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Structural-acoustic modeling of automotive vehicles inpresence of uncertainties and experimental identification

and validationJ.-F. Durand, Christian Soize, L. Gagliardini

To cite this version:J.-F. Durand, Christian Soize, L. Gagliardini. Structural-acoustic modeling of automotive vehiclesin presence of uncertainties and experimental identification and validation. Journal of the AcousticalSociety of America, Acoustical Society of America, 2008, 124 (3), pp.1513-1525. �10.1121/1.2953316�.�hal-00685108�

Structural-acoustic modeling of automotive vehicles in presence of uncertainties and experimentalidentification and validation

Jean-Francois Durand and Christian Soize∗

Universite Paris-Est, Laboratoire Modelisation et Simulation Multi echelle, FRE3160 CNRS, 5 bd Descartes,77454 Marne-la-Vallee, France

Laurent Gagliardini

PSA Peugeot-Citroen, Acoustic division, 78943 Velizy-Villacoublay, France

The design of cars is mainly based on the use of computational models to analyze structuralvibrations and internal acoustic levels. Considering the very high complexity of such structural-acoustic systems, and in order to improve the robustness of such computational structural-acousticmodels, both model uncertainties and data uncertainties must to be taken into account. In thiscontext, a probabilistic approach of uncertainties is implemented in an adapted computationalstructural-acoustic model. The two main problems are the experimental identification of the pa-rameters controlling the uncertainty levels and the experimental validation. Relevant experimentshave especially been developed for this research in order to constitute an experimental databasedevoted to structural vibrations and internal acoustic pressures. This database is used to performthe experimental identification of the probability model parameters and to validate the stochasticcomputational model.

I. INTRODUCTION

In the automotive industry, computational structural-acoustic models are nowadays intensively used to analyzethe structural-acoustic behavior of vehicles in terms ofstructural vibrations and internal acoustic levels mainlyfor the low-frequency range. The present evolution isto extend such computational models to the medium-frequency range. In this paper, we are interested inthe booming noise consisting in studying the acousticresponse at passengers’ ears induced by engine structure-borne excitations in the low-frequency band but also inthe lower part of the medium-frequency band. Note thata very few papers have been published concerning thebooming noise prediction in this frequency band using acomputational model with or without experimental com-parisons (Sol and Van-Herpe, 2001; Sung and Nefske,2001; Hamdi et al., 2005; Hayashi et al., 2000). In ad-dition, structural-acoustic analyses of cars in the high-frequency band are of a great importance for automo-tive engineering and can generally be treated by usingstatistical energy analysis and diffuse field methods forwhich numerous papers have been published in the lastdecade (see for instance Lyon and DeJong, 1995; Le Bot,2002; Gagliardini et al., 2005; Shorter and Langley, 2005;Langley, 2007). Considering the very high complexityof such structural-acoustic systems, the mean computa-tional structural-acoustic models do not allow sufficientlygood predictions to be obtained and consequently, mustbe improved in implementing a model of uncertainties inorder to increase the robustness of predictions. In thiscontext, it is necessary to validate such computational

∗Electronic address: christian.soize@univ-paris-est.fr

structural-acoustic models with experiments. It shouldbe noted that a very few complete and documented ex-perimental databases are available in the literature. Onlysome elements concerning two databases can be found inthe literature (Wood and Joachim, 1987 ; Kompella andBernhard, 1996). Nevertheless, these two experimentaldatabases cannot easily be used because there are noavailable computational structural-acoustic models asso-ciated with these databases. In order to get round thisdifficulty, a complete experimental campaign devoted tostructural vibrations and internal acoustic pressures hasbeen done for this research (Durand, 2007). This exper-imental database is presented in this paper and is usedto perform an experimental identification and to validatethe computational model. The problem related to suchpredictions with computational structural-acoustic mod-els is extremely difficult due to the over-sensitiveness ofstructural dynamical responses with respect to manufac-turing processes and due to small variabilities induced bythe presence of optional extra around a main configura-tion. Note that automotive engineering requires to pre-dict the structural-acoustic responses of a car of the sametype with optional extra using only one computationalstructural-acoustic model. This over-sensitiveness maybe seen by computation when applying design changes,but also experimentally when monitoring vehicle disper-sions (Wood and Joachim, 1987; Kompella and Bern-hard, 1996; Hills et al., 2004). Concerning the computa-tional structural-acoustic models, many uncertainties areintroduced by the mechanical-acoustical-mathematicalmodeling process due to the high complexity of the struc-ture and of the internal acoustic cavity in terms of geom-etry, boundary conditions, material properties, etc. Evenif a sophisticated structural-acoustic model is developed,model uncertainties and data uncertainties are inherentin such a computational structural-acoustic model. One

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objective of this paper is to use a representative compu-tational structural-acoustic model developed by an au-tomotive industry (Sol and Van Herpe, 2001; Durand etal., 2005a). In such a finite element model, (1) the struc-ture is discretized with a reasonable fine scale, (2) theinternal acoustic cavity is discretized with a coarse scaleadapted to the frequency band of analysis and (3) thesound-proofing schemes located at the interface betweenthe structure and the acoustic cavity is taken into accountby a very simplified model. Clearly, the sound-proofingschemes could be discretized with a fine scale using the fi-nite element method and formulations for porelastic ma-terials (Attala et al., 2001; Hamdi et al., 2005). Suchan approach has not been retained in this research. Itseems that there is no paper available in the literature de-voted to the development of a computational structural-acoustic model for cars including a model of uncertain-ties and associated with an experimental validation per-formed with a complete experimental database. In thispaper, we present such a complete computational modelincluding uncertainties modeling, experimental identifi-cation and experimental validation (Durand et al., 2004,2005a and 2005b; Durand, 2007). It is known that severalapproaches can be used to take into account uncertaintiesin computational models of complex structural-acousticsystems for the low-frequency band (interval method,fuzzy sets approach, probabilistic approach, etc). In thispaper, we have chosen to use the most efficient mathe-matical tool adapted to model uncertainties as soon asthe probability theory can be used, i.e., the probabilisticapproach. The mean computational structural-acousticmodel is constructed from the designed system (concep-tual system) using a mathematical-mechanical-acousticalmodeling process. The mean computational model whichis considered as a predictive model of the real system de-pends on parameters (or data). There are two types ofuncertainties which are data uncertainties and model un-certainties. Data uncertainties are related to the param-eters of the mean computational vibracoustic model andmodel uncertainties are induced by the modeling process.The ”parametric probabilistic approach” is the most ef-ficient and powerful method to address ”data uncertain-ties” in predictive models (see for instance Schueller, 1997and 2007, and Ghanem and Spanos, 2003 for stochas-tic finite element methods) as soon as the probabilitytheory can be used but cannot address ”model uncer-tainties”. The ”nonparametric probabilistic approach”recently proposed (Soize, 2000, 2001, 2003, 2005a and2005b) is a way to address both model uncertainties anddata uncertainties. The use of the parametric probabilis-tic approach of data uncertainties for structural-acousticanalysis of cars generally requires to introduce a verylarge number of random variables. This is due to thefact that the structural-acoustic responses are very sen-sitive to many parameters related to the geometry (suchas plate and shell thicknesses, panels curvatures, etc.),to the spot welding points, to the boundary conditions,etc. Typically, several ten thousands parameters must

be modeled by random variables. First, it should benoted that the construction of the probabilistic modelof this large number of parameters is not so easy to carryout. The experimental identification of a very large num-ber of probability distributions using measurements ofthe response of a structural-acoustic system (solving aninverse stochastic problem and consequently, solving anoptimization problem) is completely unrealistic. In ad-dition, as explained above, the parametric probabilisticapproach does not allow model uncertainties to be takeninto account and it is known that model uncertainties aresignificant in computational structural-acoustic modelsof cars. This is a reason why we propose to use the non-parametric probabilistic approach of uncertainties whichallows data uncertainties but also model uncertainties tobe taken into account. In addition, the nonparametricapproach introduces a very small number of parameters(typically 7 parameters) which controls the level of un-certainties. In this condition, the experimental identifica-tion of these parameters is realistic and can be performedby solving the stochastic inverse problem using adaptedmathematical-statistical tools. In this paper, we presentsuch an approach. It should be noted that the nonpara-metric probabilistic approach of uncertainties has beenused in the five last years for linear and nonlinear struc-tural dynamical problems. Nevertheless, this approachhas not yet been used for complex structural-acousticsystems and is presented in this paper.

