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On the Numerical and Computational Aspects of

Non-Smoothnesses that occur in Railway Vehicle

Dynamics

H. True, A. P. Engsig-Karup and D. Bigoni

DTU Informatics, The Technical University of Denmark, Richard Petersens Plads 321,

DK-2800 Kgs. Lyngby, Denmark

Abstract

The paper contains a report of the experiences with numerical analyses of rail-

way vehicle dynamical systems, which all are nonlinear, non-smooth and stiﬀ

high-dimensional systems. Some results are shown, but the emphasis is on

the numerical methods of solution and lessons learned. But for two examples

the dynamical problems are formulated as systems of ordinary diﬀerential-

algebraic equations due to the geometric constraints. The non-smoothnesses

have been neglected, smoothened or entered into the dynamical systems as

switching boundaries with relations, which govern the continuation of the so-

lutions across these boundaries. We compare the resulting solutions that are

found with the three diﬀerent strategies of handling the non-smoothnesses.

Several integrators - both explicit and implicit ones - have been tested and

their performances are evaluated and compared with respect to accuracy, and

computation time.

Keywords: Railway vehicle dynamics, Stiﬀ systems, Nonlinear dynamics,

Non-smoothness

1. Introduction

In recent years the world has seen a rapid development of theoretical

research in the area of non-smooth dynamical systems. This development is a

natural extension of the mathematical theory of nonlinear dynamical systems

that are assumed ’suﬃciently smooth’, which usually means that all partial

derivatives of second order in the dynamical system must be continuous.

The area of mathematical theory of nonlinear suﬃciently smooth dynamical

Preprint submitted to Mathematics and Computers in Simulation August 31, 2011

systems grew very fast in the 20th century. In the second half of the century

the development was strongly fueled by the growing application of digital

computers and eﬃcient numerical methods that together made it possible to

solve nonlinear dynamical problems that hitherto had not been solvable with

the known analytic solution methods.

In real life, however, the dynamical problems often do not satisfy the

’suﬃciently smooth’ criterion, and many of the mathematical results are not

valid any more. Solutions of non-smooth problems were therefore limited in

number but of important examples the theory of the clock and the motion

of a body under the action of a Coulomb type friction force ought to be

mentioned. These dynamical systems were simple one degree of freedom sys-

tems. As examples of the breakdown of the mathematical theory for smooth

dynamical systems we mention the center manifold theorem for bifurcations

and the necessary condition for existence of a bifurcation, which both do

not hold in general for non-smooth dynamical systems. Instead numerical

continuation routines must be applied to ﬁnd bifurcation points and multi-

ple attractors. This fact emphasizes the importance of accurate and reliable

numerical methods for the analysis of theoretical dynamical problems.

Nonlinear dynamical models of mechanical systems with more than two

degrees of freedom (DOF) are in general not solvable by analytic methods,

but with the development of the digital computer the impossible became

possible, and a vast amount of dynamical models of mechanical systems of

interest for the applications could then be analysed. Vehicle system dynamics

is one of such mechanical systems.

Complete dynamical models of vehicle systems are nonlinear and non-

smooth with degrees of freedom (DOFs) in the range from about 10 to above

100. It is therefore necessary to ﬁnd the solutions by numerical methods.

In the beginning of the age of numerical investigations of vehicle system dy-

namics the numerical routines were rather crude, mostly explicit formulations

with ﬁxed step length and error tolerance. No special attention was given

to the non-smoothnesses in the problem, but they often caused problems of

their own. The interest in the numerical methods were limited to the ques-

tion: Do I get an answer? If ’yes’, then ﬁne. An investigation of numerical

integration methods for vehicle dynamical problems is found in chapter 2 in

the book by Garg and Dukkipatti [1] from 1984. It describes the state of the

art at that time. The authors mainly compare explicit and implicit solution

routines and show the relative performances of several integration schemes

from that time. No attention is paid to the handling of non-smoothnesses in

2

the system.

Around the same time a production of simulation routines for modeling

and analysis of vehicle dynamical systems started around the world. New

integration routines were developed that were especially designed for vehicle

dynamical use. Characteristic for vehicle dynamical systems is the math-

ematical formulation as a diﬀerential-algebraic dynamical problem, which

is very stiﬀ. The routine DASSL deserves to be mentioned in this context.

Several of these routines are commercially available and have been further de-

veloped and used successfully in both industry and research institutes. Some

of them participated in a Manchester benchmark test [2] in 1998, where their

performances were compared.

In this article we will describe the development of the numerical han-

dling of non-smooth vehicle dynamical systems at The Technical University

of Denmark. On the background of some results with bifurcations of as well

periodic, quasi-periodic as chaotic solutions and the existence of multiple at-

tractors, we discuss the use of various numerical solution routines. In the long

period of applications we have investigated problems with discontinuous sec-

ond derivatives, discontinuous ﬁrst derivatives and discontinuous functions.

The size of the dynamical problems varies from low-dimensional test prob-

lems to high-dimensional realistic railway vehicle models. We have solved

these problems as well by ignoring the non-smoothnesses as by smoothing

them and by introduction of switching boundaries with event detection. We

have compared the solutions that resulted from the various approaches and

tried to select a solution strategy that would perform in an optimal way

for each given problem. We would like to share our experiences with other

members of the scientiﬁc and industrial community. In a ﬁnal section of

the article we shall compare the performance of several of the routines we

have used and give recommendations for their applications to non-smooth

dynamical problems on the basis of our experience.

A very informative state-of-the art article on numerical methods in vehicle

system dynamics by Arnold et.al.[3] has just been published. It is a valuable

evaluation and a description of the use of the various available integration

routines for vehicle system dynamical problems that exist today.

2. Theoretical basis for Railway Vehicle Dynamical Systems

The theoretical model of the dynamics of railway vehicles is usually for-

mulated as a dynamical multibody system under external forcing. The single

3

bodies are most often assumed rigid. Flexible bodies may appear, but the

ﬂexibilities are then often represented by a Galerkin approximation of their

characteristic frequencies of deformation in order to avoid the modeling of the

dynamics of the ﬂexible bodies by partial diﬀerential equations. The internal

forces between the bodies can be classiﬁed in two main groups: i) Spring and

damper forces and ii) Contact forces. The spring and damper forces are in

general nonlinear, and the contact forces, which always are nonlinear, can be

divided into rolling contact, sliding contact with stick/slip and impacts. All

these forces introduce non-smoothnesses in the dynamical system.

The dynamical system depends on several parameters from which the

speed, V, is usually chosen as the control parameter in a co-dimension 1

problem. In some applications other control parameters may appear e.g. in

curving, where the radius of the curve and the so-called super elevation, which

describes the slope of a cross section of the track, are important independent

parameters. All other parameters are considered constant.

If Nis the number of degrees of freedom of the multibody problem with

time, t, as the independent variable and Pa set of independent parameters,

then we obtain the 2Nstate variables xi(t;P), 1 ≤i≤2N, and the dynami-

cal system can be written as a general nonlinear initial value problem on the

form

dx

dt =F(x, t;P), t > 0 (1)

with appropriate initial conditions x(0) where Mis the number of indepen-

dent parameters. Thus, x(t)∈R2Nand P∈RM. The vector function

F(x, t;P)∈R2Nis a nonlinear and in general a non-smooth function of its

arguments.

