Reconstruction of transient vibration and sound radiation of an impacted plate

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1 INTE-NOIE 06 econsrucon of rnsen vbron nd sound rdon of n mpced ple Ln en ; Chun-Xn B ; Xo-Zhen Zhn ; Yon-Bn Zhn 4 ; Ln Xu 5,,,4,5 Insue of ound nd Vbron eserch, efe Unversy of Technoloy, 9 Tunx od, efe 0009, People s epublc of Chn ABTACT Ths pper presens n expermenl pplcon of he nerpoled me domn equvlen source mehod bsed rnsen ner-feld cousc holorphy for reconsrucn he rnsen pressure feld rded by n mpced ple nd he norml cceleron of he ple. In hs mehod, he pressure feld rded by he ple s frs modeled by se of equvlen sources posoned behnd he ple, nd hen he equvlen source srenhs ech me sep re solved by n erve solvn process nd re en s he npu o reconsruc he whole pressure feld nd he norml cceleron of he ple. An expermen of clmped recnulr seel ple mpced by seel bll s presened. The expermenl resuls demonsre h he proposed mehod s powerful ool o vsulze he rnsen vbron nd sound rdon of n mpced ple n boh he me nd spce domns. eywords: Tme domn equvlen source mehod; Trnsen ner-feld cousc holorphy; mpced ple I-INCE Clssfcon of ubecs Numbers: 4.. INTODUCTION The rnsen nose rded by mpced srucures, e.. rven nd hmmern, no only dsurbs he lvn of resdens round, bu lso brns hern mprmen o worers, nd herefore hs re enneern snfcnce o sudy hs ype of nose. nce he ple s smple srucure encounered n ndusry, lo of wor hs focused on he rnsen sound rdon from n mpced ple -. Usully, cceleromeers ched o he ple re used o mesure he norml cceleron of he ple, vn he boundry condon for furher nlyzn he rnsen sound rdon. owever, he dded mss of he cceleromeers chnes he modes of vbron nd sound rdon of he ple o cern exen, especlly for hn ple. As non-conc mesuremen echnque, nerfeld cousc holorphy NA 4, 5 cn vod hs chne by mesurn he cousc qunes rded by he ple nd reconsrucn he whole sound feld ndrecly. Especlly, he me domn NA s very useful ool for sudyn rnsen vbron nd sound rdon of n mpced ple. Bls e l.6, 7 nroduced numercl Lplce rnsform o replce Fourer rnsform n me domn holorphy TD 8 for nvesn he forwrd nd bcwrd proecons of he rnsen sound rdon from n mpced ple by mesurn he ner-feld pressure. They use n expermen wh n mpced, free Plexls ple o prove h Lplce rnsform bsed TD cn mprove he precson of recovern rnsen pressure, cceleron nd velocy snls. Oher me domn mehods lso hve he cpcy o reconsruc he rnsen sound rdon from n mpced ple, such s rel-me nerfeld cousc holorphy 9, me domn plne wve superposon mehod 0, rnsen elmholz equon les squres mehod, me domn ner-feld equvlence source mn mehod nd me domn boundry elemen mehod. ecenly, he nerpoled me domn equvlen source mehod TD-EM 4 bsed rnsen NA s presened o reconsruc he rnsen sound feld rded by n rbrrly-shped body. I enln007055@6.com cxb@hfu.edu.cn xzhenzhn@hfu.edu.cn 4 ybzhn@hfu.edu.cn 5 hf_xl07@6.com 75

2 INTE-NOIE 06 crres ou he reconsrucon drecly n boh he me nd spce domns, nd no Fourer nd Lplce rnsforms re needed n hs mehod. owever, he mehod s only exmned by smulon cse wh he reconsrucon of rnsen pressure feld. In hs pper, he mehod s furher developed no only o reconsruc he rnsen pressure feld rded by n mpced ple, bu lso o reconsruc he norml cceleron of he ple for cqurn n overll undersndn of s vbron nd sound rdon modes, nd lso wll be verfed expermenlly.. OUTLINE OF TE POPOED METOD Fure Ple n he Cresn coordne sysem o x, y, z nd b eomerc descrpon of he ple plne, he holorm plne nd he equvlen source plne E. As shown n F., ple wh he lenh of plced n he Cresn coordne sysem x, y, z L x, he wdh of Ly nd he hcness of h s o. A seel bll mpcs he ple he pon x, y 0 0 rdn rnsen sound feld. Fure b shows he eomery of he ple plne, he holorm plne nd he equvlen source plne E. There re M pons dsrbued on he plne, L mesuremen pons dsrbued on he plne nd equvlen sources plced on he plne E. Accordn o he TD-EM, he pressure p l he lh pon on he plne nd he norml cceleron m he mh pon on he plne ny me cn be expressed, respecvely, s 5 p l q l / c, l m vm n q q / c c / c, where he sers denoes he convoluon of wo me funcons, c s he sound velocy, s he medum densy, n s he un norml vecor of he plne, s he Drc Del funcon, s he evsde funcon, q s he h equvlen source srenh he me, l s he dsnce beween he lh mesuremen pon on he plne nd he h equvlen source, nd beween he mh pon on he plne nd he h equvlen source. By sen he rerded me rewren, respecvely, s c nd c l l / / s he dsnce, Eqs. nd cn be 754

