EXPERIMENTAL INVESTIGATION OF THE ADDED MASS OF THE CANTILEVER BEAM PARTIALLY SUBMERGED IN WATER

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1 D Sdl t l Espintlno istživn dodn s n gdi dloično uonno u vodu ISSN UDC/UDK 64071:51961 EXPERIENAL INVESIGAION OF HE ADDED ASS OF HE CANILEVER BEA PARIALLY SUBERGED IN WAER Di Sdl, Žln Lozin, Di Vučin h phnon of ddd (vitul) ss of subgd stuctus nlyzd nuiclly nd xpintlly in this pp h clssicl concpt of odl ddd ss is nlyzd in plll with locl ddd ss tht is vlutd using psudo-sidul foc vcto A finit lnt stuctu odl is usd to dtin th locl ddd ss h odl updt nd th odl duction tchniqus ncssy, nd thi spctiv ols dscibd nd pplid in dtining th locl ddd ss h pocdu is iplntd to th cntilv b ptilly subgd into wt Expintl odl nlysis is pfod on cntilv b in th i nd on th s b ptilly subgd in th wt his sults in two sts of odl pts, nd th cusof thi diffncs is psntd nd xplind by th ddd ss h sults obtind in this wy cn b usd s bnch fo futh study nd copison with oth siultions Kywods: ddd ss, odl ss, odl duction, ptilly subgd cntilv b Espintlno istživn dodn s n gdi dloično uonno u vodu Pliiny nots Pthodno piopćn U du s nuiči i spintlno nlizi fnon dodn (vituln) s onstuci uonn u tućinu Stnddni odl dodn odln s s nlizi uspodo s pividno dodno lolno so o s odđu oištn vto psudo-ostt sil U odđivnu lolno dodn vituln s oisti s onstuci odlin todo ončnih lnt Pocdu zhtiv popvlni odl ončnih lnt n tlu spint o i duciu odl Nčin pin popv i duci odl u nlizi lolno dodn vituln s su pizni u du t disutin nihov ulog Postup piinn n onzolu dloično uonnu u vodu Espintln odln nliz izvšn n onzoli u zu t n isto onzoli dloično uonno u vodu Ov nliz d dv st odlnih pt, uzo nihov zli dodn s o pzntin i obšnn u ovo du Rzultti dobivni n ov nčin ogu s oistiti u dli istživni o polzn toč u siulci Klučn iči: dloično uonn gd, dodn s, odln s, duci odl 1 Intoduction Elstic subgd stuctus hv wid ngining ppliction spcilly in ocn ngining Undstnding th influnc of th fluid is ipotnt to pdict th dynic bhvio of th subgd stuctus It is wll nown tht th ntul fquncis of th subgd stuctus diffnt fo thos in vcuu h ffct of fluid focs on th subgd stuctus is psntd s ddd ss [1], which dcss th ntul fquncis of th stuctus fo thos which would b sud in th vcuu his dcs in th ntul fquncis of th subgd stuctus is cusd by th incs of th intic ngy of th fluid-stuctu syst without cosponding incs in stin ngy Lindhol t l [] invstigtd th vibtions of cntilv plts in i nd wt hy copd th sults with thoticl pdictions using sipl b thoy nd thin plt thoy A copison btwn nuicl siultions using th finit lnt thod nd xpintl sunts of ptilly nd fully subgd cntilv plts is shown in [3] Diffnt nuicl nd xpintl boundy conditions w th souc of 15 % discpncy h influnc of th f sufc, subgd lngth nd boundy conditions ws invstigtd by ny schs [4-6] Nuicl siultion of th fluid-stuctu intction ffct, clcultd in ts of th gnlizd ddd ss vlus indpndnt of fquncy using th boundy intgl qution thod [7] lso shows t lst 5 % o copd with xpintl sunts Ling t l [8] doptd n piicl ddd-ss foultion to dtin th fquncis nd od shps of subgd cntilv plts A pticul xpintl invstigtion using hniči vsni 18, 4(011), ducd