Hydrodynamic lubrication of long porous bearings

Wear -

Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands

HYDRODYNAMIC

LUBRICATION

449

OF LONG POROUS BEARINGS

P. R. K. MURTI Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur (India) (Received April 2, 1971)

SUMMARY

Analytical solution is attempted for a long porous bearing, press-fitted into the housing and working with a full film of lubricant. The pressure distribution is determined by a simultaneous solution of Reynolds equation for the film and the Laplace equation for the bearing material while maintaining the continuity of pressure at the film-bearing interface. The pressure in the bearing material and film are taken in series form so that all the boundary conditions are satisfied. Using a suitably truncated series the modified Reynolds equation is then solved by the Galerkin method. Numerical calculations made with a digital computer are presented in a graphical form for pressure distribution, Sommerfeld number, coefficient of friction etc., as a function of permeability parameter, which is an appropriate nondimensional characteristic of the porous bearing. The results indicate the progressive reduction in the Sommerfeld number, and increment in nondimensional coefficient of friction parameter with increasing permeability parameter. INTRODUCTION

Porous bearings have an advantage over conventional sleeve bearings in that no external supply of lubricant is necessary for their satisfactory operation for long periods. As the journal picks up speed, the lubricant trapped in the pores of the bearing material is ejected and normally maintains a hydrodynamic film. Though these bearings have been used for a considerable time in industry, the problem of analysis and design appears to be of recent origin. Morgan and Cameron’ have given the solution for a short bearing while Cameron, Morgan and Stainsby’ experimentally verified the predicted performance, making due corrections for end leakage. However these authors closely approximated some boundary conditions while Rouleau improved3 and solved4 the same problem with exact boundary conditions. Rhodes and Rouleau5,6 analysed a short bearing with sealed ends and partial porous-metal bearings. Morgan ‘** has given a comprehensive explanation of the operation of these bearings and provided information for their successful design. Recently Capone’ has given a solution for a long bearing using the CameronMorgan assumption of a linear pressure gradient within the porous bearing material. Though Cameron2 justified the assumption for the problem of a short bearing no such justification is given by Capone 9. The problem of a long bearing is now tackled by Wear, 18 (1971) 449460

450

P. R. K. MURTI

assuming separate expressions for pressures in the bearing material and the lubricant film. The validity of the expressions is proved by so adjusting them that (1) they satisfy all the boundary conditions including the pressure continuity at the bearing-film interface (2) they satisfy the Reynolds equation in the film and the Laplace equation in the bearing material simultaneously. As a result of the present analysis is the emergence of a new nondimensional permeability parameter which differs from that of earlier authors. The relevance of this parameter for a long bearing is fully justified. NOMENCLATURE

coefficients in the series for pressure, eqn. (13) coefficients of matrix, eqns. (17), (18) coefficients in the series for pressure, cash n(B-- l), eqn. (13) radial clearance elements of column matrix, eqns. (17), (18) eccentricity e thickness of oil film h H h/c (nondimensional) pressure in film region above ambient pressure ; pc’/q Ur 1 (nondimensional) * pressure in bearing material above ambient pressure ;* p* c “/r)Ur 1 (nondimensional) velocity of lubricant in porous bearing 9 r radial variable inner radius of porous bearing rl outer radius of porous bearing r. R r/r, (nondimensional) S Sommerfeld number, ( W/qU)(c2/rf) wall thickness of porous bearing T radial velocity of fluid V radial velocity of fluid entering porous bearing at r = r 1 vo peripheral velocity of journal u radial distance measured outside from the inner surface of bearing Y* load per unit length of the bearing W WI load component along line of centres per unit length of bearing load component perpendicular to line of centres per unit length of bearing ro/r 1 (nondimensional) Laplace operator eccentricity ratio, e/c (nondimensional) E absolute viscosity rl e angular distance from line of centres (measured anticlockwise) coefficient of friction P permeability of bearing material permeability parameter, dJ attitude angle 8, r, z cylindrical coordinates

A”

Bni G

ki

fr I

Wear.18 (197 1) 449460

HYDRODYNAMIC

LUBRICATION

OF LONG

POROUS

451

BEARINGS

ANALYSIS

Figure 1 shows a long porous bearing press-fitted into a solid housing and supporting a nonporous journal through a fluid-film. In addition to all the usual assumptions of hydrodynamic lubrication for the film, the material of the bearing is

