On roll damping force of ship - effect of friction of hull and normal force of bilge keels

4 DEC.

ARCHEr

Report

of

Department of Naval Architecture

University of Osaka Prefecture

No. 00402

April

, 1979

*

On Roll Damping Force of Ship

- Effect of Hull

Surface Pressure Created by Bilge

Keels-Lab. v. Scheepsbouwkunre

Technische Hogeschoo

Deift

b.

**

Yoshiho IKEDA

Yoji HIMENO

***

KiyoShl KOMATSU

*****

and Nono TANAKA

* published in the Journal of The Kansai Society of Naval Architects,

Japan , No.165 (1977)

Research Assistant, University of Osaka Prefecture

Student of Master Course,

Associate Professor

Professor

j

H

e

G

Report of Department of Naval Architecture, University of

Osaka Prefecture , No.00402 , April 1979

Department of Naval Architecture University of Osaka Prefecture

Mozu-Umemachj , Sakai-shj

On Roll Damping Förce of Ship

-Effect of Hull Surface Pressure Created by Bilge

Keels-by Yoshiho IIOEDA , Kiyoshi KOMATSU

Yoji HIMENO and Nono TANAKA

SUMMARY

The roll damping force of bilge keels consists of two components, one due to the normal force acting on bilge keels and the other due to the hull

press-ure created by bilge keels.

Measurement of the hull pressure for several two dimensional cylinders with bilge keels were carried öut. On the basis of these experimental results,

a formula for the .ròll damping coefficient due to the hull pressure component

is proposed. The estimated values of the roll damping coefficient due to bilge keels which are derived from this formula together with the formula for the normal force component recently proposed by the authors, show fairly good agre-ements with the experimental values for two dimensional cylinders and ship

models with bilge keels. Introduction

Bilge keels are important equipments fOr reducing roll motion because of the simple form and the large roll-damping effect. Many works on the bIlge keels effect have been done since Froude's work. In Japan the prediction methods of Watanabe-Inoue 1), Tanaka 2) and Kato 3) _{are often used. However the différence}

of the results of these formulas 'is not always small , so that further works

on the bilge keel effect seems to be necessary.

In the previous works on bilge keels, the drag coefficient has been consid-ered to be the saine as that of flat plates in steady flow. However, the authors

have recently paid attention to the fact that the drag of oscillating flat plates

depends on the period parameter arid considered that the drag coefficients of

bil-ge keels should also depend on this parameter. On the basis of this point of view, the authors have recently proposed a prediction formula of the normal force component of bilge-keel roll damping in whidh the drag coefficient of bilge keels

is expressed as a funct-ion of the period parameter ).

This paper concerns the roll damping due to the pressure variation on a hull created by bilge keels which can be treated in the similar way to that of the

normal force.. The research work on the hull pressure components of the bilge keel

damping has been carried out by Sasajima 6) , Tanaka 2) and Goda 7) But theré

seems to be no measurement of the pressure on the hull except for Goda's work

so that the detailed nature of the pressure change created by bilge keels has still

been unknown.

The results of the pressure measuremeñts for several two-dmeñsional cylinders

are stated here. It is found that the pressure created by bilge keels depends oñ

the period parameter strongly as has been expected. A prediction formula for

this

hull-surface-pressure component is proposed in this paper. Finally a prediction method for the total bilge-keel effect is obtained by adding this formula to the one for the normal fOrce component proposed by the authors recently 5)

Measurement of Hull Surface Pressure

Models used in the experiments are two 2-dimensional models (Model A and Model F) and an ellipsoid model (Model G) whose cross-sections and locations of

pressure measurements are shown in Fig.l. Principal particulars of these models

are shown in Table 1. F-ig.2 shows the sketch of the pressure sênsör attached to

the hull. The two-dimensional modéls are installed perpendicular to the surface

in order to get rid of the free-surface effect and the hydrostatic pressure

eff-ect. The fiat plates are setted vertically along thé original water line of the

model to 'simulate the flow with rigid free surface condition. The ron axis of

the ellipsoid coincides with its central axis in order to get rid of the

wave-making effect. These models were forced to roll by the forced-roll equipment, and

the pressure on the hulls were measured.

The non-dimensional pressure coefficient C is obtained from the value of

the pressure P at the moment when the roIl angle e=o or the roll angular velocity

P.

