Global Sensitivity Analysis of Railway Vehicle Dynamics on Curved Tracks

Abstract

This work addresses the problem of the reliability of simulations for realistic nonlinear systems, by using efficient techniques for the analysis of the propagation of the uncertainties of the model parameters through the dynamics of the system. We present the sensitivity analysis of the critical speed of a railway vehicle with respect to its suspension design. The variance that stems from parameter tolerances of the suspension is taken into account and its propagation through the dynamics of a full car with a couple of two-axle Cooperrider bogies running on curved track is studied.
Modern Uncertainty Quantification methods, such as Stochastic Collocation and Latin Hypercube, are employed in order to assess the global uncertainty in the computation of the critical speed. The sensitivity analysis of the critical speed to each parameter and combination of parameters is then carried out in order to quantify the importance of different suspension components. This is achieved using combined approaches of sampling methods, ANOVA expansions, Total Sensitivity Indices and Low-dimensional Cubature Rules.

Full text

Proceedings of the ASME 2014 12th Biennial Conference
on Engineering Systems Design and Analysis
ESDA2014
June 25-27, 2014, Copenhagen, Denmark
ESDA2014-20529
GLOBAL SENSITIVITY ANALYSIS
OF RAILWAY VEHICLE DYNAMICS ON CURVED TRACKS
Daniele Bigoni
Dept. of Applied Mathematics and Computer Science
The Technical University of Denmark
Kgs.Lyngby, Denmark DK-2800
Email: dabi@dtu.dk
Allan P. Engsig-Karup
Hans True
Dept. of Applied Mathematics and Computer Science
The Technical University of Denmark
Kgs.Lyngby, Denmark DK-2800
ABSTRACT
This work addresses the problem of the reliability of sim-
ulations for realistic nonlinear systems, by using efficient tech-
niques for the analysis of the propagation of the uncertainties of
the model parameters through the dynamics of the system. We
present the sensitivity analysis of the critical speed of a railway
vehicle with respect to its suspension design. The variance that
stems from parameter tolerances of the suspension is taken into
account and its propagation through the dynamics of a full car
with a couple of two-axle Cooperrider bogies running on curved
track is studied.
Modern Uncertainty Quantification methods, such as Stochas-
tic Collocation and Latin Hypercube, are employed in order to
assess the global uncertainty in the computation of the critical
speed. The sensitivity analysis of the critical speed to each pa-
rameter and combination of parameters is then carried out in
order to quantify the importance of different suspension compo-
nents. This is achieved using combined approaches of sampling
methods, ANOVA expansions, Total Sensitivity Indices and Low-
dimensional Cubature Rules.
NOMENCLATURE
SSL/T Leading/Trailing Secondary Suspensions
PS Primary Suspensions
LL Leading wheel set on the Leading bogie frame
LT Trailing wheel set on the Leading bogie frame
TL Leading wheel set on the Trailing bogie frame
TT Trailing wheel set on the Trailing bogie frame
INTRODUCTION
The last couple of decades have seen the advent of
Computer-Aided Design in many areas of engineering. This al-
lows for enhanced design capabilities and the prediction and un-
derstanding of dangerous phenomena that would be difficult and
expensive to reproduce in physical experiments. The simulation
of deterministic physical systems, however, falls short in the task
of explaining the phenomena that happen in reality. One part of
the problem comes from the fact that models by definition are
simplification of the reality and the engineer in charge of mak-
ing a model bears always in mind Einstein’s words: “Everything
should be made as simple as possible, but not simpler”. This part
of uncertainty is very difficult to be dealt with and the validity of
a particular model can be assessed only through experimentation.
A second kind of uncertainty is related to the correctness of the
working conditions at which the model is applied: in this case
the model is assumed to be describing the physics accurately, but
its working conditions – the parameters involved in the model –
don’t match the reality. This kind of parametric uncertainty can
be dealt with and the continuous improvements in computational
science allows for more involved analysis of the uncertainty.
In this work we will deal with the safety analysis of a com-
plete rail car running on curved track. Railway vehicle dynam-
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(a) Front view (b) Top view
FIGURE 1: THE RAIL CAR.
ics is subject to a number of uncertainties that can affect the
rider’s safety. Some of them are external loads applied to the
system, such as track perturbations, wind gusts or different dis-
positions of the loaded goods. Others uncertainties are related
to the car design, such as the suspension characteristics and the
wheel wear.
