Components of roll damping of ship at forward speed

4 DEC. igì

ARCHIEF

Report

of

Department of Naval Architecture

University of Osaka Prefecture

No. 00404

August , 1978

Components of Roll Damping of Ship

*

at Forward Speed

by

** ***

Yoshiho IKEDA Yoji HIMENO

**** and Nor io TANAKA

* published in the Jour. _{Soc. Nay. Arch. Japan, Vol.143 (1978)}

** Research Assistant , Univ. Osaka Pref.

Associate Professor **** Professor

b.

i.

Scheepsö.Mk01e

Teciniche Hoqescno

e

)

Report of Department of Naval Architecture, University of

Osaka Prefecture , No.00404 , August 1978

Department of Naval Architecture University of Osaka Prefecture

Mozu-Umemaòhi , Sakai-shi

Components of Roll Damping of Ship at Forward Speed by

Yöshiho IKEDA , Yoji HIMENO and Noriö TANAKA

SUMMARY

A method for predicting the roll damping of a ship at forward speed is proposed. The roll damping is assumed to be divided into five components, that is, the frictional ,the wave making, the eddy making and the lift com-ponents of the hull, and the bilge keel one. It is fOund that the eddy making component of the hull decreases rapidly with forward speed, while the wave making and the lift components increase. The formulas for these three comp-onents are deduced from some theoretical considerations and the experimental

results.

The frictional component ïs estimated by Kato-Tamiya's. method. The bilge

keel component is assumed to be constant at fôrward speed, and is estimated

by the formula which the àuthors have proposed recently.

The values predicted by the present method are in fairly good agreement

with the experimental ones fôr ship models at forward speed.

1. INTRODUCTION

Concerning the roll damping of ships the theoretical treat-ment is very difficult because of the large influences of the

fluid viscosity. Many experimental works have been carried out, although an establishment of the reasonable prediction method for roll dampings does not seem to have been achieved. Recently, the authors have proposed a method in which the roll damping at zero ship speed is divided into several component dampings, the fric-tion, the wave and the eddy dampings of the naked hull in

addi-tion to the bilge keel dampingU2)3)

The idea of this method can be extended to the case of the problem at forward speed, in which, however thére appear a new component of the lifting effects of the hull and the bilge keels and furthermore the variatiOns of the other component dampings. The objective of this paper is to establish a prediction method

for the r011 damping of ships in all the range of speed and

frequency by considering the forward speed effects of those com-ponent dampings.

The results obtained seem to be satisfactly for the practi-cal purpose and the comparison between the predicted and the measured shows a fairly good agreement.

2. COMPONENTS OF ROLL DAMPING

When a ship moves in still water at a- forward speed with a certain roll frequency, the röll damping coefficient B44, as expressed in ä equivalent linear form, can be assumed tO be

divi-ded into the.five component dampings, that is, the friction _BF

the wave. B, -the eddy BE, the linear lift BL and the bilge keel cÖrnpònéntBBK. And neglecting the mutual interferences among them, we can write them in the form.

B44=BF+Bw+BE+BL+BBK (1)

Although the way of this division may not always be resonable in the point of the hydrodynamic view, the reason for this division

-1-r

is mainly because of the convenience for carrying out eçeriuents

and for deriving prediction formulas for these components.

The following is a brief outline for these componentdairçings

at forward speed, in which the dampings are all expressed

in the

form of

th

equivalent_linear roll damping B44 or its

nondimensi-onal form B44 (=B44/2g /pVB2, B the breadth,V the

displacement

volume) and the suffix 'o' means zero speed.

.1)

The frictión damping Bp

For this çomponent there have been Tamiya's

work4

and. the

'author.'.s recent reseach1

in which.it is concluded that Kato's

formula at zero speed multiplied by Tamiya's modification factor

for. an advance speed is satisfactory for the prediction of. the

friction roll damping. In this paper this Kato-Tamiya's formula

is used for BF.

-2)

The wave damping Bw

The influences.of forward speed on the wave damping of roll

motion have been attacked by Hishida6, Hanaoka7

and Watanabe8.

We can not have had, however, a successful result of their works

applied to the actual ship hull form. In this survey, through

assuming a primitive mathematical model consisting a pair of

doublets ,

the- general properties of the wave damping are shown.

Thenthe authors obtain an empirical formula containing tIse

properties, which will be stated in the chapter 5.