3 INTE-NOIE 06 p l q l, l m vm The me s dscrezed by n q q c. 4 0, 5 where,,, I, I s he ol number of me seps, A he me, he rerded me re s he me sep, nd 0 s he nl me. c nd c, respecvely. l l / / ere, new me coordne s defned, n whch boh l nd re loced. A he rerded me, he h equvlen source srenh q s nerpoled s where q q, 6 q s he h equvlen source srenh he me Lrne nerpolon funcon, whch s ven by,,, I, nd s he, ;, ; 7 0, oherwse. The dervves of he equvlen source srenh nd he Lrne nerpolon funcon wh respec o he me cn be expressed, respecvely, by q q, 8, ;, ; 0, oherwse. The subsuon of Eq. 6 no Eq., yelds pl l q. 0 Equon 0 s he nerpoled formulon of he pressure. mlrly, he nerpoled formulon of he norml cceleron m cn be expressed by l p l 9 755

4 m q. where c n. Accordn o Eqs. 0 nd, he pressures ll L mesuremen pons on he plne nd he norml ccelerons ll M reconsrucon pons on he plne he me cn be expressed, respecvely, by p p p p P, A, 4 where T L p p p P, 5 T M A, 6 T q q q, 7 L L L L L L p, 8 M M M. 9 By nvern Eq., he equvlen source srenhs he me cn be obned by. p p p p P 0 Equon 0 provdes n erve solvn process for deermnn he equvlen source srenhs ech me sep by he pressure mesured on he holorm plne. The sndrd Thonov reulrzon 6 s ppled n he solvn process ech me sep o obn he ppropre equvlen source srenhs. The solved equvlen source srenhs ech me sep cn hen be en s he npu n Eq. 4 for clculn he norml cceleron of he ple. mlrly, he pressure ny feld pon ech me sep cn lso be clculed by usn he solved equvlen source srenhs s he npu. INTE-NOIE

5 INTE-NOIE 06. EXPEIMENT Fure Expermenl seup: mcrophone rry, seel ple, pendulum, elecromne nd rer cceleromeer. The expermen ws crred ou n sem-nechoc chmber s shown n F.. A clmped seel ple wh he sze of 0.45 m 0.45 m ws mpced by seel bll o enere rnsen sound feld. An cceleromeer fxed on he bc of he ple ws se s he rer o cve he mcrophone rry recordn he pressure nformon nd noher cceleromeer recordn he norml cceleron nformon. Fure eomerc descrpon of he ple plne, he reconsrucon plne, he mesuremen plne, nd he equvlen source plne E. Four pons,, nd 4 re seleced for comprson, mred wh he symbol +. Fure shows he poson relonshps beween he ple plne, he reconsrucon plne, he holorm plne nd he equvlen source plne E. 5 5 pons were dsrbued on he plnes,,, nd E, nd he rd spce ws 0. m n boh x nd y drecons. The equvlen source plne E ws loced z m. The snl ws smpled frequency f 5.6 z provdn 8 smpln pons. e. econsrucon of he pressure feld In he expermen, he rry wh 5 5 mcrophones ws used o cqure he pressures on he holorm plne wh z 0.0 m. z 0.0 m nd he reconsrucon plne wh h For ssessn he resuls reconsruced n he me domn, four spce pons on he plne were chosen s shown n F. 4 nd her posons were 0. m, 0.4 m, 0.0 m, 0. m, 0. m, 0.0 m, 0. m, 0. m, 0.0 m, nd m, 0. m, 0.0 m, respecvely. Fure 4 shows h he reconsruced pressures hese four pons re n ood reemen wh her mesured vlues. Three me nsns wh.99 ms,.4 ms nd e r.5 ms were chosen o evlue he reconsruced resuls n he spce domn. The mesured pressure felds hese hree me nsns re shown n Fs. 5, 5b nd 5c, respecvely, nd he reconsruced pressure felds re shown n Fs. 5d, 5e nd 5f, 757

6 INTE-NOIE 06 respecvely. I cn be seen h he pressure felds vry wh he me. I s lso obvous h he pressure felds reconsruced by he proposed mehod re lmos he sme s he mesured ones, whch ndces h he proposed mehod cn be used o vsulze rnsen sound felds rded by n mpced ple effecvely dfferen me nsns. Fure 4 Tme domn wveform comprsons beween he mesured pressures sold lne nd he reconsruced pressures doed lne four pons 0. m, 0.4 m, 0.0 m, b 0. m, 0. m, 0.0 m, c 0. m, 0. m, 0.0 m nd d m, 0. m, 0.0 m. Fure 5 pl dsrbuons of he mesured pressures.99 ms, b.4 ms nd c.5 ms versus he reconsruced pressures d.99 ms, e.4 ms nd f on he reconsrucon plne..5 ms. econsrucon of he norml cceleron of he ple In he expermen, he pressure mesured on he holorm plne wh zh 0.0 m ws used o 758

