Aerostatic bearings design and analysis with the application to precision engineering: State-of-the-art and future perspectives

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Aerostatic bearings design and analysis with the application to precision engineering: State-of-the-art and future perspectives

T

Qiang Gaoa,b, Wanqun Chena,c,∗, Lihua Lua,∗∗, Dehong Huob, Kai Chengd a

School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, 150001, China School of Engineering, Newcastle University, Newcastle Upon Tyne, NE1 7RU, UK c EPSRC Future Metrology Hub, Centre for Precision Technologies, University of Huddersfield, Huddersfield, HD1 3DH, UK d School of Engineering and Design, Brunel University London, UB8 3PH, UK b

A R T I C LE I N FO

A B S T R A C T

Keywords: Air bearing Pneumatic hammer Air-induced vibration Precision engineering

Aerostatic bearings have been employed as an essential precision engineering element and enabling technology in numerous applications requiring precision and ultraprecision motions. This review paper aims at presenting the state-of-the-art in aerostatic bearings research and development with the emphasis on analytical and computational approaches for design and optimization of bearing performance, and further critically reviewing their future research directions and development trends in the coming decade and beyond. The paper is concluded with the discussion on the future trends and challenges in the aerostatic bearings research, and their applications and potential in precision engineering industries.

1. Introduction Aerostatic bearings, employing a pressurized air film with thickness at the micrometer level to support moving parts and resist external loads, have achieved a considerable performance improvement from the aspects of motion accuracy, friction, pollution and speed, compared with precision rolling bearings. Hence, aerostatic bearings have been widely adopted in various industries, such as the textile industry [1], handing and packaging [2], electronics and semi-conductors [3], metrology and ultra-precision machine tools [4], turbomachinery [5], machines for the food industry [6], and medical industry [7], etc. as illustrated in Fig. 1. The emergence of air lubrication technology can be dated back to 1828, when Willis [8] experimentally investigated the airflow status between two parallel plane surfaces. At the end of 19th century, Kingsbury [9] investigated the supporting characteristics of an air journal bearing by experiment, which validated the feasibility of gas bearing. Then, many patents about the gas bearing were issued in the early 1900s [10], such as the air thrust bearing designed by Westinghouse [11] in 1904 and the aerostatic journal bearing designed by Abbott [12] in 1916. However, in the following decades, only a few articles related with the basic principle of gas lubrication were published [13]. Fig. 2 presents the documents search results with keywords ‘air ∗

bearing or gas bearing’ in Scopus [14]. Fig. 2 a) depicts the worldwide annual quantity of articles related to air bearing since 1900. During World War II, gas lubrication technology first boomed in developed countries such as the United States, due to the demands from nuclear power and defense industries [15]. The bearing system was required to properly and continuously work under operating conditions of high precision, high temperature, low friction, and high speed. Specifically, in nuclear apparatus, the bearing is required to permit the reactor system to be sealed and left unattended for more than two decades. Moreover, the high temperature and irradiation should have no impact on the lubricant of bearing [16]. These demands cannot be satisfied by conventional rolling bearings or hydrostatic bearings, but gas bearing is considered a good candidate to meet these demands. Hence, plenty of experiments were conducted, and many data of experimental design were acquired during this period, leading to significant progress in developing gas lubrication technology. From these specialized beginnings, aerostatic bearings have been designed, manufactured and widely applied in various fields, such as high-speed dental drill [17], space-simulator [18], precision machine tools and measuring instruments [19]. It explains the reason for the first obvious ascent trend since 1960 in Fig. 2 a). From 1970s to 1990s, several monographs on the gas lubrication technology were published, which marks a mature period of the design theory of gas lubrication technology [13,16,20,21].

Corresponding author.School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, 150001, China. Corresponding author. E-mail addresses: [email protected] (W. Chen), [email protected] (L. Lu).

∗∗

https://doi.org/10.1016/j.triboint.2019.02.020 Received 28 November 2018; Received in revised form 12 February 2019; Accepted 13 February 2019 Available online 19 February 2019 0301-679X/ © 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Main advantages and application areas of aerostatic bearings.

Fig. 2. Documents search results with keywords of ‘air bearing’ in Scopus. a) Annual quantity of articles related to air bearings; b) Top ten countries on air bearings research.

significant impact on the overall performance of aerostatic bearings. The application trends of air bearing in the industry can be summarized as follows, first of all, in the machine tool industry, it is hoped that the load capacity and stiffness of the bearing can be further improved to meet the requirements of conventional machining; then, in the lithography and related industries, the air guide system is required to realize nano-level positioning control, so the micro-vibration in air bearing needs to be suppressed; at last, in the IC industry such as PCB machining, they expect to minimize air consumption while ensuring machining accuracy to reduce the production cost. However, despite the significant development of cutting-edge technologies in the design and manufacture of aerostatic bearings, the underlying design principle is still not fully understood, which greatly restricts the further performance improvement of aerostatic bearings. Therefore, to enable the research, development, and innovation of next-generation high-performance air bearing, it is necessary to comprehensively understand the underlying design principle of the air bearing. This paper aims to present a systemic review of the latest progress in aerostatic bearings research by analyzing and summarizing existing literature, which provides a comprehensive snapshot of the current situation and the development trend about the field of aerostatic

After the 1990s, the progressively increasing demands from both scientific research and industrial development for the high-precision semiconductor wafer, precision optics components, precision molds, micro parts, and microstructures etc. Which, stimulated the advance of ultra-precision machining technology, particularly ultra-precision machine tools. The demands for aerostatic bearings, one of the key components for enabling the ultra-precision machining, are also significantly increased. Design and development of high-performance air bearing have attracted increasing attention. Fig. 2 b) lists the ranking of top ten countries on the research of air bearing research. It can be seen that the United States, China, and Japan et al. take the leading position. The top ten countries in gas bearing research are also countries with high demands for ultra-precision equipment, which also confirms that gas bearings are the indispensable components in the field of ultraprecision equipment. Considering its extremely-high working requirements, including high-precision, high stability and higher speed (typically more than 10,000 rpm), etc. It remains challenging to develop high-performance aerostatic bearings. All of the static performance including load-carrying capacity and stiffness, dynamic characteristics, stability at ultrahigh speed, and the multi-physics coupling interaction effect, has a 2

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Then the pressure gradually decreases to pa from the orifice exit to the bearing clearance outlet. The load capacity FW of the bearing at the initial state is equal to the weight of moving parts W. After a load F acting upon the bearing, the clearance decreases from h0 to h1. A smaller clearance will reduce the drop of pressure through the restrictor, therefore, the pressure pd increases to pd′, which in turn ′ to balance external load F. Here, gives a higher load capacity FW ′ = W + F. A smaller clearance will lead to higher load capacity and FW vice versa.

bearings. The operating principle and classification of aerostatic bearings are introduced first, followed by a discussion on the current main analysis methods for the performance of aerostatic bearings. Subsequently, the latest research outcomes on aerostatic bearings are reviewed from the aspects of static characteristics, dynamic characteristics, and pneumatic hammer stability, and the air vortex induced vibration. Finally, the future trends and possible challenges in research, development, and innovation of aerostatic bearings are discussed. 2. Operating principle and classification of aerostatic bearings

2.3. Restrictor types of aerostatic bearings

2.1. Structural characteristics of aerostatic bearings

During working of aerostatic bearings, high-pressure air is supplied into the bearing clearance by restrictor. According to the type of restrictor, aerostatic bearings can be classified into several categories. Table 1 compares the performance of aerostatic bearings with various types of restrictors.

