2004 Paper ACMD JPombo JAmbrosio

7/28/2019 2004 Paper ACMD JPombo JAmbrosio

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Copyright 2004 by KSME

A COMPUTATIONAL EFFICIENT GENERAL WHEEL-RAIL CONTACT DETECTION METHOD

Joo PomboIDMEC Instituto Superior TcnicoAv. Rovisco Pais 1

1049-001 Lisboa, PORTUGALEmail: [email protected]

Jorge AmbrsioIDMEC Instituto Superior TcnicoAv. Rovisco Pais 1

1049-001 Lisboa, PORTUGALEmail: [email protected]

ABSTRACTThe development and implementation of an appropriate

methodology for the accurate geometric description of trackmodels is proposed in the framework of multibody dynamicsand it includes the representation of the track spatialgeometry and its irregularities. The wheel and rail surfacesare parameterized to represent any wheel and rail profilesobtained from direct measurements or design requirements.A fully generic methodology to determine, online during thedynamic simulation, the coordinates of the contact points,even when the most general three dimensional motion of thewheelset with respect to the rails is proposed. Thismethodology is applied to study specific issues in railwaydynamics such as the flange contact problem and lead andlag contact configurations. A formulation for the descriptionof the normal contact forces, which result from the wheel-railinteraction, is also presented. The tangential creep forces andmoments that develop in the wheel-rail contact area are

evaluated using: Kalker linear theory; Heuristic forcemethod; Polach formulation. The methodology isimplemented in a general multibody code. The discussion issupported through the application of the methodology to therailway vehicle ML95, used by the Lisbon metro company.

1. INTRODUCTIONIn railway vehicle dynamics, the wheel-rail interaction

plays a crucial role since the railway vehicle is guided bythe forces generated by such contact. The problems toconsider when studying the wheel-rail contact are:

a) The contact geometry, i.e., the problem of determiningthe location of the contact point on the profiled surfacesusing geometric contact constraints.

b) The contact kinematics, i.e., the problem of defining thecreepages at the point of contact.

c) The contact mechanics, i.e., the problem of determiningthe tangential creep forces and the spin creep moment.

Several authors (Kalker, 1990; Polach, 1999; Kik, 1996)studied the contact forces between the wheel and the railmaking available several computer routines for thecalculations of the tangential forces at the contact pointgiven the normal force and the relative velocities between

the contacting bodies (Kalker, 1990; Polach, 1999). Theproblem here is reduced to provide descriptions of thesurfaces in contact and of the kinematics of the bodies.

The track centerline geometry may be described bydifferent types of parametric curves such as cubic, Akima orshape preserving splines. The track description adopted usesFrenet frames that provide the appropriate referential atevery point and the definition of the cant angle variationalong the railway. A pre-processor is used to define thenominal geometry of both left and right rails based on theinterpolation of a discrete number of points, which arerepresentative of their space curves (Pombo, 2003a,b). Forthe complete representation of the track geometry, both railsare considered separate geometric entities. For efficiency, apre-processor generates a table with all track position dataand other quantities required for the multibody code asfunction of the left and right rail lengths. During a dynamicsimulation the program interpolates linearly both rails

databases to obtain the information necessary to find thewheel/rail interaction. The coordinates of the contact pointsare evaluated during the dynamic analysis by introducingsurface parameters that describe the geometry of the rail andwheel contact surfaces, each described by two surfaceparameters (Pombo, 2003b). The methodology allows theexistence of two or more points of simultaneous contactbetween the wheel and the rail in the wheel tread or flange.

The normal contact forces that develop in the wheel-railinterface are calculated using the Hertz contact force modelwith hysteresis damping to account for the dissipation ofenergy during contact (Lankarani, 1994). The creepages, or

normalized relative velocities at the contact point, are usedwith the normal contact force to determine the creep forcesand the spin creep moment. Three different methodologiesare implemented in order to calculate these tangential contactforces. These are the Kalker linear theory (Kalker,1979,1990), the Heuristic nonlinear creep force model (Shen,1983) and the Polach formulation (Polach, 1999).

The methodologies proposed here are implemented ina general multibody code that is used for the dynamicanalysis of rail-guided vehicles. Finally, the computer codeis applied to the study of a railway vehicle in what itsdynamics and stability is concerned.


