Transonic panel flutter

AlAA 93-1 476 Transonic Panel Flutter

G. A. Davis and 0. 0. Bendiksen University of California at Los Angeles Los Angeles, CA

34t h AIANAS MEIASCEIAHSIASC Structures, Structural Dynamics and Materials Conference AIANASME Adaptive Structures Forum April 19-22, 1993 / La Jolla, CA v

For permlsslon to copy or republlsh, contact the Amerlcan lnstltute of Aeronautics and Astronautics 370 L'Enfani Promenade, S.W., Washlngton, D.C. 20024

TRANSONIC PANEL FLUTTER

Gary A. Davis * and Oddvar 0. Bendiksen ** Mechanical, Aerospace and Nuclear Engineering Department University of California, Los Angeles, CA 90024-1597

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Abstract

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The finite element method was used to study the stability and aeroelastic response of thin 2-dimensional panels subjected to high speed flows at Mach numbers ranging from 0.8 to 2.5. The Euler equations were used to represent the dynamics of the unsteady flow field and nonlinear plate theory was used to model the structural behavior of the panels. The fluid equations were cast in a Lagrangian-Eulerian finite element form and solved simultaneously with the structural equations using a recently published method. Results of the study show that, in the absence of shocks. the Euler equations predict aercelastic response behaviors similar to thme obtained using potential flow methods. When structural deflections became large enough for shmks to appear, however. aerodynamic nonliearities caused a completely different response. Divergence and limit cycle flutter were observed where shocks play a significant role in the overall motion of the panel. For most panel geometries only divergence was detected in the transonic region below Mach 1. but for very thin panels, flutter was observed. At sonic velocities both divergence and flutter were found, and. at supersonic velocities. when nonlinear structural effects were present, only flutter was detected. In the Mach number range from approximately 1.4 to 1.5. flutter was found to involve the higher modes of the panel, and possibly result in chaotic motion. Finally, when structural nonlinearities were removed, a shock divergence phenomenon was observed at supersonic flows which involved rapid shock motion across the panel surface.

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1.0 Introductioii It is well-known that flutter of aircraft skin panels can occur in supersonic flows.' particu~ar~y in the presence of externally applied in-plane forces. The comequence of this instability, which is usually self-limited by structural nonlinearities. is the possibility of long tern fatigue failure of the panel. Supersonic panel flutter has been the subject of numerous studies in the Papers by Dowellp and. more recently. Reed. Hanson, and Alford? review the literature on the subject. In the subsonic and transonic speed ranges. panel flutter has been studied by Ishii and Yanagizawa' and Dowell? using potential flow theory. Recent interest in panel flutter has centered on cmpsite panels'"~''and also the effects of aerodynamic heating at supersonic Mach

Nomenclature

c D

E E, h

k L m M

p

free stream pressm pressure difference across the panel surface = poU212. free stream dynamic pressure panel generalized displacement vector panel generalized force vector time = U,$lc. nondimensional time x-direction local fluid velocity x-direction mesh velocity free stream fluid velocity y-direction local fluid velocity y-direction mesh velocity plate normal deflection streamwise coordinate normal coordinate = 1.4. ratio of specific heats = 2qc3/D,nondimensional dynamic pressure = poc/pmh , mass ratio = 0.3. Poisson's ratio fluid density plate density oscillatory frequency (radlsec)

plate chord length = Eh3/L?(l-v?. plate stiffness modulus of elasticity specific total energy of the fluid plate thickness = oc/2Uo. reduced frequency element length plate mass per unit area Machnumber pressure

num~s,l?.13.14

Although mast of the literature on panel flutter adequately considers nonlinear structural effects, there are no known studies to date which treat the nonlinear effects associated with flows at or near sonic velocity. In the high subsonic regime. where panel flutter is believed to be impassible, the formation of shocks and embedded supersonic pockets requires the use. of an aerodynamic theory with the ability to capture these flow features. Similarly. at flow velocities above, but near sonic, nonlinearities exist which invalidate most existing linear

* Graduate SNdent Researcher ** Associate Professor, Member AlAA Copyright 0 1992 by Davis and Bendiksen. Published by the AlAA wilh permission.

