A99-24781 AIAA-99-1430

CHAOTIC OSCILLATIONS OF SKIN PANELS IN HYPERSONIC FLIGHT

Miige Fermen-Coker* and Glen E. Johnsont University of Dayton, Dayton, OH 45469

ABSTRACT In recent years, the investigation of dynamical behavior

of plates under thermal loads has gained importance due to the high temperatures reached on skin panels of hypersonic vehicles. One of the potential problems associated with high-speed, high-altitude flight is the loss of predictability in the response of skim panels, as they vibrate chaotically about their thermally buckled positions. The current study addresses this problem by developing a model that enables the determination of the conditions under which the panels may exhibit chaotic behavior. Lyapunov exponents are calculated to detect the onset of chaos, and the effects of thermal moment, thermal buckling, amplitude and fi-equency of excitation, damping, thickness and the aspect ratio of panels on the chaotic response of the system are determined. It is found that introducing a temperature gradient across the thickness of the panels delays or eliminates chaos, and that it is possible to prevent chaos at the design stage by selecting the appropriate aspect ratio for the panels. It is shown that increasing damping does not necessarily yield to the elimination of chaotic behavior, as previously believed. The effects of damping are found to be highly dependent on the thickness of the panels and the excitation fi-equency.

~ INTRODUCTION In this paper, the conditions under which simply

supported plates subject to thermal and mechanical loading exhibit chaotic behavior are studied, to develop a better understanding of the effects of design variables and operating conditions on the vibratory response of skin panels on vehicles that are subject to high speed, high temperature flight conditions. Chaotic oscillations have been observed by researchers studying panel flutter, as early as 1960’s [1,2]. In 1971, Eastep and McIntosh [3]

* Graduate Student, Mechanical & Aerospace Engineering Department; currently Aerospace Engineer, SAIC, Lexington Park, MD. Member AMA. t Professor and Chair, Mechanical & Aerospace Engineering Department. Member ASME.

Copyright 0 1999 by Mtige Fe-en-Coker. Published by the American Institute of Aerodynamics and Astronautics, Inc. with permission.

analyzed nonlinear flutter and response of a thin, elastic, simply supported plate, and studied the stability of small perturbations about limit cycle oscillation. They noted the nonlbear behavior of the interaction of the buckling

-and flutter phenomena. Later in the 1980’s, Dowel1 [4] considered the chaotic region in his discussion of the classical flutter-buckling stability diagram when he re- analyzed the flutter of a buckled plate. Some more recent studies include Hopkins et. al. [5,6], Murphy et. al. [7,8], and Lee [9].

Researchers in nonlinear and chaotic dynamics frequently employ a simple nonlinear oscillator represented by Duffing’s equation with negative linear stifmess, modeling vibrations of buckled structures such as the problem studied here, to develop an intuitive understanding of how chaotic motions occur in dynamical systems with two stable equilibrium states. Dowel1 and Pezeshki [lo], studied the dynamics of a buckled beam and successfully compared their theoretical results with the experimental data provided by Moon [ 111. The complete Lyapunov spectrum versus the damping coefficient for a standard Dufflng equation was obtained by De Souza-Machado et. al. [12]. Dowel1 and Pezeshki [lo], and Pezeshki [I31 studied the occurrence of chaotic regions in the force-frequency plane for Duffing’s equation, by considering two different

Tamping levels. tie conclusions and observations made in previous research, sampled above, are unfortunately insufficient to form an idea on the effects of damping and excitation characteristics on chaos, for the physical system considered here. Also, the geometric properties of systems are implicit in previous studies, and thus the mathematical models are not appropriate for design considerations or for specifically studying the effects of varying geometric properties of a skin panel on the chaotic response of the system. Generally speaking, studies regarding control or prevention of chaotic oscillations in real-world systems have been concentrating on the use of adaptive structural concepts [14], as an extension of studies on flutter suppression [ 15,161. The current study on the other hand, aims to develop a better understanding of the effects of design variables on the chaotic response of the system, to incorporate this knowledge into the design process, and to provide insight for adaptive control applications.

