## 3 Results and Analyses
### Clearance  
The clearance between the rotor and the stator is an important design parameter of the rotor system. Smaller cleara nce of the rub-impact is beneficial to improve mechanical efficiency, but it also brings the risk of the rub-impact fault. The clearance in the real world can be changed in some cases. To investigate the effect of the clearance on the dynamic characteristics, speed bifurcation and frequency waterfall diagrams under the different clearances are drawn, as shown in Figures 28-30. By comparing Figures 28a, 29a and 30a, it can be seen that with the increase of clearance, the intervals of the quasi-periodic motion in the supercritical and critical speed regions decrease, and the maximum amplitude of the vibration decreases. Therefore, increasing the rubbing clearance is beneficial for enhancing the stability of the system. It can be seen from Figures 28b, 29b and 30b that the larger the clearance, the smaller the number of the rub-impact characteristic frequencies, and the weaker the degree of the rub-impact. Figure 28: Bifurcation diagram and frequency waterfall diagram of vibration response when \(\delta=9\times 10^{-5}\) m. (**a**) Bifurcation diagram with the change in rotating speed. (**b**) Frequency waterfall diagram. Figure 26: Axial trajectory of periodic motion under different rubbing stiffness. Figure 27: (**a**) The time domain diagram; (**b**) the spectrum diagram; (**c**) Poincare section diagram; and (**d**) the axis trajectory diagram when K = \(8.8\times 10^{7}\) N/m. The bifurcation diagram with the change in the rub-impact clearance is shown in Figure 31. It can be seen from Figure 31 that with the increase of the rub-impact clearance, the fluctuation range of the amplitude gradually narrows. The research shows that the system undergoes the quasi-period, the period-3, the quasi-period and the period-1 motions in turn. With increasing of the clearance up to 0.15 mm, the rub-impact disappears and there is a jumping phenomenon in the bifurcation diagram from the positive amplitude to negative amplitude. In order to study the specific nonlinear dynamic characteristics of the system, the rub-impact clearance 3 \(\times\) 10\({}^{-5}\) m, 6 \(\times\) 10\({}^{-5}\) m and 12 \(\times\) 10\({}^{-5}\) m are chosen to draw the corresponding time domain, the spectrum, Poincare section and the axis trajectory diagrams, as shown in Figures 32-34. It can be seen from Figure 32 that when the rub-impact clearance is 3 \(\times\) 10\({}^{-5}\) m, there is an obvious beat vibration phenomenon in Figure 32a, which shows a circular closed curve, indicating that the system is in the quasi-periodic motion state. In Figure 32d, there are annular curves; the amplitude of the frequency 1.35X generated by the rub-impact is significantly higher than that of the frequency 2X generated by the misalignment, as seen in Figure 32b. As the clearance increases, the degree of rub-impact is reduced, as shown in Figure 33.  
Figure 30: Bifurcation diagram and frequency waterfall diagram of vibration response when \(\delta=12\times 10^{-5}\) m. (**a**) Bifurcation diagram with the change in rotating speed. (**b**) Frequency waterfall diagram. Figure 29: Bifurcation diagram and frequency waterfall diagram of vibration response when \(\delta=10\times 10^{-5}\) m. (**a**) Bifurcation diagram with the change in rotating speed. (**b**) Frequency waterfall diagram. Figure 31: Bifurcation diagram with the change in rub-impact clearance.