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

AFDELING DER SCHEEPSBOUW- EN SCHEEPVAARTKUNDE

TECHNISCHE HOGESCHOOL DELFT

LABORATORIUM VOOR SCHEEPSHYDROMECHANICA

TIlE 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-na.mics,

Bulgarian Ship

Hydromechanics

,

Varria, 11 - 13 November 1982.

Report No. 546

March 1982

Ship Hydromechanics Laboratory

- Deift

Deift University of Technology Ship Hydromechanics Laboratory

Mekelweg 2

2628 CD DELFT

The Netherlands Phone 015 -786882

This report is prepared by THD/W1/NSP under contract no. 82/1528/1.3. of the Stichting Cordinatie Maritiem Ondernoek." "0 THD/WL/NSP, 1982.'

'Reproduction in whole or in port by means of prrnt, fotocopy, microfilm or in any other way a only permitted after preceding written consent of THD/WL/NSP."

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BULGARIAN SHIP HYDRODYNAMICS CENTRE

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CONFEDENCE ON BEIIAVIOUP OF

SHIPS IN PESTQICTED WATEDS

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TOM I PROCEEDINGS, Volume I

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Eleventh

Scientific and Methodological

_{Seminar on Ship Hydrodynamics}

THE DIS1RIBUTICN OF HYDROOYNAIIIC MASS

AND DAMPING OF AN GSCILLATING SHIPFOPJI

IN SHALLOW WATER

W. Beukelman, J. Gerritsma

Introduction

The depth of water has an important

influ-ence on the vertical and horizontal notions 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 cf trim, sinkage and the vertical displacement of the ship's hull as a result of the ship motions in waves. Keel clearance is of interest to ship owners and port authorities, because of the ircreasing draught of large cargo ships arid the corresponding smaller waterdepthì'draught ritio's. The safety and rnanoeuvr-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 jr 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 fer the rolling motions, viscous effects are

not irrir.'rtant in strip theory calculations, but an accurate determination of 2-dirr.ension,al damping and added mass of ship-like cross sections is necessary, as shown earlier til

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

com-parison of the distribution of hydrodynamic mass

and damping, as measured on a segmented

s'i

"io(iel

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-ae and trim, which would occur in the case of a free floating model. In addition to the heaving and pitching motions also f'rced horizontal motions in

the sway and yaw node have been carried out.

The experiments included the effects

f

forward speed, frequency of oscilliation and water-depth. A rancie of frequencies have been chosen to cover wave frequencies of interest for shio resoon-ses.

The use of a seorented ship model enables

the determination of the sectional values o' da-o-Ing and added nass This techniqie has beer. u;ed

earlier for an analogous investigation of the deeo water case

Ii)

See appendix 1.

The calculations have been carried out :iti

a computer program developed by H.Keil 13)

In this calculation the hydrodynanic pass and damping for 2-dimensional ship- like cross sec-tions are computed with potential flow theory, using a source and a linear cortinaticn of riulti-pole potentials, which satisfy the boundary

con-ditions at the free surface, the bottom, and the contour of the cross section. A Lewis transfornati on has been used to oenerate ship-like cross

sec-tions

The model.

The forced oscillation experiments have been carried out with a 2.3 meter model of the

Six-ty Series The main particulars are given in Table 1. The same model has been used earlier for the ana logous tests in deep water [1,2) . The model has

been divided in seven senments each of which was separately connected to strong beam by means of a strain gaupe dynamorneter.

WNFERENCE

flOBEMHO CAOB BERAV1OUR tF 9UPS P

B T}*EHHOM GAPBATEPE STR1CTED W&1ERS

Table 1,

Length between perpendiculars 2.258 m

Length on the waterline _LWL 2.296 m.

Beam B 0, 322 m Draught T 0.129 m Volume of displacement Q 0657rn Blockcoefficient C 0. 700 B 2

Waterplane area AWL 0572 m

Longitudinal moment of inertia _'WL o 1685m2 of waterplane

LCB forward of Lpp/2 LCF aft of Lpp/?