Aerostatic bearings are also named as externally pressurized air bearings, considering that the pressure of air film is generated by an external air supply system. Pressurized air is fed into the gap between two bearing surfaces through a specific restrictor and then discharged to the surrounding ambient from the exit edges of bearing clearance. The thin film acts as the lubricant in the clearance between stationary parts and moving parts. During the working state, the moving and stationary surfaces of air bearing do not contact, not only avoiding many problems of conventional bearings, such as wear and friction but also offering distinct merits for precision positioning. To obtain optimal performance of aerostatic bearings, the gap is required to be small enough to ensure the pressure. Generally, a clearance of 5–20 μm is adopted. From Fig. 3, it can be noted that the gap is about 4–15 times smaller comparing with the dimension of human hair. Considering the dust particle size is close to gap clearance, the air should be well filtered to ensure the bearing works properly. Moreover, sealing and cleaning of bearings are very important to prevent contamination. In addition, the form error of the mating surfaces is generally required to be better than one-tenth of clearance height, which presents a challenge to the manufacturing capability. To reach this stringent form tolerance requirement for bearing surfaces with varying shapes, including planar surfaces, cylindrical surfaces, and a spherical surface, precision machinery and methods are needed. Overall, the manufacturing and testing of air bearing have higher environmental requirements. Usually, clean and temperature controlled environments and vibration-isolated platform are required.

2.3.1. Annular orifice and simple orifice The most widely adopted restrictors are orifice restrictors because they are easy to manufacture, and plenty of design information is available to guide the design of this type of bearings. Orifice restrictor can be divided into two types [22], namely annular orifice (or inherently orifice type restrictor) and simple orifice (or pocketed orifice type restrictor), both are turbulent flow devices. The airflow passage area of annular orifice restrictor is πdh. d is the orifice diameter, and h is air film thickness. By manufacturing a small cylindrical recess with depth larger than d/4 around the orifice, a simple orifice restrictor could be acquired. Using such restrictor, the airflow passage area increase from πdh to πd2/4. It has been found that the performance of simple orifice type bearing is about thirty percent higher than that of bearing with annular orifice restrictor [16]. However, due to the recess of simple orifice restrictor, there is a large storage volume for air within the bearing, which negatively affects the stability of the bearing. Hence, the bearings with simple orifice restrictor are more prone to instability, whereas the bearing with the annular orifice is free from instability. 2.3.2. Slot restrictor The air inlet can be a narrow continuous or discrete slot with a width of several micrometers [23]. The slot restrictor is similar to orifice restrictor by replacing a small hole into rectangular slot. Slot restrictor can generate a laminar flow inside the air film, providing good stability comparing with orifice restrictor [24]. Though many available literature could provide guidelines for the design of aerostatic bearings with slot restrictor [25,26]. It remains to be rather difficult to manufacture narrow slot with the width of several micrometers. Thus, the application of aerostatic bearings with slot restrictor has been greatly limited.

2.2. Basic operating principle of aerostatic bearings Take a thrust bearing with pocket-type orifice restrictor as an example to illustrate the basic operating principle of aerostatic bearings, as shown in Fig. 4. High-pressure air flows through the feed restrictor into the clearance and then radiates to the edge of the air film where it flows into the ambient environment. h0 is the initial clearance. As the air passes through the feed orifices and enters the clearance between two bearing surfaces, the pressure at the inlet of restrictor ps falls to pd.

2.3.3. Groove restrictor The air inlet can also be specially arrayed grooves around orifice machined in one of the bearing surfaces [27]. When combining orifices with arrayed grooves, the pressure decays from orifices to the edge of the gas film can be inhibited, therefore a more uniform pressure distribution within the air film can be acquired. Consequently, the air bearing has higher stiffness and load capacity. But its manufacturing cost is usually higher than that of simple orifice feeding air bearings. 2.3.4. Porous restrictor Aerostatic porous bearings employ porous materials as a restrictor to feed the pressurized air into bearing clearance [28,29]. Since a large amount of tiny and tortuous passage within porous materials could serve as restrictors, namely porous orifices, a uniform pressure profile can be achieved inside bearing clearance. Thus, aerostatic porous bearing has substantially high stiffness, load capacity, damping, and pneumatic stability, compared with other types of aerostatic bearings

Fig. 3. Typical bearing clearance and tolerance compared with the size of a human hair. 3

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Fig. 4. Operating principle of aerostatic bearings. Table 1 Comparison of various types of restrictors. Restrictor type

Load capacity

Stiffness

Stability

Gas consumption

Manufacture

Low

Low

Fair

Small

Easy

High

High

Poor

Small

Easy

Medium

Medium

Good

Large

Medium

High

High

Good

Medium

Hard

High

High

Excellent

Large

Hard

Annular orifice

Simple orifice

Slot

Groove

Porous

theory. Then, Harrison [34] proposed Harrison's equations by combining Reynolds' equations with gas equations under the assumption of isothermal conditions, which have been known as the basic equations of gas lubrication today. After that, many flow models were proposed by to analyze the performance of aerostatic bearing. Shires [35] calculated the performance of a journal bearing based on axial flow model. A onedimensional model was adopted by regarding the journal bearing as a series of slots parallels to the axis, and the circumferential flow effects were compensated empirically. However, the experimental data acquired by Allen [36] indicated that the measured load capacity of the journal bearing is half of the calculated results, which proves the circumferential flow of journal bearing cannot be ignored. Lemon [37] proposed a simplified model for analyzing the performance of aerostatic journal bearing considering the effect of circumferential flow. The calculated performance curves agree well with the experimental curves. Mori et al. [38] compared several flow models with consideration of the inertia effect to explain the experimental results, the flow status including the entrance effects near the orifice, the growth of boundary layer and the occurrence of shock wave were discussed. Besides, depending on the restrictor type of aerostatic bearing, reasonable flow model should be adopted to obtain an accurate result. For example, the flow through the feed holes is generally considered as the flow through a nozzle. While the flow inside an aerostatic porous bearing is generally characterized by Darcy's law [39]. Recent years, to investigate the turbulent flow status inside bearing clearance, many computational fluid dynamics based turbulent models, such as k-ε and large eddy simulation, have been adopted in the research of aerostatic bearing [40].

