Performance analysis of aerostatic journal micro-bearing and its application to high-speed precision micro-spindles

Accepted Manuscript Performance analysis of aerostatic journal micro-bearing and its application to highspeed precision micro-spindles Hang Xiao, Wei Li, Zhixiong Zhou, Xiangming Huang, Yinghui Ren PII:

S0301-679X(18)30002-1

DOI:

10.1016/j.triboint.2018.01.002

Reference:

JTRI 5033

To appear in:

Tribology International

Received Date: 19 October 2017 Revised Date:

24 October 2017

Accepted Date: 1 January 2018

Please cite this article as: Xiao H, Li W, Zhou Z, Huang X, Ren Y, Performance analysis of aerostatic journal micro-bearing and its application to high-speed precision micro-spindles, Tribology International (2018), doi: 10.1016/j.triboint.2018.01.002. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Performance analysis of aerostatic journal micro-bearing and its application to high-speed precision micro-spindles Hang Xiao1, Wei Li1, 2*, Zhixiong Zhou1, Xiangming Huang1, Yinghui Ren1, 2

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1. College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China 2. Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha 410082, China *Corresponding author. Email address: [email protected] (Wei Li)

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Abstract: An aerostatic journal micro-bearing has advantages of simple structure, almost zero friction and increasable load capacity due to aerodynamic pressure effect, which makes it a possible choice for high-speed and precision micro-rotatory machines. This study comprehensively investigates the effects of restriction, structure and operation parameters on the performance of an aerostatic journal micro-bearing using numerical models. The numerical models are verified on a prototype aerostatic journal bearing. Considering both the performance requirement and current machining capacity, the restriction parameters of 8-12 µm in average bearing clearance and 0.08-0.14 mm in orifice diameter are recommended. Another focus of this study is the aerodynamic pressure effect on the performance of micro-bearing, which is thoroughly investigated from the aspects of restriction, structure and operation parameters. The aerodynamic pressure effect is outstanding only at ultra-high speeds and large eccentricities. Two prototypical high-speed precision micro-spindles are developed and their micro-tools with a diameter of 3.175 mm are directly supported by the micro-bearing. The aerodynamic pressure effect resulting from the ultra-high rotational speeds can improve the performance of micro-spindles under loading. However, the low rotational accuracy and balance quality compromise their performance. Keywords: Micro-bearing; Aerostatic pressure; Aerodynamic pressure; Micro-spindle

1. Introduction

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In manufacturing precision and ultra-precision miniature parts, micro-cutting and micro-grinding with micro-machine tools provide significant advantages not only in machining quality, efficiency and machining ability of intricate 3D geometry, but also in cost, size and energy consumption, and thus draw more and more attention [1, 2]. The micro-spindle, as a key component of micro-machine tools, requires high rotational speed (more than 500,000 rpm), excellent rotational accuracy (better than 1 µm), sufficient load capacity and compact structure [3]. To meet the above requirements, bearing of the spindle is critical, which must be able to provide high rotational accuracy, sufficient load capacity, good stiffness and excellent dynamic performance. Besides, it should be millimeter-sized. Comparing the existing types of bearings, conventional rolling element bearings in most modern production equipment are reaching their limited accuracy of about 2 µm and usually generate much heat at high speeds. The liquid-lubricated bearing has a limited rotational speed due to heat generation at high speeds. The magnetic bearing has complicated structure and thus is difficult to use in micro-spindles. The gas-lubricated bearing, due to its low heat generation, high rotational accuracy and simple structure, is a good choice for high speed and precision micro-spindles. The common gas-lubricated bearings are aerodynamic bearing, aerostatic bearing and

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bearing length non-dimensional film pressure bearing radius velocity in no-slip model load capacity eccentricity ratio length to diameter ratio, λ= L/D. gas viscosity gas density atmospheric air density coordinate in circumference coordinate in length bearing number in journal direction eccentric increment at two points ratio of throttle pressure to certain supply hole, βi = pi/p0 ratio of critical pressure

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L P R U W ε λ µ ρ ρa θ Y Λx △e βi

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Nomenclature c radial average clearance between spindle shaft and bearing d orifice diameter e eccentricity of the shaft h gas film thickness mi mass flow rate of an orifice k isentropic expansion index for air p gas film pressure pa ambient pressure p0 supplied gas pressure A orifice area D bearing diameter Fx load capacity component in horizontal direction Fy load capacity component in vertical direction H non-dimensional gas film thickness K bearing stiffness

βk

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hybrid air bearing. The aerodynamic bearing has good dynamic performance as well as larger load capacity along with higher rotational speed, but the drawback is also obvious, such as high run-out at the start or the stop process, which may accelerate the wear of such bearing and spindle shaft. Although the aerostatic bearing has good start-stop performance, but the load capacity and stiffness are highly dependent on structural dimensions, supply pressure and other parameters. The dynamic performance is inferior to that of aerodynamic bearing[4]. The hybrid air bearing combines aerostatic and aerodynamic effects and presents better performance. For instance, Zhang et al [5] investigated the hybrid gas spiral-grooved thrust micro-bearing and pointed out that it not only exhibits better steady-state and dynamic performance than the existing hydrodynamic and hydrostatic micro-bearings, but also is more stable than its hydrodynamic counterpart, especially when the frequency number is high. However, limited by most modern production equipment, hybrid gas journal micro-bearing is difficult to manufacture because of its intricate structure and high dimensional accuracy. The aerostatic bearing has good rotational accuracy and simple structure whereas its load capacity and stiffness are limited due to gas compressibility. But the micro-spindle commonly requires small cutting load. And the performance of aerostatic bearing can be greatly improved by optimizing its restriction and structure parameters as well as conditioned aerodynamic effect. It means that the optimized aerostatic bearing can provide sufficient load capacity for micro-spindle, which would be further ascertained in this study. Extensive research work has been done in the field of aerostatic journal bearings. Otsu et al [6] numerically studied the stability of the rotor supported by aerostatic journal bearing with compound restrictors and validated it in a high speed spindle. Chang et al [7] examined the discharge coefficients of inherent and pocketed orifice restrictors and showed that orifice-type restrictors is sensitive to the orifice diameter and film thickness. Lo et al [8] analyzed the film pressure distribution, friction effects, load capacity, rigidity, lubricating, and gas flow rate of high-speed aerostatic bearings. Chen et al [9] studied

