## 1 Introduction  
Harvesting energy from environmental sources has become an increasingly prominent research area in recent years due to its vast potential for various applications. Providing power to wireless electronics and sensor systems with minimal energy demands--especially in places where battery replacement is impractical--has proven highly advantageous for automation and remote monitoring. Therefore, energy harvesters have been engineered to convert ambient energy into a usable electrical form, enabling small wireless systems to operate throughout their service life. In recent developments, vibration-induced energy harvesting has emerged as a particularly attractive solution and has seen growing academic and engineering interest. Vibrational energy can be converted into electrical power through three main mechanisms: electrostatic, electromagnetic, and piezoelectric methods [1]. Among these, the piezoelectric approach stands out due to its compact size, versatility in shape design, and high energy conversion efficiency [2]. Even minor mechanical vibrations can yield a notable electrical output, eliminating the need for low-efficiency mechanical triggering systems [3]. The key functionality of piezoelectric materials lies in their ability to produce electrical charge in response to mechanical forces (direct effect) and to undergo mechanical deformation when subjected to an external electric field (inverse effect). The direct piezoelectric effect is particularly beneficial for converting mechanical vibrations into electrical energy to support intelligent and autonomous systems. The literature contains numerous reviews that explore different piezoelectric energy harvester concepts and design strategies [4; 5; 6; 7]. One of the most commonly studied designs is the cantilever beam with surface-attached piezoelectric elements, which operates under transverse vibrations. This design is valued for its structural simplicity and cost-effectiveness. These systems often exploit resonance to maximize strain in the piezoelectric material, which in turn, enhances electrical output. Furthermore, optimizing the physical shape of the harvester is considered a key area for performance enhancement. For example, optimal cantilevers width designs were explored by [PERSON] et al. [8], who conducted a parametric FEM-based study where the beam's tip width and position of the piezoelectric patch were optimized under harmonic excitation. Their results showed a 37% increase in power output compared to a baseline rectangular configuration. The study highlighted that performance gains can be achieved without changing the material or beam length, simply by adjusting the geometric distribution of strain. Their work also emphasized the synergy between geometric shaping and dynamic tuning, showing that optimal patch placement both maximizes strain and avoids nodal points of the mode shape. However, the excitation was performed continuously to the tip of the cantilever and the eigen frequencies did not remain the same. [PERSON] et al. [9] proposed a quadratic width profile for piezoelectric unimorphs and compared their performance against rectangular and triangular counterparts. FEM simulations demonstrated significant improvements in strain uniformity and output power. Their optimization was particularly notable for maintaining constant volume, which allowed a fair comparison across geometries. The study demonstrated that quadratic profiles provided better energy conversion under real-world excitation levels by reducing stress concentration while distributing deformation more evenly across the piezoelectric surface. [PERSON] et al. [10] extended this idea by introducing curved profiles into cantilever beams. Their experimental and numerical results indicated that, under a 5 g tip mass, a curved profile improved strain uniformity by 29% compared to triangular designs, which are already superior to rectangular ones.