Table 2a. Heave

Fn = 0.1 and 0.2

Table 2b. Pitch

En = 0. 1 and 0 2 x Fn = 0.2 only

Table 2c. Sway and yaw

Fn = 0 1 and 0.2

0. 011 m 0, 038 m

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 the seven

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 conditions,

The various oscillation amplitudes cover a

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

The test conditions are suntnarized 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 En.

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 ranqe of

= 1.9 - 5.8 for pitch and heave, and:

1[Jj

= 1.9 - 4.8 for sway and yaw.

Experimental results

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: (v + a22)2 + b ± + C z - d _{- e9O} 9ze9 = zz zz ze

= Fin(wt+c)

Pitch:

(I

+ aee) + b09Ô + ceee - de - e82±

= M951n(wt+c0) = 4,6,8,10,12 rad/s

r

(m) h/T 2.40 1.80 1.50 1,20 1.15 0,005 . . . . 0.020 . , . 0,030 = 4,6,8,10,12 rad/s (m) h/T 2.40 1.80 1.50 1.20 1.15 . . . ₀₀₀₅ 0.010 . 0,015 = 4,6,8,9,10 red/s

r

(m) h/T 2r4 1.8 1.5 1.2 1.15 . . . 0.010 .

ooo

0.030

Sway:

(pV + a)V +

- _{- ey} F sin(wt+c) Yaw:

- d*yY - e4,i' = M,sin(wt+c,) (1) For the Individual segments the following equations result: Heave: *

*..

* . * * * (pV + a_{zz '}z + b_zzz + c z = F

5lfl(t+c)

zz z Pitch:

(pV*x _{+ d2)}

_{+ eeó + g8e = - Fsin(wt+c)}

Sway:

(v* + a)Y + bY = Fsin(t+c)

Yaw:

(PV*xj

_{+ d) +}

_{= - Fsin(t+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 and the relative waterdepth h/I.

Iwo forwards speeds corresponding to Fn 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/I < 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-depth.

11 - 3

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 r'ass,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 shipmoel wtth 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 some of

the

ir-regularities in case of the lowest speed Fn = 0 1 and frequencies equal or below w = 6 rad/s. In

all other

cases wall effects do not seem to have influenced the experimental results.

Calculated hydrodynanic 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 dampinq 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.

Iwo versions of the strip theory have been

used.

Version 1 leads to the ordinary strip theory method, which lacks some of the syrmuetry relations

in the damping cross coupling coefficients

Version 2 includes these .additior?aI 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 and the waterdepth- draught ratio's considered.

Conclusions. Table 3.

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

HEAVE Fn 0

The calculated hydrodynamic mass for vertical notions 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 night 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 rTxtions, sway and yaw, although the differences for dampina are somewhat small-er 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.

The results of this detailed comparison

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 pur2oses. A limi ted nunter of model experiments to

de-terriine 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 (51 , see Figure 27 a+b.

Acknowledgement.

The authors are indebted to ing.A. P. de Zwaan

io carried out the computerwork to calculate the hydrodynamic forces of the oscillating ship model in shallow water.

1/5

h/I = 2.40 h/I = 1.80 h/I = 1.50 hIT = 1.20

h/I

1.15

a22 b22 a22 b22 a22 b21 a22 b2 a b2

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 6. 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

h/I = 2.40

h/I = 180

h/I = 1.50 h/I = 1.20 h/I = 1.15

yy byy ayy byy ayy byy ayy byy ayy byy 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 4755 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 43Z.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 3039 SWAY Fn = 0

Nomenclature.

Gerritsma, J. , W. Beukelman "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.

11 - 5

Keil, H., Die hydrodynamischen KrSfte bel der periodischen Bewegung zweidimensionaler Körper

an der Oberflcher Gewsser, Bericht rir.305,Institut für Schiffbau der Universitat Hamburg, 1974.

Gerritsma, J., W Beukelman and CC. 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 Doom, J. , Modelproeven en ware gro-otte metingen met m.s. "Smal Agt" (in Dutch) Report no. _{530, Ship Hydromechanics Laboratory, Deift}

Uni-versity of Technology (October 1981).

Appendix 1.

