# Optimizing Fundamental Frequencies in Axially Compressed Rotating Laminated Cylindrical Shells
## 3 Numerical Analysis
However, when the L/R ratio increased to 4, the differences in the fundamental frequencies between the shells with FF and SS ends were significantly reduced. Again, the fundamental frequencies of the laminated cylindrical shells generally decreased as the axial compressive forces increased. Furthermore, the maximum fundamental frequencies of the \(\left[\pm\theta/90/0\right]_{28}\) laminated cylindrical shells seemed to occur in the region \(40^{\circ}\leq\theta\leq 60^{\circ}\) when the axial compressive force was lower than \(0.4\) N\({}_{\rm cr}\). The maximum fundamental frequencies tended to occur in the region \(70^{\circ}\leq\theta\leq 90^{\circ}\) when the axial compressive force was larger than \(0.6\) N\({}_{\rm cr}\). Figure 6 shows the fiber angle \(\theta\) and the associated fundamental frequency \(\omega\) for \(\left[\pm\theta/90/0\right]_{28}\) laminated cylindrical shells with L/R = 4, \(\Omega=20,000\) rPM, different boundary conditions, and various axial compressive forces. Again, the fundamental frequencies of the laminated cylindrical shells decreased as axial compressive forces increased. The differences in the fundamental frequencies between the shells with FF and SS ends were significantly reduced. The maximum fundamental frequencies of these shells seemed to occur either in the region \(20^{\circ}\leq\theta\leq 40^{\circ}\) or at \(\theta\) of around \(70^{\circ}\). Figure 2: Boundary conditions of cylindrical shells. As the influence of the boundary conditions between FF ends and SS ends diminished with a larger L/R ratio, subsequent studies focused exclusively on maximizing the fundamental frequency of composite shells with fixed ends. Maximization of the Fundamental Frequency of \(\left[\pm\theta/90/0\right]_{2s}\) Laminated Cylindrical Shells with Various Lengths, Various Axial Compressive Forces, and \(\Omega=0\) RPM  
In this section, the optimization of \(\left[\pm\theta/90/0\right]_{2s}\) laminated cylindrical shells with FF ends, various lengths (R = 10 cm, 1 \(\leq\) L/r \(\leq\) 4), different axial compressive forces (0 \(\leq\) N \(\leq\) 0.8N\({}_{\rm cr}\)), and \(\Omega=0\) RPM is described. To determine the maximum fundamental frequency \(\omega\) and the corresponding optimal fiber angle \(\theta\), the optimization problem can be formulated as follows:  
\[\text{Maximize}:\ \omega(\theta) \tag{7}\]  
Figure 3: Effects of end condition and axial compressive force on the fundamental frequency of \(\left[\pm\theta/90/0\right]_{2s}\) laminated cylindrical shells (R = 10 cm, L/R = 1, \(\Omega=0\) RPM); (**a**) N = 0 N\({}_{\rm cr}\), (**b**) N = 0.2 N\({}_{\rm cr}\), (**c**) N = 0.4 N\({}_{\rm cr}\), (**d**) N = 0.6 N\({}_{\rm cr}\), (**e**) N = 0.8 N\({}_{\rm cr}\). (8)  
Before applying the golden section method [5; 22; 28; 36], the fundamental frequency \(\omega\) of the laminated cylindrical shell was computed by the Abaqus program for each 10\({}^{\circ}\) increment in the \(\theta\) angle, allowing for an approximate identification of the maximum point, as illustrated in Figures 3-6. Next, appropriate upper and lower bounds were chosen, and the golden section method was applied. The optimization process ended when an absolute tolerance (the difference between the two intermediate points) of \(\Delta\theta\leq 0.5^{\circ}\) was achieved. Figure 4: Effects of end condition and axial compressive force on the fundamental frequency of \([\pm\theta/90/0]_{28}\) laminated cylindrical shells (R = 10 cm, L/R = 1, \(\Omega=20,000\) RPM); (**a**) N = 0 N\({}_{\rm cr}\), (**b**) N = 0.2 N\({}_{\rm cr}\), (**c**) N = 0.4 N\({}_{\rm cr}\), (**d**) N = 0.6 N\({}_{\rm cr}\), (**e**) N = 0.8 N\({}_{\rm cr}\). Figure 7 shows the optimal fiber angle \(\theta_{\rm opt}\) 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.