## 2 Materials and Methods
### Axisymmetric Annular Plate Governing Equations
When the boundary conditions of the inner and outer rings of the annular plates are hinged or clamped, the boundary conditions can be expressed as \(u_{r}|_{r=r_{1}r_{2}}=0\), then we can obtain  
\[N^{R}=I_{0}\frac{\Omega^{2}}{8}\Bigg{[}(1+\mu)(r_{1}^{2}+r_{2}^{2})+(1-\mu) \frac{r_{1}^{2}r_{2}^{2}}{r^{2}}-(3+\mu)r^{2}\Bigg{]} \tag{22}\]When the boundary conditions of the inner rings of the annular plates are clamped and the boundary conditions of the outer rings are free, the boundary conditions can be expressed as \(u_{r}|_{r=r_{1}}=0,\sigma_{r}|_{r=r_{2}}=0\), then we can get  
\[N^{R}=I_{0}\frac{\Omega^{2}}{8}\Bigg{\{}(3+\mu)+\frac{(1-\mu)[(3+\mu)r_{2}^{2}-( 1+\mu)r_{1}^{2}]}{(1-\mu)r_{1}^{2}+(1+\mu)r_{2}^{2}}\frac{r_{1}^{2}}{r^{2}} \Bigg{\}}(r_{2}^{2}-r^{2}) \tag{23}\]  
According to the Hamilton principle, the stationary value of energy is zero, that is  
\[\delta\int_{I_{0}}^{t_{1}}(-U+T+H)\mathrm{d}t=0 \tag{24}\]  
substituting expressions (18)-(20) into expression (24), the unified governing equations are  
\[\frac{1}{r}\frac{\partial(rN^{\prime}{}_{r})}{\partial r}-\frac{N^{\prime}{}_ {\theta}}{r}=I_{0}\frac{\partial^{2}u}{\partial t^{2}}+v_{1}\frac{\partial^{ 2}\varphi}{\partial t^{2}}+v_{2}\frac{\partial^{3}w}{\partial r\partial t^{2}} \tag{25}\]  
\[\frac{\lambda}{r}\frac{\partial(rM^{\prime}{}_{r})}{\partial r}-\frac{\beta} {r}\frac{\partial(rP^{\prime}{}_{r})}{\partial r}+\beta\frac{P^{\prime}{}_{ \theta}}{r}-\lambda\frac{M^{\prime}{}_{\theta}}{r}+3\beta R^{\prime}{}_{r}- Q^{\prime}{}_{r}=v_{1}\frac{\partial^{2}u}{\partial t^{2}}+v_{3}\frac{ \partial^{2}\varphi}{\partial t^{2}}+v_{4}\frac{\partial^{3}w}{\partial r \partial t^{2}} \tag{26}\]  
\[ \begin{array}{l}\frac{(1-\lambda)}{r}\frac{\partial^{2}(rM^{\prime}{}_{r})} {\partial r^{2}}+\frac{\beta}{r}\frac{\partial^{2}(rP^{\prime}{}_{r})}{ \partial r^{2}}+\frac{(\lambda-1)}{r}\frac{\partial M^{\prime}{}_{r}}{\partial r }-\frac{\beta}{r}\frac{\partial p^{\prime}{}_{\theta}}{\partial r}+\frac{ \lambda}{r}\frac{\partial(rQ^{\prime}{}_{r})}{\partial r^{2}}-\frac{3\beta}{ r}\frac{\partial(rP^{\prime}{}_{r})}{\partial r}+\frac{1}{r}\frac{\partial}{ \partial r}\Big{(}rN^{R}\frac{\partial w}{\partial r}\Big{)}\\ =-v_{2}\frac{1}{r}\frac{\partial}{\partial r}\Big{(}r\frac{\partial^{2}u}{ \partial t^{2}}\Big{)}-v_{4}\frac{1}{r}\frac{\partial}{\partial r}\Big{(}r \frac{\partial^{2}\varphi}{\partial t^{2}}\Big{)}+I_{0}\frac{\partial^{2}w}{ \partial t^{2}}-v_{5}\frac{1}{r}\frac{\partial}{\partial r}\Big{(}r\frac{ \partial^{3}w}{\partial r\partial t^{2}}\Big{)}\end{array}  \tag{27}\]  
The expressions of each coefficient are  
\[(I_{0},I_{1},I_{2},I_{3},I_{4},I_{6})=\int_{-h/2}^{h/2}\rho(z,T)\Big{(}1,z,z^{2 },z^{3},z^{4},z^{6}\Big{)}\mathrm{d}z \tag{28}\]  
\[ \begin{array}{l}v_{1}=\lambda I_{1}-\beta I_{3},v_{2}=(\lambda-1)I_{1}-\beta I _{3},v_{3}=\lambda^{2}I_{2}-2\lambda\beta I_{4}+\beta^{2}I_{6}\\ v_{4}=\lambda(\lambda-1)I_{2}+\beta(1-2\lambda)I_{4}+\beta^{2}I_{6},v_{5}=( \lambda-1)^{2}I_{2}+\beta(2\lambda-1)I_{4}+\beta^{2}I_{6}\end{array}  \tag{29}\]  
When the nano-annular plates are in free linear vibration, the displacement functions \(u(r,t)\), \(w(r,t)\), \(\varphi(r,t)\) can be expressed in the form of space-time separation  
\[u(r,t)=\widetilde{u}(r)e^{iwt},w(r,t)=\widetilde{w}(r)e^{i\omega t},\varphi(r, t)=\widetilde{\varphi}(r)e^{iwt} \tag{30}\]  
where \(\widetilde{u}(r)\), \(\widetilde{\varphi}(r)\), \(\widetilde{\varphi}(r)\) are the mode functions; \(\omega\) represents the vibration frequencies of the annular plates. Combining Equations (15), (25)-(27) and (30), the mechanical variables in the governing Equations (25)-(27) can be replaced by displacement variables, and the nonlocal governing equations can be obtained after sorting.