## 2 Model Description and Theoretical Analysis
Therefore, the moment balance Equation can be used instead of the rotation Equation, in which case the moment balance Equation is  
\[\frac{\partial M_{r}}{\partial r}+\frac{M_{r}-M_{\theta}}{r}=Q_{r}, \tag{8}\]  
where \(Q_{r}\) is the transverse-shear force in the \(r\) direction. Substituting Equations (5) and (7) into Equation (8), the transverse-shear \(Q_{r}\) per unit width of the composite layer segment and the transverse-shear \(Q_{r}^{\prime}\) per unit width of the metal substrate segment can be expressed, respectively, as  
\[Q_{r}=\frac{-E_{m}h_{m}^{3}}{12(1-\sigma_{m}^{2})}\left[\frac{\partial}{ \partial r}\left(\frac{\partial^{2}w}{\partial r^{2}}+\frac{1}{r}\frac{ \partial w}{\partial r}\right)\right]-\frac{h^{3}-h_{m}^{3}}{12s_{11}^{D}(1- \sigma^{2})}\left[\frac{\partial}{\partial r}\left(\frac{\partial^{2}w}{ \partial r^{2}}+\frac{1}{r}\frac{\partial w}{\partial r}\right)\right]-\frac{ g_{31}(h^{2}-h_{m}^{2})}{4s_{11}^{D}(1-\sigma)}\frac{\partial D_{3}}{ \partial r}, \tag{9}\]  
\[Q_{r}^{\prime}=\frac{-E_{m}h_{m}^{3}}{12(1-\sigma_{m}^{2})}\left[\frac{\partial }{\partial r}\left(\frac{\partial^{2}w}{\partial r^{2}}+\frac{1}{r}\frac{ \partial w}{\partial r}\right)\right], \tag{10}\]  
Integrating the third Equation of Equation (1) over the interval \([h_{m}/2,h/2]\) for \(z\), we arrive at  
\[D_{3}=\frac{2V}{\beta_{33}(h-h_{m})}-\frac{g_{31}(h+h_{m})}{4\beta_{33}s_{11}^{ D}(1-\sigma)}\left(\frac{\partial^{2}w}{\partial r^{2}}+\frac{1}{r}\frac{\partial w}{ \partial r}\right), \tag{11}\]According to the knowledge of elastic mechanics, when the trilaminar bending ring is not subjected, the motion Equation of the composite layer segment along the \(z\)-axis can be written as  
\[\frac{Q_{r}}{r}+\frac{\partial Q_{r}}{\partial r}=\rho h\frac{\partial^{2}w}{ \partial t^{2}}, \tag{12}\]  
Substituting Equation (11) into Equation (9) and then into the motion Equation (12), the motion Equation of the composite layer segment expressed in terms of deflection \(w\) can be obtained:  
\[\
abla^{4}w-k_{1}^{4}w=0, \tag{13}\]  
where \(\
abla^{4}=\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{ \partial}{\partial r}\right)^{2}\), \(k_{1}^{4}=\frac{\rho h}{D_{c}}\omega^{2}\), \(\rho\) denotes the density of the piezoelectric materials, in which  
\[D_{c}=\frac{E_{m}h_{m}^{3}}{12(1-\sigma_{m}^{2})}+\frac{h^{3}-h_{m}^{3}}{12s_ {1}^{0}(1-\sigma^{2})}-\frac{g_{31}^{2}(h^{2}-h_{m}^{2})(h-h_{m})}{16(s_{1}^{ \mathrm{D}})^{2}(1-\sigma)^{2}\overline{\beta}_{33}}, \tag{14}\]  
Similarly, the motion Equation of the metal substrate segment can be expressed as  
\[\
abla^{4}w-k_{2}^{4}w=0, \tag{15}\]  
where \(\
abla^{4}=\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{ \partial}{\partial r}\right)^{2}\), \(k_{2}^{4}=\frac{\rho_{m}h_{m}}{D_{m}}\omega^{2}\), \(\rho_{m}\) denotes the density of the metal substrate, where  
\[D_{m}=\frac{E_{m}h_{m}^{3}}{12(1-\sigma_{m}^{2})}, \tag{16}\]  
The general solutions of Equations (13) and (15) can be respectively expressed as follows:  
\[w_{1}=A_{1}J_{0}(kr)+B_{1}Y_{0}(kr)+C_{1}I_{0}(kr)+D_{1}K_{0}(kr), \tag{17}\]  
\[Q_{r}=\frac{-E_{m}h_{m}^{3}}{12(1-\sigma_{m}^{2})}\left[\frac{\partial}{ \partial r}\left(\frac{\partial^{2}w}{\partial r^{2}}+\frac{1}{r}\frac{ \partial w}{\partial r}\right)\right]-\frac{h^{3}-h_{m}^{3}}{12s_{11}^{D}(1- \sigma^{2})}\left[\frac{\partial}{\partial r}\left(\frac{\partial^{2}w}{ \partial r^{2}}+\frac{1}{r}\frac{\partial w}{\partial r}\right)\right]-\frac{ g_{31}(h^{2}-h_{m}^{2})}{4s_{11}^{D}(1-\sigma)}\frac{\partial D_{3}}{ \partial r}, \tag{9}\]0  
where \(J_{0}\) and \(Y_{0}\) are the Bessel functions of the first and second kinds of order zero, respectively, and \(I_{0}\) and \(K_{0}\) are the modified Bessel functions of the first and second kinds of order zero, respectively.