In Section II, we present the experimental databasewhich has specially been constructed for this research.Section III deals with the mean computational structural-acoustic model and Section IV is devoted to the stochas-tic reduced computational model. Finally, in Section V,we present the structural-acoustic response of the vehi-cle for which the experimental database has been carriedout, the experimental identification of the probabilisticmodel of uncertainties and the experimental validationof the stochastic computational model.

II. EXPERIMENTS

Experiments which are described in this section havebeen performed in PSA Peugeot-Citroen facilities. Thesystem under consideration is a given vehicle for whichtwo sets of experiments have been defined. The first set ismade up of structural vibration and structural-acousticmeasurements. The second set is devoted to the acousticmeasurements inside the internal acoustic cavity.

A. Structural-vibration and structural-acousticmeasurements

The first set of experiments consists (1) of measure-ments of the structural Frequency Response Functions(FRFs) between one DOF for a given excitation force(vertical component of the force at one support of theengine) and normal accelerations to the structure at six

2

given points and (2) of measurements of the structural-acoustic FRFs between nine DOFs of excitation corre-sponding to three given point forces at supports of theengine and the acoustic pressure at the driver ears lo-cated inside the internal acoustic cavity. For structural-vibration measurements, the excitation is performed bymeans of a hammer and the responses are identifiedwith accelerometers. For the structural-acoustic mea-surements, the reciprocity method in acoustics is used.This means that the excitation is produced by an acousticsource located at the driver ears inside the internal acous-tic cavity and the acceleration responses are measured atthe nine DOFs introduced above. The frequency bandof analysis is [20, 220] Hz. The experimental databasehas been constructed using 20 cars of the same type withdifferent optional extras. The measurements have beenrealized at the exit of the assembly plant. Fig. 1 showsthe car body, the structural driving point (vertical excita-tion) and the two structural observation points (denotedby Obs4 and Obs6) for which we present experimental re-sults and for which comparisons with the computationalstructural-acoustic model is carried out. Fig. 2 showsthe front of the car body and the three components ofthe three forces applied to the points shown on this figureand corresponding to the nine DOFs of excitation. Figs.

X

ZY

Obs6

Excitation

Obs4

FIG. 1. Car body, structural excitation and structural obser-vations.

3 and 4 display the experimental structural FRFs intro-duced above for observations Obs4 and Obs6 and for the20 cars. The left figures display the moduli of the FRFsin dB and the right figures display the corresponding ex-perimental coherence functions. It should be noted thatthe coherence is not really good in the frequency band[20, 70] Hz and consequently, the level of the moduli ofthe experimental FRFs contain experimental errors. Thisexperimental difficulty is mainly induced by the hammertechnique used for the excitation when the direct methodis used to perform the experimental identification of the

F3

F2

F1

FIG. 2. Front of the car body and structural excitations forthe internal acoustic response.

structural FRFs. Note that this technique is not used toperform the experimental identification of the structural-acoustic FRFs which are identified with the reciprocalacoustic method and for which the experimental identifi-cation is correct in the frequency band [20, 70] Hz. Withthe reciprocal acoustic method, we have not obtained anydifficulties with respect to the coherence functions andthen the measurements are validated in the frequencyband [20, 220] Hz. The strategy used is then the follow-ing. The experimental structural FRFs is only used inthe frequency band [70, 220] Hz in order to identify thedispersion parameters of the nonparametric probabilisticmodel of uncertainties in the structure (and consequently,is not used in the frequency band [20, 70] Hz). The ex-perimental structural-acoustic FRFs identified with thereciprocal acoustic method is used to validate the com-putational structural-acoustic model in all the frequencyband [20, 220] Hz including the frequency band [50, 183]Hz of the booming noise. Figs. 3 and 4 show the exper-imental dispersion on the structural responses obtainedfor the 20 cars of the same type. This experimental dis-persion is due to the variability induced by the optionalextras and induced by the manufacturing process. Thisdispersion is increasing with the frequency which is incoherence with the expected results. The experimentaldispersion varies between 5 and 10 dB in the regions closeto the resonances and varies from 15 to 30 dB in theneighborhood of the antiresonances. The experimentalresults concerning the structural-acoustic FRFs betweenthe nine DOFs of the three given excitation forces andthe acoustic pressure at the driver ears are not presentedhere in order to limit the number of figures. Nevertheless,these experimental results are directly used to construct

3

the experimental booming noise synthesis which is pre-sented below.

10 dB

50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency, Hz

Coh

eren

ce

20 40 60 80 100 120 140 160 180 200 220Frequency, Hz

Mod

ulus

, dB

FIG. 3. Observation Obs4: Graphs of the modulus in dBscale of the experimental structural FRFs as a function ofthe frequency in Hertz for the 20 cars (left figure). Graphsof the corresponding experimental coherence functions (rightfigure).

10 dB

50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency, Hz

Coh

eren

ce

20 40 60 80 100 120 140 160 180 200 220Frequency, Hz

Mod

ulus

, dB

FIG. 4. Observation Obs6: Graphs of the modulus in dBscale of the experimental structural FRFs as a function ofthe frequency in Hertz for the 20 cars (left figure). Graphsof the corresponding experimental coherence functions (rightfigure).

B. Acoustic measurements inside the internal acousticcavity

The second set of experiments consists of measure-ments of the acoustic FRFs for the acoustic pressuresat 32 points inside the internal acoustic cavity whoselocations are approximatively uniformly distributed in-side the cavity and for an excitation which is an acousticsource located in the neighborhood of the front-passengerfeet. This experimental database has been constructedfor 30 different configurations of the acoustic cavity of asame car (type and position of seats, air temperature,wall boundaries between the trunk and the passengercompartment, number of passengers, etc). The frequencyband of analysis is [20, 320] Hz. In this case and for eachconfiguration, the observation is the root mean squaredBexp

rms(f) in dB scale of the acoustic pressures averagedon the 32 microphones as a function of the frequency fin Hertz. The reference pressure is Pref = 2.10−5 Pa.We then have

dBexprms(f) = 20 log10

⎛⎝

√132

∑32�=1 |pexp

� (2πf)|2Pref

⎞⎠ ,

in which pexp1 (ω), . . . , pexp

32 (ω) are the acoustic pressuresmeasured by the 32 microphones at frequency ω in rad/s.Fig. 5 displays the graphs of f �→ dBexp

rms(f) for the 30configurations of the internal acoustic cavity. This figureshows a low level of the experimental dispersion for thisquantity with respect to the different configurations. Thelevels of dispersion vary from 2 to 3 dB. It should benoted that the moduli of the corresponding experimentalFRFs at a given point (which are not presented here)have about 3 dB of dispersion in the regions close tothe resonances and have up to 6 dB of dispersion in theneighborhood of antiresonances.

