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1 Copyright © 2013 by ASME

Proceedings of the ASME 2013 Rail Transportation Division Fall Technical Conference

RTDF2013

October 15-17, 2013, Altoona, Pennsylvania, USA

RTDF2013-4713

MODERN UNCERTAINTY QUANTIFICATION METHODS

IN RAILROAD VEHICLE DYNAMICS

D. Bigoni, A.P. Engsig-Karup, and H. True

Department of Applied Mathematics and Computer Science

The Technical University of Denmark

DK-2800 Kgs. Lyngby, Denmark

ABSTRACT

This paper describes the results of the application of

Uncertainty Quantification methods to a simple railroad vehicle

dynamical example. Uncertainty Quantification methods take

the probability distribution of the system parameters that stems

from the parameter tolerances into account in the result. In this

paper the methods are applied to a low-dimensional vehicle

dynamical model composed by a two-axle truck that is

connected to a car body by a lateral spring, a lateral damper and

a torsional spring, all with linear characteristics.

Their characteristics are not deterministically defined, but

they are defined by probability distributions. The model - but

with deterministically defined parameters - was studied in [1]

and [2], and this article will focus on the calculation of the

critical speed of the model, when the distribution of the

parameters is taken into account.

Results of the application of the traditional Monte Carlo

sampling method will be compared with the results of the

application of advanced Uncertainty Quantification methods

[3]. The computational performance and fast convergence that

result from the application of advanced Uncertainty

Quantification methods is highlighted. Generalized Polynomial

Chaos will be presented in the Collocation form with emphasis

on the pros and cons of each of those approaches.

NOMENCLATURE

,

mass and inertia of the bogie

,,

suspension parameters

,,,,

,,

nonlinear spring constants used to approximate

the flange forces

,

constants determined by the sizes of the semi

axes of the contact ellipse

nominal rolling radius

conicity

INTRODUCTION

In engineering, deterministic models have been extensively

exploited to describe dynamical systems and their behaviors.

These have proven to be useful in the design phase of the

engineering products, but they always fall short in providing

indications of the reliability of certain designs over others. The

results obtained by one deterministic experiment describe, in

practice, a very rare case that likely will never happen.

However, engineers are confident that this experiment will

explain most of the experiments in the vicinity of it, i.e. for

small variation of parameters. Unfortunately, this assumption

may lead to erroneous conclusions, in particular for realistic

nonlinear dynamical systems, where small perturbations can

cause dramatic changes in the dynamics. It is thus critical to

find a measure for the level of knowledge of a dynamical

system, in order to be able to make a reasonable risk analysis

and design optimization.

Risk analysis in the railroad industry is critical for as well

the increase of the safety as for targeting investments. Railroad

vehicle dynamics is difficult to study even in the deterministic

case, where strong nonlinearities appear in the system. A lot of

phenomena develop in such dynamical systems, and the interest

of the study could be focused on different parameters, such as

the ride comfort or the wear of the components. This work will

instead focus on ride safety when high speeds are reached and

the hunting motion develops. The hunting motion is a well

known phenomenon characterized by periodic as well as

aperiodic lateral oscillations, due to the wheel-rail contact

forces, that can appear at different speeds depending on the

vehicle design. This motion can be explained and studied with

notions from nonlinear dynamics [4], combined with suitable

numerical methods for non-smooth dynamical systems [5]. It is

well known that the behavior of the hunting motion is

parameter dependent, thus good vehicle designs can increase

the critical speed. This also means that suspension components

2 Copyright © 2013 by ASME

need to be carefully manufactured in order to really match the

demands of the customer. However, no manufactured

component will ever match the simulated ones. Thus epistemic

uncertainties, for which we have no evidence, and aleatoric

uncertainties, for which we have a statistical description, appear

in the system as a level of knowledge of the real parameters [6].

