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1 Copyright © 2013 by ASME
Proceedings of the ASME 2013 Rail Transportation Division Fall Technical Conference
RTDF2013
October 15-17, 2013, Altoona, Pennsylvania, USA
RTDF2013-4713
MODERN UNCERTAINTY QUANTIFICATION METHODS
IN RAILROAD VEHICLE DYNAMICS
D. Bigoni, A.P. Engsig-Karup, and H. True
Department of Applied Mathematics and Computer Science
The Technical University of Denmark
DK-2800 Kgs. Lyngby, Denmark
ABSTRACT
This paper describes the results of the application of
Uncertainty Quantification methods to a simple railroad vehicle
dynamical example. Uncertainty Quantification methods take
the probability distribution of the system parameters that stems
from the parameter tolerances into account in the result. In this
paper the methods are applied to a low-dimensional vehicle
dynamical model composed by a two-axle truck that is
connected to a car body by a lateral spring, a lateral damper and
a torsional spring, all with linear characteristics.
Their characteristics are not deterministically defined, but
they are defined by probability distributions. The model - but
with deterministically defined parameters - was studied in [1]
and [2], and this article will focus on the calculation of the
critical speed of the model, when the distribution of the
parameters is taken into account.
Results of the application of the traditional Monte Carlo
sampling method will be compared with the results of the
application of advanced Uncertainty Quantification methods
[3]. The computational performance and fast convergence that
result from the application of advanced Uncertainty
Quantification methods is highlighted. Generalized Polynomial
Chaos will be presented in the Collocation form with emphasis
on the pros and cons of each of those approaches.
NOMENCLATURE
,
mass and inertia of the bogie
,,
suspension parameters
,,,,
,,
nonlinear spring constants used to approximate
the flange forces
,
constants determined by the sizes of the semi
axes of the contact ellipse
nominal rolling radius
conicity
INTRODUCTION
In engineering, deterministic models have been extensively
exploited to describe dynamical systems and their behaviors.
These have proven to be useful in the design phase of the
engineering products, but they always fall short in providing
indications of the reliability of certain designs over others. The
results obtained by one deterministic experiment describe, in
practice, a very rare case that likely will never happen.
However, engineers are confident that this experiment will
explain most of the experiments in the vicinity of it, i.e. for
small variation of parameters. Unfortunately, this assumption
may lead to erroneous conclusions, in particular for realistic
nonlinear dynamical systems, where small perturbations can
cause dramatic changes in the dynamics. It is thus critical to
find a measure for the level of knowledge of a dynamical
system, in order to be able to make a reasonable risk analysis
and design optimization.
Risk analysis in the railroad industry is critical for as well
the increase of the safety as for targeting investments. Railroad
vehicle dynamics is difficult to study even in the deterministic
case, where strong nonlinearities appear in the system. A lot of
phenomena develop in such dynamical systems, and the interest
of the study could be focused on different parameters, such as
the ride comfort or the wear of the components. This work will
instead focus on ride safety when high speeds are reached and
the hunting motion develops. The hunting motion is a well
known phenomenon characterized by periodic as well as
aperiodic lateral oscillations, due to the wheel-rail contact
forces, that can appear at different speeds depending on the
vehicle design. This motion can be explained and studied with
notions from nonlinear dynamics [4], combined with suitable
numerical methods for non-smooth dynamical systems [5]. It is
well known that the behavior of the hunting motion is
parameter dependent, thus good vehicle designs can increase
the critical speed. This also means that suspension components
2 Copyright © 2013 by ASME
need to be carefully manufactured in order to really match the
demands of the customer. However, no manufactured
component will ever match the simulated ones. Thus epistemic
uncertainties, for which we have no evidence, and aleatoric
uncertainties, for which we have a statistical description, appear
in the system as a level of knowledge of the real parameters [6].
