Skip to main content
Log in

Nonlinear dynamics of an SMA-pendulum system

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Pendulum is a simple system that is usually related to great discoveries in science and technology. Nonlinear dynamics of pendulum systems is related to a variety of responses being the objective of studies of oscillations, bifurcations and chaos. Smart material nonlinear effects are employed in several applications due to their adaptive behavior presenting great advantages in control strategies. This work deals with the dynamics of an SMA-pendulum system that consists of a pendulum coupled with shape memory alloy springs. A numerical investigation is carried out exploiting the temperature-dependent behavior of the SMA. Complex responses are presented and the possibility of thermal control of the system is investigated.

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

Access this article

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

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23

Similar content being viewed by others

References

  1. Pilipchuk, V.N., Vakakis, A.F., Azeez, M.A.F.: Sensitive dependence on initial conditions of strongly nonlinear periodic orbits of the forced pendulum. Nonlinear Dyn. 16(3), 223–237 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Franca, L.F.P., Savi, M.A.: Distinguishing periodic and chaotic time series obtained from an experimental nonlinear pendulum. Nonlinear Dyn. 26(3), 255–273 (2001)

    Article  MATH  Google Scholar 

  3. de Paula, A.S., Savi, M.A., Pereira-Pinto, F.H.I.: Chaos and transient chaos in an experimental nonlinear pendulum. J. Sound Vib. 294(3), 585–595 (2006)

    Article  Google Scholar 

  4. Bessa, W.M., de Paula, A.S., Savi, M.A.: Chaos control using an adaptive fuzzy sliding mode controller with application to a nonlinear pendulum. Chaos Solitons Fractals 42(2), 784–791 (2009)

    Article  MATH  Google Scholar 

  5. de Paula, A.S., Savi, M.A.: Controlling chaos in a nonlinear pendulum using an extended time-delayed feedback control method. Chaos Solitons Fractals 42(5), 2981–2988 (2009)

    Article  MATH  Google Scholar 

  6. Pan, Y., Zhou, Y., Sun, T., Er, M.J.: Composite adaptive fuzzy H\(\infty \) tracking control of uncertain nonlinear systems. Neurocomputing 99, 15–24 (2013)

    Article  Google Scholar 

  7. Suzuki, Y., Nomura, T., Casadio, M., Morasso, P.: Intermittent control with ankle, hip, and mixed strategies during quiet standing: a theoretical proposal based on a double inverted pendulum model. J. Theor. Biol. 310, 55–79 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, L., Sun, X., Jiang, H., Xu, X.: A high-performance control method of constant-controlled induction motor drives for electric vehicles. Math. Probl. Eng. 2014, 1–10 (2014)

    Google Scholar 

  9. Ju, F., Choo, Y.S., Cui, F.S.: Dynamic response of tower crane induced by the pendulum motion of the payload. Int. J. Solids Struct. 43(2), 376–389 (2006)

    Article  MATH  Google Scholar 

  10. Savi, M.A.: Nonlinear dynamics and chaos in shape memory alloy systems. Int. J. Non-Linear Mech. 70, 2–19 (2015)

    Article  Google Scholar 

  11. Bessa, W.M., de Paula, A.S., Savi, M.A.: Adaptive fuzzy sliding mode control of smart structures. Eur. Phys. J. Spec. Top. 222(7), 1541–1551 (2013)

    Article  Google Scholar 

  12. Kuribayashi, K., Tsuchiya, K., You, Z., Tomus, D., Umemoto, M., Ito, T., Sasaki, M.: Self-deployable origami stent grafts as a biomedical application of Ni-rich TiNi shape memory alloy foil. Mater. Sci. Eng. A 419(1–2), 131–137 (2006)

  13. Kim, B., Lee, M.G., Lee, Y.P., Kim, Y., Lee, G.: An earthworm-like micro robot using shape memory alloy actuator. Sens. Actuators Phys. 125(2), 429–437 (2006)

  14. Lebedev, G.A., Gusarov, B.V., Viala, B., Delamare, J., Cugat, O., Lafont, T., Zakharov, D. I.: Thermal energy harvesting using shape memory/piezoelectric composites. In: Solid-State Sensors, Actuators and Microsystems Conference (TRANSDUCERS), 2011 16th International, pp. 669–670 (2011)

  15. Machado, L.G., Savi, M.A., Pacheco, P.M.C.L.: Nonlinear dynamics and chaos in coupled shape memory oscillators. Int. J. Solids Struct. 40(19), 5139–5156 (2003)

    Article  MATH  Google Scholar 

  16. Machado, L.G., Savi, M.A.: Medical applications of shape memory alloys. Braz. J. Med. Biol. Res. 36(6), 683–691 (2003)

    Article  Google Scholar 

  17. Piccirillo, V., Balthazar, J.M., Jr. Pontes, B.R , Felix, J.L.P.: Chaos control of a nonlinear oscillator with shape memory alloy using an optimal linear control: part I: ideal energy source. Nonlinear Dyn. 55(1–2), 139–149 (2009)

  18. Arrieta, A.F., Hagedorn, P., Ertuk, A., Inman, D.J.: A piezoelectric bistable plate for nonlinear broadband energy harvesting. Appl. Phys. Lett. (2010). doi:10.1063/1.3487780

  19. Silva, L.L., Oliveira, S.A., Pacheco, P.M.L.C., Savi, M.A.: Synergistic use of smart materials for vibration-based energy harvesting. Eur. Phys. J. Spec. Top. 224(14–15), 3005–3012 (2015)

    Article  Google Scholar 

  20. Aguiar, R.A.A., Savi, M.A., Pacheco, P.M.C.L.: Experimental and numerical investigations of shape memory alloy helical springs. Smart Mater. Struct. 19(2), 25008 (2010)

    Article  Google Scholar 

  21. Lagoudas, D.C.: Shape Memory Alloys, vol. 1, 1st edn. Springer US, Boston (2008)

    MATH  Google Scholar 

  22. Paiva, A., Savi, M.A., Braga, A.M.B., Pacheco, P.M.C.L.: A constitutive model for shape memory alloys considering tensile-compressive asymmetry and plasticity. Int. J. Solids Struct. 42(11–12), 3439–3457 (2005)

    Article  MATH  Google Scholar 

  23. Enemark, S., Santos, I.F., Savi, M.A.: Modelling, characterisation and uncertainties of stabilised pseudoelastic shape memory alloy helical springs. J. Intell. Mater. Syst. Struct. (2016). doi:10.1177/1045389X16635845

  24. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Phys. D 16, 285–317 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sinou, J.-J.: Transient non-linear dynamic analysis of automotive disc brake squeal—on the need to consider both stability and non-linear analysis. Mech. Res. Commun. 37(1), 96–105 (2010)

    Article  MATH  Google Scholar 

  26. Monroe, R.J., Shaw, S.W.: Nonlinear transient dynamics of pendulum torsional vibration absorbers—part I: theory. J. Vib. Acoust. (2013). doi:10.1115/1.4007561

Download references

Acknowledgments

The authors would like to acknowledge the support of the Brazilian Research Agencies CNPq, CAPES and FAPERJ and through the INCT-EIE (National Institute of Science and Technology - Smart Structures in Engineering) the CNPq and FAPEMIG. The Air Force Office of Scientific Research (AFOSR) is also acknowledged.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Marcelo A. Savi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Costa, D.D.A., Savi, M.A. Nonlinear dynamics of an SMA-pendulum system. Nonlinear Dyn 87, 1617–1627 (2017). https://doi.org/10.1007/s11071-016-3137-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-3137-y

Keywords

Navigation