A NEW WHEEL-RAIL CONTACT MODEL FOR RAILWAY DYNAMICS

João Pombo, Jorge Ambrósio and Miguel Silva

IDMEC – Instituto Superior Técnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal

e-mail: {jpombo,jorge,miguelsilva@dem.ist.utl.pt}

ABSTRACT

The guidance of railway vehicles is determined by a complex interaction between the wheels and rails, which requires a detailed characterization of the contact mechanism in order to permit a correct analysis of the dynamic behavior. The kinematics of guidance of the wheelsets is based on the wheels and rails geometries. The movement of the wheelsets along the rails is characterized by a complex contact with relative motions on the longitudinal and lateral directions and relative rotations of the wheels with respect to the rails. A generic wheel-rail contact detection formulation is presented here in order to determine online the contact points, even for the most general three dimensional motion of the wheelset. This formulation also allows the study of lead and lag flange contact scenarios, both fundamental for the analysis of potential derailments or for the study of the dynamic behavior in the presence of switches. The methodology is used in conjunction with a general geometric description of the track, which includes the representation of the rails spatial geometry and irregularities. In this work the tangential creep forces and moments that develop in the wheel-rail contact area are evaluated using alternatively the Kalker linear theory, the Heuristic nonlinear model or the Polach formulation. The discussion on the benefices and drawbacks of these methodologies is supported by an application to the dynamic analysis of the bogie of the railway vehicle.

Keywords: Spatial Curves, Contact Point Detection, Creep Forces, Stability Critical Speed.

1. INTRODUCTION

The formulation of the wheel-rail contact problem is a complex task which has been the subject of several investigations that presented different solutions for the problem. Several authors [1-5] studied the contact forces between the wheel and the rail during the dynamic motion and, as a result of their investigations, several computer routines are now available for the calculations of the tangential forces at the contact point given the normal force and the relative velocities between the bodies [1-10].

Although the literature offers solutions for the problem of contact mechanics, detailed descriptions of the surfaces in contact and of the kinematics of the bodies, are required to make use of such solutions. Because the wheel and the rail have profiled surfaces, the prediction of the contact point location online is not simple, especially when the most general three-dimensional motion of the wheelset with respect to the rails is considered.

In this work, the computational implementation of an appropriate methodology for the accurate description of each rail, in the general case of a fully three-dimensional track, is discussed. The track geometry is described by shape preserving splines and adopts Frenet frames that provide the appropriate track referential at every point. The complete

characterization of the track model also requires the definition of the cant angle variation along the railway. It is proposed here that, for tracks with a full spatial geometry, the cant angle is measured with respect to the osculating plane [11-13].

In the methodology presented, a pre-processor is used to define the nominal geometry of both left and right rails based on the interpolation of a discrete number of points, which are representative of their space curves [14,15]. For the complete representation of the track geometry, the two rails are considered as separate geometric entities. To achieve computational efficiency, the pre-processor generates, in a tabular manner and as a function of the left and right rails arc lengths, all the track geometric data and other general quantities required by the multibody code. At every time step, during the dynamic simulations, the program interpolates both left and right rails databases in order to obtain all the necessary information to analyze the complex interaction between the wheels and rails.

Crucial to the success of any formulation for the wheel-rail contact problem is the accurate prediction of the contact points location. The coordinates of the contact points are predicted online during the dynamic analysis by introducing surface parameters that describe the geometries of the contact surfaces. The success of this computational strategy depends on the prediction of the wheel and rail surface parameters that define the points of contact. The multibody methodology can be formulated in terms of the system generalized coordinates and the surface parameters. In this approach four surface parameters are used to describe the wheel and the rail surfaces, which may have arbitrary geometries [15-17]. The proposed methodology allows the prediction of two points of simultaneous contact between one wheel and the rail by using an optimized search for possible contact points on the wheel tread and on the wheel flange. Although not implemented in this work, it would be possible to describe rail and wheels with varying profiles along the track.

The normal contact forces that develop in the wheel-rail interface are calculated using the Hertz contact force model with hysteresis damping [18-21], which accounts for the local dissipation of energy during contact. Three different methodologies are implemented, and used as alternatives to each other, to calculate these tangential contact forces. These are the Kalker linear theory [6,22,23], the Heuristic nonlinear creep force model [4] and the Polach formulation [5].

All methodologies proposed here are implemented in the general purpose multibody code DAP-3D [24] that is used for the dynamic analysis of railway and other types of rail-guided vehicles. The computer code obtained is applied to study the dynamic behavior of a railway vehicle bogie and, in the process, the benefices and drawbacks of the formulations proposed are discussed.

2. PARAMETERIZATION OF RAILWAY TRACK GEOMETRIES

For multibody analysis of railway vehicles, the track models must be defined in the form of parameterized curves. Here, a pre-processing strategy is used to obtain the accurate description of fully three-dimensional track geometries. The purpose of this railway pre-processor is to define the track model as two parameterized curves that represent the nominal geometry of the left and right rails space curves. In the pre-processor three different parametric track descriptions, based on cubic splines, Akima splines and shape preserving splines, are implemented [11-13]. The information is organized in two databases where all quantities, necessary to define the left and right rails curves, are obtained as functions of the arc length of each rail, measured from their origin. The methodology used by the railway pre-processor can be summarized as:

a) The geometry of the track centerline is parameterized using one of the piecewise cubic interpolation schemes available. The user input consists in a set of control points that

are representative of the railway geometry and the track cant angle at each nodal point. b) Regardless of the approach used to describe the railway geometry, the track centerline

is parameterized as function of the track length [11-13]. c) The track cant angle is also parameterized as a function of the track length and the

principal unit vectors, which define the reference frame associated to the track centerline after the cant angle rotation, are calculated.

d) The track irregularities, measured experimentally, can also be included in the track model. With this purpose in mind, the user only has to supply, as input data for the pre-processor, all irregularities parameters organized in a tabular manner. Then, the track irregularities are parameterized as continuous functions of the track length using the cubic spline interpolation scheme.

