Track Irregularities Model

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1. Introduction

The expected benefits of simulation in the railway field are multiple: ro-bust and optimized conception, shorter and cheaper certification procedure,better knowledge of the critical situations of the track/vehicle system, op-timization of the maintenance. However, if simulation is introduced in cer-

tification and conception processes, it has to be very representative of thephysical behavior of the system. The model has thus to be fully validatedand the simulations have to be raised on a realistic and representative set ofexcitations.

Two kinds of inputs are traditionally introduced in a railway simula-tion: the vehicle modelVand the track geometryT. Multibody simulationsare usually employed to model the train dynamics. Carbodies, bogies andwheelsets are therefore modeled by rigid bodies and are linked with connec-tions represented by rheologic models (damper, springs, ...).

Two description scales can then be distinguished for the track geometry.On the first hand, the track design, which corresponds to the mean line posi-

tion of a perfect track is decided once for all at the building of a new track foreconomical and political reasons. This description is characterized by threecurvilinear quantities: the vertical curvaturecV, the horizontal curvaturecH,and the track superelevationcL. On the other hand, for a fixed track design,the actual positions of the rails are in constant evolution, which is mostly dueto the interactions between the train, the track and the substructure. Theirregularities appearing during the track lifecycle are indeed of four types(see Figure1): lateral and vertical alignment irregularities X1 and X2 on theone hand, cant deficiencies X3 and gauge irregularities X4 on the other hand.Therefore, each rail position R/r (refers to the left rail whereas r refers to

the right rail) can be written as the sum of a mean position M/r, which onlydepends on the curvilinear abscissa of the track,s, the track gaugeE, and thethree parameters of the track design, cH,cV and cL, and a deviation towardthis mean position I/r, which only depends on the track irregularities:

R/r(s) = M/r(s) + I/r(s) , (1)

M/r(s) =ONT(s) E

2N(s), (2)

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T(s)

B(s)

N(s)

X1(s)

T(s)

B(s)

N(s)

X2(s)

T(s)

B(s)

N(s)

2X3(s)

T(s)

B(s)

N(s)

E+2X4(s)

Figure 1: Parametrization of the track irregularities (for each rail, the meanposition is represented in black, whereas the real position is in grey).

I/r(s) ={X2(s) X3(s)}B(s) + {X1(s) X4(s)}N(s), (3)

where goes with the subscript and + goes with r in the symbol ,ONT(s) = (M(s) +M/r(s))/2 is the mean position of the two rails, and(ONT(s), T(s), N(s), B(s)) is the Frenet frame.

Hence, made up of straight lines and curves at its construction, the newtrack is gradually damaged and regularly subjected to maintenance opera-tions. During their lifecycles, trains are therefore confronted to very differentrunning conditions. The track-vehicle system being strongly non-linear, thedynamic behavior of trains has thus to be analyzed not only on a few trackportions but on this whole realm of possibilities.

In reply to these expectations, the measurement train IRIS 320 has been

running continuously since 2007 over the French railway network, measuringand recording the track geometry of the main national lines. This systematicmeasurement of the railway network quality and variability makes up a veryuseful database to analyze the complex link between the train dynamics andthe physical and statistical properties of the track geometry.

Based on these experimental measurements, a complete parametrizationof the track geometry and of its variability would be of great concern in spec-ification, security and certification prospects, to monitor more systematicallythe appearing track irregularities and to be able to generate track geometries

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that are realistic and representative of a whole railway network.To this end, the random irregularities can be statistically characterized

using a one-sided power spectral density (PSD) function of the track geome-try[1,2,3]. In this case, the PSD functions of each track irregularity can beeither estimated from the available measured track geometries or from tracksafety standards such as the ones given by the Federal Railroad Administra-tion [4]. Track irregularities can then by generated from these PSD functionsfrom a time-frequency transformation based on a spectral representation (see

[5,6] for further details). In such a method, the railway track irregularitiesare thus considered as four independent Gaussian random fields, which aresupposed to be stationary and ergodic in space. By definition, these trackirregularities are however strongly dependent, and due to the specific inter-action between the train and the track, they are actually neither Gaussiannor stationary.

