Bifurcations And Hysteresis Of Varying Compliance Vibrations Of A Ball Bearing-rotor System

The time-varying compliance(VC), bearing clearance and the Hertzian contact between the rolling elements and raceways are three fundamental nonlinear factors in a rolling bearing, hence it can be taken the rolling bearing and its rotor system as a nonlinear system. Many researchers have been presented that the bearing-rotor system may possess complex dynamics responses such as super/sub-harmonic motions, bi-stabilities, and even chaos through the nonlinearities of rolling bearings, which may have important effects on the stability and fatigue life of the bearing and its rotor system. The time-varying compliance of rolling bearings is induced by the revolution of rolling elements with the bearing cage in and out of the loaded area, so it is an inevitable source of the parametrically excited factors. The studies on the VC vibration and the effect of the coupling nonlinearities of bearing clearance and the Hertzian contact have great importance for analysis of the influence of the bearing nonlinearity to the whole rotor system, and which may be also significant for the dynamical parameter optimization of the rolling bearing itself.The hysteresis and jumps induced by the nonlinearities of rolling bearings are typical phenomena of nonlinear vibrations in the rolling bearing-rotor systems, and the corresponding hysteretic impacts have direct effects on the cleavage derivative and fatigue life of the system components, for which case that the behaviors of hysteresis and jumps are given full attentions and continued studies in the theoretical and engineering fields. Besides, many researchers have done a lot of calculations to depict the various characteristics of bifurcations and chaos in the rolling bearings and their rotor systems, but few studies are presented on the physical mechanism of their behaviors. For example, few researches have been addressed on the inherent mechanism of the typical intermittency vibrations in rolling bearings. In short, the explorations to the nature of nonlinearities of the rolling bearing systems still need in process deeply.The rolling bearing is a complex strong nonlinear system with multiple piecewise functions depicting the time-varying contact and loss contact between the balls and the raceways, which leads great difficulties encountered to the traditional analytical methods. The harmonic balance and alternating frequency/time domain(HB-AFT) method is a semi-analytical method developing from the classic harmonic balance method, where the AFT technology can keep away from the analytical processing procedures such as integrations, series expansions and truncation to the complex nonlinear terms in the differential equations, so the HB-AFT method can provide the harmonic solutions of the steady-state response of the nonlinear systems, and we have given a deep investigations to the involution of the complex motions in the VC vibrations of ball bearings by this method.In this dissertation, we proceed to study on the VC vibration of the classical two-degree-of-freedom deep groove ball bearing model with nonlinearities of Hertzian contact and bearing clearance, and the following aspects are addressed on:1. With the aid of arc-length continuation, the HB-AFT method is employed to automaticaly trace the branches of periodic responses for the VC vibration, for which the Floquet stability is tried to investigate by using the Hsu algorithm. It is presented a full frequency domain strategy for solving the periodic responses and analyzing their stabilities, keeping away from the numerical integrations to the differential equation, so we can give the characteristics of the dynamical responses and their bifurcating behaviors fast and accurately.2. Contact resonance may bring soft hysteresis and jumps to the system, and the concept of Hertzian contact resonance is presented to rolling bearings, then the contact resonance of ball bearings is investigated deeply and especially the mechanism of the hysteresis and jumps and their influencing factors of VC vibration are explored in detail.3. The mechanism of the period doubling near the natural frequency in the VC vibration is studied in the view of coupling internal resonance, and further the phenomena of complex motions corresponding with the period doubling bifurcation is clarified. In addition, the influence of bearing clearance on the coupling interactions is considered.4. The evolvements of the complex behaviors of VC vibrations are completely demonstrated, then the mechanism of bifurcations of VC vibration from periodic to quasi-periodic and even chaotic motions is studied in detail, and the mechanism of the chaotic behaviours such as intermittency of VC vibrations is clarified.In conclusion, this dissertation studies the typical nonlinear behaviors of hysteretic jump, bifurcations and chaos for VC vibration in the view of nonlinear dynamics, and the corresponding dynamical method is explored. All the above works are based on the achievements at home and abroad recently and belong to the frontier tasks of this field. The characteristics of the nonlinear responses and further their mechanisms of bifurcation and chaos are tried to investigate in detail. We believe this dissertation may make contributions to clarifying the mechanism of the complex nonlinear vibrations in the rolling bearings and their rotor systems.