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:

*

cza - F

cosc pV 2 Zau) F; b

-a * * * * + F8 COSc8 *

d8 =

8 z pV x * a _* -F0 sin e -0a" * * * Fy cos * - -pV ayy yw * a _* - sinc, yy * * * F, cosc dy1p

-

_{* *} * Fq, sinc

-where pVx is the mass moment of the segment

which centre is located at a distance x from the centre of rotation. ZaOaYa and _1aaP 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-bution of the cross-sectional values of the hydrody-namic coefficients are continuous curves these

dis-Ai

waterplane area

a added mass and added mass moment of inertia

on speed, subscript for amplitude B beam

b damping coefficient on speed

Cb bi ockcoeffi ci ent

C restoring force coefficient

d _{cross-coupling coefficient for added mass} e cross-coupling coefficient for damping F forceexerted by oscillator

Fn Froude number

g restoring moment coefficient,

acceleration due to gravity

h water depth

mass moment of inertia

'wi lingitudinal moment of inertia of waterplane

L length of model

distance between oscillator legs ( l=lm) M moment exerted by oscillator

Ri added mass for zero speed

N damping for zero speed

r

amplitude of oscillation

I

draught of model

t time

V forward speed of model

x,y,z right hand coordinate system

y sway displacement z heave displacement

c _{phase angle between force or moment and motion} o pitch angle

o density of water

p yaw angle

circular frequency of oscillation

V volume of displacement of model instantaneous wave elevation

Superscripts:

asterix for value of segment indication for sectional values of

hydrodynami c coefficients References. Heave: Pitch: Sway: Yaw:

tributions can be determined from the seven mean

cross-section values.

The hydrodynamic coefficients of the whole model are

to be obtained as follows for:

Heave: _{= zazz} Pitch: a08= Ed0x

= ey = Ebx1

Similar relations are used for the sectional

values of the calculated coefficients as denoted in

appendix 2. Eb * = b = e x. d, = m'x + {2} N' V2 dm' r dN' 1 zz ee ZO 1

:

[Xj

a * *

d8 = Edze dez = Zazzxi

Iv2 dN'l * * e1, = N'x - 2Vm' - V 'x [ e =zb x. e0 = Ee ez ZZ 1 Sway: ayy = ay Yaw: a = d* .

Yaw: 2 Ypi dm' [V dN' ] = b; b - Ee* a m'x2 + 2 N'x - - x +

-

dx

Y1

w dx dx dm' 2 1V2 dN' d = d; d =

Eax

b = N'x2 - 2Vm'x - V x - - x dx 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 [4} and may be written for the motions

considered as follows: Heave: 2 dx , dm' dx [2] V2 N' V2 drn'

+[Y, -'

d' =m'x+

-zo L dx = N'x - 2Vrn' - V

Jv2

dN'

[2

_dx Pitch: ae rn'x2 + N'x V2 drn' {V dN' w w2dx be = N'x2 - 2Vrn'x - v drn'x2 [v2 dN'

_i

w dx = mx +{-Y.2.4L' dx ] = N'x - V = rn b' = N' zz

11 - 6

Sway: = rn +[_ 1& _dx] dm' 'JYY_ -in which: mx + V dN' 1 = dx _j e' =N'x - V-x dx a0 d9x be =e0x =

ax

=

bx

rn = sectional damping'

for zero speed

N' = sectional mass )

V = forward speed

= frequency of oscillation

= sectional added mass

= sectional damping on

d' = sectional mass coupling coefficient speed

= sectional damping coupling coefficient

x = longitudinal Y =sway z = heave direction a = pitch =yaw

Version 1 = coefficients excluding terms

be-tween brackets

Version 2 = coefficients including terms

be-tween brackets

From the expressions for the sectional

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.

W. Beukelman J. Gerritsma

Ship Hydromechanics Laboratory Deift University of Technology Del ft

The Netherlands

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PRINCIPLE OF MECHANICAL OSCILLATOR AND ELECTRONIC CIRCUIT

7

5

i,1i'

JiiLI

Ns/m

IPI!I!

_PJ!I'l

5,

ilIii

Figure 2: Horizontal oscillator.

