Analysis on discharge coefficients in FEM modeling of hybrid air journal bearings and experimental validation

Accepted Manuscript Analysis on discharge coefficients in FEM modeling of hybrid air journal bearings and experimental validation Laiyun Song, Kai Cheng, Hui Ding, Shijin Chen PII:

S0301-679X(17)30516-9

DOI:

10.1016/j.triboint.2017.11.002

Reference:

JTRI 4942

To appear in:

Tribology International

Received Date: 25 July 2017 Revised Date:

27 October 2017

Accepted Date: 2 November 2017

Please cite this article as: Song L, Cheng K, Ding H, Chen S, Analysis on discharge coefficients in FEM modeling of hybrid air journal bearings and experimental validation, Tribology International (2017), doi: 10.1016/j.triboint.2017.11.002. 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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Analysis on discharge coefficients in FEM modeling of hybrid

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air journal bearings and experimental validation Laiyun Songa, Kai Chenga,b, Hui Dinga,* and Shijin Chena a

College of Engineering, Design and Physical Sciences, Brunel University London, Uxbridge, Middlesex UB8 3PH, UK

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b

School of Mechatronics Engineering, Harbin Institute of Technology, P.O. Box 413, Harbin 150001, PR China

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Abstract

Solving the Reynolds equation is an efficient method to simulate the performance of the hybrid air journal bearings, and the accuracy of the method is determined by discharge coefficients (Cd). The discharge coefficients in hybrid air journal bearings are different with those

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in flat and static air bearings due to the varied film thickness in the circumference and working condition in high speeds. Taking account of these effects, a new formulation on computing discharge coefficients is proposed on the basis of the previous research. Furthermore, the

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variations in the bearing load performance are investigated using the modified Cd and the

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associated FEM method, supported by well-designed experiments.

Keywords: Hybrid air journal bearings; Discharge coefficients; Modeling and simulation; High speed air journal bearings

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1. Introduction Hybrid air journal bearings are gaining more and more applications in precision engineering due to their inherited advantages including near-zero friction, high motion accuracy and long service life

[1]

. Moreover, air as a lubricant is cheap and stable over a wide range of working

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temperatures. As a result, hybrid air bearings have advantages in ultra-high speed conditions with more than 100,000r/min in particular. To investigate the static and dynamic characteristics of hybrid air journal bearings, numerical methods like solving the Reynolds equation[2-4] and CFD [5-7]

are adopted and their precisions are verified by comparing with the respective

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simulation

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experiment results.

Compared with CFD simulation, solving the Reynolds equation is more efficient in the process of calculation. Du

[8]

studied the steady characteristics of the air journal bearings with

grooves by solving the Reynolds equation. Gao

[9]

adopted the finite element method(FEM) to

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solve the Reynolds equation and investigated the static characteristics of aerostatic journal bearings in ultra-high speeds. However, the accuracy of FEM is dominated by discharge coefficients which were set as certain values in these analyses.

[10-12]

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Traditionally, the discharge coefficient is considered as a constant ranging from 0.6-0.8 ,while some authors have taken the Cd as a function of pd /ps [13, 14], where pd and ps represent

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the downstream pressure and the supply pressure, respectively. Renn and Hsiao et al.[5] proposed a new model to calculate the mass flow rates, which indicated that the mass flow model developed from nozzle is different from the model of orifice in some situations. Belforte et al. [15, 16]

concluded an approximate formula which considered the discharge coefficients as a function of

supply pressure, air gap height and orifice size by comparing the pressure between experimental and numerical results. Neves et al. [13] investigated the influences of the discharge coefficients on aerostatic journal bearings, which proved discharge coefficients having remarkable influence on mass flow rates. Wang et al.

[14, 17]

study effects of the rotational speeds and surface waviness on

ACCEPTED MANUSCRIPT the static and dynamic performance of hybrid air journal bearings by solving Reynolds equation using FDM method, in which the discharge coefficient is a function of Pd/Ps. With the precision and efficiency of the CFD developed

[18-20]

, some researchers used CFD results to calculate

discharge coefficients for supporting further analysis. Miyatake et al.

[21]

compared

numerical

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results of volume flow rate which are derived by solving the Reynold equations and CFD simulations respectively to determine discharge coefficients, by which they studied the characteristics of aerostatic thrust bearings with small feed holes. Similarly, Chang et al.

