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Axially undefined porous journal bearings considering cavitation
1
Wear Elsevier Sequoia
SA., Lausanne
AXIALLY UNDEFINED CAVITATION
J. PRAKASH
and
SANJAY
- Printed
in the Netherlands
POROUS
KUMAR
JOURNAL
BEARINGS
CONSIDERING
VIJ
Department of Mathematics, Indian Institute of Technology, Powai, Bombay-76 (India) (Received
February
21972)
SUMMARY
An analysis is made of the performance of axially undefined porous journal bearings using the Reynolds’ boundary conditions. Expressions for angular filmextent, pressure distribution, load carrying capacity, attitude angle and coefficient of friction are obtained in closed form. The results are shown graphically for situations of practical importance.
NOMENCLATURE
c
radial clearance diameter of the journal coefficient of friction, F/W friction F h oil film thickness, C( 1+ E cos 0) film thickness at the end of pressure curve hz H porous bearing wall thickness bearing length in axial direction L pressure in the film region P dimensionless film pressure, pC2/6 @JR P P pressure in the porous matrix R journal radius S Sommerfeld number, pULR2/WC2= I/A Sh4 Sommerfeld number corresponding to minimum coefficient of friction critical Sommerfeld number S, u tangential velocity of the journal W load load component along the line of centers K load component perpendicular to the line of centers w, _. x, y, z Cartesian coordinates (12!lJ)6 1 dimensionless load, WC2/pULR2 eccentricity ratio ; angular coordinate, x = Rfl
f”
Wear, 22 (1972)
2
J. PRAKASH.
S. K. VIJ
62 angle, where the continuous film terminates p 4 @ Y
absolute viscosity of the oil attitude angle permeability of porous bearing permeability parameter, QiH/C3.
INTRODUCTION
Porous metal bearings consist of a bearing bush of porous sintered metal the pores of which are impregnated with lubricant. One of the main advantages of these bearings is that no external supply of lubricant is required for running-in. It is usually assumed that the type of lubrication found with this system is normally mixed or boundary. However, it has been shown3v5 that under certain operating conditions, full fluid film or hydrodynamic conditions are achieved. The first analytical study of this system (i.e. porous bearings running under hydrodynamic conditions) was reported by Morgan and Cameron’ who analyzed a porous journal bearing using Ocvirk’s narrow bearing approximations?. In subsequent papers3*‘, it was shown that hydrodynamic lubrication in these bearings is impossible below a certain critical Sommerfeld number which depends upon the viscous permeability of the bearing, its wall thickness, L/D ratio and the running clearance. The other limiting case, uiz. the axially undefined journal bearing was later studied by Shir and Joseph’ and Capone’. Both these analyses suffered from the major defect that a full film was assumed (Sommerfeld condition), thus giving rise to large negative pressure regions in the film, which is seldom found in bearings running under the usual operating conditions. The more realistic boundary condition describing the actual physical situation is the Reynolds boundary condition, that the pressure is positive and terminates to zero with a zero pressure gradient. Thus it is assumed that the film is continuous only in the region of positive pressure and cavitates at some position 8 = l& > rc where p = dp/d0 = 0, forming a discontinuous mixture of air, vapour and lubricant in the cavitated region (t?, < 0< 271). These conditions have been applied extensively for the pressure distribution and load capacity of non-porous journal bearings. However, to the authors’knowledge, no work has been reported which takes into account the cavitation of the fluid film in porous bearings. The purpose of this paper is to study this aspect of porous bearings. The closed form solutions for pressure distribution, load capacity, attitude angle and coefficient of friction are obtained to facilitate the study of the behaviour of various parameters. The results are compared with the earlier solutions of Capone’ who presented a numerical solution using Sommerfeld boundary condition. A significant deviation is noted. It is also found that below a critical Sommerfeld number, hydrodynamic lubrication is not possible, a result similar to that obtained by Cameron et aL3 for the case of a short journal bearing. This peculiarity of porous bearings was not noted in the earlier solutions’.‘. The analysis presented in this paper together with that of Cameron et al3 represent the two limiting cases of porous journal bearings, viz. the infinitely long and infinitely short. Both assumptions, though necessary from the analytical point of view, limit the results obtained and do not fully describe the real behaviour. However, they provide means of studying trends and relationships. Wear,22
(1972)
AXIALLY UNDEFINED
POROUS JOURNAL
3
BEARINGS
ANALYSIS
The lubricant is assumed to be incompressible, Newtonian and to have constant viscosity; the porous bearing is assumed to have uniform permeability and wall thickness, Fig. 1.
Fig. 1. Journal bearing configuration.