scl odl of Fncis tubin [9] hs shown how to dtct th odl ddd ss Continud sch on th Fncis tubin [10] includs nuicl siultions, wh th ddd ss ffct of th suounding wt hs bn vlutd by coping th spctiv fquncis in i nd wt h ddd ss ffct is physiclly xplind bsd on th ngy thoy In this pp, xpintl invstigtion- odl nlysis is pfod on th cntilv b in i nd ptilly subgd in wt Also, th b is subgd in two diffnt tns to invstigt th influnc of tn siz h odl ddd ss fo th fist fou ods is clcultd Futh sch is hding to nuicl nlysis nd dtction of th locl ddd ss which quis nowldg of syst poptis li ss nd stiffnss tics nd odl poptis i ntul fquncis In od to cop xpintl nd nuicl dt, odl duction is ncssy h ducd odl vlidtion hs bn don by coping th xpintl nd nuicl ntul fquncis h quid odl updt is cid out using optiiztion of th obctiv function by gntic lgoiths hoticl bcgound In this pp, th b ptilly subgd in wt will b clld th odifid b, nd th b tht is not subgd th gnuin b h qution of otion fo th gnuin b cn b xpssd s x Cx Kx F, (1) wh, Cnd K th ss, dping nd stiffnss 589

2 Expintl invstigtion of th ddd ss of th cntilv b ptilly subgd in wt D Sdl t l tics spctivly, xis th displcnt vcto nd F th xcittion foc vcto Fo sll vibtion plituds of th b, th fluid-stuctu intction ffcts only th stuctul ts of th otion qution In this cs th ss, dping nd stiffnss tics odifid h qution of otion fo th odifid b is obtind using qution (1) nd ppnding dditionl ts, C nd K to th gnuin ss, dping nd stiffnss tics spctivly hs ddd ss, dping nd stiffnss tics dscib th ffct of wt w x C x C C x K K x F w x K w x F h stuctu dfind using qution (1) posssss odl pts g dpd ntul fquncis ωd, ntul fquncis ω, dping tios ζ nd od shps Ψ h indx indicts th pticul vibtion od h odl pts usully dtind by xpintl odl nlysis nd thy th odl odl of th stuctu h odl odl lso consists of th odl ss, odl dping c nd odl stiffnss c Ψ Ψ Ψ Ψ CΨ KΨ h no uniqu vlus fo th odl ss, odl dping nd odl stiffnss s thy dictly ltd to th scling thod which hs bn usd to dfin th od shp Howv th tio btwn ths vlus is uniqu nd th odl pts cn b xpssd s d 1 () (3) ( 4) c ( 5) ( 6) 1 w 1 1 If th stiffnss ffct of th wt is nglctd ( = 0), which is sonbl ssuption fo ost stuctus, it is possibl to obtin th tio / his tio dscibs th pcntg of odl ddd ss nd cn b clcultd fo nown undpd ntul fquncis of th gnuin nd odifid b Howv, xpintl odl nlysis povids th dpd ntul fquncis nd dping tio, nd substituting qution (4) into qution (9) it lds to th 1 w d 1 dw 1 (9) (10) h bov qution cn b usd to find th pcntg of odl ddd ss fo ll sunt ods h iplnttion of this qution dos not qui th us of od shps Pcticlly this ns tht only on fquncy spons function is sufficint to clcult th pcntg of odl ddd ss Of cous, suing o fquncis spons functions will ld to o libl sults 3 Expintl nlysis Expintl odl nlysis is cid out using cntilv b in th i nd gin th s b ptilly subgd in wt sunts on th ptilly subgd b w cid out in two tns of diffnt sizs In this wy th influnc of th ount of wt ound th b is invstigtd, i th influnc of th distnc btwn th b nd th wll of th tn is studid h lngth of th b is 0, 65 nd coss-sction of th b is 0, 05 0, 01 h b is d of stl, Fig 1 h qutions (3) - (6) lso vlid fo th odifid b h odl ss, odl dping nd odl stiffnss of th odifid b cn b xpssd using th spctiv odl ss, odl dping nd odl stiffnss of th gnuin b nd dding th dditionl ts in siil wy s in qution () c w w w c c ( 7) If th spctiv qution (5) fo th odifid nd gnuin b is dividd w w w, ( 8) nd qution (7) substitutd into qution (8), th qution (9) is obtind Figu 1 Gnuin b 590 chnicl Gztt 18, 4(011),

