Advances in the Research and Development of Rotary Motion Driven Machine Elements - Research Article
Numerical simulation and
experimental testing of dynamic
stiffness of angular contact ball bearing
Advances in Mechanical Engineering
2018, Vol. 10(9) 1–14
Ó The Author(s) 2018
DOI: 10.1177/1687814018798973
journals.sagepub.com/home/ade
Haitao Luo1 , Jia Fu1, Changshuai Yu1, Guangming Liu1,
Wei Wang2 and Peng Wang2
Abstract
Angular contact ball bearings are widely used in the multiple rotor system, such as gear box, machine tool spindle, and
aero-engine rotors. The support stiffness is very important to the vibration of shafting. In order to obtain the dynamic
stiffness, a numerical simulation method for dynamic stiffness of angular contact ball bearings is presented. By means of
LS-DYNA software, the displacement and stress curves of the bearing’s different components are obtained successfully,
and then, the dynamic stiffness of bearing under certain working conditions also can be calculated. The maximum envelope radius of the inner ring of the measured bearing can be taken as the relative displacement change value of the inner
and outer rings of the bearing. To obtain the dynamic stiffness, the test strategy for dynamic stiffness of the angular contact ball bearings is introduced. The experimental results show that the equivalent dynamic stiffness increases with the
increase in the preload of measured bearing under certain conditions. The simulation result coincides with the experiments well. The research work can provide the basis for the design and dynamics analysis of the bearing rotor system
and have important engineering significance to improve the service performance of angular contact ball bearings.
Keywords
Angular contact ball bearing, numerical simulation, experimental testing, bearing dynamic stiffness
Date received: 2 April 2018; accepted: 13 August 2018
Handling Editor: Zengtao Chen
Introduction
Rolling bearing is widely used in high-speed rotors,
such as machine tool spindles and aero-engines, because
of their high rotation accuracy and strong bearing
capacity.1–4 The bearing stiffness is an important parameter of the bearing, which exhibits strong time-varying
characteristics and nonlinear characteristics under high
rotation speed, variable load, and other complex operating conditions.4,5 Furthermore, the bearing stiffness
directly affects the dynamic characteristics of the spindle system.5 The accuracy of bearing stiffness in rotor
dynamics modeling will directly determine the reliability
of analysis results. Therefore, it has very important significance to study the design of bearing stiffness parameters.6–11
The dynamic stiffness refers to the stiffness of the
bearing in all directions during operation, which is an
important index to measure the anti-vibration performance of the machine tool spindle structure. The larger
its value is, the greater the dynamic load required to
produce a unit amplitude for the machine spindle
1
State Key Laboratory of Robotics, Shenyang Institute of Automation,
Chinese Academy of Sciences (CAS), Shenyang, China
2
School of Mechanical Engineering and Automation, Northeastern
University, Shenyang, China
Corresponding author:
Haitao Luo, State Key Laboratory of Robotics, Shenyang Institute of
Automation, Chinese Academy of Sciences (CAS), Shenyang 110016,
China.
Email: luohaitao@sia.cn
Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License
(http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without
further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/
open-access-at-sage).
2
structure. Conversely, the smaller the dynamic stiffness,
the more severe the vibration of the structure.
Therefore, in order to ensure the spindle system has
good processing performance, the dynamic stiffness
analysis of the bearing is particularly important. Huang
Taiping12 put forward the testing principle of direct
and indirect methods for bearing dynamic stiffness test.
Chun-jie Li13 designed the dynamic stiffness test bench
for angular contact ball bearing, measured the dynamic
stiffness of the bearing at different speeds and analyzed
the influence of speed and preload on the dynamic stiffness of the bearing. And, Ali14 studied the effect of preload and rotational speed on the dynamic stiffness of
the bearing through the hammering test. For rolling
bearings, the difficulty in analysis lies in the contact
between the rolling element and the ring body. In the
no-load static condition, the contact between the rolling
element and the ring body is a point. With the increase
in the load and speed, the point contact gradually
becomes the surface contact. Meanwhile, the position,
size, shape, contact pressure, and friction distribution
of the contact area are unknown before analysis. They
all change with the change of external load and rotational speed, which is a boundary nonlinear problem.
This article designs an experimental platform for the
stiffness test of rolling bearing in view of the practical
applications for testing the dynamic stiffness of the
bearing. In addition, a finite element simulation
method for bearing stiffness is presented. The stiffness
test and finite element simulation on a certain type of
bearing are carried out, respectively, to obtain the relationship among dynamic stiffness, preload, and rotational speed, which provides a basis for the design and
dynamic analysis of the bearing rotor system.
Dynamics stiffness mathematical model of
angular contact ball bearing
The mathematical model is based on the Jones groove
control theory, and the Newton–Raphson iterative
method is used for numerical calculation. The following
assumptions are made before establishing the model:
(1) the geometry of the bearing is ideal; (2) with outer
ring fixed, inner ring rotates relative to outer ring; (3)
ignore the friction between the steel ball and the inner
and outer ring channels; (4) the interaction between
bearing components is in accordance with Hertz contact theory; and (5) excluding the influence of internal
oil film thickness and oil film resistance of the bearing.
