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Nonlinear Dynamics

, Volume 26, Issue 1, pp 87–104 | Cite as

Nonlinear Vibration of Plane Structures by Finite Element and Incremental Harmonic Balance Method

  • S. H. Chen
  • Y. K. Cheung
  • H. X. Xing
Article

Abstract

A nonlinear steady state vibration analysis of a wide class of planestructures is analyzed. Both the finite element method and incrementalharmonic balance method are used. The usual beam element is adopted inwhich the nonlinear effect arising from longitudinal stretching has beentaken into account. Based on the geometric nonlinear finite elementanalysis, the nonlinear dynamic equations including quadratic and cubicnonlinearities are derived. These equations are solved by theincremental harmonic balance (IHB) method. To show the effectiveness andversatility of this method, some typical examples for a wide variety ofvibration problems including fundamental resonance, super- andsub-harmonic resonance, and combination resonance of plane structuressuch as beams, shallow arches and frames are computed. Most of theseexamples have not been studied by other researchers before. Comparisonwith previous results are also made.

nonlinear vibration finite element method incremental harmonic balance method 

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References

  1. 1.
    Woinowsky-Krieger, S., ‘The effect of an axial force on the vibration of hinged bars’, ASME, Journal of Applied Mechanics 17, 1950, 35–36.Google Scholar
  2. 2.
    Evensen, D. A., ‘Nonlinear vibrations of beams with various boundary conditions’, AIAA Journal 6, 1968, 370–372.Google Scholar
  3. 3.
    Tseng, W. Y. and Dugundji, J., ‘Nonlinear vibrations of a beam under harmonic excitation’, ASME, Journal of Applied Mechanics 37, 1970, 292–297.Google Scholar
  4. 4.
    Bennett, J. A. and Eisley, J. G., ‘A multiple degree-of-freedom approach to nonlinear beam vibrations’, AIAA Journal 8, 1970, 734–739.Google Scholar
  5. 5.
    Prathap, G. and Varadan, T. K., ‘The large amplitude vibration of hinged beams’, Journal of Sound and Vibration 58, 1978, 87–94.Google Scholar
  6. 6.
    Prathap, G. and Varadan, T. K., ‘The large amplitude vibration of tapered clamped beams’, Computer & Structures 9, 1978, 219–222.Google Scholar
  7. 7.
    Lewanodowski, R., ‘Application of the Ritz method to the analysis of non-linear free vibrations of beams’, Journal of Sound and Vibration 114, 1987, 91–101.Google Scholar
  8. 8.
    Mei, C., ‘Nonlinear vibrations of beams by matrix displacement method’, AIAA Journal 10, 1972, 355–357.Google Scholar
  9. 9.
    Mei, C., ‘Finite element displacement method for large amplitude free flexural oscillations of beams and plates’, Computers & Structures 3, 1973, 163–174.Google Scholar
  10. 10.
    Rao, G. V., Raju, K. K., and Raju, I. S., ‘Finite element formulation for the large amplitude free vibrations of beams and orthotropic plates’, Computer & Structures 7, 1976, 169–172.Google Scholar
  11. 11.
    Raju, K. K., Sastry, B. P., and Rao, G. V., ‘A finite element formulation for the large amplitude vibration of tapered beams’, Journal of Sound and Vibration 47, 1976, 595–598.Google Scholar
  12. 12.
    Cheung, Y. K. and Lau, S. L., ‘Incremental time-space finite strip method for nonlinear structural vibrations’, Earthquake Engineering and Structural Dynamics 10, 1982, 239–253.Google Scholar
  13. 13.
    Sarma, B. S. and Varadan, T. K., ‘Lagrangian-type formulation for finite element analysis of non-linear beam vibrations’, Journal of Sound and Vibration 86, 1983, 61–70.Google Scholar
  14. 14.
    Lewanodowski, R., ‘Nonlinear free vibrations of multispan beams on elastic supports’, Journal of Sound and Vibration 32, 1989, 305–312.Google Scholar
