# Optimizing Fundamental Frequencies in Axially Compressed Rotating Laminated Cylindrical Shells
## 3 Numerical Analysis
Figure 2:1 Effects of L/R and N/N\({}_{\rm cr}\) on the maximum fundamental frequency of \([\pm\theta/90_{2}/0]_{\rm 2s}\) laminated cylindrical shells with two fixed ends (R = 10 cm, \(\Omega=20,000\) RPM); (**a**) \(\omega\) versus L/R, (**b**) \(\omega\) versus N/N\({}_{\rm cr}\), (**c**) plot of \(\omega\) versus L/R and N/N\({}_{\rm cr}\). Figure 2:2 Vibrational modes of \(\left[\pm\theta/90_{2}/0\right]_{28}\) laminated cylindrical shells with two fixed ends under optimal fiber angle (R = 10 cm, \(\Omega=20,000\) RPM). Effect of the Rotating Speed on the Optimal Fiber Angle and the Maximum Fundamental Frequency of \(\left[\pm\theta/90/0\right]_{2s}\) Laminated Cylindrical Shells  
In this section, the data obtained from Sections 3.2-3.4 are replotted to study the effect of the rotating speed. From Figure 16, we can see that the cylinder shells with higher rotating speeds had smaller optimal fiber angles than those with lower rotating speeds. In addition, the rotating speed had more effect on the maximum fundamental frequency of the laminated cylindrical shells when \(\mathrm{L/R}>2\). From Figure 17, it can be seen that the maximum fundamental frequency of the cylinder shells increased with increasing rotation speed. In addition, the maximum fundamental frequency of the cylinder shells decreased with increasing \(\mathrm{L/R}\) ratios. From Figure 18, we can see that the cylinder shells with higher rotating speeds had smaller optimal fiber angles than those with lower rotating speeds. From Figure 19, it can be seen that the maximum fundamental frequency of the cylinder shells increased with increasing rotation speed. In addition, the maximum fundamental frequency of the cylinder shells decreased with increasing \(\mathrm{N/N_{cr}}\) ratios. Finally, comparing Figure 15 with Figure 9, we can see that when rotating speed was increased, the fundamental vibration modes of the shells with high rotating speed had fewer waves in the circumferential direction. Maximization of the Fundamental Frequency of \(\left[\pm\theta/90/0\right]_{2s}\) Laminated Cylindrical Shells with Various Lengths, Various Cutouts, Various Axial Compressive Forces, and \(\Omega=\) 20,000 RPM  
In this section, optimization of \(\left[\pm\theta/90/0\right]_{2s}\) laminated cylindrical shells with FF ends, various lengths (\(\mathrm{R=10\ cm}\), \(\mathrm{L/r=2,\ 4}\)), various axial compressive forces (\(\mathrm{0\leq N\leq 0.8N_{cr}}\)), and \(\Omega=20,000\) RPM is described. In addition, the cylinder shells contained circular cutouts at the center (Figure 2), and the radius \(\mathrm{d}\) of the cutout was either \(\mathrm{0\ cm}\), \(\mathrm{2\ cm}\), or \(\mathrm{4\ cm}\), i.e., \(\mathrm{d/R=0,\ 0.2,\ 0.4}\). Figure 20 shows the optimal fiber angle \(\theta_{\mathrm{opt}}\) versus the \(\mathrm{d/R}\) ratio and \(\mathrm{N/N_{cr}}\) ratio for thin \(\left[\pm\theta/90/0\right]_{2s}\) laminated cylindrical shells with FF ends, a center cutout, \(\mathrm{L/R=2,\ 0}\), and \(\Omega=20,000\) RPM. It can be seen that the optimal fiber angle \(\theta_{\mathrm{opt}}\) of the shell fell between \(40^{\circ}\) and \(50^{\circ}\). The optimal fiber angle \(\theta_{\mathrm{opt}}\) slightly decreased with increasing \(\mathrm{d/R}\) ratios and slightly increased with increasing \(\mathrm{N/N_{cr}}\) ratios. Figure 21 shows the maximum fundamental frequency versus the \(\mathrm{d/R}\) ratio and \(\mathrm{N/N_{cr}}\) ratio for thin \(\left[\pm\theta/90/0\right]_{2s}\) laminated cylindrical shells with FF ends, a center cutout, \(\mathrm{L/R=2}\), and \(\Omega=20,000\) RPM. It can be seen that the optimal fiber angle \(\theta_{\mathrm{opt}}\) of the shell fell between \(40^{\circ}\) and \(50^{\circ}\). The optimal fiber angle \(\theta_{\mathrm{opt}}\) slightly decreased with increasing \(\mathrm{d/R}\) ratios and slightly increased with increasing \(\mathrm{N/N_{cr}}\) ratios.