# Trapezoidal Cantilevers with Irregular Cross-Sections for Energy Harvesting Systems
## 1 Introduction
Based on the results obtained during the three-stage research, the authors have concluded that when the electrical term is neglected in the mechanical equation, the resulting expression for the optimum load resistance to get the maximum power output is inaccurate, especially for systems with large electromechanical coupling. The proposed research, i.e., modeling and experimental work, concentrated on cantilever width and optimal resistance load values in order to obtain maximum power output. On the other hand, the authors neglected the possibility for improving the electrical outputs of the system by various design of the cross-section area. [PERSON] et al. conducted research related to the coupled electromechanical model, which accounts for the mechanical behavior of the structure using the Euler-Lagrange equation and piezoelectric behavior using the Gauss law [15]. On the basis of the modeling results, the authors concluded that the fundamental natural frequency and mode shape are strongly affected by varying the shape of a cantilever beam. Moreover, the authors claimed that the load resistance has a strong influence on the level of the harvested power for variably-shaped cantilevers. Modeling revealed that there are specific load resistances that give the opportunity to harvest energy at low frequency and with a higher performance than the classical harvesters. [PERSON] and [PERSON] published results on the characterization of different beam shapes for piezoelectric energy harvesting [16]. The research was based on modeling the well-established Rayleigh-Ritz method, as well as the verification of the results by experimental work. On the basis of the investigation, the authors concluded that the shape has only a small influence on efficiency, but can have an enormous effect on the maximum tolerable excitation amplitude and, thus, on the maximum output power. On the other hand, the authors neglected an opportunity in the modeling and experimental work to improve the efficiency of the system through various designs of the cross-section areas. [PERSON] et al. performed research related to the design and performance of variably-shaped cantilevers [17]. The authors developed an electromechanically-coupled model and verified the results of the modeling by experimental work. The model was based on the Euler-Lagrange equation for mechanical behavior; electrical behavior was modeled by the Gauss law equations. The target of the investigation achieved \(2.5\) mW/cm\({}^{3}\)/g\({}^{2}\) at a wide-band excitation frequency when the area for the energy harvesting system is fixed at 4 cm\({}^{2}\). The authors performed modeling and experimental work for three cases; i.e., an array of wide rectangular cantilevers and arrays of narrow rectangular and trapezoidal cantilevers. Based on the research results, the authors claimed that the best power density was obtained with wide cantilevers, but the operational bandwidth for such systems is narrow. The case with narrow cantilevers showed that power density in comparison with wide cantilevers is lower, but the operational bandwidth is much wider. The analysis of the trapezoidal cantilever case showed that such a system can be employed as a compromise in power density and bandwidth. This paper provides the results of numerical and experimental research on the piezoelectric trapezoidal cantilever beams with an irregular design of the cross-section area. Cylindrical and trapezoidal gaps were used for cross-section area modifications. The goal of the modifications was to increase strain and its distribution at the piezo ceramic layer and, as a result to, improve the power density of the energy harvesting system based on trapezoidal cantilevers.