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


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.


Energy harvesting Chaos  Piezoelectricity Nonlinear vibrations  Fractional calculus 



This project is being jointly supported by the National Natural Science Foundation of China (Grant No. 51075317), Program for New Century Excellent Talents in University (Grant No. NCET-12-0453), and International Cooperation Project in Shaanxi Province (Grant No. 2011KW-21).


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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

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