The rotor-active magnetic bearing system subjected to a periodically time-varying stiffness having quadratic and cubic nonlinearities is studied and solved. The multiple time scale technique is applied to solve the nonlinear differential equations governing the system up to the second order approximation. All possible resonance cases are deduced at this approximation and some of them are confirmed by applying the Rung–Kutta method. The main attention is focused on the stability of the steady-state solution near the simultaneous principal resonance and the effects of different parameters on the steady-state response. A comparison is made with the available published work.
Issue Section:
Research Papers
1.
Mohamed
, A. M.
, and Emad
, F. P.
, 1993, “Non-Linear Oscillations in Magnetic Bearing Systems
,” IEEE Trans. Automat. Control
, 38
, pp. 1242
–1245
. 0018-92862.
Hu
, T.
, Lin
, Z.
, Jiang
, W.
, and Allaire
, P. E.
, 2005, “Constrained Control Design for Magnetic Bearing Systems
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 127
, pp. 601
–616
.3.
Wang
, H. B.
, and Liu
, J. Q.
, 2005, “Stability and Bifurcation Analysis in a Magnetic Bearing System With Time Delays
,” Chaos, Solitons Fractals
, 26
, pp. 813
–825
. 0960-07794.
Wang
, H. B.
, and Liu
, J. Q.
, 2006, “Multiple Stability Analysis in a Magnetic Bearing System With Time Delays
,” Chaos, Solitons Fractals
, 27
, pp. 789
–799
. 0960-07795.
Jang
, M. J.
, and Chen
, C. K.
, 2001, “Bifurcation Analysis in Flexible Rotor Supported by Active Magnetic Bearing
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274, 11
(8
), pp. 2163
–2178
.6.
Jang
, M. J.
, Chen
, C. L.
, and Tsao
, M.
, 2005, “Sliding Mode Control for Active Magnetic Bearing System With Flexible Rotor
,” J. Franklin Inst.
0016-0032, 342
, pp. 401
–419
.7.
Ji
, J. C.
, Yu
, L.
, and Leung
, A. Y. T.
, 2000, “Bifurcation Behavior of a Rotor by Active Magnetic Bearings
,” J. Sound Vib.
0022-460X, 235
, pp. 133
–151
.8.
Ji
, J. C.
, and Leung
, A. Y. T.
, 2003, “Non-Linear Oscillations of a Rotor-Magnetic Bearing System Under Super Harmonic Resonance Conditions
,” Int. J. Non-Linear Mech.
0020-7462, 38
, pp. 829
–835
.9.
Inayat-Hussain
, J. I.
, 2006, “Chaos Via Torus Breakdown in the Vibration Response of a Rigid Rotor Supported by Active Magnetic Bearings
,” Chaos, Solitons Fractals
, 31
, pp. 12
–27
. 0960-077910.
Ji
, J. C.
, and Hansen
, C. H.
, 2004, “Approximate Solutions and Chaotic Motions of a Piecewise Nonlinear Oscillator
,” Chaos, Solitons Fractals
, 20
, pp. 1121
–1133
. 0960-077911.
Ji
, J. C.
, and Hansen
, C. H.
, 2001, “Non-Linear Oscillations of a Rotor in Active Magnetic Bearings
,” J. Sound Vib.
0022-460X, 240
, pp. 599
–612
.12.
Zhang
, W.
, and Zu
, J. W.
, 2003, “Nonlinear Dynamic Analysis for a Rotor-Active Magnetic Bearing System With Time-Varying Stiffness. Part I: Formulation and Local Bifurcation
,” Proceedings of ASME International Mechanical Engineering Congress and Exposition
, Washington, DC, Nov. 16–21, ASME
, New York
, pp. 631
–640
.13.
Zhang
, W.
, Yao
, M. H.
, and Zhan
, X. P.
, 2006, “Multi-Pulse Chaotic Motions of a Rotor-Active Magnetic Bearing System With Time-Varying Stiffness
,” Chaos, Solitons Fractals
0960-0779, 27
(1
), pp. 175
–186
.14.
Zhang
, W.
, and Zhan
, X. P.
, 2005, “Periodic and Chaotic Motions of a Rotor-Active Magnetic Bearing With Quadratic and Cubic Terms and Time-Varying Stiffness
,” Nonlinear Dyn.
0924-090X, 41
, pp. 331
–359
.15.
Zhang
, W.
, Zu
, J. W.
, and Wang
, F. X.
, 2008, “Global Bifurcations and Chaos for a Rotor-Active Magnetic Bearing System With Time-Varying Stiffness
,” Chaos, Solitons Fractals
, 35
, pp. 586
–608
. 0960-077916.
Zhang
, W.
, and Zu
, J. W.
, 2008, “Transient and Steady Nonlinear Response for a Rotor-Active Magnetic Bearings System With Time-Varying Stiffness
,” Chaos, Solitons Fractals
0960-0779, 38
, pp. 1152
–1167
.17.
Amer
, Y. A.
, and Hegazy
, U. H.
, 2007, “Resonance Behavior of a Rotor-Active Magnetic Bearing With Time-Varying Stiffness
,” Chaos, Solitons Fractals
0960-0779, 34
, pp. 1328
–1345
.18.
Amer
, Y. A.
, Eissa
, M.
, and Hegazy
, U. H.
, 2006, “Dynamic Behavior of an AMB/Supported Rotor Subject to Parametric Excitation
,” ASME J. Vibr. Acoust.
0739-3717, 128
, pp. 646
–652
.19.
Eissa
, M.
, Hegazy
, U. H.
, and Am
, Y. A.
, 2008, “A Time-Varying Stiffness Rotor-Active Magnetic Bearings Under Combined Resonance
,” ASME J. Appl. Mech.
, 75
, pp. 1
–12
. 0021-893620.
Eissa
, M.
, Hegazy
, U. H.
, and Am
, Y. A.
, 2008, “Dynamic Behavior of an AMB Supported Rotor Subject to Harmonic Excitation
,” Appl. Math. Model.
, 32
, pp. 1370
–1380
. 0307-904X21.
Nayfeh
, A. H.
, 1991, Introduction to Perturbation Techniques
, Wiley-Interscience
, New York
.22.
Kevorkian
, J.
, and Cole
, J. D.
, 1996, Multiple Scale and Singular Perturbation Methods
, Spinger-Verlag
, New York
.23.
Yakowitz
, S.
, and Szidaouszky
, F.
, 1992, An Introduction to Numerical Computation
, Macmillan
, New York
.24.
Isaacson
, E.
, and Keller
, H.
, 1994, Analysis of Numerical Methods
, Dover
, New York
.Copyright © 2009
by American Society of Mechanical Engineers
You do not currently have access to this content.
