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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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