3 D Sdl t l Espintlno istživn dodn s n gdi dloično uonno u vodu bl 1 sud ntul fquncis nd dping tio Gnuin b odifid b - odifid b - lg tn sll tn od d /Hz /% dw /Hz w /% dw /Hz w / % 1 18,14 0, ,93 0, ,00 0, ,07 0, ,99, ,34 1, ,57 0, ,71 0, ,66 0, ,59 0, ,94 0, ,00 0,88930 h b is dividd into tn lnts nd lvn nods h xpintl odl nlysis is pfod by th ipct tst Fo this pupos, two light cclots usd with ss of 3, gs ch to void thi influnc on th sunt dt h cclots w plcd in th thid nd sixth nod Whn suing in th tn, th lst lnt of th b is isd in th wt h distnc btwn th nd of th b nd th botto of th tn is 3 c, so th lvl of th wt in th tn is 9,5 c h lngth nd width of th lg nd sll tn 50 5 c nd 5 14 c spctivly h b is plcd in th iddl of th lg tn, Fig, nd c fo th dg in th sll tn, Fig 3 Figu 3 odifid b, sll tn Figu odifid b, lg tn odl nlysis is pfod using th ulti-fnc polynoil thod h dution of on sunt is 8,19 sconds, duing which points collctd Lin vging is usd fou tis h sults of th sunt fo th fist fou ods shown in b 1 In Fig 4, n xpl of th sud fquncy spons function is givn Fig 5 shows th gnifiction of th scond od fo Fig 4, in which th diffnc btwn th gnuin nd th odifid b cn b obsvd h sults of xpints suizd in b 1 wh th odl dping tio ξ, ccoding to th cocil softw fo xpintl odl nlysis is givn in pcntgs to b o dbl, so fo xpl ξ = 0, % hs ning of ξ = 0, h sults in b 1 w xpctd, th dpd ntul Figu 4 Expl of sud fquncy spons functions fquncis of th odifid b low nd th cosponding dping tio is high Also, coping th sults btwn th lg nd th sll tn, it is obvious tht th lg tn dps th b o nd ducs th dpd ntul fquncis Now, th odl ddd ss cn b clcultd using th qution (5) h pcntgs of odl ddd ss fo both css, lg nd sll tn, shown in b b shows tht th highst odl ddd ss is in hniči vsni 18, 4(011),

4 Expintl invstigtion of th ddd ss of th cntilv b ptilly subgd in wt D Sdl t l It is vidnt fo qution (14) tht th diffnc in fquncis is cusd by th chng in stiffnss nd ss of th stuctu Rbing tht nd K tics with vlus qul to zo xcpt fo thos tht blong to odifid lnts, qution (14) cn b ducd to odifid lnts, nd nd K cn b xpssd s functions of ss nd stiffnss tics of finit lnts K 1 n n 1 δ K δ (15) Figu 5 Zoo of on od of sud fquncy spons functions th fist od nd it gdully dcss fo both css 4 Nuicl nlysis bl Pcntg of odl ddd ss od Lg tn / % Sll tn / % 1 0,1479 0,1388 0,0971 0, ,043 0, ,0347 0,079 h finit lnt odl of th b is quid to dtin th vitul locl ddd ss on th isd pt of th b h bcgound of nuicl nlysis is bsd on psudo-sidul foc vcto [11] h qution of otion fo th odifid b is givn by () nd th cosponding chctistic qution (nglcting th dping) cn b divd nd wittn s wh δ nd δ th popotionl ss nd stiffnss odifiction fctos fo lnt spctivly, nd n is th nub of odifid lnts In this wy, th pobl is ducd to