## 7 Optimized Rectangular-Shaped Piezoelectric Energy Harvester  
The major proposed structure geometry for PEHs is the uniform, cross-sectional, rectangular-shaped beam, especially in the studies that conducted experimental investigations [24; 25; 26]. The uniform rectangular-shaped cantilever is easy to model and to manufacture, in addition to being easy to apply. The rectangular-shaped piezoelectric energy harvester is selected in this study to be the reference for evaluating the effectiveness of PEHs. In the investigations on the normalized power to the total occupied volume, the volume of the tip mass was considered as only the volume occupied by its material. However, there can be other unused volume according to the position of the tip mass on the beam, as displayed in Figure 11. This volume is relatively large compared to the volume of the transducer, which is the main part for converting vibration energy to electrical energy. If this volume is taken into account, the normalized power to the total occupied will be very low. Therefore, this position of the mass above the beam is not recommended for the optimized structure. Figure 11: Unused volume according to the position of the tip mass. Another possibility is to uniformly distribute the tip mass along the surface area of the beam so that its volume is \(l_{b}b_{b}l_{mt}\). But, this position will not also optimize the structure, because instead of fixing a uniformly distributed mass above the beam, this distributed mass can be added to the middle layer, which will increase the total thickness of the beam and improve the power output. Therefore, the uniformly distributed mass on the surface of the beam does not optimize the structure under volume constraints. For these reasons, the optimized structure is suggested as displayed in Figure 12. The width of the tip mass equals the width of the beam in order to avoid occupying additional volume in the 2-direction. The thickness of the tip mass equals the thickness of the beam in order to avoid occupying additional volume in the 3-direction. The ratio of the length of the tip mass to the length of the beam is determined according to some assumptions regarding the material of the middle layer and the piezoelectric layers. Most piezoelectric ceramics are of a density in the range of 8 g/cm\({}^{3}\). According to [10], in order to optimize the power output, highly dense materials are recommended for the middle layer. Copper and steel are suggested for the middle layer, as they are relatively highly dense materials. Hence, the density of the middle layer is approximately 8.3 g/cm\({}^{3}\). Accordingly, the average density of the beam (the middle layer and the two piezoelectric layers) is approximately 8 g/cm\({}^{3}\). In order to optimize the power output under a predefined volume, the tip mass must be made out of a very highly dense material, much higher than the average density. The densest materials currently known are in the range of 20 g/cm\({}^{3}\) (which is approximately 2.5 times the average density of the beam). Under these assumptions, and according to the suggested optimized structure in Figure 12, the mass ratio is suggested to be 2.5. The tip mass, if the mass ratio is 2.5, occupies approximately the same volume as the beam. This means that half of the beam is a bimorph piezoelectric transducer and the other half is the tip mass (\(l=l_{b}+l_{mt}\)). According to the previous investigation, the parameters of the optimized rectangular-shaped geometry are as follows: \(\rho_{mt}=20\) g/cm\({}^{3}\), \(b_{mt}=b_{b}\), \(h_{mt}=h_{b}\), \(l_{mt}=l_{b}\), and \(\beta=2.5\). A high mechanical quality factor is recommended for improving the power output. However, a very high-quality factor of more than 100 will not significantly influence the normalized power to the total occupied volume. This is due to the increase in the transversal displacement and accordingly, the volume of motion.