# Free Vibrations and Flutter Analysis of Composite Plates Reinforced with Carbon Nanotubes
## 3 Results and Discussion
### Flutter Analysis of CNT/Nanocomposite Plates  
The present analysis of CNT/nanocomposite flutter phenomena considers mainly the influence of CNT volume fractions and BC. The aeroelastic analysis of square CNT/nanocomposite plates is performed for fully simply supported (SSSS) and clamped (SCSC) structures, assuming an angle flow of 0\({}^{\circ}\) and ignoring the damping. The influence of boundary conditions and CNT volume fraction (k parameter) on the eigenvalue curves is investigated in the first part of the flutter analysis. The effect of the boundary conditions is shown in Figure 8, where the eigenvalue curves for simply supported and clamped CNT/nanocomposite plates are plotted. The contour plot of the eigenvalue curves is drawn according to Equations (21) and (22). The square nanocomposite plate with 1%vol. of nanoreinforcement and 2 mm thickness was used. In the plots, Re denotes the real part of \(r\), while Im is the imaginary part. It can be seen how the boundary conditions determined the shape and position of the eigenvalue curves. The curve for the clamped nanocomposite plate is shifted to the lower pressure value. Because eigenvalue plots are functions of the k parameter, changing this value can change the shape of the curves. The change in the eigenvalue curves affects the possible solution to the flutter problem. The comparison between the eigenvalue curve for isotropic and orthotropic square plates is shown in Figure 9. It is worth pointing out, that the parameter \(k\) is equal to 1 for the isotropic plate, while for the nanocomposite plate with 1%vol. of CNT (orthotropic plate) is much lower and equal to 0.257. For the nanocomposite plate, the eigenvalue curve is narrower and moves down. Figure 9 considers the square clamped plates (SCSC). Moreover, considering the orthotropic plates, the influence of the CNT volume fraction on the eigenvalue curve was analyzed. Figure 10 shows the results of simply supported nanocomposite plates with different volume fractions of CNTs. It can be  
Figure 7: Variation of the non-dimensional frequency of simply supported rectangular nanocomposite plates (_a/b_ = 2, _a/h_ = 200) versus the volume fraction of CNTs for the first five modes. seen that the higher the volume fraction is, the narrower the eigencurves are. However, the discrepancies are less visible for higher volume fractions of CNTs. For SSSS plates, the orthotropic eigencurves lie within the isotropic one (SSSS 0% CNT), while for SCSC plates, the movement is observed in two directions, as shown in Figure 9. Based on the results presented in Figures 8 and 10, it is evident that the boundary conditions have a higher influence on the eigenvalue curve than the volume fraction of CNT. Figure 8: The eigenvalue curves for 1% vol. CNT/nanocomposite plate for simply supported case (upper curve) and clamped case (lower curve) for \(a/b=1\), and \(h=2\) mm. Figure 9: The eigenvalue curves for isotropic plate (\(k=1\)) and 1% CNT/nanocomposite orthotropic plate (\(k=0.257\)) for \(a/b=1\), and \(h=2\) mm. The contour plot of non-dimensional pressure and non-dimensional frequency \(\lambda\) for 1% CNT/nanocomposite plate is presented in Figure 11. The increase of the real part of \(r\), the non-dimensional pressure increase, and the non-dimensional frequency \(\lambda\) decrease are visible. The vertical axis (r-Im) correspondends to a situation when the critical aerodynamic pressure \(\beta=0\). The coalescence point can be found analytically by plotting the non-dimensional pressure, non-dimensional frequency, and eigenvalue curves. The method of the analytical (graphical) derivation of the critical aerodynamic pressure for the first mode is shown in Figure 12. The graphical representation considers the case  
Figure 11: Non-dimensional pressure \(\beta\) (dashed line) and non-dimensional frequency \(\lambda\) (solid line) for 1% CNT/nanocomposite plate– \(a/b=1\), \(a/h=500\), \(h=2\) mm. Figure 10: The eigenvalue curves for various volume fractions of CNTs in nanocomposite plates–SSSS, \(a/b=1\), \(h=2\) mm.