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dimensionless load capacity and stiffness of aerostatic journal bearings at different operational conditions and geometric parameters. Besides, it has been known that the dynamic pressure effect resulting from high speeds or ultra-high speeds improves the performance of hybrid air bearings [5, 10, 11]. This effect existing in aerostatic bearings is also studied by a few researchers. Chen et al [12] investigated dynamic characteristics of ultra-precision aerostatic bearings and revealed that the dynamic stiffness increases while the damping decreases with increasing excitation frequency due to compressibility of the air. Liu et al [13] investigated the film pressure distribution and load capability of a pocket orifice externally pressurized bearing with the shaft-rotating effect considered. Nishio et al [14] investigated the static and dynamic characteristic of aerostatic thrust bearings. All above-mentioned researches have laid good foundations on the study of aerostatic journal bearings. However, the researches didn't make comprehensive and deep study on the performance of aerostatic journal micro-bearings with a millimeter size. In addition, as far as we know, the aerodynamic effect on aerostatic journal micro-bearings is rarely reported due to the inconspicuous at a low rotational speed. The micro-cutting and micro-grinding process by micro-spindles has the features of ultra-high rotational speed and low cutting force in less than 3 N [15]. When the rotational speed increases to a certain level, the aerodynamic effect is apparent in aerostatic micro-bearings, which improves the bearing performance. Therefore, it is meaningful to investigate the aerodynamic effect of aerostatic journal micro-bearings and identify how such an effect influences the load capacity and stiffness of the micro-bearings. In this study, static performance of aerostatic journal micro-bearings under various structural parameters and operation conditions is numerically analyzed. Then, the aerodynamic pressure effect on the load capacity and stiffness is thoroughly investigated. All this work mentioned above are benefit to the design of aerostatic micro-bearing. Associating with the requirements of high-speed and precision micro-spindle, performance of the micro-bearing is optimized and prototypical micro-spindles are developed. Finally, its performance is measured to validate the numerical analysis. 2. Theoretical background

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2.1. Governing equations

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Figure 1 illustrates configuration of an aerostatic journal bearing, in which the inherent orifice, pocketed orifice or porous media can be used as restrictors. But for the micro-bearing whose inner diameter is less than 10 mm, the inherent orifice may be the most appropriate choice due to its easy manufacturing. Two rows of inherent orifices are used and evenly distributed around the circumference of the bearing. Assuming that the gas mass-flow gets into the orifices completely, it can be calculated by Eqs. (1)-(2) for each orifice[16].

(a) Three-dimensional view

(b) Section view

ACCEPTED MANUSCRIPT Fig.1. Diagram of aerostatic journal bearing 2 ρa

ψi

(1)

pa

12  k 2  2k (k + 1 ) k  βi − βi , βi > βk =  + 1     k   k − 1 ψi =  12 ( k + 1 ) ( k −1 )   k  2  β i ≤ βk   ,  2  k + 1     

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k

k +1

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mi = φ Ap0

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Assumptions for the bearing numerical model are as follows:
The bearing can rapidly conduct out all generated heat and thus gas film is isothermal.
Gas viscosity is constant, which obeys the Newton law of viscosity.
Gas flow in the bearing clearance is regarded as laminar flow, and the mass flowing into the bearing is equal to that out of the bearing. At the aerostatic bearing, gas film pressures are mainly the result of the aerostatic pressure effect. If the dynamic pressure effect resulting from high speeds and the slip effect at the gas-solid interface are not considered, the governing equation for gas film can be expressed as follow: ∂ h3 ∂p2 ∂ h3 ∂p2 ( )+ ( )=0 ∂x µ ∂x ∂y µ ∂y

(3)

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Equation (3) can be applied to solving for the pressure of gas film except those around the orifice and the atmospheric boundary. However, with an increase in rotational speed, the aerodynamic pressure effect is mounting and cannot be neglected. Especially in the micro-bearings, such an effect may provide good improvement for their performance. Under this condition, the governing equation should be modified as Eq. (4). Besides, the external supply pressure, orifice loss, and mass flow may generate effect on the gas film pressure, which can be evaluated by the coefficient ofφ as shown in Eq. (1). Values of ξi at orifices are set to 1 and but to 0 at other locations. Non-dimensional form of the modified Reynolds equation is expressed as Eq. (5). p ∂ h3 ∂p2 ∂ h3 ∂p2 ∂(ph) ( )+ ( ) + 24 a ρυξi = 12U ∂x µ ∂x ∂y µ ∂y ∂x ρa

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∂ ( PH ) ∂  3 ∂P2  ∂  3 ∂P2  H  +α H  + Mξi = Λ x ∂θ  ∂θ  ∂Y  ∂Y  ∂θ

(4) (5)

Parameters in the non-dimensional equation are defined as follows:

θ=

µ R2 p x h p y 12µUR R , H = , P = , α =   , Y = , M = 24 3 2 a ρυ , Λ x = 2 . R c p0 L c p0 ρa c p0 L 2

(6)

2.2. Numerical model In this study, gas film is expanded to a plane regardless of curvature of journal bearing surface since the film thickness is far less than the bearing diameter. Fig. 2 presents the expanded gas film and meshed computational domain with periodic and atmospheric boundary conditions. The horizontal and vertical axes in Fig. 2(b) correspond to circumferential and axial directions of the bearing, respectively.

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a) Expanded gas film b) Meshed computational domain Fig.2 Meshing of expanded bearing film

F (P) =

∂ ( PH ) ∂  3 ∂P 2  ∂  3 ∂P 2  H  +α H  + Mξ i − Λ x ∂θ  ∂θ  ∂Y  ∂Y  ∂θ

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∂ ( PH ) ∂  ∂  3 ∂P  3 ∂P  =  2PH  +α  2PH  + Mξ i − Λ x ∂θ  ∂θ  ∂Y  ∂Y  ∂θ

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In order to solve the non-linear gas Reynolds equation, a nonlinear function for P is defined as Eq. (7) [17-18]. Then, principle of the Newton iteration is adopted to solving equation for gas film pressure. Meanwhile, the finite difference method (FDM) is used to solve the fluid calculation.

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According to the Newton iteration, there is:

F (P n ) + F ′(P n )(P n+1 − P n ) = 0 , n=0, 1, 2, 3 …

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where δ n = P n +1 − P n , n means the number of iteration times. Substituting the equation of Pn+1= Pn＋φδn into F (P n +1 ) =0, Eq. (9) can be obtained. F (P n +1 ) = F (P n + ϕδ n ) = 2

∂ ( P n + ϕδ n )  ∂  n  ( P + ϕδ n ) H 3   ∂θ  ∂θ  

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∂ ( P n + ϕδ n ) H ∂ ( P n + ϕδ n )  ∂  n n 3  ( P + ϕδ ) H  + Mξi − Λ x + 2α  ∂Y  ∂Y ∂θ  

Taking derivative of Eq. (9) with respect to φ and let φ=0, yields Eqs. (10)-(11) respectively.

