Vibrations of Complete Hollow Spheres with Variable Thickness: Fid-Bau Portal

Vibrations of Complete Hollow Spheres with Variable Thickness

Kang, Jae-Hoon

A three-dimensional (3D) method of analysis is presented for determining the free-vibration frequencies of complete hollow spherical shells of revolution with variable thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2D), the present method is based on the 3D dynamic equations of elasticity. Displacement components $u_{r}$ , $u_{θ}$ , and $u_{z}$ 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 complete hollow spheres 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 four-digit exactitude is demonstrated for the first five frequencies of the complete hollow spheres. Comparisons are also made between the frequencies from the present 3D method, a 2D thin-shell theory, and two other 3D analyses.