# Rotation Accuracy Analysis of Aerostatic Spindle Considering Shaft's Roundness and Cylindricity
## 2 Mathematical Model  
### Modeling of the Aerostatic Spindle  
The schematic illustration of the aerostatic journal bearing with shaft shape errors is shown in Figure 1. This bearing has two column orifice restrictors at the front and rear, and each column includes eight orifices. The compressed gas flows into the clearance between the bearing and the shaft through the orifice, and then flows out from both ends of the bearing.  
Due to the shape errors of shaft surface, the film thickness of different positions in the circumferential and axial directions of the bearing is different. In Figure 1b, \(O_{0}\) is the bearing center, \(O_{1}\) is the shaft center, and \(R\) is the bearing radius. \(h\) is the film thickness at point \(A\) on the shaft surface that can be denoted as  
\[h=R-\left|\overrightarrow{O_{0}O_{1}}+\overrightarrow{O_{1}A}\right| \tag{1}\]  
When \(\theta\) is the angle between \(\overrightarrow{O_{0}A}\) and the \(x\)-axis, \(\overrightarrow{O_{0}O_{1}}=(e_{x},e_{y})\), and \(\overrightarrow{O_{1}A}=(c_{x\theta},c_{y\theta})\), the film thickness at \(\theta\) is  
\[h_{\theta}=R-\sqrt{\left(c_{x\theta}+e_{x}\right)^{2}+\left(c_{y\theta}+e_{y} \right)^{2}} \tag{2}\]  
If the shaft is rotating clockwise at a constant angular velocity \(\omega\), then at \(t\) time,  
\[\left\{\begin{array}{c}\theta^{\prime}=\theta+\omega t+\alpha-2N\pi\\ c_{x\theta}=x_{\varphi}\\ c_{y\theta}=y_{\varphi}\end{array}\right. \tag{3}\]  
where \(\alpha\) is the initial angle of the shaft, \(N\) is the integer number of rotations of the shaft, and the values of \(x_{\theta^{\prime}}\) and \(y_{\theta^{\prime}}\) come from the measurement results of the cylindricity meter.  
By combining Equations (2) and (3), the film thickness at any angle at any time can be obtained.  
In order to study the pressure distribution of the gas film in the bearing, the circumferential cylindrical coordinates system is used to reconstruct the journal bearing. Figure 2 is the partial schematic illustration of aerostatic journal bearing in circumference cylindrical coordinates. The circumferential direction of bearing's inner surface is the \(x\)-axis and the \(y\)-axis points and is perpendicular to the bearing axis, and the \(z\)-axis is parallel to the bearing axis.  
This spindle system incorporates the following assumptions:  
Figure 1: Illustration of the aerostatic journal bearing with shaft shape errors: (**a**) axial direction; (**b**) circumferential direction.  
The spindle is cooled sufficiently--that is, the bearing, shaft and gas film is isothermal;  
There is no axial and angular movement of the shaft;  
The gas flow is laminar.  
The modified Reynolds equation for this type of aerostatic journal bearing [18] is  
\[\frac{\partial}{\partial x}\left(ph^{3}\frac{\partial p}{\partial x}\right)+\frac {\partial}{\partial z}\left(ph^{3}\frac{\partial p}{\partial z}\right)=6\eta u \frac{\partial(ph)}{\partial x}+12\eta\frac{\partial(ph)}{\partial t}+12\eta pV _{in} \tag{4}\]  
where \(p\) is the gas film pressure, \(\eta\) is the dynamic viscosity of the gas, \(u\) is the velocity of the gas in \(x\) direction, and \(V_{in}\) is the injection velocity at the orifice entrance and can be expressed by Equation (5) [19]  
\[V_{in}=-\frac{P_{s}-p}{4\eta l}[\frac{d^{2}}{4}-(x-x_{i})^{2}-(z-z_{i})^{2}] \cdot\delta_{i} \tag{5}\]  
where \(\delta_{i}=1\) at the orifice entrance, whereas it is zero elsewhere.  
By employing the dimensionless parameters  
\[x=x_{0}X,z=z_{0}Z,h=h_{0}H,t=\tau\frac{x_{0}}{u},p=p_{a}P,\Lambda=\frac{12 \eta ux_{0}}{h_{0}^{2}p_{a}},Q=\frac{24\eta x_{0}^{2}}{h_{0}^{3}p_{a}^{2}}V_{in} \tag{6}\]  
Equation (4) can be written as:  
\[\frac{\partial}{\partial X}\left(H^{3}\frac{\partial P^{2}}{\partial X}\right) +\frac{x_{0}^{2}}{2_{0}^{2}}\frac{\partial}{\partial Z}\left(H^{3}\frac{ \partial P^{2}}{\partial Z}\right)=\Lambda\frac{\partial PH}{\partial X}+2 \Lambda\frac{\partial PH}{\partial\tau}+QP \tag{7}\]