Experimental Techniques

https://doi.org/10.1007/s40799-018-0237-2

Load Distribution Calculation of a four-Point-Contact Slewing Bearing

and its Experimental Verification
R. Liu 1 & H. Wang 1 & B.T. Pang 2 & X.H. Gao 3 & H.Y. Zong 3

Received: 20 July 2017 / Accepted: 15 January 2018

# The Society for Experimental Mechanics, Inc 2018

Abstract
Analysis of the load distribution along raceways is fundamental for assessing dynamic and static capacity of the slewing bearings.
In the literature, several finite element (FE) models have been proposed to estimate the load distribution and the deformation of
slewing bearing. Although FE simulations are a useful tool for designers, there are a few experimental studies assessing accuracy
and reliability of computations. In this paper, a FE model for a single-row four-point-contact slewing bearing is built in ABAQUS
software. The key to the method is to simulate the balls under compression by traction-only nonlinear springs. Subsequently, the
static loading experiment is designed: strain gauges are mounted on the inside circumference of the inner ring to obtain load
distribution. The comparison between numerical and experimental results is discussed in detail. Remarkably, FE results are
consistent with experimental data.

Keywords Slewing bearing . Nonlinear spring . FE modelling and analysis . Load distribution

Introduction slewing bearing can be converted into the radial force Fr, axial
force Fa and tilting moment M that caused the contact loads
Slewing bearings are important parts of construction machinery, between ball and raceway. To design slewing bearings with high
wind turbines and other rotary machinery with gear wheels on static carrying capacity and reliability, the emphasis is on anal-
the outer or inner ring. Once the slewing bearing malfunctions, ysis of the load distribution and calculation of the maximal
the machine will fail. As a result, the requirement of reliability contact load. In recent years, a variety of calculation approaches
and security is very high. The static capacity and fatigue life are have been proposed. Antoine, Liao, Harris, et al. [3–6] derived
the main indices for testing slewing bearings. A large-size the distributed solving method of the contact load for the four-
slewing bearing is often subjected to heavy combined loads. contact-point slewing bearing using an analytical method; how-
The rotational speed of a slewing bearing usually ranges from ever, the geometry and the stiffness of the rings were not taken
0.1 to 5 rpm. As mentioned above, the operating condition of into consideration. On the basis of the Hertz contact theory,
such bearings is significantly different from those of convention- Zupan, Gao, et al. [7–9] developed an iterative procedure to
al bearings; thus, many theories that are suitable for conventional calculate the contact pressure over raceways for a single-row
bearings cannot be used in the design of slewing bearings [1, 2]. four-contact-point slewing bearing; however, the rings were
The main failure modes of a slewing bearing are spalling, considered as rigid. In the work of Zupan and Prebil [7], they
pitting and abrasion on the raceways. All the loads subjected to a studied the impact of the flexibility of the rings and supporting
structures using an approach similar the super-element theory to
overcome the disadvantage mentioned above; however, they did
* H. Wang not propose how to build a ball-raceway contact model and the
wanghua@njtech.edu.cn calculation cost is too high. With the development of computing
technology, a tedious calculation process can be performed by
1
School of Mechanical and Power Engineering, Nanjing Tech computers, thereby enabling the geometry parameters and the
University, Nanjing 211816, China stiffness of the raceways to be taken into consideration; such
2
Luoyang LYC Bearing Co., Ltd, Luoyang 471003, China parameters were simplified in the traditional calculation. To cal-
3
Shanghai OujiKete Slewing Bearing Co., Ltd., Shanghai 201906, culate the load distribution precisely, FEM is widely used in the
China modelling of a slewing bearing. Kania [10] and Kunc [11]
 Exp Tech

