In order to clarify the effects of support structure on a dual-rotor machine,a dynamic model is established which takes into consideration the contact force of ball bearing and the cubic stiffness of elastic support. Bearing clearance,Hertz contact between the ball and race and the varying compliance effect are included in the model of ball bearing. The system response is obtained through numerical integration method,and the vibration due to the periodic change of bearing stiffness is investigated. The motions of periodic,quasiperiodic and even chaotic are found when bearing clearance is used as control parameter to simulate the response of rotor system. The results reveal two typical routes to chaos: quasi-periodic bifurcation and intermittent bifurcation. Large cubic stiffness of elastic support may cause jump and hysteresis phenomena in resonance curve when rotors run at the critical-speed region. The modeling results acquired by numerical simulation will contribute to understanding and controlling of the nonlinear behaviors of the dual-rotor system.
We focus on the hysteretic characteristics of the varying compliance(VC) principal resonance in a ball bearing. The branches of the periodic VC response are traced by the harmonic balance method and the alternating frequency/time domain technique(HB-AFT) embedding Arc-length continuation, and the stability of these solutions is investigated by using Floquet theory. We find that the resonant response displays a swallow-tail structure due to the coupling nonlinearities between the Hertzian contact and the bearing clearance, which differs from the soft hysteresis of the non-loss Hertzian contact resonances. Furthermore, we find that period-1 VC branch cannot completely characterize the response of the system for a large bearing clearance, because multiple instability regions may occur from the cyclic fold, the secondary Hopf bifurcations, supercritical and subcritical period doubling bifurcations, in which case co-existences of period-1, period-2, and even quasi-periodic VC motions emerge in the hysteretic resonant range.
A dynamic model is established for an offset-disc rotor system with a mechanical gear coupling, which takes into consideration the nonlinear restoring force of rotor support and the effect of coupling misalignment. Periodic solutions are obtained through harmonic balance method with alternating frequency/time domain(HB-AFT) technique, and then compared with the results of numerical simulation. Good agreement confirms the feasibility of HB-AFT scheme. Moreover, the Floquet theory is adopted to analyze motion stability of the system when rotor runs at different speed intervals. A simple strategy to determine the monodromy matrix is introduced and two ways towards unstability are found for periodic solutions: the period doubling bifurcation and the secondary Hopf bifurcation. The results obtained will contribute to the global response analysis and dynamic optimal design of rotor systems.
In this paper, an investigation on the nonlinear vibration, especially on the super-harmonic resonances, in a cracked rotor system is carried out to provide a novel idea for the detection of crack faults in rotor systems. The motion equations of the system are formulated with the consideration of the additional excitation from an inertial environment as well as the forced excitation of the rotor unbalance. By using the harmonic balance method, the analytical solutions of the equations with four orders of harmonic exponents are obtained to analyze the nonlinear response of the system. Then through numerical calculations, the vibration responses affected by system parameters including the inertial excitation, the forced excitation, the crack and damping factors are investigated in detail. The results show that the occurrence of the super-harmonic resonances of the rotor system is due to the interaction between crack breathing and the inertial excitation. Correspondingly, the super-harmonic responses are significantly affected by the inertial excitation and the crack stiffness(or depth). The rotor unbalance, however, does not make apparent effects on the super-harmonic responses. Consequently, the super-harmonic resonances peaks can be viewed as an identification signal of the crack fault due to the application of the inertial excitation. By utilizing the inertial excitation, the super-harmonic response signals in rotor systems with early crack faults can be amplified and detected more easily.