Figure 1 shows the five possible displacements of the
inner ring of an angular contact ball bearing relative to
the outer ring. Figure 2 shows the position angle uk of
the kth ball. In Figure 1, u is the contact angle; and dx ,
dy , and dz are the displacements (mm) of the inner ring
relative to the outer ring in the three directions x, y, and
Advances in Mechanical Engineering
Figure 1. Relative displacement of angular contact bearing.
Figure 2. The ball position of angular contact bearing.
z, respectively; while g y and g z are the tilt angles (rad)
of the inner ring relative to the outer ring in the y and z
directions, respectively.
Figure 3 shows the relative position changes of the
ball center of the bearing and the center of curvature of
the inner and outer ring raceways at the kth ball position before and after loading. Assuming that the outer
ring is stationary, the center of curvature of the raceway is not affected by the force. When the bearing is
stressed, the bearing ball and the inner ring will be relatively deformed, and the position of steel ball center
and the inner ring channel’s center of curvature will
also change, as well as the contact angles of the inner
and outer rings. The relative distance between the center of curvature of the inner and outer ring raceway
before deformation is Bd , formulated as follows
Bd = BD = (fi + fo 1)D
ð1Þ
In the formula, fi , fo , and D, respectively, are the radius
factor of curvature of the inner ring raceway, the radius
factor of curvature of the outer ring raceway, and the
ball diameter.
Luo et al.
3
Figure 4. Force analysis diagram of ball.
Figure 3. Changes in the position of components before and
after bearing loading.
After the bearing is stressed, the positions of the center of balls and the center of curvature of the inner ring
raceway are changed in Figure 3. At this time, the radial
distance and axial distance of the center of curvature of
the inner and outer raceway are as follows
Uik = BD sin u + dx g z ric cos uk + g y ric sin uk
ð2Þ
Vik = BD cos u + dy cos uk + dz sin uk
ð3Þ
In the above formula, ric = 0:5Dm + (fi 0:5) D cos u,
Dm is the pitch circle radius of the bearing, and D is the
ball diameter of the bearing. The distances from the
center of the steel ball to the center of curvature of the
inner and outer raceway are as follows, respectively15
Dik = ri D=2 + dik = (fi 0:5)D + dik
ð4Þ
Dok = ro D=2 + dok = (fo 0:5)D + dok
ð5Þ
The radial displacement and axial displacement of
the center position of curvature of the inner raceway
before and after the bearing stress are as follows
Dicu = dx Dg z ric cos uk + g y ric sin uk
ð6Þ
Dicv = dy cos uk + dz sin uk
ð7Þ
In order to facilitate the analysis of Figure 3, Jones
introduces two new variables Uk and Vk to replace uik
and uok , and the geometric relationship can be obtained
as follows
8
Uk
>
>
> sin uok = ( f 0:5)D + d
>
>
o
ok
>
>
>
V
>
k
>
>
< cos uok = ( f 0:5)D + d
o
ok
ð8Þ
Uik Uk
>
>
>
sin
u
=
ik
>
>
( fi 0:5)D + dik
>
>
>
>
Vik Vk
>
>
: sin uik =
( fi 0:5)D + dik
In Figure 3, using the Pythagorean theorem, we can
know that
(Uik Uk )2 + (Vik Vk )2 D2ik = 0
ð9Þ
Uk2 + Vk2 D2ok = 0
ð10Þ
Figure 4 shows the bearing force of the ball during
operation. As the bearing works, the ball will make a
rotational motion around the axis of the bearing, which
will generate centrifugal inertia force. In the course of
the rotation of the ball, the gyro moment is generated
due to the constant change of the axis of rotation.
Therefore, when analyzing the force conditions of ball,
it is necessary to consider the centrifugal inertial force
and the gyro moment to which the ball is subjected.
The centrifugal force and the gyroscopic moment are
as follows
v 2
1
E
mDm v2
v
2
v v B
E
sin ak
Mgk = Jb v2
v k v
Fck =
ð11Þ
ð12Þ
In the formula, m is the quality of the steel ball, v is
the spindle speed, vE is the revolution angle of the
ball, vB is the rotation angle of steel ball, ak is the angle
between the rotation axis of the steel ball and the x-axis.
To analyze the steel ball, the equilibrium equation can
be obtained as
Qok cos uok Qok sin uok Mgk
sin uok Qik cos uik
D
Mgk
sin uik Fck = 0
+
D
ð13Þ
Mgk
Mgk
cos uok Qik sin uik +
cos uik = 0
D
D
ð14Þ
In the formula, as can be seen from Wang et al.10
4
Advances in Mechanical Engineering
3=2
ð15Þ
Table 1. NSK 60TAC120B bearing parameters.
3=2
ð16Þ
No.