Cv_1

where r,w and O represent the distance from the roil axis to the bilge keel, the

roll circular frequency and the roll amplitude respectively

The experimental results of the pressure coeffiáïent C at each location on

Model A , F and G are shown in Figs.4 to 6 , in which the positive pressure

coeff-icients in front of bilge keels are plotted with marks and the negative ones

behind bilge keels with marks O . From these figures, it is found that the

press-ure coefficient C for each hull form is constant against the ròll frequency, in

other words, the pressure created by bilge keels is in proportion to the square

of the roll frequency. As seen from Fig.4 , the effect of the roll amplitude on

the pressure coefficient does not appear in the front pressure , but appears in

the negative pressure behind bilge keels. And the effect of the roll amplitude,

i e , the period parameter , on the pressure shows a sinu.lar tendency with that

on the normal force of bilge keels -, where the period parameter is defined as

tJmaxT/D (Umax : the maximum velocity , T : period , D : the representative length)

which was used first by Kéulegan and Carpenter for arranging the experimental

data of the drag of oscillating bluff bodie.s , so it is often called K-C number.This

period parameter can be expressed as lrrOo/bBK (bBK : the breadth of the bilge

keel) for the case of bilge keels. As seen from this expression, the period

parameter depends only on the roll amplitude and not on the roll frequency.

The distribution profiles of the hull surface pressure measured are shown in

Figs.7 to 11. The values of the positive pressure in front of the bilge keel

decreases with the distance from the bilge keel, while the negative pressure behind the bilge keel remains constant for a certain distance from the bilge keel andgradually decreases. These tendencies of the shape of the pressure distri-bution seem not to depénd on the shape of cross-sections. From these figures,

we can find that the value and the distribution length of the negative pressure

depend on the period parameter , and the distributiòn length becomes longer as

the roll amplitude Oo increases. The measured values of the length of the

ñeg-ative pressure are shown in Fig.l3 , from which we can safely say that the effect of the shape of cross-sections on the distribution length S/bBK of the négative

pressure is not so large.

F-rom these measurements , it can be concluded that the positive pressure in

front of the bilge keel is independent on the period parameter , and that the

value and the distribution length of the negative pressure behind bilge keels

depend on the period parameter.

3. Prediction Method of Hull Surface Pressure Component of Bilge Keel Damping This section concerns the prediction of the hull surface pressure component of the bilge-keel damping by integrating the hull pressure assumed on the basis of

the experimental results mentioned above.

-- we can express the hull surface pressure created by bilge keels as, -

-Ppr2f2jbIC

(2)

where f is the modification factor of the flow velocity at the bilge keel,

especi-ally for full ship forms.

Integrating the pressure P over the hull surface , the roll damping moment MR due to the hull surface pressure created by bilge keels can be obtained as,

--- MR=_pr2f2OjIfCP.ldG (3)

and the non-dimensional linear damping coefficient is

-i.'C--

f

CpldG -

-S3%.

_{VB2 .k}

-where dG,l,B and V aré a girthwise length element , the moment Ieyer in a section, the beam of the ship and the displacement volume respectively. B5 is defined

as,--- - - B

f

-

B

--. (5)_

where BSMR/0 . The sûffix "s" represents the surface pressure component of bilge keel damping. The damping coefficient B _{can be obtained by prescribing- the value}

V

C and 1 in Eq. (4).

We assume the pressure coefficient C as in the following , on the basis of the experimental results shown in Chapter 2.

At first , we assume the pressure coefficient C on the front surface of the bilge

keel. As seen from the experimental results shown in Figs.7 to 17 , the positive

pressure coefficient C has its maximum at the front face of the bilge keel. So

that , we can assume that C before the bilge keel has the maximum value C at

the front face , decreases linearly with distance and then reaches zero value at

the position of free surface or the center line on the bottom surface of the hull.

The value of C , the measured values of which are shown in Fig.2 , is empirically

assumed as

C=1.2 (6)

The distribution shape of the negative pressure behind the bilge keel looks like

a trapezoid. We assume that the distribution length S of the negative pressure

can be expressed as the following equation , and that the pressure coefficient is constant in the range of the bilge keel to S/2 , and then decreases linearly

to zero at the distance of S from the bilge keel. The distribution length S can

be expressed empirically as follow,

S/bn=O.4( ,rfreo)+26 (7)

The pressure coefficient Cj at the back face of the bilge keel can be obtained in

such a manner that the value subtracting C from C should coincide with the drag coefficient CD of the normal force of the tilge keel.

C;=1.2CD=_22.

