A STUDY ON COMPLICATED ROLL MOTION OF A SHIP EQUIPPED WITH AN ANTI-ROLLING TANK

Transcription

1 607 A STUDY ON COMPLICATED ROLL MOTION OF A SHIP EQUIPPED WITH AN ANTI-ROLLING TANK Harukuni Taguchi, Hirohi Sawada and Kauji Tanizawa Naional Mariime Reearch Iniue (JAPAN) Abrac Nonlinear roll moion of a hip wih a Frahm ype ani-rolling ank (ART) onboard i inveigaed experimenally and numerically. The model experimen wa conduced for wo loading condiion, namely normal abiliy condiion and poor abiliy condiion, in regular beam wave. I i found ha he model hip wih he ART in poor abiliy condiion can exhibi irregular and complicaed roll moion even in regular wave. In order o furher inveigae complicaed roll moion of a hip wih an ART in uch condiion, ime-domain imulaion wih nonlinear equaion of coupled moion of rolling of a hip and fluid ranfer in he ank were carried ou. Bifurcaion diagram wih changing forcing frequency, Ω, and forcing ampliude, A, which correpond o wave frequency and wave eepne repecively, how ha complicaed roll moion exi in a wide range of parameer in low frequency region. Thi i alo confirmed in a conrol parameer plane, which coni of Ω and A. However in high frequency region, where complicaed roll moion wa oberved in he model experimen, uch behaviour of a hip ha no been found. Thi implie ha he numerical inveigaion reveal only one of feaure of a hip wih an ART in poor abiliy condiion and ha furher inveigaion i neceary o underand overall behaviour of uch hip in wave. 1. INTRODUCTION A roll abilizaion device, ani-rolling ank are fied o many hip. An ani-rolling ank uilie he fluid moion in he ank for decreaing he roll moion of he hip. Paive ype ani-rolling ank are deigned o have he ame naural period for he fluid ranfer in he ank a ha for he rolling of he hip, bu niney degree ou-of-phae o he roll moion. So uch ani-rolling ank achieve maximum effec in he reonance condiion. However for a hip like fiherie raining hip, whoe meacenric heigh (GM) largely varie during operaion, a paive ype of ani-rolling ank migh increae he roll moion, becaue he reulan naural rolling period deviae from he deigned one. To overcome hi drawback, frequency variable ype and acive conrol ype ani-rolling ank have been developed. When an ani-rolling ank i acivaed, he free urface momen in he ank decreae GM of he hip. Ship fied wih ani-rolling ank are deigned o have ufficien abiliy wih aking he reulan lo of GM ino accoun. However in poor abiliy condiion he reulan lo of GM migh lead he hip o dangerou iuaion. Many inveigaion o

2 608 far (e.g. [1]-[4]) were conduced for effec of ani-rolling ank on he rolling repone of hip in normal abiliy condiion. However effec of ani-rolling ank in poor abiliy condiion have no been inveigaed o much. La year auhor had a chance o conduc an experimen uing a model hip wih a Frahm ype (U-ube ype) ani-rolling ank onboard. Taking hi opporuniy we meaured he roll moion of he model hip in poor abiliy condiion in beam wave. And in order o furher udy behaviour of a hip in uch condiion qualiaively, we carried ou numerical imulaion wih nonlinear equaion of he coupled moion of rolling of a hip and fluid in an ani-rolling ank. Thi paper repor he ouline of hee inveigaion.. MODEL EXPERIMENT.1.Ouline of he Experimen Model experimen wa conduced in he eakeeping wave ank a Naional Mariime Reearch Iniue (lengh: 5m, widh: 8.0m, deph: 4.5m). A model of a mall fihing boa wa ued. Fig. 1 how he ouline of he model hip and i principal dimenion are hown in Table 1. A Frahm ype ani-rolling ank (ART) wa aached on he af deck (ee Fig. 1). The ouline of he ART i hown in Fig. and Table how i principal dimenion. The model hip wa placed a he middle of he ank and moored by weak enion cord o keep beam wave condiion and o reric free drif moion (ee Fig. 3). The roll, heave and way moion of he model hip wih he ART in regular beam wave were meaured by a video racker yem. A he arge for he video racker mall ligh were aached a he op and roo of he ma (ee Fig.1). Fig. 1 Upper ligh ART Lower ligh Model hip ued in he experimen Table 1 Principal dimenion of he model hip Lengh (overall) 130 cm Breadh (overall) 40.8 cm Draf (overall) 7.5 cm Diplacemen kg In he experimen moion of he model hip wih he ART boh acivaed and inacivaed condiion were meaured. Meauremen were conduced for a normal abiliy condiion (GM =.4 cm wih he ART inacivaed) and a poor abiliy condiion (GM = 0.74 cm wih he ART inacivaed). The reulan naural rolling period were.0 ec. for GM =.4 cm and 3.7 ec. for GM = 0.74 cm, repecively. So he naural period of rolling in he normal abiliy condiion wa he ame a ha of he ART. In order o inveigae global behaviour of he model hip, he wave period, T w, wa changed from o.6 ec. while he wave heigh, H w, wa kep conan, H w = 5 cm. Moreover meauremen were alo carried ou in wave of he period T w =.0 ec. and wih increae of he wave heigh.

