ADE664290 1..9

ADE664290 1 9 Special Issue Article Advances in Mechanical Engineering 2016, Vol 8(8) 1–9 � The Author(s) 2016 DOI 10 1177/1687814016664290 aime sagepub com Dynamic modeling method for air bearings in[.]

Advances in Mechanical Engineering

2016, Vol 8(8) 1–9

Ó The Author(s) 2016 DOI: 10.1177/1687814016664290 aime.sagepub.com

Dynamic modeling method for air

bearings in ultra-precision positioning

stages

Xiulan Bao1and Jincheng Mao2

Abstract

Air bearings have been widely used in ultra-precision positioning stages due to the property of nearly zero friction or wear Small vibration of the bearing reduces the overall moving and positioning precision of the stage and hampers its applications in fabrication facilities requiring nanometer moving and positioning precision In order to improve system precision, knowledge of the dynamic characteristics of air bearings is the first and crucial step However, it is still a chal-lenge to set up an accurate dynamic model for air bearings due to the system complexity In this article, a novel method for the dynamic modeling of air bearing is proposed, which takes into account the dynamics in both the moving direction and the supporting direction An ultra-precision positioning dual stage is investigated using the proposed dynamic model-ing method This stage has two sets of air bearmodel-ings and can be used in integrated circuit fabrication equipments Moreover, dynamic behaviors of the ultra-precision positioning dual stage are studied and compared with experimental results to validate the effectiveness and accuracy of the proposed method

Keywords

Air bearing, dynamic model, ultra-precision, positioning stage, vibration

Date received: 21 September 2015; accepted: 8 July 2016

Academic Editor: Mark J Jackson

Introduction

The ultra-precision stage is a loading platform for

pre-cision positioning in multi-directions with high speed

With the advancement of technology, ultra-precision

positioning stages are increasingly used in various

industries, such as lithography, computerized numerical

control (CNC) machine tools, micro or nano

topogra-phy measurement, and so on, to achieve positioning

motion with high speed and high precision.1,2The

per-formance of ultra-precision positioning stages directly

affects the quality and productivity of precision

machine tools As the positioning stages on high

preci-sion are increasingly required, air bearings have been

widely used in various ultra-precision positioning stages

due to their merits of near zero friction or wear and less

contamination.3 However, it has been recognized that

there exists a phenomenon of air vortices, which will

lead to small vibration with high-speed airflow.4 The small vibration of air bearing obviously reduces the overall moving and positioning precision of ultra-precision positioning stages5and even causes a kind of self-excited instability to damage the whole positioning stages.6 In order to reduce the small vibration and improve the motion stability, the study of its dynamic characteristics is both necessary and urgent.7

1

College of Engineering, Huazhong Agricultural University, Wuhan, China

2

School of Mechanical and Electrical Engineering, Wuhan Institute of Technology, Wuhan, China

Corresponding author:

Jincheng Mao, School of Mechanical and Electrical Engineering, Wuhan Institute of Technology, Wuhan, Hubei 430073, China.

Email: orchidbaoxl@mail.hzau.edu.cn

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License

(http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).

