Zhai2020 Book VehicleTrackCoupledDynamics
Zhai2020 Book VehicleTrackCoupledDynamics
Zhai2020 Book VehicleTrackCoupledDynamics
Vehicle–Track
Coupled Dynamics
Theory and Applications
Vehicle–Track Coupled Dynamics
Wanming Zhai
Vehicle–Track Coupled
Dynamics
Theory and Applications
123
Wanming Zhai
Train and Track Research Institute
State Key Laboratory of Traction Power
Southwest Jiaotong University
Chengdu, China
This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd.
The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721,
Singapore
Preface
Dynamic interaction between train and track is increasingly intensive with the rapid
development of high-speed railways, heavy-haul railways, and urban rail transits,
causing more critical and complex vibration problems. Higher train running speed
would result in severer train and track interaction, bringing more prominent
problems in terms of running safety and stability of the train moving on elastic
railway track structures. It must ensure that the train has a good ride comfort when
running at a high speed without overturn or derailment. Additionally, the greater the
wheel–axle load of a vehicle, the stronger the dynamic effect of the vehicle on track
structures, inducing more serious dynamic damage to railway tracks. This requires
mitigation of the dynamic interaction between heavy-haul train and track.
Obviously, seeking solutions to the abovementioned sophisticated dynamic inter-
action problems of the large-scale system just from the vehicle system or the track
system itself is no longer sufficient. It is necessary to conduct dedicated and
in-depth research on the dynamic interaction between rolling stock and track sys-
tems. Only with a deep and comprehensive understanding of the mechanism of
vehicle–track dynamic interaction is it possible to implement reasonable approaches
to minimize the dynamic wheel–rail interaction, to obtain optimal integrated
designs of modern rolling stocks and track structures, and eventually to ensure safe,
smooth, and efficient train operations. Owing to the fast development of compu-
tation technologies, it is realistic today to study and simulate such coupled
dynamics problems by considering the vehicle system and track system as a large
integrated system with interaction and interdependence. This is the original inten-
tion of the vehicle–track coupled dynamics theory discussed in this book.
The author proposed the concept of Vehicle–Track Coupled Dynamics for the
first time in the late 1980s. In 1991, the author completed his doctoral thesis entitled
Vertical Vehicle–Track Coupled Dynamics. In 1993, a research paper for investi-
gating the vertical interaction between vehicle and track based on the vehicle–track
coupled dynamics was published at the 13th Symposium of the International
Association for Vehicle System Dynamics (IAVSD), and then was included in a
supplement of the IAVSD journal Vehicle System Dynamics (VSD) in 1994. With
the continuous funding from the National Natural Science Foundation of China
v
vi Preface
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background of Vehicle–Track Coupled Dynamics . . . . . . . . . . 1
1.2 Academic Rationale of Vehicle–Track Coupled Dynamics . . . . 4
1.3 The Research Scope of Vehicle–Track Coupled Dynamics . . . . 7
1.4 Research Methodology of Vehicle–Track Coupled
Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 14
2 Vehicle–Track Coupled Dynamics Models . . . . . . . . . . . . . . ..... 17
2.1 On Modeling of Vehicle–Track Coupled System . . . . . . ..... 17
2.1.1 Evolution of Wheel–Rail Dynamics Analysis
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... 17
2.1.2 Modeling of Track Structure . . . . . . . . . . . . . . . ..... 21
2.1.3 Modeling of Vehicle . . . . . . . . . . . . . . . . . . . . . ..... 26
2.1.4 General Principles for Vehicle–Track Coupled
System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Vehicle–Track Vertically Coupled Dynamics Model . . . . . . . . . 29
2.2.1 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 Vehicle–Track Spatially Coupled Dynamics Model . . . . . . . . . . 56
2.3.1 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.3.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.3.3 Dynamic Wheel–Rail Coupling Model . . . . . . . . . . . . 122
2.4 Train–Track Spatially Coupled Dynamics Model . . . . . . . . . . . 136
2.4.1 Basic Principle of Train–Track Dynamic
Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
2.4.2 Train–Track Spatially Coupled Dynamics Model . . . . . 137
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
ix
x Contents
Appendices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
Chapter 1
Introduction
Railways are major transportation arteries in many countries and play a very
important role in social and economic development. The railway transportation
system is a type of wheel–rail contact transportation system (“wheel–rail system”
for short). Rolling stocks (including locomotives, passenger cars, and freight
wagons, all referred to as “vehicles” in this book) and tracks are essential com-
ponents of the railway system. The function of wheel–rail transportation is achieved
via the interaction between wheels and rails. Wheel–rail interaction is the most
significant feature that distinguishes the railway system from other types of trans-
portation systems.
For a long time, studies on railway vehicle dynamics and track structure
vibration were carried out separately. This resulted in two relatively independent
disciplines, i.e., vehicle dynamics [1, 2] and track dynamics [3, 4].
In classic vehicle dynamics [1, 2], the vehicle system is the research object while
the track structure is considered as a “rigid support foundation” (i.e., a rigidly fixed
boundary), neglecting the dynamic influence of track vibrations on the vehicle
system. Under this situation, geometric irregularities of the rail surface are treated as
external disturbances of the vehicle system. In this research field, the dynamic
behaviors of the vehicle, including the hunting stability, the running safety, the ride
comfort, etc. are investigated with the assumption that the vehicle operates on a
rigid rail surface. A basic model illustrating this is shown in Fig. 1.1.
In classic track dynamics [3, 4], the vehicle is usually simplified as external
excitation loads Peixt for the track system (the harmonic vehicle loads P are applied
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on fixed points of the track system, or applied as moving loads on the track at a
speed of v). The characteristics of the vibration response and deformation of the
track structure are analyzed correspondingly. The fundamental model of the classic
track dynamics is shown in Fig. 1.2.
Thanks to the long-term studies and practices by railway scientists all over the
world, the theories of vehicle dynamics and track dynamics are becoming more and
more complete. Significant achievements of these systematic studies have been
reported from many fields, including vehicle dynamics modeling, wheel–rail con-
tact geometry, wheel–rail creep theory, vehicle hunting stability, curve negotiation
performance, track dynamics modeling, vibration characteristics of track structure,
track loading, and deformation characteristics, etc. These research outputs have laid
the theoretical foundation in revealing and understanding vehicle dynamics per-
formances and track dynamics characteristics. These outputs have also played a
magnificent role in the development of the railway transportation systems.
The rapid development of modern railway transportation, especially the dramatic
increases of operating speed, hauling mass, and transportation density, makes the
v
iω t
Pe
Rail
Fastener
Sleeper
Ballast
dynamics problems of the railway vehicle and track systems more prominent and
complicated. In general, the higher the operating speed of the train, the stronger the
dynamic interaction between the vehicle and the track, the more prominent the
problems of running safety and ride comfort. For one thing, it must be guaranteed
that the train passes key railway sections (including horizontal curves, vertical
curves, switches, turnouts, bridge approach transitions, etc.) safely at a reasonably
fast speed or even at a high speed without overturn and derailment. In addition, it
must be ensured that the rolling stock can operate with a good ride quality and
comfort under the disturbance of track irregularities. Normally, the heavier the
gross mass of the vehicle, the stronger is the dynamic interaction between wheel
and rail, and the more detrimental are the dynamic effect of the vehicle on the
railway infrastructure. Therefore, it is critical to significantly alleviate wheel–rail
dynamic interaction by exploring efficient and economical solutions.
Here, we can take the Chinese railway as an example to illustrate this issue.
Chinese railway transportation has been in a highly loaded situation for a long time.
On the one hand, the railway network density is geographically relatively low,
however, the total transport volume is very large. This situation results in a very
high traffic density which ranks first in the world. At present, the Chinese railway is
able to complete one-fourth of the transport volume of the world’s railways with
only 6% of the world’s railway network length! On the other hand, given the need
to increase operational speeds of the passenger and freight trains to satisfy the
demands of rapid socioeconomic development, upgrades of existing railway lines,
which were originally designed and constructed with relatively low standards have
been carried out repeatedly over recent decades. From 1997 to 2007, six major
speedup projects were launched and implemented, which have increased the
maximum train operational speed from lower than 100 km/h to over 200 km/h. The
speedups have even achieved a maximum operational speed of 250 km/h. In this
way, high-speed operations were successfully achieved on those existing railway
lines in China. As a result, the transportation capacity was effectively improved.
However, the dynamic interaction between the rolling stock and infrastructure was
seriously aggravated [5]. On the one hand, the dynamic effect of running faster
trains on track structures was intensified, which directly affected the fatigue life of
the infrastructure and increased the cost for maintenance and repairs. On the other
hand, the track geometry deformation and the subgrade settlement of the railway
lines were increased, which led to increasing detrimental effects on the dynamic
behavior of running trains. In particular, the vibrations and impacts that resulted
from the damaged and worn wheel–rail interface have become even more promi-
nent, which can lead to severe safety problems of the wheel–rail system.
Therefore, it is very necessary to conduct dedicated and in-depth research on the
dynamic interaction between rolling stock and track systems. Only with a deep and
comprehensive understanding of the mechanism of vehicle–track dynamic inter-
action is it possible to achieve reasonable approaches to minimize the dynamic
wheel–rail interaction, to obtain optimal integrated designs of modern rolling stocks
and track structures, and eventually to ensure safety, smoothness, and efficiency of
train operations. In traditional disciplines, i.e., classic vehicle dynamics and track
4 1 Introduction
dynamics, the vehicle–track system was divided into two relatively independent
subsystems. In this circumstance, it is very difficult to use these theoretical tools to
solve the dynamic interaction problem of such a complex and integrated system.
Given this situation, the author proposed the new concept of “vehicle–track coupled
dynamics” from the perspective of an overall integrated vehicle and track system in
the late 1980s, and the theory was put into practice in the early 1990s [6–12]. In
1991, the author completed his doctoral thesis entitled “Vertical vehicle–track
coupled dynamics” [6]. The basic mechanism of vehicle–track coupled dynamics
was published for the first time in Chinese in 1992 [7]. In 1993, a research paper on
a coupled model that was established on the basis of vehicle–track coupled
dynamics for investigating the vertical interaction between vehicle and track was
published at the 13th Symposium of the International Association for Vehicle
System Dynamics (IAVSD). The paper was then included in a supplement of the
IAVSD journal “Vehicle System Dynamics (VSD)” in 1994 [8]. In 1996, a further
developed “Vertical and lateral vehicle–track coupled model” was published in
VSD [9]. In 2009, the “Fundamentals of vehicle–track coupled dynamics” [11]
were systematically introduced in the journal “VSD”. The first academic mono-
graph in this research field titled “Vehicle–track coupled dynamics” (First edition,
in Chinese) [10] was published in 1997. The second, third and fourth editions of
this monograph (in Chinese) were published in 2002, 2007, and 2015, respectively
[12], which became the most fundamental reference books in the field of research
on railway system dynamics and design of rolling stocks and track structures in
China, especially for high-speed railways. This book is the first English monograph
reedited from the author’s Chinese monographs.
vehicle vibration via the wheel–rail contact interface results in aggravated vibra-
tions of the track structure, which in turn deteriorates the geometry condition of the
track. It is evident that it is the dynamic wheel–rail contact force that significantly
influences the dynamic behavior of the vehicle–track system.
Furthermore, the dynamic coupling mechanism between the vehicle system and
the track system via the wheel–rail interface is illustrated in Fig. 1.4. Under the
disturbance from the wheel–rail interface, the dynamic fluctuation of the wheel–rail
contact force is stimulated correspondingly. The dynamic effect of the wheel–rail
force can be transmitted upwards resulting in vibrations of the vehicle system.
Meanwhile, the dynamic effect can also be transmitted downwards leading to
vibrations of the track structure. The vibrations of wheelset and rail can directly
result in dynamic variation of the wheel–rail contact geometry. Under the influence
of wheelset and rail vibrations, the variation of elastic compressive deformation on
the normal plane of the wheel–rail contact leads to the fluctuation of the wheel–rail
normal contact force. Meanwhile, the variation of wheel–rail creepage (depending
on the relative velocity between wheel and rail) in the tangential plane of the
wheel–rail contact results in the fluctuation of the wheel–rail tangential creep force.
The dynamic changes of the wheel–rail contact forces (wheel–rail normal force and
creep force) can, in turn, affect the vibrations of the vehicle and track systems
(including the vibrations of wheelset and rail). Actually, the coupled vibration of
the vehicle–track system results from this interactive feedback mechanism, which
ultimately determines the entire dynamic behavior of the vehicle–track system.
Obviously, the wheel–rail relationship is the essential element of the vehicle–
track coupled system. The dynamic feedback between the vehicle system and track
system is realized via the variations of the dynamic wheel–rail contact relationship,
i.e., the dynamic wheel–rail contact deformation and contact geometry due to the
vibration and deformation of wheel and rail. A typical example is given in Fig. 1.5
to clarify the influence of vehicle–track system vibrations on wheel–rail contact
relationship, as well as to further demonstrate the importance of considering track
system vibrations in certain problems. Figure 1.5 exhibits the dynamic variation of
a wheel–rail contact point position on a wheel tread of a Chinese freight wagon
negotiating a curved track with a small radius (R = 350 m). In this figure, the solid
6 1 Introduction
Vehicle system
vibration
Wheelset vibration displacement
Bogie vibration
Wheelset vibration velocity
Wheelset vibration
Sleeper vibration
Rail vibration velocity
Ballast vibration
line is the simulation result with the consideration of track vibration effects while
the dashed line is the result based on the assumption that the entire track system is
remaining stationary. The results show that the contact positions are significantly
different in these two cases. Noticeable changes in the position of wheel–rail
contact will directly lead to dramatic variations in the magnitude and direction of
the wheel–rail contact force, further affect the vibration features of the vehicle and
track systems. The measured results from the Chinese railway demonstrate that the
wheel–rail lateral forces can induce elastic rail displacements laterally, and then
correspondingly resulting in dynamic gauge widening. For example, under a
high-speed operating condition, the lateral rail displacement is about 1 mm while
the track gauge is dynamically enlarged by 1–2 mm (from a high-speed test on
Qinhuangdao–Shenyang passenger-dedicated line in China). Another example is
from the test result on the existing Chengdu–Chongqing railway [13]. When a
vehicle negotiates a small radius curve at a low speed, the lateral rail displacement
and the dynamic gauge widening on a track with concrete sleepers are 1–3 mm and
2–4 mm, respectively. For the situation of timber sleepers, the corresponding rail
displacement and gauge widening can even reach 6 mm and 10 mm, respectively.
Obviously, for these situations where intensive vehicle–track interactions were
observed, if it was assumed that the rail was absolutely stationary, the theoretical
calculation result would considerably deviate from the actual situation. Therefore,
the actual vibration effects of the elastic track system shall be taken into consid-
eration in the study of the highly interactive vehicle–track system. In other words,
the concept of vehicle–track coupled dynamics should be adopted.
1.2 Academic Rationale of Vehicle–Track Coupled Dynamics 7
Fig. 1.5 The effect of track vibration on the dynamic wheel–rail contact geometry
utilized in the studies of the dynamic safety problems (including potential derail-
ment) related to the elastic lateral movement of the rail during curve negotiation
(especially for small radius curves). The third topic of lateral vehicle–track coupled
dynamics is the safety threshold of track geometry irregularities. This is also a
research topic that needs to be undertaken for improving track maintenance stan-
dards. The vertical, alignment, cross-level and gauge irregularities have important
impacts on operational safety. It is very difficult to analyze these dynamic opera-
tional safety issues by investigating a single system (vehicle or track) as these
problems are jointly determined by the vehicle–track interactive system. Given this
situation, vehicle–track coupled dynamics can provide an appropriate theoretical
platform for comprehensive studies of the safety thresholds for different types of
irregularities. When multiple types of irregularities exist at the same location of the
track, the problems for operational safety are even more serious and the dynamic
wheel–rail interactions become more complicated. Only by the use of vehicle–track
coupled dynamics, where the entire range of interactive factors have been taken into
consideration, can the safety thresholds for the operational conditions of multiple
irregularities be obtained. In addition, the running safety (especially for high-speed
operations) of the vehicle when passing through switches and turnouts and nego-
tiating combined horizontal and vertical curves are also within the research scope of
the lateral vehicle–track coupled dynamics.
The longitudinal vehicle–track coupled dynamics is mainly related to the studies
of wheel–rail stick–slip oscillation, the wheel–rail abrasion mechanism, the cause of
rail corrugation, train longitudinal impact under traction/braking conditions and its
interaction with the railway track, the longitudinal dynamic effect of the powertrain
system on the vehicle–track system, etc. First of all, the wheel–rail system plays
fundamental roles in supporting the train on the track structure vertically as well as
in guiding and constraining the wheelset movement laterally. Furthermore, longi-
tudinal wheel–rail creep also has a key role in transforming traction or braking
torque into the longitudinal wheel–rail force to realize acceleration or deceleration
of the train. Longitudinal wheel–rail stick–slip oscillation, wheel and rail abrasion,
and rail corrugation are the main problems of the wheel–rail system during traction
or braking. New breakthroughs in investigating these traditional problems are more
likely to emerge if the viewpoint of the longitudinal vehicle–track coupled
dynamics is adopted. Second, with the increase of train running speed and hauling
mass, especially for long heavy-haul trains, longitudinal impacts between adjacent
vehicles are evident when starting, braking and correcting speed. Under these cir-
cumstances, the longitudinal impacts applied on couplers are aggravated, which
could lead to serious incidents such as decoupling, coupler breakage and even
derailment under certain conditions [15]. In curved track sections, large coupler
lateral forces can greatly exacerbate lateral dynamic interaction between vehicle and
track, causing overturn or breakage of the rail. Therefore, it is an important task to
explore the characteristic of the longitudinal impact in long heavy-haul trains and its
effect on the track. It is also important to seek effective train handling strategies for
alleviating the related problems by using longitudinal vehicle–track coupled
dynamics. Third, with the rapid development of high-speed and high-powered
10 1 Introduction
motorized vehicles, the dynamic coupled effect of motor traction and gear trans-
mission of the powertrain on the vehicle–track system is significantly increased
[16], deteriorating the working environment of related components. In fact, heat
failure of traction motor bearings, bearing cage fractures, gear tooth breakages,
gearbox cracks, oil leaks, and other serious failures could occur during the oper-
ations of the vehicle. Therefore, the investigation of the dynamic mechanism of the
key components in the powertrain and its coupled effect with the vehicle–track
system are also within the scope of longitudinal vehicle–track coupled dynamics.
The above discussion is more related to deterministic disturbances. For nonde-
terministic disturbances, stochastic vehicle–track coupled dynamics is dedicated to
this issue [17]. The stochastic vehicle–track coupled dynamics is mainly related to
the studies regarding the vibration response characteristics and evolutionary
behaviors of the vehicle–track coupled system under the excitation of stochastic
track irregularities, as well as the investigation of characteristics of the vehicle
system, track structure system and wheel–rail interaction in the frequency domain,
and the cause of the excitation sources inducing the vehicle–track coupled vibra-
tions (i.e., the cause of track irregularity formation). As a result, the study of
stochastic vehicle–track coupled dynamics provides the feasibility to restrain and
mitigate the detrimental vibrations of vehicle–track system in different situations
with a targeted approach, to improve the ride quality and passenger comfort during
train operations, and to reduce the fatigue damage of vehicle and track components
to minimize the maintenance costs.
As a new research system distinguished from the classic theories of vehicle
dynamics and track dynamics, vehicle–track coupled dynamics has a very broad
application prospect in the field of railway vehicle and track system dynamics as
well as in wheel–rail interaction. Over the past 20 years, the railway speedup
strategy was successfully implemented in the Chinese railway and many remarkable
results were achieved. Many speedup projects were carried out on existing railway
lines. However, the structures of the existing lines, which were not constructed in
accordance with the current high design standards, actually cannot be compre-
hensively upgraded in a large scale. The dynamic influences of trains on the
infrastructure are greatly intensified as the operational speed increases. Therefore,
the issues of how to reduce the dynamic wheel–rail interaction with an increased
operational speed and how to avoid serious deformation and deterioration of the
track structure in order to guarantee the operational safety have become major
concerns in the Chinese railway. It is necessary to carry out systematic studies and
propose appropriate corresponding countermeasures from the perspective of the
overall vehicle–track system. In this regard, vehicle–track coupled dynamics has
provided appropriate theoretical analysis tools [5].
High-speed and heavy-haul are the two symbolic icons in today’s railway
industry. However, both high-speed and heavy-haul transportation scenarios
aggravate the dynamic wheel–rail interactions, which means the traditional railway
systems are not well suited to these new developments. The mitigation of dynamic
wheel–rail interactions has played a key role in developing modern railway trans-
portation systems, which has contributed to economic development. In order to
1.3 The Research Scope of Vehicle–Track Coupled Dynamics 11
achieve low dynamic interactions for wheel–rail systems, optimal integrated solu-
tions for wheel–rail systems and parameters must be sought from the perspective of
system engineering where comprehensive factors of vehicle, track, and wheel–rail
interface are taken into account. This aim can be achieved by the application of
vehicle–track coupled dynamics theory. With detailed parameter and sensitivity
analyses of the overall vehicle–track system, the basic approaches in mitigating
wheel–rail interactions and the corresponding technical countermeasures can be
identified. Meanwhile, the principle of optimal integrated design and the criterion of
parameter selection for new types of rolling stocks and track structures can be
proposed to provide theoretical guidance for the designs of high-speed and
heavy-haul vehicles as well as track systems. In addition, computer simulation
systems of vehicle–track coupled dynamics can be used to predict and evaluate the
dynamic performance of new or existing designs of vehicle or track. In this case,
simulation systems can provide a critical technical platform to evaluate the safety
issues of high-speed railway design and reconstruction for existing railway speedup
projects. The simulation systems can also be used to optimize vehicle design to
achieve better dynamic performance. They can also be used for the analysis of
rolling stock overturn, derailment and other major accidents, especially for the
study of derailments caused by the wheel–rail interaction and track damage. Using
the vehicle–track coupled dynamics in derailment analysis can overcome the bias of
the classic theoretical methods, which use single vehicle or track system.
author (in the early 1990s), the conflict between required computing speed and
available computing capacity was very prominent. Therefore, the development of a
fast and practical numerical integration method, at that time, was the priority for the
implementation of vehicle–track coupled dynamics simulations on ordinary
microcomputers. Fortunately, a new fast explicit numerical integration method and
a new prediction–correction integration method were constructed by the author
[18]. The two methods have obvious advantages in numerical solutions for
large-scale dynamic problems. As they do not need to solve large-scale algebraic
equation sets at each time step, they are expected to be able to successfully solve the
issues mentioned above.
Correct selection of model parameters plays a key role in ensuring the accuracy
of numerical simulation results. For vehicle–track coupled systems, it is not very
easy to determine or identify all physical parameters accurately due to the com-
plexity of the system, especially for the track structure. This is a common problem
for many engineering calculations and analyses. However, it is not always neces-
sary to obtain completely detailed parameters for modeling. The more important
thing is to specify the key parameters that characterize the system behaviors. For the
vehicle system, many detailed and accurate methods for determination of vehicle
parameters are widely adopted. For instance, the dimensions, mass, and inertias of
the vehicle components, as well as the stiffness and damping parameters of the
suspension systems can be determined or calculated using design drawings or
technical documents or by means of relevant laboratory bench tests, etc. For the
track system, the situation is much more complicated. However, after years of
simulations and experiments, large amounts of relevant data have been collected
and the systemic parameter identification methods have been gradually developed.
First of all, the physical parameters and profiles of the rail can be determined for
specific railway lines. Second, sleeper parameters can also be accurately determined
via design standards. Third, the dynamic parameters of rail pads (stiffness and
damping) can be obtained from product documents or be identified from loading
tests in a laboratory. Fourth, for ballastless track which is widely used in high-speed
railways, the structural parameters and physical parameters of track slab, CA
mortar, and other components can be determined in more straightforward ways as
they are clearly specified in high-speed railway standards. In addition, the param-
eters of ballasted tracks, such as ballast thickness and ballast density, can also be
measured accurately or determined from design specifications. The difficult task is
the identification of the stiffness and damping parameters of the ballast and sub-
grade. The classic axle-dropping test technique [19] and the continuous measure-
ment techniques for track stiffness developed in recent years (especially, the track
elasticity test vehicles) [20–23] have provided effective approaches to measure the
overall stiffness (and even damping) of the track. It is feasible to identify accurate
stiffness parameters of ballast and subgrade from the overall track stiffness, even
though this work is not easy. The author believes that an effective indirect method
to determine ballast and subgrade stiffness is to measure the elastic modulus of
ballast and subgrade (both are easy to measure) and then calculate their stiffness
14 1 Introduction
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engineering conference, London. 2002.
23. Norman C, Farritor S, Arnold R, et al. Design of a system to measure track modulus from a
moving railcar. In: Proceedings of railway engineering conference, London. 2004.
Chapter 2
Vehicle–Track Coupled Dynamics
Models
Abstract Theoretical model is the base for the study of vehicle–track coupled
dynamics problems. In this chapter, the principle and methodology for modeling of
vehicle–track coupled systems are discussed at first. And then, three types of the-
oretical models are established: the vehicle–track vertically coupled dynamics
model, the vehicle–track spatially coupled dynamics model, and the train–track
spatially coupled dynamics model, in which typical passenger coaches, freight
wagons, and locomotives as well as typical ballasted and ballastless tracks are
included. A new dynamic wheel–rail coupling model is also established to connect
the vehicle subsystem and track subsystem. Equations of motion of the vehicle and
track subsystems are deduced and given in detail.
The earliest involvement of wheel–rail dynamic analysis dates back to 1867, when
Winkler proposed the theory of elastic foundation beam, which was quickly used
for track modeling and the deformation analysis of track under static load. In 1926,
Timoshenko applied the elastic foundation beam model to first study the dynamic
stress of the rail under vehicle loading, which is a classical method still widely used
today. In 1943, Dörr proposed that better track models should be developed to
accommodate the growth speed of the train. However, few models have been
developed to solve practical problems of wheel–rail contact. In the meantime, the
railway researchers are more concerned about the dynamic stability of moving loads
(due to rolling stock) on the beam (i.e., rail). An important reason is that in 1954,
the French National Railways (SNCF) experienced severe track sinusoidal align-
ment irregularities in the high-speed train test with a maximum speed of 330 km/h,
resulting in lateral damage to the track structure. This experience shifted the
research focus in the late 50s and early 60s to the lateral running stability of rolling
stocks. As a result, the Prud’homme limit was developed based on the experiments
carried out by SNCF between 1960 and 1965 [1].
In the 1970s, the rapid development of the railway transportation industry
greatly promoted the development of wheel–rail interaction research, especially in
the application of mathematical mechanics models to solve the practical problems
of railways. The experimental and analytical research activities on rail joint forces
carried out by the Derby Railway Technology Research Centre in the UK [2, 3] led
the wheel–rail dynamics analysis into a substantive stage. At that time, in order to
prevent and remediate the damage in the rail joint area, the British Railways took
the lead in carrying out the wheel–rail dynamic test of the vehicle passing through
the rail dipped joints, and thus defined two types of wheel–rail forces that existed
during the wheel–rail impact process: high-frequency impact force P1 and
low-frequency force P2.
Lyon [2] and Jenkins et al. [3] developed a fundamental model to analyze the
wheel–rail dynamic interaction (Fig. 2.1), and were the first to study of the effect of
some principal parameters of the vehicle and track system (such as the unsprung
mass, track stiffness) on the wheel–rail force. This model described the track as an
Euler beam supported by a continuous elastic foundation, in which the vehicle was
simplified to unsprung mass with consideration of primary suspension character-
istics. The wheel–rail contact was modeled using a nonlinear Hertzian contact
model. In 1979, Newton et al. [4] conducted track dynamics tests to study the
dynamic effect of wheel flat on the track, and made a partial improvement on the
model developed in [2, 3]. The track was modeled using Timoshenko beam, so that
the calculated rail shear strain can be directly compared against the experiments,
and the theoretical and experimental results have achieved a good consistency. In
Fig. 2.1 The most basic model for the analysis of wheel–rail dynamic interaction (Ku, Cu—
vehicle suspension stiffness and damping; Kt, Ct—track support stiffness and damping; EI—rail
bending stiffness; mr—unit length rail mass; v—vehicle speed)
2.1 On Modeling of Vehicle–Track Coupled System 19
Sprung weight
Axle spring
Unsprung weight
Wheel-rail contact
Rail
Rail support
1982, Clark et al. [5] studied the dynamic effects of vehicles traveling on corrugated
track, and adopted a continuous track model with discrete elastic supports, and
considered the influence of sleeper vibration, so that the simulation was close to the
actual track structure.
During the same period, Sato, Ahlbeck, Birmann, Gent, etc. developed a more
simplified lumped parameter model to simulate the track structure with distributed
parameter characteristics for the study of wheel–rail dynamics [6, 7]. Figure 2.2 is
the simplest lumped parameter model of a wheel–rail system in which the rail was
simplified as a concentrated mass and the substructure was simplified to a spring
and damping element.
Since the 1990s, with the rapid development of heavy-haul and high-speed
railways, especially in the booming Chinese railway industry, wheel–rail system
dynamics research was significantly active. To meet the needs of railway devel-
opment, the author of this book began to consider the vehicle and the track as a
coupled system in 1990. The author first proposed and carried out the theoretical
study of vehicle–track coupled dynamics, and successively established a series of
vehicle–track coupled models [8–14]. The vehicle–track coupled model is char-
acterized by considering various major dynamic factors of the vehicle and the track
system in detail, which is able to investigate the wheel–rail dynamic interaction
from the overall vehicle–track system. The vehicle–track lateral dynamics model
was also developed. Subsequently, researchers from many railway research insti-
tutes had carried out a large number of theoretical and applied research projects in
the field of vehicle–track coupled dynamics [15–35], especially in the application of
the vehicle–track coupled dynamics models to study the practical dynamics
20 2 Vehicle–Track Coupled Dynamics Models
The track structure is usually modeled using numerical methods (in time domain)
and analytical methods (in frequency domain). The time-domain numerical meth-
ods include the equivalent lumped parameter method, continuous beam-based
modal superposition method, finite element method, boundary element method,
discrete element method, etc. [12, 16, 39, 55, 67, 75, 78]. The equivalent lumped
parameter method simplifies the track structure into a mass–spring–damper system
with a few degrees of freedom; therefore, the calculation speed is very fast, but less
accurate. The continuous beam method includes the models of a continuous beam
(rail) supported by a continuous elastic foundation and a continuous rail beam with
discrete elastic point supports. The continuous elastic foundation beam model
simplifies the underlying foundation into a uniformly distributed foundation, while
the discrete elastic support beam models the sleeper (or slab) and foundation
individually. The continuous beam model generally uses the modal superposition
method to solve the vibration differential equations; this method is simple and
computational efficient. The finite element method discretizes the track structure
into a finite number of elements and assumes a displacement function to obtain a
unit matrix. The advantage of the finite element method is that it can model
complex track structures in detail, and consider geometric nonlinearity of the track
structure; however, this method is time consuming. The boundary element method
is mainly used for the vibration transmission and noise problem simulation of the
track structure, while the discrete element method is mainly used for modeling of
the ballast of the track system. The analytical methods for track structure modeling,
such as wavenumber finite element and Green’s function method, are mainly used
for frequency-domain calculation. The calculation efficiency is high and covers a
wide range of frequency domain. However, the disadvantage is that the nonlinear
characteristics of the track structure cannot be considered. The following sections
discuss the effect of track modeling on the interaction between the vehicle and track
based on the time domain continuous beam model and summarize the fundamental
principles of track modeling.
1. The equivalent lumped parameter model
The equivalent lumped parameter model is based on a certain equivalence principle,
transforming a track structure with a complex decentralized parameter system into a
simplified model of mass–spring–damping lumped parameters with a few degrees
of freedom. Due to its efficient calculation speed, this simplified model is widely
used by vehicle dynamics analysis software such as SIMPACK, GENSYS,
VAMPIRE, and others.
There are two common equivalent transformation principles for the equivalent
lumped parameter track model. One is to derive the equivalent mass and the
equivalent spring stiffness from the measured natural vibration frequency of the
track structure, and derive the equivalent damping coefficient from the logarithmic
22 2 Vehicle–Track Coupled Dynamics Models
Fig. 2.3 Beam model with continuous elastic foundation (m—track mass per unit length; EI—
bending stiffness of rail)
Fig. 2.4 Beam model with discrete elastic point supports (mr—rail mass per unit length; ms—
sleeper mass; kp—rail pad stiffness; ks—under-sleeper stiffness)
2.1 On Modeling of Vehicle–Track Coupled System 23
the fact that the track is supported by the sleepers and track foundation. Moreover,
the beam model with discrete elastic point supports can consider the case where the
track system parameters are nonuniformly distributed in the longitudinal direction
more conveniently. For example, it can reflect the situation where the sleeper is not
uniformly distributed at track joints, or the situation where the track supporting
stiffness is nonuniform (such as when under-track high stiffness rubber pads are laid
to reduce the force at rail joints). In addition, the beam model with discrete elastic
point supports can also consider the defects in the track structure such as loose
fasteners, hanging sleepers and track slabs.
British Railway compared the simulation results of the beam model with con-
tinuous elastic foundation and the beam model with discrete elastic point supports.
The prediction results of the two models were considered to be not much different
in the low-speed range. At high speed, the beam model with continuous elastic
foundation will overestimate the wheel–rail force, which was also supported by the
measured results (Fig. 2.5) [4]. The simulation results using the beam model with
discrete elastic point supports had good agreement with the experimental results
within the entire speed range.
3. Comparison between Euler beam model and Timoshenko beam model for rail
modeling
Two types of beam models are commonly used to describe the rail as a continuous
elastic body, namely the Bernoulli–Euler beam (Euler beam for short) model and
the Rayleigh–Timoshenko beam (Timoshenko beam for short) model. The Euler
Speed (km/h)
24 2 Vehicle–Track Coupled Dynamics Models
beam model considers the bending deformation of the rail regardless of its shear
deformation. The Timoshenko beam model introduces the shear strain of the beam
and considers the rotational inertia of the beam, so that the force analysis of the
beam is more complete and the analysis frequency is higher. The shear strain
parameters of the Timoshenko beam model are also convenient to be compared to
the measured parameters in the field. Generally, the Euler beam model could cover
a frequency range up to 500 Hz, while the Timoshenko beam model could cover a
frequency range up to 3000 Hz. However, the theory of the Timoshenko beam
model is more complex, so the computational efficiency is slightly lower; the Euler
beam model is simpler and computationally efficient, and is hence widely used in
practice.
The numerical analysis results showed that the P2 forces calculated by the
Timoshenko beam model and by the Euler beam model were almost identical, and
only the high-frequency P1 forces are slightly different. The former is 7–11% larger
than the latter (see Fig. 2.6a). The wheel–rail force comparison between the Euler
and Timoshenko beam models supported by continuous elastic foundation done by
the Derby Railway Research Centre in the UK [4] showed the difference is small at
middle to low-frequency while the difference is larger at higher frequency (see
Fig. 2.6b). Therefore, for the low-frequency wheel–rail dynamics problem, the
Euler beam rail model is generally used, which does not make the calculation
process too complicated, and can meet the accuracy requirements of the general
wheel–rail dynamic analysis. When it is necessary to pay attention to the
high-frequency wheel–rail vibration characteristics (such as the wheel–rail noise
problems), the Timoshenko beam model is used to reflect higher frequency
behavior of the track.
(b)
Timosheko beam
Dynamic/static force ratio
Euler beam
(a)
P (kN)
Timosheko beam
Euler beam
Fig. 2.6 Comparison of results from the Euler beam model and the Timoshenko beam model:
a dynamic responses of wheel–rail force and b ratio of dynamic/static wheel loads
2.1 On Modeling of Vehicle–Track Coupled System 25
Table 2.1 Comparison of numerical results for different layered track models
Indicators Triple-layer model Double-layer model Single-layer model
High-frequency force P1 (kN) 306.49 306.49 313.14
Low-frequency force P2 (kN) 226.87 253.60 288.52
Sleeper reaction force (kN) 54.43 54.46 68.19
Rail acceleration (m/s2) 1063.85 1063.83 1003.01
Sleeper acceleration (m/s2) 351.16 349.79 –a
Ballast acceleration (m/s2) 69.45 –a –a
a
This indicator cannot be calculated in the model
approximately 25% increment. This result was consistent with the conclusion in [4]
that the impact force could be overestimated using the single-layer model without
considering the effect of the sleeper and rail pad.
It is worth noting that for the slab track, or the track with sleeper or ballast based
vibration isolation (such as elastic–supporting–block track, ladder-shaped sleeper
track, and floating slab track), the vibration of the track system mainly exists in the
rail and sleeper/ballast. The two-layer discrete elastic point support beam model is
suitable for the wheel–rail dynamics analysis, which can ensure sufficient accuracy.
For the fastener vibration isolation type (or rail pad vibration isolation type) track,
since the track system vibration mainly exists on the rails, a single-layer discrete
elastic point support beam model can be used to reflect the dynamic behavior of the
rail–substructure system, which meets the requirements for engineering analysis.
For a long time, in the wheel–rail dynamics analysis model, more attention was paid
to the detailed description of the track structure, but less on the part of the vehicle
system. In fact, the vehicle and the track models are equally important due to the
strong coupling effect between the vehicle and the track, which has a significant
impact on the wheel–rail dynamics. Various simplifications of the vehicle model
can greatly reduce the workload of the analysis and calculation, but will inevitably
lead to different degrees of analysis error. Kisilowski and Knothe [84] specifically
discussed the dynamic coupling effect between adjacent wheels of a railway
vehicle, using the analysis model as shown in Fig. 2.7. The analysis results showed
that in the case of good track elasticity, the wheel displacement can be significantly
affected (approx. 37%) by the adjacent wheel; significant forced vibration around
200 Hz can be observed due to the transmission of the wheel–rail force along the
rail. Therefore, it is not reasonable to simplify the vehicle into a single wheelset
model.
The numerical analysis performed by the author in [82] showed the analysis
error caused by neglecting the influence of the adjacent wheels. The wheel–rail
2.1 On Modeling of Vehicle–Track Coupled System 27
Fig. 2.7 Model for studying the interaction of adjacent wheels on mutual dynamics
force error was around 7%, and the ballast acceleration is about 11% for
high-frequency wheel–rail impact vibration. The error was greater in the case of
sinusoidal vibration excitation. Table 2.2 lists the wheel–rail dynamic responses of
three vehicle models (single wheelset, single bogie, and complete vehicle) under
continuous sinusoidal excitation on rail surface (wavelength 250 mm, wave depth
1 mm). It can be seen from Table 2.2 that, compared with the complete vehicle
model, the single wheelset–track model without considering the interaction among
the wheelsets of the vehicle had 13% lower wheel–rail force, and 35.7% lower
ballast acceleration. The single bogie–track model considering the interaction of
two adjacent wheelsets had 9% lower wheel–rail force, and 19% lower ballast
acceleration. The main reason for the calculation error was that the dynamic action
of one wheelset and the track was transmitted to the adjacent wheelsets through the
vehicle (bogie frame and body) and the rail, hence creating a dynamic coupling
effect of the wheelsets. The track vibration had the largest effect in the range of
three sleeper spans before and after the excitation point [85], which is close to the
bogie wheelbase. Therefore, the calculation error was greatly reduced after con-
sidering the coupling effect of the two wheelsets on a bogie. Thus, the bogie (or
half-car) model was obviously superior to the single wheelset model.
Under the excitation of continuous sinusoidal track irregularity, the four
wheelsets of the vehicle would be simultaneously excited, which led to coupling
and superposition of the dynamic actions of all the four wheelsets of the vehicle.
However, the single wheelset and the bogie models could not reflect (or not fully
reflect) this coupling effect, hence, the calculation results had large errors compared
to the complete vehicle model.
Another topic should be discussed on vehicle modeling: is it necessary to
consider several vehicles or even a whole train? The traction and braking actions
cause large longitudinal forces in the inter-vehicle coupling system for the long and
heavy-haul trains, which has a significant effect on the vehicle curving performance
and running stability [86, 87]. In addition, the inter-vehicle couplers and dampers
used in high-speed trains can greatly reduce the longitudinal impulse between
adjacent vehicles in a train, and also greatly improve the ride comfort and running
safety of the train [88]. It is evident that under certain conditions, the interaction
between adjacent vehicles in a train has an important influence on the wheel–rail
dynamics. Therefore, for the long heavy-haul trains, articulated trains, and trains
using compact inter-vehicle coupling, the train–track coupled dynamics model
considering multiple vehicles is necessary, which may achieve higher calculation
accuracy than the model with only one vehicle. In general, the train–track coupled
dynamics model of a 3–5 cars can well reflect the dynamic performance and the
inter-vehicle interaction of a long train [87, 88]. For the investigation of conven-
tional vehicle dynamics problems, excluding the aforementioned three types of
trains, a single-vehicle model is sufficient.
Through the above comparative analysis, it can be found that the simplification of
the vehicle and the track structure in the modeling of the vehicle–track coupled
system may lead to the loss of the model function or the reduction of the analysis
accuracy. Therefore, the ideal model should adequately consider various factors
influencing the dynamic performance of the vehicle and track system, especially the
wheel–rail interaction characteristic. Meanwhile, the model could not be too
complicated to allow rapid calculation.
In sum, the dynamics model of the vehicle–track coupled system should comply
with the following basic principles:
2.1 On Modeling of Vehicle–Track Coupled System 29
Mc Jc c
Zc
C sz K sz
t2 t1
Mt Jt
C pz Z t2 K pz Z t1
Zw4 Zw3 Z w2 Z w1
Z 04 Z03 Z 02 Z 01
P4 P3 mr EIY P2 P1 +
8
-
8
K pi Cp i Zr
M si
K bi C bi Zs
Cw i
M bi
K wi
Zb
K fi Cf i
Mc Jc c
Zc
C sz K sz
t2 t1
Mt Jt
Z t2 Z t1
Z w4 Z w3 Z w2 Z w1
Z 04 Z03 Z 02 Z 01
P4 P3 mr EIY P2 P1 +
-
K pi C pi Zr
Ms i
K bi C bi Zs
C wi
M bi
Kw i
Zb
K fi Cf i
In the passenger vehicle model, the car body has the vertical (Zc ) and pitch (bc )
motions, the lead and rear bogie frames also have the vertical (Zt1 , Zt2 ) and pitch
(bt1 , bt2 ) motions, and the four wheelsets only have the vertical motion
(Zwi ; i ¼ 14). Thus, each passenger vehicle has 10 Degrees of Freedom (DOFs).
In the freight wagon model, both the car body and bogie frames have the vertical
and pitch motions. Since the wagon has no primary suspension, the motions of the
wheelset coincide with the motions of the bogie frame. Therefore, each wagon has
six DOFs. It is noted that the wagon model shown in Fig. 2.9 was built for a main
kind of Chinese freight wagon equipped with three-piece bogies without primary
suspension. For the other wagons equipped with primary suspension, the modeling
method can be found in Ref. [12]. And for the freight wagons with two suspensions,
its vertical coupled model should be the same as that shown in Fig. 2.8.
The double-axle locomotive model is similar to the passenger vehicle model.
The only difference is that the locomotive model has to consider the traction motor.
Figure 2.10 shows a locomotive–track vertically coupled dynamics model that
considers the dynamic effect of the traction motor system. In the model, a traction
motor (Mm , Jm) is mounted on each wheel axle via axle-hung bearings at one end
while elastically linked (Km) with the bogie frame at the other end. Each rigid motor
has only the rotational motion (bmi ). Thus, each locomotive has 14 DOFs. It is
32 2 Vehicle–Track Coupled Dynamics Models
Mc Jc c
Zc
C sz K sz
t2 t1
Mt Jt
Z t2 Z t1
K pz
C pz Km
m2
Mw Mm
Zw3 Z w2 Z w1
Zw4
Z 04 Z 03 Z02 Z 01
- P4 P3 mr EIY P2 P1 +
K pi C pi Zr
M si
K bi C bi Zs
C wi
M bi
Kwi
Zb
K fi Cf i
noted that the current locomotive model can be easily improved when the motor
suspension mode is changed. For example, when the motor is linked with the
bearing by elastic suspension, the vertical DOF of each motor should be considered,
such a model can be referred to [13].
The track sub-model shown in Figs. 2.8, 2.9, and 2.10 represent the conven-
tional ballasted track structure, consisting of the rails, the rail pads, the sleepers, the
ballast, and the subgrade. Both the left and the right rails are treated as continuous
beams (the Euler or Timoshenko beams), which are discretely supported at rail–
sleeper junctions by three layers of springs and dampers representing the elasticity
and damping of the rail pad, the ballast, and the subgrade, respectively. In order to
account for the continuity and the coupling effects of the interlocking ballast
granules, a couple of shear stiffness (Kwi) and shear damping (Cwi) is introduced
between adjacent ballast masses. In the model, mr and EIY are the mass per unit
longitudinal length and the bending stiffness of the rails, Kpi, Kbi, Kfi and Cpi, Cbi,
Cfi are the stiffness and damping of the rail pad, the ballast and the subgrade in each
rail–sleeper junction.
2.2 Vehicle–Track Vertically Coupled Dynamics Model 33
Rail
Sleeper
Ballast
Fig. 2.11 Load distribution region in continuous granular ballast (Reprinted from Ref. [90],
Copyright 2003, with permission from Elsevier.)
α
34 2 Vehicle–Track Coupled Dynamics Models
According to the ballast model shown in Fig. 2.12, the vibrating mass of ballast
under a sleeper support point could be evaluated as
4
Mb ¼ qb hb le lb þ ðle þ lb Þhb tan a þ h2b tan2 a ð2:1Þ
3
where qb is the density of ballast (kg/m3), hb is the depth of ballast (m), le is the
effective supporting length of half sleeper (m), lb is the width of sleeper underside
(m), and a is the ballast stress pervasion angle.
The supporting stiffness of a ballast mass can be determined as
2ðle lb Þ tan a
Kb ¼ h i Eb ð2:2Þ
ln llbe ððllbe þ
þ 2hb tan aÞ
2hb tan aÞ
where Ef is the K30 modulus of subgrade (Pa/m), which means the force acting on
unit area that leads to unit deformation.
The above-proposed ballast model is based on the assumption that there is no
overlapping of adjacent cone regions of ballasts. In the case of thick ballast layer,
small sleeper spacing, or big ballast stress pervasion angle, an overlapping of
adjacent ballast masses may occur, see Fig. 2.13. The above ballast model should
be modified appropriately [90]. In this case, the vibrating mass of ballast under a
rail support point could be defined as the shadowed region as shown in Fig. 2.13.
Rail
Sleeper
Ballast
Surface of subgrade
Fig. 2.13 The modified model of ballast (Reprinted from Ref. [90], Copyright 2003, with
permission from Elsevier.)
2.2 Vehicle–Track Vertically Coupled Dynamics Model 35
According to the geometry shown in Fig. 2.13, the height of the overlapping
regions is calculated by
ls lb
h0 ¼ hb ð2:4Þ
2 tan a
Kb1 Kb2
Kb0 ¼ ð2:6Þ
Kb1 þ Kb2
where
2ðle lb Þ tan a
Kb1 ¼ h i Eb ð2:7Þ
ln lb ðle þle llss lb Þ
and
To consider the continuity and the coupling effects of the interlocking ballast
granules, a couple of shear stiffness Kw and shear damping Cw should be introduced
between the adjacent ballast masses in the ballast model. The authors had studied
the coupling effects of the interlocking ballast granules on the ballast vibration in
Ref. [90]. The results show that the model will overestimate the ballast vibration
level if the ballast shearing effect is not considered—usually the acceleration of the
ballast will be at least 10% higher. The reason is that the effect of friction and
impact of ballast stones induces a counteracting motion of adjacent ballast blocks,
so that the vibration level of one ballast block will be attenuated by the adjacent
blocks. If the shearing effect is not considered, this attenuating effect is absent.
36 2 Vehicle–Track Coupled Dynamics Models
Fig. 2.14 The vertical dynamics model for the long-sleeper embedded track
Fig. 2.15 The vertical dynamics model for the elastic supporting block track
Thus, the ballast mass can vibrate more freely and its vibration level will be
overestimated.
If the analyzed track structure is a ballastless track, the above ballasted track
dynamics model can be improved correspondingly. Figures 2.14, 2.15, and 2.16
establish the vertical dynamics models of three typical ballastless track structures,
including the long-sleeper embedded track, the elastic supporting block track, and
the slab track.
The long-sleeper embedded track (or double-block track) consists of rails, fas-
tenings and rail pads, concrete sleepers, concrete slabs, and concrete base. For this
type of ballastless track, the track structure can be simply modeled as two con-
tinuous rail beams discretely supported by fastenings and rail pads (Fig. 2.14). This
is because the sleeper blocks are precast into the slab directly and there is no
elasticity between the slab and the concrete base. Thus only the vibration of the rails
is important for the wheel–rail interaction.
The elastic supporting block track consists of rails, fastenings and rail pads,
concrete blocks, block pads and rubber boots, concrete slabs, and concrete base.
Fig. 2.16 The vertical dynamics model for the typical slab track (Reprinted from Ref. [36],
Copyright 2009, with permission from Taylor & Francis.)
2.2 Vehicle–Track Vertically Coupled Dynamics Model 37
The pads provide the vertical supporting stiffness and damping for the blocks under
rails, and the rubber boots provide the lateral stiffness and damping for the blocks.
Thus, only the vibrations of rails and concrete blocks are important for the wheel–
rail interaction. The elastic supporting block track model is shown in Fig. 2.15,
where the rails modeled as continuous beams discretely supported by fastenings and
rail pads, and the concrete blocks are modeled as rigid bodies.
The slab track consists of rails, fastenings and rail pads, concrete slabs, cement
asphalt mortar (CAM) layer, and concrete base. The CAM layer has soft stiffness,
which may reduce the vibration of the subgrade. Therefore, both the vertical
vibrations of the rails and slabs have a significant effect on the vehicle–track vertical
interaction. In the model, the rails are modeled as continuous beams discretely
supported by fastenings and rail pads, while the track slabs are simplified as finite
length free beams (no constraint at two ends) continuously supported by stiffness
and damping of the CAM layer, as shown in Fig. 2.16.
The wheel–rail contact is an essential element that couples the vehicle with the
track. The key issue is the contact forces between the wheel and the rail. The
wheel–rail vertical contact force can be easily calculated using the nonlinear
Hertzian elastic contact theory. It is able to take into account the separation between
the wheel and rail.
where lc is half of the distance between two bogie centers of a vehicle (m), lt is half
of the distance between the two axles of a bogie (m); pi ðtÞ is the vertical wheel–rail
force at the ith wheelset (i ¼ 14); F0i ðtÞ is the self-excitation force at the ith
wheelset (i ¼ 14) if there is, such as the centrifugal force caused by the wheel
eccentricity.
2.2 Vehicle–Track Vertically Coupled Dynamics Model 39
For the wagon system with the three-piece bogie considered here, the equations of
motion of the wheelsets are depended on the bogie frames, which can be written as
Mt Z€t1 þ ðCsz þ 2Cpz ÞZ_ t1 þ ðKsz þ 2Kpz þ 2Km ÞZt1 Csz Z_ c Ksz Zc
Cpz ðZ_ w1 þ Z_ w2 Þ ðKpz þ Km ÞðZw1 þ Zw2 Þ þ Csz lc b_ c þ Ksz lc bc ð2:28Þ
Km ðl1 þ l2 Þðbm1 bm2 Þ ¼ Mt g
Mt Z€t2 þ ðCsz þ 2Cpz ÞZ_ t2 þ ðKsz þ 2Kpz þ 2Km ÞZt2 Csz Z_ c Ksz Zc
Cpz ðZ_ w3 þ Z_ w4 Þ ðKpz þ Km ÞðZw3 þ Zw4 Þ Csz lc b_ Ksz lc b
c c
Km ðl1 þ l2 Þðbm3 bm4 Þ ¼ Mt g
ð2:30Þ
€ þ Mm l1 Z
ðJm þ Mm l21 Þb €w1 þ Km ðl1 þ l2 Þ2 bm1
m1
ð2:36Þ
þ Km ðl1 þ l2 Þðlt l1 l2 Þbt1 Km ðl1 þ l2 ÞðZt1 Zw1 Þ ¼ 0
€ Mm l1 Z€w2 þ Km ðl1 þ l2 Þ2 b
ðJm þ Mm l21 Þb m2 m2
ð2:37Þ
þ Km ðl1 þ l2 Þðlt l1 l2 Þbt1 þ Km ðl1 þ l2 ÞðZt1 Zw2 Þ ¼ 0
€ þ Mm l1 Z
ðJm þ Mm l21 Þb €w3 þ Km ðl1 þ l2 Þ2 bm3
m3
ð2:38Þ
þ Km ðl1 þ l2 Þðlt l1 l2 Þbt2 Km ðl1 þ l2 ÞðZt2 Zw3 Þ ¼ 0
€ Mm l1 Z€w4 þ Km ðl1 þ l2 Þ2 b
ðJm þ Mm l21 Þb m4 m4
ð2:39Þ
þ Km ðl1 þ l2 Þðlt l1 l2 Þbt2 þ Km ðl1 þ l2 ÞðZt2 Zw4 Þ ¼ 0
where lc is half of the distance between bogie centers of the locomotive (m), l1 is the
distance between the motor and the axle-hung bearings (m), l2 is the distance
between the motor and the mounted point on the bogie frame (m).
42 2 Vehicle–Track Coupled Dynamics Models
It is noted that the vertical motion of the traction motor is not independent, its
displacement and velocity can be obtained by
Zmi ¼ Zwi l1 bmi
€ ð2:40Þ
Z€mi ¼ Z€wi l1 b mi
where
Jm
gm ¼ ð2:49Þ
Jm þ Mm l21
Jm Mm l1 l2
nm ¼ ð2:50Þ
Jm þ Mm l21
Mm
lm ¼ ð2:51Þ
Mw þ Mm
44 2 Vehicle–Track Coupled Dynamics Models
gm is the equivalent mass of the motor that added to the unsprung mass of the
locomotive, here it is defined as the unsprung mass contribution coefficient of the
motor system. nm is the equivalent suspension stiffness of the motor that added to
the primary suspension of the locomotive, it is defined as the primary suspension
influence coefficient of the motor system. gm and nm can clearly show the effect of
the motor system on the locomotive wheel–rail interaction [13].
The rails are usually treated as the simply supported Euler beam or Timoshenko
beam with finite length. In fact, the result by the finite length beam model is close to
that by the infinite beam model when the calculation length of the rail is sufficient
long. The criterion for selecting the calculation length l of the rail as the simply
supported beam will be proposed by numerical trials as shown in Sect. 4.4.
(1) Differential equations of rail modeled as Euler beam
Figure 2.17 shows the force analysis of a rail when it is modeled as a simply
supported Euler beam. In the figure, pi is the wheel–rail force (i ¼ 14), which
travels on the beam at train speed v, Frsi ði ¼ 1NÞ is the sleeper support force, N is
the total number of the sleepers within the calculation length l. ox is the coordinate
system fixed on the track, o0 x0 is the moving coordinate system fixed on the vehicle.
The conversion relation between the two coordinate systems is
x ¼ x0 þ x0 þ vt ð2:52Þ
p4 p3 p2 p1
where x0 is the initial coordinate of the fourth wheel in the track coordinate system,
t is the time.
The rail deflection Zr ðx; tÞ can be described as the following differential equation
@ 4 Zr ðx; tÞ @ 2 Zr ðx; tÞ XN X4
EIY þ mr ¼ Frsi ðtÞdðx x i Þ þ pj dðx xwj Þ ð2:53Þ
@x4 @t2 i¼1 j¼1
where EIy is the rail beam bending stiffness (N m2), d denotes the Dirac function,
and
Frsi ðtÞ ¼ Kpi ½Zr ðxi ; tÞ Zsi ðtÞ þ Cpi Z_ r ðxi ; tÞ Z_ si ðtÞ ð2:54Þ
X
NM
Zr ðx; tÞ ¼ Zk ðxÞqk ðtÞ ð2:58Þ
k¼1
X
NM
d4 Zk ðxÞ X
NM
EIY 4
qk ðtÞ þ mr Zk ðxÞ€
qk ðtÞ
k¼1
dx k¼1
ð2:59Þ
X
N X
4
¼ Frsi ðtÞdðx xi Þ þ pj ðtÞdðx xwj Þ
i¼1 j¼1
It is noted that the following property due to the modal orthogonality is applied in
the derivation of Eq. (2.61):
Z l
Zh ðxÞZk ðxÞdx ¼ 0 ðh 6¼ kÞ ð2:61Þ
0
By considering the character of the Dirac d function, Eq. (2.60) can be expressed as
Z l Z l
d4 Zk ðxÞ
mr €qk ðtÞ Zk2 ðxÞdx þ EIY qk ðtÞ Zk ðxÞ dx
0 0 dx4
XN X
4 ð2:62Þ
¼ Frsi ðtÞZk ðxi Þ þ pj ðtÞZk ðxwj Þ ðk ¼ 1NMÞ
i¼1 j¼1
As
Z l
1
Zk2 ðxÞdx ¼ ð2:63Þ
0 mr
Z Z l
4
l
d4 Zk ðxÞ 2 kp kpx
Zk ðxÞ dx ¼ sin2 dx
0 dx4 0 mr l l l
4 Z l
kp
¼ Zk2 ðxÞdx ð2:64Þ
l 0
1 kp 4
¼
mr l
2.2 Vehicle–Track Vertically Coupled Dynamics Model 47
The above equation is the basic form of the rail second-order ordinary differential
equation of the rail.
Substituting Eq. (2.58) into Eq. (2.54), yields
X
NM X
NM
Frsi ðtÞ ¼ Cpi Zh ðxi Þq_ h ðtÞ þ Kpi Zh ðxi Þqh ðtÞ Cpi Z_ si ðtÞ Kpi Zsi ðtÞ ð2:66Þ
h¼1 h¼1
This is the final form of the second-order ordinary differential equation of the rail
described as the Euler beam.
It should be noted that the selection of the mode number NM must comply with
the criterion that the intercepted highest frequency relating to NM is higher than the
analyzed frequency of the rail. We can also determine the NM by using the
numerical trials, which will be given in Sect. 4.4.
(2) Differential equations of rail modeled as Timoshenko beam
The Timoshenko beam model can consider the rotatory inertia of the beam cross
section and beam deformation due to the shear force. Figure 2.18 shows the
free-body diagram of a rail element dx when it is modeled as a Timoshenko beam.
In the figure, M(x, t) and Q(x, t) are the bending moment and vertical shear force
applied on the rail element, w is the rotational angle of the cross section due to the
bending moment; b is the shear angle at the neutral axis of the same cross section.
Thus, the total rotational angle of the cross section can be described as
@Zr ðx; tÞ
¼ wþb ð2:68Þ
@x
Based on the Timoshenko beam theory, the differential equations for the vertical
rail deflection Zr ðx; tÞ and the shear deformation w(x, t) are written as
48 2 Vehicle–Track Coupled Dynamics Models
pj
M
M M+ dx
x
ψ Z r
β x
Q
Q
Q+ dx
x
Frsi
Zr dx
@ 2 Zr ðx; tÞ @wðx; tÞ @ 2 Zr ðx; tÞ
mr þ jAr Gr
@t2 @x @x2
XN X
4
¼ Frsi ðtÞdðx xi Þ þ pj ðtÞdðx xwj Þ ð2:69Þ
i¼1 j¼1
@ 2 wðx; tÞ @Zr ðx; tÞ @ 2 wðx; tÞ
q r IY þ jA r Gr w EI Y ¼0 ð2:70Þ
@t2 @x @x2
where Ar is the area of the rail cross section, Gr is the rail shear modulus, j is the
shear parameter depending on the shape of the rail cross section, qr is the rail
density.
