Liquid Cargo and Its Effect on Ship Motions

Transcription

1 STAB 97, S Interntonl Conference on Stblty of Shps nd Ocen Structures, Pges 7-5, Vrn, Bulgr, September -7, 997. Reprnted: 8-- Webste: Report 9-P, September 997, Delft Unersty of Technology, Shp Hydromechncs Lbortory, ekelweg, 68 CD Delft, The Netherlnds. Lqud Crgo nd Its Effect on Shp otons J..J. Journée Delft Unersty of Technology Summry When double bottom tnk, crgo tnk or spce n rollng essel contns flud, grty wes wll pper t the surfce of ths flud. These grty wes wll cuse ectng roll moments on the essel. At lower wter depths, resonnce frequences cn be obtned wth hgh we mpltudes. A hydrulc jump or bore, whch s strongly non-lner phenomenon, trels perodclly bck nd forth between the wlls of the tnk. A theory, bsed on gsdynmcs for the shock we n gs flow, hs been used to descrbe the motons of the flud. At hgher wter depths, the behour of the flud tends to be more lner. Then, lner potentl theory, s used n strp theory shp motons computer progrms, s used to descrbe the motons of the flud n the tnk. Alble epermentl dt on the behour of the flud n free surfce nt-rollng tnks he been used to erfy the theoretcl pproches for low wter depths. Also, forced roll oscllton tests were crred out wth -D model crgo tnk of n LNG crrer for wde rnge of fllng leels. These epermentl dt he been used to erfy the theoretcl results for ll wter depths. Fnlly, shp model ws equpped wth lqud crgo tnks nd t hs been tested n bem wes t zero forwrd speed. The mesured roll dt of the model he been compred wth the results of strp theory clcultons. It hs been concluded tht the pproches gen n ths pper, bsed on theores deelopped lredy n the stes, cn be used to nclude the effect of non-scous lqud crgo n shp motons clcultons. Introducton When tnk tht contns flud wth free surfce s forced to crry out roll osclltons, resonnce frequences cn be obtned wth hgh we mpltudes t lower wter depths. Under these crcumstnces hydrulc jump or bore s formed, whch trels perodclly bck nd forth between the wlls of the tnk. Ths hydrulc jump cn be strongly nonlner phenomenon. A theory, bsed on gs-dynmcs for the shock we n gs flow under smlr resonnce crcumstnces, s gen by Verhgen nd n Wjngrden (965), hs been dpted nd used to descrbe the motons of the flud. For low nd hgh frequences nd the frequences ner to the nturl frequency, dfferent pproches he been used. A clculton routne hs been mde to connect these regons. Alble

2 epermentl dt on the behour of the flud n free surfce nt-rollng tnks, obtned from n den Bosch nd Vugts, he been used to erfy ths pproch. At hgher wterdepths, the behour of the flud tends to be more lner. Then, the lner potentl theory wth the pulstng source method of Frnk (967), s used n strp theory shp motons computer progrms, hs been used to descrbe the motons of the flud n the tnk. Forced roll oscllton tests were crred out wth - D model of crgo tnk of n LNG crrer, to mesure the ectng moments for wde rnge of fllng leels nd frequences nd to compre these wth theoretcl predctons. Fnlly, shp model ws equpped wth three of these lqud crgo tnks nd tested n bem wes t zero forwrd speed. Seerl fllng leels nd two regulr we mpltudes were used to nestgte the effect on the roll behour of the shp. The mesured dt he been compred wth the results of -D potentl clcultons. LIQUID CARGO LOADS Obsere rectngulr tnk wth length l nd bredth b, whch hs been flled untl wter leel h wth flud wth mss densty ρ. The dstnce of the bottom of the tnk boe the center of grty of the essel s s. Fgure As system nd Nottons Fgure shows -D sketch of ths tnk wth the s system nd nottons. The nturl frequency of the surfce wes n hrmonc rollng tnk ppers s the welength λ equls twce the bredth b, so: λ = b. Wth the we number nd the dsperson relton: π π g π h k = nd ω = tnh λ b b t follows for the nturl frequency of surfce wes n the tnk: ω = π g π h tnh b b. Theory of Verhgen et.l. Verhgen nd n Wjngrden (965) he nestgted the shllow wter we lods n rollng rectngulr contner, wth the center of rotton t the bottom of the contner. Ther epressons for the nternl we lods re rewrtten nd modfed n ths pper to be useful for ny rbtrry ertcl poston of the center of rotton. For low nd hgh frequences nd the frequences ner to the nturl frequency, dfferent pproches he been used. A clculton routne hs been mde to connect these regons. Low nd Hgh Frequences The hrmonc roll moton of the tnk s defned by: φ = φ sn ( ωt) In the s-system of Fgure nd fter lnerston, the ertcl dsplcement of the tnk bottom s descrbed by: z = z + y φ nd fter lnerston, the surfce eleton of the flud s descrbed by: z = s + h + ζ Relte to the bottom of the tnk, the lnersed surfce eleton of the flud s descrbed by: ξ = h + ζ y φ Usng the shllow wter theory, the contnuty nd momentum equtons re:

3 ξ t ξ + + ξ y y ξ + + g + g φ t y y = = ξ ( s + h) ω b ω + g = h π ω π ω cos ω b In these formultons, denotes the elocty of the flud n the y -drecton nd the ertcl pressure dstrbuton s ssumed to be hydrosttc. Therefore, the ccelerton n the z -drecton, ntroduced by the ectton, must be smll wth respect to the ccelerton of grty g, so: φ ω b << g The boundry condtons for re determned by the elocty produced n the horzontl drecton by the ectton. Between the surfce of the flud nd the bottom of the tnk, the elocty of the flud res between nd / coskh wth s men elocty: s / kh. Howeer, n ery shllow wter does not ry between the bottom nd the surfce. When tkng the lue t the surfce, t s requred tht: ( s + ) φ & = h t s y = ± For smll lues of φ, the contnuty equton nd the momentum equton cn be gen n lnersed form: ξ t + h = y ξ + g + g φ = t y The soluton of the surfce eleton ξ n these equtons, stsfyng the boundry lues for, s: b π ω y sn φ ω b Now, the roll moment follows from the qus-sttc moment of the mss of the frozen lqud ρ l b h nd n ntegrton of ξ oer the bredth of the tnk: φ = ρ g l b + b/ + ρ g l b / h h s + φ ξ y dy Ths delers the roll moment mpltude for low nd hgh frequences t smll wter depths: h φ = ρ g l b h s + φ + ρ g l b ω πω + πω tn ω ( s + h) ω g ω πω φ For ery low frequences, so for the lmt lue ω, ths wll result nto the sttc moment: h b φ = ρ g l b h s + + φ The phse lgs between the roll moments nd the roll motons he not been obtned here. Howeer, they cn be set to zero for low frequences nd to π for hgh frequences: ε = for: ω << ω φ φ ε φ = π for: ω >>ω φ

4 Nturl Frequency Regon For frequences ner to the nturl frequency ω, the epresson for the surfce eleton of the flud ξ goes to nfnty. Eperments showed the ppernce of hydrulc jump or bore t these frequences. Obously, then the lnersed equtons re not ld nymore. Verhgen nd n Wjngrden soled the problem by usng the pproch n gs dynmcs when column of gs s oscllted t smll mpltude, e.g. by pston. At frequences ner to the nturl frequency t smll wter depths, they found roll moment mpltude, defned by: φ l b = ρ g π 4 π ( ω ω ) b g φ 4 φ h b The phse lgs between the roll moment nd the roll moton t smll wter depths re gen by: π ε φ φ = + α for: ω < ω π ε φ φ = α for: ω > ω wth: π b( ω ω ) α = rcsn 4gφ rcsn π 96gφ b ( ω ω ) π b( ω ω ) Becuse tht the rguments of the squre roots n the epresson for ε he to be poste, the lmts for the frequency ω re t lest: ω 4gφ bπ < ω < ω + φ φ 4gφ bπ. Theory of Frnk For the clculton of the -D potentl mss nd dmpng of shp-lke cross sectons, Frnk (967) consdered cylnder, whose cross secton s smply connected regon, whch s fully or prtly mmersed, horzontlly n preously undsturbed flud of nfnte depth. Fgure Frnk s Aes System The cylnder, s gen n Fgure, s forced nto smple hrmonc moton wth rdn frequency ω, ccordng to the dsplcement equton: ( m) ( m) S = A cos ωt The superscrpt m my tke on the lues, nd 4, denotng swy, hee nd roll motons, respectely. The roll motons re bout n s through pont (, ) y n the symmetry plne of the cylnder. It s ssumed tht stedy stte condtons he been ttned. The flud s ssumed to be ncompressble, nscd nd rrottonl, wthout ny effects of surfce tenson. The moton mpltudes nd eloctes re smll enough, so tht ll but the lner terms of the free surfce condton, the knemtc boundry condton on the cylnder nd the Bernoull equton my be neglected. A elocty potentl hs to be found: { ω e t } ( m ) ( ) = Re φ m ( y ) Φ, stsfyng the equton of Lplce, the symmetry (hee) or nt-symmetry (swy 4

