Improving Capacity, Stiffness and Pneumatic Hammer Stability of Aerostatic Thrust Bearing Using Damping Orifice and Virtual Recess

 Abstract: An aerostatic thrust bearing is designed using damping orifice and virtual recess. Finite element simulations and measurements are performed and results indicate that with damping orifice and virtual recess, the maximum capacity of the designed aerostatic bearing increases from 1590 N to 2285 N and its maximum stiffness increases from 106 N/μm to 145 N/μm, while the range of pneumatic hammer decreases from 8.5 μm to 4.5 μm at 4 bar. It is therefore concluded that damping orifice and virtual recess are one of the means which can be used to improve the capacity, stiffness and pneumatic hammer stability of aerostatic


Introduction
Aerostatic bearings are widely used in ultra-precision machining equipment and measurement equipment because of their extremely high accuracy, lower friction and no wear. However, it is very difficult to design an aerostatic bearing with high capacity, stiffness and pneumatic hammer stability for a limited bearing area. Many researchers concentrate on improving the stiffness and stability of aerostatic bearings [1]. The capacity and stiffness of aerostatic bearings can be improved by increasing gas supply pressure, number of grooves [2] and orifices [3][4][5], or selecting multiple [6][7], and porous restrictors [8]. Some work has been done using RNM method about the  Wei Ma jx04319@126.com Shanghai Institute of Satellite Engineering, Shanghai 201108, China influence of the length and depth of the grooves on the static and dynamic characteristics of aerostatic bearings [9][10]. However, the increase in the gas supply pressure and the number of grooves reduces the stability of pneumatic hammer [11]. The increase in the number of orifice results in a significant increase in mass flow. Double-pad aerostatic bearing [12], multiple-micromoles aerostatic bearing [13][14][15], porous aerostatic bearing, aerostatic bearing with flexible members [16] and aerostatic bearing with active control [17] can also be used to improve the stiffness, but the utilization of these aerostatic bearings is very complicated. The characteristics of porous bearings are closely related to porous materials, which have some shortcomings, such as unstable performance, less than 8 microns working gas film, great difficulty in accurate analysis and so on. The utilization of micro-array bearings is very complicated，and its working gas film thickness is generally less than 10 microns, so it is not suitable for high working film applications. Further the stability of pneumatic hammer on aerostatic bearings has not been considered above. The effect of pneumatic hammer on the performance of aerostatic bearings is closely related to pressure fluctuation [18][19], and it can be characterized using the damping characteristics of the bearings [20][21]. Thus, the damping characteristics of aerostatic bearings are very important for the study on their dynamic stabilities. The high damping characteristics of air spring is therefore introduced into the design of an aerostatic bearing to improve its stability significantly [22]. However, pneumatic hammer is a kind of self-excited instability due to the compressibility of air, which occurs in aerostatic bearings  with recess easily. The damping of aerostatic bearings cannot be improved by increasing their bearing areas in a space-constrained situation. Therefore, we concentrate on the design of aerostatic bearings with damping orifice and virtual recess and investigate the influence of damping orifices on the pneumatic stability. A rectangular aerostatic thrust bearing is designed using damping orifices and virtual recesses so that a good balance can be achieved among the capacity, stiffness and pneumatic hammer stability. The designed aerostatic bearing is numerically and experimentally investigated to prove its effectiveness in improving its capacity, stiffness and pneumatic hammer stability.

Aerostatic Bearing Model
As shown in Figure 1 and Figure 2, aerostatic bearings can be designed with and without damping orifice and virtual recess. The structural parameters of the aerostatic bearing with damping orifice and virtual recess, such as L, H, Lo, Ld, lw, Ho, Hw, Hd, lx, and lz are shown in Figure 2.  The capacity and stiffness of aerostatic bearings can be improved by providing a ring of shallow recesses around its central land to create a high pressure region which is virtual recess named by Rowe [23]. However, because of its volume effect, pneumatic hammer occurs in the aerostatic bearing. So an array of annular orifices is introduced in the high pressure region. As shown in Figure 3, an air spring consisting of a high pressure region, annular orifices and gas supply passages is used to improve the stability of pneumatic hammer. Then the annular orifices could be named damping orifices. Table 1 shows the structural parameters of the aerostatic bearing with damping orifice and virtual recess. Then the parameters of n1, n2, d1, d2 are shown in appendix.

Figure 3
Cross-sectional view of the designed aerostatic thrust bearing Table 1 Dimensions of the designed aerostatic bearing

Numerical Calculation
Considering the effect of orifices, Generalized Reynolds equation used to analyze the capacity and stiffness of an aerostatic bearing is given by: where exists feed orifice where absents feed orifice   . The pressure Pd through orifices and the damping orifice is calculated as follows: where  is the discharge coefficient, According to the numerical calculation, the Reynolds equation for damping orifice and recess is given as [24]: (3) For an aerostatic bearing, the velocities in directions x and z are zero, that is: ww == (4) For the static characteristics of an aerostatic thrust bearing, the differential on time is zero, that is: In summary, the Reynolds equation for damping orifices and recesses could be expressed as follows: 3 3 12 0 The Reynolds equation is reconstructed by using the Galerkin residual method and is discretized by the infinite element as follows. 22 Figure 4 shows the computational meshes. The model has a triangular element and the recesses are made as linear source. By the way, the pressure in recess is considered constant.   As shown in Figure 5, there is a high pressure distribution at the center of the designed aerostatic bearing with damping orifice and virtual recess, while there is a big pressure drop at the orifice of the bearing without damping orifice and virtual recess. The capacity of the designed aerostatic bearing with damping orifice and virtual recess achieved by simulation is 795 N at 4bar with 20μm, and by measurement is 814 N. Therefore it is a good match between the simulation result and measurement result.