two unnowns p odifid lnt ( δ nd δ) Sinc qution (14) cn b syst of ov-dtind qutions, (fo nub of sud ods > nub of odifid lnts), it is ncssy to wit qution (14) in so dqut wy Using A B Ψ Ψ K Ψ Ψ δ δ1, δ,, δn δ δ, δ,, δ 1 n (16) (17) λ (18) 1,,, p wh p is th nub of sud ods, th qution (14) cn b xpssd in th fo ( K K ) ( )( ) ( Ψ Ψ ) 0, (11) A, B δ, δ λ (19) wh Δω is th diffnc in ntul fquncis nd ΔΨ th diffnc in od shps btwn th gnuin nd th odifid b Upon xpnsion, qution (11) givs ( K ) Ψ Ψ ( Ψ ( K ) Ψ ( K Ψ ) Ψ Ψ ( K ) 0 ) Ψ (1) h fist t is th chctistic qution fo th gnuin b nd thus it is qul to zo Nglcting high od ts nd p-ultiplying ll ts by yilds Ψ Ψ ( K ) Ψ Ψ 0 Ψ ( K Ψ ) Ψ (13) Agin, th fist t is th chctistic qution fo th gnuin b, hnc it is qul to zo If Ψ is ss nolizd, th qution (13) cn b siplifid to Ψ K Ψ Ψ Ψ (14) Applying th lst squ thod in solving qution (19), th unnowns cn b found fo th following xpssion 1 δ, δ A, B A, B A, B λ ( 0) In od to find th popotionl odifiction fctos ccoding to qution (0), it is ncssy to hv th finit lnt odl of th gnuin stuctu (ss nd stiffnss tics) Also, th od shps nd ntul fquncis of th gnuin stuctu quid h only dt quid fo th odifid stuctu th spctiv ntul fquncis, nd tht is th dvntg of this pocdu Anoth dvntg of th pocdu is th dtction of stiffnss odifiction If th is no significnt odifiction in stiffnss, s with subgd stuctus, this infotion cn sv s vlidtion bcus in this cs it is xpctd to b zo h dwbc of this pocdu is th quint tht two ntul fquncis nd to b nown in od to obtin th popotionl odifiction fctos fo only on lnt his ns tht it is ncssy to hv stuctu with loclizd odifiction nd finit lnt odl with sll nub of lnts h ptilly subgd b (stuctus) is vy suitbl pttn bcus th odifiction is loclizd to only th subgd 59 chnicl Gztt 18, 4(011),

5 D Sdl t l Espintlno istživn dodn s n gdi dloično uonno u vodu pt A sll nub of finit lnts cuss low solution, nd intoducs n o tht y b cucil in dtcting odifiction, popos vitul locl ddd ss o solv this pobl, in this pp, odl duction will b usd In this wy th ccucy will b tind in spit of th sll siz of th ss nd stiffnss tics du to th ducd nub of dgs of fdos (DOF) In th odl duction th nub of lnts to b subgd in wt ust b pt lss o qul to th hlf of th nub of ods tht sud 41 odl duction h syst quivlnt duction-xpnsion pocss (SEREP) [1] will b pplid In od to iplnt th duction of th odl it is ncssy to ptition th ts in th qution (1) to th st DOFs ' ' tht sud nd slv DOFs ' s' tht not sud x K K x f s s s ss xs Ks Kss xs fs (1) In th s wy, th cosponding od shps should b ptitiond Ψ Ψ () Ψ s h ltionship btwn th full st of finit lnt DOFs nd th ducd st of st DOFs cn b wittn s (3): x x, (3) wh is th tnsfotion tix nd is givn by (4): 1 Ψ Ψ Ψ Ψ (4) Ψ s h ss nd stiffnss tics of th ducd odl obtind by using th tnsfotion tix K R R K (5) hus th ducd ss nd stiffnss tics cn b usd in qution (0) h odl pts of th ducd syst qul to th odl pts of th full syst 4 Updtd finit lnt odl of th gnuin b As sttd in th pvious sction, in od to clcult th locl ddd ss, finit lnt odl of th gnuin b is quid B lnts with two DOFs p nod usd, on tnsltionl nd on ottionl h b is dividd into hundd lnts nd ft tht, th