# Identification Procedure for Design Optimization of Gravitational Electromagnetic Energy Harvesters
## 3 Results and Discussions
### Experimental and Numerical Results  
The dynamic behavior of the harvester system can be studied through the evaluation of the Frequency Response Functions (FRFs). These curves show the harvester performances when different excitation amplitudes and frequencies are applied to the system. The non-linear softening behavior of the device can be highlighted by analyzing the FRFs. The non-linearities cause the resonance frequency of the system to decrease as the external excitation amplitude increases. The FRFs evaluation is performed both numerically and experimentally for model validation. FRFs are reported in terms of the root mean square value of the generated power on the load. Numerical dynamic simulations are performed in MATLAB/Simulink environment by introducing the non-linear stiffness and electromagnetic characteristics resulting from Ansys Maxwell numerical simulations. Experimental FRFs are obtained by testing harvester prototypes on the dynamic workbench described in Section 2.3. Results are computed using the optimum resistive load described in Section 2.1 and connecting only the coil that maximizes the output power. Table 6 shows the values of the variables used to compute the FRFs both for the numerical model and experimental setup. The number of turns and \(R_{coil}\) values at this stage are not optimized for the three harvesters.  
Figure 9: Experimental setup picture.  
Simulation results need to be compared to experimental results of tests on harvester prototypes. A satisfactory superimposition of experimental and numerical FRFs leads to the validation of the proposed model. Figure 10 reports the comparison between numerical and experimental RMS power FRFs of EH1, EH2, and EH3 for different excitation amplitudes from 0.1 g to 0.3 g. For the purpose of quantifying the preciseness of these models, an error between the numerical and the experimental results in terms of power should be computed. A comparison method can be the evaluation of the relative error in terms of RMS peak power and resonance frequency between the model and the experimental ones:  
\[err_{peak}\% =\left|\frac{P_{peak,model}-P_{peak,experimental}}{P_{peak,experimental }}\right|\times 100 \tag{15}\] \[err_{freq}\% =\left|\frac{\omega_{peak,model}-\omega_{peak,experimental}}{\omega_ {peak,experimental}}\right|\times 100 \tag{16}\]  
Table 7 shows the errors obtained for EH1, EH2, and EH3. The peak errors between the model and experimental results are negligible for all the harvesters and different excitation amplitudes. The errors on the resonance frequencies are definitely low, confirming the ability of this model to identify correctly the resonance conditions. This quality is relevant considering the key role of tuning the resonance frequency to the external excitation fundamental frequencies for maximum power generation. The superimposition of the curves leads to overall fine results for all the harvesters. This positive outcome of the study allows confirming the validity of the proposed model to describe the dynamics and transducing mechanism of different-scaled GVEHs. The following step is to perform sensitivity analyses on the fully-parametrized model to define the optimization process of the devices.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Configuration** & **Coil Number** & \(R_{coil}\) (\(\Omega\)) & \(R_{load,opt}\) (\(\Omega\)) \\ \hline EH1 & 2 & 70 & 100 \\ EH2 & 4 & 100 & 150 \\ EH3 & 3 & 270 & 570 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Electric quantities used for tests.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Excitation Amplitude (g)** & \(err_{peak}\)(\%) & \(err_{freq}\)(\%) \\ \hline  & EH1 = 1.64 & EH1 = 1.64 \\
0.1 & EH2 = 3.84 & EH2 = 0 \\  & EH3 = 2.13 & EH3 = 2.94 \\ \hline  & EH1 = 1.96 & EH1 = 0 \\
0.2 & EH2 = 5.25 & EH2 = 0 \\  & EH3 = 4.04 & EH3 = 0 \\ \hline  & EH1 = 1.30 & EH1 = 1.70 \\
0.3 & EH2 = 3.16 & EH2 = 3.03 \\  & EH3 = 7.13 & EH3 = 3.33 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Peak and frequency relative errors for EH1, EH2, and EH3.