In addition there are constraint equations. For each wheel set of Ktotal

sets an equation on the form

dxk

dt =fk(x, t;P), k = 1,2, ..., K (2)

expresses that the two wheels on the axle rotate with the same angular ve-

locity (the axle is assumed rigid) or with diﬀerent velocities when an elastic

connection between the wheels is assumed. The rolling contact parameters of

the wheel/rail contact surface are calculated on the basis of the geometrical

contours of the two bodies, their relative orientations and the normal load

in the contact surface. In real life the relations are non-smooth, and must

4

be evaluated numerically and tabulated. The condition that the wheels and

rails are in contact is expressed by a set of constraint equations that com-

bine the kinematic contact variables in a nonlinear relation. These relations

together with other possible contact conditions between the bodies in the

system constitute a set of constraint equations. These reduce the number of

generalized coordinates in the problem to a value below N. Under the inﬂu-

ence of dynamical forces on the system some of these relations may become

time dependent. The sudden changes in the number of generalized coordi-

nates – for example if a wheel lifts oﬀ from the rail – introduces additional

non-smoothnesses in the system.

The wheel/rail forces in the rolling contact are explicitly formulated as

nonlinear relations between the normal and tangent forces in the contact sur-

face on one side and the deformation under normal load and the normalized

accumulated tangential strain velocities in the contact surface – the so-called

creepage – on the other. The resulting tangent forces – denoted the creep

forces – depend non-linearly on the normal forces, the wheel/rail contact

geometry and the creepage. Since the contact surface kinematic relations

depend non-smoothly on the relative orientations of the wheel and the rail

so do the wheel/rail forces. All non-smoothnesses in the dynamical problem

including those that represent sliding contact and impacts should be deﬁned

by the switching boundaries hj(x) = 0, where 1 ≤j≤J, and Jis the num-

ber of non-smoothnesses with corresponding relations. More about that in

subsection 5.2.

In this article we mainly consider equilibrium solutions of the dynamical

problems, therefore the dynamical systems become autonomous.

The dynamics of a complete wagon model running on straight track or

in canted curved tracks can be studied using the Newton-Euler formulation

of the dynamical system. Several reference frames are introduced in order to

simplify the description of the system:

•the inertial reference frame Icannot be used because the model is

quickly moving and the dynamics that need to be observed are in the

order of the 10−3−10−6m.

•atrack following reference frame Fis attached to the centerline of the

track, at the level of the height of the rails, and it moves with the

train. This reference frame can be inertial if the track is straight and

the train moves at constant speed. Otherwise the frame is not inertial

and ﬁctitious forces need to be added to the system.

5

•each body has its own reference frame, called the body following refer-

ence frame that is attached to the center of mass of the body.

•additional reference frames, called the contact point reference frames,

can be used for the modeling of wheel-rail contact forces.

For each body in the system the Newton-Euler relations hold:

n

X

i=1

I~

Fi=m~a (Newton’s Law) (3)

m

X

i=1

B~

Mi=d

dt B[J]B~ω+B~ω ×B[J]B~ω(Euler’s Law) (4)

where I~

Fiand BMiare, respectively, the forces and torques acting on the

center of mass, mand [J] are the mass and the tensor moment of inertia

respectively, ~a and ˙

~ω are the linear acceleration and the angular accelera-

tion of the bodies. The left superscripts stand for the inertial or the body

reference frame. Fictitious forces and torques are added in order to be able

to write all the equations of motion in the track following reference frame.

The simpliﬁcation of negligible torque terms leads to the following ﬁctitious

forces:

F~

Fc=

0

mhv2

Rcos(φt)i

mh−v2

Rsin(φt)i

(5)

where vis the speed of the train, Ris the radius and φtis the cant of the

track in the curve and the subscript Findicates that the force is written

in the track following reference frame. All the bodies will be subject to the

gravitational forces as well:

F~

Fg=

0

−mg sin φt

−mg cos φt

(6)

The contact forces can be split in guidance forces, determined by the normal

load and the positive conicity of the wheels, and creep forces, due to the

sliding of the wheels on the rails. For the modeling of these forces, several

6

approximations exist, that go from the use of a stiﬀ non-linear spring to the

realistic approach to the contact problem. The notation F~

FNland F~

FNrwill

be used to refer to the guidance forces on the left and the right wheel of a

wheel set. In the same way, the notation F~

FCland F~

FCrwill be used for

the creep forces. The total forces on the left and the right wheel of a wheel

set will be denoted by F~

FLand F~

FRrespectively. The torques due to these

forces will also be considered and will be denoted by B~

MLand B~

MR.

The last group of forces that are applied to all the bodies are the suspen-

sion forces. Each element of the suspension, generically called link, will be

characterized by a function fsuch that

F~

Fl

F~

Tl=fF~

bl0,F~

bl,F~vl,F~

θl,F˙

~

θl(7)

where F~

bl0is the length of the link at rest, F~

blis the deformed length of

the link, F~vlis the relative speed of the two attack points of the link, F~

θlis

the deformed angle between the bodies connected by the link and F˙

~

θlis the

angular momentum of the bodies. These quantities can be easily computed

using basic geometry, knowing the positions at which the links are connected

and the state of the dynamics. The characteristic function of the link, that is

usually non-linear, will determine the resulting forces and the torques. Each

suspension system is a collection of spring and damping elements. The total

resulting forces and torques due to the ith suspension system will be denoted

by F~

Fi

sand B~

Mi

s.

Substituting the gravitational, the centrifugal and the suspension forces in

(3) and (4), the equation of motion (EOM) of the car body can be obtained.

m¨

~x =F~

FC

g+F~

FC

c+F~

FSSl

s+F~

FSSt

s(8)

[J]˙

~ω =B~

MC

g+B~

MC

c+B~

MSSl

s+B~

MSSt

s(9)

where the superscript Cstands for the car body and SSl/t indicates the

leading and trailing secondary suspensions. Similarly, the EOM of the leading

bogie frame can be obtained:

m¨

~x =F~

FBl

g+F~

FBl

c+F~

FSSl

s+F~

FP Sll

s+F~

FP Slt

s(10)

[J]˙

~ω =B~

MBl

g+B~

MBl

c+B~

MSSl

s+B~

MP Sll

s+B~

MP Slt

s(11)

where Bstands for the leading bogie frame and P Sll/lt indicate, respectively,

the leading and trailing primary suspensions. A similar notation is used for

7

the trailing bogie frame. Since the wheel sets are spinning on the track, the

pitch angle is not relevant. However, the angular velocity is important in the

computation of the creepages as it is given by the nominal spinning speed

v

r0and the speed perturbation βdue to the odd distribution of the forces

among the wheels. The resulting equations of motion for the leading wheel

set attached to the leading bogie frame can be written as:

m¨

~x =F~

FWll

g+F~

FWll

c+~

F

F Wll

L+~

F

F Wll

R+F~

FP Sll

s(12)

Jφ¨

φ=nM

B Wll

Loφ+nM

B Wll

Roφ+

nB~

MWll

goφ+nB~

MWll

coφ+nB~

MP Sll

soφ(13)

Jχ˙

β=nM

B Wll

Loχ+nM

B Wll

Roχ(14)

Jψ¨

ψ=nM

B Wll

Loψ+nM

B Wll

Roψ+

nB~

MWll

goψ+nB~

MWll

coψ+nB~

MP Sll

soψ(15)

where Wstands for wheel set and the resulting forces are given by the sum

of gravitational, centrifugal, suspension and contact forces. The notation

{~a}istands for the ith component of the vector ~a. Similar equations can be

derived for the remaining wheel sets of the model.