50 100 150 200 250 30075

80

85

90

95

100

Frequency, Hz

Mod

ulus

, dB

FIG. 5. For the 30 configurations of the internal acousticcavity: graphs of f �→ dBexp

rms(f).

4

C. Experimental synthesis of booming noise

As explained above, the measurements presented inSubsections A and B will be used to identify the proba-bility model of uncertainties. Below we present the mea-surements which is used to validate the computationalstructural-acoustic model allowing the booming noise tobe predicted in the frequency range [50, 183] Hz corre-sponding to [1500, 5500] rpm (rotation per minute of theengine). In this paper, the booming noise is defined as themodulus of the acoustic pressure in dB[A] at the driverears for structural excitation induced by the forces deliv-ered by the engine at its 4 supports when the frequencyof rotation varies in the frequency band of analysis. Thisfunction is denoted by f �→ dBexp[A](f). Note that theforces delivered by the engine at its supports correspondto the second harmonic of rotation (for instance 1500 rpmcorresponds to a frequency equal to 50 Hz). The scaleused is the dB[A] scale which corresponds to the weight-ing of the usual dB scale by the A-weighting. Fig. 6displays the graph of f �→ dBexp[A](f) related to the ex-perimental synthesized booming noise and defined above.In this figure, it can be seen a very large experimental dis-persion induced by the variability (optional extras) andby the manufacturing process. As for the experimentalstructural FRFs, the experimental dispersion varies be-tween 5 and 10 dB in the regions close to the resonancesand varies from 15 to 30 dB in the neighborhood of theantiresonances. These experimental results are in coher-ence with the experimental structural FRFs and can beexplained by the fact that the experimental dispersion in-duced by uncertainties inside the internal acoustic cavityis relatively small with respect to structural uncertainties(see Fig. 5).

III. REDUCED MEAN COMPUTATIONALSTRUCTURAL-ACOUSTIC MODEL

The mean computational structural-acoustic model isdeveloped in the context of 3D linear elastoacoustics, isformulated in the frequency domain and is discretized byusing the finite element method. The structural-acousticsystem is made up of a damped elastic structure cou-pled with a closed internal acoustic cavity filled with adissipative acoustic fluid. The external acoustic fluid isair and its effects on the structural-acoustic system areassumed to be negligible in the low-frequency band ofanalysis which is considered in the paper and which isdevoted to the booming noise prediction. The linearresponses of the structural-acoustic system are studiedaround a static equilibrium state which is taken as thenatural state at rest. The reduced mean computationalstructural-acoustic model is derived from the mean modelusing a modal analysis.

50 100 150 200

Mod

ulus

, dB

[A]

Frequency, Hz

1000 2000 3000 4000 5000 6000

Rpm (rotation per minute)

10 dB

FIG. 6. Graph of f �→ dBexp[A](f) related to the experimen-tal synthesized booming noise.

A. Description of the structural-acoustic system

We then consider linear vibrations of a fixed dampedstructure Ωs (the car body) subjected to external loads(engine excitations), coupled with its internal cavity Ωa

(passenger compartment and trunk) filled with a dis-sipative acoustic fluid (air). We are mainly interestedin predicting the frequency responses of the structural-acoustic system in the frequency band of analysis B =[ωmin , ωmax] rad/s corresponding to Bf = [fmin , fmax]Hz with ωmin = 2π fmin and ωmax = 2π fmax. The phys-ical space R

3 is referred to a cartesian system and thegeneric point of R

3 is denoted by x = (x1, x2, x3). Thesystem is defined in Fig. 7. Let ∂Ωs = Γs∪Γ0∪Γa be theboundary of Ωs. The outward unit normal to ∂Ωs is de-noted by ns = (ns,1, ns,2, ns,3). The displacement field inΩs is denoted by u(x, ω) = (u1(x, ω), u2(x, ω), u3(x, ω)).The structure is assumed to be fixed on the part Γ0 ofthe boundary ∂Ωs. The internal acoustic cavity Ωa is thebounded domain filled with a dissipative acoustic fluid.The boundary ∂Ωa of Ωa is Γ. The outward unit nor-mal to ∂Ωa is denoted by na = (na,1, na,2, na,3) and wethen have na = −ns on ∂Ωa. The pressure field in Ωa isdenoted by p(x, ω).

5

, )(

x2

x3

(x )s

n

n

n

gsurf x

s

s

s

s

a

a

0

a.

x1

FIG. 7. Scheme of the structural-acoustic system.

B. Mean boundary value problem for thestructural-acoustic system

The equation of the structural part is written (e.g.,Trusdell, 1960; Ohayon and Soize, 1998) as

−ω2ρsui − ∂σij

∂xj= gvol

i in Ωs , (1)

with the convention for summations over repeated Latinindices, in which ρs is the mass density, σij is the stresstensor, u = (u1, u2, u3) is the displacement field of thestructure and gvol = (gvol

1 , gvol2 , gvol

3 ) is the body forcefield applied to the structure. The boundary conditionsare

σij(u)ns,j = gsurfi on Γs , σij(u)ns,j = −p ns,i on Γa , u = 0 on Γ0 ,

(2)in which gsurf = (gsurf

1 , gsurf2 , gsurf

3 ) is the surface forcefield applied to the structure. The damped structureis made up with a linear non homogeneous anisotropicviscoelastic material without memory. In the frequencydomain, the constitutive equation is written as σij =aijkhεkh + iωbijkhεkh, in which the tensor aijkh of theelastic coefficients and the tensor bijkh of the dampingcoefficients of the material depend on x, are indepen-dent of ω and have the usual properties of symmetry andpositive definiteness. The strain tensor εkh is related todisplacement field u by εkh = (∂uk/∂xh + ∂uh/∂xk)/2.Concerning the internal dissipative acoustic fluid, we usethe u − p formulation. The equation governing the vi-bration of the dissipative acoustic fluid occupying domainΩa is written as (Ohayon and Soize, 1998; Pierce, 1989;Lighthill, 1978)

ω2

ρac2a

p+iωτ

ρa∇2p+

1ρa

∇2p =τ

ρac2a∇2s−i

ω

ρas in Ωa ,

(3)for which the boundary conditions are

1ρa

(1 + iωτ)∂p

∂na= ω2u.na + τ

c2a

ρa

∂s

∂naon Γa , (4)

where ρa is the mass density of the acoustic fluid at equi-librium, ca is the speed of sound, τ is the coefficient dueto the viscosity of the fluid (the coefficient due to thermalconduction is neglected) and where s(x, ω) is the acousticsource density.