Geometric and material properties parameters
Value
1
2
3
4
5
6
7
8
9
10
11
12
Inside diameter (mm)
Outside diameter (mm)
Pitch diameter (mm)
Ball diameter (mm)
Number of balls
Ring elastic modulus (MPa)
Ferrule density (kg/m3)
Ring Poisson’s ratio
Ball elastic modulus (MPa)
Ball density (kg/m3)
Ball Poisson ratio
Contact angle (°)
60
120
93
10
20
2.06E5
7.85E3
0.3
3.2E5
3.2E3
0.25
60
Qik = Ki dik
Qok = Ko dok
In the formula, Ki and Ko are the contact deformation
coefficients of the inner ring and the outer ring, respectively. Ki , Ki affected by the position of the ball, and Ki ,
Ko is a function of dik , dok , Uk , and Vk . The dik , dok , Uk ,
and Vk can be calculated numerically using the Newton–
Raphson iterative method using formulas (9)–(12).
The force acting on the shaft by the bearing inner
ring is
8
N Mgk
>
>
> Fxi = S Qik sin uik + D cos uik
>
>
k=1
>
>
N >
>
M
>
Fyi = S Qik cos uik Dgk sin uik cos uk
>
>
>
k=1
>
<
N M
Fzi = S Qik cos uik Dgk sin uik sin uk
>
k=1
>
>
i
>
N h >
M
>
>
Myi = S ric Qik sin uik + Dgk cos uik fi Mgk sin uk
>
>
k=1
>
>
i
>
N h >
>
M
:
Mzi = S ric Qik sin uik + Dgk cos uik fi Mgk cos uk
k=1
following equivalent and assumption conditions were
adopted:
1.
2.
ð17Þ
Let d = ½dx , dy , dz , gy , gz , find the partial derivative
of d, and the fifth-order stiffness matrix of an angular
contact ball bearing is
2
∂Fxi
6 ∂dx
6
6 ∂Fyi
6
6 ∂d
6 x
6 ∂Fzi
6
½k = 6
6 ∂dx
6 ∂M
6 yi
6
6 ∂dx
6
4 ∂Mzi
∂dx
∂Fxi
∂dy
∂Fyi
∂dy
∂Fzi
∂dy
∂Myi
∂dy
∂Mzi
∂dy
∂Fxi
∂dz
∂Fyi
∂dz
∂Fzi
∂dz
∂Myi
∂dz
∂Mzi
∂dz
∂Fxi
∂g y
∂Fyi
∂g y
∂Fzi
∂g y
∂Myi
∂g y
∂Mzi
∂g y
3
∂Fxi
∂g z 7
7
∂Fyi 7
7
∂g z 7
7
∂Fzi 7
7
7
∂g z 7
∂Myi 7
7
7
∂g z 7
7
∂Mzi 5
∂g z
ð18Þ
Finite element simulation of dynamic
stiffness of angular contact ball bearings
Taking the 60TAC120B angular contact ball
bearing produced by NSK Corporation as an example,
a three-dimensional simulation model of the bearing is
established in the nonlinear explicit dynamics analysis
software LS-DYNA, and the dynamic stiffness of the
angular contact ball bearing is simulated through
the refined model processing and analysis settings under
specific working conditions. The geometric parameters
and material property parameters of 60TAC120B angular contact ball bearings are shown in Table 1.
During the three-dimensional modeling and simulation analysis of the bearing, in order to improve the
simulation accuracy and save the calculation time, the
3.
The chamfering of bearing inner and outer ring
and retainer is ignored, and hexahedral mesh is
used for meshing;
The bearing clearance and the influence of the
oil film are ignored in the analysis process. The
materials of the bearing components in the
simulation process are linear elastic materials;
It is assumed that the rigidity of the bearing
mandrel and bearing seat is infinite, and the
deformation of the bearing itself is not involved
in the analysis process;
Among them, the inner and outer rings of the bearing and the retainer use the SOLID164 solid element,
and the inner ball uses the high-order SOLID168 solid
element. Considering the rigid boundary of the inner
and outer ring and the solid unit of the bearing without
rotational freedom, a rigid shell element is applied to
the inner surface of the inner ring of the bearing to facilitate the loading and rotational speed of the bearing.
The resulting finite element model of the
60TAC120B angular contact ball bearing is shown in
Figure 5. The entire model contains a total of 141,333
units and 107,153 nodes.
Based on the above-mentioned finite element model
of the bearing, certain boundary conditions and load
conditions are set, and the rest of operating modes can
be changed accordingly:
1.
2.
3.
Boundary conditions: the rigid surface of the
bearing outer ring is a fixed constraint;
Load application: a preload of 2000 N and a
radial load of 1500 N are applied to the center
of mass of the bearing inner ring rigid surface;
Working speed: the rigid surface of the bearing
inner ring applied speed of 3000 r/min;
The analysis of the dynamic stiffness of a bearing
belongs to explicit dynamic analysis. The difference
Luo et al.
