Rail Vehicle Dynamics (eBook)

Klaus Knothe, Master in Mathematics and Civil Engineering TU Darmstadt 1963. Promotion (PhD) Mechanical Engineering, TU Berlin 1967; Habilitation Mechanical Engineering, TU Berlin 1969; Professor at TU Berlin since 1970; retired 2002; several Research Projects together with Deutsche Bahn: Lectures (among others) in Finite Element Method, Structural Dynamics and Rail Vehicle Dynamics; publication of more than 260 papers, among them 4 books. Member of the Editorial Board of Vehicle System Dynamics.Sebastian Stichel has a Master in Vehicle Engineering from the Technical University Berlin 1992 and a PhD from the same University 1996.He joined Bombardier in Västerås in 2000. From 200 to 2010 he was Manager for Vehicle Dynamics at Bombardier. Since 2010 he is Professor in Rail Vehicle Dynamics at KTH Stockholm.  

Preface 5
References 6
Contents 8
1 Introduction 16
1.1 The Basic Function of the Wheel/rail System 16
1.2 Significance of Dynamics on the Operation of Rail Vehicles 17
1.3 On the History of Research in the Field of Railway Technology Since 1800 19
1.3.1 1800--1945 19
1.3.2 A New Start After 1945: Japan and France 22
1.3.3 Research and Development Aimed at Overcoming the ``Boundaries of the Wheel/rail System'' 25
1.4 Railway Industry in Europe 26
1.5 Overview of This Book 27
1.5.1 Classification 27
1.5.2 Vertical and Lateral Vibrations 28
1.5.3 Curving Behavior 28
1.5.4 Frequency- and Time-Domain Calculations 28
References 29
2 Modeling of Vehicle, Track, and Excitation 31
2.1 Prior Considerations and Coordinate Systems 31
2.2 Vehicle Modeling 32
2.2.1 Bogie Frame Design, Primary Suspension, and Bogie Guidance 32
2.2.2 Mechanical Model of the Vehicle. Connecting Elements 37
2.2.3 Elastic Carbodies 37
2.3 Modeling of Track and Excitation 40
2.3.1 Track Modeling 40
2.3.2 Modeling of the Excitation 42
References 45
3 Modeling of Wheel/Rail Contact 47
3.1 Profile Geometry 47
3.2 Contact Kinematics Between Wheel and Rail 50
3.2.1 Contact Kinematics with Conical and Circular Profiles 51
3.2.2 Contact Kinematics for Arbitrary Profiles 57
3.2.3 On the Determination of Equivalent Contact Parameters by Means of Quasilinearization 59
3.2.4 Conversion into Equivalent Circular Profiles 61
3.2.5 Linearized Contact Kinematics with Track Irregularities 62
3.2.6 Creepage Calculation 64
3.3 Normal Contact Mechanics 65
3.3.1 Overview of the Calculation of Contact Point Stresses 65
3.3.2 Assumptions of the Normal Contact Problem 66
3.3.3 Nonelliptic Contact Patches 67
3.3.4 Hertzian Treatment of the Normal Contact Problem 67
3.3.5 Spherical and Point Contact 70
3.3.6 Ellipsoidal Contact 71
3.3.7 Contact of Rollers, Line Contact 72
3.3.8 Linearized Replacement Model 73
3.4 Tangential Contact Mechanics 74
3.4.1 Introduction to the Tangential Contact Problem 74
3.4.2 Analytical Solution for Rolling Contact (Line Contact) 77
3.4.3 Kalker's Theory of Rolling Contact for Ellipsoidal Contact 80
3.4.4 Approximate Solutions According to Vermeulen--Johnson and Shen--Hedrick--Elkins 81
3.4.5 Simplified Theory of Rolling Contact [54] 85
3.4.6 Adaptation of the Theory to Reality 90
References 91
4 Vertical Dynamics, Equations of Motion, and Free Vibrations 94
4.1 Notation and Assumptions 94
4.2 Equations of Motion with the Principle of Linear and Angular Momentum 95
4.2.1 Displacement Degrees of Freedom for a Two-Axle Vehicle 96
4.2.2 Constraints 96
4.2.3 Forces in the Spring and Damper Elements 98
4.2.4 Free-Body Diagrams of the Masses 100
4.2.5 Principle of Linear and Angular Momentum for the Formulation of the Equation System 101
4.2.6 Elimination of the Constrained Forces. Final System of Equations 102
4.3 Principle of Virtual Displacements for Rigid-Body Systems 104
4.3.1 Initial Remark 104
4.3.2 Formulation of the Principle of Virtual Displacements 105
4.3.3 Inclusion of Geometric Constraints in the Principle Of Virtual Displacement Exemplified on a Two-Axle Vehicle 107
4.4 Formalized Derivation of the Equations of Motion and the Principle ƒ 108
4.4.1 Displacement Vector with the Degrees of Freedom of the Free System 108
4.4.2 Relation Between Spring Elongation and System Displacement 109