[22]

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numerically analyzed the discharge coefficients by comparing the load capacity calculated by solving the Reynolds equations and CFD simulations and their research indicated the discharge

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coefficients are sensitive to the orifice diameter and air film thickness. In high speed working condition, Pierart et al.[23] compared the results of mass flow rates calculated by solving the Reynolds equation and CFD model so as to calculate the discharge coefficients. The modified discharge coefficients were adopted in the numerical method by which the steady state

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characteristics of an adjustable hybrid air journal bearing were investigated. But only one orifice was taken into account and the corrector factor was independent of rotational speeds. Among all the researches critically assessed above, little attention was paid on hybrid air journal bearings

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operating in high rotational speeds. Therefore, it is essential to investigate the effects of the

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rotational speed and varied air film thickness on the discharge coefficients, and furthermore their influences on performances of hybrid journal air bearing in ultra-high speeds. In this paper, modeling and analysis by solving the Reynolds equations and CFD simulations in different speeds and eccentricity ratios are presented and compared for determining the discharge coefficients in hybrid journal air bearings. On the basis of the formulation proposed by Belforte[15], a modified formulation is proposed with the consideration of journal rotational speeds and varied film thickness factor. Furthermore, the load performances of the hybrid air bearings are studied by using modified Cd FEM calculation, and the differences between the

ACCEPTED MANUSCRIPT un-modified and modified methods are presented and analyzed. Finally, experiments on testing the equilibrium positions of the rotating journal are conducted to verify the reliability of the

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modified Cd and the associated method.

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2. Numerical modeling and analysis 2.1 Mass flow rate in hybrid air journal bearing

Generally, the mass flow rate of air through an orifice is developed as an ideal nozzle. As

derived as[5]:

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Fig.1 depicts, the pressure is reduced from ps to pd through the orifice. The ideal mass flow rate is

•

mt =

πd2 4

ps

2 ρa ψs pa

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  κ 2 (κ +1)/(κ −1) 1/ 2 pd  ) ;( ≤ ) β  (  κ  ps  2 κ + 1  ψs =    κ ( pd ) 2/κ − ( pd )(κ +1)/κ   ; ( pd > β )   κ   κ − 1  p ps  s   ps  

(1)

(2)

βκ = (2 / (κ + 1))(κ +1)/κ

where d is orifice diameter, ps, pd and pa are the supply pressure, the downstream pressure and the atmosphere pressure, respectively, ρa is the density of air in the atmosphere, κ is the ratio of the specific heat and equals to 1.4 for air.

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Figure. 1. Illustration of the hybrid air journal bearing.

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However, the actual mass flow rate is different from the ideal nozzle, so we need discharge coefficients to correct the mass flow rate, then there is: •

•

mr = Cd × mt

where

is the mass flow rate in actual condition, while

and Cd represents discharge coefficient.

(3) is the theoretical mass flow rate,

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The discharge coefficient is essentially a correction factor between the theoretical result and the actual result. By comparing the experimental results with numerical results, the discharge

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coefficient is taken as a function of geometry parameters in [15]. In this paper, the conclusion of [15] for zero rotational speed situation is adopted. The Cd can be expressed as:

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Cd = 0.85(1 − e −8.2((h +δ )/d) )

(4)

where δ is the pocket depth which prevents the orifice restrictor converting to the inherent compensation restrictor. In hybrid air journal bearings, different from the aerostatic bearing in plane, the film thickness h is varied in the circumference so here h represents the average film thickness in the area directly surrounding the inlet.

2.2 Numerical method In order to study the steady-state characteristics and equilibrium positions of hybrid air journal

ACCEPTED MANUSCRIPT bearings, we need to calculate the pressure distribution in the air film which is obtained by solving the Reynolds equation. The curvature of the air film is negligible because the film thickness is quite small compared to the diameter of the journal bearing. The steady-state

∂  3 ∂p h ∂ x  ∂ x

2

 ∂  3 ∂ p2  ∂(ph) ∂(ph) + h  + Qξi = Λ x + Λz    ∂x ∂z  ∂z  ∂z 

(5)

̅ is the dimensionless air film pressure, ℎ is the dimensionless air film thickness, ̅ and

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where

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dimensionless Reynolds equation is thus shown as:

̅ are the dimensionless coordinates,

and

are

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is the mass flow factor of the orifice,

the dimensionless bearing numbers in the x- and z- direction, respectively. It is noted that where orifice exists while

=1

=0 at surface without orifice.

