A simulation study on ship manoeuvrability at low speeds

1. Introduction

With increased energy demands in recent decades, many crude oil carriers and liquefied natural gas carriers - are in service in the world. Usually these ships enter destination ports through confined and shallòw water regions. To ensure safe operation in such waters Under external forces i. e., wind and current and restricted water effects, the evãluatiòn of ship manoeuvrability in harbours has been increasingly demanded at the design stage. of a ship.

Eváluation of ship manoetivrability so far has been made mainly by çarrying out model experiments on fundamental ship manoeuvring modes such as turning -and zigzag motioit Simülation methods of ship manoeuvring have been examined by both computational studies on mathematical models of ship motion and model experiments such as captive model tests and free running model tests to provide data for improvement of accuracy of predictions. Various kinds of mathematical

models have been proposed for simulation of ship manoeuvring motion, such as the so-called MMG-model'° (MMG; Manoeuvring Model Group of the Japan Towing Tank Committee) for simulation in the ordinary speed region, and the hydrodynamic model and response model for stopping ¡notion simulation141. Further, efforts are being made to

simulate various ship manoeuvring motions in harbour such as lateral drift, short turning,, backing and so on, where lateral drift speed is higher than the ahead speed and external forces are much larger than hydrodynamic forces induced by the propeller and ruddert5".

In the present paper, a mathematical model of ship

manoeuvring motion is proposed, aiming at the estimation of ship manoeuvring with less limitation on ship speed and

motion. This is characterized by a mathematical expressiOn of hydrodynamic forces in low speed regions. An effort has been made on continuous matching of equations for low speed and ordinary speed. Various kinds of ship manoeuvring such as turning and zigzag manoeùvring' at ordinary speed, stopping, backing, short turning and lateral drift at low speed can be evaluated with acceptable accuracy from the viewpoint of

practical use.

Firstly, explanations of the mathematical expressiöns of ship manoeuvring motion and hydrodynamic forces in conditions of advancing at ordinary and low speeds, and backing will be presented. In motion prediction, the forces generated by the propeller, rudder and side thruster are considered. Secondly, results of model experiments for obtaining hydrodynamic force coefficients from the mathematical models are outlined. Free running test results are used to verify the mathematical model.

Nagasaki Research and Development Center, Technical Headquarters

r, r Ts1 U ,' T., T2 V' JTN2\ Path uf CG. TN Longitudinal force Lateral force N: Yawing moment about G

A Simulatiön Study on Ship Manoeuvrability at Low Speeds

Elichi Kobayashi* A mathematical model for estimationofship manoeuvring pefòrmance was introduced to cope with increasing demands for design stage evaluation ofsafety in ship operation while in harbour The present mathematical model enables continuous simulations of ship manoeuvre from an ahead motion through the range between null ahead speed to astern motion Good agreements ofsimulations and model tests results were obtained in ordzna?y manoeuvring motion stopping motion short turning and lateral shifting This

mathematical model is expected to be applicable to various problems not only in relatively high speed motion with a small drift angle but also in low speed motions with large drift angle, in turning with null-ahead-speed and in backing motion.

(1)

o

O x, y, z : Space fixed coordinate system

('o : vertically downwards)

Gx,y,u Body fixed coordinate system

(z: vertically downwards through G

G : Center of gravity

Fig.1 Coordinate system

Finally, some examples of simulations of manoeuvring motion under external forces are shown to demonstrate the applicabil-ity of the présent Ñethod in various kinds of manoeuvring modes.

2. Simulation model fór manoeuvring motion

2. 1 Bäsic equations of motion

Referring to the coordinate system fixed to the center of gravity of a ship as shown in F'ig.1, the basic differential equatioñs of motion aré expressed as follows

(mH-m0)û(m±my)v r

= X+ Xp + XR+ X +XE

(m+ng)i)+(m+mx)u.r

= YH+ Y,+ YR + Ys + YE

=NH+Np+NR+Ns+NE

where, X, Y and N are the hydrodynamic forces and moments in the x, y and yaw directions respectively, 'and suffixes H, P, R, S and E denote components of the hull, propeller, rudder, side thrúster and external force, respectively. The terms

related to P, R and S include interaction forces due to the hull. The structure of the present mathematical model is sununa-rized in Fig.2.

