## 2 Dynamic Modeling of Cracked Anisotropic Rotor-Bearing System  
### Modeling of Cracked Shaft  
A realistic crack in rotor is usually gapless. A gapless crack is easy to be modeled in the methods based on the theory of fracture mechanics, which is described by the variation of stiffness. For a 3 Dfinite element model, it is relatively difficult to make the crack gapless while keeping the nodes on the two crack surfaces independent. With respect to this issue, a two-step approach is developed herein, as is shown by the schematic diagram in Figure 1. Although a transverse crack is illustrated, the proposed method can be used for cracks with arbitrary shapes by adjusting the retaining nodes on the cracked surface in the pretreatment processes. In the first step, the cracked rotor is split into two parts according to the position of crack, and the common section is then divided into the connecting surface and the crack surface, respectively, where the shape and size of the crack depend on the problem of interest. As the name implies, the connecting surface is used to connect the two parts, and the crack is defined on the cracked surface. The connecting and the crack surfaces are discretized first and subsequently the left and the right parts. As the shaft and bearings are lack of anisotropy, the governing equations of motion of the system will contain periodically time-variant terms in both the fixed and rotating frames. In order to make it easier in further analysis, the rotating frame is chosen so that the periodical coefficients exist only at the degrees of freedom (DOFs) of the bearing nodes. Then, the governing equations of motion of the two rotating parts in the rotating frame are obtained independently, so that the DOFs of nodes on the connecting and the crack surfaces exist in both the two equations, which can be expressed as  
\[\mathbf{M}^{(p)}\ddot{\mathbf{u}}^{(p)}+\Omega\mathbf{C}_{cor}^{(p)}\dot{ \mathbf{u}}^{(p)}+\left(\mathbf{K}_{s}^{(p)}-\Omega^{2}\mathbf{K}_{d}^{(p)} \right)\mathbf{u}^{(p)}=\mathbf{f}_{ext}^{(p)}+\mathbf{f}_{con}^{(p)}+\mathbf{ f}_{crk}^{(p)} \tag{1}\]  
where superscript \"\(p\)\" is a label representing different parts; \(\mathbf{M}^{(p)}\) and \(\mathbf{K}_{s}^{(p)}\) are the mass and elastic stiffness matrices; \(\mathbf{u}^{(p)}\) denotes the vector of nodal displacement; \(\Omega\) is the rotating angular velocity; \(\Omega\mathbf{C}_{cor}^{(p)}\) is the Coriolis matrix; \(\Omega^{2}\mathbf{K}_{d}^{(p)}\) is the spin softening matrix due to the rotation of rotor; \(\mathbf{f}_{con}^{(p)}\), \(\mathbf{f}_{ck}^{(p)}\) and \(\mathbf{f}_{ext}^{(p)}\) represent the force vectors with respect to the DOFs of nodes on the connecting surface, the crack surface and the external excitation, respectively. In the second step, the two independent governing equations are synthesized to form that of the whole cracked shaft. The vectors \(\mathbf{u}^{(p)}\) are firstly reorganized as  
\[\mathbf{u}^{(p)}=\left\{ \begin{array}{l}\mathbf{u}_{crk}^{(p)}\\ \mathbf{u}_{con}^{(p)}\\ \mathbf{u}_{oth}^{(p)}\end{array} \right\} \tag{2}\]  
where the subscripts \"\(crk\)\", \"\(con\)\" and \"\(oth\)\" denote the nodal displacement vectors regarding the crack surface, the connecting surface and the other nodes, respectively. Then, the two parts are connected by applying the displacement compatibility conditions as  
\[\mathbf{u}_{con}^{(l)}=\mathbf{u}_{con}^{(r)} \tag{3}\]  
Figure 1: The schematic diagram for the modeling of a gapless crack. The compatibility conditions in Equation (3) will lead to a transformation, by which the governing equations of the whole cracked shaft can be obtained  
\[\mathbf{M}^{(s)}\ddot{\mathbf{u}}^{(s)}+\Omega\mathbf{C}_{cor}^{(s)}\dot{\mathbf{ u}}^{(s)}+\Big{(}\mathbf{K}_{s}^{(s)}-\Omega^{2}\mathbf{K}_{d}^{(s)}\Big{)} \mathbf{u}^{(s)}=\mathbf{f}_{ext}^{(s)}+\mathbf{f}_{crk}^{(s)} \tag{4}\]  
where the superscript \"\(s\)\" denotes the objects regarding the shaft.