4.4.3 Spring Forces and the Formulation of the Virtual Strain Energy 109
4.4.4 Mass Matrix and Formulation of the Virtual Work of the Inertia Forces 110
4.4.5 External and Constraint Forces 110
4.4.6 Equations of Motion for the Free System. Introduction of Constraints 111
4.5 Equations of Motion for Elastic Car Bodies 112
4.6 Solution for Free Vibrations 114
4.7 Exercises for This Chapter 118
4.7.1 Constraint Forces when Constraints are Met 118
4.7.2 Validity of the Rolling Condition 118
References 118
5 Forced Vertical Vibrations for Excitation with Harmonic and Periodic Track Irregularities (Frequency Domain Solution) 119
5.1 Complex Syntax 120
5.2 Vertical Vibrations on a Track with Cosine-Shaped ƒ 123
5.2.1 Track Irregularities and Base-Point Excitations 123
5.2.2 Solution for the Vertical Motion 124
5.2.3 Interpretation of the Solution for Different Suspensions Between Wheelset and Carbody 128
5.3 Vehicle on a General Periodic Track 131
5.4 Solution for a Vehicle with Elastic Carbody Modes 135
5.5 Exercises for This Chapter 136
5.5.1 Two-Axle Vehicle on Cosine Track 136
5.5.2 Two-Axle Vehicle on Generally Periodic Track 137
References 137
6 Random Vibrations due to Stochastic Track Irregularities 138
6.1 Characterization of an Irregular Track with Help of Root ƒ 138
6.2 Determination of Vehicle Response for Stochastic Track Excitation 141
6.3 Power Spectra of Track Irregularities 144
6.3.1 How to Derive Power Spectra of Track Irregularities 144
6.3.2 Power Spectra for the DB Network 146
6.4 Supplementary Comments Regarding the Relationship Between Measured Spectra of Spatial Angular Frequencies and Spectra of Angular Frequencies as Function of Time 150
6.5 Interpretation of Response Spectrum 150
References 151
7 Human Perception of Vibrations - Ride Comfort 152
7.1 Wertungsziffer According to Sperling 153
7.1.1 Periodic Vibrations 156
7.1.2 Random Vibrations 158
7.2 ISO 2631 159
7.3 CEN Standard EN 12299 162
7.3.1 Simplified Criterion for Mean Comfort: NMV 162
7.3.2 Comfort on Curve Transitions: PCT 164
7.3.3 Comfort on Discrete Events: PDE 164
7.4 Final Remarks 166
7.4.1 Measure or Simulate 166
7.4.2 Ride Comfort as System Property 167
7.4.3 Duration of Exposure of Vibration 167
7.5 Exercises for This Chapter 167
7.5.1 Calculation of Wz-Values According to Sperling 167
References 168
8 Introduction to Lateral Dynamics of Railway Vehicles 169
8.1 Preliminary Remark 169
8.2 Hunting and Klingel Equation 172
8.3 Assumptions for Derivation of the Klingel Equation 175
8.4 Determination of Equivalent Conicity with Eq.(8.13) 177
8.4.1 Final Remark to This Chapter 178
References 178
9 Derivation of Equations of Motion for Lateral Dynamics 179
9.1 Principle of Virtual Displacements for a Suspended ƒ 179
9.1.1 Evaluated System and Acting Forces 179
9.1.2 Formulation of Principle of Virtual Displacements 181
9.1.3 Determination of the Virtual Displacements 182
9.1.4 Equilibrium Conditions in the x-Direction and Around the y-Axis 184
9.1.5 Equilibrium Conditions in the y-Direction and Around the z-Axis 185
9.2 Exercises for This Chapter 186
9.2.1 Interpretation of Creep Force terms in Eq. (9.13) 186
9.2.2 Rolling Resistance Due to Spin Creepage 187
9.2.3 Equations of Motions for Forced Lateral Vibrations 187
9.2.4 Rolling Resistance in the Simplified Theory 187
9.2.5 Equations of Motion of a Suspended Wheelset with Numbers 187
9.2.6 Creep Forces Under the Assumption of a Reference State with Creep Forces 187
References 188
10 Lateral Eigenbehavior and Stability of a Wheelset on Straight Track 189
10.1 Determination of Eigenvalues and Eigenvectors 189
10.2 Root Loci Curves 191
10.3 Approximate Solution for Low Speeds 193
10.4 Stability Investigation with Hurwitz Criterion 196
10.5 Critical Speed of a Wheelset 198
10.6 Interpretation of Stability Criteria of a Single Wheelset 199
10.7 Exercises for This Chapter 203
10.7.1 Characteristic Equation 203
10.7.2 Transformation of Wheelset Equations of Motion 204
10.7.3 Graphical Illustration of the Root Loci Curves of a Suspended Wheelset and Determination of the Critical Speed 204
10.7.4 Independently Rotating Wheels 204
Reference 204
