Bending waves focusing in arbitrary shaped plate-like structures: Application to spatial audio in cars

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Nassim BENBARA, Marc RÉBILLAT, Nazih MECHBAL - Bending waves focusing in arbitrary

shaped platelike structures: Application to spatial audio in cars Journal of Sound and Vibration

-Vol. 487, p.1-20 - 2020

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Bending

waves

focusing

in

arbitrary

shaped

plate-like

structures:

Application

to

spatial

audio

in

cars

Nassim

Benbara

∗

,

Marc

Rebillat

,

Nazih

Mechbal

Processes and Engineering in Mechanics and Materials laboratory (PIMM) ENSAM, CNRS, CNAM, HESAM Université, 151 Boulevard de l’Hôpital, 75013 Paris, France

Keywords:

Spatial vibration control Bending wave focusing Multifunctional materials Advanced signal processing Inverse problems Digital twin

a

b

s

t

r

a

c

t

Advancedautomotiveaudioapplications aremoreand moredemandingwithrespectto thevisualimpactofloudspeakerswhilestillrequiringmoreandmorechannelsforhigh qualityspatialsoundrendering.Theuseofarbitraryplate-likestructuresdrivenby electro-magneticactuatorsorbypiezoelectricelementsappearsasapromisingsolutiontotackle bothissues.However,tomeetspatialrenderingaudioconstraints(i.e.tobeascloseas pos-sibletoomnidirectionalpiston-likesources),thegeneratedbendingwavesmustbefocused atagivenpositionandtoacertainextentwithinthehostplatewhichcanbeofarbitrary shape,material,and thickness.Theoretically,thismeansbeingabletoinvert the spatio-temporalwavepropagationoperatorforthegeneratedbendingwavestofitagiventarget shape.Thereare severalmethods (modal control,time-reversal, and propagatingwaves operatorinversion)thatallowtofocusbendingwavesinamedia.However,thereisscarce workontheiradaptionandperformancesassessmentinthecontextofaudioapplications. Thesemethodsdependdifferentlyontheavailableknowledgeofwavepropagationinthe plate(theoretical,partialspatialorfullspatialknowledge)andarehereinvestigatedto per-formthistask.Theirperformancesareassessedwithrespecttoseveralaspects: geomet-ricalcomplexity,thickness,andmaterialdampingofthehoststructure,numberandtype ofactuators,positionandextentofthefocusingarea.Thevariousmethodsarepresented inaunifiedtheoretical frameworkand theyarecomparedbymeansoftwokey perfor-manceindexes(focuslocalizationerrorand spatialcontrast).Anexperimentalvalidation onarelevantindustrialcaseisalsocarriedoutandlearningthroughadigitaltwininstead oftimeconsumingexperimentaldatainvestigated.Thisworkfalls withintheframework ofresearchwhichtriestobridgethegapbetweenlaboratoryresearchandindustrial de-ploymentofthiskindoftechnologies.

1. Introduction

Advancedautomotiveaudioapplicationsaremoreandmoredemandingwithrespecttothevisualimpactof loudspeak-erswhilestillrequiringmoreandmorechannelsforhighqualityspatialsoundrendering.Themainideabehindtheconcept ofspatialsoundrenderingistocalibrateeachloudspeakerofthearraytoreproducewithhighfidelitythephysics andthe acousticsoftheprimary source[1].Historically,spatialrenderingmethodssuchasstereophony(VBAP),binauralrendering

∗ _{Corresponding author.}

orholophony (ambisonicor Wave FieldSynthesis (WFS))have beenachievedby means ofelectromagnetic loudspeakers. Thereare manyconstraintsimposedby suchmethodsforthemto workproperly:thesetupmust consistofseveral loud-speakers witha flat and omnidirectionalresponse that are spatially evenly distributedto avoid spatialaliasing andable tocoverthewholerestitutionarea [2].DistributedModeLoudspeakers(DML)wereintroduced asaalternateto electrody-namicalloudspeakerswithalow visualprofile [3].The ideaisto producesoundwavesbyexciting thebending wavesof athinstiff platetroughanactuator.ThistechnologywasusedtorenderspatialsoundbytheWFSmethod[4,5].However, thistechnology suffersfromsome drawbacks becauseof thealiasing causedby jointpanels andthemodal behaviour at lowfrequencyoftheplatewhichdoesnotmeetspatialaudiorequirements.Therefore,anewtechnologyusinglargerplates calledMulti-ActuatorPanels(MAP)ormuchlarger onescalledLargeMulti-ActuatorPanels(LaMAP)hasbeendesignedto overcometheselimitations[6].Thistechnologywassuccessfullyusedin thecontextofWFS [7–9].However, some draw-backssuchasthemodalbehavioratlowfrequencies andtheinterferencesbetweensourcescausingsounddistortions and poorsoundqualitystillexistswiththistechnology.

Usingspatialsoundrenderingcoupledwithflatpanel loudspeakerappears asvery interestingparticularlyinthe auto-motiveindustry.Indeed,itwillallowtoalert directionallythedriverfromseveraldangersandtoprovidepassengers with higherquality spatialsound. Moreover,thisapproachcan potentiallydecreaseweight byreplacingelectrodynamical loud-speakerswithpiezoelectric elementsoraudioexciters andimprovedesignpurposesandvisual impact.However, insidea car,therenderingarea issmallandwithverycomplexoperationalconditions(varyingnumberofpassengers,temperature variations, etc...).Theclassical methods(VBAP,Ambisonic, WFS)focusonly onthetargetedsoundpressurefield, whichis apractical problemincarsbecauseoftheabovementioned verycomplexoperationalconditions.Ifonecan controlwhat happenonthevibratingsurfacesinsteadofrelyingonlyontheemittedsoundpressure,morerobustsoundrenderingshould beachieved.Consideringthepreviousobservations,focusingbendingwavesonarbitraryplate-likeinteriorpartsofthecar tocreateindependentsoundsources appearsasa promisingwaytocreateanarray ofaudiosources thatcan beusedin a secondtime forspatialsoundrenderingforautomotive applications.Nevertheless,carinteriorsurfacesarefarfromthe simplerectangularhomogeneousflatplatesusedwhendesigningDMLs,MAPsandLaMAPs.Asaconsequence,several addi-tionalaspectsmustbeconsideredinthatcontext:thegeometricalcomplexity,thickness,andmaterialdampingofthehost structure,thenumberandlocationofactuators,andfinallythepositionandextentofthefocusingarea.