modelled the ball and the raceway using an eight-node three-

dimensional solid element. However, the contact region’s scale
is much smaller than the whole bearing. As a result, the ap- Pr
proach is only suitable for the analysis of the contact region.
Gao [12] built a local-global model to calculate the load distri- Cid Cid
bution of two nonlinear springs only; however, the springs con-
nect the raceways directly so that the variation of the contact R d a
angle is ignored. However, the true contact angle is variable
under external loading. Daidié [13] and Smolnicki [14]
α0 a
modelled the ball-raceway contacts with a super-element
consisting of two nonlinear springs, eight rigid beam elements Ciu h h Ceu
and four rigid shell elements, and the vibration of the contact
angle is considered. This local-global analysis method is difficult
to implement because of the building of the super-element. The
result of this model is more convincing than analytical results
proposed by Zupan and Prebil [7].
However, because of the size and form of the slewing bear- Fig. 2 Cross-section of a slewing bearing
ing, few static loading experiments have been performed to
validate the consistency between simulation and engineering
practice and analyse the errors between them in a technical d and R denote the diameter of the ball and the radius of the
paper. In this paper, the static loading experiment was per- raceways, respectively, and h and a denote the horizontal ec-
formed to obtain the load distribution. The ball-raceway con- centricity and vertical eccentricity of the curvature centres,
tact behaviour was also simulated using traction-only nonlin- respectively. Each ring of the single-row four-contact-point
ear springs that have the same stiffness as that of the balls. The slewing bearing can be defined by two raceways comprised
comparison between numerical and experimental results was of the toroidal surface. Ciu, Cid, Ceu, and Ced denote the centres
discussed by calculating relative errors and RMS error. of curvature (I and e corresponding to the inner and outer ring,
respectively and u and d corresponding to the up and down
raceway, respectively). The groove curvature radius can be
Finite Element Model and Analysis calculated using (equation (1)). According to the manufac-
of the Slewing Bearing turer’s feedback in the process of production and related doc-
uments, the coefficient of raceway curvature f is 0.525. The
Parametric FE Model other parameters required in the modelling process are shown
in Table 1.
The research object is a single-row four-contact-point slewing
R
bearing marked as 012.40.1000. Figure 1 shows the structure f ¼ ð1Þ
d
and the main geometric dimensions of the slewing bearing.
The methodology presented in this paper is principally where f is the coefficient of raceway curvature, R is the cur-
based on the geometry. Figure 2 shows the cross-section of a vature radius of the raceways, and d is the diameter of the ball.
slewing bearing. Here, Pr denotes the cross-section clearance, The driven gear ring and the installation bolts holes have a
great effect on the load carrying capacity, but they have little
1001
effect on the analysis of the load distribution over raceways.
878 Based on Saint Venant’s principle [15], the driven gears and

Table 1 Specification of the parameters of the single-row four-contact-

100
90

ball slewing bearing

40 1000
90

Parameters Size

Number of balls, n 69
998 Diameter of balls, d (mm) 40
1078 Contact angle, α0 45°
1122 Coefficient of raceway curvature radius, f 0.525
Groove curvature radius, R (mm) 21
Fig. 1 Schematic of the single-row four-contact ball slewing bearing
Exp Tech

Spring Peu-Pid Spring Ped-Piu

installation bolts holes that are far from the contact pairs were
omitted to simplify computations. As the model and loads are
2a
symmetric, half of the slewing bearing rings were built in
ABAQUS software. Table 2 shows the mechanical parameters
of the slewing bearing. Ced

