Table of Contents

1 Track Dynamics Research Contents and Related Standards 1
1.1 A Review of Track Dynamics Research 1
1.2 Track Dynamics Research Contents 6
1.3 Limits for Safety and Riding Quality and Railway Environmental Standards 7
1.3.1 Safety Limit for Regular Trains 7
1.3.2 Riding Quality Limits for Regular Trains 8
1.3.3 Safety and Riding Quality Limit for Rising Speed Trains 10
1.3.4 Railway Noise Standards in China 11
1.3.5 Railway Noise Standards in Foreign Countries 11
1.3.6 Noise Limit for Railway Locomotives and Passenger Trains in China 12
1.3.7 Environmental Vibration Standards in China’s Urban Areas 14
1.3.8 Limit for Building Vibration Caused by Urban Mass Transit 18
1.4 Standards of Track Maintenance for High—Speed Railway 19
1.4.1 Standards of Track Maintenance for French High—Speed Railway 20
1.4.2 Standards of Track Maintenance and Management for Japanese Shinkansen High—Speed Railway 20
1.4.3 Standards of Track Maintenance and Management for German High—Speed Railway 21
1.4.4 Standards of Track Maintenance and Management for British High—Speed Railway 23
1.4.5 The Measuring Standards of Track Geometry for Korean High—Speed Railway （Dynamic） 24
1.4.6 Standards of Track Maintenance for Chinese 25
1.4.7 The Dominant Frequency Range and Sensitive Wavelength of European High—Speed Train and Track Coupling System 25
1.5 Vibration Standards of Historic Building Structures 28
2 Analytic Method for Dynamic Analysis of the Track Structure 37
2.1 Studies of Ground Surface Wave and Strong Track Vibration Induced by High—Speed Train 37
2.1.1 The Continuous Elastic Beam Model of Track Structure 38
2.1.2 Track Equivalent Stiffness and Track Foundation Elasticity Mociulus 40
2.1.4 Analysis of Strong Track Vibration 41
2.2 Effects of Track Stiffness Abrupt Change on Track Vibration 44
2.2.1 Track Vibration Model in Consideration of Track Irregularity and Stiffness Abrupt Change Under Moving Loads 44
2.2.2 The Reasonable Distribution of the Track Stiffness in Transition 53
3 Fourier Transform Method for Dynamic Analysis of the Track Structure 57
3.1 Model of Single—Layer Continuous Elastic Beam for the Track Structure 57
3.1.1 Fourier Transform 58
3.1.2 Inverse Discrete Fourier Transform 60
3.1.3 Definition of Inverse Discrete Fourier Transform in MATLAB 61
3.2 Model of Double—Layer Continuous Elastic Beam for the Track Structure 62
3.3 Analysis of High—Speed Railway Track Vibration and Track Critical Velocity 64
3.3.1 Analysis of the Single—Layer Continuous Elastic Beam Model 64
3.3.2 Analysis of Double—Layer Continuous Elastic Beam Model 66
3.4 Vibration Analysis of Track for Railways with Mixed Passenger and Freight Traffic 86
3.4.1 Three—Layer Continuous Elastic Beam Model of Track Structure 86
3.4.2 Numerical Simulation of Track Random Irregularity 87
3.4.3 Fourier Transform for Solving Three—Layer Continuous Elastic Beam Model of Track Structure 89
3.5 Vibration Analysis of Ballast Track with Asphalt Trackbed Over Soft Subgrade 94
3.5.1 Four—Layer Continuous Elastic Beam Model of Track Structure 95
3.5.2 Fourier Transform for Solving Four—Layer Continuous Elastic Beam Model of Track Structure 96
3.5.3 Vibration Analysis of Ballast Track with Asphalt Trackbed Over Soft Subgrade 99
References 105
4 Analysis of Vibration Behavior of the Elevated Track Structure 107
4.1 Basic Concept of Admittance 107
4.1.1 Definition of Admittance 107
4.1.2 Computational Method of Admittance 108
4.1.3 Basic Theory of Harmonic Response Analysis 109
4.2 Analysis of Vibration Behavior of the Elevated Bridge Structure 110
4.2.1 Analytic Beam Model 111
4.2.2 Finite Element Model 115
4.2.3 Comparison Between Analytic Model and Finite Element Model of the Elevated Track—Bridge 116
4.2.4 The Influence of the Bridge Bearing Stiffness 117
