ABSTRACT
I. Introduction ........................................................................................................................ 86
A. Basic Model of a Wheelset, Degrees of Freedom .................................................... 87
II. The Normal Contact........................................................................................................... 88
A. Hertzian Contact ........................................................................................................ 88
1. Ratio A/B, Relation with b/a ................................................................................ 90
2. Calculation of the Semi Axes .............................................................................. 90
3. Convexity, Concavity and Radius Sign ............................................................... 91
4. Particular Case: 2D Contact................................................................................. 91
5. Contact Pressure ................................................................................................... 91
B. Tables, Polynomial Expressions and Faster Methods............................................... 92
1. Continuous Expression of the Tables .................................................................. 92
2. Analytical Approximation of the Tables ............................................................. 92
C. Application to the Railway Field............................................................................... 93
1. Contact Plane Angle, Conicity............................................................................. 93
2. A Note on Conicity .............................................................................................. 93
3. Normal Load......................................................................................................... 93
4. Determination of the Longitudinal Curvature of the Wheel............................... 93
5. Contact Point between the Wheel and Rail Profiles ........................................... 93
6. Direct Comparison of the Antagonist Curvatures ............................................... 94
7. One or Two Wheel-Rail Pairs? .......................................................................... 94
III. The Tangent Problem......................................................................................................... 95
A. Forces and Couples on a Wheelset............................................................................ 95
1. Approximations .................................................................................................... 95
B. Tangent Forces: Rolling Friction Simple Models ..................................................... 96
1. Historical Review ................................................................................................. 96
C. Linear Expressions of the Creep Forces.................................................................... 97
D. Definition of Creepages ............................................................................................. 97
1. Quasi-Static Creepages......................................................................................... 97
2. Quasi-Static Creepages in the Railway Case....................................................... 97
3. Longitudinal Creepage ......................................................................................... 98
4. Lateral Creepage .................................................................................................. 98
5. Spin Creepage....................................................................................................... 99
6. Dynamic Formulation of the Creepages .............................................................. 99
7. Damping Terms and Stability .............................................................................. 99
8. Nondimensional Spin Creepage........................................................................... 99
E. Kalker’s Coefficients c
........................................................................................... 100
1. c
Values for Simplified Bogie Models............................................................. 101
F. Creep Forces in the Linear Domain ........................................................................ 101
1. Dependence on Load.......................................................................................... 101
2. Creepages Combinations and Saturation ........................................................... 101
3. Using Linear Models: The C110, C220 Stiffnesses .......................................... 102
4. Reduced Creepages ............................................................................................ 102
G. Saturation Laws........................................................................................................ 102
1. Vermeulen and Johnson ..................................................................................... 102
2. Kalker’s Empirical Proposition.......................................................................... 103
3. Exponential Saturation Law: CHOPAYA, Ohyama and Others....................... 103
H. Surface-Based Contact Models................................................................................ 103
1. Discretised Ellipse: FASTSIM from Kalker...................................................... 104
2. Stresses ............................................................................................................... 105
3. Linear Contact Forces, Elastic Coefficients....................................................... 105
4. Reduced Creepages in FASTSIM...................................................................... 106
5. Extensions of FASTSIM .................................................................................... 107
I. Methods Based on FASTSIM.................................................................................. 107
IV. Contact Forces in the Railway Context........................................................................... 107
A. From the Pure Dicone to the Wheelset with Real Profiles ..................................... 107
1. Equivalent Conicity............................................................................................ 107
2. Variable Conicity ............................................................................................... 108
3. Profile Measurements: Importance of the Angular Reference .......................... 108
B. Gravitational Stiffness.............................................................................................. 108
C. Flange Contact ......................................................................................................... 109
1. Flange Contact Jump and Definition of the Nominal Play in the Track .......... 109
2. Contact Jump and Load Transfer....................................................................... 109
D. Using the Contact Angle Function .......................................................................... 110
1. Localisation of Multiple Contacts on the Tread................................................ 110
2. Gravitational Centring Ability in Standard Conditions..................................... 110
E. The Diverging Effect of Spin, Influence on the Normal Load ............................... 110
F. Safety Criteria, Nadal’s Formula............................................................................. 112
G. Independent Wheel, Application to Industrial Mechanisms on Rails .................... 112
H. Modelling the Contact Jumps .................................................................................. 113
1. Hertzian Multiple Contacts ................................................................................ 113
I. Advanced Methods for Non-Hertzian Contacts ...................................................... 114
J. Wheelset Equilibrium, Explicit Scheme.................................................................. 115
1. Wheelset Equilibrium Equations ....................................................................... 115
2. Decomposition of Forces ................................................................................... 115
3. Adaptation to the Numerical Calculation .......................................................... 115
4. Equilibrium Equations........................................................................................ 116
Appendix 4.1. Kinematic Movement: The Klingel Formula ...................................................... 116
Appendix 4.2. Kinematic Hunting and Equivalent Conicity....................................................... 117
Appendix 4.3. The Circle Theory ................................................................................................ 117
Appendix 4.4. Analysis of Y/Q and Nadal’s Criteria .................................................................. 117
Nomenclature................................................................................................................................ 119
References..................................................................................................................................... 120
For more than 150 years, the wheel-rail system has provided a relatively safe system of transport.
This safety level is so high that the mechanism is generally neglected and considered as a simple
