Vertical motions of ships with different form forebody

August 1971.

LABORATORIUM VOOR

SCHEEPSBOUWKUNDE

TECHNISCHE HOGESCHOOL DELFT

VERTICAL MOTIONS OF SHIPS WITH DIFFERENT FORM OF FOREBODY

by

N. Yourkov.

Report No. 316

RVINFIONGEZNICUS OF SHIPS MTH DIFFRRINT FORM OF FOHNHOW,

by N. Yeackav:*

aiMED.

Tht, influence of the vertical priematic ooeffioient of the forebody

an

the ship motions in waves was investigated. Ship aotions and hydrodynacio

ooefficients uere calculated for the ship models uith three values of

block ocegfioient. Inside each model faiky the vertical priacatio

ooefficient had three valas, which leaded to the change of forshody

cross-motions from U-shaped to V-shaped form. The modele with V-ahapsd

fore-bode'r appeared to have advantage in heave in all loaves, while pitch ma

reduced in litAg

_{aves and inoremeed in short 'waves. This blaS causad by tha}

difference in diatribution of sectional added pass and damping along

ship's length.

Angineer Leningrad Shipbuilding University,

_{visitor of the Shipbuilding}

Laboratory, Delft.

I. Introduotion.

The influence of principal dimeneions and forebody section shape on ship behaviour in waves was inveetigated by several authors.

In

1955

Lwis (1) publiohed the reaulte of hie xperiments with two modela with a block ooeffioient CB . 0.60, both models had the same afterbody but in one case the forebody was in agreement with the Series Sixty while bathe

other ease it hed extreme V-shaped eeotiono in the forebody.

The

odels had

the Dame main dimensions and longitudinal radius of gyration. Lewis dhowed

that with regard to Ship motions a V-bow wee to be preferred except in very

abort waves. The resistance increase in the waves of the model with

the

U.ohaped Elections was in general lower exoept in yams of about 1.25 timee

the thip'e length,

Oehi (2) made experimente with two model: having different bow forms.

Paying main attention to bending memento and @lamming, ()chi found in general

a preferenoe for V-ehaped belle, although thie tendency msy reverse

_depending

on wave length and ship speed.

Taniguehi

(5)

oompared three modelo of

fishing

boats with bow sections, going-free U.ehapo to rather extreme V-shape. The models were compared on the

basio of the equal natural pitching frequency at _{zero speed; such a comparison}

ie not quite oorreot and so the final resulte aro to be discussed.

In 1961 the resulte of the experimente carried out by Swaan en Voesers wore published (4). In

thoee

experiments the main variation tested vas the eeotion Shape in the forebody. Including the Serie: Sixty, four models were tooted. They had not only the same principal dimensiono, but also the _{ourvec of}

¡motional arees were exactly the sama. The sections of the bow, however,

varied from extreme U-ehape to extreme V-ohape. The Series Sixty _{model was}

coneidered to preeent an extreme ij-ehape forebody. All the four modela had the same afterbody. So it can be etated, that in that experiment the effect of the

bow election ehape on the behaviour in waves was isolated as muoh as possible

from other factors, whioh had an important relation to _{such a behaviour. The} models were tooted at full draught

only, with

_{the exception of the two extreme}

models, whioh were aleo tested in a ballaot oondition.

It me stated from the experiments that V-shaped _{sections in the forebody}

were faveurable with regard to motions. With reopect to wetness, _{slamming and}

awed less unfavourable effeote could be expected. The internal moments were larger with V-shaped sections than with U-shaped _sections.

-1-The favourable effeot on the motions was stated to be espeoially important

for small ships, because at the same time the bending moments _{were usually}

not important for them; a V-shape was recommended for such vessels. For long

ships, where no advantage was to be expected from using V-shaped _sections, a moderate U-form was stated to be preferable.

Bengtsson

(5)

_{desoribed the experiments with four models with CB =}

_0.675

and three models with CB . 0.794 in regular head waves. The afterbodies were identical within the model families but the forebody

sections

_{were given}

different degrees of U- and V-form. A motion _{comparison between the individuals}

in the model familiee in an idealised head irregular _{sea showed minimum}

ampli-tudes for the models with V-shaped forebody _{sections. This was confirmed}

theoretioally in one oase.