HEAVE W:4 w:6 w=8 w10 W:12

Fig.3: Comparison of experimental and calculated distribution of azz for five waterdepth-draught ratios. Fn.O.lO.

w:6 w:8 w 10 :12 1.8 h -i Exp. Fn:O.1O r:O.Olm version 1 ' Calc.

+

version 2J

I

5

200 0 200 Ns/m 200 0 200 200 0 12 Fig.4: Comparison

ofb

for zz

11 - 9

HEAVE

"I'll'

-I'll"

h

T5

of experimental and calculated distribution five waterdepth-draught ratios. Fn.=0.l0.

W:1. w5 U: 8 w:1O w:1 2 U:!. w:6 w:8 w:1O w:12

'ill"

"U.

pill",

0 h_{- t8 h -15} Exp. Fn0.10 r= 0.Olm version 1) Calc.

+

version 2J

HEAVE w:5 W: 8 w 10 w:1 2 Exp. 1.5 --:115

Fig.5: Comparison of experimental and calculated distribution

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

w:4 w:6 w:8 W :10 w:12

P!

1

23

t5

6 7

iiiiii

It-I

U-

I-UI-

1 2

3j. 56 7

P

2 3

IrAiI'

1 J5 6 7

iii

FrO.2O r:OOlm version 1

Caic.

+

version 21 5 5 Ns2/m2 0

I

0 5 0 50 0

2.4

Fn:020 r0Olm

Fig.6: Comparison of experimental and calculated distribution

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

HEAVE 5 6

.18

w 1. w6 w8 w:1O w12 Exp. version 1 Caic.

+

version 2J W4 w6 w 8 w1O W:12 200 0 200 Ns/m' 0 200 0 200 0 200 0

dze -20 0 -20 N',th J

10

d -20 20 0 -20 20 U

-'I-F12

Fig.7: Comparison of experimer.tal nd calculated distribution

of d0 for five waterdepth-draught ratios. Fn.=0.10.

w-6 W:S w- 10 w:12 Exp. U-' w-6 w8 U-lU wl2 4

I

Iii!!

111

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iiii

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'Ii

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I

uiuiiu

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iiiiI

I!IVU!

-Ns/m

1I!r J.

-I

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-!IiII

1.2 -100 100 _4, 0 -100 Nsith

r

ig 1111111

-100 2.4 Fn:0.10 r0.005m

4+

PITCH :18 Exp. version ₁₁ Caic. -t- version 2J

Fig.8: Comparison of experimental and calculated distribution of e0 for five waterdepth-draught ratios. Fn.=0.l0.

-+44144

Tfl

Ii.iti

w_L. w-6 w 8 W=1O W2 0 -100 100 -100 100 0 -100

2J.

PIJCH

-4

Fig.9: Coirparisor. of experimental and calculated distributicn

of d5 _{for five waterdepth-draught ratios. Fn.=0.20.}

I7ii

umiui,

IIIi!

i!1III

uI1!u

_iIII

III

+

LIII

ii-

1

23

4

5j6

_I

k::+

IIPi!i

:1

:

+. ++

1

++p+.I +

I

Exp. Fr:O20 r0005m version 1) Caic. -f version 2J + 0 -2 2 0 -20

:i

-20 20 -20 20 0 -20 0 -20 Ns/m ₂₀

to

-20 20 0 -20 20 0 -20

100 0 0 PITCH w6 w8 W-12 u-b

Fig.10: Comparison of experimental and calculated distribution of e7A for five waterdepth-draught ratios. Fn.=0.20

w-4 w- 6 w8 u-b .1 2

UI

4

ft.!

-wypI

.1=,'

II!iI

IuIu,i

4

I!!!II

IIlIIi

tIPJ!

IuilA!

IEL

i1il

IIau.i1

IiII1!

Exp. Fn:020 r0005m version 1 Calc.

+

version 2 :t2 1 15 t8 :15 0 -1 eze 0

20 0 20 Nst/mt 0 20 0 2 0 20 20 0 20 2 * Ns /m 20 yy 0 20 0 20 0 SWAY w:4 w:6 W:5 (J-9 w:10

Fig.11: Comparison of experimental and calculated distribution

of for five waterdepth-draught ratios. Fn.=0.l0.

w:. w:6 w:8 w:9 w:10

2357

L,J.. _e,

-

-1

ui-i

!JJI!

