The distribution of hydrodynamic mass and damping of an oscillating ship form in shallow water

TECHNISCHE HOGESCHOOL DELFT

AFDELING DER SCHEEPSBOUW- EN SCHEEPVAARTKUNDE

LABORATORIUM VOOR SCHEEPSHYDROMECHANICA

THE DISTRIBUTION OF HYDRODYNAMIC MASS

AND DAMPING OF AN OSCILLATING SHIPFORM

IN SHALLOW WATER

W. Beukelman and Prof.ir. J. Gerritsma

Conference on Behaviour of Ships in

Re-stricted Water, Eleventh Scientific and

Methodological Seminar on Ship

Hydrody-namics,

Bulgarian Ship

Hydromechanics

Centre, Varna, 11 - 13 November 1982.

Report No. 546-P

March 1982

Ship Hydromechanics Laboratory

- Delft

Delft University of Technology

Ship Hydromechanics Laboratory

Mekelweg 2

2628 CD DELFT

The Netherlands Phone 015 -786882

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BULGARIAN

SHIP HYDRODYNAMICS

CENTRE

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MADBATEPE-CONFERENCE ON BEIIAVIOUR OF

SUIPS IN RESTRICTED WATERS

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PROCEEDINGS, Volume I

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Eleventh

Scientific and Methodological

Seminar

on

Ship

Hydrodynamics

THE DISTRIBUTICN OF HYDRODYNAMIC MASS AND DAMPING OF AN OSCILLATING SHIPFORM

IN SHALLOW WATER

W. Beukelman, J. Gerritsum

Introduction

The depth of water has an important

influ-ence on the vertical and horizontal motions of a ship in waves, in particular when the waterdepth is sraller than two and a half times the draught of the

vessel.

In shallow water the keel clearance depends

to a large extent on the combined effects of trim,

sinkage and the vertical displacement of the ship's

hull as a result of the ship notions in waves. Keel

clearance is of interest to ship owners and port

authorities, because of the ircreasing draught of

large cargo ships and the corresponding smaller

waterdepthidraught ratio's. The safety and manoeuvr-ability cf a ship are influenced by the amount of

keel clearance and the cost of dredging depends tc a large extent on the allowable minimum keel clearance of the largest ships considered.

A detailed knowledge of the vertical motions

of a ship due to waves in shallow water will be of

interest to assist in solving such problems.

From a technical point of view strip _theorY methods to calculate ship motions due to waves in

deep water have proved to give satisfactory results. Except fcr the rolling motions, viscous effects are

not impertant in strip theory calculations, but an

accurate determination of 2-dimensional damping and

added mass of ship-like cross sections is necessary, as shown earlier 1.1] .

In the case of shallow water the use of strip theory calculations is not obvious, because a much

larger influence of viscosity can be expected when

the keel clearance is small: In addition, the flow

conditions near the bow and the stern will differ

to a large extent from the two-dimensional flow

assumption, as used in the strip theory.

The present investigation concerns the comm

and damping, as measured on a segmented sMT, model

In shallow water, with corresponding calculated re sults using a strip theory method which takes the

finite waterdepth into account.

It should be noted that in these calculati-ons no viscosity effects have been included.

In view of the comparison with calculations the physical model has been restrained from

sink-age and trim, which would occur in the case of a free floating model. In addition to the heavinn and pitching motions also fnrced horizontal motions in

the sway and yaw mode have been carried out.

The experiments included the effects of

fonvard speed, frequency of oscilliation and water-depth. A ranpe of frequencies have been chosen to

cover wave frequencies of interest for ship

resoon-ses.

The use of a segmented ship model enables

the determination of the sectional values of damp-ing and added mass. This technique has been used

earlier for an analogous investigation of the deeo

water case El) .

See appendix I.

The calculations have been carried out with a corputer program developed by H.Keil (3) .

In this calculation the hydrodynamic r,ass and damping for 2-dimensional ship- like cross sec-tions are computed with potential flow theory,

using a source and a linear combination of

multi-pole potentials, which satisfy the boundary con-ditions at the free surface, the bottom, and the

contour of the cross section. A Lewis transformati

on has been used to penerate ship-like cross sec-tions.

The model.

