## 3 Theoretical Modeling
### Modeling of the Proposed HEH  
The electromagnetic energy harvesting part can be simplified as a 1 DOF spring-mass-damper system, as shown in Figure 1. The magnetic force is considered as a concentrated force applied on the centers of both magnets along the \(z\)-axis direction. Assume that a distributed force \(F(x,t)\) is applied on the beam. Based on the modal analysis, the corresponding modal force for the fundamental vibration mode, \(F_{1}\), can be expressed as  
\[F_{1}=\int_{0}^{L}F(x,t)\psi_{r}(x)\mathrm{d}x \tag{18}\]  
Accordingly, the modal magnetic force applied on the piezoelectric beam is given as  
\[F_{1}=F_{\mathrm{m}}(L-R,t)\psi_{r}(L-R) \tag{19}\]  
where \(R\) is the radius of the magnetic proof mass.  
Due to the low excitation frequency, the coil inductance is ignored [42]. Combing Equations (12) and (19), the electromechanically coupling governing equations of the HEH for the fundamental mode can be derived as  
\[\left\{\begin{array}{c}M\ddot{r}+C\dot{r}+Kr-\theta_{\mathrm{p}}v_{1}-F_{ \mathrm{m}}\psi_{r}(L-R)=-B_{\mathrm{f}}\ddot{u}_{\mathrm{b}}\\ M_{2}\dot{u}_{2}+C_{2}\dot{u}_{2}+K_{2}u_{2}+\theta_{\mathrm{em}}I_{2}+F_{ \mathrm{m}}=-M_{2}\dot{u}_{\mathrm{b}}\\ \theta_{\mathrm{p}}\dot{r}+C_{\mathrm{p}}\ddot{v}_{1}+v_{1}/R_{1}=0\\ \theta_{\mathrm{em}}\dot{u}_{2}-(R_{\mathrm{c}}+R_{2})I_{2}=0\end{array}\right. \tag{20}\]  
where \(M_{2}\), \(C_{2}\), and \(K_{2}\) are the lumped mass, mechanical damping, and spring stiffness of the magnetic oscillator, respectively. \(R_{\mathrm{c}}\) is the internal resistance of the coil.  
Letting  
\[\omega_{1}=\sqrt{K/M},\quad\omega_{2}=\sqrt{K_{2}/M_{2}},\quad\zeta_{1}=\frac{ C}{2\sqrt{KM}},\quad\zeta_{2}=\frac{C_{2}}{2\sqrt{K_{2}M_{2}}} \tag{21}\]  
and defining a state vector \(\mathbf{X}=\left[\begin{array}{cccc}X_{1}&X_{2}&X_{3}&X_{4}&X_{5}\end{array} \right]^{t}=\left[\begin{array}{cccc}r&\dot{r}&u_{2}&\dot{u}_{2}&v_{1}\end{array} \right]^{t}\) (\(t\) denotes the transpose of the vector), the governing Equation (20) can be written in the state space form as  
\[\dot{\mathbf{X}}=\left[\begin{array}{c}\dot{r}\\ \ddot{r}\\ \dot{u}_{2}\\ \ddot{u}_{2}\\ \dot{v}_{1}\end{array}\right]=\left[\begin{array}{c}X_{2}\\ -\omega_{1}^{2}X_{1}-2\omega_{1}\zeta_{1}X_{2}+\theta_{\mathrm{p}}X_{5}/M+F_{ \mathrm{m}}\psi_{r}(L-R)/M-B_{\mathrm{f}}\ddot{u}_{\mathrm{b}}/M\\ \phantom{-\omega_{2}^{2}X_{3}-(2\omega_{2}\zeta_{2}+C_{\mathrm{em}}/M_{2})\ X_{4}-F_{ \mathrm{m}}/M_{2}-\ddot{u}_{\mathrm{b}}}\\ \phantom{-\omega_{2}^{2}X_{2}/C_{\mathrm{p}}-X_{5}/(R_{1}C_{\mathrm{p}})} \end{array}\right] \tag{22}\]  
where \(C_{\mathrm{em}}=\theta_{\mathrm{em}}^{2}/(R_{2}+R_{\mathrm{c}})\) is the electromagnetic damping.  
The output power delivered to the external loads \(R_{1}\) and \(R_{2}\) are respectively given as  
\[P_{\mathrm{p}}=\frac{1}{T}\int_{0}^{T}\frac{v_{1}^{2}}{R_{1}}\mathrm{d}t \tag{23}\]  
\[P_{\mathrm{em}}=\frac{1}{T}\int_{0}^{T}\left(\frac{\theta_{\mathrm{em}}\dot{u} _{2}}{R_{2}+R_{\mathrm{c}}}\right)^{2}R_{2}\mathrm{d}t \tag{24}\]  
where \(T\)=\(2\pi/\omega\) is the cycle of the base excitation. \(\omega\) is the angular velocity. The total output power of the HEH is  
\[P=P_{\mathrm{p}}+P_{\mathrm{em}} \tag{25}\]