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2. PARAMETERIZATION OF RAILWAY TRACK

A pre-processor program defines the track model as twoparameterized curves that represent the nominal geometry ofthe left and right rails space curves. Parametric track

descriptions using different types of splines are available(Pombo, 2003c). The information is organized in twodatabases where all quantities, necessary to define the railscurves, are function of the arc length of each rail, measuredfrom their origin. The methodology is summarized as:a) The geometry of the track centerline is parameterized

using a piecewise cubic interpolation scheme.b) The track cant angle is parameterized as function of the

track length.c) The track centerline is also parameterized as function of

the track length (Pombo, 2003c).d) The track irregularities, measured experimentally, are

parameterized as continuum functions of the track lengthe) Define a set of control points that are representative of

the left and right rails space curves.f) Parameterize the rails space curves as a function of the arc

lengths. Account for the track cant angle and rail inclination.g) Create a database for each rail, stored with a small track

length step.A schematic representation of the methodology used

in the railway pre-processor is presented in Figure 1. Theinterested reader is referred to the work of Pombo andAmbrsio (2003a,b).

InputData to Parameterize theTrackCenterline withP iecewiseCubic Interpolation

x y z

User

InputData to Parameterize theTrackIrregularities

L ALLr ALRr LLLr LLRr h G

Parameterization

CubicSplinesParameterizationAkimaSplines

ParameterizationShape Preserve

ParameterizationCubicSplines

TrackCenterline ParameterizedasFunctionof theTrackLength

TrackIrregularitiesP arameterizedas Functionof the TrackLength

Obtainthe Nodal Points thatDefine the

SpaceCurves for the Leftand RightRails


ParameterizationAkimaSplines


Leftand RightRailsSpaceCurves

Parameterizedas Functionof theArc Lengths

TrackCantAngleParameterized asFunctionof theTrackLength

RIGHT RAIL DATABASE

L x y z tx ty tz nx ny nz bx by bz.. .. .. .. .. .. .. .. .. .. .. .. .... .. .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. .. .. .. ..

LEFT RAIL DATABASE


.. .. .. .. .. .. .. .. .. .. .. .. ..

InputData to Parameterize theTrackCenterline withP iecewiseCubic Interpolation

x y z

UserUser

InputData to Parameterize theTrackIrregularities

L ALLr ALRr LLLr LLRr h G

Parameterization

CubicSplinesParameterizationAkimaSplines



TrackCenterline ParameterizedasFunctionof theTrackLength

TrackIrregularitiesP arameterizedas Functionof the TrackLength

Obtainthe Nodal Points thatDefine the

SpaceCurves for the Leftand RightRails


ParameterizationAkimaSplines


Leftand RightRailsSpaceCurves

Parameterizedas Functionof theArc Lengths

TrackCantAngleParameterized asFunctionof theTrackLength

RIGHT RAIL DATABASE


.. .. .. .. .. .. .. .. .. .. .. .. ..

LEFT RAIL DATABASE


.. .. .. .. .. .. .. .. .. .. .. .. ..

Figure 1. FLOWCHART OF THE RAILWAY PRE-PROCESSOR.

3. WHEEL AND RAIL SURFACESThe definition of the wheel and rail needs to satisfy

three main requirements. First, the surfaces have to bedefined in a global coordinate system. Second, theparametric equations must be able to represent any spatial

configuration of the wheelsets and rails. Third, therepresentation of any wheel and rail profiles, obtained bymeasurements or design requirements, must be possible.

Let two sets of independent surface parameters beused to define the geometry of each of the wheel and rail incontact be Sr and ur, for the rail surface geometry, and Swand uw, for the wheel surface, as shown in Figure 2. Theposition vector of a contact point Q, in the wheel or railbody fixed coordinates systems, is function of the surfaceparameters only

where the subscripts (.)r and (.)w are referred to the rail andto the wheel respectively whereas the subscript (.)ws arereferred to the wheelset.

In order to account for any possible scenarios, such asa variation in the gauge or relative displacements and/orrotations of the rails due to the track irregularities, it isnecessary to define the surface of each rail independently,as depicted by Figure 3, where subscripts (.)Lr and (.)Rr arereferred to the left and right rails respectively.

Q

( )w wf u

w

r

r

ru

( )r rf u

wu

w

rs Q

ws

ws

ws

w

w

Q

( )w wf u

w

r

r

ru

( )r rf u

wu

w

Q

( )w wf u

w

r

r

ru

( )r rf u

wu

w

rs Q

ws

ws

ws

w

w

rs Q

ws

ws

ws

w

w

Figure 2. WHEEL AND RAIL SURFACE PARAMETERS.