The forced oscillation experiments have

been carried out with a 2.3 meter rpdel of the Six-ty Series. The main particulars are given in Table

1. The sare model has been used earlier for the ara logous tests in deep water [1,2] . The model has

been divided in seven segments each of which was

separately connected to strong beam by means of a

KOHOEPEEUHR CONFERENCE

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B

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ISAPBATEPE RESTRICTED WATERS

Table 1,

Length between perpendiculars

Lpp 2.258 m

Length on the water line LwL 2.296 m.

Beam B 0,322 m

Draught T 0.129 m

Volume of displacement 'V 0,0657m3

Blockcoefficient

CB 0.700 Waterpl ane area AwL 0 572 m2

Longitudinal moment of inertia I 1685m2

of waterplane

LCB forward of L/2

PP

LCF aft of L /2 PP

Table 2a. Heave

The various oscillation amplitudes _{cover a}

. Fn = 0.1 and 0.2

Table 2b. Pitch

. Fn = 0.1 and 0.2.

x Fn = 0.2 only

Table 2c. Sway and yaw

. Fn _{0, 1 and 0.2}

11

2

These dynamometers measured vertical or

hori-zontal forces only.

The test set up for vertical motions is given in Figure 1, A similar system has been used for _the horizontal motions, see Figure 2.

The instrumentation allowed the determination of in-phase and quadrature components of the verti-cal or horizontal forces on each of theseven

seg-ments when they perform forced harmonic motions with a given amplitude and frequency.

It has been shown earlier that the _influence of the gaps between segments can be neglected. [2] .

Test condi tions.

certain range, depending on the mode of motion, to study the occurrence of non-linearities,

The test conditions are summarized in Table 2.

These conditions include the waterdepth-draught ratio h/T, the oscillator amplitude _{r, the} frequency of oscillation w and the forward speed,

expressed as the Froude number Fn.

It should be noted that the distance _between the two oscillator rods (see Figure 1) is one meter.

Consequently for the pitch and the yaw modes a 0.01

meter oscillation amplitude corresponds to a 1.146

degree motion amplitude. The dimensionless _frequency covers a range of

w (E.71 = 1.9 - 5.8 for pitch and heave, and:

w

fE7=

1.9 - 4.8 for sway and yaw.

Experimental resul ts

For each of the considered _{modes of motion} the in-phase and quadrature components of the excit-ing forces has been determined. _{* These}

components

have been elaborated to the hydrodynamic mass and

hydrodynamic damping coefficient of each _segment, taking into account the amplitude and _{frequency of}

the harmonic motion.

The following expressions have been _{used in}

this respect (see Appendix 1).

Heave:

(PV aZZ)i + + _{cZZ z}

-dze - eZ96 -gZ9

e =

= Fzsin(wt+Ez)

Pitch:

(I+aee

TT )4

+bee

_6+cPe

e-d i-eOZ

_9Z - g82z = = Mesin(wt+Ee) . = 4,6,8,10,12 rad/s r (M) h/T 2.40 1.80 1.50 1,20 1,15 1 0,005 . . . . . 0,

alo

. 0.020 . . . . _0,030 w = 4,6,8,10,12 rad/s r (M) h/T 2.40 1.80 1.50 1.20 1,15 0.005 ' 0.010 .

.

0,015 w = 4,6,8,9,10 rad/s r (M) h/T 2,4 1.8 1.5 1,2 1.15 -

'

_0.010 - , .

_0,00

. 0,030 0,011 ni 0,038 m

Sway:

+ ayy)Y + byyi - dy*W - ey*i = Fy sin(wt+cy)

Yaw:

-dY Y-e*

*Y *

=Msin(wt+m ) (1)

For the individual segments the following equations result: Heave: + c z zz Fz sin(wt+cz) Pitch: (0,

Xi + dze)e + eZee + gzee = - Fesin(wt+ce)

Sway:

(.

0

+ 40Y +

= ry'sin(wt+4) Yaw:

(ovaxi + cl,*(*); + _{- F;sin(wt+c;)}

(2)

In these equations a refers to hydrodynamic

mass, b is the hydrodynamic damping coefficient and

c is a restoring force- or Moment coefficient. The coefficients d, e and g are .the

corre-sponding cross coupling coefficients. The position of a segment is denoted by X. and values of the

coef-ficients of segments are indicated by the asterix In Appendix 1 the data reduction of the

re-sults obtained from the oscillator experiments is

treated in some detail.

The coefficients a, b, d and e have been

ob-tained by integration over the length of the model of

the results of the segments.

In the Fig.3 to 26 the experimental values of hydrodynamic mass, damping and cross coupling

coef-ficients are given for pitch, heave, yaw and sway as

a function of the frequency of oscillation w and the relative waterdepth h/T.