2.2 Ilydrodynamic forces acting on a ship hull

Hydrodynamic forces acting on a ship hull are described by the f Ilowing four mathematical expressions classifiéd with

Equation of rhotiols ÑuIÑ) XH,YH,Nff

F-Propeller (P):Xp,Yp,Np Xp= (1 t) T-Yp=T(Yp,+YpR,) Ñp (Np, +Npg,)

Side thruster (S):X. Ys, Ns

t Model I

-j

Ordinary speed model J

J

Fnj Fn

JModel

II (Model I + Model Ill) Fe,rnn2 Fn Femini

-J Low speed model

-J Propeller thrust:T

-J

Lateral torce acting un ropTeller:Yp

Propeller reversing torce : YPR, NPR

Side thruster torce: Is. s. Ñu xs= (1+ axa,) Xs, s = Ts cas Os + TN sin Os

Ys=(l+oys,)Ys Ys=Tssin 8S+TNCOSOS

NS X(I+0Ns,) Ns, Ns YSXSXSIIS -J External torce (E(:XE,YE,NE

Fig.2 Structure of simulation model Speed ranges. These models are composed of two types of hydrodynamic models and' they are expressed by third and second order polynomials. The former is used in the ordinary speed región, and the latter can be used with low speeds and backing motions. In the intermediate range between the two regions, an averaged model (explained bélow) is adopted to avoid discontinuity in values of hydrodynamic forces.

Model I: Ordinary advance speed model for a speed range larger than that cm-responding to the specified Froude number Fflmjni

Model II: Averaged model

of the two mathematical

models over the speed range between Fflmini and

that corresponding to the specified Froude number Fflmm2

Model m: Low advance speed model for the speed range between zero and Fnmi,.,o

Model IV: Astern model for backing conditions 2.2.1 Ordinary advance speed model (Fn Fnmini) For the ordinary speed region, hydrodynamic forces and moments acting on a hull are expressed by the following

formulae whose

applicability has been confirmed empiricallyX2t. The longitudinal hydrodynamic force acting on a hUll XÑ is expressed as follows:

X,,=XHO + X1

where, X110 is resistance in a straight course, and expressed in terms of y and r as follows:

XHO =

R

X115=

±pLdU2Xn

V' '2_LV' ' '_.i.V' '2

fiHI.(1uv V

.fvrVr

rr r

+X'000v'4 +Xr'4

where, X,, X,,.' X,'-,' and X,-,,. are the hydrodynamic derivatives related to longitudinal motion.

When including the first and the third order terms the swaying force and moment are expressed as follows:

Model III-1

OFe<Fnnjn2 andß<ßo

Model 111-2

OFn<Fnmin2 and ßßo

Model IV

Fe<O

Rudder normal force: FN

Thruster thrust:

Thruster normal force: TN

Interaction coefficients:

0.10

0.05

- 0.05

Fig.3 Results of large drift angle tests for a VLCC model 0.05 0.1 O O YH=±pLdU2Y NH=+pL2dU2 Nk Y'ih'= Yv'+Y'rr'+Y'vuvv'3+Y,vr.v'2r + Y'urrv'r'2+ Y'rrr r'3 N =

N'0 v'+Nr'+N'050 v'3+N,vrv'2r'

where, Y,',' Y;.'

Yvr' Y,',rr' Y' N,' N-' N00' N,',r'

N and N_,, are the hydrodynamic derivatives related to yawing and swaying motions.