Uncertainty Quantification (UQ) tries to address the
question: “assuming my partial knowledge of the design
parameters, how reliable are my results?”. This work will focus
on the sensitivity of the critical speed of a railroad vehicle
model to the suspension parameters.
THE VEHICLE MODEL
This work will investigate the dynamics of the well known
simple Cooperrider truck model [2] shown in Fig. 1. The model
is composed by two conical wheel sets rigidly connected to a
truck frame, that is in turn connected to a fixed car body by
linear suspensions: a couple of lateral springs and dampers and
one torsional spring.
Fig. 1: Top view of the Cooperrider truck model.
The following equations govern this dynamical system [2]:
=22
2
,
+
,
(+)(),
=,,
2,+,
[(+)
()],
(1)
where , and are the damping coefficient and the
stiffness coefficients respectively, and are the lateral and
longitudinal creep forces and is the flange force.
The ideally stiff truck runs on a perfect straight track where
the constant wheel-rail adhesion coefficient enters the system
through the lateral and longitudinal creep-forces:
,=
(,)
(,) , ,=
(,)
(,) ,
,=
+
,
,
=()1
3()+1
27 () for ()< 3
1 for ()3
,
()=
,
where and are real numbers that are determined by the size
of the semi axes of the contact ellipse, which are constant in our
problem [7]. The creepages are given by:
=+ , =+
(+) ,
= , =+
() .
The flange forces are approximated by a very stiff non-
linear spring with a dead band:
()=exp
, 0 <
() ,
() , < 0
,
The parameters used for the analysis are listed in the
following:
=4963
= 1.5
=8135
=29200 /
=14.60 10 /
= 0.1823 10 /
= 2.710 10 /
= 0.05
= 0.4572
= 0.910685 10
= 0.60252
= 0.54219
= 6.563 10
=10
= 0.0091
= 0,1474128791 10
= 1,016261260
= 1,793756792
= 0.9138788366 10
= 0.7163
Non linear dynamics of the deterministic model
The dynamics of the deterministic model at high speed has
been investigated in [2]. The existence of a subcritical Hopf-
bifurcation has been detected at =66.61 m/s. Fig. 2 shows
the bifurcation diagram of the deterministic system. The Hopf
bifurcation point is obtained by observation of the stability of
the trivial solution using the eigenvalues of the Jacobian of the
system. The nonlinear critical speed, the fold bifurcation,
characteristic in subcritical Hopf-bifurcations, is found at
=62.02m/s using a ramping method, where the speed is
quasi-statically decreased, according to
=0 ,
if
<<
, otherwise .
(2)
The stochastic model
Let us now consider suspensions that are provided by the
manufacturer with a certain level of working accuracy. Due to
the lack of real data regarding the probability distributions of
3 Copyright © 2013 by ASME
such working accuracies, this initial study will consider
Gaussian distributions to describe them:
,
,
(std. ~ 5%)
~
,
,
(std. ~ 7%)
~,
.
(std. ~ 7%)
(3)
where the symmetry of the model is taken into consideration in
the standard deviation of the parameters and that both
represent two elements. The applicability and efficiency of the
methods presented in the next section will not be affected by
the particular choice of distribution.
Now the deterministic model is turned into a stochastic
model, where the single solution represents a particular
realization and probabilistic moments can be used to describe
the statistics of the stochastic solution.
UNCERTAINTY QUANTIFICATION
The stochastic solution of the system is now represented by
(,), where is a vector of random variables distributed
according to (3). The solution is a function that spans over a
three dimensional random parameter space. The dimension of
the parameter space is called the co-dimension of the dynamical
problem. In this work the focus will be restricted to the first two
moments of this solution, namely the mean [(,)] and
variance [(,)], but the following derivations can be used
similarly for higher moments too. Mean and variance are
defined as
()=[(,)]
=(,)() ,
()=[(,)]=(,)()
()
(4)
where () is the probability density function of the random
vector and the integrals are computed over its domain.