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EQUATIONS OF MOTION The equations of motion for the investigation of the

large deflection of elastic plates subject to thermal and mechanical or aerodynamic loading may be developed in accordance with Weiliang and Dowel1 [17], employing an assumed mode method. As discussed in detail by Fermen-Coker [ 181, the development starts with Hamilton’s extended principle. The linear stress-strain and strain-displacement relations are substituted into the strain energy expression for the system and von K&m&n’s large deflection theory, and the Kirchoff hypothesis are employed to obtain the strain energy of the heated plate in terms of displacements. The simply supported boundary conditions are then imposed on the system. Assuming only a single mode of vibration, the displacements in x, y and z directions respectively at the mid-plane, are expanded as follows:

u = hu, (x3 Y)P, (t) v = hv, (x, yhr (t)

w = h w I (x, y)q W

(1)

where, pl, ql and rl are the generalized modes for x, y and z displacements respectively, and the modal shape functions that satisfy the geometric boundary conditions are:

u, =v, =w, =sin(T)sin(%) (2)

Note that ‘a’ and ‘b’ are dimensions of the rectangular plate in x and y directions, which are parallel to the edges of the plate at the mid-plane. The derivation of the equations for multi-mode analysis of a heated plate system is also discussed in detail in [18]. The mode functions Eq. (1) are substituted into the strain energy expression and Lagrange’s equations of motion are obtained for the system. The thermal loading assumed includes: i) a uniform temperature increase at the mid- plate compared to a reference temperature state i.e. 8,,, ii) a parabolic temperature variation over the plate that is expressed in terms of plate dimensions and the temperature at the center of the plate i.e. 8,, iii) a temperature gradient across the thickness of the plate expressed in terms of plate thickness and the upper surface temperature increase i.e. 8,. 8(x,y,z)=&-t-16(x/a)(l-x/a)(y/b)(l-y/b)e,

37 (3) +ye,

where, h is the thickness of the plate, measured in the z direction that is normal to the plate. Note that 0, is the temperature increase at the z=+h/2 surface due to a thermal gradient which is assumed to be linear through the thickness, such that &=O at z=O. Once damping and

sinusoidal external excitation are added to the system, Lagrange’s equations of motion may be reduced as described in [18], to the single mode equation of motion in nondimensional form:

r;+C,r;+C,r, +C,$ =C, sin(Rr)+Cs (4)

where. c, = 47tT) c2 =4x2(1-pl)

c3 =4lr 25 Kl

, c4 =4n2$)

C5 =4n2Ftg, n=2nL 0 n

P, =c~oqJ +c, 0, C

9a4 + 2a2b2 + 9b4

K = Eh3n2(a2 + b2)o

/‘_ Eh3;?+-b’,,rt4 -9)o C- 9abx2(l - v)

FO 60=K , Ftg = 16(1 + v)o e

1 n4(h/a)2 ” Note that the effect of both the uniform temperature

term and the parabolic temperature variation term are incorporated in a single parameter, pi. When pi>], the linear stiffness term becomes negative and this corresponds to buckling temperatures for the plate. The derivatives in Eq. (4) represent the derivatives with respect to r=t/t,,, where t,=2rr/o,. The single mode equation of motion, presented as Eq. (4) is in the form of the well-known Duffig equation which may be solved numerically using a simple computer code utilizing a 4th order Runge-Kutta integration scheme, once the design variables are specified.

(5)

NUMERICAL APPROACH The detection of chaos through the computation of

Lyapunov exponents [ 19-231 has been demonstrated to be an effective tool to characterize chaotic motions by many

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researchers. One such study is performed by Pezeshki and Dowel1 [24] who calculated the exponents for 1, 2 and 4 mode modal projections of the partial differential equation governing the motionof a magnetically buckled beam, They successfully compared their numerically generated computations based on a technique presented in [20], which will also be employed here, to the exponents calculated from an experimental time history.

The single mode heated plate equation, Eq. (4), may easily be transformed into a set of three autonomous first order differential equations. Hence, the complete Lyapunov spectrum for the system consists of three exponents. The sign of the largest Lyapunov exponent determines whether or not a system is chaotic. A positive largest exponent indicates a strange am-actor, which, in turn, implies chaos [19-231. Phase planes are generated, for several points picked randomly to verify the calculated largest Lyapunov exponent estimates.

THERMAL BUCKLING EFFECT The modal potential energy of the system

represented by Eq. (4) is

U, =$C,r,4 ++C,r/ -C,r, (6)

which is obtained by realizing that sum of all the terms other than damping, excitation and acceleration must be equal to the derivative of the potential energy with respect to the generalized mode for z displacement so that Lagrange’s equation is satisfied. For simplicity, let us set CS=O for the time being, i.e. there is no difference between the temperatures of the upper and lower surfaces of the plate (e,=O). By setting the derivative of the potential energy expression with respect to rl to zero, and substimting the original expressions stated in Eq. (5) for the coefficients, the equilibrium points are obtained as:

Equation (9) indicates that, for materials with higher thermal expansion coefficient a, or higher Poisson’s ratio v, less uniform temperature increase is needed for thermal buckling to occur. Young’s modulus does not have any effect on this critical value. The thicker the plate, the later the buckling will occur as the

