Development of indirect EFBEM for radiating noise analysis including underwater problems

Transcription

1 csnk 03 It. J. Naval cht. Oca E. 03 5:39~403 Dvlopmt of dct EFBEM fo adat os aalyss clud udwat poblms Hyu-Wu Kwo Su-Yoo Ho ad J-Hu So 3 Rsach Isttut of Ma Systms E RIMSE Soul Natoal Uvsty Soul Koa Dpatmt of Naval chtctu ad Oca E Soul Natoal Uvsty Soul Koa 3 Dpatmt of Naval chtctu & Oca E Choam Natoal Uvsty Joam Koa BSTRCT: Fo th aalyss of adat os poblms mdum-to-hh fqucy as th Ey Flow Bouday Elmt Mthod EFBEM was dvlopd. EFBEM s th aalyss tchqu that appls th Bouday Elmt Mthod BEM to Ey Flow alyss EF. Th fudamtal solutos pst sphcal wav popty fo adat os poblms op fld ad d th f sufac ffct udwat a dvlopd. lso th dctvty facto s dvlopd to xpss wav s dctvty patts mdum-to-hh fqucy as. Idct EFBEM by us fudamtal solutos ad fcttous souc was appld to op fld ad udwat os poblms succssfully. Thouh umcal applcatos th acoustc y dsty dstbutos du to vbato of a smpl plat modl ad a sph modl w compad wth thos of commcal cod ad th compaso showd ood amt th lvl ad patt of th y dsty dstbutos. KEY WORDS: Ey flow bouday lmt mthod EFBEM; Radat os; F sufac ffct; Dctvty facto. INTRODUCTION Wh shps o udwat obcts mov thouh wat whos mpdac s much hh tha that of a small vbat motos of th shp body ca caus maabl oss wd fqucy as. No sl aalyss mthod of th os phomo ca b ffctvly appld to all as of os poblms. t low fqucy as th aalyss of vbato poblms s aalyzd by th covtoal mthods such as th Ft Elmt Mthod FEM ad Bouday Elmt Mthod BEM. But at hh fqucy as thos mthods qu mo computato tm ad ts ad thus altatv mthods a dd. mo may altatv mthods EF has cvd much attto. Ths mthod was toducd by Blov t al Nfs ad Su 989 mplmtd Ey Flow Ft Elmt Mthod EFFEM to solv th tasvs vbato of a bam. Wholv ad Bhad 99 dvd th y ov quato fo dvs vbat wavs of a bam. Bouth ad Bhad 99 dvd th y ov quato of a flxual wav d oly th fa-fld compot of th tasvs vbato of a mmba ad th plat. Thy xpadd th sach of EF to vbato poblms of twodmsoal stuctus. Cho 993 sachd th y bouday codto th cocto pot btw lmts of a bam by apply th wav tasmsso appoach to a complx stuctu. Pa 999 ad Pa t al. 00 dvd th y ov quato fo a -pla wav of a th plat ad studd th spatal dstbuto ad tasmsso path of vbato y fo a plat stuctu whch s coctd at a o-dtmd al. So t al. 003 xpadd th applcato of EF to Cospod autho: J-Hu So -mal: hs@choam.ac. Copyht 0x Socty of Naval chtcts of Koa. Poducto ad host by ELSEVIER B.V. Ths s a op accss atcl ud th CC BY-NC 3.0 lcs

