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Double layer porous journal bearing analysis
Mechanics of Materials 2 (1983) 233-238 North-Holland
233
D O U B L E LAYER P O R O U S J O U R N A L BEARING ANALYSIS P.D.S. VERMA
Department of Applied Sciences. Regional EngineeringCollege. Kurukshetra-132119. India Received 14 June 1983 A double layer porousjournal beating has been studied using the short bearing approximation.Taking the curvature of the bearing into consideration, expressions for the pressure distribution, load carrying capacity, coefficientof friction, and the attitude angle are derived.It is shown that the bearingcharacteristicsare improveddue to low permeabilityof the inner porous layer.
!. Introduction The investigations on porous bearings have demonstrated that the load capacity of the bearings is reduced on account of permeability. This reduction in load capacity, due to reduced seepage into the porous matrix, may be made smaller by taking a low permeability porous housing. However, smaller permeability results in the depletion of lubricant content in the porous matrix, thereby affecting the self lubricating feature of u., bearing. To avoid this, it was suggested by Marshall and Morgan, 1965-66) that a double layer porous housing in which the inner layer has a reduced pore size and lower permeability be used. Experimentally, it was found (Youssef and Euchier, 1965) that an interior layer of ultrafine carbonyl nickel powder on the bore of a conventional beating with a low permeability nickel alioy lining significantly affects the performance characteristics of the bearing. In this presentation we analyse the performance of a two layer porous bearing using the short bearing approximation. The circumferential flow, both in the film region as well as in the bearing material, has been neglected in comparison to the axial flow. The curvature of the bearing is also taken into consideration. A marked improvement is observed with low permeability inner layer. In Section 2 we formulate the problem. The solution is obtained in Section 3, followed by a discussion of the results in Section 4.
2. Formulation A two layered porous journal bearing under steady load is shown in Fig. 1. With the short bearing approximation, the modified Reynold's equation for the pressure p in the film region assumes the form
a 2 p = 1 2 ~ [ U dh a2 2 h 3 2rj dO
ck''aP'l ] 7? ~ ar ] .... '
(1)
where U is the tangential velocity of the shaft, h = c(1 + c cos 0), c the radial clearance, ~ the eccent.,icity ratio, and rj the inner radius of the bearing. The boundary conditions for p are
p(O, + ~ b ) = 0 , ?,:6%6636/83/$3.00 © 1983, ElsevierSciencePublishers B.V. (North-Holland)
(2)
P.D.S. Verma / Porous journal bearing analysis
230,
•
i
l,l L
J
Fig. 1. Sketch for journal bearing in concentricposition. 1--journal 2--film region. 3--inner porous layer. 4--interior porous layer. 5-- solid housing.
and
(a/az)p(O, o) = O.
(3)
the pressures P~ and Pz at the inner and interior layers satisfy the Laplace's equation:
a:p~/ar 2 + ( l / r )(aP,/Or ) + a'P,/az 2 = o, O2e2/ar 2 + (I/r)(Oe2/ar) + a2P2/~z2 = 0,
(4) (5)
and the boundary conditions
P2(r,O, O)=O, ( a / ~ ) P : ( r , o, o) = o,
(6) (7) (8)
(a/Oz )P2(r, O, O) = O,
(9)
(a/ar)P2(,o, O,~)=o,
(lO)
P,(r. O, +_½b) = O,
O,(aP,/Or ),=,: = 02(OV:/Or ),=,:. The corresponding matching conditions are
P,(r,, O , z ) = p ( O , z ) ,
(12)
P2(r2, O,z)= P,(r2, 0, z).
(13)
Here, b represents the length of the bearing, ~l and q~2 are the permeabilities in the inner and interior layers, respectively, r 2 - r~ is the thickness of the inner layer, and r0 - r 2 the thickness of the interior layer. To obtain the film pressure p(0, z), we need to solve (1), (4) and (5) subject to the boundary conditions (2), (3) and (6) to (13).
P.D.S. Verma / Porous journal bearing analysis
235
3. Solution
Eq. (5) subject to the conditions (7), (9) and (10) yields N
P2(r, O. z ) = 2~ C,,(a, lt,(I,r)+Ko(I,,r)) cos(l.z).
(14)
n=l
where
a,, = Kt(I,,ro)/l,(l, ro).
I, = (2n -
1).rr/b.
