Grazing phenomena and fragmented strange attractors in a harmonically forced, piecewise, linear system with impacts

Abstract

In this article, the grazing bifurcations and flows of an idealized gear transmission system with impact are investigated from the non-smooth dynamical system theory of Luo [1–3]. The necessary and sufficient conditions for such grazing bifurcations of all the generic mappings are obtained. The initial and final grazing, switching manifolds for all the generic mappings are introduced for the grazing motions. The fragmentation of strange attractors in non-smooth dynamical systems is described mathematically. The fragmentation mechanism of the strange attractors for such a non-smooth dynamical system is discussed qualitatively. Such a fragmentation of strange attractors is illustrated numerically through the switching sets. This investigation will provide a better understanding of the mathematical structures and characteristics of strange attractors in non-smooth dynamical systems. This is very useful to determine the origin of the vibration and noise in the gear transmission systems.