,rrfO0 1.2 (8)

Thus B5 can be obtained by integrating the product of the pressure assumed as above mentioned and the moment lever at each point on a hull over the hull sur-face.

Subsequently , we can derive a practical prediction formula for as in the

following manner. In order to simplify the calculation , a cross-section is

assumed to have vertical side-walls , a horizontal bottom and quadrant bilges as

shown in Fig.14 , and also a pair of bilge keels are attached at the each center of

the bilge circle. Furthermore , we replace the distribution length S of the

negative pressure with following S0 which is a constant pressure distribution

length,

S0/b,=o.3(_!L!\

_{b, )} 1.95

From the calculation under the assumptions mentioned above , the integral term in eq. (4) can be expressed as

fCpldG_d2(AC+BCp)

(IO)

In eq. (10) , A and B _{, which are functions of the period parameter and the hull}

shape , are shown in Appendix. _{The difference between the appoximated calculation}

by eq. (lO) and an exact integration of assumed _{pressure over a hull surface is}

little observed for ordinary ship hull forms.

Finally, the prediction method of the bilge keel component can be summarized

as follows. _{The bilge keel component BBK can be estimated by summing up the}

nor-mal force component BN and the surface pressure component B5 as,

BBK =BN +B5 (9) where CD is, 8 _{r2is90f2 I d'} VB2 1rbßxCD+_-(AC+BCp+)} (11) CD=r22.5 ,rfrOo+2.4

The modification factor f in eq.(l1) can be determined from _{experiments. The} experimental results of the bilge keel effects attached on two-dimensional models

(Model E,D,C and B) with various bilge radius are shown in Figs.l7 to 20. In these

figures , the broken lines show the p±edicted values in

the case of f=l.0 , and as

seen from these figures , the difference between the measured values

lines grows larger as the bilge radius decreases. The reason seems to be because the velocity increase at the bilge circle will causealargerbilge keel effect iñcrease.

When represent the velocity y of the bilge keel by fre with the modification

fac-tor f , the factor should be taken into all the velocity terms and the period

parameter terms of the bilge keel. It can be determined so that the predicted

values by eq. (il) should agree with the experimental results shown in Figs. 17 to

20.

fr1+O

where

a

is the sectional area coefficient of the section. The value f from

eq (12) is about 1.1 at most for the midship section of Series 60 , CB=O.B ship

form , so that it is taken into account only for full ship form. Though the

fac-tor f could be calculated by potential flow theory , the calculated values for

Lewis form cylinders are so large that we cannot use them , the reason of which seems to be that the difference between solid lines and broken lines shown in Figs. 17 to 20 are caused not only by the velocity increase but also by some viscous

effects. In this paper , however , this discrepancy does not affect on the

predi-ction resúlts because the factor f is determined from the experimental results. The predicted values for two-dimensional cylinders with various sectional

area coefficient

a

are shown in Figs.l5 to 16. For all

a

values , the damping

coefficient BBK increases with the breadth of the bilge keel and the rate of the

incréase becomes larger as

o

increases.

4. Comparison with Experimental Results of Forced Roll Tests.

In this chapter ,the comparisons between the predicted values by eqo (il) and

the experimental ones are made The principal dimensions of models are shown in

Table 1. End plates are setted both sides of the two-dimensional models..

In the analysis of the experiment , the moment values at the instanct 0=0

or when the roll angular velocity becomes maximum are read from the records of the

force

roil

momént , and the lineardamping coefficient B44 and the non-dimensional

ones 844 are obtained by using the following equations.

11*

(14)

0)80

44

D44pVBZ 2g

The value of 844 in eq. grating the roll moment the roll damping moment udès only the component owing equations.

(1

(15) is different from the one which is calculated by

inte-in one swinte-ing or by the fourier analysis , in the case that

has higher term., For example , if the

r9ll

damping

mcl-proportional to

obi

B4 is related to B44 by the

foil-(le

For the comparison between prdicted values arid experimental ones , we-assume

that the,rol1 damping coefficient BK due to bilge keels can be obtained by

subt-racting B of a naked hull from B4 for a hull with bilge keels. The comparisons

between predicted and measured values for two-dimensional cylinders with various

bilge radius are shown in Figs.l7 to 20. The slight modification for the factor f

are shown to.give good fits to the experients.

The effect of the roll amplitude on BK are shown in Figs.2l and 22. The

predicted value has a value at 00=0. The reason is because the roll damping due to

bilge keels consists not only the non-linear term proportional to but also the

linear term proportional to 00. The agreement between the prediction and the.

measurement is fairly good.

