# Probabilistic Sensitivity Analysis of Wear Property for MEMS Gas Bearing
## 2 Numerical Method
### Research Object and Solution Process  
The statistical method and stochastic process were adopted to investigate the sensitivity of MEMS gas bearing performance to the bearing inner wall wear in different axial and circumferential positions. Figure 7 shows the axial and circumferential distribution of wear that was considered in the study. In the axial direction, since the gas bearing is symmetrical along the central axis, the symmetric distribution wear form was adopted when considering the wear distribution. The inner wall surface of the bearing was divided into five parts along the axial direction in this study, and the serial number increases from both ends of the bearing to the middle. In the circumferential direction, the inner wall surface was equally divided into six parts along the circumferential direction from \(\varphi=0\). For the bearing given in Table 2, the attitude angle is \(40.65^{\circ}\) when the bearing inner wall is smooth. The position of the attitude angle is marked with a dashed line in Figure 7, which represents the minimum position of the gas film thickness of the bearing with the smooth wall.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Structural and Operating Parameters** & **Value** \\ \hline Bearing radius (\(r\)) & 2.4 mm \\ Bearing length (\(L\)) & 0.3 mm \\ Radius clearance (\(c\)) & 10 μm \\ Eccentricity ratio (\(\varepsilon\)) & 0.9 \\ Ambient pressure (\(p_{a}\)) & 101325 Pa \\ Gas constant (\(R_{g}\)) & 287.03 J mol\({}^{-1}\).K\({}^{-1}\).kg\({}^{-1}\) \\ Temperature (\(T_{a}\)) & 293.15 K \\ Fluid viscosity (\(\eta_{0}\)) & \(1.8\times 10^{-5}\) Pa\(\cdot\)s \\ Rotating speed (\(N\)) & \(50\times 104\) r\(\cdot\)min\({}^{-1}\) \\ Standard deviation of bearing wear (\(\sigma\)) & 0.3 μm \\ Mean value of bearing wear (\(\mu\)) & 0 \\ Exceptional data bound (\(EDB\)) & \(1\sigma\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Structural and operating parameters of MEMS gas bearing.  
Figure 7: Axial and circumferential distribution of wear. (**a**) axial wear region distribution. (**b**) circumferential wear region distribution.  
Figure 5 shows the mesh model and the boundary conditions for the MEMS gas bearing solution domain. When the wear zone to be studied was determined, the wear values were decided in accordance with the method described in Section 2.1 to simulate the wear at that position. For the static calculation, the Dirichlet boundary condition was set on both ends of the bearing, the boundary pressure was atmospheric pressure, and the coincidence conditions were set on the left and right sides of the rectangular solution domain:  
\[P(\varphi,\pm\frac{L}{2r})=1,\quad P(\varphi,\pm\frac{L}{2r})=1 \tag{35}\]  
\[P(0,\psi)=P(2\pi,\psi)\quad\psi\in[-\frac{L}{2r},+\frac{L}{2r}] \tag{36}\]  
For the dynamic calculation, the boundary type was the same as the static solution:  
\[P_{E}(\varphi,\pm\frac{L}{2r})=0,\quad P_{\Theta}(\varphi,\pm\frac{L}{2r})=0 \tag{37}\]  
\[P_{E}(0,\psi)=P_{E}(2\pi,\psi),\quad P_{\Theta}(0,\psi)=P_{\Theta}(2\pi,\psi) \quad\psi\in[-\frac{L}{2r},+\frac{L}{2r}]. \tag{38}\]