I. INTRODUCTION J. Acoust. Soc. Am. 112 (6), December /2002/112(6)/2840/9/$ Acoustical Society of America

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1 Reduction of ound trnmiion into circulr cylindricl hell uing ditributed vibrtion borber nd Helmholtz reontor Simon J. Etève ) nd Mrty E. Johnon Vibrtion nd Acoutic Lbortorie, Deprtment of Mechnicl Engineering, Virgini Polytechnic Intitute nd Stte Univerity, Blckburg, Virgini Received 4 Mrch 2002; revied 29 July 2002; ccepted 22 Augut 2002 A modl expnion method i ued to model cylindricl encloure excited by n ernl plne wve. A et of ditributed vibrtion borber DVA nd Helmholtz reontor HR re pplied to the tructure to control the interior coutic level. Uing n impednce mtching method, the tructure, the coutic cvity, nd the noie reduction device re fully coupled to yield n nlyticl formultion of the tructurl kinetic energy nd coutic potentil energy of treted cylindricl cvity. Lightweight DVA nd mll HR tuned to the nturl frequencie of the trgeted tructurl nd coutic mode, repectively, reult in ignificnt coutic nd tructurl ttenution when the device re optimlly dmped. Simultion how tht ignificnt interior noie reduction cn only be chieved by dding dmping to both tructurl nd coutic mode, which re reonnt in the frequency bndwidth of interet. In order to be independent of the zimuth ngle of the excittion nd to void unwnted modl interction, the device re ditributed evenly round the cylinder in ring. Thi tretment cn only chieve good performnce if the tructure nd the coutic cvity re lightly dmped Acouticl Society of Americ. DOI: / PACS number: Tm, At, Sk ANN I. INTRODUCTION Lrge cylindricl tructure re common in the eropce indutry. The reduction of noie trnmitted into uch encloure i prticulrly chllenging due to the high excittion level, the complex nture of the diturbnce, nd the evere m nd volume contrint impoed on the deign of the tretment. The development of lightweight tructure, mde of compoite mteril, h lowered the coutic trnmiion lo of uch tructure nd therefore further increed the coutic trnmiion problem. The trnmiion of ound into n encloed coutic cvity cn be implified into four tge: i coupling between the ernl noie nd the tructure, ii tructurl vibrtion, iii coupling between the tructurl vibrtion nd the interior fluid, nd iv the ound in the interior. Control tretment cn operte t ome or ll of thee tge in order to chieve ttenution of the interior noie level. Control device cn be ttched to the tructure in order to directly reduce the vibrtion level tge ii nd in order to reduce the tructurl coutic coupling tge i nd iii. For exmple, Hung nd Fuller 1,2 ued dynmic borber to reduce the interior ound level in n ircrft fuelge, t ingle excittion frequency, by reducing the tructurl velocity. Guigou et l. 3 ended thi work by detuning the borber in order to reduce the coupling between the tructure nd the interior ound field. For brodbnd ppliction it become difficult to reduce the tructurl coutic coupling uing lightweight tretment decoupling t one frequency tend to incree coupling t nother. Grdonio et l. 4 hd ome ucce chieving thi uing blocking Electronic mil: eteve@vt.edu me plced on the tructure but hd to ume ome knowledge of the ngle of the incident noie field. Jolly nd Sun 5 ued vibrtion borber to reduce the rdition of brodbnd noie from vibrting pnel nd Ngy nd Li 6 exmined the optimiztion of n borber tretment pplied to rditing plte uing neurl network but neither conidered rdition into cvity. The other pproch to control the ound trnmiion i to directly tret the ound in the encloure tge iv. Aborptive mteril, uch coutic blnket, perform well in the high-frequency rnge, but re unuitble for lowfrequency control due to the volume nd m contrint impoed in eropce ppliction. However, coutic ttenution cn be obtined in the low-frequency rnge by the ue of Helmholtz reontor HR. Fhy nd Schofield 7 invetigted the interction between ingle optimlly dmped HR nd n coutic mode in n encloure nd Cumming 8 ended the nlyi to reontor rry nd it effect on the ound field in cvity. Alo, Dori 9 tried to broden the functionl frequency rnge of HR by uing reontor with multiple nturl frequencie. In ll thee tudie, the coutic diturbnce i generted by n rbitrry ource ditribution inide the cvity nd i not excited by tructure. The contribution of thi work lie in the imultneou ppliction of both tructurl nd coutic control device to fully coupled tructurl-coutic ytem tge ii nd iv. Multiple optimlly dmped ditributed vibrtion borber DVA nd HR re pplied to control the ound trnmiion in cylindricl encloure over brod frequency rnge contining mny tructurl nd coutic reonnce. A conventionl modl expnion method 4 i ued to decribe the behvior of the cylindricl hell, excited by n ernl coutic plne wve, nd it coutic cvity. The 2840 J. Acout. Soc. Am. 112 (6), December /2002/112(6)/2840/9/$ Acouticl Society of Americ