Intheliterature,therearevariousmethodsthatallowtofocusbendingwavesinamedia.Thefirstmethodcalledmodal control(MC)isbasedonmodalsuperposition[10].Itallowsfocusingamonochromaticvibro-tactilepeakonplates[11,12], oranaudiosourceonarectangularplatebycomputingFIRfiltersoforder3[13].Themethodisbasedonasimpleapproach (model-based)andisextremely sensitivetomodelingerror.Thesecond methodavailableintheliteratureisbasedonthe time reversal(TR) principle, initially describedby Fink etal. in[14].It allows to focus a wave ata specific point inthe spaceby learningpartially the wavesreceived atthe actuatorspositions andinitiatedat thefocusing point. Besides,the sameauthorshowsthatthismethodcanbe adaptedforacousticpurposestocontrol preciselyacousticwave propagation inreverberatingcavitiesorinwaveguides[15].Additionally,thismethodisalsousedin[16]tofocusanaudioimpulseina room.Finally,theeaseoftheimplementationofthetimereversalencouragedmanyresearcherstouseitinhapticfeedback applications[17].Thethird methodstudied inthispaperisthespatio-temporal inversefiltering(STIF), initiallystudiedby Kahanaetal.[18]andTanteretal.[19]tofindasolutiontothelackofrobustnessofthetimereversalmethod.Firstly,the methodconsistsinlearningthepropagationofthe wavesinthewhole media,then,an inversefilteriscreatedandused asinput foreach actuator.Then, themethodwasextended tothe time domain[20],andafterexperimentally appliedin heterogeneous,absorbingandhighlyreverberatingmediassuchasahumanskull[21],ortofocusacousticwavesinrooms

[22].Finally,theinversefilteringisalsousedin[23]usingtwodifferentapproachestoradiateauniformsoundfield with a plateactuatedbyexciters atthe periphery, creatingaradiating zoneinthe middleofthe plate. Andin[24],the same authorsusetheinversemethodtocreateapistononavehicle’sroofforanarrowbassfrequencyband.Here,thepurpose ofthisstudyistotry tobridgethegapbetweenlaboratory researchandindustrialdeploymentoftheLaMAPs.Hence we proposeheretoworkonthestructuralcontrolofbendingwavesonalargefrequencyband,bycomparingthreerendering methods,accordingtoseveralparameters.

Consequently,theobjectivesofthisstudyare:

1. toadapttoaudioapplicationsthepreviously mentionedmethodsforthelocalizationofbendingwavesina defined areaofageometricallycomplexstructurewitharbitraryboundaryconditions,

2. toassesstheirperformanceswithrespecttoseveralaspects:geometricalcomplexity,thickness,andmaterialdamping ofthehoststructure,numberandtypeofactuators,positionandextentofthefocusingarea.

Thespecific requirementsforthesemethods inthepresentcontextarethat they shouldbeprecise intermsofspatial focusing andmustoperate ina widefrequency rangeto besuitable forthe audiorestitution. Besides,themethods used should also be asinsensitive as possible to noise andmodel error. Their performances are thus compared by means of two key performance indexes (focus localizationerror andspatial contrast). Finally,the methods should not relyon too muchexperimentalmeasurementforpracticalreasonsastheautomotiveindustryistargetedastheapplicativefield.These adaptedmethodswillfirstbepresentedinaunifiedformalismandthencomparednumericallytohighlighttheadvantages anddrawbacksofeachmethodbeforecomparingthemexperimentallyonarealisticindustrialcase.

Fig. 1. Arbitrary shaped-like structure with piezoelectric actuators and localized audio sources.

2. Problemstatement

LetSdenoteaplate-likestructureofarbitraryshapeandwitharbitraryboundaryconditions.Assumingthatthisstructure lies inthe (x, y) plane, the positions of the N actuators (piezoelectric or audio exciter)bonded on S are given by {(xn,

yn)}n∈[1,N]andtheout-of-planedisplacementofthebendingwavevelocityfieldisdenotedu(x,y,t)(Fig.1).

Theobjectiveisnow,foranyinputaudiosignala(t),tofocusthebendingwavefieldu(x,y,t)tofitagiventargetshape

φ

(x,y)usingtheNactuatorsthroughNFIRfiltersrn(t)appliedtotheaudioinput.Mathematically,thismeansthatwewould liketoachieve:

∀

t u

(

x,y,t

)

=

φ

(

x,y

)

a

(

t

)

. (1)

Or equivalentlyin the frequencydomain, withU(x,y, f) and A(f) the Fourier transformsof the displacement andthe audiosignalrespectively:

∀

f U

(

x,y,f

)

=

φ

(

x,y

)

A

(

f

)

. (2)

Assumingthesystemisgloballylinear(thepathbetweenanactuatoranda sensorisconsidered aslinear),the contri-butionsun(x,y,t)ofalltheNactuatorsgeneratingthefieldu(x,y,t)sumupandcanbewrittenas:

∀

t u

(

x,y,t

)

= N  n=1 un

(

x,y,t

)

= N  n=1 hn

(

x,y,t

)

∗ rn

(

t

)

∗ a

(

t

)

, (3)

where“∗” standsfortheconvolutionproductandhn(x,y,t)isthespatio-temporal impulseresponseofthenthactuator.Or equivalentlyinthefrequencydomain:

∀

f U

(

x,y,f

)

= N  n=1 Un

(

x,y,f

)

= N  n=1 Hn

(

x,y,f

)

Rn

(

f

)

A

(

f

)

. (4)

Bycombiningbothexpressions(2)and(4)ofU(x,y,f),theobjectivetoachieveisthenexpressedinthefrequencydomain independentlyoftheinputsignala(t)as:

Find

{

Rn

(

f

)

}

n∈[1,N]suchthat

∀

f,

φ

(

x,y

)

= N



n=1

Hn

(

x,y,f

)

Rn

(

f

)

. (5)

Thismeansthatthespatio-temporalpropagationoperatorsHn(x,y,f)associatedwiththeNactuatorsneedstobeinverted todesigntheNFIR filtersRn(f)abletofocusthebendingwave fieldonthetarget shape

φ

(x,y) whateverthefrequencies

fcontained within the input audio signal a(t). In practice,focusing the bending wavesthus consistsin finding for each frequencytheadequatephasesandamplitudestodrivecoherentlyeachactuatorbondedonthestructure.

3. Bendingwavesfocusingmethodsunifiedtheoreticaldescription

Inthis section, several methods able to invertthe spatiotemporal propagationoperator Hn(x, y, f) are presented. The methodsdifferdependingontheavailableknowledgeofwavepropagationintheplatetheyrelyon.Indeed,thefirstmethod reliesontheoreticalknowledge andisnamed ModalControl(MC), thesecondmethodrelieson partialspatialknowledge andisreferredasTimeReversal(TR),andthelast oneisbasedona fullspatialknowledgeandisnamedSpatio-Temporal InverseFilter(STIF).Theyareallpresentedwithinaunifiedtheoreticalframeworkforcomparisonpurpose.

The first class ofmethods presentedhere assumesthat a theoretical knowledge ofwave propagation within thehost structure is known.Forbending waves,the out-of-plane displacement u(x,y, t) ofthe hoststructure (here arectangular uniformplate)subjecttotheexternalloadp(x,y,t)canbeexpressedas[33]:

D

∇

4_u

₍

_x,y,t

₎

₊

_ρ

_hu¨

₍

_x,y,t

₎

_+cu˙

₍

_x,y,t

₎

_=p

₍

_x,y,t

₎

_{with D=} Eh3

12

(

1−

ν

2

)

. (6)

Intheaboveequations,E,

ν

,and

ρ

aretheYoung’smodulus,Poisson’sratio,anddensityofthepanelrespectively. Addi-tionally,cisthedampingconstantandhisthepanelthickness.