Figure 5. Finite element model of the dynamic stiffness of
60TAC120B angular contact ball bearings.
between explicit and implicit dynamic analyses of static
stiffness is that there is no need to set a contact element
in the explicit dynamic analysis. It only needs to specify
the contact surface, contact type, and contact-related
parameters for the model to simulate the actual contact
of the bearing rotation. Angular contact ball bearings
have three contact positions during the working process, which are the contact between the ball and the
inner raceway, the contact between the ball and the
outer raceway, and the contact between the ball and
the retainer pocket. The 60TAC120B angular contact
ball bearings have a total of 60 contact pairs. The contact locations are shown in Figure 6.
In addition, the static friction coefficient between
the rolling elements and the inner and outer ring raceways and retainer pockets is set to be 0.3E22, 0.3E22,
and 0.2E22, respectively, and the dynamic friction
coefficients are set to 0.15E22, 0.15E22, and 0.1E22,
respectively. Next, we perform analysis setup and simulation. The entire calculation time is set to 160 ms, the
number of output steps is 401 steps, and the entire calculation time is about 10 h.
The movement of each component of the bearing
In the LS-PREPOST post-processing software, view
the time history curves of the bearings in the postprocessing software LS-PREPOST. Select the node of
inner ring, the node of retainer, and the node on the
ball and draw the displacement curves during the movement of the bearing, as shown in Figure 7.
From Figure 7, it can be seen that the displacement
of the inner ring and the retainer of the angular contact
ball bearing is sinusoidal, and the displacement of the
ball shows a sinusoidal change on the basis of the displacement curve of the retainer. This is specifically
described as follows:
1.
The inner ring, balls, and retainer all show significant cycle changes during operation. The
5
Figure 6. Contact pair of 60TAC120B angular contact ball
bearings.
2.
3.
rotation period of the inner ring is 20 ms, and
the revolution period of the ball is the same as
the rotation period of the retainer, which is
about 48 ms.
The ball slides and rolls between the raceways of
the inner and outer rings of the bearing. When
the ball orbits once or the retainer rotates one
revolution, the inner ring of the bearing rotates
approximately twice.
Because the position of the selected node is
closer to the outer ring of the bearing, the maximum displacement of the inner ring of the bearing, the ball, and the node of the retainer is the
diameter of the pitch circle where these nodes
are located and is closer to the outer diameter
of the bearing.
In the same way, the nodes of the inner ring, retainer, and ball of the angular contact ball bearing are still
selected, and the speed curves during the movement of
the bearing are drawn, as shown in Figure 8.
It can be seen from Figure 8 that the speeds in the
Y- and Z-directions of the inner ring and the retainer
are sinusoidal, and the ball speed curve still shows a
sinusoidal change on the basis of the retainer speed
curve. Specifically described as follows:
1.
2.
3.
The velocity curves of the nodes on each moving component of the bearing are obviously periodic, and the period of the velocity curve is
equal to the period of the displacement curve.
When the Y-direction velocity component of the
node on the inner ring reaches the limit value of
6 12.5 m/s, the Z-direction velocity component
is zero. At this point, the Y-displacement of this
node is zero, and the Z-direction displacement
will reach the extreme value of 6 65 mm.
When the Y-direction velocity component of the
retainer reaches the limit value of 6 8 m/s, the
Z-direction velocity component is zero. At this
point, the Y-displacement of this node is zero,
6
Advances in Mechanical Engineering
(a)
(b)
(c)
Figure 7. Bearing displacement curve of the moving parts: (a) Y-direction displacement curve, (b) Z-direction displacement curve,
and (c) resultant displacement curve.
(a)
(b)
(c)
Figure 8. Bearing speed curve of the moving parts: (a) Y-direction velocity curve, (b) Z-direction velocity curve, and (c) resultant
velocity curve.
Luo et al.
7
Figure 10. Maximum equivalent stress curve of bearing’s all
components when its operation.
Figure 9. Von Mises stress between bearing components when
it is 7.2 ms: (a) inner contact stress, (b) contact stress ball,
(c) outer contact stress, and (d) contact stress slices.
4.
and the Z-direction displacement will reach the
extremum of 6 80 mm. Due to the influence of
the rotation of the ball, the contact points constantly alternate change between the inner and
outer rings.
The combined speed of the nodes on the inner
ring is about 12.5 m/s, the combined speed of
the nodes on the retainer is about 7.5 m/s, and
the combined speed of the nodes on the balls
changes sinusoidally, and the range of value
changes is about 2.5–13.5 m/s.
Equivalent stress of bearing components
In the post-processing simulation results, the stress
nephogram of each component of the bearing is displayed as shown in Figure 9.
It can be seen from the above figure that the equivalent stress of the angular contact ball bearing is concentrated in the contact position between the ball and the
inner and outer ring raceway during operation. Due to
the contact angle, the maximum stress position of the
ball appears near the contact point. According to
the Hertzian contact theory, the contact area between
the ball and the raceway is an ellipse, and the compressive stress formed between them changes along the
major and minor half axes of the ellipse, and the
smaller the stress value is toward the elliptical edge. It
can be seen from the figure that the results of the finite
element analysis are consistent with the Hertzian contact theory.