The following dimensionless parameters are defined:

24η l 2 p x z h p , z = , h = , p = , Q = 3 c 2 a ρ v% lr lr hm ps hm ps ρ a

Λx =

12η u1lc 12η w1lc , Λz = 2 hm ps hm 2 ps

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x=

(6)

The pressure square function is defined as follows: 2

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f = p (x, z) = f (x, z)

(7)

For hybrid air journal bearings, the term of x-velocity in Eq.(5) equals to 0 because the axial

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velocity can be negligible (u1=0). By substituting Equation (7) into Equation (5), the derived equation can be shown as:

∂  3∂f  ∂  3∂f h + h ∂x  ∂x  ∂z  ∂z

where λ is the bearing number,

1/2

 ∂( f h) (8)  + Qξi = λ w ∂z  is the dimensionless velocity in z- directions. Then we can

derive that:

λ=

12ηVlr w ,w = 1 2 hm ps V

(9)

ACCEPTED MANUSCRIPT The FEM method is adopted to express the boundary condition i.e. 2

① atmosphere boundary condition: p(x, L/ 2) = pa , f (x, L/ 2) = p a = f a . ② symmetric boundary condition: ∂ p (x, 0) / ∂n = 0, ∂ f (x, 0) / ∂n = 0.

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③ air film pressure at orifice: p(orifice_position) = pd , f (orifice_position) = f d

It has been proved that the symmetric boundary condition can be satisfied automatically, so

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only boundary condition ① and ③ are enforced boundary conditions.

By application of Galerkin weighted residual technique, the weak solution of Eq. (8) can be

∫

Ω

3

[h (

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obtained by:

1/ 2 ∂ f ∂δ f ∂ f ∂δ f ∂δ f + ) − Qξ iδ f − λ f h w ]d xd z = 0 ∂x ∂x ∂z ∂z ∂z

(10)

Eq. (10) has been reduced to one order, which means the linear triangular element can be

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used for the base element for the solution. The interpolation function is defined as:

f = N eT f e

f 1/2 = N eT ( f 1/2 )e

h = N eT he

(11)

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where

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δ f = N eT δ f e = δ f eT N e

N e = [Ni N j N m ]T , f e = [ fi f j f m ]T , he = [hi h j h m ]T

( f 1/2 )e = [ f 1/2i f 1/2 j f 1/2 m ]T

(12)

In the triangular element we can develop that Ni =

1 (a i + bi z + ci x ), (i = i, j , m) 2 ∆a

(13)

where ai = z j xm − zm z j , bi = x j − xm , ci = zm − z j among them the subscripts take value of i, j, m in a loop. By employing the FEM method and substituting Eq. (11) (12) (13) to Eq. (10) we can

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∆e

− ∑ λ ∫ bi w h eT N e N eT d xd z × e∈∆i

∆e

• 1 − k µ m 1 r r δi (2∆ e ) 2

1 1/ 2 e (f ) = 0 2∆ e

(14)

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∑∫

e∈∆i

(N eT h e )3 d xd z (ci ceT + bi beT ) × f e

(i = 1, 2,...q)

•

where m is the mass flow rate of the

•

•

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orifice restrictor, µr equals to mΩ / mr , where mΩ

represents the mass flow rate which flows into the analysis region; and k1 = 24η pa / (h m 3 ps 2 ρ a ) .

Wx = palc 2 ∫

L /lr

Wy = palc 2 ∫

L /lr

0

0

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Finally, the load capacity Wx and Wy can be obtained from the following

d x∫

π D /lr

0

d x∫

p sin ϕ d z

π D /lr

0

p cos ϕ d z

(15)

And the static stiffness of the hybrid air journal bearing can be expressed as

W (e+ ∆e) − W (e) (16) ∆e where e is the eccentricity ratio and ∆e is variation of the eccentricity ratio. The structure

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Kw =

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parameters of the hybrid air bearings in this investigation is shown in Table.1

Table.1. Parameters of hybrid air bearings used in this investigation Bearing number

Bearing diameter (mm) 19.01

Orifice diameter (mm) 0.12

Pocket depth (mm) 0.10

Film thickness (µm) 21.5

Orifice number

Bearing 1

Bearing length (mm) 34.813

Bearing 2

30.00

18.00

0.10

0.06

12.0

8

10

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Table.2. Convergence test of the FEM method for hybrid air journal bearings (d=0.12mm, D=19.01mm, L=34.713mm,P 0.6 , w=50000r/min) W&

Stopping criterion

5.6863 5.6874 5.6875

20.718 20.716 20.716

∑ p −∑ p ∑p n

n

n −1

< 10−8

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36 # 60 60 # 100 72 # 120

W

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nx # nz

Figure. 2. Computational domain of hybrid air journal bearing

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MATLAB® programming is employed to calculate the matrix deduced by Eq. (14). Fig. 2 depicts the computational domain of the hybrid air journal bearing. The circumference θ is

