ACOUSTIC CHARACTERISTICS OF INTERNAL SOUND FIELD IN CYLINDRICAL STRUCTURE WITH AN EXCITED END PLATE

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1 ACOUSTC CHARACTERSTCS OF NTERNAL SOUND FELD N CYLNDRCAL STRUCTURE WTH AN ECTED END LATE.Eng. Kojima A. rof. D.Eng. oriyama H. and rof. D.Eng. Ohinoya Y. Cour of Scinc and Tchnology Graduat School of Tokai Univrity Japan Dpartmnt of rim ovr Enginring Tokai Univrity Japan Abtract: n ordr to apply acoutic nrgy to indutrial fild it amplification i ntial and vibroacoutic coupling i on of bt way of amplifying acoutic nrgy. Th prnt papr dcrib vibroacoutic coupling btwn tructural vibration and th intrnal ound fild of thin tructur. A cylindrical tructur connctd two cylindr with thin plat i conidrd and coupling btwn th plat vibration and th intrnal ound fild i thortically and xprimntally invtigatd by conidring th bhavior of th plat and th acoutic charactritic of th intrnal ound fild with variation in th cylindr lngth whn an xtrnal harmonic forc i applid to on nd plat. Th acoutic charactritic ar valuatd bad on th ound prur lvl which i maximizd whil changing th pha diffrnc btwn th plat vibration. Th bhavior of th plat vibration i invtigatd by varying th pha diffrnc with th cylindr lngth. orovr th xprimnt of lctricity gnration i carrid out with pizolctric lmnt which ar intalld on th plat. Th rult rval that vibroacoutic coupling i ffctiv in incraing th acoutic nrgy and can xtract th acoutic nrgy via th plat vibration. Kyword: ROACOUSTC COULNG STRUCTURAL RATON NTERNAL SOUND FELD HASE DFFERENCE CONNECTED CYLNDERS ELECTRCTY GENERATON. ntroduction f a thin tructur having intrnal pac rciv an xtrnal priodic forc it i wll known that vibroacoutic coupling i caud btwn th tructural vibration and th intrnal ound fild. n gnral vibroacoutic coupling i conidrd to b on of phnomna which hould b avoidd from th viwpoint of tructural trngth and antinoi. On th othr hand vibroacoutic coupling i on of th availabl maur to amplify acoutic nrgy if it nrgy i applid to a ourc of om fild(). n an attmpt to control noi Chng and Nicola invtigatd coupling btwn th ound fild in an aircraft cabin and th vibration of th rar prur bulkhad(3). Thy adoptd a cylindrical tructur a th analytical modl in which th rar prur bulkhad at on nd of th cylindr wa aumd to b a circular plat. Thi analytical modl wa xamind undr a varity of condition. Th plat wa upportd at it dg by pring th tiffn of which could b adjutd to imulat variou upport condition. Th invtigation clarifid th influnc of th upport condition on th ound prur of an intrnal ound fild coupld with th vibration of th nd plat. Thy alo found a frquncy rang that gnrat an intn ound prur lvl. Almot all invtigation hav rfrd to a ingl cavity and hav conidrd coupling phnomna to b a noi control. n thi invtigation a cylindrical tructur with thin plat at both nd which partition a cavity into two caviti by th plat idntical to th nd plat i ud. Whn an xtrnal harmonic forc i applid to on nd plat coupling btwn th plat vibration and th intrnal ound fild i tudid thortically and xprimntally. Thi coupling i valuatd from th rlationhip btwn th vibration and acoutic charactritic. n th actual calculation th acoutic charactritic ar invtigatd bad on th ound prur lvl which i maximizd whil changing th pha diffrnc btwn th plat vibration. Th bhavior of th plat vibration i invtigatd by varying th pha diffrnc