Vehicle–Track coupled dynamics

Vehicle–Track

Coupled Dynamics

Wanming Zhai

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

ISBN 978-981-32-9282-6 ISBN 978-981-32-9283-3 (eBook)

https://doi.org/10.1007/978-981-32-9283-3

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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

(NSFC), the National Outstanding Young Scientist Foundation of China (received

by the author in 1995), the Ministry of Science and Technology of China (MOST),

the China Railway (former China Ministry of Railway), railway industry compa￾nies, and others, the research group (including graduate students) led by the author

carried out many follow-up research tasks, and published the first academic

monograph in this research field entitled Vehicle–Track Coupled Dynamics (First

edition, in Chinese) in 1997. Afterward, the second, third, and fourth editions of the

monograph (in Chinese) were published in 2002, 2007, and 2015 respectively,

which became the most fundamental reference books in the field of railway system

dynamics and design of rolling stocks and track structures in China, especially for

high-speed railways.

In recent years, with the great-leap-forward development of modern railway

transportation, especially for high-speed railways, the vehicle–track coupled

dynamics theory needs to address more demanding engineering requirements and

many new emerging open problems. Supported by the NSFC Major Project (Grand

No. 11790280), the NSFC Key Project (Grand No. 51735012), the Program of

Introducing Talents of Discipline to Universities (111 Project) (Grant No. B16041)

from the China Ministry of Education (MOE), the author led his group to extend the

vehicle–track coupled dynamics theory through more elaborate theoretical analysis

and more extensive investigations of field problems uncovered in practice.

Meanwhile, worldwide research on this topic has also been extremely active and

achieved much progress recently. The first English monograph re-edited from the

author’s Chinese monographs is published when the relevant field is undergoing

rapid development in terms of theoretical research and engineering practices.

The writing of this book would not be possible without the support from various

individuals and organizations. First, the author is most grateful for the continuous

support from the NSFC, the MOST, the China Railway, the MOE, etc. during the

past decades. The author also owes much gratitude to those who have participated

in the amendment of this English monograph. They are Dr. Shengyang Zhu,

Dr. Liang Ling, and Dr. Zaigang Chen from the author’s group; Dr. Yunshi Zhao,

Dr. Xiaoyun Liu, and Dr. Ilaria Grossoni from University of Huddersfield (UK),

Dr. Guoying Tian from Xihua University (China). The author would like to thank

the following scholars with special gratitude: Dr. Qing Wu and Dr. Tim Mcsweeney

from Central Queensland University (Australia), Prof. Zili Li from Delft University

of Technology (the Netherlands), Prof. Kelvin C. P. Wang from Oklahoma State

University (USA), and Prof. Manicka Dhanasekar from Queensland University of

Technology (Australia), for their extreme enthusiasm in proofreading this book.

Some calculation examples performed by Dr. Liang Ling are also gratefully

acknowledged. Finally, the author wants to thank his Ph.D. students, Mr. Yu Sun,

Ms. Yu Guo, Mr. Jun Luo, Mr. Tao Zhang, and Ms. Mei Chen, for their assistance

in carefully editing and supplying photographs, diagrams, and relevant information.

vi Preface

The author believes the publication of this English monograph on Vehicle–Track

Coupled Dynamics will be conducive to both the investigation of railway engi￾neering dynamics and the development of modern railway industry.

