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Effect of bearing deformation on the characteristics of a slider bearing
273
Wear, 78 (1982) 273 - 278
EFFECT OF BEARING DEFORMATION OF A SLIDER BEARING
G. RAMANAIAH
ON THE CHARACTERISTICS
and A. SLJNDARAMMAL
Department of Mathematics, Perarignar Anna University of Technology, of Technology Campus, Chromepet, Madras 600044 (India)
Madras Institute
(Received February 25, 1981; in revised form October 6, 1981)
Summary of the film An integral equation is presented for the determination thickness in a slider bearing with a deformable surface. First-order corrections to the load capacity and the frictional force due to bearing deformation are derived. A numerical solution is presented for a step slider bearing.
1. Introduction Generally, bearing surfaces are assumed to be rigid. However, under heavy loads the bearing surfaces may deform resulting in a variation in the bearing film thickness which may alter the load capacity and frictional force. An attempt to study the effect of bearing deformation was made by Osterle and Saibel [l] but no further work seems to have been done. In the present paper an integral equation is obtained for the film thickness when the bearing surfaces of a slider bearing are deformed. First-order corrections to the load capacity and frictional force due to bearing deformation are presented. A numerical solution is obtained for a step slider bearing. 2. Analysis The slider bearing with film thickness h(x) is shown in Fig. 1. The lubricant is assumed to be incompressible with a viscosity 1-1.The lower surface, which moves with a velocity ue, is assumed to be a semi-infinite elastic solid with a modulus of elasticity E. The upper surface, which is stationary, is assumed to be rigid. The deformed bearing is shown in Fig. 2. With the usual assumptions of hydrodynamic lubrication [ 21, the expression for the pressure distribution p(x) is P(X) = 6~40
“h-h* SF
dx
(1)
0
0043-1648/82/0000-0000/$02.75
0 Elsevier Sequoia/Printed in The Netherlands
Fig. 2. The deformed
Fig. 1. Slider bearing configuration.
bearing.
where
(2) is the film thickness at the point where the pressure is at a maximum and b is the breadth of the bearing. The bearing is assumed to be infinitely long (side leakage is neglected). The pressure distribution, in turn, deforms the bearing surface, altering the film thickness h(x). The film thickness h(x) can be obtained [3] as h(x) = h,(x) + $
/p(s) 0
d.~
In -..-L I S---X
I
where h,(x) is the film thickness in the undeformed state. Combining eqns. (1) and (3), we obtain the integral equation (4) For a given value of the undeformed film thickness h&x) we have to solve eqn. (4) to obtain h(x), from which we can obtain the load capacity w and frictional force f per unit length of the bearing: b
W=
s
p
dx
=
6~0
0
bh*-h
s ~ hs 0
xdx
(5)
and
f==w,
b4h -3h*
s 0
h2
dx
(6)
275
For convenience, the following non-dimensional quantities are introduced: X = x/b
Wh12 .=fh,
w=
6puob2 a=
H* = h*/hl
Ho = ho/h1
H = h/h,
He = hclhl (7)
6tiuob
12puob2 nEh12
where (Yis the elastic parameter. In terms of these quantities, eqns. (2) and (4) - (6) take the forms
(8) H(X)
= H,,(X)
+ a! jo
1
(9)
‘H*-H “=i
H3
(10)
xdx
and F=
$
‘4H-3H* s 0
dx
H2
(11)
The integral equation (eqn. (9)) is non-linear and too complex for an analytical solution for a given Ho. However, it may be solved by an iteration method wherein it is assumed that H = Ho(X) in the integrals of eqns. (8) and (9) to obtain H(X) which in turn can be used in these integrals to obtain a new H(X) etc. Once H(X) is determined, eqns. (10) and (11) give the dimensionless load capacity W and frictional force F. By the above procedure we can obtain the following first-order corrections due to the elastic parameter 01:
H,* = H* __H,* =
13Ho*--2Ho
s
0
w,=
w--w,=
’ (2Ho _f
0
H,dX
l 1 l-dX / 0 Ho3
HO4
3Ho*)Hc Ho4
+ H,* Ho
XdX
(13)
(14)
276
F,
_ F _
F,
’ (WHO* -
= 1
JHo)%
s
-
3H&*
dx
(15)
Ho3
60
wherein Ho*, Wo, F, are the values of H*, W, F for a rigid bearing surface with a film thickness H = Ho (X). H* may be considered as the dimensionless flow rate since it is related to the volume flow rate q per unit length by the equation H*=
% uohl
To illustrate
the procedure,
H,(X)=
A = 1
(16) we consider
the step slider bearing:
for 0 < X < B (17) forB<
X<
1
where A is the step height ratio and B is the step location. 3. Conclusions Plots of the dimensionless force F and friction coefficient
flow rate H’, load capacity W, frictional C = F/W are shown in Figs. 3 - 6 for various
0 01
I5
Fig, 3. Plots of dimensionless 0.1.
