19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007

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1 9 h INTERNATIONAL CONGRESS ON ACOUSTICS MADRID -7 SEPTEMBER 7 SOUND INSULATION CHARACTERISTICS OF A MICROPERFORATED PANEL ITH A SUBDIVIDED AIR LAYER PACS:.55.T Toyoda Masahro ; Daj Takahash B Kyoo Unv. Kasura Nshkyo-ku Kyoo Japan; masahro.oyoda@kupru.ae.kyoo-u.ac.jp C--5 Kyoo Unv. Kasura Nshkyo-ku Kyoo Japan; khs@arch.kyoo-u.ac.jp ABSTRACT A mcroperforaed-panel absorber has been nvesgaed wdely on s absorpon characerscs and s recenly recogned as one of he nex-generaon absorbng maerals because of s fber-free naure and aracve appearance. In hs sud a furher possbly of a mcroperforaed panel s nvesgaed heorecally from he vewpon of sound radaon and sound ransmsson. The absorbng mechansm of a mcroperforaed panel s Helmhol-ype resonance absorpon; herefore s ypcally backed by an ar cavy and a rgd wall. To enhance he resonance effec he ar layer s subdvded wh parons for example honeycomb srucure. Each of he subdvded cells can creae local one-dmensonal sound felds and also lead o normal ncdence no he orfces whch s he mos effecve condon for Helmhol-ype resonance absorpon. In order o evaluae he mprovemen an analycal model s consdered here. In he model a plae whch has a recangular cross secon s assumed o be smply suppored n a duc and a mcroperforaed panel s assumed o be nsalled n fron of he plae wh a subdvded ar layer. By usng hs model mprovemen of he ransmsson loss and he radaon reducon s nvesgaed. INTRODUCTION A mcroperforaed panel (MPP) has been developed and used for decades. I was frs proposed by Maa [] as a sound-absorbng maeral. Is heorecal bass was esablshed and he grea poenal was shown. Is applcaons can be made of any maeral from plasc plywood acryl glass o shee meal. Furhermore can provde wde-band absorpon whou fbrous and porous maerals whch are no desrable under severe envronmens. Therefore he MPP s recogned as one of he nex-generaon absorbng maerals. The absorbng mechansm of a MPP absorber s Helmhol-ype resonance absorpon; herefore s ypcally backed by an ar cavy and a rgd wall. In order o mprove he absorpon characerscs of MPP absorbers double resonaors wh wo MPPs were developed by Maa []. The double resonaors can broaden s absorpon band. From he vewpon of archecural acouscs s aracve ha MPP absorbers can be made of ransparen maerals lke acryl grass. Fuchs e al. [] proposed he ransparen MPP applcaons for ndoor use and Asdrubal e al. [] showed he feasbly of ransparen MPP absorbers as an oudoor nose barrer. To creae an absorbng srucure whou rgd back wall Sakagam e al. [5] proposed a double-leaf MPP (DLMPP) absorber whch consss of wo MPPs alone. I was shown ha a DLMPP srucure can provde an addonal absorpon a low frequences. In many heorecal analyses used o be assumed ha a MPP s mmoble. However some expermenal resuls of prevous works ndcaed he unexpeced absorpon peaks around low frequences [6]. Therefore some sudes focused on he panel vbraon were conduced. Sakagam e al. [7] nvesgaed heorecally he effec of he vbraon whch ofen deeroraes he effcency of he absorbng srucure. Lee e al. [8] proposed a predcon model wh he modal analyss approach and suded on he couplng effec beween a MPP and a back cavy heorecally and expermenally.

2 As lsed above MPP srucures have been nvesgaed n deal as sound absorbers. In order o develop a furher possbly of a MPP panel sound radaon and sound ransmsson characerscs of MPP srucures are suded heorecally n hs paper. An analycal model consdered here consss of a MPP and a non-rgd back wall whch have a recangular cross secon and are smply suppored n a duc. By usng hs model sound ransmsson loss of a MPP srucure s calculaed. Moreover n order o ncrease he ransmsson loss a new echnque proposed by auhors [9] s appled o he MPP srucure and he mprovemen s esmaed. ANALYTICAL MODEL Fgure shows an analycal model consdered n hs sudy. A MPP locaed a = and a back wall locaed a =d are suppored n a recangular duc wh cross secon a b. An ncden wave p s assumed o be plane wave expressed as follows wh egenfuncons ψ (x for vbraon n (mn) mode of he recangular panel suppored n he duc p = e I ψ = k ( θ cosϕ x+ snθ sn ϕ y+ cosθ ) I ψ sn k cosθ = e (Eq. ) m+ k snθ cosϕ a n+ k snθ snϕ b {( ) e + {( ) e + {( m ) ( ak snθ cosϕ) {( n ) ( bk snθ snϕ) 9 h INTERNATIONAL CONGRESS ON ACOUSTICS ICA7MADRID = sn sn (Eq. ) m x n y (Eq. ) a b where s he magnary un k s wave number of ar θ s elevaon angle and φ s amuhal angle. The me facor e -ω s suppressed hroughou where ω s he angular frequency. In he followng dscusson he ncden sde he cavy beween wo panels and he ransmsson sde are ndcaed by subscrps and respecvely. If he duc s flled wh ar he sound pressures p (x) and he parcle veloces v (x) can be expressed n erms of unknown quanes P - P + P - and P + as n= k cos k ) = { I e + P e θ p ψ (Eq. ) p + k k ) = { P e + P e n= ( ) + k x = { P e n= ψ (Eq. 5) p ψ (Eq. 6) ( ) = θ k cosθ k k v x I e P e ψ n= ρc ρω (Eq. 7) ( ) k + k k v x = { P e P e ψ ρ ω (Eq. 8) n= k v ψ + k ) = { P e n= ρω (Eq. 9) where ρ s he densy of ar and c s he sound speed. The quany k corresponds o he - drecon wave number n he (mn) mode of vbraon. h k and he wave number n he xydrecon k k can be wren as k k k = k ( k ) ( k k k ) ( k k ) ( k < k ) ( m a) + ( n b) (Eq. ) =. (Eq. )

3 p a x d b y Fgure.- An analycal model of a MPP srucure for calculang he ransmsson loss. In hs case he equaons of moon for he dsplacemen of he MPP w (x and he dsplacemen of he back wall w (x can be wren as D w hω w = p ) p ) h ω w = p d ) p y d ) ρ (Eq. ) D w ρ (Eq. ) where subscrps and ndcae he MPP and he back wall respecvel D s he flexural rgd ρ s he dens and h s he hckness. The dsplacemens w (x can be expanded n erms of unknown quanes as n= = w ψ. (Eq. ) By subsung Eqs. () (5) (6) and () no Eqs. () and () can be expressed n erms of P - P + P - and P +. Le be he acousc mpedance of he hole of he MPP whch s represened wh s ressance erm ress and reacance erm reac as = ress + reac. Then he boundary condons a each surface n Fgure are gven [9] by σ v ω (Eq. 5) ( x ) = v ) = ζw + { p ) p ) y d ) v d ) = ωw v = (Eq. 6) where σ s he perforaon rao of he MPP and ζ=-( reac / )σ. By subsung Eqs. ()-(9) and () no Eqs. (5) and (6) one can oban he smulaneous equaons n erms of P - P + P - and P +. In hs case he radaed acousc power (θφ) from he surface of he back wall can be wren as S ab = 8ρ ω { * ( θ ϕ) = Re p d ) v d ) n= Re { k 9 h INTERNATIONAL CONGRESS ON ACOUSTICS ICA7MADRID + P ds (Eq. 7) where S s he surface of he back wall. I s no necessary o calculae summaon n Eq. (7) for (mn) whch do no sasfy he nequaly k >k. The ncden acousc power (θ) s gven by ab cosθ ( θ ) =. (Eq. 8) ρ c

4 The ransmsson coeffcen τ(θφ) can be expressed by τ ( θ ϕ) ( θ ϕ) ( θ ) =. (Eq. 9) Then he mean ransmsson loss TL can be wren by TL = log = log θlm θ θ lm lm τ ( θ ) ( θ ϕ) θ lm ( θ ϕ) snθ dϕ snθ dϕ (Eq. ) sn θ sn θ dϕ where θ lm s 78 for feld ncdence or 9 for random ncdence. NUMERICAL EXAMPLE Fgure shows calculaed resuls of he ransmsson loss under feld ncdence of plane waves. In order o fnd he endences easl he ransmsson loss are gven as sldng frequency averages over a /-ocave band nerval. The cross secon s.m.5m. The cavy deph s 5mm. Parameers of he MPP are as follows: he aperure dameer s.mm; he dsance beween aperures s.5mm; he perforaon rao s.%; hckness s.mm; densy s 7kg/m (Alumnum); Young s modulus s.7 ; Posson s rao s.; and loss facor s.. Parameers of he back wall are as follows: hckness s 5mm; densy s 75kg/m (Plaser); Young s modulus s.8 9 ; Posson s rao s.; and loss facor s.. The acousc mpedance of he aperure s calculaed by Maa s approxmaon formulas []. The cases of he back wall alone and a sandard double-leaf panel are also shown n Fgure as reference. The double-leaf panel conans of he back wall and a non-perforaed panel whch has he same parameers wh he MPP excep for parameers of he aperures. For he case of he back wall alone he resuls follow he mass law excep for he dp around he crcal frequency of concdence 65H. Alhough he double-leaf panel can mprove he ransmsson loss above he concdence adverse effecs can be seen a low frequences. These adverse effecs would be caused by he mass-ar-mass resonance of he double-leaf panel. As for he case of he MPP srucure he mprovemens a hgh frequences are no greaer han he case of he double-leaf panel. Adverse effecs a low frequences however canno be seen. Alhough MPP srucures n general have a grea poenal for sound absorpon he mprovemens of ransmsson loss mgh no be sgnfcan a md frequences whch are mporan for sound nsulaon of human voces and so on because ypcal MPPs are consderably hn. Transmsson Loss [db] 8 6 Back all Alone Non-Perforaed Panel + Back all MPP + Back all Frequency [H] Fgure.- Calculaed ransmsson loss of ypcal srucures. 9 h INTERNATIONAL CONGRESS ON ACOUSTICS ICA7MADRID