>emperature is increased. The closer the plate shape is to a square, the higher are the temperatures needed to buckle the plate. For square plates, the bigger the dimensions of the plate, the lower the critical uniform temperature at the mid-plate that would buckle the plate.

where,

rl =0

‘, =*vlm

w4

~ (7)

v2 =4aZb20(l+v)[rt4(9& +4e,)-36eJ

v3 =3n6h2(a2 +b’) (8)

v4 = 3x31h]J9 a4 +2a2b2 +9b4

An aluminum plate of dimensions a=12 in., b=50 in., h=0.2 in. is considered first. The damping coefficient is set as c=O.Ol. With these values substituted into Eq. (7,8), and 8, set equal to zero, there are three possibilities: a) For 00> 13.94”F, there are three equilibrium points which results in the what is known as the twin-well potential form. b) For 00=BQ,=13.94“F, the critical thermal buckling uniform temperature increment, when 8, and 8, are set to zero. The only equilibrium point is at ri=O, in this case. c) For 00<13.94”F, the only real root is at r,=O. These results are demonstrated in Fig. 1, where the potential energy is shown as a function of e0 and ri. It is clear how the potential energy transforms from single well to double

-swell, making chaos possible for low amplitudes of excitation. The largest Lyapunov exponents are calculated by setting the excitation frequency of external sinusoidal excitation at half the natural frequency, and varying the excitation amplitude, Fo, between 1 and 100 lb.in. in increments of 1, and the uniform temperature increase at the mid-plate, 80, between 0 and 50°F in increments of 0.1 (Fig. 2). The

_s blue tones represent the negative values of the largest Lyapunov exponent for each point, and therefore define the regions of non-chaotic motion. The positive exponents, or the chaotic regions of motion, are represented with yellow and red tones. The first onset

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The critical value of the uniform temperature increment after which linear stiffness~becomes negative is the

same value after which the term in the square root above becomes positive and therefore the second and the third equilibrium points become real, constructing the twin-well potential form. In terms of the plate material and geometrical properties this value is simply

8, = n2h2(a2 +b2)+4e

cr 12a2b2a(l+v) (9)

The coefficient of 8, above is roughly -0.4. For values of e0 greater than what is given by the above expression, the fust equilibrium point at ri=O is an unstable equilibrium point, and the other two points in Eq. (7) are stable equilibrium points.

of chaos for the plate of specified dimensions is captured in Figure 3. Note that, for certain intervals of the amplitude of excitation the critical mid-plate temperature for chaos is constant, and one may determine ranges of excitation amplitude for which chaos is delayed or maybe does not exist, within the operating temperatures.

THERMAL MOMENT EFFECT A temperature differential is imposed across the

thickness of the plate, so that the upper surface temperature is higher than the lower surface temperature. Thus, a thermal moment is induced and the plate is forced to bend concave up.

The critical temperature increase above a reference temperature state, for chaos to occur, is observed to be at 14.4”F for relatively low excitation amplitudes, provided that the plate dimensions and material properties are as stated in the previous section. The potential energy of the system, converted to a 2D form by setting &=15”F to better observe the effect of increased thermal gradient, is shown in Fig. 5. The solid line displays the case where the thermal gradient is zero. The dashed and dotted lines demonstrate how the two-well form gradually transforms itself to favor a single-well form as the upper surface temperature is increased relative to the lower surface temperature. This simply indicates that as the upper surface of the plate gets hotter compared to the lower surface of the plate, the propensity of the plate to deform concave up is increased which reduces the unpredictability of the state of bifurcation when the critical thermal buckling temperature is reached at the mid-surface. This, in turn, reduces the possibility for chaos to occur.

obtained to be 0.24”F. That is, a temperature difference of approximately half a degree of Fahrenheit is needed between the upper and lower surfaces when the mid- surface temperature is increased by 15”F, in order to eliminate the two-well potential form. The next logical step is to investigate how the amplitude of the forcing function effects the motion of the system, under the influence of a thermal gradient, in addition to the temperature increase at the mid-plate. Therefore, the largest Lyapunov exponents for the ranges covered in Fig. 2, are re-calculated by setting 0,=0.5”F (Fig. 5). The small chaotic region at the bottom of the figure, which we had magnified to obtain Fig. 3 before, has disappeared. Also, note that the bigger chaotic region at the right lower comer of the figure is no longer chaotic. As the amplitude of the external excitation is increased, the influence of increased thermal gradient across the thickness becomes less and less significant. This makes sense physically since the thermal gradient increases the tendency of the plate to buckle in a certain direction as long as the amplitude of the excitation is not strong enough to compensate for this inclination. For smaller amplitudes however, the effect of thermal gradient could be quite significant in terms of delaying or eliminating chaos.

