## 2 Model Description and Theoretical Analysis
\begin{bmatrix}S_{31}(h+h_{m})V/(2s_{11}^{D}(1-\sigma) \overline{\beta_{33}})\\ 0\\ S_{31}(h+h_{m})V/(2s_{11}^{D}(1-\sigma)\overline{\beta_{33}})\\ 0\\ 0\\ 0\\ 0\end{bmatrix} , \tag{24}\]  
Complicated operations are required to solve the system of Equation (24). However, as long as the material and dimensions of the trilaminar bending ring with unequal diameters are given, the numerical solution can be obtained. When the electric end short circuits (\(V\)) = 0, the coefficient determinant (\(\det([I])\)) = 0 of the system of Equation (24) is the resonant frequency Equation of the trilaminar bending ring with unequal diameters, and the free vibration frequency (\(f\) = \(f_{r}\)) is the resonant frequency. When the electric end is open (\(Q\) = 0), the free vibration frequency (\(f\) = \(f_{a}\)) of the trilaminar bending ring is called the antiresonant frequency. Integrating the potential shift (\(D_{3}\)) over the piezoelectric ceramic area yields the following:  
\[Q=2\int_{a_{0}}^{a_{1}}2\pi rD_{3}dr=\frac{4\pi V(a_{1}^{2}-a_{0}^{2})}{(h-h_{ m})\overline{\beta_{33}}}-\frac{g_{31}(h+h_{m})\pi}{\overline{\beta_{33}}s_{11}^{ D}(1-\sigma)}\int_{a_{0}}^{a_{1}}\left(\frac{\partial^{2}w}{\partial r^{2}}+ \frac{1}{r}\frac{\partial w}{\partial r}\right)rdr, \tag{25}\]  
The system of Equation (24) becomes the system of Equation (26). \(I^{\prime}_{11}\)-\(I^{\prime}_{88}\) under different boundary conditions is given in Appendices A and B, so that the coefficients of the changed system of Equation (26) determinant \(\det([I^{\prime}])\) = 0, and the solution \(f\) = \(f_{a}\) is the antiresonant frequency of the trilaminar bending ring with unequal diameters. \[ \begin{bmatrix}I^{\prime}_{11}&I^{\prime}_{12}&I^{\prime}_{13}&I^{\prime}_{14 }&0&0&0&0\\ I^{\prime}_{21}&I^{\prime}_{22}&I^{\prime}_{23}&I^{\prime}_{24}&0&0&0&0\\ I^{\prime}_{31}&I^{\prime}_{32}&I^{\prime}_{33}&I^{\prime}_{34}&I^{\prime}_{35}& I^{\prime}_{36}&I^{\prime}_{37}&I^{\prime}_{38}\\ I^{\prime}_{41}&I^{\prime}_{42}&I^{\prime}_{43}&I^{\prime}_{44}&I^{\prime}_{45}& I^{\prime}_{46}&I^{\prime}_{47}&I^{\prime}_{48}\\ I^{\prime}_{51}&I^{\prime}_{52}&I^{\prime}_{53}&I^{\prime}_{54}&I^{\prime}_{55}& I^{\prime}_{56}&I^{\prime}_{57}&I^{\prime}_{58}\\ I^{\prime}_{61}&I^{\prime}_{62}&I^{\prime}_{63}&I^{\prime}_{64}&I^{\prime}_{65}& I^{\prime}_{67}&I^{\prime}_{67}&I^{\prime}_{68}\\ 0&0&0&0&I^{\prime}_{75}&I^{\prime}_{76}&I^{\prime}_{77}&I^{\prime}_{78}\\ 0&0&0&0&I^{\prime}_{85}&I^{\prime}_{86}&I^{\prime}_{87}&I^{\prime}_{88}\\ \end{bmatrix}\begin{bmatrix}A^{\prime}_{1}\\ B^{\prime}_{1}\\ C^{\prime}_{1}\\ D^{\prime}_{1}\\ A^{\prime}_{2}\\ D^{\prime}_{2}\\ D^{\prime}_{2}\end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{bmatrix} , \tag{26}\]  
The effective electromechanical coupling coefficient (\(k_{\textit{eff}}\)) is a physical quantity measuring the degree of mutual coupling of the energy conversion process of the transducer, from which many important characteristics of the transducer can be obtained, such as the electric limit power, acoustic limit power, and the best quality factor. Therefore, it is a comprehensive index to evaluate the transducer. Assuming that the electrical losses in bending vibration of the trilaminar bending ring with unequal diameters are small, the effective electromechanical coupling coefficient can be estimated using the following Equation according to the IEEE standard definition [27]. \[k_{eff}=\frac{f_{a}^{2}-f_{r}^{2}}{f_{a}^{2}}, \tag{27}\]  
Substituting the resonant frequency (\(f_{r}\)) and the antiresonant frequency (\(f_{a}\)) of the bending vibration of the trilaminar bending ring with unequal diameters obtained above into Equation (27), the effective electromechanical coupling coefficient of the bending vibration of the trilaminar bending ring with unequal diameters can be obtained.