Regarding resistance and propulsion the modele with _{V-shaped forebody}

seotions showed advantages in both model families

_{in waves longer}

_{than the}

model. For shorter waves the U-shaped sections _{were markedly superior in the}

CB .

0.794

_{family whilst the trend for the CB .}

_0.675

models was not olear.

These oonclusions were drawn for regular _{waves with heights of about 0.025}

to 0.030 L., For lower waves the resulte were expeoted to approach these in still water where the U formed forebody _{sections had been foundto be}

superior. In addition to the tests at load draught, afterwards three CB

_{m 0.675}

modela had been tested at a ballast draught (6). _{A motion comparison between} the three models shoWed the general advantagm of V-shaped ferebodies in _wavee equal te or longer than the model. This _{was particularly true for the}

acoele-ration at the forward part and for _{the heave reeults. For shorter waves the} U-shaped ferebedies appeared to be _{advantageous. These}

_{resultalmereAwomerel}

agreement with the loatdraught resulta (5).

The resulte of the resistance _{measurements showed that:the U-shaped}

models

were superior

in still water and in

_{waves shorter than the}

model. The clear

superiority of the V-shaped models in long waves as it-was shown in

_{(5) was}

not found to be so distinct in _{thoseballast draught teF,ts.}

All the works discussed above

were based on experimental. work. With the

progress of the oomputer technique it _{became possible to make systematic}

investigationEUsing ship motion, computer

_programs.

Ewing (7) made computations for four

ships CB = 0.70 with different form of

ferebody; the forms investigated

had similar but not identical

forebedy.sec-tion Chapes te these given by

Swaan and Vossers (4)* All the forms had the same Series Sixty afterbody.

-2-The motiene under:ooneideration - heave, pitch, relative bow

motion and

acoeleration ef bow and oton - were calculated wing a oomputer program based on the theory of Kervin-Ereukovsky with

_{added mass and damping}

coeffieientg actoording te Grim.

From the oomputed regulto it was stated that V-shaped sections resulted in cmaller motions of all five investigated variables and for the considered

ehdp longtho. This von in contradiction to the model work of Swoon and Voobers

mho found U-ohaped sections to have a favourable influence en the relativo boo motion and the V-seotion to be adventageoug in mavoo longer than the ship

length.

The oomputod reeulte indieated that ferebody section shape had little inip

flume° en pitoh and acoolaration at the otorn for tha ship length congidered.

As it tma gtated in the discussion of Emin' a work the main reason of:tho

contradiotiontetmeen computed end experimental results was mused byAising Lemia two-parameter series for determining geotional added maeprand damping,

beoauee thim method ie

givang

good remits for normal U-shopo geotions but.-failed for 7.i,ohapcd oectione.

Joopon, Wahab-and-Woortman publishod a work

(5)

where a °caparison betmeen oalculated end experimentally:determined motions'and vertieal mavo bending moments map made for a number of Seriee Sixty hulla in order te etudy the applleability oflmo caloulation-prootedureg bao ed on the strip theory. The

inveatigation ceapriee&veriationg of the block ooeffieient, _{hull proportieno4}

londitudinal radius of gyration and forebody sectional

allow.. The caloulations

mere ¡tarried out with., the equations of motion

_{introduoed by}

_{Korvin.Kroukovehr}

and witivthoseAe:.Voceera.

Itmas found that the :vault() obtained with the two calculation

_prooeduree,

in

Conepalt-Agreed;voll mith meacurements,It mas diffioult te conclude. _freak,

the moults whether the equationo of motion of_Vokonero

_{or thotao of Keryin..}

Krouhovaky provided the beet resulta. Itmag noted that the fair eorrespendence

of the ooloulatod and measured valuse of motionsmag.no conclugive proof for tho

correctnegs of.the equation.of metion used., For thig _{purpose oleo a eomparieon}

had,to be made between the calculated and the

_{measured coeffieionte in.the}

equations of motion, and between the caloulated and meaeured diatribution of the hydrodynamie:foroso over the length of the

hull. Finally, taking into

account tho imvebtiaations of Gerritoma and

_{Beukelman (9),}

_{the authare eon.}

oluded, that the equations of motion derived _{by Korvin-Kroukevsky had to be}

preferred.

In the present paper the results of the calculations of ship motions

for ship models with different block coefficients are presented, while

inside eaoh model family the forebody vertioal prismatic coefficient has three

values, which load to the change of the form of the forebody cross sections

from U-shape to V-shape.