4.2.5 The Influence of the Bridge Cross Section Model 117
4.3 Analysis of Vibration Behavior of the Elevated Track Structure 120
4.3.1 Analytic Model of the Elevated Track—Bridge System 120
4.3.2 Finite Element Model 124
4.3.3 Damping of the Bridge Structure 124
4.3.4 Parameter Analysis of the Elevated Track—Bridge System 127
4.4 Analysis of Vibration Attenuation Behavior of the Elevated Track Structure 131
4.4.1 The Attenuation Rate of Vibration Transmission 131
4.4.2 Attenuation Coefficient of Rail Vibration 135 References 136
5 Track Irregularity Power Spectrum and Numerical Simulation 137
5.1 Basic Concept of Random Process 138
5.1.1 Stationary Random Process 139
5.1.2 Ergodic 140
5.2 Random Irregularity Power Spectrum of the Track Structure 140
5.2.1 American Track Irregularity Power Spectrum 141
5.2.2 Track Irregularity Power Spectrum for German High—Speed Railways （5） 142
5.2.3 Japanese Track Irregularity Sato Spectrum 143
5.2.4 Chinese Trunk Track Irregularity Spectrum 144
5.2.5 The Track Irregularity Spectrum of Hefei—Wuhan Passenger—Dedicated Line （10） 146
5.2.6 Comparison of the Track Irregularity Power Spectrum Fitting Curves 149
5.3 Numerical Simulation for Random Irregularity of the Track Structure 156
5.4 Trigonometric Series Method 157
5.4.1 Trigonometric Series Method （1） 157
5.4.2 Trigonometric Series Method （2） 158
5.4.3 Trigonometric Series Method （3） 158
5.4.4 Trigonometric Series Method （4） 159
5.4.5 Sample of the Track Structure Random Irregularity 160
References 160
6 Model for Vertical Dynamic Analysis of the Vehicle—Track Coupling System 161
6.1 Fundamental Theory of Dynamic Finite Element Method 162
6.1.1 A Brief Introduction to Dynamic Finite Element Method 162
6.1.2 Beam Element Theory 166
6.2 Finite Element Equation of the Track Structure 172
6.2.1 Basic Assumptions and Computing Model 172
6.2.2 Theory of Generalized Beam Element of Track Structure 173
6.3 Model of Track Dynamics Under Moving Axle Loads 178
6.4 Vehicle Model of a Single Wheel With Primary Suspension System 180
6.5 Vehicle Model of Half a Car With Primary and Secondary Suspension System 182
6.6 Vehicle Model of a Whole Car With Primary and Secondary Suspension System 184
6.7 Parameters for Vehicle and Track Structure 187
6.7.1 Basic Parameters of Locomotives and Vehicles 187
6.7.2 Basic Parameters of the Track Structure 189
References 198
7 A Cross—Iteration Algorithm for Vehicle—Track Coupling Vibration Analysis 201
7.1 A Cross—Iteration Algorithm for Vehicle—Track Nonlinear Coupling System 201
7.2 Example Validation 207
7.2.1 Verification 207
7.2.2 The Influence of Time Step 210
7.2.3 The Influence of Convergence Precision 211
7.3 Dynamic Analysis of the Train—Track Nonlinear Coupling System 212
7.4 Conclusions 218
References 220
8 Moving Element Model and Its Algorithm 221
8.1 Moving Element Model 221
8.2 Moving Element Model of a Single Wheel with Primary Suspension System 224
8.3 Moving Element Model of a Single Wheel with Primary and Secondary Suspension Systems 227
8.4 Model and Algorithm for Dynamic Analysis of a Single Wheel Moving on the Bridge 231
References 234
9 Model and Algorithm for Track Element and Vehicle Element 235
9.1 Ballast Track Element Model 236
9.1.1 Basic Assumptions 236
9.1.2 Three—Layer Ballast Track Element 236
9.2 Slab Track Element Model 239
9.2.1 Basic Assumptions 239
9.2.2 Three—Layer Slab Track Element Model 240
9.2.3 Mass Matrix of the Slab Track Element 241
9.2.4 Stiffness Matrix of the Slab Track Element 242
9.2.5 Damping Matrix of the Slab Track Element 246
9.3 Slab Track–Bridge Element Model 248
9.3.1 Basic Assumptions 248
9.3.2 Three—Layer Slab Track and Bridge Element Model 248
9.3.3 Mass Matrix of the Slab Track—Bridge Element 249
9.3.4 Stiffness Matrix of the Slab Track—Bridge Element 250
9.3.5 Damping Matrix of the Slab Track—Bride Element 253