2b
Cid

Ball-Raceway Contact Simulation with Nonlinear

Springs rigid shell element
Ciu Ceu
Balls and raceways bear most of the loads when a slewing rigid beam element
bearing is in operation. The contact pairs are built when the
Y
ball was extruded by the raceways under external loads. The X
relationship between the contact force and the total deformation
is nonlinear. Similar to Daidié’s approach [13], the ball- Fig. 3 Model of the contact
raceway contact behaviour can be modelled by traction-only
nonlinear springs. Just as shown in Fig. 3, the nonlinear spring
3=2
PeuPid simulating the contact pair in the direction of CeuCid δ
Q¼K ð3Þ
between the up raceway of the outer ring and the down raceway 2
of the inner ring, and the nonlinear spring PiuPed simulates the
Where Q is the contact load, K is a constant factor (contact
contact pair in the direction of CiuCed between the up raceway
stiffness), δ is the total deformation of the contact between two
of the inner ring and the down raceway of the outer ring.
raceways and each two raceways and each rolling ball.
However, if the model only uses nonlinear springs, then the
The load-deformation relationship of the non-linear spring
vibration of the contact angle will be ignored and the local
is expressed by (equation (4)).
deformation will not be perturbed by further singular defor-
mation. To overcome the faults mentioned above, the four- F ¼ KsΔ ð4Þ
contact regions in the raceways can be tied by four rigid shell
elements, with each contacts zone being coupled to the corre- where △ is the deformation of the nonlinear spring, Ks is the
sponding centres of curvature at two nodes via two rigid beam elastic coefficient of the nonlinear spring, and F is the load
elements. The model of the contacts is shown in Fig. 3; 2a and acting on the nonlinear spring.
2b in the diagram represent the width and length, respectively, In the case of △ = δ, we have F = Q. The elastic coefficient
of the ball-raceway contact ellipse under the maximal load. of the nonlinear spring can be calculated using the following
Using this approach, the contact loads can be determined, and equation.
the variation of the contact angle can be obtained from dis-
K s ¼ 2−ð3=2Þ KΔ1=2 ð5Þ
placement of the Ced, Ceu, Cid, Ciu by using the following
(equation (2)). According to the Hertz contact theory, the value of K in this

 equation can be calculated easily.
ΔY
β def ¼ arctan ð2Þ The original length of the nonlinear spring is given by
ΔX
(equation (6)).
Where βdef is deformed contact angle, and △Yand △X is the
L ¼ 2R−2Pr −d ð6Þ
displacement of the curvature centre in the y-direction and x-
direction, respectively. where d is the diameter of the ball, Pr is the clearance, and R is
On the basis of Hertz contact theory, the load in the direc- the radius of the raceway.
tion of the ball-raceway contacts can be calculated using A ball is replaced by a pair of nonlinear springs, as shown
(equation (3)) [16]. As a result, the ball-raceway contacts in Fig. 3. Thus, half of the model requires 35 pairs of springs
can be simulated by nonlinear springs as long as the load- to simulate the ball. In the Interaction module of the
deformation between them is the same. ABAQUS software, nonlinear springs were created via the
spring/damper menu tool, with the load-deformation charac-
Table 2 Slewing bearing mechanical parameters teristics shown in Fig. 4. The connecting endpoints of the
nonlinear springs and the rigid beam elements correspond to
Part Material Elastic Modulus (MPa) Poisson’s ratio the ball-raceway contacts points.
Inner and outer rings 42CrMo 2.12 × 105 0.28
According to the Hertz contact theory, the ball-raceway
contact area is elliptical in shape, with the two parameters a
 Exp Tech

60000

50000

40000
Q (kN )

30000

20000

10000

0
0.00 0.05 0.10 0.15 0.20 0.25
(mm)
Fig. 4 Load-deformation relationship of the nonlinear spring

and b. The rigid shell elements are made by 2a × 2b- Fig. 5 Finite element model of the ball-raceway contact
rectangular shapes instead of the elliptical shape to simplify
the model. The sizes of rectangular shapes for each applied node linear hexahedral element, reduced integration) and a
load are shown in Table 3. sweep hexahedral mesh. The rings are divided, and the width

 of these elements in the contact regions (which are tied to rigid
1 −0:4091 1 =
−2
d 3 Qmax =3
1
a≈1:71  10  1− ð7Þ shell elements) is equal to the half-width of the contact ellip-
2f tical shape.


1 0:1974 1 = There are 584,073 DOFs in the FE model: this allows mesh
−2
d 3 Qmax =3
1
b≈1:52  10  1− ð8Þ independent solutions to be obtained. To facilitate the appli-
2f

 cation of the boundary conditions and the external loads, a
2 Fr 2F a 4M reference point named Rp-inner-ring in the geometric centre
Qmax ¼ þ þ ð9Þ
Zcosα0 Zsinα0 DZsinα0 of the inner ring and a reference point named Rp-outer-ring in
the geometric centre of outer ring were defined. Next, the
where Qmax is the maximal contact load, D is the diameter of
kinematic coupling can be set between the Rp-inner-ring and
raceways, Z is the number of the ball, α0 is the contact angle,
the interface of the inner ring and the installed base. Similarly,
and Fr, Fa, and M are radial, axial and tilting moment loads,
kinematic coupling can be set between the Rp-outer-ring and
respectively.
The dimensions of the rigid shell elements are given from
(equations (7) ~ (9)). Equation (9) is an empirical formula
proposed by the U. S Renewable Energy Laboratory to deter-
mine the maximal contact load [17]. Figure 5 shows the true
scale of the finite element model of the ball-raceway contact.