II Calculations.

The motions in waves and the hydrodynamic) toefficients are oaloulated

for nine ship modela;

the

main partioulars and dimensions of the modele are

given in table 1 and their body plans are premented in figure 1.

All the modela have the gana main dimensions and longitudinal radius of

gyration, while inside each model family with oonstant value of blodlc coeffioient pB 0.6o# 0.706 0.80 all the models have the same form of the

afterbodyl their curves

of sectional areas are euaatly the same. The only

parameter, which is Changed - is the forebody

vertioal priamatic coefficient,

which causes the chango of the forebody crams sections from U-shaped to

V-shaped form.

All calculations are made on the basis of

the

standard ship motion computer

program of the Delft Shipbuilding Laboratory (10) which uses the

Gerritsma-Boulcelman version of the Korvin-Krodkovelty strip

theory.

Per determining oeotional added masa and damping the multi-coefficient transformation es given by de Jong (11) is used, this method

allows to

reoeive more aeourate resulte by the oonformal transformation of different

typeo of ship's (tress-sections *o the nmit

(drag.

II Bieoueeion of the =Quito.,

a. Ship motiens.

The calculated heave and pitch amplitudes, heave and pitch phase angles are given in figure 2 - 7.

As it can be &son from the graphs the heave amplitudes are strongly dependent on the vertical priematio ooefficient of the forebody, and they are

deoreaeing with the decrease of the forobody vertical prismatic ooefficient.

This tendency can be seen in all range of wave leneths and speeds. At the

same tine the influence of U-shaped and V-shaped cross sections on heave

amplitudes is more pronounoed for the ships with low block coeffioient and less

for the

ahipm

with high block ooefficient.

The influencer of the forebody priumatic coefficient on the pitah

amplitu-dea is not so etrong aa on heave amplitudes. In general it is possible to

Day

that the ships with low forebody prismatic coefficient (v-shaped arose sections)

OSUBQa

deorease ef pitoh amplitudes in long wavem and an inorease of pitoh

amplitudes in mhort waves. But this tendency is dependent en the ship speed.

Per

low apeede (Pn . 0.15) the ehips with Ibbehaped forebody have the advantage in pitch amplitudes in wave, loneer than the ship'm length and praetioally the

memo pitch amplitudes as the thips with Voeshaped forebody in ahert wane.

With the inorease of speed the advantage of the

chips with V-shaped forebody

in pitah osa be

seen only An longer waves, while in short unveil they have higher

piteh amplitudes. The minimum wave length till which the ehipmewith V-othaped

forebody have advantage in pitoh amplitudes inoreases with the increase of

ahipf

epeed.

At the same time the difforenoe in pitoh amplitudes between the ships with U-shaped and V-shaped forebody is dependent on the ship's block

coeffi-oient, it is greater for the

chips

with low bleok ooeffioient and smaller

for the Ships with high block ooefficient.

Piteh and heave phase angles are not so influenced by the different

form of the

ahip's

forebody.

-6-b. Coefficients of the equations of motion.

The difference in calculated heave and pitch amplitudes is caused by the difference in coefficients of the equations of motion. Non-dimensional coeffioients of the equations of motion are given in figures 8 - 15.

As it is shown by the calculations, the difference, if there is some,

in the coefficient of the equations of motion is practically independent of the

ship: speed, and so all the coefficients are

given for one value of speed

Pn

a

0.20, for ether speeds the tendency is the same.

Coefficient "a", which, according

to the equations of motion, has the

form

a = .dx

is praetieally independent of the form of the ship's forebody cross sections; slight differenoe can only be

icen in the range of low frequencies, where

_the

ships with V-shaped

forebedy have higher values of "a". With

the inorease of

frequency the difference disappears.

Coeffieient "A", which has the form

A2A_

v la a a + 00e- L NIxdx v2

"doe".

zdn L

r

is bigger for the Mips with V-shaped forebody than for _{the ships with}

.U-ehaped forebedy.

The difference is dependent on the

frequency and

decreaeou with

the inereeee ef frequency. The tendency for the ooeffieients "a" _{and "A" is the'same for} all bleak coefficients.