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substituted for z, which is divided into nz parts while x is divided into nx parts. As there are two rows of orifice, the L/2 part of the domain is calculated with the symmetry boundary. By changing the nx and nz, the convergence test of the FEM method for hybrid air bearings is shown in Table 2 which indicates the validity and stability of the method. 2.3 CFD simulation As Miyatak, et al.[21] and Chang, et al.[22] reported, good agreement is achieved between CFD simulations and experimental results. Therefore, CFD simulation is employed to calculate

ACCEPTED MANUSCRIPT the same bearing with the model of FEM to determine the discharge coefficients. In this study, the FLUENT ® codes are applied to solve the three-dimension N-S equation. The rotation effects are taken into account, so the turbulence is calculated by the RNG k-ɛ model.

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Besides, the enhanced wall functions are employed to improve the accuracy of the model. Due to the asymmetry of the fluid domain in circumference and the symmetry in the axial, the model is simplified using half of the geometry with a symmetry boundary condition. As the

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Fig. 3 shows, there are 10 pressure inlets with pockets orifices which evenly distribute over circumference. The pressure of inlets and outlets are set at 6bar and 1bar, respectively. The film

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thickness is rather small and the wall beneath the air film is set as moving wall with the rotational speeds of 0-150,000 rpm. There are 300 grids in circumference while in z-direction the model is meshed in 120 grids. In this study, the number of grids in the direction of film thickness is set to 8 for the solution convergence.

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The fluid model with the boundary conditions is solved by the SIMPLE C scheme for the compressible cases. The convergence criterion in the present study is RNG < 10 −6

(17)

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In this study, to determine the discharge coefficients in hybrid air journal bearings, the mass flow rates calculated by solving the Reynolds equations and CFD simulations are compared. The

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CFD mass flow rate result, in contrast, varies under different eccentricity ratios and rotational speeds. By inputting the same parameters with CFD simulation and initial Cd calculated by Eq. (3) into FEM calculation, the mass flow rate in FEM can be calculated. And then the Cd is adjusted until limited difference (E < 1067) is reached between two calculations. The process is shown in Fig.4.

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Figure. 3. Numerical model of CFD simulations.

Figure 4. The flow chart of determining discharge coefficients

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3. Simulation results and analysis

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speeds.(bearing 1) (a)e=0.2(b) e=0.4

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Figure 5. Discharge coefficients with different eccentricity ratios, inlet positions and rotational

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Figure 6. Factors for the rates increase or decrease with every 10krpm versus dimensionless variable ha. (a) factor 1 which represents increasing rates in the upstream (b) factor 2 which represents decreasing rates in the downstream

In this study we investigated the influences of rotational speeds and varied film thickness in the circumference on the discharge coefficients in hybrid air journal bearings. The Cd in the eccentricity ratios of 0.2 and 0.4 are calculated to illustrate the speed effect, as shown in Figure 5 (a) and (b), respectively. In inlet 1 and inlet 6, where the upstream and downstream structure are symmetrical, the Cd remains constant in the varying rotational speeds. However, when the film thickness of upstream is larger than the thickness of downstream, for example inlet 2 to 5, the Cd

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the influence of rotational speeds on Cd in hybrid air bearings is linear, and in different positions and eccentricity ratios, discharge coefficients increase/decrease in different rates in terms of the rotational speeds. This is because in different positions and eccentricity ratios, the variation of the

represent the variation of the film thickness: 1/2

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d h  h0  ∆h = ⋅  d x  h(o p ) 

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converging film thickness is different. Therefore, a dimensionless variable ∆h is defined to

(18)

where h(op) is the film thickness at orifice position.

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As the rotational speed influence on Cd is linear, fac1 and fac2 are defined as the factors increasing/decreasing with rotational speeds of every 10krpm. Combined with the Eq. (3) we have:

(19)

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w  −8.2((h +δ )/d) )(1 + fac1(∆ h) × ) Cd 1 = 0.85(1 − e 10000  C = 0.85(1 − e −8.2((h +δ )/d) )(1 − fac 2(∆ h) × w )  d 2 10000

where Cd1 and Cd2 are the discharge coefficients for the inlets in upstream and the inlets in downstream, respectively.