Depending on the level of accuracy that is wanted, assumptions can be made

in order to simplify the model. For example the car body could be considered

ﬁxed if only the dynamics of the wheel sets and the bogie frames need to be

analyzed.

3. Models with impact

The dynamics of Cooperrider’s bogie model [4] has been investigated in

detail. The model is shown on ﬁgure 1. A detailed description of the model

is presented by Kaas-Petersen in [5] (notice a printing error on p.92. G

is correctly 8.08 ·1010 N/m2). The important features of the model are

that the vertical motions are assumed to be so small that the coupling with

the other degrees of freedom can be neglected, and the dynamical system

therefore is reduced to a system of 14 ﬁrst order diﬀerential equations that

describe the horizontal motion of the bogie elements. All bodies are rigid,

the wheel/rail kinematics and the spring and damper constitutive relations

8

are linearized, so the only nonlinearities in the system are the contact forces

between the wheels and the rails. There is a u|u|term in the wheel/rail

creepage/creep force relation, where udenotes the creepage. It means that

the second derivative of the relation does not exist in u= 0. The action of

the wheel ﬂange is modeled by a very stiﬀ linear restoring spring with a dead

band δ. With qdenoting the lateral displacement of a wheel set, this leads

to a non-smoothness at q=±δ, where a jump in the ﬁrst derivative occurs.

2.1 Description of the system 5

(a) Front view of the bogie

2.1 Description of the system 5

(b) Top view of the bogie

Figure 1: The Cooperrider Bogie model.

The trivial solution satisﬁes the system for all values of the speed V, but

it loses stability in a subcritical bifurcation at the speed VH. The u|u|term in

the creepage/creep force relation changes the initial growth of the bifurcating

periodic branch from a square root to a linear function (see True [6]), and the

restoring spring creates a tangent bifurcation that stabilizes the oscillation

at the lower speed the so-called critical speed VC. At higher speeds of the

bogie chaos develops (see Kaas-Petersen [5], Nordstrøm [7] and Isaksen [8]).

The problem is solved numerically. Kaas-Petersen’s continuation routine

PATH [9] is used to calculate the bifurcation diagram for the dynamical sys-

tem. PATH also calculates the eigenvalues of the Jacobian and estimates the

Floquet multipliers of the Poincar´e map in order to determine the stability

of the various branches. Its most important feature is that it uses a mix-

ture of time integration and Newton iteration to ﬁnd the periodic solutions,

9

whereby the computational work is reduced. A periodic solution is treated as

the identity under a Poincar´e map. In this way the program determines the

stable and unstable solutions with the same accuracy. The Poincar´e section

is chosen by PATH in such a way that it is ’suﬃciently transversal’ to the

phase space trajectory. For the numerical integrations the LSODA routine is

used, which automatically switches between stiﬀ and non-stiﬀ solution meth-

ods whenever needed (see Petzold [10]). PATH determines the solutions with

a relative error of 10−9.

In the points q=±δthe Jacobian is not deﬁned, and two possible ways

to handle the non-smoothness were tried. First the singularity was smoothed

by a hyperbolic cosine function around q=±δand second the singularity

was neglected and the integration simply continued across the singularity.

Since no diﬀerence in the resulting dynamics could be detected, and the

computation time was almost the same, the second way was chosen in the

numerical investigation.

Knudsen [11] and Slivsgaard [12] investigated the dynamics of a single-

axle bogie, which is essentially only one half of the Cooperrider bogie. Knud-

sen proved the existence of chaos produced by the singularity in q=±δ. For

the numerical integrations Knudsen used as well the LSODA routine as an

eighth-stage explicit Runge-Kutta pair of order ﬁve and six. It uses vari-

able time step and error control. To approximate the solution between the

integration steps an interpolant with an asymptotic error of the same order

as the global error for the numerical integration was used. The method was

developed by Enright [13]. This solver was chosen because it should be partic-

ularly well suited for the shadowing of a chaotic attractor. Knudsen observed

that the ﬂange forces changed continuously across the singularity in q=±δ

and therefore the singularity was ignored in both integration methods.

Slivsgaard [12] also used PATH and found that the bifurcation of the

periodic solution from the trivial solution is supercritical, and that the initial

growth of the periodic attractor with the speed is linear. When |q|=δ

a grazing bifurcation takes place and the motion becomes chaotic. It is

interesting to compare the result with the bifurcations in the Cooperrider

bogie model [4]. In the Cooperrider model a tangent bifurcation stabilizes the

unstable periodic branch when |q|grows through δ, but in Slivsgaard’s single-

axle bogie model the stable periodic motion becomes chaotic in a grazing

bifurcation.

All the dynamical systems described above did not include the constraint

of the rigid axle.

10

Slivsgaard [12] also investigated a model of the prototype single-axle

steered bogie for the Copenhagen S-trains. The bogie was designed by

Frederich (see ﬁgure 2). Slivsgaard used realistic rail and wheel proﬁles

that are composed of straight lines and circular arcs with diﬀerent radii.

ARGE CARE’s RSGEO routine[14] was used to calculate the ideal contact

points, which may jump when the lateral displacement qvaries, whereby

non-smoothnesses develop, and the contact points may not be uniquely de-

ﬁned in some intervals of q(double-point contact). The double-point con-

tact problem is treated by the method of Sauvage [15]. In the case of a

jump in the contact point Slivsgaard introduced a switching boundary and

treats the event with the appropriate mechanical laws on each side of the

switching boundary. Since no continuation program existed that could solve

Diﬀerential-Algebraic Equation (DAE) systems ordinary time integration in

combination with ramping of the speed was used to follow the paths in the

state-parameter space.

Figure 2: Frederich’s single-axle steered bogie.

Slivsgaard [12] introduced the constraint of the rigid axle that will ensure

that the rotational speed of the two wheels on the axle always are equal, also

when the instantaneous rolling radii on the two wheels diﬀer due to the lateral

displacement of the wheel set. Slivsgaard demonstrated that the diﬀerence

in the creep forces on the two wheels that is caused by the constraint has a

signiﬁcant eﬀect on the dynamics even at low speed with no ﬂange contact.

’The rigid axle constraint’ must therefore always be included in the dynamical

model of the rolling wheel set.

