## 3 Theoretical Modeling
### Modeling of Linear 1 DOF PeH  
When both induction coil and magnetic oscillator are removed, the HEH becomes a conventional 1 DOF linear PEH (LPEH). The LPEH consists of a bimorph piezoelectric cantilever beam and a proof mass. The electromechanical coupling model can be derived from the energy method [38,39], which is based on the modal analysis and small-signal piezoelectric material constitutive law. The proof mass is modeled as a mass point and rotation inertia is neglected. The beam is considered to be an Euler-Bernoulli beam, whose shear deformation and rotation inertia are neglected. The piezoelectric patches are assumed to be perfectly bonded with the substrate and the thicknesses of the electrodes are neglected. The electrical field is assumed to be constant across the thickness of the piezoelectric element. Geometric nonlinearity of the beam is ignored under the assumption of small deflection. Considering that the frequency of excitation source in the environment is below 200 Hz [40], the fundamental mode of the PEH is the research focus.  
Assume that the displacement of the base excitation is \(u_{\mathrm{D}}(t)\). In the Rayleigh-Ritz approach, the displacement of the PEH relative to the base \(u(x,t)\) and electrical potential for the electrode pair \(\varphi(z,t)\) can be written as  
\[u(x,t)=\psi_{r}(x)r(t) \tag{10}\]  
\[\varphi(z,t)=\psi_{\mathrm{p}}(z)v(t) \tag{11}\]  
where \(x\) is the axial position coordinate of the beam. \(\psi_{r}(x)\) is the mechanical mode shape function. \(r(t)\) is the modal coordinate. \(\psi_{v}(z)\) is the electrical potential distribution function. \(v(t)\) is the electrical temporal coordinate.  
The electromechanical coupling governing equations are simplified as  
\[\left\{\begin{array}{c}M\ddot{r}+C\dot{r}+Kr-\theta_{\mathrm{p}}v_{1}=-B_{ \mathrm{f}}\ddot{u}_{\mathrm{D}}\\ \theta_{\mathrm{p}}\dot{r}+C_{\mathrm{p}}\dot{v}_{1}+v_{1}/R_{1}=0\end{array}\right. \tag{12}\]  
where the mass \(M\), stiffness \(K\), piezoelectric coupling coefficient \(\theta_{\mathrm{p}}\), capacitive \(C_{\mathrm{p}}\), and forcing factor \(B_{\mathrm{f}}\) are obtained as follows  
\[M=\int_{0}^{L}m\psi_{r}^{2}\mathrm{d}x+M_{\mathrm{t}}\psi_{r}^{2}(L) \tag{13}\]  
\[K=\int_{V_{\mathrm{s}}}c_{\mathrm{s}}(-z\psi_{r}^{\prime\prime})^{2}\mathrm{ d}V_{\mathrm{s}}+\int_{V_{\mathrm{p}}}c_{11}^{\mathrm{E}}(-z\psi_{r}^{\prime \prime})^{2}\mathrm{d}V_{\mathrm{p}} \tag{14}\]  
\[\theta_{\mathrm{p}}=\int_{V_{\mathrm{p}}}e_{31}(-z\psi_{r}^{\prime\prime})(- \
abla\psi_{v})\mathrm{d}V_{\mathrm{p}} \tag{15}\]  
\[C_{\mathrm{p}}=\int_{V_{\mathrm{p}}}e_{33}^{\mathrm{S}}(-\
abla\psi_{v})^{2} \mathrm{d}V_{\mathrm{p}} \tag{16}\]  
\[B_{\mathrm{f}}=\int_{0}^{L}m\psi_{r}\mathrm{d}x+M_{\mathrm{t}}\psi_{r}(L) \tag{17}\]  
where the subscripts \(\mathrm{p}\) and \(\mathrm{s}\) indicate the piezoelectric element and substrate, respectively. \(V_{\mathrm{p}}\) and \(V_{\mathrm{s}}\) denote the volumes. \(M_{\mathrm{t}}\) is the proof mass. \(m\) is the mass per length in the axial direction. \(z\) indicates the distance from the neutral axis of the piezoelectric beam. \(v_{1}\) is the voltage across external resistance \(R_{1}\) for the piezoelectric patches. \(\ddot{u}_{\mathrm{b}}\) is the excitation acceleration. \(C\) denotes the mechanical damping, which can be measured by the logarithmic decrement method or bandwidth method [41]. The subscript 1 corresponds to the \(x\) direction and 3 corresponds to the \(z\) direction. \(c\), \(e\), and \(\varepsilon\) are the modulus of elasticity, piezoelectric constant, and dielectric constant, respectively. \(\
abla\) is the differential operator.