( ), ; ,l l l ls u l r w= =u u (1)


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x

P

z

y

Track centerlineRight rail

space curveLeft rail

space curve

Q

Rr

Rr

Rr

rRrG

rPG

Rrs

Lr

Lr

Lr

Lrs

Rr

P

Rr

sRRrG

( )r Rrf uRrs

nG

RrstG

Rr

Rru

r

Nodal Points

r ru

x

P

z

y



space curve

Q

Rr

Rr

Rr

rRrG

rPG

Rrs

Lr

Lr

Lr

Lrs

x

P

z

y



space curve

Q

Rr

Rr

Rr

rRrG

rPG

Rrs

Lr

Lr

Lr

Lrs

P

z

y



space curve

Q

Rr

Rr

Rr

rRrG

rPG

Rrs

Lr

Lr

Lr

Lrs

Rr

P

Rr

sRRrG

( )r Rrf uRrs

nG

RrstG

Rr

Rru

r

Nodal Points

r ru

Rr

P

Rr

sRRrG

( )r Rrf uRrs

nG

RrstG

Rr

Rru

r

Nodal Points

r ru

Figure 3. PARAMETRIZATION OF THE RAIL SURFACE.

Let a profile coordinate system be defined on each railto identify the position and orientation of any cross sectionalong the rail space curve and P be a point of contact withthe wheel. The right profile coordinate system (Rr,Rr,Rr),shown in Figure 3, translates along a rail space curve androtates about its origin. The location of the profilecoordinate system along the space curve can be defined in

such a way that the contact point P lies in its (Rr,Rr) plane.The location of the origin and the orientation of the rightrail profile coordinate system, defined respectively by thevectorrRr and the transformation matrix ARr, are uniquelydetermined using the surface parameterSr (Berzeri, 2000).The location of the contact point P on the rail surface is

+ 'P PRr Rr Rr=r r A s (2)

where 'PRrs is the position vector that defines the location

of the contact point P on the profile coordinate system. Thetransformation matrix ARr is a function of the unit vectorsthat define the moving reference frame associated to the

right rail space curve. In railway applications, the functionfrthat defines the rail profile is a function of the surfaceparameterur using a piecewise cubic interpolation scheme(De Boor, 1978). Hence, to obtainfr(ur), the user only has todefine a set of control points that are representative of the railprofile geometry, as shown in Figure 3.

The detection of the location of the contact pointsbetween two parametric surfaces requires the definition ofthe normal vector to the rail surface nrs at the point of contact

'Rrs Rr Rrs=n A n (3)

ws

z

y xQ

Left wheel

ws

ws

rwsGsQLw

G

rQG

Lw

Lw

Lw

hLw

G

Lwu

Lws ws

z

y xQ

Left wheel

ws

ws

rwsGsQLw

G

rQG

Lw

Lw

Lw

hLw

G

Lwu

Lws

Figure 4.PARAMETRIZATION OF THE WHEEL SURFACE

where nRrs={0, cosRrs, sinRrs}T is the unit vector normal to

the rail surface, defined in the profile coordinate system.This vector is obtained through the contact angle Rrs,shown in Figure 3. The contact angle is

( )1tg r RrRrsRr

df u

du

=

(4)

The surface of revolution of each wheel is generatedby a complete rotation, about the wheel axis, of the two-dimensional curve that defines the wheel profile (Shabana,2001). Figure 4 shows the left wheel with arbitrary surface

profile assembled in a wheelset. The surface geometry ofthe wheel is described using the two surface parameters Swand uw that represent the rotation of the wheel profilecoordinate system (w,w,w) with respect to the wheelsetcoordinate system (ws,ws,ws), and the lateral position ofthe contact point in the wheel profile coordinate system.The location of the origin and the orientation of thewheelset frame are defined by vector rws and matrix Aws.The global position of an arbitrary point on the wheel is

where { }120 0T

LwH=h is the local position vector

that defines the location of the profile coordinate systemwith respect to the wheelset reference frame, being H thelateral distance to the wheel profile origin. 'Q

Lws and 'P

Rws

are the local position vectors that define the location of thecontact points Q on the wheel surfaces with respect to theprofiles coordinate systems, i.e.

( )+ + 'Q QLw ws ws Lw Lw Lwr r A h A s (5)

( ){ }' 0TQ

Lw Lw w Lwu f u=s (6)


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Concave

Region

w

w

w

Flange Nodal

PointsTread Nodal

Pointsw

u

( )fw wf u( )t

w wf u

w

Left

wheel

Rw

Rw

Rw

Rwu

2t

LwtG

t

Lw ( )

f

w Lwf u

( )tw Lwf u

1t

LwtG

2f

LwtG

fLw

1f

LwtG

Concave

Region

w

w

Concave

Region

w

w

w

Flange Nodal

PointsTread Nodal

Pointsw

u

( )fw wf u( )t

w wf u

w

w

Flange Nodal

PointsTread Nodal

Pointsw

u

( )fw wf u( )t

w wf u

w

Left

wheel

Rw

Rw

Rw

Rwu

2t

LwtG

t

Lw ( )