The work will focus on the stability of a rail car equipped with
two Cooperrider bogies and running on a curved track [1] under
uncertain suspension characteristics, due to manufacturing tol-
erances. It is now well known that railway vehicles running at
speeds higher than a fixed critical speed develop what is called
the hunting motion: a sideways periodic or chaotic oscillation
that can lead to increased wheel-rail wear in the best case or to
derailment and catastrophic events in the worst case. This phe-
nomenon can be described in terms of nonlinear dynamics of the
system [2] and analyzed using suitable numerical methods for
non-smooth dynamical systems [3].
The analysis of the uncertainty of the riding safety of the ve-
hicle model will not be limited to the quantification of the total
uncertainty, but will also focus on the identification of the pa-
rameters that most influence it. We will do it from a probabilis-
tic point of view, where the safety is more or less sensitive to a
particular suspension component depending on how much of the
uncertainty is caused by it. This allows the engineer to detect the
critical components that are required to be very accurately built
by the manufacturer.
THE VEHICLE MODEL
The vehicle model chosen for this work is a complete rail car
equipped with two Cooperrider bogies and four axles with wheel
profile S1002, running on a curved track with rail profile UIC60.
The rails have a cant of 1/40. A total of 48 suspension compo-
nents connect the car body, the bogie frames and the wheel-sets.
Figure 1 shows the top and frontal view of half of the car. The
dimensions, the masses and the inertia values of the components
of the car are listed in Tab. 1, where the subscript cstands for car
body, ffor bogie frame, wfor wheel set. The dynamical system
is described using the Newton-Euler formulation:
n
∑
i=1
~
Fi=m~a,
m
∑
i=1
Mi=d
dt ([J]·~
ω) + ~
ω×([J]·~
ω),
(1)
where ~
Fiand ~
Miare the forces and torques applied on the center
of mass of the bodies, mand [J]are the mass and tensor moment
of inertia respectively, ~aand ˙
~
ωare the linear acceleration and the
angular acceleration of the bodies.
In our model we will neglect the longitudinal displacements be-
cause we will not take into account the brake and the acceleration
of the car. We will consider lateral and vertical displacement for
all the bodies in the car and we will account also for their three
possible rotations. On the wheel set the pitch angle will not be
considered and instead we will consider only its angular velocity,
to describe the rotation of the wheels. This results in a system of
66 coupled ordinary differential equations (ODEs) describing 35
degrees of freedom.
The static penetration at the contact points between wheels and
rails is obtained using the routine RSGEO [4]. These values are
tabulated and interpolated as needed during the solution of the
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system of ODEs and updated according to Kalker’s work [5] in
order to account for the additional penetration due to the dynam-
ics. The creep forces are approximated using the Shen-Hedrick-
Elkins nonlinear theory [6].
The complete deterministic system is nonlinear and non-smooth
and can be written abstractly as
d
dt u(t) = f(u,t).(2)
The model is implemented in a general framework [1] for
the simulation of railway vehicle dynamics on tangent or curved
tracks. The framework allows, among other things, to select a
variety of numerical ODE solvers and perform some analysis of
the nonlinear dynamics of the system.
Nonlinear dynamics
of the Deterministic Model
The dynamics of the complete car presented in the previ-
ous section were analyzed in [1], for trains running on tangent
and curved tracks. On tangent tracks the car undergoes a sub-
critical Hopf-bifurcation at a speed of vL=114m/s, entering a
periodic limit cycle. This sub-critical Hopf-bifurcation is char-
acterized by a significant fold, setting the critical speed of the
car to vNL =50.47m/s. On tangent tracks the Hopf-bifurcation
can be found using the Lyapunov’s second method for stability
and exploiting the fact that the center line of the track is a point
of equilibrium for the system. The critical speed is then found
using a continuation method following the periodic limit cycle
backward (i.e. decreasing the speed quasi-statically).
On curved track, the Lyapunov’s second method cannot be used
anymore because the center line is not a point of equilibrium
anymore. Thus the system of ODEs needs to be solved first ac-
celerating, to detect the Hopf-bifurcation, and then decelerating
to detect the critical speed for the curve under analysis. It is well
known now that the critical speed decreases when the train is run-
ning through a curve rather than on tangent track. Furthermore it
was found that for some combination of curve profile and vehi-
cle model, the sub-critical Hopf-bifurcation merges with the fold
into a super-critical Hopf-bifurcation: this means that the speed
where the Hopf-bifurcation occurs is also the one where the pe-
riodic limit cycle (the hunting motion) disappears when ramping
down the velocity.
