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Group Preserving Scheme for Simulating Dynamic Ship Maneuvering Behaviors Yung-Wei Chen1 , Jiang-Ren Chang2 , Wun-Sin Jhao2 and Juan-Chen Huang3

Abstract: In this paper, the group preserving scheme (GPS) is adopted to simulate the dynamic ship maneuvering behaviors. According to the mathematical model of the maneuvering ship motion, ship dynamic behaviors are affected by the ship hull resistance, rudder and propulsive propeller forces, and main engine induced force; therefore, the conventional approach uses the high order time integration to simulate the ship dynamic behaviors and results in the numerical instability. Due to the characteristics of the cone structure, Lie-algebra, and group property of the GPS, this research introduces the non-linear GPS to simulate the dynamic ship behaviors. Through the group weighting factor of the GPS, the non-linear parameters’ behaviors can be observed and the numerical stability can be guaranteed. In addition, the second order of the GPS can ensure the accuracy of the high-order numerical method for simulating ship simulation behaviors and enhance the efficiency of the numerical calculation. Finally, results of the proposed approach are compared with the sea-trial data of a 278,000 DWT ESSO OSAKA ore & tanker [Crane (1979)] further validation. Keywords: Group preserving scheme, numerical instability, ship dynamic behavior, ship maneuvering 1

Introduction

When the hydrodynamic forces during maneuvering are provided for each time step, no mathematical model is required. In such a case, only the equation of motion is used for simulations. Recently computer fluid dynamics (CFD) techniques have gradually made ship simulations possible; however, these simulation techniques cannot work to be satisfied due to the limited sea-trial data. In order to 1 Department

of Marine Engineering. of Systems Engineering & Naval Architecture. 3 Department of Merchant Marine. 2 Department

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well describe the hydrodynamic forces for each time step, mathematical models are usually used. However, the expression is not so simple because there still exist non-linear terms and the interaction of ship hull and various wave motions. Consequently, the expressions of the hydrodynamic forces in the mathematical model are assumed to depend on velocity and acceleration components. That is a well-known “quasi-steady approach”. In the figure 1 is the coordinate system for the simulation of ship maneuvering; the following equations of motion for four-degrees of freedom can be expressed as: m(u˙ − vr) = X,

(1)

m(v˙ − ur) = Y,

(2)

Izz r˙ = N,

(3)

Ixx ψ˙ = K.

(4)

Here m represent the mass of a ship, Izz is the inertia moment of yawing, Ixx is the inertia moment of rolling, X and Y denote the hydrodynamic forces in the x and y directions, and N and K denote the moments of the z and x directions acting on the gravity of the ship. For the expression of these hydrodynamic forces and moment, some polynomial functions with acceleration and velocity components are used. The coefficients of them corresponding to hydrodynamic derivatives can be obtained from [Kobayashi (2002); Yoshimura (2005)]: 1. Captive model test such as oblique towing test (OTT), rotating arm test (RAT), circular motion test (CMT) and planar motion mechanism (PMM) test. 2. Numerical calculation. 3. Identification to the free-model tests or full- scale trials. 4. Database of hydrodynamic derivatives. 2

Mathematical MODEL

In 1976-1980, Japanese research group named Maneuvering Mathematical Modeling Group (MMG) proposed a mathematical model [Ogawa and Kasai (1978); Kose, Yumuro, and Yoshimura (1981)] which is call as MMG model.

Group Preserving Scheme for Simulating Dynamic Ship Maneuvering Behaviors

2.1

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Basic equation of motion

The mathematical model used in this study is shown as equations (1) to (3). These hydrodynamic forces and moments can be divided into the following components. m(u˙ − vr) = XH + XP + XR + XW + XC ,

(5)

m(v˙ − ur) = YH +YP + XR +YW +YC ,

(6)

Izz φ¨ = NH + NP + NR + NW + NC ,

(7)

Ixx φ¨ = KH + KP + KR + KW + KC ,

(8)

where subscripts H, P, R, W and C denote hull, propeller, rudder, wind and current, respectively. The equations (5-8) represent surge motion, sway motion, yawing motion and rolling motion, respectively. 2.2