2 It. J. Naval cht. Oca E. 03 5:39~ bam-plat coupld stuctus. L t al. 008 appld EFBEM to th vbato aalyss of bam ad plat poblms. lso Wa t al. 004 appld y bouday lmt fomulato to th soud adato poblms. I ths pap th y ov quato hav sphcal wav popty s dvlopd op fld. d th dctvty ffct whch s pstd hh fqucy a poblms but ca t xpss EF s studd. d th fudamtal soluto ad y ov quato fo udwat poblms a dvlopd. Th dvlopd quato ad dctvty ffct a appld to th smpl cas ad th sults a compad wth commcal os aalyss poam SYSNOISE ad labl sults a obtad. ENERGY GOVERNING EQUTION FOR RDITING NOISE NLYSIS Ey balac quato I a laly lastc mdum th y balac quato s dvd fom th follow Eq.. Th amout of com ad out com pow thouh th sufac of a obct ad th at of cha of th total y th obct a sam. Fom ths fact y balac quato s pstd as follows: CV dv t CS ξ σ d S t CV π π dss dv wh s th total y dsty th volum of a obct CV. ξ s th dsplacmt vcto at th bouday of a obct CS. d S s a small aa vcto ppdcula to th sufac of th bouday. σ psts th stss o th sufac of th bouday. π ad π dss a xpssd as put pow ad loss pow act o a ut volum at a ut tm spctvly. Itsty dfd as th pow p ut aa flow out of a obct ca b obtad fom th stss act o th sufac of th bouday ad th vlocty of th sufac of th bouday as follows: ξ I σ t wh I s th tsty th pow p ut aa. I Eq. th fst tal o th ht had sd s wtt wth th applcato of Gauss s thom as follows: CS ξ σ d S I d S I dv 3 t CS CV Thfo th y balac quato th volum of a obct s obtad by CV dv t CV π π I dv 4 dss Fom Eq. 4 th y balac quato fo a small volum s xpssd by π π dss I 5 t

3 394 It. J. Naval cht. Oca E. 03 5:39~403 Eq. 5 s th y balac quato fo all lastc mdums stady stat ad tast stat. Fo a stady stat lastc mdum th at of y dsty wth spct to th tm s zo; th lft tm Eq. 5 s zo. Thfo th stady-stat y balac quato s pstd as follows: π dss I π 6 Fom Eq. 6 th put pow du to th xto xct foc s xpssd as th sum of th pow lost th obct ad th pow flow out th adact mdum. Ey loss quato Th loss of vbato y ca b pstd by a damp stuctu modl. Cm t al. 004 showd that y dsty loss dss du to th damp stuctu p o pod at ay pot a lastc mdum vbat at fqucy ω s popotoal to vsbl vbato y dsty. Thfo Eq. 7 s obtad by R π 7 dss R If th damp coffct s vy small th vsbl y dsty R ca b substtutd fo th tm ava valu of th total y dsty < > whch s th sum of th tc y dsty ad th pottal y dsty. Thfo Eq. 8 s v by R < > 8 wh < > mas th tm ava. Th tm ava of pow loss du o pod < π dss > ca b obtad by dvd th y loss du o pod dss by pod T b π ω. d t ca b obtad appoxmatly by us th tm ava of th total y dsty < > as follows: dss π < π dss > R ω < > T T 9 Eq. 9 s th y loss quato whch s dvd fom th applcato of stuctual damp to ay pot o a lastc mdum vbat at fqucy ω. W assumd that th tc y s th sam as th pottal y. ENERGY GOVERNING IN UNDERWTER RDITING NOISE NLYSIS Ey tasmsso quato udwat adat os aalyss Fo aalyz th adat os poblms udwat w must d th ffct of f sufac. ccod to F. ad flcto coffct of sufac sphcal wav udwat s pstd as follows: ~ ~ p xp xp 0 wh s th dct dstac btw a souc pot ad fld pot. s th dstac avd fom th souc pot to th ~ fld pot whch s flctd at th f sufac of th wat. s th wav umb d th damp f- fct. s damp loss facto ad s th wav umb. Fom th Eul s quato th vlocty of th wav ca b obtad.

4 It. J. Naval cht. Oca E. 03 5:39~ z Tasmttd wav y Mdum ρ c Mdum ρ c φ θ φ t θ φ φ x Icdt wav Rflctd wav F. Wav popaato fom mdum to mdum. ' h O B D h θ l F. Th dstac btw fld pot ad souc pot. C Thfo y dsty ad tsty ca b xpssd as follows fo a fa-fld wth low damp: h θ h θ hθ ρc h θ h θ hθ I ρc Fom Eqs. ad th follow y tasmsso quato ca b obtad as follows: c D d I ω d D 3 wh D h θ h θ h θ. Ey ov quato udwat adat os aalyss Fom th y balac quato Eq. 6 th y loss quato Eq. 9 ad th y tasmsso quato Eq. 3 th y ov quato ca b obtad as follows:

5 396 It. J. Naval cht. Oca E. 03 5:39~403 c D ω d d D < > ω < >< π > 4 Eq. 4 s th y ov quato pst th popty of a sphcal wav ad clud th ffct of th wat sufac. DERIVTION OF THE FUNDMENTL SOLUTION IN UNDERWTER RDITING NOISE NLYSIS Th fudamtal soluto G s th xact soluto fo put pow as wll as th fudamtal soluto pst th lato btw th vtual souc ad y dsty. Thfo th shap of th fudamtal soluto G pods to that of th y dsty of Eq.. ccod to ths quato th fudamtal soluto G s xpssd as follows: D G C 5 Fudamtal soluto H pst th lato btw th vtual souc ad tsty s obtad fom Eq. 4 as follows: c D d c D c D H G C C ω d D 6 ω Wh tsty bouday codto s appld to Eq. 6 th valu of C s obtad as follows: Lm 0 0 π δ d Lm 0 0 HdS 7 I th lft tm of Eq. 7 th tal of δ π ccod to ths popty ad Eq. 6 Eq. 7 s xpssd as follows: s qual to o bcaus of th popty of th Dac dlta fucto δ. c Lm CD 4π 4π ccd 0 8 h θ h θ I th ht tm of Eq. 8 D mas. Th h θ tm of D psts th ffct of th flctd wav. If adus appoachs zo ths ffct dmshs; that s D. Thfo C ca b obtad fom Eq. 8 as follows: C 4π 9 c Fom Eq. 9 w ow that th o-dtmd tat C s 4π c t th lato btw th vtual souc ad y dsty o tsty s as follows:. Thfo th fudamtal soluto ps-

6 It. J. Naval cht. Oca E. 03 5:39~ c D G π 4 ad D H π 4 0 Eq. 0 s th fudamtal solutos hav th popty of a sphcal wav ad copoat th ffct of th wat sufac. DIRECTIVITY FCTOR BEM whch s o of tadtoal os aalyss mthods shows th os s dctvty patt at hh fqucy as. Ths s causd by th dffc of wav s phas. But EF dos t show th os s dctvty patt bcaus EF dos t hav th phas fomato. So pst th os s dctvty th dctvty facto s dvlopd. Wh soucs a xstd th y s as follows: > < c c p c ρ ρ ρ L L H s wly addd. Ths quato ca b dvdd to two tms as follows: Fom Eqs. ad th follow quato ca b obtad as follows: > < c ρ 3 wh s dctvty facto. ESTBLISHMENT OF BOUNDRY INTEGRL Establshmt of bouday tal fo dct mthod I th dct mthod of th bouday lmt th al systm of ft sz s xtdd to th ft fld ad th vtual souc s assumd to xst o th bouday of th al systm. d y ad tsty th cocd fld a obtad by us fudamtal soluto as th sum of th ffcts du to vtual soucs. Basd o ths cocpt th-dmsoal poblms quatos of y dsty ad tsty a obtad th cocd fld as follows: z z z x ξ ξ ξ x x dv G ds G V S π φ ad 4

