Vibrations of hemi-spherical shells of revolution with eccentricity from a three-dimensional theory

Abstract

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies of hemi-spherical shells of revolution with eccentricity having uniform thickness. Unlike conventional shell theories, which are mathematically two-dimensional, the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u(r), u(0), and u(z) in the radial, circumferential, and axial directions, respectively, are taken to be periodic in theta and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the shells of revolution 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 three or four-digit exactitude is demonstrated for the first five frequencies of the shells of revolution. Numerical results are presented for a variety of hemi-spherical shells of revolution with eccentricity.