Based on the Ritz method, the fourth-order partial differential equations,
Eqs. (2.69) and (2.70), can also be converted into the second-order ordinary dif-
ferential equations. Considering the normalized coordinate of the rail shear defor-
mation wk(t), the shear deformation modal shape can be obtain by using the
normalized modal function of a simply supported beam
sffiffiffiffiffiffiffiffiffiffi
2 kp
Wk ðxÞ ¼ cos x ð2:71Þ
qr IY l l
X
NM
wðx; tÞ ¼ Wk ðxÞwk ðtÞ ð2:72Þ
k¼1
Substituting Eqs. (2.58) and (2.72) into Eqs. (2.69) and (2.70), we can get
2.2 Vehicle–Track Vertically Coupled Dynamics Model 49
" #
X
NM X
NM
dWk ðxÞ X
NM 2
d Zk ðxÞ
mr Zk ðxÞ€qk ðtÞ þ jAr Gr wk ðtÞ qk ðtÞ
k¼1 k¼1
dx k¼1
dx2
ð2:73Þ
X
N X
4
¼ Frsi ðtÞdðx xi Þ þ pj ðtÞdðx xwj Þ
i¼1 j¼1
" #
X
NM XNM X
NM
dZk ðxÞ
qr IY Wk ðxÞ€
wk ðtÞ þ jAr Gr Wk ðxÞwk ðtÞ qk ðtÞ
k¼1 k¼1 k¼1
dx
ð2:74Þ
X
NM
d2 Wk ðxÞ
EIY wk ðtÞ ¼ 0
k¼1
dx2
ð2:75Þ
Z Z Z
l l l
dZk ðxÞ
qr IY Wk ðxÞWk ðxÞ€
wk ðtÞdx þ jAr Gr Wk ðxÞWk ðxÞwk ðtÞdx jAr Gr Wk ðxÞqk ðtÞdx
0 0 0 dx
Z l
d2 Wk ðxÞ
EIY Wk ðxÞwk ðtÞdx ¼ 0 ðk ¼ 1NM Þ
0 dx2
ð2:76Þ
Here is applied the following properties due to the modal orthogonality in the
derivation of Eqs. (2.75) and (2.76):
Z l
Wh ðxÞWk ðxÞdx ¼ 0 ðh 6¼ kÞ ð2:77Þ
0
Z l
Zh ðxÞWk ðxÞdx ¼ 0 ð2:78Þ
0
Considering the character of the Dirac d function, Eqs. (2.75) and (2.76) can be
rewritten as
50 2 Vehicle–Track Coupled Dynamics Models
Z Z Z
l l
dWk ðxÞ l
d2 Zk ðxÞ
mr €qk ðtÞ Zk2 ðxÞdx þ jAr Gr wk ðtÞ Zk ðxÞdx jAr Gr qk ðtÞ Zk ðxÞdx
0 0 dx 0 dx2
X
N X
4
¼ Frsi ðtÞZk ðxi Þ þ pj ðtÞZk ðxwj Þ ðk ¼ 1NM Þ
i¼1 j¼1
ð2:79Þ
Z Z Z
l l l
dZk ðxÞ
€ k ðtÞ
qr IY w W2k ðxÞdx þ jAr Gr wk ðtÞ W2k ðxÞdx jAr Gr qk ðtÞ Wk ðxÞdx
0 0 0 dx
Z l
d2 Wk ðxÞ
EIY wk ðtÞ Wk ðxÞdx ¼ 0 ðk ¼ 1NM Þ
0 dx2
ð2:80Þ
As
Z sffiffiffiffiffiffiffiffiffiffiffiffiffi
l
dWk ðxÞ kp 1
Zk ðxÞdx ¼ ð2:81Þ
0 dx l mr qr IY
Z
l
d2 Zk ðxÞ 1 kp 2
Zk ðxÞdx ¼ ð2:82Þ
0 dx2 mr l
Z l
1
W2k ðxÞdx ¼ ð2:83Þ
0 qr IY
Z sffiffiffiffiffiffiffiffiffiffiffiffiffi
l
dZk ðxÞ kp 1
Wk ðxÞdx ¼ ð2:84Þ
0 dx l mr qr IY
Z
2
l
d2 Wk ðxÞ kp 1
Wk ðxÞdx ¼ ð2:85Þ
0 dx 2 l qr IY
sffiffiffiffiffiffiffiffiffiffiffiffiffi
1 E kp 2 kp 1
€ k ðtÞ þ jAr Gr
w wk ðtÞ þ wk ðtÞ jAr Gr qk ðtÞ ¼ 0 ðk ¼ 1NM Þ
qr IY qr l l mr qr IY
ð2:87Þ
2.2 Vehicle–Track Vertically Coupled Dynamics Model 51
The above two equations are the basic form of the second-order ordinary differential
equations of the rail modeled as the Timoshenko beam.
Substituting Eq. (2.66) into Eq. (2.86), we can get the detailed forms of the rail
differential equations as follows:
X
N X
NM X
N X
NM
€qk ðtÞ þ Cpi Zk ðxi Þ Zh ðxi Þq_ h ðtÞ þ Kpi Zk ðxi Þ Zh ðxi Þqh ðtÞ
i¼1 h¼1 i¼1 h¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffi
jAr Gr kp 2 kp 1 XN
þ qk ðtÞ jAr Gr wk ðtÞ Cpi Zk ðxi ÞZ_ si ðtÞ ð2:88Þ
mr l l mr qr IY i¼1
X
N X
4
Kpi Zk ðxi ÞZsi ðtÞ ¼ pj ðtÞZk ðxwj Þ ðk ¼ 1NM Þ
i¼1 j¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffi
1 kp kp 1
€ k ðtÞ þ
w ½jAr Gr þ EIY ð Þ2 wk ðtÞ jAr Gr qk ðtÞ ¼ 0
qr IY l l mr qr IY ð2:89Þ
ðk ¼ 1NM Þ
Mbi
Zbi
Fbfi
Mbi Z€bi ðtÞ þ ðCbi þ Cfi þ 2Cwi ÞZ_ bi ðtÞ þ ðKbi þ Kfi þ 2Kwi ÞZbi ðtÞ Cbi Z_ si ðtÞ Kbi Zsi ðtÞ
Cwi Z_ bði þ 1Þ ðtÞ Kwi Zbði þ 1Þ ðtÞ Cwi Z_ bði1Þ ðtÞ Kwi Zbði1Þ ðtÞ ¼ 0 ði ¼ 1NÞ
ð2:94Þ
X X
N NM
EIY kp 4
€qk ðtÞ þ Cpi Zk ðxi Þ Zh ðxi Þq_ h ðtÞ þ qk ðtÞ
i¼1 h¼1
mr l
ð2:96Þ
X
N X
NM X
4
þ Kpi Zk ðxi Þ Zh ðxi Þqh ðtÞ ¼ pj ðtÞZk ðxwj Þ ðk ¼ 1NMÞ
i¼1 h¼1 j¼1
X
N X
NM X
N X
NM
€qk ðtÞ þ Cpi Zk ðxi Þ Zh ðxi Þq_ h ðtÞ þ Kpi Zk ðxi Þ Zh ðxi Þqh ðtÞ
i¼1 h¼1 i¼1 h¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffi
jAr Gr kp 2 kp 1 X4
þ ð Þ qk ðtÞ jAr Gr wk ðtÞ ¼ pj ðtÞZk ðxwj Þ ðk ¼ 1NM Þ
mr l l mr qr IY j¼1
ð2:97Þ
where EsIs is the vertical bending stiffness of the concrete slab (N m2), Zs(x, t) is the
slab vertical deflection (m), Ms is the slab mass (kg), Ls is the length of the slab (m),
while Ks and Cs are the vertical stiffness and damping of the CAM layer underneath
the slab, n0 is the number of rail fastenings on a track slab.
By using the Ritz method, the fourth-order partial differential equation,
Eq. (2.99), can be converted into the second-order ordinary differential equation.
The modal functions of a free end beam read [92]
8
< X1 ð xÞ ¼ 1pffiffiffi
>
X2 ð xÞ ¼ 3 1 2x ð2:100Þ
>
:
Ls
Xm ð xÞ ¼ ðchbm x þ cos bm xÞ Cm ðshbm x þ sin bm xÞ ð m 3Þ
where Cm and bm are, respectively, the frequency coefficient and the function
coefficient of a beam with free–free boundary condition. Table 2.3 lists the values
of Cm and bmLs.
The vertical deflection of the concrete slab reads
X
NMS
Zs ðx; tÞ ¼ Xn ð xÞTn ðtÞ ð2:101Þ
n¼1
Substituting Eq. (2.101) into Eq. (2.99), and multiplying it by Xp(x) (p = 1–NMS),
and then integrating it for x from 0 to l. By using the properties of the modal
orthogonality and the Dirac d function, we can get
Z Ls Z Ls Z Ls
Ms €
T n ðt Þ Xn2 ð xÞdx þ cs T_ n ðtÞ Xn2 ð xÞdx þ ks Tn ðtÞ Xn2 ð xÞdx
Ls 0 0 0
Z X ð2:102Þ
Ls
d 4 X n ð xÞ n0
þ Es Is Tn ðtÞ X n ð xÞ 4
dx ¼ Frsj ðtÞXn xj
0 dx j¼1
2.2 Vehicle–Track Vertically Coupled Dynamics Model 55
Table 2.3 The frequency and function coefficient of a beam with free–free boundary condition
m 1 2 3 4 5 6
Cm – – 0.982502 1.000777 0.999966 1.000000
bmLs 0 0 4.73004 7.85320 10.9956 (2m − 3)p/2
As
Z Ls
Xn2 ð xÞdx ¼ Ls ð2:103Þ
0
Z Ls
d 4 Xn ð xÞ
X n ð xÞ dx ¼ Ls b4n ð2:104Þ
0 dx4
cs L s _ ks þ Es Is b4n Xn0
Frsj ðtÞ
T€n ðtÞ þ Tn ðtÞ þ Ls Tn ðtÞ ¼ Xn xj ð2:105Þ
Ms Ms j¼1
Ms
where G is the wheel–rail contact coefficient (m/N2/3), dZðtÞ is the elastic com-
pressing amount at the wheel–rail contact point (m).
For the wheel with cone tread, G can be chosen as
G ¼ 4:57R0:149 108 m=N2=3 ð2:107Þ
where Zwj ðtÞ is the vertical displacement of the jth wheel (m), Zr ðxwj ; tÞ is the
vertical displacement of the rail under the jth wheel (m).
It is noted that dZðtÞ\0 indicates the wheel separating from the rail, thus the
wheel–rail force pðtÞ ¼ 0:
When the rail irregularity Z0 ðtÞ is considered, the wheel–rail force can be
expressed as
1 3=2
Zwj ðtÞ Zr ðxwj ; tÞ Z0 ðtÞ
pj ðtÞ ¼ G ð2:110Þ
0 ðLoss of wheelrail contactÞ
where S is the stress coefficient (N2/3/m2) determined by the Hertzian elastic contact
theory.
If R is in the range of 0.15–0.6 m, S can be chosen by
2:49R0:251 107 ðCone-tread wheelÞ
S¼ ð2:112Þ
1:49R0:376 107 ðWorn-tread wheel)
The vehicle–track vertically coupled dynamics models established in Sect. 2.2 are
further extended to the vehicle–track spatially coupled dynamics model by con-
sidering the lateral motions of the vehicle–track system, which provide the theo-
retical tool for studying the vertical and lateral dynamic performances of the
vehicle–track coupled system.
Mc Ic y c
Zc
C sz K sz
t2 t1
Mt Ity
C pz Z t2 K pz Z t1
Zw4 Zw3 Z w2 Z w1
Z 04 Z03 Z 02 Z01
P4 P3 mr EIY P2 P1 +
8
-
8
K pv C pv Zr
Ms
K bv C bv Zs
Cw
Mb
Kw
Zb
K fv C fv
Fig. 2.21 Passenger vehicle–track spatially coupled model (side view) (Reprinted from Ref. [36],
Copyright 2009, with permission from Taylor & Francis.)
58 2 Vehicle–Track Coupled Dynamics Models
Csx Csdx
Cpx Cpy Kpy
Ksx
Kpx
Yc Yw2 Yw1
Yt1
Csy Ksy
Fig. 2.22 Passenger vehicle–track spatially coupled model (top view) (Reprinted from Ref. [36],
Copyright 2009, with permission from Taylor & Francis.)
speed. Each component of the vehicle has five DOFs: the vertical displacement Z,
the lateral displacement Y, the roll angle U, the yaw angle w, and the pitch angle b
with respect to its center of mass (the pitch for a wheelset corresponds to the
variation of rotation around its mean rotational speed). All angles are assumed to be
small, which simplifies the kinematics and the equations of motion. As a result, the
total DOFs of the passenger vehicle sub-model are 35, as shown in Table 2.6.
The typical ballasted track model consists of rails, rail pads, sleepers, ballast, and
subgrade. Both the left and right rails are treated as continuous Euler or
Timoshenko beams, which are discretely supported at rail–sleeper junctions by
three layers of springs and dampers representing the elasticity and damping of rail
pads and fastenings, ballast and subgrade, respectively. Three kinds of vibrations of
the rails are considered: vertical, lateral, and torsional. The sleeper is assumed to be
a rigid body with three DOFs of the vertical displacement, the lateral displacement,
and the roll angle displacement. Lateral springs and dampers are considered to
represent the lateral dynamic properties in the fastener system. Similarly, the lateral
springs and dampers are used to represent the elasticity and damping property
between the sleeper and the ballast in lateral direction. A five-parameter model of
ballast under each rail-supporting point is adopted [90], only the vertical motion of
the ballast mass is taken into account, see also Fig. 2.13. In order to account for the
continuity and the coupling effects of interlocking ballast granules, a couple of
shear stiffness and shear damping is introduced between adjacent ballast masses in
the ballast model. Linear springs and dampers are hired to represent the subgrade
supporting the ballast.
2.3 Vehicle–Track Spatially Coupled Dynamics Model 59
Mc Ic x
Yc
c
Zc
Ksy
Cpy Mw Iwx w
YLr Zw YRr
Cph ZLr Lr ZRr Rr
Kfv Cfv
Fig. 2.23 Passenger vehicle–track spatially coupled model (end view) (Reprinted from Ref. [36],
Copyright 2009, with permission from Taylor & Francis.)
wedges are used between the side frames and two ends of the bolsters, which are
modeled as linear springs and the Coulomb friction elements in both the vertical
and the lateral directions. The nonlinear characteristics of these suspension com-
ponents are taken into account in the models. The notations of the symbols in
Figs. 2.24, 2.25, and 2.26 are given in Table 2.7.
The freight wagon is modeled as a multibody system. Both the car body and the
wheelset have five DOFs: the vertical displacement Z, the lateral displacement Y,
the roll angle U, the yaw angle w, and the pitch angle b with respect to its center of
mass. The bolster only has one DOF: the yaw angle describing the rotation motion,
because the bolster is connected with the car body at its center bowl allowing
rotation about its vertical centerline. Three DOFs are considered to describe the
movements of each side frame, i.e., the longitudinal displacement, the lateral dis-
placement, and the yaw angle. The vertical displacement and the pitch angle of the
side frame are depended on the movements of two relating wheelsets due to the
direct contact of them in the vertical direction. Thus, the total DOFs of the wagon
model are 39, as shown in Table 2.8.
2.3 Vehicle–Track Spatially Coupled Dynamics Model 61
Mc Ic y c
Zc
C sz K sz
t2 t1
Mt Ity
Z t2 Z t1
Zw4 Zw3 Z w2 Z w1
Z 04 Z03 Z 02 Z 01
P4 P3 mr EIY P2 P1 +
8
-
8
K pv C pv Zr
Ms
K bv C bv Zs
Cw
Mb
Kw
Zb
K fv C fv
ψB1
Yc Yw2 Yw1
ψtR1 XtR1
YtR1
Mc Icx
Yc
Φc
Zc
Ksy
Csy
Ksz Csz
YtL
ZtL ZtR YtR
Yw
Mw Iwx Φw
YLr Zw YRr
Cph ZLr Φ Lr ZRr Φ Rr
Kfv Cfv
M c J c c Xc
K sz Csz Zc K sz Csz
t 2 X t2 No.2 M t J t t1 Xt1 No.1
K qy
K pz K mz Cpz Cmz
Motor-gearbox
Z t2 Z t1
g4 m4 m3 g3 g2 m2 m1 g1
w4 M m Jm M w Jw w2 w1
w3
Z w4
p4 p3 Z w3 Z w2 p2 p1 Z w1
Z m4 Z m3 Z m1
No.4 No.2 Z m2 No.1
z 04 z 03 No.3
mr EIY z 02 z 01
P4 P3 P2 P1
K pv Cpv Zr
Ms
K bv Cbv Zs
Cw
Kw Zb
K fv Cfv Mb
K sy
Cpx K sx Csdy K px
K py Cpy
L-side L-side
Motor-
Cmwy gearbox
K my
ψc ψ w2 ψ ψ t1 K mx ψm1 ψ w1
K tr m2
Xc K mwx X w2 X m2 X t1 X m1 X w1
Yc Yw2 Y Ym1 Yw1
m2
Yt1
Cmwx
K mwy
R-side R-side
Csx
Csy
Yc
Zc φc
K sy
K tr
Csy
K sz Csz
Zt φ t Yt
K mz
K my Ym Motor-
Kpz gearbox Cpz
K mwy Zm φm
Yw
Zw φw K py
Cpy Cmwy
K mwz Cmwz
Yr Yr
φr K ph φr
Zr Zr
Cph
K pv Cpv K bh
M s Js Ys
φs Cbh
K br Zs Cbr
Mb Zb Kw Cw Mb Zb
K fv Cfv
representing the motions of the rigid bodies in the locomotive dynamics system are
shown in Table 2.10.
The major components of the locomotive dynamics system are usually con-
nected via bogie suspension systems. The secondary suspension system, including
the coil springs and lateral dampers, lateral bump–stops and traction rod, is usually
used to connect the car body to the bogie frames and is able to mobilize the required
rotational stiffness. The primary suspension system in each bogie, including the coil
springs, vertical viscous dampers, lateral and vertical bump–stops and axlebox
bushings, connects the bogie frames to the four wheelsets. Nonlinear 3D spring–
damper elements were used to build the suspension model. The viscous dampers
were represented as Maxwell elements consisting of a damper and a spring in series,
in which the damping coefficient has piecewise nonlinear characteristics relating the
velocity to the force generated. The bump–stops were modeled as bilinear spring
elements.
2.3 Vehicle–Track Spatially Coupled Dynamics Model 67
Table 2.10 DOFs of the locomotive dynamics model with gear transmissions
Component Type of motion
Longitudinal Lateral Vertical Roll Pitch Yaw
Car body Xc Yc Zc /c bc wc
Bogie frame (i = 1, 2) Xti Yti Zti /ti bti wti
Wheelset (i = 1–4) Xwi Ywi Zwi /wi bwi wwi
Motor (i = 1–4) Xmi Ymi Zmi /mi bmi wmi
Rotor (i = 1–4) – – – – bri –
Pinion (i = 1–4) – – – – bpi –
Gear (i = 1–4) – – – – bgi –
K pz Cpz
K mz Cmz LOA
α0 Rbp K m Cmwz
H mw
Cmwx
Rbg Tm2 J p βp2 Tm1 J p βp1
K mwz
Motor-gearbox Wheelset
Since the traction motor and the gearbox are bolted together, the mass of the
gearbox and the lubricating oil are lumped to the traction motor and symmetrically
distributed along the transverse direction. One end of the traction motor is elasti-
cally suspended on the bogie frame through a suspender and a ball-type rubber
joint, and the other end is supported on the wheel axle by two axle-hung bearings.
Zoom-in plot of the lead bogie is depicted in Fig. 2.30, where the dash circles in a
gear transmission denote the base circles of the gears in engagement. The power
transmission path from the rotor to the pinion, then to the gear and finally to the
wheel is shown in Fig. 2.31a, and the corresponding dynamics model of the power
transmission subsystem is shown in Fig. 2.31b. The output power of the motor is
transmitted to the pinion through a torsional spring–damper element (Krp, Crp)
representing the flexibility of the shaft between the motor rotor and the pinion.
Except for the power consumed by the damping of the connecting shaft and the
pinion, the left power is transmitted from the pinion to the gear by gear teeth
engagement. Finally, the remaining power is transmitted from the gear to the left
2.3 Vehicle–Track Spatially Coupled Dynamics Model 69
Fig. 2.31 The model of the power transmission: a geometrical model, and b dynamics model
and right wheels via torsional spring–damper elements (Kgw, Cgw) representing the
flexibility of the wheel axle to generate the driving forces for the locomotive
running by wheel–rail adhesion. The elastic deformations of the connecting shaft/
coupling and the elastic deformations of the wheel axle are modeled by the torsional
spring–damping unit (Kgw, Cgw), and the compressive deformations of the engaged
gear teeth along line of action (LOA) are represented by the spring–damping ele-
ment (Km, Cm) through which the time-varying mesh stiffness and/or the gear tooth
error excitations could be considered.
4. The ballastless track model
The vehicle–track spatially coupled dynamics models shown in Figs. 2.21, 2.22,
2.23, 2.24, 2.25, 2.26, 2.27, 2.28, and 2.29 only consider the ballasted track
although the railways can contain either the ballasted track or the ballastless track.
If the analyzed track structure is a ballastless track, the above-ballasted track
dynamics model can be changed correspondingly. Figures 2.32, 2.33, 2.34, and
2.35 show the dynamics models of four typical ballastless track structures,
including the long-sleeper embedded track (systems with sleepers firmly poured
into an in situ concrete track slab), the elastic supporting block track (systems with
elastically encased supporting blocks between rails and concrete slabs), the slab
track (systems with track slabs supported by elastic cement asphalt mortar layer),
and the floating slab track (systems with the concrete floating slab supported by the
steel springs).
70 2 Vehicle–Track Coupled Dynamics Models
Fig. 2.32 Vertical and lateral dynamics model for the long-sleeper embedded ballastless track
Supporting-block
Fig. 2.33 Vertical and lateral dynamics model for the elastic supporting block ballastless track
2.3 Vehicle–Track Spatially Coupled Dynamics Model 71
Track slab
Fig. 2.34 Vertical and lateral dynamics model for the typical slab track (Reprinted from Ref. [36],
Copyright 2009, with permission from Taylor & Francis.)
The slab track widely used in high-speed railways consists of the rails, the
fastenings and rail pads, the concrete slabs, the cement asphalt mortar (CAM) layer,
and the concrete base. In the slab track model, the rails are modeled as the Euler or
Timoshenko beams with consideration of the vertical, lateral and torsional motions,
while the track slabs are described as elastic rectangle plates supported on vis-
coelastic foundation, as shown in Fig. 2.34. In the figure, Ksv and Csv are the
vertical stiffness and damping of the CAM layer, and Ksh and Csh are the lateral
stiffness and damping of the CAM layer, respectively. Because the lateral bending
stiffness of the slab is very large, it is sufficient to consider the rigid mode of the
slab vibration in the lateral direction.
The floating slab track that is commonly used in metro systems consists of the
rails, the fasteners and rail pads, the concrete floating slab, and the steel springs. In
the floating slab track model, the rails are modeled as the Euler or Timoshenko
beams with considering the vertical, lateral and torsional motions, while the track
slabs are described as elastic rectangle thick plates supported on steel springs, as
shown in Fig. 2.35. In the figure, Ksv and Csv are the vertical stiffness and damping
of the steel spring, respectively; and Ksh and Csh are the lateral stiffness and
Kph
Csh
Ksv Csv
Fig. 2.35 Vertical and lateral dynamics model for the floating slab track
72 2 Vehicle–Track Coupled Dynamics Models
X
FzfL1
Y FzfR1
FyfL1 Z
FyfR1
FxfL1 FxfR1
FRy1+NRy1
MLy1 FLy1+NLy1 M wg MRy1
FLz1+NLz1 FRz1+NRz1
FLx1+NLx1 FRx1+NRx1
MLx1 MRx1
MLz1 MRz1
2a0
damping of the steel spring. Because the lateral bending stiffness of the slab is
rather large, it is reasonable to consider the rigid mode of the slab vibration in the
lateral direction.
① When i = 1 or 2, n = 1; when i = 3 or 4, n = 2.
1
② When the subscript on the left side of the equal sign is L, the symbol, and , takes the
symbol above; while when the subscript is R, the symbol, and , takes the symbol below.
③ Similar to the situation is so appointed.
2.3 Vehicle–Track Spatially Coupled Dynamics Model 73
FztL1 FztR1
X FxtL1
FxtR1
FxfL1 FxfR1
Y FytL1
FyfL1 FytR1
FxsL1 FyfR1
Mr1 FxsR1
Z FzfL1
FzfR1
Mt g
FyfL2
FyfR2
FxfL2 FxfR2
FzfL2 FzfR2
Fig. 2.37 Free-body diagram of the lead bogie frame of a passenger car
Mc g
X
Y
Z
FytL1
FxtL2 FytL2 FxtL1
FxtL1
FztL2 FxtL2
Mr1
Mr2
FztL1
FxtR2 FxsR1 FxtR1
FxtR2
FytR2 FytR1
FztR2 FztR1
lt i1
Fxf ðL;RÞi ¼ Kpx dw wtn þ Htw btn dw wwi ð1Þ dw
Rtn
ð2:113Þ
_ _ _ i1 d lt
þ Cpx dw wtn þ Htw btn dw wwi ð1Þ dw
dt Rtn
Table 2.11 Notations for the forces in the passenger vehicle subsystem
Notation Physical meaning
FLxi, FLyi, and The x-, y-, and z-direction creep forces of the left wheel of the ith
FLzi wheelset
FRxi, FRyi, and The x-, y-, and z-direction creep forces of the right wheel of the ith
FRzi wheelset
NLxi, NLyi, and The x-, y-, and z-direction contact forces of the left wheel of the ith
NLzi wheelset
NRxi, NRyi, and The x-, y-, and z-direction contact forces of the right wheel of the ith
NRzi wheelset
MLxi, MLyi, and The x-, y-, and z-direction spin creep torque of the left wheel of the ith
MLzi wheelset
MRxi, MRyi, and The x-, y-, and z-direction spin creep torque of the right wheel of the ith
MRzi wheelset
FxfLi and FxfRi The left and right longitudinal forces at primary suspension of the ith
wheelset
FyfLi and FyfRi The left and right lateral forces at primary suspension of the ith wheelset
FzfKLi and FzfKRi The left and right vertical forces at primary suspension of the ith wheelset
FzfCLi and FzfCRi The left and right vertical damping forces at primary suspension of the ith
wheelset
FxtLi and FxtRi The left and right longitudinal forces at secondary suspension of the ith
bogie
FytLi and FytRi The left and right lateral forces at secondary suspension of the ith bogie
FztKLi and FztKRi The left and right vertical forces at secondary suspension of the ith bogie
FztCLi and FztCRi The left and right vertical damping forces at secondary suspension of the
ith bogie
FxsLi and FxsRi The left and right longitudinal forces at anti-hunting dampers of the ith
bogie
FyRi The lateral forces of the stopblock on the ith bogie
MRi Anti-roll torque of the ith bogie
i l2t
Fyf ðL;RÞi ¼ Kpy Ywi Ytn þ Htw /tn þ ð1Þ lt wtn þ
2Rtn
ð2:114Þ
l2 d 1
þ Cpy Y_ wi Y_ tn þ Htw /_ tn þ ð1Þi lt w_ tn þ t
2 dt Rtn
If the yaw dampers are equipped in the secondary suspension, the longitudinal
forces due to the yaw dampers read
Fmax vxctðL;RÞi =v0 vxctðL;RÞi \v0
FxsðL;RÞi ¼ ð2:117Þ
Fmax sign vxctðL;RÞi vxctðL;RÞi v0
where Fmax is the saturation force of the damper; v0 is the unloading velocity of the
damper; vxct is the relative velocity between two ends of the damper connecting the car
body and the bogie frame in the longitudinal direction, which can be calculated by
d lc
vxctðL;RÞi ¼ dsc w_ c dsc w_ ti þ HcB b_ c þ HBt b_ ti ð1Þi1 dsc ð2:118Þ
dt Rc
Based on the calculated suspension forces and the Newton’s second law, the
equations of motion of the vehicle components can be formulated.
76 2 Vehicle–Track Coupled Dynamics Models
Vertical motion:
€ v2
Mw Z€wi a0 / sewi / ¼ FLzi FRzi NLzi
Rwi sewi
NRzi þ FzfLi þ FzfRi þ Mw g ð2:123Þ
Roll motion:
v
€ € _
Iwx /sewi þ /wi Iwy bwi X wwi þ_ ¼ a0 ðFLzi þ NLzi FRzi NRzi Þ
Rwi
rLi FLyi þ NLyi rRi FRyi þ NRyi þ dw ðFzfRi FzfLi Þ
ð2:124Þ
Yaw motion:
Iwz w€ þv d 1 I wy
_
/ þ _
/ _ X
b
wi sewi wi wi
dt Rwi
ð2:125Þ
¼ a0 ðFLxi FRxi Þ þ a0 wwi FLyi þ NLyi FRyi NRyi
þ MLzi þ MRzi þ dw ðFxfLi FxfRi Þ þ a0 ðNLxi NRxi Þ
Rotation motion:
€ ¼ rRi FRxi þ rLi FLxi þ rRi w FRyi þ NRyi
Iwy b wi
wi ð2:126Þ
þ rLi wwi FLyi þ NLyi þ MLyi þ MRyi þ NLxi rLi þ NRxi rRi
Vertical motion:
€ € v2
Mt Zti a0 /seti /seti ¼ FztLi FzfLð2i1Þ FzfLð2iÞ
Rti ð2:128Þ
þ FztRi FzfRð2i1Þ FzfRð2iÞ þ Mt g
Roll motion:
Itx /€ þ/€
ti seti ¼ FyfLð2i1Þ þ FyfRð2i1Þ þ FyfLð2iÞ þ FyfRð2iÞ Htw
þ FzfLð2i1Þ þ FzfLð2iÞ FzfRð2i1Þ FzfRð2iÞ dw ð2:129Þ
þ ðFztRi FztLi Þds FytLi þ FytRi HBt þ Mri
Pitch motion:
d 1
Itz €
wti þ v ¼ FyfLð2i1Þ þ FyfRð2i1Þ FyfLð2iÞ FyfRð2iÞ lt
dt Rti
ð2:130Þ
þ FxfRð2i1Þ þ FxfRð2iÞ FxfLð2i1Þ FxfLð2iÞ dw
þ ðFxtLi FxtRi Þds þ ðFxsLi FxsRi Þdsc
Yaw motion:
€ ¼ FzfLð2i1Þ þ FzfRð2i1Þ FzfLð2iÞ FzfRð2iÞ lt
Ity b ti
FxfLð2i1Þ þ FxfRð2i1Þ þ FxfLð2iÞ þ FxfRð2iÞ Htw ð2:131Þ
ðFxtLi þ FxtRi ÞHBt ðFxsLi þ FxsRi ÞHBt
Vertical motion:
2
€ v /
Mc Z€c a0 / ¼ FztL1 FztR1 FztL2 FztR2 þ Mc g ð2:133Þ
sec
Rc sec
78 2 Vehicle–Track Coupled Dynamics Models
Roll motion:
€ þ/
Icx ½/ € ¼ FytL1 þ FytR1 þ FytL2 þ FytR2 HcB
c sec
ð2:134Þ
þ ðFztL1 þ FztL2 FztR1 FztR2 Þds Mr1 Mr2
Pitch motion:
Yaw motion:
d 1
Icz €
wc þ v ¼ FytL1 þ FytR1 FytL2 FytR2 lc
dt Rc
ð2:136Þ
þ ðFxtR1 þ FxtR2 FxtL1 FxtL2 Þ ds
þ ðFxsR1 þ FxsR2 FxsL1 FxsL2 Þ dsc
Table 2.12 Physical meaning of notations used in equations of motion of vehicle subsystem
Notation Physical meaning
/sewi The super elevation angle of the curve high rail where the ith wheelset locates
/seti The super elevation angle of the curve high rail where the ith bogie center locates
/sec The super elevation angle of the curve high rail where the car body center locates
Rwi The curvature radius of the track where the ith wheelset locates
Rti The curvature radius of the track where the ith bogie frame locates
Rc The curvature radius of the track where the car body locates
v Train speed
r0 Nominal contact rolling radius of the wheel
rLi and rRi The left and right contact rolling radii of the ith wheel
X Nominal rolling angular velocity of the wheel
g Gravity acceleration
dsc Half-distance between the yaw dampers on the two sides of the bogie
ds Half-distance between the secondary suspension of the two sides of the bogie
dw Half-distance between the primary suspension of the two sides of the bogie
lc Half-distance between bogie centers
lt Half-distance between the two axles of the bogie
Htw Height of the bogie center from the wheelset center
HBt Height of the secondary suspension from the bogie center
HcB Height of the car body center from the secondary suspension location
2.3 Vehicle–Track Spatially Coupled Dynamics Model 79
2. Equations of motion of the freight wagon subsystem (refer to Figs. 2.39, 2.40,
2.41, and 2.42)
The forces between the components of a railway wagon equipped with the
three-piece bogies include the spring forces, the friction forces, the wheel–rail
normal contact forces, and tangent creep forces, which are shown in Figs. 2.39,
2.40, 2.41, and 2.42 and Table 2.13. Noted that the free-body diagram of a wagon
wheelset is as the same as shown in Fig. 2.36, and not given here again.
① Longitudinal and lateral forces of the primary suspension (i = 1–4)
The three-piece bogie Z8A comprises one bolster and two side frames. There are no
spring and no viscous damping component in the primary suspensions. The side
frames directly contact with the wheel axles in the vertical direction. There are
X
MztL1 MztR1
Y
FxtL1 FxtR1
Z
F´ztL1 F´ztR1
MytL1 F´ytL1 MytR1 F´ytR1
FyfL1 FyfR1
FxfL1 FxfR1
FzfL1 FzfR1
FyfL2 FyfR2
MztL1 MztR1
MzcB1
X
FxtL1 FxtR1
Z
Mc g
X
Y
Z
MytL1
MytL2
FytL2 FytL1
MzcB2 FztL1
FytR2 FytR1
MzcB1
Kcy
Kcx
X Y
δx δy
Fig. 2.42 Modeling of the axle box connection between side frame and wheelset: a the clearances
in axle box, b the longitudinal force characteristic, and c the lateral force characteristic
longitudinal clearance dx and lateral clearance dy between the axle box and the side
frame, see Fig. 2.42a. In consideration of the combination effect of the Coulomb
friction and the clearances between the axle box and the side frame, the force–
displacement characteristics in these places are described in Fig. 2.42b and c.
According to Fig. 2.42b, the longitudinal forces at the axle box can be expressed
as
8
XxtwðL;RÞi \dx
< ls Fzf ðL;RÞi sign vxtwðL;RÞi
Fxf ðL;RÞi ¼ Kcx XxtwðL;RÞi signXxtwðL;RÞi dx ð2:137Þ
: XxtwðL;RÞi dx
þ ls Fzf ðL;RÞi sign vxtwðL;RÞi
where ls is the Coulomb friction coefficient between the axle and the side frame,
Xxtw(L,R) is the longitudinal relative displacement between the axle and the side
frame, and Vxtw(L,R)i is the longitudinal relative velocity between the axle and the
side frame, which are given by
lt
XxtwðL;RÞi ¼ XtðL;RÞn þ Htw btðL;RÞn dw wwi ð1Þi dw ð2:138Þ
Rtn
_ _ _ i d lt
vxtwðL;RÞi ¼ XtðL;RÞn þ Htw btðL;RÞn dw wwi ð1Þ dw ð2:139Þ
dt Rtn
According to Fig. 2.42c, the lateral forces at the axle box can be expressed as
8
XytwðL;RÞi \dy
< ls Fzf ðL;RÞi sign vytwðL;RÞi
Fyf ðL;RÞi ¼ Kcy XytwðL;RÞi signXytwðL;RÞi dy ð2:140Þ
: XytwðL;RÞi dy
þ ls Fzf ðL;RÞi sign vytwðL;RÞi
where Xytw(L,R)i is the lateral relative displacements between the axle and the side
frame and Vytw(L,R)i is the lateral relative velocities between the axle and the side
frame, which can be obtained by
l2t
XytwðL;RÞi ¼ Ywi YtðL;RÞn þ ð1Þi lt wtðL;RÞn þ ð2:141Þ
2Rtn
_ _ i _ l2t d 1
vytwðL;RÞi ¼ Ywi YtðL;RÞn þ ð1Þ lt wtðL;RÞn þ ð2:142Þ
2 dt Rtn
€ € v2 0
Mt ZtLi a0 /seti /seti ¼ FzfLð2i1Þ FzfLð2iÞ þ FztLi þ Mt g ð2:143Þ
Rti
€ ¼ FzfLð2i1Þ FzfLð2iÞ lt FxfLð2i1Þ þ FxfLð2iÞ Htw FxtLi HBt þ MytLi
Ity b tLi
ð2:144Þ
where HBt is height of the side frame center from the bolster center.
From Eqs. (2.143) and (2.144), the vertical forces of the left primary suspension
can be deduced:
FzfLð2i1Þ ¼ ðA1 þ B1 Þ=2
ð2:145Þ
FzfLð2iÞ ¼ ðA1 B1 Þ=2
where
8
< A1 ¼ FztLi
0 € v2 /
þ Mt g Mt Z€tLi a0 / seti Rti seti
h i ð2:146Þ
: B1 ¼ Ity b
€ þ FxfLð2i1Þ þ FxfLð2iÞ Htw þ FxtLi HBt MytLi =lt
tLi
The vertical and pitch displacements of the left side frames can be calculated by
ZtL1 ¼ ½Zw1 þ Zw2 dw ð/w1 þ /w2 Þ=2
ð2:147Þ
ZtL2 ¼ ½Zw3 þ Zw4 dw ð/w3 þ /w4 Þ=2
btL1 ¼ ½Zw2 Zw1 þ dw ð/w1 /w2 Þ=ð2lt Þ
ð2:148Þ
btL2 ¼ ½Zw4 Zw3 þ dw ð/w3 /w4 Þ=ð2lt Þ
where
8
< A2 ¼ FztRi
0 € v2 /
þ Mt g Mt Z€tRi a0 / seti Rti seti
h i ð2:150Þ
: B2 ¼ Ity b
€ þ FxfRð2i1Þ þ FxfRð2iÞ Htw þ FxtRi HBt MytRi =lt
tRi
The vertical and pitch displacements of the right side frames can be calculated by
ZtR1 ¼ ½Zw1 þ Zw2 þ dw ð/w1 þ /w2 Þ=2
ð2:151Þ
ZtR2 ¼ ½Zw3 þ Zw4 þ dw ð/w3 þ /w4 Þ=2
2.3 Vehicle–Track Spatially Coupled Dynamics Model 83
btR1 ¼ ½Zw2 Zw1 dw ð/w1 /w2 Þ=ð2lt Þ
ð2:152Þ
btR2 ¼ ½Zw4 Zw3 dw ð/w3 /w4 Þ=ð2lt Þ
(a) (b)
Fig. 2.44 Free-body diagram of the friction wedge: a bolster and side frame move face to face,
and b bolster and side frame move in direction of separation
84 2 Vehicle–Track Coupled Dynamics Models
N2 are the normal forces applied on the friction wedge when the bolster and side
frame move face to face, N10 and N20 are the normal forces applied on the friction
wedge when the bolster and the side frame move in the direction of separation. Fl
and F2 are the friction forces of the wedge when the bolster and side frame move
face to face, F10 and F20 are the friction forces of the wedge when the bolster and the
side frame move in the direction of separation. These forces are calculated as
follows:
(
Nl ¼ sin al 2 cos a
Pa
DA ð2:153Þ
N2 ¼ cos b þDAl1 sin b Pa
Fl ¼ l1 Nl
ð2:154Þ
F2 ¼ l2 N2
(
N10 ¼ sin a þDB
l2 cos a
Pa
0 bl 1 sin b
ð2:155Þ
N2 ¼ cos
DB Pa
F10 ¼ l1 N10
ð2:156Þ
F20 ¼ l2 N20
DA ¼ ð1 þ l1 u2 Þ cosða bÞ þ ðl2 l1 Þ sinða bÞ
ð2:157Þ
DB ¼ ð1 þ l1 l2 Þ cosða bÞ ðl2 l1 Þ sinða bÞ
Pa ¼ Ksz1 Z1 ð2:158Þ
where Ksz1 is the stiffness of the steel spring supporting the friction wedge, Z1 is the
compression of the wedge springs.
If the relative vertical displacement between the bolster and the side frame is
DZBt, Z1 can then be expressed as
where DZ0 is the initial compression of the coil steel spring, which can be obtained
by
1
DZ0 ¼ ðMc þ 2MB Þg=½Ksz þ 2Ksz1 =ð1 þ tanatanbÞ ð2:160Þ
4
where Ksz is the stiffness of the steel spring supporting the bolster, i.e., the sec-
ondary suspension stiffness, i.e., the secondary suspension stiffness.
Consequently, when the bolster and side frame move face to face, the secondary
suspension forces (i = 1, 2) can be calculated as follows:
2.3 Vehicle–Track Spatially Coupled Dynamics Model 85
Vertical forces
(
FztðL;RÞi ¼ Ksz ½Zc þ ð1Þi lc bc ds /c ZtðL;RÞi þ DZ0 þ 2N2 cos a þ 2F2 sin a
Fzt0 ðL;RÞi ¼ Ksz ½Zc þ ð1Þi lc bc ds /c ZtðL;RÞi þ DZ0 þ 2Pa þ 2Fl cos b 2Nl sin b
ð2:161Þ
2 dt Rc
h
i
: S0 ¼ 2l N sign _
Y _
Y tðL;RÞi þ ð1Þ i1 _
w l /_ H
l2c d 1
ytðL;RÞi 1 l c c c c cB 2 dt R c
ð2:162Þ
Friction moments
8 h i
< MytðL;RÞi ¼ l1 Nl lrs sign b_ c b_ tðL;RÞi
h i ð2:163Þ
: MzBðL;RÞi ¼ l Nl lrs sign w_ w_
1 Bi tðL;RÞi
where lrs is the distance between the center positions of two friction plates (see
Fig. 2.41) on the side frame.
When the bolster and side frame move in the direction of separation, the sec-
ondary suspension forces (i = 1, 2) becomes:
Vertical forces
(
FztðL;RÞi ¼ Ksz ½Zc þ ð1Þi lc bc ds /c ZtðL;RÞi þ DZ0 þ 2N20 cos a 2F20 sin a
Fzt0 ðL;RÞi ¼ Ksz ½Zc þ ð1Þi lc bc ds /c ZtðL;RÞi þ DZ0 þ 2Pa 2F10 cos b 2N10 sin b
ð2:164Þ
2 dt Rc
h
i
: S0 ¼ 2l N 0
sign _
Y _
Y tðL;RÞi þ ð1Þ i1 _
w l /_ H
l2c d 1
ytðL;RÞi 1 1 c c c c cB 2 dt Rc
ð2:165Þ
86 2 Vehicle–Track Coupled Dynamics Models
Friction moments
8 h i
< MytðL;RÞi ¼ l1 N10 lrs sign b_ c b_ tðL;RÞi
h i ð2:166Þ
: MzBðL;RÞi ¼ l N 0 lrs sign w_ w_
1 1 Bi tðL;RÞi
The longitudinal and lateral forces due to the coil steel springs can be calculated
by (i = 1, 2)
(
l2
FysðL;RÞi ¼ Ksy ½Yc þ YtðL;RÞi þ ð1Þi wc lc þ /c HcB þ 2Rc c
ð2:167Þ
FxsðL;RÞi ¼ Ksx ½ wBi ds þ bc HcB XtðL;RÞi þ btðL;RÞi HBt
The anti-warp moment due to the bottom frame brace system can be expressed
as (i = 1, 2)
h i
MwðL;RÞi ¼ Kwt wBi wtðL;RÞi ð2:168Þ
where Kwt is the anti-warp stiffness of the three-piece bogie, which can be obtained
from Table 2.14.
The friction moments resulting from the center plate and the side bearings on the
bolster can be calculated by
1
MzcBi ¼ Fp ls lrp þ Mc g 2Fp ls lr0 sign w_ c w_ Bi ð2:169Þ
2
where Fp is the preload of the side bearing on the bolster, ls is the friction coef-
ficient of the side bearing, lr0 is the radius of the friction moment of the center plate,
lrp is the half distance of the two the side bearings on the bolster.
To sum up, the secondary suspension forces include (i = 1, 2)
The equations of motion of the wagon components can then be formulated when
the abovementioned suspension forces are all determined.
(1) Equations of motion of wheelset (i = 1–4)
The motion equations of the wheelsets of the wagon are similar to those of the
passenger vehicle (see Eqs. (2.122)–(2.126)), which are omitted here.
(2) Equations of motion of side frame (i = 1, 2)
Longitudinal motion
Lateral motion
v2 €
Mt Y€tðL;RÞi þ þ ðr0 þ Htw Þ/ 0
seti ¼ FytðL;RÞi þ Fyf ðL;RÞð2i1Þ þ Fyf ðL;RÞð2iÞ þ Mt g/seti
Rti
ð2:172Þ
Yaw motion
d 1
Itz €
wtðL;RÞi þ v ¼ Fyf ðL;RÞð2i1Þ Fyf ðL;RÞð2iÞ lt þ MztðL;RÞi ð2:173Þ
dt Rti
€ d 1
IBz wBi þ v ¼ MzcBi MztLi MztRi ds ðFxtLi FxtRi Þ ð2:174Þ
dt Rti
where HcB is the height of the car body center from the bolster center.
Vertical motion
€ € v2
ðMc þ 2MB Þ Zc a0 /sec /sec ¼ FztL1 FztR1 FztL2
Rc
FztR2 þ ðMc þ 2MB Þg ð2:176Þ
Roll motion
h i
Icx þ 2IBx þ 2MB HcB /€ þ/
€
sec ¼ FytL1 þ FytR1 þ FytL2 þ FytR2 HcB
2
c
Pitch motion
Icy þ 2MB l2c b€ ¼ ðFztL1 þ FztR1 FztL2 FztR2 Þlc
c
ðFxtL1 þ FxtR1 þ FxtL2 þ FxtR2 ÞHcB ð2:178Þ
MytL1 MytL2 MytR1 MytR2
Yaw motion
Icz w€ þv d 1 ¼ FytL1 þ FytR1 FytL2 FytR2 lc MzcB1 MzcB2
c
dt Rc
ð2:179Þ
2.3 Vehicle–Track Spatially Coupled Dynamics Model 89
X
Y
FzfL1 Z FzfR1
Fzmesh1
Tgw Fmwx1L Fmwx1R
FyfL1 FyfR1
1
Lx
1
Lx
Rx
N
+N
M
1
+
Rx
M
M Rz1
1
Lx
1
Rx
F
F
M Lz1
Lateral forces
l2
FyfðL;RÞi ¼ Kpy Ywi Ytn þ Htw /tn þ ð1Þi1 lt wtn þ t
2Rtn
ð2:181Þ
l2 d 1
þ Cpy Y_ wi Y_ tn þ Htw /_ tn þ ð1Þi1 lt w_ tn þ t
2 dt Rtn
90 2 Vehicle–Track Coupled Dynamics Models
Fs1
FxtL2 FxtR2
FytL2 Frt1
FztL2 M tg
FytR2
FztR2
Fxm2
Fym2 Fzm2
FyfL2
FyfR2
Fxd1 FxfR2 FzfR2
FxfL2 FzfL2
Fyd1
Fzd1
X
Y
Fmwz1L Fmwz1R
Z Fmwy1L
Fmwy1R
Fmwx1L
Fmwx1R
Fx mesh1
M mg
Fxm1
Fz mesh1
Fym1
Fzm1
Vertical forces
FzfðL;RÞi ¼ Kpz Ztn Zwi þ ð1Þi lt btn dw /wi dw /tn
h i ð2:182Þ
þ Cpz Z_ tn Z_ wi þ ð1Þi lt b_ tn dw /_ wi dw /_ tn
X
Y
FztL1
Fyc1
Z FytR1
FztL2 FxtL1 FytL1 Fxc2 Fzc2 FxtR1
FxtL2 FytL2 Fsn FxtR2 FztR1
FytR2
Frc
M cg Fyd1 FztR2
Fxd1 Fzd1
FztL3
FztL4 FxtL3 Fyd2 FxtR3
FytL3 FytR3
Fxd2 FxtR4 FztR3
FytL4 Fzd2
FxtL4 FytR4
Fyc1 FztR4
Fxc2
Fzc2
Longitudinal forces
i1 L
FxtðL;RÞi ¼ Ksx Xc Xtn þ HcB bc þ HBt btn ds wc ds wtn ð1Þ ds
Rc
_ _ _ _ _ _ i1 d L
þ Csx Xc Xtn þ HcB bc þ HBt btn ds wc ds wtn ð1Þ ds
dt R c
ð2:183Þ
where
lc þ lsb ði ¼ 1; 4Þ
L¼ ð2:184Þ
lc lsb ði ¼ 2; 3Þ
where lsb is the longitudinal distance between the secondary suspension and the
bogie frame center.
Lateral forces
" #
Ytn Yc þ HcB /c þ HBt /tn þ ð1Þi1 lsb wtn
FytðL;RÞi ¼ Ksy
þ ð1Þn lc wc þ ð1Þi1 lsb wc þ 2R
2
L
2 c
3 ð2:185Þ
_Ytn Y_ c þ HcB /_ c þ HBt /_ tn þ ð1Þi1 lsb w_ tn
þ Csy 4
5
þ ð1Þn lc w_ þ ð1Þi1 lsb w_ þ L d 1
2
c c 2 dt Rc
92 2 Vehicle–Track Coupled Dynamics Models
In addition, the lateral forces generated by the lateral dampers can be calculated as
Fmax vyctn =v0 Y_ tn v \v
Fyctn ¼ yctn 0 ð2:186Þ
Fmax sign vyctn vyctn v0
where Fmax is the saturation force of the damper; v0 is the unloading velocity of the
damper; vyct is the relative velocity between two ends of the damper connecting the
car body and the bogie frame in the lateral direction, which can be obtained by
_ _ _ _ n1 _ l2c d 1
vyctn ¼ Ytn Yc þ HcB /c þ HBt /tn þ ð1Þ lc wtn þ ð2:187Þ
2 dt R c
Vertical forces
" #
Zc Ztn ds /tn ds /c þ ð1Þi1 lsb btn
FztðL;RÞi ¼ Ksz
þ ð1Þn lc bc þ ð1Þi lsb bc
" # ð2:188Þ
Z_ c Z_ tn ds /_ tn ds /_ c þ ð1Þi1 lsb b_ tn
þ Csz
þ ð1Þn lc b_ þ ð1Þi lsb b_
c c
empirical value. The equations for the traction motor suspension forces and the
axle-hung bearing forces can be calculated as follows:
Longitudinal forces
h i
Fxmi ¼ Kmx Xtn Xmi þ ð1Þi1 Hmt btn þ ð1Þi1 Hmw bmi
h i ð2:189Þ
þ Cmx X_ tn X_ mi þ ð1Þi1 Hmt b_ tn þ ð1Þi1 Hmw b_ mi
where Hmw is the vertical distance between the gravity center of the motor and that
of the wheelset, while Hmt is the vertical distance between the gravity center of the
motor and that of the bogie frame.
Lateral forces
h i
Fymi ¼ Kmy Ytn Ymi þ ð1Þi1 lm wtn þ ð1Þi l2 wmi
h i ð2:190Þ
þ Cmy Y_ tn Y_ mi þ ð1Þi1 lm w_ tn þ ð1Þi l2 w_ mi
where lm represents the longitudinal distance between the gravity center of the
bogie frame and the motor suspension position on the frame, and l2 the longitudinal
distance between the gravity center of the motor and the motor suspension position
on the frame.
Vertical forces
Fzmi ¼ Kmz Ztn Zmi þ ð1Þi lm btn þ ð1Þi lm bmi
h i ð2:191Þ
þ Cmz Z_ tn Z_ mi þ ð1Þi lm b_ þ ð1Þi lm b_
tn mi
where lmb the horizontal distance between the left and right bearing installation
location and the gravity center of the wheelset. It should be noted that the left and
right bearings are symmetrically mounted about the gravity center of the wheelset.
Rmi represents the curvature radius of the track where the ith motor locates.
94 2 Vehicle–Track Coupled Dynamics Models
Lateral forces
i l21
FmwyðL;RÞi ¼ Kmwy Ywi Ymi þ ð1Þ l1 wmi þ Hmw /mi þ
2Rmi
ð2:193Þ
_ _ i _ _ l21 d 1
þ Cmwy Ywi Ymi þ ð1Þ l1 wmi þ Hmw /mi þ
2 dt Rmi
where l1 denotes the longitudinal distance between the gravity center of the motor
and the center of the wheelset.
Vertical forces
FmwzðL;RÞi ¼ Kmwz Zmi Zwi þ ð1Þi l1 bmi lm /mi lmb /wi
h i ð2:194Þ
þ Cmwz Z_ mi Z_ wi þ ð1Þi l1 b_ lm /_ lmb /_
mi mi wi
(a) (b)
X ’c
Fyd2 Fd2
Y ’c
Xt2 Fxd2
Xc
Yc
Yt2
Fig. 2.49 Decomposition diagrams of the forces acting on the car body by the rear traction rod:
a vertical–longitudinal plane and b longitudinal–lateral plane
2.3 Vehicle–Track Spatially Coupled Dynamics Model 95
8 n
< lxtrn ¼ l0 þ ð1Þ Xc n1
> Xtn þ Hcq bc Htq btn
lytrn ¼ Yc hYtn þ ð1Þ lcq wc þ ð1Þn1 ltq wtn i ð2:195Þ
>
: l ¼ h Z Z þ ð1Þn l b þ ð1Þn1 l b
ztrn 0 c tn cq c tq tn
where Hcq and lcq, respectively, denote the vertical and longitudinal distance
between the traction point of the car body and the car body center, while Htq and ltq
represent the vertical and longitudinal distance between the traction point of the
bogie frame and the bogie frame center, respectively.
Therefore, the space distance between the traction points of the frame and the car
body ltrn and the dynamic variation of space distance of the traction rod Dltrn are as
follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ltrn ¼ l2xtrn þ l2ytrn þ l2ztrn ð2:196Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffi
Dltrn ¼ltrn l20 þ h20 ð2:197Þ
Combined with the dynamic geometric position relationship of the traction rod,
the components of the traction rod force in the x-, y-, and z-direction can be
calculated as follows:
8 n
< Fxdn ¼ ð1Þ Fdn lxtrn =ltrn
F ¼ Fdn lytrn =ltrn ð2:199Þ
: ydn
Fzdn ¼ Fdn lztrn =ltrn
Fmesh ¼ Km d þ Cm d_ ð2:200Þ
where Fmesh, Km and Cm represents the mesh force, mesh stiffness, and mesh
damping, respectively; d denotes the Dynamic Transmission Error (DTE) indicating
the relative compressive displacement of the engaged gear pairs along LOA.
96 2 Vehicle–Track Coupled Dynamics Models
During the mesh process, the gear mesh stiffness varies with time periodically
due to the variations in the number of tooth pairs in mesh and the variations of the
contact position along the tooth profile. The time-varying mesh stiffness is calcu-
lated as [98, 101]
PN
j¼1 Kj ðtÞ
Km ðtÞ ¼ PN ð2:201Þ
1þ j¼1 Kj ðtÞEij ðtÞ=FðtÞ
where N represents the number of tooth pairs in mesh, E denotes the errors of tooth
profile. The subscripts i and j denote the number of the tooth pairs, F is the total
mesh force of the gear pairs, K is the single-tooth mesh stiffness, which can be
calculated as follows [81, 99–101]:
1 1 1 1 1 1
¼ þ þ þ þ ð2:202Þ
Kj ðtÞ Kto1 ðtÞ Kff1 ðtÞ Kto2 ðtÞ Kff2 ðtÞ Kh ðtÞ
where Kto, Kff, and Kh indicates the tooth stiffness, the fillet-foundation stiffness,
and the Hertz contact stiffness, respectively. The subscripts 1 and 2 represent the
pinion and the gear, respectively. It should be noted that the gear fillet-foundation
stiffness will be altered in the presence of gear tooth root crack fault, which could be
referenced to Ref. [102].
According to the rotational direction of the pinion shown in Fig. 2.30, its
counterclockwise rotation is defined as positive, and the corresponding relative
displacement of the pinion and the gear teeth along LOA can be calculated as
d ¼ Rp bp þ Rg bg e ð2:203Þ
where Rp and Rg are the base circle radius of the pinion and the gear, respectively.
However, during the actual operation of the locomotive, the pinion vibrates with
the motor, and the gear vibrates with the wheelset. Thus, gear teeth contact loss or
even double-sided contacts may happen to the engaged gear teeth due to the drastic
speed variations [103]. Consequently, Eq. (2.203) should be revised as follows:
8
< Rp bpi þ Rg bgi þ ð1Þi DZmwi cos a0 þ ð1Þi1 DXmwi sin a0 ei ðfor drive-side contactÞ
di ¼ 0 ðfor teeth contact loss)
:
Rp bpi þ Rg bgi þ ð1Þi DZmwi cos a0 ð1Þi1 DXmwi sin a0 þ ei ðfor coast-side contactÞ
ð2:204Þ
where a0 denotes the pressure angle of the gear pair, and the symbol e is half of the
gear tooth backlash. DZmw and DXmw are the vertical and longitudinal relative
displacements of the motor and the wheelset, respectively, including the effect of
roll and yaw motions of the motor and wheelset. They are calculated as follows:
2.3 Vehicle–Track Spatially Coupled Dynamics Model 97
Based on the calculated suspension forces and the Newton’s second law, the
equations of motion of the locomotive dynamics system can be derived.
(1) Equations of motion of the wheelset (i = 1–4)
Longitudinal motion:
€w ¼ FxfLi þ FxfRi þ FLxi þ FRxi þ NLxi
Mw X
ð2:207Þ
þ NRxi þ FmwxLi þ FmwxRi þ ð1Þi1 jFxmeshi j
Lateral motion:
€ v2 €
M w Yw þ þ r0 /sewi ¼ FLyi þ FRyi FyfLi FyfRi þ NLyi
Rwi ð2:208Þ
þ NRyi FmwyRi FmwyLi þ Mw g/sewi
Bounce motion:
€ € v2
Mw Zwi a0i /sewi / ¼ FLzi FRzi NLzi NRzi
Rwi sewi ð2:209Þ
þ FzfRi þ FzfLi þ FmwzRi þ FmwzLi þ ð1Þi Fzmeshi þ Mw g
Roll motion:
v
Iwx /€ þ €
/ I wy
_
b X w_ þ ¼ a0 ðFLzi þ NLzi Þ
sewi wi wi wi
Rwi
ð2:210Þ
a0 ðFRzi þ NRzi Þ rLi ðFLyi þ NLyi Þ þ dw ðFzfRi FzfLi Þ þ MLxi
þ MRxi þ ðFmwzLi FmwzRi Þlmb rRi ðFRyi þ NRyi Þ þ lgy Fzmeshi
where lgy represents the horizontal distance between installation location of the
pinion and the gravity center of the motor.
Pitch motion:
Iwz w€ þv d 1 Iwy /_ sewi þ /_ wi b_ wi X ¼ MLzi þ MRzi
wi
dt Rwi
ð2:211Þ
þ a0 ðFLxi þ NLxi FRxi NRxi Þ þ a0 wwi ðFLyi þ NLyi FRyi NRyi Þ
þ dw ðFxfLi FxfRi Þ ðFmwxLi FmwxRi Þlmb þ lgy jFxmeshi j
98 2 Vehicle–Track Coupled Dynamics Models
Rotational motion:
€ ¼ rLi ðFLxi þ NLxi Þ þ rRi ðFRxi þ NRxi Þ þ MRyi þ MLyi
Iwy b wi
ð2:212Þ
þ rRi wwi ðFRyi þ NRyi Þ þ rLi wwi ðFLyi þ NLyi Þ þ Tgwi
where Tgw denotes the torque generated in the axle between the gear and the wheels.
(2) Equations of motion of the bogie frame (n = 1–2)
Longitudinal motion:
X
2n
€tn ¼ ð1Þn1 Fxdn
Mt X ðFxfLi þ FxfRi Þ
i¼2n1
ð2:213Þ
X
2n X
2n
þ ðFxtLi þ FxtRi Þ Fxmi Frtn
i¼2n1 i¼2n1
Lateral motion:
X2n
v2 €
Mt Y€tn þ þ ðr0 þ Htw Þ/ setn ¼ F ydn þ ðFyfLj þ FyfRj Þ
Rtn j¼2n1
ð2:214Þ
X
2n X
2n
ðFytLj þ FytRj Þ Fymj þ Mt g/seti Fsn Fyctn
j¼2n1 j¼2n1
Bounce motion:
2
X2n
€ v /
Mt Z€tn a0 /setn setn ¼ Fzdn ðFzfLj þ FzfRj Þ
Rti j¼2n1
ð2:215Þ
X
2n X
2n
þ ðFztLj þ FztRj Þ Fzmj þ Mt g
j¼2n1 j¼2n1
Roll motion:
X
2n X
2n
Itx /€ þ/€
tn setn ¼ Htw ðFyfLj þ FyfRj Þ þ dw ðFzfLj FzfRj Þ
j¼2n1 j¼2n1
X
2n X
2n
ds ðFztLj FztRj Þ Hmt Fymj Htq Fydn
j¼2n1 j¼2n1
X
2n X
2n
þ Htld Fyctn þ HBt ðFytLj þ FytRj Þ þ ldy ð1Þj1 Fzmj
j¼2n1 j¼2n1
ð2:216Þ
2.3 Vehicle–Track Spatially Coupled Dynamics Model 99
where ldy represents the horizontal distance between suspension location of the
motor on bogie frame and the bogie frame center. And Htld denotes the vertical
distance between installation location of the lateral damper on bogie frame and the
bogie frame center.