In this context, the present work proposes a method to identify, in in-verse, from measured data, the statistical properties of this vector-valued,non-Gaussian and non-stationary track irregularity random field. These databeing complete, this modeling allows us to generate numerically track geome-

tries that are physically realistic and statistically representative of the set ofavailable track measurements, while taking into account the dependenciesbetween the four kinds of track irregularities.

These tracks can then be used in any deterministic railway dynamic soft-ware to characterize the dynamic behavior of the train. Hence, this modelingcould bring an innovative technical answer to introduce numerical methodsand treatments in the maintenance and certification processes.

In Section 2, the stochastic modeling of the track geometry from experi-mental data and the hypothesis on which this modeling is based are describedin details. Section 3 presents then the spectral and statistical validation of

the method.

2. Track geometry stochastic modeling

The mean line of the track geometry being chosen at the building of anew line, this work is only devoted to the modeling of the track irregularityvector

X = (X1, X2, X3, X4) , (4)

where X1, X2, X3 and X4 are the four types of track irregularities previously

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introduced. In this work, it is supposed that the track geometry of a givenrailway network has been measured. The track irregularity vector,

X(s) = (X1(s), X2(s), X3(s), X4(s)) , s [0, S

tot]

, (5)

is assumed to be a second-order centered random field, such that:

E[X(s)] = 0, s [0, Stot], (6)

with E[] the mathematical expectation. Due to the specific interactionbetween the train and the track, it is recalled that this random field is neitherGaussian nor stationary.

To characterize the statistical properties of such a random field, this workproposes a three-step method. First, a local-global approach is introduced,which allows us to focus the stochastic modeling only on the projection ofX on the interval = [ 0, S], with S Stot. Then, a Karhunen-Love(KL) expansion is carried out, which allows us to write X as a weightedsum of deterministic spatial functions. The identification of the multidimen-

sional distribution of these weights, which are non correlated but a prioridependent random variables, is finally performed using a polynomial chaosexpansion (PCE). At last, it will be shown in what extent such a methodallows the generation of realistic track geometries that are representative ofthe measured railway network.

2.1. Local-global approach

The non Gaussian and non stationary properties of track irregularityrandom field X motivate the introduction of a local-global approach for thecharacterization of its distribution. This approach is based on the hypoth-esis that a whole railway track can be considered as the concatenation ofa series of independent track portions of same length S, for which physicaland statistical properties are the same. LengthSplays therefore a key rolein the modeling procedure, and its value has to be carefully evaluated. Inorder to choose length S such that its influence on the stochastic modelingis minimized, track portions of same length L = 10km,

z(1), . . . ,z()

,

have been collected from the available measurements of the railway networkof interest. For any value for S, we denote by

y(1)(S), . . . ,y()(S)

the

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new track geometries of total length L, which are then built from the con-catenation of track sub-sections of lengthSthat have been randomly choseninz(1), . . . ,z()

.

Three errors functions, for which definitions can be found in AppendixA, are therefore introduced in this paper to quantify the influence ofS:

a covariance error, err2cov(S): fixingSto a particular value amounts tosupposing that for|s s| S,E

X(s)X(s)T

is negligible;

a spectral error,err2spect(S): generating complete track geometries fromthe concatenation of several track portions of length S introduces anartificial periodicity and is likely to degrade the low frequency charac-terization ofX;

an estimation error, err2est(S): the higher S is, the smaller the num-ber of independent realizations for X, exp(S), can be extracted fromthe complete measurement of the railway network of interest. Thiserror is therefore directly related to the estimation accuracy of the co-variance function ofX, and to the identification precision of the PCE

coefficients, on which the modeling will then be based. With referenceto the Central Limit Theorem (see [7] for further details), we simplychoose err2est(S) = 1/

exp(S)to illustrate this phenomenon.