III"

++

!UII!

Ii-1+

PI!I!

:2/. :18 15 Fr:010 r:OOlm Exp. version 1 CaIc.

+

version 2J

FnO1O r OOim SWAY #18 w6 w 8 w 9 w1O

Fig.12: Comparison of experimental and calculated distributior.

of b for five waterdepth-draught ratios. Fn.=O.1O. w4 w6 w8 w9 Pr V 1 2 3 1. 5 617 W1O Calc.

.1

213 415 6 15 version version 1' 2) + Fxp.

0 20

Fn20

r001m + w=I. w6 w=B (Q=9

dtI1uii.h

!!UIU!

!UIIU!

ii".

!!lII!!

w=10 Exp. version 1 ? Caic. version 2J w4 w:8 w=10 +

.t4,+4

aiiii

IS.

till!

IiuIiL

"'u-I

"Iii

!UIIii

!!IlI!!

INiik

lull.

iuii

-!UIP!

!IIlII!

1ulIiu

'lull'

'lilt"

ii

!UIP!

=1B + =t5 + .=t2 + +

Fig.13: Comparison of experimental and calculated distribution of a for five waterdepth-draught ratios. Fn.=0.20.

0 Ns'm' 0 20 0 20 20 Ns'm2 0 20 o.yy 0 20

100 0 SWAY -115 wI. w6 ui 8 w 9 W: 10

Fig.14: Comparison of experimental and calculated distribution of for five waterdepth-draught ratios. Fn.=0.20.

w:4 w=6 w8 w9 w10 . . j. L

...

L.

1EI1E

I 00 Ns. 'm 0 I IOU 100 0 IOU 0 Exp. version 1) Caic.

+

version 2) Fr320 r001m

- 20 Ns/m 20 0 - 20 20 0 - 20

IIl,ra

-20 20 -

1_I

-

20liIlII

Fn:0.10 r:OOlm

20II!PV

-20W

20

1'i.

_IIMII

:12 + L:15

11 - 20

*

iIlIu

1UUP!

IUIU

-U

1 wmwi*a

w- 4 w= 6 w B w-9 w:10 Exp. 1.5

Fig.15: Comparison of experirnntal and calculated distribution

of for five waterdepth-aught ratios. Fn.=0.l0.

(& 5 u 8 w-g cül0 version 1 Caic. version 21

I +E1I

i.UiL

lIP!

ii-

_aillilli

.iIii

IPU!!

i-u

I-

i1IIII

I.

iI

Ns/rn1]flJ

I e.y 100

iliii

1i1P!ii

Fn=0.10 r:0.Olm 100

i

iii

-100 Ns/m 100 _'I

It.i&i

-10

uri.,Ji

ilp4inli

0

!!iIII

1.2 YAW * w4 w6 W8 w10 Exp. version 1] Caic.

+

version 2J

Fig.16: Coiparison of experimental and calculated distribution

of for five waterdepth-draught ratios. Fn.0.l0. wz w-6 W:8 w-9 u10

N,'

III!

20 - 20 20 - 20 20 0 - 20 -'In,

I--Ill

"WA:

0 + Nst/rn

liP!!

-20 20

"F,.---- 20 20 20

ilIlihl

2 Fn=0.20 r0.01m 4 + 4 20

iriIi

NIm

20JJI

d

ilNili.

illilli

12

-

35 -YAW w=6 w8 (9

44

h Exp. version 1 Caic.

+

version 2J

Fig.17: Comparison of experimental and calculated distribution of for five waterdepth-draught ratios. Fn.=0.20.

w-,.

w-6

w:8

w-9

10 -100 10

to

.y

_-;g

-100 100 -100 100 0 -100 100 0 -100 Ns/m 100 Y4i ₁₀₀ -100 100 -100 4 4 h _15 Exp. Fn=0.20 r=OOlm _{version 1} Calc. version 2 w-I. w-5 W 8 w-9 W:10

Fig.18: Comparison of experimental and calculated distribution

of for five waterdepth-draught ratios. Fn.=0.20. wz4 Wz8 w=9 t10

I

i1iii

l!iIII

lilt!::

!iIlI

IIui!:I

I

Ns'/m ozz 50 50 50 -10 0 1.0 1.5 2.0 h,'1. Fri =0.10

-Ns' _d11 4-A

I4

0 1.0 15 2.0 _-. o r:0.00Sml o r=0.010

.exp

-r=OQl5mJ w=1 =8 w=10 w=12

iii

250

2U

versjon 1 1 + rsion 2 JCOIC h/T h1.1.

