A FORMULATION AND PRELIMINARY RESULTS FOR SIMULATION OF SHIP MOTIONS COUPLED HEAVE, PITCH AND ROLL

Transcription

1 8 th Intenational Confeence on the Stability of Ship and Ocean Vehicle Ecuela Técnica Supeio de Ingenieo Navale A FORMUATION AND PREIMINARY RESUTS FOR SIMUATION OF SHIP MOTIONS COUPED HEAVE, PITCH AND RO inzhu Xia, Autalian Maitime College, (Autalia) Sheming Fan, Maine Deign & Reeach Intitute of China, (China) Abtact The time-domain tip theoy fo vetical and olling motion i combined to pedict the coupling of heave, pitch and oll motion though modelling the non-lineaity of hydotatic etoing foce and moment with epect to heave, pitch and oll diplacement. Coeponding compute pogam i developed. Time-domain imulation ae pefomed fo a Panamax Containe hip at diffeent heading and encounte fequencie. The coupling effect and paametic eonance ae invetigated. It i een that paametic intability occu when the hip ail in quateing wave and vey low encounte fequency. In othe computational wave condition, the effect of coupling of heave, pitch and oll ae mall.. INTRODUCTION The coupling motion of heave, pitch and oll have been dicoveed fom both expeimental obevation and theoetical imulation. An exteme phenomenon i paametic eonance. In thi cae, the enegy in heave and pitch motion may be tanfeed to oll motion mode though non-linea coupling, which lead to exceive eonant olling and tability poblem []. Exceive oll motion inceae the pobability of eaickne of the cew, educe the opeability of the onboad ytem and, in the wot cae, caue capize of the veel. Becaue of the lateal ymmety of hip hull fom, linea theoie ae unable to account fo the coupling between heave, pitch and oll. It i quite often that oll motion i teated a a ingle degee-of-feedom (DOF) dynamic poblem, wheea heave and pitch ae olved in a coupled two DOF ytem. Some peviou tudie have employed implified hydodynamic model to imulate thi poblem. Hydodynamic memoy effect due to the fee-uface wave motion i not incopoated, which give ignificant uncetaintie in the imulation. In othe tudie, the memoy effect i expeed by a time convolution. The computation of time convolution limit the pactical application, and it i difficult to extend the theoy to account fo non-linea memoy effect. A ational non-linea time-domain tip theoy wa developed in [] to pedict vetical wave load and hip epone. It wa extended to compute anti-ymmetic wave-induced hip epone [3]. Both tudie modelled the hydodynamic memoy effect by a highe-ode

2 8 th Intenational Confeence on the Stability of Ship and Ocean Vehicle Ecuela Técnica Supeio de Ingenieo Navale diffeential appoximation intead of timeconuming numeical integation. Hence the computational efficiency i maintained while the complicated non-linea phenomenon i invetigated. In the peent eeach, the tudie in [] and [3] ae meged to teat the coupling of oll motion with the vetical mode (heave and pitch) induced by additional hydotatic etoing foce and moment due to lageamplitude heave, pitch and oll diplacement. Time-domain imulation ae pefomed fo a Panamax Containe hip at diffeent heading and encounte fequencie, which ae not compaed with othe numeical o expeimental eult yet.. TIME-DOMAIN STRIP THEORY FORMUATION [,3] The Cateian coodinate ytem (Figue ) i defined a an equilibium et of axe tanlating with mean fowad peed U of the hip in the +x-diection. The oigin i at the ten. The z plane coepond to the calm wate level, and z i poitive upwad. The x-z plane i coincident with the plane of the hip hull. Figue Equilibium coodinate ytem The elative diplacement between a hip ection and wave uface can be expeed a z ( w ( ξ (, l,3,4 l l l () whee l and 3 denote the lateal and vetical movement; l4 denote the otation about the x-axi. w l ( i the diplacement component of hip ection and ξ l ( i the aveage diplacement component of the wave at the hip ection and i elated to wave elevation with the Smith coection. The time-domain ectional hydodynamic foce vecto F( coniting of the lateal, vetical and otational component without convolution may be expeed by [, 3] DI F( Dt ( B I A Dz ) Dt ( + ) t () whee I( epeent both the impulive and memoy vecto in the hydodynamic momentum; D/Dt i the total deivative with epect to time D ( U ) t, Dt t x with U being ( ) the fowad peed of the hip, ( ) A (x)and B (x) ae the o-called fequencyindependent hydodynamic coefficient matice deived by a ational appoximation fom the fequency dependent added ma m(ω) and damping coefficient N(ω) iωm kl N kl A, kl B, kl ( iω) ( iω) +, k, l,3,4 (3) whee i i the imaginay unit and ω i the wave fequency. Equation can be extended empiically to include the non-linea loading effect due to the time vaiation of the wetted hip uface by auming that the coefficient A (x) and B (x) vay with not only the longitudinal poition but alo the intantaneou ubmegence. By integating the highe-ode diffeential equation in Equation, the ectional hydodynamic foce vecto F( can be

