The non-linear vibration and dynamic instability of thin shallow shells

In this paper, the non-linear vibration and dynamic instability of thin shallow spherical and conical shells subjected to periodic transverse and in-plane loads are investigated. The Marguerre type dynamic equations used for the analysis of shallow shells, when treated by the Galerkin method, will result in a system of total differential equations in the time functions, known as Duffing and Mathieu equations, from which the various kinds of non-linear vibration and dynamic instability are determined by using numerical methods. Numerical results are presented for axisymmetric vibrations and dynamic instabilities of shallow spherical and conical shells with (a) clamped and (b) supported edge conditions. As numerical examples, non-linear vibration frequencies and instability regions for shells are determined. The effects of static load as well as static snap-through buckling on the instability are also investigated. (C) 1997 Academic Press Limited.