b is ducd to tn lnts nd lvn nods, whby in ch nod only th tnsltionl DOF is lft h dinsions of th b th s s in th sction h b is d of stl, nd its til poptis tn s: lstic odulus is 10 GP nd dnsity g/ h obtind ntul fquncis shown in th b 3 bl 3 Nuicl ntul fquncis, odl of gnuin b od Ntul fquncis / Hz 1 19,84 14, ,1 4 68,18 Coping xpintl nd nuicl ntul fquncis fo th gnuin b, b 1 nd b 3, diffnc cn b obsvd h vg diffnc is bout 10 % Such lg diffnc cn cus pobls in th dtction of th locl ddd ss h pobl cn b spcilly wll obsvd in b 1, i th diffnc btwn th gnuin nd odifid b in ntul fquncis is lss thn 5 % In this wy w ty to dtct chngs of 5 % using th dt which initilly dvit 10 % It is obvious tht th finit lnt odl of th gnuin b ust b updtd so tht its ntul fquncis cospond to thos sud 41 Finit lnt odl updt It is vy ipotnt to st th dg of th pobl poply nd slct th optiiztion pts nd thi vlus ndd to succssfully updt th odl Whn ths pts chosn, it should b tn into ccount tht thy povid ost of th infotion bout th xpintl odl As optiiztion pts, th til nd th gotic chctistics of th odl usully slctd In this cs, fiv pts will b updtd: width, hight nd lngth of th b, lstic odulus nd dnsity h hight, width nd lngth of th b slctd s updt pts bcus th sufc of th b is not idlly flt Howv, it is unlily to xpct thi significnt dvition fo th sud vlus givn in sction nd thi vlus boundd to ±5 % of thi spctiv initil vlus On th oth hnd, th til poptis lss ctin nd it is xpctd tht thi vlus y vy o hfo, th lstic odulus nd dnsity boundd to ±8 % of thi spctiv initil vlus h diffnc btwn th sud nd nuicl ntul fquncis fo th gnuin b is slctd s th obctiv function [13] f p 1 A X X, 3 (6) wh th indxadnots th nuicl ntul fquncis nd indx X th xpintlly obtind ntul fquncis h obctiv function is iniizd using gntic lgoiths Bcus of thi stochstic ntu, optiiztion by gntic lgoiths is ptd fiv tis hos dsign pts vlus tht yild th iniu obctiv function chosn s finl, b 4 h ntul fquncis of th finit lnt odl of th gnuin b obtind using th optiizd dsign pts givn in b 5 If w now cop xpintl nd nuicl ntul fquncis of th gnuin b, b 1 nd b 5, uch sll diffnc thn bfo cn b notd h vg diffnc is ust 1 % It is xpctd tht pplying this o hniči vsni 18, 4(011),

6 Expintl invstigtion of th ddd ss of th cntilv b ptilly subgd in wt D Sdl t l bl 4 Updt vlus of dsign pts Dsign pts Initil Updt Width / 0,01 0, Hight / 0,05 0, Lngth / 0,65 0, Elstic odulus / GP 10 10, Dnsity / g/ ,7788 bl 5 Nuicl ntul fquncis of th gnuin b obtind using updtd dsign pts od Ntul fquncis / Hz 1 18,14 113, , ,36 ccut nuicl odl in qution (0) will giv good sults in dtcting th locl ddd ss 43 Locl ddd ss As fosid, th finit lnt odl is ducd to tn lnts tht qul to th xpintl odl h lowst lnt t th f nd of th odifid b is isd in th wt his ns tht w should find th popotionl odifiction fctos δ nd δ, ccoding to qution (0) of on (isd) lnt o do this, it is nough to now th ntul fquncis of two ods, but fo o ccut solutions th fist fou ods w usd h ss nd stiffnss tics of th isd lnt xtctd fo th ducd finit lnt odl h obtind vitul locl ddd ss fo th odifid b in th lg nd sll tn is shown in b 6 h obtind popotionl odifiction fctos confi th xpcttion tht th is no stuctul