Railway vehicle dynamics - Université de Monshosting.umons.ac.be/html/mecara/grasmech/RailVehicles.pdf · multibody dynamics (part 2) Railway vehicle dynamics ... Heuristic modeling

1

GraSMech – Multibody – Part II

Computer-aided analysis ofmultibody dynamics (part 2)

Railway vehicle dynamics

Paul Fisette([email protected])


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


Contents


Contact geometry

Contact forces







Other topics


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


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


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


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


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


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


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


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


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


Contents


Contact geometry

Contact forces







Other topics


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


Contact forces

Creepage computation (not MBS approach !)


Lateral creepage :

Spin creepage :


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


Contact forces

Kalker’s theoryLinear theory

*

*

with :

Heuristic modeling of saturation :

Non linear theories : Duvorol, Fastsim (Kalker), Contact (Kalker)


Contents


Contact geometry

Contact forces







Other topics

10



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)



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




y ψ

=> Behavior at low and high velocity





=> Evolution of the hunting mode versus velocity(via successive modal analyses)


stable

unstable Top view (Robotran)

Rear view (Robotran)

12



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


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


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)


Contents


Contact geometry

Contact forces







Other topics

14


The « BAS2000 » bogie : a general case

Six « 3D » kinematic loopsThe « tram2000 »

Four independent wheel/rail contacts

=> No more wheelset, even artificial (left-right)


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)

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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


Contents


Contact geometry

Contact forces







Other topics

16


=> 5 dof + 1 « normal » constraint

The « wheel/rail » joint »

GraSMech – Multibody – Part II 32

Recall : coordiante partitioning reduction

Coordinate partitioning :

ODE !

17


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)


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 !!

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Wheel on rail: Constraints solution

requires

= 0 when the constraints are satisfied


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


Wheel on rail: Constraints 2d derivative

{R}

I12I

3I

O

Q

{X}{ S}{Y}

P

P

G

x

u

w

{T}


Wheel on curved rail

20


Wheel on curved rail


Creepages : a « real » MBS computation

Lateral creepage :


Spin creepage :{R}

I12I

3I

O

Q

{X}{ S}{Y}

P

P

G

x

u

w

{T}

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Wheel on rail: Kalker’s model

{R}

I12I

3I

O

Q

{X}{ S}{Y}

P

P

G

x

u

w

{T}

Kalker (linear) :


Wheel on rail: flange forces (tramways)

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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)


Contents


Contact geometry

Contact forces







Other topics : « towards » multiphysics

23


Geometrical validation

UCL

Delft


Wheelset limit cycle

UCL Delft

x x

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Bogie : « non linear » critical speed

Cycle limiteDeraillement stable

ROBOTRAN results(IAVSD benchmark #2, 1991)


The “BAS 2000” articulated bogie

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« Ping-pong » effect on a straigth track

Yaw deformation

time

“BAS 2000” : straight track behavior


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

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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


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

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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


Contents


Contact geometry

Contact forces







Other topics : « towards » multiphysics

28


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


Multibody and railway pneumatic suspension

=> Next lecture

Multibody system

Pneumatic system

Hybrid system

Documents

Railway vehicle dynamics - Université de Monshosting.umons.ac.be/html/mecara/grasmech/RailVehicles.pdf · multibody dynamics (part 2) Railway vehicle dynamics ... Heuristic modeling