Three-dimensional vibrations of thick spherical shell segments with variable thickness

Abstract

A three-dimensional method of analysis is presented for determining the free vibration frequencies and mode shapes of spherical shell segments with variable thickness. Displacement components u φ , u z , and u θ in the meridional, normal, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the φ and z directions. Potential (strain) and kinetic energies of the spherical shell segment are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Novel numerical results are presented for thick spherical shell segments with constant or linearly varying thickness and completely free boundaries. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the spherical shell segments. The method is applicable to thin spherical shell segments, as well as thick and very thick ones.