(15)
The constants (7, need to be determined. Here. K o, / o and K I, ! l denote modified Bessel functions of the second and first kinds with orders zero and one, respectively. Similarly. (4) along with conditions (6) and (8) gives N
Pt(r. 0, z)= ~ C..(fl,,lo(I.r ) +Ko(I.r)) cos(l.z),
(16)
n=l
where the constants C. and ft. are to be found. The matching condition (13) provides
C,, = C,( a.lo( i,,r2 ) + Ko( I.r2 )) /( B.lo( l.r2 ) + Ko( I,,r: )).
(17)
With p taken in the form N
rio, ~ ) =
~..
C ( a.to( t.~. ) + r,,(I,,r,)) cos(l,,~),
(18)
rl~ |
equations (2) and (12) are satisfied automatically. Eq. (11) furnishes
[ ,,,,t,( t.r2) - r,( l.r~)
(19)
where • = ¢2/*,.
(20)
From (17) and (19) we obtain
~.to( t,,r2) + ro( t,,r2) r,( t,,r2) + Oro( t,,r2)i B"= ,~.l,(/.r2)-K,(t.r2) 1
[
" . I o ( 1 - ~ ) + ro(t-"~) i,(t.,.~) _ ,lo(t,,r2)
X a.l,(i.rz)_Kl(i.r2
)
1
.
(21)
Substitution from (16) into (!) and then integrating it twice with respect to z. the use of condition (2) yields
p(O, z) = 37/U ~ -~. ~, h3r, dh dO (z 2 _ ~b2) + 12,, h----S-,,=l (fl,,l,(l.r,)- K~(l,,rl) ) cos(I,,z).
(22)
Eqs. (18) and (22) furnish N
~.. C,,A.cos(l.z)= 3*lU ._, h3r, dh de (? - 'b'-),
(23)
P.D.S. Verma / Porous journal bearing analysis
236
where
A-. = ~.lo(i.r , ) + Ko(lnr ,) - I~l(fl.l,(l,,r, ) - K,(Inr, )). •
(24)
h-l,,
Multiplying (23) by cos(I.z), then integrating over the interval (0. -~b) and substituting for A-~,and h. one obtains
I['Ur' b: 96esinO c ' = t c2 a2 (l +,cosO) 3 (2. 12q~,rt b ×
~
c3
~ (-I) "+' -
1)"
' ~.-Ko(I,,r,)
] +
fl,,lo(l.r , )
K,(i.r~)_fl,ll(i,,r,)
2
d ~r(2n- 1) (1 +ccosO)3(go(t.r,)+#,,lo(t,,r,))
]-t
1
With this value of ¢~,. equation (18) provides the pressure in the film region as N
p(O, ~ ) = E ,lvr,
.=t
96c sin 0 ( - 1) "+l cos(t:) c 2d2 (! +c cos 0) 3 .n3(2n - i) 3 b"
{
24#pjr,b[Kt__~(l,,r,__.__. ~)--~,___ll,(l.r,)] }-' × l+~c3d(2n-l)(l +,cosO)3[Ko(l-~r~)--fl.lo(I.r,)]
(25)
Let W0 and W./2 denote the components of load W along and perpendicular to the line of centres. We write ,,
fb/2 f~
Wo= W cos ~k= zr, j °
Jo P COs O dO dz. p sin OdOdz.
I'V~/2= Wsin qJ=2r I
Substituting for p(O, z) from (25) in the above expressions gives the dimensionless quantities W~ and W2 corresponding to ~ and W./2, respectively, as
r|] [bl
'
1 ct:[d]2 w, = Tdg ( ( j ~-~/ ,
Wo= 19__2_2~ L , ,
,
n=|
N
W~,/2= 192c ][~ l ~ "~ sin20 .=, r2 ( l + , ~ s O ) ~ + g ~
where r,, = (2n - l)'rr,
g~ = 12~,,r, 2b r , ( ,o,,/b ) - ~n1,( r : , / b )
c3
(26)
rJ
dr,, Ko(r.r,/b ) +~.lo(r.r,/b)
and g.+ 1 (g.+l+,) 2 g~-g.(l-,)+(l-,)2 L. -- _ _ I. 6g 2 (g.- 1 -c) 2 g~-g.(l +c)+(l +e) 2
dO,
(27)
P.D.S. Verma/ Porousjournal bearinganalysis
+ ~
< tan-
t
L ¢3-g.
3
-
tan-
237
~
.
The load capacity of the bearing in terms of the Sommerfeld number
(=(,~Vb/.,,W)(r,/c~))
s
is given by
+ w; ).
(28)
The coefficient of friction f is now furnished by f(r,/c)
= 2~2S/(1 + (2),/2
(29)
whereas the attitude angle ~ follows from ~ -- t a n - ' ( W 2 / Wi ) .
