Vibration analysis of solid ellipsoids and hollow ellipsoidal shells of revolution with variable thickness from a three-dimensional theory

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of complete ellipsoidal shells of revolution with variable thickness and solid ellipsoids. Unlike conventional shell theories, which are mathematically two-dimensional (2D), the present method is based upon the 3D dynamic equations of elasticity. Displacement components ur, uθ, and uz in the radial, circumferential, and axial directions, respectively, are taken to be periodic in θ and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the ellipsoidal shells of revolution and solid ellipsoids 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 exactness is demonstrated for the first five frequencies of the ellipsoidal shells of revolution. Numerical results are presented for a variety of ellipsoidal shells with variable thickness. Frequencies for five solid ellipsoids of different axis ratios are also given. Spherical shells and solid spheres are special cases which are included. Comparisons are also made between the frequencies from the present 3D Ritz method, a 2D Ritz method, and a 3D finite element method. The multiple degeneracies (two or more modes having the same frequency) of spherical bodies are examined by analyzing some almost-spherical ellipsoids.