Acoustic scattering reduction using layers of elastic materials 2017 Cécile Dutrion.pdf

该【Acoustic scattering reduction using layers of elastic materials 2017 Cécile Dutrion 】是由【探春文档】上传分享，文档一共【16】页，该文档可以免费在线阅读，需要了解更多关于【Acoustic scattering reduction using layers of elastic materials 2017 Cécile Dutrion 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..JournalofSoundandVibration388(2017)53–68ContentslistsavailableatScienceDirectJournalofSoundandVibrationjournalhomepage:ate/jsviAcousticscatteringreductionusinglayersofelasticmaterialsCécileDutrion,FrankSimonnOneraCentredeToulouse,2avenueEdouardBelin,30155Toulouse,FrancearticleinfoabstractArticlehistory:-HandlingEditor:,-Keywords:---Elasticmultilayermaterialsideringanexternalfluidconsistingofair,.&,,[1]intheicdomain,CummerandSchurig[2]loakingcouldbeachievedbysur-,,,forexample,aplanarwaveguidemadeofaluminum,whereacircularcavityisdrilledwithanembeddedcorrugatedstructure,inordertoproduceananisotropicmassdensity[3],loakforunderwaterultrasound[4],anda2Ddevicemadeofperforatedplasticplatesforcloakinginair[5].CaiandSánchez-Dehesa[6]studiedthefeasibilityofamultilayercoatingmadeoffluidtocurveultrasonicwavesandtofocusbeamsbehindacoolingtube,thankstothegradientofpropertiesproducedinthefluidlayers(effectofagradient-indexacousticallens).Thesecondtypeofsolutionconsistsindesigningacoatingthatdrasticallyreducesthetotalscatteringcross-,thatcanbereferredtoasscatteringcancellationtechnique,-mailaddress:frank.******@().http://dx./.-460X/&.:..,(2017)53–68loakingbyAlùandEngheta[7].Ithasthenbeenextendedtotheacousticdomain,[8,9][10,11]proposedapatternstructurewithappropriateacousticsurfaceim-,García-Chocanoetal.[12].[13][14,15],-,,-,[16]andaretakenfrom[17].?-axisincartesiancoordinatesandcorrespondstoθ=?+=22pkp0()1wherekc=ω/:,correspondingtothereflectionoftheincidentwaveonarigidcylinderofradiusb,andaradiatedpartprad,,=++increflrad(→)pscat0()2Inthefollowingparagraphs,lengthshavebeenrescaledbytheradiusoftheobstaclea,andpressuresandstressesbytheproductρ0c2,,referredtowithatildesymbol,areexpressedasfunctionsofthedi-mensionlessradialcoordinateξ=r/,forthesakeofreadibility,thee?iωtmontoallquantities,.:..,(2017)53–?incduetoaplanewaveofamplitudep?0travellinginthepositivex-directioncanberepresentedasaseriesofcylindricalBesselfunctionsofintegerorder:+∞?1if0,n=ppJn?inc0=?∑ikacoswherenεξθεnn()()n=?n=0?≥()3Thescatteredpressurep?+∞pAHn?scat=∑nn()()()kakacosξθn=0()4whereAn(ka)istheunknownscatteringcoefficientandHn(.)(ka)isdeterminedusingtheboundaryconditionontheoutersurfaceofthecoating,statingthattheacousticradialvelocityinthefluidatξN=b/?rN(ξθN,ka,)andtheunknownscatteringcoefficientAn(ka):1?p?1+∞u?rN()()()()()ξθN,ka,==∑????pJAHn?0ikakakacosnεξξθn′Nn+n′N????ka2?