# Optimizing Fundamental Frequencies in Axially Compressed Rotating Laminated Cylindrical Shells
## 3 Numerical Analysis
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, \(\mathrm{L/R=4}\), and \(\Omega=20,000\) RPM. It can be seen that the optimal fiber angle \(\theta_{\mathrm{opt}}\) of the shell usually fell between \(30^{\circ}\) and \(40^{\circ}\). The exceptions were the shells with \(\mathrm{d/R=0}\) and \(\mathrm{N/N_{cr}}\geq 0.4\). Specially, when \(\mathrm{d/R=0}\) and \(\mathrm{N/N_{cr}=0.8}\), the optimal fiber angle \(\theta_{\mathrm{opt}}\) of the shell was \(70.7^{\circ}\). Figure 23 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, \(\mathrm{L/R=4}\), and \(\Omega=20,000\) RPM. It can be seen again that the maximum fundamental frequency of the shell increased with increasing \(\mathrm{d/R}\) ratios and decreased with increasing \(\mathrm{N/N_{cr}}\) ratios. Comparing Figure 23 with Figure 21, we can see that the increase in the maximum fundamental frequency of the short shell (\(\mathrm{L/R=2}\)) due to the cutout was more significant than that of the long shell (\(\mathrm{L/R=4}\)), especially when the \(\mathrm{N/N_{cr}}\) ratio was large. Figure 2:3 Effects of rotating speed and L/R on the optimal fiber angle of \(\left[\pm\theta/90_{2}/0\right]_{28}\) laminated cylindrical shells with two fixed ends (R = 10 cm); (**a**) N = 0 N\({}_{cr}\), (**b**) N = 0.2 N\({}_{cr}\), (**c**) N = 0.4 N\({}_{cr}\), (**d**) N = 0.6 N\({}_{cr}\), (**e**) N = 0.8 N\({}_{cr}\). Figure 2:4 Effects of rotating speed and L/R on the maximum fundamental frequency of [\(\pm\)\(\theta\)/90\({}_{2}\)/0]\({}_{28}\), laminated cylindrical shells with two fixed ends (R = 10 cm); (**a**) N = 0.2 N\({}_{\rm cr}\), (**b**) N = 0.4 N\({}_{\rm cr}\), (**c**) N = 0.6 N\({}_{\rm cr}\), (**d**) N = 0.6 N\({}_{\rm cr}\), (**e**) N = 0.8 N\({}_{\rm cr}\). Figure 2:5 Effects of rotating speed and \(\rm N/N_{cr}\) on the maximum fundamental frequency of \(\rm[\pm\theta/90_{2}/0]_{28}\) laminated cylindrical shells with two fixed ends (R = 10 cm); (a) \(\rm L/R=1\), (b) \(\rm L/R=2\), (c) \(\rm L/R=3\), (d) \(\rm L/R=4\). Figure 2:6 Effects of d/R and N/N\({}_{\rm cr}\) on the maximum fundamental frequency of \(\left[\pm\theta/90_{2}/0\right]_{28}\) laminated cylindrical shells with two fixed ends and cutouts (R = 10 cm, L/R = 2, \(\Omega=20,000\) RPM); (**a**) \(\omega\) versus d/R, (**b**) \(\omega\) versus N/N\({}_{\rm cr}\), (**c**) plot of \(\omega\) versus d/R and N/N\({}_{\rm cr}\). Figure 2:7 Effects of d/R and N/N\({}_{\rm cr}\) on the optimal fiber angle of \(\left[\pm\theta/90_{2}/0\right]_{28}\) laminated cylindrical shells with two fixed ends and cutouts (R = 10 cm, L/R = 4, \(\Omega=20,000\) RPM); (**a**) \(\theta_{\rm opt}\) versus d/R, (**b**) \(\theta_{\rm opt}\) versus N/N\({}_{\rm cr}\), (**c**) plot of \(\theta_{\rm opt}\) versus d/R and N/N\({}_{\rm cr}\). Figure 2:8 Effects of d/R and N/N\({}_{\rm cr}\) on the maximum fundamental frequency of \(\left[\pm\theta/90_{2}/0\right]_{28}\) laminated cylindrical shells with two fixed ends and cutouts (R = 10 cm, L/R = 4, \(\Omega=20,000\) RPM); (a) \(\omega\) versus d/R, (b) \(\omega\) versus N/N\({}_{\rm cr}\), (c) plot of \(\omega\) versus d/R and N/N\({}_{\rm cr}\). Figures 24 and 25 present the fundamental vibration modes of thin \(\left[\pm\theta/90/0\right]_{28}\) laminated cylindrical shells with FF end conditions, a central cutout, and \(\Omega=20,000\) RPM for L/R = 2 and 4, respectively. It can be seen that as the L/R ratio increased, the fundamental vibration modes exhibited fewer waves in the shell's circumferential direction. Additionally, as the cutout size grew, the fundamental vibration modes of these cylindrical shells showed increased local disturbances near the cutout area.