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Synchronisation and Chaos in a Parametrically and Self-Excited System with Two Degrees of Freedom

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Abstract

Vibration analysis of a non-linear parametrically andself-excited system of two degrees of freedom was carried out. The modelcontains two van der Pol oscillators coupled by a linear spring with a aperiodically changing stiffness of the Mathieu type. By means of amultiple-scales method, the existence and stability of periodicsolutions close to the main parametric resonances have beeninvestigated. Bifurcations of the system and regions of chaoticsolutions have been found. The possibility of the appearance ofhyperchaos has also been discussed and an example of such solution hasbeen shown.

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References

  1. Tondl, A., On the Interaction between Self-Excited and Parametric Vibrations, Monographs and Memoranda, Vol. 25, National Research Institute for Machine Design, Prague, 1978.

    Google Scholar 

  2. Yano, S., 'Considerations on self-and parametrically excited vibrational systems',Ingenieur-Archiv 59, 1989, 285-295.

    Google Scholar 

  3. Szabelski, K. and Warmiński, J., 'The non-linear vibrations of parametrically self-excited system with two degrees of freedom under external excitation', Nonlinear Dynamics 14, 1997, 23-36.

    Google Scholar 

  4. Litak G., Szabelski, K., Spuz-Szpos, G., and Warmiński, J., 'Vibration analysis of a self-excited system with parametric forcing and nonlinear stiffness', International Journal of Bifurcation and Chaos 9, 1999, 493-504.

    Google Scholar 

  5. Szabelski, K., Litak, G., Warmiñski, J., and Spuz-Szpos, G., 'Vibrations synchronization and chaos in parametrically self-excited system with non-linear elasticity', in Proceedings of EUROMECH-2nd European Nonlinear Oscillation Conference, Prague, September 9-13, 1996, Vol. 1, L. Půst and F. Peterka (eds.), Czech Technical University, 1996, pp. 435-439.

  6. Szabelski, K. and Warmiński, J., 'The parametric self excited non-linear system vibrations analysis with the inertial excitation', International Journal of Non-Linear Mechanics 30, 1995, 179-189.

    Google Scholar 

  7. Szabelski, K. and Warmiński, J., 'The self excited system vibrations with the parametric and external excitations', Journal of Sound and Vibration 908, 1995, 595-607.

    Google Scholar 

  8. Hayashi, Ch., Nonlinear Oscillations in Physical Systems, McGraw-Hill, New York, 1964.

    Google Scholar 

  9. Litak G., Spuz-Szpos, G., Szabelski, K., and Warmiński, J., 'Vibration of externally forced Froude pendulum', International Journal of Bifurcation and Chaos 9, 1999, 561-570.

    Google Scholar 

  10. Kapitaniak, T. and Chua, L. O., 'Locally-intermingled basins of attraction in coupled Chua's circuits', International Journal of Bifurcation and Chaos 6, 1996, 357-366.

    Google Scholar 

  11. Kapitaniak, T., 'Transition to chaos in chaotically forced coupled oscillators', Physical Review E47, 1993, R2975-R2978.

    Google Scholar 

  12. Kapitaniak, T. and Chua, L. O., 'Hyperchaotic attractors of undirectionally-coupled Chua's circuits', International Journal of Bifurcation and Chaos 4, 1994, 477-482.

    Google Scholar 

  13. Kapitaniak T. and Steeb, W., H., 'Transition to hyperchaos in coupled generalized Van der Pol's equations', Physics Letters A152, 1991, 33-37.

    Google Scholar 

  14. Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.

    Google Scholar 

  15. Wiercigroch, M., 'Chaotic vibrations of a simple model of the machine tool-cutting process system', Journal of Vibration and Acoustics 119, 1997, 468-475.

    Google Scholar 

  16. Kahraman, A. and Blankenship, G. W., 'Experiments on nonlinear dynamic behaviour of an oscillator with clearance and periodically time varying parameters', Journal of Applied Mechanics 64, 1997, 217-226.

    Google Scholar 

  17. Schmidt, G., 'Interaction of self-excited, forced and parametrically excited vibrations', in Applications of the Theory of Nonlinear Oscillations-The 9th International Conference on Non-Linear Oscilations, Naukova Dumka, Kiev, 1984, pp. 310-315.

    Google Scholar 

  18. Schmidt, G. and Tondl, A., Non-Linear Vibrations, Akademie-Verlag, Berlin, 1986.

    Google Scholar 

  19. Nusse, H. E. and Yorke, J. A., Dynamics: Numerical Explorations, Springer-Verlag, New York, 1994.

    Google Scholar 

  20. Hassard, B. D., Kazarinoff, N. D., and Wan, Y. H., Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mechanics, Technical University of Lublin, Nadbystrzycka 36 str., 20-618, Lublin, Poland

    Jerzy Warmiński, Grzegorz Litak & Kazimierz Szabelski

Authors
  1. Jerzy Warmiński
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  2. Grzegorz Litak
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  3. Kazimierz Szabelski
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Warmiński, J., Litak, G. & Szabelski, K. Synchronisation and Chaos in a Parametrically and Self-Excited System with Two Degrees of Freedom. Nonlinear Dynamics 22, 125–143 (2000). https://doi.org/10.1023/A:1008325924199

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  • DOI: https://doi.org/10.1023/A:1008325924199

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