3 . 609 Fig. Table ART ued in he experimen Principal dimenion of ART Lengh L 4.4 cm Breadh B 40.4 cm Breadh of verical ube b 7.54 cm Heigh H.45 cm Heigh of horizonal ube e 1.3 cm Waer deph d.6 cm Weigh of fluid in ART W f 0.8 kg Naural period of ART T ART.0 ec. φ/ kh Inacivaed Acivaed Ω(=ω /ω ) Fig. 4 Rolling repone wih he ART acivaed and inacivaed (GM =.4 cm) normalied by he wave lope. In hi cae ypical rolling repone wih an ART were obained becaue he naural period of rolling wa he ame a ha of he ART. A hown in Fig. 4, he ART reduce rolling ampliude near he reonance frequency (Ω = ) while a higher ide region of he reonance frequency he ART give negaive effec and he rolling ampliude i increaed. Wave Breaker Mooring line Model Ship Inciden Wave Wave Maker Poor Sabiliy Condiion 0.6 Inacivaed Acivaed Fig. 3 Video Tracker Experimenal eup in he wave ank φ /kh Experimenal Reul Normal Sabiliy Condiion Fig. 4 how he meaured rolling ampliude of he hip in he normal abiliy condiion. The horizonal axi i he raio beween wave frequency and naural rolling frequency and he verical axi i he rolling ampliude Ω(=ω /ω ) Fig. 5 Rolling repone wih he ART acivaed and inacivaed (GM = 0.74 cm)

4 6 In he poor abiliy condiion he reulan GM due o he free urface momen in he ank become negaive, o he model hip i aically balanced wih abou 9 degree heeled poiion in ill waer. The experimen wa ared afer he model wa e o he aically balanced poion in he weaher ide. In hi cae he naural period of rolling differ from ha of he ART. And he wave period were e around he naural period of he ART. Fig. 5 how he meaured rolling ampliude of he hip in he poor abiliy condiion. Becaue he wave frequencie are ou of reonance condiion, he rolling repone are mall in boh he ART acivaed and inacivaed condiion, bu negaive effec of he ART i een in all wave. In order o furher inveigae behaviour of he model hip wih he ART acivaed in he poor abiliy condiion, meauremen wih increae of he wave heigh were carried ou. Fig. 6 how he ime hiorie of meaured roll angle in wave of he period T w =.0 ec. wih changing he wave heigh from 5 o 5 cm. For mall wave heigh, H w = 5 cm and cm, he rolling repone i very mall. For H w = cm and 0 cm, he model hip exhibi quie irregular and complicaed roll moion around he wo aic balanced poiion in he lee and he weaher ide. For H w = 5 cm roll moion around he lee ide aic balanced poiion wih he ame period a he wave i found. In Fig. 7 he ime hiory of meaured roll angle for H w = 0 cm in long ime duraion i hown. The roll moion eem o be chaoic. A imilar ype of complicaed rolling repone wa oberved in numerical imulaion wih a nonlinear rolling equaion for a hip in negaive GM wihou an ART [5][6] H w = 5cm (ec) H w = cm (ec) H w = cm (ec) H w = 0cm (ec) H w = 5cm (ec) Fig. 6 Meaured roll moion of he model hip (T w =.0 ec.) H w = 0cm (ec) Fig. 7 Meaured roll moion of he model hip (T w =.0 ec. and H w = 0 cm)

