ARTICLE IN PRESS
Tribology International 41 (2008) 461–472
www.elsevier.com/locate/triboint
A study of friction in worn misaligned journal bearings
under severe hydrodynamic lubrication
Padelis G. Nikolakopoulos, Chris A. Papadopoulos
Machine Design Laboratory, Mechanical Engineering Department and Aeronautics, University of Patras, Patras 26504, Greece
Received 19 April 2006; received in revised form 4 October 2007; accepted 8 October 2007
Available online 26 November 2007
Abstract
Friction occurs in all mechanical systems such as transmissions, valves, piston rings, bearings, machines, etc. It is well known that in
journal bearings, friction occurs in all lubrication regimes. However, shaft misalignment in rotating systems is one of the most common
causes of wear. In this work, the bearing is assumed to operate in the hydrodynamic region, at high eccentricities, wear depths, and
angular misalignment. As a result, the minimum film thickness is 5–10 times the surface finish, i.e., near the lower limit of the
hydrodynamic lubrication when taking into account that in the latest technology CNC machines the bearing surface finish could be less
than 1–2 mm.
An analytical model is developed in order to find the relationship among the friction force, the misalignment angles, and wear depth.
The Reynolds equation is solved numerically; the friction force is calculated in the equilibrium position. The friction coefficient is
presented versus the misalignment angles and wear depths for different Sommerfeld numbers, thus creating friction functions dependent
on misalignment and wear of the bearing. The variation in power loss of the rotor bearing system is also investigated and presented as a
function of wear depth and misalignment angles.
r 2007 Elsevier Ltd. All rights reserved.
Keywords: Journal bearings; Misalignment; Friction models; Wear
1. Introduction
Most of the moving components of mechanical systems
are lubricated with liquid or solid lubricants. The main
purpose of lubrication is to minimize the friction and
reduce the wear of the mating parts. Friction modeling is
important for all stages of the life cycle of a machine and
machine elements. During machine design, accurate friction simulation allows for performance prediction and
optimization of alternative solutions of mechanical design
materials and lubricants that affect the life cycle of the
machine element. Every machine consists of many machine
elements having moving components which are necessary
for the machine operation. The relative motion causes wear
and friction which could lead to the breakdown of the
machine. The energy loss derived from wear and friction
Corresponding author.
E-mail address: [email protected] (C.A. Papadopoulos).
0301-679X/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.triboint.2007.10.005
has significant influence on the reduction of machine
efficiency.
Operation of the shaft in the bearing housing over a long
period of time is a cause of wear, and low rotational speed,
high load and misalignment are some of the most
important factors that produce wear. Significant vibrations
are a type of motion outside the given tolerance. Many
designers, like Xie [1], indicate that this problem could be
due to bad tribological design. Tribological design must be
understood as the design of the whole tribological system
and not only of a single pair or individual machine
component. For example, unsuitable material selection of a
bearing pad in a rotating machine, during the detailed
design phase, could lead the tribological system to excessive
wear, due to harmful vibrations. As Bhushan [2] mentioned
in his book, economic losses that arise from wear and
friction was equivalent to 4% of the Gross National
Product (GNP), in the developed countries. Thus a
considerable amount of the total energy produced in the
industry was dissipated by friction and wear. Therefore, for
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Nomenclature
A
area (m2)
c
radial clearance (m)
D
journal diameter (m)
d0
dimensionless wear depth
e0
eccentricity at bearing’s mid-plane (m)
ex
bearing eccentricity in x direction (m)
ey
bearing eccentricity in y direction (m)
f
friction coefficient
f̄ ¼ fR=c normalized friction coefficient
Ffr
friction force (N)
Fx, Fy hydrodynamic component forces (N)
Fhydr
hydrodynamic force (N)
Fext
external loads (N)
Hg
power loss (W)
h
film thickness (m)
hmin
minimum film thickness (m)
9
I ½K p >
=power functional
½K V fluidity matrices
>
L ½K U ;bearing length (m)
Mx, My hydrodynamic component moments (N m)
Mhydr hydrodynamic moment (N m)
^
m
outward normal vector along the boundary
N
revolutions per second (rps)
Ni, Nj shape functions
Ob
bearing geometric center
Oj
journal geometric center
P
fluid film pressure (Pa)
reliability and longevity of machines and reduction in
production cost, wear and friction must be eliminated
or controlled properly. In this direction, tribology and
modeling tribological phenomena play a major role in the
analysis of complicated problems and derivation of the
proper solutions. The tribological bearing behavior has
been examined by many investigators as a result of various
concerns. In this work, a small segment of recent papers
concerning bearing tribological behavior is presented.
