# A Novel Piezoelectric Energy Harvester Using a Multi-Stepped Beam with Rectangular Cavities
## 4 Simulation Results and Discussions  
The harvester performance was assessed using the model derived in Section 3 and shown in Figure 1. In the proposed harvester, five steps were formed at \(L_{1}\) = 0.022 L, \(L_{2}\) = 0.187 L, \(L_{3}\) = 0.295 L, \(L_{4}\) = 0.46 L and \(L_{5}\) = 0.568 L. Patch 1, 2 and 3 were glued at \(L_{1}\), \(L_{3}\) and \(L_{6}\) where the strain was maximum. The dimensions and properties of the beam and patches are given in Tables 1-3. The change in the first two-mode frequencies with the variation in thickness (t\({}_{5}\)) and (t\({}_{6}\)) is shown in Figure 3.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Sections**} & \multicolumn{2}{c}{**Length (mm)**} & \multicolumn{2}{c}{**Width (mm)**} & \multicolumn{2}{c}{**Thickness (mm)**} \\ \cline{2-7}  & **Symbol** & **Value** & **Symbol** & **Value** & **Symbol** & **Value** \\ \hline I & \(L_{1}\) & 10 & b\({}_{1}\) & 40 & t\({}_{1}\) & 10 \\ II & \(L_{2}\)-\(L_{1}\) & 76.5 & b\({}_{2}\) & 25 & t\({}_{2}\) & 6 \\ III & \(L_{3}\)-\(L_{2}\) & 50 & b\({}_{3}\) & 40 & t\({}_{3}\) & 10 \\ IV & \(L_{4}\)-\(L_{3}\) & 76.5 & b\({}_{4}\) & 25 & t\({}_{4}\) & 6 \\ V & \(L_{5}\)-\(L_{4}\) & 50 & b\({}_{5}\) & 4 From the result, it was found that the first two-mode frequencies were adjacent with the increase in thickness ratio \(\rm t_{5}/t_{1}\) and decrease in thickness ratio \(\rm t_{6}/t_{1}\). The mode shape of the proposed harvester, proposed harvester without the cavity and conventional cantilever beam-based energy harvester for the first two consecutive modes are shown in Figure 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Description** & **Value** & **Units** \\ \hline \(\rm l_{p}\) & Length of patch 1 and 2 & 76.5 & mm \\ \(\rm b_{2}\), \(\rm b_{4}\) & Width of patch 1 and 2 & 25 & mm \\ \(\rm b_{7}\) & Width of patch 3 & 12.5 & mm \\ \(\rm t_{p}\) & Thickness of patch 1, 2 and 3 & 0.5 & mm \\ \(\rm E_{p}\) & Young’s modulus & 47.62 & \(\rm GP_{a}\) \\ \(\rm\rho_{p}\) & Density & 7500 & \(\rm Kg\,m^{-3}\) \\ \(\rm d_{31}\) & Piezoelectric charge coefficient & \(-\)265 & \(\rm pCN^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The properties of the aluminum beam. Figure 3: First and second mode frequencies; (**a**) with variation in \(\rm t_{5}/t_{1}\), (**b**) with variation in \(\rm t_{6}/t_{1}\). The output voltage (\(V_{\infty}\)) from the harvester with no load was computed using Equation (42) given in the modeling section. Figure 5 shows the change in output voltage (\(V_{\infty}\)), obtained through simulation for the first two modes of vibration. The effectiveness of the proposed harvester's performance was demonstrated by comparing the performance of the harvester having a similar structure without a cavity and also with the regular cantilever-based harvester. The analytical results revealed that in the regular cantilever-based harvester, the output was high only in the first mode of vibration and the second mode frequency was far away from the first mode. The proposed harvester had adjacent first two-mode frequencies with enhanced output in both the modes. From the simulation results, it was observed that in the proposed harvester, the second mode frequency (\(\omega_{2}\)) was 1.69 times that of the first mode frequency (\(\omega_{2}=1.69\omega_{1}\)) with a high amplitude in both the modes, whereas in the conventional cantilever-based harvester (\(\omega_{2}=6.27\omega_{1}\)), the second mode output was very low. Figure 4: Mode shape; (**a**) proposed harvester, (**b**) proposed harvester without cavity (**c**) conventional cantilever-based harvester. The load voltage and the average power were computed using Equations (39) and (43) respectively. The power (\(P_{g}\)) was differentiated with respect to the load resistance (\(R_{l}\)) to obtain the optimal load resistance. The value of the load resistance at the inflection point where \(\left(\frac{dP_{g}}{dR_{l}}=0\right)\) was the optimal load resistance.