3 8 th Intenational Confeence on the Stability of Ship and Ocean Vehicle Ecuela Técnica Supeio de Ingenieo Navale 3 ewitten a D z m Dz m Dz F( m + U ( ) Dt x Dt z Dt (4) whee m(z,z 3,z 4 ) i the added ma matix of the hip ection when the ocillating fequency tend to infinity; Dq /Dt account fo the hydodynamic memoy effect govened by the following et of diffeential equation q Dz q B q ( A + B m) t Dt,,..., q (5) The thid tem of F( in Equation 4 povide the momentum lamming foce. It vanihe in the linea poblem. Accoding to phyical udgement, thi tem i uually only taken into account when the hip ection ente into the wate. 3. CURRENT ASSUMPTIONS D Dt Thee ae ome baic aumption fo the peent poblem a follow () Neglect the effect of hoizontal motion uge, way and yaw. () Neglect hydoelatic effect on oll motion. (3) Neglect the effect of the vaiation of wetted uface on olling added ma and damping. (4) Neglect the hydodynamic coupling between oll and vetical motion. Coupling of oll motion with the vetical mode i intoduced by hydotatic etoing foce and moment in heave, oll and pitch diplacement. Unde the above aumption, the hydodynamic of the vetical and olling motion can be teated epaately. The motion q will then be coupled though hydotatic foce. 4. NON-INEAR TIME-DOMAIN HYDROEASTIC VERTICA MOTIONS [] By incopoation of the hydotatic action and olling effect in Equation 4, the total vetical non-linea extenal fluid foce acting on a hip ection can be expeed a D z3 m Dz3 Z( m + U Dt x Dt m Dz3 Dq ( ) + ρg( S + S) z Dt Dt (6) whee ρ i the denity of the wate and g i the gavitational acceleation. S denote the intantaneou immeed aea of the hip ection due to vetical motion. S epeent the incement of ubmeged aea aiing in olling motion (Figue ). dy S( x) b ( x)[ tan ( z4 )] 4 dz dy [ ( ) tan ( z4 )] dz (7) whee b (x) i the beadth at the calm wate line, dy/dz epeent the ection flae lope at the calm wate line, and z 4 i the elative diplacement between olling motion and aveage wave lop aco the ection. Figue Effect of oll on heave fo a unit ection (Incement of vetical etoing foce ÄZ g(s -S p )) The hull i modelled a a linea elatic non-

