Modeling of the Behavior of a Deep Groove Ball Bearing in Its Housing

doi:10.4236/jamp.2013.14009

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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

Email: a.a [email protected], karl.debray@univ-reims.fr

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.

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:

(1)

where

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,

−



=



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

2π

(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.

Figure 3. Bearing subjected to external load Pr [10].

12 3

2cos2 cos2

2 cos

PP PP

ββ

=+ +++

(5)

Loads and displacements are related according to hertz

theory by:

3/2



=



(6)

And

cos

δδ β

(7)

with j = 2, 3,…,n + 1.

( )

3/2

cos 1

PP j

= −





(8)

With (8), (5) becomes:

( )()

( )

[ ]

2.5 2.5

2.5

12 cos2 cos2

2 cos

n PQ

ββ



= +++…







(9)

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

PQz

= =

(11)

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.

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

L NL

ext ext

RF FF FF=−=+ −=

(12)

With

int

and

ext

are internal and external forces,

int

are internal linear and nonlinear forces. New-

ton-Raphson method is used to solve this non-linear sys-

tem, the :

[ ]

{}{ }

KR∆=u

(13 )

{}{ }

{ }

1ii

uuu

= +∆

(14)

where

[ ]

is global tangent stiffness matrix:

[ ]

{ }

int



∂







=− =−+





∂∂







(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.

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.

fatigue lifetime analysis.

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