Meshing and Loading

The complete finite element model of the slewing bearing is

shown in Fig. 6. The rings use the C3D8R element (eight-

Table 3 Load values applied on the ring and the contact areas’ size

Load Level Fa(kN) M (kNm) 2a(mm) 2b(mm)

0 0 0 0 0
1 39.7 79.3 8.144 2.071
2 60.9 152.3 9.984 2.539
3 91.5 215.9 11.252 2.862
4 129 304 12.074 3.208
5 158.2 400 13.768 3.502
Fig. 6 (a) Local finite element model; (b) complete finite element model
Exp Tech

Clamping Spring Peu-Pid

26000 Spring Ped-Piu
24000
22000
20000
18000
35

Contact force (N)

16000
Rp-inner-ring 14000
Fa
M 12000
10000
Rp-outer-ring Fr 8000
0
6000
Symmetry conditions for displacement
4000
2000
Fig. 7 Applied loads to the slewing bearing 0
0 5 10 15 20 25 30 35
Number
the interface of the outer ring and the upper slewing mechanism. Fig. 9 Contact load distribution curves over the raceways
The Rp-inner-ring has completely fixed constraints added, and
the Rp-outer-ring has three degrees of freedom displacement forces of 35 balls are presented by lines with circle and square
constraints added by restraining two rotational degrees of free- markers, which represent spring PedPiu and PeuPid respectively.
dom (UR1、UR2) and one displacement degree of freedom The curves under combined loads are clearly not symmetrical
(U3) and releasing the degree of the freedom in the direction and the nonlinear springs relative to balls from 4# to 21# are
of the tilting moment (UR3), the axial degree of the freedom both active, which is derived from the theoretical results indi-
(U1) and the radial degree of freedom (U2). Next, five models cating that balls are loaded either in the direction of PeuPid or
with the same size but different rigid shell elements and different PedPiu, but not in both of them. The reason for the above-
loads applied on the outer ring were modelled; the equivalent described phenomenon is the flexibility of the rings. In the zone
loads were sited on the Rp-outer-ring, as shown in Fig. 7. from 1# to 17#, the contact loads of spring PeuPid are higher
Table 3 shows the value of the five load levels. In this manner, than that of PedPiu, because the axial force is much greater than
the computational task was established. The computer configu- the tilting moment and because the contact load of spring
ration is AMD Opteron 2220, NVIDIA Quadro FX 5500 inde- PeuPid is decreasing and that of PedPiu is increasing via the
pendent display card, 300-gigabyte hard drive, 16 GB of mem- tilting moment weaken the effect of the axial force. In the zone
ory. It takes 1089.2 s to analyse the model. from 18# to 35#, the variation trend of the two springs is the
same as that in zone 1, but the contact load of spring PedPiu is
higher than that of spring PeuPid. In the zone, the tilting moment
Analysis of FE Results is much higher. It can also be seen that maximal contact pres-
sure of 23,665 N occurred in spring PedPiu, verifying that the
The deformation contour of the slewing bearing for the combi- tilting moment has a greater influence than the axial force. In
nation of axial force and tilting moment, with Fa = 91.5 kN and addition, the maximal variation of 12 degrees is found at the
Mn = 215.6 kNm, is shown in Fig. 8. The deformation zoom springs of the maximal contact load according to (equation (2)).
coefficient was magnified 100 times to observe the displace- The maximal contact loads are given by (equation (9)).
ment deformation clearly. The load distribution curves over the Table 4 shows the calculation results. The maximal contact
raceway were obtained by extracting the loads of the nonlinear loads obtained from the finite element model are found to be
springs, as shown in Fig. 9; the ordinate represents the number higher than those of the results obtained from (equation (9)),
of the ball, and the abscissa represents the contact loads. Each and the relative error is only 7.4%. Note that the maximum
marker in Fig. 9 represents a nonlinear spring. The contact contact load is consistent with theory.

