Free vibration analysis of shallow spherical dome by three-dimensional Ritz method

Abstract

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies of a shallow spherical dome. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components $u_{\phi }$ , $u_{\theta }$ , and u_z in the meridional, circumferential, and normal directions, respectively, are taken to be periodic in θ and in time, and the algebraic polynomials in the φ and z directions. Potential and kinetic energies of the shallow spherical domes 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. Natural frequencies are presented for different boundary conditions. The frequencies from the present 3-D method are compared with those from a 2-D exact method, a 2-D thick shell theory, and a 3-D finite element method.