Coefficients) "b" and "B",

which have

_{the form}

b

fie

dx - V

f

dM4 dX L dx

B _{N4 2dX}

_v

_{x2dx-- 2v}

_fin

_Ixdx

L dx

are bigger for the Shipe with V-shaped forebody than for the ships with

U-ehaped forebody in all

range of _{frequenoies. The differenoe is practically}

conntant for all frequencies, concerning eaoh block coefficient.

Coefficients "d" and "D", whioh have the form

d

_{m xdx +}

_{cbc -}

_dx

C(Je2 L2 L dx

We

8

-D

imtxdx

are different for the ships with U-shsped and V-shaped forebodies. The difference in these coefficients depends on the frequency and decreases with the inorease of frequency. For the ships with V-shaped forebody the ooefficient "d" is bigger, and coefficient "D" is smaller (absolute value) than for the ships with U-shaped forebody. The tendency for the coeffieients

"d" and

'fir

is the same for all block coefficients. Coefficients "e" and ',Er, which are written

e =

J(Nixdx - 2v

Lida

- v

1-datxdx

L dx

Ea

IN

IXdX V

)(

xdx

have different values for the ships with U-shaped and V-shaped forebodies in all range of frequencies and the difference is praotically _{independent of the} frequency. For the ships with V-shaped forebody ooefficients E are bigger and

coefficients "e" (absolute value) are smaller.

For the oaloulated values of exciting foroes and moments it is possible to say, that their

values

_{are practically independent of the form of the}

o. Distributioe 0

the sectional added maae and damping over the

length.

The

difference in coeffioient of the equations of motion is oaueed by difference in distribution of sectional added mess and damping over

the

length of the forward part for the ships with 15-shaped and V-shaped forebodiee.

The diotribution

of sectional added mass and damping along the length of the modele for momo frequencies is given in figures 19-21.

At low frequencieo the V-shaped cross seotione

have greater values of

emotional added noes

than U-ehaped elections of the same

area. Mith increase

of frequency the differenoe in seetional added mass between V-shaped and U-shaped oectione Immense lasso and oven the V-shaped eectiens can have

smaller valuee of sectional added mass. But thio difference in sectional

added EG000 is not oo important and co for the whole model the difference in

added mho

in small. All thie explains the differenee in the ooeffioient of the equations of motion "a" and "A" vid its dependenoe oa the frequency.

The damping in all froquenoiee io greater for V-shaped sections than

fer Vciohaped

mittens, and the difference in damping is practioally independent of the frequency. This expiable praotically the oenetent difference

in

eeeffieiente of the equations of motion "b" and "B" for all renge of frequem-step. At the same time the dictributien of the damping along the length of the =dole with V-ohaped forobodise beoones mere symmetrical and this.cemees the differonos in croes-ooupling coefficients we" and Er.

IV Conclusions.

From this series of calculations the following conclusions can be

derived-Using V-shaped seotions in the forebody instead of U-shaped sections resulta in smaller heave amplitudes of a ship in the whole range of wave lengths. At the same time it reduces the pitch amplitudes in long waves and increases them in short waves. Heave and pitch phase angles are practically

not

influnced

by

the form of the forebody sections.

The advantage of V-shaped sectioni in pitch is more clear at low speeds;

with the increase of speed the

piteh amplitudes

for a ship with V-shaped

forebody becomes bigger than for a ship with U-shaped forebody for more

longer waves.

This phenomencelwith V-shaped forebodies is more pronounced for a low block coefficient; for high values of block ooefficients it is poseible to see only some advantage in

heave amplitudes and practically

no difference in pitch amplitudes in

the

whole range of wave lengths.

The difference in ship motions for ships with V-shaped and U-shaped fore-bodiee is caused by the change in the dietribution of sectional added masa

and damping along the

ehip length. The V-shaped and U-shaped seotiens with

the same area have slight difference in added maas _{which depends on the}

frequency; the V-shaped sections have higher values of sectional damping

independent of

the frequency. This oausee the difference in the coefficients

"b" and "B" of the equations of motion and also the differenoe in cross-coup-ling coefficients.

The author is indebted to the ataff of the Delft Shipbuilding Laboratory who give him the possibility to fulfil this work during his stay in the Netherlando.

Partisularly he wishee to

thank Prof.ir. J. Garritsma, who inspired the

asoempliehment of thio work, and Mr.

W. Bedkelman, whose construotive criticiem guarded him from overdetailed investigations and whose oonstant

help allowed him te finieh this work in much a rather short period of time.

