BAOMIN WANG et al: THE EFFECT OF MATERIALS ON FATIGUE LIFE OF ANGULAR CONTACT BALL . .
The Effect of Materials on Fatigue Life of Angular Contact Ball Bearing
Baomin Wang*, Chao Gao *, Bing Du, Wu Zaixin
School of Mechanical and Electronical Engineering
Lanzhou University of Technology
Lanzhou, Gansu China
Abstract — Angular contact ball bearings are widely used in many high-speed rotating systems and play a vital role in determining
the expected life of these machineries, so it’s vital to investigate their fatigue life. Based on the dynamics theory of rolling bearing
and L-P life theory, the calculation equations for contact load and fatigue life of angular contact ball bearing were established. The
detailed calculating method and its program are given. Taking 7008A angular contact ball bearing as an example, comparing three
kinds of bearings with different materials, it’s found that, under certain load, the contact load, fatigue life of rings and bearing are
mainly affected by the contact angle and centrifugal force of balls which are related to ball density and bearing speed. Bearing
fatigue life is mainly determined by that of the outer ring, while that of inner ring has less effect on bearing fatigue life. It is also
shown that, when bearing rotational speed is relatively low, bearing fatigue life is maximum in the full steel bearing, the second in
hybrid ceramic bearings, and the minimum in full ceramic bearing. When the bearing speed is moderate, bearing fatigue life is the
maximum in the hybrid ceramic bearing, the second in full steel bearings, and the minimum in full ceramic bearing. When the
bearing speed is high, bearing fatigue life is the maximum in the hybrid ceramic bearing, the second in full ceramic bearings, and
the minimum in full steel bearing. Comparing with density, Elasticity modulus and Poisson ratio of ball and rings have relatively
less effect on the contact load, ring fatigue life, and bearing fatigue life.
Keywords - Angular contact ball bearing; Fatigue life; Materials; Effect
Harvey P. [17] analyzed the effects of extreme pressure
additives in lubricants on bearing fatigue life, proved that
some lubricant additives can result in significant detrimental
effects on bearing fatigue life and wear. Miao Xuewen [18]
divided the life prediction model of rolling bearing into three
categories, including statistical life model, based on fracture
mechanics life model and empirical life model, and analysis
of the models used today. Guan Jian [19], combined with a
new fatigue damage accumulation model, modified the
existing rolling bearing life prediction model and
investigated the influence of different factors on the fatigue
life of bearing. NENG TUNG LlAO [20] investigated the
effect of variable contact angles on the fatigue life of a ball
bearing and also evaluated the impact of ball life on the
fatigue life of a ball bearing. D. Koulocheris [21]
comparatively studied the impact of corundum particle
contaminants size on wear and fatigue life of grease
lubricated ball bearings.
All of the studies provide an important guide for
exploring the fatigue life of rolling bearing. However, it’s
should be noted that, with the rapid development of material
industry, more and more new materials, especially ceramic
materials are used in angular contact ball bearing. So the
effect of materials on the fatigue life of angular contact ball
bearings cannot be ignored. Therefore, the aim of this study
is to analysis the effect of material on the fatigue life of
angular contact ball bearing with different materials.
In this paper, basic theory on fatigue life of rolling
bearings will be presented in Section 2. The calculation
equations for contact load of angular contact ball bearing
will be proposed in Section 3. Comparative analysis about
the effect of materials on fatigue life of angular contact ball
bearings will be presented in Section 4. Section 5 will show
I. INTRODUCTION
Angular contact ball bearings are widely used in many
high-speed rotating systems due to low starting friction, high
rigidity and reliability [1-5]. However, with the development
of high speed machine, failure and damage usually happen in
angular contact ball bearings duo to high speed and
temperature lever, previous studies show that the bearing
failure accounts for about 30% of the failure in high-speed
ball bearing applications [6-11]. Hence, fatigue life is an
important criterion to measure the performance of bearings.
In this sense, it is essential to investigate the fatigue life of
angular contact ball bearings in order to prolong bearing
serve life in the design stage [12, 13].
