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Hydrodynamic Load Capacity of a Compressible Fluid Slider Bearing

Giri Agrawal*

R&D Dynamics Corporation, Bloomfield, Connecticut 06002

[Abstract] Load capacity of various foil and non-foil journal and thrust bearings can be estimated by combining results of slider bearings of various shapes. In the present paper the Reynolds’ equation for a compressible fluid is solved for a class of infinitely long slider bearings. The film is assumed to be of a power form, which encompasses a class of film shapes. Curves for pressure distribution and load capacity are plotted for low to very high compressibility numbers for different height ratios and different powers of film shape. The results can be useful in designing various types of bearings.

Nomenclature

x, y, z = coordinates u, v, w = velocity in directions of x, y, z p = pressure on bearing pa = ambient Pressure P = non-dimensional pressure ρ = density T = absolute temperature λ, µ = viscosity coefficients t = time →V =

→→→++ kwjviu

B = width of bearing in the direction of motion x = distance from leading edge X = non-dimensional distance from leading edge h = film height H = non-dimensional film height hmin= h2 = film height at trailing edge h1 = film height at leading edge

I. Introduction In many high speed turbomachines with gas as a working fluid, it is advantageous to use hydrodynamic gas bearings. However load capacity and stability are major concerns for the design. Load capacity of hydrodynamic journal and thrust gas bearings can be approximately predicted by combining the results of load capacity of slider bearings. This paper presents a numerical scheme to solve non-linear second order Reynolds’ equations for a class of infinitely long slider bearing operating on compressible fluid film, in order to calculate pressure distribution and load capacity. Analyzing the load capacity of infinitely long slider is close and analogous to predicting load capacity of finite length bearing since the end leakage of the hydrodynamic bearings in many cases are small and the pressure distribution is almost constant across the length of the bearing. The film thickness is assumed to be of a power form which takes into account various film shapes. In high speed machinery, one may encounter very high values of compressibility numbers, hence values ranging from 10 to 8000 are considered for calculation. All the results in this paper are presented as a set of non-dimensional curves, because such curves can be used as a tool by an engineer engaged in the design of hydrodynamic gas bearings with compressible fluid of various types as working fluid. *President, R&D Dynamics Corporation, 15 Barber Pond Road, Bloomfield, Connecticut, AIAA Senior Member.

1 American Institute of Aeronautics and Astronautics

43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit 8 - 11 July 2007, Cincinnati, OH

AIAA 2007-5095

Copyright © 2007 by R&D Dynamics Corporation. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

In Section II the principle of hydrodynamic bearing is explained using a straight slider. The governing equation for pressure distribution in a hydrodynamic bearing is derived from the equation of momentum, continuity and state by assuming valid assumptions applicable to hydrodynamic bearings. The bearing parameters are non-dimensionalized to arrive at non-dimensionalized governing equation. In Section III the equation for calculating film thickness is explained. The film thickness of the hydrodynamic bearing is calculated using power of film shape, minimum film height and height ratios. The film shapes are plotted for various order of power of the film shape. The film shapes showed there are useful in designing hydrodynamic bearings. In Section IV the numerical scheme for solution of the non-linear second order Reynolds’ equation is derived. The greatest difficulty encountered in solving the steady Reynolds’ equation for gas films is due to the non-linearity, the generalized Newton algorithm type quasi-linearization described here is very effective and the solution converges fast even for a large compressibility number. In this kind of linearized numerical method a transformation is done by assuming 22 HP=Ψ , where P is the non-dimensional pressure and H is the non-dimensional film height. In Section V the results of the solution are plotted for various compressibility numbers, film height ratio and film power. FORTRAN language is used in programming the solution of Reynolds’ equation. The pressure distribution along the bearing width are plotted for different compressibility numbers. The pressure distribution is integrated and the effective load carrying capacity of the bearing is plotted. The plots are non-dimensionalized to arrive at universal plots for designing hydrodynamic bearings. In Section VI necessary conclusions are drawn.

