INFLUENE OF MPING ON THE ROLL MOTION OF SHIPS Er PESMN, niz YRKTR and Mtin TYLN STRT This papr analyzs th ffct of daping on nonlinar roll otion of ships advancing in ba sas. s it is known, roll daping is a vry iportant paratr in stiating ship rsponss in cal watr and wavs. Thrfor, it has bn studid by any rsarchrs in diffrnt ways. Howvr, it ss that it has bn far fro bing coplt and uch work nds to b don towards thorough undrstanding of th phnonon. This work stiats daping cofficints of a ship by using availabl thods. aping cofficints ar brokn down into various catgoris such as friction, wav aking, ddy aking tc. Influnc of forward spd is also considrd in th analysis. Frquncy doain solution of roll otion quation is thn found incorporating linar and nonlinar daping cofficints. Nonlinaritis ar introducd in daping and rstoring trs in th quation. It has bn concludd that daping plays a vry iportant rol on th roll otion of a ship and rducs pak aplituds to a considrabl lvl. Thrfor, it ust b tratd with utost car sinc it dictats otion aplituds dirctly.. INTROUTION Roll otion is th ost iportant phnonon for ships, coupld with a fw othrs, which ay lad to capsizing. Thrfor, any rsarchrs hav bn studying it in all aspcts to find satisfactory answrs to th physical phnonon. aping on th othr hand is th ost iportant paratr in roll otion quation aong othrs sinc it controls agnitud of th aplituds. Howvr, it is th ost difficult paratr to stiat bcaus of its coplx natur. Thr ar svral diffrnt coponnts in roll daping such as wav daping, lift daping, friction daping, ddy aking daping and bilg kl daping. Th abov-ntiond fors of daping wr prsntd by Ikda t al (97 in a vry coprhnsiv analysis of daping study. H introducd pirical thods in ordr to stiat various daping coponnts. Hino (9 analyzd any aspcts of roll daping by conducting a sris of odl xprints towards bttr undrstanding of roll daping. Schitk (97 also st out intrsting ways to prdict roll daping of naval ships. Haddara and uing (99 phasizd that inviscid daping was a function of forward spd. Haddara and Zhang (99 carrid out xtnsiv xprintal work with fishing vssl odls to prdict daping charactristics of ships. Thy suggstd a odification to Ikda s lift daping forula basd on th rsults of th xprints. Exprintal invstigation has bn carrid out by othr rsarchrs as wll. This work aks us of th availabl roll daping thoris to solv a nonlinar roll otion odl. It also utilizs nurical siulation tchniqus to apply th chosn odl to a sapl ship. Th dtails of th roll otion odl, daping and othr hydrodynaic charactristics and sapl odl ar givn in th following sctions.. SMPLE VESSEL Th sapl vssl is a id-siz, twin scrw fishing vssl. Th charactristics and body plan of th ship ar givn blow: L P 6...6. pth 7.. raft.. (loadd Δ 6 tons GM.7..9 M. LF.. (aft L.7. (aft Th ship is fittd with a st of bilg kls, shaft brackts and fin stabilizrs. Various loading conditions and corrsponding hydrostatic and hydrodynaic charactristics of th ship hav bn usd throughout th analysis. Th fin stabilizrs and ruddrs wr not considrd in th analysis bcaus of thir functionalitis. That ans thy ar controlld activly to ract to opposing forcs and ont whn ndd. Othr appndags on th othr hand, work as passiv anti rolling dvics and cannot b controlld lik othrs. Various hydrostatic and stability charactristics of th ship wr coputd by using a corcial softwar packag. Fin staer PEŞMN niz YRKTR Mtin TYLN Istanbul Tchnical Univrsity, Faculty of Naval rchitctur and Ocan Enginring 69 Maslak, Istanbul Turky. Istanbul Tchnical Univrsity, Faculty of Naval rchitctur and Ocan Enginring 69 Maslak, Istanbul Turky. Istanbul Tchnical Univrsity, Faculty of Naval rchitctur and Ocan Enginring 69 Maslak, Istanbul Turky.