In her argument Slivsgaard [12] compared the bifurcation diagram with

the constraint with the bifurcation diagram without the constraint. Although

11

Figure 3: Bifurcation diagram for Frederich’s bogie.

the bifurcation diagram without the constraint (see ﬁgure 3) does not rep-

resent the realistic dynamics of the railway vehicle, it shows an interesting

nonlinear dynamic feature. At V= 48.29 m/s a symmetry-breaking bifur-

cation leads to a period adding sequence for decreasing speed (see ﬁgure

4a). The asymmetric limit cycle undergoes a period doubling bifurcation at

V= 47.92 m/s, and immediately hereafter a band of chaos exists. This band

disappears in a period three solution. A blow-up of the subsequent bifurca-

tions is shown on ﬁgure 4b. Due to the symmetry of the dynamical problem

the period adding sequence consists of periodic symmetric and asymmetric

windows separated by chaotic bands. Figures 4 show for decreasing speed an

asymmetric period one solution, a symmetric period three, an asymmetric

period two, a symmetric period ﬁve solution etc. Figure 4b shows clearly the

windows with the two asymmetric period two solutions and the symmetric

period ﬁve solution. The largest Lyapunov exponent at V= 47.59 m/s in

the chaotic regime was calculated and it converged to 0.43. A close investi-

gation of the wheel/rail contact at V= 47.59 m/s revealed that the wheel

in its oscillation crosses a jump in the curvature of the rail surface from 300

mm to 80 mm (see ﬁgure 5), when the amplitude of the lateral oscillation

of the wheel set crosses 4.34 mm. The jump in curvature creates a jump in

the shape and size of the contact ellipse, which again makes the creep forces

jump and introduce a non-smoothness in the dynamical system. Slivsgaard

[12] repeated the calculation of the bifurcation diagram with the constraint

12

of the rigid axle and it turns out that the period adding sequence disappears,

and only a tangent bifurcation is left at V= 47.2 m/s (compare with ﬁgure

3).

(a) Detail of ﬁgure 3.

(b) Detail of ﬁgure 4a.

Figure 4: Detail of ﬁgure 3.

Slivsgaard [12] solved the dynamical system either with LSODA or with

a fourth order Runge-Kutta solver. Both solvers use variable error and step

size control. The use of the two diﬀerent solvers will be discussed in section

6.

13

Figure 5: The jump of the contact ellipse.

The investigation of a complete wagon model has recently been performed

by Bigoni in [16]. The model employed two Cooperrider bogies attached to a

car body and four wheel sets with proﬁle S1002. Figure 6 shows the design of

(a) Front view (b) Top view

Figure 6: Design of the Cooperrider bogie attached to a car body.

the model and the location of the suspension elements. The original Cooper-

rider bogie uses torsional springs and dampers in the secondary suspension.

They have been substituted by yaw springs and dampers. The suspension

elements can be linear or non-linear.

14

The rail proﬁle UIC60 with cant 1/40 combined with the wheel proﬁle S1002

cause the appearance of multiple contact points for certain displacements of

the wheel sets. These are approximated by a single patch using the method

proposed by Sauvage [15]. The static parameters for the computation of the

contact forces have been obtained using the RSGEO [14] routine. The nor-

mal load can be found using the Hertz’s contact theory [17] and adjusting

the value with the additional penetration due to the dynamics using Kalker’s

work [18]. The creep forces were found using the Shen, Hedrick and Elkins

non-linear theory [19].

Using the formulation of the multibody problem introduced in section 2, a

system of 66 coupled ﬁrst-order diﬀerential equations has been obtained. The

system can be simpliﬁed using superposition when only suspension elements

with linear characteristic function are used. Also the computation of the

Jacobian can be sped up using the analytical values for the parts that have

linear functions and using diﬀerence approximation for the the wheel sets,

where the contact forces are the only non-linear part of the system. These

simpliﬁcation cannot be performed if the model employs non-linear suspen-

sion elements.

On a straight track, for a adhesion coeﬃcient µ= 0.15, the linear critical

speed of the subcritical Hopf bifurcation that determines the loss of stabil-

ity of the stationary motion has been found to V=110m/s. The non-linear

critical speed at V=51m/s has been obtained by an adiabatic decrease of

the velocity. However, front-rear asymmetrical dynamical behaviors have

been observed for the leading and the trailing part of the wagon. On tightly

curved tracks, when the cant is not suﬃcient to compensate for the centrifu-

gal acceleration, the wheel sets move to ﬂange contact and the dynamics is

stabilized. However, the lateral forces on the wheel-rail contact points are

increased and the possibility of ﬂange climbing is increased. The Hopf bi-

furcation appears only in the realistic speed interval in curves with low cant

deﬁciency, i.e. when the curve is either wide or canted enough to compensate

for the centrifugal acceleration. The subcritical Hopf bifurcation found on

straight track can, in some cases depending on the parameters, change into

a supercritical Hopf bifurcation.

The dynamical problem was solved numerically using the Explicit Singly

Diagonal Implicit Runge-Kutta (ESDIRK) method with appropriate initial

conditions for increasing values of the speed. The ESDIRK method by

Nielsen-Thomsen (ESDIRK34 NT1) [20] is a Runge-Kutta method of or-

der 3 for the solution of stiﬀ systems of ODE’s and index one DAE’s. The

15

type of method is a 4-stage generalized linear method that is reformulated

in a special semi-implicit Runge-Kutta method. The error estimation is by

imbedding a method of order 4 based on the same stages as the method and

the coeﬃcients are selected for ease of the implementation. The method has

4 stages and the stage order is 2. For purposes of generating a dense output

and for initializing the iteration in the internal stages a continuous extension

is derived. The method is A-stable.

4. Models with Dry Friction Contact

In mechanical systems with dry friction contact, with stick/slip between

some bodies in the system, the degrees of freedom of the system will vary

with the changes of the acting dry friction force vector. Such a system is often

referred to as a structure varying system or a structural variant system. In

these systems the switching boundaries that were mentioned in section 2

must be introduced in the state space in order to deﬁne the location of the

non-smoothnesses. At the switching boundaries the switch conditions must

be formulated in order to deﬁne the initial conditions for the continuation

of the integration of the dynamical system in the appropriate domain of the

state space. In this section only one-dimensional dry friction forces occur.

Figure 7: The single-axle bogie with lateral dry friction damper (left) and

with lateral and yaw dry friction damper (right).

Our ﬁrst dynamical model of a railway vehicle with dry friction dampers

with stick/slip was set up to investigate the interaction between the nonlinear

dry friction damping and the nonlinear wheel/rail creep forces. Therefore the

16

model should be so simple that the dynamical features easily can be related

to this interaction without interference from other sources. True and Asmund

[21] therefore started the analysis with a model of a modiﬁcation of half the

Cooperrider bogie. Figure 7 illustrates the model. The stiﬀ spring model of

the action of the wheel ﬂange in the original Cooperrider bogie was left out,

and the linear wheel/rail kinematic relation and the linear characteristic of

the spring was kept in place. This of course might result in unrealistically

large amplitudes of the lateral motion of the wheel set.

The modeling of the stick/slip action in the dry friction is crucial. In order

to control the jump from stick to slip in the friction relation a new heuristic

smooth transition was developed and tried on some simple test cases. The

results were satisfactory, and the dry friction model was therefore adopted

for the vehicle model.

Figure 8: Bifurcation diagrams for the single-axle bogie with and without

lateral dry friction damper.