2 tudy, the tndrd Donnell Muhtri theory, even though inccurte for the low circumferentil wve number, give ufficient firt pproximtion of the reonnt frequencie. The hpe of the (n,m ) tructurl mode hpe n m the upercript or ubcript ignifie tht the vrible refer to the tructure i given by n m z, 1 n m in n z L com inm, 1 FIG. 1. Tet cylinder mounted in n infinite bffle nd excited by n ernl coutic plne wve treted with HR nd DVA. optimiztion of the DVA tretment bed on the tudie by Johnon et l. 10,11 i ended by nlogy to the HR tretment. All the element re then fully coupled uing n impednce mtching method to compute the interior coutic ttenution provided by the noie reduction device. II. THEORY In thi ection the nlyticl formultion of the problem i introduced. The ytem being modeled i hown in Fig. 1. It i contituted of imply upported cylinder embedded in n infinite rigid bffle excited by n coutic plne wve. To control the vibrtion of the cylinder, DVA cn be ttched nywhere on it urfce except the top nd bottom dik, which re not excited by the erior coutic field due to the preence of the bffle. HR cn be plced nywhere inide the cylinder, but in order to mintin imple model for the coutic cvity they re umed to lie outide nd to couple to the encloed fluid t the circumference of the tructure. The behvior of the tructure nd the coutic cvity i decribed uing modl pproch. The ytem i put in mtrix form, nd uing n impednce mtching method i then fully coupled with the noie reduction device. Once the necery component of the model re defined, the expreion for the vibrtion nd interior coutic repone of the ytem i derived. A. Cylinder tructurl model The dynmic behvior of thin cylindricl hell h generted multitude of theory bed on different umption nd pproximtion. The comprion of thee different theorie h lo been ubject to vluble tudie, uch the work done by Lei. 12 For the purpoe of thi work, the cylinder i umed to be thin, iotropic, nd mde out of homogeneou mteril whoe mechnicl propertie re djuted in order to mtch the behvior of n experimentl compoite prototype. Therefore within the frmework of thi where the mode order re 1n n mx,0m m mx, L i the length of the cylinder, n m i the normliztion fctor of the (n,m ) mode, uch tht S ( n m ) 2 dss, nd S denote the urfce of the cylinder. Becue of their ymmetry, cylinder hve circumferentil mode whoe orienttion depend on the excittion loction. Therefore, the cylinder i conidered to hve two orthogonl circumferentil mode of the me order m : one ine nd one coine. Thee two mode re independent nd lthough they hve the me reonnt frequency, the incident coutic field excite them differently. Thi i equivlent to ingle mode with n orienttion ngle tht chnge depending on the excittion. The out-of-plne velocity w cn be decribed modl ummtion by w,z, v n m n m N,z, 2 where v n m i the complex mode mplitude, nd N i the totl number of tructurl mode conidered. Once the tructurl nturl frequency n m i obtined, the modl mobility of the cylinder A n m i derived uing the econd-order ytem eqution: i A n m 2 M n m 2 2 n m i n m, 3 where M i the m of the cylinder, n m the modl dmping rtio, nd A n m denote modl velocity over modl force. In the imultion, the modl dmping rtio re djuted to be repreenttive of the obervble dmping level in rel compoite cylinder. In the ce of tructure vibrting in dene fluid uch ubmrine, the effect of fluid loding or rdition loding on the tructure dynmic mut be tken into ccount. In the preent ce, the ernl rdition loding cn be neglected becue of the low denity of ir. The complex mplitude v n m of ech tructurl mode i obtined by multiplying the mobility of the cylinder A n m by the totl modl force F n m pplied to the cylinder. Writing thi in mtrix form yield va F, 4 where A i n N N digonl mtrix of modl mobilitie obtined with Eq. 3, F i n N 1 vector of modl force, nd v i n N 1 vector of tructurl modl velocitie. Prt of the force exerted on the cylinder i due to the incident coutic plne wve. In order to clculte the ernl coutic preure cting on cylinder, it i necery to ccount for the cttering cued by the cylinder. The er- J. Acout. Soc. Am., Vol. 112, No. 6, December 2002 S. J. Etève nd M. E. Johnon: Control of ound trnmiion into cylinder 2841