1.1. Friction and wear in journal bearings
The friction and the wear in journal bearings attracted
the attention of many investigators due to its importance,
during the operation of the machines. Muskat and Morgan
[3] presented a general theory of flooded journal bearings
and bearings with a circumferential groove. It was found
that the journal eccentricities and friction coefficients, for
fixed Sommerfeld variable, are greater for bearings of finite
length than for that of infinite length, the difference
increasing with decreasing bearing length. On the other
hand the load-carrying capacities, for fixed eccentricity, of
the bearings of finite length are much less than that for the
infinitely long bearing. Greenwood and Tripp [4] gave a
lubricant shear flow vector (m3 s1)
the volume flow (m3 s1)
generalized coordinate
journal radius (m)
bearing radius (m)
Sommerfeld number
moment Sommerfeld number
fluid film boundary segment
fluid film boundary
moment of shear stress for element i (N m)
effective lubricant temperature (1C)
inlet lubricant temperature (1C)
velocity of journal surface parallel to the film
(m s1)
{U}
velocity vector (m s1)
Vi
velocity of element i (m)
x, y, z spatial coordinates (m)
{Q}
q
qi
Rj
Rb
S
Sm
Sq
s
Ti
Teff
Tin
U
Greek letters
d0
maximum wear depth (m)
e ¼ e0/c eccentricity ratio
y
circumferential bearing coordinate (rad)
Z, z:
axial bearing coordinates (m)
m
lubricant viscosity (Pa s)
j0
attitude angle (rad)
cx, cy misalignment angles in x and y direction (rad)
c̄x ¼ cx L=c normalized misalignment angle
c̄y ¼ cy L=c normalized misalignment angle
o
angular velocity (rad s1)
general theory of contact between two rough plane
surfaces. They concluded that any model of contact
between surfaces, both of which are assumed to be rough,
can be simulated by a model in which only one surface is
rough.
Dufrane et al. [5] investigated worn steam turbines and
measured them during overhaul periods to determine the
extent and nature of the wear. They established two models
of wear geometry for use in further analysis of the effect of
wear on hydrodynamic lubrication. These worn models,
used by many investigators, are not of circular type. The
first of the proposed models is based on the concept of
imprinting itself in the bearing, and the second one is based
on a hypothetical abrasive wear model with the worn arc at
a radius larger than the journal. They concluded that there
is an optimum film thickness, as wear progresses that may
explain the difference between bearings that wear in versus
those that wear out. If the initial wear results in a film
thickness sufficient to prevent further wear, the bearing
wears in. If the film thickness reaches his optimal value and
is not sufficient to prevent further wear then the bearing
wears out.
Using numerical simulations, Benson et al. [6] have
studied the dynamics of a ‘‘mini-Winchester’’ magnetic
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P.G. Nikolakopoulos, C.A. Papadopoulos / Tribology International 41 (2008) 461–472
recording slider during contact with a hard rotating disk.
An on-line solution of the Reynolds equation was used to
calculate the air-film pressure and a ‘coefficient-of-restitution’ model was used to describe intermittent slider/disk
contacts. Studies are made to identify system configurations which reduce the possibility of a ‘head crash’ during
contact start/stop.
The research reported in the Rachoor’s dissertation [7] is
concerned with the development of friction models for
lubricated contacts, for different dynamic velocity conditions including uni-directional and bi-directional oscillations. In Rachoor’s study, two different tribological
situations, such as conformal and nonconformal contacts,
have been chosen. de Marchi [8], in his dissertation deals
with reports on methods for identifying combinations of
friction, backlash, and compliance in mechanical drive
mechanisms.
In Xie’s work [1] some of the published results on
tribology design coming from the Theory of Lubrication
and Bearing Institute of Xi’an Jiatong University (TLB)
are collected and discussed. Results in the design of rotorbearing–sealing-gear systems, thrust-bearing systems, piston-rings–cylinder liner systems, active magnetic bearings
and a smart tribo-material are introduced. In this work,
Xie emphasizes the importance and the new possibilities
provided by the development of systems engineering of
tribo-systems. The contents, target and characteristics of
tribology design are discussed.
Computer simulations of mechanical systems with
friction are difficult because of the strongly nonlinear
behavior of the friction force near zero sliding velocity.
Tariku and Rogers [9] proposed two improved friction
models for stick–slip motions. One model is based on the
force-balance method and the other model uses a spring–
damper during sticking. The extensive test results, of their
experiments, show that the new force–balance algorithm
gives much lower sticking velocity errors compared to the
original method and that the new spring–damper algorithm
exhibits no spikes at the beginning of sticking.
