PHYSICS-BASED NUMERICAL METHOD FOR ANALYZING MID-FREQUENCY ACOUSTIC RADIATION

Transcription

1 ICV4 Cais Austalia 9-2 July, 2007 PHYIC-BAED UMERICAL METHOD FOR AALYZIG MID-FREQUECY ACOUTIC RADIATIO Abstact tephe D. O Rega aval uface Wafae Cete, Cadeock Divisio Cadeock, MD 2087 tephe.orega@avy.il As idustial oise souces have iceased yea by yea, toleace fo oise levels by idusty, goveet, ad the geeal populatio has deceased. Eve oe stiget oise egulatios ae divig up the cost of desig ad aufactuig. To educe this cost, desiges look to the VH couity fo sophisticated ueical tools to pedict oise ove a boad fequecy age. This wok focuses o id-fequecy acoustic aalysis tools. Bouday, fiite, ad ifiite eleet ethods have pove useful fo solvig the 3D acoustic wave equatio at low fequecies. Fo high fequecy, localized foulatios such as plae-wave ad ay-tacig ethods have bee applied. These ethods ae ipactical, iaccuate, o both fo teatig id-fequecy pobles. Low-fequecy ethods ake vey lage deads o copute pocessig while high-fequecy ethods do ot accout fo olocal effects. eveal ivestigatos [-4] have poposed applyig a hybid low/high fequecy appoach to solve the id-fequecy poble. This appoach essetially applies atheatical fuctios that atch the fluid ipedace at the low ad high eds of the fequecy spectu ad povide a atheatically sooth bidge betwee the. The dawback to this appoach is that it does ot addess the physics uique to id-fequecy, ad theefoe it caot povide vey accuate solutios. The goal of this wok is to develop a ethod fo efficietly addessig a class of idfequecy vibatio. This class deals with stuctual sufaces chaacteized by a odeate ube of ajo sectios i which the spatial waveube vibatio cotet is badliited ad kow a pioi. A physics-based stategy will be followed to captue the fluid-stuctue iteactio chaacte that is peculia to the id-fequecy age. Ackowledgeet This wok is fuded by I-House Laboatoy Idepedet Reseach (ILIR) poga of the Office of aval Reseach. The autho is gateful fo the sposoship povided by D. Joh Bakyoub, Diecto of Reseach, aval uface Wafae Cete, Cadeock Divisio. The autho would also like to ackowledge the suppot ad ecouageet give by his supeviso D. Matthew Cau ad divisio head D. Paul hag.

2 ICV4 9-2 July 2007 Cais Austalia. ITRODUCTIO The diffeetial equatios goveig acoustic adiatio wee deived i the late 9 th cetuy [5],6]. Coplete, closed-fo aalytical solutios wee tactable aily fo the siplest of geoeties, fo exaple the sphee, as othe geoeties poduce itegads usuitable fo aalytical itegatio. Ivestigatos iteested i fidig solutios fo oe coplex geoeties focused o low- ad high-fequecy ages fo which asyptotic appoxiatios geatly siplified the itegads. With the advet of coputes, ueical techiques expaded the geoetic coplexity fo which solutios could be foud, but still eaied withi the fequecy estictio of low o high fequecies. While eve iceasig speed ad stoage capacity of coputes cotiues to aise the uppe boud fo low-fequecy ethods, fo ay pactical pobles thee is a lage gap i the fequecy age fo which asyptotic ethods will ot suffice fo the foeseeable futue. eveal ivestigatos [-4] have developed so-called hybid ethods, by which low- ad high-fequecy solutios ae bidged atheatically. Fequecy-doai polyoials ae ceated that atch the acoustic ipedaces asyptotically at low ad high fequecies, while eaiig sooth i the id-fequecy age. This ethod has foud soe pactical applicatio i tie-doai shock-wave pobles, i which the spectal cotet is doiated by the low ad high fequecy. Howeve, fo pobles with pedoiatly id-fequecy excitatio, these techiques have ot poved accuate. Asyptotic ethods (ad thei hybid offspig) do ot wok well i id-fequecy because id-fequecy physics is sigificatly diffeet fo low- ad high-fequecy physics. I id-fequecy pessue at ay suface poit is highly affected by otio i a fiite eighbouhood suoudig that poit. This is quite diffeet fo low-fequecy, whee the otio of evey suface poit ca sigificatly cotibute to the suface pessue, ad fo high-fequecy, whee oly the otio of the suface poit itself cotibutes to its pessue. This diffeece i spatial ifluece caot be captued by followig the hybid appoach of applyig a weighted aveage of the low- ad high-fequecy solutios. The difficulty i developig a geeal ueical ethod fo id-fequecy acoustic pobles has bee that a wide age of suface velocity waveubes poses isatiable deads fo pocessig ad stoage. Howeve, thee is a lage class of acoustic adiatio pobles fo which the suface velocity waveube cotet is faily aowbad. Vibatios of lage shells fo exaple. By liitig the scope to pobles with badliited suface velocity waveubes, soe pactical pogess ca be ade. This pape pesets the foulatio of a ueical ethod fo aalysig acoustic adiatio fo spatially-badliited suface vibatios. 2. FORMULATIO The suface pessue P geeated by a suface vibatig i a ifiite acoustic space is expessed by the Helholtz Itegal Equatio (), P( x, k) = ρ, (a) [ G(, k) V ( y, k) H (, k) P( y k) ] d( y) x y (b) ikc ik G(, k) e 2π (c)