5 NUMERICAL INVESTIGATION In order o inveigae global rolling repone of a hip wih an ani-rolling ank in poor abiliy condiion qualiaively, we carried ou numerical imulaion wih equaion of he coupled moion of rolling of a hip and fluid in an ani-rolling ank. In he inveigaion we reaed he coupled moion a a nonlinear dynamical yem. In hi paper, a he fir ep of he numerical inveigaion, we examine he coupled moion wih bifurcaion diagram and a conrol parameer plane where ome feaure of global rolling repone of a hip wih an ani-rolling ank may be exhibied Coupled Moion of Rolling of a Ship and Fluid in an Ani-Rolling Tank Uiliing coordinae yem hown in Fig. 8, linear coupled equaion of moion of rolling of a hip and fluid in an ani-rolling ank can be expreed a follow [7]. d φ dφ d θ J + B + K φ+ J + K θ = K ϕ d d d d φ d θ dθ J + K φ+ J + B + K θ = K ϕ d d d () 1 ( ) where φ i he hip roll angle, θ i he ank waer lope angle, ϕ i he wave lope angle, i he ime, J i he momen of ineria of hip, B i he damping coefficien of hip, K i he reoring coefficien of hip: W GM (W i he diplacemen of hip), J i he momen of ineria of fluid in he ank, B i he damping coefficien of fluid in he ank, K i he reoring coefficien of fluid in he ank: W GG 0 (GG 0 i he virual rie of he cenre of graviy due o free urface in he ank), J i he coupling coefficien of hip and fluid in he ank. Fig. 8 Coordinae yem Moreover he wave lope angle, ϕ, i generally expreed in following ime dependen form. ϕ = ak inω (3) where a i he wave ampliude, k i he wave number and ω i he wave frequency. In order o examine he global rolling repone in poor abiliy condiion, i wa conidered ha nonlineariy in reoring momen of a hip hould be included in he equaion of coupled moion. In hi inveigaion we ued he following approximaed expreion for a abiliy curve of a hip in poor abiliy condiion (lighly poiive GM). φ φ WGZ = K φ 1 + r( ) φv φv where φ v i he vanihing angle of abiliy and K i defined a menioned above: K =W GM. If he value of r i pecified, boh GZ max and φ max are deermined auomaically in hi form, however ypical characeriic of nonlineariy in reoring momen can be obained qualiaively. 4 ( 1 r) ( ) (4) Uing hi expreion in reoring erm, equaion (1) i rewrien a follow.

6 61 J Normalizing each angle by φ v and he ime by ω, he naural rolling frequency of a hip (ω = (K /J ) 1/ ), and uing ime dependen form of he wave lope, equaion (5) and () are convered o d φ ' λ d θ ' + φ ' + + µ d ν d 4 { 1 + rφ' (1 + r) φ' } d φ' dφ' + κ + φ ' d d d θ ' + ν + λθ ' = d ν ν d φ + B d dφ φ + K φ 1 + r( ) d φv + J Ain Ω λ = Ain Ω ν λ dθ ' λ + θ ' ν d ν where = ω, φ = φ/φ v, θ = θ/φ v, κ = B /(J ω ), λ = K / K, ν = J /J, ν = J /J, µ = B /(J ω ), A = ak/φ v, and Ω = ω/ω. Table 3 Value of fixed coefficien ued in he numerical inveigaion κ 0.60 µ 0.6 λ 1.5 r 6.0 ν ν 0.37 ( 1 r) d θ + K θ ϕ = K d ( 6) ( 7) In he numerical inveigaion coefficien A, he forcing ampliude, and Ω, he forcing frequency, were varied while he oher coefficien were fixed. The value of fixed coefficien are ummarized in Table 3. The value of κ, µ, ν, and ν, are cloe o hoe of he experimen. And he value of r i φ 4 ( ) φv (5) deermined by he lea quare approximaion of abiliy curve of he model hip. Moreover in order o inveigae an exremely poor abiliy condiion, λ wa e o 1.5. For A 0, which correpond o ill waer, from equaion (6) and (7) we ge he following relaion in eady ae. φ ' r = 0 () 8 4 { 1 + φ ' (1 + r) φ ' } + λθ ' φ ' +θ ' = 0 () 9 Subiuing equaion (9) ino equaion (8) give u equaion (). (1 λ ) φ ' + r φ ' From equaion () we can underand ha when λ i larger han 1, he uprigh condiion become unable and he hip i aically balanced in heeled condiion. 3..Bifurcaion Diagram 3 (1 + r) φ ' 5 = 0 ( ) Fig. 9 how a bifurcaion diagram for A = 0. wih changing Ω. Thi bifurcaion diagram i obained a follow. Fir we olve he equaion (6) and (7) numerically wih he 5h order Runge-Kua mehod for Ω = 1 wih he iniial condiion, φ (0) = φ 0, θ (0) = -φ 0, φ (0)/d = d θ (0)/d = 0, which correpond o he aically heeled condiion in ill waer. Afer he ranien repone die ou, we plo he repone a every one forcing cycle wih he ame phae angle a much a 0 cycle. Then increaing Ω by mall amoun, Ω = 1, we coninue he ime inegraion and afer exincion of he ranien repone, we imilarly plo he repone a every one cycle. By coninuing hee procedure he bifurcaion diagram, Fig. 9, i obained. I i conidered ha a bifurcaion diagram wih changing Ω correpond o eady ae rolling repone obained by a model experimen wih he wave eepne kep conan bu he wave frequency gradually changed. And in hi cae