4 4 8 th Intenational Confeence on the Stability of Ship and Ocean Vehicle Ecuela Técnica Supeio de Ingenieo Navale unifom Timohenko beam. The aumption of mall igid-body motion and tuctual ditotion allow model upepoition of the diplacement [, 4] w( n w ( x) p ( (8) whee w i the th unit dy mode hape, and p the th genealized pincipal coodinate of motion. In thi analyi only the vetical epone poblem i conide. Theefoe and coepond to heave and pitch motion, while,3,,n epeent elatic ditotion of the hull gide. The numbe of elatic mode of the hip-beam i taken a 3, i.e. n4. The equation of motion fo the hip hull can be witten a n [ a & p ( + b p& ( + c p ( ] ( Z( µ g) wd,,..., n. (9) Heeafte ovedot denote diffeentiation with epect to time. In thi equation, Z( i the total fluid action a epeented by Equation 6, µ(x) the longitudinal ditibution of ma of the hip tuctue, the hip length, and a, b, c ae the component of the genealized tuctual ma, damping and tiffne matice a b δ ( µ w w + I yθ θ ) dx δ a ω α / π c δ aω () whee δ i the Konecke delta function, I y the longitudinal ditibution of moment of inetia of the tuctue, θ the th mode angula function of the hip hull gide beam in the x-z plane induced by vetical bending, α the th mode logaithmic damping decement of the tuctue in vacuum, and ω the th mode eigenfequency of the dy hull. In ode to peeve the othogonality between the igidbody mode (heave and pitch), it i neceay that the mode hape w (x) and x-axi inteect at the cente of ma of beam. It i aumed that the co damping coefficient b ( ) can be neglected. By ubtituting Equation 6 into Equation 9, the equation of the fluid tuctue inteaction can be obtained a ( a + A) p& ( + ( b + B) p& ( + ( c + C) p( R + W + P + Q( q) q& ( q( + Ø with the initial condition p ( ) p& () q( ) () () In the above equation, p i the genealized coodinate vote, p{p,p,,p n } T, q i a vote ued in the integation of the highe-ode hydodynamic equation, q{q,q,,q } T ; A,B,C ae the genealized fluid added ma, damping and etoing foce coefficient, which ae function of t and p in the non-linea poblem, R(t;p) epeent the genealized combined action of the intantaneou hydotatic etoing foce and the tuctual gavitational foce, W(t;p) i the genealized non-linea wave diffaction foce, P ( t; p, p& ) i the genealized momentum lamming foce; Q(q) i the genealize memoial hydodynamic foce

5 8 th Intenational Confeence on the Stability of Ship and Ocean Vehicle Ecuela Técnica Supeio de Ingenieo Navale 5 A ( t; p) B C ( t; p) U ( t; p) U R ( t; p) g W ( t; p) P ( t; p, p& ) f m m V ( t; p, p& ) z3 Q ( q),,,,..., n. (3) In the above equation, pime indicate diffeentiation with epect to x. V i the elative vetical velocity between the ection and the wave uface. The matice Φ and Ψ in Equation ae, epectively, expeed by Ö( t; p) mw w dx [ ρ( S Umw ( ξ& Uξ ) ( q& w ' m( w w ' ' mwwdx Λ Λ Λ Λ Μ ( A ( A Ø( t; p, &p ) ( A ' w w ) dx Umw w + S ) µ ] w dx ' [ mw ( & ξ U & ξ ) + Umw ( ξ& Uξ )] dx f m w dx + U B B B B ' mww V < + mb ) V + mb ) V Μ + mb ) V V ' + Uq w ) dx + Uq w (4) (5) 5. NON-INEAR TIME-DOMAIN RO MOTION The mathematical model employed in thi tudy to decibe the non-linea oll motion i of the fom [5,6] ( I x + A44 ) φ + B( φ) + C( φ, z4 q& Φ & & ) W + Q( q) q Ψ with the initial condition φ ( ) φ& () q( ) (6) (7) whee I x i the ma moment of inetia with epect to x-axi, A 44 i added inetia in oll. φ i the genealized coodinate coeponding to oll. The damping moment i given a [6] B ( φ &) B φ& φ& φ& 44 + B444 (8) whee B 44 i the linea damping moment coefficient, and B 444 i the quadatic damping moment coefficient. The damping coefficient can be obtained fom fee decay expeiment, in which the model i eleaed fom a given inclination angle to feely oll in calm wate with no fowad peed. It i aumed that the hip i unde uncoupled oll motion duing the fee decay expeiment. Then uing appopiate cuve paametic identification technique, the coefficient B 44 and B 444 can be obtained by fitting equation to the ecoded fee decay expeiment data [7] ( I xx + A )& φ + B44φ& + B444φ& φ& + GMφ 44 (9) whee i weight of wate diplaced by the hip. GM i metacentic height. In peent eeach, linea damping coefficient