5 nd roll) condton, the free surfce condton, the bottom condton, the knemtc boundry condton t the cylndrcl surfce nd the rdton condton. Frnk hs gen potentl functon, bsed on pulstng sources nd stsfyng the boundry condtons. Usng erler work of Wehusen nd Ltone, he defned the comple potentl t z of pulstng pont source of unt strength t the pont ζ n the lower hlf plne, s gen n Fgure. Tke the -s to be concdent wth the undsturbed free surfce. Let the cross sectonl contour C of the submerged porton of the cylnder be n the lower hlf plne nd the y -s, poste upwrds, beng the s of symmetry of C. Select N + ponts ( ξ, η ) of C to le n the fourth qudrnt. Connect these N + ponts by successe strght lnes. Then, N strght-lne segments re obtned whch, together wth ther reflected mges n the thrd qudrnt, yeld n ppromton to the gen contour s shown n Fgure. The coordntes, length nd ngle ssocted wth the j th segment re dentfed by the subscrpt j, wheres the correspondng qunttes for the reflected mge n the thrd qudrnt re denoted by the subscrpt j, so tht by symmetry ξ j = ξ j nd η j = η j for j N +. Potentls nd pressures re to be eluted t the mdpont of ech segment nd for N the coordntes of the mdpont of the th segment re: ξ + ξ+ π + η+ = nd y = In the trnsltonl modes, ny pont on the cylnder moes wth the elocty: ( ) ( ) = A ω sn ωt for swy = A ω sn ωt for hee ( ) ( ) The length of the th segment nd the ngle mde by ths segment wth the poste -s re: s α = ( ξ ξ ) + ( η η ) + η = rctn ξ + + η ξ + In here α s defned s π / α + π /. If the denomntor s negte, dependng on the sgn of the numertor, π hs to be dded or subtrcted, so tht α wll be defned s π α + π. The outgong unt ector norml to the, s: cross secton t the th mdpont ( ) y n = + sn α j cosα where nd j re unt ectors n the drectons nd y, respectely. The roll moton s llustrted n Fgure nd consderng pont (, y ) on C, n nspecton of ths fgure yelds: R θ = + ( y y ) y y = rctn In here, θ s defned s π / θ + π. If the denomntor s negte, dependng on the sgn of the numertor π hs to be dded or subtrcted, so tht θ wll be defned s π θ + π. By elementry two-dmensonl knemtcs, the unt ector n the drecton θ s: τ so tht: 4 = S ( ) ( 4) = ωa = sn θ τ R + j cosθ ( 4) R ( sn θ j cosθ ) sn ωt 5