PUrther the author thanks Mr. A. Veraluis

for

hie constant assistance and help in oarrying out all the computer work.

Refergnoes.

Lewis, E.V.: "Ship speeds

in irregular seas".

Trans. Soc. Naval Arohiteots and Marine Engineers, 1955.

Oohi, K.: "Investigation

on the

_{influenoe of ship form upon the} etrength of ships going in waves".

Journal Soc. Naval Arohltects of.Japan, 19570 v. 100 and v. 101. Taniguohi, K.: "Tests; of fishing boat models in waves".

F40 publioation; Fishing boats of the world, 2, London, 1960.

Swarm, W.A. and Vossers G.: "The effoot-ef the forobody _section

ehape on ship behaviour in waves". I.S.P., 1961, Vol. 8, N 83.

Bs:lesson, B.G.: "Influenoe of V and U shaped forebody _{sections on}

motions and propulsion of ships in waves".

Publioations of the Swedish State ahipbuiLdling experimental tank, 1962, N 49.

Bengteson, B.G.: "Influence of V and U Shaped _{forebody sections}

_on

motions and propulsion of Ships in waves at ballast draught".

Publioations of the Swodish State ahipbuilding _{experimental tank, 1965,}

N 56.

7,

Swing, J.A. _{$ "The effect of speed, forebody shape and weight} distri-bution on ship motions".

Transactions of the Royal Institution of _{Naval Arohitects,}

_{1967, vol. 109.}

Jooaen, W.P.A., Whhab, R.

_{and Woortman, J.J.: "Vertical motions and}

bon-ding moments in regular wavesP. I.S.P., 19680 vol. 15, N 161.

Gorritoma, J. and Beukelman, W.: "Ths _{distribution of the hydrodynamio}

forces on a heaving and pitohing _{ship model in still water".}

Fifth symposium Naval Hydrodynamics, _1964.

Smith, W.E.: "Computation of pitoh and heave motions for _{arbitrary chip} forme".

I.S.P., 1967, vol. 14

Jong, B. de, 0Computation of the

hydrodynamic coefficients of oscillating oylindere".

Delft Shipbuilding Laboratory, report N

_1744.

-pemencla*ure.

abedeg

ABCDEG

CB Fa

Fn

IL

K

27T/X

Kyy

Mn

ml W4 Yw

coefficients of the equations of motion

for heave and pitch

area of waterplane

block coefficient wave force

Wave force amplitude Froude number

acceleration of gravity

longitudinal moment of inertia of waterplane area wave number

longitudinal radius

of inertia of

the modal length between perpendiculars

wave moment

wave moment amplitude

sectional added mass

sectional damping

draught of the modal

forward speed of the model

half width

of

waterline

heave displacement

heave amplitude

phase angle between the motions and the waves instantaneous wave elevation

wave amplitude pitoh angle pitch amplitude wave length density of water volume of displacement circular frequency

circular frequenoy of encounter

-Tabl- 1.

Model

dimensions and Particulars.

