## 2 Discussion about the Importance of Volume Consideration
The geometrical parameters for the PEHs are displayed in Table 4.  
The two governing equations of the PEHs according to [23] are as follows:  
\[m_{eq}\ddot{x}(t)+d_{m}\dot{x}(t)+c_{eq}x(t)+\alpha v_{p}(t)=F_{eq}, \tag{1}\]  
\[\alpha\dot{x}(t)-C_{p}\dot{v}_{p}(t)=\frac{v_{p}(t)}{R}, \tag{2}\]  
where \(c_{eq}\) is the equivalent stiffness of the beam, \(m_{eq}\) is the equivalent mass of both the beam and the tip mass, \(d_{m}\) is the mechanical damping coefficient, \(R\) is the applied load resistance, \(v_{p}\) is the output voltage, \(\alpha\) is the equivalent coupling factor [N/V or C/m], and \(C_{p}\) is the capacitance of the two piezoelectric layers. The transfer function between the input displacement and the output displacement of the bimorph piezoelectric energy harvester is obtained as expressed by Equation (3a). At excitation frequency around the first eigenfrequency, the transfer function is obtained as expressed by Equation (3b). The power output of the bimorph piezoelectric energy harvester are obtained by Equation (4), as derived in [10]. \[H_{xu}=\frac{x(t)}{u(t)}=\left(\frac{0.375+\beta}{0.2357+\beta}\right)\frac{1} {1-\left(\frac{\omega_{m}}{\Omega}\right)^{2}-\frac{\alpha^{2}C_{p}}{m_{eq}( \frac{1}{R^{2}}+C_{p}^{2}\Omega^{2})}-2\zeta\left(\frac{\omega_{m}}{\Omega} \right)j-\frac{\alpha^{2}}{m_{eq}\Omega R(\frac{1}{R^{2}}+C_{p}^{2}\Omega^{2 })}j}, \tag{3a}\] \[H_{xu}\big{|}_{\Omega=\omega_{u}}=\left(\frac{0.375+\beta}{0.2357+\beta}\right) \frac{-1}{\frac{-\alpha^{2}C_{p}}{m_{eq}(\frac{1}{R^{2}}+C_{p}^{2}\omega_{u} ^{2})}+2\zeta_{m}j+\frac{\alpha^{2}}{m_{eq}\omega_{u}R(\frac{1}{R^{2}}+C_{p}^ {2}\omega_{u}^{2})}j},\] (3b) \[P=\frac{\alpha^{2}\Omega^{2}u^{2}\left(\frac{0.375+\beta}{0.2357+\beta}\right) ^{2}}{\left(\frac{1}{k}+C_{p}^{2}\Omega^{2}R\right)\left(\left(1-\left(\frac{ \omega_{m}}{\Omega}\right)^{2}-\frac{\alpha^{2}C_{p}}{m_{eq}(\frac{1}{R^{2}} +C_{p}^{2}\Omega^{2})}\right)^{2}+\left(2\zeta_{m}\left(\frac{\omega_{m}}{ \Omega}\right)+\frac{\alpha^{2}}{m_{eq}\Omega R(\frac{1}{R^{2}}+C_{p}^{2} \Omega^{2})}\right)^{2}\right)^{2}\right)}}, \tag{4}\]  
where \(\zeta_{m}\) is the mechanical damping ratio, \(\omega_{n}\) is the resonance frequency, and \(\Omega\) is the excitation frequency. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(I\) [mm] & \(b\) [mm] & \(h\) [mm] & \(h_{sp}\) [mm/mm] & \(\beta\) [g/g] \\ \hline A & 40 & 9 & 1.5 & 0.5/0.5 & 1 \\ B & 40 & 9 & 0.66 & 0.22/0.22 & 0 \\ C & 40 & 4.5 & 0.66 & 0.22/0.22 & 0 \\ D & 20 & 4.5 & 0.11 & 0.037/0.037 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Geometrical parameters of the PEHs. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Piezoelectric Layers** & **Non-Piezoelectric Layer** \\ \hline Mechanical data & \(\begin{array}{c}s_{11}^{E}=14.2\times 10^{-12}\text{ m}^{2}/\text{N},\\ \rho_{p}=8.1\times 10^{03}\text{ kg}/\text{m}^{3}\end{array}\) & \(\begin{array}{c}E_{s}=2\times 10^{11}\text{ N}/\text{m}^{2},\\ \rho_{s}=8\times 10^{03}\text{ kg}/\text{m}^{3}\end{array}\) \\ \hline Piezoelectric data & \(\begin{array}{c}d_{31}=315\times 10^{-12}\text{ C}/\text{N},\\ \varepsilon_{33}^{2}=3.98\times 10^{-08}\text{ F}/\text{m}\end{array}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Material properties of the bimorph piezoelectric energy harvesters. There are three significant applied load resistances as investigated in [10]. Each of the three load resistances provides the maximum power capability at a corresponding excitation frequency, as displayed in Table 5. The optimal load resistance at resonance frequency (\(R_{op\_res}\)) and at anti resonance frequency (\(R_{op\_anti}\)) provide the same maximum power output at constant input acceleration. The applied load resistance of \(R_{d}\) results in the maximum electrical damping to the circuit. It is worth mentioning that the maximum power output occurs at \(\omega_{d}\) not \(\omega_{n}\) when the applied load is \(R_{d}\).