,---y_-_____ 20
25
n
30
31
flow rate H* vs. step height ratio A. .~~~~~;~~~
01 =
Fig. 4. Plots of dimensionless Fig. 3.
load capacity
W vs. step height ratio A : symbols
as for
‘0
277
01 15
I
20
25
A
I
I
30
35
‘0
Fig. 5. Plots of dimensionless
friction
F us. step height ratio A : symbols as for Fig. 3.
Fig. 6. Plots of dimensionless
friction
coefficient
C us. step height ratio A : symbols
as for
Fig. 3.
values of the step height ratio A and step location B values of 0.25, 0.5 and 0.75. Deformation increases the flow rate and decreases the load capacity. The frictional force decreases because of deformation but the more important parameter, the friction coefficient, increases with the elastic parameter (Y. Although numerical results are presented for a step slider, the analysis may be used to obtain corrections for the bearing characteristics of any type of slider bearing,
Nomenclature A b
B ; F h h*
ho
hl
step height ratio in the step slider breadth of the bearing riser location in the step slider modulus of elasticity of the bearing frictional force dimensionless frictional force defined in eqn. (7) film thickness film thickness at the maximum pressure film thickness for a rigid bearing minimum film thickness
278 h, H H* HO Ho* HC HC* P 9 s S uo ; Wo WC X,Y X a! IJ
correction to ho dimensionless film thickness h*lhl holhl value of H* for a rigid bearing h,lhl H* -Ho* pressure volume flow rate per unit length dummy variable in eqn. (4) dummy variable in eqn. (9) velocity of the bearing surface load capacity per unit length dimensionless load capacity defined in eqn. (7) dimensionless load capacity for the rigid surface correction to WO Cartesian coordinates xl5 elastic parameter defined in eqn. (7) coefficient of viscosity
References 1 F. Osterle and E. Saibel, ASLE Trans., I (1958) 213. 2 A. Cameron, Basic Lubrication Theory, Ellis Horwood, Chichester, 1976. 3 S. Timoshenko, Strength of Materials, Part II, Van Nostrand, New York, 1940.
Wear, 78 (1982) 273 - 278
EFFECT OF BEARING DEFORMATION OF A SLIDER BEARING
G. RAMANAIAH
ON THE CHARACTERISTICS
and A. SLJNDARAMMAL
Department of Mathematics, Perarignar Anna University of Technology, of Technology Campus, Chromepet, Madras 600044 (India)
Madras Institute
(Received February 25, 1981; in revised form October 6, 1981)
Summary of the film An integral equation is presented for the determination thickness in a slider bearing with a deformable surface. First-order corrections to the load capacity and the frictional force due to bearing deformation are derived. A numerical solution is presented for a step slider bearing.
1. Introduction Generally, bearing surfaces are assumed to be rigid. However, under heavy loads the bearing surfaces may deform resulting in a variation in the bearing film thickness which may alter the load capacity and frictional force. An attempt to study the effect of bearing deformation was made by Osterle and Saibel [l] but no further work seems to have been done. In the present paper an integral equation is obtained for the film thickness when the bearing surfaces of a slider bearing are deformed. First-order corrections to the load capacity and frictional force due to bearing deformation are presented. A numerical solution is obtained for a step slider bearing. 2. Analysis The slider bearing with film thickness h(x) is shown in Fig. 1. The lubricant is assumed to be incompressible with a viscosity 1-1.The lower surface, which moves with a velocity ue, is assumed to be a semi-infinite elastic solid with a modulus of elasticity E. The upper surface, which is stationary, is assumed to be rigid. The deformed bearing is shown in Fig. 2. With the usual assumptions of hydrodynamic lubrication [ 21, the expression for the pressure distribution p(x) is P(X) = 6~40
“h-h* SF
dx
(1)
0
0043-1648/82/0000-0000/$02.75
0 Elsevier Sequoia/Printed in The Netherlands
Fig. 2. The deformed
Fig. 1. Slider bearing configuration.
bearing.