First the bifurcation diagram of the model with linear dampers was cal-

culated. The dampers were laid out in such a way that the dissipation in

one period of the oscillation would be approximately the same as the dissipa-

tion of the dry friction damper. Then the bifurcation diagram for the same

17

model but now with a lateral dry friction damper and no yaw damper was

calculated. The two bifurcation diagrams were plotted for comparison on the

ﬁgure 8. It is interesting to note that with the dry friction damper the bifur-

cation disappears and a periodic oscillation with a low amplitude exists down

to very low speeds. The amplitude of the oscillation increases fast with the

speed near and on the other side of the bifurcation point. Such a behavior is

known from stochastic dynamical systems, and probably reﬂects the erratic

nature of the stick/slip mechanism in the dry friction damper. We also found

that the amplitude of the oscillation at speeds below the bifurcation point

depends on the initial condition of the dynamical problem. At speeds below

the bifurcation point there exists an entire set of equilibrium solutions to the

dynamical problem but on ﬁgure 8 only one amplitude of one representative

periodic motion out of the entire set is shown.

Then the same model but with an added yaw damper was investigated.

It turns out that the dry friction yaw damper destabilizes the motion at

almost all speeds, because the stick/slip mechanism in combination with

any asymmetry in the initial conditions and the lack of a restoring force

will initiate a drift yaw velocity that leads to a growth of the numerical

value of the yaw angle and thereby to a breakdown of the computations.

The phenomenon is known in mechanics e.g. in vibrational transportation

of objects. For certain combinations of dry friction characteristics of the

two dry friction dampers, however, a bounded solution was found in certain

speed ranges. The motion then was chaotic, and in one case the chaos was

illustrated by a series of pictures of plane domains of attraction for a selection

of speed values on the ﬁgures 9 and 10. Figure 10 is a blow-up of a part of

ﬁgure 9 to illustrate the fractal structure of the domains of attraction.

The dynamical system was solved numerically at discrete values of a grow-

ing speed with appropriate initial conditions. An explicit Runge-Kutta 5/6’th

order solver with variable step length and error control was used for the in-

tegrations of the system.

The unrealistic assumption of no wheel ﬂanges or other motion limiters

were of course contributing to the breakdown of the calculations of the dy-

namics of the model. True and Trepacz [22] therefore introduced a realistic

wheel/rail kinematic relation in the model and repeated the investigations.

The kinematic contact problem was solved by use of ARGE CARE’s RS-

GEO routine[14], but again only the resultant tangent forces were taken into

account in the model. In order to simplify the dynamics the horizontal com-

ponent of the normal forces in the contact surface was kept constant, which

18

Figure 9: A projection of the domain of attraction for the speeds V=43, 45,

47 and 49 m/s (top-left to bottom-right) on the plane spanned by the lateral

displacement of the bogie frame versus the lateral displacement of the axle.

Black points belong to the domain of attraction of the bounded solution

and the white points belong to the domain of attraction of the unbounded

solution.

Figure 10: A blow-up of a domain of attraction for V= 47 m/s. It illustrates

the fractal character of the domain.

19

of course is an unrealistic assumption.

The dynamics was stabilized by theses changes of the model, but speed

intervals, where the calculations break down, still existed. The bounded

motions were again chaotic. On ﬁgure 11 we have plotted the lateral creep

force versus the longitudinal creep force for the two speeds V= 10 and

45 m/s respectively. The scatter of the points and the curves are typical

indications of chaotic motions. Since the symmetry in the model is broken

by the stick/slip in the dampers, ﬁgure 11 only shows half of the trajectories

in the state plane. The other half is obtained by a reﬂection of the trajectories

ﬁrst in the x-axis and afterwards in the y-axis.

Figure 11: Chaotic attractors. Left: V= 10m/s, right: V= 45m/s

In a real 2-axle freight wagon the motion of the axle box relative to the

car body will be limited by a plate (see ﬁgure 12 ). In the lateral direction

the plate acts as a linear spring with a spring constant of 1500 kN/m and

a dead band of 20 mm. In the longitudinal direction the plate acts as an

elastic impact with E= 2.1·1011 and a dead band of 22.5 mm. Eis Young’s

modulus for steel. This very stiﬀ restoring force makes the dynamical system

so stiﬀ that the computation time becomes unacceptably high. We therefore

approximated the impact by an ideally elastic one, where the yaw speed

of the wheel set is the same before and after the impact, but its direction

is reversed. We have compared some computations with either assumption

and found that the dynamics remain the same, but the computation time of

course increases strongly, when the impact is computed with E. If we were

interested in ﬁnding the impact forces, then it would have been necessary to

use the detailed model of the impact.

20

Figure 12: The axle-guidance.

Figure 13: Illustration of the chaotic motion of the attractor.

The limiting plate has almost no inﬂuence on the lateral dynamics, but it

keeps the wheel set from derailment by limiting the maximum yaw motion.

The motion is chaotic, see ﬁgure 13, where the maximum amplitudes of the

lateral oscillations of the wheel set are plotted versus the speed of the vehicle.

The dynamical system was solved numerically, initially with MATLAB,

but then using an explicit Runge-Kutta/Cash/Karp 5/6th order solver with

adaptive step size and error control. The speed of the computations with the

Runge-Kutta method was around 1000 times faster than when MATLAB

was used. MATLAB was, however, used for the post-processing. The time

of the ideally elastic impact, when the yaw speed changes direction, was

21

Figure 14: Periodic motion at V= 5 m/s.

Figure 15: ’Mildly chaotic’ motion at V= 8.75 m/s.

22

Figure 16: Chaotic motion at V= 20 m/s.

approximated by the time in the time stepping sequence when the axlebox

had penetrated the guiding plate. In the case of full elastic impact the

instants, when the axlebox hit the plate and when it left the plate again,

were calculated more accurately by a Newton iteration. In the time interval

of the impact the forces on the axlebox were supplemented by the elastic

reaction forces of the plate.

In the work by True and Brieuc [23], the important inﬂuence of the nor-

mal rail/wheel contact forces on the dynamics, which were missing in the

models by True and Trepacz [22] and True and Asmund [21], was taken into

account. The guiding plate in the model by True and Trepacz [22] was, how-

ever, dropped. The wheel/rail contact model is now a realistic one, and the

kinematic contact problem was solved by Xia’s program WRKIN. The com-

putations no longer break down in this case. The dynamics was investigated

in the speed interval 5 < V < 40 m/s. It was found, that the dynamics was

very complicated and depended strongly on the speed. The results are sum-

marized in table 1, where it is seen that periodic as well as chaotic motion

occur in several speed intervals. As examples we show on ﬁgure 14 state space

23

Figure 17: The stick motion at V= 18.5 m/s. A state of complete stick

starts near t= 12s.

portraits for 5 m/s with periodic motion, on ﬁgure 15 state space portraits for

8.75 m/s with ’mildly chaotic’ motion and on ﬁgure 16 state space portraits

for 20 m/s with fully developed chaos. The existence of chaos was veriﬁed

either by a demonstration of the sensitivity of the solutions to inﬁnitesimal

initial disturbances or by a calculation of the largest Lyapunov exponent.

The problem is non-smooth so Brieuc had to develop a new method for the

calculation of the largest Lyapunov exponent. An interesting feature is the

existence of two very narrow speed intervals at V= 18.5 (printing error in

the reference) and 37 m/s. Notice that the ratio between the two speeds is

2. After a short growing chaotic transient the wheel set gets locked at these

speed values, when the lateral displacement reaches 9 mm and the yaw is

around 0.01 radians ∼0.6 degrees (see ﬁgure 17).

The dynamical problem was solved numerically using the ESDIRK34 NT1

24

method already used in section 3.