Yaw motion:
X2n h i
Itz w€ þv d 1 ¼ lt ð1Þj1 FyfLj þ ð1Þj1 FyfRj
tn
dt Rtn j¼2n1
X
2n X
2n
þ ð1Þn ltq Fydn þ lsb ð1Þ j FytLj þ ð1Þ j FytRj lm ð1Þj1 Fymj
j¼2n1 j¼2n1
X
2n X
2n X
2n
dw ðFxfLj FxfRj Þ þ ds ðFxfLj FxfRj Þ ldy ð1Þj1 Fxmj
j¼2n1 j¼2n1 j¼2n1
ð2:217Þ
Pitch motion:
2n h
X i
€ ¼ lt
Ity b ð1Þj1 FzfLj þ ð1Þj1 FzfRj þ ð1Þn1 ltq Fzdn þ ð1Þn1 Htq Fxdn
tn
j¼2n1
X
2n X
2n X
2n
H tw ðFxfLj þ FxfRj Þ HBt ðFxtLj þ FxtRj Þ Hmt Fxmj
j¼2n1 j¼2n1 j¼2n1
X
2n X
2n
þ lsb ð1Þ j FztLj þ ð1Þ j FztRj þ lm ð1Þj1 Fzmj
j¼2n1 j¼2n1
ð2:218Þ
Lateral motion:
v2 €
Mm Y€mi þ þ ðr0 þ Hmw Þ/ semi ¼ Fymi þ FmwyRi þ FmwyLi þ Mm g/semi
Rmi
ð2:220Þ
where /semi denotes the super elevation angle of the curve high rail where the ith
motor center locates.
100 2 Vehicle–Track Coupled Dynamics Models
Bounce motion:
€ € v2
Mm Zmi a0 /semi / ¼ Fzmi FmwzRi FmwzLi þ ð1Þi1 Fzmeshi þ Mm g
Rmi semi
ð2:221Þ
Roll motion:
Imx /€ þ/€
mi semi ¼ ðFmwzLi FmwzRi Þlmb
ð2:222Þ
ðFmwyRi þ FmwyLi þ Fymi ÞHmw lgy Fzmeshi
Pitch motion:
Imz w€ þv d 1 ¼ ð1Þi1 ðFmwyRi þ FmwyLi Þl1
mi
dt Rmi ð2:223Þ
i
þ ð1Þ Fymi l2 lgy Fxmeshi ðFmwxLi FmwxRi Þlmb
Yaw motion:
X
2 X
2 X
4
€c ¼
Mc X ð1Þn1 Fxcn Fxdn ðFxtLj þ FxtRj Þ Frc ð2:225Þ
n¼1 n¼1 j¼1
Lateral motion:
€ v2 €
Mc Y c þ þ ðr0 þ Htw þ HBt þ HcB Þ/sec
Rc
X 2 X
4 ð2:226Þ
¼ ðFycn Fydn þ Fsn þ Fyctn Þ þ ðFytLj þ FytRj Þ þ Mc g/sec
n¼1 j¼1
2.3 Vehicle–Track Spatially Coupled Dynamics Model 101
Bounce motion:
2
X2 X4
€ v /
Mc Z€c a0 / ¼ Mc g ðFzdn Fzcn Þ ðFztLj þ FztRj Þ
sec sec
Rc n¼1 j¼1
ð2:227Þ
Roll motion:
X
4 X
4
Icx /€ þ/
€ ¼ HcB ðF ytLj þ FytRj Þ d s ðFztLj FztRj Þ
c sec
j¼1 j¼1
ð2:228Þ
X
2
ðHcg Fycn Hcq Fydn þ Hcld Fyctn þ Hs Fsn Þ
n¼1
where Hcg represents the height of the coupler from the car body center; Hcld
denotes the height of installation location of the lateral damper on car body from the
car body center; Hs represents the height of the secondary lateral stopper from the
car body center.
Pitch motion:
2 h
X i
€ ¼ lsb
Icy b ð1Þn1 FztLn þ FztRn þ FztLðn þ 2Þ þ FztRðn þ 2Þ
c
n¼1
X
2 X
4
þ lc FztLn þ FztRn FztLðn þ 2Þ FztRðn þ 2Þ HcB ðFxtLj þ FxtRj Þ
n¼1 j¼1
X
2 2 h
X i
þ Hcg ð1Þn1 Fxcn þ ð1Þn Hcq Fxdn þ ð1Þn1 lcg Fzcn þ ð1Þn1 lcq Fzdn
n¼1 n¼1
ð2:229Þ
where lcg denotes longitudinal distance between the coupler and the car body
center.
Yaw motion:
2 h
X i
Icz w€ þv d 1 ¼ lsb ð1Þn1 FytLn þ FytRn þ FytLðn þ 2Þ þ FytRðn þ 2Þ
c
dt Rc n¼1
X
2 X2
þ lc FytLn þ FytRn FytLðn þ 2Þ FytRðn þ 2Þ þ ð1Þn lcg Fycn
n¼1 n¼1
X
4 2 h
X i
þ ds ðFxtLj FxtRj Þ þ ð1Þn1 lcq Fydn þ ð1Þn1 lc Fsn þ ð1Þn1 lc Fyctn
j¼1 n¼1
ð2:230Þ
102 2 Vehicle–Track Coupled Dynamics Models
€ ¼ Tmi Trpi
Jri b ð2:231Þ
ri
where Trp represents the torque generated in the shaft connecting the rotor and the
pinion due to their relative rotational displacements. It can be calculated as
€ ¼ Trpi Fmi Rp
Jri b ð2:233Þ
pi
€ ¼ Tgwi Fmi Rg
Jgi b ð2:234Þ
gi
where
@ 4 Zr ðx; tÞ @ 2 Zr ðx; tÞ XN X4
EIY þ mr ¼ FVi ðtÞdðx x i Þ þ Pj dðx xwj Þ
@x4 @t2 i¼1 j¼1
ð2:236Þ
@ 4 Yr ðx; tÞ @ 2 Yr ðx; tÞ XN X4
EIZ þ mr ¼ FLi ðtÞdðx xi Þ þ Qj dðx xwj Þ
@x 4 @t 2
i¼1 j¼1
ð2:237Þ
2.3 Vehicle–Track Spatially Coupled Dynamics Model 103
@ 2 /r ðx; tÞ @ 2 /r ðx; tÞ XN X4
qr I0 Gr It ¼ Msi dðx xi Þ þ Mwj dðx xwj Þ
@t 2 @x 2
i¼1 j¼1
ð2:238Þ
in Eqs. (2.236)–(2.238), EIy and EIz are the rail bending stiffness to the y-axle and
to the z-axle, respectively; I0 is the torsional inertia of the rail, Gr is the rail shear
modulus; Zr(x, t), Yr(x, t), and Ur(x, t) are the vertical, lateral, and torsional dis-
placements of the rail, respectively; FVi(t) and FLi (t) are the vertical and lateral
dynamic forces at the ith rail-supporting point; mr is the rail mass per unit length, q
is the rail density; Pj(t) and Qj(t) are the vertical and lateral forces of the jth wheel
acting on the rail; Msi(t) and Mwj(t) are the moments acting on the rails due to the
forces FVi(t) and FLi (t) and due to the forces Pj(t) and Qj(t), respectively; d(x) is the
Dirac delta function; xsi is the coordinate of the ith sleeper, xwj is the coordinate of
the jth wheel, and N is the number of sleepers under the rail.
To solve the fourth-order partial differential equations of rails with time-stepping
integration methods, it is necessary to transform Eqs. (2.236)–(2.238) into a series
of second-order ordinary differential equations in terms of the generalized coordi-
nates. This could be done by means of the Ritz’s method and results are given as
follows:
8
> PN P 4
>
> €
q ðtÞ þ EIY kp 4
q ðtÞ ¼ F Z ð x Þ þ Pj Zk ðxwj Þ ðk ¼ 1NV Þ
>
>
Vk m r l Vk Vi k si
>
> i¼1 j¼1
< 4 PN P 4
€qLk ðtÞ þ EI Z kp
qLk ðtÞ ¼ FLi Yk ðxsi Þ þ Qj Yk ðxwj Þ ðk ¼ 1NL Þ
>
>
mr l
>
>
i¼1 j¼1
>
> 2 PN P 4
>
: €qTk ðtÞ þ G r It kp
qTk ðtÞ ¼ Msi Hk ðxsi Þ þ Mwj Hk ðxwj Þ ðk ¼ 1NT Þ
q I0 l
r
i¼1 j¼1
ð2:239Þ
where qzk(t), qyk(t), and qtk(t) are the kth vertical, lateral, and torsional mode time
coordinates, respectively; l is the calculated length of the rail; and Zk, Yk, and Uk are
the rail vertical, lateral, and torsional mode functions, described as
8 qffiffiffiffiffi
>
> Zk ðxÞ ¼ m2r l sin kpx
>
< qffiffiffiffiffi
l
Yk ðxÞ ¼ mr l sin l
2 kpx
ð2:240Þ
>
> qffiffiffiffiffiffiffi
>
: H ðxÞ ¼ 2 kpx
k q I0 l sin l
r
Figure 2.50 illustrates the free-body force diagram of the right rail, where Or is
the twisting center of the rail, e is the distance from the contact point to the central
104 2 Vehicle–Track Coupled Dynamics Models
line of the rail, hr is the height from the contact point to the central of rail torsion,
a is the height from the rail bottom to the central of rail torsion, b is half distance of
two supporting forces under the rail bottom, FV1i and FV2i are the vertical forces
between the rail and the sleeper.
FV1i and FV2i can be expressed as
8 h i
< FV1i ¼ 1 Kpv ½Zr b/r Zs ðd bÞ/s þ 1 Cpv Z_ r b/_ r Z_ s ðd bÞ/_ s
2 2
h i
: FV2i ¼ 1 Kpv ½Zr þ b/ Zs ðd þ bÞ/ þ 1 Cpv Z_ r þ b/_ Z_ s ðd þ bÞ/_
2 r s 2 r s
ð2:241Þ
where Zr and /r are the vertical and torsional displacements of the rail; Zs and /s are
the vertical and torsional displacements of the sleeper.
FVi ¼ FV1i þ FV2i
ð2:242Þ
FLi ¼ Kph ðYr Ys a/r Þ þ Cph ðY_ r Y_ s a/_ r Þ
The rail vertical, lateral, and torsional displacements at the time t can then be
expressed as
2.3 Vehicle–Track Spatially Coupled Dynamics Model 105
8
>
> PNV
>
> Zr ðx; tÞ ¼ Zk ð xÞqVk ðtÞ
>
>
>
<
k¼1
PNL
Yr ðx; tÞ ¼ Yk ð xÞqLk ðtÞ ð2:244Þ
>
>
>
>
k¼1
>
> P
NT
>
: /r ðx; tÞ ¼ Hk ð xÞqTk ðtÞ
k¼1
where NV, NL and NT are the total mode numbers of the vertical, lateral and
torsional mode functions of the rail selected in the calculation.
(2) Differential equations of rail modeled as Timoshenko beam
When the rail is modeled as a Timoshenko beam, the differential equations of the
vertical motion are the same as shown in Eqs. (2.69) and (2.70), while the differ-
ential equations of the lateral motion are similar to those of the vertical motion, and
the differential equation of the torsional motion is the same of that as shown in
Eq. (2.238).
The fourth-order partial differential equations of the Timoshenko beam rail can
be transformed into a series of second-order ordinary differential equations in terms
of the generalized coordinates by using the Ritz’s method, which are given as
follows:
Vertical motion (k = 1–NV)
8 qffiffiffiffiffiffiffiffiffiffi
> PN P4
< €qVk ðtÞ þ jz Ar Gr m1 kpl 2 qVk ðtÞ jz Ar Gr kpl m q1 I wVk ðtÞ ¼ FVi Zk ðxsi Þ þ Pj Zk ðxwj Þ
r r r Y
> h 2 i q ffiffiffiffiffiffiffiffiffi
ffi i¼1 j¼1
:w € Vk ðtÞ þ jzqArIGy r þ qE kpl wVk ðtÞ jz Ar Gr kpl mr q1 IY qVk ðtÞ ¼ 0
r r r
ð2:245Þ
> h 2 i q ffiffiffiffiffiffiffiffiffi
ffi i¼1 j¼1
:w jAG
€ Lk ðtÞ þ yq IrZ r þ qE kpl wLk ðtÞ jy Ar Gr kpl mr q1 IZ qLk ðtÞ ¼ 0
r r r
ð2:246Þ
where jz and jy are the vertical and lateral shear parameters for the rail cross
section.
106 2 Vehicle–Track Coupled Dynamics Models
Based on the modal superposition principle, the vertical and lateral displace-
ments (Zr, Yr) and rotation displacements (wzr, wyr) of the rail can be written as
Vertical
8
> PNV
>
< Zr ðx; tÞ ¼ Zk ðxÞqVk ðtÞ
k¼1
ð2:248Þ
>
> P
NV
: wzr ðx; tÞ ¼ Wzk ðxÞwVk ðtÞ
k¼1
Lateral
8
> P
NL
>
< Yr ðx; tÞ ¼ Yk ðxÞqLk ðtÞ
k¼1
ð2:249Þ
>
> P
NL
: wyr ðx; tÞ ¼ Wyk ðxÞwLk ðtÞ
k¼1
Torsional
X
NT
/r ðx; tÞ ¼ Hk ðxÞqTk ðtÞ ð2:250Þ
k¼1
where the normalized shape functions of a simply supported beam are given by
Vertical
8 qffiffiffiffiffi
< Zk ðxÞ ¼ 2 kp
m l sinð l xÞ
qrffiffiffiffiffiffiffi ð2:251Þ
: Wzk ðxÞ ¼ 2 kp
q IY l cosð l xÞ
r
Lateral
8 qffiffiffiffiffi
< Yk ðxÞ ¼ 2 kp
m l sinð l xÞ
qrffiffiffiffiffiffiffi ð2:252Þ
: Wyk ðxÞ ¼ 2 kp
q IZ l cosð l xÞ
r
Torsional
sffiffiffiffiffiffiffiffiffi
2 kp
Hk ðxÞ ¼ sinð xÞ ð2:253Þ
qr I0 l l
FLrVi FRrVi
X
FLsLi FRsLi
Z
FLsVi 2d FRsVi
i and the left and right rails; FLrLi and FRrLi are the lateral forces between the sleeper
i and the left and right rails; MLri and MRri are the moments acting on the sleeper
i from the left and right rails; FLsVi and FRsVi are the vertical supporting forces due
to the equivalent ballast bodies; FLsLi and FRsLi are the lateral forces acting on the
sleeper from the ballast; d is half length of the sleeper.
The equations of vertical, lateral and rotational motions of the sleeper i read
Vertical motion
Lateral motion
Rotational motion
MLri ¼ bðFLrV1i FLrV2i Þ
ð2:258Þ
MRri ¼ bðFRrV1i FRrV2i Þ
FLsLi ¼ Kbh Ys þ Cbh Y_ s
ð2:259Þ
FRsLi ¼ Kbh Ys þ Cbh Y_ s
108 2 Vehicle–Track Coupled Dynamics Models
X
FLb1i FRb1i
FLbRi i F RbL i
FLb2 i Y FRb2i
Z
Z bLi Z bRi
FLbf i FRbf i
Fig. 2.52 Force diagram of equivalent ballast bodies under a sleeper: a left side, and b right side
In Eq. (2.257), ZLb and ZRb are the vertical displacements of the left and right
ballast blocks.
(4) Equations of motion of the ballast
The ballast bed is modeled as equivalent rigid bodies as shown in Fig. 2.52, where
FRb1i, FRb2, FLb1i, FLb2i, FLbRi, and FRbLi are the vertical shear forces between the
neighboring ballast bodies, FRbfi and FLbfi are the vertical forces between the ballast
bodies and the subgrade. The interactive influence of the left and right equivalent
ballast bodies are taken into account. Only the vertical motion of each ballast body
is considered.
The vertical motion equations of the ballast body i read
Left:
Right:
FLb1i ¼ Kw ZLbi ZLbði þ 1Þ þ Cw Z_ Lbi Z_ Lbði þ 1Þ ð2:264Þ
FLb2i ¼ Kw ZLbi ZLbði1Þ þ Cw Z_ Lbi Z_ Lbði1Þ ð2:265Þ
FLbRi ¼ Kw ðZLbi ZRbi Þ þ Cw Z_ Lbi Z_ Rbi ð2:266Þ
FLrLi FRrLi
FLsLi FRsLi
FLsVi FRsVi
Fig. 2.53 Free-body diagram of concrete supporting blocks: a left side, and b right side
where
E h3
Ds ¼ s s ð2:277Þ
12 1 m2s
2.3 Vehicle–Track Spatially Coupled Dynamics Model 111
ð2:279Þ
ZZ
qs hs 2
T¼ x W 2 ðx; yÞdxdy ð2:280Þ
2
S
Furthermore, the plate modal function can be can be further equivalent to the
product of the Euler–Bernoulli beam functions along the length and width of the
plate, respectively.
where the Amn is the modal coefficient; Xm(x) and Yn(y) are Euler–Bernoulli beam
functions along the length and width directions, respectively, which are given
by [92]
8
< X1 ð xÞ ¼ 1pffiffiffi
>
X2 ð xÞ ¼ 3 1 2x
>
:
Ls
Xm ð xÞ ¼ coshðam xÞ þ cosðam xÞ bm ½sinhðam xÞ þ sinðam xÞ ð m 3Þ
ð2:282aÞ
8
< Y1 ð yÞ ¼ 1pffiffiffi
>
Y2 ð yÞ ¼ 3 1 W2ys ð2:282bÞ
>
:
Yn ð yÞ ¼ coshðen yÞ þ cosðen yÞ nn ½sinhðen yÞ þ sinðen yÞ ðn 3Þ
where Ls and Ws are the length and width of the plate, respectively; am and en are
the frequency coefficients corresponding to Xm(x) and Yn(y), which can be calcu-
lated by
112 2 Vehicle–Track Coupled Dynamics Models
(
a3 ¼ 4:73004
Ls
ð2:283aÞ
am ¼ 2m3
2Ls p m4
(
e3 ¼ 4:73004
Ws
ð2:283bÞ
en ¼ 2n3
2Ws p n4
bm and nn are the mode coefficients corresponding to Xm(x) and Yn(y), which can be
calculated by
(
b3 ¼ 0:982502
ðam Ls Þcosðam Ls Þ ð2:284aÞ
bm ¼ cosh
sinhðam Ls Þsinðam Ls Þ m4
n3 ¼ 0:982502
ðen Ws Þcosðen Ws Þ ð2:284bÞ
nn ¼ cosh
sinhðen Ws Þsinðen Ws Þ n4
where d denotes the variation symbol; T and U denote the kinetic and potential
energy for a dynamic system, respectively; t1–t2 is an arbitrary integration time
period.
By substituting Eqs. (2.279)–(2.280) into Eq. (2.285), homogeneous lin-
ear equations with respect to the modal coefficient Amn can be arranged in the form
qs hs 2 Ds
xmn B1m B1n ½B2m B1n þ B1m B2n þ 2ðms B3m B3n þ ð1 ms ÞB4m B4n Þ Amn ¼ 0
2 2
ð2:286Þ
where B1m, B1n, B2m, B2n, B3m, B3n, B4m, B4n are the integral constants with respect
to the Euler–Bernoulli beam functions, which take the form
Z Ls Z Ws Z Ls
B1m ¼ Xm2 ð xÞdx B1n ¼ Yn2 ð yÞdy B2m ¼ Xm002 ð xÞdx
0 0 0
Z Ws Z Ls Z Ws
B2n ¼ Yn002 ð yÞdy B3m ¼ Xm00 ð xÞXm ð xÞdx B3n ¼ Yn00 ð yÞYn ð yÞdy
0 0 0
Z Ls Z Ws
B4m ¼ Xm02 ð xÞdx B4n ¼ Yn02 ð yÞdy
0 0
ð2:287Þ
2.3 Vehicle–Track Spatially Coupled Dynamics Model 113
Based on Eq. (2.286), the natural frequency of the thin plate is given by
sffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ds B2m B1n þ B1m B2n þ 2½ms B3m B3n þ ð1 ms ÞB4m B4n
xmn ¼ ð2:288Þ
qs hs B1m B1n
After obtaining the natural frequency of the thin plate, the classical modal
damping can be introduced into the vibration equation of the thin plate, and the
forced vibration of the damped thin plate (q 6¼ 0) can be deduced as follows.
According to Ritz method, the solutions of the dynamic vertical displacements
can be separately expanded as
X
Nx X
Ny Nx X
X Ny
wðx; y; tÞ ¼ Wmn ðx; yÞTmn ðtÞ ¼ Xm ð xÞYn ð yÞTmn ðtÞ ð2:289Þ
m¼1 n¼1 m¼1 n¼1
Nx X
X Ny
T€mn ðtÞ þ x2mn Tmn ðtÞ qs hs Wmn ðx; yÞ qðx; y; tÞ ¼ 0 ð2:291Þ
m¼1 n¼1
Furthermore, by taking the classical modal damping into account and combining
with the actual force distribution of the track slab as shown in Fig. 2.54, the forced
vibration equations of the damped concrete slab according to the thin plate theory
can be recast as
114 2 Vehicle–Track Coupled Dynamics Models
hs
PrVi
PrLi
X Ls
O
Y
FsLj
Ws FsVj
hP P b i
Np
i¼1 PrVi ðtÞXm xpi Yn ypi Nj¼1 FsVj ðtÞXm xbj Yn ybj
T€mn ðtÞ þ 2fmn xmn T_ mn ðtÞ þ x2mn Tmn ðtÞ ¼
qs hs B1m B1n
ð2:294Þ
X
Np X
Nl
qs Ls Ws hs€ys ¼ PrLi 2 FsLj ð2:295Þ
i¼1 j¼1
where PrLi is the lateral force at the ith rail fastener, FsLj is the lateral dynamic force
at the jth slab supporting point; Nl is the the total node numbers of the slab.
The equation of torsional motion reads
X
Np X
Nl
€ ¼
Jsz / PrLi dpi 2 FsLj dbj ð2:296Þ
s
i¼1 j¼1
where dpi is the longitudinal distance of the ith fastener to the slab center, dbj is the
longitudinal distance of the jth supporting point to the slab center, Jsz is the moment
of inertia of the sleeper in roll, which has a value of Jsz = qsLshsW3s /12.
2.3 Vehicle–Track Spatially Coupled Dynamics Model 115
where E is the modulus of elasticity, G is the shear modulus, I is the area moment of
inertia of cross section, q is the density, A is the cross-sectional area, and jr is the
shear factor.
After some algebraic manipulations, the bending slope w or the deflection w can
be eliminated from Eqs. (2.297a) and (2.297b), respectively.
@4w @2w EIq @ 4 w Iq2 @ 4 w
EI 4 þ qA 2 Iq þ þ ¼0 ð2:298aÞ
@x @t Gjr @x2 @t2 Gjr @t4
@4w @2w EIq @ 4 w Iq2 @ 4 w
EI 4 þ qA 2 Iq þ þ ¼0 ð2:298bÞ
@x @t Gjr @x2 @t2 Gjr @t4
Let
pffiffiffiffiffiffiffi
where X is the mode function of w, W is the mode function of w, i is 1; x is the
angular eigenfrequency of the beam, L is the length of the beam, and n is the
nondimensional coordinate.
Substituting the Eq. (2.299) into Eq. (2.298a, 2.298b), the solutions of the mode
functions X(n) and W(n) can be obtained as [105]
baA1 baA2
W¼ sinh ban þ cosh ban
L½1 b s ða þ r Þ
2 2 2 2 L½1 b s2 ða2 þ r 2 Þ
2
ð2:301Þ
bbA3 bbA4
2 sin bbn þ cos bbn
L 1þb s b r
2 2 2 L 1 þ b s2 b2 r 2
2
where
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a 1 4
¼ pffiffiffi ðr 2 þ s2 Þ þ ðr 2 s2 Þ þ 2
b 2 b ð2:302Þ
I EI qAL4 2
r 2 ¼ 2 s2 ¼ b 2
¼ x
AL Gjr AL2 EI
W 0 ð 0Þ ¼ 0 W 0 ð 1Þ ¼ 0
X 0 ð0Þ X 0 ð 1Þ ð2:303Þ
W ð 0Þ ¼ 0 W ð 1Þ ¼ 0
L L
b h 2 i
2 2 cosh ba cos bb þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 r 2 r 2 s2 þ 3r 2 s2 sinh ba sin bb
1 b2 r 2 s 2
¼0
ð2:304Þ
The roots of the above transcendental equation indicate the values of a series of
frequency parameters b1, b2, …bk, …, and the corresponding mode functions can be
determined accordingly.
1
X ¼ cosh ban þ kd sinh ban þ cos bbn þ d sin bbn ð2:305Þ
f
2.3 Vehicle–Track Spatially Coupled Dynamics Model 117
ba bakd
W¼ sinh ban þ cosh ban
L½1 b2 s2 ða2 þ r 2 Þ L½1 b2 s2 ða2 þ r 2 Þ
bb bbd
sin bbn þ cos bbn ð2:306Þ
fL 1 þ b2 s2 b2 r 2 L 1 þ b2 s2 b2 r 2
k2
¼ k1 sinh ban þ k1 kd cosh ban sin bbn þ k2 d cos bbn
f
in which
a a2 þ r 2 cosh ba cos bb
k¼ f¼ d¼ ð2:307Þ
b a2 þ s 2 k sinh ba f sin bb
It should be noted that the free–free beam modes given by Eqs. (2.305) and
(2.306) do not constitute the complete expressions of the mode functions of the
Timoshenko beams having a pair of opposite edges free. The above bending modes
provide the third and higher trail functions and can be supplemented by the first
two-order rigid mode functions representing the translation and rotation degrees of
freedom.
8
< X1 ¼ 1pffiffiffi
X2 ¼ 3ð1 2nÞ
: X ¼ cosh b an þ kd sinh b an þ 1 cos b bn þ d sin b bn k3
k k2 k2 f k2 k2
ð2:308Þ
8
< Wx1 ¼ 0
Wx2 ¼ 1
: W ¼ k sinh b an þ k kd cosh b an k2 sin b bn þ k d cos b bn k3
xk 1 k2 1 k2 f k2 2 k2
ð2:309Þ
Now consider the vibration equations of Mindlin plate. Different from the
classical thin plate, the inclusion of shear effects in Mindlin plate theory means that
the cross-sectional rotations bx, by can no longer be expressed solely in terms of the
deflection w of the plate median surface, thus, three independent quantities namely
w, bx, and by are introduced to represent the deformations of the plate. The three
differential equations that govern the vibration of the undamped Mindlin plate can
be expressed as [104]
@ 2 w @ 2 w @bx @by @2w
Gjr hs þ 2 þ þ qs h s þ qðx; y; tÞ ¼ 0 ð2:310aÞ
@x2 @y @x @y @t2
118 2 Vehicle–Track Coupled Dynamics Models
2
2
Ds @ bx @ 2 by @ bx @ 2 by @w @2b
ð1 m s Þ þ þ ð1 þ m s Þ þ Gjr hs bx þ qs J 2x ¼ 0
2 @x 2 @y 2 @x 2 @x@y @x @t
ð2:310bÞ
2
2
Ds @ by @ 2 by @ by @ 2 bx @w @ 2 by
ð1 m s Þ þ þ ð 1 þ m s Þ þ Gjr h s b þ q J ¼0
2 @x2 @y2 @y2 @x@y y
@y s
@t2
ð2:310cÞ
where w is the vertical displacement of the plate; bx and by are the cross-sectional
rotations in x- and y-directions, respectively; J is the plate rotary inertia per unit
length with the value of J = h3s /12; and other parameters can be referred to the
previous section.
Analogously, free vibration of the plate (q = 0) is studied first to obtain the plate
natural frequencies. The three independent quantities, namely, w, bx, and by, take
the following forms by introducing the separation variable method:
where W(x, y) is the dynamic deflection function of the plate; Ux(x, y) and
Uy(x, y) are the dynamic rotation functions; and x is the angular eigenfrequency of
the plate.
Based on Eq. (2.311a–2.311c), the strain energy U and the kinetic energy T of
the plate can be expressed as [106]
ZZ "
#
Ds @Ux 2 @Uy 2 @Ux @Uy 1 ms @Ux @Uy 2
U¼ þ þ 2ms þ þ dxdy
2 @x @y @x @y 2 @y @x
ZZ "
2 #
S
2
Ghs @W @W
þ þ Ux þ þ Uy dxdy
2jr @x @y
S
ð2:312aÞ
ZZ ZZ
q s hs 2 qs J 2
T¼ x W 2 dxdy þ x U2x þ U2y dxdy ð2:312bÞ
2 2
S S
Moreover, the modal functions of the Mindlin plate corresponding to the mnth
natural frequency xmn can be further equivalent to the product of the Timoshenko
beam functions along the length and width of the plate, respectively.
2.3 Vehicle–Track Spatially Coupled Dynamics Model 119
where the Amn, Bmn, Cmn are the modal coefficients representing the arbitrary ampli-
tudes of deflection and rotation, the four unidirectional functions Xm(x), Yn(y), Wxm(x),
and Wyn(y) are those Timoshenko beam functions derived in Eqs. (2.263)–(2.264),
which are appropriate to the vibration of thick plates with free boundary conditions.
The free vibration of the thick plates satisfies the Hamilton principle of conserva-
tive systems. Substitution of Eqs. (2.312a, 2.312b) and (2.313a–2.313c) into
Eq. (2.285) yields homogeneous linear equations with respect to the modal coefficients
Amn, Bmn, Cmn that, after some algebraic manipulation, can be arranged in the form
2
Gjr hs ðB3m B1n þ B1m B3n Þ qs hs x2mn B1m B1n Gjr hs B5m B1n
6
6 Gjr hs B5m B1n Ds B4m B1n þ Gjr hs B2m B1n þ Ds ð1ms Þ
B2m B3n qs Jx2mn B2m B1n
4 2
Ds ð1ms Þ
Gjr hs B1m B5n Ds ms B6m B6n þ 2 B5m B5n
32 3
Gjr hs B1m B5n Amn
Ds ms B6m B6n þ Ds ð1m sÞ 76 7
2 B5m B5n 54 Bmn 5 ¼ 0
Ds ð1ms Þ
Ds B1m B4n þ Gjr hs B1m B2n þ 2 B3m B2n qs Jx2mn B1m B2n Cmn
ð2:314Þ
where B1m, B1n, B2m, B2n, B3m, B3n, B4m, B4n, B5m, B5n, B6m, and B6n are the integral
constants with respect to the Timoshenko beam functions, which take the form
Z Ls Z Ws Z Ls
B1m ¼ Xm2 ð xÞdx B1n ¼ Yn2 ð yÞdy B2m ¼ W2xm ð xÞdx
0 0 0
Z Ws Z Ls Z Ws
B2n ¼ W2yn ð yÞdy B3m ¼ Xm02 ð xÞdx B3n ¼ Yn02 ð yÞdy
0 0 0
Z Ls Z Ws Z Ls
B4m ¼ W02
xm ð xÞdx B4n ¼ W02
yn ð yÞdy B5m Wxm ð xÞXm0 ð xÞdx
¼
0 0 0
Z Ws Z Ls Z Ws
B5n ¼ Wyn ð yÞYn0 ð yÞdy B6m ¼ W0xm ð xÞXm ð xÞdx B6n ¼ W0yn ð yÞYn ð yÞdy
0 0 0
ð2:315Þ
In order to obtain the nonzero solutions of the modal coefficients, the coefficient
determinant of Eq. (2.314) must be zero
Dmnð11Þ Dmnð12Þ Dmnð13Þ
Dmnð21Þ Dmnð22Þ Dmnð23Þ ¼ 0 ð2:316Þ
Dmnð31Þ Dmnð32Þ Dmnð33Þ
120 2 Vehicle–Track Coupled Dynamics Models
Equation (2.316) is an univariate cubic equation related to the x2, for any set of
(m, n), three sequent frequencies can be solved x(k)
mn (k = 1, 2, 3, m = 1, 2, 3, …,
n = 1, 2, 3, …), namely low, medium, and high ones in which the low one rep-
resents the flexural frequency and the other two represent thickness–shear
frequencies.
Once a series of frequencies are obtained, the proportional relationship between
the modal coefficients can be determined by substituting them back to Eq. (2.314)
Dmnð21Þ Dmnð23Þ Dmnð21Þ Dmnð23Þ
Dmnð31Þ Dmnð33Þ Dmnð31Þ Dmnð33Þ
Bmn ¼ Amn Cmn ¼ Amn ð2:317Þ
Dmnð22Þ Dmnð23Þ Dmnð23Þ Dmnð22Þ
Dmnð32Þ Dmnð33Þ Dmnð33Þ Dmnð32Þ
X
Nx X
Ny X
3 Nx X
X Ny X
3
ðkÞ ðkÞ
wðx; y; tÞ ¼ Wmn ðx; yÞTmn ðt Þ ¼ Aðmn
kÞ ðkÞ
Xm ð xÞYn ð yÞTmn ðt Þ
m¼1 n¼1 k¼1 m¼1 n¼1 k¼1
ð2:318aÞ
Nx X
X Ny X
3 Nx X
X Ny X
3
bx ðx; y; tÞ ¼ Uðxmn
kÞ ðkÞ
ðx; yÞTmn ðtÞ ¼ Bðmn
kÞ ðkÞ
Wxm ð xÞYn ð yÞTmn ðt Þ
m¼1 n¼1 k¼1 m¼1 n¼1 k¼1
ð2:318bÞ
Nx X
X Ny X
3 Nx X
X Ny X
3
by ðx; y; tÞ ¼ Uðymn
kÞ ðkÞ
ðx; yÞTmn ðt Þ ¼ ðkÞ
Cmn ðkÞ
Xm ð xÞWyn ð yÞTmn ðt Þ
m¼1 n¼1 k¼1 m¼1 n¼1 k¼1
ð2:318cÞ
2
2
Ds @ Ux @ 2 Uy @ Ux @ 2 Uy @W
ð 1 ms Þ þ þ ð 1 þ ms Þ þ Gjr hs Ux þ
2 @x2 @y2 @x2 @x@y @x
¼ qs Jx2 Ux
ð2:319bÞ
2
2
Ds @ Uy @ 2 Uy @ Uy @ 2 Ux @W
ð 1 ms Þ þ þ ð 1 þ m s Þ þ Gj r h s U y þ
2 @x2 @y2 @y2 @x@y @y
¼ qs Jx Uy 2
ð2:319cÞ
X
Nx X 3 h
Ny X i
T€mn
ðkÞ
ðtÞ þ xðmn
kÞ2 ðkÞ ðkÞ
Tmn ðtÞ qs hs Wmn qðx; y; tÞ ¼ 0 ð2:320aÞ
m¼1 n¼1 k¼1
Nx X
X 3 h
Ny X i
T€mn
ðkÞ
ðtÞ þ xðmn
kÞ2 ðkÞ
Tmn ðtÞ qs JUðxmn
kÞ
¼0 ð2:320bÞ
m¼1 n¼1 k¼1
Nx X
X 3 h
Ny X i
T€mn
ðkÞ
ðtÞ þ xðmn
kÞ2 ðkÞ
Tmn ðtÞ qs JUðymn
kÞ
¼0 ð2:320cÞ
m¼1 n¼1 k¼1
the three modal coordinates can be decoupled into the following three independent
equations ultimately.
RR ðkÞ
S qðx; y; tÞW ðx; yÞdxdy
T€mn ðtÞ þ xmn Tmn ðtÞ ¼ RR h
ð kÞ ðk Þ2 ð kÞ mn
i k ¼ 1; 2; 3
ðkÞ2 ðkÞ2 ðkÞ2
S qs h s W mn þ q s J U xmn þ U ymn dxdy
ð2:322Þ
Furthermore, by taking the classical modal damping into account and combining
with the actual force distribution of the floating slab as shown in Fig. 2.55, the
forced vibration equations of the damped floating slab according to the Mindlin
plate theory can be recast as a series of second-order ordinary differential equations
122 2 Vehicle–Track Coupled Dynamics Models
hs
PrVi
FsVj
x
Ls
O
y
z Ws
Fig. 2.55 The floating slab subjected to the vertical forces from fasteners and steel springs
T€mn
ðkÞ kÞ _ ðkÞ
ðtÞ þ 2fmn xðmn Tmn ðtÞ þ xðmn
kÞ2 ðkÞ
Tmn ðtÞ
" #
Pp
N P Nb
PrVi ðtÞXm xpi Yn ypi FsVj ðtÞXm xbj Yn ybj ð2:323Þ
i¼1 j¼1
¼
ðkÞ2 ðkÞ2
qs hs B1m B1n þ qs J Bmn B2m B1n þ Cmn B1m B2n
where fmn denotes the damping ratio of the floating slab; xpi and ypi are the x-
coordinate and y-coordinate of the rail-supporting points; xbj and ybj are the x-
coordinate and y-coordinate of the steel springs; Np and Nb are the total number of
rail fasteners on the slab and the total number of steel springs under the slab,
respectively; PrVi is the vertical force at the ith rail fastener, FsVj is the vertical steel
spring force at the jth slab supporting point; and k = 1, 2, 3 denote the vertical
motion, torsional motion in x- and y-directions, respectively.
For the lateral motion and torsional motion of the floating slab in the x-y plane,
the equations of motion can be referred to Eqs. (2.295)–(2.296).
The wheel–rail coupling model is an essential element that couples the vehicle
subsystem with the track subsystem at the wheel–rail interfaces. Unlike the early
wheel–rail contact model used in the classical vehicle system dynamics, in which
the rails are assumed to be fixed without any movement, the proposed dynamic
wheel–rail coupling model in this book for analysis of the three-dimensional
vehicle–track coupled dynamics problems could consider three kinds of rail
motions in the vertical, lateral, and torsional directions.
2.3 Vehicle–Track Spatially Coupled Dynamics Model 123
(a) (b)
Left rail
Xw
O Ow
X
Yw
Right rail Y δL δR
Z XRc
X Lc
Ow φw Zw ORc
O Lc YRc
Xw
Yw ψw Z Lc
Zw YLc ZRc X Rr
XLr
YRr
YLr O Rr
OLr
Z Rr
Z Lr
Fig. 2.56 Definitions of coordinate systems: a absolute coordinate system and b wheelset and rail
coordinate systems (Reprinted from Ref. [36], Copyright 2009, with permission from Taylor &
Francis.)
8 9 2 38 9
< iRc = cos ww sin ww 0 <i=
jRc ¼ 4 cosðdR /w Þ sin ww cosðdR /w Þ cos ww sinðdR /w Þ 5 j
: ; : ;
kRc sinðdR /w Þ sin ww sinðdR /w Þ cos ww cosðdR /w Þ k
ð2:326Þ
8 9 2 38 9
< iLr = 1 0 0 <i=
jLr ¼ 40 cosð/rL þ /0 Þ sinð/rL þ /0 Þ 5 j ð2:327Þ
: ; : ;
kLr 0 sinð/rL þ /0 Þ cosð/rL þ /0 Þ k
8 9 2 38 9
< iRr = 1 0 0 <i=
jRr ¼ 4 0 cosð/rR /0 Þ sinð/rR /0 Þ 5 j ð2:328Þ
: ; : ;
kRr 0 sinð/rR /0 Þ cosð/rR /0 Þ k
where dL, dR are the left and right wheel–rail contact angles; /rL, /rR are the roll
angles of the left and right rails; /0 is standard rail cant.
2. Dynamic wheel–rail spatially coupling model
(1) Wheel–rail contact geometry calculation
Trace curve method is a common computation method of wheel–rail spatial contact
geometry. According to the trace curve method [107], the wheel–rail spatial contact
points are only on a spatial curve, which is named trace curve, and thus the contact
points could be searched out through seeking a trace curve. Therefore, it becomes a
one dimensional scanning problem.
The detailed procedure for solving the wheel–rail contact geometry relationship
is described as follows:
Step 1 Obtain the lateral displacement, yaw angle and roll angle of wheelset, as
well as the lateral displacements, vertical displacements and the torsion angles of
rails by solving the vehicle and track system equations with numerical integration
method.
Step 2 Based on the discrete datum of wheel and rail profiles and the instanta-
neous motions of the wheelsets, calculate the contact trace curve of the wheels in
3D spaces by using the trace curve method. Figure 2.57 schematically shows the
wheel–rail spatial contact geometry relationship, where OR is the right wheel–rail
contact point, B is the center of the rolling circle, A is the intersection point of the
common normal line of the contact surfaces and the wheel axle centerline, Ow is the
center of the wheelset, O′R, B′, A′, O′w are, respectively, the projections of OR, B, A,
Ow on the O-XY plane, O′wB′ is the projections of OwB on the O-XY plane, O″w is
the projection point of O′w on the OY axis.
From Fig. 2.57, the wheel–rail contact point OR in the absolute coordinate
system O-XYZ can be calculated as
2.3 Vehicle–Track Spatially Coupled Dynamics Model 125
8
< x ¼ xB þ lxRRw tand
> R
y ¼ yB 1lw2 ðl2x ly tandR þ lz mÞ ð2:329Þ
> x
: z ¼ zB Rw2 ðl2 lz tandR ly mÞ
1lx x
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where m ¼ 1 l2x ð1 þ tan2 dR Þ; lx, ly, lz can be calculated by
8
< lx ¼ cos /w sin ww
l ¼ cos /w cos ww ð2:330Þ
: y
lz ¼ sin /w
The coordinates of the rolling circle center B in the absolute coordinate system are
8
< x B ¼ dw l x
y ¼ d w l y þ Yw ð2:331Þ
: B
zB ¼ dw lz
where dw is the lateral distance of the rolling circle from wheelset center.
Thus, the series of the wheel rolling circles can be determined with the changing
of dw, and then the wheel–rail contact trace curve can be determined.
Step 3 Create the discrete datum of profiles of the wheels and rails by dealing
with the given profiles (measured or artificially generated) based on the spline
interpolation algorithm, here the rail deformation and the surface defects are
combined into the rail profile.
126 2 Vehicle–Track Coupled Dynamics Models
relationship, we have DZLwj0 = DZRwj0 = DZwj0. Therefore, the left and right
wheel–rail vertical relative displacements can be expressed as
dZLj ¼ Zwj ðtÞ ðDZLwjt DZwj0 Þ
ð2:332Þ
dZRj ¼ Zwj ðtÞ ðDZRwjt DZwj0 Þ
Obviously, when dZLcj or dZRcj is less than zero, which indicates that the wheel
at left or right side separates from the rail, the left or right side wheel–rail normal
contact force is then equal to zero.
When vehicles move on an irregular track, three types of wheel–rail contact
states as shown in Fig. 2.59 are possible. In some particular conditions, the
instantaneous loss of contact between one side wheel of a wheelset and the rail
would appear (Fig. 2.59b), and worse still, it might happen to both sides of the
wheelset under some extreme circumstances (Fig. 2.59c).
(3) Wheel–rail creep force model
The tangential wheel–rail creep forces are calculated first by use of Kalker’s linear
creep theory [108] and then modified by Shen–Hedrick–Elkins nonlinear model
[109, 110]. Unlike the classical wheel–rail contact model, in which the rails are
assumed to be fixed without any movement, the current model is capable of con-
sidering the rail motions in vertical, lateral and torsional directions when the creep
forces are calculated.
Figure 2.60 shows the schematic diagram of wheel–rail contact in coordinate
system ORc-XRcYRcZRc, where ORc is the wheel–rail contact point. Here, the right
side is taken as the example. The wheelset moves along the axis ORcXRc, which is
coincident with the axis ORcX. ORcYRc lies in the wheel–rail contact plane, ORcZRc
indicates the normal direction of the wheel–rail contact plane. The intersection
Fig. 2.59 Wheel–rail contact states: a constant contact, b instantaneous loss of contact at one
side, and c loss of contact at two sides
128 2 Vehicle–Track Coupled Dynamics Models
X, XRc
Y Rc
O Rc
Y
0
+φ
rR
−φ
δR
Z ZRc
angle between the axis ORcYRc and the axis ORcY is dL + /rL + /0 for the left
wheel–rail contact point, and dR-/rR + /0 for the right wheel–rail contact point.
In wheel–rail contact spot coordinate ORc-XRcYRcZRc, the longitudinal, lateral,
and spin creepages are defined as
8 v v
< nx ¼ w1 v r1
ny ¼ vw2 v r2
ð2:334Þ
: v
Xw3 Xr3
n/ ¼ v
where v is the wheelset speed moving on rails, vw1, vw2 and Xw3 are the speed of
contact ellipse on the wheel along ORcXRc axis, ORcYRc axis and around ORcZRc
axis, respectively. vr1, vr2 and Xr3 are the speed of contact ellipse on the rail along
ORcXRc axis, ORcYRc axis and around ORcZRc axis, respectively.
(1) Calculation of longitudinal and lateral creepages
The translational motion speed of wheelset center in the absolute coordinate system is
vow ¼ X_ ow i þ Y_ ow j þ Z_ ow k ð2:335Þ
2.3 Vehicle–Track Spatially Coupled Dynamics Model 129
The absolute angle speed of the wheelset in the absolute coordinate system is
expressed as
2 3T 8 9
/_ w cos ww ðX þ b_ w Þ cos /w sin ww > <i> =
6 _ 7
xw ¼ 4 /w sin ww þ ðX þ b_ w Þ cos /w cos ww 5 j
: >
> ;
ðX þ b_ w Þ sin /w þ w_ w k
2 3T 8 9 ð2:336Þ
xwx > <i> =
6 7
¼ 4 xwy 5 j
: >
> ;
xwz k
The vectors from the left and right contact points to the wheelset center RL, RR
can be described as
2 3T 8 9
R0xL < iw >
> =
6 R0 7
RL ¼ 4 yL 5 jw
: >
> ;
R0zL kw
2 0 3 8 9
RxL cos ww R0yL cos /w sin ww þ R0zL sin /w sin ww T ><i> =
6 7
¼ 4 R0xL sin ww þ R0yL cos /w cos ww R0zL sin /w cos ww 5 j ð2:337Þ
: >
> ;
R0yL sin /w þ R0zL cos /w k
2 3T 8 9
RxL > <i> =
6 7
¼ 4 RyL 5 j
: >
> ;
RzL k
2 3T 8 9
R0xR < iw >
> =
6 R0 7
RR ¼ 4 yR 5 jw
: >
> ;
R0zR kw
2 0 3 8 9
RxR cos ww R0yR cos /w sin ww þ R0zR sin /w sin ww T >
<i> =
6 R0 sin w þ R0 cos / cos w R0 sin / cos w 7 ð2:338Þ
¼ 4 xR w w w w w5 j
yR zR
: >
> ;
R0yR sin /w þ R0zR cos /w k
2 3T 8 9
RxR > <i> =
6 7
¼ 4 RyR 5 j
: >
> ;
RzR k
130 2 Vehicle–Track Coupled Dynamics Models
The relative speeds of the contact points with respect to the wheelset center read
vRðL;RÞ ¼ xw RðL;RÞ
i j k
¼ xwx xwy xwz
RxðL;RÞ RyðL;RÞ RzðL;RÞ ð2:339Þ
2 3 8 9
xwy RzðL;RÞ xwz RyðL;RÞ T ><i> =
6 7
¼ 4 xwz RxðL;RÞ xwx RzðL;RÞ 5 j
: >
> ;
xwx RyðL;RÞ xwy RxðL;RÞ k
According to the speed synthesis theorem, the absolute speeds of the wheelset
and the rails at the contact points are obtained:
2 3T 8 9
X_ ow þ xwy RzðL;RÞ xwz RyðL;RÞ < i =
6 7
vwðL;RÞ ¼ 4 Y_ ow þ xwz RxðL;RÞ xwx RzðL;RÞ 5 j ð2:340Þ
: ;
Z_ ow þ xwx RyðL;RÞ xwy RxðL;RÞ k
2 3T 8 9 2 3T 8 9
0 <i= 0 <i=
vrðL;RÞ ¼ 4 dY_ rðL;RÞ 5 j þ 4 Y_ rðL;RÞ þ hr /_ rðL;RÞ 5 j ð2:341Þ
: ; : ;
dZ_ rðL;RÞ k Z_ rðL;RÞ k
where Y_ rðL;RÞ ; Z_ rðL;RÞ ,/_ rðL;RÞ are the lateral, vertical, and rotational speeds of rails,
dY_ rðL;RÞ ; dZ_ rðL;RÞ are the lateral and vertical relative speeds induced by track
irregularities.
Thus, the differences of absolute speeds of the wheelset and rails at the contact
points are calculated as
2 3T 8 9
X_ ow þ xwy RzðL;RÞ xwz RyðL;RÞ >i>
6 7 < =
Dv0ðL;RÞ ¼6 _Yow þ xwz RxðL;RÞ xwx RzðL;RÞ Y_ rðL;RÞ dY_ rðL;RÞ hr /_ rðL;RÞ 7
4 5 >j>
: ;
Z_ ow þ xwx RyðL;RÞ xwy RxðL;RÞ Z_ rðL;RÞ dZ_ rðL;RÞ k
2 3T 8 9
DvxðL;RÞ > <i> =
6 Dv 7
¼ 4 yðL;RÞ 5 j
: >
> ;
DvzðL;RÞ k
ð2:342Þ
Because the creepage is defined in the wheel–rail contact spot coordinate system,
the absolute speed differences should be transformed into the contact spot coordi-
nate system. The transformation between the wheel–rail contact spot coordinate and
the absolute coordinate complies with
2.3 Vehicle–Track Spatially Coupled Dynamics Model 131
8 9 2 31 8 9
<i= cos ww sin ww 0 < iLc =
j ¼ 4 cosðdL þ /w Þ sin ww cosðdL þ /w Þ cos ww sinðdL þ /w Þ 5 j
: ; : Lc ;
k sinðdL þ /w Þ sin ww sinðdL þ /w Þ cos ww cosðdL þ /w Þ kLc
ð2:343Þ
8 9 2 31 8 9
<i= cos ww sin ww 0 < iRc =
j ¼ 4 cosðdR /w Þ sin ww cosðdR /w Þ cos ww sinðdR /w Þ 5 j
: ; : Rc ;
k sinðdR /w Þ sin ww sinðdR /w Þ cos ww cosðdR /w Þ kRc
ð2:344Þ
Thus, the relative speed differences of the wheelset and rails at the contact points
in the wheel–rail contact spot coordinate system are calculated as
2 3T 2 31 8 9
DvxL cos ww sin ww 0 < iLc >
> =
6 7 6 7
DvL ¼ 4 DvyL 5 4 cosðdL þ /w Þ sin ww cosðdL þ /w Þ cos ww sinðdL þ /w Þ 5 jLc
>
: >
;
DvzL sinðdL þ /w Þ sin ww sinðdL þ /w Þ cos ww cosðdL þ /w Þ kLc
2 3T 8 9
Dv1L > < iLc >
=
6 7
¼ 4 Dv2L 5 jLc
>
: >
;
Dv3L kLc
ð2:345Þ
2 3T 2 31 8 9
DvxR cos ww sin ww 0 >
< iRc >
=
6 7 6 7
DvR ¼ 4 DvyR 5 4 cosðdR /w Þ sin ww cosðdR /w Þ cos w sinðdR /w Þ 5 jRc
>
: >
;
DvzR sinðdR /w Þ sin ww sinðdR /w Þ cos w cosðdR /w Þ kRc
2 3T 8 9
Dv1R > < iRc >
=
6 7
¼ 4 Dv2R 5 jRc
>
: >
;
Dv3R kRc
ð2:346Þ
where
1 rðL;RÞ
vðL;RÞ ¼ vþ v cos ww ð2:348Þ
2 r0
In Eq. (2.348), r0 is the nominal rolling radius of wheel, while r(L,R) are the
actual rolling radii of the left and right wheels.
132 2 Vehicle–Track Coupled Dynamics Models
The relative differences of the angular speeds of the wheelset and rails in the
absolute coordinate system are described as
2
3T
/_ w cos ww X þ b_ w cos /w sin ww /_ rðL;RÞ 8 9
6 7 > i>
6
7 < =
Dx0ðL;RÞ ¼6 _ _
/w sin ww þ X þ bw cos /w cos ww 7
6 7 >j>
4
5 : ;
k
X þ b_ w sin /w þ w_ w ð2:350Þ
2 3T 8 9
DxxðL;RÞ > <i> =
6 7
¼ 4 DxyðL;RÞ 5 j
: >
> ;
DxzðL;RÞ k
When the angular speed differences are transformed into the contact spot
coordinate system, yield
2 3 2 31
DxxðL;RÞ T cos ww sin ww 0
6 7 6 7
DxðL;RÞ ¼ 4 DxyðL;RÞ 5 4 cos dðL;RÞ /w sin ww cos dðL;RÞ /w cos ww sin dðL;RÞ /w 5
DxzðL;RÞ sin dðL;RÞ /w sin ww sin dðL;RÞ /w cos ww cos dðL;RÞ /w
8 9 2 3 8 9
< iðL;RÞc >
> = Dx1ðL;RÞ T >< iðL;RÞc >
=
6 7
jðL;RÞc ¼ 4 Dx2ðL;RÞ 5 jðL;RÞc
>
: >
; >
: >
;
kðL;RÞc Dx3ðL;RÞ kðL;RÞc
ð2:351Þ
Thus, the spin creepage at the left and right wheel–rail contact points read
Dx3ðL;RÞ
n/ðL;RÞ ¼ ð2:352Þ
vðL;RÞ
8
< Fx ¼ f11 nx
Fy ¼ f22 ny f23 n/ ð2:353Þ
:
Mz ¼ f23 ny f33 n/
in which Gwr is the combined shear modulus of the wheel and rail materials, a and
b are the semi-axil lengths of the wheel–rail contact patches, Cij are the Kalker’s
creep coefficients [108], which are depended on the ratios of the semi-axil lengths
of the wheel–rail contact patches.
The combined shear modulus Gwr and the combined Poisson ratio m for the
wheel and rail materials are given as
(
Gwr ¼ G2G w Gr
w þ Gr
Gw mr þ Gr mw ð2:355Þ
m ¼ 2Gw Gr Gwr
where E is the Young’s modulus of wheel–rail material, Gw and Gr are the shear
modulus of the wheel and the rail, mw and mr are the Poisson ratios of the wheel and
the rail.
The semi-axil lengths of the wheel–rail contact patches a and b can be calculated by
8
>
< a ¼ ae ðNRw Þ
1=3
where N is the wheel–rail normal force at the contact point, ae and be are parameters
dependent on q/Rw given in [111].
If q/Rw 2,
8
1=3
>
< ae ¼ 0:1506m q 103 ðm2 =NÞ
1=3
Rw
1=3 ð2:358Þ
>
: be ¼ 0:1506n q 1=3
Rw 103 ðm2 =NÞ
134 2 Vehicle–Track Coupled Dynamics Models
If q/Rw > 2,
8
1=3
>
< ae ¼ 0:1506n q 1=3
Rw 103 ðm2 =NÞ
1=3 ð2:359Þ
>
: be ¼ 0:1506m q 1=3
Rw 103 ðm2 =NÞ
where m = 0.3 and E = 21011 N/m2 are assumed for the common wheel–rail
contact condition, and q is given by
1 1 1 1 1
¼ þ þ ð2:360Þ
q 4 Rw rw rr
where Rw is the rolling radius of wheel, rw is the curve radius of wheel tread, rr is
the curve radius of rail head.
2=3
q
ae be ¼ 22:68mn 109 ðm2 =N)2=3 ð2:361Þ
Rw
According to the Hertzian elastic contact theory, m and n are depended on the
coefficient b
q 1 1 1
b ¼ arccos ð2:362Þ
4 Rw rw rr
Creep Slip
Pure creep
The resultant force of the longitudinal and lateral linear creep forces reads
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F¼ Fx2 þ Fy2 ð2:363Þ
F0
e¼ ð2:364Þ
F
where
8
2
3
<
0 fN F
13 F
þ 1 F
ðF 3fNÞ
F ¼ fN fN 27 fN ð2:365Þ
:
fN ðF [ 3fNÞ
In Eq. (2.365), f is the Coulomb friction coefficient between the wheel and the
rail.
The modified global tangent creep forces/torque between the wheel and the rail
can then be obtained by
8 0
< Fx ¼ e Fx
F 0 ¼ e Fy ð2:366Þ
: y0
Mz ¼ e Mz
Unlike the classical wheel–rail contact model, the above new spatial wheel–rail
coupling model eliminates the following assumptions: (1) the wheel and the rail are
in contact all the time, (2) the rails are assumed to be fixed without any movement,
and (3) the wheels and rails are assumed to be rigid bodies. Thus, the current model
is capable of considering three kinds of rail motions in vertical, lateral and torsional
directions, and dealing with the situation that the wheel loses its contact with the rail
existing in practical railway operations. Therefore, it is well suitable for analyzing
the dynamic behavior of wheel–rail interactions.
136 2 Vehicle–Track Coupled Dynamics Models
The main components of the train and the track system as well as their inter
relationship are shown in Fig. 2.62. The running status of the adjacent vehicles is
likely to be different when the train traction/braking control is implemented, or
when the train passes through curves and ramps. In these conditions, various
postures of the coupler and draft gear systems, such as a large tilt angle of coupler
and/or a mismatch of coupling heights, would inevitably emerge, which will cause
detrimental dynamic inter-vehicle interactions. The generated in-train forces could
be transmitted to the wheelsets through the bogie suspension systems, which is
likely to affect the wheel–rail contact relations and the vibrations of track structures.
In reverse, the track vibrations will have an effect on the vehicle dynamic responses,
and finally influence the working conditions of the coupler and the draft gear
packages. Therefore, the coupler and draft gear subsystem, the train subsystem and
the track subsystem are closely interrelated.
Fig. 2.62 Basic principle of dynamic interaction between train and track [87]
2.4 Train–Track Spatially Coupled Dynamics Model 137
Fig. 2.63 Heavy-haul train–track spatially coupled dynamics model (side view)
Based on the basic principle of the train–track interactions introduced in Sect. 2.4.1,
the train–track coupled dynamics model is established. The structure characteristics
of the train and the track need to be recognized first so as to simulate the dynamic
behaviors. The basic components of a certain type of railway vehicles are usually
definite. For example, a locomotive is generally composed of the car body, bogie
frames, traction motors, wheelsets and suspension systems, while, a freight wagon
usually consists of the car body, side frames, bolsters, wheelsets and suspension
systems. For the ballasted track structure which is most widely used in the
heavy-haul railways, it also has the normative form. Figure 2.63 shows a schematic
diagram of the typical heavy-haul train–track coupled dynamics model, in which
the locomotives are distributed in different positions of the train (with the
Distributed Power mode), and the track is the commonly used ballasted track.
The relevant dynamics models of the locomotive, the freight wagon, and the
ballasted track could be referenced to Sect. 2.3 or be found in some literatures
[10–13, 112]. Specially, the forces such as the traction force, the braking force, the
coupler force and the running resistance should be also considered in the vehicle
model when the longitudinal motions are concerned. Furthermore, the possible
large creepage between the wheel and the rail needs to be taken into account. Their
detailed calculation methods are given in the following subsections.
1. Calculation of wheel–rail forces
Large creepage may appear at the wheel–rail contact interface under the train
driving or braking conditions. Knothe et al. [113] pointed out that the variable
relationship between the wheel–rail tangential force and the creepage could be
depicted by introducing the dynamic wheel–rail friction coefficient into the
Vermeulen–Johnson model or the Shen–Hedrick–Elkins model, so as to calculate
the wheel–rail forces accurately in the case of large creepage. This method is also
adopted in our dynamics modeling. Figure 2.64 shows the flow chart for the general
calculation of the wheel–rail forces where the large creepage is considered.
