Nonlinear Dynamics

, Volume 80, Issue 4, pp 1705–1719 | Cite as

Chaos in the fractionally damped broadband piezoelectric energy generator

  • Junyi Cao
  • Shengxi Zhou
  • Daniel J. Inman
  • Yangquan Chen
Original Paper

Abstract

Piezoelectric materials play a significant role in harvesting ambient vibration energy. Due to their inherent characteristics and electromechanical interaction, the system damping for piezoelectric energy harvesting can be adequately characterized by fractional calculus. This paper introduces the fractional model for magnetically coupling broadband energy harvesters under low-frequency excitation and investigates their nonlinear dynamic characteristics. The effects of fractional-order damping, excitation amplitude, and frequency on dynamic behaviors are proposed using the phase trajectory, power spectrum, Poincare map, and bifurcation diagram. The numerical analysis shows that the fractionally damped energy harvesting system exhibits chaos, periodic motion, chaos and periodic motion in turn when the fractional order changes from 0.2 to 1.5. The period doubling route to chaos and the inverse period doubling route from chaos to periodic motion can be clearly observed. It is also demonstrated numerically and experimentally that the magnetically coupling piezoelectric energy harvester possesses the usable frequency bandwidth over a wide range of low-frequency excitation. Both high-energy chaotic attractors and large-amplitude periodic response with inter-well oscillators dominate these broadband energy harvesting.

Keywords

Energy harvesting Chaos  Piezoelectricity Nonlinear vibrations  Fractional calculus 

