## 2 Theory and Formulation
### Vibration Study
#### 4.2.1 Parametric Vibration Study
Figure 5: Convergence of nondimensional natural frequency of MHCAP with b/a = 4, h/b = 0.1, T\({}_{\rm i}\) = 273 [K], T\({}_{\rm o}\) = 300 [K], weight fraction of CNT (W\({}_{\rm CNT}\) = 0.02), volume fraction of carbon fiber (V\({}_{\rm F}\) = 0.2), \(\theta\) = \(\pi\)/4, UTR, \(\beta\) = 1, FG-A, K\({}_{\rm w}\) = 0 [M N/m\({}^{3}\)] and K\({}_{\rm p}\) = 0 [M N/m] for various boundary conditions. (**a**) Simply-Simply; (**b**) Clamp-Clamp; (**c**) Clamp-Simply. Figure 6: The effect of Winkler–Pasternak coefficient on the nondimensional natural frequency of MHCAP with clamp-clamp boundary conditions for different outer radius to inner radius ratios and h/b = 0.1, \(UTR\), T\({}_{\rm i}\) = 273 [K], T\({}_{\rm o}\) = 300 [K], \(\theta\) = \(\pi\)/4, W\({}_{\rm CNT}\) = 0.02, V\({}_{\rm F}\) = 0.2, FG-A, \(\beta\) = 1, (**a**) K\({}_{\rm p}\) = 0, and (**b**) K\({}_{\rm w}\) = 0. (**a**) Winkler coefficient; (**b**) Pasternak coefficient. Figure 8 illustrates the effects of the applied temperature on the upper surface on the vibration performance of the annular plate for different outer radius to inner radius ratios. Increasing the temperature gradient resulted in a weaker vibration performance as a consequence of a decreasing structural stiffness. Choosing a higher outer to inner radius caused a minimum natural frequency at higher temperature b/a. Accordingly, we concluded that plates with higher b/a ratios resulted in a better vibration response. The influence of the volume fraction V\({}_{\rm F}\) and orientation angle \(\theta\) of carbon fibers on the free vibration is plotted in Figure 9. The vibration response of the plate was symmetric about \(\theta=\frac{\pi}{2}\). Therefore, we analyzed the vibration performance of the plate for \(0\leq\theta\leq\frac{\pi}{2}\). Increasing V\({}_{\rm F}\) improved the vibration characteristics of the plate when \(0\leq\theta\leq 0.425\pi\) and increasing \(\theta\) decreased the natural frequency. The effect of the volume fraction of carbon fibers on the vibration behavior of the plate became less pronounced with increasing orientation angle. The natural frequency of the plate tended to constant values regardless of the volume fraction of carbon fibers at \(\theta=0.425\pi\). For plates with \(\theta>0.425\pi\), increasing V\({}_{\rm F}\) had a negative effect on the vibration response of the plate. The effects of the CNTs' weight fraction W\({}_{\text{CNT}}\) on the vibration behavior of the annular plate with respect to different orientation angles can be found in Figure 10. An increasing W\({}_{\text{CNT}}\) improved the vibration response of the plate and was more pronounced for lower values of \(\theta\). The CNT weight fraction barely influenced the vibration performance of MHCAP when carbon fibers were arranged in an orientation of \(\frac{\pi}{2}\). Figure 11 shows the effect of the outer radius to inner radius ratio b/a on the natural frequency of the plate with and without elastic foundation and different orientation angles of carbon fibers. The effect of \(\beta\) involved in Equations (29) and (30) on the vibration performance of the MHCAP is analyzed in Figure 12 for different values of corresponding temperature of the top surface. Choosing \(\beta>1\) in Equation (29) yielded a nonlinear temperature gradient, while \(\beta=1\) in Equation (29) yielded a linear temperature gradient. Rising \(\beta\) increased the natural frequency. However, the gradient of the ascending trend of the diagrams decreased gradually until it tended near to zero at \(\beta>9.5\). The temperature played a more important role on the vibration performance of the plate for lower \(\beta\).