Figure 2 shows an example of a bifurcation analysis for the
car running through a curve with radius 1600mand with the track
super-elevated on the outer rail of 110mm. Both the bifurca-
tion point and the folding point cannot be detected precisely, but
we can design a criteria based on the qualitative observation of
the data. Using a sliding window Fourier analysis of the lat-
eral displacement of the different components, and adjusting for
the fact that the train is running on a curved track, we can de-
Parm. Value Unit Parm. Value Unit
r00.425 [m]a0.75 [m]
h10.0762 [m]h21.5584 [m]
l10.30 [m]l20.30 [m]
l30.30 [m]x10.349 [m]
v10.6488 [m]v20.30 [m]
v30.30 [m]v40.3096 [m]
s10.62 [m]s20.6584 [m]
s30.68 [m]s40.759 [m]
u17.5 [m]u21.074 [m]
mc44388.0 [kg]Icx 2.80 ·105[kgm2]
Icy 5.0·105[kgm2]Icz 5.0·105[kgm2]
mf2918.0 [kg]If x 6780.0 [kgm2]
If y 6780.0 [kgm2]If z 6780.0 [kgm2]
mw1022.0 [kg]Iwx 678.0 [kgm2]
Iwy 80.0 [kgm2]Iwz 678.0 [kgm2]
K1 1823.0 [kN/m] K2 3646.0 [kN/m]
K3 3646.0 [kN/m] K4 182.3 [kN/m]
K5 333.3 [kN /m] K6 903.35 [kN/m]
D1 20.0 [kNs/m] D2 29.2 [kNs/m]
D6 166.67 [kNs/m]
TABLE 1: DIMENSIONS, MASS, INERTIA AND SUSPEN-
SION PARAMETERS OF THE RAIL CAR.
fine a detection criteria for the end of the hunting motion, based
on the remaining power in the signal kYk: a threshold of 10−5
was found to be a good indicator of the disappearance of the
hunting motion. The application of such criteria can be seen in
Fig. 2b. The legend in the figure stands for the different bod-
ies: CB=“Car Body”, LB=“Leading Bogie frame”, TB=“Trailing
Bogie Frame”, LBLW=“Leading Wheel-set of the Leading Bogie
frame”, and so on.
The Stochastic Model
In the previous model we made the unrealistic assumption
that we knew exactly the parameters involved in the system.
From now on we will admit that the suspension parameters are
not exactly known, but we can describe them with probability
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(a) Bifurcation diagram
10 1 5 20 2 5 3 0 35
Sp e e d (m /s)
10 -11
10 -10
10 -9
10 -8
10 -7
10 -6
10 -5
10 -4
10 -3
Y
Crit ic a l Sp e e d = 2 3. 47 m /s
CB
LB
TB
LBLW
LBTW
TBL W
TBTW
CB
LB/TB
TBLW/TBTW
LBLW/LBTW
(b) Critical speed detection criteria
FIGURE 2: NONLINEAR DYNAMICS OF THE RAIL CAR ON CURVED TRACK.
distributions. With this setting we want to model the realistic
case where manufacturing fluctuations are present in the suspen-
sion components.
In a rigorous setting, the distribution of such parameters should
be assessed from collected data. Several approaches, that make
different assumptions, are available in order to construct a prob-
ability distribution from data. One of the most popular is the
Kernel Smoothing [7, Ch. 6].
Due to the lack of data, in this work the probability distribu-
tions of the suspension parameters will be assumed to be Gaus-
sian around their nominal values with a standard deviation of
5%. This assumption does not undermine the applicability of the
method to other settings, where other distributions might be more
suitable. We let Zbe the d-dimensional vector of random vari-
ables {zi∼N(µi,σi)}d
i=1describing the suspension parameters,
where dis called the co-dimension of the system. The stochastic
dynamical system that we will aim to solve is then of the form
d
dt u(t,Z) = f(u,t,Z),(0,T]×Rd.(3)
With this system we will investigate the critical speed vNL (Z)
and the sensitivity of it with respect to Z.
SENSITIVITY ANALYSIS
Sensitivity analysis is used to identify the input parameters
that affect the model output in the biggest amount. This analysis
provides a useful tool to engineers in both the design phase and
in the risk analysis phase of the production.
