## 2 Theory and Formulation
### Verification  
In order to verify the present approach, the nondimensional natural frequency of the annular plate obtained in this study was compared with those reported in [83] for different axisymmetric vibration modes, thickness to outer radius ratios h/b, inner radius to outer radius ratio a/b, and three boundary conditions (see Tables 2-4). The discrepancy between our results and the results of [83] came from the difference in the shear deformation theory [83] and HSDT here. Nonetheless, the discrepancy between the results decreased with increasing modes, and h/b and a/b ratios. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(\frac{\text{a}}{\text{b}}\)} & & \multirow{2}{*}{\(\frac{\text{h}}{\text{b}}\)} & \multicolumn{4}{c}{**Axisymmetric Vibration Mode Number**} \\ \cline{5-8}  & & & **1** & **2** & **3** & **4** \\ \hline \multirow{8}{*}{0.1} & Ref. [83] & 0.001 & 14.485 & 51.781 & 112.99 & 198.44 \\  & Present & 0.001 & 13.624 & 50.302 & 111.86 & 198.61 \\  & Ref. [83] & 0.050 & 14.324 & 50.409 & 107.25 & 182.55 \\  & Present & 0.050 & 13.528 & 49.109 & 106.09 & 182.08 \\  & Ref. [83] & 0.100 & 13.874 & 46.947 & 94.670 & 151.91 \\  & Present & 0.100 & 13.254 & 46.061 & 93.794 & 151.36 \\  & Ref. [83] & 0.150 & 13.218 & 42.630 & 81.519 & 124.92 \\  & Present & 0.150 & 12.838 & 42.182 & 81.036 & 124.24 \\  & Ref. [83] & 0.200 & 12.450 & 38.337 & 70.224 & 104.20 \\  & Present & 0.200 & 12.326 & 38.243 & 70.075 & 104.04 \\ \hline \multirow{8}{*}{0.2} & Ref. [83] & 0.001 & 16.780 & 63.370 & 140.60 & 248.62 \\  & Present & 0.001 & 16.652 & 63.003 & 140.90 & 250.68 \\  & Ref. [83] & 0.050 & 16.618 & 61.512 & 132.21 & 224.50 \\  & Present & 0.050 & 16.512 & 61.164 & 131.99 & 225.32 \\  & Ref. [83] & 0.100 & 16.164 & 56.912 & 114.70 & 182.50 \\  & Present & 0.100 & 16.118 & 56.621 & 114.14 & 181.93 \\  & Ref. [83] & 0.150 & 15.491 & 51.317 & 97.337 & 147.50 \\  & Present & 0.150 & 15.529 & 51.108 & 96.780 & 146.33 \\  & Ref. [83] & 0.200 & 14.688 & 45.866 & 82.963 & 121.66 \\  & Present & 0.200 & 14.819 & 45.751 & 82.567 & 121.21 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of nondimensional natural frequency of the annular plate for different axisymmetric vibration mode number, inner radius to outer radius ratio, and thickness to outer radius ratio for simply-simply boundary condition. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{Carbon} & \(\text{E}_{11}^{\text{F}}\) & \(\text{E}_{22}^{\text{F}}\) & \(\text{G}_{12}^{\text{F}}\) & \(\text{\rho}^{\text{F}}\) & \(\
u^{\text{F}}\) & \(\alpha_{11}^{\text{F}}\) & \(\alpha_{22}^{\text{F}}\) \\  & [Gpa] & [Gpa] & [Gpa] & [kg/m\({}^{3}\)] & & & \(\left[\times 10^{-6}/\text{k}\right]\) & [\(\times 10^{-6}/\text{k}\)] \\  & 233.05 & 23.1 & 8.96 & 1750 & 0.2 & & \(-0.54\) & 10.08 \\ \hline \multirow{2}{*}{Epoxy Matrix} & \(\text{E}^{\text{m}}\) & & \(\text{v}^{\text{m}}\) & & & \(\alpha^{\text{m}}\) & \\  & [Gpa] & & & \(\left[\text{kg}/\text{m}^{3}\right]\) & & & \(\left[\times 10^{-6}/\text{k}\right]\) & \\  & 3.51 & & 0.34 & & 1200 & & 45 & \\  & \(\text{E}_{11}^{\text{F}}\) & \(\text{E}_{22}^{\text{CNT}}=\text{E}_{33}^{\text{CNT}}\) & \(\text{G}_{12}^{\text{CNT}}=\text{G}_{13}^{\text{CNT}}\) & \(\
u_{12}^{\text{CNT}}\) & \(\rho^{\text{CNT}}\) & \(\alpha^{\text{CNT}}\) & \(\Gamma^{\text{CNT}}\) & d\({}^{\text{CNT}}\) & t\({}^{\text{CNT}}\) \\  & [Tpa] & [Tpa] & [Tpa] & [kg/m\({}^{3}\)] & [\(\times 10^{-6}/\text{k}\)] & [\(\mu\)m] & [nm] & [nm] \\  & 5.6466 & 7.0800 & 1.9445 & 0.175 & 1350 & 3.4584 & 25 & 1.4 & 0.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of the multiscale hybrid nanocomposite annular plate [64].