Chengdu, China Wanming Zhai

December 2018

Preface vii

Contents

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

3 Excitation Models of Vehicle–Track Coupled System ........... 151

3.1 Excitation Input Method ............................. 151

3.1.1 Fixed-Point Method .......................... 152

3.1.2 Moving-Vehicle Method ....................... 153

3.1.3 Tracking-Window Method ..................... 153

3.2 Impact Excitation Models ............................ 156

3.2.1 Impact Model of Wheel Flat .................... 156

3.2.2 Model of Rail Dislocation Joint ................. 161

3.2.3 Model of Dipped Rail Joint .................... 163

3.2.4 Impact Model of Turnout ...................... 163

3.2.5 Other Impulsive Excitation Models ............... 166

3.3 Harmonic Excitation Models .......................... 167

3.3.1 Displacement Input Model of Harmonic Excitation.... 168

3.3.2 Input Method of Common Track Irregularities ....... 175

3.3.3 Input Function of Periodic Harmonic Force ......... 177

3.4 Excitation Model of Track Dynamic Stiffness Irregularity ..... 178

3.4.1 Stiffness Irregularity at Track Transition Sections ..... 179

3.4.2 Track Stiffness Irregularity at Turnout Section ....... 181

3.4.3 Modeling of Rail Infrastructure Defects ............ 182

3.5 Excitation Model of Random Track Irregularity ............ 183

3.5.1 Track Irregularity PSDs of United States

of America ................................. 185

3.5.2 Track Irregularity PSDs of Germany .............. 187

3.5.3 Track Irregularity PSDs of China ................ 188

3.5.4 Comparison of Typical Track Irregularity PSDs ...... 193

3.5.5 Numerical Simulation Method for Random Track

Irregularity Time-Domain Samples Transformed

from Track Irregularity PSDs ................... 196

References ............................................ 201

4 Numerical Method and Computer Simulation for Analysis

of Vehicle–Track Coupled Dynamics ....................... 203

4.1 Time Integration Methods for Solving Large-Scale Dynamic

Problems ........................................ 203

4.2 New Simple Fast Explicit Time Integration Method: Zhai

Method ......................................... 205

4.2.1 Integration Scheme of Zhai Method............... 205

4.2.2 Stability of Zhai Method ....................... 206

4.2.3 Accuracy of Zhai Method ...................... 208

4.2.4 Numerical Dissipation and Dispersion ............. 208

4.2.5 Numerical Examples for Verification .............. 211

4.3 Application of Zhai Method to Analysis of Vehicle–Track

Coupled Dynamics ................................. 214

x Contents

4.3.1 Numerical Integration Procedure ................. 215

4.3.2 Determination of Time Step of Zhai Method ........ 216

4.4 On Some Key Issues in Solving Process of Vehicle–Track

Coupled Dynamics ................................. 218

4.4.1 Determination of Calculated Length of Track and

Mode Number of Rail ........................ 218

4.4.2 Solving Technique for the Train–Track Coupled

Dynamics ................................. 219

4.5 Computer Simulation of Vehicle–Track Coupled Dynamics.... 223

4.5.1 Vehicle–Track Vertically Coupled Dynamics

Simulation ................................. 223

4.5.2 Vehicle–Track Spatially Coupled Dynamics

Simulation ................................. 225

4.5.3 Train–Track Spatially Coupled Dynamics

Simulation ................................. 225

References ............................................ 228

5 Field Test on Vehicle–Track Coupled System Dynamics......... 231

5.1 Field Test Methods of Vehicle–Track Coupled System

Dynamics ........................................ 231

5.1.1 Field Test Methods of Vehicle Dynamics .......... 232

5.1.2 Field Test Methods of Track Dynamics ............ 233

5.2 Typical Dynamics Tests of Vehicles Running on Tracks ...... 237

5.2.1 Dynamic Test for a Typical High-Speed Train

on Slab Track .............................. 237

5.2.2 Dynamic Test for a Typical Freight Vehicle

on Ballasted Track ........................... 243

5.3 Typical Vehicle–Track Dynamic Interaction Tests........... 246

5.3.1 Wheel–Rail Interaction Test with a High-Speed

Train on Qinshen Passenger Dedicated Line ........ 246

5.3.2 Track Dynamics Test with a 10,000-Tonne

Heavy-Haul Train on Daqin Line ................ 251

5.3.3 Wheel–Rail Interaction Test on a Small-Radius

Curve in Mountain Area Railway ................ 253

References ............................................ 258

6 Experimental Validation of Vehicle–Track Coupled Dynamics

Models .............................................. 259

6.1 Experimental Validation on the Vehicle–Track Vertically

Coupled Dynamics Model ............................ 259

6.1.1 Comparison of Vehicle Vibrations Between

Theoretical and Measured Results ................ 260