2.2.2 Low .ádvance speed model (O Fn < Fflnoin2)

In the low speed region the drift angle generally becomes relatively large. In the first step, the so-called current force model was considered, in which the tesistarice coefficients are non-dimensionalized by thesquare of the ship's velocity. Using some trial simulations, this model can be expanded to include

hydrodynamic forces and moments acting on a hull as

described by three components such as the terms depending only on y or r and coupling terms of y and r as follows:

X,,, =XHO+XH1 XHO=R

YN =YH-v+YH-r+YÑ-vr

NR =NHv+NHr+NJvr

Hydrodynamic forces and moments X11 ,, YH,' NH which are due to the lateral velocity y are expressed in both cases; one is a case where the drift anglefi is less than the specified angle fib, and the other case is where fi is larger than ßj. For

ß, 45 degrees wäs adopted in the present study to attain

smoother matching of the two regions. XH_ y' YR_V and Nj_y

are expressed as follows:

(a) ß<ß

XHV=±pLd(X'vo-v2) YHU =+pLd(YUy+Y'vyvvI) (8) NHV=+pL2d(N'yUy+N',)yvIvI) 0.4 0.8 N'PR O A Measured O for Simulation 0.2

Fig.4 Results of propeller reversing test

for a VLCC model 0.8 J 0.05--0.1 } (4) } (5) } (6) J' (7) O A Measured - tor Simulation 0.10 XHI is

(b)

j9ßo

X

=+c.LdUfxii

Y =±PLdU2fYH (9)

NH-V =+PL2dU2fNH

where, coefficients ¡X,," fYH and ¡NR are expressed as

functions of drift angle ß (sin p = - v/U). Examples of ¡VN and INH are shown in Fig.3. Hydrodynamic forces and

moments due to r and coupling of y and r are expressed as follows:

XH_r =+pL3d(X'rrr2)

YH_r

=-4-pL2d(YrUr±LYrrrD

(io)

NH_r =±pL3d(NUr+LN.rHrI)

XH_VT=±pL2d(X'vrvr)

YR-VT =+pL2d(Y'vrvr) (11)

NH_yr =+ pL3d(N' vr)

In fòrmulae (8), (10) and (11) X'' X'vr' X',,,' Y'yy' v'vr' Y'rr'

N','

N'vr' and N',,, are second order hydrodynamic deriva-tives.

2.2.3 Averaged model (Fnmm2 Fn < Fnmmi)

In the Speed range between Fflmjn2 and Fflmjni, both the ordinary advance speed model and the low speed model are matched in the following manner,

XH XH

'X,,-Y11

=gi

YR +g2 YR (12)

NH averaged N11 ordinary NH low

where, Fn Fflmln2 g1 = Fnmrni Fflmjn2 FnminvFn g2 - Fflmjni - Fflmln2

In this paper, Fnmini=O.Ol and FflmjnO.005 were adopted to attain smoother matching of the two speed regions.

2.2.4 Astern model (Fn <O)

A similar type model to the one for low advance speed is adopted also for the astern motion. Namely, hydrodynamic fòrces and moments acting on the hull are expressed as follows: X,,- =XHO+XHL X110=R XHI XHV+XHr+XHVr YH =YH.V+YHr+YH-vr NH = NHV+NHr+NH-Vr

V _1

_H-VP

1.1.1121_./XH -v

_{H-V 2P}

__ 1 ¡ .1.1121_JYH NH-V = +PL2dU2fNH

X1_r =+pL3d(X"rr2)

YHr

=pL2d(Y'Ur+LY'rr.rHl)

NH-r

±pLd(N'»Ur+LN"rrrIH)

XH_VT=+pL2d(X' vr) YHVT _{=±pL2d(Y"VTvr)} NH_Vt=+pL3d(N" vr)

where, X' Xr' X' Y'.' Y' Y',,.' N',.' N',, and

denote hydrodynamic derivatives during astern motion. 2.3 Hydrodynamic forces induced by propeller

Hydrodynamic forces and moment induced by propeller operation are expressed as follows:

x=(it,)7

Y E(YP+YpR) (19)

N =(NpI+NPRI)

where, p and Ñ denote the lateral force and moment acting ('3)

on the propeller, and YpR and NPR are lateral force and

moment acting on the hull due to reversal of the propeller. denote summation with respect to number of propellers. Nondimensionalized YPR and are obtained empirically as

functions of propeller advance constant J as shown in Fig.4 -Propeller thrust T is expressed as follows:

T=PDF4nP2KT (20)

and KT is given by polynomial expression on the basis of the open water test and captive model test results. Changes of

inflow velocity toward the propeller due to yawing and

swaying motions are expressed as follows:

Up=u{(1 - w) + (cV V+ c r'+ C3

+c4vr+c5r2)}

(21)

where, c1, c2, c3, c4 and c, are empirical coefficients. 2.4 Hydrodynamic forces due to rudder

Hydrodynamic forces and moments due to the rudder are expressed as follows:

XR= (1 + axj) i

YR

=(i+ay,)?R

(22)

NR

=Xi +

aN)ÑR

where, XR, í,- and ÑR express hydrodynamic forces caused by

the rudder itself, and ax' ay and a,,' are coefficients which express the interactiòn betweeñ the hull and rudder. XR YR and NR are expressed in terms of rudder normal force FN as follows: XR=FN sin 8

=F,,-cosô

ÑR= YRXRXRYR and FN is expressed by FN±pARUR2fRSifl(8e) (24)

As an expression of IR' the following empirical formula was adopted in terms of the rudder aspect rätio:

IR=6.13À/(A+2.25) (25)

UR and ôe are expressed by:

UR2 =UR2 + VR2

8R =tan(vR/uR)

(26)

8 =88R

where, VR consists of two components. One component depends on the propeller slip ratios and the -other depends on the variation of stern flow direction y' + 1',,- r' as follows:

VR=

kpsUR+Uf(v'+1rer')

(27)

where, k and l'R are the empirical coefficients and f is a

function of the rudder inflow direction. The longitudinal velocity component u,, is expressed by;

UR

uI1+81cK1/(J2)

(28)

where, L. and K denote coefficients determined by experiments on propeller slipstream.

2.5 Hydrodynamic forces due to side thruster

Hydrodynamic forces due to side thruster are expressed as follows:

Xs(1+axs1) s'Si,

Ys

=(1+ays) Vj,

(29) Ns=E(1+aNgI) Ñ1, cos 8s+TNsinôs =TSsin8S+TNcos8s (30)

ÑsisxsJ(sys

where, 2, and Ñ express hydrodynamic forces generated

by the side thruster itself, and

axs' ays and aNs are coefficients which express the interaction between the hull and

3

Table 1. Featured of ship models

the thruster. As defined in Fig.1, i's' TN and s denote the thrust of the side thruster, lateral force normal to i's and azimuth angle ab6ut the vertical aus, respectively; X' Ys' X

and y coordinates of the center of the impeller. In the case of a tunnel type thruster, 6's is equal to 90 degrees and TN is assumed to be zero.

3 Model test to obtain hydrodynamic coefficients 3.1 Tested model

Hyclrodynamjc coefficients and other empirical factors were deterrriined to have consistency in computation for various kinds of rnanoeuvring motion. Extensive model experiments necessary for the above have been conducted in the Nagasaki Experimental Tank of MHI, such as circular motion tests,

oblique towing tests, rudder angle tests, large drift angle tests, propeller reversing tests, planar motion mechanism tests and so on. The following ship models were tested to obtain hydrodynamic coefficients applicable to various kinds of

manoeuvring motions. Single screw VLCC Single screw LNGC

Twin screw shallow draught module carrier Research vessel with azimuth thrusters

Principal particulars of the ship models are shown in Table

1.

3.2 Hydrodynarnic characteristics

Using the captive model test, hydrodynamic coefficients were obtained as follows:

my : pure swaying test pure yawing test

ax'

ay' a

: rudder angle test

Yv' N' Y,.' N' Yvvv etc. : oblique towing test and circular motion

test-fxa' fyji' INN: large drift angle test

£,x : rudder angle test at various propeller loadings c,-4 : oblique towing test and circular motion test

YpR(J), NPR(J) : propeller reversing test at various

propeller loadings axs' ays' aNs : side thruster test.