[30,31]. Moreover, owing to the existence of a large amount of porous orifice, the porous media surface could provide better bearing protection. When air supply failure occurs, the bearing could be allowed to adjust spontaneously, without damaging the support surface. However, due to its special material properties, during the machining process, the porous material is easy to be clogged. Moreover, it may release very tiny particles under working conditions, which is not desirable in clean machining environments such as equipment for manufacturing the semiconductor wafer. In general, all types of restrictor have their advantages and disadvantages. In order to further enhance the performances of aerostatic bearings comprehensively, different restrictor methods could be combined as compound restrictors, which will be discussed in Section 4.

3. Performance analysis methods of aerostatic bearings To develop high-quality aerostatic bearings, it is imperative to accurately predicting its performance with varying design parameters and operating conditions at the design stage, particularly predicting the pressure profile inside the clearance resulting from the injection of airflow at various source points [20]. For this reason, it is necessary to determine methods of reasonable analysis. The earliest efforts to theoretically calculate the performance of gas bearings can be dated back to the late 19th century [32]. In 1886, Osborne Reynolds [33] proposed the famous Reynolds' equations by combining the simplified Navier-Stokes equations with a continuity equation, which provided mathematical foundations for the lubrication 4

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algorithm, compared with FEM. According to the researches of Lo et al. [50] and Liu et al. [51], the performance of aerostatic bearings, such as pressure distribution, loading capacity, stiffness, volume flow rate, can be estimated conveniently by FDM. But the disadvantages of FDM are also obvious, such as poor computational efficiency and convergence. To overcome this problem, Li et al. [52] put forward a novel method to reduce the iteration times and enhance the insensibility for initial conditions by correcting the pressure ratios of the orifice in each iteration loop. In addition to orifice type aerostatic bearings, the performance of other types of aerostatic bearings can also be investigated by FEM or FDM, such as slot type [53], grooved type [54] and porous type [55,56]. When using FEM or FDM to solve the Reynolds equations of an air film, the size of the restrictor is usually ignored and considered as a point, and the actual mass flow rate of an orifice is generally corrected by a constant, called discharge coefficient, according to its ideal mass flow rate. It indicates that the discharge coefficient plays a significant role which affects the accuracy of FEM and FDM dramatically [57]. Many works have been done to investigate the discharge coefficient of the restrictor [58]. The discharge coefficients with varying orifice diameter and film thickness acquired by Chang et al. [59,60] are shown in Fig. 6. It clearly shows that the discharge coefficient decreases dramatically with the decrease of air film thickness and the increase of orifice size. Belforte et al. [61] proposed an experimental formula to predict the discharge coefficient considering the Reynolds number and the geometry of air feeding system. To accurately analyze the performance of an aerostatic journal bearing, Song et al. [62] modified Belforte's formula by considering the rotational speed and varying film thickness along the circumferential direction. Zhang et al. [63] proposed a numerical method to study the discharge coefficients of inherent orifice restrictor with consideration of the effects of airflow status and geometrical parameters. In most cases, FEM and FDM could predict the performance of aerostatic bearings with adequate accuracy. However, for simple orifice type restrictor, the pressure inside the recess keeps constant. Simplifying the orifice dimension as a point may lead to large calculation errors, especially in small air film thickness regions. Besides, as this approach cannot provide details of airflow status, thus it could not be applied to investigate the turbulent flow status inside air film.

Currently, several numerical methods (including engineering simplification algorithm (ESA), finite element method (FEM) and finite difference method (FDM), computational fluid dynamics (CFD), and multi-physics coupling method (MPCM)) and experimental method have been applied to investigate the performance of aerostatic bearings. 3.1. Engineering simplification algorithm In the calculation process using ESA, the two-dimensional flow within the air film is simplified into a one-dimensional flow by using a linear gas source assumption, which significantly simplifies the solution process of the Reynolds equation. Liu et al. [21] systemically investigated the performance of aerostatic journal and thrust bearings through ESA, the detailed process of calculation is shown in their monography. Colombo et al. [41–43] investigated the performance of rectangular aerostatic thrust bearings with single and multiple holes by proposing a lumped parameters model. The accuracy of the proposed model was verified using FEM and FDM. It was found that the analysis process using ESA is easy to operate, and the existing design curve could greatly simplify the design and calculation process. However, due to the deviation between the assumed linear gas source and the actual gas flow, ESA could produce a large error for a gas bearing with a small gap. In addition, the ESA can only be used to calculate the bearing in the static state. When considering the actual working speed of the bearing, the ESA will no longer be effective. 3.2. Finite element & finite difference methods Finite element and finite difference methods have become the two main methods for solving the gas-lubricated Reynolds equation [44–46]. Finite element method, owing to its high calculation accuracy and flexibility for irregular bearing geometry, has been widely used in the performance estimation of aerostatic bearings. Liu et al. [21,47,48] detailed the FEM-based algorithm for analyzing the performance of aerostatic journal and thrust bearings. To improve the stability and convergence of this algorithm, the pressure of orifice exit is corrected by using the proportional division method in each iteration. Moreover, to improve the calculation efficiency, only a basic sector of the aerostatic thrust bearing with the angle of π/n (n is orifice number) is discretized, see Fig. 5 a). And the pressure distribution of the air film is shown in Fig. 5 b). However, due to the complex meshing, the calculation process and iterative algorithm, it is hard to formulate the calculation program as a general program [49]. As for FDM, it is rather conveniently-used for performance estimation of aerostatic bearings for less complex meshing and iterative

3.3. Computational fluid dynamics Since the late 1970s, due to the advent of the high-performance computer along with the development of accurate numerical algorithms

Fig. 5. Analyzing the pressure distribution of a thrust bearing based on FEM. a) Computational mesh of the air film; b) Dimensionless pressure distribution of the air film. 5

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capacity and mass flow rate. Wang et al. [72] analyzed the flow status of a new type spherical aerostatic bearing by adopting CFD simulation, and a similar phenomenon was observed comparing with that of Eleshaky's [70] study. In some cases, CFD provides an efficient tool to explain the deviation between the experimental results and theoretical results [73]. Though CFD has so many merits, it is rather time-consuming. A typical CFD analysis process generally consists of preprocessing, solving, and post-processing. During the preprocessing, several meshes with different resolutions should be prepared to ensure the calculation results are independent of mesh resolution, which could take about eighty percent of the total time for discretizing fluid domain in preprocessing. Moreover, the parameterization in CFD simulation is rather difficult, so any changes in the geometrical parameters of the aerostatic bearing will lead to the recalculation of the simulation. In addition, the CFD method may encounter the convergence problem in some complex cases. Fig. 6. Effect of the orifice diameter and film thickness on discharge coefficient [59].

