Strojniški vestnik – Journal of Mechanical E [619169]

Strojniški vestnik – Journal of Mechanical E
ngineering 55(2009)2, 141-147
Paper received: 09.01.2009
UDC 621.8 Special issue Paper accepted: 12.03.2009
*
Corr. Author's Address: University of Salerno, Department of Mechanical Engineering, via Ponte don Melillo,
84084 Fisciano (SA), Italy, [anonimizat]
141
Effects of Couple Stresses on the Unsteady Performance
of Finite Lubricated Bearings
Adolfo Senatore
1,*
– Alessandro Ruggiero
1
– Vladeta Jevremovi
ü
2
– Valentin Nedeff
3
1
Department of Mechanical Engineering, University of Salerno, Italy
2
High Technical Mechanical School, Trstenik, Serbia
3
Faculty of Engineering, Un
iversity of Bacau, Romania
Based upon the Stokes micro-continuum theory, the
influence of the behaviour of journal bearings
with couple stress fluids on the dynamics of rotor-systems havs been studied by several Authors during
the last decade.
This paper is a part of a general approach
aimed at examining the pe
rformance of the couple
stress lubricants used to minimize the friction losses in steady operating conditions. Its purpose is to
illustrate a method of fo
rmulating the steady/unsteady fluid film
forces for the “infinitely long” and
“finite” lubricated couple stress journa
l bearings with clo
sed-form solutions,
assuming the micro-
continuum Stokes model. Th
e model allows the advantage of mini
mising the computational time required
for the analysis dynamic states of couple stress jour
nal bearings without any significant loss of accuracy,
while the analytical form of
the solution involves a better read
ability of the parameter effects on the
system unsteady behaviour.
© 2009 Journal of Mechanical En
gineering. All rights reserved.
Keywords: journal bearings, frictio
n, lubricants, stability of lubricants
0 INTRODUCTION
Hydrodynamic journal bearings are
commonly used for supporting rotating shafts
subjected to high radial
loads. Applications can
be seen in a wide variety of machines where
satisfactory performances are necessary for
proper functioning, such as pumps, turbines,
compressors, etc. The design of the journal
bearings focuses the first analysis on the static
characteristics such as hydrodynamic film force,
load-carrying capacity and friction coefficient.
Under certain external unexpected disturbances
the bearing system involves self excited
oscillating behaviors. The problem of oil film
instability is of primary
importance in high
speed rotating machines.
It is well known that
the dynamic performance of a rotor on
lubricated bearings, in fact, is strongly affected
by the fluid film
characteristics. The instability
occurs when the speed exceed a certain value
and appears as self excited orbital motions
induced by action of fluid dynamic forces. The
fluid film forces rise up directly by the gap oil
film pressure field wh
ich is essentially related to
the lubricant viscos
ity of the used lubricant.
It is well known that
the additives are
typically added to petroleum oils to modify the
physical properties such as pour point, foaming
or viscosity–temperature behaviour, chemical
actions such as detergency, oxidation, or
corrosion and to improve wear and extreme
pressure resistance.
With reference to long-chain organic
compounds additives, e.g., the length of the
polymer chain may be a million times the
diameter of a water molecule, the experimental
studies showed good load enhancement and
friction reduction effects due to their presence
[1] to [3].
The increasing use of complex fluids as
lubricants has received widespread interests
owing to the development of modern machine
elements. Common complex fluids are polymer-
thickened oils, lubrican
ts with various additives,
synthetic fluids, liquid crystals and bio-fluids.
Experimental investigatio
ns have also shown
that the use of complex fluids can decrease the
sensitivity to shear rate change, improving the
stabilization of lubricating properties.
According to the observation in strip squeeze
film flow, polymer thickened oil gives
significant load enhancement as compared to a
Newtonian one under similar conditions [3].
In the first work about the short journal
bearing by Oliver [4], the presence of dissolved

