railway vehicles: wheel/rail model · 2007. 3. 20. · chapter 1 railway vehicles: wheel/rail model...

RAILWAY VEHICLES: WHEEL/RAIL

MODEL

Jean-Claude Samin and Paul Fisette

March 20, 2007

Contents

1 Railway vehicles: wheel/rail model 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Wheel/rail kinematic model . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Contact model of a wheel on a straight track . . . . . . . 31.2.2 Contact of a wheel on a curved track (with constant radius) 12

1.3 Wheel/rail contact forces and torques . . . . . . . . . . . . . . . 131.3.1 Wheel/rail contact kinematics . . . . . . . . . . . . . . . . 131.3.2 Wheel/rail contact forces . . . . . . . . . . . . . . . . . . 14

1.4 Applications in railway dynamics . . . . . . . . . . . . . . . . . . 151.4.1 Geometrical contact between a S1002 wheelset and UIC60

rails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4.2 Limit cycle of a rigid wheelset at constant speed . . . . . 171.4.3 BAS2000 bogie . . . . . . . . . . . . . . . . . . . . . . . . 171.4.4 Tramway 2000 . . . . . . . . . . . . . . . . . . . . . . . . 21

iii

Chapter 1

Railway vehicles: wheel/railmodel

1.1 Introduction

Vehicles - on road or on track - certainly represent one of the most importanttypes of application of the multibody approach. In the case of railway vehicles,an arduous aspect of the modeling phase results from the contact between thewheels and the track. Indeed, from a purely geometrical point of view, locatingthe contact point between a wheel and a rail becomes complicated since both areprofiled, and from a dynamical point of view, the large number of parameters(shape of the profiles in contact, contact pressure, relative contact velocity,physical properties of the materials, ...) leads to complex theories such as thosedeveloped by Kalker [8].

The classical approach for the geometrical problem is dedicated to conven-tional railway vehicles whose ”rolling elements” consist of wheelsets. The con-tact between a wheelset and a track has been abundantly developed in theliterature (see for instance [5], [2]); the corresponding models take advantage ofthe fact that the left and right wheels are rigidly linked by a common axle andhave, generally, identical profiles. From a dynamical point of view, the classicalapproach often splits the whole vehicle (and the corresponding mathematicalmodel) into several sub-systems, the wheelset being one of them. The couplingbetween the latter and the rest of the vehicle is then modeled by linear springsand dampers (primary suspension) which allow relative motions with up to 6degrees of freedom.

This classical approach is no longer suitable for new bogie designs, such asfor instance the so-called ”BAS 2000” bogie developed by the Belgian companyB.N.- Eurorail (see figure 1.1); indeed this bogie consists of an articulated framewith independent wheels, meaning that the left and right wheels are no longermounted on a common axle; moreover, the front and rear wheels of the bogiemay have different geometrical dimensions and profiles.

1

2 CHAPTER 1. RAILWAY VEHICLES: WHEEL/RAIL MODEL

Figure 1.1: Bogie ”BAS 2000”

For that reason, the classical approach is unsuitable because in the presentcase, each wheel of the vehicle should be considered separately. This sectiontherefore presents an appropriate model for a single wheel/rail contact (which,being more general than a wheelset model, can deal with any railway applica-tion).

In order to take the wheel/rail contact constraints into account, we mayconsider as in [1] that the track belongs to the multibody system, the wheel/railcontact then being modeled as an internal joint. Since this approach increasesthe size of the model, a better efficiency is obtained by restricting the multibodysystem to the vehicle itself and by considering the kinematic contact constraintsand the corresponding contact forces and torques as external, for each individualwheel [4]. One of the advantages of this method is that the auxiliary variables,which are needed to solve the geometrical problem, are easily eliminated fromthe constraint equations and from their time derivatives. In this way, the dy-namics of the vehicle is described by a set of equations of motion which dependonly on the generalized coordinates relating to the multibody representation ofthe vehicle: this considerably reduces the CPU time requirements.

When relative coordinates are used, most of the constraints encounteredin multibody systems result from kinematic loops (for instance, the BAS 2000articulated bogie contains five independent loops closed by connecting rods).As we shall see, each wheel/rail contact also defines a kinematic loop which isclosed by the track and must also be taken into account when formulating theequations.

1.2. WHEEL/RAIL KINEMATIC MODEL 3

1.2 Wheel/rail kinematic model

Considering the previous definition of the ”independent wheel”, it is clear thateach wheel/rail contact must be analyzed separately. An isolated wheel movingalong on a straight track has five d.o.f. which could be naturally described by:

• the lateral and longitudinal displacements of the wheel center of mass G,

• the yaw and roll angles of the wheel,

• the rotation angle of the wheel around its axis of symmetry.