For every inlet in hybrid air bearing, the conditions of the e=0.2, 0.3, 0.4, 0.5 and 0.6 are calculated and presented in the Figure 6 (a) and (b). With the increasing ∆h, the fac1 and fac2 increase exponentially and in large eccentricity ratio the factor increases to 3% for every 10krpm, which means the discharge coefficient could change as much as 45% in the angular velocity of

ACCEPTED MANUSCRIPT 150krpm. As a result, an approximate function for fac1 and fac2 were formulated with the single variable Δh as followed:

 fac1 = 0.82 ×10−3 ⋅ e2.16⋅∆h  −3 2.21⋅∆h  fac2 = 1.06 ×10 ⋅ e

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(20)

By substituting Eq. (20) to Eq. (18), the final expression for discharge coefficients in hybrid air bearings can be deduced with consideration of the influence of the rotational speeds and

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varied film thickness.

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Furthermore, the results of CFD simulation, FEM calculation with un-modified Cd and FEM calculation with modified Cd are compared and discussed in terms of pressure distribution, load capacity and attitude angle. Figure. 7 depicts the pressure distribution in circumference profile (at x=L/4) using CFD simulations, modified Cd FEM calculation and un-modified Cd FEM calculation. It can be seen that the pressure in the upstream is lower than that in the downstream.

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It could be explained as following: In upstream, the flow from the injector and the flow in the air film are in opposite direction while in downstream two flows are in concert. Therefore, the

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velocity profiles in upstream/downstream are enhanced/diminished which lead to the pressure difference. As the Figure. 7 depicts, FEM with un-modified Cd underestimates the upstream

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pressure and overestimates the downstream pressure without considering the effect of the varied film thickness. After employing the modified Cd to FEM, the pressure profile is almost equal with the result of the CFD which proves the validity of the method. It is noted that FEM cannot simulate the condition of the pressure drop region, as shown in the zoomed view.

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Figure 7. Pressure distribution in circumference profile at x=L/4 using CFD, modified Cd FEM and un-modified Cd FEM.

Figure 8. Load performance with different eccentricity ratios and rotational speeds using CFD, modified Cd FEM and un-modified Cd FEM for two different bearings. (a) load capacity versus rotational speeds in Bearing 1 (b) attitude angle versus rotational speeds in Bearing 1 (c) load capacity versus rotational speeds in Bearing 2 (d) attitude angle versus rotational speeds in Bearing 2

ACCEPTED MANUSCRIPT Figure 8(a) shows the load capacities of the two different hybrid air journal bearings when the rotational speed increases in the condition of e= 0.4 and 0.6. As the figure shows, the FEM model with un-modified Cd overestimates the load capacity comparing with the CFD results. The attitude angles of the Bearing 2 reach peak values and then decrease rapidly with the grow of the

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rotational speeds. It means in the small film thickness the pressure difference between upstream and downstream caused by dynamic effect are weakened after reaching certain speeds. After applying the modified Cd FEM method, the differences between two calculations of load capacity

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and attitude angle are evidently reduced for both bearings.

In the high speed condition of 150k rpm, the differences of the load capacity between the

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CFD simulation and FEM calculation in two different bearings are highlighted in Table 3. After applying the modified Cd in FEM, the differences are further decreased in both hybrid air journal bearings, which indicates the FEM model with modified Cd being a more accurate and general model for hybrid air journal bearings. In high eccentricity ratios, the adjustment using modified

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Cd has more obvious effect in high speed condition. Table 4 shows the attitude angle differences are also decreased obviously after applying the modified Cd in FEM in the rotational speed of 150k rpm in two different bearings. This can be further explained that the modified Cd results in

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the decline of the pressure difference between upstream and downstream which leads to the

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reduced attitude angle of the hybrid air journal bearings, as shown in the Figure. 7. It can be found that the differences of load capacity in Bearing 2 are smaller than those in Bearing 1. This is because the effect of modified discharge coefficients is weakened in the smaller orifice number.

Table. 3. Difference of load capacity between CFD and FEM in different eccentricity ratios Rotation speed

Bearing 1 Bearing 2

Eccentricity ratio FEM with un-modified Cd FEM with modified Cd FEM with un-modified Cd FEM with modified Cd

0.2 10.1% 8.7% 8.4% 5.2%

0.4 16.7% 10.6% 9.3% 3.7%

0.6 13.7% 7.3% 11.3% 5.1%

ACCEPTED MANUSCRIPT Table. 4. Difference of attitude angle between CFD and FEM in different eccentricity ratios Rotation speed bearing 1

0.2 32.3% 20.5% 7.5% 4.3%

0.4 31.0% 25.2% 7.8% 4.4%

0.6 9.4% 3.4% 17.7% 8.9%

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bearing 2

Eccentricity ratio FEM with un-modified Cd FEM with modified Cd FEM with un-modified Cd FEM with modified Cd

