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)

Copyright © 2013 SciRes. 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 zianou@gmail.com, 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