3.4. Multi-physics coupling method

such as finite volume method [64], CFD has offered an alternative way to study fluid dynamics problems [65]. Especially in recent two decades, with the rapid progress in commercial CFD software, such as FLUENT [66] and ANSYS CFX [67], CFD has become an increasingly popular tool to investigate the performance of aerostatic bearings. Based on CFD, the details of flow status inside air film, such as the pressure contour, the velocity vector and streamline, could be observed. Fox example, Yoshimoto et al. [68] conducted a numerical study on the pressure distribution of an aerostatic thrust bearing. Pressure depression phenomenon [69] and turbulent airflow region were observed in bearing clearance. Eleshaky et al. [70] numerically investigated the pressure depression phenomenon near the orifice outlet by adopting CFD simulation. The computational grid is shown in Fig. 7 a). The calculation results indicate that there is a shock region where coherent structures were observed and airflow transits from supersonic flow to subsonic flow, as shown in Fig. 7b) and c). Computational fluid dynamics also show better flexibility and accuracy for analyzing the performance of aerostatic bearings with different types of the restrictor and geometrical parameters of air film. Especially, it shows superiority compared to FEM and FDM in investigating the impact of varying dimension and shape of air film on its performance. For example, Chen et al. [71] analyzed the influence of recess shape on the bearing performance based on CFD simulation, which indicates that the recess shape has an important impact on load

During past decades, driven by the massive demands from various engineering industries, such as electronics, biomedical, and IT components [74,75], micromachining technology has fast advanced. For example, in PCB micro drilling, the aerostatic spindle is expected to work at the rotational speed of higher than 100,000 rpm [76]. In such working conditions, viscous heat of the aerostatic bearings and the heat generated by the built-in motor will cause large temperature gradient between the internal heat source and cooling system, which in turn affects the performance of aerostatic bearings. Moreover, in applications large load capacity or high stiffness is also required, so the pressurized air film force is able to cause deformation of bearing plate, sequentially change the clearance height and the static characteristics of aerostatic bearings [77]. Besides, the multi-physics coupling interactions among structural deformation, temperature distribution, dynamics, fluid pressure and electromagnetic, could also greatly influence the performance of aerostatic bearings, such as static characteristic, dynamic characteristic, thermal error, and stability etc. [78,79]. Thus, increasing attention has been paid on this issue in recent years [80–82]. Chien et al. [83] numerically analyzed the temperature distributions of a high-speed aerostatic spindle considering the cooling system. To facilitate the multi-physics coupling analysis of an aerostatic bearing system, Li [79], Liu [84] and Gao [85] et al. proposed several different modeling methods to simulate the multi-physics coupling effect, respectively. Take Gao's [85] research as an example, a multi-physics integrated

Fig. 7. Computational mesh of air film and analysis on simulation results [70]. a) Computational mesh of air film; b) Velocity vector of air film; c) Shock region. 6

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Fig. 8. Multi-physics integrated modeling of a high-speed aerostatic spindle system [85]. a) Spindle configuration; b) Cooling system; c) Temperature distribution of shaft; d) Temperature distribution of cooling system.

aerostatic bearing. Especially at the early stage, it was the principal method due to the insufficient development of theoretical modeling methods. With the development of the flow model and numerical method, it has been widely employed to verify the numerical results and various test rigs were developed [89]. For example, Li et al. [90] numerically investigated the performance of aerostatic thrust bearing with varying geometrical parameters. To verify the accuracy of the numerical model, the load capacity curve of aerostatic bearing was tested. Belforte et al. [61] designed a test apparatus for testing the pressure profile of orifice restricted aerostatic thrust bearing, and a formula to calculate the discharge coefficients of inherent orifice restrictor and pocketed orifice restrictor were proposed based on experiential results. Some test rigs with similar configuration were also developed by Yoshimoto [68] and Bhat [91] for their own research. Recently, Zhou et al. [92] proposed a novel pressure distribution test method in aerostatic thrust bearings by using pressure-sensitive film and its feasibility and accuracy were verify experimentally. In addition, in certain research cases [93], experimental results provide essential information for numerical modeling. There were also some test rigs designed for testing the performance of aerostatic bearing from the perspective of dynamic characteristics. Yu et al. [94] developed an experiment bench for testing the nonlinear dynamic stiffness of aerostatic bearings under external perturbation. The experimental results indicate that the dynamic stiffness and damping coefficients of the aerostatic bearing are nonlinear frequency dependent due to the compressibility of gas. Song et al. [62] measured the horizontal and vertical position of an aerostatic journal bearing by laser displacement sensor under varying rotating speed, which proves that the attitude angles of aerostatic journal bearing increase with the increase of rotating speed. Belforte et al. [95] carried out an experimental investigation to study the stability of a high-speed rotor supported by an aerostatic bearing. It was indicated that no whirling instability occur under the operating speed up to 60,000rmp. It is noteworthy that the visualization technology combined with numerical simulation has been adopted in some scholars’ research in recent years. For example, to investigate the thermal behavior of a high-speed aerostatic spindle, Gao et al. [85] compared the temperature field of an aerostatic spindle system acquired by a thermal imager and multiphysics coupling simulation results. Yoshimura et al. [96] developed a visualization apparatus to observe the turbulent flow near the outlet

model for analyzing a high-speed air-spindle is proposed based on CFD and FEM. The configuration of a spindle is illustrated in Fig. 8 a) and its cooling system is demonstrated in Fig. 8 b). The interaction phenomenon between structural deformation, temperature distribution, and the electromagnetic field is studied. The temperature distributions of the shaft and cooling system are shown in Fig. 8 c) and Fig. 8 d). The change of bearing clearance caused by thermal deformation of the shaft, thrust plate and sleeve can be considered to predict the actual working status of air bearing based on the proposed simulation model. Lu and Gao et al. [86–88] conducted theoretical and experimental investigations on the static performance of aerostatic bearing with consideration of structural deformation caused by air film pressure. The influence of some crucial structural dimensions on the static performance, such as load carrying capacity and stiffness, is investigated to provide theoretical guidance for the design of structural parts. The above researches indicate that the MPCM is capable to accurately predict the influence of complex multi-physics fields coupling interactions on the performance of aerostatic bearings. However, the modeling process using MPCM is very sophisticated, and the data exchange between each sub-model may lead to the convergence problem. Besides, the high computational cost also limits the application of this approach greatly. Table 2 briefly compares the advantage and disadvantage of each performance analysis methods of aerostatic bearings, which should be taken into full consideration for the sake of sufficient accuracy and efficiency. 3.5. Experimental method The experimental method is an important method on the research of Table 2 Comparison of different modeling methods. Methods