ividing Equation ( throughout by (I + δi, and substituting th valus of cofficints, and ar substitutd rspctivly, it taks th for; && φ + b & φ + b & φ & φ + ω L λω N α cosω t φ φ + φ + φ ( whr; ΔGM ω φ I + δi ωφ GM ( (6 Figur. ody plan of th sapl vssl.. ROLL MOTION MOEL In gnral, quation of nonlinar roll otion can b writtn as follows; & φ + ( & φ, φ + ( φ M ( t ( In th abov quation, ovr dot dnots diffrntiation with rspct to ti. Mor spcifically, Equation ( ay b xprssd as follows undr th influnc of rgular sinusoidal wavs; ( I δi ω + && φ + ( & φ, φ + ΔGZ( φ α I cosω t In th litratur, rprsntation of daping and rstoring trs has bn handld diffrntly. Th rstoring tr which is chosn as an odd-ordr polynoial is also nonlinar in natur. Rstoring tr in th quation ay appar as cubic or quintic polynoials dpnding on th charactr of th GZ curv in qustion. Sotis, vn highr ordr polynoials ay bst rprsnt a particular GZ curv,.g. a svnth dgr polynoial. Howvr, highr ordr polynoials caus sowhat bulky anipulations throughout th solution sch. If an quation of nonlinar roll otion is to b stablishd undr th abov-ntiond assuptions, on ay nd up with th following for: ( I δi + Δ + && φ + & φ + & φ & φ L N ( φ + φ + φ ω α I cosω t ( ( b b L N ωφ φ v GM (7 L ( I + δi N (9 I + δi Evaluation of linar and nonlinar daping trs b L and b N ar assssd by Ikda s and Hino s approach. Mor laborat valuation of th daping trs and thir ffcts on th otion charactristics ar supplid in th following sctions. Othr rlatd cofficints of th roll otion quation ar givn in Equations ( through (. Equation ( incorporats various ffcts of ship s dynaic and nvironntal paratrs including daping, rstoring and wav xcitation. Th righting ars curv is approxiatd as a quintic GZ φ + φ + φ polynoial. officints, and of th polynoial ar dtrind by a nubr of dynaic and static charactristics of th GZ curv naly, tacntric hight GM, angl of vanishing stability φ v, and ara undr th curv as follows: ( GZ d GM ( dφ ( GMφ ( 6 ( GMφ v ( v Solution of th quation can b ad by a prturbation thod, Taylan (99, 996. Nurical siulation ay
also b carrid out asily by th ordr Rung-Kutta solvr.. LOING N ENVIRONMENTL ONITIONS asically, thr ar two diffrnt displacnt valus usd in th analysis as 6 tons and 99 tons. Howvr, thr tacntric hight valus.,.6 and.7 wr takn into account. Thrfor, four diffrnt stability conditions ar considrd in th systatic analysis, as follows;. Δ 6 tons, GM... Δ 6 tons, GM.6.. Δ 6 tons, GM.7.. Δ 99 tons, GM.7. s was ntiond arlir, th vssl is fittd with various appndags. Only bilg kls wr takn into account for siplicity. Gotric dtails of th bilg kls ar givn blow; Lngth.. pth.. In ordr to s th ffct of bilg kls, daping valus and corrsponding roll aplituds wr calculatd with and without th appndags. Thus, ight diffrnt tst conditions in total hav bn analyzd in total. s far as nvironntal conditions ar concrnd, linar sinusoidal wav charactristics wr usd disrgarding any phas lag btwn th wavs and th otion. It is obvious that wav slop plays an iportant rol on th right hand sid of th quation. Thr distinct wav hight/wav lngth ratios as /, / and /6 ar slctd within th scop of study. Finally, spd of th ship has varid btwn and knots in ordr obsrv th influnc of spd on roll daping and vntually on roll aplituds.. MPING EVLUTION Estiation of roll daping onts or cofficints ar xtrly abiguous owing to highly nonlinar natur of th otion. lthough xtnsiv thortical rsarch, xprintal work and nurical siulation studis hav bn conductd on th attr, it is still far fro bing coplt. Unfortunatly, it is not fasibl to stiat roll daping corrctly fro radiation thory only. Viscosity of th fluid plays an iportant rol in prdicting roll otion of ships. Haddara and Zhang (99 hav carrid out sris of odl tst to show th influnc of forward spd in roll daping. Thrfor, contributions fro various sourcs and intraction btwn th ak prdiction of roll daping a difficult task. Ikda (97 has brokn down roll daping into fiv constitunts: a. Friction b. Lift c. Wav d. Eddy aking. ilg kl s a rsult of his work, pirical forulas wr introducd to stiat diffrnt coponnts listd abov. Furthror, it is or appropriat to divid abovntiond coponnts of roll daping as linar and nonlinar as statd in