Exact spectral element model for rectangular membranes subjected to transverse vibrations

Abstract

This study proposes an exact frequency-domain spectral element model for a rectangular membrane subjected to arbitrary boundary conditions. The frequency-domain general solution of a finite rectangular membrane element is first assumed in the trigonometric Fourier series with the spatial coordinates x and y. After all the trigonometric Fourier series terms that vanish at the four boundary edges are eliminated, the general solution is expressed in the spectral (i.e., discrete Fourier transform) forms with respect to the spatial coordinates x and y. By using the projection method, the spatial-domain spectral coefficients of the general solution are related to the spatial cosine/sine series coefficients of the functions specified for the geometric and natural boundary conditions. Lastly, the spectral element matrix (i.e., dynamic stiffness matrix) is formulated from the force–displacement relationships. Owing to the spectral representation of the general solution in both temporal and spatial coordinates, the well-known fast Fourier transform algorithm can be applied for efficiently obtaining the dynamic responses and wave propagations in a membrane, in both the time and spatial domains. The high accuracy of the proposed spectral element model for the rectangular membranes is verified via comparison with existing solution techniques, such as the exact theory, modal analysis method, and finite element method.