Sóme examples of the test results for the VLCC model are shown in Figs.3, 4 and 5.

4. VerificatiOn of the mathematical model

Verification of the mathematical model of ship manoeuvring motion was made by comparison of typical computed ship motion and free running model test results. In this study, the applicability of each mathematical model mentioned previous-ly was examined as follows.

(1) Ordinary manoeuvring motion

To verify the validity of the mathematical model in the ordinary manoeuvring motion with relatively large advance

0.2- VLCC

O D

+ Iv'0, 0.1,0.2,0.3,0.40.5 0.5 0.1 - o 0.1

r

7,

Fig.5 Results of circular motion test for a VLCC

model

speed, numerical simulations for turning and zigzag manoeu-vre test were carried out for the above mentioned four ships. The computed results of steady turning characteristics in relation to the non-dimensional rate of tùrn r' and speed drop ratio U/U0 versus rudder angle are compared with the results of free running model tests as shown in Fig.6. It can be clearly seen from these figures that the simulation results are in good agreement with the model test results. From these results it may be said that the present

mathemat-ical model can be applied to the manoeuvring motioñ at

ordinary advance speeds. Stopping and backing motion

Fig.7 shows-the simulatibñ results of stopping and backing. motion of the VLCC and the LNG carrier in comparison with the model test results. The simulations were continuously carried out from the ahead motion, through the range of null-ahead-speed, to astern motion. Good agreement can be seen in the ship's trajectories, lateral deviätion and backing directioñ. Thus, the present mathematical model has also been validated in the conditions of stopping manoeuvres.

Low speed motion by side thruster

The computed short turning motion using a bow thruster are compared with the model test results as shown in Fig.8. Good agreement is seen in the trajectories about the center

VLCC LNGC Module carrier Research vessel Scale 1/70.7 1/53.2 1125.3 1/26.0 Lpp(Ship) (m) 325 266 152 143 (Model) (m) 4.60 5.00 6.00 5.50 Lpp/B 6.13 6.18 - 4.00 6.09 Bld - 2.44 4.03 8.42 3.57 -0.4 o 0.4 0.8 - 0.8 -0.4 o 0.4 0.8 + + 4.--*--*,

U/u0 40-30-20-10

¡o io

20 8 (deg) U/U0 VLCC 1.0 - 1.0 Module Carrier 1.0 - Computed O A Measured 40 30 20 lO 0 10 20 30 40 40 30 20 10 8 (deg) LNGC U/Uo LNGC 1.0 40-40-30-20-10 r.° 10 20 30 40

/

8 (deg) 1.0 - Computed 1.0 O A Measured ,, 1.0 Research Vessel 1.0 - Computed O A Measured 0 2 4 6 xo/Lm,

Fig.7 Stopping and backing loci computed and

measured for VLCC and LNGC models

-

Computed

O A Measured Fig.6 Turning ability computed and measured for models

VLCC

Fig.8 Comparison of loci computed and measured for a LNGC model in short turning by use of bow thruster

Research Vessel

LNGC

Fig.9 Comparison of loci computed and measured for a ship model in various harbour ma-noeuvring by use of azimuth thrusters

5 Computed

II

iIí1tiI\

Measured ns35rps 4 . £& L00 Computed

Ai

.=-35rps Measured AIiTII Computed

'I

p0

II'

Computed Measured

-I11u1i

ivi

r

t,

-_

Measured

-,

2_IIÍÍl,U---

8 10 12

Module Carrier

L1,

Fig.1O Computed locus for a ship model in short turning by use of twin pro-pellers and bow thruster

of gravity. According to the results of simulation and measurement, it was also confirmed that the proposed mathematical model coüld be applied to low speed motion using the side thruster.

Low speed motiön by azimuth thruster

The simulations of low speed motion were carried out for a research vessel with azimuth thrusters at thebow and stern as shown in Fig.9. The top figure shows the trajectory of simulated lateral shift motion withazimuth angle of bow and stern thruster at 90 degrees starboard in comparison with the model test. By use of bow and stern thrusters with the

azimuth angle of the bow thruster at 90 degrees port and the stem thruster at 90 degrees starboard, short turning can be achieved in which both the lateral and longitudinal velocity nearly equal zero. The middle figure shows the trajectories

computed and measured in such a short turning test.

