# Trapezoidal Cantilevers with Irregular Cross-Sections for Energy Harvesting Systems
## 2 The Design of Trapezoidal Cantilevers with an Irregular Design of the Cross-Section  
Cantilevers used for energy harvesting usually vibrate at the first out-of-plane bending mode. The cantilever beam with a uniform cross-section area has linear characteristics of the strain distribution along the length of the beam. Therefore, the generated output voltage is proportional to the strain inducted in the piezo ceramic. The strain of the piezo ceramic layer can be expressed as follows:  
\[\varepsilon(x)=\frac{m(\ddot{y}+\ddot{z})h}{2Y_{Piezo}I(x)}(2l_{b}-l_{e}) \tag{1}\]  
where \(\varepsilon(x)\) is strain; \(m(\ddot{y}+\ddot{z})\) is a vertical force that is acting on the free end of the cantilever with a seismic mass; \(h\) is the distance between the centers of the cantilever and the piezo electric layer; \(Y_{Piezo}\) is the Young modulus of the piezoelectric material; \(l_{b}\) is the length of the beam; \(l_{e}\) is the length of the piezoelectric material, which is covered by an electrode; \(I(x)\) is the second moment of inertia as a function of distance \(x\) from the fixed end of the cantilever. The function of the second moment of inertia can be expressed as shown below:  
\[I(x)=2\bigg{[}\frac{w(x)t_{c}^{3}}{12}+w(x)t_{c}h^{2}\bigg{]}+\frac{\eta_{s}w( x)t_{s}^{3}}{12} \tag{2}\]  
where \(w(x)\) is the width of the cantilever as a function of distance \(x\) from the fixed end of the cantilever; \(t_{c}\) is the thickness of the piezo ceramic layer; \(h\) is the distance between the centers of the cantilever and the piezo electric layer; \(t_{s}\) is the thickness of the cantilever; \(\eta_{s}\) is the ratio between the [PERSON]'s modulus of the piezo electric and cantilever materials, and it can be expressed as \(\eta_{s}=Y_{Piezo}/Y_{Cantilevers}\). Therefore, the relation between voltage and strain can be written as follows [2]:  
\[U=\frac{d_{31}Y_{Piezo}b}{C_{p}}\int\limits_{0}^{l}\varepsilon(x)dx \tag{3}\]  
where \(Y_{Piezo}\) is the [PERSON]'s modulus of the piezo ceramic; \(C_{p}\) is the capacitance of the piezo ceramic layer; \(b\) is the thickness of the piezo ceramic layer; \(d_{31}\) is the piezoelectric coefficient; \(\varepsilon(x)\) is the strain in the piezo ceramic layer; \(l\) is the length of the piezo ceramic layer. Based on Equation (3), it can be concluded that the integral of the strain function must be increased in order to increase the output voltage. One of the ways to obtain that is to change the strain function, so that it would be a constant or close to the constant along the length of the piezo ceramic layer. Such a strain function can be obtained by modifications of the cross-section area, herewith modifications of the second moment of inertia in the cantilever. Therefore, trapezoidal cantilever beams with an irregular cross-section are proposed and analyzed. The principle design of the trapezoidal cantilevers with an irregular cross-section area is given in Figure 1. It consists of a bronze plate, a piezo ceramic plate and a seismic mass. The piezo ceramic plate has a constant thickness and length of 0.5 mm and 30 mm, respectively. Furthermore, each cantilever has a constant length equal to 70 mm; therefore, the ratio of the piezo ceramic length and cantilever length is 0.42. Moreover, each cantilever has a seismic mass, and its weight is 8.71 g. The seismic mass was used to reduce the resonant frequency of the cantilever. Cantilevers were clamped to the host through the plastic support structure. The cross-section areas of two cantilevers were modified (Figure 1a,b). The modifications were made by employing cylindrical and trapezoidal gaps. Such types of gaps were chosen based on their geometrical functions. A conventional design of the cross-section area (Figure 1c) was chosen in order to obtain reference data for comparison. The proposed improvements of the cross-section area could provide almost a uniform strain distribution along the length of the piezo ceramic plate.