Nonlinear Dynamics of Magnetic Bearing Systems

Magnetic bearings use magnetic forces to support various machine components. Because of the non-contact nature of this type of suspension, magnetic bearing technology offers a number of significant advantages over conventional bearings, such as rolling element and fluid film bearings. An active magnetic bearing basically consists of an electromagnetic actuator, position sensors, power amplifiers, and a feedback controller. All of these components are characterized by nonlinear behavior and therefore the entire system is inherently nonlinear. However, in simulations of the dynamic behavior of magnetic bearing systems, the nonlinearities are usually neglected to simplify the analysis and only linear models are used. Moreover, many control techniques currently used in magnetic bearing systems are generally designed by ignoring nonlinear effects. The main reason for simplification is the intractability of the complexity of the actual model. In fact, the inherent nonlinear properties of magnetic bearing systems can lead to dynamic behavior of a magnetically suspended rotor that is distinctly different from that predicted using a simple linearized model. Therefore, the nonlinearities should be taken into account. This literature review is focused on the nonlinear dynamics of magnetic bearing systems and it provides background information on analytical methods, nonlinear vibrations resulting from a rotor contacting auxiliary bearings, and other active topics of research involving the nonlinear properties of magnetic bearing systems, such as nonlinear self-sensing magnetic bearings and nonlinear control of magnetic bearings. The review concludes with a brief discussion on current and possible future directions for research on the nonlinear dynamics of magnetic bearing systems.

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

[2]  Jinchen Ji,et al.  Dynamics of a Jeffcott rotor-magnetic bearing system with time delays , 2003 .

[3]  Alan Palazzolo,et al.  Detailed ball bearing model for magnetic suspension auxiliary service , 2004 .

[4]  E. H. Maslen,et al.  Feasible whirl or rotors in auxiliary bearings , 1995 .

[5]  Mary Kasarda.  An overview of active magnetic bearing technology and applications , 2000 .

[6]  Colin H. Hansen,et al.  Effect of External Excitations on a Nonlinear System with Time Delay , 2005 .

[7]  Myounggyu Noh,et al.  Self-sensing magnetic bearings using parameter estimation , 1997 .

[8]  Chris R. Fuller,et al.  The Effect of Actuator and Sensor Placement on the Active Control of Rotor Unbalance , 2003 .

[9]  Tetsuya Iwasaki,et al.  Robustness Limitations in Self-Sensing Magnetic Bearings , 2006 .

[10]  Richard Markert,et al.  Improvements of the integration of active magnetic bearings , 2002 .

[11]  H. Troger,et al.  Nonlinear stability and bifurcation theory , 1991 .

[12]  Li Feng,et al.  Steady-State Dynamic Behavior of a Flexible Rotor With Auxiliary Support From a Clearance Bearing , 1999 .

[13]  Sheng Zeng.  Motion of AMB rotor in backup bearings , 2002 .

[14]  Paul E. Allaire,et al.  Constrained Control Design for Magnetic Bearing Systems , 2005 .

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

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

[17]  Matthew T. Cole,et al.  Multi-State Transient Rotor Vibration Control Using Sampled Harmonics , 2002 .

[18]  Colin H. Hansen,et al.  Analytical approximation of the primary resonance response of a periodically excited piecewise non-linear–linear oscillator , 2004 .

[19]  Andrew J. Provenza,et al.  Test and Theory Correlation Study for a Flexible Rotor on Fault-Tolerant Magnetic Bearings , 2002 .

[20]  Colin H. Hansen,et al.  Approximate solutions and chaotic motions of a piecewise nonlinear–linear oscillator , 2004 .

[21]  E. E. Swanson,et al.  AMB rotor drop initial transient on ball and solid bearings , 1995 .

[22]  Peter J. Fleming,et al.  On-line evolution of robust control systems: an industrial active magnetic bearing application , 2001 .

