TIME DOMAIN SIMULATIONS OF A COUPLED PARAMETRICALLY EXCITED ROLL RESPONSE IN REGULAR AND IRREGULAR HEAD SEAS

8 th Iteratioal Coferece o 349 TIME DOMAIN SIMULATIONS OF A COUPLED PARAMETRICALLY EXCITED ROLL RESPONSE IN REGULAR AND IRREGULAR HEAD SEAS S. Ribeiro e Silva, T. Satos ad C. Guedes Soares Uit of Marie Techology ad Egieerig, Techical Uiversity of Lisbo Istituto Superior Técico, Av. Rovisco Pais, 149-1 Lisboa, (Portugal) Abstract I this study dyamic stability problems i head seas are ivestigated with the use of a timedomai, o-liear umerical model of ship's motios i 5 degrees-of-freedom (sway, heave, roll, pitch, ad yaw). The results preseted i this paper have show that the ship ca be subjected to the wave-iduced parametric resoace i regular ad also i irregular waves. A quasi-static approach is adopted to determie the o-liear restorig coefficiets i heave, roll, ad pitch motios i waves, i which calculatios are made usig a pressure itegratio techique over the istataeous submerged hull. Dyamic ad hydrodyamic effects i waves for a give ecouter frequecy are icluded i the 5 degrees-of-freedom resposes calculatios, which are based o the potetial flow strip theory. Comparisos betwee computed values ad applicable experimetal results demostrated the usefuless of the techique proposed. 1. INTRODUCTION Sice the early fifties [1], parametric resoace has bee already bee studied ad discussed by several ivestigators ad safety authorities. It has bee foud that a ship headig ito a twodimesioal logitudial wave system will experiece periodic variatios i her trasverse stability, which uder certai coditios will result i proouced rollig. This pheomeom has bee cosidered mostly of theoretical iterest ad less worthy of practical cocer. However, recet evidece of parametric rollig i head seas o a post-paamax C11 class cotaier ship [] received wide ad reewed attetio, demostratig the practical importace of this pheomeo. A moohull ecouterig waves with legth early equal to the ship legth will have sigificat variatios o waterplae area relatively to still water coditio. The rightig arm decreases if a wave crest is amidships ad icreases whe a trough is ear amidships. I this dyamic situatio, the ship motio should be described by coupled roll, pitch ad heave, ad hece the restorig forces ad momets should iclude effects relevat to these motios. I referece [3], it was demostrated that both liearised ad o-liear theories could be used to predict parametric rollig i regular head waves. O the liear model (i the form of Mathieu s equatio) stability variatios were evaluated from the liearised rightig arm curves with the wave crest varyig logitudially alog the ship hull. However, this model was ot accurate eough to predict ship's roll respose magitude uder wave-

35 8 th Iteratioal Coferece o iduced parametric resoace coditios, sice deck submergece effect o restorig characteristics of the vessel could ot be icluded ad therefore the limit cycle behaviour could ot be obtaied. A o-liear umerical model of parametric resoace takig ito cosideratio deck submergece ad other o-liearities o restorig momet of ships i regular waves was also proposed [3]. I that model a quasistatic approach was adopted to study the roll motio, where oly the variatios o trasverse stability i regular waves were cosidered. For that purpose, a ucoupled roll equatio, which icluded the effects of heave ad pitch resposes i regular waves, ad immersed hull variatios due to wave passage o roll restorig term, was used to describe the parametrically excited roll motios. While good agreemet i terms of limited respose behaviour was foud betwee the time domai simulatio of roll motio i logitudial regular waves ad the existig experimetal data, simulatios of parametric rollig i irregular head waves as preseted i literature by some authors [, 4, 5] could ot be performed usig that 1 degree-of-freedom (dof) model.. EQUATIONS OF MOTION I this work urestraied rigid body motios of a sleder vessel with advacig speed are cosidered. The dyamics of oscillatory ship motios is govered by Newto s secod law, which represet the equilibrium betwee the iteral forces due to iertia, gravity, ad the exteral forces actig o the ship, give by: [ M ]{ } = [ F ] & (1) These forces [ F ] ad motios { } may be represeted o a coordiate system (see Fig. 1) fixed with respect to the mea positio of the ship, X = ( x, y, z), with z i the vertical upward directio ad passig through the cetre of gravity of the ship, x alog the logitudial directio of the ship ad directed to the bow, ad y perpedicular to the latter ad i the port directio. The origi is i the plae of the udisturbed free surface. The traslatory displacemets i the x, y, ad z directios are respectively the surge 1, the sway, ad the heave 3. The rotatioal displacemets about the x, y, ad z axes are respectively the roll 4, the pitch 5, ad the yaw 6. To overcome these shortcomigs a o-liear model, coupled i the 5 dof s has bee developed to simulate the time domai resposes of a ship i ui-directioal logcrested irregular waves. Although this oliear time domai simulatio does ot accout for all o-liear hydrodyamic pheomea, the code is ow more sophisticated ad capable of predictig parametric rollig resposes i a sceario more closely related to the ship s coditios that may be foud at sea.