A straightforward way of computing the moments of the
solution is to approximate the integrals as:
()
()=1
,()
,
()
()=1
1 ,()()
,
(5)
where ()
are realizations sampled randomly from the
probability distribution of . This is the Monte-Carlo (MC)
method and it has a probabilistic error of 1
.
Even though the MC methods are really robust and
versatile, such a slow convergence rate is problematic, when
the solution of a single realization of the system is
computationally expensive. Alternative sampling methods are
the Quasi Monte-Carlo methods (QMC). These can provide
convergence rates of ((log )/), where is the co-
dimension of the problem. They use low discrepancy sequences
in order to uniformly cover the sampling domain. Without
presumption of completeness, in this work only the Sobol
sequence will be considered as a measure of comparison with
respect to other advanced UQ methods. QMC methods are
known to work better than MC methods when the integrand is
sufficiently smooth, whereas they can completely fail on an
integrand of unbounded variation [8]. Furthermore, randomized
versions of the QMC method are available in order to improve
the variance estimation of the method.
Stochastic collocation method (SCM)
Collocation methods require the residual of the governing
equations to be zero at the collocation points ()
, i.e.
,()=,(),(0, ]
(0)=,= 0 .
(6)
Fig. 2: Non-linear dynamics of the deterministic system. The subcritical Hopf-bifurcation is highlighted and the critical speed is
determined exactly at
=66.61 /
. The ramping method is then used in order to detect the non-linear critical speed at
=62.02 /
.
4 Copyright © 2013 by ASME
Then an approximation (,) of (,) is found as an
expansion in a set of Hermite polynomials, which are suitable
for approximations of the Gauss distribution functions:
(,)=
() ()
||
,
=1
(,) ()()
=1
,
()
()
()
,
(7)
where we used a cubature rule with points and weights
(),()
. The points ()
are the set of parameter
values for which deterministic solutions must be computed.
Cubature rules with different accuracy levels and sparsity exist.
In this work simple tensor product structured Gauss cubature
rules will be used. These are the most accurate but scale with
(), where is the number of points in one dimension and
is the co-dimension. The fast growth of the number of
collocation points with the dimensionality goes under the name
of “the curse of dimensionality” and can be addressed using
more efficient cubature rules such as Smolyak sparse grids [9].
UNCERTAINTY QUANTIFICATION IN RAILROAD
VEHICLE DYNAMICS
Uncertainty quantification is recently gaining much
attention from many engineering fields and in vehicle dynamics
there are already some contributions on the topic. In [10] a
railroad vehicle dynamic problem with uncertainty on the
suspension parameters was investigated using MC method
coupled with techniques from Design of Experiments.
Here SCM will be applied to the simple Cooperrider truck
[2] in order to study its behavior with uncertainties, and the
results will be compared to the ones obtained by the MC and
QMC methods.
These methods belong to the class of non-intrusive
methods for Uncertainty Quantification. This means that they
only require a deterministic method to compute the quantity of
interest (QoI) for different parameters. In this work this is the
ramping method to detect the critical speed.
The focus of this work is on the determination of the
nonlinear critical speed with uncertainties, so the investigation
of the stochastic dynamics with respect to time will be
disregarded here. Fig. 3 shows the SCM method applied to the
model with 1D uncertainty on parameter , for the
determination of the first two moments of the nonlinear critical
speed. The estimation done by the SCM is already satisfactory
at low order and little is gained by increasing it. This means
that the few first terms of the expansion (7) are sufficient in
approximating the nonlinear critical speed distribution.
Fig. 4 shows the SCM method applied to the same problem
with 1D uncertainty on the torsional spring stiffness . Again
the first few terms in expansion (7) are sufficient in order to
give a good approximation of the nonlinear critical speed
distribution. It is worth noting that the torsional spring stiffness
has an higher influence on the critical speed than .
Fig. 5 shows the SCM method on the problem with
uncertainty on parameters , and . Again, a low-order
SCM approximation is sufficient to get the most accurate
solution.