3 nl preure t the cylinder urfce P i function of the zimuth nd elevtion ngle i nd i, repectively, nd of the frequency of the wve, hown in Fig. 1. Since the cylindricl bffle i umed to be infinite, the cttering i only function of ngle. The imple phe dependence in the z direction cn then be clculted from the xil wve number in ir k z k in i, where k/c, nd c i the peed of ound in the fluid. From More nd Ingrd, 13 the preure round the cylinder expreed in term of cylindricl wve i the um of the incident field preure P i nd the cttered field preure P : P,z,, i, i P i,z,, i, i P,z,, i, i. 5 The mplitude of ech cttered cylindricl wve i derived uing hrd wll boundry condition. The preure ditribution P on the urfce of the cylinder due to n incident plne wve of mplitude P 0 i thu expreed ummtion of cylindricl wve of circumferentil mplitude P m, 13 P,z,, i, i P 0 e ik z zt m0 P m, i co m i. The erior modl force F n m 6 tht excite ech mode i thu obtined by integrting the ernl preure over the tructurl mode hpe: F n m, i, i S n m P,z,, i, i ds, where the tructurl mode hpe n m re given in Eq. 1. Note tht ny kind of force cn be decompoed on the tructurl modl be uing thi method. B. Structure coutic cvity coupling In thi ection, the element of the coutic cvity model re derived uing imilr pproch for the previouly derived tructurl model, fter which the coupling mechnim between the two model re preented. 1. Cylinder coutic model Uing Beel function 13 J m, the coutic mode hpe for circulr cylindricl encloure of rdiu R re given by nmp r,,z 1 J m k mp nmp r inmco com nz L, 7 8 where nmp i the normliztion fctor uch tht V ( nmp ) 2 dvv, V i the volume of the cvity, nd the upercript ignifie tht the vrible refer to the coutic cvity. The circumferentil wve number k mp re derived from the hrd wll boundry condition (/r)j m (k mp r) rr 0. The reonnt frequency of the n,m,p mode i thu given by nmp k 2 n k 2 mp, where k n n/l i the xil modl wve number. A with the tructurl mode, the circumferentil orienttion of n coutic mode reult from the combintion of two independent orthogonl mode, one ine nd one coine, of the me order m. At ny point (r,,z) inide the cylinder, the coutic preure p(r,,z,) i pproximted by the modl ummtion: pr,,z, p nmp nmp r,,z, 9 N where p nmp i the complex mode mplitude, nd N the totl number of coutic mode conidered. The coutic modl impednce of the encloed fluid A nmp defined modl preure over modl coutic ource trength i given by A nmp c 2 i 2 V nmp 2 2i nmp nmp, 10 where i the ir denity nd nmp the modl dmping rtio. Thi dmping i incorported to ccount for the borption of the coutic tretment tht i uully preent in rel ppliction but i typiclly very mll t low frequencie. Once n N 1 vector of coutic modl ource trength u i defined, the N 1 vector of coutic modl preure p cn be expreed by pa u, 11 where A i the N N digonl mtrix of modl coutic impednce of the cylinder clculted from Eq. 10. Once the component of the tructurl nd coutic model re defined, the two model re coupled together, decribed below. 2. Structurl coutic ptil coupling The coupling coefficient C between tructurl nd n coutic mode i computed by integrtion of the product of their hpe over the cylinder urfce t rr, 2 C 0 L n m 0,z nmp R,,zR d dz. 12 Due to the orthogonlity of the ine nd coine function, the tructurl coine circumferentil mode couple only with the coine circumferentil coutic mode nd likewie with the ine mode. The coupling coefficient hve the dimenion of urfce m 2. Thee coefficient cn thu form n N N coupling mtrix C, whoe element re the reult of the integrl in Eq. 12, Cn,m,p,n,m 2RLJ mrk mp n 1 n 1 n m n m nmp n 2 n 2 m,m, 13 where the Kronecker delt ymbol m,m i zero if mm nd unity if mm nd the Neumnn ymbol m equl 1 if m0 nd equl 2 if m0. Therefore, to obtin coupling coefficient different thn zero, the circumferentil order m nd m of the tructurl nd coutic mode mut be equl nd the xil order n nd n, mut define n odd even or even odd combintion. Due to the (n 2 n 2 ) term in the denomi J. Acout. Soc. Am., Vol. 112, No. 6, December 2002 S. J. Etève nd M. E. Johnon: Control of ound trnmiion into cylinder

4 FIG. 2. Coupling between the cylinder nd the noie reduction device: HR nd DVA. ntor of Eq. 13, mode with gretly different xil mode order will be poorly coupled. A hown by Grdonio et l., 4 the propertie of the coupling coefficient C determine the number of mode required in the imultion. The number of coutic mode cn be reduced to thoe whoe reonnt frequencie lie inide bnd lightly lrger thn the one of interet, however, the tructurl mode tht re well coupled to the coutic mode mut be included, even though their reonnt frequencie lie well outide the bnd of interet. The mtrix C repreent the link between the tructurl nd coutic model. By men of it phyicl dimenion m 2, it convert the N 1 tructurl modl velocity vector v into n N 1 modl coutic ource trength vector u, or, reciproclly, it convert the N 1 modl coutic preure vector p into n N 1 internl modl force vector F int : ucv, 14 F int C T p; 15 C T denote the trnpoe of C nd the minu ign ued in Eq. 15 yield poitive F int in the inwrd norml direction. C. Coupling of the noie reduction device In thi ection we preent the modeling of the noie reduction device nd their coupling to the tructurl nd coutic model. 