## 4 Mathematical Model of Air Film Load-Bearing Characteristics
The gas is flowing in an isothermal laminar flow, and the equation of state is:  
\[\frac{p}{\rho}=\frac{p_{0}}{\rho_{0}} \tag{5}\]  
where \(p_{0}\) is the atmospheric pressure and \(\rho_{0}\) is the atmospheric density. Combining Equations (3)-(5), a simplified Reynolds equation is obtained when the gas flow state is laminar:  
\[\frac{\partial}{\partial x}\left(h^{3}\frac{\partial\mathrm{p}}{\partial x} \right)+\frac{\partial}{\partial z}\left(h^{3}\frac{\partial\mathrm{p}}{ \partial x}\right)=0 \tag{6}\]  
The air film load-bearing capacity is calculated as follows:  
\[W=\int_{S}pds \tag{7}\]  
Figure 5: Structure of air flotation sleeve and gas flow diagram. (**a**) Section structure view. (**b**) Bottom structure view. where \(W\) is the air film's load-bearing capacity and \(S\) is the effective area of the air film. Equation (6) represents a second-order partial differential expression characterizing pressure variation with respect to geometric position. Finding its analytical solution poses a considerable challenge. In the interest of algorithmic simplification, this study employs an approach described in the existing literature [18]. This involves an assumption that the pressure distribution along the radial direction adheres to linearity, facilitating an approximation of the air film's load-bearing capacity. The structure of the air flotation sleeve is shown in Figure 5. The optimum thickness of the axial air film and the corresponding load-bearing capacity are approximately calculated. Input:\(L=D\), \(l=L/4,\ R_{1}=D/2,\ R=D/4,\ R_{2}=D/8\), \(r_{1}=R+d,\ r_{2}=R-d\), number of throttle orifice \(n=6\), throttle orifice diameter \(d=0.2\ mm,\ b=1/6\ (b=p_{1}/p_{0})\), supply pressure \(p_{1}=6\times 10^{5}\)Pa. To calculate the optimal thickness of the axial air film and its corresponding load-bearing capacity, the main steps are as follows. 1. \(f_{1},\ f_{2},\ f_{3}\) are calculated. These are the gas passage coefficient, working medium coefficient, and gas film structure coefficient, respectively. \[\left\{ \begin{array}{c}f_{1}=\frac{\phi An}{h\beta}\\ f_{2}=\frac{12\mu}{\pi}\sqrt{\frac{2}{\eta\varphi_{0}}}\\ f_{3}=\frac{\ln\left(\frac{R_{1}}{r_{1}}\right)\times\ln\left(\frac{r_{2}}{R_ {2}}\right)}{\ln\left(\frac{r_{1}}{r_{1}}\right)+\ln\left(\frac{r_{2}}{R_{2}} \right)}\end{array} \right.\] (8) The air film flow medium is air, so \(f_{2}=268\times 10^{-6}\) mm. 2. \(\varepsilon\) is calculated. \[\varepsilon=f_{1}\times f_{2}\times f_{3}\] (9)
3. According to \(\varepsilon\), the corresponding \(\beta\) value is found from the universal design curve (Figure 2-1, attached in the references [18]). 4. The optimal thickness of the axial air film \(h_{1}\) is calculated. \[h_{1}=\left(\frac{\phi Anf_{2}f_{3}}{2.4\ g}\right)^{\frac{1}{3}} \tag{10}\]  
where \(\phi\) is the flow coefficient, generally taken as 0.8, \(A\) is the cross-sectional area of the throttle orifice, and g is a constant related to \(\sigma\), when \(\sigma=1/6,\ g=2.284\). The axial air film load-bearing capacity \(W\) is calculated. \[W=\pi R_{*}^{2}p_{1}(\beta-\sigma) \tag{11}\]  
where  
\[R_{*}^{2}=\left(R_{1}^{2}+R_{1}r_{1}+r_{1}^{2}-R_{2}^{2}-R_{2}r_{2}-r_{2}^{2} \right)/3 \tag{12}\]  
For the radial load-bearing air film, its structural configuration and force application characteristics align with those of conventional cylindrical air bearings. As such, the optimal thickness of the radial load-bearing air film can be readily determined through a table look-up method, leveraging the general design guidelines established within the air flotation bearing industry. These guidelines, typically accounting for factors such as air film stiffness and processing feasibility, provide a comprehensive assessment. The study cited in [18] (p. 72), specifically Tables 2-4 (b), detailing cylindrical air bearing performance calculations, facilitates this determination.