.3)

The eddy damping BE. and the lift damping BL

These ter-ms considerd here are caused by the separati.ng

vis-cous.flow around the naked ship hull. Yumuro and others9

have

firstly pointed out that the lifting effect of the hull on the

roll damping is considerably great at forward speed, and they bave

proposed an empirical formula which will be mentioned laterand

will be modified by the authors.

The damping moment including the nonlinear eddy component

can be written in the form,

M8=B1+B2ÔIOj (2)

where

represents the angular.velocity of roll. Here we define.

the first linear term B1 as the lift damping coefficient BL and

the second nonlinear term as the eddy damping BE (=8wOoB2/3ff),

in which w and O

stand for the frequency and the roll amplitude

respectively, and the coefficient is expressed as an equivalent

lirìeaì form. These dampings will be discussed in the chapter 3

and,4

-4)

The bilge keel damping B

This term can further be sub].vided into two parts; the term

BN dué to the normal pressure on the bilge keel and Bs due to the

hull pressure variation. Although both of these terms are slightly

affected by the presence of a forward speed as. seen in the Yuasa

and others' work10) and the measurement by the authors11), the total

effects of the forward speed on the bilge keel damping BBK can be

neglected according to the recent experiients-2)43'l4. In this

paper, therefore, the value of BBK is to be replaced to -its value

BBKQ at zero forward speed, the prediction formula of which has

recently been obtained by the aUthors ])2).

-2-3. FORWARD SPEED EFFECT ON EDDY DAMPING

In the absence of forward speed, the eddy damping represents the nonlinear force caused by the two-dimensional separation at the keel or near the bilge circle, while in the forward motion it corresponds to the nonlinear part of the hydrodynamic lift force on the hull. For wings of low aspect ratio this nonlinear effect is often considered according to the concept of the cross-flow

model15). In the steady wing motion with a constant incidence

angle

,

the value of the cross-flow drag is often determined as

the one at ci=90° 15),16)

On the other hand, when the body moves forward with a roll or sway motion, we have further to take into account of the effe-cts of the amplitude ratio, i.e., the period parameter and the

reduced frequency K (=Lw/U , L the body length, w the circular

frequency and U the forward speed). Fig.l shows a schematic view of such a motion of the body, where the condition l/K=O corres-ponds to the case of oscillating bluff body and the condition

l/K= the case of a steady forward motion at a smallincidence angle. In the case l/K=, the drag coefficient of a flat plage of a

small aspect ratio is often estimated by its value at a=90 which

depends largely on the aspect ratio as shown in Fig.2. When 1/K =0, however, the drag coefficient is known to take a greater value

than that in the steady cross motion. Its value increases as the amplitude ratio decreases while it does not depend on the aspect ratio. Therefore the difference of the drag coefficient between the cases l/K=0 and l/K= becomes large as the aspect ratio incr-eases and the amplitude ratio decrincr-eases. On the contrary, the bilge keels which have very small aspect ratio and relatively large amplitude ratio, are little affected by a forward speed, which agrees well with the experimental facts.

Fig.3 shows an influence of forward speed on the eddy dairping of a flat plate in a swaying motion, the value of which is obtai-ned by subtracting the steady lift component from the measured data. The variation of the eddy damping shows the same tendency as stated above against the change of the aspect ratio and the amplitude a/b (a the sway amplitude, b the span). And the eddy damping tends to a certain constant when the value 1/K exceeds l/w since in that condition, the locus of the eddy separating at the leading edge no longer intersects again to the hull as can be seen in Fig.4.

For the actual ship hull form, the drag coefficient at 1/K0 is little influenced of the amplitude ratio as pointed out by the authors3). Thus we only need to consider of the reduced frequency. The experimental results for the ship models (Series 60, parent forms 0.6,0.7,0.8 CB) are also plotted in Fig.3. Although the measured values are somewhat scattering, we can recognize the

general tendency that the eddy damping of actual ship forms

decreases rapidly with increasing forward speed until it becomes zero at large value of 1/K. From these experimental data, we can

deduce an prediction formula for the forward speed effect of the

eddy roll damping of ships in the following form,

(0.04K)2 BE/Bo=(oo4K)2

where _{BE0 represents the value at zero speed which can be obtained}

by the authors' formula3).