Using the usual assumptions of lubrication theory with the additional assumption that the flow in the porous matrix satisfies Darcy’s law, the equations governing the pressure distribution in the film region and the porous matrix have been derived by Morgan and Cameron’. These equations are:
for the fluid film region, and
a2p a2p a9 &T+ayZ+z=
o
(2)
for the porous region, where (aP/ay),,,, is the pressure gradient at the bearing surface and can be obtained by solving eqn. (2) under suitable boundary conditions. A simplifying assumption is incorporated similar to Morgan and Cameron’. This approximation consists essentially of assuming that the pressure in the porous region can be replaced by the average pressure with respect to the bearing thickness. As discussed by Pinkus and reported by Morgan5 the exact solution and the solution obtained by using this approximation are practically identical for the values of the parameters which correspond to the majority of bearings. It has been shown that for H/D< 0.2, the _rror is insignificant. Under this assumptiun, the modified Reynolds equation (1) reduces to ;[(h3+12@H) Wear, 22 (1972)
$1+$
hh3+12@H)$]
= 6,dJ 2
4
J. PRAKASH,
S. K. VIJ
For the case of an axially undefined journal bearing, the equation further simplifies to
ECose)3+12(Y]~ = - 6p URE sin B
I
c2
Boundary conditions are p=Oatti=O dLOat8=8 d% p=OatB=8
2
6)
2
Pressure distribution
Solving eqn. (4) under the boundary conditions eqn. (5), the pressure in the film region is obtained as A ((1 +a)2-&2)+
tan-l{
/Stan:
,/Etani
} - JGcotz
‘- tan-’ 12(F+,/‘=))+ jEtan;
+ JGcoti-
{2(m-F)]* I
JEtan;+
,/Gcoti+
(61
{2(,,/EG-F)it J
where, ECOSe~fCIfl 3a2s -
A=-B =
_
~2C~~~2+j(1+C()+(2C(-1)COS0)2~E-(@2-~+1) 3cx2&
D = ~~cos~~+{(1+_~)-(2a-1)cost)~)8+(a~-ai-f) 3a2 E E = &“-(2-a)e+(a’-a+l) F = (a’-a+l)-E2 G = &2+(2-a)&f(a2-a+-l)
The angle e2 at which the film breaks down and the cavitation region starts is given by S&) = 0
(8)
Load capacity
The load carried by the bearing can be found by integrating the pressure Wear,22
(1972)
AXIALLY
UNDEFINED
POROUS
JOURNAL
5
BEARINGS
around the bearing from tJ= 0 to 8 = e2 and resolving along and normal to the line of centers. Thus the load component along the line of centers is e2 w, = w cos $I =
s
p cos 0 LRdt3 0
On substituting for the pressure from eqn. (6) and integrating, it gives
(9) Normal to the line of centers, the load component
W, is
02 WY = W sin 4 =
p sin 8 LR d6 0
or w _
4A(l +c()
3puL.R’ 7-
Y
{(l+a)*-E2}* I
L tan 2
A(JE+JG)-&(BIJE-D/JG) {2(F+p)}’
_
A (JG-
- ,/G
cot (10)
{2(F+,/=)}*
& +E(W/E+D/,/G)In
JEtan:
+ JGcot:
-
{2($=-F))*
JE tan:
+ JG
+ {2(m-F)}*
cot $
{2(,,b%-F)J’
il
The dimensionless total load carrying capacity is A = WC2/p ULR2 where w = ( w,’ + w;,+
(11)
Attitude angle The attitude angle 4 can be found from eqns. (9) and (10). It is given by 4=tan-‘(WY/W,)
(12)
CoefJicient offriction Since the oil film is assumed to cavitate beyond the end of the pressure curve, the friction is made up of two parts. The first part is due to the oil film extending from 8=0 to 0= e2 and the second part is because of the cavitated region from 19=0, to 8=2z. The friction for these two parts can be obtained in a way similar to Flobergg therefore, Wear,22
(1972)
J. PRAKASH, S. K. VIJ
6
F=[ ~;(y,de+g
sin $1 +
[\I:
(;
y,)de
]
(13)
The coefficient of friction is obtained from eqn. (13) on integrating and dividing by the load. Thus, 1
(f)f=;sin++i(
~~c(~-E~)+(~~c-I~&(E+cos~,)+E(~+Ecos~~)~~~~,
(1 - &‘)*
I (14)
where sin & = JiY?
sin e,i(
1 +E cos e,).
Solutions for the case of Sommerfeld boundary condition
In this case the boundary conditions eqn. (5) are replaced by p=O
at e=o
p=O at e=2n 1
As is well known, these boundary conditions result in large negative pressure regions. Since the solutions given by Capone’ were not in the closed form, the expressions for pressure ~stribution, load capacity, attitude angle and coefficient of friction are given in this section without the mathematical details. Pressure distribution
(B’/,/E-D’I,h4
JE tan i + ,/G cot 4 - {2(,/%%-F
if I
,n
+ 4{2(m-
F)j*
(16) JEtan!