II. Theory

In a hydrodynamic bearing, pressure is generated by sliding of a surface relative to another close by surface. A typical slider bearing is shown in Fig. 1 where one of the surface is fixed and the other surface moves with a velocity (u). Pressure in the slider is generated by the wedging action. Study of a simple slider bearing is the most fundamental for understanding both journal and thrust bearings. The governing equaalong with the equationthe boundaries of the gadissipated, hence the flucan be neglected. Follow Equations of momentum

Figure 1. Slider Bearing Principle


tions for a laminar continuum flow are given by the well known Navier-Stokes equations of state, energy and continuity. In hydrodynamic bearings, the high thermal conductivity of s film relative to that of the thin gas film allows any heat generated in the film to be readily id flow can be considered isothermal. Because of the thinness of the gas film, the body force ing are the five governing equations:

:

)1()(2

xu

xp

DtDu

∂∆∂

++∆+∂∂

−= µλµρ

)2()(2

yv

yp

DtDv

∂∆∂

++∆+∂∂

−= µλµρ

)3()(2

zw

zp

DtDw

∂∆∂

++∆+∂∂

−= µλµρ

Equation of continuity:

)4(0)( =+∂∂ →

Vdivt

ρρ

Equation of state:

)5(ρCp = Where C is a constant, and

zw

yv

xu

∂∂

+∂∂

+∂∂

=∆

In order to derive a governing equation for pressure in a hydrodynamic bearing the following assumptions are made:

1. Height of fluid film y is very small compared to span x and z.

2. There is no variation of pressure across the fluid film, hence 0=∂∂

yp

3. Compared to the two velocity gradients yu∂∂ and

yw∂∂ , all other velocity gradients are negligible.

4. In case of steady-state flow 0=∂∂t

.

From above assumptions one-dimensional Reynolds’ equation for compressible fluid is derived as described by Gross [1]:

)6()(6)( 3 hpdxdu

dxdpph

dxd µ=

By defining non-dimensional parameters:

)7(BxX =

)8(minhhH =

)9(ap

pP =

Equation (6) becomes

)10()()( 3 HPdxd

dXdPPH

dxd

Λ=

Where, )(62

min

numberlitycompressibphBu

a

µ=Λ

3

American Institute of Aeronautics and Astronautics

Equation (10) is a non-linear second order differential equation, which is highly unstable at high compressibility numbers and does not easily converge to solution using numerical methods.

III. Film shape & Thickness Figure 2 shows the powethe film height at trailingshape is

The film shape equation

The Equation (13) deedge is always unity. DepIn high speed machines important parameters. He

Figure 2. Power Shaped Film


r shaped hydrodynamic fluid film, where “h1” is the film height at leading edge and “h2” is edge and “h” is the film height at any distance “x” from the leading edge. Equation of film

γ)1()( 212 Bxhhhh −−+=

can be non-dimensionalized by dividing every term with minimum film height “h2”

)11(2

1

min

11 h

hhhH ==

)12(12

2

min

22 ===

hh

hhH

)13()1()(1 21γXHHH −−+=

fines film thickness at different position along the bearing. The film thickness at trailing ending on the power of the film shape the thickness varies from leading to trailing edge. with high compressibility numbers different film shapes and large height ratios become ight ratios (H1/H2) ranging from 1 to 30 are considered for calculating the load capacity.

Figure 3 shows the film shape and thickness for a range of power from 1 to 10 for a height ratio of 10.

This class of power scalled a straight slide

The Reynolds’ Edifficult to obtain byhowever since thereconverge and one runnew variable is deΨ

The numerical sctechnique described binto a set of finite equations are solved Differentiating (14),

Or

Substituting (15) in (

0

1

2

3

4

5

6

7

8

9

10

11

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

X

H

Film Height

Height Ratio (H1/H2) =10

γ=1

γ=2

γ=6

γ=4

γ=8

γ=10

Figure 3. Class of Power Shaped Fluid Film


hapes are very useful in designing the shape of bearing. The film shape with a power of unity is r bearing.

