## 5 Rotordynamic Performance Analysis of GFPBs  
The rotordynamic performance analysis with respect to static and dynamic performances of the GFPBs are conducted. The structural model of the GFPBs is coupled with the in-house computation tool anchored in solving the transient Reynolds equation for an isothermal iso-viscous ideal gas and the perturbation method of GFPBs, which was established from [PERSON]'s work [29]. An unsteady compressible Reynolds equation for an isothermal and isoviscous ideal gas is expressed as  
\[\frac{\partial}{\partial X}\bigg{(}ph^{3}\frac{\partial p}{\partial X}\bigg{)} +\frac{\partial}{\partial X}\bigg{(}ph^{3}\frac{\partial p}{\partial X}\bigg{)} =6\mu R\Omega\frac{\partial(ph)}{\partial X}+12\mu\frac{\partial(ph)}{ \partial t} \tag{13}\]  
where \(\mu\) is the air viscosity, \(\Omega\) is the rotor speed, and \(t\) is the time; \(p\) and \(h\) are gas-film pressure and thickness, respectively. The fluid-film thickness for the GFPB is expressed in terms of the journal center positions and bearing radial clearance and deflection of the bump foil plus polymer layer (\(\zeta_{0}\)) as  
\[h=C_{r}+e_{X}cos\theta+e_{Y}sin\theta+\zeta_{0} \tag{14}\]  
where the deflection occurs because of the compliance of the bump foil plus polymer layer, i.e., \(\zeta_{0}=\big{(}p_{avg}-p_{a}\big{)}/k_{GFPB}^{*}\), \(p_{avg}\) is the average fluid-film pressure over the bearing length, \(p_{a}\) is the ambient pressure, and \(k_{GFPB}^{*}\) is the complex structural stiffness of the GFPB expressed as \(k_{GFPB}^{*}=k_{GFPB}(1+j\gamma_{GFPB})\), where \(\gamma_{GFPB}\) is the structural loss factor, and \(j\) is the imaginary unit. A common approach to consider dynamic effects using static stiffness and loss factors of bearing structures is to use complex stiffness. Note that \(\zeta_{0}\) is mechanical deflection of an equivalent spring of the bump foil plus the polymer layer caused by gas-film pressure distribution. \begin{table}
\begin{tabular}{c c c c c c} \hline  & \(\mathbf{C_{r}}\) & \(\mathbf{C_{1}}\) & \(\mathbf{C_{2}}\) & \(\mathbf{C_{3}}\) & \(\mathbf{R^{2}}\) \\ \hline \multirow{3}{*}{Prediction} & 230 \(\upmu\)m & \(1.25\times 10^{-6}\) & \(9.42\times 10^{-8}\) & \(6.53\times 10^{-2}\) & 0.99 \\  & 250 \(\upmu\)m & \(1.16\times 10^{-6}\) & \(4.02\times 10^{-7}\) & \(4.38\times 10^{-2}\) & 0.99 \\  & 270 \(\upmu\)m & \(1.08\times 10^{-6}\) & \(6.31\times 10^{-7}\) & \(2.47\times 10^{-2}\) & 0.99 \\ \hline \multirow{2}{*}{Test} & 200 \(\upmu\)m & \multirow{2}{*}{\(1.25\times 10^{-6}\)} & \multirow{2}{*}{\(3.48\times 10^{-5}\)} & \multirow{2}{*}{\(3.03\times 10^{-2}\)} & \multirow{2}{*}{0.98} \\  & (designed) & & & & \\ \hline \multicolumn{5}{c}{Curve Fitting Function: _f(y)_ = \(C_{1}y^{3}+C_{2}y^{2}+C_{3}y\)} \\ _f(y)_: Static Load in [N], _y_: Displacement in [\(\upmu\)m], \(\mathbf{R^{2}}\): Correlation Coefficient \\ \hline \end{tabular}
\end{table} 
Table 2: Curve fitting functions of the static load versus the displacement from the prediction and test results. The structural loss factor of the GFPB can be easily estimated from a series combination relation of the bump foil and polymer layer. Thus, the structural loss factor of the GFPB is defined as  
\[\frac{1}{\gamma_{GFPB}}=\frac{k_{GFPB}}{k_{p}\gamma_{p}}+\frac{k_{GFPB}}{k_{b} \gamma_{b}} \tag{15}\]  
where \(\gamma_{p}\) and \(\gamma_{b}\) denote the loss factor of the polymer layer and bump foil, respectively. Generally, the structural loss factor of the bump foil is estimated as a constant value from 0.05 to 0.2 [29]. In this study, a value of 0.1 is used for the foil structures. Meanwhile, the polymer layer is assumed to have a loss factor of 0.49 [22], which is much higher than the bump foil has. Figure 6 displays how the structural stiffness and loss factor of the GFPB, which constitute the complex structural stiffness \(k_{GFPB}^{*}\), vary depending on thickness of the polymer layer. It is assumed that the polymer layer has \(r_{in}\) = 20.92 mm, and the outer radius is varied to a value corresponding to analysis thickness.