Nonlinear Dynamics

, Volume 51, Issue 4, pp 529–539 | Cite as

Efficient computation of quasiperiodic oscillations in nonlinear systems with fast rotating parts

  • Frank Schilder
  • Jan Rübel
  • Jens Starke
  • Hinke M. Osinga
  • Bernd Krauskopf
  • Mizuho Inagaki
Original Paper


We present a numerical method for the investigation of quasiperiodic oscillations in applications modeled by systems of ordinary differential equations. We focus on systems with parts that have a significant rotational speed. An important element of our approach is that it allows us to verify whether one can neglect gravitational forces after a change of coordinates into a corotating frame. Specifically, we show that this leads to a dramatic reduction of computational effort. As a practical example, we study a turbocharger model for which we give a thorough comparison of results for a model with and without the inclusion of gravitational forces.


Invariant tori Noise Oil-whirl Quasiperiodic oscillation Rotordynamics Turbocharger Unbalance oscillation Vibration 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bonello, P., Brennan, M.J., Holmes, R.: An investigation into the non-linear dynamics of an unbalanced flexible rotor running in an unsupported squeeze film damper bearing. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 217, 955–971 (2003)CrossRefGoogle Scholar
  2. 2.
    Childs, D.: Turbomachinery Rotordynamics. Wiley, New York (1993)Google Scholar
  3. 3.
    Crandall, S.H.: Rotordynamics. In Kliemann, W., et al. (eds.) Nonlinear Dynamics and Stochastic Mechanics. Dedicated to Prof. S. T. Ariaratnam on the Occasion of his Sixtieth Birthday, CRC Mathematical Modelling Series, pp. 3–44. CRC, Boca Raton, FL (1995)Google Scholar
  4. 4.
    Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Yu.A., Sandstede, B., Wang, X.: Auto97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont). Technical report, Concordia University (1997), available at: URL: http://cmvl.
  5. 5.
    Gasch, R., Nordmann, R., Pfützner, H.: Rotordynamik, 2nd edn. Springer, Berlin Heidelberg New York (2002)Google Scholar
  6. 6.
    Ge, T., Leung, A.Y.T.: Construction of invariant torus using Toeplitz Jacobian matrices/fast Fourier transform approach. Nonlinear Dyn. 15(3), 283–305 (1998)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Glazier, J.A., Libchaber, A.: Quasi-periodicity and dynamical systems: an experimentalist's view. IEEE Trans. Circuits Syst. 35(7), 790–809 (1988)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Hayashi, Ch.: Nonlinear Oscillations in Physical Systems, McGraw-Hill Electrical and Electronic Engineering Series. McGraw-Hill, New York (1964)Google Scholar
  9. 9.
    Holt, C., San Andres, L., Sahay, S., Tang, P., La Rue, G., Gjika, K.: Test response and nonlinear analysis of a turbocharger supported on floating ring bearings. ASME J. Vib. Acoust. 127, 107–115 (2005)CrossRefGoogle Scholar
  10. 10.
    Kijimoto, Sh., Matsuda, K., Kanemitsu, Y.: Stability-optimized clearance configuration of fluid-film journal bearings. In: Proceedings of ASME International Design Engineering Technical Conference & Computers and Information in Engineering Conference, 24–28 September, DETC2005-84671, Long Beach, CA (2005)Google Scholar
  11. 11.
    Kuznetsov, Yu.A.: Elements of Applied Bifurcation Theory, Vol. 112, Applied Mathematical Sciences, 3rd edn. Springer-Verlag, New York (2004)Google Scholar
  12. 12.
    Meirovitch, L.: Elements of Vibration Analysis, 2nd edn. Wiley, New York (1986)Google Scholar
  13. 13.
    Muszynska, A.: Whirl and whip — rotor/bearing stability problems. J. Sound Vib. 110(3), 443–462 (1986)CrossRefGoogle Scholar
  14. 14.
    Muszynska, A.: Tracking the mystery of oil whirl. Sound Vib. 21(2), 8–12 (1987)Google Scholar
  15. 15.
    Muszynska, A.: Stability of whirl and whip in rotor/bearing systems. J. Sound Vib. 127(1), 49–64 (1988)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Nayfeh, A.H.: Nonlinear Interactions. Wiley-Interscience, New York (2000)MATHGoogle Scholar
  17. 17.
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995)MATHGoogle Scholar
  18. 18.
    Nelson, H.D., McVaugh, J.M.: The dynamics of rotor-bearing systems using finite elements. ASME J. Eng. Ind. 98(2), 593–600 (1976)Google Scholar
  19. 19.
    Newkirk, E.L., Taylor, H.D.: Shaft whirling due to oil action in journal bearings. Gen. Electr. Rev. 28(7), 559–568 (1925)Google Scholar
  20. 20.
    Schilder, F., Osinga, H.M., Vogt, W.: Continuation of quasi-periodic invariant tori. SIAM J. Appl. Dyn. Syst. 4(3), 459–488 (2005) (electronic)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Schilder, F., Peckham, B.B.: Computing Arnol′d tongue scenarios. J. Comput. Phys. 220(2), 932–951 (2007)Google Scholar
  22. 22.
    Schilder, F., Vogt, W., Schreiber, S., Osinga, H.M.: Fourier methods for quasi-periodic oscillations. Int. J. Numer. Methods Eng. 67(5), 629–671 (2006)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Cambridge University Press, Perseus Publishing, New York (2000)Google Scholar
  24. 24.
    Vance, J.M.: Rotordynamics of Turbomachinery. Wiley, New York (1988)Google Scholar
  25. 25.
    Wagner, B.B., Ginsberg, J.H.: The effect of bearing properties on the eigenvalues of a rotordynamic system. In: Proceedings of ASME International Design Engineering Technical Conference & Computers and Information in Engineering Conference, 24–28 September, DETC2005-84787, Long Beach, CA (2005)Google Scholar
  26. 26.
    Wang, C.C.: Nonlinear dynamic behavior and bifurcation analysis of a rigid rotor supported by a relatively short externally pressurized porous gas journal bearing system. Acta Mech. 183(1–2), 41–60 (2006)MATHCrossRefGoogle Scholar
  27. 27.
    Yamamoto, T., Ishida, Y.: Linear and Nonlinear Rotordynamics. Wiley, New York (2001)Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • Frank Schilder
    • 1
  • Jan Rübel
    • 2
  • Jens Starke
    • 3
  • Hinke M. Osinga
    • 1
  • Bernd Krauskopf
    • 1
  • Mizuho Inagaki
    • 4
  1. 1.Bristol Center for Applied Nonlinear Mathematics, Department of Engineering MathematicsUniversity of BristolBristolUK
  2. 2.Interdisciplinary Center for Scientific Computing (IWR) and Institute of Applied MathematicsUniversity of HeidelbergHeidelbergGermany
  3. 3.Department of MathematicsTechnical University of DenmarkKongens LyngbyDenmark
  4. 4.Toyota Central Research and Development Laboratories, Inc.NagakuteJapan

Personalised recommendations