Bank effect on ship maneuverability in a channel with varying width

by

Hironori YASUKAWA. Member

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Repnnted from TRANSACTIONS OF

THE WEST-JAPAN SOCIETY OF

NAVAL ARCHITECTS

Bank Effect on Ship Maneuverability in a Channel with Varying Width

by

Hironori YASUKAWA*,

Member

Summary

Theoretical investigation is made into bank effect on ship maneuverability in a channel

with varying width. First. a method to calculate asymmetric forces such as lateral force

and yawing moment acting on a ship in the proximity of arbitrary shaped bank is

introduced. Calculation is made by use of the present method and comparison s made with the experiments. And it is confirmed that the present method is a useful mean for obtaining

the characteristics of the asymmetric forces. Next, simulation study is made in order to investigate the bank effect on the ship maneuvering motion in the channel with varying

width. As a result, it is shown that in the channel with varying width course.keeping is relatively difficult due to the effect of the asymmetric forces, and potential hazards of collision and grounding are considerably large.

Nagasaki Experimental Tank, Mitsubishi Heary Industries. Ltd. 1.

Introduction

When a ship moves parallel to a bank it experiences a lateral force toward and a turning momentaway

from the bank[1j. The forces and moments (we call them hereafter asymmetric forces) arise from the asymmetric flow around the ship hull due to the presence of the hank. 1f the length of the bank is short in comparing with the ship length, transient forces occur when the ship passes the ends of the bank. These

transients may be larger than the resultant steady state forces obtained near an infinitely long bank [9] [10]. Therefore in restricted water such as a channel with varying width, the hazards of collision and grounding

will increase.

A brief review of the past analytical work on the transient asymmetric forces will be given first. In 1980,

Yeung and Tan presented a method to calculate the asymmetric forces acting on a ship moving near an irregular shaped bank[16J by extending a method for interaction forces between ships[15]. Similar work

was also given by Davis[2]. _{Thereafter, by applying the Yeungs method. Tan and Hsiung et al. calculated}

the asymmetric forces acting on ships in the proximity of various shaped bank[12][6]. _{Further, Kijima and} Qing investigated bank effect on maneuvering motion for a ship moving near a wedged shaped bank by a simulation taking the asymmetric forces into account[7]. In the Yeungs method the bank shape is dealt with theoretically by applying the conformal mapping method. However, it is difficult to obtain explicitly the mapping function for a complicated shaped bank such as a channel with varying width.

In this paper, first, a method to calculate the asymmetric forces is introduced by a combination with slender body theory and panel method. In the present method the bank is represented by source (or sink) panels which are distributed on the bank surface. Therefore the arbitrary shaped bank can be dealt with

as a channel with varying width, simulation study is made taking the effect of the asymmetric forces into

account. As a result, it is shown that in the channel with varying width course-keeping is relatively difficult due to the effect of the asymmetric forces, and potential hazards of collision and grounding are considerably large.

2.

Asymmetric Forces on a Ship moving in the Proximity of Arbitrary Shaped Bank

2. 1 Problem formulation

Let us consider the problem of a ship moving in a straight course with constant speed U in the proximity

of arbitrary shaped bank as shown in Fig. 1. It is assumed that the bank wall is vertical and water depth h

is constant. Two coordinate systems, Oo-XoyoZo fixed with respect to the space and o-xyz in steady translation

with the forward velocity of the ship are employed. The x-axis coincides with the positive direction of the ship's forward velocity. The z = O and ¿o = O planes coincide with the undisturbed free-surface, and the a-axis is taken positive upward. The ship direction O is taken between Xo- and x-axes.

In order to simplify the problem, the following assumptions are made regarding the order of magnitude:

The breadth of the ship B and the draft a' are small relative to the ship length L:

d/L = O(e), B/L = O(e),

where e is slenderness parameter. The water depth h is shallow:

h/L = O(e).

(c) The bank is a large distance away from the ship:

YO -X0

N

Ship Wake

x7U

o

Fig. i Coordinate systems

\

S/L = 0(1).

where Sp means the distance between ship center line and the bank wall.

On the above assumptions. the governing equation in flow field far away from the ship (far-field) is

expressed as[13]

(1_Fnh)2+

=0, (1)

where represents velocity potential for far-field, F,h the Froude number based on the water depth. Fh is

assumed to be small since the ship speed is not high in the restricted water. Then eq. (1) is expressed as two-dimensional Laplace's equation:

a2a2

_=0.

