## 2 Lateral Stiffness of an Aerostatic Slide
\(k_{\text{ss}}\) can be deduced by Equation (5), as shown in:  
\[k_{\text{ss}}=\frac{\partial F_{3}}{\partial W_{3}}=\frac{1}{\lambda} \tag{13}\]  
When \(h_{3\text{R}}-h_{3\text{L}}\) tends towards 0, \(d\) tends towards 0, and vice versa. The two tendencies are identical. Combining Equations (10), (12) and (13) gives:  
\[k_{\text{s}}=\frac{2k_{\text{fs}}k_{\text{ss}}}{k_{\text{fs}}+k_{\text{ss}}} \tag{14}\]It can be seen that \(k_{\text{s}}\) can be directly induced if \(k_{\text{fs}}\) and \(k_{\text{ss}}\) are known. The lateral stiffness of the aerostatic slide has a functional relationship with that of the lateral bearings and solid structure. The lateral bearing can be regarded as a linearly elastic body in the film thickness direction when its film thickness changes a little, which is similar to a spring in the lateral direction. The lateral supporting surface deforms under the lateral supporting force, and the relationship between \(W_{3}\) and \(F_{3}\) is linear, which is similar to a spring deforming under an external force. In fact, Equation (14) can be obtained directly if the lateral bearings are seen as two springs and the slide is seen as an assembly consisting of a rigid body and two springs, as shown in Figure 4. One end of the spring representing the lateral bearing connects the guideway, and the other end connects the spring representing the slide. The stiffness constants of the springs representing the lateral bearing and slide are \(k_{\text{fs}}\) and \(k_{\text{ss}}\), respectively. The structure of the slide and the two lateral bearings is elastic in the lateral direction. When a lateral force is applied on the slide, the two lateral floating plates will deform to counteract a part of the force, and the film thicknesses of the two lateral bearings will change to counteract the rest of the force. Therefore, the lateral stiffness of the aerostatic slide derives from both the solid structure and the supporting bearings. Therefore, this paper proposes a theory for the stiffness of the aerostatic spindle and slide, which is that the stiffness derives from both the solid structure and the supporting bearings and can be obtained by seeing the supporting bearings and the solid structure as springs. The lateral stiffness will be calculated based on Equation (14) to verify the proposed theory. The film thickness of the lateral bearing can be calculated by the transient FSI method, which is 20.61 \(\upmu\)m when the slide bears no load; namely, \(F_{\text{n}}\) is the gravity of the slide. The lateral support forces with the lateral film thickness, \(h_{3}\), in the range of 18 \(\upmu\)m to 25 \(\upmu\)m can be calculated by the CFD method, which are shown in Figure 5. The functional relationship between \(F_{3}\) and \(h_{3}\) can be obtained through 2-order polynomial fitting, as shown in Equation (15). The fitting curve is shown as the red curve in Figure 5.  
\[F_{3}=15.273{h_{3}}^{2}-945.11h_{3}+16227.2 \tag{15}\]  
\(k_{\text{fs}}\) is the derivative of the opposite number of \(F_{3}\) with respect to \(h_{3}\), as shown in:  
\[k_{\text{s}}=\lim_{d\to 0}\frac{2}{\frac{h_{3\text{R}}-h_{3\text{L}}}{f(h_{3 \text{L}})-f(h_{3\text{R}})}+\lambda} \tag{10}\]0  
\(k_{\text{fs}}\) can be obtained by substituting \(h_{3}\) = 20.61 into Equation (16), which is 315.56 N/\(\upmu\)m. \(k_{\text{ss}}\) can be obtained based on Equation (13), which is 346.62 N/\(\upmu\)m. Therefore, \(k_{\text{s}}\) is 330.4 N/\(\upmu\)m according to Equation (14), which is significantly less than double \(k_{\text{fs}}\). \[k_{\text{s}}=\lim_{d\to 0}\frac{2}{\frac{h_{3\text{R}}-h_{3\text{L}}}{f(h_{3 \text{L}})-f(h_{3\text{R}})}+\lambda} \tag{10}\]1 Schematic of the aerostatic slide with lateral equivalent springs. The tests for the lateral stiffness of the aerostatic slide are conducted to verify the calculation. The lateral displacements of the slide under four external lateral forces are measured to obtain the lateral stiffness.