On Vibrations of Shallow Spherical Shells

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Abstract
The problem of the axi-symmetrical vibrations of a shallow spherical shell is reduced to two simultaneous differential equations for the tangential and normal components of displacement. The solution of these equations is obtained in terms of Bessel functions. With this solution the third-order determinant for the frequencies of a shell segment with clamped edge is given but not evaluated. Instead, an approximate value for the lowest frequency is calculated by means of the Rayleigh-Ritz procedure [Eq. (40)]. It is found that very little curvature is needed to modify appreciably the corresponding flat-plate frequency. The approximations which are made are those customary for shallow shells: (1) omission of the transverse shear term in the tangential force equilibrium equation, and (2) relations between couples and changes of curvatures as in the theory of flat plates.

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