Theoretical Considerations of Some Nonlinear,Aspects of Hypersonic Panel Flutter. S C McIntosh,J I Lerner,Fourth Annual Report,NASA Grant NGR 05 020 102. 1 September 1968 to 31 August 1969,TABLE OF CONTENTS. I INTRODUCTION I, II EFFECT OF AERODYNAMIC NONLINEARITIES ON PANEL STABILITY 2. 2 1 Determination of an Amplitude Sensitive,Stability Boundary. 2 2 Comparison with Experiment, III EFFECT OF AN UNSTEADY BOUNDARY LAYER ON PANEL FLUTTER 9. 3 1 Some Relevant Parameters 9,3 2 Two Dimensional Unsteady Viscous Flow Past an. Oscillating Panel Formulation of the Problem l 11, 3 3 Illustration of the Method Incompressible Steady. Flow Past a Semi Infinite Wavy Wall 20,3 4 Critique of Present Efforts and Discussion. of Alternative Formulations 27,IV CONCLUDING REMARKS 32. REFERENCES 33,FIGURES 35,NOMENCLATURE,a Panel chord or wavy wall wavelength. ak Modal amplitude for transverse displacement,k Dimensionles modal amplitude i h. A Wavy wall amplitude,bo bR Modal amplitudes for in plane displacement. c Sound speed, cI Constant in momentum integral equation for wavy wall. fl a 1R See Eq 3 49,Cp Pressure coefficient 2 p p pU2. D Plate modulus Eh3 12 1 V E here is modulus,of elasticity. ETotal energy See Eq 2 5,E Dimensionless total energy a E Dh. h Panel thickness,h Enthalpy,h0 Total enthalpy h u2 v 2 2. k Reduced frequency wa U,k0 Wave number 27T a, K Running spring constant panel in plane restraint spring. M Free stream Mach number, N Number of assumed modes for panel transverse displaceent. p Pressure, Ap Static pressure difference across panel positive if. cavity pressure exceeds free stream static pressure. Ap Dimensionless static pressure difference Ap a Dh. Pr Prandtl number,q Free stream dynamic pressure P U2 2. q Heat transfer rate, R Reynolds number based on inverse wave number u Vk0. Applied in plane load,R Dimensionless applied in plane load Ra 2 D. s Sheltering coefficient,j Panel axial displacement. u Local velocity component in x direction,U Free stream speed. v Local velocity component in y direction,V Viscous effect on pressure see Eq 3 53. w Panel middle surface transverse displacement,w Dimensionless panel transverse displacement w h. x Axial coordinate,x0 Lenght of flat plate starting section. y Transverse coordinate Sec III,z Transverse coordinate Sec II. Z Dimensionless transverse coordinate y ys A, aConstant in momentum integral equation for wavy wall. 2 a2 ia see Eq 3 49,In plane restraint parameter KI K Eh a l 2. al Momentum thickness parameter A see Eq 3 46, a2 Displacement thickness parameter 6 A see Eq 3 47. Dimensionless surface velocity gradient see Eq 3 48. Y Gas constant for free stream y 1 4,6 Boundary layer edge. 61 Density defect thickness see Eq 3 25,6 a Mass displacement thickness see Eq 3 22. 6 Enthalpy displacement thickness see Eq 3 26,A AI Thickness of velocity boundary layer. A2 Thickness of thermal boundary layer, e Dimensionless wavy wall amplitude parameter k 0A. 7 Displacement surface i 61,0 91 Momentum defect thickness see Eq 3 23. 02 Enthalpy defect thickness see Eq 3 24, x Dimensionless dynamic pressure parameter 2q a3 ID. Dimensionless mass ratio p a pmh,1 Viscosity,V Poisson s ratio. V Kinematic viscosity A p,p Mass density,Panel mass. T Dimensionless time t D Pmha,TShearing stress,Velocity potential. P Perturbation velocity potential,W Frequency, Derivative of dimensionless quantity with respect to T. Subscripts,e Boundary layer edge also inviscid values on the. displacement surface, s Surface values on oscillating panel or wavy wall. O Free stream values,I INTRODUCTION, This report presents a summary of the fourth year s research activity. under NASA Grant NGR 05 020 102 monitored technically by the Nonsteady. Phenomena Branch of Ames Research Center The research program has been. divided into two more or less independent areas, 1 A study of the effects of aerodynamic nonlinearities on panel. flutter at hypersonic speeds This study was begun in an attempt to ex. plain certain nonlinear panel behavior observed experimentally it has. led to more general considerations of the impact of nonlinear aerodynamic. loading on panel stability and postcritical response. 