Periodic and Chaotic Motions of a Rotor-Active Magnetic Bearing with Quadratic and Cubic Terms and Time-Varying Stiffness

In this paper, we use the asymptotic perturbation method to investigate nonlinear oscillations and chaotic dynamics in a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time varying in a periodic form. Because of considering the weight of the rotor, the formulation on the electromagnetic force resultants includes the quadratic and cubic nonlinearities. The resulting dimensionless equations of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions are a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found that there exist period-3, period-4, period-6, period-7, period-8, quasiperiodic and chaotic modulated amplitude oscillations in the rotor-AMB system with the time-varying stiffness. It is seen from the numerical results that there are the phenomena of the multiple solutions and the soft-spring type and the hardening-spring type in nonlinear frequency-response curves for the rotor-AMB system. The parametric excitation, or the time-varying stiffness produced by the PD controller is considered to be a controlling force which can control the chaotic response in the rotor-AMB system to a period n motion.

[1]  X. Wang,et al.  Nonlinear dynamics of a magnetically supported rotor on safety auxiliary bearings , 1998 .

[2]  James A. Yorke,et al.  Dynamics: Numerical Explorations , 1994 .

[3]  Wei Zhang,et al.  Analysis of Nonlinear Dynamics for a Rotor-Active Magnetic Bearing System With Time-Varying Stiffness: Part I — Formulation and Local Bifurcations , 2003 .

[4]  A. Maccari,et al.  The Kadomtsev–Petviashvili equation as a source of integrable model equations , 1996 .

[5]  Alan Palazzolo,et al.  STABILITY AND BIFURCATION OF ROTOR MOTION IN A MAGNETIC BEARING , 1998 .

[6]  W. Zhang,et al.  Local and global bifurcations of valve mechanism , 1994 .

[7]  Abdelfatah M. Mohamed,et al.  Nonlinear oscillations in magnetic bearing systems , 1993 .

[8]  Kon Well Wang,et al.  An Energy-Based Parametric Control Approach for Structural Vibration Suppression via Semi-Active Piezoelectric Networks , 1996 .

[9]  Lawrence N. Virgin,et al.  Nonlinear Behavior of a Magnetic Bearing System , 1995 .

[10]  Kon Well Wang,et al.  Parametric Control of Structural Vibrations via Adaptable Stiffness Dynamic Absorbers , 1996 .

[11]  Further studies on nonlinear dynamics of one dimensional crystalline beam , 1996 .

[12]  A. Maccari,et al.  Approximate Solution of a Class of Nonlinear Oscillators in Resonance with a Periodic Excitation , 1998 .

[13]  Wei Zhang,et al.  Nonlinear oscillations and chaotic dynamics of an antisymmetric cross-ply laminated composite rectangular thin plate under parametric excitation , 2005 .

[14]  Andrew Y. T. Leung,et al.  BIFURCATION BEHAVIOR OF A ROTOR SUPPORTED BY ACTIVE MAGNETIC BEARINGS , 2000 .

[15]  Colin H. Hansen,et al.  Non-linear oscillations of a rotor in active magnetic bearings , 2001 .

[16]  A. Maccari,et al.  Modulated motion and infinite-period homoclinic bifurcation for parametrically excited Liénard systems , 2000 .

[17]  Anil K. Bajaj,et al.  Bifurcations in a parametrically excited non-linear oscillator , 1987 .

[18]  A. Maccari.  Dissipative bidimensional systems and resonant excitations , 1998 .

[19]  Roland Siegwart.  Design and application of active magnetic bearings (AMB) for vibration control , 1992 .

[20]  A. Maccari.  A model system for the behavior of two non-linearly coupled oscillators , 1998 .