6 The norml components of the elocty, re: t the mdpont of the th segment ( ) y ( ) ( ) = ωa ( ) ( ) = + ωa ( ) ( 4) = + ωa Defnng: R ( sn θ sn α + cosθ cosα ) sn ωt n ( m) sn α sn ωt cosα sn ωt = A ( m) ( m) ω sn ωt then, consstent wth the preously mentoned notton, the drecton cosnes for the three modes of moton re: n n n ( ) ( ) ( ) = sn α = + cosα = + sn θ sn α + cosθ cosα A set of two coupled ntegrl equtons re ppled by Frnk t the mdponts of ech of the N segments nd s ssumed tht oer n nddul segment the comple source strength remns constnt, lthough t res from segment to segment. Then, the set of coupled ntegrl equtons becomes set of N lner lgebrc equtons n the unknowns. ( m) The hydrodynmc pressure p long the cylnder s obtned from the elocty potentl by mens of the lnersed ( m) equton of Bernoull, where p nd ( m) p re the hydrodynmc pressures n phse wth the dsplcement nd n phse wth the elocty, respectely. The potentl s well s the pressure s functon of the oscllton frequency ω. The hydrodynmc force or moment per unt length on the cylnder, necessry to sustn the osclltons, s the ntegrl of ( m) ( m) p n oer the submerged contour of the cross secton C. It s ssumed tht the pressure t the th mdpont s the men pressure for the th segment, so tht the ntegrton reduces to summton, whence: N N ( m ) ( m ( ) ) ( m) ( ) ω = N ( m ) ( m ( ) ) ( m) ( ) ω = = = p p, y, ω, y, ω n n s s for the potentl mss nd dmpng forces or moments, respectely. Frnk's method s sutble for the computton of the potentl mss nd dmpng of symmetrc -D shpes, n or below the surfce of flud. Ths method hs been ncorported n lot of -D shp moton computer codes, ll oer the world. Strtng from the keel pont of the cross secton, the nput dt of the off sets he to be red n n upwrd order. Then, the (outwrd) norml on the elements of the cross secton wll be defned to be poste n the drecton of the flud outsde the cross secton. Esly, ths method cn be used to clculte the lner lods due to potentl flud n n osclltng symmetrcl tnk too. Strtng from the ntersecton of the free surfce wth the tnk wll, the offsets of the tnk he to be red n downwrds order, so n n opposte drecton s hs to be done for the cross sectons of shp. When dong ths, the (nwrd) norml on the elements of the cross secton of the tnk wll be defned to be poste n the drecton of the flud n the tnk. Then, the clculted potentl mss nd dmpng delers the n nd out phse prts of the lods due to the mong lqud n the tnk. Wth ths, the n nd out phse prts of the -D ectton forces nd moments bout the orgn n the wter surfce of the flud n rectngulr tnk re found wth: 6

7 X X X X X X c s c s 4c 4s = ω = ωn = ω = ωn = ω ( ) ( ) ( ) ( ) ( 4) 4 h b + ρg bh s + + ( 4) = ωn Ths ery smple pproch cn be crred out esly wth mny estng -D (nd lso -D) shp motons computer progrms. Howeer, one should keep n mnd tht n the clculton routne of Frnk, the ngles α nd θ he to be defned well n ll four qudrnts.. Vldtons Three dfferent ldtons were performed to erfy the ldty of usng Frnk s pulstng source method for obtnng the moments cused by the motons of lquds n osclltng tnks. Frstly, results of etense model eperments on rectngulr free surfce nt-rollng tnks, crred out n the pst by Vn den Bosch nd Vugts (966), he been used. oment n phse (-). -. Free surfce tnk s/b = -.4 h/b =.8 φ =. oment out phse (-) Frequency (-). -. Verhgen Frnk Eperment B&V Frequency (-) Fgure Roll oments of Free Surfce Tnk wth Bottom below G oment n phse (-). -. Free surfce tnk s/b = +. h/b =.8 φ =. oment out phse (-) Frequency (-). -. Verhgen Frnk Eperment B&V Frequency (-) Fgure 4 Roll oments of Free Surfce Tnk wth Bottom boe G Fgure nd Fgure 4 show few comprsons between clculted nd mesured n nd out phse prts of the frst hrmonc of the roll moments for ery smll wter depths n the tnk. The comprsons re gen here for one fllng h / b =.8 nd one roll mpltude leel ( ) ( φ =. =.9 ). Two postons of the bottom wth respect to the rotton pont,.e. the center of grty G of the essel, he been tken: 4 per cent of the tnk wdth below the s of rotton ( / b =.4) ( s / b = +.) s nd per cent boe t. The roll moment hs been mde non-dmensonl by ddng ths moment through ρ glb. The nondmensonl frequency prmeter hs been obtned by ddng the frequency through g / b. Outsde the nturl frequency re, the fgure shows good greement between Frnk's method nd the eperments. But, for frequences close to the nturl frequency, ery poor predcton hs been found. Becuse of the ppernce of hydrulc jump or bore t these frequences, the lnersed equtons re not ld nymore. The clculted phse lgs between the roll moments nd the roll motons he step of 8 degrees t the nturl frequency, whle the clculted roll moment mpltudes go to nfnty. Becuse of dstncton between frequences close to the nturl frequency nd frequences fr from t, the shllow wter method of Verhgen nd n 7

