AIAA-97-1686-CPANALYTICAL, NUMERICAL AND EXPERIMENTAL STUDY
OF THE ACOUSTIC BEHAVIOR OF A RECTANGULARCAVITY WITH ABSORBING TREATMENT ON A WALL
Christian Y. Glandier*, Carole E. Floc'h*, Mohamed A. Hamdi*** Daimler-Benz A.G. F1M/GA, HPC G201,70546 Stuttgart, Germany, [email protected]
Phone: (49) 711 17 55 866 - Fax: (49) 711 17 52 420** Professor, Universite" de Technologic de Compiegne, 60200 Compiegne, France,
Scientific Advisor, Straco S.A. rue des fonds Pernants, 60200 Compiegne, France, [email protected]: (33) 3 44 30 43 60 - Fax: (33) 3 44 86 87 77
AbstractThe acoustic behavior of a rigid-walled rectangularenclosure with a porous absorbing material on one ofits sides is investigated experimentally, as well asusing an analytical method (modal expansion) and anumerical method (Boundary Element Formulation).The investigation is not limited to the lowerresonance frequencies, but extends up to the mid-frequency range. The frequency band of interestincludes more than 180 resonance frequencies. Thematerial is considered locally reactive. The BEM andanalytical results agree very well. A good agreementis also obtained for the comparison betweencomputations and experiments.
Nomenclaturec0 speed of sound in the acoustic medium/ frequencyG(x,y) Green's functionk wavenumberkl eigenvalue for the n* cavity modeLx Ly Lt dimensions of the rectangular cavityn outward pointing unit normalnx ny «j modal indexesp acoustic pressureQ volume velocity of a sound sourcer(x,y) distance between points x and yS surface bounding the enclosure50 acoustically rigid surface51 surface treated with a porous absorbing
material52 rigid surface with imposed normal acoustic
velocityu imposed piston normal velocityV internal volume of the enclosurex, y coordinate vectors
x,y,z Cartesian coordinatesP specific normal admittance of an absorbing
materialA Laplacian operator
—— normal derivative
8 ,̂ Kronecker delta (1 if m = n , 0 if m # n)8 (x) Dirac delta function (1 if x = 0, 0 if x * 0)
, modal normalization constantsacoustic wavelengthangular frequency (2nf)
(»„ resonance frequency of the n* acousticmode
*Pn(x) n* acoustic mode of the cavity with rigidboundaries
Po density of the acoustic medium (air)£ viscous damping factor
1. IntroductionInterior acoustics is a major concern for car
manufacturers, as well as for the whole transportationindustry as it stands as a symbol for quality andcomfort. The automotive engineer faces a difficultchallenge. Automobiles are complex products whichevolve rapidly along with customer expectations,technical progress and regulations, thereforedevelopment time and costs have to be reduced to theminimum. Weight is also a major concern. Numericalsimulation has become a crucial tool forunderstanding problems and solving them in a quickand efficient manner by assessing the virtues of alarge number of potential solutions in a short periodof time.
In the past ten years, the development ofadequate numerical formulations and supercomputershas allowed tremendous progress to be achieved inthe field of vibroacoustics. Structural Finite Element(FE) car models are commonly used to compute thevibroacoustic behavior of the body in white coupled
with the car cabin, in the low frequency range (up to200 Hz)1. Efforts have been made to incorporate theeffect of absorbing materials (seats, roof liner, mats,etc...). However, these complete FE models onlyapply to low frequencies. The structural complexity ofthe body makes it extremely difficult to extend thecomputations to the medium frequency range. Verylittle has been done to study the effect of absorbingmaterials in the interior for this frequency range. Abetter understanding of these effects might lead toacoustical improvements as well as to weightreductions. The same considerations also apply to theaerospace and railway industries.