In the past decades, performances of air bearings

have been investigated theoretically and

experimen-tally The general rules for dynamic parameters of the

air bearing were analyzed, and dynamic design

princi-ples of air bearings were proposed to reduce the

syn-chronization error of dual stages.8 Performances of

air bearings and effects of the recess shape, orifice

dia-meter, gas film thickness, and so on on the load

capac-ity and mass flow rates were also investigated.9,10The

design method and preload technique for air bearings

were proposed to improve the load capability and

stiffness.11The aforementioned works mainly focused

on the effects of restrictor design parameters and

operating conditions on the dynamic performance of

air bearings Some scholars also studied the model

and dynamic characteristics of the air bearings A fast

modified three-dimensional (3D) flow mode from the

Darcy equation and the modified Reynolds equation

is deduced and used the finite element method to solve

for the gas pressures distribution.12 The resistance

network method (RNM) which takes into account the

equilibrium of the mass flow rate and the squeeze film

effect is developed, and dynamic behaviors such as

variation in film gap and stability range of a grooved

aerostatic bearing are also analyzed.13 They mainly

studied the own dynamics of air bearing, without

con-sidering the impact of air bearing dynamic

character-istics on supported motion stages And the established

model of air bearing is too large to be used in dynamic

model for supported motion stages Few studies dealt

with the dynamic characteristics of supported motion

stages with air bearings,14,15 but most of them

mod-eled the bearing as a spring in the support direction

This approach is simple enough to use in the overall

modeling of supported motion stages, but the static

and dynamic tilt characteristics of air bearings are

ignored, which are also important for the realization

of ultra-precision.16 Accurate dynamic modeling of

the air bearing including the effects of dynamics in

both directions and the dynamic characteristics of

supported motion stages affected by air bearings are

to be studied

The rest of this article is organized as follows: in

section ‘‘Description of the dual-stage system,’’ an

ultra-precision positioning dual-stage system with two

sets of air bearing is introduced In section ‘‘Dynamic

modeling of the bearing system,’’ a dynamic modeling

method for air bearings is proposed In section

‘‘Dynamic modeling of the stage,’’ using the proposed

dynamic modeling method, a dynamic model with 24

degrees of freedom for the ultra-precision positioning

dual stage is established In section ‘‘Experiments,’’

comparison with experimental results is made to

vali-date the proposed dynamic modeling method Finally,

conclusions are drawn in section ‘‘Conclusion.’’

Description of the dual-stage system Figure 1 illustrates the ultra-precision positioning dual-stage system, which contains two air bearing systems, one named the coarse air bearing system and another named air-foot Accordingly, the ultra-precision posi-tioning dual-stage system can be divided into a coarse stage and a fine stage The coarse air bearing system is used for supporting the coarse stage, and the air-foot is used for supporting the fine stage

In the ultra-precision positioning dual-stage system, the coarse stage is used for coarse and large range motion while the fine stage for precision and small range motion As for the coarse air bearing system, the coarse stage also has two linear motors to achieve movement in the x and y directions Each linear motor consists of a stator and a slider In the coarse stage, the slider of the x direction linear motor is also used as a stator of the y direction linear motor The specific structure of the coarse air bearing system is shown in Figure 2 The linear motor stator of the x direction is fixed on the ground through bolts The linear motor sli-der of the x direction is suspended in the linear motor stator of the x direction by eight air bearings and can move in the x direction The actuator of the fine stage

is a planar motor, the stator of which is linked with the linear motor slider of the y direction and suspended in the linear motor stator of the y direction by eight air bearings Hence, the fine stage can be driven in the x and y directions The coarse air bearing system is a combination of 16 air bearings which support the linear motor slider in the x and y directions

In Figure 1, the 3D size of linear motor stator of the x direction is 585 mm 3 210 mm 3 65 mm, the 3D size of linear motor slider of the x direction is

Figure 1 Schematic diagram of the ultra-precision positioning dual-stage system.

330 mm 3 280 mm 3 209 mm, and the length of the

linear motor slider of the y direction is 900 mm The size

of the granite base is 740 mm 3 566 mm 3 100 mm,

the diameter of air-foot is 234 mm, the orifice diameter

is 0.15 mm, the length of planar motor is 310 mm, and

the 3D size of the table is 288 mm 3 319 mm 3 33 mm

A schematic diagram of the air-foot is shown in

Figure 3 The air-foot has an annular air bearing, a

cir-cular vacuum preloading chamber, and two circir-cular

ambient pressure grooves which are used as vents The

annular air bearings have 12 compound restrictors

con-sisting of small orifices and rectangular chambers

sym-metrically Compressed gas flows through the feed

holes to the chambers and then fills the air gap and

flows out to the atmosphere by the vent finally The preloading device which is at the center of the air-foot generates suction by vacuum The advantage of vacuum preloading is that it creates a preloading force on the bearings without adding mass As for the coarse air bearing system, the fine stage mainly consists of a gran-ite base, a planar motor, and a table The grangran-ite base

is used to ensure the flatness of gas flow boundary The planar motor which is supported by the air-foot can drive the fine stage to move in the x and y directions and rotate around the z direction The workpiece requiring ultra-precision at high speed is placed on the table, which is mounted on the air-foot directly

