A numerical method for analysis of free vibration of spherical shells.

Abstract

The Holzer method, which was developed for the determination of the torsional frequencies of a vibrating shaft, has been generalized to the analysis of free vibration of spherical shells. This method consists of repeatedly assuming a value for the natural frequency and attempting to solve the governing two-point boundary-value problem as a set of initial-value problems until all the boundary conditions can be satisfied. To avoid the loss of accuracy in the numerical integration of the governing ordinary differential equations, a method for the suppression of the extraneous solutions has been developed. In general, this method involves significantly less computational work than the usual method of breaking the steep shell into a system of short shells and satisfying the compatibility requirements at the boundaries of these shells. Numerical results are obtained for clamped spherical shells continuous at the apex. It is concluded that the Holzer method is in fact a very accurate method for the determination of frequencies and mode shapes of spherical shells; however, it is not applicable to extremely thin shells since extraneous solutions grow larger and oscillate more rapildy as the shell becomes thinner.