This set of coordinates would however require solving a preliminary nonlineargeometrical problem, because the position of the wheel/rail contact point cannotbe known in advance since the rail and the wheel are profiled. This is the reasonwhy a set of auxiliary geometrical variables is used, which explicitly refer to theposition of the contact point.

1.2.1 Contact model of a wheel on a straight track

Definition of the kinematic quantities

Considering the wheel and the straight track illustrated in figure 1.2, let us firstdenote Q the contact point on the wheel side and P the contact point on therail side respectively. Denoting by {I} the inertial frame located at the fixedpoint O, we define the frames {Y} and {X} as follows1:

[Y] = RM [I] is the material wheel attached frame,

[X] = RG [I] is the geometrical wheel attached frame.

Note the difference between the latter two frames. The first, {Y}, is attachedto the physical wheel which rotates around its axis of symmetry according tothe speed of the vehicle. The second, {X}, does not take this rotational speedinto account and is attached to a so-called frozen or geometrical wheel whichwould be rigidly attached to its bearing.

Further, we define

[S] = Tϑ[X], with Tϑ (ϑ) = R2(ϑ)∆=

cosϑ 0 − sinϑ0 1 0

sinϑ 0 cosϑ

(1.1)

which is such that the wheel point of contact Q belongs to the {S2, S3} plane;

[T] = Tβ [S], with Tβ (β) = R1(β)∆=

1 0 00 cosβ sinβ0 − sinβ cosβ

(1.2)

1as we did for the wheel/ground model in chapter ??


{R}

I12I

3I

O

Q

{X}{ S}{Y}

P

P

G

x

u

w

{T}

Figure 1.2: Wheel and straight track

which is such that the unit vectors T1 and T2 belong to the wheel tangent planeat the contact point Q.

The position of the mass center and of the wheel contact point are given bythe following vectors:

x =−−→O G and w =

−−→GQ = [S]T

0w

ρ(w)

(1.3)

where ρ(w) is the wheel profile function with2 β(w) = arctan(ρ0(w)). Since thefrozen wheel is considered to be a part of the multibody system, the positionof point G and the orientation of the {X} frame depend only on the multibodygeneralized coordinates q (see figure 1.3). On the other hand, two auxiliary vari-

2where ρ0(w) stands for dρdw.


I1

2I

3I

xx( )q

GR q( )

Figure 1.3: Location of the geometrical wheel in the multibody system.

ables, w and ϑ, have been used to locate point Q and to specify the orientationof its related frame {T}:

[T] = TβTϑRG(q)[I]

u∆= x + w = [I]T

u1u2u3

= [I]T

x(q) +¡RG(q)

¢T ¡Tϑ(ϑ)

¢T 0w

ρ(w)

Geometrical constraints

In order to ensure the geometrical contact between the profiled wheel and rail:

• point Q must belong to the rail surface:

h1(q, w,ϑ)∆= u3 − µ(u2) = 0 (1.4)

where µ is the describing function of the rail profile in the {I2, I3} plane.

• unit vector T3 must be normal to the rail surface:

h2(q, w,ϑ)∆= T3 . R1 = 0 (1.5)

h3(q, w,ϑ)∆= T3 . R2 = 0 (1.6)


where we have introduced a new frame {R}, defined as follows:

[R] = Tα [I] with Tα = R1(α)∆=

1 0 00 cosα sinα0 − sinα cosα

(1.7)

and which is such that the unit vectors R1 and R2 belong to the tangentplane to the rail surface at contact point P . The angle α must be consistentwith the rail profile at the contact point and therefore satisfy the relation

tanα = µ0(u2) with µ0(u2)∆=

dµ

du2(1.8)

The constraints 1.5 and 1.6 are orientation constraints for the wheel, andit can be shown (see [9]) that the associated Lagrange multipliers are two puretorques (along the tangent directions) applied by the rail at contact point Q.However, the wheel and rail being considered as rigid, and assuming that theirrespective profiles are not conformal (i.e. the contact patch area reduces topoint Q), the rail is physically unable to apply such pure torques to the wheel.As a consequence, it can be shown (see [9]) that these Lagrange multipliers areidentically equal to zero.

