## 5 Numerical Study
### Deterministic Analysis and Validation
#### 5.1.3 Stability Analysis  
Next, the stability threshold is determined using the new implementation verified above. The parameters of the rotor-bearing system are adopted from Table 1. The misaligned angles are set as \(\alpha=0^{\circ},\gamma=0.004^{\circ}\). Figure 6 shows the run-up curve and borderline calculated for both misaligned and aligned situations. These curves are used to determine the stability threshold of the rotor-bearing system. The run-up curves in both situations show that the eccentricity ratio \(\varepsilon\) decreases as the rotation speed \(\Omega\) increases. This implies that the journal center \(O_{j}\) moves towards the bearing center \(O_{b}\) (Figure 1) as the rotation speed increases.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Description** & **Value** & **Description** & **Value** \\ \hline \hline \multirow{2}{*}{Bearing radius (m)} & 0.03 & Viscosity (\(\mathrm{Ns/m^{2}}\)) & \(9\times 10^{-3}\) \\ \multirow{2}{*}{Bearing length (m)} & 0.066 & rotation speed (RPM) & 3000 \\ \multirow{2}{*}{Clearance (m)} & 0.03 \(\times\) 10\({}^{-3}\) & Eccentricity ratio & 0.8 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameters of HDB provided by [PERSON] and [PERSON] [20].  
Figure 5: Dimensionless stiffness and damping coefficients and comparison with references [45; 46].  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Description** & \multicolumn{2}{c}{\(\alpha=\mathbf{\Phi_{0}=0^{\circ}}\)} & \multicolumn{2}{c}{\(\alpha=\mathbf{\Phi_{0}=90^{\circ}}\)} \\  & \(\gamma\)/\(\circ\) & **0** & **0.004** & **0.007** & **0** & **0.01** & **0.02** \\ \hline \multirow{2}{*}{Pmax (Mpa)} & [PERSON] and [PERSON] [20] & 33.06 & 39.6 & 63.58 & 33.06 & 32.95 & 34.95 \\  & Present study & 32.73 & 39.29 & 62.2 & 32.78 & 29.41 & 30.48 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Maximum pressure taking into consideration the misalignment effect.  
In the case of the borderline, the rotation speed increases as the eccentricity ratio increases. Moreover, it is observed that the gradient \((d\Omega/d\varepsilon)\) increases with the increment of the eccentricity ratio.  
By comparing the results from the misaligned and aligned conditions, it is observed that misalignment has an obvious influence on the run-up curve. The impact of misalignment varies over different parts of the borderline. It is observed that the impact is negligible when the eccentricity ratio remains small (approximately \(\varepsilon<0.4\)), whereas it is evident for larger eccentricity ratios (approximately \(0.4<\varepsilon<0.7\)). Furthermore, the intersections between the respective run-up curves and borderlines, i.e., the stability thresholds, are 212.51 rad/s and 212.38 rad/s under misaligned and aligned situations, respectively. To investigate the influence of the misaligned angle on the threshold, the thresholds are calculated under various misaligned angles, as listed in Table 6. It is found that the thresholds in the misaligned situation are greater than those in the aligned situation. In addition, the threshold increases as the misaligned angle increases. This implies that the misalignment improves the stability of the rotor-bearing system, which results in the same conclusion as that given by [PERSON] [46].