## 2 Modelling and Validation of a Two-Disc Rotor with a Breathing Crack  
In order to analyze the characteristics of a two-disc rotor with a breathing crack, the finite element method is adopted to establish the model, in which the breathing crack is simulated by the CCLP (crack closure line position) breathing model proposed by [PERSON] et al. [55]. Since the method has been elaborated in the previous work by the authors [56], here, for the conciseness of the paper, the modelling procedure will just be briefly introduced. Further, validation of the established model is carried out by modal experiment and comparison with previous published work. Two-node Timoshenko beam elements with six degrees-of-freedom (three translations and three rotations) per node are used to model the rotor-bearing system. Figure 1 shows the two-disc cracked circular rotor-bearing system, the cracked shaft element, and the finite beam element model. To establish the model of the cracked rotor, the key point is to model the crack appropriately and calculate the stiffness matrix of the crack element that can be calculated by the strain energy release rate approach [56]. After that, through assembling the cracked and un-cracked elements, the finite element model of the rotor can be obtained. Though the method is universal for various crack angles, only the transverse crack is considered in this paper in order to simplify the problem. Figure 1: Models of a cracked rotor-bearing system. (**a**) The cracked two discs rotor-bearing system; (**b**) the cracked shaft element; (**c**) the finite beam element model. For the rotor in Figure 1a, two discs are considered as rigid bodies, which have three translational and three rotational inertias, and they are added to the mass matrix elements at the corresponding degrees-of-freedom. The gyroscopic effect of the two discs is also included. The bearings are simplified as stiffness and damping systems, both constrain 2 lateral degrees-of-freedom. The rotating speed of the rotor is \(\Omega\). By assembling the system matrix of the cracked element and un-cracked elements, the finite element beam model shown in Figure 1c can be established. Denote \(\mathbf{q}_{i}\) as displacement vector of node \(i\) having 6 degrees-of-freedom:  
\[\mathbf{q}_{i}=\left\{x_{i}\,,y_{i}\,,z_{i}\,,\theta_{xi}\,,\theta_{yi}\,, \theta_{zi}\right\}^{\text{T}} \tag{1}\]  
The equations of motion in the stationary coordinate system can be written as:  
\[\mathbf{M}\ddot{\mathbf{q}}+\left(\mathbf{C}+\Omega\mathbf{G}_{\text{y}} \right)\dot{\mathbf{q}}+\mathbf{K}(t)\mathbf{q}=\mathbf{F}_{\text{e}}+\mathbf{ F}_{\text{g}}+\mathbf{F}_{\text{ex}} \tag{2}\]  
\[\mathbf{q}=\left\{\mathbf{q}_{1},\ \mathbf{q}_{2},\ \ldots\ \mathbf{q}_{i},\ldots,\ \mathbf{q}_{n}\right\}^{\text{T}} \tag{3}\]  
where \(\mathbf{M}\) is the system mass matrix; \(\mathbf{C}=a\mathbf{M}+b\mathbf{K}\) is system damping matrix considering the Rayleigh damping; \(\mathbf{G}_{\text{y}}\) is system gyroscopic matrix; \(\mathbf{K}\) is system stiffness matrix that will be updated as the crack breathes; \(n\) is the total node number (in the following numerical calculation, and the rotor is evenly divided into 60 elements, hence \(n\) is equal to 61); \(\mathbf{F}_{\text{e}}\) is the excitation due to eccentricity of discs; \(\mathbf{F}_{\text{g}}\) is excitation due to the gravitational force; and \(\mathbf{F}_{\text{ex}}\) is external excitation during operation. Parameters of the rotor-bearing system are shown in Table 1, where \(a\) and \(b\) are calculated by assuming modal damping ratios of the first two modes being 0.005 and 0.01. To validate the established finite beam element model which called beam model below, on one hand, the intact rotor is established by ABAQUS software, on the other hand, modal experiment by an impact hammer is carried out. The tested rotor is shown in Figure 2, in which two orthogonal eddy current transducers are used to obtain the response excited by the hammer.