Two forwards speeds corresponding to En = 0.1 and Fn = 0.2 have been considered.

In general the experiments indicate a rather

good linearity with regard to the amplitudes of

mo-tion, except some minor non-linearities at the

small-est waterdepth.

Mass and damping coefficients of heave and

pitch increase with decreasing waterdepth for all

considered frequencies, in particular for h/T < 1.5. For the lateral motions, sway and yaw, the

hydrodynamic mass coefficients decrease with

decreas-ing waterdepth, whereas the dampdecreas-ing coefficients de-crease slightly or are almost independent of

water-'00)i '004'

(pV + a )z + b z

zz zz

The distribution of the hydrodynamic mass and

damping along the length of the model is given in the Figures 3 to 18 for heave, pitch, sway and yaw, as a function of frequency, waterdepth and forward speed.

The distribution of the hydrodynamic mass, expressed as a percentage of the total hydrodynamic mass,is not greatly influenced by the waterdepth, but for the

distribution of the damping coefficients a

signifi-cant shift of larger damping values towards the fore body of the shipmoliel mi-th decreasing waterdepth is

observed.

For low frequencies of oscillation, combined with low forwards speeds wall effects or oscillation in the models own wave-system could have influenced

the measurements. This could explain sore of the

ir-regularities in case of the lowest speed En = O. 1. and frequencies equal or beloww = 6 rad/s. In all other cases wall effects do not seem to have influenced the

experimental results.

Calculated hydrodynamic mass and damping

The measured mass and damping values have been compared with the corresponding . calculated values,

according to the numerical procedure as given by Keil

(3) This concerns the coefficients a, b, d and e

for the four considered modes of motion, as well as

the distribution of these quantities along the length

of the model.

The results are shown in the Figures 3 to 26. In the strip theory the added mass and damping values at zero speed of advance are used to compose

the coefficients of the equations of motion. The

ex-pressions for the sectional coefficients for heave

and pitch as derived in [4] are given in Appendix 2 together with an analogous extension for sway and yaw.

Two versions of the strip theory have been

used.

Version 1 leads to the ordinary strip theory

method, which lackt some of the symmetry relations

in the damping cross coupling coefficients.

Version 2 includes these additional terms. In general the calculated results according to both

versions agree rather well except for the sectional

Values- of the coefficients near the ends of the ship

form

For the integrated values of mass and

damp-ing the differences between version 1 and 2 may be neglected.

For zero forward speed the calculated values

of added mass and damping are presented in table 3 for heave and sway, the different frequencies

Table 3.

Calculated added mass and damping for heave and sway at zero speed

HEAVE Fn =

The calculated hydrodynamic mass for vertical motions agrees very well with the experimental

valu-es for the ship on forward speed. For the damping coefficients the agreement at the lower relative wa-ter depths and higher frequencies is less

satisfac-tory, which might be due to viscous influence. The

same phenomena though less pronounced is found for the case of deep water [1, 2] .

This applies also to the horizontal motions,

sway and yaw, although the differences for damping

are somewilat smaller than for the vertical motions.

A reasonable agreement is found for the

dis-tribution of mass and damping along the length of

the shipmodel, except in those cases where wall

ef-fect could have influenced the experimental results, as discussed above.

Conclusions.

The results of this detailed coeparison of

measured and calculated mass and damping values for

vertical and horizontal motions indicate that strip

theory methods, using potential theory to determine

11 - 4

hydrodynamic mass and damping can be of value for

the calculation of ship response due to waves in

shallow water, at least for engineering purposes. A limited number of model experiments to

de-termine the amplitude response of heave and pitch in shallow water and the comparison with calculated

motions confirm this conclusion to a certain extent

for the vertical motions [5] , see Figure 27 a+b.