As mentioned above, the two air bearing systems are supporting the coarse stage and the fine stage, and they will influence the performance of the ultra-precision positioning dual-stage system directly

Dynamic modeling of the bearing system The general form of the air bearing

Figure 4(a) shows the general form of the air bearing, which is made of ceramic or hard aluminum alloy The air channels are processed, and the restrictor is installed inside the air bearing In the air bearing system, the precision granite or marble table is usually used as a support base The air bearings are placed on the sup-port base and carry a motion stage to achieve move-ment of single direction or multi-directions The air bearing working surface are the support base and bot-tom surface of air bearing The air gap between the two surface is very small When an air bearing works, the lubricating gas flows through the restrictor c into the recess of the bearing and then fills the air gap before

Figure 2 Structure of the coarse air bearing system.

Figure 3 Schematic diagram of the air-foot.

flowing into the ambient As shown in Figure 4(b), the

air supply pressure is p0 in the restrictor c, the air

pres-sure becomes pd in the recess and then gradually

decreases in the air film gap until being the ambient

pressure pa This is the air film between two surfaces

that supports the bearing and achieves lubrication

Modeling method of the air bearing

In order to investigate the effects of small vibration of

air bearing on the motion stage, a dynamic modeling

method for air bearings is presented in this section In

practice, the motion stage is supported by an air

bear-ing system composed by multiple air bearbear-ings As

shown in Figure 5, a moving part is supported on a

support base by an aerostatic bearing system The

support base is denoted by body A, and the moving

part denoted by body B Define So(O XYZ) denoting

the absolute frame, define SoA(OA XAYAZA), and

SoB(OB XBYBZB) denoting local frames fixed on body

Aand body B, respectively

In the conventional multi-rigid-body modeling, the

two ends of spring are fixed on the two rigid bodies,

respectively The acting positions of the spring on two

rigid bodies are constant The direction of the spring

stiffness changes with the relative pose of two rigid

bod-ies In the air bearing, the gas film makes the supported

stage move on the granite base with near zero friction

When the supported stage has planar motion on the

support base, support forces are always vertical upward, and the acting positions of the spring of the support base are varying The simplified spring in conventional multi-rigid-body modeling method is not applicable According to the characteristics of the air bearing, a signal air bearing can be simplified as a sliding spring which has the same direction with line OAZA in body frame SoA, and location of the supported force is deter-mined by the body B In other words, a signal air bear-ing can be modeled as a slidbear-ing sprbear-ing whose direction

is determined by the stator (body A), and applied posi-tion of spring force is determined by the slider (body B) In Figure 5, the moving body B accomplishes movement from the initial position 1 to the final posi-tion 2 A signal air bearing is simplified as a sliding spring h One end of the sliding spring h is fixed on the moving body B through point m The other end of the sliding spring h slides on the fixed body A through point n to ensure that the direction of the spring force

is parallel with coordinate OAZA of body frame SoA l is the effective deformation of the sliding spring h between the initial position 1 and the final position 2 LetoP1 andoP2denote the position vectors of point m

in the absolute coordinate So for the initial position 1 and the final position 2, respectively Let uA and uB denote the deflection angles of the body frame SoAand body frame SoB, respectively, which are relative to the absolute frame So Then, the sliding spring deforma-tion l can be expressed as follows

l =ðoP2oP1Þ  sin uð B uAÞ ð1Þ

As mentioned above, a single air bearing can be modeled as a sliding spring which has only nonzero stiffness in the normal direction which represents the effect of the finite area of pressurized air The air bear-ing system can be modeled as a combination of distrib-uted sliding springs, and each one of them represents a

Figure 4 Structure schematic diagram of the air bearing.