Input Data to Parameterize the Track Centerline with Piecewise Cubic Interpolation

x y z ϕ … … … … … … … …

User

Input Data to Parameterize the Track Irregularities

L ALLr ALRr LLLr LLRr Δh ΔG… … … … … … … … … … … … … …

Parameterization Cubic Splines

Parameterization Akima Splines

Parameterization Shape Preserving Parameterization

Cubic Splines

Track Centerline Parameterized as Function of the Track Length

Track Irregularities Parameterized as Function of the Track Length

Obtain the Nodal Points that Define the Space Curves for the Left and Right Rails

Parameterization Cubic Splines

Parameterization Akima Splines

Parameterization Shape Preserving

Left and Right Rails Space Curves Parameterized as Function of the Respective Arc Length

Track Cant Angle Parameterized as Function of the Track Length

RIGHT RAIL DATABASE

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

LEFT RAIL DATABASE

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

Figure 1: Schematic representation of the railway pre-processor

e) Calculate the coordinates of two sets of control points that are representative of the left and right rails space curves.

f) Parameterize each rail space curve as a function of its arc length using one of the piecewise cubic interpolation schemes available. Define the reference frame associated to each rail space curve taking into account the track cant angle and the rail inclination.

g) Create an output database for each rail where all parameters necessary to define the geometric characteristics of the spatial curve of the rail are stored. Each database has a track length step as small as desired by the user. During dynamic analysis the rails databases are used by the multibody code to find the necessary geometric information, at each time step, by performing linear interpolations on the data contained in the tables.

A schematic representation of the methodology used in the railway pre-processor is presented in Figure 1. The detailed explanation of the formulation inherent to each one of the procedures reported in the flowchart is too lengthy to be described in this text. However, it must be emphasized here that the track length step used for the railway database must be similar, or smaller, than the ratio between the length of the track covered by one wheel of the vehicle during a single time step of the integrator of the equations of motion and the velocity of the vehicle. In this form, the use of the linear interpolation for the rail curves or any other type of interpolation leads to the same results. The interested reader is referred to the work of Pombo and Ambrósio [14,15,25] for a more detailed discussion of this subject.

3. DEFINITION OF THE WHEEL AND RAIL SURFACES

In the multibody contact model required to solve the problem of the wheel-rail interaction, it is necessary to define in a parametric form the geometry of the surfaces that are in contact. This definition needs to satisfy two main requirements. First, the surfaces have to be defined in a global coordinate system since the equations of motion of the multibody system are written with respect to the global inertial frame. Second, this definition needs to be general in the sense that the parametric equations must be able to represent any spatial configuration of the wheelsets and rails as well as any wheel and rail profiles obtained by direct measurements or by design requirements. The approach followed here is to consider the surfaces of the wheels and rails as sweep surfaces obtained by dragging plane curves on spatial curves. As a result, the problem of describing the surface reduces to the problem of defining plane curves, which are the cross sections of the wheel and rail.

In the formulation developed for the contact between a rigid wheel and a rigid rail, four independent surface parameters are used to define the geometry of the surfaces in contact. The surface parameters rs and ru are used to describe the geometry of the rail surface, while

ws and wu are the surface parameters used to define the wheel surface, as shown in Figure 2 [26,27]. The position vector of a contact point Q in the body coordinate system of the wheel or rail is a function of the surface parameters only. Such position vectors are defined as:

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

In all that follows the quantities with subscripts (.)r and (.)w refer to the rail and to the wheel, respectively, whereas the quantities with subscript (.)ws refer to the wheelset.

rs

ws

wsζ

wsξ( )w wf u

wζ

wη

rζ

rη

ru

( )r rf u

wu

Q Q

wξ

wζ

Figure 2: Wheel and rail surface parameters

3.1. Rail surfaces in a general track

In order to account for any possible scenarios, such as a variation in the gauge or relative displacements and/or rotations of the rails due to the track irregularities, it is necessary to define the surface of each rail independently. Figure 3 shows the left and right rails with an arbitrary geometry and surface profiles. The surface of each rail can be obtained as the envelope of the surfaces generated by the two-dimensional curve that defines the rail profile when it is moved along the rail space curve [28]. In all that follows, the quantities with subscripts (.)Lr and (.)Rr refer, respectively, to the left and right rails.

Rrζ

Rrη

P

Rrζ

Rrη

x

Rrξ

rRr s R

Rr

P

r R Rrs

( )r Rrf u

z

y

Lrζ

Lrη Lrξ

Track centerline Right rail

space curve Left rail space curve

Lrs

Rrsn

RrstRrγ

Rru

Q

Figure 3: Parameterization of the rail surface

In a general case, the surface geometry of the rail is described using the two surface parameters rs and ru . The arc length of the rail space curve, denoted by rs , defines the rail cross-section on which the contact point lies. The parameter ru defines the lateral position of the contact point in the rail profile coordinate system (ξr,ηr,ζr). Furthermore, ru is used to define the rail profile curve at each cross section, as shown in Figure 2 and Figure 3.

On each rail, a profile coordinate system is defined identifying the position and orientation

of any cross section along the rail space curve. Assume that the right rail has a contact point P with one of the wheels. The right profile coordinate system (ξRr,ηRr,ζRr) of this contact is shown in Figure 3. This frame translates along the right rail space curve and rotates about its origin. The location of the profile coordinate system along the space curve can be defined in such a way that the contact point P always lies in the (ηRr,ζRr) plane. The left rail profile coordinate system is defined in a similar way.

The location of the origin and the orientation of the right rail profile coordinate system, defined respectively by the vector rRr and the transformation matrix ARr, are uniquely determined using the surface parameter rs [29]. Using this description, the global position vector of an arbitrary point P on the rail surface is written as:

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

where 'PRrs is the local position vector that defines the location of the contact point P on the rail surface with respect to the profile coordinate system. Note that due to the above-mentioned description of the rail geometry, the following relations hold:

( ) ( ) ( ){ } ; ; ' 0 TP

Rr Rr Rr Rr Rr Rr Rr Rr r Rrs s u f u= = =r r A A s (3)

where fr is the function that defines the rail profile. The transformation matrix ARr can be expressed in terms of the three unit vectors tRr, nRr and bRr that define the moving reference frame associated to the right rail space curve. Hence, the transformation matrix is [24]:

( ) ( ) ( ) ( ) Rr Rr Rr Rr Rr Rr Rr Rr Rrs s s s= = ⎡ ⎤⎣ ⎦A A t n b (4)

The unit vectors are expressed uniquely in terms of the rail arc length, i.e., as function of the surface parameter Rrs . The Cartesian components of these vectors are obtained from the respective rail database previously described.