Equation (. ccording to this classification, lift and wav daping will b rgardd as linar whras friction, ddy aking and bilg kl daping will b rgardd as nonlinar daping. Th ffcts of ach daping coponnt ar invstigatd on th sapl fishing vssl by using nurical siulation. Epirical forulations by Ikda (97 and by Hino (9 hav bn utilizd in th siulation. Th procdur and rlvant forulation to stiat daping coponnts for this particular fishing vssl is supplid in th ppndix. asd on th nurical siulation, th daping charactristics of th sapl ship ar shown in Figurs and. aping officints....... bh 6 6 V(knot Figur. Linar and nonlinar daping cofficints with rspct to vlocity. aping officints....... 6 6 V (knot Figur. Non-dinsional daping cofficints coponnts with rspct to vlocity. b b L b N
It is intrsting to not that Figurs and dpict siilar trnds as indicatd by Ikda and Hino. 6. EFFET OF MPING ON ROLL MPLITUES Th ain purpos of dtrining daping valus is to analyz thir ffcts on roll aplituds. Thrfor, svral paratrs such as ship displacnt and GM, wav charactristics and spd wr altrd systatically and quation of otion was solvd both in ti and frquncy doains. Sinc cobination of thos variabls yildd so any diffrnt altrnativs and solutions in turn only a fw ar includd in this papr du to spac constraints. Evn though, a sapl of graph is givn for ach ain paratr, siilar trnds hav bn obtaind for othr altrnativ siulations. Roll Rspons plitud (g 6 6 6......6.7..9..... Frquny (rad/sc GM. GM.6 GM.7 Figur 6. Effct of GM for Δ6 tons, V knot and H/L/. Figur. shows th ffct of spd for a spcific loading condition. % incras is obsrvd btwn pak aplituds for knots with copard to knots. It should b notd that daping valus ar also suscptibl to spd variations. Figur copars th roll aplituds for a particular loading condition with and without bilg kls. Th bilg kls dap th otion for about % at rsonanc frquncy. Th ffct ay b diffrnt in agnitud for diffrnt tst conditions. Roll plitud Rspons (dg H/L/ H/L/ H/L/6 Roll Rspons plitud (dg...6.. Frquncy (rad/sc knot knot knot Figur. Effct of spd for H/L/, Δ6 tons and GM.77 with bilg kls....6.. Frquncy (rad/sc Figur 7. Effct of H/L for V knot, Δ6 tons and GM.77 with bilg kls. Figur 6 dpicts th ffct of changing GM for a constant displacnt. Incrasing GM not only rducs pak aplituds but also shifts rsonant frquncis to highr valus with no forward spd. Incrasd wav stpnss has a noticabl ipact roll aplituds in th ngativ sns. bout % rduction is xpctd going fro / to /6, Figur 7. Effct of incrasd displacnt on th pak roll aplituds for th highst wav slop is shown in Figur. Roll plitud Rspons (dg...6.. Frquncy (rad/sc With bilg Without bilg Figur. Effct of bilg kls for H/L/6, Δ 6 tons and GM.77. Roll rspons aplitud (dg...6... Frquncy (rad/sc Figur. Roll rspons aplitud whn H/L/, Δ99 tons and GM.77 without bilg kls.
plitud (rad... -. -. 6 6 ti(t Figur 9. Ti history of roll aplituds at ω. rad/sc for H/L/, Δ 99 tons, GM.77 and without bilg kls. Figur. Phas diagra at ω. rad/sc for H/L/, Δ 99 tons, GM.77 and without bilg kls. Furthror, ti doain solutions and phas diagras of th roll otion quation ay also b obtaind for ach cas analyzd. Figur 9 and Figur ar xapls of such spcific solutions. 7. ONLUSIONS.... -. -. -. -. -..... -. -. -. & φ In this papr th ffct of daping, wav and stability charactristics and wav paratrs on th roll otion of ships hav bn invstigatd by nurical siulation tchniqus for a sapl vssl. It has bn shown that, wav stpnss plays an iportant rol on spcially rsonant pak aplituds rgardlss of variation of othr paratrs. Th diffrnc in th pak roll aplituds ay b in th rang of % to % btwn th lowst and highst wav stpnss valus. Incrasing th spd fro to knots also rducs pak roll aplituds by about %. It should b notd that daping valus ar also affctd by incrasd spd in this annr. It is not possibl to gt a linar trnd in this annr. Stability charactristics which ay b translatd into diffrnt wight and cntrs that a ship undrgos during hr voyag also influnc roll otion of a ship. Shift in rsonant frquncy can b sn along with th chang in pak roll aplituds. φ s xpctd, bilg kls ar convntional and siplst for of roll rduction dvics. Thy ay b xpctd to rduc roll aplituds by %-% for this particular