Figure 5 Modeling for an air bearing system.

single air bearing, which indicates three single air

bear-ings in the air bearing system in Figure 5 As shown in

Figure 6, two air bearing systems in the ultra-precision

positioning dual-stage system are modeled as two sets

of multiple distributed sliding springs, which represent

as blue spring in this figure In this way, the established

air bearing model can reflect the tilt characteristics of

actual air bearing system and can be used in the

dynamic modeling of ultra-precision positioning dual

stage

Dynamic modeling of the stage

In order to verify the proposed modeling method of air

bearing, a dynamic model of the ultra-precision

posi-tioning dual stage is established, in which two air

bear-ing systems are modeled usbear-ing the method described in

section ‘‘Dynamic modeling of the bearing system.’’

According to the proposed modeling method, an air

bearing can be simplified as a sliding spring In the

ultra-precision positioning dual stage, the 12 air

bear-ings of the air-foot can be simplified as 12 distributed

sliding springs mentioned above, to ensure the fine

stage floats on the granite base with no horizontal

fric-tion and high vertical stiffness The 16 air bearings in

the coarse air bearing system can also be simplified as

16 distributed sliding springs to support the linear

motor slider in the x and y directions

The influence of vibration caused by excitation of

the air bearing system and mechanical structure

domi-nates the performance of the ultra-precision positioning

dual stage Because these effects are within the

ultra-precision positioning dual stage, the stage needs to be

divided into a finite number of components This

num-ber has to be relatively small so as to come up with a

low dimensional description of the ultra-precision

posi-tioning dual stage On one hand, this number should be

as small as possible in order to keep track of the basic

mechanisms causing the dynamic behavior of the ultra-precision positioning dual stage; on the other hand, this number should be large enough to be able to describe all the relevant phenomena with sufficient details In order to be able to describe the aforementioned effects, the ultra-precision positioning dual stage is split up into four components or bodies Body 1 contains the granite base and the linear motor stator of the x direction Body 2 contains only the linear motor slider of the x direction or the linear motor stator of the y direction Body 3 contains the linear motor slider of the y direc-tion, the planar motor, and the table of the fine stage Body 4 contains only the air-foot of the fine stage The dynamic model of the ultra-precision positioning dual stage is shown in Figure 6 The blue springs represent air bearings, and the black springs fixed at both ends reflect the structural flexibility

There are 24 generalized coordinates qi(i = 1,., 24) which are chosen to be equal to the global orientation

of each body, that is, q(t) =½q0

1, q0

2, q0

3, q0

4T, where

q0i=½xio, yio, zio, ai, bi, giT (i = 1, 2, 3, 4); xio, yio, zio (i = 1, 2, 3, 4)—translation in the x, y, and z directions

of body i; and ai, bi, gi(i = 1, 2, 3, 4)—rotation in the

x, y, and z directions of body i Furthermore, the gener-alized velocities are chosen to be equal to the time deri-vatives of the generalized coordinates, that is,

vi(t) = _qi(t) (i = 1,., 24)

The vibration differential equations are derived according to the Newton–Euler method The resulting equations are in the matrix form

M _v tð Þ + Cv tð Þ + Kq tð Þ = F ð2Þ where M is the generalized mass matrix, C(q, t) is the damping matrix with only structural damping, K(q, t) is the stiffness matrix which has structural stiffness and air bearing stiffness, and F is the external excitation caused by air vortices

In equation (2), M, C(q, t), and K(q, t) are 24 3 24 matrices Clearly, the stiffness matrix K(q, t) is not diag-onal, which means the stiffness of air bearings is coupled with other structural stiffness

The rigid body masses are calculated based on the shape and size of corresponding components The structural spring stiffness is obtained through the finite element analysis of the corresponding components in the fine stage In order to obtain the spring stiffness of air bearings, the following procedures are proposed:

1 According to the structural properties of the coarse air bearing system and air-foot, we estab-lish a 3D flow model of the air bearing, set proper boundary conditions and properties

of the fluid, select the appropriate solver for iterative calculation, and use computational

Figure 6 Dynamic model of the ultra-precision positioning

dual stage.

fluid dynamics (CFD) software Fluent to obtain

numerical solutions of the gas pressure

distribution

2 Obtain the load capability of each air bearing

The load capability can be calculated by

inte-grating the pressure in the lubricating film area

The air bearing in the air-foot is a ring bearing,

the load capacity of which can be calculated as

follows

W = 2p

ð

R 2

R 1

P rdr + PapR21 PapR22 ð3Þ

where P is the pressure in the lubricating film area, Pa

is the atmospheric pressure, R1 is the inner diameter of

the ring bearing, and R2 is the external diameter of the

ring bearing

The air bearing in the coarse air bearing system is a

rectangular bearing, and its load capacity can be

calcu-lated as follows

W = ð

A

P Pa

where A is the regional area of the gas film

3 Calculate the air bearing stiffness According to

the above steps, the load capacity W can be

cal-culated when the gas film thickness is h0 Then,

the gas film thickness is increased by a small

increment Dh, and the load capacity W0 is

recal-culated The stiffness of an air bearing can be

obtained as follows

kw=W

0 W

There are three specifications of air bearings in the

ultra-precision positioning dual stage: 12 air bearings in

the air-foot, 8 vertical air bearings, and the 8 horizontal air

bearings in the coarse air bearing system Each type of the

spring has the same stiffness Through the above processes,

the stiffness of a vertical air bearing and a horizontal air

bearing in the coarse air bearing system can be identified

as 75 and 66 N/mm, respectively The stiffness of an air

bearing in the air-foot can be identified as 100 N/mm

The damp effect of aerostatic bearing is

squeezed-film damping mainly, and its damping can be calculated

by the following equation

c = ∂W

The damping ratio of a vertical air bearing and a

horizontal air bearing in the coarse air bearing system

can be identified as 0.0097 and 0.0117, respectively The

damping ratio of an air bearing in the air-foot can be identified as 0.0074

The ultra-precision positioning dual stage modeled

by four bodies is represented by a 24th-order dynamic model This dynamic model can be used for analysis in the simulation software package Simulink, by trans-forming it into C-code and using the so-called s-func-tions defined in Simulink Besides time-domain analysis, frequency-domain analysis can also be performed

Experiments

To confirm the validity of the simulation result, a series

of experiments of the ultra-precision positioning dual stage are conducted The standard drop hammer tests

on specimens have been carried out The locations of the excitation point and sensor arrangement are shown

in Figure 7 In the drop hammer test, the hammer equipped with a rubber head is used, and an accelera-tion sensor (CA-YD-106 and CA-YD-117) is placed at the test position to obtain the vibration responses The main technical specifications of CA-YD-106 were presented as follows: sensitivity at 20°C 6 5°C is 2.7 pC/m/s2, transverse sensitivity is 5%, maximum allowable acceleration is 1.96 3 104m/s2, frequency range is 0.5–12 kHz, and the weight of the sensor is

15 g The main technical specifications of CA-YD-117 were presented as follows: sensitivity at 20°C 6 5°C is

50 pC/m/s2, transverse sensitivity is 5%, maximum allowable acceleration is 1.5 3 103m/s2, frequency range is 0.2–3 kHz, and the weight of the sensor is 50 g Seven groups of experiments are performed, each group for four times, and average data values are taken The excitation point and measurement point for each experiment are shown in Table 1 A modal hammer (ENDEVCO 2302-10, 500 lb range, frequency range of

8 kHz, and sensitivity of 10 mV/lb) is used to supply an impulse force signal Using modal test and analysis

Figure 7 Location of excitation points and measurement points.