The. comparjsons for three-dimensional ship forms (Model H,I,J and K) in Fig.23

also show good agreements in all cases. The improvement on predicting the effeôt

of the roll amplitude is achieved compared with Watanabe-Inoue's method l) and Tanaka's method 2) _{From the 'agreement between predicted and experimental value}

by Tasai 'and Takaki 8) shown in Fig.24 , it can be safely said that the

assump-tion that the roll damping due to bilge keels arà proporassump-tional to the square of

the roll frequency is resonable.

bilge keel component are shown in Fig.25. The effectofbilge keels is almost the same as the one at zero advance speed , ii' the Froude number. range under 0:2,

but the effect of the advance speed increases in a small-amplitúde range at high

Froude number. The tendency of décreasing non-linear damping component arid

increasin? linear one is also shown in experimental results by Yamanouchi 9) and

Tanaka As seen from Fig 25 , however _{, we can safely use the predicted values}

at zero advance speed even at a high advanpe speed range iii the moderate roll ampl-itude over 10 deg.

5. Conôlusions

The authors have proposed a prediction meto of thé surface-pressure compo-nent of the bilge keel damping based on the experimental results of _the surface-pressure measurement _{The comparisons between predicted and experimental results}

of roll damping due to bilge keels have also been mâde. The following conclusIons

can be obtained. ..

The pressure on a ship hull surface created by bilge keels is In proportion _to

the square of the roll frequency , and the values of the negative pressure behind

the bilge keel depend on the. period parameter. '

A predIction method of the huil-surfâce-pressure component is presentêd by

assuming the pressure distribution on the hull on the basis of pressure' measuements.

The predicted values of bilge' keels which _{are obtained bysuperposing the normal}

force component and the hull-surface-pressure _{component are in good agreément with}

the experìmental results for two-dimensional cylinders.

The predicted roll damping fOr three-dimensional actual ship formi are in good

agreement with the experimental results

The prediction method proposed by the authors can safely be used in the presence

of the advance speed except for small roll amplitude _{and high advancé spéed ran'qe.} Acknowledgment

The present work-was carried out as a part of the research work of the l6lth Research Committee of the Shipbuilding Research _{Association of Japan. The authors} would like to express their gratitude to Prof. Schoichi Nakamura , Osaka University,

the chairman of this committee and to the committee members for their valuable

sugg-essions and discussions. _{A part of this work was supported by the Scientific}

Research Fund of the Ministry of Education. 'The _{computer at University of Osaka}

Prefecture was used for the calculation.

Reference .

Y.Watanabe, S.Inoue and T.Murahashj : The modification of Rolling Resistance. for

Full Ships , Jour, of Soc. of Naval Arch. of West Japan

, Jo.27(l964) ,p.69

N.Tanaka : AStudy on the Bilge Keel , Part l-Part 4 , Jour. Soc. of Naval Arch.

Japan, No.101,103,105,109 (1957,1958,1959,1961)

H.Kato : Effects on Bilge Keels on the Rolling Ships,

Jour. Soc.of Naval Arch. Japan, Vol.117 (1965), p.93

N.Tanaka : Bilge Keel , Proceeding of the Symposium on Seakeeping,

Soc. Naval

Arch. Japan (1969) , p.143

Y.Ikeda, Y.Himeno and N.Tanaka _{On Roll Damping Force of Ship -Effect of}

Hull

Friction of Hull and Normal Force of Bilge _{Keels-, JOUE, of The Kansai Soc.}

of Naval Arch. Japan, No.161 (1976), p.4]. .

H. Sasajima : Effect of Bilge Keel in Rolling of Ship,

Jour. Soc. f Naval

Arch. Jàpan, Vol.86 (1954), p.285

-K.Goda and T.Miyamoto : Measurement of Hydrodynamic Pressure on Two-Dimensional

Ship Models Heaving 'and 'Rolling _{with Large Amplitude}

, Jour, of Soc. of Naval Arch. of West Japan, No.49(1974)., p.17

Japan Ship Research AssociatiOn, SR.161 Committee :' Ón thePrediction of Ship Performance in Wavé, Rep.275(1977)