Nevertheless, these orientation constraints will allow us to find, in an explicitway, the value of the auxiliary variables w and ϑ corresponding to a given valueof the generalized coordinates q. Equation 1.5 can be developed by expressingthe scalar product in the inertial frame; it leads to

T3 . R1 =¡

0 0 1¢Tβ (β) Tϑ (ϑ)RG(q)

100

= 0

Therefore

RG(3,1) cosϑ + RG

(1,1) sinϑ = RG(2,1) tanβ(w) (1.9)

This typical equation has two well-known solutions in (sinϑ, cosϑ) one of which(cosϑ < 0) is irrelevant in our case because it corresponds to a contact pointlocated on the upper part of the wheel.In the same way, equation 1.6 can be developed as follows:

T3 . R2 =¡

0 0 1¢Tβ (β)Tϑ (ϑ) RG(q)

0cosαsinα

= 0

Assuming that cosα and cosβ are non zero (zero would correspond to a contacton a purely vertical tangent plane), this leads to

tanα = −RG(1,2) sinϑ−RG

(2,2) tanβ(w) + RG(3,2) cosϑ

RG(1,3) sinϑ−RG

(2,3) tanβ(w) + RG(3,3) cosϑ

(1.10)


where angle α must also satisfy relation 1.8.In order to solve the geometrical constraints 1.4, 1.9 and 1.10, we could

blindly resort to the Newton/Raphson algorithm as explained in section ??.By experience, for some practical wheel/rail pairs, the nonlinearity of the ori-entation constraints 1.9 and 1.10 can lead to a numerical divergence of theNewton/Raphson formula ??.

To circumvent this problem, the following numerical procedure is thus car-ried out for each wheel:

1. computation of the rotation matrix RG(q) for a given multibody configu-ration by means of the recursive kinematics of section ?? (see figure 1.3),

2. choice of a circle line on the wheel profile: w, ρ(w),β(w),

3. computation of ϑ from 1.9,

4. computation of tanα from 1.10 and computation of tanα from 1.8

5. choice of a new circle (step 2) until¯tanα(1.10) − tanα(1.8)

¯< ², a nu-

merical threshold3, using a dichotomic method, which takes advantage ofthe fact that on one side of the ”assumed” contact point, the slope of thewheel profile 1.10 is larger than that of the rail 1.8, contrary to the otherside (see figure 1.4).

“assumed” contact pointon the wheel surface

“assumed” contact pointon the rail surface

(7.8)

(7.8)

(7.10)

α

α

α

α(7.10)

Figure 1.4: Geometry of the wheel and rail contact point.

After computing the auxiliary variables (w,ϑ), we may consider equation 1.4as an external constraint linking the generalized coordinates q of the multibody

3² = 10−8...10−9


system. This constraint, which expresses that the wheel has only five d.o.f. withrespect to inertial space, is implicit and nonlinear. It will thus be taken intoaccount by means of the Lagrange multipliers technique and the coordinatepartitioning method (see section ??) for which the generalized coordinates qmust be partitioned into dependent (v) and independent (u) variables.

Figure 1.5 clearly makes the distinction between two interwoven iterativeprocess:

1. the dichotomic procedure explained above, applied to each wheel sepa-rately, which provides a new estimate for the auxiliary variables w,ϑ fora given configuration of the system (u, v);

2. the Newton/Raphson procedure, applied to the complete set of multibodyconstraints (i.e. including loop and user constraints), which provides a newestimate for the dependent variables v.

wheel 4

wheel 3

wheel 2

wheel 1

...

h(uu v u, ( )

,ˆ vu v, , ww ,, )

= 0 ?yes

no

Newton Raphson procedure

Dichotomic procedure

h(u,v=v v+∆ˆv, w, ˆ)= 0

ϑ ϑ

ϑ

Figure 1.5: General scheme for the geometrical solution of the contact.

As shown in figure 1.5, the set of constraints 1.4 associated with each wheelof the vehicle — denoted h — is solved by a Newton-Raphson procedure, for whichthe Jacobian matrix dh

dvT is needed:

dh

dvT=

∂h

∂vT+

∂h

∂w

∂w

∂vT+

∂h

∂ϑ

∂ϑ

∂vT(1.11)

As will be seen later, the quantities ∂h∂w and ∂h

∂ϑ are equal to zero when theconstraints are satisfied. Thus, insofar as the initial conditions of the Newton-Raphson procedure are close to the final solution, a good convergence is obtainedby ”freezing” the auxiliary variables (w, ϑ) within this procedure and thus re-stricting the Jacobian to the first term of 1.11: ∂h

∂vT.

Although it is more costly than a global Newton/Raphson method whichwould also iterate on the orientation constraints 1.9, 1.10, the procedure summa-rized in figure 1.5 is far more reliable in terms of convergence for the wheel/rail


contact problem, especially when highly nonlinear profiles are considered (seeexample of section 1.4.1).

