Squeeze films in an axially undefined porous journal bearing considering cavitation

271

Wear, 45 (1977) 271 - 275 @ Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands

Short Communication

Squeeze films in an axially undefined cavitation

porous journal

bearing considering

UMA SRINIVASAN

Department (Received

of ~~thern~t~~~, Indian institute September

of Technology,

Bombay (India)

3, 1976; in final form January 17, 1977)

An analysis is made of the performance of squeeze films in an axially undefined porous journal bearing using the Reynolds boundary condition. Expressions for the pressure distribution and the load capacity are obtained in a closed form. Introduction Squeeze films in porous journal bearings were considered by Prakash and Vij [l] assuming a full lubricant film (SommerfeId condition) which is seldom found in bearings in service. In the present analysis a more realistic boundary condition, the Reynolds condition, for the pressure distribution is considered. It is assumed that the pressure is positive and falls to zero with a zero pressure gradient. The film i‘s continuous only in the region of positive pressure and it cavitates at some value 0s of 0 (where 0s > n) at which p = dp/d6’ = 0. A discontinuous mixture of air, vapour and the lubricant is formed in the cavitated region (0, < 0 < 2n) [ 21. Analysis

Using the usual assumptions of conventional lubrication theory and taking the porosity of the bearing into account, the Reynolds equation for the film region reduces to

$ ](h’

(h3 + 129H) $

(1)

which is similar to that obtained by Morgan and Cameron [ 31 and that obtained by Capone [4]. With an axially undefined porous journal bearing the equation further simplifies to + 12#H) -f-](h”

2

I

= 6&U

dh dx+

12pv

(21

212

For the case of squeeze films between

non-rotating

surfaces

V=~=C~cose

U=O

(3)

and eqn. (2) takes the form + E co&

dp

+ 12$

--& = I

The boundary

conditions

cos B (4)

C2

I

are

=o

atB=O

dp/d6’

=0

attI=e2

p

=o

ate =e2

p

12yR2(dc/dt)

(5)

Pressure distribution Solving eqn. (4) with respect to the boundary (5), the pressure distribution is obtained as -

conditions

given by eqn.

PC2

’ =

2pR2(de/dt)

=A-

=

{(~+Ecos~)~-~~(~+~cos~)+cY~)(~+E+cY)~

I[ In

cl26

(1 +

E

e + aj2 ((1 + g2 --a(1

cos

&a(1

+2fitan-l

I2a

2(Blfl+

a2)

-COST)

2

tan

Dlfi)

+ W)}1’2

i -

--a(2+~+~~0~e)+2(i+~)(i+~cose)

4~ sin e2 - ((1 + e)2 - 9}1/2

+ (2(F

+ E) +

+ I

-1

Ij::“,T:)“‘tani$)/ _1

7~

*tan

z + tan

(e/2)

+ ---cot

(2 (F + m)}l’s

(e/2)

_ I)

Blfl--DIG - {2(W

-F)}l’2

x

-tan

(e/2) + fl

@tan

(e/2)

+JGcot

cot (e/2) + {2(&?G (e/2)-

- F)}l12

{2(~-F)}1/2

where E=e2-(2-++e2-e+1 F=e2-~+l-~2 G=E~+(~-(Y)E+(Y~-(Y+~

B = -E(E + 2a - 1) sin e2 D = E(E- 201 + 1) sin e2 The angle f12 where the film breaks down and cavitation p(e2) = 0.

begins is given by

(0)

273

Load capacity The load W is given as w =-

e’p cos e LR de s 0

Substituting obtained as

(8)

for p and integrating,

the non-dimensional

load w is

-WC2

WE

8n/~LR3 (dc/dt)= 1 =((1 +cx) 2 _gy* 7TCY2e2i

tan-'

~(~~~~~)"'tarp)j-

- (l+E+(Y)@+(l+CY-f)@ 2 {2(F + I,/x)}“* X

fltan

i+tan-’

(e,/2)

+c+cw)*-(l+a-+/G F)}l’*

&!7 tan (e,/2)

+ flcot

(e2/2) + flcot

E sin e2

+-ln

4

[

G

+-esine2 2 X tan-l

+

cot (es/2) I) x

4{2(m--

fltan

=

-fl

{2(F + m)}l’*

( +(l

X

(e2/2) + {2(m-

F)}l/*

(e2/2) - {2(&%7-

F)}l/* I +

+ E cos e2j2 - ~(1 + c cos e,) ‘(’(l+~c0se2+~)2{(l+~)2-~(l+~)+~2)

1

+ a*}(1 + E +a)* +

X

1

2a2 --a(2

fi~a(i

- cos e,)

+ E + f cos e,) + 2(1 + e)(l + E cos e,) I

f(v)

(9)

Response time For a constant load W, the time taken by the journal centre to move from E = 0 to E = e1 is obtained by integrating eqn. (9) with respect to time. Thus T=

wc*t

87r/.lLR3

= “~(~,~) de so

(10)

274 L

4

3

82

3

3

3

3.

001

I

0.001

I

!

0.1

001

0

3 Fig. 1. Angular extent 62 of the oil film us. the permeability values of the eccentricity ratio E.

parameter

J/ for various

Results Figure 1 shows the angular extent B2 of the film uersus the permeability parameter $ for different values of the eccentricity ratio E. An increase in $ results in an increase in 19swhich implies that the cavitation region is reduced by increasing the permeability, by increasing the wall thickness or by decreasing the clearance. The extent of the film increases with a decrease of e for all values of $, i.e. the cavitation region is reduced by decreasing the eccentricity ratio. Nomenclature C h H L P

radial clearance C(1 + f cos O), porous bearing axial length of pressure in the

oil film thickness wall thickness the bearing Cilm region

215 PC2

, non-dimensional

pressure

2pR2 de/dt radius of the journal time taken by the journal centre to move from E = 0 to e = EI WC2t/8npLR3, time parameter tangential velocity of the journal velocity of approach load capacity WC2 8npR3L

, non-dimensional

load

deldt

Cartesian coordinates (12$)1’3 eccentricity ratio angular coordinate (x = R6) angle where the continuous film terminates absolute viscosity of the lubricant permeability of the porous bearing QH/C3, permeability parameter

References 1 J. Prakash and S. K. Vij, Squeeze films in porous metal bearings, J. Lubr. Technol., 94 (4) (1972) 302 - 305. 2 L. Floberg, Boundary conditions of cavitation regions in journal bearings, ASLE Trans., 4 (1961) 282 - 286. 3 V. T. Morgan and A. Cameron, Mechanism of lubrication in porous metal bearings, Proc. Conf. on Lubrication and Wear, London, 1957, Inst. Mech. Eng., London, 1957, paper 89, pp. 151 - 157. 4 E. Capone, Lubrication of axially undefined porous bearings, Wear, 15 (1970) 157 170.