machines-10-01024

machines
Article
A Coupling Model of High-Speed Train-Axle Box Bearing and
the Vibration Characteristics of Bearing with Defects under
Wheel Rail Excitation
Qiaoying Ma 1,2 , Yongqiang Liu 2,3, * , Shaopu Yang 2 , Yingying Liao 2,4
1
2
3
4
*
Citation: Ma, Q.; Liu, Y.; Yang, S.;
Liao, Y.; Wang, B. A Coupling Model
of High-Speed Train-Axle Box
Bearing and the Vibration
Characteristics of Bearing with
Defects under Wheel Rail Excitation.
Machines 2022, 10, 1024.
and Baosen Wang 1,2
School of Traffic and Transportation, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures,
Shijiazhuang Tiedao University, Shijiazhuang 050043, China
School of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
School of Civil Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
Correspondence: [email protected]
Abstract: A three-dimensional vehicle-axle box bearing coupling model is established. The model
can calculate the contact force in three directions and obtain the dynamic response of axle box
bearing under the real vehicle running environment. The load distribution on the double row tapered
roller bearing and the vehicle is analyzed, and the co-simulation is conducted by comprehensively
considering the force transmission between vehicle and bearing. Taking into account the great
impact of defects on the bearing, three different types of bearing defects are added into the model,
respectively. The simulation results verify the effectiveness of the model. The model is also verified by
using the rolling and vibrating test rig of single wheelset. It is concluded that the simulation results
show good agreement with experimental results. The influence of track irregularity on the system
motion state is studied by using axis trajectory and vibration RMS (Root Mean Square value). The
results show that the influence of track irregularity and wheel flat scar on axle box bearing cannot be
ignored. The RMS of acceleration will change greatly due to the existence of defects. Wheel flat scar
will greatly interfere with the extraction of bearing defect, but it can be selected at high speed and
low frequency to monitor the existence of wheel flat scar, and select low speed and high frequency to
monitor the existence of bearing defect. The research results are helpful to the detection of wheel flat
scar and axle box bearing defect.
https://doi.org/10.3390/
machines10111024
Keywords: high speed train; axle box bearing; defect; track irregularity; wheel flat scar
Academic Editor: Xuesong Jin
Received: 27 September 2022
Accepted: 28 October 2022
1. Introduction
Published: 4 November 2022
Axle box bearing is a key component of a high-speed train bogie. It bears highfrequency alternating loads during operation, including radial force, axial force and continuous impact vibration. When the vehicle is running, the performance of axle box bearing
is an important factor affecting the safe operation of trains; the failure of axle box bearing
occurs on multiple types of EMUs, fatigue spalling of inner ring, outer ring and rollers are
major defect forms [1]. These failures will cause economic losses to a certain extent. However, up to now, we have not fully mastered the key research and development technology
of high-speed train axle box bearings [2]. Therefore, it is of great engineering significance
and economic value to conduct in-depth research on axle box bearings.
Analysis on the bearing dynamic model experiences a long period. Many experts
and scholars have done a lot of work on traditional bearings and have obtained many
achievements. Jones [3] first proposed the bearing quasi-static model, which can define
the load and position of each rolling element, as well as the displacement of inner ring
relative to its outer ring, this innovation work laid the foundation of bearing dynamics.
After that, Walters [4] established a dynamic model considering the rolling element for the
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Machines 2022, 10, 1024. https://doi.org/10.3390/machines10111024
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Machines 2022, 10, 1024
2 of 34
first time and analyzed the vibration characteristics of the bearing when it moved at high
speed. Using this model, the displacement and speed of the rolling element at any time can
be calculated. Harris [5] added inertia force and moment on the basis of Walters’s model
and considered the slip of inner and outer raceways, which made the dynamic research
of rolling bearings more accurate. Gupta [6–8] studied the non-equilibrium dynamics
of rolling bearing cage and obtained the motion state of the cage and rolling elements
under the conditions of static load, non-equilibrium load and radial load, respectively.
Since then, many scholars have studied the dynamic performance of rolling bearings in
various aspects. Kogan et al. [9] proposed a three-dimensional bearing dynamic model
that can simulate various bearing faults. By using this model, the bearing with axial
deformation of the outer ring was simulated and tested, and the effectiveness of the model
was verified through experiments. Tong and Hong [10] established a 5-DOF tapered roller
bearing model and studied its characteristics under combined load and high speed. It was
concluded that the radial displacement and moment load are almost linear, while the axial
displacement and moment load are nonlinear and gradually decrease with the increase
of load. Shi and Liu [11] established an improved dynamic equation of cylindrical roller
bearing, considered the interaction of various components of the bearing, and evaluated
the influence of radial load, axle speed and clearance on bearing vibration, respectively.
Yang et al. [12] established a mechanical analysis model of the double row tapered roller
bearing, studied its resistance torque, contact load and fatigue life under different loads. It
was concluded that external radial, thrust loads and angular misalignment often lead to a
significant reduction in bearing fatigue life, while the speed tends to increase it. Liu [13]
established a axle box bearing model, discussed the distribution law of internal load when
the bearing bore radial and axial loads at the same time. It can be found that with the
increase of the radial load, the contact load of the double row rollers will also increase,
while when the axial load increased, the contact load of one row of rollers increased and
that of the other row decreased.
When the bearing fails, some methods are needed to extract the fault. In the process of
fault diagnosis, the bearing vibration data is collected and analyzed to achieve effective
and preventive maintenance of the bearing at the same time. Many scholars have carried
out a lot of research in this direction. Eren et al. [14] proposed a real-time fault diagnosis
method of induction bearing based on machine learning. They adopted an adaptive 1D
Convolutional Neural Network classifier, which can automatically learn the best features
from the original bearing vibration data. Its structure was simple and compact with high
accuracy. The validity and feasibility of this method were verified by experimental data.
Entezami et al. [15] introduced several signal processing technologies in detail, and summarized the latest development of axle box bearing condition monitoring system. The authors
though that the number of monitoring systems applied to axle box bearing was relatively
limited, and their technology and performance still had a large space for development.
Amini et al. [16] proposed a customized onboard AE condition monitoring system to monitor defective railway bearings. Through a large number of experiments and field data, it
has been proved that acoustic emission signal envelope analysis can effectively monitor
bearing fault in the real world conditions. Papaeliasa et al. [17] introduced a novel condition monitoring system based on high-frequency acoustic emission and vibration analysis.
Its method was simple and cost less and its effectiveness was proved by experimental
data under actual conditions. Fan et al. [18] proposed a statistical condition monitoring
and fault diagnosis method based on tunable Q-factor wavelet transform, constructed the
Shewhart control charts on multiscale wavelet coefficients and proved the effectiveness
and superiority of the method. Yi et al. [19] used three EEMD-based steady-state indexes to
characterize the stable state of the bearing, which proved that the proposed state detection
and fault diagnosis methods can effectively identify different bearing faults.
The above research has proposed many effective methods. However, they only focus
on the condition monitoring and diagnosis of bearings, but cannot explain the cause
of bearing failure from the mechanism, as well as the process of bearing performance
Machines 2022, 10, 1024
3 of 34
degradation and the mechanism of failure degradation. These problems can be studied
from the perspective of dynamic analysis.
Therefore, in order to study the failure mechanism of bearing, many scholars have
used dynamic models to conduct in-depth analysis of high-speed train axle box bearing.
Yang et al. [20] established a rotor-bearing system model with local defects in the raceway,
and analyzed the influence of resonance characteristics and rotor eccentric excitation on
the motion state of the system. Tu et al. [21] established the explicit dynamic finite element
model of the bearing, and studied the contact characteristics between the rolling element
and the fault, as well as the fault characteristics in the bearing vibration signal. Luo et al. [22]
established a 4-DOF rolling bearing model with inner and outer rings composite fault, and
studied the vibration response characteristics of the defect-ball-defect model. Liu et al. [23]
respectively added outer ring fault, inner ring fault and roller fault to the bearing rotor
system of high-speed train, obtained the vibration response of the system under variable
speed conditions. It was concluded that the vibration acceleration amplitude of the bearing
outer ring was positively correlated with the axle speed of the bearings. Du et al. [24]
established a 3-DOF dynamic model of double tapered roller axle box bearing of high-speed
train, and analyzed the change law of contact stress at the defect position of each element
under different fault degrees. Liu [25] established a dynamic model of the rotor-bearing
pedestal system, took into account the excitation of the contact area caused by the fault
and the complex contour of the outer raceway etc. The results showed that the additional
excitation zone of the inner raceway was larger than that of the outer raceway, and it will
decrease with the increase of the roller load. Patel and Upadhyay [26] presented a nonlinear
dynamic model of a cylindrical roller bearing-rotor system with 9 degrees of freedom and
established a combined defect model. The nonlinear dynamic analysis of the model can
predict the behavior of the bearing-rotor system. Singh et al. [27] established a dynamic
model of rolling bearing with outer raceway defects. The finite element software package,
LS-DYNA, was used for numerical solution, and the dynamic contact force between the
rolling element and the raceway was analyzed in depth. Petersen et al. [28] studied the
defective double row bearing model and proposed a method to calculate the radial load
distribution under different stiffness. It was helpful to understand the variable stiffness
excitation in defective bearings, and can be used in the research of other nonlinear defective
bearing models. Sawalhi and Randall [29] studied the acceleration signal generated when
the rolling body entered and exited the peeling defect. Two signal enhancement methods
were analyzed, and the entrance and exit of the defect area were simulated.
It can be found that the above research only separates the bearing model for analysis
and discussion, ignoring its coupling with the carbody, bogie, wheelset and other systems,
which makes the research results deviate from the actual situation to a certain extent.
To solve this problem, some scholars have proposed a series of more complete bearing
coupling systems on the basis of bearing model research. Wang et al. [30] established a
vehicle bearing coupling model, studied the thermal characteristics of the bearing when
the track is uneven. In addition, the key influencing factors of the bearing operating
temperature were analyzed. Wang et al. [31] established a coupling dynamic model of axle
box bearing and vehicle-track system, considered the dynamic performance of axle box
bearing under complex dynamic excitation. Liu et al. [32] established the locomotive-track
space coupling dynamics model in which the dynamic effect of traction power transmission
was considered. The dynamic characteristics of motor bearings at the driving end and non
driving end were studied, the results showed that the loaded region of the motor bearing
at the driving end is larger than that at the non-driving end. Ma et al. [33] established a
17-DOF vertical dynamic model, including carbody, bogie, axle box bearing and wheelset,
discussed the changes of vehicle dynamic performance and vibration characteristics during
the early fault evolution of axle box bearing. Some scholars have also studied the axle box
bearing fault model based on the coupling model. Niu [34] established a vehicle dynamics
simulation model, analyzed the vibration characteristics of the axle box under different
bearing defects. In addition, the influence of different excitation types on the vibration
Machines 2022, 10, 1024
4 of 34
characteristics of the bearing was analyzed. Liu et al. [35] established a dynamic calculation
model of vehicle with early defects in axle box bearing and analyzed the impact of bearing
defects on the vertical vibration characteristics of vehicle. Liu and Du [36] established a
vehicle model considering bearings, analyzed the impact of bearing defects on the dynamic
performance of high-speed train. Wang et al. [37] established a longitudinal and vertical
dynamic model of railway vehicle considering the inner and outer raceway faults of axle
box bearing. The change of bearing fault index with the degree of damage was calculated.
It can be found that the longitudinal vibration characteristics were suitable for inner ring
fault identification, and the vertical vibration characteristics were suitable for outer ring
fault identification. Lu et al. [38] established a coupling model of railway vehicle and axle
box bearing with defects. The theoretical rolling trajectory of roller passing through the
defect was analyzed and deduced. It was concluded that the location of defect points on
the outer raceway affected the intensity of defect impact.
As the vehicle affected by various external excitation factors, such as turnout, track
joint, track irregularity and wheel wear; some scholars have discussed the vehicle dynamic
response under external excitation. Cheli and Corradi [39] studied the influence of track
irregularity on vehicle vibration, and the results showed that the vibration caused by track
irregularity had a great impact on vehicle comfort. Based on the time-varying nonlinear
contact load of axle box bearing, Li et al. [40] established a vehicle-track space coupling dynamic model to study the vibration characteristics of the vehicle under the track irregularity.
Liu and Zhai [41] studied the wheel rail interaction caused by polygonal wheel at high
speed, it was shown that the vertical wheel rail contact force would fluctuate greatly under
the polygonal wheel. Wu et al. [42] calculated the influence of wheel polygon wear order
on wheel rail force and analyzed the influence of wear parameters on vibration response of
vehicle components. Yang et al. [43] analyzed the measured vibration acceleration signal of
the gearbox line; the results showed that the wheel rail excitation frequency may cause resonance with the gearbox structure. Liu et al. [44] established an elastic vehicle-track model,
it can be found that the flat scar excitation had a great impact on the vertical vibration of the
axle box. Chudzikiewicz et al. [45] put forward new indexes for evaluating railway track
quality and carried out a field test and an evaluation of the results through the Rail Vehicle’s
and Rail Track Monitoring System. The results showed that the proposed method can
effectively monitor the track state. Bogacz and Frischmuth [46] studied the rolling motion
of a polygonized railway wheel on the rail, analyzed the variation of contact point velocity
under wheel polygon. The authors studied the contact point and vertical acceleration trajectories of rigid and viscoelastic models, respectively, and the results showed that the latter
description was more accurate. Bogacz and Kurnik [47] established a wheel-tire mechanical
model to study the motion stability of railway wheel and evaluated the influence of the
beam curvature and residual stress on the phase velocity and its critical value.
Axle box bearing are also affected by external excitation. Wang et al. [48] studied
the influence of wheel polygon wear and nonlinear wheel rail force on the contact force
between rollers and raceway; the results showed that the influence of high-order wheel
polygon wear on the force was more significant than that of low-order wheel polygon wear.
Liu et al. [49] analyzed the dynamic response and slip phenomenon of motor bearing under
irregularities; the results showed that locomotive vibration caused by track irregularity
and gear meshing will lead to alternating loads, result in high dynamic contact and friction
between rollers and raceways. Zha et al. [50] studied the impact of flat scar impact on the
bearing outer ring force; it was concluded that the impact load in the loaded zone was
larger than that in the unloaded zone, which was distributed symmetrically. The closer to
the symmetry axis, the greater the impact load on the raceway.
However, in the above study, the coupling relationship between vehicle and axle box
bearing does not consider the spatial transmission characteristics of force, the model is
mostly solved by a lot of differential equation, which simplifies the internal force of the
vehicle and has low calculation efficiency. At the same time, most of the coupling models
focus on the vehicle, and there is less research on the axle box bearing itself. Furthermore,
Machines 2022, 10, 1024
closer to the symmetry axis, the greater the impact load on the raceway.
However, in the above study, the coupling relationship between vehicle and axle box
bearing does not consider the spatial transmission characteristics of force, the model is
mostly solved by a lot of differential equation, which simplifies the internal force of the
vehicle and has low calculation efficiency. At the same time, most of the coupling models
5 of 34
focus on the vehicle, and there is less research on the axle box bearing itself. Furthermore,
the research on the influence of external excitation factors on axle box bearing is very limited, especially in the analysis of vibration characteristics when the bearing with defects.
To address
these on
problems,
an improved
vehicle-bearing
coupling
modeling
methodisisvery
the research
the influence
of external
excitation factors
on axle
box bearing
proposed
inespecially
this paper.
Universalcharacteristics
Mechanism (UM)
MATLAB/Simlimited,
in By
thecombining
analysis of vibration
whenwith
the bearing
with defects.
ulink,
coupling
between
the vehicle
and thevehicle-bearing
bearing is realized,
and the
internal method
forces is
Tothe
address
these
problems,
an improved
coupling
modeling
proposed
this
paper.rail
Bycontact
combining
Universal
Mechanism
with
of the
vehicle in
and
wheel
are fully
considered.
This (UM)
model
canMATLAB/Simulink,
realize fast calthe coupling
between the
vehicle
and the
bearing is
internal
forces
of the
culation
and conveniently
apply
external
excitation.
Inrealized,
addition,and
thethe
bearing
outer
ring
vehicle
and
wheel
rail
contact
are
fully
considered.
This
model
can
realize
fast
calculation
fault, inner ring fault and roller fault are added to the coupling model, respectively. The
andisconveniently
apply
external
excitation.
In running
addition,conditions,
the bearingand
outer
ring fault,
model
simulated and
analyzed
under
the actual
verified
by theinner
ringand
fault
and roller
added
to theSince
coupling
model, respectively.
The
model is
rolling
vibrating
test fault
rig of are
single
wheelset.
the vibration
of the axle box
bearing
simulated
and
underofthe
running
conditions,
by the rolling
is closely
related
toanalyzed
the movement
theactual
vehicle
and the
rail, makesand
useverified
of the advantage
and
vibrating
test rig
single
wheelset.
Since theresponses
vibrationofofthe
theaxle
axlebox
boxbearing
bearing is
of the
coupling
model,
theofpaper
studies
the dynamic
closely
related
to
the
movement
of
the
vehicle
and
the
rail,
makes
use
of
the
advantage
under the impact of track irregularity and wheel flat scar, analyzes the dynamic perfor- of
theof
coupling
model,
thedefects
paper studies
the dynamic
responses
the axle box
bearing
under
mance
the bearing
with
and summarizes
some
bearingofvibration
laws.
The vithe
impact
of
track
irregularity
and
wheel
flat
scar,
analyzes
the
dynamic
performance
bration mechanism of axle box bearing under external excitation is revealed in time and
of the bearing
and summarizes
some
vibration
laws.
The vibration
frequency
domain with
so asdefects
to provide
some reference
for bearing
the research
of fault
diagnosis
of
axle box
external
excitation is revealed in time and frequency
axlemechanism
box bearingofunder
realbearing
vehicle under
running
environment.
domain so as to provide some reference for the research of fault diagnosis of axle box
bearing underofreal
vehicle running
environment.
2. Establishment
Vehicle-Bearing
Coupling
Model
2.1. High-Speed
EMU Model
2. Establishment
of Vehicle-Bearing Coupling Model
In the
paper a EMU
vehicle
model of a high-speed EMU is established through the multi2.1.
High-Speed
Model
body dynamics
simulation
software,
Mechanism
as shown
in Figure
1.
In the paper
a vehicle
modelUniversal
of a high-speed
EMU (UM),
is established
through
the multiThe body
modeldynamics
includes 1simulation
carbody, 2software,
bogies, 4 Universal
wheelsets and
8
axle
boxes.
The
carbody,
bogie
Mechanism (UM), as shown in Figure 1.
and The
wheelset
6 degrees
of freedom,
respectively,
axleboxes.
box has
3 carbody,
degrees of
modelhave
includes
1 carbody,
2 bogies,
4 wheelsetsand
andthe
8 axle
The
bogie
freedom
(translation
in
X,
Y
and
Z
directions).
In
the
process
of
modeling,
all
and wheelset have 6 degrees of freedom, respectively, and the axle box hasstructural
3 degrees of
components
regardedin
asX,
rigid
bodies
without considering
theirofelastic
deformation.
freedom are
(translation
Y and
Z directions).
In the process
modeling,
all structural
components are regarded as rigid bodies without considering their elastic deformation.
Carbody
Wheelset
Bogie
Axle box
Figure
1. Dynamic
model
of the
train.train.
Figure
1. Dynamic
model
ofhigh-speed
the high-speed
Invehicle
the vehicle
model,
the primary
suspension
includes
4 axle
box positioning
devices,
In the
model,
the primary
suspension
includes
4 axle
box positioning
devices,
4
coil
springs
and
4
vertical
dampers
etc.
The
secondary
suspension
includes
2
air
springs,
4 coil springs and 4 vertical dampers etc. The secondary suspension includes 2 air springs,
4
yaw
dampers,
lateral
stops,
traction
rods,
and
anti-rolling
torsion
bars,
etc.
4 yaw dampers, lateral stops, traction rods, and anti-rolling torsion bars, etc.
2.2. Axle Box Bearing Model
In this paper, the dynamic model of double row tapered roller bearing is established
through MATLAB/Simulink in which the displacement of inner ring, outer ring and rollers
in X, Y and Z directions are considered. For the convenience of research, the modeling
process simplifies the whole bearing motion system as follows:
(1)
Regardless of the stiffness and damping of elastohydrodynamic lubrication, the
nonlinear factors in the system include the nonlinear contact force between the roller
and the outer raceway, the roller and the inner raceway, the roller and inner ring
flange, and the radial clearance between rollers and the raceways;
ers in X, Y and Z directions are considered. For the convenience of research, the modeling
process simplifies the whole bearing motion system as follows:
Machines 2022, 10, 1024
(1) Regardless of the stiffness and damping of elastohydrodynamic lubrication, the nonlinear factors in the system include the nonlinear contact force between the roller and
the outer raceway, the roller and the inner raceway, the roller and inner ring flange,
6 of 34
and the radial clearance between rollers and the raceways;
(2) The outer ring is fixed, the inner ring and the wheel axle rotate around the Z axis
synchronously. At the same time, without considering slipping or creeping, the mo(2) tion
Theofouter
ring
is fixed, the
innerrolling.
ring and the wheel axle rotate around the Z axis
rollers
is supposed
as pure
synchronously. At the same time, without considering slipping or creeping, the
motion
of rollers
is supposed
as Bearing
pure rolling.
2.2.1. Force
Balance
of Tapered
Roller
The
tapered
roller
bearing, Roller
composed
of outer ring, inner ring, rolling elements and
2.2.1.
Force
Balance
of Tapered
Bearing
cage, has a complex behavior. Its raceway is conical and has strong bearing capacity,
The tapered roller bearing, composed of outer ring, inner ring, rolling elements and
which can bear the combined radial and axial load. The basic structure of double row
cage, has a complex behavior. Its raceway is conical and has strong bearing capacity, which
tapered roller bearings is shown in Figure 2a. Take out a roller for analysis, as shown in
can bear the combined radial and axial load. The basic structure of double row tapered
Figure 2b. Qi represents the contact force between the roller and inner raceway, Qo repreroller bearings is shown in Figure 2a. Take out a roller for analysis, as shown in Figure 2b.
sents the contact force between the roller and outer raceway, Qf represents the contact
Qi represents the contact force between the roller and inner raceway, Qo represents the
force between the roller and the inner ring flange, αi, αo and αf are their contact angles,
contact force between the roller and outer raceway, Qf represents the contact force between
respectively. When the roller is in equilibrium, its forces
satisfy the following balance
the roller and the inner ring flange, αi , αo and αf are their contact angles, respectively. When
equation:
the roller is in equilibrium, its forces satisfy the following balance equation:
 Qo sin α o − Qi sin α i − Q f sin α f = 0
Q
(1)
 o sin αo − Qi sin αi − Q f sin α f = 0
Qo cos α o − Qi cos α i − Q f cos α f = 0
(1)
Qo cos
αo − Qi cos αi − Q f cos α f = 0
(a)
(b)
(c)
Figure2.2.Schematic
Schematicdiagram
diagramofofdouble
doublerow
rowtapered
taperedroller
rollerbearing:
bearing:(a)
(a)bearing
bearingstructure;
structure;(b)
(b)force
force
Figure
balance
balancediagram
diagramofofsingle
singleroller;
roller;(c)
(c)displacement
displacementprojection
projectionrelation.
relation.
Thecontact
contactload
loadQQo oofofthe
theouter
outerraceway
racewayisisused
usedasasthe
thereference
referencevariable,
variable,ititcan
canbe
be
The
obtainedfrom
fromEquation
Equation(1)
(1)that:
that:
obtained
sin((α
αo ++
α fα) )
sin
Q
o
Qii =
= QQo o sin(αio+α f ) f ==cci Q
i Qo
sin (α i + α f )
sin(αo −α f )
Q f = Qosin α − α = c f Qo
sin((αio+α f )f )
Qf = Qo
= c f Qo
sin (α i + α f )
where ci and cf can be expressed as:


 ci =

c f =
(2)
(2)
sin(αo +α f )
sin(αi +α f )
sin(αo −α f )
(3)
sin(αi +α f )
From the above derivation, the relationship between the three contact forces is obtained.
If Qo is calculated, Qi and Qf can be solved by using the balance Equation (2).
Machines 2022, 10, 1024
7 of 34
2.2.2. Relationship between Force and Displacement
The contact between the tapered roller and the raceway is linear contact. Based on the Hertz
contact theory, Palmgren [51] gave an empirical formula for calculating the elastic approach:
"
1 − ν12
1 − ν22
+
δ = 3.81
πE1
πE2
#0.9
Q0.9
l 0.8
(4)
where Q represents the roller force and l represents the roller length. E and v represent the
elastic modulus and Poisson’s ratio, respectively. For steel bearings, their values can be
taken as:
ν1 = ν2 = 0.3, E1 = E2 = 2.06 × 105 N/mm2
So Equation (4) can be derived as:
δ = 3.84 × 10−5
Q0.9
l 0.8
(5)
The deformation of the roller in contact with the inner and outer raceways can be
obtained by:

 δ = 3.84 × 10−5 Q0.9
o
o
l 0.8
(6)
0.9
0.9 0.9
Q
 δ = 3.84 × 10−5 i = 3.84 × 10−5 ci Qi
i
0.8
0.8
l
l
Due to the different contact angles of the inner and outer raceways, the total contact
deformation of the roller cannot be obtained by directly adding the deformation of the two
raceways. Take the contact deformation of the outer raceway as a reference, and project
the contact deformation of the inner raceway to the normal direction of the outer raceway,
then the total contact displacement of the roller in the normal direction can be obtained, as
shown in Figure 2c. It can be deduced as:
δn = δo + δi cos(αo − αi )
(7)
Substituting Equation (6) into Equation (7) to get:
δn =
3.84 × 10−5 0.9
1
+
c
cos
(
α
−
α
)
Q0.9
o
i
i
o
l 0.8
(8)
The expression of contact force between roller and outer raceway can be obtained:
Qo = Kne δn1.11
(9)
where Kne represents the total stiffness coefficient at the contact of the outer raceway:
Kne =
−1.11
3.84 × 10−5 1 + ci 0.9 cos(αo − αi )
0.8
l
(10)
2.2.3. Bearing Load Distribution Calculation
When the bearing is loaded in X, Y and Z directions, the inner ring will displace
relative to the outer ring, which are δx , δy and δz , respectively. From the stress analysis
of tapered roller bearing, it can be seen that there is only one contact load Qo between
the roller and the outer raceway. Therefore, the roller-inner ring system can be studied
as a whole part, as shown in Figure 3. In this way, the contact force between each roller
and the outer raceway Qoi are decomposed in the three directions of X, Y and Z, and then
superimposed to balance with the external load.
Machines 2022, 10, 1024
When the bearing is loaded in X, Y and Z directions, the inner ring will displace relative to the outer ring, which are δx, δy and δz, respectively. From the stress analysis of
tapered roller bearing, it can be seen that there is only one contact load Qo between the
roller and the outer raceway. Therefore, the roller-inner ring system can be studied as a
whole part, as shown in Figure 3. In this way, the contact force between each roller and
34
the outer raceway Qoi are decomposed in the three directions of X, Y and Z, and8 of
then
superimposed to balance with the external load.
Figure
Figure3.3.Contact
Contactdiagram
diagramof
ofsingle
singlerow
rowbearing.
bearing.
Suppose
i , i,ititcan
Supposeatattime
timet,t,the
theposition
positionangle
angleofofthe
theroller
rolleri iisisθθ
canbe
beobtained
obtainedby:
by:
2π2π
θi =θω
(i −i −11))
c tω+
i =
ct +
NN (
(11)
(11)
where
the rotation
rotation angular
angularspeed
speedofofthe
thecage,
cage,
andNNisisthe
thenumber
number of
of single
single row
whereωωcc is
is the
and
row
rollers.
It
can
be
expressed
as:
rollers. It can be expressed as:
1 1  DD
2πn

