Natural frequencies of an inhomogeneous square thin plate (Q2047529)

The purpose of this paper is to analyze the inhomogeneity effect on the natural transverse vibrations of square thin plates using asymptotic methods, when the thickness and/or stiffness of plate can be assumed almost constant. The results of calculations by asymptotic formulas, which are determined by the perturbation method, are compared with the numerical results obtained by finite element method using the COMSOL software package. The objective of the study is to obtain asymptotic formulas describing the effect of inhomogeneity of a thin plate thickness or stiffness parameters on its natural frequencies. In the study, it is assumed that the geometric and physical parameters of the plate are smooth functions of coordinates, constant in one direction and close to constant in the other. The asymptotic formulas make it possible to determine, which of the two multiple frequencies corresponding to wave numbers \(n\) and \(m\) varies faster with the variations of the small parameter. Only the homogeneous boundary conditions at the plate edges (\(\xi^* = 0\) or \(\xi^*=1\)) of the simply supported (S) or clamped (C) type are considered in this paper.