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We present an approach to global sensitivity analysis aiming at the reduction of its computational cost without compromising the results. The method is based on sampling methods, cubature rules, high-dimensional model representation and total sensitivity indices. It is applied to a half car with a two-axle Cooperrider bogie, in order to study the sensitivity of the critical speed with respect to the suspension parameters. The importance of a certain suspension component is expressed by the variance in critical speed that is ascribable to it. This proves to be useful in the identification of parameters for which the accuracy of their values is critically important. The approach has a general applicability in many engineering fields and does not require the knowledge of the particular solver of the dynamical system. This analysis can be used as part of the virtual homologation procedure and to help engineers during the design phase of complex systems.
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February 19, 2014 17:10 Vehicle System Dynamics manuscript
Vehicle System Dynamics
Vol. 00, No. 00, Month 200x, 1–16
RESEARCH ARTICLE
Sensitivity analysis of the critical speed in railway vehicle
dynamics
D. Bigoni, H. True and A.P. Engsig-Karup
DTU Compute, The Technical University of Denmark, 303B Matematiktorvet, DK-2800
Kgs. Lyngby, Denmark
(Received 00 Month 200x; final version received 00 Month 200x)
We present an approach to global sensitivity analysis aiming at the reduction of its compu-
tational cost without compromising the results. The method is based on sampling methods,
cubature rules, High-Dimensional Model Representation and Total Sensitivity Indices. It is
applied to a half car with a two-axle Cooperrider bogie, in order to study the sensitivity of
the critical speed with respect to the suspension parameters. The importance of a certain
suspension component is expressed by the variance in critical speed that is ascribable to it.
This proves to be useful in the identification of parameters for which the accuracy of their
values is critically important. The approach has a general applicability in many engineering
fields and does not require the knowledge of the particular solver of the dynamical system.
This analysis can be used as part of the virtual homologation procedure and to help engineers
during the design phase of complex systems.
Keywords: Reliability analysis, Uncertain dynamics, Vehicle safety, Bifurcation analysis
1. Introduction
The past couple of decades have seen the advent of computer simulations for the
study of deterministic dynamical systems arising in any field of engineering. The
reasons behind this trend are both the enhanced design capabilities during pro-
duction and the possibility of understanding dangerous phenomena. However, de-
terministic dynamical systems fall short in the task of giving a complete picture of
reality: several sources of uncertainty can be present when the system is designed
and thus obtained results refer to single realizations, that in a probabilistic sense
have measure zero, i.e. they never happen in reality. The usefulness of these simu-
lations is however proved by the achievements in Computer-Aided Design (CAD).
The studies of stochastic dynamical systems allow for a wider analysis of phenom-
ena: deterministic systems can be extended with prior knowledge on uncertainties
with which the systems are described. This enables an enhanced analysis and can
be used for risk assessment subject to such uncertainties and is useful for decision
making in the design phase. In the railway industry, stochastic dynamical systems
are being considered in order to include their analysis as a part of the virtual
homologation procedure [1], by means of the framework for global parametric un-
certainty analysis proposed by the OpenTURNS consortium. This framework splits
the uncertainty analysis task in four steps:
(a) Deterministic modeling and identification of Quantities of Interest (QoI) and
source of uncertainties
Corresponding author. Email: dabi@dtu.dk
ISSN: 0042-3114 print/ISSN 1744-5159 online
c
200x Taylor & Francis
DOI: 10.1080/0042311YYxxxxxxxx
http://www.informaworld.com
February 19, 2014 17:10 Vehicle System Dynamics manuscript
2
(a) Front view (b) Top view
Figure 1. The half-wagon equipped with the Cooperrider bogie.
(b) Quantification of uncertainty sources by means of probability distributions
(c) Uncertainty propagation through the system
(d) Sensitivity analysis
Railway vehicle dynamics can include a wide range of uncertainty sources. Sus-
pension characteristics are only known within a certain tolerance when they exit
the manufacturing factory and are subject to wear over time that can be described
stochastically. Other quantities that are sub ject to uncertainties are the mass and
inertia of the bodies, e.g. we dont know exactly how the wagon will be loaded, the
wheel and track geometries, that are subject to wear over time, and also external
loadings like wind gusts.