In this paper, we are mainly interested in theacoustic behavior of the cabin in the presence ofabsorbing material, for the medium frequency range.No vibroacoustic coupling with an elastic structure istaken into account in order to focus on the absorption.A boundary element (BEM) formulation2 is used tosolve the acoustic problem (I-DEAS Vibro-Acoustics™ software3). Similar investigations werecarried out in the past4 but they were limited to the50 - 200 Hz range.
The aim of the present study is to validatesimulations in the simple case of a rectangularenclosure with one absorbing panel. This configura-tion has been chosen because of the availability of ananalytical solution and the relative simplicity ofsetting up a well controlled experiment.
A BEM model of the cavity was built and usedfor computations from 100 to 1000 Hz. An analyticalmodel based on a modal superposition approach5'6 isalso presented. The sound field is expressed as aweighted sum of rigid modes. In the model, theabsorbing material is taken into account for modaldamping as well as for inter-modal coupling. Thismodel has the advantage of providing more physicalinsight into the problem.
2. Problem under studyThe cavity under study (see Fig. 1) has the
following dimensions: 1.45 m x 1.05 m x 0.75 m. Itsvolume is approximately one third of the interiorvolume of a European family car. Five of its walls areconsidered acoustically rigid and one panel is treatedwith absorbing material. In the models, the acousticmaterial is considered locally reacting. The acousticexcitation is provided by wall mounted 56 mmdiameter sources having a known volume velocity Q(area integral over the source surface of the outwardnormal velocity). This type of source can be modeledas a piston.
The following section is dedicated to settingup the theoretical basis for the modeling of theproblem and its solution using two methods.
•xcftetion position(known voJum*wkxlty) 82
Fig. 1 Configuration under study.
2.1 Boundary value formulation of theproblem
Let us consider a rectangular enclosure withdimensions LxxLy\ Lz (see Fig. 1).The walls are acoustically rigid, except for the x = Lxplane which is covered with a porous absorbingmaterial of spatially constant known specific acousticadmittance J3. Excitation is provided by an idealizedsound source placed on one wall acting like a rigidpiston with known normal velocity u.
Considering harmonic solutions for theacoustic pressure p of the form p = p (x,y,z) e~'°" ,the problem is completely described by the followingset of four equations:• The homogeneous Helmholtz equation in the
acoustic medium in the absence of sources:Ap+k*p=0 (1)
The boundary conditions are:• The zero normal velocity condition on the
acoustically rigid walls SQ:
= 0 (2)
The local reaction impedance condition on thetreated wall Si. Local reaction assumes that therelation between pressure and normal velocity atthe point under consideration is independent of thesurrounding acoustic environment5.
(3)
The continuity equation for the acoustic velocityat the source, on S2:
(4)
The problem can also be described in integralform by the Kirchhoff-Helmholtz equation whichexpresses the sound pressure at any point x in the
volume V with respect to the sound pressure and itsnormal derivative on the surface S of the enclosure:
(5)
withy = 1y = Vay = 0
for x e V, interior volume of the cavityfor x e Sfor x e (VuS) : the acoustic sources donot radiate outside the cavity
S = S0 + Si + S2 surface enclosing the cavitywhere G(x,y) denotes a Green's function whichsatisfies the inhomogeneous Helmholtz equation inthe enclosed acoustic medium in the followingmanner:AG(x,y)+*2G(x,y) = -
2.2 Modal approach
(6)
2.2.1 Theoretical backgroundIn this approach, the sound field is described
as a theoretically infinite sum of the rigid modes ofthe enclosure (zero normal velocity condition onevery wall). This is similar to solving the problemusing rigid acoustic cavity modes computed by finiteelement software.
The n"1 mode, defined by a triplet of integermodal indexes (nx, «>, nt) has the followingexpression:
cos cos y cos(7)
The corresponding eigenvalue kn2 is given by the
formula
(8)
,/£„ £„ £„ is a normalization constant such that the•y nx nv nz
orthogonality relation yields
n(x)Vm(x)dV = V8nm . (9)
This normalization with respect to the volume Vimplies that e^ =1 if «,= 0 and £„. =2 if n, & 0, fori - x, y, z.These modes are orthonormal eigenfunctions whichsatisfy the homogeneous Helmholtz equation (1)inside the cavity, so thatA¥,,+ * n
2 ¥ , ,=0 . (10)It can also be proved that these functions constitute acomplete set, i.e., that any well-behaved function/(x)
in the cavity can be approximated as a linearcombination of the ^(x) . If/(x) is a complex func-tion, the summation will involve complexcoefficients.