Uncertainty Quantification (UQ) tries to address the

question: “assuming my partial knowledge of the design

parameters, how reliable are my results?”. This work will focus

on the sensitivity of the critical speed of a railroad vehicle

model to the suspension parameters.

THE VEHICLE MODEL

This work will investigate the dynamics of the well known

simple Cooperrider truck model [2] shown in Fig. 1. The model

is composed by two conical wheel sets rigidly connected to a

truck frame, that is in turn connected to a fixed car body by

linear suspensions: a couple of lateral springs and dampers and

one torsional spring.

Fig. 1: Top view of the Cooperrider truck model.

The following equations govern this dynamical system [2]:

=22

2

,

+

,

(+)(),

=,,

2,+,

[(+)

()],

(1)

where , and are the damping coefficient and the

stiffness coefficients respectively, and are the lateral and

longitudinal creep forces and is the flange force.

The ideally stiff truck runs on a perfect straight track where

the constant wheel-rail adhesion coefficient enters the system

through the lateral and longitudinal creep-forces:

,=

(,)

(,) , ,=

(,)

(,) ,

,=

+

,

,

=()1

3()+1

27 () for ()< 3

1 for ()3

,

()=

,

where and are real numbers that are determined by the size

of the semi axes of the contact ellipse, which are constant in our

problem [7]. The creepages are given by:

=+ , =+

(+) ,

= , =+

() .

The flange forces are approximated by a very stiff non-

linear spring with a dead band:

()=exp

, 0 <

() ,

() , < 0

,

The parameters used for the analysis are listed in the

following:

=4963

= 1.5

=8135

=29200 /

=14.60 10 /

= 0.1823 10 /

= 2.710 10 /

= 0.05

= 0.4572

= 0.910685 10

= 0.60252

= 0.54219

= 6.563 10

=10

= 0.0091

= 0,1474128791 10

= 1,016261260

= 1,793756792

= 0.9138788366 10

= 0.7163

Non linear dynamics of the deterministic model

The dynamics of the deterministic model at high speed has

been investigated in [2]. The existence of a subcritical Hopf-

bifurcation has been detected at =66.61 m/s. Fig. 2 shows

the bifurcation diagram of the deterministic system. The Hopf

bifurcation point is obtained by observation of the stability of

the trivial solution using the eigenvalues of the Jacobian of the

system. The nonlinear critical speed, the fold bifurcation,

characteristic in subcritical Hopf-bifurcations, is found at

=62.02m/s using a ramping method, where the speed is

quasi-statically decreased, according to

=0 ,

if

<<

, otherwise .

(2)

The stochastic model

Let us now consider suspensions that are provided by the

manufacturer with a certain level of working accuracy. Due to

the lack of real data regarding the probability distributions of

3 Copyright © 2013 by ASME

such working accuracies, this initial study will consider

Gaussian distributions to describe them:

,

,

(std. ~ 5%)

~

,

,

(std. ~ 7%)

~,

.

(std. ~ 7%)

(3)

where the symmetry of the model is taken into consideration in

the standard deviation of the parameters and that both

represent two elements. The applicability and efficiency of the

methods presented in the next section will not be affected by

the particular choice of distribution.

Now the deterministic model is turned into a stochastic

model, where the single solution represents a particular

realization and probabilistic moments can be used to describe

the statistics of the stochastic solution.

UNCERTAINTY QUANTIFICATION

The stochastic solution of the system is now represented by

(,), where is a vector of random variables distributed

according to (3). The solution is a function that spans over a

three dimensional random parameter space. The dimension of

the parameter space is called the co-dimension of the dynamical

problem. In this work the focus will be restricted to the first two

moments of this solution, namely the mean [(,)] and

variance [(,)], but the following derivations can be used

similarly for higher moments too. Mean and variance are

defined as

()=[(,)]

=(,)() ,

()=[(,)]=(,)()

()

(4)

where () is the probability density function of the random

vector and the integrals are computed over its domain.

