## 4 Parametric Investigation
The UD and FG-A beams show intermediate behavior, highlighting the tunability of aeroelastic characteristics through tailored CNT distributions. Moreover, Figure 5 shows the time history of the system for varying CNT volume fractions. As the volume fraction increases, both the frequency and amplitude of oscillations increase, a consequence of the enhanced material stiffness imparted by the CNT reinforcement. These results demonstrate the effectiveness of higher CNT content in increasing structural rigidity and improving resistance to aeroelastic instabilities. Another key aspect affecting the structural vibrations and oscillations is the free stream air velocity, as analyzed in the parametric time response of the beam in Figure 6, for different free stream air velocities. The results from this figure show that both the vibration frequency and lateral deflection increase with airspeed, suggesting that the aerodynamic loading amplifies the structural response. However, this amplification is relatively linear compared to more substantial changes observed with Mach number variations, as visible from Figure 7. In this last figure, indeed, a noticeable increase in both the frequency and amplitude of oscillations occurs as the Mach number rises, indicating a strong aerodynamic coupling. Compared to Figure 6, these results suggest that the Mach number exerts a more dominant influence than air velocity alone, due to nonlinear compressibility effects captured by the third-order piston theory. Figure 8 focuses on the influence of slenderness ratio on the nonlinear frequency response. As the slenderness ratio decreases (i.e., the beam becomes stubbier), the system becomes stiffer, leading to higher oscillation frequencies. However, beyond a certain threshold, the system exhibits a damped harmonic response with amplitude modulation--a signature of beating phenomena. This indicates the proximity of the excitation frequency to the system's damped natural frequency, resulting in a complex dynamic behavior. As last parametric investigation, we assess the role of the thickness ratio on the time history response, as visible in Figure 9. Based on this figure, as the thickness ratio increases, the system clearly shows a marked increase in vibration frequency and dynamic stiffness. However, the damping rate diminishes, leading to larger amplitude oscillations and sustained lateral deflections. This behavior suggests a trade-off between increased stiffness and reduced damping, which must be carefully balanced in design applications involving high thickness ratios. While the proposed numerical model has been validated against classical theoretical results, the incorporation of experimental data from physical tests on CNT-reinforced beams under aeroelastic loading would be an important step to further substantiate the predictions. Due to the complexity and limited availability of such data under supersonic flow conditions, this remains an area of active investigation for future work. In addition, the current study has solely focused on aerodynamic forces and does not account for thermal effects. This simplification was made to isolate the influence of aerodynamic loads and structural parameters such as CNT volume fraction, slenderness, and thickness ratios under supersonic flow using the modified third-order piston theory. However, we recognize the importance of thermal loads, especially in high-speed environments where significant temperature gradients can arise due to aerodynamic heating. As pointed out in Ref. [55], thermal stresses can considerably alter the stiffness and damping characteristics of composite structures, thereby influencing their vibration and stability behavior. These effects are particularly relevant for FG and thick composite beams, where thermal gradients can lead to significant stress redistribution across the thickness. Given the potential coupling between aerodynamic and thermal loads, a more complete aero-thermoelastic model would provide deeper insight into the dynamic behavior of CNT-reinforced composites.