Hydrodynamic forces and moment acting on the hull

As mentioned above, steady hydrodynamic forces with XH and YH and moments with NH and KH are the functions of u, v, r, and β . Here r and β denote yawing rate and drift angle. In the total force model, these functions are described as the following polynomials using Taylor expansion, for example. 0 04 0 0 0 02 0 1 2 2 0 02 XH = mx u+(m ˙ x +Xvr )vr + ρL V (Xvv v +Xrr v r +Xrr r +Xvvvv v )+X(u). (9) 2

0 0 0 0 0 0 1 YH = −my v˙ − mx ur + ρL2V 2 (Yβ β +Yr r +YNL +YRoll ). 2

(10)

0 0 0 0 0 0 1 NH = −Jzz r˙ + ρL3V 2 (Nβ β + Nr r + NNL + NRoll ). 2

(11)

KH = −Jxx φ¨ − N(φ˙ ) − mgGZ(φ ) −YH ZH .

(12)

These total force models can easily describe the steady hydrodynamic forces, and has been widely used. The coefficients in equations (9), (10), (11) and (12) are called hydrodynamic derivatives and can be obtained by some model tests [Kobayashi (2002)] using scaled model. These mathematical models can be usually applied for the simulation of the specified ship.

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Hydrodynamic forces induced by propeller

The hydrodynamic forces induced by the propeller are expressed as follows: 2πJPP n˙ = QE + QP .

(13)

XP = (1 − tP )ρn2 D4P KT (JP ).

(14)

QP = 2πJPP n˙ − ρn2 D5P KQ (JP ).

(15)

where t is thrust deduction factor, n˙ is propeller revolution, DP is propeller diameter, JP is propeller advance ratio, JPP is added polar moment of inertia of propeller, QP and QE are the moment of propeller and engine, and KT and KQ are the thrust coefficient and torque of propeller. 2.4

Hydrodynamic force and yaw moment induced by rudder

In order to considering the effect of hydrodynamic forces acting on hull, the forces caused by propeller and rudder are estimated using the following mathematical model, and are expressed as below: XR = −FN sin δ ,

(16)

YR = −(1 + αH )FN cos δ ,

(17)

NR = −(1 + αH )xR FN cos δ ,

(18)

KR = (1 + αH )zR FN cos δ ,

(19)

where FN is rudder normal force, δ is rudder angle, and xR , αH and zR are the interactive coefficients between rudder and hull. FN is rudder normal force and can be described as following: FN = 0.5ρ

6.13λ ARVR2 sin αR , λ + 2.25 0

(20)

αR = δ − γβR ,

(21)

γ = Cp ·Cs ,

(22)

where λ is aspect ratio of rudder, AR is rudder area, VR is effective rudder inflow, αR is effective rudder inflow angle, γ is flow-rectification coefficient, Cp is flowrectification coefficient of propeller, and CS is flow-rectification coefficient of ship form.

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For describing the effect of propeller stream, the longitudinal and lateral inflow velocity of rudder can be described as the followings: p VR = V (1 − wR ) 1 +C1 g(s), (23) g(s) =

ηk[2 − (2 − k)s]s , (1 − s)2

η = D p /H,

(25)

0.6(1 − wP , 1 − wR s (1 − s)2 , CP = 1 + 0.6η(2 − 1.4s)s k=

s = 1−

(24)

u(1 − wP ) , nP

0.5 , 0.45 0 0.5 or CS = 0.5, if βR > , 0.45 0

(26) (27) (28)

0

CS = 0.45βR , if βR ≤

0 0

0

βR = β − 2x r ,

(29) (30)

where H is rudder height, wR is effective rudder wake fraction, s is slip ratio, P is propeller pitch. 3

Group Preserving Scheme

The group-preserving scheme [Liu (2001); Chen, Liu, and Chang (2007)] is a scheme that can preserve the internal symmetry group of the considered system. Although previously we do not know what kind symmetry group of the general nonlinear dynamical systems is, yet Liu (2001) has embedded them into the augmented dynamical systems, which concern with not only the evolutions of the state variable itself but also with the evolution of its magnitude. That is, for the general dynamical system of n ordinary differential equations: x˙ = f(x,t),

x ∈ R,t ∈ R,

(31)

