# Vibration Energy Harvesting from Plates by Means of Piezoelectric Dynamic Vibration Absorbers
## 3 Calculated Results
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Structural layer length & 70 mm \\ Structural layer thickness & 1 mm \\ Structural layer width & 25 mm \\ Structural layer elastic modulus & 24 GPa \\ Structural layer density & 1920 kg m\({}^{-3}\) \\ Piezoelectric patch length & 28 mm \\ Piezoelectric patch thickness & 0.18 mm \\ Piezoelectric patch width & 14 mm \\ Piezoelectric patch elastic modulus & 30.24 GPa \\ Piezoelectric patch density & 7400 kg m\({}^{-3}\) \\ Piezoelectric patch capacitance & 30.8 nF \\ Piezoelectric constant \(e_{31}\) & 5.02 pC m\({}^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the M2814P2C-01 cantilever harvester. A small clamp is needed to fix the cantilever base to the plate. If the clamp is made of polymeric material, its mass is about 0.018 kg. Before tuning the harvester to the plate, the effect of the mass of the clamp on the natural frequency of the first plate mode was evaluated by means of the mathematical model of Section 2, setting \(M=0\) kg. The clamp mass has a very small effect on the natural frequency of the first mode, which moves from 16.5 Hz to 16.4 Hz. Harvester tuning is then carried out by means of Equation (9) by setting the term on the right-hand side to zero (free vibration analysis). The undamped natural frequency is given by:  
\[\omega_{n}=\sqrt{K/M} \tag{37}\]  
where \(K\) and \(M\) are given by Equations (13) and (14), respectively. Since all the terms in the expressions of \(K\) and \(M\) are known except \(M_{t}\), the undamped natural frequency (\(\omega_{n}\)) is set equal to the natural frequency of the first mode of the plate, and the tip mass that tunes the harvester is found: the calculated value is \(M_{t}=0.051\) kg. The calculations reported in this section aim to show the validity of the CPDVA and the importance of tuning and the location of the device. The focus is on voltage generation. Figure 5 makes a comparison between the acceleration FRF at the center of the plate and the acceleration FRF at the same point when the CPDVA is tuned to the first mode of the plate and is mounted at the center of the plate. Since the dimensions of the patch are very small compared to the dimensions of the plate, the effect of the patch is negligible, and simulations show that the difference between the first natural frequency calculated with and without the patch is about \(3\times 10^{-3}\) Hz. To calculate the FRFs, a distributed load with constant amplitude and variable frequency (\(q_{z}(t)\)) is assumed to excite the plate. As in the other cases considered in the scientific literature (e.g., beams [22]), the CPDVA eliminates the original resonance peak of the plate and generates a couple of new peaks: the first at a lower frequency and the second at a higher frequency. The amplitudes of the new acceleration peaks are significantly lower than the one of the original resonance peak of the bare plate. Figure 6 makes a comparison between the OC voltage FRF of the CPDVA tuned to the first mode of the plate and mounted at the center of the plate and the OC voltage FRF of a piezoelectric patch directly bonded under the center of the plate. The first mode of the plate corresponds to the bending of a beam in the \(x\)-\(z\) plane; hence, when the first mode is excited, the curvature at the center of the plate is large and the electro-mechanical coupling between the patch and the plate is large. Indeed, for the first mode, Equation (16) can be generalized for the patch bonded on the plate as:  
\[\theta=-e_{3,1}b_{p}h_{pcp}\int\limits_{x_{1}}^{x_{2}}\frac{\partial^{2}\Phi_ {1}}{\partial x^{2}}dx \tag{38}\]  
where \(h_{pcp}\) is the distance between the center of the piezoelectric layer and the neutral axis of the plate, and \(x_{1}\), \(x_{2}\) are the coordinates of the extremities of the patch. This piezoelectric patch has the same properties and sizes of the piezoelectric layer included in the M2814P2C-01 harvester.