For the chosen railway network, based on these sets of track geometriesof same lengths L = 10km, errors err2cov(S), err

2spect(S) and err

2est(S) are

represented in Figure2. When S increases, it can be verified that err2cov(S)and err2spect(S) decrease whereas err

2est(S) increases. LengthShas thus to

be chosen as the right balance between these three error functions.

For confidentiality reasons, the exact value ofS is however not given inthis work, and the spatial quantities will be normalized by length S in thefollowing.

Under this local-global hypothesis, exp = 1, 730 track portions of samelengthSare now extracted from the complete railway network of total lengthStot. These measurements are supposed to be exp independent realizationsof random field X, which defines the maximum available information for thestochastic modeling. These portions are in practice discretized at a spatial

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103

102

101

100

Values ofS

errors(%)

err2cov(S)

err2spect(S)

err2est(S)

Figure 2: Graphs of errors err2cov(S), err2spect(S) and err

2est(S) for the

computation of the local-global length S.

steph that is sufficiently small for the statistical and spectral information ofX to be completely taken into account. The experimental discretized data

are thus assumed to be complete. Hence, random field X can be identifiedto its NS-dimension discretized projection X, with NS = 4(S/h+ 1), suchthat:

X=

X1X2X3X4

, Xt=

Xt(0)Xt(h)

...Xt(S)

, 1 t 4. (7)

The random vectorXis still called irregularity vector, whereas measuredtrack portions are denoted by x1, ,x

exp

and will now be considered asexp independent realizations of random vector X. Therefore, the covariancematrix,[CXX ], ofXcan be estimated, for

exp sufficiently large, by:

[CXX ] =EXXT

[CXX ] = 1

exp

expi=1

x(i)x(i)T

.(8)

Approximation[

C11]of[C11] =E

X1X

T1

is presented in Figure3. This

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Abscissa s Abscissa s

[C11]

0

SS

Abscissa sC

ovariance(106S2) C111C221C331C441

0

0-5

5

S

Figure 3: Graphs of projections of[CXX ].figure emphasizes a quasi symmetry along the first bisector. The first column,Cij1, of matrices [Cij ] = [Cij1 CijNS/4] can thus be used to condense thecovariance information of each block. Figure3compares also the four first

columns,

C

tt

1 , 1 t 4, of the diagonal blocks of[

CXX ]. It can be noticed

that the covariance matrices are very different from one track irregularity toanother.

2.2. KL and PCE expansions

The objective of the stochastic modeling is to identify in inverse the sta-tistical properties of irregularity vector X from the exp independent real-izations {x1, ,x

exp}. A two-step approach is introduced in this work to

characterize the distribution ofX. First, a spatial and statistical representa-tion is achieved, which is based on a Karhunen-Love (KL) expansion. Thisvery efficient method, which has first been introduced by Pearson [8] in dataanalysis, has been applied in many works for the last decades (see, for in-stance [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]).It is indeed particularly interesting as it allows the uncorrelation of the pro-jection coefficients ofXon the KL vector basis, while optimally compactingthe signal energy. The Karhunen-Love expansion (see [28, 29] for furtherdetails) and more precisely the Principle Component Analysis of centeredrandom vector Xcan be written as:

X=

NSk=1

kuk k, (9)8

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in which12 NSare the eigenvalues and u1, ,uNSare

the associated eigenvectors, which are solutions of the eigenvalue problem:

[CXX ]u=u, (10)in which matrix[CXX ]is defined by Eq. (8). Projection basis u1, ,uNSis chosen orthonormal in the sense that, for all k and :

uTuk =kl, (11)where kl is the Kronecker symbol, equal to one ifk= and zero otherwise.In the same manner than the track irregularity vector is the concatenation offour random sub-vectors corresponding to the four track irregularities, suchthat X = (X1,X2,X3,X4), the components ofuk can be divided in foursub-vectors of same dimension NS/4, which are denoted byuk1,uk2,uk3 anduk4. It has to be noticed that, for a particular eigenvectoruk, if the norms ofat least two sub-vectors, (ukt )Tukt and (ukr )Tukr , for 1 t < r 4, are nonnegligible, then, a spatial dependency is introduced in the modeling between

the track irregularities t and r. We compare in Figure4 several graphs ofeigenvectorsuk. For a better visualization, the mean values of the differentsub-vectors, which are zero, are deliberately translated.