Acknowledgement

The authors are indebted to ing.A.P. de Zwaan who carried out the computerwork to calculate the

hydrodynamic forces of the oscillating ship model in

shallow water. ' 1/s h/T = 2.40 h/T . 1.80 h/T . 1.50 h/T = 1.20 h/T 1.15 a zz bzz azz bzz a zz bzz azz bzz a zz bzz 4 46.6 399.1 58.6 464.8 78.5 514.3 149.4 583.7 183.1 598.0 6 50.2 317.7 6e.4 378.7 82.2 427.7 152.8 498.7 186.5 513.5 8 57.5 208.4 69.4 261.9 88.7 309.3 158.6 381.4 192.1 396.5 9 62.4 155.1 74.2 200.5 93.3 244.3 162.6 314.4 196.0 329.2 10 67.6 111.0 79.6 145.2 98.5 182.3 167.2 246.4

2005

260.4-12 76.3 55.0 89.8 69.2 109.1 88.1 177.8 127.6 210.9 136.8 1/s h/T = 2.40 h/T . 1.80 h/T = 1.50 h/T = 1.20 h/T = 1.15 a YY b YY a YY b YY a YY b YY a YY b YY a YY b YY 4 78.7 188.4 73.1 259.3 66.3 323.0 50.0 428.5 45.0 453.5 6 53.2 364.9 43.6 386.9 36.2 412.9 24.9 463.0 22.3 475.5 8 24.8 450.5 21.8 435.6 18.5 434.1 13.5 450.8 12.3 455.4 9 15.6 433.4 14.6 422.8 13.1 419.4 10. 4 430.7 9.7 432.1 10 10.5 396.2 10 2 392.4 9.6 391.5 8.5 402.8 8.1 400.1-12 7.7 313.2 7.6 314.2 7.5 317.2 7.7 331.8 7.8 303.9 SWAY En =

Nomenclature.

waterplane area

added mass and added mass moment of inertia

on speed, subscript for amplitude

beam

damping coefficient on speed

blockcoefficient

restoring force coefficient

cross-coupling coefficient for added mass

cross-coupling coefficient for damping

forceexerted by oscillator

Froude number

restoring moment coefficient,

acceleration due to gravity

water depth

mass moment of inertia

lingitudinal moment of inertia of waterplane

length of model

distance between oscillator legs ( 1.1m)

moment exerted by oscillator

added mass for zero speed

damping for zero speed

arplitude of oscillation

draught of model

time

forward speed of model

x,y,z

right hand coordinate system

Y

sway displacement

heave displacement

phase angle between force or moment and motion

pitch angle

density of water

yaw angle

circular frequency of oscillation

volume of displacement of model

instantaneous wave elevation

Superscripts:

asterix for value of segment

indication for sectional values of

hydrodynamic coefficients

References.

Gerritsma, J

, W. Beukelmen "The

Distri-bution of the Hydrodynamic Forces on a

Heaving and

Pitching Ship Model in Still Water", 5th Office

_of

Naval Research Symposium 1964, Bergen,

_Norway

Gerritsma, J., W.Beukelman,

_Analysis

_of

the Modified Strip Theory for the

,

_calculation

_of

Ship Motions and Wave Bending Moments",

Interna-tional Shipbuilding Progress, 1967.

Keil, H., Die hydrodynamischen

KrUte bei

der periodischen Bewegung zweidimensionaler

Körper

an der Oberflächer Gewisser, Bericht nr.305,Institut

fUr Schiffbau der Universita Hamburg, 1974.

Gerritsma, J., W!

Beukelman and

C.C.

Gla-nsdorp, The Effect of Beam on the

_Hydrodynamic

Characteristics of Ship Huuls, 10th

Office of Naval

Research Symposium, 1974, Boston, U.S.A.

Van Doorn, J., Modelproeien en ware

gro-otte metingen met m.s.

_{"Smal Agt" (in Dutch)}

_Report no.

530, Ship Hydromechanics

_Laboratory,

_Delft

Uni-versity of Technology (October 1981).

Appendix I.

Experimental determination of

mass and

damp-ing with a segmented model.

For the four modes of motions

_{considered the}

hydro-dynamic coefficients of the

segments

_{are determined}

after substitution

_{of the in-phase and quadrature}

component of the measured sectional

_{force into the}

equation of motion of the segment

(2).

In this way it can be shown

_{that for:}

Heave:

Pitch:

Sway: Yaw: a

-zz

C za - Fz cosez

zaw

.

g 6

*

ze a + F8 COSEe

.

d pV . ze _{* ea}w'

_*

1

-Fe sin Ee

eze

-0aw 07

whereovxi

the

ma.is

ss moment of the segment

which

_{centre is located at a distance x. from the}

centre of rotation.

za,ea,ya and

aare the amplitudes

of the related motions.

