## 2 Theory and Formulation
### Vibration Study
#### 4.2.1 Parametric Vibration Study  
Table 5 shows the effects of the distribution patterns and CNT weight fraction on the nondimensional natural frequency of MHCAP. The plate based on FG-A (shown in Figure 3c) exhibited the best vibration response among the CNT patterns. Increasing the volume fraction of carbon fibers also yielded higher frequencies. The vibration results obtained by GDQM are diagramed in Figure 5 for various numbers of grid points and different boundary conditions. According to the figures, choosing _N = 10_ as the number of grid points provided adequate accuracy to the results. Figure 6 depicts the effects of the outer radius to inner radius ratio on the vibration performance of the annular plate for different values of Winkler-Pasternak elastic substrate coefficients. Apparently, the Winkler coefficient played a considerable role in the vibration response of the plate with increasing outer radius to inner radius ratio (b/a). Moreover, the minimum value of the natural frequency of the plates resting on Winkler substrate with higher stiffness occurred at lower b/a. For example, the minimum frequency of the plate with Winkler coefficient (K\({}_{\rm w}\) = 300 (MN/m\({}^{3}\))) was associated with b/a = 2.1 while the corresponding minimum frequency of K\({}_{\rm w}\) = 100 (MN/m\({}^{3}\)) occurred at b/a = 2.3. Furthermore, K\({}_{\rm w}\) barely affected the vibration behavior of the plate when b/a < 1.6. Figure 6b shows the minimum frequency of the system occurred at b/a = 2.3 and then increased uniformly for all the values of the Pasternak coefficient. The frequencies for various Pasternak coefficient (K\({}_{\rm p}\) ) barely changed for different ratios b/a. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{\(\mathbf{V_{F}}\)} & \multirow{2}{*}{
\begin{tabular}{c} **CNT** \\ **Distribution** \\ \end{tabular} } & \multicolumn{5}{c}{\(\mathbf{W_{CNT}}\)} \\ \cline{3-7}  & & **0.02** & **0.04** & **0.06** & **0.08** & **0.1** \\ \hline \multirow{4}{*}{0.2} & FG-X & 2.7059 & 2.9976 & 3.2255 & 3.4195 & 3.5921 \\  & FG-V & 2.7869 & 3.0596 & 3.2498 & 3.4018 & 3.5325 \\  & FG-A & 2.8217 & 3.1038 & 3.3006 & 3.4589 & 3.5958 \\  & FG-UD & 2.8089 & 3.0889 & 3.2854 & 3.4439 & 3.5812 \\ \hline \multirow{4}{*}{0.3} & FG-X & 2.9599 & 3.2604 & 3.4808 & 3.6596 & 3.8132 \\  & FG-V & 3.0806 & 3.3817 & 3.5783 & 3.7265 & 3.8479 \\  & FG-A & 3.1203 & 3.4291 & 3.6294 & 3.7809 & 3.9058 \\  & FG-UD & 3.1070 & 3.4156 & 3.6170 & 3.7695 & 3.8950 \\ \hline \multirow{4}{*}{0.4} & FG-X & 3.1711 & 3.4722 & 3.6836 & 3.8486 & 3.9859 \\  & FG-V & 3.3139 & 3.6278 & 3.8245 & 3.9670 & 4.0796 \\ \cline{1-1}  & FG-A & 3.3563 & 3.6766 & 3.8753 & 4.0191 & 4.1332 \\ \cline{1-1}  & FG-UD & 3.3437 & 3.6662 & 3.8675 & 4.0132 & 4.1283 \\ \hline \end{tabular}
\end{table} 
Table 5: Effect of CNT pattern on the nondimensional natural frequency of the simply-simply multiscale hybrid composite annular plate (MHCAP) with outer radius to inner radius ratio (b/a = 4), thickness ratio (h/b = 0.2), temperature at bottom surface (T\({}_{\rm i}\) = 273 [K]), temperature at top surface (T\({}_{\rm o}\) = 300 [K]), orientation angle of fibers (\(\theta\) = \(\pi\)/4), Winkler coefficient (K\({}_{\rm w}\) = 0 [N/m\({}^{3}\)]), Pasternak coefficient (K\({}_{\rm P}\) = 0 [N/m\({}^{3}\)]), axial load (P = 0 [N]), uniform temperature rise (UTR), and \(\beta\) = 1 for different weight fractions of CNT and volume fractions of carbon fiber. The influence of the temperature rise on the vibration response of the MHCAP is depicted in Figure 7. The sinusoidal temperature rise (STR) exhibited the highest natural frequency of the plate, while the uniform temperature rise (UTR) led to the weakest vibration response. The corresponding frequencies of UTR was close to those of STR. The temperature rise became more influential on the vibration behavior of plates for higher b/a.