The friction coefficient is important for the determination of wear loss conditions at journal bearings. Ünlü et al.
[10] determined the friction coefficient at CuSn10 Bronze
radial bearings, by a new approach as experimental and
artificial neural networks method. In experiments, effects
of bearings have been examined at dry and lubricated
conditions and at different loads and velocities.
1.2. Wear and misalignment in journal bearings
The role of the misalignment, in the dynamic behavior of
the journal bearings and particularly in the rate of the wear
progress has been examined in a number of works. The
methods of finite differences and finite elements are widely
used for the solution of the equation of Reynolds. Goenka
[11] described a finite element formulation with low
computational cost for transient analysis of journal
bearings. This formulation can be used for partial or
463
full-arc bearings with oil-supply hole, and oil-feed grooves,
with tapered or misaligned journal, and with elliptical or
eccentric bearings. An important feature of this analysis is
relatively low computing cost. Nikolakopoulos and Papadopoulos [12] presented an analysis of misaligned journal
bearing operating in linear and nonlinear regions. The
finite element method (FEM) was used to find the solution
of the Reynolds equation. After the solution was obtained,
they calculated the linear and nonlinear dynamic properties
for the misaligned bearing depending on the developed
forces and moments as functions of the displacements and
misalignment angles. The effects of misalignment on the
linear and nonlinear plain journal bearing characteristics
were analyzed and presented.
Bouyer and Fillon [13] present a study dealing with
experimental determination of the performance of a
100 mm diameter journal bearing with an applied misalignment torque. They found that the bearing performances were greatly affected by the misalignment. The
maximum pressure in the mid-plane was decreased by 20%
for the largest misalignment torque, while the minimum
film thickness was reduced by 80%. The misalignment
caused more significant changes in bearing performance
when the rotational speed or load was low. The hydrodynamic effects were then relatively small and the bearing
offered less resistance to the misalignment.
Liu et al. [14] presented a finite element model for
mixed lubrication of journal-bearing systems operating in
adverse conditions. The asperity effects on contact and
lubrication at large eccentricity ratios are modeled. In
the model system, the elastic deformation due to both
hydrodynamic and contact pressure and the cavitation
of the lubricant film are considered. Finally, the influence
of waviness depth, secondary roughness, and external
force and shaft speed on the mixed lubrication were
discussed.
Fillon and Bouyer [15] present a thermohydrodynamic
analysis of a worn plain journal bearing, in continuation of
the above presented work, done in 2002 [13]. They reported
that defects caused by wear of up to 20% have little
influence on bearing performance, whereas above this value
(30–50%) they can display an interesting advantage: A
significant fall in temperatures, due to the tendency of the
bearing to go into the footprint created by the wear. Thus,
they concluded that the worn bearings present not only
some disadvantages but also some advantages, such as
lower temperature, since in certain cases of significant
defects due to wear, the geometry approaches that of a lobe
bearing.
Pierre et al. [16] present in detail a three-dimensional
thermo-hydrodynamic approach to consider thermal
effects and also to take into account the lubricant film
rupture and reformation phenomena by conservation of
mass flow rate. An experimental validation was also carried
out by comparison with measurements extracted by their
experimental apparatus for various operating conditions
and misalignment torques.
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464
Boedo and Booker [17] investigated the transient and
steady-state behavior of grooveless (angularly) misaligned
bearings using finite element formulations of the completed
two-dimensional Reynolds equation. It was found that
misaligned bearings have infinite load and moment
capacity as the endplane minimum film thickness approaches zero, under transient journal squeeze motion
and under steady load and speed conditions. These
results differ markedly from finite capacity trends
reported previously in both numerical and experimental
studies.
Sun et al. [18] developed a special test bench for the
study of lubrication performance of cylindrical journal
bearings. The effects of journal misalignment as a result of
shaft bending under load were studied. They observed
changes at distribution and value of oil film pressure, oil
film thickness and oil temperature of journal bearing due to
journal misalignment.
1.3. The present work
In the present work, an analytical model is developed in
order to determine the relationship among the friction
force (and consequentially the coefficient of friction) the
misalignment angles, and wear depth in circular journal
bearings. The Reynolds equation is solved numerically,
using FEM in the hydrodynamic region, with the minimum
film thickness at 5–10 times that of the surface finish which
is near the lower limit of the hydrodynamic lubrication
when taking into account that in the latest technology
CNC machines, the surface finish is prepared with roughness less than 1–2 mm. The friction force is calculated in the
equilibrium position as the result of each friction force
arising from the hydrodynamic shear stresses in the fluid
lubricant film acting on each element of the bearing
surface. The friction coefficient is presented versus misalignment angles and wear depths for different Sommerfeld
numbers, creating functions of friction coefficient with
respect to wear and misalignment condition of the bearing.
