## 4 Numerical Results
### Effects of the Anisotropy in Bearings  
The effects of the anisotropy in journal bearings are presented in this section. The stiffness coefficient \(k_{yy}\) is set to be \(2\times 10^{8}\,\mathrm{N}/\mathrm{m}\). The anisotropy in the damping coefficients is ignored, and \(c_{xx}=c_{yy}=1\times 10^{3}\,\mathrm{Ns}/\mathrm{m}\). For the sake of brevity, the node on the axis of the shaft at the crack section is called the central node in the following parts. Figure 6 shows the response amplitudes of the central node versus the rotating speed. Two kinds of models are considered, including an uncracked rotor supported by isotropic bearings and three cracked models. The crack depth is 20 mm. As can be seen in the figures, the existence of crack leads to significant changes in the forced responses of the rotor system. The most obvious difference of the cracked rotor and uncracked rotor is that the breathing behavior of the crack leads to the 1/2 and 1/3 subharmonic resonance of the system. This is easy to understand and is reported in a vast of investigations. Besides, the anisotropy in the principle and the cross-coupling stiffness coefficients significantly affects the response of the system, in both the amplitudes and the resonant frequencies. As is reported in previous paper [18], the amplitude curves of the isotropy case in the frequency regions near the 1, 1/2 and 1/3 critical speeds contain only one resonant peak. The introduction of anisotropy in the cross-coupling stiffness coefficient leads to the split of the isolated resonant peaks, and the anisotropy in the principle stiffness coefficients results in more obvious split. It can be easily understood that the decrease of stiffness in the horizontal direction reduces the corresponding natural frequency. Moreover, the amplitudes of the resonant peaks are also affected by the anisotropy of bearings, which are the results of the complex interactions of the anisotropy of bearings and the nonlinear breathing behavior of the cracked rotor. Figure 7 shows the evolution of the whirling orbits of the central node versus the parameters of \(\alpha_{k}\) at the speed of 3800 r/min, where \(\beta_{k}\) is set to be 0.2 and the unbalance force is 10% of gravity. The rotating speed is selected to be 3800 as it is in or near the resonant region of the 1/2 critical speed, at which the response is more sensitive to the variation of system parameters. As can be seen in the figures, the variation of \(\alpha_{k}\) significantly affects the whirling orbits, especially when \(\alpha_{k}\) is smaller than 1.0. The orbits gradually change from petals to twisty heart shapes as \(\alpha_{k}\) increases. These interesting evolution processes of whirling orbits may come from two aspects of reasons. First, the variation of \(\alpha_{k}\) changes the resonant regions, then the rotating speed is within the resonant region at some cases while is not at the other cases. Second, the different level of anisotropy affects the weight of each harmonic component, resulting in the different shapes in orbits. Figure 6: The response amplitudes of the central node of the anisotropic rotor-bearings system with a 20-mm crack versus the rotating speed: (**a**) along the horizontal direction; (**b**) along the vertical direction. From the results shown above, it can be seen that the periodically time-variant characteristic caused by the anisotropy of bearings has significant influence on the vibration of the cracked rotor-bearing system. Then, its effects on the system's eigenvalues are discussed. The eigenvalues of periodically time-variant systems obtained by the [PERSON]'s method are often appeared in groups. If \(\lambda_{1}\) is one eigenvalue of the system, then \(\lambda_{1}=\pm 2h\Omega\), \(h=1,2,3,\cdots N_{h}\) will be also its eigenvalues, where \(N_{h}\) denotes the retained number of harmonics. If the eigenvalues are all illustrated in one figure, it will be disordered. Therefore, only the central eigenvalue \(\lambda_{1}\) is included to avoid the clusters of eigenvalues.