Dynamic Behaviour of Plates Subjected to a Flowing Fluid

Transcription

1 SES RNSCIONS on FLID MECHNICS Y. Krboua and. Lak Dnamc Bavour o la Subjcd o a Flowng Flud Y. KERBO ND.. LKIS Mcancal Engnrng Dparmn Écol olcnqu o Monral C.. 679, Succural Cnr-vll, Monral, Qubc, HC 7, CND aoun.lak@polml.ca brac: - Elac rucur ubjcd o a lowng lud undrgo a condrabl cang n r dnamc bavour and can lo r abl. In arcl w dcrb dvlopmn o a lud-old n lmn o modl pla ubjcd o lowng lud undr varou boundar condon. mamacal modl or rucur dvlopd ung a combnaon o a brd n lmn mod and Sandr ll or. mmbran dplacmn ld appromad b blnar polnomal and ranvral dplacmn b an ponnal uncon. Flud prur prd b nral, Corol and cnrugal lud orc, wrn rpcvl a uncon o acclraon, vloc and ranvral dplacmn. Brnoull quaon or lud-old nrac and a paral drnal quaon o ponal low ar appld o calcula lud prur. mprmabl condon nur conac bwn m o pla and lud. Ma and rgd marc or ac lmn ar calculad b ac ngraon. Calculad rul ar n raonabl agrmn w or analcal or. K-ord: - Vbraon, Fn lmn, la, onal low, Flud rucur nracon, Crcal vloc. Inroducon Sm o pla ubjcd o lud low ar on ound n conmporar ndur uc a nuclar racor and aropac. Gnrall ndur rqur g ra o lud low and low pla ckn. ndr condon, lng o pla cv rucur bcom vr ucpbl o alur. Earlr work n ld wr carrd ou on ngnrng racor (ER) m conng o man n pla ackd n paralll w narrow cannl bwn pla o l coolan low roug. Mllr [] wa r o prn a orcal anal prdcng crcal low vloc or dvrgnc. H anal appl a mod o nural qulbrum wrb prur and pla rorng orc ar balancd, ladng o a drvaon o crcal vloc o low or varou p o uppor. I mporan o undrln a moon o a pla cd b lud low dplac narb lud, and n lud racv moon ma urr dorm pla. Ecv lud racv moon a a cran low vloc ovr urac o pla rrrd o a dvrgn vloc, wc ma alo b condrd a crcal low vloc [8]. Ronbrg and Youngdal [] av ormulad a dnamc modl dcrbng moon o a ul pla n a paralll pla ambl. ound a good agrmn bwn rul o dnamc modl and a o nural qulbrum ud b Mllr []. r paralll pla ambl wr d b Gronngr and Kan [] o nvga low-nducd dlcon o ndvdual pla. modl owd a adjacn pla alwa mov n oppo drcon a g low ra, caung alrna opnng and clong o cannl. dcd a voln dnamc nabl a.9 m Mllr collap vloc. aumpon o Mllr [], and Ronbrg and Youngdal [] wr am. lnard prur drop pron ung onl a r-ordr appromaon. ambgan [9] rand cond-ordr rm n an amp o a r nlunc on abl. cond-ordr rm gnra an addonal abl crron n orm o an uppr bound on amplud o qua-ac dlcon or abl ocllaon. H drvd a nw pron or crcal vloc. Smar [7 and 8] prormd analcal and prmnal nvgaon on an MR-p la-pla ul lmn. prmnal rul [7] ow a or low vloc pla wll dorm a a rul o ac prur drnc n cannl bwn pla. g lud vloc a g- ISSN: Iu, Volum, prl 8