The traditional approach to a sensitivity analysis is to investigate
the partial derivatives of a Quantity of Interest (QoI) with respect
to the parameters. The directions with the highest gradients will
be considered the most influential. Due to the locality of deriva-
tives, this method goes under the name of local sensitivity analy-
sis and it reduces to the computation of finite difference formulas
around the nominal values of the parameters.
In this work we will instead look at the global sensitivity: the
most influential parameters in the system are represented by the
ones that give the biggest contribution to the total variance of the
model output. This approach is not restricted to small perturba-
tions, but it takes into account the uncertainty on the parameter
values.
Uncertainty Quantification (UQ)
The solution of (3) is u(t,Z), varying in the parameter
space. The random vector Zis defined in the probability space
(Ω,F,µZ), where Fis the Borel set constructed on Ωand µZ
is a probability measure (i.e. µZ(Ω) = 1). In uncertainty quan-
tification we are interested in computing the density function of
the solution and/or its first moments, e.g. mean and variance:
µu(t) = E[u(t,Z)]ρZ=ZΩdu(t,z)dFZ(z),
σ2
u(t) = Var [u(t,Z)]ρZ=ZΩd(u(t,z)−µu(t))2dFZ(z),
(4)
where ρZ(z)and FZ(z)are the probability density function
(PDF) and the cumulative distribution function (CDF) respec-
tively. Several techniques are available to approximate these
high-dimensional integrals. In the following we present the two
main classes of these methods.
Sampling based methods. The most known sampling
method is the Monte Carlo (MC) method, which is based on the
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law of large numbers. Its estimates are:
µu(t)≈¯
µu(t) = 1
M
M
∑
j=1
ut,Z(j),
σ2
u(t)≈¯
σ2
u(t) = 1
M−1
M
∑
j=1ut,Z(j)−¯
µu(t)2,
(5)
where nZ(j)oM
j=1are realizations sampled randomly with respect
to the probability distribution Z. The MC method has a proba-
bilistic error of O(1/√M), thus it suffers from the work effort
required to compute accurate estimates (e.g. to improve an es-
timate of one decimal digit, the number of function evaluations
necessary is 100 times bigger). However the MC method is very
robust because this convergence rate is independent of the co-
dimension of the problem, so its useful to get approximate esti-
mates of very high-dimensional integrals.
Sampling methods with improved convergence rates have been
developed, such as Latin Hypercube sampling and Quasi-MC
methods. However, the improved convergence rate comes at the
expense of several drawbacks, e.g., the convergence of Quasi-
MC methods is dependent of the co-dimension of the problem
and Latin Hypercube cannot be used for incremental sampling.
Cubature rules. The integrals in (4) can also be com-
puted using cubature rules. These rules are based on a polyno-
mial approximation of the target function, i.e. the function de-
scribing the relation between parameters and QoI, so they have
super-linear convergence rate on the set of smooth functions.
Their applicability is however limited to low-co-dimensional
problems because cubature rules based on a tensor grid suffer
the curse of dimensionality, i.e. if mis the number of points used
in the one dimensional rule and dthe dimension of the integral,
the number of dpoints at which to evaluate the function grows as
O(md). They will however be presented here because they rep-
resent a fundamental tool for the creation of high-dimensional
model representations that will be presented in the next section.
Let Zbe a vector of independent random variables (i.e. Z:Ω→
Rd) in the probability space (Ω,F,µZ), where Fis the Borel
set constructed on Ωand µZis the measure associated to Z.
By the independence of Z, we can write Ωas a product space
Ω=×d
i=1Ωi, with product measure µZ=×d
i=1µi. For A⊆Rd,
we call FZ(A) = µZ(Z−1(A)) the distribution of Z.