+ ,/Gcott
+ {~(Q’%?--F)]~
where, A, = _ l+a-A,e 3a2E
D, _ -A,~~+{(1+a)+(2a-l)A~)~+(a~-al-l)
(17)
3a2e
and t+a
A = #+a)‘--&*I* 1
+ ((l+a)E-(a2-a+1)~/~E-f(l+~)~+(a*-a+I)~/~G w
+ VfmP
(E+2a-l)/JE-(E-2a+l)/JG W+x/=)l*
Weur, 22 ( 1972)
AXIALLY UNDEFINED
POROUS
JOURNAL
7
BEARINGS
E, F and G are the same as given in eqn. (7). Load capacity w, = 0 A =
wYc2 m
- ((1+7;z21t
+
1 (18)
(B’/JE-D’/JG){2(F+~))~ 2x
Attitude angle
+=tan-’
c> W
$
(19)
X
Coeficient offriction
(20) RESULTS AND DISCUSSIONS
Figures 2 to 9 show the effect of oil flow in the porous journal bearing, on the film extent, pressure distribution, load carrying capacity, attitude angle and coefficient of friction. As pointed out earlier, the theoretical design parameter to be applied to account for the effect of porosity in thin walled bearings (H/D < 0.1) is Y= @H/C3 ; a parameter which is very sensitive even to small clearance changes. 4.3
4.2
-
4.0 -
3.9 62 3.8
-
3.7 Ei0.7 3.6
-
3.5
-
3.4
-
3.3 1 0.001
I 0.1
0.01
0
‘21
Fig. 2. Angular extent of oil-film, e2 VS.permeability parameter, \p for various values of eccentricity ratio, E. Wear, 22 (1972)
8
J. PRAKASH,
09-
0.8 -
F 0.7 -
0.6 -
o.s0.4 -
0.3 -
0.2 -
0.1 -
T
5
3
e
Fig. 3. Pressure
distribution
for various
values of permeability
distribution
for various
values of eccentricity
parameter.
1.6
-POROUS ----SOLID
0.E
Fig. 4. Pressure
Wear,22 (1972)
ratio, E,
I
Y
S. K. VIJ
AXIALLY
UNDEFINED
POROUS
JOURNAL
9
BEARINGS
The results are obtained for various values of eccentricity ratio, E, and design parameter, Y. Figure 2 is a graph of angular extent of the film for different values of eccentricity ratio E. It is observed that an increase in Y results in an increase in o2 which means, other parameters being fixed, that the cavitation region is reduced by increasing the permeability. The same effect can also be obtained by keeping permeability fixed and increasing the wall thickness or decreasing the clearance. Also, the extent of the film decreases with increase of a for all values of Y. However, this effect is more pronounced at low values of Y. The effect of increasing the permeability (other parameters being fixed) is to reduce the pressure, shift the point of maximum pressure towards the start of the film (Fig. 3) and to reduce the load carrying capacity (Figs. 56). The decrease in pressure and load capacity is, however, more for large eccentricity ratios (Figs. 4-6). It is also observed that the porous bearing behaves similarly to a solid bearing for 12Y x 0.001 and the effect of porosity is not significant up to 12Y z 0.05. After this, there is a rapid decrease in the load capacity (Fig. 5). The curves corresponding to Sommerfeld solution’ are also drawn in Figs. 5 and 6. It is seen that the difference is quite significant for all values of permeability parameter, Y. However, in the limit E-+ 0, both the solutions tend to be identical. One more interesting feature is noteworthy as E--P1, the load capacity for the porous bearings tends to a finite value whereas for the non-porous bearings, it increases indefinitely. Figure 7 shows the variation of coefficient of friction with the permeability parameter Y for various values of E.It indicates that other factors being equal, porous bearings generally have a higher coefficient of friction than solid bearings. However,
I
-REYNOLDS ----SOMMERFELD
Fig. 5. Dimensionless Wear, 22 (1972)
load, A us. permeability
parameter,
Y for various
values
of eccentricity
ratio,
E.
10
J. PRAKASH.
S. K. VIJ
1000.0 r -REYNOLDS -- __ SOMMERFELD
Fig. 6. Dimensionless
load, A us. eccentricity
ratio,
E for various
values
of permeability
parameter,
Y.
