A symplectic analytical wave based method for the wave propagation and steady state forced vibration of rectangular thin plates

Abstract

A semi-analytical method is used to investigate the wave propagation characteristics and steady state forced vibration response for rectangular thin plates. By way of a rigorous but simple derivation, the governing differential equations for transverse vibration of rectangular thin plates are first converted into Hamiltonian canonical equations. Following the method of separation of variables, a symplectic eigenproblem is formed whose solution gives analytically the dispersion equation and the wave mode shape. Using the wave modes, i.e. the wave propagation parameters and wave shapes, and combining the directly excited waves, the wave propagation within the plate and the wave reflection at the boundary, the forced response of the plate can be computed in the wave domain with high precision and high efficiency. The present method is based on the basic elasticity equations of the plate, and can give the symplectic analytical solutions for the wave modes for any combination of simple boundary conditions along the plate edges. The present method eliminates the limitation of the traditional analytical wave propagation method which can only obtain wave modes for plates with two opposite edges simply supported. In contrast to numerical wave propagation methods, the present method provides symplectic analytical solutions for the wave modes, and hence the computations are of high precision and well conditioned. Also, continuously distributed external forces can be easily considered. In the numerical examples, the wave propagation characteristics are analyzed for plates with three different combinations of boundary conditions, i.e. with two opposite edges either both simply supported, or both clamped, or one simply supported and the other clamped. The steady state forced responses are also computed for plates excited by point forces, as well as for line and area distributed forces, for the three combinations of boundary conditions. Comparison of the present results with analytical results, wave finite element and finite element results validates the high accuracy and high efficiency of the present method.