4. Discussion of the results In Figs. 2 and 3, ( S ( b / d ) 2 ) - ~ is used as the load parameter and it is designated as the load number. Fig. 2 represents results relating to the load number as a function of the porous housing thickness ratio, ~=r2--rl r o - r2
for(=0.4,
~=10,
r0 = 0 . 6
and
r k=0.5.
The calculations are sketched for two values of ~l ( = rhlr, c - 3 ) = 0.01 and ~t = 0.1. It may be observed that as R- increases, so does the load number. Further, an increase in the permeability parameter ~ ~ of the inner layer implies a decrease of the load number keeping constant value of the ratio ~. The variation of the load number with the permeability ratio parameter ~ is plotted in Fig. 3 for different values of the thickness
/
'
' ~-oo;' =1o
I
rJ,t.iJ t~
~q -0.5 -0.6
:E:
n¢ II1
r,..
4
_,,
o
v~o 0.~1i
~-,,
r°" °'sl
~
] j
1
o .J
~
~
Z
g
5
8
fl
Fig. 2.
Fig. 3.
10
:p
15
20
P.D.S. Verma / Porous journal bearing analysis
238
e -
I..5
0/.
~-I0 ro ° 0 6
I"1 =
0.5
11
'
'
t' rl C
to'O,6
I
I 4
~
~-O01 3
£
--9
e
Fig. 4.
Fig. 5.
ratio R. The load number is observed to increase with increasing R and decreasing O. Fig_ 4 exhibits a graphical representation of the variation of the friction coefficient with the thickness ratio R for values of ~ = 0.1 and 6't = 0.01. The results sh6w th,~t the coefficient of friction increases when ,~ decreases, in Fig. 5 we display the variation of friction coefficient against O for different values of R, indicating an increase in the coefficient of friction wh,:n ~ is int'reased. All these results demonstrate that a low permeability inner layer increases the load capacity while reducing the coefficient of friction of the porous bearing.
References Marshall, P.R. and V.T. ~org~-n (1965-66), "Review of porous metal bearing developmem", Proc. Inst. Mech. Engrs. 180 (7) 154.
Youssef, H. and M. Euchier (1965), "Production and properties of a new porous bearing", in: Pro('. Conf. Powder Metallurgy. American inst. of Mining Engrs., NY, Pdpcr 53.
233
D O U B L E LAYER P O R O U S J O U R N A L BEARING ANALYSIS P.D.S. VERMA
Department of Applied Sciences. Regional EngineeringCollege. Kurukshetra-132119. India Received 14 June 1983 A double layer porousjournal beating has been studied using the short bearing approximation.Taking the curvature of the bearing into consideration, expressions for the pressure distribution, load carrying capacity, coefficientof friction, and the attitude angle are derived.It is shown that the bearingcharacteristicsare improveddue to low permeabilityof the inner porous layer.
!. Introduction The investigations on porous bearings have demonstrated that the load capacity of the bearings is reduced on account of permeability. This reduction in load capacity, due to reduced seepage into the porous matrix, may be made smaller by taking a low permeability porous housing. However, smaller permeability results in the depletion of lubricant content in the porous matrix, thereby affecting the self lubricating feature of u., bearing. To avoid this, it was suggested by Marshall and Morgan, 1965-66) that a double layer porous housing in which the inner layer has a reduced pore size and lower permeability be used. Experimentally, it was found (Youssef and Euchier, 1965) that an interior layer of ultrafine carbonyl nickel powder on the bore of a conventional beating with a low permeability nickel alioy lining significantly affects the performance characteristics of the bearing. In this presentation we analyse the performance of a two layer porous bearing using the short bearing approximation. The circumferential flow, both in the film region as well as in the bearing material, has been neglected in comparison to the axial flow. The curvature of the bearing is also taken into consideration. A marked improvement is observed with low permeability inner layer. In Section 2 we formulate the problem. The solution is obtained in Section 3, followed by a discussion of the results in Section 4.
2. Formulation A two layered porous journal bearing under steady load is shown in Fig. 1. With the short bearing approximation, the modified Reynold's equation for the pressure p in the film region assumes the form
a 2 p = 1 2 ~ [ U dh a2 2 h 3 2rj dO
ck''aP'l ] 7? ~ ar ] .... '
(1)
where U is the tangential velocity of the shaft, h = c(1 + c cos 0), c the radial clearance, ~ the eccent.,icity ratio, and rj the inner radius of the bearing. The boundary conditions for p are
p(O, + ~ b ) = 0 , ?,:6%6636/83/$3.00 © 1983, ElsevierSciencePublishers B.V. (North-Holland)
(2)
P.D.S. Verma / Porous journal bearing analysis
230,
•
i
l,l L
J
Fig. 1. Sketch for journal bearing in concentricposition. 1--journal 2--film region. 3--inner porous layer. 4--interior porous layer. 5-- solid housing.
and
(a/az)p(O, o) = O.
