Vibrations of Complete Hollow Spheres with Variable Thickness

Abstract

A three-dimensional (3D) method of analysis is presented for determining the free-vibration frequencies of complete hollow spherical shells of revolution with variable thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2D), the present method is based on the 3D dynamic equations of elasticity. Displacement components u(r), u(theta), and u(z) in the radial, circumferential, and axial directions, respectively, are taken to be periodic in. and in time, and algebraic polynomials in the r- and z-directions. Potential (strain) and kinetic energies of the complete hollow spheres 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 of the complete hollow spheres. Comparisons are also made between the frequencies from the present 3D method, a 2D thin-shell theory, and two other 3D analyses. (C) 2015 American Society of Civil Engineers.