8 th Iteratioal Coferece o 351 depedet coefficiets of added mass ad dampig are computed by the Frak s close fit method, ad the sectioal diffractio forces are evaluated usig the Haskid-Newma relatios [6]. Fig. 1 - The right-had tri-axial coordiate system ad six modes of ship motio, ad defiitio of the ship headig agle. I geeral, the forces actig o the ship s hull cosist of cotrol forces from rudder, ad active fis, evirometal forces from wid ad waves ad reactio forces due to ship motios. I this study water is assumed icompressible, iviscid (although viscous effects are cosidered whe roll dampig is calculated), ad deep. Whe the problem of wave-iduced parametric rollig is cocered, oly the forces due to wave excitatio ad reactio forces due to wave-iduced ship motios are take ito accout. Other forces are assumed to be cacelled by each other, which meas, the ship ad relative course of the ship to the wave directio is kept costat durig the time domai simulatio for a give loadig coditio. Hece, the wave excitatio forces cosist of icidet wave forces (or Froude- Krylov forces), diffractio forces, ad the reactio forces of restorig forces ad radiatio forces. Surge motio, is assumed to be fixed. β A quasi-static approach is adopted to calculate the o-liear restorig coefficiets i heave, roll, ad pitch motios i waves, i which calculatios are made over the istataeous submerged hull. At this poit it should be metioed that a more sophisticated waveiduced parametric roll model could be adopted, where added masses ad dampig coefficiets would be also calculated with cosideratio to the istataeous submerged hull body uder the wave surface. However, the hydrodyamic compoet of the parametric excitatio is isigificat i compariso with quasi-hydrostatic excitatio caused by the icidet wave potetial ad the wave-iduced heave ad pitch motio. The hydrodyamic compoet depeds upo the overall submerged hull form, while the quasi-static hydrostatic compoet is strogly depedet upo wave passage ad hull-shape (i.e. variatios about the still waterlie). For this reaso ad others associated with larger computatioal efforts oly the quasihydrostatic compoet of the parametric excitatio is take ito accout at this stage. I particular the roll added iertia ad radiatio coefficiets are take to be liearly proportioal to the roll acceleratio ad velocity, respectively. Hece, the hydrodyamic memory effect due to roll motio ad cosequetly its effect, expressed as roll dampig, is practically egligible at frequecies lower tha.5 [rad/s] (see Fig. ). I a approximate way the radiatio ad wave excitatio forces are calculated at the equilibrium waterlie usig a stadard strip theory, where the two-dimesioal frequecy-

35 8 th Iteratioal Coferece o Roll Hydrodyamic Coefficiets η w [ x cos β ( c U cos β ) t] a = η cos k (4) w A44/(ML^) B44/(M.L^.sqrt(g/L)) 5.E-4 4.E-4 3.E-4.E-4 1.E-4.E+.5.5.75 1 W [rad/s] Fig. Added mass ad dampig coefficiets of roll motio. Uder the assumptios preseted before all hydrodyamic forces are liear, ad combiig these with the mass forces oe obtais six liear coupled differetial equatios of motio, give by: 6 = j 1 {( M + A )& ( ) } k, j = 1,...,6 kj kj + B + C t = F j kj& j kj j k Here the subscripts A44 B44 () k are associated with forces i the k -directio due to motios i the j -mode ( k =1,, 3 represet the surge, sway ad heave directios, ad 4, 5, 6 represet roll, pitch ad yaw directios). M are the compoets of the mass matrix for the ship, A ad B are the added mass ad dampig kj kj coefficiets, C kj (t) are the hydrostatic (time depedet) restorig coefficiets, ad F k are the complex amplitudes of the excitig forces, where the forces are give by the real part of i et F e ω. k If the ship travels alog a prescribed path β at a iitial steady velocity U (see Fig. 1), she will ecouter the regular wave crests with a frequecy of ecouter, give by: ω e = ω ku cos β (3) j kj I irregular seas, it is possible to describe the equatios of motio give by the sum of siusoidal waves yieldig to a irregular wave profile give by: ω x cos β = N g a η w η w cos (5) = 1 ω + ω U cos β t ε g where N is the umber of compoet waves, ω the circular frequecy, ε the radom a phase agle, ad η the amplitude of the -th w compoet waves which are give by the wave spectrum S (ω ). Therefore, i this study parametrically excited roll respose i regular ad irregular head seas is ivestigated where liear ideal flow hydrodyamic added mass, dampig coefficiets ad wave excitatio forces are cosidered ad o-liearities are itroduced oly via hydrostatic terms time depedece associated with wave passage ad hull-shape, ad o-liear roll dampig. It has bee foud for example by Foseca ad Guedes Soares [7, 8] that this type of approach describes well o-liear motios. 