Strojniški vestnik – Journal of Mechanical
Engineering 55(2009)
2, 141-147
Senatore, A. – Ruggiero, A.
– Jevremovic, V. – Nedeff, V.
142
polymer in the lubricant produces load
enhancement and friction reduction.
Since the classical continuum theory
neglects the fluid particles size, this approach is
not suitable for describing the rheological
behaviour of these kinds of non-Newtonian
complex fluids. However, the micro-continuum
theory takes into account the intrinsic motion of
material constituents; it is
developed by polar
theory of complex fluids characterized by
classical Cauchy stresses as well as by couple
stresses resulting from the spin of
microelements in fluids
(Ariman and Sylvester
[5], and Stokes [6]).
In particular, the Stokes micro-
continuum theory [6] is a generalization of
conventional theory which allows the study of
the polar effects such as the presence of couple
stresses, body couples and non-symmetric
tensors, involving the ro
tational velocity field
with the dimensional effect of the particles.
Such an approach can be found for other
applications, as in th
e following list: peristalsis
mechanisms by Srivastava [7] and Shehawey
and Mekheimer [8]; line contacts by Das [9];
rolling elements by Sinha and Singh [10] as
well as by Bujurke and Naduvinami [11];
externally pressurized bearings by Lin [12];
squeezing films by Bujurke and Jayaraman [13]
and Lin [14]; slider bearings by Ramaniah [15];
finite bearings by Lin [16] and Chiang et al.
[17]; short journal bearings by Naduvinamani et
al. [18], Chiang et al. [19], Ruggiero and
Senatore [20].
Focusing on the literature about finite
bearings with couple stress fluid, both the
papers [16] and [17] study the performance of
these tribological components considered in
steady state conditions thro
ugh implementation
of finite difference schemes; the latter also takes
into account the surface roughness effect.
However, an approximate closed form
analysis for the finite journal bearings
considering the gap cavitation zone due to the
unsteady operating c
onditions was not known.
Therefore, it has inspired
further interests toward
the journal bearings with couple stress fluids.
In this paper, the inference of couple
stress fluid property on the film forces in
unsteady operating conditions of infinitely long
and finite journal be
arings is investigated.
The unsteady Reynolds equation
governing the film pressure is achieved through
the Stokes equations of
motion for accounting
the couple stress effects resulting from the flow
behaviour of non-Newtonian complex fluids.
Based on the product function approach, the
pressure solution for the
infinitely long bearing
in an approximate closed-form description has
been extended to Fi
nite Journal Bearing (FJB)
configuration. The outco
mes allow the
availability of analytical expr
essions for very
fast assessment on large motion unsteady
behaviour of shafts rotating on oil bearings with
couple stress fluids.
1 OIL FILM PRESSURE MODEL
The system here analysed consists of a
rigid, symmetric and balanced rotor supported
by equal cylindrical bearings. Symmetry about
the rotor middle plane allows limiting the
analysis to one of the two halves into which the
system is subdivided by
the above mentioned
plane.
Based upon the classical conception of
hydrodynamics, the Stok
es model allows for the
inspection of polar effects such as the presence
of couple stresses, body couples and non-
symmetric tensor. This couple stress fluid is a
peculiar case of a non-Newtonian lubricant and
takes account of particle-size effects of the
blending additives with a large molecule [6].
Isothermal conditions will be assumed to prevail
throughout the present investigation. Couple
stresses might be expected to appear in
noticeable magnitudes in liquids containing
additives with a large molecule.
These couple stresses may be significant
particularly under lubr
ication conditions where
thin films usually exist. Couple stresses
introduce non-linear terms
in the relationship
between shear stresses and velocity gradients.
As a result the lubricant should be considered as
non-Newtonian and it’s characterized by two
constants, the shear viscosity

and the couple
stress property.
The continuity and momentum equations
governing the motion of an incompressible
coupled stress fluid under the Stokes’
assumptions are [6]:

Strojniški vestnik – Journal of Mechanical
Engineering 55(2009)
2, 141-147
Effects of Couple Stresses on the Unsteady Performance of Finite Lubricated Bearings
143
V
V
C
F
V
4
2
Ș
ȝ
ȡ
2
1
ȡ
p
t
D
D
ȡ
’

’

u
’


’

(1)
0

˜
’
V
.
(2)
Where the vectors V, F and C represent
the velocity, the body fo
rce per unit mass, and
body couple per unit mass, respectively;

