1. Modelling of Rolling Contact in a Multibody Environment Delft University of Technology Design Engineering and Production Mechanical Engineering Workshop on Multibody System Dynamics, University of Illinois at Chicago, May 12, 2003 Arend L. Schwab Laboratory for Engineering Mechanics Delft University of Technology The Netherlands

2. Contents -FEM modelling -Wheel Element -Wheel-Rail Contact Element -Example: Single Wheelset -Example: Bicycle Dynamics -Conclusions

3. 4 Nodal Coordinates: 2D Truss Element 3 Degrees of Freedom as a Rigid Body leaves: 1 Generalized Strain: Rigid Body Motion  Constraint Equation FEM modelling

4. Generalized Nodes: Position Wheel Centre Contact Point Euler parameters Rotation Matrix: R(q) Wheel Element Rigid body pure rolling: 3 degrees of freedom In total 10 generalized coordinates Impose 7 Constraints Nodes

5. Radius vector: Rotated wheel axle: Normal on surface: Surface: Holonomic Constraints as zero generalized strains Strains Wheel Element Elongation: Lateral Bending: Contact point on the surface: Wheel perpendicular to the surface Normalization condition on Euler par:

6. Non-Holonomic Constraints as zero generalized slips Wheel Element Slips Generalized Slips: Velocity of material point of wheel at contact in c: Longitudinal slip Lateral slip Two tangent vectors in c: Radius vector: Angular velocity wheel:

7. Generalized Nodes: Position Wheel Centre Contact Point Euler parameters Rotation Matrix: R(q) Wheel-Rail Contact Element Rigid body pure rolling: 2 degrees of freedom In total 10 generalized coordinates Impose 8 Constraints Nodes

8. Wheel-Rail Contact Element Strains Local radius vector: Normal on Wheel surface: Wheel & Rail surface: Two Tangents in c: Distance from c to Wheel surface: Distance from c to Rail surface: Wheel and Rail in Point Contact: Normalization condition on Euler par: Holonomic Constraints as zero generalized strains

9. Wheel-Rail Contact Element Slips Wheel & Rail surface: Two Tangents in c: Non-Holonomic Constraints as zero generalized slips Velocity of material point of Wheel in contact point c: Generalized Slips: Lateral slip: Longitudinal slip: Spin: Normal on Rail Surface: Angular velocity wheel:

10. Single Wheelset Example Klingel Motion of a Wheelset Wheel bands: S1002 Rails: UIC60 Gauge: 1.435 m Rail Slant: 1/40 FEM-model : 2 Wheel-Rail, 2 Beams, 3 Hinges Pure Rolling, Released Spin 1 DOF

11. Single Wheelset Profiles Wheel band S1002 Rail profile UIC60

12. Single Wheelset Motion Klingel Motion of a Wheelset Wheel bands: S1002 Rails: UIC60 Gauge: 1.435 m Rail Slant: 1/40 Theoretical Wave Length:

13. Single Wheelset Example Critical Speed of a Single Wheelset Wheel bands: S1002, Rails: UIC60 Gauge: 1.435 m, Rail Slant: 1/20 m=1887 kg, I=1000,100,1000 kgm 2 Vertical Load 173 226 N Yaw Spring Stiffness 816 kNm/rad FEM-model : 2 Wheel-Rail, 2 Beams, 3 Hinges Linear Creep + Saturation 4 DOF

14. Single Wheelset Constitutive Critical Speed of a Single Wheelset Linear Creep + Saturation according to Vermeulen & Johnson (1964) Tangential Force Maximal Friction Force Total Creep

15. Single Wheelset Limit Cycle V cr =130 m/s Limit Cycle Motion at v=131 m/s Critical Speed of a Single Wheelset

16. Bicycle Dynamics Example FEM-model : 2 Wheels, 2 Beams, 6 Hinges Pure Rolling 3 DOF Bicycle with Rigid Rider and No-Hands Standard Dutch Bike

17. Bicycle Dynamics Root Loci Stability of the Forward Upright Steady Motion Root Loci from the Linearized Equations of Motion. Parameter: forward speed v

18. Bicycle Dynamics Motion Full Non-Linear Forward Dynamic Analysis at different speeds Forward Speed v [m/s]: 0 5 10 11 14 18

19. Conclusions Proposed Contact Elements are Suitable for Modelling Dynamic Behaviour of Road and Track Guided Vehicles. Further Investigation: Curvature Jumps in Unworn Profiles, they Cause Jumps in the Speed of and Forces in the Contact Point. Difficulty to take into account Closely Spaced Double Point Contact.