the increase in the coefficient of friction is more for large eccentricity ratios. Also plotted in this figure are the corresponding Sommerfeld curves. Again a significant deviation from the present results is indicated. The attitude angle C#Jalso changes appreciably with Y as shown in Fig. 8. The deviation is, however, more pronounced for large values of Y. For 12 Y x 0.001 there is no appreciable difference upto E= 0.8. The fact that A is finite when E= 1 explains why the attitude angle C$is not zero. It is found that C$approaches 90” asymptotically as Y increases indefinitely. This corresponds to the case of an infinitely porous bearing which, of course, can carry no load because all the lubricant will bleed away into the porous wall. Figure 9 illustrates the relationship between the coefficient of friction and the Sommerfeld number, S for an inlinitely long porous journal bearing. The curves corresponding to the solid bearing and the idealized Petroff bearing are also shown. It is to be noted that as the Sommerfeld number, S is reduced (for a fixed Y), the system suddenly becomes unstable and the friction rises rapidly. This leads to the concept of a critical Sommerfeld number, S, below which hydrodynamic lubrication is not possible. In addition to the critical value (S,), the curves also show a minimum value (S,). In bearing design, this is of greater importance as it can readily be calculated. The same phenomena was also noted by Morgan’ for the case of infinitely short Wear, 22 (1972)
AXIALLY UNDEFINED
POROUS
JOURNAL
BEARINGS
11
3.5 --------
REYNOLDS SOMMERFELO
3.0 -
2.5 -
2.0 -
0.5 -
DO_ 0.001
I 0.1
a01 ‘2Y
Fig. 7. Coefficient of friction, (R/C')fus. permeability parameter, !I’for various values ofeccentricity ratio, E.
Fig. 8. Attitude angle, 4 0s. eccentricity ratio, E for various values of permeability parameter, y, Wear, 22 (1972)
12
J. PRAKASH,
S. K. VIJ
bearings. It is also observed that as the value of Y decreases, the curves tend towards that of a non-porous bearing. In Fig. 10, S, and S, are plotted against Y. The regions of hydrodynamic and boundary lubrication are indicated. It is found that the Sommerfeld number corresponding to the minimum friction increases with Y.
!p .I
-.. 01
.a
/
0.01
1.0
0.1
1 .O
S i l,‘,,
Fig. 9. Coeffkient
of friction, (R/C)f
US.Sommerfeld
number,
S for various values of permeability
parameter.
LO
Fig, 10. Sommerfeld number corresponding number, S, us. permeability parameter, $. Wear, 22 (1972)
to minimum
coefficient
of friction,
SM and critical Sommerfeld
AXIALLY
UNDEFINED
POROUS
JOURNAL
13
BEARINGS
APPENDIX
The eqn. (4) on integration gives y
P’
[I+A2]
(21)
where (cos B+A,)dB z = t
(If&
B’I&+D’I,/G
+
A’ = {(l+c+EZ)f
cos ey+c?
tan_,
2{2(F+&)}+
+
W&W&
&tan;
+ JG
cot;
-
{2(&%F))+
jEtan;
+ JG
cot;
+ {2(--F))’
,n
4(2(&-F)}+
and the expressions for A’, B’ and D’ are as given in eqn. (17). A, and A, are constants of integration. The indefinite integrals for load components are obtained as W=~ WJLRZ x C2
-2Esin
B(I+A,)+
c L
(1+s+a)‘(Esin4:
+ 2Fsin*~cos2~
+ Gcos4i)
+ gin G(~+Ecos~+~)~
~+ECOS~-LY
(24
and
+
-
A’(+
+ JG) - s@‘I& {2(F+m)}*
- D/,/G)
tan _ 1
G/G - JE)+ WI@ +D’I,/G)ln 4(2(*-F)}+
JE
tan i + ,,/G cot:
JE
tan i + JG
-
{2(@-F))f (23)
cot p + {2(@-F)}f
Equations (21) to (23) will facilitate the extension of the present work to partial arc-type bearings. REFERENCES 1 V. T. Morgan and A. Cameron, Mechanism of lubrication in porous metal bearings, Lubrication and Wear, London 1957, Inst. Mech. Engrs., Paper 89 (1957) 151-157. Wear, 22 (1972)
Proc. Conf
on
14
J. PRAKSH, S. K. VIJ
2 F. W. Ocvirk, Short bearing approximation for full journal bearings, NACA TN 2808, 1952. 3 A. Cameron, V. T. Morgan and A. E. Stainsby, Critical conditions for hydrodynamic lubrication of porous metal bearings, Proc. Inst. Mech. Engrs., 176 (1962) 761-770. 4 W. T. Rouleay Hydrodynamic lubrication of narrow press-fitted porous metal bearings, Trans. ASME, .I. Basic Eng., 85 (1963) 123-128. 5 V. T. Morgan, Hydrodynamic porous metal bearings, Lubrication Eng., 20 (1964) 448-455. 6 V. T. Morgan, Closure to discussion on porous metal bearings, Lubrication Eng., 21 (1965) 33. 7 C. C. Sbir and D. D. Joseph, Lubrication of porous bearings-Reynold’s Solution, Trans. ASME, J. Appl. Mech., 33 (1966) 761-763. 8 E. Capone, Lubrication of axially undefined porous bearings, Wear, 15 (1970) 157-170. 9 L. Floberg, Boundary conditions of cavitation regions in journal bearings, ASLE Trans., 4 (1961)282-286. Wear, 22 (1972)
Wear Elsevier Sequoia
SA., Lausanne
AXIALLY UNDEFINED CAVITATION
J. PRAKASH
and
SANJAY
- Printed
in the Netherlands
POROUS
KUMAR
JOURNAL
BEARINGS
CONSIDERING
VIJ
Department of Mathematics, Indian Institute of Technology, Powai, Bombay-76 (India) (Received
February
21972)
SUMMARY
An analysis is made of the performance of axially undefined porous journal bearings using the Reynolds’ boundary conditions. Expressions for angular filmextent, pressure distribution, load carrying capacity, attitude angle and coefficient of friction are obtained in closed form. The results are shown graphically for situations of practical importance.