4. LIFT DAMPING

4.1 PREDICTION METHOD FOR LIFT DAMPING

The roll damping noment MT caused by the lifting force on

e hùIl

cán

be expresséd in

the

form,

ML=+PCLSU2IR (4_)

whereCL, 1R and

s represent the lift coefficient, the moment lever cörresponding to the distance from the roll axis to the apparent center of the lifting pressure on the hull, and the product of the ship length and the draft (=Lxd), respectively. The lift cóefficient CL is assumed to be proportional to a certain représentative incidence a, that is, CL=kNao, where a0 is

ftther

defined as =i0O/U, the lever l

standing 'for the distance from the 'ròll axis to the point where a=a0. Then Eq.(4) becomes,

ML=fPSU2kN()1R

=f

SUkN1ola (5)

where the term

kN

represents the slope constant for CL..

Yumuro and ôtheis' assumptions for the terms _{'R' kN} ze

expressed. as the following form,

10=OG±0.5d (6) lJ?=0G+0.3d (7) kN=kz+(4.1B/LO.0). (8) -. . . O , CO.92 where k=2d/L, = 0.1, 0.92<CM0. 97 0.3, 0.97<CMO.99

where. 0G represents the distance from the. still water level to the roll axis G being positive toward downward. The slope cons-tänt kN is given by the formula often used in the field of ship manoeuveringl8). These Yumuro's assumption may not be adequate for all cases. The following is more detailed discussions about

the

uonnt

levers ₁₀ and _-R

For the actual ship forms, the lift damping depends little on the roll frequency w as stated

by Yuasa and others10), so

that wé

can

safely consider the problem as that of the steady

flow past a twisted wing for which the incidence angle varies in

the

vertical direction. Defining Ap(x,y)

and

w(=ywe0) as. the

pressure load distribution and the downwash velocity of the twisted 'wing (y the coordinate downward) and further expressing Ap(x,Y) and w(=a1U) as those for an inclined flat plate with incidencea1, we can apply the reverse flow theorem19) to these

two wings, and obtain the following relatiOns.

f4p(x.w)ivdxdv,=_f Aj5(,y)wdxdy (9)

-

aif4P(x)axdy=fy4(z»y)azdy

(10)

Eq. (10) represents a relation between the lifting force and the

heel moment Ms of the obliquely. towed hull. Multiplying _1R

both sides of Eq. (10), and rearranging, we can obtain the form,

ML=--!4JM (11)

-Let is be the distance from origin to the center of the lifting forces acting on the inclined flat plate, that is, the obliquely towed hull, we obtain

and

kNls= pU2A

5-(20)

The lever i is determined by the position of the roll axis

which gives

e5=o.

For example, from Fig.(4) we can find 1 =O.3d.

The comparisons of the value kNis between the measured anä the predicted by Eqs. (8) and (14) are shown in Table 1. The agreement

seems to be allowable.

4.2 MEASURE'ENT ÓF LIFT DAMPING

From Eq. (1) it becomes

BE+BL=B44BFBW (21)

and therefore the left hand side representing the compont danping dueto viscous pressure effects can be evaluatèd experimentally

by substituting B and Bw from the total damping measured. And

the 'term BE vanisFies at hiqh speed range so that only the term

MSO=fPSUkNOGLSÔ (17)

Substituting Eqs. (13)and(17) into Eq. (16), we obtain the

form,

(18)

where MLO represents the toll moment in the case OG=O. The

values of the constants in q. (18) can be modified by

experi-ments as will be seen later.

The product kNis can be measured by the oblique towing test

in, which the heel angle O i-s made free.

M5WGMO5 (19)

In Eq. (19) W is the displacement of the ship model. From Eqs. (12)and(19) we can deduce the form,

9 W( I,M

M3fP SU2kNaIIS (12)

and then the fòrm

ML=--PSUkNls1pß (13)

Comparing Eq. (13) with Eq. (5), it can be found that l=l, which gives án interesting explanation that the roll moment ML

due, to the lifting force can be obtained by the steady heel

moment Ms of the obliquely towed hull multiplied by 1R' in which the incidence angle takes the value o=lSO/U. The values of l and 'R for the ordinary ship hull forms âre determined by experi-ments, as follows in the case OG=O,

l0=ls(=O.3d ) 9) (14)

1R0.5d (15)

where these values appear to be just interchanged by each other term in Yumuro's formula, and therefore give the same value for

the moment _ML!

