## 2 Governing Equations and Dynamic Coefficients
### Dynamic Coefficients  
According to [PERSON] [45], the bearing forces are described by eight dynamic coefficients referring to the displacement and velocity. To calculate the dynamic coefficients, the bearing forces are characterized by reserving the first-order terms of a Taylor series expansion around the equilibrium position, i.e., [8; 50]  
\[ \begin{cases}F_{x}=F_{x0}+\left(\frac{\partial F_{x}}{\partial x}\right)_{0}x+ \left(\frac{\partial F_{x}}{\partial y}\right)_{0}y+\left(\frac{\partial F_{x }}{\partial\dot{x}}\right)_{0}\dot{x}+\left(\frac{\partial F_{x}}{\partial y} \right)_{0}\dot{y},\\ F_{y}=F_{y0}+\left(\frac{\partial F_{y}}{\partial x}\right)_{0}x+\left(\frac{ \partial F_{y}}{\partial y}\right)_{0}y+\left(\frac{\partial F_{y}}{\partial \dot{x}}\right)_{0}\dot{x}+\left(\frac{\partial F_{y}}{\partial\dot{y}}\right) _{0}\dot{y},\end{cases}  \tag{11}\]  
in which \(F_{x0}\) and \(F_{y0}\) represent the bearing forces at the equilibrium position. The coefficients of displacement and velocity components are defined as dynamic coefficients in the form of  
\[ \begin{cases}k_{xx}=\left(\frac{\partial F_{x}}{\partial x}\right),\;k_{xy}= \left(\frac{\partial F_{x}}{\partial y}\right),\;k_{yx}=\left(\frac{\partial F _{y}}{\partial x}\right),\;k_{yy}=\left(\frac{\partial F_{y}}{\partial y} \right),\\ d_{xx}=\left(\frac{\partial F_{x}}{\partial\dot{x}}\right),\;d_{xy}=\left( \frac{\partial F_{x}}{\partial\dot{y}}\right),\;d_{yx}=\left(\frac{\partial F _{y}}{\partial\dot{x}}\right),\;d_{yy}=\left(\frac{\partial F_{y}}{\partial \dot{y}}\right),\text{respectively}.\end{cases}  \tag{12}\]  
Conventionally, the following dimensionless form is adopted in the stability analysis  
\[ \begin{cases}[K_{xx}\quad K_{xy}\quad K_{yx}\quad K_{yy}]=\frac{c}{W}[k_{xx} \quad k_{xy}\quad k_{yx}\quad k_{yy}],\\ [D_{xx}\quad D_{xy}\quad D_{yx}\quad D_{yy}]=\frac{c\Omega}{W}[d_{xx}\quad d_ {xy}\quad d_{yx}\quad d_{yy}].\end{cases}  \tag{13}\]  
Similar to the bearing forces, the pressure of the oil-film under dynamic conditions is expanded as  
\[P=P_{0}+\Delta P=P_{0}+P_{X}X+P_{Y}Y+P_{\dot{X}}\dot{X}+P_{\dot{Y}}\dot{Y}, \tag{14}\]  
where \(P_{0}\) represents the pressure at the equilibrium position; \(\Delta P\) indicates the increment of pressure caused by displacement and velocity under dynamic conditions; and \(P_{X}\), \(P_{Y}\), \(P_{\dot{X}}\), and \(P_{\dot{Y}}\) are the corresponding perturbation pressures with respect to the displacement and velocity vectors \(\{X,Y,\dot{X},\dot{Y}\}\). Further, the thickness of the oil-film is written as  
\[H=H_{0}+\Delta H=H_{0}+\dot{Y}\text{cos}\Phi+\dot{X}\text{sin}\Phi. \tag{15}\]  
By substituting Equations (14) and (15) into the Reynolds equation, five dimensionless equations are obtained as  
\[ \begin{split}&\frac{\partial}{\partial\Phi}\left(H^{3}\frac{ \partial P_{s}}{\partial\Phi}\right)+\frac{\partial}{\partial Z}\left(H^{3} \frac{\partial P_{s}}{\partial Z}\right)\\ &=\begin{cases}\frac{\partial H}{\partial\Phi}&\text{for}\;P_{s}=P_{0 },\\ \text{cos}\Phi-3\frac{\partial}{\partial\Phi}\left(H^{2}\text{sin}\Phi\frac{ \partial P_{0}}{\partial\Phi}\right)-3\frac{\partial}{\partial Z}\left(H^{2} \text{sin}\Phi\frac{\partial P_{0}}{\partial Z}\right)&\text{for}\;P_{s}=P_{X}, \\ &-\text{sin}\Phi-3\frac{\partial}{\partial\Phi}\left(H^{2}\text{cos}\Phi\frac{ \partial P_{0}}{\partial\Phi}\right)-3\frac{\partial}{\partial Z}\left(H^{2} \text{cos}\[PERSON]\frac{\partial P_{0}}{\partial Z}\right)&\text{for}\;P_{s}=P_{Y}, \\ &2\text{sin}\[PERSON]&\text{for}\;P_{s}=P_{\dot{X}},\text{ and}\\ &2\text{cos}\[PERSON]&\text{for}\;P_{s}=P_{\dot{Y}}.\end{cases}\end{split}  \tag{16}\]  
The dynamic coefficients are estimated by integrating the perturbation pressures as\[ \begin{bmatrix}k_{xx}&k_{yx}\\ k_{xy}&k_{yy}\\ d_{xx}&d_{yx}\\ d_{xy}&d_{yy}\end{bmatrix} =-C_{b}\int\limits_{-\frac{1}{2R}}^{\frac{L}{2R}}\int_{0 }^{2\pi}