References

  1. 1.
    Pearson, M.R., Eaton, M.J., Pullin, R., Featherston, C.A., Holford, K.M.: Energy harvesting for aerospace structural health monitoring systems. J. Phys. 382, 012025 (2012)Google Scholar
  2. 2.
    Qing, X., Chan, H., Beard, S.J.: An active diagnostic system for structural health monitoring of rocket engines. J. Intell. Mater. Syst. Struct. 17(7), 619–628 (2006)CrossRefGoogle Scholar
  3. 3.
    Ihn, J., Chang, F.: Detection and monitoring of Hidden fatigue crack growth using a built-in piezoelectric sensor/actuator network: I. Diagn. Smart Mater. Struct. 13(3), 609–620 (2004)CrossRefGoogle Scholar
  4. 4.
    Lynch, J.P., Loh, K.: A summary review of wireless sensors and sensor networks for structural health monitoring. Shock Vib. Digest 38(2), 91–128 (2006)CrossRefGoogle Scholar
  5. 5.
    Lu, K.C., Loh, C., Yang, Y., Lynch, J.P., Law, K.H.: Real-time structural damage detection using wireless sensing and monitoring system. Smart Mater. Struct. 4(6), 759–778 (2008)CrossRefGoogle Scholar
  6. 6.
    Zhao, X., Gao, H., Rose, J.L.: Active health monitoring of an aircraft wing with embedded piezoelectric sensor/actuator network: I. Defect detection, localization and growth monitoring. Smart Mater. Struct. 16(4), 1208–1225 (2007)CrossRefGoogle Scholar
  7. 7.
    Leland, E.S., Wright, P.K.: Resonance tuning of piezoelectric vibration energy scavenging generators using compressive axial preload. Smart Mater. Struct. 15(5), 1413–1420 (2006)CrossRefGoogle Scholar
  8. 8.
    Hu, Y., Xue, H., Hu, H.: A piezoelectric power harvester with adjustable frequency through axial preloads. Smart Mater. Struct. 16(5), 1961–1966 (2007)CrossRefGoogle Scholar
  9. 9.
    Rhimi, M., Lajnef, N.: Passive temperature compensation in piezoelectric vibrators using shape memory alloy-induced axial loading. J. Intell. Mater. Syst. Struct. 23(15), 1759–1770 (2012)CrossRefGoogle Scholar
  10. 10.
    Lallart, M., Anton, S.R., Inman, D.J.: Frequency self-tuning scheme for broadband vibration energy harvesting. J. Intell. Mater. Syst. Struct. 21, 897–906 (2010)CrossRefGoogle Scholar
  11. 11.
    Eichhorn, C., Tchagsim, R., Wilhelm, N., Woias, P.: A smart and self-sufficient frequency tunable vibration energy harvester. J. Micromech. Microeng. 21(10), 104003–11 (2011)CrossRefGoogle Scholar
  12. 12.
    Mann, B.P., Sims, N.D.: Energy harvesting from the nonlinear oscillations of magnetic levitation. J. Sound Vib. 319, 515–530 (2009)CrossRefGoogle Scholar
  13. 13.
    Burrow, S., Clare, L., Carrella, A., Barton, D.: Vibration energy harvesters with nonlinear compliance. In: Proceedings of SPIE Smart Structures/NDE Conference, pp. 3–10 (2008)Google Scholar
  14. 14.
    Ramlan, R., Brennan, M.J., Mace, B.R., Kovacic, I.: Potential benefits of an on-linear stiffness in an energy harvesting device. Nonlinear Dyn. 59, 545–558 (2009)CrossRefGoogle Scholar
  15. 15.
    Stanton, S.C., McGehee, C.C., Mann, B.P.: Reversible hysteresis for broadband magnetopiezoelastic energy harvesting. Appl. Phys. Lett. 96, 174103 (2010)CrossRefGoogle Scholar
  16. 16.
    Daqaq, M., Stabler, C., Qaroush, Y., Seuaciuc-Osorio, T.: Investigation of power harvesting via parametric excitations. J. Intell. Mater. Syst. Struct. 20(5), 545–557 (2009)CrossRefGoogle Scholar
  17. 17.
    Shahruz, S.: Increasing the efficiency of energy scavengers by magnets. J. Comput. Nonlinear Dyn. 3, 1–12 (2004)Google Scholar
  18. 18.
    Cottone, F., Vocca, H., Gammaitoni, L.: Nonlinear energy harvesting. Phys. Rev. Lett. 102, 1–4 (2009)CrossRefGoogle Scholar
  19. 19.
    Erturk, A., Hoffmann, J., Inman, D.: A piezo-magneto-elastic structure for broadband vibration energy harvesting. Appl. Phys. Lett. 94, 254102–3 (2009)CrossRefGoogle Scholar
  20. 20.
    Gammaitoni, L., Neri, I., Vocca, H.: Nonlinear oscillators for vibration energy harvesting. Appl. Phys. Lett. 94, 164102–2 (2009)CrossRefGoogle Scholar
  21. 21.
    Stanton, S.C., McGehee, C.C., Mann, B.P.: Nonlinear dynamics for broadband energy harvesting: investigation of a bistable piezoelectric inertial generator. Phys. D 239, 640–653 (2010)CrossRefMATHGoogle Scholar
  22. 22.
    Erturk, A., Inman, D.J.: Broadband piezoelectric power generation on high-energy orbits of the bistable Duffing oscillator with electromechanical coupling. J. Sound Vib. 330, 2339–2353 (2011)CrossRefGoogle Scholar
  23. 23.
    Masana, R., Daqaqa, M.F.: Energy harvesting in the super-harmonic frequency region of a twin-well oscillator. J. Appl. Phys. 111, 044501–044511 (2012)CrossRefGoogle Scholar
  24. 24.
    Twiefel, J., Westermann, H.: Survey on broadband techniques for vibration energy harvesting. J. Intell. Mater. Syst., Struct (2013) Google Scholar
  25. 25.
    Harne, R.L., Wang, K.W.: A review of the recent research on vibration energy harvesting via bistable systems. Smart Mater. Struct. 22(2), 023001 (2013)CrossRefGoogle Scholar
  26. 26.
    Zhou, S., Cao, J., Erturk, A., Lin, J.: Enhanced broadband piezoelectric energy harvesting using rotatable magnets. Appl. Phys. Lett. 102, 173901 (2013)CrossRefGoogle Scholar
  27. 27.
    Kumar, G.S.; Prasad, G.: Piezoelectric relaxation in polymer and ferroelectric composites. J. Mater. Sci. 28(9), 2545–2550 (1993)Google Scholar
  28. 28.
    Hartley, T.T., Lorenzo, C.F.: A frequency-domain approach to optimal fractional-order damping. Nonlinear Dyn. 38(1–4), 69–84 (2004)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Maia, N.M.M, Silva, J.M.M., Ribeiro, A. M. R.: On a general model for damping. J. Sound Vib. 218(5), 749–767 (1998)Google Scholar
  30. 30.
    Machado, J.A.T., Galhano, A.: Fractional dynamics: a statistical perspective. ASME J. Comp. Nonlinear Dyn. 3(2), 1–5 (2008)Google Scholar
  31. 31.
    Rossikhin, Y.A., Shitikova, M.V.: Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. ASME J. Appl. Mech. Rev. 63(1), 1–52 (2010)Google Scholar
  32. 32.
    Vinogradov, A.M., Schmidt, V.H., Tuthill, G.F.: Damping and electromechanical energy losses in the piezoelectric polymer PVDF. Mech. Mater. 36(10), 1007–1016 (2004)CrossRefGoogle Scholar
  33. 33.
    Cattin, D., Oboe, R., Dahiya, R.S., Valle, M.: Identification and validation of fractional order dynamic model for a piezoelectric tactile sensor. In: Proceeding of the 11th IEEE International Workshop on Advanced Motion Control, Nagaoka, Japan. pp. 430–435 (March 2010)Google Scholar
  34. 34.
    Galucio, A.C., Deu, J.F., Ohayon, R.: A Fractional derivative viscoelastic model for hybrid active–passive damping treatments in time domain—application to sandwich beams. J. Intell. Mater. Syst. Struct. 16(1), 33–45 (2005)CrossRefGoogle Scholar
  35. 35.
    Ducharne, B., Zhang, B., Guyomar, D., Sebald, G.: Fractional derivative operators for modeling piezoceramic polarization behaviors under dynamic mechanical stress excitation. Sens. Actuator A 189, 74–79 (2012)CrossRefGoogle Scholar
  36. 36.
    Chen, Y.Q., Moore, K.L.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuits Syst. 49(3), 363–367 (2002)CrossRefMathSciNetGoogle Scholar
  37. 37.
    Ma, C., Hori, Y.: The time-scaled trapezoidal integration rule for discrete fractional order controllers. Nonlinear Dyn. 38, 171–180 (2004)CrossRefMATHGoogle Scholar
  38. 38.
    Machado, J.A.T.: Fractional derivatives: probability interpretation and frequency response of rational approximations. Commun. Nonlinear Sci. Numer. Simul. 14(9–10), 3492–3497 (2009)CrossRefGoogle Scholar
  39. 39.
    Cao, J., Xue, S., Lin, J., Chen, Y.: Nonlinear dynamic analysis of a cracked rotor-bearing system with fractional order damping. J. Comp. Nonlinear Dyn. 8, 031008–14 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Junyi Cao
    • 1
  • Shengxi Zhou
    • 1
  • Daniel J. Inman
    • 2
  • Yangquan Chen
    • 3
  1. 1.Research Institute of Diagnostics and CyberneticsXi’an Jiaotong UniversityXi’anChina
  2. 2.Department of Aerospace EngineeringUniversity of MichiganAnn ArborUSA
  3. 3.School of EngineeringUniversity of CaliforniaMercedUSA

Personalised recommendations