We' can proceed further to obtain the èffect of the distance 0G. The roll about an arbitrary point can be divided into the

r011 and the sway about the axis 0, which gives the relations,

ML=MLU"°' Mso

¡OG

- - (16)

BL remains. It is difficult, however, to evaluate the term Bw at present time so that it is necessary to made the effects of the term Bw negligibly small in certain ways.

Two methods are attempted here; the method A is to cover the water surface with a suitable plate in a circulating water channel (Fig.l) in order to prevent the wave arising. The other way (Method B) is to ca±ry out the roll test in a quite low

frequency' as to neglect the, wave effects.

The test results by these methods are shown in Figs.6 arid 7. In the rangé. where F1>O.l5, that is, BE seems to vanish, the

difference of the results between the methods A and B is not large Fig 8 shows the results for several ship forms, and the agreement between predicted and measured values is considerably good, which means that the assumption of 1R=O Sd seems to be valid for these

cases.

For the effect of 0G, Eq. (18) obtained theoretically can be modified through considerating the experimental results, as seen

in.Fig.9, to the following form.

(22) .

Fig.lO alsoshows an comparisòn of the value of Eq.(22) and the measured one which includes the Tasai and Takaki'è dáta for the

.

'3incargo ship model. ,. . I.

From these discussion, it is concluded that the lift danang

fOr ördinary

ship

fOrms can be predicted by Eqs. (22),(14),(15)

in a plausible accuracy.. For the flat

ship

and the ballast

condi-.tion, that is, for smali-dráft hull forms, however, the further study will be needed.

5. WAVE DAMPING AT FORWARD SPEED

It is quite diffiçult at present, time 'to establish rigorously

the théoretical treatnient for thewave roll damping f án

ordin-ary ship hull form at forward speed. Here we present an simple mathematical model and its comparison with experiments, on the basis of which an empirical formula will be derived.

In order to represent the flow due to a roll motion oUa

ship forin, we can introduce a two doublets system into the

uniform flow as shown in Fig.11. The doublets are situated at the points (1,0,-d) and (-1,0,-d) ,and are horizontally directed. Then

the strength of the oscillatory doublets can be determined so as to be proportional to the roil velocity 'rO, where r is thedistance f rpm the roll axis to the position of the doublets and O the ang-ulr've1ocity of roll givenby the expressiön,

O=O0sinwt (23)

The wave damping is approxImately deduced' from the wave energy loss at infinite down-stream after Hishidats work6),

' .

' Bq {(aó+b6)FB+2aZb3Fz} (24)

where the 'constants a and b is proportional to the strengths of the doublets" located at bow and stern and the detailed expressions for the functions FB and F1 will be given in the Appendix. The fIrst term in Eq. (24) can be called the basic term whiòh expresses the sum of the individual contributiön öf the each doublet, while 'the second the interference term between the two doublets. Fig.12

shows ari example of the results of the calculation in the case

where a=b=0 1 and d/21=0 2125 It is found that there is a

domi-nant' hump near the point 2=l/4 and a monotonöus decrease in the

range öf larger values. The, contribution of the interference

-6-term is not large compared with that of the basic -6-term.

.

To confirm.these fundamental properties of the. wave damping the forced roll tests for flatplate models as shown in Fig.13

are carried out In the nodels 2 to 4 in Fig 13 the contribution

of the midship part to the wave damping is negligible, so that these models correspond to a pair of doublets in a unifòn flow.. In the experiment 0G is kept constant (OG/d=-0.176), and. the data of the roll moment 4at the time when 0=0 are read and transformed to the linear damping coefficiént BZ4 using the

föllowing relation.

('T1(_P,B2

_"lB

Ifr'

44) (25)

The results are shown in Figs..14 to 17, in which the lift component BL is estimated by the present method with kNIS values measured at oblique towing test. and the wave damping is calculated assuming that the strengths of the doublets corresponds to those

givenb

"Ordinary Strip Method" at. zero Froude number. In these

figures the data at low speed are largely influenced by the eddy

damping.

Comparing with these experimental results and taking these circumstances into accoûnt1 it can be said that the present siuple mathematical model expresses the fundamental feature of. the wave damping considerably well.

Now let us consider to apply these results to actual ship fö. The measured values of the wave damping, as shown in Fig.18, can be obtained by subtracting the component dampings estimated by the present method from the total damping measured. From Fig 18 we can recognize that there appear the saine trends as that of the simple

model mentioned above Therefore, we can evaluate an empirical

function as shown in Fig 19 retaining the natures of the basic term of Eq. (24) in order to reptesent the wave damping of the

ordinary ship hull forms The values of the parameter A1 and A2

in Fig.19 can be determined by the experiments as shown in Figs. 20 and 21.

