Complex Dynamics Analysis of Non-Smooth Electromagnetic-Controlled Rotating Mechanical Rubbing System with Time Delay

With the aim of ensuring the safe operation and enhanced longevity of rotor-stator systems, this research seeks to characterize their non-smooth frictional dynamics under time-delayed electromagnetic governance and to delineate the existence criteria for self-sustained oscillations caused by dry friction. Given that electromagnetic forces provide a non-contact method for dynamic balancing, this paper proposes a non-smooth feedback control strategy based on time-delayed electromagnetic dynamic balancing control. A corresponding Filippov-type time-delayed electromagnetic regulation rotor-stator friction model is established to systematically analyze the non-smooth dynamic behaviors under the adjustment of time-delayed electromagnetic forces. Leveraging non-smooth system theory, the study ascertains the presence of subsystem equilibrium points and rigorously derives the associated conditions for Hopf bifurcation. Through the application of the multi-scale method, the tongue-shaped fractal structure and multi-stability motion patterns of the system are elucidated. Subsequently, using switching flow theory, the conditions required and sufficient for the occurrence of time-delayed crossing motion, grazing bifurcation, and sliding motion are definitively established. Numerical simulations demonstrate various typical non-smooth dynamic behavioral characteristics of the rotor mechanical system, with a focus on the complex influence of the synergistic effect of time delay and system parameters on the relative sliding time. In particular, this paper introduces a method for determining the critical rotational speed and an explicit analytical formula for the initiation of dry friction backward whirl (DFBW) in the time-delayed electromagnetic regulation system, building upon the critical conditions of sliding motion. Numerical simulations demonstrate various typical non-smooth dynamic responses of this system, highlighting the complex impact of the synergistic effect between time delay and system parameters on the relative sliding time. This provides an important theoretical foundation for dynamic design, stability optimization, and active control of rotor machinery.