Speed intervals [-;-] m/s Attractor type

[5; 8.71],[9.5; 11],[21.5; 24] Periodic, small amplitude

[13; 19],[24.5; 28] Periodic, large amplitude

[11.135; 12.765],[19.5; 21],[25; 36.5] Chaotic

[8.72; 9.32] Mildly chaotic, Type 1

Only for V= 32 m/s and V= 37.5 m/s Mildly chaotic, Type 2

Only for V= 18.5 m/s and V= 37 m/s Complete stick

Table 1: Summary of results from tests due to True and Brieuc [23] where

the inﬂuence of the normal rail/wheel contact forces on the dynamics was

investigated for speeds 5 < V < 40 m/s.

5. Realistic Railway Vehicle Models

5.1. The 4-axle Hopper Wagon on Three-piece Freight Trucks

Figure 18: A 4-axle Chinese Hopper wagon.

Xia [24, 25] investigated the dynamics of a 4-axle empty Chinese Hopper

wagon on a straight track. The wagon (see ﬁgure 18) runs on two ’Three-

piece freight trucks’ (bogies) (see ﬁgure 19) that are the most used bogies

worldwide due to their simplicity, robustness and low price. The dynamics,

25

Figure 19: Three-piece freight truck (bogie).

however, leaves something to be desired. The dynamical model has 81 de-

grees of freedom (DOF) and is loaded with ’non-smoothnesses’. First there

are the non-smoothnesses in the wheel/rail kinematic relations that we have

seen earlier in this work. In addition – and that is unique for this design – all

the damping is performed by dry friction with stick/slip between plane sur-

faces under a dynamically varying normal load. The axle boxes are ﬁt with

adapters that carry the bogie frames. The adapters can slide longitudinally

under the bogie frames with dry friction contact between stops that limit

their relative horizontal motion (see ﬁgure 20). In the only (the secondary)

suspension system between the bolster and the car body (see ﬁgure 21) the

vertical as well as the lateral damping of the relative motion are performed

by dry friction with stick/slip between spring loaded wedge shaped blocks

that are called ’snubbers’. Since the occurrence of stick or slip between the

snubbers depends both on the normal pressure and the resulting shear force

between the contacting surfaces the contact forces establish a non-smooth

coupling between the horizontal and vertical components of the forces and

26

Figure 20: The contact between the end of a frame and an adapter.

Figure 21: A cross-section of the wedge dampers in the Three-Piece Freight

Truck.

thereby also between the horizontal and vertical dynamics. Under the in-

ﬂuence of the dynamic forces the blocks may separate from the bolster or

from the side frame, which is the source of another non-smoothness in the

dynamical system. The rolling between the car body and the bogie frames is

limited by bumper stops that are modelled as very stiﬀ vertical springs with

a dead band. The friction forces on the surfaces of the bumper stops are inte-

grated into the non-smooth yaw friction torque on the car body and bolsters.

Xia used the smoothened heuristic dry friction model that was used in the

works in [21, section 4]. He extended the application to two-dimensional dry

27

friction forces on a plane. Xia introduced a friction direction angle, which

replaces the sign function used in the one-dimensional dry friction analysis.

The wheel/rail kinematics was calculated by his own routine WRKIN. For a

description of the total model and the detailed formulation of the dynamical

system the interested reader is referred to Xia’s thesis [26].

Figure 22: Bifurcation diagrams for the Chinese Hopper wagon. Left for

increasing speed, right for decreasing speed.

Xia’s main results were described in the two bifurcation diagrams on

ﬁgure 22. The left diagram was made for growing speed and the right one

for decreasing speed. The hysteresis is clearly visible. Below V= 16 m/s

the equilibrium solutions found may be a set valued stationary motion or a

combination of set valued stationary and periodic motions. A typical result

for such a motion is shown on ﬁgure 23. At the supercritical bifurcation from

the ’zero’ solution on the left diagram a stable periodic solution develops. It

only exists in a short speed interval after which it changes into a chaotic

motion. For decreasing speed the chaotic attractor is found all the way down

to 21 m/s, where it disappears - probably in a crisis. The maximum speed of

the car in normal use is below 30 m/s ∼108 km/h. On ﬁgure 24 we show the

chaotic lateral displacements of the four wheel sets at V= 29 m/s. As far as

it is possible the results have been compared with tests of the dynamics of

a real hopper car on a railway line, and the test results agree well with the

theoretical values.

Xia used MATLAB for his calculations. The calculations were therefore

very time consuming. The bifurcation diagram on ﬁgure 22 needed one week

28

Figure 23: The motions of the leading wheelset of the leading bogie at speed

V= 20 m/s. Top left: The longitudinal displacement, top right: The lateral

displacement, bottom left: The yaw angle and bottom right: The roll angle

of shared computer time on the cluster of the DTU Informatics department(!)

The entire dynamical system with its constraint equations is a diﬀerential-

algebraic system with index-3. The system was, however, transformed into an

index-1 system by a diﬀerentiation with respect to time of the algebraic stick-

constraint equations in the system. The index-1 system was then integrated

in the domains where the state variables changed continuously by the Runge-

Kutta solver ode45 from MATLAB because it is eﬀective. The system is stiﬀ,

and ﬁrst the ode45 solver was used, and if it failed then an implicit method

was used. Due to the discontinuities each step of the integration of the system

proceeded in eight steps with a loop. The details can be seen in Xia [26],

where also the detailed derivation of the switch conditions are found.

29

Figure 24: The lateral displacements of the wheel sets at the speed V= 29

m/s as a funtion of the distance after the transients are negligible. From top

to bottom: The leading wheel set in the leading bogie, the trailing wheel

set in the leading bogie, the leading wheel set in the trailing bogie and the

trailing wheel set in the trailing bogie.

5.2. The 2-axle Freight Wagon with a Standard UIC-suspension

Mark Hoﬀmann investigated the dynamics of two-axle European freight

wagons with the UIC standard suspension [27, 28, 29]. One wagon is shown

on ﬁgure 25, and its long wheelbase of 10 m distinguishes the wagon from

the majority of two-axle wagons. The construction data were given to us

from The German Railways, DB AG, in Minden. The UIC suspension (see

ﬁgure 26) consists of two double links that connect the car body with a leaf

spring that rests on an axle box. The links act as a pendulum suspension in

both the lateral and longitudinal direction with combined rolling and sliding

friction with stick/slip in the bearings. When the lateral displacement of

30

Figure 25: The Hbbills 311 wagon.

Figure 26: The UIC standard suspension.

a link becomes large, then the lower link will hit the bracket and the pen-

dulum length will be halved for the further motion. The leaf spring damps

the vertical motions through dry friction sliding with stick/slip between the

steel leafs of the spring, and it also acts with a restoring force on the vertical

motion through bending of the leafs. The mathematical model of the leaf

springs that are used on the wagons was formulated by Fancher et al. [30].

The dissipated work is measured by the areas of the hysteresis loops created

31

Figure 27: The links of the UIC standard suspension.

in the dry friction surfaces by the dynamics. Piotrowski [31] formulated the

mechanical and mathematical models for the action of the links (see ﬁgure

27) on the basis of measurements of the behaviour of a real suspension in his

laboratory. They are shown on ﬁgures 28a and 28b. Piotrowski also gave

values for the parameters in his models. Hoﬀmann has demonstrated how ac-

(a) Longitudinal link model (b) Lateral link model

Figure 28: The link models.

