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1 Thank you fo downloading this doument fom the RMIT ReseahR Repositoy Citation: Davy, J, Mahn, J, Kaji, L, Waeing, R and Pease, J 205, 'Compaison of theoetial peditions of adiation effiieny with expeimental measuements', in Malolm J. Coke, Maek Pawelzyk, Fanesa Pedielli, Eleonoa Caletti, Segio Luzzi (ed.) Poeedings of the 22nd Intenational Congess on Sound and Vibation (ICSV22 205), Feaa, Italy, 2- See this eod in the RMIT Reseah Repositoy at: Vesion: Aepted Manusipt Copyight Statement: Copyight Intenational Institute of Aoustis and Vibation (IIAV), 205 Link to Published Vesion: PLEASE DO NOT REMOVE THIS PAGE

2 COMPARISON OF THEORETICAL PREDICTIONS OF RADIATION EFFICIENCY WITH EXPERIMENTAL MEASUREMENTS John Lauene Davy Shool of Applied Sienes, RMIT Univesity, GPO Box 2476, Melboune, Vitoia 300, Austalia Jeffey Mahn National Reseah Counil of Canada, 200 Monteal Rd, Ottawa ON KA 0R6, Canada Luboš Kajči Soundthem, Aoustis and Physis of the Buildings, Batle 9, 8255 Shlattingen, Switzeland Robin Waeing and John Pease Univesity of Cantebuy, Mehanial Engineeing, Pivate Bag 4800, Chisthuh 840, New Zealand The measuement and pedition of adiation effiieny is impotant in many aeas of aoustis. It has eently beome an even moe impotant topi, beause the adiation effiieny needs to be known in ode to pedit the flanking sound tansmission using EN2354 when the fequeny is below the itial fequeny of one o moe of the walls involved in the flanking sound tansmission path. This pedition is moe diffiult fo lightweight walls beause they usually onsist of a omplex onstution. This pape uses Maidanik s and Leppington s appoximate fomulae to pedit the esonant adiation effiieny of a finite size panel and Davy s appoximate fomulae to pedit the foed adiation effiieny when a finite size panel is exited by a diffuse sound field. The adiation effiieny of an infinite size panel exited by a point o line foe is pedited using Hekl s appoximate fomulae. The damping loss fato of the panel is needed in ode to alulate the magnitude of the esonant vibation of the panel elative to the point o line nea field vibation of the panel o elative to the vibation of the panel foed by an inident diffuse sound field on one side of the panel. The theoies used in this pape assume that the finite size panel is mounted in an infinite baffle and the peditions ae fo the powe adiated on one side of the panel. This pape ompaes the theoetial peditions of adiation effiieny with expeimental measuements made in sound insulation laboatoies. The sufae veloities of the panels wee measued with a lase veloimete o aeleometes. The sound powe adiated on one side of the panel was detemined using sound intensity measuements o evebeation oom measuements of sound powe that utilised sound pessue measuements and evebeation time measuements. ICSV22, Floene (Italy) 2-6 July 205

3 The 22 nd Intenational Congess on Sound and Vibation. Intodution The adiation effiieny of a wall needs to be known in ode to make many peditions in aoustis. Fo example, the attempt to extend the EN 2354 flanking sound tansmission standads [, 2] to fequenies below the itial fequeny of the walls involved has highlighted this need to know the adiation effiienies of the walls. This pape pesents appoximate fomulae fo the adiation effiienies of etangula panels and ompaes the peditions of these fomulae with expeimental measuements. 2. The pedition of the adiation effiieny of a etangula panel This setion gives appoximate fomulae fo the single sided adiation effiienies of a simply suppoted etangula panel mounted in an infinite igid baffle. Define thee empiial onstants [3]. () n 2 (2) w.3 (3) 0.24 Define the length of the side of an equivalent squae panel as [3]: 4S (4) 2a U whee S is the aea of the panel and U is the peimete of the etangula panel. Using the wave numbe: 2 f (5) k whee is the speed of sound in ai and f is the fequeny, alulate some intemediate values [3]: w if w 2ka 2ka (6) p if w 2 ka (7) h 2 2ka 3 h (8) p 2 (9) q 2 ks Calulate the non-esonant diffuse field exited adiation effiieny [3] q h h q (0) n ln ln p p q p p q Calulate some moe intemediate values [3]: f () t f (2) if f f g t 0 if f f ICSV22, Floene, Italy, 2-6 July 205 2