where
(2) is the film thickness at the point where the pressure is at a maximum and b is the breadth of the bearing. The bearing is assumed to be infinitely long (side leakage is neglected). The pressure distribution, in turn, deforms the bearing surface, altering the film thickness h(x). The film thickness h(x) can be obtained [3] as h(x) = h,(x) + $
/p(s) 0
d.~
In -..-L I S---X
I
where h,(x) is the film thickness in the undeformed state. Combining eqns. (1) and (3), we obtain the integral equation (4) For a given value of the undeformed film thickness h&x) we have to solve eqn. (4) to obtain h(x), from which we can obtain the load capacity w and frictional force f per unit length of the bearing: b
W=
s
p
dx
=
6~0
0
bh*-h
s ~ hs 0
xdx
(5)
and
f==w,
b4h -3h*
s 0
h2
dx
(6)
275
For convenience, the following non-dimensional quantities are introduced: X = x/b
Wh12 .=fh,
w=
6puob2 a=
H* = h*/hl
Ho = ho/h1
H = h/h,
He = hclhl (7)
6tiuob
12puob2 nEh12
where (Yis the elastic parameter. In terms of these quantities, eqns. (2) and (4) - (6) take the forms
(8) H(X)
= H,,(X)
+ a! jo
1
(9)
‘H*-H “=i
H3
(10)
xdx
and F=
$
‘4H-3H* s 0
dx
H2
(11)
The integral equation (eqn. (9)) is non-linear and too complex for an analytical solution for a given Ho. However, it may be solved by an iteration method wherein it is assumed that H = Ho(X) in the integrals of eqns. (8) and (9) to obtain H(X) which in turn can be used in these integrals to obtain a new H(X) etc. Once H(X) is determined, eqns. (10) and (11) give the dimensionless load capacity W and frictional force F. By the above procedure we can obtain the following first-order corrections due to the elastic parameter 01:
H,* = H* __H,* =
13Ho*--2Ho
s
0
w,=
w--w,=
’ (2Ho _f
0
H,dX
l 1 l-dX / 0 Ho3
HO4
3Ho*)Hc Ho4
+ H,* Ho
XdX
(13)
(14)
276
F,
_ F _
F,
’ (WHO* -
= 1
JHo)%
s
-
3H&*
dx
(15)
Ho3
60
wherein Ho*, Wo, F, are the values of H*, W, F for a rigid bearing surface with a film thickness H = Ho (X). H* may be considered as the dimensionless flow rate since it is related to the volume flow rate q per unit length by the equation H*=
% uohl
To illustrate
the procedure,
H,(X)=
A = 1
(16) we consider
the step slider bearing:
for 0 < X < B (17) forB<
X<
1
where A is the step height ratio and B is the step location. 3. Conclusions Plots of the dimensionless force F and friction coefficient
flow rate H’, load capacity W, frictional C = F/W are shown in Figs. 3 - 6 for various
0 01
I5
Fig, 3. Plots of dimensionless 0.1.
,---y_-_____ 20
25
n
30
31
flow rate H* vs. step height ratio A. .~~~~~;~~~
01 =
Fig. 4. Plots of dimensionless Fig. 3.
load capacity
W vs. step height ratio A : symbols
as for
‘0
277
01 15
I
20
25
A
I
I
30
35
‘0
Fig. 5. Plots of dimensionless
friction
F us. step height ratio A : symbols as for Fig. 3.
Fig. 6. Plots of dimensionless
friction
coefficient
C us. step height ratio A : symbols
as for
Fig. 3.
values of the step height ratio A and step location B values of 0.25, 0.5 and 0.75. Deformation increases the flow rate and decreases the load capacity. The frictional force decreases because of deformation but the more important parameter, the friction coefficient, increases with the elastic parameter (Y. Although numerical results are presented for a step slider, the analysis may be used to obtain corrections for the bearing characteristics of any type of slider bearing,
Nomenclature A b
B ; F h h*
ho
hl
step height ratio in the step slider breadth of the bearing riser location in the step slider modulus of elasticity of the bearing frictional force dimensionless frictional force defined in eqn. (7) film thickness film thickness at the maximum pressure film thickness for a rigid bearing minimum film thickness
278 h, H H* HO Ho* HC HC* P 9 s S uo ; Wo WC X,Y X a! IJ
correction to ho dimensionless film thickness h*lhl holhl value of H* for a rigid bearing h,lhl H* -Ho* pressure volume flow rate per unit length dummy variable in eqn. (4) dummy variable in eqn. (9) velocity of the bearing surface load capacity per unit length dimensionless load capacity defined in eqn. (7) dimensionless load capacity for the rigid surface correction to WO Cartesian coordinates xl5 elastic parameter defined in eqn. (7) coefficient of viscosity
References 1 F. Osterle and E. Saibel, ASLE Trans., I (1958) 213. 2 A. Cameron, Basic Lubrication Theory, Ellis Horwood, Chichester, 1976. 3 S. Timoshenko, Strength of Materials, Part II, Van Nostrand, New York, 1940.