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The European Physical Journal Special Topics

, Volume 222, Issue 7, pp 1637–1647 | Cite as

Analysis of chaotic non-isothermal solutions of thermomechanical shape memory oscillators

  • Grzegorz Litak
  • Davide Bernardini
  • Arkadiusz Syta
  • Giuseppe Rega
  • Andrzej Rysak
Regular Article

Abstract

Shape memory materials exhibit strong thermomechanical coupling, so that temperature variations occur during mechanical loading and unloading. In previous works the nonlinear dynamics of pseudoelastic oscillators subject to an harmonic force has been studied and the possibility of non-regular chaotic responses has been thoroughly documented. Instead of the standard Lyapunov exponent treatment, the statistical 0–1 test based on the asymptotic properties of a Brownian motion chain was successively applied to reveal the chaotic nature of trajectories in the special case in which temperature variations were neglected. In this work, the 0–1 test is applied to fully non-isothermal trajectories. To improve its reliability the test has been applied to the time-histories of maxima and minima of each trajectory, in each component. The obtained results have been validated and confirmed by the corresponding Fourier spectra. Non-regular solutions with different levels of chaoticity have been analyzed and their qualitative difference is reflected by the different values to which the control parameter K asymptotically converge.

Keywords

Shape Memory Alloy European Physical Journal Special Topic Fourier Spectrum Energy Harvest Mean Square Displacement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    D.C. Lagoudas (ed.), Shape Memory Alloys: Modeling and Engineering Applications (Springer, Berlin, 2008)Google Scholar
  2. 2.
    M. Schwartz (ed.), Encyclopedia of Smart Materials, Vols. 1,2 (Wiley and Sons, New York, 2002)Google Scholar
  3. 3.
    M.A. Savi, P.M.C.L. Pacheco, Int. J. Bif. Chaos 12, 645 (2002)CrossRefGoogle Scholar
  4. 4.
    L.G. Machado, M.A. Savi, P.M.C.L. Pachecom, Shock Vibration 11, 67 (2004)CrossRefGoogle Scholar
  5. 5.
    D. Bernardini, G. Rega, Math. Computer Model. Dyn. Syst. 11, 291 (2005)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    W. Lacarbonara, D. Bernardini, F. Vestroni, Int. J. Solids Struct. 41, 1209 (2004)CrossRefMATHGoogle Scholar
  7. 7.
    D. Bernardinim, G. Rega, Int. J. Non-Lin. Mech. 45, 933 (2010)CrossRefGoogle Scholar
  8. 8.
    D. Bernardini, G. Rega, Int. J. Bif. Chaos 21, 2769 (2011)CrossRefMATHGoogle Scholar
  9. 9.
    D. Bernardin, G. Rega, Int. J. Bif. Chaos 21, 2783 (2011)CrossRefGoogle Scholar
  10. 10.
    D. Sado, M. Pietrzakowski, Int. J. Non-Lin. Mech. 45, 859 (2010)CrossRefGoogle Scholar
  11. 11.
    V. Piccirillo, J.M. Balthazar, B.R. Pontes Jr., Nonlinear Dyn. 59, 733 (2010)CrossRefMATHGoogle Scholar
  12. 12.
    B.C. dos Santos, M.A. Savi, Chaos, Solitons Fractals 40, 197 (2009)ADSCrossRefGoogle Scholar
  13. 13.
    D. Bernardini, T.J. Pence Mathematical Models for Shape Memory Materials Smart Materials (CRC press, Taylor and Francis, 2009), p. 20.17Google Scholar
  14. 14.
    G.A. Gottwald, I. Melbourne, Proc. R. Soc. Lond. A 460, 603 (2004)ADSCrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    G.A. Gottwald, I. Melbourne, Physica D 212, 100 (2005)ADSCrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    I. Falconer, G.A. Gottwald, I. Melbourne, K. Wormnes, SIAM J. App. Dyn. Syst. 6, 95 (2007)MathSciNetGoogle Scholar
  17. 17.
    G. Litak, A. Syta, M. Wiercigroch, Chaos, Solit. Fract. 40, 2095 (2009)ADSCrossRefGoogle Scholar
  18. 18.
    B. Krese, E. Govekar, Nonlinear Dyn. 67, 2101 (2012)CrossRefGoogle Scholar
  19. 19.
    D. Bernardini, G. Rega, G. Litak, A. Syta, Proc. IMechE. Part K: J Multi-body Dyn. 227, 17 (2013)CrossRefGoogle Scholar
  20. 20.
    G. Litak, S. Schubert, G. Radons, Nonlinear Dyn. 69, 1255 (2012)CrossRefMathSciNetGoogle Scholar
  21. 21.
    C. Piccardi, S. Rinaldi, Int. J. Bifurcat. Chaos 13, 1579 (2003)CrossRefMATHGoogle Scholar
  22. 22.
    Y.F. Yang, X.G. Ren, W.Y. Qin, Nonlinear Anal. 68, 582 (2008)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    H. Kantz, T. Schreiber, Non-linear time series analysis (Cambridge University Press, Cambridge 1997)Google Scholar
  24. 24.
    A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Physica D 16, 285 (1985)ADSCrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    H. Kantz, Phys. Lett. A 185, 77 (1994)ADSCrossRefGoogle Scholar
  26. 26.
    L.G. Machado, D.C. Lagoudas, M.A. Savi, Int. J. Solids Struct. 46, 1269 (2009)CrossRefMATHGoogle Scholar

Copyright information

© EDP Sciences and Springer 2013

Authors and Affiliations

  1. 1.Faculty of Mechanical Engineering, Lublin University of TechnologyLublinPoland
  2. 2.Dipartimento di Ingegneria Strutturale e Geotecnica, Sapienza Universita di RomaRomaItaly

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