4 The 22 nd Intenational Congess on Sound and Vibation whee f is the itial fequeny of the panel [3]. if g p n n n g q (3) 2 if p g 0 n n n h g q If f < f, alulate σ [4]. (4) 2 3/2 2 a f ( t) [( t)ln ( t ) / ( t ) 2 t] Note that Maidanik s [5] exta low fequeny tem has not been used. Calulate the esonant adiation effiieny. min, 2 if f f (5) 2 if f f Calulate anothe intemediate value (6) 4t whee η is the total in situ damping loss fato. Note that is the atio of the esonant vibational enegy to the non-esonant vibational enegy level of a panel whih has been exited by a diffuse sound field [6]. The vaiable is also the atio of the powe adiated by the esonant vibational fields to the powe adiated by the vibational nea fields fo a panel exited by point foes ating at ight angles to the panel [7]. Calulate the aibone diffuse field exited adiation effiieny (see equation () of [8]). n if f f (7) a if f f Calulate the adiation effiieny of a panel exited by point foes ating at ight angles to the panel (see equation (28) of [8]). if f f (8) p if f f Calulate the atio of the powe adiated by the esonant vibational fields to the powe adiated by the vibational nea fields fo a panel exited by line foes ating at ight angles to the panel [7]. f (9) l 2 f Calulate the adiation effiieny of a panel exited by line foes ating at ight angles to the panel. if f f (20) l l if f f 3. Expeimental ompaison The adiation effiienies wee measued fo panel of size.546 by 0.95 m mounted between a ICSV22, Floene, Italy, 2-6 July 205 3

5 The 22 nd Intenational Congess on Sound and Vibation evebeation oom and a semi-anehoi oom. The aveage adiated sound intensity was measued on the semi-anehoi oom side of the panel. The aveage veloity of the panel was measued with aeleometes. The panels wee exited by a diffuse sound field in the evebeation oom to measue σ a. To measue σ, anothe simila panel was attahed at ight angles to the oiginal panel in the evebeation oom and this panel was exited by a shake. The damping loss fatos wee measued using the stutual evebeation time [9]. 0 0 σ theoy σ exp Figue. Compaison of the theoetial and expeimental diffuse field exited (σ a ) and esonant adiation (σ ) effiienies fo a.6 mm thik steel panel measuing.546 by 0.95 m. The expeimental 95% onfidene limits ae shown σ theoy 0 σ exp Figue 2. Compaison of the theoetial and expeimental diffuse field exited (σ a ) and esonant adiation (σ ) effiienies fo a 4 mm thik medium density fibe boad panel measuing.546 by 0.95 m. The expeimental 95% onfidene limits ae shown. Figues to 3 ompae the pedited and measued values fo a.6 mm thik steel panel, 4 mm medium density fibe boad panel and a 0 mm gypsum plaste boad panel, espetively. Only σ a was measued in the ase of the gypsum plaste boad panel. The pedited values ae usually out- ICSV22, Floene, Italy, 2-6 July 205 4

6 The 22 nd Intenational Congess on Sound and Vibation side the 95% onfidene limits, but thee is vey ough ageement exept in the ase of σ fo the medium density fibe boad panel. These panels all had elatively high itial fequenies Figue 3. Compaison of the theoetial and expeimental diffuse field exited (σ a ) effiienies fo a 0 mm thik gypsum plasteboad panel measuing.546 by 0.95 m. The expeimental 95% onfidene limits ae shown Figue 4. Compaison of the theoetial and expeimental diffuse field exited (σ a ) and point exited adiation (σ p ) effiienies fo an 80 mm oss timbe laminated panel measuing 4.8 by 2.89 m. Thee ae two expeimental measuements of the point exited adiation effiieny. Figue 4 ompaes the pedited and measued values fo an 80 mm oss laminated timbe panel measuing 4.8 by 2.89 m mounted between two evebeation ooms. Its adiated powe was measued in a evebeation oom and its aveage veloity was measued using a lase veloimete. Its damping loss fato was measued using stutual evebeation time. It was exited with a diffuse sound field fo σ a and with a shake fo σ p. Thee wee two measuements of σ p [0]. Again thee was vey ough ageement between theoy and expeiment. This is a easonable outome sine the two measuements of σ p wee only in ough ageement with eah othe. This is a wall with a lowe itial fequeny than the fist thee walls onsideed in this pape. 0. σp theoy σp exp σp exp2 ICSV22, Floene, Italy, 2-6 July 205 5