32

Figure 29: A comparison between the mathematical model and the measured

hysteresis loop.

curately the measured hysteresis loop in the laboratory can be approximated

by Piotrowski’s model when the model parameters are chosen appropriately

(see ﬁgure 29). The wheel sets are restrained by a guidance plate with a

dead band of 22.5 mm in the longitudinal and 20 mm in the lateral direction.

The action of the guidances is explained in section 4 by True and Trzepacz.

Hoﬀmann handles the non-smoothnesses in the dynamic problem through a

deﬁnition of the switching boundaries and event detection. In Hoﬀmann’s

model the car body and the axles all have their own degrees of freedom, and

the calculation of the instances of events when a trajectory hits a switching

boundary therefore becomes much more elaborate than was the case in our

earlier examples.

The non-smoothness is due to the nature of of the interacting forces i.e.

stick-slip transitions in the suspension model, impacts between the axle box

and axle guidance and discontinuities in the contact parameters for the wheel-

rail contact. Classical solvers are all based on the existence of the derivatives

of the function F(see section 2). The non-smoothnesses tend to have the

following eﬀect on the numerical method: 1) The numerical solution is sim-

ply inaccurate because the progress of the solution is based on non-existing

derivatives of F. This is a common situation for constant step size integration

schemes. 2) The simulation time is unacceptably high because the step size

is forced down near the non-smooth points in order to satisfy the speciﬁed

33

error tolerance. This happens when integration schemes with variable step

size and error control are applied, but it is due to the lack of smoothness of

the local error. The interested reader is referred to Hoﬀmann’s thesis [27] for

a deeper discussion of the solution of this problem.

(a) Without event location (b) With event location

Figure 30: The time steps on the hysteresis loop.

Hoﬀmann illustrated the importance of the location of the events. He

investigated a model hysteresis loop and plotted the discrete solution points

that were calculated by the ESDIRK34 NT1 solver with step and error con-

trol and event location and compared the result with the discrete solution

points that were calculated with the same solver but without event location

(see ﬁgure 30). The comparison between the ﬁgures clearly demonstrates the

increase in the number of steps without the event location, which results in

a larger computational eﬀort. It should be noted that the number of distin-

guishable points in the left hand corner on ﬁgure 30a, may be misleading,

because several points may be lying so densely that the eye cannot separate

them.

It is also evident from ﬁgure 30a that the computation time would increase

enormously if a solver with constant step size had been applied. Such a solver

will namely need a step size that is determined by the density of the points

in the corners in order to satisfy the given error tolerance. Since the step

size is constant the solver must use the same step size also in the integration

along the linear sections.

34

Hoﬀmann compared the dynamics of the diﬀerent types of freight wagons

with UIC standard suspension. His results were presented on time series

plots and bifurcation diagrams. The dynamics is very complicated with set

valued stationary as well as periodic, multi-periodic and chaotic motions.

He found subcritical and supercritical bifurcations into the various kinds of

behavior caused by shear force instabilities and nonlinear resonances as well

as symmetry breaking bifurcations. The interested reader is referred to the

references [27, 28, 29].

The dynamical system is integrated with ESDIRK34 NT1 already men-

tioned in section 3.

The solution to the initial value problem is found by a piecewise integra-

tion strategy where each smooth section is integrated separately. The isolated

events are located during the integration and treated independently. It is cru-

cial to locate the non-smooth events during the integration. The events are

determined by root ﬁnding of the event functions that deﬁne the switching

boundaries between the diﬀerent states of the model. For the details of the

procedure the interested reader is referred to Hoﬀmann [27, section 3.2] .

Newton-Raphson’s method needs the Jacobi matrix of the dynamical sys-

tem. In our case it is a sparse matrix with 68 ·68 = 4624 elements of

which very many are zero. Therefore the dependencies of the function F

are identiﬁed before the integration starts, and only the non-zero elements

are computed. The entries in the Jacobi matrix are computed in a column-

wise fashion because the relative kinematics and interacting forces that are

computed for the relative perturbations related to xjcan be reused for all

non-zero elements in the j’th column.

6. Discussion of numerical methods and challenges

The formulation of railway vehicle dynamical systems based on the phys-

ical principles (3), (4) can be expressed in the form of a general initial value

problem (1). In general there will be no closed form solution except the triv-

ial state solution obtained at low speed and the models are typically both

nonlinear and are subject to non-smoothness (fx. in wheel-rail contact and

suspension forcing). From a practitioners viewpoint, to solve such systems

numerically demands the use of suitable numerical methods for the control

of robustness, accuracy and eﬃciency. These properties are essential and

without them it can be diﬃcult to establish improved insight into the crit-

ical model behavior. A general class of numerical methods for solving (1)

35

that have good support for local error estimation (for use with step size con-

trollers) and event detection for non-smooth problems is the one-step/multi-

stage Runge-Kutta methods.

The general class of m-stage Runge-Kutta (RK) methods for advancing

(1) a single time step ∆tn=tn+1 −tnis given as

gi=xn+ ∆tn

m

X

j=1

aijF(gj, tn+cj∆tn;P)

xn+1 =xn+ ∆tn

m

X

j=1

bjF(gj, tn+cj∆tn;P) (16)

The coeﬃcients of a convergent numerical scheme are typically given in terms

of a Butcher Tableau [32] deﬁned in terms of A∈Rm×m,b∈Rmand c∈Rm.

A Runge-Kutta method is said to be order pif the local truncation error

behaves asymptotically as O(∆tp) for ﬁxed step sizes. For computations one

should only use methods which has order p > 1 due to accuracy concerns.

This rules out the Euler’s explicit method. Local errors committed during

one time step can be estimated by comparing the computed approximate

solution xn+1 to one computed using an embedded Runge-Kutta method.

This local error estimate can for eﬃciency reasons be based on the same

intermediate Runge-Kutta stage values using the following formula

E(m)= ∆tn

m

X

j=1

djF(gj, tn+cj∆tn;P) (17)

where d∈Rm. If the local error estimate is used for variable step size control,

it is often possible to signiﬁcantly improve eﬃciency over ﬁxed step size time

integration by using a variable step size controller that tries to maintain a

constant accuracy level. The use of step size control has the added advantage

that at the same time robustness is improved because thereby exponential

growth of errors that may develop due to choices of step size will not be

permitted.

The local error should be compared to user-deﬁned acceptable error tol-

erances, respectively, absolute aand relative rlevels of accuracy. In case

of non-smoothness such local error estimates may become unreliable because

local smoothness and asymptotic behavior of the solution is assumed. For

this reason, several time steps may be rejected before the time step sizes have

36

been reduced suﬃciently for the error to be acceptable, in which case eﬀort

is wasted but the accuracy is maintained.

Dynamical systems for railway vehicles can exhibit signiﬁcant stiﬀness

due to the presence of widely diﬀerent dynamical time scales in the models.

For stiﬀ systems, stability and not accuracy imposes a constraint on valid

choices of the step sizes and may require signiﬁcant reductions in the step

sizes for securing stability. Explicit numerical schemes have bounded stability

regions and therefore they may incur a performance penalty in such cases -

in particular when the step size is governed by stability needs rather than

accuracy. For this reason, it is customary to choose implicit solvers, which

formally have large absolute (linear) stability regions. In practice, implicit

Runge-Kutta methods require for time step ﬁnding a suﬃciently accurate

root of the nonlinear system G(z) = 0 for the unknown z=xn+1 ∈R2N×2N.