2 SES RNSCIONS on FLID MECHNICS Y. Krboua and. Lak amplud lur vbraon obrvd. lur do no appar blow a mnmum avrag war vloc rrrd o a lur vloc, wc appromal qual o wo m Mllr vloc o ambl. In analcal ud, Smar [8] ndcad a a pla ambl caracrd b wo vloc; Mllr vloc and lur vloc. On planaon o dnamc nabl a cng rqunc o ull-dvlopd urbuln low approac n-lud naural rqunc o pla. orcall, undr condon amplud o pla vbraon bcom larg. avr and nn [] udd dnamc bavour o a ngl la pla, on d o wc pod o g low vloc o a av lud uc a war. amnd varaon o naural rqunc accordng o ra o low. concludd a or a gvn ma rappor, nural on o abl ollowd b a on o ac nabl. r ag pla quckl rurn o nural abl, wc connu unl occurrnc o dnamc nabl. Kornck al. [9] condrd a la panl o nn wd and n lng mbddd n an nn rgd plan w unorm ncomprbl ponal low ovr uppr urac. udd pla wr conrand (clampd or mpl uppord) along r ladng and ralng dg. ca o a panl clampd a ladng dg and r a ralng dg wa nvgad bo orcall and prmnall. oband rul dmonra a a panl d a ladng and ralng dg lo abl b dvrgnc (ac nabl), wl canlvrd panl lo abl b lur. Or nvgaor av udd lud low c on dnamc bavour o rcangular pla;.. I [6], Dowll [] and Holm [5]. Mor rcnl, Km and Dav [8] dvlopd an analcal modl o a m o n rcangular la pla. r modl wa ud o nvga ac and dnamc nabl o m. Guo and adou [] udd orcall abl o rcangular pla w r d-dg n nvcd cannl low. rad pla a on dmnonal and cannl low a wo-dmnonal. Galrkn mod wa mplod o olv pla quaon, wl Fourr ranorm cnqu wa mplod o oban prurbaon prur rom ponal low quaon. nvgad vr pobl combnaon o clacal uppor a ladng and ralng dg o pla. concludd a dvrgnc and coupld mod lur ma occur or pla w an p o nd uppor, wl ngl mod lur onl ar or nonmmrcall uppord pla. Guo and adou [] av alo conducd a orcal ud o dro-lac nabl o rcangular paralllpla ambl. condrd pla a wodmnonal, w a n lng, and low ld wa aumd o b nvcd and r-dmnonal. wo p o nabl wr ound; ngl-mod lur and coupld-mod lur. alo concludd a rqunc a a gvn low vloc dcra a apc rao ncra and cannl g o pla-wd rao dcra. purpo o papr o dvlop a oldlud n lmn o ud dnamc rpon o a rcangular pla ubjcd o ponal low. nw n lmn prm u o oban low a wll a g rqunc o lud-rucur m w prcon or an combnaon o boundar condon wou cangng dplacmn ld. n lmn appld o mula a numbr o pla and o paralll pla ubjcd o lowng lud. mamacal modl or rucur dvlopd ung a combnaon o n lmn mod and Sandr ll or. vloc ponal and Brnoull quaon ar adopd o pr lud prur acng on rucur. Sold n lmn gomr o man urac o rcangular pla and co-ordna m ud or anal ar own n Fg..b. pcal our-nod lmn and nodal dgr o rdom ar own n Fg..a. Eac nod a dgr o rdom conng o n-plan and ou-o-plan dplacmn componn and r paal drvav.. Equlbrum quaon and dplacmn uncon o dvlop qulbrum quaon or rcangular pla, Sandr quaon or clndrcal ll ar ud aumng radu o b nn, θ= and rdθ =d. Bo mmbran and bndng c ar akn no accoun n or. I wor o no a Sandr ll or bad on Lov r appromaon or bu lad o ro ran or ca o rgd bod moon. dvlopd dplacmn uncon ror a convrgnc crra or propod n lmn. ISSN: Iu, Volum, prl 8

3 SES RNSCIONS on FLID MECHNICS Y. Krboua and. Lak qulbrum quaon o a rcangular pla accordng o Sandr or can b wrn a a uncon o dplacmn componn w rpc o rrnc urac: V V ( ) = (.a) V V ( ) = (.b) ( ) 55 = (.c) Nod, { δ } Nod j, { δ { } { j } δ =, V,,, } Y (a) (b) Fg. : (a) Gomr and dplacmn ld o a pcal lmn, (b) Fn lmn dcraon o a rcangular pla No a bo crcumrnal and longudnal brd lmn ud n dnamc anal o vrcal Lak and adou [] and oronal Lak and Slman [] opn clndrcal ll wr dvlopd bad on ac oluon o qulbrum quaon. approac ruld n a vr prc lmn wc lad o a convrgnc and l numrcal dcul rom compuaonal pon o vw. ncouragd u o dvlop a nw n lmn ung am approac or dnamc anal o rcangular pla. Y Nod l, { δ } l,,, la F. E. Nod k, { δ k } B Gnrall, ac oluon o qulbrum quaon or ca o rcangular pla dcul. o ovrcom w prn n-plan mmbran dplacmn componn n rm o blnar polnomal and ou-o-plan bndng dplacmn componn b an ponnal uncon. Hnc, dplacmn ld ma b dnd a ollow: (,,) = C C C C (.a) B B V (,,) = C5 C6 C7 C (.b) 8 B B π B ω (, ) C j = j= 9, (.c) wr and V rprn n-plan dplacmn componn o mddl urac n and Y drcon, rpcvl, ranvral dplacmn o mddl urac, and B ar pla dmnon n and Y drcon, ω naural rqunc o pla (rad/c), a compl numbr and C j ar unknown conan. Equaon (.c) can b dvlopd n alor r a ollow: (,,) = C9 C C C C B B C C 5 C 6 C 7 C 8 () B 6 B B 6B C9 C C C 6 B B 6 B B C C B 6 B can wr dplacmn, V and n mar orm: V = [ R]{ C} () wr [R] a mar o ordr () n wc componn ar and rm o Equaon (.a,.b and ) wou unknown conan ( ppnd) and { C } vcor or unknown conan. componn o la vcor can b drmnd ung wn-our dgr o rdom or a pla lmn a own n Fg.. dplacmn vcor o ac lmn gvn a: {} δ = { δ } { } { } { } }, δ j, δ k, δ l (5) Eac nod,.. nod, po a nodal dplacmn vcor compod o ollowng rm: ISSN: Iu, Volum, prl 8

