Bursting oscillations of a geometrical nonlinear system with a third-order catastrophe point

Abstract

Bursting oscillations of a geometrical nonlinear system with a third-order catastrophe point are investigated. The third-order catastrophe point is a high order cusp point at which the geometrical nonlinear system can achieve the third-order quasi-zero stiffness for vibration isolation. The symmetry of triple-wells, double-wells and single-wells near the catastrophe point is broken due to the introduction of a slow-varying parameter, which results in the appearance of asymmetrical triple-wells, double-wells and single wells. Complex evolution dynamics of both the non-autonomous system and the generalized autonomous system are studied through a natural connection that treats the low frequency excitation as a slow-varying parameter. Bursting oscillations are found and the mechanisms are revealed through the analysis of asymmetrical global dynamics. The relations between the isolation performance and bursting oscillations are studied through the coordinates corresponding to the force transmissibility. Results showed that both the increase of geometrical parameters and the introduction of the nonlinear damping with multiple roots could result in the appearance of more quiescent states. A novel step-up curve of inter spike intervals for quiescent states is found which corresponds to the fish-scale curve of inter spike intervals for spiking states. A smaller excitation amplitude can be used to generate the inter spike intervals of spiking states which means that vibrations can be observed earlier with the increase of the small stiffness and small restoring force regions near the origin. Low transmissibility can be achieved when the system trajectory reaches its amplitude at the quiescent states for any situation near the third-order catastrophe point.