3 ICV4 9-2 July 2007 Cais Austalia ik H ( x, y, k) e (d) 2π whee x ad y ae the positio vectos of the field ad souce poits, espectively, V is the outwad oal velocity, ad is the uit oal vecto diected ito the fluid. ueical aalysis equies the discetizatio of the itegal equatio to fo a coplete set of algebaic equatios. The ueical appoach developed i this pape fo id-fequecy aalysis is a extesio of the bouday eleet aalysis (BEA) ethod that has bee developed fo lowfequecy pobles. Hece it is istuctive to suaize BEA to set the cotext fo idfequecy aalysis. 2. Low Fequecy Bouday Eleet Aalysis Fo low-fequecy aalysis, we patitio the suface ito a esh of suface eleets ad suppose the esh spacig sall eough that the oal velocity ad pessue ae appoxiately costat ove each eleet. The the Helholtz Itegal Equatio tasfos ito a set of discete algebaic equatios. [ G ( k) V ( k) H ( k) P ( k) ], =, K P ( f ) =, = G ( k) G(, k) d( y) d( x),, =, K, (2b) H ( k) H (, k) d( y) d( x),,,, = K (2c) i which is the th eleet suface aea. Asseblig the suface eleet pessues, velocities, aeas, ad associated coefficiets ito atices poduces a atix equatio that ca be solved fo the pessues fo kow distibutios of velocities: { P} = [ G][ ]{ V} [ H ][ ]{ P} = [ I ] + [ H ][ ] [ G][ ]{ V} (2d) The ueical itegatios ivolved to copute the atix coefficiets (2b) ad (2c) equie special algoiths eovig sigulaities fo the diagoal copoets (coicidet field ad souce eleets). This sigulaity-eoval step is oe of two ajo ueical tasks i settig up the bouday eleet poble. The secod ajo task is ivetig the atix [[] I + [ H ][ ] i (2d). pecial cae ust be take i liitig the eleet size so this atix is o-sigula. 2.2 Modifyig The Discete Helholtz Itegal Equatio Fo Mid-Fequecy If we take a bouday eleet esh suitable fo low-fequecy aalysis ad aise the fequecy, the assuptio of costat oal velocity ad pessue ove a eleet begis to fail as the acoustic wavelegth deceases to the eleet spa. (2a)

4 ICV4 9-2 July 2007 Cais Austalia Figue : uface eleet geoety Coside the geoety of a geeal suface eleet depicted i Figue. Let the 3D space positio vecto y of a poit o the suface eleet be specified by two suface coodiates α q ( α =,2). The ajo coditio iposed by this ethod is to estict the oal velocity to be siusoidally-distibuted ove the eleet, V ( y, k) V ( k) exp[ ik ( y y0 )], y, =, K, (3a) whee y 0 is the 3D space vecto of the positio of a efeece poit, ad k is the velocity waveube 3D space vecto, o the th suface, espectively. The velocity aplitude ca be foud by itegatig the velocity with the sae siusoidal weightig agitude but with opposite phase, [ ] d( y), =, V ( k) = V ( y, k) exp ik ( y y ), 0 K (3b) The ajo assuptio is that the pessue ove the eleet has the sae siusoidal distibutio, P( y, k) P ( k) exp[ ik ( y y0 )], y, =, K, (4) Isetig equatios (3) ad (4) ito equatios (2b) ad (2c) odifies the coefficiets of the discete HIE atices. ˆ G ( k; k, k ) G(, k) exp[ ik ( y y0 )] d( y)exp[ ik ( x x0 )] d( x) (5a) Hˆ ( k; k, k ) H(, k) exp[ ik ( y y0 )] d( y)exp[ ik ( x x0 )] d( x) (5b) i which hatted quatities efe to id-fequecy coefficiets. Copaiso of Equatio (5) with Equatio (2) shows the id-fequecy coefficiets appoach the low-fequecy coefficiets as the velocity waveubes decease to zeo. 2.4 Evaluatig uface Itegals The ube of eleets ust be sall eough fo the atix ivesio i equatio (2d) to be pefoed accuately without excessive deads o copute eoy. This pactical liitatio equies eleet sizes of the sae ode as fo low-fequecy BEA. A futhe step fo keepig the ethod pactical is to use a esh suitable fo both low- ad id-fequecy aalyses. To distiguish betwee 3D space ad 2D suface coodiates, idices fo space coodiate values use Roa lettes ad idices fo suface coodiate values use Geek idices.

