Dynamic Characteristics And Electricity Generation Performance Of A Bistable Piezo-magneto-elastic Energy Harvester

Vibration-based piezoelectric energy harvesting has drawn accelerating interest over the last decade due to its application in supplying energy to low power consuming micro-electronic devices. Bistable piezoelectric energy harvester can induce large-amplitude vibration in a considerable wide range of low excitation frequency. Thus, it is considered as one of the most promising harvesting system for practical application. Until now, research on the bistable piezoelectric energy harvester still remains much worthwhile work for that the studies have been started presently. The responses of the harvesting system are influenced by many factors. The key point of the study is to explore the conditions leading to large-amplitude motions which can maximize the output power. It should be noticed that multiple nonlinear factors coexist in the system, such as magnetic nonlinearity, piezoelectric material nonlinearity and geometrical nonlinearity. Such nonlinearities can lead to multiple types of complex responses, which have direct influence on the efficiency of energy output. In the present paper, we focus on a bistable cantilevered piezoelectric energy harvester(BCPEH). A variety of approaches, such as theoretical modeling and analysis, numerical simulation and experimental research, are applied to investigate the bifurcation characteristics and complex responses of the system, to clarify the influence of system parameters and excitation conditions on motions and outputs respectively, and to study the manifestation and influence of these nonlinearities on the dynamic response of the bistable system. The main results are as follows.Valid mathematic model is a premise of predicting and analyzing the system’s dynamic characteristics. However, most of the models in the present literature are lumped-parameter ones and the nonlinear magnetic force are often regarded as a constant. To develop a reliable mathematical model, the repulsive potential model based on magnetic dipole of two permanent magnets is developed. Then a piezo-magneto-elastically coupled distributed-parameter model for the BCPEH is established by using the generalized Hamilton principle. Based on the distributed-parameters model, static bifurcation analysis is carried out to detect the magnet spacing range corresponding to bistable state. When the magnet spacing d is less than the bifurcation value, the system is a bistable one whose linear stiffness is negative. And with the decreasing of d, the wells are deeper and wider. Besides, the magnitude order of each term in equations varies with the change of d, which leads to the change of nonlinear strength. The bistable system with cross-well motions has strong nonlinearity.To characterize the periodic response of the BCPEH, we apply the method of harmonic balance to solve the approximate analytic solution. The results demonstrate that the high-energy attractor coexists with the low-energy attractor during the low frequency range. In the high frequency region, only the intrawell solution exists. With increasing excitation level, the frequency range corresponding to the interwell solution enlarges. Furthermore, high-energy solutions also come along with the multi-solutions. The realization of high-energy solution depends on the initial conditions. Subsquently, Melnikov theory is used to verify the condition of cross-sectional intersect of the homoclinic orbits and to predict the threshold for the occurrence of chaos. The results indicate that chaotic motion is inferred to occur within a low frequency range(5~30Hz) with exceptionally high likelihood.Then the dynamic responses, bifurcation characteristics and the output performance of the BCPEH are investigated with different essential parameters of the system by numerical simulations. It is found that large-amplitude limit cycle oscillation could lead to an optimal energy output. Several dynamical phenomena are discovered in BCPEH for the first time, such as the symmetric breaking bifurcation, coexistence of period-doubling bifurcation and intermittency to chaos, and phase-amplitude-modulation. Bifurcations cause power output not assuredly increasing with respect to the input energy. During the sequence of periodic-doubling bifurcations and the initial intermittency, the generating efficiency of the system is not influenced obviously. Whereas when the system is fully chaotic, the output is reduced significantly. In addition, the operating condition, such as frequency and amplitude of excitation, should be taken into account when choosing the optimal distance between the magnets. When the system is at monostable-into-bistable transition region(d is less than but near the bifurcation value) and under weak excitation, it is easier for the system to cross wells and exhibit optimal performance. Moreover, the system has an optimal matched load resistance corresponding to the maximum power output.A cantilevered piezoelectric energy harvesting device with adjustable magnet spacing is designed and manufactured. The response of the system with different magnet spacing and under different excitation conditions are measured experimentally. In the experiment, a variety of response types and symmetric breaking bifurcation are observed. Moreover, the qualitative trend of period-doubling bifurcation and intermittency routes to chaos is also observed. The conclusions of large-amplitude limit cycle oscillation leading to the optimal energy output and the effect laws of bifurcations on the power output are investigated experimentally. Through a disturbance applied to simulate the change of the initial conditions, coexistence of the multi-attractors and dependency of the high-energy solution’s realization on the initial conditions is verified. The effects of the magnet spacing and load resistance on the response characteristic are also experimentally examined. In addition, some new nonlinear phenomena are discovered in the experiment, and we hypothesize that they are caused by the piezoelectric material nonlinearity and geometrical nonlinearity.The influences of piezoelectric material nonlinearity on the primary resonance response of the piezo-elastic energy harvester(monostable) are revealed. In the region of primary resonance, the monostable system exhibits softening behavior due to the nonlinearity, coexistence of two different attractors and jumping behaviors are observed. Subsequently, considering both piezoelectric material and geometrical nonlinearities, we investigate their effects on the dynamic response of the BCPEH. The results show that the two nonlinearities result in asymmetric stiffness. Phase trajectories and potential curves lose symmetry, and the depths of the two potential wells are no longer equal. Hence, crossing between different potential wells needs different energies, which results in asymmetrical responses, such as the motion response prone to falling into unilateral potential well. That is the explanation for the phenomenon observed in the experiments.Finally, Based on the stochastic resonance, the generation mechanism of large-amplitude motion under weak excitation is explored. The results of numerical analysis show that it is possible to obtain the maximum power generation efficiency of the bistable system under low-frequency weak periodic and weak stochastic excitation when the condition of stochastic resonance is met.Above all, nonlinear dynamic characteristics of the BCPEH are investigated. Considering the major nonlinear factors, the present work focuses on the bifurcation, complex response and new nonlinear phenomena of the bistable system, especially their impact on the electrical output performance. The occurrence condition and parameter range of large-amplitude limit cycle oscillation are also obtained. The conclusions can provide some theoretical foundations for designing and optimizing BCPEHs.