# Vibration Energy Harvesting from Plates by Means of Piezoelectric Dynamic Vibration Absorbers
## 2 Mathematical Model
### Harvester Lumped Element Model  
The CPDVA is a cantilever beam partially covered by a piezoelectric layer. The cantilever is fixed to the plate at coordinates \((x_{a},y_{a})\), and a tip mass is fixed at the cantilever tip. Figure 3 represents the harvester and its geometrical parameters. The harvester is modeled through a single-degree-of-freedom (SDOF) approach; the variable that describes the configuration of the harvester is the tip displacement with respect to the harvester base (\(u_{r}(t)\)). The equation of motion of the harvester subject to base excitation is:  
\[M\ddot{u}_{r}(t)+Ci_{r}(t)+Ku_{r}(t)=-\alpha M\ddot{w}(x_{a},y_{a},t) \tag{9}\]  
where \(M\), \(C\) and \(K\) are the lumped mass, damping and stiffness of the harvester, respectively; \(\alpha\) is a corrective factor for base excitation in SDOF models [1,10]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mode Number (\(i\))** & \(m\) & \(n\) & **Frequency (Hz)** \\ \hline
1 & 2 & 0 & 16.49 \\
2 & 2 & 1 & 23.89 \\
3 & 3 & 0 & 45.45 \\
4 & 3 & 1 & 56.40 \\
5 & 2 & 2 & 61.27 \\
6 & 4 & 0 & 89.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculated natural frequency and \(m\)-\(n\) combination for the first six plate modes. Figure 3: Scheme of the CPDVA (**left**) and harvester cross section (**right**). Figure 2: Calculated modes of vibration of the plate. The lumped parameters are calculated following the approach used and validated in [26]. First of all, the displacement of the cantilever along the beam axis is expressed as the product between a shape function \(g(\xi)\) and the relative tip displacement:  
\[u(\xi,t)=g(\xi)u_{r}(t) \tag{10}\]  
where \(\xi\) is the coordinate along the beam axis: \(\xi\) is zero at the clamp location and increases moving to the tip. The static deformed shape of a cantilever with a concentrated load on its tip is assumed to approximate the shape of the first mode of the harvester. Therefore, \(g(\xi)\) is defined as:  
\[g(\xi)=\frac{3\xi^{2}L-\xi^{3}}{2L^{3}} \tag{11}\]  
where \(L\) is the length of the cantilever. Moreover, with this assumption, the ratio between tip rotation (\(\varphi\)) and tip displacement is:  
\[\frac{\varphi}{u_{r}}=\frac{3}{2L} \tag{12}\]  
Using the Lagrangian approach, the following expressions for the lumped parameters of the harvester are found:  
\[M=\frac{33}{140}(m_{s}+m_{p})+M_{t}\left(1+\frac{3w_{t}}{4L}\right)^{2}+\frac {M_{t}}{8L}\left(h_{t}^{2}+w_{t}^{2}\right) \tag{13}\]  
\[K=\frac{3E_{h}I_{h}}{L^{3}}+\frac{\theta^{2}}{C_{p}} \tag{14}\]  
\[C=2\zeta\sqrt{KM} \tag{15}\]  
where \(m_{s}\) and \(m_{p}\) are the masses of the structural and piezoelectric layers of the cantilever, respectively; \(M_{t}\), \(w_{t}\) and \(h_{t}\) are the mass, width and height of the tip mass; \(E_{h}I_{h}\) is the equivalent bending stiffness of the beam; \(C_{p}\) is the capacitance of the piezoelectric layer; \(\theta\) is the electro-mechanical coupling coefficient; and \(\zeta\) is the damping coefficient of the harvester. It is worth noting that stiffness \(K\) depends on the backward piezoelectric effect as well. In open circuit conditions, the electro-mechanical coupling coefficient is defined by the following equation ([20]):  
\[\theta=-e_{3,1}b_{p}h_{pc}\int\limits_{\xi_{1}}^{\xi_{2}}\frac{d^{2}g(\xi)}{d \xi^{2}}d\xi \tag{16}\]  
where \(e_{3,1}\) is the piezoelectric constant of the material; \(b_{p}\) is the width of the piezoelectric layer; \(h_{pc}\) is the distance between the neutral axis of the composite beam and the center of the piezoelectric layer; and \(\xi_{1}\), \(\xi_{2}\) are the coordinates that identify the extremities of the piezoelectric layer along the beam axis, see Figure 3:0  
Finally, the open-circuit voltage produced by the harvester is expressed as [26]:  
Figure 3:1  
Piezoelectric energy harvesters provide alternating voltage (AC), but most electronic devices require a DC voltage input. Hence, piezoelectric energy harvesters are equipped with specific converting systems [27].