7 613 A = 0. correpond o he wave eepne of abou 1/3 in he model experimen menioned in ecion. φ '(Tn) Ω Fig. 9 Bifurcaion diagram wih changing Ω (A = ) From Fig. 9 i i found ha in low frequency region, 0.4 < Ω < 0.93, very complicaed roll moion eem o exi. To illurae hee complicaed rolling more clearly, ime hiorie of φ, he hip roll angle, and θ, he ank waer lope angle, ogeher wih phae porrai in (φ -d φ /d) and (θ -dθ /d) for Ω = 0.4 and 0.7 are hown in Fig.. Roll moion for Ω = 0.4, 0.7 migh be chaoic. φ '(Tn) A In he model experimen complicaed roll moion were oberved a relaively high wave frequency, which correpond o Ω = 1.85, however under he condiion for Fig. 9 only imple roll moion wih he ame period a he forcing period i found in high frequency region. Fig. 11 how bifurcaion diagram for Ω = wih changing A. A bifurcaion diagram wih changing A i conider o correpond o eady ae rolling repone obained by a model experimen wih keeping he wave frequency conan bu changing he wave heigh gradually. From Fig. 11 i i found ha rolling repone change in very complicaed way a he forcing ampliude increae. The period doubling bifurcaion begin a abou A = 0.16 and i lead o chaoic roll moion. A he forcing ampliude i increaed furher, chaoic moion change o a ubharmonic moion of order 3 a abou A = and hi ubharmonic moion coninue o exi unil he forcing ampliude reache a abou 0.. Then he roll moion become chaoic again and a abou A = 0.8 i change o imple moion wih he ame period a he forcing period. Fig. 11 Bifurcaion diagram wih changing A (Ω = )

8 614 φ' φ' dφ'/d θ' φ' dφ'/d θ' φ' dθ'/d dθ'/d θ' θ' (a) Ω = 0.4 (b) Ω = 0.7 Fig. Time hiorie of φ, he hip roll angle, phae porrai in (φ -d φ /d), ime hiorie of θ, he ank waer lope angle, and phae porrai in (θ -dθ /d) obained by numerical imulaion (from op)

