On the study of characteristics of ship motion by a forced oscillation method

On the Study of Characteristics of Ship Motion

by a Forced OscIllation Method

by

Hiraku Tanaka

&

Hiromitsu Kitagawa

Ship Research Institute

i 96+

Contribution to 2nd I.S.S.C., Committee 2b-I

SEEI A4 NO. 5

lab. y.

Scheepsbouwkunile

Tcchnischa

Hogeschool

SEKIREI A 4 NO. 5

On the Study of Characteristics of

Ship

Motion by a Forced Oscillation Method

by H. Tanaka & H. Kitagawa

1. Introduction

The physical properties of the longitudinal shIp motions in waves such as exciting force, virtual mass,

damping force, cros-couplIng force, etc. were studied

by means of the forced oscillation test. The authors intend to examine the equations of the longitudinal ship

motions with moderate amplitudes in waves and contribute

to study on the performance In waves.

The equations of the longitudinal ship motions In waves are shown as follows

abcZ

;

o.5(t3)

(i)

where z vertical displacement of centre of gravity

of ship

= pitching angle

H,M = amplitudes of exciting force and moment

phase differences with resnect to the ship motion

Eq.(].) seer.- to be generally accepted at present as the most useful equation since Korvin-Kroukovsky and

Jacobs have fully investigated the character of the

equations' . In the equation, however, the effect

of the non-linearity of pitching and heaving Is omitted and no experimental investigation has been carried out to determine the coefficients of the equations as the

functions of frequency and speed of ship.

Recently CumrnIns proposed other different mathematical

model and he showed that it was necessary to consider

the ship motions for infinite past for exact understand-Ing of ship motions.

Although the Eq.(l) are Insufficient f ror'

theore-tical point of view, the equations have been used in almost all the studies of pitching and heaving of ships

in waves as fundamental equations, because the Eq.(l) are simple and practical. As

It

has already been

mentioned by Korvin-Kroukovsky that the experimental

results poInted out shortcoming of the erivatlons were

little known yet. Then

it Is extremely expected to

make progress in the field of the experImental technique. 1

SEKIREI A4 NO.5

2. Forced Oscillation Test

It is difficult to obtain the physical or theoretical properties of longitudinal ship motions from the model

experiments In waves because the exciting force and moment due to waves are quite comp1Iated. A number of forced oscillation tests, of which methods are te excite a model ship in calm water and examine th response of the model ship, have been äe'ised.

These experiments may be classified In the following two groups

Imparting the known sinusoidal heaving force and pitching moment on a model and measuring the longitudinal motions

(GerrItsma' , Motora4 ),

Imparting the known sinusoidal longitudinal ship

motion on a model and measuring the heaving force

and pitching moment

(Go1ovato , GerrjtsmaC , Present rnethád ).

The inethod-(b) is more convenient for the purpose of

authors' investigation than the method-(a). trsIng the

method-(b), the arnplIttide of the motions are kept constant at any frequency. No experiment has been fully carried out by the rnethod-(b) because of the difficulties of the techniques.

The techniques of the eperiments had been so

rn-proved by using the analog computer that

the

authors ob-tained various interesting results.

The princinle* and results of the experiments of the longitudinal ship motion by the present method are dis-cribed in the following.

3.

PrInciple

Let a model ship be sinusoidally irnpartea heaving

and pitching with

frequency w

and then heaving force, H, and pitching moment,

N, with non-linìearltles of n-th

order are shom as follows,

i

M =

co(wt

) Ç

The amplitudes of the In-phase

and 90-out-of-phase

corn-ronerìts for n-th order against the displacement are shown as follows, respectIvely,

H ca M cOS31

(3)

i'i c6, ,t4,, ,'ii f3,.

*

The similar Idea has been already reported by Tucerman

of and recently reported by Zunderdrop and

Buiterihek8

SEKIREI A 4 NO. 5

Each

term of

Eq.(3)

is determined by computing the

follow-Ing

equations with the analog computer Installed on the

3 towing carriage.