In the past decade, a great deal of work has been done in
understanding the fatigue life of rolling bearing. Early in the
1950s, Lundberg and Palmgren [14] took comparative
analysis about the deformation of the bearing under different
loads and rotating speed, but did not consider the impact of
the friction and lubrication on bearing life. Jones [15]
analyzed the bearing statics and established the static model
of bearing, then analyzed the bearing contact load in actual
working condition, the results are in agreement with the
results obtained by using Hertz contact theory,and the
contact fatigue life of bearing is also calculated. Oswald,
Fred B. [16] analyzed the beneficial effect of residual
stresses on rolling-element bearing fatigue life in the
presence of high hoop stresses for three bearing steels; these
additional stresses were superimposed on Hertz principal
stresses to calculate the inner race maximum shearing stress
and the resulting fatigue life of the bearing. Affected by
many factors, bearing fatigue life is very complex, Nixon,
DOI 10.5013/IJSSST.a.17.22.17
17.1
ISSN: 1473-804x online, 1473-8031 print
BAOMIN WANG et al: THE EFFECT OF MATERIALS ON FATIGUE LIFE OF ANGULAR CONTACT BALL . .
the effect of materials on the fatigue life of angular contact
bearings. Study results will provide useful guidelines for
determining the bearing material.
Q co 
98.1(
(7)
II. BASIC THEORY ON FATIGUE LIFE OF ROLLING BEARINGS
A.
o 
Bearing fatigue life
Qo  (
Lh 
j 1
L10 106
60  n
CONTACT LOAD IN ANGULAR CONTACT BALL BEARING
A. Deformation consistent equations
Figure.1 illustrates the relative geometrical relationships
between the ball and the inner race curvature assuming that
the outer raceway is fixed according to outer raceway control
theory. Without rotation and preload, the out raceway
curvature center Oo and the inner raceway curvature center
Oi are aligned with the ball center Ob, the outer contact
angle is equal with the inner contact angle. However, with
the increase of rotation speed, centrifugal force and
gyroscopic moment are developed, which causes the ball
'
center to migrate from Ob to Ob , and the inner raceway
(3)
(4)
3 1/ 3
ij
Q )
(5)
2) The fatigue life of bearing outer ring
Defining Lo as the fatigue life of bearing outer ring，is
determined by the following formula:
'
curvature center to migrate from O i to Oi , as a result, inner
contact angle and outer contact angle no longer remain
constant. Where, δa, δr and θ are axial, radial, and angular
displacement of inner ring under load respectively.
Q
L o  ( co ) 3
Qo
(6)
Where, Qco is the rated dynamic load of outer ring, and
determined by the following formula.
DOI 10.5013/IJSSST.a.17.22.17
Lh . Thus, the
III. THE CALCULATION EQUATIONS AND SOLUTION FOR
(2)
Where, Qci is the rated dynamic load of inner ring，Qi is
the equivalent dynamic load of inner ring,

Q oj10 / 3 ) 0 .3
(11)
Where, n is the rotating speed, unit r/min.
According to the above analysis, obtaining contact load
between balls and rings is the key in obtaining the fatigue
life of angular contact ball bearings.
Qci 3
)
Qi
Z
j 1
convenient to express life with a rotation time
life calculation formula can be transformed as:
(1)
Where, Qc is the rated dynamic load of the rolling
element，Q is the contact load of the rolling element.
The fatigue life of bearing inner ring
Defining Li as the fatigue life of bearing inner ring，it is
determined by the following formula:
1
Qi  (
Z
Z
(10)
By the above method, bearing ring fatigue life and
bearing whole fatigue life can be obtained, but in practice, it
is difficult to express the life using the total number of
revolutions. If the rotational speed is constant, it is more
Q
L10  ( c )3
Q
D cos  i
i 
dm

L10  ( Li 1.11  L01.11 ) 0.9
The calculation of rolling bearing fatigue life
2f
(1i )1.39 i 0.3 1.8 1/ 3
Qci  98.1( i )0.41
(
) D Z
2 fi 1 (1i )1/ 3 cosi
1
Z
(9)
3) Bearing fatigue life
In order to determine the whole fatigue life of rolling
bearing, the lives of inner and outer rings must be considered
together. According to the product law,
The fatigue life of rolling bearings which withstand
normal load Q and point contact with raceway is determined
by the following formula:
Li  (
D cos  o
dm
(8)
Qo is the equivalent dynamic load of outer ring. Because
the inner ring is rotational and outer ring is stationary, the
formula Qo is different with Qi,
Generally, rolling bearing life refers to the fatigue life of
bearing, which are the working hours until the fatigue
spalling appears on the surface of bearing. When bearing
moves in the condition of load, the load transfers from a
bearing ring to another through the rolling element, and
appropriate stress distribution will be appeared inside the
metal material. Contact stress has a significant influence on
bearing contact fatigue and wear, to a large extend it also
determine the bearing life. Even if the load is small, the
contact stress can also be quite large as the small contact
area. The analysis of contact load is the basis for analyzing
the fatigue life of rolling bearing and must be firstly
calculated.
