Nonlinear Ship Rolling and Capsizing

Transcription

1 UDC Marta PEDIŠIĆ BUČA Ivo SENJANOVIĆ Nonlnear Shp Rollng and Capszng Orgnal scentfc paper The exstng level of shp safety rules s presented and analysed, and gudelnes for upgradng and mprovng the rules are gven. An uncoupled equaton of shp rollng s set and the methods for solvng nonlnear shp rollng n regular and rregular waves are presented. Calculatons of rollng for a partcular shp n regular and rregular waves are done usng the harmonc acceleraton method. Nonlnear response phenomena are analysed. Capszng probablty s calculated by means of nonlnear dynamcs (basn eroson technque). Based on these calculatons new crtera for shp stablty are recommended, based on the use of the shp survvablty dagrams. Keywords: shp safety, stablty, rollng, capszng, nonlnear dynamcs, chaos theory, shp survvablty Authors addresses: Brodarsk nsttut, Zagreb Faculty of Mechancal Engneerng and Naval Archtecture, Zagreb Receved (prmljeno): Accepted (Prhvaćeno): Open for dscusson (Otvoreno za raspravu): Nelnearno ljuljanje prevrtanje broda Izvorn znanstven rad U radu je dan prkaz postojeće razne sgurnost brodova defnrane pravlma, te smjernce moguće nadogradnje poboljšanja pravla. Postavljena je raspregnuta jednadžba ljuljanja broda te je dan prkaz metoda za rješenje nelnearnog ljuljanja prevrtanja broda na harmonjskom valu morskm valovma. Proveden je proračun ljuljanja odabranog broda na harmonjskom valu na morskm valovma prmjenom metode harmonjskog ubrzanja. Analzran su nelnearn fenomen u odzvu. Proračun vjerojatnost prevrtanja broda proveden je prmjenom prstupa nelnearne dnamke (erozja bazena). Temeljem ovh zračuna predlažu se nov krterj o stabltetu broda korštenjem djagrama prežvljavanja broda. Ključne rječ: sgurnost broda, stabltet, ljuljanje, prevrtanje, nelnearna dnamka, teorja kaosa, sposobnost prežvljavanja broda Introducton Safety aganst capszng s of great mportance for every shp. Capszng could be regarded as a rare event, but the consequences of such event are fatal (loss of shp and the crew). Practce shows that even shps whch satsfy all the exstng rules are exposed to the rsk of capszng. The exstng level of shp safety s defned by numerous natonal and nternatonal rules. Tendences for the mprovement of shp safety are orented towards: - shft from determnstc towards probablstc approach, - untng the knowledge of desgners and drect experences of shp crew members, - takng nto account specfc characterstc of certan shp types, - takng nto account mportant nfluental factors mssng from the exstng rules. Consderng the fact that a shp besdes demands on stablty has to satsfy many other crtera, t s necessary to nclude real condtons a shp could be faced wth, and that defntely ncludes probablty n calculaton. A great effort s done to establsh smple and avalable methods for the estmaton of shp safety applcable durng the desgn process and shp servce. Suggestons concernng the crtera for and regulatons on shp safety should satsfy: avalablty of nput data, smplcty for applcaton, relablty of the results and the possblty of the smple analyss of the results. Revew of shp capszng cases One of the most dsastrous cases s defntely the capszng of the passenger RO-RO shp Estona n 994 n the Baltc Sea whch caused 85 human casualtes. After the wave rpped off the bow gate, capszng of the shp followed wthn only 5 mnutes. Table Accdents of merchant shps under the flag of Great Brtan from 994 to 00, >00 gt Tablca Nesreće trgovačkh brodova pod zastavom Velke Brtanje od 994. do 00., većh od 00 gt Year Capszng Total number of accdents (006)4,

2 NONLINEAR SHIP ROLLING AND CAPSIZING Table Accdents of fshng boats under the flag of Great Brtan from 994 to 00 Tablca Nesreće rbarskh brodova pod zastavom Velke Brtanje od 994. do 00. Year Capszng Percentage of capszng(%) Total number of accdents Statstcal data about marne accdents accordng to the yearly reports of MAIB (Marne Accdent Investgaton Branch) [] are shown n Tables and. Relevant statstcal data about the accdents of shps under the flag of Japan accordng to the Japanese report on marne accdents [] are shown n Table 3. Table 3 Accdents of shps under the flag of Japan durng 00 Tablca 3 Nesreće brodova pod zastavom Japana tjekom 00. Shp type Number of capszed shps Capszng