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∂ ( P n + ϕδ n ) d ∂  ∂δ n   F (P n + ϕδ n ) = 2  δ nH 3 + ( P n + ϕδ n ) H 3 dϕ ∂θ  ∂θ ∂θ    n n n  ∂ (δ nH ) ∂  n 3 ∂ ( P + ϕδ ) n n 3 ∂δ   δ H + 2α + ( P + ϕδ ) H − Λx ∂Y  ∂Y ∂Y  ∂θ  

d ∂  ∂P n ∂δ n  F (P n + ϕδ n ) = 2  δ nH 3 + P nH 3  dϕ ∂θ  ∂θ ∂θ  ϕ =0 n ∂ ( δ nH ) ∂  n 3 ∂P n n 3 ∂δ  δ + 2α H + P H − Λ   x ∂Y  ∂Y ∂Y  ∂θ

According to Taylor series about P, F(Pn＋φδn) can be expanded as Eq. (12).

(10)

(11)

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F (P + ϕδ ) = F (P ) + ϕδ F ′(P n ) + n

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Taking derivative of Eq. (12) with respect to φ and letting φ=0, yield Eq. (13). d = δ F ′(P n ) + ϕ F ′′(P n )δ 2 + ⋯ F (P n + ϕδ n ) dϕ ϕ =0

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Since δn is far less than Pn+1 and Pn, neglecting high-order small amounts in Eq. (13) such as ϕF ′′(P n )δ 2 and combining it with Eq. (8), result in Eq. (14). d F (P n + ϕδ n ) = δ F ′(P n ) = −F (P n ) dϕ ϕ =0

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∂ ∂θ

n n ∂ ( δ nH )  n 3 ∂P n ∂  n 3 ∂P n n 3 ∂δ  n 3 ∂δ  δ α δ H + P H + 2 H + P H − Λ =     x ∂θ ∂θ  ∂Y  ∂Y ∂Y  ∂θ 

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Then, combining Eq. (11) with Eq. (14), obtain Eq. (15). Finally, the solving equation can be described as Eq. (16), which is a representation of Eq. (15) in the form of discretization.

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∂ ( P nH ) ∂  n 3 ∂P n  ∂  n 3 ∂P n  −2  P H  − 2α P H  − Mξi + Λ x ∂θ  ∂θ  ∂Y  ∂Y  ∂θ

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∂ δ i, jHi, j ∂Pi, j ∂δ i, j  ∂Pi,j ∂δ i, j  ∂  ∂  3 3 + Pi, jHi,3j + Pi, jHi,3j =  δ i, jHi, j  + 2α  δ i, jHi, j  − Λx ∂θ  ∂θ ∂θ  ∂Y  ∂Y ∂Y  ∂θ

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∂ Pi, jHi, j ∂Pi,j  ∂Pi, j  ∂  ∂  3 −2  Pi, jHi,3j  − 2α  Pi, jHi,j  − Mξi + Λ x ∂θ  ∂θ  ∂Y  ∂Y  ∂θ

)

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In order to solve Eq. (16), the finite difference formulas are used such as Eqs. (17)-(18). Substituting the variants of Pi,j、Hi,j and δi,j into Eqs. (17)-(18), Eq. (16) can be simplified into Eq. (19). Moreover, Eq. (20) is a convergence equation to improve the convergent rate of iterative computation. Eqs. (21)-(26) are the detailed expression of coefficients in Eq. (19). ∂Fi, j Fi+1, j − Fi-1, j Fi, j+1 − Fi, j-1 (17) = or ∂t 2∆t 2 ∆t ∂ 2Fi, j Fi+1, j + Fi-1, j − 2Fi, j Fi, j+1 + Fi, j-1 − 2Fi, j (18) = or 2 2 ∂t 2 ( ∆t ) ( ∆t )

δ i, j = − ( Ai, jδ i+1, j + Bi, jδ i-1, j + Di, jδ i, j+1 + Ei, jδ i, j-1 − Gi, j ) C i, j

(19)

Pn+1 = Pn + ωδ n Where,

(20)

Ai,j =

(

Hi,j3 Pi+1, j − Pi−1, j

Bi, j = −

( ∆θ )

Hi,3j Pi+1, j − Pi−1,j

( ∆θ )

2 i,j i, j

i+1,j

− Hi−1, j

2 ( ∆θ )

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) + 3H P ( H

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2 i, j i, j

i+1, j

3 i,j i, j 2

( ∆θ )

2

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) + 2H P

− Hi−1, j

2 ( ∆θ )

2

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−

3 i, j i,j 2

P

( ∆θ )

Λ x Hi,j 2∆θ +

Λ x Hi, j 2 ∆θ

(21)

(22)

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 Hi+1,j − Hi−1, j Pi+1, j − Pi−1, j α Hi+1, j − Hi−1, j Pi+1, j − Pi−1,j Ci, j = 3Hi,2j  + 2 2  2 ( ∆θ ) 2 ( ∆Y )  Pi+1,j + Pi−1, j − 2Pi, j Pi, j+1 + Pi,j−1 − 2Pi,j  + 2Hi,3j  + α  2 2 ( ∆θ ) ( ∆Y )    1 α  Λ x ( Hi+1,j − Hi-1,j ) − 4Hi,3jPi, j  + −  ( ∆θ )2 ( ∆Y )2  2∆θ    Hi,3j Pi, j+1 − Pi, j−1 3Hi,2jPi, j Hi, j+1 − Hi, j−1 2Hi,3jPi, j   Di, j = α  + + 2 2 2  2 ( ∆Y ) 2 ( ∆Y ) ( ∆Y )   Hi,j3 Pi,j+1 − Pi,j−1 3Hi,j2 Pi,j Hi,j+1 − Hi,j−1 2H 3 P  i, j i, j  Ei,j = α  − − + 2 2 2  2 ( ∆Y ) 2 ( ∆Y ) ( ∆Y ) 

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 Pi+1, j + Pi−1,j − 2Pi, j Pi,j+1 + Pi, j−1 − 2Pi,j  α Gi,j = 2Hi,j3  +  2 2 ( ∆θ ) ( ∆Y )    Hi+1, j − Hi−1,j Pi+1,j − Pi−1,j Hi,j+1 − Hi,j−1 Pi,j+1 − Pi,j−1   α + 3Pi,jHi,2j  + 2 2   2 ( ∆θ ) 2 ( ∆Y ) (26)  Pi+1,j + Pi−1, j − 2Pi,j Pi,j+1 + Pi,j−1 − 2Pi,j  + 2Pi,jHi,3j  +α  2 2 ( ∆θ ) ( ∆Y )    Pi+1,j − Pi−1,j Hi,j+1 − Hi,j−1  + Λ x Hi,j + Pi,j  2 ( ∆θ ) 2 ( ∆θ )   In the meshed domain, calculations for node pressures are divided into three kinds. Specifically, node pressures at the atmospheric boundary are a constant, the downstream pressures of orifices need to combine Eqs. (19)-(20) with Eqs. (1)-(2), other node pressures except orifices and atmospheric boundary are calculated by Eqs. (19) and (20). In the computation process, an iterative method such as the Gauss-Seidel method is used, which is an efficient method to solve linear equations. All calculated node pressures are updated by Eq. (20) at each iteration loop. The value for relaxation coefficient ω is selected from the computational tests. In the iterative process, a relative precision criterion is adopted. When all node pressures converge to the criterion n +1 n n -6 of ( P -P ) P ≤ 10 , the calculation terminates.