The coefficients of the segments divided

by

the

length of the segment give the

_{mean hydrodynamic}

cross-section coefficients. Assuming

_{that the}

distri-Awl a Cb e F. Fn 9

to be obtained as follows for: Heave: a = Ea zz zz b = Eb* zz zz

de =Ze

d = Ea x. eZ zz e = Ee ze ze Sway: a_YY = Ea Yaw: a = Ed x. YY Y* b = Eb* YY YY b = EeY*x. d = Ed* Yg

4

eyq, = Ee e = Ebyo YYx

Similar relations are used for the sectional

values of the calculated coefficients as denoted in

appendix 2.

Appendix 2.

Expressions according to version 1 and

ver-sion 2 of the strip theory for the hydrodynamic

mass-, damping- and cross coupling coefficients.

The expressions for the sectional values of

the hydrodynamic coefficients are derived from

app-endix 1 in Ell] and may be written for the motions

considered as follows: Heave: V dN'] azz = m + [--5 dx dm' b' = N' - V -a zz * bee = Eeze x. eeZ = Ebzz x. d = Ea x. YY r V V2 dm' d' = m'x + 1.2.1

T N,2

-za w dx 2 e'e = N'x - 2Vm'z - V dm'x_dx - V b' _{= N'x2 - 2Vm'x - V} dm x2 _y_22 dN'x ee dx d' = m'x

+{V dN'

ez

x

w dx dm' e'z = Nix - V ---x e _dx Yaw: dm' Pitch: a68 =

Ed x

b'YY = N' - V d' in which: b' V2 dm' dN' = m'x + [2] .H4(--7 N' - + x dx w dx dm dtq e' = N'x - 2Vm' - V ' x -[-j V2 l dx .2 dx 2 V V2 dm' V dN' a'

_*0

=m1x-4-2.7pCx ---6

x + dx

w'

dx .2 dx , dm' 2 2 dN' = Nix2 - 2Vm x -Vg4,x - --2 -- x w dx V dN' =m'x +[ x

0

2

w dx dm' e' = N'x - V ---x *Y dx m' = sectional damping

for zero speed

N' _{= sectional mass} V = forward speed

= frequency of oscillation = sectional added mass = sectional damping

t. on = sectional mass coupling coefficient _speed = sectional damping coupling coefficient x = longitudinal Y = sway z = heave direction e = pitch * = yaw

xl Version 1 _{= coefficients excluding terms}

be-tween brackets

Version 2 = coefficients including _terms

be-tween brackets

From the expressions for the sectional

coef-ficients the following relations _{may be derived:}

= d' x OP

4

= e' x 44

4

= a' x YY = b' x *Y YY

11 - 6

ee ea d' eZ 9x ze x ze = a' x zz b' x zz Pitch: , 2 a'e = m x e V + 2-2 N'x V2 dm' w2 dx V dN ' x2 dx

tributions can be determined from the seven mean

Sway:

cross-section values.

m' The hydrodynamic coefficients of the whole model are

dx

dN'

:2 dx

In appendix 1 the same relations are used for

the measured values to obtain the not directly

mea-sured coefficients.

The values of the hydrodynamic

mass-,damping-and cross coupling coefficients for the whole

model

are obtained by _integration of the sectional

values

over the model-length.

N. Beukelman

J. Gerritsma

Ship Hydromechanics Laboratory

Delft University of Technology

Delft

The Netherlands

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HEAVE w=4

IFER

mime

nummol

Irma

w=6 w.8 w=10 w=12

Fig.3: Comparison of experimental and calculated distribution

of azz for five waterdepth-draught ratios. Fn.=0.10.

W.6 w=8 w=10 w=12 h _15 Exp. Fn,010 r=001m

version 1

Calc.

version 2

=12 -y-115

200 200 Ns/m2 0 200 b'zz o 200 O 200 o

L-12

T

HEAVE

111111

MEE

WWI

AIM

Pirard

RPM

1 2* 4 5 6 7 Th-115 -0.1= 41=

,8

w=10 w=12

w8

w =1 0 wr.12

h.

h

Exp.

Fn..0.10 r 0.01m

version 1

Calc.

version 2

Fig.4: Comparison of experimental and calculated distribution

of b

_zz

for five waterdepth-draught ratios. Fn.=0.10.

50 o 50 50 o 50 o 5 50

L.21.

T HEAVE

1

10

u:=10 u.,12

Exp.

version 1

Calc. -f-

version 2

tal.6 W.8 W=10 w=12

Insi

111114

MI

it

5

hF

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IIIIIN

MIMI

ICES

pm

.m...,..

