## 2 Mathematical Model of Aerostatic Bearing
### Engineering Calculation Method  
The engineering calculation method usually expands the gas film of the aerostatic bearing into a plane for analysis. According to the number of single row orifice \(n\), it is divided into \(n\) equal parts; each width is \(2\pi R/n\), and the following assumptions are made:  
1. The gas film thickness is very small relative to the diameter of the journal. The curvature of the gas film can be ignored. It is considered that the thickness of the gas film in each equal area is a fixed value;
2. The gas film thickness is tens of microns. It can be considered that the gas pressure and density distribution in the direction of the gas film thickness shows no change;  
Figure 1: (**a**) Structure diagram of aerostatic bearing; (**b**) orifice distribution diagram. 3. Lubricating gas does not slide on the solid surface, and the velocity of gas on the solid surface is the same as that on the solid surface;
4. As the lubricating gas is air and the density is small, the effect of volume force can be ignored, and the gas flow in the lubrication process is assumed to be laminar flow;
5. It is considered that the pressure between the two orifices in each equipartition area is equal, and the pressure is equal to the pressure after gas throttling. It is assumed that there is no gas flow between each equipartition area. The gas film expansion diagram is shown in Figure 2. The direction of the gas flowing to the end face is the \(x\)-axis, the direction of the gas film thickness is the \(y\)-axis, and the circumferential direction is the \(z\)-axis. The gas film thickness at the orifice represents the gas film thickness in an equal area. According to the assumption that the gas in each equal area shows only one-dimensional motion flowing axially to the bearing outlet, the gas momentum equation can be expressed as:  
\[\left\{ \begin{array}{l}\frac{\partial p}{\partial x}=\mu\frac{\partial^{2}u }{\partial y^{2}}\\ \frac{\partial p}{\partial y}=0\\ \frac{\partial p}{\partial z}=0\end{array} \right. \tag{1}\]  
The gas mass flow \(m_{i1}\) of gas flowing out of the \(i\)th region can be obtained using Equation (2):  
\[m_{i1}=b\cdot m_{ix}=\frac{2\pi R}{n}\int_{0}^{h_{i}}\rho udy \tag{2}\]  
where \(m_{ix}\) is the mass flow per unit width section perpendicular to the \(x\)-axis. The gas film thickness \(h_{i}\) of the \(i\)th region can be obtained using Equation (3):  
\[h_{i}=h_{av}(1-\varepsilon\cos\varphi_{i}) \tag{3}\]  
where \(h_{av}\) is the average gas film thickness; \(\varepsilon\) is eccentricity; and \(\varphi_{i}\) is the position of the orifice on the circumference of the bearing. The gas mass flow \(m_{i2}\) of gas flowing into the \(i\)th region can be obtained using Equation (4):  
\[m_{i2}=AP_{s}\phi\sqrt{\frac{2\rho_{s}}{P_{s}}}\psi_{i} \tag{4}\]  
where \(A\) is the cross-sectional area of the orifice; \(\phi\) is the gas flow coefficient, and its value is 0.8; and \(\psi_{i}\) is the flow function of the orifice, which is determined by the air supply pressure and orifice outlet pressure. \[\psi_{i}=\left\{ \begin{array}{l}\left[\frac{\delta}{2}\left(\frac{2}{k+1} \right)^{(k+1)/(k-1)}\right]^{1/2},\left(\frac{P_{0i}}{P_{s}}\leq\beta_{k} \right)\\ \left\{\frac{k}{k-1}\left[\left(\frac{P_{0i}}{P_{s}}\right)^{2/k}-\left(\frac {P_{0i}}{P_{s}}\right)^{(k+1)/k}\right]\right\}^{1/2},\left(\frac{P_{0i}}{P_{s }}>\beta_{k}\right)\end{array} \right. \tag{5}\]  
where \(P_{s}\) and \(P_{0i}\) are the gas supply pressure and the orifice outlet pressure of the \(i\)th region, respectively, and \(\beta_{k}\) is the critical pressure ratio; the orifice throttling process can be regarded as an adiabatic process, where \(k\) is the adiabatic index, with a value of 1.4, so the critical pressure ratio is 0.528.