Investigation of Sway, Roll and Yaw Motions of a Ship with Forward Speed: Numerical modeling for flared up conditions

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1 h Inernaonal onference on Hgh Performance Marne Vehcles 8- November usrala Invesgaon of Sway Roll and Yaw Moons of a Shp wh Forward Speed: Numercal modelng for flared up condons S.K. Das Inernaonal Insue of Informaon Technology Pune Inda samrd@square.ac.n S.N. Das enral Waer & Power Research Saon Pune Inda sndas_cwprs@redffmal.com P.K. Sahoo usralan Marme ollege Launceson usrala P.Sahoo@me.amc.edu.au bsrac The paper deals wh he numercal modelng of sway roll and yaw moons of a shp for flared up condons wh zero or nonzero forward speeds n snusodal waerway. To compue hydrodynamc forces we employ nonlnear roll resorng characerscs and speed dependen srp heory ha are obaned from he Frank s close-f mehod. The governng equaons are solved numercally by usng Runge-Kua-Gll mehod wh adapve sep sze adusmen algorhm. In order o nvesgae he effec of nonlnear resorng n roll numercal expermens have been carred ou for a Panamax onaner shp under he acon of snusodal wave of perodcy. sec wh varyng wave hegh and speed. To emulae he sof sprng behavour nonlnear resorng momen s represened by an odd order polynomal of roll angle where he correspondng coeffcens are obaned by analyzng he resuls of numercal expermens. Ths modelng approach provdes an mporan gudelne o undersand he role of varous parameers whle flared up condons does occur ogeher wh s conrollng mechansms. Keywords Sway Roll Yaw Numercal modelng Flared up moons Froude-Krylov force.. Inroducon In shp moon sudes he analyss of large amplude nonlnear rollng s mporan o undersand capszes dynamcs. Que ofen he moon responses are flared-up o an exen when roll-resorng momen poses serous sably problem whle he shp moves wh forward speed. For such roll analyss lnear approxmaon s no longer vald (haacharyya 978) and as a resul obanng closed form soluon becomes dffcul. In he pas several researchers Haddara (97) Robers (98) ardo e al.(98) Nayfeh and Khder (98) and Vrgn (987) have analyzed he effecs of nonlnear resorng momens of a rollng shp. Haddara (97) developed an analycal mehod o oban an approxmae soluon correspondng o nonlnear rollng equaon of a shp n random waves. Vrgn (987) and ardo e al. (98) have examned he nfluence of nonlnear shp rollng n regular seas by applyng Poncare mappng echnques o nclude chaoc moon and perurbaon analyss respecvely. Nayfeh and Khder (98) obaned a second order approxmae soluon for nonlnear harmonc roll moon usng perurbaon analyss as well as numercal mehod o oban lm cycles. Robers (98) esmaed he roll response process by makng comparson beween smulaon resuls and heorecal predcons. However mos of he earler analyses were resrced o sudy of uncoupled rollng n beam waves. In hs paper we examne he behavour of nonlnear roll resorng for coupled sysem (sway-rollyaw) of a shp movng wh consan forward speed n snusodal waves. To smulae he sof sprng acon lnear cubc and qunuple dependence of roll angle s consdered on exendng he mahemacal modelng approach gven by Das and Das () for a saonary shp. Usng he srp heory approach of Salvesen e al. (97) he seconal coeffcens are negraed along he