Figure. 9. Load performance of Bearing 1with different rotational speeds, film thickness and eccentricity ratios using modified Cd FEM.(a) load capacity versus rotational speeds in different eccentricity ratios (h=22µm) (b) attitude angle versus rotational speeds in different eccentricity ratios (h=22µm) (c)load capacity versus rotational speeds in different film thickness (e=0.2) (d) attitude angle versus rotational speeds in different film thickness (e=0.2) (e) stiffness versus eccentricity ratios in different speeds (h=22µm)(f) stiffness versus eccentricity ratios in different film thickness (e=0.2)

ACCEPTED MANUSCRIPT The load performances of hybrid air journal bearing calculated by modified Cd FEM method under various rotational speeds, eccentricity ratios and film thicknesses are illustrated in Figure 9. Figure 9(a) shows that the load capacity of hybrid journal bearing increases with increasing rotational speeds. Besides, with the increasing eccentricity ratios, the growth rate of the load

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capacity increases especially in high eccentricity ratios. Meanwhile, the attitude angles in different eccentricity ratios are calculated with the increasing rotational speeds, as depicted in Figure 9 (b). At the low eccentricity ratios like 0.2, 0.4 and 0.6, the attitude angle grows first and

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then remains a constant or decline slightly with the increasing angular velocity. However, at high eccentricity ratios like 0.8 and 0.9, the critical rotational speed where attitude angle reaches its

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peak value is smaller than that of in the low eccentricity ratio and the decline is more obvious. The similar phenomenon is observed and explained in

[9]

. Differently, by adopting the modified

Cd FEM method, the disparities between different eccentricity ratios are enlarged and it can be observed that the attitude angle at the eccentricity ratio of 0.4 is obviously smaller than any other

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conditions in Figure 9(b). It can be explained as following: The attitude angle is the ratio of the x-direction load capacity to the total load capacity. The total load capacity increases with the growth of the eccentricity ratios. Meanwhile, by the adjustment of the Cd, the difference between

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the upstream and downstream is attenuated which leads to the decline of the x-direction load

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capacity. When the eccentricity ratio is smaller than 0.4(or alike)，the difference is not obvious and the low load capacity is the dominating effect. As the eccentricity ratio exceeds 0.4(or alike), the pressure difference becomes substantial and dominates the attitude angle of the hybrid air bearing in high speeds.

Furthermore, the load capacity and attitude angle in different film thickness are investigated with the increasing rotational speeds in the eccentricity ratio of 0.2, as shown in Figure.9 (c) and (d). With the decline of the film thickness, the load capacity increases due to the rise of the whole pressure field. Also, the combined effects of the load capacity and the pressure difference

ACCEPTED MANUSCRIPT between the upstream and downstream determine the attitude angle of the hybrid air journal bearings. Figure 9 (e) depicts the hybrid air bearing static stiffness in different rotational speeds with the increasing eccentricity ratios. At the low rotational speed, the air bearing stiffness first

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increases and then drops to zero with increasing eccentricity ratios. As the rotational speed exceeds 10000 rpm, the stiffness increases with the growth of the eccentricity ratio. It can also be observed that when the eccentricity ratio is larger than 0.5, the stiffness grows faster, and the rates

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of stiffness grow is positive correlated with the rotational speeds of the journal. Figure 9 (f)

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shows the stiffness of the hybrid air journal bearing decreases as the film thickness increase. Also,

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the stiffness has a faster tendency over the eccentricity ratio of 0.5 in any film thickness.

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4. Experimental setup, experiment trial and results

In ultra-high speed condition, it is difficult to apply precise loads on the rotating journal and

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measure the loads and the corresponding displacements. As a result, the mass of the journal itself is taken as a constant load, and the equilibrium positions on the conditions of various rotational speeds are measured to validate the FEM model of modified Cd. The geometry parameters of the hybrid journal bearing tested in this paper are shown in the Table 1(Bearing 1). The mass of the rotor is equal to 0.24 kg. The front journal bearing and the rear journal bearing both have two rows of orifice and are in the same geometry parameters, so in fact each bearing loads the half weight of the journal.

ACCEPTED MANUSCRIPT Figure. 10 shows the experimental devices which are designed to measure the equilibrium positions both in horizontal and vertical direction with the variations of rotational speeds. The gas filter eliminates particles, water and oil vapors in the air, and the volume flow rates are tested by the gas flow meter. The supply pressure is set by the pressure reducing valve and the rotational

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speeds are controlled by variable-frequency drive. The equilibrium position is measured by a non-contact laser displacement measurement: Keyence-LK-H020, whose measuring range and

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resolution are ±3mm and 0.01µm, respectively.