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are discretized using deforming triangular Jinite elements which follow the movement of the panel. The pressure below the panel, where no flow OCCUIS, is set equal to free stream pressure for all cases considered here. The panel itself is modeled using two-dimensional plate (beam) finite elements. which are intended to represent a thin. isotropic plate undergoing cylindrical bending. The cylindrical bending assumption approximates a plate of large width in relation to chord length and neglects variations of normal displacement in the spanwise direction. Rigidly clamped as well as simply supported boundary conditions are used at the panel leading and trailing edges, and axial motion of the ends is taken as zero. This introduces a nonlinear term in the equations which accounts for mid-surface stretching as the plate deflects laterally. Figure 1 shows a schematic of the simply supported plate and associated coordinate system.

aerodynamic theories, particularly when structural deformations become large.

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There are several aerodynamic theories that attempt to caphire the nonlinear features of transonic flows. namely the transonic small disturbance and the nonlinear full potential equations. These are limited because they do not account for the production of vorticity and entropy across shocks, although special procedures have been devised which attempt to correct these deficieucies.15 In order to properly account for these effects, within the framework of the conservation laws. the Euler equations must be used. In the present paper we perform a fully nonlinear study on the dynamic behavior of thin panels subjected to both supersonic and transonic flow fields on one surface. Time-marchiag flutter calculations for the Mach number range of 0.8 through 2.5 are presented. which clearly show the existence of oscillating shock waves and supersonic pockets as the panel deforms. Furthermore, the method used to perform this study involves a novel computational scheme whereby the fluid and the smcture are treated as one continuum using finite elements in a mixed Lagrangian-Eulerianformulation.16 The objectives of the present study are twefold. First we wish to explore the behavior of thin panels subjected to transonic and supersonic flows on one surface. The transonic speed range will be emphasized, that is. high subsonic flow from Mach 0.8 to low supersonic flows as high as Mach 1.5; althougb solutions are also presented for supersonic Mach numbers as high as 2.5. Second. we will demonstrate the utility of a novel computational method which will be used to perform this study, wherein both the fluid and structural solutions are carried out simultmeously using the finite element method.

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Fig. 1: Panel Geometry

2.1 Structural Equations The partial differential equation which governs the plate lateral deflection. w can be written as,

.

Here, N , , is the initial in-plane loading due to external forces, and N, is the in-plane force induced by deflections of the panel. N, is given by,

2.0 Aeroelastic Model The aeroelastic model consists of a mathematical representation of the panel structure and fluid. coupled through pressures and boundary conditions at the. panel surface. It has been showd6 that the structural and fluid equations are related in that they both represent conservation laws, the difference being that the fluid equations are cast in a mixed Lagrangian-Eulerian formulation and the shuctural equations are in a Lagrangian form. The entire fluid-structure system can thedore be treated as one continuum dynamics problem by switching from a mixed Lagrangian-Eulerian description to a pure Lagrangian formulation as the solution scheme passes from the fluid into the structure. Thus the basic problem becomes one of paforming the structural and Buid solutions in a simultaneous fashion to obtain the aeroelastic respow of the panel. The Euler equations are used to describe the twodimensional flow field above the panel. These equations

Equations (1) and (2) are the two dimensional equivalent of the von Karman plate equations. These equations have been discretized using standard Jinite element techniques" and the resulting matrix equations are,

I m l t q } + ( [ k l + [ k , I ) t q } = {Ql

(3)

Here the Iumwd mass matrix is used. which for a single given by,

I n 0 0 0 L2/24 0 0 0 1/2 0

2

0

0 0 0

0 L2/24]

u'

The linear part of the stiffness matrix can be written as,

r2

[kl =

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6L -12

6L

6D 6L 4L2 -6L 2L2

1

-12 -6L 12 -6L 6L 2L2 -6L 4L2 Here the region of integration S ( t ) is not required to be a material region: thus it may deform in any arbitrary manner. The specific total energy, E,, is defined as the sum of the specific internal energy, e , and the specific kinetic energy. and can be written as,

Finally, the nonlinear stiffness due to in-plane forces induced by normal deflections is given by,

where,

Eh Nx =

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fwllr r36

E, = e

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3L -36 3~ 3L 4L2 -3L -L2 1-36 -3L 36 -3LJ 3L -L2 -3L 4L2

+ z(u2 + v2)

(10)

Finally. an equation of state is specified to relate the flow variables to the static pressure.