ξkan()5n=0InordertocalculateAn(ka),andthusp?scat,itisnecessarytostudythevibrationsofthecoatingunderplanewaveexcitation,andtoevaluatetheradialdisplacementu?rN(ξθN,ka,).Thisisdoneusingthesemi-,ponentsare,inpolarcoordinates,?σ?rrj11?σ?rjθjjj++()σσρ?rr??θθ=??jka,2u?r?ξξ?θξ?σ?jj12?σ?rθθθ++σρ?rjθθ=??jka,2u?j?ξξ?θξ()6whereρ?posingthelayerj,and()uu??r,,Hooke'slawcanbewrittenas?σ?j??cc??jj0??εj??rr??1112??rr??σ?θθj?=?cc??jj220??εθθj????12????σ?j??00c?j??2εj??rθ??33??rθ?()7Thecoefficientsc?:j?u?rjεrr=,?ξj1?u?θju?rjεθθ=+,ξθξ?1?1?u?j?uu?jj??εjr=????+θθ?????.rθ2?ξθξξ???()8Inthecaseoforthotropicmaterial,thesystemobtainedbysubstitutionofEq.(7)andEq.(8)intoEq.(6)[18],Hasheminejad[19]andJamalietal.[20].(6),(7)and(8)maybewrittenasfollows:??jjj=Μ??ξ()9with?j=uu??j,,,jσσ??jjthestatevector,andΜja4iveninAppendixB.(θθrrrr)Thestatevector?jisexpandedintermsofunknownmodalcoefficientsas:..,(2017)53–68?u?j??un?j()()ξθsin??θ??θ,n??u?j?+∞?un?rnj,()()ξθcos??j=?r?=∑??.?σ?rrj?n=0?σξθ?rrnj,()()cosn????????????σ?j??σξθ?jsinn?rθ?rnθ,()()?()10TheorthogonalityoftrigonometricfunctionsgivesdVnjjj=Ρn()ξVn,dξ()11j??jjjjTjwhereVn=()uuθθ,n,,,rn,,σσ??rrnrn,(ξ)=ξj?1,,es:VVnj()ξξξξξ=?exp,????()()()j???111Ρnjj????njj()12whereξj?1≤≤ξξjandjN=…1,2,,.Applyingthisequationatξ=ξj,itispossible,foreachlayer,torelatethestatevectorattheoutersurfaceVnj(ξj)tothestatevectorattheinnersurfaceVnj(ξj?1).Continuityconditionsatallinterfacesbetweenlayersenabletorelatethestatevariablesattheoutersurfaceofthecoating()ξN=ba/tothestatevariablesatitsinnersurface()ξ0=1,VTVN()()ξξNn=10,()13nnwiththeglobalmodaltransfermatrixTn1TPn=∏exp,????h?jj()ξj?1????njN=()14andh??rN(ξN),andconsequentlyofthepressureintheexteriorfluid:??J′()kaξN???pJ?0ikanεξn????n()?nHn()kaξ?????+∞??Hn′()kaξN??p?()ξθ,ka,=∑??cos.()nθn=0?urnN,()ξN??+kaHn()kaξ??Hn′()kaξN?(),fordifferentfluidsandboundarycon-[21]andSkeltonandJames[22].Forexample,,forarigidcylinderandforacylinderwithauniformcoatinginpolyvinylbutyral(PVB)parisonwithanalyticformulationsofRomanovandTolokonnikov[21]appearssatisfactoryforthetwopulsationratesillustratedhere(ω=4andω=8).,thatconstitutetheoptimizationvariables,are,foreachlayer:thethicknessh,Young'smoduliinthedifferentdirectionsEr,Eθ,Ez,theshearmodulusGrθ,Poisson'sratioνrθ,νθzandνrz,themassdensityρ,-layercoatingstolimitthenumberofoptimizationvariables.:..,(2017)53–,parisonbetweenmodeland[21].icalgorithmSOGAfromtheDAKOTAsoftwarehasbeencoupledwiththevibro-,1definedastheratioofthescatteringcross-sectioninthe“cloaked”caseσsc(cloaked)tothescatteringcross-sectioninthe“uncloaked”caseσsc(uncloaked),σsc()cloakedFobj,1=()uncloaked.(16).σscThescatteringcross-sectionσscisdefinedbyσ=Iscd,Ssc∫SI()17incwhereSrepresentsasurfaceenclosingtheobstacle,andwhereIscandIincaretheacousticactiveintensities,perpendiculartothesurfaceelementdS,=1Repˉ2()()-absorptionofthefluidmedium,,,.(17).Thescatteringgainσgainprovidedbythecoatingismeasuredindecibelasfollows:?σ()cloaked?σgain=10log?sc?10?()uncloaked??σsc?()