Page  00000001 A FINITE DIFFERENCE PLATE MODEL Stefan Bilbao Sonic Arts Research Centre Queen's University Belfast Maarten van Walstijn Sonic Arts Research Centre Queen's University Belfast ABSTRACT Iripp a flekio dffeene shm fDh hh6 ecbighr babt paetd, ah inwhet (shmcefficei a d i6f bdaycia dq eihb~bd~ ad ead ceip*i ad a di csi6 feats s has ýi exhhad eadtbah Sd eoiae paeid. 1. INTRODUCTION Irdit, waeta cdend btum b Babka nfih dffeenee sbhfi oi ba ihll a dbibapci calfeaks6 hie asfPqpiepdedt dai ýpad pbak ptihtk bdaythbad i o ad cW t Suhd+ e) 1hyae c h*s ifer a ide age 6ldsta cpW ad kp *Peat hkeldstal chene (ie., h Asrchnaldei fm a kb alde). Emar cWBleigedflEaý WHiEase ap aRibeccifeabb ewmilewabh~ ad ae cpmabb ýe td brtch is (nrhasdal [1]). Tindoiz h" Misrf i hald F btcbh beerpead befe, j4 b3Schdietal [2] ad Lahg etal [3]; tkpkthe ýsain ata d i4&WtWdett ad is6 pctalcuepiptu h)1gad tEl Wphiiaddo w petaneftwed bcdh frh fih dffeepue shm [O, L] ad fdin V4 h bhinpab Hee, antr K 2 idefind t > 0. V2 ih Lacigipad by by 2 Eh2 p(1 - v2) he E, h, p ad v ae Yg'sdib bk e de ad Piamepckfd O (al and chthe). Th Orb hk pantr c reta chtih dgihsda taWd deed, f chk b K, k eqdesbesbeh beh&i a haa The ae thigiv ab b k trficefficit a cih galecay at 6b ahoad htktefficie2 a trtsbiathtlfqies Th tm f (x, t) ebta d4 tpina*, ad dphydbad. Th delabe canbe c1eed bbe a fngemahb6 htfca if k ttphldbk Thecd deiimdepdetPDE (1) eils Wledoie., athldocee adubk ( x(, 0y,). Itadiph spcficahtudbataybdayk Ow cdamiwd chpa Icprade g scabd chpd ad Wp cdogiby c ihge u(X, y,O) (2a) (2b) Bu 0 2 n -0 -0 a ae -ies a phldei* hilta gibday Fee bdagri doibe mdffictitcocp n k cas 6 W bWdthbe dbad be. Ih dec 2. A THIN PLATE MODEL A dI e & Abb fuhlEal i aja isa ht h chsal K&cf del [4]: 92U _K2V4U+C2V2u-2a +b, V2U+f (X, Y, t) 09t2 09t 09t (1) Hef u(x, y, t) ih xe [L deflecy defind erkh echghrega x E [OLx], y (E 3. FINITE DIFFERENCE SCHEME Iridert (1) nerakwcamk a 6 a fih dffeene a fitdefi4 a gl fa& a peg# anau.h u(x, y, t) atcdabs X =,Ai, y = Aj, ad t = nT. Hee, Ax ad,ae ah gl pchi h x ad y decoepckifad T hiin Ap(1/ T h asnat). Nbe, ilpth htA x ad A, ae, igemal dffeei wdefin hpaner a by a = Aj/Ay.

Page  00000002 3.1. Difference Operators ad he iw aklefind Sbdad aih fitad ecd in dffeeklpatae ghiby KT A2x cT Ax bjT A2 x 62Un 6to~n 6_n -i~j 1 2~ i(u 1 n - il. t2U uai z3 09t2 1 + uT 2At 73 u nI4) - u" ) atu at At i - Dffeene shm(3) ifn1ecd deraccat iipce ad fitderacca ib* dei th E 6 h pab st-, aceayi'derb bbdiaw akiS he h trh 4 6t- iigeeala alphaowdda epct Rha tnbhu a de1kieffect n accacyasa h Th fitad ecd pa~tbe ap ceted ad ecd-deraccat; h Hl, a bachd dffer ene pa4 ifitdieraccat. Secd-der ceted dffeene aotwcd dewksih x ad y deichai gyiby 3.2.1. Stability 62 n xO iJ 6 n yo i,3 - u-2U A2x i+1,y ziy +n 92~ + -li 09X2) + Unj _0) 92 U ila09Y2 Iih ca 6 a 6 fih 1phleta ci dhiffictfcbucanbe dethd IA EcLk ee k1h Et6 pChlc 8e1 aIibCiS Itig Iby 1 2 inj+1 - 2u4 ad ih Laocirad bhinpatan be aoitd by A21 1+ a2 - 2bT + c2T2 + Iý(2bIT + c2T2)2 H+ 16T22 2 (5) 62 =_620+ 20 2 +o y 4 62 62 4 3.2. An Explicit Finite Difference Scheme Atuawaybhh h abe pab ffoi-cifi pce cbpth(1), tgeth fPW e fih dffeene sbbm 6t2U = _K262 62 U C262 u-2u6tou+bj6t- 62 