11 Lateral Eigenbehaviour and Stability of Bogies 205
11.1 Numerical Calculation of Eigenvalues and Critical Speed 205
11.2 Analytic Approximations for Bogies 211
11.2.1 Coordinate Transformation to Introduce Generalized Degrees of Freedom 213
11.2.2 Bogie with Infinitely High Bending and Shear Stiffness 221
11.2.3 Realization of Infinitely High Bending Stiffness or Shear Stiffness 225
11.2.4 Bogies with Infinite Shear Stiffness 226
11.2.5 Bogies with Infinite Bending Stiffness 229
11.2.6 Bogies with Finite Bending and Shear Stiffness 230
11.3 Exercises for This Chapter 232
11.3.1 Equations of Motion of a Bogie 232
11.3.2 Equations of Motion of a Freely Rolling Wheelset at Low Speed 233
11.3.3 Equations for Bending and Shear Stiffness 233
References 233
12 Lateral Eigenbehavior and Stability of Bogie Vehicles 235
12.1 Stability of a Train with Two Vehicles 235
12.2 General Conclusions Regarding the Stability ƒ 238
12.2.1 Theory and Simulation of Bogie Vehicle Hunting 239
12.2.2 Bogie Hunting 242
12.2.3 Influence of Friction Yaw Damping on Bogie Hunting (Nonlinear Stability Investigation) 243
12.3 Suggestions for Further Work for This Chapter 245
12.3.1 Influence of Bending and Shear Stiffness on the Stability of Bogie Vehicles 245
12.3.2 Stability of a Vehicle with Independently Rotating Wheels 245
12.3.3 Friction and Viscous Yaw Damping 245
References 245
13 Introduction to Non-linear Stability Investigations 246
13.1 Preamble 246
13.2 Nonlinear Critical Speed 247
13.3 Fourier Decomposition of Nonlinear Limit Cycle Motions: The Method of Urabe and ReiTer 249
13.4 Simplified Investigation of Nonlinear Stability with the Method of Quasilinearization 252
13.5 Limits of Fourier Decomposition 254
13.6 Nonlinear Stability Investigations in the Time Domain 254
13.7 Ideas for Further Study in this Chapter 255
13.7.1 Stability Investigation for the Boedecker Vehicle 255
References 255
14 Quasistatic Curving Behavior 258
14.1 Historical Introduction 258
14.2 General Remarks 259
14.3 A Single Wheelset in a Curve 260
14.3.1 Free Wheelset in a Curve (Kinematic Curving) 260
14.4 Wheelset Guided in Track Following Frame 261
14.5 Curving of Bogies and Entire Vehicles 268
14.5.1 Curving According to Uebelacker and Heumann 269
14.5.2 Curving of Bogies with Suspension 273
14.6 Wear Calculation in the Wheel--Rail Contact 275
14.7 Exercises for This Chapter 278
14.7.1 Directions of Creep Forces for Different Wheelset Positions 278
14.7.2 Deviation of Angle of Attack or Lateral Displacement of Wheelsets 279
14.7.3 Curving of Single Wheelset 279
References 279
15 Determination of Load Collectives for Vehicle Components 282
15.1 Introduction 282
15.2 General Procedure 283
15.3 Stress Calculation in Components 284
15.3.1 Finite Element Calculation in Each Time Step 285
15.3.2 Stress Calculation with the Help of Transformation Matrices 285
15.4 Determination of Load Collectives 288
15.4.1 Determination of Sustainable Stresses 288
15.4.2 Cycle Counting Methods to Determine Load Collectives 291
15.4.3 Conversion from a Two-Parameter to a One Parameter Collective 296
15.4.4 Superposition to Total Load Collective 298
15.5 Damage Accumulation: Proof of Strength 299
15.5.1 Damage Accumulation Hypotheses 299
15.5.2 Concepts for Variable-Amplitude Fatigue Strength Proof for Rail Vehicles 301
15.6 Exercises for This Chapter 302
15.6.1 Transformation Matrix Between Degrees of Freedom in a Multibody Simulation Model and Stresses in the Bogie Frame 302
15.6.2 Determination of Collective of Spring Forces with Spectral Analysis 302
References 302
16 Appendix 305
16.1 List of Symbols 305
16.2 Coordinate Systems 310
16.3 Fundamentals of Contact Mechanics 312
16.3.1 Hertz Contact Mechanics 312
16.3.2 Contact Equation 314
16.3.3 Basic Equations for the Tangential Contact Problem According to Carter 316
16.4 Function ? for the Vermeulen--Johnson Solution 319
16.5 Basic Equations of the Simplified Theory of Rolling Contact 320
16.6 Stability Criteria of Characteristic Equations with Hurwitz Criterion 321
16.7 Critical Speed of Single Wheelset Taking Nondiagonal Elements of Creepage Damping Matrix into Account 322
References 323
Index 324