# Free Vibrations and Flutter Analysis of Composite Plates Reinforced with Carbon Nanotubes
## 2 Materials and Methods
### Equations of Motion  
An equilibrium position in a structure occurs, according to [PERSON]'s principle, when  
\[\int_{t_{1}}^{t_{2}}(\delta U+\delta W-\delta K)dt=0, \tag{9}\]  
where \(\delta U\), \(\delta W\), and \(\delta K\) are the virtual strain energy, the virtual work conducted by the external forces, and the virtual kinetic energy, respectively, defined as [50; 51]  
\[\delta U=\int_{\Omega}\sigma_{ij}\delta\epsilon_{ij}d\Omega, \tag{10a}\] \[\delta W=\int_{S}\Delta p\delta wdS,\] (10b) \[\delta K=\int_{\Omega}\rho\Bigg{[}\bigg{(}\frac{\partial u}{\partial t}\bigg{)} ^{2}+\bigg{(}\frac{\partial v}{\partial t}\bigg{)}^{2}+\bigg{(}\frac{\partial w }{\partial t}\bigg{)}^{2}\Bigg{]}d\Omega, \tag{10c}\]  
where \(\Omega\), \(S\), and \(\rho\) denote the volume, surface, and density of the nanocomposite plate, respectively, \(\Delta p\) is the aerodynamic pressure, and \(t\) is the physical time. The aerodynamic pressure \(\Delta p\) of a supersonic airflow acting on the nanocomposite plate is estimated based on first-order piston theory, where a linear function of the lateral deflection of a plate is used. Piston theory is the most commonly used method for elucidating the aerodynamic pressure acting on a body in a supersonic or hypersonic flow [21; 52]. First-order piston theory is applicable to calculate the aerodynamic pressure for the supersonic airflow (\(\sqrt{2}\leq M_{\infty}\leq 5\)) [43]. This theory provides a computationally simple aerodynamic model for calculating aerodynamic loads, assuming that the local pressure is related to the local normal component of the air velocity, analogous to the piston moving in a cylinder [42; 53]. The aerodynamic pressure field \(\Delta p\) is described in terms of the out-of-plane displacement \(w\), the air stream density \(\rho_{\infty}\), the air velocity \(V_{\infty}\), and Mach number \(M_{\infty}\) of the plate exposed to the supersonic airflow parallel to the \(x\)-axis as follows [43; 54]:  
\[\Delta p=-\Lambda\frac{\partial w}{\partial x}-\mu\frac{\partial w}{\partial t}, \tag{11}\]  
\[\Lambda=\rho_{\infty}V_{\infty}^{2}/\sqrt{M_{\infty}^{2}-1},\ \mu=\rho_{\infty}V_{\infty} \Big{(}M_{\infty}^{2}-2\Big{)}/\Big{(}M_{\infty}^{2}-1\Big{)}^{3/2}\]  
\(\Lambda\) is the aerodynamic pressure parameter owing to the inclination of the plate concerning the flow angle direction, while represents the aerodynamic damping [43]. Since the aerodynamic damping always stabilizes the flutter [55], the aerodynamic pressure is given by  
\[\Delta p=\frac{\rho_{\infty}V_{\infty}^{2}}{\sqrt{M_{\infty}^{2}-1}}\frac{ \partial w}{\partial x}, \tag{12}\]  
[PERSON]'s principle can be used to derive the equations of motion for a system. [PERSON]'s system of equations leads to fourth-order differential linear equations. Since for the flat plate, the membrane and bending strains are uncoupled for the free vibration and flutter analysis, the governing relations are reduced to the study of out-of-plane deflections \(w\) only [47]. For the plate exposed to the supersonic airflow parallel to the \(x\)-axis (the airflow angle is equal to \(0^{\circ}\)) the fundamental relation can be written as [46]  
\[ \begin{array}{c}D_{11}\frac{\partial^{2}w}{\partial x^{2}}+2(D_{12}+2D_{66}) \frac{\partial^{2}w}{\partial x^{2}\partial y^{2}}+D_{22}\frac{\partial^{2}w} {\partial y^{4}}-\Big{(}N_{x}^{0}\frac{\partial^{2}w}{\partial x^{2}}+N_{y}^{ 0}\frac{\partial^{2}w}{\partial y^{2}}\Big{)}+\rho l\frac{\partial^{2}w}{ \partial t^{2}}+\\ \Lambda\frac{\partial w}{\partial x}+\mu l\frac{\partial w}{\partial x}=0, \end{array}  \tag{13}\]The problem is reduced to finding out the eigenvalues and corresponding mode shapes of the system for a given value of the aerodynamic pressure parameter \(\Lambda\).