In addition, it can be seen from Figure 9(a)–(c), the
maximum equivalent stress of the inner ring, outer ring,
and ball at the same moment is not the same. Among
them, the maximum stress of the ball is 252 MPa, followed by the inner ring of 241.8 MPa, and the lowest
stress is 135.4 MPa of the outer ring. The relatively
large stress position on the ball occurs in the contact
area with the inner and outer rings, and the ball stress
in the bearing area is greater than the ball stress in the
non-bearing area. The inner ring and the outer ring are
the main elements that are directly contacted and carried in the bearing. The stress distribution is closely
related to the stress distribution of the ball. From the
slicing diagram of the bearing in Figure 9(d), it can also
be seen that the relatively large stresses in the inner and
outer rings are also distributed in the contact area with
the ball, and the stress in the load bearing area is greater
than that in the non-load bearing area. The other parts
of the ring body without apparent stress distribution.
The stress distribution area mainly focuses on the contact surface and presents a distinct oval.
The time curve of the maximum equivalent stress of
the bearing inner ring, outer ring, and ball during normal operation of the angular contact ball bearing is
shown in Figure 10. It can be seen from the above that
the equivalent stress curve of the components in the
bearing changes randomly, which also indicates that
the stress variation of the components during the movement of the bearing is strongly nonlinear. In addition,
the maximum stress on each component at a certain
moment is also different, from big to small, namely,
ball, inner circle, and outer ring. This also verifies the
analysis results in Figure 9.
Because the accuracy of the finite element model
establishment plays a decisive role in the final analysis
result, establishing an accurate and feasible finite element model is the premise of the simulation analysis.
From the comparison of the above bearing stress cloud
diagram and the equivalent stress time curve of each
component, it can be seen that the establishment of the
8
Advances in Mechanical Engineering
Test of dynamic stiffness of angular
contact ball bearing
The ability of a bearing to resist deformation under an
external load is called the stiffness of the bearing. The
rigidity of the bearing is the ratio of the amount of
change in the applied load and the relative displacement between the outer ring and the outer ring of the
bearing under certain conditions.16,17
Principle of dynamic stiffness testing
Figure 11. Displacement variation between inner and outer
rings of angular contact ball bearings.
whole finite element simulation model satisfies the
requirements, so the result of the finite element simulation analysis is credible finally.
Dynamic stiffness simulation curve of bearing
After the simulation is completed, the displacement
curve between the rigid faces of the inner and outer
rings of the angular contact ball bearing in the direction of radial load is shown in Figure 11.
It can be seen from the figure that there is instability
in the rotation of the bearing, and the displacement curve
shows obvious fluctuations. In spite of this, since the
rotational speed and the external load are constant, the
displacement data in the direction of the radial load can
still be approximated as a linear relationship. By fitting
the data of the simulation curve, it can be seen that the
displacement curve becomes approximately a straight
line within the time range in which the bearing operation
reaches the steady state. The value of the ordinate represented by this straight line is also the value of the bearing
at the speed and given operating conditions. The amount
of radial displacement change is a constant value.
Therefore, the radial stiffness of the bearing only needs
to be removed with a radial load in this direction.
When the bearing reaches steady state, the radial
displacement between the inner and outer rings is
5:38 3 104 mm. Under the working conditions of
bearing preload of 2000 N, radial load of 1500 N, and
bearing rotation speed of 3000 r/min, the radial
dynamic stiffness of 60TAC120B angular contact ball
bearings is 5:38 3 106 N=mm. This value is smaller
than the theoretical calculation of the axial dynamic
stiffness of the bearing, which is mainly due to the fact
that the finite element simulation considers the interaction between the components of the bearing. It is more
accurate than the theoretical calculation.
Synchronous excitation. Using a rolling bearing as a single
degree of freedom system, the dynamic characteristics
can be expressed as18
Z ðvÞ = K ðvÞ mv2 + iC ðvÞv
ð19Þ
In the formula, K(v) is the equivalent stiffness of the
rolling bearing; m is the equivalent parametric mass of
the rolling bearing; v2 is the inertial term of the equivalent stiffness of the rolling bearing; v is the angular frequency of the rotor operation; and C(v) is the
equivalent damping coefficient of the rolling bearing.
When the rolling bearing is in operation, the radial
load synchronous with the rotation speed is directly
applied, and the corresponding displacement generated
at the center of the bearing is measured at the same
time
Z ðvÞ =
Q sinðvtÞ
d sinðvt cÞ
ð20Þ
In the formula, Q is the rotational radial force acting
on the bearing inner ring synchronous with the running
speed of the bearing; d is the radial displacement of the
bearing center; c is the phase difference between the displacement and the force vector.
The use of unbalanced mass excitation to apply
rotational radial load in synchronism with the rotational speed is achieved by adding unbalanced mass to
the experimental stage rotor system, that is, an excitation disk is mounted near the rolling bearing, and a
screw constituting an eccentric mass is added to the
excitation disk, so that synchronous excitation of the
bearing can be realized. Then, measure the excitation
displacement of the main shaft system under the action
of the eccentric excitation force, and then, convert the
equivalent displacement generated by the bearing center under equivalent load according to the geometric
relationship. The equivalent dynamic stiffness of
the tested bearing is calculated from equation (1).