f

w Lwf u

( )tw Lwf u

1t

LwtG

2f

LwtG

fLw

1f

LwtG

Left

wheel

Rw

Rw

Rw

Rwu

2t

LwtG

t

Lw ( )

f

w Lwf u

( )tw Lwf u

1t

LwtG

2f

LwtG

fLw

1f

LwtG

Figure 5. WHEEL PROFILE AND PARAMETRIZATION

To use the multibody contact model to solve theproblem of wheel-rail contact it is necessary to devise astrategy to determine the location of the contact pointsbetween two parametric surfaces. This formulation requiresthat the parametric surfaces are convex. Therefore, thewheel profile is represented by two independent functions

t

wf and f

wf that parameterize the wheel tread and flange,

respectively. The search for the location of the contact

points requires the definition of two tangent vectors to thewheel surface, tw1 and tw2, at the point of contact

where 1 {1 0 0}l T

Lw =t and 2 {0 cos sin }

l l l T

Lw Lw Lw =t . The

quantities with superscripts (.)t and (.)f are referred to thewheel tread and flange respectively. The contact angle is

4. WHEEL-RAIL CONTACT MODELThe location of the contact points between the wheel and

the rail is complicated since both are profiled. Furthermore, thelarge amount of parameters that include the shape of thesurfaces in contact, relative contact velocities, contact forces,and physical properties of the materials, unavoidably lead tocomplex theories to find the contact forces.

t sjG

x

z

y

(j)

(i)

t tjG

t wiG

dG

n jG

ni

G

( )p ,u wG

( )q ,s tG

tu

i

G

t sjG

x

z

y

(j)

(i)

t tjG

t wiG

dG

n jG

ni

G

( )p ,u wG

( )q ,s tG

tu

i

G

Figure 6. CONTACT POINTS ON TWO SURFACES

4.1 Wheel-Rail Contact DetectionConsider two generic surfaces i and j depicted in

Figure 6 defined by the parametric functions p(u,w) andq(s,t), respectively. The minimum distance between the twopatches p(u,w) and q(s,t) is given by

( ) ( ), ,u w s t = d p q (9)

The tangent vectors ui

t , wi

t , sjt andt

jt to theparametric surfaces, shown in Figure 6, are defined as:

( ) ( ) ( ) ( ), , , ,; ; ;u w s t i i j j

u w u w s t s t

u w s t

= = = =

p p q qt t t t (10)

The normal unit vectors to the parametric surfaces,are

;s tu wj ji i

i ju w s t

i i j j

= =t tt t

n nt t t t

(11)

For the wheel-rail contact problem the equations thatdefine the candidates to contact points, represented insituations Figure 7 (a), (b) and (c)II, arei) The surfaces normals ni and nj at the candidates to

contact points have to be parallel:

0

0

T u

j i

j i T wj i

= =

=

n t

n n 0 n t (12)

ii) The vectord has to be parallel to the normal vectorni:

iii) The contact condition specifies that:

1 1' ; ,l l

Lw ws Lw Lwl t f= =t A A t

2 2' ; ,l l

Lw ws Lw Lwl t f= =t A A t

(7)

( )1tg ; ,l l

w Lwl

Lw l

Lw

df ul t f

du

= =

(8)

0

0

T u

i

i T w

i

= =

=

d td n 0

d t (13)

0Tj d n (14)


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(j)

(i)njG

niGd = 0

G G

(j)

I II III

(i)

d = 0G G

njG

niG

d = 0G G

dG

njG nj

G

niG

niG

njG ni

G

(j)

(i) dG

(j)

(i)njG

niGd = 0

G G

(j)

(i)

(j)

(i)njG

niGd = 0

G G

(j)

I II III

(i)

d = 0G G

njG

niG

d = 0G G

dG

njG nj

G

niG

niG

(j)

I II IIII II III

(i)

d = 0G G

njG

niG

d = 0G G

dG

njG nj

G

niG

niG

njG ni

G

(j)

(i) dG

njG ni

G

(j)

(i) dGdG

Figure 7. A) NO CONTACT; B) CONTACT AT A SINGLE POINT; C)

CONTACT WITH PENETRATION

The geometric conditions in equations (12) and (13) are

four nonlinear equations with four unknowns, which are thesurfaces parameters u, w, s and t. In the computationalimplementation, the information of a previous time step isused as initial estimate for the solution search of the equations.

4.2 The Two Point Contact ScenarioMore than one pair of contact points can develop

between the wheel and the rail wheel, as shown in Figure 8.Let two different functions t

wf andf

wf parameterize, thewheel tread and flange, respectively, as shown in Figure 5.The formulation used to look for the candidates to contactpoints is fully independent for the wheel tread and for thewheel flange surfaces.