The coefficients{1, , NS}, such that:

k=XTukk , (12)

are moreover uncorrelated random variables, for which mean is zero andvariance is equal to one. Based on the eigenvalues decrease, random vectorX can then be approximated by its truncated expansion with N

terms,

X(N), such that:

X=

Nk=1

kuk k, (13)for which the truncation residual error, X X(N), is evaluated by thenormalized L2 error,

2KL(N), such that:

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0

0

0.1

0.2

0.3

Abscissas S

u11 u12 u13 u14

0

0

0.1

0.2

0.3

Abscissas S

u51 u52 u53 u54

0

0

0.1

0.2

0.3

Abscissas S

u101 u102 u103 u104

0

0

0.1

0.2

0.3

Abscissas S

u251 u252 u253 u254

Figure 4: Graphs of four particular eigenvectorsu1,u5,u10 andu25(103S).

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0 500 1000 1500 200010

8

106

104

102

100

N

2 K

L(N

) 2KL = 0.1 N = 280

2KL = 0.05 N = 495

2KL(N)

2

KL

= 0.01 N = 940

Figure 5: Graph of2KL(N).

2KL(N) =

X X(N)22

X22= 1

kN

k

X22, (14)

where for all second order random vector Z,

Z22 = EZTZ

. (15)

The evolution of error 2KL(N) with respect to N is plotted in Fig-

ure 5. The higher N, the more precise the characterization of the trackgeometry, but the more difficult the characterization of the random vector =

1, . . . , N

. As a good compromise, the truncation parameter N is

fixed to the value940 in the following, which corresponds to an error thresh-old of 1% for2KL.

The second step of the modeling ofXis the characterization of the mul-tidimensional probability density function (PDF) of, which is denoted byp. Two kinds of methods can be used to build such a PDF: the directand the indirect methods. Among the direct methods, methods based on

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Information Theory and the Maximum Entropy Principle (MEP) have beendeveloped (see [30] and [31]) to directly compute p from the only availableinformation of random vector, as the solution of an optimization problem.On the other hand, the indirect methods allow the construction of the PDFp of the considered random vector from a transformation t of a knownN-dimensional random vector :

= t(), p = T(p), (16)

defining a transformation T between p and the known PDF p of. Theconstruction of the transformation t is thus the key point of these indirectmethods. Among these methods, the polynomial chaos expansion (PCE)methods, which have first been proposed by Wiener [32], and spread tocomputational sciences by Ghanem and Spanos [33, 28], are based on a di-rect projection of random vector on a chosen polynomial Hilbertian basisBorth = { j(), 1 j } of all the second-order random vectors with valuesin RN , such that:

=

+

j=1

y(j)

j (). (17)

In the last decades, this very promising method has therefore been appliedin many works (see [34,35,36,37,38,39,40,41,42] for more details about theinverse PCE identification from experimental data). For practical purposes,the sum that is defined by Eq. (17) has however to be truncated. Wetherefore define

1(1, . . . , Ng), , N(1, . . . , Ng)

as theN-dimensional

subset ofBorth that gathers the polynomial functions of total degree inferiorto p, which are normalized with respect to the PDF p1,...,Ng of the Ng firstelements of random vector , such that:

chaos(N) =N

j=1

y(j)j (1, . . . , Ng), (18)

j(1, . . . , Ng) =N

q=1

cqj (q)1

1 (q)Ng

Ng , 0

(q)1 +. . .+

(q)Ng

p, (19)

