Nonlinear Dynamics

, Volume 91, Issue 4, pp 2319–2330 | Cite as

Chaotic characteristic of a linear oscillator coupled with vibro-impact nonlinear energy sink

  • Tao Li
  • Claude-Henri Lamarque
  • Sébastien Seguy
  • Alain Berlioz
Original Paper
  • 165 Downloads

Abstract

The chaotic characteristic of a system with vibro-impact nonlinear energy sink is studied here. An analytical method is developed to calculate Lyapunov exponent. The mechanism by which impact results in chaos is further clarified rather than only by the calculation of Lyapunov exponent. In addition, an approach to identifying Lyapunov exponents from experimental data is proposed, and the estimated results are consistent with numerical results.

Keywords

Chaos Lyapunov exponent Vibro-impact Nonlinear energy sink Targeted energy transfer Impact damper 

Notes

Acknowledgements

The authors acknowledge the French Ministry of Science and the Chinese Scholarship Council under Grant No. 20130449 0063 for their financial support.

References

  1. 1.
    Ibrahim, R.A.: Vibro-Impact Dynamics: Modeling, Mapping and Applications, vol. 43. Springer, Berlin (2009)MATHGoogle Scholar
  2. 2.
    Vakakis, A.F., Gendelman, O., Bergman, L., McFarland, D., Kerschen, G., Lee, Y.: Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, vol. 156. Springer, Berlin (2008)MATHGoogle Scholar
  3. 3.
    Lee, Y., Nucera, F., Vakakis, A., McFarland, D., Bergman, L.: Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibro-impact attachments. Phys. D 238(18), 1868–1896 (2009)CrossRefMATHGoogle Scholar
  4. 4.
    Gendelman, O.V.: Analytic treatment of a system with a vibro-impact nonlinear energy sink. J. Sound Vib. 331, 4599–4608 (2012)CrossRefGoogle Scholar
  5. 5.
    Gendelman, O., Alloni, A.: Dynamics of forced system with vibro-impact energy sink. J. Sound Vib. 358, 301–314 (2015)CrossRefGoogle Scholar
  6. 6.
    Gendelman, O., Alloni, A.: Forced system with vibro-impact energy sink: chaotic strongly modulated responses. Proc. IUTAM 19, 53–64 (2016)CrossRefGoogle Scholar
  7. 7.
    Gourc, E., Michon, G., Seguy, S., Berlioz, A.: Targeted energy transfer under harmonic forcing with a vibro-impact nonlinear energy sink: analytical and experimental developments. J. Vib. Acoust. 137(3), 031008 (2015)CrossRefGoogle Scholar
  8. 8.
    Li, T., Seguy, S., Berlioz, A.: Dynamics of cubic and vibro-impact nonlinear energy sink: analytical, numerical, and experimental analysis. J. Vib. Acoust. 138(3), 031010 (2016)CrossRefGoogle Scholar
  9. 9.
    Li, T., Seguy, S., Berlioz, A.: On the dynamics around targeted energy transfer for vibro-impact nonlinear energy sink. Nonlinear Dyn. 87(3), 1453–1466 (2017)CrossRefGoogle Scholar
  10. 10.
    Li, T., Seguy, S., Berlioz, A.: Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation. Nonlinear Dyn. 87(4), 2415–2433 (2017)CrossRefGoogle Scholar
  11. 11.
    Li, T., Gourc, E., Seguy, S., Berlioz, A.: Dynamics of two vibro-impact nonlinear energy sinks in parallel under periodic and transient excitations. Int. J. NonLinear Mech. 90, 100–110 (2017)CrossRefGoogle Scholar
  12. 12.
    Pennisi, G., Stephan, C., Gourc, E., Michon, G.: Experimental investigation and analytical description of a vibro-impact NES coupled to a single-degree-of-freedom linear oscillator harmonically forced. Nonlinear Dyn. (2017).  https://doi.org/10.1007/s11071-017-3344-1 Google Scholar
  13. 13.
    Müller, P.C.: Calculation of Lyapunov exponents for dynamic systems with discontinuities. Chaos Soliton Fract. 5(9), 1671–1681 (1995)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lamarque, C.H., Bastien, J.: Numerical study of a forced pendulum with friction. Nonlinear Dyn. 23(4), 335–352 (2000)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Stefanski, A.: Estimation of the largest Lyapunov exponent in systems with impacts. Chaos Soliton Fract. 11(15), 2443–2451 (2000)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Stefański, A., Kapitaniak, T.: Estimation of the dominant Lyapunov exponent of non-smooth systems on the basis of maps synchronization. Chaos Soliton Fract. 15(2), 233–244 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    De Souza, S.L., Caldas, I.L.: Calculation of Lyapunov exponents in systems with impacts. Chaos Soliton Fract. 19(3), 569–579 (2004)CrossRefMATHGoogle Scholar
  18. 18.
    Jin, L., Lu, Q.S., Twizell, E.: A method for calculating the spectrum of Lyapunov exponents by local maps in non-smooth impact-vibrating systems. J. Sound Vib. 298(4), 1019–1033 (2006)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institut Clément Ader (ICA), CNRS-INSA-ISAE-Mines Albi-UPSUniversité de ToulouseToulouseFrance
  2. 2.LTDS UMR CNRS 5513Université de Lyon, École Nationale des Travaux Publics de l’ÉtatVaulx-en-Velin CedexFrance

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