Localization of natural oscillations of thin elastic gaskets

Abstract

We study natural oscillations of thin homogeneous isotropic gaskets of constant or variable thickness whose bases are rigidly fixed. It is shown that the traditional two-dimensional model, namely the plane problem of the elasticity theory in the longitudinal section with the Dirichlet condition at the boundary, gives correct results for eigenfrequencies of the this spatial solid only for the plate of a constant thickness with clamped lateral surface. In other cases the asymptotic analysis provides another models of reduced dimension, in particular, ordinary differential equations, while modes of natural oscillations enjoy concentration near the lateral side or some points on the boundary.