B.
2 f o 0.41 (1   o )1.39

)
( o ) 0.3 D1.8 Z 1 / 3
2 fo  1
(1   o )1 / 3 cos  o
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BAOMIN WANG et al: THE EFFECT OF MATERIALS ON FATIGUE LIFE OF ANGULAR CONTACT BALL . .
Qij sin aij  Qej sin aej 
Qij cos aij  Qej cos aej 
2 M gj
Db
2 M gj
Db
cos aej  0
(14)
sin  ej  Fcj  0
(15)
The gyroscopic moment Mgj and the centrifugal forces
Fcj are calculated respectively as follows:

Db3d mcj2
12
1
M ej  JDbbjcj sin 
2
Fcj 
(17)
Contact force Qij and Qej can be derived from Hertz
contact analysis and are represented by
Figure 1. Relation position of ball and race curvature center.
Qij  K ij ij3 / 2 

Qej  K ej ej3 / 2 
The following equations are developed from Figure.1.
A
(12)


 x aj   A rj  x rj   f i  0.5D b  ij  0
2
aj
2

(18)
Where Db is rolling element diameter, dm is bearing
pitch diameter, J is rotary inertia of rolling element, ωc is
orbital speed of rolling element, ωb spin speed of rolling
element, β is position angle of rolling element.
2

x aj2  x 2rj  f e  0.5D b  ej  0
2
(13)
Where Xaj is the axial projection of distance between the
j-th ball center and the curvature center of outer channel, Xrj
is the radial projection of distance between the j-th ball
center and the curvature center of outer channel, Aaj is the
axial distance between the centers of curvature of the inner
and outer rings channel, Arj is the radial distance between
the centers of curvature of the inner and outer rings channel.
B.
(16)
C.
Bearing load analysis
The force equilibrium equations of the inner ring of the
bearing in the axial and radial directions, the moment
equilibrium for the bearing can be obtained:
Z
Fa   Qijsinαij  0
j 1
Analysis of ball load
(19)
Z
Fr   Qijcosαij  0
Fig.2 illustrates the force equilibrium of bearing j-th ball.
j 1
(20)
Z
M y   QijR isinαij  0
j 1
(21)
Where Fa, Fr and My are axial load, radial load and
moment respectively.
D.
So far, a group of nonlinear equations can be obtained
including Esq. (12) ~ (15) and Esq. (19) ~ (21). In order to
analyze the effect of materials on the fatigue life of angular
contact ball bearing, solving this nonlinear equation set is
necessary. It should be noted that Esq. (12) ~ (15) and (19) ~
(21) can be established for every ball, so the total number of
equations is 4z+3, where z denotes the number of balls.
Then, given Fa, Fr and My as input conditions, using
Newton-Raphson numerical iterative solution method this
nonlinear equation set can be solved. The corresponding
calculation program was developed; the flow chart is shown
in Fig.3.
Figure 2. The force equilibrium of bearing ball.
The j-th rolling element is subjected to centrifugal force
Fcj, gyroscopic moment Mgj, outer ring contact load Qej and
inner ring contact load Qij, frition force Fij and Fej, the force
equilibrium equations in the horizontal and vertical
directions for the j-th rolling element can be obtained.
DOI 10.5013/IJSSST.a.17.22.17
Equations solution
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BAOMIN WANG et al: THE EFFECT OF MATERIALS ON FATIGUE LIFE OF ANGULAR CONTACT BALL . .