A straightforward way of computing the moments of the

solution is to approximate the integrals as:

()

()=1

,()

,

()

()=1

1 ,()()

,

(5)

where ()

are realizations sampled randomly from the

probability distribution of . This is the Monte-Carlo (MC)

method and it has a probabilistic error of 1

.

Even though the MC methods are really robust and

versatile, such a slow convergence rate is problematic, when

the solution of a single realization of the system is

computationally expensive. Alternative sampling methods are

the Quasi Monte-Carlo methods (QMC). These can provide

convergence rates of ((log )/), where is the co-

dimension of the problem. They use low discrepancy sequences

in order to uniformly cover the sampling domain. Without

presumption of completeness, in this work only the Sobol

sequence will be considered as a measure of comparison with

respect to other advanced UQ methods. QMC methods are

known to work better than MC methods when the integrand is

sufficiently smooth, whereas they can completely fail on an

integrand of unbounded variation [8]. Furthermore, randomized

versions of the QMC method are available in order to improve

the variance estimation of the method.

Stochastic collocation method (SCM)

Collocation methods require the residual of the governing

equations to be zero at the collocation points ()

, i.e.

,()=,(),(0, ]

(0)=,= 0 .

(6)

Fig. 2: Non-linear dynamics of the deterministic system. The subcritical Hopf-bifurcation is highlighted and the critical speed is

determined exactly at

=66.61 /

. The ramping method is then used in order to detect the non-linear critical speed at

=62.02 /

.

4 Copyright © 2013 by ASME

Then an approximation (,) of (,) is found as an

expansion in a set of Hermite polynomials, which are suitable

for approximations of the Gauss distribution functions:

(,)=

() ()

||

,

=1

(,) ()()

=1

,

()

()

()

,

(7)

where we used a cubature rule with points and weights

(),()

. The points ()

are the set of parameter

values for which deterministic solutions must be computed.

Cubature rules with different accuracy levels and sparsity exist.

In this work simple tensor product structured Gauss cubature

rules will be used. These are the most accurate but scale with

(), where is the number of points in one dimension and

is the co-dimension. The fast growth of the number of

collocation points with the dimensionality goes under the name

of “the curse of dimensionality” and can be addressed using

more efficient cubature rules such as Smolyak sparse grids [9].

UNCERTAINTY QUANTIFICATION IN RAILROAD

VEHICLE DYNAMICS

Uncertainty quantification is recently gaining much

attention from many engineering fields and in vehicle dynamics

there are already some contributions on the topic. In [10] a

railroad vehicle dynamic problem with uncertainty on the

suspension parameters was investigated using MC method

coupled with techniques from Design of Experiments.

Here SCM will be applied to the simple Cooperrider truck

[2] in order to study its behavior with uncertainties, and the

results will be compared to the ones obtained by the MC and

QMC methods.

These methods belong to the class of non-intrusive

methods for Uncertainty Quantification. This means that they

only require a deterministic method to compute the quantity of

interest (QoI) for different parameters. In this work this is the

ramping method to detect the critical speed.

The focus of this work is on the determination of the

nonlinear critical speed with uncertainties, so the investigation

of the stochastic dynamics with respect to time will be

disregarded here. Fig. 3 shows the SCM method applied to the

model with 1D uncertainty on parameter , for the

determination of the first two moments of the nonlinear critical

speed. The estimation done by the SCM is already satisfactory

at low order and little is gained by increasing it. This means

that the few first terms of the expansion (7) are sufficient in

approximating the nonlinear critical speed distribution.

Fig. 4 shows the SCM method applied to the same problem

with 1D uncertainty on the torsional spring stiffness . Again

the first few terms in expansion (7) are sufficient in order to

give a good approximation of the nonlinear critical speed

distribution. It is worth noting that the torsional spring stiffness

has an higher influence on the critical speed than .

Fig. 5 shows the SCM method on the problem with

uncertainty on parameters , and . Again, a low-order

SCM approximation is sufficient to get the most accurate

solution.

