# Vibration Energy Harvesting from Raindrops Impacts: Experimental Tests and Interpretative Models
## 5 Interpretative Mathematical Model
### Force Model for the Wet Spoon
#### 5.2.1 Determination of the Cylinder Areas  
The dimensions of the two unknown areas representing the cavity and the crown can be evaluated using an energy conservation approach [26]. In this analysis, the maximum depth of the cavity \(y_{i_{max}}\) and the maximum height of the crown \(y_{e_{max}}\), which occurs when the cavity depth is maximum, are assumed to be known. The first can be equated within a good approximation to the water film thickness \(h\), since in medium film the residual film thickness (i.e., the water layer under the cavity) is small. The latter, \(y_{e_{max}}\), can be inferred from experimental data if available, or can be computed exploiting the models available in the literature [28; 29; 30].  
When the maximal values of the crown height and the cavity depth are reached, only a portion \(\alpha\) of the initial total energy of the drop \(E_{t,d}\) is converted to the gravity \(E_{g,w}\) and surface \(E_{s,w}\) energy of the cavity and the crown; it holds that:  
\[E_{g,w}+E_{s,w}=\alpha E_{t,d} \tag{19}\]  
where the total drop energy is the sum of the kinetic energy \(E_{k,d}\) and the surface energy \(E_{s,d}\):  
\[E_{t,d}=E_{k,d}+E_{s,d}=\frac{1}{12}\rho_{w}\pi D_{d}^{3}v_{d}^{2}+\sigma\pi D _{d}^{2} \tag{20}\]  
with \(\sigma\) indicating the water surface tension. Based on the work of [PERSON] et al. [26], who numerically investigated the energy conversion process during the impact process for different film depths, for a medium film during the crown stabilization phase (i.e., the one in which the maximum crown height is reached) the coefficient \(\alpha\) in Equation (19) varies from 0.5 to 0.6.  
The gravitational potential energy of the crown and the cavity, i.e., of the displaced water mass, in Equation (19) is:  
\[E_{g,w}=g\rho_{w}\bigg{(}A_{e}\frac{y_{e_{max}}^{2}}{2}+A_{i}\frac{y_{i_{max}}^{2 }}{2}\bigg{)} \tag{21}\]  
where as the surface energy \(E_{g,w}\) is given by:  
\[E_{g,w}=\sigma\pi(D_{e}y_{e_{max}}+D_{i}(y_{e_{max}}+y_{i_{max}})) \tag{22}\]  
By taking advantage of Equation (16) and of the areas' definitions, it is possible to derive the following relations:  
\[\frac{A_{e}}{A_{i}}=\frac{\pi(D_{e}^{2}-D_{i}^{2})}{\pi D_{i}^{2}}=\frac{D_{e}^ {2}}{D_{i}^{2}}-1=\frac{y_{i_{max}}}{y_{e_{max}}}\to D_{i}=D_{e}\sqrt{ \frac{y_{e_{max}}}{y_{e_{max}}+y_{i_{max}}}}, \tag{23}\]  
The cavity and crown energies can be rewritten as:  
\[E_{g,w}=g\rho_{w}A_{e}\frac{y_{e_{max}}^{2}}{2}\bigg{(}1+\frac{A_{e}}{A_{i}} \bigg{)}=\frac{\pi}{8}g\rho_{w}y_{e_{max}}y_{i_{max}}D_{e}^{2} \tag{24}\]  
\[E_{g,w} = \sigma\pi y_{e_{max}}\bigg{(}1+\sqrt{1+\frac{y_{i_{max}}}{y_{e_{ max}}}}\bigg{)}D_{e} \tag{25}\]  
When Equations (20), (24) and (25) are substituted into Equation (19), a quadratic equation in the only unknown \(D_{e}\) is obtained:  
\[\frac{\pi}{8}g\rho_{w}y_{e_{max}}y_{i_{max}}D_{e}^{2}+\sigma\pi y_{e_{max}} \bigg{(}1+\sqrt{1+\frac{y_{i_{max}}}{y_{e_{max}}}}\bigg{)}D_{e}-\alpha\bigg{(} \frac{1}{12}\rho_{w}\pi D_{d}^{3}v_{d}^{2}+\sigma\pi D_{d}^{2}\bigg{)}=0 \tag{26}\]  
Once Equation (26) is solved, the other unknown, i.e., \(D_{i}\), can be computed by means of Equation (23).