Journal of Applied Mathematics and Physics, 2013, 1, 45-50
http://dx.doi.org/10.4236/jamp.2013.14009 Published Online October 2013 (http://www.scirp.org/journal/jamp)
Modeling of the Behavior of a Deep Groove
Ball Bearing in Its Housing
Ayao E. Azianou1 ,2, Karl Debray1, Fabrice Bolaers1, Philippe Chiozzi2, Frédéric Pall eschi2
1Groupe de Recherche en Sciences de l’Ingénieur, Université de Reims Champagne Ardenne, EA 4694 URCA, Moulin de la Housse,
51687 Reims Cedex 2, France
2Valeo Electric Systems, 2 Rue André Boulle, 94000 Créteil, France
Received August 2013
ABSTRACT
Load distribution in deep groove ball bearing has been studied in this work. A deep groove ball bearing model is pro-
posed basing on geometry specific measurement. Two approaches (finite element method and semi-analytical) have
been used to determine the distribution of an external radial force applied. These two approaches have been compared
in terms of computation time and precision. At the second point, a deformable complex bearing housing has been inte-
grated in the FEM ball bearing model to assess the influence of its deformation on load distribution.
Keywords: Ball Bearing; Finite Element; Semi-Analytical; Load Distribution; Housing
1. Introduction
Automotive alternators are rotating machines whose
main role is to convert the mechanical energy of rotation
into electrical energy for powering electrical and elec-
tronic components of the car. With the evolution of
technology, new generations of alternators have been
developed. These alternators have complex shapes and
are equipped with stop-and-start systems to reduce noise,
fuel consumption and the emission of greenhouse gases.
An alternator is mainly composed of a stator and a rotor
guided in rotation by two ball bearings. The transfer of
loads from one ring to another is possible by means of
the balls in contact with the rings. Load distribution in
bearings is an important factor in the proper functioning
of the ball bearing and its fatigue lifetime estimation.
Depending on the type of external load, all the balls don’t
have the same contribution to the load transfer. Another
important parameter involved in the load distribution is
the internal clearance within the ball bearing.
Several authors have worked on load distribution
problem. The first work was done by Stribeck [1] for
external radial loading. He has determined in his formu-
lation the maximum force on a ball. Although this for-
mulation has been used for a long time to calculate the
static bearing capacity, it do es not explicitly consider the
value of the clearance. Sjovall [2] established a formula-
tion taking into account the value of the clearance where
the maximum force is a function of an integral that holds
his name.
Jones [3], based on the Hertz contact theory [4] pro-
posed an analytical approach to determine the relative
motion of the rolling over rings. This approach was im-
proved by Harris and was called Jones-Harris method
(JHM) [5]. The relative displacements at the contact ball-
rings due to external forces were determined by Newton-
Raphson method. From the relation between force and
displacement, the forces are obtained from the displace-
ments o btained by Hertz formulation. Other authors used
Finite Element Method (FEM) to solve the problems of
contact within the bearing and determine load distribu-
tion. By using the finite element method to determine
coefficients used in load-displacement relation, Yuan
Kang et al. [6] modified the method of Jones-H ar - ris
which was called modified Jones-Harris method
(MJHM). Bourdon et al. [7] replaced, in a modeling of
deep groove ball bearings, balls by specific elements to
study the bearing behavior.
Most of these studies considered that the bearings are
mounted in a rigid housing. This assumption is not rea-
listic when we consider the case of a new generation of
alternator where the housing is complex and deformable.
This paper assesses the influence of housing deformation
on load distribution in the bearing by two different ap-
proaches: a finite element approach and a semi-analytical
approach where the rolling elements are replaced by user
elements. A prior study has been done by using these two
approaches in the case of rigid housing.
The geometry of ball bearing is important in load dis-
tribution modeling, especially curvature radii of the