Let

θ

m(x, y) be the theoretical orthogonal modesassociated with the structure under studywhich can be computed analyticallyorthroughFEMmodels.Thedisplacementcanbewrittenasadecompositioninthemodeshapebasis:

u

(

x,y,t

)

=∞

m=1

um

(

t

)

θ

m

(

x,y

)

, (7)

wherethecoefficientsum(t)denotethemodalamplitudes.Byreinjectingthedisplacementexpression(7)inthemovement

Eq.(6),andbyprojectingtheobtainedequationontheMfirstmodes,theorthogonalitypropertyallowstogetthedecoupled equationforeachmodaldisplacementum(t):

∀

m∈[1,M],u¨m

(

t

)

+cu˙m

(

t

)

+

(

2

π

fm

)

2um

(

t

)

= 1 Sm pm

(

x,y,t

)

with fm= 1 2

π



D

ρ

hk 2 m. (8)

Intheaboveequation,fmistheeigenfrequencycorrespondingtothemthmodeandk2mthewavenumberdependingon thegeometry.Additionally,Smistheprojectiontermandpm(x,y,t)themodalforcegivenby:

Sm=  S

ρ

h

θ

m

(

x,y

)

2dxdy and pm

(

x,y,t

)

=  S p

(

x,y,t

)

θ

m

(

x,y

)

dxdy. (9)

Wecanthenwriteinthefrequencydomain,atthenthactuator:

∀

m∈[1,M],

∀

f,Un,m

(

f

)

=_S 1

m

((

2

π

fm

)

2+

(

i2

π

f

)

c+

(

i2

π

f

)

2

)

Pn,m

(

x,y,f

)

. (10)

Incaseofnopunctualactuation(wheretheareaisdenotedSact),themodalforceisgivenby:

Pn,m

(

x,y,f

)

=



 Sact bn

(

x,y

)

θ

m

(

xn,yn

)

dxdy



Rn

(

f

)

A

(

f

)

=Bn,mRn

(

f

)

A

(

f

)

, (11)

wherebn(x,y)isashapefunctioncharacterizingthespatialdistributionofeachactuator. Finally,bysummingoverallmodesandactuators,oneobtains:

∀

f, U

(

x,y,f

)

= N  n=1



M  m=1 Bn,m

θ

m

(

x,y

)

Sm

((

2

π

fm

)

2+

(

i2

π

f

)

c+

(

i2

π

f

)

2

)



Rn

(

f

)

A

(

f

)

= N  n=1 Hn

(

x,y,f

)

Rn

(

f

)

A

(

f

)

. (12)

Thisconstitutesatheoreticalexpressionofthespatio-temporalbendingwavepropagationoperatorHn(x,y,f)thatneeds tobeinverted.

Now,bydecomposingthetarget

φ

(x,y)onthehoststructuremodeshapes

θ

m(x,y)as:

φ

(

x,y

)

= M  m

φ

m

θ

m

(

x,y

)

with

φ

m=  S

φ

(

x, y

)

θ

m

(

x,y

)

dxdy. (13)

Wecanwritethefollowingequationafterprojectingoneachmode

φ

m:

∀

f,

∀

m

φ

m= N  n



Bn,m S

θ

m

(

x,y

)

dxdy Sm

(

2

π

fm

)

2+

(

i2

π

f

)

c+

(

i2

π

f

)

2





Rn

(

f

)

. (14)

withthenumberofmodesMchosen equaltothenumberofactuatorsorsuperior(potentialmatrixconditioningproblem duringtheinversion).Theprevious equationcanbewritteninthematrixformas



=H

(

x,y,f

)

R

(

f

)

.WheninvertingH(x, y,f),itgives: Rn

(

f

)

=



B−1m

S

θ

m

(

x,y

)

dxdy







Sm

(

2

π

fm

)

2+

(

i2

π

f

)

c+

(

i2

π

f

)

2



φ

m, (15)

whereBm−1representthemthlineoftheinvertedmatrixB−1.Goingbacktothetimedomain,itcanbeshown[13]that

suchequationcanbeexpressedinthediscrete-timedomainasfollows:

Rn

(

z

)

=Jn+Knz−1+Lnz−2. (16)

WhereJn,KnandLncanbeexpressedusingthemodalparameters(km,

θ

m(xn,yn))aswellasthetargetshapeparameters

φm

.

The method described above offers the advantage of not needing any prior experiments. Indeed, only the analytical knowledgeofthemodelisneeded.Inaddition,thecomputationcostislowbecausetheFIRfilterorderis3andthemethod isnotsensitivetonoise,becausenopriormeasuresaredone.Finally,themethodallowstoreproducearbitrarytargetshapes. However,theadvantageslistedbeforeareavailable onlyforverysimplestructures. Indeed,onlysimplestructures suchas rectangularorcircularplateswithboundaryconditionsknownanalyticallycanbeused.Additionally,foranaudiopurpose thefrequencybandwidthisingenerallargeandthenumberofmodestobeaddressedmightbehigh.However,thetheory assumesthat thenumberofmodescontrolledisequaltoora bithigherthanthenumberofactuatorssothatthematrix canbeinverted,orpseudo-invertedandmatrixconditioningissuescanbeavoided,butitwillnotbethecasepractically.

3.2.MethodsrelyingonapartialspatialknowledgeofHn(x,y,f)

Letusnowassumethatapartialspatialknowledgeofthepropagationoperatorisavailable.Moreprecisely,itisassumed herethatthespatio-temporaloperatorisknownforanexcitationlocatedatonespatialposition(xi,yi)andmeasurements performedatactuatorpositions{(xn,yn)}n∈[1,N].Thispropagationoperatorimpulseresponseisdenoted



_

h

(

xn,yn,t

)



n∈[1:N]. Moreover,themeasuredimpulseresponsebetweentheactuatornandthereceivingpointattheposition(x_i,y_i)inthetime domainishn(xi,yi,t)suchthat:

∀

n∈[1,N]hˆ

(

xn,yn,t

)

=hn

(

xi,yi,t

)

. (17) According to the spatialreciprocity principle [15], the FIR filters are chosen by reversing the time, i.e. by applying the transformationt−→−t:

∀

n∈[1,N]rn

(

t

)

=ˆh

(

xn,yn,−t

)

. (18)

Inthefrequencydomain,bytakingtheFouriertransformofrn(t),oneobtains:

Rn

(

f

)

=  _+∞ −∞ _h

₍

_xn,yn,−t

₎

_e−i2πf t_{dt =} u=−t _+∞ −∞ _h

₍

_xn,yn,u

₎

_ei2πf u_du =conj



 _+∞ −∞ _h

₍

_xn,yn,u

₎

_e−i2πf u_du



=conj[H

(

xn,yn,f

)

]. (19)

Throughthisprocedure,oneshouldbeabletofocusthebendingwavesintheneighborhoodofthelocation(xi,yi).Using previousnotation,thismeansthatthetargetshapetobereconstructedcorrespondsheretoaDiracfunctionlocatedat(xi,

y_i),i.e.,

φ

(

x_,y

)

₌

δ

(

x_{− xi,}y_{− yi}

)

.

According to the theory, this method is based on the time invariance principle of partial derivative equations in an undampedmedia.Indeed,ifastructureisexcitedatapointAandthewavemeasuredatthepointB,thereforetheexcitation at B and measurement at A gives the same response function. Even ifthe invariance ensure the filters robustness, the approximationinthinandlowdampedstructureismade.Themethodfurthermoreneedstobeextendedtoawidertarget shape byconsidering time reversal onadjacent pointsbecause forthemoment there isno control ofthe width. Finally, thecomputationcostmustbediscussed.Theoretically,impulseresponsesmustbemeasuredbetweeneachfocuspointand actuator. Ifthe position of the focus area location is known, the numberof initial measurements will be very low. The methodmayalsobesensitivetotheexperimentalnoise.