Results and Discussion
Comparison of the designed aerostatic bearings with and without damping orifice and virtual recess on load capacity and stiffness at 4 bar is shown in Figure 6. It could be seen in Figure 6 that with damping orifice and virtual recess, the maximum capacity and stiffness of the designed aerostatic bearing increase from 1590 N to 2285 N, and from 106 N/μm to 145 N/μm at 4 bar respectively. Therefore, the capacity and stiffness of the designed aerostatic bearing are significantly improved by adding damping orifice and virtual recess.
The influence of supply pressure, diameter of orifice and number of damping orifices on the designed aerostatic bearing with damping orifice and virtual recess is analyzed. The relationship between supply pressure and static characteristic of the designed aerostatic bearing is shown in Figure 7. As shown in Figure 7(a) and 7(b), with the same supply pressure, the capacity of the designed aerostatic bearing gradually decreases but the mass flow gradually increases as the film thickness increases. For the same film thickness, the capacity of the designed aerostatic bearing and the mass flow increase as the supply pressure increases. It can be seen from Figure 7(c) that, the stiffness of the designed aerostatic bearing increases as the supply pressure increases. The maximum stiffness of 987 N/μm occurs at supply pressure of 3 bar and a film thickness of 14 μm. The maximum stiffness of 197 N/μm occurs at supply pressure of 5 bar and a film thickness of 11 μm. This means that the maximum stiffness of the designed aerostatic bearing moves toward the smaller film thickness The relationship between orifice diameter and static characteristic of aerostatic bearing at 3.5 bar is shown in Figure 8.  c) Relationship between stiffness and film thickness for different orifice diameter Figure 8 Relationship between orifice diameter and static characteristic of aerostatic bearing at 3.5 bar As shown in Figure 8, the maximum capacity is approximately constant, and the capacity and mass flow of the designed aerostatic bearing gradually decrease and the maximum stiffness gradually increases with the decrease in the orifice diameter for the same film thickness. The maximum mass flow is 1.41 L/min for the orifice diameter of 0.1 mm and the maximum mass flow is 15.96 L/min for the orifice diameter of 0.4 mm. In conclusion, the lower mass flow and higher stiffness of the designed aerostatic bearing can be achieved for a small film thickness by reducing the orifice diameter, which needs a bearing surface with average roughness of 0.1 microns.
The relationship between number of damping orifices and static characteristic of aerostatic bearing at 3.5 bar is shown in Figure 9. It can be seen from Figure 9 that the maximum load capacity is 1985 N and the maximum stiffness is 133.7 N/μm at n1=0, while the maximum load capacity is 1990 N and the maximum stiffness is 120.6 N/μm at n1=10. Then compare with the influence of supply pressure on the static characteristic of the bearing, adding damping orifices will not cause significant change of load capacity and stiffness.

Experimental Verfication
The designed aerostatic bearing with damping orifice and virtual recess is shown in Figure 10. It is fabricated with stainless steel to verify the theoretical analysis above.

Figure 10
Structure of designed aerostatic thrust bearing A servo electric cylinder is used to apply the external loads, and a force sensor is used to detect the applied loads. A spherical decoupling is used between the designed aerostatic bearing and the force sensors. Two inductive displacement sensors are used to detect the relative distance between the aerostatic bearing and granite, which is the film thickness. The pressure sensor is arranged as close as possible the aerostatic bearing to avoid the impact of pipeline on the supply pressure. The experimental apparatus is shown in Figure 11. Experiments are run to investigate the load capacity and pneumatic hammer stability of the aerostatic bearing with and without damping orifices and virtual recesses at 4 bar.

Figure 11
Experimental set-up for the designed aerostatic thrust bearing Figure 12 Load capacity of designed aerostatic bearing with 10 damping orifices at 4 bar Figure 13 Load capacity of designed aerostatic thrust bearing without damping orifices at 4 bar As shown in Figure 12 and 13, there is a difference of less than 6% between experimental and theoretical load capacities at 4 bar. Pneumatic hammer occurs from 18 μm to 22.5 μm in the designed aerostatic bearing with 10 damping orifices, and from 13 μm to 21.5 μm in the designed aerostatic bearing without damping orifices. So the damping orifice can be used to improve the stability of pneumatic hammer in an aerostatic bearing.

Conclusions
An aerostatic thrust bearing is designed using damping orifice and virtual recess to achieve a balance among the load capacity, stiffness and pneumatic hammer stability. It can be seen from the presentation above that: (1) With damping orifice and virtual recess, the maximum load capacity of the designed aerostatic bearing increases from 1590 N to 2285 N and its maximum stiffness increases from 106 N/μm to 145 N/μm at 4 bar. So the static characteristics of the designed aerostatic bearing are significantly improved. (2) The optimum stiffness characteristics of an aerostatic thrust bearing can be achieved by decreasing the number of damping orifice, the orifice diameter, and increasing the supply pressure and the number of recess. (3) The difference between the experimental and the theoretical results shows that the theoretical analysis is valid. Measurement results indicated that the range of pneumatic hammer decreases from 8.5 μm to 4.5 μm at 4 bar for the designed aerostatic bearing with damping orifice and virtual recess. Therefore it can be concluded that without significant change in load capacity and stiffness, the pneumatic hammer stability of an ·9· aerostatic thrust bearing can be improved by adding damping orifice.