The elastohydrodynamic lubrication of heavily loaded journal bearing having non-cylindrical axial geometry

Elastohydrodynamics '96 / D. Dowson et al. (Editors)

0 1997 Elsevier Science B.V. All rights reserved.

675

'Ihe elastohydmdynanic lubrication of heavily loaded joumal bearing having non-cylindrical axial geometry Hasscan E. Rasheed Arab Academy for Science & Technology, Alex'andria, Egypt The elatsto~ydmdynamicanalysis of bearings having non-cylindrical geometrical configurations in the axial direction is presented in this paper. Several geometrical configurations having concave, convex and wedge shaped surfaces are examined. The effects of changing the shape and radius to length ratio of the journal are investigated. The results show a marked increase in the load carrying capacity over that of the plain cylindrical bearing. A decrease in the friction variable is also found. These effects become pronounced for relatively long bearings. Among the geometries selected the concave shaped bearing is found superior to all other geometries.

1. INTRODUCTION The analysis of bearings having elastic liners and of several geometrical configurations have been reported in different ways in the literatures [1-5]. Some investigators treated the elastohydrodynamic lubrication of non-circular geometrical shapes of the bearing liner especially those having elliptical and multiple lobes configurations [6-71. Others examined the effect of geometrical changes caused by either the detlection of the journal or the angular misalignment together with the elastic deformation of the liner [X-91. On the other hand the recent advancement in numerically controlled machine tools has made the accurate machining of rather complex shapes a feasible task. This has encouraged some researchers tc) attempt to study the performance of bear i n g s h av i n g ti on - c y lindr ical ax i a1 configurations and running under normal loading conditions considering the journal and the liner as rigid elements [ 10- 121. Fortunately improved performance characteristics over the plain cylindrical bearings are reported. However to (he outhor's knowledge nothing has been reported concerning the elastohydrodyn~unic analysis of those types of bearings. It is the aim of this paper to fill this gap.

pressure. The system of coordinates used is shown in Fig. 1. The fluid inertia forces are considered negligible compared to viscous forces. The lubricant flow in the bearing is assumed laminar, incompressible and the viscosity of the lubricant is assumed to vary along the fluid film depending solely on the film pressure as is always the case for heavily loaded bearings. The equations governing the lubricant flow in a dimensionless form may be written as,

aU*+ (-) R av' + ae c a

(2)($+ c15

dlw * ) = 0

h(<) is a geometry variable defined as

2. ANALYSIS To obtain the performance characteristics of a bearing of a r b i t r q axial shape a curvilinear system of ctxjrdinates is to adopted. 'I'he analysis proposed by El-Gamal [ 121 is extended in the present work to include the det'ormation of the bearing liner and the variation oi the lubricant viscosity with film

and H* (<) is an arbitrary function describing the bearing geometrical configuration. The preori selection of the geometrical shape of the bearing dictates the form of the function H* and

676

I3g. 1 The curvilinear coordinate system. conscqucntly the variable h. 'I'hree geometrical configurations are selected for the analysis, namely thc concave, convcx and wcdgc shapes (sec Fig. 2). 'Ihese geometrical configurations can simply be described by ;t single function H* which may be written a s : H' = a,<+a?<'

Concave Fig. 2 Thc bearing geometrical configurations.

in which al and a? xe constants. their values depend on bearing geometry, as follows: al=O , a:! = 46 * for bearing of concave geometry a]= 46 * ,a2 = - 46 * for bearing of convex geometry al= 26 * , a2 = 0 for wedgc shaped geometry For completion ; a] = a? = 0 for bearing of plain cylindrical shape.