C. Mean computational structural-acoustic model

The finite element method (Zienkiewicz and Taylor,2000) is used to solve numerically the above boundaryvalue problem. We consider a finite element mesh ofthe structure Ωs and of the internal acoustic fluid Ωa.Let us = (us

1, . . . , usns

) be the complex vector of the ns

degrees of freedom (DOFs) of the structure accordingto the finite element discretization of the displacementfield u. Let pa = (pa

1, . . . , pa

na) be the complex vector of

the na DOFs of the acoustic fluid to the finite elementdiscretization of the pressure field p. The finite elementdiscretization of the boundary value problem in terms ofu and p (Ohayon and Soize, 1998), defined by Eqs.(1) to(4) with the Dirichlet condition on Γ0 yields the meancomputational structural-acoustic model,

[[As

ns(ω)] [Cns,na

]ω2[Cns,na

]T [Aana

(ω)]

] [us(ω)pa(ω)

]=

[f s(ω)fa(ω)

], (5)

where [Asns

(ω)] is the dynamical stiffness matrix of thedamped structure in vacuo which is a symmetric (ns×ns)complex matrix such that

[Asns

(ω)] = −ω2[M sns

] + iω[Dsns

] + [Ksns

] ,

in which [M sns

], [Dsns

], [Ksns

] are the mass, damping andstiffness matrices of the structure which are positive-definite symmetric (ns × ns) real matrices. In Eq.(5),[Aa

na(ω)] is the dynamical stiffness matrix of the dissipa-

tive acoustic fluid which is a symmetric (na×na) complexmatrix such that

[Aana

(ω)] = −ω2[Mana

] + iω[Dana

] + [Kana

] ,

in which [Mana

], [Dana

], [Kana

] are the ”mass”, ”damp-ing” and ”stiffness” matrices of the acoustic cavity withfixed coupling interface. The matrix [Ma

na] is a positive-

definite symmetric (na×na) real matrix, [Dana

] and [Kana

]are positive-semidefinite symmetric (na ×na) real matri-ces whose ranks are na − 1 . The matrix [Cns,na ] is thestructural-acoustic coupling matrix which is a (ns × na)real matrix.

D. Reduced mean computational structural-acoustic model

The projection of the mean computational structural-acoustic model on the structural modes in vacuo andon the acoustic modes of the acoustic cavity with fixedcoupling interface yields the reduced mean computa-tional structural-acoustic model. The structural modes

6

in vacuo and the acoustic modes of the cavity with fixedcoupling interface are calculated by solving the two gen-eralized eigenvalue problems

[Ksns

]ψ = λs[M sns

]ψ (6)

[Kana

]φ = λa[Mana

]φ . (7)

The eigenvectors verify the usual orthogonal properties(Bathe and Wilson, 1976; Geradin and Rixen, 1994;Ohayon and Soize; 1998). The structural displacement iswritten as

us(ω) = [Ψ] qs(ω) , (8)

in which [Ψ] is the (ns×n) real matrix whose columns areconstituted of the n structural modes associated with then first positive eigenvalues (the n first structural eigenfre-quencies). The internal acoustic cavity has one constantpressure mode and m − 1 acoustic modes. The internalacoustic pressure is written as

pa(ω) = [Φ] qa(ω) , (9)

in which [Φ] is the (na × m) real matrix whose columnsare constituted (1) of the constant pressure mode asso-ciated with the zero eigenvalue and (2) of the acous-tic modes associated with the positive eigenvalues (them − 1 first acoustical eigenfrequencies). It should benoted that the constant pressure mode is kept in orderto model the quasi-static variation of the internal fluidpressure induced by the deformation of the coupling in-terface (Ohayon and Soize, 1998). Using Eqs. (8) and(9), the projection of Eq. (5) yields the reduced meanmatrix model of the structural-acoustic system

[[As

n(ω)] [Cn,m]ω2[Cn,m]T [Aa

m(ω)]

] [qs(ω)qa(ω)

]=

[Fs(ω)Fa(ω)

], (10)

where [Cn,m] = [Ψ]T [Cns,na][Φ], [Fs(ω)] = [Ψ]T [f s(ω)]

and [Fa(ω)] = [Φ]T [fa(ω)]. The generalized dynamicalstiffness matrix [As

n(ω)] of the damped structure is writ-ten as

[Asn(ω)] = −ω2[Ms

n] + iω[Dsn] + [Ks

n] ,

in which [Msn] = [Ψ]T [M s

ns][Ψ] and [Ks

n] = [Ψ]T [Ksns

][Ψ]are positive-definite diagonal (n × n) real matrices andwhere [Ds

n] = [Ψ]T [Dsns

][Ψ] is, in general, a positive-definite full (n×n) real matrix. The generalized dynam-ical stiffness matrix [Aa

m(ω)] of the dissipative acousticfluid is written as

[Aam(ω)] = −ω2[Ma

m] + iω[Dam] + [Ka

m] ,

in which [Mam] = [Φ]T [Ma

na][Φ] is a positive-definite

diagonal (m × m) real matrix and where [Dam] =

[Φ]T [Dana

][Φ] and [Kam] = [Φ]T [Ka

na][Φ] are positive-

semidefinite diagonal (m×m) real matrices of rank m−1.

IV. STOCHASTIC REDUCED COMPUTATIONALSTRUCTURAL-ACOUSTIC MODEL

As explained in the Introduction, both data uncertain-ties and model uncertainties can be taken into account byusing the nonparametric probabilistic approach of uncer-tainties. Such an approach (Soize, 2000, 2001 and 2005a)has been used in linear and non linear structural dynam-ics. Nevertheless, this approach has not yet been used forcomplex structural-acoustic systems which requires theuse of extended results concerning random matrix the-ory in order to take into account model uncertainties forthe structural-acoustic coupling operator. Such extendedresults have been recently proposed by (Soize, 2005b).

A. Constructing the stochastic model

The use of the nonparametric probabilistic approach(Durand, 2007) of both model uncertainties and data un-certainties for the structure, the acoustic cavity and thestructural-acoustic coupling consists (1) in modeling thegeneralized mass [Ms

n], damping [Dsn] and stiffness [Ks

n]matrices of the structure by random matrices [Ms

n], [Dsn]

and [Ksn] whose dispersion parameters are denoted by

δMs , δDs and δKs respectively; (2) in modeling the gener-alized mass [Ma

m], damping [Dam] and stiffness [Ka

m] ma-trices of the acoustic cavity by random matrices [Ma

m],[Da

m] and [Kam] whose dispersion parameters are denoted

by δMa , δDa and δKa respectively; (3) in modeling thegeneralized structural-acoustic coupling matrix [Cn,m] bya random matrix [Cn,m] whose dispersion parameter isdenoted by δC . The explicit construction of the proba-bility distribution of these random matrices is given by(Soize 2000, 2001 and 2005b; Durand, 2007) for randommatrices [Ms

n], [Dsn], [Ks

n], [Mam], [Da

m], [Kam] and [C].