IV. Numerical Solution

quation (10) is highly non-linear. Its solution for given film height (H) as described in (13) is directly applying any numerical method. An iteration process can be used for the solution;

is sudden pressure drop near the trailing edge of the slider, the iteration process does not s into various types of numerical instability. In order to improve the numerical convergence a fined as,

)14(22HP=Ψ . heme discussed here is called generalized Newton algorithm type quasilinearization iteration y Coleman and Snider [2], using this technique the non-linear Reynolds’ equation is converted

difference equations by using a grid system along the width of the bearing. The resulting using the column method as described by Castelli and Pirvics [3].

dXdHHPH

dXdPP

dXd 22 22 +=Ψ

)15(213

dXdH

dXdH

dXdPPH Ψ−

Ψ=

10),

)16(21 2

1

ΨΛ=⎥⎦⎤

⎢⎣⎡ Ψ−

ΨdXd

dXdH

dXdH

dXd

Equation (16) is of a general form,

)17(FL =Ψ Where F is a function of Ψ and P . Comparing Equation (16) and (17)

⎥⎦⎤

⎢⎣⎡ Ψ−

Ψ=Ψ

dXdH

dXdH

dXdL

21

dXd

dXdF Ψ

Ψ

Λ=ΨΛ=

21

21

12

Equation (17) can be solved by Newton algorithm iteration process where,

( ) [ ] [ ][ ] [ ] )18(,, )()1()()1()()()()(1 rrrrrrrrr PPPFPFPFL −∂∂

+Ψ−ΨΨΨ∂∂

+Ψ=Ψ +++

Where the superscript (r+1) is the (r+1)th iteration and superscript (r) corresponds to (r)th iteration. Substituting “F” in Equation (18) and differentiating with respect toΨ and P ,

[ ] )19(1211

2)()1(

)(

23

)(

)1(

21

)(

)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡Ψ−Ψ

Ψ

Ψ

−Ψ

Ψ

Λ=Ψ +

++ rr

r

r

r

r

r

dXd

dXdL

Define , in (19), Ψ=Ψ + )1(r φ=Ψ )(r

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−Ψ−ΨΛ

=Ψ−

φφφ

φdXd

dXdL 2

3

21 2

112

Or

)20(2

221 2

1

2

223

21

2

2

dXd

dXHd

dXd

dXd

dXdH

dXdH φφφφφ

−−−Λ

=Ψ⎥⎥⎦

⎤

⎢⎢⎣

⎡−Λ+

Ψ⎥⎥⎦

⎤

⎢⎢⎣

⎡Λ+−

Ψ


Equation (20) can be written in the form of finite difference for an ith point:

)21()(1424

)(

)2(2)(4

2

2)(

41

)1()1()()(

)1()1()()1(

)1()()1()1()1(3)(

)()(

)()1()1(

)()1(

−+−+

−

−+−+

−++

−Λ

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ Λ

+∆

−+

∆Ψ+

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧+−

∆−−

Λ+

∆

−Ψ+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ Λ

−−∆

−∆

Ψ

iiii

iiii

iiiii

i

ii

iii

ii

XHH

XH

HHHXX

H

HHXX

H

φφφφ

φφφ

φ

Where and means value of )(iH )(iΨ H and Ψ at the ith point.


For a problem with N number of points including the end points and NN actual points to find pressures where NN = N-2, the iteration process can be denoted as a grid system as shown in Fig. 4,

Figure 4. Finite Difference Grid System

In order to solve the finite difference Equation, (21) is re-written taking into account the end points, all the ith points are incremented to represent (i+1)th point,

NNi

HHHX

HHHX

HHHX

iiii

iiii

ii

i

iiii

iiiii

,1

)22()(142

)4(4

1

)(4

)(2

2)4(

41

)()2()1()1(

)()1()2()1(

)()2(3)1(

)2()1()()(

)1()2()1()()1(

=

−Λ

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ Λ

+−+∆

Ψ+

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧−

Λ+−−

∆−

Ψ+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ Λ

−−+∆

Ψ

+++

++−

+

+

++

++++

φφφφ

φφφ

φ

The boundary condition for solving the finite difference equation is that the pressure values at the beginning point and the end point are unity.

