# Trapezoidal Cantilevers with Irregular Cross-Sections for Energy Harvesting Systems
## 1 Introduction  
The power supply for wireless sensors and low power electronics is an important research topic driven by the disadvantages of conventional power supply methods, such as short life time, high cost of service and environmental pollution [1,2]. Therefore, there is the need to replace conventional power sources with harvested energy sources provided suitable energy exists in the application environment. One promising solution is kinetic energy harvesting technology that can provide permanent power supply and ensure low cost service. Moreover, this technology is environmentally friendly [3]. Kinetic energy harvesting can be carried out by electromagnetic, electrostatic, piezoelectric and triboelectric transducers. Electromagnetic, electrostatic and triboelectric transducers have low power density and a complex design. In addition, electrostatic transducers need an external power supply. Hence, these characteristics have a negative impact on the practical application of such technologies [4]. Piezoelectric energy harvesting transducers are more suitable for practical applications because of their simple design, high power density and low cost. Therefore, piezoelectric energy harvesting becomes the most promising power supply technology for wireless electronic devices [5]. In most cases, a piezoelectric kinetic energy harvesting device consists of a bimorph cantilever beam with one or two piezoelectric layers. Usually, structural motions of the host excite a cantilever beam. Structural motions of the host induct dynamic strains in piezoelectric layers; therefore, an alternating voltage is generated on the electrodes of the piezoelectric layers [6]. The best output performance of the piezoelectric energy harvesting devices can be achieved by matching the vibrationfrequency of the host and the natural frequency of the cantilever; i.e., when the system operates in resonant mode [7]. Numerous theoretical and experimental investigations related to piezoelectric energy harvesting have been performed in recent years. Usually, two common models are used for the theoretical investigation of piezoelectric energy harvesting systems: the lumped parameter model and the equivalent circuit model [8; 9; 10]. The lumped parameter model analyzes only the mechanical behavior of the system, and it is highly simplified; therefore, such a model is highly inaccurate. The equivalent circuit model describes the mechanical and electrical conjunction of the system and provides a good understanding of the electromechanical behavior of the energy harvesting system. There are many papers related to research on cantilevers used for energy harvesting. [PERSON] et al. analyzed three different designs of the cantilever beams with the goal of obtaining the highest power density [11]. It was concluded that one of the most promising designs is a cantilever beam with a trapezoidal shape. The analysis of the results showed that this design gives more than 30% higher power output compared to the cantilever beams with rectangular or triangular shapes. [PERSON] et al. conducted modeling and experimental research dedicated to two types of cantilever beams: trapezoidal and reversed trapezoidal cantilevers [12]. The research revealed that both proposed geometries of the cantilevers allow one to considerably increase the output power in comparison with rectangular cantilevers. The author has achieved a specific value of power per volume that is close to the power requirements of wireless sensors. [PERSON] and [PERSON] conducted research on the modeling and experimental verification of electromechanically-coupled beams with varying cross-sectional areas [13]. The results of the modeling and experimental verification of the proposed model were reported and compared to [PERSON] and [PERSON]'s reported results [14].