2 longudnal axs of he shp by applyng Frank and Salvesen s close f mehod (97) based on he expermenal resuls of Vugs (98). To oban he realzaon of roll responses n coupled condons governng equaons are solved numercally wh he varaon of shp speed. Ths enables us o examne he sensvy of nal condons and flared-up condons for a conaner shp.. Problem formulaon caresan co-ordnae sysem (xyz) fxed wh respec o he mean poson of he shp s consdered wh z-axs acng n he vercal upward drecon and he orgn O les n he undsurbed free surface. I s assumed ha he shp s a rgd and slender body symmerc abou x-z plane and he cenre of gravy G s locaed a (z o z c ) where z OO and z c O G (Fgure ). Fgure : Moon and oordnae Sysem defnon of a floang body The shp s exced by monochromac waves of frequency and he force componens generaed by he propeller and wnd are negleced. The ranslaory dsplacemens along he x y and z drecons can be descrbed knemacally n erms of surge ( ) sway ( ) and heave ( ) and he angular dsplacemens of he roaonal moon abou he same se of axes are roll ( ) pch ( ) and yaw ( ). In shp moon sudes frequency response analyss correspondng o a Fourer approach can be convenenly appled Tck (99). Owng o complex neracons beween he hull and shp generaed waves; he governng equaons can be wren n he form of negro-dfferenal equaons whch poses enormous dffculy n solvng ummns (9). Such dffculy can be convenenly avoded by consderng he occurrence of shp moon under he acon of regular waves. Ths reduces he negro-dfferenal equaon no ordnary dfferenal equaon (ODE). Followng he approach of Tck (99) for coupled sysem n hree-degrees of freedom sway-roll-yaw moons can be wren as: M ( ) ( ) e ( ) ( ) e ( ) ( ) e Χ + Χ + Χ k k k k k k k D F e ( ) ( ) () where Χ ( ) k s he dsplacemen F ( ) s he wave force wh amplude D ( ) ; M ( ) k

3 ( ) and ( ) are he vrual mass dampng and resorng momens correspondng o he k k wave frequency respecvely. Now defnng e k ( ) Χ k ( ) f F e k ( ) k ( ) k () one can oban M ( ) () ( ) () ( ) () D ( ) f k k k k k k k + + () I s apparen ha he moon varables ( ) excng force f () and wave frequency ( ) descrbed n equaon () are complex quanes and hese can be expressed as algebrac sum of real and magnary pars. ccordngly he forcng funcon f () becomes ( ) ( ) ( ) R + I F e F ( ) e R f e I () For smplcy we assume he magnary par of wave frequency ( I ) o be equal o zero yeldng f F e R ( ) ( ) () The moon responses and forcng funcons can also consdered as sum of real and magnary pars: R + and I F F R + F () I onsderng only he real par of moon response and excng momen for a gven wave frequency he equaon of moon for coupled sway-roll-yaw can be descrbed as Hoof (98): d + d + d ] F (7) [ where he operaor d s gven by d d d + d + (8) d F () s he wave excng force or momen M + s he vrual mass momen of nera () ( ) and ( ) are he cross-coupled coeffcens lke added mass dampng and resorng n he drecon due o any moon n he drecon. Usng equaons (7) and (8) he governng equaons can be expressed n he followng marx form: { [ M ] + [ ] } [ ] + [ ][ ] + [ ][ ] [ F ] (9) The coeffcen marces can be expressed as [ M ] M Mz c Mz c I I I I [ ]

4 [ ] [ ()] r ( ) [ ] [ ] ( ) ( ) F [ ( )] [ F ] F () ( ) F ( ) where M s he mass of he shp and he componens n he marces [ ] and [ ] ndcae me dervaves. F ( ) F ( ) and F ( ) are he wave forces or momens for sway roll and yaw. I s he momen of nera n he h mode and I s he produc of nera. In hs formulaon k he added mass and dampng coeffcens are frequency dependen however can accoun speeddependen varaons. These are compued by negrang wo-dmensonal seconal coeffcens correspondng o known wave frequency along he lengh of he body Salvesen e al. (97). In he presen sudy we nvesgae flared up condons when he resorng momen s nonlnear. Here we consder he resorng momen r s havng funconal dependence on roll angle and can be expressed as an odd order polynomal of [(Dalzell (978) haacharryya (978) ardo e al.(98) and Nayfeh and Khder (98)]. r ( + + () ) GZ ( ) where s he dsplacemen n wegh GZ ( ) s he rghng arm and are coeffcens. The equaon () represens nonlnear resorng and can emulae he behavour of sof sprng acon. To represen lnear roll resorng one may consder and as zero yeldng r ( ) GZ ( ) () where ρ g GM () Here s he dsplaced volume GM s he meacenrc hegh ρ s he densy of waer and g s he acceleraon due o gravy. The wave excng forces and momens are expressed n snusodal form as F ( ) F sn( + θ ) () where F F and F are he ampludes of he sway excng force roll excng momen and yaw excng momen respecvely θ s he phase angle and s he encounerng wave frequency. The ampludes of he sway excng force roll excng momen and yaw excng momen can be obaned as per Salvesen e al. (97) F α ρ ( f + h ) dξ () F α ρ ( f + h ) dξ ()

