## 2 Model Description and Theoretical Analysis  
The structure of the trilaminar bending ring with unequal diameters studied in this article is shown in Figure 2. The structure consists of three layers: the middle layer is the metal substrate ring, and the top and bottom layers are the piezoelectric ceramic rings. The inner diameters of the metal substrate and piezoelectric rings are equal, and the outer diameter of the metal ring is larger than the outer diameter of the piezoelectric ring for engineering assembly convenience. Both piezoelectric ceramic rings are polarized in the thickness direction (as indicated by the arrows in Figure 2a) and exhibit isotropic behavior in the transverse direction. The top and bottom surfaces of the piezoelectric ceramic ring are covered by electrodes. The piezoelectric ceramic rings are electrically connected in parallel, as shown in Figure 2a. A cylindrical coordinate system (\(r\), \(\theta\), \(z\)) is established to facilitate the following discussion. Note that the \(r\)-axis is in the direction of 1, the \(\theta\)-axis is in the direction of 2, and the \(z\)-axis is in the direction of 3. Take the \(r\theta\)-plane as the middle plane of the ring, and the \(z\)-axis is perpendicular to the middle plane. The inner radius, outer radius, thickness of the piezoelectric ceramic and the metal substrate, as well as the total thickness of the trilaminar bending ring, are denoted by \(r_{0}\), \(r_{1}\), \(h_{p}\), \(r_{2}\), \(h_{m}\), and \(h\), respectively. As shown in Figure 2a, the trilaminar bending ring with unequal diameters consists of two parts. This first part is an equal-diameter trilaminar composite ring composed of a piezoelectric ceramic and metal substrate, and the second part is a metal substrate ring. Assuming that the small deflection problem of the thin plate is discussed, that is, the outer diameter of the trilaminar bending ring is much larger than its thickness, and the deflection of the trilaminar bending ring is much smaller than its thickness. Therefore, [PERSON]'s hypotheses can be used to analyze the bending vibration characteristics of the trilaminar bending ring. [PERSON]'s hypotheses include the following three: The first hypothesis is the straight line perpendicular to the midplane before the deformation of the thin plate, which is still a straight line and perpendicular to the midplane of the thin plate after the deformation, so the strain component \(S_{rz}=S_{\theta z}=S_{z}=0\). The second hypothesis is on any line perpendicular to the thin plate, the displacement in the \(z\) direction of the thin plate is equal, so the deflection \(w=w(r,\theta)\). The third hypothesis is that the points inside the midplane of the thin plate have no displacement parallel to the midplane, so the displacement of each point inside the midplane of the thin plate is in the \(r\) and \(\theta\) directions \(u=v=0\). Furthermore, assuming that the problem under discussion is axisymmetric, that is, considering that the force on the thin plate is symmetric to the \(z\)-axis and the bending of the thin plate is symmetric to the \(z\)-axis, we can correlate \(u=u(r)\), \(v=0\), \(w=w(r)\). In the composite layer segment, the piezoelectric Equations of the piezoelectric ceramic [27] are  
\[\left\{ \begin{array}{l}T_{1}=\frac{1}{s_{11}^{D}(1-\sigma^{2})}S_{1}+\frac{ \sigma}{s_{11}^{D}(1-\sigma^{2})}S_{1}-\frac{1}{s_{11}^{D}(1-\sigma^{2})}D_{3 }\\ T_{2}=\frac{\sigma}{s_{11}^{D}(1-\sigma^{2})}S_{1}+\frac{1}{s_{11}^{D}(1- \sigma^{2})}S_{1}-\frac{1}{s_{11}^{D}(1-\sigma^{2})}D_{3}\\ E_{3}=-\frac{s_{31}}{s_{11}^{D}(1-\sigma)}S_{1}-\frac{s_{31}}{s_{11}^{D}(1- \sigma)}S_{2}+\vec{\beta}_{33}\end{array} \right.