(

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Boundary conditions for the calculations are expressed as follows:
The atmospheric boundary conditions: P 1, j = PN+1, j=Pa/P0
The periodic boundary conditions: P i,1 = P1,M The load capacity of the bearing can be obtained by the integration of gas film pressure. The horizontal and vertical load capacities are Eqs. (27) and (28), respectively. L

2π

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0

L

2

0

0

∫ p ( y ,θ ) cosθ dθ dy π = ∫ ∫ p ( y ,θ ) sinθ dθ dy

Fx = ∫ Fy

(27) (28)

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Bearing stiffness can be expressed as Eq. (30). W ( e + ∆e) −W ( e − ∆e) ∆W = 2∆e 2∆e

(30)

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K=

where △e is a perturbation item around eccentricity e and △W is the difference of load capacity between the two adjacent eccentric points. 3. Calculation results and discussion

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In the following study, pressure distribution, load capacity and stiffness of aerostatic journal micro-bearings are estimated by various supply pressures, orifice diameters, eccentricities and so on. The aerodynamic effect of aerostatic journal micro-bearings is also investigated and the influences on load capacity and stiffness are acquired. All this work will be good instruction to the micro-bearing design. 3.1. Aerostatic pressure and corresponding performance

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By an external gas supply, the bearing clearance is filled with gas and generates gas film as shown in Fig. 3. Under the condition of unloading, the aerostatic pressure in the circumferential direction is almost equal. Once the bearing shaft is loaded, it will be eccentric and the distribution of film thickness and pressure are changed as shown in Fig. 4. The pressure around thin gas film increases while that of thick gas film decreases. The pressure difference between the thin and thick gas films generate load capacity and support the external load.

(a) (b) Fig. 3. Film thickness (a) and pressure distribution (b) under no load

(a) (b) Fig. 4. Film thickness (a) and pressure distribution (b) under load

ACCEPTED MANUSCRIPT 3.1.1. Effect of bearing clearance and orifice (namely restriction parameters)

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Fig.5 shows the change of load capacity and stiffness with restriction parameters including average bearing clearance and orifice diameter at a supply pressure of 0.5 MPa and an eccentricity of 1µm. The micro-bearing has a diameter of 3 mm and a length of 27 mm. Two rows with a total of 12 orifices are located at L/4 and 3L/4, respectively. It can be seen that within a certain extent, the load capacity increases with the increasing average bearing clearance whereas it decreases with the increasing orifice diameter. Moreover, the smaller the orifice diameter, the larger the effect of average bearing clearance on the load capacity. For instance, the load capacity at an orifice diameter of 0.06 mm increases to the max load capacity of 0.5 N when the average bearing clearance increases to 12 µm. In contrast, it only increases to 0.076 N at the orifice diameter of 0.14 mm. In Fig.5(b), stiffness decreases with the increase of average bearing clearance due to the compressibility of air. For example, as the average clearance at the orifice of 0.14 mm increases from 4 µm to 12 µm, the corresponding stiffness decreases from 2.74 mN/µm to about 1 mN/µm. Then, difference of the stiffness at the orifices is minor below the clearance of 8 µm. For instance, the stiffness at orifices of 0.06-0.14 mm with the clearance of 6 µm are 1.91469 mN/µm, 1.92249 mN/µm, 1.92424 mN/µm, 1.92544 mN/µm,1.92508 mN/µm, respectively. But to the clearances of 8-14 µm, the stiffness are slightly enlarged with the increase of orifice diameters. As it shown, the maximum increase of stiffness is just 0.18 mN/µm as the orifice diameters increase from 0.06 mm to 0.14 mm. Limited by processing equipment and technique, most of manufacturing enterprises only reach 8µm in average bearing clearance and 0.08 mm in orifice diameter. Meanwhile, considering both load capacity and stiffness, the restriction parameter of 8-12µm in the average bearing clearance and 0.08-0.14 mm in the orifice diameter are recommended for precision aerostatic journal micro-bearings. Based on above analysis, the restriction parameter of 10 µm in average bearing clearance and 0.1 mm in orifice diameter are designed for the micro-spindle in this study.

(a) Load capacity (b) stiffness Fig. 5. Performance vs. average bearing clearance and orifice diameter

Moreover, the number of orifices may affect the performance of the micro-bearings. Under the existing equipment conditions, it is difficult to fabricate the orifice with high forming accuracy and high position accuracy, which can be seen in the following prototype micro-spindle. The bearing shaft at high pressure is pressed to approach the bearing wall due to the flow deviation of the orifices. More orifices may compensate this limitation to some extent and benefit the stability of the bearing. However, no more than 8 orifices are usually provided in the micro-bearings due to their small structure. In this study, a micro-bearing with 6 orifices is analyzed.

ACCEPTED MANUSCRIPT 3.1.2. Effect of length and diameter of bearing (namely structure parameters)

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Figure 6 shows the change of load capacity and stiffness with the bearing length and diameter at a supply pressure of 0.5 MPa and an eccentricity of 1µm. The restriction parameters are 0.1 mm in orifice diameter and 10 µm in average bearing clearance. Two rows with a total of 12 orifices are located at L/4 and 3L/4, respectively. It is obvious that load capacity has a maximum value at different structure parameters. For instance, a maximum value of 3.1 N is obtained at a length of 16 mm and a diameter of 9 mm. The stiffness increases with the bearing length and diameter, which means that the large structure is beneficial to the stiffness. However, the longer length does not provide better load capacity in micro-bearings. In fact, the load capacity is influenced by bearing length and bearing diameter synchronously. To conventional aerostatic journal bearings, the ratio of length to diameter (L/D) is an important design parameter and it usually ranges from 0.5 to 2 [19]. In this study, the L/D ratios corresponding to the optimal load capacity with bearing diameters of 3 mm, 5mm, 7mm and 9 mm are around 1.61, 1.67, 1.71, 1.78, respectively. It can be concluded that the optimal L/D ratio is raised with the increase of bearing diameter and the optimal range is within 1.6-1.8 for the micro-bearing at the diameters of 3-9 mm.

