## 3 Results  
### Results of the Optimal Design of the Substructure  
To improve the electromechanical efficiency of the piezoelectric energy harvester, the cantilever beam geometry was optimized by adjusting the thickness distribution while constraining the structure to operate effectively at the first natural frequency. Rayleigh damping was applied as a modeling technique to represent energy dissipation within the simulation. The primary design goal was to enhance strain energy concentration in the piezoelectric active layer, thereby increasing the electric polarization and energy output. The base geometry consisted of a cantilever beam with a length of 100 mm and a constant width of 10 mm. Finite element modeling (FEM) was used for structural optimization, employing 20 bending beam-type elements to discretize the structure. The selected material was Engineering Resin Tough 2000, a homogeneous photopolymer offering stable and repeatable mechanical properties. The optimization was carried out using the gradient projection method with constraints ensuring that the first eigenfrequency remained within a target range of 21-23 Hz. The final optimized configuration achieved a resonant frequency at 22.2 Hz, with harmonic response analysis indicating a peak electrical output at this frequency. The resulting optimal geometry consists of two functionally distinct sections. The first segment, extending from the clamped base to approximately 0.58 L, exhibits a linear taper from 1.25 mm down to 0.95 mm. This taper promotes a uniform strain distribution in the active region, maximizing the piezoelectric response. The second segment, from 0.58 L to the free end, features a non-linear increase in thickness, reaching 2.14 mm. This section behaves as a distributed inertial mass, enhancing vibrational energy concentration and increasing strain level at the first segment. The optimal thickness profile is shown in Figure 3, with panel (a) displaying the plane view and panel (b) displaying the axonometric representation (with thickness exaggerated 2\(\times\) relative to beam length for clarity). To assess how the optimized geometry enhances strain energy concentration and overall conversion efficiency, the axial strain distribution for the first vibration mode was computed for the optimal design. The resulting strain field is illustrated in Figure 4, where the optimal shapes' (picture a) normal strain field is compared to uniform shapes' (picture b) one. As outlined in the optimization results, the optimal cantilever is divided into two synergistic structural regions. Uniform shapes cantilever works as expected for a classic cantilever--the largest strain appears at the clamped point of the cantilever. A more detailed distribution of strain along the optimized cantilever is provided in Figure 5, allowing for a better explanation of the results. The first section of the cantilever, nearest the fixed support, exhibits a modest linear taper and maintains a near-uniform  
Figure 4: First eigenform and normal strain field of the cantilever: (**a**)—optimal shape thickness, (**b**)—constant thickness. Figure 3: Substructure of the optimal thickness shape for first eigenfrequency: (**a**) plane view, scale 1, (**b**) thickness scale exaggerated \(\times\)2 relative to length. strain along its length. This profile ensures that the active piezoelectric material experiences consistently elevated strain, thereby optimizing electrical output. It is followed by the second segment of the beam--which begins at approximately 0.58 L and extends to the free end--featuring a non-linear increase in thickness. Although this segment is located farther from the strain-intense clamped boundary and may initially appear less effective for energy harvesting, it still plays a crucial role. Its tailored mass distribution significantly enhances dynamic amplification, improving overall performance.