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Meccanica

, Volume 45, Issue 1, pp 7–22 | Cite as

Nonlinear behavior of a rotor-AMB system under multi-parametric excitations

  • M. Kamel
  • H. S. Bauomy
Article

Abstract

A rotor-active magnetic bearing (AMB) system subjected to a periodically time-varying stiffness with quadratic and cubic nonlinearities 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 Runge-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.

Keywords

Rotor-active magnetic bearing Time-varying stiffness Multi-parametric excitations Stability Multiple-valued solutions Jump phenomenon 

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References

  1. 1.
    Zhang W, Zu JW (2003) Nonlinear dynamic analysis for a rotor-active magnetic bearing system with time-varying stiffness. Part I: Formulation and local bifurcation. In: Proceedings of 2003 ASME international mechanical engineering congress and exposition, Washington (DC), November 16–21, 2003. ASME, New York, pp 631–640 Google Scholar
  2. 2.
    Zhang W, Yao MH, Zhan XP (2006) Multi-pulse chaotic motions of a rotor-active magnetic bearing system with time-varying stiffness. Chaos, Solitons Fractals 27:175–186 CrossRefGoogle Scholar
  3. 3.
    Zhang W, Zhan XP (2005) Periodic and chaotic motions of a rotor-active magnetic bearing with quadratic and cubic terms and time-varying stiffness. Nonlinear Dyn 41:331–59 MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Zhang W, Zu JW, Wang FX (2008) Global bifurcations and chaos for a rotor-active magnetic bearing system with time-varying stiffness. Chaos, Solitons Fractals 35:586–608 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Zhang W, Zu JW (2008) Transient and steady nonlinear response for a rotor-active magnetic bearings system with time-varying stiffness. Chaos, Solitons Fractals 38:1152–1167 CrossRefMathSciNetGoogle Scholar
  6. 6.
    Ji JC, Yu L, Leung AYT (2000) Bifurcation behavior of a rotor by active magnetic bearings. J Sound Vib 235:133–151 CrossRefADSGoogle Scholar
  7. 7.
    Ji JC, Hansen CH (2001) Non-linear oscillations of a rotor in active magnetic bearings. J Sound Vib 240:599–612 CrossRefADSGoogle Scholar
  8. 8.
    Ji JC, Leung AYT (2003) Non-linear oscillations of a rotor-magnetic bearing system under super harmonic resonance conditions. Int J Non-linear Mech 38:829–835 MATHCrossRefGoogle Scholar
  9. 9.
    Jang MJ, Chen CL, Tsao M (2005) Sliding mode control for active magnetic bearing system with flexible rotor. J Franklin Inst 342:401–419 MATHCrossRefGoogle Scholar
  10. 10.
    Inayat-Hussain JI (2007) Chaos via torus breakdown in the vibration response of a rigid rotor supported by active magnetic bearings. Chaos, Solitons Fractals 31:912–927 CrossRefGoogle Scholar
  11. 11.
    Zhu C, Robb DA, Ewin DJ (2003) The dynamics of a cracked rotor with an active magnetic bearing. J Sound Vib 265(3):469–487 CrossRefADSGoogle Scholar
  12. 12.
    Malvano R, Vatta F, Vigliani A (2001) Rotordynamic coefficients for labyrinth gas seals: single control volume model. Meccanica 36:731–744 MATHCrossRefGoogle Scholar
  13. 13.
    Vatta F, Vigliani A (2007) Asymmetric rotating shafts: an alternative analytical approach. Meccanica 42:207–210 MATHCrossRefGoogle Scholar
  14. 14.
    Francesco S (2009) Rotor whirl damping by dry friction suspension systems. Meccanica 43:577–589 Google Scholar
  15. 15.
    Zhang W, Li J (2001) Global analysis for a nonlinear vibration absorber with fast and slow modes. Int J Bifurc Chaos 11:2179–2194 CrossRefGoogle Scholar
  16. 16.
    Zhang W, Tang Y (2002) Global dynamics of the cable under combined parametrical and external excitations. Int J Non-linear Mech 37:505–526 MATHCrossRefGoogle Scholar
  17. 17.
    Amer YA, Hegazy UH (2007) Resonance behavior of a rotor-active magnetic bearing with time-varying stiffness. Chaos, Solitons Fractals 34:1328–1345 CrossRefGoogle Scholar
  18. 18.
    Eissa M, Amer YA, Hegazy UH, Sabbah AS (2006) Dynamic behavior of an AMB/supported rotor subject to parametric excitation. ASME J Vib Acoust 182:646–652 Google Scholar
  19. 19.
    Eissa M, Hegazy UH, Amer YA (2008) A time-varying stiffness rotor-active magnetic bearings under combined resonance. J Appl Mech 75:1–12 Google Scholar
  20. 20.
    Eissa M, Hegazy UH, Amer YA (2008) Dynamic behavior of an AMB supported rotor subject to harmonic excitation. Appl Math Model 32:1370–1380 MATHCrossRefGoogle Scholar
  21. 21.
    Amer YA, Hegazy UH (2008) A time-varying stiffness rotor-active magnetic bearings under parametric excitation. J Mech Eng Sci Part C 223:447–458 Google Scholar
  22. 22.
    Nayfeh AH (1991) Introduction to perturbation techniques. Wiley-Interscience, New York Google Scholar
  23. 23.
    Kevorkian J, Cole JD (1996) Multiple scale and singular perturbation methods. Springer, New York MATHGoogle Scholar
  24. 24.
    Yakowitz S, Szidaouszky F (1992) An introduction to numerical computation. Macmillan, New York Google Scholar
  25. 25.
    Isaacson E, Keller H (1994) Analysis of numerical methods. Dover, New York Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Engineering Mathematics, Faculty of Electronic EngineeringMenoufia UniversityMenoufEgypt
  2. 2.Department of Mathematics, Faculty of ScienceZagazig UniversityZagazigEgypt

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