1. DVA nd HR modeling A hown in Fig. 2, DVA conit of ditributed pring typiclly mde of coutic polyurethne fom on which i plced ditributed m. Therefore, DVA cn be conidered vibrtion borber cting over urfce re d. The rection force of the DVA induced by the velocity of the tructure repreent the DVA impednce Z d : i 2 d 2 d 2 d Z d m d 2 d 2, 16 2i d d where d i the nturl frequency, m d the m, nd d the dmping rtio of the DVA. The term Z d () re grouped into n N d N d digonl DVA impednce mtrix Z d, where N d i the number of DVA. A HR conit of rigid wll cvity of volume V h nd neck of cro ection re h nd length l h, hown in Fig. 2. If ll it dimenion re mll compred to the coutic wvelength, HR cn be modeled n coutic equivlent to mechnicl vibrtion borber, where the ir in the cvity ct like pring, nd the ir in the neck like lumped m. The interior rdition m effect i included by correction fctor 14 dded to l h, yielding n equivlent neck length l e. The ernl rdition loding i ccounted for by the ummtion of the N coutic mode t the HR throt. Since thi i ner-field effect, the convergence of the velocity mplitude t the throt of the HR with increing N w checked. At reonnce, the HR throt velocity converge quickly only 0.16 db mgnitude difference between N 76 nd N 273 occur. Thu, 76 i et lower bound for N. Uing econd-order pring-m ytem eqution, the HR coutic dmittnce, expreed volume velocity over preure, i given by h i Y h l e 2 h 2 2i h h, 17 where h c h /V h l e i the HR reonnt frequency, nd l e l h 0.85 h / for qure necked reontor. The dmittnce term Y h () re grouped in n N h N h digonl HR dmittnce mtrix Y h, where N h i the number of HR. In order to couple DVA nd HR to the tructurl nd coutic model decribed in the previou ection, the velocity nd preure input to the impednce nd dmittnce of the DVA nd HR, repectively, re expreed uing modl ummtion t the loction of the device. Thi loction on the urfce of the cylinder with repect to prticulr mode hpe node or ntinode define level of ptil coupling between the device nd the different mode, hown in the n ection. 2. Structure DVA nd coutic HR ptil coupling The coupling between DVA nd the cylinder i obtined by integrting ech tructurl mode hpe over the DVA rectngulr urfce of ttchment d b t it deired loction ( 0,z 0 ) on the cylinder. The contct urfce between the cylinder nd the DVA i umed to be flt. Normlized by d, the dimenionle coupling coefficient n m, re given by n m 0,z 0, d 1 z d 0 b/2 z0 0 n,zr d dz, b/2 0 m 18 where in 1 (/2R). The tructurl mode in the m lyer of the DVA itelf re not tken into ccount; it i umed tht the DVA pplie uniformly ditributed norml force on the cylinder. The coefficient n m form, fully populted N d N mtrix tht couple N d DVA to N tructurl mode. In imilr mnner, the coupling between HR nd the encloed fluid i computed by integrting ech coutic mode hpe over the qure re ( h ) of the reontor throt t it deired loction (r 0, 0,z 0 ) in the cylinder. In the imultion the reontor re umed to lie outide the cyl- J. Acout. Soc. Am., Vol. 112, No. 6, December 2002 S. J. Etève nd M. E. Johnon: Control of ound trnmiion into cylinder 2843

5 inder nd to couple t the circumference (r 0 R). Thi i conidered to be comprble to plcing HR inide the cvity, long the HR dimenion re mll compred to the coutic wvelength nd hve mll volume. By mking thi umption, the mode hpe remin unchnged, nd thi implifie the imultion coniderbly. A with the DVA coupling, the HR throt i umed flt over h, nmp 0,z 0, h 1 z h 0 /2 z0 /2 0 nmp R,,zR d dz The coefficient nmp form fully populted N h N mtrix,, coupling N h HR to N coutic mode. Uing the different element previouly defined, the velocity nd coutic repone of the fully coupled ytem i derived in the n ection. D. Mtrix formultion of the coupled ytem The coupling between ll the component of the ytem i chieved uing n impednce mtching method. Therefore, the modl force F DVA exerted on the tructure by the DVA i expreed function of the mtrix, it trnpoe T, the digonl mtrix Z d, nd the tructurl modl vibrtion vector v: F DVA T Z d v. 20 Auming the velocity ditribution in the throt of the HR to be uniform over the urfce h, HR ct coutic piton ource. Therefore, the totl coutic modl ource trength of the coupled ytem u i the um of two quntitie, u h nd u. The modl