Parametric resonance in cylindrical shells: a case study in the nonlinear vibration of structural shells

The aim of this work is to demonstrate in a tutorial fashion how recent ideas and methods of bifurcation theory and nonlinear dynamics have improved the understanding of structural buckling under dynamic loads and vibration of shells under parametric excitation. The paper focuses on geometrically nonlinear forced vibrations of circular cylindrical shells. The emphasis is on fundamental issues and differences between results obtained by linear and nonlinear analysis. Analytical and numerical results for shell models are presented and discussed in the light of nonlinear mode interaction and parametric resonance. The main conclusion from the case studied is that linear theory provides only incomplete and in some cases inaccurate results, when the vibration amplitude becomes comparable to the shell thickness.