with th cylindr lngth. Th thortical rult ar vrifying xprimntally through an xcitation xprimnt uing an xprimntal apparatu that i mulatd by th analytical modl. orovr th xcitation xprimnt for th lctricity gnration i carrid out with pizolctric lmnt a th applocation of coupling phnomna.. Thortical and xprimntal mthod. Analytical modl Th analytical modl conidrd hrin conit of two caviti with thr circular nd plat a hown in Fig.. Th plat ar upportd by tranlational and rotational pring ditributd at contant intrval and th upport condition ar dtrmind by thir rpctiv pring tiffn T T T 3 and R R R 3 whr th uffix and 3 indicat plat and 3 rpctivly. Although th dimnion (i.. th radiu a and th thickn h) of th plat which hav Young modulu E and oion ratio ν and th cylindr lngth L ar fixd th cylindr lngth L i widly varid. Th ound fild i aumd to b cylindrical and to hav th am radiu a th plat and th lngth of th ound fild ar aumd to vary with th cylindr lngth. Th boundary condition ar conidrd to b tructurally and acoutically rigid at th latral wall btwn th tructur and th ound fild. Th coordinat ud ar th radiu r th angl θ btwn th plan of th plat and th cro-ctional plan of th caviti and th ditanc z and z along th rpctiv cylindr ax. Th harmonic point forc F i applid to plat at r dividd by a and θ i fixd at dg. Th natural frquncy of th plat i ud a it xcitation frquncy.. Coupling quipmnt btwn plat and cavity n ordr to formulat th plat motion Hamilton principl bad on th variational principl i applid to th analytical modl and th flxural diplacmnt w w and w 3 on plat and 3 ar xprd by ubtituting Eq. () for th plat mod hap into Eq. () and ar xpandd ovr thr t of uitabl trial function: w = w w 3 = = = n= m= = n= m= = n= m= 3 3 m = in( nθ π / )( r / a) () Fig. Configuration of th analytical modl () 9

2 whr n m and ar rpctivly th circumfrntial ordr th radial ordr and th ymmtry indx for th z and z ax of th vibration mod with rpct to th plat vibration. n addition and 3 ar th cofficint to b dtrmind ω i th angular frquncy of th harmonic point forc on th plat t i th lapd tim and φ φ and φ 3 ar th pha of th rpctiv plat vibration. Th quation of plat motion ar obtaind by finding th xtrmum of Hmilton function in trm of Eq. (). Th acoutic modal hap Y npq and Y npq inid th caviti and and tho angular ronanc frquncy ω npq (whr th indx n p and q indicat th circumfrntial radial and longitudinal ordr rpctivly) ar dfind a Y Y = in( nθ π / )( λnpr) co{ ( qπ / L ) } (3) npq z = in( nθ π / )( λnpr) co{ ( qπ / L ) } (4) npq z npq { λ ( qπ / L) } / ω = (5) c np whr J n i th nth-ordr l function and λ np i th pth olution of an ignvalu function for a circular ound fild having (np) mod dividd by th radiu and c i th pd of ound in th cavity. Th boundary condition btwn both plat vibration and ach ound fild on th rpctiv plat urfac ar found by auming continuity of vlociti on th plat: z = c = ρcω w = ρcω w c z = z = L c = ρcω w = ρcω w3 c z = L whr c and c ar rpctivry th ound prur ρ c i th fluid dnity in th cavity and u i th unit normal to th corrponding plat urfac uch that c / u and c / u ar on th latral wall of th cylindr. orovr in th prnt analyi w aum that thr i no ound ourc and hnc by uing th abov boundary condition and Grn function for a non-uniform cavity with non-rigid wall. c and c can b xpandd in trm of th acoutic mod coniting of a trio of ordr n p and q a follow: npqy c = wda wda = A A c = npq p= q= npq npqy wda w3da3 = A3 A p= q= npq (6) (7) (8) npq m' { K m' ( ) ω m' } afn T R m' = a a jφ jφ ω A ( ' ' ) = c m' = p= q= npq ( ωnpq jηcωnpqω ω ) jφ jφ ω A3 ( ' 33' ) ' = = = ( ω jη ω ω ω ) c c m p q npq m' = p= q= npq m' { K3m' ( ) ω 3m' } afn T3 R3 m' = a a jφ jφ ω A3 3 ( ' 33' ) = ( ω jη ω ω ω ) ' 3' jφ () jφ3 () whr K m' K m' K 3m' and m' m' 3m' ar componnt of th ymmtrical tiffn and ma matric rpctivly bcau th indic m and m' hav th rlation of th tranpoition in th matric. n addition η p i th tructural damping factor of th plat and F n i a cofficint that i dtrmind by indic n and. On th right-hand id of Eq. () th firt and cond trm xpr th point forc and coupling btwn both plat vibration and th ound fild rpctivly whra th right-hand id of Eq. () and.() contain only th coupling trm. Hr A and A 3 ar th total ara of th plat that touch ach ound fild and c and c ar th volum occupid by th rpctiv caviti. orovr and 3 th firt and cond uffix of which man th plat and th cavity ar th patial coupling cofficint drivd from th plat and acoutic modal hap and η c i th acoutical damping factor inid th caviti. Furthrmor inc ' ' and 3' can b obtaind from th abov quation th bhavior of plat vibration and th ound fild undr vibroacoutic coupling can b dtrmind..3 Exprimntal apparatu and mthod Figur how th xprimntal apparatu ud in th prnt tudy. Th cylindrical tructur conit of a tl cylindr with circular aluminum nd plat that ar 3 mm thick. Th cylindr ha innr radiu of 53 mm and th lngth can b varid from 5 to mm in ordr to mulat th analytical modl. A harmonic point forc xcit only on nd plat. Thi forc i applid to th plat by a mall vibrator th amplitud of which i controlld to b N. Th xcitation frqunci ar th natural frquncy f of th plat corrponding to th () mod obtaind from th xprimntal modal analyi. Th poition of th point forc r i whr G i Grn function A A and A 3 ar th urfac ara of th rpctiv plat and npq npq and npq npq ar rpctivly th prur cofficint to b dtrmind and th modal gnralizd ma of th nth acoutic mod for ach cavity. Whn th plat i xcitd by point forc th coupling quation btwn both plat vibtration and ach ound fild ar xprd a(4) (9) { K m' ( ) ω m' } m' = = F ω A c af n m' T R a a jφ jφ ( ' ' ) ( ω jη ω ω ω ) m' = p= q= npq ' jφ () :ibration gnrator :Load cll 3:Acclration nor 4:Condnr microphon 5:Amplifir 6:FFT analyzr Fig. Configuration of th xprimntal apparatu

3 φ [dg] 8 5 φ[dg] φ xp φmax 5 5 L[mm] φ min φ xp φ min 5 5 (a) ha diffrnc (a) Whn diplacmnt i minimizd L pv L p [d] 3 L p φ[dg] 8 5 φ xp φ max L pv φ max L[mm] 5 5 (b) Sound prur lvl (b) Whn diplacmnt i maximizd Fig. 3 Comparion btwn thortical and xprimntal rult for pha diffrnc and ound prur lvl at th ingl cylindr Fig. 4 Comparion btwn thortical and xprimntal pha diffrnc at ingl ylindr normalizd by radiu a and i t to r /a =.4. n th prnt tudy th main charactritic of th plat vibration undr conidration i th pha diffrnc btwn th plat vibration. Thrfor acclration nor ar intalld on thr plat to maur thi pha diffrnc. n ordr to timat th intrnal acoutic charactritic th ound prur lvl in th cavity i maurd uing a condnr microphon with a prob tub th tip of which i locatd in th vicinity of th non-xcitd plat and th cylindr wall which i th approximat location of th maximum ound prur lvl. 3. Rult and dicuion 3. Fundamntal coupling charactritic n th prnt tudy th plat ar aumd to b aluminum