In the figures 3-5, left, we have compared the convergence
of the SCM method with that of the MC method. Therefore the
number of evaluations was prescribed. It is also of interest to
compare the computation time of the methods expressed by the
CPU time. For the comparison we used the calculated mean
values of the critical speed as the basis for the comparison. For
the SCM method the iteration process was ended when the
second decimal remained constant. The mean values in the MC
and QMC methods change however a good deal as shown in
the figures 3-5, left. Therefore, for the comparison a window
with 20 iterative values, which is glided over the number of
iterations was used. When the second decimal of the average of
Fig. 3: SCM on the model with 1D uncertainty on parameter
compared with MC and QMC. Left, estimation of mean and
variance of the nonlinear critical speed. Right, histograms of NL critical speeds obtained using 500 MC simulations of model (1)-(2)
and
10
realizations using the approximated stochastic solution (7) with only 2 function evaluations. The standard deviation is
shown as a shaded confidence interval, blue for SCM and red for MC. The two confidence intervals are overlapping almost exactly.
5 Copyright © 2013 by ASME
the iterated values in the window remained constant, when the
widow was pushed one more step, then the iterations were
stopped, and the CPU time was stored.
Table 1 shows the final results obtained with the chosen
accuracy, using the three methods, Monte-Carlo (MC), Quasi-
Monte-Carlo (QMC) and Stochastic Collocation (SCM). We
can observe that the variances in the multi-dimensional cases
are almost equal to the sum of the single-dimensional cases.
This means that the effect of the nonlinear interactions between
the three elements of the suspension is small with the variances
chosen in this problem.
CONCLUSIONS
Manufacturing tolerances have been introduced into the
dynamical investigations of vehicles. A new method, the
Stochastic Collocation Method (SCM) is applied as a tool for
“Uncertainty Quantification”, and the accuracy and
computational effort is compared with that of Monte-Carlo
(MC) and Quasi-Monte-Carlo (QMC) methods. The
“Uncertainty Quantification” methods are applied to the
estimate of the calculated critical speed of a railroad vehicle
model. The critical speed is delivered as a mean value with
variance. The results show that under the condition of the same
accuracy the convergence rate of the SCM outperforms the
rates of as well the MC as the QMC methods. Table 1 shows
that the CPU time and thus the computational effort by
application of the SCM is much smaller than the computational
effort by application of the MC or QMC methods. By all three
methods the total computational effort is larger than the effort
by a deterministic computation, because the same dynamical
system must be solved repeatedly only with different parameter
values. Under these conditions it is however possible to reduce
the total elapsed time significantly by straightforward
application of parallel computing. The dynamics of the vehicle
model is calculated in the process, but the results are not shown
here due to the limited space. A very simple model was chosen
in order to demonstrate the superiority of SCM over the MC
Fig. 4: SCM on 1D uncertainty on parameter k6 compared with MC and QMC. Left, estimation of mean and variance of the non-
linear critical speed. Right, histograms of NL critical speeds obtained using 500 MC simulations of model (1)-(2) and
10
realizations using the approximated stochastic solution (7) with only 2 function evaluations. The standard deviation is shown as a
shaded confidence interval, blue for SCM and red for MC. The two confidence intervals are overlapping almost exactly.
Fig. 5: SCM on 3D uncertainty compared with MC and QMC. Left, estimation of the mean and variance of the non-linear critical
speed. Right, histograms of nonlinear critical speeds.
6 Copyright © 2013 by ASME
and QMC methods. By using the same distributions for the
characteristics of the two lateral springs and dampers the effect
of the loss of symmetry in a real vehicle was not investigated
here. SCM can be 100 times faster than MC for low co-
dimensional problems, but for high co-dimensional problems
SCM methods suffer from the “curse of dimensionality”. The
computational effort of the SCM grows very fast with the
number of independent parameters. In a realistic vehicle model
that number easily surpasses 20. Therefore the work continues
with an investigation of the application of statistical methods
that may reduce the computational effort by singling out the
parameters that have the most important influence on the
wanted result of the dynamical problem. Some early results are
shown in [11].