In this work the QoI will be the critical speed of a fixed half-wagon with respect
to uncertain suspension components – step (a). The deterministic and stochastic
models will be presented in section 2. Step (b) requires measurements of the input
uncertainty that are not available to the authors, so the probability distribution
of the suspension components will be assumed to be Gaussian, without losing the
generality of application of the methods used in (c) and (d). Techniques for Un-
certainty Quantification (UQ) will be presented in section 3.1. They have already
been applied in [2] and [3] to perform an analysis of Uncertainty propagation –
step (c). They will turn useful also in section 3.2 and 3.3 for the sensitivity analy-
sis technique to be presented – step (d). This is based on Total Sensitivity Indices
(TSI) obtained from the ANOVA expansion of the function associated to the QoI
[4]. Section 4 will contain the results of such analysis.
2. The Vehicle Model
In this work we will consider a fixed half wagon equipped with a Cooperrider
bogie [5], running on tangent track with wheel profile S1002 and rail UIC60. The
position of the suspension components is shown in fig. 1. The original design of the
Cooperrider bogie included a torsional spring among the secondary suspensions,
connected vertically from the geometrical center of the bogie to the car body, in
order to counteract the yaw motion. The design used in this work substitute such
spring with two yaw springs that execute an equivalent torsional resistance to the
original model. Thus, the spring K6 and the yaw damper D6 are mounted in parallel
in this setting. See table 1 and 2 for the list of parameters of the model used in this
work. In [6] a framework for the simulation of the dynamics of complete wagons
February 19, 2014 17:10 Vehicle System Dynamics manuscript
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running on straight and curved tracks has been implemented and tested based on
the Newton-Euler formulation of the dynamical system:
n
i=1
Fi=ma,
m
i=1
Mi=d
dt ([J]·ω)+ω ×([J]·ω),
(1)
where Fiand Miare respectively the forces and torques acting on the bodies, m
and [J] are the mass and inertia of the bodies, a is the acceleration and ω is the
angular velocity of the bodies.
In this work the wagon will be fixed in order to alleviate the lateral oscillations
during the hunting motion that would, in some cases, break the computations.
The mathematical analysis and the generality of the methods proposed are not
weakened by this assumption, even if the results may change for different settings.
Since we are considering a wagon running at quasi-constant speed, the longitudinal
motion of the bodies has been neglected in the model. The motion of the bogie
frame is then modeled using lateral, vertical and angular degrees of freedoms, with
the following equations of motion:
m¨
x =F
FBl
g+F
FBl
c+F
FSSl
s+F
FPS
ll
s+F
FPS
lt
s,
[J]˙
ω =B
MBl
g+B
MBl
c+B
MSSl
s+B
MPS
ll
s+B
MPS
lt
s,
(2)
where the upper left superscript identifies the reference frame (F, track following,
B, body following) on which the forces are applied, the right superscript identifies:
Bl, the leading bogie frame, SSl, the secondary suspension of the leading bogie
frame, PS
ll/lt, the leading/trailing primary suspensions of the leading bogie frame.
The right subscripts g, c, s refer instead to the gravity, centrifugal (not used for
this work) and suspension forces.
The equations of motion for the wheel sets are given by:
m¨
x =F
FWll
g+F
FWll
c+
F
FW
ll
L+
F
FW
ll
R+F
FPS
ll
s
Jφ¨
φ=M
BW
ll
Lφ+M
BW
ll
Rφ+
B
MWll
gφ+B
MWll
cφ+B
MPS
ll
sφ
Jχ˙
β=M
BW
ll
Lχ+M
BW
ll
Rχ
Jψ¨
ψ=M
BW
ll
Lψ+M
BW
ll
Rψ+
B
MWll
gψ+B
MWll
cψ+B
MPS
ll
sψ
(3)
where the same notation for (2) was used, Wstands for wheel set and the additional
Land Rsubscripts indicate the left and right forces on the axle due to the wheel-
rail contact forces. The pitch motion of the wheel set is substituted by the angular
velocity perturbation βdue to the odd distribution of the forces among the wheels.
The wheel-rail interaction is modeled using tabulated values generated with the
routine RSGEO [7] for the static penetration at the contact points. These values
February 19, 2014 17:10 Vehicle System Dynamics manuscript
4
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]
mf2918.0 [kg]Ifx 6780.0[kgm2]
Ify 6780.0[kgm2]Ifz 6780.0[kgm2]
mw1022.0 [kg]Iwx 678.0[kgm2]
Iwy 80.0[kgm2]Iwz 678.0[kgm2]
Table 1. Dimension (see fig. 1), mass and inertia values for the comp onents of the Cooperrider model. The sub-
script fstands for b ogie frame, whereas wstands for wheel set. The nominal values of the suspension components
are listed in the first column of table 2.