Let [H ] be anyone of these random matrices. The proba-bility distribution of such a random matrix [H] dependson its mean value [H ] = E{[H]} where E is the mathe-matical expectation and depends on its dispersion param-eter δH which must be taken independent of the matrixdimension. An algebraic representation of random ma-trix [H ] has been developed and allows independent re-alizations to be constructed for a stochastic solver basedon the Monte Carlo numerical simulation. This algebraicrepresentation is recalled below. For random matrices[Ms

n], [Dsn], [Ks

n], [Mam], [Da

m] and [Kam], random ma-

trix [H] is then a symmetric positive-definite (or positive-semidefinite) real-valued random matrix and [H] is writ-ten as [H ] = [LH ]T [GH ] [LH ] in which [H ] = [LH ]T [LH ]and where [GH ] is the random matrix germ. Note that if[H ] is positive-semidefinite than the factorization is nota Cholesky factorization and is obtained by another al-gebraic algorithm. When [H] is the rectangular matrix[Cn,m] (see Soize 2005b), using the following polar de-composition [Cn,m] = [U ][T ] with [U ]T [U ] = [ I ] where[T ] is a positive-definite matrix which can then be factor-ized as [T ] = [LC ]T [LC ], the random matrix [Cn,m] can

7

then be written as [Cn,m] = [U ][LC ]T [GH ] [LC ] in which[GH ] is another random matrix germ. The stochastic re-duced model of the uncertain structural-acoustic systemfor which the reduced mean model is defined by Eqs. (8)to (10) is written, for all ω fixed in the frequency bandof analysis as

U s(ω) = [Ψ]Qs(ω) , P a(ω) = [Φ]Qa(ω) , (11)

in which the �n-valued random variable Qs(ω) =

(Qs1(ω), . . . , Qs

n(ω)) and the �m-valued random variableQa(ω) = (Qa

1(ω), . . . , Qam(ω)) are the solution of the fol-

lowing random matrix equation[

[Asn(ω)] [Cn,m]

ω2[Cn,m]T [Aam(ω)]

] [Qs(ω)Qa(ω)

]=

[Fs(ω)Fa(ω)

], (12)

in which the random complex matrices [Asn(ω)] and

[Aam(ω)] are defined by [As

n(ω)] = −ω2[Msn] + iω[Ds

n] +[Ks

n] and where [Aam(ω)] = −ω2[Ma

m]+ iω[Dam]+ [Ka

m].Let GH be the positive-definite symmetric (ν × ν) ran-dom matrix representing [GMs

n], [GDs

n], [GKs

n], [GMa

m],

[GDam

], [GKam

] or [GCn,m ]. The following algebraic rep-resentation of random matrix [GH ] allows independentrealizations of [GH ] to be constructed. Random matrix[GH ] is written [GH ] = [LH ]T [LH ] in which [LH ] is arandom upper triangular (ν × ν) real matrix whose ran-dom elements are independent random variables definedas follows:(1) For j < j′, the real-valued random variable [LH ]jj′ iswritten as [LH ]jj′ = σνUjj′ in which σν = δH (ν +1)−1/2

and where Ujj′ is a real-valued Gaussian random variablewith zero mean and variance equal to 1. The parameterδH controlling the dispersion level of random matrix [H]is such that δH =

√E{‖[GH ] − [I]‖2

F }/ν in which [I]is the identity matrix and where the subindex F corre-sponds to the Frobenius norm.(2) For j = j′, the positive-valued random variable[LH ]jj′ is written as [LH ]jj′ = σν

√2Vj in which σν is

defined above and where Vj is a positive-valued gammarandom variable whose probability density function pVj

with respect to dv is written as

pVj (v) = �R+(v)1

Γ(ν+12δ2

H+ 1−j

2 )v( ν+12δ2

H

− 1−j2 )

e−v .

B. Confidence region of the random responses

Let ω �→ W (ω) from B into R be a random ob-servation of the structural-acoustic system in the fre-quency domain for which nr independent realizationsω �→ W (ω, θ1), . . . , ω �→ W (ω, θnr) are computed us-ing the stochastic model presented in Subsection IV.Awith the Monte Carlo numerical simulation. The con-fidence region associated with the probability level αfor the random function {W (ω), ω ∈ B} is constructedby using the method of quantiles (Serfling, 1980). Forfixed ω in B, let FW (ω) be the cumulative distribution

function (continuous from the right) of random variableW (ω) which is such that FW (ω)(w) = P (W (ω) ≤ w).For 0 < p < 1, the p-th quantile or fractile of FW (ω) isdefined as ζ(p) = inf{w : FW (ω)(w) ≥ p}. Then the up-per envelope w+(ω) and the lower envelope w−(ω) of theconfidence region are defined by w+(ω) = ζ((1 + α)/2)and w−(ω) = ζ((1−α)/2). The estimation of w+(ω) andw−(ω) is performed by using the sample quantiles.

V. STRUCTURAL-ACOUSTIC RESPONSE OF THEVEHICLE

In this section, we present the experimental identifi-cation of the dispersion parameters of the nonparamet-ric probabilistic approach of uncertainties for vibroa-coustic analysis of the vehicle for which the experi-mental database has been presented in Section II. Wethen present the validation of the structural-acousticmodel including uncertainties in comparing the identifiedstochatic computational model with experiments (Du-rand, 2007).

A. Description of the mean computationalstructural-acoustic model and experimental comparisons

The mean computational structural-acoustic model ofthe vehicle is a finite element model with 978, 733 struc-tural dofs and 8, 139 acoustic pressure dofs in the inter-nal acoustic cavity. Fig. 8 displays the finite elementmesh of the structure and Fig. 9 shows the finite elementmesh of the acoustic cavity. Finally, Fig. 10 deals withthe finite element mesh of the computational structural-acoustic model. The reduced mean computational modelis constructed using n = 1, 723 elastic modes of the struc-ture in vacuo and m = 57 acoustic modes of the inter-nal acoustic cavity with rigid walls. These values of nand m have been deduced from a mean-square conver-gence analysis of the random response. Concerning thedamping of the mean computational structural-acousticsmodel, the generalized damping matrix of the structureand of the internal acoustic cavity are assumed to be di-agonal matrices for all frequencies in B. Note that sucha diagonal assumption introduces model uncertainties inthe mean computational model which are taken into ac-count by the nonparametric probabilistic approach. Con-sequently, the damping part in the mean model is de-scribed in terms of damping rates which have been fixedto their nominal values. As it can be seen on the ex-perimental frequency response functions, the structuraldamping is significant and the internal acoustic cavitydamping is relatively high due to the presence of absorb-ing materials. For the structural FRFs, Figs. 11 and12 display the comparisons between the mean computa-tional structural-acoustic model predictions and the ex-periments for the moduli of the FRFs in dB concern-ing observations Obs4 (Fig. 11) and Obs6 (Fig. 12) and

8

FIG. 8. Finite element mesh of the structure: 978, 733 struc-tural dofs.

FIG. 9. Finite element mesh of the acoustic cavity: 8, 139acoustic pressure dofs.

for the 20 cars. For the root mean square of the acous-tic pressures averaged on the 32 microphones inside theinternal acoustic cavity (see Subsection II.B), Fig. 13displays the graphs of f �→ dBexp

rms(f) for the 30 con-figurations of the internal acoustic cavity and the cor-responding graph f �→ dBrms(f) calculated with themean computational model. Fig. 14 displays the graphof f �→ dBexp[A](f) related to the experimental synthe-sized booming noise (see Subsection II.C) and the corre-sponding graph f �→ dB[A](f) calculated with the meancomputational model. Figs. 11 to 14 show that there aresignificant differences between measurements and numer-ical simulations which are not due to the variabilities ofexperiments (optional extras and manufacturing process)but are mainly due to model uncertainties. As explainedin Section I, these figures clearly show that both data un-certainties and model uncertainties have to be taken intoaccount in the mean computational structural-acousticmodel in order to improve the predictability and the ro-

FIG. 10. Finite element mesh of the computationalstructural-acoustic model.

bustness of the predictions.

20 40 60 80 100 120 140 160 180 200 220Frequency, Hz

Mod

ulus

, dB

10 dB

FIG. 11. Comparisons of the mean computational model re-sults with the experiments. Graphs of the modulus in dB scaleof the structural FRF for observation Obs4: experiments forthe 20 cars (grey lines), mean computational model (thicksolid line).

