# Experimental Study and Parameter Optimization of a Magnetic Coupled Piezoelectric Energy Harvester
## 2 Modeling and Analysis
### Modeling
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Meaning** & **Symbol** & **Meaning** \\ \hline \(M\) & Equivalent mass & \(H\) & Height \\ \(C\) & Equivalent damping & \(W\) & Width \\ \(K\) & Equivalent stiffness & \(L\) & Length \\ \(m\) & Weight of the mass & \(\rho\) & Density \\ \(C_{p}\) & Capacity of piezoelectric material & \(F_{mag}\) & Magnetic force \\ \(V_{p}\) & Output voltage & \(B_{r}\) & Flux density of the magnet \\ \(R\) & Load resistance & \(\mu_{air}\) & Vacuum permeability \\ \(\Theta\) & Electromechanical coupling coefficient & \(r\) & Radius of the magnet \\ \(\mu\) & Compensation factor & \(h\) & Thickness of the magnet \\ \(E\) & Elastic Modulus of the beam & \(m_{mag}\) & Weight of the magnet \\ \(I\) & Moment of inertia & \(d\) & Spacing of the magnets \\ \(w\) & Deflection & \(D\) & Spacing of the beams’ fixed ends \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter list. assumed as always vertically aligned. Based on these assumptions, the approximate magnetic force is simplified [37]:  
\[F_{mag}(d)=\frac{6\pi B_{r}^{2}r^{4}h^{2}}{\mu_{air}d^{4}}. \tag{5}\]  
Suppose \(d_{0}\) is the static spacing of the magnets, \(\Delta d(t)=z_{1}(t)-z_{2}(t)\) is the change of the spacing over time, and the magnetic force is expressed as follows:  
\[F_{mag}(t)=\frac{6\pi B_{r}^{2}r^{4}h^{2}}{\mu_{air}(\Delta d(t)+d_{0})^{4}}. \tag{6}\]  
The \(d_{0}\) is different from the spacing of the fixed ends \(D\), and the analysis for \(d_{0}\) will be expanded below. The tip deflection of the cantilever beam is affected by the gravity of the beam itself and the concentrated mass of the tips. Therefore, based on the superposition theorem, the deflection equations of the two beams can be established as:  
\[E_{1}I_{1}w_{1}^{{}^{\prime\prime}}(x)=\big{(}m_{1}g+m_{mag}-F_{mag}\big{)}(L- x)+\frac{m_{p}g+\rho BLH_{1}g}{2L}x^{2}, \tag{7}\]  
\[EI_{2}w_{2}^{{}^{\prime\prime}}(x)=\big{(}m_{2}g+m_{mag}+F_{mag}\big{)}(L-x)+ \frac{\rho BLH_{2}g}{2L}x^{2}. \tag{8}\]  
Since the length \(L_{p}\) of the piezoelectric material is very close to the length \(L\) of the cantilever beam, it is assumed that the two parameters are equal in the modeling process. The elastic modulus of the piezoelectric composite beam is:  
\[E_{1}=\frac{H_{1}E+H_{p}E_{p}}{H_{1}+H_{p}}. \tag{9}\]  
The calculation of the moment of inertia requires firstly determining the position of the neutral plane which is shown as:  
\[H_{m}=\frac{0.5H_{1}^{2}+n\big{(}H_{1}+0.5H_{p}\big{)}H_{p}}{H_{1}+nH_{p}}, \tag{10}\]  
where \(n=E\)/\(E\) is the ratio of the modulus. The moment of inertia is:  
\[I_{1}=\frac{1}{12}WH_{1}^{3}+WH_{1}(H_{m}-0.5h_{1})^{2}+\frac{n}{12}WH_{p}^{3 }+nWH_{p}\big{(}H_{1}-H_{m}+0.5H_{p}\big{)}^{2}. \tag{11}\]  
The deflection static equation can be obtained by solving Equations (7) and (8) with the boundary condition (12), and the end deflection expression can be obtained. \[w_{1}(0)=0;w_{1}^{\prime}(0)=0;w_{2}(0)=0;w_{2}^{\prime}(0)=0. \tag{12}\]  
The static magnetic force can be expressed as:  
\[F_{mag}=\frac{6\pi B_{r}^{2}r^{4}h^{2}}{\mu_{air}}\frac{1}{\big{(}D-w_{1}(L)+ w_{2}(L)-2h\big{)}^{4}}. \tag{13}\]  
Equation (13) is solved jointly with the end deflection equations of the two beams to obtain the static deflection \(w_{1}(L)\) and \(w_{2}(L)\). Therefore, the dynamic magnetic force expression is:  
\[F_{mag}(d)=\frac{6\pi B_{r}^{2}r^{4}h^{2}}{\mu_{air}d^{4}}. \tag{5}\]0  
So far, the lumped parameter model of the magnetic coupled piezoelectric energy harvester has been established. Using the ode45 solver of MathWorks' Matlab software, the Runge-Kutta algorithm can be used to solve the ordinary differential equation of Equation (2) numerically, and the output voltages at different excitation frequencies can be obtained under simulation conditions.