Skip to main content
SpringerLink
Log in

Nonlinear Dynamics of a Cantilever Beam Carrying an Attached Mass with 1:3:9 Internal Resonances

  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper the nonlinear response of a base-excited slender beam carrying an attached mass is investigated with 1:3:9 internal resonances for principal and combinationparametric resonances. Here the method of normal forms is used to reduce the second order nonlinear temporal differential equation of motion of the system to a set offirst order nonlinear differential equations which are used to find the fixed-point, periodic, quasi-periodic and chaotic responses of the system.Stability and bifurcation analysis of the responses are carried out and bifurcation sets are plotted. Many chaotic phenomena are reported in this paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. To, C. W. S., ‘Vibration of a cantilever beam with a base excitation and tip mass’, Journal of Sound and Vibration 83, 1982, 445–460.

    Google Scholar 

  2. Fung, E. H. K. and Shi, Z. X., ‘Vibration frequencies of a constrained flexible arm carrying an end mass’, Journal of Sound and Vibration 203, 1997, 373–387.

    Google Scholar 

  3. Hamdan M. N. and Dado, M. H. F., ‘Large amplitude free vibrations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia’, Journal of Sound and Vibration 206, 1997, 151–168.

    Google Scholar 

  4. Handoo, K. L. and Sundarajan, V., ‘Parametric instability of a cantilevered column with end mass’, Journal of Sound and Vibration 18, 1971, 45–53.

    Google Scholar 

  5. Lee, H. P., ‘Stability of a cantilever beam with tip mass subjected to axial sinusoidal excitations’, Journal of Sound and Vibration 183, 1995, 91–98.

    Google Scholar 

  6. Saito, K., Sato, H., and Otomi, K., ‘The parametric response of a horizontal beam carrying a concentrated mass under gravity’, ASME, Journal of Applied Mechanics 45, 1978, 643–648.

    Google Scholar 

  7. Saito, H. and Koizumi, N., ‘Parametric vibrations of a horizontal beam with a concentrated mass at one end’, International Journal of Mechanical Sciences 24, 1982, 755–761.

    Google Scholar 

  8. Zavodney, L. D. and Nayfeh, A. H., ‘The non-linear response of a slender beam carrying a lumped mass to a principal parametric excitation: theory and experiment’, International Journal of Non-Linear Mechanics 24, 1989, 105–125.

    Google Scholar 

  9. Mustafa, G. and Ertas, A., ‘Dynamics and bifurcations of a coupled column-pendulum oscillator’, Journal of Sound and Vibration 182, 1995, 393–413.

    Google Scholar 

  10. Kar, R. C. and Dwivedy, S. K., ‘Nonlinear dynamics of a slender beam carrying a lumped mass with principal parametric and internal resonances’, International Journal of Non-Linear Mechanics 34, 1999, 515–529.

    Google Scholar 

  11. Dwivedy, S. K. and Kar, R. C., ‘Dynamics of a slender beam with an attached mass under combination parametric and internal resonances Part I: Steady state response’, Journal of Sound and Vibration 221, 1999, 823–848.

    Google Scholar 

  12. Dwivedy, S. K. and Kar, R. C., ‘Dynamics of a slender beam with an attached mass under combination parametric and internal resonances Part II: Periodic and chaotic responses’, Journal of Sound and Vibration 222, 1999, 281–305.

    Google Scholar 

  13. Dwivedy, S. K. and Kar, R. C., ‘Nonlinear dynamics of a slender beam carrying a lumped mass under principal parametric resonance with three-mode interactions’, International Journal of Non-Linear Mechanics 36, 2001, 927–945.

    Google Scholar 

  14. Dwivedy, S. K., ‘Nonlinear dynamics of a parametrically excited slender beam with an attached mass’, Ph.D. Dissertation, Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, India, 1999.

    Google Scholar 

  15. Nayfeh, A. H. and Mook, D.T., Nonlinear Oscillations, Wiley-Interscience, New York, 1979.

    Google Scholar 

  16. Cartmell, M., Introduction to Linear, Parametric and Non-Linear Vibrations, Chapman and Hall, London, 1990.

    Google Scholar 

  17. Szemplińska-Stupnicka, W., The Behaviour of Non-Linear Vibrating Systems, Vols. 1 and 2, Kluwer, Dordrecht, 1990.

    Google Scholar 

  18. Nayfeh, A. H. and Balachandran, B., ‘Modal interactions in dynamical and structural systems’, ASME Applied Mechanics Reviews 42, 1989, S175–S201.

    Google Scholar 

  19. Tezak, E. G., Mook, D. T., and Nayfeh, A. H., ‘Nonlinear analysis of the lateral response of columns to periodic loads’, ASME, Journal of Machine Design 100, 1978, 651–655.

    Google Scholar 

  20. Chin, C.-M. and Nayfeh, A. H., ‘Three-to-one internal resonance in parametrically excited hinged-clamped beams’, Nonlinear Dynamics 20 1999, 131–158.

    Google Scholar 

  21. Nayfeh, A. H., ‘The response of two-degree-of-freedom systems with quadratic non-linearities to a parametric excitation’, Journal of Sound and Vibration 88, 1983, 547–557.

    Google Scholar 

  22. Nayfeh, A. H. and Jebrill, A.E.S., ‘The resonance of two degree-of-freedom systems with quadratic nonlinearities to parametric excitation’, Journal of Sound and Vibration 88, 1983, 547–557.

    Google Scholar 

  23. Tso, W. K. and Asmis, K.G., ‘Multiple parametric resonance in a nonlinear two degree of freedom system’, International Journal of Non-Linear Mechanics 9, 1974, 269–277.

    Google Scholar 

  24. Bux, S. L. and Robert, J. W., ‘Non-linear vibratory interactions in systems of coupled beams’, Journal of Sound and Vibration 104, 1986, 497–520.

    Google Scholar 

  25. Nayfeh, A. H., ‘The response of multi-degree-of-freedom systems with quadratic non-linearities to a harmonic parametric resonance’, Journal of Sound and Vibration 90, 1983, 237–244.

    Google Scholar 

  26. Chen, S. H., Cheung, Y. K., and Lau, S. L., ‘On internal resonance of multi-degree-of-freedom systems with cubic nonlinearity’, Journal of Sound and Vibration 128, 1989, 13–24.

    Google Scholar 

  27. Vakakis, A. F., Manevitch, L. I., Mikhlin, Y. V., Pilipchuk, N. V., and Zevin, A. A., Normal Modes and Localization in Nonlinear Systems, Wiley-Interscience, New York, 1996.

    Google Scholar 

  28. Nayfeh, A. H. and Balachandran, B., Applied Nonlinear Dynamics, Wiley-Interscience, New York, 1995.

    Google Scholar 

  29. Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer, New York, 1983.

    Google Scholar 

  30. Nayfeh, A. H., Method of Normal Forms, Wiley-Interscience, New York, 1993.

    Google Scholar 

  31. Valeev, K. G., ‘On the danger of combination resonance’, Journal of Applied Mathematics and Mechanics 6, 1963, 1745–1769.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology, Guwahati, 781 039, India

    S. K. Dwivedy

  2. Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, 721 302, India

    R. C. Kar

Authors
  1. S. K. Dwivedy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. C. Kar
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dwivedy, S.K., Kar, R.C. Nonlinear Dynamics of a Cantilever Beam Carrying an Attached Mass with 1:3:9 Internal Resonances. Nonlinear Dynamics 31, 49–72 (2003). https://doi.org/10.1023/A:1022128029369

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1022128029369

Search

Navigation