Three-dimensional vibration analysis of solid and hollow hemispheres having varying thicknesses with and without axial conical holes

Abstract

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of solid and hollow hemispherical shells of revolution of arbitrary wall thickness having arbitrary constraints on their boundaries. Unlike conventional shell theories, which are mathematically two-dimensional, the present method is based upon the 3D dynamic equations of elasticity Displacement components u(phi), u(z), and u(theta) in the meridional, normal, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in theta, and algebraic polynomials in the phi-direction and z-direction. Potential (strain) and kinetic energies of the hemispherical shells are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies 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 solid and hollow hemispheres with linear thickness variation. The effect on frequencies of a small axial conical hole is also discussed. Comparisons are made for the frequencies of completely free, thick hemispherical shells with uniform thickness from the present 3D Ritz solutions and other 3D finite element ones.