1. JOURNAL 3. POROUS BEARING 2. FILM REGION 4. SOLID HOUSING WALL THICKNESS OF BEARING = T Fig. 1. Porous

journal

bearing.

assumed to be homogeneous-and isotropic, the flow of the lubricant in the bearing laminar and satisfies Darcy’s equation q= -

T!vp*

(1)

rl

From eqn. (1) the velocity of flow of the lubricant into the bearing material, uO, is given by,

Combining eqn. (1) with the continuity equation for incompressible-flow v.q=o

(3)

it is found that p* satisfies Laplace equation, v=p* = 0

(4)

For a long bearing ap*/Jz = 0, and eqn. (4) takes the form

a=p* a2p* r; de2 + ar2 =

i --

o

(5)

where Y* -=

r1

=+l

(6) Wear, 18 (1971) 449460

452

P. R. K. MURTI

and

Since the maximum value of y* is the thickness of the wall of the bearing, ‘I:it is noted that T/r, @ I, a condition that is satisfied for the usual commercial sintered metal bearings. The governing differential equation for pressure in the film region can be derived’ starting from Navier-Stokes equations and allowing for the flow of the lubricant into the porous bearing material. This equation is referred to as the modified Reynolds equation and is given as

Substituting eqn. (2) in eqn. (7) and neglecting axial flow in the case of a long porous bearing

(8) Introducing the following dimensionless variables H =;

p*=

= l+ecosH

$$

and

PC2 P = ~ vUr1

(9)

eqn. (8) takes the form

The boundary conditions for the pressures in the film and bearing are

P(O) = P(2n) p* (0) = P*(2rc) P(O) = P*(O, rl)

(11)

The third of these conditions indicates pressure continuity at the film-bearing interface Wear,18(1971)449%460 ,

HYDRODYNAMIC

LUBRICATION

OF LONG POROUS BEARINGS

453

and the last indicates lack of flow due to press Iit at the bearing outer surface. The pressure distributions in the film and bearing are obtained by a simultaneous solution of eqns. (5) and (lo), satisfying the boundary conditions (11). SOLUTION OF THE EQUATIONS

Let P* = f

A,(sin nt?)/cosh n/I (1 - :)I

(12)

n=1

where A, are the unknown coefficients to be determined subsequently. This expression satisfies eqn. (6) and the boundary conditions (11). P(0) = P*(0, rl) = f A, {cash n(/?- l)}(sin no) PI=1

p(e) = f A,C,(sinnB)

(13)

n=l

Now the modified Reynolds equation (10) is solved by Galerkin method. An excellent exposition of this method is given by Kantorovich and Krylov” and Mikhlin”. Introducing expression (13) in the modified Reynolds equation (lo), the residual L(P) is obtained which is identically zero if it is orthogonal to all terms of the series given by eqn. (13), over the entire domain 0=2?l i=1,2,3 ,... . L(P)(sin ie)de = 0 (14) I e=o The residual L(P)is given by L(P) =

(15)

t II=1

where f n PI

=[(1 + ”

+

i

(-n2

(3&4&3)

sinno)

1

;(sin(n+l)0+sin(rt--1)s))

-

[i -

i(sin(n+l)8--sin(n-l)H)_ i

2 -

+g

II

G(sin(n+2)0+sin(n-2)0))

I( s

-2

~(sin(n+2)fJ-sin(n-2)$~] i

E3

+4 [{ -

z(sin(n+3)0+

sin(n-3)0)J

-3(i(sin(n+3)B-sin(n-3)8)j] - 1214$ n (sin n

ne) , D,= sinhn@-1) Wear, 18 (1971) W-460

454

and ,q(B)= - 61: sin 0 So that eqn. (14) becomes *I)= 2n 2

A,C,f,(@(sin

i0)dtJ =

y(Q)(sin i0)dt)

i = 1, 2, 3, . . .