/-!e

77T2 o r / Û T

ct

A4t""dt

(Li.) C)

of the equations of the longitudinal ship motions are

Using the measured values ( Eq.(3) ), the coefficents determined.

It IS

assumed

that the motions of ships contain the non-linearities of the 2nd and 3rd order as shown In the

following equatins, '1

bCDE

z (5.) z, =

= C/JIJ

When the motions of a ship Is shown as follows, Z,. co +

(6)

q) - (V

-the relations between -the measured values (

Ej.(3)

) and

the coefficients of the Eq.(5.) Is shown as follows,

for n=l,

[(-a _{7 e}

[(,-d- L

c3t"

M,t'

d-,w.7e

_[(p)LEJtT

çt

for nr= 2, =

[ç+

M2

= ±

+ &A

2F

t

J

For instance, taking pure heaving motion (iO,

the Eq.(7) becomes as follows,

=

Mccs(3

SEKIREI A4 NO. 5

The coefficients of the restoring terms, e1 , C3 ,

and F , are determined by the inclining experiments of the model ship. The other coefficients, a, b, D and E, can be obtained from the Eq.(9). Similar results are also obtained in the case of the nure pitching motion.

+. _Equipments

The equipments of the forced oscillation test at

Ship Research Institute ar schematically shown in Fig. 1. The exciter consists of two Scotch yokes, link mechanism and two three-component dynamometers (resistance, lift,

nd lateral force). The Scotch yokes are connected to a model ship through a couple of struts and the dynamo-meters.

The phase angles and the strokes of the Scotch yokes

are adjustable, with whiCh the ship motions such as nure pitching, pure heaving or their combined motions can be

imparted. The computing devices are synchi'oriizd th a ship ¡notion as shown in Fig. 1.

The restoring and exciting forces and moments are also easily determined with the system.

. Experiments

- I Models

A couple of ship model having great difference

dis-placement length ratio and shape of the frame lines are

used in this experiments. The models are a tanker model of 2. meters in length (M.No.1367) and a high

speed cargo ship model of 3.0 meters in length (M.No.l23w of which the properties are riven in Table i and body

plans in Fig. 2.

- 2 Exciting force and moment

The excitIng force and moment measures In regular waves are shown In Figs. 3 to

6.

The e'rcitlng force

and moment calculated by aid of the Froude-Krilov ho-theIs are shown in FIgs. ₃ and +.

It can be seen from these figures that the excitlng

force and moment are less dependent from the Froude nnber

in this speed range.

- 3 Restoring force and moment

SEKIREI A 4 NO. 5

polynoininals of 3rd-order with resrect to the displace-ments z, and .has in Eq.(D, these coeffIcients are shown

in Figs. J)+ and 19.

The effect of the model speed on the restoring func-tions are little known yet excert In the Gerritsrna's in-vestigations. The authors' experiments resulted In the

fact that the coefficients of restoring functions were

appro mated by inear equation with respect to the model

speed without much error.

Amount of the cross-coupling and the non-linear terms of the restoring forces and moments obtained from the in-clining tests of the cargo ship model are shown in Table 2 as reference. In the buoyant forces and moments re-suiting in very small ship motions, amount of the cross-coupling terms and the non-linear ternis seem not to he significant, but In large anirl itude motIon, they might be superior.

- I. _{Results of forced oscIllation tests}

In the prent experiments, the amplitudes of heaving

and pitching are so small that the motions seem to be within the linear range. The bodily sinkage and trIm of th models were adjusted to be the same as those at the

free running In calm water at the corresponding sîeed. As explained in the Tre7icus section, In order to determine the coefficients of Ea.(5) except the restoring terms, the amplitudes of the in-phase and 90-out-of-phase components as Eo.(3) must be determined.

Those components obtained from the experiments of the

cargo ship model that

IS

Iinrrted nure pitching at Fraude mmiber 0.2', as ari example, are shown In Fig. 7. From

those components, the resnonse functions of the heaving and pitching are also easIly determined. The non-dimensional response functions presented in terms of the

force

or moment at w= 0, are shown in Figs. to il for the tanker model and cargo ship model.

