3.2.3. Effect of Curvature Ratio

The effect of curvature ratio *Rx*/*Ry* on non-dimension frequency of the panels is investigated in this subsection. The geometrical dimensions of the panels are taken as *a*/*b* = 1, *a*/*h* = 20, *Rx*/*a* = 5. It can be seen from Figure 4a,b, that the non-dimension frequencies of panels decrease with the increase of curvature ratio from −3 to −1, and increase with the value of curvature ratio bigger than −1 for different numbers of layers and different CNT volume fractions. Moreover, the values of ω are at minimum when *Rx*/*Ry* = −1 shows that the curvature effect can be suppressed if the shell panels have both negative and positive curvature.

**Figure 4.** Effect of *Rx*/*Ry* of FG-CNTRC shell panels (*a*/*b* = 1, *a*/*h* = 20, *Rx*/*a* = 5, FG-X): (**a**) For different number of layers, *V*∗ *CNT* = 0.17; (**b**) for different CNT volume fractions.

## 3.2.4. Effect of Thickness Ratio

The SPH shell panel was chosen to study the effect of thickness on the free vibration response of the FG-CNTRC doubly curved shell panel. For this purpose, another non-dimensional frequency is defined as [34]:

$$
\hat{\phi} = \omega a \sqrt{\frac{\rho^m}{E^m}}\tag{29}
$$

$$PCF = \left(\frac{\hat{\alpha}\_{FG} - \hat{\alpha}\_{lID}}{\hat{\alpha}\_{lID}}\right) \times 100\tag{30}$$

The effect of thickness ratio, *h*/*a*, on the non-dimensional frequency of the FG-CNTRC panels is shown in Figure 5. This figure indicates that with all types of CNT distribution, the panels become stiffer with the increase of the thickness ratio, as a result, the non-dimensional frequency of the FG-CNTRC panels increase. Besides, the influence of the thickness ratio, *h*/*a*, on the percentage change of frequency (PCF) of the SHP panel is depicted in Figure 5b. It is observed that FG-X panels show positive effectiveness while other FG-CNTRC panels show the negative effects concerning uniformly distribution (UD) panels. The highest percentage change of frequency of an FG-X panel and FG-O panel are about 14.5% and −15.2%, respectively.

**Figure 5.** Effect of *h*/*a* ratio on free vibration of FG-CNTRC shell panels ((a/b = 1; *Rx* = *Ry* = *R*; *V*∗ *CNT* = 0.17; (0/90)): (**a**) For the frequency parameter ωˆ = ω*a* - ρ*<sup>m</sup> Em* ; (**b**) for the (*PCF*).

#### 3.2.5. Effect of Aspect Ratio

Figure 6a,b show the effects of the aspect ratio (*a*/*b*) on the vibration of FG-CNTRC. Here, we take *a*/*b* = 1; *Rx* = *Ry* = *R*; *R*/*a* = 5; *V*<sup>∗</sup> *CNT* = 0.17 and (0/90).

**Figure 6.** Effect of aspect ratio (*a*/*b*) on free vibration of FG-CNTRC shell panels (*a*/*b* = 1; *Rx* = *Ry* = *R*; *V*∗ *CNT* = 0.17; (0/90)): (**a**) For the frequency parameter ωˆ = ω*a* - ρ*<sup>m</sup> Em* ; (**b**) For the percentage change of frequency (PCF).

Figure 6a reveals that the non-dimensional frequencies of all four types of doubly curved panels decrease uniformly by increasing aspect ratio. In other words, the stiffness of doubly curved panels will be reduced as the aspect ratio increases. Figure 6a states that the PCF of the FG-CNTRC panels remains unchanged with the increase of aspect ratio.