NOMENCLATURE
c
radial clearance diameter of the journal coefficient of friction, F/W friction F h oil film thickness, C( 1+ E cos 0) film thickness at the end of pressure curve hz H porous bearing wall thickness bearing length in axial direction L pressure in the film region P dimensionless film pressure, pC2/6 @JR P P pressure in the porous matrix R journal radius S Sommerfeld number, pULR2/WC2= I/A Sh4 Sommerfeld number corresponding to minimum coefficient of friction critical Sommerfeld number S, u tangential velocity of the journal W load load component along the line of centers K load component perpendicular to the line of centers w, _. x, y, z Cartesian coordinates (12!lJ)6 1 dimensionless load, WC2/pULR2 eccentricity ratio ; angular coordinate, x = Rfl
f”
Wear, 22 (1972)
2
J. PRAKASH.
S. K. VIJ
62 angle, where the continuous film terminates p 4 @ Y
absolute viscosity of the oil attitude angle permeability of porous bearing permeability parameter, QiH/C3.
INTRODUCTION
Porous metal bearings consist of a bearing bush of porous sintered metal the pores of which are impregnated with lubricant. One of the main advantages of these bearings is that no external supply of lubricant is required for running-in. It is usually assumed that the type of lubrication found with this system is normally mixed or boundary. However, it has been shown3v5 that under certain operating conditions, full fluid film or hydrodynamic conditions are achieved. The first analytical study of this system (i.e. porous bearings running under hydrodynamic conditions) was reported by Morgan and Cameron’ who analyzed a porous journal bearing using Ocvirk’s narrow bearing approximations?. In subsequent papers3*‘, it was shown that hydrodynamic lubrication in these bearings is impossible below a certain critical Sommerfeld number which depends upon the viscous permeability of the bearing, its wall thickness, L/D ratio and the running clearance. The other limiting case, uiz. the axially undefined journal bearing was later studied by Shir and Joseph’ and Capone’. Both these analyses suffered from the major defect that a full film was assumed (Sommerfeld condition), thus giving rise to large negative pressure regions in the film, which is seldom found in bearings running under the usual operating conditions. The more realistic boundary condition describing the actual physical situation is the Reynolds boundary condition, that the pressure is positive and terminates to zero with a zero pressure gradient. Thus it is assumed that the film is continuous only in the region of positive pressure and cavitates at some position 8 = l& > rc where p = dp/d0 = 0, forming a discontinuous mixture of air, vapour and lubricant in the cavitated region (t?, < 0< 271). These conditions have been applied extensively for the pressure distribution and load capacity of non-porous journal bearings. However, to the authors’knowledge, no work has been reported which takes into account the cavitation of the fluid film in porous bearings. The purpose of this paper is to study this aspect of porous bearings. The closed form solutions for pressure distribution, load capacity, attitude angle and coefficient of friction are obtained to facilitate the study of the behaviour of various parameters. The results are compared with the earlier solutions of Capone’ who presented a numerical solution using Sommerfeld boundary condition. A significant deviation is noted. It is also found that below a critical Sommerfeld number, hydrodynamic lubrication is not possible, a result similar to that obtained by Cameron et aL3 for the case of a short journal bearing. This peculiarity of porous bearings was not noted in the earlier solutions’.‘. The analysis presented in this paper together with that of Cameron et al3 represent the two limiting cases of porous journal bearings, viz. the infinitely long and infinitely short. Both assumptions, though necessary from the analytical point of view, limit the results obtained and do not fully describe the real behaviour. However, they provide means of studying trends and relationships. Wear,22
(1972)
AXIALLY UNDEFINED
POROUS JOURNAL
3
BEARINGS
ANALYSIS
The lubricant is assumed to be incompressible, Newtonian and to have constant viscosity; the porous bearing is assumed to have uniform permeability and wall thickness, Fig. 1.
Fig. 1. Journal bearing configuration.