having a Young modulu E of 7 Ga and a oion ratio ν of.33. Th plat hav a radiu a of 53 mm and a thickn h of 3 mm and th cylindr lngth vari from to mm. Th upport condition of th plat which hav flxural rigidity D [= Eh 3 /{( ν )}] ar xprd by th non-dimnional tiffn paramtr T n (= T a 3 /D = T a 3 /D) and R n (= R a/d = R a/d). n thi analyi T n and R n ar t to 8 and rpctivly to mak th valu cho to th actual upport condition in th xprimntal apparatu and th valu ar idntical for all of plat. Th point forc F i t to N and i applid to r /a =.4 on th plat th am a th xprimnt. Th ronanc frquncy of th cylindrical ound fild i rprntd by f npq { i.. th natural frquncy corrponding to th (npq) mod }. Figur 3(a) how th thortical and xprimntal pha diffrnc btwn th vibration of th two plat a function of cylindr lngth L. Th xcitation frquncy f i t to 8 Hz in ordr to induc th () mod for a plat in thi analyi. Th xprimntal valu of f i 8 Hz. n th thortical rult φ max at which th ound prur lvl i maximizd i approximatly 87 dg at L = mm and dcra gradually with incraing L up to approximatly L = 46 mm whr φ max uddnly incra to ovr 9 dg and thn dcra with incraing L again. Thi bhavior of φ max i rpatd in a imilar mannr a L incra to L = mm. n th xprimntal rult th valu of φ xp rmain approximatly contant at 8 dg whn L i xtndd from 55 to 5 mm but fall rapidly to approximatly dg and rmain clo to dg for valu of L up to 75 mm. yond L = 75 mm φ xp again incra rapidly to approximatly 8 dg. Th pha diffrnc thn rmain almot contant up to L = mm. Figur 3(b) how th variation in L pv avragd ovr th ntir ound fild corrponding to φ max in th analyi and in th ound prur lvl L p maurd in th xprimnt. ak in L pv for φ max appar at L = 6 3 and 84 mm at which φ max i approximatly 9 dg. Th pak ar caud by th () () and (3) mod rpctivly. ak in L pv indicat that vibroacoutic coupling btwn th plat vibration and th ound fild i promotd. Not that f and f npq mut b approximatly qual for pak in L pv to appar. Hr L p incra ignificantly at 55 and 85 mm. n addition φ xp xhibit rapid chang around th valu of L and th φ max appar at th midpoint of th rapid chang in φ xp. Sinc L pv i caud by vibroacoutic coupling th magnitud of th flxural diplacmnt w and w i alo ignificant in tudying th ffct of w and w on L pv. n Fig. 4(a) th pha diffrnc whn w and w ar minimizd dnotd a φ min and φ min rpctivly ar conidrd. Hr φ min i contant at dg for L ranging from to 48 mm and incra abruptly up to 8 dg

4 φ[dg] L pv L p [dg] L [mm] 4 8 φ xp (a) ha diffrnc (b) Sound prur lvl Fig.5 Comparion btwn thortical and xprimntal rult for pha diffrnc and ound prur lvl at connctd cylindr at L = 49 mm. Thn φ min rmain contant at 8 dg up to L = 6 mm dcra gradually with L and rturn to dg at L = 4 mm dcraing omwhat abruptly in th vicinity of L = 8 mm. yond L = 4 mm φ min i again contant at dg up to L = 69 mm and thi bhavior i rpatd a L incra to L = mm. On th othr hand φ min xhibit gradual and abrupt chang that ar imilar to th chang in φ min. Howvr th chang occur in th altrnat rang of th chang in φ min and ar in th oppoit dirction to th chang in φ min. For xampl a gradual incra occur btwn L = and 6 mm and an abrupt dcra occur in th vicinity of L = 8 mm. oth φ min and φ min hift btwn and 8 dg with th chang in L and intrct at approximatly 9 dg and nar th lngth at which L pv pakd in Fig. 3(b). n addition φ xp hift btwn and 8 dg with th chang