3. ROLL DAMPING ASSESSMENT The dampig coefficiet ca be obtaied from free decay experimets, i which the model is released from a give icliatio agle to freely roll i calm water with o forward speed. Usig adequate istrumetatio a decay curve data poits as show i Fig. 3 were obtaied. where the surface elevatio of a regular wave is give by:

8 th Iteratioal Coferece o 353 Roll Amplitude [deg] 3 1-1 4 6 8 - -3-4 Time [sec] phi phi_id Fig. 3 Ship model free decay ad o-liear simulatio curves. The usig appropriate parametric idetificatio techiques, the coefficiets B 441 ad B 44, ca be obtaied by fittig equatio 6 to the recorded free decay experimet data. ( M + A )& + B & + B & & + C (6) = 44 44 4 441 4 44 4 4 44 4 where B 441 is the liear dampig coefficiet, ad B is the quadratic dampig coefficiet. 44 Because of the limited umber of cycles of roll decay traces, the eergy balace method is adopted. This method is based o the cocept that the rate of chage of the total eergy i roll motio is equal to the rate of eergy dissipated by the roll dampig. As show i equatio 6, it is also assumed that the ship is uder ucoupled roll motio durig the free decay experimets. Accordig to referece [9], the equivalet liearised roll-dampig coefficiet is therefore related to the dissipated eergy ad is give by: 8 = + ω (7) 3π a b b b 44 eq 441 44 4 44 Very good agreemet is foud betwee the 'idetified' curve obtaied from umerical itegratio, give i Fig. 3 by the cotiuous lie, ad the experimetal data poits acquired durig free decay test. From previous studies [3] it is kow that whe roll dampig is tued to model test results, a very good correlatio of the roll motio ca be achieved betwee model tests ad the umerical aalyses. Particularly, the magitude of the roll respose durig parametric rollig is dictated i large part by the amout of viscous dampig i the roll degree of freedom. To accout for these dampig effects i this study, it was decided to compare the empirical roll dampig established from the free-decay model tests with a applicable aalytical method. Hece, it was foud that the roll-dampig method of compoets, preseted i referece [1], could also be used to accurately calculate the total roll-dampig coefficiet at zero advace speed. 4. INSTANTANEOUS NON-LINEAR RESTORING FORCES AND MOMENTS Istataeous restorig forces ad momets are calculated from the exact determiatio of the ship s displacemet ad its cetre of buoyacy at each step. The relatioship betwee the wave surface ad the ship s hull i the seaway was established takig ito accout the resposes of the ship i the five degrees-of-freedom as well as the ship speed ad headig. I fact, i this coditio of excitatio of motio the ship basically has o respose i sway ad yaw modes but the computer code allows for it. As demostrated i referece [3], cosiderig a moohull i logitudial regular waves about the same legth as the ship's legth variatios of istataeous restorig forces ad momets are due to sectioal beam variatios dy dx, which deped o ship's hull vertical flare dy dz (see Fig 4). Hydrostatic forces ad momets calculatios are made usig the pressure itegratio techique over the ship hull, rather tha usig area ad volume itegratio of the ship offsets. The origial theoretical approach to the pressure itegratio techique outlied by referece [11] has bee adopted i cojuctio with a practical method to geerate the paels required to calculate the hydrostatic pressure