chng in stiffnss, i th popotionl odifiction fcto is qul to zo his ns tht th wt dos not psnt significnt sistnc popotionl to th b displcnts h ss popotionl odifiction fcto is high fo th b subgd in th lg tn thn in th sll tn his ws xpctd nd gs with sults obtind fo th odl ddd ss in b bl 6 Locl ddd ss, popotionl odifiction fctos od Lg tn Sll tn δ / % 0,0000 0,000 δ / % 0,31 0,547 5 Conclusion h nifsttion of ddd ss fo th b subgd in wt is invstigtd xpintlly nd nuiclly h obtind sults of th odl ddd ss nd locl ddd ss show th s tndncy, th ddd ss ffct is o doinnt in th lg tn thn in th sll tn h poposd pocdu fo dtining th locl ddd ss is vy usful bcus byond th ddd ss it lso givs th ddd stiffnsss tht cn b usd s contol pts It is shown tht th in pobl y is fo th istch btwn th finit lnt odl nd th xpintl odl, which is h solvd by pplying th odl updt h obtind popotionl odifiction fctos fo stiffnss cn b usd s pocss contol h sults fo ddd ss obtind in this xpint will b usd s bnch fo futh nuicl invstigtion 6 Rfncs [1] Bnnn, C E A Rviw of Addd ss nd Fluid Intil Focs, Dptnt of th Nvy, Pot Hun, (198) USA ( ) [] Lindhol, U S; Kn, D D ; Chu W H; Abson, H N Elstic vibtion chctistics of cntilv plts in wt // Jounl of Ship Rsch 9, 1(1965), pp 11- [3] Volcy, GC; ol P; Buu, ; nid, K So studis nd schs ltd to th hydo-lsticity of stl wo // Pocdings of th 1nd Euoch Colloquiu on nuicl nlysis of th dynic of ship stuctus, Pis, 1979, pp [4] uthuvppn, G; Gnsn, N; Vluswi, Not on vibtion of cntilv plt isd in wt // Jounl of Sound nd Vibtion, 63, (1979), pp [5] Fu, Y; Pic, W G Intction btwn ptilly o totlly isd vibting cntilv plt nd th suounding fluid // Jounl of Sound nd Vibtion, 118, 3(1987), pp [6] Kw, K Hydolstic vibtion of cicul plts // Jounl of Sound nd Vibtion, 01, 3(1997), pp [7] Egin, A; Ugulu, B Lin vibtion nlysis of cntilv plts ptilly subgd in fluid // Jounl of Fluids nd Stuctus, 17, 7 (003), pp [8] Ling, C C; Lio, C C; i, Y S; Li, W H h f vibtion nlysis of subgd cntilv plts // Ocn Engining, 8, 9, (001), pp [9] Rodiguz, C G; Egusquiz, E; Escl, X; Ling, Q W; Avlln, F Expintl invstigtion of ddd ss ffct on Fncis tubin unn // Jounl of Fluids nd Stuctus,, 5(006), pp [10] Ling, Q W; Rodiguz, C G; Egusquiz, E; Escl, X; Fht, ; Avlln, F Nuicl siultion of fluid ddd ss ffct on fncis tubin unn // Coputs nd Fluids, 36, (007), pp [11] G, ; Lui, E Stuctul dg idntifiction using syst dynic poptis // Coputs nd Stuctus, 83, (005), pp [1] O'Cllhn, J; Avitbl, P; Ri, R Syst quivlnt duction xpnsion pocss // Pocdings of VII Intntionl odl Anlysis Confnc, Ls Vgs, 1989, pp 9-37 [13] Schwz, B; Richdson, FEA odl Updting Using SD // Pocdings of XXV Intntionl odl Anlysis Confnc, Olndo, 007, pp 18- Authos' ddsss: Di Sdl, PhD Fculty of Elcticl Engining, chnicl Engining nd Nvl Achitctu, Univsity of Split Ruđ Bošović 3, 1000 Split, Coti -il: disdl@fsbh Žln Lozin, PhD Fculty of Elcticl Engining, chnicl Engining nd Nvl Achitctu, Univsity of Split Ruđ Bošović 3, 1000 Split, Coti -il: zlnlozin@fsbh Di Vučin, PhD Fculty of Elcticl Engining, chnicl Engining nd Nvl Achitctu, Univsity of Split Ruđ Bošović 3, 1000 Split, Coti -il: divucin@fsbh 594 chnicl Gztt 18, 4(011),