(a) Bifurcation diagram (b) Critical speed detection criteria
Figure 2. Left: complete bifurcation diagram where the folding point is detected by continuation (ramping)
method from the periodic limit cycle. Right: criterion for the determination of the critical speed based on
the power of the lateral oscillations in a sliding window. LB, LLW and LTW stand for the bogie frame,
the leading wheel set and the trailing wheel set respectively.
are then updated using Kalker’s work [8] for the additional penetrations. The
creep forces are approximated using Shen-Hedrick-Elkins nonlinear theory [9]. The
complete deterministic system [6] can be written abstractly as
d
dtu(t)=f(u,t).(4)
It is nonlinear, non-smooth, and it has 28 degrees of freedom.
2.1. Nonlinear dynamics of the deterministic model
The deterministic dynamics of the complete wagon with a couple of Cooperrider
bogies were analyzed in [6]. The stability of the half-wagon model considered in
this work is characterized by a sub-critical Hopf-bifurcation at vL= 114m/s ,as
it is shown in fig. 2(a), and a critical speed vNL =50.47m/s . The critical speed
is found using a continuation method from the periodic limit cycle detected at a
speed greater than the Hopf-bifurcation speed vL. In order to save computational
time, we try to detect the periodic limit cycle at speeds lower than vLperturbing
the system as described in [10]. This is the approach that we will take during
February 19, 2014 17:10 Vehicle System Dynamics manuscript
5
all the computations of critical speeds in the next sections. The criterion used
in order to detect the value of the critical speed is based on the power of the
lateral oscillations in a 1ssliding window of the computed solution. In particular,
a threshold is selected – in this case a strict threshold of 1011 was used – and the
critical speed is defined as the speed at which the power of the lateral displacement
of all the components fall below such threshold. Fig. 2(b) shows how this criterion
is applied.
2.2. The stochastic model
In the following we will assume that the suspension characteristics are not determin-
istically known. Rather, they are described by probability distributions stemming
from the manufacturing uncertainty or the wear.
If experimental information is available, then some standard distributions can be
assumed and an optimization problem can be solved in order to determine the sta-
tistical parameters of such distributions (e.g. mean, variance, etc.). Alternatively
the probability density function of the probability distribution can be estimated
by Kernel smoothing [11, Ch. 6].
Due to the lack of data to the authors, in this work the probability distribu-
tions associated with the suspension components will be assumed to be Gaussian
around their nominal value, with a standard deviation of 5%. We define Zto be
the d-dimensional vector of random variables {zi∼N(μi
i)}d
i=1 describing the
distributions of the suspension components, where dis called the co-dimension of
the system. The stochastic dynamical system is then described by
d
dt u(t, Z)=f(u,t,Z),(0,T]×Rd.(5)
3. Sensitivity analysis
Sensitivity analysis is used to describe how the model output depends on the input
parameters. Such analysis enables the user to identify the most important param-
eters for the model output. Sensitivity analysis can be viewed as the search for the
direction in the parameter space with the fastest growing perturbation from the
nominal output.
One approach of sensitivity analysis is to investigate the partial derivatives of the
output function with respect to the parameters in the vicinity of the nominal out-
put. This approach goes by the name of local sensitivity analysis, stressing the fact
that it works only for small perturbations of the system.
When statistical information regarding the parameters is known, it can be embed-
ded in the global sensitivity analysis, which is not restricted to small perturbations
of the system, but can handle bigger variability in the parameter space. This is the
focus of this work and will be described in the following sections.
