A study on accuracy of porous journal air bearing
Introduction
Air bearings are widely used in ultra-precision machine tools such as diamond turning lathes. There are two main types of air bearings in practice including orifice and porous air bearings. Porous air bearing has easier structure, higher angular stiffness and lower cost than traditional orifice air bearing [1,2]. Therefore, many companies and researchers have focused on the porous air bearing.
A representative product example is the porous air spindle produced by Professional Instruments Company, whose total motion error is less than 6.4 nm throughout entire speed range of 500–10,000 rpm with a cutting accuracy of 35 nm and surface finish of 0.16 nm [3]. Another representative product example is the porous air spindle of Nanotech 250 produced by Moore Nanotechnology Systems with less than 12.5 nm motion error throughout entire speed range of 50–10,000 rpm. The obvious accuracy advantage and simple structure are the reasons why porous air bearings can be used more in ultra-precision machine tools.
In the existing research, Knapp et al. developed an approximate stiffness model for porous graphite air bearing thorough experiment and finite element method [4]. Cui et al. analyzed the influences of processing errors on static performance of porous air bearing, finding that the stiffness can be raised by increasing bearing geometric errors in spite of concavity errors [5]. Ohtsuka et al. studied the impact of wave shaped flats on hydrodynamic lubrication performance of a porous thrust bearing, finding that the optimum height of wave 15 μm provides a better hydrodynamic pressure [6]. For porous air bearings, the static and dynamic characteristics seem to be the focus of existing research rather than accuracy, and the study on accuracy of porous air bearings is rarely reported.
Actually, porous air bearings have many structures, and their accuracy theory must be different. According to the continuity of material, they can be divided into discrete and continuous bearings. According to the shape of bearing, they can be divided into journal, thrust and linear bearings. The porous air linear bearing is used to achieve linear motion for air guideway. The porous air journal and thrust bearings are used to achieve rotary motion for air spindle and rotary table. The porous air guideways are usually equipped with several discrete porous linear air pads. The porous air spindle and rotary table are usually equipped with continuous porous journal and thrust bearings. Among them, some researchers have studied the accuracy of porous air guideway.
Park et al. used a five degrees of freedom model by transfer function to predict error motions of an air guideway equipped with several discrete porous air pads, finding that the error motions predicted by transfer function model agree well with the experimental result [7]. They also used magnetic preload pads to improve the motion accuracy of the air guideway with discrete porous air pads, finding that the vertical, pitch and roll motion errors can reduced greatly [8]. However, the discrete structure of air guideway with several porous air pads is different from the continuous structure of porous journal and thrust bearings. This cannot provide a research idea for continuous porous journal and thrust air bearings. Although there are few studies on the accuracy of them, similar phenomenon existing in other types of air and hydrostatic bearings has been studied.
Yabe et al. studied the error motion of a orifice thrust air bearing, finding that the impact of part errors on performances is very small resulted from the error averaging effect [9]. Kane et al. designed a new type of self-compensated air bearing, finding that the radial motion error 0.05 μm is one-fiftieth of the part accuracy 2.5 μm [10]. Cappa et al. studied the motion accuracy of orifice journal air bearing, finding that the accuracy could be availably improved by increasing the feedhole number [11]. Zhang et al. used the approximate accuracy model to study the accuracy of hydrostatic journal bearing based on Reynolds equation, finding that the bearing with recess number 6 has better accuracy than those with recess numbers 3, 4, 5 [12]. Although the porous journal and thrust air bearings seem to be equivalent to the situation where the recess numbers of air or hydrostatic bearings are very large, their error averaging effect must have their own specific rule for their bearing structures are continuous.
Due to the continuity of porous material, it requires partial differential equations, not just Reynolds equation, to solve the distribution of air pressure inside and on the bearing, and then to calculate air film force and error motions. Therefore, the accuracy prediction of continuous porous journal and thrust air bearings is much more complicated than traditional orifice air and hydrostatic bearings. Since the accuracy of porous journal air bearing is the main factor which determines the accuracy of the porous spindle and rotary table compared with the porous thrust air bearing, this article only focuses on this type of bearing.
In this article, an accuracy model for continuous porous journal air bearing is firstly established to predict error motion based on partial differential equations. Then, averaging coefficient is defined to quantitatively characterize the error averaging ability and reveal the relationship between roundness error, error motion and structure parameter. The averaging coefficients for the shaft wave numbers 2~15 and bush wave numbers 2~7 are calculated to find the main influencing factor for motion accuracy. The content discussed is of significant importance for design and prediction of motion accuracy for the porous journal air bearings used widely in ultra-precision machine tools.
Section snippets
Structure and accuracy model
As shown in Fig. 1(a), the porous journal air bearing is made up of a bearing bush and a shaft. The bush is mounted in a mounting sleeve with a fastened cover and the bush material is porous graphite, or other. The air film will be formed in the gap of bush and shaft after the air enters from the annular air inlet. If the inner hole of bush and outer circle of shaft are ideal shapes, the circumferential air film is of equal thickness . As a result, the shaft center will coincide with the bush
Discussion
In order to discuss the influences of bush and shaft roundness errors on error averaging ability of porous journal air bearing, the averaging coefficients under the condition that the shaft has roundness errors with wave numbers 2~15 and the bush has roundness errors with wave numbers 2~7 are calculated first. Then the influence law for all kinds of roundness errors is summarized. Before the discussion on error motions and averaging coefficients, the parameters for calculation and results of
Examples of potential applications and limitation of accuracy model
The theory of this article has many potential applications for porous journal air bearing such as prediction of error motion, structural optimization and selection of parts grinding method. Despite this, the accuracy model also has its own limitation. The following is a brief discussion.
Conclusion
In order to predict the error motion of continuous porous journal air bearing, an accuracy model is built in this article to reveal the relationship among error motion, roundness error and structure parameter based on partial differential equations under quasi-static conditions. Averaging coefficient is defined to quantitatively characterize the error averaging ability of porous journal air bearing.
The theoretical analysis finds that whether the bush and shaft roundness errors match is the
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was financially supported by National Natural Science Foundation of China (Grant No. 51905056) and the Fundamental Research Funds for the Central Universities (Grant No. 2018CDXYJX0019).
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