Flange contact(Lead contact)

Treadcontact


Treadcontact

t

Rrs

f

Rrs

f

Rws

0tRw

s =

f

Rw

tRw ws

f

Rw

t

Rw ws


Treadcontact


Treadcontact


Treadcontact

t

Rrs

f

Rrs

f

Rws

0tRw

s =

f

Rw

tRw ws

f

Rw

t

Rw ws


Treadcontact

t

Rrs

f

Rrs

f

Rws

0tRw

s =

f

Rw

tRw ws

f

Rw

t

Rw ws

t

Rrs

f

Rrs

f

Rws

0tRw

s =

f

Rw

tRw ws

f

Rw

t

Rw ws

Figure 8. LEAD AND LAG CONTACT IN WHEEL-RAIL

This methodology allows finding multiple wheel-railcontact points locations and to study the lead and lag contactconfigurations. Since the methodology used to look forcandidates to contact points is fully independent for thewheel tread and for the wheel flange, the contact point in the

flange does not have to be located in the same plane as thecontact point in the wheel tread, as shown in Figure 8. Thisallows the analysis of derailment or of the effect of switches.

4.3 Normal Contact Forces in the Wheel-RailThe normal contact forces are calculated using the

contact force model proposed by Lankarani and Nikravesh(Lankarani, 1995), which requires the amount of penetration,the relative velocity between the contact point and thematerial properties of the wheel and rail. The direction of thenormal forces is determined from the wheel and rail profile

data. The evaluation of the normal contact force is

( )2 n( )

3 1= 1+

4

eN K

(15)

where is the indentation, n=1.5 is the parameter used formetal to metal contact, K is the Hertzian constant thatdepends on the surface curvatures and the elastic propertiesof contacting bodies, e is the coefficient of restitution, isthe velocity of indentation and ( ) is the velocity ofindentation at the initial instant of contact, both evaluated as

the projection of the relative velocity vector of the point ofcontact on the vector normal to the contact surfaces.

4.4 Tangential Contact ForcesKnowing the normal contact forces that develop

between the wheel and rail and the creepages, i.e., therelative velocities, it is possible to calculate the tangentialcontact forces using one of the models available in theliterature. Three models are presented here in order to allowfor a comparative study between them to be developed.

The Kalker Linear evaluates the longitudinal F andlateral F components of the creep force and the spin creep

momentM, that develop in the wheel-rail contact region as

11

22 23

23 33

0 000

cF

F G ab c ab c

M ab c ab c

=

(16)

where G is the combined shear modulus of rigidity of wheeland rail materials and a and b are the semi-axes of thecontact ellipse. The parameters cij are the Kalker creepageand spin coefficients, obtained in references (Kalker, 1990;Garg, 1984). The quantities , and represent the


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longitudinal, lateral and spin creepages at the contact point,respectively. For sufficiently small values of creep and spin,the linear theory of Kalker is adequate to determine the creepforces. For larger values, this formulation is no moreappropriated since it does not include the saturation effect of

the friction forces, i.e., it does not assure that F N .The Heuristic Nonlinear Force Model involves

calculating the creep force expected from the Kalker lineartheory and modifying it by a factor that takes into accountthe limiting creep force (Shen, 1983). First, the resultantcreep force of Kalker linear theory is calculated

2 2F F F = + (17)

where the notation (.) now means that the quantities areobtained with the Kalkers linear theory. The limiting

resultant creep force is determined by:2 3

1 1if 3

3 27

1 if 3

F F FF N

F N N N N

F N

+ = >

(18)

where is the friction coefficient. The new resultant creepforce F is used to calculate the tangential forces as:

;F F

F F F F F F

= =

(19)

In the Heuristic method the spin creep moment M isneglected. This theory gives more realistic values for creepforces outside the linear range than the Kalkers lineartheory. For high values of spin, the Heuristic theory canlead to unsatisfactory results (Andersson, 1998).

In the Polach Nonlinear Force Model the longitudinaland lateral components of the creep force are (Polach, 1999)

; SC C C

F F F F F

= = + (20)

where F is the tangential contact force caused bylongitudinal and lateral creepages, C is the modified

translational creepage, which accounts the effect of spincreepage, and FS is the lateral tangential force caused byspin creepage. The Polach algorithm takes, as input, thecreepages , and , the normal contact force N, thesemi-axes a and b of the contact ellipse, the combinedmodulus of rigidity of wheel and rail materials G, thefriction coefficient and the Kalker creepage and spincoefficients cij. The Polach algorithm is suitable to studythe tangential contact forces that develop in the wheel-railinterface. This method allows the calculation of fullnonlinear creep forces and takes spin into account.