These rigid modes can be used to construct aGreen's function of the problem such that
y ) =0 on every wall.
It has the following expression6:
Applying the boundary conditions from Eqs (2) to (4)to the integral equation (5) and utilizing the propertyof the Green's function (zero normal derivative on theboundary), leads to the expression
yp(\)=\ G(x,y)jtop0u(y)JS2
G(x,y)jkj3(y) p(y)dSl(y)(12)
Since the eigenmodes form a complete set, theacoustic pressure can be expressed as a sum of therigid modes
Xx)=!>,,¥,,(x). (13)n= 0
Inserting this expression and the expression of theGreen's function (11) into Eq. (12) yields a system ofequations in terms of the an coefficients
1a. =
(14)
In our case, u and p are constant with respect to y andcan thus be taken out of the integral.
This leads to the following coupled algebraicsystem for the an coefficients:
(15)
with
If m = n, Dm corresponds to the damping termof mode n. It can be shown7 that the real part ofDm provides damping and that its imaginary partis responsible for a frequency shift.If m it n, Dnn, is a coupling term between mode nand m. In the case under study,Dm * 0 if ny = nz
cn is the projection of the excitation on the n*mode.
2.2.2 Computational considerationsEquation (15) can be put into matrix form for a
finite number of modes as follows:[B-jkD](a(a>)} = {c} (18) I
2{a} vector of the modal contributions an
B diagonal matrix, such that Bnm — 8^ ( k 2 - k2) t{c} vector of the modal excitations cn JD matrix of the Dnm coefficients.
The modal approach requires the determina-tion of the number of modes which ensure theconvergence of the series in equation (13). Thisnumber increases with damping and inter-modalcoupling.
For the computations which follow, it wasfound that taking into account the rigid modes havingresonances up to 2000 Hz, a total of 1145 modes,provided a good convergence up to 1200 Hz.
2.3 BEM formulation2.3.1 Theoretical background
Now, we use the free field Green's function in Eq. (5).
It is expressed as G(x,y)= (19)4;rr(x,y)
where r (x,y) is the distance between the excitation, amonopolar source placed at position y and theobservation point x in the three dimensional space /?3.
In a bounded domain, as in our case, whenonly the internal sound field matters and the externalconditions have no influence on the solution, the mostappropriate method for solving the problem is the so-called "direct" or "trace" approach8. It consists ofsolving the problem in terms of the pressure p (x) andits normal derivative on the internal boundary S.
For completeness, it should be emphasizedthat a more general approach called "indirect" or"jump" can be used to solve simultaneously theinterior and exterior problems, for the case ofidentical acoustic media in both domains. This time,the unknowns are the pressure jump and the velocityjump on the boundary. This method is particularlyappropriate if the external and internal sound fieldsinfluence each other, as for example in the case of anelastic boundary or of an open cavity9.
For x e 5, Eq. (5) and its normal derivative yieldrespectively:
(20-b)
where the singular integrals corresponding to the firstand second derivative of the free space Green'sfunction G are taken respectively in the sense ofCauchy's principal value and in the sense ofHadmard's finite part.