8 Wjngrden shows good predcton t ll frequences..5. LN G tnk h = 45 %.5 Secondly, forced roll oscllton eperments he been crred out wth -D model of crgo tnk of n LNG crrer. A sketch of ths :5 model of the tnk s gen n Fgure 5. o ment n ph se (-) omen t ou t ph se (-) Frnk Eper ment Fre que ncy (-) Freq uency (-) Fgure 7 Roll oments of n LNG Tnk wth 45 % Fllng Leel.5 LN G tnk h = 9 %.5. o ment n ph se (-).5 omen t ou t ph se (-) Frnk Eper ment Fgure 5 odel of n LNG Tnk At fllng leels of 5, 45, 7, 9, 97.5 nd per cent of the depth of the tnk, the ectng roll moments he been mesured for rnge of oscllton frequences nd one roll mpltude ( φ =. = 5.7 ) of the tnk. Becuse of the shpe of ths tnk, strong non-lner behour ws epected t the lowest nd hghest freesurfce leels. Fgure 6 through Fgure show the mesured nd predcted n-phse nd outof-phse prts of the frst hrmonc of the roll moments of the LNG tnk s functon of the frequency. o ment n ph se (-).5..5 LN G tnk h = 5 % Fre que ncy (-) omen t ou t ph se (-) Verhg en Frn k Ep ermen t Freq uency (-) Fgure 6 Roll oments of n LNG Tnk wth 5 % Fllng Leel Fre que ncy (-) Freq uency (-) Fgure 8 Roll oments of n LNG Tnk wth 7 % Fllng Leel omen t n p hse (-).5..5 LNG t nk h = 9 % Freq uen cy (-) ome nt o ut p hse (-) Frnk Epe rment Fre quen cy (-) Fgure 9 Roll oments of n LNG Tnk wth 9 % Fllng Leel ome nt n p hse (-).5..5 LN G t nk h = 97.5 % Freq uency (-) ome nt o ut p hse (-) Frnk Eper ment Freq uency (-) Fgure Roll oments of n LNG Tnk wth 97.5 % Fllng Leel 8

9 omen t n p hse (-).5..5 L NG tnk h = % Freq uen cy (-) ome nt o ut p hse (-) Frn k Ep ermen t Fre quen cy (-) Fgure Roll oments of n LNG Tnk wth % Fllng Leel The roll moment hs been mde nondmensonl by ddng ths moment through ρ glb. The non-dmensonl frequency prmeter hs been obtned by ddng the frequency through g / b. Ecept t the nturl frequency of the flud n the tnk, frly good predcton hs been found wth Frnk's method. Agn, the shllow wter method of Verhgen nd n Wjngrden ges good predcton for ll frequences t the lowest fllng leel of the tnk. Thrdly, fully flled rectngulr tnk hs been obsered. It s obous tht the rto between the effecte nd the solfed mss for swy or hee of the flud n fully flled tnks s.. The rto between the effecte nd the solfed moments of nert for roll of the flud n fully flled rectngulr tnk s functon of the spect rto h / b of the tnk ws gen by Grhm nd Rodrques nd publshed lter by Slermn nd Abrmson (966) s: c e = + 5 h π h h + + b b b πh tnh ( n + ) b 5 n= + ( n ) Ths epresson hd been obtned from results of spce ehcle studes crred out by NASA. The contrbutons of frequences hgher thn the frst order re ery smll nd cn be neglected. Clcultons he been performed wth the -D computer code SEAWAY of Journée (996), tht ncludes Frnk's pulstng source method. Also, -D clcultons he been crred out wth the DELFRAC computer code of Pnkster (996). The results re gen n Fgure. Rto of hydrodynmc nd solfed msses Swy / Hee Roll (Grhm) Roll (-D) Roll (Frnk) Hee (F rnk) Swy (Frnk) 4 5 Aspect rto of tnk, h/b Fgure oment of Inert of Fully Flled Tnk Fgure shows frly good greement between these results. For smll nd lrge spect rtos detons cn be epected, cused by the lmted number of 6 lne elements n SEAWAY or pnels n DELFRAC on the contour of hlf the cross secton of the tnk. Shp otons To nestgte the effect of free surfce (lqud crgo) tnks on the roll motons of shp, three tnks s gen n Fgure 5 were buld n :6 model of n LNG crrer. The body pln of ths essel s gen n Fgure. Fgure Body Pln of LNG Crrer Sttc heel tests nd roll decy tests n stll wter nd roll moton mesurements n regulr bem wes wth two dfferent 9