(2)

ox_

ay-Thus, the problem in the far field can be dealt with as the two-dimensional problem with respect to x-y plane.

The velocity potential can be represented by distribution of source (or sink) and vortex along the ship

center line, and source on the bank line as:

'L12

£12

(P, t) ₌

c(Q. t)G(P. Q)d+J

(Q, t)Gv(P.

Q)dE+f(Q t)G(P. Q)dc.

-L12

(3)

Here. and ;' represent the strengths of source and vortex along the ship center line respectively, and p the

where P means field point (xo. yo), Q the source or vortex point along the ship center line (Ea,27o) and Q the

source point ori the bank line (E, .

Making a Taylor expansion of for small value of y and neglecting the higher order terms, the inner solution of eq. (3) can be obtained as:

lit ₌

_2fLI2'

t)

logxjd

t)Od LfLJ2 t)d

I ír12

_{r(. t)}

-'

dEl

i

1y1--oix, t)IyI

+-f(Q. t)G(P', Q)dc

'fui, t)&(P', Qc)a'c}y

(7)

where P' means the point on the ship center line n space fixed coordinate.

On the other hand, the outer solution for the inner problem has been obtained as [14] [161 strength of source on the bank line. G, Gv and G are represented respectively as follows:

C(P, Q) = lOg(xo_o)2±(yo_o)2, (4)

(5)

G(P, Q) = tan'( :I:)

= 4h

Iy+ V*(y±Ca)f(x, t),

(8)

where 0 means velocity potential for inner region, S the double body sectional area, C,5 the blockage

coefficient [12], V* the cross flow velocity between the ship hull and sea bottom, f the arbitrary constant in

the two-dimensional problem formulation.

The outer solution for the inner problem and the inner solution for the outer problem are required to match in the intermediate region. By matching terms of similar nature in egs. (7) and (8). the following equations are obtained:

UdS

2h

1 (LI2

V*CaJ

ï(E,t)de, (lo)

= t)

-(QC, t)-KP', Q)dc.

(11)

Eliminating V" from egs. (10) and (11), and substituting eq. (9), the following integral equation can be obtained

as:

2C8

L?'

t)d_-f'(E. t)

= t)

oGC(p _Q)dc.

(12)

Further, the solution of eq. (12) should be augmented by the additional conditions that the pressure is

continuous across the wake vortex, and that Kelvin's theorem is satisfied as follows:

y(x, t) = 7(x) for

-

<x < L/2.

(13)

L [2

t)d

= 0. (14)

The boundary condition of the bank wall is expressed as:

t) =0,

(15)

where n denotes outward normal of the bank surface and P the value on the bank line. Substituting eq. (3) into eq. (15). the following condition is obtained as:

2

fL/Z(Q

t)°2(P. Q)dEf 7(Q, t) oGv(p Q)dE+

Í(Q,

t)°(P, Q)dc

=0. (16)

LYfl Ic üfl

IL12 Ofl

By solving eqs. (12)(14) and (16), where ' and are unknown values, the asymmetric forces can be calculated at each time steps. In this paper, eqs. (12)(14) and (16) are solved numerically by a method combined with vortex lattice [6] and panel method [3]. The detailed numerical procedure is explained in

the Appendix.

The asymmetric forces acting on a ship can be calculated from following equations. By applying

Bernoulli's theorem, the linearized pressure is given by

p=

(17)

(x, t)

fL2

t)d.

Therefore the pressure jump J across the ship center line is obtained as:

L!2

Jp(x. t) =

-{7f y(, t)d-- Uy(x, t)}.

Integrating eq. (19) over the length of ship, lateral force Y and yawing moment N are obtained as:

Y(t) = _hfp(x. t)dx.

N(t) = hf

t)d.

In this paper, the non-dimensional asymmetric forces for Y and N are defined as: Cy = Y/(l/2pU2Bd),

C, = N/(l/2pU2LBd).

2. 2

Verification of the Present Method

Calculation was made for a 280.000tdw ore/oil carrier, which was used by Norrbin for studying bank effects on ships [9] [101. Table i shows principal particulars of the ship

blockage coefficient C8 was estimated by using a

small-keel-clearance formula by Taylor [12].

