Flow induced nonlinear vibrations of rectangular plates

This study concerns the flow-induced large-amplitude vibrations of two- and three-dimensional rectangular plates which are clamped on all edges. It is focused on understanding the cause-and-effect relationships for limit cycle oscillations of aircraft panels. The partial differential equations governing the motion of the panel are obtained from von Kármán non-linear plate theory. For a given plate, the normal modes of vibration of the corresponding linear problem (describing small deflections) are constructed. These normal modes form a complete set of eigenfunctions and are used to develop a spectral method of solution for general non-linear vibrations of the panel. The problems are thereby reduced to solving a set of time-dependent coupled non-linear ordinary differential equations.; First, free vibrations of two- and three-dimensional plates were considered. The effect of various boundary conditions and different problem formulations was assessed. It is concluded that the conventional formulation using the Airy stress function is not suitable to represent the conditions of zero displacements on all edges. An alternative formulation is developed in terms of equations for all three displacements.; The response of two- and three-dimensional plates to the motion of a single recti-linear vortex in incompressible flow is studied for various physical parameters. It is shown that the passage of a vortex over the plate can induce complex vibrations and large deformations of the plate.; The motion induced by a convected array of vortices or a periodic pressure wave is also addressed. It is shown that near the linear resonance frequency of the plate, the non-linear response of the plate is a complex large amplitude motion consisting of a high frequency vibration that is modulated.; The problem of supersonic flow over a plate is also considered for plates which are simply supported or clamped on all edges. Piston theory is a leading-order relation obtained from linearized inviscid supersonic flow theory (for large Mach numbers) and it is used to model the interaction between the plate and the flow. It is shown that complex vibrations of a clamped panel which in some cases are chaotic, can occur.