Suppression of nonlinear panel flutter at supersonic speeds and elevated temperatures

Abstract

Coupled structure-electrical nonlinear panel flutter equations of motion are derived using the finite element method for composite panels with embedded piezoelectric layers subjected to thermal loads. The von Karman large-deflection strain-displacement relations, quasisteady first-order piston theory aerodynamics, quasisteady thermal stress theory, and linear piezoelectricity theory are used. Following a modal transformation and reduction, a set of coupled nonlinear modal equations of motion is obtained. By using the linear optimal control design for the linearized modal equations, an optimal shape and location of piezoelectric actuators can be determined. Numerical simulations based on the nonlinear modal equations show that the maximum flutter-free dynamic pressure with the linear optimal control can be increased as high as six times of the critical dynamic pressure. The panel's largeamplitude limit-cycle motions, as well as periodic and chaotic motions at moderate temperatures, are shown to be completely suppressed within the maximum flutter-free dynamic pressure region. Flutter suppression on composite panels with different aspect ratios, boundary conditions, and thermal effects are also studied. The results reveal that piezoelectric actuators are effective in nonlinear panel flutter suppression. Pa Q,R q U, V, W [ X Nomenclature extension, coupling, and bending panel stiffness matrices system and element aerodynamic influence matrices panel length and width modal aerodynamic damping matrix air-panel mass ratio over Mach number electric displacement and field electromechanical coefficients modal force vector system and element aerodynamic damping matrices nondimensional aerodynamic damping thickness of piezoelectric layer and panel modal stiffness matrix system and element linear stiffness matrices first-order nonlinear stiffness matrices second-order nonlinear stiffness matrices feedback control gain matrix modal mass matrix system and element mass matrices Mach number system and element load vectors aerodynamic pressure penalty matrices modal amplitude vector dynamic pressure time displacement system and element nodal displacement vectors state space vector jc ¥ = width of piezoelectric material placed at the leading edge [a] = thermal expansion coefficients AT = temperature change £, Y = strain components [©]/» [®L/ = linear and nonlinear electromechanical coupling matrices [K] = curvature vector A = nondimensional dynamic pressure /x = air-panel mass ratio p = density a,r = stress components X I / , <p = modal matrix and normal mode coo = reference frequency Subscripts a = air b = bending cr = critical m = membrane A7 = thermal 9 = nonlinear 0 = electrical

Nonlinear finite element modeling of vibration control of plane rod-type structural members with integrated piezoelectric patches

“…Batra et al [69,70] dealt with shape and vibration control of plates at finite deformations taking into account also nonlinear constitutive equations for piezoelectric patches. Nonlinear flutter suppression has been considered by Lai et al [71], Zhou et al [72,73] based on classical von Kármán large deflection plate theory, and by Shen and Sharpe [74] using discrete Kirchhoff theory finite elements. FE analysis for piezoelectric laminates under strong electric field was performed by Zhang et al [75], Rao et al [76].…”

Application of aeroelastic modes on nonlinear supersonic panel flutter at elevated temperatures

“…For isotropic or orthotropic rectangular plates under an arbitrary non-zero yawed supersonic flow, 36 or 6 · 6 NMs are needed [4]; for laminated anisotropic rectangular plates even at zero yaw angle, 36 or fewer NMs are needed [4]. The large number of NMs or coupled nonlinear modal equations is not only computationally costly for flutter analysis, but also causes certain complexity and difficulty in controller design for flutter suppression [5,6].…”

Influence of the elastic parameters of a fluttering plate on its post-critical behavior

“…By these methods it was possible to reduce the mathematical problem to a system of nonlinear ordinary differential equations in time, which was solved by numerical integration. Other researchers utilized the finite element method (FEM) [Kikuchi 1986;Reddy et al 1988] to integrate over the plate or shell surface and to derive a system of ordinary differential equations in-time [Xue and Mei 1993;Dixon and Mei 1993;Zhou et al 1994;Zhou et al 1995;Zhou et al 1996]. The Galerkin method was also used by [Abbas et al 1993] to examine the problem of nonlinear aerothermoelasticity of panels in supersonic airflow.…”

“…Accurate dynamic analysis with actuators present was performed by the use of FEM. Furthermore, the effects of thermal loads were taken into account by some of the above mentioned authors [Xue and Mei 1993;Zhou et al 1994;Zhou et al 1995;Zhou et al 1996;Abbas et al 1993].…”

“…A large amount of research was also conducted in the field of piezoelectric actuators to suppress large amplitude limit-cycle flutter [Zhou et al 1995;Zhou et al 1996]. Accurate dynamic analysis with actuators present was performed by the use of FEM.…”