For verification of the present method the

asym-metric forces were calculated in the proximity of a

circular obstacle. Fig. 3 shows the notations for the circular obstacle problem. This calculation has been

performed by applying the conformal mapping method by Tan[11]. In the present method the circular obstacle is divided into line segments of finite numbers. Effect

of the segment numbers on the numerical solution was

it X XX XX X. X

Fig. 2 Body plan of a ship [9]

and Fig. 2 the body plan. The

Table i _{Principal particulars of a ship}

ui

.-_f

LLIU

11L__1IN

N

IUI1.IID1

Full Scale Model Lpp (m) 321.56 5.024 B (ml 54.56 0.852 d (m) 21.67 0.339 17 (m3) 1.192 312.200 Cb 0.8213 Cp 0.8231 Cm 0.9978

SP

yo

Ship Trajectory

/ Ship

Fig. 3 Notations for circular obstacle problem

0.010 0.005 0.000 0.005 h/d = 1.1 S0/L = 0.5 r0/L = 1.0 Attraction

1

Bow-Out Present O Conformal Maoping Bow-In S1! L

Fig. 4 Variation of calculated results due to panel numbers

examined first for calculating with three different segment numbers as .V = 16. 36 and 72. The water-depth

ratio was chosen to be h / d = 1. 1. the separation Sp ¡ L = 0.5 and the radius of the circular obstacle r0 / L = 1. 0. Fig. 4 shows the variation of calculated results due to panel numbers. The solution for N = 36 and 72 shows good agreement with that by conformal mapping method. However, in the solution for \c = 16,

oscillations occur due to lack of the segment numbers. Thus it is confirmed that by use of the present method

reasonable solution can be obtained when the panel numbers are sufficient.

Next, the asymmetric forces were calculated when a ship moves parallel to a protruded bank (as

illustrated in Figs. 5 and 6). and were compared with Norrbin's experiments[10]. Figs. 5 and 6 show the comparison of the asymmetric forces acting on the ship in transient motion past short and long bank. The

lengths of the banks were equal to 3L and 6L respectively. The calculated results are indicated in relation

to an infinitely long bank steady-state values of the asymmetric forces. The present calculations show that at the ends of the bank the transient asymmetric forces are larger than the steady-state forces. Qualitative tendency of hamp as ®, ®, ® in the force curve and ® in the moment curve, and hollow as ® in the moment curve shows good agreement with the experiments. However, it seems that the fluctuation of the calculated

transient forces at the ends of the bank is larger than the experiments.

Fig. 7 shows the comparison of variation of pressure load distributions in transient motion past short bank with Norrbin's experiment [10]. It is noted that Norrbin measured the distributions of lateral force

acting on a "waterline cylinder" of the 280.000tdw ore/oil carrier [10]. The calculated results are

quantita-tively about three times smaller than the experiments (scaling ratio of the calculated results is different from

that of the experiments in Fig. 7). However, qualitative tendency of the variation of the pressure load

2

2

i

o

- Cal.

I

2

X0/L

X0/L

hId =2.31

Sp/L = 0.145

Lb/L =3.0

Bb/L = 1.0

. 2

\i4

6

8

Fig. 5 Comparison of asymmetric forces acting on a ship in transient motion past a short bank

(above : experiment by Norrbin. below : present calculation)

It

Lb

7

/////////////////////////////

4-J o

/

2CaL

2

X0/L

h/d =2.31

S/L = 0.145

Lb/L=6.0

,I\ Bb/L=1.0

\\

L®A

'

\

I I

®I\'

2

4

8

Fig. 6 Comoarision of symmetric forces acting on a ship in transient motion past a long bank

(above : experiment by Norrbin. below: present calculation)

Experiment by Norrbin

(i)

+LICp (0.15)

Present

Calculation

(3)

(4)

>

(2)-(4)

.t,/,,/t,///I///e/t/'/ ,,(,/J/Jt//,,//JJJfl//,/ li

''t'. 'J , J -J 'er 'Ji 'J''('IJJtt ((e 'eel

ti e,,,JJ,Jt

(6)

(7)

-(8)

(7)

JF,/J/,(JfJ,eJ#/JJ/JJJJ,nI JJJJPFIJ

(8)-/ e' /,J//,///,Ji/t,,/,,,,,JJ//; //J//JJJ/e'ZJJJ(/fJJJFJJJJ,'"JJ?JJJJJ'JJJJ/JJJ/JJJ/JJ

,

r J//JJJJJJJJ/J/ (Je'/J''JJ''J'J//''J/JJJ//J

Fig. 7 Comparison of change of pressure load distributions in transient motion past a short bank

distributions shows good agreement with the experiments. lt may be said that the present method is a useful

mean for understanding the features of the asymmetric forces.