in L and approximatly corrpond with φ min and φ min. Howvr φ xp i diffrnt from φ min and φ min btwn L = 5 and 55 mm btwn L = and mm and btwn L = 7 and 85 mm. Figur 4(b) how variation in th pha diffrnc φ max and φ max with L whn w and w rpctivly ar maximizd. Not that φ max and φ max bhav in th oppoit mannr to φ min and φ min and o ar ignificantly diffrnt from φ xp in th almot ntir rang of L. Howvr φ max and φ max corrpond with φ xp in th abov rang of L in which φ min and φ min diffr from φ xp. Th rapid chang in φ xp imply that φ xp i clo to φ max if pha diffrnc ar obrvd in th vicinity of th valu of L around which L pv and L p ar maximizd. 3. Coupling phnomna of connctd cylindr Hr connctd cylindr having two caviti into which th abov cylindr partition th cavity by th plat ar tudid. n th analyi th lngth L i fixd to 6 mm at which th ound prur lvl L pv pak in th cavity whil L rang from φ C L pv φ xp L p L p φ C L pv 5 5 L [mm] mm to mm in th cavity. n th actual xprimnt L rmain 6 mm lik th analytical modl whra L i in th rang btwn 5 mm and 4 mm bcau th total lngth L of th connctd cylindr i rtrictd by th dimnion of th xprimntal apparatu and th firt and cond pak of L p appar in that rang. Figur 5(a) how th thortical rult with th pha diffrnc φ max btwn th vibration of th plat and and φ max btwn tho of th plat and 3 a function of L. L pv and L pv pak at φ c and φ c rpctivly lik φ max at which L pv pakd. φ c i in th proximity of 9 dg ovr th ntir rang of L inc th firt pak of L pv i maintaind du to L = 6 mm vn if L i widly changd. φ c vari priodically with L and xhibit a trnd imilar to φ max in Fig.3(a) bing omwhat rratic in comparion with φ max of th ingl cylindr. n thi figur th pha diffrnc φ xp and φ xp maurd btwn th plat vibration with which th rpctiv ound fild ar coupld ar alo plottd to dmontrat th abov thortical rult. φ xp hift toward th in-pha id rmaining approximatly contant lik φ max. Th xcntric bhavior of φ xp i caud by intability of pha diffrnc whn th ound prur lvl i maximizd o that it i o difficult to confirm xprimntally uch a pha diffrnc. Dcraing abruptly with incraing L btwn 65 mm and 69 mm and incraing abruptly btwn 9 mm and 9 mm φ xp hift btwn both id of in-pha and out-of-pha with th conqunc that φ xp i almot uniform in th rang btwn both abrupt hift having a light variation. Such a chang in φ xp with L i imilar to that of φ xp in Fig.3(a) inc φ xp ha not only abrupt hift but intrction with φ c at around 9 dg bing omwhat gntl in abrupt hift and compltly oppoit with rpct to th apparanc in th viciniti of th in-pha and out-of pha id. Figur 5 (b) how th ound prur lvl corrponding to th pha diffrnc in Fig.5 (a). Th thortical rult L pv and L pv corrpond to φ c and φ c rpctivly. For th cylindr lngth of 6 mm at which L pv i alway maximizd L pv ha a priodic variation and maintain a rlativly grat valu ovr th ntir rang of L. aking at L =6 3 and 84 mm L pv how th am bhavior a th ingl cylindr in which L pv i about 5 d largr than L pv. Th acoutic charactritic inid th cavity bring th priodic variation to th ound fild of th cavity influncing that of th cavity via th vibration of th plat. Th xprimntal rult L p and L p corrpond to φ xp and φ xp rpctivly. L p rmain almot contant in th rang of L ovr which it i maurd falling a littl in th viciniti of L = 6 mm and 4 mm which L p pak clarly at tho lngth. L p and L p bhav in a mannr imilar to L