## 2 Description of the Cracked Rotor System  
This section briefly presents the modeling of the cracked element based on the notion of stiffness reduction and the breathing mechanism. The rotor system under study is also described. For the interested reader, the complete modeling of the rotor system was explained previously in [40], and the cracking model used in this study was proposed and discussed previously in [16]. The mechanical system under study consists of a two-bearing flexible cracked rotor, as shown in Figure 1. The system is composed of a shaft with a circular cross-section, 0.5 m in length and 0.01 m in diameter, with two rigid circular discs located at the middle of the shaft and at a quarter of the left side, respectively. The rotor is supported by two flexible supports, one on each extremity. It is excited by an out-of-balance force on both discs. The rotor shaft is discretized with 20 Euler beam elements, where each node has four degrees of freedom (dof). The values of the rotor parameters are given in Table 1.  
Considering that the presence of one transverse crack induces local flexibility due to strain energy concentration in the vicinity of the tip of the crack under load [13; 14], it can  
Figure 1: Model of a rotor system with one crack. be assumed that the reduction in the second moment of area \(\Delta I\) at the location of the crack is given by  
\[\Delta I=I_{0}\left(\frac{\dfrac{R}{I}\Big{(}1-\
u^{2}\Big{)}F(\mu)}{1+\dfrac{R}{ I}\Big{(}1-\
u^{2}\Big{)}F(\mu)}\right) \tag{1}\]  
where \(I_{0}\) is the second moment of the area of the cross-section of the healthy rotor. \(R\), \(l\), and \(\
u\) are the beam radius, element length of the section, and Poisson ratio, respectively. \(F(\mu)\) defines a nonlinear compliance function which can be obtained from a series of experiments with chordal cracks [13; 14]. \(\mu=\frac{k}{R}\) is the non-dimensional crack depth and \(h\) defines the depth of the crack, as illustrated in Figure 1. It then follows that the local flexibility due to the presence of one crack leads to an additional amount of stiffness denoted by \(\mathbf{K}_{crack}\) at the crack. The complete expressions of the stiffness matrix \(\mathbf{K}_{crack}\) are given in [16]. In addition, due to the rotation of the rotor system, it can be assumed that the crack will open and close once per revolution. The function describing this periodic opening and closing of the crack, called \"breathing\", can be approximated by a cosine function \(g(t)\)[16], by assuming that the gravity force is much greater than the imbalance force (i.e., the cracked rotor rotates under the load of its own weight). This results in the expression of a simple crack breathing mechanism  
\[g(t)=\frac{1-\cos\omega t}{2} \tag{2}\]  
where \(\omega\) defines the rotational speed of the rotor. For \(g(t)=1\), the crack is fully open, and for \(g(t)=0\), there is no effect due to the crack on the dynamic behavior of the rotor system (i.e., the crack is totally closed, and the global cracked rotor stiffness is equal to the stiffness of the healthy rotor). Finally, the equations of the rotor with a breathing crack can be written as:  
\[\mathbf{M}\ddot{\mathbf{x}}+\mathbf{D}\dot{\mathbf{x}}+(\mathbf{K}-g(t) \mathbf{K}_{c})\mathbf{x}=\mathbf{f}+\mathbf{q} \tag{3}\]  
where \(\mathbf{x}\), \(\dot{\mathbf{x}}\), and \(\ddot{\mathbf{x}}\) are the displacement, velocity, and acceleration vectors. \(\mathbf{K}\) and \(\mathbf{M}\) are the stiffness and mass matrices of the complete uncracked rotor. \(\mathbf{f}\) and \(\mathbf{q}\) are the gravitational force and the imbalance, respectively. The matrix \(\mathbf{D}\) combines the effects of the shaft's internal damping and gyroscopic moments.