Constraint derivatives

In order to obtain the Jacobian matrix corresponding to the geometrical con-straint 1.4 associated with the wheel/rail contact, the partial derivatives of theconstraint function h1(q, w,ϑ) with respect to the generalized coordinates q areneeded. Let us calculate the time derivative of 1.4:

h1 =d

dt

³u . I3

´− d

dt

³µ(u . I2)

´=

.u . I3 − µ0 (u2)

.u . I2 = 0

Using equations 1.7 and 1.8 and assuming, as previously explained, that α isdifferent from ±π

2 , it can be written

h1 =1

cosα

³.u . R3

´= 0

This finally leads to the relation

.u . R3 = 0 (1.12)

which simply expresses that the geometrical contact point has no velocity com-ponent along the direction normal to the contact plane. Since this velocityconstraint is equivalent to the contact constraint 1.4 (assuming compatible ini-tial conditions), we can now extract from 1.12 the Jacobian matrix associatedwith the Lagrange multipliers in the dynamical equations.The velocity vector

.u can be further developed as follows:

.u =

.x +

.w =

.x+

o

w +ωS × w (1.13)

where

o

w = [S]T

01

ρ0(w)

w =³S2+ρ0(w)S3

´w

with ρ0(w) = tanβ and ωS = ωX + ϑX2

and where ωS and ωX are the angular velocities of the {S} and {X} framesrespectively.By using equations 1.1, 1.2 and 1.3, we can show that

S2+ ρ0(w)S3 =1

cosβT2 and ϑ X2 × w = ϑ ρ(w)T1

Equation 1.12 can then be rewritten:µ.x +

w

cosβT2 + ωX × w+ϑ ρ(w)T1

¶. R3 = 0


Since the constraints 1.5 and 1.6 imply that R3 is normal to the plane {T1, T2},this expression reduces to: ¡ .

x + ωX × w¢

. R3 = 0 (1.14)

Since vector ωX is the angular velocity of the ”frozen” wheel, we observe thatthe term between brackets in the left hand side of equation 1.14 representsthe velocity of a point attached to the ”frozen” wheel when it is located atQ. Through linearity of

.x and ωX with respect to the multibody generalized

velocities q, equation 1.14 has the desired form:

J (q, w(q),ϑ(q)) q = 0 (1.15)

since it does not depend on the auxiliary velocities w and ϑ. As mentionedbefore, ∂h

∂w and ∂h∂ϑ are thus equal to zero in equation 1.11 when the constraints

are satisfied.Therefore, this matrix J can be used as the Jacobian matrix associated with

the constraint 1.4 of the multibody system. Furthermore, it can easily be shownthat the corresponding Lagrange multiplier is equal to the normal componentof the contact force (see [9]) whose value is indispensable for the calculation ofthe tangent forces.

Since the differential-algebraic system ?? formed by the dynamical and con-straint equations will be solved using the coordinate partitioning method (sec-tion ??), the second time derivatives of the constraints are also needed. Let usthus calculate the time derivative of 1.12:

d

dt

³.u . R3

´=

..u . R3 +

.u .³ωR × R3

´(1.16)

where ωR is the angular velocity of the {R} frame. As mentioned above, thevelocity vector

.u can be written as:

.u =

.x +

.w with

.w = ωX .w+ ϑ ρ(w)T1 +

w

cosβT2 (1.17)

The acceleration vector..u can then be developed as follows:

..u =

..x + ˜ω

X.w+ ωX .

.w + ωT .

µϑ ρ(w)T1 +

w

cosβT2

¶+

d

dt

³ϑ ρ(w)

´T1 +

d

dt

µw

cosβ

¶T2

where ωT is the angular velocity vector of the {T} frame.Since, from the constraints 1.5 and 1.6, T3 and R3 are continuously aligned,

the following property holds:

ωT . R3 = ωR . R3


Using the latter and substituting 1.17 into 1.16, we obtain³..x + ˜ω

X.w+ ωX .

.w− ωR .

¡ .x + ωX .w

¢´. R3 = 0 (1.18)

which does not depend on the second time derivatives ϑ, w of the auxiliaryvariables: their computation is thus superfluous4.

Developing.w as in 1.17, the final form of 1.16 becomesµ

..x +

³˜ωX

+ ωX . ωX´

.w + ωX .

µϑ ρ(w)T1 +

w

cosβT2

¶¶. R3

− ¡ωR .¡ .x + ωX .w

¢¢. R3 = 0 (1.19)

where ωR = α I1 from 1.7.This last equation 1.19 corresponds to the general form

Jq +

µ∂J

∂qTq +

∂J

∂ww +

∂J

∂ϑϑ

¶= 0 (1.20)

from which the velocities w and ϑ of the auxiliary variables should now beeliminated. For this purpose, by differentiating the auxiliary constraints 1.5and 1.6 with respect to time, we obtain

ωT . R2 = 0 and ωT . R1 = α (1.21)

where:

ωT = ωX + ϑX2 + βS1 = ωX + ϑX2 +ρ00(w)

1 + (ρ0(w))2 w S1

and, from 1.8,

α =d

dt(arctan (µ0(u2))) =

Ãµ00(u2)

