## 5 Conclusions  
This work focuses on the semi-analytical response of aero-elastic oscillations for cantilever beams reinforced by CNTs, considering four different distributions of the reinforcement phase in the polymeric matrix. A modified third-order piston theory was adopted successfully to model the supersonic air flow rather than a first-order piston theory. The nonlinear dynamic equation was derived from [PERSON]'s principle, including the Von Karman nonlinearities. Such an equation was discretized into a nonlinear ordinary differential equation by applying the [PERSON] decomposition technique, here solved analytically according to the MTS method. A large systematic investigation has focused on the influence of the reinforcement patterns and volume fraction on the fundamental frequency, free air, and Mach number of the system. From the numerical investigation, the main conclusions can be summarized as follows:  
* Composite cantilever beams with _FG-X_ reinforcements feature the highest frequency, followed by beams with a uniform reinforcement (_UD-Beam_) and _FG-A_ pattern. In all cases, composite structures with an _FG-O_ reinforcement pattern show the lowest fundamental frequency.
* The fundamental frequency increases for an increased volume fraction. The effect of the volume fraction on the fundamental frequency is more noticeable at lower slenderness ratios.
* The _CNTR_ composite cantilever beam with a _FG-X_ reinforcement pattern shows the highest lateral deflection and the highest frequency response, followed by _UD-Beam_ and _FG-A_ reinforcement pattern. However, the _CNTR_ composite cantilever beam with a _FG-O_ reinforcement pattern exhibits the lowest lateral dynamical deflection and the lowest frequency response.
* The frequency response increases with the volume fraction of CNTs, due to an overall increased stiffness of the system.
* The frequency response of the system increases by decreasing the slenderness ratio, due to its increased stiffness, up to a certain point, after which we note a reverse behavior with a damped harmonic envelope. This means that the damped beating phenomena occur because the excitation frequency is near the damped natural frequency.
* The frequency response and the lateral dynamic deflection of the system increase for an increased velocity of the free stream air or an increased Mach number. The effect of Mach number is even more pronounced.
* For an increased thickness ratio, the nonlinear frequency response increases significantly. However, the damping rate decreases, which results in an increased lateral deflection.
* Please consider that our findings, especially under varying thickness ratios and high Mach numbers, may be limited by the lack of shear and rotary inertia considerations, as assumed by the Euler-Bernoulli theory. Dynamic modes, indeed, may be influenced by coupled bending and shear effects, especially for short or thick beams subjected to high-speed aerodynamic loading, making the simplifications of Euler-Bernoulli theory more questionable. For these reasons, the Timoshenko theory and higher-order shear deformation theories will be explored in the next work to analyze thick beams and high-speed aeroelastic environments.  
Conceptualization, F.T. and R.D.; Methodology, R.D.; Validation, M.A., F.T. and R.D.; Formal analysis, F.T. and R.D.; Investigation, M.A., F.T. and R.D.; Writing--original draft, M.A.; Writing--review & editing, F.T. and R.D. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
**Conflicts of Interest:** The authors declare no conflict of interest.