# Probabilistic Sensitivity Analysis of Wear Property for MEMS Gas Bearing
## 2 Numerical Method  
### Mathematical Model of Wear  
The wall wear that is investigated in this paper mainly consists of two parts: firstly, as shown in Figure 1, in the manufacturing process of the bearing, the surface of the bearing has various types of irregularities due to the influence of various uncontrollable factors. This surface structure forms the geometric characteristics of the gas bearing's inner surface, which includes the surface roughness, waviness, and the surface profile. This paper only considers the influence of surface roughness and supposes that there is no systematic error in bearing manufacturing--that is, the waviness is 0. Secondly, because of the wear and tear effect of the bearing in operation, pits or protrusions on the inner wall of the bearing appear. This surface structural change will also lead to changes in the gas film morphology of the bearing and furthermore affects the gas bearing's performance. When the wear dimension of the bearing's inner surface is close to the machined roughness, the two can be considered together. In this investigation, the comprehensive study was carried out by combining these two kinds of surface topography, and for convenience, \"wear\" was adopted to comprehensively describe them.  
Figure 2 shows the microstructure of wear. Similar to the definition of roughness, the following two parameters were introduced to describe the wear mathematical model of the bearing's inner wall:  
\[Ra =\frac{1}{l}\int_{0}^{l}\bigl{|}y(x)\bigr{|}dx \tag{1}\] \[Rz =\bigl{|}y_{\max}-y_{\min}\bigr{|} \tag{2}\]  
When the wall wear is discrete value, the definition of \(Ra\) gives:  
\[Ra=\frac{1}{n}\sum_{i=1}^{n}\bigl{|}y_{i}\bigr{|} \tag{3}\]  
During the operation of gas bearings, the wear of the inner surface is mainly caused by the erosion of gas. With the action of gas force, the adhesion and shedding of microscopic particles constantly occurs on the inner wall of the gas bearing, resulting in the occurrence of pits and protrusions on the  
Figure 1: Three types of surface topography.  
Figure 2: Microstructure of wear.  
inner wall of the bearing. In order to simplify the description, without considering the movement of the wall contour profile that is caused by wear, assume that the wall wear is a small value and the mean value of wear is 0. \(Ra\), which can be further written as:  
\[Ra=\frac{1}{n}{\sum_{i=1}^{n}}\big{|}y_{i}-0\big{|}=\sigma(y) \tag{4}\]  
where \(\sigma(y)\) represents the standard deviation of the wear degree. Furthermore, the distribution of the surface wear value is assumed to have a normal distribution. When analyzing the problem from a statistical point of view, the probability density of the wall wear value is:  
\[prd(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{(x-y)^{2}}{2\sigma^{2}}}=\frac{1}{ \sqrt{2\pi}}e^{-\frac{x^{2}}{2}} \tag{5}\]  
where \(\mu\) is the mean value of the normal distribution and \(\mu=0\), \(\sigma\) is the standard deviation. Figure 3 shows the probability density of the normal distribution, and the probability density function is:  
\[PRB(x)=\int\limits_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{-\frac{(x-y)^{2}}{2 \sigma^{2}}}=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}} \tag{6}\]  
The stochastic process is introduced into the process of modeling the degree of wear at different circumferential or axial positions, and all wear data follows the above normal distribution. In the simulation process, it is impossible to take extreme values considering the actual wear situation. According to the [PERSON] criterion, the simulated wear data is considered a coarse error when the absolute value of the wall wear is greater than \(3\sigma\), and this data should be eliminated. Applying this concept to the investigation of this paper, in order to make the surface wear distribution more uniform, it is considered that the coarse error occurs when the wear value is larger than \(\sigma\)--that is, when the absolute value of the wear value is less than \(\sigma\).