Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

Nonlinear Dynamics Analysis of a Continuum Rotor through Generalized Harmonic Balance Method

Journal of Applied Nonlinear Dynamics 5(1) (2016) 1--31 | DOI:10.5890/JAND.2016.03.001

Haiyang Luo$^{1}$,$^{2}$; Yuefang Wang$^{1}$,$^{2}$

$^{1}$ Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China

$^{2}$ State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian 116024, China

Download Full Text PDF



The dynamics of a continuum rotor system excited by nonlinear oilfilm force and electromagnetic force is investigated through the Generalized Harmonic Balance Method. The governing differential equation of the continuum rotor system is reduced through the Galerkin’s approximation method. Firstly, the motion in Y-direction is considered as the response of a planar continuum rotor. The analytical periodic motion is obtained and the bifurcation and stability is determined. Secondly, the dynamical model of a spatial continuum rotor is studied. The stability and bifurcation of the analytical periodic motion is obtained and compared to numerical results to show the accuracy of the Generalized Harmonic Balance Method.


The authors are grateful for supports from Free Exploration Project of State Key Laboratory of Structural Analysis for Industrial Equipment (Grant S14204), Liaoning Provincial Program for Science and Technology (Grants 201303002, 2014028004), and the State Key Development Program for Basic Research of China (Grant 2015CB057300).


  1. [1]  Yamamoto, T., Ishida, Y. and Ikeda, T. (1982), Non-linear forced-oscillations of a rotating shaft carrying an unsymmetrical rotor at the major critical speed, Bulletin of the JSME-JAPAN Society of Mechanical Engineers, 25(210), 1969-1976.
  2. [2]  Shaw, S. W. (1988), Chaotic dynamics of a slender beam rotating about its longitudinal axis, Journal of Sound and Vibration, 124(2), 329-343.
  3. [3]  Ishida, Y. Nagasaka, I. Inoue, T. and Lee, S. W. (1996), Forced oscillations of a vertical continuous rotor with geometric nonlinearity, Nonlinear Dynamics, 11(2), 107-120.
  4. [4]  Luczko, A.(2002), A geometrically non-linear model of rotating shafts with internal resonance and self-excited vibration, Journal of Sound and Vibration, 255(3), 433-456.
  5. [5]  Chasalevris, A. C., Papadopoulos, C. A. (2009), A continuous model approach for cross -coupled bending vibrations of a rotor-bearing system with a transverse breathing crack, Mechanism and Machine Theory, 44(6), 1176-1191.
  6. [6]  Khanlo, M., Ghayour, M. and Ziaer-Rad, S. (2011), Chaotic vibration analysis of rotating, flexible, continuous shaft-disk system with a rub-impact between the disk and the stator, Communications in Nonlinear Science and Numerical Simulation, 16(1), 566-582.
  7. [7]  Khadem, S. E., Shahgholi, M. and Hosseini, S. A. A. (2011), Two-mode combination resonances of an inextensional rotating shaft with large amplitude, Nonlinear Dynamics, 65(3), 217-233.
  8. [8]  Hosseini S. A. A. (2013), Dynamic stability and bifurcation of a nonlinear in-extensional rotating shaft with internal damping, Nonlinear Dynamics, 74(1-2), 345-358.
  9. [9]  Guo, D., Chu, F. and Chen, D. (2002), The unbalanced magnetic pull and its effects on vibration in a three-phase generator with eccentric rotor, Journal of Sound and Vibration, 254(2), 297-312.
  10. [10]  Wang, Y. F., Huang, L. H. and Li, Y. (2007), Nonlinear vibration and stability of a Jeffcott rotor under unbalanced magnetic excitation, International Journal of Nonlinear Sciences and Numerical Simulation, 8(3), 375-384.
  11. [11]  Muszynska, A. (2005), Rotordynamics, Taylor & Francis: Boca Raton.
  12. [12]  Szeri, A. (2010), Fluid film lubrication, Cambridge University Press: Cambridge, UK.
  13. [13]  Zhao, J. Y., Linnett, I. W. and Mclean, L. J. (1994), Stability and bifurcation of unbalanced response of a squeeze film damped flexible rotor, Journal of Tribology, 116(2), 361-368.
  14. [14]  Pagano, S. Rocca, E. and Russo, R. (1995), Dynamic behaviour of tilting-pad journal bearings, Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology, 209(4), 275-285.
  15. [15]  Adiletta, G., Guido, A. R. and Rossi, C. (1996), Chaotic motions of a rigid rotor in short journal bearing, Nonlinear Dynamics, 10(3), 251-269.
  16. [16]  Zheng, T. and Hasebe, N. (2000), Nonlinear dynamic behaviors of a complex rotor-bearing system, Journal of Applied Mechanics, 67(3), 485-495.
  17. [17]  Chang-Jian, C. W. and Chen, C. K. (2007), Chaos and bifurcation of a flexible rub-impact rotor supported by oil film bearings with nonlinear suspension, Mechanism and Machine Theory, 42(3), 312-333.
  18. [18]  Hwang, J. L. and Shiau, T. N. (1991), An application of the generalized polynomial expansion method to nonlinear rotor bearing systems, Journal of Vibration and Acoustics, 113(3), 299-308.
  19. [19]  Kim, Y. B. and Noah, S. T. (1996), Quasi-periodic response and stability analysis for a non-linear Jeffcott rotor, Journal of Sound and Vibration, 190(2), 239-253.
  20. [20]  Qin, W. Y., Chen, G. R. and Ren, X. M. (2004), Grazing bifurcation in the response of cracked Jeffcott rotor, Nonlinear Dynamics, 35(2), 147-157.
  21. [21]  Villa, C. V. S., Sinou, J. J. and Thouverez, F. (2005), The invariant manifold approach applied to nonlinear dynamics of a rotor-bearing system, European Journal of Mechanics –A/Solids, 24(4), 676-689.
  22. [22]  Tong, H. Z., Yang, F. H. and Chen, L. H. (2009), Global bifurcations for a rotor-active magnetic bearings system, Industrial Engineering and Engineering Management, 2009. IEEM 2009. IEEE International Conference on. IEEE, 2124-2127.
  23. [23]  Ding, Q. and Zhang, K. P. (2012), Order reduction and nonlinear behaviors of a continuous rotor system, Nonlinear Dynamics, 67(1), 251-262.
  24. [24]  Luo, H. Y. and Wang, Y. F. (2012), Nonlinear vibration of a continuum rotor with transverse electromagnetic and bearing excitations, Shock and Vibration, 19(6), 1297-1314.
  25. [25]  Paez-Chavez, J. and Wiercigroch, M. (2013), Bifurcation analysis of periodic orbits of a non-smooth Jeffcott rotor model, Communica in Nonlinear Science and Numerical Simulation, 18(9), 2571-2580.
  26. [26]  Luo, A. C. J. (2012), Continuous dynamical systems, Higher Education Press: Beijing.
  27. [27]  Huang, J. Z. and Luo, A. C. J. (2015), Periodic motions and bifurcation trees in a buckled, nonlinear Jeffcott rotor system, International Journal of Bifurcation and Chaos, 25(1), 155002-1-155002-34.
  28. [28]  Meirovitch, L. (2001), Fundamentals of Vibrations, McGraw-Hill: Singapore.
  29. [29]  Wan, F. Y., Xu, Q. Y. and Li, S. T. (2004), Vibration analysis of cracked rotor sliding bearing system with rotor–stator rubbing by harmonic wavelet transform, Journal of Sound and Vibration, 271(3-5), 507-518.