pv and L pv and thn th pak of L p and L p appar at around 9 dg whr φ c and φ xp intrctd. n thi thortical approach bcau th acoutic nrgy i not aumd to pa through th plat th traiion at th plat influnc mor trongly th ound fild of th cavity than that at th othr plat o that th pak of L p bcom dullr than tho of L pv around which chang in φ xp wr mor gradual than tho in φ c. Th xprimntal rult dmontrat that th thortical timation i jutifid to om xtnd. 3.3 Exprimnt of lctricity gnration Hr an lctricity gnration i conidrd a th application of thi invtigation. Th lctricity i gnratd by pizolctric lmnt which ar intalld on both nd plat of th ingl cylindr. Figur 6 how voltag E and E of th lctricity upplid from th rpctiv plat vibration a function of th cylindr lngth L. Th cylindr lngth rang from mm to 35 mm in which th cond pak of th ound prur lvl L p appard and i changd at intrval of mm to tudy gnration charactritic in dtail. E do not vary gratly inc th contant point forc i applid to th plat whil E i maximizd in th vicinity of L= 5 mm at which L p rachd th cond pak. Th valu of E i almot qual to th valu of E or mor than that at around L = 5mm bing much blow that xcpt nar it lngth.

5 E [].8.6 E E α [m/ ] α α Fig. 6 oltag of lctricity gnratd by pizolctric lmnt at ingl cylindr Fig. 7 Acclration of nd plat at ingl cylindr n particular th bhavior of E i imilar to that of L p. Naturally th plat vibration rlat trongly to th ound fild in charactritic coupld with th ound fild. Thrfor th acclration α and α ar maurd on th plat and rpctivly imultanouly with maurmnt of th voltag. Figur 7 how variation in α and α with L. α and α bhav in a mannr imilar to E and E rpctivly and incra in α in th that rang in which L p and E incra ar rmarkabl. Th rult rval that acoutic nrgy which hould b fundamntally diipatd into th nviront can b xtractd by vibroacoutic coupling. 4. Concluion n thi invtigation th connctd cylindr with th thin plat wa conidrd and vibroacoutic coupling btwn th plat vibration and th intrnal ound fild wa thortically and xprimntally tudid. n particular a th application of thi invtigation th xprimnt of lctricity gnration wa carrid out with th pizolctric lmnt. Th thortical tudy how that th ound prur lvl i maximizd whn th pha diffrnc i clo to 9 dg and in th xprimnt th ound prur lvl i maximizd in proc of rapid chang in a pha diffrnc with varying th cylindr lngth. Th rult dmontrat th accuracy of th thortical rult. n th xprimnt of lctricity gnration whn vibroacoutic coupling i promotd it i confirmd that voltag on th id without a point forc i almot qual to voltag on th othr id or mor than that. Th rult rval that vibroacoutic coupling can xtract th acoutic nrgy via th plat vibration. 5. Rfrnc () K. Hayamizu AndoR. and TakfujiY. Simultanou providing dvic of baband and carrir ignal uing ound-gnratd lctricity (Japan) obil ultimdia Communication No.5-8 (5-5) pp () L. Chng and Nicola J. Radiation of ound into a cylindrical nclour from a point-drivn nd plat with gnral boundary condition Journal of th Acoutical Socity of Amrica ol.9 No.3 (99) pp (3) L. Chng Fluid-tructural coupling of a plat-ndd cylindrical hll: vibration and intrnal ound fild Journal of Sound and ibration. ol.74 No.5 (994) pp (4) H. oriyama odal charactritic of ound fild in cylindrical nclour with nd plat upportd by latic urround (Japan) Th Journal of th Acoutical Socity of Japan ol.58 No.6 () pp