## 2 Structure and Mathematic Model
### Modeling of the Nonlinear Magnetic Model  
In this paper, the magnetic dipole theory is used to establish a nonlinear magnetic model. Figure 3 is a diagram of the magnetic force model of IBPEH, in which \(r_{ij}\) represents the direction vector of the magnetic force that magnet A receives. As can be seen from the figure, in the IBPEH system, magnet A is subjected to the magnetic force of magnets B, C, and D. For the BPEH system, magnet A is only subjected to the magnetic force of magnet B.  
Figure 3: Magnetic force model of IBPEH.  
Figure 2: Restoring force model of the composite beam.  
The repulsion force of magnet A by magnet B is  
\[\begin{split} F_{m1}=\frac{h_{0}M_{A}S_{A}M_{B}S_{B}}{4\pi}\ \left\{-\frac{ \sigma_{1}}{\left(\
icefrac{{\sigma_{1}}{{(\vec{\ell}^{2}+\sigma_{1}^{2})}^{3}}}{ 2}}+\frac{\sigma_{1}}{\left[\
icefrac{{\sigma_{2}}}{{(\vec{\ell}_{h}+\vec{\ell} )}^{2}+\sigma_{2}^{2}}\right]^{\frac{3}{2}}}+\frac{\sigma_{2}}{\left[\
icefrac{{ \sigma_{2}}}{{(\vec{\ell}-\vec{\ell}_{h})}^{2}+\sigma_{2}^{2}}\right]^{\frac{3 }{2}}}\right.\\ \left.-\frac{\sigma_{2}}{\left(\vec{\ell}^{2}+\sigma_{2}^{2}\right)^{ \frac{3}{2}}}\right\}\end{split} \tag{3}\]  
The repulsion force of magnet A by magnet C is  
\[\begin{split} F_{m2}=\frac{\mu_{0}M_{A}S_{A}M_{B}S_{C}}{4\pi}\ \left\{-\frac{\sigma_{3}}{\left(\
icefrac{{\sigma_{1}}}{{( \vec{\ell}+\vec{\ell})}^{2}+\sigma_{2}^{2}}\right)^{\frac{3}{2}}}+\frac{\sigma _{3}}{\left[\
icefrac{{\sigma_{3}}}{{(\vec{\ell}_{h}+\vec{\ell})}^{2}+\sigma _{2}^{2}}\right]^{\frac{3}{2}}}\right.\\ \left.+\frac{\sigma_{4}}{\left[\
icefrac{{\sigma_{1}}}{{(\vec{\ell }+\vec{\ell}-\vec{\ell}_{h})}^{2}+\sigma_{2}^{2}}\right]^{\frac{3}{2}}}-\frac{ \sigma_{4}}{\left(\
icefrac{{\sigma_{1}}}{{(\vec{\ell}+\vec{\ell})}^{2}+\sigma _{2}^{2}}\right)^{\frac{3}{2}}}\right\}\end{split} \tag{4}\]  
The repulsion force of magnet A by magnet D is  
\[\begin{split} F_{m3}=\frac{\mu_{0}M_{A}S_{A}M_{D}S_{D}}{4\pi}\ \left\{-\frac{\sigma_{5}}{\left(\
icefrac{{\sigma_{1}}}{{( \vec{\ell}+\vec{\ell})}^{2}+\sigma_{5}^{2}}\right)^{\frac{3}{2}}}+\frac{\sigma _{5}}{\left[\
icefrac{{\sigma_{1}}}{{(\vec{\ell}_{h}+\vec{\ell})}^{2}+\sigma _{5}^{2}}\right]^{\frac{3}{2}}}\right.\\ \left.+\frac{\sigma_{6}}{\left[\
icefrac{{\sigma_{1}}}{{(\vec{ \ell}+\vec{\ell}-\vec{\ell}_{h})}^{2}+\sigma_{2}^{2}}\right]^{\frac{3}{2}}}- \frac{\sigma_{6}}{\left(\
icefrac{{\sigma_{1}}}{{(\vec{\ell}+\vec{\ell})}^{2}+ \sigma_{6}^{2}}\right)^{\frac{3}{2}}}\right\}\end{split} \tag{5}\]  
For the BPEH system, the magnetic force received by magnet A is \(F_{m1}\), while in the IBPEH system, the total magnetic force received by magnet A is  
\[F_{m}=F_{m1}+F_{m2}+F_{m3} \tag{6}\]  
In the formula,\(\sigma_{1}=y(t),\sigma_{2}=y(t)+h_{a}\theta,\sigma_{3}=y(t)+d_{g},\sigma_{4}=y (t)+h_{a}\theta+d_{g},\sigma_{5}=y(t)-d_{g},\sigma_{6}=y(t)+h_{a}\theta-d_{g}\). The displacement of the free end of the composite beam in the y-axis direction at time t is \(y(t)\). \(M_{A},M_{B},M_{C}\), and \(M_{D}\) are the magnetization of magnets A, B, C, and D, respectively. \(S_{B},S_{C},S_{D}\), and \(S_{A}\) are the surface areas of magnets B, C, D, and A facing each other, and the deflection angle \(\theta\) of magnet A and displacement \(y(t)\) satisfy the following relationship:  
\[\theta(t)=\arctan\left(y^{\prime}(t)\right) \tag{7}\]  
where \(y^{\prime}(t)=\frac{\partial y(t)}{\partial x}\).  
The potential energy generated between the magnets in the piezoelectric energy harvester system can be obtained by integrating the magnetic force. For the BPEH system, the potential energy generated between the magnets is  
\[U_{m1}=\int F_{m1}dy \tag{8}\]  
For the IBPEH system, the potential energy generated between the magnets is  
\[U_{m2}=\int F_{m}dy \tag{9}\]