Normal Axisymmetrical Modes of Oscillation of a Circular Plate with a Non-Linear Foundation

Abstract

Several methods have been proposed for constructing the nonlinear normal modes of continuous systems. All of them need the linear eigenfunctions of the problem as a first step. In this paper the case when these eigenfunctions and eigenvalues are not at hand but their approximations, is studied. The method is applied to the cak:uIalion of lbe non-linear axisymmetric frequency coefficients and amplitude-dependent mode shapes of a simply supported circular plate. The plate is partially founded on a cubic non-linear foundation. The approach here presented combines the optimized Rayleigh-Ritz method with the invariant manifold techniques. The approach is applicable to a wide variety of systems. In such cases where the exact solution of the linear system is known the first step may be overridden and the recently developed Shaw and Pierre method may be aplied.