2πn
ωc =
cos
, ωsω=
(12)
1−
cos α
ωc2= 1 −
α ω
ωss，
(12)
s = 60
D
2  Dmm
60

where
wherennand
andωωs sisisthe
therotational
rotationalangular
angularspeed
speedofofthe
thewheel
wheelaxle,
axle,the
theunits
unitsare
arer/min
r/minand
and
rad/s,
respectively.
In
order
to
be
consistent
with
the
vehicle
coordinate
system,
take
X
and
rad/s, respectively. In order to be consistent with the vehicle coordinate system, take X
Zand
directions
as the as
radial
of the bearing,
and the Y
direction
as the axial
Z directions
the direction
radial direction
of the bearing,
and
the Y direction
asdirection.
the axial
As
the
rotation
of
the
axle
and
the
effect
of
the
radial
clearance,
the
radial
deformation
of
direction. As the rotation of the axle and the effect of the radial clearance, the radial deeach
roller
is
different.
Set
the
radial
clearance
at
the
roller
i
as
h
,
when
the
bearing
bears
i
formation of each roller is different. Set the radial clearance at the
roller i as hi, when the
radial
load,
theradial
radialload,
contact
produced
by theproduced
roller i can
expressed
as: be
bearing
bears
thedeformation
radial contact
deformation
bybethe
roller i can
expressed as:
δri = δxi sin θi + δzi cos θi − hi
(13)
h
h
hi =
sin θθii + δ zi cos θ i − hi
δ ri2 =−δ2xi cos

(13)

h h
−
θ i displacement components of all rollers
When the bearing bears axial hload,
the cos
axial
i =

2 2
are the same. At this time, the contact deformation of each roller is δy .
The
contact
deformation
three
directions
is projected
to thecomponents
normal direction
of the
When
the bearing
bearsin
axial
load,
the axial
displacement
of all rollers
outer
raceway,
respectively,
and
the
total
contact
deformation
between
the
roller
i
and
the
are the same. At this time, the contact deformation of each roller is δy.
outer raceway can be obtained as follows:
δni = δri cos αo + δyi sin αo
(14)
The double row tapered roller bearing located in the axle box of the train adopts the
back-to-back installation method. It is defined that in each row of bearings, the displacement
that increases the roller load is positive, and the displacement that decreases the roller load
is negative. The absolute value of displacement and corresponding load of bearings in the
second row is the same as that in the first row, but the direction is different. The force on the
outer raceway is shown in Figure 4a; for two rows of rollers, the radial displacement is the
same as the direction of the load, and the axial displacement is opposite to that. The balance
equation of double row bearing can be obtained by superimposing the corresponding loads
of two rows of rollers.
the corresponding loads of two rows of rollers.
The contact deformation of each row of rollers along the normal direction can be obtained by:
δ n1i = δ ri cos α o + δ yi sin α o

δ n 2i = δ ri cos α o − δ yi sin α o
Machines 2022, 10, 1024
(a)
(15)
9 of
34
(b)
Figure
roller
bearing:
(a)(a)
contact
force
between
rollers
Figure 4.
4. Contact
Contactforce
forceanalysis
analysisofofdouble
doublerow
rowtapered
tapered
roller
bearing:
contact
force
between
and
outer
raceway;
(b)
decomposition
of
contact
force.
rollers and outer raceway; (b) decomposition of contact force.
The contact deformation of each row of rollers along the normal direction can be
2.2.4. Bearing Balance Equation
obtained by:
According to the force oftapered
bearing and the load displacement relationδn1i = δroller
ri cos αo + δyi sin αo
(15)
ship between rollers and raceways,
the
overall
balance
of bearing can be estabδn2i = δri cos αo −
δyi sin equation
αo
lished. Taking the force analysis of the outer raceway as an example, under the external
load in
the X,Balance
Y and ZEquation
directions, the roller will produce contact deformation. From the
2.2.4.
Bearing
above analysis of the relationship between force and displacement, it can be seen that the
According to the force of tapered roller bearing and the load displacement relationship
contact force between the roller and the outer raceway is expressed as:
between rollers and raceways, the overall balance equation of bearing can be established.
Taking the force analysis of the outerQraceway
as1.11an example,
the
δ ni > 0 under the external load in (16)
oi = K neδ ni
X, Y and Z directions, the roller will produce contact deformation. From the above analysis
the contact
force force
between
roller and the
inner
ii and
the contact
of theThen,
relationship
between
andthe
displacement,
it can
beraceway
seen thatQthe
contact
force
fi
can
be
obtained
through
Equation
force
between
the
roller
and
the
inner
ring
flange
Q
between the roller and the outer raceway is expressed as:
(2).
Qoithe
= balance
Kne δni 1.11equation
δni > 0of force, the components of Q(16)
As shown in Figure 4b, form
oi in
the X, Y and Z directions can be expressed as:
Then, the contact force between the roller and the inner raceway Qii and the contact
force between the roller and the inner ring flange Qfi can be obtained through Equation (2).
As shown in Figure 4b, form the balance equation of force, the components of Qoi in
the X, Y and Z directions can be expressed as:

 Qoxi = Qoi cos αo cos θ
Q = Qoi sin αo
 oyi
Qozi = Qoi cos αo sin θi
(17)
i
The balance equation of the bearing can be obtained by balancing the internal contact
force of the bearing with the external force. They are nonlinear equations with three relative
displacements as variables, and the contact load of each roller can be obtained. Then, the
total contact force between the roller and the outer raceway in the X, Y and Z directions
can be expressed as:

2 N
N


F
=
Q
=
( Q1oxi + Q2oxi ) cos αo cos θi
∑
∑
∑
ox

moxi


m =1 i =1
i =1


2 N
N
Foy = ∑ ∑ Qmoyi = ∑ Q1oyi − Q2oyi sin αo

m =1 i =1
i =1



2 N
N


 Foz = ∑ ∑ Qmozi = ∑ ( Q1ozi + Q2ozi ) cos αo cos θi
m =1 i =1
i =1
(18)
 Fox =  Qmoxi =  ( Q1oxi + Q2 oxi ) cos α o cos θ i
m =1 i =1
i =1

2 N
N

 Foy =  Qmoyi =  ( Q1oyi − Q2 oyi ) sin α o
m =1 i =1
i =1

2 N
N

 Foz =  Qmozi =  ( Q1ozi + Q2 ozi ) cos α o cos θ i
m =1 i =1
i =1

Machines 2022, 10, 1024
(18)
10 of 34
Similarly, the total contact loads of the roller and the inner race raceway Fi, the roller
and the
inner the
ringtotal
flange
Ff, and
components
in inner
the X,
Y and
Z directions
canand
be
Similarly,
contact
loadstheir
of the
roller and the
race
raceway
Fi , the roller
obtained.
the inner ring flange F , and their components in the X, Y and Z directions can be obtained.
f
2.2.5.
2.2.5. Bearing
Bearing Defect
Defect Model
Model
As
the
bearing
rotates
high
speed,
when
bearing
or overloads,
may
As the bearing rotates atat
high
speed,
when
the the
bearing
slipsslips
or overloads,
it mayitcause
cause
wear
on
the
surface.
Once
the
wear
intensifies,
it
will
gradually
evolve
into
defects
wear on the surface. Once the wear intensifies, it will gradually evolve into defects on
on
inner
raceway,
outer
raceway
rollers
surface,
such
as peeling,
pitting
or crackthethe
inner
raceway,
outer
raceway
andand
rollers
surface,
such
as peeling,
pitting
or cracking,
ing,
which
will
cause
hidden
dangers
to
the
operation
safety
of
the
bearing.
which will cause hidden dangers to the operation safety of the bearing.
The
The excitation
excitation generated
generated by
by the
the roller
roller at
at the
the defect
defect can
can be
be divided
divided into
into two
two cases,
cases,
which
are
related
to
the
size
of
the
defect.
One
is
that
the
width
of
the
defect
is
smaller
which are related to the size of the defect. One is that the width of the defect is smaller than
than
the diameter
the roller,
and
the roller
only contacts
the of
edge
the defect.
Take
the
the diameter
of theofroller,
and the
roller
only contacts
the edge
theof
defect.
Take the
outer
outer
ring
defect
as
an
example,
as
shown
in
Figure
5a,b,
in
this
case,
the
bearing
will
ring defect as an example, as shown in Figure 5a,b, in this case, the bearing will generate
generate
impact excitation.
other
that of
thethe
width
ofisthe
defect
is the
larger
than the
diimpact excitation.
The other The
is that
the is
width
defect
larger
than
diameter
of the
ameter
thetime,
roller,the
at roller
this time,
will notthe
only
contact
edge but
of the
defect,
but
roller, atofthis
will the
not roller
only contact
edge
of thethe
defect,
also
sink into
also
sink into
ofcontact
the defect
to shown
contactin
it,Figure
as shown
5c.the
In this
case,will
the
the bottom
of the
thebottom
defect to
it, as
5c. in
In Figure
this case,
bearing
bearing
not only
generate
impact
but also
harmonic excitation.
not onlywill
generate
impact
excitation
butexcitation
also harmonic
excitation.
(a)
(b)
(c)
Figure 5. Bearing
Bearingdefect
defecttype:
type:(a)(a)
defect
width
than
roller
diameter
without
contact
defect
width
lessless
than
thethe
roller
diameter
without
contact
the the
bottom; (b)
defect
width
less
than
the
roller
diameter
and
contact
the
bottom;
(c)
the
defect
width
(b) defect width less than the roller diameter and contact the bottom; (c) the defect width
greater
than the
the roller
roller diameter.
diameter.
greater than
early
defect
of rolling
bearing
occurs,occurs,
its areaits
is generally
small, therefore,
In fact,
fact,when
whenthe
the
early
defect
of rolling
bearing
area is generally
small,
this paper this
onlypaper
studiesonly
the studies
case thatthe
thecase
defect
width
is smaller
than
rollerthan
diameter.
therefore,
that
the defect
width
is the
smaller
the roller
When the roller rolls over the defect, a certain amount of deformation H0 will be
diameter.
released,
which
will change
thethe
contact
deformation
between
roller andHthe
raceway.
When
the roller
rolls over
defect,
a certain amount
of the
deformation
0 will
be reAt
this time,
(14)the
cancontact
be transformed
into:between the roller and the raceway. At
leased,
whichEquation
will change
deformation
this time, Equation (14) can be transformed into:
δni = δri cos αo + δyi sin αo − λH0
(19)
where λ is a switching value.
When there is a defect on the outer raceway, as shown in Figure 6, the maximum value
of H0 can be expressed as:
Hmax =
D
−
2
s
D
2
2
−
2
L
2
where D is the roller diameter. The deformation H0 can be calculated by:
H0 =

D
2

H
−
r 2
D
2
−
2
L
2
H ≥ Hmax
H < Hmax
(20)
where D is the roller diameter. The deformation H0 can be calculated by:
2
2
D
D
L
 −   −  
H0 =  2
 2  2

H
Machines 2022, 10, 1024
H ≥ H max
(20)
H < H max
11 of 34
Figure6.6.Schematic
Schematicdiagram
diagramofofouter
outerraceway
racewaydefect.
defect.
Figure
Theparameter
parameterλλcan
canbe
beexpressed
expressedas:
as:
The
Machines 2022, 10, x FOR PEER REVIEW
λ=
(
)
θ−
, 2π << Φ
−j θ,j 2π
Φout
θout
1 
(
)
1modmod
out θ
out
λ=
0 other
0 other
(21)
12 (21)
of
35
L
Φout = arcsin
(22)
L
D
Φ out = arcsin out
(22)
D 2πof <the
Φ inouter ring.
θdiameter
(
where Φout is the defect angle and D
is the
1,
in − θ j ) , out
outmod
λ=
(24)
When
a defect
on the
as shown in Figure 7, the position of the
where
Фoutthere
is theisdefect
angle
andinner
D
the diameter
 out israceway,
0, other of the outer ring.
defectWhen
changes
at all
theinner
innerraceway,
ring rotates
with the
axle. The
maximum
there
is atimes
defectsince
on the
as shown
in Figure
7, the
positionvalue
of the
of H0 can be expressed as:
defect changes at all times since the inners
ringL rotates with the axle. The maximum value
ωs⋅ t2
2θin =
Φ in = arcsin ，
(25)
of H0 can be expressed as:
L
D
D
D
in
−
Hmax =
−
2
2 2
2 2
D diameter
 D  of L

where Фin is the defect angle and D
in is the
the
H max = −   −   inner ring.
2
 2  2
(
)
At this time, H0 can be calculated by:
2
2
D
D
L
 −   −  
H0 =  2
 2  2

H
H ≥ H max
(23)
H < H max
The parameter λ can be expressed as:
Figure7.7.Schematic
Schematicdiagram
diagramof
ofinner
innerraceway
racewaydefect.
defect.
Figure
At
this there
time, H
be calculated
by:the impact will occur both when the defect posiWhen
is 0acan
defect
on the roller,
tion contacts the outer or theinner r
raceway; the contact condition is shown in Figure 8.
2 2
D
Assuming that the roller k is defective,
so
D at timeL t, the roller rotates with the cage and its
−
− 2
H ≥ Hmax
2
H0 as:
= 2
(23)
position can be expressed