REFERENCES
[1]
N. Cooperrider, "The hunting behavior of conventional
railway trucks,"
ASME Journal of Engineering and
Industry, vol. 94, pp. 752-762, 1972.
[2]
H. True and C. Kaas-Petersen, "A Bifurcation Analysis of
Nonlinear Oscillations in Railway Vehicles," in
Proc. 8th
IAVSD-IUTAM Symposium on Vehicle System Dynamics
,
Lisse, 1984.
[3]
D. Xiu, Numerical Methods for Stochastic Computations:
A Spectral Method Approach, Princeton: Princeton
University Press, 2010.
[4]
H. True, "On the Theory of Nonlinear Dynamics and its
Applications in Vehicle Systems Dynamics,"
Vehicle
System Dynamics , vol. 31, pp. 393-421, 1999.
[5]
H. True, A. P. Engsig-Karup and D. Bigoni, "On the
Numerical and Computational Aspects of Non-
Smoothnesses that occur in Railway Vehicle Dynamics,"
Mathematics and Computers in Simulation, 2012.
[6]
S. F. Wojtkiewicz, M. S. Eldred, R. V. Field, A. Urbina and
J. R. Red-
Horse, "Uncertainty Quantification In Large
Computational Engineering Models,"
American Institute
of Aeronautics and Astronautics , vol. 14, 2001.
[7]
P. J. Vermeulen and K. L. Johnson, "Contact of
no
nspherical elastic bodies transmitting tangential forces,"
Journal of Applied Mathematics, vol. 31, pp. 338-
340,
1964.
[8]
W. J. Morokoff and R. E. Caflisch, "Quasi-Monte Carlo
Integration," Journal of Computational Physics ,
vol. 122,
no. 2, pp. 218-230 , 1995.
[9]
K. Petras, "Smolyak cubature of given polynomial degree
with few nodes for increasing dimension,"
Numerische
Mathematik , vol. 93, no. 4, pp. 729-753 , 2003.
[10]
L. Mazzola and S. Bruni, "Effect of Suspension Parameter
Uncertainty on t
he Dynamic Behaviour of Railway
Vehicles," Applied Mechanics and Materials ,
vol. 104, pp.
177-185 , 2011.
[11]
D. Bigoni, H. True and A. P. Engsig-Karup, "Sensitivity
analysis of the critical speed in railway vehicle dynamics,"
in 23rd International
Symposium on Dynamics of Vehicles
on Roads and Tracks, Qingdao, China, 2013.
MC
QMC
SCM
2
σ
#fE
CPUt
µ
2
σ
#fE
CPUt
µ
2
σ
#fE
CPUt
6
k
62,26
1,64
169
~
24h
62,24
1,47
152
~
21 h
62,23
1,55
2
~
10m
4
k
62,23
0,14
17
~
2,5h
62,25
0,14
22
~
3 h
62,25
0,14
2
~
11m
2
D
62,23
0,02
9
~
1 h
62,25
0,02
4
~
30m
62,25
0,03
2
~
11m
46
,kk
62,22
1,53
148
~
21 h
62,22
1,62
152
~
22 h
62,28
1,69
4
~
36m
26
,Dk
62,18
1,72
216
~
30 h
62,24
1,50
142
~
20 h
62,28
1,57
4
~
37m
24
,Dk
62,25
0,17
25
~
3,5h
62,25
0,16
25
~
3,5h
62,30
0,17
4
~
35m
246
,, Dkk
62,18
1,68
221
~
32 h
62,23
1,63
154
~
22 h
62,23
1,72
8
~
1 h
Table 1: Estimated mean and variance of the nonlinear critical speed using MC, QMC and SCM. The three methods are compared in
terms of number of function evaluations (#fE) and computation time (CPUt).
µ