Fig. l

shows th

2nd harmonics of the force and

moment obtained from the

forced

nure pitching

evperiment

of the tanker model at Froude niber 0.l, and as

re-ference the linear terms obtained from the saine experIment are shown in FIg. 12.

The non-dtmensIonal coefficients of Ec.() such as

vIrtual mass coefficient, damping force coefficient, etc. are shown against the non-dimensional frequency in Fics.l

to 23. Those coefficients of the cargo ship model are

also shown against Froude niber in Figs. 2+ and 2.

6.

Discussion

The figures of the uncoupled re3nonse functions of the heavIng and pitching, as shown in Figs. 8 to 11, seem

SEKIREI A 4 NO. 5

for heaving motion

function of 2nd order although the danping force and

roment, b and B, are extrerel:r Influenced by the ieriod

of ship oscillation.

This IrnDiles that the dairiDin

force and moment are not surerlor terms evcet by the

range of resonance period.

The F1. 26 shows the

indivdua1 terirs of heaving force of cargo shIp model,

a, b, cz, d

, eqi' ,

fq

which Is obtained from the

pure heaving and pitching experiment with i cm and

53

minute

amplitude, resDectively.

The arnp1itues

correspond to the Ship motions in waves ofA7L=O.9 at

Froude number 0.25,

The cros-coup1ing response function are different

from the linear frequency resnonse function of 2nd order

because the virtual mass and damping force in

cross-ccupling terms vary with the ship speed and frequency of

shIp motion.

In hIghr speed or higher frecuency than the critical

value of (wV/n- 0.25), the effect of the ship speed on the

coefficients of the eouations of the motions are rela-.

tivel: small.

On the ether hand, in lower speed or

lower frquency than that, the coefficIents of the

equa-tions are extremely varied with ship speed.

Ts imtlies that the characteristics of lor.gitudinal shir

motions in the lower sDeed or frequency than the critical

value of (v/ = 0.25) are quite different freni those In

higher speed or higher frequency than that.

The virtual mass, virtual moment of Inertia, damning

force and moment, a, A, b and B, obtaIned from Motera's

diagrarn4 derived from the experirrent by means of the

impact method are shown in Figs. 15, 17, 20 and 22.

Nemants theoretical results

for damping force and

moment are not significantly different from the authors'

resul ts.

Korvin-Kroukovsky and Jacobs

have shown

theore-tically a relatIon among the each coefficient of Eq.(1)

and ship speed.

The results of this Investigation are

not coincident with their relation, but the following

characteristics can confirm with this investigation that

the coefficient e and E are proportioned to ship speed

and the coefficient d and D ( dD )

are very small.

As mentioned in the previous section, Eq.(l) or

(5)

can not completely express the resnonse of the ship

motion to the e7citing force and moment because these

coefficients are closely denendent on the shIp motions

themselves.

The vector diagram is rather convenient

for Illustrating the characteristics of the shin motion

as Fig. 27.

The ship motIons calculated by using the result of

the forced oscillation tests are compared with measured

ship motions.

The solution of Ea.(l) Is given by

Korvin-Kroukov-sky

,

as follows

z0e1=

Me&

Fel-ll4eU7

for pitching motion where F and e are exciting forces and moments due to waves, respectively. Taking the euations of the

longitudinal ship motions s En.(1),

Z'=

a--ib4c

±Lf

II? SEK(REI A 4 NO. 5 2

D1E

or more generally,

a) ;

uncoupled

resoonse function for heaving force, cross-coupling terms of

resnonse

function for heaving force,

ff _;

uncoupled

res'nonse function for pitching moment,

; cross-coupling terms of response function for

pitching moment.

In this exneriment, for instance, and correspond to the heaving force and pitch.ng moment of the pure heaving motion, respectively. From these

data,

the heaving

and pitching motion can be directly determined tri dis-regard with any equation of the motions.