Using the usual assumptions of lubrication theory with the additional assumption that the flow in the porous matrix satisfies Darcy’s law, the equations governing the pressure distribution in the film region and the porous matrix have been derived by Morgan and Cameron’. These equations are:
for the fluid film region, and
a2p a2p a9 &T+ayZ+z=
o
(2)
for the porous region, where (aP/ay),,,, is the pressure gradient at the bearing surface and can be obtained by solving eqn. (2) under suitable boundary conditions. A simplifying assumption is incorporated similar to Morgan and Cameron’. This approximation consists essentially of assuming that the pressure in the porous region can be replaced by the average pressure with respect to the bearing thickness. As discussed by Pinkus and reported by Morgan5 the exact solution and the solution obtained by using this approximation are practically identical for the values of the parameters which correspond to the majority of bearings. It has been shown that for H/D< 0.2, the _rror is insignificant. Under this assumptiun, the modified Reynolds equation (1) reduces to ;[(h3+12@H) Wear, 22 (1972)
$1+$
hh3+12@H)$]
= 6,dJ 2
4
J. PRAKASH,
S. K. VIJ
For the case of an axially undefined journal bearing, the equation further simplifies to
ECose)3+12(Y]~ = - 6p URE sin B
I
c2
Boundary conditions are p=Oatti=O dLOat8=8 d% p=OatB=8
2
6)
2
Pressure distribution
Solving eqn. (4) under the boundary conditions eqn. (5), the pressure in the film region is obtained as A ((1 +a)2-&2)+
tan-l{
/Stan:
,/Etani
} - JGcotz
‘- tan-’ 12(F+,/‘=))+ jEtan;
+ JGcoti-
{2(m-F)]* I
JEtan;+
,/Gcoti+
(61
{2(,,/EG-F)it J
where, ECOSe~fCIfl 3a2s -
A=-B =
_
~2C~~~2+j(1+C()+(2C(-1)COS0)2~E-(@2-~+1) 3cx2&
D = ~~cos~~+{(1+_~)-(2a-1)cost)~)8+(a~-ai-f) 3a2 E E = &“-(2-a)e+(a’-a+l) F = (a’-a+l)-E2 G = &2+(2-a)&f(a2-a+-l)
The angle e2 at which the film breaks down and the cavitation region starts is given by S&) = 0
(8)
Load capacity
The load carried by the bearing can be found by integrating the pressure Wear,22
(1972)
AXIALLY
UNDEFINED
POROUS
JOURNAL
5
BEARINGS
around the bearing from tJ= 0 to 8 = e2 and resolving along and normal to the line of centers. Thus the load component along the line of centers is e2 w, = w cos $I =
s
p cos 0 LRdt3 0
On substituting for the pressure from eqn. (6) and integrating, it gives
(9) Normal to the line of centers, the load component
W, is
02 WY = W sin 4 =
p sin 8 LR d6 0
or w _
4A(l +c()
3puL.R’ 7-
Y
{(l+a)*-E2}* I
L tan 2
A(JE+JG)-&(BIJE-D/JG) {2(F+p)}’
_
A (JG-
- ,/G
cot (10)
{2(F+,/=)}*
& +E(W/E+D/,/G)In
JEtan:
+ JGcot:
-
{2($=-F))*
JE tan:
+ JG
+ {2(m-F)}*
cot $
{2(,,b%-F)J’
il
The dimensionless total load carrying capacity is A = WC2/p ULR2 where w = ( w,’ + w;,+
(11)
Attitude angle The attitude angle 4 can be found from eqns. (9) and (10). It is given by 4=tan-‘(WY/W,)
(12)
CoefJicient offriction Since the oil film is assumed to cavitate beyond the end of the pressure curve, the friction is made up of two parts. The first part is due to the oil film extending from 8=0 to 0= e2 and the second part is because of the cavitated region from 19=0, to 8=2z. The friction for these two parts can be obtained in a way similar to Flobergg therefore, Wear,22
(1972)
J. PRAKASH, S. K. VIJ
6
F=[ ~;(y,de+g
sin $1 +
[\I:
(;
y,)de
]
(13)
The coefficient of friction is obtained from eqn. (13) on integrating and dividing by the load. Thus, 1
(f)f=;sin++i(
~~c(~-E~)+(~~c-I~&(E+cos~,)+E(~+Ecos~~)~~~~,
(1 - &‘)*
I (14)
where sin & = JiY?
sin e,i(
1 +E cos e,).
Solutions for the case of Sommerfeld boundary condition
In this case the boundary conditions eqn. (5) are replaced by p=O
at e=o
p=O at e=2n 1
As is well known, these boundary conditions result in large negative pressure regions. Since the solutions given by Capone’ were not in the closed form, the expressions for pressure ~stribution, load capacity, attitude angle and coefficient of friction are given in this section without the mathematical details. Pressure distribution
(B’/,/E-D’I,h4
JE tan i + ,/G cot 4 - {2(,/%%-F
if I
,n
+ 4{2(m-
F)j*
(16) JEtan!