The variation in power loss of the rotor bearing system is
also investigated and presented as a function of wear depth
and misalignment angles.
2. Bearing model formulation using FEM
In this paper, the bearing is considered to be rigid rather
than elastic because the misalignment and the wear are the
main targets here. The journal bearing is assumed to
operate in the steady-state situation. The flow is chosen to
be laminar and an isothermal regime is also assumed. The
wear is produced by misalignment forces. The geometry of
the worn bearing follows the model introduced by Dufrane
et al. [5]. The geometry is shown in Fig. 1, where, Ob is the
bearing center, Oj is the journal center, Rb is the bearing
radius, Rj is the journal radius, ex and ey are the bearing
eccentricities in the x and y directions, respectively, and L
is the bearing length. The external vertical load is
considered to be constant.
Fig. 1. (a) Geometry of bearing; (b) of misalignment angles; (c) of journal bearing; and (d) of the worn bearing.
ARTICLE IN PRESS
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465
2.1. Numerical model
the unknown (generally interior) nodal pressures subject to
the nonnegativity constraint:
As shown previously by Huebner [19] in the FEM
formulation, the true pressure distribution P in an
incompressible lubricant film minimizes the discretized
power functional I(P),
Z 3
qIðPÞ
q
h
¼
rP hU rP dA
qP
qP A 24m
)
Z
PX0.
QP ds
þ
¼ 0,
ð1Þ
Sq
where h is the film thickness, m the viscosity of the fluid, U
the velocity of the journal surface parallel to the film, A the
integration surface, Sq is the boundary segment, and s is the
fluid film boundary.
This differentiation leads to the following equation in
matrix form:
K p fPg ¼ q ½K U fU g ½K V fV g.
(2)
Here q is the volume flow, U is the journal surface
velocity, V is the squeeze velocity, and [Kp], [KU] and [KV]
are the element fluidity matrices such that:
Z
h3 qN i qN j qN i qN j
þ
K Pij ¼ dA,
12m qx qx
qz qz
Z A
qN i
N j dA,
h
K U ij ¼
qx
A
Z
qN i
K V ij ¼
N j dA,
h
ð3Þ
qz
A
Z
qi ¼
QN i ds,
(5)
2.2. Equilibrium position
The hydrodynamic forces and moments developed in the
bearing can be evaluated by integrating the fluid pressure
over the entire bearing area
Z L=2 Z y2
Pðy; zÞR cos y dy dz,
Fx ¼
Z
L=2
L=2
Z
y1
y2
Pðy; zÞR sin y dy dz,
Fy ¼
Z
L=2
Z
y2
Pðy; zÞzR sin y dy dz,
Mx ¼ Z
ð6Þ
y1
L=2
y1
L=2
L=2
Z
y2
Pðy; zÞzR cos y dy dz,
My ¼
L=2
ð7Þ
y1
where y1 and y2 mark the extend of the pressure field,
which depends on the boundary conditions applied. For a
given external loading condition, the static equilibrium
position of the journal center can be found by equating the
hydrodynamic forces with the externally applied load. A
two-dimensional Newton–Raphson search technique is
applied for the calculation of the equilibrium position of
the journal center, when the sum of the hydrodynamic
forces and the external loads is equal to zero:
X
ðF hydr þ F ext Þ ¼ 0.
(8)
Sq
h2
^
Q ¼ h Ū rP m,
12m
ð4Þ
^ is the outward normal vector
where Q is the flow vector, m
along the boundary, and Ni’s are shape functions given by
ðai þ bi y þ ci zÞ
,
2A0
where a1 ¼ y2z3y3z2, b1 ¼ z2z3, c1 ¼ y3y2, a2 ¼ y3z1
y1z3, b2 ¼ z3z1, c2 ¼ y1y3, a3 ¼ y1z2y2z1, b3 ¼ z1z2,
c3 ¼ y2y1 are the interpolation coefficients and A0 is the
determinant:
1 y1 z1 ZZ
1
1
A
A0 ¼ 1 y2 z2 ¼ dy dz ¼
2
Rb
1 y3 z 3 N i ðy; zÞ ¼
with A the area of the triangular element. The coordinate y
is defined from the negative x-axis, and the Reynolds
equations are developed in the coordinate system (x ¼ Rby,
z) as shown in Fig. 1. In the FEM analysis, the global
fluidity matrices and flow vector can be numerically
assembled from element matrices and vectors in the usual
manner. In its discretized form, the problem is solved by
minimizing the system functional of Eq. (1), with respect to
2.3. Hydrodynamic operational parameters
The basic hydrodynamic operational parameters which
are used in the present analysis are (a) the Sommerfeld
number S that is given by
mNDL R 2
S¼
(9)
F hydr c
and (b) the moment Sommerfeld number Sm described by
mNDL2 R 2
,
(10)
Sm ¼
M hydr c
where m is the oil viscosity, N the journal rotational speed
in rps, D the journal diameter, L is the bearing length,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F hydr ¼ F 2x þ F 2y , and M hydr ¼ M 2x þ M 2y . The geometric quantity L/D is also of great importance as a
geometric parameter of the journal bearing. The moment
Sommerfeld number is introduced when misalignment
appears. The moment components Mx and My increase
as the misalignment angles become greater. Consequentially, greater misalignment angles cx, cy imply a lower
moment Sommerfeld number.