10 we mpltudes were performed t zero forwrd shp speed. These tests were crred out n Towng Tnk I of the Shp Hydromechncs Lbortory of the Delft Unersty of Technology wth length of 4 meter, bredth of 4. meter nd wter depth of.5 meter. The model ws plced trnsersl n the towng tnk t hlf the length of the tnk nd the spcng between the model nd the tnk wlls ws bout.7 meter. The model ws free to moe n 6 degrees of freedom nd the roll dmpng wes could propgte oer long dstnce before they were reflected to the model by the tnk-ends. The mn dmensons of the shp re gen n Tble. LNG Crrer Length bpp L pp m 64. Bredth B m.4 Drught d m.6 Trm t m. Volume m 568 Block coeffcent C b Length of blge keels l bk m 69.7 Heght of blge keels h bk m. Gyrdos for yw k zz m 45.4 Tble n Dmensons of Shp The length of ech of the three crgo tnks ws.45 meter nd the dstnce of the bottom of the tnks boe the shp's bse lne ws. meter. Wth the ecepton of these mdshps crgo tnks, 5 nd 7, ll other crgo tnks re supposed to be flled up to 97.5 per cent of the nner tnk heght wth stowge fctor of. ton/m. Ths ws smulted n the eperments by sold bllst weghts nd n dptton of the rdus of nert for roll of the shp. For the crgo tnks, 5 nd 7 three lodng condtons he been chosen: Condton I: frozen lquds ( %) The three crgo tnks re eqully flled up to 45% of the nner tnk heght wth homogeneous frozen lqud crgo wth stowge fctor of. ton/m, smulted by sold bllst weghts. Condton II: lquds ( %) Ths condton s smlr to condton I fter meltng the crgo, so the three tnks re prtlly flled wth wter. Condton III: lquds (45-7-5%) Ths condton s smlr to condton II, but the fllng leels of the three tnks re 45%, 7% nd 5%, respectely. The results of the sttc heel ngle tests nd the roll decy tests re gen n Tble. Condton I II III m KG m G m GG m k m T φ-mes. s.7.. T φ-clc. s.8.8 Tble Lodng Condtons The mesured non-dmensonl roll dmpng coeffcents κ re presented n Fgure 4. Roll dmp ng coeff c ent κ (-).5 LNG crrer Frozen crgo LN G c rrer Tnk : 45 %. 4 Tnk 5: 45 % Tnk 7: 45 %... Iked et.l. Epermen t e n rol l mpl tu de φ (deg).5 LNG crrer.4... Tnk : 5 % Tnk 5: 45 % Tnk 7: 7 % Fgure 4 Roll Dmpng Coeffcents These dt he been compred wth predcted lues obtned wth the sememprcl method of Iked. For condton I, wth frozen lqud crgo, ery good greement hs been found. For condtons II nd III, wth lqud crgo, the predcted roll dmpng coeffcents re somewht underestmted t smller roll ngle mpltudes. But t lrger roll ngle mpltudes, whch re of nterest n more dngerous crcumstnces, the fgure shows frly good greement.