9 Complicaed Roll Moion Region in a Conrol Plane To illurae he parameer dependency of complicaed roll moion, a conrol plane, which coni of Ω and A, conrol parameer of he equaion (6) and (7), i divided ino wo region a hown in Fig. 1. In Fig. 1 parameer Ω and A in black region lead o complicaed rolling while parameer in whie region lead o imple moion wih he ame period a he forcing period. Thi figure i obained a follow. We olve he equaion (6) and (7) wih 5h-order Runge-Kua mehod from he iniial condiion, which correpond o he aically heeled condiion in ill waer, φ (0) = φ 0, θ (0) = -φ 0, φ (0)/d = d θ (0)/d = 0, up o forcing cycle for a combinaion of Ω and A and examine wheher he la 0 cycle roll moion exhibi complicaed feaure or no. By repeaing hee procedure for parameer a fine grid poin of he conrol plane, we obain Fig. 1. The conrol plane in Fig. 1 i divided ino 50 x 300 grid poin. A Ω Fig. 1 Complicaed roll moion region in Ω-A conrol plane ( : in he model experimen) From Fig. 1 i i found ha complicaed rolling exi in a wide range of parameer Ω and A. And a een in he bifurcaion diagram wih changing A, Fig. 8, here i he limi for complicaed rolling in larger forcing ampliude ide. In Fig. 1 he condiion, in which complicaed roll moion wa oberved in he model experimen, are hown wih. Thee condiion are ou of he complicaed roll moion region defined by he numerical inveigaion. In general, nonlinear dynamical yem can exhibi more han one ype of repone for he ame parameer depending on he iniial condiion. So he region of complicaed rolling in he conrol plane migh be changed for he differen iniial condiion. To clarify he dependency on iniial condiion inveigaion ino he iniial value plane obained wih imilar procedure o he conrol plane i neceary Some Dicuion on Numerical Inveigaion A hown in Fig 9-1 i i found ha roll moion of a hip wih an ART, expreed by equaion (6) and (7) could exhibi complicaed repone in a wide range of conrol parameer Ω and A, which correpond o he wave frequency and he wave eepne, in relaively low frequency region. However complicaed roll moion in high frequency region, which wa oberved in he model experimen, i no found in hi numerical inveigaion. Thi implie ha only one of feaure of roll moion of a hip wih an ART in poor abiliy condiion i clarified wih hi numerical inveigaion. In order o underand overall behaviour of uch hip, furher numerical inveigaion including examinaion of dependency of

10 616 complicaed rolling on he parameer λ and on he iniial condiion i neceary. 4. CONCLUSION Roll moion of a hip wih a Frahm ype ani-rolling ank onboard wa inveigaed experimenally and numerically. A a reul i i found ha he model hip wih poor GM migh exhibi complicaed behaviour in beam wave of relaively high frequency compared wih he naural frequency of rolling. Meaured daa how ha he roll moion in uch condiion migh be chaoic. Numerical inveigaion wih equaion of he coupled moion of rolling of a hip and fluid in an ani-rolling ank how ha roll moion in poor abiliy condiion could exhibi complicaed repone in a wide range of conrol parameer Ω and A, which correpond o he wave frequency and he wave eepne, in relaively low frequency region. However in he numerical inveigaion only imple roll moion wih he ame period a he forcing period i found in high frequency region. In order o deepen our underanding of he behaviour of a hip wih an ART in poor abiliy condiion, furher inveigaion hould be coninued. 5. ACKNOWLEDGEMENTS The experimen of hi work wa conduced a a funded reearch from an ART maker Sabilo Corporaion. The auhor would like o acknowledge he repreenaive direcor Mr. N. Maumura for hi ponorhip o hi work. 6. REFERENCE [1]. Y. Waanabe: On he Deign of Ani-Rolling Tank, Journal of he Sociey of Naval Archiec of Japan, Vol. 46, pp -3, 1930 (in Japanee). []. J. H. Chadwick: On he Sabilizaion of Roll, Tranacion of he Sociey of Naval Archiec and Marine Engineer, Vol. 63, pp 37-80, [3]. G. J. Goodrich: Developmen and Deign of Paive Roll Sabilier, Tranacion of he Royal Iniuion of Naval Archiec, Vol. 111, pp 81-95, [4]. Y. Takaihi: A Model Experimen on he Effec of an Ani-Rolling Tank of Ship in Oblique Sea, Journal of he Kanai Sociey of Naval Archiec, Japan, Vol. 141, pp 45-53, 1971 (in Japanee). [5]. M. Kan and H. Taguchi: Chao and Fracal in Loll Type Capize Equaion, Tranacion of he We-Japan Sociey of Naval Archiec, No. 83, pp , 199 (in Japanee). [6]. K. Tanizawa and S. Naio: An Applicaion of Fully Nonlinear Numerical Wave Tank o he Sudy of Chaoic Roll Moion, Inernaional Journal of Offhore and Polar Engineering, Vol. 9, No., pp 90-96, [7]. S. Waanabe: Reducion of Rolling, () Ani-Rolling Tank, Proceeding of he 1 Sympoium on Seakeeping, The Sociey of Naval Archiec of Japan, pp 8-179, 1969 (in Japanee).