## 2 Basic Theory of the TPOD Method and the Rotor Model with Coupling Faults
### Discussion of the Dynamical Characteristics  
Figure 3 shows the four amplitude-frequency curves of the rotor systems with different faults: the original system without faults; the cracked fault; the rub-impact fault and the coupling fault. 1/2 sub-harmonic vibration occurs in the cracked and coupling system at the frequency of 2\(\omega_{c}\), \(\omega_{c}\) is the natural frequency. The amplitude of the coupling system is apparently higher than the system with only crack fault, which indicates that the rub-impact will increase the response amplitude of the crack. The amplitude of the system with rub-impact bumps up at the frequency of 1000 rad/s(nearby 3\(\omega_{c}\)), this result illustrates rub-impact will aggravate the complex motion of the system. Figure 4 shows the bifurcation diagrams of the rotor systems with different faults: (a) the oil-film fault; (b) the crack fault; (c) the rub-impact fault; (d) the coupling fault. It is clear that the bifurcation behaviors vary with different rotor faults. The obvious period-doubling bifurcation occurs at the second order natural frequency of the cracked fault system. The period-doubling bifurcation also occurs in the coupling fault system, but the amplitude is much larger. When the rotating speed is 1000 rad/s (nearby 3\(\omega_{c}\)), the amplitude of the rub-impact fault system suddenly increases, which indicates that the rub-impact fault can give rise to the complex motion of the system. Figure 3: Comparison of amplitude–frequency curves. When \(\omega=350\) rad/s (nearby the primary resonance), the time history curves of the four systems are drawn clearly in Figure 5. The curves contain both the transient and steady-state processes. The transient process of the oil-filmed system (Figure 5a) vanishes in about two periods, the other three systems (Figure 5b-d) need three periods. This indicates that the stability of the system with faults can't be kept in comparison to the normal system. Meanwhile, in Figure 5d, the steady-state signal of the coupling fault is not standard sinusoidal, which has a visible difference in the other three systems. Figure 4: Bifurcation diagrams. In Figure 6, it shows the trajectories of the orbit of shaft center of the four systems. The motion curves in (a) and (c) are all elliptical. The major axis and minor axis of ellipse of the oil-filmed system is longer than the rub-impact system. The trajectories of the orbit of shaft center of cracked (b) and coupling (d) faults are the closed curves similar to ellipse, but the bottom is a little narrow. The crack fault is the main reason to change the ellipse of trajectories of the orbit of shaft center as closed curves similar to the ellipse near the primary resonance. We use the comparison of the phase portraits (Figure 7) to verify the results in Figure 6 based on the same initial values and rotating speed. In Figure 7a,c, it is easy to discover that the phase portraits are both the closed curves similar to circle, and the diameter of the oil-filmed system is larger. In the other two figures (Figure 7a,c), the curves are both irregularly closed. Figure 5: Time history curves of \(x_{1}\) when \(\omega=350\) rad/s. (**a**) The oil-filmed system. (**b**) The cracked system. (**c**) The rub-impacted system. (**d**) The coupled system. Figure 6: Trajectories of the orbit of shaft center of \(o_{1}\) when \(\omega=350\) rad/s. (**a**) The oil-filmed system. (**b**) The cracked system. (**c**) The rub-impacted system. (**d**) The coupled system. Figure 7: The phase portraits of \(o_{1}\) when \(\omega=350\) rad/s. (**a**) The oil-filmed system. (**b**) The cracked system. (**c**) The rub-impacted system. (**d**) The coupled system. **Remark 1**. : _The crack and coupling faults have greater effect to the motion state near the primary resonance. Moreover, the rub-impact fault has more obvious effect to the vibration amplitude at the frequency of primary resonance.