The studies have beer carried out with the cargo ship model and the data of the motions in waves were supplied by the self-propulsion tests in longitudinal

regular waves with the similar model of 5 meters in length

A good agreement Is demonstrated between the amril-tudes of motions as measured and as calculated with

coupling e.r'fect in Figs. 2P and 29. Without the cross-coupling terms, the discrepancies between measured and

calculated motions are not serious as in tie erritsma'

exnerimentbO)

The difference between measured and calculated rhase angles are small but not so agreed

as

in

the

amnUtudes

of motions. hese discrepancies are mainly due to

experimental errors in obtaining excIting forces and moments.

The phase

lags

seem to be considerably sensitive to

comtutatonal errors.

It will be

concluded

that the

linear

differenttal

eauatlons

wIth the couDling effects give a suffIciently

accurate expression of

moderate heavIng

and pitching motions of

a ship in longit'd1nal

waves.

f,,'

LI. I

Golovato,

Gerrltsrra

Tuckerrnan,

Zunderdorp

Newman, J

Gerritsma

Referer: es

Korvin-Kroukovsky, B.V. and Jacobs, R.J.

_;

"PItching

and Heaving Motions of a Ship In Reu1ar

Waves", T.SNAME, 1957.

Curnrnlns, W.E.

;

"The Impulse Re3por1se Function and

Ship Motions", Schiff stechnik ,Vol .19,

JUne, l62.

Gerrtsma, J.

; "Exprimenta1 Determination of Damping,

Added 'ss an-i Added Mass Mortent of Inertia

of a Shiprnodel", ISP, Vol.1k, No.38, 1957.

(1+)

_Motora,

_3.

;

'On the Measurement of Added Mass and

Added Moment of Inertia for Ship Notions

(Part 14.

:

Pitching Motion) and (Part 5

:

Heaving Motion)t1, J. of' Zosen Kiokal,

Vol.107, July, 1960.

P.

"A Study of the Forces and Moments on

a

eavIng Surface Ship",

fl

_{Report 107+,}

September, 1957.

J. and Beukclman, W.

"DIstribution of

Damping and Added Mass along the Lnßth

of' a Shlpltorlel", 3tuiecentrum T.N.U. vcc'

'che.:psbouw en Navigatle, Report No.+9S,

March, 1963.

R.G.

_;

"A Phase-Component Measure System",

TMB Report 119, Apr11, 1958.

and Buiterthek

_;

"Oscillator-Techniques at

the Shipbuilding Laboratory", Technical

tniversity, Deift, Report No.111, November,

i 9)3

.N.

_;

"The Damping and Wave Resistance of a

PitchIng and Hetving Ship", J.SR, June, 1961

,

J.

_;

"An Experimental Analysis of

Ship-motions In Longitudinal Regular Waves",

ISP, Vol.5, No.52, December, 198.

Tasaki, R. and Kltagawa, H.

_;

"On Self-propulsion Test

In Waves with High Speed Liner Models",

Abstract Note of the 22nd Generai Meeting

of TTRI(at present, Ship Research Institute)

Novernbr, 1961.

SEKIREI A 4 NO. 5

Table i

Characteristics of Models

Longitudinal radIus of gyration in air,

of Lpp, ka 25.00 25.00

1367

1523w

Model number

Length between perpendiculars, Lpp, m 2.500 3.000

Breadth, B, m 0. ₅₇ 0.1+05

Det,th, D, in 0.197 0.21+8

Draft, d, ni

0.11+5

0.l,+5

Displacement, V, m3 0.101+

0.099

Displacement-length ratio, V/Lpp3 x lO3

6.628

3.669

Block coefficient, CB 0.800

0.562

Prismatic coefficient, Cp

0.808

0.608

Midship section area coefficient, _CM

0.990

0.926

Waterplane area coefficIent,

0.868

0.728

Longitudinal centre of buoyancy,

Table 2 Comparison of restoring terms (M.No.1523w)

ci. z0

i

i

where

restoring heaving force

restoring pitching moment - f3q) +

#

Z 0.01 m and

I=

0.+°

; Froude number 0.2

CZ:

0.02 0.01 2 C2q)0 0.01 0 per cent of c«z0 fi P0 0.03 per cent of C F z0 0.1 f2q) 0.01 0 -1-3 z0

j

0.01 r2 _Z0 0.01

'14

2

-j

i

STRAIN GAGE A MP LIF IR E s

ANALOG

COMPUTER F f C : ADDER.

