Two Special Finite Elements for by wuyunyi

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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
Contents

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

2D Truss Element

4 Nodal Coordinates:
x  ( x1, y1, x2 , y2 )

3 Degrees of Freedom as a Rigid Body leaves:
1 Generalized Strain: l  l  l0

l    x2  x1 2   y2  y1 2  l0  ε  D(x)
Rigid Body Motion  ε  0            Constraint Equation
Wheel Element                                     Nodes

Generalized Nodes:
w  ( wx , w y , wz )   Position Wheel Centre

q  (q0 , q1, q2 , q3 , ) Euler parameters

Rotation Matrix: R(q)

c  (c x , c y , cz )   Contact Point

In total 10 generalized coordinates

Rigid body pure rolling: 3 degrees of freedom

Impose 7 Constraints
Wheel Element                                 Strains

Holonomic Constraints as zero generalized strains  ε  D( x )  0

Elongation:

1  (r  r  r02 ) /( 2r0 )
Lateral Bending:
 2  ew  r
Contact point on the surface:
 3  g (c )
Wheel perpendicular to the surface

Radius vector: r  c  w                    4  (r  e w )  n
Rotated wheel axle: e w  R(q) ew
Normalization condition on Euler par:
Surface: g ( x )  0
 5  q0  q  q  1
2

Normal on surface: n  g (c )
Wheel Element                               Slips
Non-Holonomic Constraints as zero generalized slips  s  V( x )x  0

Velocity of material point of wheel
at contact in c:

vc  w  ω  r


Generalized Slips:

Longitudinal slip

s1  a  vc
Radius vector: r  c  w                  Lateral slip
Two tangent vectors in c:
s2  b  c

a  (r  e w ) , b  n  (r  e w )
Angular velocity wheel: ω
Wheel-Rail Contact Element
Nodes

Generalized Nodes:
w  ( wx , w y , wz )   Position Wheel Centre

q  (q0 , q1, q2 , q3 , ) Euler parameters

Rotation Matrix: R(q)

c  (c x , c y , cz )   Contact Point

In total 10 generalized coordinates

Rigid body pure rolling: 2 degrees of freedom

Impose 8 Constraints
Wheel-Rail Contact Element
Strains
Holonomic Constraints as zero
generalized strains  ε  D(x )  0

Distance from c to Wheel surface:
1  g w ( r )
Distance from c to Rail surface:

 2  g r (c )
Wheel and Rail in Point Contact:

 3  nw  a r
 4  nw  b r
Wheel & Rail surface: g w ( x )  0 , g r (x)  0
Normalization condition on Euler par:
Local radius vector: r  RT (c  w)
Normal on Wheel surface: nw  Rg w                           5  q0  q  q  1
2

Two Tangents in c: a r , br
Wheel-Rail Contact Element
Slips
Non-Holonomic Constraints as zero
generalized slips  s  V(x )x  0


Velocity of material point of Wheel in
contact point c:
vc  w  ωw  (c  w)


Generalized Slips:
Longitudinal slip:

s1  a r  vc
Lateral slip:

Wheel & Rail surface: g w ( x )  0 , g r (x)  0                  s2  br  v c
Two Tangents in c: a r , br                                Spin:
Normal on Rail Surface: nr  g r                                  s3  nr  ωw
Angular velocity wheel: ωw
Single Wheelset
Example

Klingel Motion of a Wheelset

Wheel bands: S1002
Rails: UIC60
FEM-model :
Gauge: 1.435 m
2 Wheel-Rail, 2 Beams, 3 Hinges
Rail Slant: 1/40
Pure Rolling, Released Spin    1 DOF
Single Wheelset
Profiles

Wheel band S1002             Rail profile UIC60
Single Wheelset
Motion

Klingel Motion of a Wheelset

Wheel bands: S1002
Rails: UIC60
Theoretical Wave Length:
Gauge: 1.435 m
br0 (  w   r )          b
Rail Slant: 1/40             2                                           14.463 m
        w         (b   r sin  )
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    FEM-model :
Vertical Load 173 226 N            2 Wheel-Rail, 2 Beams, 3 Hinges
Yaw Spring Stiffness 816 kNm/rad   Linear Creep + Saturation    4 DOF
Single Wheelset
Constitutive

Critical Speed of a Single Wheelset
Linear Creep + Saturation according to Vermeulen & Johnson (1964)

F        Tangential Force                abGCii vi
w                Total Creep
f Fz      Maximal Friction Force            3 fFz
Single Wheelset
Limit Cycle
Limit Cycle Motion at v=131 m/s

Critical Speed of a Single Wheelset

Vcr=130 m/s
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
Bicycle Dynamics
Root Loci
Stability of the Forward Upright Steady Motion

Root Loci from the Linearized Equations of Motion.   Parameter: forward speed v
Bicycle Dynamics
Motion
Full Non-Linear Forward Dynamic Analysis at different speeds

Forward
Speed
v [m/s]:

18
14
11
10
5
0
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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