Finally we can state the present formula for the wave damping of ships at forward speed in the follöwing forms,

A, i +l.2e-2ed (26) A, =0. 5+E'-°e2Cd (27)

B/B,rr0. 5[{(A2+i)

+ (A,-1) tanh2G(Q-0. 3)} +(241_A,_1)e-'50(Q-O.25)'] (25)

,where Bw0 represents the wave damping at zero forward- speed which can be obtained for example,by the "Ordïnary Strip Method".

6.

COMPARISON WITH EXPERIMENT

The total roll damping estimated by the present method are

now compared with the results of the model experiments. Fig.22

.shows -the example in the case of naked hull .(cf.Table _{1). It can}

be recognized that the lift and the wave damping prevails in the -high speed range. Fig.23 to 26 include the cases with bilge keels,

the effect of. which is predicted by applying the authors' formula at. zero Froude nuniber to the forward speed cases without any

corrèction. Figs.27 and.28 show an expression for the. dependency

on the röll frequency.

-From these comparisons we can -conclude that the present formulas, for the component dampings can give, an satisfactory measure for the roll damping of an ordinary ship form at forward speed.. However,.it maybe nöted that there still remain several

problems, for instance, on. the _{case of small draft, also on the}

-7-..more rigorous treatxnent.ofthe lift and the wavedampings and

SOOfl. . .

7.

CONCLtJSIONS -.

-The concept. of the component dainpings for roll motion of-.

ships at forward speed are-established. For each component,

theoretical and expérimental discussions are made, through

which

the empirical formulas are derived. The conclusions can

be listed

in the following.

.

The eddy damping of the naked hull decreases

rapidly with

forward speed until it becömes negligibly small at large

value of

1/K(=U/L).The rate of thé decreasé little depends on the hull form.

For the lift damping,. a modified formula to the

usual one is

presented.

. . . -

-Ftom. a simple mathematical modeling, the fundamental natures

of the wave damping are clarified and a practical formula is

-proposed.

The total damping obtained by the present method are in

plausible agreements with the results of. the model tests.

REFERENCE

Y.iJedá,. Y.Hîmeno and N.Tànáka: ON RoliDamping Force of Ship-Effect of Friction of Hull and Normal Force of Bilge Keels- , Jour of The Kansai

Soc of Naval Arch Japan, No 161 (1976) , p 41

Y Ikeda, K Komatsu, Y Himeno arid N Tanaka On Roll Damping Force of Ship

-Effect of Hull Surface Pressure Created by Bilge Keels- , Jour of The KänsaiSoc. ofNaval Arch. Japan, No.165 (1977), p.3].

Y Ikeda, Y Himeno and N Tanaka On Eddy Making Component of Roll Damping

Force ön Naked Hull , Jour. Soc. of Naval Arch. Japan, Vol 142 (1977), p.54

S Tamiya and T Komura Topics on Ship Rolling Characteristics with Advance

- Speéd, Jour. Söò. of NavaÏ Arch. Japan, Vol.132 (1972), p.159

H.Kato : On the frictional resistance to

th

rolling ships, Jour. Soc. of

Naval Arch. Japan, Vol.102 (1958), p.115

T. Hishida : Studies on the wave-making resistance for rolling of ships

(Report No.6 The effects of motion ahead on wave resistances for the rolling)

Jàur. Soc. of Navál Arch. Japan, Vol.87 (1955), p.67

-T.Hanaoka : On. Michell's Method of Solving the Non-Uniform Wave-Making

Phenomena of a Ship, Proceedings, Joint Meeting of the Japan Soc. Applied

Mechanics, No.5 (1955)

I Watanabe On the Effect of the Forward Velocity on the Roll Damping Moment,

PapeÏ of Ship Reseach Ïnstitute, No5l, (Feb.1977)

A.Ymuro and I.Mizutani : A Study on Anti-Rolling Fins (2),

Ishikawajima-Harima Engineering Revïew, Vôl.l0,No..2 (1970), p.107

K.Yuasa, M.Fujino and S.Motora New Approach to Hydrodynamic Forces on

.-0scil1ating Low -Apect Ratio Wings, Jour. Soc. of Naval Arch....Japan,Vo1.142

-tl-977), .p65.- - . - .

