## 3 Stability Analysis
### Borderline and Stability Threshold  
This paper considers a rotor-bearing system, as shown in Figure 3.  
Taking into account the flexibility of the rotor and referring to the coordinates shown in Figure 3, the motion is expressed as [51; 52]  
\[\begin{cases}m\ddot{x}_{d}+k_{s}(x_{d}-x_{b})=me_{m}\Omega^{2}\mathrm{sin} \Omega t,\\ m\ddot{y}_{d}+k_{s}(y_{d}-y_{b})=me_{m}\Omega^{2}\mathrm{cos}\Omega t+W,\end{cases} \tag{20}\]  
Figure 3: Diagram of a rotor-bearing system and coordinates in the \(x\)-\(y\) plane.  
where \(m\) represents the mass of the rotor and disk; \(e_{m}\) denotes the eccentricity of the unbalance mass; \(k_{s}\) is the parameter used to describe the flexibility of the rotor; \([x_{b},y_{b}]\) and \([x_{d},y_{d}]\) represent the displacement components of the journal and disk center, respectively; and \(\{\bar{x}_{d},\bar{y}_{d}\}\) stands for the acceleration components of the disk center. In addition, combining with the linear bearing forces gives us [51]  
\[\begin{cases}k_{s}(x_{d}-x_{b})-2(F_{x0}+k_{xx}x_{b}+k_{xy}y_{b}+d_{xx}\dot{x}_ {b}+d_{xy}\dot{y}_{b})=0,\\ k_{s}(y_{d}-y_{b})-2(F_{y0}+k_{yx}x_{b}+k_{yy}y_{b}+d_{yx}\dot{x}_{b}+d_{yy} \dot{y}_{b})=0.\end{cases} \tag{21}\]  
The solutions of the homogeneous Equation (20) can be written in the form of  
\[x_{d}=X_{s}\mathrm{e}^{\lambda t},\quad y_{d}=Y_{s}\mathrm{e}^{\lambda t}, \quad x_{b}=X_{b}\mathrm{e}^{\lambda t},\quad y_{b}=Y_{b}\mathrm{e}^{\lambda t}. \tag{22}\]  
The characteristic equation for the system described by Equation (20) together with Equation (21) is then obtained by referring to Equation (22), as  
\[\alpha_{i}\lambda^{6}+\alpha_{5}\lambda^{5}+\alpha_{4}\lambda^{4}+\alpha_{3} \lambda^{3}+\alpha_{2}\lambda^{2}+\alpha_{1}\lambda+\alpha_{0}=0, \tag{23}\]  
in which \(\alpha_{i}(i=0,1,\ldots,6)\) are polynomials, as shown in Appendix A. The eigenvalue comprises the real part \(a\) and imaginary part \(\omega\) as  
\[\lambda=a\pm\mathrm{j}\omega. \tag{24}\]  
At stability boundary, \(\lambda\) is purely imaginary, i.e., \(\lambda=j\omega\) and \(a=0\). Thus, the rotation speed on the borderline denoted by \(\Omega_{b}\) is determined by [51]  
\[\Omega_{b}=\omega_{n}\sqrt{\frac{\beta_{1}\beta_{2}\beta_{3}^{2}}{(\beta_{1}^ {2}-\beta_{1}\beta_{3}\beta_{4}+\beta_{0}\beta_{3}^{2})(\beta_{1}+\
u\beta_{3} )}}, \tag{25}\]  
where \(\omega_{n}=\sqrt{k_{s}/m}\) stands for the natural frequency of the flexible shaft, \(\beta_{i}\) (\(i=0,1,\ldots,4\)) represent polynomials as shown in Appendix A, and \(\beta_{i}\) (\(i=0,1,\ldots,4\)) are functions of the dimensionless dynamic coefficients shown in Equation (13). Meanwhile, it is known from Section 2 that dynamic coefficients are functions of the eccentricity ratio \(\varepsilon\). It is therefore possible to conclude that \(\Omega_{b}\) is a function of \(\varepsilon\) by combining with Equation (25). Using this approach, it is possible to determine the borderline.  
Finally, the deterministic stability analysis procedure is summarized as Algorithm 1.  
```
1: Define the parameters of the rotor-bearing system for initialization;
2: Give an eccentricity ratio \(\varepsilon_{0}\) to determine the thickness distribution;
3: Solve Equation (16) to obtain the static pressure \(P_{0}\) and perturbation pressures \(\{P_{X},P_{Y},P_{X},P_{Y}\}\);
4: Obtain the dimensionless oil-film force \(F\) using Equation (19) to determine \(\Omega_{\mathrm{st}}\) as Equation (18);
5: Calculate the dimensionless dynamic coefficients using Equations (17) and (13) to determine \(\Omega_{b}\) based on Equation (25);
6: Develop the run-up curve and borderline by repeating steps (2-5) under different eccentricity ratios;
7: Obtain the stability threshold \(\Omega_{\mathrm{th}}\) from the intersection of the run-up curve and borderline.
```  
**Algorithm 1** Deterministic stability analysis of rotor-bearing system