For stiﬀ problems this is typically done using Newton-Raphson’s iterative

method, which can be expressed as a two-recurrence in the compact form

zk+1 =zk+δk, δk=zk−z=−M−1rk, k = 0,1, ... (18)

where M=I−∆tJis an iteration matrix, Iis the identity matrix, J=

∂G(z)/∂z|z=zkis the Jacobian matrix for the nonlinear system and rk=

G(zk) is the residual of the system in the k’th iteration. For non-smooth

problems, the Jacobian matrix can cease to be unique and this can be the

cause of numerical problems if event detection is not used [27]. Reduction in

solution eﬀort per time step of the Runge-Kutta method is typically achieved

by exploiting properties in the coeﬃcients of Aand/or using inexact approx-

imations to Min the inner solve step for determining δk. A class of Runge-

Kutta schemes that is subject to the idea of minimizing the work eﬀort per

step and also have good stability properties are the ESDIRK methods. A

suitable stopping criteria is based on making sure that the errors are suﬃ-

ciently small such that the local truncation error of the chosen RK method

is dominant, i.e. smaller than the acceptable user-deﬁned tolerance level.

A number of pre-packaged scientiﬁc solvers for semi-discrete equations

exist (e.g. see www.netlib.org) but details will not be given, however, they

will be referenced in the following where appropriate.

Performance is another key concern for practical use of solvers. It can be

useful to evaluate the performance of a numerical scheme in terms of algo-

rithmic and numerical eﬃciencies. The algorithmic eﬃciency is measured in

terms of iteration counts (successful/failed steps and function evaluations),

37

0 5 10 15 20

−4

−2

0

2

4

6

8

10 x 10−5 ESDIRK Nielsen−Thomsen 3rd adv. − 4th emb. − tolabs=10−4

Time (s)

Lateral Displacement Leading Wheel set (m)

(a) ESDIRK34 NT1 (a= 10−4)

0 5 10 15 20

−5

−4

−3

−2

−1

0

1

2

3

4

5x 10−3 ESDIRK Nielsen−Thomsen 3rd adv. − 4th emb. − tolabs=10−5

Time (s)

Lateral Displacement Leading Wheel set (m)

(b) ESDIRK34 NT1 (a= 10−5)

0 2 4 6 8 10 12 14 16 18 20

−5

−4

−3

−2

−1

0

1

2

3

4

5x 10−3

Time (s)

Lateral Displacement Leading Wheelset (m)

Runge−Kutta−Fehlberg 3rd adv. − 4th emb. − tolabs=10−4

(c) ERKF34 (a= 10−4)

0 2 4 6 8 10 12 14 16 18 20

−4

−3.5

−3

−2.5

−2

−1.5

Time (s)

Step Size (log10)

(1) ERKF34 − tolabs=10−4

(2) ERKF34 − tolabs=10−5

(3) ESDIRK34 NT1 − tolabs=10−5

(2)

(3)

(1)

(d) Step size histories

Figure 31: Computed results for lateral displacement of leading wheelset of

Cooperrider’s bogie model for diﬀerent user-deﬁned absolute tolerance levels.

Using ESDIRK34 NT1 it is found that a) the transient behavior is fully

damped for a= 10−4and b) periodic oscillations (hunting) are captured for

a= 10−5. With ERKF34 it is found that c) periodic oscillations (hunting)

are captured already at a= 10−4.

and the numerical eﬃciency is a direct measure of wall clock time. To com-

pare alternative methods the step size history needs to be taken into account

and a fair comparison can be done by using the same step size control for

each method together with a speciﬁcation of the same acceptable tolerance

38

εaSolver CPU Time # Fun. ev. # Jac. ev. # Acc. # Rej.

10−4ERKF34 15.34s 74505 12617 6009

ESDIRK34 NT1 1.47s 1181 112 90 20

10−5ERKF34 21.25s 130389 22989 9613

ESDIRK34 NT1 467.12s 428957 34911 25525 9385

Table 2: Performances of the RKF34 and ESDIRK34 NT1 for solving a

transient analysis of 20 s of a Cooperrider model hunting. The table shows

the absolute tolerance used, the method’s names, the wall clock time, the

number of function evaluations, the number of Jacobian evaluations, the

number of accepted steps and the number of rejected steps. ESDIRK34 NT1

with tolerance 10−4fails in detecting the hunting phenomenon.

level. As an example, a recent investigation of the dynamics of the Coop-

errider’s bogie model shown in ﬁgure 5 has been performed on a straight

track at V= 40 m/s (not hunting) and V= 120 m/s (hunting) using two

diﬀerent RK methods with same step size control and diﬀerent tolerance lev-

els. In Figure 31 we present computed results obtained with the package

SDIRK [33] which includes an PI step size control strategy (e.g. see [34]).

The basic version of the package contains the ESDIRK34 NT1 method by

Nielsen-Thomsen [20] and the code has been extended to include an Explicit

Runge-Kutta-Fehlberg method ERKF34 [35], for use in combination with

the existing PI controller to make comparisons fair. A detailed breakdown

of important performance characteristics is given in Table 2. It is noticeable

that the hunting phenomenon can be captured using the explicit ERKF34

but not the implicit ESDIRK34 at a tolerance level a= 10−4. The rea-

son is that the implicit method exhibit strong numerical damping of these

high frequency modes at this tolerance level. With a reduced tolerance level

a= 10−5the implicit ESDIRK34 has reduced numerical damping of the

hunting modes and captures the phenomenon. However, it has a wall clock

time which is close to 22 times larger as a result of more work per step com-

pared to the explicit solver for this tolerance level. This result challenges the

wide use of implicit methods instead of explicit methods. It also highlights

the importance of tuning the (usually user-deﬁned) tolerance level to be able

to resolve a physical phenomenon of interest. It demonstrates that explicit

solvers from a performance viewpoint can be more attractive for both eﬃ-

cient and accurate analysis than an alternative implicit method of similar

39

formal accuracy.

7. Lessons learned

It is highly recommended to employ numerical schemes for dynamic Rail-

way vehicle simulations which employ variable step size control for control

of local errors (targets eﬃciency, robustness and accuracy), introduce the

relevant switching boundaries in the model formulations (targets accuracy

and eﬃciency) and make use of event location for the numerical solution

of non-smooth dynamical problems (targets accuracy and eﬃciency). Final

results should be subject to convergence tests to rule out the possibility of

errors, which may arise from the choice of too relaxed tolerance levels. For

numerical investigation of chaotic dynamics we have experienced that explicit

solvers may have an advantage over implicit solvers, because for accurate re-

sults the step size is bound by accuracy rather than stability requirements

and the explicit methods require less work per step for same formal order of

accuracy.

The time spent with the formulation of the root ﬁnding method for de-

termination of the events and of the laws that apply in the events is a cheap

investment in a numerical routine that then will operate much faster and

yield reliable results. If however the switching boundaries lie very close to-

gether in the state space other strategies may apply, see e.g. Studer[36],

where a modiﬁed scheme is applied.

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