Large Amplitude Vibrations of Circular Plate on a Uniform Elastic Foundation

Abstract

Herrmann’s equations, the dynamic analogues of the von Karman equations, are solved for a circular plate on a linear elastic foundation by assuming a series solution of the separable form involving unknown time functions. The spatial functions include both regular and modified Bessel functions and are chosen to satisfy the linear mode shape distributions of the plate as well as the usual edge conditions. Total differential equations governing the symmetric plate motions are derived using the Galerkin averaging techniques for a spatially uniform load. By extending the concept of normal modes to nonlinear plate vibrations, comparisons between normal mode response and single mode response, as functions of the first mode amplitude, are shown for different values of the elastic foundation parameter. Results are obtained for plates with simply supported and clamped edges and with both radially moveable and immoveable edges. These results are used to discuss the limitations of single-mode response of circular plates, both with and without an elastic foundation.