+ ,/Gcott
+ {~(Q’%?--F)]~
where, A, = _ l+a-A,e 3a2E
D, _ -A,~~+{(1+a)+(2a-l)A~)~+(a~-al-l)
(17)
3a2e
and t+a
A = #+a)‘--&*I* 1
+ ((l+a)E-(a2-a+1)~/~E-f(l+~)~+(a*-a+I)~/~G w
+ VfmP
(E+2a-l)/JE-(E-2a+l)/JG W+x/=)l*
Weur, 22 ( 1972)
AXIALLY UNDEFINED
POROUS
JOURNAL
7
BEARINGS
E, F and G are the same as given in eqn. (7). Load capacity w, = 0 A =
wYc2 m
- ((1+7;z21t
+
1 (18)
(B’/JE-D’/JG){2(F+~))~ 2x
Attitude angle
+=tan-’
c> W
$
(19)
X
Coeficient offriction
(20) RESULTS AND DISCUSSIONS
Figures 2 to 9 show the effect of oil flow in the porous journal bearing, on the film extent, pressure distribution, load carrying capacity, attitude angle and coefficient of friction. As pointed out earlier, the theoretical design parameter to be applied to account for the effect of porosity in thin walled bearings (H/D < 0.1) is Y= @H/C3 ; a parameter which is very sensitive even to small clearance changes. 4.3
4.2
-
4.0 -
3.9 62 3.8
-
3.7 Ei0.7 3.6
-
3.5
-
3.4
-
3.3 1 0.001
I 0.1
0.01
0
‘21
Fig. 2. Angular extent of oil-film, e2 VS.permeability parameter, \p for various values of eccentricity ratio, E. Wear, 22 (1972)
8
J. PRAKASH,
09-
0.8 -
F 0.7 -
0.6 -
o.s0.4 -
0.3 -
0.2 -
0.1 -
T
5
3
e
Fig. 3. Pressure
distribution
for various
values of permeability
distribution
for various
values of eccentricity
parameter.
1.6
-POROUS ----SOLID
0.E
Fig. 4. Pressure
Wear,22 (1972)
ratio, E,
I
Y
S. K. VIJ
AXIALLY
UNDEFINED
POROUS
JOURNAL
9
BEARINGS
The results are obtained for various values of eccentricity ratio, E, and design parameter, Y. Figure 2 is a graph of angular extent of the film for different values of eccentricity ratio E. It is observed that an increase in Y results in an increase in o2 which means, other parameters being fixed, that the cavitation region is reduced by increasing the permeability. The same effect can also be obtained by keeping permeability fixed and increasing the wall thickness or decreasing the clearance. Also, the extent of the film decreases with increase of a for all values of Y. However, this effect is more pronounced at low values of Y. The effect of increasing the permeability (other parameters being fixed) is to reduce the pressure, shift the point of maximum pressure towards the start of the film (Fig. 3) and to reduce the load carrying capacity (Figs. 56). The decrease in pressure and load capacity is, however, more for large eccentricity ratios (Figs. 4-6). It is also observed that the porous bearing behaves similarly to a solid bearing for 12Y x 0.001 and the effect of porosity is not significant up to 12Y z 0.05. After this, there is a rapid decrease in the load capacity (Fig. 5). The curves corresponding to Sommerfeld solution’ are also drawn in Figs. 5 and 6. It is seen that the difference is quite significant for all values of permeability parameter, Y. However, in the limit E-+ 0, both the solutions tend to be identical. One more interesting feature is noteworthy as E--P1, the load capacity for the porous bearings tends to a finite value whereas for the non-porous bearings, it increases indefinitely. Figure 7 shows the variation of coefficient of friction with the permeability parameter Y for various values of E.It indicates that other factors being equal, porous bearings generally have a higher coefficient of friction than solid bearings. However,
I
-REYNOLDS ----SOMMERFELD
Fig. 5. Dimensionless Wear, 22 (1972)
load, A us. permeability
parameter,
Y for various
values
of eccentricity
ratio,
E.
10
J. PRAKASH.
S. K. VIJ
1000.0 r -REYNOLDS -- __ SOMMERFELD
Fig. 6. Dimensionless
load, A us. eccentricity
ratio,
E for various
values
of permeability
parameter,
Y.