TABLE Ⅱ. MATERIAL PARAMETERS OF STEEL GCR15 AND CERAMIC
SI3N4
IV. CALCULATION RESULTS AND ANALYSIS CONCLUSION
Using the above calculation method and program, 7008A
angular contact ball bearing is taken as the study object of
fatigue life evaluation, the geometric parameters of 7008A
Angular contact ball bearing is shown in Table.1.The fatigue
life of three kinds of bearings, full steel bearing with steel
rings and ball, hybrid ceramic bearing with ceramic ball and
steel rings, full ceramic bearing with ceramic rings and ball,
are comparatively analyzed, material parameters of steel
GCr15 and ceramic Si3N4 is presented in Table.2. Bearing
cage is made of nylon material; the axial and radial loads are
both 500N. Bearing is lubricated with grease, grease
composition is listed in Table.3.
parameters
Elasticity modulus, E /GPa
Poisson ration, υ
Density, ρ / kg  m-3
Specification
Li-12-OH(23%)
Base oil content
PAO(85%)+Est.(15%)
Polymer
(mm2/s)
17.6
A. The contact load of bearing
When the bearing is pressure preloaded, the change of
contact load between ball and outer ring is shown in Fig.4.
Fig.4 shows that the contact load between the ball and outer
ring increases with the increase of rotational speed, it is
because that the contact angle of ball on outer ring decrease
with the rotational speed, in order to balance the load the
contact load between balls and outer ring increase with the
rotational speed, furthermore, centrifugal force is another
more important reason that causes contact load increase with
rotational speed. It’s also found that, when the rotating speed
is lower than 9000r/min, the contact load between ball and
outer ring is the maximum in full ceramic bearing and the
minimum in full steel bearing, the decrease of outer contact
load is mainly resulted from contact angle change. When the
bearing speed is between 9000r/min-11500r/min, the contact
load between ball and ring is the maximum in full ceramic
bearing and the minimum in hybrid ceramic bearing, which
is resulted from both contact angle and centrifugal force.
When bearing speed is higher than 11500r/min, the contact
load between ball and ring is the maximum in full steel
bearing and the minimum in hybrid ceramic bearing, which
is mainly resulted from centrifugal force because contact
angle change is small in high speed region.
Set the initial value of δa, δr, θ
Set the initial value of
χ1j, χ2j, δij, δoj
Solving the equations of (24) ~(27)
Correct
initial value
Correct
initial value
Calculating the value of αij and αoj
Is the iterative process
convergent?
Calculating the contact load
Item
Thickener content
Oil viscosity at 40
Set the initial value of αij and αoj
Solving the equations of (33) ~（35
）
Test convergence condition
Si3N4
310
0.26
3200
TABLE Ⅲ. GREASE COMPOSITION
Input bearing structure parameters
Is the iterative process
convergent?
GCr15
206
0.29
7850
Correct
initial value
Output calculation results
Figure 3. The calculation flow chart of contact load.
TABLE I. THE GEOMETRIC PARAMETERS OF 7008A ANGULAR CONTACT
BALL BEARING
Item
Inner diameter d/mm
Outer diameter D/mm
Width B/mm
Pitch circle diameter dm/mm
Curvature radius of inner ring groove ri /mm
Curvature radius of outer ring groove ro /mm
Ball diameter Db/mm
Number of steel balls
Initial contact angular α/(°)
Specification
40
68
15
54
0.55
0.53
7.91
11
15
Figure 4. The change of the contact load between ball and outer ring with
the rotational speed.
DOI 10.5013/IJSSST.a.17.22.17
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BAOMIN WANG et al: THE EFFECT OF MATERIALS ON FATIGUE LIFE OF ANGULAR CONTACT BALL . .
The change of contact load between ball and inner ring is
shown in Fig.5, the figure shows that the contact load
between the ball and inner ring decreases with the increase of
rotational speed, it is because that the contact angle of ball on
inner ring increases with the rotational speed, in order to
balance the external load the contact load between ball and
inner ring increase with the rotational speed. It’s also found
that, when the rotating speed is lower than 8500r/min, the
contact load between ball and inner ring is the maximum in
full steel bearing and the minimum in full ceramic bearing.
When the bearing speed is between 8500r/min-10000r/min,
the contact load between ball and ring is the maximum in
hybrid ceramic bearing and the minimum in full ceramic
bearing. When bearing speed is higher than 10000r/min, the
contact load between ball and ring is the maximum in hybrid
ceramic bearing and the minimum in full steel bearing.
ball bearing outer ring is the minimum and the hybrid
ceramic contact ball bearing is the maximum.
Figure6. The change of the fatigue life of the outer ring with the rotational
speed.
