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1
GraSMech – Multibody – Part II
Computer-aided analysis ofmultibody dynamics (part 2)
Railway vehicle dynamics
Paul Fisette([email protected])
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
Kluwer Academic Publishers, 2003
Chapter 9
2
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 geometry
Wheelset with conical wheel on « knife-edge » rails
Rolling radii :
Equivalent conicity :
Rolling radii difference (left-right) :
Roll angle :
( In practice : δ0 ~ 1/20e … 1/40e )
3
GraSMech – Multibody – Part II
Contact geometry
Wheelset kinematic motion (~ first order kinematics)
Hypothesis : pure rolling motion (no slip)
Kinematic relations (first order)
Constraints (in y and ψ) :
Lateral (y) :
Yaw (ψ) :
Motion of a wheelset initially centered on the track
Combined lateral-yaw motion of a wheelset
GraSMech – Multibody – Part II
Contact geometry
2 d.o.f. kinematic equations (= constraints at « velocity level »)
Periodic Solution :
frequency :
Wheelset kinematic yaw (under pure rolling assumption)
Wheelset kinematic motion (next)
Wave length :(Klingel formula, 1883 !)
4
GraSMech – Multibody – Part II
Contact geometry
Wave length :
Bogie : yaw kinematic motion (similar reasoning)
l : half wheelset base
l0 : half wheelset length
Wheel/rail with circular profiles (as a matter of interest)
More complex kinematics
Roll angle :
vertical disp.:
equiv. conicity : 4 bar-mechanism
GraSMech – Multibody – Part II
Contact geometry
=> Longitudinal shift of the contact point ( « shift angle » θ) due to wheelset yaw + wheel conicity
Wheel/rail contact : yaw angle influence on contact geometry
Insignificant for the tread contact point (angle ~ 0.05 rad)
Fundamental for the flange contact point (angle > 1 rad) : see latter on
θ
5
GraSMech – Multibody – Part II
Contact geometry
Real wheel profiles : numerical somutionProfile definition (analytical form or numerical measurements)
ex. UIC 60 rail (piecewise circular), S1002 wheel (piecewise polynomials)
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
6
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)
7
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 !!!!!
8
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
9
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
11
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)
12
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
13
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
Independent wheels Wheelset = (axle + 2 « locked » indep. wheels)
15
GraSMech – Multibody – Part II
The « BAS2000 » bogie : a general case
A wheel = a leaf body which is constrained on a rail (inertial body)
The BAS2000 cannot be modeled as a bogie
… but as a mechanism with wheels ! => relative coordinates approach suitable
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
16
GraSMech – Multibody – Part II
=> 5 dof + 1 « normal » constraint
The « wheel/rail » joint »
GraSMech – Multibody – Part II 32
Recall : coordiante partitioning reduction
Coordinate partitioning :
ODE !
17
GraSMech – Multibody – Part II
A Single Wheel on a Rail : 5 d.o.f.
Q
i
h
k
G
θ
xj
_
w
ρ (w)
...
jj
3-D Contact Constraints :
1. Point Q belongs to Rail Surface
2. Wheel / Rail Surfaces Tangent at Point Q
=> Lagrange Multipliers Technique
2 Auxiliary variables :w : lateral position
θ : Shift angleθ
w
β(w)
GraSMech – Multibody – Part II
Robust solution of contact constraints
A dichotomic method “embeded into” the Newton-Raphson algorithm
Wheel on rail: Constraints solution
rail profile
λ2 = 0 !!
λ3 = 0 !!
18
GraSMech – Multibody – Part II
Wheel on rail: Constraints solution
requires
= 0 when the constraints are satisfied
GraSMech – Multibody – Part II
Wheel on rail: Constraints derivative
{R}
I12I
3I
O
Q
{X}{ S}{Y}
P
P
G
x
u
w
{T}
α
cos α = + π/2_
h(q) satisfied
λ1 = FN
19
GraSMech – Multibody – Part II
Wheel on rail: Constraints 2d derivative
{R}
I12I
3I
O
Q
{X}{ S}{Y}
P
P
G
x
u
w
{T}
GraSMech – Multibody – Part II
Wheel on curved rail
20
GraSMech – Multibody – Part II
Wheel on curved rail
GraSMech – Multibody – Part II
Creepages : a « real » MBS computation
Lateral creepage :
Longitudinal creepage :
Spin creepage :{R}
I12I
3I
O
Q
{X}{ S}{Y}
P
P
G
x
u
w
{T}
21
GraSMech – Multibody – Part II
Wheel on rail: Kalker’s model
{R}
I12I
3I
O
Q
{X}{ S}{Y}
P
P
G
x
u
w
{T}
Kalker (linear) :
GraSMech – Multibody – Part II
Wheel on rail: flange forces (tramways)
22
GraSMech – Multibody – Part II
Wheel on rail: flange forces
Influence of the shift angle
No wheel yaw With wheel yaw
For the flange contact force, in terms of tangent slip, the tread contact (« creepage ») can be seen as a pure rolling motion (~ICR)
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 : « towards » multiphysics
23
GraSMech – Multibody – Part II
Geometrical validation
UCL
Delft
GraSMech – Multibody – Part II
Wheelset limit cycle
UCL Delft
x x
24
GraSMech – Multibody – Part II
Bogie : « non linear » critical speed
Cycle limiteDeraillement stable
ROBOTRAN results(IAVSD benchmark #2, 1991)
GraSMech – Multibody – Part II
The “BAS 2000” articulated bogie
25
GraSMech – Multibody – Part II
« Ping-pong » effect on a straigth track
Yaw deformation
time
“BAS 2000” : straight track behavior
GraSMech – Multibody – Part II
The “Tram 2000” tramway
The "Tram 2000"
Model :Complete TRAM 2000 in Entry Curving :
- 4 Sub-Systems- 74 Variables- 32 Constraints- 42 Degrees of Freedom
Carbody A
Front BAS 2000 Bogie
Carbody C Carbody B
Conventionnal Bogie Back BAS 2000 BogieV
Velocity : 25 km/hInitial Conditions : Tramway Centered on the Straight Track
26
GraSMech – Multibody – Part II
Result :
Wheel / Rail Alignment in a Low Radius Curve (R = 50 m)
α
V
wheel
right rail
Wheel / Rail Angle of Attack
“Tram 2000” tramway - resultsCarbody A
Front BAS 2000 Bogie
Carbody C Carbody B
Conventionnal Bogie Back BAS 2000 BogieV
GraSMech – Multibody – Part II
Comparison of bogie behaviors
BW : Rigid bogie with wheeslets
BI : Rigid bogie with independent wheels
BA : BA2000 articulated « bogie »
Bogies :
Situation : entry curving
BW
BI
BA
27
GraSMech – Multibody – Part II
Comparison of bogie behaviors
Steady state equilibrium : tread and flange lateral forces
- « Crab-wise »behavior
- « Crab-wise »behaviorBW
BI
BA
- Low longitudinal tread forces
- Distorted, no crab-wise- Dissipated power minimized
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 : « towards » multiphysics
28
GraSMech – Multibody – Part II
Railway bogies actuated by threephase induction motors
Problem:Torque oscillations lead to fatigue stress in the chassis
Fz
Fz
Cm
Cm
• Fz in the -motor to chassis-joint
• electromechanicaltorque Cm
Multibody and railway “electro” dynamics
=> Next lecture
GraSMech – Multibody – Part II
Multibody and railway pneumatic suspension
=> Next lecture
Multibody system
Pneumatic system
Hybrid system