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Modelling of Rolling Contact in a Multibody Environment

Delft University of Technology Design Engineering and Production Mechanical Engineering. Modelling of Rolling Contact in a Multibody Environment. Arend L. Schwab Laboratory for Engineering Mechanics Delft University of Technology The Netherlands.

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Modelling of Rolling Contact in a Multibody Environment

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

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

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

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

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

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

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

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

  9. Wheel-Rail Contact Element Slips Non-Holonomic Constraints as zero generalized slips Velocity of material point of Wheel in contact point c: Generalized Slips: Longitudinal slip: Lateral slip: Wheel & Rail surface: Spin: Two Tangents in c: 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 kgm2 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 Limit Cycle Motion at v=131 m/s Critical Speed of a Single Wheelset Vcr=130 m/s

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

  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]: 18 14 11 10 5 0

  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.

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