(3)
the pressures P~ and Pz at the inner and interior layers satisfy the Laplace's equation:
a:p~/ar 2 + ( l / r )(aP,/Or ) + a'P,/az 2 = o, O2e2/ar 2 + (I/r)(Oe2/ar) + a2P2/~z2 = 0,
(4) (5)
and the boundary conditions
P2(r,O, O)=O, ( a / ~ ) P : ( r , o, o) = o,
(6) (7) (8)
(a/Oz )P2(r, O, O) = O,
(9)
(a/ar)P2(,o, O,~)=o,
(lO)
P,(r. O, +_½b) = O,
O,(aP,/Or ),=,: = 02(OV:/Or ),=,:. The corresponding matching conditions are
P,(r,, O , z ) = p ( O , z ) ,
(12)
P2(r2, O,z)= P,(r2, 0, z).
(13)
Here, b represents the length of the bearing, ~l and q~2 are the permeabilities in the inner and interior layers, respectively, r 2 - r~ is the thickness of the inner layer, and r0 - r 2 the thickness of the interior layer. To obtain the film pressure p(0, z), we need to solve (1), (4) and (5) subject to the boundary conditions (2), (3) and (6) to (13).
P.D.S. Verma / Porous journal bearing analysis
235
3. Solution
Eq. (5) subject to the conditions (7), (9) and (10) yields N
P2(r, O. z ) = 2~ C,,(a, lt,(I,r)+Ko(I,,r)) cos(l.z).
(14)
n=l
where
a,, = Kt(I,,ro)/l,(l, ro).
I, = (2n -
1).rr/b.
(15)
The constants (7, need to be determined. Here. K o, / o and K I, ! l denote modified Bessel functions of the second and first kinds with orders zero and one, respectively. Similarly. (4) along with conditions (6) and (8) gives N
Pt(r. 0, z)= ~ C..(fl,,lo(I.r ) +Ko(I.r)) cos(l.z),
(16)
n=l
where the constants C. and ft. are to be found. The matching condition (13) provides
C,, = C,( a.lo( i,,r2 ) + Ko( I.r2 )) /( B.lo( l.r2 ) + Ko( I,,r: )).
(17)
With p taken in the form N
rio, ~ ) =
~..
C ( a.to( t.~. ) + r,,(I,,r,)) cos(l,,~),
(18)
rl~ |
equations (2) and (12) are satisfied automatically. Eq. (11) furnishes
[ ,,,,t,( t.r2) - r,( l.r~)
(19)
where • = ¢2/*,.
(20)
From (17) and (19) we obtain
~.to( t,,r2) + ro( t,,r2) r,( t,,r2) + Oro( t,,r2)i B"= ,~.l,(/.r2)-K,(t.r2) 1
[
" . I o ( 1 - ~ ) + ro(t-"~) i,(t.,.~) _ ,lo(t,,r2)
X a.l,(i.rz)_Kl(i.r2
)
1
.
(21)
Substitution from (16) into (!) and then integrating it twice with respect to z. the use of condition (2) yields
p(O, z) = 37/U ~ -~. ~, h3r, dh dO (z 2 _ ~b2) + 12,, h----S-,,=l (fl,,l,(l.r,)- K~(l,,rl) ) cos(I,,z).
(22)
Eqs. (18) and (22) furnish N
~.. C,,A.cos(l.z)= 3*lU ._, h3r, dh de (? - 'b'-),
(23)
P.D.S. Verma / Porous journal bearing analysis
236
where
A-. = ~.lo(i.r , ) + Ko(lnr ,) - I~l(fl.l,(l,,r, ) - K,(Inr, )). •
(24)
h-l,,
Multiplying (23) by cos(I.z), then integrating over the interval (0. -~b) and substituting for A-~,and h. one obtains
I['Ur' b: 96esinO c ' = t c2 a2 (l +,cosO) 3 (2. 12q~,rt b ×
~
c3
~ (-I) "+' -
1)"
' ~.-Ko(I,,r,)
] +
fl,,lo(l.r , )
K,(i.r~)_fl,ll(i,,r,)
2
d ~r(2n- 1) (1 +ccosO)3(go(t.r,)+#,,lo(t,,r,))
]-t
1
With this value of ¢~,. equation (18) provides the pressure in the film region as N
p(O, ~ ) = E ,lvr,
.=t
96c sin 0 ( - 1) "+l cos(t:) c 2d2 (! +c cos 0) 3 .n3(2n - i) 3 b"
{
24#pjr,b[Kt__~(l,,r,__.__. ~)--~,___ll,(l.r,)] }-' × l+~c3d(2n-l)(l +,cosO)3[Ko(l-~r~)--fl.lo(I.r,)]
(25)
Let W0 and W./2 denote the components of load W along and perpendicular to the line of centres. We write ,,
fb/2 f~
Wo= W cos ~k= zr, j °
Jo P COs O dO dz. p sin OdOdz.
