, Volume 54, Issue 1–2, pp 311–320 | Cite as

Attenuation of Sommerfeld effect in an internally damped eccentric shaft-disk system via active magnetic bearings

  • Abhishek Kumar Jha
  • Sovan Sundar DasguptaEmail author


Eccentric shaft-disk system with internal damping driven by a non-ideal power source exhibits Sommerfeld effect characterized by nonlinear jump phenomena of amplitude and rotor speed upon exceeding a critical power input around the critical speed. This effect causes instability in high speed rotors. So the diminution of such effect is extremely important in order to smooth running of the rotors at high speeds. The aim of this paper is to attenuate the Sommerfeld effect of an internally damped unbalanced flexible shaft-disk system via linearized active magnetic bearings. The shaft-disk system is excited through a brushed DC motor which acts as a non-ideal energy source. The characteristic equation of fifth order polynomial in rotor speed is obtained through energy balance of supplied power and the mechanical power dissipated at steady-state condition. Using MATLAB simulations, amplitude frequency responses are obtained close to system resonance for several values of bias current of active magnetic bearings. Thus the Sommerfeld effect is found to be attenuated as bias current increases gradually. The complete disappearance of Sommerfeld effect is also reported when the bias current reaches a specific value under certain conditions. A few numerical results are validated with established results when bias current is made to zero.


Sommerfeld effect AMB system Non-ideal source Internal damping Shaft-disk system PD controller 



