Up to lowest 100 frequencies of rectangular plates using Jacobi polynomials and TSNDT

Abstract

This paper finds up to lowest 100 free vibration frequencies of isotropic and linearly elastic plates with thickness to side length ratio varying from 0.001 to 0.5 by using the Ritz method, a third-order shear and normal deformable plate theory (TSNDT), and weighted Jacobi polynomials as admissible functions that are mutually orthogonal and exactly satisfy essential boundary conditions. The numerical method is stable even for 18th degree polynomials as basis functions. It is shown that this approach requires fewer degrees of freedom than those in the traditional finite element method (FEM) to find converged lowest 100 frequencies and the corresponding mode shapes. No shear correction factor is employed in the TSNDT. The presently computed results agree well with those from either analytical or numerical solutions of the corresponding 3-dimensional linearly elastic problems obtained with the commercial FE software, Abaqus. Furthermore, results for plates made of an incompressible material can be computed by setting Poisson's ratio = 0.49.