[23]  Xinghuo Yu,et al.  Bifurcation Control: Theory and Applications , 2003 .

[24]  R. G. Kirk,et al.  Reduction of subsynchronous vibrations in a single-disk rotor using an active magnetic damper , 2004 .

[25]  Clifford R. Burrows,et al.  Towards fault-tolerant active control of rotor–magnetic bearing systems , 2004 .

[26]  G. Schweitzer.  Magnetic bearings : proceedings of the first international symposium, EHT Zurich, Switzerland, June 6-8, 1988 , 1989 .

[27]  Alan Palazzolo,et al.  NON-LINEAR FUZZY LOGIC CONTROL FOR FORCED LARGE MOTIONS OF SPINNING SHAFTS , 2000 .

[28]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[29]  W. Wonham.  Linear Multivariable Control: A Geometric Approach , 1974 .

[30]  S. M. Tamer,et al.  Experiments in robust control of rotor unbalance response using magnetic bearings , 1997 .

[31]  Lichuan Li,et al.  Linearizing magnetic bearing actuators by constant current sum, constant voltage sum, and constant flux sum , 1999 .

[32]  Chong-Won Lee,et al.  Design and control of active magnetic bearing system with Lorentz force-type axial actuator , 2006 .

[33]  Roland Siegwart,et al.  Proceedings of the Seventh International Symposium on Magnetic Bearings , 1994 .

[34]  Alan Palazzolo,et al.  Optimized Realization of Fault-Tolerant Heteropolar Magnetic Bearings , 2000 .

[35]  Hannes Bleuler.  A Survey of Magnetic Levitation and Magnetic Bearing Types , 1992 .

[36]  H. Antosiewicz,et al.  Differential Equations: Stability, Oscillations, Time Lags , 1967 .

[37]  Jinchen Ji,et al.  Stability and Hopf bifurcation of a magnetic bearing system with time delays , 2003 .

[38]  Matthew T. Cole,et al.  Fault detection and tolerance in synchronous vibration control of rotor—magnetic bearing systems , 2001 .

[39]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[40]  Joga Dharma Setiawan,et al.  Adaptive Compensation of Sensor Runout for Magnetic Bearings With Uncertain Parameters: Theory and Experiments , 2001 .

[41]  Richard Markert,et al.  Influence of cross-axis sensitivity and coordinate coupling on self-sensing , 2001 .

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

[43]  Jinchen Ji,et al.  Stability and bifurcation in an electromechanical system with time delays , 2003 .

[44]  Carl R. Knospe,et al.  Experiments in the control of unbalance response using magnetic bearings , 1995 .

[45]  Jinchen Ji,et al.  Drop Dynamics of a High-Speed Unbalanced Rotor in Active Magnetic Bearing Machinery* , 2000 .

[46]  T. Mizuno,et al.  Self-Sensing Magnetic Suspension Using Hysteresis Amplifiers , 1996 .

[47]  Guangyoung Sun.  Rotor drop and following thermal growth simulations using detailed auxiliary bearing and damper models , 2006 .

[48]  Livija Cveticanin.  Free vibration of a Jeffcott rotor with pure cubic non-linear elastic property of the shaft , 2005 .

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

[50]  J. Hale.  Oscillations in Nonlinear Systems , 1963 .

[51]  Alan Palazzolo,et al.  The Fault-Tolerant Control of Magnetic Bearings With Reduced Controller Outputs , 2001 .

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

[53]  Andrea Tonoli,et al.  Analysis of losses due to rotor vibrations in a high-Tc superconducting flywheel system , 1998 .

[54]  M. C.,et al.  STABILITY AND BIFURCATION OF ROTOR MOTION IN A MAGNETIC BEARING , 1998 .

[55]  Ting-Jen Yeh,et al.  Sliding control of magnetic bearing systems , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[56]  Colin H. Hansen,et al.  Hopf Bifurcation of a Magnetic Bearing System with Time Delay , 2005 .