RNg

j (x)n(x)p1,...,Ng (x)dx= jn. (20)

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Given values ofNandNg, characterizingpamounts finally to identifyingthe deterministic projection vectors

y(1), . . . ,y(N)

from the available infor-

mation for . According to Eq. (12), this available information correspondsto the exp independent realizations of chaos(N),{1, ,

exp}, which can

be deduced from theexp independent realizations ofX,{x1, ,xexp

}, as:

ik= 1

kxiTuk, 1 i exp, 1 k N. (21)

In[40,41], it has been shown that a good approach to identify such coef-ficients in high dimension (that is to say when N is high) is to search themas the arguments that maximize the likelihood of random vector chaos(N)at the experimental points {1, ,

exp}.

Finally, the last step of this identification is the justification of the valuesof the truncation parameters N and Ng. In this prospect, the log-errorfunction err(N, Ng) is introduced to quantify the amplitude of the residualerror of the PCE truncation, chaos(N), such that:

err(N, Ng) =

Nk=1

errk(N, Ng), (22)

errk(N, Ng) =

BIk

log10(pk(xk)) log10 pchaosk (xk) dxk, (23)

where BIk is the domain bounding the experimental values ofk, pk andpchaosk are the PDFs ofk and

chaosk (N) respectively. Truncation parameters

N and Ng can thus be chosen with respect to a given error threshold forerr(N, Ng).

For our study, is a N-dimension random vector, whose componentsare independent and uniformly distributed between -1 and 1. According toFigure6, where err(N, Ng) is plotted against N for different values ofNg,truncation parametersNgand Nare chosen equal to3and3, 276respectively,which corresponds to the maximal polynomial order p= 25 for the reducedpolynomial basis,

1(1, . . . , Ng), , N(1, . . . , Ng)

.

2.3. Generation of a whole track geometry

Once truncation parameters N, N, Ng have been identified accordingto convergence analysis, and PCE projection coefficients

y(j), 1 j N

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0 1000 2000 3000 4000 5000200

250

300

350

400

Dimension N

err(N,

Ng

)

Ng = 3Ng = 4

Figure 6: Convergence analysis for the truncated PCE of.

have been computed with the advanced algorithms described in[41] and [40],the track irregularity random field is completely characterized and can finallybe estimated from Eqs. (13) and (18) as:

X

Nk=1

kuk Nj=1

y(j)k j(1, . . . , Ng). (24)

For each realization of random vector (1, . . . , Ng), a representative andrealistic track geometry of length S can finally be generated. Thanks tothe local-global approach, described in Section 2.1, a whole track geometryof length Stot = NTS (NT can be smaller or greater than

exp), Xtot, cantherefore be constructed from NT independent copies X

(1), ,X(NT) oftrack irregularity vector X, such that Xtot = (X(1), , X(NT)).

Therefore, independent realizationsXtot(1), , X

tot()

ofXtot

can be generated from NT realizations of irregularity vector X. However,

for each particular realization Xtot(m)ofXtot, a particular attention has tobe paid to the interface between the different realizations ofX. Indeed, thesejunctions have to guarantee the continuity of the track irregularity vectorand at least the continuity of its first and second order spatial derivativesin order to avoid an artificial perturbation of the train dynamics. Splineinterpolations on a length corresponding to the minimal wavelength of themeasured irregularities are then used to fulfill these continuity conditions.

Therefore, the proposed stochastic modeling allows us to generate realistic

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30

20

10

0

Abscissas

Trackirre

gularity(103S)

Xtot1 (1) Xtot2 (1) X

tot3 (1) X

tot4 (1)

S/2 S 3S/2

Figure 7: Extract of a simulated track geometry.

track geometries of length S

tot

= NTSthat are representative of the wholenetwork, and which take into account the spatial and statistical dependenciesbetween the different track irregularities. As an illustration, a particularextract of length S of complete track geometry Xtot (1) is represented inFigure7. This graph is centered at abscissa s = 3S/2, that is to say at a

junction between the two first realizations of random vectorX. The fourcomponents ofXtot(1) are represented in the same graph, but their valuesare translated to allow a better visualization of the results.

