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MODELING OF THE SOUND TRANSMISSION LOSS OF LEAKS AND OPENINGS. Montreal, QC H3A3C2, Canada 2 LASH/DGCB, Ecole Nationale des TPE

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ICSV14 Carns Australa 9-1 July, 007 ODELING OF THE SOUND TRANSISSION LOSS OF LEAKS AND OENINGS Franck Sgar 1,, Hugues Nelsse 1 an Nourene Atalla 3 1 IRSST, Servce e la recherche, 505 B e asonneuve ontreal,
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ICSV14 Carns Australa 9-1 July, 007 ODELING OF THE SOUND TRANSISSION LOSS OF LEAKS AND OENINGS Franck Sgar 1,, Hugues Nelsse 1 an Nourene Atalla 3 1 IRSST, Servce e la recherche, 505 B e asonneuve ontreal, QC H3A3C, Canaa LASH/DGCB, Ecole Natonale es TE Rue aurce Aun, Vaulx-en-Veln Ceex, France 3 GAUS, Department o echancal Engneerng, Unversty o Sherbrooke 500 B e l Unversté, Sherbrooke, QC J1KR1, Canaa Abstract The arborne soun transmsson perormance o structures may be strongly aecte by the presence o apertures. These apertures mght be esgne on purpose to allow or the crculaton o matter (machnery enclosures) or or the passage o wres an ppes through the structure (vehcles, bulngs). They are then reerre to as openngs. They coul also be unwante or not part o the ntal esgn ue to ba assembly an/or mountng contons. They are then calle leaks. These two kns o apertures er by ther sze relatve to the acoustc wavelength. To prect the nsulatng perormance o a structure nvolvng apertures, t s oten requre to know the oblque ncence or use el soun transmsson loss (TL) o the aperture. Numerous moels exst or the normal ncence TL but rgorous moels or oblque ncence an use el exctatons have rarely been consere. The use el TL s usually obtane usng a correcton actor apple to the normal ncence TL whch ranges rom 0b to 5B epenng on the authors. The purpose o ths paper s to propose a general ecent moel base on the escrpton o the el nse the aperture n terms o propagatng an evanescent acoustc moes together wth an ecent computaton o moal raaton mpeance matrces to prect the oblque ncence an use el TL o rectangular an crcular apertures o nte epth. The moel s valate by comparson wth exstng expermental ata an moels. Numercal results llustratng the normal ncence, oblque ncence an use el TL o apertures are then prove an the relatonshps between these ncators are scusse. 1. INTRODUCTION The arborne acoustc transmsson perormance o enclosures strongly epens on the presence o leaks an openngs. The leaks are the result o a ba assembly an enote apertures whose sze s small compare to the acoustc wavelength. The openngs have generally larger mensons an are rather esgne on purpose to allow or the crculaton o matter between the external envronment an the machnery through the enclosure. Durng the acoustc esgn o enclosures usng smple analytcal or energy base moels, prectng the eect o an aperture on the overall acoustc transmsson can become crucal an requres the knowlege o the aperture use el soun transmsson loss. Despte ts mportance, relatvely ew authors have ealt wth ths problem an there s lttle consensus on whch ormulatons work best. In prevous stues, apertures were consere as rectangular, slt or crcular shapes wth neglgble or nte thckness snce these geometres are representatve o most practcal cases. In a recent paper, the authors have presente a comprehensve lterature revew an comparsons o the prncpal approaches avalable n the lterature. They have ponte out several ssues () the lack o a moel that woul prect accurately an raply the use el soun transmsson loss n the acoustcal requency range whatever the sze o the aperture; () the relevance o usng the normal ncence transmsson loss (wth or wthout correctons) to prect the use el TL () the lack o an ecent tool or prectng the broaban use el soun transmsson loss o both crcular an rectangular apertures n the requency range covere n most nose control applcatons (100Hz-5000Hz); (v) the lack o numercal results regarng the use el soun transmsson loss o openngs (large apertures). To aress these questons, they have ntrouce a general an rgorous ormulaton to prect the oblque ncence an use el soun transmsson loss (TL) o rectangular an crcular apertures o nte epth. The evelope tool s smple, userrenly, ecent an can easly be utlze to prove nput ata to SEA coes or nstance. In the present paper, atonal valaton an numercal results are prove. The rest o the paper s organze as ollows. Frstly, the theoretcal approach evelope n [1] s brely recalle. The moel s base on the escrpton o the el nse the aperture n terms o propagatng an evanescent acoustc moes together wth an ecent computaton o moal raaton mpeance matrces. Seconly, comparsons between exstng expermental results an calculatons are gven. Fnally, numercal results llustratng the normal ncence, oblque ncence an use el TL o apertures are presente an the relatonshps