percentage (%) Total number of shp accdents Passenger Merchant Pleasure boats Fshng boats Revew of the exstng rules about the shp safety Besdes the rules of the nternatonal organsatons (IMO, SOLAS), the rules whch defne the shp safety n general (capszng ncluded) could be the natonal rules, the classfcaton socetes rules and the rules establshed by Port State Control. The nternatonal rules represent recommendatons, and the natonal rules could make these recommendatons oblgatory. The rules accordng to [3, 4] are dvded n the followng categores:. Determnstc rules from emprcal and statstcal data (IMO Resoluton A.67, 968; based on Raholas work (939)),. Probablstc rules based on probablstc calculatons (IMO Resoluton A.56, 985; based on Yamagatas work (959)), 3. Performance based rules based on model testng and numercal smulaton (IMO SOLAS 95 Resoluton). Practce showed the need for establshng the rules specalsed for certan shp types [5]. The example of such rules s Weather stablty crteron for fshng boats IMO Torremolnos Conventon (for shps longer than 4 m) from 993. Besdes the rules a shp has to satsfy before delvery, nstructons to the shp crew are of great mportance. An example of such nstructons s Gudance to the Master for avodng dangerous stuatons n followng and quarterng seas (MSC/Crc.707) based on Takash work (98) [6]. Practcal stablty crtera appled for shp desgn comprse statc stablty. The crteron of statc stablty s represented by the statc stablty arm moment curve whch s used as the man estmator of shp's ablty to resst capszng. Besdes ths man requrement, there are addtonal requrements concernng the requred amount of ntal metacentrc heght, mnmal amount of stablty arm and heelng angle correspondng to the maxmum stablty arm. Dynamc stablty crteron prescrbes that the work of the restorng moment n the calm sea should be larger than the work of the heelng moment. As obvous, for complcated dynamcal stuatons statc calculaton s done. Only a nonlnear dynamcal approach corresponds to the realstc stuaton. An approprate selecton of external nfluences s also of great mportance. Although the selecton of maxmum values would be on the safe sde, t could greatly nfluence other mportant shp propertes. The exstng crtera are mostly smple for use, but wth an unknown level of safety. Crtera ganed through statstcal methods are no more relable when changes and nnovatons appear n the desgn. Although t s consdered that the weather crteron sgnfcantly mproved the safety of shp servce, certan tems are becomng questonable when the range of applcablty of the rules s extended outsde the ntal borders of the rules [3]. Crtera used n the desgn process and gudance for shp handlng concernng shp stablty n bad weather condtons are treated as separate and dfferent ssues. The desgner should provde the followng nformaton: how to use the shp n the best way, how to dentfy and avod the rsk, how to mtgate the rsk whch s notced. Such experence should contan two dfferent types of nformaton: change of rsk at the sea as a functon of shp's headng and speed and the nstructons concernng when t s safe to conduct manoeuvre whch could contan capszng rsk. The crew provdes valuable nformaton n ths area, whch could systematcally mprove the exstng nstructons. The advance n numercal smulaton and probablstc analyss s enablng the development of nstructons for shp handlng based on rsk. Every change n rules accordng to IMO should provde the same average level of safety as the exstng rules. The advance n IMO rules s slow wth the average tme nterval of 0 to 30 years between the scentfc approval and practcal applcaton. When the mprovement of the rules s consdered, accordng to [3] t s mportant to take nto account: smplcty, economc mplcaton, connecton wth engneerng practce of shp desgn. IMO has made the revson of the rules for stablty of ntact shp at SLF 45 (Sub-Commttee on Stablty, Load Lnes and Fshng Vessels) n 00 and SLF 46 n 003. The concluson s that some changes n the weather crteron and the ncluson n the rules so far excluded phenomena (wnd acton, manoeuvrablty...) are nevtable [3]. 4 Shp rollng and capszng One of the man causes of shp capszng n waves s the loss of stablty at rollng. Shp ressts to capszng wth her restorng moment and the capablty of dsspaton of energy va dampng. 3 57(006)4, 3-33