Accuracy

Modeling difficulty

Computational efficiency

Convergence

Details of flow status

ESA FEM & FDM CFD MPCM

Low Medium High High

Easy Medium Medium Hard

Good Fair Fair Poor

Good Fair Fair Poor

No No Yes Yes

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genetic algorithm is adopted to search the optimal designs. Generally, the design of aerostatic bearing is rather difficult for ordinary engineers because they are lack of specialist knowledge and experience in this area. To facilitate the design process of aerostatic bearings for them, some expert systems for aerostatic bearing design are developed to bridge the gap between inexperienced designers and the design experience [106]. For example, two expert systems were proposed by Rowe et al. [107] and Liang et al. [108], which are able to guide the unexperienced designer based on built-in design knowledge. Considering the important impact of the restrictor on the static performance of aerostatic bearings, not only the geometrical parameters of restrictor, but also the types of restrictor were optimized to improve the performance of aerostatic bearings. Park et al. [109] designed an air journal bearing with novel slot restrictor. The slot clearance has unequal width along the circumferential direction. The experiment results indicated that the proposed unequal slot clearance bearing offered better performance than conventional slot-restricted bearings. Ise et al. [110] proposed an aerostatic journal bearing withrectangular slot along its axial direction. It was experimentally verified that this new slot not only contributes to the improvement of bearing stiffness, but could also eliminate the whirling vibration at high rotational speed over 22,000 rpm. Cui et al. [111] studied the performance of an aerostatic porous annular thrust bearing and found that the damping ratio at the natural frequency of the aerostatic porous bearing system is improved. Belforte et al. [112] investigated the permeability of porous media. The influence of grain size of porous media on its porous resistances is experimentally and theoretically studied. Cui et al. [93] investigated the impact of circumferential waviness, taper, concavity, and convexity on the static characteristics of porous aerostatic journal bearings. The film thickness was first modeled based on the experimental results, as shown in Fig. 10, and its influence on the static performance of bearing was investigated numerically. Wang et al. [113–115] optimized the aerostatic porous bearing by combing genetic algorithm and hypercubedividing method. In addition, several novel types of restrictor and compound restrictors have been developed to further enhance the bearing performance of bearings. By using microfabrication technology, Fan et al. [116] fabricated multiple-microhole bearing, as shown in Fig. 11a and b). It was found that the prototype of air bearing exhibited good performance, though the parameter identification and process control were very time-consuming. Belforte et al. [117,118] investigated the influence of a circumferential groove on the static performance of an aerostatic thrust bearing with orifice restrictor, as shown in Fig. 11 c). The results indicate that

region of aerostatic bearing. Generally, the experimental results have high reliability. However, the precondition is that the design of a test apparatus is reasonable and measuring instruments have adequate precision. Besides, the manufacturing of the test apparatus and the testing process may quite costly and time-consuming. Nevertheless, the experimental method will play an important role in various aspects of aerostatic bearing research in the future. 4. State-of-the-art of aerostatic bearing research 4.1. Static characteristics Static stiffness and load carrying capacity are the two main static characteristics of aerostatic bearings. Compared with rolling bearings and hydrostatic bearings, the static characteristics of aerostatic bearings are much weaker. The static characteristics of aerostatic bearings are influenced by various factors [97], including the amount and layout of restrictor [98], the geometrical dimensions of restrictor [90], restrictor shape [99], the operating condition [100] like the pressure of supply source and rotating speed, and the waviness error of bearing surfaces [101]. Therefore, the prediction and improvement of the static performance of aerostatic bearings in the design stage are an important direction of aerostatic bearing research. Comparing with conventional rolling bearings, aerostatic bearings are still not standardized currently. The design of aerostatic bearings is usually performed with certain specifications and restrictions. Moreover, the performance of aerostatic bearings has many influence factors. For enabling the optimal design of aerostatic bearings, many researches have been carried out [102]. For instance, Bhat et al. [91] optimized the design variables of an aerostatic flat pad bearing, such as the pressure of air source and amount of orifices, a good trade-off between stiffness, load carrying capacity, and the flow rate were achieved. Chang et al. [103] improve the stiffness of an aerostatic bearing by optimizing its orifice diameter and pocket size. To save calculation time, a modified particle swarm optimization (MPSO) algorithm is proposed. To avoid the occurrence of local optimal value in the iteration process, the mutation of the genetic algorithm is adopted in proposed MPSO, thereby increasing the convergence speed, as shown in Fig. 9 a). Chan et al. [104] designed a new multi-objective algorithm (MOA) by combining particle swarm optimization and Pareto methods. The efficiency of MOA and Hypercube dividing method (HDM) is compared for convex and nonconvex problems, as shown in Fig. 9 b). Gao et al. [105] improve the performance of an aerostatic bearing using a parametric CFD modeling method, and multi-objective optimization

Fig. 9. The efficiency analysis of optimization algorithm proposed by Chang [103] and Chan [104]. a) Efficiency comparison of MPSO and PSO; b) Efficiency comparison of MOA and HDM. 8

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Fig. 10. Manufacturing errors of aerostatic porous journal bearing [93].

instability, Huang and Chang [121], integrated an axial passive magnetic bearing into the aerostatic bearing. With the magnetic bearing, not only the bearing capacity can be improved, but also the pneumatic hammer effect can be significantly damped. Extensive researches have been performed on the static performance of aerostatic bearings with varying type of restrictors, but most of these researches were focused on orifice restricted aerostatic bearings. The design principle for those with other types of restrictors is still not sufficiently elucidated. Currently, several novel types of restrictors and compound restrictors have been designed and proved to be effective to enhance the performance of aerostatic bearings, which points out a new direction to the research and development of high-performance air bearings.

the stiffness and the load carrying capacity were improved significantly by the grooves, especially at low clearance, with little air flow increases. Du et al. [54] studied the performance of aerostatic journal bearing with pressure-equalizing grooves. A systematical investigation of the influence of groove length, groove depth, and the location and layout of grooves on the load capacity and stiffness were conducted. The investigations conducted by Belfort and Du are meaningful for design of aerostatic bearings by combining the groove type restrictor and orifice restrictor together. By combining the herringbone grooves and orifice, Stanev et al. [119] investigated the influence of the grooves on the air bearing performance, as shown in Fig. 11 d). It is found that the grooves are significantly useful to improve its resistance to highspeed whirl instability. To improve the title stiffness of the thrust bearing, a feed-hole restrictor and a groove compensation restrictor were combined by Nakamura and Yoshimoto [27], the result indicated that aerostatic thrust bearing with compound restrictors can have larger tilt moments than bearing with feed-hole restrictors. For improving the performance of the prototype air-bearing spindle, Mizumoto et al. [120] proposed an accuracy improvement method by using an active aerodynamic bearing, the maximum controllable rotational speed achieved 700 Hz (42,000/min). In order to eliminate the pneumatic

4.2. Dynamic characteristics and pneumatic hammer stability 4.2.1. Dynamic characteristics In order to acquire a feasible design of aerostatic bearing, the dynamic characteristics of aerostatic should also be assessed to avoid the occurrence of instability. Many researches have been reported about investigate the dynamic stiffness and damping of aerostatic bearings

Fig. 11. Novel design of restrictor and compound restrictor [116–119]. a) Multiple-microhole bearing; b) Pressure distribution of multiple-microhole bearing. c) Thrust bearing with circumferential grooves; d) Journal bearing with herringbone grooves. 9

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aerostatic bearing manufacturers usually limit their recommended pressure to 5 bar, which limits the performance of aerostatic bearings. Clarifying the formation mechanism of pneumatic hammer vibration is essential to eliminate the bearing vibration instability. In 1970, Powel [16] described this phenomenon in his monograph. It is believed that the 180° phase difference of air film motion was the main cause for pneumatic hammer vibration. The influence of orifice diameter and recess depth on pneumatic hammer vibration were studied through experiments. However, the generation of the phase difference has still not been exactly explained, and the stability boundary condition of pneumatic hammer vibration is not determined. In 1999, Wang et al. [13] summarized three generation conditions for pneumatic hammer vibration, including size of gas volume, excitation vibration source and internal pressure. It was pointed out that pneumatic hammer resonance is prone to occur if the interference frequency was near the natural frequency of the bearing structure. However, due to an oversimplified single degree of freedom vibration model, it is difficult to clearly describe the specific vibration process.