ource trength produced by the HR u h i function of the coutic modl preure p, nd the coutic modl ource trength due to the tructure u given by Eq. 14 i function of v, uu h u T Y h pcv. 21 Uing Eq. 11, the coutic modl preure vector p due to the totl coutic ource trength u become pa T Y h pcv. 22 The totl force F exciting the cylinder i the um of the ernl coutic force F, the internl coutic force F int given by Eq. 15, nd the recting force of DVA F DVA given by Eq. 20. Expnding the vector F in Eq. 4 into thee three component, the tructurl modl velocity vector v become va F C T p T Z d v. 23 Eqution 22 nd 23 re two coupled mtrix eqution defining the behvior of the fully coupled ytem. Solving thi ytem of two eqution yield v nd p function of the ernl coutic modl force F, Auming the interior coutic pce to be reltively uncoupled from the tructure, nd o neglecting the internl coutic force F int in comprion to the ernl coutic force F, the expreion for v nd p cn be implified to 26 Severl imultion uing different dmping rtio for the tructure nd the coutic cvity, with different configurtion of DVA nd HR, hve hown tht the difference in the obtined noie reduction uing Eq. 26, 27 inted of Eq. 24, 25, repectively, re negligible. A i hown in the implified eqution 26 nd 27, the vibrtion of the cylinder i only ffected by the DVA; however, the internl coutic field repreented by p i modified by both HR nd DVA. In order to obtin n verge ound preure level independent of the loction inide the cylinder, the totl time verge coutic potentil energy E p i computed E p 1 4c 2 V p,r,,z 2 dv If the modl expreion for the preure given by Eq. 9 i ubtituted into Eq. 28, the orthonorml propertie of the mode llow the coutic potentil energy 15 to be computed uing p nd it Hermitin trnpoe p H : E p, i, i V 4c 2 p nmp, i, i 2 V N 4c 2 p H p. 29 Similrly, the totl tructurl kinetic energy i ued n indictor of the verge vibrtion level of the cylindricl 2844 J. Acout. Soc. Am., Vol. 112, No. 6, December 2002 S. J. Etève nd M. E. Johnon: Control of ound trnmiion into cylinder

6 TABLE I. Geometry nd phyicl propertie ued in the numericl imultion. Denity of ir Speed of ound in ir c Denity of tructure Young Modulu E Length of cylinder L Rdiu of cylinder R Thickne of cylinder 1.19 kg m m kg m 3 3.5e P 2.81 m 1.23 m 0.01 m tructure. Due to the orthogonlity nd normliztion of the tructurl mode, the totl tructurl kinetic energy E k cn be expreed by umming the qure of the N tructurl modl velocitie, which i equl to the product of v by it Hermitin trnpoe v H, E k, i, i 1 2 M N v n m, i, i MvH v. 30 III. NUMERICAL SIMULATION The previouly developed nlyticl formultion i pplied to cylinder whoe geometry nd phyicl propertie re ummrized in Tble I. The incident coutic wve excite the tructure with n elevtion ngle i 70 nd n zimuth ngle i 0. The tructurl nd coutic cvity dmping rtio re et to 1% in order to be repreenttive of the dmping level encountered below 200 Hz in typicl compoite cylindricl encloure. A. Bre cylinder repone to incident coutic field In thi ection we preent the min ound trnmiion mechnim in greement with the detiled tudy by Grdonio et l. 4 The coupling between the ernl field nd the tructurl mode chrcterize the excittion of the cylinder. Thi ernl coutic tructurl coupling i repreented by the ernl modl force F n m, which, fter integrtion of Eq. 7, reduce to 2Re it n /L F n m, i P 0 n k 2 m z n /L 2 1 n e ik z L 1P m, i. 31 The circumferentil mplitude P m denote the cttering of the wve by the cylinder, nd i plotted in Fig. 3 for different m function of frequency. Thi mplitude P m, behve like high-p filter whoe cut-on frequency incree with the circumferentil mode order m, except for the brething mode m 0, which h mximum vlue t 0 Hz. The xil component repreented by the term n /L/k 2 z (n /L) 2 (1) n e ik z L 1 i the Fourier wve number trnform of the xil mode hpe in(n /L). It repreent the ptil coupling between the xil wve number in ir k z k in i nd the xil modl wve number k n n /L in the cylinder. Thi coupling i therefore chrcterized by min lobe ner the coincidence frequency between k z nd k n nd idelobe of decying mplitude. However, for n 1, the xil modl wve number k n repreent FIG. 3. Mgnitude of the circumferentil mplitude P m of F n m due to n incident plne wve of 1 P. ( i 70, i 0 ) function of the excittion frequency for m 0,1,2,3. b Mgnitude of the Fourier wve number trnform of in(n /L) function of the normlized xil wve number in ir for n 1,2,3,4. only hlf of wve long the length of the cylinder nd o h it min lobe t k z 0. Thi xil component of F n m i plotted in Fig. 3b function of the normlized xil wve number in ir. Although 36 