1 + (µ0(u2))2

!.u . I2

where.u is given by 1.17 as a linear function of ϑ and w. By substituting the

latter two expressions into 1.21, we finally obtain a 2 by 2 linear system ofequations with the formµ

a11 a12a21 a22

¶µw

ϑ

¶=

µb1b2

¶q (1.22)

Since this system is invertible analytically, the auxiliary velocities w and ϑ canbe eliminated from 1.20 in order to obtain the second time derivative associatedwith the constraint 1.4:

J (q,w(q),ϑ(q)) q + J³q, q, w(q, q), ϑ(q, q), w(q),ϑ(q)

´q = 0 (1.23)

4contrary to what it is suposed in [10].


1.2.2 Contact of a wheel on a curved track (with constantradius)

These developments can be extended to curved track by considering for eachwheel (see figure 1.6) a local straight track whose center line, tangent to thecurved track at point C, is perpendicular to the vertical plane Π which containsthe track center of curvature O and the center of mass G of the wheel.

I1

I1

ˆ

ˆ

I

I

2

2

Q

O

C

C

C

G

xx

curved track centerline

local straight trackcenterline

local tangent track (left rail)

Figure 1.6: Wheel on a curved track with constant radius.

Let us define the vector u∗ ∆= −−→C Q. The geometrical constraints express that:

1. point Q belongs to the (straight) rail surface:

h∗1(q,w,ϑ) = u∗ . IC3 − µ³u∗ . IC2

´= 0 (1.24)

where µ is the describing function of the local straight track rail profile,defined in the {IC} frame,

2. the surfaces of the wheel and rail must be tangent at the contact point:

h∗2(q, w,ϑ) = h2 with h2 given by 1.5 (1.25)

h∗3(q, w,ϑ) = h3 with h3 given by 1.6 (1.26)

where w, ϑ are the auxiliary variables already defined for the straighttrack.

1.3. WHEEL/RAIL CONTACT FORCES AND TORQUES 13

When evaluated in extreme cases (high curvature, important yaw angle ofthe wheel, high inclined wheel profiles, ...), the geometrical error due to the as-sumption of a local straight track is less than 5.10−6 meters. The developmentsrelated to the time derivatives of these constraints are thus similar to those pre-sented above for a straight track, all calculations essentially being made withrespect to the {IC} moving frame.

1.3 Wheel/rail contact forces and torques

1.3.1 Wheel/rail contact kinematics

As for the wheel/ground contact forces presented in section ??, wheel/rail con-tact forces rely on specific kinematic quantities measured at the material pointof the wheel which coincides with the geometrical contact point Q, previouslycomputed by solving the geometrical constraints 1.4, 1.5, 1.6. This contact pointis in fact a simplified geometrical representation of a small elliptical contact sur-face which can be divided into a rolling and a slipping region. The relative sizeand the shape of these regions in the ellipse strongly depend on the linear andangular velocities of contact: a detailed discussion of this very complex phe-nomenon can be found in the literature (ex. [8], [5]). As our model relies on apoint contact — which coincides with the ellipse center — the above-mentionedrolling and slipping phenomena must be condensed in a unique kinematic con-cept, denoted as creepages. They thus represent the deviations from a purerolling motion of the wheel on the rail and they are at the root of contact forcemodels, such as those proposed by Kalker in [8].Referring to figure 1.2, the creepages of the material contact point which co-incides with the geometrical contact point Q are defined with respect to thecontact plane {T1, T2} as follows:

• The longitudinal creepage ξx is the longitudinal velocity of the materialcontact point divided by the longitudinal velocity5 of the wheel center:

ξx∆=

¡ .x + ωY × w

¢. T1

.x . R1

(1.27)

• The lateral creepage ξy is the lateral velocity of the material contact pointdivided by the longitudinal velocity of the wheel center:

ξy∆=

¡ .x + ωY × w

¢. T2

.x . R1

(1.28)

• The spin creepage ξsp is the angular velocity of the wheel in the directionnormal to the contact plane, divided by the longitudinal velocity of the

5Depending on the authors, the denominator can slighty differs, being for instance therolling circumferential velocity or the mean of the latter and of the forward wheel velocity, ...


wheel center:

ξsp∆=

ωY . T3.x . R1

(1.29)

The two first are dimensionless while the spin has the dimension of length−1.