To reflect the large creep speed and consider the effect of the rail vibrations, the
creepage calculation formulas are given as [87]
138 2 Vehicle–Track Coupled Dynamics Models
8 V1 Vr1
< nx ¼ V 1
>
V V V
ny ¼ 2 Vr21 Ry ð2:367Þ
>
:
nz ¼ X3 X
V1
r3
where the symbols V1, V2 and X3 are successively the instantaneous running speed,
the lateral speed, and the spin speed at the wheel mass center; while, Vr1, Vr2, and
Xr3 are the peripheral speed, lateral speed, and spin speed at the wheel contact
point; and VRy represents the lateral speed induced by the rail vibration and track
irregularity variation.
According to the field test results of the wheel–rail friction coefficient under the
braking conditions, the wheel–rail friction coefficient has a descending trend as the
increasing of the creep speed. The empirical formula of the variable friction
coefficient is expressed as [114]
fstat
fkin ¼ ð2:368Þ
1 þ 0:23jvs j
where, the symbols, fstat and fkin, represent the static and the dynamic friction
coefficients, respectively; vs denotes the wheel–rail creep velocity; and fstat is set to
be 0.45 and 0.25 for the dry and the wet rail surface conditions, respectively. Then,
2.4 Train–Track Spatially Coupled Dynamics Model 139
the wheel–rail creep forces are calculated by introducing the dynamic friction
coefficient into the Shen–Hedrick–Elkins model [110].
2. Calculation of train driving and braking forces
The train tractive force is an external force which can be adjusted as required in
practical train operations. For the locomotive with the property of step speed reg-
ulation, its traction characteristics are usually represented by a number of traction
force curves which are related to the driver controlling handle positions, as shown
in Fig. 2.65. It can be seen that the traction force can be determined by the loco-
motive running speed and the handle position. It should be noted that the maximum
driving force is limited by the starting current and the wheel–rail adhesions during
the startup process.
While for the locomotive with the stepless speed regulation, the traction force is
dependent on the running speed and the given percentage, which can be calculated
as
Ft ¼ f ðvÞ n% ð2:369Þ
where, f(v) denotes the maximum traction force, n% is the percentage of control.
In the braking process, the electric braking and the brake shoe braking are the
two major types used in the heavy-haul train. For the electric braking which is only
applied in the locomotive, it can be regarded as the inverse process of the traction.
Thus, the braking force is also limited by the motor power and the wheel–rail
adhesion. In the case of the step speed regulation, the electric braking force can be
also calculated by interpolating in the electric braking force curves according to the
locomotive running speed and the braking handle position. However, if the loco-
motive has the property of stepless speed regulation, the braking force is dependent
on the running speed and the given percentage of control, which is very similar to
Eq. (2.369).
However, for the brake shoe braking approach, the action force of the brake shoe
braking is from the compressed air (see Fig. 2.66), which distributes in both the
140 2 Vehicle–Track Coupled Dynamics Models
locomotives and the freight wagons. In this case, the braking force can be calculated
by the product of the brake shoe pressure and the friction coefficient of the contact
interface. The brake shoe pressure K is calculated as [115]
pdz2 pz gz cz nz
K¼ ðkNÞ ð2:370Þ
4nk 106
where dz is the diameter of the brake cylinder (unit: mm); pz is the air pressure in the
brake cylinder (unit: kPa); ηz is the computational transmission efficiency of
foundation brake gear; cz is the braking leverage; and nz and nk are the numbers of
cylinder and brake shoes, respectively. The air pressures in the brake cylinders of
the wagons and the locomotives could be obtained from the train pneumatic braking
tests. And the friction coefficient of brake shoe could be obtained from the
experiments or the relevant railway occupation standards.
3. Dynamic model of coupler and draft gear system
For the heavy-haul locomotives and freight wagons in China, the nonrigid auto-
matic couplers with a self-centering ability are commonly used. There is usually a
free clearance between two couplers connected with each other, which permits them
to move relatively in the vertical direction. And the coupler could also sway within
a small angle range relative to its draft key in the horizontal direction.
In the longitudinal train dynamics, a pair of connected couplers are usually
considered as an entirety without mass. The typical simplified mathematic model
[116] is shown in Fig. 2.67. The symbols Kbuf and Cbuf represent the stiffness and
the damping of the draft gear, respectively. And Ks denotes the structural stiffness
of the car body. Kbuf and Cbuf are not two constants, but have the nonlinear
characteristics. For convenience in simulation, the stiffness and the damping of the
draft gear are usually described by the hysteresis curves. It should be noted that the
inter-vehicle interactions in the 3D train model will be affected comprehensively by
the coupler forces in the longitudinal, the lateral, and the vertical directions, which
is different from the coupler forces in the traditional train longitudinal dynamic
models. For the typical coupler and draft gear package used in China, the coupler
forces can be calculated as below.
2.4 Train–Track Spatially Coupled Dynamics Model 141
Fig. 2.67 Dynamic model of heavy-haul coupler and draft gear system
where vf is the switching speed between the loading and the unloading conditions;
and F0 and Fd represent, respectively, the spring force and the damping force of the
coupler and draft gear system.
(2) Coupler lateral force
The coupler lateral force includes two parts, namely, the lateral component of the
coupler force and the lateral force induced by the coupler restoring moment. The
values of these two forces are related to the magnitude of the coupler swing angle in
the horizontal plane. In the simulation, the connected two couplers are regarded as a
rigid-straight bar without relative rotations. The swing angle of the coupler relative
to the car body is defined as positive when it rotates around its draft key clockwise
from the top view, otherwise it is negative.
When the train negotiates a curved track, the coupler swing angle will be
affected by the line alignment. The coordinates of the track centerline are shown in
Fig. 2.69. For the ith vehicle, the positions of the front and rear center pins are
determined by the coordinates (xti1, yti1) and (xti2, yti2), respectively. Then, the
coordinate values of the car body mass center are obtained as
xci ¼ ðxti1 þ xti2 Þ=2
ð2:372Þ
yci ¼ ðyti1 þ yti2 Þ=2
When the centerline of the draft keys in the car body has displacement D and the
yaw angle wc, the coordinates of the front and the rear draft keys in the absolute
coordinate system are expressed as
Fig. 2.69 Geometric relationship between the couplers and the vehicles in a curved track [87]
2.4 Train–Track Spatially Coupled Dynamics Model 143
Fig. 2.70 Calculation of lateral coupler force [87]: a coupler restoring torque and b coupler force
analysis
xdi1 xci sin wc cos wc sin wc L
¼ þD þ ð2:373Þ
ydi1 yci cos wc cos wc cos wc 0
where D = yci-/cihcgi. L equals lcg and −lcg for the front and rear draft keys,
respectively. The coordinates of the draft keys C11, C12, C21 and C22 of the adjacent
two vehicles can be finally calculated by Eq. (2.373). Connecting these four points,
three two-dimensional vectors r1, r2, and rc are obtained. Then the intersection
angles of these vectors can be obtained accordingly. The swing angles of the
couplers relative to the centerlines of the front and the rear vehicles are calculated as
8 h i
< a1 ¼ arccos r1 rc sign½kðrc Þ k ðr1 Þ
jr1 jjrc j
h i ð2:374Þ
: a2 ¼ arccos r2 rc sign½kðrc Þ k ðr2 Þ
jr2 jjrc j
where k(r) is the corresponding gradient of the vector r in the global reference
system.
Actually, the coupler swing angle could not increase continuously, and its
amplitude is limited by the structures of the coupler and the draft gear system. Once
the coupler angle exceeds the free swing limit (dy), a restoring torque will be
generated to resist its swing motion. The restoring torques can be classified into two
main forms according to their action principles (see Fig. 2.70a). One is called the
rigid stop characteristic with the restoring torque changing linearly with the vari-
ation of the swing angle. The other one is called the nonlinear impedance char-
acteristic, which indicates that the restoring torque depends on the compression
amount and the load of the draft gear. More specifically, when the coupler swing
angle exceeds the free limit value, the coupler needs to first overcome the resistant
torque caused by the initial pressure of the draft gear, and then it can compress the
draft gear.
The coupler lateral forces acting on the draft keys can be calculated according to
the force equilibrium conditions (see Fig. 2.70b):
144 2 Vehicle–Track Coupled Dynamics Models
where Lcp is the length of the two connecting couplers, and M1 and M2 represent the
coupler restoring moments.
(3) Coupler vertical force
The coupler vertical force is mainly caused by the friction force occurring at the
coupler contact interface, which can be simplified as a nonlinear spring–friction
element with the coupler clearance [116]. Figure 2.71 shows the schematic diagram
of the calculation model for the coupler vertical force. The vibrations of the vehicle
system are the main reasons that induce the vertical motion of the coupler head.
And the vertical and pitching motions of the car body have the primary influence.
The relative displacement and speed between the adjacent vehicles in the vertical
direction are calculated as
(
Dz ¼ zc1 þ lcg1 bc1 zc2 lcg2 bc2
ð2:376Þ
D_z ¼ z_ c1 þ lcg1 b_ z_ c2 lcg2 b_
c1 c2
When the longitudinal relative displacement is smaller than the coupler slack dfc,
the friction force in the vertical direction is Fcz = 0. Otherwise, two different cases
need to be clarified as follows:
(1) If jFcz j jl0 Fcx j, then
Fcz ¼ kz Dz ð2:377Þ
where vr is the switching speed, and l0 and ld are the static and dynamic friction
coefficients, respectively.
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Chapter 3
Excitation Models of Vehicle–Track
Coupled System
Generally, the wheel–rail system excitations can be input into vehicle–track cou-
pled dynamics model by three methods, namely the fixed-point method,
moving-vehicle method, and tracking-window method. In the analysis of vehicle–
track coupled dynamics, the three methods could be selected according to practical
situation and requirement.
© Science Press and Springer Nature Singapore Pte Ltd. 2020 151
W. Zhai, Vehicle–Track Coupled Dynamics,
https://doi.org/10.1007/978-981-32-9283-3_3
152 3 Excitation Models of Vehicle–Track Coupled System
The fixed-point method assumes that the vehicle does not move on the track, while
the geometry irregularities of the wheel and rail surfaces move at the train speed in
the direction opposite to the train’s traveling direction [1]. The irregularities are
input to the vehicle–track coupled system through each wheel–rail contact point.
This input method is also called the moving-irregularity method, as shown in
Fig. 3.1.
When employing the fixed-point method, the shortest track model can be
adopted in the calculation (by just having the effective calculation length described
in Sect. 4.4). Therefore, this method has the prominent advantages of high calcu-
lation speed and efficiency. However, its shortcomings are also obvious. Instead of
considering the longitudinal movement of the vehicle along the track, this method
considers the wheel–rail excitations moving opposite to the train’s traveling speed,
which is approximate for the simulation and cannot reflect the dynamic interaction
between discrete supporting sleepers and wheelsets that runs along the tracks. When
track structures have defects and uneven supporting stiffness, the fixed-point
method cannot simulate the complicated dynamic effects of the structures during
vehicles’ passing.
Then, for what kind of problems should we use the fixed-point method? It is
particularly suitable for analyzing the vehicle–track coupled dynamics problem
when the dynamic parameters of the track structure is uniform along the longitu-
dinal direction, which usually corresponds to some common problems in the most
general cases, such as the riding comfort, lateral stability, and curve negotiation
when vehicles move along elastic track structures, as well as the dynamic param-
eters optimization of the vehicle and track systems. Therefore, the fixed-point
method is still a simple and useful method that is widely used by railway engi-
neering dynamics researchers all over the world.
Train speed v
Geometry irregularities Z0
In the moving-vehicle method, the wheel–rail system excitation is input into the
vehicle–track coupled system while the vehicle moves along the track, as shown in
Fig. 3.2. Obviously, it is the most realistic input method.
However, the moving-vehicle method still has its inconvenience. Due to the fact
that this method fully considers the spatial position of the vehicle that moves along
the track, a sufficiently long track is required in the model. This would lead to a leap
increase of model degrees of freedom and computational efforts.
Then, in what case should the moving-vehicle method be employed? The author
holds that it is necessary to apply this method when track defects and infrastructure
uneven stiffness (namely, dynamic stiffness excitation) are involved. These include
vehicle–track interaction at places with failed fasteners, unsupported sleepers,
turnouts, or transition zones (subgrade–bridge transition, ballast track–ballastless
track transition). In these cases, the track concerned should not be long, so that it is
quite convenient and efficient to adopt this input method.
Under the excitations by the rail joints, wheel flats, rail corrugations, and other
local track irregularities, the required effective calculation length of track models
could be limited, thus it is suitable to adopt the moving-vehicle method for these
cases. However, if the variation of infrastructure stiffness is considered, the required
calculation length of track models might be very large, and a large number of rail
modes will be required in the moving-vehicle method, so that it is quite difficult to
efficiently perform a vehicle–track coupled dynamics simulation. Fortunately, the
tracking-window method described below can be employed for this case.
the track modeling. Therefore, only a finite length of track structure under and
around the vehicle should be considered in the simulation of the dynamic behavior
of the vehicle–track coupled system. In the vehicle–track model using the
tracking-window method, a tracking calculation window moving with the vehicle is
set for the vehicle–track interaction, where only a finite length lTW of the track
structure under and around the vehicle is considered. It can reflect the dynamic
behavior of the track structure under a running vehicle through calculation per-
formed only in the moving window.
For the tracking-window method, two key points should be noted: (1) The
vibration of track structure within the tracking window is taken into consideration
and the vibration of track structure outside the tracking window is assumed to be 0;
(2) The vibration characteristic of the track structure possesses an “inheritance
relationship” at the instant of the tracking window moving forward. To efficiently
reflect the “inheritance relationship” of track vibration in the tracking-window
method, the calculation framework is shown in Fig. 3.3, where a unit slab bal-
lastless track is taken as an example. The calculation steps are illustrated as follows:
(1) Input vehicle parameters, track parameters and other parameters required in the
vehicle–track interaction simulation, including the length lTW and the moving
distance lm of the tracking window;
(2) Set the number of sub-windows Nw = 1 and set the initial position of the
vehicle on the track in the first sub-window, where the initial position of the jth
wheelset is represented as xwj0;
O x
Rail
(i-1)lM
v
Initial position End position
lM
O (i) x(i)
ith sub-window
z(i)
v
Initial position End position
lM
(i+1)
x(i+1)
O
(i+1)th sub-window z(i+1)
lTW
(3) The vehicle moves forward from the initial position, and the dynamics of the
vehicle–track coupled system is calculated based on the traditional
moving-vehicle method, in which the position of each wheelset on the track
should be obtained. It can be calculated that the moving distance of the vehicle
at the kth integral step in the sub-window is xmk = xm(k−1) + vDt, and the
position of the jth wheelset on the track can be expressed as xwjk = xwj0 + xmk;
(4) If xm(k−1) < lm and xmk lm, that is, when the moving distance of the vehicle at
the kth integral step is greater than or equal to the distance lm, the tracking
window moves forward a distance of lm. Meanwhile it satisfies:
(i) The track vibration in the overlap part of the two adjacent windows must
be the same (marked by red in Fig. 3.3), i.e., the “inheritance relationship”
is satisfied. At the same time, the track vibration in the nonoverlapping
part (marked by blue in Fig. 3.3) is assumed to be 0;
(ii) As the window moves, the moving distance of the vehicle in the updated
sub-window is xmk = xmk − lm;
(5) Set the sub-window number Nw = Nw + 1, and then go to Step (3). Continue the
cyclic calculation until the calculation time or the calculation distance satisfies
the termination condition.
For ballasted tracks on embankment that including rails, fastenings, sleepers,
ballast, and subgrade, the length lTW and the moving distance lm of the window are
determined by the sleeper number Ns covered in the tracking window, which is
obtained by lTW = Ns ls (ls is the sleeper spacing). The moving distance lm is
advised to be a sleeper spacing. For ballastless slab tracks on embankment that
includes rails, fastenings, slabs, and subgrade, the length lTW and the moving
distance lm of the window are determined by the slab number Nsb covered in the
tracking window, which is obtained by lTW = Nsb lsb (lsb is the slab length). The
moving distance lm is advised to be the length of a unit slab. For the ballasted and
ballastless tracks on multi-span simply supported bridges, the length lTW and
moving distance lm of the window are determined by the number of bridge span Nb
covered in the tracking window, which is obtained by lTW = Nb lb (lb is the
length of a bridge span). The moving distance lm is advised to be the length of a
bridge span. When the calculation length of the tracking window is selected, the
vehicle should be set at the center of the tracking window if possible, and the
distance from the vehicle to the two ends of the tracking window should be suf-
ficient to ensure that the vibration of the track components outside the calculation
window is negligible.
It is noted that the tracking-window method is able to carry out fast calculation
of a vehicle moving forward on an infinite track with intricate structures, including
the discrete sleepers, slabs, simply supported bridges, transition zones between
bridge and embankment, track substructure defects, etc. It overcomes the
time-consuming problem of the moving-vehicle method. Another advantage of the
tracking-window method is that it can deal with all kinds of track structures neatly
such as rail–slab–bridge/embankment structure, rail–sleeper–ballast–bridge/
156 3 Excitation Models of Vehicle–Track Coupled System
When a wheel passes a dipped joint, dislocation joint, joint gap, or surface spalling
of rails, the abrupt change of the instantaneous rotation center of the wheel would
lead to a vertical impact velocity to the track, which disappears instantly when the
wheel runs away from these locations, resulting in a sudden impact and vibrations
of the wheel–rail system. Similarly, the same impact vibrations will appear when a
wheel with flats moves along a rail. These excitations are defined as impact exci-
tations; they are often input into the wheel–rail system as impact velocities, or as
impact displacements.
During train operation, wheel local scratch and spalling (shown in Fig. 3.4) may
occur due to various reasons (braking, wheel spin and slip). These phenomena are
collectively known as wheel flats. Wheel flats will induce special dynamic effect
during rolling, and the function mechanisms of new and old wheel flats are quite
different. An ideal new flat is similar to the chord of a wheel circle, as shown in
Fig. 3.4, while an old flat is the wear result of a new flat. An old wheel flat can be
(a) (b)
O v
simply described by a cosine curve (as shown in Sects. 3.3.1). Here, the impact
mechanism of new wheel scratches will be mainly introduced [2].
1. Impact mechanism of wheel flat and its critical impact velocity
As shown in Fig. 3.5, when a wheel rolls at the start point A of a wheel flat at a low
speed, the wheel will rotate around point A until the entire flat surface impacts the
rail surface, and then the wheel rotates immediately around point B, further exerting
a dynamic load on the track until the wheel restores to its normal rolling state.
When the wheel rolls at point A at a high speed, it will leave the rail surface and
rotate through the air as it moves forward with an inertial motion, and then fall
down and make contact with the rail surface at point B, resulting in an impact force
on the track, as shown in Fig. 3.6.
Clearly, as the rolling speed increases, the impact characteristics of wheel flats
will inevitably have a sudden change at a critical running velocity. When the wheel
rolls from location (a) to location (b) shown in Fig. 3.5 at an angular velocity of
x = v/R (v is the running speed, and R is the wheel radius), the elapsed time t1
equals to the time t2 needed for the wheel to fall the distance h. Then the critical
state will occur, that is
u=2 Ru
t1 ¼ ¼ ð3:1Þ
x 2vcr 0
sffiffiffiffiffi
2h
t2 ¼ ð3:2Þ
l
where l is the falling acceleration of the wheel. According to Newton’s second law,
M1 þ M2
l¼ g ð3:3Þ
M2
where M1 and M2 are the sprung mass and unsprung mass of the primary sus-
pension, respectively; M1g is the car body load on the wheel applied through the
axle box, as shown in Fig. 3.7.
Setting t1 = t2, and substituting
u 1 2
h ¼ R 1 cos Ru
2 8
into Eq. (3.2) yields the critical running velocity of the wheel flat
pffiffiffiffiffiffi
vcr0 ¼ lR ð3:4Þ
u L
v01 ¼ v sin ¼ v ð3:5Þ
2 2R
where L is the length of the flat. The other part is the instantaneous impact com-
ponent in the opposite direction of the wheel vertical velocity, which is produced at
the moment when the rail hinders the wheel rotating around point B, expressed as
L
v02 ¼ c v ð3:6Þ
2R
where c is the coefficient for the transformation from rotational inertia to recipro-
cating inertia [3].
Therefore, the impact velocity can be obtained as
L
v0 ¼ ð 1 þ cÞ v ð3:7Þ
2R
Obviously, the impact velocity at low speed is proportional to the flat length
L and the running speed v, and is inversely proportional to the wheel radius R.
3. Impact velocity equation for high speed (v > vcr0)
For v > vcr0, the impact velocity also consists of two parts, which are the falling
speed from the air to the rail surface
v01 ¼ lt ð3:8Þ
and the vertical component of the wheel center velocity due to the rotation
In Fig. 3.6, the elapsed time for the angular u h of the wheel rotating from the
location (a) to the location (b) can be given as
u h ðu hÞR
t¼ ¼ ð3:10Þ
x v
1 lðu hÞ2 R2
x ¼ lt2 ¼ ð3:11Þ
2 2v2
160 3 Excitation Models of Vehicle–Track Coupled System
and
u L=R ð3:14Þ
Therefore,
rffiffiffi
L l
h¼ pffiffiffiffiffiffi ð3:15Þ
v þ lR R
L
t¼ pffiffiffiffiffiffi ð3:16Þ
v þ lR
and thus
lL
v01 ¼ pffiffiffiffiffiffi ð3:17Þ
v þ lR
rffiffiffi
cvL l
v02 ¼ pffiffiffiffiffiffi ð3:18Þ
v þ lR R
As can be seen form Eq. (3.19), the impact velocity at a high speed is also
proportional to the wheel flat length L, while it slightly decreases with an increase
of the velocity, and finally, it goes to a constant value
rffiffiffi
l
v0 ¼ lim v0 ¼ cL ð3:20Þ
v!1 R
3.2 Impact Excitation Models 161
A rail dislocation joint refers to an abnormal joint which has height difference
between the adjacent rail surfaces of a rail joint. According to the vehicle running
direction, the rail dislocation joints can be divided into forward and backward
dislocation joints. As shown in Fig. 3.8a, a backward dislocation joint corresponds
to the case that the wheel runs from a lower rail surface to a higher one, while a
forward dislocation joint is presented in Fig. 3.8b, where h is the height difference.
It is known from Fig. 3.8a that the impact velocity of the backward dislocation
joint can be expressed as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffi
R 2 ð R hÞ 2 2h
v0 ¼ v sin h ¼ v v ð3:21Þ
R R
Setting the critical impact velocity as vcr0, a wheel with an initial horizontal
speed of vcr0 is supposed to fall down at the point B, then
162 3 Excitation Models of Vehicle–Track Coupled System
Fig. 3.9 Schematic of wheel passing through a forward dislocation joint at a low speed and
b high speed
pffiffiffiffiffiffiffiffi
vcr 0 t0 ¼ jACj ¼ 2Rh ð3:23Þ
By substituting Eq. (3.24) into Eq. (3.23), the critical impact velocity is
pffiffiffiffiffiffi
vcr 0 ¼ lR
which is the same as that of the wheel flat as shown in Eq. (3.4).
It is noted that the above analysis is based on the assumption of rigid rail. In fact,
the rail is elastically supported, so that the impact velocity is smaller than that of
rigid support, according to Ref. [4]. Therefore, Eq. (3.21) should be modified as
rffiffiffiffiffi
meq 2h
v0 ¼ v ð3:25Þ
mw R
where mw is the wheel mass and meq is the equivalent impact mass of the track,
written as
3.2 Impact Excitation Models 163
4=3 1=3
3 5 pffiffiffi 4EI
meq ¼ m C C 2 ð3:27Þ
4 4 KH
where KH is Hertz linear contact stiffness at wheel–rail interface and m is the rail
(including the sleepers) mass per unit length.
For a general wheel–rail contact condition
Dipped rail joints are the most common impact excitation sources for jointed track
lines, which present themselves as a dip at the rail joints, as shown in Fig. 3.10. The
impact velocity of a dipped rail joint is usually expressed by the product of the joint
angles a1, a2 and the vehicle speed v.
where 2a is the total angle of the dipped joint (see Fig. 3.10).
When a vehicle passes a turnout in the through route, the vehicle–turnout inter-
action mainly presents vertical impact and vibrations at the crossing frog, while
lateral interaction between the vehicle and turnout becomes dominant when the
vehicle passes via the divergent route.
1. Vertical impact model of turnout frog
The vertical impact at a turnout crossing mainly occurs at the fixed frog. As shown
in Fig. 3.11, there is a harmful gap between the point rail and the wing rail, leading
to a discontinuous rolling path of the wheel and inducing the wheel–frog impact.
When a wheel rolls from the wing rail to the point rail, the wheel rolling radius
becomes increasingly smaller as the wheel moves away from the wing rail, causing
a descent of the wheel gravity center. To avoid collision between the wheel and the
point rail, the rail surface height at this location decreases dramatically, and then
increases gradually to the normal height. In this way, the wheel gradually rolls back
to the original height when it is completely on the point rail.
The traveling path of the wheel center on the frogs was investigated by mea-
suring the wear of the wheel tread and frog in the former Soviet Union. China
Academy of Railway Sciences also measured the vertical irregularity of the
Hadfield steel railway frogs and obtained a similar traveling path of the wheel
center. This irregularity can be described with the schematic curve as shown in
Fig. 3.12, where the first half part is sinusoidal, and the latter part is triangular. The
mathematical expression can be written as
8
>
> h0 sinð2px=L0 Þ ð0 x x1 Þ
<
hd ðx x1 Þ=ðx2 x1 Þ ðx1 \x x2 Þ
Z0 ¼ ð3:30Þ
>
> h ðx xÞ=ðx3 x2 Þ ðx2 \x x3 Þ
: d 3
0 ðx [ x3 Þ
Table 3.1 Relationship Wear level Slight wear Medium wear Severe wear
between vertical irregularity
parameters and wear level of Wear value/mm 2–4 4–6 >6
fixed frog h0/mm 0.7 0.8 0.9
hd/mm 3.7 4.3 7.1
where the irregularity amplitude h0 and hd are related to the wear level of frogs.
Table 3.1 lists the values from measurement [5].
For the No. 12 Hadfield steel frogs of CN60 rail commonly used in Chinese
railways, L0 = 0.87 m, x1 = 0.87 m, x2 = 1.276 m, x3 = 1.816 m. The irregularity
function can be simplified as
8
>
> h0 sinð70:222xÞ ð0 x 00:87Þ
<
20:463hd ðx 00:87Þ ð00:87\x 10:276Þ
Z0 ¼ ð3:31Þ
>
> 10:852hd ð10:816 xÞ ð10:276\x 10:816Þ
:
0 ðx [ 10:816Þ
Impact point
Running direction
166 3 Excitation Models of Vehicle–Track Coupled System
Impact point
Running direction
For the straight switch rail, b is the angle between the straight switch rail and the
stock rail. For the curved switch rail, b is determined by the equation
pffiffiffiffiffiffiffiffiffiffiffi
b¼ 2d=R ð3:33Þ
where d is the gap between the outside wheel flange and the switch rail; according
to the limit size method or probability theory, d is usually set to be 47 mm; R is the
radius of the turnout.
Therefore, when a vehicle passes through the divergent route, the excitation
between the wheelset and switch rail can be simply applied to the lateral velocity of
the wheelset by an instantaneous impact velocity v0L. Indeed, the lateral wheel–rail
interaction is very complicated for a vehicle passing through the divergent route and
the excitation description for the entire turnout still needs further investigation.
Welds of continuous welded rails can induce the same impact effect as that of the
backward dislocation joints due to the convex weld surface (Fig. 3.15) caused by
poor welding process. The impact velocity can be found with Eq. (3.25).
The excitation models of rail surface spalling (Fig. 3.16) and wide rail joint gaps
(Fig. 3.17) are similar to that of wheel flat by simply replacing the L in Eqs. (3.7)
and (3.19) with the spalling length L0 or the rail gap width H.
where
2pv
x¼ ð3:35Þ
L
For the excitation with multiple waves, the variable t in the displacement input
function (3.34) should satisfy
nL
0t ð3:36Þ
v
point rail, it rolls at a smaller rolling radius. When the wheel totally rolls on the
point rail, the wheel gradually rises due to the heights of the point rail gradually
ascending to the same level as the stock rail. As a result, the irregularity in the
vertical plane will be presented as shown in Fig. 3.22. For the Chinese
No. 12 turnout with 60AT rail, the irregularity wavelength at the point rail
L 2 m, and the wave depth a can be determined according to the wear level,
which are 2–4 mm for slight wear, 4–6 mm for medium wear, and above 6 mm for
severe wear. The vertical irregularity for the movable-point frog is similar to that of
the turnout switch area, the only difference is that the change rate of the irregularity
at the point rail is larger than that of the switch rail, namely, its wavelength is
smaller. For the Chinese No. 12 movable-point frog of CN60 rail, L 1 m.
Wheel out-of-roundness is also a typical harmonic irregularity, which includes
the wheel tread local defect and wheel polygon. For wheel local defect due to
eccentric tread wear (Fig. 3.23), the harmonic irregularity function as illustrated in
Fig. 3.19 can be adopted. Considering the fact that the wheel local defect is a
periodic irregularity at wheel–rail interface, it can be expressed as
(1 2pv 2pR
2 a 1 cos L mod t; v v v
mod t; 2pR
Z0 ðtÞ ¼
L
ð3:37Þ
0 v [ v
mod t; 2pR
where mod is the remainder function, R is the wheel radius, and L and a are the
wavelength and wave depth of the wheel local defect, respectively. Generally,
L = 250–800 mm, a = 0.4–3.5 mm.
Wheel polygon refers to periodic radial deviation formed by wheel nonuniform
wear. Figure 3.24 shows a real wheel polygon of a Chinese high-speed train, and
Fig. 3.25 shows the field measured wheel radial profile of a Chinese high-speed
train, which indicates that this wheel has obvious polygonal features. Similar to the
O
R
a
L
wheel local defect, wheel polygon is also a periodic irregularity along the wheel
circumference, which can be represented by the following equation using Fourier
series:
172 3 Excitation Models of Vehicle–Track Coupled System
(a) 0
(b)
0.1 330 30 0.10
0.00
-0.2 270 90
-0.1 -0.05
240 120
0.0
-0.10
210 150 0 500 1000 1500 2000 2500
0.1
(mm) 180 Wheel circumference (mm)
(c) (d)
0.03
23rd
/2
0.02
Amplitude (mm)
24th
Phase (rad)
1st
0
0.01
- /2
0.00
-
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Order Order
Fig. 3.25 Field measured wheel polygon. a Polar diagram of wheel radial deviation; b expansion
of wheel radial deviation along its circumference; c amplitude of each order of wheel polygon;
d phase of each order of wheel polygon
X
1 h v i
Z0 ðtÞ ¼ Ai sin i t þ ui ð3:38Þ
i¼0
R
where i is the order of the wheel polygon, Ai is the amplitude of the ith harmonic
wave, ui is the corresponding phase, Ai and ui can be obtained by performing
discrete Fourier transform of the measured radial deviation of the wheel along its
circumference. In the equation, A0 is the overall deviation of the test data from the
nominal wheel that has no radial deviation; it is negligible for the vehicle–track
system dynamics. Figure 3.25 shows the amplitude and phase distributions as
functions of the order of wheel polygon presented in Fig. 3.24. As can be seen, the
1st-, 23rd-, and 24th-order harmonic waves are the most obvious components; they
are the high-order wheel polygons and have some eccentric wheel wear. The
amplitudes of the other harmonic waves are quite small. The phase distributes
randomly versus the order.
In fact, most wheel polygons present the main harmonic components within the
40th order; components with higher order usually have a small amplitude.
3.3 Harmonic Excitation Models 173
Moreover, due to the wheel–rail contact filtering effect, the high order components
have a negligible effect on the vehicle–track coupled dynamics. Considering the
first N-order dominant components and ignoring the component A0, Eq. (3.38) can
be further written as
X
N h v i
Z0 ðtÞ ¼ Ai sin i t þ ui ð3:39Þ
i¼1
R
>
> 2 að1 cos xtÞ 0 t Lv
< L
L þ LB
Z0 ðtÞ ¼ 0
> v \t v ð3:40Þ
>
:1
L þ L
L þ LB 2L þ LB
2 a 1 cos x t v v \t
B
v
Fig. 3.27 Sketch for a wheel passing an extremely short-wave harmonic irregularity
Table 3.2 Critical wavelength Lcr of impulsive excitation induced by harmonic irregularity
Depth a/mm 0.5 1.0 1.5 2.0 2.5 3.0 4.0 5.0
Lcr/mm 40.98 57.93 70.93 81.88 91.52 100.22 115.65 129.23
3.3 Harmonic Excitation Models 175
Depth (mm)
Wavelength (mm)
Y Gauge
Left rail
Right rail
Alignment irregularity
Z
Railway line center
Left rail Height irregularity
X
Right rail
Cross level irregularity
The alignment irregularity is the lateral deviation of the railway center line due
to lateral deviations of left and right rails, expressed as
1
yt ¼ ðyl þ yr Þ ð3:42Þ
2
where yl and yr are the horizontal coordinates of the left and right rails, respectively.
The gauge irregularity is the gauge variation due to lateral deviations of the left
and right rails, written as
gt ¼ y l y r g0 ð3:43Þ
1
zt ¼ ðzl þ zr Þ ð3:44Þ
2
where Zl and Zr are the vertical coordinates of the left and right rails, respectively.
The cross-level irregularity is the height difference between the left and right
rails due to rail vertical deviation, defined as
Dzt ¼ zl zr ð3:45Þ
In addition, there are two particular kinds of track irregularities, namely the
torsion irregularity and the superposed irregularity. The torsion irregularity refers to
the torsion of left and right rail surfaces with respect to the track plane, namely,
firstly the left rail surface is higher than the right rail surface, and then inverse state
appears, commonly known as the track triangle twist, and vice versa. The super-
posed irregularity refers to the case that the vertical and lateral irregularities
simultaneously appear at the same location of railway lines.
The above track geometry irregularities can be input as system excitation by
using displacement function. For the common detection method of the irregularities
given in the code of Chinese railway maintenance and repair, the geometry devi-
ation over a certain chord length is generally adopted to measure the irregularities.
Therefore, the input of different track irregularities (Fig. 3.30a–f) can be realized by
simply applying the single-wave cosine irregularity (as shown in Eq. 3.34) with the
same or opposite directions, being in phase or out of phase, to one or two of the
rails. In the figure, L is the wavelength and A is the wave depth.
3.3 Harmonic Excitation Models 177
(a) (b)
2A 2A A A
(c) (d)
(e) (f)
y 2A
z
y
Fig. 3.30 Input models of track irregularities. a Alignment irregularity; b gauge irregularity;
c height irregularity; d cross-level irregularity; e torsion irregularity; f superposed irregularity
If a wheel center does not coincide with its geometry center with an eccentricity, as
shown in Fig. 3.31, an unbalanced inertia force with a constant value pointing
outwards would be produced, expressed as
178 3 Excitation Models of Vehicle–Track Coupled System
M0 2
F0 ¼ Mw x2w r0 ¼ x ð3:46Þ
g w
where Mw is the mass of the wheel, xw is the angular velocity of wheel, r0 is the
eccentricity, and M0 is the static moment of eccentricity.
v
xw ¼ ð3:47Þ
R
M0 ¼ Mw gr0 ð3:48Þ
The vertical component of the inertia force is the periodic harmonic force due to
the wheel eccentricity
v 2 v
F0 ðtÞ ¼ Mw r0 sin t ð3:49Þ
R R
or
M0 v 2 v
F0 ðtÞ ¼ sin t ð3:50Þ
g R R
vehicle passes over a stiffness irregularity. Vibrations due to the stiffness irregu-
larity will over time lead to local permanent deformation of tracks, worsening track
geometry irregularity, and in return, intensifying wheel–rail interactions. Generally,
the cause of track stiffness irregularity mainly includes loose or disabled rail fas-
teners, voided sleepers, ballast hardening or loosening, and transition zones (be-
tween subgrade and bridge, subgrade and culvert, subgrade and tunnel, and ballast
track and ballastless track), etc.
Fig. 3.32 DSB Strandmoelle bridge–subgrade transition (bridge abutment is on the right)
Bending angle
Rail
Abutment
Under-sleeper pad
Wooden sleeper
Half-sleeper Ballast
Bridge pier
Subgrade
23 wooden sleepers were used in the subgrade transition zone. As it can be seen that
from the subgrade to the bridge abutment, not only the infrastructure supporting
stiffness changed significantly, but also the sleeper supporting conditions were quite
different, leading to a large stiffness variation of the tracks at the subgrade–bridge
transition. In this case, differential settlement in the transition zone was quite
obvious, presenting a rail bending section with a length of 30 m and a bending
angle of 4‰. This substantially increased the maintenance and repair work of the
track structure, which adversely made the ballast looser and caused larger track
deformation. This typical dynamic stiffness irregularity of track may not be
neglected in the vehicle–track coupled dynamics analysis, and the discrete sup-
ported track model has to be employed to describe in details the track stiffness
parameters at the transition.
To avoid sudden stiffness change, a certain length of transition zone can usually
be set between the subgrade and bridge abutment, see Fig. 3.34. It is able to render
3.4 Excitation Model of Track Dynamic Stiffness Irregularity 181
Bottom layer
Transition
Embankment
a gradual change of the stiffness in a certain range, and to reduce the variation rate
of height difference of rail surface (bending angle of rail surface). As a result, it
alleviates vibrations of trains and tracks and reduces track dynamic interactions,
ensuring the operation safety and stability of trains. The variation of the subgrade
stiffness in the transition zone can be simply illustrated with the piece-wise linear
curve shown in Fig. 3.35, where the bridge abutment stiffness is assumed to be
n times of the subgrade stiffness. This case can be extended by analogy to the
transition zones of subgrade–culvert, subgrade–tunnel, etc.
The special configuration of turnouts determines the fact that track stiffness at
turnouts is larger than that in normal tracks. Figure 3.36 shows the elastic track
deformation at a turnout in the exit section of Linying station on Zhengzhou–
Wuchang railway line measured by an inspection vehicle for track deformation [7].
Being the inverse of the stiffness, the deformation indirectly reflects the track
stiffness variation. As can be seen, the track elastic deflection in the turnout section
(especially at the frog and the switch rail) is one third smaller than that in the
normal track, and it changes gradually at both ends of the turnout. The reasons that
182 3 Excitation Models of Vehicle–Track Coupled System
Amplitude (mm)
Distance (km)
Fig. 3.36 Rail deformation (corresponding to track stiffness irregularity) at a turnout section
cause the larger and varied stiffness at the turnout mainly include different length of
turnout sleepers that leads to different supporting areas of sleepers, more rails and
complicated cross section of the railheads that cause a significant change of the rail
bending stiffness, uneven tamping of ballast bed, etc.
2. Voided sleeper
Subgrade settlement, track deflection, and tamping could cause a local hidden pit of
ballast bed, inducing a void under sleeper, as shown in Fig. 3.38. If the sleeper void
dgap is large, the ballast bed at this location will completely lose its function.
Correspondingly, Kbi = Cbi = 0 can be used in the model. If dgap is not large so that
the ballast bed still has partial bearing capacity, namely the sleeper still makes
contact with the ballast bed and interaction forces are induced, the following
equation can be employed instead of normal interaction force between sleeper and
ballast bed.
Kb Zs ðtÞ Zb ðtÞ dgap ðZs Zb dgap 0Þ
Fbs ðtÞ ¼ ð3:51Þ
0 ðZs Zb dgap \0Þ
where Kb is the ballast stiffness; Zs and Zb are the dynamic displacements of the
sleeper and the ballast, respectively.
3. Hardening or loosening ballast bed
During ballast bed laying and maintenance and due to natural conditions during
operation, etc., ballast hardening or loosening could occur at certain sections, which
leads to significant variations of stiffness and damping of ballast bed. The stiffness
and damping at the corresponding supporting point can be expressed as
Kbi0 ¼ gk Kbi
0 ð3:52Þ
Cbi ¼ gc Cbi
where ηk and ηc are coefficients of variations in the stiffness and damping of ballast
bed, respectively. For different cases, ηk and ηc can be 0.1–10.
In view of this, intensive research was carried out by CARS in the late 1990s on
Chinese railway track irregularities. Data were acquired on about 40,000 km of the
railway main lines all over the country, mainly by train-borne measurement with the
track inspection cars and partially by on-track measurement. The data were filtrated,
classified and analyzed statistically; the track irregularity PSDs (including height,
cross-level, and alignment, as well as some long wavelength track irregularities) of
Chinese railway main lines (heavy-haul lines, speedup lines, high-speed test lines,
with various track structures, super-large bridges, etc.) were proposed [10].
The large-scale launch and operation of the Chinese new high-speed railway
lines in recent years have put the measurement and study of high-speed railway
track irregularity PSDs high on the agenda. To meet the need for research and
maintenance of the Chinese high-speed railway, CARS together with Southwest
Jiaotong University (the research group of the author), based on the statistical
analysis of high-speed railway ballastless track irregularity data measured by the
high-speed track inspection cars from Beijing–Tianjin, Wuhan–Guangzhou,
Zhengzhou–Xi’an, Shanghai–Hangzhou, Shanghai–Nanjing, and Beijing–Shanghai
high-speed railways, put forward the first Chinese standard on high-speed railway
ballastless track irregularity PSDs in October 2014 [11]. Then in June 2016, CARS
et al., based on the statistical analysis of high-speed railway ballasted track irreg-
ularity data measured from Hangzhou–Shenzhen, Nanning–Guangzhou,
Nanchang–Fuzhou, Hengyang–Liuzhou, Jinan–Qingdao, and Hefei–Wuhan
high-speed railways, published the PSDs of ballasted track irregularities of Chinese
high-speed railways [12].
Typical track irregularity PSDs are listed below, and the features are clarified
through analysis and comparison, so as for the readers to conveniently choose the
appropriate random track irregularity excitation model for the vehicle–track coupled
dynamics analysis.
Av /2v2 /2 þ /2v1
Sv ð/Þ ¼
ð3:53Þ
/4 /2 þ /2v2
186 3 Excitation Models of Vehicle–Track Coupled System
(2) Alignment
Aa /2a2 /2 þ /2a1
Sa ð/Þ ¼
ð3:54Þ
/4 /2 þ /2a2
(3) Cross-level
Ac /2c2
Sc ð/Þ ¼
ð3:55Þ
/ þ /2c1 /2 þ /2c2
2
(4) Gauge
Ag /2g2
Sg ð/Þ ¼ ð3:56Þ
/2 þ /2g1 /2 þ /2g2
where S(/) is the track irregularity PSD [m2/(1/m)]; / is the spatial frequency of
track irregularity (1/m); A is the roughness constants (m); /1, /2 are the cut-off
frequencies (1/m).
The roughness constants and cut-off frequencies of the six track classes are listed
in Table 3.3. In this table, the allowable maximum traffic speeds for the different
track classes are also shown according to the FRA safety standard.
German railway divides the track irregularity PSDs into two categories, namely
“low disturbance” and “high disturbance”, which are described as unified form in
the following contents [14]:
(1) Height
Av X2c
Sv ðXÞ ¼ ð3:57Þ
ðX þ X2r ÞðX2 þ X2c Þ
2
(2) Alignment
Aa X2c
Sa ðXÞ ¼ ð3:58Þ
ðX2 þ X2r ÞðX2 þ X2c Þ
(3) Cross-level
Av b2 X2c X2
Sc ðXÞ ¼ ð3:59Þ
ðX2 þ X2r ÞðX2 þ X2c ÞðX2 þ X2s Þ
where the units of height and alignment PSDs are m2/(rad/m); As the cross-level
irregularity was measured by the inclination angle, the unit for Sc(X) becomes
1/(rad/m); X is the spatial frequency of track irregularity (rad/m); Xc, Xr and Xs are
the cut-off frequencies (rad/m); Av and Aa are the roughness constants (m2 rad/m);
b is half the distance between two the rolling circles of a wheelset (m), normally it is
set as 0.75 m.
(4) Gauge
No equation was given in Ref. [14] for the gauge PSD; it just stipulated that the
range of gauge variation be between −3 and 3 mm. In general, the PSD expressions
for gauge and cross-level irregularities are in similar form, so the equation of the
gauge PSD can be expressed as
Ag X2c X2
Sg ðXÞ ¼ m2 =ðrad=mÞ ð3:60Þ
ðX2 þ X2r ÞðX2 þ X2c ÞðX2 þ X2s Þ
The roughness coefficients and cut-off frequencies are shown in Table 3.4, in
which Ag are the reference values calculated under the assumption that the gauge
variation is between −3 and 3 mm.
“Low disturbance” is suitable for German high-speed railways whose opera-
tional speeds are 250 km/h and above, while “high disturbance” is suitable for
German ordinary railways whose speeds are below 250 km/h.
188 3 Excitation Models of Vehicle–Track Coupled System
Table 3.4 Roughness coefficients and cut-off frequencies for German track irregularity PSDs
Track class Xc (rad/m) Xr (rad/m) Xs (rad/m) Aa (m2 rad/m) Av (m2 rad/m) Ag (m2 rad/m)
Low 0.8246 0.0206 0.4380 2.119 10−7 4.032 10−7 5.32 10−8
disturbance
High 0.8246 0.0206 0.4380 6.125 10−7 1.08 10−6 1.032 10−7
disturbance
To date, there is still no complete system of track irregularity PSD standards, which
can represent the various track geometry states in China. But as mentioned above,
many studies have been carried out by the institutions on the track irregularity
PSDs; the PSD formula of various track types have been given according to the
measurement results. These are introduced as follows:
1. Track irregularity PSD of Chinese conventional main lines
This track irregularity PSD reflects the track geometry state of the Chinese existing
conventional main lines after the speedup renovation. A same analytic expression is
adopted for track height, cross-level, and alignment PSDs, but with different
coefficient values [10].
Aðf 2 þ Bf þ CÞ
Sðf Þ ¼ ð3:61Þ
f4 þ Df 3 þ Ef 2 þ Ff þ G
Table 3.5 Characteristic parameters of the track irregularity PSDs for Chinese existing speedup
main lines
Parameter A B C D E F G
Height (left) 1.1029 −1.4709 0.5941 0.8480 3.8016 −0.2500 0.0112
Height (right) 0.8581 −1.4607 0.5848 0.0407 2.8428 −0.1989 0.0094
Alignment (left) 0.2244 −1.5746 0.6683 −2.1466 1.7665 −0.1506 0.0052
Alignment (right) 0.3743 −1.5894 0.7265 0.4353 0.9101 −0.0270 0.0031
Cross-level 0.1214 −2.1603 2.0214 4.5089 2.2227 −0.0396 0.0073
3.5 Excitation Model of Random Track Irregularity 189
employed for the high-speed railway with operational speed between 200 and
250 km/h. The track irregularity PSDs of the high-speed railways were obtained
based on the field measurement data of typical ballastless and ballasted tracks.
Piece-wise fitting by the power function was adopted for the PSDs and each
wavelength segment of the PSD curves had the same expression as follows [11, 12]:
A
Sðf Þ ¼ ð3:62Þ
fn
where the unit of S(f) is mm2/(1/m); f is the spatial frequency(1/m), A and n are the
fitting coefficients. The fitting coefficients of the mean ballastless and ballasted track
irregularity PSDs for the Chinese high-speed railways are given in Table 3.6 and
Table 3.7, respectively. There are four segments for the fitting coefficients, their
corresponding cut-off spatial frequencies and wavelengths are shown in Tables 3.8
and 3.9. These parameters are suitable for ballastless track with operational speed
between 300 and 350 km/h and ballasted track with operational speed between 200
and 250 km/h. Note that the ballastless track irregularity PSD is applicable to the
spatial frequency range of 0.005–0.5 m−1 and the corresponding wavelength range
of 2–200 m; and the ballasted track irregularity PSD is applicable to the spatial
frequency range of 0.01–0.5 m−1 and the corresponding wavelength range of
2–100 m.
Previous study has shown that track irregularity PSDs estimated from a large
amount of irregularity data approximately obey the v2 distribution with two degrees
of freedom. The values given in Tables 3.6 and 3.7 correspond to the so-called
mean PSDs. For different track geometrical states, the percentile PSDs can be
estimated according to the mean track irregularity PSDs of the high-speed railways.
Sa ð f Þ ¼ C Sð f Þ ð3:63Þ
Table 3.6 Fitting coefficients for the mean ballastless track irregularity PSDs of the Chinese high-speed railways
Type of irregularity Segment 1 Segment 2 Segment 3 Segment 4
A n A n A n A n
Height 1.0544 10−5 3.3891 3.5588 10−3 1.9271 1.9784 10−2 1.3643 3.9488 10−4 3.4516
Alignment 3.9513 10−3 1.8670 1.1047 10−2 1.5354 7.5633 10−4 2.8171 – –
Cross-level 3.6148 10−3 1.7278 4.3685 10−2 1.0461 4.5867 10−3 2.0939 – –
Gauge 5.4978 10−2 0.8282 5.0701 10−3 1.9037 1.8778 10−4 4.5948 – –
3 Excitation Models of Vehicle–Track Coupled System
Table 3.7 Fitting coefficients for the mean ballasted track irregularity PSDs of the Chinese high-speed railways
Type of irregularity Segment 1 Segment 2 Segment 3 Segment 4
A n A n A n A n
Height 8.1981 10−2 1.6284 4.6880 10−5 3.4194 3.8215 10−2 1.2306 2.4609 10−4 4.1663
Alignment 2.5934 10−4 2.6607 1.3245 10−1 0.8041 7.1498 10−4 3.2979 – –
3.5 Excitation Model of Random Track Irregularity
Cross-level 9.9714 10−3 1.6103 5.5743 10−2 1.1336 1.6833 10−3 3.1315 – –
Gauge 1.3635 10−1 0.6903 2.1384 10−2 1.5653 8.1641 10−4 4.1305 – –
191
192 3 Excitation Models of Vehicle–Track Coupled System
Table 3.8 Spatial frequencies and corresponding wavelengths at the segment points for the
ballastless track irregularity PSDs of the Chinese high-speed railways
Type of Point separating first and Point separating second Point separating thirrd and
irregularity second segments and third segments fourth segments
Spatial Spatial Spatial Spatial Spatial Spatial
frequency wavelength frequency wavelength frequency wavelength
(1/m) m (1/m) m (1/m) m
Height 0.0187 53.5 0.0474 21.1 0.1533 6.5
Alignment 0.0450 22.2 0.1234 8.1 – –
Cross-level 0.0258 38.8 0.1163 8.6 – –
Gauge 0.1090 9.2 0.2938 3.4 – –
Table 3.9 Spatial frequencies and corresponding wavelengths at the segment points for the
ballasted track irregularity PSDs of the Chinese high-speed railways
Type of Point separating first and Point separating second Point separating thirrd and
irregularity second segments and third segments fourth segments
Spatial Spatial Spatial Spatial Spatial Spatial
frequency wavelength frequency wavelength frequency wavelength
(1/m) m (1/m) m (1/m) m
Height 0.0155 64.5 0.0468 21.4 0.1793 5.6
Alignment 0.0348 28.7 0.1232 8.1 – –
Cross-level 0.0270 37 0.1735 5.8 – –
Gauge 0.1204 8.3 0.2800 3.6 – –
Table 3.10 Transformation coefficients between the mean and percentile track irregularity PSDs
of the Chinese high-speed railways
Transformation Percentile (%)
coefficient 10.0 20.0 25.0 30.0 50.0 60.0 63.2 70.0 75.0 80.0 90.0
C 0.105 0.223 0.288 0.357 0.693 0.916 1.000 1.204 1.386 1.609 2.303
where the unit of S(f) is mm2/(1/m) and f is the spatial frequency (1/m). Its
applicable wavelength range is 0.01–1 m.
To evaluate high-frequency vibrations caused by short-wavelength rail surface
irregularity in high-speed railways, the Train and Track Research Institute at
Southwest Jiaotong University (the author’s research group) carried out field
measurements in Tianjing–Qinhuangdao high-speed railway line using the
“muller-BBM” rail surface roughness gauge. The test section was located at the
China Railway Track System (CRTS) II ballastless track on a bridge, as shown in
Fig. 3.39. The result is shown in Fig. 3.40, which could only be regarded as a
sample of the short-wavelength rail vertical irregularity for Chinese high-speed
railway.
3.5 Excitation Model of Random Track Irregularity 193
10
-10
1 0.1 0.01
Wavelength (m)
Two kinds of Chinese track irregularity PSDs are introduced in the previous section
based on the measured track geometric parameters, namely the track irregularity
PSDs of conventional railway speedup lines and of the high-speed railway lines.
However, it is not clear what their essential features are and what the differences are
compared to the typical track irregularity PSDs? It is necessary to answer these
questions to facilitate the rational selection of the random track irregularity dis-
turbances for the dynamics analysis. In Ref. [16], the differences of vehicle dynamic
performances under the excitation of Chinese conventional main lines and typical
foreign track irregularity PSDs were compared through dynamics simulation. Here,
from the viewpoint of track irregularity PSD characteristics itself, the differences
between the Chinese track irregularity PSDs and the American, German track
irregularity PSDs are analyzed and compared.
194 3 Excitation Models of Vehicle–Track Coupled System
(a) (b)
2
10
2
10
PSD [mm2(1/m)]
PSD [mm2(1/m)]
0 0
10 10
-2 -2
10 10
-4 -4
10 10
30 10 1 30 10 1
Wavelength (m) Wavelength (m)
Fig. 3.41 Comparison of the PSDs for conventional railways: a alignment and b height
3.5 Excitation Model of Random Track Irregularity 195
(a) (b)
104 104
102 102
PSD [mm2(1/m)]
PSD [mm2(1/m)]
100 100
10-2 10-2
Fig. 3.42 Comparison of the PSDs for high-speed railways: a alignment and b height
196 3 Excitation Models of Vehicle–Track Coupled System
For the height irregularity (Fig. 3.42b), the Chinese ballastless track irregularity
PSD is generally better than both the German high and low disturbance PSDs in the
whole wavelength range of 2–200 m, especially for the wavelength from 10 to
100 m. The Chinese ballasted track PSD is slightly poorer than the Chinese bal-
lastless track PSD, but it is obviously better than German low and high disturbance
PSDs in the wavelength range of 2–100 m, especially for the wavelengths from 10
to 60 m. Similarly, it can be deduced that the vertical passenger ride comfort should
be good under the excitation of the Chinese ballastless or ballasted track
irregularities.
3. Conclusions
The geometrical state of the Chinese conventional railway lines is generally more
irregular than the FRA 5th and 6th class tracks. The amplitudes are obviously
higher, especially when the wavelengths are shorter than 20 m. But for the
wavelength above 20 m, the PSDs of the Chinese conventional main lines are better
than that of the FRA 5th class track, and are better than that of the FRA 6th class
track when the wavelength is above 25 m.
The ballastless and ballasted track irregularity PSDs of the Chinese high-speed
railways are obvious smoother than the German low disturbance track irregularity
PSD in the wavelength range of 2–200 m and 2–100 m, respectively, which is more
noticeable for the longer wavelength irregularity PSDs above 10 m. The compar-
ison shows that the Chinese high-speed tracks have rather excellent geometrical
smoothness.
It can be seen from above that the power spectral density function is usually
employed for representing the random track irregularity. However, for the nonlinear
vehicle–track coupled dynamics model established in Chap. 2, time-domain input is
generally adopted as system excitation for convenience of numerical simulation.
Therefore, it is necessary to convert the random track irregularity PSD functions
into spatial irregularity samples varying with distance (time-domain samples can be
obtained accordingly). For this purpose, proper time–frequency conversion method
should be used, and the accuracy of the method is crucial for the true representation
of an actual track spatial geometry state.
At present, there are several existing methods to realize time–frequency con-
version, such as the quadratic-filtering method [17], trigonometric-series method
[18], white-noise filtering method, and so on. These methods have various problems
when they are applied to the numerical simulation of random track irregularities.
For example, suitable filters have to be designed for different track irregularity types
3.5 Excitation Model of Random Track Irregularity 197
in the case of the quadratic-filtering method, which thus lacks versatility. Here a
new algorithm based on frequency-domain PSD equivalence is introduced [19], in
which the spectral amplitude and random phase are first obtained according to the
random track irregularity PSDs and then the track irregularity time-domain samples
are computed through Inverse Fast Fourier Transform (IFFT).
According to Ref. [19], the PSD Sxx(k) has a definite relationship with the signal
spectrum at the discrete sampling points as
( )( X )
1XN 1
2p 1 N 1 2p
Sxx ðkÞ ¼ xs exp i k s xj exp i k j
N s¼0 N N j¼0 N
ð3:65Þ
1 1
¼ 2 jDFT½xs j2 ¼ 2 ½X ðk ÞX ðkÞ
N N
fmin fmax f
o
N0 Nf Nr/2 Nr
198 3 Excitation Models of Vehicle–Track Coupled System
As the time sequence X(k) is a random process, the spectrum phase will
incorporate randomness. Suppose nn is the independent phase sequence with zero
mean value and |nn | = 1. Because the Fourier transform of a real sequence will be a
complex sequence (the real part is even symmetry, the imaginary part is odd
symmetry), nn should be complex. Therefore,
Simulated PSD
Analytical PSD
PSD (cm2/Hz)
Z0(t) (cm)
t (s) f (Hz)
Fig. 3.44 An example on the time/frequency conversion of random track irregularity based on the
frequency-domain PSD equivalence. a Simulation results of random track irregularity time series;
b comparison of simulated and analytical PSD
3.5 Excitation Model of Random Track Irregularity 199
track irregularity PSDs of the Chinese high-speed railways (as shown in Eq. (3.62))
are carried out, respectively, below. The track irregularity samples as functions of
distance are obtained, as shown in Figs. 3.45 and 3.46, where the wavelength range
of the ballastless track irregularities is from 2 to 200 m, and the wavelength range
(a)
3
Amplitude (mm)
2
1
0
-1
-2
-3
0 400 800 1200 1600 2000
Distance (m)
(b)
3
Amplitude (mm)
2
1
0
-1
-2
-3
0 400 800 1200 1600 2000
Distance (m)
(c)
5.0
Amplitude (mm)
2.5
0.0
-2.5
-5.0
0 400 800 1200 1600 2000
Distance (m)
(d) 5.0
Amplitude (mm)
2.5
0.0
-2.5
-5.0
0 400 800 1200 1600 2000
Distance (m)
Fig. 3.45 Numerical simulation of random track irregularity samples of ballastless track of
Chinese high-speed railway. a Alignment irregularity (left rail); b alignment irregularity (right
rail); c height irregularity (left rail); d height irregularity (right rail)
200 3 Excitation Models of Vehicle–Track Coupled System
(a)
3
2
Amplitude (mm)
1
0
-1
-2
-3
0 400 800 1200 1600 2000
Distance (m)
(b)
3
2
Amplitude (mm)
1
0
-1
-2
-3
0 400 800 1200 1600 2000
Distance (m)
(c)
4
Amplitude (mm)
-2
-4
0 400 800 1200 1600 2000
Distance (m)
(d)
4
Amplitude (mm)
-2
-4
0 400 800 1200 1600 2000
Distance (m)
Fig. 3.46 Numerical simulation of random track irregularity samples of ballasted track of Chinese
high-speed railway. a Alignment irregularity (left rail); b alignment irregularity (right rail);
c height irregularity (left rail); d height irregularity (right rail)
3.5 Excitation Model of Random Track Irregularity 201
of ballasted track irregularities is from 2 to 100 m. It can be seen in Fig. 3.45 that
for the ballastless tracks of the Chinese high-speed railways, the amplitude of the
height irregularity changes within the range of −5 to 5 mm and the amplitude of the
alignment irregularity varies between −3 and 3 mm, showing very good track
geometrical condition. For the ballasted tracks shown in Fig. 3.46, the amplitude of
the height irregularity changes within the range of −4 to 4 mm and the amplitude of
the alignment varies between −3 and 3 mm, also showing pretty good track geo-
metrical condition.
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7. China Academy of Railway Sciences, Zhengzhou Railway Bureau. Research report on the
comprehensive operation test of the high-speed train with maximum speed of 240 km/h on
Zhengzhou–Wuhan railway. TY-1345. Beijing: China Academy of Railway Sciences; 1998
(in Chinese).
8. Luo L. Track random excitation functions. China Railw Sci. 1982;13(1):74–110 (in Chinese).
9. Research Group on Random Vibration of Changsha Railway Institute. Research on random
excitation functions of rolling stock/track system. J Chang Railw Inst. 1985;(2):1–36 (in
Chinese).
10. Railway Engineering Research Institute of China Academy of Railway Sciences. Study on the
track irregularity power spectrum density of Chinese main lines, TY-1215. Beijing: China
Academy of Railway Sciences; 1999 (in Chinese).