[122–124]. To investigate the pneumatic hammer and rotationally induced instability of pressurized gas journal bearings, Lund [125] proposed a perturbation method to analyze the dynamic coefficients. Bhat et al. [126] investigated the dynamic performance of flat pad aerostatic bearing with inherent orifice-type restrictor. The dynamic characteristics of air bearing with varying geometrical parameters and varying operating conditions were investigated numerically. Negative-stiffness was observed when the aerostatic bearing has an orifice diameter of 0.05 mm and a gap height of 15 μm, which indicates that pneumatic hammer tends to occur at small orifice diameter and large gap height. Miyatake and Nishio et al. [127,128] investigated the dynamic characteristics of inherently compensated aerostatic bearing with orifice diameter less than 0.05 mm. According to their research, dynamic stiffness and damping coefficient can be improved by using small feed holes, and the maximum dynamic stiffness was decline with the increase of feed holes’ number. Besides, it suggests that the impact of the bearing surface roughness cannot be ignored when analyzing the dynamic characteristics of aerostatic thrust bearings. Otsu et al. [129] studied the dynamic characteristics of an aerostatic porous journal bearing with a surface restricted layer. Their research indicated that surface restricted layer could improve the dynamic stiffness of aerostatic porous journal bearing, and porous bearing with larger permeability can achieve higher dynamic performance. To enhance the dynamic performance of aerostatic bearing, Al-Bender and Aguirre et al. [82,130] proposed an active compensation strategy by adopting piezoelectric actuators to dynamically control the gap shape of the aerostatic bearing. A serial of works on theoretical modeling and experimental verification was conducted, which proves that the active compensation strategy is efficient to improve the dynamic performance of aerostatic bearing. Mori et al. [131,132] first reported that the inertia effect of air flow has a dramatic impact on the squeeze damping and whirl instability of inherently compensated bearings. To take the inertia effect of air flow into account in their modeling, the Reynolds equations were modified by considering the contribution from the inertia forces. The numerical results indicated that the inertia effect could increase the whirl ratio and damping coefficient of the aerostatic bearings. Belforte et al. [133] also proposed a modeling method for analyzing the dynamic behavior and the instability problem of aerostatic journal bearing considering the inertial effects. Their research suggested that the inertial effect cannot be ignored when modified Reynolds number is larger than 1. By adopting the Mori's method, Otsu et al. [134] investigated the stability of an aerostatic bearing with compound restrictors considering the inertia effect. The above research indicated that the inertia effect of air flow has a great influence on the stability of aerostatic bearings. The above researches are meaningful to provide guidance for the design of aerostatic bearing from the perspective of dynamic performance.

4.2.2.1. Methods for studying the pneumatic hammer stability. At present, accurately estimating the dynamic performance of air bearing has attracted extensive attention, and several methods have been employed to predict the pneumatic hammer stability of aerostatic bearings as following. (1) Stable boundary method Stable boundary method adopts the dynamical equations of the aerostatic bearing system, and flow continuity condition of air film under small disturbance to deduce the stability criterion of bearings. The pneumatic hammer stability of aerostatic bearings under certain condition could be quickly predicted using this method. Thus, this method is widely adopted to analyze the pneumatic hammer stability of various types of aerostatic bearings [137–139]. Mohamed et al. [140] studied the pneumatic hammer stability of a circular aerostatic thrust bearing, and stability maps were acquired to predict the stability range of aerostatic bearing with varying geometrical parameters. Du et al. [141] investigated the pneumatic hammer stability of an aerostatic bearing with a circumferential groove. The stable boundary of bearing with varying air film thickness, orifice diameter, and air supply pressure is acquired to ensure the stability of designed bearing. However, the influence of the bearing mass is ignored in stable boundary method, hence the calculation results are not accurate enough. (2) Dynamic stiffness and damping method Some researchers determined the stability of aerostatic bearings based on its damping characteristic. Bhat [126] et al. studied the dynamic characteristics of an inherently compensated thrust bearing using FEM. The influence of orifice diameter, supply pressure and gap height on the dynamic performance of aerostatic bearing is acquired. Cui et al. and Chen et al. [142,143] obtained the dynamic stiffness and dynamic damping of aerostatic thrust bearing by using finite volume method with dynamic grid technique. However, the modeling process is complex and time-consuming. Overall, the dynamic stiffness and damping method is a frequency-domain analysis method which can quantitatively calculate the dynamic characteristic of aerostatic bearing. However, the structural stiffness and damping are not taken into consideration by this method. Therefore, inaccurate results may be acquired by this method in some cases.

4.2.2. Pneumatic hammer stability The vibration of aerostatic bearings has an important influence on bearing performance, due to low damping of the air film. The vibrations of aerostatic bearings can be summarized into the two types: one is micro-vibration, with small amplitude ranging from several nanometers to several tens of nanometers, see 4.3. The other is pneumatic hammer vibration with large amplitude and continuous whistling. The pneumatic hammer vibration is a kind of self-excited vibration, which is generated by the self-movement of the bearing system. Thus it does not require continuous excitation from the outside to maintain certain amplitude [135,136]. It will destroy the stability of the air-floating support, cause the gas film lubricating layer to fail, and damage the bearing surfaces. For aerostatic bearings in industrial applications, self-excited vibration has induced most serious problems. However, the physical causes for this vibration are still not elucidated sufficiently. Pneumatic hammer vibrations are prone to occur at high air supply pressure. So,

(3) Graphic method Graphic method is a qualitative method to investigate the pneumatic hammer stability of aerostatic bearing. By solving the time-dependent vibration model of aerostatic bearing, the transient vibration 10