software LMS to collect excitation signals and response

signals, with a sampling resolution of 0.5 Hz, the

aver-age frequency response functions are calculated from 20

sampling data The experimental results for tests 1–7

are shown in Figure 8

In test 1 and test 2, the excitation point is located on

the stator of the y direction linear motor, and the

mea-surement points are located on the stator and slider of

the y direction linear motor, respectively The direction

of excitation and measurement are in the z direction In

test 3 and test 4, the excitation point is located on the

slider of the y direction linear motor; the measurement

points are located on the stator and slider of the y

direction linear motor, respectively; and the direction

of excitation and measurement are the same as those in

test 1 and test 2 The excitation and measurement

points of the four tests are interchangeable; therefore,

the common peak frequencies of the two frequency

response functions in both tests are adopted According

Table 1 Locations of excitation points and sensors for each experiment.

Test number Excitation point Sensor location Sensor type Test 1 Excitation point 1 Measurement point 1 CA-YD-106 Test 2 Excitation point 1 Measurement point 2 CA-YD-106 Test 3 Excitation point 2 Measurement point 1 CA-YD-117 Test 4 Excitation point 2 Measurement point 2 CA-YD-117 Test 5 Excitation point 3 Measurement point 3 CA-YD-117 Test 6 Excitation point 4 Measurement point 4 CA-YD-117 Test 7 Excitation point 5 Measurement point 5 CA-YD-106

Table 2 Simulation and experimental modal of the ultra-precision positioning dual stage.

Simulation frequency (Hz) Experiment frequency (Hz) Modal shape Test number Error (%)

78 80 Rotation around y-axis Test 6 2.5

128 140 Rotation around z-axis Test 5 8.6

149 148 Rotation around y-axis Test 1 0.7

Test 2 Test 3 Test 4

300 317 Rotation around x-axis Test 5 5.3

416 439 Rotation around y-axis Test 1 5.2

Test 3

490 494 Rotation around x-axis Test 7 0.8

591 589 Rotation around z-axis Test 6 0.5

724 739 Rotation around y-axis Test 7 2.0

754 809 Rotation around x-axis Test 1 6.8

Test 2 Test 3

896 872 Translational motion along z-axis Test 2 2.8

912 964 Rotation around x-axis Test 7 5.4

937 972 Rotation around y-axis and x-axis Test 2 3.6

Test 3 Test 4

981 1003 Rotation around y-axis Test 7 2.2

Figure 8 Experimental results for tests 1–7.

to the position and direction of the excitation points,

the frequencies corresponding to the vibration mode

for rotation around x-axis and y-axis and translational

motion along z-axis of coarse stage are relatively easy

to be represented in tests 1–4 The frequencies

corre-sponding to the vibration mode for rotation around

x-axis and z-x-axis of coarse stage are relatively easy to be

excited in test 5 The frequencies corresponding to the

vibration mode for rotation around y-axis and z-axis of

coarse stage are relatively easy to be excited in test 6

The frequencies corresponding to the vibration mode

for rotation around x-axis and y-axis of fine stage are

relatively easy to be excited in test 7 Comparison of

simulation results and experimental results is shown in

Table 2 It can be seen from Table 2 that the relative

error ratio of the main peak frequencies is within 10%,

and the simulation results of the dynamic model and

experimental results are consistent The simulation

modes of the ultra-precision positioning dual stage on

78 and 128 Hz are shown in Figures 9 and 10,

respectively

Conclusion

In this article, a novel dynamic modeling method for

air bearing is proposed, which can simultaneously

reveal the moving direction dynamics, and the tilt

char-acteristics of bearings The proposed method models a

signal air bearing as a sliding spring with force

direc-tion determined by stator and locadirec-tion by the slider

and models a system of air bearings as a combination

of distributed sliding springs And each spring has only

nonzero stiffness along the normal axis which

repre-sents the effect of the finite area of pressurized air An

ultra-precision positioning dual stage which contains

multiple air bearings is presented, and the system

struc-ture and bearing distribution are also introduced The

proposed dynamic modeling method has been applied

successfully for an ultra-precision positioning dual

stage which contains multiple air bearings An analytic

dynamic model of an ultra-precision positioning dual

stage with air bearings is established Model parameters

of the dynamic model are obtained through the finite

element analysis Experimental results demonstrate that the proposed modeling method for air bearings is accurate and effective The proposed dynamic model can quantitatively describe the ultra-precision posi-tioning dual stage accurately and can be successfully used for controller design or dynamic optimization in the future