For each independent dimension of Ωwe can construct orthog-
onal polynomials {φn(zi)}Ni
n=1,i=1,...,d, with respect to the
probability distribution Fi, where FZ=×d
i=1Fi[8]. The tensor
product of such basis forms a basis for
L2
FZ=f:I⊆Rd→R


ZI
f2(z)dFZ(z) = Var[f(Z)] <∞
(6)
that means that there exists a projection operator P
N:L2
FZ→PN
such that for any f∈L2
FZ, and with the notation i= (i1,...,id)∈
[0,...,N1]×. .. ×[0,...,Nd],
f≈P
Nf=
N1,...,Nd
∑
i=0
ˆ
fiΦi,ˆ
fi=
(f,Φi)L2
FZ
kΦik2
L2
FZ
,(7)
where Φi=∏k∈iφk,kfk2
L2
FZ
= ( f,f)L2
FZ
and
(f,g)L2
FZ
=ZRdf(z)g(z)dFZ(z)(8)
In the following we will be marginally interested in the approx-
imation (7) of the QoI function. However the fast – possibly
spectral – convergence of such approximation is inherently con-
nected with the convergence in the approximation of statistical
moments, because µf=ˆ
f0and σ2
f=∑iˆ
f2
i−ˆ
f2
0[9].
From the orthogonal polynomials used in the construc-
tion of (7), the 1-dimensional Gauss quadrature points
and weights zji,wjiNi
jican be derived using the Golub-
Welsch algorithm [8]. Gauss quadrature points and weights
zj1,..., jd,wj1,..., jdN1,...,Nd
j1,..., jd=1for the tensor product space can be
obtained as tensor product of one dimensional cubature rules (see
fig. 3a), obtaining the following approximations for (4):
µu(t)≈¯
µu(t) =
N1
∑
j1···
Nd
∑
jd
ut,zj1,..., jdwj1,..., jd
σ2
u(t)≈¯
σ2
u(t) =
N1
∑
j1···
Nd
∑
jdut,zj1,..., jd−¯
µu(t)2wj1,..., jd
(9)
Gauss quadrature rules of order Nare accurate for polynomi-
als of order up to degree 2N−1. This high accuracy comes
at the expense of the curse of dimensionality due to the use of
tensor products in high-dimensional integration. This effect can
be alleviated by the use of Sparse Grid technique proposed by
Smolyak [10] that uses an incomplete but accurate version of
the tensor product. However, in the following section we will
see that we can often avoid working in very high-dimensional
spaces.
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(a) Tensor grid (b) cut-HDMR grid
FIGURE 3: EXAMPLE OF THE DISTRIBUTION OF THE POINTS IN TENSOR CUBATURE RULES AND cut-HDMR ACCOUNT-
ING FOR 2nd ORDER INTERACTIONS.
High-Dimensional Model Representation (HDMR)
High-dimensional models are very common in practical ap-
plications, where a number of parameters influence the dynam-
ical behavior of a system. These models are very difficult to
handle, in particular if we consider them as black-boxes where
we are only allowed to change parameters. One method to cir-
cumvent these difficulties is the HDMR expansion [11], where
the high-dimensional function f:Ω→R,Ω⊆Rdis represented
by a function decomposed with lower order interactions:
f(x)≡f0+∑
i
fi(xi) + ∑
i<j
fi,j(xi,xj) + ···+f1,2,...,d(x1,...,xd).
(10)
This expansion is exact and exists for any integrable and mea-
surable function f, but it is not unique. There is a rich variety
of such expansions depending on the projection operator used
to construct them. The most used in statistics is the ANOVA-
HDMR where the low dimensional functions are defined by
fA
0≡PA
0f(x) = ZΩ
f(x)dµ(x),
fA
i(xi)≡PA
if(x) = ZΩi
f(x)∏
i6=j
dµj(xj)−PA
0f(x),
fA
i1,...,il(xi1,...,xil)≡PA
i1,...,ilf(x) =
ZΩi1,...,il
f(x)∏
k/∈{i1,...,il}
dµk(xk)−
∑
k1<···<kl−1∈{i1,...,il}
PA
k1,...,kl−1f(x)−
. . . −∑
k∈{i1,...,il}
PA
kf(x)−PA
0f(x),
(11)
where Ωi1,...,il⊆Ωis the hypercube excluding indices i1,...,il
and µis the product measure µ(x) = ∏d
i=1µi(xi). This expansion
can be used to express the total variance of f, by noting that
D≡E(f−f0)2=∑
i
Di+∑
i<j
Di,j+···+D1,2,...,d,
Di1,...,il=ZΩi1,...,ilfA
i1,...,il(xi1)2∏
k∈{i1,...,il}
dµk(xk),
(12)
where Ωi1,...,il⊆Ωis the hypercube including indices i1,...,il.
However, the high-dimensional integrals in the ANOVA-HDMR
expansion are computationally expensive to evaluate.