U+ f (3) Tiranbe umsarnieckiih gHi f&h a, he adkik do aih(w Mefin hp4 iatd acd4 ib it i Ott) flo key ht hn bdaycdhae aWd, hbyrci dbbecosecinceay fhraa* ekd bdethinbha gkibdaria daibd ta gkffik dffeene smhdegadesh kb~bd ginbue. We aki n fhrc bd rhrhtin icarbe ek debat igemal ad htw iw beud io1binh ajad6 at bat~ibdadkthshm We ]iwfred hkbicdtasa bd \A, ad A y (kh a), hkgHipc4~ he fcadiakaftiHblthk h a n at (I/ T) be find fdah tt Th p 6 h4 A x dhtfnAy i iitfecbghr ink~ tHJ& die hk gh Ly tarkgeiher6 pýie., wih A_ = Lx/Nx ad A y-=Ly/Ny fikger Ny. Thbe4 adl, t adsugen(fnk (6xnb " dui kraldid tBt a asch t1 asoý (ie., t tidibr t et a = Lx/Ly) ad A xaschdth bd gin by(5) ask. n+1 vi.. |k|+|1|<2 +,q |k|+|1| /3|k|,lUi+k,j+1l (4) L, ad N, ad |k|,||n-U k~lH+- 2f f (iAx, jAy, nT) (b h f i fiaraj* be dad i-ct3.2.4 bevad 1,o1 /32,0 012 'Yo, 0 ',Yo 1 2 - 2pu2 (3H+4a2 +H3a4) -2(A2 H- V)(I H-a2) 4pu2 (1H+ a2) H-A2 H- V a 2 (4U2 (IH+a2) H-A2 H-+) =-2/12a 2 2 S1-t2T =-1 + 2v (I + a2) +uT 3.2.2. Initialization Dffeene shm(4), ecd derily eis h hih6 h gl fa& bp n = 0 ad n 1h i u(x, y,O0) ad ((x, y, 0 1n 2,3 at in 4hl 1cdi ). We carnhdiit u0, = y, 0) = -V - -02 btfi e# 6 uv jeiskimcae.

Page  00000003 Ahi a Owfipeed4 itat u, = u(iA, jAY, ) + T U (iAx, JAy, 0) blia fitleraccat a* Hghr deraccat hlhiobe accfrl by taig phldei- fna It~igthth 84 6 kh la-hs bh gi fakfi& tt bed byk foi fiia f; ihhr d f bh hlcdhad aehtae!1petit debbbi hkth erhimtijhh afbrk fiktk-&p ad yf(t) a h panid cdiits6 h ne 6 gh g(t) ati t. Sp, atmE gt t = nT, wa i Lf = wfL (nTl)/Alx ad jn n - i f yf= Lf(nT)/Ay] ad EX, = Xfr(nT)/Ax - f ad n yf = (nT)/Ay -jr, h ne aybe bbak pad tfnghg gH by ^ a dy f f3 ^ 3t+1,jf nn j 4,nn" +1 n 19+1,jf+1 S(1 - Ef)(l - ef)gn(nT) = EX,f(l - E,f)g(nT) S(1 - e",f)e",g(nT) Sfn~,f(T) = 6, f6yfg(T)cf (T 3.2.3. Boundary Conditions Asedn abe, wcamage ht Lx Nx ad Ly/Ay = Ny, ea4 htnagl fh 0u idehl bdaesi=,... 0,.., N,. Aqiak dffeene smb(4) ePi ata gingil accestabs6 h gif fcafath phnbhpattyhlbpawii x aD y dec w ned ib dfh shmatcm i bdaym bpiWyie., fD i= 0, 1,Nx - 1,N,, feal j = 0, 1, N - 1, N fual i. Ci1e4 hfied cd4 2), tcaiht h cdh u = 0aybe efed bim dffeene shmby# Un = 0 atgH t diecdi bdaytibtceas co" * Leffe t nig h hki Nwi1era gH ladjicetth bdaynITs i, j 1. Chakfr44), at S H- + 1 1i4 accest i _ l hl iled ch gil. Fh dffeene a* ýff deik bdaed(2) kH, inT cjAx /a7 = 5, I = On auk cae hthne cditidli falkl bphl eunib b bdayh eiTa mci(apk) pad4 Mt egWlbecnaceay Ohr4#deme pad bkgiesae 6 ca B. A sahrgalJ u atppcditbs xo(t), yo(t) carbe Batd hk1lbbar Athhti cod Ohs. ILe., f wet i = Lxo(nT)/Ax] ad jo = Lyo(nT)j/A, ad E, = zxo(nT)/A - in ad E, = y(nT)/A - o", bm carbih jad 12 'Uo EX1o YO io Hi +,jg,o yo i 1, H (1 - 0 H-0 Obkasafffeethahisdle aybe coi iAy A pthiA e phk effectiaciwd bya aRtrkt 6k h obabeht b ab hbhbadstir ek g effec$ in en, irarf asa eyihhfi sand ýi[5] ie h sara bikrpi at 6 badt6 k hm n n i,1 2,1 _ n vi. - n i,-i = -4 _2 (6) (7) IrdertaIe bdayrdka ay begifr44), ad nbce ahne 6 a gH aibbi fihe# h bdayb3a 6h cdkgimabe. We doebat, Wek ht h ink fckb~ae bymwkas ie., a dffeene sn uhttb iabbeh ib h b m dnicarbecmkb deih ajatd5 bdadi fkhraakhnyb e ek~ctiW-ie4d. 3.2.5. Sound Examples Areacoka d EiginFga 1 