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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-

1 Copyright c

2014 by ASME

(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

2 Copyright c

2014 by ASME

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

3 Copyright c

2014 by ASME

(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

4 Copyright c

2014 by ASME

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

M−1

M

∑

j=1ut,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,wjiNi

jican be derived using the Golub-

Welsch algorithm [8]. Gauss quadrature points and weights

zj1,..., jd,wj1,..., jdN1,...,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

ut,zj1,..., jdwj1,..., jd

σ2

u(t)≈¯

σ2

u(t) =

N1

∑

j1···

Nd

∑

jdut,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.

5 Copyright c

2014 by ASME

(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,...,ilfA

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).

6 Copyright c

2014 by ASME

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.

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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.

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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

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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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