## 3 Structural Deflection Model of GFPBs  
A structural deflection model for GFPBs is established by coupling bump-foil elasticity with the structural elasticity model of polymer layers. Figure 3 presents the coordinate systems for the deflection model with equivalent springs of GFPBs. The origin of rectangular coordinate system \(X\), \(Y\) is located at the center of the bearing, and the circumferential coordinate system \(\theta\) is defined as counterclockwise from the negative \(X\)-axis. Each single bump foil and polymer layer is modeled as circumferentially distributed equivalent springs. The springs correspond to the number of single bump foils (\(N_{b}\)), which are circumferentially located at the peak of the bump arc; thus, they are located at \(\theta_{i}\), where \(i=1-N_{b}\). Considering the force equilibrium of the applied static loads (\(F_{X,Y}\)) on the journal and sum of the reaction forces of the equivalent springs (\(F_{X,R}\), \(F_{Y,R}\)) in the \(X\)- and \(Y\)-directions leads to  
\[F_{X}-F_{X,R}=0,\ F_{Y}-F_{Y,R}=0 \tag{9}\]  
where the sum of the linear reaction force is calculated from the summation of the reaction forces of each spring (\(F_{X,i}\), \(F_{Y,i}\)) in each direction, i.e., \(F_{X,R}=\sum_{i=1}^{N_{b}}F_{X,i}\) and \(F_{Y,R}=\sum_{i=1}^{N_{b}}F_{Y,i}\). The reaction force of each spring is determined using its stiffness coefficient and deformation. The spring deformation is determined from the gap between the journal and bearing surface (\(\tau\)) at the spring location \(\theta_{i}\), as follows:  
\[\tau(\theta_{i})=C_{r}+e_{X}cos\theta_{i}+e_{Y}sin\theta_{i} \tag{10}\]  
where \(C_{r}\) is the bearing radial clearance, and \(e_{X}\) and \(e_{Y}\) are the journal center positions in the \(X\)- and \(Y\)-directions, respectively. Note that when the gap becomes negative at a certain angle of the spring location, the springs at the location are engaged to generate reaction forces, whereas the springs located at the gap with positive values do not induce the reaction forces. If the gap is negative, i.e., the shaft and bearing surface come into contact, the deformation of the spring at \(\theta_{i}(\delta_{i})\) is expressed as  
\[\delta_{i}=-\tau(\theta_{i})\ if\ \tau(\theta_{i})<0. \tag{11}\]  
Thus, the reaction force of each spring is  
\[F_{i}=K_{GFPB}\delta_{i}\ if\ \tau(\theta_{i})<0. \tag{12}\]  
where the reaction force becomes zero at \(\theta_{i}\) if \(\tau(\theta_{i})>0\), and \(K_{GFPB}\) is the equivalent structural stiffness coefficient of the GFPB, i.e., \(K_{GFPB}=K_{GFPB}\times\) unit area (=bump pitch \(\times\) bearing length). The reaction force components in the \(X\)- and \(Y\)-directions are \(F_{X,i}=-F_{i}cos\theta_{i}\) and \(F_{Y,i}=-F_{i}sin\theta_{i}\), respectively. The \(K_{GFPB}\) is estimated from a series combination of the bump and polymer springs; it is expressed as \(1/k_{GFPB}=1/k_{b}+1/k_{p}\). The stiffness coefficient per unit area of the equivalent bump foil springs (\(k_{b}\)) is determined from [PERSON]'s formula [9] under free-free end condition, whereas the coefficient of the equivalent polymer springs (\(k_{p}\)) is calculated from the structural elasticity model of the polymer layer, Equation (6). Note that this simple concept that bump foil and polymer layer springs are connected serially allows us to understand complex structural mechanism of GFPBs easily. On the other hand, [PERSON] et al. [24] modeled the polymer layer in the gas bearing as complex rheological  
Figure 3: Coordinate systems for the structural deflection model with equivalent springs of the GFPB. elements with the discrete springs and dampers to consider viscoelastic behavior of the polymer layer. The computational procedure is as follows. First, the gap is calculated at each angular position of the spring location using the current journal center position, and then the reaction forces of each spring are determined. Second, the force equilibrium in Equation (7) provides the updated shaft center position by using the Newton-Raphson method.