Japan Ship Reseach Association., SRl6l Coitteé On. the predict-ion of Ship

-Performance in Wave, .Rep.No.254 (1976), p.27 .

-l2) N.Tanaka,- Y.Himeno, Y.Ogura- and K.Masuyama Free rolling test at forward

speed, Jour. of The KansaiSoc. of Naval Arch. No.146 (1972), p.63

Y.Yamanouchi : On the analysis- of the ship oscillations among waves-Part 1,

- Jour.. Soc. of NaVal Arch. Japan, Vol.1Q9 (1961), p.169 - .

-A.Taniguchi and M.Shibata : Rolling tests in free running, Jour.. of Soc.- of

Naval Arch. of West Japan,, No.14 (1957), p.49

-see T.Harnamoto : Theorectical Background of Mathematical Modeling on Ship

Maneuvrability, Bulletin of the Soc. of Naval Arch. of Japan, No.577

(1977), p.322

I.Tani

: Flow Theory, 3rd ed.., Iwanami Book Co. (1967), p.249

H.Kato-: Effécts of bilge keels-on the rolling of ships, Jour. Soc.Nava].

Arch. Japán, Vol.117 (1965), p.93

-ed. Soc. Naval Arch. Japan : Hand Book of Naval Arch. ,Vol.l, Corona Book

Co. (1960), p.696

Hydrodynamics Research Group, SRI : Note on Ship Hydrodynamics-No.8,

Bulletin of the Soc of Naval Arch of Japan, No 566 (1976), _p 407

Japan Ship Research Association, SR161 Committee : On the. Prediction of Ship Performance in Wave, Rep.No.275 (1977), p.1.7

APPENDIX

The functions FB and F1 in Eq. (24) in thïs paper are expressed in the forms, - . (A-1)

F1=øø2

. (A-2) where. d c2c1 0 \ C1/

IO'[(i___ (cz-_-ci)Iexp( ¡ 2F,cos20

lSin2(4Z_::1o) ( c2+c1 \ d c2+c1 +(1+-l_)(co+ci)sexp(

_TvS2o,Ì

2' C2C1 \ C1 4F,cos0) xsign( o)] sin20 --ci=.Jl+4QcosO, c21+2QcosO,

Q=-.

F=-L

g 011 12r

(-i)

.gn(x)_{ i (x)-O) SI [2r_c0s_1

(.<i)

-

i (z<O)

In the absence of forward speed, the following represeñtation is possible because the free-surface condition becomes simple.

(A-3)

where

(2(O 5k0cos0)),

{;}

=16kexP(_4-ko)f' SifllOj.O(0 5k0cosö)JdO

2w2

*0= ¡

_g

When carrying the integration in Eq. (A-3), it js necessary to consider of the singularity of the order 1/-taj-e at the point 8=Oi. The range of the integration can be divided into two parts,

O<O01=c

_and

el-eO<Ol, where e is a certain small quantity. in

the former range

Simpson's first formula is used for the numerical

integration, and in the latter area, an analytical integration is

made as follows i r0 2' f dQ2\ _2"

'

øc=-J,1_.-_exP_4T-)cos

--xsinO/i-- QdO i

'

d Q2 \ /

Q \

-=--(' -j--)

We can obtain on the saine procedure.

(A-5)

Table i Princip al dimensions of, models and heel moment coefficients

(1)kNlo/d : esthnated from (8) and (14)

(2)kwlo/d : nasured

Fig. i Schematic view of roll (sway) motion of ship 'at forward speed

2.0 Co 1.5 1.0 0.0 Kato's formula"1 0:1

'.Ä.R.

0. (aspect ratio)

Fig. 2 Drag cOefficient for flat plate

0.0 A.R..O.04. a-l.O

N

O.O6.a/b-1.0 n.e...008

f.i7

o _{prvsvn t} s o A _£ O Sanes 60, CQ.6 Ot Seiios.60. Cj.0.7 A SOnes 60. C50.B

2 meaSured main value

far aaayieg flat plptaa by Ososa ct.ai.'"

A _{meanurad mean valuo}

far roiling flat plato

Fig. 3 Decrease of eddy making damping with forward speed

lo

Fig. 4 Heel moment in oblique towing

0.005 Series 60, Cg.0.6 )uithoat B.C.) 0.004 measured . o t Method A (C0.53.O0.0.175) _o O Method S )50.29.8,-O.262) O 0.003 à -s

.