H
H < Hmax
2π
θ k1 = ωc t + ( k − 1)
(26)
N
In addition, the defect will also rotate with the roller itself, and the location of the
defect can be expressed as:
θ k 2 = ωr t
where ωr is the rotation speed of the roller, it can be calculated as follows:
(27)
Machines 2022, 10, 1024
12 of 34
The parameter λ can be expressed as:
λ=
1, mod θin − θ j , 2π < Φin
0, other
Φin = arcsin
L
, θin = ωs · t
Din
(24)
(25)
where Φin is the defect angle and Din is the diameter of the inner ring.
When there is a defect on the roller, the impact will occur both when the defect
position contacts the outer or the inner raceway; the contact condition is shown in Figure 8.
Assuming that the roller k is defective, so at time t, the roller rotates with the cage and its
position can be expressed as:
2π
Machines 2022, 10, x FOR PEER REVIEW
13 of 35
(26)
θk1 = ωc t +
( k − 1)
N
Figure
Figure8.8. Schematic
Schematicdiagram
diagramof
ofroller
rollerdefect.
defect.
In
addition,
the defect
also
rotate with
with the
the inner
rollerraceway,
itself, andthe
the
location of
the
When
the defect
of thewill
roller
contacts
maximum
defordefect
be expressed
as: ring can be obtained by:
mationcan
released
by the inner
θk2 = ωr t
(27)
2
2
D
D
 be
L
in calculated
where ω r is the rotation speed of H
the
roller,in it
as follows:
− can
max =
 2  − 2 
2

  !
2
D
Dm
When the defect of theωroller
contacts
with
the
outer
raceway,
the maximum defor1−
cos α
ω
(28)
r =
s
2D
Dm
mation released by the outer ring can be obtained by:
2
2
When the defect of the roller contacts with the inner
raceway,
the maximum deformaDout
D  L
− by:out  −  
H max
tion released by the inner ring can
be =obtained
2
 2  2
s
Din 2 by:L 2
At this time, the deformation HD0 incan be calculated
−
−
Hmax =
2
2 
2
2 2
Din
Din   L 
H ≥ H max


− 
−



2 the
2
 2outer
 raceway,
When the defect of the roller contacts with
the maximum deforma
tion released by the outer ring can be
obtained
by:
2
2
H0 = 
H ≥ H max
(29)
D
D  L
 sout −  out  −  
 2  22   22
Dout 
Dout
L
H < H max
Hmax =
−
−
H

2
2
2
The parameter λ can be expressed as:



3π 
1, mod   θ k 2 −  , 2π  < Φ roller
2 




λ=
π


1, mod   θ k 2 −  , 2π  < Φ roller
(30)
Machines 2022, 10, 1024
13 of 34
At this time, the deformation H0 can be calculated by:
H0 =











Din
2
Dout
2
−
−
r
r
Din
2
2
Dout
2
−
2
2
−
L
2
2
L
2
H
H ≥ Hmax
H ≥ Hmax
(29)
H < Hmax
The parameter λ can be expressed as:

 1, mod θk2 − 3π
2 , 2π < Φroller
λ=
1, mod θk2 − π2 , 2π < Φroller

0, other
Φroller = arcsin
L
D
(30)
(31)
where Φroller is the defect angle.
2.3. Vehicle-Bearing Coupling Model
The movement and internal action of each component of axle box bearing are very
complex. In this paper, when the bearing model is coupled with the vehicle model, the
bearing model is simplified by considering the contact between the roller and the inner
raceway, the contact between the roller and the outer raceway, and the contact between
the roller and the inner ring flange. As the contact between the bearing inner ring and the
axle is an interference fit, so the two are considered as a whole for research. The contact
between the bearing outer ring and the axle box is also an interference fit, and the two are
also studied as a whole part.
The inner ring and outer ring of the bearing are coupled with the vehicle through the
interaction of forces. The bogie frame is connected to the axle box through the primary
suspension and the node point of rotaty arm. On the one hand, the force generated by the
primary suspension (Ft ) and the node point of rotaty arm (Fa ), respectively, exert on the
outer ring of the bearing through the axle box. The wheel-rail force (Fw ) is generated by the
contact between the wheelset and track exerts on the inner ring of the bearing through the
axle. On the other hand, the contact force generated by the roller-outer raceway (Fo ) exerts
on the axle box of the vehicle model through the outer ring. The contact force is generated
by the roller-inner raceway (Fi ) and the contact force generated by the roller-inner ring
flange (Ff ) exert on the wheelset of the vehicle model through the inner ring. In this way,
the coupling of bearing model and vehicle model is realized through the transmission of
forces, which is shown in Figure 9.
Machines 2022, 10, 1024
the contact between the wheelset and track exerts on the inner ring of the bearing through
the axle. On the other hand, the contact force generated by the roller-outer raceway (Fo)
exerts on the axle box of the vehicle model through the outer ring. The contact force is
generated by the roller-inner raceway (Fi) and the contact force generated by the rollerinner ring flange (Ff) exert on the wheelset of the vehicle model through the inner ring. In
14 of 34
this way, the coupling of bearing model and vehicle model is realized through the transmission of forces, which is shown in Figure 9.
Fax
Fix
Ftx
Fiy
Fty Fay
Outer ring
Bearing
Ftz
Fiz
Foy
Wheelset shaft
Faz
Foz
Fwz
Fox
Inner ring
Fwy
Axle box
Fwx
Figure9.9.Coupling
Coupling
relationshipbetween
betweenvehicle
vehicleand
andbearing.
bearing.
Figure
relationship
The corresponding relationship between the vehicle and the bearing coordinate system
The corresponding relationship between the vehicle and the bearing coordinate syscan be expressed as: the longitudinal direction of the vehicle corresponds to the X direction
tem can be expressed as: the longitudinal direction of the vehicle corresponds to the X
of the bearing; the lateral direction of the vehicle corresponds to the Y direction of the
direction of the bearing; the lateral direction of the vehicle corresponds to the Y direction
bearing; and the vertical direction of the vehicle corresponds to the Z direction of the
of the bearing; and the vertical direction of the vehicle corresponds to the Z direction of
bearing.
the bearing.
Based on the above analysis, the force exerted by the vehicle on the outer ring of the
Based
above
analysis,
thesuspension
force exerted
the
vehicle
onofthe
outer
ringthe
of force
the
bearing ison
thethe
force
of the
primary
andby
the
node
point
rotaty
arm,
bearing
is
the
force
of
the
primary
suspension
and
the
node
point
of
rotaty
arm,
the
force
exerted by the vehicle on the inner ring of the bearing is the wheel-rail force. The motion
exerted
by the
vehicle
on theofinner
ring of can
the be
bearing
wheel-rail
force.law.
The motion
equations
of each
element
the bearing
givenisbythe
Newton’s
second
equations
of
each
element
of
the
bearing
can
be
given
by
Newton’s
second
law.
Motion equation of bearing outer ring:
Motion equation of bearing outer ring:

..
.
c x + k x = F + F + Fax
 mo x.. +
mo 
x +oc.o x + koo x = Foxox+ Ftx tx
+F
(32)
mo y + co y + k o y = Foy + Fty +axFay
..
.
 
+ FFtytz ++FF
y c+oczo +
y +k okzo y== FFozoy +
(32)
ayaz + m o g
z o+
mom
 
mo z + co z + ko z = Foz + Ftz + Faz + mo g
Motion equation of bearing inner ring:
Motion equation ofbearing
inner
ring:
..
.
 mi x.. + ci x. + k i x = − Fix + Ff x + Fwx
m y + ci y + k i y = − Fiy − Ff y + Fwy
(33)
.
 i ..
mi z + ci z + k i z = − Fiz − Ff z + Fwz + mi g
Motion equation of bearing rollers:

..
.
 mr x.. + cr x. + kr x = − Fox + Fix + Ff x
mr y + cr y + kr y = − Foy + Fiy + Ff y
..
.

