## 3 Governing Equations
### Dynamic Stability of Bearings Based on the Linear Model  
According to Figure 7, the dynamic stability of noncircular journal bearings in the linear model is primarily determined by the equivalent stiffness and damping coefficients. These coefficients, which characterize the lubricating film, are derived from solving the governing equations. Equation (17) describes the calculation of dynamic force components generated by the lubricant film due to the perturbation behavior of the rotor center  
\[ \begin{bmatrix}\overline{F}_{X}\\ \overline{F}_{Y}\end{bmatrix} =\frac{2C_{m}^{2}}{b/\omega R^{3}L} \begin{bmatrix} F_{X}\\ F_{Y}\end{bmatrix} =\sum_{n=1}^{N}\int_{-1}^{+1}\int_{\theta_{1}^{n}}^{\theta_{2} }\overline{P}_{n} \begin{bmatrix}cos\theta\\ sin\theta\end{bmatrix} d\theta d\overline{z} \tag{17}\]  
The vector representation of fluid film forces in a dynamic state can be expressed as the sum of steady-state equilibrium forces and unstable disturbance forces, as follows:  
\[ \begin{bmatrix}\overline{F}_{X}\\ \overline{F}_{Y}\end{bmatrix} = \begin{bmatrix}\overline{F}_{X0}\\ \overline{F}_{Y0}\end{bmatrix} + \begin{bmatrix}\Delta\overline{F}_{X}\\ \Delta\overline{F}_{Y}\end{bmatrix}  \tag{18}\]  
In Equation (18), \(\left(\Delta\overline{F}_{X},\Delta\overline{F}_{Y}\right)\) represent the components of the unsteady disturbance force affecting the rotor in the dynamic state along the coordinate axes. In the linear dynamic analysis model, these perturbed forces \(\left(\Delta\overline{F}_{X},\Delta\overline{F}_{Y}\right)\) are considered as the linear functions of the dynamic components of rotor displacement \(\left(\overline{X}^{\prime},\overline{Y}^{\prime}\right)\) and velocity \(\left(\dot{\overline{X}}^{\prime},\dot{\overline{Y}}^{\prime}\right)\). \[ \begin{bmatrix}\Delta\overline{F}_{X}\\ \Delta\overline{F}_{X}\end{bmatrix} =- \begin{bmatrix}\overline{S}_{XX}& \overline{S}_{XY}\\ \overline{S}_{YX}&\overline{S}_{YY}\end{bmatrix}\begin{Bmatrix}\overline{X}^{ \prime}\\ \overline{Y}^{\prime}\end{Bmatrix} - \begin{bmatrix}\overline{B}_{XX}&\overline{B} _{XY}\\ \overline{B}_{YX}&\overline{B}_{YY}\end{bmatrix}\begin{Bmatrix}\dot{\overline{X}} ^{\prime}\\ \dot{\overline{Y}}^{\prime}\end{Bmatrix}  \tag{19}\]  
In the above equation, the parameters \(\overline{S}_{ij}\) and \(\overline{B}_{ij}\) where \(i,j=X,Y\) represent the equivalent stiffness and damping coefficients of the lubricating film in the linear dynamic model. By referencing Equations (14), (17), and (19) and separating the variables, the dimensionless forms of the equivalent stiffness and damping components of the lubricating fluid film are derived based on the following relationships. \[ \begin{bmatrix}\overline{S}_{XX}&\overline{S}_{XY}\\ \overline{S}_{YX}&\overline{S}_{YY}\end{bmatrix} =\sum_{n=1}^{N}\int_{-1}^{+1} \int_{\theta_{1}^{n}}^{\theta_{2}^{n}} \begin{bmatrix}\overline{P}_{X}\\ \overline{P}_{Y}\end{bmatrix}\begin{Bmatrix}cos\theta&sin\theta\end{Bmatrix} d \theta d\overline{z} \tag{20}\]  
\[ \begin{bmatrix}\overline{B}_{XX}&\overline{B}_{XY}\\ \overline{B}_{YX}&\overline{B}_{YY}\end{bmatrix} =\sum_{n=1}^{N}\int_{-1}^{+1} \int_{\theta_{1}^{n}}^{\theta_{2}^{n}} \begin{bmatrix}\overline{P}_{X}\\ \overline{P}_{Y}\end{bmatrix}\begin{Bmatrix}cos\theta&sin\theta\end{Bmatrix} d \theta d\overline{z} \tag{21}\]  
Figure 7: Vibration modeling of the lubricating fluid film in a two-lobe bearing system. The equations describing the dynamic displacement of the rotor center supported by hydrodynamic two-lobe journal bearings during periodic rotational movement around the static equilibrium point along the coordinate axes \((X,Y)\) are expressed as follows:  
\[\overline{M}_{f} \begin{pmatrix}\frac{\vec{\Sigma}^{\prime}}{\vec{\Sigma}^{ \prime}}\\ \frac{\vec{\Sigma}^{\prime}}{\vec{Y}}\end{pmatrix} + \begin{bmatrix}\overline{B}_ {XX}&\overline{B}_{XY}\\ \overline{B}_{YX}&\overline{B}_{YY}\end{bmatrix}\begin{B}\frac{\vec{\Sigma}^{ \prime}}{\vec{Y}}\\ \overline{Y}^{\prime}\end{B} +