5 F α ρ ξ ( f + h ) dξ (7) whereα s he amplude of he ncden wave force and dffracon force respecvely. f and h represen he D seconal Froude-Krylov. Modellng for nonlnear resorng and expermen The governng equaon (9) comprsed of sway roll and yaw equaons s analycally nracable owng o he presence of nonlnear roll resorng erm r. In he absence of nonlnear erm he governng equaons can be solved analycally. The dealed descrpon of he analycal mehod by consderng lnear dampng and lnear resorng momen for wo and hree degrees of freedom and whou consderng speed dependen seconal coeffcens can be obaned from he nvesgaons of Das and Das (a). However for uncoupled roll wh nonlnear added mass and dampng Das e al. (b) have obaned numercal soluon when he shp s eher saonary or movng wh a forward speed of knos. In he presen sudy he coupled sway-rollyaw governng equaons are solved numercally o ge he effec of nonlnear roll resorng on oher moons. In order o solve hese equaons numercal negraon based on Runge-Kua mehod has been adoped Press e al. (99). In hs case hree-second order ordnary dfferenal equaons (9) are ransformed no sx frs order ordnary dfferenal equaons assgnng approprae nal condons: φ (8) φ (9) φ () F φ ( - M z ) c φ φ φ ( ) /( + M ) φ φ () φ [ F ( Mz c ) φ - ( I ) φ φ ] / φ φ ( + I ) F φ φ ( I ) φ φ ( ) / + I φ φ ( ) () () The sysem of equaons (8) o () wh prescrbed condons poses well-defned nal value problem whch are beng solved hrough sep-by-sep negraon procedure. s he roll moon s coupled wh sway and yaw mplc dependence of hese moons on nonlnear resorng s explored wh he varaon of nal condon wave hegh and shp s speed. The varaon of shp s speed s also accouned n he formulaon o examne he rollng behavour. pplyng Runge-Kua-Gll mehod wh sep-sze adusmen algorhm desred accuracy s acheved. The rghng-arm curve or he GZ curve s represened here by a ffh order polynomal; G Z ( ) + + () where and + I + I + I ()

6 haacharryya (978) dscussed nonlnear resorng momen by expressng as an odd order polynomal of roll angle. onsderng hrd order polynomal n roll nonlnear resorng was and were obaned from he approxmaon of he rghng smulaed where he coeffcens momen curve from an equaon of he form: G Z ( ) ( + ) () In general he sably curve or he GZ curve can be obaned from he physcal expermen. On fng hs curve wh he polynomal descrbed n () one can deermne he correspondng coeffcens. In he absence of such physcal expermen he represenaon of resorng momen becomes dffcul. Neverheless aemps have been made for such represenaon hrough a seres of numercal expermen o supplemen expermenal resul. To emulae he sprng acon one can assgn suable values of and correspondng o parcular ype of vessel based on wo prmary characerscs: () hard sprng > and () sof sprng < Hoof (98). The coeffcens of resorng momen may be obaned from he approxmaon of rghng-momen curve usng he polynomal approxmaon haacharryya (978). Wrgh and Marsfeld (98) Fea and Jones (98) and ass (98) ncluded all he erms n he polynomal of he resorng momen where as ardo e al. (98-98) consdered only he lnear and cubc erms. n mporan aspec of sudyng nonlnear resorng s o deermne he nfluence of he nal condons. Ofen such nonlneary may lead o flare-up condon owng o he ndrec npus caused by he neracons beween dfferen moon componens n hgher degree of freedom. Fgure : ody plan of a Panamax onaner shp For numercal expermen compuaons are performed n me-doman for a Panamax onaner shp under he acon of snusodal wave of perodcy. sec wh varyng wave heghs acng beam o he shp hull when he shp s eher saonary or havng consan forward speed (U). The man parculars of he onaner shp and body plan (Wang ) are gven n Fgure. The seconal coeffcens for added mass dampng Froude-Krylov force and dffracon force correspondng o he wave perod. sec are compued from he expermenal resuls of Vugs (98) and Frank s closef mehod (97). Ths has been shown n Table. To sar compuaon he nal me sep for numercal negraon s specfed as. sec.