9.4 Vehicle Element Model 254
9.4.1 Potential Energy of the Vehicle Element 256
9.4.2 Kinetic Energy of the Vehicle Element 260
9.4.3 Dissipated Energy of the Vehicle Element 260
9.5 Finite Element Equation of the Vehicle—Track Coupling System 261
9.6 Dynamic Analysis of the Train and Track Coupling System 262
References 269
10 Dynamic Analysis of the Vehicle—Track Coupling System with Finite Elements in a Moving Frame of Reference 271
10.1 Basic Assumptions 272
10.2 Three—Layer Beam Element Model of the Slab Track in a Moving Frame of Reference 272
10.2.1 Governing Equation of the Slab Track 273
10.2.2 Element Mass， Damping， and Stiffness Matrixes of the Slab Track in a Moving Frame of Reference 276
10.3 Vehicle Element Model 289
10.4 Finite Element Equation of the Vehicle—Slab Track Coupling System 289
10.5 Algorithm Verification 290
10.6 Dynamic Analysis of High—Speed Train and Slab Track Coupling System 292
References 299
11 Model for Vertical Dynamic Analysis of the Vehicle—Track—Subgrade—Ground Coupling System 301
11.1 Model of the Slab Track—Embankment—Ground System Under Moving Loads 301
11.1.1 Dynamic Equation and Its Solution for the Slab Track—Subgrade Bed System 302
11.1.2 Dynamic Equation and Its Solution for the Embankment Body—Ground System 305
11.1.3 Coupling Vibration of the Slab Track—Embankment—Ground System 307
11.2 Model of the Ballast Track—Embankment—Ground System Under Moving Loads 309
11.2.1 Dynamic Equation and Its Solution for the Ballast Track—Subgrade Bed System 310
11.2.2 Coupling Vibration of the Ballast Track—Embankment—Ground System 312
11.3 Analytic Vibration Model of the Moving Vehicle—Track—Subgrade—Ground Coupling System 313
11.3.1 Flexibility Matrix of Moving Vehicles at Wheelset Points 313
11.3.2 Flexibility Matrix of the Track—Subgrade—Ground System at Wheel—Rail Contact Points 316
11.3.3 Coupling of Moving Vehicle—Subgrade—Ground System by Consideration of Track Irregularities 317
11.4 Dynamic Analysis of the High—Speed Train—Track—Subgrade— Ground Coupling System 318
11.4.1 Influence of Train Speed and Track Irregularity on Embankment Body Vibration 318
11.4.2 Influence of Subgrade Bed Stiffness on Embankment Body Vibration 320
11.4.3 Influence of Embankment Soil Stiffness on Embankment Body Vibration 321
References 322
12 Analysis of Dynamic Behavior of the Train， Ballast Track， and Subgrade Coupling System 323
12.1 Parameters for Vehicle and Track Structure 323
12.2 Influence Analysis of the Train Speed 324
12.3 Influence Analysis of the Track Stiffness Distribution 326
12.4 Influence Analysis of the Transition Irregularity 330
12.5 Influence Analysis of the Combined Track Stiffness and Transition Irregularity 336
References 340
13 Analysis of Dynamic Behavior of the Train， Slab Track， and Subgrade Coupling System 341
13.1 Example Validation 342
13.2 Parameter Analysis of Dynamic Behavior of the Train， Slab Track， and Subgrade Coupling System 344
13.3 Influence of the Rail Pad and Fastener Stiffness 345
13.4 Influence of the Rail Pad and Fastener Damping 347
13.5 Influence of the CA Mortar Stiffness 350
13.6 Influence of the CA Mortar Damping 353
13.7 Influence of the Subgrade Stiffness 355
13.8 Influence of the Subgrade Damping 359
References 364
14 Analysis of Dynamic Behavior of the Transition Section Between Ballast Track and Ballastless Track 365
14.1 Influence Analysis of the Train Speed for the Transition Section Between the Ballast Track and the Ballastless Track 366
14.2 Influence Analysis of the Track Foundation Stiffness for the Transition Section between the Ballast Track and the Ballastless Track 369
14.3 Remediation Measures of the Transition Section between the Ballast Track and the Ballastless Track 372
References 376
15 Environmental Vibration Analysis Induced by Overlapping Subways 377
