# Vibration Energy Harvesting from Raindrops Impacts: Experimental Tests and Interpretative Models
## 5 Interpretative Mathematical Model
### Force Model for the Wet Spoon  
When the raindrop impinges onto a spoon that is full of water, the harvester vibrates at the same frequency at which it would vibrate if the spoon were empty and equipped with a solid mass equal to the water mass. Although the resonance frequency is equal in the two cases, the vibration amplitude of the harvester with the wet spoon is much larger, i.e., the water in the spoon acts as a force amplifier. Although the harvester dynamics are highly affected by the dynamics of the water in the spoon, the vice-versa does not hold: the video analysis discussed in Section 4 showed that the motion of the water is scarcely affected by the vibration of the harvester. Thanks to this experimental evidence, in a first simplified-modeling attempt, it is reasonable to model the water dynamics neglecting the harvester motion, i.e., considering the spoon as fixed (see Figure 11). The phenomena related to the impact of a drop on a water surface are affected by the water film depth, \(h\), significantly [26]. In particular, the impacts can be divided into three types according to the dimensionless film thickness (\(h^{*}=h/2R_{d}\)): drop impacts on shallow (\(h^{*}\leq 0.1\)), medium (\(0.1<h^{*}<2\)), and deep (\(h^{*}\geq 2\)) films. Regardless of the film depth, the drop impact on film leads to the formation of a crater below the free surface level of the water, and a crown above it at the circumference of the crater [27]. If the drop impacts on a medium film--as in the case analyzed--a cylindrical crater with flat-bottom is formed in the water film [26]. Starting from these results regarding the impact of a drop on a medium film, the complex fluid dynamics of the splash is here simplified in order to infer an easy and computationally inexpensive model describing the macroscopic effect that the water movement has on the harvester. The simplified splashing mechanism is schematically represented in Figure 12. It is assumed that, as the drop impacts the water, a mass of water included in a cylinder of diameter \(D_{i}\) starts moving downwards, forming the crater, and concurrently the same amount of water -- included in a hollow cylinder of internal diameter \(D_{i}\) and external diameter \(D_{e}\) -- begins to move upwards, forming the crown. Therefore, this continuity equation holds:  
\[\rho_{w}A_{e}\dot{y}_{e}(t)=\rho_{w}A_{i}\dot{y}_{i}(t), \tag{12}\]  
where \(A_{e}=\dfrac{\pi}{4}\big{(}D_{e}^{2}-D_{i}^{2}\big{)}\), and \(A_{i}=\dfrac{\pi}{4}D_{i}^{2}\) are the areas of the crown and of the crater, respectively, wheres \(\dot{y}_{e}\) and \(\dot{y}_{i}\) are their velocities. The linear momentum \(p(t)\) of the water after the impact of the drop can be expressed as:  
\[p(t)=(m_{i}-\rho_{w}A_{i}y_{i}(t))(-\dot{y}_{i}(t))+(m_{e}+\rho_{w}A_{e}y_{e}( t))(\dot{y}_{e}(t)), \tag{13}\]  
Figure 11: Full spoon model. Figure 12: Schematic of the simplified splashing mechanism: (**a**) 3D view; (**b**) 2D view. where \(m_{e}=\rho_{w}A_{e}h\) and \(m_{i}=\rho_{w}A_{i}h\) are the masses contained in the external and the internal cylinders, respectively, before the impact, i.e., when \(y_{i}=y_{e}=0\) and \(\dot{y_{i}}=\dot{y_{e}}=0\).