# Trapezoidal Cantilevers with Irregular Cross-Sections for Energy Harvesting Systems
## 3 Numerical Investigation of Modified Trapezoidal Cantilever Beams
The analysis of the graph showed that the highest strain was achieved when the trapezoidal cantilever base width was 10 mm. This value was obtained when the excitation frequency was 85 Hz, and it is almost equal to the natural frequency of the beam (Figure 4.). The obtained strain value is directly related to cantilevers' stiffness. As can be noticed in Figure 5, the highest value of the strain was obtained at the lowest base width, when the cross-section design is unmodified. The difference of the strain values at the fixed end of the cantilever and at \(L_{Piezo}\) is 25.23%. Constant decreasing of the strain shows that the piezo ceramic is strained unevenly and inefficiently; moreover, the standard sample deviation of strain values is 7.4 \(\times\) 10\({}^{-4}\). According to the results of the parametric investigation, it can be concluded that the electrical outputs will be at the highest level when the base width is 10 mm. Next, the parametric sweep was conducted for the trapezoidal cantilever base \(L_{B2}\). The cantilever cross-section area was modified by cylindrical gaps. The frequency range was set according to the optimization problem in Equation 6. The strain characteristics are presented in Figure 6.  
The analysis of the graph revealed that the highest strain value was achieved when \(L_{B2}\) was equal to 12 mm. This value was obtained when the excitation frequency was 90 Hz, and it matches the natural frequency of the cantilever (Figure 3). According to this, it can be said that the cantilever stiffness characteristic has the lowest value when the base width is 12 mm; considering this, it can be concluded that modifications decreased the stiffness of the cantilever, and this ensured approximately the same strain as in the conventional cantilever, but in a larger volume of the piezo ceramic. The graph shows that the strain characteristics along the piezo ceramic layer have a nonlinear behavior and constantly decrease. The difference of the strain values at the fixed end of the cantilever and at \(L_{Piezo}\) is 12.16%. This is more than two-times lower in comparison with the conventional cantilever, and it shows that the strain distribution becomes more, even than in the case before; moreover, the standard sample deviation of the strain is 9.3 \(\times\) 10\({}^{-4}\). Such results can be explained by the nonlinear  
Figure 5: Strain characteristics of the conventional trapezoidal cantilever; freq–frequency in Hertz. inertia moment in the cantilever cross-section. The nonlinearity of the inertia moment is caused by cross-section modifications that ensure more effective usage of the piezo ceramic layer; hence, the electrical characteristics of the cantilever will be improved. The third parametric sweep was conducted for the trapezoidal cantilever with trapezoidal gaps. The frequency range was set as in the previous two cases. The obtained strain characteristics are given in Figure 7. The analysis of the graph revealed that higher strain and its best distribution in the piezo ceramic were achieved when the width of the base was equal to 16 mm. Moreover, the strain increased significantly, and it is evenly distributed along \(L_{Piezo}\). The difference of the strain values at the fixed end and \(L_{Piezo}\) is 14.16%, and it is 1.5-times lower in comparison with the conventional cantilever. The standard sample deviation of strain is 19.6 \(\times\) 10\({}^{-4}\). According to these results, it can be concluded that modifications of the cross-section area improved the strain and its distribution characteristics. Considering this, the electrical characteristics will be improved, as well. Summarizing this part of the numerical investigation, it can be concluded that modifications of the cross-section area have improved the strain level and its distribution. The analysis of the differences in strain values along \(L_{Piezo}\) showed that the strain distribution was increased more than 1.5-times in comparison with the conventional one.