Wedge

Convex

677 The integration o f Eqns. 1 with respect to q twice and making spatial averaging across the film thickness lead to a Reynolds - like equation of the following form,

”[ + (6’ h

1

- I[*)(

91” R

a0

(2)

The boundary conditions are,

on the value o f the local pressure. M a n y investigators dealing with bearings having elastic bushings considered the viscosity to be pressure dependent only, see [ 3 ] and 141. I-lowever ;I simple viscosity pressure relation is used in the present analysis and is in the form,

j

(4)

p* = HXP(tr*p*

The radial and tangential load components are,

w:

edf j ” ~ [ i + ( t i *- H*)(L / R ) ]

j”

112

=-2

0

0

p* cos8 d 8 d <

JP’ [=o , -

a(

-0

,

5=1/2 , p*=o

112 eeff

with h* = l+&cos€)+Ii: where 11: is the change in the lubricant film thickness due to elastic deformation of the bearing liner under h i d pressure. I n the present analysis i t is assumed that the jounial is rigid and experience no elastic deformation under pressure. In general the determination of hz requires the solution of the three - dimensional elasticity e q u a t i o n s of equilibrium a n d compatibility. However. when the ratio of the thickness of the bearing bush to the radius of the 0.1) and Poisson’s ratio bearing is small (t/Rl, I ranging between 0.3 and 0.4, it is shown by Hooke et a1 [ l ] :und lain and Sinhasan [ 5 ] that a simple plane stress model may approximate satisfactorily the coinplcx three - dimensional model. The model is based on the assumption that a plane stress condition exists and the tangential strains are considered negligible. The same models results from the assumption of plane strain condition and the neglection of tangential stresses. LJsing this model the resulting deformation may be written as.

It remains here to consider the viscosity variation with pressure. As for heavily loaded bearings. the viscosity varies along the film dcpcnding very much

j”

w;=2J 0

h[1+(6’-II’)(L/I<)]

0

p‘ sin0

t l ~ )d

j

The resultant load and the attitude angle are.

and $ = tan-’(Wr / WT) The frictional force and the normal force acting on the journal surface are given by.

”)[

I / ? 2iT

F* = 2

j”tp*( 0

0

arl

)[

1 + (6’ - 11’ L]] R

q=h’

d0 dl;

and the friction variable may bc calculated from,

m

c = F* / ~

1,

618 3. RESULTS AND DISCUSSION I t is expected. as is always the case in el~stohytlrodyu~unic problems, that a decrease in the load w r y i n g capacity W* due to elastic deformation of the liner will take place for any geometrical shape (see Figs. 3a and h) . But it is to be noted that the dccrcase in W* is overwhelmed by the marked increase in W ” due to changing the geometrical shape from cylindrical to any of the geometries considered. ‘This is clearly demonstrated from Fig. 3 when compared to Fig. 4 respectively. The percentage changes in the load carrying capacity for the shapes considered relative to the plain cylindrical case are given in Figs. 4a. b,c and d versus R/L for different values of E . They all show an increase in the load carrying capacity ranging from about 5% to surprisingly 80%. The longer h e bearing the larger the percentage increase i n W“ especially at low eccentricity ratios. The figures show that the concave geometry is superior to all other geometries over the entire range of R/L and E . And this is pronounced for relatively long bearings and lower vducs of E. The percentage change in the friction variable is also plotted (see Figs. 5a. b, c and d). ‘The concave and wedge - shaped geometries show a decrease i n fK/c especially for long bearings and at low cccentricity ratios. The convex geometry gives a dccreasc i n tR/c for longer bearings and an increase i n lR/c for IUL appsoximately over 0.6. The largest decrease in fR/c is always given by the concave geometry especially for long bearings. The percenrage change in the attitudc angle $I is also given in 1:igs. ha, h, c and d. A decrease in the attitude angle relative to that of the plain cylindrical hearing is ohtained for all geometries considered ranging lrom practically zero to approximately 15% . The percentage decrease in 0 is practically of the siune order for all geometries especially at larger values of E, The maximum decrease in 0 is found to be approximately at R/L = 0.5 for all geometries and any value ol’ E.