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial sup-port for the research, authorship, and/or publication of this article: This study was supported by the Scientific Research Funds of the Educational Commission of Hubei Province (grant no Q20151519), the Scientific Research Funds of the Transportation Commission of Hubei Province (grant no 201472122), and the Fundamental Research Funds for the Central Universities of China (grant no 2016PY017 and 2013QC007).

References

1 Wang FJ, Ma ZP, Gao WG, et al Dynamic modeling and control of a novel XY positioning stage for semiconductor packaging T I Meas Control 2015; 37: 177–189.

2 Sen R, Pati C, Dutta S, et al Comparison between three tuning methods of PID control for high precision posi-tioning stage J Metrol Soc India 2015; 30: 65–70.

3 Otsu Y, Somaya K and Yoshimoto S High-speed stabi-lity of a rigid rotor supported by aerostatic journal bear-ings with compound restrictors Tribol Int 2011; 44: 9–17.

Figure 9 Simulation modes of the ultra-precision positioning

dual stage on 78 Hz.

Figure 10 Simulation modes of the ultra-precision positioning dual stage on 128 Hz.

4 Dupint R Robust rotor dynamics for high-speed air

bearing spindles Precis Eng 2015; 40: 7–13.

5 Aoyama T, Koizumi K, Kakinuma Y, et al Numerical

and experimental analysis of transient state

micro-bounce of aerostatic guideways caused by small pores.

Ann CIRP 2009; 58: 367–370.

6 Ye YX, Chen XD, Hu YT, et al Effects of recess shapes

on pneumatic hammering in aerostatic bearings Proc

IMechE, Part J: J Engineering Tribology 2010; 224:

231–237.

7 Wardle FP, Bond C, Wilson C, et al Dynamic

character-istics of a direct-direct drive air-bearing slide system with

squeeze film damping Int J Adv Manuf Tech 2010; 47:

911–918.

8 Ye YX, Chen XD and Luo X Dynamic design of

aero-static bearings in dual gas-lubricated stages China Mech

Eng 2010; 21: 757–761.

9 Chen XD and He XM The effect of the recess shape on

performance analysis of the gas-lubricated bearing in

optical lithography Tribol Int 2006; 39: 1336–1341.

10 Chen XD, Chen H, Luo X, et al Air vortices and

nano-vibration of aerostatic bearings Tribol Lett 2011; 42:

79–183.

11 Zhang CP and Liu Q Development of high-performance linear motor stage based on static-pressure air-bearing sli-der J Mech Sci Tech 2006; 25: 1212–1216.

12 Tian Y Static study of the porous bearings by the simpli-fied finite element analysis Wear 1998; 218: 203–209.

13 Chen MF and Lin YT Static behavior and dynamic sta-bility analysis of grooved rectangular aerostatic thrust bearings by modified resistance network method Tribol Int 2002; 35: 329–338.

14 An CH, Zhang Y, Xu Q, et al Modeling of dynamic characteristic of the aerostatic bearing spindle in an ultra-precision fly cutting machine Int J Mach Tool Manu 2010; 50: 374–385.

15 Chen XD, Bao XL, He XM, et al Vibration analysis of ultra-precision stage with air-bearing J Huazhong Univ Sci Tech 2007; 35: 5–8.

16 Yoshimoto S, Tamura J and Nakamura T Dynamic tilt characteristics of aerostatic rectangular double-pad thrust bearings with compound restrictors Tribol Int 1999; 32: 731–738.