## 4 Numerical Results  
Based on the proposed method, the effects of the anisotropic and speed-dependent characteristics of bearings on the eigenvalues and the forced response of the cracked rotor-bearing system are presented. The model shown in Figure 3 is employed as an example, where the rotor model contains 13,368 elements and 56,413 nodes. The dimensions of the shaft are as follows. The length and radius of the shaft equal 1 and 0.05 m, respectively. The thickness and outer radius of the disk are 0.075 and 0.15 m, respectively. The length of the center of the disk to the left side of the shaft is 0.6125 m. The material of the rotor is an steel with Young's modulus \(E=210\,\mathrm{GPa}\), density \(\rho=7850\,\mathrm{Kg/m^{3}}\) and [PERSON]'s ratio \(\
u=0.3\).  
The first 60 eigenvectors are retained during model order reduction, as the 60 th natural frequency has been much larger than the excitation frequencies. The retained harmonics in truncating Equations (21) and (27) is \(h\in[-7,7]\). In this paper, the anisotropy in the stiffness coefficients of bearing is considered, and two dimensionless parameters are defined to quantify the anisotropy patterns  
\[\alpha_{k}=\frac{k_{xx}}{k_{yy}},\quad\beta_{k}=\frac{k_{xy}}{k_{yy}} \tag{29}\]  
where \(\alpha_{k}\) and \(\beta_{k}\) are referred to as the anisotropic parameters of principal and cross-coupling stiffness coefficients, respectively.  
First, the proposed method is validated. Although the full order model may not be perfectly consistent with the realistic rotor, it can be modified by employing the model updating method, which  
Figure 3: The finite element model of a cracked rotor-bearing system.  
is not the focus of this paper. The proposed method can be validated, to some extent, by comparing the results obtained by the reduced order model with that by the full order model. Figure 4 shows the relative errors of the first 50 eigenvalues at the rotating speeds of 3800 and 5000 r/min, where the anisotropy parameters \(\alpha_{k}=0.5\) and \(\beta_{k}=0.2\). As can be seen in Figure 5, the maximum relative error of the real parts is less than 0.5% and that of the imaginary parts is less than 0.1%. Although the real parts possess larger relative errors, the absolute values of errors of the real parts are actually quite small as the real parts are much smaller that the imaginary parts.  
Then, the forced responses obtained by the two models are compared, which is shown in Figure 5. The external excitations consist of gravity and unbalance force. In order to reveal the effects of the level of unbalance and get the response characteristics of the cracked rotor at different rotating speeds, the unbalance force is defined by its ratio to the gravity of rotor. where the unbalance force is set to be 10% of gravity and is along the crack opening direction. The crack with a depth of 20 mm, which is 20% of the diameter, is employed as a representative. It can be seen in the figures that the amplitudes obtained by the reduced order model agree well with that by the full order model, where the maximum relative error is less than 1%. Then, the reduced order model is employed in the following sections.  
Figure 4: The relative errors of the first 50 damped eigenvalues of the anisotropic rotor-bearing system with a 20-mm crack at the speeds of 3800 and 5000 r/min: (**a**) the real parts, (**b**) the imaginary parts.  
Figure 5: The response amplitudes of nodes on shaft axis of the rotor-bearing system with a 20-mm crack at the speeds of 3800 and 5000 r/min: (**a**) along the horizontal direction; (**b**) along the vertical direction. The labels “FEM” and “ROM” denote the full order model and the reduced order model, respectively.