1.3.2 Wheel/rail contact forces

Let us first write the contact forces and torques in vector form as follows:

Fw = Flong T1 + Flat T2 + Fvert T3 and Mw = Mspin T3

According to Kalker’s linear theory ([8], [5]), for small creepages, the followinglinear relationship holds: Flong

FlatMspin

=

−k11 0 00 −k22 −k230 k23 −k33

ξxξyξsp

(1.30)

where:

k11 = (ab)Gshc11

k22 = (ab)Gshc22

k33 = (ab)2Gshc33

k23 = (ab)3/2Gshc23

in which:

• Gsh is the combined shear modulus of rigidity of wheel and rail materials;

• a, b respectively denote the longitudinal and lateral semi-axes of the con-tact ellipse. The product (ab) depends — nonlinearly — on the normal forceFvert and on the contact curvatures, the Poisson coefficients σPois and theYoung moduli of elasticity EY of the materials in contact;

• cij are the Kalker’s coefficients, which he tabulated as a function of thePoisson coefficients σPois and the a/b ratio.

Let us point out that the normal force Fvert is a constraint force which isalready taken into account via the Lagrange multiplier λ associated with thenormal constraint 1.14. For the numerical simulation its value, which is requiredfor the computation of the Kalker’s coefficients, can be reasonably picked upfrom the previous time integration step. Indeed, computing the current valueof Fvert (= λ) would require a global iterative procedure on the whole set ofequations, since creep forces and torque in relation (1.30) nonlinearly dependon Fvert.

1.4. APPLICATIONS IN RAILWAY DYNAMICS 15

Finally, as for the wheel/ground contact, creep forces and torque values mustbe saturated when large creepages occur. Various models can be found in theliterature (see [5]): the most elementary ones only deal with the saturation ofthe forces with respect to the longitudinal and lateral creepages. Refinementscan be found in models (as developed by Kalker for instance) which carefullytake the spin creepage influence into account, especially when the contact occursin the wheel flange region where ξsp can become very large.

1.4 Applications in railway dynamics

This section first presents some simulation results for conventional vehicleswhich, by comparison with those obtained by means of classical methods, val-idate the single wheel/rail model of the previous section. The model of a nonconventional vehicle, equipped with BAS 2000 articulated bogies (depicted infigure 1.1), is then described and illustrated with some typical simulation re-sults.

1.4.1 Geometrical contact between a S1002 wheelset andUIC60 rails

The geometrical particularity of the standardized ”S1002” wheel profile (seefigure 1.7) on a standardized ”UIC60” rail profile (see figure 1.8) with a 1

40 cantslope is the following: when the wheelset is progressively displaced laterally from

0.500m.

1.5 m.

“S1002” wheelset

piecewise polynomial profile

Figure 1.7: S1002 wheel profile

its central position on the track to the extreme position corresponding to thederailment limit, the location of the contact point on the wheel and rail profilesdoes not change continuously, but three jumps occur6.

6assuming new profiles which perfectly fit the standardized norms, and assuming the wheeland rail are perfectly rigid.


1.435 m

1:40

0.014 m.

“UIC60” railway track

piecewise circular profile

Figure 1.8: UIC60 rail profile

These discontinuities obviously affect the rolling radii (see figure 1.9) andcause modeling difficulties (for the contact geometry, the computation of con-tact forces and the numerical analysis). In the literature (see [6] for instance),the locations of these jumps (versus the lateral wheelset displacement) vary,depending on the geometrical modeling approach which is used. In some ex-treme cases, the first contact jump simply disappears because of the smoothingtechniques used in describing the wheel and rail profiles. A rigorous approachto this geometrical problem is presented in [11]. Figure 1.9 compares the resultsobtained in [11] for a rigid wheelset with those obtained with the present singlewheel/rail contact model, in terms of variations of the difference between leftand right rolling radii, given as a function of the lateral displacement of thewheelset. One may observe on this figure the three discontinuities which clearlyillustrate the contact jump phenomenon, typical of the S1002 wheelset / UIC60rail pair.

1.4.2 Limit cycle of a rigid wheelset at constant speed

Reference [7] makes an extensive study of the limit cycles and chaotic motionsof (rigid) wheelsets and (classical) bogies. Such behaviors are of great practi-cal importance. Indeed, one essential characteristic of a railway vehicle is itscritical speed. The latter, denoted vcr, is generally determined by means of amodal analysis of the vehicle running at constant speed on a straight track.It corresponds to the speed for which an eigenmode involving lateral and yawmotions of the wheelsets (or the bogies) becomes unstable and could provokethe derailment of the vehicle. However, as it results from a linear analysis, thecritical speed vcr does not provide any information on the occurrence of limitcycles which, even if they are stable, are unacceptable from a practical point ofview. Such limit cycles appear for speeds above a certain limit velocity denotedvlim . Since vlim < vcr, the limit speed vlim should be considered in practice


[m]

[mm] [mm]

[mm]

yw yw

∆ r∆ r [m]

[mm] [mm]

[mm]

yw yw

∆ r∆ r

Figure 1.9: Rolling radius difference ∆ r between left and right wheels versuswheelset lateral displacement yw (left : results from [4] - right : results from[11]

Figure 1.10: Wheelset limit cycle (left: from [4] - right: from [7])

as the effective critical speed. A nonlinear dynamical analysis (using numericalintegration) of the vehicle at constant speed is one of the means to detect thepresence of limit cycles. Figure 1.10 shows an illustrative example of such ananalysis, performed on the right via a classical wheelset approach [7] and onthe left by using the present wheel/rail contact model imbedded in a multibodyrepresentation of the wheelset [4]. By looking at this figure, we see that such abehavior cannot be accepted during an actual vehicle ride!

