Numerical investigation of static and dynamic characteristics of aerostatic thrust bearings with small feed holes

doi:10.1016/j.triboint.2010.01.002

Tribology International

Volume 43, Issue 8, August 2010, Pages 1353-1359

https://doi.org/10.1016/j.triboint.2010.01.002 Get rights and content

Abstract

Recently, laser beam machining and micro drills have made it easier to manufacture small feed holes of less than 0.1 mm diameter. Accordingly, aerostatic bearings with these small feed holes have become commercially available for improving bearing performance. In this work, an aerostatic annular thrust bearing with small feed holes of less than 0.05 mm diameter was treated and the static and the dynamic characteristics of this type of bearing were investigated numerically. In numerical calculations, computational fluid dynamics (CFD) was used to determine discharge coefficients for a small feed hole and the finite difference method (FDM) was used to obtain the bearing characteristics. In addition, the characteristics of this type of bearing were compared with those of aerostatic thrust bearings with typical compound restrictors to show clearly the features of this type of bearing.

Introduction

Because of their low friction and high accuracy of motion, aerostatic bearings have been applied successfully to various precision devices such as precision machine tools, precision measuring equipment and lithography associated production equipment. There are many types of restrictors used in aerostatic bearings, such as orifice-type, surface-compensated, compound, slot and porous. Among these, the orifice-type restrictor has been often used in actual devices because of its simplicity. There are two types of orifice-type restrictors: one is an orifice restrictor with a feed pocket and the other is an inherently compensated restrictor formed only by the feed hole. The mass flow rate of an orifice restrictor with a feed pocket is mainly affected by the diameter of the orifice, $π d_{s}^{2} / 4$ , whereas that of the inherently compensated restrictor is related to both the diameter of the orifice and the bearing clearance, πd_sh. As Kazimierski and Trojnarski [1] show numerically and experimentally, an aerostatic bearing with orifices with a feed pocket could have static stiffness values 20–30% larger than those for bearings with inherently compensated restrictors. In addition, to obtain the bearing performance numerically, the discharge coefficients of the orifice have to be determined. However, the airflow around these orifice-type restrictors is very complicated and, therefore, the discharge coefficients are usually assumed based on experimental results. Accordingly, many researchers have studied the discharge coefficients of an orifice-type restrictor and the static and dynamic characteristics of aerostatic bearings with orifice-type restrictors associated with discharge coefficients.

Belforte et al. [2] investigated experimentally discharge coefficients of these orifice-type restrictors. They reported that, for an orifice restrictor with a deep pocket, the flow characteristics could be described by using two discharge coefficients for the orifice and the pocket edge. Renn and Hsiao [3] studied the mass flow rate of aerostatic bearings with pocketed orifice-type restrictors experimentally and numerically by using computational fluid dynamics (CFD). They found that CFD simulation results show good agreement with experimental data. Li and Ding [4] also studied the bearing performance by using CFD to solve numerically airflow within an orifice and the bearing clearance. They reported that the simulations could predict the experimental load capacity of orifice-type aerostatic bearings very well.

Recently, it has become easy to drill small feed holes of less than 0.1 mm diameter using laser beam machining and micro drills. If the diameter of an orifice becomes smaller than 1/4 of the bearing clearance, it is expected that higher stiffness can be obtained without forming a feed pocket, because the restricted area is not dependent on the bearing clearance. Accordingly, aerostatic bearings with orifice-type restrictors using these small feed holes have become commercially available. This type of aerostatic bearing usually has numerous small feed holes of less than 0.05 mm diameter without a feed pocket, which makes the restrictor easier to manufacture than the pocketed orifice-type restrictor. However, to our knowledge, there are few reports on aerostatic bearings with such small feed holes.

The objectives of this paper were to clarify the discharge coefficients of a small feed hole of less than 0.05 mm diameter without a feed pocket by using CFD and to investigate the static and dynamic characteristics of this type of aerostatic bearings by using the finite difference method (FDM), which adopts the discharge coefficients obtained by CFD. In addition, by comparing aerostatic bearings with typical compound restrictors, the usefulness of aerostatic bearings with small feed holes was confirmed.

Section snippets

Aerostatic thrust bearings with small feed holes

Recently, aerostatic bearings with small feed restrictors of less than 0.1 mm diameter became commercially available. These bearings are considered to have relatively higher stiffness and damping coefficient because they can be operated at small bearing clearances, of less than 10 μm and no feed hole has a pocket. Fig. 1(a) shows the geometrical configuration of the aerostatic thrust bearing with small feed holes. Generally, this bearing has numerous feed holes on the bearing surface to avoid a

Discharge coefficients of aerostatic thrust bearings with small feed holes

To numerically obtain the static and dynamic characteristics, discharge coefficients in a small feed hole must be determined. As Renn and Hsiao reported, CFD simulations can predict well experimental discharge coefficients. Therefore, CFD simulations were adopted to determine the discharge coefficients in this paper.

In Fig. 1(a), the simulation model of the bearing with a small feed hole for CFD is shown as a hatched area. This model is three dimensional as shown in Fig. 1(c) and a part of the

Static and dynamic characteristics of aerostatic bearings with small feed holes

The static and dynamic characteristics of the aerostatic annular thrust bearings with small feed holes were investigated using FDM. In calculations of the dynamic characteristics, the ordinary perturbation method was adopted as described elsewhere [10].

Fig. 6 shows the relationship between bearing clearances and load capacities for the bearings with small feed holes of 0.03 and 0.05 mm diameter. As seen in this figure, the load capacity increased with the number of feed holes. The maximum load

Comparison with aerostatic bearings with typical compound restrictors

Fig. 9 shows the dynamic characteristics of an aerostatic thrust bearing with typical compound restrictors (see Fig. 1(b)). It was assumed that the number of feed holes in the bearings with compound restrictors was n=8, the feed holes were 0.1 and 0.15 mm in diameter and the depth of the shallow groove was 15 μm. In the calculations, the FDM described elsewhere [8], [9] was used and the discharge coefficient C_D was assumed to be 0.8. As seen in this figure, the aerostatic thrust bearings with

Conclusions

In this paper, the static and dynamic characteristics of aerostatic annular thrust bearings with small feed holes of less than 0.05 mm diameter were investigated numerically. CFD was used to determine the discharge coefficients of a small feed hole and the finite difference method was used to clarify the static and dynamic characteristics of aerostatic thrust bearings with small feed holes. As a result, the following conclusions might be drawn.

1.
In aerostatic bearings with small feed holes,

Notation

b	damping coefficient
C_D₁, C_D₂	discharge coefficients at two positions in a small feed hole
d_s	diameter of a small feed hole
g_d	depth of a circular groove
g_w	width of a circular groove
h₀	average bearing clearance
h	bearing clearance
k_s	static stiffness
K_s	dimensionless static stiffness= $h_{0} k_{s} / (p_{a} π r_{0}^{2})$
k_d	dynamic stiffness
n	number of feed holes
p	pressure
p_a	ambient pressure
p_s₀	supply pressure
p_s₁	pressure in a small feed hole
p_s₂	pressure at the boundary between the bearing clearance and a small feed hole
P	dimensionless

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There are more references available in the full text version of this article.

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