sapl ship. Finally, th abov assssnt rvals that roll daping is a vry critical paratr in otion charactristics of a ship. Thrfor aningful stiation of roll daping ay lad to or accurat prdiction of roll charactristics of a ship. In principl, it ay b xplaind as th nrgy balanc btwn daping and xcitation forcs. Sinc wav and wind xcitation can not b controlld, on ay play around th daping forcs to nhanc stability and otion qualitis of ships to an xtnt. It ay b incorporatd in th prsnt wathr critrion towards bttr valuation of dynaic stability of ships. REFERENES. Hino, Y., Prdiction of Ship Roll aping-stat of th rt, Rsarch Projct Rport No. 9, Univrsity of Michigan, 9.. Haddara, M.R., Effct of Forward Spd on th Roll aping of Thr Sall Fishing Vssls, Journal of OME, Vol. 6, pp. -, 99.. Ikda, Y., Hino, Y. and Tanaka, N., Prdiction Mthod for Ship Roll aping, Rport No. of partnt of Naval rchitctur, Univrsity of Osaka Prfctur, 97.. hakrabarti, S., Epirical alculation of Roll aping for Ships and args, Ocan Enginring, Vol., pp.9-9,.. uing,., Haddara, M.R. and Graha, R., Exprintal Invstigation of Roll aping haractristics of a stroyr Modl, ST 9, pp.69-66, Napoli Italy, 99. 6. Schitk, R.T., Ship Sway, Roll and Yaw Motions in Obliqu Sas, Transactions SNME, Vol. 6, pp.6-6, 97. 7. ontnto, G., Francscutto,. and Piciullo, M., On th Effctivnss of onstant officints Roll Motion Equation, Ocan Enginring, Vol. (7, pp.97-6, 996.. Taylan, M., Th Effct of Nonlinar aping and Rstoring in Ship Rolling. Ocan Enginring Vol.7, pp.9-9,. 9. Taylan, M., Nonlinar Roll Motion of Ships in a Wavs. ulltin of th Istanbul Tchnical Univrsity 9, pp.9-79, 996.. Taylan, M., Solution of th Nonlinar Roll Motion Modl by a Gnralizd syptotic Modl, Ocan Enginring Vol.6, pp.69-, 999.. Zborowski,. and Taylan, M., Evaluation of Sall vssls Roll Motion Stability Rsrv for Rsonanc onditions. SNME Spring Mting/STR Syposiu Nw Orlans, U.S., pp. S- -7, 99.
PPENI Following Ikda (97 and Hino (9, th following pirical xprssions hav bn utilizd in th calculation of daping cofficints; Skin Friction: S L(.7 + r (.7 +. S + * KG L π (.6 +.9( ρ. +.6(. ( 6 ν ρ ( r φ. R ν f ω.. ( rn +.. rn V.ρ r S f ( +. ω φ w f π f L F Lift aping: SL L LO. LR. KG If M <.9 κ If M >.97 κ. If.97 > M >.9 κ k N l (.9 7(. 9 M M 6 π + κ.. L L ρ SLV KN LO LR +. Wav aping: ω ξ d g Vω τ g LR +.7 LO LR + ξ. W. d + ξ ξd. d W ξd [( + + ( tanh(( τ. ] + ( Eddy Making aping: RL and H O σ ( τ. is th bilg radius and is th function of, H O H O H O ' σ ' σ,, M and H ar functions of xtinction cofficints fro fitting th xtinction curv in roll with a third dgr polynoial with rspct to roll angl;.6x (σ F + σ ( + tanh( F F γ P R σ ( cos( σπ. ( (.. cos(πσ π ( R ' σ ' M F M + H H O γ. γ (.7 + 7 R M F RL + FRL + F H O x ρ PR this is th ffct unit lngth F RL is th valu of non-dinsional ddy aking cofficint along th ship; K E V.ωL φ ω π + ilg-kl aping: + K KN KH K
Th noral forc coponnt pr unit lngth is writtn as; KN ρr π cb b K Equivalnt drag forc; ω RL bk. +. πr f cb f +. RL 6(σ f Th prssur coponnt of daping pr unit lngth du to hull surfac was obtaind fro th prssur asurnt on -dinsional hull surfac, which was causd by th prsnc of th bilg kls. KH ρrcb ωrl f π b K.. +. πrcb frl and ar th functions which dpnd on bilg-circl radius, ship dinsions and th dinsions of th stations pr unit lngth. NOMENLTURE L : Ship lngth : a of ship : raft : lock cofficint M : Mid-sction ara cofficint GM : Mtacntric hight GZ : Righting ar L : Longitudinal cntr of buoyancy LF : Longitudinal cntr of floatation KG : Vrtical position of cntr of gravity I : Inrtia of ship δi : ddd inrtia of ship Δ : isplacnt ρ : nsity of sa watr α : Maxiu wav slop υ : Kinatic viscosity ω : Frquncy ω : Encountr wav frquncy φ : Roll aplitud R : Rynolds nubr f : Friction cofficint W : Radiation daping RL : ilg radius l : Lift daping W : Wav daping E : Eddy aking daping F : Skin friction daping : ilg kl daping K L N b K f S r cb : Linar daping cofficint ( L l + W : Nonlinar daping cofficint ( N E + F + K : radth of th bilg kl : orrction factor to tak into account for th incras in th flow spd at th bilg kl : Wttd Surfac ra : a of any station : raft of any station : ra of any station : Th an distanc fro th roll axis