# Free Vibrations and Flutter Analysis of Composite Plates Reinforced with Carbon Nanotubes
## 3 Results and Discussion
### Free Vibration Analysis of CNT/Nanocomposite Plates  
Various geometric parameters and volume fractions are considered in the free vibration analysis of CNT/nanocomposite plates. Plates with thicknesses of 2 mm and 5 mm, assuming the width-to-thickness ratios \(a/h=200\) (\(h=5\) mm) and \(a/h=500\) (\(h=2\) mm) with different aspect ratios of the plates, i.e., \(a/b=0.5\); 1; 1.5; and 2, were studied. Figure 3 shows the variation of the non-dimensional fundamental frequency of various simply supported nanocomposite plates as a function of the plate aspect ratio. The increase in the aspect ratio results in the decrease in the fundamental frequency for all analyzed volume fractions of the nanotubes. [PERSON] et al. [37] also observed a similar degradation of the nanocomposite non-dimensional frequency with increasing plate aspect ratios. The results are in good agreement. [PERSON] et al. [37] reported a 58% reduction in non-dimensional frequency comparing SSSS square (\(a/b=1\)) and rectangular plates (\(a/b=2\)), and current predictions indicate a 64% reduction. As the volume fraction of CNTs grows, the natural frequency increases, as shown in Figure 4. For the square plate (a/b = 1), a 2.7-fold improvement in frequency was observed when the volume fraction of CNTs was increased from 1% to 5%. It can  
that the higher the aspect ratio, the smaller the effect of the volume fraction of CNTs on the frequencies. For the rectangular plates, comparing 5% to 1% CNTs for a/b = 1.5 and a/b = 2, there is a 90% and 46% increase in the fundamental frequency, respectively. This is a much lower value than the 2.7-fold increase for the square nanocomposite plate. Figure 5 shows the fundamental frequency of the SSSS plate as a function of the aspect ratio. The different volume fractions of CNTs and plate thicknesses have been considered. The thinner the nanocomposite plate, the lower the frequency observed. However, the discrepancies tend to worsen as the aspect ratio of the plate increases. The higher the volume fraction, the higher the frequency. The square plate's increase is approximately 65% when comparing 1% and 5% volume fractions. Comparable observations were presented by [PERSON] et al. [19] for FG CNT/nanocomposite plates with uniform distribution of CNTs where comparable aspect ratios were used. Figures 6 and 7 show the non-dimensional frequency vs. CNT volume fraction of SSSS square and rectangular (\(a/b=2\)) nanocomposite plates for the first five modes. An observation of the trends and results shown in Figures 3-7 reveals some interesting phenomena. For all modes (assuming \(n=1\)), the frequencies are higher for the square plate. The increase in frequency is also observed as the volume fraction of CNTs increases. As expected, the increase in volume fraction causes the increase of non-dimensional frequencies (Figure 6), and the increase in the aspect ratio shows the decrease in eigenfrequencies (Figure 3). Similar thickness-to-width ratios of thin CNT/nanocomposite plates (\(h/a=0.005\) and \(h/a=0.002\)) and ranges of aspect ratios analyzed (0.5, 1.0, 1.5, 2.0) were studied by [PERSON] et al. [19]. [PERSON] et al. [19] also observed a decrease in frequency with an increasing aspect ratio. It should be noted here that [PERSON] et al. [19] analyzed much higher volume fractions of CNTs (11%,14%, and 17%) compared to the present analysis, which focused on small fractions (1%, 3%, 5%). The choice of such volume fractions of CNTs was dictated by descriptions of experimental work on the difficulty of obtaining uniform distribution of CNTs in the matrix structure at higher fractions and the tendency of CNTs to form agglomerates at higher volume fractions. The proposed volume fractions of CNTs were adopted based on the experimental work of [PERSON] et al. [8], where nanocomposites with low fractions of  
Figure 4: Variation of the non-dimensional fundamental frequency of the simply supported nanocomposite plate as a function of the volume fractions of CNTs for the various aspect ratios of the plates (\(h=5\) mm, \(m=n=1\)).