4. CON C I, U S I 0 N S It is concluded here that the new geometrical hearing con tigurations considered, all give considerahle increase in the load carrying capacity over the plain cylindrical bearing. This is pronounced most for relatively long bearings and low eccenuicity ratios. A decrease in the friction

‘3

e=0.4

c,=o.o

.-

C

C

0 ..w 0

3

-0

2 al

cn 0

w

c

al 0 L

al

a

3-

- Concave

______

Convex Cylindrical

01

I

000

0.50

1 .oo

Radius to length ratio, R/L

91

k C .C

0 .c 0

3

U

2 Q)

cn 0

c C

al

2 al

a

Wedge Concave Convex Cylindrical

01 0.00

I

0.50

1 .oo

Radius t o length ratio, R/L

Fig.3 The percentage reduction in W* due to the deformation of the bearing liner for different geometries.

>

&=0.2

(a )

80

c ~-

c,=0.02

a,

? 0

60

(b)

.-C

a,

rn 0

G

-~

......

40

-- _

20

-..-_-. ._ ._ --. -.._ --.___

'--

0.50

1

.oo

0 0.00

a,

60

L

0

.-c or 0

a,

1 .oo

I

c,=0.02

\\

a,

7

80

J

I

0.50 Radius t o length ratio, R/L

50

5

\

\

Radius t o length ratio, R/L

'r

&=0.4

c,=0.02

a

.....

Wedge Concave Convex

0 00

a, 0 L a,

\

679

40

Y

a 20

0

0.00

20

0.50 Radius t o length ratio, R/L

1 .oo

0' 0.00

I

0.50

1 .oo

Radius t o length ratio. R/L

Fig. 4 The percentage increase in the dimensionless load carrying capacity, W*, for different bearing geometries relative to that of the plain cylindrical bearing, versus FUL for different eccentricity ratios E .

680

.-C

~=0.4

.-C

c

8

co=o.02 \

4.

0. 0.0

r

Radius t o length ratio, R/L

12,

Radius to length ratio, R/L

1

c=0.6

co=o.02

Radius t o length ratio, R/L

1;ig.S

Radius t o length ratio, R/L

'lhc pcrcentagc reduction in the friction variable, fWc, for different geometries relaiive to thc plain cylindiical journal bearing, versus IUL for different eccentricity ratios E.

68 1 16

16

9.c_

&=0.2

c,=0.02

a,

:

L

9.-C

12

e=0.4

al

(0

?!

V a,

V

aJ

n

n

a,

al

m

0,

0

7

0

8

2 2 al a

V

L

a,

a

-

n v

0.00

-

~

4

6

- - Wedge - Concave

Wedge Concave

.__ Convex ___

1 .oo

0.50

Radius to length ratio, R/L

9-

.-c

~=0.6

a,

c,= 0.02

12

v)

O

?! 0

V

a,

n aJ

a,

m

m

0

8

Z

a,

8

Y

V

L

a,

al

a

a - Concave

______

4

0

0.00

0.50

- - Wedge .Concave

4

Convex

Radius to length ratio, R/L

Fig.6

c,=0.02

12

al

n

a,

~=0.8

a,

L

0

1 .oo

1E

9 c ._

7

Convex

0.50

0.00

16

;

______

4 t

Radius to length ratio, R/L

Lo

c,=0.02

12

1 .oo

0 0.00

1

0.50

1 .oo

Radius t o length ratio, R/L

The percentage decrease in the attitude angle, $, for different bearing geometries relative to rhar of the plain cylindrical karing, versus IUL for different eccentricity ratios E.

682 variable is also found for the concave and wedge shaped geometries. A similar decrease in the friction variable is present for relatively long hearings having convex geometry. The concave geometry is found superior to all other geometries and is best recommended tor the design of heavily loaded journal hearings.

Bearing surface axial lengths: L = 25 mm, SO mm and 100 inm I>ubricantdimensionless viscosity index : 01* = 0.006843 Lubricant inlet temperature. T(,= 60 C Bearing liner thickness, t = 2.5 mm

REFERENCES

N OM E N C I, A T U R E c C,, E h* I> p*

radial clearance (in) Elasticity coel'l'icient Young's modulus (Pa) * dimensionless film thickness (11 = h/c) axial length along bearing surface (in) dimensionless lluid pressure p*+ op,, li'/c?) I< minimum shaft raduis (in) K,,minimum hearing liner radius (in) t * bearing * .*' liner thickness (m) u , v , w dimensionless velocity components in directions 0, q , respectively (u* = u / COR v * = v / 0112 . w * = w / (OR) Greek symlwls IY.

