Available online at www.sciencedirect.com
Procedia Engineering 37 (2012) 174 – 178
The Second SREE Conference on Engineering Modelling and Simulation (CEMS 2012)
Contact Analysis on Large Negative Clearance
Four-point Contact Ball Bearing
You Hui-yuana, Zhu Chun-xia,Li Wu-xingb, a*
a
Department of Mechanical Engineering, Luoyang Institute of Science and Technology, Luoyang,471023,China
b
Luoyang Building Material Machine Factory, Luoyang,471003,China
Abstract
Based on the theory of Hertz Contact, this paper presents a contact analysis method for large negative clearance
four-point contact ball bearing. Take turntable bearing 3-645K for example, normal contact stresses between steel ball
and inner and outer ring raceway are calculated under different negative clearance when bearing idling. The results
show that contact loads of four contact points of the bearing increase with the absolute value of negative clearance,
tiny changes of the negative clearance have a great effect on contact stresses.
© 2012 Published by Elsevier Ltd. Selection
Open access under CC BY-NC-ND license.
Keywords: Hertz theory; four-point contact ball bearing; negative clearance; contact stress
1.
Introduction
The subject investigated of this paper is a kind of large four-point contact ball bearing, it is a kind of
turntable bearing too. These bearings are widely used in crane, wind generators, radar as well as heavy
machinery such as excavators. The bearing we studied is used on radar. According to special working
condition of the radar, it requires a certain starting friction torque. Therefore, negative clearance bearing
must be used. In other words, interference fit is required between steel ball and inner and outer ring
raceway. At this time, contact area between steel ball and raceways will produce contact stress. Contact
stress has an important impact on bearing friction torque, contact fatigue and wear, which largely
determines bearing life. Calculation of contact stress is the basis of analysis of rolling bearing. At present,
* Corresponding author. Tel.: +8613949217563
E-mail address: [email protected]
1877-7058 © 2012 Published by Elsevier Ltd. Open access under CC BY-NC-ND license.
doi:10.1016/j.proeng.2012.04.222
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You Hui-yuan et al. / Procedia Engineering 37 (2012) 174 – 178
domestic and foreign theoretical analysis of large negative
clearance four-point contact ball bearing is a blank.
We use Hertz elastic contact theory to analyze negative
clearance four-point contact ball bearing and analysis method of
contact stress of the bearing under no load is given. Contact
stresses, shapes and sizes of contact areas of turntable bearing
3-645K (As is shown in Figure 1) are calculated. All these can
provide a theoretical basis and important data for bearing design.
2. Basic equation of elastic contact
Fig.1 Turntable bearing 3-645K
Elastic contact is to study two or more elastomers under
external load resulting from local stresses and deformation. The
sizes of contact areas and the magnitude of contact stresses are related to initial gaps, frictions and
magnitude of loads of the contact areas when two objects contact in the boundary. The sizes of contact
areas change constantly with loads in the process of load. That is, the boundary conditions are constantly
changing. They can not restore and are not reversible. The uncertainty and irreversibility makes it no
longer linear relationship between contact stress and external load. The states of stresses are related to the
order of load[1]. Therefore, contact problem is a kind of highly nonlinear behaviour, it has been always
one of the difficulties in nonlinear problems. The contact between steel ball and inner and outer ring
raceway of bearing is the same problem and it belongs to typical state nonlinear problem.
Elastic contact has the following two basic equations.
1)Balance equation
(1)
V ( x, y )dxdy Q
³³
Ac
In this equation, Q represents load, ı represents contact stress and Ac represents contact area.
2) Compatibility of deformation equation
V ( x' , y ' )dx' dy'
1
G z ( x, y )
/ ³³
SE Ac ( x x' )2 ( y y' )2
E'
1 P12 1 P22
E1
E2
(2)
(3)
In this equation, Z represents initial clearance of two contact objects, į represents the amount of
elastic deformation, E1 and E2 respectively represent young's modulus of the two objects, ȝ1 and ȝ2
respectively represent two elastic Poisson's ratio.
Solving these two equations is the basis of elastic contact problem.
3. Hertz elastic contact theory
Hertz contact theory is a basic and classical theory of contact deformation and stress calculation of
the elastomer. Hertz theory is founded on the following assumptions [2]:
1) Material of objects in contact with each other should be homogeneous and isotropic.
2) Load must be perpendicular to contact surface. The contact surface is completely smooth,
regardless of friction between the object and the surface.
