Vibrations of deep and shallow hemi-spheroidal domes with non-uniform thickness having a top cut-out

Abstract

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies of hemi-spheroidal domes with non-uniform thickness having a top cut-out. 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 edges of the shell may be free or may be subjected to any degree of constraint. Displacement components ur, uθ, and uz in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic simple polynomials in the r and z directions. Potential (strain) and kinetic energies of the hemi-spheroidal domes with non-uniform thickness having a top cut-out 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 algebraic polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Numerical results are presented for shallow and deep hemi-spheroidal domes with non-uniform thickness having a top cut-out of five values of aspect ratios, which are free at the inner edge and free or fixed at the bottom of the domes. The frequencies from the present 3-D Ritz method are compared with those from other 3-D analysis and a 2-D thin shell theory by previous researcher.