B. Methodology and assumptions for the experimentalidentification of the dispersion parameters

The methodology and the assumptions used to identifythe dispersion parameters δMs , δDs , δKs for the struc-ture, δMa , δDa , δKa for the internal acoustic cavity andδC for the structural-acoustic coupling interface are thefollowing.

(1) The dispersion parameters δMa , δDa and δKa ofthe internal acoustic cavity are identified using the ex-perimental database defined in Subsection II.B. For thisidentification it is assumed that δMa = δDa = δKa . First

9

20 40 60 80 100 120 140 160 180 200 220Frequency, Hz

Mod

ulus

, dB

10 dB

FIG. 12. Comparisons of the mean computational model re-sults with the experiments. Graphs of the modulus in dB scaleof the structural FRF for observation Obs6: experiments forthe 20 cars (grey lines), mean computational model (thicksolid line).

50 100 150 200 250 30075

80

85

90

95

100

Frequency, Hz

Mod

ulus

, dB

FIG. 13. Comparisons of the mean computational model re-sults with the experiments. Graphs of the root mean squareof the acoustic pressures averaged in the cavity in dB scale:experiments for the 30 configurations (grey lines), mean com-putational model (thick solid line).

this hypothesis allows the computational cost to be de-creased. Secondly, different analyses have shown that theconfidence regions are not very sensitive to the value ofdispersion parameter δDa . This is due to the fact thatthe modal density is relatively high in the frequency bandof analysis and in addition, the internal acoustic cavityhas a large dissipative factor due to the presence of theabsorbing materials as explained above. Finally, a sensi-tivity analysis of the responses has been carried out withrespect to δMa and δKa and has shown that the confi-dence regions were very sensitive to the values of the dis-persion parameters but that δMa = δKa could be written.

50 100 150 200Frequency, Hz

Mod

ulus

, dB

[A]

1000 2000 3000 4000 5000 6000

Rpm (rotation per minute)

10 dB

FIG. 14. Comparisons of the mean computational model re-sults with the experiments. Graphs of the modulus of thesynthesized booming noise in dB[A] scale: experiments forthe 20 cars (grey lines), mean value of the experiments (thinsolid line), mean computational model (thick solid line).

The method used to identify the dispersion parameter δa

such that δa = δMa = δDa = δKa is the maximum likeli-hood method (Spall, 2003 ; Walter and Pronzato, 1997).

(2) The dispersion parameters δMs , δDs and δKs forthe structure are identified using the structural frequencyresponse functions (and not the structural-acoustic fre-quency response functions) of the experimental databasedefined in Subsection II.A. Therefore, the stochasticcomputational structural-acoustic model must be used.Then, for the identification of these dispersions parame-ters, dispersion parameters δMa , δDa and δKa are fixed tothe identified experimental value δa and the value of δC isfixed to a nominal value. This last hypothesis is reason-able because the confidence regions of the random struc-tural responses are not sensitive to the acoustic coupling(note that the structural acoustic responses are sensitiveto δC , but we are presently speaking about the identi-fication of δMs , δDs and δKs using only the structuralresponses). Concerning δDs the situation is similar tothe case of the internal acoustic cavity, but for the struc-ture, δDs has been fixed to a nominal value which resultsfrom previous studies. In addition, it should be notedthat the confidence regions of the structural responsesare a little sensitive to δDs compared with the high sen-sitivity of these responses with respect to δMs and δKs .Concerning the experimental identification of δMs and

10

δKs the assumption δs = δMs = δKs (similarly to the as-sumption used for the internal acoustic cavity) has beenreplaced by the following one δs =

√δ2Ms + δ2

Ks . Thischoice results from computational tests and is much moreefficient and more representative than the other one. Forcomputational cost reasons (see below), the maximumlikelihood method could not be used and consequently,has been substituted by the mean-square method witha differentiable objective function (Spall, 2003 ; Walterand Pronzato, 1997).

(3) For the identification of the dispersion param-eter δC of the structural-acoustic coupling interface,the structural-acoustic frequency response functions ofthe experimental database defined in Subsection II.Care used. Therefore, the stochastic computationalstructural-acoustic model must be used. Then, for theidentification of δC , dispersion parameters δMa , δDa , δKa ,δMs , δDs and δKs are fixed to their identified values (seeabove). For similar reasons to those given above, themaximum likelihood method cannot be used and shouldthen be substituted by the mean-square method. Un-fortunately, the corresponding objective function is notsufficiently sensitive to the value of δC and the maximumlikelihood method is not efficient. So we have identifiedthe value of δC using a trial method.

C. Experimental identification of the dispersion parametersof the internal acoustic cavity

As explained above, the maximum likelihood methodis used to identify the dispersion parameter δa introducedin Subsection V.B(1). The random observation Y and itscorresponding experimental quantity Y exp used for thisidentification is defined by

Y =∫

Bf

dBrms(f) df , Y exp =∫

Bf

dBexprms(f) df ,

in which dBrms(f) corresponds to dBexprms(f) defined in

Subsection II.B for the experiments and which is writtenas

dBrms(f) = 20 log10

⎛⎝

√132

∑32�=1 |P a

j�(2πf)|2

Pref

⎞⎠ ,

in which j1, . . . , j32 are the DOFs corresponding to themeasurement points. Let Y exp(η1), . . . , Y exp(η30) be theindependent experimental realizations corresponding tothe 30 configurations of the internal acoustic cavity. Letδa �→ L(δa) be the log-likelihood function defined by

L(δa) =30∑

ν=1

log10 pY (δa, Y exp(ην)) ,

in which y �→ pY (δa, y) is the probability density func-tion of random variable Y . For all fixed δa, the right-hand side of the above equation is estimated by the

Monte Carlo method with the stochastic computationalstructural-acoustic model presented in Subsection IV.A.The maximum likelihood method consists in solving thefollowing optimization problem

δopta = argmax

δa

L(δa) .

This method needs an accurate estimation ofpY (δa, Y exp(ην)) which has been estimated with20, 000 independent realizations for the Monte Carlosimulation. Instead of using an optimization algorithmto solve the above optimization problem, the values ofthe log-likelihood function have directly calculated in8 values of δa in order to construct an approximationof its graph on the interval of interest. Fig. 15 displaysthe approximation of the graph of δa �→ L(δa) allowingthe optimal value δopt

a to be estimated. For this opti-

Para

met

erFu

nctio

n (

a )

aParameter opta

FIG. 15. Graph of δa �→ L(δa)

mal value of δopta , Fig. 16 compares the experimental

measurements with the computational results for theroot mean square dBexp

rms(f) of the acoustic pressuresaveraged on the 32 microphones inside the internalacoustic cavity. In this figure, (1) the 30 grey linesrepresent the experimental measurements, (2) the upperand lower thick solid lines represent the upper andlower envelopes of the confidence region calculated fora probability level of 0.96, (3) the mid thin solid linerepresents the mean value of the random response ofthe stochastic reduced computational model, and (4)the dashed line represents the response of the reducedmean computational model. It can be seen that thethe experiments belongs to the confidence region with aprobability level of 0.96 that validates the acoustic partof the stochatic computational model.

D. Experimental identification of the dispersion parametersof the structure

As explained in Subsection V.B(2), the mean-squaremethod with differentiable objective function is used.

