Nonlinear Dyn DOI 10.1007/s11071-014-1320-6 ORIGINAL PAPER Chaos in the fractionally damped broadband piezoelectric energy generator Junyi Cao · Shengxi Zhou · Daniel J. Inman · Yangquan Chen Received: 28 November 2013 / Accepted: 18 February 2014 © Springer Science+Business Media Dordrecht 2014 Abstract Piezoelectric materials play a significant role in harvesting ambient vibration energy. Due to their inherent characteristics and electromechanical inter- action, the system damping for piezoelectric energy harvesting can be adequately characterized by frac- tional calculus. This paper introduces the fractional model for magnetically coupling broadband energy harvesters under low-frequency excitation and inves- tigates their nonlinear dynamic characteristics. The effects of fractional-order damping, excitation ampli- tude, and frequency on dynamic behaviors are proposed using the phase trajectory, power spectrum, Poincare map, and bifurcation diagram. The numerical analy- sis shows that the fractionally damped energy har- vesting 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 J. Cao (B ) · S. Zhou Research Institute of Diagnostics and Cybernetics, Xi’an Jiaotong University, Xi’an 710049, China e-mail: caojy@mail.xjtu.edu.cn D. J. Inman Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109-2140, USA e-mail: daninman@umich.edu Y. Chen School of Engineering, University of California, Merced, 5200 North Lake Rd., Merced, CA 95343, USA e-mail: ychen53@ucmerced.edu also demonstrated numerically and experimentally that the magnetically coupling piezoelectric energy har- vester 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 1 Introduction Vibration energy harvesting has been acknowledged as a potentially promising approach to power wire- less sensor nodes in structural health monitoring due to recent advances in miniaturized fabrication of sensors with low cost and low power consumption [1–6]. The varying excitation frequency coming from monitored machinery due to changing load and ambient tempera- ture in practice will result in the inefficiency of the tradi- tional linear resonant energy harvester. Consequently, the generated power output will be decreased largely. This issue has motivated several research groups to develop frequency tuning approaches to match the exci- tation and resonance frequencies, such as axial preload [7, 8] and passive and active stiffness tuning [9–11]. Additionally, the application of nonlinear dynamic phe- nomena to enhance its efficient operation over a wider 123