Two Special Finite Elements for

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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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