3.1. Uncertainty Quantification (UQ)
The solution of (5) is u(t, Z), varying in the parameter space. The random vector Z
is defined in the probability space (Ω,F
Z), where Fis the Borel set constructed
on Ω and μZis a probability measure (i.e. μZ(Ω) = 1). In uncertainty quantification
we are interested in computing the density function of the solution and/or its first
February 19, 2014 17:10 Vehicle System Dynamics manuscript
6
moments, e.g. mean and variance:
μu(t)=E[u(t, Z)]ρZ=Ωd
u(t, z)dFZ(z),
σ2
u(t)=Var [u(t, Z)]ρZ=Ωd
(u(t, z)μu(t))2dFZ(z),
(6)
where ρZ(z)andFZ(z) are the probability density function (PDF) and the cumu-
lative distribution function (CDF) respectively. 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 law of large numbers. Its estimates are:
μu(t)¯μu(t)= 1
M
M
j=1
ut, Z(j),
σ2
u(t)¯σ2
u(t)= 1
M1
M
j=1 ut, Z(j)¯μu(t)2,
(7)
where Z(j)M
j=1 are realizations sampled randomly with respect to the probability
distribution Z. The MC method has a probabilistic error of O(1/M), thus it
suffers from the work effort required to compute accurate estimates (e.g. to improve
an estimate 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-dimensionality of the problem, so its useful to get
approximate estimates 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-dimensionality of the problem and
Latin Hypercube cannot be used for incremental sampling.
Cubature rules
The integrals in (6) can also be computed using cubature rules. These rules
are based on a polynomial approximation of the target function, i.e. the function
describing 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 d points
at which to evaluate the function grow as O(md) . They will however be presented
here because they represent 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. ZRd)inthe
probability space (Ω,F
Z), where Fis the Borel set constructed on Ω and μZ
is the measure of Z. By the independence of Z, we can write Ω as a product
space Ω = ×d
i=1Ωi, with product measure μZ=×d
i=1μi.ForARd,wecall
FZ(A)=μZ(Z1(A)) the distribution of Z.
February 19, 2014 17:10 Vehicle System Dynamics manuscript
7
For each independent dimension of Ω we can construct orthogonal polynomials
{φn(zi)}Ni
n=1,i=1,...,d, with respect to the probability distribution Fi,where
FZ=×d
i=1Fi[12]. The tensor product of such basis forms a basis for
L2
FZ
f:IRdR
I
f2(z)dFZ(z)=Var[f(Z)] <(8)
that means that there exists a projection operator PN:L2
FZPNsuch that for
any fL2
FZ, and with the notation i=(i1,...,i
d)[0,...,N
1]×...×[0,...,N
d],
fPNf
N1,...,Nd
i=0
ˆ
fiΦi,ˆ
fi(f,Φi)L2
FZ
Φi2
L2
FZ
,(9)
where Φi=kiφk,f2
L2
FZ
=(f,f)L2
FZand
(f,g)L2
FZ=Rd
f(z)g(z)dFZ(z) (10)
In the following we will be marginally interested in the approximation (9) of the QoI
function. However the fast – possibly spectral – convergence of such approximation
is inherently connected with the convergence in the approximation of statistical mo-
ments, because μf=ˆ
f0and σ2
f=iˆ
f2
iˆ
f2
0[13].
From the orthogonal polynomials used in the construction of (9), the 1-
dimensional Gauss quadrature points and weights {zji,w
ji}Ni
jican be derived
using the Golub-Welsch algorithm [12]. Gauss quadrature points and weights
{zj1,...,jd,w
j1,...,jd}N1,...,Nd
j1,...,jd=1 for the tensor product space can be obtained as tensor
product of one dimensional cubature rules (see fig. 3(a)), obtaining the following
approximations for (6):
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
(11)
Gauss quadrature rules of order Nare accurate for polynomials of order up to de-
gree 2N1. 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 Grids techniques proposed by Smolyak [14] that
use an incomplete version of the tensor product. However, in the following section
we will see that we can often avoid working in very high-dimensional spaces.
3.2. High-Dimensional Model Representation (HDMR)
High-dimensional models are very common in practical applications, where a num-
ber of parameters influence the dynamical behaviors 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 circumvent these difficul-
ties is the HDMR expansion [15], where the high-dimensional function fR,
February 19, 2014 17:10 Vehicle System Dynamics manuscript
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(a) Tensor grid (b) cut-HDMR grid
Figure 3. Example of points’ distribution for tensor cubature rules (left) and points’ distribution for the
cut-HDMR grid accounting for 2nd order interactions (right).
