# Nonlinear Dynamics of a Piezoelectric Bistable Energy Harvester Using the Finite Element Method
###### Abstract  
The conversion of ambient mechanical vibrational energy into electrical energy through piezoelectric devices has received an increasing attention in recent years. The main challenges are to develop efficient devices that operate over a wide frequency range, adapting to diverse environmental energy sources. This work presents a framework for the analysis of a nonlinear vibration-based energy harvesting devices combining the nonlinear finite element method with a reduced-order model, which provides a broader dynamical investigation. On this basis, a flexible tool is developed, allowing the multimodal analysis of nonlinear systems. A bistable piezoelectric energy harvesting device is investigated considering the influence of multimodal and nonlinear effects on the system performance. Bistability is due to magnetic interactions among magnets and the beam tip, modeled by cubic nonlinearities. Numerical simulations show the influence of vibration sources on the dynamics and performance of the device. Nonlinear effects furnish rich dynamics, presenting periodic and chaotic responses. All these effects can be combined to enhance energy harvesting capacity. a pacs: 12.30.-c, 12.30.-d, 12.30. Different design configurations can be imagined in order to reach this goal. [PERSON] and [PERSON] [19] proposed a pizza-shaped irregular structure while [PERSON] and [PERSON] [20] proposed a star-shaped structure exploiting the same idea. In addition, different configurations can be employed to aggregate multidirectionality to the device. In this regard, [PERSON] and [PERSON] [20] showed that the incorporation of pendulum masses can propitiate the multidirectional aspects necessary to deal with direction uncertainty on the energy sources. Dual-beam design combining multiple degrees of freedom and compactness is explored by [PERSON] and [PERSON] [21; 22]. Nonlinearities are another attempt to increase the performance of energy harvesting devices. In this regard, multistable energy harvesting devices attracted attention due to their unique characteristics, creating different equilibrium configurations that can be explored to increase energy harvesting capacity [21; 22; 23]. Among various piezoelectric energy-harvesting structures, permanent magnets are often used in conjunction with these structures by exploiting magnetic coupling to design monostable [24; 25], bistable [26; 27] and multistable [28; 29; 30; 31] nonlinear energy harvesters. [PERSON] et al. [32] employed magnets in vibration energy harvesters to investigate both monostable and bistable configurations at various excitation levels to improve the performance of conventional linear energy harvesters. The primary advantage of a bistable system over a monostable system is that inter-well motion can cause a significant deformation, resulting in higher output power. However, these configurations are associated with energetic barriers that need to be overcome to reach interesting behaviors for energy harvesting [33; 34]. Consequently, the energy source needs to have an energy level high enough for a good harvesting performance since low energy levels impose situations where the system cannot overcome energetic barriers, presenting a poor performance [34]. Multistable energy harvesters with magnetic coupling have been investigated to facilitate overcoming these barriers when compared with bistable configurations, generating high energy output over a broader range of frequencies [28; 35]. The main drawback is the added complexity in system dynamics and energy conversion. Nonlinear energy harvesting devices are investigated for different types of ambient excitation, such as harmonics [36; 37] and random [38; 39], showing a considerable increase in energy generation. The nonlinear magnetic force is a critical factor influencing the energy scavenging efficiency and the dynamic behavior of vibration-based systems. The magnetic force can primarily be obtained from experimental measurement, numerical simulation, and analytical calculations [29].