-t

: SIGN CHANGER J : INTEGRATOR

)::::::

PITCHINS MOMENT GEAR IN M

SIN- COS POTENTIOMETER

C,

of forced oscillation test

IN

PHOTO

-LAMP TRANSISTOR

'p

M.No. 1367

M.No. 1523w

2

w -J

90

w (I.)

I

Q-o

400

E

3300

z

I-0I00

X

w

200-FROUDE- KRILOV

.

AI A

_Fn

O s

Fn

0.I5

2

WAVE LENGTH /MODEL LENGTH, AIL

90

o

3 w

z

D--90

200

E 5150

I-z

u00

o

(D

o50

o

L FROUDE- KRL0V L

-s---I 2

WAVE LENGTH/MODEL LENGTH, AIL

Fig. 1

Exciting moment

(M.No.1367)

Ai _{L F}

=0

$80 w 9O

z

w Cr,

z

Q-45

'E O

800

w C.)

0600

u-z

j:400

X

w

200

I I I Is

F=o

V 0.15 D

0.25

o

0.35

Fig. 5

Exciting force

_(M.No.1523w)

I 2 3

Q)

D45

w

-J.

Q

z

w (I,

45

I

Q-'E o,

90

-90

3

320

-

I-z

240

o

Q 160 k-L) >< w

80

I I I

Fn =0

ç;' 0.15 D 0.25 o 0.35

Fig. 6

Exct1ng monent

(M.No.15'23w)

2 3

2 o E

z

PURE PITCH

.-I

wVI

F

= 0.25

I I I 2

4

6 8

CIRCULAR FREQUENCY, w

s s' s' s' 'o IO 12

Fig. 7 In-phase and 9Otoutof_phase components

of response function of pitching (M.No.1523w)

AMPLITUDE A HSINo(, MSIN(3

26'30"

A o HSINX, H COSc, MSINf3 M COS f3 53' 0" 26' 30" HCOSX, MCOSf3 53' ON

PURE HEAVE

w VL

g 4 F = 0.15

0.25

*

--I I 0.2

0.4

0.6

0.8

1.0 1.2

a / W0

Fig. 8-a

Response function of heaving

(M.No .1367)

4.0

3.0

2.0

I.0

1.6 1.2 Ó.8

0.4

Q) w -J

z

135 o

PURE HEAVE

2

4

6 8 CIRCULAR FREQUENCY) W

Fig. 8-b

Phase angle of resronse

functIon of heaving (M.No.1367)

45

Q) t

90

w

_-j

wv-I

(r,

z

g-4

w

35 CI)

I

Q- F

=0.15

025

80

8.0 6.0

4.0

2.0 1.6 1.2 0.8 0.4 I I

PURE

PITCH

r 1- I I

Fig. 9-a

Response function of pitching

(M .No .1367)

1.2

I.0

I I I I I i

o

f45

w -J (D

z

w 135

(r)

I

Q-80

Q,

45

Fig. 9-b

Phase angle of resDonse function

of pitching

(M.No.1367)

8

35

0

2

4

6

4.0

3.0 2.0

IO

1 .6 1.2 0.8 0.4

PURE HEAVE

wV

I

g-4

F

= 0.15

0.25

0.35

i I I I I I I I I I

Fig. 10-a

Response function of heaving (M.No.1523w)

0.2 0.4 0.6 0.8 1.0 1.2 1.4

& w

45

& Q)

V

w -J

o

z

135 w (t)

I

Q-180

45

w -J (D

z

Lu (t)

I

°-135 180 o 2 F

=0.15

0.25

0.35

4 6 8

CIRCULAR FREQUENCY, w s

Fig. 10-b

Phase angle of response function

of heavIng

(M.No.15'23w)

1.6 1.2 0.8 0.4 r i f t r i i r i

PURE PITCH

t'lo.