(a) Load capacity (b) stiffness Fig. 6. Performance vs. bearing length and diameter

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Figure 7 shows the change of load capacity and stiffness with restriction parameters. Within the recommended restriction parameters of 8-12 µm in average bearing clearance and 0.08-0.14 mm in orifice diameter, the load capacity of micro-bearings mainly depends on the structure parameters and obviously increases with the increasing bearing diameter. The change of stiffness in Figs.7(a)-(c) are the same in Fig.5(b) with a characteristic that the larger the bearing diameter, the greater the max stiffness. The optimized restriction parameters also improve their performance. Within the orifice diameters of 0.06-0.14 mm and the bearing diameters of 3-9 mm as well as the average bearing clearances of 4-14 µm, the max load capacity does not always occur at the max average bearing clearance of 14 µm. With the increase of bearing diameter, the max load capacity happens with the decrease of the average bearing clearance in Fig.5 and Fig.7. For example, the max load capacity at bearing diameters of 5 and 7 mm occurs at the average clearances of 12 and 10 µm, respectively. The max stiffness appears at the minimum average clearance of 4µm. Fig.7(d) is a summary of the max load capacity and max stiffness in Fig.5 and Fig.7(a)-(c). The max load capacity in bearing diameters of 3-9 mm is decreased with the increase of orifice diameter. Differences of the max stiffness at certain bearing diameter are minor with different orifice diameters.

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(b) Bearing diameter D = 7 mm

(c) Bearing diameter D = 9 mm

(d) Summary of the max load capacity and stiffness

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In Fig.8, the micro-bearing is 3 mm in diameter and other restriction parameters are the same as Fig. 6. It can be seen from Figs.8(a)-(b) that the load capacity is closely related with supply pressure. Both the load capacity and stiffness increase with the increasing supply pressure. Not only that, the larger the eccentricity, the longer the increasing range of load capacity. It can also be noted that the eccentricity rather than eccentricity ratio is used in this study. In the conventional gas bearings, the eccentricity ratio rather than eccentricity is paid more attention. But in precision micro-machines, the eccentricity is related with the radial run-out of precision micro-spindles. It means that the eccentricity directly presents the error motion and thus is a critical design parameter. Figs.8(c)-(d) come from the results of Figs.8(a)-(b). It can be seen from Fig.8(c) that the load capacity has an approximate linear relationship with eccentricity. The load capacity has an approximate linear relationship with eccentricity. The load capacity with an eccentricity of 1 µm is 0.2 N at a supply pressure of 0.8 MPa. When the eccentricity increases to 5 µm, the load capacity increases to 1.2 N. However, the larger the eccentricity, the more the error motion, which can severely affect the accuracy of the micro-bearing and micro-spindle. Besides, the stiffness decreases with the increasing eccentricity. Accordingly, the eccentricity of 1 µm should be emphasized and thus is adopted in this study.

(b) stiffness

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(c) Load capacity (d) stiffness Fig. 8. Performance vs. supply pressure and eccentricity

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Orifice diameter

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Performance of aerostatic journal micro-bearings

L/D ratio

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Eccentricity

Fig.9 . Influence factors of the bearing performance

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According to above analysis, influence factors of the bearing performance are summarized in Fig.9. The influence degrees are shown from large to small in the clockwise direction. Specifically, the performance of the aerostatic journal micro-bearings mainly depends on their structure parameters, especially bearing diameter. The restriction parameters can greatly improve their performance. The operation parameters should be chosen according to the application situation. Besides, the L/D ratio of micro-bearings is different from that of conventional bearings. And eccentricity ratio is not a suitable design parameter for precision micro-bearings. The performance graphs in Figs.(5)-(9) make it easier to design and use aerostatic journal micro-bearings for engineer designers. 3.2. Aerodynamic pressure effect and corresponding performance

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When the bearing rotates, gas may flow along the circumferential direction due to its viscous action. The circumferential gas flow may further increase the pressure at high pressure area and lead to the increase of load capacity of the bearing, which is the action of aerodynamic pressure effect. It can be observed from Fig. 10. At w = 0 r/min, the pressure distribution is shown in Fig.10 (a). The bumps are pressures of the orifices, the max pressure is observed at the orifice which locates at the minimum gas film clearance. The pressure trend is gently changed. In Fig.10 (b), change of the film pressure is apparent. It can be seen that pressure distribution has an obvious increase from the location of the minimum film clearance to the max one. Then, the pressure decreases from the location of the max film clearance to the minimum one. The phenomenon is caused by the aerodynamic effect when the rotational speed increases to w = 200, 000 r/min. Therefore, the aerodynamic pressure effect can be measured by the enlarged load capacity Wd and enlarged stiffness Kd relative to that resulting from aerostatic pressure effect and expressed as: Wd = Wt - Ws (31) Kd = Kt - Ks (32) where Wt is the total load capacity, Ws the load capacity resulting from aerostatic pressure effect, Kt the total stiffness, Ks the stiffness resulting from aerostatic pressure effect.

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(a) w =0 r/min (b) w =200, 000 r/min Fig. 10. Pressure distribution at different rotational speeds 3.2.1. Effect of restriction parameters

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Fig. 11 shows the change of enlarged load capacity and stiffness at a speed of 200,000 rpm with average bearing clearance and orifice diameter, in which other parameters are the same as that in Fig. 5. The enlarged load capacity decreases with the increasing average bearing clearance whereas it increases with the increasing orifice diameter, which is opposite with that of static load capacity as shown in Fig. 5. It means that restriction parameters have opposite effects on aerodynamic pressure and static pressure from the aspect of load capacity. However, the effect of restriction parameters on aerodynamic pressure is very small, especially when the average bearing clearance is more than 10 µm. When the average bearing clearance exceeds 12 µm, the enlarged load capacity is almost negligible. The enlarged stiffness decreases with the increasing average bearing clearance but is barely affected by orifice diameter. It can be seem from the Fig.11 (b) that difference of the enlarged stiffness is very minor and the curves by different orifice diameters seem to be coincident with each other. Nevertheless, the stiffness is improved. At the recommend restriction parameters of 10 µm in average bearing clearance and 0.1 mm in orifice diameter, the load capacity increases by 0.22 mN whereas the stiffness increases by 3.7 mN/µm. It follows that the restriction parameters have some effect on aerodynamic pressure, but this effect at the restriction parameters of 10-14 µm in average bearing clearance and 0.08-0.12 mm in orifice diameter is very small.

