## 3 Governing Equations
### Dynamic Stability of Bearings Based on the Linear Model
\begin{bmatrix}\overline{S}_{XX}&\overline{S}_{ XY}\\ \overline{S}_{YX}&\overline{S}_{YY}\end{bmatrix}\begin{B}\frac{\vec{\Sigma}^{ \prime}}{\vec{Y}}\\ \overline{Y}^{\prime}\end{B} =0 \tag{22}\]  
In Equation (22), the parameter \(\overline{M}_{f}=M_{f}\big{(}2C_{m}^{3}\omega/\mu R^{3}L\big{)}\) represents the dimensionless mass of the rotor. By substituting the values and derivatives of the parameters \(\overline{X}^{\prime}\) and \(\overline{Y}^{\prime}\) from Equation (10) into Equation (22), the resulting expression can be derived as follows:  
\[ \begin{bmatrix}-\overline{M}_{f}\gamma^{2}+\overline{S}_{XX}+i\gamma \overline{B}_{XX}&\overline{S}_{XY}+i\gamma\overline{B}_{XY}\\ \overline{S}_{YX}+i\gamma\overline{B}_{YX}&-\overline{M}_{f}\gamma^{2}+ \overline{S}_{YY}+i\gamma\overline{B}_{YY}\end{bmatrix}\begin{B}\overline{X }^{\prime}\\ \overline{Y}^{\prime}\end{B} =0 \tag{23}\]  
A non-trivial solution of the Equation (23) is possible if the coefficient matrix is singular. Consequently, the characteristic equation of the linear dynamic model, under the assumption of limit cycle disturbances affecting the displacement of the rotor center at the instability threshold, is expressed as:  
\[ \begin{pmatrix}-\overline{M}_{f}\gamma^{2}+\overline{S}_{XX}+i\gamma\overline{ B}_{XX}\end{pmatrix}\begin{pmatrix}-\overline{M}_{f}\gamma^{2}+\overline{S}_{ YY}+i\gamma\overline{B}_{YY}\end{pmatrix} - \begin{pmatrix}\overline{S}_{XY}+i \gamma\overline{B}_{XY}\end{pmatrix}\begin{pmatrix}\overline{S}_{YX}+i\gamma \overline{B}_{YX}\end{pmatrix} =0 \tag{24}\]  
Separating the real and imaginary parts of Equation (24) and setting them equal to zero yields two distinct Equations, (25) and (26). By assuming an initial value for the parameter \(\gamma\) and calculating the dynamic coefficients corresponding to it from Equations (20) and (21), the necessary parameters for analyzing the subsequent equations are provided. \[\overline{M}_{f}\gamma^{2}=\frac{\big{(}\overline{S}_{XX}\overline{B}_{YY}+ \overline{S}_{YY}\overline{B}_{XX}-\overline{S}_{XY}\overline{B}_{YX}- \overline{S}_{YX}\overline{B}_{XY}\big{)}}{\big{(}\overline{B}_{XX}+\overline{ B}_{YY}\big{)}} \tag{25}\]  
\[\gamma^{2}=\frac{\big{[}\big{(}\overline{S}_{XX}-\overline{M}_{f}\gamma^{2} \big{)}\big{(}\overline{S}_{YY}-\overline{M}_{f}\gamma^{2}\big{)}-\overline{S }_{XY}\overline{S}_{YX}\big{]}}{\big{(}\overline{B}_{XX}\overline{B}_{YY}- \overline{B}_{XY}\overline{B}_{YX}\big{)}} \tag{26}\]  
The equations above serve as a criterion for determining the margin of dynamic instability of the system based on the dimensionless mass parameter of the rotor and the whirl frequency ratio of the rotor center \((\overline{M}_{f},\gamma)\). By determining the final value of the corrected whirl frequency ratio and completing the steps of the mentioned iteration process in the flowchart of Figure 8, the stability field of the investigated rotor-bearing system can be determined using the following equation, based on the linear dynamic model. \[\overline{M}_{C}=\frac{\big{(}\overline{S}_{XX}\overline{B}_{YY}+\overline{ S}_{YY}\overline{B}_{XX}-\overline{S}_{XY}\overline{B}_{YX}-\overline{S}_{YX} \overline{B}_{XY}\big{)}}{\big{[}\big{(}\overline{B}_{XX}+\overline{B}_{YY} \big{)}\gamma^{2}\big{]}} \tag{27}\]  
In Equation (27), \(\overline{M}_{C}\) represents the dimensionless critical mass parameter of the rotor, which positions the system at the threshold of instability. By starting with an estimated value for \(\gamma\), the critical mass parameter and the whirl frequency ratio at the dynamic instability threshold of the modeled rotating system can be determined through iterative calculations.