U
is
the density,
p
is the pressure,
μ
is the shear
viscosity and
ҏ
K
is a material constant
responsible for the couple stress fluid property;
the following assumptions have been made: thin
fluid film, body forces and body moments are
absent and fluid inertia is small as compared to
the viscous shear.
Then the field equations governing the
motion of the lubricant given in cartesian
coordinates reduce to:
4
4
2
2
y
u
Ș
y
u
ȝ
x
p
w
w

w
w

w
w
(3)
0
y
p

w
w
(4)
4
4
2
2
y
w
Ș
y
w
ȝ
z
p
w
w

w
w

w
w
(5)
uvw
0.
xy z
www

www
(6)
The boundary conditions at the bearing
surface:
0
)
z
,
0
,
x
w(
)
z
,
0
,
x
v(
)
z
,
0
,
x
u(

(7.1)
uw
0
22
22
y0 y0
.
yy

ww

ww
(7.2)
While the boundary conditions at journal
surface are described by:
U
)
z
,
h
,
x
u(

(8.1)
V
)
z
,
h
,
x
v(

(8.2)
0
)
z
,
h
,
x
w(

(8.3)
uw
0
22
22
yh yh
.
yy

ww

ww
(8.4)
Integrating the (3) and (5) by applying the
above boundary condition
s, the velocity
components can be derived as:
»
»
»
»
¼
ș
«
«
«
«
¬
ă
¸
¸
¸
¸
¹
·
¨
¨
¨
¨
©
§
¸
¹
·
¨
©
§




w
w


"
"
"
2
h
cosh
2
h
y
2
cosh
1
2
h)
y
(
y
x
p
ȝ
2
1
h
y
U
u
2
»
»
»
»
¼
ș
«
«
«
«
¬
ă
¸
¸
¸
¸
¹
·
¨
¨
¨
¨
©
§
¸
¹
·
¨
©
§




w
w

"
"
"
2
h
cosh
2
h
y
2
cosh
1
2
h)
y
(
y
z
p
2
ȝ
1
w
2
where
"
is the characteristic length of
additives:
P
K

"
(11)
The measurement methods and procedures
for
"
have been proposed in [6]. However, the
available published data ju
st give its theoretical
value. In (11),
K
has the dimensions of
momentum. Integrating
the continuity equation
(6) with respect to
y
using the velocity
components
u
and
w
with boundary conditions
(7.1), (8.2) and (8.3), with reference to the
bearing in Fig. 1, the modified form of the
Reynolds equation can be derived [16]:

¸
¸
¹
·
¨
¨
©
§

T
w
w

¸
¹
·
¨
©
§
w
w
w
w

¸
¹
·
¨
©
§
T
w
w
T
w
w
V
2
h
R
U
ȝ
6
z
p
h;
g
z
p
;
h
g
R
1
2
"
"
(12)
with:

h
gh ; h h t a n h .
3
2
12 2
2
§·
 
¨¸
¨¸
©¹
"" "
"
(13)
The journal speeds are given as:
Ȧ
R
U

t
h
V
w
w

.
(14)
The governing model for the hydrodynamic
lubrication in the shaft-b
earing wedge is the
dimensionless form of (12):
(
9
)
(10)

Strojniški vestnik – Journal of Mechanical
Engineering 55(2009)
2, 141-147
Senatore, A. – Ruggiero, A.
– Jevremovic, V. – Nedeff, V.
144
P
e
O
W
x
y
r
t
M
T
L
z
D
Fig. 1.
Finite journal bearing scheme

t
h
2
h
z
p
IJ
h;
g
z
p
IJ
h;
g
w
w

T
w
w

¸
¹
·
¨
©
§
w
w
w
w

¸
¹
·
¨
©
§
T
w
w
T
w
w
(15)
where:

¸
¹
·
¨
©
§



IJ
2
h
tanh
IJ
2
h
IJ
12
h
IJ
h;
g
2
3
.
(16)
In (15) and (16) these dimensionless
variables have been introduced:
ref
0
2
ref
p
p
p
p
C
R
6
ȝ
p


¸
¹
·
¨
©
§

Z
ș
cos
İ
1
C
)
(
h
)
h(


T

T
C
IJ
"

L
z
z

(17)
For the journal b
earings with couple stress
fluids, analytical solu
tions of the Reynolds
equation are not general
ly achievable and
numerical methods must be involved; this is the
case of the ‘infinitely long’ bearing, with the
partial differential equation (15) reduced as
follows, for which an exact solution can’t be
obtained:

ș
cos
İ
2
2
1
h
p
)
;
h
(
g



M

T
w
w

»
¼
ș
«
¬
ă
T
w
w
W
T
w
w
(18)
In this equation, the
function for accounting
the couple stress effect:

¸
¹
·
¨
©
§



IJ
2
h
tanh
IJ
2
h
IJ
12
h
IJ
h;
g
2
3
(19)
replaced by:

3
3
)
IJİ
İ
cos
ș
1
(
)
IJİ
)
(
h
(
h;
IJ
g
~




T

(20)
allows a closed-form in
tegration of an
approximation of (18), while numerical
calculations show that (20) gives good results
within the normal operating
conditions of this
tribological pair [20].
In this way, the differential equation for the
infinitely long bearing can be integrated to give
the following expressions:

I
c
I
2
–
I
)
2
1
(
;
,
,
,
p
3
1
2
1
L

H
I


W
I
H
H
T




(21)
where:
T
W
T

³
d
)
;
h
(
g
~
)
(
h
I
1
(21)
T
W
T

³
d
)
;
h
(
g
~
sin
I
2
(22)
.
(;)
3
1
I
d
gh
T
W

³

The constant
c
1
is calculated by analysing
the pressure discontinuity at
T
=
S
and the
right/left limits.
Then, the unstead
y infinitely long bearing
approximate solution
can now be written as:

2
2
2
L
cos
1
2
4
2
sin
cos
2
2
2
1
;
,
,
,
p
T
H

HW

H
H

HW

W

H
H

T
T
H

HW

H
M


W
I
H
H
T




.
(23)

Strojniški vestnik – Journal of Mechanical
Engineering 55(2009)
2, 141-147
Effects of Couple Stresses on the Unsteady Performance of Finite Lubricated Bearings
145
2 PRODUCT FINITE JOURNAL BEARING
SOLUTION
The flow correction factor for modifying
the fluid film pressure fo
r the infinitely long
bearing with couple stress fluid (23) is analysed
through the characteristic scalar value
O
and the
end leakage ‘shape’ function
s
, given for the
bearing in unsteady
conditions, respectively, by
[21]:

T
W
T
»
¼
ș
«
¬
ă
T
W

W
M
H
H
O
³
³
S

D
D
S

D
D
d
p
;
h
g
~
d
d
dp
;
h
g
~
;
,
,
2
L
2
L


(24)

Cosh , , ;
,,,;, /
Cosh , , ;
L
2 z
D
s
zL D 1 .
L
D
OHHMW
HHMW
OHHMW
ăș
«»
¬¼

ăș
«»
¬¼



(25)
The following product solution produces
the complete unsteady
oil film pressure for
finite length configuration in approximate
closed form (not presented for the sake of
briefness); the following integrations on the
thrust domain provide the forces acting on the
journal, which are also analytical expressions:

,,,,; ,,,;
,,,;, /
FL
p
zp
sz LD
TH H I W T H H I W
HHMW




(26)

^`
/
/
,,;, /
cos
,,,,;, /
sin
r
t
12
F
12
f
LD
f
p
zL D d d z
DS
D
HHIW
T
TH H I W T
T


­½

®¾
¯¿


³³




(27)
The angle

D
which defines the fluid film
boundaries in (24) and (27) is evaluated by
using the following relationships, according to
[22]:

cos sin
sin cos
2
2
21 2 0
21 2 0 .
HH DH M D
HH DH M D
­
 
°
®
  t
°
¯


(28)
3 RESULTS
The following graphs show the
dimensionless oil film forces in the rotating
system frame for three aspect ratios (L/D = 0.5,
1, 2) where the ‘short’ and ‘infinitely long’
bearing solutions lack in accuracy and a finite
bearing solution is required.
Two typical coupl
e stress parameter
values (
W
) as well as the newtonian case (
W
= 0)
are plotted.
Figs. 2 and 3 depict the effects of the
couple stress parameter on the oil film forces
f
r
and
f
t
for different static equilibrium eccentricity
ratio
H
. Figs. 4 and 5 show the maps achieved
through the analytical knowledge of oil film
forces in unsteady o
il film response (journal
‘squeeze’).
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.0001
0.001
0.01
0.1
1

f
r
H
2
L/D

1
0. 5
0
W

0.1
0..
2
Fig. 2.
Oil film force f
r
in the rotating system frame:
steady operating conditions
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.001
0.01
0.1
1

f
t
H
2
L/D
1
0
.
5
0
W
0.1
0.2
Fig. 3.
Oil film force f
t
in the rotating system
frame: steady operating conditions