6 6 8 th Intenational Confeence on the Stability of Ship and Ocean Vehicle Ecuela Técnica Supeio de Ingenieo Navale B 44 wa etimated by empiical fomula [5] B44 d ( I x + A44) C44,.8 d( β ).4 () d(β). wa ued in the calculation. The quadatic damping moment coefficient B 444 wa neglected. The etoing moment i deived unde the aumption that vetical motion due to amplitude wave ae mall, uch that elative vetical diplacement at a point of the length of the hip may be taken a the um of two effect of heave and pitch motion. The oll etoing moment in wave i then given by [6] 3 C( φ, z4 ) C44φ + C444φ + C44zφ () whee C 44 and C 444 epeent linea and thid ode oll etoing coefficient [5] b x C ( ) 44 g [ ρ( + zbs) µ z g ] dx GM () whee z b i vetical poition of buoyancy cente of the ection. C 444 can be obtained fom the hydotatic cuve of the hip [8]. It i neglected in peent calculation. C 44z epeent time-dependent vaiation of hull etoing chaacteitic due to heave motion of a ection (Figue 3), to econd ode [6], ρg dy ( x) z dz C44 z [ b gb ( x)] z3 (3) whee the definition of b (x) and dy/dz ae the ame a in Equation 7. z g i vetical coodinate of hip cente of gavity, z 3 i the elative vetical diplacement between a hip ection and the wave uface. dx Figue 3 Vaiation of ectional beam with elative vetical diplacement (Roll hydotatic moment coefficient change with the change of inetia, aea cente and ubmeged volume) In equation 7, q( i a vaiation ued in the integation of the highe-ode hydodynamic equation; W( i the genealized wave foce and Q(q) i the genealized hydodynamic memoy foce. 6. NUMERICA METHODS [, 3] The genealized non-linea diffeential equation fo the coupling motion of heave, pitch and oll include Equation (), (), (6) and (7). Thi et of equation i integated in the time domain by uing the Adam pedictocoecto cheme. The fouth-ode Runge- Kutta method i ued to pedict the function value at the peviou fou time tep. The econd-ode cental diffeence cheme i ued to detemine the eigenvalue and pincipal mode of the fee-fee non-unifom Timohenko beam. The mode of diplacement w, bending lope, heaing angle, bending moment M, heaing foce V and the natual fequencie ω can be imultaneouly obtained though an iteative pocedue with vey mall computational effot. A multi-paamete confomal tanfomation technique i ued to olve the vetical twodimenional fequency-domain hydodynamic coefficient (added ma and added damping) of the ection at eveal daught. The

