## 3 Theory and Modeling
### Principal and Modeling of Piezoelectric Transducer Component
The linear elastic small deformation model of the piezoelectric component and axial stress along the length direction of \(x\) in a steady state, when the tip mass at the active end on the right is subjected to the static force \(F\) of 0.4 N in the form of the volumetric load based on analysis of Equation (8), is shown in Figure 10.  
From the stress distribution contour map in Figure 10, it can be observed that the extension of the fixed end reduces the level of stress under the same excitation, consistent  
Figure 10: Simulation model of variable cross-section piezoelectric cantilever beam and axial stress along the length direction: (**a**) constant cross-section cantilever beam and (**b**) non-uniform cross-section cantilever beam. with theoretical analysis based on Equation (8). And under the linear elastic deformation model, stress and strain are proportional, which exhibit a similar distribution pattern. The widths change \(\Delta b\) of the upper and lower edges are limited to within 20 mm, with the initial condition being a rectangular cantilever beam width of 20 mm. The widths of the fixed end and the movable end are also denoted as \(b_{1}\) and \(b_{2}\), respectively. Let the change in size of the fixed or movable end be denoted as \(\Delta b\), add or subtract \(\Delta b\) from \(b_{1}\), or \(b_{2}\) will cause the open-circuit voltage and the natural frequency of the piezoelectric cantilever beam to change. Changing \(b_{1}\) or \(b_{2}\) could transform the cantilever beam from a rectangular structure to a trapezoidal structure, as shown in Figure 9, where the width of the fixed end is typically greater than that of the active end in a trapezoidal cantilever beam. This is because the fixed end experiences greater stress than the active end, and a narrower fixed end is detrimental to the structural stability. Therefore, this paper adopts the approach of \(b_{1}+\Delta b\) or \(b_{2}-\Delta b\) to form the variable cross-section trapezoidal cantilever beam model in Figure 10. Use the same piezoelectric cantilever beam simulation model in Figure 10 to calculate the open-circuit voltage under the action of the static force \(F\) and the modal analysis to calculate the natural frequency. Keep the excitation and tip mass the same, with the results shown in Figure 11.  
The simulation results in Figure 11 indicate that during the process of increasing the fixed end \(b_{1}\), the open-circuit voltage output of the piezoelectric component gradually decreases, while the natural frequency of the piezoelectric cantilever beam structure gradually increases. Conversely, during the process of decreasing the active end \(b_{2}\), the opposite characteristics are observed, which aligns with the analysis of the aforementioned theoretical model. And all the simulation results converge successfully, achieving the tolerance requirement of \(1\times 10^{-6}\) after thirteen iterations. Taking the process of increasing the fixed end in Figure 11 as an example, under the condition that the applied force \(F\) remains constant, the piezoelectric cantilever beam undergoes linear elastic small deformations. Then, according to the analysis using Equations (8)-(10), the reduction in stress and strain distribution is offset by the increase in the surface area of the piezoelectric layer, resulting in nearly unchanged total surface charge on the piezoelectric component's surface. Meanwhile, the increased surface area of the piezoelectric layer leads to an increase in the equivalent capacitance between the upper and lower surfaces, causing a decrease in the open-circuit voltage. Consistent with the theoretical model, this does not imply that strain changes the equivalent capacitance, but rather that widening \(b_{1}\) increases  
Figure 11: Simulation result of changing width of fixed end \(b_{1}\) or active end \(b_{2}\) by \(\Delta b\) for non-uniform cross-section cantilever beam open–circuit voltage \(U_{\text{P}}\) under the action of \(F\) and natural frequency \(\omega_{n}\). Initial value \(b_{1}\) and \(b_{2}\) = 20 mm.