## 3 Results and Analyses
### Misalignment Angle  
The misalignment of the power turbine rotor system will also affect the vibration response. In order to study the influence of the misalignment of the rotor on the vibration, the  
Figure 8: (**a**) The time domain diagram; (**b**) the spectrum diagram; (**c**) Poincare section diagram; and (**d**) the axis trajectory diagram when \(\omega\) = 1100 rad/s.  
Figure 10: (**a**) The time domain diagram; (**b**) the spectrum diagram; (**c**) Poincare section diagram; and (**d**) the axis trajectory diagram when \(\omega\) = 1785 rad/s.  
Figure 7: (**a**) The time domain diagram; (**b**) the spectrum diagram; (**c**) Poincare section diagram; and (**d**) the axis trajectory diagram when \(\omega\) = 640 rad/s.  
Figure 9: (**a**) The time domain diagram; (**b**) the spectrum diagram; (**c**) Poincare section diagram; and (**d**) the axis trajectory diagram when \(\omega\) = 1650 rad/s.  
misalignment angle \(\alpha=5^{\circ}\), \(8^{\circ}\) and \(10^{\circ}\) are selected as the control parameter, and the bifurcation and frequency waterfall diagrams are shown in Figures 11-13. In Figure 11a, when the rotating speed is \(586\) rad/s, the system enters the quasi-periodic motion from the period-\(1\) motion, i.e., the state of motion changes, the speed is accordingly called the switching speed, and the switching speeds for \(\alpha=8^{\circ}\) and \(10^{\circ}\) are similarly \(588\) rad/s and \(593\) rad/s respectively, as shown in Figures 12a and 13a. It can be seen from Figures 11a, 12a and 13a that when the misalignment angle increases, the vibration stability near half of the switching speed slumps, the maximum amplitude of the vibration response decreases, and the range of the quasi-periodic motion widens. Figures 11b, 12b and 13b show that with the increase of the misalignment angle, the maximum amplitude of the resonance decreases slightly.  
Chaos has intrinsic randomness, initial value sensitivity, and the characteristics of irregular order; therefore, a variety of tools is applied to judge chaos. Under the misalignment and rub-impact fault, when the speed \(\omega\) is 600 rad/s, the bifurcation diagram of the amplitude with the misalignment angle is shown in Figure 14, which shows that the rotor system has experienced the quasi-periodic, the period-3, the period-6, the period-12, the period-4 and the chaotic motions. The time domain, the spectrum, Poincare and the axis orbit diagrams under the misalignment angles \(\alpha=7^{\circ}\), \(9^{\circ}\), \(11^{\circ}\), \(14^{\circ}\) and \(16^{\circ}\) respectively are shown in Figures 15-19. It can be shown from Figures 15-19 that with the increase of the misalignment angle, the amplitude of the 2X frequency gradually magnifies up to the first largest peak from the third largest peak in the spectrum diagrams. The other diagrams, such as the time domain, Poincare and the axis orbit, verify this point. The results show that the frequency of 2X is the characteristic frequency of the misalignment fault. When \(\alpha=11^{\circ}\), Figure 17c,d shows the system is in chaos, which is so unpredictable that this misalignment angle should be avoided in design.