It is proposed, in this investigation, that the rail profile is parameterized as function of the surface parameter ru using a piecewise cubic interpolation scheme. For this purpose, three different interpolation methods, using cubic splines [30-34], Akima splines [30,34-36] and shape preserving splines [30,31,34,37,38], are implemented. Hence, to obtain ( )r rf u the user only has to define a set of control points that are representative of the rail profile geometry, as shown in Figure 4. Note that this methodology is general in the sense that it allows changing the rail profile during simulation if needed. It also allows performing the dynamic analysis of railway vehicles using rail profiles obtained from direct measurements or by design requirements.

rζ

rη

Nodal Points ( )r rf u

ru

Figure 4: Rail profile parameterization using piecewise cubic interpolation schemes

The multibody contact model used to solve the problem of wheel-rail contact requires the definition of the normal vector to the rail surface nrs at the point of contact. For the right rail this normal vector is defined as:

'Rrs Rr Rrs=n A n (5)

where { }' 0 , sin , cos TRrs Rrs Rrs= −n γ γ is the unit normal vector to the right rail surface

defined in the profile coordinate system (ξRr,ηRr,ζRr). This vector is obtained through the contact angle Rrsγ , measured in the cross-sectional plane, defined between the vector tangent to the right rail surface Rrst , at the contact point, and the local axis ηRr, as shown in Figure 3. The contact angle is obtained by differentiating the function fr, which defines the rail profile, with respect to the non-generalized surface parameter Rru associated to the right rail, as:

( )1 tg r RrRrs

Rr

df udu

γ − ⎛ ⎞= ⎜ ⎟

⎝ ⎠ (6)

In all that follows, the quantities with subscript (.)Lrs refer to the left rail surface whereas the quantities with subscript (.)Rrs refer to the right rail surface. All the expressions presented here for the right rail are also valid for the left rail. It is important to note that the locations of the profiles coordinate systems (ξRr,ηRr,ζRr) and (ξLr,ηLr,ζLr) are, respectively, given by the coordinates Rrs and Lrs measured on right and left space curves and that these frames remain fully independent from each other.

3.2. General wheel surface

In order to account for any spatial configuration of the wheelset and any wheel profile, which can be obtained from direct measurements, it is necessary to define the surface of each wheel independently. Figure 5 shows the left and right wheels assembled in a wheelset and with arbitrary surface profiles. The surface of revolution of each wheel can be obtained as the envelope of the surfaces generated by a complete rotation, about the wheel axis, of the two-dimensional curve that defines the wheel profile [28]. In what follows the quantities with subscripts (.)Lw and (.)Rw refer, respectively, to the left and right wheels.

z y x

wsζ

wsη

wsξ

rwss P

Rw

r PP

Rwζ

Rwη

Rwξ

hRw

Rwu Rws

sQLw

rQ Q

Lwζ

Lwη Lwξ

hLw

Lwu Lws

Left wheel

Right wheel

Figure 5: Parameterization of the wheel surface

In the general case, the surface geometry of the wheel can be described using the two surface parameters ws and wu . The surface parameter ws represents the rotation of the wheel profile coordinate system (ξw,ηw,ζw) with respect to the wheelset coordinate system (ξws,ηws,ζws), i.e., it defines the rotation of the contact point, as depicted in Figure 2. The parameter wu defines the lateral position of the contact point in the wheel profile coordinate system, i.e., represents the coordinate of vector 'ws on the wheel reference axis ηw, as shown in Figure 5. Note that wu also characterizes the wheel profile curve at each cross section.

On each wheel, a profile coordinate system is defined by identifying the orientation of the cross section where the contact point is located. Assume that the left wheel has a contact point Q and the right wheel has a contact point P. The left and right profile coordinate systems of these contacts are shown in Figure 5. These frames rotate about the wheel axis with respect to the wheelset coordinate system in such a way that the contact points Q and P always lie in the (ηLw,ζLw) and (ηRw,ζRw) planes respectively.

The location of the origin and the orientation of the wheelset reference frame (ξws,ηws,ζws) are defined, respectively, by the vector rws and the transformation matrix Aws. Using this description, the global position vectors of an arbitrary point on the surface of both left and right wheels can be written as:

( )( )

+ + ' + + '

+ + ' + + '

Q Q Q QLw ws ws ws ws ws ws ws Lw Lw Lw

P P P PRw ws ws ws ws ws ws ws Rw Rw Rw

= = =

= = =

r r s r A s r A h A s

r r s r A s r A h A s (7)

where Lwh and Rwh are local position vectors that define the location of left and right wheels profiles coordinate systems with respect to the wheelset reference frame, given by:

{ } { }1 12 2 0 0 ; 0 0 T T

Lw RwH H= = −h h (8)

where H is the lateral distance between wheel profile origins. LwA and RwA , appearing in equation (7), are the transformation matrices that describe the orientation of the left and right wheels coordinate systems with respect to the wheelset reference frame, written as [24,39]:

cos 0 sin cos 0 sin

0 1 0 ; 0 1 0sin 0 cos sin 0 cos

Lw Lw Rw Rw

Lw Rw

Lw Lw Rw Rw

s s s s

s s s s

− −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= = −⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

A A (9)

and 'QLws and 'PRws are the local position vectors that define the location of the contact points Q and P on the wheel surfaces with respect to the profiles coordinate systems, respectively, as:

( ){ } ( ){ }' 0 ; ' 0 T TQ P

Lw Lw w Lw Rw Rw w Rwu f u u f u= =s s (10)

where fw is the function that defines the wheel profile. In this investigation, it is proposed that the wheel profile is parameterized as a function of

the surface parameter wu using a piecewise cubic interpolation scheme, as described for the rail surface. Hence, to obtain ( )w wf u the user only has to define a set of control points that are representative of the profile geometry of the wheel tread and wheel flange, as shown in Figure 6(a). Note that this procedure is general in the sense that it allows changing the wheel profile without making modifications in the computer code. It also permits the dynamic analysis of railway vehicles using wheel profiles obtained from direct measurements or by design requirements.

wζ

wu Flange Nodal Points

( )fw wf u

wη

( )tw wf u

Tread Nodal Points

(a)

wζ

wη

Concave Region

(b)

Figure 6: Wheel profile: a) Parameterization using cubic interpolation schemes; b) Concave region

In the multibody contact model used to solve the problem of wheel-rail contact, it is necessary to devise a strategy to determine the location of the contact points between the parametric surfaces. The formulation proposed here requires that the parametric surfaces are convex. Therefore, when parameterizing the wheel profile, it is necessary to avoid the geometric description of the small concave region in the transition between the wheel tread and the wheel flange, which is depicted in Figure 6(b). In order to avoid this difficulty, the wheel profile is represented by two independent functions t

wf and fwf that parameterize,

respectively, the wheel tread and the wheel flange. Therefore, the concave region is neglected and the wheel surface is made of two convex regions, as shown in Figure 6(a).