2. 3 Asymmetric Foi-ces in a Channel with Varying Width

Using the present method, the characteristics of the asymmetric forces acting on a ship moving in a channel with varying width were theoretically examined. Calculations were made for three cases.

when left side bank only exists (Cal. L),

when right side bank only exists (Cal. R),

when both left and right side banks exist (Cal. B).

The bank geometry is illustrated in Fig. 8. The nallowest channel width is O.5L (about 2.95 B) in Cal. B. Fig. 8 shows the asymmetric forces acting on a ship moving in a channel with varying width. In Cal. L

the ship experiences attractive force and bow.out moment rejecting her toward the left hank, and this tendency is basically same with the asymmetric forces in case of an infinitely long bank [i]. _{The force and} moment have almost constant values except near the ends of the protruded bank (xo/L = O and 4). In Cal. R the ship experiences attractive force and bowout moment rejecting her toward the right bank. _{lt is shown}

that when the midship is just about at the tip of the right bank (xo/L = 1). the force and moment become

maximum. In Cal. B while the ship approaches to entrance of the channel (xo/L

= Ol). it experiences

attractive force and bow-in moment pulling her toward the right bank. This patten of the asymmetric forces

indicates that the ship has a dangerous tendency to move toward the right bank during the approach. After

the transit when the midship is just about at the tip of the right bank (xo/L = 1), the force quickly reaches a maximum value of a repulsive nature with respect to the right bank. _{Thus, the calculated results of the}

asymmetric forces indicate that the ship must be handled carefully when the ship approachesto the entrance of the channel with varying width.

3.

_{Ship Maneuverability in a Channel with Varying Width}

3. 1

Outline of Srnulation Method

Next, effect of the asymmetric forces on ship maneuvering motion was investigated by a simulation study. A ship maneuvering motion in the proximity of a wedged shaped bank wall has been investigated _by

Kijima and Qing [7]. _{In this paper, therefore, emphasis is placed on maneuvering motion in a channel with} varying width. _{Outline will be first shown of the simulation method where the effect of the asymmetric}

forces has been taken into account. _{Force and moment (X5, Y5, N5) acting on a ship moving near the} irregular bank is expressed as:

Ks =XK+XR±XP+XB,

Ys= Y+YR+Yp+Y8, (24)

N5 = NH ±NR+ N + N3.

Here. subscript H, R, P, and B means ship hull, a rudder, a propeller and component of the asymmetric forces respectively. _{Mutual interaction between the terms with the subscript H, R. and P, and with subscript B is} assumed to be small. _{Then, a mathematical model for the simulation in infinite water region can be used for}

0.15

0.10

0.05

>-ci

0.05

0.10

0.15

₁

L)

C

0O

E

0.04

0.02

0.00

0.02--

-0.04---U.0-1

Ye

N O Yo

h/d = 1.3

y-'

1/

"U

///////I//////////////////////////////////////////////////4

i

2

3

Xo / L

--

Left Bank

(CaL L)

Right Bank (Cal R)

Both Banks (Cal. B)

/''

I I

0

1

2

3

X0/L

7

J

Fig. S Asymmetric forces acting on a ship moving in a channel with varying width

5

6

5 6

where 0 and 0-2 are gain constants, and Uo denotes

initial velocity.

Fig. 10 shows the simulation results of the ship

and steered motions through in the channel. The

gain constants were chosen to be 0 10 and 02 = 2(Cal.1), and C = 3 and 02= 2(Cal.2). The

water-depth ratio was chosen to be h/d = 1.3 and initial

velocity U0 = 1.Oknot. Despite the rudder control is made for course keeping, it approaches to the right bank due to the action of bow-in moment pulling her

toward the right bank around the entrance of the channel (x0/L = 0-2). When the midship is just

around the

tip of the right bank

(x0/L 2), her

direction changes toward the left bank. After that,

n Cal. 1 the ship can move near the center line of the

channel. but in Cal. 2 it approaches to the left bank gradually around .ro/L 3--5. The maximum

rud-o Present Calculation Experiment by Norrbin h/d= 2.31 Spa / L 0.24 S00/LsC 0 1 2 x0/L

Fig. 9 Comparison of ship motion between

present simulation and experiment

model [8] was employed.