The boundary conditions (2) - (4) can bewritten in more general form as
U(JC) (21)
where, velocity u is zero everywhere except on 82 andadmittance (3 is zero everywhere except on Si.Substituting Eq. (21) into the difference between(20-b) and (20-a), we obtain the following boundaryequation:
(22)jp0c0k
where the kernels K(x,y) and H(x,y) are respectivelygiven by
2 dGK(x,y) = k p(x)G(x,y) + jkff(x)-——
dG
dn;*j3(x)G(x,y)
(22-a)
(22-b)
The boundary integral equation (22) can besolved using the symmetric variational formulationproposed by Hamdi10, as described below. Eq. (22) ismultiplied by an arbitrary test function q(x) regular onthe boundary S and is integrated over S, yielding:
= q(\)u(\)dS(x) ( 2 3 - b ) possible source positionsmicrophone arrays
absorbing materialon this panel
and
B(q,u)=\ q ( x ) u ( y ) H ( x , y ) d S ( y ) d S ( x ) (23-c)JSxS
are the linear coupling operators between theprescribed velocity u and the trial function q(\) on S.
The discretization by the Boundary ElementMethod (BEM) leads to the symmetric linearalgebraic system,
= jkpacu[±C-B]{u} (24)
where D is a complex full symmetric matrix and[jC-fi] is a rectangular (non-symmetric) complexmatrix. {/?} and {«} are the nodal vectorscorresponding respectively to the unknown pressureson S and to the prescribed normal velocities on S\.
System (24) can be solved using efficient linearalgorithms, and the acoustic pressure inside the cavitycan be computed using the Kirchhoff-Helmholtzequation (5).
The above formulation was implemented byStraco S.A. in the Rayon-3D solver which is part ofthe SDRC I-DEAS Vibro-Acoustics package.
2.3.2 Computational considerationsUnlike standard FE schemes which require a
meshing of the cavity volume, a surface mesh of thesurface S is sufficient for the BEM. The meshingcriterion requires the characteristic dimension of theelements to be smaller than one quarter of theacoustic wavelength A,. This criterion is independentof the damping provided by the absorbing material.
In order to reduce the computation times, twomeshes were generated: one for the 100 -600 Hzband (1149 nodes and 880 elements) and a finer onefor the 600- 1000 Hz band (1777 nodes and 1769elements).
3. ExperimentThe previously described enclosure was built
out of 50 mm thick concrete panels. The wall in thex - Lx plane is treated with a porous material. Fig. 2shows the experimental setup. The acoustic sourcescan be fastened to the exterior walls and emit into56 mm diameter circular holes. A total of 30 of theseholes are available. When not used, they are pluggedwith plaster inserts. Some of these source positionsare shown in Fig. 2.
3.1 Absorbing materialThe absorbing material used in this study is a
30 mm thick porous material of type "Illtec". Itsacoustic admittance under normal incidence, shown in
Fig. 2 Experimental setup.
Fig. 3, was measured in a Bruel & Kjaer 100 mmdiameter impedance tube, using a two-microphonetechnique". For the 100 to 200 Hz range the resultswere found to be very sensitive to the experimentalconditions.
Fig. 3 Specific admittance spectrum of the 30 mm thick"Illtec" absorbing material.
3.2 MeasurementsThe acoustic field in the cavity was
investigated using three linear arrays of Wmicrophones of type Bruel & Kjaer 4165 placedalong the edges of the cavity (see Fig. 2), with a totalof 29 microphones. Several of the possible sourcepositions were used in the investigations.Multichannel processing was carried out on a LMSdata-acquisition system, using white noise excitation.The frequency resolution is 0.2 Hz.
4. Analysis of the results
4.1 Mode shapesA comparison of the simulated mode shapes
for the cavity with all rigid walls and the casepresently under study shows that the absorbingmaterial modifies the shapes in the following manner:• For modes having no x dependence in the rigid
rigid walls (231 Hz) with absorbing material(226.2 Hz)
rigid walls (420.1 Hz)
b) mode (1,2,1)
with absorbing material(411.7 Hz)
Fig. 4 Comparison between the mode shapes with andwithout absorbing material.
case (i.e., nx = 0), the corresponding dampedmodes exhibit an x dependence: the surfaces ofequal sound pressure are no longer parallel to the(x,y) or to the (x,z) plane (see Fig. 4 a).
• The nodal lines in the (y,z) plane are shiftedtowards the treated wall (see Fig. 4 b).