5. APPLICATION TO A RAILWAY VEHICLEThe ML95 trainset, shown in Figure 9, is used by the

Lisbon metro company (ML). The ML95 trainset is anelectrical three-car unit composed of two powered endvehicles with driving cabs, and a intermediate vehicle,

represented in Figure 10.

Figure 9. THE ML95 TRAINSET.

Each ML95 trailer vehicle is composed of one carbodywhere the passengers travel, supported by two bogiesthrough the secondary suspension, which is set to minimizethe vibrations induced by the track on the passengercompartment. The bogies are the subsystems that, throughthe wheelsets, are in contact with the track and include theprimary suspension, which is the main responsible for thesteering capabilities and stability behavior.

Trailer Motor

Figure 10. SCHEMATIC REPRESENTATION OF THE ML95.

Each trailer bogie of the ML95 consists of one frame,two wheelsets, four axleboxes and the mechanical elementsthat compose the primary suspension. The bogie frame issupported by the axleboxes through eight metal-rubbersprings of the Chevron type. The vertical displacementsof the primary suspension are limited by bumpstops andliftstops, shown in Figure 11.

Axlebox

ChevronspringsLiftstop

Bumpstop

WheelsetBogieframe

Axlebox

ChevronspringsLiftstop

Bumpstop

WheelsetBogieframe

Figure 11. PRIMARY SUSPENSION OF THE TRAILER BOGIE.


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Airspring

Vertical damperLiftstop Bogie frame

Carbody Airspring

Vertical damperLiftstop Bogie frame

Carbody

Figure 12. SECONDARY SUSPENSION OF THE TRAILER BOGIE.

The trailer vehicle carbody is supported by fourairsprings, and with each one there is a vertical Chevronbumpstop assembled in series. In parallel with the airsprings,four vertical hydraulic dampers and four vertical liftstopsdevices, shown in Figure 12, are mounted. The connectionbetween the carbody and each bogie is done by a shaft,

which guarantees an appropriate and stable rotation of thebogie with respect to the carbody. The mechanical elementsthat assure the connection between each bogie and thecarbody are mounted between a center plate and the bogieframe, as shown in Figure 13.

Centerplate

Transversalbumpstop

Tractionrod

Transversaldamper

Pivot shaftsupport

Bogieframe

Centerplate

Transversalbumpstop

Tractionrod

Transversaldamper

Pivot shaftsupport

Bogieframe

Figure 13. BOGIE-CARBODY CONNECTION OF THE VEHICLE.

The transmission of traction and braking effortsbetween each bogie and the carbody is done by traction rods.The lateral stabilization of the carbody needs two transversalhydraulic dampers, between the center plate and bogie frame.The characteristics of the ML95 trainset are shown in Table 1.

The model of the railway vehicle leads to representationof the multibody model shown in Figure 14. The mass andinertia properties of system components have been suppliedby the manufacturer company. For bodies with no dataavailable, the mass and inertia properties are estimated based

on their geometry. The location and type of kinematic jointsis also obtained with the manufacturers information.

The first simulation scenario used to apply themethodology developed corresponds to a straight track withno irregularities in which a vehicle with new wheels runs.The starting position of the vehicle is such that the model ismisaligned by 2 mm relative from the track centerline. Theresults of the simulation, for several velocities, arerepresented by the lateral displacements of the wheelsetspresented in Figure 15. In all cases the Polach formulationhas been used to calculate the creep forces.

Table 1. MAIN CHARACTERISTICS OF THE ML95 VEHICLE

Traffic velocity range 40 60 Km/hMinimum curve radius on track 60 mTrack gauge 1.435 mWheel rolling radius new 0.43 mBogie wheelbase 2.1 mBogie center distance 11.1 mWheelset weight 1109 KgBogie weight 4200 KgWeight of the carbody 11160 KgFloor height 1.155 mVehicle height 3.523 mVehicle width 2.789 mVehicle length 15.30 m

Rear Wheelset

(Rigid body)

Rear Bogie

Carbody (Rigid body)

Revolutejoints

Rear Axleboxes

(Rigid body)

PrimarySuspension

Rear Bogie Frame(Rigid body)

Front Wheelset

(Rigid body)

Revolutejoints

Front Axleboxes

(Rigid Body)

PrimarySuspension

Secondary

Suspension

Front

Bogie

Rear Wheelset

(Rigid body)

Rear Bogie

Carbody (Rigid body)

Revolutejoints

Rear Axleboxes

(Rigid body)

PrimarySuspension

Rear Bogie Frame(Rigid body)

Front Wheelset

(Rigid body)

Revolutejoints

Front Axleboxes

(Rigid Body)

PrimarySuspension

Secondary

Suspension

Front

Bogie

Figure 14. VEHICLE MULTIBODY MODEL.

