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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

3

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=ma,

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 10−11 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,σ

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,μ

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

M−1

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. Z:Ω→Rd)inthe

probability space (Ω,F,μ

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.ForA⊆Rd,wecall

FZ(A)=μZ(Z−1(A)) the distribution of Z.

February 19, 2014 17:10 Vehicle System Dynamics manuscript

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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:I⊆Rd→R

I

f2(z)dFZ(z)=Var[f(Z)] <∞(8)

that means that there exists a projection operator PN:L2

FZ→PNsuch that for

any f∈L2

FZ, and with the notation i=(i1,...,i

d)∈[0,...,N

1]×...×[0,...,N

d],

f≈PNf

N1,...,Nd

i=0

ˆ

fiΦi,ˆ

fi(f,Φi)L2

FZ

Φi2

L2

FZ

,(9)

where Φi=k∈iφk,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

i−ˆ

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 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 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 f:Ω→R,

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).

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).(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

0≡PA

0f(x)=Ω

f(x)dμ(x),

fA

i(xi)≡PA

if(x)=Ωi

f(x)

i=j

dμj(xj)−PA

0f(x),

fA

i1,...,il(xi1,...,xil)≡PA

i1,...,ilf(x)=Ωi1,...,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),

(13)

where Ωi1,...,il⊆Ω 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

D≡E(f−f0)2=

i

Di+

i<j

Di,j +···+D1,2,...,d,

Di1,...,il=Ωi1,...,ilfA

i1,...,il(xi1)2

k∈{i1,...,il}

dμk(xk),

(14)

February 19, 2014 17:10 Vehicle System Dynamics manuscript

9

where Ωi1,...,il⊆Ω 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

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),

(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!

(d−i)!i!(s−1)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

l≤n, (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

10

can be expressed by

TS(i)=1−S¬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 q≤1,the eﬀective dimension of the integrand fis an integer Lsuch that

0<|t|≤L

Dt≥qD, (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)

PSLL_LEFT_K2

PSLL_LEFT_K3

PSLL_RIGHT_K2

PSLL_RIGHT_K3

PSLT_LEFT_K2

PSLT_LEFT_K3

PSLT_RIGHT_K2

PSLT_RIGHT_K3

(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

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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(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, 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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