# Identification Procedure for Design Optimization of Gravitational Electromagnetic Energy Harvesters
## 3 Results and Discussions
### Design Optimization
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Sensitivity Analysis** & **Fixed Magnet** & **Moving Magnet** \\ \hline Fixed magnet external diameter & \(\begin{array}{c}D_{fix,ext}=\textbf{12-40 mm 1}\\ D_{fix,int}=\textbf{10 mm}\\ H_{fix}=\textbf{12 mm}\end{array}\) & \(D_{mocv,ext}=\textbf{30 mm}\) \\ \hline Fixed magnet external diameter & \(\begin{array}{c}D_{fix,int}=\textbf{10 mm}\\ H_{fix}=\textbf{12 mm}\end{array}\) & \(D_{mocv,int}=\textbf{10 mm}\) \\ \hline Fixed magnet radial thickness & \(\begin{array}{c}D_{fix,ext}=\textbf{20 mm}\\ \textbf{2}_{fix,int}=\textbf{0-14 mm}\\ H_{fix}=\textbf{12 mm}\end{array}\) & \(D_{mocv,int}=\textbf{10 mm}\) \\ \hline Fixed magnet height & \(\begin{array}{c}D_{fix,ext}=\textbf{20 mm}\\ D_{fix,int}=\textbf{10 mm}\\ H_{fix}=\textbf{6-30 mm}\end{array}\) & \(D_{mocv,int}=\textbf{10 mm}\) \\ \hline Moving magnet external diameter and height 2 & \(\begin{array}{c}D_{fix,ext}=\textbf{20 mm}\\ D_{fix,int}=\textbf{10 mm}\\ H_{fix}=\textbf{6-12/18}\end{array}\) & \(\begin{array}{c}D_{mocv,ext}=\textbf{15-40 mm}\\ D_{mov,int}=\textbf{1}/\textbf{2}\end{array}\) \\ \hline Moving magnet internal diameter & \(\begin{array}{c}D_{fix,ext}=\textbf{20 mm}\\ D_{fix,int}=\textbf{10 mm}\\ H_{fix}=\textbf{12 mm}\end{array}\) & \(D_{mocv,ext}=\textbf{30 mm}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Sensitivity analyses descriptions. it is the sum of the air gap and the fixed magnet height, hence the tube length needs to grow to allow a proper stroke to the proof mass. The radial thickness of the fixed magnet influences the tuning of the stiffness in a similar way to the external diameter parameter. In the end, for the purpose of tuning the resonance frequency, it is better to variate the internal or external diameters of the fixed magnet, keeping the height at the minimum possible value in order to reduce the tube length and volume. The second sensitivity analysis is performed on the moving magnet. External diameter and height vary together keeping their ratio at a fixed value equal to 1. This study is carried out three times using the fixed magnet of each of the three configurations EH1, EH2, and EH3 in order to insert the experimental results and compare them with the simulation data. The results of the analysis are reported in Figure 12. The trend linear curves for the three configurations interpolate quite well with the experimental results of the real devices. With increasing fixed magnet height, the slope of the trend curve decreases, hence the influence of the moving magnet volume on the resonance frequency and air gap reduces. In all three cases, increasing the moving magnet volume causes the resonance frequency to grow and consequently the air gap to reduce. The influence of the moving magnet on the resonance frequency formulation (17) is not only in terms of magnet field intensity, as happens for the fixed magnet, but also in terms of the mass at the numerator. A sensitivity analysis is performed also on the internal diameter of the moving magnet since an internal guide is used for the reasons explained in Section 2.1. The outcome is not particularly relevant as the influence on the studied variables of natural frequency and air gap is negligible. Figure 11: Fixed magnet sensitivity analysis. Moreover, the moving magnet sensitivity analysis should not be based only on the mechanical characteristics but also on the electromagnetic ones, as it influences the generated power. The expression of the linearized electrical power in resonance condition is the following [32]:  
\[P_{e}=\frac{m\zeta_{e}\gamma_{0}^{2}\omega_{n}^{3}}{4\zeta_{tot}^{2}} \tag{20}\]  
where:  
* \(\zeta_{e}\) is the electromagnetic damping factor;
* \(\zeta_{tot}\) is the sum of electromagnetic and viscous damping factors;  
The moving magnet volume influences power in terms of mass and damping factors, both electromagnetic and viscous as it changes the magnetic flux linkage and the friction values.