(a) Enlarged load capacity (b) Enlarged stiffness Fig. 11. Aerodynamic pressure effect vs. average bearing clearance and orifice diameter 3.2.2. Effect of structure parameters Fig. 12 shows the change of enlarged load capacity and stiffness at a speed of 200,000

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rpm with bearing length and diameter, in which other parameters are the same as that in Fig. 6. It can be seen that both the enlarged load capacity and stiffness increase with the increasing bearing length and diameter. Especially when the bearing length is more than 16 mm, the enlarged load capacity and stiffness increase more quickly. It is the reason that the large structure increases both linear velocity and action area, and thus enhance the load capacity and stiffness. This means that the structure parameters have obvious effect on aerodynamic pressure and large ones can enhance aerodynamic pressure effect. However, the increase of aerodynamic pressure effect is limited. Taking the micro-bearing with a diameter of 9 mm of example, the load capacity resulting from aerodynamic pressure effect increases by only 0.33 mN when its bearing length increases to 28 mm. But the load capacity resulting from static pressure in Fig. 6 decreases by 0.7 N from the maximum of 3.1 N. It demonstrates that although aerodynamic pressure effect can improve the performance of the micro-bearing, but be mainly determined by the aerostatic pressure.

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(a) Enlarged load capacity (b) Enlarged stiffness Fig. 12. Aerodynamic pressure effect vs. bearing length and diameter 3.2.3 Effect of operation parameters

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Fig. 13 shows the change of enlarged load capacity and stiffness at a speed of 200,000 rpm with supply pressure and eccentricity, in which other parameters are the same as that in Fig. 8. Both the enlarged load capacity and stiffness decrease with the increasing supply pressure. The phenomenon indicates that higher supply pressure cannot provide better aerodynamic pressure effect. But in fact, higher supply pressure generates more static load capacity and greatly increases the total load capacity. For instance, when the supply pressure increases from 0.5 MPa to 0.7 MPa, the enlarged load capacity at an eccentricity of 5 µm decreases from 0.35 N to 0.05 N whereas the static load capacity increases from 0.38 N to 0.85 N. Both the enlarged load capacity and stiffness increase with the increasing eccentricity. However, the enlarged load capacity and stiffness at an eccentricity of 1µm are very small.

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(a) Enlarged load capacity (b) Enlarged stiffness Fig. 13. Aerodynamic pressure effect vs. supply pressure and eccentricity

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The aerodynamic pressure effect is highly related with rotational speed and necessarily increases with the increasing rotational speed, which can be seen from Fig. 14. The micro-bearing has a bearing diameter of 3 mm, 12 orifices with a diameter of 0.1 mm and an average bearing clearance of 10 µm. It can be seen that both the enlarged load capacity and stiffness increase with the increasing rotational speed, but only when the rotational speed is more than 100,000 rpm. It follows that the aerodynamic pressure effect improves the performance of aerostatic bearing and the higher the rotational speed, the greater the enlarged load capacity and stiffness. However, the load capacity and stiffness at the eccentricity of 1 µm hardly increase with the increasing rotational speed. It implies that for the application to ultra-precision micro-spindles, the aerodynamic pressure effect is limited. The static performance mainly depends on the aerostatic pressure effect. But the aerodynamic pressure effect plays an important role at high speeds as eccentricities are more than 3 µm, which is benefit to the dynamic stability of ultra-precision micro-spindles.

(a) Enlarged load capacity (b) Enlarged stiffness Fig. 14. Aerodynamic pressure effect vs. rotational speed and eccentricity

4. Experimental verification and application 4.1. Verification of numerical models The numerical models are proposed for the performance calculation of aerostatic journal micro-bearings. Moreover, it also can be used to analyze the performance of conventional aerostatic journal bearings. Thus, an experiment for the numerical models verification is carried out in a conventional aerostatic journal bearing pre-existing in our lab as shown in Fig.15. Restriction parameters are 0.2 mm in orifice diameter and

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Pull force

Spindle supported by aerostatic bearings

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approximate 12 µm in average bearing clearance. There are two rows with 8 orifices for each row. Structure parameters are 25 mm in bearing diameter and 50 mm in bearing length. As shown in Fig.15, the dynamometer provides pull force and the micrometer measures the shaft displacement. Meanwhile, the load capacity is equal to the corresponding pull force. To get exact results, the experiment is carried out three times at a supply pressure of 0.5 MPa and the average result is taken. It should be noted that performance at the static state is tested and compared with theoretical results as shown in Table. 1. It is shown that with the increasing shaft eccentricity, the errors of load capacity and stiffness increase and are up to 13.4% and 12.6%, respectively. The error is inevitable due to the assumptions of numerical models, machining error of bearing and experimental error, but the increase of error with the increasing shaft eccentricity is mainly caused by the experimental error. However, the results show the theoretical analysis is in good agreement with experimental one and has proved the validity of the proposed numerical models to some extent.

Displacement

Fig. 15. Experiment setup of aerostatic journal bearing

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Table. 1 Load capacity and stiffness of the experimental and numerical analysis Displacement

Load capacity

Difference of

Stiffness

Difference

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(N)

load capacity

(N/µm)

of stiffness

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7.59

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3

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22.15

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7.41

11.6

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25.79

29.78

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6.45

7.39

12.6

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4.2. Application in micro-spindles The micro-tools used in micro-cutting and micro-grinding usually have a tip with a diameter of less than 1mm and a shank with a diameter of 3.175 mm. And its length is less than 40 mm. When the machining efficiency of micro-cutting and micro-grinding is comparable with that of mechanical macro-machining, the rotational speed must be increased as the tool diameter decreases. For instance, to realize a typical macro-milling cutting speed of 200 m/min using the micro-tool with a tip diameter of 0.2 mm, a spindle rotational speed of 250,000 rpm should be achieved. Currently, many micro-parts with structure size of less than 10 mm are demanded to reach machining precision of micron level, so the micro-spindle should have a tool radial run-out of submicrometer. To meet the requirements, our team proposes a novel design concept of the separate spindle and

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tool and one-piece tool/rotor structure for high-speed and precision micro-spindles as shown in Fig. 16. A radial impulse turbine is used to provide high speeds. The turbine shaft and the tool are decoupled by a monolithic flexible coupling and then the tool shank is directly supported by aerostatic journal micro-bearing. It can minimize the run-out of the tool at high speeds. Additionally, a shape memory alloy (SMA)-based clamp consisting of an SMA ring and a collet is proposed for tool clamping [20].

