## 4 Parametric Investigation  
A numerical investigation of the nonlinear oscillation for the selected structural components subjected to aero-elastic load is herein performed based on a modified third-order piston theory, and the following assumptions: \(\rho_{\infty}=1.29\left(\frac{k_{\rm g}}{m^{3}}\right)\), \(190\left(\frac{m}{s}\right)\leq u_{\infty}\leq 300\left(\frac{m}{s}\right)\), and \(2\leq M_{\infty}\leq 6\). The CNTRC beam is characterized by the following geometric and material properties at 300 \(k\)[36]: \(L=1.2\) m, \(b=0.15\) m, \(h=0.01\) m, \(\Lambda=0.001\) m, \(E_{11}^{CNT}=5.6466\) TPa, \(E_{22}^{CNT}=7.0800\) TPa, \(G_{12}^{CNT}=1.9445\) TPa, \(c^{CNT}=0.175\), \(\rho^{CNT}=1400\) kg/m\({}^{3}\), \(E^{p}=3.52\) GPa, \(v^{p}=0.34\), and \(\rho^{p}=1150\) kg/m\({}^{3}\). The numerical analysis starts by evaluating the accuracy of the proposed method, based on a comparative study against the literature. To this end, the fundamental frequencies of a homogeneous cantilever beam (made of pure polymer) are computed in Table 2 for different values of slenderness ratio, while comparing them with Ref. [53] based on a simple Euler-Bernoulli beam theory. From Table 2, it is worth noticing that the current results match accurately predictions from Ref. [53], confirming the expected accuracy of the method here proposed. In Figure 2, we illustrate the variation in the fundamental frequency as a function of the slenderness ratio for different CNT reinforcement patterns. It is evident that the fundamental frequency decreases significantly with increasing slenderness ratio, particularly for values below 60, due to the reduced structural stiffness. Among the reinforcement schemes, \begin{table}
\begin{tabular}{c c c c}  & _Fundamental Frequency_\(\omega_{L}\)\(\frac{rad}{s}\) \\ \cline{2-4} \(\mathbf{L}\) & \multicolumn{2}{c}{**Present Work**} \\ \cline{2-4}  & \(\mathbf{\hat{\theta}}\)=0 & \(\mathbf{\hat{\theta}}\)\(\
eq\)**0** & **Rao**[53] \\ \hline
25.0 & 302.1178 & 284.1192 & 284.1192 \\ \hline
50.0 & 75.5294 & 71.0298 & 71.0298 \\ \hline
75.0 & 33.5686 & 31.5688 & 31.5688 \\ \hline
90.0 & 23.3116 & 21.9228 & 21.9228 \\ \hline
100.0 & 18.8824 & 17.7575 & 17.7575 \\ \hline
120.0 & 13.1128 & 12.3316 & 12.3316 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparative results for fundamental frequencies of a cantilever beam. ( \(u_{\infty}=0\)). the FG-X pattern demonstrates the highest frequency response, followed by the UD and FG-A patterns. The FG-O configuration consistently yields the lowest frequency. This trend underscores the significant impact of reinforcement grading on dynamic stiffness and aeroelastic stability. At the same time, Figure 3 presents the influence of CNT volume fraction on the fundamental frequency for various slenderness ratios. An increase in the CNT volume fraction enhances the stiffness of the composite, which leads to a noticeable increase in the natural frequency. This effect is especially pronounced at lower slenderness ratios, where the structure is more sensitive to reinforcement distribution due to its greater susceptibility to bending. These findings confirm the critical role of CNT content in tailoring vibrational performance. Figure 4 shows the time history of the lateral dynamic displacement for beams with different reinforcement patterns under a supersonic airflow. The beam with a FG-X rein  
Figure 3: Variation in the fundamental frequency with the slenderness ratio for different volume fractions. \(FG-X\) pattern. Figure 2: Variation in the fundamental frequency vs. slenderness ratio for different patterns. \(V^{*}_{CNT}=0.17\). forcement exhibits the largest amplitude of oscillation and the highest vibration frequency, indicating a stiffer and more responsive structure. Conversely, the FG-O pattern results in the smallest dynamic deflection and the lowest frequency, confirming the inferior stiffness and dynamic performance of this configuration.