1.4.3 BAS2000 bogie

The development of a single-wheel contact model — instead of a wheelset model —within the context of a multibody approach is motivated by applications whichinvolve articulated bogies like the BAS 2000 represented in figure 1.1. The


open-loop multibody model of this bogie, sketched in figure 1.11, contains 24elementary joints (either revolute or prismatic) distributed as follows:

• 6 (fictitious) joints connecting the ground and the crossbeam of the bogie,

• 10 revolute joints within the articulated structure of the bogie itself,

• 4× 2 joints (one prismatic and one revolute) connecting the wheels to thebogie.

The open-loop structure of the bogie was obtained by cutting six independentkinematic loops: five of these must be closed by connecting rods and the lastone by a ball joint (see section ??). These kinematic loops imply a set of 8independent closure constraints. In addition, each wheel induces an additionalconstraint due to its contact with the rail. The constrained system thereforehas 24− 8− 4 = 12 degrees of freedom:

• 6 d.o.f. (translations + rotations) of the bogie with respect to the ground(conferred via the 6 first joints),

• 2 d.o.f. associated with the chassis deformation (around a vertical and atransverse axis respectively),

• 4× 1 d.o.f. for the wheel suspensions (between each wheel bearing and itscarrying beam).

Joint and externally applied forces (and torques) are modeled as follows.

1. The (external) interactions with the carbody through the secondary sus-pension are modeled by six force/torque components applied to the maincrossbar. This allows us to model the complete vehicle by means of thesub-system segmentation technique explained in section ??.

2. Stiffness is introduced in the revolute joints of the bogie. It correspondsto the elasticity introduced into some joints in order to prevent excessiveyaw deformation of the bogie.

3. The tangent contact forces on the wheels are computed according toKalker’s theory [8].

4. Since this particular bogie is designed to run with cylindrical wheels, theguidance along the track is ensured mainly by second contacts occurringbetween the wheel flange and the rail. Due to the lateral clearance betweenthe wheels and the rail gauges (3.9 mm for the BAS 2000), these contactsare intermittent. The kinematic multibody tools of section ?? are thusused to detect this contact which is approximated as indicated in figure1.12.


excentric

excentric

longitudinal beam (rear)

main crossbeam

ground

ball joint

main crossbeam

rear link

x

x

z

z

y

y

longitudinal beam (rear)

connecting rodconnecting rod

connecting rod

connecting rod

front wheels

rear wheels

longitudinal beam (front)

longitudinal beam (front)

longitudinal beam (front) longitudinal beam (front)

Figure 1.11: Multibody model of BAS2000 bogie


1

2

wheel

rail

flange

w

(w)β

ρ( )w

Figure 1.12: Second wheel-rail contact.

Since this second contact occurs on the wheel flange, we assume that itslateral position w on the wheel profile is constant. In the wheel-rail contactmodel of section 1.2, the values w, ρ(w) and β(w) are thus kept constant whilethe so-called shift angle ϑ is still evaluated from 1.9. Indeed, due to the largevalue of β in the flange area (more than one radian), the shift angle rapidlybecomes significant in the presence of yaw.

Figure 1.13: Simulation of the front BAS 2000 bogie

The normal contact force FN at the second contact on the rail is evaluatedby considering an elastically restrained rail (as shown in figure 1.12). Such acontact model does not introduce a constraint, since the wheel is allowed to”penetrate”7 into the rail. Considering the high level of creepage occurring atthis contact point, the model for the contact tangent force applied to the wheel

7The lateral stiffness of a rail results mainly from its roll deformation, or, in some situations,from the global transverse motion of the roadbed with respect to the inertial frame.


is dry friction: FT = µFN , oppositely aligned with the direction of the velocityof the material point of the wheel which is in contact with the rail. Note thatfor a large shift angle ϑ, the vector of the contact point velocity (and thus ofthe tangent force) has a large vertical component which can significantly loador unload the wheel (see [3]).