7 The 22 nd Intenational Congess on Sound and Vibation Figues 5 to 7 ompae theoy and expeimental fo two 3 mm gypsum plaste boad double leaf avity wood stud walls mounted at ight angles to eah othe to fom an L shape [0]. The 50 by 00 mm wood studs wee plaed at 600 mm entes. Thee was no sound absobing mateial in the wall avities exept fo the ase of the ight hand gaph in fig 7. The measuement tehniques wee the same as those desibed fo figs. to 3. To obtain easonable ageement, the width of the wall in the theoetial peditions was set equal to the stud spaing following Maidanik s eommendation fo ibbed panels Figue 5 shows σ a and σ p fo a wall measuing 4.68 by 2.4 m mounted between a evebeation oom and a semi-anehoi oom. This wall was exited by a diffuse sound field and a shake. Beause the studs and sews an be egaded as point and line onnetions between the two wall leaves, the theoetial values of σ, σ p and σ l ae plotted. The theoetial value of σ l gives the best ageement with the expeimental value of σ σ theoy σp exp σp theoy σl theoy Figue 5. Compaison of the theoetial and expeimental diffuse field exited (σ a ) and point exited adiation (σ p ) effiienies fo a 3 mm gypsum plaste boad double leaf avity wood stud wall measuing 4.68 by 2.4 m. The 50 by 00 mm wood studs wee plaed at 600 mm entes. Thee was no sound absobing mateial in the wall avity. The expeimental 95% onfidene limits ae shown. 0 σ theoy σm exp σp theoy σl theoy Figue 6. Compaison of the theoetial and expeimental esonant adiation effiienies fo a 3 mm gypsum plaste boad double leaf avity wood stud wall measuing by m. The 50 by 00 mm wood studs wee plaed at 600 mm entes. Thee was no sound absobing mateial in the wall avity. The wall was exited via its ight angled onnetion with anothe wall. This othe wall was exited by a diffuse sound field (σ a ) and mehanially by a tansvese point foe (σ m ). The expeimental 95% onfidene limits ae shown. ICSV22, Floene, Italy, 2-6 July 205 6

8 The 22 nd Intenational Congess on Sound and Vibation Figue 6 shows σ fo the wall measuing by m mounted in the semi-anehoi oom and onneted at ight angles to the pevious wall. It was exited by the fist wall whih was exited by a diffuse sound field (σ a ) o by a shake (σ m ). Again the theoetial value of σ l gives the best ageement with the expeimental value of σ σp theoy σpos exp σpfs exp σpos2 exp σpfs2 exp σpos3 exp σpfs3 exp 0 0. σp theoy σpos exp σpfs exp σpos2 exp σpfs2 exp Figue 7. Compaison of the theoetial and expeimental point exited adiation effiienies (σ p ) fo a 3 mm gypsum plaste boad double leaf avity wood stud wall measuing by m. The 50 by 00 mm wood studs wee plaed at 600 mm entes. Thee was no sound absobing mateial in the wall avity fo the esults in the left hand gaph while thee was sound absobing mateial in the wall avity fo the esults shown in the ight hand gaph σa2 exp Figue 8. Compaison of the theoetial and expeimental diffuse field exited (σ a ) effiienies fo a 7 mm thik plywood panel and a 2 mm thik plywood panel measuing 4.8 by 2.4 m. Figue 7 shows the ase of the seond avity wall when it was exited at diffeent positions by a shake. The notations pos and pfs denote exitation on and off a stud loation espetively. Thee is little diffeene between on stud and off stud exitation exept at low fequenies. The ageement between theoy and expeiment is bette with these lage and moe omplex walls and is bette fo wall with sound absobing mateial in its wall avity. Figue 8 shows the ompaison between theoy and expeiment fo σ a fo 7 mm and 2 mm plywood panels measuing 4.8 by 2.4 m. The Young s modulus and damping loss fato wee detemined by measuements on beams. The Young s modulus was the geometi mean of the values in the oss ply and paallel ply dietions. The measuement tehnique was the same as that used to ICSV22, Floene, Italy, 2-6 July 205 7

