# Nonlinear Dynamics of a Piezoelectric Bistable Energy Harvester Using the Finite Element Method
## 3 Numerical Simulations  
Numerical simulations are developed to show the system dynamics that define the energy harvesting capacity. The employed models are initially compared with other numerical and experimental tests available in the literature, promoting a model verification. Afterward, the reduced-order model is of concern in order to observe the general characteristics of the system dynamics. The fourth-order Runge-Kutta method is employed considering time steps less than 0.001 s, defined after a convergence analysis. A bifurcation diagram is employed to provide a global comprehension of the system dynamics. This diagram is built by considering a stroboscopic view of the response under slow quasi-static variation in a parameter, using a Poincare map. Figure 5 presents the bifurcation diagram varying the base displacement (\(w_{0}\)) that represent the external stimulus of Equations (9) and (10), defined as \(w_{b}=w_{0}\)sin(\(\Omega\tau\)), with a constant frequency excitation of 70 Hz. There is a noticeable period-1 response followed by bifurcations that reaches a chaotic region and, following this, another period-1 response is reached. Three different types of responses can be identified in terms of energy harvesting capacity, essentially related to motion amplitude and therefore, with the dynamical response. These responses are associated with external stimulus, characterized by forcing amplitude and frequency. The first one is the intra-well behavior, related to low energy stimulus (0.15 mm, 70 Hz), where the system response is characterized by oscillations limited to only one equilibrium point (Figure 6a). The second behavior is a chaotic response (0.4 mm, 70 Hz) presented in Figure 6b: a solution associated with a strange chaotic attractor that visits all equilibrium points. Finally, the third kind of behavior is an inter-well response (1.0 mm, 70 Hz), where both stable equilibrium points are visited. The reduced-order model scenarios furnish the dynamic perspective of the energy harvester, guiding the more complex investigation of the FE model, which is now in focus. System performance is monitored by considering output voltage. The finite element model uses a structured mesh composed of 6000 elements. Time steps are determined as a function of the excitation frequency (\(\Omega\)), with at least 40 points per excitation period (\(T=1/\Omega\)) ensuring good convergence. Figure 5: Bifurcation diagram built with the reduced-order model. Once again, an excitation frequency of 70 Hz is adopted and initially, consider a base excitation of 0.15 mm amplitude. Figure 7a presents the system response showing a periodic pattern with small intra-well oscillation, generating an rms voltage about 0.8 V. A chaotic-like response is obtained by increasing the base excitation amplitude for 0.4 mm, Figure 7b, an inter-well response where external energy is enough to visit both stable equilibrium points, and generating an rms voltage of 2.2 V. The Poincare section of this response shows a strange attractor illustrated in Figure 7b. A new increase in the external energy is observed in the presence of a periodic pattern. Figure 6: Reduced-order model analysis considering an excitation frequency of 70 Hz. Phase space and displacement time-history for different excitation amplitudes: **(a)** Intra-well response (0.15 mm); **(b)** chaotic response (0.4 mm); and **(c)** inter-well response (1.0 mm). The blue and gray solid lines represent the time-history, while the black and blue markers represent the Poincaré’s section. amplitude to 1 mm is in response to a periodic inter-well that generates an output voltage of 4.6 V. It should be pointed out that the different kinds of response are directly associated with the energy harvesting capacity, clearly establishing the connection between collected energy and dynamics.