 Open Access
 Total Downloads : 191
 Authors : Shaochuan Li , Yanshuang Wang
 Paper ID : IJERTV6IS090069
 Volume & Issue : Volume 06, Issue 09 (September 2017)
 Published (First Online): 13092017
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Calculation Method of Dynamic Load Bearing Curve of Doublerow Fourpoint Contact Ball Bearing
Shaochuan Li
College of Mechanical Engineering Tianjin University of Technology and Education
Tianjin 300222, China
Yanshuang Wang
College of Mechanical Engineering Tianjin University of Technology and Education
Tianjin 300222, China
AbstractOn the basis of the static analysis of doublerow fourpoint contact ball bearings, the relationship between the basic rated life of bearings and the dynamic loads of bearings were deduced. According to the definition of dynamic load bearing curve of bearing, the dynamic bearing curve of doublerow fourpoint contact ball bearing was plotted. The calculation case and application case of dynamic bearing curves of doublerow fourpoint contact ball bearings were also given.
Key words DoubleRow FourPoint Contact Ball Bearing; Dynamic Load; Bearing Curve

INTRODUCTION
Wind energy as a renewable clean energy has been paid attention all over the world, and the wind power generation technology developed rapidly. Yaw bearings and variable pitch bearings are the key components of wind turbines, and the structure is mostly doublerow fourpoint contact ball bearings with inner ring or outer ring with teeth. The wind turbines in the work are usually subjected to combined loads (axial force, radial force and overturning moment) and impact loads, which requires that the yaw bearings and the variable pitch bearings have sufficient carrying capacity. The dynamic bearing curve of the bearing shows the maximum dynamic load bearing can bear under the premise of the given life, which is of great significance for the selection of the bearing and the load .
At present, the research of bearing load curve mainly focuses
[14,15].In this paper, we deduced the mathematical relation between the basic rated life and the dynamic load of doublerow fourpoint contact ball bearing on the basis of static analysis, introduce the drawing method of dynamic load curve, and give the calculation and application case.

EXACT SOLUTION OF CONTACT FORCE
The structure of the doublerow fourpoint contact ball bearing is shown in Figure 1:
Fig.1 Stucture of doublerow fourpoint contact ball bearing
Build the bearing coordinate system as shown in Figure 2. x axis along the bearing axis direction, r is the inner diameter direction, each ball position angle j can be expressed as:
j=2 ( j1)/(z/2) (j=1,2,3… Z/2) where Z is the number of steel balls in double row bearings.
x
y
r
j
on the static load bearing curve[19]. In 2012, Wang Hong and o z
others gave the theoretical calculation formula of bearing
capacity and life estimation of multi row roller slewing bearing, and introduced simplified drawing method of bearing dynamic and static bearing capacity curve based on Hertz
contact theory, LundbergPalmgren fatigue life theory and the special geometric structure characteristics and working conditions of multi row roller slewing bearing[10]. This method is generally used to quickly test whether the load is available, but the calculation results are not accurate enough.
Abroad, GÃ¶ncz P presented calculation model of dynamic and static bearing capacity of three row roller bearing and analyzed the static bearing capacity of large doublerow
fourpoint contact ball bearing[1113]. Kania L et al analyzed
Fig.2 Coordinated system of the bearing
The contact pairs mainly subjected to axial force symmetry for contact 1 (upper), contact 3 (below), the other two contact pairs, respectively called contact 2, contact 4.
Before being subjected to an external load, when take axial clearance ua into account, the center of curvature of the inner and outer groove of any pair of steel ball contacts A can be obtained by the following formula:
1
the bearing capacity of cross roller slewing bearings, and gave the bearing curves of bearings under different cross angles
A fi fe 1DW
2 ua cos0
(1)
Where fi is the radius coefficient of the inner raceway curvature; fe is the curvature radius factor of the outer raceway; Dw is the diameter of the steel ball; 0 is the contact angle of initial position.
When the axial clearance ua =0, the curvature center distance of the inner and outer groove A0 can be obtained by the following formula:
the radial displacement and the inclination angle of the inner ring when the inner ring bears the axial dynamic load Fa, the radial load Fr and the overturning moment dynamic load M; the radius of curvature of the raceway Ri = 1/2 dm(fi0.5) Dw cos01/4ua(cos0)2;
Where dm is the diameter of the bearing pitch circle; j is
the position angle of steel ball.
A0 fi fe 1DW
(2)
After the displacement of the inner ring, the contact angle
Suppose the outer ring is fixed and the inner race rotates,
kj of contact pairs k (k=1,2,3,4) at position j is:
the outer load acted on the inner ring is shown in Figure 3.
Where Fa is the axial dynamic load, Fr is the radial load, M is
arcsin( Asin 0
a