Figure 10. Experiment bench for testing the equilibrium position of high-speed spindle with hybrid air journal bearings. (a) photo of test bench (b) schema of measurement instrumentation, rotation speed control and gas supply circuit.

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Figure 11. Equilibrium position of rotating spindle in different rotational speeds under supply pressure of 5bar. (a) Horizontal equilibrium position. (b) Vertical equilibrium position.

In the high-speed condition, the position of the journal is vibrating at the equilibrium position, so the horizontal and vertical equilibrium position is taken as the average of positions in the time of 30s under the sampling frequency of 20 kHz. The data under various conditions are measured five times to eliminate the influence of machining errors of the auxiliary part,

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installation errors of sensor, fluctuations of the gas supply, etc. In this experiment, the supply pressure is set to 5bar for the hybrid air bearings. Figure. 11

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depicts the equilibrium positions of the journal with variation of the rotational speed of 0-150krpm acquired by un-modified Cd FEM, modified Cd FEM and experiment data. It is noted

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that the error bar form is shown because of the vibration of the journal in high speed condition. Figure 12 (a) and (b) shows the horizontal and vertical equilibrium position, respectively. It can be found that the modified Cd FEM model is in good agreement with the experiment data, which proves the validity of the proposed modified Cd and the associated FEM model.

5. Conclusions In this paper, the discharge coefficients in hybrid air journal bearings are examined and determined by solving the Reynolds equation while in comparison with the results of the CFD

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and validate the method. The following conclusions are therefore summarized:

(1) The effect of rotational speed increases the discharge coefficients of upstream inlets while the discharge coefficients of downstream inlets are decreased. The influences are both

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linear in the conditions of having the large journal eccentricity ratios and high rotational speeds, while the difference can be up to 45%.

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(2) The varied air film thickness also has the influences on the discharge coefficients. By introducing a dimensionless variable ∆h, an exponential relationship can be concluded between the rates of Cd increasing/decreasing with every 10k rpm and ∆h.

(3) By introducing the modified Cd and the associated FEM model, the differences of the

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loading capacity and attitude angle between the CFD simulations and FEM calculations decrease in rotational speeds of 0-150k rpm. The effect of modified Cd FEM method on the load

circumference.

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performance is more obvious in the hybrid air journal bearings with larger orifice number in the

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(4) With the increase of the journal rotational speed and the decrease of the air film thicknesses, the loading capacity of the hybrid air journal bearing increases and in high eccentricity ratios the increase is much obvious. (5) The attitude angles increase first and then remain constant or decline with the increasing angular velocity. In the medium eccentricity ratio (e.g. e=0.4 or alike), the attitude angle is obviously smaller than other condition because of the combined effects of loading capacity and modified Cd.

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Acknowledgements The authors thank for the funding support for this research by the National Natural Science

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Foundation of China (Grant No. 51435006).

References

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[1] Cheng K, Huo DH. Mirco cutting: fundamentals and applications.: West Sussex Wiley; 2013. [2] Su JCT, Lie KN. Rotation effects on hybrid air journal bearings. Tribol Int. 2003;36:717-26. [3] Lo CY, Wang CC, Lee YH. Performance analysis of high-speed spindle aerostatic bearings. Tribol Int. 2005;38:5-14. [4] Jia CH, Pang HJE, Ma WS, Qiu M. Dynamic Stability Prediction of Spherical Spiral Groove Hybrid Gas Bearings Rotor System. J Tribol-Trans ASME. 2017;139:12. [5] Renn JC, Hsiao CH. Experimental and CFD study on the mass flow-rate characteristic of gas through orifice-type restrictor in aerostatic bearings. Tribol Int. 2004;37:309-15. [6] Belforte G, Raparelli T, Trivella A, Viktorov V, Visconte C. CFD Analysis of a Simple Orifice-Type Feeding System for Aerostatic Bearings. Tribol Lett. 2015;58:8. [7] Gao SY, Cheng K, Chen SJ, Ding H, Fu H. CFD based investigation on influence of orifice chamber shapes for the design of aerostatic thrust bearings at ultra-high speed spindles. Tribol Int. 2015;92:211-21. [8] Du JJ, Zhang GQ, Liu T, To S. Improvement on load performance of externally pressurized gas journal bearings by opening pressure-equalizing grooves. Tribol Int. 2014;73:156-66. [9] Gao SY, Cheng K, Chen SJ, Ding H, Fu HY. Computational design and analysis of aerostatic journal bearings with application to ultra-high speed spindles. Proc Inst Mech Eng Part C-J Eng Mech Eng Sci. 2017;231:1205-20. [10] Nakamura T, Yoshimoto S. Static tilt characteristics of aerostatic rectangular double-pad thrust bearings with compound restrictors. Tribol Int. 1996;29:145-52. [11] Chen MF, Lin YT. Static behavior and dynamic stability analysis of grooved rectangular aerostatic thrust bearings by modified resistance network method. Tribol Int. 2002;35:329-38. [12] Li Y, Zhou K, Zhang Z. A flow-difference feedback iteration method and its application to high-speed aerostatic journal bearings. Tribol Int. 2015;84:132-41. [13] Neves MT, Schwarz VA, Menon GJ. Discharge coefficient influence on the performance of aerostatic journal bearings. Tribol Int. 2010;43:746-51. [14] Wang XK, Xu Q, Wang BR, Zhang LX, Yang H, Peng ZK. Numerical Calculation of

