II - PARAMETRIC ROLLING. The description of ship rolling in a purely longitudinal sea can be obtained by considering the following mathematical model:

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2 in had rgular wavs with wavlngth qual to modl lngth at watrlin. Svral wav hights wr tstd to invstigat th dtaild structur of th thrshold for aramtric rolling, its dndnc on th tuning ratio and th roll motion amlitud abov thrshold. This last is an imortant itm sinc usually w rad that abov thrshold w can hav roll instability, but rarly w rad in th rorts that this rolling motion, whn xcitd, usually is boundd and rarly unboundd, this last fact aaring as a kind of scond thrshold. Th roll dcays at sd wr also rcordd togthr with th vrtical acclration and th itch angl to rconstruct th shi rlvant aramtrs for th simulation of th hnomnon. In articular, frqunt cass of grn watr on dck wr obsrvd at th highst sa stat (wav stnss=/) wr obsrvd. Th onst of vry larg roll amlitud lading to casiz was rstrictd to th highst sa stat and to th lowr shi sds. Th corrlation of ths rsults to th dndnc of roll daming on shi sd and btwn th thrsholds and th rlativ variations of righting arm in wavs comutd by matching th rsults of stri thory and a hydrostatic calculations was satisfactory. Th roblm of th rdiction of roll amlitud abov thrshold was solvd by mans of a nonlinar modlling taking into account th nonlinar faturs of th righting arm and th nonlinar faturs of th diffrnc btwn maximum and minimum valu of th righting arm in wavs, which was sn to disaar at larg angls. Th comarison btwn xrimntal and aroximat rturbativ solution of th quation of motion was satisfactory. II - PRMETRIC ROLLING Th dscrition of shi rolling in a urly longitudinal sa can b obtaind by considring th following mathmatical modl: I ' D(, ) R(, t) = () whr I is th virtual momnt of inrtia, D th daming, R = GZ( ) th rstoring and thr is no xlicit forcing trm du to th wav action. ctually, all trms bcom tim dndnt, but xlicit tim dndnc is usually rtaind only in th righting momnt R, bing th othr of minor ntity. oth D and R ar non-linar and dislay this fatur at larg amlituds of th motion. If w considr th onst of aramtric rolling, that is if w rstrict for th momnt to th analysis of th stability of th solution ( t), th Eq. can b linarisd, artly simlifying th roblm: I ' M GM(t) = () Considring a sinusoidal tim variation of th transvrsal mtacntric hight (othrwis w can considr th first trm of its Fourir sris) with amlitud δ GM around th avrag valu GM and, dividing as usual by I, on has: δgm cos GM ( t ε) µ = ()

3 which is an quation of th Mathiu ty []. Th has ε can b nglctd without rjudic for th subsqunt analysis and with a chang of th tim scal t = t' (rtaining th sam nam) Eq. can b transformd into th mor familiar form: µ δgm cos GM = ( t) () with µ = µ /. µt Canclling th daming by mans of a linar transformation, which introducs an µ amlitud chang in th roll amlitud, with µ =, and considring that th natural frquncy of th slightly damd rolling dam w obtain GM δ cos = GM ( t) () This quation is a diffrntial quation with riodic cofficints. Floqut thory [] indicats ±σt ψ that its solutions can b ut in th form (t) whr ψ (t) is a riodic function and σ is th charactristic xonnt". Th solutions of undamd Mathiu quation ar thus divrging if σ and stabl if σ=. If linar daming is rsnt, th situation is qualitativly and quantitativly modifid, i.. th solutions of damd Mathiu quation and th shi bhaviour will b: Condition Mathiu qn solution Shi vrtical osition quilibrium Effct of a rturbation to Shi vrtical osition both µ ±σ > Divrging Stabl Growing in tim µ = σ = Stabl Unstabl Stabl in tim both µ ±σ < Dcaying Unstabl Dcaying in tim Ths conclusions ar valid to discuss th stability of quilibrium and hnc initial stability. Thy ar valid to discuss roll amlitud bounddnss and hnc dynamic shi stability (or stability in th larg) as far as th shi rolling can b dscribd by a linar mathmatical modl lik Eq. abov. From a ractical oint of viw, w know that this is no longr tru at larg angls, whr Eq. has to b substitutd by a much mor comlicatd non-linar modl, which will b solvd by mans of a rturbation mthod. Th ossibility of onst of th dangrous hnomnon of aramtric rolling is thus tid to th simultanous vrification of th following conditions: th ratio of th ncountr wav frquncy to th natural roll frquncy is clos to th condition: with n intgr; (6) n th riodic variation of mtacntric hight du th combind ffct of shi motions and wav is sufficintly larg;

