# Nonlinear Dynamics of a Piezoelectric Bistable Energy Harvester Using the Finite Element Method
###### Abstract
While experimental measurements limit the description of nonlinear magnetic force when system parameters change, numerical simulations can accurately estimate magnetic force with parametric variation, but it requires considerable computational cost. Although nonlinear magnetic forces in energy harvesting systems are calculated employing magnetic dipole method, equivalent magnetizing current model, or magnetic charge model, some researchers express magnetic forces by fitting odd-order polynomials for some specific situations. In particular, theoretical analysis of a Duffing-type system is a relatively common modeling alternative to investigating energy harvesters with magnetic coupling [34; 36; 38; 40]. This strategy accurately represents nonlinear dynamical systems, considerably reducing the modeling complexity. The literature presents analytical and numerical solutions of piezoelectric energy harvesting devices based on simple structures. However, the modeling of complex structures is related to difficulties to accurately describe their behaviors. In this regard, the finite element method (FEM) is an alternative for the analysis of these energy harvesting systems. Various attempts have been used to model piezoelectric energy harvesters through FEM [41; 42; 43; 44]. [PERSON] et al. [41] presented a coupled piezoelectric-circuit model to analyze the power output of the system. [PERSON] et al. [42] compared analytical and finite element (FE) models with experimental results presenting good agreements. Concerning nonlinear devices, [PERSON] and [PERSON] [43] proposed a technique to study devices with magnetic interaction by using a nonlinear spring element to model magnetic forces. Recently, the authors investigated a three degrees-of-freedom system to harvest electrical energy from wideband, low-frequency, and low-amplitude ambient vibrations. [PERSON] and [PERSON] [19; 20] employed the FEM to investigate novel energy harvesting designs as pizza-shaped and star-shaped structures. Although many efforts have been made, the design of efficient energy harvesting devices capable of adapting to different ambient vibration sources remains a challenging task. 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, allowing a broader dynamical investigation. The device harvester is a bistable piezoelectric beam where the nonlinear behavior is due to magnetic interactions exerted by magnets in the beam tip. Ambient vibration sources are represented by harmonic base excitation. The use of the finite element method defines a flexible representation that allows multimodal analysis over a wide frequency range. Since the nonlinear dynamics perspective is essential for the design of the energy harvester, this framework allows a broader representation of the system, exploiting the better possibilities for proper operational conditions. Numerical simulations of the finite element model are developed using ANSYS 17 software and results show a qualitative agreement with reduced-order model described by ordinary differential equations. The system presents rich dynamical responses of the device, including chaos. The framework shows the importance to employ a proper model to describe the system dynamics, which can identify enhanced operational conditions.