Free vibrations of solid and hollow hemi-ellipsoids of revolution from a three-dimensional theory

Abstract

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of solid and hollow hemi-ellipsoids of revolution (the hollow ones being shells) with and without an axially circular cylindrical hole. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. The ends of the shell may be free or may be subjected to any degree of constraint. Displacement components u(r), u(theta), and u(z) in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in theta, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the hemi-ellipsoidal shells of revolution are formulated, and the Ritz method is used to solve the eigen-value 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 shallow and deep hemi-ellipsoidal shells of revolution. Numerical results are presented for a variety of hollow hemi-ellipsoidal shells with and without an axially circular cylindrical hole, which are either shallow or deep. Frequencies for five solid hemi-ellipsoids of different depth are also given. Comparisons are made between the frequencies from the present 3-D Ritz method and a 2-D thin shell theory. (C) 2004 Elsevier Ltd. All rights reserved.