I'V~/2= Wsin qJ=2r I
Substituting for p(O, z) from (25) in the above expressions gives the dimensionless quantities W~ and W2 corresponding to ~ and W./2, respectively, as
r|] [bl
'
1 ct:[d]2 w, = Tdg ( ( j ~-~/ ,
Wo= 19__2_2~ L , ,
,
n=|
N
W~,/2= 192c ][~ l ~ "~ sin20 .=, r2 ( l + , ~ s O ) ~ + g ~
where r,, = (2n - l)'rr,
g~ = 12~,,r, 2b r , ( ,o,,/b ) - ~n1,( r : , / b )
c3
(26)
rJ
dr,, Ko(r.r,/b ) +~.lo(r.r,/b)
and g.+ 1 (g.+l+,) 2 g~-g.(l-,)+(l-,)2 L. -- _ _ I. 6g 2 (g.- 1 -c) 2 g~-g.(l +c)+(l +e) 2
dO,
(27)
P.D.S. Verma/ Porousjournal bearinganalysis
+ ~
< tan-
t
L ¢3-g.
3
-
tan-
237
~
.
The load capacity of the bearing in terms of the Sommerfeld number
(=(,~Vb/.,,W)(r,/c~))
s
is given by
+ w; ).
(28)
The coefficient of friction f is now furnished by f(r,/c)
= 2~2S/(1 + (2),/2
(29)
whereas the attitude angle ~ follows from ~ -- t a n - ' ( W 2 / Wi ) .
4. Discussion of the results In Figs. 2 and 3, ( S ( b / d ) 2 ) - ~ is used as the load parameter and it is designated as the load number. Fig. 2 represents results relating to the load number as a function of the porous housing thickness ratio, ~=r2--rl r o - r2
for(=0.4,
~=10,
r0 = 0 . 6
and
r k=0.5.
The calculations are sketched for two values of ~l ( = rhlr, c - 3 ) = 0.01 and ~t = 0.1. It may be observed that as R- increases, so does the load number. Further, an increase in the permeability parameter ~ ~ of the inner layer implies a decrease of the load number keeping constant value of the ratio ~. The variation of the load number with the permeability ratio parameter ~ is plotted in Fig. 3 for different values of the thickness
/
'
' ~-oo;' =1o
I
rJ,t.iJ t~
~q -0.5 -0.6
:E:
n¢ II1
r,..
4
_,,
o
v~o 0.~1i
~-,,
r°" °'sl
~
] j
1
o .J
~
~
Z
g
5
8
fl
Fig. 2.
Fig. 3.
10
:p
15
20
P.D.S. Verma / Porous journal bearing analysis
238
e -
I..5
0/.
~-I0 ro ° 0 6
I"1 =
0.5
11
'
'
t' rl C
to'O,6
I
I 4
~
~-O01 3
£
--9
e
Fig. 4.
Fig. 5.
ratio R. The load number is observed to increase with increasing R and decreasing O. Fig_ 4 exhibits a graphical representation of the variation of the friction coefficient with the thickness ratio R for values of ~ = 0.1 and 6't = 0.01. The results sh6w th,~t the coefficient of friction increases when ,~ decreases, in Fig. 5 we display the variation of friction coefficient against O for different values of R, indicating an increase in the coefficient of friction wh,:n ~ is int'reased. All these results demonstrate that a low permeability inner layer increases the load capacity while reducing the coefficient of friction of the porous bearing.
References Marshall, P.R. and V.T. ~org~-n (1965-66), "Review of porous metal bearing developmem", Proc. Inst. Mech. Engrs. 180 (7) 154.
Youssef, H. and M. Euchier (1965), "Production and properties of a new porous bearing", in: Pro('. Conf. Powder Metallurgy. American inst. of Mining Engrs., NY, Pdpcr 53.