## 4 Normalized Power to the Volume  
In order to improve the effectiveness of PEHs, the maximum possible power per unit volume must be generated. For this purpose, the normalized power to the volume is studied. First, the normalized power to the volume of the beam is investigated. The volume of the beam is \(V_{b}=lbh\). Then, the impact of the volume of the tip mass is investigated. The resonance frequency is kept constant by increasing the total thickness simultaneously with the tip mass. The volume of the tip mass is \(V_{mt}=l_{mt}b_{mt}h_{mt}\), where \(l_{mt}\) is the length of the tip mass, \(b_{mt}\) is the width of the tip mass, and \(h_{mt}\) is the thickness of the tip mass. Finally, the normalized power to the total occupied volume is investigated by taking into account the volume of motion. The volume of motion is expressed as \(V_{motion}=2 lbx\), where \(x\) is the tip displacement (\(x_{\text{max}}\) and \(x_{\text{min}}\) are considered for the investigation). The properties of the PEH B from Figure 1 are considered for these. Figure 5 displays the normalized power to the volume of the beam (\(\text{NPV}_{\text{b}}\)) at two operating conditions: the corresponding \(R_{op\_res}\) at excitation frequency of \(\omega_{n}\) and the corresponding \(R_{d}\) at excitation frequency of \(\omega_{d}\). The resonance frequency is kept constant at 155 Hz, while the width is varied. The total thickness is simultaneously varied with the free length (no tip mass). For a given material and the same mass ratio between the tip mass and the mass of the beam, the normalized power to the volume of the beam is constant at the corresponding \(R_{d}\) and \(\omega_{d}\) or the corresponding \(R_{op}\) and \(\omega_{n}\). The normalized power to the volume of the beam and the tip mass (\(\text{NPV}_{\text{b+mt}}\)) is investigated. The volume of the tip mass varies according to the density of its material, and accordingly, the normalized power \(\text{NPV}_{\text{b+mt}}\) varies depending on the density of the tip mass (\(\rho_{mt}\)). The investigations include variant densities of the tip mass. The resonance frequency is kept constant at 155 Hz by varying the total thickness  
Figure 4: The impact of the applied load resistance and excitation frequency on (**a**) the output power (**b**) the tip displacement. Figure 5: Normalized power to the volume of the beam (\(f_{n}\) constant at 155 Hz, free length varies from 4.41 to 9.62 cm for the corresponding total thickness varies from \(h=0.08\) to 0.38 cm, respectively, constant mass ratio at \(\beta=0\), mechanical damping ratio of \(\zeta_{m}=0.01\), and input displacement of \(u=0.8\)\(\mu\)m). simultaneously with the mass ratio between the tip mass and the mass of the beam. The free length is constant at 8.49 cm. The investigations will consider two operating conditions: the corresponding \(R_{op}\) and \(\omega_{n}\) and the corresponding \(R_{d}\) and \(\omega_{d}\). Figure 6 displays the \(\mathrm{NPV_{b+mt}}\). If the density of the tip mass is less than the average density of the bimorph piezoelectric cantilever (average density \(\rho_{av}=(2h_{p}\rho_{p}+\rho_{s}h_{s})/(2h_{p}+h_{s})\)), an increase in the tip mass results in a decrease in the \(\mathrm{NPV_{b+mt}}\), despite the fact that the resonance frequency is being kept constant and the total thickness is simultaneously increasing with the increase of the mass ratio. If the density of the tip mass is in the range of the average density, the \(\mathrm{NPV_{b+mt}}\) increases very slightly with a large increase in the tip mass and simultaneously in the total thickness. If the density of the tip mass is much greater than the average density, a simultaneous increase in the tip mass with the total thickness results in an increase in the \(\mathrm{NPV_{b+mt}}\). A simultaneous increase in the length and the thickness optimizes the power output under volume constraints compared with a simultaneous increase in the mass ratio and the thickness when the tip mass is made out of a less dense material.