9 The 22 nd Intenational Congess on Sound and Vibation obtain figs to 3. Again thee is ough ageement between theoy and expeiment exept at the lowest fequenies. 4. Conlusions This pape has pesented appoximate fomulae fo pediting the esonant, point exited, line exited and diffuse field exited adiation effiienies fo a simply suppoted etangula panel mounted in an infinite baffle. These fomulae have been ompaed with the measued adiation effiienies of single leaf panels and double leaf avity stud walls. Thee is ough ageement between the theoetial fomulae and the expeimental measuements. Supisingly the ageement is slightly bette fo the moe ompliated double leaf avity stud walls. Fo avity stud walls it is neessay to use the stud spaing athe the wall length when pediting the theoetial adiation effiieny. The damping loss fato is an impotant paamete beause it defines the elative amplitude of the esonant and non-esonant adiation vibational fields in the adiating wall panels. The esonant adiation effiieny depends on the bounday onditions of the adiating wall panel below the itial fequeny. The exat bounday onditions of the wall panels ae not nomally known vey auately. Ineased damping below the itial fequeny will deease the elative amplitude of the esonant vibation field and thus would be expeted to inease the auay of the theoetial peditions. REFERENCES ISO 572-:2005(E) Building aoustis - Estimation of aousti pefomane of buildings fom the pefomane of elements - Pat : Aibone sound insulation between ooms, Intenational Oganization fo Standadization, Geneva, Switzeland, (2005). 2 ISO 572-2:2005(E) Building aoustis - Estimation of aousti pefomane of buildings fom the pefomane of elements - Pat 2: Impat sound insulation between ooms, Intenational Oganization fo Standadization, Geneva, Switzeland, (2005). 3 Davy, J. L. The foed adiation effiieny of finite size flat panels whih ae exited by inident sound, Jounal of the Aoustial Soiety of Ameia, 26 (2), , (2009). 4 Leppington, F. G., Boadbent, E. G. and Heon, K. H. The aousti adiation effiieny of etangula panels, Poeedings of the Royal Soiety of London Seies a-mathematial Physial and Engineeing Sienes, 382, , (982). 5 Maidanik, G. Response of ibbed panels to evebeant aousti fields, Jounal of the Aoustial Soiety of Ameia, 34 (6), , (962). 6 Coke, M. J. and Pie, A. J. Sound tansmission using statistial enegy analysis, Jounal of Sound and Vibation, 9 (3), , (969). 7 Davy, J. L. Sound tansmission of avity walls due to stutue bone tansmission via point and line onnetions, Jounal of the Aoustial Soiety of Ameia, 32 (2), 84-82, (202). 8 Davy, J. L., Mahn, J. P., Guigou-Cate, C. and Villot, M. The pedition of flanking sound tansmission below the itial fequeny, Jounal of the Aoustial Soiety of Ameia, 32 (4), , (202). 9 Mahn, J. P. and Pease, J. R. The alulation of the esonant sound edution index fo use in EN2354, Euonoise 202, Pague, Czeh Republi, 0-3 June, (202). 0 Kajči, L., Hopkins, C., Davy, J. L. and Töbs, H.-M. Aibone sound tansmission of a osslaminated timbe plate with othotopi stiffness, Euonoise 202, Pague, Czeh Republi, 0-3 June, (202). ICSV22, Floene, Italy, 2-6 July 205 8