tructurl mode with circumferentil order high 13 reonte below 200 Hz, Fig. 3 how tht only mode with circumferentil order m 3 nd xil order n 5 re well excited by the incident field. Thi i confirmed by Fig. 4, which how tht the tructurl kinetic energy of the bre cylinder, computed uing ll the mode reonting in the bnd, i dominted by only three lower-order mode. Thi filtering effect llow reduction in the number of tructurl mode necery to obtin convergence of the imulted reult. The internl coutic repone, plotted on top of the tructurl kinetic energy in Fig. 4 i compoed of both coutic reonnce nd tructurl reonnce well excited by the incident coutic field. The level of coupling between tructurl nd n coutic mode i both ptil nd frequency relted. In Fig. 4, the 1,2 tructurl mode i well coupled to the 0,2,0 coutic mode due to mximum of the ptil FIG. 4. Overly of the tructurl kinetic nd totl coutic potentil energy curve due to n incident plne wve ( i 70, i 0 ), with tructurl mode order bold itlic nd coutic mode order. J. Acout. Soc. Am., Vol. 112, No. 6, December 2002 S. J. Etève nd M. E. Johnon: Control of ound trnmiion into cylinder 2845

7 coupling coefficient C(0,2,0,1,2) given by Eq. 13, nd lo due to the proximity of the two reonnce, 112 Hz for the 1,2 mode, nd 137 Hz for the 0,2,0 mode. Therefore, the 0,2,0 coutic mode i reponible for nerly ll of the repone in the Hz rnge. Similrly, in the Hz rnge, the coupling between the 2,3 tructurl nd the 1,3,0 coutic mode i reponible for the mjority of the interior noie. B. Noie reduction mechnim In thi ection we preent the HR nd DVA effect on the ytem nd the trtegie developed to optimize the noie ttenution. The mechnim of both HR nd DVA i bed on the dynmic vibrtion borber ytem. 16 Conider vibrtion borber of m m d nd nturl frequency d ttched to m-pring ytem with m m nd nturl frequency n. Tuning the vibrtion borber uch tht d n plit the reonnce of the ytem into two new reonnce of imilr mplitude on either ide of n. The bigger the m rtio, the frther prt the two reonnce of the coupled ytem pper. By dding dmping to the borber i.e., between the ytem nd the borber m, both new reonnce re well dmped nd ignificnt brodbnd ttenution cn be chieved. Depending on the type of excittion, everl formul for dmping rtio 16,17 led to optiml vibrtion reduction. For wide bnd rndom excittion, the optiml dmping rtio d opt derived by Korenev nd Reznikov 16 i expreed opt d , 32 where m d /m. Once coupled to continuou tructure of m M, the effective m rtio i weighted by the normlized mode hpe qured t the borber loction ( 0,z 0 ), n m 0,z 0 m 2 d M. 33 The coupling between HR of volume V h nd n coutic mode of n encloure of volume V obey the me mechnim. By nlogy with den Hrtog optimized dynmic borber, 17 Fhy nd Schofield 7 derive n optiml HR dmping level opt h olution of h opt 4 4 nmp 1 2 h opt h opt 2 10, 34 where nmp i the dmping of the encloure nd the effective volume rtio, given by FIG. 5. Influence of the dmping rtio of the DVA, d, on the tructurl kinetic energy of the cylinder, the optiml dmping rtio d opt i given by Eq. 32. b Influence of the dmping rtio of the HR, h, on the coutic potentil energy of the cylinder; the optiml dmping rtio h opt i given by Eq. 34. nmp r 0, 0,z 0 V 2 h V. 35 In relity, the dmping of the reontor i creted by vicou loe of the ir moving in the neck. Therefore the dmping cn be djuted by plcing mll mount of porou mteril in the HR throt. In DVA, the dmping i produced by tructurl loe in the coutic fom it compree. Uing different type of fom led to different level of dmping for the DVA. In both ce, the mount of vibrtion ttenution i wek function of the dmping rtio, nd thu mll vrition bout the optiml level only mrginlly degrde the performnce of the HR nd DVA. To illutrte the mechnim of the noie reduction device, Fig. 5 how the effect of the DVA nd HR dmping rtio on the tructurl kinetic energy nd the coutic potentil energy b repone of the cylinder, repectively. With low dmping, the two new mode re both firly lightly dmped nd only mll brodbnd noie reduction i chieved. Alterntively, if the dmping i too high, the device become uncoupled from the tructurl/coutic mode nd no longer diipte energy. In both ce the device re plit into everl identicl unit ditributed evenly round the circumference nd tuned to the reonnt frequency of the trgeted mode: 112 Hz for the 13 DVA nd 61 Hz for the 5 HR. Multiple device re ued for two reon. Firt, uing ymmetric ring of borber llow the tretment to be independent of the zimuth ngle i of the incident field tht i umed to be unknown. Conequently, only the xil mode hpe component i ued in the computtion of the effective m or volume rtio, which i weighed by 1/ m ince hlf the m or volume of the device i effectively cting on circumferentil mode different thn zero. Second, the device ct dicontinuitie tht cn couple mode together by hifting energy from one circumferentil mode to nother. For intnce, TABLE II. Amplitude of the 1,2 nd 3,8 mode normlized by the 1,2 mode mplitude of the bre cylinder t112hzfor3dvand13dva ring tretment tuned to Hz. Mode order frequency Bre cylinder Normlized mode mplitude t 112 Hz db Ring of 3DVA Ring of 13 DVA 1, Hz , Hz J. Acout. Soc. Am., Vol. 112, No. 6, December 2002 S. J. Etève nd M. E. Johnon: Control of ound trnmiion into cylinder

