## 3 Dynamic Force Coefficients of a Single-Oil Wedge Hydrodynamic Bearing  
According to the study of [PERSON] (1954), the coefficients of stiffness and damping of a hydrodynamic bearing can be expressed as  
\[K_{ij}=-\frac{\partial F_{i}}{\partial X_{j}}\ ;\ C_{ij}=-\frac{\partial F_{i}}{ \partial\dot{X}_{j}}i,\ j=X,\ Y. \tag{3}\]  
The first subscript represents the direction of the corresponding stiffness, and the second subscript represents the direction of displacement. The stiffness coefficients (\(K_{XX}\), \(K_{YY}\)) and (\(K_{XY}\), \(K_{XX}\)) are called the direct stiffness term and coupled stiffness term. Figure 5 shows the directions of the different stiffness and damping terms. The inertia or additional mass coefficient \(\{M_{ij}\}_{ij}=X\),\(Y\) can be defined as \(M_{ij}=-\frac{(\partial\dot{x}_{j})}{\left(\partial\dot{X}_{j}\right)};i,\ j=X,Y\). The term \(\left\{\ddot{X},\ddot{Y}\right\}\) is the acceleration of the journal center. Thus, the reactive force of the bearing can be written as  
\[\left[ \begin{array}{c}F_{X}(t)\\ F_{Y(t)}\end{array} \right]=\left[ \begin{array}{c}F_{X_{o}}\\ F_{Y_{o}}\end{array} \right]-\left[ \begin{array}{cc}K_{XX}&K_{XY}\\ K_{YX}&K_{YY}\end{array} \right]\left[ \begin{array}{c}\Delta X\\ \Delta Y\end{array} \right]-\left[ \begin{array}{cc}C_{XX}&C_{XY}\\ C_{YX}&C_{YY}\end{array} \right]\left[ \begin{array}{c}\Delta\dot{X}\\ \Delta\dot{Y}\end{array} \right], \tag{4}\]  
where \(F_{X_{0}}=F_{o}=1/2W\) and \(F_{Y_{0}}=0\). Then, the dynamic equation for the rotor-bearing system under a small vibration condition can be expressed as  
\[\left[ \begin{array}{cc}M&0\\ 0&M\end{array} \right]\left[ \begin{array}{c}\Delta\ddot{X}\\ \Delta\ddot{Y}\end{array} \right]+\left[ \begin{array}{cc}C_{XX}&C_{XY}\\ C_{YX}&C_{YY}\end{array} \right]\left[ \begin{array}{c}\Delta\dot{X}\\ \Delta\dot{Y}\end{array} \right]+\left[ \begin{array}{cc}K_{XX}&K_{XY}\\ K_{YX}&K_{YY}\end{array} \right]\left[ \begin{array}{c}\Delta X\\ \Delta Y\end{array} \right]=Mu\Omega^{2}\left[ \begin{array}{c}cos\Omega t\\ sin\Omega t\end{array} \right]. \tag{5}\]  
Figure 5: Directions of stiffness and damping of a hydrodynamic bearing. The dimensionless form factor can be written as  
\[k_{ij}=K_{ij}\frac{c}{F_{o}};c_{ij}=C_{ij}\frac{c\Omega}{F_{o}}i,\ j=X,\ Y; \tag{6}\]  
where \(F_{o}\) is the static load applied at the bearing along the \(X\)-direction. Then, the dimensionless stiffness and dimensionless damping can be expressed as  
\[k_{XX}=4 Ss^{2}\pi\left[\frac{\varepsilon_{o}}{\left(1-\varepsilon_{o}^{2} \right)^{2}}sin^{2}\phi_{0}+\frac{3\pi\varepsilon_{o}^{2}}{4\left(1- \varepsilon_{o}^{2}\right)^{2}\sqrt{1-\varepsilon_{o}^{2}}}cos\phi_{0}sin\phi _{0}+\frac{2\varepsilon_{o}\left(1+\varepsilon_{o}^{2}\right)}{\left(1- \varepsilon_{o}^{2}\right)^{3}}cos^{2}\phi_{0}\right] \tag{7}\]  
\[ \begin{array}{ll}k_{XY}&=4 Ss^{2}\pi\left[\frac{\pi\left(1+2\varepsilon_{o}^{2 }\right)}{4\left(1-\varepsilon_{o}^{2}\right)^{2}\sqrt{1-\varepsilon_{o}^{2} }}sin^{2}\phi_{0}+\frac{\varepsilon_{o}\left(1+3\varepsilon_{o}^{2}\right)}{4 \left(1-\varepsilon_{o}^{2}\right)^{3}}cos\phi_{0}sin\phi_{0}\right.\\ &\left.+\frac{\pi}{4\left(1-\varepsilon_{o}^{2}\right)^{3}}cos^{2}\phi_{0} \right]\end{array} \]  
\[ \begin{array}{ll}k_{YY}=4 Ss^{2}\pi\left[\frac{\varepsilon_{o}}{\left(1- \varepsilon_{o}^{2}\right)^{2}}cos^{2}\phi_{0}-\frac{3\pi\varepsilon_{o}^{2}} {4\left(1-\varepsilon_{o}^{2}\right)^{2}\sqrt{1-\varepsilon_{o}^{2}}}cos\phi _{0}sin\phi_{0}+\frac{2\varepsilon_{o}\left(1+\varepsilon_{o}^{2}\right)}{ \left(1-\varepsilon_{o}^{2}\right)^{3}}sin^{2}\phi_{0}\right]\end{array} \]  
\[ \begin{array}{ll}k_{YX}=4 Ss^{2}\pi\left[-\frac{\pi\left(1+2\varepsilon_{o}^{2 }\right)}{4\left(1-\varepsilon_{o}^{2}\right)^{2}\sqrt{1-\varepsilon_{o}^{2} }}cos^{2}\phi_{0}+\frac{\varepsilon_{o}\left(1+3\varepsilon_{o}^{2}\right)}{ 4\left(1-\varepsilon_{o}^{2}\right)^{3}}cos\phi_{0}sin\phi_{0}-\frac{\pi}{4 \left(1-\varepsilon_{o}^{2}\right)\sqrt{1-\varepsilon_{o}^{2}}}sin^{2}\phi_{0 }\right],\end{array}