mr z + cr z + kr z = − Foz + Fiz − Ff z + mr g
(34)
The characteristic frequency of each element of the bearing can be obtained by:
V0
2πR
(35)
1
D
n
(1 −
cos(αo ))
2
Dm
60
(36)
fs =
fc =
N
D
n
(1 −
cos(αo ))
2
Dm
60
2 !
1 Dm
D
=
1−
cos αo
fs
2 D
Dm
f vc =
f roller
(37)
(38)
Machines 2022, 10, 1024
15 of 34
N
D
f out =
1−
cos αo f s
2
Dm
D
N
1+
f in =
cos αo f s
2
Dm
(39)
(40)
where R represents the wheel radius, fs , fc , f vc , f roller , f out and f in represent the axle rotation
frequency (ARF), cage rotation frequency (CRF), VC vibration frequency (VCF), roller fault
characteristic frequency (FCFR), outer ring fault characteristic frequency (FCFO) and inner
ring fault characteristic frequency (FCFI), respectively.
3. Co-Simulation and Model Verification
During the co-simulation of UM and MATLAB/Simulink, the vehicle model is output
as an s-function to realize the data exchange between the two models. The calculation
Machines 2022, 10, x FOR PEER REVIEW
16 of 35
process is shown in Figure 10; the parameters of double row tapered roller bearing are
shown in Table 1.
Start
Set the vehicle parameters
Set the bearing parameters
Defective
Set working condition
Defect location and initial angle
Calculation
No defect
Judge whether the roller enters the defect area
Node point
of rotaty
army force
Primary
force
Wheel rail
force
Calculate the total deformation of outer raceway
Calculate the contact force of outer raceway
Calculate the contact force between the inner
raceway and the inner ring flange
Input to bearing model
Co-simulation
End
Figure 10.
10. Flow
Flow chart
chart of
of co-simulation
co-simulation solution.
solution.
Figure
Table 1. Structural parameters of double row tapered roller bearings.
Table 1. Structural parameters of double row tapered roller bearings.
Parameters
Parameters
Inner race diameter(Di)
Inner race diameter
(Ddiameter(D
i)
Outer race
o)
Outer race
diameter
(D
) diameter(D)
Average rolleroof
Average roller of diameter (D)
Number of single row rollers(N)
Number of single row rollers (N)
Effective length of rollers(l)
Effective length of rollers (l)
m)
Pitch
Pitch diameter
(Ddiameter(D
m)
Inner race
contact
Inner race contact
angle
(αi ) angle(αi)
Outer race
contact
Outer race contact
angle
(αo ) angle(αo)
Flange
contact
Flange contact
angle
(αf ) angle(αf)
Radial clearance
Radial(e)clearance(e)
Value
130
240
26.5
17
45
185
7.5
10
14
50
Value
130
240
26.5
17
45
185
7.5
10
14
50
Unit
Unit
mm
mmmm
mmmm
mm
/
/
mm
mm
mmmm
◦
°
◦
°
◦
°
µm μm
Table 2. Characteristic frequency of bearing.
ARF
19.22Hz
CRF
8.26 Hz
VCF
140.34 Hz
FCFO
140.34 Hz
FCFI
186.43 Hz
FCFR
65.76 Hz
Machines 2022, 10, 1024
16 of 34
In the simulation, the vehicle runs straight, the wheel tread is S1002CN new tread,
and the rail is CN_ Rail_ 60 profile. Set the running speed of the vehicle to 200 km/h. In
this paper, all the bearing faults used in the simulation are through groove faults, with a
width of 1 mm and a depth of 1 mm. The characteristic frequency of the bearing at this
time is shown in Table 2.
Table 2. Characteristic frequency of bearing.
ARF
CRF
VCF
FCFO
FCFI
FCFR
19.22 Hz
8.26 Hz
140.34 Hz
140.34 Hz
186.43 Hz
65.76 Hz
Figure 11 shows the vibration acceleration of a normal bearing without track irregularity. The periodic shock signal generated by VC vibration can be seen from the figure, and
the shock period is T = 0.0071 s. The VCF can be clearly observed in the frequency spectrum,
Machines 2022, 10, x FOR PEER REVIEW
17 of 35
which is 140.4 Hz (approximately 1/T); the double-frequency can also be observed, and the
value is consistent with the theoretical result.
Figure11.
11.Outer
Outerring
ringvertical
verticalacceleration
acceleration of normal
(b)(b)
frequency
spectrum.
Figure
normalbearing:
bearing:(a)
(a)time
timehistories;
histories;
frequency
spectrum.
Figure 12 shows the vibration acceleration of a bearing with outer raceway defect. The
period
caused
by defect
T = 0.0071
s, and the FCFO
is 140.4with
Hz (approximately
1/T).
Figure
12 shows
the is
vibration
acceleration
of a bearing
outer raceway defect.
The period caused by defect is T = 0.007 1 s, and the FCFO is 140.4 Hz (approximately 1/T).
Figure 11. Outer ring vertical acceleration of normal bearing: (a) time histories; (b) frequency spectrum.
Machines 2022, 10, 1024
17 of 34
Figure 12 shows the vibration acceleration of a bearing with outer raceway defect.
The period caused by defect is T = 0.007 1 s, and the FCFO is 140.4 Hz (approximately 1/T).
Figure12.
12.Outer
Outerring
ringvertical
verticalacceleration
acceleration
bearing
with
outer
raceway
defect:
time
histories;
Figure
ofof
bearing
with
outer
raceway
defect:
(a)(a)
time
histories;
(b)
frequency
spectrum.
(b) frequency spectrum.
Machines 2022, 10, x FOR PEER REVIEW
18 of 35
Figure1313shows
showsthe
the vibration
vibration acceleration
inner
raceway
defect.
The
Figure
acceleration of
ofaabearing
bearingwith
with
inner
raceway
defect.
impact
period
caused
by
defect
is
T
=
0.0054
s,
and
the
FCFI
is
186.2
Hz
(approximately
The impact period caused by defect is T = 0.0054 s, and the FCFI is 186.2 Hz (approximately
1/T).As
Asthe
theinner
innerraceway
racewaydefect
defectrotates
rotateswith
withthe
theaxle,
axle,its
itsimpact
impacteffect
effectisismodulated
modulatedby
by
1/T).
theARF,
ARF, and
and the
the rotation
rotation period
period is
is TT ==0.0536
0.0536s.s.
the
Figure13.
13.Outer
Outerring
ringvertical
verticalacceleration
accelerationof
ofbearing
bearingwith
withinner
innerraceway
racewaydefect:
defect:(a)
(a)time
timehistories;
histories;
Figure
(b)
(b)frequency
frequencyspectrum.
spectrum.
In Figure 13b, some sidebands can be observed on both sides of FCFI and its doublefrequency, which the amplitude gradually attenuates. The interval between sidebands is
approximately 19.1 Hz (approximately 1/T), which is equal to the ARF exactly.
Figure 14 shows the acceleration and its spectrum diagram when the roller has a de-
Machines 2022, 10, 1024
18 of 34
Figure 13. Outer ring vertical acceleration of bearing with inner raceway defect: (a) time histories;
(b) frequency spectrum.
In
InFigure
Figure13b,
13b,some
somesidebands
sidebandscan
canbe
beobserved
observedon
onboth
bothsides
sidesofofFCFI
FCFIand
andits
itsdoubledoublefrequency,
which
the
amplitude
gradually
attenuates.
The
interval
between
sidebands
frequency, which the amplitude gradually attenuates. The interval between sidebandsisis
approximately
approximately19.1
19.1Hz
Hz(approximately
(approximately1/T),
1/T),which
whichisisequal
equaltotothe
theARF
ARFexactly.
exactly.
Figure
14
shows
the
acceleration
and
its
spectrum
diagram
when
the roller
has a deFigure 14 shows the acceleration and its spectrum diagram when
the roller
has
fect.
The
impact
period
caused
by
roller
defect
is
T
=
0.0152
s,
and
the
FCFR
is
a defect. The impact period caused by roller defect is T = 0.0152 s, and the 65.79Hz
FCFR is
(approximately
1/T). As the1/T).
rollerAs
rotates
withrotates
the cage,
thethe
magnitude
the impactof
gen65.79 Hz (approximately
the roller
with
cage, theofmagnitude
the
erated
by
the
fault
is
modulated
by
the
cage
frequency.
It
can
be
seen
that
its
period
is
T
=
impact generated by the fault is modulated by the cage frequency. It can be seen that its
0.121
6
s
in
Figure
14a.
As
shown
in
Figure
14b,
a
sideband
with
gradually
decreasing
period is T = 0.1216 s in Figure 14a. As shown in Figure 14b, a sideband with gradually
amplitude
be observed
onobserved
both sides
the sides
FCFRofand
double-frequency;
the interdecreasingcan
amplitude
can be
onof
both
theits
FCFR
and its double-frequency;
val
is
approximately
8.3
Hz
(approximately
1/T),
which
is
equal
to
the
CFR.
the interval is approximately 8.3 Hz (approximately 1/T), which is equal to the CFR.
Machines 2022, 10, x FOR PEER REVIEW
Machines 2022, 10, x FOR PEER REVIEW
19 of 35
19 of 35
Figure
14. Outer
vertical
acceleration
ofacceleration
bearing with roller
defect: (a)
timeroller
histories; (b) fre-(a) time histories;
Figure
14. ring
Outer
ring
vertical
bearing
with
Figure
14. Outer
ring verticalofacceleration
of
bearing withdefect:
roller defect: (a) time histories; (b) frequency spectrum.
quency spectrum.
(b) frequency spectrum.
When the outer raceway of the bearing has a defect, the vertical and lateral acceleraWhenof
the
outer
raceway ofa the
bearing has
a defect,
thelateral
vertical
and lateral acceleratheand
outer
raceway
the
bearing
defect,
vertical
and
acceleration
tion of theWhen
axle box
wheelset
are shown
in Figureshas
15 and
16. A the
small
amplitude
vition of the axle box and wheelset are shown in Figures 15 and 16. A small amplitude viof the
boxinand
wheelset
shown
in Figures
15 and 16.
small
vibration
bration
canaxle
be seen
the vertical
andare
lateral
directions.
It is indicated
thatA
when
theamplitude
vebration can be seen in the vertical and lateral directions. It is indicated that when the vehiclecan
andbe
theseen
bearing
are
coupled
together,
the
vibration
of
the
bearing
is
transmitted
to
in the vertical
and
is indicated
that when
vehicle
and
hicle and
thelateral
bearingdirections.
are coupledIttogether,
the vibration
of thethe
bearing
is transmitted
to
the vehicle.
In addition,
the FCFO
can be detected
in both of
the horizontal
and
dithe bearing
are coupled
together,
the vibration
is vertical
transmitted
to the vehicle.
the vehicle.
In addition,
the FCFOthe
can bearing
be detected
in both the horizontal
and vertical directions
in the frequency
spectrum,
which
furtherin
verifies
the
ofand
the coupling
In addition,
the FCFO
can be
both
theeffectiveness
horizontal
vertical
directions of
inthe
thecoupling
rections
in detected
the frequency
spectrum,
which further
verifies
the effectiveness
model.
frequency spectrum,model.
which further verifies the effectiveness of the coupling model.
(a)
(b)
(a)
(b)
Figure 15. Vertical acceleration of axle box and wheelset: (a) time histories; (b) frequency spectrum.
Figure 15. Vertical acceleration of axle box and wheelset: (a) time histories; (b) frequency spectrum.
Figure 15. Vertical acceleration of axle box and wheelset: (a) time histories; (b) frequency spectrum.
2.5
2
1.5
10-3
1×fout
2.5
2×fout
3×f out
4×f
10-3
Axle box
1×fout
Wheelset
2
5×fout
2×fout
Axle box
Wheelset
3×f out
5×f
Machines 2022, 10, 1024
(a)
(b)
19 of 34
Figure 15. Vertical acceleration of axle box and wheelset: (a) time histories; (b) frequency spectrum.
10-3
1×fout
2.5
2
2×fout
1.5
Axle box
Wheelset
3×f out
4×f out5×fout
1
0.5
0
0
200
(a)
400
600
Frequency(Hz)
800
1000
(b)
Figure16.
16.Lateral
Lateralacceleration
accelerationofofaxle
axlebox
boxand
andwheelset:
wheelset:(a)
(a)time
timehistories;
histories;(b)
(b)frequency
frequency
spectrum.
spectrum.
20 of 35
Figure
Machines 2022, 10, x FOR PEER REVIEW
The above analysis shows that the vehicle-bearing coupling model established in
this paper
cananalysis
reflect the
vibration
characteristicscoupling
of the system
and can be
The above
shows
that the vehicle-bearing
model established
inused
this for the
simulation
analysis
of
vehicle
and
bearing
system.
paper can reflect the vibration characteristics of the system and can be used for the simulation analysis of vehicle and bearing system.
4. Model Validation by Experiment
4. Model
Validation
by Experiment
In order
to further
verify the correctness of the model established in the paper, experi-
ments
carried
out on
the the
rolling
and vibrating
test rig
of single in
wheelset.
The
In are
order
to further
verify
correctness
of the model
established
the paper,
ex-test rig is
periments
are carried
on the rolling
andframe,
vibrating
test rig of
single wheelset.
mainly
composed
ofout
actuator,
reaction
wheelset,
hydraulic
station,The
railtest
wheel and
rig exciter,
is mainly
composed
of actuator, reaction
frame,
station,
wheel
its
etc.
It is a multi-purpose
test-bed
to wheelset,
simulatehydraulic
the operation
ofrail
high-speed
train,
and its can
exciter,
etc. Itto
is study
a multi-purpose
test-bed
to simulate characteristics
the operation ofof
high-speed
which
be used
the vibration
transmission
wheelsets, gears,
train,boxes,
which can
be used
to study
vibration
characteristics
of wheelsets,
axle
bearings
and
otherthe
systems.
It transmission
can carry out
vibration test,
traction braking
gears, axle boxes, bearings and other systems. It can carry out vibration test, traction braktest, lateral stability test, etc. It can be loaded through the action of horizontal and vertical
ing test, lateral stability test, etc. It can be loaded through the action of horizontal and
actuators to simulate abnormal wheel-rail contact conditions, such as track irregularity,
vertical actuators to simulate abnormal wheel-rail contact conditions, such as track irregwheel
polygon, rail corrugation, etc. The maximum axle load of the test rig is 25 t, the
ularity, wheel polygon, rail corrugation, etc. The maximum axle load of the test rig is 25 t,
maximum
torque
is 5000
N·N·m,
m, the
is 500
500 km/h,
km/h,and
andthe
the
the maximum
torque
is 5000
themaximum
maximumrunning
running speed
speed is
ex-excitation
frequency
range
is
0–500
Hz.
The
site
layout
of
the
test
is
shown
in
Figure
17.
citation frequency range is 0–500 Hz. The site layout of the test is shown in Figure 17.
Reaction frame
Actuator
Sensor
Wheelset
Rail wheel
Axle box
Hydraulic
station
Rail excitation
(a)
(b)
Figure 17.
rolling and
and vibrating
vibrating test
test rig
rig of
of single
single wheelset;
wheelset; (b)
(b) ensor
ensor arrangement.
arFigure
17. Site
Site layout
layout of
of the
the test:
test: (a)
(a) rolling
rangement.
The objects of this experiment are three kinds of bearings with faults in different
The objects
of thisraceway
experiment
threeraceway
kinds of of
bearings
with faults
popositions.
The inner
andare
outer
the bearing
usedinindifferent
the experiment
are
sitions. The inner raceway and outer raceway of the bearing used in the experiment are
processed into through groove faults, with a width of 1 mm and a depth of 1 mm. The roller
processed into through groove faults, with a width of 1 mm and a depth of 1 mm. The
is processed as a pitting fault. Set the running speed as 200 km/h. The wheel is subject
roller is processed as a pitting fault. Set the running speed as 200 km/h. The wheel is subject to sinusoidal excitation in which the axial loading amplitude is 1 mm; the vertical
loading amplitude is 2 mm, and the loading frequency is 5 Hz. Select appropriate positions at the left and right axle boxes as measuring points to collect their lateral and vertical
acceleration signals. Set the sampling frequency to 25.6 kHz and the sampling time to 60
Machines 2022, 10, 1024
20 of 34
to sinusoidal excitation in which the axial loading amplitude is 1 mm; the vertical loading
amplitude is 2 mm, and the loading frequency is 5 Hz. Select appropriate positions at the
left and right axle boxes as measuring points to collect their lateral and vertical acceleration
Machines 2022, 10, x FOR PEER REVIEW signals. Set the sampling frequency to 25.6 kHz and the sampling time
21 to
of 60
35 s. The correctness
of the model is verified by comparing the experimental results with the simulation results.
Figure 18 shows the frequency spectrum of the vertical vibration acceleration of the
Tableouter
3. Comparisons
between
the theoretical
and simulation
results.and its double-frequency can be found in
ring. The
bearing
characteristic
frequency
Bearing Type
Value
(Hz) because
Experimental
(%) is not large enough, the
theSimulation
figure, but
perhaps
the sizeValue
of the(Hz)
roller faultDifference
processing
Normal bearing
140.34
148.2
5.6 results and the simulation
FCFR is not
obvious. The comparison
between the experimental
Bearing with outer race fault
3.25
results is 140.34
listed in Table 3, and the144.9
errors are within a reasonable
range, which further
Bearing with inner race fault
186.43
192.2
3.09
proves
the
correctness
of
the
model
established
in
this
paper.
Bearing with roller fault
131.52
135.5
3.03
FigureFigure
18. Frequency
spectrum spectrum
of outer ring
acceleration:
normal bearing;
bearingbearing; (b) bearing
18. Frequency
ofvertical
outer ring
vertical(a)acceleration:
(a)(b)
normal
with outer
raceway
fault;
(c)
bearing
with
inner
raceway
fault;
(d)
bearing
with
roller
fault.
with outer raceway fault; (c) bearing with inner raceway fault; (d) bearing with roller fault.
Machines 2022, 10, 1024
21 of 34
Table 3. Comparisons between the theoretical and simulation results.
Bearing Type
Simulation Value (Hz)
Experimental Value (Hz)
Normal bearing
140.34
140.34
186.43
131.52
148.2
144.9
192.2
135.5
Bearing
with outer race fault
2022, 10, x FOR
PEER REVIEW
Bearing with inner race fault
Bearing with roller fault
Difference (%)
5.6
22 of 353.25
3.09
3.03
5. Track Irregularity Spectrum and Wheel Flat Scar Model
5. Track Irregularity Spectrum and Wheel Flat Scar Model
When the train runs on the track, it will be affected by various external disturbances,
When the train runs on the track, it will be affected by various external disturbances,
resulting in relatively strong vibration. The vibration will transmitted upward through the
resulting in relatively strong vibration. The vibration will transmitted upward through
wheel rail interface [52], which will have a certain impact on the vibration of the axle box
the wheel rail interface [52], which will have a certain impact on the vibration of the axle
bearing. Therefore, how the vibration amplitude and frequency characteristics change needs
box bearing. Therefore, how the vibration amplitude and frequency characteristics change
further analysis and discussion. In order to analyze the axle box bearing more accurately
needs further analysis and discussion. In order to analyze the axle box bearing more
and comprehensively, the paper will study the vibration of the bearing under the wheel rail
accurately and comprehensively, the paper will study the vibration of the bearing under
excitation.
the wheel rail excitation.
Track irregularity is one of the main factors of vehicle vibration, which is an unavoidTrack irregularity is one of the main factors of vehicle vibration, which is an unavoidable track excitation. Various track irregularities can be directly added to the vehicle for
able track excitation. Various track irregularities can be directly added to the vehicle for
simulation in UM. In the paper, the German high interference track spectrum is selected as
simulation in UM. In the paper, the German high interference track spectrum is selected as
an input, which comes
from the calculation of UM internal program, as shown in Figure 19.
an input, which comes from the calculation of UM internal program, as shown in Figure 19.
0.01
0.01
0.005
0.005
0
0
–0.005
–0.005
Left rail
Right rail
–0.01
0
200
400
600
Distance(m)
Left rail
Right rail
800
(a)
1000
–0.01
0
200
400
600
Distance(m)
800
1000
(b)
Figure 19. Track irregularity
spectrum:
(a) vertical;
(b) horizontal.
Figure 19. Track
irregularity
spectrum:
(a) vertical; (b) horizontal.
Wheel defects will
increase
thewill
impact
of wheel
rail and
of vehicle
Wheel
defects
increase
the impact
ofaffect
wheelthe
railsafety
and affect
the safety of vehicle
and rail. Wheel and
flat scar
a common
wheel
shape defect,
which
will
causewhich
a special
rail. isWheel
flat scar
is a common
wheel
shape
defect,
willtype
cause a special type
of intermittent pulse
excitation source
at the wheel
rail contact
interface
Flat scars
can [53]. Flat scars
of intermittent
pulse excitation
source
at the wheel
rail[53].
contact
interface
be divided into new
and
old ones.
newold
flatones.
scar is
formed,
it will
produce
an acute
can be
divided
intoOnce
newaand
Once
a new
flat scar
is formed,
it will produce an
acute
andscar
become
old flat
scar wear
after wheel
tread wear
and plastic
angle and become
anangle
old flat
afteranwheel
tread
and plastic
deformation.
Asdeformation. As
in the
Figure
20, of
L0 the
is the
length
of the
flat
scar, L1 islength
the minimum
shown in Figureshown
20, L0 is
length
new
flat scar,
L1new
is the
minimum
of the length of the
flat scar that
guarantees
convex
form
ofdthe
wheel,
and
is theThe
depth of them. The
old flat scar thatold
guarantees
the convex
formthe
of the
wheel,
and
is the
depth
ofdthem.
geometric
dimension
of flat scarby
is Equations
determined
by Equations
geometric dimension
of flat
scar is determined
(41)–(43)
in UM.(41)–(43) in UM.
Z rfx
p
2
L0 = 8Rd − d L
8Rd − d2
0 =
d
πx
Zr f x =
d
π x2  1 + cos 2 L
= 1 + cos  2
1