beOderpantrchesasgkrik ca~ Inhcae, h gil i26 hwe4d tA sb d, at44.1 klý i a~in22.6 sea 2 GHzhh Th c~o sta eMkhd E (it feqny72 H. Finrk ri~bs6 dffeetbhs(eflectd iih h 6 sad th ckinkibbe gficafabr x 34, ad h 61 K), h gl 3.2.4. Input/output 3.2.6. Computational Complexity On eanb e* h Ik ihhtii hlcdoabriklB a 6 a d-k 4 fah f(xy, t). Fuek adck tfHfhytcXert aXyiy)e, ie., f(X, y, t) = g(t)6(x - xf(t), y - y (t)); e, Irk geealcas, fi x 7 A y, shm(4) e is h W pr gHi fa pr inp Fo eaciap iia4 h ek gil vei g) aLxL /,A ad a her bixmcd eid ibe 9 aLLy/AT. Xf(t)

Page  00000004 x10 output waveform 0 0.1 0.2 0.3 0.4 0.5 0.6 time (s) output spectrum 0.7 0.8 0.9 1 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 frequency (Hz) Figure 1. Sound output (top) and spectrum (bottom) for a clamped steel plate (K = 15.26), with side lengths L, = Im and L, = 1.3m, excited by an initial condition in the form of a raised 2D cosine, centered at the plate center, with a width of 20 cm. The output is read from a point at coordinates x = 5L,/8, y = L,/2. Sample rate is 44.1 kHz. iprtnipke hherb gHi saksdec~h her6 descoI. The ae adsagesad dadshgesberaip pch da1leaflgees ad bdaedW ad aabab 4 tLTI 4nnFh dffee e saheffem 6 he ekk btiaddtl c1eank Rhasciral Abkr ad dj* canbecn4Ahat. 4. CONCLUSIONS AND FUTURE DIRECTIONS T~hl ak~txpd he bcfn thacbaph ca1bitCae 6h am dler 6 agtle as dal1[1], ad i mgeal i eian hta) tdeostrn pcu6 6 d-al feqiesad hps (hhcanbe eyepk h ah Age eip* ad b) taybe etded b delh anmy bhairbs (php6 k uKamreY[6]); bar oae tvldeMed bydralcis Abildhikt6phhffin a he-diblaa# fpaksiahbb (as t cas iB Sh Lab, atSARC, iBefao. 5. REFERENCES [1] J.-M. Adrien. The missing link: Modal synthesis. In G. DePoli, A. Picialli, and C. Roads, editors, Representations of Musical Signals, pages 269-297. MIT Press, Cambridge, MA, 1991. [2] S. Schedin, C. Lambourg, and A. Chaigne. Transient sound fields from impacted plates: Comparison between numerical simulations and experiments. Journal of Sound and Vibration, 221(32):471-490, 1999. [3] C. Lambourg, A. Chaigne, and D. Matignon. Time domain simulation of impacted plates. part ii. numerical model and results. Journal of the Acoustical Society of America, 109(4):1433-1447, 2001. [4] K. Graff. Wave Motion in Elastic Solids. Dover, New York, New York, USA, 1975. [5] B. Verplank, M. Mathews, and R. Shaw. Scanned synthesis. In Proceedings of the 2000 International Computer Music Conference, pages 368-371, Berlin, Germany, 2000. C1e4 h cas 6 ip ht (5) i afied heo(hht eat tipcte), whthw 9 a2)KT2 jOxacd. Fia qe bel1k 6 le lghl md 6 bbs2 cgd amn 4 ht a=1, hattauakl.4 108 pahprecd, b4kv by easchapik capb a ggahk age peo Dectfik dffeene d&e kialk eifbnchmcthd asbe mcpifn tihdeadI 4hrrtcy izpr thrdali* Itisotbt k fly a fih dffeene shinfta hark ia4tmarahybe hrith-pce fph her6 dcal feqpisbe4 hf h ý 6 h Ak; iheE k fih dffer ene shmbehEsasa dalRior Tadi bldal esal 2 in Osprde, prtntp(f a de bbe dd asa t e4. Th fil dffeene sbmayey4 au bttgficage.ifi petrae, 6 iprgfl c = b1 = 0, ad as aLL,/2(1+ x [6] C. Touze, O. Thomas, and A. ric nonlinear forced vibrations plates. part i. theory. Journal tion, 258(4):649-676, 2002. Chaigne. Asymmetof free-edge circular of Sound and Vibra