## 2 Structure and Mathematical Model of CTEHS
### Mathematical Model of CTEHS
#### 2.2.3. Motion Equation of CTEHS  
In this research, the motion equation of CTEHS considering the gravity effect is established based on the Lagrange equation. The Lagrange function of the system is as follows  
\[L_{a}=T_{l}+T_{m}+W_{p}-U_{r}-U_{m}-U_{g} \tag{5}\]  
where \(T_{l}\) is the kinetic energy of the combined beam and PVDF, \(T_{m}\) is the kinetic energy of magnet 1, \(W_{p}\) is the electric energy produced by the PVDF, \(U_{r}\) is the resilient potential energy of the combined beam and PVDF, \(U_{m}\) is the magnetism potential energy between three magnets, and \(U_{g}\) is the gravitational potential energy of the magnet 1. The following are the specific manifestations of these energies:  
\[T_{l}=\frac{1}{2}\big{(}\rho_{p}A_{p}+\rho_{b}A_{b}\big{)}\int_{0}^{L}\!\! \left[\frac{\partial u(x,t)}{\partial x}+\dot{y}(t)\right]^{2}\!\!dx \tag{6}\]  
\[T_{m}=\frac{1}{2}M_{\rm e}\bigg{\{}\left[\frac{\partial u(x,t)}{\partial t} \right]_{x=L}+\dot{y}(t)\bigg{\}}^{2}+\frac{1}{2}I_{\rm e}\bigg{[}\frac{ \partial^{2}u(x,t)}{\partial t\partial x}\bigg{]}_{x=L}^{2} \tag{7}\]  
\[W_{p}=\frac{1}{4}e_{31}k_{p}\big{(}h_{b}+h_{p}\big{)}V(t)\bigg{[}\frac{ \partial u(x,t)}{\partial x}\bigg{]}_{x=L}+\frac{1}{2}C_{p}V^{2}(t) \tag{8}\]  
\[U_{r}=\int F_{r}\mathrm{d}u(L,t) \tag{9}\]  
\[U_{m}=\int F_{m}\mathrm{d}u(L,t) \tag{10}\]  
\[U_{g}=F_{g}u(L,t)=M_{\rm e}gu(L,t) \tag{11}\]  
where \(\rho_{b},\rho_{p}\) are the material densities of the combined beam and PVDF, \(A_{b},A_{p}\) are the cross-section areas of the combined beam and PVDF, and \(h_{b},h_{p}\) are the thicknesses of the combined beam and PVDF, respectively. \(\dot{y}(t)\) is the vibration velocity of the base, \(M_{\rm e},I_{\rm e}\) are the mass and moment of inertia of the end magnet 1, respectively, \(V(t)\) is the response voltage of PVDF, \(e_{31},C_{p}\) are the piezoelectric constants and equivalent capacitance of PVDF, respectively.  
Figure 3. Resilience fitting results.  
The end vibration displacement \(u(x,t)\) of the combined beam can be expressed according to the [PERSON] method as  
\[u(x,t)=\sum_{i=1}^{N}\psi_{i}(x)\gamma_{i}(t) \tag{12}\]  
where \(\psi_{i}(x),\gamma_{i}(t)\) express the \(i\)-th mode shape function and modal coordinates of the combined beam, respectively. The research in this paper is mainly aimed at low-frequency vibration. Therefore, we only take into account the 1 st-order modalities of the beam, and the specific expression of the mode shape function \(\psi(x)\) is  
\[\psi(x)=1-\cos\left(\frac{\pi x}{2L}\right) \tag{13}\]  
Combining all the above equations, the electromechanical coupling motion equation of the system considering gravity can be obtained as follows:  
\[\left\{\begin{matrix}M\ddot{\gamma}(t)+C\dot{\gamma}(t)+F_{r}-\Theta V(t)+F_{ m}+F_{g}=-\beta\ddot{y}(t)\\ \Theta\dot{\gamma}(t)+V(t)/R+C_{p}\dot{V}(t)=0\end{matrix}\right. \tag{14}\]  
where  
\[M=\left(\rho_{p}A_{p}+\rho_{b}A_{b}\right)\int_{0}^{L}\psi^{2}(x)\mathrm{d}x+M _{\mathrm{e}}\psi^{2}(L)+I_{\mathrm{e}}\left(\psi^{\prime}(L)\right)^{2} \tag{15}\]  
\[\Theta=\frac{1}{2}\epsilon_{31}k_{p}\left(l_{p}+h_{b}\right)\psi^{\prime}(L) \tag{16}\]  
\[\beta=\left(\rho_{p}A_{p}+\rho_{b}A_{b}\right)\int_{0}^{L}\psi(x)\mathrm{d}x+M _{\mathrm{e}}\psi(L) \tag{17}\]  
\[\ddot{y}(t)=Acos(2\pi ft) \tag{18}\]  
In Equation (18), \(A\) is the acceleration amplitude of the external vibration, and \(f\) is the frequency of the external vibration.