# Optimizing Fundamental Frequencies in Axially Compressed Rotating Laminated Cylindrical Shells
## 3 Numerical Analysis  
In the numerical analysis, laminated cylindrical shells were modeled with eight-node S8R shell elements. Each node of the element had three degrees of freedom for displacements and three for rotations. The element stiffness matrix was computed using a reduced integration rule, with hourglass stiffness control applied for stability [29]. The accuracy of the shell element in the Abaqus program for frequency analysis has been validated by the authors [5; 21; 34], showing a high level of agreement between the numerical results and both analytical solutions and experimental data. Thus, the shell element in Abaqus was deemed sufficiently precise for the vibration analysis of laminated cylindrical shells. Laminated Cylindrical Shells with Various End Conditions, Axial Compressive Forces, and Rotating Speeds  
This section analyzes composite laminated cylindrical shells with two types of end conditions (shown in Figure 2): fixed at both ends (denoted FF) and simply supported at both ends (denoted SS). These boundary conditions were defined in the cylindrical coordinate system, where x denotes the longitudinal direction, y the circumferential direction, and z the radial direction (Figure 1). The radius R of the shell was 10 cm, while the length L was either 10 cm or 40 cm. The laminate layup of the shells was \(\left[\pm\theta/90/0\right]_{2g}\), with each ply having a thickness of 0.125 mm. The axial compressive force ranged from 0 to 0.8N\({}_{\mathrm{cr}}\), where N\({}_{\mathrm{cr}}\) is the critical buckling load of the laminated cylindrical shell. The rotational speed \(\Omega\) was either 0 or 20,000 RPM. The lamina in analysis was composed of graphite/epoxy. Its material properties were taken from [PERSON] [35], namely, E\({}_{11}=128\) GPa, E\({}_{22}=11\) GPa, \(\
u_{12}=0.25\), G\({}_{12}=G_{13}=4.48\) GPa, G\({}_{23}=1.53\) GPa, and \(\rho=1500\) Kg/M\({}^{3}\). In this investigation, no symmetry simplifications were applied to the shells. Based on prior experience and current results of convergence studies [3; 5; 36], \(160(32\times 5)\), \(320(32\times 10)\), \(480(32\times 15)\), and \(640(32\times 20)\) shell elements were chosen to model the laminated cylindrical shells with L/R ratios equal to 1, 2, 3, and 4, respectively, in the subsequent analyses. Figure 3 shows the fiber angle \(\theta\) and the associated fundamental frequency \(\omega\) for thin \(\left[\pm\theta/90/0\right]_{28}\) laminated cylindrical shells with L/R = 1, \(\Omega=0\) RPM, different boundary conditions, and various axial compressive forces. From the figure, we can observe that the fundamental frequencies of the laminated cylindrical shells with FF ends were usually higher than those of the shells with SS ends. The fundamental frequencies of the laminated cylindrical shells generally decreased as the axial compressive forces increased. Additionally, the maximum fundamental frequencies of the \(\left[\pm\theta/90/0\right]_{28}\) laminated cylindrical shells appeared to occur when the value of \(\theta\) fell within the range of \(40^{\circ}\) to \(60^{\circ}\). Figure 4 shows the fiber angle \(\theta\) and the associated fundamental frequency \(\omega\) for thin \(\left[\pm\theta/90/0\right]_{28}\) laminated cylindrical shells with L/R = 1, \(\Omega=20,000\) rPM, different boundary conditions, and various axial compressive forces. From the figure, we can observe similar trends as those in Figure 3. Comparing Figure 4 with Figure 3, we can see that the fundamental frequencies of the laminated cylindrical shells increased as the rotating speeds \(\Omega\) increased. Figure 5 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, \(\Omega=0\) RPM, different boundary conditions, and various axial compressive forces. From the figure, we can observe that the fundamental frequencies of the laminated cylindrical shells with FF ends were again higher than those of the shells with SS ends.