4 th roll daming is sufficintly small. s far as th first instability zon is concrnd (n=), a thrshold valu: δgm = GM ( µ ) (7) for th onst of aramtric rolling, is found, which rducs to GM δ < < GM whn µ =, whil it givs a minimum thrshold valu (8) δgm 8µ = µ = GM µ = (9) in roximity of th xact synchronism condition =. Whn a nonlinar daming trm is considrd,.g. viscous daming, a thrshold similar to Eq. 7-9 can still b obtaind []. III - ROLL MOTION MPLITUDE OVE THRESHOLD Th thrshold hnomnon is known sinc th arly fiftis. On th othr hand, th mathmatical modlling of th roll motion abov thrshold is still subjct of discussions. nonlinar aroach, basd on a sris of xrimnts conductd on th scal modl of a dstroyr is hr rsntd. Th had sa condition was tstd du to th fact that xrimnts hav bn conductd in th towing tank of th Dartmnt of Naval rchitctur of th Univrsity of Trist. Th limitd lngth of th towing tank ( m) maks ossibl xrimnts in following wavs only at vry low sd. In Tabl. th main dimnsions and mchanical data of th shi modl ar rsntd. Th righting arm in full scal in calm watr is rrsntd in Fig.. Lb Loa T KG GM (at zro sd) µ.7 µ.896 (.±.) m (.6±.) m (.7±.) m (.8±.) m (6±) N (.±.) m (.±.) m (.±.) rad/s Tabl. Main data of th scal modl of th dstroyr. Two sris of xrimnts hav bn conductd:

5 - roll dcay in calm watr with forward sd to obtain an stimat of roll frquncy and of roll daming dndnc on vlocity. Th rsults hav bn analysd in trms of quivalnt linar daming, which rsultd to b wll aroximatd by a cubic function of th forward sd: µ = µ µ v ; - analysis of th stability of vrtical osition vrsus xcitation of aramtric rolling with forward sd in had wavs with wavlngth qual to shi lngth at watrlin. Thr h w diffrnt wav stnsss hav bn tstd: s w = =,,. λ.8 w.7.6 GZ, GM (m) fi (dg) Fig.. Righting arm in calm watr. Th dtails of xrimntal stu and th xrimntal rsults ar rortd in [-]. In Fig. and Fig., th righting arm with crst/through amidshis ar comard with th calm watr rsult for th two xtrm wav stnsss. Th righting arms hav bn calculatd in th fr trim hydrostatic quilibrium hyothsis. Looking for a mathmatical modlling of th aramtric roll, w first considr a modl basd on on ordinary diffrntial quation dscribing isolatd rolling motion with th inclusion of on or mor tim dndnt trms dscribing th intraction with th wavs. Th ossibility of using concntrat aramtr modls to simulat larg amlitud motions has bn th subjct of many discussions in th ast. Th conclusion is that th sarability of th diffrnt contributions (addd mass, daming, rstoring and forcing) is ossibl only in th rsnc of small amlitud rolling motion [6]. Th sam is tru for th couling btwn rolling motion and th othr latral motions. From a ractical oint of viw, an xtnsiv sris of masurmnts conductd on svral scal modls in bam wavs of small and so small stnss, with roll amlituds attaining dg in svral cass, indicatd that th ossibility of a rliabl dscrition basd on isolatd roll motion diffrntial quation gos far byond th xntanc [7-9]. This is somthing similar to th qustion of th validity of th rsults of rturbation mthods alid to nonlinar roll motion which, although basd on th hyothss of small rturbation aramtr (connctd usually with amlituds <<), comar rasonably wll with xact numrical rsults xtnding to vry larg amlituds. In th following, thrfor, w will try again th sam rout avd with th following assumtions: - sarability of calm watr and wav actions;