2 An attempt to determine theoretically the effects of a turbu. lent boundary layer on the aerodynamic loading of an oscillating panel. It has already been demonstrated experimentally that the critical flutter. dynamic pressure does depend on the thickness of the boundary layer over. the panel at least for the lower supersonic free stream Mach numbers. The present study is aimed at improving existing theories to the extent. of including all important effects while avoiding unnecessary complica. tion in representing the boundary layer, In Section II it is shown how nonlinear aerodynamic loading can cause. an amplitude sensitive instability where the panel is unstable in a para. meter region that is a stable one on the basis of linear theory if the. initial excitation is severe enough Also discussed is an attempt to. reproduce qualitatively the nonlinear experimental behavior alluded to. above Section III describes in some detail the fundamental parameters. that govern the unsteady boundary layer problem and presents a description. of an integral formulation with a simple example Section IV concludes. this report with an outline of plans for continuing the research. II EFFECT OF AERODYNAMIC NONLINEARITIES,ON PANEL STABILITY. 2 1 Determination of an Amplitude Sensitive Stability Boundary. References I and 2 present the derivation of the equations of motion. for the panel on hinged supports of Fig 1 along with an assessment of. the importance of various nonlinear aerodynamic terms for parameter ranges. of practical interest The panel middle surface displacement i x t is. approximated by a series of assumed modes in space that satisfy the geo. metric boundary conditions and the Rayleigh Ritz technique is used to. derive equations of motion in time for the modal amplitudes These. equations are then integrated by computer to produce the panel motion. versus time With variables and parameters as defined in the Nomencla. ture these equations are,1222 k R 2k2 a n22,a k ak 3 42. a kn l l l X 2 Ap,2 k 2 n2 n 2 M k kir,yl k h S mn k2 m2n l l kmfl aa. 4 am l k rn mn 2 fk2 mi n 2 mn,V l Mh X ma A na mamn 0. m n l m n71,m n k m n k,k 1 2 N 2 1, In these equations are two aerodynamic nonlinear terms which come from. the piston theory aerodynamic terms proportional to U x 2 and. 6 6x 6 at These two terms are the ones found to be significant. for the parameter ranges of interest, In Ref 2 it was also demonstrated that energy levels capable of. causing amplitude sensitive instability could be generated by unstable. panel motion near the linear stability boundary In view of these results. it was decided to condider how the linear stability boundary presented. say in the X RS plane for other parameters fixed would be changed. for a given level of initial excitation, The stability of nonlinear nonconservative systems has been the. subject of several papers in the past few years a recent example is the. one by Dimantha and Roorda Ref 3 Their basic method of analysis is. the direct method of Liapunov with Zubov s procedure for constructing. the Liapunov functional It is not clear whether or not this method can. be successfully applied to determine the panel stability boundary discussed. above however the ideas of Ref 3 do at least suggest a meaningful. approach to the problem In principle one can determine a stability. boundary for the panel in terms of initial values of the modal amplitudes. and their time derivatives for fixed values of the system parameters. This boundary can be viewed as a hypersurface S in the phase space made. up of the modal amplitudes and their time derivatives Clearly the ori. gin of this phase space represents the panel in its flat undisturbed. equilibrium position Any combination of initial values that plots on. one side of S will result in unstable panel motion while a combination. that plots on the other side of S will result in stable panel motion. Dimantha and Roorda then propose to calculate the minimum total panel. energy on S this minimum value E will determine a hypersphere that. just touches S at one or more points and everywhere else is on the. stable side of S Hence E1 provides an upper bound for the initial. energy of the disturbance such that the resultant panel motion is stable. One can then calculate the maximum total panel energy on S this value. E2 determines another hypersphere that touches S from its unstable. side and it provides a lower bound on the initial disturbance energy in. the panel for unstable motion The reader is referred to Ref 3 for the. details Suffice it to say here that these ideas suggest determining a. nonlinear stability boundary in the X Rx plane for constant initial. energy and comparing this boundary with the linear stability boundary. which is independent of the initial conditions, The total energy of the panel system is obtained from Eqs 2 4 and. 2 5 of Ref 1,a 2 ra Eh u22,E Pmh f 3 dx J l x,D x b Kb0 1. Into this expression are substituted the assumed mode series for i and u. i x t F t sin s,u x t bR b0 t E bk t a, Then bR bo and bk are eliminated in favor of ak and other system. parameters by making use of the panel equilibrium equations in the absence. of aerodynamic loading see Ref 1,2 l Zk a 2 4,1 Iri Z ra. M n117 m n7 l n,m n7k m n k, After rearranging and cancelling some terms the equation for the energy. 