4 SES RNSCIONS on FLID MECHNICS Y. Krboua and. Lak { } = {,V,,,, ( δ } (6) ) wr and V ar nodal n-plan dplacmn componn and rprn nodal dplacmn componn normal o mddl urac a own n Fg..a. B nroducng Equaon (.a,.b and ) no rlaon (5), lmnar dplacmn vcor can b dnd a: {} δ = [ ]{ C} (7) vcor { C} n Equaon (7) wll b n rplacd b gnrald dplacmn vcor o a quadrlaral n lmn. dplacmn ld ma b dcrbd b ollowng rlaon: V = [ R][ ] {} δ = [ N]{} δ (8) wr mar [N] o ordr ( ) dplacmn ap uncon o n lmn and rm o mar [] - ar gvn n ppnd.. Knmac Rlaon Sran-dplacmn rlaon or rcangular pla ar gvn a Sandr [5]: ε ε V ε V = (9) κ κ κ ( ) Subung dplacmn componn dnd n Equaon (8) no ran-dplacmn rlaonp (9), on oban an pron or ran vcor a a uncon o nodal dplacmn. {} ε = [ Q][ ] {} δ = [ B]{} δ () wr mar [Q] o ordr (6 ) gvn n ppnd.. Conuv Equaon r-ran rlaonp o an oropc rcangular pla dnd a ollow: { σ} = [ ]{ ε} () wr [] lac mar or an oropc pla and no bndng-mmbran couplng prn ( ppnd). Subung Equaon () no Equaon () rul n ollowng pron or r nor: { σ} [ ][ B]{} δ = () ma and n marc or on n lmn can b prd a: [ ] = [ B] [ ][ B] k ds (.a) S [ m ] [ N ] [ N]dS = ρ (.b) wr S lmn urac ara, pla ckn, ρ maral dn and [], [N] and [B] ar dnd n Equaon (, 8 and ), ubung m no Equaon (.a and.b) w oban: [ k ] [ ] ] [ ] [ ][ ] [ ] = Q Q dd (.a) [ m ] = [ ] S [ ] [ R] [ R] dd [ ] ρ (.b) wr and ar dmnon o an lmn accordng o and Y coordna, rpcvl. ngral ar calculad analcall ung Mapl mamacal owar. Flud-old nracon lud prur acng upon rucur gnrall prd a a uncon o ou-o-plan dplacmn and drvav.. vloc and acclraon. r rm ar rpcvl known a cnrugal, Corol and nral orc [6]. lud marc wll b combnd w old marc a ollow: [ M ] [ M ]]{ && δ } [ C ] [ C ]]{ & δ } (5) [ K ] [ K ]]{ δ } = { } wr [ M ], [ C ] and [ K ] ar global marc o ma, dampng and rgd o lac pla, K rprn nral, Corol [ M ], [ ] C and [ ] and cnrugal orc o ponal low and { δ } global dplacmn vcor. lmnar marc o old ar calculad n Equaon ().. Flud-old n lmn lud-old modl dvlopd bad on ollowng po: () lud low ponal; () vbraon lnar (mall dormaon); () lud man vloc drbuon ( ) conan ISSN: Iu, Volum, prl 8

5 SES RNSCIONS on FLID MECHNICS Y. Krboua and. Lak acro a pla con and (v) lud ncomprbl. akng aumpon no condraon, vloc ponal mu a Laplac quaon. rlaon prd n Caran m b: φ φ φ = (6) wr φ ponal uncon. Brnoull quaon gvn b: φ V = (7) r ρ = wr lud prur, lud vloc and ρ lud dn. componn o lud vloc along, Y, drcon, rpcvl ar dnd b: φ V = V = φ V = φ (8) wr man vloc o lud n - drcon. Fg. dpc a lud-old n lmn ubjcd o lowng lud on uppr urac. Inroducng Equaon (8) no (7) and nglcng non-lnar rm w can wr dnamc prur a old-lud nrac a ollow ( Fg. ): φ φ = ρ = (9) l V r = Fg. : Flud-old n lmn mprmabl condon nur conac bwn ll and lud. ould b: φ = () = ollowng para varabl rlaon aumd or ponal vloc uncon: φ (,,,) = F( ) S(,,) () wr F() and S(., ) ar wo para uncon o b dnd. j k Flowng lud Sold-lud nrac ollowng pron ma b dnd b nroducng Equaon () no () S(,, ) = () df( ) d For and n n lmn doman ( Fg. ) ponal and prur a nrac ar coupld b ranvr movmn o pla (,,) and drvav. Equaon () dcrb uncon S(,,) n rm o ranvr movmn o pla wc l var a a uncon o pla gomr and m. ror, movmn o lud a an pon on nrac (ncludng boundar and ) nmal lnkd o movmn o dg o rucur. n, ubung Equaon () no (), rul n ollowng pron or ponal uncon: F( ) φ (,,, ) = () df() d onl unknown uncon n Equaon () F(). B nroducng Equaon () no Equaon (6), w oban ollowng drnal quaon o cond ordr: d F( ) F( ) = () d wr: = π B oluon o la quaon : F() = (5) wr and ar unknown conan. B ubung Equaon (5) no Equaon () ponal uncon bcom: ( ) ( ) φ,,, = (6) df( ) d and ar wo conan o b drmnd ung lud boundar condon. no a a old-lud nrac (=) w av mprmabl condon () wc common or all ca. cond lm o lud (= ), w av a boundar condon corrpondng o r a rgd wall, an lac pla or an nn lud lvl. For ac ca w av a dnc oluon. Blow w wll calcula prur o lud acng on onl on d (= ) o pla. I lud prur ac on wo d, oal dnamc prur wll b a combnaon o prur corrpondng o lud boundar condon a bo op and boom (= ) urac o pla. prnd approac ma b adapd o modl curvd rucur ubjcd o lowng lud ISSN: Iu, Volum, prl 8