  15. 15.
    Leung, A. Y. T. and Fung, T. C., ‘Nonlinear steady state vibration of frames by finite element method’, International Journal for Numerical Methods in Engineering 28, 1989, 1599–1618.Google Scholar
  16. 16.
    Lewandowski, R., ‘Non-linear, steady-state vibration of structures by harmonic balance/finite element method’, Computers & Structures 44, 1992, 287–296.Google Scholar
  17. 17.
    Lewandowski, R., ‘Non-linear free vibrations of beams by the finite element and continuation methods’, Journal of Sound and Vibration 170, 1994, 577–593.Google Scholar
  18. 18.
    Lewandowski, R., ‘Computational formulation for periodic vibration of geometrically nonlinear structures, Part 1: Theoretical background; Part 2: Numerical strategy and examples’, International Journal of Solids and Structures 34, 1997, 1925–1964.Google Scholar
  19. 19.
    Ribeiro, P. and Petyt, M., ‘Non-linear vibration of beams with internal resonance by the hierarchical finiteelement method’, Journal of Sound and Vibration 224, 1999, 591–624.Google Scholar
  20. 20.
    Lau, S. L. and Cheung, Y. K., ‘Amplitude incremental variational principle for nonlinear vibration of elastic systems’, ASME, Journal of Applied Mechanics 48, 1981, 959–964.Google Scholar
  21. 21.
    Ferri, A. A., ‘On the equivalence of the incremental harmonic balance method and the harmonic balance-Newton Raphson method’, ASME, Journal of Applied Mechanics 53, 1986, 455–457.Google Scholar
  22. 22.
    Lau, S. L., Cheung, Y. K., and Wu, S. Y., ‘A variable parameter incrementation method for dynamic instability of linear and nonlinear elastic systems’, ASME, Journal of Applied Mechanics 49, 1982, 849–853.Google Scholar
  23. 23.
    Lau, S. L., Cheung, Y. K., and Wu, S. Y., ‘Incremental harmonic balance method with multiple time scales for aperiodic vibration of nonlinear system’, ASME, Journal of Applied Mechanics 50, 1983, 871–876.Google Scholar
  24. 24.
    Lau, S. L., Cheung, Y. K., and Wu, S. Y., ‘Nonlinear vibration of thin elastic plates, Part 1: Generalized incremental Hamilton's principle and element formulation; Part 2: Internal resonance by amplitude-incremental finite element’, ASME, Journal of Applied Mechanics 51, 1984, 837–851.Google Scholar
  25. 25.
    Cheung, Y. K., Chen, S. H., and Lau, S. L., ‘Application of the incremental harmonic balance method to cubic non-linearity systems’, Journal of Sound and Vibration 140, 1990, 273–286.Google Scholar
  26. 26.
    Lau, S. L. and Yuen, S. W., ‘The Hopf bifurcation and limit cycle by the incremental harmonic balance method’, Computer Methods in Applied Mechanics and Engineering 91, 1991, 1109–1121.Google Scholar
  27. 27.
    Lau, S. L. and Zhang, W. S., ‘Nonlinear vibrations of piecewise-linear systems by incremental harmonic balance method’, ASME, Journal of Applied Mechanics 59, 1992, 153–160.Google Scholar
  28. 28.
    Lau, S. L. and Yuen, S. W., ‘Solution diagram of nonlinear dynamic-systems by IHB method’, Journal of Sound and Vibration 167, 1993, 303–316.Google Scholar
  29. 29.
    Lau, S. L., ‘Incremental harmonic balance method for nonlinear structural vibrations’, Ph.D. Thesis, University of Hong Kong, 1982.Google Scholar
  30. 30.
    Shi, Y., Lee, R., and Mei, C., ‘A finite element multimode method to nonlinear free vibrations of composite plates’, AIAA Journal 35, 1997, 159–166.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • S. H. Chen
    • 1
  • Y. K. Cheung
    • 2
  • H. X. Xing
    • 1
  1. 1.Department of MechanicsZhongshan UniversityGuangzhouPeople's Republic of China
  2. 2.Department of Civil EngineeringUniversity of Hong KongHong Kong

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