Fig.19: Comparison of experimental and calculated coëfficints

for heave as a function of waterdepth-draught ratio. Fn.=0.l0.

1.0 1.5 2.0

I

Ns/m a Ns' d ze 10 1.5 2.0 Fn:0.20 h/T O r:0.005 m o r:0J10m

a r0.05m

exp.

-wL w=6 W=8 w=12

Fig.20: Comparison of experimental and calculated coëfficints

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

NE

J-.

Iversioni +v&sion2 CaiC. h/T wL. w6

(8

w:10 w12 1.5 2.0 1.0 1.5

Ns'/m oyy Ns' dy 0 25 0 2 0 25 0 25 0 0 1.0 15

h -

20 Fn=0i0

-

20 h/T o r:0.01 m }ex

__

W:4 :5 W: 9 versOn1 1 +veron2 CdC.

:1

+ w:5

Fig.21: Comparison of experimental and calculated coëfficiënts for sway as a function of waterdepth-draught ratio. Fn.0.1O.

++ + 4+ +

1

W1O 10 15 20 10 0 -1 -1 10 0 _1

I

Ns'/ m Qyy Ns' d 0 10 15 20 h Fn020 o r0.O1mo rO.02m A

r03m

- 0-+ ex p. Ns 100 w10 0 100 0 10 versionl +ersion2 Ca1C. 1.5 2.0 h,1 (i): 4 W5 W8 W9 W:10

Fig.22: Comparison of experimental and calculated coëfficiënts for sway as a function of waterdepth-draught ratio. Fn.=0.20

W4 100 0

w5

-100 100

Nsm Ns

IL

4 4. W :6 w:6 w:1O w:12 Ns . versxx 1

}

cdc. + version 2

-Fig.23: Comparison of experimental and calculated coëfficiënts for pitch as a function of waterdepth-draught ratio. Fn.0.l0

10 15 20 to 15 20

h/T

Fn:0.10 -*- from exp.

1.0 1.5 2.0

p

N?m Gee Ns d8 1.0 15 20 h/T

Fn=0.20 a-- from exp.

10 + h/T . version 1

}

cdc. + version 2 h/T 20

-Fig.24: Comparison of experimental and calculathd coëfficiënts for pitch as a function of aterdepth-draught ratio. Fn.=0.

)

N?m °W41 Nst

dy

Fn:0.10 c.--from exp.

w4

w:6 w:5

w9

w10 w:4

w6

w:8 w=9 w10 AW

'

-Fig.25: Comparison of experimental and calculated coëfficiënts for yaw as a function of waterdepth-draught ratio. Fn.0.lO. 1.0 15 20 h/T

.

version 1 } coic + version 2

-

h/T

Nsm Ns'

dy

1.0 Fn:0.20 -0-- from exp. 1+ 15 h/T 20 + W: 4 w:6 w:8 w:9 w :5 w:S w:9 w:10 AW 100 Nsm

b00

100 100 0 10 0 15 h/T version 1 l .coIc + version 2 j 20 w:4 w:6 w:8 w:9 w:10

Fig.26: Comparison of experimental and calculated coëfficiënts for yaw as a function of waterdepth-draught ratio. Fn.=0.20.

+ +

0.8 0.6 0.4 0.2 1.0 0.8

/Ko

0.2 0

We

-Fig.27a:Heave amplitude response.

We

Fig. 27 b: Pitch amplitude response.

0

0

Fn. :013 h :0.156

\\\\0

h/T:1.2

o

Exp. Ca(c.

0

Fn. =013 h 0.156

h/T1.2

0

o

Exp. Caic. 0 5.0 10.0 5.0 10.0