In the figures 3-5, left, we have compared the convergence

of the SCM method with that of the MC method. Therefore the

number of evaluations was prescribed. It is also of interest to

compare the computation time of the methods expressed by the

CPU time. For the comparison we used the calculated mean

values of the critical speed as the basis for the comparison. For

the SCM method the iteration process was ended when the

second decimal remained constant. The mean values in the MC

and QMC methods change however a good deal as shown in

the figures 3-5, left. Therefore, for the comparison a window

with 20 iterative values, which is glided over the number of

iterations was used. When the second decimal of the average of

Fig. 3: SCM on the model with 1D uncertainty on parameter

compared with MC and QMC. Left, estimation of mean and

variance of the nonlinear critical speed. Right, histograms of NL critical speeds obtained using 500 MC simulations of model (1)-(2)

and

10

realizations using the approximated stochastic solution (7) with only 2 function evaluations. The standard deviation is

shown as a shaded confidence interval, blue for SCM and red for MC. The two confidence intervals are overlapping almost exactly.

5 Copyright © 2013 by ASME

the iterated values in the window remained constant, when the

widow was pushed one more step, then the iterations were

stopped, and the CPU time was stored.

Table 1 shows the final results obtained with the chosen

accuracy, using the three methods, Monte-Carlo (MC), Quasi-

Monte-Carlo (QMC) and Stochastic Collocation (SCM). We

can observe that the variances in the multi-dimensional cases

are almost equal to the sum of the single-dimensional cases.

This means that the effect of the nonlinear interactions between

the three elements of the suspension is small with the variances

chosen in this problem.

CONCLUSIONS

Manufacturing tolerances have been introduced into the

dynamical investigations of vehicles. A new method, the

Stochastic Collocation Method (SCM) is applied as a tool for

“Uncertainty Quantification”, and the accuracy and

computational effort is compared with that of Monte-Carlo

(MC) and Quasi-Monte-Carlo (QMC) methods. The

“Uncertainty Quantification” methods are applied to the

estimate of the calculated critical speed of a railroad vehicle

model. The critical speed is delivered as a mean value with

variance. The results show that under the condition of the same

accuracy the convergence rate of the SCM outperforms the

rates of as well the MC as the QMC methods. Table 1 shows

that the CPU time and thus the computational effort by

application of the SCM is much smaller than the computational

effort by application of the MC or QMC methods. By all three

methods the total computational effort is larger than the effort

by a deterministic computation, because the same dynamical

system must be solved repeatedly only with different parameter

values. Under these conditions it is however possible to reduce

the total elapsed time significantly by straightforward

application of parallel computing. The dynamics of the vehicle

model is calculated in the process, but the results are not shown

here due to the limited space. A very simple model was chosen

in order to demonstrate the superiority of SCM over the MC

Fig. 4: SCM on 1D uncertainty on parameter k6 compared with MC and QMC. Left, estimation of mean and variance of the non-

linear critical speed. Right, histograms of NL critical speeds obtained using 500 MC simulations of model (1)-(2) and

10

realizations using the approximated stochastic solution (7) with only 2 function evaluations. The standard deviation is shown as a

shaded confidence interval, blue for SCM and red for MC. The two confidence intervals are overlapping almost exactly.

Fig. 5: SCM on 3D uncertainty compared with MC and QMC. Left, estimation of the mean and variance of the non-linear critical

speed. Right, histograms of nonlinear critical speeds.

6 Copyright © 2013 by ASME

and QMC methods. By using the same distributions for the

characteristics of the two lateral springs and dampers the effect

of the loss of symmetry in a real vehicle was not investigated

here. SCM can be 100 times faster than MC for low co-

dimensional problems, but for high co-dimensional problems

SCM methods suffer from the “curse of dimensionality”. The

computational effort of the SCM grows very fast with the

number of independent parameters. In a realistic vehicle model

that number easily surpasses 20. Therefore the work continues

with an investigation of the application of statistical methods

that may reduce the computational effort by singling out the

parameters that have the most important influence on the

wanted result of the dynamical problem. Some early results are

shown in [11].

