A time-mode approach to nonlinear vibrations of orthotropic thin shallow spherical shells

Abstract

A new analytical time-mode-assumption-based approach to nonlinear (large amplitude) axisymmetrical free vibrations of orthotropic thin shallow spherical shells is presented in this paper. The concept of the drift in theory of nonlinear dynamics is first adapted to modelling the asymmetry of vibration amplitudes of the shells. The nonlinearly-coupled algebraic-differential eigenvalue equations determining vibration frequencies and the drift are formulated with the variational method. Asymptotic solutions are obtained with a successive iteration method suggested by the author. The evolution of nonlinearities within a wider range of vibration amplitudes of the shell vibration with various parameters and boundary conditions is extensively investigated. It is generally concluded that very shallow spherical shells behave like flat circular plates when subject to free vibrations, while the vibration response for the moderately shallow spherical shells is complicated by the occurrence of both softening and hardening nonlinearities in different ranges of vibration amplitudes, and shallow spherical shells always tend to behave more like flat circular plates when subject to much larger amplitudes.