Furthermore, the bottQm figure shows the turning motions with an azimuth angle of the bow thruster at 45 degrees port and the stern thruster at 45 degrees starboard. Fairly good agreement is seen between computed motion and model test

results. From these results it was found that the present mathematical model can also be applied to such conditions with larger lateral velocity in comparison with longitudinal velocity and a turning motion without advance speed.

ShOrt turning by twin propellers and side thruster Fig.1O shows the computed locus in short turning by use of twin propellers and a bow thruster for the module carrier. In this simulation, both rudders are set to zero helm position,

arid a port side propeller is running ahead while the

starboard side propeller is running astern. The lateral force is generated by a bow thruster. Various kinds of ship manoeuvring operations are possible in the case of wide beam ship with twin rudders, twin propellers and a thruster. The preserit mathematical model enables computation of mañoeuvring motion in such complicated operations.

Oblique running by side thruster

Fig.!! shows the computed loci of the LNG carrier in parallel shift by use of a propeller, a rudder and a bow

thruster. To accomplish this parallel shift simulation effectively, an automatic side thruster control algorithm is added to the present mathematical model. The simulation results of oblique running are shown in the upper part of Fig.

11.

In this simulation, the rudder angle is set to 35 degrees port and the propeller revolutions are kept constant at the upper or the lower limit for the prescribed period, and the side thruster revolutions are controlled automatically to keep the yaw angle

30 20 10 . _.-. Wind VA0-4.Om/se Current Lt v0 = 0.2 ko O Current 10 20o LNGC VA04.Om/sec V,0= 0.2 kn 20 30 LNGC Vs0 2.0 ko Without bow thruster Operation With bow thruster operatión

o-- Designated course - Simulation

50 60

7080

40

Fig.12 Results of automatic harbour manoeuvring

simulation

zero. The lower part of Fig.11 shows lateral shifting with a rudder angle 35 degrees port and mean propeller revolutions are set to -0.22 rps arid the side thruster is controlled in the same manner as the above These methods of simúlation fOr

parallel shift manoeuvres can be adopted to satisfy the

requirements of the various kirids of evaluation of berthing manoeuvres.

As mentioned in this section, the proposed mathematical model has been proved to be applicable to various modes of ship manoeuvring motion over a wide speed range including astern motion.

5. Applications

In the foregoing sections the wide applicability of this

mathematical model for estimating manoeuvring was demon-strated. Of the applications of this model, the evaluation of manoeuvrability of ships can be carried out for safèty opera-tion under external forces such as wind, current, shallow water :effects etc. In this section, some examples of ship manoeuvring

simulations based on the present mathematical model are

shown. including the motion control algorithm.

Fig.12 shows the results of an automatic harbour manoeuvr-ing simulation in which three cases of simulation condition are shown. The first is under the conditiúns of no wind and no current with 2 knots of advance speed without bow thruster operation. Thesecorid is under steady wind and current forces without bow thruster operation, and the third is under steady wind and current forces with bow thruster operation. Such kinds of simulation are useful for the investigation of

auto-jj

(I pF np,nOrps flpm 0.22rps LT4, np Figli Computed bd in lateral shift by Use of propel-ler

and bow

thruster

Research Vessel

Y /L

Fig.13 Computed locus of a reseach vessel in

course keeping under wind force

matic harbour manoeuvrings under external forces.

Fig.13 shows the computed locus of a research vessel on a course maintaining 1 knot advance speed by use of twin propellers and bow and stern thrusters under steady wind

conditions. In this simulation an automatic rudder and thruster control algorithm is adopted and control gains are obtained by

the optimum regulator theory.