The layout of the experimental bench is shown in
Figure 12.
The synchronously rotating radial force produced by
the eccentric mass is
Luo et al.
9
Figure 12. Schematic diagram of bearing stiffness test principle.
Figure 13. Bearing dynamic stiffness test bench.
Q = ma ev2
ð21Þ
In the formula, ma is the additional unbalanced mass; e
is the mass eccentricity of the additional unbalanced
mass. Due to the approximate method used, the synchronous unbalanced force applied to the bearing under
test is not directly located in the symmetry plane of the
bearing, so the necessary data conversion is needed.
The synchronous radial force acting on the tested bearing is
Qe =
QL1
ma ev2 L1
=
2ð1 + L1 + L2 + L3 =2Þ
2ð1 + L1 + L2 + L3 =2Þ
ð22Þ
L1 is the distance between balance mass and left support
bearing, L2 is the distance between the balance mass
and the tested bearing, and L3 is the distance between
tested bearing and right support bearing.
Radial displacement. As can be seen from Figure 13, the
displacement sensor is located close to the bearing
under test to measure the radial displacement of the
inner ring of the bearing. At the same time, in order to
more accurately characterize the radial displacement of
the bearing, it is necessary to remove the influence of
the bending deformation of the main shaft itself. The
equivalent displacement is18
de = dc ds
ds = dst = 1 r2
dst =
ma ev2 L21 ðL2 + L3 =2Þ2
3EI ðL1 + L2 + L3 =2Þ
ð23Þ
ð24Þ
ð25Þ
In the formula, dc is the vibration displacement
obtained by the test, ds is the dynamic deflection caused
by the imbalance of the spindle, dst is static deflection,
and r is the ratio of operating speed to critical
speed. The radial stiffness of the rolling bearings at
each speed is
Kd =
Qe
QL1
=
de
2(1 + L1 + L2 + L3 =2)de
ð26Þ
Dynamic stiffness test bench design
In order to verify the correctness of the simulation analysis results, a bearing dynamic stiffness test bench was
designed. The test stand is mainly composed of the following parts: motor, accelerator, coupling, excitation
disk, preload adjustment nut, preload device, precision
adjustment slide, bearing assembly under test, and eddy
current displacement sensor. The base of the test stand
is fixed on the concrete foundation by damping rubber,
which is used to reduce the system vibration caused by
the unbalanced excitation of the excitation disk during
the high-speed rotation of the rotor. According to the
above structural design scheme, the dynamic stiffness
test bench for angular contact ball bearings is shown in
Figure 13.
Wherein, the motor has a speed regulating function,
and the current motor rotating speed can be displayed
in real time through the control panel. Since the rated
speed of the motor is low, the high-speed test range of
the bearing to be tested are not satisfied, and therefore,
an accelerator is used to further increase the working
speed of the rotor system. The exciting disk is the applicator of the radial force of the bearing to be tested.
Different eccentric masses are installed on the exciting
disk to generate synchronous radial forces of different
sizes. The pre-tensioning device is used to apply different preloads to the tested bearing, wherein the preload
adjustment nut can be screwed in and out, and the preload can be accurately converted by the pitch and rotation angle. The precision adjustment slider is a set of
linear pairs used to adjust the installation accuracy of
the entire test rig horizontally and vertically (including
parallelism, verticality, and coaxiality). The bearing
assembly to be tested is used to locate and fix the tested
bearing. Its internal structure can be adapted to the
installation of different types of angular contact ball
bearings, which is versatile. The two eddy current displacement sensors are orthogonally distributed in the
horizontal and vertical directions and are mounted
near the inner ring of the tested bearing through a fixed
bracket for monitoring the range of displacement in the
10
Figure 14. Bearing preload adjusting device.
horizontal and vertical directions during bearing
rotation.
Among them, the bearing preload adjustment device
is shown in Figure 14. The preload device is mainly
composed of several parts, namely, adjusting nut,
radial sliding bearing, sliding block, fixed support, pretensioned adjusting spring, angular contact ball bearing, and spindle.
Its working principle can be described as follows:
when the preload adjusting nut rotates to the right side,
the radial ball bearing of the thrust ball moves inwards,
thereby compressing the preload adjusting spring. And,
the other end of the spring also acts uniformly on the
outer ring of the tested bearing through the sliding
block. The inner ring of the tested bearing is positioned
by the mandrel, so that the compression force of the
spring acts as a preload and directly acts on the tested
bearing. In order to provide different bearing preloads
in different ranges, six pre-adjusting springs are used in
parallel to obtain the preload applied value by giving
the pitch and rotation angle of the pre-tensioned nut
and calculating the compression of the spring.19–23
The test stand adopts a servo motor and the model
number is 130SJT-M150D. The rated speed of the
motor is 2500 r/min, the maximum speed is 3000 r/min,
the rated torque is 15 N m, and the maximum torque is
30 N m, which can meet the test speed requirement.
Among them, the accelerator growth ratio is 4 to ensure
that the bearing to be tested working rotating speed
adjustment range of 0–10,000 r/min. Preload adjustment spring model is YB3.5 3 18 3 25, elastic coefficient k = 83.7 N/mm, and maximum compression
amount is 5.85 mm. Six groups of springs are used,
design maximum compression is 5.5 mm, and the maximum axial force that can be provided is 2762 N, which
meets test requirements.