Fig. 16. Design concept for high-speed precision micro-spindles

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Based on the accomplished analysis, optimal restriction parameters of the micro-bearing for a 3.175-mm-diam and 38-mm-long micro-tool are orifice diameter of 0.08-mm and average bearing clearance of 6-µm. Since the structure parameters are small and the average bearing clearance is strictly required, the bearing parts are difficult to manufacture directly using precision machine tools. In reality, the bearing and bearing shaft are artificially ground with continuous measurement, and then matched. This procedure needs good experience of the worker. Even so, a minimum bearing clearance of 10 µm is achieved with the equipment and artificial experience of our research team. Therefore, the aerostatic journal micro-bearing with average bearing clearance of 10 µm and orifice diameter of 0.1 mm is employed. Besides, there are two rows with 6 orifice for each row. A prototype micro-spindle with a shaft diameter of 7 mm is manufactured as shown in Fig.17. The radial motion error of the turbine shaft increases with the rotational speed. When the rotational speed achieves about 120,000 rpm, the shaft is locked. But due to the accommodation of the monolithic flexible coupling, the radial motion error of the tool shaft is around 3.31 µm and it can be regarded as the accuracy of the micro-bearing. Because the tool shaft has small diameter and radial motion, it is difficult to measure its load capacity and corresponding eccentricity at high-speeds. The static load capacity is measured. In the experiment, the small weights manufactured by our team are loaded on the tool shaft by copper wire as shown in Fig. 18. A weight supported by the tool shaft under no rotation is the static load capacity and the result is shown in Fig. 19 and Table. 2. The load capacity increases with the eccentricity, which is consistent with the theoretical analysis. However, the experimental results are larger than that obtained by the established theoretical models. It is attributed to the monolithic flexible coupling. The free end of the micro-tool are loaded whereas the other end is connected with the monolithic flexible coupling. Although the coupling is flexible, it still provides small supporting force for the micro-tool and results in large experimental load capacity. Moreover, the prototype micro-spindle is disassembled. It is found that the turbine shaft wears obviously as shown in Fig. 20. And the gas flow rate of the orifices are different. Not only that, the nonuniformity of the bearing clearance is another critical issue. These manufacturing and assembly errors, which are not taken into account in the established models, also aggravate the deviation of experimental and theoretical results.

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Fig. 17. Initial prototype micro-spindle

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Fig. 18. Experiment setup of prototype micro-spindle

Fig. 19. Static load capacity of initial micro-spindle Table.2 Static load capacity of micro-tool

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Load capacity (N) Experimental

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Fig. 20. Turbine shaft Based on the previous experience on design and manufacturing, another prototype micro-spindle is developed as shown in Fig. 21. It is still driven by a radial impulse turbine, but its structure is improved and optimized. The micro-tool is supported by the

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same micro-bearing. However, its machining accuracy is improved. The developed micro-spindle has a compact structure with a 30-mm outer diameter and a 95-mm length. Its maximum speed exceeds 240,000 rpm. The radial run-out of the tool shaft decreases to 2.79 µm. The load capacity is also measured and shown in Fig. 22. It is obvious larger than that of the initial prototype. It follows that machining accuracy is critical to the micro-bearing. Besides, the radial run-out under a load of 0.5 N decreases by about 1.53 µm when its rotational speed increases to 200,000 rpm. It can conclude that the dynamic pressure effect resulting from high speeds appears and improves the performance of micro-bearing and corresponding micro-spindle, but only at large eccentricities and ultra-high rotational speeds. It should be noted that the experimental result can qualitatively validate the connected analysis. If very precision value of load capacity and stiffness is demanded, especially at high speed conditions, the experiment may be conducted with the aid of ultra-precision measuring instruments and innovative measuring method.

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(a) Rotational speed (b) Radial run-out Fig. 21. Performance measurement of improved prototype micro-spindle

Fig. 22. Static load capacity of improved micro-spindle

In the improved micro-spindle, the friction instead of locking occurs on the turbine shaft at the maximum speed of 240,000 rpm. In addition to the machining accuracy, the balance of bearing shaft is another critical issue. It means that the bearing shaft should be balanced to enhance the rotational accuracy of micro-bearing and the rotational speed of the micro-spindle. According to the balance quality grade of G2.5 for machine tool spindles based on ISO and ANSI standards, an imbalance of 1.25 µg•mm is recommended for a 21 g spindle shaft rotating at 400,000 rpm-500,000 rpm. But common commercial balancing machines achieve a minimum residual unbalance of 50 µg•mm. Therefore, the balance of micro-shaft has become an important and challenging research topic.

ACCEPTED MANUSCRIPT 5. Summary and conclusions

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The numerical models for the performance calculation of aerostatic journal millimeter-scale micro-bearings are established based on the modified Reynolds equation. The dynamic pressure effect resulting from high rotational speeds and eccentricity is considered in the models. The static load capacity and stiffness are calculated and fully analyzed from the aspects of restriction parameters, structure parameters and operation parameters. Then from the same aspects, the aerodynamic pressure effect is also thoroughly investigated and the corresponding load capacity and stiffness are obtained. To verify the numerical models, static load capacity and stiffness of an aerostatic journal bearing are measured and compared with theoretical results. Since the testing bearing is not a micro-bearing, the measured results verify the proposed numerical models to some extent. Based on above analysis, a prototype micro-spindle is developed and a micro-bearing with a diameter of 3.175 mm is applied to support the micro-tool. Due to the accommodation of monolithic flexible coupling, the static load capacity of the micro-tool is measured and can be regard as the performance of the micro-bearing. However, the poor machining precision of miniature parts badly affects prototype performance. Therefore another improved prototype micro-spindle is developed and the load capacity of micro-bearing is measured. Its maximum speed exceeds 240,000 rpm and the radial run-out of the tool shaft decreases to 2.79 µm, which demonstrates the great improvement in the bearing performance. Based on above study, the following conclusions can be drawn:
The established numerical models enable performance analysis of aerostatic journal micro-bearings. Especially the aerodynamic pressure effect can be directly and accurately obtained by numerical results. The presented graphs of performance change with restriction, structure and operation parameters provide a comprehensive understanding and design reference for engineers.
The performance of aerostatic journal micro-bearings mainly depends on its structure parameters and obviously increases with the increasing bearing diameter. The bearing length is ascertained along with bearing diameter for optimal performance. The restriction parameters including orifice diameter and average bearing clearance can greatly improve their performance. Considering both the performance requirement and current machining capacity, the restriction parameters of 8-12 µm in average bearing clearance and 0.08-0.14 mm in orifice diameter are recommended. The operation parameters including supply pressure and eccentricity can be ascertained according to the application situation.
The aerodynamic pressure effect is resulted from rotational speed, but still affected by restriction and structure parameters. The aerodynamic pressure effect increases with the eccentricity and structure parameters. And it is outstanding at ultra-high speeds and large eccentricities.
Although the aerodynamic pressure effect existing in high-speed aerostatic journal micro-bearings can improve their performance, it only plays a supporting role. Therefore, the structure parameters of the micro-bearings need to be determined and optimized according to static performance, which plays a major role.
The high-speed precision micro-spindle can use a micro-bearing to support the micro-tool. However, the aerodynamic pressure effect resulting from rotational speeds has little improvement on the load capacity and stiffness of the micro-tool unless the high rotational speed and large eccentricity are applied. Moreover, the low accuracy of miniature parts and poor balance quality of micro-shaft also compromise its performance. And the balance of micro-shaft has become an important and challenging research topic.