8 TABLE III. Acoutic mode below 160 Hz. Mode order n,m,p Reonnt frequency Hz 1,0,0 61 0,1, ,1, ,0, ,2, ,1, ,2, ring of N d DVA trgeting mode of circumferentil order m i reditribute the energy to ll m iqn d mode, where q i n integer. Thu, DVA tretment i likely to excite tructurl mode tht re not forced by the incident coutic field. The cloer the reonnt frequency of the m iqn d mode i to the DVA tuning frequency, the higher the excittion. Therefore, lrge number of DVA per ring enure wek modl coupling ince only mode with gretly different circumferentil order, which in mot ce implie gretly different reonnt frequencie, cn interct. A n exmple, Tble II preent the performnce on the trgeted 1,2 tructurl mode of two different tretment, both weighing 2% of the totl m of the cylinder. A ring of 13 DVA led to n ttenution of 18 db t the reonnce, where ring of 3 DVA only reduce it by 5 db nd incree the mplitude of the 3,8 mode, which i brely excited by the ernl coutic field. C. Control of the Hz bnd with DVA nd HR In thi ection we preent n exmple of tretment deigned to control the interior coutic level from 50 to 160 Hz. In thi frequency bnd, the encloure preent only even coutic mode lited in Tble III. The firt three re well eprted nd o re trgeted individully by three independent ring of HR, where trgeting two out of the lt four mode enure good reduction. Becue the highet circumferentil order below 230 Hz i m3, five HR per ring i ufficient to obtin negligible excittion of higher-order mode. A ring of 13 DVA tuned to 112 Hz i ued to trget the tructurl 1,2 mode. Ech ring of device i plced on TABLE IV. Noie reduction device chrcteritic for the tretment ued in Fig. 6. Trgeted mode Ring of 13 DVA totl m1.6 Kg Tuning frequency Hz M/M % % d cm 2 d opt % z m 1, Trgeted mode Tuning frequency Hz Ring of 5 Hr totl volume0.8 m 3 Volume/V % % h cm 2 h opt % z m 1,0, ,1, ,1, ,2, ,2, FIG. 6. Acoutic potentil energy inide the cylinder excited by 1 P. Plne wve before nd fter tretment. n xil ntinode of the trgeted mode hpe to mximize the effective volume rtio or m rtio. The totl volume of the HR repreent 6% of the cvity volume V, nd the totl m of the DVA 2% of the cylinder m. The chrcteritic of thi tretment re detiled in Tble IV. Uing et of 76 coutic mode nd 36 tructurl mode, including the two orthogonl mode of the me circumferentil order, the effect of the HR nd DVA on the coutic potentil energy i plotted in Fig. 6. Although the optimlly dmped HR reduce the coutic reonnce by more thn 10 db, lmot hlf of the energy trnmitted in the Hz bnd i due to the 1,2 tructurl reonnce, hown in Fig. 6. The DVA tretment i therefore necery to improve the noie reduction. Once the dmping, the frequency, nd the loction of the device re optimized, the performnce of tretment cn only be improved by increing both the totl m of DVA nd the totl volume of HR, hown in Tble V. Note tht for ech ce, the dmping i optimized with the new m nd volume rtio. A explined previouly, the DVA nd HR noie reduction mechnim i bed on dding dmping to hrp tructurl nd coutic reonnce reponible for the mjority of the interior noie. Therefore, uch tretment cn be dpted to different type of excittion by trgeting in ech different ce the unfvorble reonnce. However, the performnce of thee device i directly relted to the mount of dmping initilly preent in the tructure nd in the coutic cvity. TABLE V. Attenution in the Hz bnd of the coutic potentil energy uing the tretment decribed in Tble IV for different totl m of DVA nd totl volume of HR with optiml dmping rtio computed ccordingly Hz ttenution db Totl volume of HR % of cvity volume Totl m of DVA % of cylinder m 3% 6% 9% 12% 1% % % J. Acout. Soc. Am., Vol. 112, No. 6, December 2002 S. J. Etève nd M. E. Johnon: Control of ound trnmiion into cylinder 2847

