## 3 Numerical Analysis
### Verification  
#### 3.2.1 Validation of Literature and FEM  
The model of the shaft-disk system is validated, and Table 3 demonstrates the literature comparison results of the shaft-disk system with the S-S boundary, whose structural parameters are \(L\) = 0.35 m, \(L_{d}\) = 0.37 \(L\), \(r\) = \(R_{i}\) = 0.01 m, \(R_{o}\) = 0.1 m, and \(h_{d}\) = 0.002 m and whose material parameters are \(E\) = 200 GPa, \(\mu\) = 0.266, and \(\rho\) = 7860 kg/m\({}^{3}\). The comparative values of the intrinsic frequencies are very similar, which indicates that the constructed shaft-ring plate model is equally reliable. Table 4 compares the coupled natural frequencies and modes of the shaft-disk rotor of the fiber woven material, and the coupled natural frequencies of the steel rotor are also given in the table. The parameter settings of the disk are the same as those in Table 4, except that \(L_{d}\) = 0.09 m and \(h_{d}\) = 0.006 m. The shaft parameters \(L_{1}\) = 0.08 m, \(L_{2}\) = 0.02 m, \(L_{3}\) = 0.04 m, and \(L_{4}\) = 0.1 m correspond to the radii \(r_{1}\) = \(r_{4}\) = 0.01 m, \(r_{2}\) = 0.02 m, and \(r_{3}\) = 0.015 m. It can be seen that: the 1 st- and 4 th-order coupled modes are dominated by the displacement deformation of the disk, and the deformation of each structure is generated in the 2 nd- and 3 rd-order modes, among which the displacements of the disk in the 2 nd order are mainly determined by the coupling points on the shaft. The frequency deviation does not exceed 2.12%, while the modes are basically consistent, which directly reflects the accuracy of the present model. In addition, comparing the steel rotor, it is found that the use of fiber woven material leads to a reduction in the rotor's 1 st coupling frequency, and at the same time, it can reduce the mass of the ring plate by 82.6%, which helps to achieve the goal of light weight. Considering that there is almost no vibration displacement of the shaft in the 1 st-order coupled mode of the system, its coupling characteristics are not obvious, so \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Natural Frequency Order**} & \multicolumn{8}{c}{**Number of Shaft Units**} \\  & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline
1 & 565.25 & 565.05 & 565.03 & 565.03 & 565.03 & 565.03 & 565.03 \\
2 & 646.35 & 646.35 & 646.35 & 646.35 & 646.35 & 646.35 & 646.35 \\
3 & 696.87 & 696.87 & 696.87 & 696.87 & 696.87 & 696.87 & 696.87 \\
4 & 793.65 & 793.65 & 793.65 & 793.65 & 793.65 & 793.65 & 793.65 \\
5 & 1588.52 & 1588.52 & 1588.52 & 1588.52 & 1588.52 & 1588.52 & 1588.52 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Convergence of the natural frequency (Hz) for fiber woven composite rotors with respect to the number of shaft units. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Natural Frequency Order**} & \multicolumn{8}{c}{**Number of Nodes in the Disk Unit**} \\  & **4** & **6** & **8** & **10** & **12** & **14** & **16** \\ \hline
1 & 572.86 & 567.67 & 565.23 & 565.03 & 565.01 & 565.00 & 565.01 \\
2 & 666.23 & 649.99 & 646.77 & 646.35 & 646.28 & 646.28 & 646.28 \\
3 & 811.67 & 702.49 & 697.43 & 696.87 & 696.78 & 696.74 & 696.73 \\
4 & 824.67 & 794.71 & 793.82 & 793.65 & 793.63 & 793.62 & 793.64 \\
5 & 1601.52 & 1589.62 & 1588.52 & 1588.36 & 1588.32 & 1588.31 & 1601.52 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Convergence of the natural frequency (Hz) for fiber woven composite rotors with respect to the number of nodes in the disk unit.