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.