3.3.MethodsrelyingonacompletespatialknowledgeofHn(x,y,f)

Letusnowassumethat acompletespatialknowledgeofthepropagationoperatorisavailable.Moreprecisely,itis as-sumedherethat thespatio-temporaloperatorisknownforanyactuatorn∈[1,N]andthat measurementsareperformed overacompletegridofKpointsonthestructureforpositions{(x_k,y_k)}_k∈[1,K].Mathematically,whatisknowninthe fre-quencydomainis{Hn(xk,yk,f)}n∈[1:N],k∈[1,K].

Theproblemtobesolvedcanthenberewrittenonthemeasurementspatialgridas:

φ

(

xk,yk

)

= N



n=1

Hn

(

xk,yk,f

)

Rn

(

f

)

. (20)

Thus,inthematrixform:

Fig. 2. (a) Dimensions of the plate and (b) meshes of the studied plate and (c) ribbed plate.

Table 1

First eigen frequencies and damping ratios of the plate. Index Eigen frequency (Hz) Damping (%)

Mode (1,1) 127 4.2 Mode (2,1) 222 4,7 Mode (1,2) 292 3.8 Mode (2,2) 379.4 2.1 Mode (3,1) 379.9 2,1 Mode (3,2) 530 3

Theresultingmatrixcanfirstbedirectlypseudo-inversedinthefrequencyrangeofinterestbykeepingonlylargeenough eigenvalues.ThisleadsdirectlytotheFIRfiltersRn(f):

R

(

f

)

=H

(

f

)

+



, (22)

where“+” denotesthepseudo-inversionoperator.

Itisimportanttonotethatthepseudo-inversionstepissensitivetonoise.Hence,duringthepseudo-inversionstep,the lowestsingularvaluesarecanceledtodealwiththosematrixconditioningissues.Moredetailsareavailablein[21].

This methodoffers severaladvantagesin comparisonwith theothers evoked previously.Firstly, the methodis robust because it allows acquiring information about the wave propagation in the whole media. Additionally, the filter size is chosenbytheuser.Hence,themodeltakesintoaccounttherealstructureanditscomplexity,witheventuallosses,andthe notionofmodaldensityisnotcrucial.Finally,thismethodisveryinteresting foritseaseofimplementation.Nevertheless, thepointsevoked aboveare alsosources ofdrawbacks.Indeed,itisnecessary torealizeseveralexperimentsbetweenall thepointsofthestructureandtheactuators,thiscomplicatesnumericallythemethod.Moreover, themethodissensitive tothenoise,mainlyduringthepseudo-inversionstep.Thus,itcouldneedaregularizationstepwhichisnumericallyheavy.

4. Numericalstudyoftheinvestigatedfocusingmethods

4.1. Structuresunderstudy

In order to test the previous methods andto compare their performances,a simple anda ribbed clamped plate are chosen.Polypropyleneisselectedhereasitisamaterialoftenusedforcarpassengergarnishments.Thedimensionsofthe plate are givenin Fig. 2a.Besides, ribs withlength of 1 cmand thicknessof 3 mm are added to approach asmuch as possiblethe surfacesusedintheautomotive industry.Theribsare stiffeningtheplate, shiftingthe eigenmodestohigher frequenciesandintroducinglocalizedmodes.

The materialused ispolypropylene,witha Youngmodulus ofE=1.1GPa,a Poisson’sratioof

ν

=0.33anda density of

ρ

=990 kg/m3_{. Mode shapesof the structures are computed thanks to a finite element model implemented on the}

softwareSDT(SDTools©,Structural Dynamics Toolbox[25]) andan experimentalmodalanalysiswasperformedjointlyto identifymodal dampings,available inTable1.The modalidentificationwasachievedby usinga sine sweepinput signal. AfterthetransferfunctionshavebeenacquiredthankstoapiezoelectricactuatorandalaserDopplervibrometersensor,a pole/residueparametrizationwasmadetoidentifythe modalcharacteristicsofinterest(the testrigisdescribedprecisely inSection5.1).Thestudiedplateisrelativelydampedandthereforedifferentfromthemostfrequentlyhandledcasesinthe literaturewhichinvolveverylowdampedstructuressuchasmetalplatesorsmartphoneglasses[12,13,17].

The plateis modeled with shell elements. The bending wave focusing algorithms are developed andimplemented in MatlabwithSDTsoftwarejointly.ThemeshesareavailableFig.2bandc.Throughouttheremainingofthisstudy,theaudio signaltobereconstructedisasweepsinewithadurationof2s,spanningfrequenciesfrom100Hzto800Hz,sampledat 44.1kHz.Thechosenfrequencyrangeallowstoexcitethe6firstmodes

In the following sections, there will first be a parametric studyaimed at comparing the three methods theoretically studiedpreviously(Section3)byvaryingseveralparameters.Theparameterstobestudiedare:

• thedampingratioandthethickness,becausethecardoorpanelmaterialismadeupofpolypropylene,withvariable thicknessandcoating,

• thenumberofactuators,becauseonlyasmallnumberofexciterscanbeintegratedforreasonofcompactness, • thegeometriccomplexity(ribbedplateversussimpleplate),

• thepositionandthesizeofthetargetshape.

Thethreemethodsdescribedpreviouslyare appliednumerically,andcomparedtoacasewherenocontrolisimposed (nocontrol),thatconsistsinactuatingonlythenearestactuatortothetargetshape.

4.2.Keyperformanceindicatorsfortheparametricstudy

Fouroriginal key performance indicatorswillbe used inorder tocompare the3 methodspreviously presented when varyingdamping,thenumberofactuators andthe geometricalcomplexityonthestructure understudy.Theseindicators are describedbelow. Keep in mind that u(x, y, t) isthe out-of-plane displacement ofthe plate andU(x,y, f) its Fourier transform.

Localizationerror:givenareconstructedtargetshape,thefirstcriterionisthemeasurementbetweenthecenterofthe targetshapeandthemaximumamplitudeofthereconstructedshape.Thus,thelowerthedistanceis,themoretheshape islocalizedpreciselyfromaspatialpointofview.

Spatialcontrast:thesecondcriterion isthecomputationofthecontrastbetweentheenergyinthereconstructedarea (SR)andenergyofthetotalarea(ST=Lx× Ly),sothat:

C

(

f

)

= SR



U

(

x,y,f

)

|

2 dS ST



U

(

x,y,f

)

|

2_dS. (23)

IfC

(

f

)

=1,thenall theenergyispresentinthetargetshape andnothingoutside,andifC

(

f

)

=0,thenall theenergy isoutsidethetargetshape.

Normalizedmeanquadraticdisplacement:thenormalizedmeanquadraticdisplacementamplitudeofbendingwaves inthefrequencyrangeofinterestisdefinedby:

<

|

U

(

x,y

)

|

2>= 1

fmax− fmin

 fmax

fmin

|

U

(

x,y,f

)

|

2df, (24)

Itprovidesavisualoverviewofhowwellthealgorithmsareabletofocusbendingwavesinthetargetedfrequencyrange. Mean quadratic phase variation: this KPI corresponds to the average quadratic phase difference between the point (xmax,ymax)atwhichthemaximumoftheamplitudeisreachedandthecurrentpoint(x,y):

(

x,y

)

= 1 fmax− fmin  fmax fmin phase



U

(

x,y,f

)

U

(

xmax,ymax,f

)



2_{df. (25)}

Itprovidesavisualoverviewofhowwellthealgorithmsare abletofocusbendingwaveswithspatiallycoherentphasein thetargetedfrequencyrange.