It is observed that for vehicle forward velocities under60 m/s the wheelset undergoes lateral decaying oscillationsand returns to the center of the track. This indicates a stablerunning of the vehicle. At sufficiently high speeds, 70 m/sfor instance, the lateral oscillations increase and the vehiclederails. By running the vehicle at intermediate speeds it isobserved that the critical speed of the vehicle is 60.5 m/s.

-0.009

-0.006

-0.003

0.000

0.003

0.006

0.009

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Traveled Distance [m]

LateralDisplacement[m]

V =30 m/s (108 Km/h) V =50 m/s (180 Km/h)V =60 m/s (216 Km/h) V =70 m/s (252 Km/h)

Figure 15. LATERAL DISPLACEMENT OF THE FRONT WHEELSET.


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23750

23800

23850

23900

23950

24000

24050

24100

24150

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

Time [s]

VerticalWheelForce[N]

Left Wheel (Kalker)

Left Wheel (Heuristic)

Left Wheel (Polach)

Figure 16. VERTICAL FORCES ON ONE WHEEL FOR A VEHICLE

VELOCITY OF 50 M/S FOR THE THREE CREEP MODELS

To appraise the difference between the different forcemodels, several vehicle simulations are done for a velocity of

50 m/s. The results for the vertical and lateral wheel forcesare represented in Figures 16 and 17, respectively. It isobserved that for all creep models the results obtained aresimilar. This is expected results since there are no highcreepages at the speeds considered.

To evaluate the performance of different creep modelsthe vehicle is simulated, for velocities of 10 and 20 m/s, ina scenario where the track has a curved geometry, as shownin Figure 18. The dashed lines represent transition segmentswith 50 m. For both simulation velocities the contact forcesobtained with Kalker linear theory lead to the lift of theouter wheel of the front wheelset, as shown in Figure 19.Derailment does not occur due to the flange contact. Since

high creepages are involved the Kalker linear theory isinappropriate to compute the creep forces. It is suggestedthat for cases that involve curved tracks or running speedscloser to the critical speeds the Heuristic or the Polachcreep force models should be selected. Based on resultsnot shown in this paper, due to space restrictions, it can beshown that the Polach nonlinear force model is superior,and more efficient from the computational point of view,than the other two force models considered.

-400

-300

-200

-100

0

100

200

300

400

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

Time [s]

LateralWheelForce[N]

Left Wheel (Kalker) Right Wheel (Kalker)

Left Wheel (Heuristic) Right Wheel (Heuristic)

Left Wheel (Polach) Right Wheel (Polach)

Figure 17. LATERAL FORCES ON ONE WHEEL FOR A VEHICLE

VELOCITY OF 50 M/S FOR THE THREE CREEP MODELS

1 125L m=

2 100L m=

3 125L m=

200R m=

1 125L m=

2 100L m=

3 125L m=

200R m=

Figure 18. CURVED TRACK GEOMETRY

Figure 19. LIFTING OF THE RIGHT FRONT WHEEL WHEN USING

THE KALKER LINEAR THEORY

Another aspect worth checking concerns the differentparametrization schemes for the rail curves. All results ofthe contact force models are dependent on their geometriccorrectness. Two cubic parameterization schemes aretested: Cubic Splines, which is probably the most popularinterpolation scheme used; Shape Preserving Splines,which maintain the curvature of the segments faithful tothat of the original data. Several simulations of the vehicle

are carried for the curved track parameterized with the twotypes of splines. In all simulations only the Polach forcemodel is used. Selected results for the vertical and lateralforces in the wheel are presented in Figures 20 and 21.

-10000

-5000

0

5000

10000

15000

20000

25000

30000

0 3 6 9 12 15 18 21 24 27 30

Time [s]

LateralWheelForce[N]

Left wheel - Cubic Splines Track

Right wheel - Cubic Splines Track

Left wheel - Shape Preserv Track

Right wheel - Shape Preserv Track

Figure 20. LATERAL CONTACT FORCES ON THE LEADING

WHEELSET FOR A FORWARD VELOCITY OF 10 M/S

In figure 20 it can be observed that the Cubic splineslead to a response that contains very pronounced peaks fortimes close to 14 and 19 sec. These type of peaks arealways present in interpolations involving cubic splines andlead to a perception of the system response associated with


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high perturbations. However, these perturbations have nophysical content and, therefore, the interpolation of curvedtracks based on cubic splines can be misleading. These typeof peaks are not present when the track is obtained with theshape preserving splines interpolation. Note also that the

vertical wheel forces obtained for tracks interpolated bythese two schemes exhibits the same isolated high peaksthat lead to the same conclusions. These problems are notobserved with tangent tracks because all interpolatingsplines lead to similar curves.

