# Probabilistic Sensitivity Analysis of Wear Property for MEMS Gas Bearing
## 2 Numerical Method
### Solution of the Lubrication Equation Considering the Rarefaction Effect  
The structural schematic diagram of the gas bearing is shown in Figure 4. For MEMS gas bearing, the scale of the molecular free path cannot be negligible relative to the characteristic film thickness. Previous studies have shown that the flow state in the bearing clearance is a typical slip flow [26], and the simulation accuracy is not good enough to describe the flow state using the traditional Reynolds equation. According to the theory of [PERSON]. [PERSON] and [PERSON] [11; 12], the lubrication equation can be obtained by the Boltzmann equation based on the theory of molecular dynamics under any \(Kn\) number  
Figure 3: Probability density of the normal distribution. Extend the lubrication equation to two-dimensional space and transfer to the cylindrical coordinate system:  
\[\frac{\partial}{\partial\varphi}\Bigg{[}PH^{3}\overline{Q}_{p}(D_{0}PH)\frac{ \partial P}{\partial\varphi}\Bigg{]}+\frac{\partial}{\partial\psi}\Bigg{[}PH^{3 }\overline{Q}_{p}(D_{0}PH)\frac{\partial P}{\partial\psi}\Bigg{]}=\Lambda\frac {\partial(PH)}{\partial\varphi}+2\Lambda\frac{\partial(PH)}{\partial\tau} \tag{7}\]  
where \(D_{0}\) is the characteristic form of \(D\) (inverse Knudsen number), \(\tau\) is the dimensionless time item, defined as \(\tau=\omega\times t\), \(\omega\) is the journal rotating speed, \(\overline{Q}_{p}\) is the relative Poiseuille flow rate, which can be expressed as:  
\[\overline{Q}_{p}(D)=-\frac{Q_{P}(D)}{Qcon} \tag{8}\]  
where \(Qcon\) is the Poiseuille flow rate of continuous flow and \(Q_{p}(D)\) is the Poiseuille flow rate of rarefied flow, which can be obtained by Equation (9) according to the FK model:  
\[Q_{p}=\left\{ \begin{array}{ll}D/6+1.0162+1.0653/D-2.1354/D^{2}&(5\leq D)\\ 0.13852D+1.25087+0.15633/D-0.00969/D^{2}&(0.15\leq D<5)\\ -2.22919D+2.10673+0.01653/D-0.0000694/D^{2}&(0.01\leq D<0.15)\end{array} \right. \tag{9}\]  
Equation (7) describes the flow state of the gas bearing in the transient state considering the rarefaction effect. For the steady process, the static lubrication equation can be gained by discarding the time-dependent term:  
\[\frac{\partial}{\partial\varphi}\Bigg{[}PH^{3}\overline{Q}_{p}(D_{0}PH)\frac{ \partial P}{\partial\varphi}\Bigg{]}+\frac{\partial}{\partial\psi}\Bigg{[}PH^ {3}\overline{Q}_{p}(D_{0}PH)\frac{\partial P}{\partial\psi}\Bigg{]}=\Lambda \frac{\partial(PH)}{\partial\varphi} \tag{10}\]  
In this paper, the finite element method (FEM) is used to solve Equations (7) and (10). The solution is based on two assumptions: (1) The film pressure does not change in the radial direction; (2) The flow in the film clearance is isothermal. As shown in Figure 5, the cylindrical solution domain is expanded into a rectangular domain and the whole solution domain is discretized by a four-node quadrilateral element, in which the transient and static lubrication equations are solved. \(\mathbf{\Phi}\) is adopted as the interpolation function as well as the weight function in Galerkin integral. Equation (10) can be transformed in any finite element region as:  
\[\iint\limits_{\Omega}\mathbf{\Phi}\Bigg{[}\frac{\partial}{\partial\varphi} \Bigg{[}PH^{3}\overline{Q}(D)\frac{\partial P}{\partial\varphi}\Bigg{]}+ \frac{\partial}{\partial\psi}\Bigg{[}PH^{3}\overline{Q}(D)\frac{\partial P}{ \partial\psi}\Bigg{]}-\Lambda\frac{\partial(PH)}{\partial\varphi}\Bigg{]}d \varphi d\psi=0 \tag{11}\]  
Figure 4: Schematic diagram of a typical bearing-rotor system.