4.3.Parametricstudyforthebendingwavefocusing 4.3.1. Influenceofdampingandthicknessoftheplate

Thecasethat isconsidered iswithan actuatorconfiguration(3× 2), asizeshapeof(0.25Lx,0.3Ly) anda targetshape positioncenteredat(0.3Lx,0.6Ly),asshowninFig.4d.TheFig.3showsthecontrast(Fig.3a,c,e)andspoterror(Fig.3b,d, f) ofthethreealgorithmsandforthecasewithoutanycontrol(Fig.3g,h)accordingtosimultaneouslydampingvariations (from1%to6%)andthicknessvariations(from1mmto5mm).ItcanbeseenthattheSTIFmethodlocalizeswavesalways very efficiently (high contrast andlow spot error), andfar better withhigh thicknesses. For low level of thickness, the contrastincreaseswithdamping,butafter3 mm,thedampinghaslesseffectsthan thickness.Conversely, theTRmethod contrastishigherandspoterrorlowerwithlow thickness(1mm to2mm). Abovethisvalue ofthickness,the abilityto focusispoorer.Forthesetwothicknesses,theperformanceincreaseswithdamping.FortheMC,thecontrastincreasesfrom 0.4to 0.7withthe dampingratio.For theno control case, the platebehaves accordingto its modalshapes, so that the contrastisalwaysaround 0.4andthespoterrorvariesrandomly.Accordingto thesecurves,theSTIFisperforming much betterthanthenocontrolcase,theMCisbetterforhigherdampingandthickness,andtheTRforthinnerplates.

4.3.2. Effectofactuatordispositionandthickness

Thissubsectionaimstostudytheeffectofthenumberofactuatorsforseveralthicknesses.Havingmoreactuatorscould be interesting because one can control only as many modes as the number of actuators in the modal control context. However,intheperspectiveofapplyingthesemethodstorealsystems,anoptimizationofthenumberofactiveelements isnecessary.TheFig.4showstheselecteddispositionsoftheactuators,whichareevenlyspacedalongxandy.TheFig.5

Fig. 3. Contrast and spot error ((a)(b) STIF, (c)(d) TR, (e)(f) MC, and (g)(h) No control) according to the damping: ( 1%) ( 1.5%) ( 2%) ( 2.4%) ( 2.8%) ( 3.3%) ( 3.7%) ( 4.2%) ( 4.6%) ( 5%) ( 5.5%) ( 6%).

Fig. 5. Contrast and spot error ((a)(d) STIF, (b)(e) TR, and (c)(f) MC) according to the actuator number and disposition, respectively : ( 4(4 × 1)) ( 5(5 × 1)) ( 6(3 × 2)) ( 8(4 × 2)) ( 9(3 × 3)) ( 10(5 × 2)) ( 12(4 × 3)) ( 15(5 × 3)) ( 16(4 × 4))).

According to the Fig.5a, for the STIFmethod, increasing the number of actuators allows to increase contrast and to diminishthespoterror,whateverthethickness.Indeed,forthe twoconfigurationswherethereisonlyalinealongx,the contrastgoesfrom0.6to0.8.Moreover,whentheconfigurationhasatleasttwolinesofactuator,thecontrastiscomprised between0.8 to 1. This result can be explained by the fact that more information is available during the learning step. Conversely,fortheTR,thereisbetterlocalizationbetween1mmand2mmofthickness.Forthesethicknessvalues,there arebetter resultswhenthe numberofactuatorsincreases,becausethereis moreinformationavailable inthepropagator operation.Inaddition,inthe caseofmodalcontrol,thecontrastincreaseingeneralwiththewidthofeachconfiguration. Particularly,from3mm,thespoterroroftheconfiguration(3× 2)decreasesdrasticallyfrom10%to3%,andthecontrastis around0.5.Therefore,thetargetshapeiswelllocatedbutwithsomesmallamplitudesaroundwhichincreasesthecontrast. Additionally,the configuration (4 × 4) exhibits the highestcontrast, that is 0.6 to 0.8 from2.5 mm until 5mm. This is becausethenumberofcontrolledmodesis16,whichincreasestheaccuracyofthemethodinthereducedfrequencyband [100Hz;800Hz].Inconclusion,theSTIFmethodincreaseslocalizationabilitieswithmoreactuators,theTRmethodisalmost notinfluencedbythenumberofactuators,andtheMCmethoddependshighlyontheactuatorconfiguration.

4.3.3. Effectofthetargetshapespositionandsize Targetshapeposition

This subsection allows to study the size and the position of the target shape. Considering the results shown

Section4.3.2andthesmallnumberofactuatorsduetopracticallimitations,theactuatordispositionusedhereisthe(3× 2). Thetargetshapeextensionhereisalso(0.3Lx,0.6Ly)andthecoordinatesofthecenterarevaryingfrom0.2Lx to0.5Lx along thexdimensionand0.5Lyto0.75Lyalongtheydimension,andcanbeseenFig.6a.

TheFig. 7exhibits thedifferent contrastsforvarious damping ratiosandthicknesses. The contrastobtainedby target shapeisalwaysabove7.5andthespoterroralwaysunder10%,therefore,theSTIFmethodisnotinfluencedby thetarget shapelocationandisalwaysrobustinthiscase. Conversely,from2.5,theTRcontrastissuperiorto0.7andthespoterror inferiorto15%fortheconfigurations(3,4,7,8)wherethetargetshapeiscenteredaroundthecenteroftheplate.Whenthe targetshapegetsawayfromthecenteroftheplate,theresultsoflocalizationaredeteriorated.Furthermore,the configura-tion(2,3,4)givesthehighestcontrastfortheMCmethod,above2.5mm.

Targetshapesize

Differenttargetshapesizesaccordingtoxandyalternativelyare compared,asshowninFig.6b.Thesizesvariesfrom (0.25)to(0.4)accordingtoxandyrespectivelyandiscenteredat(0.3,0.6).Therearetotally16shapes,morerectangularor square.Themainideaistostudytheinfluenceoftheshape.

TheFig.8presentsthecontrastandspoterrorforeachmethodaccordingtothethicknessandthetargetshapepositions. The STIF methodalso shows herethe best results with a contrast above 7.5 foreach position andincreasing along the thickness.Moreover,thespoterrorisalsolow,alwaysbelow10%.Thisresultexhibittherobustnessofthemethodbecause itcanlocalizeforanyshapes.Conversely,theTRmethodshowsbetterresultsforthicknessesfrom1mmto2mm,where the contrast is around 0.5, after it, even if the contrast increases, the spoterror also become much higher. Finally, the

Fig. 6. (a) Positions of the centers and (b) sizes of the target shapes for the parametric study.

Fig. 7. Contrast and spot error ((a)(d) STIF, (b)(e) TR, and (c)(f) MC) according to the target shape center, respectively at : ( 1(.2,.5)) ( 2(.3,.5)) ( 3(.4,.5)) ( 4(.5,.5)) ( 5(.2,.6)) ( 6(.3,.6)) ( 7(.4,.6)) ( 8(.5,.6)) ( 9(.2,.75)) ( 10(.3, .75)) ( 11(.4,.75)) ( 12(.5,.75)).

contrastforMC isaround0.3to 0.6,dependingonthe size.Thebiggerthe size,themorethecontrast increasesandthe spoterrordiminishes.