Y.Yamariouchi _{On the Analysis of the Ship Oscillations}

among Waves, Part 1,

Your. Soc. of 'Naval Arch. Japan , Vol.109 (1961), p.169

N.Tanalça, Y.Himeno, YOgurà and _{K.Ì'lasuyama : Free Rolling Test at Forward}

Speed'., Jour, of The K'ansaj. Soc. of'Naza1 _{Arch., No.146 (1972), p.63}

-5-Appendix

A and B in equation (10) can be obtained as follows

A= (m3 +ni4)m, - m

+(l_rni)2ms_m2)

- 3(HoO.215m,) 6(1-0.215m,) + m1(m3m5 +m4m,) n1=R/d, ,n2=OG/d

in3=1m1m2, m4=H,-1n1

0. 414Ff0+O.O651'fl (0.382H,+0.0106)mi (H,.-0.215m,)(1-0.?lSmi) 0. 414H + 0. 0651mi (0.382+0. 0106H0) m1 (H0-0.215m1)(1-0.215mi) J S,/d-0. 25,rm1 (S0>0. 252rR) R= m7=, O

(S,0. 25rR)

m7+0.414m, (So>0.25rR) m7 +p'T (1cos(Sô/R)} ,n1(So0. 25rR)

Bilge radius R and the distance r between the roll axis and the bilge keel can b

expressed as, (R<d. R<B12) d (H,1, R/d>1) B/2 (H01, R/ d>H0) 4f H0 -

_{2 IdJ}

0 _.I\.&10 I

2 Idi

-6-Fig. 1. Measuring points of pressure.

Table 1. Particular of models.

-7-Fig. 3. Schematic view f test- condition.

-1

MODEL F

Roll axis

b3,0.0O8m A C, io frost of 5.0.

00 -0.17-Orad O C, behind O.K.

0.3 0.6 0 0.9 0 3 0.6 0 O9

Fig. 5. Hull surface pressure created by

bilge keels (Model F).

Model C

C _A : in front of S.K.

2

o behind B.K.

Fig. 6. Hull surface pressure created by

bilge keels (Model G).

Model note - L (ro) B (ro) d Cm) (m') H0 (B/2d) e --.. B.K. (in -X ro) Roll. axis

r.. u A bilge radius 3cm -0.5 0.25 0.i 0.0123 1.25- 0.985 0.005 x 0.45 0.009 X 0.45 0.015 X 0.45 0 0 0

B bilge radius i cm 0.8 0.28 0.112 0.02501 1.25 0.997 O.005X 0.8 0

a 2 cm 0.8 0.28 0.112 0.02495 l.25 0.9945 0.010 X 0.8 'b -b 3 cm 0. 0.28 0l22. 0.02702- 115 0.989. 0.010 X 0.8 0 E Series 60 ,CBO.6 os 5 0.8 0.237 0.096 0.01775 1.232 0.977 0.010 X 0.8 0 F Series 60_{os 7 0.8 0.398 0.]j3 0.0549 1.036} 0.893 0.008 X 0.8 0 B G ellipsoid 1.5 0.30 - 0.15 0.03534 -, 0.010 X 0.6 0 SR 108 container (single screw) 1.75 0.254 0.095 0.0241- 0.0045 X 0.44 0 I Series 60 ,Co=0.6 1.8 0.2365 0.096 00247 0.0054 x'0.63 O C8-0.7 1._8- 0.2570 0.1028 0.-0331 - - 0.0054 x 063 O K, CB=O.8 1.8 0.2769 0.1108 Ò.0439 - 0.0054 X 0.63 0 A B AA A A

00

0 -0 A A AA o - o D A A o O 1 0 D AA A A A 0.3 0.5 1.0 -1 -2 00 O o o o -3 Adhesive 2.8inxs Amplifier Sensor

r

Cap

Fig. 2. Sketch of pressure sensor.

Cl, -2 00 -3 Cp A F -i--2 o o o C1,1 -1 Cpi -1 Cpi -1

MODEL Â CI o -2 cp 2 O -2 c O -2 -4 C, in front of B.K.(80 0.199) z C behind B.K.(80 0.199) z C, in front of B.K.(90 0.295)

Fig. 7. Pressure distribution on hull.

measúred

JiadelA

b 0.009m o O- 0.199

c A 0. 0.295

Fig. 8. Pressüre d stribution onhull.

Fig. .9. Pressure distributión on hull.

-8-O C, behind B.K.(90= 0.295)

co

Fig. 10. Pressure dstribution on hull.

Fig. 11. Pressure distribution on hull.

measured O 5Odé1 A a lodel F model G O ₈ C1.2 A

-

D G

: '

- . .. .

i.