The change of the fatigue life of the inner ring is shown
in Fig.7, it’s can be seen that affected by the change of
contact load between ball and inner ring, the fatigue life of
the inner ring increase with the increase of rotational speed.
Furthermore, when the rotating speed is less than
10000r/min, the fatigue life of inner ring is the maximum in
full ceramic angular contact ball bearing, the second in
hybrid ceramic angular contact ball bearing, and the
minimum in full steel bearing. When the bearing speed is
between 10000r/min -14000r/min, the fatigue life of full
steel angle contact ball bearing is less than that of full
ceramic angular contact ball bearing, but higher than that of
the hybrid ceramic angle contact ball bearing. When bearing
speed is higher than 14000r/min, the fatigue life of the full
steel angular contact ball bearing outer ring is the minimum
and the fatigue life of hybrid ceramic bearing is the
maximum.
Figure 5. The change of the contact load between ball and inner ring with
the rotational speed.
According to Fig. 4 and Fig. 5, it’s found that the change
of contact load between ball and rings is mainly effected by
the change of contact angle and centrifugal force of ball
which are related to ball density. Elasticity modulus and
Poisson ration has effect on the change of contact area with
the increase of bearing speed, so they has effect on contact
load, however, their effect are relatively less.
B.
The fatigue life of bearing
The fatigue life of the outer ring is shown in Fig.6, it’s
found that the fatigue life of the outer ring decreases with the
increase of speed. Affected by the change of contact load
between ball and outer ring, when the rotating speed is less
than 9000r/min, the fatigue life of the full steel angular
contact ball bearing outer ring is the maximum, the full
ceramic angular contact ball bearing is the minimum, and
hybrid ceramic angular contact ball bearing takes the second
place. When the bearing speed is between 9000r/min12000r/min, the fatigue life of full steel angle contact ball
bearing is less than that of full ceramic angular contact ball
bearing, but higher than that of the hybrid ceramic angle
contact ball bearing. When bearing speed is higher than
12000r/min, the fatigue life of the full steel angular contact
DOI 10.5013/IJSSST.a.17.22.17
Figure7.The change of the fatigue life of the inner ring with
the rotational speed.
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ISSN: 1473-804x online, 1473-8031 print
BAOMIN WANG et al: THE EFFECT OF MATERIALS ON FATIGUE LIFE OF ANGULAR CONTACT BALL . .
The change of bearing fatigue life is shown in Fig. 7. It’s
found that bearing fatigue life is mainly determined by
fatigue life of outer ring, fatigue life of inner ring has
relatively less effect on bearing fatigue life. It’s also found
that, when the rotating speed is less than 9000r/min, bearing
fatigue life is the maximum in the full steel bearing, the
second in hybrid ceramic bearings. When the bearing speed
is between 9000r/min-14000r/min, the fatigue life of full
steel bearing is less than that of hybrid ceramic bearing, but
higher than that of full ceramic angle contact ball bearing.
When bearing speed is higher than 14000r/min, the fatigue
life of the full steel angular contact ball bearing outer ring is
the minimum and that of hybrid ceramic contact ball bearing
is the maximum.
bearing fatigue life is the maximum in the hybrid ceramic
bearing, the second in full ceramic bearings, and the
minimum in full steel bearing.
3) Comparing with density, Elasticity modulus and
Poisson ration of ball and rings has relatively less effect on
the contact load, ring fatigue life, and bearing fatigue life.
ACKNOWLEDGMENT
This work was support by the National Science
Foundation of China (Grant No.51165024), and Gansu
Province Science Foundation of China (Grant
No.1208RJZA131).
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Figure7. The change of whole fatigue life with the rotational speed.
[9]
V. CONCLUSION
[10]
Comparing three kinds of 7008A angular contact ball
bearings with different materials, the following conclusion
can be obtained.
1) Under certain load, the contact load is mainly affected
by the change of contact angle and centrifugal force of ball
which are related to ball density and bearing speed. Fatigue
life of bearing inner and outer rings is determined by the
contact load between ball and rings.
2) Bearing fatigue life is mainly determined by fatigue
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When the rotating speed is relatively low, bearing fatigue life
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full ceramic bearing. When the bearing speed is moderate,
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in full ceramic bearing. When the bearing speed is high,
DOI 10.5013/IJSSST.a.17.22.17
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