Some simulation results obtained with this model are illustrated here. Thefront carbody of a tramway, supported by a single BAS 2000 bogie (see figure1.13) is assumed to be driven at constant speed along a straight track by therest of the vehicle: this means that the pivot C between the first and secondcarbodies travels at constant speed along the track-centerline. Under these con-ditions, the resulting behavior of the yaw angle (measured at point C) betweenthe front carbody and the direction of the track is then observed. As could beexpected with cylindrical wheels, the curves plotted in figure 1.14 show that thevehicle bounces laterally between the flanges of the left and right rails. The am-plitude of this bouncing phenomenon increases with the clearance between thewheel and rail gauges. For small clearances (ex.: 2mm), this bouncing behaviordisappears and the vehicle tends to remain in permanent contact on the left orright side, depending on the initial perturbation.

Figure 1.14: Unstable behavior of BAS 2000 bogie: wheel/rail bouncing

1.4.4 Tramway 2000

The complete vehicle (represented in figure 1.15) consists of three articulatedcarbodies, two BAS 2000 bogies (mounted in opposite directions) at the ends and


Figure 1.15: The tramway 2000

one conventional bogie (denoted B4×4) carrying the central carbody. Accordingto the subsystem segmentation technique presented in section ??, the multibodymodel of this complex system involves:

- 4 sub-systems (one for the carbodies and one for each bogie),- 74 joint variables in the open-loop multibody representation,- 33 constraints due to 17 kinematic loops and 12 wheel-rail contacts.

The complete system thus has 41 degrees of freedom. Since it is designed forurban transportation, the vehicle must be able to travel over tracks featuringtight curves. During curving, the yaw angle formed by a wheel and the rail(denoted attack angle, see figure 1.16) is a critical parameter: the smaller thisattack angle, the lower the creep friction at the contact points. Reducing theangle of attack therefore reduces energy dissipation and wear. Figure 1.18 in-dicates that the operational behavior of the BAS 2000 bogies during the curveentry depicted in figure 1.17 is much better (nearly one order of magnitude)than that of the conventional B4×4 bogie in terms of this angle of attack.

rail

αv

Figure 1.16: Wheel/rail attack angle


Figure 1.17: Tramway 2000: curve entry simulation

Figure 1.18: Comparaison of wheel attack angles during curving, for a conven-tional and a BAS 2000 bogie (from [3])

Bibliography

[1] Chatelle, P., J. Duponcheel, and J.-C. Samin: 1984, ‘Investigation onNon-Conventional Railway Systems through a Generalized Multibody Ap-proach’. In: K. Hedrick (ed.): The Dynamics of Vehicles on Roads and onRailway Tracks. Lisse, pp. 43—57.

[2] de Pater, A.-D.: 1988, ‘The Geometrical Contact between Track andWheelset’. Vehicle System Dynamics 17, 127—140.

[3] Fisette, P., K. Lipinski, and J.-C. Samin: 1995, ‘Dynamic Behaviour Com-parison Between Bogies : Rigid or Articulated Frame, Wheelset or Inde-pendent Wheels’. Supplement to Vehicle System Dynamics 25, 152—174.

[4] Fisette, P. and J.-C. Samin: 1994, ‘A New Wheel/Rail Contact Model forIndependent Wheels’. Archive of Applied Mechanics 64, 180—191.

[5] Garg, V.-K. and R.-V. Dukkipati: 1984, Dynamics of Railway Vehicle Sys-tems. Toronto: Academic Press.

[6] Gasch, R., D. Moelle, and K. Knothe: 1984, ‘The Effect of non-Linearitieson the Limit-Cycles of Railway Vehicles’. In: K. Hedrick (ed.): The Dy-namics of Vehicles on Roads and on Railway Tracks. Lisse, pp. 207—224.

[7] Jaschinski, A.: 1990, ‘On the Application of Similarity Law to a ScaledRailway Bogie Model’. Ph.D. thesis, Delft University of Technology, Delft,The Netherlands.

[8] Kalker, J.-J.: 1990, Three dimensional Elastic Bodies in Rolling Contact.Dordrecht: Kluwer Academic Publishers.

[9] Samin, J.-C.: 1984, ‘A Multibody approach for Dynamic Investigation ofRolling System’. Ingenieur Archiv 54, 1—15.

[10] Shabana, A.-A., M. Berzeri, and J.-R. Sany: 2001, ‘Numerical Procedurefor the Simulation of Wheel/Rail Contact Dynamics’. Journal of DynamicSystems, Measurement and Control 123(2), 168—178.

[11] Yang, G.: 1993, ‘Dynamic Analysis of Railway Wheelsets and CompleteVehicle Systems’. Ph.D. thesis, Delft University of Technology, Delft, TheNetherlands.

25

railway vehicles: wheel/rail model · 2007. 3. 20. · chapter 1 railway vehicles: wheel/rail model...

Documents