## 3 Analytical Model of Prototypes
 \tag{10}\]  
In Equation (10) \(\omega_{i}\) is the undamped natural frequency of the \(i\)-th mode of vibration and \(\zeta_{i}\) is modal damping. Coefficient \(\chi_{i}\) is the backward modal electromechanical coupling term [6]. The electrical equation and the DVA equation after expansion in modal coordinates become:  
\[C_{pu}\frac{dv(t)}{dt}+\frac{v(t)}{R}=\sum_{i=1}^{\infty}\phi_{i}\frac{d\eta_ {i}(t)}{dt} \tag{11}\]  
\[m_{eq}\bigg{(}\frac{d^{2}u_{r}(t)}{dt^{2}}+\sum_{i=1}^{\infty}\phi_{i}(L)\frac {d^{2}\eta_{i}(t)}{dt^{2}}+\frac{d^{2}w_{b}(t)}{dt^{2}}\bigg{)}=-k_{a}u_{r}(t) -c_{a}\frac{du_{r}(t)}{dt} \tag{12}\]  
In Equation (11) \(\varphi_{i}\) is the modal electromechanical coupling term [6]. The forced response of the harvester with DVA can be calculated by solving the system of ordinary differential Equations (10)-(12) considering a finite number of modes. On the one hand the aim of this analytical model is to give a first prediction of the response of the harvesters with trimming devices, more detailed analyses will be carried out with the FE approach. On the other hand, some preliminary results [35] showed that the second mode of the harvester has a natural frequency much higher than the first and a very small effect on low frequency dynamics. For these reasons, only the first mode of vibration (\(i=1\)) is considered in Equations (10)-(12). The FRFs of the prototype can be calculated considering the harmonic base excitation:  
\[\frac{d^{2}w_{b}(t)}{dt^{2}}=a_{b0}e^{iwt} \tag{13}\]and looking for harmonic solutions for the first mode of the harvester, DVA displacement, and voltage:  
\[\eta_{1}(t)=\eta_{10}e^{i\omega t}\ u_{r}(t)=u_{r0}e^{i\omega t}\ v(t)=v_{0}e^{i\omega t} \tag{14}\]  
\(a_{b0}\), \(\eta_{10}\), \(u_{r0}\) and \(v_{0}\) are complex constants. The system composed of Equations (10)-(12) becomes a linear-algebraic system, which can be solved numerically by means of MATLAB (Version R2014a, MathWorks, Natick, MA, USA, 2014). At first the analytical model was used to calculate the FRFs between the open circuit voltage and base acceleration of the three prototypes, because the open circuit voltage is an important figure for the design of the energy conversion system of the harvester [36]; results are presented in Figure 3.  
When the most massive cantilever DVA (A1) is added to the harvester, the original resonance peak (125 Hz) is substituted by two new resonance peaks at 97.1 and 161.5 Hz respectively, the interval between the two resonances is 64.4 Hz. The first peak is slightly lower than the peak of PPA 1001 alone, whereas the second peak is rather small. If the DVAs with smaller masses are considered (A2 and A3), the frequency interval between the two new peaks decreases, the height of the first peak decreases, whereas the height of the second peak increases. Table 1 summarizes the resonance frequencies and the corresponding amplitudes in resonance of the three prototypes. The phases of the modal displacement of the harvester (\(\eta_{10}\)) and of the displacement of the DVA (\(u_{r0}\)) with respect to base acceleration give useful information about the functioning of the system. Figure 4 shows that at the frequency of the first resonance peak of the harvester with DVA, the two displacements essentially have the same phase with respect to base acceleration, therefore the harvester and the DVA move in phase when the first mode of vibration is excited in resonance. At the frequency of the second resonance peak there is a phase shift of about 180\({}^{\circ}\) between the modal response of the harvester and the displacement of the DVA, therefore the DVA moves in opposition with respect to the harvester when the second mode of vibration is excited in resonance. The analytical model makes it possible to analyze the effect of the trimming frequency (\(f_{t}\)) of the cantilever DVA on the performance of the harvester.