:BIM

MIN

MIMI

EMI

EMI

6

wpmr

L

gm,

....

i nn

4h...1

tir 1

mi Ill

IIIINISIIIIIEWINI

-T-h_115

Fig.5: Comparison of experimental and calculated distribution

of a

for five waterdepth-draught ratios. Fn.=0.20.

zz

Ns/rn' 5

'zz

o 5 O 50 o

Fn .0.20 r .0.01m

HEAVE

12,18

T

11=15

Fig.6: Comparison of experimental

and calculated distribution

of bzz for five waterdepth-draught ratios. Fn.=0.20.

Exp.

version

11.

Calc. version 2

o 2 O 2 -2 h_=2.4 -12-. 12 T h w_6 W.8 w_10 L-12 Exp. Fn. 010 r.ono5m

version 1

Calc.

version 2

12_, 5 T

Fig.7: Comparison of experimental ..nd calculated distribution

of d

five waterdepth-draught ratios. Fn.=0.10.

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mil

ir.

_,....

Mr=

MIME

NOMIIMMIEVW1INtA

Mill

_tom

1

.

Í

KIN

mi Ili LE i

EWA

-no

CM

lewd

w- t. w-6 W.8 w-10 (.12

eZe 100 o -100 100 -100 100 -100

10:qmpop

Nsim 1°

HOPI

e.ze

-1111111111

-1°411116

o -h-r24 Fn=0.10 r= 0.005 m PITCH h

T-w-i. w_6 111=8 W=7.2 Exp. version 1}. Calc.

version 2

Fig.8: Comparison of experimental and calculated distribution

of ez, for five waterdepth-draught ratios. Fn.=0.10.

w-4 w-6

4,8

Ws10 2

.11/4-1

.

Wri

. 44

Il

.i.

-7.

I

íiîiíU

iro-rali

MINI

IIIIIPARAMMIIIILMNIta

2 o -2 2 o

d

-20 -2o -20 20 C -20 Ne'm 2 0 20 o -2 2 O -20 11=12

Fig.9: Comparison of experimental and calculated distra.ution

ofz6 for five waterdepth-draught ratios. Fn.=0.20.

PI1CH 4. ±1=18

Exp.

Fn:020 r = 0005m

version

Calc.

version 2

W=10 W-12 ca-t w-6 W=8 w=10

WM III

Illaltria

Fam111E

!-MINIA

omen

IINEADI

ifirligil

i!!1III

mum

EMI

...

wpm

ihdril

MI

M.

ihrAT,

5 6

I I I

f

¡AIM I 1611.

111111=11

libiltrark...

SIMAd._

1011112A.1.11

Illbala.

rril

MI

Aft,

ism II

11 NMI

MAI

_iii

seme

NM

hiv

5

Ill

....V.

++! r +

i.

,

_.

II

SWAM

--

.

n 6 7

o

T + T

Fig.10: Comparison of experimental and calculated distribution

of e7, for five waterdepth-draught ratios. Fn.=0.20

PITCH 4+ w_6 w_8 4,10 W.12 w-6 w.8 41-10 U.12

kol

Fria_

I;

NM

MIN

mom

Il

MI5

IPPIPI

1111911111

MA

_{sull ilk,}

FA.

New

tom

_No

...

mil

bled

1111117

liaiti

um

rithwai

111111/1"

1111112

litirdil

RYA"

k 'IF

1 1 2 1 3 4 5 1 6 7 T 21.

t_=Is

Exp.

Fn=020 r.0.005m

₀

version 1

Calc.

version 2

10 o o tm-roo o o -1

20 o 20 o 20 o 20 O SWAY

4:115

Fig.11: Comparison of experimental and calculated distribution

of a

for five waterdepth-draught ratios. Fn.=0.10.

YY W.10

MIMI

RI

am

Milmnrarriiiiii

P1111...aziac

Fill.0

t

I 1

'

I

7411

st

1

yr.-i

+.4..

al

r _

--I

II.

04.

, .

...I

*4'1

I

:

ilh

rik

....

r -

1

...._ , 1

.---amike

aiiras.---.. aiiras.---.. . aiiras.---..

,

...saga

2 3 4 5 7 +

+ t I,

.

*.. 1 ...o.

ma.._.

.... AnallO

411611...P1111.).- IZMa

2 3

,

5 5 7

4....

t

+

..

ERIN

NM

mom74166:..._....

;IL.

