Nonlinear study of a rotor–AMB system under simultaneous primary-internal resonance

Abstract A rotor–active magnetic bearing (AMB) system subjected to a periodically time-varying stiffness with quadratic and cubic non-linearities under multi-parametric excitations is studied and solved. The method of multiple scales is applied to analyze the response of two modes of a rotor–AMB system with multi-parametric excitations and time-varying stiffness near the simultaneous primary and internal resonance. The stability of the steady state solution for that resonance is determined and studied using Rung–Kutta method of fourth order. It is shown that the system exhibits many typical non-linear behaviors including multiple-valued solutions, jump phenomenon, hardening and softening non-linearities and chaos in the second mode of the system. The effects of the different parameters on the steady state solutions are investigated and discussed also. A comparison to published work is reported.

[1]  J. Cole,et al.  Multiple Scale and Singular Perturbation Methods , 1996 .

[2]  Wei Zhang.  Global and chaotic dynamics for a parametrically excited thin plate , 2001 .

[3]  H. Keller,et al.  Analysis of Numerical Methods , 1969 .

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

[5]  Ming-Jyi Jang,et al.  Sliding mode control for active magnetic bearing system with flexible rotor , 2005, J. Frankl. Inst..

[6]  Y. A. Amer,et al.  Dynamic Behavior of an AMB/Supported Rotor Subject to Parametric Excitation , 2006 .

[7]  Y. A. Amer,et al.  A Time-Varying Stiffness Rotor Active Magnetic Bearings Under Combined Resonance , 2008 .

[8]  David A. Robb,et al.  The dynamics of a cracked rotor with an active magnetic bearing , 2003 .

[9]  Y. A. Amer,et al.  A time-varying stiffness rotor-active magnetic bearings system under parametric excitation , 2008 .

[10]  Wei Zhang,et al.  Periodic and Chaotic Motions of a Rotor-Active Magnetic Bearing with Quadratic and Cubic Terms and Time-Varying Stiffness , 2005 .

[11]  Andrew Y. T. Leung,et al.  Non-linear oscillations of a rotor-magnetic bearing system under superharmonic resonance conditions , 2003 .

[12]  Wei Zhang,et al.  Multi-pulse chaotic motions of a rotor-active magnetic bearing system with time-varying stiffness , 2006 .

[13]  Wei Zhang,et al.  Global Analysis for a nonlinear Vibration Absorber with Fast and Slow Modes , 2001, Int. J. Bifurc. Chaos.

[14]  P. Yu,et al.  Global Dynamics of a Parametrically and Externally Excited Thin Plate , 2001 .

[15]  Z. C. Feng,et al.  Global Bifurcations in Parametrically Excited Systems with Zero-to-One Internal Resonance , 2000 .

[16]  J. I. Inayat-Hussain.  Chaos via torus breakdown in the vibration response of a rigid rotor supported by active magnetic bearings , 2007 .

[17]  A. Nayfeh.  Introduction To Perturbation Techniques , 1981 .

[18]  Y. A. Amer,et al.  Resonance behavior of a rotor-active magnetic bearing with time-varying stiffness , 2007 .

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

[20]  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 .

[21]  Y. A. Amer,et al.  Dynamic behavior of an AMB supported rotor subject to harmonic excitation , 2008 .

[22]  Wei Zhang,et al.  Transient and steady nonlinear responses for a rotor-active magnetic bearings system with time-varying stiffness , 2008 .

[23]  J. Zu,et al.  Global bifurcations and chaos for a rotor-active magnetic bearing system with time-varying stiffness , 2008 .

[24]  Wei Zhang,et al.  Global dynamics of the cable under combined parametrical and external excitations , 2002 .