6 SES RNSCIONS on FLID MECHNICS Y. Krboua and. Lak uc a urbn blad. o accompl local marc mu b ranormd o global m bor amblng no global marc []... Flud-old n lmn ubjc o lowng lud w nn lvl o lud n lowng lud g on and/or undr pla ( and/or ) nn ( Fg. ), w aum a vr ar rom pla ponal null. boundar condon wrn a ollow: φ = (7) ±, l, Fg. : Flud-old n lmn ubjcd o lowng lud w nn g In ordr o avod an nn ponal, conan o Equaon (6) mu b null. Equaon () prm u o calcula cond conan. ponal pron bcom: F ( ) φ(.,, ) = (8) nroducon o Equaon (8) no rlaon (9), rul n ollowng pron or prur uncon ρ = (9) or: = ().. Flud-old n lmn ubjc o lowng lud boundd b rgd wall own n Fg., lud low bwn a rgd wall and an lac pla. provd anor boundar condon a = wn mprmabl condon akn no accoun. boundar j, Flowng lud δ k k { } Fludold nrac condon adopd b Lamb [], McLaclan [] and prd b: φ = () l = Rgd wall Flowng lud ( ) Flud-old nrac Fg. : Flud old n lmn n conac w lowng lud boundd b a rgd wall ng Equaon () and () w can calcula conan and corrpondng o la boundar condon. Subung conan no (6) w oban: F ( ) ( ) φ(.,, ) = ( ) () Rplacng Equaon () no (9), corrpondng dnamc prur bcom: ρ ( ) = () ( ) or: = ().. Flud-old n lmn ubjc o lowng lud boundd b lac pla n lud low roug wo paralll lac pla ( Fg. 5) wo ranvr vbraon mod, npa and ou-o-pa, ould b condrd. mprmabl condon a old-lud nrac rman am or bo mod wl boundar condon a = cang accordng o mod o vbraon (n-pa or ou-o-pa).... In-pa mod In ca o n-pa mod boundar condon a lud lm = prd a ollow []: ISSN: Iu, Volum, prl 8

7 = = φ (5) Fg. 5: Flud-old n lmn n conac w lowng lud boundd b an lac pla Smlarl, and can b calculad b nroducng Equaon (6) no rlaon () and (5). ubuon o conan n Equaon (6) nabl u o dvlop ollowng pron or ponal: ( ) = ) ( ) (,,, φ (6) B rplacng Equaon (6) no (9) w oban ollowng pron or prur: = ρ (7) or: = (8)... Ou-o-pa mod boundar condon or ou-o-pa mod prd a ollow []: = = φ (9) Hr agan, and can b calculad ung Equaon (6, 9 and ). ponal uncon ma b wrn a ollow: = ) ( ) ( F ( ) ).,, ( φ () ng Equaon () and (9) w oban ollowng dnamc prur: = ) ( ) ( ρ () or: l Flud-old nrac Flowng lud ( ) Elac all = (). Calculaon o Flud-Inducd Forc lmnar vcor o lud-nducd orc prd b: { } [ ] { } = v ds N F () wr [ ] N ap uncon mar o n lmn dnd n Equaon (8), { a nor prng prur appld b lud on pla and S lmnar lud-rucur nrac ara. v } B placng mar [ o Equaon (8) no Equaon (), lmn load vcor bcom: N] { } [ ] [ ] [ ] { } = v ds R F () dnamc prur o Equaon (9,, 7 and ) ma b rwrn a: = (5) wr (=, ), dpnd on boundar condon ( Equaon,, 8 and ) and onl non-ro componn n prur nor{ } v. Subung Equaon (.c) no (5) prur pron bcom: = π π (6) ranvral dplacmn can b parad rom Equaon (8) a ollow: [ ][ ] {} δ R = (7) wr [R ] a () mar gvn n ppnd. Subung Equaon (7) no (6), w oban ollowng pron or prur: { } [ ][ ] {} {} {} = δ π δ π δ R v & & & (8) B combnng Equaon () and (8) lmn load vcor gvn b ollowng rlaon: SES RNSCIONS on FLID MECHNICS Y. Krboua and. Lak ISSN: Iu, Volum, prl 8