7 8 th Intenational Confeence on the Stability of Ship and Ocean Vehicle Ecuela Técnica Supeio de Ingenieo Navale 7 ingulaity ditibution method i ued to detemine the two-dimenional fequencydomain hydodynamic coefficient fo oll motion. Equation 3 i exactly atified at - fequencie. In ode to achieve the bet appoximation within a given fequency ange, the fequency-independent hydodynamic coefficient A and B ae obtained by minimizing K { Re( ε ) + Im( ε ) } with (4) in k ε ( iω ) [ A, k + ( mk ) B, k ], k 3,4 ω (5) whee ω ae dicete cicula fequencie within the given hip-dependent fequency ange. Fom numeical tet it i found that, in ode to each good ageement between the oiginal and the appoximated fequencydomain hydodynamic coefficient, a thidode diffeential appoach (3) i neceay and ufficient fo mot ectional hape except fo oll of the exteme U-haped ection. In thi cae, the twin peak appea in the fequency epone of damping. It i difficult to appoximate coectly fo the whole fequency ange by thid-ode (o even highe-ode) diffeential appoach. In ode to get a moe eaonable appoximation, added damping wa only appoximated in highe fequency ange becaue the damping in low fequency i much malle than that in high fequency. 7. CACUATIONS FOR A PANAMAX CONTAINER SHIP Time-domain imulation ae pefomed fo a Panamax Containe Ship. A detailed deciption of the expeiment including the body plan and the main paticula of the hip can be found in Tan [9]. The uncoupled eult have been compaed with model tet. The ageement i acceptable fo the linea oll motion and non-linea vetical mode [3,5]. Conideing the hamonic chaacte of time vaying tem, the equation and 7 ae ecognized a foming a et of coupled Mathieu equation. Thi type of ytem of odinay diffeential equation with peiodic coefficient may be, in geneal, ubected to paametic eonance fo encounte fequencie cloe to [] mω e ωi ± ω, o,ω i, m,,..., i, 3,4,5 (6) Fo the Panamax Containe Ship, the natue fequencie of heave, oll and pitch ae ω 3.73 ad/, ω 4.5 ad/, and ω 5.73ad/, epectively. Following the above analye, the imulation ae conducted fo thee wave heading: β5, 9, 65 degee (8 degee denote head wave). Fo each wave heading, two encounte fequencie, which pobably lead to paametic eonance, ae conideed. The ange of wavelength to hip length atio i: λ/.5.5. The wave amplitude wa kept contant at 5 m. Ship peed i 4.5 knot. The computational eult ae howed in Figue 4 to 9. Fo each time eie, uncoupled eult ae alo given. Figue 4 and 5 give the eult fo wave heading β5 and encounte fequencie ω e. ad/ (4ω e ω 3 -ω 4 ) and ω e.6 ad/ (3ω e ω 3 -ω 4 ). In Figue 4, amplitude of uncoupled oll epone ae of the ode of 4 degee; the oll epone fom the coupled et of equation i of the ode of 3 degee. Much lage coupled oll motion ae found in thi condition. Coupled heave and pitch ae alo vey diffeent fom the uncoupled olution, but thee motion ae mall in amplitude. In Figue 5, coupled oll motion i

8 8 8 th Intenational Confeence on the Stability of Ship and Ocean Vehicle Ecuela Técnica Supeio de Ingenieo Navale not vey diffeent fom uncoupled motion. Coupled heave and pitch motion ae nealy the ame a uncoupled motion. Figue 6 and 7 peent the eult fom the beam wave. The encounte fequencie ae ω e.49 ad/ (ω e ω 3 +ω 4 ) and ω e.366 ad/ (3ω e ω 3 +ω 4 ). In thi cae, coupled pitch motion change a lot. The effect of coupled motion ae bigge in lowe fequency than that in highe fequency. Both coupled and uncoupled pitch motion ae mall in amplitude. Coupled heave and oll motion ae not vey diffeent fom uncoupled motion. Figue 8 and 9 how the eult fo the bow wave (wave heading β65 ). The encounte fequencie ae ω e.98 ad/ (ω e ω 3 +ω 4 ) and ω e.48 ad/ (ω e ω 3 -ω 4 ). The coupled heave and pitch motion ae nealy the ame a uncoupled motion. The mean value of coupled oll motion ae biaed. It may be due to the bia of pitch motion. Both coupled and uncoupled oll motion ae mall. It i een fom thee compaion that paametic intability occu when the hip ail in quateing wave and vey low encounte fequency. In othe computational wave condition, the effect of coupling of heave, pitch and oll ae mall.