2 
 L1  
πL0
L1 =
2
π L0
L1 =
2
(41)
(42)
(41)
(42)
(43)
(43)
Machines 2022, 10, x FOR PEER REVIEW
Machines 2022, 10, 1024
Machines 2022, 10, x FOR PEER REVIEW
23 of 35
23 of 35
22 of 34
Figure
20. Schematic
diagram
of new
and oldscar.
flat scar.
Figure 20.
of new
flat flat
scar scar
and old
Figure
20. Schematic
Schematicdiagram
diagram
of new
flat
scar
andflat
old flat scar.
6.
DynamicResponse
Response
Bearing
under
Track
Irregularity
6. Dynamic
of of
Bearing
under
Track
Irregularity
6. Dynamic Response of Bearing under Track Irregularity
Set the
thevehicle
vehiclespeed
speedtoto300
300
km/h,
German
interference
track spectrum as
Set
km/h,
taketake
the the
German
highhigh
interference
track spectrum
Set
the
vehiclethe
speed
to 300
km/h,
take
the
German
high
interference
track
spectrum
an
input,
simulate
normal
bearing
and
the
bearings
with
outer
raceway,
inner
raceway
as an
input,
simulate
the
normal
bearing
and
the
bearings
with
outer
raceway,
inner
raceas
an
input,
simulate
the
normal
bearing
and
the
bearings
with
outer
raceway,
inner
way
and
roller
defects,
respectively.
The
results
of
the
vertical
acceleration
of
the
outer
and roller defects, respectively. The results of the vertical acceleration of the outerracering
way
andshown
roller
defects,
respectively.
The
results
of the
vertical
acceleration
the outer
ringshown
are
Figure21,
21,
where WOI
without
irregularity
and WIand
represents
are
in in
Figure
where
WOIrepresents
represents
without
irregularity
WIofrepresents
withare
irregularity
as as
blew).
Through
comparison
of track
and
ring
shown in(same
Figure
21,
where
WOI the
represents
without
irregularity
and
WInon
represents
with
irregularity
(same
blew).
Through
the comparison
ofirregularity
track irregularity
and non
track
irregularity,
it
can
be
seen
that
the
vertical
acceleration
of
normal
bearing
changes
with
Through
the comparison
of of
track
irregularity
non
trackirregularity
irregularity,(same
it can as
beblew).
seen that
the vertical
acceleration
normal
bearing and
changes
greatly,
while the acceleration
of defective
bearings
change
slightly. Extract
the maximum
track
irregularity,
it can be seen
the
vertical
acceleration
of normal
bearing
changes
greatly,
while the acceleration
of that
defective
bearings
change
slightly.
Extract
the maximum
acceleration amplitude of normal bearing and defective bearing, respectively, and make
greatly,
whileamplitude
the acceleration
of defective
change
slightly.respectively,
Extract the maximum
acceleration
of normal
bearingbearings
and defective
bearing,
and make
quantitative analysis, as shown in Table 4. It can be seen from the comparison results that
acceleration
amplitude
ofshown
normal
bearing
and
defective
bearing,
respectively,
and make
quantitative
analysis,
as
in
Table
4.
It
can
be
seen
from
the
comparison
results
that
the maximum amplitude of normal bearing has changed greatly under the condition of
quantitative
analysis,
as
shown
in
Table
4.
It
can
be
seen
from
the
comparison
results
that
the
maximum
amplitude
of
normal
bearing
has
changed
greatly
under
the
condition
of
track irregularity, while that of defective bearings has changed little. This is because the
the
maximum
amplitude
of normal
bearing
has irregularity
changed
greatly
under
theiscondition
of
track
irregularity,
while
that
defective
bearings
has changed
little.
because the
vibration
of the normal
bearing
isofslight,
and
the
track
will cause
itsThis
amplitude
track
irregularity,
while
that
of is
defective
bearings
hasirregularity
changed
little.
isits
because
the
vibration
ofgreatly,
the normal
slight,
and
the
will This
cause
amplitude
to fluctuate
whilebearing
the defective
bearing
willtrack
produce
large vibration
under
the
to
fluctuate
greatly,
while
the the
defective
bearing
willirregularity
produce
vibration
under
the
vibration
of defect,
the
normal
bearing
is slight,
and
theirregularity
track
causeThis
its amplitude
effect
of the
which
causes
effect of
track
to not large
bewill
obvious.
is
because
of
the normal
bearing
isbearing
slight,
and
the
track irregularity
cause
effect
ofthe
thevibration
defect, which
causes
the effect
of track
irregularity
to notvibration
bewill
obvious.
This
is
to
fluctuate
greatly,
while
the defective
will
produce
large
under
the
its amplitude
to greatly
The
defective
bearing
willthe
produce
anot
large
because
offluctuate.
thecauses
normal
bearing
and
tracktoirregularity
will cause
effect
of the
the vibration
defect,
which
the
effect is
ofslight,
track irregularity
bevibration
obvious.
This its
is
under thethe
effect
of the defect,
causes
the effect
of track
irregularity
be
not obvious.
amplitude
tovibration
greatly
fluctuate.
The
defective
will
produce
large
vibration
because
of thewhich
normal
bearing
isbearing
slight,
and
the tracktoairregularity
will under
cause
the
effect
of
the
defect,
which
causes
the
effect
of
track
irregularity
to
be
not
obvious.
its amplitude to greatly fluctuate. The defective bearing will produce a large vibration
under the effect of the defect, which causes the effect of track irregularity to be not obvious.
40
20
40
300
WOI
WI
100 300
WOI
WI
20
–100 100
Machines 2022, 10, x FOR PEER REVIEW
0
–20
200
WOI
WI
0 200
0
–20
WOI
WI
200
0.2
0.4
0.6
–200
0.8
Time(s)
24 of 35
0
0.2
0.4
–100
(a)
0.2WOI
0.4
WI
0.8
(b)
0.6
–200
0.8
Time(s)
100
0.6
Time(s)
500
WOI
WI
0.2
0.4
0.6
0.8
Time(s)
(a)
(b)
0
0
–100
–500
0.2
0.4
0.6
0.8
0.2
0.4
0.6
Time(s)
Time(s)
(c)
(d)
0.8
Figure 21.
of of
outer
ringring
vertical
acceleration:
(a) normal
bearing;bearing;
(b) bearing
Figure
21. Time
Timehistories
histories
outer
vertical
acceleration:
(a) normal
(b) with
bearing with
outer raceway defect; (c) bearing with inner raceway defect; (d) bearing with roller defect.
outer raceway defect; (c) bearing with inner raceway defect; (d) bearing with roller defect.
Table 4. Comparison before and after input track irregularity.
Type of Bearing
WOI(m·s−2)
WI(m·s−2)
Normal bearing
Bearing with outer race
fault
1.8
38.8
Variation of Amplitude (%)
2055.56
172.2
191.9
11.44
(c)
(d)
Figure 21. Time histories of outer ring vertical acceleration: (a) normal bearing; (b) bearing with
outer raceway defect; (c) bearing with inner raceway defect; (d) bearing with roller defect.
Table 4. Comparison before and after input track irregularity.
Machines 2022, 10, 1024
23 of 34
Type of Bearing
WOI(m·s−2)
Variation of Amplitude (%)
2055.56
WI(m·s−2)
Normal
bearing
1.8 track irregularity.38.8
Table 4.
Comparison
before and after input
Bearing with outer race
Typefault
of Bearing
Bearing with inner race
Normal
faultbearing
Bearing with outer race fault
Bearing with roller fault
Bearing with inner race fault
Bearing with roller fault
172.2
191.9
11.44 of
Variation
Amplitude (%)
144.11.8
150.5
38.8
172.2
460.3
144.1
460.3
191.9
550.2
150.5
550.2
4.37
2055.56
11.44
19.53
4.37
19.53
WOI (m·s−2 )
WI (m·s−2 )
Figure 22 shows the comparison of the axis trajectory under four conditions with and
without track irregularity. It can be seen that the axis trajectory changes greatly under the
Figure 22 shows the comparison of the axis trajectory under four conditions with and
action
of track irregularity, whether it is a normal bearing or a defective bearing. So, it can
without track irregularity. It can be seen that the axis trajectory changes greatly under the
be further explained that the impact of track irregularity on system vibration cannot be
action of track irregularity, whether it is a normal bearing or a defective bearing. So, it can
ignored. Therefore, through the vehicle-bearing coupling model established in the paper,
be further explained that the impact of track irregularity on system vibration cannot be
the study can be closer to the real running environment of axle box bearing.
ignored. Therefore, through the vehicle-bearing coupling model established in the paper,
the study can be closer to the real running environment of axle box bearing.
0
0
WI
WOI
WI
WOI
–0.2
–0.4
–0.5
–0.6
–0.8
–1
–1
–1.2
Machines 2022, 10, x FOR PEER REVIEW
–3
–2
–1
0
1
2
Longitudinal displacement(μm)
3
25 of 35
–1.4
–3
–2
–1
0
1
2
Longitudinal displacement(μm)
(a)
3
(b)
0
WI
WOI
–5
–10
–15
–3
–2
–1
0
1
2
3
Longitudinal displacement(μm)
(c)
(d)
Figure22.
22.Axis
Axistrack
trackof
ofbearing
bearingouter
outerring:
ring: (a)
(a) normal
normal bearing;
bearing;(b)
(b) bearing
bearingwith
with outer
outer raceway
racewayfault;
fault;
Figure
(c) bearing with inner raceway fault; (d) bearing with roller fault.
(c) bearing with inner raceway fault; (d) bearing with roller fault.
Inorder
ordertotostudy
studythe
theinfluence
influence
speed
vibration
of each
element
of the
bearIn
ofof
speed
onon
thethe
vibration
of each
element
of the
bearing,
ing,
the
vehicle-bearing
system
at
different
speeds
under
the
action
of
track
irregularity
the vehicle-bearing system at different speeds under the action of track irregularity isis
simulated.Figure
Figure23
23shows
showsthe
theRMS
RMSvalues
valuesof
ofthe
thevertical
verticaland
andlateral
lateralaccelerations
accelerationsof
ofthe
the
simulated.
outer
ring,
inner
ring
and
roller,
respectively.
It
can
be
seen
that
with
the
increase
of
speed,
outer ring, inner ring and roller, respectively. It can be seen that with the increase of speed,
theRMS
RMSvalues
valuesof
ofthe
thethree
threeelements
elementsgradually
graduallyincrease,
increase,the
thevalue
valueof
ofthe
theinner
innerring
ringisisthe
the
the
largest,and
andthe
thevalue
valueof
ofthe
theroller
rollerisisthe
thesmallest.
smallest.ItItisisindicated
indicatedthat
thatthe
thevibration
vibrationof
ofthe
the
largest,
innerring
ringisisrelatively
relativelyintense,
intense,followed
followedby
bythe
theouter
outerring
ringand
androller
rollerininthis
thisstate.
state.
inner
ing, the vehicle-bearing system at different speeds under the action of track irregularity is
simulated. Figure 23 shows the RMS values of the vertical and lateral accelerations of the
outer ring, inner ring and roller, respectively. It can be seen that with the increase of speed,
the RMS values of the three elements gradually increase, the value of the inner ring is the
largest, and the value of the roller is the smallest. It is indicated that the vibration24of
the
of 34
inner ring is relatively intense, followed by the outer ring and roller in this state.
Machines 2022, 10, 1024
(a)
(b)
Figure23.
23.RMS
RMSofofouter
outerring
ringacceleration
accelerationofofnormal
normalbearing:
bearing:(a)
(a)vertical;
vertical;(b)
(b)lateral.
lateral.
Figure
Whenthe
theouter
outerraceway
raceway of the bearing
values
is shown
in FigWhen
bearinghas
hasaadefect,
defect,the
theRMS
RMS
values
is shown
in
ure 24.24.
With
thethe
increase
of of
speed,
thethe
RMS
value
of of
each
element
gradually
increase
and
Figure
With
increase
speed,
RMS
value
each
element
gradually
increase
tend
to be
It may
be that
in the
process
of speed
increasing,
the the
defect
causes
relaand
tend
to stable.
be stable.
It may
be that
in the
process
of speed
increasing,
defect
causes
Machines 2022, 10, x FOR PEER REVIEW
26 of 35
tively large
vibration
of of
each
element
of of
thethe
bearing.
Under
thethe
current
fault
size,
thethe
acrelatively
large
vibration
each
element
bearing.
Under
current
fault
size,
celeration of
reaches aa maximum
maximumvalue
valueand
andstops
stopswhen
whenititreaches
reachesa a
acceleration
of the
the bearing
bearing gradually reaches
certainvalue.
value.When
Whenthe
thespeed
speedisishigher
higher(200–300
(200–300km/h),
km/h),the
theRMS
RMSofofacceleration
accelerationdecreases
decreases
certain
The
RMS ofbecause
the
defective
bearings
is greater
than that
of slowness
the
normal
bearing,
because
slightly,
possibly
the
reflects
the
and
ofof
the
slightly,
possibly
because
thespeed
speed
reflects
thequickness
quickness
and
slowness
themovement,
movement,
the
vibration
of
the
outer
ring
will
increase
under
the
effect
of
the
defect,
which
will
corwhile
whilethe
theacceleration
accelerationreflects
reflectsthe
thechange
changeofofthe
thespeed.
speed.When
Whenthe
thespeed
speedreaches
reachesa acertain
certain
respondingly
increase
the
RMS
of
the
defective
bearings.
Compared
with
normal
bearing,
value,
change
of speed
willwill
be relatively
stable,
so theso
corresponding