8 SES RNSCIONS on FLID MECHNICS Y. Krboua and. Lak [ ] [ R] [ R ][ ] {} && δ [ ] ds π { F} = [ ] ] [ R] [ R ][ ] {} & δ ds (9) [ ] ] [ ] [ ][ ] {} π R R δ ds No a orc nducd b a lowng lud a uncon o acclraon, vloc and dplacmn o old n lmn. From Equaon (9) w can para addd marc nducd b lowng lud, rpcvl dcrbng nral, Corol and cnrugal c a ollow: [ m ] [ ] [ ] [ R] [ R ][ ] d = (5.a) π [ c ] = [ ] π [ k ] = [ ] [ ] [ R] [ R ][ ] d [ ] [ R] [ R ][ ] d (5.b) (5.c) Dnamc qulbrum rqur a combnaon o la r lmnar marc w corrpondng marc gvn n Equaon (). Egnvalu problm rcangular pla ubdvdd no a r o quadrlaral n lmn uc a ac o m a mallr rcangular pla ( Fg..b). poon o nodal pon o lmn ar con n uc a wa a local and global coordna ar paralll. n n-ou compur cod a bn dvlopd o abl rucural marc o ac lmn bad on quaon dvlopd ung orcal approac. global marc mnond n Equaon (5) ar oband b uprmpong marc or ac ndvdual lmn. r applng boundar condon marc ar rducd o quar marc o ordr 6*N-NC, wr N numbr o nod n rucur and NC numbr o conran appld. gnvalu problm olvd b man o quaon rducon cnqu. Equaon (5) ma b rwrn a ollow: [] [ M ] [ M ] [ C] wr: {&& δ } {& δ } [ M ] [ ] [] [ K] {& δ } { δ } = {} (5) [ M ] = [ M ] [ M ], [ C ] = [ C ], [ K ] = [ K ] [ K ], { δ } global dplacmn vcor and rucural dampng nglcd. gnvalu problm gvn b: [ DD ] Λ [ I ] = (5) wr : [ I ] [ ] [ ] [ ] [ ] [ ] DD =, Λ = / ω and [ I ] K M K C dn mar. 5 Rul and dcuon prcon o n lmn mod dpnd on numbr o lmn ud o dcr pcal problm. r o calculaon ror o drmn rqu numbr o lmn or a prc drmnaon o naural rqunc. varaon o r v rqunc vru numbr o n lmn o rcangular pla mpl uppord on our d plod n Fg. 6 and ow mnmum rqurd numbr o lmn o aur a convrgnc n drmnng bo low and g rqunc. valu o maral and gomrcal propr ud n calculaon ar: Young modulu E=96Ga, maral dn ρ=786 kg/m, oon rao ν=., ckn =.5 mm, =69.6 mm and B=.8mm. Eg lmn ar ucn o calcula wo r mod, wra or or mod convrgnc rqur a la wn v lmn. numbr o lmn rqurd b prn mod muc lowr an a o or ng approac. In all o ollowng ampl 6 lmn ar ud, wc aur a rul wll b ndpndn o m. In ordr o ow a dvlopd modl provd accura rul, calculaon wr prormd on am pla ud n convrgnc. r naural rqunc ar ld n abl along w analcal rul and NSYS oupu daa. I can b n a prn mod gv arl good rul compard o ac oluon and commrcal n lmn cod n nv ud a bn conducd o old n lmn n vacuum n rrnc [7]. Fr vbraon o rcangular pla wr oband or a var o boundar condon and pla dmnon rao (/B). compud naural ISSN: Iu, Volum, prl 8

9 SES RNSCIONS on FLID MECHNICS Y. Krboua and. Lak rqunc wr compard o o oband b or or [] and rom prmn. rul wr n vr good agrmn. Frqunc (H) mod mod mod mod mod Numbr o lmn Fg. 6: v r naural rqunc o a ourdd mpl uppord pla a a uncon o numbr o lmn abl Naural rqunc (H) o a pla mpl uppord a our d Mod rn modl NSYS (ll 6) nalcal oluon La In ollowng ampl, w prn om calculaon o old-lud modl n ca o pla ubjcd o lowng lud. o pu rul n non-dmnonal orm, ollowng paramr ar dnd: ρ B ψ = (5) ρ ρ ω = B ω (5) K ρ = B (55) K wr ψ ma rao, ω dmnonl rqunc, and dmnonl vloc. r ampl a n pla clampd on wo oppo dg ( Fg. 7) ubjcd o lowng lud on uppr and lowr urac. lud lvl ( and/or ) aumd o b nn. corrpondng dnamc prur would b wc prur calculad n Equaon (9). gomrc rao and dmnonl paramr or rucur ar: ψ =.9, /, /, / B = Numrcal rul wr ud o plo curv own n Fg. 8. no a pla bcom ncrangl vulnrabl o ac nabl a ra o low ncra. Bond crcal vloc w pc occurrnc o a larg dlcon o pla [8]. Dmnonl rqunc la Cro con Fg. 7: la clampd on wo oppo dg ubjcd o lowng lud ( and Y ) mod mod mod Dmnonl vloc Fg. 8: Varaon o rqunc ϖ vru lud vloc or pla clampd on wo oppo dg ubjcd o lowng lud. In ordr o nvga c o boundar condon on crcal vloc valu, am pla condrd n r ampl udd agan, bu m w wo oppo dg mpl uppord ( Fg. 9) nad o wo clampd dg. varaon o dmnonl rqunc or r r mod vru dmnonl vloc o lud own n Fg.. I can b n a crcal vloc or r r mod ar lowr an o o clampd pla. I can b concludd a clampd pla ar mor abl an mpl uppord pla ISSN: Iu, Volum, prl 8

10 SES RNSCIONS on FLID MECHNICS Y. Krboua and. Lak wc n good agrmn w obrvaon o Km and Dav [8]. Dmnonl rqunc Fg. 9: la mpl uppord on wo oppo dg ubjcd o lowng lud ( and la ) mod mod mod Dmnonl vloc Fg. : Varaon o rqunc ϖ vru lud vloc or a pla mpl uppord on wo oppo dg ubjcd o lowng lud. la Fg. : Canlvrd pla ubjcd o lowng lud ( and ) av calculad crcal vloc corrpondng o r r mod n ca o a canlvrd pla ( Fg. ) wc a am maral and gomrcal paramr a o o wo prvou ampl. ca o a canlvrd pla ubjcd o lowng lud on ncounrd n pracc. In abl w av compard crcal vloc o a canlvrd pla w o o pla a ar mpl uppord and clampd on wo oppo dg ( Fg. 8 and ). conclud 8 a canlvrd pla mor vulnrabl o ac nabl abl : Dmnonl crcal vloc ( ) o pla w varou boundar condon Dmnonl crcal Boundar vloc = B ρ K condon Mod Mod Mod Clampd on wo oppo dg (Fg. 6) Smpl uppord on wo oppo dg (Fg. 8) Canlvrd (Fg.) aralll-pla ambl ar on ud n powr nuclar racor. Man n pla ar ackd n paralll and bwn pla ar narrow cannl o l coolan low roug ( Fg. ). ll pla av am and ar unorml drbud. n cannl g rlavl low, knc nrg ravl roug lud rom on pla o anor. Vbraon o pla mod drbuon o prur and vloc along cannl. ror lud n cannl nrac mulanoul w bo uppr and lowr pla. mnond prvoul, pla vbra accordng o wo mod; n-pa and ou-o-pa. dnamc prur or ac ca dnc. I a bn provn a dnamc bavour o paralll-pla ambl clampd a wo laral wall can b ucnl prdcd ung onl on pla wc vbra n oppo drcon rlavl o adjacn pla []. modl o Gronngr and Kan [] owd a adjacn pla alwa mov n oppo drcon a g low ra, caung alrna opnng and clong o cannl. condon provd lowr crcal vloc []. Mllr [] drvd rlaon prng crcal vloc o an ngnrng racor m. For ca o a la pla clampd on wo oppo dg ( Fg..a), dvlopd ormula : 5E Mllr = (56) ρ ( ν )B wr E Young modulu, ρ lud dn, cannl g, ν oon cocn, pla ckn and B pla wd. ISSN: Iu, Volum, prl 8