11. Kang X, Zhai WM, Liu XB, et al. PSD of ballastless track irregularities of high-speed railway.
Standard of National Railway Administration of People’s Republic of China: TB/T
3352-2014. Beijing: China Railway Publishing House; 2014 (in Chinese).
12. Li GQ, Gao L, Zhai WM, et al. PSD of ballast track irregularities of high-speed railway.
Standard of China Railway Corporation: Q/CR 508-2016. Beijing: China Railway Publishing
House; 2016 (in Chinese).
13. Garg VK, Dukkipati RV. Dynamics of railway vehicle systems. Ontario: Academic Press
Canada; 1984.
14. Munich Research Center of German Federal Railway. ICE technology assignment for
inter-city express train; 1993.
15. Wang L. Random vibration theory of rail/track structure and its application in the rail/track
vibration isolation. Ph.D. thesis. Beijing: China Academy of Railway Sciences; 1988 (in
Chinese).
202 3 Excitation Models of Vehicle–Track Coupled System
16. Chen G, Zhai WM, Zuo HF. Comparing track irregularities PSD of Chinese main lines with
foreign typical lines by numerical simulation computation. J China Railw Soc. 2001;23
(3):82–7 (in Chinese).
17. Ontes RK, Enochson L. Digital time series analysis. New York: Wiley; 1972.
18. Katsu H. Random vibration analysis. Beijing: Seismological Press; 1977.
19. Chen G, Zhai WM. Numerical simulation of the stochastic process of railway track
irregularities. J Southwest Jiaotong Univ. 1999;34(2):138–42 (in Chinese).
20. Zhai WM, Xia H. Train–track–bridge dynamic interaction: theory and engineering
application. Beijing: Science Press; 2011 (in Chinese).
Chapter 4
Numerical Method and Computer
Simulation for Analysis of Vehicle–
Track Coupled Dynamics
Abstract As can be seen from Chap. 2, the vehicle–track coupled system belongs
to a large-scale dynamic system including strong nonlinearities. It is impossible to
theoretically solve dynamic response for such a complicated system. Time-stepping
integration provides the best way for the numerical solution of the equations of
motion of the vehicle–track coupled dynamics system. This chapter discusses the
application of time integration methods to the analysis of vehicle–track coupled
dynamics, focusing on the application of a new simple fast time integration method
(Zhai in Int J Numer Meth Eng. 39(24):4199–214, 1996 [1]), and introduces
associated computer simulation programs.
Step-by-step time integration methods are widely used to solve the equations of
multi-degree of freedom systems in mechanical and structural dynamics, especially
in nonlinear system dynamics. For large-scale dynamic problems, as is frequently
the case in modern dynamic analysis in practical engineering problems, the cal-
culation efficiency is always a crucial concern, especially in the beginning of the
study of the vehicle–track coupled dynamics. At that moment, the computer
technology was not advanced. Therefore, it is necessary to develop more efficient
time integration algorithms for large-scale complex system analysis.
Basically, there are two general classes of algorithms for dynamic problems:
implicit and explicit. When implicit algorithms, such as the Newmark-b method [2]
and the Wilson-h method [3], are applied to large-scale dynamic problems, the
computational time and the cost increase dramatically with the degrees of freedom
of system because large-scale simultaneous algebraic equations must be solved in
each time step although large time steps are permitted due to their good numerical
stabilities. On the contrary, explicit schemes tend to be inexpensive. Hoff and
Taylor [4] have pointed out that if lumped mass and damping matrices are used, an
explicit scheme probably consists of pure vector operations. This is very convenient
© Science Press and Springer Nature Singapore Pte Ltd. 2020 203
W. Zhai, Vehicle–Track Coupled Dynamics,
https://doi.org/10.1007/978-981-32-9283-3_4
204 4 Numerical Method and Computer Simulation for Analysis …
for computers with vector processors and the disadvantage of the explicit schemes
in the aspect of conditional stability can be effectively alleviated through a vec-
torized implementation. Therefore, these explicit algorithms become more com-
petitive on large-scale problems compared to the more stable implicit algorithms.
As we know, advances have been made to improve explicit methods. A state of
the art in high-order explicit schemes for use in structural dynamic applications has
been summarized in [4] by Hoff and Taylor in 1990. As indicated by these authors,
it seems that the second-order accurate central difference method still remains the
most popular explicit algorithm although several new developed explicit methods
have their own advantages. These newly developed methods usually have lower
stability than the central difference method, and some of them need more than one
function evaluation (calculation of the internal forces), that is the most expensive
part in nonlinear problems which are discretized by finite elements, finite differ-
ences or boundary elements.
When applied to nonlinear system dynamics, the central difference method is
still supposed to solve a set of linear algebraic equations in each step unless the
problem is an ideal one which satisfied the following conditions: (a) the mass
matrix is diagonal, and (b) the damping matrix can be neglected or is proportional
to the mass matrix. However, condition (b) is difficult to meet for practical engi-
neering problems, while condition (a) can usually be observed. For the problem of
vehicle–track coupled dynamics, the damping is not only non-negligible but also
complicated and generally, the damping matrix is not proportional to the mass
matrix, while the mass matrix is usually diagonal or is easily diagonalized. Thus,
there is no obvious advantage for the central difference method to solve the vehicle–
track coupled dynamics problem because a set of large-scale linear algebraic
equations has to be solved in each time step.
Fortunately, a new simple explicit method was developed by the author in the
early 1990s [1], originally aiming to fast solve long train longitudinal dynamics
problem at that time. This new explicit method has at least the same stability limit
as the central difference method, and needs only simple vector operations in each
time step as long as the mass matrix of the solved system is diagonal, no matter
what form the damping matrix of the system is. The computational efficiency is
greatly enhanced. It has been found that it is fast, convenient, and economical to use
this explicit method to analyze large-scale nonlinear dynamic problems in engi-
neering. Therefore, it is suggested to employ this new simple fast explicit method to
numerically solve the vehicle–track coupled dynamics response.
It is worth mentioning that this explicit method has been positively reviewed and
widely adopted as so-called “Zhai method” or “Zhai algorithm” in international
journals on engineering computation method, mechanics, vibration and structural
engineering during the past decade [5–18]. Representatives are as follows: Rio on
Advances in Engineering Software [5], Rezaiee–Pajand on Engineering
Computations [6], Zhang on Journal of Sound and Vibration [7], Chen on ASME
Journal of Vibration and Acoustics [8] and on Journal of Sound and Vibration [9],
Zhou on Vehicle System Dynamics [10], Banimahd on Proceedings of the
4.1 Time Integration Methods for Solving Large-Scale Dynamic Problems 205
€ þ CX_ þ KX ¼ F
MX ð4:1aÞ
or
MA þ CV þ KX ¼ F ð4:1bÞ
where M, C, and K are the mass, damping, and stiffness matrices, respectively; F is
the vector of applied loads of the system (a given function of time, F ¼ FðtÞ); X,
V and A (i.e., X, X, _ and X) € are the vectors of displacements, velocities, and
accelerations, respectively. The initial-value problem consists of finding a function
X ¼ XðtÞ, which satisfies Eq. (4.1a, 4.1b), and the initial conditions
Xð0Þ ¼ X0
ð4:2Þ
Vð0Þ ¼ V0
Inspired by the well-known Newmark-b implicit method, the new explicit inte-
gration scheme for approximate solutions of Eqs. (4.1a, 4.1b) and (4.2) is con-
structed as follows:
Xn þ 1 ¼ Xn þ Vn Dt þ ð12 þ wÞAn Dt2 wAn1 Dt2
ð4:3Þ
Vn þ 1 ¼ Vn þ ð1 þ uÞAn Dt uAn1 Dt
206 4 Numerical Method and Computer Simulation for Analysis …
where Xn , Vn , and An are the approximations to X(t = nDt), V(t = nDt) and
A(t = nDt), respectively; Dt is the time step, and w and u are free parameters that
control the stability and numerical dissipation of the algorithm.
Substituting Eq. (4.3) into Eq. (4.1a, 4.1b) at time step t = (n + 1)Dt
~n þ 1
An þ 1 ¼ M1 F ð4:5Þ
where
h i
~ n þ 1 ¼ Fn þ 1 KXn ðC þ KDtÞVn ð1 þ uÞC þ 1 þ w KDt An Dt
F
2 ð4:6Þ
þ ðuC þ wKDtÞAn1 Dt
To investigate the stability of Zhai method, we need only consider the linear
homogeneous form of Eq. (4.4) without damping for the single-degree of freedom
case as
an þ 1 þ x 2 x n þ 1 ¼ 0 ð4:8Þ
where
pffiffiffiffiffiffiffiffiffi
x¼ k=m ð4:9Þ
4.2 New Simple Fast Explicit Time Integration Method: Zhai Method 207
where
X ¼ xDt ð4:11Þ
1þZ
k¼ ð4:13Þ
1Z
and then the requirement for stability is simply that Re(Z) 0. According to the
Routh–Hurwitz criterion, stable steps can be derived, which have been given in
Table 4.1 in detail.
It can be seen from Table 4.1 that the range of stable steps is very wide. When
u = w = 1/2, the stability limit is Dt = 2/x, which is the same as that of the central
difference method. And if u = 1/8 and w = 1/4, the stable step range is
qffiffi qffiffi
5 Dt x
2 6 2 4
x 3, in which the maximum stable step is larger than 2/x.
1€
An1 ¼ An A_ n Dt þ A n Dt
2
ð4:15Þ
2
where a supposed dot denotes a time derivative. Substituting Eq. (4.15) into
Eq. (4.3) yields
8
> 1
< Xn þ 1 ¼ Xn þ Vn Dt þ An Dt2 þ wA_ n Dt3 wA
1 €
n Dt þ OðDt Þ
4 5
2 2 ð4:16Þ
>
: Vn þ 1 ¼ Vn þ An Dt þ uA_ n Dt2 1 uA
€ n Dt3 þ OðDt4 Þ
2
If w = 1/6 the order of accuracy of E(X) is O(Dt4), and if u = 1/2 the order of
accuracy of E(V) will be O(Dt3). Otherwise, the orders of accuracy decrease to
O(Dt3) and O(Dt2), respectively. Obviously, the Zhai method has the same order of
accuracy as that of the implicit Newmark-b method.
In order to show numerical dissipation and dispersion of the Zhai algorithm, the
analytical procedure used by Hilber et al. [19] is adopted here. Rewriting Eq. (4.10)
as follows:
where
8
>
> P1 ¼ 1
1
wþ
1 2
>
> X
< 2
2
1 ð4:19Þ
>
> P ¼ 1 þ þ u 2w X2
> 2
> 2
:
P3 ¼ ðu wÞX2
4.2 New Simple Fast Explicit Time Integration Method: Zhai Method 209
X
3
xn ¼ ci kni ð4:20Þ
i¼1
when
4
X\ ð4:22Þ
2u þ 1
k1 and k2 become two complex conjugate eigenvalues of Eq. (4.12), called prin-
cipal roots, which satisfy ∣k1,2∣ 1. Let’s write k1,2 as
f iÞ
k1;2 ¼ P Qi ¼ exp½Xð ð4:23Þ
where
8
>
> P ¼ Pp1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
<
Q ¼ P2 P21
¼ tan1 ðjQ=PjÞ ð4:24Þ
>
> X
:
f ¼ lnðP2 þ Q2 Þ=ð2XÞ
In this case, the solution of Eq. (4.18) could be written in the form
xn ¼ expðfxt
n Þ½c1 cosðxt
n Þ þ c2 sinðxt
n Þ ð4:25Þ
where
¼ X=Dt
x
ð4:26Þ
tn ¼ nDt
Δt/T
AD ¼ 1 expð2pfÞ
1 ð4:27Þ
TD ¼ ðT TÞ=T ¼ X=X
Δt/T
In order to examine the actual integration accuracy and stability of the Zhai method,
three numerical examples including a linear, a piecewise linear, and a nonlinear
problem are employed and analyzed in this section.
1. Example I
In the linear case, we consider the following initial-value problem:
8
< €x þ x ¼ 0
x_ ð0Þ ¼ 0 ð4:28Þ
:
xð0Þ ¼ 1
When the time step is sufficiently small, e.g., Dt = 0.05 s, the numerical result with
the Zhai method is almost the same as the exact solution. Comparison of the Zhai
method with the trapezoidal rule and with the Houbolt method is shown in Fig. 4.3,
with a time step of Dt = 0.5 s.
212 4 Numerical Method and Computer Simulation for Analysis …
Τime (s)
Fig. 4.3 Some integrated results with time step Dt = 0.5 s for linear case (Reprinted from Ref.
[1], Copyright 1996, with permission from John Wiley & Sons.)
To examine the stability, we consider the explicit scheme with the same value, 0.5,
both for u and w. According to Table 4.1, the critical stable step is Dtcr = 2/x = 2 s
for this example, which is well proved in Fig. 4.4. A little increment of Dt from 2.0 s
to 2.01 s causes a divergent result.
-2
-4
0 5 10 15 20 25
Time (s)
4.2 New Simple Fast Explicit Time Integration Method: Zhai Method 213
2. Example II
Consider the piecewise linear system such as
8
< €x þ kx ¼ 0
x_ ð0Þ ¼ 10 ð4:29Þ
:
xð0Þ ¼ 0
where the variation of parameter k is given in Fig. 4.5. Numerical results obtained
by the Zhai method are analyzed and compared for different values of u and w. It is
shown that if u = c = 1/2, the best accuracy and stability of the new explicit scheme
is obtained. Therefore, the optimum integration parameters of the Zhai method
(u = w = 1/2) will be adopted in the following calculation including the simulation
of vehicle–track coupled dynamics.
Table 4.2 lists the comparison of the Zhai method (u = w = 1/2) with the exact
solution for two kinds of time steps: Dt1 = 0.005 s and Dt2 = 0.05 s. It can be seen
from Table 4.2 that when the time step is set to 0.005 s, the integration results of
the Zhai method are almost the same as the exact solution. When the time step
increases to 0.05 s, the maximum error is only 2.1%.
Displacement (m)
1
-1
-2
0.0 0.1 0.2 0.3 0.4 0.5
Time (s)
3. Example III
Consider a nonlinear system with cubically hardening spring quoted in Ref. [20]
8
< €x þ 100x þ 1000x3 ¼ 0
x_ ð0Þ ¼ 60 ð4:30Þ
:
xð0Þ ¼ 0
The same time step, Dt = 0.015 s, employed by Park in Ref. [20] is adopted
here. Numerical response processes with the Zhai method, the Newmark-b method
(b = 1/4), the Houbolt method, and the Park method are plotted in Fig. 4.6. It can
be seen that the response by the Zhai method traces the actual behavior more
closely than that by the Newmark method, and much more closely than those by the
Park method and by the Houbolt method, see Ref. [1].
Up till now, the Zhai method has been widely applied to analysis of railway
dynamics problems, especially for vehicle–track system dynamics [7, 11, 16, 17,
21–24], for train–track–bridge dynamic interaction [25–30], and even for train–
track–tunnel–soil dynamic interactions [10]. Nowadays, it becomes the most
common method for analyzing the vehicle–track coupled dynamics problems.
4.3 Application of Zhai Method to Analysis of Vehicle–Track Coupled Dynamics 215
MV AV þ CV VV þ KV XV ¼ FV ð4:31Þ
MT A T þ C T V T þ K T X T ¼ F T ð4:32Þ
where MV is the mass matrix of the vehicle; CV and KV are the damping and the
stiffness matrices which can depend on the current state of the vehicle subsystem to
describe nonlinearities within the suspension; XV, VV, and AV are the vectors of
displacements, velocities, and accelerations of the vehicle subsystem, respectively;
FV is the vehicle subsystem load vector representing the nonlinear wheel–rail
contact forces determined by the wheel–rail coupling model and the external forces
including gravitational forces and forces resulting from the centripetal acceleration
when the vehicle is running through a curve; MT is the mass matrix of the track
structure; CT and KT are the damping and the stiffness matrices of the track sub-
system; XT, VT, and AT are the vectors of displacements, velocities, and acceler-
ations of the track subsystem; and FT is the load vector of the track subsystem
representing the nonlinear wheel–rail forces.
The main solution procedure for the vehicle–track coupled dynamics consists of
the following six steps:
Step 1 Calculate the displacement XV,n+1 and the velocity VV,n+1 of the vehicle
subsystem at the time (n + 1)Dt by using Eq. (4.3) based on the states of the system
at time nDt and time (n − 1)Dt;
Step 2 Calculate the displacement XT,n+1 and the velocity VT,n+1 of the track
subsystem at the time (n + 1)Dt by using Eq. (4.3) based on the states of the system
at time nDt and time (n − 1)Dt;
Step 3 Estimate the wheel–rail normal contact forces and creep forces with the
wheel–rail coupling model based on the calculated displacements and velocities of
wheels and rails from steps 1 and 2;
Step 4 Calculate the load vectors FV,n+1 and FT,n+1 at the time (n + 1)Dt from the
wheel–rail contact forces and from the external forces due to the curvature,
superelevation, etc., for the current position of each body of the vehicle;
216 4 Numerical Method and Computer Simulation for Analysis …
Step 5 Compute the acceleration AV,n+1 of the vehicle subsystem from Eq. (4.31) at
time (n + 1)Dt:
AV;n þ 1 ¼ M1
V ½FV;n þ 1 CV VV;n þ 1 KV XV;n þ 1 ð4:33Þ
Step 6 Compute the acceleration AT,n+1 of the track subsystem from Eq. (4.32) at
time (n + 1)Dt:
AT;n þ 1 ¼ M1
T ½FT;n þ 1 CT VT;n þ 1 KT XT;n þ 1 ð4:34Þ
Because the mass matrices MV of the vehicle subsystem and MT of the track
subsystems are diagonal matrices, no algebraic equations have to be solved to
obtain the inverse mass matrices M−1 −1
V and MT required in Eqs. (4.33) and (4.34).
Thus, computational efficiency is greatly enhanced.
One of the most important issues for the Zhai method to be applied to analyzing the
vehicle–track coupled dynamics problem is the determination of the integration
time step. How to obtain the critical stable time step of the Zhai method for such a
complicated engineering dynamics system with strong nonlinearities? How to
determine a rational time step in order to ensure the numerical solution with suf-
ficiently high accuracy? The author believes that the numerical trial method is able
to provide the possibility to deal with these problems.
A meticulous numerical trial has been carried out for the Zhai method to solving
dynamic responses of the vehicle–track coupled system. The variation of system
dynamic responses with the change of integration time step was carefully investi-
gated. Each key dynamics indices of the whole coupled system were observed.
Here is just shown some results of an example taken for the vertical vehicle–track
coupled dynamic system in the case of a freight car passing over a dipped rail joint
on a traditional ballasted track. Figure 4.7 shows the variations of the calculated
vertical wheel–rail forces with the time step Dt, in which P1 represents the
high-frequency wheel–rail impact force and P2 denotes the low-frequency wheel–
rail force. Figure 4.8 shows the variations of the calculated wheel–rail system
accelerations with the time step Dt, where aw is the wheelset vertical vibration
acceleration as representative of the vehicle subsystem and ab is the ballast vertical
vibration acceleration as representative of the track subsystem. It can be seen
from both figures that the critical stable time step of the Zhai method is
4.3 Application of Zhai Method to Analysis of Vehicle–Track Coupled Dynamics 217
P (kN)
with the time step Dt
Δte Δtcr
Δt (ms)
Δte Δtcr
Δt (ms)
Dtcr = 1.5 10−4 s when used for solution of the present vertical vehicle–track
dynamics problem. In order to get a high calculation accuracy, however, the
actually adopted effective time step Dte is suggested to be smaller than the critical
time step Dtcr, for example, Dte = 1.0 10−4 s.
As to the lateral dynamics problem of the vehicle–track coupled system,
numerical trial results indicate that the effective time step Dte could be 1.0 10−4 s
for the passenger coach and track system and 5.0 10−5 s for the freight wagon
and track system. Although the time step has to be very small in order to reflect
high-frequency wheel–rail contact vibration, the computational efficiency is quite
high due to the needless of solving a large-scale set of algebraic equations at each
time step. A common micro-computer is enough to implement the simulation of the
vehicle–track coupled dynamics.
218 4 Numerical Method and Computer Simulation for Analysis …
There are two undetermined issues arising from the established vehicle–track
coupled dynamics models in Chap. 2. One is the calculated length of track, l, in the
actual simulation. The other is the mode number of rail, NM, adopted in the cal-
culation. Here, the numerical trial method will be used again to determine these two
values.
Taking the vertical vehicle–track coupled dynamics model as an example, the
numerical trial is carried out for the track length under the same case as used in
above Sect. 4.3.2, i.e., a freight car passing over a dipped rail joint on a traditional
ballasted track. Figures 4.9 and 4.10 give the numerical trial results of the influence
l (m)
l (m)
4.4 On Some Key Issues in Solving Process of Vehicle–Track Coupled Dynamics 219
of the length on the vertical wheel–rail forces and on track accelerations, where ar
and as are the vertical accelerations of rail and sleeper. It is observed from these two
figures that the track length has obvious influence on calculated dynamic responses,
however, little influence can be found if the calculated track length is larger than
100 m. Therefore, it is enough for the calculated track length to be set as 100 m in
the simulation of vertical vehicle–track interaction. For the lateral dynamics
problem of the vehicle–track coupled system, numerical trial reveals the similar
results of the vertical problem. Thus a conclusion can be made: the track length
could be set as 100 m for simulation of general vehicle–track coupled dynamics
problems due to local short-wavelength track irregularities such as rail joints, void
sleeper, and defects on the wheel and rail surfaces. In these cases, the moving
distance of a vehicle needed in calculation could be short and the excitation can be
input with the moving-vehicle method as introduced in Sect. 3.1.2 of Chap. 3.
100 m is also long enough for the calculated track length if the excitation is input
by the fixed-point method as shown in Sect. 3.1.1 of Chap. 3. For long-distance
track simulation with the moving-vehicle method, we can adopt the tracking win-
dow method to deal with the infinite length of the track. More detail can be referred
to Sect. 3.1.3 in Chap. 3.
The mode number of the rail is important to describe the rail vibration behavior.
Different rail mode numbers reflect different frequency components of the vibration.
The higher the rail mode number selected, the higher the rail vibration frequency
included. In order to felicitously simulate the vehicle–track system dynamics
behavior, especially the high-frequency wheel–rail contact vibration, the maximum
mode number of rail should be selected sufficiently high. Generally, it is required
that the highest frequency reflecting by the maximum mode number of rail should
be at least twice as high as the concerned rail frequency in simulation. Rational
maximum mode number can also be determined by use of the numerical trial
method. Our investigation indicates that there is a matching relation between the
rational rail mode number and the calculated length of track
NM ¼ Nl ð4:35Þ
where NM is the maximum mode number of rail and Nl is the number of rail
fastening supporting points within the calculated track length. If the calculated track
length increases, the rail mode number should increase accordingly.
In the train–track spatially coupled dynamics model which has been introduced in
Sect. 2.4 of Chap. 2, there exists a huge number of DOFs. To achieve the fast
solution on the dynamic responses of this large-scale system, a fast explicit
numerical integration algorithm, namely the Zhai method [1], is adopted. However,
220 4 Numerical Method and Computer Simulation for Analysis …
Fig. 4.11 Flow chart for calculation of train–track coupled dynamics model
In order to clarify this question, the simulation experiments for two short trains
with the freight wagon marshaling mode and the locomotive–wagon marshaling
mode are carried out to determine the minimum value of the number N required.
2. Determination of the required number of the vehicles modeled as a three-
dimensional model
Two cases of a heavy-haul train formation are considered: (1) the short train is only
composed of the freight wagons, as shown in Fig. 4.13. The vehicles at both the
train ends are modeled as the single-mass model, while other wagons are modeled
as the 3D model; (2) for the combined train, the locomotives are usually distributed
at the head, the middle and the end positions. The dynamic behaviors of the
locomotive may be more sensitive to the coupler force than the wagons due to its
softer horizontal suspension stiffness. For these reasons, once the focused vehicle in
the train is a locomotive, all of the centralized locomotives at this position should be
considered as the detailed 3D model. Then the key question arises that whether it is
necessary to use the 3D model to simulate the neighboring freight wagons of the
locomotives. Figure 4.14 displays the dynamic model of the marshaling train used
for this case.
In the numerical experiments, the heavy-haul locomotive HXD2 and freight
wagon C80, which are widely used in China are applied for the analysis under the
braking conditions. The dynamics indices of the focused freight wagon are
extracted to illustrate the effect of the number of the vehicles represented by the 3D
model. For example, the lateral wheelset forces for the two cases are shown in
Fig. 4.15.
The principle of using the 3D model for the dynamic analysis of a heavy-haul
train can be concluded based on the results of numerical experiments as: (1) if the
maximum coupler force occurs in the freight wagon which is far away from the
Fig. 4.14 Dynamic model of a train composed by locomotives and freight wagons
4.4 On Some Key Issues in Solving Process of Vehicle–Track Coupled Dynamics 223
Fig. 4.15 Wheelset lateral force of the first axle in the focused vehicle with different number of
3D model: a case 1, and b case 2
locomotives, at least three wagons around the maximum coupler force should be
modeled as the 3D models; (2) if the maximum coupler force appears in the
locomotives, the centralized distributing locomotives and adjacent two freight
wagons in this area ought to be simulated simultaneously by the 3D models.
The first simulation software is called VICT, which was developed by the author in
early 1990s based on the vehicle–track vertically coupled dynamics model and used
for analyzing the vertical dynamic interactions between railway vehicles and tracks,
especially for evaluating dynamic effects of vehicles on track structures.
Figure 4.16 gives the flowchart of the VICT simulation system.
224 4 Numerical Method and Computer Simulation for Analysis …
Start
No Yes
No
Simulation stop
t=t+Δt
t T?
Yes
End
Vehicle parameters
System pa rameter s input subroutine
Pre-processing module
Track parameters
Standard profile
Wheel /rail profiles input subroutine
Measured profile
Track spectrum
Track irregularity input subroutine
Measured irregularity
Zhai method
Numerical integration subroutine
implementation
Passenger vehicle
Locomotive
Contact geometry
Creep force
Ballast track
Track dynamics solution subroutine
Ballastless track
Running safety
Ride comfort
Track deformation
Start
Stability
No Yes
Loss of wheel-rail contact ?
No
End of simulation t=t+Δt
t T?
Yes
End
Fig. 4.18 Flow chart of TTISIM simulation system (Reprinted from Ref. [31], Copyright 2009,
with permission from Taylor & Francis.)
228 4 Numerical Method and Computer Simulation for Analysis …
References
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engineering. Int J Numer Meth Eng. 1996;39(24):4199–214.
2. Newmark NM. A method of computation for structural dynamics. J Eng Mech Div ASCE.
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References 229
Abstract Field test and theoretical analysis are the two fundamental investigation
approaches in the field of vehicle–track coupled dynamics. Field test is the essential
step in validating the dynamics model and the simulation system. The validated
dynamics model and simulation system can then be used to optimize the dynamic
performance of the system, leading to a shortened designing phase and a reduced
need for the costly field test. Furthermore, field test is also the last step in examining
the reliability of the vehicle–track system design. Section 5.1 introduces the field
test methods including field test methods of vehicle dynamics and track dynamics.
Typical tests that were carried out by the author and his team are then presented,
which includes dynamic performance tests for typical high-speed passenger train
and freight train (Sect. 5.2), as well as vehicle–track dynamic interaction test for a
high-speed train on slab track and track dynamics tests for a heavy-haul train
passing a rail joint and for an ordinary train negotiating a small-radius curve
(Sect. 5.3).
© Science Press and Springer Nature Singapore Pte Ltd. 2020 231
W. Zhai, Vehicle–Track Coupled Dynamics,
https://doi.org/10.1007/978-981-32-9283-3_5
232 5 Field Test on Vehicle–Track Coupled System Dynamics
Field tests of vehicle dynamics covers the areas of running stability, running safety
and ride comfort.
For low-speed operation tests in China, the test methods should follow standards
GB5599-85 ‘Railway vehicles—Specification for evaluation the dynamic perfor-
mance and accreditation test’ [1] and TB/T2360-93 ‘Identification method and
evaluation standard for dynamic performance test of railway locomotives’ [2]. For
high-speed train tests, the test methods should follow “Code for testing of
high-speed electric multiple unit on completion of construction” [3] and
TB10761-2013 “Technical regulations for dynamic acceptance for high-speed
railways construction” [4]. The test methods for low and high-speed vehicles are
generally similar, although the acceptance criteria are different.
1. Hunting stability
Bogie lateral accelerations are normally measured to assess the vehicle running
stability. A 0.5–10 Hz bandpass filter will be applied to the measured acceleration
signal. The vehicle is classified as unstable if the acceleration magnitude reaches or
exceeds the threshold (normally 8–10 m/s2) for 6 consecutive peaks, otherwise, it is
classified as stable.
2. Running safety
Vehicle running safety can be assessed using the derailment coefficient, the wheel
unloading rate, and the rollover factor. These parameters can all be calculated using
measurements from instrumented wheelsets. Instrumented wheelsets can (continu-
ously or discretely) measure wheel vertical force P and lateral force Q, hence the
derailment coefficient can be computed as Q/P and the wheel unloading rate is
worked out as
where DP is the wheel load variation relative to P0 and P0 is the static wheel load.
The rollover factor is used to evaluate the possibility of rollover of a vehicle
under the action of crosswind, lateral vibrations and centrifugal forces. It is assessed
with the wheel load variation for each wheel on one side of the vehicle defined as
D ¼ Pd =P0 ð5:2Þ
where Pd is the dynamic load of the wheel. The rollover safety can only be assessed
with the rollover factors for all wheels on one side of the vehicle. If the rollover
factors for all wheels on one side of the vehicle reach or exceed the threshold
(normally 0.8), the vehicle is assumed dangerous (more details can be found in [1]).
5.1 Field Test Methods of Vehicle–Track Coupled System Dynamics 233
Fig. 5.1 Sensors arrangement for passenger car body acceleration measurement (unit: mm)
3. Ride comfort
Vehicle ride comfort is assessed by the ride comfort indices, including vertical and
lateral ride comfort indices, which are calculated from measured vertical and lateral
accelerations of the car body.
Low-frequency accelerometers are used to measure the vertical and lateral car
body accelerations. For passenger vehicles, the sensors for measuring the vertical
and lateral car body accelerations should be mounted on the floor level and at the
position 1000 mm away from the center pivots, as shown in Fig. 5.1. For freight
vehicles, the sensors should be mounted on the floor level (in case of an empty car)
or under-frame level (in case of a loaded car) and within 1000 mm away from the
center pivot as shown in Fig. 5.2. For locomotives, the sensors should be mounted
on both ends of the traction beam along the centreline of the under-frame. For
driver cab, the sensors should be mounted at the center of the floor in the cabs.
Centre
pivot
1-Center sill, 2- Vertical acceleration sensor, 3- Lateral acceleration sensor, 4-installed steel plate
Fig. 5.2 Sensors arrangement for freight car body acceleration measurement
direction should form an angle of 45°. The mounting positions and the electrical
bridge diagram are shown in Fig. 5.3.
To measure lateral wheel–rail force, strain gauges should be placed on the top
surface of the rail foot, 20 mm away from its outer edge, and 110 mm away from
the mid-point between two sleepers. The strain gauge orientation and the track
longitudinal direction should form an angle of 45°. The mounting positions and the
electrical bridge diagram are shown in Fig. 5.4.
2. Track structural deflection
Track structural deflection is also a key indicator for assessing vehicle–track
interaction, which includes rail vertical and lateral displacements, gauge widening
and sleeper (or slab) vertical and lateral displacements.
Neutral axis
Neutral axis
Fig. 5.3 Mounting positions and the electrical bridge diagram for measuring vertical wheel–rail
force (unit: mm)
5.1 Field Test Methods of Vehicle–Track Coupled System Dynamics 235
Fig. 5.4 Mounting positions and the electrical bridge diagram for measuring lateral wheel–rail
force (unit: mm)
Leaf spring displacement sensor can be used to measure the dynamic dis-
placements of track components conveniently. The sensor should be made to suit
the test site conditions (space-wise) for easy installation. A nonuniform
cross-section cantilever with uniform strength can be made from spring steel as
shown in Fig. 5.5. Two strain gauges mounted at the same position on each side of
the cantilever and form an electrical bridge as shown in Fig. 5.5, where R0 is a
standard resistance of 120 Ω in normal cases. The dynamic movement of the free
end of the cantilever (in contact with the measured component) can be calculated
from the deflection of the cantilever. This sensor can be calibrated using feeler
gauge at the test site.
It is essential to ensure that the measuring datum is securely fixed when mea-
suring track displacement. Displacement piles (steel rods) can be used as the
Strain gauges
Fig. 5.5 Leaf spring displacement sensor with nonuniform cross section and the corresponding
electrical bridge diagram of strain gauges
236 5 Field Test on Vehicle–Track Coupled System Dynamics
Table 5.1 Recommended accelerometer specifications for test of track structural vibration
Test object Rail Sleeper Slab Ballast
Working frequency (Hz) 0–5000 0–1000 0–2000 0–500
Measuring range (g) 1000 100 100 20
absolute datum for the ballasted track system. The displacement sensors can be
fixed on to the pile with various suitable ways to measure the track vertical and
lateral displacements (see details in Sect. 5.3). For slab track system or ballasted
track sections on bridges or in tunnels, the displacement sensors can be fixed
accordingly to measure the relative displacements between rail and slab, and
between slab and bridge beam.
3. Track structural vibration
Track structural vibration is resulted from vehicle–track interaction, which can be
measured using piezoelectric accelerometers. The recommended accelerometer
specifications for rail, sleeper (or slab) and ballast can be found in Table 5.1.
For ballasted track systems, the accelerometers could be mounted at the loca-
tions marked in Fig. 5.6. The accelerometer mounted on rail at point A (20 mm
away from the edge of rail foot) measures rail vibration; the accelerometer mounted
on sleeper at point B (250 mm away from the center line of the rail) measures
sleeper vibration; the accelerometer embedded in ballast at point C (150 mm under
the bottom of the sleeper along the rail center line) measures ballast acceleration
Long-term monitoring for a CRH high-speed train was carried out by State Key
Laboratory of Traction Power, Southwest Jiaotong University from August 2010, to
get a better understanding of vehicle vibration in operation and to provide funda-
mental knowledge for high-speed train running safety. This section presents mea-
surement results of the vibration accelerations of the CRH high-speed train running
at 350 km/h on a slab track in Ref. [7].
238 5 Field Test on Vehicle–Track Coupled System Dynamics
The first to fourth cars of this high-speed train are tested. Figures 5.8 and 5.9
show the onsite photos of different sensors installed on the corresponding com-
ponents. Car body acceleration sensor is installed on the lower surface of car body
floor (Fig. 5.8), and the bogie acceleration sensor is fixed on the top surface of
bogie frame near the primary coil spring (Fig. 5.9), the axle-box acceleration sensor
is installed on the upper side of the axle-box end cover (Fig. 5.9). In order to obtain
the vehicle vibration characteristics in a wide frequency range, the sampling fre-
quency for the axle-box acceleration sensor is set to be 5000 Hz while others are
2000 Hz.
1. Vibration characteristics of the high-speed vehicle system
The vibration of the main components of the high-speed train can be measured
during field tests, which can then be used to assess the stability and ride
Axle-box acceleration
test sensor
5.2 Typical Dynamics Tests of Vehicles Running on Tracks 239
(a) (b)
0.10 0.005
0.004
0.05
0.003
0.00
0.002
-0.05
0.001
-0.10 0.000
0 2 4 6 8 10 0.1 1 10 100
Time (s) Frequency (Hz)
Fig. 5.10 Test results of car body vertical acceleration at train speed of 350 km/h: a time history
and b frequency spectrum
performance of the vehicle. The vibration measurements of the car body, bogie, and
axle box of a car of the high-speed train running at 350 km/h will be discussed in
this section.
Figures 5.10 and 5.11 show the test results of the vertical and the lateral
acceleration responses of the car body, respectively. Both the results in time and
frequency domains are displayed in the figures. As shown in Fig. 5.10a, the peak
value of the car body vertical acceleration is less than 0.075 g (g = 9.8 m/s2). It is
indicated in Fig. 5.10b that there are three distinct dominant frequencies in the
vertical vibration of the car body. The first main frequency is near 1 Hz, which
represents the natural vibration frequency of the car body’s vertical suspension. The
second main frequency is about 10 Hz, close to the first-order natural vibration
frequency of the car body vertical bending. The third main frequency is 34 Hz,
which reflects the forced vibration induced by the excitations from the wheel, such
as the first-order out-of-round of the wheel, wheelset dynamical unbalance, local
defects in wheel profile and other possible factors. Figure 5.11 shows that the
lateral vibration acceleration of the car body is usually less than 0.05 g. There also
(a) (b)
0.10 0.008
Acceleration spectrum (g/Hz)
Lateral acceleration (g)
0.05 0.006
0.00 0.004
-0.05 0.002
-0.10 0.000
0 2 4 6 8 10 0.1 1 10 100
Time (s) Frequency (Hz)
Fig. 5.11 Test results of car body lateral acceleration at train speed of 350 km/h: a time history
and b frequency spectrum
240 5 Field Test on Vehicle–Track Coupled System Dynamics
(a) (b)
4 0.10
0.08
2
0.06
0
0.04
-2
0.02
-4 0.00
0 2 4 6 8 10 0.1 1 10 100
Time (s) Frequency (Hz)
Fig. 5.12 Test results of bogie frame vertical acceleration at train speed of 350 km/h: a time
history and b frequency spectrum
exist three distinct dominant frequencies. The first main frequency is 1.9 Hz, which
is determined by the natural vibration frequency of the suspension system. The
second one is 13 Hz, corresponding to the natural frequency of the car body tor-
sional vibration. The third one is 34 Hz, which indicates that the forced vibration
induced by wheel perimeter also affects the car body’s lateral vibration. In short, the
car body works at a low vibration level when the train runs at high speed. Both its
vertical and lateral accelerations are far less than the limit of 2.5 m/s2 defined by the
“Code for Testing of High-speed Electric Multiple Unit on Completion of
Construction” [3]. It indicates that the high-speed train can be in service with
excellent ride comfort. A significant reason is the high geometry quality of the
ballastless track in Chinese high-speed railway lines.
Figures 5.12 and 5.13 are the measured results of the vertical and the lateral
vibration accelerations for the bogie frame, respectively. It can be found in
Fig. 5.12 that the bogie frame vibrates more violently than the car body in the
vertical direction. Its vertical vibration acceleration varies within the range of
(a) 4 (b)
0.20
Acceleration spectrum (g/Hz)
Lateral acceleration (g)
0.16
2
0.12
0
0.08
-2
0.04
-4 0.00
0 2 4 6 8 10 0.1 1 10 100
Time (s) Frequency (Hz)
Fig. 5.13 Test results of bogie frame lateral acceleration at train speed of 350 km/h: a time
history and b frequency spectrum
5.2 Typical Dynamics Tests of Vehicles Running on Tracks 241
(a) 15
(b)0.5
10
0.4
5
0.3
0
0.2
-5
-10 0.1
-15 0.0
0 2 4 6 8 10 0.1 1 10 100 1000
Time (s) Frequency (Hz)
Fig. 5.14 Test results of axle-box vertical acceleration at train speed of 350 km/h: a time history
and b frequency spectrum
±2.5 g. In the frequency domain, the vibration energy distributes mainly in the
range of 15–35 Hz which contains the low-order elastic modal frequencies of the
bogie frame. Especially, the forced vibration induced by the wheel perimeter
dominates a large component at 34 Hz. Figure 5.13 indicates that the maximum
value of the lateral acceleration in the bogie frame is also less than 2.5 g, and its
main frequencies are similar to those of the vertical vibration.
The time history of the axle-box vertical acceleration and its corresponding
frequency spectrum are shown in Fig. 5.14, and those of the axle-box lateral
acceleration are displayed in Fig. 5.15. It can be seen from Fig. 5.14 that during
full-speed operation the axle-box vibrates strongly and the peak value of its vertical
acceleration is approximately 13 g. Due to the large contact stiffness between the
wheel and the rail, the high-frequency contact vibration in wheel–rail interface
excited by the track irregularity is liable to be transmitted to the axle-box. The
vertical vibration distributes in a wide frequency range below 700 Hz, which covers
two distinct dominant frequency ranges. The first one is the frequency range of
(a) (b)
15 0.30
Acceleration spectrum (g/Hz)
Lateral acceleration (g)
10 0.25
5 0.20
0 0.15
-5 0.10
-10 0.05
-15 0.00
0 2 4 6 8 10 1 10 100 1000
Time (s) Frequency (Hz)
Fig. 5.15 Test results of axle-box lateral acceleration at train speed of 350 km/h: a time history
and b frequency spectrum
242 5 Field Test on Vehicle–Track Coupled System Dynamics
30–50 Hz, which is mainly related to the vibrations induced by wheel perimeter and
the elastic vibration of the bogie frame. The second one is the high-frequency range
of 350–500 Hz, which reflects the high-frequency Hertzian contact vibration
occurring in the wheel–rail interface, as well as the elastic vibration of the wheelset.
As shown in Fig. 5.15, the axle-box lateral vibration is weaker than its vertical
vibration (Fig. 5.14). The lateral vibration energy mainly distributes in the fre-
quency range of 290–650 Hz. The vibration acceleration below 290 Hz is much
lower, and there is a very small peak value at 34 Hz, which indicates that the
periodic excitation caused by wheel perimeter also affects the axle-box lateral
vibration.
The measured results indicate that the vibration was significantly dissipated
when passed through the primary and secondary suspensions to the bogie and to car
body. The axle-box acceleration is reduced to 1/3–1/5 of its original value when
transferred to the bogie; which is further reduced by an order of magnitude when
transferred to the car body. The high-frequency components above 50 Hz are
eliminated by the primary and secondary suspension systems.
It is worth noting that the measured results discussed above include the elastic
modes of the car body, bogie, and wheelset. However, such elastic modes are not
included in rigid body dynamics simulations, whereas such features should be
modeled in elastic body dynamics simulations.
2. Evaluation of high-speed vehicle running stability
The lateral stability of a high-speed vehicle running on a track can be evaluated
using the tested response of lateral vibration acceleration of the bogie frame as
mentioned above. The basic principle is, by monitoring the vibration acceleration
responses of the bogie frame, to evaluate whether a continuous lateral oscillation of
the bogie can decay or not. The assessment of vehicle lateral stability was not
defined in China in the early days [1, 2], however, the evaluation method of vehicle
running stability is now explicitly given in Chinese Code [3, 4]. A bandpass filter
between 0.5 and 10 Hz needs to be applied to the measured bogie lateral acceler-
ation, and the vehicle running stability could be evaluated through counting if the
acceleration peaks reach or exceed 8–10 m/s2 for 6 consecutive times. If it is the
case, the vehicle is regarded as losing its stability.
The above method is applied to the bogie acceleration measurement discussed in
the above section. The filtered signal from the time history of the bogie frame lateral
acceleration in Fig. 5.13a is plotted in Fig. 5.16, where the peak is much lower than
8 m/s2, proving that the high-speed train has outstanding stability.
The above measurements and assessment indicate that the bogie and car body
vibrations are isolated and dissipated by the primary and secondary suspension
systems to a very low level. Hence, the high-speed train has remarkable ride
comfort and stability performance running at 350 km/h on the slab track.
5.2 Typical Dynamics Tests of Vehicles Running on Tracks 243
-0.2
-0.4
0 2 4 6 8 10
Time (s)
This section will discuss a field test regarding a less usual freight vehicle behavior.
A freight vehicle dynamic test was carried out by the author and his team on the
Shuohuang railway line in June 2004, which captured the lateral hunting movement
of a freight vehicle with three-piece bogies (Z8A) [9].
The measured freight wagon was in empty condition, and well maintained.
66 C64 wagons were towed by two SS4B locomotives, and the measurements were
taken on the 45th wagon (number 12591).
The lateral acceleration of the car body, bogie side frame, and wheelset, as well
as the lateral displacement between the car body and side frame were measured.
The car body accelerometer was mounted on the wagon floor (empty car); the side
frame accelerometer was mounted on a rigid plate on the right-hand side of the front
bogie (Fig. 5.17); the wheelset accelerometer was mounted on the right-hand side
axle box of the first wheelset (Fig. 5.17).
The lateral displacement between the car body and side frame was measured
using the string potentiometer. One end of the string was fixed on the car body of
the wagon, and the other end on a steel frame fixed perpendicular to the side frame,
ensuring the string horizontal, as shown in Fig. 5.18.
The lateral accelerations of the car body, side frame, and axle box, and the lateral
displacement between the car body and side frame were measured at speeds of 60,
70, and 75 km/h. The test results show that the C64 empty wagon experienced
severe hunting movement at 75 km/h, which was demonstrated by the extraordi-
narily large periodic acceleration wave (Fig. 5.19). The maximum lateral acceler-
ation of the car body reached at 0.77 g, which greatly exceeded its safety value,
0.5 g, allowed for the freight car used in Chinese Railways. The side frame also
experienced severe lateral vibration (Fig. 5.20), and peaked at 10.81 g. The lateral
displacement between the wagon and side frame was also significantly increased
compared to normal working conditions, and the vibration magnitude changed
periodically as shown in Fig. 5.21.
0.8
Lateral acceleration (g)
0.4
0.0
-0.4
-0.8
0 4 8 12 16 20
Time (s)
Fig. 5.19 Lateral acceleration of wagon (at the center plate) measured at 75 km/h
5.2 Typical Dynamics Tests of Vehicles Running on Tracks 245
15
10
Lateral acceleration (g)
-5
-10
-15
0 4 8 12 16 20
Time (s)
20
Lateral displacement (mm)
15
10
-5
-10
-15
0 4 8 12 16 20
Time (s)
Fig. 5.21 Lateral displacement between the car body and side frame measured at 75 km/h
It can be concluded from the experimental results that the lateral dynamical
system is very close to the unstable state at the speed of 75 km/h. Figure 5.22
depicts the measured frequency spectra of the car body lateral accelerations, in
which the hunting frequency of the car body is 2.69 Hz.
The results above were in good agreement with those obtained by the vehicle–
track coupled dynamics analysis. In our analysis, the theoretical critical speed of
such vehicle under empty condition is 78 km/h [9]. Therefore, running the C64
empty wagon at a speed of 75 km/h is at the stability margin and deemed unsafe.
246 5 Field Test on Vehicle–Track Coupled System Dynamics
Frequency (Hz)
Fig. 5.22 Frequency content of wagon lateral acceleration (at the center plate) measured at
75 km/h
Qinshen Passenger Dedicated Line is the first passenger dedicated line built in
China from Qinhuangdao to Shenyang, with a designed speed of 200 km/h,
including a 66.8 km high-speed test section with a designed speed of 300 km/h.
A series of high-speed train tests were organized by the Ministry of Railways at
the end of 2002, and set the train speed record at that moment in China, reaching
321.5 km/h. The author led the wheel–rail dynamic interaction tests on the slab
track system on Shuanghe Bridge, and on the ballasted track system on Xing-Yan
Bridge. This section will discuss the dynamics test on the slab track system as an
example.
This test is to assess the risk of high-speed wheel–rail interaction and the ade-
quacy of the slab track system design. The vertical and lateral wheel–rail forces, the
rail pad force, vertical displacement between the rail and slab, lateral rail dis-
placement and gauge widening, as well as the vertical accelerations of the rail and
slab were measured in the test. The layout of the sensors is shown in Fig. 5.23, and
Figs. 5.24, 5.25, 5.26, and 5.27 are the photos taken at the test site.
The “China Star” high-speed test train, which is comprised of 2 motor cars and 4
trailer cars, was used in this test. The wheel–rail dynamic interaction indices were
measured when the test train passed the slab track section on Shuanghe Bridge at
160, 180, 200, 220, and 225 km/h, respectively, as shown in Fig. 5.28. As
examples, the measured results at the speed of 200 km/h are shown in Figs. 5.29,
5.30, 5.31, 5.32, 5.33, 5.34, 5.35, and 5.36. The characteristics of dynamic impacts
on the track components can be clearly seen in these graphs, where the motor car
had a much larger impact on the track than the trailer cars due to its higher axle
5.3 Typical Vehicle–Track Dynamic Interaction Tests 247
Shenyang Qinghuangdao
Lateral wheel-rail force Vertical wheel-rail force Vertical rail acceleration Vertical slab acceleration
Vertical rail displacement Vertical slab displacement Lateral slab displacement Rail supporting force
Longitudinal slab strain Lateral slab strain
Fig. 5.23 Sensor layout of main test section (at the middle of the second span of the bridge)
80
Vertical force ( kN)
40
-40
0 1 2 3
Time (s)
Fig. 5.29 Measured vertical wheel–rail force when the high-speed train passed at 200 km/h
20
Lateral force (kN)
-20
-40
0 1 2 3
Time (s)
Fig. 5.30 Measured lateral wheel–rail force when the high-speed train passed at 200 km/h
60
Rail supporting force (kN)
40
20
0 1 2 3
Time (s)
Fig. 5.31 Measured rail supporting force when the high-speed train passed at 200 km/h
load. The measured results proved that all the measured values were significantly
lower than the safety thresholds. The vertical dynamic wheel–rail force peaked at
111.34 kN, which is much lower than the threshold of 300 kN set in ‘On-Bridge
Ballasted Track Design Specification for Qinshen Passenger Dedicated Line’. The
lateral wheel–rail forces were generally lower than 30 kN, in compliance with the
250 5 Field Test on Vehicle–Track Coupled System Dynamics
0.6
0.2
0.0
-0.2
0 1 2 3
Time (s)
Fig. 5.32 Measured vertical rail displacement when the high-speed train passed at 200 km/h
Lateral displacement (mm)
0.4
0.0
-0.4
-0.8
0 1 2 3
Time (s)
Fig. 5.33 Measured lateral rail displacement when the high-speed train passed at 200 km/h
Vertical displacement (mm)
0.08
0.04
0.00
-0.04
0 1 2 3
Time (s)
Fig. 5.34 Measured vertical slab displacement when the high-speed train passed at 200 km/h
thresholds (78 kN for the power car, 51.16 kN for the trailer car). The derailment
coefficient was 0.27 and the wheel unloading rate was 0.29, which were much lower
than their safety thresholds.
5.3 Typical Vehicle–Track Dynamic Interaction Tests 251
100
-100
-200
0 1 2 3
Time (s)
Fig. 5.35 Measured vertical rail acceleration when the high-speed train passed at 200 km/h
12
Vertical acceleration (g)
-6
-12
0 1 2 3
Time (s)
Fig. 5.36 Measured vertical slab acceleration when the high-speed train passed at 200 km/h
The Datong–Qinhuangdao Line (Daqin Line for short) is the first heavy-haul
coal-transportation line in China. Two 10,000 t heavy-haul train tests were taken
during the early 1990s to improve the coal-carrying capacity. This section will
introduce the test carried out in October 1993 on the Upper Line at K318+70, and
the photo in Fig. 5.37 was taken during the test. The test aimed to provide fun-
damental data to study the dynamic impact of the 10,000 t train on the track
structure.
The test focused on the rail, sleeper and ballast vibrations when the heavy-haul
train passed a rail joint with a 0.5 mm height difference. The test vehicle in the
10,000 t heavy-haul train is the Chinese freight wagon C61 equipped with 3-piece
bogies (type Z8A), and the axle load was 21 t. Accelerometers were arranged as in
Fig. 5.6 in Sect. 5.1.2. Figures 5.38, 5.39, and 5.40 plot the vibration acceleration
responses of the rail, sleeper, and ballast when the wagon passed the rail joint with a
speed of 52 km/h.
Our measured results indicated that the dynamic impact of the heavy-haul freight
vehicle was not significantly increased, comparing to normal vehicles. The
252 5 Field Test on Vehicle–Track Coupled System Dynamics
Time (s)
Time (s)
5.3 Typical Vehicle–Track Dynamic Interaction Tests 253
Time (s)
acceleration peaks for the rail, sleeper, and ballast were 76.12 g, 13.16 g, and
2.64 g, respectively, because the wagon used for both heavy-haul and typical
freight trains were the same, despite the heavy-haul train had much more wagons.
The track structure will experience much more cycles of the dynamic impact due to
the extra wagons, therefore the accumulated damage was much higher, requiring
more frequent maintenance on the track.
Chongqing Chengdu
Fig. 5.41 Plan sketch of K444 test site at a small-radius curve on Chengdu–Chongqing line
(a)
Chongqing Chengdu
(b)
Chongqing Chengdu
Lateral wheel–rail force, Vertical wheel–rail force, Lateral rail displacement, Vertical rail displacement,
Fig. 5.42 Sensor layout of each test section in K444 test site on Chengdu–Chongqing line: a the first
test section (with IIIb-C type concrete sleeper) and b the second test section (with timber sleeper)
Fig. 5.45 Freight train passed the test sections with a small-radius curve on Chengdu–Chongqing
Line
Lateral wheel-rail force (kN)
Time (s)
Fig. 5.46 Measured lateral wheel–rail force at the outer rail side
5.3 Typical Vehicle–Track Dynamic Interaction Tests 257
Time (s)
Fig. 5.47 Measured lateral rail displacement at the outer rail side
Lateral sleeper displacement (mm)
Time (s)
Fig. 5.48 Measured lateral sleeper displacement at the outer rail side
Table 5.2 Measured gauge widening induced by different vehicles at both test sections (unit:
mm)
Vehicle type Locomotive Freight wagon Passenger coach
(SS3) (C62A) (YZ22)
Wooden sleeper 11.8 6.37 3.84
track
Concrete sleeper 3.02 2.15 1.69
track
gauge widening on the wood sleeper section was 11.8 mm, and 3.02 mm on the
concrete sleeper section. The gauge widening can be greatly reduced if the concrete
sleepers were adopted. Therefore, the lateral wheel–rail interaction on small-radius
curves could be severe, and the technical measures discussed in [6] must be applied
to ensure the train running safety.
258 5 Field Test on Vehicle–Track Coupled System Dynamics
Nowadays, the wood sleeper tracks on the small-radius curves in the moun-
tainous areas railways have been completely strengthened with the concrete
sleepers, and the train speed has been raised accordingly.
References
© Science Press and Springer Nature Singapore Pte Ltd. 2020 259
W. Zhai, Vehicle–Track Coupled Dynamics,
https://doi.org/10.1007/978-981-32-9283-3_6
260 6 Experimental Validation of Vehicle–Track Coupled Dynamics Models
Table 6.1 Comparison between measured and simulated car body vertical accelerations
Test Vertical Wavelength k 10 12 12 24 24
condition irregularity (m)
Wave depth 10 9 9 16 20
a (mm)
Vehicle speed v (km/h) 160 135 150 160 160
Peak value of measured result (g) 0.12 0.06 0.08 0.12 0.13
Peak value of simulated result (g) 0.104 0.078 0.085 0.096 0.120
6.1 Experimental Validation on the Vehicle–Track Vertically … 261
(a) (b)
30 30
Measured result
Measured result
25 25 Simulated result
Simulated result
20 20
15 15
10 10
5 5
0 0
0 20 40 60 80 100 0 20 40 60 80 100
Speed (km/h) Speed (km/h)
Fig. 6.1 Comparison between simulated and measured vehicle axle-box accelerations of a C62A
car and b C75 car
are especially concerned [1]. These test data could be applied to check the cor-
rectness of the simulated impulsive accelerations of the vehicle wheelsets. As
shown in Fig. 6.1, the simulated axle-box vibration accelerations are compared with
the measured results for the C62A and the C75 cars, respectively. Here, the vehicles
were running under different speeds on 50 kg/m rails through the dipped rail joints
with a dipped angle of 0.02 rad.
Acceleration of track components is the main index for track structure vibration.
The validation analysis is performed based on the tests carried out on the Datong–
Qinhuangdao heavy-haul railway line, focusing on the dynamic effect of 10,000 t
trains on the track structure. The tested track conditions consist of 60 kg/m rail of
U74 type; J-2 concrete sleeper with mass of Ms = 251 kg and with 1840 sleepers per
kilometer; spring bar fastener of x type, common rail pads with thickness of 10 mm
and with stiffness of Kp = 7.8 107 N/m; ballast bed with thickness of 450 mm.
The tested vehicles are of the C61 type for coal transportation with three-piece
bogies and with 21 t axle load. These cars have similar parameters with the C62A
type cars. In the test, the train was running over a rail joint with a height difference
of 0.5 mm in reverse direction at a speed of 52 km/h. The measured and theoret-
ically simulated peak values of the impulsive vibration accelerations are compared
for the rail, the sleeper, and the ballast. The measured and simulated results shown
in Table 6.2 indicate a good agreement.
In addition, the measured results from Ref. [2] are also referenced here for
validation. The simulated results using the vehicle–track vertically coupled
dynamics model are compared with the measured results obtained from the
262 6 Experimental Validation of Vehicle–Track Coupled Dynamics Models
Table 6.2 Measured and simulated impulsive vibration accelerations of track at a rail joint on
Datong–Qinchuangdao railway line
Comparison index Rail acceleration Sleeper acceleration Ballast acceleration
(g) (g) (g)
Measured peak 76.12 13.16 2.64
value
Simulated peak 79.58 12.62 2.65
value
regression equation in Ref. [2], as shown in Fig. 6.2. These results are for the
common wagons of 21 t axle load running reversely over a rail joint with a height
difference of 1 mm. It can be seen that the theoretical and the measured results
coincide well with each other in general, especially for the rail acceleration, and
then the sleeper acceleration.
Further, the simulated ballast vibration is especially validated due to its partic-
ularity. The validation data are mainly from the vibration attenuation test of track
structure performed on the Chengdu–Kunming railway line, where the ballast
vibration acceleration was measured by the author [3]. The track conditions of the
test section are as follows: buffer area of the continuous welded line with 60 kg/m
rail; concrete sleeper of 69 type with 1840 sleepers per kilometer; high elastic rail
pads; ballast bed with the thickness of 450 mm and common limestone ballast.
There is an apparent rail joint depression at the test position, the wavelength of
which is measured to be about 80 mm with a wave depth of about 0.4 mm.
Figure 6.3a displays a field measured impact waveform of ballast acceleration
when a Chinese main kind of freight wagon (C62A loaded car) was passing over the
rail joint at a speed of 60 km/h. Figure 6.3b gives the simulated result using VICT
based on the vehicle–track vertically coupled dynamics model. In the simulation,
the ballast parameters were derived by using the ballast modified model (see
100
50
0
0 20 40 60 80 100
Speed (km/h)
6.1 Experimental Validation on the Vehicle–Track Vertically … 263
(a) (b)
6
6
4 4
2 2
0 0
-2 -2
-4 -4
0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20
Time (s) Time (s)
Fig. 6.3 Comparison between a simulated and b measured time histories of ballast vibration
acceleration (Reprinted from Ref. [3], Copyright 2003, with permission from Elsevier.)
Fig. 2.13). According to the measured results of the Chinese railway track
parameters, the ballast density is qb = 1800 kg/m3, the ballast elasticity modulus Eb
= 110 MPa and the subgrade modulus Ef = 90 MPa/m (K30 value). Thus, it can be
derived by Eqs. (2.5)–(2.9) that the mass of ballast is Mb = 531.4 kg, the ballast
stiffness Kb = 137.75 MN/m and the subgrade stiffness Kf = 77.5 MN/m. This
simulated ballast stiffness is close very much to the stable value (about 140 MN/m)
measured on a Chinese conventional railway line by China Academy of Railway
Sciences [3]. Consequently, it indicates that the modified model of ballast could
reasonably predict the vertical supporting stiffness of granular ballast bed, which is
a key parameter for analysis of ballast vibration.
It can be seen from the comparison between Fig. 6.3a, b that, the measured and
the simulated results coincide with each other very well for both the waveform and
the amplitude; namely, it is 4.69 g for the measured result and 4.97 g for the
simulated result. The corresponding spectral density curves of the ballast acceler-
ation are shown in Fig. 6.4, from which the spectral features from the theoretical
modeling results agree well with the field measured results except for some minor
detailed discrepancies. The main frequency range of the measured ballast acceler-
ation is 70–100 Hz, while that of the theoretical result is 80–110 Hz.
