## 3 Results
### Results of the Optimal Design of the Substructure
Second segment is working as an additional inertia concentration, making the harvester enhance resonance behavior, indirectly increasing strain near the root of the cantilever. The optimization was carried out using an external routine written by the authors in Fortran. The optimized form was found in 180 iterations. Starting from an initial uniform strain of 0.92 mm thickness cantilever, the objective function improved significantly within the first 40 iterations. This early phase demonstrated a steep optimization gradient, indicative of the algorithm's strong responsiveness to initial design changes. From iteration 40 onward, the objective function followed a nearly linear trend with minor fluctuations, suggesting diminishing sensitivity as the design approached an optimal configuration. After 150 iterations, the changes in both strain and frequency values remained within a negligible tolerance band. The eigenfrequency of the optimized design settled at 22.2 Hz, aligning well with the target range of 21-23 Hz. During the optimization path, brief deviations outside the frequency bounds were observed, particularly between iterations 80 and 110, where the frequency temporarily dropped below 20 Hz or exceeded 24 Hz in isolated instances. However, these deviations were self-correcting due to the frequency constraint mechanism embedded in the optimization model. While shortening the length of the high-strain active segment might suggest a potential performance loss, this is more than compensated for by the beneficial inertial effects of the mass-enhancing tail section. Together, the segments work in tandem to achieve superior performance under resonant excitement. The resulting design allows for more effective energy harvesting at the first eigenfrequency. Figure 6 illustrates the normalized axial strain distributions for the first three vibration modes of a uniform-thickness cantilever beam. The strain patterns are shown using three colors: blue for the first mode, green for the second, and red for the third. Figure 7 presents the strain distribution for the cantilever beam with optimized thickness. As in the previous figure, the first three modes are represented by same colors. The strain mode of the first eigen frequency is a classical example for the uniform cantilever; however, in optimal, it is linear from 0 to around 58 mm of the cantilever and later switches to a non-linear pattern. The green line, representing the second mode, is a curved line for the uniform cantilever shape, but in the optimized one, it is straight from the beginning, till 58 mm of length, later  
Figure 5: Normal strain distribution along the optimal shape thickness cantilever under first-mode vibration. turning into non-linear. Also, it is important to mention that the node for this moved from 20 mm from the beginning in uniform to 30 mm in the optimal shape cantilever. Red line, representing third mode, has nodes in 17 mm and 50 mm in uniform shape, which switches to 18 mm and 58 mm in the optimal shape cantilever. A transient dynamic analysis was conducted to evaluate the time-dependent strain behavior of the optimized cantilever configuration relative to a beam with uniform thickness. Figure 8 presents a comparative profile of the integrated axial strain along the upper surface of both designs when subjected to harmonic base excitation at their respective first natural frequencies. Although the axial strain values are dimensionless by definition, their integration over the cantilever length results in a metric value with units of meters, representing the cumulative deformation experienced by the active material layer. Figure 8: Time-dependent axial strain generation along the upper surface of cantilever beam subjected to harmonic base excitation at the first natural frequency. The blue curve corresponds to the uniform-thickness design, the red curve represents the optimized thickness configuration. Figure 6: Normalized strain modes of the uniform thickness substructure: blue—first, green—second, red—third. Black—uniform design cantilever.