15.1 Vibration Analysis of the Ground Induced by Overlapping Subways 378
15.1.1 Project Profile 378
15.1.2 Material Parameters 378
15.1.3 Finite Element Model 380
15.1.4 Damping Coefficient and Integration Step 381
15.1.5 Vehicle Dynamic Load 382
15.1.6 Environmental Vibration Evaluation Index 383
15.1.7 Influence of Operation Direction of Uplink and Downlink on Vibration 384
15.1.8 Vibration Reduction Scheme Analysis for Overlapping Subways 386
15.1.9 Vibration Frequency Analysis 389
15.1.10 Ground Vibration Distribution Characteristics 390
15.2 Vibration Analysis of the Historic Building Induced by Overlapping Subways 391
15.2.1 Project Profile 391
15.2.2 Finite Element Model 392
15.2.3 Modal Analysis of Building 393
15.2.4 Horizontal Vibration Analysis of the Building 395
15.2.5 Vertical Vibration Analysis of the Building 400
15.3 Conclusions 405
References 406
Index 409

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With increasing train speed， axle loads， traffic density， and the wider engineering applications of new vehicles and track structures， the interaction between vehicle and track has become more complex.Accordingly， the train running safety and stability are more affected by increased dynamic stresses.Dynamic loads acted on the tracks by vehicles fall into three categories： moving axle loads， dynamic loads of fixed points， and moving dynamic loads.Axle loads， due to their moving loading points， are dynamic loads for the track—subgrade—ground system， though irrelevant to the vehicle dynamics and constant in size.As soon as the speed of moving axle loads approaches the critical velocity of the track， the track is subjected to violent vibration.Dynamic loads of fixed points come from the impacts of vehicles passing by the fixed irregularity on track， such as rail joints， the welding slag of continuously welded rail and turnout frogs.Moving dynamic loads are induced by wheel—rail contact irregularities.The dynamics analysis of the train—track system lays a good foundation for investigating the complicated wheel—rail relationship and interaction mechanism， which provides essential references for guiding and optimizing vehicle and track structure designs.
1.1 A Review of Track Dynamics Research
Up to the present， scholars both at home and abroad have achieved a wealth of research findings about the establishment of track dynamics models and numerical methods.Studies on track dynamics models have experienced a development process from simple to complex.Historically， moving loads and vehicle structures have been the earliest practical issues in structure dynamics， especially in the train—track system.Knothe and Grassie （1–3） published several papers concerning track dynamics and vehicle—track interaction in frequency domain.Mathews （4， 5） found out solutions to the dynamic problems of any moving loads on infinite elastic foundation beam by means of Fourier transform method （FTM） and the moving coordinate system.The Fourier transform method （FTM）， as a method of frequency domain analysis， was applied by Trochanis （6）， Ono and Yamada （7） in some related studies.Jezequel （8） simplified the track structure into infinite Euler–Bernoulli beam on elastic foundation， regarding the train loads as concentrated force of uniform motion and considering its rotation and transverse shear effects.Through modal superposition， Timoshenko （9） figured out the governing differential equation for moving loads on simply supported beams in time domain， while Warburton solved this equation by using analytical methods， and he also proved the deflection of beams would reach a maximum under moving loads at a certain speed （10）.Cai et al.（11）， by modal superposition， studied the dynamic response of infinite beams over periodic bearings under moving loads.