II

-,II

i

WV

I

4

F

0.15

0.25

0.35

i r r i t I i P I I _{I i} 0.2 0.4 0.6 0.8 1.0 1.2

a)

/ W0

Fig. 11-a

Response function of pitching (M.No.1523w)

1.4

8.0

6.0

4.0

2.0 o o o

\.

PURE PITCH

\\\

180 I i L. 0 2

4

6 8 IO CIRCULAR FREQUENCY, w

Fig. 11-b

Phase angle of response function

of pitching (M.No.1523w)

F = 0.15

0.25

0.35

\\\\

N

45

cr' Q, V

ç90

w -J (9

z

135 w (J)

I

û-180

o

Q, V

o45

w -J (9

z

90

I

û-35

Q)

o

45

w -J

o

z

w

35

4

I

w L) 2

o

o 0 M1

Fn = 0.15

0 2 4 CIRCULAR FREQUENCY, w 8

Fig. 12

_{Force and monient of' linear terms}

Q)

o

d-45

w -J

z

₉₀

uJ (J-)

I

Q-I 35 E -180 N

f-z

0.6

04

N 0.2

M2

2ND HARMONICS F

=015

PURE PITCH

WV I

g-.4

o

_. s

O 2 4 6 CIRCULAR FREQUENCY, w

Fig. 13

Force and moment of 2nd harmonics

900

800

700

Fig. 1+

Restoring coefficients (M.No.1367)

C1 I I I I

kg-m

-kg-m-rad'

kg-rad

F1

kg-m-m''

-I I I I I I I o LO

20

VM m- s

40

30 20 5 3 2 6 4 2

2 o 8 6

4

o

ag/L

MOTOR A S D IAGRAM Fn AMP. 0.15

icm

o 0.25

icm

Fig. 15 Virtual nass and dauiping coefficients

(M .No .1367)

o 2 3

0.4

0.2

-0.2

-0.4

-0.6

1.2

0.8

0.4

d g /L L

wVI

4

FIr. 16

Cross-coupling coefficients

(M.

No .1367)

F AMP

0.15 2613011 Â 0.15 53' o"

0.4

0.3

02

0.!

2

Ag/(O.25L) A

WV I

-4

B /U/LL2

-- MOTORA'S DIAGRAM

Fig. 17

Virtual inonent of inertia and

damping coeffic1ents (1.No.1367)

F AMP. A _{0.15 26'30} A _0.15 53' 0" ° 0.25 26'3d' 2 3

4

w,/ L/g

0.6

0.4

0.2

-0.2

-0.4

-0.8

Fn AMP. 0.15

icm

0 0.25

icm

I 2 3

WJL/g

Fig. 18

Cross-coupling coefficients

000

900

800

450

400

350

40

30

20

o

o

kqmrcd'

cl

kg.m

kgrad

F1

kgmm

1.0 VM

ms

2.0

Fig. 19

Restoring coefficients (M.No.1523w)

I0

60

40

8

6

4

2

o

lo

8

6

4

2

o

2

Fig. 20

Virtual mass and damping coefficients (N.No.1523w)

Non-dimensional Coefficients (Pure Heave)

Heaving Amplitude =

1 cm

ag/A

'h

I'

MARKS F D

0.15

0

0.25

IA D

_0.35

I I I I I o D A t I - --- -.! .. D

Io

F

= 0.35

F

=0.25

(s)V

=0.15

bF

IA

b.gL

01 . -I

Z.

I.

MOTORAS DIAGRAM D o V A I A I

II.