4.4. Bendingwavefocusingdetailedexamples 4.4.1. Simplestructure

Inthissubsection,anexampleisdetailedinordertoillustrate thevariousmethodpresentedinthefrequencydomain thankstotheparametricstudy.ThetargetshapeandtheactuatordispositionareprovidedinFig.4d.

The spoterrorandthe contrastinthe frequencyband [100Hz;800 Hz] foreach methodforthe simplestructure are plottedinFig.9.Firstly,wecanobservethattheSTIFmethodallowsachievingaveryhighlocalizationperformance,in com-parisonwithothers.Indeed,itshowsaverylowspoterror,whichisaround12%ofthetotallengthinthe[100Hz;220Hz] region,andlower than10%above220Hzapproximatelyuntil540Hz(Fig.9).Itisimportanttonoticethatthisfrequency

Fig. 8. Contrast and spot error ((a)(d) STIF, (b)(e) TR, and (c)(f) MC) according to the target shape size: ( 1(.25,.25)) ( 2(.25,.3)) ( 3(.25,.35)) ( 4(.25,.4)) ( 5(.3,.25)) ( 6(.3,.3)) ( 7(.3,.35)) ( 8(.3,.4)) ( 9(.35,.25)) ( 10(.35, .3)) ( 11(.35,.35)) ( 12(.35,.4)) ( 13(.35,.25)) ( 14(.35, .3)) ( 15(.35,.35)) ( 16(.35,.4)).

Fig. 9. (a) Spot error and (b) contrast for the simple plate.

correspondstothelimitwherethemodaldensity(numberofmodes/Hz)isveryhigh.Thus,abovethisfrequencytheplate couldnotbe describedwithamodalbehavioranymore [7].Moreover,thecontrastdropsdrasticallyatmodalfrequencies, withapeakoftheerrorat220Hzandasmallaround380Hz(Fig.9).Itcorrespondstotheeigenmodeswhichinfluences largelythetarget shape pattern.Indeed,thisphenomenon isnatural, butcanbe enhancedby theplacement ofactuators toincrease thecontrollabilityofthose specificmodes. Therefore,the STIFmethodcanachieve a preciselocalizationfrom 100Hzto540Hz.Forhigherfrequencies,thealgorithmdoesnotfocusvibrations anymore.Itisinterestingto noticethat thereisthe samebehaviorin thefrequencyband fortheMC andforthecasewithoutcontrol, alsohighlyinfluenced by eigenmodes.Besides, accordingto thespot errorandcontrastcurves, theMC method canfocus on thewhole frequency

band,but moreefficientin thebandwidth between380Hz and500Hz approximately,andwitha highinfluence ofthe

fourth(2,2) mode. However, the no control caseismore influenced by the (1,1)mode butdoes not focuson the whole frequencyband. The timereversal (TR)alsooperates ina narrow band,with amaximal contrast of0.4, anda veryhigh

Fig. 10. Reconstructed shapes with (a) STIF, (b) TR, (c) MC and (d) without control.

Fig. 11. Reconstructed shape phases with (a) STIF, (b) TR, (c) MC and (d) without control.

influenceofthefirstmodeshapewhichdeterminesthefinalshape.Thisisduetothehighdampinginthematerialandthe highthickness.

An overviewofthe resultsoflocalizationobtained inthefrequency band ofinterest isplottedin Fig.10 throughthe normalizedmeanquadraticdisplacementamplitudeandinFig.11throughthemeanquadraticphasevariation.FromFig.10, it canbe seenthat thevibration amplitudeisonaverage muchmore localizedby usingtheSTIFmethodthan anyother

Fig. 12. (a) Spot error and (b) contrast for the plate with ribs.

Fig. 13. Reconstructed shapes with (a) STIF, (b) TR, (c) MC and (d) without control.

method.FromFig.11,itcanbe concludedthatthe STIFmethod additionallyguaranteesalmostno phasevariationwithin thetargetarea,meaningthat allthepointsinthatarea arevibratingspatiallycoherently.Thisisalsotrueinthiscasefor theTRmethod,truetoalowerextentforMC,andnottrueatallforthenocontrolcase.

4.4.2. Effectofthecomplexity(RIBs)

Thestudiedstructureisnowtheplateasintroducedabovewithribsfixedonit.Themainideaistoseewhichmethod willstillworkonamorecomplexstructurerepresentativeofwhatcan beencounteredinpracticalindustrialapplications. Moreover, themodal frequencies are higherthan before, from154 Hz to 590Hz,and the modalamplitudes are located betweenribs.

TheFig.12showsthespoterrorandthecontrast.Onthefrequencybandbetween100Hzand800Hz,theSTIFmethod localizesthe energyin thetarget area, similarly tothe simple casewith6 actuators(Fig.13a). Additionally,the contrast dropsat frequencies wheremodesappear. However, the highervaluesof theerror between270 and380Hz correspond tohighamplitudevibrations on actuatorsaround thetarget area butthe shape isstill reproduced withhighcontrast.In addition,theTRmethodcanfocusthewavesintheselectedfrequencyband(Fig.13b),butitdoesnotmaintainthecorrect targetshapeforeachfrequency.Furthermore,theMCmethodlocalizestheenergyalternativelyclosetothetargetshapeand betweenribs(Fig.13c),butdoesnot workwellsincethemode shapesarenot conventionalflexuralmodesbutamixture ofglobalandlocal modes. Finally,without controlstrategy, the energyis alsoconfined betweentheribs nearthetarget shapeand showsbetterprecision than MC andTR (Fig. 13d). Theinitial learningof thedynamicbehavior allows having betterfocusingprecisionbyconsideringthelocalmodesinthedynamics,inawiderfrequencyband.Whenthenumberof actuatorsishigher,itincreasesthelocalizationcapabilities.

Fig. 14. Reconstructed shape phases with (a) STIF, (b) TR, (c) MC and (d) without control.

Fig.14showsthemeanquadraticphasevariationforthevariousapproaches.ItcanbeseenagainthattheSTIFmethod guaranteesalmostnophasevariationwithinthetargetarea,meaningthatallthepointsinthatareaarevibratingspatially coherently.Thisishowevernotanymore trueinthiscasefortheTRmethod,theMC method,andthe nocontrolcaseas theysufferfromthepresenceoftheRIB.

