Analytical Solution of the Problem of Free Vibrations of a Plate Lying on a Variable Elastic Foundation

Abstract

AbstractAn analytical solution of the problem of free bending vibrations of rectangular plates with Levy boundary conditions lying on a continuous variable elastic foundation, which is described by the Winkler model, is given. An exact solution of the differential equation of free vibrations of plates when the bedding coefficient is an arbitrary continuous function of one variable is found. The quadratures for numerical realization of the found solutions are derived. The formulas for dynamic state parameters, which allow one to investigate free vibrations of the plates under any boundary conditions at two parallel edges, are obtained. The dependence of the frequency of free vibrations of the system under consideration on its other parameters is established in the analytical form. Computational formulas for determining the spectrum of frequencies of free vibrations of the plates are obtained. The general form of frequency equation and formulas for the main forms of vibrations corresponding to the three cases of boundary conditions are established. Frequency spectra of free vibrations of hinged plates resting on a variable elastic base for four different laws of bedding coefficient changes are determined. It is shown that in the case of constant bedding coefficient the frequencies calculated by the author’s method practically coincide with the frequencies calculated by the known exact formula.KeywordsPlateVariable foundationWinkler modelAnalytical solutionVibrationsFrequency