Two Special Finite Elements for Modelling Modelling of Rolling Contact in a Multibody Environment

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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
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Contents

• -FEM modelling
• Wheel Element
• Wheel-Rail Contact Element
• Example Single Wheelset
• Example Bicycle Dynamics
• Conclusions

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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
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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
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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
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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
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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
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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
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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
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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
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Single Wheelset
Profiles
Wheel band S1002
Rail profile UIC60
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Single Wheelset
Motion
Klingel Motion of a Wheelset
Wheel bands S1002 Rails UIC60 Gauge 1.435
m Rail Slant 1/40
Theoretical Wave Length
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Single Wheelset
Example
Critical Speed of a Single Wheelset
Wheel bands S1002, Rails UIC60 Gauge 1.435 m,
Rail Slant 1/20 m1887 kg, I1000,100,1000
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
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Single Wheelset
Constitutive
Critical Speed of a Single Wheelset
Linear Creep Saturation according to Vermeulen
Johnson (1964)
Tangential Force Maximal Friction Force
Total Creep
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Single Wheelset
Limit Cycle
Limit Cycle Motion at v131 m/s
Critical Speed of a Single Wheelset
Vcr130 m/s
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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
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Bicycle Dynamics
Root Loci
Stability of the Forward Upright Steady Motion
Root Loci from the Linearized Equations of
Motion. Parameter forward speed v
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Bicycle Dynamics
Motion
Full Non-Linear Forward Dynamic Analysis at
different speeds
Forward Speed v m/s
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10
5
0
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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.