An alternative expansion is the cut-HDMR, that is built by su-
perposition of hyperplanes passing through the cut center y=
(y1,...,yd):
fC
0≡PC
0f(x) = f(y),
fC
i(xi)≡PC
if(x) = fi(xi)−PC
0f(x),
fC
i1,...,il(xi1,...,xil)≡PC
i1,...,ilf(x) =
fi1,...,il(xi1,...,xil)−
∑
k1<···<kl−1∈{i1,...,il}
PC
k1,...,kl−1f(x)−
. . . −∑
k∈{i1,...,il}
PC
kf(x)−PC
0f(x),
(13)
where fi1,...,il(xi1,...,xil)is the function f(x)with all the re-
maining variables set to y. This expansion requires the evalu-
ation of the function fon lines, planes and hyperplanes passing
through the cut center (see fig. 3b).
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If cut-HDMR (13) is a good approximation of fat order L, i.e.
considering up to L-terms interactions in (10), such an expansion
can be used for the computation of ANOVA-HDMR in place of
the original function. This reduces the computational cost dra-
matically: let dbe the number of parameters and sthe number
of samples taken along each direction (being them MC samples
or cubature points), then the cost of constructing cut-HDMR in
terms of function evaluations is
L
∑
i=0
d!
(d−i)!i!(s−1)i(14)
Total Sensitivity Indices
The main task of Sensitivity Analysis is to quantify the sen-
sitivity of the output with respect to the input. In particular it is
important to know how much of this sensitivity is accountable
to a particular parameter. With the focus on global sensitivity
analysis, the sensitivity of the system to a particular parameter
can be expressed by the variance of the output associated to that
particular input.
One approach to this question is to consider each parameter sep-
arately and to apply one of the UQ techniques introduced. This
approach goes by the name of one-at-a-time analysis. This tech-
nique is useful to get a first overview of the system. However,
this technique lacks an analysis of the interaction between input
parameters, which in many cases is important.
A better analysis can be achieved using the method of Sobol [11].
Here single sensitivity measures are given by
Si1,...,il=Di1,...,il
D,for 1 ≤i1<·· · <il≤n,(15)
where Dand Di1,...,ilare defined according to (12). These express
the amount of total variance that is accountable to a particular
combination i1,...,ilof parameters. The Total Sensitivity Index
(TSI) is the total contribution of a particular parameter to the total
variance, including interactions with other parameters. It can be
expressed by
T S(i) = 1−S¬i,(16)
where S¬iis the sum of all Si1,...,ilthat do not involve parameter
i.
These total sensitivity indices can be approximated using sam-
pling based methods in order to evaluate the integrals involved in
(12). Alternatively, [12] suggests to use cut-HDMR and cubature
rules in the following manner:
1. Compute the cut-HDMR expansion on cubature nodes for
the input distributions (see fig. 3b),
2. Derive the approximated ANOVA-HDMR expansion from
the cut-HDMR,
3. Compute the Total Sensitivity Indices from the ANOVA-
HDMR.
This approach gives the freedom of selecting the level of accu-
racy for the HDMR expansion depending on the level of interac-
tion between parameters. The truncation order Lof the ANOVA-
HDMR can be selected and the accuracy of such expansion can
be assessed using the concept of “effective dimension” of the
system: for q≤1,the effective dimension of the integrand fis an
integer Lsuch that
∑
0<|t|≤L
Dt≥qD,(17)
where tis a multi-index i1,...,iland |t|is the cardinality of such
multi-index. The parameter qis chosen based on a compromise
between accuracy and computational cost.
SENSITIVITY ANALYSIS
ON RAILWAY VEHICLE DYNAMICS
The sensitivity analysis of a dynamical system with respect
to its parameters is a computationally expensive task and this
cost increases dramatically with the number of parameters. We
will adopt the collocation approach presented earlier, thus we
will need to obtain an ensemble of solutions. This ensemble is
formed by the solutions to the Initial Value Problem IVP (3) for
different realizations of the parameters. Each solution is com-
puted using the program DYnamics Train SImulation (DYTSI)
developed in [1] with the Explicit Runge-Kutta-Fehlberg method
ERKF34 [13]. An explicit solver has been used in light of the
analysis performed in [3], where it was found that the hunting
motion could be missed by implicit solvers, used with relaxed
tolerances, due to numerical damping. In particular implicit
solvers are frequently used for stiff problems, like the one treated
here, because their step-size is bounded by accuracy constraints
instead of stability. However, the detection of the hunting motion
requires the selection of strict tolerances, reducing the allowable
step-sizes and making the implicit methods more expensive than
the explicit ones. Since the collocation approach for UQ involves
the computation of completely independent realizations, this al-
lows for a straightforward parallelization of the computations on
clusters. Thus, 25 nodes of the DTU cluster have been used to
speed up the following analysis.
