Mechanism of a Nonsmooth Bifurcation in a Piecewise Nonlinear Rotor–Stator Rubbing System of the Filippov Type

When adopting a viscous friction model in a piecewise nonlinear rotor–stator rubbing system, it becomes a Filippov system with smoothness of degree one. The nonautonomous system exhibits an interesting nonsmooth induced bifurcation phenomenon, that is, as the rotating speed increases, a quasi-periodic response emerges just after a periodic orbit of the linear sub-system touches the switching manifold, while the nonlinear sub-system possesses no regular periodic solution nearby. To investigate the mechanism of the nonsmooth induced bifurcation, the system is first transformed to the rotating coordinate system to get an equivalent autonomous system. Then, further transformations are performed to transform the system to an equivalent linear normal form in the neighborhood of the boundary equilibrium of the linear sub-system on the straightened switching manifold. After that, the Poincaré return map is constructed, and through numerical and analytical methods, it is shown that the tangency points of the two subsystems are mostly visible–invisible folds and invisible–invisible folds, and the system generates a limit cycle crossing the switching manifold. Through this analysis, the mechanism of the nonsmooth bifurcation in the original nonautonomous rotor–stator rubbing system is clarified. Moreover, the approximate model not only can accurately predict the maximum amplitude of quasi-periodic responses, but also reveal the coexistence of quasi-periodic and periodic motions. This highlights the usefulness of the proposed analytical approach for handling nonsmooth systems with engineering relevance.