ΩRdis 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).(12)
This expansion is exact and exists for any integrable and measurable 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
0PA
0f(x)=Ω
f(x)(x),
fA
i(xi)PA
if(x)=Ωi
f(x)
i=j
j(xj)PA
0f(x),
fA
i1,...,il(xi1,...,xil)PA
i1,...,ilf(x)=Ωi1,...,il
f(x)
k/∈{i1,...,il}
k(xk)
k1<···<kl1∈{i1,...,il}
PA
k1,...,kl1f(x)
...
k∈{i1,...,il}
PA
kf(x)PA
0f(x),
(13)
where Ωi1,...,ilΩ is the hypercube excluding indices i1,...,i
land μis the product
measure μ(x)=d
i=1 μi(xi). This expansion can be used to express the total
variance of f,bynotingthat
DE(ff0)2=
i
Di+
i<j
Di,j +···+D1,2,...,d,
Di1,...,il=Ωi1,...,ilfA
i1,...,il(xi1)2
k∈{i1,...,il}
k(xk),
(14)
February 19, 2014 17:10 Vehicle System Dynamics manuscript
9
where Ωi1,...,ilΩ is the hypercube including indices i1,...,i
l. However, the high-
dimensional integrals in the ANOVA-HDMR expansion are computationally ex-
pensive to evaluate.
An alternative expansion is the cut-HDMR, that is built by superposition of hy-
perplanes passing through the cut center y=(y1,...,y
d):
fC
0PC
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<···<kl1∈{i1,...,il}
PC
k1,...,kl1f(x)
...
k∈{i1,...,il}
PC
kf(x)PC
0f(x),
(15)
where fi1,...,il(xi1,...,xil) is the function f(x) with all the remaining variables set
to y. This expansion requires the evaluation of the function fon lines, planes and
hyperplanes passing through the cut center.
If cut-HDMR (15) is a good approximation of fat order L, i.e. considering up to
L-terms interactions in (12), such expansion can be used for the computation of
ANOVA-HDMR in place of the original function. This reduces the computational
cost dramatically: 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!
(di)!i!(s1)i(16)
3.3. Total Sensitivity Indices
The main task of Sensitivity Analysis is to quantify the sensitivity 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 separately and to
apply one of the UQ techniques introduced in section 3.1. This approach goes by
the name of one-at-a-time analysis. This technique 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 [16]. Here single
sensitivity measures are given by
Si1,...,il=Di1,...,il
D,for 1 i1<···<i
ln, (17)
where Dand Di1,...,ilare defined according to (14). These express the amount of
total variance that is accountable to a particular combination i1,...,i
lof param-
eters. The Total Sensitivity Index (TSI) is the total contribution of a particular
parameter to the total variance, including interactions with other parameters. It
February 19, 2014 17:10 Vehicle System Dynamics manuscript
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can be expressed by
TS(i)=1S¬i,(18)
where S¬iis the sum of all Si1,...,ilthat do not involve parameter i.
These total sensitivity indices can be approximated using sampling based methods
in order to evaluate the integrals involved in (14). Alternatively, [4] suggests to use
cut-HDMR and cubature rules in the following manner:
(1) Compute the cut-HDMR expansion on cubature nodes for the input distri-
butions (see fig. 3(b)),
(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 accuracy for the HDMR
expansion depending on the level of interaction between parameters. The trun-
cation 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 q1,the effective dimension of the integrand fis an integer Lsuch that
0<|t|≤L
DtqD, (19)
where tis a multi-index i1,...,i
land |t|is the cardinality of such multi-index. The
parameter qis chosen based on a compromise between accuracy and computational
cost.
