## 5 Model Parameters and Numerical Simulation
### Vibration Response of the Coupled System with Rubbing Fault  
For the multi-faults case (i.e., unbalance, misalignment and rotor-stator rub coexist), the selection of the clearance was completed with an approximation of the realistic friction of the rotor into the stator wall that takes into account the local nonlinear deformations proposed in Equation (36). The dynamic response in orbit patterns, lateral direction and its FFT spectral are shown in Figure 9. As another parameter of study, the orbit shapes revealing the contact between the rotor and the stator of each shaft varied from \(5.127\times 10^{-3}\)m for the driveshaft to \(1.127\times 10^{-2}\)m for the driven shaft in various amplitude levels, as shown in Figure 9a,b. As the contact stiffness increased, the orbits of the driveshaft fluctuated highly with multiple rebounds represented by the irregularity of inside loops (Figure 9a), which causes the disturbing orbit to diverge. However, the driven shaft orbit only increases in amplitude with tight and regular circular loops. As the contact stiffness increases, the critical speeds of both shafts also increase in frequency and amplitude. From the above plots, as illustrated in Figure 9c, it can be seen that there is an increase in the critical speeds of both shafts when recurring frictional contact occurs, leading to several fluctuating peaks around the main frequency. The higher harmonic resonance at \(1\times\) order (47.21 Hz) can be observed in the two unbalanced rotors. However, it is worth mentioning from the FFT spectral plot that the higher sup-harmonic frequencies, for example, \(2\times\), \(4\times\), \(6\times\), etc., along with the \(1/2\times\) subharmonic, which usually arise due to the rubbing presence as indicated in [10; 18], do not appear in this case. It is evident that the decomposition by FFT is not satisfactory enough as some irregular and  
Figure 9: Dynamic response of driveline rotors with driveshaft-to-stator rubbing (**a,b**) orbit of the input and output unbalanced shafts (**c**) Fast Fourier Transform spectrum response (**d**) transient deflection response of shafts. imprecise fluctuations exist around the critical frequency levels. Figure 9d illustrates the impact of contact on driveshaft deflection by the rise of compressed low frequencies with subsynchronous broadband noise related to rubbing. There is a significant influence of the coupling joint on the overall dynamic characteristics of the connected rotor system where the monitoring of the waveform and frequencies vibration of the driven shaft vibration disappears significantly. From the components of the 3-D speed-frequency spectrum, the phenomenon of rotor-stator contact can be found in the marginal spectral and peaks distribution (Figure 10a). However, there are a few complex interference components (e.g., the embraced part by the lightly blue peaks line at the starting operation (10 Hz) in Figure 10a), introduced by the unbalance excitation. The higher frequencies observed at the frequency above 50 Hz are excited by misalignment and contact rubbing. These observations of accumulated peaks of the frictional driveshaft are also seen in the marginal spectral in Figure 9d. The angular velocity profile of the driveshaft shows approximate, periodic occurrences of low excitation peaks at higher amplitudes as the characteristics of the rub (Figure 10b). Moreover, it is visible that there exist considerable lags over time and periods of rub occurrences between the FFT spectrum (Figure 9c) and the 3-D speed-frequency (Figure 10a) due to the inherent drawbacks in the coupling joint (misalignment). The results of this section demonstrate that the occurrence of shocks due to the reduction of lateral clearance of the driveshaft-stator rubbing is the specific source of noise that generates irregular operation of the drive unit and contributes to the abrupt variation in frequency vibration (change in angular velocity during one revolution).