Dynamic stiffness test data analysis
The data analysis uses Siemens LMS high-resolution
signal acquisition system. The time-domain signals
Advances in Mechanical Engineering
collected by the eddy current displacement sensors of
the tested bearing positions are shown in Figure 15(a)
and (b). It can be seen that the waveforms presented by
the two sensors show periodic changes with a phase difference of 90°. Due to the accuracy and nonlinear
effects of the test bench, the time-domain waveforms
collected by the two sensors are not smooth.
In order to more clearly show the level of displacement of the bearing being measured under a given operating condition, take a certain period out of the above
time-domain waveform to draw the axis trajectory, as
shown in Figure 16. In the figure, the dashed line is the
true axial trajectory profile. It can be seen that the true
axial trajectory curve is also approximately circular due
to the error and nonlinearity of the test bench. In order
to facilitate the calculation of the bearing stiffness, its
circumcircle is fitted here, and the maximum envelope
radius is taken as the radial displacement of the bearing
to be tested under this working condition.
According to the stiffness calculation formula, the
dynamic stiffness value of the tested bearing under
the current working condition can be obtained. Since
the actual working conditions of the bearing are different, the dynamic stiffness of the 60TAC120B angular
contact ball bearing under different working conditions
is obtained by changing the rotational speed of the
bearing under test and adjusting the magnitude of its
preload. Set the adjusting range of the bearing preload
is 0–2500 N, the interval is 500 N, the changing interval
of the bearing speed is 400–2400 r/min, and the interval
is 400 r/min. The changing interval of the bearing speed
of revolution is 400–2400 r/min, and the interval is
400 r/min. The dynamic stiffness of the tested bearing
under all operating conditions is drawn into a form, as
shown in Table 2.
From Table 2 above, the specific data of the radial
stiffness of the bearing to be tested can be known. In
order to facilitate the observation, they are drawn into
corresponding graphs, with the rotation speed and the
preload as the abscissa, and the radial dynamic stiffness
of the bearing as the ordinate. The curves of the radial
dynamic stiffness of the bearing with the change of
speed of revolution and preload are shown in Figure
17(a) and (b).
From Figure 17(a), it can be known that the
dynamic stiffness of the angular contact ball bearing
becomes larger with the increase in the preload of the
bearing, but when the preload reaches a certain degree,
the change of bearing stiffness is not obvious.
From Figure 17(b), it is known that when the preload is fixed, the radial dynamic stiffness of the bearing
is increased with the increase in the speed. When the
rotation speed is low, the change of the dynamic stiffness of the bearing with the preload is not obvious, but
when the rotation speed is high, the change range of
stiffness with preload increases obviously. This is
Luo et al.
11
(a)
(b)
Figure 15. Test data and axis trajectory: (a) X-direction displacement time-domain curve and (b) Y-direction displacement
time-domain curve.
Table 2. Radial stiffness of bearing (106 N/mm).
Preload (N)
0
500
1000
1500
2000
2500
Speed (r/min)
400
800
1200
1600
2000
2400
2.46
4.14
4.82
5.50
5.89
6.85
5.47
8.21
9.30
10.39
11.49
12.58
7.25
10.40
12.32
13.77
15.22
16.67
8.87
13.31
14.70
16.50
18.63
19.44
12.28
15.07
16.33
17.60
20.50
22.23
13.50
17.47
19.14
21.12
23.10
24.57
Figure 16. Test data and axis trajectory.
mainly because with the increase in the rotational
speed, the dynamic body in the bearing to be tested will
expand obviously under the action of centrifugal force.
The contact area between the rolling element and the
ferrule increases gradually, and the corresponding contact angle will also change, which eventually leads to
the increase in the radial stiffness of the bearing.
In order to verify the consistency of the bearing stiffness of the angular contact ball bearing to be tested
during loading and unloading, the change range of the
bearing rotation speed is set to 0–2500 r/min, and
the adjusting range of the preload is 0–2500 N, and the
loading and unloading stiffness tests are carried out,
respectively. When the preload of the bearing is set at
1500 N, the load and unload dynamic stiffness curves
of the tested bearing with the change of rotational
speed are shown in Figure 18, and its data are shown in
Table 3. Similarly, when the rotation speed of the bearing is set at 1200 r/min, the load and unload dynamic
stiffness curves of the tested bearing with the change of
preload are shown in Figure 19, and its data are shown
in Table 4.
It can be seen from Figure 18 and Table 3, under the
condition of constant preload, the radial stiffness of
angular contact ball bearings increases with the increase
in the rotational speed during loading and unloading.