A. E. AZIANOU ET AL.
46
raceways of the inner and outer ring. These values are
not indicated by bearing suppliers an d they are difficultly
determinable by conventional measurement methods. We
have developed a new methodology to measure these
values using a three-dimension measuring machine.
2. Method for Determining Deep Groove
Ball Bearing Geometry
2.1. Removing Bearing Component to Access the
Raceway
The method is to reconstruct the geometry and determine
the radii of curvature from points equally distributed on
the raceway. Because of the fact that ball bearings deep
groove are rigid (non-demountable), a way to access
raceways has to be found. The ball bearings are divided
in two parts (Figure 1) by an electrical discharge ma-
chining (EDM). EDM was used because of its advantag-
es. Cutting is performed by removing small particles of
the material of electric arc between tool and workpiece.
There is no contact between the workpiece and the tool,
so no deformation of the bearing which may influence
the measurement results.
2.2. Determination of Ball Bearing Geometry
The parts of bearing ring are mounted and locked in a
vee fixed on the magnetic table of the measuring ma-
chine. Bearing raceway has a shape of a h alf torus with a
small and a big radius. The plane in which points are
probed the big curvature radius is the median plane to
planes P1 and P2 from the ring shoulder (Figure 2) . The
small radius is scanned in the plane perpendicular to the
median plane. From the probed points coordinates, the
curvature radii are obtained by the least squares method.
With the values obtained by measurement, the geome-
try of the bearing is reconstructed numerically.
3. Modeling of Ball Bearing Behavior
A ball bearing is composed by the outer ring, inner ring,
the balls, and the cage whose role is to maintain the dis-
tance between balls. In the present analysis the effect of
the cage is neglected. During loading, balls in contact
with the rings deform both inner and outer rings. Because
of the fact that the contact surfaces in ball bearing are not
known, contact pro blem is complex. Her tz was the first to
solve the problem by considering contact between two
ellipsoids. He established a relation between load and
displacement. Based on his work, the relation between the
displacement on each ball and the force that produces it is
given by:
s
QK δ =
(1)
where
δ
is the relative approach between the two rings,
(a) (b)
Figure 1. Determination of bearing geometry. (a) Bearing
disassembled; (b) Palpating rings.
Figure 2. The principle of probing points
K is the rigidity of the ball and raceway contact and s is
the load deflection exponent. s value is 3/2 for ball bear-
ing. The rigidity is related to the raceway geometry and
material property [8].
The load distribution in a ball bearing is an important
parameter in fatigue lifetime calculation. In mechanisms,
bearings can be submitted to many solicitations: radial,
axial, moments or combination of axial and radial loads.
Rolling elements (balls) don’t behave in the same way. In
the case of r a di a l loading, only s ome of the balls a r e unde r
load [1,9].
In the configuration showed in Figure 3, the number of
balls under l oad is given by:
21mn= +
(2)
With,
1
4
−

=

z
n INT
(3)
where n is the parameter that designates the number of
pairs of balls transferring load in addition to the ball 1
(Figure 3) and z is the number of balls. Balls are distri-
buted equitably on the raceway circumference and the
angle between balls is
β
.
2π
z
β =
(4)
By using static equilibrium and projecting load on each
ball in the three directions, the relation between loads is
given by:

A. E. AZIANOU ET AL.
47
Figure 3. Bearing subjected to external load Pr [10].
12 3
2cos2 cos2
2 cos
r
n
PP PP
Pn
ββ
β
=+ +++
(5)
Loads and displacements are related according to hertz
theory by:
3/2
11
nn
P
P
δ
δ

=

(6)
And
1
cos
j
j δδ β =
(7)
with j = 2, 3,…,n + 1.
So
( )
( )
3/2
1
cos 1
j
PP j β = − 

(8)
With (8), (5) becomes:
( )()
( ) [ ]
2.5 2.5
1
2.5
1
12 cos2 cos2
2 cos
r
PP
n PQ
ββ
β

