## 5 Conclusions  
This study proposed and validated a gradient-based optimization framework for the thickness distribution of cantilever piezoelectric energy harvesters, constrained to maintain a prescribed first eigenfrequency. The optimized geometry, characterized by a linear taper from 0.95 mm to 1.25 mm across 58% of its span and a non-linear increase to 2.14 mm at the free end, achieved a fundamental resonance of 22.2 Hz--closely aligning with the uniform of 20 Hz. Compared to a conventional uniform-thickness beam, the optimized configuration yielded a 36% increase in integrated strain magnitude (from 2.2 \(\times\) 10\({}^{-6}\) m to 3.0 \(\times\) 10\({}^{-6}\) m) under base excitation.  
Two independent experimental setups were employed for empirical validation. In the first, a non-contact laser vibrometry system (Polytec 3D) was used to characterize the dynamic behavior of stereolithographically fabricated specimens. The measured fundamental frequencies of 20.6 Hz (optimized) and 21.6 Hz (uniform) confirmed simulation predictions and manufacturing precision within 5%. The second experiment used a first resonant base excitation setup, where the optimized cantilever generated a 1.4 V signal, compared to 1.07 V for the uniform design, under a matched acceleration of 14 m/s\({}^{2}\)--representing an approximate 30% gain in experimental voltage output. This directly confirms the predicted improvements in energy harvesting efficiency due to optimized geometry.  
This work contributes to the existing literature by demonstrating that non-linear thickness shape optimization, unlike more commonly studied width variation, can significantly enhance strain localization and electrical output while preserving the original eigenfrequency. The demonstrated gain in voltage output, achieved without shifting resonance characteristics, marks a notable advancement in the structural optimization of piezoelectric harvesters. While this research focused on optimizing the geometry for the first eigenfrequency, the same approach could be extended to higher-order modes, such as the second or third eigenfrequency, to support future multimode or broadband energy harvesting applications.  
In conclusion, the proposed thickness optimization methodology significantly improves the performance of cantilever piezoelectric energy harvesters in low-frequency applications. The strong agreement between numerical and experimental results confirms the robustness and manufacturability of the approach. These findings highlight the potential of geometric tailoring as a scalable and effective design strategy for enhancing energy conversion in self-powered sensing, IoT nodes, and structural health monitoring systems.  
Conceptualization, R.G.; methodology, R.G.; validation, P.S. and R.G.; resources, P.S.; data curation, P.S.; writing--original draft, P.S. and R.G.; writing--review and editing, P.S. and R.G.; visualization, P.S. and R.G.; supervision, R.G.; project administration, [PERSON]. All authors have read and agreed to the published version of the manuscript.  
This research received funding from the Research Council of Lithuania (LMTLT), agreement No. S-MIP-24-46.  
Not applicable.  
The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.