Nonlinear dynamics for broadband energy harvesting: Investigation of a bistable piezoelectric inertial generator
Introduction
Vibratory energy harvesting has become a prominent research endeavor, owing to both its application potential and technical challenge. Energy harvesting devices primarily rely upon the physics of electromagnetic induction and piezoelectricity to convert ambient mechanical motion into usable forms. Some of the significant applications include wireless sensor systems [1], [2], [3], self-powered microelectronics [4], [5], [6], autonomous battery recharging [7], [8], and many other applications (e.g. see the review articles by Anton et al. [9] and Beeby et al. [10]). Initial concept demonstrations from nearly twenty years ago and much of the literature today lean heavily on optimally tuned linear resonant systems for energy extraction. Although resonance yields markedly increased response in a linear system, efficient operation demands an excitation source that is well characterized by a single harmonic. In recent years it has become apparent that continued success demands enhanced capability to draw energy from more complex and realistic multi-frequency and stochastic environments. Since linear devices exhibit resonance within a very narrow frequency range, they are ill-suited to answer this challenge and research has begun to explore active and semi-active strategies to increase their efficiency [11], [12], [13].
Within the past few years, however, dynamicists have begun to demonstrate the efficacy of nonlinear systems to overcome modern challenges in vibratory energy harvesting. One of the first experimental investigations of an energy harvester specifically designed to exhibit a nonlinear response was described in Ref. [14]. In particular, the authors showed how magnetic levitation could be used to extend device bandwidth through a hardening frequency response. A phenomenologically similar device based on piezoelectric energy conversion in Ref. [15] was shown to be capable of either a softening or a hardening response as a consequence of magnetic interactions with the elastic potential of an electroelastic beam. The nonlinear response was additionally shown to exceed the power output and bandwidth of the equivalent linear system. In both devices, the nonlinearity also enables tuning to counteract manufacturing imperfections. Similar bandwidth widening and increased power from hardening phenomena were studied with a different device in Ref. [16]. Softening frequency response characteristics in a parametrically forced piezoelectric device with structural nonlinearities have been reported by Daqaq et al. [17], who also carefully investigated the influence of electromechanical coupling parameters. Many of these systems demonstrate the advantages of monostable Duffing oscillators for increased bandwidth. For stochastically excited systems, however, this type of nonlinear design may be less desirable [18].
The bistable Duffing oscillator (commonly referred to as the double-well oscillator or the Duffing–Holmes oscillator) has also been investigated for energy harvesting. A bistable mechanical beam was numerically investigated by Shahruz [19], who focused on large amplitude interwell oscillations to illustrate potential benefits of a nonlinear approach to energy harvesting. The system is shown to achieve amplitudes four times greater than its linear counterpart while stochastically excited. Later, energy harvesting experiments with a noise activated bistable piezoelectric device were carried out in Ref. [20] and a more theoretical discussion was presented in Ref. [21]. Contemporaneous to this investigation, Erturk et al. [22] experimentally resurrected the ferromagnetic buckling beam in Ref. [23] with piezoelectric laminates. Forced oscillations were studied and the existence of a high energy large orbit attractor was experimentally verified. In a forthcoming article, Mann and Owens [24] experimentally and numerically evaluate potential well escape phenomena in a bistable electromagnetic harvester. Their investigation identifies similar attributes and challenges in the nonlinear design as discussed in this work.
In this paper, we explore in depth the variety of multiple attractors that may exist across a broad frequency range. Specifically, a bistable energy harvester is modeled and tested to demonstrate this phenomena. The energy harvester is comprised of a discontinuously laminated piezoelectric beam with a nonlinear boundary condition imposed by repelling permanent magnets. A mathematical model is derived from energy principles and numerically simulated across a range of excitation amplitudes and frequencies. Since many nonlinear energy harvesting designs are increasingly incorporating permanent magnets [14], [19], [21], [20], [22], [25], we suggest and apply an analytical formulation to describe these nonlinear forces. We additionally present the effect of a bifurcation parameter as either a fixed or adaptable tuning parameter.
The content of this paper is organized as follows. In Section 2 the bistable configurations is illustrated and modeled. The Euler–Lagrange equations are used to derive a discretized model, where we employ basis functions for a discontinuously laminated beam. In addition, an analytical expression for the potential energy field between repelling magnets is derived. Section 3 presents the results of numerical simulations and experimental results are given in Section 4. Conclusions are discussed in Section 5.
Section snippets
Nonlinear energy harvester: Description and modeling
Fig. 1 illustrates the nonlinear energy harvester that is modeled, numerically investigated and experimentally tested. The system is a driven bistable oscillator comprised of a linearly elastic piezoelectric cantilever with a nonlinear magnetic force at the free end. The cantilever end mass is a Neodymium permanent magnet that is oriented with opposite polarity to the field of a fixed magnet. The free magnet (magnet ) may either be pushed toward or withdrawn away from the cantilever system
Numerical investigation of power generation from nonlinear oscillations
This section investigates the forced response of energy harvester and the power delivered to an electrical load. Eqs. (17a)–(17b) were simulated across a range of excitation parameters to demonstrate broad frequency response, multiple attractors, and tuning capability.
Experimental results
This section describes the experimental equipment and tests performed to obtain both quantitative comparison between the theoretical model and qualitative agreement with numerical simulation predictions. A primary objective of the experimental tests was to verify the existence of multiple attractors and hysteresis.
Summary and conclusions
This paper investigated the merits of harvesting energy from a bistable nonlinear oscillator. Most importantly, a broad frequency response was numerically predicted and experimentally validated. A full nonlinear model was derived with an analytical magnet formulation and proper modal expressions to account for discontinuous piezoelectric laminates. In addition, methods of reaching multiple attractors was discussed through perturbations or mechanical tuning or impulse type perturbations.
Acknowledgments
The authors would like to acknowledge financial support from Dr. Ronald Joslin through an ONR Young Investigator Award.
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