Ri cos j )
(4)
the the overturning moment dynamic
k j
j
Ak
The inner ring is in equilibrium under the action of external load and normal contact load Qkj, and the forces acting on the inner ring are shown in Figure 4
Fig.3 External applied loads on bearing
load, dm is the pitch circle diameter of the bearing. dc is the center distance between the two rows of ball bearings of doublerow fourpoint contact ball bearing.
When the doublerow fourpoint contact ball bearing is
Fa M
Fr
Q2
j
j
j
j
Q1 Q4 Q3
subjected to an external load, the inner ring is displaced, and the center of curvature of the groove of all pairs of contacts has changed. The center of curvature between the inner and outer groove of the contact pairs k (k=1,2,3,4) at the position angle j is Akj:
Fig.4 Forces acting on inner raceway
Asin
0

a

Ri cos j
2 2
1
A
A cos
cos 2
1 j
0 r j
0.5dc cos j
1
Fig.5 the diagram of raceway groove of bearing
Asin
0
– a
– Ri cos j
2 2
According to mechanical equilibrium equation:
2
j
A
A cos
cos 2
0
r
j
2 Q
1 j
Q
sin
1 j
sin
– Q
2 j
– Q
sin
2 j
– F
0
sin a
0.5dc cos j
(3)
j 0
3 j
3 j
4 j
4 j
Asin

R cos
2 Q1
cos 1
– Q2
cos 2
j r
1
2 2
0 a i j
j
Q3
j
cos 3
j
– Q4
j
cos 4
cos – F 0
A
A cos
cos 2
j 0
j
2 Q
j
sin
j j
– Q sin
(5)
3 j
0 r j
1 d
1 j
1 j
2 j
2 j
cos
– 0.5d cos
2 m Q
sin
– Q sin j
1
c
j
j 0
3 j
3 j
4 j
4 j
Asin –
– R cos
2 2
1 2 Q1 j
0j
dc
cos 1
– Q2
cos 2
j
j
j
j
cos j – M 0
j
j
A
0
A cos
a i j
cos 2
2
Q3 j
cos 3
– Q4
cos 4
4 j
0 r j
– 0.5d cos
Where Qkj is the normal contact load of contact pairs k at
c
j
position j; dc is the center distance between the two rows of ball bearings of doublerow fourpoint contact ball bearing.
Where a, r and respectively are the axial displacement,
Qkj can be get according to the Hertz contact theory:
K
1.5 , 0
The equivalent rolling load on the outer raceway k is:
Qk
n k j
k j
0
(6)
j 0,
2
0.3
k j
Q 1 Q10/ 3
(10)
evk
Z
k j
Where, Kn is the total load deformation constant of the rolling body and the inner and outer rings; kj is the total elastic contact deformation between the steel ball and the inner and outer raceway, along the direction of contact pairs k, at position j:
j 0
Where Qkj is the contact load of steel ball.

Rated Life Calculation of Inner Ring.
j
k
Ak A0
(7)
The rated life of each raceway on the inner ring is:
j
3
According to the given geometric parameters of the bearing and an initial value of inner ring displacement (a, r,
L10ik
(Qci
/ Qek )
(11)
), A, A0 and Akj can be calculated through formula 1~3. Put the values of A, A0 and Akj into the formula 7 to obtain kj, Then, Qkj and kj are calculated by formula 6 and 4,
Rated life of inner ring is:
10ik
4 0.9
respectively. Put Qkj and kj into the formula 5, while
L10i
L10/ 9
(12)
making Fr=0, Fa and M for continuous values, according to formula 5, using the NewtonRaphson method, to obtain the final value of bearing inner ring displacements (a, r, ) under each working conditions (Fa, M, Fr). By formula 6, the
k 1

Rated Life Calculation of Outer Ring.
The rated life of each raceway on the outer ring is:
normal contact load Qkj of each position angle of the bearing
L (Q / Q )3
(13)
is obtained.
10ek
ce evk


CALCULATE THE BASIC RATED LIFE OF THE BERARING
The raceway of doublerow fourpoint contact ball
bearings is a typical peach shaped groove. The steel ball has
Rated life of outer ring is:
10e 10ek
4
L ( L10/ 9
k 1
) 0.9
(14)
four contact points with the inner and outer raceway, which correspond to four channels. Name the four channels as channel 1, 2, 3, 4, as shown in Figure 5.
The rated life of doublerow fourpoint contact ball bearings L10 can be obtained by fitting the rated life of the inner ring and the rated life of the outer ring:

Basic Dynamic Load of Bearing.
L L10/9 L10/ 9 106
(15)
For doublerow fourpoint contact ball bearings, the rated
10 10i
10e
dynamic load of the rings Qci(e) is:


APPLICATION CASE
The structural parameters and material parameters of a
Q 98.1
2 fi(e)
0.41
1 1.39
certain type doublerow fourpoint contact ball bearing are
ci (e)
D
0.3
2 f
i (e)
1
1 1/ 3
(8)
shown in Tab 1:
W
TABLE.1 Parameters of a Doublerow Fourpoint Contact Ball Bearing
W
d
Z 1/ 3 3.624D1.4
m
parameter values
In the formula, the symbol i stands for the inner ring, and the symbol e stands for the outer ring; and are correction factors for doublerow fourpoint contact ball bearings.

Basic Equivalent Dynamic Load of Bearing.
As the outer ring is fixed and the inner race rotates, the equivalent rolling load on the raceway k of the inner race is:
Pitch diameter of ball set dm [mm] 2215 Ball diameter DW [mm] 44.45
The center distance of double row steel ball dc
69
[mm]The radius of curvature of inner channel ri [mm] 23.34
1 2
1/3
3
(9)
The radius of curvature of outer channel re [mm] 23.34
Z
Qek Q
0
k j
j
The number of balls Z 128Ã—2 Poisson ratio of ball and ferrule v 0.3
Axial play ua [mm] 0.01 Elastic modulus of ball and bearing rings E [Gpa] 207
The bearing's structural parameters, material parameters
and an initial value (0, 0, 0) of the displacement of the inner ring are substituted into above calculation method to obtain the rated life of the bearing L10. The axial dynamic load Fa and overturning moment dynamic load M which conform to the L1030000< are extracted as the points on the coordinate system. This example takes =0.01 and obtains a series of points, as shown in Figure 6.
The method of using the dynamic load bearing curve to determine whether the bearing selected meets the life requirement under given load are as follows:

Users provide the axial dynamic load FaA, overturning moment MA and rated life LA of the doublerow fourpoint contact ball bearings. In this example, the doublerow fourpoint contact ball bearing has a rotational speed of 0.1r/min, and the required service life is 175200 hours. That is to say LA=1051200 r. The axial dynamic load and overturning moment respectively are: FaA=1100kN and MA=500kNÂ·m.

In the dynamic load bearing curve, find the coordinate point corresponding to the given load FaA and MA, and use the point "A" to indicate. In this example, the coordinates of point "A" are ( FaA , MA ), in which FaA=1100kN and MA=500kNÂ·m.

Connect the coordinate system origin and point "A", extend above line and crossed the dynamic load bearing curve to point "B". As shown in Figure 8, find the coordinate value of "B" point. In this example, the coordinates of the point "B" are ( FaB , MB ), where FaB=4000kN and MB=1800kNÂ·m.

Calculate load factor fL. In this example, the load factor
fL=FaA/FaB=4000/1100=3.636.

Calculate the life of the selected doublerow fourpoint contact ball bearings at a given FaA and M conditions:
L=30000f 3. In this case, L=30000 f 3=1442524.4 r.
L L
Fig.6 Force combination position points of doublerow fourpoint contact ball bearings
Connected above points, the dynamic load bearing curves of the doublerow fourpoint contact ball bearings are obtained, as shown in Figure 7.
Fig.7 Dynamic load bearing diagram of double row four point contact ball bearing
Fig.8 Schematic diagram of "A" and "B" points

Determine whether the condition LLA is established, if it is established, the designed bearings can meet the life requirements under the given load FaA and MA, otherwise, the designed bearings cant meet the life requirements. In this example, L=1442524.4>LA=1051200, so the doublerow fourpoint contact ball bearing selected can meet the requirements of life under the given load FaA=1100kN and M=500kNÂ·m.



CONCLUSION
This paper deduced the relationsip between the basic rated life and the dynamic load of bearing based on the static model of doublerow fourpoint contact ball bearing. The bearing curve of bearing dynamic load is drawn, axial dynamic load Fa as the abscissa, overturning moment dynamic load M as the ordinate, according to the definition of dynamic load bearing curve of rolling bearing. The point on the curve can be understood as the dynamic load that the bearing can bear when the bearing life is given value. The calculation of dynamic carrying curve of doublerow fourpoint contact ball bearing provides a basis for the selection and application of such bearings.
ACKNOWLEDGMENT
This research is supported by the National Science Foundation of China(No. 51475143) and Tianjin Natural Science Foundation (No.16JCYBJC18900).
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