(16)

./ R=O

from which i’sobtained f

(171

A,C,B,i=Di

n= 1

where

*2n B,i =T: f,(@(sin I .o

iO)dO

and rtx

i = 1, 2, 3, . . . g(B)( sin i0)dO 08) 1 Equation (17) mvolves the solution of simultaneous linear equations of infinite order for an exact solution. However, to obtain an approximate solution, the series in eqn. (13) is truncated to N terms and the finite order simultaneous equations thereby obtained are solved to determine the unknown coefficients A,, (a= 1, . . ., N). In this manner, once the coefficients are evaluated, the pressure distribution in the bearing material and film region can be obtained by eqns. (12) and (13) respectively. Di =

LOAD CARRYING CAPACITY

Considering only the pressure force per unit length on the journal due to the film and neglecting the shear-stress contribution to the total force, PcosOd0=0

(19)

P sin @de = TU-~A~C~

(20)

so that the total force on the journal is given by (23) and the attitude angle (22) FRICTION FORCE ON THE JOURNAL

It was shown by Cameron and WoodI that for both solid and porous bearings, the nondimensional friction parameter, pr,/‘c, is given by CL’1 -= c

271 f : sin Cp S(1 -s2)

Wear, 18 (1971) 449-460

(23)

HYDRODYNAMIC

LUBRICATION

8 v

.\

Km-

36

72

BEARINGS

144

ISa

453

_

IQ8 8

POROUS

DEGREES

$=.OOl,

E= 0.4

0

IN

OF LONG

IN DEGREES

8

IN DEGREES

Fig. 2. (For caption see 0uerleujJ Wear, I8 f1971) 449-460

456

P 5-

OO

IO@

72

36

8

IN

144

180

DEGREES

30E = 0.9 BE

1.2

20-

P IO -

0 0

36

72

108 8

IN

144

I80

DEGREES

60 d

* 0.95

fs-

I.2

40-

30-

P 20-

IO-

oc 0

-

36

72 8

Fig. 2. Pressure

distribution

Wear, 18 (1971) 449460

108 IN

I44

DEGREES

at constant

eccentricity

ratio, s, for different values of permeability

parameter,

3.

HYDRODYNAMIC LUBRICATION OF LONG POROUS BEARINGS

457

PRESENTATION AND DISCUSSION OF THE RESULTS

The results are presented in graphical form (Figs. 2-6). Pressure distribution

It is seen from Fig. 2 that for a given eccentricity ratio, E, the film pressure, P, decreases and becomes more even with increasing value of the permeability parameter $. The position of maximum pressure shifts away from the position of minimum film thickness. For E< 0.95, the curves for $ = 0.001 and $ =0 (i.e. a solid bearing) are indistinguishable while for s=0.95, the difference is negligible. Hence a porous bearing with +=O.OOl is comparable in performance to a solid bearing. Jl+=.001 LO.08

-

0.6 6 0.4 0.2 -

01 0

’

1

.08

I

I

I

.I6

I

.24

I

I

.52

I 5

Fig. 3. Plot of E against l/S for different values of permeability parameter, $.

-------

IO TERMSIN EO. 3

TERMS

IN

EQ.

s 20

IO

0 I

Fig. 4. Plot of S against $ for different values of eccentricity ratio, E.

Load carrying capacity This is shown in Fig. 4. Beyond a certain range of permeability parameter rl/, the Sommerfeld number S falls rapidly with +. However this range of $ itself is dependent on the eccentricity ratio, E, and diminishes with increasing E. But in all cases the bearing has very little load carrying capacity for 1.0 < Ic/< 10 and no load carrying capacity for * > 10. Wear, 18 (1971) 449-460

458

P. R. K. MURTI

Figure 4 also indicates that the 3 term solution is indistinguishable from the 10 term solution for EC 0.7. Again for E< 0.90, there is negligible difference between the performance of a solid bearing ($ = 0) and that of a porous bearing with J/ = 0.001. Suffkient scope is thus available for the designer to select a porous bearing with the advantage of self-lubrication and the performance of a solid bearing. Friction

in the bearing

The nondimensional friction parameter pr r/c is plotted against permeability parameter +!Iin Fig. 5. p-,/c increases rapidly with I/I, beyond a certain range. The effect is more pronounced at higher values of the eccentricity ratio, F.

01 l65

IO

IO

16’

IO

Fig. 5. Plot of (~LT,~c.I,le,llnst IJI for different

1.0

values of eccentricity

ratio, E.