4. Sensitivity Analysis on Railway Vehicle Dynamics
The study of uncertainty propagation and sensitivity analysis through dynamical
systems is a computationally expensive task. In this analysis we adopt a colloca-
tion approach, where we study the behaviors of ensembles of realizations. From the
algorithmic point of view, the quality of a method is measured in the number of
realizations needed in order to infer the same accuracy in statistics. Each realiza-
tion is the result of an Initial Value Problem (IVP) computed using the program
DYnamics Train SImulation (DYTSI) developed in [6], where the model presented
in section 2 has been set up and the IVP has been solved using the Explicit Runge-
Kutta-Fehlberg method ERKF34 [17]. An explicit solver has been used in light of
the analysis performed in [18], where it was found that the hunting motion could
be missed by implicit solvers, used with relaxed tolerances, due to numerical damp-
ing. 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 meth-
ods more expensive than the explicit ones. Since the collocation approach for UQ
involves the computation of completely independent realizations, this allows 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 characterization of the
probability distribution of the QoI. Since the complete model has co-dimension 24,
a traditional sampling method, among the ones presented in section 2, is the most
suited for the task of approximating the integrals in eq. (6). Fig. 4(a) shows the
histogram of the computed critical speeds with respect to the uncertainty in the
February 19, 2014 17:10 Vehicle System Dynamics manuscript
11
(a) (b)
Figure 4. Left: histogram of the critical speed obtained using Latin Hypercube sampling and the estimated
density function (KDE) obtained using Kernel Smoothing. Right: pie plot of the Total Sensitivity Indices
on the reduced stochastic model, where only the most influential components are analyzed. (See table 2
for an explanation of the notation)
suspension components. In order to speed up the convergence, we used 200 samples
generated with the Latin Hyper Cube method [19]. Kernel smoothing [11] has been
used to estimate the density function according to this histogram. The estimated
mean and variance are ¯μv=51.83m/s and ¯σv=4.07m2/s2. It is important to keep
in mind that the first two moments do not account for all the information about
the distribution of the QoI unless it is Gaussian. As shown in fig. 4(a), the distri-
bution is not Gaussian and the ensemble spans approximately 14m/s! However, in
this particular case, the outliers appear only in the upper end of the distribution,
whereas the lower end is fairly well defined by the ensemble.
4.1. One-at-a-time analysis
When each suspension component is considered independently from the others,
the estimation problem in (6) is reduced to the calculation of a 1-dimensional inte-
gral. This task can be readily achieved by quadrature rules that have proven to be
computationally more efficient on problems of this dimensionality than sampling
methods [3]. Fourth order quadrature rules have been used to approximate the
variances due to the single components. The convergence of this method enables
a check of accuracy through the decay of the expansion coefficients of the target
function [3].
The second column in Table 2 lists the results of such analysis. The amount
of variance described by this analysis is given by the sum of all the variances:
ˆσ=1.47m2/s2. This quantity is far from representing the total variance of the
stochastic system, suggesting that interactions between suspension components
are important. Anyway the method is useful to make a first guess about which
components are the most important: the critical speed of the railway vehicle model
analyzed in this work shows a strong sensitivity related to the longitudinal springs
(K2) in the trailing wheel set.
4.2. Total Sensitivity analysis
The technique outlined in section 3.3 can fulfill three important tasks: taking into
account parameter interactions, performing the analysis with a limited number of
realizations and enabling an error control in the approximation. In a first stage
we consider the full stochastic model and we construct a cut-HDMR expansion
which takes into account 2nd order interactions and describes the target function
February 19, 2014 17:10 Vehicle System Dynamics manuscript
12
Suspension Nom. Value One-at-time ANOVA ANOVA-Ref.
¯σv¯σvTSI ¯σvTSI
PSLL LEFT K1 1823.0kN/m 0.00 0.03 0.01
PSLL LEFT K2 3646.0kN/m 0.06 0.18 0.06 0.18 0.09
PSLL LEFT K3 3646.0kN/m 0.02 0.13 0.04 0.14 0.07
PSLL RIGHT K1 1823.0kN/m 0.00 0.05 0.02
PSLL RIGHT K2 3646.0kN/m 0.06 0.17 0.06 0.22 0.11
PSLL RIGHT K3 3646.0kN/m 0.03 0.17 0.06 0.10 0.05
PSLT LEFT K1 1823.0kN/m 0.00 0.02 0.01
PSLT LEFT K2 3646.0kN/m 0.54 1.71 0.56 1.29 0.63
PSLT LEFT K3 3646.0kN/m 0.14 0.20 0.07 0.11 0.05
PSLT RIGHT K1 1823.0kN/m 0.00 0.05 0.02
PSLT RIGHT K2 3646.0kN/m 0.55 1.73 0.56 1.22 0.59
PSLT RIGHT K3 3646.0kN/m 0.03 0.13 0.04 0.17 0.08
SSL LEFT K4 182.3kN/m 0.00 0.01 0.00
SSL LEFT K5 333.3kN/m 0.00 0.01 0.00
SSL LEFT K6 903.35kN/m 0.00 0.02 0.01
SSL LEFT D1 20.0kNs/m 0.00 0.02 0.01
SSL LEFT D2 29.2kNs/m 0.02 0.04 0.01
SSL LEFT D6 166.67kNs/m 0.00 0.02 0.01
SSL RIGHT K4 182.3kN/m 0.00 0.01 0.00
SSL RIGHT K5 333.3kN/m 0.00 0.00 0.00
SSL RIGHT K6 903.35kN/m 0.00 0.02 0.01
SSL RIGHT D1 20.0kNs/m 0.00 0.03 0.01
SSL RIGHT D2 29.2kNs/m 0.02 0.04 0.01
SSL RIGHT D6 166.67kNs/m 0.00 0.02 0.01
Table 2. Nominal values of the suspension components, variances and Total Sensitivity Indices of the critical
speed, obtained using the One-at-a-time analysis, the ANOVA expansion of the complete model and the more
accurate ANOVA expansion of the reduced model. The naming convention used for the suspensions works as
follows. PSL and SSL stand for primary and secondary suspension of the leading bogie respectively. The following
L and T in the primary suspension stand for leading and trailing wheel sets. The last part of the nomenclature
refers to the particular suspension components as shown in Fig. 1.
