# Trapezoidal Cantilevers with Irregular Cross-Sections for Energy Harvesting Systems
## 3 Numerical Investigation of Modified Trapezoidal Cantilever Beams
The characteristics of the material properties are given in Table 2.  
The results of the modal analysis of each cantilever are given in Figure 4 and compared in Table 3. The aim of this investigation was to find the natural frequencies of the first out-of-plane bending mode for each beam. The next step of the investigation was the base harmonic excitation case. The aim of this investigation was to obtain output voltage, current, average power and strain conducted in the \(YY\) direction. The strain distribution at the middle line of the trapezoidal beam was analyzed (Figure 4). The parametric sweep of cantilever base length \(L_{B}\) was performed in order to obtain the highest strains at the middle line of the trapezoidal beam. The optimization problem for the cantilevers' parameters can be written as follows:  
\[\left\{ \begin{array}{l}\int\limits_{0}^{L_{\text{piezo}}}\varepsilon_{YY}^{ PZT}\rightarrow\text{max}\\ \omega_{\text{min}}\leq\omega\leq\omega_{\text{max}}\\ L_{\text{Tip}}\rightarrow\text{const}\\ L_{\text{min}}\leq L_{B1}\leq L_{\text{max}}\\ L_{\text{min}}\leq L_{B2}\leq L_{\text{max}}\\ L_{\text{min}}\leq L_{B3}\leq L_{\text{max}}\end{array} \right. \tag{6}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material Properties** & **Beryllium Bronze** & **Modified Lead Zirconate Titanate PIC255** \\  & **DIN 2.1247** & **(Physik Instrumente, Lindenstraße, Germany)** \\ \hline Density (kg/m\({}^{3}\)) & 8360 & 7800 \\ Young’s modulus (N/m\({}^{2}\)) & \(10^{10}\) & - \\ Poisson’s ratio & 0.34 & - \\ Isotropic structural loss factor & 0.002 & 0.004 \\ Relative permittivity & - & In the polarization direction \(\varepsilon_{\text{Q}}\text{T}/\varepsilon_{0}=1200\) \\  & & Perpendicular to polarity \(\varepsilon_{11}\text{T}/\varepsilon_{0}=1500\) \\ Elastic stiffness coefficient \(c_{3D}\) (N/m\({}^{2}\)) & - & \(16.6\times 10^{10}\) \\ Dielectric loss factor tan \(\delta\) (\(10^{-3}\)) & - & 20 \\ Coupling factor \(\text{k}_{31}\) & - & 0.35 \\ Piezoelectric voltage coefficient \(\text{g}_{31}\) & - & \(-11.3\) \\ (\(10^{-3}\) Vm/N) & & \(-11.3\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties used for modeling. Figure 4: Results of modal analysis: (**a**) conventional trapezoidal cantilever beam, \(\omega_{n}=80.6\) Hz; (**b**) trapezoidal cantilever beam with cylindrical gaps, \(\omega_{n}=89.5\) Hz; (**c**) trapezoidal cantilever beam with trapezoidal gaps, \(\omega_{n}=119.1\) Hz. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cantilever Type** & **Calculated Natural Frequency (Hz)** & **Measured Resonant Frequency (Hz)** \\ \hline Conventional cantilever & 80.6 & 70.05 \\ Cantilever with cylindrical gaps & 89.5 & 79.68 \\ Cantilever with trapezoidal gaps & 119.1 & 101.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of numerical and experimental results. where \(\varepsilon\) is the strain in the piezo ceramic layer; \(\omega_{\text{min}}\) is the lowest excitation frequency; \(\omega_{\text{max}}\) is the highest excitation frequency; \(L_{\text{min}}\) is the minimal width of the base; \(L_{\text{max}}\) is maximum width of the base; \(L_{B1}\), \(L_{B2}\) and \(L_{B3}\) are the optimal base widths of conventional cantilevers and cantilevers modified by cylindrical and trapezoidal gaps. The optimal width of the base will be determined from the viewpoint of the maximum value of strain in the piezo ceramic. \(L_{Tip}\) is the width of the cantilever's tip; it is constant for all cases. The value of \(L_{Tip}\) is given in Table 2.  
During the parametric sweep, the following values were used: \(\omega_{\text{min}}\) = 50 Hz, \(\omega_{\text{max}}\) = 500 Hz, \(L_{\text{min}}\) = 10 mm, \(L_{\text{max}}\) = 30 mm. The first parametric sweep was made for \(L_{B1}\). It is the base width of the conventional trapezoidal cantilever. The results of the parametric sweep are given in Figure 5. The parametric sweep step was set to 2 mm.