# Experimental Study and Parameter Optimization of a Magnetic Coupled Piezoelectric Energy Harvester
## 2 Modeling and Analysis
### Modeling  
In order to facilitate the analysis of structural characteristics, the modeling process is described in detail below. The main body of the structure is still a conventional cantilever and adds coupling of the magnetic field. The amplitude and bandwidth characteristics of the first-order response of the system dynamics are focused in this paper. The lumped parameter model can be established and calculated rapidly, and the first-order dynamic response can be accurately predicted under the action of the compensation factor [35]. Therefore, this paper plans to select the lumped parameter method for  
Figure 2: Schematic diagram of the magnetic coupled piezoelectric energy harvester. modeling. Different from [PERSON]'s model [32], a more precise model is established that considers the influence of the static position of the magnets. Some of the parameters used in the modeling process are listed in Table 1 and marked in Figure 2. For ease of description, the lower corner 1 represents the piezoelectric cantilever beam, the lower corner 2 represents the auxiliary cantilever beam and the lower corner p represents the piezoelectric material. The electromechanically coupled lumped-parameter model of the linear cantilever energy harvester is expressed as [36]:  
\[\left. \begin{array}{ll}Mz^{\prime\prime}(t)+Cz^{\prime}(t)+Kz(t)+\Theta V_{p} (t)=\mu Mz^{\prime\prime}_{0}(t)\\ C_{p}V^{\prime}_{p}(t)-\Theta z^{\prime}(t)=-V_{p}(t)/R\end{array} \right\}, \tag{1}\]  
where \(z(t)\) is the relative displacement of the mass and \(z_{0}(t)\) is the base motion. The directions of the dipoles are assumed as always vertically aligned. After introducing nonlinear magnetic force, a two-degree-of-freedom nonlinear piezoelectric dynamic model can be obtained:  
\[\left. \begin{array}{ll}M_{1}z^{\prime\prime}_{1}(t)+C_{1}z^{\prime}_{1}(t)+K _{1}z_{1}(t)+\Theta V_{p}(t)+QF_{mag}(t)=\mu_{1}M_{1}z^{\prime\prime}_{0}(t) \\ M_{2}z^{\prime}_{2}(t)+C_{2}z^{\prime}_{2}(t)+K_{2}z_{2}(t)-QF_{mag}(t)=\mu_{ 2}M_{2}z^{\prime\prime}_{0}(t)\\ C_{p}V^{\prime}_{p}(t)-\Theta z^{\prime}_{1}(t)+V_{p}(t)/R=0\end{array} \right\}, \tag{2}\]  
where \(Q\) is the system factor, \(Q=1\) when the beams are attracted, and \(Q=-1\) when repulsed. The parameters in the equation are analyzed in detail below, taking into account the influence of piezoelectric materials, static position and other influencing factors. To include the complete contribution of the piezoelectric layer in the system dynamics, the effect of the mass and the inertia of the piezoelectric layer are considered in the proposed model. The equivalent mass and stiffness of the two beams can be expressed as:  
\[M_{1}=\frac{33}{140}(m_{p}+\rho BLH_{1})+m_{1}+m_{mag};M_{2}=\frac{33}{140} \rho BLH_{2}+m_{2}+m_{mag}, \tag{3}\]  
\[K_{1}=\frac{3E_{1}I_{1}}{L^{3}};K_{2}=\frac{3 EI_{2}}{L^{3}}. \tag{4}\]  
\(\mu\) is the correction factor of lumped parameter forcing function [35]. \(\Theta\) can be obtained by short circuit current method [32]. C can be obtained by logarithmic decay method or viscous damping simplified empirical estimation method. Two flat cylindrical magnets of the same size are used on the tip of the beams. To approximate the magnetic force, the force is assumed as a dipole-dipole interaction and the directions of the dipoles are