Nonlinear Dynamics

, Volume 78, Issue 4, pp 2439–2457 | Cite as

Vibration suppression of a time-varying stiffness AMB bearing to multi-parametric excitations via time delay controller

  • M. Eissa
  • M. Kamel
  • A. Al-Mandouh
Original Paper


The applications of active magnetic bearings are growing in industry due to its amazing advantages in reducing friction losses. In this research, the vibration of a two-degree-of-freedom rotor, active magnetic bearings system is suppressed via a nonlinear time delay controller at the confirmed worst resonance case. The selected resonance case is the simultaneous primary and sub-harmonic resonance case. The main aim of this paper was to study the effects of the nonlinear, time delay controller on the behavior of the vibrating system. The multiple time scale perturbation technique is applied to obtain an approximate solution to the second-order approximation. The steady-state solution is obtained around the worst resonance case. The stability of the system is studied applying both frequency response equations and phase-plane method. The worst resonance case is confirmed applying numerical technique. The effects of the different parameters on the steady-state response of the vibrating system are investigated. The obtained approximate solution is validated numerically. Some recommendations are given regarding the design of such system. At the end of the work, a comparison is made with the available published work.


Rotor–active magnetic bearings (AMB)-control Time delay Multi-parametric excitation Stability Resonance cases Perturbation method 


  1. 1.
    Kamel, M., Bauomy, H.S.: Nonlinear oscillation of a rotor-AMB system with time varying stiffness and multi-external excitations. J. Vib. Acoust. 131(3) (Apr 22 2009).Google Scholar
  2. 2.
    Kovačič, G., Wiggins, S.: Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation. Phys. D 57(1), 185–225 (1992)MATHMathSciNetGoogle Scholar
  3. 3.
    Zhang, W., Tang, Y.: Global dynamics of the cable under combined parametrical and external excitations. Int. J. Non-Linear Mech. 37(3), 505–526 (2002)CrossRefMATHGoogle Scholar
  4. 4.
    Ji, J.C., Hansen, C.H.: Non-linear oscillations of a rotor in active magnetic bearing. J. Sound Vib. 240(4), 599–612 (2001)CrossRefGoogle Scholar
  5. 5.
    Ji, J.C., Yu, L., Leung, A.Y.T.: Bifurcation behavior of a rotor supported by activemagnetic bearings. J. Sound Vib. 235, 133–151 (2000)CrossRefGoogle Scholar
  6. 6.
    Ji, J.C., Leung, A.Y.T.: Non-linear oscillations of a rotor-magnetic bearing system under super harmonic resonance conditions. Int. J. Non-Linear Mech. 38, 829–835 (2003)CrossRefMATHGoogle Scholar
  7. 7.
    Amer, Y.A., Hegazy, U.H.: Resonance behavior of a rotor-active magnetic bearing with time-varying stiffness. Chaos Solitons Fractals 34, 1328–1345 (2007)CrossRefGoogle Scholar
  8. 8.
    Amer, Y.A., Eissa, M., Hegazy, U.H.: Dynamic behavior of an AMB/supported rotor subject to parametric excitation. ASME J. Vib. Acoust. 128(5), 646–652 (2006)CrossRefGoogle Scholar
  9. 9.
    Amer, Y.A., Eissa, M., Hegazy, U.H.: A time-varying stiffness rotor active magnetic bearings under combined resonance. J. Appl. Mech. 75, 1–12 (2008)Google Scholar
  10. 10.
    Eissa, M., Hegazy, U.H., Amer, Y.A.: Dynamic behavior of an AMB supported rotor subject to harmonic excitation. J. Appl. Math. Model. 32(7), 1370–1380 (2008)CrossRefMATHGoogle Scholar
  11. 11.
    Zhang, W., Zhan, X.P.: Periodic and chaotic motions of a rotor-active magnetic bearing with quadratic and cubic terms and time-varying stiffness. Nonlinear Dyn. 41(4), 331–359 (2005)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Zhang, W., Yao, M.H., Zhan, X.P.: Multi-pulse chaotic motions of a rotor-active magnetic bearing system with time-varying stiffness. Chaos Solitons Fractals 27(1), 175–186 (2006)CrossRefGoogle Scholar
  13. 13.
    Zhang, W., Zu, J.W., Wang, F.X.: Global bifurcations and chaos for a rotor-active magnetic bearing system with time-varying stiffness. Chaos Solitons Fractals 35(3), 586–608 (2008)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Wang, H., Liu, J.: Stability and bifurcation analysis in a magnetic bearing system with time delays. Chaos Solitons Fractals 26, 813–825 (2005)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Jiang, W., Wang, H., Wei, J.: A study of the singularities for magnetic bearing systems with time delays. Chaos Solitons Fractals 36(3), 715–719 (2008)CrossRefMATHGoogle Scholar
  16. 16.
    Wang, H., Jiang, W.: Multiple stabilities analysis in a magnetic bearing system with time delays. Chaos Solitons Fractals 27(3), 789–799 (2006) Google Scholar
  17. 17.
    Balachandranand, B., Nayfeh, A.H.: Observation of modal interactions inresonantly forced beam-mass. Nonlinear Dyn. 2, 77–117 (1991)CrossRefGoogle Scholar
  18. 18.
    Saeed, N.A., Eissa, M., El-Ganini, W.A.: Nonlinear oscillations of rotor active magnetic bearings system. Nonlinear Dyn. 74, 1–20 (2013)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Sayed, M., Kamel, M.: Stability study and control of helicopter blade flapping vibrations. Appl. Math. Model. 35, 2820–2837 (2011)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973)MATHGoogle Scholar
  21. 21.
    Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981)MATHGoogle Scholar
  22. 22.
    Kevorkian, J.K., Cole, J.D.: Multiple Scale and Singular Perturbation Methods. Spinger, New York (1996)CrossRefMATHGoogle Scholar
  23. 23.
    Nayfeh, A., Mook, D.: Nonlinear Oscillations. Wiley, New York (1995)CrossRefGoogle Scholar
  24. 24.
    Yakowitz, S., Szidarovszky, F.: An Introduction to Numerical Computations, 2nd edn. Macmillan, New York (1990)Google Scholar
  25. 25.
    Isaacson, E., Keller, H.: Analysis of Numerical Method. Dover Edition, New York (1994)Google Scholar
  26. 26.
    Kamel, M., Bauomy, H.S.: Nonlinear study of a rotor-AMB system under simultaneous primary-internal resonance. Appl. Math. Model. 34(10), 2763–2777 (2010)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Engineering Mathematics, Faculty of Electronic EngineeringMenoufia UniversityMenoufEgypt
  2. 2.Faculty of Science in TureifNorthern Border UniversityArarKingdom of Saudi Arabia

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