= +++…


+=

(9)
1
P
is the maximum load on ba ll,
[ ]
maxr
PPQ=
(10)
These relations are valid for ball bearings with zero
clearance submitted to radial load. The maximum load
is deducted by the relation:
max tr
rSP
P
PQz = =
(11)
t
S
is called Stribeck Number, and it is around 4.37.
Although it is valid for bearing with zero clearance
mounted in a rigid housing, Stribeck suggested attribut-
ing to his constant a value of 5.0 to take into considera-
tion the presence of clearance.
Figure 4. 3D bearing model loaded.
Figure 5. Ball replaced by “ball element”.
Stribeck relation was one of first mathematical formu-
lations that d eals with load distribution in bearings. Other
formulations were developed; all of them consider in its
assumptions a rigid housing. Realistic models that con-
sider housing deformation are proposed. First, two ap-
proaches are studied in the case of rigid housing and
compared to mathematical formulations. Secondly, the
deformable housing is integrated in the model to evaluate
its deformation influenc e .
3.1. Finite Element Method Model
A three-dimension FEM bearing model is used for the
determination of load distribution. The model consists of
an inner ring, an outer ring and seven balls. A radial ex-
ternal load of 5000N is applied on inner ring. The geome-
try of the bearing was obtained from the method de-
scribed in 2 .
Assumptions: The material of the bearing is applied in
the elastic area; a frictionless contact between the balls
and the bearing rings is considered. The influence of the
cage is neglected. A quasi-static analysis is considered.
Because of low deformation of balls, they are represented
by rigid spheres.
The main steps of the analysis of bearing contact are:
(1) 3D construction using ball bearing geometry, (2) De-
finition of material properties of components: (Table 1),
(3) Definition of contact properties (4) Application of the
necessary boundary conditions and loading, (5) Resolu-
tion of contact problem , (6) T r e a tment of results.
FEM ball bearing model has 89134 node and 25354
elements. Materials properties for balls and rings are de-
fined in Table 1.
3.2. Semi-Analytical Approach
The approach is based on Bourdon and De Mul’s mod-

A. E. AZIANOU ET AL.
48
els. In this approach, balls are replaced by two nodes
elements that proper ties are computed. Outer and inner
rings are meshed like in finite elements method. Be-
cause of the fact that solid elements have only three
degrees of freedom (3 translations), shell elements are
merged on raceways to enable rotation degrees of
freedom of “ball elements”. Five degrees of freedom (3
translations and 2 rotations) are considered, the rotation
around the shaft is fr ee.
The global static equilibrium is reached when the re-
sidual vector {R} is cancelled. In a revolving machine,
the bearings have a nonlinear behavior due to the Hertz
contact, but other components essentially have linear
behavior, so the internal force vector can be separated
into two parts:
{ } {} {}
{} {}
{ } { }
intint int
0
L NL
ext ext
RF FF FF=−=+ −=
(12)
With
int
F
and
ext
F
are internal and external forces,
int
NL
F
int
L
F
are internal linear and nonlinear forces. New-
ton-Raphson method is used to solve this non-linear sys-
tem, the :
[ ] {}{ }
T
KR∆=u
(13 )
{}{ }
{ }
1ii
uuu
+
= +∆
(14)
where
[ ]
T
K
is global tangent stiffness matrix:
[ ]
{ }
{ }
int
NL
L
TT
NL
F
R
KK
uu