6.1.2 Comparison Between Theoretical and Measured

Vibrations of Track Structure ................... 261

Contents xi

6.1.3 Comparison Between Computed and Measured

Wheel–Rail Dynamic Forces .................... 264

6.1.4 Conclusions ................................ 266

6.2 Experimental Validation of the Vehicle–Track Spatially

Coupled Dynamics Model ............................ 267

6.2.1 Experimental Validation by Field Test

on Beijing–Qinhuangdao Speedup Line ............ 267

6.2.2 Validation by High-Speed Train Running Test

on Qinshen Passenger Dedicated Line ............. 271

6.2.3 Validation by Derailment Experiment for Freight

Train Running on Straight Line .................. 273

6.2.4 Experimental Validation by Wheel–Rail Dynamic

Interaction Test on a Small Radius Curve

of Mountain Railway ......................... 275

6.2.5 Conclusions ................................ 276

6.3 Experimental Validation of the Train–Track Spatially

Coupled Dynamics Model ............................ 276

6.3.1 Validation by Measured Coupler Longitudinal

Forces of a Heavy-Haul Combined Train Under

Braking Conditions........................... 277

6.3.2 Validation by Tested Train Dynamic Characteristics

Under Electric Braking Conditions ............... 277

6.3.3 Validation by Measured Results of Heavy-Haul

Train Curving Performance ..................... 281

6.3.4 Conclusions ................................ 283

References ............................................ 283

7 Computational Comparison of Vehicle–Track Coupled

Dynamics and Vehicle System Dynamics .................... 285

7.1 Comparison of Computational Results on Vehicle Hunting

Stability ......................................... 285

7.1.1 Numerical Calculation Method of Vehicle

Nonlinear Hunting Stability .................... 285

7.1.2 Comparison of Calculated Critical Speeds Between

the Coupled Model and the Traditional Model ....... 288

7.1.3 Summary .................................. 289

7.2 Comparison of Calculation Results on Vehicle Ride

Comfort ......................................... 290

7.3 Comparison of Calculation Results on Curving

Performance ...................................... 292

7.3.1 Comparison of Vehicle Passing Through a Small

Radius Curved Track at Low Speed .............. 292

7.3.2 Comparison of Vehicle Passing Through a Large

Radius Curved Track at High Speed .............. 295

xii Contents

7.4 Conclusions ...................................... 296

References ............................................ 297

8 Vibration Characteristics of Vehicle–Track Coupled System ..... 299

8.1 Steady-State Response of Vehicle–Track Interaction ......... 299

8.1.1 Steady-State Response Due to Sleeper Span ......... 300

8.1.2 Track Steady-State Response Under Moving

Vehicle ................................... 301

8.1.3 Steady-State Curving Response .................. 303

8.2 Dynamic Response of Vehicle–Track Interaction Due

to Local Geometry Defects ........................... 305

8.2.1 Dynamic Response to Vertical Impulsive Defects..... 305

8.2.2 Dynamic Response to Lateral Impulsive Defects ..... 313

8.2.3 Dynamic Response to Vertical Local Harmonic

Geometry Defects............................ 314

8.2.4 Dynamic Response to Lateral Local Harmonic

Geometry Defects............................ 318

8.3 Dynamic Response of Vehicle–Track Interaction to Cyclic

Geometry Defects .................................. 321

8.4 Dynamic Response of Vehicle–Track Interaction Due

to Failure of System Component ....................... 323

8.4.1 Dynamic Response to Disabled Lateral Dampers

on a High-Speed Bogie........................ 323

8.4.2 Dynamic Response to Fracture of Fastener Clips ..... 324

8.4.3 Dynamic Response to Unsupported Sleepers ........ 325

8.5 Dynamic Response of Vehicle–Track Interaction to Random

Irregularities ...................................... 327

8.5.1 Vibration Characteristics of the Car Body .......... 330

8.5.2 Vibration Characteristics of the Bogie Frame ........ 331

8.5.3 Vibration Characteristics of the Wheelset ........... 331

8.5.4 Characteristics of the Wheel–Rail Forces ........... 331

8.5.5 Vibration Characteristics of the Rail .............. 332

8.5.6 Vibration Characteristics of the Track Slab ......... 332

8.6 Dynamic Response Due to Railway Infrastructure

Settlement ....................................... 333

8.6.1 Dynamic Response Due to Differential Subgrade

Settlement ................................. 333

8.6.2 Dynamic Response Due to Differential Ballast

Settlement ................................. 341

References ............................................ 346

Contents xiii

9 Principle and Method of Optimal Integrated Design

for Dynamic Performances of Vehicle and Track Systems ....... 347

9.1 Principle of Optimal Integrated Design for Dynamic