4.5. Discussion

4.5.1. Aboutthespatialfocusingcapabilitiesoftheinvestigatedmethods

The previous parametricstudyallows understanding the influenceofdamping,thickness, actuator dispositionandthe targetshapeforeachmethod.TheMCapproachisverysensitivetothethicknessanddampingbecauseitchangesthespread ofthemodesusedinthefrequencybandofinterest.Indeed,thelargerthenumberofactuatorsis,thelargerthenumberof controlledmodeswillbe,andthebettertheresultswillbe.However,thedispositionofactuatorsisalsoimportantforthis methodbecauseitcanexcitemorespecificmodesthanothersiftheyareplacedonavibrationnodeoranti-node(Fig.10c). Ontheotherside, theSTIFmethodisbasedonthelearningofthepropagationoperatorsbetweeneachobservationpoint andactuator.Hence,thismethodiseasiertoimplementthantheMCbutrequiresmuchmorecomputationtime.Thismeans thattheSTIFallowsfocusingfarbetterbendingwavesinthefrequencybandofinterest,asshowninFig.10a.Besides,these resultsallowunderstandingthattheSTIFmethodisalwaysveryrobustbecauseofthelargeamountofinformationknown aboutthepropagationoperator,thus,thelargerthenumberofactuatorsthebetterthealgorithmsresultsare.Inopposition, theTRmethodbasedonapartialknowledgerequireslesscomputationtimethantheSTIF,butinthiscaseislessefficient than STIFandgivespoorerlocalizationprecision (Fig. 10b).Indeed,the TRuses afew informationavailable onactuators, needstobeusedinalightlydampedmediabecauseitnecessitatesseveralreflectionsonboundaries,thereforeonalightly dampedandthinplate.ThisisareasonwhytheTRoperatesbetterforthinnerplatesthanthickerplates.Finally,whenthere is no control,the energy isspread around the plateandthere is no localization(Fig.10d). In addition, previous section analyzed energyfocusing inthetarget area aswell asthe globalinphase movementof thetarget shape forthevarious investigatedmethodsandthetwoinvestigatedcases.Forthesimplecase,STIF,TRandMCmethodsprovidedadequateand spatiallycoherentfocusing,butfortheRIBcase,onlytheSTIFmethodwasabletodoso.

Fig. 15. Frequency response between the input sweep and the center of the target shape maximum vibration, for (a) STIF, (b) TR, (c) MC and (d) No control.

4.5.2. Aboutthetemporalfocusingcapabilitiesoftheinvestigatedmethods

Afterstudyingthebendingwavefocusingalgorithmsintermsofspatialabilities,thissubsectionillustratesthetemporal phasepropertiesassociatedwiththevariousinvestigatedmethods.Indeed,thetimedomaincharacteristicsofasignalmust bepreciselycontrolledforanyaudioapplication.Inpractice,wefacehereissuessimilartotheonesfacedwhenusingother conventionalorpanel-basedloudspeakers. Theseissuesare associatedtoi)signal processing, ii)hardwarelimitationsand iii) propertiesof thematerial (dampingandmodalbehavior). Therefore,andasforanyother audiorendering device,we arenot fullyreproducingthephase alongthe wholespectrum butcome ascloseaspossibleto it.Tolimitthe impactof signal processingrelatedphase distortions, FIR filterswhich possesslinearphase andthus induce only aglobal and fre-quencyindependentdelayhavebeenchosen toimplementthespatialfocusingmethods. Tolimit theimpactofhardware relatedphasedistortion,hardwarededicatedtoaudiohasbeenused.Andfinally,wecannotlimittheimpactofthe vibrat-ingmaterialduetoexistingmodesandhavetolivewithitasitisthecasefortraditionalloudspeakers(membranemodes inthehighfrequencyrange)orforpanel loudspeakers(panelmodesinthelowfrequencyrange). Inthatsense, the pro-posedapproachinthussimilarregardingphaseissuesthananyotherhigh-fidelityaudioapproachanddonotintroduceany additionalphasemismatchasFIRfiltershavebeenretainedforsignalprocessing.

Asallthefilteringprocessandthenaturalfilteringbythehoststructureareconsideredhereaslinear,thewaveformof theresponseinthe targetedareaisa linearlyfilteredsweepsignal whentheconsidered inputisa sweptsine excitation. More precisely, according to Eq.(1), all the spatialpoints within the target area should have a vibration corresponding exactlytothissweptsinesignalexcitation.However,inpractice,duetosignalprocessing,hardwarelimitations,andmodal responseofthestructurethat areunavoidableinanyaudiorenderingdevice(eitheran electrodynamicalloudspeakerora panel-basedone),we arenot fullyachievingthisgoalbutcomeascloseaspossibletoit asshownin Fig.15.Thisfigure representsthe amplitudeof thefrequency response betweenthe sweep input audiosignal andthe displacement on the pointwiththemaximumamplitudeofthetarget shapeforthe simplesimulatedplate. FortheSTIF,theamplitudeofthe frequencyresponseis comprisedbetween0 dBand−2 dB until500 Hz,where thelocalizationisnot anymore correctly achievedasdiscussedearlier.Furthermore,therearedipsthatcan beobservedattheresonancefrequencieswhichshows thatthemodesarestillinfluencingthewholeplateresponse.Inaddition,thesamefigureexhibitsthephaseafterremoving thelineartrendcorrespondingtotheglobaldelayinducedbytheFIRfilters.Theinfluenceofmodesisstillapparentinthe phaseresponse, withphasejumps atresonancefrequencies. Itisimportant tonoticethat inthissimulation case,the TR

Fig. 16. (a) Plate with actuators position in red and (b) experimental setup used for this study. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

methodperformsextremely wellwithrespecttotemporalphaseissues.Finally,compensatingforsuchphase mismatchis somethingverycommonintheaudioindustryandshouldbemanagedaposteriorthroughadequateequalizationfilters. 5. Experimentalvalidation

5.1. Experimentalsetup

Tovalidatethepreviousmethods,anexperimentaltestcampaignwascarriedout.Thetestedstructureisthe polypropy-lene clampedplatewiththesamedimensionsandpropertiesmentionedbefore.Thismaterial iswidelyusedinthe auto-motiveindustry asdoorcarpanel garnishment.Theplateisactuatedbysixmoving coilactuatorsTEAX14C02-8 amplified witha BerhingerEPQ304 Europoweramplifiersandthesignalsare managedthanks toa RMEFireface 802soundcard and Matlabalsoforsignalsgenerations.ThevelocityfieldoftheplateisscannedbyavibrometercomprisedofanOFV-303head scanningheadanditsOFV-3001controller,throughaNIDaqUSB9234acquisitioncard.Thetestrigmadeupofisavailable inFig.16a.

The first step consistsin learningthe dynamicofthe structure. Tothat end, a first modalexperimental analysis was madebyexcitingthestructure.Eachexciterissuccessivelyactuatedwithalogarithmicsinesweepfrom100Hzto2kHz, during5s andsampled at44.1kHz. The velocity field is acquiredby a 72point gridspanningthe plate(Fig.16b). After the learningstep,thefrequencyresponse functionsare computed, thepropagationoperator isgenerated andinvertedso thattheSTIFandTRfiltersarecomputedbyaroutinedevelopedinMatlabaccordingtotheprocessdescribedSection3.3. Moreover, fortheSTIFandtheTR,thefilters obtainedabovewithnumericallearningphasewill beused tocomparethe relevanceofusingadigitaltwin.RegardingtheMCfilters,anidentificationprocedureisperformedusingSDToolsSoftware (resultsare presentedinTable 1) tocompute thefilters.The targetshapesizeisalso(0.25Lx,0.3Ly),aswellasthecenter locatedat(0.3Lx,0.6Ly).

5.2. Experimentalvalidationonasimpleplate 5.2.1. Resultsusingexperimentaldata

Firstly, eveniftheparametric studyshownthat thelargerthe numberofactuatorsisthe morethealgorithms are ef-ficient,wechose tousetheminimumpossiblenumberofactuatorstofittorealconditions.Indeed,thecongestion,mass andpowersupplyareseveralfactorsforcingtoreducethenumberofactuators.