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Rotation Effects on Hybrid Air Journal Bearings. Tribol Trans. 2017;60:195-207. [15] Belforte G, Raparelli T, Viktorov V, Trivella A. Discharge coefficients of orifice-type restrictor for aerostatic bearings. Tribol Int. 2007;40:512-21. [16] Belforte G, Colombo F, Raparelli T, Trivella A, Viktorov V. A new identification method of the supply hole discharge coefficient of gas bearings. Southampton: Wit Press; 2010. [17] Wang XK, Xu Q, Huang M, Zhang LX, Peng ZK. Effects of journal rotation and surface waviness on the dynamic performance of aerostatic journal bearings. Tribol Int. 2017;112:1-9. [18] Chen XD, He XM. The effect of the recess shape on performance analysis of the gas-lubricated bearing in optical lithography. Tribol Int. 2006;39:1336-41. [19] Li YT, Ding H. Influences of the geometrical parameters of aerostatic thrust bearing with pocketed orifice-type restrictor on its performance. Tribol Int. 2007;40:1120-6. [20] Zhou YJ, Chen XD, Chen H. A hybrid approach to the numerical solution of air flow field in aerostatic thrust bearings. Tribol Int. 2016;102:444-53. [21] Miyatake M, Yoshimoto S. Numerical investigation of static and dynamic characteristics of aerostatic thrust bearings with small feed holes. Tribol Int. 2010;43:1353-9. [22] Chang SH, Chan CW, Jeng YR. Numerical analysis of discharge coefficients in aerostatic bearings with orifice-type restrictors. Tribol Int. 2015;90:157-63. [23] Pierart FG, Santos IF. Steady state characteristics of an adjustable hybrid gas bearing Computational fluid dynamics, modified Reynolds equation and experimental validation. Proc Inst Mech Eng Part J-J Eng Tribol. 2015;229:807-22.

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Cd d e f

Discharge coefficient Orifice diameter(mm) Eccentricity ratio Approximate solution Pressure square function

f

Dimensionless square atmosphere and downstream pressure

fa , f d

hm

Average air film thickness(µm)

i, j, m

lc

subscript Characteristic length(mm)

L

Bearing length(mm)

u1 w1

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Supply pressure(Pa) Mass flow factor of orifice

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u, w v%

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p Q

Actual r mass flow rate(kg/s) Theoretical mass flow rate(kg/s) Dimensionless air film pressure Atmosphere pressure(Pa) Downstream pressure(Pa) Dimensionless atmosphere and downstream pressure

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Wx , Wy x, z ∆a ∆e

λ δ η ξ ρ

Velocity components in the x and z directions(m/s)

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pa , pd

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ℎ

Air film thickness(µm) Dimensionless air film thickness

h

p p; p<

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Nomenclature

Dimensionless velocity components in x and z direction

Flow velocity of orifice

Reference velocity(m/s) Load capacity in x and y direction(N) Dimensionless coordinates Element area Vibration of eccentricity ratio Bearing number Pocket depth(µm) Air dynamic viscosity Delta function Air density(kg/m3)

ACCEPTED MANUSCRIPT Air density at atmosphere pressure(kg/m3) Coefficient of flow through orifice

Λ , Λ κ

Dimensionless bearing numbers in the x- and z- direction Specific heat ratio of air(=1.4)

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ρ; ψ

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Highlights

The highlights of the study “Analysis on discharge coefficients in FEM modeling of

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hybrid air journal bearings and experimental validation” are as follows:

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Analysis of the influence of the rotation effect and varied film thickness on discharge coefficients A new formulation for computing discharge coefficients with the consideration of rotation effect and varied film thickness Analysis of static characteristic differences between modified Cd and un-modified Cd with associated FEM method. Load performance of hybrid air bearings in high speed condition using modified Cd FEM method