2t

Rwt

RwζRwη

RwξRwu

Lwζ

Lwη Lwξ

Lwu

Left wheel

Right wheel

1t

Rwt

tRwγ

2t

Lwt

tLwγ ( )f

w Lwf u

( )fw Rwf u

( )tw Lwf u

1t

Lwt

2f

Lwtf

Lwγ

1f

Lwt

( )tw Rwf u

2f

Rwt f

Rwγ

1f

Rwt

Figure 7: Tangent vectors to the left and right wheel surfaces

The formulation used to find the location of the contact points requires the definition of two tangent vectors to the wheel surface, 1wt and 2wt , at the point of contact. For the left and right wheels the first tangent vector, represented in Figure 7, is defined as:

1 1

1 1

' ; ,

' ; ,

l lLw ws Lw Lwl lRw ws Rw Rw

l t f

l t f

= =

= =

t A A t

t A A t (11)

where vectors { }1' 1 0 0 TlLw =t and { }1' 1 0 0 Tl

Rw = −t for ,l t f= , being the quantities with superscripts (.)t and (.)f referred to the wheel tread and to the wheel flange, respectively. The second tangent vector, shown in Figure 7, is:

2 2

2 2

' ; ,

' ; ,

l lLw ws Lw Lwl lRw ws Rw Rw

l t f

l t f

= =

= =

t A A t

t A A t (12)

where vectors { }2' 0 cos sinTl l l

Lw Lw Lw=t γ γ and { }2' 0 cos sinTl l l

Rw Rw Rw=t γ γ , for ,l t f= , and l

Lwγ and lRwγ are the contact angles, measured in the cross-sectional plane. These angles

are defined between the vectors tangent to the left and right wheel surfaces 2wt , at the contact point, and the corresponding local axis Lwη and Rwη , as shown in Figure 7. The contact angles are obtained by differentiating the functions t

wf and fwf , which define the wheel tread and

flange profiles, with respect to the surface parameters Lwu and Rwu , as:

( ) ( )1 1 tg ; tg ; ,

l l l lw Lw w Rwl l

Lw Rwl lLw Rw

df u df ul t f

du duγ γ− −

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟= = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(13)

4. WHEEL-RAIL CONTACT MODEL

From a purely geometrical point of view, the identification of the contact points between the wheel and the rail is a complex task since both of the surfaces are profiled. Furthermore, from a dynamical point of view, the large amount of parameters that include the shape of the surfaces in contact, relative contact velocities, contact forces, and physical properties of the materials, unavoidably lead to complex contact algorithms. Besides the complexity of the equations, the wheel-rail contact is characterized by severe non-linearities and often very stiff behavior [40]. A multibody methodology that leads to an accurate description of the wheel-rail contact problem and that is computationally efficient is proposed here.

4.1. Algorithm for the wheel-rail contact problem

The majority of the investigations dealing with railway systems and with the wheel-rail contact problem do not focus on the methodologies used to calculate the tangential (creep) forces that develop in the wheel-rail interface. Several authors [1-5] have presented different solutions for this fundamental problem and, as a result, several computer routines are now available for the calculation of the wheel-rail tangential forces given the normal force and the relative velocities between the bodies at the contact point [1-10].

Most of the attention is directed to identifying the proper data required by contact routines, which plays a significant role in the results. Since the wheel and the rail have profiled surfaces, the first problem that arises in these studies is the accurate prediction of the contact points location. This problem has to be solved online, for every time step, during dynamic analysis. Once the coordinates of the contact points are determined, the normal contact forces that develop in the wheel-rail interface have to be calculated. For this purpose, a Hertz contact force model with hysteresis damping [18-21], which accounts for the dissipation of energy during contact, is applied. The creepages are obtained in the process of calculating the contact points, and afterwards one of several computer routines can be applied to evaluate the creep forces. In this work, three distinct methodologies are implemented to calculate the tangential contact forces that develop in the wheel-rail interface: Kalker linear theory [6,22,23]; Heuristic nonlinear creep force model [4]; Polach formulation [5].

A new computer algorithm, developed in the framework of the multibody methodology, is proposed here to deal with the wheel-rail contact problem. This algorithm, presented in the flow-chart of Figure 8, is summarized in the following steps:

a) Set the initial conditions for the system generalized coordinates ( )0tq and velocities ( )0tq and define the initial estimate for the surface parameters ( )0

rs t , ( )0ru t , ( )0

ws t and ( )0

wu t associated to each wheel-rail pair. b) Solve a system of nonlinear equations in order to obtain the surfaces parameters that

define the coordinates of the contact points associated to each wheel-rail pair.

c) Calculate the normal contact forces that develop in the wheel-rail interfaces and the dimensions of the contact areas.

d) Evaluate the creepages and calculate the tangential creep forces and moments that develop in the wheel-rail interface using either the Kalker linear theory, the Heuristic nonlinear creep force model or the Polach nonlinear formulation.

e) Add the contact forces and moments, associated to each wheel, to the vector of external applied forces of the system. Apply a multibody formulation in order to obtain the new generalized positions and velocities of the system for time step t+Δt.

f) Update the system time variable and use the information of the previous time step as initial guess for the non generalized surface parameters associated to each wheel-rail contact pair.

g) Go to step b) and repeat the whole process until the final of the analysis is reached. The application of the described methodology demands the use of an appropriate

formulation for the dynamic analysis of multibody systems. The explanation of such formulation, which is out of the scope of this text, can be found in references [24,41,42]. The remaining steps of the proposed procedure are described hereafter.

4.2. Contact points detection between wheel and rail surfaces

A two step methodology to determine the coordinates of the contact points between two parametric surfaces using geometric relational properties is proposed here. First, the geometric equations are defined and solved in order to find the surface parameters that define the coordinates of the candidates to be contact points between the surfaces. The second step consists in the calculation of the distance between the candidate points and evaluation of the penetration condition in order to check if the points are, in fact, in contact or not.

MULTIBODY FORMULATION

Solve System

Nonlinear Equations

Begin

End

Initialize

0t t=

( )0tq

( )0tq

Estimate

( )0rs t

( )0ru t

( )0ws t

( )0wu t Normal

Contact Forces

Creep Forces and Moments

Obtain

( )rs t ; ( )ru t

( )ws t ; ( )wu t

Coordinates of Contact Points

Polach Theory

Heuristic Model

Obtain

( )t Δt+q

( )t Δt+q endt Δt t+ >

Update t t Δt= +

No

Yes

Kalker Theory

Dimensions of Contact Area

Calculate Creepages

Figure 8: Schematic representation of the algorithm proposed to study the wheel-rail contact problem

Consider two generic surfaces i and j, depicted in Figure 9, defined by the parametric functions p(u,w) and q(s,t), being u, w, s and t the surfaces parameters. According to Mortenson [39], the minimum distance between the two patches p(u,w) and q(s,t) is given by vector d, calculated as:

( ) ( ), ,u w s t= −d p q (14)

while fulfilling the condition to be aligned with the normal vectors of the surfaces:

0

0

0

0

T ui

i T wi

T sj

j T tj

⎧ =× = ⇔ ⎨

=⎩⎧ =⎪× = ⇔ ⎨ =⎪⎩

d td n 0

d t

d td n 0

d t

(15)