3) Contact object only has elastic deformation. The deformation obeys Hooke's law and cannot
exceed the elastic limit.
4) The size of contact surface is small in comparison with radius of curvature of the contact surface.
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You Hui-yuan et al. / Procedia Engineering 37 (2012) 174 – 178
Hertz theory discusses the status of two objects with curvature in contact with each other when
external force acts (As is shown in Figure 2). Two elastomers
˄V1ǃV2˅separately have different radius of curvature on two
main planes˄P1ǃP2˅. When the value of load Q is zero they
contact with each other at one point. The contact area changes into
an ellipse when the load increases gradually. As contact area is a
part of the elastomer and far less than radius of the elastomer, so
we can use a semi-infinite plane to analyze local deformation.
Curvature of the curve which passes through the contact point and
contains the intersection of coordinate surface of normal line and
curved face changes along with the coordinate plane when two
objects with arbitrary curved face come into contact. Maximum
and minimum curvatures of which are called principal curvatures.
Principal curvatures are positive or negative. The convexity is
Fig.2 Model of Hertz contact theory
positive while the concavity is negative. Plane where the principal
curvature exists is called principal plane. Principal curvature function and curvature are calculated as
follows.
( U V 1P1 U V 1 P 2 ) ( U V 2 P1 U V 2 P 2 )
F (U )
¦U
(4)
¦U
UV 1P1 UV 1P 2 UV 2 P1 UV 2 P 2
(5)
In these two formulasˈȡ is principal curvature of the contact object, it is reciprocal of the radius
RV1P1, RV1P2, RV2P1, RV2P2 respectively. Concave curvature of the bearing raceway groove is a negative
value. Assuming that young's modulus of the objects are E1 and E2, Poisson's ratios are ȝ1 and ȝ2. We can
calculate sizes of semi-major axis and semi-minor axis of the contact ellipse as follows:
a
ª 3Q
a «
¬« 2¦ U
*
§ 1 P12 1 P22 ·º
¨¨
¸»
E2 ¸¹¼»
© E1
2
1
1
3
(6)
1
2
2
3
ª 3Q § 1 P
1 P ·º
¨¨
¸»
b* «
E2 ¸¹»¼
«¬ 2¦ U © E1
(7)
In the centre of contact area, relative displacements of the two objects are respective w1 and w2, the
amount of elastic deformation į can be calculated as follows:
G w1 w2
(8)
b
2
1
3
º
2 K (e) ª 1 § 3 1 P12 1 P22 · 2
(9)
G
) ¸¸ Q ¦ U »
« ¨¨ (
*
Sa « 8 © 2 E1
E2 ¹
»
¬
¼
Maximum contact stress is calculated as follows:
(10)
3Q ( ¦ U1 )2
3Q
V max
2
2Sab 3
§ 1 P12 1 P22 ·
¸
2S 3 ¨¨
E2 ¸¹
© E1
In above formulas a* and b* are coefficient of semi-major axis and semi-minor axis of the contact
ellipse which are related to the main curvature function f(ȡ) [3], 2k(e) / ʌa* is Hertz coefficient, k(e) is
complete elliptic integral of the first kind.
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You Hui-yuan et al. / Procedia Engineering 37 (2012) 174 – 178
The steel balls of turntable bearing 3-645K are made of GCr15, the inner and outer ring are made of
42CrMo.These two kinds of materials are similar in characteristic, therefore the formula of contact
calculation can be simplified, the sizes of contact areas can be calculated as follows:
Q 13
(11)
)
a 0.0236a * (
U
¦
b 0.0236b* (
Q
¦U
)
1
(12)
3
Here follows the formula of calculate maximum Hertz contact stress and the amount of elastic
deformation.
3Q
2Sab
4 2 K ( e)
2.79 u 10
Q2 ¦ U
Sa *
(13)
V max
G
>
@
1
3
(14)
4 Use Hertz theory to solve contact problem of turntable
bearing 3-645K
Turntable bearing 3-645K is a negative clearance four-point
contact ball bearing. The status of steel ball contact with the
inner and outer raceway at zero clearance is shown in figure 3.
Due to the negative clearance, elastic deformation between steel
ball and raceways has occurred when bearing idling. Contact
areas are ellipses after the deformation. Now the amount of
deformation between steel ball and raceways is known.
Therefore, it needs to be derived relationship between size of the
contact areas and the contact stress and the amount of elastic
deformation.