∂
∂ 


=− =−+
 
∂∂



(15)
The steps to solve the problem of this approach are
defined in Figure 6. At each iteration increment, the
Figure 6. Flow chart of load determination of deep groove
ball bear i ngs.
Table 1. Material properties for ball bearing.
Bearing
components
Properties
Material
Young module
(Mpa) Density
(kg/m3) Poisson
ratio
Inner and
outer ring Bearing steel 210000 7800 0.285
Balls Bearing steel 210000 7800 0.285
obtained displacement vector {} allows to compute
the bearing’s tangent stiffness matrix. The linear part
of the stiffness matrix does not change in course of
iterations. The displacement of elements is initialized
in the beginning of the analysis. The equilibrium of
each ball element is solved, an d their stiffn ess matrices
are calcula ted.
The methodology is used to study load distribution in a
deep ball bearing. Rings geometry is determined by me-
thod used in 2. A radial load is applied in the
3.3. Housing Deformation Influence: FEM
Approach
Previous analyses are valid for rigid housing. Alternator
bearing housing (Figure 7), due to his complex shape
may influence the load distribution. It has been integrated
numerically in FEM model to evaluate its deformation on
load distribution in the deep groove bearing. FEM me-
thod is chosen for this study to be closer to reality be-
cause balls are geometrically represented.
The analysis is focused on the radial loading case be-
cause the front bearing of the alternator is essentially
loaded radially. The model consists of a deep groove ball
bearing and alternator housing. The connection between
the outer ring and the housing is complete. The appropri-
ate boundary conditions are applied. The external load of
5000N is radially applied in the geometric center of the
bearing (Figure 8).
Materials properties ar e defined in Tab le 2. The model
consist s of 12 1425 nodes and 39 6840 element s .
4. Results and Discussion
Load distribution obtained by the two approaches has
been compared to analytical formulation in rigid housing
case. Among seven balls, only three participate in the
load transfer. This is in accordance with literature for-
mulation. A radar chart (Figure 9) presents radial loads
on three balls. The maximal load is on ball number 0.
For the two approaches, we note that results are con-
sistent (Table 3) and closer to analytical formulation,
deviation is less than 3 percent when we compare the two
approaches. The semi-analytical approach presents an
advantage of computation time. It is four times faster
than FEM approach.

A. E. AZIANOU ET AL.
49
Figure 7. Schematic view of automotive alternator.
Figure 8. Complex shape of bearing hou s i ng.
Figure 9. Load distribution.
Table 2. Material properties for ball bearing and its hous-
ing.
Bearing
components
Properties
Material
Young module
(Mpa) Density (kg/m3) Poisson ratio
Inner and
outer ring Bearing steel 210,000 7800 0.285
Balls Bearing steel 210,000 7800 0.285
Housing Aluminium alloy 70,000 2700 0.3
FEM approach is compared for two cases (Figu re 10):
rigid and deformable housing. Although the maximal
load is on ball number 0, external radial load is not dis-
tributed by the same way compared to rigid housing
(Table 4).
The complexity of housing shape and its material
properties (aluminum alloy) lead a load distribution dif-
ferent from a rigid housing. Housing influence on load
distribution can be an important factor that affects bear-
ing durability estimation.
Figure 10. Load distribution considering deformable hous-
ing
Table 3. Loads on balls values.
Balls
Radial loading,
Clearance = 0.011 mm (Pr = 5000 N)
FEM Semi-Analy
tical Deviation (%) Analytical formulation
Ball 0 3102.50 3142.51 −1.28 3098.07
Ball 1 1529.52 1489.61 2.61 1525.23
Ball 1’ 1529.52 1489.60 2.61 1525.23
Table 4. Influence of housing deformation.
Balls
Radial loading,
Clearance = 0.011 mm (Pr = 5000 N)
Rigid housi ng Deformable housing
FEM
Ball 0 3102.50 2435.72
Ball 1 1529.52 2032.2
Ball 1’ 1529.52 2100.06
5. Conclusions
In this study, two approaches: a FEM and a semi ana-
lytical approach are used for load distribution calculation
in statically radial loading. These results are consistent
and closer to analytical formulations in the case of rigid
housing. Semi-analytical approach is seen to be cheaper
in term of computation time.
Using FEM, a complex deformable housing is nu-
merically integrated in a ball bearing to study the load
distribution consid ering. The nu merical results sh ow that
housing deformation has effect on load distribution. Be-
cause load distribution is an important parameter in ball
bearing durability, the housing has to be considered in

A. E. AZIANOU ET AL.
50
fatigue lifetime analysis.
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