Performances of Vehicle and Track Systems .............. 347

9.2 Method of Optimal Integrated Design for Dynamic

Performances of Vehicle and Track Systems .............. 349

9.2.1 Dynamic Design Method for Vehicle System

Based on the Optimal Integrated Design Principle .... 349

9.2.2 Dynamic Design Method for Track System Based

on the Optimal Integrated Design Principle ......... 350

9.3 Case Study I: Optimal Design of Suspension Parameters

of a Heavy-Haul Locomotive .......................... 351

9.3.1 Operation Safety Analysis of HXD2C Prototype

Locomotive Through Small Radius Curves ......... 352

9.3.2 Optimization Scheme to Improve Curve Negotiation

Performance of HXD2C Heavy-Haul Locomotive ..... 353

9.3.3 Application of HXD2C Heavy-Haul Locomotive

After Design Optimization ..................... 356

9.4 Case Study II: Design of a Steep Gradient Section

of a High-Speed Railway ............................ 359

9.4.1 Engineering and Research Background ............ 359

9.4.2 Comparison of High-Speed Running Performance

Between Long Tunnel Scheme and Bridge–Tunnel

Scheme for Shazai Island ...................... 361

9.4.3 Comparison of High-Speed Running Performance

Between Long Tunnel Scheme and Bridge–Tunnel

Scheme for Haiou Island ...................... 363

9.4.4 Comparison and Selection Between Shazai Island

Scheme with Long Tunnel and Haiou Island with

Long Tunnel ............................... 364

9.4.5 Project Implementation and Operation Practice ....... 365

References ............................................ 366

10 Practical Applications of the Theory of Vehicle–Track

Coupled Dynamics in Engineering ......................... 367

10.1 Redesign of Dynamic Performance of a Speedup

Locomotive ...................................... 367

10.1.1 Engineering Background ....................... 367

10.1.2 Simulation on Abnormal Lateral Vibration

of SS7E Locomotive Prototype .................. 368

10.1.3 Technical Proposal for Improving the Lateral

Vibration Performance of SS7E Locomotive ......... 370

10.1.4 Practical Performance and Application Status

of the Improved SS7E Speedup Locomotive ......... 373

xiv Contents

10.2 Reducing Rail Side Wear on Heavy-Haul Railway Curves .... 375

10.2.1 The Problem of Rail Wear on Curves of Heavy-Haul

Railways .................................. 375

10.2.2 Design Methodology of Rail Asymmetric-Grinding

Profiles for Curves ........................... 378

10.2.3 Numerical Implementation for Design of Rail

Asymmetric-Grinding Profiles on a Practical

Railway Curves ............................. 381

10.2.4 Engineering Practice and Implementation Effect ...... 385

10.3 Safety Control of the Coupler Swing Angle of a Heavy-Haul

Long Train ....................................... 389

10.3.1 Engineering Background ....................... 389

10.3.2 Analysis of Wheel–Rail Dynamic Interaction

with Large Coupler Free Swing Angle ............. 390

10.3.3 Effect of Coupler Free Swing Angle on Heavy-Haul

Locomotive Running Safety and Its Safety Design .... 392

10.4 Application and Practice for Design of Fuzhou–Xiamen

Shared High-Speed Passenger and Freight Railway .......... 394

10.4.1 Engineering and Research Background ............ 394

10.4.2 Effect of Key Parameters of Horizontal Curve

on Dynamic Performance of High- and Low-Speed

Trains .................................... 396

10.4.3 Optimal Integrated Design of Horizontal and Vertical

Profiles for the Shared Passenger and Freight

Railway ................................... 398

10.4.4 Dynamic Effects of High- and Low-Speed Trains

on Track Structures .......................... 400

10.4.5 Technical Measures for Mitigating Dynamic Effects

of Freight Train on Shared Passenger and Freight

Railway Track .............................. 403

10.4.6 Project Implementation and Practical Operation

Effect..................................... 405

References ............................................ 406

Appendices.................................................. 407

Contents xv

Chapter 1

Introduction

Abstract To better understand vehicle–track coupled dynamics which is a new

theoretical system, it is necessary for readers to understand the following questions.

What is the background under which the theory was proposed? What is the academic

rationale of the theory? What are the research scopes and research methodologies? In

this chapter, the author will give detailed explanations of these questions.

1.1 Background of Vehicle–Track Coupled Dynamics

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 Pe

ixt

for the track system (the harmonic vehicle loads P are applied

© Science Press and Springer Nature Singapore Pte Ltd. 2020

W. Zhai, Vehicle–Track Coupled Dynamics,

https://doi.org/10.1007/978-981-32-9283-3_1

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