We can embed it into the following (n + 1)-dimensional augmented dynamical system: #    "  0n×n f(x,t) d x x ||X|| = fT (x,t) = (32) ||x|| dt ||x|| 0 ||X||

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Consequently, we have an (n + 1)-dimensional augmented system: ˙ = AX X With a constraint of equation (33), where " # 0n×n f(x,t) ||X|| A = fT (x,t) 0 ||X||

(33)

(34)

Satisfying AT g + gA = 0

(35)

Remarkably, the original n-dimensional dynamical system of equation (31) in E n can be embedded naturally into an augmented (n + 1)-dimensional dynamical system of equation (33) in Mn+1 . Although the dimension of the new system is raised by one, it shows that the new system has the advantage of devising grouppreserving numerical scheme as follows [Liu (2001)]: X`+1 = G(`)X` ,

(36)

where X`+1 denotes the numerical value of X at the discrete time t` , and G(`) ∈ SO0 (n, 1) is the group value at time t` . Inserting the above Cay [τA(`)] for G(`) into equation (36) and ranking its first row, we obtain x`+1 = x` + hw(`)f` w(`) =

||x` ||2 + τf` · x` ||x` ||2 + τ 2 ||f` ||2

(37) (38)

Where is called a weighting factor. In the above, x` denotes the numerical value of x at the discrete time t` , τ is one half of the time increment, i.e., τ = h/2, and more precisely, f` is f (x` ,t` ). 4

Numerical Examples

According to the above-mentioned MMG model, maneuvering ship motions can be predicted by the computer simulation. The principal particulars are listed in Table 1. Hydrodynamic derivatives and coefficients for the simulation are listed in Table 2. According to turning circle test, simulated ship motions, ψ, r, the nondimensional velocity ratio ,and ship trajectory, under turning right rudder 35◦ are

Group Preserving Scheme for Simulating Dynamic Ship Maneuvering Behaviors

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Tab ble 1 Generral Data for a Sample Ship S

Table 1: General Data for a Sample Ship

Lenngth betweenn perpendicculars, L, m

343

Length perpendiculars, L, m Breadth molde d, B,between m 53 343 Breadth molded, B, m 53 Draaft in full loaad, T, m 21.76 Draft in full load, T , m 21.76 Block coefficieent 0.831 Block coefficient 0.831 Dispplacement, metric tons 27313.5 Displacement, metric tons 27313.5 Lonngitudinal position of center of graavity, xcg, m 10.3 Longitudinal position of center of gravity, xcg, m 10.3 119.817 Movvable rudde er area,rudder AR, m2area, AR, m2 Movable 119.817 Ruddder span, Sp Sp, m 13.85 Rudder span, Sp, m 13.85 Enggine type: DieselDiesel Engine type: Maxxim continu uouscontinuous rating of o engine M MCR, RPM MCR, RPM 82 82 Maxim rating of engine Nom minal contin nuous continuous rating g of engine NCR, RPM M NCR, RPM 81 81 Nominal rating of engine Taable 2 Hydro odynamic Coefficients C [Chu and[8]Tseng (1988)] Table 2: Hydrodynamic Coefficients

Sy Symbol

Value

Symbol Value 0 -0.001276 6 Xvv -0.001276 0 0.0003 Xrr 0.0003 0.045241 0 Xvvvv 0.045241

Symbbol

Symbol 0 Yβ γ 0 Yγ 0 Yβ β 0 Yγγ

Value V

Symbol

Value Symbol 0 0.0 022335 0.022335 Nrr 0 0.0045843 0.0045843 Nβ 0.02255 0 0.02255 Nr -0.0 0022275 0 -0.0022275 Nrrβ

Value

Value -0.0009 976 -0.000976 0.00944 447 0.0094447 -0.0036 6063 -0.0036063 -0.0050 014 -0.005014

Figure 1:e Co-ordinate Figure 1 Co-ordin natesystem system m

shown in figs. 2-5. In additional, ship motions, ψ, r, the non-dimensional velocity ratio ,and ship trajectory, under turning left rudder 35◦ are shown in figs. 6-9. In order to further test maneuvering stability, we compute heading change and yaw10

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ing rate by the zig-zag test. Under rudder at 10/-10 and 20/-20 by zig-zag test, the rudder angle, yawing rate and heading change of ESSO OSAKA are shown in figs. 10-13. In summary, the method presented is a more effective and convenient approach to simulate maneuvering ship motion problems.