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vibrational energy, thereby inhibiting pneumatic hammer vibration. Several novel designs were proposed such as rubber O-ring [152], Sixsmith [153], dual gas film [154,155], tangential orifices [156] and combined types, as shown in Fig. 13. Rubber O-ring could improve the stability of aerostatic bearings dramatically, which has been experimentally verified. But, it is not suitable for high or low temperature environments. To overcome this limitation, a novel design of aerostatic bearings with stabilizing chambers was proposed by Sixsmith in 1959 [149]. By adjusting the diameter of exit orifice, chamber volume, or both, the pneumatic hammer stability and critical instability speed of aerostatic spindle could be enhanced. But the air supply pressure and air consumption of Sixsmith type aerostatic bearings are higher than conventional ones, and a lower load capacity is acquired. Moreover, the manufacturing of those bearings were complex. Nevertheless, it provided an efficient method to keep the bearing system stable in extremely-high temperature case or low temperature case that rubber O-ring cannot be adopted. In the 1960s, a novel design with dual air film was proposed, which also provided an alternative solution to overcome the restriction of rubber O-ring [157]. The stability of aerostatic bearing system can be improved dramatically by using dual air film. Tangential orifices are first proposed by Ales Tondl in 1967 [158]. He investigated the instability of rotor-aerostatic bearing system at high rotating speed. The results indicated that the tangential orifices could improve the stability dramatically. In addition, the critical instability speed is higher about thirty percent when the rotating direction of the rotor is opposite to the tangent direction of orifice supply direction. In the above methods, the Sixsmith type is currently seldom adopted due to high air supply pressure and high air consumption. Rubber Oring, dual air film and tangential orifices are three common ways to enhance the stability of the aerostatic bearing system. Chen et al. designed a novel aerostatic bearing by combining the rubber O-ring, dual air film and tangential orifices [159], it further improved the stability of the aerostatic bearing system. Wardle et al. [160,161] proposed a novel design of oil-lubricated squeeze film damper to enhance the dynamic stiffness of an aerostatic bearing which adopted in a turning machine, and the surface roughness could be improved obviously by the proposed design. Until now, no consensus has been reached on the cause of pneumatic hammer vibration. There are various causes for pneumatic hammer vibration, including the compressibility of gas, the negative damping of the system, the interference force, and the phase difference of air film motion. The generation process of pneumatic hammer vibration is still not described. Many key issues in the formation mechanism of air hammer vibration, such as the negative effect of turbulent flow, the influence of structural mass, stiffness and damping, the influence of fluid-structure interaction and squeeze film effect, are not clearly elucidated. Therefore, the underlying mechanism of pneumatic hammer vibration needs further research and exploration.

behaviors of bearing can be acquired, which could be utilized to determine its stability. Chen et al. [144,145] acquired the time-varying vibration displacement of X-shaped groove aerostatic bearing by adopting a resistance network method and finite element method, and a bearing was considered unstable if its vibration amplitude grows gradually under a small instantaneous impulse. Kong et al. [146] investigated the transient vibration behavior of an aerostatic bearing using single degree vibration model. It indicated that the air film thickness has an important effect on the pneumatic hammer stability, and the pneumatic hammer stability could be improved by reducing the supply pressure. However, the graphic method can only qualitatively determine the pneumatic hammer stability of aerostatic bearing as it only considers the vibration behavior of aerostatic bearing in the time domain. (4) Experimental method Some researchers investigated the pneumatic hammer performance experimentally. Talukder et al. [147] investigated various factors affecting the occurrence of pneumatic hammer experimentally, including recess depth, orifice diameter and bearing mass. However, they found that it was difficult to acquire repeatable results due to the influence of external damping and support stiffness. Kong et al. [148] experimentally studied the pneumatic hammer phenomenon of aerostatic thrust bearing. They considered that the resonant of bearing system induced by the turbulent airflow and the fluid-structure interaction effect between bearing structure and air film is the main reason for pneumatic hammer. Ye et al. [149] investigated the pneumatic hammer vibration mode of aerostatic bearing system with varying recess shapes experimentally, and found that the recess geometry also has important impact on the stable region of bearing. Experimental method can acquire the vibration behavior of bearing structure, but it is difficult to explain the mechanism of pneumatic hammer instability theoretically.

4.2.2.2. Suppression strategies of pneumatic hammer vibration. (1) Structural optimization Optimizing the bearing structural parameters have been found to be effective to reduce the occurrence of pneumatic hammer. Mizumoto et al. [150] reduced the size of the orifice and changed the structure of the orifice, which suppressed the formation of the pneumatic hammer vibration. Ding et al. [90] compared the pressure distribution of annular orifice and simple orifice throttling, and found that annular orifice throttling is less prone to pneumatic hammer vibration. Ma et al. [151] found that the pneumatic hammer stability of aerostatic thrust bearing could be improved by reducing the depth of the recess and increasing the number of damping orifice, as shown in Fig. 12. (2) External damping Considering that the pneumatic hammer vibration is a form of energy release, thus, introducing the external damping can consume

Fig. 12. Aerostatic thrust bearing with recess using damping orifices a) The designed aerostatic thrust bearing; b) designed aerostatic thrust bearing. 11

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Fig. 13. Methods to improve the damp of aerostatic bearing system [148–152].

Akhondzadeh et al. [170] investigated the vibration of aerostatic journal bearing with variable depth pocket experimentally. It was noted that the air pocket geometry has an important influence on the vibration of air spindle, and pyramidical pockets show better performance for reducing vibrations. Li et al. [171,172] investigated the timevarying two-dimensional flow status of an aerostatic thrust bearing by adopting the large eddy simulation (LES). A complicated turbulent flow region with many vortices is observed near the outlet of orifice, and in each vortex, the pressure decreases gradually from the edge of vortex to its center. The further researches of Li et al. [173,174] indicated that the air pressure fluctuation could first increase and then declined with the increasing of air film thickness. Zhu et al. [175] numerically investigated the transient three-dimensional flow status of aerostatic bearing adopting the LES. The timevarying turbulent structures at different moments are compared, as demonstrated in Fig. 16. Vortex shedding phenomenon was observed near the orifice outlet. There are local pressure minima for each vortex and its location corresponds to the vortex center. The transient change of locations and shapes of these vortexes will directly induce pressure fluctuation inside the air film. Gao et al. [176] investigated the airinduced vibration of the aerostatic spindle of an ultra-precision flycutting machine tool and its impact on the machined surface. It was found that the amplitude and spatial period of high-frequency ripples agreed well with those of tool tip vibration induced by air pressure fluctuation. To suppress this kind of vibration induced by turbulent flow inside the air film, Chen et al. [177] proposed a restrictor with arrayed microhole, and Li et al. [178] designed a disturbance structure inside the air chamber. Both the experimental and numerical investigations showed that the novel restrictors could evidently suppress the formation of the vibration, compared with conventional restrictors. Yoshimura et al. [96] studied the nano-fluctuation of an aerostatic thrust bearing with T-shaped groove-type surface restrictor. They found that the main reason for the nano-fluctuation of surface restriction aerostatic bearings is not the inside turbulent airflow, but the unsteady