9

V-bow

model number

model designation and

condition U-bow

2 ti V-bow

3

V-bow

4 U-bow

5

UV-bow

6

V-bow

7

U-bow

8 UV-bow 1 2

Displacement

kg

V

Length between

57.095

57.095

57.095

66.48o

66.480

66.430

75.946

75.946

75.946

perpendiculars

_2.26

_2.26

_2.26

_2.26

_2.26

2.26

2.26

2.26

2.26

3 4

Breadth

Draught _{m T}

0.323

0.129

0.323

0.129

0.323

0.129

0.323

0.129

0.323

0.129

0.323

0.129

0.323

0.129

0.323

0.129

0.323

0.129

5

6

Block coefficient

CB

Midship section

o.600

0.600

_0.600

_0.700

_0.700

_0.700

_o.800

_o.800

_0.800

7

coefficient

Prismatic coefficient,

0.976

0.976

0.976

0.976

_0.976

_0.976

_0.976

_0.976

_0.976

afterbody

_CpA

_0.791

_0.791

0.791

0.835

0.835

0.835

0.863

0.863

0.863

Prismatic coefficient,

forebody

CPF

0.907

0.857

0.817

0.952

0.908

0.870

_0.957

_0.926

_0.903

9

Prismatic coefficient,

total hull

_Cp

_0.849

_0.824

_0.804

0.893

0.871

0.852

0.910

0.894

0.883

10

_{Naterplane coefficient}

afterbody

_CdA

_0.808

_o.8o8

_0.808

0.829

0.829

0.829

0.886

0.886

0.886

11

_{Naterplane coefficient}

forebody

CWF

0.636

0.674

0.706

0.763

_0.795

_0.833

_0.892

_0.920

_0.942

12

_{Naterplane coefficient}

total hull

Cd

0.722

0.742

0.757

0.796

0.812

0.831

_0.889

_0.903

_0.914

13

Centre of effort of

14

waterplane

Centre of buoyancy

-0.082

-0.037

-0.067

-0.037

-0.051

-0.037

-0.037

0.012

-0.021

0.012

-0.002

0.012

-0.003

0.0440

0.011

0.0440

0.020

0.0440

Table 1.

model number

model designation and condition

1 U-bow -) ,_

UV-bow

3 7-bow 4 U-bow 5

UV-bow

6 V-bow 7 U-bow

8

UV-bow 9 V-bow 15 16

Longitudinal radius of

gyration

KYY/L

Half angle of entrance

on load waterline

_deg.

0.25

8.5

0.25 9.0 0.25 9.5 0.25 16.o 0.25 19.5

0.25

24.0

0.25 k2.5 0.25

58.0

0.25

69.0

.

U- bow

ii

Cb.060

Cb

Cb=ON

Fig 1 _{BODY PLANS OF THE INVESTIGATED}

MODELS.

(\

V-bow V- bow U - bow UV-bow U - bow UV-bow UV-bow _{V- bow}

2.4

2.0

'1.6

Z a

a

1.2

ea

0.8

0.4

1.6

1.2

KCa

0.8

0.4

1

CB

= 0.60

U

- bow

UV- bow

V- bow

Fig.2

Eteave and pitch amplitudes for Fn

- 0.20.

1 6

ZaL

1.2

0. 4

2.4 H

2.0 H

0.8

0.4

CB

= 0.70

U - bow

UV- bow

V- bow

vr175-4::

Fig.3

, .

-1.6

ZaL

1.2

2.4

0,8

0.4

2.0

F g.4

CB

=

0.70

U bow

UV- bow

V- bow

0.2

0.4

0.6

0.8

_1.0

₁₂

0.4

o

0

0.2

0.4

Fig.5

CB

= 0.70

2.4

U - bow

.1a

UV- bow

2.0

1.6

r----Za4a

₁₂

r

0.8

0.6

0.8

1.0

irL7

_40

_80

_120

auc" 111

_160

Ez

_200

Ce

_240

_280

_320

CB= 0.70

02

0.4

0.6

0.8

1.0

1.2

2.4

2.0

0.8

0.4

o

0.4

CB

=

0.80

U bow

UV- bovv

V- bow

0.2

Fig.7

0.4

0.6

0.8

1.0 1.2 _V^

25

2.0

1

CB =0.70

25

I ... ....

I.

0

1

2

₃

₄

₅

w

V

L/g 1

'''...411151"'

Figs

Co e.T.'fi c i. en

t...-.1

ad. ...3. et:.. .'

rtt.).v.s "a" or,.ii

.,

v''itled

rntiou .,7::)E,Teti.

t

_{. '.',:-F} i..n_c7,:-..

..,..

T

r

_L

3

4

w

Fig"

+

1.0

_{U bow}

UV bow

0.5

V bow

2.01---\

CB

= 0.60

0.125L

0.100E

0.075

M

0.050H

11025

-/

/

L

2

3

4

5

₀

1

2

3

₄

₅

oj

1/ Lh

GO

1/177

Fig. 9

:tmpinfT "o4 a

igtmninr 91:mPnt,

_

_{t?z,e'fficient;-for Fn}

_'Y

7

1.0

_{U bow}

U V bow

-CI I T

2.5

05L--1 mom. em

U bow

UV bow

V bow

T T

CB

=0.70

0125

m 0.050

0.025

$ I

j

I I

I

J

_{_L_}

I O 1

2

3

1,

5

₀

₁

₂

₃

i

5

w

1/ Lig "

.. ''''' 1 li I a iI..

