# Rotation Accuracy Analysis of Aerostatic Spindle Considering Shaft's Roundness and Cylindricity
## 4 Results and Discussions  
### Simulation Results  
First of all, an ideal shaft without shape errors is used for the calculation to verify the correctness of the numerical model and the simulation program; Figure 5 shows the result of dimensionless pressure distribution when the shaft rotates 180\({}^{\circ}\) at 6000 r/min. It can be seen that the pressure distribution on the shaft surface is very regular, which is consistent with the calculation results in reference [22]. In this ideal case, the gas film force acting on the shaft is zero.  
Figure 6a-d show the dimensionless pressure distribution of shaft 1 at 6000 r/min when it rotates 90\({}^{\circ}\), 180\({}^{\circ}\), 270\({}^{\circ}\) and 360\({}^{\circ}\) with the same coordinate scales, respectively. Due to the roundness and cylindricity errors, the gas pressure on the shaft surface varies greatly. As the shaft rotates, the pressure changes obviously. Thus, the shaft will produce an unbalanced gas film force, which changes both in size and direction, and then affects the rotary accuracy of the spindle.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Bearing diameter (\(D\)/mm) & 32 \\ Bearing length (\(L\)/mm) & 100 \\ Nominal radius clearance (\(h_{0}\)/mm) & 0.01 \\ Orifice diameter (\(d\)/mm) & 0.16 \\ Orifice length (\(l\)/mm) & 2 \\ Column number of feeding orifices & 2 \\ Number of orifices on each column & 8 \\ Atmospheric pressure (\(p_{atm}\)/Pa) & \(1.013\times 10^{5}\) \\ Supplied pressure (\(p_{s}\)/\(P_{atm}\)) & 4 \\ Gas dynamic viscosity (\(\eta\)/Pa-s) & \(18.448\times 10^{-6}\) \\ Shaft Material & Ti-6 Al-4V \\ Shaft density (\(\rho_{shaft}\)/kg/m\({}^{3}\)) & \(4.51\times 10^{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Spindle parameters.  
Figure 5: Dimensionless pressure distribution of ideal shaft.  
Figure 7a-d are box charts of the surface pressure distribution of shaft 1, shaft 2, shaft 3, and shaft 4 at different speeds with the same coordinate scales. With the increase in rotating speed, the distribution of pressure value becomes more and more divergent. By comparing the interquartile range (IQR) and the range within 1.5 IQR of the dimensionless pressure in Figure 7a-d, it can be observed that the pressure fluctuation of shaft 2 is significantly greater than that of the other three shafts.  
Figure 6: Dimensionless pressure distribution of shaft 1 at 6000 r/min at different angles: (**a**) 90\({}^{\circ}\); (**b**) 180\({}^{\circ}\); (**c**) 270\({}^{\circ}\); (**d**) 360\({}^{\circ}\).  
Figure 7: Box charts of surface pressure distribution: (**a**) shaft 1; (**b**) shaft 2; (**c**) shaft 3; (**d**) shaft 4.  
If the pressure is known, the gas film force acting on the shaft can be calculated using Equation (27) [23]:  
\[\left(\begin{array}{c}F_{x}\\ F_{y}\end{array}\right)=\int_{0}^{L}\int_{0}^{2\pi}\left(p-p_{atm}\right)\left( \begin{array}{c}\cos\theta\\ \sin\theta\end{array}\right)rd\theta dz \tag{27}\]  
Figure 8a shows the gas film force (with 5 revolutions) of shaft 1 at 6000 r/min. With the rotation of the shaft, the direction and size of the force change, and the shaft adopts an unsteady state. Plotting the average force of the shaft at different speeds in Figure 8b, it can be seen that with the increase in speed, the force on the shaft increases basically. The force of shaft 2 at each speed is significantly higher than that of the other three shafts. According to the force from large to small, the order is as follows: shaft 2 > shaft 1 > shaft 4 > shaft 3.