knowledge, these three loops may still be seen, 25
-20 -0.3
-0.1
0.1
0.3
0.7
0.5
0.9
I.1
I.3
I.5
W
Figure
8. Phase plane plot: effect of static pressure
differential.
A = 150; R, = -4~‘;
P = 50.
CHAOTIC
FLU-ITER
OF A BUCKLED
PLATE
341
somewhat dimly to be sure, at R, = -47r*. The most remarkable result occurs, however, at P = 50 and 56.25 (see Figure 8). Here the motion has a very distinct, non-chaotic character with two loops in evidence: that is, the static pressure differential has suppressed the chaotic character of the motion and one of the three loops which appear for smaller P. For P = 62.5 the motion returns to a chaotic state with no readily discernible pattern and for P 2 68.75, the limit cycle degenerates to a point and all motion ceases.
6. POINCARE
PLOTS
To gain further insight into the nature of the motion, an alternative format for presentation of the data was considered: i.e., a Poincare plot. For present purposes such a plot is one where the continuous phase plane diagram is sampled at discrete time intervals. Of course, since the time histories here were generated by a digital computer and are thus necessarily discrete, all of the previous phase plane diagrams were, strictly, Poincare plots. However, previously the time interval used was short compared to any time intervals characteristic of the motion, so that the diagrams were effectively continuous. The effect of systematically increasing this time interval was examined for two example cases, A = 150 and R, = -3~~ or -4~~. The former, R, = --37~~, is a periodic motion and as the sampling time interval becomes equal to the period of the motion, the Poincare plot reduces to a single point. The latter, R, = -41r’, is a chaotic motion where no such reduction is expected, but other interesting insights may be obtained. A unit time interval was selected to be AT = O-02. Results were obtained for R, = -3~~ and time intervals of 1, 2, 5, 10, 20, 34, and 40 units. For a small number of units the complete character of the phase plane diagram is evident; for 34 units essentially a single point is obtained indicating that this is the period of the periodic motion. Similar results were obtained for R, = -4~~. Because the motion is now chaotic the character of the motion becomes very difficuit to ascertain as the sampling time interval is increased. This is emphasized when the sampling points in the phase plane are unconnected. No period is found in which the phase plane diagram is reduced to a single point. An alternative Poincare plot may be constructed by specifying one of the dynamical variables and taking slices in variable space of the remaining co-ordinates. For example, one could specify A; = 0 and examine, at those times when this is true, the values of Ai, AZ, AL, As A;, . . . . However for even four modes, the remaining variable space has dimension 7 = (4 x 2) - 1. Hence it would be difficult to display and interpret such results. To reduce the dimension of this Poincare plot, one might proceed as follows (which is suggested by physical considerations). Consider a particular physical velocity, W’, and the corresponding displacement, W, for each such W’ which appears in the phase plane. (Of course, W is a linear combination of the A,, and W’ of the AA. Hence they are suitable lower dimensional representations of the individual co-ordinates, A,, and A;.) Furthermore the range of W, call it A W, (for each such W’) is a measure of the system chaos. A strictly periodic system would have a range of zero: i.e., for each such W’ there is a single W. Conversely if all W are possible between Wmin and W,,, the system motion might be termed completely chaotic. The larger R =A W/( W,, - Wmin), the more chaotic the motion. In Figures 9(a) and (b) Poincare plots are shown for W’= 0, A = 150 and R,.T’ = - 3 and - 6, respectively. The former results display the expected repetitive, periodic character for an oscillation with two dominant frequencies. For the latter a pattern appears to
342
.
.
.
.
.
.
.
. .
. .
.
. ..-*. .
Figure
9. PoincarC
.
.-*.
.
....*
plot. A = 1.50; W’=O.
be present, but it has a far more complicated suggestively, large scale turbulent motion.
7. EFFECT
OF NUMBER
.._-. _~----.....~ . -. .-.-...
.
.
.. . ..-.. ._ (a) Rx/r’=
character.
OF MODES
.
..
-3;
(b) R,/n2=
-6.
One might call this pattern,
RETAINED
Two mode calculations were made which gave qualitatively similar results to those with four modes. Based upon the present results and earlier studies [l, 41 with up to six modes, it is expected that four mode calculations will give quantitatively accurate results for the parameter combinations studied here.
8. CONCLUSIONS
Chaotic motions occur for certain combinations of in-plane compressive load and fluid flow velocity. These evolve continuously from simpler motions with changes in these two parameters. Phase plane plots effectively display the results; conventional Poincare plots are less useful, though extensions of these appear fruitful. Results from the qualitative theory of non-linear differential equations (differential dynamics) were helpful in motivating the present study and interpreting the results obtained. Finally there are several questions which frequently are asked about this work. These are dealt with here. How many modes are required to give the chaotic motion, how many are required to give essentially converged quantitative results, and what is the largest number that has been used? The answers are, respectively, two, four (for the range of non-dimensional parameters studied in this paper) and as many as 20 (the maximum number of modes employed being determined by computing cost). Four modes were normally used in the present calculations. Is the time step size used in the numerical simulation small enough to insure numerical stability of the results? The usual estimates of step size required for numerical stability for the corresponding linear system were made based upon the highest frequency mode and also numerical testing was done with various step sizes. These results were in good agreement and showed that for a sufficiently small time step, no numerical instability occurred and the results for the time simulation were closely repeatable.