_{W Vf I y}

_g

1

-31M°'

o)

2.5H

1

2.0'

1.5r-CB

= 0.80

jj

'

\\,

\

\

i I 1 i

L

1

I

1

I

0

1

2

3

7:

₅

w

fr.../g-5

0.150

0i25

0125

--61 0100[

_,

m 0.075

-I

0.050

N

NN.

r-t

0.025.

L

O 1

2

3

4

5

w

0.7.7g"

U bow

ioH

U V bow

0.5

V bow

Damping "1).

,tto.moy; "...3"

ö4ffjc er.,.t :77

F-1

07-050\

\

025

13\ig:

o

2

3

4

w

FA"

Fig.12

71-.1;3

CB

= 030

01.7t

050\--

025\--ck>.

C.

0

U

_{- bow}

U V bow

V

- bow

====11111 101IN

-0.25

0.00 40=00

Q50

025

0.50

Fi g.13

Coup1ir

I

I

U bow

-- U V bovv

V bow

...MDawn.

CB =0.60

0.50

I I _{I I I} O 1

2

3

₄

₅

w

VT/7

0.50

AQ25

e_

a25

0.50

T-

I

bow

UV bow

bow

....W.I.

CB

= 0.70

0.50

2

3

w

lrETg

--2111°

0.50

025

,

T

.- 11.)

0

a_

-Q25

-0.50

U- bow

- U V- bow

V- bow

...

...

..mms

CB= 0.80

0.50

0.25

-0.50

1

2

3

4

5

W

1\177

125-ro

l'-f)

0.75

-cn

a_

12 0.50

--

0.251--Fig. 16

7- 7

U

- bow

- - - - U V - bow

- - - -

- -

V

- bow

,

L_

0

1

2

3

4

₅

w

CB= 0.60

100

0.50

-I 1

2

3

Lt

5

w

1/7.7g-1

1.25r

1.00

a

0.75

en

LL 0.50

0.25

Fi g.17

U V bow

V bow

U bow

2

3

4

w

Lig

5

CB =0.70

IP

100

0.75

0.50

0.25

o

1

2

3

W

Lig

125

025

Fig.18

U -bow

- - - - U V -bow

-

V- bow

3

4

w11F:17

f

CB

= 0.80

3

4

4

1

o

"E

4 csicue.) 3

a2

.x

.E

o 4

3

2

1 3 2

0

2

4

6

8

10 12

14

16 18

20

CB= 0.60

2

2

1 10

5

o

o

20

15 10

5

2

Fig.19

_{Distribution of mt and N' ovzr the 1,ndth of the ship model.}

4

6

8 10 12 14 16 18

₂₀

w =5

'

rad/sec.

U-bow.

--- UV-bow.

.N

_,

,

,

_\\

.-4-bow.

_

_

_

w=7rad/sec.

\

_

w=9rad/sec.

w=5rad sec.

..._

...

_...

.

\

% \

_

\ \

\\*\

w=7rad/sec.

- ----,

\

_

.,

_\\.

\

.,

..

w=9rad sec.

_

_

..

1

o

4

3

2

1

o

5

3

2

= 5 rad/isec.

6.)=7rad/sec.

w=9rad sec.

2

4

6

U-bow.

_{UV-bow. .-4-bow.}

10

12

14 16 18

20

Cg= 0.70

2

2

1 1

"E2

'1,15

.T10

z5

o

20

15 10

5

Flg.20

_{Distribution of}

_m'

_and

_N'

over the length of the ship

_m,Jdel.

J

W =

_{5rad sec.}

)

i

,

!Al

,..)=7rad/sec.

\

, \

1

\

....

AI

w=9rad/sec.

111

li

4 lb,

,

-,

.,

--III

4

8

10

12

14 16 18

20

4

3

2

1

3

2

1

0

2

4

6

8

10

12

14 16 18

20

C

B

= 0.80

25

20

15 10 5

o

5

0

20

15 10

5

0

26

Fig. 21

Distribution of

m'

and

_{g' over the length of the ship model.}

6

8

10

12

14 16 18

20

w

=

w

=

7

w= 9

All

Adi

/Ai

5 rad/sec.

rad/

rad/sec.

p

U-bow.

---

_{UV-bow. ..i-bow.}

%

',..

1111

1111,

w

=

5 rad sec.

41

,

v

%

IlL

w =7 rad/sec.

4 111

RI

Ilk

1

.

Il