3 Accordng to [7], the followng modes of capszng are known (dvson by ITTC): statc loss of stablty ( surf-rdng ) dynamc loss of stablty - dynamc rollng - parametrc exctaton - resonant exctaton - mpact exctaton - bfurcatons - broachng. If the rollng on beam waves s consdered, t s possble to gnore the couplng of rollng and the other degrees of freedom of shp moton. Shp rollng n beam waves could be regarded as a smple pendulum. For solvng the rollng problem the lnear and nonlnear approach could be used. The lnear approach s appled for small rollng ampltudes, and the problem s solved n the frequency doman by the spectral analyss. Shp rollng wth larger ampltudes (and such stuatons lead to capszng) represents a nonlnear problem. In ths case the nonlnear equaton of rollng s set n order to foresee the shp s nonlnear response. The equaton of rollng could be set for regular or rregular waves. The problem of shp rollng could be analysed n the frequency and tme doman or by applyng the methods of dynamcs of nonlnear systems. The analyss n the tme doman s more acceptable for ths problem than the analyss n the frequency doman. A dsadvantage s the need for conductng a great number of realzatons n order to determne the probablty of capszng. For the analyss of dynamc behavour of nonlnear systems.e. nonlnear oscllatons shp rollng s ncluded, the followng methods are developed: - perturbaton method [8] - method of multple tme scales [9] - methods of tme averagng [0] - Krylov-Bogolubov-Mtropolsky method [] - harmonc balance method [] - tme ntegraton by Runge-Kutta or harmonc acceleraton method [3] - Galerkn method [0]. Lately, more attenton s gven to the approach of the drect determnaton of the probablty of capszng wthout the repetton of random realzatons n the tme doman. The probablstc approach demands solutons of numerous problems whch nclude the calculaton of capszng n the gven tme nterval. In ths case, the descrpton of the sea state s also probablstc. Accordng to [4], the methods whch tackle the mentoned problems are as follows: - analyss n phase plane [Sevastanov 979, Umeda 99] - dynamc heel angle determnaton [Dudzak 978, Belansk 994] - stablty moton determnaton [Prce 975] - nonlnear dynamcs methods (.e. basn eroson method) [Hseh 994, Ln and Solomon 995] - dscretsed lnear method [Belenky 994] - Markov process theory [Roberts 98]. It s mportant to know the possble modes of capszng and the nterrelatonshps of certan nfluence parameters based on the results of model testng and numercal smulatons. Researches usng mathematcal models n conjuncton wth the theoretcal development n the dynamcs of nonlnear systems lead to mproved understandng and nsght to the nature of the shp capszng process. 5 Revew of nonlnear phenomena The man characterstc of nonlnear systems s the possblty of occurrence of sgnfcant changes n the behavour of the systems due to small varatons of one of the parameters. Accordng to [5], shp rollng wth monoharmonc exctaton enables the nsght n the seres of the followng nonlnear phenomena: - shft of the natural frequency - multple responses wth jumps - symmetry breakng - resonant phenomena: superharmonc, subharmonc, supersubharmonc response - bfurcatons - transton to chaos through cascade - fractal boundary - chaotc attractor - chaos. The nonlnear system could have a polyharmonc rregular response to monoharmonc exctaton whch s called determnstc chaos. Consderng the shp rollng n sea waves, transfer to chaos.e. fractal boundary of stablty area could be observed and the so called basn of eroson could be formed. The am s to establsh whch combnatons of parameters (rollng angles, wave heghts, shp headng, shp speed, draught and so on) can cause the shp capszng. For certan combnatons of parameters t s desrable to determne the rsk of capszng n a fast and effcent way. Although the stuaton of monoharmonc exctaton s not realstc, the development of nonlnear phenomena observed n ths way could serve as a precursor of possble capszng.e. to pont out stuatons for whch the calculaton wth polyharmonc exctaton should be done [6]. 6 Calculaton procedure Accordng to [7,8], the nonlnear equaton of rollng n general form reads (6.) (I xx + δi xx ) - vrtual moment of nerta (sum of nerta moments of the shp mass and added mass of surroundng water) D(θ, θ) - nonlnear dampng moment R(θ) M(t) ( I + δi ) θ + D( θ, θ) + R( θ) = M( t) xx xx - nonlnear restorng moment - outer exctaton moment To determne the value of the moment of nerta f the rollng perod s known, the followng expresson accordng to [4] s used: T = π Ixx + δ I gmg xx (6.) 57(006)4,

