## 3 Numerical Results and Discussion
### The Influence of Sprung Eccentricity  
In order to evaluate the influence of the static sprung eccentricity, the nonlinear dynamic characteristics of the rotor system under different static sprung eccentricities were computed and analyzed. The parameters for computation were the same as described in Section 3.3. The results are presented in Figures 8-15. In these figures, \(e_{xx}=x_{01}/c_{1}\) is the normalized static sprung eccentricity for support I, in which \(x_{01}\) is the static offset in the \(x\) direction for support I and \(c_{1}\) represents the radial clearance of the SFD for support I. The following conclusions can be reached from Figures 8-15:  
1. The periodicity of the motion of the rotor system is prone to be affected by the static sprung eccentricity, as shown in Figure 8. The bifurcation diagrams shown in Figure 8a,c,e,g suggest that for the same unbalanced configuration, quasi-periodic and chaotic motion are prone to occur at larger static sprung eccentricity. For the rotor system studied in this work, the operating speed regime within which the periodicity is strongly affected is 2500-3800 rad/s. Furthermore, by comparing Figure 8a-h, it can be seen that a larger unbalance would be beneficial in suppressing the non-periodic motion of the rotor system. Figure 8e shows an obvious bifurcation at 2800 rad/s, while Figure 8f indicates that the rotor system executes periodic motion in the whole  
Figure 7: Horizontal response of support I in 1000–4500 rad/s under different unbalanced configuration: (**a**) \(e_{2}=0.1-1.0\) mm; (**b**) \(e_{4}=0.1-1.0\) mm. operating speed range. The same conclusion can also be reached by comparing Figure 8g,h. 2. The static sprung eccentricity would lead to the occurrence of the 0X and 2X frequency components in the waterfall diagram, as shown in Figure 9. This phenomenon has also been reported in Reference 4. Comparing Figure 5, Figure 9, it can be seen that the larger the static sprung eccentricity, the larger the amplitude of 0X, 1X and 2X. However, Figure 9g,h indicate that a larger unbalance would suppress the 0X component. Figure 9b,d,f,h also suggest that the 0X component would fluctuate under larger static sprung eccentricity. 3. The amplitude of the 1X component increases with an increasing amount of balance while that of the 0X component decreases, as shown in Figures 10-13. On the one hand, Figures 10a, 11a, 12a and 13a show that the amplitude of the 0X component decreases with an increasing amount of balance; on the other hand, by comparing these figures, it can be seen that the larger the static sprung eccentricity, the larger the amplitude of the 0X component. In Figure 10a, the maximum amplitude of the 0X component with \(e_{2}=0.1\)mm and \(e_{sX}=0.1\) is approximately 1.4 \(\times\) 10\({}^{-5}\) m; in Figure 12a, the maximum amplitude of the 0X component with \(e_{2}=0.1\) mm and \(e_{sX}=0  
Figure 9: _Cont._  
Figure 10: Horizontal response of support I in 1000–4500 rad/s with \(e_{2}=0.1-1.0\) mm, \(e_{\rm{s}x}=0.1\): (a) OX; (b) IX. Figure 11: Horizontal response of support I in \(1000-4500\) rad/s with \(e_{2}=0.1-1.0\) mm, \(e_{\rm{xx}}=0.2\): (a) 0X; (b) 1X. Figure 12: _Cont._  
Figure 14a indicates that the further the distance between the components of the rotor and the static eccentric SFD, the smaller the amplitude of the 0X frequency component. From Figure 14a, it can be seen that the amplitude of the 0X component for support II is the smallest, while that of support I is the largest. Figures 6a and 14b show that the operating deflection shape around 3665 rad/s coincides with the second mode shape shown in Figure 3b with and without static sprung eccentricity. Figure 12: Horizontal response of support I in 1000\(-\)4500 rad/s with \(e_{2}=0.1-1.0\) mm, \(e_{\mathrm{s}x}=0.3\): (a) 0X; (b) 1X. Figure 13: Horizontal response of support I in 1000\(-\)4500 rad/s with \(e_{2}=0.1-1.0\) mm, \(e_{\mathrm{s}x}=0.35\): (a) 0X; (b) 1X.