*

lubricant dimensionless viscosity index cIc = a(op,,R' / c')

6" ciimensionless iiiaximum axial variation of bearing geometry (6* = 6 I L) E

eccentricity ratio

- q dimensionless coordinates along and normal to

one side of the hearing surface respxtively (C=S/l. , q = y / c ) 0 cmrdinnte in tmgential direction (rod) p" dimensionIess fluid viscosity (p*= p/p,,) pO inlet fluid viscosity (Pa s) I) Poisson's ratio w journal angular velocity (fads)

DATA FOR T H E 1,URRICANT AND 0PE H A'L'I N ( ; C 0 N DIT I 0 N S Bearing radial clearance , c=lW pm Journal miniinuin radius. R=25 mm Rotational spccd. N=25 rev/h Maximum axial variations 0 1 geometrical shape; 6 =2.5 mm. 5 mm and 10 mm

Hooke C.J., Brighton D.K. and O'Donoghuc J.P. The effect of elastic distortions o i l thc performance of thin shell hearings. l'roc. Inst. Mech. Eng., vol. 181 (Put 3D). 1066-7 . pp 6 3 - 69. 2 . Benjamin M.K. and (';istclli V . A . 'l'hcorctical investigation of comp 1i ;in t su r fac c .journal bearings. ASME 'I'rans. , Journal o f lubrication technology , vol. 0 3 . N o 1 , 1071 . pp. 191 - 201. 3. Conway H.D. and Lee H.C. 'Ihe iinalysis of the lubrication of a flexible journal hearing. ASME Trans., Journal of lubrication technology, vol. 97 , No 4, 1975. pp. 599 604. 4. Jain S.C.,Sinhasan R. :in11 Singh D . V . Elastohydrodynamic analysis of ;I cylindrical journal bearings shell. Wear, vol. 78 , 1982, pp 325 - 335. 5 . Jain S.C. and Sinhasaii I<. I'reformance o f flexiblc shell journal bearings with variable viscosity lubricants. Tribology h i t . , vol. 16 , No 6 , 1983. pp. 331 - 339. 6. Prabhakaran Nair K., Sinhasan I<. and Singh D.V. EHD effects in elliptical joum;ll bearings. W e a r , 118 ( 2 ) . 1987 , pp 120 - 146. 7 . Prabhakaran Nair K., Sinhas;iii I<. aiid Singh D.V. A study of 1JHD effects in the three - lobe journal bearings. Tribology 1111,. 30 ( 3 ) , 1987, pp. 125 - 132. 8. Pinkers 0. and Bupara S.S. Aiiiilysis 0 1 misaligned groove .journal bearings. ASME, Journal of Lubr. Tech.. vol. 1 0 1 . 1070, pp. 503 - 509. 9. El-Gamal HA. , Awad T.. IIelmy A. and ElFahham I.M. I3fect of shaft misalignment of the performance of hcavily loatlcd journal hearings. Fourth I i i t . ('ont'. o f 1;luid Mechanics (IC'FM4) April, 28-30. vol 111, 1992, pp 560 - 5x2. 10. Leurig P.S., Groighed I.A. :iiicl Wilkinson T.S. An analysis o f the steady and dyan;unic characteristics o f ;I spherical hydrodynamic

1.

683 journal bearing. Jouinal of Tribology. vol. 1 1 1, 1980 , pp. 459 - 467. 11. l
12. El-Gamal H.A. The effect of axial geometrical variations on sliding eleinenr bearing characteristics. XI National ('onlcrcnce 011 Industrial Tribology, Ian 22 - 25 . 1005 . New Delhi, India.