4 NONLINEAR SHIP ROLLING AND CAPSIZING n the form: T ( Ixx + δ Ixx ) = ( ) gmg (6.3) π T - rollng perod - dsplacement mass MG - metacentrc heght. Accordng to [9], the total dampng moment could be regarded as the sum of lnear D L and nonlnear D N contrbuton D = D ( V, ω) + D ( V, a) L (6.4) V - speed a - exctaton ampltude ω - frequency. The dampng moment could be represented n numerous ways, n ths work the lnear formulaton and the formulaton of the cubc polynomal are used If the expresson s dvded by I, t s obtaned (6.5) (6.6) d and d 3 are the relatve dampng coeffcents. The restorng moment s functon of the underwater form of the shp hull. Accordng to [7] t s usually represented as the odd polynomal of roll angles (f the shp s symmetrc and n uprght poston), such as (6.7) In ths work, polynomals of 3 rd, 5 th and 7 th degree are used n the form (6.8) k are the relatve restorng coeffcents. Coeffcents of ths polynomal are ganed n such a way, that they approxmate the curve of stablty arm n the best possble way. Accordng to [0] for the dfferental equaton of rollng n the case of a damaged shp t s suggested (6.9) α D and β D are the coeffcents whch descrbe the reducton of the restorng moment. The outer wave exctaton could be approxmated by a harmonc snusodal functon, but the attenton should be pad to the fact that ths knd of approxmaton s not approprate for realstc calculatons N D( θ ) = Dθ + Dθ 3 d( θ ) = dθ + d θ m R( θ) = Kθ + K θ + K θ K m θ m r( θ) = kθ + k θ + k θ k m θ 3 θ θ + D Rθ( α θ )( β θ ) = M( t) D D Accordng to [], the expresson for the ampltude reads HW a = Iαωπ 0 0 (6.) lw α 0 - effectve wave slope ω 0 - wave frequency H w - wave heght l w - wave length. If the shp speed V and headng angle χ are taken nto account HW Mt () = Iαωπ 0 0 snχcos( ωet) (6.) lw ω ω ω V e = cos χ - encounter frequency. g Accordng to [], the equaton for polyharmonc exctaton s gven N Mt () = a cs n cos( nωt+ ε n ) n= N (6.3) c S - spectral coeffcent, normalzed ampltude of the exctaton harmoncs ε - random phase angle, probablty of occurrence s equally dvded n the range from 0 to π. When the expresson for M(t) s dvded by I, t s obtaned Mt () f() t = I The fnal form of nonlnear equaton of rollng yelds n θ θ + d + kθ = f() t = = (6.4) The equaton s solved by the harmonc acceleraton method. Accordng to ths method [7], the equaton of rollng s presented n the form (6.5) ξ - nondmensonal dampng coeffcent The rght part of the equaton s represented as the so called pseudoexctaton ψ() t = f() t + ( ξω) θ dθ + ω θ kθ (6.6) The expresson for the acceleraton s represented n a harmonc form n θ + ξωθ + ω θ = ψ(, t θ, θ ) = 3, = 35,, Mt () = acosωt (6.0) θ = ucosω t + vsnω t (6.7) 34 57(006)4, 3-33

5 u and v are unknowns. Accordng to [7], the equaton s solved usng the followng algorthm (6.8) {Y} - response vector (dsplacement, velocty and acceleraton) {O} - loadng vector [U] - transfer matrx. Coeffcents n ths expresson depend on the frequency, tme step and nondmensonal dampng [3,6,7]. It s necessary to set the ntal condtons n the form θ( 0) = θ0 (6.9) θ ( 0) = θ (6.0) 0 The soluton of the equaton gves the rollng angles n the tme doman. The exctaton values are chosen and tme ntegraton s performed. Estmaton of the probablty of shp capszng for certan outer condtons and ntal condtons s conducted usng the method of basn eroson. For ths purpose the fgure of the dscretsed phase space s determned []. Ths area s dvded nto parts. The trajectory of each pont s calculated untl the tme moment n whch the decson whether the pont belongs to the basn s made. Eroson coeffcent s number of cases (wth specfed ntal condtons) whch reman n the basn after a certan number of teratons. 