## 3 Results
### Regression  
Although it is possible to obtain the direct solution of the time-dependent problem in the model described above, the process is significantly time consuming because the air flow problem must be solved at every step. Conversely, the air-bearing forces and moments can be calculated using the interpolated stiffness and damping coefficients. In this study, each dynamic coefficient for each combination of \(\varepsilon_{x}\), \(\varepsilon_{y}\), \(\hat{t}_{x}\), and \(\hat{t}_{y}\) was first fitted with a quadratic polynomial of \(t_{x}\) and \(t_{y}\), and then each coefficient of the polynomial was fitted with a linear function of \(\varepsilon_{x}\), \(\varepsilon_{y}\), \(\hat{t}_{x}\), and \(\hat{t}_{y}\). The accuracy of the fitting is expressed in the form of determination coefficients, as presented in Table 2. The accuracy is relatively good, although it can be improved for \(\widetilde{K}_{tx,ty}\), \(\widetilde{D}_{tx,tx}\), \(\widetilde{D}_{tx,ty}\), and \(\widetilde{D}_{ty,tx}\) by providing more points and increasing the fitting order in \(\varepsilon_{x}\) and \(\varepsilon_{y}\) degrees of freedom. The fitting results are presented in Tables 3-10. An example of the table interpretation is given below for \(K_{tx,tx}\):  
\[\widetilde{K}_{tx,tx}=\left(1.09533+1.49129\varepsilon_{x}+0.177 19\varepsilon_{y}-0.02161\hat{T}_{x}\right)+\] \[\left(-0.30745-3.94666\varepsilon_{x}+0.10915\hat{T}_{x}+0.00478 \hat{T}_{y}\right)\] \[\left(-0.15003+0.09324\varepsilon_{x}-1.16823\varepsilon_{y}+0.0 1582\hat{T}_{x}+0.0114\hat{T}_{y}\right)\] \[\left(-0.3232+2.45267\varepsilon_{x}+2.42515\varepsilon_{y}+0.16 265\hat{T}_{x}-0.28434\hat{T}_{y}\right)\] \[\left(7.99445\varepsilon_{x}-0.26854\varepsilon_{y}-0.35236\hat{T }_{x}+0.14364\hat{T}_{y}\right)\] \[\left(0.34256+0.41188\varepsilon_{x}+1.09915\varepsilon_{y}-0.1 3117\hat{T}_{x}+0.17535\hat{T}_{y}\right)\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **1** & \(\varepsilon_{x}\) & \(\varepsilon_{y}\) & \(\hat{T}_{x}\) & \(\hat{T}_{y}\) \\ \hline
1 & 0.10973 & 0 & 0.51941 & \(-0.00802\) & \(-0.02162\) \\ \(T_{x}\) & \(-0.13722\) & \(-0.006\) & \(-0.80745\) & 0 & \(-0.02419\) \\ \(T_{y}\) & 0 & \(-1.68577\) & \(-0.42286\) & 0.08025 & 0.193 \\ \(T_{x}T_{y}\) & 0.22904 & 3.62763 & \(-2.10164\) & 0 & 0.74463 \\ \(T_{x}^{2}\) & \(-0.1213\) & 1.02097 & 1.42966 & 0.06678 & \(-0.15351\) \\ \(T_{y}^{2}\) & 0.50093 & 3.62374 & \(-2.3118\) & 0 & \(-0.34029\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coefficients of determination for dimensionless dynamic coefficient fitting. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Stiffness (kN-m/rad)** & & & **Damping (kN-m-s/rad)** \\  & \(\widetilde{K}_{tx,tx}\) & \(\widetilde{K}_{tx,ty}\) & \(\widetilde{K}_{ty,tx}\) & \(\widetilde{K}_{ty,ty}\) & \(\widetilde{D}_{tx,tx}\) & \(\widetilde{D}_{tx,ty}\) & \(\widetilde{D}_{ty,tx}\) & \(\widetilde{D}_{ty,tx}\) \\ \hline R\({}^{2}\) & 0.99 & 0.77 & 0.97 & 0.96 & 0.89 & 0.83 & 0.91 & 0.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coefficients of determination for dimensionless dynamic coefficient fitting. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(\mathbf{1}\) & \(\varepsilon_{x}\) & \(\varepsilon_{y}\) & \(\hat{T}_{x}\) & \(\hat{T}_{y}\) \\ \hline \(1\) & \(0.0097\) & \(0.00358\) & \(0\) & \(0.00156\) & \(0\) \\ \(T_{x}\) & \(-0.02028\) & \(-0.05093\) & \(0\) & \(-0.00117\) & \(0\) \\ \(T_{y}\) & \(-0.00168\) & \(-0.00288\) & \(-0.01683\) & \(0.00023\) & \(0\) \\ \(T_{x}T_{y}\) & \(-0.01595\) & \(-0.04394\) & \(0.09286\) & \(0.00158\) & \(0\) \\ \(T_{x}^{2}\) & \(0.03404\) & \(0.19438\) & \(0\) & \(-0.00225\) & \(0\) \\ \(t_{y}^{2}\) & \(0.01708\) & \(0.04876\) & \(-0.01605\) & \(-0.0015\) & \(0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Fitting results for \(\widehat{D}_{tx,ty}\).