3. Validation of the track geometry stochastic modeling

As presented in Introduction, a complete parametrization of the physicaland statistical properties of the track geometry would be of great interest inconception, certification and maintenance prospects. Several validations ofthe proposed stochastic modeling are thus presented in this section, in orderto allow a relevant investigation of the dynamic interaction between the trainand the track.

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0 2 4 6 80

50

100

150

200

level u(106S)

Meannum

berofupcrossings

Nmesup (X1, u , S )

Ngenup (X1, u , S )

Nmesup (X2, u , S )

Ngenup (X2, u , S )

Nmesup (X3, u , S )

Ngenup (X3, u , S )

Nmesup (X4, u , S )

Ngenup (X4, u , S )

Figure 8: Validation of the generation of track geometries with respect tothe number of upcrossings.

3.1. Statistical and frequency validation of the local track geometry stochastic

modeling

In this section, 5, 000 track irregularities vectors of total length S aregenerated from the track stochastic modeling developed in Section 2. For1 i 4, we use Nmesup (Xi, u , S ) and N

genup (Xi, u , S ) to denote the mean

numbers of upcrossings (see [43] for more details about the upcrossings)of the level u by Xi over the length S, which have been evaluated fromthe exp = 1, 730 available measured track geometries and from the 5, 000generated track geometries of lengthSrespectively. In the same manner, for1 i 4, letP SDmes

i andP SDgen(X

i)be the mean power spectral densities

ofXi that have been computed from the former 1, 730 measured and 5, 000generated track geometries of length S.

A good agreement between the quantities corresponding to the measuredand to the generated track geometries can be seen in Figures 8and9. Thisallows us to validate the local stochastic modeling over a length S from astatistical and frequency point of view.

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108

106

104

102

100

Wavelength (103S)

PowerSpectralDensity

P SDmes(X1)P SDgen(X1)P SDmes(X2)P SDgen(X2)P SDmes(X3)P SDgen(X3)P SDmes(X4)P SDgen(X4)

1 10 102

Figure 9: Validation of the generation of track geometries with respect tothe frequency content.

3.2. Dynamical validation of the global track geometry stochastic modeling

This section aims now at validating the complete modeling from a dy-namical point of view. In this prospect, a measured track design of length5km around a horizontal curve of a high speed line is considered, for whichhorizontal and vertical curvatures, cH and cV, and track superelevation, cL,are represented in Figure 10. track irregularities of total length 5km,z(1), . . . ,z()

, are then extracted from the available measured track geom-

etry. These track irregularities have also been chosen around a horizontalcurve. In parallel, track irregularities,

Xtot(1), , X

tot()

, of totallength Stot = 5km are generated from the local-global stochastic modelingdescribed in Section2.

Coupled to the model of a train, these two sets of track conditions cannow be used in any rigid-multibodies railway software to characterize thetrain dynamic behavior. For our study, it is supposed that a normalizedmodel of a TGV train is available, for which mechanical parameters havebeen accurately identified, and a commercial code, which is called Vampire,has been used to perform the railway simulations.

The validation of the global stochastic modeling is then based on thefrequency and the statistical analysis of eight dynamical quantities of interest:

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0 1000 2000 3000 4000 5000

0

0.1

0.2

Abscissas(m)

cH (km1)

cV (km1)

cL (m1)

Figure 10: Track Design of the simulated railway track.

the vertical and horizontal accelerations of the first bogie of the motorcar,Q1 = zM C and Q2= yMC;

the vertical and horizontal accelerations of the second bogie of thesecond passenger car,Q3= zP C and Q4= yP C;

the left and right transverse contact forces at the first wheelset of thefirst bogie of the motor car, Q5 = Y

MC and Q6 = Y

rMC;

the left and right transverse contact forces at the second wheelset ofthe second bogie of the second passenger car, Q7 =Y

P CandQ8=Y

P C.