Note that the inner products between d and the surfaces tangent vectors are equivalent to the cross product between d and the vectors normal to the surfaces. The tangent vectors u

it , wit ,

sjt and t

jt to the parametric surfaces, shown in Figure 9, are defined as:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

, , , ; ,

, , , ; ,

u u w wi i i i

s s t tj j j j

u w u wu w u w

u ws t s t

s t s ts t

∂ ∂≡ = ≡ =

∂ ∂∂ ∂

≡ = ≡ =∂ ∂

p pt t t t

q qt t t t

(16)

In this formulation ni and nj are unit normal vectors to the parametric surfaces, defined as:

( ) ( ) , ; ,

s tu wj ji i

i i j ju w s ti i j j

u w s t≡ = ≡ =t tt tn n n n

t t t t (17)

Note that the tilde placed over a vector indicates that the components of the vector are used to generate a skew-symmetric matrix, which is used to define the vector product.

tsj

x

z

y

(j)

(i)

ttj

twi

dn j

ni

( )p ,u w

( )q ,s t

tui

tsj

x

z

y

(j)

(i)

ttj

twi

dn j

ni

( )p ,u w

( )q ,s t

tui

Figure 9: Candidates to contact points between two parametric surfaces

The minimum distance conditions in equation (15) are not sufficient to find the candidates to contact points between wheel and rail surfaces because they do not cover all possible situations that may occur in the wheel-rail contact problem. These possible situations are sketched with detail in Figure 10 where i and j represent the wheel and rail surfaces, respectively. When there is no penetration between the bodies, depicted by Figure 10(a) and (b), the minimum distance conditions are applied to find the surface parameters that define the coordinates of the candidates to become contact points. If penetration occurs, the contact point on one body has to be located inside the volume of the other body, as shown in Figure 10(c). In this case, the contact points have to be selected from the points belonging to the volume of intersection between the two solids. The contact points are defined as those that correspond to maximum indentation, i.e. the points of maximum elastic deformation, measured along the normal to the contact patch. In the situation described by Figure 10(c), the minimum distance conditions in equation (15) are verified for the three cases I, II and III. However, only the points represented in case II should be considered to be candidates to contact points since they are the points that lead to maximum penetration. Therefore, for the wheel-rail contact problem, new geometric equations must be set in order to find, as candidates to contact points, only the points represented in situations described by Figure 10(a), (b) and case II of Figure 10(c). The new geometric equations are defined as:

i) The surfaces normals ni and nj at the candidates to contact points have to be parallel. This condition means that nj has null projections over the tangent vectors u

it and wit :

0

0

T uj i

j i T wj i

⎧ =⎪× = ⇔ ⎨ =⎪⎩

n tn n 0

n t (18)

ii) The vector d, which represents the distance between the candidates to contact points, has to be parallel to the normal vector ni. This condition is mathematically written as:

0

0

T ui

i T wi

⎧ =× = ⇔ ⎨

=⎩

d td n 0

d t (19)

The geometric conditions represented by equations (18) and (19) are 4 nonlinear equations with 4 unknown surfaces parameters u, w, s and t. This system of equations provides solutions for the location of the candidates to contact points that have to be sorted out. Notice that the formulation proposed here to find the candidates to contact points is limited to convex parametric surfaces. In fact, if one or both of the parametric surfaces are concave, multiple solutions can be obtained for the geometric conditions given by equations (18) and (19).

(a)

(j)

(i) d

n j ni

(b)

(j)

(i)

d = 0

n j

ni (j)

I

d = 0

n j

ni

d = 0 d

(c)

n j n j

ni ni

II III

(i)

Figure 10: Wheel-rail contact situations: a) No contact, i.e., wheel lift; b) Contact at a single point,

without penetration; c) Contact with penetration

The second step of the methodology consists in the calculation of the distance between the candidates to contact points and the verification if the penetration condition is met. The distance between the candidates to contact points is given by:

d = d (20)

The penetration condition states that the contact between the surfaces exists and, therefore, the candidates to contact points are real contact points if:

0Tj ≤d n (21)

Otherwise, there is no penetration and the candidate points are discarded from further considerations in the current time step.

The selection criteria proposed here to determine the location of the contact points between two parametric surfaces is fully generic and can be applied to the specific problem of the wheel-rail contact. The only condition is that the parameterized wheel and rail surfaces must be convex.

By introducing surface parameters that describe the geometry of the contact surfaces, the coordinates of the contact points can be predicted online, during dynamic analysis, even when the most general three-dimensional motion of the wheelset with respect to the rails is considered. Since the wheel and rail surfaces have complex geometries, the position of the wheel-rail contact points cannot be known in advance. Therefore, during dynamic simulations, the calculation of the surface parameters requires the solution of the preliminary system of nonlinear equations outlined in Figure 8. The numerical implementation of this methodology leads to an efficient algorithm since the information of the previous time step is used as initial guess to find the solution of the nonlinear equations and, therefore, only few iterations are required to obtain the solution.

4.3. The two points of contact scenario

The methodology proposed here allows the detection of two points of contact between wheel and rail surfaces. This strategy takes advantage of the fact that the wheel profile is represented by two different functions t

wf and fwf that parameterize, respectively, the wheel

tread and the wheel flange, as shown in Figure 6(a). All ingredients of the formulation used to look for the candidates to contact points are fully independent for the wheel tread and for the wheel flange surfaces. With this approach, the user is free to choose if the investigation on the flange contact problem is important, though its solution is made at the cost of additional computations. In the dynamic analysis of railway vehicles on straight tracks or in large radius curved tracks at balanced speeds [43,44], the flange contact is unlikely to occur, unless the train runs close to its critical speed. For such cases, the user may choose to only look for the contact points between wheel tread and rail surfaces. In cases for which the flange contact phenomenon is likely to occur, such as in small radius curved tracks, the user can choose to look for the contact points between the wheel flange and the rail as well.

Another important feature of the methodology proposed here to determine the wheel-rail contact points location is that it allows for the study of the lead and lag contact configurations. In fact, since the methodology used to look for the candidates to contact points is fully independent for the wheel tread and for the wheel flange, the contact point in the flange does not have to be located in the same plane as the contact point in the wheel tread. This feature allows for the analysis of the derailment or for the study of the effect of switches in the railroad, among others.

tRrs

Flange contact (Lead contact)

Tread contact

fRrs

fRws

0tRws =

fRwζ

tRw wsζ ζ≡

fRwξ

tRw wsξ ξ≡

Flange contact (Lead contact)

Tread contact

Figure 11: Two points of contact in different planes – Lead contact configuration in the right wheel

Figure 11 shows the location of the second point of contact at the right wheel in case of lead contact. As it can be observed, the diametric section that contains the wheel flange contact point makes an angle f

Rws with the diametric section that contains the wheel tread contact point. Also, the section of the rail that contains the second point of contact is located at a distance f

Rrs from the origin of the rail spatial curve, while the section that contains the tread contact point is located at a distance t

Rrs . Lag contact occurs when the value of fRws is

negative and f tRr Rrs s< [26,27]. In the method used in this investigation, the search process is

able to detect the lead and lag contact and, consequently, the values of the geometric parameters that correspond to contact points.