....aft,

min

Fr, .010 001m

Exp.

version 1 Calc.

version 2

20 0 2 Ns/mi O o 1 2 2 wr w=6 W78 ws9 11=18

L,15

T wr:L. (A.1: o (J.10

21-12 T S WAY 418 6 w.8 w.10 1 2 I 1

41 5

5 1 7 L.-15 T

Fig.12: Comparison of experimental and calculated distribution

of b

for five waterdepth-draught ratios. Fn.=0.10.

YY

4,6

w=8 41.9 w.10

=Eli

---,'

'Mild

gilial

FAN"

Fn .010 r. 001m

Fxp.

version Calc.

version 2

20 o 20 SWAY

4,8

w.8 1,10 w=1.

I

111"29

UPI

NI

rami.ohimmasm

lo

i m

il

4.6_,....am,

pl I

al

r.hailailAM

I

Alm

6.6

PI

mom

_w...i.

all.mow

II

III

it

mminaiiimata

I

BEIM

am OWAti.

all.mw

IIII

mumiiiiiiimumn

Ali

II

111

WM

umagrimiaelisio

IR!!!

u 11

1

ram-MaiiiiatlIMINI

..1,4FP:4

II

Fn..0.20 r=0,01m Exp.

version 1

Calc.

version 2

h

Fig.13: Comparison of experimental and calculated distribution

of

aYY

for five waterdepth-draught ratios. Fn.=0.20.

mom

w=6 taB

mun

20 O 2 20

I

20 a'ry o 20 20

10' O

12

3

L.5 7

L.21. 7 h 2 SWAY hi-18 T-h --115 -W=L. W=8 co.8 W=9 w=10

Fig.14: Comparison of experimental and calculated distribution

cf h

for five waterdepth-draught ratios. Fn.=0.20.

YY

WrL. W=8 w=9

...eliMi

_Alma

L.

- -

.

AIIIIIk-1111111PAWAMMIIIMMI

Fn =0 20 r .001m Exp.

version

Calc.

version 2

2 2 2 - 20 20 0 -20 20 o -20 21. Fr) .0.10 r= 001m

FEIN

Ne/;

Ìuuuui

0

Emu'e

-20 Y,

2011MMINE

-O-.12 t t ±-=115

Fig.15: Comparison of experimental and calculated distribution

of d for five waterdepth-,f_:aught ratios. Fn.=0.10.

11

20

Exp. w_4 w_6 wr8 w-9 u.10 version 1}. Calc

version 2

11111211111

MINIIMp

iikamitima

dill!

1111111111

NMI

riim

I

MIIIIIIIMIIINIffilea

II

EMI

MEIN

WWI

imiquile

11-111111

EN

¡1

immaria

mum.

mum

t._

IMP

IMP

WM

1"

is

Will& 20 o -20 20 W-4 w-6 co.-8 w_9 w-10

Fr1=M

r.001m

YAW

-b-115 h

wimp

INIVE

marmEMB

MUM

mocips

_""m"imma

w-4 _ w-8

.10

mommonl

Malki

III

12E2

RIVAMPR

Exp .

version 1

Calc.

version 2

Fig.16: Comparison of experimental and calculated distribution

of e

for five waterdepth-draught ratios. Fn.=0.10.

ill

W.

UP

-NNW*

VD

011"!

_ir

k

Rd=

tom.2....

1

mmammummm

w 4 w-I3 w-9 .10

MESsaki

8

Mir

o O ₀ 0

ammii

20

35

-YAW

L=1.15

Fig.17: Comparison of experimental and calculated distribution

of d

for five waterdepth-draught ratios. Fn.=0.20.

11 - 22

W-4 W-6 w.9

WAIME

WIN

RIM

11111111

WNW

w w-6 W.8 W-9

oistizolo

JIM

oingN 0

ey,

0

imp

MOM

EOM

0 .b..= h

=is

Exp. Fn 0.20 1,0.01in

version 1

Calc.

version 2

-b-20 20

10 10 tyy -100 100 -100 100 o -100 100 o -100 Nsim 100 21. Fh.0.2 0 r.0.01n,

Aw.

h .15 T-Exp.

version 1

version2

w-6 w-9 w-10

Fig.18: Comparison of experimental and calculated distribution

of e

for five waterdepth-draught ratios. Fn.=0.20.

w-6 w.8 c.10 Calc.

111

ILLS

WON

lige:

1

Mil

MIMI

mEld

E

tammium,..

Mild

Ï

MIMI

mgmi

...