7 Table : D-seconal values Wave Seconal coeffcens Frequency Rad/sec Perod sec Sway added mass a Roll added mass a Swayroll added mass a Sway dampng b Roll dampng b Swayroll dampng b Sway excng force f + h Roll excng momen f h. Model Resuls and Dscusson From he revew of he earler leraures s observed ha mos of he researchers deal wh uncoupled roll moon wh lnear or nonlnear dampng and cubc or qunc represenaon of nonlnear resorng. These researchers obaned he resorng momen coeffcens for cubc and qunc erms from he approxmaon echnque of he rghng momen curve. Wrgh and Marshfeld (98) obaned he resorng coeffcens for cubc ( ) and qunc ( ) erms whch are. and. respecvely for hgh freeboard. In he compuaon of nonlnear shp roll dampng ass and Haddara (988) assumed he value of as.. ardo e al. (98) examned he nfluences of dampng effecs on he shp rollng moon n regular beam seas. He expressed he rghng momen up o hrd order polynomal and obaned he value of as.9 whch gves bes f for rghng arm curve when he shp s fully loaded. ardo e al. (98) examned he nonlnear resonance for rollng of a shp n beam seas where he encouner frequency s an neger mulple or a sub-mulple of he naural frequency of he sysem. They solved he governng equaon for wo dfferen suaons where -.7 and and found ha for < he correspondng rghng arm curve a frs reaches a relave maxmum and hen goes o zero where as for for > he sysem s no sable. Wave Hegh (m) > he curve ncreases monooncally n he whole range. Ths ndcaes ha Table : Inal values correspondng o wave heghs Inal condons a sec Sway Roll Yaw U knos Remarks U knos FU M FU M FU M FU FU To model he flared up condons he values of and mus be assgned properly. fer havng numercal smulaon for varous nal condons as menoned n Table he numercal values for and are obaned as -.8 and -.9 respecvely. Ths corresponds o he behavour of a sof sprng. The performance of he shp wh and whou forward speed and ncrease of wave hegh are shown n Table. In hs able FU and M represen he abbrevaed form for flared-up and moon connue respecvely. If he moon s flared-up hs ndcaes ha he

8 sysem s unsable. Snce he roll moon s havng resorng propery he nal values peranng o sway and yaw have no nfluence n he coupled condon. Hence he roll nal condons and he values of and are havng grea nfluence on moon me-hsores. FU moon flared-up M moon connue To llusrae non-lnear roll resorng whle coupled moons are consdered several runs were aken by varyng envronmenal condons shp speed and nal condons (I). The resuls shown n hese fgures were obaned from smulaon sudes for wave heghs m m m and m wh nal roll angles...8 and. degrees. We specfy he nal condons of sway and yaw correspondng o Das and Das (a). Fgs. (a)-(c) exhb he comparson of harmonc behavour of sway roll and yaw whle nal roll angle s se o.. Wh he ncrease of wave hegh from m o m moon ampludes ncrease preservng her perodces. However furher ncrease of wave hegh beyond he regme of small amplude harmonc response ( m) oscllaons become unsable. Such flarng up of harmonc response s caused due o he propagaon of non-zero nal condon. The correspondng roll angle s found o be ± degree whou aenuaon (Fgure b). Sway response (m) omparson of Sway responses when Roll I. deg and U knos Wave h: m Wave h: m Wave h: m 8 Tme (sec) Fgure : (a) Sway Response omparson of Roll responses when Roll I. deg and U knos Roll response (deg) - Wave h: m Wave h: m Wave h: m 8 - Tme (sec) Fgure : (b) Roll Response

