Free vibration of a class of solids with cavities

Abstract

The Ritz method is used to obtain an eigenvalue equation for the free vibration of a class of solids. Each solid is modelled by means of a segment which is described in terms of Cartesian coordinates and is bounded by the yz, zx and xy orthogonal coordinate planes as well as by two curved surfaces which are defined by polynomial expressions in the coordinates x, y and z. Simple algebraic polynomials which satisfy the boundary conditions at the five surfaces of the segment are used as trial functions. By exploiting symmetry, the range of problems which can be treated is substantially broadened and includes a variety of problems of significant interest in structural analysis, such as thick or very thick shells with various shaped cavities. In order to demonstrate the accuracy of the approach, natural frequencies are given for a sphere with a spherical cavity (a thick spherical shell) as calculated by using the present analysis and by using an exact formulation. The versatility of the approach is then demonstrated by the treatment of several other hollow solids of differing geometry, including a thick cylinder with end plates, a cubic box, a cube with a spherical cavity and a cylinder with a conical inclusion.