The first step in the analysis of a stochastic system is the char-
acterization of the probability distribution of the QoI. Since the
complete model has co-dimension 48, a traditional sampling
method is the best suited for the task of approximating the in-
tegrals in eq. (4). In order to speed up the convergence, we used
samples generated with the Latin Hyper Cube method [14]. Fig.
7 Copyright c
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(a) Histogram of the critical speed. (b) Convergence of the Latin Hyper Cube
FIGURE 4: APPLICATION OF THE LATIN HYPER CUBE TO OBTAIN THE TOTAL VARIANCE.
4a shows the histogram of the computed critical speeds with re-
spect to the uncertainty in the suspension components. We can
notice a big clustering of outliers around v≈33m/s. This is an
indicator of a discontinuity in the parameter space. In partic-
ular for such combinations of suspension parameters, the vehi-
cle recovers its stability soon after starting ramping the speed,
indicating the merger of the sub-critical Hopf-bifurcation with
the fold to a super-critical Hopf-bifurcation. The convergence
of the method was checked using a the magnitude of change in
the first two estimated moments as shown in figure 4b. Kernel
smoothing [7] has been used to estimate the density function ac-
cording to this histogram. The estimated mean and variance are
¯
µv=27.12m/sand ¯
σ2
v=3.77m2/s2.
One-at-a-time analysis
When each suspension component is considered indepen-
dently from the others, the estimation problem in (4) is reduced to
the calculation of a 1-dimensional integral. This task can be read-
ily achieved by quadrature rules that have proven to be compu-
tationally more efficient on problems of this dimensionality than
sampling methods [15]. Fourth order quadrature rules have been
used to approximate the variances due to the single components.
For the 48 parameters describing the suspensions, this leads to
the solution of 48×4+1=193 Initial Value Problems. The con-
vergence of this method enables a check of accuracy through the
decay of the expansion coefficients of the target function [15].
Figure 5a shows the contribution that each suspension compo-
nent gives to the total variance of the model output. The nomen-
clature of the components is partly explained in the nomen-
clature section at the beginning of the paper: for example,
PSTT RIGHT K2 stands for the right suspension K2 (see fig. 1)
in the primary suspension connecting the trailing wheel set to the
trailing bogie frame. We notice that the analysis doesn’t explain
the whole variance, but only half of it. This means that some of
Suspension One-at-time ANOVA
σ2
vTSI
PSLT LEFT K2 0.08 0.09
PSLT RIGHT K2 0.08 0.09
PSTT LEFT K2 0.17 0.24
PSTT RIGHT K2 0.17 0.24
SSL LEFT D6 0.59 1.07
SSL RIGHT D6 0.59 1.07
SST LEFT D6 0.04 0.15
SST RIGHT D6 0.04 0.15
TABLE 2: SENSITIVITIES OF THE MOST RELEVANT SUS-
PENSIONS.
the variance must be explained by the combined contribution by
several parameters.
This first analysis is anyway useful to get a first selection of the
most relevant suspensions in the system. Table 2 shows the value
of the variance due to the most relevant components. The re-
maining components contribute less than 0.02m2/s2each.
Total Sensitivity Analysis
The calculation of the total sensitivity analysis through the
use of the cut-HDMR representation and of high order quadra-
ture rules, allows to take into account the interaction between
parameters and at the same time limits the amount of computa-
tions required exploiting the fast convergence of the quadrature
rules.
8 Copyright c
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(a) One-at-a-time analysis. (b) ANOVA analysis
FIGURE 5: APPLICATION OF THE ONE-AT-TIME AND ANOVA ANALYSIS.
The complete sensitivity problem involves 48 parameters. For
the full sensitivity analysis using cut-HDMR truncated at sec-
ond order interaction and with second order quadrature rules, this
would result in 1 +2×48 +4×48
2=4609 solutions of the de-
terministic problem. Even if this is affordable in approximately
4 days using 25 nodes of the DTU cluster, we decided to use the
fact that the One-at-a-time analysis already provided a good in-
dication of which components would be the most relevant. We
use this information to perform a more accurate Total Sensitivity
Analysis on the eight suspension components identified before.