9 Yaw response (m)..8. omparson of Yaw responses when Roll I. deg and U knos Wave h: m Wave h: m Wave h: m. 8 Tme (sec) Fgure : (c) Yaw Response Zero forward speed Roll response (deg) - Wave h m Roll I. deg Wave h m Roll I.8 deg Wave h m Roll I. deg Tme (sec) Fgure : omparson of roll responses due o dfferen wave heghs and nal condons The sway and yaw response also show he dvergence n numercal soluon obaned n Fgs. (a) and (c). To undersand relave conrbuon of nal dsurbance and wave hegh smulaons were carred for varous combnaons of wave hegh and nal condon and hree ypcal cases are llusraed here for comparson; () m wave hegh and I ±. degree () m wave hegh and I ±.8 degree and () m wave hegh and I ±. degree when forward speed s absen (Fgure ). These crcal parameers form wave hegh-i-speed marx from whch moon sably can be obaned for a parcular wave frequency. The aenuaon of roll amplude and hereby conrol of roll moon for all me s noced as he speed of he shp s ncreased from knos o knos (Fgure ). We analyze relave conrbuon of lnear cubc and ffh order erms of roll resorng

10 momen r as specfed n equaon () and hese are exhbed n Fgs. (a)-(c) for wave hegh I and speed varaons. In he case of zero forward speed shp fals o resore and he order of roll resorng momen ncreases from O ( ) Newon-meer o hgher order ndcang oscllaory dvergence. However hs oscllaon s conrolled due o consan forward speed ( knos) or equvalenly hgher values for Froude number (whn sub-crcal range). Fgure 7 shows he oal roll resorng for hree ypcal cases for comparson where he combned effec of lnear cubc and ffh order erm behaves lke a sof sprng. Furher ncrease of wave hegh he roll moon s unbounded leadng o capsze of shp even wh small nal dsurbance I. deg wave hegh s m and forward speed s knos (Fgure 8). Roll response (deg) - - Wave h m Roll I. deg U knos U knos 8 Tme (sec) Roll resorng momen (N-m).E+.E+.E+.E+ -.E+ Fgure : omparson of roll responses when he shp s eher saonary or havng knos speed Lnear Roll resorng: wave h m Roll I. deg -.E+ 8 Tme (sec) U knos U knos Fgure : (a) omparson of roll resorng: Lnear order of roll for wave heghs and I ubc Roll resorng: wave h m Roll I.8 deg Roll resorng momen (N-m).E E+ U knos U knos -.E+ Tme (sec) Fgure : (b) omparson of roll resorng: ubc order of roll for wave heghs and I

11 Ffh order Roll resorng:wave h m Roll I. deg Roll resorng momen (N-m).E+.E+ -.E+ -.E+ -.E+ U knos U knos -.E+ Tme (sec) Fgure : (c) omparson of roll resorng: Ffh order of roll for wave heghs and I Toal roll resorng: U knos.e+.e+.e+ -.E+ -.E+ -.E+ -.E+ Tme (sec) Wave h m Roll I. deg Wave h m Roll I.8 deg Wave h m Roll I. deg Fgure 7: omparson of oal roll resorng Wave h m Roll I. deg Roll response (deg) U knos U knos 8 Tme (sec). oncluson Fgure 8: omparson of roll responses for m wave hegh and I. deg Ths modellng approach descrbed n hs paper provdes an mporan gudelne o undersand he role of parameers for sablzng undesred roll oscllaons and resorng mechansm. The dampng momen consdered n hs paper s lnear n form however can be expressed as nonlnear o accoun