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The critical value of 0, after which one of the wells disappears can be found by setting the derivative of U,, presented in Eq. (6) to zero. The resulting equation is a cubic equation, the solution of which leads us to Eq. (10) below.

8 “”

=n2h2(a2 +b2)+4e 12a2b2a(l+v)

(10)

a=13xlO” /“F, p=.l/386, v=l/3) and setting a=12 in., b=50 in., h=0.2 in., <=O.Ol, and 8,=0, as before, the excitation frequency is varied between 0.03 to 1.5 times the natural frequency of the system. The force- frequency plane for &=12”F does not contain any significant chaotic regions. As the mid-plate temperature of the plate is increased beyond the critical thermal buckling temperature to 15°F however, several chaotic regions in the form of patches stretched across the force-frequency plane following traces of second order curves (Fig. 6). Note that, for a slightly buckled plate, it is more likely to observe chaos at low frequencies. It seems that it is possible to find regions within the force-frequency planes that are chaos free, if the operating temperatures are known.

~1~ = J [12aa2b20,,(1+v)-n2h2(a2 +b2)13 (11)

w7 =72a2b21ah(l+v)\J9a4 +2a2b2 +9b4 Substituting the geometrical and material properties of the plate, as stated before, into this expression, and also setting 80=15”F as it is in our current example of Fig. 5, the critical upper surface temperature increase is

More buckled states at &=25’F and 8,-,=49’F, are considered (Figures 7, 8). The curves of fragmented pockets of chaotic regions seem to be pushed up to higher magnitudes of the forcing amplitude, as e0 is increased. At 8,,=49”F, which corresponds to the small chaotic region shown in the lower right comer of Fig. 2, a chaotic band at the lower edge of the force-frequency plane is observed, covering the whole range of frequencies studied, and expanding between F0 equal to

EXCITATION AMPLITUDE AND FREQUENCY Choosing an aluminum plate (E=lO’ lb./in.,

0 to 10, as shown in Fig. 8. Note also that the pockets of chaotic regions at low frequencies seem to be pushed up further in the two-dimensional plane, and mostly disappear for higher amplitudes for frequency ratios between 0.4 to 1.3.

The force-temperature plane originally displayed in Fig. 2 was obtained by setting the frequency ratio to. 0.2, 0.4, 0.7 and 0.9 this time, as shown in Figures 9 to 12 respectively. Overall, these results indicate that there are more chaotic regions at lower frequencies compared to higher frequencies. It seems that the effect of varying the frequency may be causing a bigger impact on the middle portion of this region, i.e. moderate amplitudes, such that very low and very high amplitudes do not display any significant changes until very high frequencies are reached. The mid-plate temperatures do not seem to have any impact on determining the effects of varying the excitation frequency. Note that, it is possible to find chaos-free regions in the Bo-Fo plane for certain frequencies. Fig. 11 for o/0,=0.7 for example, displays a very large chaos-free region for amplitudes between 10 and 40 Ib.in., for the whole range of mid-plate temperatures considered. The chaos delaying effect of thermal moments for low excitation amplitudes is tested and verified for the complete frequency range studied.

DAMPING EFFECT The eigenvalues of interest for the system represented by Eq. (4) are:

4.2 =

-c, f cf -‘tc, -12c,r/ Cl21 2

where C,, C2 and Cj are as stated in Eq. (5). Note that there are three possibilities to consider, depending on the value of the term inside the square root, being greater than, equal to, or less than zero. In more convenient terms, these possible cases are overdamped, critically damped and underdamped respectively. The value of C,, corresponding to the case of critical damping, is obtained in terms-of geometric and material specifications and also the uniform mid-plate temperature components: --

(13)

where, vz and yr3 are as expressed in Eq. (8). Considering the region shown in Fig. 2, the

damping ratio is increased from 0.01, first to 0.05 and then to 0.1. The results are displayed in Fig 13 and Fig. 14 respectively. Increasing the damping coefficient eliminated the small chaotic region at the lower right

comer of Figure 2, as demonstrated by these two figures.

The force-frequency plane originally shown in Figure 6 is regenerated for two other damping ratios. Figure 15 is obtained by setting {=O.OS and Figure 16 by setting c=O.l, where 8,,=15”F and 8,=0 for both cases. We observe from these two figures that o/0,=0.5 is a particularly troublesome frequency ratio, when the other parameters of the problem are specified as such. These results indicate that increasing damping is more effective in eliminating chaos for lower frequency ratios, regardless of the excitation amplitude. Figure 16 is particularly helpful in realizing this effect.