5-- 9 estimated : mami estimated k i /a using measured 0.002 0.501 -001 00

0.01 _{attack angle 0.0698rad}

0.0 _{0._1 02}

0.3 OG/d

s

Fig. 5 Schematic view of free roll test in circulating water channel (Method A)

0.0 .0.1 0.2 F0

Fig. 6 Roll damping coefficient of eddy making and lift components

-Model L x B x d (m) 2 00x0 364x0 119 _ V (m3) 0 0693 CB 0 802 ';' CM 0 995 (1) kNlo/d 0 175 . (2) kNlo/d 0 204 SR 98 fäker withbuib SR 159 tanker without bulb 2.00x0.33x0.l21 0.0646 0.802 0.990 0.171_ 0.181 SR 108 oentainer with:bulb 1 75x0 254x0 095_ 0 0241 0 572 0 970 - 0 119 0 111 Se1es60,'Cz06 i 80x0 237x0 096 0 0247 0 60 0 977 0 148 0 107 Series _{6OPCBO '} without bulb 1 80x0 257x0 103 0 0331 0 70 0 986 0 156 0 133 Series 6O,CB=O.8 1.80x0.277x0.ii]. 0.0439 0.80 0.994 0.172 0.168 cargo sh.lp,C5= 2.00xO.319x0.130 0.0592 0.7119 0.991 0.178 0.188 2M6

1/K0.O 1/K=1/r PLd2U Series 60. CBO.7

0.004 BC+OL 0.003 0.002 : Method A (G)0.53) £ Method A (0-0.47) 0.001 O Method B (61-0.39)

estiosted A Method B (il-0.35)

-: Using estimated k 10/d using measured k510/d .0 0 0.003 B5 0.0 0.003 00 0.003 00 0.004 0.002 0.0 Series 60. C50.B (sithout U.K.) A A A o o o O

Fig. 7 Roll damping coefficient of eddy making and lift components

Fig. 8 Roll damping coefficient of eddy making and lift components (Method B)

\\

\ 1.0 A A S A measured O Series 60. CB-O.7 2.0 A Series 60. C5-0.8 cargo ship model

: SR 98 tanker Y000ro s formula - present formula (22) :e.(1$)

0

/

-0.6 -0.4 -0.2 0 0 0.2 0.4 0.6 - ac/d

Fig. 9 Effect of roll axis on lift component

0.005 BE+BL 0.0 Bw (kg-m) 0. 000]_ 0.0 Bw 0.000L 0.0 0.0002, 0.0001. 0.0

cargo ship model cc50.7119)

reti axis c (OG/d0.108)

a measured by Takaki and (3m model) O measured (2m modei)__ 1. doublet

4-r6 is

estimated_{using Sti}

k5L0/d using Me2

4

0.2 P

Fig. 10 Roll damping coefficent ofeddy making and lift components for cargo ship model

y

N- roll axis

5_ rß0ws

Fig. 11 Coordinate system

r

basic term calculated for doublet model basic term Series 60. CBO.7 roll axis : o ...-(1 0.32 - :using :usi3sg timated usthieted k5i0/d meusured Ic5l/d O: measured SR 98 tanker roll axis r O SR 108 container ship 0.0 0.1 0.2 F5 0.0 0.). 0.0 _0.5 0(-wu/g0 Fig. 12 Examples of calculated results _for

dóublet model

M;d!1 lOOms Ñodel 4 B44 0.5V (kg-m) o 00 0. 844 00mm 3005m

.1 j4

I I E

j

J

-.-

_J l J'-300mm -JOOmm 150mm 200mm

O T )Oints of the dotoblets Fig. 13 Profiles of flat plates

Q

Fig. 14 Roll damping coefficient B.fói. Model i 0.0 B:, 00. iSrad w = 5 (1/seC) o measured 0 =1/4 o o 0.0 0.2 1000mm Model'l 00-0.l5rad w = 5)1/sec) O: measured 0 1/4 0001 044 (kg-roO 0.10 0.0

Fig. 16 Roll damping coefficiçnt B'4 for Model 3 0. Oi. 0.0 0. 1C

0

844 MOdel 3 e0mo.lsrad w= 5 (1/sec) 0d

9°

0°='==

B w a '9 w = 6(1/Sec)