(7)

p = peh-1)

(11)

In a manner similar to the structural formulation. we use the finite element method to obtain a set of ordinary differential equations describing the flow variables at discrete points thronghont the flow field. The results of this discretization process. details of which are given in Reference 18. is a set of equations of the form,

Although the mass and stiffness terms are shown above in matrix form, the solution technique used here does not involve assembling the individual element matrices into large system matrices. as is the common practice in bite element analysis. Instead, individual terms were added to the appropriate equations as the solution praxdure marched from one node to the next in a manner identical to the way the fluid equations were solved. Thus the full s!mctural equation set was solved simnltaneously with the fluid equations using the same algorithm for both. A normal mode representation of the structure was not attempted.

where W,represents the vector of flow variables. defined in Eq. (9a). at the node point j Here M,,and Q, are the consistent mass matrix and global flux vector, respectively. A stable solution to Eq. (12) is not possible because the discretization scheme produces an underdissipative equation set for problems of this type. Thus we add an artificial dissipation term that effectively damps nnmencal oscillations yet is conservative in that it adds no mass, momentum, or energy to the system. Here we follow the approach of Jameson and coworkrs '9,20.21.22 in using an adaptive dissipation operator. Thus the discretized form of Eq. (12) is written as,

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2.2 Aerodynamic Equations The two-dimensional unsteady Euler equations were used to represent the flow field dynamics. These equations, which are the basic statements of conservation of mass, momentum, and energy, in the absence of viscous forces. can be written in a mixed LagrangianEulerian form as,

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X(AfrIw,)+ Q,- D, = 0 (13) Z,*I where the dissipation D, is a blend of second and fourth differences of the flow variables. The second difference operator is used only in the vicinity of shock waves and is w e d off in smooth regions of the flow to preserve the second order spatial accuracy of the scheme. The boundary condition used at the panel surface consists of the requkment of flow tangency and is simply a statement that no flow should pass through an element edge that is adjmnt to the panel. Far field bonndary conditions must be considered carefully because of the hyperbolic nature of the Euler equations and the fact that we are using a 6nite computational domain to simulate a physical problem involving an infinite half-plane.

where.

3

diagonalized. the time stepping prmdnre becomes.

If flow conditions are specified at the outer edge of the region, a partially reflective surface is created that echoes some of the energy in outgoing waves and eventually contaminates the solution. A nonreflective boundary condition is employed. which allows outgoing waves to pass without reflection. The boundary condition is based on incoming and outgoing characteristics of the flow in the direction normal to the far field boundary as described by Jameson and Mavriplis?' A mesh deformation scheme is desired which efficiently moves grid points in a smooth fashion as the panel deforms. It is particularly important to use a scheme which is as simple as possible to avoid excessive computational expense. In our present work, grid points on the outermost boundary remain fixed while interior points are allowed to move as the panel deforms. The movement of grid points is accomplished by assigning to each interior point a scale factor and a master node located on the. panel. The movement of master nodes is determined by the solution of the structural equations, and the movement of a l l interior points assigned to a given master node is just the displacement of the master node multiplied by the scale factor. Grid points located in the immediate vicinity of theii corresponding master nodes are assigned scale factors c l m to unity while grid points on the far field boundary have factors which are zero. These factors are computed only once at the start of the solution and vary linearly with distance from the panel. Thus the mesh movement scheme is very simple and requires only a few computations to determine new grid point coordinates &r a given panel motion.

W,@' = w."

1 1 W,n" = w,'5'

Here. Mb,. is the i I h diagonal term from the lumped mass matrix and the coefficients a, are given by, 1 1 3 1 a,=. u + = ~ a5=l The structural equations, given by Eq. (3). are in the form of a second order system which can be split into two 6rst order equation sets using standard techniqueP. This places the structural equations in the same form as the fluid equations and allows the solution routine given by Eq. (14) to be used. The dissipation terms are not needed for the structural equations since t h e ' d discretization procedure produces a stable equation set. A further simplification occurs because the mass matrix for the struchue is constant, thus eliminating the need for recalculation after each mesh movement.

