# Parametric and Experimental Modeling of Axial-Type Piezoelectric Energy Generator with Active Base
## 5 Experimental Probe
### Method of Natural Modeling of the Oscillatory Processes of PEG  
The sequence for determining the parameters of PEG characteristics was as follows. At the first stage, a preliminary assessment of the difference in natural frequencies obtained in the experiment and simulation was carried out. By formula (11), the discrepancy of natural frequencies was calculated:  
\[\Delta=[(\omega_{\rm Exp}-\omega_{\rm Sol})/\omega_{\rm Sol}]\times 100\%. \tag{11}\]  
Table 6 shows a comparison of the first and second natural harmonics of oscillations, with an experimental value of the mass of the load of 3.71 g at its various locations. The active load for each PE was taken, both in the experiment and in the calculation of 10 k\(\Omega\). The analysis of the obtained results showed a satisfactory coincidence of the values of the natural frequencies (the difference was no more than 5%). First, the frequency response of the piezoelectric generator was recorded, then the dependence of the output voltage and power of the PEG at the resonant frequency on the value of the active electrical load and proof mass on the value of the output voltage of the PEG and frequency characteristics was investigated. In the test PEG sample, fixed on a vibrating plate, transverse bending vibrations were sequentially excited in the frequency range from 0 to 1000 Hz (the generator voltage amplitude did not change in this case) value of electrical load resistance Rl from 10 k\(\Omega\) to 2 M\(\Omega\). The output power was calculated for the obtained voltage characteristics for 1, 2 and 3 PE. Figure 12 shows examples of visualization of the process of measuring the output characteristics of a PEG during sweeping. This analysis reveals the parameters of the output voltage on the PE electrodes at different loads. During the sweep, two points were chosen to obtain the estimated parameters of the PEG operation: (i) 39 Hz and (ii) 107 Hz. At a frequency of 107 Hz, the oscillation frequencies of the system elements were found, causing a resonant increase in oscillations. For a frequency of 39 Hz, the oscillation amplitude of the rigid base of the generator (4) was 0.266 mm, while the oscillation amplitude of the base (7) center was 1.683 mm. For a frequency of 107 Hz, the oscillation amplitude of the rigid base of the generator (4) was 0.035 mm, while the oscillation amplitude of the base (7) center was 0.356 mm. The results are shown in Tables 7 and 8. Data analysis shows that the maximum output voltage of 1.31 V for cylindrical PE occurs at a frequency of 39 Hz and a load of 2 M\(\Omega\). At the same time, for PE in the form of plates, the highest voltage is achieved at an active load of 150 k\(\Omega\) and a frequency of 39 Hz. Figures 13 and 14 show the graphs of the dependence of the output voltage and peak output power for all types of PE used. Analysis of the output power shows that with a large value of resistance, it drops. In this case, the maximum output power can be obtained at a load of up to 51 k\(\Omega\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Location of Proof Mass L\({}_{\rm m}\), mm**} \\ \hline  & 150 & 170 & 190 & 210 & 230 \\  & & \multicolumn{4}{c}{**1 natural frequency, Hz**} \\ Experiment \(\omega_{\rm Exp}\) & 255.0 & 258.0 & 267.0 & 288.0 & 302.0 \\ Calculation \(\omega_{\rm Sol}\) & 252.2 & 253.0 & 261.2 & 274.8 & 292.2 \\ \(\Delta\), \% & 1.1 & 2.0 & 2.2 & 4.8 & 4.0 \\  & & \multicolumn{4}{c}{**2 natural frequency, Hz**} \\ Experiment \(\omega_{\rm Exp}\) & 581.0 & 576.0 & 560.0 & 563.0 & 587.0 \\ Calculation \(\omega_{\rm Sol}\) & 564.4 & 547.5 & 538.1 & 546.0 & 574.2 \\ \(\Delta\), \% & 2.9 & 5.2 & 4.1 & 3.1 & 2.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of the experimental and numerical values 1 and 2 of the natural frequencies of PEG oscillations at an active load of PE of 10 k\(\Omega\) and a load value of 3.71 g at different positions.