## 2 Discussion about the Importance of Volume Consideration
The electrical damping due to the effect of the piezoelectric energy harvesting can be represented as \(d_{e}\), where \(d_{e}=2\zeta_{e}m_{eq}\omega_{n}\) and \(\zeta_{e}\) is the electrical damping ratio. The equation of motion of the beam can be rewritten as the following:  
\[m_{eq}\ddot{\chi}(t)+(d_{m}+d_{e})\dot{x}(t)+c_{eq}x(t)=F_{eq}. \tag{5}\]  
By taking the Laplace transform and replacing the excitation frequency by the resonance frequency in Equation (5), the transfer function \(H_{xu}\) at resonance is obtained in terms of the mass ratio between the tip mass and the mass of the beam, the electrical damping ratio, and the mechanical damping ratio, as expressed by Equation (6). \[H_{xu}|_{\Omega=\omega_{n}}=\left(\frac{0.375+\beta}{0.2357+\beta}\right)\frac {-1}{2(\zeta_{m}+\zeta_{e})j}. \tag{6}\]  
By equating Equation (3b) with Equation (6), the electrical damping ratio as a function of the system parameters is obtained as follows:  
\[\zeta_{e}=\frac{\alpha^{2}}{2m_{eq}\omega_{n}\left(\frac{1}{k}+C_{p}\Omega j \right)}. \tag{7}\]  
Figure 2 displays the impact of the applied load resistance normalized to \(R_{d}\) on the electrical damping ratio. The properties of the PEH considered for the investigation are imported from Table 3. The damping ratio is assumed to be 0.01 (as measured experimentally in [10]). The free length is 39.11 mm. The maximum electrical damping occurs at \(R/R_{d}=1\) (which means \(R=R_{d}\)). \begin{table}
\begin{tabular}{c c} \hline
**Resistance** & **Frequency** \\ \hline \(R_{op\_res}=\frac{2\zeta_{e}m_{eq}\omega_{n}m_{eq}}{\sqrt{\alpha^{4}+4\zeta_{ e}^{2}\omega_{n}^{2}m_{eq}^{2}\zeta_{e}^{2}}}\) & \(\omega_{n}=\sqrt{\frac{c_{eq}}{m_{eq}}}\) \\ \(R_{op\_anti}=\frac{\sqrt{\alpha^{4}+4\zeta_{e}^{2}\omega_{n}^{2}\omega_{n}^{2} \zeta_{e}^{2}}\omega_{ani}^{2}}{2\zeta_{m}m_{eq}\omega_{n}^{2}\zeta_{e}^{2} \omega_{ani}^{2}}\) & \(\omega_{anti}=\sqrt{\omega_{n}^{2}+\frac{\alpha^{2}}{m_{eq}\zeta_{e}}}\) \\ \(R_{d}=\frac{1}{C_{p}\omega_{d}}\) & \(\omega_{d}=\sqrt{\omega_{n}^{2}+\frac{\alpha^{2}}{2m^{2}\zeta_{e}}}\) \\ \hline \end{tabular}
\end{table} 
Table 5: Significant applied load resistances and corresponding frequencies for maximum power. Figure 2: The impact of the applied load \(R\) normalized to \(R_{d}\) on the electrical damping ratio \(\zeta_{e}\). For investigating the power output of the PEHs A, B, C, and D, the applied load resistance is the corresponding optimal load resistance \(R_{op\_res}\) for each of them at resonance operation condition and is 1.86 k\(\Omega\), 0.811 k\(\Omega\), 1.62 k\(\Omega\), and 0.811 k\(\Omega\), respectively. The PEHs are assumed to be excited by a constant input displacement of 0.8 \(\upmu\)m. Figure 3 displays the power output of the four piezoelectric energy harvesters. The PEH A, with the largest geometrical parameters, supplies the highest power output. The power output of B, with no tip mass and a reduced thickness, supplies a much lower power output than A. The power output of C is half of the power output of B (C has the same geometrical parameters as \"B\" but half of its width). The power output of D is much less than other PEHs. Despite the fact that the four PEHs are tuned to the same resonance frequency, there is a significant difference in the maximum power output between them. This highlights the importance of volume consideration for a better estimation of the effectiveness of the developed structures and the broadband techniques, which require a considerable volume compared to the volume of the transducer. For a more accurate estimation of the effectiveness of PEHs, the total occupied volume must be considered, including the volume of motion, not only the volume of the transducer. The tip displacement is an important factor to investigate in order to calculate the volume of motion.