# Vibration Energy Harvesting from Plates by Means of Piezoelectric Dynamic Vibration Absorbers
## 3 Calculated Results  
The first mode of vibration of a plate has the largest probability of being stimulated by environmental sources of excitation (wind, traffic and rain) for two reasons: the frequency and the modal shape. Typically, environmental sources are characterized by power spectral densities (PSDs) showing the maximum amplitudes in the low frequency range. Turbulence of grazing flows used to collect energy from artificial piezoelectric grass [28] shows relevant PSD amplitudes below 50 Hz. Wind turbulence PSDs used for the design of civil structures show maximum amplitudes below 1 Hz [29; 30; 31]. The PSDs of traffic-induced vibrations show their main peaks below 20 Hz [32]. The first mode of vibration of a clamped-clamped-free-free plate has the simplest modal shape (Figure 2). Higher-order modes show an increasing number of nodal lines that define different zones of the plate that vibrate in phase opposition. A spatially uniform pressure fluctuation on the surface of the plate can excite the first mode of vibration but is not able to transfer much energy to higher-order modes with different areas that vibrate in phase opposition. To highlight this concept, the FRF between the acceleration of the point at the center of the plate and the distributed load \(q_{z}(t)\) was calculated by means of the mathematical model of Section 2. The FRF magnitude, which is plotted in Figure 4, shows that only the first and sixth modes of vibration are excited by the distributed load. The other modes are not excited at all because there is a complete cancellation between the pressure loads applied in the areas of the plate that move in opposition. Owing to the dependence on the frequency squared (Equation (32)), the acceleration FRF shows the highest peak for the sixth mode. If the sizes of the plate have an order of magnitude of 1 m, they are much smaller than the typical length scale of wind turbulence that excites buildings [31]; therefore, pressure can be considered spatially uniform. For pressure fluctuations related to turbulence having smaller length scales, the possible excitation of higher-order modes has to be evaluated using more sophisticated methods like the acceptance integral [33]; nevertheless, also in this case, the first mode is the most excited [33]. In many practical conditions, the plates can be excited by vibrations coming from the supports [34]; also in this case, the lower-order modes are the most excited. For the above-mentioned reasons, this research focuses on vibration energy harvesting from the first mode of vibration of the plate, and the CPDVA is tuned to the natural frequency of this mode and is located at the center of the plate, which is an anti-node for the first mode of vibration. The mathematical model of the CPDVA is developed starting from the properties of a commercial cantilever harvester M2814P2C-01 built by Smart Material. This harvester has a piezoelectric layer made of macro fiber composite [35] and has a structural layer made of glass fiber composite (FR4). Its properties are summarized in Table 3. The transfer function of the M2814P2C-01 was experimentally evaluated with the impulsive method [36] and showed a resonance peak at 146.69 Hz. Therefore, the addition of a large tip mass is needed to tune the harvester to the first natural frequency of the plate (16.5 Hz). Figure 4: The FRF (magnitude) between the acceleration at the center of the plate and the uniform distributed load.