## 2 Dynamic Modeling of Cracked Anisotropic Rotor-Bearing System
### Modeling of Anisotropic Speed-Dependent Journal Bearings  
The eight-coefficient model [4] is commonly used for the modeling of journal bearing, which considers the cross-coupling interaction between vertical and horizontal motions and can accurately describe the dynamic characteristic of bearings. Therefore, it is used herein to establish the dynamic model for anisotropic journal bearings, where the speed-dependent characteristic is considered. Figure 2 shows the diagram for the model, which contains four stiffness parameters and four damping parameters. As these coefficients are all defined in the fixed reference frame, they should be transformed into the rotating frame. First, the reaction force of the journal bearings in the fixed frame can be expressed as  
\[\mathbf{f}_{bf}=\left\{ \begin{array}{c}f_{x}\\ f_{y}\end{array} \right\}=\left[ \begin{array}{cc}k_{xx}(\Omega)&k_{xy}( \Omega)\\ k_{yx}(\Omega)&k_{yy}(\Omega)\end{array} \right]\left\{ \begin{array}{cc}u_{x} \\ u_{y}\end{array} \right\}+\left[ \begin{array}{cc}c_{xx}(\Omega)&c_{xy}(\Omega) \\ c_{yx}(\Omega)&c_{yy}(\Omega)\end{array} \right]\left\{ \begin{array}{c}\dot{u }_{x}\\ i_{y}\end{array} \right\}=\mathbf{K}_{bf}\mathbf{u}_{bf}+\mathbf{C}_{bf}\dot{ \mathbf{u}}_{bf} \tag{5}\]  
where \(k_{xx}\), \(k_{xy}\), \(k_{yx}\), \(k_{yy}\), \(c_{xx}\), \(c_{xy}\), \(c_{yx}\) and \(c_{yy}\) are the stiffness and damping coefficients; \(u_{x}\) and \(u_{y}\) represent the transverse displacements. When the speed-dependent characteristic is considered, the eight coefficients are depending on the rotating speed. Then, the displacements and bearing reaction forces are transformed into the rotating frame  
\[\mathbf{u}_{bf}=\mathbf{R}\mathbf{u}_{br},\quad\mathbf{f}_{bf}=\mathbf{R}^{\rm T} \mathbf{f}_{br},\quad\mathbf{R}=\left[ \begin{array}{cc}\cos\Omega t&-\sin \Omega t\\ \sin\Omega t&\cos\Omega t\end{array} \right] \tag{6}\]  
where \(\mathbf{R}\) denotes the rotation transformation matrix and \(\mathbf{R}^{-1}=\mathbf{R}^{\rm T}\); \(\mathbf{u}_{br}\) and \(\mathbf{f}_{br}\) represent the displacement and reaction force vectors in the rotating frame, respectively. By substituting Equation (6) into Equation (5) and reasonable simplification, it can be obtained that  
\[\mathbf{f}_{br}=\left[\mathbf{R}^{\rm T}\mathbf{C}_{bf}\frac{d}{dt}(\mathbf{R })+\mathbf{R}^{\rm T}\mathbf{K}_{bf}\mathbf{T}_{R}\right]\mathbf{u}_{br}+ \left(\mathbf{R}^{\rm T}\mathbf{C}_{bf}\mathbf{R}\right)\dot{\mathbf{u}}_{br} =\mathbf{K}_{br}\mathbf{u}_{br}+\mathbf{C}_{br}\dot{\mathbf{u}}_{br} \tag{7}\]  
As the matrix \(\mathbf{R}\) contains functions of time and the obtained terms cannot be completely turned time-independent, the matrices \(\mathbf{K}_{br}\) and \(\mathbf{C}_{br}\) will contain periodical terms of the sine or cosine functions of \(2\Omega t\). By expressing the trigonometric functions as complex exponential functions, the matrices \(\mathbf{K}_{br}\) and \(\mathbf{C}_{br}\) can be expressed as the periodically time-variant form  
\[ \begin{array}{l}\mathbf{K}_{br}=\mathbf{K}_{bc}(\Omega)+\mathbf{K}_{bn}( \Omega)e^{-j2\Omega t}+\mathbf{K}_{bp}(\Omega)e^{j2\Omega t}\\ \mathbf{C}_{br}=\mathbf{C}_{bc}(\Omega)+\mathbf{C}_{bn}(\Omega)e^{-j2\Omega t }+\mathbf{C}_{bp}(\Omega)e^{j2\Omega t}\end{array}  \tag{8}\]  
where \(\mathbf{K}_{bc}\), \(\mathbf{K}_{bp}\), \(\mathbf{K}_{bn}\), \(\mathbf{C}_{bc}\), \(\mathbf{C}_{bp}\) and \(\mathbf{C}_{bn}\) represent the time-independent coefficient matrices. An isotropic bearing requires the two principle stiffness coefficients, \(k_{xx}\) and \(k_{yy}\), to be identical, as well as the damping coefficients \(c_{xx}\) and \(c_{yy}\). Meanwhile, the two cross-coupling stiffness and damping coefficients needs to be opposite in sign. If these two conditions are satisfied, the matrices \(\mathbf{K}_{br}\) and \(\mathbf{C}_{br}\) will no longer contain periodically time-variant terms. However, it is often too strict for realistic bearings to satisfy these constraints, so anisotropic bearings are more common cases.