Equation (21) is of the form , these set of finite difference simultaneous linear equations with a coefficient matrix [A] being band structured can be solved using Gauss elimination method.

RA =Ψ

For example a slider bearing solved for 7 pressure solution points will have 5 unknown intermediate points whereas the end points are unity, will have a matrix structure as:

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ΨΨΨΨΨ

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

5

4

3

2

1

5

4

3

2

1

5,54,5

5,44,43,4

4,33,32,3

3,22,21,2

2,11,1

00000

0000000

RRRRR

AAAAA

AAAAAA

AA

By solving for Ψ , pressure value at each solution point corresponding to a film thickness is found by using the relation (14). The total load capacity of the bearing is calculated by integrating the load capacity at each point using Simpson’s rule.

V. Results In order to perform bearing analysis the film thickness is assumed using Section III. Once the film height at every point along the width of the bearing is known, the Reynolds’ equation is solved as in Section IV and pressure values are calculated. The distance along the bearing is normalized as X and the pressure values are normalized by dividing the pressures by ambient pressure. The analysis is done for over a range of compressibility number for different height ratios and power of film shape. The pressure value at any given distance on a bearing is dependent on film height at that point (H), compressibility number of the bearing (Λ), height ratio (H1/H2) and the shape of the film which is a function of power (γ). Figures 5 through 12 show the pressure distribution for different height ratios and film shapes. The pressure distribution shown in the figures are integrated and the load capacity are plotted against height ratios in Figs 13 through 16. Figure 17 shows the classic compressibility effect of hydrodynamic bearings, where the load capacity of the bearings flattens out after certain values for a given bearing configuration. In other words in

)(62

min

numberlitycompressibphBu

a

µ=Λ , keeping all other terms constant and increasing the speed (u) the load

capacity flattens out after certain speed of the bearing, this is due to the compressibility of the hydrodynamic films, whereas in an incompressible fluid film the relation between speed and load capacity is linear. The load carrying capacity of the bearing increases with the power of film shape and height ratio. As the power of film shape increases, the pressure distribution is uniform over the length of the bearing. The rate of increase in load capacity with film power is not as predominant compared to increase in height ratio. Figure 18 shows the effect of power of film shape on load carrying capacity for a certain height ratio and compressibility number. The curve flattens out approximately after a power of four.


0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2

Λ=4000

Λ=500

Λ=1000

Λ=100Λ=50Λ=10

Height Ratio = 15.0Power of Film Shape = 1.0

NO

RM

ALI

ZED

PR

ESSU

RE

(P)

NORMALIZED DISTANCE (X)

Λ=8000

Figure 5. Pressure Distribution Plot

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2

Λ=4000

Λ=500

Λ=1000

Λ=100Λ=50Λ=10


NO

RM

ALI

ZED

PR

ESSU

RE

(P)


Λ=8000



0

2

4

6

8

10

12

14

0 0.2 0.4 .6 0.8 1 1.20

Λ=4000

Λ=500

Λ=1000

Λ=100Λ=50Λ=10


NO

RM

ALI

ZED

PR

ESSU

RE

(P)


Λ=8000


0

2

4

6

8

10

12

14

0 0.2 0.4 0.6 0.8 1 1.2

Λ=4000

Λ=500

Λ=1000

Λ=100Λ=50Λ=10


NO

RM

ALI

ZED

PR

ESSU

RE

(P)


Λ=8000



0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1 1.2

Λ=4000

Λ=500

Λ=1000

Λ=100Λ=50Λ=10


NO

RM

ALI

ZED

PR

ESSU

RE

(P)


Λ=8000


0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1 1.2

Λ=4000

Λ=500

Λ=1000

Λ=100Λ=50Λ=10


NO

RM

ALI

ZED

PR

ESSU

RE

(P)