acceleration
value
value,the
the
change
of speed
be relatively
stable,
the corresponding
acceleration
the
RMS
has
changed
greatly, so it is necessary to further analyze the bearing with defects.
will
decrease
slightly.
value
will
decrease
slightly.
60
Outer ring
Inner ring
Roller
50
40
30
20
10
0
0
50
100 150 200
Speed(km h–1)
250
300
(a)
(b)
Figure24.
24.RMS
RMSofofouter
outerring
ringacceleration
accelerationofofbearing
bearingwith
withouter
outerraceway
racewayfault:
fault:(a)
(a)vertical;
vertical;(b)
(b)lateral.
lateral.
Figure
Normal
and bearings
withisinner
raceway,
outer
andbearing,
roller defects
are
The
RMS bearing
of the defective
bearings
greater
than that
of raceway
the normal
because
simulated
respectively.
25 shows
the
variation
of RMS
values
of vertical
and correlateral
the
vibration
of the outerFigure
ring will
increase
under
the effect
of the
defect,
which will
acceleration
of outer
conditions.
As Compared
the speed with
increases,
thebearing,
RMS value
spondingly
increase
thering
RMSunder
of thefour
defective
bearings.
normal
the
shows
a
gradually
increasing
trend
from
the
overall
situation.
When
the
bearing
is
normal,
RMS has changed greatly, so it is necessary to further analyze the bearing with defects.
the RMS
value
of theand
outer
ring increases
approximately
linearly.
When
inner
ring are
and
Normal
bearing
bearings
with inner
raceway, outer
raceway
and the
roller
defects
outer ringrespectively.
of the bearing
have25
defects,
values
rapidly
and gradually
tend
simulated
Figure
showsthe
theRMS
variation
ofincrease
RMS values
of vertical
and lateral
to be stable.of
When
roller
has four
defect,
the RMS value
rapidly atthe
lowRMS
speed,
and
acceleration
outerthe
ring
under
conditions.
As theincreases
speed increases,
value
gradually
tends toincreasing
be stable trend
at high
speed.
It can be
found that
thethe
dynamic
shows
a gradually
from
the overall
situation.
When
bearingresponse
is normal,of
the
RMS value
the outer
ring increases
approximately
linearly.
When
the inner ring and
bearing
showsofdifferent
characteristics
when
the defect in
different
components.
120
100
Normal bearing
Outer ring fault
Inner ring fault
Roller fault
12
10
Normal bearing
Outer ring fault
Inner ring fault
Roller fault
Figure 24. RMS of outer ring acceleration of bearing with outer raceway fault: (a) vertical; (b) lateral.
Normal bearing and bearings with inner raceway, outer raceway and roller defects are
simulated respectively. Figure 25 shows the variation of RMS values of vertical and lateral
of 34
acceleration of outer ring under four conditions. As the speed increases, the RMS25value
shows a gradually increasing trend from the overall situation. When the bearing is normal,
the RMS value of the outer ring increases approximately linearly. When the inner ring and
outerring
ringofofthe
thebearing
bearinghave
havedefects,
defects,the
theRMS
RMSvalues
valuesincrease
increaserapidly
rapidlyand
andgradually
graduallytend
tend
outer
bestable.
stable.When
Whenthe
theroller
rollerhas
hasdefect,
defect,the
theRMS
RMSvalue
valueincreases
increasesrapidly
rapidlyatatlow
lowspeed,
speed,and
and
totobe
graduallytends
tendsto
tobe
bestable
stableatathigh
highspeed.
speed.ItItcan
canbe
befound
foundthat
thatthe
thedynamic
dynamicresponse
responseofof
gradually
bearing
components.
bearingshows
showsdifferent
differentcharacteristics
characteristics when
when the
the defect in different components.
Machines 2022, 10, 1024
120
Normal bearing
Outer ring fault
Inner ring fault
Roller fault
100
80
10
8
60
6
40
4
20
0
Normal bearing
Outer ring fault
Inner ring fault
Roller fault
12
2
0
50
100
150
200
Speed(km h–1)
250
300
0
0
50
Machines 2022, 10, x FOR PEER REVIEW (a)
100
150
200
Speed(km h–1)
250
300
27 of 35
(b)
Figure25.
25.RMS
RMSofofouter
outerrace
raceacceleration:
acceleration:(a)
(a)vertical;
vertical;(b)
(b)lateral.
lateral.
Figure
Based
on
above
by of
comparing
theand
vibration
responses
under
Figure
26the
shows
theanalysis,
RMSvalues
values
ofthe
thevertical
vertical
and
horizontal
contact
forcesdifferent
between
Figure
26
shows
the
RMS
horizontal
contact
forces
between
defects,
it and
can
concluded
that when
the
defect
location
appears
on
the
roller,
the impact
theroller
roller
andbe
the
outerraceway
raceway
four
cases.
The
RMSvalues
values
thevertical
vertical
contact
forces
the
the
outer
ininfour
cases.
The
RMS
ofofthe
contact
forces
on
the
vibration
of
the
bearing
is
greater,
followed
by
the
defect
located
in
the
outer
ring,
are
arerelatively
relativelyclose
closeand
andstable,
stable,and
andthat
thatofofthe
thelateral
lateralcontact
contactforces
forceshave
havelittle
littledifference,
difference,
and
thegradually
impact
isincrease
minimal
when
the
defect is
located
in
the
inner
ring.
which
increase
ofof
speed
tend
totobe
which
gradually
increasewith
withthe
the
increase
speedand
and
tend
bestable.
stable.
104
6
600
5
500
4
400
3
300
2
Normal bearing
Outer ring fault
Inner ring fault
Roller fault
1
0
0
50
100
150
200
–1
Speed(km h )
(a)
250
300
200
Normal bearing
Outer ring fault
Inner ring fault
Roller fault
100
0
0
50
100
150
200
Speed(km h–1)
250
300
(b)
Figure
Figure 26.
26. RMS
RMS of
of outer
outer raceway
raceway contact
contact force:
force: (a)
(a) vertical;
vertical; (b)
(b) lateral.
lateral.
7. Dynamic
Response
of analysis,
Bearing under
Flat Scarthe
Impact
Based on
the above
by comparing
vibration responses under different
defects,
it can
be flat
concluded
that for
when
the defectshort
location
appears
on the
roller,
the impact
As the
new
scar exists
a relatively
time,
this paper
only
considers
the
on the vibration
of the
is greater,
followed
by scar
the defect
in the
ring,
influence
of the old
flat bearing
scar in the
study. Take
the flat
heightlocated
as 0.1 mm,
theouter
length
of
and
the
impact
is
minimal
when
the
defect
is
located
in
the
inner
ring.
the old flat scar after long-term wear is set to 90 mm. Set the vehicle speed as 300 km/h
and input the German high interference track spectrum. The time history of the accelera7. Dynamic Response of Bearing under Flat Scar Impact
tion of the inner ring under four conditions are shown in Figures 27–30. As the inner ring
As the new
forbearing
a relatively
short
time,
this paper
only considers
the
is connected
withflat
thescar
axle,exists
it is the
element
most
directly
disturbed
by the wheel
influence
ofthe
the response
old flat scar
in the
study.
the flat for
scaranalysis.
height asThe
0.1 results
mm, theshow
length
of
and
rail, so
of the
inner
ringTake
is selected
that
the
old
flat
scar
after
long-term
wear
is
set
to
90
mm.
Set
the
vehicle
speed
as
300
km/h
and
when the vehicle is running at a high speed, the flat scar impact will have a great impact
on the bearing vibration in both the lateral and vertical directions. Judging from the amplitude change of acceleration, the bearing vibration caused by flat scar impact is far
greater than that caused by bearing defects, which indicates that it may be difficult to
detect the bearing defects when the wheel set has a flat scar.
Figure 26. RMS of outer raceway contact force: (a) vertical; (b) lateral.
7. Dynamic Response of Bearing under Flat Scar Impact
As the new flat scar exists for a relatively short time, this paper only considers the
influence of the old flat scar in the study. Take the flat scar height as 0.1 mm, the length
of
26 of 34
the old flat scar after long-term wear is set to 90 mm. Set the vehicle speed as 300 km/h
and input the German high interference track spectrum. The time history of the acceleration of the inner ring under four conditions are shown in Figures 27–30. As the inner ring
input
the German
spectrum.
time
history
of the acceleration
is connected
withhigh
the interference
axle, it is thetrack
bearing
elementThe
most
directly
disturbed
by the wheel
ofand
the inner
ring
under
four
conditions
are
shown
in
Figures
27–30.
As
innershow
ring that
is
rail, so the response of the inner ring is selected for analysis. Thethe
results
connected with the axle, it is the bearing element most directly disturbed by the wheel and
when the vehicle is running at a high speed, the flat scar impact will have a great impact
rail, so the response of the inner ring is selected for analysis. The results show that when
on the bearing vibration in both the lateral and vertical directions. Judging from the amthe vehicle is running at a high speed, the flat scar impact will have a great impact on the
plitude change of acceleration, the bearing vibration caused by flat scar impact is far
bearing vibration in both the lateral and vertical directions. Judging from the amplitude
greater than that caused by bearing defects, which indicates that it may be difficult to
change of acceleration, the bearing vibration caused by flat scar impact is far greater than
detect the bearing defects when the wheel set has a flat scar.
that caused by bearing defects, which indicates that it may be difficult to detect the bearing
defects when the wheel set has a flat scar.
Machines 2022, 10, 1024
4000
800
600
2000
400
200
0
0
–200
–2000
–400
–600
–4000
0
0.2
0.4
0.6
0.8
0
1
0.2
0.4
0.6
0.8
1
Time(s)
Time(s)
Machines 2022, 10, x FOR PEER REVIEW
(a)
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(b)
Figure
Time
history
inner
ring
acceleration
normal
bearing:
vertical;
lateral.
Figure
27.27.
Time
history
of of
inner
ring
acceleration
of of
normal
bearing:
(a)(a)
vertical;
(b)(b)
lateral.
4000
800
600
2000
400
200
0
0
–200
–2000
–4000
–400
–600
0
0.2
0.4
0.6
0.8
0
1
0.2
0.4
0.6
0.8
1
Time(s)
Time(s)
(a)
(b)
Figure28.
28. Time
of inner
ring ring
acceleration
of bearing
with outer
raceway
(a) defect:
vertical;
Figure
Timehistory
history
of inner
acceleration
of bearing
with
outer defect:
raceway
(b) lateral.
(a) vertical; (b) lateral.
4000
800
600
2000
400
200
0
0
–200
–2000
–400
–4000
–600
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Time(s)
Time(s)
Time(s)
Time(s)
(a)
(a)
(b)
(b)
Figure 28. Time history of inner ring acceleration of bearing with outer raceway defect: (a) vertical;
Figure
28. Time history of inner ring acceleration of bearing with outer raceway defect: (a) vertical;
(b)
lateral.
27 of 34
(b) lateral.
Machines 2022, 10, 1024
4000
4000
2000
2000
0
0
–2000
–2000
–4000
–4000 0
0
0.2
0.2
0.4
0.6
0.4
Time(s)0.6
0.8
0.8
1
1
Time(s)
800
800
600
600
400
400
200
200
0
0
–200
–200
–400
–400
–600
–600
0
0
0.2
0.2
0.4
0.6
0.4
Time(s)0.6
0.8
0.8
1
1
Time(s)
(a)
(a)
(b)
(b)
Figure 29. Time history of inner ring acceleration of bearing with inner raceway defect: (a) vertical;
Figure
29. Time
history
of inner
ring acceleration
of bearing
with inner
raceway
(a) vertical;
Figure
29.
Time
history
of inner
ring acceleration
of bearing
with
inner defect:
raceway
defect:
(b)
lateral.
(b)
lateral.
(a) vertical; (b) lateral.
4000
4000
2000
2000
0
0
–2000
–2000
–4000
–4000 0
0
0.2
0.2
0.4
0.6
0.4
Time(s)0.6
Time(s)
(a)
(a)
0.8
0.8
1
1
800
800
600
600
400
400
200
200
0
0
–200
–200
–400
–400
–600
–600
0
0
0.2
0.2
0.4
0.6
0.4
Time(s)0.6
0.8
0.8
1
1
Time(s)
(b)
(b)
Figure 30. Time history of inner ring acceleration of bearing with roller defect: (a) vertical; (b) lateral.
The contact frequency between the flat scar and the rail is equal to the SFR [37].
Figures 31–34 show the frequency characteristics of the inner ring acceleration in four cases.
In the low frequency section of 0–600 Hz, there are obvious SFR and its double-frequency.
When the bearing defect exists, the FCFO and FCFR can be found in Figures 32 and 34,
respectively, but the FCFI cannot be found in Figure 33. In addition, the fault characteristic
frequency of the bearing in the low frequency section will be covered by the SFR, while
that in the high frequency section will gradually appear, especially in the lateral direction.
Therefore, when there is flat scar on the wheel with high vehicle speed, it is difficult to
detect the inner raceway defect. In addition, it may be relatively easier to detect the bearing
defect by selecting the lateral direction and the high frequency section.
respectively, but the FCFI cannot be found in Figure 33. In addition, the fault characteristic
respectively, but the FCFI cannot be found in Figure 33. In addition, the fault characteristic