4.3. Air-induced vibration With the demands for the performance improvement of ultra-precision equipment such as optical lithography machines, the movement and position accuracy of aerostatic spindle and guideways are expected to reach sub-nanometer order [162]. During the analysis of the static performance of aerostatic bearings, the flow status inside air film is commonly considered as a steady flow. However, Kawai et al. found that a small vibration of the aerostatic slide in the nanometer order is always observed even though the slider keeps stationary, and air turbulence is considered as the reason for this nano-meter order vibration [163]. Zhang et al. [164] and Kim et al. [165] pointed out that the micro-vibration induced by unsteady flow of fluid bearing could deteriorate the positioning accuracy of moving parts. Hence, the turbulent flow inside air film should be considered. In 2006, Aoyama et al. [166] for the first time numerically and experimentally studied the airflow status inside the bearing clearance of an aerostatic bearing with an annular orifice. Turbulent flow is found near the outlet of the orifice by CFD simulation, which is regarded as the main reason for this kind of small vibration of aerostatic guideways. To suppress the vortex flow, improved design with a round corner at the outlet of the orifice is proposed and further verified experimentally. Subsequently, it was found that a micro bounce happens as the slider arrives the small pores of the guideway [167], as shown in Fig. 14. After Aoyama's work, the unsteady flow of aerostatic bearings and its influence on the vibration of moving parts have attracted increasingly attention. Chen et al. [168,169] found that the strength of air vortices has a close relationship with the design and operating conditions, such as depth of recess, shapes of recess, diameter of orifice, and pressure of air source. The strength of air vortices leads to the strength of vibrationincrease with the increase of orifice restrictor diameters and pressure of air source. It was also pointed out that, comparing with cylindrical recesses, aerostatic bearing with spherical recesses has lower vibration without decreasing load capacity, as shown in Fig. 15. 12

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Fig. 14. Vibration of aerostatic guideways caused by small pores [167]. a) Structure of the air pad; b) The shape of a small pore; c) Result of the measurement.

restrictor show excellent stability, load capacity, and stiffness, but the underlying design principle of those classes of bearing are not fully understood. Therefore, more efforts should be made to formulate the design methodology of aerostatic bearings with the slot, grooved and porous restrictors. In addition, considering the advantages and disadvantages of each type of restrictor, a single throttling form is difficult to meet current complex requirements for the air bearing. Compound restrictor should be further investigated, to improve the comprehensive performance of the aerostatic bearings. Secondly, at present, the cause of pneumatic hammer vibration is still not elucidated clearly. Various causes, including the compressibility of gas, the negative damping of the system, the interference force, and the phase difference of motion, are regarded to induce air hammer vibration. Some critical issues, such as the negative effect of the microscopic turbulent airflow inside the bearing, the influence of bearing mass, and structural stiffness and damping, the influence of fluidstructure interaction considering squeeze film effect, and how the rotor is converted from stable to unstable motion under certain operation condition, remain unclear. The generation mechanism of air hammer vibration needs further research and exploration. The advent of multiphysics coupling simulation method may provide a powerful tool to take the fluid-structural interaction, the turbulent flow status, and the squeeze film effect into consideration simultaneously in future research. Thirdly, the turbulent flow inside air film is generally recognized as the reason for nanometer order vibration, however, the negative impact of air-induced vibration on positioning accuracy of feed drive system of machine tool and surface quality of the machined surface is still not clearly described. Besides, the negative influence of the unsteady flow of air film on the pneumatic hammer vibration and high-speed stability also needs further investigation. It remains to be challenging to further enhance the positioning accuracy of aerostatic bearings by suppressing or eliminating the air vortex, reducing nano-vibration of aerostatic bearing supported parts, and improving the motion stability.

airflow at the bearing outlet. The unsteadiness of outlet airflow could induce the pressure fluctuation inside the air film, which in turn induces the fluctuation of supported object. The experimental investigation indicated that the fluctuation amount of spool increased nearly proportion to the increase in Reynolds number at the bearing outlet. It means that the Reynolds number should be taken into consideration to reduce the nano-fluctuation of supported object. To summarize, the above researches indicated the turbulent flow inside air film and the outlet of air film are reasons for nanometer level vibration. However, the existing literature mainly focuses on the turbulent flow status inside air film. Though some new designs were proposed to suppress this kind of vibration, the impact of nanometer order vibration on the positioning accuracy of moving parts and machined surface quality is still not fully understood. Besides, the unsteady flow of air film may have a negative influence on the pneumatic hammer stability and high-speed stability which should be further investigated. 5. Air bearing research challenges and future perspectives Although numerous theoretical and experimental works have been conducted in the field of aerostatic bearings, some key problems are still not fully understood. Despite the current achievements, the future work on the research of aerostatic bearings is proposed and summarized as follows. Firstly, many researches on the orifice-type aerostatic bearings have been conducted, and massively available design information could guide the design of this kind of air bearing. However, orifice-type restrictor may not the best option to meet the higher demands of load capacity, stiffness and high-speed stability for future aerostatic bearings, due to the poor stability of bearing with a pocked-type restrictor, and low load capacity and stiffness of bearing those with annular orifice restrictor. Bearings with slot restrictor, grooved restrictor, and porous

Fig. 15. Airflow status near the orifice with different recess shapes [168]. a) Cylindrical recess; b) Spherical recess; c) No recess. 13

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Fig. 16. Iso-surfaces of instantaneous vorticity and pressure [175]. a) Iso-surfaces of instantaneous vorticity; b) Iso-surfaces of instantaneous pressure.

Fourthly, the interactions among fluid flow, heat transfer, structural deformation, electro-magnetic, acoustics and vibration have great influence on the performance of air bearing. Compared with FEM, FDM, and CFD, multi-physics coupling simulation method, however, is more competitive and powerful to investigate the complex phenomenon of aerostatic bearings, such as pneumatic hammer and high-speed instability. However, its sophisticated modeling process and high computational cost limit its application greatly. Though many MPC models have been developed, such as the models of Gao and Li et al. [79,85,88], they can only be adopted for specific cases, and hence lack of flexibility and versatility. Considering the rapid progress of algorithms and commercial software [179], multi-physics coupling modeling method will be promising in performance evaluation for the design and research of aerostatic bearings, which therefore should be further investigated.

principle. But, the poor stability of such kind of aerostatic bearings limits its application in ultra-high speed applications. The underling design principle for aerostatic bearings with slot restrictor, grooved restrictor, porous restrictor, and compound restrictor has not been clearly elucidated, which needs further investigation. (2) The formation mechanism of air hammer vibration is still not fully determined. Some critical issues, including the negative effect of the microscopic turbulent airflow, the influence of bearing mass, and structural stiffness and damping, the influence of fluid-structure interaction considering squeeze film effect, and how the rotor is converted from stable to unstable motion under certain operation condition, need further investigation in future research. (3) A consensus has been reached on the cause of nanometer order vibration that the turbulent flow inside air film is the vibration source. More works should be conducted to clarify the negative impact of air induced vibration on the pneumatic hammer vibration and high-speed stability of aerostatic spindle, the positioning accuracy of the feed drive system of machine tool and the surface quality of the machined surface. (4) The rapid development of multi-physics coupling simulation algorithms and commercial software provides an efficient tool to study the complex phenomenon of aerostatic bearings, such as pneumatic hammer and high-speed stability. However, research on coupling modeling method of aerostatic bearings is insufficient, which needs more investigation in future work.

6. Conclusions This paper aims to provide a systematic and comprehensive analysis of latest research progress of aerostatic bearings. The literature on aerostatic bearings is reviewed from the perspectives of static characteristic, pneumatic hammer, stability, dynamic characteristic, and air-induced vibration. Some important conclusions can be drawn as follows: (1) Plenty of research has been carried out on orifice feeding aerostatic bearings, which contributes to the establishment of design 14

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Acknowledgment

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