EFFECTS OF PLATE GEOMETRY Using Equation (9) and varying the width of the

plate between 12 and 52 inches in increments of 4, f& is obtained as a function of the aspect ratio of the plate (Fig. 17). Note that the critical temperature increase varies significantly when the aspect ratio is around 1, which of course corresponds to square plates. The closer a plate is to being a square, the more sensitive are the critical buckling temperatures to the changes in the aspect ratio of the plate. This effect is increased if the plate dimensions are smaller. Considering several plates

-of the same aspect ratio but different sizes, the difference between the corresponding critical temperatures of each, becomes less as the dimensions

.-get larger. pi.

In other words, for larger plates the significance of the changes in the aspect ratio of plates

Xmuch less; compared to smaller plates, in terms of the -critical buckling temperatures. Also, as the plate dimensions get larger, the critical buckling temperatures drop. In other words, not surprisingly, chaos may be observed at lower temperatures as the plate sizes get larger.

Force-aspect ratio planes corresponding to et,= 15, 20, and 49°F are generated by calculating the largest Lyapunov exponent for several different sizes of plates, keeping the thickness constant, and are displayed in Figure (18-20) respectively. Fig. 18, along with Fig. 17 indicate that, at &,=15”F, and F0=86 Ib.in., for a plate of

Pimensionsm12x25, pre-buckling chaos exists. It seems that for plates of less rectangularity, chaos is expected to appear at smaller forcing amplitudes, until a certain aspect ratio of possibly close to 1 is reached. This would indicate that the critical value of forcing amplitudes, after which chaos may be observed before the potential energy transforms from single-well to double-well, is reached sooner, as plates of smaller aspect ratios are considered.

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At &,=20’F, the chaotic regions appear after this critical aspect ratio of 1.4 for a 12 in. thick plate, as predicted by Figure 17 is reached (Figure 19). Note that the regions that are arranged in an almost ring- system-like pattern start to curve upwards as the shape of the rectangular plate transforms into a square. Thus, we do not observe pre-buckling chaos. For &,=49”F, we see chaos mostly around the square plate, as shown in Figure 20. The response of the plates with lengths 2 to 3 times their widths is free from chaos, even though they are thermally buckled.

Employing Eq. (IO), one may determine the critical 8, values for a wide range of aspect ratios at a pre-specified value of I$,, and observe that as the plate dimensions get larger, much higher thermal gradient values are required to eliminate the two-well potential form. For large plates, the sensitivity of the critical temperature difference imposed between upper and lower surfaces of the plate is increased with respect to the relative dimensions of the plate. However, for high aspect ratio rectangular plates with shorter side smaller than or equal to 12 in. in this specific case, the thermal moment generated by the temperature differential across the plate’s thickness may prove to be efficient in delaying or minimizing chaos.

The next step is to study the effects of plate thickness on the chaotic response of the system. So far, we have been only considering plates of 12-m. thickness. Force- frequency planes are generated by setting t&,=13, 15, 20 and 49°F respectively (Figures 21-24). Recall that for h=0.2, t&=13”F corresponds to a pre-buckling state. Hence in Figure 21, there are no chaotic regions or points corresponding to h=O.2 in. for the range of amplitudes studied. This value of thickness also marks the point after which we do not observe chaos at all for the given set of conditions at this temperature. It seems, for thinner plates, it is more likely to observe heavily chaotic regions for much wider ranges of amplitudes.

The force-temperature plane originally presented in Fig. 2 is regenerated for a thinner (h=0.15 in.) and a thicker (h=O.3 in.) plate. The results are shown in Figures 25 and 26 respectively. As anticipated, the number and sizes of chaotic regions are increased for the thinner plate, with a definite shift to the left, indicating that chaos is observed for smaller values of &. We may also infer that chaos is more likely to occur at lower amplitudes as the plate’s thickness is reduced. Note that some chaotic regions observed at very low amplitudes in Fig. 2, do not exist in Fig. 25. Predictably, the number of chaotic regions is significantly reduced for the thicker plate. Unpredictably however, the only region left in this wide

range of amplitudes and temperatures, corresponds to relatively lower forcing amplitudes, and most interestingly the shape of it bears a striking resemblance to the shape of the familiar chaotic region originally presented in Figure 3.

The curves of chaotic patches are spread all across the force-frequency for a thinner plate of 0.15-m. thickness (Fig. 27), as opposed to Figure 6 where the thickness was 0.2 in. Both figures were generated for 80=150F. The curves of chaotic regions in the force- frequency plane are pushed upward to higher values of the forcing amplitude as we continue increasing O0 beyond the critical buckling temperatures.

