Natural frequencies and bending mode shapes of thin perforated plates using a variational method

Abstract

A theoretical method to predict eigenfrequencies and mode shapes of a perforated plate was developed by building its Hamiltonian as a functional of the displacement field. The displacement functional base was chosen to be the natural base of a nonperforated plate having the same geometrical and mechanical characteristics as the perforated one with simply supported boundary conditions. Limiting the base vectors to a finite number, the extremization of the Hamiltonian was transformed into a Rayleigh-Ritz problem. Formulas of this analytical development were implemented on a computer to solve the eigenvalue problem for any kind of rectangular perforations. Numerical singularities were encountered, regarding the holes' shapes and sizes, and vector sets. They were removed by using some simple physical considerations. The program developed for this study was tested by treating special cases, well known in the literature. For example, the case of a full plate with free boundary conditions was simulated by introducing a crown of holes around the edges of the plate, setting it free of its supports. Several cases of perforated plates were also computed. This work is the first part of a more general study on sound radiation from plates with holes and perforations.