[57]  Sung-Chong Chung,et al.  Integrated design of radial active magnetic bearing systems using genetic algorithms , 2002 .

[58]  John Y. Hung,et al.  Nonlinear control of a magnetic bearing system , 2003 .

[59]  M. S. de Queiroz,et al.  Nonlinear control of a large gap 2-DOF magnetic bearing system based on a coupled force model , 1997 .

[60]  Joga Dharma Setiawan,et al.  Synchronous Sensor Runout and Unbalance Compensation in Active Magnetic Bearings Using Bias Current Excitation , 2002 .

[61]  Richard Markert,et al.  Improvements in the integration of active magnetic bearings , 2000 .

[62]  Zongli Lin,et al.  Modeling of a High Speed Rotor Test Rig With Active Magnetic Bearings , 2006 .

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

[64]  Guanrong Chen,et al.  Bifurcation Control: Theories, Methods, and Applications , 2000, Int. J. Bifurc. Chaos.

[65]  Hiroki Okubo,et al.  Aseismic Vibration Control of Flexible Rotors Using Active Magnetic Bearing , 2002 .

[66]  C. Hayashi,et al.  Nonlinear oscillations in physical systems , 1987 .

[67]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[68]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[69]  Takeshi Mizuno,et al.  Self-sensing magnetic bearing control system design using the geometric approach , 1995 .

[70]  Weihua Jiang,et al.  Multiple stabilities analysis in a magnetic bearing system with time delays , 2006 .

[71]  D. Vischer,et al.  A New Approach to Sensor-less and Voltage Controlled AMBs Based on Network Theory Concepts , 1990 .

[72]  S Zeng.  Modelling and experimental study of the transient response of an active magnetic bearing rotor during rotor drop on back-up bearings , 2003 .

[73]  Youngbae Kim,et al.  A MULTIPLE HARMONIC BALANCE METHOD FOR THE INTERNAL RESONANT VIBRATION OF A NON-LINEAR JEFFCOTT ROTOR , 1997 .

[74]  G. Schweitzer.  Fourth International Symposium magnetic bearings : proceedings , International Center for Magnetic Bearings ETH Zurich, Switzerland, August 23-26, 1994 , 1994 .

[75]  Lie Yu,et al.  Effect of Thrust Magnetic Bearing on Stability and Bifurcation of a Flexible Rotor Active Magnetic Bearing System , 2003 .

[76]  Chong-Won Lee,et al.  ON-LINE IDENTIFICATION OF CURRENT AND POSITION STIFFNESSES BY LMS ALGORITHM IN ACTIVE MAGNETIC BEARING SYSTEM EQUIPPED WITH FORCE TRANSDUCERS , 1999 .

[77]  Leon O. Chua,et al.  Practical Numerical Algorithms for Chaotic Systems , 1989 .

[78]  Uhn Joo Na,et al.  Fault tolerance of homopolar magnetic bearings , 2004 .

[79]  Matthew T. Cole,et al.  Fault-tolerant control of rotor/magnetic bearing systems using reconfigurable control with built-in fault detection , 2000 .

[80]  R. G. Kirk.  Evaluation of AMB Turbomachinery Auxiliary Bearings , 1999 .

[81]  Colin H. Hansen,et al.  An approximate solution for the super-harmonic resonance response of a piecewise nonlinear-linear oscillator , 2005 .

[82]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[83]  Takashi Hikihara,et al.  DYNAMICAL BEHAVIOR OF FLYWHEEL ROTOR SUSPENDED BY HYSTERETIC FORCE OF HTSC MAGNETIC BEARING , 1999 .

[84]  Reza Langari,et al.  Robust fuzzy control of a magnetic bearing system subject to harmonic disturbances , 1997, IEEE Trans. Control. Syst. Technol..