7 Illustratve example k+ { Y} = [ U]{ Y} + { O} Ψ + For the calculaton of shp rollng the form of the ferry (see Table 4 and Fgure ) tested n Brodarsk Insttute n 997 s chosen [3]. The model testng results enable a comparson wth the results ganed numercally, and thus also valdaton of the appled procedure. Table 4 Characterstcs of the analysed shp Tablca 4 Značajke analzranog broda Shp type : FERRY FOR VEHICLES AND PASSENGERS Length between perpendculars/ Breadth L pp /B Length on waterlne/breadth L WL /B 3.63 Breadth/Draught B/d 4.8 Longtudnal poston of the centre of buyoancy LCB 5% L pp Vertcal poston of the centre of buoyancy ZCB 56.7%d Dfference of vertcal postons of the centre of buoyancy and centre of gravtaton ZCG-ZCB m Waterlne coeffcent C WL Maxmum secton coeffcent C M Block coeffcent C B (L WL ) k + Fgure Theoretcal frames, bow and stern contour Slka Teoretska rebra, pramčana krmena kontura 7. Coeffcents n the equaton of rollng The vrtual moment of nerta s defned accordng to the expresson (6.3) I = 968 tm shp wthout the blge keels I = tm - shp wth blge keels Coeffcents of the restorng moment are obtaned n a way presented n Chapter 6. The results are shown at Fgures and 3 and Tables 5 and 6. Fgure Statc stablty dagram for the ntact shp and ts approxmatons Slka Djagram statčkog stabltteta njegove aproksmacje Table 5 Coeffcents of the relatve restorng moment for the ntact shp Tablca 5 Koefcjent relatvnog povratnog momenta za neoštećen brod 3rd degree polynomal 5th degree polynomal 7th degree polynomal k (s - ) k 3 (s - ) k 5 (s - ) k 7 (s - ) Fgure 3 Comparson of the statc stablty curves for the ntact and damaged shp Slka 3 Usporedba krvulja statčkog stablteta za oštećen neoštećen brod The coeffcents of the relatve restorng moments for the damaged shp are gven n Table 6. The shp s tested wth and wthout the blge keels and the calculatons are done for both varants. The model s manufactured n a scale Λ=5 (:5). 57(006)4,

6 NONLINEAR SHIP ROLLING AND CAPSIZING Table 6 Coeffcents of the relatve restorng moment for the damaged shp Tablca 6 Koefcjent relatvnog povratnog momenta za oštećen brod k (s - ) k 3 (s - ) k 5 (s - ) 5th degree polynomal The lnear coeffcents of dampng are determned on the bass of the roll decay test, see Fgures 4 and 5, and the cubc coeffcent s calculated n such way that the soluton of the equaton of rollng wth the method of harmonc acceleraton gves the best approxmaton (usng the method of least squares). Shp wthout blge keels d = s - d 3 = s Shp wth blge keels d = s - d 3 =3.765 s H /3 = 3.75 m sgnfcant wave heght χ = 90 0 (beam waves) V shp = 0.0 m/s The effectve wave slope was taken for all calculated condtons as follows r=.53 For the numercal ntegraton wth the harmonc acceleraton method the followng parameters are used T = s - tme perod equal to the tme perod of the lowest harmonc δt = s - tme step Fgure 6 Response spectra comparson for the ntact shp Slka 6 Usporedba spektara odzva za neoštećen brod Fgure 4 Roll decay dagram - ntact shp Slka 4 Djagram stšavanja ljuljanja neoštećen brod Fgure 5 Roll decay dagram - ntact shp wth blge keels Slka 5 Djagram stšavanja ljuljanja za neoštećen brod s ljuljnm koblcama The monoharmonc snusodal exctaton and the polyharmonc exctaton descrbng the Adratc Sea spectrum are used accordng to [4]. 7. Comparson of the expermental results and numercal smulaton by the method of harmonc acceleraton Comparson of the results obtaned by the model testng and by the harmonc acceleraton method s done for the followng values of exctaton Fgure 7 Response spectra comparson for the ntact shp wth blge keels Slka 7 Usporedba spektara odzva za neoštećen brod s ljuljnm koblcama Comparson of spectra (Fgures 6 and 7) ponts to satsfyng correlaton of the expermentally and numercally obtaned results,.e. smmlarty of spectra values. 7.3 Calculaton of rollng n regular waves For the calculaton of rollng and capszng of the ntact shp n harmonc wave, the equaton of rollng n the so called Duffng form s used (lnear dampng, nonlnear restorng moment): θ θ 3 + d + k θ k θ = acosωt 3 (7.) 36 57(006)4, 3-33