14000

19000

24000

29000

34000

39000

0 3 6 9 12 15 18 21 24 27 30

Time [s]

Vertic

alWheelForce[N]

Left wheel - Cubic Spline Track Right wheel - Cubic Spline Track

Left wheel - Shape Preserv Track Right wheel - Shape Preserv Track

Figure 20. VERTICAL CONTACT FORCES ON THE LEADING

WHEELSET FOR A FORWARD VELOCITY OF 10 M/S

6. CONCLUSIONA new methodology was proposed to identificaty the

contact points between the wheel and the rail. Thisprocedure allows that two simultaneous contact points to beidentified, allowing to study track switches and problemsinvolving derailment. The application of the procedures tothe simulation of a railway vehicle in different scenariosmade it possible to identify its critical velocity and toevaluate the virtues and drawbacks of different trackinterpolation schemes and creep force models. The resultsshow that the use of cubic splines for the rails and wheelsleads to spurious oscillations on the contact forces. It isconcluded that shape preserving splines is the mostadvantageous cubic polynomial interpolation process.Among the three creep force models tested it was shown

that the Polach nonlinear force model is the only one that issuitable for all simulations carried. The Kalker linear forcemodel fails when the tangential forces reach their saturationlevel. The Heuristic model leads to less accurate results forflange contact, when there are high creepages involved.

ACKNOWLEDGMENTThe support of Fundao para a Cincia e Tcnologia

(FCT) through the grant BD/18180/98, is gratefullyacknowledged.

REFERENCESAndersson, E., Berg, M. and Stichel, S. Rail Vehicle

Dynamics, Fundamentals and Guidelines, Royal Institute

of Technology (KTH), Stockholm, Sweden, 1998.Berzeri, M., Sany, J. and Shabana, A., Curved Track

Modeling Using the Absolute Nodal Coordinate Formulation,Technical Report MBS00-4-UIC, Department of MechanicalEngineering, University of Illinois, Chicago, 2000.

De Boor, C. A practical guide to splines, Springer-Verlag, New York, New York, 1978.

Garg, V. and Dukkipati, R.,Dynamics of Railway VehicleSystems, Academic Press, New York, New York, 1984.

Kalker, J., Book of Tables for the Hertzian Creep-Force Law, Report of the Faculty of Technical Mathematicsand Informatics No. 96-61, Delft University of Technology,

Delft, The Netherlands, 1996.Kalker, J., The Computation of Three-DimensionalRolling Contact with Dry Friction, Numerical Methods inEngineering, 14(9), 1293-1307, 1979.

Kalker, J., Three-Dimensional Elastic Bodies inRolling Contact, Kluwer Academic Publishers, Dordrecht,The Netherlands, 1990.

Kik, W. and Piotrowski, J. A Fast, ApproximateMethod to Calculate Normal Load at Contact BetweenWheel and Rail and Creep Forces During Rolling, 2nd MiniConference on Contact Mechanics and Wear of Rail/Wheel

System, TU Budapest, Budapest, Hungary, 1996Lankarani, H. M. and Nikravesh, P. E., Continuous

Contact Force Models for Impact Analysis in MultibodySystems,Nonlinear Dynamics, 5, 193-207, 1994.

Polach, O., A fast wheel-rail forces calculation computercode, Vehicle System Dynamics, Sup 33, pp. 728-739. 1999

Pombo, J. and Ambrsio, J. A General Track Modelfor Rail Guided Vehicles Dynamics, VII Congresso deMecnica Aplicada e Computacional, April 14-16, vora,Portugal, pp.47-56, 2003a.

Pombo, J. and Ambrsio, J., "A Wheel-Rail ContactModel for Rail Guided Vehicles Dynamics", ECCOMASConference Multibody 2003 - Advances in Computational

Multibody Dynamics, July 1-4, Lisbon, Portugal, 2003b.

Pombo, J. and Ambrsio, J., General Spatial CurveJoint for Rail Guided Vehicles: Kinematics and Dynamics",Multibody Systems Dynamics, 9, 237-264, 2003c.

Shabana, A., Berzeri, M. and Sany, J., Numericalprocedure for the simulation of wheel/rail contact dynamics,Journal of Dynamic Systems Measurement and Control-

Transactions of the Asme, 123(2), 168-178, 2001.Shen,Z.,Hedrick,J. and Elkins,J., Comparison of Alternative

Creep Force Models for Rail Vehicle Dynamic Analysis, Proc. of8th IAVSD Symp on Dynamics of Vehicles on Road and Tracks,Cambridge, Massachussetts 591-605, 1983.

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