I I I

o

3

4

5

6

WJL/g

0.2

-0.2

-0.4

0.6

-0.8

1.5

LO

0.5

o O

CL)VI

o

Non-dimensional Coupling Coefficients

0.25

O ¿p

d.g /!L

o---

o

O-Pitching

Amplitude

o

265'

o

53.0'

e/gL/ L\L

o O I

2

3

4

5

6

7

WIL / g

05

04

0.3

02

o.'

Do

I

Non-dimensional Coefficients (Pure Pitch)

Ag/(O.25L)2

,,

o W0JLIg

2

3

CO.\J L/g

MOTORSA'S DIAGRAM

Fig. 22

Vrtua1 moment of inertia and damping coefficients

(M,No .1 523w)

MARKS F AMPLITUDE

0.15

26.5

015

53.0'

IC

_o

_0.25

_26:5'

0

0.25

53.0'

¿

0.25

10_461 O

0.35

26.5'

En =

0.35

wV

En

= 0.25

0.6

Fn

= 0.15

0.75

0.50

0.25

o

o

0.5

-1.0

-1.5

-2.0

0 1° 2 3

CQÇIL/g

4

5 6

Fig. 23

Cross coupling coefficients (M.No.15'23w)

Non-dimensional Coupling Coefficients

Fn = 0.25

l4aving Amplitude =

I cm

Dg/LL

t I

-

s _o

4)V_I

_g-4

o s o . s o o . o o o s

EgL/At

t

4

3 2 o

o

o 5 4 3 2 0.2 0.I

-0.I

o

0.6

0.4

0.2

-0.2

-0.4

0 0.1 Q2 0.3 F 0.4 0.5

Fig. 2+

Coefficients of equation of motion as

a function of Froude number (M.No.123w)

I

Circular Frequency, W= 5.9s'

-ag/i

LWVI

r_g

4

bjgL/

f

.---

o

.

from Fn constant

test results

dg/AL

t o o o o

egL/L

o,

o

o o

i o0 I I i i I

0. I

0--0. I

0.2

O

0.2

-0.4

o

o cP

oj

Circular Frequency, a) = 5.9 s'

A/(O.25L)2

WV-i

g

4

0.3

E 1gL /L

0.2

0.3

0.4

0.5

F

Fig. 25

Coefficients of equation of ìnotcn as

a function of Froude number (I.No.1523w)

0.2

0.1

0.2

B

JgL/LL

o results o

from Fn

constant

test

Dg/SL

24

20H

16 12

4

8

4

o

F

= 0.25

ia+b+czI = IHI

z0 =

0.01 m

/

/

dz

/

/

WV_I

g 4

id+ e4+f1'pi=

IH'I

= 53'

/

/

Fig. 26

Components of heaving force (M.No.1523w)

0

2

4

6 8 IO 12

CIRCULAR FREQUENCY, W

o

CRITICAL FREQUENCY OR SPEED

ATWV

g FORCE OR MOMENT PHASE ANGLE INCREASING C I R CU LA R FREQUENCY RESTORING FORCE OR MOMENT

Fig. 27

yDia1 vector diarari of response

function for 1onituina1 ship motion

90

O

90--

-90--

-180-

-270-

1.0-

0.8-

0.6-

0.4-0.2

0-

8-

4-

2-00

---.

-i.

V

n

O.l5

EXPERIMENT COUPLED MOTION - 1J"COUPLED MOTION

2(P

deg. 0.5 1.0 1.5 2.0 2.5 3.0

WAVE LENGTH/MODEL LENGTH,

A/L

Fig. 28

Coniparlson of co!nuted and measured motion

amplitudes and phases (M.No.]523w)

90

0

-90

-180

- 90

-180

-270

1.0

0.8

0.6 0.4 0.2

o

Io

8 6

4

2 I I i I En

= 0.25

Ez

deg.

deg.

2Z/Hw

EXPERI MENT COUPLED MOTION UNCOUPLED MOTION

o

I I I I I 0 0.5 1.0 1.5 2.0 2.5

WAVE LENGTH/MODEL LENGTH, A/L

3.0

Fig. 29

ComparIson of computed and measured motion

amplitudes and phases (M.Nc.1523w)