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466
2.4. Geometry of worn bearing
The geometry of the worn bearing surface used in the
present analysis is the well-known model presented by
Dufrane et al. [5]. The film thickness is given by superposition of film thicknesses as is mentioned by Nikolakopoulos and Papadopoulos [12] and Dufrane et al. [5] for the
abrasive bearing wear model as it is depicted in Fig. 1d
hðy; zÞ ¼ c þ e0 cos y
h
i
þ z cy cosðy þ j0 Þ þ cx sinðy þ j0 Þ þ dh. ð11Þ
Here,
dh ¼ cðd 0 1 cos yÞ
(12)
with d0 ¼ d0/c the dimensionless wear depth (which
corresponds to the percentage of the wear depth in relation
to the radial clearance), and d0 is the maximum wear depth.
Eq. (12) describes the change in the film thickness due to
the bearing wear, and is applicable between the angles
relevant to the worn region, otherwise dh ¼ 0. The worn
zone is assumed to be centered on the vertical load
direction.
2.5. Friction model
The friction force Ffr can be calculated using the
following formula:
ZZ h qP mU
dA.
(13)
F fr ¼
2 qx
h
A
Here it is assumed that the friction force results only
from shearing of the fluid. The integration of the shear
stress over the entire bearing area A, yields the total friction
force, as described by Eq. (13), operating across the fluid
film. The total friction force is calculated in the equilibrium
position, as the result of each element’s friction force. In
Eq. (13) the sign (+) refers to journal friction and the ()
to that on the bearing.
Integrating the right-hand side of Eq. (13) over the
complete journal or bearing surface and dividing the results
by the external load, the friction coefficient is calculated by
the relation:
f ¼
F fr
.
F ext
(14)
In this paper, the friction coefficient is calculated in
reference to the journal. The two terms of Eq. (13) are
given in discretized form in [11], expressing the shear stress
in the fluid film. The shear stress in the fluid film for each
unit of bearing length in the bearing is the sum of the
product of one-half of the film thickness times the pressure
gradient and the product of viscosity times the ratio of
velocity to film thickness.
A finite element grid of size 50 9 is used in order to
obtain the solution including the geometry of the worn
bearing, while a 27 9 finite element grid can be used in the
case without wear. Fifty elements are used for the
circumferential dimension and nine elements are used in
the longitudinal bearing dimension. The grid is generated
with linear triangular elements with shape functions
defined in Section 2.1. Several checks were made to ensure
internal consistency of the FEM program, including the
bearing’s worn geometry. The 50 9 finite element grid is
the optimized proposed grid able to achieve an accurate
solution including the worn geometry. A two-dimensional
Newton–Raphson search method is applied in order to
calculate the journal equilibrium position. The calculated
hydrodynamic forces are compared with the applied
external loads, and equilibrium is achieved when the sum
of the external loads and the produced hydrodynamic
forces equals zero. In the equilibrium position, the
eccentricity e0 and the attitude angle j0 are the variables
in the Newton–Raphson method and are determined for a
given set of misalignment angles, cx, cy. As practical in the
algorithm, the horizontal and the vertical hydrodynamic
forces at the equilibrium are compared in each step with a
tolerance factor which is 0.001 N or less. If the vertical and
horizontal forces are equal to or less than the tolerance
factor then the equilibrium point is obtained. If the
comparison criterion is not fulfilled, the equilibrium point
is far away from the real actual point and wrong values of
the eccentricity and attitude angle could occur. As a result,
wrong bearing performances, in respect to load carrying
capacity, friction coefficient, or heat generation could be
calculated for the given bearing characteristics. In order to
avoid this, the equilibrium position has to be determined
with very high accuracy.
