## 4 Results and Discussion
(**a**) \(K_{xx}\) vs. \(\Omega\); (**b**) \(K_{xy}\) vs. \(\Omega\); (**c**) \(K_{yx}\) vs. \(\Omega\); (**d**) \(K_{yy}\) vs. \(\Omega\). To investigate the effect of eccentricity ratio on dynamical coefficients of Boltzmann model for \(\Omega=2\), \(\Lambda=3\) and \(B/D=0.5\), comparisons of stiffness and damping coefficients at different values of Knudsen number from \(K_{n}=0.065\) to \(K_{n}=2.1667\) are shown in Figures 11 and 12. It can be observed that the dynamic stiffness coefficients rise markedly with increasing eccentricity ratio. The reason is that the larger eccentricity ratio \(\varepsilon\) yields the smaller gas film thickness \(H\) of gas-lubricated microbeearings, which leads to the increase in the Poiseuille flow rate. For small Knudsen number flows \(K_{n}<0.325\), the damping coefficients \(D_{xy}\), \(D_{yx}\) and \(D_{yy}\) decrease while \(D_{xx}\) remains nearly constant as the eccentricity ratio increases. The increase in Knudsen number, which indicates that the rarefaction effect tends to be stronger. With the exception of \(D_{yx}\), the dynamic coefficients gradually increase when \(K_{n}>0.325\). So the increase of direct damping coefficients result in a more stable state of the microbearing system. Figure 4:0 _Cont._  
Figure 4:1 Effect of perturbation frequency \(\Omega\) on dynamic damping coefficients of gas microbeearings. (**a**) \(D_{xx}\) vs. \(\Omega\); (**b**) \(D_{xy}\) vs. \(\Omega\); (**c**) \(D_{yx}\) vs. \(\Omega\); (**d**) \(D_{yy}\) vs. \(\Omega\). Figure 4:2 Dynamic damping coefficients of Boltzmann model plotted as function of eccentricity ratio for various Knudsen numbers. (**a**) \(D_{xx}\) vs. \(\varepsilon\); (**b**) \(D_{yy}\) vs. \(\varepsilon\); (**c**) \(D_{yx}\) vs. \(\varepsilon\); (**d**) \(D_{yy}\) vs. \(\varepsilon\). Figures 13 and 14 show the relation between the dynamic coefficients and the aspect ratio \(B\)/\(D\) for various Knudsen numbers with \(\Omega\) = 3, \(\varepsilon\) = 0.6 and \(\Lambda\) = 5. It is found that all the stiffness coefficients increase significantly with the increase of \(B\)/\(D\). The variations of damping coefficients is more complicated. The reason for this is the ultra-thin gas film is stiffened at higher aspect ratio and can reduce the dominant energy dissipation in such narrow spacing. For \(K_{n}\) < 0.325, the dimensionless damping coefficients \(D_{xx}\), \(D_{xy}\) and \(D_{yy}\) first increase with increasing aspect ratio, reach a maximum and then begin to fall, while \(D_{yx}\) first decreases and then increases with increasing \(B\)/\(D\). As the gaseous rarefaction effect is enhanced, the stiffness coefficients show a slightly more gradual changes  
at \(K_{n}\) = 2.1667 which means they are insensitive to the aspect ratio. The direct damping coefficients increase linearly with the aspect ratio up to \(B/D\) = 1 and \(D_{yx}\) decreases marginally. Figure 4:3 _Cont._  
Figure 4:4 Stiffness coefficients of Boltzmann model against the aspect ratio \(B/D\) for various Knudsen numbers. (**a**) \(K_{xx}\) vs. \(B/D\); (**b**) \(K_{xy}\) vs. \(B/D\); (**c**) \(K_{yx}\) vs. \(B/D\); (**d**) \(K_{yy}\) vs. \(B/D\).