## 1 Introduction  
The influence of the faults in the rotating machines on the associated dynamic behavior has been a focus of attention for many researchers. The presence of the faults may lead to a dangerous and catastrophic effect on the dynamic behavior of rotating structures and cause serious damage to the rotating machineries. In general, the faults in the rotor system contain the crack fault [1; 2; 3; 4; 5], the rub-impact of rotor-to-stator fault [6; 7; 8; 9], the pedestal looseness fault [10; 11; 12; 13], the ball bearing fault [14; 15], the misalignment fault, etc. [16; 17; 18; 19]. On the basis of the common faults in the rotor systems, the coupling faults (the looseness and rub-impact, the crack and rub-impact, etc.) usually occur in the actual rotor systems. The amplitude varies when the cracked rotor is operating, which will lead to the rub-impact with other units of the rotating machine. However, if the rub-impact fault occurs, the crack of shaft or impeller may occur on account of the harsh operating conditions and the strong influence. Few researchers studied the rotor systems with the coupling faults, since they are more complex than the systems with only single fault [20]. The dynamical characteristics of the rotor systems with the complex coupling faults should be further explored. The rotor systems with faults are high-dimensional and non-linear [21], they are difficult to study the qualitative characteristics and the calculations are extremely expensive. The situations will be more complicated if the rotor has compound faults like a crack and anisotropic supports [22]. Consequently, the identification of mass unbalance [23], which is critical for the safe operation of the machine, can be a great challenge. The low-dimensional models should be provided to represent the high-dimensional ones. The common order reduction methods include the center manifold method [24; 25; 26], the inertial manifold method [27; 28; 29], the Galerkin method [30; 31; 32], the Lyapunov-Schmidt (L-S) method [33; 34], the POD method [35; 36; 37; 38; 39; 40], and other order reduction methods [41; 42; 43], were summarized by [PERSON] and [PERSON] in their applied studies of non-linear dynamics, mechanical systems [44; 45]. [PERSON] [26] used center manifold theory to analyze a model of gene transcription and protein synthesis, which consists of an ordinary differential equation coupled to a delay differential equation. The concept of an inertial manifold for non-linear evolutionary equations, in particular for ordinary and partial differential equations was introduced in [PERSON]. [28]. [PERSON] proposed [31] a new method of integrating evolution differential equations--the non-linear Galerkin method. The existence and multiplicity of spatially nonhomogeneous steady-state solutions were obtained by means of Lyapunov-Schmidt reduction [33]. [PERSON]. [40] introduced tensorial calculus techniques in the framework of POD to reduce the computational complexity of the reduced non-linear terms. The POD method is an effective and powerful method for data analysis aimed at obtaining low-order modes of the original system. On the basis of the order reduction accuracy, the POD method can not only reduce the DOFs of a system greatly but also improve the computational efficiency [46]. Many researchers applied the POD method in the rotor systems to study the non-linear dynamical characteristics. [PERSON] et al. [47] proposed the TPOD method based on the inertial manifold method and applied it in the 23-DOFs rotor system, the efficiency of the order reduction method was verified via the comparison of the bifurcation diagram and the relative error. The POD method was applied in the rotor-bearing system to discuss the singularity and the prime vibration of the reduced system [48]. [PERSON] et al. [49] applied the TPOD method to reduce a 7-DOFs rotor system supported by a pair of ball bearings with pedestal looseness to a 2-DOFs rotor system.