# Vibration Energy Harvesting from Raindrops Impacts: Experimental Tests and Interpretative Models
## 5 Interpretative Mathematical Model
### Model Validation
#### 5.3.2 Full Spoon  
The determination of the force exciting the harvester provided with a spoon full of water requires the knowledge of some system parameters: the maximum crown height and the instant at which it is reached--the instant at which the cavity collapses. As discussed in Section 5.2, such parameters can be approximated through the models available in the literature or inferred from experimental data, as in the present case (see Figure 16). \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Symbol** & **Dry Spoon** & **Wet Spoon** \\ \hline Load resistance & \(R\) & 305 k\(\Omega\) \\ Beam width & \(b\) & 20.3 mm \\ Beam length & \(L\) & 45.5 mm \\ Drop impact position & \(L+L_{d}\) & 59.5 mm \\ Force distance & \(h\) & 1 mm \\ Eigenfunction derivative & \(d\phi_{1}(x)/dx|_{x=L}\) & 936.00 & 601.36 \\ Tip mass & \(m_{t}\) & 1.28 g & 4 g \\ Undamped natural angular frequency & \(\omega_{1}\) & 262.00 rad/s & 166.46 rad/s \\ Modal damping & \(\zeta_{1}\) & 0.55\% & 1.0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model parameters. Figure 14: Footage of the experimental test of a drop impacting on a dry spoon. \(D_{d}\) drop diameter. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Coefficient of restitution & \(\varepsilon\) & 0 \\ Drop diameter & \(D_{d}\) & 2.01 mm \\ Drop mass & \(m_{d}\) & 4.25 ug \\ Drop impact velocity & \(v_{d}\) & 4.04 m/s \\ Impact duration & \(\tau\) & 0.497 ms \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters for the computation of the impact force acting on the dry spoon. The same indoor experiment shown in Figure 16 was simulated in MATLAB. Therefore, a drop with a diameter of 1.9 mm falling from a height of 1 m was considered for the computation of the energy balance in Equation (19), and hence for the determination of the crown (\(A_{e}\)) and the cavity (\(A_{i}\)) areas. The equivalent mass-spring-damper system representing the motion of the crown (see Section 5.2.2) was found using the data in Figure 16 and by solving the nonlinear Equation (28) through the fsolve function in MATLAB. The resulting parameters of the equivalent system, together with all the other parameters involved in the force computation, are listed in Table 3. The equivalent force that acts on the harvester due to the impact of a raindrop on the spoon full of water is shown in Figure 17.  
Figure 16: Footage of the experimental test of a drop impacting on the wet spoon and corresponding parameters needed for the force model determination. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Drop diameter & \(D_{d}\) & 1.9 mm \\ Drop mass & \(m_{d}\) & 3.59 ug \\ Drop impact velocity & \(v_{d}\) & 4.01 m/s \\ Film thickness & \(h\) & 3 mm \\ Water surface tension & \(\sigma\) & 0.0728 N/m \\ Transformed energy fraction & \(\alpha\) & 0.55 \\ Maximum crown height & \(y_{emax}\) & 2.2 mm \\ Crown section & \(A_{e}\) & 62.08 mm\({}^{2}\) \\ Cavity section & \(A_{i}\) & 42.98 mm\({}^{2}\) \\ Displaced water mass & \(m\) & 0.315 mg \\ Equivalent crown stiffness & \(k\) & 6.66 Nm \\ Equivalent crown damping coefficient & \(c\) & 0.0462 N/m \\ Initial crown velocity & \(\dot{y}_{e0}\) & 0.668 m/s \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters for the computation of the force acting on the wet spoon. Figure 15: Comparison between the experimental (red) and simulated (blue) harvester voltage in the time (**a**) and frequency (**b**) domain for a drop impacting on a dry spoon. The voltage generated by the harvester excited by this force is represented with a blue line in Figure 18. The same figure represents the measured voltage with a red line. Comparing the FFT diagrams of the two signals (Figure 18b), it is apparent that the simulated voltage correctly matches the oscillation frequency. The time histories of the simulated and experimental voltage in Figure 18a are very similar.