## 2 Lateral Stiffness of an Aerostatic Slide
If the wedge angle of the lower supporting bearing increases from 0 to 200 \(\upmu\)rad, its supporting force only increases by 1.16%, and thus the influence of wedge angle on the supporting forces can be ignored. The supporting force from the bearing can be obtained by calculating the CFD model of the bearing, whose film thickness is \(h_{0}\) plus \(W\). The left and right lateral film thicknesses, \(h_{\mathrm{3L}}\) and \(h_{\mathrm{3R}}\), can be expressed as:  
\[ \begin{cases}h_{\mathrm{3L}}=h_{0}+W_{\mathrm{3L}}-d\\ h_{\mathrm{3R}}=h_{0}+W_{\mathrm{3R}}+d\end{cases}  \tag{3}\]  
\(d\) can be obtained from Equation (3), as shown in:  
\[d=\frac{h_{\mathrm{3R}}-h_{\mathrm{3L}}+W_{\mathrm{3L}}-W_{\mathrm{3R}}}{2} \tag{4}\]  
Figure 3: Schematic of the deformation of the slide and supporting bearing. (**a**) Deformation of slide; (**b**) supporting bearing before solid deforms; (**c**) supporting bearing after solid deforms. Figure 2: Schematic of the forces on the slide. When the external lateral force is 0, \(F_{3\text{L}}\) is equal to \(F_{3\text{R}}\). In this case, \(W_{3\text{L}}\) is equal to \(W_{3\text{R}}\). The relationship between \(W_{3\text{L}}\) or \(W_{3\text{R}}\) and the supporting forces can be obtained by the finite element method, which is shown as Equation (5). In this equation, \(W_{3}\) represents \(W_{3\text{L}}\) or \(W_{3\text{R}}\), and \(F_{3}\) is equal to \(F_{3\text{L}}\) and \(F_{3\text{R}}\). \[W_{3}=-5.084\times 10^{-4}F_{1}+6.866\times 10^{-4}F_{2}+2.885\times 10^{-3}F_{3} \tag{5}\]  
The functional relationship in Equation (5) is linear because the solid deformation is small and the structural material is linear. The change in \(F_{3\text{L}}\) hardly changes \(W_{3\text{R}}\), and the change in \(F_{3\text{R}}\) hardly changes \(W_{3\text{L}}\). When the external lateral force is small, Equation (5) can be rewritten as:  
\[\left\{ \begin{aligned} & W_{3\text{L}}=-5.084\times 10^{-4}F_{1}+6.866 \times 10^{-4}F_{2}+2.885\times 10^{-3}F_{3\text{L}}\\ & W_{3\text{R}}=-5.084\times 10^{-4}F_{1}+6.866\times 10^{-4}F_{2}+2.885 \times 10^{-3}F_{3\text{R}}\end{aligned} \right. \tag{6}\]  
Subtracting \(W_{3\text{R}}\) from \(W_{3\text{L}}\) in Equation (6) gives Equation (7), where \(\lambda\) is 2.885 \(\times\) 10\({}^{-3}\). \[W_{3\text{L}}-W_{3\text{R}}=\lambda(F_{3\text{L}}-F_{3\text{R}}) \tag{7}\]  
The supporting force from the bearing decreases with the increase in film thickness. The relationship between the supporting force and the film thickness can be obtained by the CFD method. The design parameters of the left and right lateral bearings are identical. The functional relationship between \(F_{3\text{L}}\) and \(h_{3\text{L}}\),\(f\), is equal to that between \(F_{3\text{R}}\) and \(h_{3\text{R}}\), as shown in:  
\[\left\{ \begin{aligned} & F_{3\text{L}}=f(h_{3\text{L}})\\ & F_{3\text{R}}=f(h_{3\text{R}})\end{aligned} \right. \tag{8}\]  
Combining Equations (1), (2), (4), (7) and (8) yields:  
\[k_{\text{s}}=\lim_{d\to 0}2\frac{f(h_{3\text{L}})-f(h_{3\text{R}})}{h_{3 \text{R}}-h_{3\text{L}}+\lambda[f(h_{3\text{L}})-f(h_{3\text{R}})]} \tag{9}\]  
Equation (9) can be written as:  
\[k_{\text{s}}=\lim_{d\to 0}\frac{2}{\frac{h_{3\text{R}}-h_{3\text{L}}}{f(h_{3 \text{L}})-f(h_{3\text{R}})}+\lambda} \tag{10}\]  
For a lateral supporting bearing whose film thickness is \(h\), its thrust stiffness, \(k_{\text{fs}}\), is the derivative of the opposite number of its supporting force with respect to \(h\), as shown in:  
\[k_{\text{fs}}=\lim_{\Delta h\to 0}-\frac{f(h+\Delta h)-f(h)}{\Delta h} \tag{11}\]  
Substituting for \(h\) and \(\Delta h\) in Equation (11) using \(h=h_{3\text{L}}\) and \(\Delta h=h_{3\text{R}}-h_{3\text{L}}\) gives:  
\[k_{\text{fs}}=\lim_{(h_{3\text{R}}-h_{3\text{L}})\to 0}\frac{f(h_{3\text{L}})-f(h_{3 \text{R}})}{h_{3\text{R}}-h_{3\text{L}}} \tag{12}\]  
The lateral structural stiffness, \(k_{\text{ss}}\), is defined as the derivative of \(F_{3}\) with respect to \(W_{3}\). The change in \(F_{3}\) barely changes \(F_{1}\) and \(F_{2}\).