## 3 Numerical Analysis
### Parametric Studies  
Based on the constructed rotor model, the vibration characteristics of a rotating fiber woven composite rotor with a simply supported boundary are investigated. Unless otherwise stated, the system geometrical parameters and boundary settings in the following examples are the same as those in Table 4.  
Firstly, the effect of the hole types on the rotor's Campbell diagram is explored. In order to avoid an influence of the hole's area on the structure, the number of holes \(N\) is set as 4 uniformly, and at the same time, the shape is set to make the area of the single holes as similar as possible, with the specific dimensions as follows: round holes: \(R_{h}\) = 0.06 m, \(r_{h}\) = 0.025 m; sector-shaped holes: \(R_{1}\) = 0.045 m, \(R_{2}\) = 0.075 m, \(d_{h}\) = 0.015 m; curved holes: \(R_{1}\) = 0.043 m, \(R_{2}\) = 0.077 m, \(\varphi\) = \(\pi/6\). The Campbell diagrams of the 2 nd and 3 rd coupled natural frequencies of the system are shown in Figure 11. It is clear that the hole type has an effect on the 2 nd forward (solid line) and backward (dashed line) coupled frequency values of the system, especially for the latter. This is due to the fact that this order mode is dominated by the shaft and there is a huge difference between the modes of the disk with and without holes at the same time. The 3 rd natural frequency is affected by the vibration of the shaft and the disk at the same time, and the overall frequency shows the law of \"No hole > Round hole > Curved hole > Sector hole\". The corresponding 3 rd critical speed has the same trend. Figure 12 exemplifies the effect of the total fiber volume fraction \(V_{\text{f}}\) on the vibration characteristics of the system, with the system pore parameters set as above. In conjunction with Figure 3, the Young's modulus, shear modulus, and density of the fiber woven material itself continue to increase as \(V_{\text{f}}\) grows from 0.28 to 0.34, with a small decrease in Poisson's ratio. The joint change of these parameters makes the overall stiffness of the rotor systems \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Hole Type** & **Method** & **1** & **2** \\ \hline \hline \multirow{4}{*}{Sector hole} & \multirow{4}{*}{Present} & \multirow{4}{*}{Present} & \multirow{4}{*}{Present} \\  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: _Cont._with different open hole types increase, so that the 3 rd coupling frequency shows a similar linear reduction law: the growth rate of the 3 rd coupling forward frequency gradually slows down, and the deceleration of the backward frequency gradually grows. The comparison between the subplots reveals that rotors with different hole types have different coupling frequency reduction rates, among which the reduction rate of the rotor with sector holes is the largest. Taking the circular hole pattern with \(R_{h}\) = 0.06 m and \(r_{h}\) = 0.02 m as an example, Figures 13-15 demonstrate the effects of the radius \(r_{h}\), the number of holes \(N\), and the hole eccentricity \(R_{h}\) on the Campbell diagrams of the 2 nd and 3 rd coupling frequencies. Figure 13 shows the effect of the round hole radius \(r_{h}\) on the natural frequency of the system when \(N\) = 4 and \(R_{h}\) = 0.06 m. The change in \(r_{h}\) can also be regarded as the change in the hole area, which is reflected as the change in stiffness in the overall structure. From the figure, it is found that when \(r_{h}\) is raised from 0 to 0.03 m, the 2 nd frequency increases gradually and the 3 rd decreases continuously. The backward frequency deceleration of the former mutates to 0 at different rotational speeds, and the mutation rotational speed increases with the increase in \(r_{h}\).