REFERENCES

[1]

N. Cooperrider, "The hunting behavior of conventional

railway trucks,"

ASME Journal of Engineering and

Industry, vol. 94, pp. 752-762, 1972.

[2]

H. True and C. Kaas-Petersen, "A Bifurcation Analysis of

Nonlinear Oscillations in Railway Vehicles," in

Proc. 8th

IAVSD-IUTAM Symposium on Vehicle System Dynamics

,

Lisse, 1984.

[3]

D. Xiu, Numerical Methods for Stochastic Computations:

A Spectral Method Approach, Princeton: Princeton

University Press, 2010.

[4]

H. True, "On the Theory of Nonlinear Dynamics and its

Applications in Vehicle Systems Dynamics,"

Vehicle

System Dynamics , vol. 31, pp. 393-421, 1999.

[5]

H. True, A. P. Engsig-Karup and D. Bigoni, "On the

Numerical and Computational Aspects of Non-

Smoothnesses that occur in Railway Vehicle Dynamics,"

Mathematics and Computers in Simulation, 2012.

[6]

S. F. Wojtkiewicz, M. S. Eldred, R. V. Field, A. Urbina and

J. R. Red-

Horse, "Uncertainty Quantification In Large

Computational Engineering Models,"

American Institute

of Aeronautics and Astronautics , vol. 14, 2001.

[7]

P. J. Vermeulen and K. L. Johnson, "Contact of

no

nspherical elastic bodies transmitting tangential forces,"

Journal of Applied Mathematics, vol. 31, pp. 338-

340,

1964.

[8]

W. J. Morokoff and R. E. Caflisch, "Quasi-Monte Carlo

Integration," Journal of Computational Physics ,

vol. 122,

no. 2, pp. 218-230 , 1995.

[9]

K. Petras, "Smolyak cubature of given polynomial degree

with few nodes for increasing dimension,"

Numerische

Mathematik , vol. 93, no. 4, pp. 729-753 , 2003.

[10]

L. Mazzola and S. Bruni, "Effect of Suspension Parameter

Uncertainty on t

he Dynamic Behaviour of Railway

Vehicles," Applied Mechanics and Materials ,

vol. 104, pp.

177-185 , 2011.

[11]

D. Bigoni, H. True and A. P. Engsig-Karup, "Sensitivity

analysis of the critical speed in railway vehicle dynamics,"

in 23rd International

Symposium on Dynamics of Vehicles

on Roads and Tracks, Qingdao, China, 2013.

MC

QMC

SCM

2

σ

#fE

CPUt

µ

2

σ

#fE

CPUt

µ

2

σ

#fE

CPUt

6

k

62,26

1,64

169

~

24h

62,24

1,47

152

~

21 h

62,23

1,55

2

~

10m

4

k

62,23

0,14

17

~

2,5h

62,25

0,14

22

~

3 h

62,25

0,14

2

~

11m

2

D

62,23

0,02

9

~

1 h

62,25

0,02

4

~

30m

62,25

0,03

2

~

11m

46

,kk

62,22

1,53

148

~

21 h

62,22

1,62

152

~

22 h

62,28

1,69

4

~

36m

26

,Dk

62,18

1,72

216

~

30 h

62,24

1,50

142

~

20 h

62,28

1,57

4

~

37m

24

,Dk

62,25

0,17

25

~

3,5h

62,25

0,16

25

~

3,5h

62,30

0,17

4

~

35m

246

,, Dkk

62,18

1,68

221

~

32 h

62,23

1,63

154

~

22 h

62,23

1,72

8

~

1 h

Table 1: Estimated mean and variance of the nonlinear critical speed using MC, QMC and SCM. The three methods are compared in

terms of number of function evaluations (#fE) and computation time (CPUt).

µ