6 - singl dgr of frdom; - alicability of rturbation mthod to obtain rliabl aroximat analytical solutions; - ffct of longitudinal wav on addd mass and daming ngligibl with rsct to th ffct on righting arm..6. calm Sw=. GZ (m). through.. crst fi (dg) Fig.. Righting arm in calm watr and in th rsnc of a / stnss longitudinal wav..6. calm Sw=. through GZ (m).. crst fi (dg) Fig.. Righting arm in calm watr and in th rsnc of a / stnss longitudinal wav. Th only contribution of th longitudinal wav will thrfor b on righting arm: diffrnt modls for th dscrition of this action will b roosd in th following. W hav th intrinsic nonlinarity in th angl of th righting arm (that is rsnt also in calm watr) and th tim variation of th righting arm dnding on th longitudinal wav assing along th shi and its dirct ffcts (vrtical motions). Ths two ffcts can b considrd in "could" and "uncould" modls.

7 Th analysis of th curvs rlativ to crst and through of th wav amidshis rvals that th righting arm oscillation is givn by th linar aroximation: GM cos t () δgm with = at small inclinations, thn it grows to a maximum valu and finally vanishs GM at an angl max. On this basis, a arabolic variation of th amlitud of this oscillation was assumd. III. - Uncould modls Th following mathmatical modl was slctd for this rliminary nonlinar aroach: µ δ [ ( ) cos t] α α... = with <. In Eq., th rrsntation of th righting momnt through a olynomial was usd: () m R = R I' = α α... () Th valus of th cofficints α, α, α7,... can b obtaind by mans of a last squar fit to th hydrostatic calculation rsults. In this study only th cubic nonlinar trm will b rtaind. Posing again: t t' and rtaining th sam nam for tim: = µ δ [ ( ) cos t] α = with: µ as abov and δ = δ α = α =. Th sam mathmatical modl, with aramtric xcitation rrsntd by only has bn usd by othr authors (s for instanc []). In th first instability zon n= and th solution has th form: () ( t) sin t cos t () with and slowly varying amlituds. Driving, substituting in Eq. (8) and using th auxiliary condition: cos t sin t = () a systm of algbraic quations is obtaind for and. vraging ovr on riod, th following volutionary systm is obtaind for th avragd tim drivativs and :

8 ( ) ( ) ( ) ( ) (6) α δ µ = α δ µ = Th stationary solution C = can thn b obtaind by solving for and th abov systm with th osition = =. Sinc this is quit comlicatd for insrtion in a aramtr idntification tchniqu for a nonlinar systm in th rsnc of bifurcations [], in this ar a simlifid aroach was ffctivly usd, basd on th us of an avrag -valu: av = (7) at th gnric itration of th zro-sarching rocdur usd to solv th algbraic quation giving C. doting a constant -valu, indd, th abov algbraic systm rducs quit asily to a singl algbraic scond dgr quation for C, which can b asily solvd. III. - Could systm Now th ffcts of th wav assing along th shi and th intrinsic nonlinarity of th righting arm ar could: ( ) [ ] ( ) cos t = α δ µ (8) y alying th sam rocdur as bfor, th volutionary quations of th solution ar givn by: ( ) ( ) ( ) ( ) α α α δ µ = α α α δ µ = (9) 8 6 6

9 Which givs a fully nonlinar systm of algbraic quations in and for th stationary solution. IV - COMPRISON WITH EXPERIMENTL RESULTS ND CONCLUSIONS Th xrimntal rsults ar givn in Fig. to 6. Th simulation was obtaind by using Eq. 6 with th simlifying hyothsis (7). Th aramtr valus usd wr: - th xrimntally masurd valus for daming; - th valu of obtaind from fr trim hydrostatic analysis; whras th valus of and α hav bn stimatd by mans of a nonlinar rgrssion to th xrimntal valus (Paramtr Idntification Tchniqu). Fn Sw=/ Roll mlitud (dg) v (m/s) Fig.. Exrimntal rsults vrsus simulation basd on Eq. 6-7 for th cas s w =/. Fn Sw=/ Roll mlitud (dg) v (m/s) Fig.. Exrimntal rsults vrsus simulation basd on Eq. 6-7 for th cas s w =/.