frequency of the bearing in the low frequency section will be covered by the SFR, while
frequency of the bearing in the low frequency section will be covered by the SFR, while
that in the high frequency section will gradually appear, especially in the lateral direction.
that in the high frequency section will gradually appear, especially in the lateral direction.
Therefore, when there is flat scar on the wheel with high vehicle speed, it is difficult to
Therefore, when there is flat scar on the wheel with high vehicle speed, it is difficult to
detect the inner raceway defect. In addition, it may be relatively easier to detect the beardetect the inner raceway defect. In addition, it may be relatively easier to detect the28bearof 34
ing defect by selecting the lateral direction and the high frequency section.
ing defect by selecting the lateral direction and the high frequency section.
Machines 2022, 10, 1024
300
300
250
250
200
200
150
150
100
100
50
50
0
00
0
f
fss
20
20
150
f
150
fss
150
150
100
100
100
100
50
5050
50
0
00
0 200
250
300
200 260 280
250 300 320300
260 280 300 320
500
500
1000 1500 2000
1000 1500 2000
Frequency(Hz)
Frequency(Hz)
2500
2500
3000
3000
f
fss
10150
10150
100
100
5 50
5 50
0
0 0
0 200
200 260
260
15
15
10
10
5
5
0
00
0
500
500
f
fss
250
300
250 300 320
300
280
280 300 320
1000 1500 2000
1000 1500 2000
Frequency(Hz)
Frequency(Hz)
(a)
(a)
2500
2500
3000
3000
(b)
(b)
Figure 31. Frequency spectrum of inner ring acceleration of normal bearing: (a) vertical; (b) lateral.
Figure
Figure 31.
31. Frequency
Frequency spectrum
spectrum of
of inner
inner ring
ring acceleration
acceleration of
of normal
normal bearing:
bearing: (a)
(a) vertical;
vertical; (b)
(b) lateral.
lateral.
300
300
250
250
200
200
150
150
100
100
50
50
0
00
0
f
fss
150
150
150
fout
150
fout
100
100
100
100
50
50
50
500
fout
200
240
00
fout
200
240
0
2×fout
2×fout
3×fout
260 280 300 320
3×fout
260 280 300 320
500
500
1000 1500 2000 2500 3000
1000 1500 2000 2500 3000
Frequency(Hz)
Frequency(Hz)
(a)
(b)
(b)
Machines 2022, 10, x FOR PEER REVIEW(a)
30 of 35
Figure 32.
Frequency
spectrum
ofofinner
ring
acceleration
of bearing
with
outer
raceway
defect:
(a)
Figure
32. Frequency
Frequencyspectrum
spectrumof
inner
ring
acceleration
bearing
with
outer
raceway
defect:
Figure
32.
inner
ring
acceleration
of of
bearing
with
outer
raceway
defect:
(a)
vertical;
(b)
lateral.
(a) vertical;
lateral.
vertical;
(b) (b)
lateral.
(a)
(b)
Figure
ring
acceleration
of of
bearing
with
inner
raceway
defect:
(a)
Figure 33.
33. Frequency
Frequencyspectrum
spectrumofofinner
inner
ring
acceleration
bearing
with
inner
raceway
defect:
vertical; (b) lateral.
(a) vertical; (b) lateral.
20
15
10
fs
20
150
froller
100
10
2×froller
50
0
050
3×froller
froller
100
150
(a)
Machines 2022, 10, 1024
(b)
Figure 33. Frequency spectrum of inner ring acceleration of bearing with inner raceway defect: (a)29 of 34
vertical; (b) lateral.
20
15
fs
20
150
froller
100
10
2×froller
10
froller
50
0
050
100
3×froller
5
0
260
0
500
280
150
300
320
1000
1500
Frequency(Hz)
(a)
2000
(b)
Figure 34. Frequency spectrum of inner ring acceleration of bearing with roller defect: (a) vertical;
Figure 34. Frequency spectrum of inner ring acceleration of bearing with roller defect:
(b) lateral.
(a) vertical; (b) lateral.
In order to further explore the vibration law of the faulty bearing with flat scar imIn order
to further
explore theof
vibration
the faulty of
bearing
with with
flat scar
impact,
pact,
the frequency
characteristics
the innerlaw
ringofacceleration
the bearing
outer
thering
frequency
characteristics
of
the
inner
ring
acceleration
of
the
bearing
with
outer
defect at different speeds are analyzed. As shown in Figures 35–40, when the vehicle ring
defect
at
speeds
arebeanalyzed.
in obvious,
Figures but
35–40,
the vehicle
speed isdifferent
50–150 km/h,
it can
found that As
the shown
SFR is not
therewhen
are obvious
speed
is 50–150
km/h, it can be found
thatatthe
SFR isspeed.
not obvious,
therespeed
are obvious
FCFO
and its double-frequency,
especially
a lower
When thebut
vehicle
is
200km/h,
thedouble-frequency,
amplitude of SFR begins
to exceed
that
of FCFO,
the When
FCFO isthe
obvious
in the
FCFO
and
its
especially
at
a
lower
speed.
vehicle
speed
is
Machines 2022, 10, x FOR PEER REVIEW
31 of 35
Machines 2022, 10, x FOR PEER REVIEW
31 of 35
but not very
obvious
into
theexceed
verticalthat
direction.
When
vehicle
speed
is in the
200lateral
km/h,direction
the amplitude
of SFR
begins
of FCFO,
thethe
FCFO
is obvious
250 km/h–300
thevery
presence
of FCFO
is weak.
lateral
direction km/h,
but not
obvious
in the
vertical direction. When the vehicle speed is
250 km/h–300 km/h, the presence of FCFO is weak.
(a)
(a)
(b)
(b)
Figure 35. Inner ring acceleration at speed of 50km/h: (a) vertical; (b) lateral.
Figure 35. Inner ring acceleration at speed of 50km/h: (a) vertical; (b) lateral.
Figure
35. Inner ring acceleration at speed of 50 km/h: (a) vertical; (b) lateral.
(a)
(a)
(b)
(b)
Figure 36. Inner ring acceleration at speed of 100km/h: (a) vertical; (b) lateral.
Figure
36.36.
Inner
ring
speedofof100km/h:
100 km/h:
(a) vertical;
(b) lateral.
Figure
Inner
ringacceleration
acceleration at speed
(a) vertical;
(b) lateral.
(a)
(b)
Figure 36. Inner ring acceleration at speed of 100km/h: (a) vertical; (b) lateral.
30 of 34
Amplitude
Amplitude
Machines 2022, 10, 1024
Machines 2022, 10, x FOR PEER REVIEW
32 of 35
Machines 2022, 10, x FOR PEER REVIEW
32 of 35
(a)
(b)
Figure
Figure 37.
37. Inner
Inner ring
ring acceleration
acceleration at
at speed
speed of
of 150km/h:
150 km/h:(a)
(a)vertical;
vertical;(b)
(b)lateral.
lateral.
80
fs
60
fs
60
fout
40
fout
40
20
0
fs
50
0
1000
0
1000
fout
2×fout
fout
2×fout
50
0
2×fout 0
0
2×fout 0
20
0
fs
100
100
200
300
100
200
300
2000
3000
Frequency(Hz)
2000
3000
Frequency(Hz)
(a)
4000
5000
4000
5000
Amplitude
Amplitude
100
80
(b)
(a) 38. Inner ring acceleration at speed of 200km/h: (a) vertical;
(b) (b) lateral.
Figure
Amplitude
Amplitude
Amplitude
Amplitude
Figure 38.
38. Inner
at speed
speed of
of 200
200km/h:
Figure
Inner ring
ring acceleration
acceleration at
km/h:(a)
(a)vertical;
vertical;(b)
(b)lateral.
lateral.
(a)
(b)
(a) 39. Inner ring acceleration at speed of 250km/h: (a) vertical;
(b)
Figure
Figure
39. Inner ring acceleration at speed of 250 km/h: (a) vertical;(b)
(b)lateral.
lateral.
Figure 39. Inner ring acceleration at speed of 250km/h: (a) vertical; (b) lateral.
(a)
Machines 2022, 10, 1024
(b)
Figure 39. Inner ring acceleration at speed of 250km/h: (a) vertical; (b) lateral.
(a)
31 of 34
(b)
Figure40.
40.Inner
Innerring
ringacceleration
accelerationat
atspeed
speedof
of300
300km/h:
km/h: (a)
Figure
(a) vertical;
vertical; (b)
(b) lateral.
lateral.
canbe
beseen
seenfrom
fromthe
theoverall
overall
trend
that
growth
of the
caused
by flat
the
ItItcan
trend
that
thethe
growth
raterate
of the
SFRSFR
caused
by the
flat scar
excitation
is greater
that
the FCFO
caused
the bearing
outdefect
ring defect
scar
excitation
is greater
than than
that of
theofFCFO
caused
by thebybearing
out ring
with
the gradual increase of the speed. The SFR is easily detected in the range of 0–2200 Hz in
the vertical direction and 0–1000 Hz in the horizontal direction. The FCFO in the lateral
direction is obvious. When the speed exceeds 200 km/h, the influence of bearing defect is
greater than that of flat scar. When the speed is in the range of 100–200 km/h, it is relatively
easier to distinguish the impact caused by wheel flat scar and that caused by bearing defect.
From the above analysis, it can be concluded that when worn wheel flat scar and
bearing outer raceway defect coexist, wheel flat scar is relatively easier to detect in the low
frequency section. The higher the speed, the easier it is to detect. The bearing defect is
relatively easier to detect in the high frequency section. The lower the speed, the easier it is
to detect, especially in the lateral direction with low speed.
8. Conclusions
This paper establishes a three-dimensional dynamic model of a high-speed train-axle
box bearing coupling system and adds the defects of outer raceway, inner raceway and
roller to the model, respectively. The model is verified by simulation and experiment,
which fully proves the effectiveness of the model established in the paper. The coupling
model is applied to analyze the vibration response of the bearing. Based on the above
analysis, some important conclusions can be drawn, which are as follows.
Through the co-simulation of MATLAB/Simulink and UM software and base on the
force coupling between vehicle and bearing, the dynamic model of high-speed train-axle
box bearing coupling system is established. The model can easily and efficiently solve the
dynamic response of vehicle and bearing components, which lays a good foundation for
further analysis and research into them.
(1)
(2)
The inner raceway defect, outer raceway defect and roller defect models are added
to the vehicle bearing coupling system, respectively. The corresponding vibration
responses are obtained by model simulation. The effectiveness of the model is verified
by the comparison between the simulation and the experimental results;
Considering the real running conditions, the influence of track irregularity on normal bearing and defective bearings are analyzed. The results show that the track
irregularity has a greater impact on the acceleration amplitude of normal bearing,
and a smaller impact on that of defective bearings. However, whether the bearing is
defective or not, its axis trajectory will change greatly, which shows that the influence
of wheel rail excitation on axle box bearing cannot be ignored. This conclusion further
reflects the superiority of the model proposed in this paper in analyzing the influence
of actual factors on dynamic response of bearing;
Machines 2022, 10, 1024
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(3)
(4)
The vibration characteristics of the system under different train speeds are studied. The
results show that with the increase of train speed, the RMS value of each element of
normal bearing increases gradually. When the bearing has defects, the RMS values
gradually increase but tend to be stable. When the roller has a defect, the vibration of the
bearing is relatively intense, followed by outer raceway defect and inner raceway defect;
When there is a flat scar on the wheel, it may have a great impact on the vibration of
the bearing. When the worn wheel flat scar and bearing defect exist at the same time,
wheel flat scar is relatively easier to detect in the low frequency section, and bearing
defect is relatively easier to detect in the high frequency section, but it is difficult to
detect the inner raceway defect. It is possible to obtain better results by selecting the
lateral direction to detect the bearing defect when the speed is low enough. The defect
of the outer ring and rollers of the bearing are relatively easy to detect, while that of the
inner ring is relatively difficult. When the speed is in the range of 100–200 km/h, the
impact caused by wheel flat scar and bearing fault may be easier distinguished. Based
on the establishment of the three-dimensional vehicle-bearing coupling model, the
paper focuses on the dynamic analysis of the bearing with early defects and that other
defect types still need to be further explored and verified. The research conclusion of
this paper may be helpful to the detection of bearing defect and wheel flat scar.
Author Contributions: Conceptualization, methodology, validation, and writing—review and editing, Q.M.; project administration, Y.L. (Yingying Liao) and Y.L. (Yongqiang Liu); supervision, S.Y.;
data curation, B.W. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by the, National Key R&D Program (Grant No. 2020YFB2007700),
National Natural Science Foundation of China (Grant No. 11790282; 12032017; 12002221 and
11872256), S&T Program of Hebei (Grant No. 20310803D), Natural Science Foundation of Hebei
Province (Grant No. A2020210028) and Graduate Innovation Funding Program of Hebei Education
Department (Grant No. CXZZBS2022119).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
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