In the same manner than in Section 3.1, for 1 i 8, we are inter-ested in the mean power spectral densities of Qi and the mean numbersof upcrossings of the level u by Qi over the length S

tot = 5km, which arerespectively denoted byP SDmes(Qi)and N

mesup (Qi, u , S

tot)when these quan-tities are computed from the measured track geometries and P SDgen(Qi)andNgenup (Qi, u , S

tot)when these quantities are computed from the generatedtrack geometries. The comparisons between these quantities are representedin Figure11. In can therefore be deduced from these figures that the stochas-tic modeling method used in this work can make sure that the frequency and

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0 1 2 3 4 5 6 7 80

1000

2000

3000

4000

5000

6000

level u(103S s2)Meannu

mberofupcrossings

Nmesup (Q1, u , S )

Ngenup (Q1, u , S )

Nmesup (Q3, u , S )

Ngenup

(Q3

, u , S )

103

102

101

Wavelength (103S)

Power

SpectralDensity

P SDmes(Q1)P SDgen(Q1)

P SDmes(Q3)

P SDgen(Q3)

10 102

0 2 4 6 8 100

2000

4000

6000

8000

level u(103S s2)Meannumberofupcrossings

Nmesup (Q2, u , S )

Ngenup (Q2, u , S )

Nmesup (Q4, u , S )

Ngenup (Q4, u , S )

105

100

101

102

103

104

Wavelength (103S)

PowerSpectralDensity

P SDmes(Q2)

P SDgen(Q2)

P SDmes(Q4)

P SDgen(Q4)

10 102

0 10 20 30 40 500

2000

4000

6000

8000

10000

level u(kN)Meannumberofupcrossings

Nmesup (Q5, u , S )

Ngenup (Q5, u , S )

Nmesup (Q6, u , S )

Ngenup (Q6, u , S )

Nmesup

(Q7, u , S )

Ngenup (Q7, u , S )

Nmesup (Q8, u , S )

Ngenup (Q8, u , S )

102

101

100

Wavelength (103S)

PowerSpectralDensity P SDmes(Q5)

P SDgen(Q5)

P SDmes(Q6)

P SDgen(Q6)

P SDmes

(Q7)P SDgen(Q7)

P SDmes(Q8)

P SDgen(Q8)

10 102

Figure 11: Spectral and statistical analysis of the dynamical quantities ofinterest Q1,Q2,. . ., Q8.

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[Rzz (s, s)] =

1

n=1

z(n)(s)z(n)(s)T, (25)

[Ryy (s, s, S)] =

1

n=1

y(n)(s, S)y(n)(s, S)T, (26)

z =1

n=1

P SD (z(n)), y(S) =1

n=1

P SD (y(n)(S)), (27)

where P SD (z) = (P SD(z1), . . . , P S D(zP)) is the power spectral densityestimation of any RP-valued function z = (z1, . . . , z P). For any value ofS in[0, L], errorserr2cov(S)and err

2spect(S), which have been introduced in Section

2.1are then defined by:

err2cov(S) =[Rzz ] [Ryy (S)]2M/ [Rzz ]

2M, (28)

err2spect(S) =z y(S)2V/ z

2V , (29)

where, for all (P P) matrix-valued function [R], and for all RP-valuedfunction ,

[R]2M=

L0

L0

Tr

[R(s, s)][R(s, s)]T

dsds, (30)

2V =

+1/L

(f)T(f)df. (31)

Hence, on the first hand, err2cov(S) corresponds to a covariance error,which quantifies the approximation introduced by the asumption that E

X(s)X(s)T

is negligible. On the other hand, err2spect(S) can be seen as a spectral error,which characterizes the impact ofSon the frequency content of the trackirregularities.

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Track Irregularities Model

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