## 3 Results and Analyses
### Stiffness of Rub-Impact  
In the misalignment-rubbing coupling system, the rubbing force is one of the nonlinear sources of the rotor system. As a factor affecting the rubbing force, rubbing stiffness has an important influence on the vibration response. Keeping the rub-impact clearance unchanged, the misalignment angle is 3\({}^{\circ}\), and the bifurcation diagrams and spectrum diagrams under stiffness \(K=1.3\times 10^{7}\) N/m, 1.5 \(\times\) 10\({}^{7}\) N/m and 1.8 \(\times\) 10\({}^{7}\) N/m are drawn, as shown in Figures 20-22. It can be seen from Figures 20(a), 21(a) and 21(a) that with the increase of the stiffness for the rub-impact, the number of the chaotic zone increases from zero to two; and the range of the chaos is widening, i.e., the larger the stiffness, the larger the unstable quasi-periodic motion interval. In addition, the corresponding critical rotational speeds are 561 rad/s, 567 rad/s and 579 rad/s respectively. This is because increasing the stiffness of the rub-impact will lead to an increase in the critical speed. It is found from Figures 20(b), 21(b) and 21(b) that with the increase of the rub-impact stiffness, the maximum amplitude of the speed frequency decreases.  
Figure 19: **(a)** The time domain diagram; **(b)** the spectrum diagram; **(c)** Poincare section diagram; and **(d)** the axis trajectory diagram when \(\alpha=16^{\circ}\).  
Figure 17: **(a)** The time domain diagram; **(b)** the spectrum diagram; **(c)** Poincare section diagram; and **(d)** the axis trajectory diagram when \(\alpha=11^{\circ}\).  
The bifurcation diagram of the rub-impact stiffness under the misalignment-rubbing coupling fault is drawn with the misalignment angle = 6\({}^{\circ}\), as shown in Figure 23. It can be seen that with the increase of stiffness, the system goes through the period-1, the quasi-period, the period-8, the quasi-period, the period-5, the quasi-period, the period-2, the period-4, the multi-period, the period-3, the chaotic motions, intermittent inverted bifurcation, paroxysmal bifurcation, and so on.  
Figure 21: Bifurcation diagram and frequency waterfall diagram of vibration response when K = 1.5 \(\times\) 10\({}^{7}\) N/m. (**a**) Bifurcation diagram with the change of rotating speed. (**b**) Frequency waterfall diagram.  
Figure 20: Bifurcation diagram and frequency waterfall diagram of vibration response when K = 1.3 \(\times\) 10\({}^{7}\) N/m. (**a**) Bifurcation diagram with the change in rotating speed. (**b**) Frequency waterfall diagram.  
When the stiffness is less than 2 \(\times\) 10\({}^{7}\) N/m, the system is in the period-1 motion because of the weak rub-impact characteristics under the small stiffness; the axis orbit is an ellipse, and there is a point in the Poincare section, as shown in Figure 24. With the increase of stiffness up to 3.5 \(\times\) 10\({}^{7}\) N/m, the rub-impact characteristics are more and more obvious, and the vibration response is more and more complex. The system enters the quasi-periodic motion, and the beat vibration phenomenon is obvious, as shown in Figure 25; when the rubbing stiffness is 5.5 \(\times\) 10\({}^{7}\) N/m, 6.3 \(\times\) 10\({}^{7}\) N/m, 6.8 \(\times\) 10\({}^{7}\) N/m and 7.3 \(\times\) 10\({}^{7}\) N/m, the system is in the period-8, the period-5, the period-2 and the period-4 motion respectively, the axis trajectory is shown in Figure 26; when the rub-impact stiffness increases up to 8.8 \(\times\) 10\({}^{7}\) N/m the system enters chaotic state whose characteristics are shown in Figure 27.  
Figure 23: Bifurcation of rub-impact stiffness under misalignment-rubbing coupling fault.  
Figure 24: (a) The time domain diagram; (b) the spectrum diagram; (c) Poincare section diagram; and (d) the axis trajectory diagram when K = 1.5 \(\times\) 10\({}^{7}\) N/m.