Nonsmooth Behavior of Sliding Bifurcations in a General Piecewise Smooth Rotor/Stator Rubbing System

Abstract

Sliding bifurcations of different types are defined to characterize the topological features of trajectories around the switching manifolds of nonsmooth dynamical systems. In this paper, the stick-slip transitions, which are related to the dynamical scenarios of sliding bifurcations, of the self-excited dry friction backward whirls of a general rotor/stator rubbing system, are investigated. The four-degree-of-freedom piecewise smooth rotor/stator rubbing model is said to be general because it takes into account main factors in the rotor/stator rubbing systems, including both the dynamics of the rotor and the stator as well as the dry friction and the flexibility on the contact surfaces. The switching manifold that separates two discontinuous vector fields is defined as the curved hypersurface in a nine-dimensional extended state space where the relative velocity at the contact points equals zero. After deriving the formulae defining the sliding regions and their boundaries for the piecewise smooth system, two extreme cases with rigid and soft contact surfaces are theoretically analyzed and confirmed to correspond respectively to a continuous pure rolling with full sliding region and a continuous crossing without sliding region on the switching manifold. Furthermore, three types of sliding bifurcations, namely, crossing-sliding, grazing-sliding and switching-sliding, are observed in the dry friction backward whirls of the present model in a semi-analytical way. Moreover, hybrids of the three kinds of the sliding solutions in one period of oscillation are also identified with the variation of system parameters and initial conditions. The main scenarios of the switching transition of sliding bifurcations in the self-excited dry friction backward whirl of the general rotor/stator rubbing system with the variation of system parameters are also summarized.