The terms with subscript B, X8, Y8 and N8. are assumed to be expressed as follows

X8 1/2p BC/U2 k1C(x0, yo) sinO,

Y8 = 1/2p BC/Li" k1Cy(xo. Yo) cosO, (25)

N8 = l/2p BdLU k2C(x0.yo).

where Cr and C are the non-dimensional asymmetric forces when the ship moves straight with constant

speed in the channel, k1 and k2 the correction factors of Cr and Cv. TheCr and C can be predicted using the present method described above. However, for saving the computational time, the Cr and C.-, were estimated from their tables which are provided using the present method in advance.

3. 2

Ship Maneuvering motion in a Channel with Varying Width

For verification of the present method, a simulation was made of the maneuvering motion for an

unsteered ship in response to the interference with the protruded short bank as illustrated in Fig. 5, and compared with free'running model test result performed by Norrbin [101. Maneuvering forces such as hydrodynamic derivatives, added masses and rudder force were estimated from the model test results with

various water depth for Esso Osaka ship hull form [41. Fig. 9 shows the comparison of ship motion between

present simulation and experiment. The present simulation result shows good agreement with the experi ment for ship trajectory, heading angle O and drifting angle . Thus, it was confirmed that the present

simulation has sufficient accracy from the practical view point.

Next, a simulation of maneuvering motion for a ship moving in a channel with varying width was made.

For steering in the channel the automatic control of a rudder angle J is made according to the formula

40 30 20

20

30

40

1

0 1 2 3 4 5 I I h/d = 1.3

Cal. i: G =10, G2=2

CaL2 : G1=3, G2=2

6

x0/L

Fig. lo Simulation results of ship and steered motions through a channel with varying width der angle in Cal. i becomes larger than 30 (deg). Larger rudder angle than conventional operation may be

necessary for the safety navigation in the channel. Thus, it is shown that in the channel with varying width

course-keeping is difficult due to the effect of the asymmetric forces in any case, and potential hazards of

collision and grounding are considerably large.

4. Concluding Remarks

In this paper, theoretical investingation is made into bank effect on ship maneuverability in a channel

with varying width. First, a method to calculate asymmetric forces such as lateral force and yawing moment

acting on a ship in the proximity of arbitrary shaped bank is introduced. Calculation vas made by use of the present method and comparison was made with experiments. The present calculations show good

agreement with the experiments for qualitative tendency of the asymmetric force and moment curves, and it

is confirmed that the present method is a useful mean for obtaining the characteristics of the asymmetric

forces. Further, using the present method, theoretical examinations were made of the behavior of the

asymmetric forces in the channel with varying width. The calculated results of the asymmetric forces indicate that the ship must be handled carefully when the ship approaches to the entrance of the channel.

Next. simulation study was made in order to investigate the bank effect on ship manevering motion in

the channel. As a result, it is shown that conrse-keeping is difficult due to the effect of the asymmetric forces, and potential hazards of collision and grounding are considerably large in the channel with varying

width.

For improvement of the prediction accuracy of the asymmetric forces, further study on the effect of the

ship's thickness may be needed, since the thickness effect is not exactly taken into consideration in the present method, which is based on two-dimensional thin wing theory. Such improvement will be left to the future wo r k

Acknowledgments

The author would like to express his sincere gratitude to Prof. K. Kijima of Kvushu University for his

continuous encouragement and guidance. Thanks are also due to Mr, H. Kasai. Senior Project Manager of Ship and Ocean Engineering Laboratory of Nagasaki Research and Development Center, MHI for his

valuable discussion. Thanks are also extended to all the members of the Nagasaki Experimental Tank for

their cooperation.

References

Beck. R. F.: Forces and Moment on a Ship Moving in a Shallow Channel, Journal of Ship Research,

Vol. 21, No. 2 (1977), pp. 107-119.

Davis, A. M. J.: Hydrodynamic Effects of Fixed Obstacles on Ship in Shallow Water. Journal of Ship

Research. Vol. 30, No. 2 (1986), pp. 94-102.

Hess, J. H. and Smith. A. M. O.: Calculation of Potential Flow about Arbitrary Bodies. Progress in

Aeronautical Science. Vol.8. Pergamon Press 1966).

Hirano, M., Yumuro, A., Nonaka, K. and Kobayashi, H.: Bulletin of the Society of Naval Architects

of Japan. Vol. 668 (1985), pp. 45-57.

Hsiung, C. C. and Gui, Q.: Computing Interaction Forces and Moments on a Ship in Restricted

Waterway, I. 5. P. , Vol. 35, No. 403 (1988), pp. 219-254.