These observations are valid for the BEM as well asfor the analytical model.
4.2 Comparison between the transferfunctions
The spectra presented here correspond to thetransfer function between the volume velocity Q atthe source and the pressure p at a microphone in theenclosure.
4.2.1 Comparison between the BEM andanalytical models
An excellent agreement between the twomodels is observed. Typical examples of the obtainedresults are presented in Fig. 5.
In the upper frequency range, the BEMcomputed spectra tend to exhibit a few sharper peaksthan with the analytical approach.
4.2.2 Comparison between computationsand experiment
A good agreement between computations andexperiments is obtained. Fig. 6 provides an exampleof typical results.
- a) 100 - 600 Hz range
too
600 650 700 750 800 900 950 1000
b) 600 -1000 Hz range
Fig. 5 Comparison between BEM and analytical results for atypical case.
The shape of the transfer functions, as well asthe positions of the resonance frequencies are in verygood agreement, even for fairly high modal densities.However, the higher amplitude of the resonant peaksin the models seems to indicate that the damping isunderestimated or that the volume velocity isincorrectly measured at the resonances.
An underestimation of the damping has severalpossible causes:
• Some of them are inherent to the experimentalconditions:* The concrete walls are not perfectly rigid and
vibrate when the acoustic source is activated.This accounts for energy dissipation.However, for completeness it should be notedthat no structural resonance of these panelswas found to strongly influence the acousticfield.
* The 30 holes at the possible excitationpositions are not tightly plugged and may leak.
Fig. 6 Comparison between BEM computations and experi-ments.
• Another possible reason is that the locally reactingmodel of the material's admittance does notdescribe correctly the behavior of the materialunder non-normal incidence.
The latter hypothesis is supported by the factthat the levels agree much better for "x" axial modes(ny = 0, nz=0, i.e., modes implying waves impingingnormal to the material) than for other modes. Thesemodes have their resonance at 118, 237, 357, 477,597, 711 and 826 Hz, respectively. Modes 1-5 arevisible in Fig. 6.
This phenomenon is better observed in Fig. 7.It presents a comparison of the mean square averagefor the 29 microphone positions of the pIQ transferfunctions for the BEM results and experimentalresults. The excitation is provided by five sourceswith identical volume velocity placed on the wallopposite to the absorbing material. The sourcepositions can be seen on Fig. 2. This source configu-ration excites almost exclusively ";c" axial modes.
100 200 300 400 500 600 700 800 900frequency (Hz)
Fig. 7 Transfer functions averaged over 29 microphonepositions for an excitation privileging "x" axial modes.Comparison between BEM and experiments.
The resonance frequencies of these modes are shownby arrows on the graph.The levels agree within less than 1.5 dB for njf= 2, 3,4, 5, 6 and 7. For the (1,0,0) mode, the difference is4 dB. This greater discrepancy might be explained bythe lower precision of the impedance measurement atlow frequencies as pointed out in 3.1. For non-axialmodes, larger discrepancies are observed. This obser-vation allows us to have a reasonable confidence inthe measurement of the volume velocity.This seems to be in agreement with the conclusions ofwork by Bliss12 who showed that the assumption oflocal reaction can be inadequate for thin porousmaterials of moderate flow resistance with rigidbacking.
The following section is dedicated to finding asimple damping compensation.
4.2.3 Simple scheme for correcting thedamping
The introduction of a £ = 0.001 viscousdamping in the modal approach improves the resultsgreatly for the 600 to 1000 Hz frequency band13.Below 600 Hz, the effect is less satisfactory. Fig. 8gives an example of the obtained improvement.
This viscous damping can be inserted intoequation (18) as a supplementary Dv diagonal realmatrix added to matrix D. As argued in 2.2.1, thismodification reduces the amplitude of the resonanceswithout shifting the peaks in frequency. The Dvmterm can be expressed as follows14:
• 600 Hz range, with C = 0.001 modal damping d) 600 - 900 Hz range, with £ = 0.001 modal damping
Comparison between experimental and computational results. Influence of a 0.001 modal damping factor.
