## 4 Results and Discussions
### Validation of the Mechanical System
#### 4.1.1 Case 1--Resistive Load  
In this case (resistive load), the value of maximum power was determined for comparison with analytical values, and, consequently, the circuit power (ROpt) for the frequency used was determined.  
In this experiment, the excitation frequency was fixed, and the value of the load resistance was varied to obtain the electrical voltage and output current in the piezoelectric transducer. At the resonance frequency, an ROpt of 25 k\(\Omega\) and 50 k\(\Omega\) was obtained for the PZT-5H and P5-13B transducers, respectively. In comparison with the analytical result, errors of 9.46% and 9.48% were obtained for the PZT-5H and P5-13B transducers, respectively (Figure 7 and Table 4).  
Figure 6: Validation of the natural frequencies of PZTs: (**a**) analytical and (**b**) experimental.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Piezo** & **Fn—Analytical** & **Fn—Experimental** & **Fn—Experimental** & **Error (\%)** & **Error (\%)** \\  & & **(Analyzer)** & **(Oscilloscope)** & **(Analyzer)** & **(Oscilloscope)** \\ \hline PZT-5H & 18.3 Hz & 18.0 Hz & 18.8 Hz & 1.63 & 2.73 \\ P5-13B & 31.1 Hz & 32.25 Hz & 31.9 Hz & 4.50 & 2.57 \\ \hline \end{tabular}
\end{table}
Table 3: Natural frequencies and ROpt.  
In Figure 7, it can be observed that there is conformity between the analytical and experimental curves up to the value of 0.5 \(\times\) 105 \(\Omega\), after which the values diverge, explaining the high error. This error is associated with the fact that the resistances do not reach the maximum generated voltage as the piezos' generation capacity has practically been reached. In Figure 8 and Table 5, the results of the voltage curves generated as a function of the generated power for the transducers are detailed.  
The comparison between the experimental and analytical values of the maximum powers generated by the circuit with a load resistor, with values of 25 k\(\Omega\) and 50 k\(\Omega\), was below 5%, validating the model for Case 1--resistive load.  
Figure 8: Resistance vs. power curves—resistive circuit: (**a**) PZT-5H and (**b**) P5-13B.  
Figure 7: Resistance vs. voltage curves—resistive circuit; (**a**) PZT-5H and (**b**) P5-13B.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Voltage** & **PZT-5H** & **P5-13B** \\ \hline Analytical & 24.2 V & 23.93 V \\ Experimental & 21.9 V & 21.66 V \\ Error & 9.46\% & 9.48\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: Maximum voltages—resistive circuit.  
The errors obtained from the analytical and experimental comparison for the maximum generated powers were 2.29% and 6.09%, associated with the resistance values of 17.5 k\(\Omega\) and 75 k\(\Omega\) for the PZT-5H and P5-13B transducers, respectively. This is consistent with expectations due to variations in the electrical parameters of the electrical components.  
However, pronounced distortions were observed in piezo P5-13B, which showed the lowest generated potential due to its lower coupling factor, a smaller volume of piezoelectric material, and lower deformation capacity due to the dimensions of the beam to which the piezoelectric material is attached, along with greater difficulty in securing the beam.