3.0 Solution Scheme The solution technique differs from classical modal procedures in that the fluid and structural equations are solved simultaneously. using the same solution algorithm for both. The approach is unique in that individual element matrices for the fluid and the structure are assembled as the procedure marches from one node to the next, therefore eliminating the need to form assembled (global) mass or stiffness matrices. The discretized form of the Euler equations with artificial dissipation terms. as given by Eq. (13). consists of a system of first order nonlinear ordinary differential equations, the solution of which can be obtained by a number of schemes. The method outlined here uses an explicit five-stage Runge-Kuaa procedure. wherein the dissipation terms am evaluated only at the fust two stages and are kept frozen for the remaining three. In thihis work a simplification to the procedure was performed to avoid the unnecessary computational effort of using the consistent mass matrix. Studies by the authors" have shown no significant advantage to using a consistent mass matrix. A diagonalized lumped mass matrix can be obtained from the consistent mass mahix by summing all the terms in a particular row and placing the result on the diagonal of that row. With M,, thus

4.0 Results and Discusion

Solutions were computed for unsteady eansonic and supersonic flows over two-dimensional panels with either pinned or clamped ends. Computations in the fluid domain were carried out using the unstructured grid shown in far field and doseup views in Fig. 2. The mesh consists of 1804 node points of which 50 are on the deforming panel surface and 28 are on the far field boundary, which is set at a radius of 25 panel chords. The linite element model of the smctural panel used in this study cmsisted of 10 equal length plate (beam) elements. It was felt that this model would be able to capture the complex motion that might be induced by the nonlinear aerodynamics. yet still retain enough simplicity so that the computational expense would not be excessive. In all cases studied, the externally applied normal force. Nz0, was set to zero, however, the nonlinear in-plane stretch term. N,, was retained. v

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Also shown in Fig. 3a are stability boundaries obtained using third order piston** and quasi-steady= aerodynamic theories. These c w e s were computed by the authors using the same structural model as was used with the Euler equations. Note that the piston theory solution is within a few percent of the Euler result at Mach 2.5 but. as expected. overpredicts the flutter speed by a wide margin as the Mach number is decreased. The quasi-steady result, on the other hand, matches the Euler solution quite well at low supersonic Mach numbers and underpredicts stability for higher speeds. In the limit, as Mach number increases. it can be seen that the quasisteady results approach piston theory. Another observation from Fig. 3a is that. if a conservative estimate of flutter speed is desired for Mach numbers above 1.5. one is probably safe in using the quasi-steady aerodynamic theory. Finally, Fig. 3a shows a typical operating line for an aluminum panel at sea level. Comparison of the operating l i e with the stability boundary shows that, for p4.1, unstable behavior is expected at all Mach numbers. It is interesting to note that this represents the lowest operating l i for an aluminum panel of this mass ratio. At higher altitudes the operating line moves farther upward, since at a gjven Mach number. the dynamic pressure parameter, h. is a function of pdpi. An enlarxement of the stabilitv boundary for the transonic region i s presented in Fig.-3b and shows ~ t a ~ the results of the various canputations needed to establish the cnrve. As expected, the lowest point on the boundary occu~sat Mach one. Divergence was observed at Mach numbers below sonic, flutter was observed above sonic, and both flutter and divergence were observed at Mach one. Although not shown in the figure. results were obtained for all Mach numbers shown, at dynamic pressnres equal to those on the panel operating line of Fig. 3a. For panels of this mass ratio, oscillatory flutter was not obseked at any subsonic points. Within the dynamic pressure range studied, the divergence instability which transitions to flutter with increasing dynamic pressure was found at Mach one only. We suspect, however, that for p4.1 there is a small region above sonic in which divergent behavior can be found and also a small region below sonic where flutter will be found. Flutter was observed in the high subsonic region with very thin panels at mass ratios other than 0.1. Figure 4 presents a stability plot for clamped panels in a slightly different format allowing for direct determination of the thichess ratio necessary to preclude flutter. The results of Nelson and Cunningham4 are also shown here for comparative purposes. Good agreement exists at Mach 1.6, however at Mach 1.3 the thickness ratio required to prevent flutter increases by about 35 percent over the linear results. From the figure, it is clear that the thickness ratio peaks in the Mach number range from 1.2 to 1.3. and, if one is to prevent panel flutter in supersonic flows, this should be the design point.

(b) ~ i 2~ ~ . Closeup view.