Rail displacement is another index revealing the track dynamics properties. It
should be pointed out that the rail displacement in the transition zone between
subgrade and bridge is especially remarkable and is used here for the model vali-
dation. As introduced in Sect. 3.4.1 of Chap. 3, the dynamic performance degra-
dation problem of the transition zone between the subgrade and the Strandmoelle
bridge of the Danish State Railways (DSB) had perplexed the railway maintenance
department for a long period. So, the DSB carried out several dynamic tests suc-
cessively in 1990, 1995, and 1997. Regarding this problem, some collaboration
between the Technical University of Denmark and the author was carried out to
calculate the rail displacement in the subgrade–bridge transition zone by using the
264 6 Experimental Validation of Vehicle–Track Coupled Dynamics Models
0.05
0.00
0 100 200 300 400
Frequency (Hz)
Table 6.3 Comparison of measured and simulated rail displacements in the subgrade–bridge
transition zone
Position (see Subgrade zone Transition zone Bridge abutment zone
Fig. 3.33) (half-sleeper zone) (wooden sleeper zone) (long sleeper zone)
Measured 0.7 1.0 0.4
value (mm)
Simulated 0.72 1.06 0.41
value (mm)
VICT software; it demonstrated that the simulated and the measured results illus-
trated in Table 6.3 agree well with each other [4].
Figure 6.5a displays measured time history of the vertical wheel–rail force induced
by a wheel flat of a wagon equipped with the Chinese Z8A type of bogie. The
vehicle speed through the test section was 27 km/h. The test data were acquired at
Luoyang Eastern railway station by using a wheel flat detection device developed
by the Research Institute of Zhengzhou Railway Administration. This wheel flat
with a length of 52.8 mm and a depth of 1 mm is typical of an old worn one. It is
approximated by a cosine function. The simulated results using the VICT software
are shown in Fig. 6.5b. Comparison between Fig. 6.5a and b reveals that the the-
oretical simulation is capable of reproducing the waveform of the actual dynamic
interaction force caused by the wheel flat [5].
Wheel–rail interaction forces, P1 and P2, are important indexes that reveal the
wheel–rail vertical impact action when a vehicle passes over impulsive excitations
6.1 Experimental Validation on the Vehicle–Track Vertically … 265
(a) (b)
300 300
Vertical wheel–rail force (kN)
200 200
150 150
100 100
50 50
0 0
0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10
Time (s) Time (s)
Fig. 6.5 Comparison between a measured and b simulated vertical wheel–rail forces caused by a
typical wheel flat
such as rail joints. Comparison between measured and simulated wheel–rail impact
forces is made in Fig. 6.6 when a C62A vehicle was running reversely through a rail
joint with a height difference of 1.5 mm. The simulated and the measured P2 forces
are compared in Fig. 6.7 for the C75 and C62A cars, respectively, when they passed
through a dipped rail joint (2a = 0.02 rad) with various speeds [1]. It is seen from
the two figures that the simulated values coincide with the measured results quite
well in the entire speed range.
The vertical wheel–rail forces excited by rail corrugations are now compared for
harmonic excitations. The tests were carried out at the curved section K38+25 (with
a radius of 600 m) of the Fengtai–Shacheng railway line, since there existed
apparent short pitch rail corrugations with a wavelength of about 200 mm. The
maximum values of the vertical wheel–rail forces were measured for rail corru-
gations with wave depth of 0 mm (after grinding), 0.5 mm, 0.7 mm, and 1.1 mm,
respectively. The measured and the simulated results are compared in Fig. 6.8,
indicating a good agreement between them.
150
100
Measured result
50 Simulated result
0
0.0 0.3 0.6 0.9 1.2 1.5
Wave depth (mm)
6.1.4 Conclusions
In order to validate the vehicle–track spatially coupled dynamics model and the
corresponding software TTISIM, especially to validate the dynamic wheel–rail
coupling model, a number of experimental studies of a variety of aspects are
introduced in this section for comparison purpose. These mainly include the
dynamics tests on the speedup lines or the high-speed lines performed by China
Academy of Railway Sciences and Southwest Jiaotong University. They are as
follows:
(1) Freight train derailment test in the straight section of Beijing ring railway test
line in December 1999;
(2) Field test on Beijing–Qinhuangdao railway line after the reconstruction for train
operation speeds up to 200 km/h on the first try in December 2000;
(3) First high-speed train running test in China on Qinhuangdao–Shenyang pas-
senger dedicated line in December 2002;
(4) Wheel–rail dynamic interaction test in a curve with a small radius in Chengdu–
Chongqing mountain railway line in April 2003.
During the period of December 510, 2000, a field test was performed for train
operation speed raised up to 200 km/h from previous 120 km/h on the existing
Beijing–Qinhuangdao railway line by Beijing Railway Bureau together with China
Academy of Railway Sciences. Indexes of the train operation safety on straight
sections, curved sections and turnout zones were measured in the speed range of
160–210 km/h. The measured results of the train passing through a curved section
at speed of 160 km/h and results of the train running at a straight section with speed
of 200 km/h are adopted to validate the vehicle–track spatially coupled dynamics
model and the corresponding software TTISIM.
The test line consisted of 60 kg/m rails, type II concrete sleepers and a common
ballast bed used in Chinese conventional railway lines. The measured results in the
representative curved section (K121+233*K121+575) are selected for the vali-
dation of the vehicle–track lateral dynamic interaction. This curved section had a
circular part of 142.25 m with a radius of 1200 m; the length of the transition curve
was 100 m and the superelevation was 100 mm. The normal operation speed limit
was 120 km/h and the actual speed was, however, up to 160 km/h in this test. Due
to the fact that there is no track irregularity data measured on this line, the track
irregularity data measured on a similar line, i.e., the Zhengzhou–Wuhan line, is
used in the simulation. The test vehicle was a 4-axle double-deck passenger car.
268 6 Experimental Validation of Vehicle–Track Coupled Dynamics Models
(a) (b)
30 30
10 10
0 0
-10 -10
-20 -20
0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350
Running distance (m) Running distance (m)
Fig. 6.9 Comparison of a measured and b simulated lateral wheel–rail forces on the curved
section
(a) (b)
140 140
Outer side Inner side
Vertical wheel–rail force (kN)
100 100
80 80
60 60
40 40
20 20
0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350
Running distance (m) Running distance (m)
Fig. 6.10 Comparison of a measured and b simulated vertical wheel–rail forces on the curved
section
The measured and the simulated results of wheel–rail interaction force, the car
body acceleration and the derailment coefficient for the test train passing over the
curved section under the speed of 160 km/h are given in Figs. 6.9, 6.10, 6.11, and
6.12, respectively. In general, the simulated dynamic responses are very similar to
the measured results. The results in Fig. 6.9 show that the maximum lateral wheel–
rail force is 20 kN for the test and 25 kN for the calculation. The vertical wheel–rail
forces shown in Fig. 6.10 also display a good agreement between the measured and
the simulated results with maximum values of 128.5 kN for the test and 122.6 kN
for the calculation, and the minimum values of 37.7 kN for the test and 40.6 kN for
the calculation. For the car body vibration acceleration in Fig. 6.11, the maximum
value of the measured result is greater than that of the simulated result for both the
lateral and the vertical vibrations, while the trends of the time response curves are
the same. Finally, the derailment coefficients computed from the lateral and vertical
6.2 Experimental Validation of the Vehicle–Track Spatially Coupled … 269
(a) (b)
0.10 0.2
Lateral car body acceleration (g)
Fig. 6.11 Comparison of measured and simulated car body accelerations on the curved section:
a lateral acceleration, and b vertical acceleration
0.1
0.0
Measured result
Simulated result
-0.1
0 50 100 150 200 250 300 350
Running distance (m)
wheel–rail forces, as shown in Fig. 6.12, also coincide with each other due to the
good agreement between the measured and the simulated wheel–rail forces.
In addition, the measured results in the straight section (downstream K110
+158*K110+508) of the test line are also employed for validation. The measured
and the simulated results of the wheel–rail interaction force, the car body accel-
eration and the derailment coefficient under the speed of 200 km/h are shown in
Figs. 6.13, 6.14, 6.15 and 6.16, respectively. It can be seen that the simulated
response waveforms coincide well with the measured results. Variations of the
wheel–rail dynamic forces are small when the test vehicle runs on the straight
section at a high speed. Both the measured and the simulated lateral wheel–rail
forces vary within 15 kN (see Fig. 6.13a), and the vertical wheel–rail forces vary
between 55 and 115 kN (see Fig. 6.13b). For the car body vibration, the lateral
acceleration responses in Fig. 6.14a are in the range of −0.05 to 0.05 g, while the
measured vertical acceleration in Fig. 6.14b is greater than that of the simulated
result. A good agreement between the measured and the simulated derailment
coefficients can be observed in Fig. 6.15.
270 6 Experimental Validation of Vehicle–Track Coupled Dynamics Models
(a) (b)
30 140
10 80
60
0
40
-10 20
0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350
Running distance (m) Running distance (m)
Fig. 6.13 Comparison of measured and simulated wheel–rail forces on the straight section:
a lateral force, and b vertical force
(a) (b)
Lateral car body acceleration (g)
0.10 0.2
Measured result Measured result
Simulated result Simulated result
0.05 0.1
0.00 0.0
-0.05 -0.1
-0.10 -0.2
0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350
Running distance (m) Running distance (m)
Fig. 6.14 Comparison of measured and simulated car body accelerations on the straight section:
a lateral acceleration, and b vertical acceleration
In all, the simulated wheel–rail forces and the car body vibrations agree well
with the measured results for both the curved and the straight sections. There exist,
of course, inevitably some minor discrepancies between the simulated and the
measured results due to the fact that it is impossible for the track irregularities used
in the simulation (from Zhengzhou–Wuhan line) to be completely the same as those
of the test line (the Beijing–Qinhuangdao line).
At the end of 2002, the Chinese first high-speed train running test was performed on
the Qinshen (Qinhuangdao–Shenyang) passenger dedicated line, including both a
wheel–rail dynamic interaction test (specified in Sect. 5.3.1 of Chap. 5) and a train
dynamics test. Key dynamic performance indexes such as the lateral wheel–rail
force, the derailment coefficient, and the lateral and vertical car body ride comfort
indexes were measured. In this subsection, the dynamic performance indexes
measured when the “Chinese Star” high-speed test train was passing through the
curved test section near K197+412 are employed to validate the theoretical cal-
culation results. The design parameters of the test track were as follows: curve
radius of 4000 m, transition curve length of 390 m, and superelevation of 115 mm.
The speeds of the test train passing through the test region were 160 km/h, 180 km/
h, 200 km/h, 220 km/h, and 225 km/h, respectively. The actual track irregularities
measured on the Qinshen passenger dedicated line are applied as the excitation to
the wheel–rail system in the simulation. The simulated and the measured values of
the lateral wheel–rail force, lateral wheelset force, derailment coefficient, wheel
unloading ratio, and ride comfort index of the high-speed vehicle are compared in
Figs. 6.16, 6.17, 6.18, 6.19, and 6.20.
It can be seen by the comparisons in Figs. 6.16, 6.17, 6.18, 6.19, and 6.20 that
the theoretical calculation results coincide satisfactorily with the measured results,
30
20
10
0
160 180 200 220 240
Speed (km/h)
272 6 Experimental Validation of Vehicle–Track Coupled Dynamics Models
30
20
10
0
160 180 200 220 240
Speed (km/h)
0.3
0.2
0.1
0.0
160 180 200 220 240
Speed (km/h)
0.4
0.3
0.2
0.1
0.0
160 180 200 220 240
Speed (km/h)
especially for the ride comfort indexes (see Fig. 6.20). The discrepancies between
the simulated and the measured results only appear at some places, e.g., at the speed
of 200 km/h for the lateral wheelset force and at the speeds of 160 and 180 km/h
for the wheel unloading ratio. Consequently, the vehicle–track spatially coupled
6.2 Experimental Validation of the Vehicle–Track Spatially Coupled … 273
(a) (b)
Simulated result Simulated result
2.0 2.0
1.5 1.5
1.0 1.0
160 180 200 220 240 160 180 200 220 240
Speed (km/h) Speed (km/h)
Fig. 6.20 Ride comfort indexes versus speed: a lateral, and b vertical
At the early stage in the implementation of the Chinese railway speedup project
(around 1997), a new problem presented, i.e., empty freight wagons derailed on
some straight sections of the speedup lines. It threatened train operation safety
seriously. Therefore, a freight train derailment experiment was performed in the
straight section of the Beijing ring test line by the previous Ministry of Railways
together with China Academy of Railway Sciences from September 18, 1999 to
January 26, 2000. The dynamic responses of both the vehicle and the track systems
were measured for a comprehensive evaluation of system dynamic performance.
The measured results indicated that apparent hunting motions occurred when the
empty C62 wagons equipped with the Z8A three-piece bogies were running in the
straight-line section at the speed of 78 km/h, which was higher than before. The
hunting motion could cause a very drastic fluctuation of the lateral wheel–rail force,
such as the measurement result shown in Fig. 6.21a with the maximum values
beyond 50 kN. The vehicle–track spatially coupled dynamics model was adopted to
calculate the dynamic responses under the same conditions as those in the test. The
test track conditions included: 60 kg/m rails, concrete sleepers, and a common
ballast bed. The track irregularity for such line might be similar to that of FRA 5th
track class, and thus this irregularity spectrum was used in the calculation. Time
history of the theoretically simulated lateral wheel–rail force is displayed in
Fig. 6.21b. Similar hunting motion can be found in the theoretical result with the
274 6 Experimental Validation of Vehicle–Track Coupled Dynamics Models
(a) (b)
80 80
Lateral wheel-rail force (kN)
40 40
20 20
0 0
-20 -20
0 50 100 150 200 250 300 0 50 100 150 200 250 300
Running distance (m) Running distance (m)
Fig. 6.21 Lateral wheel–rail force of empty C62 wagon running at 78 km/h on straight line:
a measured result, and b simulated result
maximum lateral wheel–rail force up to 50 kN, agreeing well with the measured
result.
The corresponding vertical wheel–rail force responses from measurement and
calculation are shown, respectively, in Fig. 6.22. The measured vertical wheel–rail
force varies between 5 and 80 kN, which is close to the simulated result varying in
the range of 0–70 kN. In addition, both the measured and the simulated results
reveal that serious wheel unloading occurs, which brings about a huge possibility of
the derailment, seriously threatening vehicle operation safety.
It can be concluded that the vertical and lateral wheel–rail force responses
obtained by the theoretical calculation using the vehicle–track spatially coupled
dynamics model agree very well with those from the field tests. It is also inevitable
that some discrepancies exist between the theoretical results and the measured ones
due to too many complicated external influencing factors, such as the differences in
(a) (b)
120
Vertical wheel-rail force (kN)
120
80 80
40 40
0 0
-40 -40
0 50 100 150 200 250 300 0 50 100 150 200 250 300
Running distance (m) Running distance (m)
Fig. 6.22 Vertical wheel–rail force of empty C62 wagon running at 78 km/h on straight line:
a measured result, and b simulated result
6.2 Experimental Validation of the Vehicle–Track Spatially Coupled … 275
the track irregularities and the wear of the vehicle components, causing the devi-
ations of the actual parameters from the simulated ones. However, both the theo-
retical results and the measured results were able to reveal the same essence,
namely, drastic hunting motions would happen to the empty C62 wagons with the
Z8A three-piece bogies running on straight lines at the speed of 78 km/h.
Table 6.4 Maximum values of measured and simulated rail displacements (unit: mm)
Vehicle type Vertical displacement Lateral displacement Dynamic gauge
widening
Measured Simulated Measured Simulated Measured Simulated
Locomotive 2.08 2.55 3.45 3.37 4.98 4.10
Freight wagon 1.72 1.82 3.04 2.31 2.78 3.38
Passenger car 1.63 1.46 1.99 1.59 1.92 1.95
276 6 Experimental Validation of Vehicle–Track Coupled Dynamics Models
sleeper agree also well with the measured results. In general, the vehicle–track spa-
tially coupled dynamics model is capable of analyzing the track structure vibration
performance (including the lateral dynamic performance) in various operation con-
ditions with good accuracy. This is because the model adopts the dynamic wheel–rail
coupling relationship and fully considers the track structure vibrations.
6.2.5 Conclusions
In this section, the train–track spatially coupled dynamics model are validated by
comparing simulated results with those from the field measurements, with special
emphasis on the validation of the longitudinal dynamic interaction between
6.3 Experimental Validation of the Train–Track Spatially Coupled … 277
vehicles. The detailed validation under several conditions can be found in [7] and
are briefly presented in the following.
(a)
(b)
Longitudinal coupler force(kN)
Fig. 6.23 Comparison of measured and simulated longitudinal coupler forces of 20,000 t
combined train under braking conditions: a emergency braking, and b full service braking
position under pulling coupler force, and the swing amplitudes decrease with the
increase of the pulling force. Conversely, the couplers will tilt to one side as the
coupler compression force increases to a certain magnitude. The calculation results
in Fig. 6.25b indicate that the variation of the coupler swing angle follows similar
trend as the measured results in Fig. 6.25a. It can be found that the swing angles of
6.3 Experimental Validation of the Train–Track Spatially Coupled … 279
(a)
5
4
Coupler swing angle (°)
-1
-2
-1500 -1000 -500 0 500 1000 1500
Longitudinal coupler force(kN)
(b)
5
4
Coupler swing angle (°)
-1
-2
-1500 -1000 -500 0 500 1000 1500
Longitudinal coupler force(kN)
the middle and the rear couplers are slightly below 4° and 3°, respectively. Overall,
the simulated and the measured results coincide well with each other.
The lateral wheelset forces obtained from the field test and the simulation are
compared and displayed in Fig. 6.26 [7]. The results reveal that the lateral wheelset
forces present statistically a unidirectional increase when the compression coupler
force increases. The measured wheelset force is slightly lower than the simulated
280 6 Experimental Validation of Vehicle–Track Coupled Dynamics Models
(a)
25
20
Lateral wheelset force (kN)
15
10
-5
-10
-15
-1000 -800 -600 -400 -200 0 200 400 600 800 1000 1200
Longitudinal coupler force(kN)
(b)
40
30
Lateral wheelset force (kN)
20
10
- 10
- 20
- 1000 - 800 - 600 - 400 - 200 0 200 400 600 800 1000
Longitudinal coupler force(kN)
Fig. 6.26 Comparison of a measured and b simulated lateral wheelset forces of the locomotive
under the electric braking condition
value, which is most likely caused by the differences in the data processing method
and the track irregularities. It is worth pointing out that the measured data are closed
to the average values of the simulated results.
6.3 Experimental Validation of the Train–Track Spatially Coupled … 281
(a) (b)
Vertical wheel-rail force (kN)
6.3.4 Conclusions
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Chapter 7
Computational Comparison of Vehicle–
Track Coupled Dynamics and Vehicle
System Dynamics
© Science Press and Springer Nature Singapore Pte Ltd. 2020 285
W. Zhai, Vehicle–Track Coupled Dynamics,
https://doi.org/10.1007/978-981-32-9283-3_7
286 7 Computational Comparison of Vehicle–Track Coupled …
vD vA vB
Train speed
In the interval vD < v < vB, the stability of the vehicle system depends on the
amplitude of the external disturbance. With the continuous increase of train speed,
the vehicle system will firstly exhibit the limit cycle vibration at point D when
subjected to a large initial disturbance, whereas the limit cycle vibration at point
A will first occur when subjected to a small initial disturbance. The corresponding
speed vD at point D is called the nonlinear critical hunting speed and vA at point A is
called the linear critical hunting speed. For the investigation of the stability of
railway vehicle system in field practice, the nonlinear critical hunting speed vD is
generally considered.
2. Numerical method to determine the critical hunting speed
In consideration of the fact that the vehicle–track coupled dynamics theory deals
with a very large-scale dynamic system involving strong nonlinearities, it is
impossible to obtain the system stability with any analytical method. Therefore, a
direct numerical method based on the characteristic of time histories of vehicle
components is proposed to determine the critical hunting speed of a vehicle running
on a track. The simple fast time-integration method (i.e., the Zhai method) is
adopted to numerically solve the system equations and to obtain time responses of
vehicle components due to initial disturbance. The time histories of lateral dis-
placements of vehicle components are monitored as the train speed increases
step-by-step from very low value. If the time responses of the lateral displacements
induced by large initial disturbance decay with time and regain their equilibrium
states at one speed, the vehicle system is stable at this speed. Then another cal-
culation is needed to increase the speed, until the critical situation occurs where at
least one of the time histories of the lateral displacements does not decay to its
equilibrium state and sustains variation almost with the same amplitude. According
to Fig. 7.1, the speed at this critical situation is regarded as the nonlinear critical
hunting speed. It should be noted that a little decrease of the speed from the
nonlinear critical hunting speed will result in the occurrence of the stable
7.1 Comparison of Computational Results on Vehicle Hunting Stability 287
equilibrium state of the system. Therefore very small step of the speed should be
used in the calculation when the speed approaches the nonlinear critical hunting
speed.
Due to the occurrence of hunting instability, the motion of each rigid body of the
vehicle system shows the same form of bifurcation and limit cycle characteristics,
which means the Höpf bifurcation and saddle-node bifurcation will simultaneously
appear on the motion of a rigid body under the same speed [2]. Therefore, the
critical speed of the vehicle system can be determined according to the time history
and phase plane graph of a rigid body, for instance, the first wheelset of the vehicle.
It is very important to use a large initial lateral disturbance so as to ensure that
the large vibration of the vehicle system can be excited and the nonlinear critical
hunting speed is ascertained. In order to inflict a large initial lateral disturbance to a
railway vehicle system, it is suggested to use the actual track irregularity spectra of
the railway line on which the analyzed vehicle will operate. Once the lateral
vibration of the vehicle is completely induced, the track irregularities should be cut
off in the calculation so that the running vehicle can vibrate freely without any
external excitation [3]. This initial disturbance obtained by this manner is larger
enough and closer to the real operating condition than the commonly used distur-
bance using a small lateral displacement given to the leading wheelset. It is
noticeable that the initial condition may also affect the dynamic behavior of the
nonlinear vehicle system. So, the length of the irregular track section used in the
calculation should be sufficiently long so as to be able to make each component of
the vehicle system be fully excited.
3. Numerical example
The computation of the critical hunting speed of a typical Chinese high-speed
vehicle is illustrated here as an example. The vehicle first runs on an elastic track
with random track irregularity (200 m long), and then moves on to the straight track
without irregularity. The lateral motion of the first wheelset is investigated. By
increasing the vehicle speed continuously, it can be found that the lateral dis-
placement of the wheelset at speed of 361 km/h gradually attenuates to the equi-
librium position (zero position) as shown in Fig. 7.2a, while it starts the periodic
motion with a constant amplitude as the speed increases to 362 km/h as shown in
Fig. 7.2b. In the corresponding phase plane graph (Fig. 7.3), the phase locus of the
lateral velocity of the wheelset at speed 361 km/h converges to the singular point of
the limit cycle in Fig. 7.3a. While at the vehicle speed of 362 km/h, the wheelset
lateral velocity no longer attenuates to zero but tends to achieve a stable limit cycle
as shown in Fig. 7.3b. Thus, 362 km/h is the actual critical hunting speed of this
vehicle.
288 7 Computational Comparison of Vehicle–Track Coupled …
(a) (b)
15 15
V=361 km/h V=362 km/h
10 10
5 5
0 0
-5 -5
-10 -10
-15 -15
0 2 4 6 8 0 2 4 6 8
Time (s) Time (s)
Fig. 7.2 Time history of wheelset lateral vibration near the critical state of vehicle instability:
a equilibrium solution, and b periodic solution
(a) (b)
0.2 0.6
Lateral wheelset velocity (m/s)
0.0 0.0
-0.1 -0.3
-0.2 -0.6
-15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15
Lateral wheelset displacement (mm) Lateral wheelset displacement (mm)
Fig. 7.3 Phase plane graph of wheelset lateral vibration near the critical state of vehicle
instability: a equilibrium solution, and b periodic solution
Based on the above-mentioned method for computing the nonlinear critical speed,
the theories of vehicle–track coupled dynamics and the traditional vehicle dynamics
are applied to determine the nonlinear critical speeds of four typical vehicles
employed in China, a high-speed passenger car, a freight car equipped with tra-
ditional Z8A bogie, a Chinese speedup locomotive, and a metro power car.
Table 7.1 compares the results.
It can be seen from Table 7.1 that:
7.1 Comparison of Computational Results on Vehicle Hunting Stability 289
Table 7.1 Comparison of computed critical speeds based on the coupled model and the
traditional model (unit: km/h)
Vehicle type Coupled model Traditional model
High-speed passenger car 362 391
Loaded freight car 113 118
Speedup locomotive 162 176
Metro car on floating slab track 188 210
(1) For the high-speed passenger car, the critical speed calculated through the
coupled model is 362 km/h, while that of the traditional model is 391 km/h
which is 8.01% larger.
(2) For the loaded freight car, the calculated critical speed of the traditional model
is 118 km/h, which is 4.42% larger than that of the coupled model, 113 km/h.
(3) For the speedup locomotive, the calculated critical speed of the coupled model
is 162 km/h while that of the traditional model is 176 km/h, which shows an
increase of 8.64%.
(4) For the metro car running on floating slab track, the calculated critical speed of
the coupled model is 210 km/h while that of the traditional model is 188 km/h,
which shows an increase of 11.70%.
It can be further concluded that lower critical speeds are obtained by the coupled
model with comparison to the case of the traditional model. The reason for the
reduction in critical speeds of vehicles calculated by the vehicle–track coupled
model is due to the consideration of track elasticity and damping in the coupled
dynamics model. Since the rail has a lateral degree of freedom in the coupled
model, the lateral vibrations of the left and right rails relieve the lateral constraint on
wheelsets. The wheelsets are, therefore, prone to hunting, which subsequently leads
to a reduction in the critical speed with respect to the traditional model accounting
for rigid track structure. Field experience concurs with this conclusion.
7.1.3 Summary
(1) The critical speed of vehicle calculated by the traditional model is higher than
that of the coupled model. The difference between these results by the two
models is around 10%, which is consistent with the comparison results reported
in Refs. [3, 4].
(2) Although there is no significant difference between the two models’ simulation
results, the traditional model exaggerates the lateral constraint of rails on
wheelsets due to the assumption of the rigid track. The critical speed of the
vehicle system is thus overestimated. The calculation result of the traditional
model, therefore, has less safety margin, which must be considered in the actual
design.
290 7 Computational Comparison of Vehicle–Track Coupled …
To evaluate the simulation results of vehicle ride comfort based on the two models,
three types of commonly used Chinese vehicles are considered, i.e., the high-speed
passenger car, the speedup locomotive, and the fast freight car. The operation
speeds of these vehicles are 300 km/h, 160 km/h, and 120 km/h, respectively. The
adopted PSDs of random track irregularities are the low disturbance spectrum of
German high-speed railway for the high-speed passenger car and the track spectrum
of Chinese speedup trunk line for the speedup locomotive and the freight car.
Tables 7.2 and 7.3 compare the simulation results between the coupled model
and the traditional model, in terms of the maximum lateral and vertical vibration
accelerations of the car body (at the floor where 1 m away from the center plate in
the lateral direction) and the corresponding ride comfort indexes.
For a detailed analysis of the difference in car body vibration simulation between
the two models, the time histories of car body acceleration of the fast freight car are
illustrated, as shown in Figs. 7.4 and 7.5.
The above simulation results indicate that the obtained vehicle vibration char-
acteristics based on the coupled model and the traditional model show good
agreement both in the waveform and the amplitudes, and both for vertical and
lateral vibrations. The difference between the two models in lateral vibration is
slightly larger than that in vertical vibration, and the ride comfort index calculated
by the traditional model is a little higher compared with the coupled model.
Table 7.2 Car body accelerations between the coupled model and the traditional model (unit: m/
s2)
Vehicle type Coupled model Traditional model
High-speed passenger car Lateral 0.49 0.50
Vertical 0.73 0.76
Speedup locomotive Lateral 1.56 1.60
Vertical 1.07 1.08
Fast freight car Lateral 4.21 4.59
Vertical 4.51 4.54
Table 7.3 Ride comfort indexes between the coupled model and the traditional model
Vehicle type Coupled model Traditional model
High-speed passenger car Lateral 2.33 2.36
Lateral 2.61 2.63
Speedup locomotive Lateral 3.19 3.20
Vertical 2.90 2.91
Fast freight car Lateral 3.62 3.64
Vertical 3.84 3.87
7.2 Comparison of Calculation Results on Vehicle Ride Comfort 291
(a) (b)
5.0 5.0
Lateral acceleration (m/s 2)
0.0 0.0
-2.5 -2.5
-5.0 -5.0
0 1 2 3 4 5 6 0 1 2 3 4 5 6
Time (s) Time (s)
Fig. 7.4 Lateral car body accelerations of the fast freight car obtained from: a the coupled model,
and b the traditional model
(a) (b)
5.0 5.0
Vertical acceleration (m/s2)
2.5 2.5
0.0 0.0
-2.5 -2.5
-5.0 -5.0
0 1 2 3 4 5 6 0 1 2 3 4 5 6
Time (s) Time (s)
Fig. 7.5 Vertical car body accelerations of the fast freight car obtained from: a the coupled model,
and b the traditional model
The good consistency of the results calculated by these two models is owing to
the effective insulation of the high-frequency vibration via the primary and sec-
ondary suspensions. Influence of track vibration on car body vibration above the
secondary suspension is significantly attenuated as a result.
On this account, both the vehicle–track coupled dynamics model and the tra-
ditional vehicle dynamics model can be adopted for analyzing and evaluating the
ride comfort of vehicle system. Since the track vibration is omitted in the traditional
model, it is more simple and efficient. However, to take account of the vibrations of
the vehicle components below the secondary suspension (i.e. the bogie frames,
wheelsets, traction motors, etc.), the results of the two models are still different [5].
292 7 Computational Comparison of Vehicle–Track Coupled …
In the small radius curve case, 250 m is selected as the radius of the curve without
loss of generality. The transition curve length is 80 m and the superelevation of the
outer rail is 120 mm. A traditional Chinese passenger train with a running speed of
60 km/h is used as the train platform. The fifth-grade American track spectrum is
set as the random track irregularity, which can simulate the track state with small
radius curves in most cases.
The simulation results are compared in terms of the lateral wheel–rail force, the
vertical wheel–rail force, the dynamic gauge widening and other curving safety
indices.
1. Lateral wheel–rail force
Figure 7.6 shows the time histories of the lateral wheel–rail forces obtained by the
two models. It is obvious that the result of the traditional model is larger than that of
(a) (b)
Lateral wheel-rail force (kN)
120
Lateral wheel-rail force (kN)
120
80 80
40 40
0 0
-40 -40
0 5 10 15 20 0 5 10 15 20
Time (s) Time (s)
Fig. 7.6 Comparison of the calculated lateral wheel–rail forces (outer rail) between a the coupled
model and b the traditional model
7.3 Comparison of Calculation Results on Curving Performance 293
-1
10
-3
10 Coupled model
Traditional model
-5
10
0.3 1 10 100 400
Frequency (Hz)
Fig. 7.7 Comparison of the PSDs of the lateral wheel–rail forces (outer rail) between two models
the coupled model, especially on the circular curve. The maximum value from the
traditional model is 127.49 kN, which is 18.2% larger than the value 107.82 kN
from the coupled model.
The PSDs of the lateral wheel–rail forces are shown in Fig. 7.7. The PSDs of the
lateral wheel–rail force obtained from the two models show good agreement when
the frequency is below 20 Hz. For frequency larger than 20 Hz, a significant dif-
ference occurs. Particularly at a frequency near 200 Hz, the result obtained from the
traditional model is much larger. The reason lies in the consideration of track
stiffness and damping effect in the coupled model, which can absorb the vibration
with medium–high frequencies.
2. Vertical wheel–rail force
Figure 7.8 illustrates the time histories of the vertical wheel–rail forces. Compared
with the coupled model, the result from the traditional model is much larger. The
maximum force from the coupled model is 123.15 kN, while that from the tradi-
tional model increases by 10.1% to 135.54 kN.
(a) (b)
140 140
Vertical wheel-rail force (kN)
120 120
100 100
80 80
60 60
40 40
0 5 10 15 20 0 5 10 15 20
Time (s) Time (s)
Fig. 7.8 Comparison of the calculated vertical wheel–rail force (outer rail) between a the coupled
model and b the traditional model
294 7 Computational Comparison of Vehicle–Track Coupled …
Fig. 7.9 Comparison of the PSDs of the vertical wheel–rail forces (outer rail) between two
models
Figure 7.9 compares the corresponding PSDs of the vertical wheel–rail forces.
There is no distinct difference between the two models in the frequency range
below 40 Hz. With the increase of frequency, the result from the traditional model
shows an enlarged tendency with respect to the coupled model, particularly in the
frequency range higher than 100 Hz. These differences should be also caused by the
track elasticity and damping effect considered in the coupled model.
3. Dynamic gauge widening
In practice, when railway vehicle passes through curved tracks, especially small
radius curves, there will be elastic deformations and vibrations stimulated on track
components due to the combined effect of vertical and lateral wheel–rail forces,
including the torsion deformations of the rails (overturn) and the dynamic gauge
widening (the difference of lateral vibration displacements between the left and
right rails). These dynamic changes may pose threats to the running safety of
vehicles. The traditional vehicle dynamics model is unavailable for dynamic
deformations of track structures and their influence on the dynamic wheel–rail
interaction due to the lack of consideration of track system vibration.
Taking the dynamic gauge widening as example, the calculated result of the
vehicle–track coupled dynamic model is illustrated in Fig. 7.10. It shows that the
gauge varies dynamically during the whole process of the vehicle passing through
the curve. The maximum value of the dynamic gauge widening, in this case, is
1.32 mm. Relevant field test results can be found in Sect. 5.3.3 of Chap. 5.
Results in Ref. [6] indicated that the gauge widening had a large influence on the
wheel–rail contact geometry. That’s why the traditional model without considera-
tion of rail dynamic deformation and gauge widening effect shows a distinct dif-
ference in simulating the wheel–rail contact geometry compared with the coupled
model. The calculated wheel–rail forces and creep forces are therefore different. It
provides another explanation of the difference in the calculated wheel–rail dynamic
forces between the two models.
7.3 Comparison of Calculation Results on Curving Performance 295
1.5
0.5
0.0
-0.5
0 5 10 15 20
Time (s)
Fig. 7.10 Variation of dynamic gauge widening calculated by the coupled model
The above results are very similar to those reported in Ref. [7], where another
small radius curve is analyzed (R = 287 m).
In calculation, the radius of the curve is 6000 m. The transit curve length is 180 m
and the superelevation of the outer rail is 70 mm. A high-speed motor car named
“China star” is simulated with a speed of 250 km/h. The low disturbance spectrum
of German high-speed railway is used to describe the random track irregularity.
Several curving performance indices calculated by the coupled model and the
traditional model are compared in Table 7.4. The lateral wheel–rail force, vertical
wheel–rail force, and the lateral wheelset force from the traditional model are
28.03 kN, 180.79 kN, and 18.18 kN, respectively, which are 15%, 28.32%, and
15.28% larger than those from the coupled model. The calculated derailment
Table 7.4 Comparison of calculated dynamic indexes between the coupled model and the
traditional model
Dynamic index Coupled model Traditional model
Lateral wheel–rail force (kN) 24.38 28.03
Vertical wheel–rail force (kN) 140.89 180.79
Wheelset lateral force (kN) 15.77 18.18
Derailment coefficient 0.29 0.36
Wheel unloading rate 0.31 0.46
Dynamic gauge widening (mm) 0.51 –a
a
This index cannot be obtained from the traditional model
296 7 Computational Comparison of Vehicle–Track Coupled …
coefficient and wheel unloading rate from the traditional model are also larger than
those from the coupled model. The dynamic gauge widening calculated by the
coupled model is 0.51 mm, which is not available from the traditional model.
It can be concluded that for the case of a vehicle passing a large radius curve
with a high speed, the calculated dynamics indices from the traditional model are
much larger than those from the coupled model. The difference between the two
models attains 15–20% and even 30%.
7.4 Conclusions
following two facts: (a) the track system can absorb part of the vibration energy
induced by the wheel–rail interaction; (b) the dynamic gauge widening actually
exists due to the lateral motion of rails, which can result in the change of the
wheel–rail contact geometry and eventually influence the wheel–rail contact
forces. However, these dynamic effects are not considered in the traditional
vehicle dynamics model.
References
1. Hans T, Kaas PC. A bifurcation analysis of nonlinear oscillation in railway vehicles. In:
Proceedings of the 8th IAVSD symposium. 1983. p. 320–9.
2. Wu P, Zeng J. A new method to determine linear and non-linear critical speed of the vehicle
system. Rail Veh. 2000;38(5):1–4 (in Chinese).
3. Zhai WM, Wang KY. Lateral hunting stability of railway vehicles running on elastic track
structures. J Comput Nonlinear Dyn ASME. 2010;5(4):041009-1*9.
4. Gialleonardo ED, Braghin F, Bruni S. The influence of track modelling options on the
simulation of rail vehicle dynamics. J Sound Vib. 2012;331:4246–58.
5. Zhai WM. Vehicle-track coupled dynamics. 2nd ed. Beijing: China Railway Press; 2002 (in
Chinese).
6. Wang KY, Zhai WM, Cai CB. Effect of the wheel–rail profile and system parameters on the
wheel-rail space contact geometry relation. Rail Veh. 2002;40(2):14–8 (in Chinese).
7. Zhai WM, Wang KY, Cai CB. Fundamentals of vehicle–track coupled dynamics. Veh Syst
Dyn. 2009;47(11):1349–76.
Chapter 8
Vibration Characteristics
of Vehicle–Track Coupled System
© Science Press and Springer Nature Singapore Pte Ltd. 2020 299
W. Zhai, Vehicle–Track Coupled Dynamics,
https://doi.org/10.1007/978-981-32-9283-3_8
300 8 Vibration Characteristics of Vehicle–Track Coupled System
The periodic discrete sleepers along railway lines cause uneven support stiffness,
which inevitably results in periodic fluctuations of wheel–rail interaction when a
rail vehicle travels on the track. Figures 8.1, 8.2 and 8.3 show the periodic
steady-state responses of wheel–rail forces and rail deformation due to the sleeper
span in three typical vehicle–track systems.
Figures 8.1 and 8.2 illustrate the steady-state responses of the wheel–rail force
and rail deformation when a passenger vehicle (axle load 14 t) runs on ballasted and
ballastless tracks (rail pad stiffness 60 MN/m and 25 MN/m, respectively) at an
operating speed of 300 km/h. For the two types of high-speed railway tracks, the
periodic fluctuation amplitudes of the vertical wheel–rail force caused by sleeper
span are approximately 2 kN and 1.5 kN, respectively. Figure 8.2b also depicts that
the periodic discrete geometry characteristics of the track slabs in the ballastless
track are reflected in the steady-state response of rail deformation. Figure 8.3 shows
the steady-state responses of wheel–rail force and rail deformation when a freight
(a) 70.5
70.0
Wheel-rail force (kN)
69.5
69.0
68.5
68.0
67.5
67.0
0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4
Track coordinate (m)
(b) 0.979
0.978
Rail deformation (mm)
0.977
0.976
0.975
0.974
0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4
Track coordinate (m)
Fig. 8.1 Steady-state responses of a high-speed vehicle running on a ballasted track: a wheel–rail
interaction force, and b rail deformation
8.1 Steady-State Response of Vehicle–Track Interaction 301
(a) 70.5
Rail
Track slab
70.0
Lslab=6.5m
Wheel-rail force (kN) 69.5
69.0
68.5
68.0
Lsp=0.65m
67.5
0.00 3.25 6.50 9.75 13.00 16.25 19.50
Track coordinate (m)
(b) 0.994
Rail
0.993
Track slab
Rail deformation (mm)
0.991
0.990
0.989
0.988 Lsp=0.65m
0.987
0.00 3.25 6.50 9.75 13.00 16.25 19.50
Track coordinate (m)
Fig. 8.2 Steady-state responses of a high-speed vehicle running on a ballastless track: a wheel–
rail interaction force, and b rail deformation
wagon (axle load is 25 t) travels on a heavy-haul railway ballasted track (rail pad
stiffness 160 MN/m) at a speed of 80 km/h. For the freight vehicle–track system,
the periodic fluctuation amplitude of the wheel–rail force is approximately 2.5 kN.
The above results indicate that the steady-state response of the rail deformation
decreases as the rail pad stiffness increases and increases as the axle load increases.
It is also observed that the steady-state periodic fluctuation amplitude of the wheel–
rail force increases with the increase in rail pad stiffness and axle load. This is
consistent with the phenomenon that the wheel–rail interaction caused by the
sleeper span is conspicuous for the Harbin–Dalian high-speed railway system in
winter.
Figures 8.4, 8.5 and 8.6 illustrate the steady-state responses of the rail deformation
and the rail–sleeper interaction force of high-speed and freight vehicle–track
302 8 Vibration Characteristics of Vehicle–Track Coupled System
(a) 125
124
Wheel-rail force (kN)
123
122
121
120
0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4
Track coordinate (m)
(b) 0.875
0.870
Rail deformation (mm)
0.865
0.860
0.855
0.850
0.845
0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4
Track coordinate (m)
Fig. 8.3 Steady-state responses of a freight wagon running on a ballasted track: a wheel–rail
interaction force, and b rail deformation
systems when the vehicle runs over a sleeper. Figures 8.4 and 8.5 show the track
steady-state responses due to a passenger vehicle (axle load 14 t) traveling on
ballasted and ballastless tracks (rail pad stiffness 60 MN/m and 25 MN/m,
respectively) at an operating speed of 300 km/h. Steady-state responses of the
ballasted track components under a freight wagon (axle load 25 t) running on a
heavy-haul railway (rail pad stiffness 160 MN/m) at speed of 80 km/h are shown in
Fig. 8.6.
From these figures, it can be seen that the peaks of the rail deformation and the
rail–sleeper interaction force exhibit correspondence to the four wheelsets of the
respective vehicles. For the high-speed vehicle–track systems, the vertical rail
deformation and the rail–sleeper interaction force are approximately 1 mm and 25
kN, respectively; while for the freight wagon–track system, the two indices are
0.85 mm and 53 kN, respectively. This means a larger rail support stiffness results
in a smaller rail deformation, and a larger axle load leads to a larger rail–sleeper
interaction force.
8.1 Steady-State Response of Vehicle–Track Interaction 303
(a) 0.6
0.4
0.2
Rail deformation (mm)
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time (s)
(b) 30
25
Rail/sleeper force (kN)
20
15
10
-5
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time (s)
Fig. 8.4 Steady-state responses of a high-speed vehicle passing a ballasted track section: a rail
deformation, and b rail–sleeper interaction force
When a rail vehicle negotiates a curved track, large lateral forces are generated
between the wheels and rails due to the centrifugal force and the change of track
geometry. These lateral forces, in combination with small vertical forces, may cause
wheel climbing and reduce ride comfort as the vehicle negotiates the curve. In this
section, the steady-state curving performances of a high-speed passenger vehicle
and a freight wagon are presented, as shown in Figs. 8.7 and 8.8, respectively.
Figure 8.7 illustrates the steady-state responses of the lateral and vertical wheel–
rail forces, wheelset displacements, and lateral car body displacement of a
high-speed vehicle negotiating a large radius curved track at a speed of 300 km/h.
The high-speed curved track had a circular curve radius of 7000 m, a transition
curve length of 670 m, an arc length of 400 m, and a super-elevation of 100 mm.
The results show that the wheel–rail forces and the lateral vehicle displacements
increase when the vehicle crosses the transition curve, and their peak responses
304 8 Vibration Characteristics of Vehicle–Track Coupled System
(a) 0.6
0.4
0.2
Rail deformation (mm)
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time (s)
(b) 30
25
Rail/sleeper force (kN)
20
15
10
-5
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time (s)
Fig. 8.5 Steady-state responses of a high-speed vehicle passing a ballastless track section: a rail
deformation, and b rail–sleeper interaction force
appear at the circular curve. The lead wheelset has the largest lateral displacement
of about 4 mm, and the lateral car body displacement increases to about 20 mm.
Figure 8.8 depicts the steady-state responses of the wheel–rail forces and lateral
vehicle displacements of a freight wagon negotiating a small radius curved track at
a speed of 80 km/h. The curved track had a curve radius of 250 m, a transition
curve length of 80 m, an arc length of 100 m, and a super-elevation of 120 mm. It
shows that the steady-state responses of the wheel–rail forces and lateral vehicle
displacements of the freight wagon are similar to that of the high-speed vehicle,
while the former has worse curving performance. The maximum lateral and vertical
wheel–rail forces of the lead wheelset attain 25 kN and 160 kN, respectively, while
the lateral wheelset and car body displacements have peak values of 10 mm and
28 mm, respectively. It can be seen that the curve with a smaller radius makes the
vehicle curving performance poor.
8.2 Dynamic Response of Vehicle–Track Interaction … 305
(a) 0.6
0.4
0.2
Rail deformation (mm)
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Time (s)
(b) 70
60
Rail/sleeper force (kN)
50
40
30
20
10
0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Time (s)
Fig. 8.6 Steady-state responses of a freight wagon traveling passing a ballasted track section:
a rail deformation, and b rail–sleeper interaction force
There are multiform vertical impulsive excitation sources in the wheel–rail system,
such as the rail weld joint, the wheel flat, and so on. These defects cause sudden
transient impact loads to the wheel–rail system, resulting in intense vibration to the
vehicle–track coupled system. In this section, the dynamic responses of railway
vehicles passing rail dipped joints and weld joint irregularities on a ballasted track
are provided as examples to illustrate the vibration characteristics of the vehicle–
track coupled system under the excitation of vertical impulsive defects.
Figure 8.9 depicts the dynamic vertical wheel–rail force of a freight wagon
negotiating a dipped rail joint (see Fig. 3.10 in Chap. 3) at a speed of 80 km/h.
A total dip angle of 2a = 0.02 rad was adopted in the simulation. Figure 8.9 shows
306 8 Vibration Characteristics of Vehicle–Track Coupled System
(a) (b)
2.0 76
0.5 70
0.0 68
-0.5 66
-1.0 64
0 400 800 1200 1600 2000 0 400 800 1200 1600 2000
Track coordinate (m) Track coordinate (m)
(c) (d)
Lateral wheelset displacement (mm)
Fig. 8.7 Steady-state curving performance of a high-speed vehicle negotiating a large radius
curved track: a lateral wheel–rail force, b vertical wheel–rail force, c lateral wheelset displacement,
and d lateral car body displacement
that the time history of the wheel–rail impact force caused by the dipped rail joint
exhibits two dominating peaks, which correspond to the P1 and P2 forces defined in
the BR standard [1]. The P1 force (short peak force) is a high-frequency force in the
range of 200–1000 Hz, which is related to the natural frequency of the Hertzian
contact between the unsprung mass of the vehicle and the rail. The P1 force usually
appears immediately (0.5 ms) after the wheel–rail impact and exists only for a very
short duration. Therefore, the P1 force will not be transmitted to the sprung mass of
the vehicle system and the rail infrastructures. However, it can potentially damage
the local contact region of the wheel tread and the railhead. On the contrary, the P2
force (delayed peak force) is a low-frequency force in the range of 30–200 Hz,
which is relevant to the vibration characteristics of the vehicle–track coupled sys-
tem. The P2 force usually acts for a relatively longer duration and hence affects the
dynamic behavior of the vehicle sprung mass and the rail infrastructure. Therefore,
the P2 force has the potential to damage most of the vehicle and track components
located in the vicinity of wheel–rail contact zone. As shown in Fig. 8.9, the P1 and
P2 forces caused by a dipped rail joint with 2a = 0.02 rad to a freight wagon are
approximately 265 kN and 190 kN, respectively.
Figure 8.10 shows the time histories of the vertical displacements of the
wheelset and track system induced by the dipped rail joint. It can be seen that the
wheel–rail displacements decay slowly and exhibit a low-frequency vibration,
8.2 Dynamic Response of Vehicle–Track Interaction … 307
(a) (b)
30 180
10
140
0
120
-10
100
-20
-30 80
0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400
Track coordinate (m) Track coordinate (m)
(c) (d)
12 30
10
8 25
6 Wheelset 1 20
4 Wheelset 2
2 Wheelset 3 15
0 Wheelset 4
10
-2
-4 5
-6
0
-8
0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400
Track coordinate (m) Track coordinate (m)
Fig. 8.8 Steady-state curving performance of a freight wagon negotiating a small radius curved
track: a lateral wheel–rail force, b vertical wheel–rail force, c lateral wheelset displacement, and
d lateral car body displacement
300
P1
250
Vertical wheel-rail foce (kN)
P2
200
150
P0
100
50
0 2 4 6 8 10
Time (ms)
Fig. 8.9 Vertical wheel–rail force caused by dipped rail joint in heavy-haul railway system
which has a similar characteristic to the P2 force. Besides, the maximum dis-
placements of the wheelset and the track components appear at 4–6 ms after the
wheel–rail impact. This also indicates that the P2 force has a very important effect
308 8 Vibration Characteristics of Vehicle–Track Coupled System
(a) (b)
2.0
Vertical wheelset displacement (mm)
2.0
0.5 Zs
0.5
Zb
0.0
0.0
0 10 20 30 40 50 0 10 20 30 40 50
Time (ms) Time (ms)
Fig. 8.10 Vertical displacements of a freight wagon wheelset and b track under the impact of
dipped rail joint
on the dynamic response of the vertical displacements of the wheelset and track
system. It is also evident from Fig. 8.10 that the impact vibration is transmitted
from wheel–rail interface to rail infrastructure, which leads to that the vertical
displacements of the wheelset, the rail, the sleeper, and the ballast (see Zw, Zr, Zs,
Zb) decrease gradually.
The dynamic responses of the vertical accelerations of the wheelset and track
system induced by the dipped rail joint are presented in Fig. 8.11. It shows that the
time history of the vertical wheelset acceleration is basically consistent with that of
the wheel–rail force, while the rail acceleration peak exists for a very short duration,
which is similar to the P1 force. The maximum vertical accelerations of the wheelset
and the rail are approximately 30 g and 300 g, respectively, the latter is about 10
times larger than the former. It is also observed that the vertical acceleration
amplitudes of the rail, the sleeper, and the ballast decrease gradually, their ratio
remains approximately 90:7:1. The predominant frequencies of the vertical accel-
erations of the rail, the sleeper, and the ballast also decrease gradually; their ratio
remains approximately 6:2:1. It is obvious that the vibration amplitude and fre-
quency of track parts are highly related to the track structure parameters (such as
mass, property of fastenings and rail pad, ballast damping, etc.). However, for
different types of track structures, their vibration characteristics remain similar.
Figure 8.12 shows the dynamic vertical wheel–rail force of a high-speed vehicle
negotiating a dipped rail joint at a speed of 300 km/h. It can be seen that the
dynamic characteristics of the high-speed wheel–rail force are similar to that shown
in Fig. 8.9, but it decays slower than that of the freight wagon. Figure 8.12 shows
the P1 and P2 forces caused by a dipped rail joint with 2a = 0.006 rad to the
high-speed vehicle are approximately 210 kN and 160 kN, respectively.
The time histories of the vertical displacements of the high-speed wheelset and
the track affected by the dipped rail joint are shown in Fig. 8.13, respectively. The
results also show that the dynamic vertical displacements of the wheelset and track
system are closely related to the P2 force. It is also evident from Fig. 8.13 that the
8.2 Dynamic Response of Vehicle–Track Interaction … 309
(a) (b)
40 400
Vertical wheelset acceleration (g)
10 0
-100
0
-200
-10 -300
0 2 4 6 8 10 0 2 4 6 8 10
Time (ms) Time (ms)
(c) (d) 6
30
Vertical sleeper acceleration (g)
2
10
0
0
-2
-10 -4
-20 -6
0 4 8 12 16 20 0 4 8 12 16 20
Time (ms) Time (ms)
Fig. 8.11 Vertical accelerations of freight wagon–track system under the impact of dipped rail
joint: a wheelset, b rail, c sleeper, and d ballast
150
100
P0
50
0 2 4 6 8 10
Time (ms)
impact vibration is transmitted from the wheel–rail interface to the rail infrastruc-
ture, which leads to that the vertical displacements of the wheelset, the rail, the
sleeper, and the ballast (see Zw, Zr, Zs, Zb) decrease gradually.
310 8 Vibration Characteristics of Vehicle–Track Coupled System
(a) (b)
Vertical wheelset displacement (mm) 2.5 2.5
Zs
1.0 1.0
Zb
0.5 0.5
0.0 0.0
0 10 20 30 40 50 60 0 10 20 30 40 50 60
Time (ms) Time (ms)
Fig. 8.13 Vertical displacements of a wheelset and b track under the impact of dipped rail joint
(a) (b)
20 400
Vertical wheelset acceleration (g)
-5 -300
0 2 4 6 8 10 0 2 4 6 8 10
Time (ms) Time (ms)
(c) (d)
15 10
Vertical sleeper acceleration (g)
5
0
0
-5
-5
-10 -10
0 5 10 15 20 0 5 10 15 20
Time (ms) Time (ms)
Fig. 8.14 Vertical accelerations of high-speed vehicle and track under the impact of dipped rail
joint: a wheelset, b rail, c sleeper, and d ballast
δ
λ
δ
of d1 = 0.2 mm, and a secondary short wave with a wavelength of k = 0.1 m and
amplitude of d2 = 0.1 mm. It can be seen that the rail weld geometry irregularity
can induce a large impact to the high-speed wheel–rail system. The peak dynamic
responses of the wheel–rail force and the accelerations appear after the wheel
passing over the superimposed short-wavelength irregularity. It means that the
maximum dynamic responses of the wheel–rail system due to the rail weld
312 8 Vibration Characteristics of Vehicle–Track Coupled System
160
140
100
80
60
40
20
0
0 2 4 6 8 10 12 14 16
Time (ms)
Fig. 8.16 Vertical wheel–rail force caused by rail weld irregularity in high-speed railway system
irregularity are mainly affected by the irregularity with a shorter length. Figure 8.16
shows the maximum force caused by a rail weld irregularity with a short wave
(wavelength k = 0.1 m and amplitude d2 = 0.1 mm) to the high-speed wheel–rail
system is approximately 135 kN (about 1 time of the static axle load), while the
minimum wheel–rail force is only 5 kN. The results indicate that the
short-wavelength irregularity existed in the high-speed railway weld zone can
largely exacerbate the wheel–rail contact force and result in heavy unloading to the
wheels, which can not only damage the local contact region of the wheel–rail
system, but also adversely affect the running safety of high-speed trains.
The dynamic responses of the vertical accelerations of the high-speed wheel–rail
system induced by the rail weld irregularity are shown in Fig. 8.17. It shows that
the dynamic response of the vertical wheel–rail acceleration is basically consistent
with that of the wheel–rail force. The maximum vertical accelerations of the
(a) (b)
10.0 200
Vertical wheelset acceleration (g)
7.5 150
5.0 100
2.5
50
0.0
0
-2.5
-50
-5.0
-7.5 -100
-10.0 -150
0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16
Time (ms) Time (ms)
Fig. 8.17 Vertical accelerations of a wheelset and b rail under impact of rail weld irregularity
8.2 Dynamic Response of Vehicle–Track Interaction … 313
wheelset and the rail are approximately 7.5 g and 150 g, respectively, the latter is
about 20 times of the former. This means that the wheel–rail impacts caused by the
rail weld joint irregularities mainly damage the track system.
Turnout is a very important component in railway track structures, which turns the
running direction of trains from one railway line to another. When a high-speed
train crosses the rail discontinuity in a turnout, strong lateral and vertical impacts
between the wheels and rails occur [3, 4]. These impacts can potentially damage the
local contact region of wheels and turnout components, and then shorten their
service life. In this section, the dynamic response of a high-speed railway vehicle
passing the rail discontinuity in a turnout is taken as an example to illustrate the
vibration characteristics of vehicle–track coupled system under the excitation of
lateral impulsive defects at railway turnout.
Figures 8.18 and 8.19 depict the dynamic responses of a high-speed wheel–rail
system under the lateral impact of a high-speed wheel crossing a switch point rail
(see Fig. 3.13 in Chap. 3). In the simulation, an angle of attack b = 0.006 rad was
considered, which generated a lateral impact velocity of 0.5 m/s to the wheel. From
Figs. 8.18 and 8.19, it is observed that the wheel–rail lateral impact at switch zone
has a very important effect on the dynamic behavior of the high-speed train–track
system, especially the lateral wheelset motion. Figure 8.18 shows that the maxi-
mum lateral wheel–rail force caused by the switch point impact is larger than 30
kN. This means such a strong lateral impact can induce wheel flange climbing the
railhead. At the same time, the lateral wheelset displacement exhibits a
low-frequency harmonic response and decays slowly, as shown in Fig. 8.19. The
maximum lateral wheelset displacement attains approximately 8 mm. The above
20
10
Left wheel
-10
0.0 0.2 0.4 0.6 0.8 1.0
Time (s)
314 8 Vibration Characteristics of Vehicle–Track Coupled System
(a) (b)
15 1.5
Lateral wheelset displacement (mm)
0 0.5
Left rail
-5
0.0
-10
-15 -0.5
0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
Time (s) Time (s)
Fig. 8.19 Lateral displacements of high-speed a wheelset and b rail under lateral impact at rail
switch
results show that the lateral impact at a railway switch can potentially trigger the
hunting motion and wheel flange climbing behavior, which threatens the running
safety of high-speed trains. Figure 8.19 shows that the lateral impact can also cause
large deformation of rails; the lateral rail deformation attains a maximum value of
approximately 1.1 mm. The dynamic response of lateral rail deformation is basi-
cally consistent with that of the lateral wheel–rail force is shown in Fig. 8.18.
The vertical local harmonic geometry defects usually exist at rail joint zones,
bridge–subgrade transition zones, rail infrastructure local settlement zones, and so
on. Figures 8.20, 8.21 and 8.22 present the dynamic responses of a high-speed
vehicle–track coupled system under the impact of a vertical local harmonic
geometry defect (see Fig. 3.19 in Chap. 3) with a short wavelength of L = 2 m and
an amplitude of a = 2 mm at a speed of 300 km/h.
It shows that the vibration characteristics of the vehicle–track coupled system
induced by the vertical local harmonic geometry defect are quite different from that
caused by the vertical impulsive defect. The dynamic vertical wheel–rail force
caused by the local harmonic defect (see Fig. 8.20) no longer contains the P1, P2
forces as shown in Figs. 8.9 and 8.12, but exhibits two peaks of Pmax, Pmin forces.
Under the excitation of the local harmonic defect, the wheel–rail force also shows a
harmonic fluctuation, as shown in Fig. 8.20. It should be noted that the Pmin force is
very important for the evaluation of train running safety. If the Pmin force reduces to
zero, the wheel will lose contact with the rail. Such a situation will be a threat to the
operating safety of high-speed trains. The Pmax force shown in Fig. 8.20 is similar
8.2 Dynamic Response of Vehicle–Track Interaction … 315
180
150 Pmax
90
P0
60
30
Pmin
0
0.00 0.03 0.06 0.09 0.12
Time (s)
Fig. 8.20 Vertical wheel–rail force induced by a short local harmonic defect in high-speed
railway system
(a) (b)
4 3
Vertical wheelset displacement (mm)
3 Zr
2
2
Zs
1 1
Zb
0
0
-1
0.00 0.03 0.06 0.09 0.12 0.00 0.03 0.06 0.09 0.12
Time (s) Time (s)
Fig. 8.21 Vertical displacements of a wheelset and b track induced by a short local harmonic
geometry defect
to the P2 force presented in Figs. 8.9 and 8.12, which is a low-frequency force and
usually acts for a relatively long duration. The dynamic wheel–rail force induced by
local harmonic geometry defects can affect the dynamic behavior of the vehicle
sprung mass and the rail infrastructure significantly. Therefore, the dynamic
responses of the vehicle and track systems also exhibit harmonic fluctuations as the
wheel–rail force does as shown in Figs. 8.21 and 8.22.
Figure 8.21 shows the time histories of the vertical displacements of the
wheelset and track system induced by the local harmonic geometry defect. It can be
seen that the wheel–rail displacements decay slowly and exhibit a low-frequency
vibration, which have similar characteristics to that shown in Fig. 8.13. It is also
316 8 Vibration Characteristics of Vehicle–Track Coupled System
(a) (b)
10 10
Vertical vehicle acceleration (g)
-5 -5
-10 -10
0.00 0.03 0.06 0.09 0.12 0.00 0.03 0.06 0.09 0.12
Time (s) Time (s)
Fig. 8.22 Vertical accelerations of a vehicle and b rail induced by a short local harmonic
geometry defect
evident that the impact vibration is transmitted from the wheel–rail interface to the
rail infrastructure, which leads to that the vertical displacements of the wheelset,
rail, sleeper, and ballast (see Zw, Zr, Zs, Zb) decrease gradually.