The remaining suspension coefficients are set to their nominal
values. Of course this refinement is susceptible to errors if the
underlying function is particularly pathological.
The cut-HDMR representation is truncated at second order inter-
actions, with fourth order quadrature rules. The construction of
such surrogate requires the computation of 1+4×8+16 ×8
2=
481 solutions to the deterministic problem (2). Figure 5b and ta-
ble 2 show the Total Sensitivity Indices for the suspension param-
eters. The total variance represented by this analysis is sufficient
to explain all the variance of the model output, indicating that the
effective dimensionality – see (17) – of the model is L=2. Actu-
ally, the total variance computed using the cut-HDMR represen-
tation exceeds the total variance computed using the Latin Hyper
Cube method. This is due to both the additional computational
noise introduced by the heuristic for the detection of the criti-
cal speed and a discontinuity in the parameter space that makes
the sub-critical Hopf-bifurcation merge with the fold to create
a super-critical Hopf-bifurcation, as shown in the histogram in
figure 4a.
Discussion of the obtained results
The sensitivity analysis of the rail car, running at hunt-
ing speed on a track with a curve radius of 1600mand super-
elevation of 110mm, reveals that the key parameters determining
the critical speed are the yaw dampers in the secondary suspen-
sions and the yaw springs in the trailing primary suspensions in
the leading and trailing bogie frames. These components are ex-
pected to have an important role in the steering of the car in the
curve and the yaw dampers were historically introduced to stabi-
lize the dynamics of rail cars.
However, we remind the reader that these results are strongly
conditioned by the choice of the distributions describing the sus-
pension parameters. If different distributions are used, maybe
based on the observation of the manufacturing uncertainty of the
suspension coefficients, the results could change drastically.
Remarks on uncertainty quantification
and sensitivity analysis
The first question that an engineer performing analysis of
a stochastic model has to wonder about is whether the uncertain
input parameters considered are independent from a probabilistic
point of view (we remind that the events A,Bare independent if
P(A∩B) = P(A)P(B)) or at least uncorrelated. In motivating our
example of the uncertainty on the suspension components, we
mentioned that their values are uncertain at the manufacturing
time. This uncertainty is even more relevant after thousands of
running kilometers, due to the wear. However the two cases are
slightly different: in the first case the value of each component
can be considered independent and uncorrelated from the others,
instead in the second case the wear on each of the components
cannot be considered independent from the others, because they
9 Copyright c
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undergo coupling dynamics! A variety of techniques exist to deal
with this problem, in order to find a map from the high dimen-
sional correlated random variables to a lower dimensional set of
uncorrelated ones. This however goes beyond the scope of this
work. We refer the reader to [9] for a short introduction to the
problem.
CONCLUSIONS
Sensitivity analysis is of critical importance in a wide range
of engineering applications. The traditional approach of local
sensitivity analysis is useful in order to characterize the behav-
ior of a dynamical system in the vicinity of the nominal values
of its parameters, but it fails in describing wider ranges of vari-
ations. The global sensitivity analysis aims at representing these
bigger variations and at the same time it embeds the probability
distributions of the parameters in the analysis. This enables the
engineer to take decisions, such as improving a design, based on
the partial knowledge of the system.
Wrongly approached, a global sensitivity analysis can turn to be a
computationally expensive or even prohibitive task. In this work
a collection of techniques are used in order to accelerate such
analysis for a high-co-dimensional problem. Each of the tech-
niques used allows for a control of the accuracy, e.g., in terms
of convergence rate for the cubature rules and the “effective di-
mension” of the model. This makes the framework flexible and
easily adaptable to problems with more diversified distributions
and target functions.
The analysis performed on the complete car running in the curve
with radius 1600mand super-elevation 110mm showed that the
steering suspension components account for most of the variance
of the system, meaning that their coefficient values must be care-
fully monitored.
It is important to notice that the same settings for global sen-
sitivity analysis can be used for the investigation of different
Quantities of Interests, such as wear in curved tracks, angle of
attack etc., once they have been properly defined. Furthermore,
the “non-intrusive” approach taken allows the engineer to use
closed software for the computations. The machinery for sensi-
tivity analysis needs only to be wrapped around it, without addi-
tional implementation efforts.
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