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Nonlinear Dynamics

, Volume 47, Issue 1–3, pp 235–248 | Cite as

Nonlinear vibration analysis by an extended averaged equation approach

  • Nguyen Dong Anh
  • Ninh Quang Hai
  • Werner Schiehlen
Original Article
  • 90 Downloads

Abstract

The paper presents an extended averaged equation approach to the investigation of nonlinear vibration problems. The proposed method is applied to some free and self-excited oscillators, the Duffing's forced oscillators including main resonance, subharmonic resonance and super harmonic resonance. The results in analyzing the vibration systems with arbitrary non-linearity show advantages of the method.

Keywords

Free oscillation Self-excited oscillation Main resonance Subharmonic resonance Super harmonic resonance Averaged 

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References

  1. 1.
    Nayfeh, A.H.: Perturbation Methods. John Wiley, New York (1973)Google Scholar
  2. 2.
    Bogoliubov, N.N., Mitropolskii, Yu. A.: Asymptotic Methods in the Theory of Nonlinear Oscillations, 4th ed., nauka, Moscow (1974)Google Scholar
  3. 3.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. John Wiley, New York (1979)Google Scholar
  4. 4.
    Schmidt, G., Tondl, A.: Nonlinear Vibration. Cambridge University Press, Cambridge (1986)Google Scholar
  5. 5.
    Roseau, M.: Vibrations in Mechanical Systems. Springer Verlag, Berlin (1989)Google Scholar
  6. 6.
    Rega, G.: Non-linear vibrations of suspended cables. Part I: Modeling and analysis. Appl. Mech. Rev. 57, 443–478 (2004)CrossRefGoogle Scholar
  7. 7.
    Rega, G.: Non-linear vibrations of suspended cables. Part II: Deterministic phenomena. Appl. Mech. Rev. 57, 479–514 (2004)CrossRefGoogle Scholar
  8. 8.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York (1990)Google Scholar
  9. 9.
    Mitropolskii, Yu. A., Nguyen Van Dao, Nguyen Dong Anh.: Nonlinear oscillations in the systems of arbitrary order. “Naukova” (Science), Kiev, (1992) (in Russian).Google Scholar
  10. 10.
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995)MATHGoogle Scholar
  11. 11.
    Lawrence, N. Virgin: Introduction to Experimental Nonlinear Dynamics, A Case Study in Mechanical Vibration. Cambridge University Press, New York (2000)Google Scholar
  12. 12.
    Roberts, J.B., Spanos, P.D.: Random Vibration and Stochastic Linearization. John Wiley, New York (1990)Google Scholar
  13. 13.
    Anh, N.D., Schiehlen, W.: An approach to the problem of closure in the non-linear stochastic Mechanics. Int. J. Meccanica 29, 109–123 (1994)CrossRefGoogle Scholar
  14. 14.
    Wojtkiewicz, S.F., Spencer, B.F., Bergman, L.A.: On the cumulant-neglect closure method in stochastic dynamics. Int. J. Non-Linear Mech. 31(5), 657–684 (1996)CrossRefGoogle Scholar
  15. 15.
    Anh, N.D., Hai, N.Q.: A technique of closure using a polynomial function of gaussian process. Probabilistic Eng. Mech. 15, 191–197 (2000)Google Scholar
  16. 16.
    Anh, N.D., Hai, N.Q.: A Technique for solving nonlinear systems subject to random excitation. IUTAM symposium on recent developments in non-linear oscillations of mechanical systems. Kluwer Academic Publishers, 217–226 (2000)Google Scholar
  17. 17.
    Sanders, J.A., Verhulst, F.: Averaging methods in nonlinear dynamical systems. Springer-Verlag, New York (1985)MATHGoogle Scholar
  18. 18.
    Jack Hale: Theory of functional differential equations. Springer, Verlag, New York, Heidelberg, Berlin (1977)Google Scholar
  19. 19.
    Roy, R.V.: Averaging method for strongly nonlinear oscillations with periodic excitation. Int. J. Nonlinear Mech. 29, 737–753 (1994)CrossRefGoogle Scholar
  20. 20.
    Bogaevsky, V.N., Povzner, A.Y.A.: Algebraic methods in the non-linear theory of perturbations, “ Nauka” (Science), Moscow, 1987 (in Russian).Google Scholar
  21. 21.
    Grigoriu, M.: Applied non-gaussian processes. PTR Prentice Hall, Englewood Cliffs, NJ (1995)Google Scholar
  22. 22.
    Lutes, L.D., Sarkani, S.: Stochastic analysis of structural and mechanics. Prentice-Hall, Englewood Cliffs, NJ (1997)Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Nguyen Dong Anh
    • 1
  • Ninh Quang Hai
    • 2
  • Werner Schiehlen
    • 3
  1. 1.Institute of MechanicsHanoiVietnam
  2. 2.Head of Mathematics DepartmentHanoi Architectural UniversityHanoiVietnam
  3. 3.Institute of Engineering and Computational MechanicsStuttgart UniversityStuttgartGermany

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