Λ=8000



0

2

4

6

8

10

12

14

16

18

20

0 0.2 0.4 0.6 0.8 1 1.2

Λ=4000

Λ=500

Λ=1000

Λ=100Λ=50Λ=10


NO

RM

ALI

ZED

PR

ESSU

RE

(P)


Λ=8000


0

2

4

6

8

10

12

14

16

18

0 0.2 0.4 0.6 0.8 1 1.2

Λ=4000

Λ=500

Λ=1000

Λ=100Λ=50Λ=10


NO

RM

ALI

ZED

PR

ESSU

RE

(P)


Λ=8000



0

0.5

1

1.5

2

2.5

3

0 5 10 15 20 25 30 35

Power of Film Shape=1.0

Λ=4000

Λ=500

Λ=1000

Λ=100Λ=50Λ=10

NO

RM

ALI

ZED

LO

AD

(W/B

*L*P

a)

HEIGHT RATIO (H1/H2)

Λ=8000

Figure 13. Load Capacity Plot

0

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 35


Λ=4000

Λ=500

Λ=1000

Λ=100Λ=50

Λ=10

NO

RM

ALI

ZED

LO

AD

(W/B

*L*P

a)


Λ=8000



0

2

4

6

8

10

12

0 5 10 15 20 25 30 35


Λ=4000

Λ=500

Λ=1000

Λ=100Λ=50Λ=10

NO

RM

ALI

ZED

LO

AD

(W/B

*L*P

a)


Λ=8000


0

2

4

6

8

10

12

0 5 10 15 20 25 30 35


Λ=4000

Λ=500

Λ=1000

Λ=100Λ=50Λ=10

NO

RM

ALI

ZED

LO

AD

(W/B

*L*P

a)


Λ=8000



0

2

4

6

8

10

12

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

NO

RM

ALI

ZED

LO

AD

(W/B

*L*P

a)

COMPRESSIBLITY NUMBER (Λ)


Figure 17. Classic Effect of Compressibility on Load Capacity

0

2

4

6

8

10

12

1 3 5 7 9 11

NO

RM

ALI

ZED

LO

AD

(W/B

*L*P

a)

POWER OF FILM SHAPE (γ)

Height Ratio = 20.0Compressibility Number = 4000.0

Figure 18. Effect of Power of Film Shape on Load Capacity



VI. Conclusion

1) The numerical method is successful in converging at high compressibility numbers and predicted the load capacity of the hydrodynamic slider bearings.

2) For high speed machines which use compressible fluids the power shape is very useful to design bearings for higher load capacity compared to straight slider bearings.

3) The greater the power of film shape is, the better the load carrying capacity. After a value of 4 the increase in load capacity of the bearing is not as high as compared to the increase from 1 to 4.

4) The higher the compressibility number is, the better the load carrying capacity. The load capacity curve starts to flatten out after a compressibility number of 4000 and shows only slight increase in load capacity at higher values.

5) The greater the height ratio of the bearing is, the higher the load carrying capacity. Approximately after a height ratio of 20 the load capacity starts to flatten out, resulting only slighter increase in load capacity.

6) From the plots given bearing engineers can understand the effect of various factors on load carrying capacity and can easily design a successful hydrodynamic bearing with some testing and adjustments to the design.

Acknowledgments

The author would like to express his sincere thanks to Mr. Sam Rajendran for providing necessary assistance in compiling the paper and computer programming.

References

1Gross, W.A., et al., “Fluid Film Lubrication,” J.Wiley and Sons, Inc., (1980). 2Coleman, R. and Snider, A.D., “Linearization for Numerical Solution of the Reynolds Equation,” Journal of Lubrication

Technology, 91, p506 (1969). 3Castelli,V. and Privics,J., “Equilibrium Characteristic of Axial-Groove Gas Lubricated Bearings,” Journal of Lubrication

Technology, 89, p177 (1967).

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