Vibration Analysis of Shallow Spherical Domes with Non-Uniform Thickness

Abstract

A three-dimensional (3D) method of analysis is presented for determining the natural frequencies of shallow spherical domes with non-uniform thickness. Unlike conventional shell theories, which are mathematically two dimensional (2D), the present method is based upon the 3D dynamic equations of elasticity. Displacement components [Formula: see text], [Formula: see text], and [Formula: see text] in the meridional, circumferential, and normal directions, respectively, are taken to be periodic in [Formula: see text] and in time, and algebraic polynomials in the [Formula: see text] and z directions. Potential (strain) and kinetic energies of the shallow spherical domes with non-uniform thickness are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Natural frequencies are presented for different boundary conditions. The frequencies from the present 3D method are compared with those from a 2D exact method, a 2D thick shell theory, and a 3D finite element method by previous researchers.