4.4. Normal contact forces in the wheel-rail interaction

In this investigation, it is assumed that the contact area between the wheel and the rail has an elliptic shape. The dimension of the contact ellipse is determined using the Hertz's contact theory [2,45]. The surface parameters are used to find the principal curvatures of the wheel and rail at the contact point that are used to calculate the semi-axes of the contact ellipse. The dimensions of the contact ellipse and the normal contact force are required in order to calculate the creep forces.

The evaluation of the normal contact force includes the Hertzian component, which is a function of the indentation, and a hysteresis damping force component, proportional to the velocity of indentation. The expression of the normal contact force used in this investigation is given by [18-21,46]:

( )2

n( )

3 1 = 1+

4e

N K δ δδ −

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

(22)

where δ is the indentation, n=1.5 is the parameter used for metal to metal contact, K is the Hertzian constant that depends on the surface curvatures and the elastic properties of contacting bodies [46,47], e is the coefficient of restitution, δ is the velocity of indentation and ( )δ − is the velocity of indentation at the initial instant of contact. The velocity of indentation δ is evaluated as the projection of the relative velocity vector of the bodies at the point of contact on the vector normal to the contact surfaces.

The Hertz force model with hysteresis damping, defined in equation (22) to calculate the normal contact forces that develop in the wheel-rail interface is not suited to describe the problems when the dynamic analysis starts with effective contact between the wheel tread and the rail head. In this case, the value of the indentation velocity at the initial instant of contact

( )δ − is unknown. To overcome this problem, the ratio ( )δ δ − appearing in equation (22) is set to be the unity, while the maximum velocity of penetration is not reached,. When the maximum penetration velocity ( )maxδ is obtained, the ratio ( )δ δ − is considered to be

( )maxδ δ until there is contact loss. A schematic representation of this approach is shown in Figure 12.

( )maxδ

δ

0δt

( ) 1δδ −

= ( ) ( )max

δ δδ δ−

=

N Nmax

loading

unloading

energy loss

( ) 1δδ −

=

N0

δmaxδ0δ

Figure 12: Hertz contact force model with hysteresis damping. The dashed line represents the contact force condition when the analysis starts with contact between wheel and rail. The solid line describes

the normal contact conditions during the analysis.

The advantage of the contact force model with hysteresis damping, defined in equation (22) over the classical Hertz force model, given by n= N K δ , is that it accounts for the dissipation of energy occurring during contact, as represented in Figure 12. From the physical point of view this energy dissipation reflects the physical phenomena associated to the energy loss due to micro deformations. In what the numerics are concerned the energy dissipation helps to increase the numerical stability of the integration algorithm.

4.5. Tangential contact forces using the Kalker linear theory

The Kalker linear theory [6,22,23] is implemented here to study the wheel-rail contact. The longitudinal Fξ and lateral Fη components of the creep force and the spin creep moment Mφ that develop in the wheel-rail contact are expressed as:

11

22 23

23 33

0 0 G 0

0

cFF a b c ab cM ab c ab c

⎡ ⎤⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎢ ⎥= −⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪−⎢ ⎥ ⎩ ⎭⎩ ⎭ ⎣ ⎦

ξ ξ

η η

φ

υυφ

(23)

where G is the combined shear modulus of rigidity of wheel and rail materials and a and b are the semi-axes of the contact ellipse that depend on the material properties and on the normal contact force N. The parameters cij are the Kalker creepage and spin coefficients that are tabulated [1,6,44,45] and only depend on the combined Poisson’s ratio ν of wheel and rail materials and on the semi-axes ratio a/b of the contact ellipse. The quantities ξυ , ηυ and φ represent, respectively, the longitudinal, lateral and spin creepages at the contact point. The creep forces and the spin creep moment, together with the normal contact forces, define the contact loads resulting from the wheel-rail interaction. Note that for sufficiently small values of creep and spin, the linear theory of Kalker, embodied in equation (23), is adequate to determine the creep forces. For larger values, this formulation is no more appropriated since it does not include the saturation effect of the friction forces.

4.6. Tangential contact forces using the Heuristic nonlinear creep force model

The Heuristic nonlinear model [4] is also implemented in this work. First, the unlimited resultant creep force of Kalker linear theory is calculated as:

2 2 F F Fυ ξ η′ ′ ′= + (24)

where the notation (.)′ means that the quantities are obtained with the Kalker’s linear theory. Then, the limiting resultant creep force is determined by:

2 31 1 if 3 3 27

if 3

F F FN F NF N N NN F N

υ υ υυ

υ

υ

μ μμ μ μ

μ μ

⎧ ⎡ ⎤′ ′ ′⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ′⎪ − + ≤⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟= ⎨ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦⎪ ′ >⎩

(25)

where μ is the friction coefficient. The new resultant creep force Fυ is used to calculate the tangential forces as:

; ′ ′= =′ ′

F FF F F FF Fυ υ

ξ ξ η ηυ υ

(26)

In the Heuristic method the spin creep moment Mφ is neglected. This theory gives more realistic values for creep forces outside the linear range than the Kalker’s linear theory [44].

4.7. Tangential contact forces using the Polach nonlinear creep force model

The Polach formulation [5], is the third alternative creep force model that is implemented here. The longitudinal and lateral components of the creep force that develop in the wheel-rail contact region are expressed as:

; = = + SC C C

F F F F Fξ ηξ η η

υ υ φυ υ υ

(27)

where F is the tangential contact force caused by longitudinal and lateral creepages, Cυ is the modified translational creepage, which accounts the effect of spin creepage, and SFη is the lateral tangential force caused by spin creepage. The Polach algorithm requires as input the creepages ξυ , ηυ and φ , the normal contact force N, the semi-axes a and b of the contact ellipse, the combined modulus of rigidity of wheel and rail materials G, the friction coefficient μ and the Kalker creepage and spin coefficients cij. As in the Heuristic method, in the Polach formulation the spin creep moment is neglected.

This method allows the calculation of full nonlinear creep forces and takes spin into account. Because of its short calculation time, the Polach algorithm can be used instead of simpler formulations, in order to improve the accuracy, or as a substitute for more complex theories, in order to save computation time.