11

50 100 150 200 250 30075

80

85

90

95

100

Frequency, Hz

Mod

ulus

, dB

FIG. 16. Comparisons of the stochastic computational modelresults with the experiments. Graphs of the root mean squareof the acoustic pressures averaged in the cavity in dB scale:experiments for the 30 configurations (grey lines), mean com-putational model (dashed line), mean value of the randomresponse (mid thin solid line), confidence region: upper andlower envelopes are the upper and lower thick solid lines.

The dispersion parameters δMa , δDa , δKa , δDs and δC

are fixed to the values identified above (see V.B(1) and(2)). Consequently, we have to solve the optimizationproblem defined by

δopts = arg min

δs

J(δs) ,

in which the objective function J(δs) depends only on δs

and is defined by

J(δs) = 2(1−γ) |||Z(δs)−m(δs)|||2+2γ ||m(δs)−Zexp||2B ,

with γ = 0.5. In this equation, the random observa-tion Z(ω, δs) = (Z1(ω, δs), . . . , Z6(ω, δs)) of the stochas-tic computational model related to the six structural ob-servations (see Subsection II.A) is such that

Zj(ω, δs) = log10(ω2 |U s

kj(ω, δs)|) ,

and its mean valuem(ω, δs) = (m1(ω, δs), . . . , m6(ω, δs))is

m(ω, δs) = E{Z(ω, δs)} .

The corresponding experimental observation is denotedby Zexp(ω) = (Zexp

1 (ω), . . . , Zexp6 (ω)) and its mean value

is

Zexpj (ω) =

120

20∑�=1

Zexpj (ω, η�) .

Finally, the norms are defined by

|||Z(δs)−m(δs)|||2 = E{6∑

j=1

∫B

|Zj(ω, δs)−mj(ω, δs)|2 dω} .

||m(δs) −Zexp||2B =6∑

j=1

∫B

|mj(ω, δs) − Zexpj (ω)|2 dω .

The first norm represents the variance of the computa-tional model due to uncertainties and the second normrepresents the bias between the experiments and thestochastic model. In this objective function, B = 2π Bf

with Bf = [100, 180] Hz and the frequency resolution is0.5Hz. For each evaluation of the objective function, thestochastic reduced computational model is solved usingthe Monte Carlo method with 1000 independent realiza-tions and corresponding to a mean-square convergence ofthe second-order stochastic solution. Since each evalua-tion of the objective function requires about 500 hours ofCPU time (the computations have been realized with 20CPU yielding an elapsed time of 25 hours for each eval-uation of the objective function), we have used a trialmethod consisting in computing the cost function for 10values of the couple (δMs , δKs). It can then be deducedthe approximation of the graph of δs �→ J(δs) for the 10

values of δs =√

δ2Ms

+ δ2Ks

. Fig. 17 displays this approx-imation of the graph of δs �→ J(δs) allowing the optimalvalue δopt

s to be estimated. From δopts , it can then be

deduced the optimal values δoptMs

and δoptKs

of δMs and δKs .For this optimal values δopt

Msand δopt

Ksof δMs and δKs , and

s

s

s

Func

tion

J(

)

Parameteropt

FIG. 17. Graph of δs �→ J(δs)

for the values of the other dispersion parameters fixed totheir values previously identified, Figs. 18 and 19 com-pare the experimental measurements with the computa-tional results for the moduli of the FRFs in dB concern-ing observations Obs4 (Fig. 18) and Obs6 (Fig. 19) andfor the 20 cars. In these figures, (1) the 20 grey linesrepresent the experimental measurements, (2) the upperand lower thick solid lines represent the upper and lowerenvelopes of the confidence region calculated for a prob-ability level of 0.96, (3) the mid thin solid line representsthe mean value of the random response of the stochasticreduced computational model, and (4) the dashed linerepresents the response of the reduced mean computa-tional model. It can be seen that the experiments belongto the confidence region with a probability level of 0.96

12

that validates the structural part of the stochatic com-putational model.

20 40 60 80 100 120 140 160 180 200 220Frequency, Hz

Mod

ulus

, dB

10 dB

FIG. 18. Comparisons of the stochastic computational modelresults with the experiments for observation Obs4. Graphs ofthe moduli of the FRFs in dB scale: experiments for the 20cars (grey lines), mean computational model (dashed line),mean value of the random response (mid thin solid line), con-fidence region: upper and lower envelopes are the upper andlower thick solid lines.

20 40 60 80 100 120 140 160 180 200 220Frequency, Hz

Mod

ulus

, dB

10 dB

FIG. 19. Comparisons of the stochastic computational modelresults with the experiments for observation Obs6. Graphs ofthe moduli of the FRFs in dB scale: experiments for the 20cars (grey lines), mean computational model (dashed line),mean value of the random response (mid thin solid line), con-fidence region: upper and lower envelopes are the upper andlower thick solid lines.

E. Experimental validation of the stochastic computationalstructural-acoustic model for the booming noise

In this subsection, we compare the experiments for thesynthesized booming noise of the database (see Subsec-

tion II.C) with the predictions given by the stochasticcomputational structural-acoustic model. For the calcu-lation, the dispersion parameters δMa , δDa , δKa , δMs ,δDs , δKs and δC are fixed to their optimal values whichare either fixed or experimentally identified as explainedin the previous subsections. The Monte Carlo stochas-tic solver is used with 1500 realizations. A mean-squareconvergence analysis of the stochastic response has beencarried out (Durand, 2007) and convergence is reachedfor this number of realizations. Fig. 20 compares theexperimental measurements with the computational re-sults for the booming noise in dB[A] scale. The 20 thingrey lines represent the experimental measurements ofthe booming noise for the 20 cars and the thick greyline the mean value of the experiments. The upper andlower thick solid lines represent the upper and lower en-velopes of the confidence region calculated for a proba-bility level of 0.96. The mid thin solid line representsthe mean value of the random response of the stochasticreduced computational model. The mid dashed line rep-resents the response of the reduced mean computationalmodel. Taking into account the complexity of the vibroa-coustic model, the obtained results validate the stochas-tic computational model and demonstrate its capabilityto predict experimental measurements knowing that thedispersion parameters of model uncertainties have beenidentified.

40 60 80 100 120 140 160 180 200Frequency, Hz

Mod

ulus

, dB

[A]

1000 2000 3000 4000 5000 6000

Rpm [rotation per minute]

10 dB[A]

FIG. 20. Comparisons of the stochastic computational modelresults with the experiments for the booming noise. Graphs ofthe moduli of the FRFs in dB[A] scale: experiments for the 20cars (thin grey lines), mean value of the experiments (thichgrey line), mean computational model (dashed line), meanvalue of the random response (mid thin solid line), confidenceregion: upper and lower envelopes are the upper and lowerthick solid lines.