7 398 It. J. Naval cht. Oca E. 03 5:39~403 I x H x ξ φ ξ ds ξ H x z π z dv z 5 S wh V s assumd as th cocd fld th dmsos ad S s assumd as th bouday sufac compass th cocd fld. Th fudamtal soluto G ad H has th sam shap as that of Eq. 0. x dcats a fld pot of th cocd fld. ξ mas th posto of th vtual souc o th bouday. z s th locato of put pow. φ ξ ca b obtad by us mas th vtual souc o th bouday sufac. Th vtual souc o th bouday sufac φ ξ Eqs. 4 ad 5 accod to th popty of th bouday codto. If x appoachs th bouday sufac th bouday tal of th ht tm Eqs. 4 ad 5 has a sula tal whos pot s ctd o th bouday sufac. Ths pot s th dtty of x. Thfo th tal tm of th Eq. 4 s dvdd to th follow tms as follows: S V G x ξ φ ξ ds ξ G x ξ φ ξ ds ξ G x ξ φ ξ ds ξ 6 Sε SSε wh f th bouday of th cocd fld s a smooth cuv sufac th bouday sufac S ε s a hmsph wth th small adus ε ct o th bouday pot. d th bouday sufac S Sε dcats th pat whch s subtactd fom th bouday sufac S fom total bouday sufac S. Th fst ht tm of th Eq. 6 s as follows: ε Sε G x ξ φ ξ ds ξ φ ξ ds ξ S 4 c ε π Sε 4π c ε φ ξ ds ε ξ 7 πε. So Eq. 8 s obtad by ta- wh f th adus ε appoachs zo th aa of th bouday sufac th lmt of Eq. 7 as follows: S ε bcoms lm ε ε φ ξ ds ξ lm φ ξ φ ξ 8 ε 0 S ε 4π c ε ε 0 c c d f adus ε appoachs zo th bouday tal tm wth spct to th bouday sufac S Sε s placd wth th bouday tal wth spct to th total bouday sufac S. Eq. 7 s xpssd about o pot o th smooth bouday sufac as follows: ξ φ ξ G x ξ ξ dsξ G x z x dv z c φ S π 9 V PPLICTION EF s dvlopd by assumptos whch a hh damp valu ad mdum-to-hh fqucs. ccod to ths assumptos th afld vasct wavs ca b lctd. So EF s sult s ot ood amt wth a classcal soluto at low damp valus bcaus th ffct of afld wavs wll b affctd o th vbato phoma low damp systm. Thfo EFBEM wll b appld to th hh damp systms. To vfy th accuacy of th dvlopd wos th bouday tal of dct EFBEM was appld to th adat os poblms of stuctus wth th vbato of a smpl plat modl ad a smpl sph modl op fld ad udwat. d th sult of th dct EFBEM was compad wth that of SY- SNOISE. Fs. 3 ad 4 show y dsty dstbuto spctvly th z-dcto by SYSNOISE ad dct EFBEM wh th plat whch sz s m m s locatd at x-y pla wh f s Hz v s 0. m/s ad s 0. Fld pots whch sz 7 m 3 m s locatd x-z pla. Thouh EFBEM ca d th damp loss facto a mdum a damp loss facto of zo was usd so that EFBEM could b compad wth SYSNOISE. F. 5 shows a smpl sph stuctu whch adus s 0.5 m wth a ufom vbato op fld. Wh f s Hz v s 0. m/s ad s 0 Fs. 6 ad 7 show th y dsty dstbutos spctvly th z-dcto by SYSNOISE ad dct EFBEM wh th sph s locatd ud th fld

8 It. J. Naval cht. Oca E. 03 5:39~ pots. Fld pots sz s m m. Fs. 8 ad 9 show th y dsty dstbuto th x-z pla pdctd by SYS- NOISE ad dct EFBEM wh th sph s st at th ct of th fld pots. Fom Fs. 3 4 ad 6-9 w ca s that th sults of dct EFBEM a wth thos of SYSNOISE op fld. F. 0 shows th y dsty dstbuto th z- dcto pdctd by SYSNOISE ad dct EFBEM wh th sph s locatd a th wat sufac. Thouh th dffc of th sults btw SYSNOISE ad dct EFBEM s about db th patts of th sults a th sam. F. shows a smpl ba typ stuctu wth a ufom vbato op fld. Wh f s 63 Hz v s 0. m/s ad s 0 F. shows th y dsty dstbutos spctvly th z-dcto by SYSNOISE ad dct EFBEM wh th ba typ stuctu s locatd abov th fld pots. Thouh th dffc of th sults btw SYSNOISE ad dct EFBEM s about 5 db th patts of th sults a th sam. d F. 3 shows th y dsty dstbuto wh f s Hz v s 0. m/s ad s 0. Bcaus th ba typ stuctu s la SYSNOISE ca t aalyz os aalyss but EFBEM ca aalyz thos ov Hz. a b F. 3 Th y dsty dstbuto z-dcto wh th plat s vbat: a SYSNOISE b Idct EFBEM SPL F. 4 Th y dsty dstbuto z-dcto wh th plat s vbat: * SYSNOISE; o Idct EFBEM. F. 5 Ufomly vbat 3-dmsoal sphcal stuctu.