5. APPLICATION CASES

The methodology presented earlier for the dynamic analysis of railway vehicles is implemented in the general purpose multibody computer program DAP-3D [24]. This program has been developed for the dynamic analyses of spatial multibody systems and contains all the ingredients required to support the implementation of a specialized wheel-rail

contact module. The methodology is discussed based on two examples where a single wheelset and a single bogie, running on a straight track, are studied. The wheelset and the bogie considered here are the same that fit the ML95 trainset, which is used by the Lisbon metro company for the transportation of passengers. A detailed description of the multibody model of the bogie is presented in the work of Pombo and Ambrósio [25,48].

5.1. Single wheelset running on a straight track

The track considered here has a straight geometry with 300 m length and no irregularities. The track model is built by the railway pre-processor, represented in Figure 1, in order to obtain the left and right rails databases. A distance between control points of 1 m is used and the travel distance stepping adopted for the databases construction is 0.1 m. The cubic spline interpolation scheme is used to parameterize the wheel and rail profiles as well as the left and right rails space curves. The simulation scenario is represented in Figure 13.

Figure 13: Single wheelset running on a straight track

The stability theory of railway vehicles, an unsuspended wheelset is always unstable [43,44]. However, the creep forces act as damping forces that lead to some energy dissipation ensuring the existence of a range of velocities where the wheelset experiences a stable running. When a wheelset is running on a tangent track with an initial lateral displacement with respect to the track centerline, the hunting motion arises, as shown in Figure 14. According to Klingel and Boedecker, the equation that describes the wave length Lw of the sinusoidal motion for an unsuspended wheelset is written as [44]:

0 02 weq

b RL πλ

= (28)

where b0 is half the lateral distance between contact points, R0 is the wheel rolling radius and λeq is the equivalent conicity. However, because the Klingel equation (28) is purely kinematical and the creep and inertia forces play a role in the dynamical equilibrium the wavelength observed in the dynamic analysis differs slightly from that predicted by the Klingel’s equations. In order to study the hunting motion, at the initial time of analysis the wheelset is assembled in the track model with an initial misalignment of 2 mm to the right side with respect to the track centerline, as depicted in Figure 14.

Figure 15 presents the evolution of the lateral displacement of the wheelset center of mass with respect to the track centerline. These results are obtained for different initial forward velocities of the wheelset using the Polach formulation as creep force model. For the velocity of 2 m/s the creep forces act as damping forces and the hunting amplitude remains constant, leading to the conclusion that the critical speed of the unsuspended wheelset is close to 2 m/s

It is observed that, for all other velocities, the amplitude of the wheelset oscillations increase leading to an unstable running. Additionally, for speeds of 15 and 20 m/s the amplitude of the sinusoidal motion grows to a limit cycle where the wheelset oscillations are limited by the flange contact. These results allow concluding that the possible lateral displacement of the wheelset until flange contact is reached is approximately 7 mm.

2 mm

Figure 14: Initial misalignment of the wheelset to study the hunting motion

-0.009

-0.006

-0.003

0.000

0.003

0.006

0.009

0 50 100 150 200 250 300

Traveled Distance [m]

Late

ral D

ispl

acem

ent [

m]

V0 = 2 m/s V0 = 10 m/s V0 = 15 m/s V0 = 20 m/s

Figure 15: Lateral displacement of the wheelset center of mass for initial forward velocities of 2, 10,

15 and 20 m/s, using the Polach formulation

The geometric characteristics of the wheelsets that fit the ML95 trainset are: b0 = 0.75 m; R0 = 0.43 m; and λeq = 0.05. Therefore, according to equation (28), the wave length of the wheelset is 15.9 m. This result is in agreement with Figure 15 where it is observed that the wheelset wave length is approximately 16 m for speeds of 2, 10, 15 and 20 m/s.

5.2. Single bogie running on a straight track

Consider now the bogie of the ML95 trainset running on the same straight track without irregularities used for the single wheelset. The simulation scenario is represented in Figure 16.

The wheelset is a basic element of the railway vehicle steering system. When the wheelset is incorporated into a vehicle, its hunting behavior influences the motion of the whole system.

Following a disturbance, the vehicle will undergo a decaying oscillation, and it will return to the centre of the track. As the speed is increased, the oscillations decay rate is reduced. At sufficiently high speeds, the amplitude of the oscillations grows after a disturbance and, eventually, leads to a limit cycle where the oscillations are limited by the flange contact and the derailment may occur [43,49]. The lowest speed at which the natural oscillations of the vehicle have zero decay rate is known as the critical speed.

Figure 16: Single bogie running on a straight track

In order to study the hunting motion, at the initial time of analysis the wheelsets of the bogie are assembled in the track model with an initial misalignment of 2 mm with respect to the track centerline, as depicted in Figure 14. In Figure 17 the lateral displacement of the front wheelset of the bogie, with respect to the track centerline, is presented. These results are obtained for initial forward speeds of 30, 38, 39 and 40 m/s and using the Polach formulation for the creep forces. For speeds of 30 and 38 m/s it is clear that, after the initial misalignment of 2 mm, the wheelset presents a decaying oscillation motion that leads it to the centre of the track. For the velocity of 40 m/s, the initial amplitude of the wheelset sinusoidal motion increases and an unstable running is obtained. For the velocity of 39 m/s, the hunting motion of the wheelset has zero decay rate, leading to a harmonic oscillation. Such results allow concluding that the critical speed of the ML95 bogie is 39 m/s, or 140 Km/h.

-0.004

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

0.005

0 40 80 120 160 200 240 280

Traveled Distance [m]

Late

ral D

ispl

acem

ent [

m]

V = 30 m/s (108 Km/h) V = 38 m/s (137 Km/h)

V = 39 m/s (140 Km/h) V = 40 m/s (144 Km/h)

Figure 17: Lateral displacement of the front wheelset of the bogie for forward velocities of 30, 38, 39

and 40 m/s, using the Polach formulation

-0.005

-0.004

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

0.005

0 40 80 120 160 200 240 280

Traveled Distance [m]

Late

ral D

ispl

acem

ent [

m]

V = 130 Km/h (Kalker) V = 130 Km/h (Heuristic) V = 130 Km/h (Polach)

Figure 18: Lateral displacement of the front wheelset of the bogie for a forward velocity of 36 m/s,

using the three creep force models

With the purpose to investigate the sensitivity of the maximum stability speed on the creep force model used, the hunting motion of the bogie is also studied using the Kalker linear theory and the Heuristic nonlinear model. Figure 18 presents the evolution of the lateral displacement of the front wheelset for an initial forward speed of 36 m/s (130 Km/h) and using the three creep force models. For the Polach and Kalker force models it is evident that the wheelset motion undergoes a decaying oscillation, leading it to the centre of the track. When the heuristic model is used to calculate the creep forces, the wheelset presents an unstable running as a consequence of its increasing sinusoidal motion. These results show that the maximum stability speed, when using the Heuristic model, is lower than when using the other force models. In Table 1 the values for critical speed of the ML95 bogie, obtained according to the methodology presented here, are presented for the three creep force models.