[85]  Martin Aenis,et al.  Active Magnetic Bearings for the Identification and Fault Diagnosis in Turbomachinery , 2000 .

[86]  M. Wiercigroch,et al.  Non-linear dynamic interactions of a Jeffcott rotor with preloaded snubber ring , 2004 .

[87]  Matthew T. Cole,et al.  Control of multifrequency rotor vibration components , 2002 .

[88]  J. Cooper,et al.  HOPF BIFURCATION ANALYSIS OF A ROTOR/SEAL SYSTEM , 2002 .

[89]  Guangyoung Sun.  Auxiliary Bearing Life Prediction Using Hertzian Contact Bearing Model , 2006 .

[90]  Jiaqi Liu,et al.  Stability and bifurcation analysis in a magnetic bearing system with time delays , 2005 .

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

[92]  L. Chua,et al.  Chaos via torus breakdown , 1987 .

[93]  Matthew T. Cole,et al.  Vibration control of a flexible rotor/magnetic bearing system subject to direct forcing and base motion disturbances , 1998 .

[94]  G. Schweitzer,et al.  Active magnetic bearings : basics, properties and applications of active magnetc bearings , 1994 .

[95]  D. Dane Quinn,et al.  Active health monitoring in a rotating cracked shaft using active magnetic bearings as force actuators , 2006 .

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

[97]  Carl R. Knospe,et al.  Active magnetic bearings for machining applications , 2004 .

[98]  Eric H. Maslen,et al.  Fault tolerant magnetic bearings , 1999 .

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

[100]  Colin H. Hansen,et al.  Local bifurcation Control in a Rotor-Magnetic Bearing System , 2003, Int. J. Bifurc. Chaos.

[101]  S. Harsha.  Nonlinear dynamic analysis of a high-speed rotor supported by rolling element bearings , 2006 .

[102]  Colin H. Hansen,et al.  Forced phase-locked response of a nonlinear system with time delay after Hopf bifurcation , 2005 .

[103]  Huajun Xie,et al.  Steady-state dynamic behavior of an auxiliary bearing supported rotor system , 1995 .

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

[105]  Jinchen Ji,et al.  Dynamics of a piecewise linear system subjected to a saturation constraint , 2004 .

[106]  Juan Shi,et al.  Synchronous disturbance attenuation in magnetic bearing systems using adaptive compensating signals , 2004 .

[107]  Roy S. Smith,et al.  Magnetic Bearing Measurement Configurations and Associated Robustness and Performance Limitations , 2002 .

[108]  D.J. Inman,et al.  Damage detection of a rotating cracked shaft using an active magnetic bearing as a force actuator - analysis and experimental verification , 2005, IEEE/ASME Transactions on Mechatronics.

[109]  H. Bleuler,et al.  Self-sensing active magnetic levitation , 1993 .

[110]  Matthew T. Cole,et al.  Rotor vibration with auxiliary bearing contact in magnetic bearing systems Part 2: Robust synchronous control for rotor position recovery , 2003 .

[111]  R. Gordon Kirk,et al.  Transient Response Technique Applied to Active Magnetic Bearing Machinery During Rotor Drop , 1996 .

[112]  Matthew T. Cole,et al.  Rotor vibration with auxiliary bearing contact in magnetic bearing systems Part 1: Synchronous dynamics , 2003 .

[113]  Dong-il Dan Cho.  Experimental results on sliding mode control of an electromagnetic suspension , 1993 .

[114]  R. Seydel.  Practical Bifurcation and Stability Analysis , 1994 .

[115]  Paul E. Allaire.  Proceedings of the Third International Symposium on Magnetic Bearings : July 29-31, 1992, Radisson Hotel at Mark Center, Alexandria, Virginia , 1992 .

[116]  D. Howe,et al.  Simplified characteristics of active magnetic bearings , 2002 .

[117]  Richard Markert,et al.  Simulation of nonlinear effects on magnetically suspended rotors , 1995 .