between these ncators are scusse.. THEORY Fgure 1. Scheme o an aperture lle wth an absorbng materal nserte nto a rg planar bale Conser a rectangular or crcular aperture o area S an epth nserte nto a rg planar bale an excte acoustcally by an oblque plane wave wth ncence angles (θ,ϕ ) as llustrate n Fg.1. The total acoustc el p 1 n meum 1 s gven by : p = p + p + p = p + p (1) 1 R r b r j t wth the conventon p() t = p e ω. p s the ncent acoustc pressure (o ampltue A ), p b s ( ) the blocke acoustc pressure ( jk θ ϕ x + θ ϕ p = Ae y cos k cosθ z ), p r s the acoustc ( ) 0 sn cos sn sn b 0 pressure raate by the ront ace o the aperture (exctaton se) an p the acoustc pressure raate n meum by the back ace o the aperture. p r an p are classcally gven by Raylegh s ntegrals. The acoustc el nse the aperture s note p =p 1 +p. The aperture can be lle wth an acoustc materal o characterstc mpeance Z an complex propagaton constant k. For smple shapes such as rectangular an crcular cross sectons, the acoustc el nse the aperture can be expane analytcally n terms o propagatng an evanescent moes: ( ) ( ) ( ),, jk z jk z p xyz = Ae + Be φ xy, () ( ) ( ) where or a rectangular cross secton o sze a b φ pπ qπ a b ( x, y) = φ ( x, y) = cos ( x+ a) cos ( y+ ) raus a, pq pq pq k k k λ = =, ( ) ( ) a pπ qπ k = kpq = k a b, b an or a crcular cross secton o λpqr π φ x, y = φpqs r, γ = Jp sn pγ + s a, λ pq beng the zeroes o the ervatve o Bessel uncton o orer p ( p' ( pq) J λ = 0) an s beng a symmetry nex equal to 0 or 1. The next step s to wrte own the bounary contons on both ses o the aperture (contnuty o pressure an normal partcle velocty). Substtutng Eq() an Raylegh s φ x, y an ntegratng ntegrals or p r an p nto these equatons, multplyng them all by ( ) over the surace o the ront an back aces o the aperture leas to the ollowng lnear system: [ ] [ ] { Ψ } 1 Ψ A { F } = (3) [ Ψ3] [ Ψ4] { B } {} 0 where t has been assume that meum 1 an are entcal ( ρ 1 = ρ = ρ0 an c 1 = c = c0 ) an wth jk S k jk Ψ 1, = N e δ + e Z (4) Z k jk jk, Ne δ e Z Z k S k Ψ = (5) S k Ψ = N δ Z (6) 3, Z k wth ( ) φ ( ) F = pb xy, xys, S jk Z S N = φ ( x, y) S( ) S S k Ψ = N δ + Z (7) 4, Z k Z = φ ( xygxyx, ) (,,, y) φ ( x, y) S( ) S( ) S S F s the generalze orce actng on moe, Z s the aperture cross moal raaton mpeance between moes an an N s the square norm o moe. For a rectangular secton, Z can be ecently calculate usng a change o varables [] whch transorms the quaruple ntegral nto a ouble ntegral that can be resolve wth a Gauss ntegraton scheme. For a crcular cross secton, ths quaruple ntegral can be reuce to a smple one the problem s solve n the wavenumber oman. All the etals are prove n [1]. Once the system has been solve, the transmtte acoustc power or oblque ncence exctaton can be calculate as: 1 1 t * * Π ( θ, ) ϕ = R N * kcd (9) ρω = ( + ) ( ) wth C A B an D = A + B. The oblque ncence transmsson coecent τ ( θ, ϕ ) s ene as: t Π ( θ, ϕ) τθ (, ϕ) = (10) Π nc ( θ, ϕ ) A nc cosθ S nc where Π ( θ, ϕ), the ncent power, s gven by Π ( θ, ϕ) =. ρ0c0 Usng the prevous equatons, the oblque ncence transmsson coecent then reas : ρ0 * * (, ) τ θ ϕ = R N k C D * k cosθρ A S (11) 0 (8) The use el soun transmsson coecent can then be calculate numercally usng a Gauss ntegraton scheme : τ π θ lm ( ) τ θ, ϕ snθ cosθ θ ϕ = 0 0 πsn θ (1) lm where the lmt angle θlm s taken equal to 78 (el ncence) n all the ollowng numercal examples. The soun transmsson loss (TL) s nally calculate usng TL = 10 log 10 ( τ ). 3. RESULTS 4.1 Valatons Fgure. Oblque ncence (θ =45,ϕ =0 ) transmsson losses o a rectangular aperture (b/a=1/, /a=1), a crcular aperture (/a=/3) an a slt (/b=) as a uncton o normalze requency ka wth aeq = 4ab π the equvalent raus or rectangular shape an a = aor crcular shape (bottom axs) an k0b or slts (top axs) In [1], the propose approach has been valate or varous conguratons (rectangular an crcular cross secton apertures an slt-shape) usng exstng analytcal ormulas when avalable [3,4,5] or numercal moels such as ark an Eom s [6] or FE/BE. As an llustraton, Fgure splays the transmsson losses calculate usng the present approach together wth numercal moels (FE/BE an ark & Eom s metho) or a rectangular an a crcular aperture excte by an oblque ncence plane wave. The oblque ncence transmsson loss o a slt shape aperture calculate usng echel s analytcal moel [5] an the present approach s also plotte. Note that or the analytcal moel the slt s assume o nnte breath whereas n the present approach t s consere o nte length wth a=400b. A perect agreement s observe or all the teste cases or example FE/BE results conce wth the present approach. The convergence stuy carre out n [1] has ncate that t was sucent to keep moes up to the maxmum requency o nterest to nsure the convergence o the soluton convergence wthn a 0.1B error. In ths paper, atonal valaton results are