11 SES RNSCIONS on FLID MECHNICS Y. Krboua and. Lak ng our numrcal modl, w av calculad ou-o-pa vbraon o an nrnal pla or paralll-pla ambl own n Fg..a. Rgd wall Flowng lud Rgd wall Flowng lud Clampd pla,, (a) Smpl uppord pla,, (b) Fg. : Engnrng racor (ER) m ubjcd o lowng lud (a) paralll pla clampd o laral wall (b) paralll pla mpl uppord a laral wall corrpondng dnamc prur would b wc prur calculad n Equaon (). Fg. ow varaon o dmnonl crcal vloc o r ou-o-pa mod a a uncon o cannl g o pla lng rao compud b prn mod and b Mllr analcal ormula. B amnng Fg. clar a crcal vloc or a gvn pla can b ncrad b ncrang cannl g. In am gur w can a a low lud g good agrmn ound bwn numrcal and analcal rul, owvr or g lud lvl w obrv a condrabl dcrpanc. can b pland b ac a Mllr ormula drvd pccall or paralll pla m w vr low rao ( /). On or and, mporan o no a bond a cran lud g, ncrang or don av an nlunc on dnamc bavour o a pla ubjcd o a lowng lud. wll b conrmd n ollowng ampl. Dmnonl crcal vloc 8 6 Mllr rn mod,,,,,5,6,7 Cannl g o pla lng (/) Fg. : Dmnonl crcal vloc ( ) o an nrnal pla n an ER m clampd on wo oppo d ( Fg..a) vru cannl g o pla lng rao ( /), ψ =.9, / B =. Dmnonl crcal vloc 5 Mllr rn mod,,,,,5,6,7 Cannl g o pla lng (/) Fg. : Dmnonl crcal vloc ( ) o an nrnal pla n an ER m mpl uppord on wo oppo d ( Fg..b) vru cannl g o pla lng rao ( /), ψ =.9, / B =. Mllr [] a alo drvd a rlaon prng crcal vloc o an ngnrng racor m wn paralll pla ar mpl uppord on wo oppo dg ( Fg..b), dvlopd ormula : 5E Mllr = (57) ρ B ( ν ) wr E Young modulu, ρ lud dn, lud lvl on pla, ν oon cocn and B wd o pla. ISSN: Iu, Volum, prl 8

12 SES RNSCIONS on FLID MECHNICS Y. Krboua and. Lak av ud old-lud n lmn dvlopd n work o calcula crcal vloc corrpondng o r mod wn adjacn mpl uppord pla mov n oppo drcon (ou-o-pa mod). rul ar oband or drn cannl g. Fg. prn dmnonl crcal vloc compud or nrnal mpl uppord pla n an ngnrng racor. agrmn bwn our rul and o calculad b Mllr ormula vr good, pcall or low cannl g. dnamc bavour o rucur ma b nluncd b cangng lud lvl and/or lud boundar condon. Varaon o crcal vloc a a uncon o lud g and lud boundar condon ar vrd n ollowng ampl. wll drmn lm valu o lud lvl ( lm ) bond wc ncra o lud lvl don av an nlunc on dnamc bavour o rucur. nall condrd ca o a pla clampd on wo laral d placd n a cannl o rgd wall ( Fg. 5). pla ubjcd o lowng lud on bo uppr and lowr urac. corrpondng prur wc prur calculad n Equaon (). av graduall ncrad lud g and calculad or ac ( ) valu corrpondng crcal vloc or r wo mod. Fg. 6 ow a nall crcal vloc ncrad b ncrang rao ( /). Howvr, r a lm valu or rao bond wc an ncra n lud lvl don cang crcal vloc. For ca o a pla ubjcd o lowng lud boundd b wo rgd wall lm valu o rao.5. Rgd wall r wo mod o an nrnal pla n a paralllpla ambl ( Fg..a.). dnamc prur appld on ac d gvn b Equaon (). own n Fg. 7, crcal vloc r ncra a cannl g ncra. n rao / rac a valu o, dmnonl crcal vloc rman conan vn w ncra cannl g. Dmnonl crcal vloc Mod Mod 5,5,5,5,5,5,55,65 Flud lvl o pla lng (/) Fg. 6: Crcal vloc o a ngl pla clampd on wo oppo dg ubjcd o lowng lud boundd b wo rgd wall ( Fg. 5) vru lud g o pla lng rao ( /), ψ =.9, B = Dmnonnl crcal vloc Mod Mod,5,,5,5,65,8,95,,5 Elac la Flow cro con Fg. 5: la ubjcd o lowng lud boundd b wo rgd wall av alo udd c o cannl g on crcal vloc corrpondng o Cannl g o pla lng rao (/) Fg. 7: Crcal vloc o an nrnal pla n paralll-pla ambl clampd on wo oppo dg ubjcd o lowng lud boundd b wo lac pla wc vbra n ou-o-pa mod vru cannl g o pla lng rao ( /), ψ =.9, / B =. 6 Concluon old-lud n lmn dvlopd or dnamc anal o pla ubjcd o dnamc ISSN: Iu, Volum, prl 8