354 8 th Iteratioal Coferece o distributio uder either a regular or irregular wave profile [1]. Therefore, i this model o-liearities are cosidered i the heave, roll, ad pitch restorig terms, takig ito accout all the istataeous variatio of the hull-shape i waves icludig evetually the occurrece of deck submergece. Fig. 4 Variatio of sectioal beam o logitudial waves. 5. EXPERIMENTS IN REGULAR HEAD SEAS Tests were coducted at the small towig tak of the UCL [13], equipped with a flap type wavemaker ad a towig carriage. A typical fast form of a large refrigerated cotaier ship was selected as a subject of the experimetal studies o parametric rollig. The body lies are preseted i Fig. 5. The model was of woode ad GRP costructio, the hull is vertical sided amidships above the waterlie ad completely uappeded ad upropelled. The pricipal particulars of the model are give i Table 1. The model was costraied i yaw, sway, ad surge. This 1:1 scaled model was istrumeted so that traces of its dyamic respose i waves were produced. I the experimets with o advace speed, the ship s model loadig coditio was maitaied ad set to be compliat with the applicable IMO itact stability criteria i force at that time [14]. Durig these rus, wave amplitude ad frequecy were varied i order to ivestigate the sesitivity of the scaled model to parametric resoace durig which capsize ca be iduced. The i order to obtai parametric resoace with advace speed, the roll stiffess of the model was adjusted. This variatio o trasverse metacetric height was achieved displacig vertically the existig masses o the stud mast. Durig the ru alog the tak with regular waves of wavelegth about the model's legth, low cycle resoace was obtaied by varyig the forward speed of the carriage. Roll decremet tests were performed for differet iitial agles, ad icliig experimets were coducted to assess dampig ad trasverse stability characteristics, respectively. It was foud that rollig was iduced oly i a few coditios. As show i Fig. 7.a (scaled by a factor 1:1), the most proouced rollig occurred whe the model was adjusted to a roll period twice the period of the wave. As ca be see from Fig. 7.a, the roll agle icreases o each successive swig up to defied limited amplitude, ad is at half the wave frequecy. I Fig. 7.b, it ca be see that the pitch motio was regular ad of the same frequecy as the waves. Heavig motio was also regular ad had the same periodicity of the waves. The sway ad yaw motios were very small ad irregular. Fig. 5 Body lies of the refrigerated cotaier ship model.

8 th Iteratioal Coferece o 355 L BP L OA B WL D T WL W Table 1 - Refrigerated cotaier ship model particulars. = 18. cm = 13.4 cm = 18.1 cm = 11.5 cm = 7.68 cm = 1.7 Kg C M =.973 C P =.578 C W =.731 S W = 319 cm T GM t =. sec =.17 cm Scale = 1:1 Iitially, several umerical simulatios were carried out at the same load coditio utilised for the UCL model tests for a umber of combiatios of speeds ad wave coditios. Wave Surface Elevatio [m] 4 3 1 - Wave Surface Elevatio at Fore PP 4 6 8 1 1 14 16-1 Wave 6. NUMERICAL RESULTS As metioed, the strip theory computer program was used to perform the calculatio of the added mass ad dampig coefficiets, ad wave excitatio forces ad momets for a give wave ecouter frequecy at the equilibrium waterlie. With respect to roll dampig coefficiet, viscous compoets were the added to wave-makig compoet i order to obtai the total roll-dampig coefficiet. These are fed ito equatios, which are the itegrated to simulate a oliear time-domai wave-iduced parametric resoace coditio. For this purpose the computer code preseted herei was developed to icorporate ship motios calculatios, istataeous restorig forces ad momets calculatios, ad the to perform a umerical itegratio of the equatios usig a 4th order Ruge-Kutta algorithm o a step-by-step basis. Numerical results of motios excited by logitudial regular waves obtaied from simulatios showed that sway ad yaw were isigificat i compariso with heave, roll, ad pitch resposes. The pitch ad roll resposes were the used for compariso with time series recorded i the experimetal programme for the model. Eta_3 [m] Eta_4 [deg] Eta_5 [deg] -3-4 Heave Displacemet 1.5 1.5 4 6 8 1 1 14 16 -.5-1 -1.5 - Roll Agle 4 3 1 4 6 8 1 1 14 16-1 - -3-4 Pitch Agle 1 5 4 6 8 1 1 14 16-5 -1 eta_3 eta_4 eta_5 6.1. Simulated istability i regular head seas -15 Fig. 6 - Numerical simulatio i regular head seas ( H W = 6 [m] ad T W = 14 [s]).