5 IMPROVEMENT FOR SOUND INSULATION In order o ncrease he ransmsson loss of he MPP srucure he ar-cavy subdvson echnque [9] s appled o he model. The echnque can be acheved by nserng parons lke honeycomb srucure o he ar cav and glung hem ghly o he MPP and he back wall so as o gve he same moon wh he back wall o he MPP. In hs case Eq. (5) (8) () and () should be replaced wh followng equaons. + k k ) = { P e + P e p ψ (Eq. ) v n= + k k ) = { P e P e ψ (Eq. ) n= ρc w w x y ρ hω w x y = p x p x + p x d p w D = (Eq. ) ( ) ( ) ( ) ( ) ( ) d ) (Eq. ) In hese equaons neracon effecs resulng from dfferences beween dsplacemens of he MPP and he back wall are negleced because he hckness and he flexural rgdy of he back wall are suffcenly greaer han hose of he MPP. Calculaed resuls of he ransmsson loss and he absorpon coeffcen under feld ncdence of plane waves are shown n Fgure and Fgure respecvely. The grea mprovemens a md frequences can be seen n boh hese fgures. These effecs can be obaned because each of he subdvded cells creae local one-dmensonal sound felds and also lead o normal ncdence no he orfces whch s he mos effecve condon for Helmhol-ype resonance absorpon. Furhermore he same moon of he MPP and he back wall would cancel he undesrable effec of he vbraon of he MPP [7]. Therefore he energy loss n he ar cavy s ncreased and boh of he ransmsson loss and he absorpon coeffcen are mproved. Transmsson Loss [db] 8 6 MPP + Back all MPP + Subdvded Ar Layer + Back all Frequency [H] Fgure.- Calculaed ransmsson loss of he proposed MPP srucure. Absorpon Coeffcen MPP + Back all MPP + Subdvded Ar Layer + Back all Frequency [H] Fgure.- Calculaed absorpon coeffcen of he proposed MPP srucure. 9 h INTERNATIONAL CONGRESS ON ACOUSTICS ICA7MADRID 5

6 CONCLUSION In hs sud a furher possbly of a MPP s nvesgaed heorecally from he vewpon of sound nsulaon. By usng he analycal model consdered here he ransmsson loss of a ypcal MPP srucure s esmaed. In order o mprove he performance of sound nsulaon he ar layer s subdvded wh parons for example honeycomb srucure. Each of he subdvded cells can creae local one-dmensonal sound felds and also lead o normal ncdence no he orfces whch s he mos effecve condon for Helmhol-ype resonance absorpon. Furhermore he same moon of he MPP and he back wall would cancel he undesrable effec of he vbraon of he MPP [7]. Therefore he energy loss n he ar cavy s ncreased and boh of he ransmsson loss and he absorpon coeffcen are mproved. The wde use of hs srucure for boh sound nsulaon and sound absorpon s ancpaed. ACKNOLEDGMENT Ths sudy was suppored by Program for Improvemen of Research Envronmen for Young Researchers from Specal Coordnaon Funds for Promong Scence and Technology (SCF) commssoned by he Mnsry of Educaon Culure Spors Scence and Technology (MEXT) of Japan. References: [] D. Y. Maa: Theory and desgn of mcroperforaed panel sound-absorbng consrucons. Scena Snca 8 (975) 55-7 [] D. Y. Maa: Mcroperforaed panel wde-band absorber. Nose Conrol Engneerng Journal 9 (987) 77-8 [] H. V. Fuchs X. Zha: Acrylc-glass sound absorbers n he plenum of he Deuscher Bundesag. Appled Acouscs 5 (997) -7 [] F. Asdrubal G. Pspola: Properes of ransparen sound-absorbng panels for use n nose barrers. Journal of he Acouscal Socey of Amerca (7) - [5] K. Sakagam M Mormoo. Koke: A numercal sudy of double-leaf mcroperforaed panel absorbers. Appled Acouscs 67 (6) [6] J. Lee G.. Swenson: Compac sound absorbers for low frequences. Nose Conrol Engneerng Journal 8 (99) 9-7 [7] K. Sakagam M. Mormoo M. Yar: A noe on he effec of vbraon of a mcroperforaed panel on s sound absorpon characerscs. Acouscal Scence and Technology 6 (5) -7 [8] Y. Y. Lee E.. M. Lee C. F. Ng: Sound absorpon of a fne flexble mcro-perforaed panel backed by an ar cavy. Journal of Sound and Vbraon 87 (5) 7- [9] M. Toyoda D. Takahash: Reducon of acousc radaon by mpedance conrol wh a perforaed absorber sysem. Journal of Sound and Vbraon 86 (5) 6-6 [] M. Toyoda M. Tanaka D. Takahash: Reducon of acousc radaon by perforaed board and honeycomb layer sysems. Appled Acouscs 68 (6) h INTERNATIONAL CONGRESS ON ACOUSTICS ICA7MADRID 6