Strojniški vestnik – Journal of Mechanical
Engineering 55(2009)
2, 141-147
Senatore, A. – Ruggiero, A.
– Jevremovic, V. – Nedeff, V.
146
-0.2
-0.1
0
0.1
0.2
0.3
-2
-1
0
f
r
H

W
L/D=1
Fig. 4.
Oil film force f
r
in the rotating system
frame: unsteady operating conditions
-0.2
-0.1
0
0.1
0.2
0.3
1
1.5
H

L/D=1
W
f
t
Fig. 5.
Oil film force f
t
in the rotating system
frame: unsteady operating conditions
4 CONCLUSIONS
The present solution
scheme for the
evaluation of the oil film forces provides
satisfactory results in th
e unsteady oil film
forces computation for infinitely long and finite
journal bearings with couple stress fluids,
mainly for typical values of couple stress
parameter for which the approximation
introduced in (20) allows an enough accurated
evaluation of the integrals (22).
The proposed approach allows
generating approximated closed-form oil film
expressions as function of the eccentricity ratio,
eccentricity and attitude angle variation rates.
The knowledge of the analytical form of the
fluid film response is very useful to reduce
drastically the calculation time in the computer
simulations of flexible rotors supported by
several bearings with non-newtonian fluids.
The effects of couple stresses result in a
significant increase of th
e unsteady action of
fluid film forces for a give
n journal centre radial
speed.
The linearized stability analysis can be
seen as an effortless application of the present
work outcomes: in fact, th
e derivatives of the
unsteady expressions of th
e oil film forces lead
to the stable/unstable
onset values and stability
map for each aspect ratio.
5 NOMENCLATURE
C
Radial clearance
C
Body couple per unit mass
D=2R
Bearing diameter
F
Body force per unit mass
f
r
,
f
t
Dimensionless oil film force (rot.
system frame)
f
ref
=
p
ref
RL
Reference oil film force
g,
g
~
Dimensionless functions
C
h
h
/

Dimensionless oil film thickness
"
Characteristic additives length
L
Bearing length

ref
0
p
p
p
p
/


Dimensionless pressure

2
ref
C
/
R
6
p
PZ

Reference pressure
p
L
Infinitely long bearing film
pressure
p
F
Finite bearing film pressure
s
End leakage axial ‘shape’ function
t
Time
u, v, w
Velocity components (circum.,
radial, axial directions)
U
Journal rotational speed
V
Journal radial speed
V
Fluid Velocity
W
Journal load
z
Axial coordinate
D
Pressure field boundary angle
C
/
e

H
Eccentricity ratio
M
Attitude angle
K
Couple stress oil property
O
Characteristic scalar value
P
Oil shear viscosity
T
Circumferential coordinate

Strojniški vestnik – Journal of Mechanical
Engineering 55(2009)
2, 141-147
Effects of Couple Stresses on the Unsteady Performance of Finite Lubricated Bearings
147
W
Couple stress parameter
Z
Journal angular speed
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The behaviour of lubricants
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Effect of oil
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[3] Oliver, D.R., Shahidullah, M.
Load
enhancement effects by polymer-thickened
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Journal of
Non-Newtonian Fluid Mechanics 1983,
vol. 3, p. 93-102.
[4] Oliver, D.R.
Load enhancement effects due
to polymer thickening in a short model
journal bearings
. Journal of Non-
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185–196.
[5] Ariman, T., Sylvester, N.D.
Microcontinuum fluid mechanics, a review
,
Int. J. Eng. Sci. 1973, vo
l. 11, p. 905-930.
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Couple stresses in fluids
.
Phys. Fluids 1966; Vo
l. 9, p. 1709–1715.
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Peristaltic transport of
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Couple stresses in perista
ltic transport of
fluids
, J. Phys. D: Appl. Phys 1994, vol.
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Elastohydrodynamic lubrication
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act: couple stresses fluid
model
, STLE Tribology Transactions 1997,
vol. 40, p. 353-359.
[10] Sinha, P., Singh, C.
Couple stresses in the
lubrication of rolli
ng contact bearings
considering cavitation
, 1981, vol. 67, p.
85-91.
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The
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, Int. J. Mech. Sci. 1990,
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[12] Lin, J.R.
Static and dynamic
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circular step thrust bearings lubricated
with couple stress fluids
, Tribology
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