13 SES RNSCIONS on FLID MECHNICS Y. Krboua and. Lak prur nducd b ponal low. rucural mamacal modl dvlopd bad on a combnaon o n lmn mod and Sandr ll or. n-plan and ou-o-plan dplacmn componn ar modlld ung blnar polnomal and ponnal uncon, rpcvl. gnral quaon o dplacmn uncon ar drvd rom qulbrum quaon o a rcangular pla. ma, dampng and n marc corrpondng o a old and lud ar drmnd b ac analcal ngraon or ac lmn. lud prur drvd rom a ponal; a uncon o acclraon, vloc and ranvr dplacmn o pla, rpcvl known a nral, Corol and cnrugal c. Svral pla w varou boundar condon wr udd. rqunc o vbraon wr calculad or ac man vloc o low and or ac lud g, unl crcal vloc wa racd. Eablmn o crcal vloc vr mporan or dgn o pla m ubjcd o lud low. no a boundar condon and lud lvl on pla rongl nlunc dnamc bavour o pla. dvlopd lmn can b ud or anal o rcangular pla w an boundar condon conrar o prvou analcal mod wc wr dvlopd or parcular ca. crcal vloc calculad ung our lmn agr wll w o oband ung analcal ormula drvd b Mllr, pcall or low lud g. lm valu o lud g wa calculad or a pla ubjcd o lowng lud boundd b wo rgd wall and or an nrnal pla n an ngnrng racor m. lud-old n lmn dvlopd n work can b adapd o ud dnamc bavour o rucur w mor compl orm ubjcd o lowng lud orc. Morovr, wor o no a prn mod can b ud or rcangular pla, r unorm or non-unorm (ckn or or gomrc dconnu) ubjcd o an boundar condon. Rrnc: [] E.H. Dowll, Nonlnar ocllaon o a lurng pla II, I journal Vol. 5, No., 967, pp [] R.D. Gronngr, J.J. Kan, Flow nducd dlcon o paralll la pla, Nuclar Scnc and Engnrng, Vol.6, 96, pp [] C.Q. Guo, M.. adou, Sabl o rcangular pla w r d-dg n wodmnonal nvcd cannl low, SME, Journal o ppld Mcanc, Vol 67, No.,, pp [] C.Q. Guo, M.. adou, nal o drolac nabl o rcangular paralllpla ambl, SME, rur Vl and png Dvon, Vol.89, 999, [5].J. Holm, Burcaon o dvrgnc and lur n low-nducd ocllaon: a n dmnonal anal, Journal o Sound and Vbraon, Vol. 5, No., 977, pp 7-5. [6]. I, rolac nabl o mpl uppord panl n ubonc low. I journal, Vol. 65, 965, pp. 77. [7] Y. Krboua,.. Lak, M. oma L. Marcoullr, Hbrd mod or vbraon anal o rcangular pla, Nuclar Engnrng and Dgn, Vol.7 No.8 7, pp [8] G. Km, D.C. Dav, Hdrodnamc nabl n la-pla-p ul ambl, Nuclar Engnrng and Dgn, Vol.58, No., 995, pp. -7. [9]. Kornck, E.H. Dowll, J. O'Brn, On arolac nabl o wo-dmnonal panl n unorm ncomprbl low, Journal o Sound and Vbraon, Vol. 7, No., 976, pp [] K.H. Jong, G.H. Yoo, S.C. L, Hdrolac vbraon o wo dncal rcangular pla, Journal o Sound and Vbraon, Vol.7, No.- 5,. pp [] H. Lamb, On Vbraon o an Elac la n Conac w ar, rocdng o Roal Soc o London, Vol.98, No. 69, 9, pp [] N.. McLaclan, ccon o nra o lbl dc vbrang n a lud, rocdng o cal Soc, Vol., No., 9, pp [] D.R. Mllr, Crcal low vloc or collap o racor paralll-pla ul ambl, SME, Journal o Engnrng or owr Sr, Vol. 8, No., 96, pp [] G.S. Ronbrg, C.K. Youngdal, mpld dnamc modl or vbraon rqunc and crcal coolan low vloc or racor ISSN: Iu, Volum, prl 8

