The asymptotic perturbation method is used to analyze the nonlinear vibrations and chaotic dynamics of a rotor-active magnetic bearing (AMB) system with 16-pole legs and the time-varying stiffness. Based on the expressions of the electromagnetic force resultants, the influences of some parameters, such as the cross-sectional area Aα of one electromagnet and the number N of windings in each electromagnet coil, on the electromagnetic force resultants are considered for the rotor-AMB system with 16-pole legs. Based on the Newton law, the governing equation of motion for the rotor-AMB system with 16-pole legs is obtained and expressed as a two-degree-of-freedom system with the parametric excitation and the quadratic and cubic nonlinearities. According to the asymptotic perturbation method, the four-dimensional averaged equation of the rotor-AMB system is derived under the case of 1 : 1 internal resonance and 1 : 2 subharmonic resonances. Then, the frequency-response curves are employed to study the steady-state solutions of the modal amplitudes. From the analysis of the frequency responses, both the hardening-type nonlinearity and the softening-type nonlinearity are observed in the rotor-AMB system. Based on the numerical solutions of the averaged equation, the changed procedure of the nonlinear dynamic behaviors of the rotor-AMB system with the control parameter is described by the bifurcation diagram. From the numerical simulations, the periodic, quasiperiodic, and chaotic motions are observed in the rotor-active magnetic bearing (AMB) system with 16-pole legs, the time-varying stiffness, and the quadratic and cubic nonlinearities.
In this paper, the nonlinear dynamics of a rotor-active magnetic bearing system with 16-pole legs and the time varying stiffness is investigated. The magnetic forces are obtained through an electromagnetic theory. The motion governing equation is derived by using Newton law. The resulting dimensionless equation of motion for the rotor-AMB system with 16-pole legs and the time varying stiffness is presented with the two-degree-of-freedom system including parametric excitation, the quadratic and cubic nonlinearities. The averaged equations of the rotor-AMB system are obtained by using the method of multiple scales under the case of the primary parametric resonance and 1/2 sub-harmonic resonance. The numerical results show that there exist the periodic, quasi-periodic and chaotic motions in the rotor-active magnetic bearing system. Since the weight of the rotor effect the system, it is also found that there are the different shapes of motion on the two directions of the rotor-AMB system. The parametric excitation, or the time-varying stiffness produced by the PD controller has great impact on the system. Thus, the complicated dynamical response in the rotor-AMB system can be controlled through adjusting the parametric excitation.
In this paper, we use the asymptotic perturbation method to analyze the nonlinear dynamics of a rotor-active magnetic bearing (AMB) system with 16-pole legs. The motion governing equation is derived by using classical Newton law. The resulting dimensionless equation of motion for the system is expressed as a two-degree-of-freedom system including the parametric excitation, quadratic and cubic nonlinearities. The asymptotic perturbation method is used to obtain the averaged equation when the primary resonance and 1/2 sub-harmonic resonance are taken into consideration. From the averaged equations obtained, numerical simulations are presented to investigate the modulation of vibration amplitudes of the rotor-AMB system. Based on a specific set of parameters, it is found that there exist the periodic, quasi-periodic and chaotic motions in the modulated amplitude of the rotor in the system.
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