3 1 N 2 3 4N 2j 2 2,Dh k l k 1 1,j R k 2 4 E Zk4 2 2 5. It was decided at this point to choose an energy level corresponding. to supercritical panel motion near the linear stability boundary in the. X R K plane with the other system parameters fixed and to use this energy. level to determine initial conditions There are of course many differ. ent combinations of the ak and k that will give the same E it was. further decided to set a 0 a2 0 and to let these be the only. nonzero components of the initial state vector This particular choice. was a purely arbitrary one and some consideration will be given later to. determining more realistic forms of the initial conditions Figure 2. presents the variation of a1 0 a2 0 with Rx for E 1750 and. The other system parameters were then chosen and a new stability. boundary was determined as is shown in Fig 3 The values of the system. parameters are given in the caption This boundary was obtained by inte. grating Eqs 2 1 with initial conditions determined as discussed above. and observing whether or not the calculated panel motion persisted or died. out past the initial transient Note that there are some uncertainties. in the stability boundary near RS,0 where it appears that the bound. ary has two branches Clearly the boundary should somehow be a closed. one at this time there are insufficient data to show just how the closure. takes place The panel flutter motion versus time is shown in Fig 4. for X 295 and RS 0 Values of the other parameters are given in. the caption The linear stability boundary gives the familiar value of. 343 for X so the critical value of X is decreased by 14 The flutter. mode shapes corresponding to the panel of Fig 4 are given in Figs 5 12. The mode shapes and time history are very similar to those discussed in. Ref 2 and are representative of the flutter motion all along the nonlin. ear stability boundary of Fig 3 A strong traveling wave component is. evident in the flutter mode even for positive tensile values of Rx. The remarks in Ref 3 concerning the accuracy of the numerical compu. tation and the validity of the assumptions underlying the theoretical. development of the equations of motion are also applicable here The. accuracy of the numerical integration was checked and it was verified. that the unstable panel motion did not violate the moderate rotation. assumption,2 2 Comparison with Experiment, Reference 4 describes some high Mach number panel flutter experiments. where in plane tension was used to stabilize the panels during tunnel. starting The tension was reduced until flutter was observed and then. increased to stabilize the panel before it was damaged It was found. that in certain cases the final value of the tension had to be much great. er than the initial value in order to stabilize the panel. The aerodynamic nonlinear effects on stability discussed in Sec 2 1. above suggest a possible explanation for this behavior It was decided. to attempt a qualitative comparison by looking for an amplitude sensitive. instability with Eqs 2 1 and parameters corresponding to experimental. conditions from Ref 3 The following parameters were chosen Rx 160. X 2000 M 10 A 0 1 Ap 0 and h a 0 00054 These values of. RS X M and Ap give a point just on the stable side of the. linear stability boundary for a panel on hinged supports The experi. mental edge conditions would be much better represented by clamped supports. but in fact the theoretical differences between stability boundaries for. these two edge conditions are not great at the relatively large value of. 160 for R see Ref 4 The remaining unknown parameter is a Vari. ous initial amplitudes were used appropriate to flutter amplitudes. observed experimentally w 10 in some cases and a was varied to. see if these initial amplitudes would produce an unstable panel motion. The only unstable motion that could be produced was an oscillatory but. divergent one for values of a near 10 3,These results indicate that. nonlinear aerodynamic influences are not the cause of the experimentally. observed behavior since the value of a needed and the unstable motion. calculated are not consistent with the experimental setup or observations.

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