356 8 th Iteratioal Coferece o The, as show i Fig. 6 the same wave coditios (correspodig to the most critical wave-iduced parametric rollig coditio) were used i the simulatios as i the model tests. Plotted are wave surface elevatio, ad heave, roll, ad pitch displacemets durig low-cycle parametric resoace. I Fig. 7.a wave surface elevatio ad ship s roll motio records from simulatio ad experimets are compared, ad, obviously the parameters ivolved i the equatios are equivalet to those used i model calculatios. I what cocers logitudial plae motios good agreemet is agai obtaied betwee umerical simulatios ad experimetal results, as compared i Fig. 7.b for pitch motio. Roll Amplitude [deg] & Wave Amplitude [m] 4 3 1 - -3-4 Experimetal vs Numerical Roll Respose Time Histories 4 6 8 1 1 14-1 Noli. Roll [deg] Wave [m] Exp. Roll [deg] Wave_Tq [m] Tim e [sec] Fig. 7.a - Compariso of umerical itegratio with UCL experimetal results i regular waves roll motio. Pitch Agle[deg] & Wave Amplitude [m] 15 1 5-5 -1-15 Pitch Motio Compariso 5 1 Time [sec] Wave_Tq Pitch_Exp Fig. 7.b - Compariso of umerical itegratio with UCL experimetal results i regular waves pitch motio. Wave Pitch It ca be see that the mai features of parametric roll pheomeo are predicted by the computer programme. Like the model test results, there is a period with o rollig, ad the roll agle icreases o each successive swig up to defied limited amplitude, ad is at half the wave frequecy. Also, it ca be see from Fig. s 6 ad 7 that roll agles icreased from a few degrees to over 3º i oly five roll cycles, ad at this stage that the model was pitchig to agles of about 5º, respectively. Moreover, there are two pitch cycles for each roll cycle, ad the model is always pitched dow by the bow at maximum roll. A questio of particular iterest i o-liear systems is the existece of closed trajectories, as such trajectories imply periodic motio, ad evetually a stable state kow as the limit cycle. As show i Fig. 8 for the wave-iduced parametric rollig coditio the respose is practically siusoidal ad the phase plae plot is early a elliptical spiral. Roll Velocity [deg/sec] 15 1 5-4 -3 - -1 1 3 4-5 -1-15 Roll [deg] Fig. 8 Roll phase diagram i regular waves. As show i a previous study [3] with a liear formulatio the roll motio will ot have a limit cycle behaviour. The o-liear model proposed here captures the time variatio of the roll restorig momet, which is the most importat effect that cotributes parametric resoace at the iitial trasiet stage. It also takes ito cosideratio the effect of deck submergece o restorig momets at the steady state stage, which prevets roll agles from buildig up to ifiity.