2.6. Power loss calculation
The hydrodynamic friction power, the rate of the work
done on the film and dissipated by it, is given by Goenka
[11] by the expression:
Hg ¼
Ne X
2
X
ðF hydr;i V i þ T i oi Þj ,
(15)
i¼1 j¼1
where i ¼ 1 to Ne (number of elements), j ¼ 1 for the
journal, and 2 for the bearing, Fhydr,i is the hydrodynamic
force acting on the film due to element i, Vi is the velocity
of element i, Ti is the moment of shear stress for element i,
and oi the angular velocity of element i. The discretization
procedure of Eq. (15) is also described in [11].
2.7. Minimum film thickness
Another useful parameter which indicates the bearing
lubrication regime is the minimum film thickness hmin,
which occurs at one of the two ends of the bearing.
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467
The minimum film thickness for misaligned journal
bearings is defined by
Table 1
Bearing characteristics
hmin jz¼L=2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ c ðe0 sinðj0 Þ cy L=2Þ2 þ ðe0 cosðj0 Þ þ cx L=2Þ2 ,
Parameter
Parameter’s value and value range
Journal diameter, D
Bearing lengths, L
Clearance, c
Rotational speed, o
Misalignment angles, cx ¼ cy
50.8 mm
50.8 or 25.4 mm
65 mm
400 rad/s
From 255.91 to 1279.53 mrad for
L/D ¼ 0.5
From 127.95 to 639.76 mrad for
L/D ¼ 1.0
From 0.1 to 0.5
ð16Þ
hmin jz¼L=2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ c ðe0 sinðj0 Þ cy L=2Þ2 þ ðe0 cosðj0 Þ cx L=2Þ2 .
ð17Þ
In Fig. 2, the variation of the minimum film thickness is
illustrated with respect to moment Sommerfeld number,
for wear depth d0 ¼ 0.5, and for two different Sommerfeld
numbers, S ¼ 0.15 and 0.0564, respectively.
It is easy to observe that the bearing in the case study
operates under severe lubrication conditions due to a small
minimum thickness which falls between 5 and 18 mm, while
taking into account that the dimensionless minimum film
thickness is between 0.025 and 0.25 and the radial clearance
is 65 mm, as mentioned in Table 1.
Dimensionless misalignment angles,
c̄x ¼ cx L=c; c̄y ¼ cy L=c
Wear depth, d0
External load variation, Fext
From 0.1 to 0.5
From 500 to 4000 N for L/D ¼ 1
(to create Figs. 9a–c)
2.8. Algorithm validation
In this section, two validation diagrams are presented
showing that the results of the present algorithm are in
good agreement with the results coming from the literature.
In Fig. 3, a comparison of the results from the present
wear model and those by Fillon and Bouyer [15] are
illustrated. The geometrical data used are: bearing diameter
100 mm, bearing length 100 mm, radial clearance 75 mm,
rotational speeds 1000 and 10,000 rpm, and external
loads 5000 and 10,000 N, as depicted in Fig. 3. Since in
the Ref. [15] a thermo-hydrodynamic analysis is considered
for a Newtonian lubricant (no grease, or additives in the
lubricant) and constant load, the lubricant viscosity is
Fig. 3. Eccentricity ratio as a function of relative wear depth.
calculated in an effective temperature as follows:
T eff ¼ T in þ
Fig. 2. The influence of the moment Sommerfeld number on the minimum
film thickness variation, for d0 ¼ 0.5.
DT
,
2
(18)
where Tin is the lubricant inlet temperature (40 1C) and DT
is the difference between inlet and outlet lubricant
temperature. Since the outlet temperature is 100 1C, the
effective temperature obtained is 70 1C and the viscosity
used for the validation is 0.01 Pa s.
In Fig. 4, the results concerning the friction coefficient
are compared with the respective results from analytical
solutions found in Ref. [3]. The bearing is considered to be
aligned in Ref. [3] and the analysis performed is dimensional. In order to get the validation results, the following
bearing characteristics are used: bearing diameter 25.4 mm,
radial clearance 65 mm, bearing length 25.4 mm and
viscosity 0.012 Pa s and rotational speed 400 rad s1. The
friction coefficient is normalized by the ratio of bearing
diameter over clearance and the Sommerfeld number is
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Regarding the numerical methods presented by Fillon
and Bouyer [15], and Muskat and Morgan [3], the
maximum percentage difference of the present method is
8% and 4.1%, respectively. Thus the proposed numerical
method is in very good agreement with both methods of
comparison.
3. Results
Fig. 4. Comparison of normalized friction coefficient resulting from
present analysis and that of Ref. [3].