11 Non-dmensonl roll mpltude (-) Clcultons SEAWAY Lqud crgo, ζ =. m Lqud crgo, ζ =.85 m Frozen crgo, ζ =. m Frozen crgo, ζ =. m Condton II LNG crrer Tnk : 45 % Tnk 5: 45 % Tnk 7: 45 % Condton I We frequency (rd/s) Fgure 5 Roll otons of LNG Crrer, Condtons I nd II Non-dmensonl roll mpltude (-) Clcultons SEAWAY Lqud crgo, ζ =. m Lqud crgo, ζ =.75 m Frozen crgo, ζ =. m Frozen crgo, ζ =. m Condton III LNG crrer Tnk : 45 % Tnk 5: 7 % Tnk 7: 5 % Condton I We frequency (rd/s) Fgure 6 Roll otons of LNG Crrer, Condtons I nd III Fgure 5 nd Fgure 6 show comprson of the mesured nd predcted roll mpltudes t two dfferent we mpltudes for ech lodng condton. For lodng condton I, the non-potentl roll dmpng hs been obtned by the method of Iked. The rdus of nert for roll of the shp's mss hs been obtned from the mesured nturl roll perod of.7 seconds nd clculted hydromechnc coeffcents. The fgure shows good greement between eperments nd predctons. For the lodng condtons II nd III, the non-potentl roll dmpng hs been obtned by the method of Iked, Hmeno nd Tnk (978) too. The rdus of nert for roll of the shp's mss hs been obtned from the rdus of nert of the shp wth the frozen lqud crgo of condton I nd theoretcl correcton for meltng ths crgo. The detng olume of lqud crgo n condton III hs been ccounted for too. The ectng roll moments due to lqud crgo he been obtned wth Frnk's method. So, for the lodng condtons II nd III, the roll motons he been clculted wthout usng ny epermentl dt of these lodng condtons. Tble shows ery good greement between the predcted nd the mesured nturl roll frequences; the deton s less thn %. Neertheless some oerestmton t hgher frequences, Fgure 5 nd Fgure 6 show ery good greement between the predcted nd the mesured response mpltude opertors for roll. Howeer, t my be noted tht for the lodng condtons II nd III, the nturl roll frequency of the shp s bout hlf the lowest nturl frequency of the flud n the three crgo tnks. When these frequences re close to ech other, non-lner effects cused by the bore or the hydrulc jump t the surfce of the flud n the tnks wll ply much more mportnt role. 4 Conclusons From the clcultons nd the eperments, the followng conclusons cn be drwn:. At ery low fllng leels of the tnk, the method of Verhgen nd n Wjngrden predcts the ectng roll moments frly good. Becuse ths theory s gen for shllow wter only, the method fls for hgher fllng leels.. Wth the ecepton of frequences ner to the nturl frequency of the flud n the tnk, the potentl theory of Frnk predcts the ectng roll moments frly good for ll fllng leels of the tnk.. An ddton of these roll moments n the rght hnd sde of the equton of motons of the shp, results n good predctons of the roll motons of the shp. 4. The non-lner roll dmpng of the shp plys n mportnt role n these roll motons. Lnerston results n

12 good predcton of the hrmonc roll motons. The pulstng source method of Frnk, s ncluded n mny strp-theory shp motons computer progrms, cn be used esly to nclude the effect of non-scous lqud crgo n shp moton clcultons. Also, -D clculton technques for clcultng potentl mss nd dmpng of shps cn be used for ths purpose. 5 References Vn den Bosch nd Vugts (966) J.J. n den Bosch nd J.H. Vugts, Roll Dmpng by Free Surfce Tnks, Report 8-S, 966, NSRC-TNO, Delft, The Netherlnds. Frnk (967) W. Frnk, Oscllton of Cylnders n or below the Free Surfce of Deep Fluds, Report 75, October 967, NSRDC, Wshngton D.C., U.S.A. Iked, Hmeno nd Tnk (978) Y. Iked, Y. Hmeno nd N. Tnk, A Predcton ethod for Shp Rollng, Report 45, December 978, Deprtment of Nl Archtecture, Unersty of Osk Prefecture, Jpn. Journée (996) J..J. Journée, The Behour of Shps n Sewy, Report 49, y 996, Shp Hydromechncs Lbortory, Delft Unersty of Technology, The Netherlnds. Pnkster (996) J.A. Pnkster, Hydrodynmc Aspects of Flotng Offshore Structures, Report 5, y 996, Shp Hydromechncs Lbortory, Delft Unersty of Technology, The Netherlnds. Slermn nd Abrmson (966) S. Slermn nd H.N. Abrmson, The Dynmc Behor of Lquds n ong Contners, Edted by H.N. Abrmson, Chpter : Lterl Sloshng n ong Contners, Report NASA SP-6, 966, Scentfc nd Techncl Informton Dson, Ntonl Aeronutcs nd Spce Admnstrton, Wshngton D.C., USA. Verhgen nd n Wjngrden (965) J.H.G. Verhgen nd L. n Wjngrden, Non-lner Osclltons of Flud n Contner, Journl of Flud echncs, 965, Volume, Prt 4, Pges