## 4 Conclusions  
A recently developed time domain technique has been applied to an industrial turbine engine fitted with squeeze film dampers to study the nonlinear dynamic characteristics of the rotor system and the influence of static eccentric SFDs. The conclusions are listed below:  
1. The periodicity of the motion of the rotor system is prone to be affected by the static sprung eccentricity. Without the sprung eccentricity, the rotor would execute periodic orbital motion under different unbalanced configurations. However, the jump phenomenon can be observed when \(e_{4}\geq 0.7\) mm. With sprung eccentricity, obvious bifurcation, which indicates non-periodic motion, can be observed from the bifurcation diagram.
2. Due to the nonlinearity of the SFDs, the rotational speed at which the peak response occurs can be different from the critical speeds identified from the Campbell diagram. The peak response speed range would shift under different unbalanced configurations.
3. The operating deflection shape of the rotor system at a continuous operating speed of 3663 rad/s is strongly influenced by the amount and the axial position of the unbalance. This indicates that the response of the rotor system is much more sensitive to the unbalance of compressor II than compressor IV.
4. The static sprung eccentricity would lead to the occurrence of the 0X and 2X frequency components in the waterfall diagram. The amplitude of the 1X component increases as the amount of balance increases, while that of the 0X component decreases. The larger the static sprung eccentricity, the larger the amplitude of 0X, 1X and 2X.
5. The further the distance between the components of the rotor and the static eccentric SFD, the less affected the components are by the static eccentricity of the SFD.
6. Due to the static sprung eccentricity, the center for the orbit of support I would be offset from the center of the SFD and thus make the SFD inefficient/ineffective or even lead to the occurrence of rub impact.  
Figure 19: Results obtained with method in this work: (**a**) orbit of the inner rotor disk \(-\omega=188\) rad/s; (**b**) orbit of the outer rotor disk \(-\omega=188\) rad/s; (**c**) orbit of the inner rotor disk \(-\omega\)= 230 rad/s; (**d**) orbit of the outer rotor disk \(-\omega\)= 230 rad/s.  
Therefore, we propose to improve the structure of the spring butt-bearing seat and add a centering surface between them. The thickness of the spokes of the cage in the vertical direction can also be optimized to reduce the influence of the rotor's own gravity.  
Conceptualization, S.Z. and F.W.; methodology, S.L.; software, F.W.; validation, F.W., X.M. and Y.L.; formal analysis, S.L.; investigation, S.Z. and F.W.; resources, F.W.; data curation, X.L.; writing--original draft preparation, S.L.; writing--review and editing, S.Z. and S.L.; visualization, X.L.; project administration, X.M. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Jiangxi Provincial Natural Science Foundation grant number 20202 BAB211005 and the Aero Science Foundation of China grant number 201928056001.  
Not applicable.  
The data presented in this study are available on request from the corresponding author. Data are unavailable due to privacy.  
The authors declare no conflict of interest.