Nonlinear Dynamics

, Volume 70, Issue 2, pp 979–998 | Cite as

On non-linear dynamics of a coupled electro-mechanical system

  • Radoslav DarulaEmail author
  • Sergey Sorokin
Original Paper


Electro-mechanical devices are an example of coupled multi-disciplinary weakly non-linear systems. Dynamics of such systems is described in this paper by means of two mutually coupled differential equations. The first one, describing an electrical system, is of the first order and the second one, for mechanical system, is of the second order. The governing equations are coupled via linear and weakly non-linear terms. A classical perturbation method, a method of multiple scales, is used to find a steady-state response of the electro-mechanical system exposed to a harmonic close-resonance mechanical excitation. The results are verified using a numerical model created in MATLAB Simulink environment. Effect of non-linear terms on dynamical response of the coupled system is investigated; the backbone and envelope curves are analyzed. The two phenomena, which exist in the electro-mechanical system: (a) detuning (i.e. a natural frequency variation) and (b) damping (i.e. a decay in the amplitude of vibration), are analyzed further. An applicability range of the mathematical model is assessed.


Non-linear system Method of multiple scales Analytical modeling Electro-mechanical system Backbone curve Envelope curve 



Financial support by InterReg IV A—the Silent Spaces Project is gratefully acknowledged.


  1. 1.
    Behrens, S., Fleming, A.J., Moheimani, S.O.R.: Passive vibration control via electromagnetic shunt damping. IEEE/ASME Trans. Mechatron. 10(1), 118–122 (2005) CrossRefGoogle Scholar
  2. 2.
    Bishop, R. (ed.): The Mechatronics Handbook. CRC Press, Boca Raton (2002) Google Scholar
  3. 3.
    Cheng, D.: Field and Wave Electromagnetics, 2nd edn. Addison-Wesley, Reading (1989) Google Scholar
  4. 4.
    Cheng, T., Oh, I.: Vibration suppression of flexible beam using electromagnetic shunt damper. IEEE Trans. Magn. 45(6), 2758–2761 (2009) CrossRefGoogle Scholar
  5. 5.
    Fitzgerald, A.E., Kingsley, C., Umans, S.D.: Electric Machinery, 6th edn. McGraw-Hill, Boston (2003) Google Scholar
  6. 6.
    Hinch, E.J.: Perturbation Methods. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1991) zbMATHGoogle Scholar
  7. 7.
    João, P., Sadowski, N.: Electromagnetic Modeling by Finite Element Methods, 1st edn. Marcel Dekker, New York (2003) Google Scholar
  8. 8.
    Kovacic, I., Brennan, M.: The Duffing Equation: Nonlinear Oscillators and Their Behaviour. Wiley, Chichester (2011) zbMATHCrossRefGoogle Scholar
  9. 9.
    Krause, P., Wasynczuk, O.: Electromechanical Motion Devices. McGraw-Hill, New York (1989) Google Scholar
  10. 10.
    Kreyszig, E.: Advanced Engineering Mathematics, 9th edn. Wiley, Haboken (2006) Google Scholar
  11. 11.
    Nayfeh, A.H.: Perturbation Methods. Wiley Classics Library, 1st edn. Wiley-VCH, New York (2000) zbMATHCrossRefGoogle Scholar
  12. 12.
    Nayfeh, A.H.: Resolving controversies in the application of the method of multiple scales and the generalized method of averaging. Nonlinear Dyn. 40(1), 61–102 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Paulitsch, C., Gardonio, P., Elliott, S.J.: Active vibration control using an inertial actuator with internal damping. J. Acoust. Soc. Am. 119(4), 2131–2140 (2006) CrossRefGoogle Scholar
  14. 14.
    Rao, S.: Mechanical Vibrations, 4th edn. Prentice Hall, New Jersey (2004) Google Scholar
  15. 15.
    Schmidt, G., Tondl, A.: Non-Linear Vibrations. Cambridge University Press, Cambridge (1986) CrossRefGoogle Scholar
  16. 16.
    Thomsen, J.J.: Vibrations and Stability. Advanced Theory, Analysis, and Tools, 2nd edn. Springer, Berlin (2003) Google Scholar
  17. 17.
    Yeadon, W., Yeadon, A.: Handbook of Small Electric Motors. McGraw-Hill, New York (2001) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of Mechanical and Manufacturing EngineeringAalborg UniversityAalborgDenmark

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