CHAOTIC
FLUTTER
OF A BUCKLED
PLATE
343
Is the result truly random or are the time histories repeatable from one run to the next for a given set of initial conditions? The results are repeatable from one computer run to the next for the same initial conditions. Hence calling the motion chaotic is perhaps a better choice of term than random. Do the results depend upon the initial conditions? Apparently not. Initial displacements in the first modal amplitude, Al, from 0.1 to 2 were chosen with no perceived difference in the steady state oscillations and their character. How do you know that the steady state oscillation has been reached and the transient has decayed? The time for the transient oscillation to decay can be computed for the corresponding linear system and also numerical experiments were performed for varying time simulation record lengths. Results from these are consistent and show that for a sufficiently long time simulation the transient has decayed. Typically, the maximum time of the simulation was T = 18 and the (conservative) time at which the steady results were taken to be reached was 9 (corresponding to, typically, 20 cycles of oscillation for the transient to decay and 20 cycles of steady state motion). Do similar results obtain when a plate of finite width which bends in two directions is considered? Yes. Perhaps the most profound and difficult question is why do chaotic motions occur? Here a partial, intuitive answer is given. The appearance of chaotic motion seems to arise as a consequence of the presence of two parameters, in this case flow velocity and mechanical in-plane load, which govern two distinct types of instability, in this case flutter (sometimes called Hopf bifurcation) and Euler buckling (sometimes called static bifurcation). In the corresponding linear model, flutter occurs as a result of the coalescence of two eigenfrequencies with increasing flow velocity and buckling occurs upon the vanishing of an eigenfrequency with increasing compressive mechanical in-plane load. Near the point in the parameter space of flow velocity and mechanical in-plane load where the flutter and buckling stability boundaries merge, chaos appears. As the mechanical in-plane load increases, the range of flow velocity for which chaotic motion occurs also increases. Also see the discussion by Holmes and Marsden in references [9] and [lo]. ACKNOWLEDGMENT
The author would like to thank the reviewers for several helpful suggestions.
REFERENCES 1. E.H. DOWELL,H.C.CURTISS,JR.,R.H.SCANLAN andF. SISTO 1979 AModern Course in Aeroelasticity. Leyden, The Netherlands: Sitjhoff-Noordhoff. 2. E. H. DOWELL 1975 Aeroelasticity of Plates and Shells. Leyden, The Netherlands: Noordhoff. 3. E. BREITBACH 1977 AGARD Report R-665. Effectsof structural nonlinearities on aircraft vibration and flutter. 4. E. SIMIU and R. H. SCANLAN 1978 Wind Effects on Structures. New York: John Wiley and Sons. 5. R. L. BISPLINGHOFF and H. ASHLEY 1962 Principles of Aeroelasticity. New York: John Wiley and Sons. 6. R. L. BISPLINGHOFF, H. ASHLEY and R. L. HALFMAN 1955 Aeroelasticity. Cambridge, Massachusetts: Addison-Wesley Publishing Co. 7. Y. C. FUNG 1955 An Introduction to the Theory of Aeroelasticity. New York: John Wiley and Sons. 8. R. H. SCANLAN and R. ROSENBAUM 1951 Aircraft Vibration and Flutter. New York: Macmillan Company.
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E. H. DOWEL1
9. P. J. HOLMES 1977 Journal of Sound and Vibration 53, 471-503.
10. 11. 12. 13. 14. 15. 16.
Bifurcations to, divergence and flutter in flow-induced oscillations: a finite dimensional analysis. P. J. HOLMES and J. MARSDEN 1978 Automatica 14, 367-384. Bifurcations to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis. J. MARSDEN and M. MCCRACKEN 1976 The Hopf Bifurcation and Its Application, Springer Applied Mathematics Series No. 19. Berlin: Springer-Verlag. S. SMALE 1967 Bulletin of the American Mathematical Society 73, 747-817. Differentiable dynamical systems. E. H. DOWELL 1966 American Institute of Aeronautics and Astronautics Journal 4, 12671275. Nonlinear oscillations of a fluttering plate, Part I. E. H. DOWELL 1967 American Institute of Aeronautics and Astronautics Journal 5, 18561862. Nonlinear oscillations of a fluttering plate, Part II. C. C. LIN 1955 The Theory of Hydrodynamic Stability. Cambridge University Press. C. S. VENTRES and E. H. DOWELL 1970 American Institute of Aeronautics and Astronautics Journal 8, 2022-2030. Comparison of theory and experiment for nonlinear flutter of loaded plates.