presente. eq Fgure 3. Comparsons between measure an calculate normal ncence an use el transmsson losses n the case o a crcular aperture (a=5.08cm, =7.6cm) as a uncton o requency. Fgure 4. Comparsons between measure an calculate normal ncence an use el transmsson losses n the case o a rectangular aperture (a=18cm, b=4.5cm, =30.48cm ) as a uncton o normalze requency ka wth a = 4ab π the equvalent raus. eq Fgures 3 an 4 splay a comparson o calculatons usng the present approach an soun transmsson measurements between two reverberant rooms carre out by Wlson an Soroka [3] an Sauter an Soroka [4] or crcular an rectangular shape apertures. Both the calculate normal ncence an use el soun transmsson loss are plotte. It s seen that there s a goo agreement between the moel an the expermental ata. In partcular, Fgures 3 an 4 show that the use el moel reprouces very well the peaks an troughs observe n the measurements except maybe at low requences. In ths requency range, the acoustc el s nee probably not use enough or the use el exctaton moel to be val. In aton, these gures ncate as emonstrate n [1] that () the normal ncence moel s able to capture most o the physcs () the erence between normal ncence an use el exctatons s small (o the orer o 1 to B on average an up to 3B locally or the cases nvestgate here). 4.1 Normal ncence versus use el soun transmsson loss Fgure 5. Narrow ban erences between normal ncence an use el soun transmsson losses or nvestgate apertures calculate usng the present approach as a uncton o normalze requency Fg.5 plots the narrow ban calculate erences between an TL 0 as a uncton o normalze requency ka TL ( ) or multples sze o rectangular an crcular apertures whose characterstcs are presente n the legen. At low requences ka 0.3, the erence between TL an TL ( 0) s equal to.b when θ lm =78 s use. At hgh requences TL ( ) ka 7, the erence between an TL 0 s less than 1B. In between, the erences oscllate an can reach up to approxmately 5B or very thck apertures. When the thr-octave ban average results are consere, the same observatons reman but n the m requency zone, the erence between an TL 0 s less than B [1]. These TL ( ) conclusons are n agreement wth the observatons mae by prevous authors [3,4] on the erences between expermental results (use el exctaton) an ther normal ncence moel. 4. CONCLUSIONS In ths paper, a general an ecent numercal metho to prect the use el soun transmsson loss o bale apertures o rectangular an crcular cross-sectons has been ntrouce. The wave el nse the aperture s escrbe n terms o propagatng an evanescent acoustc moes an the acoustc raaton o the aperture s taken nto account wth a moal raaton mpeance matrx whose calculaton s carre out usng ecent numercal algorthms. The couple problem s solve n terms o moal contrbuton actors rom whch the transmsson loss can be evaluate. The evelope tool s smple, portable, oes not requre meshng tools such as those employe n BE/FE technques an can easly be utlze to prove nput ata to SEA coes or nstance. The precton tool has been valate by comparsons wth exstng analytcal or numercal moels n varous conguratons [1]. Ths paper has prove atonal valaton normaton by comparng numercal results an expermental exstng ata n the case o rectangular an crcular apertures. The use el moel allows or a better agreement wth expermental results. Results regarng the narrow ban use el soun transmsson loss have been conronte to the classcal normal ncence soun transmsson loss moels. It has been observe that a smple normal ncence transmsson loss moel or apertures can be use wth a correcton actor o about B at low requences. The maxmum erence between the narrow ban use el an normal ncence transmsson losses s expecte to be less than 5B at meum requences, an 1B at hgh requences. For average ban ncators, the 5B are reuce to B. REFERENCES [1] F. Sgar, H. Nelsse an N. Atalla, On the moelng o the use el soun transmsson loss o nte thckness apertures, accepte n Journal o the Acoustcal Socety o Amerca (007). [] H. Nelsse, O. Besln an J. Ncolas, A generalze approach or the acoustc raaton rom a bale or unbale plate wth arbtrary bounary contons, mmerse n a lght or heavy lu. Journal o Soun an Vbraton, 11(), pp (1998). [3] G.. Wlson an W.W. Soroka, Approxmaton to the racton o soun by a crcular aperture n a rg wall o nte thckness, Journal o the Acoustcal Socety o Amerca, 37(), pp , (1965) [4] A. Sauter an W.W. Soroka, Soun transmsson through rectangular slots o nte epth between reverberant rooms, Journal o the Acoustcal Socety o Amerca, 47(1) art 1, pp.5-11 (1970) [5] F.. echel, The acoustc sealng o holes an slts n walls, Journal o Soun an Vbraton, 111(), pp , (1986) [6] H.H. ark, H.J. Eom, Acoustc scatterng rom a rectangular aperture n a thck har screen, Journal o the Acoustcal Socety o Amerca,, 101(1), pp , (1997)
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