## 1 Introduction  
Harvesting the energy of structural vibrations is becoming increasingly significant as a viable approach to providing an autonomous and independent power source for numerous low-power electronic devices. This process involves converting mechanical energy from ambient vibrations into electrical energy using various transduction processes, such as piezoelectric, electromagnetic, and electrostatic processes. Using different transduction techniques including piezoelectric, electromagnetic, and electrostatic techniques, the mechanical energy from ambient vibrations is transformed into electrical energy in this process. The increasing need for long-term, trouble-free power solutions for things like wireless sensors, wearable electronics, and Internet of Things (IoT) systems, where it can be difficult or impossible to replace or recharge batteries, motivates this strategy. Energy harvesting, which makes use of ambient vibrations in the environment, provides a sustainable and reliable power source, lessening the need for traditional power plants and paving the way for more sustainable, environmentally conscious infrastructure. Therefore, it a great option for situations that need continuous, long-term operation, applications in hostile conditions, or in isolated areas. Piezoelectric materials produce an electrical charge when subject to mechanical pressure. This characteristic renders them well suited for capturing energy from surrounding vibrations. Scientists have focused their efforts on enhancing the effectiveness and energy generation capability of piezoelectric energy harvesters by making improvements in materials and developing creative structural designs [1; 2; 3; 4; 5]. In their study, [PERSON] and [PERSON] [6] employed zinc oxide nanowires to fabricate piezoelectric nanogenerators with a high energy output. Their findings highlight the potential of nanostructured materials to enhance energy conversion efficiency, mostly attributed to their large surface area-to-volume ratio. To increase the power output, [PERSON] et al. [7] investigated the use of cantilever-based piezoelectric harvesters with tip mass. It was found that the output power and resonance frequency range of these devices could be increased by adjusting the tip mass, dimensions, and position. In actual situations where vibration frequencies may vary, such a modification can increase the device's efficiency. Altering the form of the cantilever beam is an alternative method used to improve the efficiency of piezoelectric energy harvesters. Various geometries, such as trapezoidal, triangular, or serpentine shapes, offer more flexibility, can reduce resonance frequencies, and enhance strain distribution [8; 9; 10; 11]. In their study, [PERSON] et al. [12] investigated how different beam forms impact the efficiency of piezoelectric energy harvesters. The researchers found that a trapezoidal beam design resulted in greater power output compared to a rectangular beam, mostly because of improved stress distribution over the length of the beam. [PERSON] et al. [13] present a low-frequency piezoelectric MEMS harvester with a serpentine structure, optimized using a neural network and genetic algorithm. The design achieves a 121.7 Hz resonant frequency and 0.73 \(\upmu\)W power output, demonstrating efficient performance in a compact device. The integration of piezoelectric materials into various structural systems has been a significant area of research. [PERSON] et al. [14] developed a flexible piezoelectric composite by aligning PZT particles within a PDMS matrix using dielectrophoresis. The structured material shows enhanced dielectric and piezoelectric properties, making it suitable for sensors and energy harvesting applications across various surfaces. [PERSON] et al. [15] developed a hybrid energy harvester combining piezoelectric and electromagnetic mechanisms within an inside design. The device efficiently captures low-frequency biomechanical energy from walking, optimizing power generation for wearable electronics. [PERSON] et al.