Figure 2:Figu Comparisons of of numerical results seadata trial ESSO for ure 2 Compparisons numerical n rresults andand sea s trial a of data ESSO of O OSAKO forOSAKO the ◦ the yawing varying with time right rudder 35 35condition yawinng varying with w at time att right rudder cond dition Figuure 2 Compparisons of numerical n rresults and sea s trial dataa of ESSO O OSAKO for the yawinng varying with w time att right rudder 35 cond dition

Figgure 3 Com mparisons off numerical results and d sea trial data d of ESSSO OSAKO O for the yawing y rate varying witth time at riight rudder 35 conditiion

Figure 3: Fig Comparisons of numerical and of for gure 3 Com mparisons off numericalresults results and d seasea trialtrial data d data of ESS SO ESSO OSAKO OOSAKO for the yawing y rate varying th time rudder at riight rudder 35 conditiion the yawing rate varying with time atwitright 35◦ condition 11 11

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Figure 4: Comparisons results data ESSO OSAKOfofor Figuure 4 Comp parisons of numerical n of numerical rresults andand sea s sea trialtrial data a of of ESSO O OSAKO r the the non-dimensional velocity ratio varying with time at right rudder 35◦ condition non-d dimensional velocity atiorresults varying time aofright der 35forcondi Figuure 4 Compparisons of numerical n rat andwith sea s trial dataaat ESSOrudd O OSAKO the ition non-ddimensional velocity rat atio varying with time at a right ruddder 35 condiition

FigureFigu 5: ure Comparisons of numerical results ands sea trial data of ESSO OSAKO for 5 Compparisons of numerical n reesults and sea trial dataa of ESSO O OSAKO for the ◦ the simulated turning circle track at right rudder 35 condition simulaated turning g circle tracck at right ru udder 35 condition.

Figuure 5 Compparisons of numerical n reesults and sea s trial dataa of ESSO O OSAKO for the simulaated turning g circle tracck at right ru udder 35 condition. 12

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Figure 6: Comparisons ofnnumericalreesults results and sea trial data of ESSO OSAKO for Figuure 6 Compparisons of numerical and sea s trial◦ dataa of ESSO O OSAKO for the theFigu rudder varying time re atesults left and rudder 35 data condition ure 6 angle Compparisons ofwith numerical n sea s trial a of ESSO O OSAKO for the rudder angle vary ying with tim me at left ru udder 35 condition c rudder angle vary ying with tim me at left ru udder 35 condition c

Figuure 7 Compparisons of numerical n and sea s trial dataa of ESSO O OSAKO for the Figure 7: Comparisons ofnumerical numericalrrresults results and sea trial data of ESSO OSAKO for Figuure 7 Comp parisons of n results and sea s attrial data a of ESSO O OSAKO for the ◦ yawin ng angle vel locity varyin ng with tim me left rud dder 35 con ndition the yawing angle velocity varying with time at left rudder 35 condition

yawinng angle vellocity varyinng with tim me at left rud dder 35 conndition 13 13

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Figuure 8 Compparisons of numerical n reesults and sea s trial dataa of ESSO O OSAKO for the non-ddimensional velocity rration vary ying with time at leeft rudder 35

Figure 8: Comparisons of numerical results and sea trial data of ESSO OSAKO for condittion Figu ure 8 Comp parisons of numerical n reesults andwith sea s trial of ESSO OSAKO for the the non-dimensional velocity ration varying timedata ataleft rudder O 35◦ condition non-ddimensional velocity rration vary ying with time at leeft rudder 35 condittion

Figure 9: results sea data trial ofO ESSO OSAKO for Figu ureComparisons 9 Compparisons of of numerical nnumerical reesults and and sea s trial a ofdata ESSO OSAKO for the ◦ the simulated turning circle tracktrack atkleft rudder condition simula ated turning g circle at left rud dder35 35 cond dition 14

Figuure 9 Compparisons of numerical n reesults and sea s trial dataa of ESSO O OSAKO for the simulaated turning g circle trackk at left rud dder 35 cond dition 14