through 2nd order polynomials, computing the realizations for up to 2-dimensional
cubature rules. The ANOVA-HDMR expansion of the cut-HDMR expansion can be
quickly computed, due to the low dimensionality of the single terms in (15). At this
point, the Di1,...,ilvalues in (14) can be obtained and the effective dimensionality
of the target function, given by (19) for q=0.95, is found to be L=2.This
confirms that the 1 and 2 order interactions are sufficient to describe most of the
variance. The third and fourth columns of Table 2 list the total variances induced
by each parameter, including interactions with other parameters, and the Sobol
total sensitivity indices (TSI).
Once the first approximation of the sensitivities is obtained, the parameters with
the lowest sensitivity indices can be fixed to their nominal values and we can
perform a more accurate analysis of the remaining stochastic system. Longitudinal
and vertical springs (K2 and K3) in the primary suspensions have shown to be
very influential for the critical speed of the analyzed model, thus a new cut-HDMR
expansion, with 2nd order interactions and 4nd order polynomial approximation is
constructed. The resulting total variances and total sensitivity indices are listed
in the fifth and sixth column of Table 2. A visual representation of the sensitivity
indices is shown in the pie chart in Fig. 4(b).
The results obtained by the one-at-a-time analysis are confirmed here by the total
sensitivity analysis, but we stress that the latter provide a higher reliability because
February 19, 2014 17:10 Vehicle System Dynamics manuscript
13
(a) Critical Speed vs D6 (b) Maximum Nadal’s Ratio vs D6
Figure 5. Critical speed and Maximum Nadal’s Ratio with respect to the yaw damping coefficients on the
left and right side of the bogie frame. We can see that the value of the critical speed is not significantly
affected by the value of the yaw damping coefficient for the mean value chosen for sensitivity analysis
(1.66 ·105Ns/m). However if the yaw damping coefficient is lowered too much, the intensity of the lateral
oscillations increase, as shown by the growing Nadal Ratio. The missing values in the Critical speed plot are
due to the oscillations being so big, that the model exit the computational domain for which the employed
contact model works. The missing values in the Nadal’s Ratio plot are both due to the computations
exiting the domain of the contact model and due to the vertical force being zero (lifting) at some instants
during the ramping of the speed for the computation of the critical speed.
they describe a bigger part of the total variance of the complete stochastic system.
4.3. Discussion of the obtained results
Even if the results obtained are formally correct, the interpretation of such results
can raise some questions. A railway engineer might wonder why the yaw dampers
D6 are not listed among the most important by the sensitivity analysis. The yaw
dampers in the secondary suspension are known to provide stability to the vehicle
ride, helping to increase its critical speed. This result is true also with the vehicle
model considered here, in fact low values of D6 cause a drastic worsening of the ride
stability. However, the total sensitivity indices embed the probability distributions
of the uncertain parameters in the global sensitivity analysis: the impact of a
component is weighted according to these distributions. Thus we say that the yaw
damper has little influence on the riding stability with respect to the distributions
chosen. A change in the distributions can dramatically change these results, thus
particular care should be taken with the quantification of the source of uncertainty.