Figure 8.22 depicts the dynamic responses of the vertical accelerations of the
high-speed vehicle system and ballast layer induced by the local harmonic geom-
etry defect. It also shows that the dynamic response of the vertical wheelset
acceleration is consistent with that of the wheel–rail force, while the car body
acceleration is very small. It is also observed that the vertical acceleration ampli-
tudes of the rail, sleeper and ballast decrease gradually, their ratio remains
approximately at 2.1:1.6:1. The dynamic response of the ballast vertical accelera-
tion is also similar to that of the vertical wheel–rail force, as shown in Fig. 8.22b.
This means that the impact caused by the local harmonic geometry defect has a
great influence on the vibration of the rail infrastructure.
Figures 8.23, 8.24, and 8.25 illustrate the high-speed wheel–rail dynamic
responses under the impact of a vertical local harmonic geometry defect (see
Fig. 3.19 in Chap. 3) with a long wavelength of L = 20 m and an amplitude of
a = 5 mm at a speed of 300 km/h. Figure 8.23 shows that the wheel–rail force
induced by a long-wavelength local harmonic defect no longer exhibits harmonic
fluctuation as shown in Fig. 8.20. The reason should be that a local geometry defect
with a wavelength larger than the distance between two axles of a bogie can trigger
a significant pitch motion of the bogies, which reduce the wheel–rail impact force
and then decrease the vibration amplitude of high-speed vehicle–track coupled
system. From Fig. 8.23, it can also be seen that the wheel–rail force induced by a
long vertical local harmonic geometry defect is small. However, this type of track
defect has a significant effect on the car body vibration and ride quality, as shown in
Fig. 8.24. Figure 8.25 demonstrates that the dynamic vertical displacement of the
wheelset is very close to the shape of the local geometry defect, but the dynamic
8.2 Dynamic Response of Vehicle–Track Interaction … 317
80
70
65
60
55
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Time (s)
Fig. 8.23 Vertical wheel–rail force induced by a long local harmonic geometry defect in
high-speed railway system
(a) 6
Wheelset
4
Vertical acceleration (m/s )
2
Bogie frame
2
-2
-4
0.0 0.1 0.2 0.3 0.4 0.5
Time (s)
(b) 1.0
Vertical car body acceleration (m/s )
2
0.5
0.0
-0.5
-1.0
0.0 0.2 0.4 0.6 0.8 1.0
Time (s)
Fig. 8.24 Vertical accelerations of a bogie and b car body induced by a long local harmonic
geometry defect
318 8 Vibration Characteristics of Vehicle–Track Coupled System
(a) 15
-3
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Time (s)
(b) 1.2
1.1
Vertical rail displacement (mm)
1.0
0.9
0.8
0.7
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Time (s)
Fig. 8.25 Vertical displacements of a wheelset and b rail induced by a long local harmonic
geometry defect
response of the vertical rail deformation is basically consistent with that of the
vertical wheel–rail force is shown in Fig. 8.23.
Right
-5
-10
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Time (s)
Figures 8.26, 8.27 and 8.28 present the dynamic responses of a high-speed
vehicle–track coupled system under the impact of a lateral local harmonic geometry
defect (see Fig. 3.19 in Chap. 3) with a wavelength of L = 10 m and an amplitude
(a) 8
6
Wheelset
Lateral acceleration (m/s )
2
4
Bogie frame
2
0
-2
-4
-6
-8
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time (s)
(b)
1.2
Lateral car body acceleration (m/s )
2
0.8
0.4
0.0
-0.4
-0.8
-1.2
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Time (s)
Fig. 8.27 Lateral accelerations of a bogie and b car body induced by a lateral local harmonic
geometry defect
320 8 Vibration Characteristics of Vehicle–Track Coupled System
(a)
10
-5
-10
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Time (s)
(b) 0.3
Lateral rail deformation (mm)
0.2 Right
0.1
0.0
-0.1
Left
-0.2
-0.3
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Time (s)
Fig. 8.28 Lateral displacements of a wheelset and b rail induced by a lateral local harmonic
geometry defect
The cyclic geometry defects exist widely in the wheel–rail system, such as the rail
corrugation, the wheel polygonization (wheel harmonic wear or wheel periodic out
of roundness), etc. These cyclic geometry defects cause periodically forced vibra-
tion to the vehicle–track coupled system. In this section, the dynamic responses of
railway vehicles running on ballasted and ballastless tracks with wheel polygo-
nization or rail corrugation are taken as examples to illustrate the vibration char-
acteristics of the vehicle–track coupled system under the excitation of cyclic
geometry defects.
Figures 8.29, 8.30, and 8.31 illustrate the dynamic responses of high-speed
vehicle–track coupled systems subjected to the impact of cyclic geometry defects at
a speed of 300 km/h. In the simulation, a ballasted track (rail pad stiffness 60 MN/m)
and a ballastless track (rail pad stiffness 25 MN/m) were considered. Specifically,
Fig. 8.29 exhibits the time history of the dynamic wheel–rail force of a high-speed
passenger vehicle (axle load is 14 t) containing polygonized wheels (see Fig. 3.24 in
Chap. 3). The rolling circle of the polygonal wheels has 22 waves, which means the
wavelength of the polygonal wear is 122.73 mm. Figures 8.30 and 8.31 show the
160
120
80
40
0
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Track coordinate (m)
160
120
80
40
0
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Track coordinate (m)
322 8 Vibration Characteristics of Vehicle–Track Coupled System
120
80
40
0
0.0 0.4 0.8 1.2 1.6 2.0 2.4
Track coordinate (m)
-5
-10
0 1 2 3 4 5
Time (s)
324 8 Vibration Characteristics of Vehicle–Track Coupled System
-5
-10
0 1 2 3 4 5
Time (s)
It is evident from Figs. 8.32 and 8.33 that the failure of bogie secondary lateral
dampers can largely increase the lateral displacements of the vehicle system and
reduce the lateral stability of the high-speed vehicle. If only one lateral damper is
disabled, the lateral dynamics and stability of the high-speed vehicle are only
deteriorated a little, which does not affect the running safety. However, if two lateral
dampers in a bogie are disabled, the lateral wheelset displacement largely increases
and the hunting motion of the bogie is likely to occur. If all the 4 lateral dampers are
disabled at the same time, the lateral displacements of both the wheelset and the car
body are continually amplified. This means that the vehicle has already lost the
lateral stability and it operates at great risk to derail. The above results show that the
failure of vehicle dampers (such as lateral and yaw dampers) would greatly affect
the dynamics performance and running safety of high-speed trains. Therefore,
real-time monitoring of the service status of these key dampers and components is
very important for ensuring the safe operation of high-speed trains.
Fatigue fracture of fastener clips (see Fig. 3.37 in Chap. 3) has been observed
frequently in track section with short-pitch rail corrugations, serious rail weld
defects, etc., which greatly affect the running safety and ride comfort of rail
vehicles. When the clip of a rail fastener fractures, the fastening system can no
longer provide the lateral constraint to the rail, although the vertical support will
prevail. In such a situation, the rail lateral deformation and vibration largely
increase (see Fig. 8.34). The drastic fluctuation of wheel–rail force and the wheel
load reduction commence (see Fig. 8.35) when a vehicle passes over the track
section with the disabled fastener clips.
Figures 8.34 and 8.35 show the rail deformation and the dynamic vertical
wheel–rail force of a high-speed vehicle negotiating a curved track section (same as
considered in Sect. 8.1.3) with three adjacent disabled fastener clips at a speed of
300 km/h. It is observed that the lateral rail deformation increased approximately
8.4 Dynamic Response of Vehicle–Track Interaction Due to Failure … 325
Disabled Disabled
-1.2 -0.8
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time (s) Time (s)
Fig. 8.34 Rail deformation due to fracture of fastener clips: a vertical and b lateral
(a) (b)
78 10
Vertical wheel-rail force (kN)
Disabled
Lateral wheel-rail force (kN)
Disabled
77 8
76 Normal Normal
6
75
4
74
2
73
72 0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time (s) Time (s)
Fig. 8.35 Wheel–rail forces due to fracture of fastener clips: a vertical and b lateral
fivefold, with the associated enlargement of the vertical deformation. The vertical
and lateral wheel–rail forces on the track with three adjacent disabled fastener clips
increased by 5% and 1 time compared to the normal track. Due to the disabled
fastener clips, the vertical and lateral accelerations of the rail increased by 30% and
5 times, respectively. The above results show that the failure of fastener clips can
greatly aggravate the wheel–rail interaction, enlarge the rail deformation and
vibration, and cause heavy damages to the railway tracks, which affect the ride
comfort and running safety of rail vehicles.
Unsupported sleepers or hanging sleepers (see Fig. 3.38 in Chap. 3) are very
common in the ballasted track due to the nonuniform settlement of the ballast bed.
326 8 Vibration Characteristics of Vehicle–Track Coupled System
-1
-2
-3 Unsupported sleeper
-4
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Time (s)
The unsupported sleepers lower the rail support stiffness, increase the discontinuity
to track system. It can also cause wheel load reduction and increase the rail
deformation substantially under the vehicle operation, as shown in Figs. 8.36 and
8.37.
Figures 8.36 and 8.37 depict the dynamic vertical wheel–rail force and the rail
deformation of a freight wagon negotiating a straight track section with two adja-
cent unsupported sleepers at a speed of 80 km/h. It can be seen that the vertical
wheel–rail force on the track with two adjacent unsupported sleepers increase by
10% compared to the normal track, and the rail deformation increase about two
times. This means that the hanging sleepers can aggravate the wheel–rail interaction
and damage the track structures. It also reduces the ride comfort and influences the
safe operation of rail vehicles.
8.5 Dynamic Response of Vehicle–Track Interaction to Random Irregularities 327
-4
Fig. 8.38 Frequency 10
spectrum of the vertical car
-6
body acceleration 10
PSD (g /Hz)
-8
10
2
-10
10
-12
10
-14
10
0.1 1 10 100
Frequency (Hz)
-4
Fig. 8.39 Frequency 10
spectrum of the lateral car
-6
body acceleration 10
PSD (g /Hz)
-8
10
2
-10
10
-12
10
-14
10
0.1 1 10 100
Frequency (Hz)
328 8 Vibration Characteristics of Vehicle–Track Coupled System
PSD (g /Hz)
-6
10
2
-8
10
-10
10
-12
10
0.1 1 10 100
Frequency (Hz)
-2
Fig. 8.41 Frequency 10
spectrum of the lateral bogie
-4
frame acceleration 10
PSD (g /Hz)
-6
10
2
-8
10
-10
10
-12
10
0.1 1 10 100
Frequency (Hz)
10-4
10-6
10-8
10-10
0.1 1 10 100 1000
Frequency (Hz)
10-8
10-10
10-12
10-14
0.1 1 10 100 1000
Frequency (Hz)
8.5 Dynamic Response of Vehicle–Track Interaction to Random Irregularities 329
PSD (kN2/Hz)
10-2
10-4
10-6
10-8
0.1 1 10 100 1000
Frequency (Hz)
10-3
10-4
10-5
10-6
10-7
10-8
0.1 1 10 100 1000
Frequency (Hz)
10-5
10-6
10-7
10-8
10-9
10-10
0.1 1 10 100 1000
Frequency (Hz)
330 8 Vibration Characteristics of Vehicle–Track Coupled System
PSD (g2/Hz)
10-4
10-6
10-8
10-10
0.1 1 10 100 1000
Frequency (Hz)
10-8
10-10
10-12
10-14
0.1 1 10 100 1000
Frequency (Hz)
Figures 8.38 and 8.39 show the dynamic responses of the vertical and lateral
accelerations of the car body in the frequency domain. As shown in Fig. 8.38, the
frequency peaks of the vertical acceleration of the car body are in the range of 0.5–
40 Hz. The first frequency peak is near 1 Hz, which represents the natural vibration
frequency of the car body vertical and pitching motions [5, 6]. The other frequency
peaks around 4, 7 and 10 Hz are mainly related to vibration modes of the bogie
system. The frequency peaks around 30 Hz mainly reflect the forced vibration
induced by the excitations from the vertical wheel–rail interaction. Figure 8.39
indicates that the distinct frequency peaks of the lateral acceleration of the car body
are mainly in the range of 0.5–30 Hz. The first two frequency peaks are around 0.7
and 2.0 Hz, which are determined by the natural vibration frequency of the car
body lateral and yawing motions [7]. The other frequency peaks near 4, 10, 13, 19
and 25 Hz reflect the vibration modes of the bogie system and the forced vibration
induced by the excitations from the lateral wheel–rail interaction.
8.5 Dynamic Response of Vehicle–Track Interaction to Random Irregularities 331
Figures 8.40 and 8.41 depict the frequency spectrums of the vertical and lateral
accelerations of the bogie frame. As shown in Fig. 8.40, the frequency peaks of the
vertical vibration of the bogie frame are in the range of 0.5–40 Hz. The frequency
peaks around 1, 7 and 30 Hz mainly reflect the vibration modes of the bogie system
and the forced vibration induced by the excitations from vertical wheel–rail inter-
action. Figure 8.41 shows that the distinct frequency peaks of the lateral acceler-
ation of the bogie frame are also in the range of 0.5–40 Hz. The frequency peaks
around 2, 10, and 25 Hz are affected by the natural vibration frequency of the bogie
system and the forced vibration induced by the excitations from the lateral wheel–
rail interaction.
Figures 8.42 and 8.43 show the dynamic responses of the vertical and lateral
accelerations of the wheelset in the frequency domain. As presented in Fig. 8.42,
the vibration frequency of the vertical acceleration of the wheelset is mainly in the
range of 10–800 Hz. Due to the large contact stiffness between the wheel and the
rail, the high-frequency contact vibration in the wheel–rail interface excited by the
track irregularity is transmitted to the wheelset. Therefore, the frequency peaks of
the vertical wheelset acceleration appeared at 30–50 Hz and 400–800 Hz mainly
reflect the forced vibration induced by the excitations from the vertical wheel–rail
interaction. Figure 8.43 indicates that the distinct frequencies of the lateral wheelset
acceleration are mainly in the range of 1–100 Hz. The frequency peaks near 1.5, 8,
15, 25, and 230 Hz reflect the vibration modes of the bogie system and the forced
vibration induced by the excitations from the lateral wheel–rail interaction. It is
evident from Figs. 8.42 and 8.43 that the frequency spectrums of the vertical and
lateral wheelset accelerations are quite different, which is determined by the char-
acteristics of the vertical and lateral wheel–rail forces applied to the wheelset. The
vertical force is the Hertzian contact force generated by the elastic deformation at
the wheel–rail interface, while the lateral force is the creep force due to the relative
slip between the wheel tread and the railhead. Obviously, the nonlinear charac-
teristics of the Hertzian elastic contact force and the creep force are different.
Figures 8.44 and 8.45 show the frequency spectrums of the vertical and lateral
wheel–rail forces. Due to the high running speed and the large contact stiffness
between the wheel and the rail, the high-frequency contact vibration in the wheel–
332 8 Vibration Characteristics of Vehicle–Track Coupled System
rail interface excited by the track irregularity is significant. The dynamic response
of the vertical wheel–rail force is in a wide frequency range below 1000 Hz, which
covers three distinct dominant frequency ranges, as shown in Fig. 8.44. The first
main frequency is near 1 Hz, which represents the natural vibration frequency of
the vertical car body suspension. The secondary frequency peak appears around
34 Hz, which is mainly related to the coupled vibration of the wheel–rail system.
The third one is the high-frequency peak near 700 Hz, which reflects the
high-frequency Hertzian contact vibration occurring at the wheel–rail interface. It
can be seen that the frequency peak around 34 Hz has the maximum value. This
means the wheel–rail coupled (or resonant) vibration has the biggest contribution to
the vertical wheel–rail interaction force.
Figure 8.45 indicates that the lateral wheel–rail force is much smaller than the
vertical one. The dynamic response of the lateral wheel–rail force is mainly dis-
tributed in the frequency range below 100 Hz. The frequency peaks near 0.6, 2 and
25 Hz reflect the vibration modes of the vehicle system and the forced vibration
induced by the excitations from the lateral track irregularity. The above results
show that the frequency spectrums of the vertical and lateral wheel–rail forces are
quite different. Therefore, a clear understanding of the dynamic characteristics of
the wheel–rail force is very important for developing design methods to minimize
the wheel–rail interaction.
Figures 8.46 and 8.47 illustrate the dynamic responses of the vertical and lateral
accelerations of the rail in the frequency domain. From these figures, it can be
observed that the vertical and lateral rail vibration frequencies are distributed in a
wide range of frequency up to thousand Hz due to the high-frequency contact
vibration at the wheel–rail interfaces. The lateral rail vibration is weaker than its
vertical vibration. The lateral rail acceleration has several distinct frequency peaks
in the range of 30–1000 Hz, while the frequency peaks of the vertical acceleration
are in higher frequency range due to the vertical rail support stiffness being higher
than the lateral stiffness. This is also because the vertical wheel–rail force is larger
than the lateral force, as shown in Figs. 8.44 and 8.45. The above results are
consistent with the experimental results reported in Ref. [8].
Figures 8.48 and 8.49 show the frequency spectrums of the vertical and lateral
accelerations of the track slab. It can be seen that the frequency peaks of the vertical
acceleration of the track slab are mainly in the range of 300–700 Hz, while the
vibration acceleration below 300 Hz is much lower. The lateral acceleration
8.5 Dynamic Response of Vehicle–Track Interaction to Random Irregularities 333
frequency is distributed in a wide frequency range of 30–500 Hz, which covers two
distinct dominant frequency ranges of 30–140 Hz and 300–500 Hz. It is obvious
that the vertical and lateral vibration characteristics of the track slab are quite
different due to the differences in the wheel–rail interface and the rail vibrations
upon the track slabs.
The aforementioned simulation results demonstrate that the primary and sec-
ondary suspension systems can effectively attenuate the vehicle system vibrations.
Both the vertical and the lateral vibration amplitudes of the wheelset, the bogie
frame, and the car body decrease successively when the wheel–rail vibration is
transmitted upwards from the wheelset to the car body. The high-frequency
vibrations of the bogie frames, as well as the car body high-frequency vibration
above 40 Hz, are effectively suppressed. The wheel–rail vibration is also trans-
mitted downwards from the rail to the track slab. Both the vertical and the lateral
vibration amplitudes of the wheel–rail force, the rail, and the track slab decrease
successively. However, for the high-speed railway tracks, the high-frequency
vibrations at the wheel–rail interface can also have important effects on the dynamic
behavior of rail infrastructure.
It is obvious that the vibration characteristics and frequency spectrums of the
vehicle–track coupled system are highly related to the train and track structure
parameters. But for different types of trains and track structures, their vibration
characteristics are similar [9, 10]. Therefore, the simulated vibration characteristics
of other train and track structures are not shown here.
The superior integrity and stability have made the ballastless track being a preferred
track form in high-speed railways. The use of the ballastless track on soil subgrade
is still a major concern in view of the post-construction settlement, especially the
differential subgrade settlement. Since soft soil ground with large compressibility
and low permeability is widely distributed in China [11], the differential settlement
on soft soil subgrade along the high-speed railway lines is a common occurrence
due to the nonuniform soil properties, variation of groundwater-level and other
defects of subgrade [12, 13]. Relevant monitoring has shown that in some regions,
subgrade settlements of high-speed railways can be quite serious, sharply changing
the original gradient of the railway line such that speed restrictions are warranted.
Due to the limited adjustability of the ballastless track, the control of differential
settlement becomes a key parameter of significance in the high-speed railways.
334 8 Vibration Characteristics of Vehicle–Track Coupled System
M c Jc c v
Zc
Ksz Csz
M t Jt t2 t1
Zt2 Zt1
Kpz Cpz
Fig. 8.50 Vehicle–slab track coupled dynamics model with differential subgrade settlement
ð8:1Þ
and the base; Crk, Csk and Dbk are the addition items derived from track weight,
which can be calculated using
8 pffiffiffiffiffiffi
g
>
> Ck ¼ 2ml
kp ð1 8 cos kpÞ
>
<
< ls ;
> for k ¼ 1
ð8:2Þ
>
> Dk ¼ ms g 0; h i for k ¼ 2
>
: >
: sinhðbk ls Þ þ sinðbk ls Þ k ls Þ cosðbk ls Þ
b bk Gk coshðb
bk bk ; for k [ 2
k
where Zb(x, t) is the vertical displacement of the concrete base; Z0(x) is the dif-
ferential subgrade settlement. Thus, the force of the subgrade spring at each node
can be written as
Fbi ðtÞ ¼ Hdi kb ½Zb ðxi ; tÞ Z0 ðxi Þ þ cbi Z_ b ðxi ; tÞ ð8:4Þ
Commonly, the cosine curve is utilized as the typical settlement pattern as shown
in Fig. 8.51. In the settlement section, the subgrade displacement Z0(xi) is defined
as
A
s xi 2
1 + cos 2p ; for 2l 2s \xi \ 2l þ
l s
Z0 ðxi Þ ¼ 2 2 ð8:5Þ
0; others
80 0.3
70 0.0
60 -0.3
50 -0.6
0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.2 0.4 0.6 0.8 1.0 1.2
Time (s) Time (s)
Fig. 8.52 Vehicle dynamic responses induced by differential subgrade settlement: a vertical
wheel–rail force, and b car body acceleration
8.6 Dynamic Response Due to Railway Infrastructure Settlement 337
9.5 0
9.0 -10
0.0 0.22 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
Time (ss) Time (s)
(c)
Fig. 8.53 Track dynamic responses induced by differential subgrade settlement: a rail displace-
ment, b fastening force, and c track–subgrade contact force
From Fig. 8.53a, it can be seen that the initial rail displacement is not zero
because the track settles along with the subgrade under the action of self-weight
before the vehicle’s arrival. However, due to the high bending stiffness of the
concrete track, the track deflections vary as the ballastless track generally loses
contact with the subgrade at some positions. The magnitude of the rail deflection
above the center point of the settlement section, in this case, is a little lower than
10 mm. During the vehicle’s passage, the wheelset-induced rail displacement is
about 1 mm, so the amplitude of rail vibration is close to 11 mm.
Figure 8.53b shows the dynamic rail–slab contact force (i.e., fastening force).
There are four peaks corresponding to the four wheelsets of the vehicle on the
curve. The resistance force of the fastener at the settlement center also has an initial
value caused by the differential settlement, but it is quite small compared to the
vehicle-induced fastening force. The initial fastening force is only 0.6 kN, while
that caused by the four wheelsets increases to about 22 kN.
Figure 8.53c shows the variation of the track–subgrade contact force during the
entire process of the passage of the vehicle. Different from the fastening force
338 8 Vibration Characteristics of Vehicle–Track Coupled System
connecting the rail and the slab, it shows that the contact force between the track
and the subgrade is more affected by the differential settlement. In Fig. 8.53c, two
apparent waves corresponding to the two bogies of the vehicle move over time, and
two stationary wave crests located between −7.5 and 7.5 m are ascribed to the
differential settlement section. Three unsupported areas along the track corre-
sponding to the hanging sleepers around the settlement center are evident; in these
areas, the track is separated from the subgrade and near the settlement boundaries
the track is slightly arched. The track–subgrade contact forces in these unsupported
areas are 0 as shown in the figure. Before the vehicle moves into the settlement
zone, the wheelset-induced track–subgrade contact force is about 14 kN, and the
maximum force caused by the settlement is 11 kN. When the wheelset moves close
to the settlement area, the track–subgrade contact force shows significant variation.
For those unsupported areas, the contact between track and subgrade is reestab-
lished and lost again periodically during the passage of the vehicle. The track–
subgrade contact forces at two settlement boundaries at ±7.5 m display stress
concentrations significantly amplified by the moving vehicle. The peak force due to
the superposition of the dynamic vehicle impact and the stationary settlement effect
is approximately 20 kN.
(2) Influence of settlement wavelength
Figure 8.54 shows the effect of the settlement wavelength on the dynamic
responses of the coupled system. The settlement wavelength has been increased
from 10 to 30 m, and the settlement amplitude is set as 10 mm.
The variations of the dynamic indexes are nonlinear with an increase in the
settlement wavelength. From the vertical wheel–rail force, wheel unloading rate and
car body acceleration shown in Fig. 8.54a, b, it is clear that the settlement wave-
length of 10 m exacerbates the wheel–rail interaction and the vehicle vibration the
most. The maximum vertical wheel–rail force at 10 m is about 116.56 kN (in-
creasing by about 70%) and the wheel unloading rate is 0.29. The maximum car
body acceleration is 0.39 m/s2. With the settlement wavelength expands to 15 m
and larger, the settlement-induced maximum vertical wheel–rail force drops below
80 kN, and both the wheel–rail interaction and the car body acceleration gradually
decline with the increase in settlement wavelength.
For the rail displacement at the settlement center in Fig. 8.54c, the settlement
wavelength of 10 m also leads to an abnormal response. The initial rail deflection is
only 7.06 mm, resulting from the significant contact failure between the track and
subgrade around the settlement center. When the vehicle moves on, the rail
vibration is intensified, and the maximum rail displacement increases to 11.63 mm.
When the settlement wavelength is 15 m, the initial rail deflection is almost close to
the subgrade deformation with a slight difference, so the vehicle-induced maximum
rail displacement becomes normal, which is only 1.15 mm larger than the initial
displacement. With the progressive increase in wavelength, the rail displacements
remain stable, due to the track being unsupported.
The track–subgrade contact force at the settlement center in Fig. 8.54d shows
that the initial force for the settlement wavelength of 10 m is 0, which means that
8.6 Dynamic Response Due to Railway Infrastructure Settlement 339
80 0.1 0.2
60 0.0 0.1
10 15 20 25 30 10 15 20 25 30
Settlement wavelength (m) Settlement wavelength (m)
(c) 14 (d) 20
15
12
10
10
5
8
0
6 -5
10 15 20 25 30 10 15 20 25 30
Settlement wavelength (m) Settlement wavelength (m)
Fig. 8.54 Influence of settlement wavelength on system dynamic responses: a vertical wheel–rail
interaction, b car body acceleration, c rail displacement, and d track–subgrade contact force
the initial contact between the track and the subgrade here is lost. Due to the
hanging track structure, the vehicle-induced dynamic track–subgrade contact force
is smaller, which is only 8.86 kN. With the increase of the settlement wavelength,
the unsupported areas are gradually eliminated, and the difference between the
vehicle-induced maximum track–subgrade contact force and the settlement-induced
initial track–subgrade contact force stabilize around 10 kN.
The above results indicate that the dynamic performance of the vehicle–slab
track system suffers slightly from the wide-range settlement. Besides, there exists a
particular range of settlement wavelength which may excite the resonance of the
vehicle structure to an extent and will significantly exacerbate the wheel–rail
interaction. For the specific conditions considered in this analysis, it is around
10 m.
(3) Influence of settlement amplitude
Similarly, the influence of the settlement amplitude is illustrated in Fig. 8.55, with a
constant settlement wavelength of 15 m.
Figures 8.55a, b show almost linear growth of the vertical wheel–rail force,
wheel unloading rate and car body acceleration as the settlement amplitude
340 8 Vibration Characteristics of Vehicle–Track Coupled System
60 0.0 0.0
10 15 20 25 30 10 15 20 25 30
Settlement amplitude (mm) Settlement amplitude (mm)
(c) (d)
35 20
15
25
10
20
5
15
0
10
5 -5
10 15 20 25 30 10 15 20 25 30
Settlement amplitude (mm) Settlement amplitude (mm)
Fig. 8.55 Influence of settlement amplitude on system dynamic responses: a vertical wheel–rail
interaction, b car body acceleration, c rail displacement, and d track–subgrade contact force
amplifies from 10 to 30 mm. It implies that the wheel–rail interaction and the ride
comfort will be likely to exceed the critical limit if the settlement amplitude con-
tinues to worsen.
In Fig. 8.55c, it is visible that the initial rail displacement induced by the dif-
ferential settlement without vehicle load also increases with the settlement ampli-
tude but gradually slows down. For the settlement case of 30 mm/15 m, the initial
rail deflection at the settlement center is 26.44 m. It indicates that the unsupported
area between the track and the subgrade is increasing significantly. Additionally,
the wheel–rail force is increasing, so the corresponding dynamic rail displacement
shows a corresponding increase, and the difference between the maximum rail
displacement and the initial displacement gradually increases from 1.09 to
4.82 mm.
It can be seen from Fig. 8.55d that at the settlement center, the initial track–
subgrade contact force is always nil, it indicates that there is always a separation
between the track and the subgrade. Due to the high rigidity of track structure, the
gap between the track and the subgrade around the settlement center widens with
the increasing settlement amplitude. Consequently, the dynamic impact from the
vehicle load here is significantly attenuated as depicted by the red dotted line.
8.6 Dynamic Response Due to Railway Infrastructure Settlement 341
The settlement of the ballasted railway track mainly depends on the plastic
deformation of the subgrade and the degradation of the ballast accounting for train
dynamic loading as well as the combination of geologic conditions and ambient
conditions [15]. When the external factors are different along the track, the dif-
ferential settlement of the subgrade and ballast occurs [16]. The rail and the sleepers
settle along with the ballast due to gravity. The rail surface deflection finally turns
into a track irregularity [17]. With an increase in the ballast settlement, the sepa-
ration of the sleepers from the ballast bed may appear in local areas, which will lead
to the formation of unsupported sleepers [18]. Track irregularity and unsupported
sleepers resulting from the settlement have a significant influence on the dynamic
response of the vehicle–track coupled system.
1. Vehicle–track coupled dynamics model with differential ballast settlement
Since ballast is composed of granular material, the relationship between the sub-
grade settlement and the ballast settlement is affected by many complex factors. An
improved vehicle–track coupled dynamics model [19] that concerns the effect of the
differential ballast settlement is shown in Fig. 8.56.
Similar to the differential subgrade settlement model proposed in Sect. 8.6.1,
three key points are included in this ballast settlement model: (1) the weight of rail
and sleeper; (2) no-tension springs accounting for the compression-only ballast
support; (3) the differential ballast settlement as a given boundary condition and
additional forces to keep the ballast blocks in the given settlement positions when
Mc Jc c
Csz Ksz Zc
t2
Mt Jt
t1
Cpz Zt2
Kpz
Z t1
Settled ballast
Fig. 8.56 Vehicle–track coupled dynamics model with differential ballast settlement
342 8 Vibration Characteristics of Vehicle–Track Coupled System
the track system is in equilibrium. Taking these three key points into consideration,
the equations of motion of the track subsystem can be modified as follows:
(1) Vertical rail motion
Taking the weight of rail into account, the equation of the vertical rail motion can
be written as
X X
EI kp 4 N 4
€qk ðtÞ þ qk ðtÞ ¼ Frsi ðtÞYk ðxi Þ þ pj ðtÞYk xwj þ Ck ; ðk ¼ 1NM Þ
mr l i¼1 j¼1
ð8:6Þ
with
pffiffiffiffiffiffiffiffiffi
2mr l g
Ck ¼ ð1 cos kpÞ ð8:7Þ
kp
where Fbsi(t) is the contact force between the ith sleeper and the ballast. As the
ballast support is modelled by the non-tension springs and only the fully supported
or unsupported sleepers are considered, Fbsi(t) can be described as
Fbsi ðtÞ ¼ Hdi ðtÞ kb ½Zsi ðtÞ Zbi ðtÞ þ cb Z_ si ðtÞ Z_ bi ðtÞ ð8:9Þ
€bi ðtÞ þ ð2cw þ cf ÞZ_ bi ðtÞ þ ð2kw þ kf ÞZbi ðtÞ cw Z_ bði1Þ ðtÞ cw Z_ bði þ 1Þ ðtÞ
Mb Z
kw Zbði1Þ ðtÞ kw Zbði þ 1Þ ðtÞ Fbsi ðtÞ Fbi0 ¼ 0 ði ¼ 1N Þ;
ð8:11Þ
8.6 Dynamic Response Due to Railway Infrastructure Settlement 343
where Fbi0 is the additional force applied to the ith ballast block to keep the ballast
blocks in the given settlement position, which can be calculated by
Fbi0 ¼ kb Hdi0 ðZbi0 Zsi0 Þ þ kf Zbi0 þ kw 2Zbi0 Zbi1;0 Zbi þ 1;0 ði ¼ 1N Þ
ð8:12Þ
where Zbi0 is the given ballast settlement, Zsi0 is the vertical displacement of the
sleeper induced by the ballast settlement when the whole track system is in equi-
librium state without vehicle load. Detailed description of the solution method to
calculate Zbi0 and Zsi0 can be seen in Ref. [9].
The cosine curve shown in Fig. 8.51 is used to describe the ballast settlement.
Zbi0 is defined as
A 2p
2 1 þ cos s xi 2 ; for 2l 2s \xi \ 2l þ
l s
Zbi0 ¼ 2 ð8:13Þ
0; others
70 0.0
60 -0.3
50 -0.6
0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.2 0.4 0.6 0.8 1.0 1.2
Time (s) Time (s)
Fig. 8.57 Vehicle dynamic responses induced by differential ballast settlement: a vertical wheel–
rail force and b car body acceleration
1.0
10.0
0.5 9.5
9.0
0.0
Transient contact 8.5
-0.5 8.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
Time (s) Time (s)
Fig. 8.58 Track dynamic responses induced by differential ballast settlement: a sleeper–ballast
gap and b rail displacement
about 0.5 mm. The unsupported sleeper is transiently in contact with the ballast
during the successive passages of two adjacent bogies through the settlement zone.
It can be seen from Fig. 8.58a that the ballast settlement and moving vehicle are
two main factors influencing the contact state of the unsupported sleeper during the
vehicle’s passage. Figure 8.58b shows the response of the rail displacement. The
initial rail displacement is about 9.5 mm due to the track deflection under the action
of self-weight. During the two bogies passing through the settlement zone, the
amplitude of the rail vibration increases to 10.5 mm under the combined influence
of the ballast settlement and the wheel–rail contact force. Due to lack of effective
vibration reduction measures for the unsupported sleepers, the vibration of rail
attenuates very slowly relative to that shown in Fig. 8.53a.
When the vehicle passes through the settlement zone, the contact states between
the sleepers and the ballast progressively vary due to the settlement effect.
Figure 8.59 shows the sleeper–ballast contact state induced by the differential
ballast settlement and moving vehicle. In this figure, the horizontal axis shows the
8.6 Dynamic Response Due to Railway Infrastructure Settlement 345
Sleeper number
10
-10
-20 th
4 wheelset
rd
3 wheelset
-30
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Time (s)
time range, and the vertical axis shows the sleeper number. The two black lines
represent the boundary of the ballast settlement, and the middle area is the settle-
ment zone. The four blue lines depict the locations of the four wheelsets at different
times. The running direction of the vehicle is from sleeper −30 to sleeper +30. The
full contact between sleeper and ballast is shown in white color, while the
unsupported state is red-colored. It can be seen from Fig. 8.59 that there are three
unsupported areas corresponding to the hanging sleepers at the settlement center
zone and two arched areas near the settlement boundaries where the track is slightly
arched. The initial unsupported sleepers instantaneously maintain contact with the
ballast when the wheelset moves over, and immediately recover to the unsupported
state after the wheelset moves off.
The sleeper–ballast contact force induced by both the differential ballast settle-
ment and the moving vehicle is shown in Fig. 8.60. The variations of the contact
force between the sleepers and the ballast of the settlement area are similar to the
variations of the track–subgrade contact force that depicted in Fig. 8.53c. For each
sleeper, the contact force reaches a peak value when one of the four wheelsets pass
10
Sleeper number
40.00
0 30.00
-10 20.00
-20 10.00
-30 0.000
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Time (s)
346 8 Vibration Characteristics of Vehicle–Track Coupled System
through, as shown by the four tilted strips in the figure. The sleepers with larger
contact forces are at the boundary of the settlement zone. This is because the
sleepers beside the settlement boundary are all unsupported, and those
well-supported sleepers at the boundary need to carry the gravity and the inertial
forces of the rail and the nearby sleepers. At the boundaries of the settlement zone,
the maximum sleeper–ballast contact force is 63.0 kN, which is about 54.7 kN at
the non-settlement area.
References
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forces. Railw Eng J. 1974;3(1):2–16.
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high-speed railways. Proc Inst Mech Eng Part F: J Rail Rapid Transit. 2018;232(1):249–61.
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Southwest Jiaotong University; 1997 (in Chinese).
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Southwest Jiaotong University; 2000 (in Chinese).
5. Wang TF. Vehicle system dynamics. Beijing: China Railway Press; 1994 (in Chinese).
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Canada; 1984.
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CRH train at speed of 350 km/h. Int J Rail Transp. 2015;3(1):1–16.
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Memorandum. Derby: British Rail Research; 1974.
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Chengdu: Southwest Jiaotong University; 2000 (in Chinese).
11. Wang BL. Subgrade and track engineering of high-speed railway. Shanghai: Tongji
University Press; 2015 (in Chinese).
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railway under varying water levels. Int J Rail Transp. 2014;2(4):205–20.
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vibrations. Int J Rail Transp. 2016;4(4):229–46.
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differential subgrade settlement. Struct Eng Mech. 2018;66(1):15–25.
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geo-inclusions—experimental evidence and DEM simulation. Int J Rail Transp. 2017;5
(2):63–86.
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Rail Rapid Transit. 2001;215(4):289–300.
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vehicle track coupled systems. Int J Struct Stab Dyn. 2018;18(7):1850091-1–29.
Chapter 9
Principle and Method of Optimal
Integrated Design for Dynamic
Performances of Vehicle and Track
Systems
It is well known that the operation of railway transportation networks relies on the
interactions between wheel and rail. Dynamic wheel–rail forces play a key role in
this function, since they are the main causes of vibrations, impacts, fatigue, and
damage in the vehicle and track systems. They also result in degradation and failure
of the wheel–rail system. Therefore, mitigating dynamic wheel–rail interactions is
crucial to ensuring long-term efficient operations of railway transportation.
In order to alleviate the dynamic interactions between vehicles and tracks, the
author has proposed the optimal integrated design concept for the dynamic per-
formance of vehicles and tracks [1]. The “optimal integrated design” refers to the
concept that the designs for the railway vehicle system and the track structure
system should be adapted and adjusted with full consideration of each other. In this
way, optimal results for the overall dynamic performance of the whole system can
be achieved.
In order to achieve optimal integration of the dynamic performances of vehicle
and track systems, the systematic design concept must be adopted. The vehicle
system and the track system should be regarded as an integrated system with their
interactive and coupled behaviors included. A comprehensive optimized design
should be aimed at the satisfaction of overall vehicle–track system dynamic
© Science Press and Springer Nature Singapore Pte Ltd. 2020 347
W. Zhai, Vehicle–Track Coupled Dynamics,
https://doi.org/10.1007/978-981-32-9283-3_9
348 9 Principle and Method of Optimal Integrated Design …
Fig. 9.1 Flowchart of the optimal integrated design principle for dynamic performances of
vehicle and track systems
It is necessary to note that, if the design subject is the vehicle system, its
dynamic impact indexes on the track system can be set as the dynamic wheel–rail
force, the dynamic deformation of the track structure due to dynamic wheel loads,
etc. In contrast, if the design subject is the track system, its dynamic influence
indexes on the vehicle system can be specified as the vehicle’s running performance
indexes, such as the ride comfort, running safety, hunting stability and so on.
Based on the integrated design principle described in Sect. 9.1, the dynamic design
methods for the optimal integrated designs of vehicle and track systems can be
realized using the simulation platform of vehicle–track coupled dynamics.
As shown in Fig. 9.2, the initial design scheme of the railway vehicle system (the
parameters of the vehicle dynamics model), together with its operational conditions
from the track (the parameters of the track structure and those of track horizontal
alignment and vertical profile), should be the input of the simulation system for
vehicle–track coupled dynamics analysis. The dynamic responses of the vehicle can
be analyzed and predicted by using the simulations, including vehicle hunting
stability, ride comfort and curve negotiation performance. Meanwhile, the dynamic
loading indexes of the vehicle on the track (dynamic wheel–rail force, dynamic
track deformation, and dynamic track stress) can be obtained. According to the
criteria for vehicle dynamic performance and dynamic wheel–rail interaction, the
dynamic response indexes can be evaluated comprehensively and the feasibility of
the vehicle design can be determined correspondingly. If the design cannot be
accepted, the inferior indexes and their related sensitive parameters of the vehicle
(such as suspension stiffness and damping, unsprung mass, etc.) can be identified
and revised. In this way, optimized parameters can eventually be determined and
used in the simulation design platform for dynamic performance analysis and
evaluation. This design process can be implemented repeatedly until an optimized
design is achieved. The final vehicle design should satisfy the vehicle’s dynamic
performance requirements and the track loading requirements simultaneously.
350 9 Principle and Method of Optimal Integrated Design …
Fig. 9.2 Dynamic design method for railway vehicle system based on the optimal integrated
design principle
The dynamic design procedure of the track system based on the principle of the
optimal integrated design concept is shown in Fig. 9.3. The initial design of the
track (horizontal alignment and vertical profile parameters and the track structural
parameters), together with the conditions of the vehicles to be operated (vehicle
dynamics parameters and operating speed) should be regarded as the inputs of the
simulation system for vehicle–track coupled dynamics analysis. The dynamic
responses of the track structure, including track vibration, loading and deformation
characteristics, can be predicted and analyzed. Meanwhile, the vehicle’s running
performance indexes on the designed track (such as ride comfort index and dynamic
wheel–rail safety index) can be obtained. According to relevant evaluation criteria
(or standards) for the performances of the vehicle and track, the dynamic response
indexes can be assessed in a comprehensive way and the feasibility of the track
design can be investigated. If the design is unacceptable, the unsatisfactory per-
formance indexes and relevant sensitive design parameters (such as minimum curve
radius, cant, transition curve length, track stiffness parameters, etc.) should be
identified and revised. The structural parameters optimized can be utilized in the
simulation platform for dynamic performance analysis and evaluation. The design
process should be carried out repeatedly until the optimized design is satisfactorily
obtained. An optimized design means both track dynamic performance and running
behavior of the vehicles on the designed track can be guaranteed.
9.3 Case Study I: Optimal Design of Suspension Parameters … 351
Fig. 9.3 Dynamic design method for track system based on the optimal integrated design
principle
The simulation conditions are the same as that of the field test: the curve radius is
300 m; the rail height difference is 120 mm; the track structure is of the existing
common ballasted track with the rail mass of 60 kg/m, and the sleeper is made from
concrete; the rail irregularities of the Chinese speedup main railway line are applied
as the excitations; the locomotive is running at the speed of 70 km/h.
The time histories of the lateral wheel–rail force and the derailment coefficient
when the locomotive is negotiating the curve are presented in Fig. 9.4. The lateral
wheel–rail dynamic force increases gradually when the locomotive is moving into
the transition curve. When the locomotive runs at the positions connecting the
transition curve and the circular curve, the dynamics indexes representing the
locomotive curve negotiation performance, reach their maximum values.
Specifically, the mean value of the lateral wheel–rail force is up to 90 kN with a
maximum wheel–rail force of 123 kN, and the corresponding maximum value of
the derailment coefficient has reached 0.91 which has exceeded the threshold value
of 0.9 given in the Chinese standard TB/T 2360-93 [2].
In practice, it has been also found that, in the line service test of the HXD2C
prototype locomotive, the maximum outside (on the high rail) lateral wheel–rail
force of the guiding wheelset exceeded 110 kN. The derailment coefficient has
instances where the value is beyond the threshold, although most values of this index
are below 0.9. Results from both theoretical calculation and field test indicate that
intensified lateral wheel–rail dynamic interactions are observed when the HXD2C
prototype locomotive is negotiating the small radius curves, which brings great
challenges to the safe operation of the locomotive and this issue must be alleviated.
(a) (b)
150 1.2
Lateral wheel-rail force (kN)
120
Derailment coefficient
0.9
90
0.6
60
0.3
30
0 0.0
-30 -0.3
100 200 300 400 500 600 100 200 300 400 500 600
Running distance (m) Running distance (m)
Fig. 9.4 Calculated wheel–rail dynamic indexes for the HXD2C locomotive running safety:
a lateral wheel–rail force, and b derailment coefficient
9.3 Case Study I: Optimal Design of Suspension Parameters … 353
There exist many potential ways to improve the locomotive curve negotiation
performance. However, based on the practical locomotive conditions and theoret-
ical analysis, our research group had found that the previously designed axle-box
positioning stiffness of the HXD2C locomotive is high, and reducing the longitu-
dinal and the lateral axle-box positioning stiffness has an evident effect on the
alleviation of the lateral wheel–rail dynamic interaction. Consequently, improve-
ment of the axle-box positioning stiffness is regarded as the key link of the fol-
lowing optimization process.
The designed and the measured axle-box positioning stiffness of the HXD2C
prototype locomotive are shown in Table 9.1. It can be seen that the designed
longitudinal and lateral axle-box positioning stiffness values are 199 kN/mm and
6.89 kN/mm, respectively. However, the measured practical values for these are
236 kN/mm and 10.01 kN/mm, respectively. This indicates that the practical
axle-box positioning stiffness values are higher than the designed values, namely,
with the ratio of 1.19 and 1.45 for longitudinal and lateral positioning stiffness,
respectively.
Variations of the simulated lateral wheel–rail force versus the locomotive pri-
mary suspension stiffness when the locomotive is negotiating the curve are dis-
played in Fig. 9.5. The results indicate that reduction of longitudinal and lateral
stiffness of the primary suspension can effectively reduce the lateral wheel–rail
dynamic interaction in small radius curves. Further, the variation of the lateral
wheel–rail force is more sensitive to the change of lateral stiffness. Consequently,
optimizing the primary suspension lateral stiffness will be the major measure to
reduce the lateral wheel–rail forces.
It can be seen from Fig. 9.5a that a smaller primary suspension lateral stiffness
will benefit the reduction of lateral wheel–rail force. However, smaller stiffness
could also deteriorate locomotive lateral stability. Thus, the principle for selecting
primary suspension lateral stiffness is that a smaller stiffness value is preferred in
the premise of guaranteeing good motion stability. Besides, it is worthy of notice in
Fig. 9.5a that the lateral wheel–rail force varies gently with the lateral stiffness in
the region of soft stiffness below 2.65 kN/mm, while a prompt increase in the lateral
force can be observed when the lateral stiffness is greater than 2.65 kN/mm. Hence,
the reasonable primary suspension lateral stiffness of the HXD2C locomotive should
be 2.65 kN/mm in theory.
Table 9.1 Designed and measured values of HXD2C locomotive axle-box positioning stiffness
Positioning stiffness Vertical (kN/mm) Lateral (kN/mm) Longitudinal (kN/mm)
Designed value 1.71 6.89 199
Measured value 1.41 10.01 236
354 9 Principle and Method of Optimal Integrated Design …
(a) (b)
150 150
125 125
100 100
75 75
50 50
0 2 4 6 8 10 0 50 100 150 200
Lateral stiffness of the primary suspension (kN/mm) Longitudinal stiffness of the primary suspension (kN/mm)
Fig. 9.5 Lateral wheel–rail force variation versus a lateral, and b longitudinal stiffness of primary
suspension
For the determination of primary longitudinal stiffness, the results in Fig. 9.5b
show that the lateral wheel–rail force decreases with the reduction of lateral stiff-
ness. However, the sensitivity of the lateral force to lateral stiffness is low.
Consequently, the primary suspension longitudinal stiffness should not be too small
due to the consideration of its high traction force delivery function and so as to also
guarantee the locomotive motion stability. In addition, the structural shape of the
rubber joint will affect the longitudinal and lateral stiffness of the primary sus-
pension simultaneously as it is an independent entirety in the structural design of
the primary suspension. In other words, there exists a certain relation of interde-
pendence between the lateral and longitudinal stiffness of the primary suspension.
Based on the aforementioned analysis, the theoretical longitudinal stiffness of the
locomotive primary suspension is selected as 40 kN/mm after the determination of
lateral stiffness value. Further, due to the complexity of the rubber joint manu-
facturing technique and limitations of rubber spring manufacturing conditions, the
final primary suspension stiffness values for practical usage are determined as
longitudinal stiffness is 52 kN/mm while the lateral stiffness is 2.6 kN/mm.
In order to verify the reasonability of the stiffness parameter optimization,
comparisons of the locomotive dynamic performance are made using the primary
suspension stiffness parameters before and after optimization. The time histories of
the lateral wheel–rail forces and derailment coefficient before and after parameter
optimization are illustrated in Figs. 9.6 and 9.7. In these simulations, the locomo-
tive is also negotiating the small curve with a radius of 300 m at the speed of
70 km/h. A conclusion can be drawn from the results that the lateral wheel–rail
force and the derailment coefficient have significantly decreased after parameter
optimization. The greatest decrement ratios are 20% and 18.7% for the lateral
wheel–rail force and the derailment coefficient, respectively.
The smaller primary suspension stiffness after parameter optimization may affect
locomotive motion stability. Therefore, the nonlinear critical speed of the HXD2C
locomotive running on elastic track has been further verified. The lateral dis-
placement responses of the first wheelset are shown in Fig. 9.8 where the loco-
motive is running on a straight line at the speed of 237 km/h and 238 km/h,
9.3 Case Study I: Optimal Design of Suspension Parameters … 355
150
90
60
30
0
-30
100 200 300 400 500 600
Running distance (m)
Fig. 9.6 Comparison of HXD2C locomotive lateral wheel–rail force during curve negotiation
before and after parameter optimization
After optimization
0.9
0.6
0.3
0.0
-0.3
100 200 300 400 500 600
Running distance (m)
Fig. 9.7 Comparison of HXD2C locomotive derailment coefficient during curve negotiation
before and after parameter optimization
(a) (b)
Lateral wheelset displacement (mm)
20
Lateral wheelset displacement (mm)
20
v =238km/h
v =237km/h
10 10
0 0
-10 -10
-20 -20
0 100 200 300 400 500 0 100 200 300 400 500
Running distance (m) Running distance (m)
Fig. 9.8 Determination of nonlinear critical speed of HXD2C locomotive after optimization
design: a stable solution, and b periodic solution
respectively. The results indicate that the nonlinear critical speed of the locomotive
after parameter optimization is 238 km/h which is much higher than its designed
speed (120 km/h). The optimized parameters can meet the requirement of running
stability.
356 9 Principle and Method of Optimal Integrated Design …
120
Before improvement
Lateral wheel-rail force (kN)
80
60
40
20
0
0 5 10 15 20 25 30 35 40
Curve curvature (×10-4/m)
Fig. 9.9 Variation of lateral wheel–rail force versus curve curvature before and after the
improvement of the primary suspension parameters
9.3 Case Study I: Optimal Design of Suspension Parameters … 357
3.5
Before improvement
3.0 After improvement
Fig. 9.10 Vertical stability of HXD2C locomotive before and after improvement of locomotive
design
smaller radius curve. Thus, the curve negotiation performance, especially for small
radius curves, has been improved greatly after the improved design of the HXD2C
locomotive.
At the same time, the locomotive running stability was again examined in this
test so as to check whether the change of primary suspension stiffness will bring any
adverse effect to the locomotive operation quality. Variations of the vertical and
lateral stability indexes of the locomotive on a straight line versus the running speed
are presented in Figs. 9.10 and 9.11, respectively. It can be seen that the locomotive
running stability is changed slightly after improvement. The running stability of the
locomotive is in a good state for both unimproved and improved designs. The
maximum vertical stability index shown in Fig. 9.10 is about 3.1 which is of the
3.0
Before improvement
After improvement
2.5
Lateral stability indicator
2.0
1.5
1.0
0.5
0.0
0 20 40 60 80 100 120 140 160
Running speed (km/h)
Fig. 9.11 Lateral stability of HXD2C locomotive before and after improvement of locomotive
design
358 9 Principle and Method of Optimal Integrated Design …
“fine” level according to testing standards. The lateral stability index versus running
speed is shown in Fig. 9.11 and it reaches its maximum value of 2.75 when the
locomotive runs at the speed of 120 km/h. For this case, the lateral stability index is
also in the range of the “fine” level, while, in the other running speed range, the
lateral stability index is in the “excellent” level. Thus, a conclusion can be drawn
that the change in the primary suspension stiffness of the improved design will not
cause apparent variations to the HXD2C locomotive running stability.
The aforementioned results of the full-scale locomotive running test indicate that
the curve negotiation performance of the HXD2C heavy-haul locomotive has been
improved effectively through the design optimization process. Specifically, the
lateral wheel–rail force decreases significantly, and the running stability of the
locomotive has also been guaranteed. This improvement can benefit the operation
of this locomotive in small radius curves.
2. Practical application of the HXD2C heavy-haul locomotive
The HXD2C locomotive after the improved design has been put into mass pro-
duction (see Fig. 9.12). Currently, there have been 250 locomotives produced and
deployed to Xinxiang locomotive depot. This type of locomotive has been the
prime freight locomotive for the railway lines, such as the Houyue line and the
Xinhe line, and the locomotives are operating with good performance. The
development of this locomotive has met the requirements during capacity expan-
sion and speedup development of conventional Chinese railway lines.
9.4 Case Study II: Design of a Steep Gradient Section of a High-Speed Railway 359
Haiou Island
Shazai Island
(a)
Shiziyang area
8
20
20
8
3 20
32 34 36 38 40 42 44 46
(b)
Shiziyang area
34
20
12
3
32 34 36 38 40 42 44 46
Fig. 9.14 Vertical profiles of two design schemes in the section of Shazai Island: a long tunnel
scheme, and b bridge–tunnel scheme
design speed was set at 300 km/h. However, could a high-speed train safely and
smoothly pass through such steep gradients at a speed of 300 km/h? This was the
first challenging problem relating to the compatible design of train and track sub-
systems in construction engineering of this Chinese high-speed line, and was a
major technical problem that involved the economy of line-selection technology
and the running safety and ride comfort of high-speed trains which had to be solved
during the design stage.
Designated by the design institution of the Guangzhou–Shenzhen–Hong Kong
high-speed railway, we used the dynamic design method for track system based on
the optimal integrated design principle presented in Sect. 9.2.2 to solve the
abovementioned compatible design problem. Complete dynamic simulations were
carried out for the running safety and ride comfort analysis of high-speed trains
passing at 300 km/h through the horizontal alignment and vertical profile sections
of the proposed four line-selection schemes in the Shiziyang area. Through safety
evaluation and scheme comparison according to the evaluation code of vehicle
dynamic performance, the best scheme suggestion was finally put forward [4].
9.4 Case Study II: Design of a Steep Gradient Section of a High-Speed Railway 361
(a)
11
Shiziyang area
12
20
30
3
32 34 36 38 40 42 44
(b)
Shiziyang area
30
12
30
32 34 36 38 40 42 44
Fig. 9.15 Vertical profiles of two design schemes in the section of Haiou Island: a long tunnel
scheme, and b bridge–tunnel scheme
In accordance with the design speed of 300 km/h for the Guangzhou–Shenzhen–
Hong Kong high-speed railway and the practice of high-speed trains internationally,
the Germany ICE350 high-speed EMU1 was adopted in the simulation analysis, and
the low interference track spectrum of the German high-speed railway was
employed as the excitation input.
When the high-speed train passes Shazai Island through the long tunnel scheme
and bridge–tunnel scheme, the indexes of running safety and ride comfort show
different characteristics. Figures 9.16 and 9.17 present the variation of derailment
coefficient and lateral acceleration of the car body, respectively, for both schemes.
1
ICE350 high-speed EMU is the improved vehicle of ICE3. The top speed is 350 km/h. The train
consists of 8 passenger cars and 431 passenger places in total.
362 9 Principle and Method of Optimal Integrated Design …
(b)
Derailment coefficient
1.0
0.5
0.0
-0.5
-1.0
32 34 36 38 40 42 44 46
Location DK (km)
Fig. 9.16 Variation of derailment coefficient of high-speed train for a bridge–tunnel and b long
tunnel design schemes of the Shazai Island
(a)
Lateral acceleration (g)
0.2
0.1
0.0
-0.1
DK 36+390 DK38+539
-0.2
32 34 36 38 40 42 44 46
Location DK (km)
(b)
Lateral acceleration (g)
0.2
0.1
0.0
-0.1
-0.2
32 34 36 38 40 42 44 46
Location DK (km)
Fig. 9.17 Lateral car body acceleration of high-speed train for a bridge–tunnel and b long tunnel
design schemes of the Shazai Island
As can be seen in Fig. 9.16, for the bridge–tunnel scheme, with the high-speed train
running at location DK36+390 (grade change point from a vertical curve to a 34‰
downhill slope), the derailment coefficient increases abruptly from the normal state
under the combined influence of the vertical profile with a slope of 34‰ and
9.4 Case Study II: Design of a Steep Gradient Section of a High-Speed Railway 363
7000 m radius horizontal transition curve. This process continues to the location of
DK38+539 (grade change point between a 12‰ downhill slope and a 3‰ uphill
slope). Clearly, the maximum value of the derailment coefficient reaches 0.98,
which is larger than the dynamic safety limit of the high-speed train (0.8), while the
derailment coefficient for the long tunnel scheme remains smaller than the safety
limit. As can be seen from Fig. 9.17, the lateral car body acceleration in the bridge–
tunnel scheme has abnormal fluctuations at the location of DK36+390–DK38+539.
The peak acceleration is 0.15 g, exceeding the ride comfort limit of the high-speed
train (0.1 g). This performance index of the bridge–tunnel scheme is larger than that
of the long tunnel scheme by about 66.7%. The lateral running stability of the
bridge–tunnel scheme also increases to 3.0 which just reaches the edge of the
“qualified” level, while this value is found to be 2.46 for the long tunnel scheme
which belongs to the “excellent” level. Additionally, the wheel unloading rates for
the long tunnel scheme and bridge–tunnel scheme are 0.47 and 0.64, respectively.
Obviously, the former one is smaller than the qualified limit of 0.65, and the latter
one is close to the qualified limit [4].
As can be seen, the requirements for indexes of running safety and ride comfort
of the high-speed train can be satisfied for the long tunnel scheme of Shazai Island.
However, the running safety and ride comfort of the high-speed train cannot be
ensured for the bridge–tunnel scheme.
When the high-speed train passes Haiou Island through the long tunnel scheme and
bridge–tunnel scheme, the dynamic behavior of the high-speed train shows similar
characteristics with that for Shazai Island. The calculated maximum value of
dynamic responses, such as indexes of running safety and ride comfort, are sum-
marized in Table 9.2.
By comparing the various safety indexes, the maximum derailment coefficient is
0.53 for the long tunnel scheme, which is smaller than the safety limit of 0.8. The
maximum derailment coefficient for the bridge–tunnel scheme is 0.92, which
exceeds the safety limit. The maximum wheel unloading rate for the long tunnel
scheme is 0.41, which is smaller than the limit value of 0.65, and this value is found
to be 0.98 for the bridge–tunnel scheme which is, therefore, not a qualified scheme.
The maximum overturning coefficient is calculated as 0.61 for the long tunnel
scheme, which is lower than the limit value of 0.8, whereas this maximum value is
1.0 for the bridge–tunnel scheme, which is greater than the safety limit.
For the comparison of stability indexes, it can be seen that the lateral stability
index of the car body for the long tunnel scheme is 2.44 which is indicating an
excellent level, while the same index is found to be 2.94 (just a qualified level) for
364 9 Principle and Method of Optimal Integrated Design …
Table 9.2 Peak dynamic responses of two design schemes for Haiou Island
Dynamic performance indexes Long tunnel scheme Bridge–tunnel scheme
Lateral wheel–rail force (kN) 27.89 30.97
Derailment coefficient 0.53 0.92
Wheel unloading rate 0.41 0.98
Overturning coefficient 0.61 1.00
Lateral car body acceleration (g) 0.08 0.16
Vertical car body acceleration (g) 0.05 0.07
Lateral stability index 2.44 2.94
Vertical stability index 2.25 2.33
the bridge–tunnel scheme. The vertical stability indexes are all excellent for the
long tunnel scheme and the bridge–tunnel scheme.
As can be seen from above analyses, when the high-speed train passes through
Haiou Island at a speed of 300 km/h, all safety indexes and comfort indexes for the
long tunnel scheme can meet the requirements of high-speed operation and have
sufficient safety margins. However, for the bridge–tunnel scheme, under the com-
bined effect of the horizontal curve with a 7000 m radius and the vertical profile
with a slope of 30‰, the running safety index of the train exceeds the allowable
limit for part of the operation, and the ride comfort index is greatly reduced and
does not meet the requirements.
It can be seen from the above analysis that the running safety and ride stability
indexes of the bridge–tunnel scheme for Shazai Island and Haiou Island are all
worse than those of the long tunnel scheme for these two islands. The bridge–tunnel
scheme cannot meet the requirements for running safety and ride comfort, while the
long tunnel scheme for the two islands can satisfy these requirements. Therefore,
the long tunnel scheme is recommended as the solution.