The variation of the friction coefficient and power loss
with moment Sommerfeld number for several values of
wear depth parameter, for different L/D ratios, and for
several Sommerfeld numbers S, are shown in Figs. 5a, b,
6a, b, 7a, b, 8a and b. In Table 1, the input data of the
bearing are illustrated. In Table 2, the variation of the
normalized friction coefficient with minimum film thickness, moment Sommerfeld number, and wear depth
is presented. The results are taken for S ¼ 0.15 and
L/D ¼ 0.5. In Table 3, the variation of the normalized
Fig. 5. The variation of the normalized friction coefficient as a function of
moment Sommerfeld number, for L/D ¼ 0.5 and (a) S ¼ 0.15 and (b)
S ¼ 0.3.
calculated from Eq. (9) with the external load varied from
500 to 4000 N. The grid used in both validation cases is the
50 9 consisting of triangular elements.
Fig. 6. The dependency of the normalized friction coefficient on the
moment Sommerfeld number, for L/D ¼ 1, and (a) S ¼ 0.15, and (b)
S ¼ 0.3.
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Fig. 7. Power loss versus moment Sommerfeld number and wear depth for
L/D ¼ 0.5, and (a) S ¼ 0.15 and (b) S ¼ 0.3.
friction coefficient with the minimum film thickness,
moment Sommerfeld number, and wear depth is also
presented. The results of this table correspond to S ¼ 0.3
and L/D ¼ 0.5. In both Tables 2 and 3, the results are
obtained using the bearing characteristics of Table 1, and
by putting the parameters in dimensional format. It is
observed that the friction coefficient increases as the
minimum film thickness is decreased.
It is clearly observed from Figs. 5a and 6a, that for
S=0.3 and both L/D ratios, the friction coefficient
increases as the misalignment angles and the wear depth
become greater. For L/D=1 and especially for S=0.15, the
friction coefficient is decreased as the wear depth varies
from 0.0 to 0.3, and then increased as the wear depth varies
from 0.3 to 0.5. The same phenomenon, a decrease in the
friction coefficient for small wear depths and an increase
in bigger wear depth parameters, is also observed for
L/D=0.5 and S=0.15. This could be explained by the fact
469
Fig. 8. Power loss versus moment Sommerfeld number and wear depth for
L/D ¼ 1, and (a) S ¼ 0.15 and (b) S ¼ 0.3.
Table 2
Friction coefficient variation as a function of minimum film thickness, for
S ¼ 0.15, L/D ¼ 0.5 and several worn and misalignment conditions
c̄x ¼ c̄y
d0
Sm
hmin/c
f̄ ¼ fR=c
0.1
0.1
0.1
0.1
0.4
0.4
0.4
0.4
0.0
0.1
0.3
0.5
0.0
0.1
0.3
0.5
4.718
5.149
6.720
8.408
1.154
1.264
1.691
2.172
1.780e01
1.880e01
2.210e01
2.490e01
8.120e04
1.930e03
1.870e02
4.210e02
4.529
4.487
4.395
4.402
4.809
4.735
4.565
4.533
that the present analysis follows the Dufrane et al. [5] worn
bearing geometry, in which the minimum film thickness is
increased by the wear depth, as it varies from 0 to 0.15, and
decreased with bigger wear depth ratios.
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470
Table 3
Friction coefficient variation as a function to minimum film thickness for
S ¼ 0.3, L/D ¼ 0.5 and several worn and misalignment conditions
c̄x ¼ c̄y
d0
Sm
hmin/c
f̄ ¼ fR=c
0.1
0.1
0.1
0.1
0.4
0.4
0.4
0.4
0.0
0.1
0.3
0.5
0.0
0.1
0.3
0.5
16.18
17.14
21.07
23.41
4.09
4.39
5.48
6.13
0.298
0.306
0.334
0.360
0.097
0.102
0.127
0.158
7.676
7.663
7.666
7.798
7.868
7.850
7.833
7.957
Generally, as depicted in Figs. 5a and 6b, that for a fixed
set of Sommerfeld number, moment Sommerfeld number
and wear depth, the friction coefficients are higher for the
ratio L/D=0.5 than those of the ratio L/D=1. The reason
for higher friction coefficients for L/D=0.5 obviously lies
in the higher eccentricities for a certain wear, moment
Sommerfeld number, and Sommerfeld number. The power
loss increases when the wear depth is increased. It is also
observed that for bigger ratios L/D, the power loss is
bigger for the same Sommerfeld number. Both, friction
coefficient and power loss are clearly increased when
misalignment angles increase.