Secondly,here, the resultsare presentedas thesame format asabovewith thesame configurationmadeavailable in

Section 4.4.1.TheFig.17bpresentsthecontrastandspoterrorofthereconstructedtarget shape.TheFig.18 presentsthe meanquadraticvelocityprofileoftheplatebetween100Hzand800Hz.Until500Hz,thecontrastoftheSTIFmethodis above0.6andafterdecreasesslightly,exceptforsomefrequencies.Indeed,thosedropscorrespondstomodeshapesofthe plateandasseeninthenumericalexample,thismethodissensibletomodeshapes.Moreover,thespoterrorisbelow15% until500Hzexceptforfrequencydropswhereitincreases.After500Hz,themethodislessefficient,thecontrastdecreases between0.2and0.5andtheerrorincreaseshighly.Inaddition,thecontrastobtainedbyMCoscillatesbetween0.2and0.8 butmostofthetimeisabove0.4. Asaconsequence,theaveragedoperativedeflectionshapebetween100Hzand800Hz shows efficientcapabilities on focusing. The time reversaldoesn’t locate the bending wavesin this case, because ofthe highlydampedmedia,thethicknessandthelownumberofactuators.Whennocontrolisapplied,themaximumofenergy

Fig. 17. (a) Contrast and (b) spot error for the experimental study.

Fig. 18. Experimental reconstructed shapes with (a) STIF, (b) TR, (c) MC and (d) without control.

islocalized (mostlyuntil 300Hz, andafterathigher frequencies)around thetarget shape andthus theactuator evenif thereisahighamountofenergyaround(Fig.18d).TheresultsareverysimilartothenumericalresultspresentedinFig.9. Themethodis lessefficientafter500Hz fortheSTIFexperimentally aswell asnumerically.Moreover, thereis thesame evolutionpatternforthethreeothermethods.

5.2.2. Resultsusingadigitaltwin

Thissubsectionallowstoinvestigatetheuseofadigitaltwininsteadoflearningonarealstructure.Indeed, experimen-talcampaignsinduce heavy setuppreparation, costs alotofprocessingtime andalarge amountofdatato postprocess. Moreover,itintroducebiasduetonoises.TheMCmethodisnotshownherebecausethefiltersarecomputedaccordingto theknowledgeofthemodalproperties,andhere,themodeshapesbasedonanalyticmodel,FEM,andexperimentleadsto thesamefilters.

Hence, the Fig.19 shows thecomparison betweenthe previous experimental results andthe results obtainedby us-ingnumericalfilters producedby simulation.TheSTIFcontrastishighasfortheexperimental resultuntil300 Hzwhich correspondstoaverytypicalmodeshape.Thereafter,thecontrastiscomprisedbetween0.2and0.5after300Hzandis ap-proximatelythesameasexperimentalSTIFafter500Hz.However,theTRgeneratedexperimentallyshowsaslightlybetter contrastthantheTRgeneratednumerically,butstilllowcomparedtoothermethods.Theyexhibitaswellthesame recon-structedshape andcontrast patternasthenumericalstudybefore.The velocityprofiles inthefrequencyband ofinterest areavailableFig.20.

Fig. 19. (a) Contrast and (b) spot error for the numerical vs experimental generated filters for STIF and TR.

Fig. 20. Experimental reconstructed shapes with numerical generated filters (a) STIF and (b) TR.

Fig. 21. (a) The door car panel studied (b) and the zone of interest.

The results exhibit the possibility to learn withthe numerical model, which can present several benefits mentioned above.Hence,theSTIFmethodisnotverysensitivetonoiseandthefiniteelementmodelcanbetranslatedintothe exper-imentalmodelforthelearningstep.

5.3. Applicationonadoorcarpanel

Now, theSTIF methodis applied on a doorcar panel to focusvibrations on a specific area allowing to generatelow frequencies. Here, 4 PUI Audio ASX05408-HD-R electromagnetic actuators are used to focus vibrations, and will allow to avoid thelow frequency modalbehavior. The same experimental setup is used,withhere 102pointsstuck on the plate toscan thevelocity profile,withthe samelearningstepprocedure. Thedoorcarpanel,thereflectivepointsdenotingthe sensingareaandtheactuatorlocationareavailableFig.21.AnumericalstudyontheFEMmodelshowsthatthereismore than300modesinthe[20Hz,200Hz]frequencyband.Thus,themodaldensityisveryhigh.TheTRandMCarenotused becauseofthesmallnumberofactuatorsusedhere,duetothesmallareaavailable.

The velocity profileforthe actuatorsdriven separately are showninFig.22. Moreover,the Fig.23shows thevelocity profileofthereconstructedshapethankstotheSTIFmethodbetween100Hzand500Hz.Themethodcanthuslocalizethe vibrationsbetweenactuatorsquiteaccuratelytocontroltheshapeofthedoorcarpanelandincreasethesoundreproduction capabilities.

Fig. 22. Experimental reconstructed shapes with only (a) actuator 1, (b) actuator 2, (c) actuator 3 and (d) actuator 4 active.

Fig. 23. Experimental reconstructed shape with the STIF method.

6. Conclusion

Toconclude,threefocusing wavemethods weretestedandadaptedforbendingwavesinthe caseofsound reproduc-tion.Firstly,there werecompared accordingto anumericalmodel,andaftervalidatedonan experimental case.Also, the STIFwasreproduced ona doorcarpanel andshowsvery goodresultsatlow andhighfrequencies,wherethe vibrations staylocatedaroundtheactuator.Withinthecontextofautomotiveindustryandspatialsoundrendering,itisobviousthat thespatio-temporal inversefilteringmethod iseasiertoimplementandgivesbetter resultscompared toothers,even on a complexstructure. Indeed, the filtersare computed without prior knowledge aboutboundaries ortheoretical dynamic behavior. However,the computationcost ishigher,because thestructure dynamicmust belearned before.In contrast,it wasshownthat the learning stepcan be made thanks to a digitaltwin andgives inthis casevery good results.Above a certain frequency where themethods focuspoorly, only exciting the actuator located nearthe target shape wouldbe enough.Mainlyforhighlydampedstructuresandathighfrequencies.Finally,severalparametersasthemodaldensity,the numberofactuatorsandthecharacteristicsofthetargetshapehavebeenstudiedastheyinfluenceonthelocalization re-sults.Thoseparametersmustbetakenintoaccount,regardlessofthecomplexityofthestructure.Asaconsequence,some prospectsflowingfromthisworkcanbeunderlined.Firstly,thecomparisonofthosemethodsandthedrawbacksassociated willallowconsideringactive controltoincrease therobustnessincaseofcomplexstructuresandsubjectedtoseveral per-turbations(operational conditions...).Secondly,theuseofdigital twinswillallowto easethestudyofcomplexstructures andlearningphases.Thiswillservetoenhancetheindustrializationofthiskindoftechnologyinautomotiveindustry.

DeclarationofCompetingInterest

The authors declare that they have noknown competing financial interests orpersonal relationshipsthat could have appearedtoinfluencetheworkreportedinthispaper.

CRediTauthorshipcontributionstatement

NassimBenbara:Methodology,Software,Validation,Writing-originaldraft.MarcRebillat:Conceptualization, Method-ology,Software,Supervision,Writing-review&editing.NazihMechbal:Fundingacquisition,Supervision,Writing-review &editing.

Acknowledgments

This work wasfinancially supported by the French NationalResearch Agency(ANR, contract ANR-17-CE33-0004). The authorswish tothank ChristianBolzmacher fromCEA andJean-Christophe ChamardfromPSAfortheir helpinproviding thecarandsomehardwareforexperiments.

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