To better show this fact, we looked for the relation of the critical speed with
respect to the yaw dampers, for values below the mean value used for sensitivity
analysis (1.66 ·105Ns/m). We selected a range between [1.0·105,1.5·105]Ns/m,
and looked at the value of the critical speed. Figure 5(a) shows such response
surface: the critical speed is not significantly changing when the yaw damping is
high, as it is the case for the nominal value used in sensitivity analysis, but it
increases drastically when the yaw damping is lowered too much. Unfortunately
this doesn’t mean that the car will run more safely. On the contrary, figure 5(b)
shows that the maximum Nadal’s ratio, obtained while decreasing the speed in
the continuation method for the detection of the critical speed, increases while
lowering the yaw damping parameters. This suggests that the lateral oscillations
become more violent and less compensated by the vertical forces. The missing
values in the figures 5(a) and 5(b) are due to the lateral oscillations being outside
the range of applicability of the contact model employed. Additionally, figure 5(b)
February 19, 2014 17:10 Vehicle System Dynamics manuscript
14
has some missing values due to the lifting of a wheel, leading to zero vertical forces.
This example suggests some observations on the extent to which sensitivity anal-
ysis should be used: it provides a measure of how much a QoI depends on a pa-
rameter, when the parameter value is not exactly known. In principle, from a risk
management perspective, we would like the QoI not to be sensitive to any param-
eter – i.e. the change in QoI should be little with respect to the parameter, like
the yaw damper in the flat part of figure 5(a). The fact that a QoI is sensitive to a
certain parameter, doesn’t mean that this will be dangerous, but it must lead to a
more detailed investigation. Furthermore, in real cases of virtual homologation we
must look at several QoIs, as the previous example showed for the Critical Speed
and the maximum Nadal’s ratio.
4.4. 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 inde-
pendent from a probabilistic point of view (we remind that the events A, B are
independent if P(AB)=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, and are even more uncertain af-
ter 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, be-
cause they undergo coupling dynamics! This doesn’t mean we can do nothing, but
we need first to find a map from the correlated random variables, to some lower
dimensional uncorrelated random variables. If the distributions are Gaussians, a
simple Cholesky factorization of the correlation matrix will be sufficient as a map.
In this case uncorrelation implies independency and we are well set for the ap-
plication of the methods presented. If the distributions are non Gaussian, then
additional care should be paid to the particular problem at hand and one possible
solution is the application of the Rosenblatt transformation [13].
The second remark regards the influence of the selection of the Quantity of Interest
in uncertainty quantification and sensitivity analysis. In section 2.1 the continua-
tion method used to estimate the critical speed was presented and the threshold
used to determine the end of the hunting motion was chosen in a conservative
way, as it is shown in fig. 2(b). Thus, the value of the computed critical speed
will depend also on the deceleration chosen for the continuation method, i.e. the
computed critical speed will be exact in the limit when the deceleration goes to
zero. Of course, the exact computation of the critical speed is not computationally
feasible. With the limited computational resources available, we then chose a fixed
deceleration coefficient for the continuation method, and thus we introduced nu-
merical uncertainty in the computations. Furthermore, the value has been found
to be numerically accurate up to the first decimal digit, due to different choices
of initial conditions and the tolerances set in the time steppers (these can have
a large effect, considering the long time integration needed for this problem and
the accumulation of rounding errors). Therefore, the variance expressed from the
analysis is given both by the variance due to the stochastic system and the vari-
ance introduced by the computation of the QoI. This is, however, a conservative
consequence, meaning that a decision taken on the basis of the computed results is
at least as safe as a decision taken using the “exact results”. A test performed with
different initial conditions showed that the sensitivity values found are qualitatively
February 19, 2014 17:10 Vehicle System Dynamics manuscript
REFERENCES 15
accurate up to the first decimal digit.
5. Conclusions
Sensitivity analysis is of critical importance in a wide range of engineering ap-
plications. The traditional approach of local sensitivity analysis is useful in order
to characterize the behavior of a dynamical system in the vicinity of the nominal
values of its parameters, but it fails in describing wider ranges of variations, e.g.,
caused by long-term wear. 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, 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
techniques used allows for a control of the accuracy, e.g., in terms of convergence
rate for the cubature rules in section 3.1 and the “effective dimension” in section
3.3. This makes the framework flexible and easy to be adapted to problems with
more diversified distributions and target functions.
The analysis performed on the half wagon equipped with a Cooperrider bogie shows
a high importance of the longitudinal primary suspensions, and this reflects the
connection between hunting and yaw motion.
It is important to notice that the same settings for global sensitivity 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 sensitivity analysis needs only to be wrapped
around it, without additional implementation efforts.
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