10 Fn casiz Sw=/ Roll mlitud (dg) Fig. 6. Exrimntal rsults vrsus simulation basd on Eq. 6-7 for th cas s w =/. Th comarison is satisfactory. It is worth noting th cas sw=/, whr th jum in th simulatd amlitud is clos to th zon whr th xrimnts gav vry larg amlituds with casizing tndncy (th modl was rstraind not to rach xcssiv roll amlituds). This jum is tid to th fact that crosss zro at an inclination of about dg, and thn rcovrs du to th invrsion of th curvs. This bhaviour, although rsulting from an analysis basd on som strong assumtions, is nvtthlss uzzling. Th stimatd aramtr valus ar in qualitativ agrmnt and follow th trnd givn by hydrostatic calculations, but gnrally th good simulation was obtaind with highr valus of α and lowr valus of. This can b du to th vrtical motions of th shi and to th artial quivalncy of th trms corrsonding to th two factors in th rturbativ aroach. Th comarison with th full Eq. 6 and with th could systm Eq. 9 is in rogrss du to th nd to introduc in th Paramtr Idntification Tchniqu th solution of th abov systms of quations. CKNOWLEDGEMENT This rsarch has bn dvlod with th financial suort of INSEN undr contract "Study of th Roll Motion in Longitudinal Wavs" in th fram of INSEN Rsarch Plan -. REFERENCES v (m /s) [] Hayashi, C., "Nonlinar Oscillations in Physical Systms", McGraw Hill, Nw York, 96. [] Hsih, D. Y., "On Mathiu Equation with Daming", J. Math. Phys., Vol., 98, [] Francscutto,., n Exrimntal Invstigation of a Dangrous Couling twn Roll Motion and Vrtical Motions in Had Sa, Procdings th Intrnational Confrnc on

11 Hydrodynamics in Shi Dsign and joint nd Intrnational Symosium on Shi Manouvring Hydronav 99 Manouvring 99, Gdansk, 999, [] Francscutto,., "n Exrimntal Invstigation of Paramtric Rolling in Had Wavs", To aar on Intrnational Journal Offshor Mchanics and rctic Enginring,. [] Francscutto,. "Nonlinar nalysis of th Dangrous Couling twn Roll Motion and Vrtical Motions in Had Sa", Procdings th Intrnational Scintific and Profssional Congrss on Thory and Practic of Shibuilding in Mmoriam Prof. Loold Sorta, Rijka, Novmbr,. -. [6] R. Kishv, S. Sasov, "Scond-Ordr Forcd Roll Oscillations of Shi-Lik Contour in Still Watr", Proc. Int. Symosium SMSSH, Varna, Vol., 98, [7] Francscutto,., Studio Torico-Srimntal dll ccoiamnto dl Moto di Rollio con ltri Moti Nav Fondamntali. Part I: Risultati Srimntali, INSEN Tchnical Rort n. 8, 999. [8] Francscutto,., Studio Torico-Srimntal dll ccoiamnto dl Moto di Rollio con ltri Moti Nav Fondamntali. Part II: Idntificazion Paramtrica, INSEN Tchnical Rort n. 9, 999. [9] Francscutto,., "On th couling btwn roll-hav-sway in bam wavs", in raration. [] Umda, N., Hamamoto, M., "Casiz of Shi Modls in Following/Quartring Wavs: Physical Exrimnts and Nonlinar Dynamics", Phil. Trans. R. Soc. London, Vol. 8,, [] Contnto, G., Francscutto,.: ifurcations in Shi Rolling: Exrimntal Rsults and Paramtr Idntification Tchniqu, Ocan Enginring, Vol. 6, 999,. 9-.