7 and for the calculatons for the damaged shp, the equaton n the followng form s used θ θ d + k θ + k θ + k θ = acosωt 3 5 (7.) The values of exctaton parameters are used n the way that the results manfest the nonlnear phenomena. Stablty of the soluton s proved by applcaton of Floquet s theorem [5]. Fgure Multple perods, chaos and capszng Slka Všestruk perod, kaos prevrtanje ω = ω 0 a , ω = Fgure 8 Regular harmonc response to the harmonc exctaton Slka 8 Pravlan odzv na harmonjsku uzbudu ω = ω 0 a 0 Fgure 9 Symmetry breakng Slka 9 Poremećaj smetrje ω = ω , 0.666, θ max = rad ω = a , , postve ampltude ω = θ max = rad, negatve ampltude θ mn = rad Fgure 0 Perod doublng Slka 0 Udvostručenje peroda ω ω 0 = a , ω = Based on ths example, t s possble to follow the development of nonlnear phenomena whch lead to the chaotc response and fnally to shp capszng. Frst, regular harmonc response to the harmonc exctaton s presented, see Fgure 8. By ncreasng the ampltude of exctaton the symmetry breakng,.e. the occurrence of changes of the values of postve ampltudes related to negatve values occurs, see Fgure 9. After that, the perod doublng,.e. occurrence of two dfferent perods happens, see Fgure 0. After that, multple perods, chaotc motons, and fnally capszng occur, see Fgure. Very small changes of the exctaton ampltude values could cause such changes n the response. Results of the repeated numercal calculatons could be represented n the ntal values plane (exctaton values kept constant, ntal values varable) or n the exctaton plane (ntal values constant, exctaton ampltude and frequency varable). The calculaton for each pont s done n a certan tme perod. If durng ths perod capszng occurs, the pont s drawn n a dfferent colour (depends on the tme perod to capsze, see Fg. for legend of the colours). If capszng does not occur, the pont s drawn n blue. All the ponts whch represent the stuaton when the capsze does not occur form the so called safe basn. Fgure Legend for values n Fgures 3-7 Slka Legenda za vrjednost na slkama (006)4,

8 NONLINEAR SHIP ROLLING AND CAPSIZING Fgure 3 Safe basn n the exctaton plane, ntact shp Slka 3 Bazen sgurnost u ravnn uzbude, neoštećen brod θ = 0, θ = 0 Fgure 6 Safe basn n the ntal value plane, ntact shp wth blge keels Slka 6 Bazen sgurnost u ravnn početnh uvjeta, neoštećen brod s ljuljnm koblcama a=0.705, ω=ω 0 Fgure 4 Safe basn n the exctaton plane, ntact shp wth blge keels, Slka 4 Bazen sgurnost u ravnn uzbude, neoštećen brod s ljuljnm koblcama θ = 0, θ = 0 In Fgures 3-6 safe basns n exctaton and ntal value plane for the ntact shp wth and wthout blge keels are shown. The safe basn of the ntact shp n the exctaton plane shows the sgnfcant nfluence of the blge keels. The capszng area for shps wth blge keels s much smaller and the transton to capszng condton s much sharper. For the shp wthout the blge keels ths transton s of extensvely fractal nature. The form of the safe basn n the ntal values plane for the shp wth and wthout the blge keels s smlar, but the safe basn for the shp wthout the blge keels shows pronounced dependence of capszng on the perod of exposure to the exctaton. Such llustratons n the exctaton plane or ntal value plane can serve as orentatonal values of safe and unsafe area for shps, and at the same tme ther fractal nature shows that a small change of the external condton can cause the transton of shp from rollng to capszng area. 7.4 Calculaton of rollng and capszng n sea waves Beck [6] and conclusons of the 3rd ITTC Conference [7] pont out the great mportance of the development of such analyss. In Fgures 7, 8 and 9 the safe basns n the plane of ntal condtons wthout external exctaton are shown. The basn area for the shp wthout the blge keels s 75% of the basn area for Fgure 5 Safe basn n the ntal value plane, ntact shp Slka 5 Bazen sgurnost u ravnn početnh uvjeta, neoštećen brod a=0.075, ω=ω 0 Fgure 7 Safe basn n the ntal value plane, ntact shp Slka 7 Bazen sgurnost u ravnn početnh uvjeta, neoštećen brod 38 57(006)4, 3-33

9 the shp wth the blge keels, whch ndcates the great nfluence of the blge keels on the stablty aganst capszng. The basn area for the damaged shp s 7.