Table 1: Critical speed of the ML95 bogie using the three creep force models

Critical Speed Creep Force Model (m/s) (Km/h) Kalker Linear Theory 39 140 Heuristic Model 35 126 Polach Formulation 39 140

Table 2 presents some parameters resulting from the dynamic analyses of the bogie for

initial forward velocities of 20 and 36 m/s, using the Kalker, Heuristic and Polach creep force laws. These parameters include the simulation time for the dynamic analyses, the required CPU time, in a computer with a Pentium 4 – 2.5 GHz processor with 256 Mb of RAM, the traveled distance along the track by the left and right wheels of each wheelset and the existence of flange contact. For the cases presented, the CPU time required for the dynamic analyses has an insignificant sensitivity to the creep force model used. This is not surprising because all computational costs, associated to the calculation of the wheel-rail contact forces, are nearly the same regardless of the creep force law adopted.

Another aspect that is shown in Table 2 is that the traveled distances along the track by the left and right wheels of the wheelsets are the same, except for the results obtained with the forward velocity of 36 m/s and using the Heuristic force model. In this case, the traveled

distances are not equal for the wheels of the bogie. Such results are explained by the fact that, according to Table 1, the velocity of 36 m/s is above the critical speed of the bogie when using the Heuristic method. Consequently, hunting instability arises and yaw movements of the wheelsets with respect to the track centerline are obtained.

Table 2 – Parameters resulting from the dynamic analyses performed in a straight track for bogie forward speeds of 20 and 36 m/s

Traveled Distance (m) Force Model

Simulation Time

CPU Time Wheelset Left Wheel Right Wheel

Flange Contact

Initial Forward Speed of 20 m/s (72 Km/h) Rear 270.163 270.163 No Kalker

Linear 13.5 s 5m 16s

Front 270.163 270.163 No Rear 270.163 270.163 No

Heuristic 13.5 s 5m 16s Front 270.163 270.163 No Rear 270.163 270.163 No

Polach 13.5 s 5m 15s Front 270.163 270.163 No

Initial Forward Speed of 36 m/s (130 Km/h) Rear 270.170 270.170 No Kalker

Linear 7.5 s 4m 42s

Front 270.170 270.170 No Rear 270.174 270.175 No

Heuristic 7.5 s 4m 37s Front 270.172 270.173 No Rear 270.172 270.172 No

Polach 7.5 s 4m 30s Front 270.172 270.172 No

In Figure 19 the lateral contact forces on both wheels of the front wheelset are shown.

Only the results for the leading wheelset of the bogie are presented since it often gives the highest creep forces and causes the highest amount of wear [44]. As shown, there is a good agreement between the three creep force laws, being the forces calculated with the Heuristic method slightly higher than the ones calculated with the others force models. The good correlation between the results can be explained by the fact that for a bogie running in the tangent track at low speeds only small values of creep and spin are involved. Therefore, all theories are appropriate to compute the creep forces. The decreasing oscillations on the lateral contact forces are a direct result of the bogie stable running for this forward speed.

-400

-300

-200

-100

0

100

200

300

400

0 40 80 120 160 200 240 280

Traveled Distance [m]

Late

ral W

heel

For

ce [N

]

Left Wheel (Kalker) Right Wheel (Kalker)

Left Wheel (Heuristic) Right Wheel (Heuristic)

Left Wheel (Polach) Right Wheel (Polach)

Figure 19: Lateral contact forces on the wheels of the front wheelset for a forward velocity of 20 m/s

The lateral contact forces on the wheels of the front wheelset are shown in Figure 20 for a forward velocity of 36 m/s, using the three creep force models. The increasing oscillations on the lateral contact forces calculated with the Heuristic model result from running at a velocity above the critical speed of the bogie that, according to Table 1, is 35 m/s. When using the Kalker and the Polach methods, the lateral contact forces exhibit a decaying oscillation as a consequence of the bogie stable running for this velocity.

-600

-400

-200

0

200

400

600

0 40 80 120 160 200 240 280

Traveled Distance [m]

Late

ral W

heel

For

ce [N

]

Left Wheel (Kalker) Right Wheel (Kalker) Left Wheel (Heuristic)Right Wheel (Heuristic) Left Wheel (Polach) Right Wheel (Polach)

Figure 20: Lateral contact forces on the wheels of the front wheelset for a forward velocity of 36 m/s

6. CONCLUSIONS

A new general formulation for the accurate prediction of the contact points location on the wheel and rail surfaces has been proposed and implemented. The coordinates of the contact points are predicted online during the dynamic analysis by introducing the surface parameters that describe the geometry of the contact surfaces. This method is applied to study specific problems inherent to the railway dynamics such as the two points of contact scenario. The methodology to look for the candidates to contact points is fully independent for the wheel tread and for the wheel flange. Consequently, the contact point in the flange does not have to be located in the same plane as the contact point in the wheel tread. This is a relevant issue, especially when studying lead and lag contact configurations that occur on switch transitions or when dealing with high angles of attack. This formulation also allows for investigations related with hunting instability and prediction of wheel climbing, which are very important to study derailment phenomena.

The methodology used in this work for the parameterization of the wheel and rail surfaces and for the description of the wheel-rail contact phenomenon is general since it is able to represent any spatial configuration of the wheels and rails and any wheel and rail profiles, even the ones obtained from direct measurements. Because the wheels are treated separately, this approach allows dealing with railway vehicles either with conventional wheelsets, like trains, or with independent wheels, such as in many of the trams in operation.

The application cases presented showed that, when running on tangent tracks with low speeds, the results obtained with the three creep force models are very similar. This is explained by the fact that, in such conditions, small values of creep and spin are involved and, therefore, all creep force models are adequate to compute the contact forces. Despite the good correlation between the results obtained at low speeds with the different creep models, the

contact forces calculated with the Heuristic method are slightly higher than the ones calculated with the others force models, leading to the observation that the maximum stability speed of the bogie, calculated using the Heuristic model, is lower than the ones predicted by the Kalker and Polach methods.

ACKNOWLEDGEMENTS

The support of Fundação para a Ciência e Tecnologia (FCT) through the grant with the reference BPD/19066/2004 made this work possible and it is gratefully acknowledged. The authors want to gratefully acknowledge the valuable discussions had with Prof. José Escalona, University of Seville, and with Prof. Ahmed Shabana, University of Illinois at Chicago.

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[13] Pombo, J. and Ambrósio, J. (2002) "Development of a Roller Coaster Model", Proceedings of the Métodos Numéricos en Ingeniería V, (J. Goicolea et al.), SEMNI, Madrid, Spain, June 3-6,

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