8 th Iteratioal Coferece o 357 6.. Simulated istability i irregular head seas For the simulatios i irregular waves, both radomly distributed amplitude ad phase compoets are utilised to geerate a icidet wave realizatio. More specifically, the modellig of the icidet wave spectrum is made by a fiite umber of harmoic waves where a lower ad a upper limit for the wave frequecy, ω mi ad ω Max are defied. The cotiuous icidet wave spectrum is therefore discretised by a umber of harmoic wave compoets N of frequecy ad amplitude: wc ω = ω +. ω (8) mi seed a η =. S ( ω ). ω w (9) where: ω Max ω ω = mi (1) N wc The phase agles of the regular waves ε are also radomly distributed i the etire rage ε = [, π ], ad is give by: ε = seed. ε (11) Although, the wave eergy of the discretised wave systems resultig from the above approach equals the wave eergy of the icidet irregular seaway adopted, this wave realizatio i space is strogly depedet upo the seed umber seed provided as iput for geeratio of radom waves. I this case, the icidet wave is described by a JONSWAP spectrum S JS (ω ), with a peak ehacemet factor γ = 3.33, a sigificat wave height H = 6 [m], ad a peak period T s p = 14 [s] (see Fig. 9). Fig. 9 JONSWAP spectrum (γ = 3.33, H = 6 [m], ad T p = 1 [s]). S JS ( ω) = K. S PM ( ω). φ JS ( ω) (1) where: 3 3 4π H - = S 1 16π 1 S ( ) exp PM ω 4 5 is 4 4 TZ ω TZ ω the eergy desity fuctio of the Pierso- Moskowitz spectrum; 1 ω p ω exp σω - ( ) p 1.5 ( ω) = 1 e.l γ γ φ JS is the fuctio of the JONSWAP spectrum; - K is a factor such as H S K = S ( ω ) dω = PM. 16 The wave spectrum is approximated by a wave system cosistig of 41 elemetary waves, ad as far as the reproductio of aperiodicity for the modelled seaway is cocered, this is clearly achieved (see Fig. 1). The simulatio of ship motios is agai performed for a ship headig of 18º ad a zero advace speed coditio. Therefore added mass ad dampig hydrodyamic coefficiets are calculated for the cosidered values of peak spectral frequecy ad sigificat wave height. Sice parametric rollig is a resoat pheomeo, it was foud that cotrarily to the regular wave s sceario i irregular seas ships might ot be proe to regularly exhibit parametric rollig due to waves groupiess effect. More specifically, i irregular seas the s

358 8 th Iteratioal Coferece o required sychroisatio betwee waves ad roll respose period might ot be sustaied log eough due to variatios of the wave realizatio i time ad space. I Fig. 1, the simulatio records of wave surface elevatio, heave, roll, ad pitch vessel resposes i the auto-parametric rollig sceario are preseted. Fig. 11, shows a combied view of the roll ad pitch resposes i irregular waves, where roll agles exceedig 3º to each side, after 13 secods are evidet. Comparisos of motio time histories betwee the preset predictios ad model tests are ot possible because free-ruig experimets i irregular waves have ot yet bee carried out. However, the couplig betwee maximum roll ad bow dow pitch observed i the regular waves model tests is also detected i irregular waves. I Fig. 1 the same result is preseted i the form of roll phase diagram. Eta_3 [m] Wave Surface Elevatio [m] Wave Surface Elevatio at Fore PP 14 1 1 8 6 4-4 6 8 1 1 14 16-4 -6 Heave Displacemet 3 1 4 6 8 1 1 14 16-1 - -3 Roll Agle 4 3 Wave eta_3 As i regular waves, the roll motio builds to large amplitudes ad whe either the wave period is chaged or the wave height dimiishes, the parametric resoace disappears. Therefore, parametric rollig ca occur i irregular seas coditios provided agai there is sufficiet ecoutered eergy ear twice the atural roll period. Although computer code does ot accout for all the hydrodyamic pheomea, the code is capable of predictig parametric rollig i head seas ad ca be used to predict if a vessel is proe to exhibit parametric rollig i a early desig stage. Eta_4 [deg] Eta_5 [deg] 1 4 6 8 1 1 14 16-1 - -3-4 Pitch Agle 1 8 6 4-4 6 8 1 1 14 16-4 -6-8 -1-1 eta_4 eta_5 Fig. 1 Numerical simulatios i irregular waves.

