## 6 Conclusions  
This paper has presented a stability analysis of a rotor-bearing system taking into consideration the misalignment effect and uncertainty in a finite-length HDB. Determination of the stability threshold is guided by the Routh-Hurwitz method. The gPC expansion is used to quantify the impact of uncertainty. The characteristics of the HDB used to determine the stability threshold are calculated by numerical solving of the governing equations. The accuracy of the results is verified by comparison with those published in previous papers. The misalignment effect is investigated by introducing two misaligned angles in the expression of thickness of the oil-film. The results show that misalignment has a remarkable effect on the characteristics of the HDB and the run-up curve. The impact of misalignment on the borderline is negligible for a small eccentricity ratio, but evident for a larger eccentricity ratio. Moreover, the stability threshold increases as the misaligned angles increase. To study the influence of uncertainties on the stability threshold, clearance and viscosity are assumed as random parameters with predefined distribution. Using the collocation method, the gPC expansion is constructed by solving the responses at a series of specific collocation points using the deterministic code. MC simulation is implemented as a validation. The results show that the stability threshold becomes uncertain since the random input parameters can affect the static and dynamic characteristics of the system. By comparing the evaluated statistical properties of random outputs, it is apparent that gPC expansion is able to approximate the random parameters with reasonable accuracy. The results also illustrate that gPC expansion is more efficient, since it requires far fewer sample realizations.  
In this study, the bearing force is considered linear and an analytical rotor is employed for the stability analysis. These are the main limitations of this study, which might limit its significance for direct industrial application. However, according to the previous studies, it is suggested that the developed framework is suitable to understand the performance of the rotor-bearing system. In addition, the interdisciplinary approach of combining stability analysis, misalignment effect, and uncertainty analysis using a gPC approach builds a bridge between these subjects. In the future, the nonlinear behavior of physical quantities and a complex rotor model will be simulated and investigated experimentally to close the gap between the purely numerical approach and a real physical model.  
Conceptualization, X.S.; methodology, X.S.; formal analysis, X.S.; writing--original draft preparation, X.S.; writing--review and editing, X.S., K.K.S. and S.M. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Figure 11: PDF of stability threshold by gPC expansion and MC simulation.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Acknowledgments:** [PERSON] is acknowledged for his suggestions regarding paper writing. The first author wishes to acknowledge the support received from the China Scholarship Council (File No. 201806450023).  
**Conflicts of Interest:** The authors declare no conflict of interest.