Fig. 8 Deformation contour of

the slewing bearing
 Exp Tech

Table 4 Comparison of G2
the maximum contact Module Qmax(N) relative error
load obtained by
different models FEM 22,665 0
Equation (9) 21,105 7.4%

69# 1# 2#
Experimental Validation

As mentioned above, the load distribution and the maximal

contact were obtained using finite element simulations and an
empirical formula. However, numerical results must be vali- 53# 18#
dated experimentally.

Static Loading Experiment

Because of the form of the slewing bearing, the contact load 36#
cannot be obtained directly. Hence, resistances strain gauges
were mounted on the inside circumference of the inner ring.
The strain gauges’ type is BE120-3AA produced by Tongda
Trading Company. The grid size is 2.8 × 2 mm, the standard G1
resistance is 120 Ω, and the sensitivity coefficient is 2.08 ±
Fig. 11 Layout diagram of the strain gauge measuring points
1%. By measuring strains with strain gauges, the load distri-
bution can be obtained indirectly. The static loading experi-
ment was conducted on the slewing bearing test stand.
Figure 10 shows a picture of the slewing bearing test stand. upwards and G2 provided pressure downwards. The loading
The acquisition unit is DH3815N-3 static strain test system provided by the cylinders can be divided into five levels, as
produced by DongHua Testing Technology Co., Ltd. It uses shown in Table 3. Note that all load values are below limit loads.
USB port to communicate with computer. There are 25 acqui- There are 69 steel structure strain gauges mounted internal
sition channels during experimental tests. Every acquisition profile of the inner ring. These gauges were glued onto the
channel can sample continuously at the rate of 2 Hz. inner circle of the inner ring according to the position of the
Before testing, the balls, separate units and grease were balls. Figure 11 shows the strain gauge measuring points in
allowed to operate at high speed based on the accelerated theory. The deformation direction of the strain gauges is the
experimentation approach to achieve the normal working con- same as the axial direction of the slewing bearing. Resistance
dition. Next, axial force Fa and tilting moment M were applied strain gauges are sensitive to environmental temperature var-
on the outer ring by controlling the output pressure of cylin- iations. It will lead to the change of resistances and error will
ders G1 and G2. It can be seen that G1 provided tension be produced. Besides, the materials of rings and strain gauges

Fig. 10 Test stand for the slewing

bearing

G2 Slewing bearing G1
Exp Tech

Fig. 12 Actual layout diagram of

the strain gauge measuring points

have different thermal expansion coefficients. The strain gra- theoretical results. Ring deformation is minimum in the zone
dient between strain gauges and rings will also lead to errors, near the driver, causing the strain gauges to be more easily
so temperature compensation is necessary. The temperature affected by other factors independent of the experiment. In the
self - compensated strain gauges are used to reduce error. third zone, which is from 25# to 45# near cylinder G2, the
However, the strain gauges cannot be mounted evenly spaced strain gauges suffer from heavy pressure because the tilting
in practice because of the existence of nozzles, nameplate and moment strengthens the effect of the axial force.
other components. As a result, the positions of the measuring
points were adjusted appropriately. Figure 12 shows the strain
gauge measuring points in practice.
Experimental Validation of FE Model
Analysis of Experimental Data In relation to the strain gauges’ pasted position in the experi-
ment, contact strains in the finite element model can be ex-
Under combined loading of the axial force and the tiling mo-
tracted from the inner circle of the inner ring. Figure 14 shows
ment, the experimental results of 1#-69# strain gauges were
the strain distribution curves, with the strain gauges’ position
collected by the DH3815N-3 instrument. The strain distribu-
on the horizontal axis and the strain value on the vertical axis.
tion curves under five levels loads are shown in Fig. 13; the
The five curves denote the strain distribution of the five levels
number of strain gauge measuring points is indicated on the
loads.
horizontal axis, and the strain value is indicated on the vertical
Comparing the above diagrams in Fig. 13 and Fig. 14, one
axis. It can be seen that the change trend of fives curves is
can notice that experimental data agree with FE simulations.
consistent. The larger the loads, the higher the strain. The
Contact load distribution is found to be basically consistent
curves can be divided into three zones to analyse the results
and the heavy load is distributed in the 25#-45# strain gauges
carefully. In the first zone, which is from 1# to 10# and 59# to
(around cylinder G1), the lower loads are distributed in the 1#-
69# near cylinder G1, the strain gauges suffer from light pres-
10# and 59#-69# strain gauges (around cylinder G2), and the
sure because the tilting moment weakens the effect of the axial
loads in the rest of the positions are smaller. The maximal
force. In the second zone, which is from 15# to 25# and 49# to
56#, the strain suffers from tension that deviates from the 0 10 20 30 40 50 60 70
0
60
40 -20
20
0 -40
Strain (µe)
Strain (µe)

