Nonlinear Dynamics

, Volume 60, Issue 4, pp 513–524

Analytical study of the nonlinear behavior of a shape memory oscillator: Part II—resonance secondary

  • V. Piccirillo
  • J. M. Balthazar
  • B. R. PontesJr.
Original Paper
  • 102 Downloads

Abstract

In Part II of this work, we investigated the dynamics of the shape memory oscillator, in the particular case of the secondary resonances. We used the equation of motion developed in Part I (Piccirillo et al., Nonlinear Dyn., 2009). The method of multiple scales is used to obtain an approximate solution to the governing equations of motion. To examine subharmonic and superharmonic resonances, we need to order the excitation so that it appears at same time as the free-oscillation part of the solution. Firstly, the analysis is made for the superharmonic resonance where we find the frequency-response curves and these curves show the influences of the damping, nonlinearity, and amplitude of the excitation. Results showed that it occurs in the jump phenomena, bifurcation saddle-node, and motions periodic the period-2. In the subharmonic resonance, we note that it does not occur in the jump phenomena, but on the other hand, we found the regions where the nontrivial solutions of the subharmonic resonance exit. The frequency-response curves show the behavior of the oscillator for the variation of the control parameters. Numerical simulations are performed and the simulation results are visualized by means of the phase portrait, Poincaré map, and Lyapunov exponents.

Shape memory alloy Nonlinear dynamic Method of multiple scales Bifurcations Resonances Jump phenomena Frequency-response 

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References

  1. 1.
    Piccirillo, V., Balthazar, J.M., Pontes Jr., B.R.: Analytical study of the nonlinear behavior of a shape memory oscillator: Part I—primary resonance and free response at low temperatures. Nonlinear Dyn. (2009). doi:10.1007/s11071-009-9573-1
  2. 2.
    Van Hunbeeck, J.: Non-medical applications of shape memory alloys. Mater. Sci. Eng. A 273–275, 134–148 (1999) Google Scholar
  3. 3.
    Duering, T., Pelton, A., Stockel, D.: An overview of nitinol medical applications. Mater. Sci. Eng. A 273–275, 149–160 (1999) Google Scholar
  4. 4.
    Lagoudas, D.C., Rediniotis, O.K., Khan, M.M.: Applications of shape memory alloys to bioengineering and biomedical technology. In: Proceedings of 4th International Workshop on Mathematical Methods in Scattering Theory and Biomedical Technology, October, Perdika, Greece (1999) Google Scholar
  5. 5.
    Denoyer, K.K., Erwin, R.S., Ninneman, R.R.: Advanced smart structures flight experiments for precision spacecraft. Acta Astron. 47, 389–397 (2000) CrossRefGoogle Scholar
  6. 6.
    Pacheco, P.M.C.L., Savi, M.A.: A non-explosive release device for aerospace applications using shape memory alloys. In: Proceedings of XIV the Brazilian Congress of Mechanical Engineering (COBEM 1997–ABCM), Bauru, Brazil (1997) Google Scholar
  7. 7.
    Garner, L.J., Wilson, L.N., Lagoudas, D.C., Rediniotis, O.K.: Development of a shape memory alloy actuated biomimetic vehicle. Smart Mater. Struct. 9(5), 673–683 (2001) CrossRefGoogle Scholar
  8. 8.
    Webb, G., Wilson, L., Lagoudas, D.C., Rediniotis, O.: Adaptive control of shape memory alloy actuators for underwater biomimetic applications. AIAA J. 38(2), 325–334 (2000) CrossRefGoogle Scholar
  9. 9.
    Rogers, C.A.: Intelligent materials. Sci. Am. (September), pp. 122–127 (1995) Google Scholar
  10. 10.
    Paiva, A., Savi, M.A.: An overview of constitutive models for shape memory alloys. Math. Probl. Eng. 2006, 1–30 (2006) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Birman, V.: Review of mechanics of shape memory alloy structures. Appl. Mech. Rev. 50, 629–645 (1997) CrossRefGoogle Scholar
  12. 12.
    Bernardini, D., Vestroni, F.: Non-isothermal oscillations of pseudoelastic devices. Int. J. Non-Linear Mech. 38, 1297–1313 (2003) MATHCrossRefGoogle Scholar
  13. 13.
    Lacarbonara, W., Bernardini, D., Vestroni, F.: Nonlinear thermomechanical oscillations of shape memory devices. Int. J. Solids Struct. 41, 1209–1234 (2004) MATHCrossRefGoogle Scholar
  14. 14.
    Lagoudas, D.C., Khan, M.M., Mayes, J.J., Henderson, B.K.: Pseudoelastic SMA spring element for passive vibration isolation: Part II—simulation and experimental correlations. J. Intell. Mater. Struct. 15, 443–470 (2004) CrossRefGoogle Scholar
  15. 15.
    Savi, M.A.: Pacheco, P.M.C.L.: Chaos and hyperchaos in shape memory systems. Int. J. Bifurc. Chaos 12(3), 645–657 (2002) CrossRefGoogle Scholar
  16. 16.
    Savi, M., Sá, M.A.N., Paiva, A., Pacheco, P.M.C.L.: Tensile-compressive asymmetry influence on shape memory alloy system dynamics. Chaos Solitons Fractal 36(20), 828–842 (2008) CrossRefGoogle Scholar
  17. 17.
    Falk, F.: Model free energy, mechanics, and thermodynamics of shape memory alloys. Acta Metall. 28, 1773–1780 (1980) CrossRefGoogle Scholar
  18. 18.
    Machado, L.G., Lagoudas, D.C., Savi, M.A.: Lyapunov exponents estimation for hysteretic system. Int. J. Solids Struct. 46(6), 1269–1598 (2009) CrossRefGoogle Scholar
  19. 19.
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computation and Experimental Methods. Wiley, New York (1995) CrossRefGoogle Scholar
  20. 20.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979) MATHGoogle Scholar
  21. 21.
    Von Kármán, T.: The engineer grapples with nonlinear problems. Bull. Am. Math. Soc. 46, 615–683 (1940) CrossRefGoogle Scholar
  22. 22.
    Lefschetz, S.: Linear and nonlinear oscillations. In: Modern Mathematics for the Engineer, pp. 7–30. McGraw-Hill, New York (1956) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • V. Piccirillo
    • 1
  • J. M. Balthazar
    • 1
    • 2
  • B. R. PontesJr.
    • 1
  1. 1.Department of Engineering MechanicalUNESP—Sao Paulo State UniversityBauruBrazil
  2. 2.Department of Statistic, Applied Mathematical and ComputationUNESP—Sao Paulo State UniversityRio ClaroBrazil

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