## 3 Analytical Model of Prototypes  
The prototypes are equipped with cantilever DVAs, which are beams with tip masses and can be modeled with a lumped element approach. Therefore, it is possible to develop an analytical model of the prototype by coupling the distributed parameter model of the harvester [6; 7] with the lumped element model of the cantilever DVA. Figure 2 shows a scheme of the model. \(w_{b}(t)\) is the displacement of the base of the harvester in the transverse direction (\(y\)), base rotation is neglected since the testing equipment was designed in order to generate only base displacement. \(w_{r}(x,t)\) is the displacement of any point \(x\) along the harvester axis with respect to the base, \(u_{r}(t)\) is the displacement of the mass of the cantilever DVA with respect to the harvester tip (\(w_{r}(L,t)\)). The partial differential equation of the harvester (having length \(L\)) is:  
\[ \begin{array}{c}(EI)_{eq}\frac{\partial^{4}w_{r}(x,t)}{\partial x^{4}}+c_{s }I\frac{\partial^{5}w_{r}(x,t)}{\partial x^{4}\partial t}+m\frac{\partial^{2} w_{r}(x,t)}{\partial t^{2}}+\phi v(t)=-m\frac{\partial^{2}w_{b}(t)}{\partial t^{2}}+F(t) \delta(x-L)+\\ M(t)\frac{\delta(x-L)-\delta(x-(L-h))}{h}\end{array}  \tag{4}\]  
In Equation (4) \(\left(EI\right)_{eq}\) is the equivalent bending stiffness of the composite cross-section, \(m\) is the mass per unit length of the harvester and \(c_{s}\) is the strain-rate damping coefficient [6], air damping is neglected because usually it has a small effect on harvester performance [7]. \(\theta\) is the piezoelectric coupling term [6] and \(v(t)\) is generated voltage. At the right-hand side of Equation (4) there are the forcing terms. The first term is due to base acceleration, whereas the last terms are caused by the force \(F(t)\) and the torque \(M(t)\) exerted by the cantilever DVA on the harvester; \(\delta(x)\) is the Dirac delta function. It is worth noticing that the loads due to the cantilever harvester are calculated with the same approach adopted in [7] to calculate the inertia load caused by a tip mass. Figure 1: Cantilever DVA (Dynamic Vibration Absorber) for harvesters. Figure 2: Model of the coupled system. The piezoelectric layer can be considered a current \(i(t)\) source in parallel with its internal capacitance \(C_{pu}\)[6] and it is connected to an electrical load that can be represented by a resistance \(R\). The application of the Kirchoff's laws to this circuit gives the following differential equation:  
\[C_{pu}\frac{dv(t)}{dt}+\frac{v(t)}{R}=i(t) \tag{5}\]  
The ordinary differential equation of the DVA is:  
\[m_{\omega}\bigg{(}\frac{d^{2}u_{r}(t)}{dt^{2}}+\frac{\partial^{2}w_{r}(L,t)}{ \partial t^{2}}+\frac{d^{2}w_{b}(t)}{dt^{2}}\bigg{)}=-k_{a}u_{r}(t)-c_{a}\frac {du_{r}(t)}{dt} \tag{6}\]  
constant \(c_{a}\) in Equation (6) is DVA damping. The inertia force and torque exerted by the cantilever DVA on the harvester are:  
\[F(t)=-m_{eq}\bigg{(}\frac{d^{2}u_{r}(t)}{dt^{2}}+\frac{\partial^{2}w_{r}(L,t)} {\partial t^{2}}+\frac{d^{2}w_{b}(t)}{dt^{2}}\bigg{)} \tag{7}\]  
\[M(t)=-m_{eq}L_{a}\bigg{(}\frac{d^{2}u_{r}(t)}{dt^{2}}+\frac{\partial^{2}w_{r}( L,t)}{\partial t^{2}}+\frac{d^{2}w_{b}(t)}{dt^{2}}\bigg{)} \tag{8}\]  
According to the proportional damping assumption, the equations of motion can be solved with the modal expansion approach [7]. Displacement \(w_{r}(x,t)\) is represented by a series of eigenfunctions:  
\[w_{r}(x,t)=\sum_{i=1}^{\infty}\phi_{i}(x)\eta_{i}(t) \tag{9}\]  
In Equation (9) \(\phi_{i}(x)\) is the \(i\)-th mass-normalized eigenfunction (mode of vibration) and \(\eta_{i}(t)\) is the \(i\)-th modal coordinate. Modal expansion transforms Equation (4) into this set of ordinary differential equations:  
\[ \begin{array}{c}\frac{d^{2}\eta_{i}(t)}{dt^{2}}+2\zeta_{i}\omega_{i}\frac{d \eta_{i}(t)}{dt}+\omega_{i}^{2}\eta_{i}(t)+\chi_{i}\eta_{i}(t)=\int_{0}^{L}-m \frac{d^{2}w_{b}(t)}{dt^{2}}\phi_{i}(x)dx+F(t)\phi_{i}(L)+\\ M(t)\frac{\phi_{i}(L)-\phi_{i}(L-h)}{h}\;i=1\ldots\end{array}