0 10 20 30 40 50 60 70
-20
Level 1
-40 -60
Level 2
Level 1 Level 3
-60 Level 4
Level 2
-80 Level 5
-80 Level 3
Level 4
-100 Level 5 -100
-120
Fig. 13 Strain distributions measured experimentally Fig. 14 Strain distribution obtained by FEM
 Exp Tech

Fig. 15 Comparison of result of 30

(a) Level 1; (b) Level 2; (c) Level 20 experiment
3; (d) Level 4; (e) Level 5 20
experiment simulation
simulation 10
10

Strain (µe)
0

Strain (µe)
0 10 20 30 40 50 60 70
0
0 10 20 30 40 50 60 70 -10
Number
-10 -20

-30
-20
-40

-30 -50
(a) (b)

40
experiment
40
simulation experiment
20 simulation
20

Strain (µe)
Strain (µe)

0 0
0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70
Number -20 Number
-20
-40

-40 -60

-80
-60
-100
(c) (d)

60
40 experiment
simulation
20
Strain (µe)

0
0 10 20 30 40 50 60 70
-20
Number
-40
-60
-80
-100
-120

(e)

Table 5 Relative error on maximum contact strain

Loads levels Maximum strain of Maximum strain of Error Table 6 RMS error on
the simulation (με) the experiment (με) the strains Load levels RMS error

Level 1 19 26 26.9% 1 32.25

Level 2 38 43 11.6% 2 54.39
Level 3 53 60 11.6% 3 91.80
Level 4 75 84 10.7% 4 120.92
Level 5 100 112 10.7% 5 143.14
Exp Tech

strain is distributed in 34# or 35# in both experiment and error is of the order of just 10%. RMS errors also are
simulation, in agreement with the analytical result proposed reasonably small. Hence, the proposed model can be
by Zupan and Prebil. Figures 13 and 14 are combined into used for analysing the load distribution in a slewing
Fig. 15(a) through (e) to determine relative errors. Table 5 bearing.
shows the relative error of the maximum load in the simulation (2) The load distribution curves of the slewing bearing under
and the experiment. The maximum relative error of the max- external loads were plotted. It was found that the large-
imum contact load is 26.1% in the five levels. The others are loads zone was close to the G2 cylinder, where tilting
all below 12%. However, note that the maximum strains in the moment strengthens the effect of the axial force. The
simulation are all less than those in the experiment, even effect of tilting moment M is greater than the effect of
though the stiffness of the rings and the variation of the contact axial force Fa on the contact loads. The fault diagnosis
angle are considered. This becomes evident when the external and lifetime analysis should mainly consider high load
loads are small. zone. The load distribution over the raceways can pro-
The root mean square (RMS) error can reflect the precision vide an important foundation for designing slewing
of the measured data well and is sensitive to the maximum and bearings.
minimum error in a set of data: the smaller the value, the higher
the accuracy. The RMS error is defined by (equation (10)). Acknowledgements The authors gratefully acknowledge the support pro-
vided by the National Natural Science Foundation of China (51105191,
Table 6 shows the calculation results: the larger the loads, the
51375222), the project of Jiangsu Province Six Talent Peaks (GDZB-033),
larger the RMS error. Compared to the results of Wang et al. the Shanghai Sailing Program (16YF1408500) and the China Postdoctoral
[18] who report a 458 RMS error, the present FE model is very Science Foundation (Project No.2015 M580632).
accurate.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 2
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