## 2 Basic Theory of the TPOD Method and the Rotor Model with Coupling Faults
### Discussion of the Dynamical Characteristics
The variations of the dynamical characteristics in Figures 5-7 can provide theory guidance to the fault diagnosis of the non-linear rotor-bearing system._  
Similar to the discussions before, we study the time histories, trajectories of the orbit of shaft center, phase portraits of the four different systems at the frequency of 1/2 sub-harmonic vibration (730 rad/s) in Figures 8-10. In Figure 8, the time histories contain the transient process and the stable motion state, the four systems vary from transient to stable state after \(15\pi\), so the three different faults have little effect to the dynamical characteristics of the systems near the frequency \(2\omega_{c}\). The signal is similar to sinusoidal after the oil-filmed (Figure 8a) and cracked system (Figure 8c) leads to periodic motion. The signals of the other two systems (Figure 8b,d) are more complex relative to oil-filmed and cracked systems. Figure 9 analyzes the trajectories of the orbit of shaft center of the four different system when \(\omega=730\) rad/s. The motion curves of the oil-filmed and rub-impact systems (Figure 9a,c) are similar to ellipse. In comparison to Figure 9a,c, double-loop curves are embodied in the cracked and coupling systems. There is one intersection in the cracked system and two intersections in the coupling system, this result indicates that the crack fault is a key factor in the appearance of double-loop curves at \(2\omega_{c}\). The rub-impact fault may affect the topological structures of the double-loop curves. As is clearly seen in Figure 10a,c, the phase portraits of the oil-filmed and rub-impact systems are the curves similar to ellipse. The curves in Figure 10b,d are both double-loop, the two loops are closer in the cracked system relative to the coupling system. This result also verifies that the crack fault can lead to double-loop curves, the rub-impact fault affects the topological structures of phase portraits. Figure 9: Trajectories of the orbit of shaft center of \(o_{1}\) when \(\omega=730\) rad/s. (**a**) The oil-filmed system. (**b**) The cracked system. (**c**) The rub-impacted system. (**d**) The coupled system. **Remark 2**. : _In Figures 8-10, the crack fault affects more to the rotor system at the frequency of \(2\omega_{c}\). The signal of cracked fault system is more complex, the double-loop curves occur in both trajectories of the orbit of shaft center and phase portraits, the period-doubling bifurcation occurs in Figure 4_b. The rub-impact fault will not affect the natural structures of the cracked characteristics. Hence, the dynamical characteristics near the frequency of \(2\omega_{c}\) can be regarded as the important judgments to verify the crack fault of the rotor-bearing system._  
In Figure 11, the four figures (Figure 11a-d) show the time histories of the rotor-bearing models at the frequency of \(3\omega_{c}\). There exists a certain difference between the four faults of the rotor system. The difference of the time history can provide theory guidance to the fault detection of the rotor system. Figure 10: The phase portraits of \(o_{1}\) when \(\omega=730\) rad/s. (**a**) The oil-filmed system. (**b**) The cracked system. (**c**) The rub-impacted system. (**d**) The coupled system. As is shown in Figure 12, it shows the orbit of shaft center diagrams of the oil-film, crack, rub-impact, coupling fault when \(\omega=1020\) rad/s (the third order natural frequency). Based on the constraint of the displacement of rub-impact fault, the orbit of shaft center is easier than the other three curves. However, the displacement is the largest, which indicates that the rub-impact fault arouses the larger motion near the third order frequency. The orbit of shaft center of the crack fault is more complex than that of the oil-film fault. The complexity of the orbit of shaft center of the coupling fault is situated between the oil-film fault and crack fault. Figure 11: Time history curves of \(x_{1}\) when \(\omega=1020\) rad/s. (**a**) The oil-filmed system. (**b**) The cracked system.