% of the basn area of the ntact shp wthout the blge keels. The shp s behavour at the headng angles χ=0-80, sgnfcant wave heghts H /3 =3.75 m and 5 m and speeds V=, 3 and 4 m/s s observed. For the ntact shp wth the blge keels there s no change n the safe basn area at any of the observed condtons. For the ntact shp wthout the blge keels at the speed V= m/s, there s no basn eroson greater than 0.3%. For the angles π χ= -π there s no sgnfcant basn eroson for the nvestgated speeds and wave heghts. At the speed V=3 m/s there s no observable eroson at the wave heght H /3 =3.75 m, but at the wave heght H /3 =5 m mld basn eroson occurs n-between 5π/8-7π/8. At the speed V=4 m/s the development of basn eroson s smlar at the both nvestgated wave heghts, only the values are somewhat larger for the wave heght H /3 =5 m, see Fgures 0 and. The nfluence of the speed ncrement s greater than the nfluence of the sgnfcant wave heght ncrement. Fgure 8 Safe basn n the ntal value plane, ntact shp wth blge keels Slka 8 Bazen sgurnost u ravnn početnh uvjeta, neoštećen brod s ljuljnm koblcama Fgure Safe basn n the ntal value plane, ntact shp, area reduced 7.5% Slka Bazen sgurnost u ravnn početnh uvjeta, neoštećen brod, površna smanjena 7,5% V=4 m/s, H /3 =3.75 m, χ=70 0, The damaged shp shows also almost no senstvty for the π headng angles χ= -π for all nvestgated speeds and wave π heghts. At the angles χ=0- basn eroson occurs. At the wave heght H /3 =3.75 m and speeds V= and 3 m/s there s no Fgure 9 Safe basn n the ntal value plane, damaged shp Slka 9 Bazen sgurnost u ravnn početnh uvjeta, oštećen brod Fgure 0 Safe basn n the ntal value plane, ntact shp, area reduced 0.3% Slka 0 Bazen sgurnost u ravnn početnh uvjeta, neoštećen brod, površna smanjena 0,3% V=3 m/s, H /3 =5 m, χ=70 0 Fgure Safe basn n the ntal value plane, damaged shp, area reduced 54.4% Slka Bazen sgurnost u ravnn početnh uvjeta, oštećen brod, površna smanjena 54,4% V=4 m/s, H /3 =3.75 m, χ=55 0 Fgure 3 Safe basn n the ntal value plane, damaged shp area reduced 86.8% Slka 3 Bazen sgurnost u ravnn početnh uvjeta, oštećen brod, površna smanjena 86,8% V=4 m/s, H /3 =5 m, χ= (006)4,

10 NONLINEAR SHIP ROLLING AND CAPSIZING sgnfcant basn eroson. At H /3 =5.00 m and V=3 m/s the eroson s sgnfcant for the headng angles range.5π/8-5π/8. At the speed V=4 m/s eroson s mld n the headng angle range χ=6π/8-9π/8, but the safe basn almost dsappears for the headng angles 3π/8-6π/8, see Fgures and 3. If the damaged shp wth the presumpton of only lnear dampng s consdered n the specfed condtons, the safe basns have a hghly fractal structure, and ther area s much smaller compared to the area of the safe basns when the cubc dampng coeffcent s ncluded n the equaton. Ths mples the great mportance of takng nto account the nonlnearty of dampng. The development of basn eroson n ths case s shown n Fgures 4-7. Fgure 7 Safe basn n the ntal value plane, assumpton of lnear dampng Slka 7 Bazen sgurnost u ravnn početnh uvjeta, pretpostavka lnearnog prgušenja V=4 m/s, H /3 =5 m, χ= Shp survavblty probablstc dagram Fgure 4 Safe basn n the ntal value plane for damaged shp, wthout external exctaton, assumpton of lnear dampng Slka 4 Bazen sgurnost u ravnn početnh uvjeta za oštećen brod, bez vanjske uzbude, pretpostavka lnearnog prgušenja The rato of the areas of the basns of eroson wth and wthout wave exctaton can be used as the measure of shp survval probablty estmaton. Constructon of such dagrams gves the probablty of capszng as the functon of the followng three parameters: shp speed, headng angle, sgnfcant wave heght. The navgaton condtons n whch there s mmnent danger of shp capszng are defned by these parameters. Fgures 8 and 9 represent such dagrams for the ntact and damaged shp respectvely. Fgure 5 Safe basn n the ntal value plane, basn area reduced 8.%, assumpton of lnear dampng Slka 5 Bazen sgurnost u ravnn početnh uvjeta, površna ba zena smanjena 8,%, pretpostavka lnearnog prgušenja V= m/s, H /3 =3.75 m, χ=55 0, Fgure 6 Safe basn n the ntal value plane, basn area s 97.% reduced, assumpton of lnear dampng Slka 6 Bazen sgurnost u ravnn početnh uvjeta, površna bazena smanjena 97,%, pretpostavka lnearnog prgušenja V=4 m/s, H /3 =5 m, χ=75 0, Fgure 8 Shp survvablty dagram for the ntact shp Slka 8 Djagram prežvljavanja broda, neoštećen brod V=4 m/s, H /3 =5.00 m Ths dagram represents the unfcaton of the numercal method of harmonc acceleraton and the knowledge on nonlnear dynamcs (006)4, 3-33

