# Dynamic Behavior of Twin-Spool Rotor-Bearing System with Pedestal Looseness and Rub Impact
## 3 Analysis
### Dynamic Analysis of the System with Coupled Failure  
This subsection examines the impact of the looseness-rub coupled failure on the vibration response of the twin-spool rotor-bearing system by comparing it with the vibration response under the looseness failure. We analyze how the vibration response of the system changes under different rotational speeds and degrees of loosening, and we derive the vibration characteristics of the looseness-rub coupled fault. The loosening gap is 0.1 mm, the loosening stiffness is \(2.5\times 10^{7}\) N/m, and the loosening damping is 400 N \(\cdot\) s/m to simulate a slight looseness failure; the rub impact gap is 0.8 mm and the rub impact stiffness is \(2\times 10^{7}\) N/m to simulate a rub impact fault. The effect of the coupled failure on the vibration response of the system was comparatively analyzed under these fault parameters, as shown in Figure 19. Figure 19 shows the shaft centerline orbit and spectrum at the LP compressor at a LP rotor speed of 560 rad/s. The system was operated at a speed of fL = 0.5 rpm. At this speed, the system is in periodic motion with \(f_{L}=89.13\) Hz and \(f_{H}=115.86\) Hz as the main frequencies, and the change of the shaft centerline orbit of the system before and after the occurrence of the looseness failure is not obvious. But multiplier frequencies of the HP rotor rotating frequencies, \(2f_{H}\) and \(3f_{H}\), appeared in the frequency spectrum after the looseness failure, as shown in  
Figure 17: Spectrum and time history of LP compressor at speed 1034 rad/s. (**a**) Time history (faultless). (**b**) Spectrum (faultless). (**c**) Time history (rub impact failure). (**d**) Spectrum (rub impact failure). Figure 19b. Considering the influence of rub impact faults on the basis of looseness failures, the shaft centerline orbit and spectrum of the system under the influence of coupling faults are shown in Figure 19c,d. The red circle in the shaft centerline orbit diagram indicates the rub impact initial gap, and it can be seen that the shaft centerline orbit of the LP compressor partially exceeds the rub impact gap and a slight local rub impact fault occurs. At the same time by the impact of local rub impact, the area of the circle surrounded by the shaft centerline orbit also increased. In addition, after the coupled failure, there is an obvious and chaotic low-frequency component in the spectrum. The loosening gap is 0.2 mm, the loosening stiffness is \(1\times 10^{7}\) N/m, and the loosening damping is 400 N \(\cdot\) s/m to simulate a serious looseness failure; the rub impact gap is 1.56 mm, and the rub impact stiffness is \(1.8\times 10^{7}\) N/m to simulate a rub impact fault. Figure 20 shows the shaft centerline orbit and spectrum at the LP compressor at 1034 rad/s rotor speed. At this speed, the system is in simple harmonic motion with \(f_{\rm L}=164.57\) Hz, and the shaft centerline orbit at the LP compressor is no longer in a regular circular shape after the looseness failure occurs; there is an obvious \(3f_{L}\) frequency component in the spectrum, and a relatively small amount of spurious wave \(f_{b}\) appears in the low-frequency band (Figure 20b). After considering the influence of the looseness-rubbing coupling fault, the shaft centerline orbit and spectrum are shown in Figure 20a,d. The system has already produced the local rub impact phenomenon, the multiplier component in the spectrum increases slightly, and the \(f_{b}\) frequency component and the low-frequency component caused by loosening increase significantly. To examine the dynamic behavior of the twin-spool rotor system affected by the looseness-rub coupled failure, the evolution law of coupled fault and their impact on the vibration behaviors of the entire machine is investigated under various levels of loosening stiffness. The variable of loosening stiffness is chosen due to its heightened sensitivity within the system's looseness failure parameters.