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Numerical determination of the contact(s) point(s) = f (y, ψ)
Pre-computation (look-up tables => f (y, ψ) )
In-line computation (Newton-Raphson, dichotomic method, … see Part II)
Computation of the left/right rolling radii r and contact angles δ :
y y
∆ r ∆ δ
GraSMech – Multibody – Part II
Contact geometry
Double (flange) contact point
Tramway « clear » double contact (tread + flange)
Train the contact point « moves » from tread to flange (with possible jump !!)
Always present
intermittent
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GraSMech – Multibody – Part II
Contact geometry
Double (flange) contact point
Tramway « clear » double contact (tread + flange)
Train the contact point « moves » from tread to flange (with possible jump !!)
Always present
GraSMech – Multibody – Part II
Contact geometry
Contact point « jump »Exists on theoretical « new » profiles (ex. S1002 wheelset on UIC60 rail)
on produces :
Exists also on worn profile (but their location change with wear !)
New rail :
Slightly worn rail :
ROBOTRAN results(IAVSD benchmark #1, 1991)
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GraSMech – Multibody – Part II
Contents
Part I : Wheel-rail contact in railway dynamics
Contact geometry
Contact forces
Wheelset dynamic behavior
Part II : Railway dynamics - multibody approach
Multibody representation
Wheel/rail contact – independent wheel model
Flange contact model
Validations - Applications
Other topics
GraSMech – Multibody – Part II
Contact forces
Coulomb’s law :No slip =>
Slip =>
Creepage (« between slip and pure rolling »)
Longitudinal creepage :
Lateral creepage :
Spin creepage :
with : Point of contact assumption !!!!!
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GraSMech – Multibody – Part II
Contact forces
Creepage computation (not MBS approach !)
Longitudinal creepage :
Lateral creepage :
Spin creepage :
GraSMech – Multibody – Part II
Contact forces
Contact surface
Non uniform pressure distribution
Contact ellipse (a, b) Hertz (but + creepage)
A, B, m, n depend on
contact curvature radii
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GraSMech – Multibody – Part II
Contact forces
Kalker’s theoryLinear theory
*
*
with :
Heuristic modeling of saturation :
Non linear theories : Duvorol, Fastsim (Kalker), Contact (Kalker)
GraSMech – Multibody – Part II
Contents
Part I : Wheel-rail contact in railway dynamics
Contact geometry
Contact forces
Wheelset dynamic behavior
Part II : Railway dynamics - multibody approach
Multibody representation
Wheel/rail contact – independent wheel model
Flange contact model
Validations - Applications
Other topics
10
GraSMech – Multibody – Part II
Wheelset dynamic behavior
Linear model
Hypotheses :
« suspended » wheelset
Linearized equations of motion :
Solution :
=> « Linear » Critical speed VLim:
« Hunting » eigenmode
Eigenvalue λ versus speed :
Real (λ)
Im (λ)
or.: Jashinski thesis (Delft)
GraSMech – Multibody – Part II
Wheelset dynamic behavior
Linear model : example
ROBOTRAN/MBsysLab model :http://www.prm.ucl.ac.be/FDP/Tutorial/Medium/Bogie/Introduction.html
=> Comparison between linear – non linear model
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GraSMech – Multibody – Part II
Wheelset dynamic behavior
Linear model : example
y ψ
=> Behavior at low and high velocity
ROBOTRAN/MBsysLab model :http://www.prm.ucl.ac.be/FDP/Tutorial/Medium/Bogie/Introduction.html
GraSMech – Multibody – Part II
Wheelset dynamic behavior
Linear model : example
=> Evolution of the hunting mode versus velocity(via successive modal analyses)
ROBOTRAN/MBsysLab model :http://www.prm.ucl.ac.be/FDP/Tutorial/Medium/Bogie/Introduction.html
stable
unstable Top view (Robotran)
Rear view (Robotran)
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GraSMech – Multibody – Part II
Wheelset dynamic behavior
Non-linear modelNotion of limit cycle (ex. van der Pol differential equation
Application : limit cycle of a « suspended » wheelset :Lateral wheelset displacement y y y
.
ω0 , ξ > 0
• small y => negative damping the system stores energy• large y => positive damping the system looses energy=> Limit cycle : oscillatory solution in which the 2 energies compensate
« damping term »
ROBOTRAN results ROBOTRAN results
GraSMech – Multibody – Part II
Contact geometry
Critical speed VcrLimit cycle amplitudes versus system velocity (system = wheelset, amplitude = y)
Numerical determination of the critical speed Vcr and VLim
Time simulation from high speed (with lateral « safety bumps ») to low speed
ROBOTRAN results (IAVSD benchmark #2, 1991)
y
speed
speed
y
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GraSMech – Multibody – Part II
Conclusions (part I)
Key points
Contact geometry : a difficult problem to solve
Contact forces : idem thanks ! Mr Kalker (et al.)
Dynamic behavior of a wheeset on a straight track : critical speeds
Other aspects
Curving dynamics : « crab-wise » bogie behavior
Derailment criteria (Y/Q wheel force ratio)
Wear
Track irregularities => noise, comfort, …
Full train : different morphologies, tilting trains, pneumatic suspensions, …
next lecture (N. Docquier)
GraSMech – Multibody – Part II
Contents
Part I : Wheel-rail contact in railway dynamics
Contact geometry
Contact forces
Wheelset dynamic behavior
Part II : Railway dynamics - multibody approach
Multibody representation
Wheel/rail contact – independent wheel model
Flange contact model
Validations - Applications
Other topics
14
GraSMech – Multibody – Part II
The « BAS2000 » bogie : a general case
Six « 3D » kinematic loopsThe « tram2000 »
Four independent wheel/rail contacts
=> No more wheelset, even artificial (left-right)
GraSMech – Multibody – Part II
MBS : wheel or wheeset - does’nt matter!
In relative coordinates, a wheel is a leaf body to which forces apply