The application of Hamiltonian dynamics and averaging to nonlinear shell vibration

Abstract

Internal auto-parametric instabilities in the nonlinear vibration of undamped and unloaded cylindrical shells are discussed. The focus is on the coupling between a few simple modes that can combine to break the in–out symmetry and give an energetically favourable pattern of deformation. When the ratio of the natural frequencies is close to a resonance, internal auto-parametric instability triggers energy transfer between some of the modes. A Rayleigh–Ritz discretization of the von Kármán–Donnell equations leads to the Hamiltonian and transformation into action-angle coordinates followed by averaging provides readily a geometric description of the modal interaction. It is established that the interaction should be most pronounced between concertina and chequerboard modes with no energy transfer between the chequerboard modes. A simple mechanical system that exhibits similar dynamics is the extensible (spring) spherical pendulum.