The damping effect is re-examined considering two different plates of sizes 12x31x0.2 and 12x12x0.2, in addition to the 12x50x0.2 plate we have already presented. Within the context of the results generated, it seems that the effect of increasing damping is amplified, in terms of delaying chaos to higher mid- plate temperatures, as the plate dimensions get smaller.

In addition to the 0.2-m thick plate of 12x50, a thinner plate with h=0.15 in. and a thicker plate with h=0.3 in. are also studied, to investigate the combined effects of damping and thickness. A force-temperature plane corresponding to Fig. 2 is generated for the h=O. 15 plate, keeping all the other parameters the same, as shown in Figure 28. An excellent example, demonstrating the unpredictable effect of varying damping of the system, is obtained when we increase the damping coefficient from c=O.Ol to c=O.l, for the 0.3 in. thick plate and regenerate Fig. 26. The initially small chaotic region became quite large as seen in Fig 29, covering a much wider range of excitation amplitudes, when the damping of the system is increased; a most undesirable effect.

CONCLUSIONS Chaotic regions of motion are obtained through the

computation of the Lyapunov exponents, for a sinusoidally excited heated plate system modeling skin panels on hypersonic, transatmospheric vehicles. The effects of variations in the design variables on the chaotic response of the system are studied, and several important observations were made, including: i) Chaos is delayed or eliminated in the presence of a thermal moment acting on the system; an effect that is reduced gradually as the excitation amplitude is increased. ii) Chaotic oscillations are more likely to occur at low excitation frequencies. iii) As damping is increased, chaotic regions of motion are observed to correspond to higher amplitudes of excitation, while not necessarily getting smaller in size. iv) Increasing damping seems to be more effective in eliminating chaos for low

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excitation frequencies and becomes more complex and unpredictable as the thickness is varied. The discovery of cases for which increased damping results in larger chaotic regions in the force-temperature and force-frequency planes questions the validity of employing adaptive control to suppress chaos through additional damping. v) It is possible to-find chaos-free aspect ratios, when all the other variables are specified. vi) As the thickness is reduced, finding chaos-free excitation amplitude and frequency combinations becomes more difftcult.

ACKNOWLEDGEMENTS The computational work presented here was -

performed on Ohio Supercomputer Center’s CRAY Y- MPBE machine. The authors wish to thank the OSC personnel, especially Leslie Southern, Al Stutz and Larry Cooper for their assistance and cooperation.

REFERENCES 1. Kobayashi, S., “Two Dimensional Panel Flutter 1. Simply Supported Panel,” Trans. Japan Sot. Aero. Space Sci., 5 (8), 1962, pp. 90-102. 2. Dowell, E. H., “Nonlinear Oscillations of a Fluttering Plate,” AIAA Journal, 4(7), 1966, pp. 1267-1275. 3. Eastep, F. E., McIntosh, S. C., Jr., “Analysis of Nonlinear Panel Flutter and Response Under Random Excitation or Nonlinear Aerodynamic Loading,” AIAA J., Vol. 9, No. 3, March 1971, pp. 41 l-418. 4. Dowell, E. H., “Flutter of a Buckled Plate as an Example of Chaotic Motion of a Deterministic Autonomous System,” J. Sound & Vibration, 85(3), 1982, pp. 333-344. 5. Hopkins, M. A., Dowell, E. H., “Limited Amplitude Panel Flutter With a Temperature Differential,” AIAA Structures, Structural Dynamics, and Materials Conference, Paper No. 94-1486, 1994. 6. Hopkins, M. A., “Nonlinear Response of a Fluttering Plate Subject to Supersonic Aerodynamic, Thermal, and Pressure Loads,” Ph.D. i’ksis, Duke University, 1994. 7. Murphy, K. D., Virgin, L. N, Rizzi, S. A., “Characterizing the Dynamics Response of a Thermally Loaded Acoustically Excited Plate,” 1 of Sound & Vibration, 196(5), 1996, pp. 635-658. 8. Murphy, K. D., Virgin, L. N, Rizzi, S. A., “Experimental Snap-Through Boundaries for Acoustically Excited Thermally Buckled Plates,” Experimental Mechanics, 36(4), 1996, pp. 3 12-3 17. 9. Lee, J., “Large-Amplitude Plate Vibration in an Elevated Thermal Environment,” Appl. Mech. Rex, Vol. 46, No. 11, Part 2, November 1993.