James, R. M.: On the Remarkable Accuracy of the Vortex Lattice Method. Computer Methods in

Applied Mechanics and Engineering, Vol. 1, No. 1 (1972), pp. 59-79.

Kijima. K. and Qing, H.: Maneuvering Motion of a Ship in the Proximity of Bank Wall, Journal of the Society of Naval Architects of Japan. Vol. 162 (1987), pp. 125-132.

Kobayashi. E. and Asai, S.: A Simulation Study on Ship Maneuverability at Low Speeds, Proc. RINA mt. Conf. on Ship Maneuverability-Prediction and Achievement, London (1987), Paper No. 10.

Norrbin, N. H.: Bank Effects on a Ship Moving Through a Short Dredged Channel. Proc. 10th Symposium on Naval Hydrodynamics, Cambridge. Mass. (1974), pp. 52-63.

[io] Norrbin, N. H.: A Method for the Prediction of the Maneuvering Lane of a Ship in a Channel of

Varying Width, Symposium on Aspects of Navigability of Constraint Waterways, including Harbor

Entrances. Delft (1978). pp. 1-16.

[ii] Tan, W. T.: Unsteady l-Ivdrodynamic Interaction of Ships in the Proximity of Fixed Obstacles. MIT

Report 79-1 (1979).

Taylor, P. J.: The Blockage Coefficient for Flow about an Arbitrary Body Immersed in a Channel. Journal of Ship Research, Vol. 17, No. 2 (1973), pp. 97-105.

where

points respectively.

forE > x,

(A-3)

(1966). pp. 81-95.

Yasukawa. H.: A Theoretical Study of HydrodynamicDerivatives on Ship Maneuvering in Restricted Water, Journal of the Society of Naval Architects of Japan, Vol. 163 (1988), pp. 119-129.

Yeung, R. W.: On the Interactions of Slender Ships in Shallow Water. Journal of Fluid Mechanics. Vol.

85. Part 1 (1978). pp. 143-159.

Yeung, R. W. and Tan, W. T.: Hydrodynamic Interactions of Ships with Fixed Obstacles, Journal of

Ship Research. Vol. 24. No. 1 (1980), pp. 50-59.

Appendix Numerical Procedure

A ship is first divided into M segments of equal length.zix. and the ship's moving distance for one time

step is defined as i.. Bank surface is also divided into Nd segments. Within each of these segments the

vortex and source strengths are assumed to be constant.

Using these discretedistribution assumptions, eqs. (12) and (16) can be transformed into a system of linear

simultaneous equations as:

M

-

k_{ß(k) y.ik)±}

= 0, (A-1)

M k

Er.ik 7,1(k) + Fmj y(k) +

IImn,

= G,n. (A-2)

J=t

i=1.2

,M

m=1,2.

,N

xi -

forE < x,

=

f

aGc

_Q)dc,

_(A-S) = 2k 0.,C8(Pcm, Qc)z1x}. (A-9)

Here, the vortex strength within the jth element of the ship at time tk is denoted by

y1,

j = 1, 2. , M,

and the vortex strength in the wake element i = 1, 2. . k, where

j = k

corresponds to the wake

element nearest the trailing edge of the ship. Further, the source strength within the n'th segment of the bank wall is denoted by n = 1. 2, , N. The - and - mean the value of the wake and the source

The additional equations (13) and (14) can be written in discretized form as follows:

ik) ik-i) _for

_i=1,2.

ki

k = 2, 3,

M k

E7k.Jx±E

k)J _{= 0.}

tjsing eq. (A-10), eqs. (A-1), (A-2) and (A-li) are rewritten as follows:

i! Vc k-i

E A1 yik+Bik),c

±

E D,(pk> = -

E B1I (k_l) n=I J=I if Vc k-t E Emj' 7(k) + F,kk '(kt1+ E [2r,,,(k)1(k) 0,,,tkt

_{- E}

(k-l1 flL M k-t

Eytixi 7k1X =

-1=1,2,

.M m = 1, 2, . _Vc k = 2, 3.

From eqs. (A-12)--(A-14) the matrix of linear equations for ;' (M+1 numbers) and p, (N numbers) can be

constructed. By solving the matrix at each time step, the asymmetric forces on a ship can be calculated. In

the present calculations, the vortex and control points on the ship hull were chosen to be 1/4 and 3/4 of the

segment length respectively.

(A-10>

(A-11)