ACCEPTED MANUSCRIPT Given the above, the load capacity and stiffness of aerostatic journal micro-bearings with inherent orifices are small and decrease with the bearing diameter greatly. For the application in high-speed precision micro-spindles, if the cutting force is required to be higher and more stable, the aerostatic porous bearings might be more suitable, on which an on-going research is currently conducted by the authors and the test results will be reported in the near future. Acknowledgements

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The presented work are funded by the National Science Foundation of China (51505140, 51375156), China Postdoctoral Science Foundation (2016T90749, 2015M570676), Natural Science Foundation of Hunan Province, China (2016JJ3037) and Open Research Fund of Key Laboratory of High Performance Complex Manufacturing, Central South University. The authors acknowledge the financial supports. Moreover, thanks to Professor Bi Z for his guidance. References

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[1] Aurich JC, Carrella M, Walk M. Micro grinding with ultra small micro pencil grinding tools using an integrated machine tool. CIRP Ann-Manuf Techn 2015;64(1):325-328. [2] Yoon HS, Ehmann KF. Dynamics and stability of micro-cutting operations. Int J Mech Sci 2016;115:81-92. [3] Zhou ZX, Li W, Song TJ, Huang XM. Performance requirements and research state of micro-spindles for micro-cutting (in Chinese). Chin J Mech Eng 2011;47(19):149-157. [4] Rowe WB. Hydrostatic and hybrid bearing design. Elsevier 2013. [5] Zhang XQ, Wang XL, Liu R, Zhang YY. Modeling and analysis of micro hybrid gas spiral-grooved thrust bearing for microengine. J Eng Gas Turb Power 2013;135(12): 122508(1-8). [6] Otsu Y, Somaya K, Yoshimoto S. High-speed stability of a rigid rotor supported by aerostatic journal bearings with compound restrictors. Tribol Int 2011;44(1):9-17. [7] Chang SH, Chan CW, Jeng YR. Numerical analysis of discharge coefficients in aerostatic bearings with orifice-type restrictors. Tribol Int 2015;90:157-163. [8] Lo CY, Wang CC, Lee YH. Performance analysis of high-speed spindle aerostatic bearings. Tribol Int 2005;38 (1):5-14. [9] Chen YS, Chiu CC, Cheng YD, Influences of operational conditions and geometric parameters on the stiffness of aerostatic journal bearings. Precis Eng 2010;34 (4):722-734. [10] Bou-Said B, Chaomleffel JP. Hybrid journal bearings: theoretical and experimental results. J Tribol 1989;111(2):265-269. [11] Su JCT, Lie KN. Rotation effects on hybrid air journal bearings. Tribol Int 2003;36(10):717-726. [12] Chen XD, Zhu JC, Chen H. Dynamic characteristics of ultra-precision aerostatic bearings. Adv Manuf 2013;1(1):82-86. [13] Liu ZS, Zhang GH, Xu HJ. Performance analysis of rotating externally pressurized air bearings. Proc IMechE Part J: J Eng Tribol 2009;223(4): 653-663. [14] Nishio U, Somaya K, Yoshimoto S. Numerical calculation and experimental verification of static and dynamic characteristics of aerostatic thrust bearings with small feedholes. Tribol Int 2011;44(12):1790-1795. [15] Vogler MP, Kapoor SG, Devor RE. On the modeling and analysis of machining performance in micro-endmilling. Part II: Cutting force prediction. J Manuf Sci Eng 2004;126 (4):695-705. [16] Liu D, Liu YH, Chen SJ. Aerostatic lubrication. Harbin Institute of Technology 1990. [17] Wang N, Chang SH, Huang HC. Comparison of iterative methods for the solution of compressible-fluid Reynolds equation. J Tribol 2011;133(2): 021702(1-7). [18] Wang NZ, Chang CY. An application of Newton's method to the lubrication analysis of air-lubricated bearings. Tribol T 1999;42 (2):419-424. [19] Chen YS, Chiu CC, Cheng YD. Influences of operational conditions and geometric parameters on the stiffness of aerostatic journal bearings. Precis Eng 2010;34(4):722-734. [20] Li W, Zhou ZX, Zhang B, Xiao YY. A micro-coupling for micro mechanical systems. Chin J

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Mech Eng 2016;29(3):571-578.

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Highlights




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This study establishes the numerical models for the performance calculation of aerostatic journal micro-bearings. Dynamic pressure effect of the bearing is investigated with different rotational speeds and eccentricities. The presented graphs of performance changed with restriction, structural and operation parameters provide a comprehensive understanding and design reference for engineers. The numerical models are verified on a prototype aerostatic journal bearing. Considering both the performance requirement and current machining capacity, the restriction parameters of 8-12 µm in average bearing clearance and 0.08-0.14 mm in orifice diameter are recommended. According to the theoretical analysis, a prototypical micro-spindle is developed with aerostatic micro-bearing in diameter of 3.175 mm. Due to the accommodation of a monolithic flexible coupling, the static load capacity of the micro-tool can be regard as the performance of the micro-bearing. However, the performance is badly affected by the poor machining precision. Therefore, another improved prototype micro-spindle is developed. Its maximum speed exceeds 240,000 rpm and radial run-out of the tool shaft decreases to 2.79 µm, whose performance is greatly improved. The load capacity of the micro tool at high-speeds is measured. It indicates that the aerodynamic pressure effect resulting from rotational speeds has little improvement on the load capacity and stiffness of the micro-tool unless the speed and eccentricity are greatly increased. The static performance mainly depends on the aerostatic pressure effect. But the aerodynamic pressure effect plays an important role at high speeds as eccentricities are more than 3 µm, which is benefit to the dynamic stability of ultra-precision micro-spindles. According to our experiment, the low accuracy of miniature parts and poor balance quality of micro-shaft compromise the performance of the spindle and the micro-bearing. Balance of micro-shaft is also critical and becomes an important and challenging research topic.

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