# Optimizing Fundamental Frequencies in Axially Compressed Rotating Laminated Cylindrical Shells
## 3 Numerical Analysis
From Figure 7a-c, we can see that when L/R \(\leq\) 3, the optimal fiber angle \(\theta_{\rm opt}\) of the cylindrical shell seemed to be less influenced by the N/N\({}_{\rm cr}\) ratio and oscillated between 45\({}^{\circ}\) and 60\({}^{\circ}\). When L/R = 4, the optimal fiber angle \(\theta_{\rm opt}\) of the cylindrical shell seemed to be influenced by N/N\({}_{\rm cr}\) ratio significantly and scattered between 35\({}^{\circ}\) and 80\({}^{\circ}\). Figure 8 shows the maximum fundamental frequency \(\omega\) versus the L/R ratio and N/N\({}_{\rm cr}\) ratio for thin \(\left[\pm\theta/90/0\right]_{\rm 2s}\) laminated cylindrical shells with FF ends. From Figure 8a, we can observe that the optimal frequency \(\omega\) decreased with increasing of L/R ratios. From Figure 8b, we can observe that the optimal frequency \(\omega\) also decreased with increasing of N/N\({}_{\rm cr}\) ratios. From Figure 8c, we can see that the laminated cylindrical shells with larger L/R ratios and larger N/N\({}_{\rm cr}\) ratios had the lowest optimal frequency \(\omega\). In addition, the  
Figure 9 displays the fundamental vibration modes of thin \(\left[\pm\theta/90/0\right]_{28}\) laminated cylindrical shells with fixed-fixed ends under optimal fiber orientation with L/R ratios of 1 and 4. We can see that as the L/R ratio increased, the fundamental vibration modes of these cylindrical shells exhibited fewer waves in the circumferential direction. Additionally, when the L/R ratio was low (e.g., L/R = 1), the fundamental vibration modes of shells under higher axial compressive forces showed fewer waves in the circumferential direction than those under lower axial compressive forces. Figure 7: Effects of L/R and N/N\({}_{\rm cr}\) on the optimal fiber angle of \(\left[\pm\theta/90_{2}/0\right]_{\rm 2S}\) laminated cylindrical shells with two fixed ends (R = 10 cm, \(\Omega=0\) RPM); (**a**) \(\theta_{\rm opt}\) versus L/R, (**b**) \(\theta_{\rm opt}\) versus N/N\({}_{\rm cr}\), (**c**) plot of \(\theta_{\rm opt}\) versus L/R and N/N\({}_{\rm cr}\). Figure 8: Effects of \(\rm L/R\) 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, \(\Omega=0\) RPM); (**a**) \(\omega\) versus \(\rm L/R\), (**b**) \(\omega\) versus \(\rm N/N_{cr}\), (**c**) plot of \(\omega\) versus \(\rm L/R\) and \(\rm N/N_{cr}\). Figure 9: 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, \(\Omega=0\) RPM). Maximization of the Fundamental Frequency of \(\left[\pm\theta/90/0\right]_{2s}\) Laminated Cylindrical Shells with Various Lengths, Various Axial Compressive Forces, and \(\Omega=\) 10,000 RPM  
In this session, optimization of \(\left[\pm\theta/90/0\right]_{2s}\) laminated cylindrical shells with FF ends, various lengths (R = 10 cm, \(1\leq\mathrm{L}/\mathrm{r}\leq 4\)), various axial compressive forces (\(0\leq\mathrm{N}\leq 0  
Figure 11: Effects of L/R and N/N\({}_{\rm cr}\) on the maximum fundamental frequency of \([\pm\theta/90_{2}/0]_{28}\) laminated cylindrical shells with two fixed ends (R = 10 cm, \(\Omega=10,000\) 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 12: 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, \(\Omega=10,000\) RPM). Figure 2:0 Effects of \(\rm L/R\) and \(\rm N/N_{cr}\) on the optimal fiber angle of \(\left[\pm\theta/90_{2}/0\right]_{28}\) laminated cylindrical shells with two fixed ends (\(\rm R=10\) cm, \(\rm\Omega=20,000\) RPM); (**a**) \(\rm\theta_{opt}\) versus \(\rm L/R\), (**b**) \(\rm\theta_{opt}\) versus \(\rm N/N_{cr}\), (**c**) plot of \(\rm\theta_{opt}\) versus \(\rm L/R\) and \(\rm N/N_{cr}\).