10. Dowell, E. H., Pezeshki, C., “On the Understanding of Chaos in Dufftng’s Equation Including a Comparison with Experiment,” J Applied Mech., Vol. 53, March 1986, pp. 5-9. 1 I. Moon, F. C., “Experiments on Chaotic Motions of a Forced Nonlinear Oscillator: Strange Am-actors,” J, Sound h Vib., 85(3), 1982, pp. 333-344. 12.De Souza-Machado, S., Rollins, R. W., Jacobs, D. T., Hartman, J. L., “Studying Chaotic Systems Using Microcomputer Simulations and Lyapunov Exponents,” Am. J. Phys., 58 (4) April, 1990. 13.Pezeshki, C., “An Examination of Chaotic Motion

~for the Buckled Beam,” Ph.D. Thesis, Duke University: 1987. 14.Hall II, E. K., Hanagud, S. V., “Active Control of Chaotic Systems by Using Adaptive Structural Concepts,” AIAA-91-I062-CP, pp. 2179-2189. 15.Scott, R. C., Weisshaar, T. A., ‘Controlling Panel Flutter Using Adaptive Materials,” AIAA-91-1067-CP, pp.22 18-2229. 16.Pak, C. G., Friedmann, P. P., Livne, E., “Transonic Adaptive Flutter Suppression Using Approximate Unsteady Time Domain Aerodynamics,” AIAA-9I- 0986~CP, pp. 1832-I 854. 17. Weiliang, Y., Dowell, E. H., “Limit Cycle Oscillations of a Fluttering Cantilever Plate,” ALU Journal, Vol. 29, No. 11, November 199 1, pp. 1929- 1936. 18. Fermen-Coker, M., “Chaotic Vibrations of Heated Plates,” Ph.D. Thesis, University of Dayton, 1998. 19. Schuster, H. G., “Deterministic Chaos: An Introduction,” VCH Publishers, 1989. 20. Wolf, A., Swift, J. B.. Swinney, H. L., Vastano, J. A., “Determining Lyapunov Exponents from a Time Series,” Physica, 16D, 1985, pp. 285-3 17. 21. Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J. M., “Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems: A Method for Computing All of Them,” Meccanica, 15, 1980, pp. 21-30. 22.Baker, G. L., Gollub, J. P., ‘Chaotic Dynamics: An Introduction,” Cambridge University Press, 1990. 23.Korsch, H. J., Jodl, H.-J., “Chaos: A Program Collection for the PC,” Springer-Verlag, 1994. 24.Pezeshki, C., and Dowell, E. H., “Generation and Analysis of Lyapunov Exponents for the Buckled Beam,” Int. J. Non-Linear Mechanics, Vol. 24, No. 2, 1989, pp. 79-97.

1937

02

0 15

0.1 -

OLt5-

O-

offi- . /

Fig. 1 Transition of Potential Energy from single-well Fig. 4 Thermal gradient effect: rl vs.U,I for various to double-well as e. is increased. values of 8,.

1W

90

60

m al 40 40

30

20

10

L 0 10 a0 30 40 xl

50 t to

40 40 5 5

30 30 20 20 1 1

10 10 .5 .5 s s

0 0 10 10 al al 30 30 40 40 n n

Fig. 2 Largest Lyapunov Exponent depicted as a function Fig. 5 Largest Lyapunov Ekpcnent depicted as a function of of e. in “F (x-axis) & Fe in Ibin. (y-axis). 9,~ in T (x-axis) & Fe in Ib.in. (y-axis). (a=12 b=50, h=0.2, &J.Ol, &o,,=O.5,8,=0) (a=l2, b=50, h=0.2, ~=O.Ol. ~n/~,,=0.5, 8,=0.5)

Fig. 3 Largest Lyapunov Exponent depicted as a function Fig. 6 Largest Lyapunov exponent depicted as a function of 0,~ in “F (x-axis) & Fo in Ib.in. (y-axis). of o/co, (x-axis) & F0 (y-axis) (a=12 b=50, h=0.2, ~=O.Ol, cA.J,=O.~, O,=O) (a=12 b=50, h=O.2 ~=O.Ol, t+l5,0,=0)

1938

f

2 im

5 90

80 1

70 5 81

0 50

5 40

1 30

a0 5

lu 2

0

Fig. 7 Largest Lyapunov exponent depicted as a Fig. 10 Largest Lyapunov exponent depicted as a function of o/o, (x-axis) & F,(y-axis). function of El0 (x-axis) & FO (y-axis). (a=12 b=.50. h=0.2, c=O.Ol, t&,=25. 8,=0) (a=12.b=50, h=0.2, c=O.Ol, 0/0.1,,=0.4, CI,=O)

80

m- =~

60

50

40

30

im

90

80

m

60

m

40

33

Fig. 8

im

m

80

M

60

xl

‘lo

30

m

10

Largest Lyapunov exponent depicted as a Fig. 11 Largest Lyapunov exponent depicted as a function of w/w, (x-axis) & F,, (y-axis). function of Cl0 (x-axis) & F0 (y-axis). (a=12 b=50, h=O.2, c=O.Ol, $349, O,=O) (a=12 b=50, h=0.2, c=O.Ol, 0/~,,=0.7, C&=0)

im

90

m

m Q

xl 40 30

al 10

0

Fig. 9 Largest Lyapunov exponent depicted as a Fig. 12 Largest Lyapunov exponent depicted as a function of 80 (x-axis) & F0 (y-axis). function of 0” (x-axis) & F0 (y-axis).