the increase in the coefficient of friction is more for large eccentricity ratios. Also plotted in this figure are the corresponding Sommerfeld curves. Again a significant deviation from the present results is indicated. The attitude angle C#Jalso changes appreciably with Y as shown in Fig. 8. The deviation is, however, more pronounced for large values of Y. For 12 Y x 0.001 there is no appreciable difference upto E= 0.8. The fact that A is finite when E= 1 explains why the attitude angle C$is not zero. It is found that C$approaches 90” asymptotically as Y increases indefinitely. This corresponds to the case of an infinitely porous bearing which, of course, can carry no load because all the lubricant will bleed away into the porous wall. Figure 9 illustrates the relationship between the coefficient of friction and the Sommerfeld number, S for an inlinitely long porous journal bearing. The curves corresponding to the solid bearing and the idealized Petroff bearing are also shown. It is to be noted that as the Sommerfeld number, S is reduced (for a fixed Y), the system suddenly becomes unstable and the friction rises rapidly. This leads to the concept of a critical Sommerfeld number, S, below which hydrodynamic lubrication is not possible. In addition to the critical value (S,), the curves also show a minimum value (S,). In bearing design, this is of greater importance as it can readily be calculated. The same phenomena was also noted by Morgan’ for the case of infinitely short Wear, 22 (1972)
AXIALLY UNDEFINED
POROUS
JOURNAL
BEARINGS
11
3.5 --------
REYNOLDS SOMMERFELO
3.0 -
2.5 -
2.0 -
0.5 -
DO_ 0.001
I 0.1
a01 ‘2Y
Fig. 7. Coefficient of friction, (R/C')fus. permeability parameter, !I’for various values ofeccentricity ratio, E.
Fig. 8. Attitude angle, 4 0s. eccentricity ratio, E for various values of permeability parameter, y, Wear, 22 (1972)
12
J. PRAKASH,
S. K. VIJ
bearings. It is also observed that as the value of Y decreases, the curves tend towards that of a non-porous bearing. In Fig. 10, S, and S, are plotted against Y. The regions of hydrodynamic and boundary lubrication are indicated. It is found that the Sommerfeld number corresponding to the minimum friction increases with Y.
!p .I
-.. 01
.a
/
0.01
1.0
0.1
1 .O
S i l,‘,,
Fig. 9. Coeffkient
of friction, (R/C)f
US.Sommerfeld
number,
S for various values of permeability
parameter.
LO
Fig, 10. Sommerfeld number corresponding number, S, us. permeability parameter, $. Wear, 22 (1972)
to minimum
coefficient
of friction,
SM and critical Sommerfeld
AXIALLY
UNDEFINED
POROUS
JOURNAL
13
BEARINGS
APPENDIX
The eqn. (4) on integration gives y
P’
[I+A2]
(21)
where (cos B+A,)dB z = t
(If&
B’I&+D’I,/G
+
A’ = {(l+c+EZ)f
cos ey+c?
tan_,
2{2(F+&)}+
+
W&W&
&tan;
+ JG
cot;
-
{2(&%F))+
jEtan;
+ JG
cot;
+ {2(--F))’
,n
4(2(&-F)}+
and the expressions for A’, B’ and D’ are as given in eqn. (17). A, and A, are constants of integration. The indefinite integrals for load components are obtained as W=~ WJLRZ x C2
-2Esin
B(I+A,)+
c L
(1+s+a)‘(Esin4:
+ 2Fsin*~cos2~
+ Gcos4i)
+ gin G(~+Ecos~+~)~
~+ECOS~-LY
(24
and
+
-
A’(+
+ JG) - s@‘I& {2(F+m)}*
- D/,/G)
tan _ 1
G/G - JE)+ WI@ +D’I,/G)ln 4(2(*-F)}+
JE
tan i + ,,/G cot:
JE
tan i + JG
-
{2(@-F))f (23)
cot p + {2(@-F)}f
Equations (21) to (23) will facilitate the extension of the present work to partial arc-type bearings. REFERENCES 1 V. T. Morgan and A. Cameron, Mechanism of lubrication in porous metal bearings, Lubrication and Wear, London 1957, Inst. Mech. Engrs., Paper 89 (1957) 151-157. Wear, 22 (1972)
Proc. Conf
on
14
J. PRAKSH, S. K. VIJ
2 F. W. Ocvirk, Short bearing approximation for full journal bearings, NACA TN 2808, 1952. 3 A. Cameron, V. T. Morgan and A. E. Stainsby, Critical conditions for hydrodynamic lubrication of porous metal bearings, Proc. Inst. Mech. Engrs., 176 (1962) 761-770. 4 W. T. Rouleay Hydrodynamic lubrication of narrow press-fitted porous metal bearings, Trans. ASME, .I. Basic Eng., 85 (1963) 123-128. 5 V. T. Morgan, Hydrodynamic porous metal bearings, Lubrication Eng., 20 (1964) 448-455. 6 V. T. Morgan, Closure to discussion on porous metal bearings, Lubrication Eng., 21 (1965) 33. 7 C. C. Sbir and D. D. Joseph, Lubrication of porous bearings-Reynold’s Solution, Trans. ASME, J. Appl. Mech., 33 (1966) 761-763. 8 E. Capone, Lubrication of axially undefined porous bearings, Wear, 15 (1970) 157-170. 9 L. Floberg, Boundary conditions of cavitation regions in journal bearings, ASLE Trans., 4 (1961)282-286. Wear, 22 (1972)