# Trapezoidal Cantilevers with Irregular Cross-Sections for Energy Harvesting Systems
## 3 Numerical Investigation of Modified Trapezoidal Cantilever Beams
Moreover, as shown above in Figures 6 and 7, the strain characteristics' curve became nonlinear in comparison with Figure 5. The nonlinearity of the strain characteristics was caused by modifications of the cross-section area of the cantilevers. Figure 6: Strain characteristics of the trapezoidal cantilever with cylindrical gaps; freq–frequency in Hertz. Table 2.0 Strain characteristics of the trapezoidal cantilever with trapezoidal gaps. freq–frequency in Hertz. The modifications ensure the decreasing of the second moment of inertia, and thus, strain in the modified part of the cantilever is increased; therefore, the piezo ceramic is employed more efficiently. The highest level of strain and its distribution were obtained for the cantilever with trapezoidal gaps. The comparisons of the strain distribution functions are given in Table 2.1 This summary reveals that the strain distribution in the modified cantilevers has a more constant behavior for the piezo ceramic in comparison with the conventional one. The summary of the strain average values shows that the strain level in modified cantilevers was increased more than 2.7-times compared to the conventional cantilever. According to this summary, the geometrical parameters of the trapezoidal cantilever beams were chosen for further investigation; the values are given in Table 2.2  
Numerical investigation related to the electrical characteristics of the trapezoidal cantilevers was performed. Resistance load was included in an FEM model. A resistor was connected in parallel to the piezo ceramic layer. The other parameters of the model were set as in the previous cases. The value of optimal resistive load \(R_{opt}\) was chosen according to Equation (4). Firstly, the conventional trapezoidal cantilever was analyzed. A frequency response study was employed for this investigation. The frequency range from 50 Hz to 500 Hz was analyzed. The results of the electric outputs as a function of excitation frequency are presented in Table 2.3 It can be noticed that the highest values of the electrical outputs were obtained at the resonant frequency of the trapezoidal cantilever. The output voltage on the electrical load is equal to 2.8 V. Moreover, the current is equal to 2.6 \(\upmu\)A, and the average power of the system is 6.6 \(\upmu\)W.  
Table 2.4 Strain functions and average values of strain: (**a**) conventional cantilever; (**b**) cantilever with cylindrical gaps; (**c**) cantilever with trapezoidal gaps. Table 2.5 Electrical characteristics of the conventional trapezoidal cantilever. Figure 10 represents the electrical characteristics of the trapezoidal cantilever with trapezoidal gaps. It can be noticed that the electrical characteristics of the system reach the highest values at the resonant frequency of 80.6 Hz. Voltage generated by a modified trapezoidal cantilever is equal to 9 V, and it is 3.2-times larger in comparison to the conventional beam. Current generated by a cantilever is equal to 8 \(\upmu\)A and is 3.1-times higher in comparison with the conventional beam. The average output power is equal to 80 \(\upmu\)W, and it is 12.1-times higher in comparison with the conventional beam. Summarizing the results of this investigation, it can be concluded that the modification of the cross-section gives a positive impact on the electrical characteristics of the beam. Figure 11 represents the electrical characteristics of a trapezoidal cantilever with cylindrical gaps. It can be noticed that the output voltage is equal to 8.3 V at a resonant frequency of 90 Hz, and it is 2.96-times higher in comparison with the conventional cantilever. The analysis of the output current and average power shows that the highest values are also in the same frequency. Current reached 7.2 \(\upmu\)A, and average power become equal to 59.76 \(\upmu\)W; these are 2.76- and 9.05-times higher in comparison with the conventional cantilever. In summary, it could be said that the modification performed for a trapezoidal cantilever has a notable impact on the electrical characteristics.