Nonlinear Dynamics

, Volume 72, Issue 3, pp 671–682 | Cite as

A comparison of stability computational methods for periodic solution of nonlinear problems with application to rotordynamics

  • Loïc Peletan
  • Sébastien Baguet
  • Mohamed Torkhani
  • Georges Jacquet-Richardet
Original Paper


In this paper a comparative study of five different stability computational methods based on the Floquet theory is presented. These methods are compared in terms of accuracy and CPU performance. Tests are performed on a set of nonlinear problems relevant to rotating machinery with rotor-to-stator contact and a variable number of degrees of freedom, whose periodic solutions are computed with the Harmonic Balance Method (HBM).


Harmonic Balance Method Stability Floquet theory Rotor nonlinear dynamics Rotor-stator contact 



This work was partially supported by the French National Agency (ANR) in the framework of its Technological Research COSINUS program (IRINA, project ANR 09 COSI 008 01 IRINA).


  1. 1.
    Sundararajan, P., Noah, S.T.: An algorithm for response and stability of large order non-linear systems—application to rotor systems. J. Sound Vib. 214(4), 695–723 (1998). doi: 10.1006/jsvi.1998.1614 CrossRefGoogle Scholar
  2. 2.
    Nataraj, C., Nelson, H.D.: Periodic solutions in rotor dynamic systems with nonlinear supports: a general approach. J. Vib. Acoust. 111(2), 187–193 (1989). doi: 10.1115/1.3269840 CrossRefGoogle Scholar
  3. 3.
    Von Groll, G., Ewins, D.J.: The harmonic balance method with arc-length continuation in rotor/stator contact problems. J. Sound Vib. 241(2), 223–233 (2001). doi: 10.1006/jsvi.2000.3298 CrossRefGoogle Scholar
  4. 4.
    Friedmann, P., Hammond, C.E., Woo, T.H.: Efficient numerical treatment of periodic systems with application to stability problems. Int. J. Numer. Methods Eng. 11(7), 1117–1136 (1977). doi: 10.1002/nme.1620110708 MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Friedmann, P.: Numerical methods for determining the stability and response of periodic systems with applications to helicopter rotor dynamics and aeroelasticity. Comput. Math. Appl. 12(1, Part A), 131–148 (1986). doi: 10.1016/0898-1221(86)90091-X CrossRefGoogle Scholar
  6. 6.
    Gaonkar, G.H., Simha Prasad, D.S., Sastry, D.: On computing Floquet transition matrices of rotorcraft. J. Am. Helicopter Soc. 26(3), 56–61 (1981). doi: 10.4050/JAHS.26.56 CrossRefGoogle Scholar
  7. 7.
    Sinha, S., Wu, D.H.: An efficient computational scheme for the analysis of periodic systems. J. Sound Vib. 151(1), 91–117 (1991). doi: 10.1016/0022-460X(91)90654-3 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Wu, D.H., Sinha, S.: A new approach in the analysis of linear systems with periodic coefficients for applications in rotorcraft dynamics. J. R. Aeronaut. Soc. 1, 8–16 (1994) Google Scholar
  9. 9.
    Liaw, C.Y., Koh, C.G.: Dynamic stability and chaos of system with piecewise linear stiffness. J. Eng. Mech. 119, 1542–1558 (1993) CrossRefGoogle Scholar
  10. 10.
    Raghothama, A., Narayanan, S.: Bifurcation and chaos in geared rotor bearing system by incremental harmonic balance method. J. Sound Vib. 226(3), 469–492 (1999). doi: 10.1006/jsvi.1999.2264 CrossRefGoogle Scholar
  11. 11.
    Zheng, T., Hasebe, N.: An efficient analysis of high-order dynamical system with local nonlinearity. J. Vib. Acoust. 121(3), 408–416 (1999). doi: 10.1115/1.2893995 CrossRefGoogle Scholar
  12. 12.
    Kim, Y.B., Noah, S.T.: Quasi-periodic response and stability analysis for a non-linear Jeffcott rotor. J. Sound Vib. 190(2), 239–253 (1996). doi: 10.1006/jsvi.1996.0059 CrossRefGoogle Scholar
  13. 13.
    Shen, J., Lin, K., Chen, S., Sze, K.: Bifurcation and route-to-chaos analyses for Mathieu-duffing oscillator by the incremental harmonic balance method. Nonlinear Dyn. 52(4), 403–414 (2008) MATHCrossRefGoogle Scholar
  14. 14.
    Rook, T.: An alternate method to the alternating time-frequency method. Nonlinear Dyn. 27, 327–339 (2002). doi: 10.1023/A:1015238500024 MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Sinha, S.K.: Dynamic characteristics of a flexible bladed-rotor with coulomb damping due to tip-rub. J. Sound Vib. 273(4–5), 875–919 (2004). doi: 10.1016/S0022-460X(03)00647-3 CrossRefGoogle Scholar
