Nonlinear Dynamics Of A Rotor Supported By Active Magnetic Bearings

Active Magnetic Bearings (AMDs) have been widely used in the engineering, for example, the mechanical engineering, the aeronautic and astronautics engineering. Because most of the components in AMBs are of the nonlinear characteristics, the dynamics in AMBs is very complicated. The electromagnetic force is a nonlinear function with respect to the displacement of the rotor and the controlling electric current. The nonlinear electromagnetic force may cause the large oscillations of the rotor in some parameter regions. Thus, the studies on the properties of the nonlinear dynamics and the stability for the rotor-AMBs system play an important role in the engineering. There are abundant and complicated dynamical behaviors in the rotor-AMBs system, such as the local and global bifurcations and the chaotic dynamics. In this dissertation, we investigate the nonlinear dynamics in the rotor-AMBs system. The two-degree-of-freedom nonlinear system with cubic nonlinearities will be used to explore the bifurcations and chaotic dynamics in the rotor-AMBs system with eight pole pairs. The results obtained by the dissertation show that there exist the chaotic motions in some parameter regions.The research contents and the major results obtained in this dissertation are as follows.(1) We give a review on the researches for the rotor-AMBs system. The applications and developments of the rotor-AMBs system in the engineering are showed in recent years. In particularly, we present the results for the studies of nonlinear dynamics in the rotor-AMBs system. We also point out the trend of the rotor-AMBs system in future and the necessity of the study on the nonlinear dynamics of the rotor-AMBs system.(2) Due to the effect of the rotor weight, the equations of motion in the horizontal and vertical directions and the nonlinear properties are different. Therefore, a rigid model with two-degree-of-freedom is established after the rotor weight is considered. Because there are the eight pole pairs, the capacity supporting load in AMBs is greatly increased. The dimensionless equations of the rotor-AMBs in horizontal and vertical directions are also obtained.(3) We investigate the nonlinear dynamics of the rotor-AMBs system in 1/3 and 1/2 subharmonic resonances. Using the multiple method of scale, the averaged equations of the rotor-AMBs system are obtained. The amplitude-frequency response equations and the local bifurcations are respectively analyzed in the two resonant cases. The numerical simulations are given to obtain the amplitude-frequency response curve.(4) The global bifurcations and chaotic dynamics are investigated when the rotor-AMBs system has the time-varying stiffness. The multiple method of scale is used to obtain the averaged equations in primary parameter resonance. From the averaged equations, the theory of normal form is applied to find the explicit formulas of normalform associated with a double zero and a pair of pure imaginary eigenvalues with the aid of the Maple program. Based on the normal form obtained above, a global perturbation method is utilized to give the analysis for the global bifurcations and chaotic dynamics of the rotor-AMBs system. The global bifurcations analysis indicates that there exist the heteroclinic bifurcations and the Silnikov-type homoclinic orbit in the averaged equations. These mean that the chaotic motions can occur in the rotor-AMBs system with the time-varying stiffness. The numerical simulations verify the analytical prediction.