12
Advances in Mechanical Engineering
(a)
(b)
Figure 17. Experimental curve of radial dynamic stiffness of 60TAC120B bearing: (a) dynamic stiffness changes with preload and
(b) dynamic stiffness changes with speed.
Table 3. The radial stiffness of loading and unloading varies with the rotation speed (106 N/mm).
Condition
Load
Unload
Speed (r/min)
400
800
1200
1600
2000
2400
5.5
5.1
10.3
9.6
13.7
13.2
16.5
16.5
17.6
18.3
21.1
20.8
Figure 18. The radial stiffness of loading and unloading varies
with the rotation speed of 60TAC120B bearing.
The bearing stiffness obtained from the two processes is
basically consistent. The error is less than the measured
data by an order of magnitude, which can be neglected,
and the consistency is good.
It can be seen from Figure 19 and Table 4 that under
the condition of constant speed, the radial stiffness of
the angular contact ball bearing increases with the
increase in the preload during the loading process and
the unloading process. The bearing stiffness obtained
during the loading and unloading process is basically
consistent. The error is less than an order of magnitude
of the measured data and can be ignored. It also verifies the consistency of the test data and the reliability of
the test bench design.
In order to verify the accuracy of the simulation
results of angular contact ball bearings, the simulation
analysis curve and the test curve are compared. Taking
the bearing rotation speed as 1600 r/min, and the
change curve of the radial dynamic stiffness of the
bearing with the preload force is shown in Figure 20.
Similarly, taking the preload force as 1500 N, and the
change curve of the bearing radial dynamic stiffness
with the rotation speed is shown in Figure 21.
Through the comparison of simulation and test
curves, it can be seen that the radial dynamic stiffness
of 60TAC120B angular contact ball bearings increases
with the increase in the preload when the rotational
speed is constant, and the variation range is 9E6–
2.2E7 N/mm, as shown in Figure 20; under the condition of constant preload, the radial stiffness also
increases with the increase in the rotation speed, and
the range of change is 5E6–2.4E7 N/mm, as shown in
Figure 21. But the test data are slightly lower than the
simulation data, which is mainly due to the machining
Luo et al.
13
Table 4. The radial stiffness of loading and unloading varies with the preload (106 N/mm).
Condition
Load
Unload
Preload (N)
0
500
1000
1500
2000
2500
7.2
6.9
10.4
10.0
12.3
11.9
13.7
13.0
15.2
15.4
16.6
16.7
Figure 19. The radial stiffness of loading and unloading varies
with the preload of 60TAC120B bearing.
Figure 21. Bearing radial dynamic stiffness curve.
bench for testing the dynamic stiffness of angular contact ball bearings is designed. A finite element simulation method for dynamic stiffness analysis of bearings
is presented. Through simulation analysis and experimental comparison, the relationship between the equivalent dynamic stiffness of the 60TAC120B angular
contact ball bearing and the preload and working speed
is studied. The specific conclusions are as follows:
1.
Figure 20. Bearing axial dynamic stiffness curve.
accuracy and installation error of the test bench, and
the acquisition error of sensors. However, the influence
of these environmental factors on stiffness is not considered in the simulation.
2.
Conclusion
In this article, the testing method of dynamic stiffness
of angular contact ball bearings is studied. The test
Comparing the dynamic stiffness simulation
and experimental curves of 60TAC120B angular contact ball bearings, we can see that
the dynamic stiffness of the bearing is only
1E7 N/mm when the rotation speed of the
bearing is 1600 r/min and the preload is 0 N.
When the preload rises to 2500 N, the dynamic
stiffness of the bearing is close to 2E7 N/mm,
which is twice as high as that without preload.
Under certain rotational speed, the equivalent
dynamic stiffness increases with the increase in
the preload. When the preload reaches over
2000 N, the dynamic stiffness value changes
little with the increase in the preload.
When the preload of the bearing is 1500 N and
the working range of rotational speed is 400–
2400 r/min, the dynamic stiffness of the bearing
varies from 5E6 to 2E7 N/mm. Under certain
preload, the dynamic stiffness of angular
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Advances in Mechanical Engineering
3.
contact ball bearings increases with the increase
in the rotational speed, and the speed of stiffness increase slows down with the gradual
increase in the rotational speed.
Overall, the simulation analysis results are consistent with the experimental test results, but the
stiffness simulation data of the local position
are slightly larger than the experimental value,
this is because there being the geometric gap
between the ball and the ring raceway of bearing. Nevertheless, the work in this article can
provide some data support for online adjustment of the pre-tightening force of angular contact ball bearings and selection of the working
speed of bearings, which has important engineering significance for improving the service
performance of angular contact ball bearings.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of this
article.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this
article: This research was supported by the National Natural
Science Foundation of China (grant no. 51505470) and Youth
Innovation Promotion Association, Chinese Academy of
Sciences (CAS).
ORCID iD
Haitao Luo
https://orcid.org/0000-0001-8865-6466
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