  16. 16.
    Petrov, E.P., Ewins, D.J.: Analytical formulation of friction interface elements for analysis of nonlinear multi-harmonic vibrations of bladed disks. J. Turbomach. 125(2), 364–371 (2003). doi: 10.1115/1.1539868 CrossRefGoogle Scholar
  17. 17.
    Lau, S.L., Cheung, Y.K., Wu, S.Y.: A variable parameter incrementation method for dynamic instability of linear and nonlinear elastic systems. J. Appl. Mech. 49(4), 849–853 (1982). doi: 10.1115/1.3162626 MATHCrossRefGoogle Scholar
  18. 18.
    Cameron, T.M., Griffin, J.H.: An alternating frequency/time domain method for calculating the steady-state response of nonlinear dynamic systems. J. Appl. Mech. 56(1), 149–154 (1989) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Hsu, C.S.: Impulsive parametric excitation: theory. J. Appl. Mech. 39(2), 551–558 (1972) MATHCrossRefGoogle Scholar
  20. 20.
    Cardona, A., Lerusse, A., Géradin, M.: Fast Fourier nonlinear vibration analysis. Comput. Mech. 22(2), 128–142 (1998) MATHCrossRefGoogle Scholar
  21. 21.
    Peeters, M., Viguié, R., Sérandour, G., Kerschen, G., Golinval, J.C.: Nonlinear normal modes, part ii: toward a practical computation using numerical continuation techniques. Mech. Syst. Signal Process. 23(1), 195–216 (2009). doi: 10.1016/j.ymssp.2008.04.003 CrossRefGoogle Scholar
  22. 22.
    Ibrahim, S., Patel, B., Nath, Y.: Modified shooting approach to the non-linear periodic forced response of isotropic/composite curved beams. Int. J. Non-Linear Mech. 44(10), 1073–1084 (2009). doi: 10.1016/j.ijnonlinmec.2009.08.004 CrossRefGoogle Scholar
  23. 23.
    Bauchau, O.A., Nikishkov, Y.G.: An implicit transition matrix approach to stability analysis of flexible multi-body systems. Multibody Syst. Dyn. 5, 279–301 (2001). doi: 10.1023/A:1011488504973 MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Pernot, S., Lamarque, C.H.: A wavelet-Galerkin procedure to investigate time-periodic systems: transient vibration and stability analysis. J. Sound Vib. 245(5), 845–875 (2001). doi: 10.1006/jsvi.2001.3610 CrossRefGoogle Scholar
  25. 25.
    Pernot, S., Lamarque, C.H.: A wavelet-balance method to investigate the vibrations of nonlinear dynamical systems. Nonlinear Dyn. 32, 33–70 (2003). doi: 10.1023/A:1024263917587 MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Lazarus, A., Thomas, O.: A harmonic-based method for computing the stability of periodic solutions of dynamical systems. C. R., Méc. 338(9), 510–517 (2010). doi: 10.1016/j.crme.2010.07.020 MATHCrossRefGoogle Scholar
  27. 27.
    EDF R&D: Code_Aster: a general code for structural dynamics simulation under gnu gpl licence (2001). URL
  28. 28.
    Jones, E., Oliphant, T., Peterson, P.: Scipy: Open source scientific tools for Python. (2001). URL
  29. 29.
    Lalanne, M., Ferraris, G.: Rotordynamics Prediction in Engineering. Wiley, New York (1998) Google Scholar
  30. 30.
    Hahn, E.J., Chen, P.Y.P.: Harmonic balance analysis of general squeeze film damped multidegree-of-freedom rotor bearing systems. J. Tribol. 116(3), 499–507 (1994). doi: 10.1115/1.2928872 CrossRefGoogle Scholar
  31. 31.
    Sarrouy, E., Sinou, J.J.: Advances in Vibration Analysis Research, Chap. Non-Linear Periodic and Quasi-Periodic Vibrations in Mechanical Systems—On the Use of the Harmonic Balance Methods, pp. 419–434. InTech, Rijeka (2011). doi: 10.5772/15638 Google Scholar
  32. 32.
    Gabale, A.P., Sinha, S.: Model reduction of nonlinear systems with external periodic excitations via construction of invariant manifolds. J. Sound Vib. 330(11), 2596–2607 (2011). doi: 10.1016/j.jsv.2010.12.013 CrossRefGoogle Scholar
  33. 33.
    Newmark, N.M.: A method of computation for structural dynamics. J. Eng. Mech. Div. 85(EM3), 2596–2607 (1959) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Loïc Peletan
    • 1
  • Sébastien Baguet
    • 1
  • Mohamed Torkhani
    • 2
  • Georges Jacquet-Richardet
    • 1
  1. 1.CNRS, INSA-Lyon, LaMCoS UMR5259Université de LyonVilleurbanne cedexFrance
  2. 2.LaMSID UMR EDF-CNRS-CEA 2832EDF R&DClamart CedexFrance

Personalised recommendations