## 3 Results and Analyses  
### Rotating Speed  
The rotating speed will change when the rotor system changes the operation mode and conditions or starts and stops. As an important parameter of rotating machinery, rotational speed has a significant impact on the dynamic characteristics of the system. When the rubbing clearance \(\delta\) = 8 \(\times\) 10\({}^{-5}\) m, the rub-impact stiffness \(K\) = 2 \(\times\) 10\({}^{7}\) N/m, the friction coefficient \(\mu\) = 0.1, and the misalignment angle \(\alpha\) = 3\({}^{\circ}\), the bifurcation diagram of the vibration amplitude for the rotor with the change of the rotating speed under the misalignment fault is shown in Figure 4. It can be seen from Figure 4 that the vibration of the first-order critical speed occurs in the system, and the critical speed is 499 rad/s. After the speed increases to the critical speed, the amplitude of the vibration jumps from the positive to the negative direction. In addition, due to the small misalignment angle, the vibration at half the critical speed is not excited. The bifurcation diagram under the misalignment-rubbing coupling fault is shown in Figure 5, and the motions of the different speed intervals are shown in Table 2. It can be seen from Figure 5 that the system undergoes the period-1 motion, the quasi-period motion, the period-1 motion, the period-2 motion, the period-3 motion, the quasi-periodic motion and the period-1 motion in turn under the coupling fault. By comparing Figures 4 and 5, it can be found that when the misalignment and rubbing faults exist at the same time, the critical speed of the system increases by 54% from 499 rad/s to 770 rad/s, and the Hopf bifurcation occurs; comparing the misalignment fault system, a new quasi-periodic motion of the coupling fault system appears in the range of 1726 rad/s\(\sim\)1832 rad/s. Figure 4: Bifurcation diagram of vibration amplitude with the change in rotating speed under the misalignment fault. When the rotating speed of the coupled fault system \(\omega\) is 400 rad/s, 640 rad/s, 1100 rad/s, 1650 rad/s and 1780 rad/s respectively, the time domain, the spectrum, Poincare section and the axis trajectory diagrams are shown in Figures 6-10. It can be seen from Figure 6d that the system has a rub-impact at this rotating speed. Due to the weak rub-impact, there are no other frequency components except the rotational frequency, as shown in Figure 6b. The system is in period-1, which is consistent with Figure 6a,c. The frequency components of 0.25X, 0.5X, 0.75X, 1.5X, 2X and 1.27X in Figure 7b are not conventions. The beat vibration is obvious in Figure 7a,c presents a similar circular curve; the corresponding axis orbit is shown in Figure 7d. It can be seen from Figure 8b that there are two dominant frequency components, 0.5X and 1X, which are in accord with Figure 8c; the amplitude of 0.5X is larger than that of 1X. From Figure 8d, the axis orbit is evolved from the 8-type trajectory; there are two peaks in Figure 8a, which correspond to the two closed circle-like curves in Figure 8d. There is a X/3 frequency division in Figure 9a; the Poincare section is divided into three independent scatter points, as seen in Figure 9c. However, when the rotating speed is 1785 rad/s, the system enters the quasi-periodic motion again. Figure 10c presents a curve like shamrock, and the axis trajectory is not disorderly, as shown in Figure 10d. \begin{table}
\begin{tabular}{c c c c} \hline
**Speed Range (rad/s)** & **Motion Type** & **Speed Range (rad/s)** & **Motion Type** \\ \hline
1-584 & Period-1 & 584–770 & Quasi-periodic \\
770–1018 & Period-1 & 1028–1225 & Period-2 \\
1225–1571 & Period-1 & 1571–1726 & Period-3 \\
1726–1832 & Quasi-periodic & 1832–3000 & Period-1 \\ \hline \end{tabular}
\end{table} 
Table 2: The motion types for the different speed ranges. Figure 5: Bifurcation diagram of vibration amplitude with the change in rotating speed under misalignment and rubbing faults.