# Vibration Energy Harvesting from Raindrops Impacts: Experimental Tests and Interpretative Models
## 5 Interpretative Mathematical Model  
The cantilever harvester used in this experimental study was modeled according to the system schematic shown in Figure 10. This schematic depicts a composite cross section, including the active piezoelectric layer made of PZT 5H and the structural layer made of steel. The harvester was modeled with the distributed parameter approach of [24], as modified in [20] in order to take into account the impulsive force and moment caused by raindrop impact on the spoon. The mechanical problem is described by means of the following partial derivative equation:  
\[EI\frac{\partial^{4}w(x,t)}{\partial x^{4}}+c_{s}I\frac{\partial^{5}w(x,t)}{ \partial x^{4}\partial t}+m\frac{\partial^{2}w(x,t)}{\partial t^{2}}+c_{a} \frac{\partial w(x,t)}{\partial t}+\theta v(t)=F_{t}(t) \tag{3}\]  
where:  
* \(w(x,t)\) is the transverse displacement of any point along the cantilever. * \(I\) is the equivalent area moment of inertia of the composite section. * \(EI=\dfrac{b}{3}\Big{[}E_{s}\Big{(}p_{pb}^{3}-p_{sb}^{3}\Big{)}+Y_{11}^{E} \Big{(}p_{pt}^{3}-p_{pb}^{3}\Big{)}\Big{]}\) is the bending stiffness of the composite cross section [25], with \(E_{s}\) and \(Y_{11}^{E}\) denoting the Young's modulus of the substructure and of the PZT layers, respectively, and \(p_{pt}\), \(p_{pb}\), and \(p_{sb}\) indicating the positions from the neutral axis of the PZT layer top, of the PZT layer bottom, and of the substructure layer bottom, respectively (see Figure 10b). * \(b\) is the width of the beam. * \(L\) is the cantilever length. * \(c_{s}\) and \(c_{a}\) are the strain rate and air damping coefficients, respectively, which are assumed to satisfy the proportional damping criterion. * \(\theta=-\dfrac{Y_{11}^{E}d_{31}b}{2h_{p2}}\Big{(}p_{pt}^{2}-p_{pb}^{2}\Big{)}\) is the backward piezoelectric coefficient that couples the mechanical problem with the electrical problem; with \(d_{31}\) piezoelectric constant, and \(h_{p2}\) thickness of piezoceramic layer. Figure 9: Drop impact on a full spoon. Time evolution of voltage and camera pictures at selected instants, showing the main phases of water motion. The impact of the drop on the harvester is modeled as a concentrated force \(F(t)\) acting on the system at distance \(L+L_{d}\) from the fixed base (see Figure 10a), which corresponds to the middle point of the spoon. Such a force can be transported at the cantilever end \((x=L)\) by using an equivalent bending moment, represented by a pair of opposite forces located at distance \(h\) and with a magnitude \(F^{\prime}(t)=F(t)L_{d}/h\). Therefore, the equivalent force acting on the harvester tip \(F_{t}(t)\), at the right-hand side of Equation (3), is:  
\[F_{t}(t)=\big{(}F(t)+F^{\prime}(t)\big{)}\delta(x-L)-F^{\prime}(t)\delta(x-(L-h )) \tag{4}\]  
where \(\delta(x)\) represent [PERSON]'s delta. The electrical problem is described by the first-order ordinary differential equation of the circuit shown in Figure 10c. The charge generation caused by the piezoelectric effect is represented by an equivalent current source \(i(t)\), whereas the capacitive effect related to the electric field in the PZT layer is simulated by a capacitance \(\mathbb{C}_{pu}\). The external electrical load is schematized by a simple resistance \(R\):  
\[C_{pu}\vartheta(t)+\frac{1}{R}v(t)=i(t) \tag{5}\]  
Equations (3) and (5) are solved by means of the modal expansion method [25], by setting:  
\[w(x,t)=\sum_{i=1}^{\infty}\phi_{i}(x)\eta_{i}(t) \tag{6}\]  
The absolutely and uniform convergent series of the mass-normalized eigenfunctions (\(\phi_{i}(x)\)), representing the vibratory motion of the harvester (\(w(x,t)\)), has been truncated at the first term, i.e., \(w(x,t)=\sum_{i=1}^{\infty}\phi_{i}(x)\eta_{i}(t)\approx\phi_{1}(x)\eta_{1}(t)\), since this approximation is sufficient in most harvesting applications [20].