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(a) F~ ~ field view, ~(t,)

Solutions were begun with the flow field in a uniform state with all flow variables set equal to the far field conditions. The panel was initially set in the zero displacement position and given an initial velocity such that, in the absence of aerodynamic forces and nonliiear stiffness .. effects, .. .it would oscillate in its first mode with a smau amplituae. Panel limit cycle amplitudes were obtained by calculating the time histories over a period long enough for a steady state response to be reached.' Stability boundaries were determined in a similar manner, by noting the dynamic pressure at which the l i t cycle amplitude tended towards zero. 4.1 Linear Panel Stability Boundary A comparison of the linear panel stability boundary obtained using Euler equation aerodynamics to the potential flow results of h e l l g is shown in Fig. 3a, for simply supported boundary conditions. It can be seen from the figure that the Fader solution agrees well with the potential flow model throughout the speed range studied, with the exception that the transonic bucket extends to a slightly higher Mach number than shown by the linear results. As will be discussed later, it was found that this region was rich in nonlinear behavior, even at the small vibration amplitudes near the stability limit.

5

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direction of the flow. The motion of the panel is highly irregular and is iatluenced by shock formation and oscillation over a large portion of the panel.

4.2 Panels at Transonic Mach Numbers Below Unity Solutions were computed for simply supported panels exposed to flow at speeds close to. but below. sonic. At these Mach numbers and at low dynamic pressures the character of the response is highly damped with only a few cycles necessary for the panel motion to decay to near zero amplitude. As dynamic pressm is increased the solution changes to one in which the initial oscillations are damped, but the resulting equilibrium position is nonzero. This is shown in Fig. 5. which presents the time history response of an aluminum panel undergoing divergence at Mach 0.9. A plot of local Mach number and static deformed shape of the panel is presented in Fig. 6 and clearly shows the existence of a shock at about the 70 percent chord position. The initial condition for this case consisted of a negative (away from the flow) panel velocity such that, in the absence of aerodynamic forces and structural nonlinearities, the amplitude of free vibration would be about 20 pemnt of the panel thickness. As can be seen from the figure the initial motion overshoots this amplitude by almost a factor of 3 indicating a net negative force is being imparted to the panel during this time. This suggests the existence of a second (negative) equilibrium position. The. negative equilibrium position was found by giving the panel a positive initial velocity and allowing the response to proceed to a steady state solution, The response time history is shown in Fig. 7 and is very similar to that of Fig. 5, but in the opposite direction and with slightly lower amplitude. The local Mach number and deflected shape for the negative equilibrium position are presented in Fig. 8 and, as expected. show a surface free of shocks. At Mach numbers below 0.95, the aeroelastic behavior of the panel was found to be much the same. The negative equilibrium position was always found to be shock free,with a peak deflection slightly lower than for divergence in the positive direction. Shocks were not always present for positive divergence cases. however. Generally, deflections on the order of 1.5 percent chord were necessary for shocks to form at high transonic Mach numbers. Figure 9 presents the peak divergence amplitude for a simply supported panel as a function of thickness ratio and shows the effect of shock formation during positive divergence. The solutions for Mach 0.8 are shock free and the deflections are nearly symmetric. At Mach 0.9 and low thickness ratios, however. the positive equilibrium position was found to be inauenced by the presence of a shock with an amplitude almost 20 percent greater than in the negative position. As Mach n u m b was increased to 0.95 and above, oscillatory flutter was found in panels with small thickness ratios. A typical time history for a simply supported panel at M 4 . 9 5 with a thickness ratio of h/c=.001 is presented in Fig. 10. Note the rather large value of wlh indicating that structural nonliiearities are significant. During a single period of vibration the panel undergoes traveling wave motion with the waves moving in the

b

4.3 Panels at the Speed of Sound Results of computations carried out on both simply supported and clamped panels subjected to sonic flow velocities show that both negative and positive divergence, as well as limit cycle flutter. is possible. In a manner similar to the subsonic cases, highly damped oscillatory motion is observed at low values of dynamic pressure. Increasing dynamic pressure above the stability threshold produces a response which consists of an initial oscillation followed by a relatively slow divergence to an equilibrium position. Further increases in dynamic pressure beyond a second threshold value result in l i t cycle flutter. The extent of the divergence phenomenon is presented in Figs. 3b and 4. and clearly show the transition from stable motion to divergence and finally to flutter. Figure 11 presents a plot of the peak deflection of simply supported panels as a function of dynamic pressure showing the regions in which divergence and flutter were found. The negative peak was nearly always greater than the positive peak for both divergence and flutter, A typical response plot for a simply supported panel undergoing divergence in the oegative direction is shown in Fig. 12 and the final static deflected shape and surface Mach number plot are given in Fig. 13. Note that a shock has formed at approximately the quarter chord location on the panel. The negative equilibrium position’& seems to be more dominant and. at some values of dynamic pressure. we were unable to find the positive position as indicated by the short dashed line in the c w e of Fig. 11. Both clamped and simply supported panels exhibited flutter at Mach one above certain values of dynamic pressure. Figures 14a and 14b present typical response. time histories for panels showing the highly nonliiar. yet periodic. motion which was found to be characteristic of flutter at this Mach number. Note that steady state oscillation has been reached within one cycle indicating a seong instability. Deflected shapes and snrface Mach number plots for a simply supported panel during a complete cycle of motion are presented in Fig. 15 and clearly show the formation and movement of shocks as well as the traveling wave character of the panel vibration. The vibration mode shape is very similar to the subsonic flutter mode discussed in the previous section and appears to move in the direction of the flow. 4.4 Panels at Transonic Mach Numbers Above Unity At Mach 1.1 and low dynamic pressures the response. consisted of damped oscillatory motion, which became m a e lightly damped as the flutter boundary was approached. Above the flutter boundary the response reached its limit cycle amplitude relatively quickly, depending on the proximity to the boundary. For exam-