To determine the best design scheme, the dynamic performance indexes for the
Shazai Island scheme and the Haiou Island scheme that can meet the requirements
of high-speed operation are listed in Table 9.3.
As can be seen from Table 9.3, when the high-speed train passes through the
Shazai Island when adopting the long tunnel scheme, the key safety indexes
(derailment coefficient and lateral wheel–rail force) are all smaller than those of
Haiou Island when adopting the long tunnel scheme. The wheel unloading rates and
overturning coefficients of the two schemes are quite close to each other, and there
is no obvious difference in the ride comfort index. Therefore, the best design
9.4 Case Study II: Design of a Steep Gradient Section of a High-Speed Railway 365
Table 9.3 Comparison of high-speed running performance indexes of the long tunnel scheme for
Shazai Island and Haiou Island
Dynamic performance indexes Shazai Island Haiou Island
Lateral wheel–rail force/kN 18.85 27.89
Vertical wheel–rail force/kN 97.67 105.21
Derailment coefficient 0.37 0.53
Wheel unloading rate 0.47 0.41
Overturning coefficient 0.69 0.61
Lateral car body acceleration (g) 0.09 0.08
Vertical car body acceleration (g) 0.06 0.05
Lateral stability index 2.46 2.44
Vertical stability index 2.24 2.25
scheme we finally recommend was the long tunnel scheme crossing Shazai Island,
which provided scientific guidance for the design department’s decision-making.
The above-recommended scheme was adopted for the practical project of the
Shiziyang section of Guangzhou–Shenzhen–Hong Kong high-speed railway
(Fig. 9.18), which solved the complicated problem for line selection with large
slopes that was first encountered in Chinese high-speed railway design at that time.
On December 18, 2005, the Guangzhou–Shenzhen section of the high-speed
railway line officially started construction, and the construction of Shiziyang tunnel
began on November 9, 2007, and was completed on March 12, 2011. The total
length of this tunnel is 10.8 km and it carries the fastest trains of any underwater
railway tunnel in the world.
On December 26, 2011, the Guangzhou–Shenzhen–Hong Kong high-speed
railway was officially opened. From the experiment results before opening and the
practical operation experience, the high-speed trains have a good running perfor-
mance which fully satisfies the requirements of high-speed running safety and ride
Guangzhou Shenzhen
Tunnel Tunnel
References
1. Zhai WM. Principle, method and engineering practice of optimal integrated designs of railway
vehicle and track. China Railw Sci. 2006;27(2):60–5 (in Chinese).
2. TB/T2360-93. Evaluation method and standard of dynamic performance test of railway
locomotive. Beijing: China Railway Publishing House; 1993 (in Chinese).
3. Zhai WM, Wang KY, Yang YL, et al. Applications of the theory of vehicle–track coupling
dynamics to the design of modern locomotives and rolling stocks (in Chinese). J China Railw
Soc. 2004;26(4):24–30.
4. Zhai WM, Wang KY. Evaluation of running safety and ride comfort of trains passing through
the Pearl river section of Guangzhou-Shenzhen-Hong Kong high-speed railway line. Report
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Chapter 10
Practical Applications of the Theory
of Vehicle–Track Coupled Dynamics
in Engineering
© Science Press and Springer Nature Singapore Pte Ltd. 2020 367
W. Zhai, Vehicle–Track Coupled Dynamics,
https://doi.org/10.1007/978-981-32-9283-3_10
368 10 Practical Applications of the Theory of Vehicle–Track Coupled …
At the end of 2002, the prototype of the locomotive went through an operational
assessment of 200,000 km. However, it was found in the trial operation that the
locomotive lateral swaying vibration was much intensified in the speed range of
100–150 km/h. In this speed range, the lateral acceleration index exceeded the limit
value specified in the Chinese national standard. However, the lateral vibration of
the locomotive running at other speeds was relatively moderate, and the vertical
running stability was kept in a good state for the complete running speed range.
Thus, the abnormal lateral vibration problem of the locomotive had to be solved
before commencing mass production.
Under this situation, commissioned by the manufacturer (Datong Electric
Locomotive Company), the author’s team applied the theory of vehicle–track
coupled dynamics to perform in-depth analysis and studies on this practical engi-
neering problem. The locomotive suspension parameters were optimized with the
consideration of lateral dynamic interactions between the locomotive and the track.
Using this method, an effective solution was proposed without changing the
locomotive structure.
The simulation software TTISIM was used to simulate and analyze the lateral
vibration performance of the SS7E locomotive prototype running on tangent track.
The range of locomotive running speed for the simulation was 70–180 km/h with
an interval of 10 km/h. The operational conditions of the Chinese mainline railway
were applied and the common ballast track structure was used. The rail size was
60 kg/m, and the rail was discretely supported by concrete sleepers.
Variations of the maximum lateral displacements of the car body and the bogie
versus locomotive running speed are presented in Figs. 10.1 and 10.2, respectively.
It can be seen that the lateral vibrations of the car body near the speed of 100 km/h
are quite drastic, and similar for those of the bogie near the speed of 120 km/h.
Lateral car body displacement (mm)
12
11
10
8
50 100 150 200
Locomotive running speed (km/h)
Fig. 10.2 Variation of lateral bogie displacement versus locomotive running speed
12 (a) 12 (b)
6 6
(mm)
(mm)
0 0
-6 -6
-12 -12
0 200 400 600 800 1000 0 200 400 600 800 1000
Locomotive running speed (km/h) Locomotive running speed (km/h)
Fig. 10.3 Time histories of lateral car body displacement for the locomotive running at the speed
of: a v = 80 km/h, and b v = 180 km/h
370 10 Practical Applications of the Theory of Vehicle–Track Coupled …
12 (a) 12 (b)
6 6
(mm)
(mm)
0 0
-6 -6
-12 -12
0 200 400 600 800 1000 0 200 400 600 800 1000
Locomotive running speed (km/h) Locomotive running speed (km/h)
Fig. 10.4 Time histories of lateral bogie displacement for the locomotive running at the speed of:
a v = 120 km/h, and b v = 180 km/h
15
Lateral car body displacement (mm)
Csdx1200kN·s/m Csdx1300kN·s/m
Csdx1400kN·s/m Csdx1600kN·s/m
12 Csdx1800kN·s/m Csdx2000kN·s/m
damping increases from 1200 to 2000 kN s/m. In particular, it provides the best
performance to resist the lateral car body vibration in the low-speed range when the
damping coefficient reaches 2000 kN s/m.
Figure 10.6 displays the effect of the damping coefficient of the yaw absorber on
the lateral vibration of the bogie frame. When the locomotive runs at a speed lower
than 140 km/h, the lateral displacement of the bogie frame could be effectively
reduced by increasing the yaw absorber damping, especially at the speed of
120 km/h when the damping increases to be 1400 kN s/m or greater. Additionally,
the maximum value of the lateral bogie frame displacement will generally reduce
with the increase of the damping when the locomotive speed lies in the range of
140–180 km/h, because the lateral bogie frame displacement actually increases in
some cases (1400 kN s/m, for example) in this speed range. However, the dis-
placement value will increase with the increase of the damping for minor condi-
tions. In general, the lateral bogie frame vibrations of the locomotive could be
effectively suppressed when the damping is 1800 kN s/m or greater.
Furthermore, the effects of the lateral stiffness of the secondary suspension on
the lateral displacements of the car body and bogie frame are presented in
Figs. 10.7 and 10.8, respectively. It can be seen that the decrease of the secondary
suspension lateral stiffness can not only reduce the lateral car body displacement in
the entire speed range (see Fig. 10.7), but also can effectively resist the lateral bogie
frame vibration in the speed range of 100–140 km/h (see Fig. 10.8).
Based on the aforementioned results, two improved designs are proposed.
Comparisons of the lateral car body and bogie frame vibrations are made between
the proposed designs and the original design, and the results are shown in Fig. 10.9.
It can be seen that the abnormal lateral vibration of the SS7E prototype locomotive
has been suppressed effectively through the application of the improved designs.
Comparisons between the two proposed designs reveal that both of them have
similar performance with regard to mitigating the lateral vibration of the bogie
frame, while the second design is better for resisting the lateral car body vibration in
the low-speed range.
Lateral bogie displacement (mm)
13 Csdx1200kN·s/m Csdx1300kN·s/m
Csdx1400kN·s/m Csdx1600kN·s/m
12
Csdx1800kN·s/m Csdx2000kN·s/m
11
10
8
50 100 150 200
Locomotive running speed (km/h)
Fig. 10.6 Effect of the yaw damping on bogie frame lateral vibration
372 10 Practical Applications of the Theory of Vehicle–Track Coupled …
15
Fig. 10.7 Effect of secondary suspension lateral stiffness on lateral car body displacement
Lateral bogie displacement (mm)
13 Ksy 0.657MN/m
Ksy 0.5MN/m
12
11
10
8
50 100 150 200
Locomotive running speed (km/h)
Fig. 10.8 Effect of secondary suspension lateral stiffness on lateral bogie frame displacement
Lateral car body displacement (mm)
15
(a) Original design 13 (b) Original design
1st improved design 1st improved design
2nd improved design 12 2nd improved design
12
11
9 10
9
6
8
50 100 150 200 50 100 150 200
Locomotive running speed (km/h) Locomotive running speed (km/h)
Fig. 10.9 Locomotive lateral vibration performances for different designs: a car body, and
b bogie frame
10.1 Redesign of Dynamic Performance of a Speedup Locomotive 373
Due to the limitation from time requirements of the fifth Chinese railway speedup
plan, the first improved design was finally applied into practice by the locomotive
manufacturer although the second design had a better performance than the first
one. In order to assess the practical performance of the locomotive after the initial
improvement, the line running tests on the riding comfort of the SS7E locomotive
were performed on the Longhai and Jingguang lines by Southwest Jiaotong
University together with Datong Electric Locomotive Company. The tested results
were then compared with that of the prototype locomotive before improvement as
tested by China Academy of Railway Sciences.
1. Comparison of locomotive lateral vibrations before and after improvements
The statistical test results for the lateral car body vibration acceleration and the
lateral stability index before and after improvement are presented in Figs. 10.10 and
10.11, respectively. In these figures, the locomotive running speed ranges from 0 to
160 km/h.
It can be seen from Fig. 10.10 that the lateral car body vibration accelerations of
the locomotive before improvement have seriously exceeded the limit value of
0.25 g in the speed range of 80–160 km/h, and most of them are even up to 0.25–
0.35 g with the maximum value of 0.37 g. However, for the locomotive after
improvement, the lateral car body vibrations are reduced greatly with most of the
values below 0.2 g which meets the eligibility requirements. It should be noted that
0.4
Lateral car body acceleration (g)
Before improvement
After improvement
0.3
0.2
0.1
0.0
0 20 40 60 80 100 120 140 160
Running speed (km/h)
Fig. 10.10 Variation of lateral car body vibration acceleration versus locomotive running speed
374 10 Practical Applications of the Theory of Vehicle–Track Coupled …
4
Before improvement
Lateral car body stability index
After improvement
3
0
0 20 40 60 80 100 120 140 160
Running speed (km/h)
Fig. 10.11 Variation of lateral car body stability index versus locomotive running speed
the lateral car body vibrations after improvement have been suppressed for the
speed range of 80–160 km/h, while the amplitudes of the lateral car body vibrations
are increased for the speed range of 30–70 km/h. However, it has to be pointed out
that the increases were very small, and the vibrations were still below 0.15 g which
belongs to an “excellent” level. Consequently, the locomotive after the improve-
ment has a good lateral operational quality in the entire designed speed range.
Figure 10.11 shows that the car body stability index of the improved locomotive
was reduced when compared with the results before improvement for the entire
speed range. This is especially true for the higher speed range (100–160 km/h) with
a maximum value of 3.0, which is ranked in the “good” level. While, the maximum
car body stability index before improvement is up to 3.6, which belongs to the
disqualification grade.
2. Practical application status of the SS7E speedup locomotive
The improved SS7E locomotive, as shown in Fig. 10.12, had been put into mass
production ahead of schedule in 2003 to meet the time requirement from the fifth
Chinese railway speedup plan that began on April 18, 2004. Up to now, about 140
SS7E locomotives have been manufactured and put into service. They have become
the main passenger locomotives for the speedup of existing railway lines such as the
Longhai and Jingguang railways. It has demonstrated that these locomotives have
run on the two lines safely for a long period, which brings considerable social and
economic benefits.
10.2 Reducing Rail Side Wear on Heavy-Haul Railway Curves 375
The wear of wheels and rails, especially the wheel flange wear and rail side wear on
curves, is a long-standing problem of heavy-haul railways. With the increasing train
axle load and transport capacity, the wheel–rail interaction is inevitably aggravated.
376 10 Practical Applications of the Theory of Vehicle–Track Coupled …
Severe wheel–rail dynamic interaction will induce severe wear of wheels and rails,
especially in curved track sections.
This section takes the Shuohuang railway (one of the Chinese major heavy-haul
railway lines) as an example. The rail wear state and characteristics of the railway
are discussed in this section. The Shuohuang heavy-haul railway is a dedicated line
for coal transportation from west to east. At present, the annual traffic volume of
this railway has exceeded 300 million gross tonnes (MGt). Severe rail wear occurs
on curves of this railway. Onsite wear measurements were carried out for rails on
curves with three different radii, namely 500, 600, and 1000 m [1]. The measure-
ment locations of rail wear are illustrated in Fig. 10.13a while Fig. 10.13b shows
the statistics of the measurements of rail side wear and vertical wear.
It is shown from Fig. 10.13b that the predominant wear is the side wear of outer
rails on small radius curves. The outer rail side wear increases rapidly with the
decrease of curve radius, and reaches a maximum of 22.14 mm on the 500 m radius
curve. As the curve radius increases to 1000 m, the rail side wear drops down
greatly; the maximum value is 7.05 mm. The values of the rail wear were measured
at a traffic volume of about 355 MGt. Compared with the outer rail, the inner rail
has much less side wear (only 1–2 mm). For the vertical wear, there are small
differences between the inner rail and outer rail. The vertical wear is much smaller
than the side wear on the outer rail of sharp curves.
On sharp curves, a common defect that occurs on inner rails is the spalling
defect. This defect is mainly found distributed along the center region of the rail
crown, as shown in Fig. 10.14a. On the outer rail, severe side wear is found and the
profile nearly conforms to the wheel flange shape (see Fig. 10.14b, c). The wheel–
rail contact under this condition usually leads to an increase in load on the gauge
side of the outer rail. With a continuous expansion in railway traffic volume, the
wear rate of wheels and rails presents an increasing tendency. Therefore, it is of
significance to seek technical measures to alleviate wear at the wheel–rail contact
interface in heavy-haul railways.
Numerous studies have proved that rail profile grinding is an effective measure
to slow down the wear and to prolong rail service life. In the 1970s, a rail grinding
(a) (b)
W1
16mm
W2
37.5mm
W1 vertical wear
W2 side wear
Fig. 10.13 Rail wear on Shuohuang heavy-haul railway curves: a measurements, and b statistics
(Reprinted from Ref. [1], Copyright 2014, with permission from Taylor & Francis.)
10.2 Reducing Rail Side Wear on Heavy-Haul Railway Curves 377
(a) (b)
(c) 10
0
Gauge side
Height (mm)
-10
Original profile
-20 Inner rail profile
Outer rail profile
-30
-40
-50
-40 -30 -20 -10 0 10 20 30 40 50 60
Width (mm)
Fig. 10.14 Rails on sharp curve of Shuohuang heavy-haul railway: a rail defect of inner rail,
b rail defect of outer rail, and c measured rail profiles (Reprinted from Ref. [1], Copyright 2014,
with permission from Taylor & Francis.)
Field surveys show that the wear of inner and outer rails presents an obvious
asymmetric feature which is caused by the asymmetries of wheel–rail contact points
and wheel–rail interactive forces on the inner and outer rails. In order to alleviate
the rail side wear, rail asymmetric-grinding for the outer and inner rails may be an
effective way to change the wheel–rail contact status and eventually improve the
dynamic performance of wheel–rail interaction. For this purpose, it becomes very
important to accurately understand the wheel–rail contact relationships and the
vehicle–track interaction characteristics on curved sections. On curved sections, the
vehicle–track dynamic interaction is more complicated than that on tangent tracks
due to the wheel–rail contact points on curves generally being distributed over a
wider area on both wheel and rail profiles. In order to make a detailed study on
vehicle–track interactive performance, it is necessary to synthetically consider the
vehicle system, the track system, and the wheel–rail contact interface. To this end,
the vehicle–track coupled dynamics theory and its simulation techniques, described
in Chaps. 2 and 4, provide an effective tool in the design process of rail asymmetric-
grinding profiles. Figure 10.15 gives an overview of the design process of rail
asymmetric-grinding profiles for curved tracks [1].
The implementing procedures illustrated in Fig. 10.15 are explained as follows.
First, the actual profiles of wheels and rails are measured in the field. Second, the
wheel–rail contact geometry is analyzed and assessed using the measured wheel
and rail profiles. Third, dynamic simulation of the whole vehicle–track system is
carried out using the theory of vehicle–track coupled dynamics. In the simulations,
the measured wheel and rail profiles, as well as the real operational conditions, are
used. Next, the dynamic performance of the wheel–rail interaction is evaluated
Fig. 10.15 Design process of rail asymmetric-grinding profiles based on analysis of wheel–rail
interaction (Reprinted from Ref. [1], Copyright 2014, with permission from Taylor & Francis.)
10.2 Reducing Rail Side Wear on Heavy-Haul Railway Curves 379
the wheel and Zone E is the contact region between the far-field side of both the rail
and the wheel.
On curves, the side wear of the outer rail mainly occurs in Zones A and B. There
are several principles that can be used to reduce the outer rail side wear through the
grinding of rail profiles:
• Grind Zone A to avoid contact between the wheel flange and the gauge corner.
• Modify Zone B to reduce the discontinuous distribution of the contact points and
to provide conformal contact.
• Modify Zone C to make the rail crown radius of curvature slightly smaller than
that of the wheel tread so that the wheel–rail contact stress can be reduced.
• Grind Zone D to avoid contact in this zone and move contact points to Zone
C. In this way, the rolling radius of the outer wheel is increased and the
self-steering ability can be improved on curves.
For the inner rail, the contact points are mainly centered along the top of the rail
[3]. In order to reduce the side wear of the outer rail and restrict the vertical wear on
the inner rail simultaneously, Zones B, C and D on the inner rail also need to be
modified:
• Modify Zones B and C to avoid contact in the gauge corner, and to make contact
points in Zone C move toward Zone D, resulting in the decrease of the rolling
radius of the inner wheel.
• Modify Zone D to ensure contact points to be distributed in Zone C or
D uniformly.
After the modification of rail profiles, the wheel–rail contact geometry should be
analyzed and evaluated. Through the analysis of static wheel–rail contact geometry
including both contact point distributions and contact geometry parameters of rail,
the appropriate asymmetric-grinding profiles can be obtained. Among them, the
RRD between the outer and inner wheels is the most important parameter affecting
the curve negotiation performance of railway vehicles.
By a compatible design of wheel–rail contact geometry, the rail-grinding profiles
with a good contact geometry state may be obtained. However, another important
issue must be paid attention to, namely the wheel–rail dynamic interaction per-
formance of a heavy-haul freight vehicle passing through a curved track with the
redesigned rail-grinding profiles. Here, the lateral wheel–rail force is the key factor
which is mainly responsible for rail side wear. On the curved track, the wheelset is
steered by the lateral forces acting on the wheels. The steering mainly depends on
the longitudinal creep forces (FxLi, FxRi) and the lateral creep forces (FyLi, FyRi), as
shown in Fig. 10.17 [7]. If the wheel and rail profiles are redesigned to be able to
provide enough RRD, the creep forces will push the wheelset to adjust itself to the
radial position without wheel flange contact. Conversely, small RRD results in large
angles of attack, and the wheel flange usually contacts with the rail gauge side.
Once the wheelset is steered by the wheel flange, severe rail side wear may happen.
10.2 Reducing Rail Side Wear on Heavy-Haul Railway Curves 381
Fig. 10.17 Basic wheel–rail forces acting on bogie in curve (Reprinted from Ref. [1], Copyright
2014, with permission from Taylor & Francis.)
In the actual track structure, rails are connected with sleepers by fasteners while
sleepers are supported by the ballast. All the components form an elastic-damping
vibration system. Subjected to a large lateral and vertical wheel–rail forces on
curved track, the lateral, vertical, and torsional vibration displacements of rails
cannot be ignored, especially the lateral and torsional displacements of the outer
rail. Undoubtedly, rail movements will affect wheel–rail contact geometry as well
as the dynamic forces. Therefore, the track vibrations should be taken into account
in solving the wheel–rail dynamic interaction on curves. Here, the wheel–rail
coupled model established under the framework of vehicle–track coupled dynamics
in Chap. 2 is used. The wheel–rail coupled model is capable of considering rail
vibrations in lateral, vertical and torsional directions.
Based on the techniques of the above methodology and analysis, the rail
asymmetric-grinding profiles can be redesigned for curves. As an example, the rail
asymmetric-grinding profiles were redesigned for a curve with 600 m radius on the
Chinese Shuohuang heavy-haul railway. On this curve, the superelevation of the
outer rail is 75 mm, and the length of the transition curve is 140 m. The rail cant is
382 10 Practical Applications of the Theory of Vehicle–Track Coupled …
1:20. The track structure includes 75 kg/m size rail, type III sleepers, type II elastic
fasteners and a ballast bed with the thickness of 0.3 m. The main vehicle running on
the line is a freight wagon of type C70 that is using the Chinese LM wheel profile.
All of the vehicle and track components comply with the Chinese railway industry
standards.
Based on the standard profile of 75 kg/m size rail, a number of asymmetric-
grinding profiles were redesigned. Among them, a pair of rail asymmetric profiles
with lower wheel–rail interaction and lesser grinding requirement was selected to be
the recommended profiles for rail grinding. Figure 10.18 gives the redesigned
asymmetric-grinding profiles for the inner and outer rails. Comparing with the inner
rail profile, the grinding range of the outer rail profile is larger and its grinding
zones are more widely spread over the rail gauge face, gauge corner, and rail crown.
The inner rail profile is mainly ground in the local region of the rail crown.
Using the profile data, the RRD functions of the wheelset before and after rail
grinding are calculated, as shown in Fig. 10.19. Here, the wheelset lateral dis-
placement ranges from −15 to 15 mm. The lateral wheelset displacement is defined
as negative if it moves laterally toward the outer rail and positive if toward the inner
rail. In the case of the LM wheel profile matching with the redesigned rail profiles,
there exists obvious asymmetric characteristic in the RRD curve (Fig. 10.19). This
is due to the asymmetries of the left and right rail profiles. When the lateral
(a) 3.00
0.10
0.23
0.17
0.25
0.17
Gauge side
(b) 3.0
0.11
0.11
0.22
0.16
0.22
0.34
0.14
0.26
0.32
0.48
0.58
0.57
0.46
0.37
0.22
0.43
0.25
0.12
0.07
0.91
20.53
1.35
1.45
Standard rail profile 1.26
0.82
2.0
Gauge side
Fig. 10.18 Redesigned rail-grinding profiles for a inner and b outer rails of the 600 m radius
curve of Shuohuang heavy-haul railway (Reprinted from Ref. [1], Copyright 2014, with permis-
sion from Taylor & Francis.)
10.2 Reducing Rail Side Wear on Heavy-Haul Railway Curves 383
displacement of the wheelset varies within −6 to −10 mm, the RRD of the rede-
signed rail profiles is larger than that of the standard rail profile. In this case, the
steering ability of the wheelset can be improved. The modification of rail profiles
will influence the contact angle difference of the wheelset simultaneously. The
analysis results in Fig. 10.20 indicate that, if the wheelset moves laterally from −6
to −10 mm, the contact angle difference value of the redesigned rail profile is larger
than that of the standard rail profile. For the redesigned rail profile with a larger
contact angle difference, the lateral component of the wheel–rail normal force can
provide appropriate steering ability, which prevents the wheelset moving exces-
sively towards the outer rail. Consequently, the probability of contact between the
wheel flange and the rail gauge side will be reduced.
The distributions of wheel–rail contact points reflect the regions where the wheel
and rail frequently contact with each other. In order to reduce the rail side wear, the
number of contact points on the rail gauge face should be limited to as few as
possible. Figure 10.21 compares the wheel–rail contact point distributions on the
Fig. 10.21 Wheel–rail contact point distributions of LM wheel profile with standard rail profile
and redesigned rail-grinding profiles (Reprinted from Ref. [1], Copyright 2014, with permission
from Taylor & Francis.)
surfaces of the standard rail profile and the designed rail-grinding profiles. In
comparison, the lateral displacement of the wheelset is from 0 to −15 mm. It can be
seen from Fig. 10.21 that the wheel–rail contact geometry property is greatly
improved after rail grinding. For the outer rail, the contact points spread over a
wider zone after the rail grinding. According to Fig. 10.16, the wheel and outer rail
rarely contact in Zone A and Zone D. Even if a large lateral displacement of the
wheelset occurs, the contact is still at the gauge corner, not the rail gauge face. The
contact points in Zone B distribute continuously if the rails are ground. For the inner
rail, the contact points distribute more uniformly in Zone C after the grinding. It is
proven theoretically that the rail profiles after asymmetric grinding could meet the
demand of the wheel–rail contact geometry required in the design criteria.
To evaluate the wheel–rail dynamic interaction after rail grinding, the vehicle–
track coupled dynamics simulation system described in Chap. 4 is used. The curve
negotiation performances of a freight wagon on a section of track with the standard
and ground rail profiles are analyzed, respectively. In the simulation, the measured
track irregularities are used as the input excitation of the system. Figure 10.22
compares the wheel–rail dynamics responses at a normal curve negotiation speed of
70 km/h before and after rail grinding. The results in Fig. 10.22a, b show that,
compared with the standard rail profile, the use of the redesigned rail profile reduces
the maximum lateral displacement of the wheelset by 14.6% and the maximum
wheelset angle of attack by 24.5%. Figure 10.22a indicates that the redesigned rail
profiles are helpful to prevent the wheelset moving laterally with large displace-
ment, so that the probability of wheel flange contact can be reduced. Figure 10.22b
reflects that the wheelset with the redesigned rail profiles tends to be at the radial
position of the curve. Figure 10.22c and d show that, after rail grinding, the
maximum values of the lateral wheel–rail force and the frictional power of the outer
rail are reduced by 26.7% and 27%, respectively.
10.2 Reducing Rail Side Wear on Heavy-Haul Railway Curves 385
-15 -0.002
-20 -0.004
0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700
Running distance (m) Running distance (m)
(c) (d)
60 6
Lateral wheel-rail force (kN)
20 3
2
0
1
-20 0
0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700
Running distance (m) Running distance (m)
Fig. 10.22 Comparison of dynamics indexes before and after rail grinding: a lateral wheelset
displacement, b angle of attack, c lateral wheel–rail force, and d frictional power (Reprinted from
Ref. [1], Copyright 2014, with permission from Taylor & Francis.)
It is concluded from the above theoretical analysis that, if the redesigned rail
asymmetric-grinding profiles are adopted, the wheel–rail contact geometry and the
wheel–rail dynamics performance will be effectively improved, and the rail side
wear on heavy-haul railway curves can be significantly reduced.
In order to validate the actual effect of the method proposed in this chapter, the
design example of the rail asymmetric-grinding profiles illustrated in Sect. 10.2.3
was put into practice on the Shuohuang heavy-haul railway.
1. Asymmetric grinding of rail profiles on curve
A test section was chosen at a curve with a radius of 600 m on the Shuohuang
railway. Using the redesigned rail profiles in Fig. 10.18, the inner and outer rails on
the test section were ground by the RR48-HP4 grinding train in August 2008.
386 10 Practical Applications of the Theory of Vehicle–Track Coupled …
(a) (b)
0.65
1.02
1.18
0.85
1.56
1.60
1.61
1.42
1.45
1.71
1.55
1.25
1.35
1.55
8.0 5.5
5.5
Fig. 10.23 Redesigned rail profiles for corrective grinding in the test curve: a inner rail, and
b outer rail (Reprinted from Ref. [1], Copyright 2014, with permission from Taylor & Francis.)
Fig. 10.24 Dynamic performance measurements: a track structure vibration, and b wheel–rail
forces (Reprinted from Ref. [1], Copyright 2014, with permission from Taylor & Francis.)
10.2 Reducing Rail Side Wear on Heavy-Haul Railway Curves 387
In the test, some key dynamic performance indexes such as the wheel–rail
vertical force, wheel–rail lateral force, track gauge dynamic widening and rail
displacements were obtained before and after rail grinding. As an example,
Fig. 10.25 compares the wheel–rail forces and the outer rail displacements mea-
sured before and after the corrective grinding. With the same conditions of the
vehicle and the track, the average lateral wheel–rail force decreased obviously after
grinding. The average lateral and vertical rail displacements also decreased by 26%
and 24%, respectively, after grinding. This clearly demonstrates that, when the rails
in the test section were ground according to the designed schemes, the wheel–rail
contact conditions were improved so that the lateral wheel–rail dynamic interaction
was alleviated.
(a)
80 200
Lateral force before rail grinding
60 150
50 125
40 100
30 40 50 60 70
Running speed (km/h)
(b)
5
Lateral displacement before rail grinding
Lateral displacement after rail grinding
4
Rail displacement (mm)
0
30 40 50 60 70
Running speed (km/h)
Fig. 10.25 Comparisons of wheel–rail dynamics indexes of outer rail before and after rail
grinding in test section: a lateral and vertical wheel–rail forces, and b lateral and vertical rail
displacements (Reprinted from ref. [1], Copyright 2014, with permission from Taylor & Francis.)
388 10 Practical Applications of the Theory of Vehicle–Track Coupled …
Circular curve
Fig. 10.26 The measuring point arrangement of rail profiles in track test section (Reprinted from
Ref. [1], Copyright 2014, with permission from Taylor & Francis.)
In order to solve this practical problem and improve operational safety of the
heavy-haul trains, the author and team were commissioned by the manufacturer of
the heavy-haul locomotives to conduct comprehensive investigations and studies on
this problem. The analyzed results indicate that the cause for the rail overturning is
the occurrence of a large lateral wheel–rail forces which may come from the lateral
component of longitudinal coupler force. Under this situation, the established
locomotive–track spatially coupled dynamics model is employed for detailed
simulation and longitudinal coupler forces are also considered in the model. Based
on this model, the effect of longitudinal coupler forces on lateral dynamic inter-
actions of the locomotive–track coupled system is analyzed. The influencing
mechanism of the coupler free swing angle on the locomotive operational safety
demonstrates that there is a causal relationship between them. On this basis, a limit
value for the coupler free swing angle design is then proposed and applied in the
final coupler system design to successfully solve this significant practical problem.
The main study results are going to be introduced as follows.
The coupler may swing in the horizontal plane when it is under compressive
actions. The maximum possible swing angle is defined as the coupler free swing
angle u which is illustrated in Fig. 10.29. This coupler free swing angle is limited
physically due to the action of various kinds of auxiliary elements, such as the
secondary lateral stop.
The FT prototype coupler imported to China was applied in the locomotive
shown in Fig. 10.28 which depicts how the locomotive was derailed in a
heavy-haul train test. This type of coupler has a large free swing angle, and it is
likely to generate relatively large coupler angles with a maximum value exceeding
10° under longitudinal compressive forces. First, the wheel–rail dynamic interaction
performance of the locomotive with the prototype coupler is analyzed when the
coupler swing angle is 8°. Note that this swing angle is still smaller than the
3000
2000
1000
-1000
-2000
86.7 86.8 86.9 87.0 87.1 87.2 87.3 87.4 87.5 87.6
Running distance (km)
maximum coupler angle that the coupler can reach. For the longitudinal coupler
force, its value is assigned with 1500 kN (far smaller than its maximum value)
according to the tested coupler longitudinal force results of the heavy-haul loco-
motive (see Fig. 10.30).
The calculated lateral wheel–rail force results are presented in Fig. 10.31.
During the simulation, the heavy-haul locomotive is running on tangent track at its
maximum operational speed of 80 km/h. Here, the coupler swing angle is set to be
8°. It can be seen that most of the lateral wheelset forces are greater than 120 kN;
and the maximum value is even up to 174.7 kN which is far higher than the safety
threshold of 79.3 kN [8].
In addition, the corresponding calculated derailment coefficient is given in
Fig. 10.32. The results indicate that there are so many occurrences with amplitudes
larger than 0.9, and the maximum value of 1.2 was observed during the operational
process. This does not meet the required limit value of 0.9 specified in the Chinese
railway standard entitled “Identification methods and evaluation standard for the
test of railway locomotive dynamics performance” (TB/T2360-93) [9].
The analyzed results indicate that the lateral wheel–rail interactions are much
intensified, and the safety indexes, such as the lateral wheelset force and the derail-
ment coefficient, significantly exceed their safety thresholds. This locomotive with
this type of coupler with a large swing angle cannot meet the requirements for safe
120
80
40
0
0 100 200 300 400 500
Running distance (km)
392 10 Practical Applications of the Theory of Vehicle–Track Coupled …
Derailment coefficient
derailment coefficient 1.2
0.9
0.6
0.3
0.0
0 100 200 300 400 500
Running distance (km)
The analyzed results in the last subsection demonstrate that the possible reason for
the heavy-haul locomotive derailments is the large coupler free swing angle which
generates a large lateral coupler force component. The large lateral force component
can cause abnormal lateral wheel–rail dynamic interaction forces and rail over-
turning. Thus, the author and team further investigate the relationship between the
coupler free swing angle and locomotive running safety. The study was conducted
to determine the safety threshold of the coupler free swing angle and supply some
scientific guidance for improving the coupler design.
The maximum values of the calculated lateral wheelset force under various
coupler free swing angles are shown in Fig. 10.33. In these simulations, the sim-
ulated locomotive is braking on tangent track at the speed of 80 km/h that is the
practical maximum operational speed when the locomotive is used for 20,000 t
heavy-haul trains. The lateral wheelset force increases nonlinearly with the increase
of the coupler free swing angle. When the coupler free swing angle is less than 3°,
the lateral wheelset force has a slight increase with the increase of the coupler free
angle. In the angle range of 3–6°, the lateral wheelset force increases promptly with
the coupler free swing angle, especially the lateral wheelset force is beyond its
safety threshold when the coupler free swing angle is larger than 3.7°.
The corresponding maximum values of the locomotive derailment coefficient are
displayed in Fig. 10.34. Similarly, the derailment coefficient also increases non-
linearly with the increase of coupler free swing angle. When the coupler free swing
angle is less than 4°, the derailment coefficient has a gradual increase with the
growth of the coupler free swing angle. When the coupler free swing angle exceeds
4°, the derailment coefficient increases rapidly with the increase of the coupler free
10.3 Safety Control of the Coupler Swing Angle of a Heavy-Haul Long Train 393
120
Safety threshold
100
80
60
40
20
0 1 2 3 4 5 6 7 8 9
Coupler free swing angles ( ° )
1.0
0.8
0.6
0.4
0.2
0 1 2 3 4 5 6 7 8 9
Coupler free swing angles ( ° )
swing angle, and it reaches the safety threshold when the coupler free swing angle
is increased to 4.9°.
It can also be seen from the analysis that both lateral wheelset force and derail-
ment coefficient indicate that the locomotive running safety could meet the safety
requirements when the coupler free swing angle is smaller than 3.7°. Considering a
reasonable safety margin, it is suggested that the maximum free swing angle of the
heavy-haul locomotive coupler should be controlled to no more than 3°.
This proposal had subsequently been applied to the improved design of the
prototype coupler by the locomotive manufacturer. This newly designed coupler
has a maximum coupler free swing angle of 3°. It had been put into practice in July
394 10 Practical Applications of the Theory of Vehicle–Track Coupled …
2008. The practical application demonstrates that the operational safety of the
heavy-haul locomotive can be guaranteed using the improved coupler.
Fuzhou
Fujian
Province
High-speed railway
Xiamen
transportation must be taken into consideration. Specifically, this line should not
only satisfy the operational requirement of high-speed passenger train running at a
maximum speed of 200–250 km/h, but also meet the operational requirement of
freight trains with a hauling mass of 3500 t. This railway line was the first shared
high-speed passenger and freight railway in China.
Shared high-speed passenger and freight railways are only employed in a few
countries such as Germany, Italy, etc.; the design of this kind of railways was still a
challenging task for railway engineers. On the one hand, it lacked simple design
criteria for the determination of various parameters (minimum curve radius,
superelevation, transition curve length, longitudinal slope) of horizontal and vertical
profiles for the high and low speed shared line because there was an insufficient
theoretical basis for the influence of these parameters on the overall running safety
and ride comfort at both high and low speeds. On the other hand, the dynamic
effects of high-speed passenger trains and low-speed freight trains on track struc-
tures are quite different. When a freight train with a larger axle load runs at a low
speed, and a high-speed passenger train with a smaller axle load runs at high speed,
how can the dynamic effects on the track structures for the two cases be reconciled?
Is it possible to realize a compatible design? Quite a few studies with regard to these
issues have been published.
For this practical engineering problem, the author and team collaborated with the
design institution of Fuzhou–Xiamen Railway (China Railway Eryuan Engineering
Group Co. Ltd.) to comprehensively investigate the running safety and stability [10]
of the track–vehicle system, as well as the dynamic interaction between vehicle and
track [11] by combining the line design parameters with vehicle dynamic perfor-
mance using the vehicle–track coupled dynamics theory and the principle and
method for optimal integrated design of vehicle and track systems in Chap. 9. The
design work was completed at the beginning of 2003, thus realizing the opti-
mization of the line design and safety pre-evaluation of the Fuzhou–Xiamen shared
passenger and freight railway. The design work also provided the necessary theo-
retical basis for the development of design standards for shared high-speed pas-
senger and freight railways in China.
As examples, the determination of parameters for horizontal curves, the
parameter matching of horizontal and vertical sections, and the evaluation of
dynamic interaction between vehicle and track will be briefly discussed for the
speeds of 200/120 km/h. More detailed research on dynamic performance under
other speeds can be found in Refs. [10, 11].
Based on the actual conditions at that time, the “China Star” high-speed test train
and the 120 km/h speedup freight train were selected as the high- and low-speed
operational conditions, respectively. For a clearer expression, the motor car and
trailer car of the “China Star” high-speed train is referred to as ZH-D and ZH-T,
respectively. HJ-120 and ZK4 are referred to as speedup freight locomotive and
freight vehicle are referred to as HJ-120 and KZ4, respectively.
396 10 Practical Applications of the Theory of Vehicle–Track Coupled …
Existing studies show that the important horizontal curve parameters are mainly the
minimum curve radius, superelevation and transition curve length, which have
important influences on the dynamic performance of high- and low-speed trains.
According to the design requirements, the minimum curve radius is divided into
four different levels under each speed matching scheme, and the superelevation is
considered as a constant value that complies with existing standards. For the speed
match of 200/120 km/h, the abovementioned parameters are listed in Table 10.1.
Using the track irregularity that was measured from the high-speed test section in
the Zhengzhou–Wuhan railway, the dynamic performance indexes of train running
safety and stability, and the wheel–rail wear index are calculated and summarized in
Table 10.2. This table shows the results when the “China Star” train with a speed of
Table 10.1 Parameters of horizontal curves under the speed match of 200/120 km/h
Curve radius (m) Level 1 Level 2 Level 3 Level 4
4,000 2,800 2,200 1,800
Transition curve length (m) 120 180 240 240
Superelevation (mm) 70 110 145 150
Table 10.2 Dynamic performance of passenger and freight train passing through track curves
under speed matching scheme of 200/120 km/h
Speed (km/h) Passenger car 200 Freight car 120
Curve radius range (m) 4,000 (Level 1), 2,800 (Level 2), 2,200 (Level 3), 1,800
(Level 4)
Vehicle type ZH-D ZH-T HJ-120 ZK4
Transition curve length (m) 130 200 260 270
Lateral wheel–rail force (kN) 34.58–47.70 26.91–38.59 31.42–43.36 30.73–31.92
Vertical wheel–rail force (kN) 134.0–154.4 99.98–113.8 152.2–169.9 138.4–143.8
Lateral wheelset force (kN) 35.39–42.49 28.52–37.19 41.01–54.83 29.74–34.92
Rollover coefficient 0.44–0.53 0.46–0.73 0.27–0.43 0.29–0.36
Derailment coefficient 0.27–0.34 0.27–0.44 0.25–0.27 0.26–0.29
Wheel unloading rate 0.45–0.53 0.57–0.79 0.44–0.58 0.30–0.38
Wheel–rail wear index (N m/m) 108.8–158.4 89.18–144.9 99.21–227.9 106.2–128.7
Lateral car body acceleration (g) 0.11–0.15 0.08–0.13 0.10–0.24 0.23–0.24
Vertical car body acceleration (g) 0.07–0.09 0.06–0.06 0.06–0.09 0.20–0.20
Lateral stability index 2.36–2.45 1.88–2.08 2.09–2.48 2.47–2.65
Vertical stability index 1.98–2.11 1.76–1.85 1.66–1.92 2.32–2.43
10.4 Application and Practice for Design of Fuzhou–Xiamen Shared High-Speed … 397
200 km/h and the speedup freight train with a speed of 120 km/h passes through
curved tracks with different radius levels. More detailed data can be seen in ref-
erence [10]. Conclusions that can be drawn from Table 10.2 are as follows:
(1) The safety indexes of all cases are within the range of qualified and allowable
values. Some indexes, such as the lateral and vertical wheel–rail forces and the
derailment coefficient have large safety margins, and most of the stability
indexes are of excellent levels.
(2) Dynamic performance indexes all increase to a certain extent with the decrease
of curve radius. For example, when the curve radius reduces from 4,000 to
1,800 m, the lateral wheel–rail force increases by about 10 kN. The increase of
vertical wheel–rail force is relatively small, with a maximum increase of about
11%. For the safety indexes, the increments are significant. Specifically, the
rollover coefficient and wheel unloading rate increase by 20–35%, the wheel–
rail wear index grows by 30–220%, and the lateral and vertical car body
accelerations rise by 20–40%.
(3) For the “China Star” high-speed test train, the lateral wheel–rail force, wheel–
rail wear index, lateral and vertical car body accelerations of the ZH-D motor
car are all larger than those of the ZH-T trailer car. However, the rollover
coefficient, derailment coefficient, and wheel unloading rate of the ZH-D are all
smaller than those of the ZH-T.
(4) When the speedup freight train passes through the track curves at the speed of
120 km/h, the safety indexes and wheel–rail wear index of the HJ-120 (except
for the derailment coefficient) are all larger than those of the ZK4, whereas the
lateral and vertical stability index of the HJ-120 are smaller compared with
those of the ZK4.
(5) The lateral wheel–rail interaction force and wear index of the HJ-120 are
greater than those of the ZH-D. However, the rollover coefficient and derail-
ment coefficient of the former are smaller. In several curved tracks, the lateral
car body acceleration of the former are about 60% larger than that of the latter.
The vertical car body accelerations of the two cases are almost the same.
(6) The vertical wheel–rail interaction force of the speedup train is 30% greater
than that of the ZH-T. The rollover coefficient, derailment coefficient, wheel
unloading rate, and the wear index of the former are smaller. The lateral and
vertical car body accelerations of the former are 1–2 times larger than those of
the latter.
In summary, the safety indexes of the freight train are smaller than those of the
passenger train, but they all meet the safety limit requirement. The ride comfort
indexes of the motor car are larger than those of the trailer car for the passenger
train. The result for the freight train is these indexes are higher for the wagons than
for the locomotives, but they are all at an excellent level. The ZH-D has the largest
wheel–rail wear index, followed by the HJ-120, the ZH-D, and the ZK4.
In conclusion, from the viewpoints of train running safety and ride comfort, the
minimum radius of the horizontal curve can be selected as 1,800 m. Accordingly,
398 10 Practical Applications of the Theory of Vehicle–Track Coupled …
the minimum transition curve length can be set as 240 m for a shared high-speed
passenger and freight railway under the speed matching scheme of 200/120 km/h.
Whether the vertical curves are compatible with the horizontal curves or not is an
important research question in the matching design of horizontal and vertical
profiles. The setting rules described in relevant standards issued by the Ministry of
Railways limit the overlapping of horizontal and vertical profile curves. However,
these rules mainly consider the workload of the railway line and the associated
maintenance difficulty; the rules do not specifically account for train running safety
and ride comfort. Obviously, in the case of overlapped horizontal and vertical
curves, if the dynamic performance indexes meet the running safety and comfort
requirements on the shared high-speed passenger and freight railway line, this
design will greatly reduce the project cost and generate a good economic benefit.
This section takes a length of the Fuzhou–Xiamen high-speed railway as an
example, and the effects of key parameters of vertical and horizontal curves on the
dynamic performance of a high-speed passenger train and low-speed freight train
are investigated. The purpose of the investigation is to guide a reasonable matching
design of the railway horizontal and vertical profile curves. Only the locomotive
that has a larger dynamic interaction on the track structure is adopted in the cal-
culation [10]. The 200 km/h high-speed train is represented by the ZH-D, and the
120 km/h speedup freight train is represented by the HJ-120.
A schematic diagram of the railway horizontal and vertical profiles is shown in
Fig. 10.36, and the corresponding curve parameters are listed in Table 10.3 where
i is the vertical curve slope; L1 and L2 are the slope lengths; DL1 and DL2 are the
shortest lengths of transition curves that meet the requirements of train running
safety and ride comfort; Rs is the radius of the vertical curve; ls is the length of the
vertical curve; R is the radius of the horizontal curve; l is the transition curve length
Table 10.3 Parameters of horizontal and vertical profile curves for 200/120 km/h speed match
design
Vertical curve Horizontal curve
i (‰) L1 (m) L2(m) Rs (m) R (m) l (m) h (mm)
6 400 420 15,000 1800 240 150
In this section, dynamic effects of high- and low-speed trains on track structures of
the Fuzhou–Xiamen shared passenger and freight railway are compared using
vehicle–track coupled dynamics simulations. Safety assessments are further con-
ducted according to the corresponding evaluation criteria.
1. Dynamic interaction analysis under the excitation of random track
irregularities
Under the excitation of random track irregularities, the vertical dynamic effects on
track structures induced by passenger and freight train loads are summarized in
Table 10.5.
As it can be seen from the table, the vertical wheel–rail contact force, rail
supporting force, and ballast bed surface stress induced by the HJ-120 running at
the speed of 120 km/h are greater than those induced by the ZH-D running at the
speed of 200 km/h. For example, the vertical wheel–rail contact force increases by
about 15%, the ballast bed surface stress increases by around 10%. Other dynamics
indexes of the train vehicles in the table are very close to each other. The vertical
wheel–rail force of the ZK4 is smaller than that of the ZH-T; the ZK4 and ZH-T
show almost the same level in the rail supporting force, ballast bed surface stress,
and rail displacement. The ballast bed acceleration and subgrade surface stress are
found to be smaller under low-speed freight train operations.
Compared with the evaluation criteria for track dynamic effect, it can be known
that all the maximum values of the vertical wheel–rail dynamic responses do not
exceed the safety control limits.
2. Dynamic interaction analysis under the excitation of turnout
Studies show that the vertical dynamic effect is more severe when locomotives pass
through turnouts compared with that of other vehicles. Figure 10.37 compares the
Table 10.5 Dynamic effect indexes of track structures induced by passenger and freight train
loads
Speed (km/h) Speed of passenger Speed of freight train
train 200 120
Locomotive vehicle type ZH-D ZH-T HJ-120 ZK4
Vertical wheel–rail force (kN) 164.9 130.6 191.5 125.3
Rail supporting force (kN) 55.9 44.0 64.5 47.0
Wheel–rail contact stress (MPa) 1041.0 1014.4 1024.9 1033.0
Ballast bed surface stress (MPa) 0.275 0.216 0.317 0.231
Subgrade surface stress (MPa) 0.089 0.073 0.090 0.069
Ballast bed acceleration (m/s2) 16.51 17.74 7.63 6.39
Vertical rail displacement (mm) 1.44 1.12 1.67 1.23
10.4 Application and Practice for Design of Fuzhou–Xiamen Shared High-Speed … 401
ZH-D (200km/h)
200 HJ-120 (120km/h)
150
100
50
0
0 1 2 3 4 5
Running distance (m)
Fig. 10.37 Comparison of vertical wheel–rail force induced by ZH-D and HJ-120 passing
through turnout
wheel–rail dynamic response when the ZH-D and HJ-120 pass through a
movable-point turnout with different running speeds. As can be seen from the
figure, the dynamic force induced by the HJ-120 passing the turnout at 120 km/h is
significantly greater than that induced by the ZH-D passing the turnout at 200 km/
h; an increase of 20% can be identified for the former case, but it still meets the
safety operational standard.
3. Dynamic interaction analysis under the excitation of rail joint
As shown in Fig. 10.38, the wheel–rail force induced by the ZH-D passing through
the rail joint at the speed of 200 km/h is much larger than that induced by the
HJ-120 with the speed of 120 km/h. Correspondingly, the rail supporting force,
wheel–rail contact stress and ballast–bed surface stress have similar characteristics
[11]. This is due to the fact that wheel–rail interaction is quite sensitive to train
running speeds, and an increase of the running speed would result in a dramatic
increase of wheel–rail impact force. The dynamic effects of passenger and freight
vehicles basically have effects to those of the locomotives (see Fig. 10.39).
Generally, the rail displacement and subgrade surface stress induced by freight
locomotives are greater than those induced by a motor car of the high-speed pas-
senger train. This is mainly due to the fact that the axle load of the HJ-120 (23 t) is
larger than that of the ZH-D (19.5 t). However, simulation results show that the
amount of increase is not significant at around 10–20% [11].
All dynamic behavior indexes under the speed match design can meet the safety
operational standard.
402 10 Practical Applications of the Theory of Vehicle–Track Coupled …
250
ZH-D (200km/h)
HJ-120 (120km/h)
Vertical wheel-rail force (kN)
200
150
100
50
0
0.0 0.5 1.0 1.5
Running distance (m)
Fig. 10.38 Comparison of vertical wheel–rail contact force induced by ZH-D and HJ-120 passing
through rail joint
200
ZH-D (200km/h)
HJ-120 (120km/h)
Vertical wheel-rail force (kN)
150
100
50
0
0.0 0.5 1.0 1.5
Running distance (m)
Fig. 10.39 Comparison of vertical wheel–rail contact force induced by ZH-T and ZK4 passing
through rail joint
10.4 Application and Practice for Design of Fuzhou–Xiamen Shared High-Speed … 403
The above results show that the dynamic effects of the HJ-120 at the speed of
120 km/h on track structures are generally greater than that of the ZH-D at the
speed of 200 km/h. Therefore, it is necessary to study how to reduce the dynamic
interactions between speedup freight trains and track. This section gives an example
of the parameter optimization scheme of railway line design based on dynamic
analysis and economic feasibility studies.
1. Improved design scheme for horizontal curve parameters
To illustrate the mitigation effect of the improved design scheme on lateral wheel–
rail forces, Table 10.6 lists the comparison of lateral wheel–rail forces of the
HJ-120 between the original scheme (minimum curve radius R = 1800 m, transi-
tion curve length l = 240 m, superelevation h = 150 mm) and the improved
scheme (R = 2800 m, l = 180 m, h = 110 mm) under the mixed operational mode
of 200/120 km/h. It can be seen that when the minimum horizontal curve radius
changes from 1800 to 2800 m, the lateral wheel–rail force drops from 44.39 to
37.23 kN, a reduction of about 16%; the lateral wheelset force decreases by around
17%; the derailment coefficient decreases from 0.28 to 0.26; the wheel–rail wear
index declines by 26%; and the lateral rail displacement reduces from 1.42 to
1.28 mm. Clearly, the improved design scheme is proven to be an effective measure
for dynamic interaction mitigation.
2. Improved design scheme for track parameters
To illustrate the reduction effect of the improved design scheme on vertical wheel–
rail contact forces, Table 10.7 lists the comparison of vertical wheel–rail contact
forces between the original scheme (rail pad stiffness of 60–80 MN/m) and the
Table 10.6 Comparison of lateral dynamic interaction between speedup freight locomotive and
track after improvement of horizontal curve parameters
Dynamic performance index Lateral Lateral Derailment Wear Lateral rail
wheel– wheelset coefficient index displacement
rail force force (N m/ (mm)
(kN) (kN) m)
Original scheme of horizontal 44.39 54.92 0.28 230.05 1.42
curve parameters (R = 1800 m,
l = 240 m, h = 150 mm)
Improved scheme of horizontal 37.23 45.72 0.26 170.31 1.28
curve parameters (R = 2800 m,
l = 180 m, h = 110 mm)
Decline in level of index (%) 16 17 7 26 10
404
10
Table 10.7 Comparison of vertical dynamic interaction between speedup freight locomotive and track after the improvement of rail pad stiffness
Dynamic performance Vertical Rail Wheel–rail Surface stress of Surface stress Acceleration Vertical rail
index wheel–rail supporting contact stress ballast bed of subgrade of ballast bed displacement
force (kN) force (kN) (MPa) (MPa) (MPa) (g) (mm)
Original rail pad 203.8 69.3 1046.5 0.341 0.096 6.95 1.81
stiffness (Kp = 60–80
MN/m)
New rail pad stiffness 198.1 63.6 1036.1 0.313 0.088 6.35 2.16
(Kp = 50–60 MN/m)
Decline level of index 2.8 8.2 1.0 8.2 8.3 8.6 −19
(%)
Practical Applications of the Theory of Vehicle–Track Coupled …
10.4 Application and Practice for Design of Fuzhou–Xiamen Shared High-Speed … 405
improved scheme (rail pad stiffness of 50–60 MN/m) when the HJ-120 locomotive
passes through a No. 18 turnout at the speed of 120 km/h. As can be seen from the
table, all wheel–rail dynamic indexes decrease to a certain extent with the decrease
of rail pad stiffness, especially for the infrastructure. For example, the rail sup-
porting force drops by 8.2%, the acceleration and stress of ballast bed and subgrade
stress reduce by 8–10%. This is of great importance to relieve the dynamic effect of
freight trains on the track structure on a shared passenger and freight railway line.
The above research results were applied directly in the design of the 200–250 km/h
Fuzhou–Xiamen shared high-speed passenger and freight railway. The proposed
design schemes and technical standards were adopted based on practical engi-
neering concepts which strongly supported the design and construction of China’s
first shared high-speed passenger and freight railway in 2003 at a time when the
corresponding design standard was limited. The research results provided the the-
oretical basis for the development of the design standards of shared high-speed
passenger and freight railways in China.
The construction of Fuzhou–Xiamen high-speed railway began on September
30, 2005, and was completed on July 20, 2009. It officially opened to traffic on
April 26, 2010 (Fig. 10.40). The upgraded operation shortened the running time on
that route from 13 h to around 1 h and 40 min. The operational practice has proven
Fig. 10.40 Practical operation of Fuzhou–Xiamen shared railway for high-speed passenger and
freight trains
406 10 Practical Applications of the Theory of Vehicle–Track Coupled …
that the high and low speed (passenger and freight) trains can achieve the desired
effect and run safely and smoothly on the same corridor.
The four engineering application examples presented in this chapter has
demonstrated the effectiveness of the vehicle–track coupled dynamics theory. With
the development of the modern railway, especially for the higher speed passenger
transportation and heaver freight transportation, the operational environment of
trains appears to be more and more complicated, and the dynamic interaction
between the vehicles and the tracks becomes more intensified. The vehicle–track
coupled dynamics theory will play more active roles in the modern railway train
and track design process.
References
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Appendices
This appendix provides the geometry and parameters of the railway vehicles and
tracks adopted in the simulations in this book, which may be helpful for researchers
who want to analyze the vehicle–track coupled dynamics.
© Science Press and Springer Nature Singapore Pte Ltd. 2020 407
W. Zhai, Vehicle–Track Coupled Dynamics,
https://doi.org/10.1007/978-981-32-9283-3
408 Appendices
2.50 m
17.50 m
Bogie frame
Axle box
Anti-yaw damper
Wheelset
Traction motor Primary vertical spring
Gear box
Primary vertical damper
Considering the usage of the parameters in the simulation examples in Chaps. 8 and
10, the configuration and main parameters of two typical Chinese three-piece
freight wagons, namely C80 and C62, are presented in this appendix. The maxi-
mum operating speed of the wagons can attain 80 km/h. The configuration of the
C80 freight wagon is shown in Fig. B.1, while its main parameters are listed in
Table B.1. The main parameters of the C62 freight wagon are listed in Table B.2.
1.83 m
8.20 m
Side frame
Side bearing
Primary suspension
Wheelset
Secondary suspension
Cross sustaining device
Table B.1 Main parameters of the freight wagon C80 with three-piece bogies
Notation Parameter Valuea Unit
Mass/inertia
Mc Car body mass 91,838 kg
Mt Side frame mass 745 kg
MB Bolster mass 497 kg
Mw Wheelset mass 1,171 kg
Icx Mass moment of inertia of car body about X-axis 2.163 105 kg m2
Icy Mass moment of inertia of car body about Y-axis 0.996 106 kg m2
Icz Mass moment of inertia of car body about Z-axis 0.984 106 kg m2
IBz Mass moment of inertia of bolster about Z-axis 258 kg m2
Ity Mass moment of inertia of side frame about Y-axis 188 kg m2
Itz Mass moment of inertia of side frame about Z-axis 173 kg m2
Iwx Mass moment of inertia of wheelset about X-axis 700 kg m2
Iwy Mass moment of inertia of wheelset about Y-axis 140 kg m2
Iwz Mass moment of inertia of wheelset about Z-axis 700 kg m2
Primary suspension
Kpx Stiffness coefficient along X-axis 13.0 MN/m
Kpy Stiffness coefficient along Y-axis 11.0 MN/m
Kpz Stiffness coefficient along Z-axis 160.0 MN/m
Secondary suspension
Ksx Stiffness coefficient of secondary suspension along X-axis 3.127 MN/m
Ksy Stiffness coefficient of secondary suspension along Y-axis 3.127 MN/m
Ksz Stiffness coefficient of secondary suspension along Z-axis 4.235 MN/m
Kpx Stiffness coefficient of a cross joint bar 14.8 MN/m
Dimension
lc Semi-longitudinal distance between bogies 4.10 m
lt Semi-longitudinal distance between wheelsets in bogie 0.915 m
dw Lateral semi-span of primary suspensions 0.9905 m
ds Lateral semi-span of secondary suspensions 0.9905 m
R0 Wheel radius 0.42 m
a
Loaded case
412 Appendices
Table B.2 Main parameters of the freight wagon C62 with three-piece bogies
Notation Parameter Valuea Unit
Mass/inertia
Mc Car body mass 77,000 kg
Mt Side frame mass 330 kg
MB Bolster mass 470 kg
Mw Wheelset mass 1,200 kg
Icx Mass moment of inertia of car body about X-axis 1.0 105 kg m2
Icy Mass moment of inertia of car body about Y-axis 1.2 106 kg m2
Icz Mass moment of inertia of car body about Z-axis 1.07 106 kg m2
IBz Mass moment of inertia of bolster about Z-axis 190 kg m2
Ity Mass moment of inertia of side frame about Y-axis 100 kg m2
Itz Mass moment of inertia of side frame about Z-axis 80 kg m2
Iwx Mass moment of inertia of wheelset about X-axis 740 kg m2
Iwy Mass moment of inertia of wheelset about Y-axis 100 kg m2
Iwz Mass moment of inertia of wheelset about Z-axis 740 kg m2
Secondary suspension
Ksx Stiffness coefficient of secondary suspension along X-axis 4.14 MN/m
Ksy Stiffness coefficient of secondary suspension along Y-axis 4.14 MN/m
Ksz Stiffness coefficient of secondary suspension along Z-axis 5.32 MN/m
Dimension
lc Semi-longitudinal distance between bogies 4.25 m
lt Semi-longitudinal distance between wheelsets in bogie 0.875 m
R0 Wheel radius 0.42 m
a
Loaded case
Appendices 413
The typical Chinese ballasted tracks are adopted in the simulation examples in
Chaps. 8, 9 and 10. Thus, their configuration and main parameters are presented in
this appendix. The configuration of the Chinese high-speed ballasted track is shown
in Fig. C.1, while its main parameters are listed in Table C.1. The main parameters
of the Chinese heavy-haul railway ballasted track are listed in Table C.2.
Ballast 0.35 m
Subgrade upper layer 0.70 m
8.15 m
Rail
Sleeper
0.35 m Ballast
2.60 m
0.70 m
Subgrade upper layer
Rail
Fastener
Track slab
CAM layer
Concrete base
Subgrade
3.25 m
2.50 m
Rail
Fastener
Subgrade