If we now want to introduce it as an admittancecondition on the rigid walls, the correspondingadmittance is:
2c0RCO (26)
where R represents the surface integral over the rigidwall of a squared eigenmode *Pn.
S rigid(27)
In the case of oblique modes (all modal indexesdifferent from zero) which corresponds to mostmodes above 600 Hz,
z ) . (28)
This formulation is simple to implement and givesfairly good results, as can be seen in Fig. 8.
5. SummaryAn investigation of a rigid-walled enclosure
with one absorbing panel was carried out. The BEMand modal approaches were found to agree in a nearlyperfect manner. Comparison with experiments alsoprovided a good agreement. The shapes of thetransfer functions are accurately computed, however,the computations exhibit higher amplitudes at reso-nances. This phenomenon seems to be due to the non-locally reacting behavior of the material underconsideration. In the 600 - 1000 Hz frequency range,the introduction of a viscous damping term£ = 0.001. enables a correction of the amplitude. Thisdamping can be introduced into the BEM in the formof a real-valued admittance applied to the non treatedwalls.
A crucial advantage of the BEM formulation isthat it does not require a volume mesh of the acousticmedium, but only a surface mesh of the objectsimmersed in it or bounding it. The meshing criterion
is independent of damping. This is also a veryimportant point when considering the problem of aheavily damped car interior.
AcknowledgmentsThe authors would like to express their thanks toDr. R. Helber, Dr. J.M. Auger and to F. Doncker forfruitful discussions.
References
porous materials", J. Acoust. Soc. Am. 71(3), pp.533-545, March 1982
13 C.E. Floc'h, "Etude analytique, nume'rique etexp6rimentale d'une cavite" paralle'le'pipe'dique, enpresence de mate'riau absorbant", Memoire deDiplome d"Etudes Approfondies, Universite deTechnologic de Compiegne, France, 1996
I4P.A. Nelson & S.J. Elliot, "Active control ofsound", Ed. academic press, UK, 1992
1 H. Burfeindt, "Schalldruckberechnung im PKWInnenraum", Computergestutzte Berechnungsver-fahren in der Fahrzeugdynamik, edited by Prof.Dr.-Ing. H.P. Willumeit, VDI Verjag, Dusseldorf1991, pp. 279-303
2 M.A. Hamdi, "Formulation Variationnelle parEquations Integrates pour le calcul de ChampsAcoustiques Proches et lointains", Doctoral Thesis,Universitd de Compiegne, France 1982
3 "I-DEAS Vibro-Acoustics User's manual", I-DEASMaster Series 3.0, 1996.
4 A. Bardot, "Modelisation acoustique bassesfrequences de cavity's en presence d'absorbants surparois rigides ou elastiques", Doctoral Thesis,University de Technologic de Compiegne, France1992
5 A.D. Pierce, "Acoustics, an Introduction to itsPhysical Principles and Applications", McGraw-Hill, NewYorkl981,Ch. 6
6 P.M. Morse and K.U. Ingard, "Linear AcousticTheory", Handbuch der Physik, vol. 11, pt. 2(Akustik I), edited by S. Fliigge, Springer, Berlin,1961, pp. 60
7 P.M. Morse and K.U. Ingard, "Theoreticalacoustics", McGraw-Hill, New York 1981
8 M.A. Hamdi, C. Lesueur, "Rayonnementacoustique des structures, interaction fluide-structure", Chap. 9, Collection de la DER, EDF,ed. Eyrolles, France 1988
9 G. Elias, P. Malbdqui, "Scattering by an opensphere: exact solution and comparison with aBoundary Integral Method", ", J. Acoust. Soc. Am.93(2), pp 609-616, 1993
10 M.A. Hamdi, J.M. Ville, "Sound radiation fromducts: theory and experiment", Journal of Soundand Vibration, Vol. 107, pp. 232-242, June 1986.
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