6

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4

ple. a simply supported panel at Mach 1.1 with a dynamic pressure 25 percent above the stability limit, reaches its limit cycle amplitude in about 5 cycles. At low dynamic pressures the panel motion can be classified as single mode flutter with no traveling wave motion. AS dynamic pressure approaches the operating l i of Fig. 3a the response changes to a nonlinear nature and traveling wave motion is again observed. Figure 16 shows a time history plot of the limit cycle behavior for a simply supported panel at Mach 1.1 operating on the limit l i of Fig. 3a. and clearly shows the nonlinear character of the motion. When the Mach number was increased to 1.2 the panel behaved in much the same manner with the exception that the instability became weaker, with more cycles necessary for the panel to reach steady state l i t cycle amplitude. The nonlinear character of the time traces. although still present, became less pronounced. Figure 17 shows a time history plot of the same panel shown in Fig. 16, only at M=1.2. which clearly shows the diminished role of the aerodynamic nonlinearities. For this case we found no shocks on the panel surface,but only weak oblique shocks which formed intermittently at the leading and trailing edges. A comparison of the peak limit cycle amplitudes found in the present study with the linear potential theory results presented by Dowell' is shown in Fig. 18 and illustrates the effect of nonlinear aerodynamics. Here the Euler solution predicts a larger peak deflection than the linear themy. At Mach 1.3 the instability became weaker still, requiring a significant amount of time to reach limit cycle amplitude. The nonlinear appearance of the response time history plot was about equal to that observed at Mach 1.2, however, in some cases a higher frequency was present along with the basic low frequency signal. The character of the response changed markedly at Mach 1.4. This is somewhat expected since it is in this region that the flutter dynamic pressure rises sharply with Mach number. In the early stages of the solution the panel mode shape appears to change from cycle to cycle. Over a relatively long time period the solution progresses from a low frequency, single mode response. originated by the initial conditions to a complex, but periodic. high frequency oscillation. Figure 19 shows a typical time history of this motion for the three quarter chord point on a simply supported panel. The frequency content of the panel response is presented in Eg. 20 and shows a definite shift from a single low frequency signal to a much more complex waveform consisting of several frequencies. An expanded time history plot of a several cycles of the solution during steady state motion is given in Fig. 21 and shows the periodicity of the solution. The deflected shape and pressure coefficient of the panel during a single time point is given in Fig. 22 and shows a pattern which is similar to the fifth mode shape. At other times during the vibration cycle a four lobe deflected shape was observed which is characteristic of the panel

