Advertisement

Nonlinear Dynamics

, Volume 32, Issue 1, pp 1–13 | Cite as

A New Method for Approximate Analytical Solutions to Nonlinear Oscillations of Nonnatural Systems

  • B. S. Wu
  • C. W. Lim
  • L. H. He
Article

Abstract

This paper deals with nonlinear oscillations of a conservative,nonnatural, single-degree-of-freedom system with odd nonlinearity. Bycombining the linearization of the governing equation with the method ofharmonic balance, we establish approximate analytical solutions for thenonlinear oscillations of the system. Unlike the classical harmonicbalance method, the linearization is performed prior to proceeding withharmonic balancing thus resulting in linear algebraic equations insteadof nonlinear algebraic equations. Hence, we are able to establish theapproximate analytical formulas for the exact period and periodicsolution. These approximate solutions are valid for small as well aslarge amplitudes of oscillation. Two examples are presented toillustrate that the proposed formulas can give excellent approximateresults.

nonnatural system nonlinear oscillation odd nonlinearity linearization harmonic balance method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.Google Scholar
  2. 2.
    Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley, New York, 1981.Google Scholar
  3. 3.
    Nayfeh, A. H., Problems in Perturbation, Wiley, New York, 1985.Google Scholar
  4. 4.
    Schmidt, G. and Tondl, A., Non-Linear Vibrations, Cambridge University Press, Cambridge, 1986.Google Scholar
  5. 5.
    Hagedorn, P., Non-Linear Oscillations, Clarendon, Oxford, 1988 [translated by Wolfram Stadler].Google Scholar
  6. 6.
    Mickens, R. E., Oscillations in Planar Dynamic Systems, World Scientific, Singapore, 1996.Google Scholar
  7. 7.
    Landa, P. S., Regular and Chaotic Oscillations, Springer, Berlin, 2001.Google Scholar
  8. 8.
    Cheung, Y. K., Chen, S. H., and Lau, S. L., 'A modified Lindstedt-Poincaré method for certain strongly nonlinear oscillators', International Journal of Non-Linear Mechanics 26, 1991, 367–378.Google Scholar
  9. 9.
    Wu, B. and Zhong, H., 'Summation of perturbation solutions to nonlinear oscillations', Acta Mechanica 154, 2002, 121–127.Google Scholar
  10. 10.
    Mickens, R. E., 'Comments on the method of harmonic balance', Journal of Sound and Vibration 94, 1984, 456–460.Google Scholar
  11. 11.
    Delamotte, B., 'Nonperturbative method for solving differential equations and finding limit cycles', Physical Review Letters 70, 1993, 3361–3364.Google Scholar
  12. 12.
    Liao, S. J. and Chwang, A. T., 'Application of homotopy analysis method in nonlinear oscillations', ASME Journal of Applied Mechanics 65, 1998, 914–922.Google Scholar
  13. 13.
    Wu, B. S. and Li, P. S., 'A method for obtaining approximate analytic periods for a class of nonlinear oscillators', Meccanica 36, 2001, 167–176.Google Scholar
  14. 14.
    Wu, B. and Li, P., 'A new approach to nonlinear oscillations', ASME Journal of Applied Mechanics 68, 2001, 951–952.Google Scholar
  15. 15.
    Lim, C. W., Wu, B. S., and He, L. H., 'A new approximate analytical approach for dispersion relation of the nonlinear Klein-Gordon equation', Chaos 11, 2001, 843–848.Google Scholar
  16. 16.
    Wu, B. S., Lim, C. W., and Ma, Y. F., 'Analytical approximation to oscillation of a nonlinear conservative system', International Journal of Non-Linear Mechanics 38, 2002, 1037–1043.Google Scholar
  17. 17.
    Lim, C. W. and Wu, B. S., 'A new analytical approach to the Duffing-harmonic oscillator', Physics Letters A, 2003, in press.Google Scholar
  18. 18.
    Lim, C. W. and Wu, B. S., 'A modified Mickens procedure for certain nonlinear oscillators', Journal of Sound and Vibration 251, 2002, 202–206.Google Scholar
  19. 19.
    Meirovitch, L., Methods of Analytical Dynamics, McGraw-Hill, New York, 1970.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • B. S. Wu
    • 1
  • C. W. Lim
    • 2
  • L. H. He
    • 3
  1. 1.Department of MathematicsJilin UniversityChangchunP.R. China
  2. 2.Department of Building and ConstructionCity University of Hong KongKowloonHong Kong, P.R. China
  3. 3.Department of Modern MechanicsUniversity of Science and Technology of ChinaHefeiP.R. China

Personalised recommendations