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Figure 10:10Comparisons numerical results andtrial seadata ESSO OSAKO Figuure Com mparisons ofofnumerical l results and d sea d trial of data ESSSOofOSAKO O for for the heading change varying with time at 20/20 zig-zag test the heading chaange varyinng with timee at 20/20 ziig-zag test Figuure 10 Com mparisons of numericall results and d sea trial data d of ESSSO OSAKO O for the heading chaange varyinng with timee at 20/20 ziig-zag test

Figuure Com mparisons ofofnumerical l results and d sea d trial of data ESSSOofOSAKO O for Figure 11:11Comparisons numerical results andtrial seadata ESSO OSAKO the y yawing rate varying wi ith time at 20/20 2 zig-za ag test for theFigu yawing rate varyingowith timel atresults 20/20 zig-zag ure 11 Com mparisons f numerical and d sea trialtest data d of ESSSO OSAKO O for

the yawing y rate varying wiith time at 20/20 2 zig-zaag test 15

15

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Figure 12: Comparisons of numerical resultsd sea andtrial seadata trial data of ESSOOOSAKO Figuure 12 Com mparisons of numericall results and d of ESSSO OSAKO for for the heading change varying with time at 10/10 zig-zag test the heading h chaange varyingg with time at 10/10 zig g-zag test Figuure 12 Com mparisons of numericall results and d sea trial data d of ESSSO OSAKO O for the heading h chaange varyingg with time at 10/10 zig g-zag test

Figuure 13 Com mparisons of numericall results and d sea trial data d of ESSSO OSAKO O for the yawing y ratee varying wiith time at 10/10 1 zig-zaag test

Figure 13:13Comparisons numerical results seadata ofOSAKO ESSO Figuure Com mparisons oof f numerical l results and d and sea trial dtrialofdata ESSSO O OSAKO for for the yawingtherate varying with time at 10/10 zig-zag test y yawing ratee varying wiith time at 10/10 1 zig-zaag test 16

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Conclusions

In this paper, we have successfully applied the GPS to simulate ship maneuvering behaviors. Since the ship dynamic behaviors are affected by those non-linear forces such as the ship hull resistance, rudder and propulsive propeller forces and induced engine force, the non-linear GPS is adopted to simulate the dynamic ship behaviors and ensure the numerical stability. Through the group weighting factor of the GPS, the non-linear parameters’ behaviors can be observed and the numerical instability can be overcome. In addition, the second order of the GPS can ensure the accuracy of the high-order numerical method for simulating ship simulation behaviors. Finally, results of the proposed approach are compared with the sea-trial data of turning circle test in a 278,000 DWT ESSO OSAKA ore & tanker for further validation. Acknowledgement: The authors would like to express his thanks to the National Science Council, Taiwan for their financial support under contract number: NSC102-2218-E-019-001 and NSC-99-2221-E-019-053-MY3. References Chen, Y. W.; Liu, C. S.; Chang, J. R. (2007): A chaos time step-size adaptive numerical scheme for non-linear dynamical systems. Journal of Sound and Vibration. Chu, F. S.; Tseng, K. T. (1988): Discussions on the prediction method for ship maneuvering performance, 1988. Crane, C. L. (1979): Maneuvering trials of a 278,000-dwt tanker in shallow and deep waters. Trans. SNAME, vol. 87. Kobayashi, H. (2002): The specialist committee on esso osaka final report and recommendations to the 23rd ittc. In In 23rd International Towing Tank Conference, Proceedings of the 23rd ITTC, volume II, pg. 581. Kose, K.; Yumuro, A.; Yoshimura, Y. (1981): Concrete of mathematical model for ship maneuverability. In 3rd Symposium in ship maneuverability, pp. 27–80. SNAJ. Liu, C. S. (2001): Cone of nonlinear dynamical system and group preserving schemes. International Journal of Non-Linear Mechanics, vol. 36, pp. 1068. Ogawa, A.; Kasai, H. (1978): On the mathematical model of maneuvering motion of ship. ISP, vol. 25, no. 292.

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Yoshimura, Y. (2005): Mathematical model for maneuvering ship motion. In Workshop on Mathematical Models for Operations Involving Ship-ship Interaction, Tokyo.