1

Imma

TIM

RIPPI

_,01411

MI"

Ada

WniMIOR

w.**

go.

E row

1

milimmmmimim

i

misig

Mill

MORI

Ltrisipm

mamma

Or"

MA

ailLmlid

iglIME11.0

I

Ne/m

1

°zz

10 0 Fn=010 4-O r=0.005 rn} r=0.010 m _exp r=0.015m w.4 (ù=6 =8 W.10 w=12

11 - 24

verson 1 + versron 2 calc. 10 15 20 1.0 1.5 20 hiT W.4 w.12

Fig.19: Comparison of experimental and calculated coëfficiënts

Ns'im a zz Fn=0 20 h/T r= 0.005 m

r=0.010m}exp

r=0.015m w=4 w=6 W=8 w=12 1.0 version 1 +verspon2 cdc 1.5

htr

20 (4=4 w=6 W=8 w=12

Fig.20: Comparison of experimental and calculated coëfficrénts

Ns /m a YY Ns'

dyw

Fn=010 r= 0.01 m r= 002m r= 0.03m

:Lai

w=6 4)=8 w=9 w=10 250 250 Ns/rn 250 YY 250 10 15 20 h/T h/T w=5 w=8 W=9 w=10 w=4 w=5 41=8 UJ=9 w=10

Fig.21: Comparison of experimental and calculated cafficients

for sway as a function of waterdepth-draught ratio. Fn.=0.10

version 1

exp

Ns' 20 10 o 20 10 o 20 10 o 20 10 Fnr020 O r=0.01m O r= 0.02m

a r=013m

WAY

6 Wrg w.10 10 15 20 h/T Oversion1 I exp calc + versoon 2 100 o -100 100 Ns 0 -100 eyto 100 o 100 o 10 1.5 2.0 W.6 .10 w=6 wrg W.10

Fig.22: Comparison of experimental and calculated coëfficiënts

for sway as a function of waterdepth-draught ratio. Fn.=0.20

Jo

-Ns/m

°

bYY 250 o 25 o 250 250 o 250 o w=g w.10 u).4

Nem

a es

Nst

dez

Fn=0.10

o from exp.

version

11

_caic

+ version 2 w=4 w=6

w,8

w-,10 w=12

Fig.23.

Comparison of experimental and calculated coëfficiënts

for pitch as a function of waterdepth-draught ratio. Fn.=0.10

1.0 15 2.0 10 15 20

Hs m

Ns'

dez

++

,

Fn.0.20

o-- from exp.

UT-21

w=4 w=12 w=4 w=6 w.8 w=10 w.12 version 1 + version

2alc

15 h/T 20

Fig.24: Comparison of experimental and calculatbd cobfficiënts

for pitch as a function of Waterdepth-draught ratio. Fn.=0.

10 15

h/T

20

-NS' dwy 20 o 20 20 o

Fn=0.10

-o- from exp.

20°-h/T (.4=5 w=8 u=10

AW

versen 1 colt + version 2

RUM

MEE

Ms

Fig.25: Comparison of experimental and calculated coEffici'énts

for yaw as a function of waterdepth-draught ratio. Fn.=0.10.

1.0 15 20 10 15 20

hiT

o

Nsi WY 2 10 2 20 20 15 20 hAr hiT Fn=0.20 from exp. 29 0

_'

20 w=6

w.8

w-9

w.10

AW

1 100 100 o 1 o 10 15 20 h/T version 1 } colc. + version2 u=6 w=5 w=9 wr10

Fig.26:

Comparison of experimental and calculated coëfficiënts

for yaw as a function of waterdepth-draught ratio. Fn.=0.20.

:t -

-ty

t

* _

**

*

-

-* e _

-**

4

-4,

_

-

₊

+-.61,...

a

-++

-

+ 1,-49 .-____4

;

_e _

-

+ +

44-- ;

± 4--_ _ ,

,

20 tv o 20 20-o

w.t.

w=6 cor8 wx9 w .10 0 20 up4 w=5 w=8 w=9 w=10

0.8 0.6

04

0.2

04

0.2

O

o

50

Fn.=0.13

11=0.156

h/T=1.2

o

Exp.

-Calc.

we

Fig.27a:Heave amplitude response.

we

Fig.27b Pitch amplitude response.

100

o

Fn.=0.13

h =0.156

h/T=1.2

o

o

Exp.

Calc.

o

o o

O 5.0 10.0

0

O

10

08