## 4 Numerical Results
### Effects of the Speed-Dependent Characteristic of Journal Bearings  
The speed-dependent characteristic is a widely existed phenomenon in journal bearings, which is caused by the complex fluid dynamic behaviors of lubricating oil. Nevertheless, this characteristic has not been paid enough attention in the dynamic analysis of cracked rotors. In previous sections, the stiffness and damping coefficients are all assumed to be fixed. In this section, the effects of the speed-dependent characteristic on the vibration of the system are discussed. Figure 9 illustrates the variation of the stiffness and damping coefficients of a journal bearing. The tilting-pad type bearings are used, and the stiffness and damping coefficients are obtained by the MADYN2000 software. The type of lubrication oil is VG46, and the load is set to be 1125 N during the simulations. It can be seen that the stiffness and damping coefficients change remarkably versus rotating speed. A fixed case is also employed, where \(k_{yy}=2\times 10^{8}\,\mathrm{N/m}\), \(c_{xx}=c_{yy}=1\times 10^{3}\,\mathrm{Ns/m}\) and \(c_{xy}=c_{yx}=0\). The coefficients \(k_{yy}\) and \(c_{xx}\) of the fixed case are selected according to the mean values of the speed-dependent ones in the speed range [0, 10,000], where \(k_{xx}\) and \(k_{xy}\) are determined by \(\alpha_{k}\) and \(\beta_{k}\). Figure 8: The Campbell diagram of the rotor-bearings system with a 20-mm crack. \(\blacktriangledown\) and \(\blacklozenge\) denote the results of \(\alpha_{k}=1.0\), \(\beta_{k}=0\) and \(\alpha_{k}=0.5\), \(\beta_{k}=0.2\), respectively. Figure 9: The speed-dependent coefficients of a journal bearings versus the rotating speed: (**a**) stiffness coefficients, (**b**) damping coefficients. Figure 10 shows the response amplitudes of the cracked rotor-bearing system with fixed and speed-dependent coefficients. The depth of the crack and the unbalance force are same as those in previous section. The parameter \(\alpha_{k}\) of both cases equals 1.0, and the parameter \(\beta_{k}\) of the fixed case is 0.2. As can be seen in the figures, although the stiffness and damping coefficients of the fixed and speed-dependent cases are obviously different, the amplitudes of the two cases agree well in the speed ranges far from the resonant regions, such as in the ranges with the speed smaller than the 1/2 critical speed. Nevertheless, the differences in the amplitudes of the two cases become quite obvious when the speed is close to the resonant regions. Of course, the difference in the stiffness and damping coefficients has some impacts, whereas the parametric excitation effect caused by the anisotropy of bearings plays a more important role. For the speeds close to the resonant regions, the parametric excitation effect is much stronger than those far away from the resonant regions, so that the dynamic responses are more sensitive to the changes of the system's parameters. The abovementioned characteristics will also be reflected in the whirling orbits. Figure 11 shows the orbits of the central node of the fixed and the speed-dependent cases at ten typical rotating speeds. It can be seen in the figures that the evolution characteristics of the orbits are similar to the variation of response amplitudes. At low rotating speeds, the orbits of the two cases are very similar in shape and size, as the first three figures show. When the rotating speed is within the resonant region of the 1/2 critical speed, such as the speed of 3800, 4000, 4125 and 4300 rpm, the orbits become obviously different, not only in the shape but also in the size. After passing through the resonant region, the orbits are both gradually turning into circular shape, until the speed reaches the resonant region of the 1\(\times\) critical speed, at which the orbits become elliptical shapes. Figure 10: The response amplitudes of the central node of the cracked anisotropic rotor-bearing system with fixed and speed-dependent coefficients: (**a**) along the horizontal direction; (**b**) along the vertical direction.