7 INTE-NOIE 06 reconsruc he norml cceleron on he ple plne wh z s 0 m, nd he norml cceleron on he ple plne ws lso mesured by one cceleromeer o serve s he rue vlue for comprson. Fure 6 Tme domn wveform comprsons beween he mesured norml ccelerons sold lne nd he reconsruced norml ccelerons doed lne he four pons 0. m, 0.4 m, 0 m, b 0. m, 0. m, 0 m, c 0. m, 0.m, 0 m nd d m, 0. m, 0 m. Fure 7 pl dsrbuons of he mesured norml ccelerons.99 ms, b.4 ms nd c.5 ms versus he reconsruced norml ccelerons d.99 ms, e.4 ms nd f.5 ms on he ple plne. Jus s he reconsrucon of he pressure feld, four pons 0. m, 0.4 m, 0 m, 0. m, 0. m, 0 m, 0. m, 0. m, 0 m, nd m, 0. m, 0 m were chosen o show he comprsons beween he me domn cceleron wveforms reconsruced by he proposed mehod nd hose mesured vlues, 759

8 INTE-NOIE 06 s shown n F. 6. I cn be seen h he reconsruced norml ccelerons hose four pons re close o her mesured vlues. Fnlly, he comprsons beween he norml ccelerons reconsruced by he proposed mehod nd hose mesured he sme me nsns wh.99 ms,.4 ms nd.5 ms re shown n F. 7, whch demonsres h he proposed mehod cn lso vsulze he rnsen norml cceleron dsrbuon on he ple plne effecvely. 4. CONCLUION The nerpoled TD-EM bsed rnsen NA ws ppled o relze he reconsrucon of he rnsen vbron nd sound rdon of n mpced ple. The reconsrucon formuls of boh he pressure rded by he ple nd he norml cceleron of he ple were deduced by pplyn he Lrne nerpolon o equvlen source srenhs. An expermen wh n mpced seel ple s he source ws crred ou o vlde he proposed mehod. The expermenl resuls demonsre h he mehod no only cn be used o reconsruc he rnsen pressure snls nd he rnsen norml cceleron snls drecly n he me domn, bu lso cn be used o vsulze boh he pressure nd norml cceleron modes of he ple n he spce domn. In prculr, he norml cceleron of he ple s obned by non-conc mesuremen n he proposed mehod, nd herefore s ble o vod he dded mss effec ssoced wh he norml cceleron mesured drecly by he cceleromeer. ACNOWLEDEMENT Ths wor ws suppored by Nonl Nurl cence Foundon of Chn rn Nos nd EFEENCE. Ay A, Lch M. ound rdon from n mpc-exced clmped crculr ple n n nfne bffle. J Acous oc Am. 98;74: Troccz P, Woodcoc, Lvlle F. Acousc rdon due o he nelsc mpc of sphere on recnulr ple. J Acous oc Am. 000;085: Chne A, Lmbour C. Tme-domn smulon of dmped mpced ples. I. Theory nd expermens. J Acous oc Am. 00;094: Mynrd JD, Wllms E, Lee Y. Nerfeld cousc holorphy: I. Theory of enerlzed holorphy nd he developmen of NA. J Acous oc Am. 985;784: Verones WA, Mynrd JD. Nerfeld cousc holorphy NA II. olorphc reconsrucon lorhms nd compuer mplemenon. J Acous oc Am. 987;85: Bls JF, oss A. Forwrd proecon of rnsen sound pressure felds rded by mpced ples usn numercl Lplce rnsform. J Acous oc Am. 009;55: Bls JF, oss A. Bcwrd propon of sound felds rded by mpced ples usn rnsen couscl holorphy pproch. Proc INTE-NOIE 8; -6 Auus 009; Ow, Cnd ld J. Tme domn couscl holorphy nd s pplcons. ound nd Vbron. 00;5: Thoms J, ruler V, Pllsseur, Pscl JC, oux JCL. el-me ner-feld cousc holorphy for connuously vsulzn nonsonry cousc felds. J Acous oc Am. 00;86: Zhn XZ, Thoms J, B CX, Pscl JC. econsrucon of nonsonry sound felds bsed on he me domn plne wve superposon mehod. J Acous oc Am. 0;4: Wu F, Lu, nd Bw M. econsrucon of rnsen cousc rdon from sphere. J Acous oc Am. 005;74: B M, Ln J. ource denfcon sysem bsed on he me-domn nerfeld equvlence source mn: Fundmenl heory nd mplemenon. J ound Vb. 007;07: Jn W, Ih J. blzon of me domn cousc boundry elemen mehod for he exeror problem vodn he nonunqueness. J Acous oc Am. 0;: Zhn XZ, B CX, Zhn YB, Xu L. Trnsen nerfeld cousc holorphy bsed on n nerpoled me-domn equvlen source mehod. J Acous oc Am. 0;0: ropp W, vensson UP. Tme domn formulon of he mehod of equvlen sources. Ac Acusc. 995;: Wllms E. eulrzon mehods for ner-feld couscl holorphy. J Acous oc Am. 00;04: