Three-Dimensional Vibration Analysis of a Hermetic Capsule with Variable Thickness

Abstract

The natural frequencies of a hermetic capsule of a circular cylinder closed with hemispheroidal caps at both ends and having variable thickness are determined by the Ritz method using a three-dimensional analysis. However, in the traditional shell analysis, mathematically two-dimensional thin-shell theories or higher-order thick-shell theories, which make very limiting assumptions about the displacement variation through the shell thickness, have usually applied. Although most researchers have used three-dimensional shell coordinates that are normal and tangent to the shell midsurface, the present analysis is based upon the circular cylindrical coordinates. Using the Ritz method, the Legendre polynomials, which are mathematically orthonormal, are used as admissible functions instead of ordinary simple algebraic polynomials. The potential and kinetic energies of the hermetic capsule are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the Legendre polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. The frequencies from the present three-dimensional method are compared with those from other three-dimensional approaches and two-dimensional shell theories by previous researchers. The present three-dimensional analysis is applicable to very thick shells as well as thin shells.