machines-10-01024

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.
https://doi.org/10.3390/
machines10111024
Academic Editor: Xuesong Jin
Received: 27 September 2022
Accepted: 28 October 2022
Published: 4 November 2022
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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 and Baosen Wang 1,2
1School of Traffic and Transportation, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
2State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures,
Shijiazhuang Tiedao University, Shijiazhuang 050043, China
3School of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
4School 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.
Keywords: high speed train; axle box bearing; defect; track irregularity; wheel flat scar
1. Introduction
Axle box bearing is a key component of a high-speed train bogie. It bears high-
frequency alternating loads during operation, including radial force, axial force and contin-
uous 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. How-
ever, 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
Machines 2022,10, 1024. https://doi.org/10.3390/machines10111024 https://www.mdpi.com/journal/machines

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 summa-
rized 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 moni-
tor 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 condi-
tion 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 dy-
namic 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 res-
onance 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 trajec-
tories 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 5 of 34
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 problems, an improved vehicle-bearing coupling modeling method is
proposed in this paper. By combining Universal Mechanism (UM) with MATLAB/Simulink,
the coupling between the vehicle and the bearing is realized, and the internal forces of the
vehicle and wheel rail contact are fully considered. This model can realize fast calculation
and conveniently apply external excitation. In addition, the bearing outer ring fault, inner
ring fault and roller fault are added to the coupling model, respectively. The model is
simulated and analyzed under the actual running conditions, and verified by the rolling
and vibrating test rig of single wheelset. Since the vibration of the axle box bearing is
closely related to the movement of the vehicle and the rail, makes use of the advantage of
the coupling model, the paper studies the dynamic responses of the axle box bearing under
the impact of track irregularity and wheel flat scar, analyzes the dynamic performance
of the bearing with defects and summarizes some bearing vibration laws. The vibration
mechanism of axle box bearing under external excitation is revealed in time and frequency
domain so as to provide some reference for the research of fault diagnosis of axle box
bearing under real vehicle running environment.
2. Establishment of Vehicle-Bearing Coupling Model
2.1. High-Speed EMU Model
In the paper a vehicle model of a high-speed EMU is established through the multi-
body dynamics simulation software, Universal Mechanism (UM), as shown in Figure 1.
The model includes 1 carbody, 2 bogies, 4 wheelsets and 8 axle boxes. The carbody, bogie
and wheelset have 6 degrees of freedom, respectively, and the axle box has 3 degrees of
freedom (translation in X,Yand Zdirections). In the process of modeling, all structural
components are regarded as rigid bodies without considering their elastic deformation.
Machines 2022, 10, x FOR PEER REVIEW 5 of 35
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,
the research on the influence of external excitation factors on axle box bearing is very lim-
ited, especially in the analysis of vibration characteristics when the bearing with defects.
To address these problems, an improved vehicle-bearing coupling modeling method is
proposed in this paper. By combining Universal Mechanism (UM) with MATLAB/Sim-
ulink, the coupling between the vehicle and the bearing is realized, and the internal forces
of the vehicle and wheel rail contact are fully considered. This model can realize fast cal-
culation and conveniently apply external excitation. In addition, the bearing outer ring
fault, inner ring fault and roller fault are added to the coupling model, respectively. The
model is simulated and analyzed under the actual running conditions, and verified by the
rolling and vibrating test rig of single wheelset. Since the vibration of the axle box bearing
is closely related to the movement of the vehicle and the rail, makes use of the advantage
of the coupling model, the paper studies the dynamic responses of the axle box bearing
under the impact of track irregularity and wheel flat scar, analyzes the dynamic perfor-
mance of the bearing with defects and summarizes some bearing vibration laws. The vi-
bration mechanism of axle box bearing under external excitation is revealed in time and
frequency domain so as to provide some reference for the research of fault diagnosis of
axle box bearing under real vehicle running environment.
2. Establishment of Vehicle-Bearing Coupling Model
2.1. High-Speed EMU Model
In the paper a vehicle model of a high-speed EMU is established through the multi-
body dynamics simulation software, Universal Mechanism (UM), as shown in Figure 1.
The model includes 1 carbody, 2 bogies, 4 wheelsets and 8 axle boxes. The carbody, bogie
and wheelset have 6 degrees of freedom, respectively, and the axle box has 3 degrees of
freedom (translation in X, Y and Z directions). In the process of modeling, all structural
components are regarded as rigid bodies without considering their elastic deformation.
Carbody
Bogie
Axle box
Wheelset
Figure 1. Dynamic model of the high-speed train.
In the vehicle 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 yaw dampers, lateral stops, traction rods, and anti-rolling torsion bars, etc.
Figure 1. Dynamic model of the high-speed train.
In the vehicle 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 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,Yand Zdirections 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;