The appropriate functions for the normalized friction
coefficient for worn-misaligned journal bearings could be
taken from Figs. 5a, b, 6a, and b, using an appropriate
fitting approximation. As an example, one could write the
functions obtained for L/D=0.5 and for d0=0.1 and 0.5,
as well as for L/D=1 and d0=0.3. Using the definition
f̄ ¼ fR=c, and the second-order exponential fitting rule, the
following functions are taken:
f̄ ¼ 7:66003 þ 7:97969 eSm =0:88392
8
L=D ¼ 0:5;
>
>
<
þ 7:69711eSm =3:34136 for S ¼ 0:3;
>
>
: d ¼ 0:1;
ð19Þ
ð20Þ
0
f̄ ¼ 6:85014 þ 12:7252 eSm =1:55292
8
L=D ¼ 1;
>
>
<
þ 0:11354 eSm =7:0420 for S ¼ 0:3;
>
>
: d ¼ 0:5:
f̄ ðS; S m Þ ¼
3 X
3
X
C ij Si Sjm
i¼0 j¼0
8
0:07pSp1:5;
>
>
<
0:07pSp1:5;
>
>
: 0:07pSp1:5;
0:6pSm p42 for d 0 ¼ 0:0;
1:0pSm p37 for d 0 ¼ 0:3;
0:1pSm p33 for d 0 ¼ 0:5:
ð22Þ
Here Cij, i ¼ 0, y, 3, j ¼ 0, y, 3, are the coefficients of
the cubic surface function.
From the contour diagrams of Fig. 9, that are general
and applicable to any bearing, it can be observed that, for
S ¼ 1 and Sm ¼ 15, the normalized friction coefficient are:
at point P1, f̄ ð1; 15Þ ¼ 20, at P2, f̄ ð1; 15Þ ¼ 21, and at P3,
f̄ ð1; 15Þ ¼ 23, for d0 ¼ 0, 0.3, and 0.5, respectively. Thus
the friction increases as the wear depth increases. In general
the normalized friction coefficients appear to be slightly
influenced by the misalignment (expressed here by the
moment Sommerfeld number). For small Sommerfeld
numbers (high loads) such as S ¼ 0.1–0.6, the friction
contours appear to be almost independent from the
Sommerfeld number (moment), whereas for S ¼ 0.6–1.0
(small loads) the influence of misalignment becomes larger
than in the previous case. In this figure more bend curves
indicate bigger misalignment effect.
4. Conclusions
0
f̄ ¼ 7:79173 þ 4:34792 eSm =1:46147
8
L=D ¼ 0:5;
>
>
<
þ 0:25790 eSm =6:16632 for S ¼ 0:3;
>
>
: d ¼ 0:5;
Sommerfeld number, S, and moment Sommerfeld number,
Sm, for relative wear depths of 0% (Fig. 9a), 30% (Fig. 9b)
and 50% (Fig. 9c), and for L/D ¼ 1 are illustrated. Cubic
surfaces have been used to interpolate the data points in
order to construct the surfaces and the contours. The cubic
surfaces are obtained, for d0 ¼ 0, 0.3, 0.5, and have the
following form:
ð21Þ
0
Similar equations could be obtained for the rest of
the curves showing the variation of the friction coefficient
with the moment Sommerfeld number and wear depth. In
Fig. 9, the variations of the friction coefficient versus force
The following conclusions may be drawn from the above
analysis regarding the friction coefficients of worn misalignment journal bearings operating under severe hydrodynamic conditions. The friction coefficient is increased, in
general, with increasing wear depth as well as misalignment
and Sommerfeld number. The friction coefficient and
consequently the power loss, are strongly dependent upon
the misalignment angle, giving higher friction values as the
moment Sommerfeld number is decreased. The power loss
is increased with wear depth. It is also observed that, for
larger ratios L/D, the power loss is larger, for the same
Sommerfeld number.
The exact variation of the friction coefficient is strongly
dependent on the wear model and bearing slenderness ratio
L/D. This means that the friction coefficient is dependent
on the wear depth, which affects the worn region, the
misalignment angles, and the slenderness ratio L/D, as
it is typically depicted in Eqs. (19)–(21). Friction coefficient
functions related by the moment Sommerfeld number
(or misalignment angles) and by the Sommerfeld number
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471
Fig. 9. Contours of the normalized friction coefficient, depending on the two Sommerfeld numbers (force and moment), for L/D ¼ 1 and wear depth: (a)
d0 ¼ 0, (b) d0 ¼ 0.3, and (c) d0 ¼ 0.5.
and wear depth, could be obtained from the respective
Figs. 5a, b, 6a, b, and 9a–c, using the appropriate fitting
equations.
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