6.0 r

0

.04

Fig. 6. Plot of (prI/c) limit case. Wear. 18 (1971) 449QW

against

.08

.I2

l/S for different

.I6

values

.20

of permeability

.24

parameter

.28

$ including

.32

the Petrof

HYDRODYNAMIC

459

LUBRICATION OF LONG POROUS BEARINGS

Figure 6 shows the plot of prI/c against l/S for different values of the permeability parameter $, including the Petrof limit case. The curve for $ = 0.001 is characteristic of that of a solid bearing whilst with increasing $ larger load carrying capacity involves higher friction. In all the cases the Petrof limit is approached as s+o. Eccentricity

ratio

Eccentricity ratio, E, diminishes with l/S at all values of permeability parameter II/, as indicated in Fig. 3. Significance

of permeability

parameter,

I) = @r,lc3

The nondimensional permeability parameter $ which enters the second term of eqn. (10) is appropriate for a long porous bearing since it includes the effects of (1) the permeability @ of the sintered metal (2) the circumferential length of the bearing through radius rl and (3) the operating clearance, c, of the bearing. The only physical dimension of the bearing that does not explicitly enter this parameter is the wall thickness of the bearing, T, but its effect is implicit in the term fi contained in the second term in eqn. (10). The designer has ample scope to select the most appropriate permeability parameter through different combinations of @, rI and c. CONCLUSIONS

The load carrying capacity of porous bearings progressively decreases and friction force increases with the permeability parameter-a nondimensional characteristic of the bearing. There is a more even distribution of pressure in the film as the permeability parameter increases. By carefully selecting the permeability parameter for a given application, porous bearings of comparable performance to solid bearings can be designed to exploit the advantage of self-lubrication. The design aspects based on the last conclusion is the subject of another paper. ACKNDWLEDGEMENT

The author is grateful to Prof. B. M. Belgaumkar for suggesting the problem and to Prof. R. Mishra, Head of the Department of Mechanical Engineering for constant encouragement. REFERENCES 1 V. T. MORGAN AND A. CA~~ERON,Proc. 1957 Conference on Lubrication

2 3 4 5 6 7

Engrs., London, (1957) 151-157. A. CAMERON, V. T. MORGAN AND A. E. STAINSBY, Proc. Inst. Mech. 761-110. W. T. ROULEAU, J. Basic Eng., 84 (1962) 205-206. W. T. ROULEAU, J. Basic Eng., 85 (1963) 123-128. C. A. RHODE AND W. T. ROULEAU, Wear, 8 (1965) 474486. C. A. RHODES AND W. T. ROULEAU, J. Basic Eng., 88 (1966) 5360. V. T. MORGAN, Lubrication Eng., 20 (1964) 448455.

and Wear, Proc. Inst. Mech. Engrs.

(London),

176 (1962)

Wear, 18 (197 1) 449460

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P. K. I<. MIJRTI

8 V. T. MORGAN,Tribology, 2 (1969) 107-l 15. 9 E. CAPONE, Wear, I5 (1970) 157-170. 10 A. E. SCHEIDIGGER,The Physic.~ ofFion, through Porous Media, The MacMillan Co., New York. 1957, pp. 54-69. 11 L. V. KANTOROVICH AND V. I. KRYLOV, Approximate Methods of Higher Analysi.?, Inter Science. New York, 1958, pp. 240-357. 12 S. G. MIKHLIN, Variational Methods in Mathematicul Physics, Pergamon, London, 1964. I3 A. CAMERONANDMRS.W. L. WOOD, Proc. Inst. Mesh. Engrs. (London), 161 (1949) 59-72.

APPENDIX ACCURACY OF THE RESULTS AND CONVERGENCE OF THE METHOD

Numerical calculations were carried out on an IBM 1620 digital computer for s=0.2, 0.4, 0.7, 0.8, 0.9, 0.95. It was found that for fixed values of It/ and fi, very rapid convergence was achieved for E-C0.7 while the convergence slowed down for E > 0.7. For example, (Fig. 4) with E= 0.2, 0.4, the results of 3 terms in eqn. (13) were indistinguishable from those of 10 terms in eqn. (13). For E >0.7, 10 terms were quite adequate since for the really high value of E= 0.95 and $ = 0.001, the 10 and 15 term solutions differed only in the second decimal place. Again it was found that for given E, convergence became very rapid for increasing values of $. With the 10 term solution, in the test case of E= 0.95 and $ = 0.001, coefficient A,, was 1% of coefficient A,. This percentage rapidly decreased with decreasing t: and increasing $. Wear, 18 (1971) 449-460