## 3 Normal Stiffness of an Aerostatic Slide  
The lateral stiffness of the aerostatic slide follows the proposed theory well. The proposed theory should be valid for the stiffness of the aerostatic slide in other directions. This section will prove the theory of normal stiffness. According to the proposed theory, the two upper supporting bearings are seen as two springs whose stiffness, \(k_{\text{fn1}}\), is the thrust stiffness of the upper bearing. Similarly, the two lower supporting bearings are seen as two springs whose stiffness, \(k_{\text{fn2}}\), is the thrust stiffness of the lower bearing. The slide is seen as an assembly consisting of a rigid body and four springs that connect to the four springs representing the upper and lower bearings, as shown in Figure 8. The stiffness of the springs connecting to the springs representing the upper bearings, \(k_{\text{sn1}}\), is the derivative of the supporting force from the upper bearing with respect to \(W_{1}\). The stiffness of the springs connecting to the springs representing the lower bearings, \(k_{\text{sn2}}\), is the derivative of the supporting force from the lower bearing with respect to \(W_{2}\). The normal stiffness of the aerostatic slide can be directly obtained from Figure 8, as shown in Equation (17), which will be verified to prove that the proposed theory is also valid for the normal stiffness of the aerostatic slide. \[k_{\text{n}}=\frac{2k_{\text{fn1}}k_{\text{sn1}}}{k_{\text{fn1}}+k_{\text{ sn1}}}+\frac{2k_{\text{fn2}}k_{\text{sn2}}}{k_{\text{fn2}}+k_{\text{sn2}}} \tag{17}\]  
When the external normal force, which is \(F_{\text{n}}\) in Figure 2, changes, the slide will displace in the normal direction to make the slide stable again. Supposing the change of \(F_{\text{n}}\) is \(\Delta F_{\text{n}}\)  
Figure 8: Schematic of the aerostatic slide with normal-equivalent springs. Figure 7: Measured and calculated lateral displacements under different external lateral forces. and the corresponding displacement of the slide is \(d\), the normal stiffness of the aerostatic slide, \(k_{\text{n}}\), can be expressed as:  
\[k_{\text{n}}=\lim_{d\to 0}\frac{\Delta F_{\text{n}}}{d} \tag{18}\]  
If \(\Delta F_{\text{n}}\) is positive, the slide will displace in the gravity direction. Then the upper supporting force \(F_{1}\) will increase and \(F_{2}\) will decrease to counteract \(\Delta F_{\text{n}}\). Let the changes in \(F_{1}\) and \(F_{2}\) be \(\Delta F_{1}\) and \(\Delta F_{2}\), respectively. \(\Delta F_{\text{n}}\) is equal to \(\Delta F_{1}\) minus \(\Delta F_{2}\). Hence Equation (18) can be rewritten as:  
\[k_{\text{n}}=\lim_{d\to 0}\frac{\Delta F_{1}-\Delta F_{2}}{d} \tag{19}\]  
The supporting forces from the single upper and lower bearings are \(F_{1}/2\) and \(F_{2}/2\), respectively. Because \(\Delta F_{1}\) and \(\Delta F_{2}\) are very small, \(k_{\text{sn1}}\) and \(k_{\text{sn2}}\) can be expressed as Equation (20), where \(\Delta W_{1}\) and \(\Delta W_{2}\) are the changes of \(W_{1}\) and \(W_{2}\) due to the changes of \(F_{1}\) and \(F_{2}\). \[ \begin{cases}k_{\text{sn1}}=\frac{\Delta F_{1}}{2\Delta W_{1}}\\ k_{\text{sn2}}=\frac{\Delta F_{1}}{2\Delta W_{2}}\end{cases}  \tag{20}\]  
The changes in the film thicknesses of the upper and lower supporting bearings, \(\Delta h_{1}\) and \(\Delta h_{2}\), can be obtained based on \(\Delta W_{1}\), \(\Delta W_{2}\), and \(d\), as shown in:  
\[ \begin{cases}\Delta h_{1}=\Delta W_{1}-d\\ \Delta h_{2}=\Delta W_{2}+d\end{cases}  \tag{21}\]  
\(k_{\text{fn1}}\) and \(k_{\text{fn2}}\) are the opposite numbers of the derivatives of \(F_{1}/2\) and \(F_{2}/2\) with respect to \(h_{1}\) and \(h_{2}\), respectively, as shown in:  
\[ \begin{cases}k_{\text{fn1}}=\lim_{\Delta h_{1}\to 0}-\frac{\Delta F_{1}}{2 \Delta h_{1}}\\ k_{\text{fn2}}=\lim_{\Delta h_{2}\to 0}-\frac{\Delta F_{2}}{2\Delta h_{2}} \end{cases}  \tag{22}\]  
Substituting for \(\Delta h_{1}\) and \(\Delta h_{2}\) in Equation (21) using Equation (22), and \(\Delta W_{1}\) and \(\Delta W_{2}\) using Equation (20), gives:  
\[