# Vibration Energy Harvesting from Plates by Means of Piezoelectric Dynamic Vibration Absorbers
## 3 Calculated Results
Figure 6 highlights that the CPDVA generates much more voltage than the patch because not only are the two peaks of CPDVA higher than the peak of the patch, but also, the bandwidths around resonances are larger. The latter effect can be very useful in the presence of broadband random excitation. In the previous simulations, the CPDVA was mounted at the center of the plate (\(x_{a}=a/2\)\(y_{a}=b/2\)), where the first mode of vibration shows the largest amplitudes (see Figure 2). In the simulations reported in Figure 7, the CPDVA is moved from the best location along the longitudinal direction (\(x\)). It is worth noting that since the first mode of the plate is symmetric, leftward and rightward displacements have the same effect on the coupling between the CPDVA and this mode of the plate. A harmonic distributed load with constant amplitude and variable frequency (\(q_{z}(t)\)) is assumed to excite the plate. The FRF between the acceleration of the point of the plate just above the CPDVA and the distributed load shows that the coupling between the CPDVA and the plate decreases when the CPDVA is moved from the center of the plate, i.e., when \(x_{a}<a/2\). Indeed, the peaks of the FRF become closer and their amplitudes decrease. Eventually, when the CPDVA is located near the nodal line, the two peaks tend to merge into a unique peak. Figure 5: Comparison between the magnitude of calculated FRF of the acceleration at the center of the plate and the uniformly distributed load: plate with patch and plate with CPDVA. Figure 6: Comparison between the magnitude of calculated FRF of the generated OC voltage and the uniformly distributed load: plate with patch and plate with CPDVA. The FRF between the generated OC voltage and the distributed load shows that the bandwidth that includes the two resonance peaks decreases when the CPDVA is moved from the central position. This is a negative effect when the plate is excited by broadband random pressure fluctuations. The effect of the CPDVA position on the peaks of the OC voltage FRF is more complex. If the lateral displacement is small (e.g., \(x_{a}=a/4\)), the amplitudes of the two peaks decrease. The maximum amplitude is reached when the two peaks merge into a unique peak (\(x_{a}=a/10\)). Eventually, peak amplitude decreases when the CPDVA is close to the nodal line of the plate mode. This behavior can be exploited if the plate is excited by a harmonic load having a well-defined frequency. In Figure 8, the CPDVA is moved from the center of the plate in the lateral direction (\(y\)). This displacement does not affect the coupling with the first mode of vibration of the plate and with the other modes for which \(n=0\) (see Table 2 and Figure 2), but it alters the coupling with the modes with \(n\
eq 0\), e.g., the second mode. Therefore, this effect can be found only when the CPDVA is coupled with a plate. The FRF between the acceleration of the plate point above the CPDVA and the distributed load shows small modifications in the position and height of the two main resonance peaks, but when the CPDVA is moved towards the anti-node of the second mode (\(y_{a}=0\)), a minor resonance peak at the frequency of the second peak appears. This phenomenon happens because when \(y_{a}\) decreases, there is coupling between the CPDVA and the second mode as well. Since this coupling alters the shape of the second mode, the uniformly distributed load is able to cause a small excitation of the second mode. The excitation of the second mode slightly alters the effect of the CPDVA on the first mode, and the OC voltage FRF shows small variations in the position and height of the main resonance peaks. For \(x_{a}=a/2\), \(y_{a}=0\), the second peak becomes higher. Figure 9 deals with the effect of CPDVA tuning; in this case, the harvester is always at the center of the plate, which is excited by a harmonic distributed load (\(q_{z}(t)\)). Figure 7: FRF (magnitude) of the acceleration at the center of the plate (**top**) and FRF of the OC voltage generated by the CPDVA (**bottom**).