fourth mode. At Mach 1.5. and at values of dynamic pressure above, but clok to the stability l i t . the initial panel response was damped, indicating what appears to be stable behavior. After a period of vibration at very low amplitudes the panel became unstable and limit cycle flutter was observed. For simply supported panels the motion is predominantly fifth mode and the frequency of vibration is nearly identical to the fifth natural frequency of the panel oscillating in vacuum. Elgure 23 shows a time history of the deflection at the panel midchord position which clearly shows the initial decay followed by the unstable behavior. An explanation of this phenomenon is that, at Mach 1.5. the first mode is stable while the fifth mode is unstable. As mentioned previously, all of the simulations were begun with panel initial velocities in a first mode pattern. Thus, the initial motion consists of almost entirely first mode response which decays to near zero amplitude. After a period of time, components from any unstable mode, which may have been introduced by aerodynamics or simply numerical noise. begin to grow because of their unstable nature. Another interesting phenomenon was found at Mach 1.5 for dynamic pressures close to the operating line given in Fig. 3a. Here we observe an initial increase in the first mode response followed by a gradual decay and shift to a higher frequency. In a manner completely different than observed elsewhere, a clean, steady state signal was never reached, even for long time simulations. Figure 24a presents a sample time history of the motion at the panel three quarter chord point. The frequency content of the signal changes in a seemingly erratic fashion as the solution progresses and the deflected shape appears to have components similar to modes 8 through 10 of a freely vibrating panel. In addition, an underlying low frequency component is present, as can be obwved in the expanded time trace shown in Fig. 24b. Panel oscillations similar to this have been reported for buckled plates,% and have been cited as examples of chaotic motion. At this time we are not prepared to classify the response as chaotic, as we feel further study is necessary. We do believe however, that chaos is a distinct possibility in the vicinity of this Mach number. 4.5 Shock Divergence Phenomenon

We conclude our discussion by describing a shock divergence phenomenon which we have observed when conducting simulations in the supersonic regime with the nonlinear in-plane stretch term removed from the panel equations. This configuration represents a simply supported panel in which the ends are free. to move axially. Cases were run above the stability boundary with the intent of determining if the nonlinear aerodynamic theory used in the our pmsent study, would cause l i t cycle behavior without the aid of a nonlinear shucture. Results show an unstable, first mode oscillation which

7

grows to an amplitude of about 3 percent of the panel chord and then suddenly diverges in the negative direction with such amplitude that the fluid grid becomes entangled and the solution halts. Investigation of the flow field at various time points during this motion shows that. as the panel moves in the negative direction, a normal shock forms near the trailing edge. The increased pressure behind the shock forces the panel further downward which causes the shock to move fonvard. Movement of the shock increases the panel area exposed to the high pressure region and causes further downward motion. The instability occurs very rapidly and was observed at Mach numbers from sonic up to 2.0.

8.

Near Mach 1.5. and at dynamic pressures above but near the stability limit. simply supported panel motion consists of almost entirely @th mod, response. The frequency of vibration is equalJ,ct that of the fifth mode of a plate in a vacuum. At Mach 1.5. and at high dynamic pressures a h i t cycle response was not achieved. even for long run times. The frequency response of the motion varied with time and consisted of many components. The response appeared chaotic.

Acknowledgments This research was supported by NASA Dryden Research Center Grant FDF NASA/A NCC 2-374. with Dr. K. Cupta as grant monitor.

5.0 Concluding Remarks

References E. H., Aeroelasrrcify of Plates and Shells. Noordhoff International Publishing, The Netherlands. 1975. 2 Dugundji, J.. "Theoretical Considerations of Panel Flutter at High Supersonic Mach Numbers." AIAA Journal, Vol. 4, July 1966. pp. 1257-1266. Dowell. E. H.. "Nonlinear Oscillations of a Fluttering Plate." AIAA Journal, Vol. 4, July 1966. pp. 1267-1275. Nelson, H. C.. and cunoingham, H. J., 'Theoretical Investigation of Flutter of Two-dimensional Flat Panels with One Surface Exposed to Supersonic Potential Flow." NACA Langley Research Center. TR1280, 1956. Dowell. E. H.. 'Theoretical and Experimental P a n e f a Flutter Studies ih the Macb Number Range 1.0 to 5.0." AIAA Journal, Vol. 3. Dec. 1965. pp. 2292-2301. Dowell. E. H.. 'F'anel Flutter: A Review of the Aercelastic Stability of Plates and Shells," A I M Journal, Vol. 8, March 1970, pp. 385-399. Reed, W. H.. Hanson, P. W.. and Alford. W. J., "Assessment of Flutter Model Testing Relating to the National Aero-Space Plane." NASP Contractor Report 1002. July 1987. Isbii, T. and Yanagizawa, M., "Experimental and Theoretical Studies of Transonic Panel Flutter." National Aerospace Laboratory Report TR-74. Tokyo. 1964, (NASA 1T F-11,314). Dowell. E. H.. "Nonlinear Oscillations of a Fluttering Plate. II,"AIAA Journal, Vol. 5. Ckt. 1%7. pp. 18561862. lo Shiau, L. C., "Flutter of Composite Laminated Plates with Delamination," AIAA Paper 92-2131. April 1992. l 1 Gray. C. E. and Mei. C.. "Large-Amplitude Finite Element Flutter Analysis of Composite Panels in Hypersonic Flow." AIAA Paper 92-2130. April 1992. Abbas, J. F., Ibrahim, R. A.. and Gibson, R. F., "Nonlinear Flutter of Orthotropic Composite Panel Under Aerodynamic Heating," AIAA Paper 92-2132. April 1992.