14 SES RNSCIONS on FLID MECHNICS Y. Krboua and. Lak paralll pla ul ambl, Nuclar Scnc and Engnrng, Vol., No., 96, pp. 9-. [5] J.L. Sandr, n mprovd r appromaon or or n ll, 959, NS R-. [6]. Slman.. Lak, Vbraon anal o anoropc opn clndrcal ll ubjcd o a lowng lud, Journal o Flud and Srucur, Vol., No., pp. -. [7] G.E. Smar, Sac and dnamc drolac nabl n MR-p ul lmn, Nuclar Engnrng and Dgn, Vol. 7, No. 6, 968, pp [8] G.E. Smar, Sac and dnamc drolac nabl n MR-p ul lmn -, Nuclar Engnrng and Dgn, Vol. 9, No., 969, pp. 5-. [9] M.. ambgan, Scond-ordr c a rlad o crcal coolan low vloc and racor paralll pla ul ambl, Nuclar Engnrng and Dgn, Vol. 5, No., 967, pp [] D.S. avr,.e. nn, drolac abl o a la pla, ranacon o SME. Sr E, Journal o ppld Mcanc, Vol. 7, No., 97, pp [] Y. Krboua,.. Lak, M. oma, L. Marcoullr, M H. ooran, Dnamc nal o Coupld-lud Srucur Sm, SES ran. on ppld and orcal Mcanc, Vol., No., 6, pp65-7. [].. Lak, M.. adou, Fr vbraon o clndrcal ll parall lld w lqud, Journal o Sound and Vbraon, Vol. 9, No., 97, pp. -5. []. Slman,.. Lak, Vbraon anal o anoropc opn clndrcal ll ubjcd o a lowng lud, Journal o Flud Srucur, Vol. 997, pp.. [].. La, r vbraon o rcangular pla, Journal o Sound and vbraon, Vol., No., 97, pp R (,9 ) = R (, ) = R (,) = B R (, ) = R (, ) = B R (, ) = B R (,5 ) = 6 R(,6 ) = B R(,7 ) = B R (,8 ) = B R(,9 ) = 6 B R (, ) = B R (,) = 6 B R (, ) = B R (,) = B R (, ) = 6 B. Mar [R ] () R (, j ) = R(, j ) or j= o R (, j ) = or = o and j= o. Non-ro lmn o Mar [Q] Q (, ) = Q (, ) = B Q (,7 ) = B Q (,8 ) = B Q (, ) = B Q (, ) = B Q (,6 ) = Q (,8 ) = B Q(, ) = Q (,5 ) = Q(,6 ) = B Q(,9 ) = B Q(, ) = B Q(, ) = B Q(, ) = 6 B Q(, ) = 6 B Q( 5, ) = B Q( 5,7 ) = B Q( 5,8 ) = B Q( 5, ) = B Q( 5,) = B Q( 5, ) = B Q( 5, ) = 6 B Q( 5, ) = 6 B Q(6, ) = B Q( 6,6 ) = B Q(6,7 ) = B Q( 6,9 ) = B Q(6, ) = B Q(6,) = B Q( 6, ) = B Q( 6, ) = B Q(6, ) = B ppnd. Non-ro lmn o mar [R] () R (,) = R (, ) = R (, ) = B R (, ) = B R (,5 ) = R (,6 ) = R (,7 ) = B R (,8 ) = B.5 Elac mar lac mar [ ] o ordr (66). In ca o oropc maral non vanng rm o lac mar ar: = = D, = = K, = = νd, 5 = 5 = νk 55 = ( ν )D, 66 = ( ν )K wr K = E ( ν ) and D = E ( ν ), ISSN: Iu, Volum, prl 8

15 SES RNSCIONS on FLID MECHNICS Y. Krboua and. Lak.6 Non-ro lmn o mar [] - (,) =, (,) = (,) = (,9 ) = B ( 7, ) = (, ) = (7, )) = (, ) = (, 9) = B ( 5, ) = (7, ) = B ( 8, ) =, (6, ) = (6,8 ) = ( 8, ) = B, ( 8,8 ) = (7, ) = B ( 9, ) =, (, ) =, (, ) = (,9 ) = 6 ( 8, ) = (, ) = (,6 ) =.B (, ) = (, ) = (,) = (,5 ) = (,5 ) = (, ) = (, ) = = = (5, ) = (5,9 ) (5, ) = (5, ) 6 (6,5 ) = (6,) = 6 B (6,6 ) = (6, ) = B (7, ) = (7,6 ) = (7, ) = 6 B (7, ) = B (8, ) = (8,) = (8,5 ) = ( 8, ) = (9,5 ) = (9,) = (9,6 ) = ( 9, ) = (, ) = (, ) = B (,5 ) = B 6 B B 6 B (, ) = (,7 ) = (,9 ) = ( 8, ) = B B (,6 ) = (, ) = B (,) = (,6 ) = (, ) = (, ) = 6 B (, ) = (, ) = (,6 ) = (, ) = (, ) = (,) = 7 B (, ) = B 6 B (,9 ) = (, ) = 6 B (,5 ) = (, ) = (, ) = 8 B (,6 ) = (, ) = B (, ) = (,8 ) = (,5 ) = (,) = (, ) = (,) = 7 B (, ) = (,9 ) = (, ) = 8 B (,5 ) = (, ) = 6 B (,6 ) = (, ) = B (, ) = (,6 ) = (, ) = (,7 ) = (,8 ) = (,5 ) = (,) = (, ) = (,) = B (, ) = (,9 ) = (, ) = 7 B (,5 ) = (, ) = (, ) = 7 B (,6 ) = (, ) = 6 B (,6 ) = (,7 ) = (,8 ) = (,5 ) = (,) = (, ) = (,6 ) = (,7 ) = (,8 ) = ISSN: Iu, Volum, prl 8