9 400 It. J. Naval cht. Oca E. 03 5:39~403 a b F. 6 Th y dsty dstbuto z-dcto wh th sph s vbat: a SYSNOISE b Idct EFBEM. SPL F. 7 Th y dsty dstbuto z-dcto wh th sph s vbat: * SYSNOISE; o Idct EFBEM. a b F. 8 Th y dsty dstbuto x-z pla wh th sph s vbat: a SYSNOISE b Idct EFBEM.

10 It. J. Naval cht. Oca E. 03 5:39~ SPL F. 9 Th y dsty dstbuto x-dcto wh th sph s vbat: * SYSNOISE; o Idct EFBEM. a b F. 0 Th y dsty dstbuto z-dcto wh th sph s vbat: a SYSNOISE b Idct EFBEM. F. Ufomly vbat ba typ stuctu.

11 40 It. J. Naval cht. Oca E. 03 5:39~403 a b F. Th y dsty dstbuto z-dcto wh ba typ stuctu s vbat: a SYSNOISE b Idct EFBEM. F. 3 Th y dsty dstbuto z-dcto wh ba typ stuctu s vbat. CONCLUSION Ths pap aalyzd adat os poblms th mdum-to-hh fqucy a. Th y ov quato ad fudamtal solutos hav th sphcal wav popty w dvlopd. d th dctvty whch was th popty of hh fqucy aalyss but could t pst xst EF was dvlopd. lso th fudamtal solutos pst th ffct of th wat sufac w dvlopd fo th udwat adat os poblms. Last dct EFBEM apply BEM to EF was dvlopd fo ths aalyss. To vfy th dvlopd wos th sults of th dct EFBEM w compad wth thos of commcal cod SYSNOISE op fld. Fo plat th dctvty was pstd smla shaps SYSNOISE ad dvlopd dct EFBEM. d th dctvty was vashd fo sph modl. lso th ffct of th wat sufac was xamd by th compaso of th sults btw dct EFBEM ad SYSNOISE fo a udwat cas. Ths compasos showd satsfactoy sults ad dvlopd dct EFBEM s appld to ba typ stuctu at hh fqucy. CKNOWLEDGEMENTS Ths sach was suppotd by Basc Scc Rsach Poam thouh th Natoal Rsach Foudato of Koa NRF fudd by th Msty of Educato Scc ad Tcholoy R R

12 It. J. Naval cht. Oca E. 03 5:39~ REFERENCES Blov V.D. Ryba S.. ad Tataovs B.D Popaato of vbatoal y absob stuctus. Sovt Physcs coustcs 3 pp.5-9. Bouth O.M. ad Bhad R.J. 99. Modls of spac-avad tcs of plats. mca Isttut of oautcs ad stoautcs Joual 303 pp Cho P.E Ey flow aalyss of coupld stuctus. PhD. Pudu Uvsty. Cm L. Hcl M. ad Ptsso B..T Stuctu-bo soud. Bl: Sp-Vla. L H.-W. Ho S.-Y. Pa D.-H. ad Kwo H.-W Ey flow bouday lmt mthod fo vbato aalyss of o ad two dmso stuctus. Shoc ad Vbato 5 pp Nfs D.J. ad Su S.H Pow flow ft lmt aalyss of dyamc systms : basc thoy ad applcato to bams. Joual of Vbato coustcs Stss ad Rlablty Ds pp Pa D.H Vbato pow flow aalyss of coupld plats ad box-typ stuctus. M.S. Thss. Soul Natoal Uvsty. Pa D.-H. Ho S.-Y. Kl H.-G. ad Jo J.-J. 00. Pow flow modl ad aalyss of -pla wavs ft coupld th plats. Joual of Soud ad Vbato 444 pp So S.-H. Ho S.-Y. ad Kl H.-G Pow flow aalyss of focd bam-plat coupld stuctus. Joual of Soud ad Vbato 595 pp Wa. Vlahopoulos N. ad Wu K Dvlopmt of a y bouday lmt fomulato fo comput hh-fqucy soud adato fom coht tsty bouday codtos. Joual of Soud ad Vbato 78- pp Wohlv J.C. ad Bad R.J. 99. Mchacal y flow modls of ods ad bams. Joual of Soud ad Vbato 53 pp.-9.