5 ICV4 9-2 July 2007 Cais Austalia As desiged, the itegads i equatio (5) will be highly vayig ove the suface eleet, thus pesetig a challege fo efficiet, accuate itegatio. Pefoig puely ueical itegatio would offe little coputatioal savigs ove eely pefoig lowfequecy BEA ove a uch fie esh. Istead we pusue a stategy of iceasig the aout of itegatio that ca be pefoed aalytically ad ake use of pe-tabulated fuctios. Exaie the double suface itegal (ove souce ad field eleet sufaces) i [ ] Equatio (5). The itegads (, k) exp ik ( y y ) ad (, k) exp ik ( y y ) G 0 [ ] H 0 the diffeetial suface aea d(y α ) ca be expessed explicitly i tes of q. These aalytical expessios cotai poducts of coplex expoetials of polyoials i geeal fo [ M 0 + M ( q ) + M 2 ( q ) + M 3 ( q ) + M 4 ( q )( q ) + M 5 ( q ) +L], ad α q of the expi (6) i which M M,, K ae costats. Closed-fo aalytical itegatio is ot possible fo 0, M 2 these tes, so these itegals ae pe-coputed ueically fo a ube of paaetes ad stoed i lookup tables. The secod itegatio, ove the field suface, also ivolves these types of expessios ad the lookup tables would be used hee as well. 2.5 tatus The autho has developed algoiths fo defiig suface eleets ad extactig the suface geoetic ifoatio equied fo expessig the itegads i tes of suface coodiates. To facilitate applicatio to geeal pobles, the equatios ae foulated i geeal cuviliea coodiates. Rules of teso calculus [7] ae applied to expess ad copute vecto quatities ad thei spatial deivatives. The autho is usig the pogaig laguage MATLAB with special ephasis o usig the eleets of MATLAB ost aki to the C++ pogaig laguage such as class data stuctues. The poject eached its fist ilestoe of beig capable of pefoig stadad lowfequecy bouday eleet aalysis. The secod ilestoe, scheduled fo Octobe 2007, is to pefo id-fequecy aalysis o a subeged sphee. ubsequet ilestoes i 2008 ad 2009 ae to pefo ad validate id-fequecy aalyses o a vaiety of sufaces. 4. COCLUIO A ethod fo solvig the id-fequecy acoustic adiatio poble fo a spatiallybadliited suface velocity has bee peseted. Code developet is udeway, with expected copletio of a id-fequecy aalysis capability fo spheical sufaces by Octobe 2007, cylidical ad plaa sufaces i 2008, ad geeal sufaces i 2009.

6 ICV4 9-2 July 2007 Cais Austalia REFERECE [] Gees, IL, 978, Doubly asyptotic appoxiatios fo tasiet otios of subeged stuctues, J. Acoust. oc. A. 64, [2] X. Zhao,. Vlahopoulos, Mid-Fequecy Vibatio Aalysis of ystes Cotaiig oe Type of Eegy based o a Hybid Fiite Eleet Foulatio, AE Pape o , AE Joual of Passege Ca Mechaical ystes, 200, Vol. 0, pp [3] X. Zhao,. Vlahopoulos, A Hybid Fiite Eleet Foulatio fo Mid-Fequecy Aalysis of ystes with Excitatio Applied o hot Mebes, Joual of oud ad Vibatio, Vol. 237(2), Octobe 9, 2000, pp [4] P. J. hote, R. Lagley, Modelig ethods fo id-fequecy vibo-acoustics, /C ad L/V Dyaic Evioets Wokshop, El egudo, CA, Jue [5] J. W.. Rayleigh, The Theoy of oud, st editio, 877. [6] Lab, Hoace, Teatise o the Matheatical Theoy of the Motio of Fluids, Cabidge Uivesity Pess, 879. [7] McCoell, A.J., Applicatios of Teso Aalysis, Dove Publicatios, Ic., ew Yok, 957.