Transcript Two Special Finite Elements for Modelling Modelling of

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  ( w x , 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
Rotated wheel axle: e w  R(q) ew
Surface: g ( x )  0
Normal on surface: n  g (c )
 4  (r  e w )  n
Normalization condition on Euler par:
 5  q02  q  q  1
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:
  ωr
vc  w
Generalized Slips:
Longitudinal slip
s1  a  vc
Radius vector: r  c  w
Two tangent vectors in c:
a  (r  e w ) , b  n  (r  e w )
Angular velocity wheel: ω
Lateral slip
s2  b  c
Wheel-Rail Contact Element
Nodes
Generalized Nodes:
w  ( w x , 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
Wheel & Rail surface: g w ( x )  0 , g r (x )  0
Local radius vector: r  RT (c  w)
Normal on Wheel surface: nw  Rg w
Two Tangents in c: a r , br
 4  nw  b r
Normalization condition on Euler par:
 5  q02  q  q  1
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:
  ωw  (c  w)
vc  w
Generalized Slips:
Longitudinal slip:
s1  a r  vc
Lateral slip:
Wheel & Rail surface: g w ( x )  0 , g r (x )  0
Two Tangents in c: a r , br
Normal on Rail Surface: nr  g r
Angular velocity wheel: ωw
s2  br  v c
Spin:
s3  nr  ωw
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
Single Wheelset
Wheel band S1002
Rail profile UIC60
Profiles
Single Wheelset
Motion
Klingel Motion of a Wheelset
Wheel bands: S1002
Rails: UIC60
Theoretical Wave Length:
Gauge: 1.435 m
Rail Slant: 1/40
  2
br0 (  w   r )

w
b
(b   r sin  )
 14.463 m
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
f Fz
Tangential Force
Maximal Friction Force
w 
abGCii vi
3 fFz
Total Creep
Single Wheelset
Limit Cycle
Limit Cycle Motion at v=131 m/s
Critical Speed of a Single Wheelset
Vcr=130 m/s
Bicycle Dynamics
Bicycle with Rigid Rider and No-Hands
Standard Dutch Bike
FEM-model :
2 Wheels, 2 Beams, 6 Hinges
Pure Rolling
 3 DOF
Example
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.