.

.

B _.E .

:.tta4t

-òo ¿6-O

-

-..-. O

00000.

e $ e. .c.

,

%

--.-..-.

ra*

00

b 0.009m 33K . 0

00

0 o O e . . .- . - - .

-o0 b00 O

0.2 O4 0.6 0.8 0 4 0.6 0.8 w 0.4

Fig. 4. Hull Surface pressure created by bilge keels (Módel A).

à.0 0.005 0.01 0.015 b35

Fig. 12. Pressure coefficient in fröñt of

-bilge keéls.

0.6 0.8

0.06 0.04 0.02 0.0 0.02 0.0 0.0 1(odl.25 6o0. l75racl ì =0.4 Roll axis : O H0=l.25 a = 0.997 BBK .60o0.l75rad a = 0.920 0.04 0.4 Roll Axis : O a = 0.997 a = 0.990 o = 0.970 o = 0.950 a = 0.920 Model E O : measured e0=0.I7Srad bBKO. 01m 0=0.4 (irre0) +2. Gb05

Fig. 13. Distribution length of negative

pressure.

negativc pressure

Fig. 14. Assumed pressure distribution.

0.05

o

b3 K/B

Fig. 15. Calculated values of roll damping coefficient due to bilge keels.

s 0.1 00 00 asir 0.05 0.0 - present formula present formula (f=l.0) 00 -. 0.0 0.0 o : measured OØ=O . 227rad bBK=O. 01m

Fig. 19. Roll damping coefficient

BK Model B

present 0 0.751

formula 0 0.536 Model C

Fig. 21. Roll damping coefficient ax.

Model F present O = 0.609 formula O = 0.560 Model C present formula present formula (f=l .0) o present formüla present foriñula (fol.0) present formula present formula (f=l .0) measured O r S 0751 a r S = 0.536 measured or 0 0.689 a r O = 0.560 measured megative o e modelA model F BBK Model D Q r measured rossore f Io,oç 0.05 60=O.l7lrad b =0.Olm 5K' os a-o o o 0.0 - 0.5

Fig. 18. Roll damping coefficient

0.0 0.5 - 1.0 _.0

0.1 0.2 8,(rad) 0.3

Fig. 17. Roll damping coefficient _{Fig. 22. Roll damping coefficient}

1.0 w 1.0 0.5-0.0 _w O measured =0 . 17 S'rad bBK=O.00Sm 0.05 0.0 0.5 Id 1.0 force. corn-0.0 0.05

Fig. 16. Comparison between normal component and hull pressure

ponent. -.0 0.1 0.2 80(rad) 0.3 o o.ö 5.0 10.0 15.0 b55 B5( 0.0 0.01 B55 0.005

0.01 BK 0.005 0.02 Bar 0.01 0.0 Model II

SR 108 container ship

(single screw) measurèd : _{= 0.894} O : = 0.397 Model J

Series 60

, C5 0.7 méasured O : eO.496

present formula

C5 0.7119 cargo abip model

A 8=15°

o o1O° measured by TOSai

o 0 .5° and Takagi. precent

formula 815e010°

6Q 5°

Fig. 24. Roll dampiñg coefficient 2oK due

to bilge keels. 0.005 formula 0.. 01 0.005 -10-= Model I

Series 60

, C 0.6 measured B O: = 0.575

(3=wJ7)

o o o O

present formula

Model I

Series 60

, CB=O.8 measured O: = 0.533 o

_{present formula}

o G: F,0.3' 0.0 0.005 0.0 VQQj o : A.: 0: Módel I Scrjea 60 . C5O.6 P0.0 -F0.2 measurèd F00.3 0 o A O ₀ 00

0-0

9 a Model J Serica 60 . C9=0.7 O A O A

Bar Model K_{Series 60 . C5O.0} 02

0.01 _O A o neadured 0: 0.005 o A A: FO.2 0.005 0.0 .0 measured A B o:1'O.O 0 0

A:j'0.25

O : F00.35 0.0 0..l 0.2 9 0.3 0.0 0.1

02

0..3 (rad) 0.0 0.1 0.2 80 0.3 0.0 0.1 0.2 8 0.3

Fig. 23. Roll damping èoefficient due to bilge keels.

0.0 0.1 0.2 90(rad) 0 3

Fig. 25. Effect of advance spe on roll

damping due to bilge keels.

1.5 . 0.5 0.0 0.01 BBK 0.005