As shown in Figs.12 and 13, the present model is expected to be applicable to various problems such as the evaluation of side thruster performance in low Speed conditions under external forces. Furthermore, it will be used to design a ship control system such as for course keeping or dynamic position-ing in oceans or in harbours.

6. Concluding remarks

In the presentpaper, a mathematical model for manoeuvring motion was introduced to cope with increasing demands for design stage evaluation of safety in ship operation in harbouis. The conclusions obtained in this study are as follows:

The present mathematical model enables continuous

simulations of ship manoeuvres from ahead motion, through the range of null-ahead-speed, to astern motion. In ordinary motion such as turning and zigzag manoeuvres, the simula-tion results showed good agreement with model test results. In the stopping manoeuvres, good agreement was observed between the simulation and the model test. Furthermore, good agreements were also found in short turning and lateral shift.

The present model enables manoeuvring simulations with twin ruddeìs; twin propellers and bow and stern thrusters.

The present model is expected to be applicable to various kinds of problems not only in relatively high speed motion with a small drift angle but in low speed motion with a.large drift angle, in turning under null-ahead-speed and in backing motions.

References

Kose K., On a New Mathematical Model of Manoeuvring Motions of a Ship and its Applications, International

Ship-building Progress Vol.29 No.336 (August 1982)

Ogawa A., and Kasai H., On the Mathematicâl Model of Manoeuvring Motion of Ships, International Shipbuilding

Progress Vol.25 No.292 (December 1978)

Fûjino M., and Kirita A., On the Manoeuvrability of Ships

while Stopping by Adverse Rotation of Propeller (2nd reptrt),

Journal of the Kansai Society of Naval Architects of Japan

No.169 (1978)

Yoshimura y., and Nomoto K., Modeling of Mànoeuvring Behaviour with a Propeller Idling, Boosting and Reversing, Journal of the Society of Naval Architects of Japan Vol.144

(1978)

Kobayashi E., and Asai S., A Study on Mathematicàl Model

for Manoeuvring Motion at Low Speed, Journal of the Kansái Society of Naval Architects of Japan No.193 (1984)

Kose K., Hinata H., Hashizume Y. and Futagawa E., On a

Mathematical Model of Manoeuvring Motions of Ships in Low

Speeds, Journal of the Society of Naval Architects of Japan

Vol.155 (1984)

Kose K., On a New Mathematical Model of Manoeuvririg MotiOns of Ships, Proceedings of the Third International

Conference on Marine SimulatiOn, MARSIM 84 (1984)

Fujii H., and Tsuda T., Experimental Researches on Rudder Petíormance (2nd report), Journal of the Society of Naval

Architects of Japan Vol.110 (1961) Nomencláture

AR : effective rudder area

B : breadth of ship

d : draught of ship

propeller diameter Fn : Froude number

FN : rudder normal force

fR : gradient of rudder normal force

g : acceleration of gravity

Izz : mass moment of inertia

J

: propeller advance constant : added mass moment of inertia

K : thrust coefficient of propeller

L : length bétween perpendiculars

m mass ofaship

added mass in x-direction m-y : added mass in y-direction

propeller revolutions

ns : impeller revolutions of side thruster

P : propeller pitch

r

: rate of turn

r'

: non-dimensional rate of turn (= r/(U/L)J

yaw acceleration T : propeller thrust

t : thrust deduction factor

Ts : thrust of side thruster

lateral force normal to T5 U ship velocity (=v/u2+v2)

u : longitudinal component of ship velocity

û : longitudinal acceleration in x-direction Up inflow velocity to propeller

UR : x-component of UR

y : lateral component of ship velocity

VR y-component of. UR 7

iL

V5 ihn

;

4

₄

Wind VA=lOm/sec

f',

I

1

-2 0 2 8 6

nondimensional lateral component of ship velocity w : wake fractioi

(v/U)

ß

: drift angle C

sin(v'))

i' swaying acceleration in y-direction rudder angle

x coordinate of rudder position : effective rudder angle

xs x coordinate of thruster position : azimuth angle about vertical axis of side thruster

YR : y coorinate of rudder position : aspect ratio of rudder