___-o r=.-'

. ¿ 0.0 0.2

yl

J<asured

0.4 F0 044 _{Model 4} 0.10V 90=0.l5rad (kg-m) = 5 (1/5CC) o B 1W .,____ j...measured -. T o: forward : reverse t-: 01/4

H-0.4

F0-Fig. 17 Roll damping coefficient B4 for Model 4 Series 60, C5O.7 measured _: Io : O = 0.599 LS : O = 0.496 toO ml/4 o 0.2 - 0.3

Fig. 18 Wave making component of roll damping for ship model

0.0 0.5 O

1.0-Fig. 19 Effect of advance speed on wave Fig. 15 Roll damping coefficient B4 for making component

Model 2

12

-e0=o.l5rad w = 6 (1/sec) _o o Oc° o&t 0.009 B 0.004 o o 0.0 0.2 0.0 _e=o.15rad Model 2 600mm -200mm 0.0 0.4 F 0.2 0.0 0.0 044 0.10 0.0 0.]. 044 (kg-m) o10 0.05 o 1.0 0.0 0.10 0.05-0.0 t. 0.4 F5

30 20 10

0.0

measured

O i D : SR1O8 tainer snip

: SR159 tanker O : Series6ü,C5=O.6 O Series6O,CB=O.7 calculated for doublet model (d=0. 17m) : present formula (26) 0.00 0.005 0.0

-

13

-measured D: SR108 mtainer ship A : 5R159 tanker O: Series60,C5ß.6 O : Serios60,CQ.7 calculated for doublet del (d=0.17m) present formula (27)

Series 60, C=0.6 Series 60, CBO.6

Series 60, CB=O.7 roll axis :

(without O.K.) 80=O.l7Srad

o & =0.90

°

0 1344

O

SR 108 container ship(single screw) (without B.K.) _o O roll axis : O O

800.115rad O

0.0 0.5 djg)

Fig. 20 AL value for ship forms

0.0 0.1 0.2 0.3 F1 0.0 0.1 0.2 0.3 F0

Fig. 22 Roll damping coefficient Ê _{for ship models at forward speed}

0.0 0.5

Fig. 21 _{A2 value for Ship forms}

0.0

044

44 0.010. 0 005 estimated (with B.K. estimated - (a,ithout B.K.) 0.015 Series 60, C8=0.6 0.02 roll axis a O 8=0. l75rad = 0.6 0.0 0.0 0 . 0.2 0.3

Fig. 23 Roll damping coefficient Ê for ship model st forward speed

B44 0.015 0.010 0.005 measured A with B.K. estimated O without B.K.

- :usthg estimated kNlo/d

:USiflg measured k510/d Series60, C5=0.7 roll axis a Ô e=o. l7Srad = 0.60 A measured A a with B.K. estimated - O : without B.K.

- a

using estimated k510/d using measured k510/d 4.0 0.0 0.1 0.2 0.3 F5

Fig. 2-4 Roll damping coefficient Ê:4 for ship

modêl at forward speêd

0. 02 Series 60, C50.8 roll axis : O 00=0.l75rad =0.53-3 0.01 estimated (withOUt B.K.) o estimated (withãut B.K o A estimated 0.01 A _{(with S.S.)} A o estimated(with .-K.) measured A : with .K. O without B.K. es tima ted

-: using estimated kNlo/d

0.0 0.1 0.2 0.3 F

Fig. 25 Roll damping coefficient B4 for ship model at forward speed

0.01 00 0.005 0.0 0.005

14

-cargo ship model (CBO.7119) B

"

roll axis O 8=0.175rad O 0.50 measured A a with S.S. o: without S.S.

Fig. 26 Roll damping coefficient Ê:4 for cargo

ship model st forward speed

0.0

544

0.0

estimated

using estimated k510/d

cargo ship model (C=0.71l9)

roll axis : G (OG/d=0.108)

800.l57rad P50.l5 measured A: with 5.5. O: without S.S. estimated -: using estimated k510/d 0.5

cargo ship model (C90.71l9) roll axis a G (OG/d0.108)

O0. lS7rad Fa'O.2 0.01 measured A: with S.S. o without O.K. estimated (with 5.5.) o estimated (without S.S.) S A Ba 1.0 .4:

Fig. 27 Roll damping coefficient Ê4 for cargo

ship model at FO 15

estimated

-a using estimated k510/d

0.0 0.5 10 O

Fig. 28 Roll damping coefficient Ê for cargo ship model at F5=O.2

0 3

0.1

--0.2

F5