10 m

(a=12 b=50, h=O.2, &I.Ol, m&,=0.2. e,=o) (a=12. b=50. h=0.2, c=O.Ol, o/t&=0.9, O,=O)

1939

Fig. 13 Largest Lyapunov exponent depicted as a function of O0 (x-axis) & F0 (y-axis). (a=12, b=50, h=0.2, <=O.OS. 0/~,,=0.5, e,=O)

0 10 20 30 40 !w

Fig. 14 Largest Lyapunov exponent depicted as a function of O0 (x-axis) & F0 (y-axis). (a=l2. b=50, h=0.2, <=O. 1, o/&=0.5, &=O)

1m

90

60

M

60 50 40

3l 20 10

02 0.4 06 08 I 12 1.4

Fig. 15 Largest Lyapunov exponent depicted as a Fig. 18 Largest Lyapunov Exponent depicted as a function of o/o, (x-axis) & FO(y-axis). function of b/a (x-axis) & F0 (y-axis)

Fig. 16 Largest Lyapunov exponent depicted as a function of O/O, (x-axis) & Fo(y-axis). (a=12 b=50, h=0.2, C,=O.l, eC15, e,=o)

m

20

&12 10 aAJ

0 aa2 05 I 15 2 25 3 35 4 45 5

Fig. 17 b/a (x-axis) vs. 00,, (y-axis)

(ALa= b=SO. h=0.2, c=O.OS, t&,=15,8,=0) (a=l2, h=O.Z <=O.Ol, o/0,=0.5, eo=15, e,=o)

1940

Fig. 19 Largest Lyapunov Exponent depicted as a Fig. 22 function of b/a (x-axis) & FO (y-axis) (a=l2, h=0.2, ~=0.01.0/~,,=0.5, &20, O,=O)

im

-1

15

-2 1 2 3 4

Fig. 20 Largest Lyapunov Exponent depicted function of b/a (x-axis) & F. (y-axis)

Fig.

as a Fig.

Largest Lyapunov Exponent depicted as a function of h (x-axis) & F. (y-axis) (a=Q b=50, &I.Ol. m&=0.5, &=15, e,=o)

23 Largest Lyapunov Exponent depicted as a function of h (x-axis) & F. (y-axis) .

(a=12 h=0.2, t+o.oi,d~=o.5, e&9, e,=o) (a=12 b=50, i+O.Ol, o/m&.5, t&20. e,=o)

0.3 0.4

im

90

m

?o

a,

M

40 ;

3l

al

10, =f P:

0.15 02 025 0.3 035

21 Largest Lyapunov Exponent depicted as a Fig. 24 Largest Lyapunov Exponent depicted as a function of h (x-axis) & F. (y-axis). function of h (x-axis) & Fo (y-axis) (a=l2, b=50, ~=O.Ol: 0/6&I.5. Cb=l3, e,=o) (a=l2. b=50. c=O.Ol, 6.Aq70.5.&F49, e,=o)

1941

Fig.

Fig.

Fig.

100 -. ~ -22 25

90 2

Eo 15

40

25 Largest Lyapunov Exponent depicted as a function of (3, (x-axis) & F0 (y-axis) (a=l2, b=50, h&15. &+O.Ol, w/0,=0.5.8,=0)

26. Largest Lyapunov Exponent depicted as a function of 0, (x-axis) & F0 (y-axis) (a=l2, k50, h=O.3, c=O.Ol, 0/~,,=0.5, O,=O)

02 0.4 06 06 1

27 Largest Lyapunov Exponent depicted as a function of o/o, (x-axis) & FO (y-axis) (a=12. b=50, h=0.15, &=O.Ol, $=15, t&=0)

Fig.

Fig.

im

50

80

m 69 50 40 xl al 10

0 10 20 30 40 50

28 Largest Lyapunov exponent depicted as a function of O. (x-axis) & FO (y-axis) (a=l2, b=50, h=O. 15, c=O.l, w/t0,,=0.5, O,=O)

im

90

80

70

60

50

40

sl

20

10

0 10 al 40 Yl

29 Largest Lyapunov exponent depicted as a function of Cl,, (x-axis) & F0 (y-axis) (a=l2, b-50, h=O.3, [=O.l, w/0,=0.5. O,=O)

1942