13

VI. CONCLUSION

In this paper, we have presented and validated amethodology to analyze a very complex structural-acoustic system in the low- and medium-frequencyranges. Both data uncertainties and model uncertain-ties have been taken into account in the computationalmodel. The nonparametric probabilistic approach intro-duces a small number of dispersion parameters which canthen be experimentally identified in solving optimizationproblems. In the context of the automative industry, alarge experimental database has been constructed andhas been presented in this paper. This database hasbeen used for the identification of the probabilistic model.An experimental validation has been proposed for eachstep of the identification. The global experimental val-idation of the stochastic computational model has beenobtained. The comparisons are good enough taking intoaccount the complexity of the system and taking into ac-count that the mean computational model is unique torepresent all the variabilities induced by optional extrasand manufacturing process. In addition, this mean com-putational model has voluntary not been updated withthe measurements (this is the practical situation encoun-tered by engineering in such an automative industry).With respect to the real structural-acoustic system, thedynamical behavior of the sound-proofing schemes (in-sulation materials) existing in the real cars (and in par-ticular in the cars for which the experimental databasehas been constructed) has not been taken into account inthe mean computational model (but has been taken intoaccount as pure mass subsystems). Such a model couldbe improved in including such a dynamical model of thesound-proofing schemes.

VII. REFERENCES

Attala, N., Hamdi, M.A., and Panneton, R. (2001).”Enhanced weak integral formulation for the mixed (u, p)poroelastic equations,” J. Acoust. Soc. Am. 109(6),3065-3068.

Bathe, K.J. and Wilson, E.L. (1976) Numerical Meth-ods in Finite Element Analysis, Prentice-Hall, Inc., En-glewood Cliffs, New Jersey.

Durand, J.F., Gagliardini, L., and Soize, C. (2004).”Random modeling of frequency response functions ofcars,” International Conference on Modal Analysis Noiseand Vibration Engineering (ISMA 2004) proceedingsCDROM ISBN 90-73802-82-2.

Durand, J.F., Gagliardini, L., and Soize, C. (2005a).”Nonparametric modeling of the variability of vehiclestructural-acoustic behavior,” SAE 2005 Noise and Vi-bration Conference proceedings CDROM ISBN 0-7680-1657-6.

Durand, J.F., Gagliardini, L., and Soize, C. (2005b).”Nonparametric modeling of structural-acoustic couplinguncertainties,” EuroDyn 2005 Conference proceedings

CDROM ISBN 85-85769-03-03.Durand, J.F. (2007). ”Structural-acoustic modeling

of automotive vehicles in presence of uncertainties andexperimental identification and validation,” PHD Thesis,Universite de Marne-la-Vallee, France, 10 mai 2007.

Gagliardini, L., Houillon, L., Borello, G., andPetrinelli, L. (2005). ”Virtual SEA-FEA-based model-ing of mid-frequency structure borne noise,” Sound andVibration 39(1), 22-28

Geradin, M. and Rixen, D. (1994) Mechanical Vibra-tions, Wiley, Chichester, U.K.

Ghanem, R. and Spanos, P.D (2003). Stochastic Fi-nite Elements: A spectral approach revised edition, DoverPublications, New-York.

Hamdi, M.A., Zhang, C., Mebarek, L., Anciant, M.,and Mathieux, B. (2005). ”Analysis of structural-acoustic performances of a fully trimmed vehicle usingan innovative subsystem solving approach faciliting thecooperation between carmakers and sound-package sup-pliers,” EuroDyn 2005 Conference proceedings CDROMISBN 85-85769-03-03.

Hayashi, K., Yamaguchi, S., and Matsuda, A. (2000).”Analysis of booming noise in light-duty truck cab,” J.of the Society of Automotive Engineers 21, 255-257.

Hills, E., Mace, B., and Ferguson, N.S. (2004).”Statistics of complex built-up structures,” InternationalConference on Modal Analysis Noise and Vibration En-gineering (ISMA 2004) proceedings CDROM ISBN 90-73802-82-2.

Jaynes, E.T. (1957) ”Information theory and statis-tical mechanics,” Physical Review 106(4)620-630 and108(2)171-190.

Kompella, M.S. and Bernhard, R.J (1996). ”Varia-tion of structural-acoustic characteristics of automotivevehicles,” J. of Noise Control Engineering, 93-99.

Langley, R.S (2007). ”On the diffuse field reciprocityrelationship and vibrational energy variance in a randomsubsystem at high frequencies,” J. Acoust. Soc. Am.121(2), 913-921.

LeBot, A. (2002). ”Energy transfer for high frequen-cies in built-up structures,” J. Sound Vib. 250(2), 247-275.

Lighthill, J. (1978) Waves in Fluids, Cambridge Uni-versity Press, MA.

Lyon, R.H. and Dejong, R.G (1995). ”Theory andapplications of Statistical Energy Analysis,” edited byButterworth-Heinemann, Boston.

Ohayon, R., and Soize, C. (1998). Structural Acous-tics and Vibration, Academic Press, San Diego.

Pierce, A.D. (1989). Acoustics: An Introduction toits Physical Principles and Applications, Acoust. Soc.Am. Publications on Acoustics, Woodbury, NY, USA,(originally published in 1981, McGraw-Hill, New York).

Schueller, G.I (1997). ”A state-of-the-art report oncomputational stochastic mechanics,” Probab. Eng.Mech., 12(4), 197-321.

Schueller, G.I (2007). ”On the treatment of uncer-tainties in structural mechanics and analysis,” J. Comp.

14

Struct., 85, 235-243.Serfling, R .J. (1980). Approximation Theorems of

Mathematical Statistics, John Wiley & Sons.Shannon, C.E. (1948). ”A mathematical theory of

communication,” Bell System Technology Journal 27379-423 and 623-659.

Shorter, P.J., and Langley R.S. (2005). ”Vibro-acoustic analysis of complex systems,” J. Sound Vib.288(3), 669-699.

Soize, C. (2000). ”A nonparametric model of randomuncertainties for reduced matrix models in structural dy-namics,” Probab. Eng. Mech., 15(3), 277-294.

Soize, C. (2001). ”Maximum entropy approach formodeling random uncertainties in transient elastodynam-ics,” J. Acoust. Soc. Am. 109(5), 1979-1996.

Soize, C. (2003). ”Random matrix theory and non-parametric model of random uncertainties in vibrationanalysis,” J. Sound Vib. 263, 893-916.

Soize, C. (2005a). ”A comprehensive overview of anon-parametric probablistic approach of model uncer-tainties for predictive models in structural dynamics,”J. Sound Vib. 288(3), 623-652.

Soize, C. (2005b). ”Random matrix theory for model-

ing random uncertainties in computational mechanics,”Comp. Meth. Appl. Mech. Eng. 194(12-16), 1333-1366.

Sol, A., and Van-Herpe, F. (2001). ”Numerical predic-tion of a whole car vibroacoustic behavior at low frequen-cies,” SAE 2001 Noise and Vibration Conference proceed-ings CDROM ISBN 0-7680-0775-5.

Spall, J.C. (2003). Introduction to Stochastic Searchand Optimization, John Wiley and Sons, Hoboken, NewJersey.

Sung, H.S., and Nefske, D.J. (2001). ”Assessment of avehicle concept finite element model for predicting struc-tural vibration,” SAE 2001 Noise and Vibration Confer-ence proceedings CDROM ISBN 0-7680-0775-5.

Trusdell, C. (1960). ”The Elements of ContinuumMechanics,” Springer-Verlag, Berlin.

Walter, E., and Pronzato, L. (1997). Identification ofParametric Models from Experimental Data, Springer.

Wood, L.A., and Joachim, C.A (1987). ”Interior noisescatter in four-cylinder sedans and wagons,” Int. J. ofVehicle Design 8, 428-438.

Zienkiewicz, O. C., and Taylor, R. L. (2000). TheFinite Element Method, Fifth edition, Vol. 1 to 3,Butterworth-Heinemann, Oxford.

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