8 th Iteratioal Coferece o 359. Roll [deg] & Pitch [deg] 4 3 1-1 - -3-4 Roll ad Pitch Time Histories 4 6 8 1 1 14 16 eta_4 eta_5 Time [sec] Fig. 11 Combied view of roll ad pitch motios i irregular waves. Roll Velocity [deg/sec] 15 1 5-4 -3 - -1 1 3 4-5 -1-15 Roll [deg] Fig. 1 Roll phase diagram i irregular waves. 7. CONCLUSIONS I this study a o-liear umerical model is utilised to simulate parametric resoace i both regular ad irregular waves. I additio, special attetio has bee give to the usefuless of model experimets coducted uder coditios as realistic as possible to validate the theoretical approach proposed. The mai cocludig remarks are summarised as follows: a) It has bee cofirmed that a statically stable ship ecouterig waves of her ow legth ad a frequecy twice her atural roll frequecy will experiece a wave-iduced parametric rollig situatio. Therefore, uder these coditios, roll agles exceedig 3º to each side ca be produced rapidly, resultig sometimes i cargo losses ad ship s damage, or evetually i capsize. b) Whe roll dampig is tued to model test results, a very good correlatio of the roll motio ca be achieved betwee model tests ad the umerical aalyses. Therefore, i this study, the liear ad quadratic dampig momets are liearised by meas of the eergy balace method ad expressed by a equivalet liear roll-dampig coefficiet, which compares well with the aalytical method of compoets. c) A o-liear umerical model of parametric resoace takig ito cosideratio deck submergece ad other o-liearities o restorig forces ad momets of ships at sea is the proposed. Good agreemet is foud betwee the time domai simulatio of ship s resposes i a logitudial seaway ad experimetal data. Moreover, oly represetig coupled heave, roll, ad pitch resposes i log-crested irregular waves parametric rollig ca be properly simulated. d) I irregular waves the occurrece of parametric rollig ca be highly depedet o the iput seed umber ad a large umber of simulatios ca be required to obtai a parametric resoace coditio with roll agles exceedig 3º. 8. REFERENCES [1] Kerwi, J. E., Notes o Rollig i Logitudial Waves, Iteratioal Shipbuildig Progress, Vol., Nº 16, pp. 597-614 (1955). [] Frace, W. N.; Levadou, M.; Treakle, T. W.; Paullig, J. R.; Michel, R. K. ad Moore, C., A Ivestigatio of Head-Sea Parametric Rollig ad Its Ifluece o Cotaier Lashig Systems, Marie Techology, Vol. 4, Nº 1, pp. 1-19 (3). [3] Ribeiro e Silva, S. ad Guedes Soares, C., Time Domai Simulatio of Parametrically

36 8 th Iteratioal Coferece o Excited Roll i Head Seas, Proceedigs of the 7th Iteratioal Coferece of Ships ad Ocea Vehicles (STAB ), Vol. B, pp. 65-664, Laucesto, Tasmaia, Australia (). [4] Hua, J., A Study of the Parametrically Excited Roll Motio of a RoRo Ship i Followig ad Headig Waves, Iteratioal Shipbuildig Progress, Vol. 39, pp. 345-366 (199). [5] Fracescutto, A. ad Bulia, G., Noliear ad Stochastic Aspects of Parametric Rollig Modellig, Proceedigs of the 6th Iteratioal Ship Stability Workshop, Webb Istitute (). [6] Salvese, N.; Tuck, E. O. ad Faltise, O., Ship Motios a d Sea Loads, Trasactios of SNAME, Vol. 78, pp. 5-87 (197). [7] Foseca, N. ad Guedes Soares, C., Time - Domai Aalysis of Large-Amplitude Vertical Ship Motios ad Wave Loads, Joural of Ship Research, Vol. 4, º, pp. 139-153 (1998). [8] Foseca, N. ad Guedes Soares, C., Compariso of Numerical ad Experimetal Results of No-Liear Wave Iduced Vertical Ship Motios ad Loads, Joural of Marie Sciece ad Techology, Vol. 6, pp.139-4 (). [9] Lloyd A. R. J., Seakeepig: Ship Behaviour i Rough Weather, Ellis Horwood Publishers (1989). [1] Ikeda, Y; Himeo, Y. ad Taaka, N., A Predictio Method for Ship Roll Dampig, Uiversity of Osaka Prefecture, Report Nº 45 (1978). [11] Schalck, S. ad Baatrup, J. "Hydrostatic Stability Calculatios by Pressure Itegratio", Ocea Egieerig, Vol. 17, pp. 155-169 (199). [1] Satos, T. A. ad Guedes Soares, C. G., RoRo Ship Damage Stability Calculatios Usig the Pressure Itegratio Techique, Iteratioal Shipbuildig Progress, Vol. 18, Nº, pp. 169-188 (1). [13] Ribeiro e Silva, S., Desig Criteria ad Guidace to Avoid Parametric Rollig, MSc i Naval Architecture Dissertatio Project Report, Uiversity College Lodo (1998). [14] Iteratioal Maritime Orgaisatio, Co de of Itact Stability for all Types of Ships Covered by IMO Istrumets. Aex to Report of the Ad Hoc Itact Stability Workig Group. Doc. SLF 36 / 3 (1991).