9 TABLE VI. HR-DVA tretment performnce with different initil coutic nd tructurl dmping rtio. Acoutic nd tructurl dmping rtio % The higher the dmping rtio of the ytem, the le reduction the tretment cn provide. Thi i illutrted by Tble VI, which diply the noie ttenution provided by the tretment decribed in Tble IV over the Hz bnd for different tructurl nd coutic initil dmping rtio of the cylinder. When the tructure nd the coutic cvity re lightly dmped, i.e., tructurl nd coutic dmping rtio below 3%, the DVA nd HR cn provide ignificnt noie ttenution. The performnce of combined HR/DVA tretment i lo robut with repect to the elevtion ngle of the ernl coutic plne wve. A n exmple, imultion how tht the tretment detiled in Tble IV, which i deigned for n elevtion ngle of 70, chieve ttenution between 6.2 nd 8.2 db in the Hz bnd for different elevtion ngle between 30 nd 80. Thi i to be expected ince the min mechnim of the control i dmping nd not modl retructuring, which i more enitive to the primry excittion. IV. CONCLUSIONS Noie ttenution from Hz db Thi work evluted the bility of combined DVA/HR tretment to reduce the ound trnmiion in n encloed cylindricl hell excited by n ernl plne wve. Uing modl expnion nd n impednce mtching method, the tructure, the interior coutic field, nd the noie reduction device were fully coupled, leding to the nlyticl formultion of the tructurl kinetic energy nd the coutic potentil energy of treted cylindricl cvity. The nlyi how tht t low frequencie, the tructurl vibrtion i only dominted by few lower-order mode becue of the coupling between the plne wve nd the cylinder. The fvorble coupling between thee mode nd the cvity generte n coutic repone compoed of both coutic nd tructurl reonnce. A conequence, ignificnt reduction of the interior coutic level cn only be chieved by uing DVA nd HR imultneouly. A n exmple, n overll reduction of 7.7 db in the Hz bnd i obtined by uing DVA weighing only 2% of the cylinder m nd HR repreenting 6% of it volume. Thi reult w obtined by tuning the device to the nturl frequency of the trgeted mode nd by uing optiml dmping rtio for both DVA nd HR. The device were ued in ring to void unfvorble modl interction nd to obtin tretment independent of the zimuth ngle of excittion. Such tretment i lo robut to vrition in the elevtion ngle of excittion it i bed on dding dmping to hrp tructurl nd coutic reonnce, nd not on reducing the tructurl coutic coupling. In concluion, thi work h hown tht lightweight DVA nd mll HR tretment cn ignificntly reduce the ound trnmiion in n encloure long the tructure nd the cvity re lightly dmped, which i uully the ce t low frequency in eropce ppliction. ACKNOWLEDGMENTS We cknowledge Vibro-Acoutic Science nd Fuller Technologie for upporting prt of thi work. 1 Y. M. Hung nd C. R. Fuller, The effect of dynmic borber on the forced vibrtion of cylindricl hell nd it coupled interior ound field, J. Sound Vib. 200, Y. M. Hung nd C. R. Fuller, Vibrtion nd noie control of the fuelge vi dynmic borber, J. Vibr. Acout. 120, C. Guigou, J. P. Millrd, nd C. R. Fuller, Study of globlly detuned borber for controlling ircrft interior noie, 4th Interntionl Congre on Sound nd Vibrtion, St. Peterburg, Rui, June P. Grdonio, N. S. Ferguon, nd F. J. Fhy, A modl expnion nlyi of the noie trnmiion through circulr cylindricl hell tructure with blocking me, J. Sound Vib. 244, M. R. Jolly nd J. Q. Sun, Pive tuned vibrtion borber for ound rdition reduction from vibrting pnel, J. Sound Vib. 191, K. Ngy nd L. Li, Control of ound noie rdited from plte uing dynmic borber under the optimiztion by neurl network, J. Sound Vib. 208, F. J. Fhy nd C. Schofield, Note on the interction between Helmholtz reontor nd n coutic mode of n encloure, J. Sound Vib. 72, A. Cumming, The effect of reontor rry on the ound field in cvity, J. Sound Vib. 154, A. Dori, Control of coutic vibrtion of n encloure by men of multiple reontor, J. Sound Vib. 181, M. E. Johnon, C. R. Fuller, nd P. Mrcotte, Optimiztion of ditributed vibrtion borber for ound trnmiion into compoite cylinder, Proceeding of the 7th AIAA/CEAS Aerocoutic Conference, My 2001, p H. Omn, M. E. Johnon, C. R. Fuller, nd P. Mrcotte, Interior noie reduction of compoite cylinder uing ditributed vibrtion borber, in Ref. 10, p A. W. Lei, Vibrtion of hell, NASA-SP-288, 1973, Chp. 2, pp M. More nd K. U. Ingrd, Theoreticl Acoutic McGrw-Hill, New York, 1986, Chp. 8, p. 400; Chp. 9, p L. E. Kinler, A. R. Frey, A. B. Coppen, nd J. V. Snder, Fundmentl of Acoutic, 4th ed. Wiley, New York, 2000, Chp. 10, pp P. A. Nelon nd S. J. Elliott, Active Control of Sound Acdemic, New York, 1992, Chp. 10, pp B. G. Korenev nd L. M. Reznikov, Dynmic Vibrtion Aborber Wiley, New York, 1993, Chp. 1, p J. P. den Hrtog, Mechnicl Vibrtion Dover, New York, 1985, Chp. 3, pp J. Acout. Soc. Am., Vol. 112, No. 6, December 2002 S. J. Etève nd M. E. Johnon: Control of ound trnmiion into cylinder