9 8 th Intenational Confeence on the Stability of Ship and Ocean Vehicle Ecuela Técnica Supeio de Ingenieo Navale 9 Figue 4 Simulated motion (wave heading β5º, encounte fequency ωe (ω3-ω4)/4) Figue 5 Simulated motion (wave heading β5º, encounte fequency ωe (ω3-ω4)/3)

10 8 th Intenational Confeence on the Stability of Ship and Ocean Vehicle Ecuela Técnica Supeio de Ingenieo Navale Figue 6 Simulated motion (wave heading β9º, encounte fequency ωe (ω3+ω4)/) Figue 7 Simulated motion (wave heading β9º, encounte fequency ωe (ω3+ω4)/3)

11 8 th Intenational Confeence on the Stability of Ship and Ocean Vehicle Ecuela Técnica Supeio de Ingenieo Navale Figue 8 Simulated motion (wave heading β65º, encounte fequency ωe (ω3+ω4)) Figue 9 Simulated motion (wave heading β65º, encounte fequency ωe (ω3-ω4))

12 8 th Intenational Confeence on the Stability of Ship and Ocean Vehicle Ecuela Técnica Supeio de Ingenieo Navale 8. CONCUDING REMARKS The time-domain tip theoy fo vetical and olling motion ha been extended to pedict the coupling of heave, pitch and oll motion due to additional hydotatic etoing foce and moment with epect to lage-amplitude heave, pitch and oll diplacement. Time-domain imulation wee pefomed fo a Panamax Containe hip at diffeent heading and encounte fequencie. The coupling effect and paametic eonance wee invetigated. Acknowledgement: The autho would like to thank D Zhaohui Wang of the Clough Engineeing td (Autalia) fo poviding he compute pogam on oll motion imulation and fo valuable dicuion and help in the coue of thi wok. The autho ae gateful fo the financial uppot unde a UWA Small Gant of The Univeity of Weten Autalia awaded to. Xia and a Cente of Excellence funding allocated to the Cente fo Maine Science and Technology of Cutin Univeity of Technology by the Weten Autalia tate govenment. 9. REFERENCES [] Neve, M.A.S., Valeio,., Paametic eonance in wave of abitay heading, 7 th Intenational Confeence on Stability of Ship and Ocean Vehicle, Tamania, Autalia, ,. [] Xia,., Wang, Z, enen,., Non-linea wave load and hip epone by a timedomain tip theoy, Maine Stuctue (3), -3, 998. [3] Wang, Z., Hydoelatic analyi of high peed hip. Ph.D. Thei, Depatment of Naval Achitectue and Offhoe Engineeing, Technical Univeity of Denmak,. [4] Bihop, R.E.D., Pice, W.G., Hydoelatict of hip, Cambidge Univeity Pe, 979. [5] Wang, Z, enen,., Xia,., Pediction of wave-induced olling epone by a timedomain tip theoy, PRADS, Shanghai,. [6] Neve, M.A.S., Peez, N.A., Valeio,., Stability of mall fihing veel in longitudinal wave, Ocean Engineeing, 6, , 999. [7] Silva, S.R., Soae, C.G., Time domain imulation of paametically excited oll in head ea, 7 th Intenational Confeence on Stability of Ship and Ocean Vehicle, Tamania, Autalia, ,. [8] Ouyang, R., Application of bifucation theoy and numeical method in tability of maine vehicle, Ph.D. Thee, Shanghai iao Tong Univeity, (in Chinee), 999. [9] Tan, S.G., Wave load meauement on a model of a lage containe hip, Netheland Ship Reeach Cente, Repot 73S, 97. [] Ceai,., Aymptotic behavio and tability poblem in odinay diffeential equation, Thid Edition, Spinge- Velag, 65-8, 97.