12 vscous dampng and varaons n he mass momen of nera. The mporan fndngs of hs sudy elucdae ha forward speed conrols he moon oscllaon and dampens he nal dsurbance.. References ass D.W. (98): On he response of based shps n large amplude waves Inernaonal Shpbuldng Progress Vol. 9 pp -9 ass D.W. and Haddara M.R. (988): Nonlnear models of shp roll dampng Inernaonal Shpbuldng Progress Vol. pp - haacharyya R. (978): Dynamcs of Marne Vehcles John Wlley & Sons New York ardo. escha M. Francescuo. and Nabergo R. (98): Effecs of he angle-dependen dampng on he rollng moon of shps n regular beam seas Inernaonal Shpbuldng Progress Vol. 7 pp-8 ardo. Francescuo. and Nabergo R. (98): Ulraharmoncs and subharmoncs n he rollng moon of a shp: seady-sae soluon Inernaonal Shpbuldng Progress Vol. 8 pp - ardo. Francescuo. and Nabergo R. (98): On dampng models n free and forced rollng moon Ocean Engneerng Vol. 9 pp 7-79 ardo. Francescuo. and Nabergo R. (98): Nonlnear rollng response n a regular seas Inernaonal Shpbuldng Progress Vol. pp -8 ummns W.E. (9): The mpulse response funcon and shp moons Schffsechnk. 9 pp- 9. Das S. N. and Das S. K. (): Deermnaon of oupled Sway Roll and Yaw Moon of a Floang body n Regular Waves Inernaonal Journal of Mahemacs and Mahemacal Scences Vol. pp 8-97 Das S. N. and Das S. K. (): Mahemacal Model for oupled Roll and Yaw Moons of a Floang ody n Regular Waves under Resonan and Non-resonan ondons ppled Mahemacal Modellng Vol. 9 pp 9- Das S. K. and Das S. N. (a): Modellng and analyss of coupled nonlnear oscllaons of a floang body n wo degrees of freedom ca Mechanca Vol. 8 pp - Das S. N. Sahoo P.K. and Das S. K. (b): Deermnaon of roll moon for a floang body n regular waves Proceedngs of he IMechE Par M: Journal of Engneerng for he Marme Envronmen (n press) Dalzell J.F. (978): noe on he form of shp roll dampng Journal of Shp Research Vol. pp 78-8 Fea G. and Jones D. (98): Paramerc excaon and he sably of a shp subeced o seady heelng momen Inernaonal Shpbuldng Progress Vol. 8 pp -7 Frank W. and Salvesen N. (97): The Frank lose-f Shp-Moon ompuer Program Repor 89 NSRD Washngon D..

13 Haddara M.R. (97): On nonlnear rollng of shps n random seas Inernaonal Shpbuldng Progress Vol. pp Hoof J.P. (98): dvanced Dynamcs of Marne Srucures John Wlley & Sons New York. Nayfeh.H. and Khder.. (98): Nonlnear rollng of shps n regular beam seas Inernaonal Shpbuldng Progress Vol. pp -9 Press W.H. Teukolsky S.. Veerlng W.T. and Flanuery.P. (99): Numercal Recpes n Forran 77 ambrdge Unversy Press New York Robers J.. (98): omparson beween smulaon resuls and heorecal predcons for a shp rollng n random beam waves Inernaonal Shpbuldng Progress Vol. pp 8-8 Salvesen N. Tuck E.O. and Falnsen O.M. (97): Shp moons and sea loads Trans. Socey of Naval rchecs and Marne Engneer Vol. 78 pp -87 Tck L.J. (99): Dfferenal equaons wh frequency-dependen coeffcens Journal of Shp Research Vol. pp - Vrgn L.N. (987): The nonlnear rollng response of a vessel ncludng chaoc moons leadng o capsze n regular seas ppled Ocean Research Vol. 9 pp 89-9 Vugs J.H. (98): The hydrodynamc coeffcens for swayng heavng and rollng cylnders n a free surface Repor 9 Shpbuldng Lab. Delf Unversy of Technology Delf Wrgh J.H.G. and Marshfeld W.. (98): Shp roll response and capsze behavour n beam seas Trans. Royal Insue of Naval rchecs Vol. pp 9-8 Wang Z. H. (): Hydroelasc nalyss of Hgh-Speed Shp PhD Thess Dep Naval rchecure and Offshore Engneerng Techncal Unversy of Denmark Lyngby