## 2 Lateral Stiffness of an Aerostatic Slide  
The studied aerostatic slide is from an ultra-precision machine tool, which is shown in Figure 1. Two lateral aerostatic bearings support the slide in the lateral direction and the other four aerostatic bearings in the normal direction. The lateral and normal directions are the horizontal and vertical directions in Figure 1b, respectively. Under external and supporting forces, the displacement and deformation of the slide in the normal and lateral directions can change the shapes and dimensions of the six supporting bearings and thus affect the stiffness of the aerostatic slide. In the cutting process, the external lateral force on the slide is mainly from the cutting force, which is small. Its effect on the lateral stiffness of the aerostatic slide can be ignored. Besides, the linear motor counteracts the external force in the feed direction. Only the external force in the normal direction, \(F_{\text{n}}\), and the supporting forces from the bearings should be considered. The external moment is small and can also be ignored. Therefore, for the upper or lower bearings, the supporting forces from the left and right bearings are equal. The upper and lower supporting forces, \(F_{1}\) and \(F_{2}\), are from the two upper and lower bearings, which are applied on the contact surfaces between the slide and supporting bearings in the normal direction. Hence, the supporting forces from the single upper and lower bearings are \(F_{1}/2\) and \(F_{2}/2\), respectively. If a very small external lateral force, \(F_{\text{s}}\), is applied on the barycenter of the slide as shown in Figure 2, the slide will move a distance, \(d\). Then the film thickness of the left lateral bearing, \(h_{\text{3L}}\), decreases, and thus the left lateral supporting force, \(F_{\text{3L}}\), increases. On the contrary, the film thickness of the right lateral bearing, \(h_{\text{3R}}\), increases and the right lateral supporting force, \(F_{\text{3R}}\), decreases. The resultant forces of \(F_{\text{3L}}\) and \(F_{\text{3R}}\) will change from 0 to a force that is equal in magnitude but opposite in direction to \(F_{\text{s}}\), making the slide static again. The relationship between \(F_{\text{s}}\) and the two lateral supporting forces is shown as:  
\[F_{\text{3L}}=F_{\text{3R}}+F_{\text{s}} \tag{1}\]  
Figure 1: Configuration of the studied aerostatic slide. (**a**) Structure of slide and guideway; (**b**) structure of slide and its supporting bearing. The lateral stiffness of the aerostatic slide, \(k_{\mathrm{s}}\), is the derivative of \(F_{\mathrm{s}}\) with respect to \(d\), which can be written as:  
\[k_{\mathrm{s}}=\lim_{d\to 0}\frac{F_{\mathrm{s}}}{d} \tag{2}\]  
When high-pressure air is supplied to the bearings, the slide will deform under the six supporting forces, as shown in Figure 3a. \(W_{1}\) and \(W_{2}\) are the deformation values of the upper and lower supporting surfaces' center lines parallel to the feed direction in the film thickness direction. \(W_{\mathrm{3L}}\) and \(W_{\mathrm{3R}}\) are the deformation values of the left and right lateral supporting surfaces' center lines. The shapes of the six air films will change from cuboid to wedge, which can be defined by two parameters for every air film. One is the deformation value of the supporting surfaces' center line parallel to the feed direction in the film thickness direction, \(W\). The other one is the wedge angle, \(\theta\). The shapes of the supporting bearing before and after the slide deforms are shown in Figure 3b,c, where \(h_{0}\) represents the designed film thickness. The solid deformation of the supporting surface can be seen as the superposition of the supporting surface moving along the film thickness direction by \(W\) and rotating around the center line by \(\theta\). \(W\) changes the film thickness of the aerostatic bearing and \(\theta\) changes the shape. In fact, the wedge angles of the six supporting bearings are less than 200 \(\upmu\)rad and hardly change the supporting forces.