The finite element method was used to obtain unsteady flow solutions for a panel interacting with a high speed flow on one surface. Results of the study show several interesting phenomenon. The main conclusions which can be. drawn from this study are as follows. 1. At transonic Mach numbers below the speed of sound, aeroelastic instability is typically a divergence to a positive or negative equilibrium position. Shock formation is possible when the panel diverges in the positive direction only. Flutter was found below Mach 1, but only for very 2. thin panels operating at high subsonic Mach numbers. The flutter mode is highly nonlinear and contains traveling waves, dominated by shmk formation and oscillation over a large region of the panel. At the speed of sound, both divergence and flutter 3. were observed. The divergence phenomenon occurred at low dynamic pressures and was similar to subsonic divergence. with the exception that shocks were found in the negative as well as in the positive equilibrium positions. Divergence in the positive direction appeared to be less stable. with weak shocks at the leading and trailing edges of the panel. During negative divergence a normal shock attached to the panel surface. 4. Panel flutter at the speed of sound occurs at dynamic pressures above those for divergence and is characterized by shock formation and motion on the panel surface. The panel response consists of a traveling wave moving in the direction of the flow and is highly nonlinear in appearance. Above the speed of sound and up to about Mach 5. 1.3. the panel response has the same n d i e a r appearance as at Mach one. 6. Near Mach 1.4 the panel reaches a limit cycle amplitude after a period of vibration that involves mode shape change to the fourth or fifth vibration mode.

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Xue. D. Y. and Mei C.. "Finite Element Nonlinear Panel Flutter with Arbitrary Temperatures in Supersonic Flow." AJAA Paper 92-2129. April 1992. l4 Resende. H. B.. 'Temperalure and Initial Curvature Effects in Low-Density Panel Flutter." AIAA Paper 922128. April 1992. l5 Batina. J. T.. "Unsteady Transonic Small-Disturbance Theory Including Entropy and Vorticity Effects," Jourml of Aircraft, Vol. 26, June 1989, pp. 531-538. l6 Bendiksen. 0. 0.. "A New Approach to Computational Aercelasticity," AIAA Paper 91-0939. April 1991. Mei. C.. "A Finite-Element Approach for Nonlinear Panel Flutter," AIAA Journal, Vol. 15, Aug. 1977, pp. 1107-1110. Davis. G. A,, and Bendiksen, 0. 0.. "Unsteady Transonic Euler Solutions Using Finite Elements." AIAA Paper 92-2504. April 1992. 19 Jameson, A,. and Baker. T. J.. "Solution of the Euler Equations for Complex Conl?gurations." AIAA Paper 83-1929, Proc. AIAA 6th Computational Fluid Dynamics Conference, Danvers. MA. 1983. pp. 293-302. 2o Jameson, A,, Schmidt. W., and Turkel. E., "Numerical Solutions of the Euler Equations by Finite Volume Methods using Runge-Kutta Time Stepping Schemes." AIAA Paper 81-1259, June 1981. ?' Jameson, A,. and Mavriplis. I). J., '%iOite Volume Solution of the Two-Dimensional Euler Equations on a Regular Triangular Mesh." AIAA Paper 85-0435. Jan. 1985. 22 Mavriplis. D. J., "Accurate Multigid Solution of the Euler Equations on Unsbuctured and Adaptive Meshes." AIAA Journal. Vol. 28. Feb. 1990. pp. 213-221. 23 Ashley. H. and Zartarian. G., "piston Theory - A New Aerodynamic Tool for the Aeroelastician." Journal of the Aeronautical Sciences, Vol. 23, Dec. 1956. pp. 1109-1118. 24 Fung. Y. C.. "On Two-Dimensional Panel Flutter." Journal of the Aeronautical Sciences, Vol. 25. March 1958. pp. 145-160. 25 Dowell, E. H., "Flutter of a Buckled Plate as an Example of Chaotic Motion of a Deterministic Autonomous System," Journal of Sound and Vibration, Vol. 85. March 1982. pp. 333-344. l3

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