Modelling of Rolling Contact in a
Multibody Environment
Delft University of TechnologyDesign Engineering and Production
Mechanical Engineering
Workshop on Multibody System Dynamics, University of Illinois at Chicago , May 12, 2003
Arend L. SchwabLaboratory for Engineering Mechanics
Delft University of TechnologyThe Netherlands
Contents
-FEM modelling
-Wheel Element
-Wheel-Rail Contact Element
-Example: Single Wheelset
-Example: Bicycle Dynamics
-Conclusions
4 Nodal Coordinates:
2D Truss Element
),,,( 2211 yxyxx
3 Degrees of Freedom as a Rigid Body leaves:
1 Generalized Strain:
)(02
122
12 xDε lyyxxl
Rigid Body Motion Constraint Equation
0ε
0lll
FEM modelling
Generalized Nodes:
Position Wheel Centre
Contact Point
Euler parameters
Rotation Matrix: R(q)
),,( zyx wwww
Wheel Element
),,,,( 3210 qqqqq
),,( zyx cccc
Rigid body pure rolling: 3 degrees of freedom
In total 10 generalized coordinates
Impose 7 Constraints
Nodes
)2/()( 02
01 rr rr
re w2
)(3 cg
ner )(4 wRadius vector:
Rotated wheel axle:
Normal on surface:
wcr
Surface:
ww eR(q)e
0)( xg
)(cn g
Holonomic Constraints as zero generalized strains
StrainsWheel Element
0xDε )(
Elongation:
Lateral Bending:
Contact point on the surface:
Wheel perpendicular to the surface
Normalization condition on Euler par: 12
05 qqq
Non-Holonomic Constraints as zero generalized slips
Wheel Element
0xxVs )(
Slips
Generalized Slips:
cs va1
cb 2s
Velocity of material point of wheel at contact in c:
rωwv c
Longitudinal slip
Lateral slip
Two tangent vectors in c:
)( , )( ww ernbera
Radius vector: wcr
Angular velocity wheel: ω
Generalized Nodes:
Position Wheel Centre
Contact Point
Euler parameters
Rotation Matrix: R(q)
),,( zyx wwww
Wheel-Rail Contact Element
),,,,( 3210 qqqqq
),,( zyx cccc
Rigid body pure rolling: 2 degrees of freedom
In total 10 generalized coordinates
Impose 8 Constraints
Nodes
Wheel-Rail Contact ElementStrains
Local radius vector:
Normal on Wheel surface:
)( wcRr T
ww gRn
Wheel & Rail surface: 0)( , 0)( xx rw gg
Two Tangents in c: r , bar
)(1 rwg
rw an 3
Distance from c to Wheel surface:
Distance from c to Rail surface:
Wheel and Rail in Point Contact:
Normalization condition on Euler par: 12
05 qqq
)(2 crg
rw bn 4
Holonomic Constraints as zero generalized strains 0xDε )(
Wheel-Rail Contact ElementSlips
Wheel & Rail surface: 0)( , 0)( xx rw gg
Two Tangents in c: r , bar
Non-Holonomic Constraints as zero generalized slips 0xxVs )(
Velocity of material point of Wheel in contact point c:
)( wcωwv wc
Generalized Slips:
crs va 1
Lateral slip:
crs vb 2
wrs ωn 3
Longitudinal slip:
Spin:
Normal on Rail Surface: rr gn
Angular velocity wheel: wω
Single WheelsetExample
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
Single WheelsetProfiles
Wheel band S1002 Rail profile UIC60
Single WheelsetMotion
Klingel Motion of a Wheelset
Wheel bands: S1002
Rails: UIC60
Gauge: 1.435 m
Rail Slant: 1/40
Theoretical Wave Length:
m 463.14)sin(
)(2 0
rw
rw
b
bbr
Single WheelsetExample
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
Single WheelsetConstitutive
Critical Speed of a Single Wheelset
Linear Creep + Saturation according to Vermeulen & Johnson (1964)
Tangential Force
Maximal Friction Force zF f
F
z
iii
fF
vabGCw
3 Total Creep
Single WheelsetLimit Cycle
Vcr=130 m/s
Limit Cycle Motion at v=131 m/s
Critical Speed of a Single Wheelset
Bicycle DynamicsExample
FEM-model :
2 Wheels, 2 Beams, 6 Hinges
Pure Rolling 3 DOF
Bicycle with Rigid Rider and No-Hands
Standard Dutch Bike
Bicycle DynamicsRoot Loci
Stability of the Forward Upright Steady Motion
Root Loci from the Linearized Equations of Motion. Parameter: forward speed v
Bicycle DynamicsMotion
Full Non-Linear Forward Dynamic Analysis at different speeds
Forward
Speed
v [m/s]:
05
1011
14
18
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.