This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Sommerfeld A (1902) Beiträge zum dynamischen ausbau der festigkeitslehe. Phys Z 3:266–286zbMATHGoogle Scholar
  2. 2.
    Timoshenko SP (1955) Vibration problems in engineering. Princeton, Hew JerseyzbMATHGoogle Scholar
  3. 3.
    Blekhman II (2000) Vibrational mechanics: nonlinear dynamic effects, general approach, applications. World Scientific, SingaporeCrossRefGoogle Scholar
  4. 4.
    Kononenko VO (1964) Vibrating systems with limited excitation. Nauka, Moscow (in Russian) Google Scholar
  5. 5.
    Nayfeh AH, Mook DT (1979) Nonlinear oscillations. Wiley, New YorkzbMATHGoogle Scholar
  6. 6.
    Ogly Alifov AA, Frolov KV (1990) Interaction of non-linear oscillatory systems with energy sources. Taylor & Francis, LondonGoogle Scholar
  7. 7.
    Felix JL, Balthazar JM (2009) Comments on a nonlinear and non-ideal electromechanical damping vibration absorber, Sommerfeld effect and energy transfer. Nonlinear Dyn 55:1–11CrossRefzbMATHGoogle Scholar
  8. 8.
    Bolla MR, Balthazar JM, Felix JL, Mook DT (2007) On an approximate analytical solution to a nonlinear vibrating problem, excited by a nonideal motor. Nonlinear Dyn 50(4):841–847CrossRefzbMATHGoogle Scholar
  9. 9.
    Warminski J, Balthazar JM, Brasil RM (2001) Vibrations of a non-ideal parametrically and self-excited model. J Sound Vib 245:363–374ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    Felix JL, Balthazar JM, Brasil RM (2005) On tuned liquid column dampers mounted on a structural frame under a non-ideal excitation. J Sound Vib 282:1285–1292ADSCrossRefGoogle Scholar
  11. 11.
    Balthazar JM, Mook DT, Weber HI, Brasil RM, Fenili A, Belato D, Felix JL (2003) An overview on non-ideal vibrations. Meccanica 38(6):613–621CrossRefzbMATHGoogle Scholar
  12. 12.
    Dimentberg MF, McGovern L, Norton RL, Chapdelaine J, Harrison R (1997) Dynamics of an unbalanced shaft interacting with a limited power supply. Nonlinear Dyn 13(2):171–187CrossRefzbMATHGoogle Scholar
  13. 13.
    Samantaray AK, Dasgupta SS, Bhattacharyya R (2010) Sommerfeld effect in rotationally symmetric planar dynamical systems. Int J Eng Sci 48(1):21–36CrossRefGoogle Scholar
  14. 14.
    Samantaray AK (2009) On the non-linear phenomena due to source loading in rotor—motor systems. Proc Inst Mech Eng Part C J Mech Eng Sci 223(4):809–818CrossRefGoogle Scholar
  15. 15.
    Dasgupta SS, Samantaray AK, Bhattacharyya R (2010) Stability of an internally damped non-ideal flexible spinning shaft. Int J Non-Linear Mech 45(3):286–293CrossRefGoogle Scholar
  16. 16.
    Samantaray AK, Dasgupta SS, Bhattacharyya R (2010) Bond graph modelling of an internally damped non-ideal flexible spinning shaft. J Dyn Syst Measur Control 132(6):061502–061509CrossRefGoogle Scholar
  17. 17.
    Dasgupta SS, Rajamohan V (2017) Dynamic characterization of a flexible internally damped spinning shaft with constant eccentricity. Arch Appl Mech 87(10):1769–1779CrossRefGoogle Scholar
  18. 18.
    Dasgupta SS, Rajan JA (2018) Steady-state and transient responses of a flexible eccentric spinning Shaft. FME Trans 46(1):133–137CrossRefGoogle Scholar
  19. 19.
    Belato D (1998) Nao-linearidades no Eletro Pêndulo Doctoral dissertation. MSc Dissertation, State University of Campinas, BrazilGoogle Scholar
  20. 20.
    Felix JL, Balthazar JM, Brasil RM, Pontes BR (2009) On lugre friction model to mitigate non-ideal vibrations. J Comput Nonlinear Dyn 4(3):034503CrossRefGoogle Scholar
  21. 21.
    Castão KA, Goes LC, Balthazar JM (2011) A note on the attenuation of the sommerfeld effect of a non-ideal system taking into account a MR damper and the complete model of a DC motor. J Vib Control 17(7):1112–1118CrossRefzbMATHGoogle Scholar
  22. 22.
    Piccirillo V, Tusset AM, Balthazar JM (2014) Dynamical jump attenuation in a non-ideal system through a magnetorheological damper. J Theor Appl Mech 52(3):595–604Google Scholar
  23. 23.
    Maslen EH, Schweitzer G (eds) (2009) Magnetic bearings: theory, design, and application to rotating machinery. Springer, BerlinGoogle Scholar
  24. 24.
    Ji JC, Hansen CH (2001) Non-linear oscillations of a rotor in active magnetic bearings. J Sound Vib 240(4):599–612ADSCrossRefGoogle Scholar
  25. 25.
    Ji JC, Leung AY (2003) Non-linear oscillations of a rotor-magnetic bearing system under superharmonic resonance conditions. Int J Non-Linear Mech 38(6):829–835CrossRefzbMATHGoogle Scholar
  26. 26.
    Zhang W, Yao MH, Zhan XP (2006) Multi-pulse chaotic motions of a rotor-active magnetic bearing system with time-varying stiffness. Chaos Solitons Fract 27(1):175–186ADSCrossRefzbMATHGoogle Scholar
  27. 27.
    Eissa MH, Hegazy UH, Amer YA (2008) Dynamic behavior of an AMB supported rotor subject to harmonic excitation. Appl Math Model 32(7):1370–1380CrossRefzbMATHGoogle Scholar
  28. 28.
    Kamel M, Bauomy HS (2010) Nonlinear behavior of a rotor-AMB system under multi-parametric excitations. Meccanica 45(1):7–22CrossRefzbMATHGoogle Scholar
  29. 29.
    Bauomy HS (2012) Stability analysis of a rotor-AMB system with time varying stiffness. J Frankl Inst 349(5):1871–1890MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Saeed NA, Kamel M (2016) Nonlinear PD-controller to suppress the nonlinear oscillations of horizontally supported Jeffcott-rotor system. Int J Non-Linear Mech 87:109–124CrossRefGoogle Scholar
  31. 31.
    Wu R, Zhang W, Yao MH (2017) Nonlinear vibration of a rotor-active magnetic bearing system with 16-pole legs. In: ASME 2017 international design engineering technical conferences and computers and information in engineering conference 2017 Aug 6. American Society of Mechanical Engineers, pp V006T10A037–V006T10A037Google Scholar
  32. 32.
    Ran S, Hu Y, Wu H (2018) Design, modeling, and robust control of the flexible rotor to pass the first bending critical speed with active magnetic bearing. Adv Mech Eng 10(2):1687814018757536CrossRefGoogle Scholar
  33. 33.
    Jung D, DeSmidt H (2018) A new hybrid observer based rotor imbalance vibration control via passive autobalancer and active bearing actuation. J Sound Vib 415:1–24ADSCrossRefGoogle Scholar
  34. 34.
    Genta G (2007) Dynamics of rotating systems. Springer, BerlinGoogle Scholar
  35. 35.
    Dasgupta SS (2011) Sommerfeld effect in internally damped shaftrotor systems. PhD dissertation, IIT KharagpurGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mechanical EngineeringVITVelloreIndia

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