# Trapezoidal Cantilevers with Irregular Cross-Sections for Energy Harvesting Systems
## 3 Numerical Investigation of Modified Trapezoidal Cantilever Beams  
Numerical investigation of the cantilevers with conventional and two irregular cross-section designs was performed. A geometrical optimization problem for each cantilever was set as presented in Equation (6). The geometrical parameters and a side view of each investigated cross-section area design are presented in Figure 2 and Table 1, respectively. As shown in Figure 2, the cross-section modifications were made only under the piezo ceramic. The length of the modified cross-section area is equal to the length of the piezo ceramic \(L_{Piezo}\), i.e., 30 mm. This was made in order to obtain a constant strain and its distribution in this part of the cantilever. Figure 1: Design of the cantilever: (**a**) cross-section area with cylindrical gaps; (**b**) cross-section area with trapezoidal gaps; (**c**) conventional design of the cross-section area; 1, plastic clamping frame; 2, piezo ceramic; 3, cantilever beam; 4, seismic mass. Figure 2: Geometrical characteristics of cross-section areas: (**a**) cross-section area with trapezoidal gaps; (**b**) cross-section area with cylindrical gaps; (**c**) conventional cross-section area. At first, modal analysis was performed in order to obtain the modal shapes and natural frequencies of each cantilever. After that, a study based on harmonic excitation of the base was performed in order to obtain the electrical and mechanical characteristics of each cantilever. Finite Element Modeling (FEM) software Comsol Multiphysics 5.2 (Comsol Inc., Burlington, MA, USA) was used for investigation. A FEM model was built, and boundary conditions were set, as well (Figure 3). Mechanical boundary conditions were set as follows: host structure acceleration was set to 0.5 m/s\({}^{2}\); the weight of the seismic mass was equal to 8.71 g; gravity force was included, as well; the resistive load connected in parallel with the piezo ceramic was set as an electrical boundary condition. The value of the resistive load for each cantilever was calculated as presented below [2]:  
\[R_{opt}=\frac{1}{\omega_{n}C_{p}} \tag{4}\]  
where \(\omega_{n}\) is the natural frequency of the beam and \(C_{p}\) is the capacitance of the piezoelectric layer. The equation related to piezoelectric layer capacitance can be written as shown below [2]:  
\[C_{p}=\frac{\epsilon_{33}\epsilon_{0}A}{t_{c}} \tag{5}\]  
where \(\epsilon_{33}\) is the dielectric constant, \(\epsilon_{0}\) is the permittivity of free space, and it is equal to 8.9 \(\times\) 10\({}^{-12}\) (F/m), \(A\) is the area of an electrode and \(t\) is the distance between two electrodes. Finally, material properties were added to the model. Beryllium bronze DIN (German Institute for Standardization) 2.1247 was used as a base layer of the trapezoidal beam and soft modified lead zirconate titanate PIC255 piezo ceramic (Physik Instrumente, Lindenstrasse, Germany) for active layer \begin{table}
\begin{tabular}{l l l} \hline
**Parameter** & **Value (mm)** & **Explanation** \\ \hline \(L\) & 70 & Length of the trapezoidal cantilever beam. \\ \(L_{Plexo}\) & 30 & Length of the piezo ceramic layer. \\ \(L_{Tlp}\) & 4.5 & Width of the trapezoidal cantilever beam tip. \\ \(L_{B1}\) & 16 & Width of the trapezoidal cantilever beam base. \\ \(L_{B2}\) & 12 & Width of the trapezoidal cantilever beam base. \\ \(L_{B3}\) & 10 & Width of the trapezoidal cantilever beam base. \\ \(L_{b}\) & 2 & Width of the trapezoidal gap base. \\ \(L_{S}\) & 1.2 & Width of the trapezoidal gap base. \\ \(D_{S}\) & 2 & Diameter of the cylindrical gap \\ \(D_{b}\) & 1.2 & Diameter of the cylindrical gap \\ \(H_{Plexo}\) & 0.5 & Thickness of the piezo ceramic layer \\ \(H\) & 2 & Thickness of the trapezoidal cantilever beam \\ \hline \end{tabular}
\end{table} 
Table 1: Geometrical parameters of the trapezoidal cantilevers. Figure 3: FEM (Finite Element Modeling) model of the trapezoidal cantilever beam. PZT-Lead Zirconate Titanate. modeling. The isotropic loss factor for beryllium bronze was defined.