Fig.3 Model of contact of Turntable bearing
When clearance of the bearing is zero, the steel ball diameter
3-645K at zero clearance
d=15.875mm, the inner ring raceway radius of curvature ri
=8.41mm and diameter of flute Di =405mm. Same parameters of the outer ring raceway are ro=8.57mm
and Do=429.2mm. This time, the steel ball and the inner and outer ring raceway respectively have two
points of contact. Contact angles of the steel ball and the inner and outer ring are as follows:
Dm
arcsin
gm
2( rm d )
2
45o
˄m=i ៪ o)
(15)
For convenience, we assume that by changing ball diameter to change bearing clearance while the
remaining dimensions remain unchanged.The bearing clearance is negative when increasing the ball
diameter. Contact areas between the steel ball and inner and outer ring raceway are four ellipses. Because
inner and outer ring raceways are symmetrical geometric shapes, two contact ellipses of the inner ring
raceway are exactly the same. So does the outer ring raceway.
When the steel ball diameter d1 = 15.92mm, normal total amount elastic deformation between the steel
ball and the raceways is as follows:
(16)
G G i G o d1 d 0.045 mm
Radial negative clearance is as follows:
G1 Do Di ˄
2 d ho hi )
0.064mm
(17)
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You Hui-yuan et al. / Procedia Engineering 37 (2012) 174 – 178
2
rm ( rm d ) 2 sin 2 D m ( rm d ) cos D m d
2
2
2 ˄m=i ៪ o˅
Load of the steel ball of turntable bearing 3-645K can be obtained by formula (14)
hm
(18)
3
Q
2
ª
GSa *
1
4º
2
« 2.79 u 2 K ( e) u 10 » ( ¦ U )
¬
¼
(19)
(20)
QicosĮi=QocosĮo
The sum of curvature is Ȉȡ=0.1372 when steel ball contact with inner ringˈthe function of principal
curvature F(ȡ)=-0.9023, then a*=3.12˗b*=0.459˗2K(e)/ʌa*=0.677.
The sum of curvature is Ȉȡ=0.1298 when steel ball contact with outer ringˈthe function of principal
curvature F(ȡ)=-0.8629ˈthen a*=2.68˗b*=0.496˗2K(e)/ʌa*=0.730.
We can calculate sizes of the inner and outer of contact areas and maximum contact stresses by
formula (16), (19), (20) and (11), (12), (13). Similarly, we can follow same method to calculate the same
parameters with different negative clearances, the results are listed in Table 1.
Table 1 Contact sizes and maximum contact stresses of different negative clearances
7.9375
7.940
7.945
7.950
7.955
7.960
Total amount of normal elastic deformation˄mm ˅
0
0.005
0.015
0.025
0.025
0.045
Radial negative clearance˄mm ˅
0
0.007
0.021
0.035
0.049
0.064
major axis radius ˄mm ˅
0
0.72
1.25
1.60
1.89
2.14
minor axis radius ˄mm ˅
0
0.11
0.20
0.24
0.28
0.32
maximum contact stresses ˄MPa˅
0
754
1242
1779
2083
2389
major axis radius ˄mm ˅
0
0.92
1.07
1.40
1.65
1.87
minor axis radius ˄mm ˅
0
0.11
0.20
0.26
0.31
0.35
maximum contact stresses ˄MPa˅
0
845
1451
1837
2155
2481
steel ball radius˄mm ˅
Steel ball contact with
the inner ring
Steel ball contact with
the outer ring
5. Conclusion
1) This paper obtains contact analysis method for large negative clearance four-point contact ball
bearing based on Hertz contact theory, take turntable bearings 3-645K for example, normal contact
stresses of the steel ball and the inner and outer ring raceway are calculated under different negative
clearances when the bearing is idling.
2) The results show that the contact loads of four contact points of bearing increase with the absolute
value of negative clearances, small changes of the negative clearance have a great impact on the contact
stresses.
References
[1]Li Run-fang, Gong Jian-xia, Contact Problems in Numerical Method and Application of Machine Design, 1991
[2]Wan Chang-sen, Analysis of Rolling Bearings, 1987
[3] Harris TA, Rolling Bearing Analysis, John Wiley and Sons, 1984.
[4] G.Lundberg˂A.Palmgren, Dynamic Capacity of Rolling Bearings, Acta Polytech, 196 (194)