Parametrically Excited Vibrations and Auto-Parametric Resonance in Cylindrical Shells

Abstract

Abstract Vibrations of cylindrical shells parametrically excited by external axial forcing or by internal auto-parametric resonances are considered. A Rayleigh-Ritz discretization of the von Kármán-Donnell equations through symbolic computations leads to low dimensional models of shell vibration. After applying methods and ideas of modern dynamical systems theory, complete bifurcation diagrams are constructed and analyzed with an emphasis on modal interactions and their relevance to structural behaviour. In the case of free shell vibrations, the Hamiltonian and a transformation into action-angle coordinates followed by averaging provides readily a geometric description of the interaction between concertina and chequerboard modes. It was established that the interaction should be most pronounced when there are slightly less than 2 N square chequerboard panels circumferentially, where N is the ratio of shell radius to thickness. The two mode interaction leads to preferred vibration patterns with larger deflection inwards than outwards, and at internal resonance, significant energy transfer occurs between the modes. The regular and chaotic features of this interaction are studied analytically and numerically.