11 Fgure 9 Shp survvablty probablstc dagram for the damaged shp Slka 9 Djagram prežvljavanja broda, oštećen brod H /3 =5.00 m 8 Concluson In order to rase the level of shp safety t s necessary to work systematcally on mprovng shp desgn methods, gudance for shp handlng and rules and recommendatons concernng the shp safety. The numercal method (harmonc acceleraton method) n tme doman for solvng the nonlnear equaton of rollng s presented n ths paper. Based on the obtaned results, the evaluaton of the probablty of capszng for specfc shp condtons s done usng the method of basn eroson. The shp survvablty dagram s presented as the fnal result (probablty of capszng dependng on shp speed, sgnfcant wave heght and the headng angle). Such dagram represents: - contrbuton to the stablty crtera connected wth the functon of the probablty of shp survval, - contrbuton to the defnton of non-permssble condtons for shp s voyage. Up to date achevements n ths area pont to the mportance of takng nto account nonlnearty n the problem and also the mportance of the probablstc approach. The suggested procedure should be mproved n several ways: - rollng should be coupled wth other degrees of freedom of shp moton - quanttatve defnton of shp characterstcs n the damaged condton - varaton of restorng moment (functon of headng angle) - possbltes of drect determnaton of probablty (reducton of great number of numercal smulatons requred). References [] Avalable at. (last access June 004). [] Avalable at. (last access June 004). [3] FRANCESCUTTO, A.: Intact Shp Stablty: The Way Ahead, Marne Technology, 4(004), p [4] KOBLYNSKI, C.J., KASTNER, S.: Stablty and Safety of Shps, Volume I: Regulaton and Operaton, Elsever Ocean Engneerng Book Seres, 003. [5] JOHNSON, B., WOMACK, J.: On Developng a Ratonal and User-frendly Approach to Fshng Vessel Stablty and Operatonal Gudance, Proceedng of 5th Internatonal Workshop on Stablty, 5:.4.-, Treste, 00. [6] ALMAN, P.R., McTAGGART, K.A., THOMAS, W.L.: Heavy Weather Gudance and Capsze Rsk, Proceedng of 5th Internatonal Workshop on Stablty,..-8, Treste, 00. [7] : The Specalst Commttee on Stablty: Fnal Report and Recommendatons to the nd ITTC, Proceedngs of the nd ITTC Conference, Seoul and Shanga, 999. [8] CARDO, A., FRANCESCUTTO, A., NABERGOJ, R.: Nonlnear Rollng Response n a Regular Sea, Internatonal Shbuldng Progress, 8, 99, p.04-09,. [9] NAYFEH, A.H., KHDEIR, A.A.: Nonlnear rollng of shps n regular beam seas, Internatonal Shpbuldng Progress, 379, 986, p [0] SCHMIDT, G., TONDL, A.: Non-lnear Vbratons, Cambrdge Unversty Press, Akademe Verlag Berln, 986. [] FRANCESCUTTO, A., CONTENTO, G.: Bfurcaton n shp rollng : expermental results and parameter dentfcaton Technque, Ocean Engneerng, 6999, p.095-3,. [] SENJANOVIĆ, I., FAN,Y.: Some Advances of the Harmonc Balance Method, Journal of Sound and Vbraton,9(996), p ,. [3] SENJANOVIĆ, I.: Harmonc Acceleraton Method for Dynamcal Structural Analyss, Computers & Structures 8 (984), p [4] BELENKY, V.L.: Probablstc Approach for Intact Stablty Standards SNAME Transactons, 08:3-46, 000. [5] THOMPSON, M.T., STEWART, H.B.: Nonlnear Dynamcs and Chaos, John&Wley and Sons, 986. [6] SENJANOVIĆ, I., FAN, Y.: Dynamc Analyss of Shp Capszng n Regular Waves, Brodogradnja, 4(994), p [7] SENJANOVIĆ, I., PARUNOV, J., CIPRIĆ, G.: Safety Analyss of Shp Rollng n Rough Sea Chaos, Soltons & Fractals, 8(997)4, p [8] SENJANOVIĆ, I., CIPRIĆ, G., PARUNOV, J.: Nonlnear Shp Rollng and Capszng n Rough Sea Brodogradnja, 44(996), p.9-4. [9] TAYLAN, M.: The effect of nonlnear dampng and restorng n shp rollng, Ocean Engneerng, 7, 000, p [0] SENJANOVIĆ, I., FAN, Y. : Numercal Smulaton of Shp Capszng n Irregular Waves, Chaos, Soltons & Fractals 5(995)5, p [] SENJANOVIĆ, I., CIPRIĆ, G., PARUNOV, J.: Survval Analyss of Fshng Vessel Rollng n Rough Sea, EUROMECH-nd European Nonlnear Oscllaton Conference, Prague, 996. [] SCOLAN, Y.M.: Analyss of drect and parametrc exctaton wth the Melnkov method and the technque of basn eroson, Proceedngs of 5th Internatonal Workshop on Stablty, 5:4.8.-0,Treste, 00. [3] : Shp Rollng Measurements for a Ferry, 5353-M (n Croatan), Techncal Report, Brodarsk nsttut, Zagreb, 997. [4] TABAIN, T.: Standard Wave Spectrum for the Adratc Sea (n Croatan), Jadranska meteorologja 43, 998, p.-3. [5] SENJANOVIĆ, I., FAN, Y., PARUNOV, J.: Transton from regular to chaotc oscllatons of cubc dynamcal systems, st Congress of Croatan Socety of Mechancs, Pula, 994, p [6] BECK, R.B.: Modern Computatonal Methods for Shps n Seaway, SNAME Transactons, 09:-5, 00. [7] : Proceedngs of the 3rd ITTC The Specalst Commtttee on Predcton of Extreme Shp Motons and Capszng (Fnal Report and Recommendatons to the 3rd ITTC) Volume II, Vence, (006)4,