Motions, wave loads and added resistance in waves of two Wigley Hull forms

IN WAVES OF TWO .WIGLEY HULL FORMS.

Prof. ir. J. Gerritsma Report No. 804

Motions,, wave loads and added res;istance in waves of two Wig.iey hull forms.

Prof. ir. J. Ge.rritsma

1.

Introduction

Model experiments with two mathematically defined hull forms

to determjne the motions, wave loads and added resistance in

regular head waves has been carri.ed out _{In addition forced} oscillation experiments, in the heave and pitch mode have been performed to determine hydr'odynamic _{mass a.n4 damping.} Th'e

culations of the.dyn'ami.c. behaviour of such hul.l forms in waves.

2.. The models

The hull form of both models is givenby':'

= (i-c2).(l2).(l+o.2E2) + ç2(lç'8)(l2)4

Two length-beam ratio's have been considered: L/B = 5 a.n4 L/B 10

The main dimensions of both models _{are given in Table 1 and}

a body plan is given in Figure 1 for the

L/B 10 model.

3'.

Model experiments

3. 1 Model 241

.3."l.lMotjon 'characteristiCs and''addedres:js'tan'ce

The. 'mot'ionamplitudev and' phase characteristics have been measured -n r;eguia'r.h':ead waves.. In case of the

narrow .beam model nr..241 th.re.e model speeds have been considered..,: corresponding to Fn = 0.2, 0.3 and 0.4. The measured results for pitch.,, heave and added

resis-tance are given in Table 2.

As a reference fo.r the added r'ess.tance values the

still water, r'e,is'ta'nce is. given in table 3.

The amplitude and phase characteristics of heave and

pitch are given in the Figures 2 and 3.

The measured added resistance _{is very small relative} to the corresponding stil.1 water values and a few neg'atie values' a:re observed'. 'These are not included

in Figure 4, where the added resistance is plotted on a' base. of wave I ength ratio.

The accuracy of the added resistance measurements

_with this iiärr'ow b'eam mo'de.l-i,s r-a'the r po.o r ,a:n.d these data

sh.o-u-1-d_be cansidered with care.

-height of aproximately a factor 2. The observed motion

amplitudes are almost linear with respect to

wave

height within this range, as shown in th.e Table 2.

3.1.2 Wave loads

In Tables 4,

5 and 6,

as well as in the Figures 5,

6,

and 7 the heave force and the pitch moment as measured

on the restrained model are given as a function of

wave length rat:io for three forward 'speeds.

Also the

phase lags of heave force and pitch moment

are

inclu-ded in these figures.

i.-l.-3Hydrodynamicuiass'anddampjng

Forced oscillation tests have -been, carried out to

det:er.min'e -added mass ;r

_of

inertia -and:t'he co.upi'inig .co effi'c i ent's-i.n:theequat ions

of'motion of heave and pitch.

Also 'in, this 'case'"three

_'have-been

con-s i.dered, acon-s well a'ria:n'geof frequenciecon-s.'

"

-The equations of motion are given by:

+ bz + cz + d99 + eO + gO

F (heave)

a090 + b06

_{+ c990 + do,z + 'e9'.z + g9z}

_{M (pitch)}

The coefficients a, b,

d and e are tabulated in

dimen-:-sionless form jn the Tables 7 and 8 and the

corres

ponding Figures '8

-

15.

The experiments have been carried out for

two

ampli-tudes of motion':

Za = 0.025 m.,

0.050 m and 0a = 0.026 rad, 0.052 rad.

A very, satisfactory iiñea-ri-ty.'.-in-:this--.ra'n'ge could be

ob se;ryed. The ,non

iinen:s.j,ona1. coé ffici'ént.s - ua're defined

as follows:

=.

a/pV

=

* (L/g)½'/p:V

do/pVL

ee,z *

(L/g),½/pVL

ä99/pVL2

b90 * (L/g)½/pVL2

dz0/pVL

ezO * (L/g),½/pgL

= -- 2 -= =

abo

=

bbo

₌

d9

=

e9

=

3.2 Model 253.

A s,iinular set of experiments has been carried out for model 253, which has. the same form coefficients. as

model 241.

Model 253 has. a length-beam ratio. _{. L/B = 5.}

However only one model speed, corresponding to F-n =. 0.2 has been considered in this case, because for

h.ighe,r Froude numbers t-he waves generated by th,e model

are. quit.e extreme.

The measured pitch and heave, as well

a:s added

regis-tance is given in.Table 9.

For Fn .= 0.2 the still, water resis-t'ance is 5.98 N, _as

a reference f-or'. the -.e's'istan'ce values in waves. -The corresponding. figu-re.s are.:

Figure L6 'for -heave.-

-Figure 17 Lor pitch

Figure 1-8 .-for...adde-d resistance

Wave force and moment, as mea.sured on the -r'estrai.ne-.d model is given in Table 10 and Figure 19.

Finally the experimental values fo.r added mass., added.

mass- moment of inertia, damping- and c-ross. coupling coefficients are tabulated in Table 11 and 12.

The. corresponding figures give the values for. t-he coefficient1s of the equation's of motion _{Orfl a base' of} frequency w'

.Figur.e 20:, added. mass atid damping coefficient. -for heave

Figure- .21-::--cro.ss .:coupli-ng'..-coef.fi.c.i.ents-. for heave

Figur-e .22 :-.'added. mass. moment of inertia a-nd dampi.ng-coeffi'cient.-for-pitc.h

.Figure .2-3: _{cross c-oupiing- coefficients .for p-itch}

4.

Acknowledgement

The experiments and the data reduction- have been carried out

Main dimensions of models Modelnr. 241 253 L (ni) 3.00 _3.00 B (in) _{0.30 0.60} T (rn) 0.1875 H _0.1875 V (rn3) 0.09462 0.18920 (rn) 0.75 _0.75 GK (rn) 0.1875 _0.1875 CM 0.909 0.909 L/B 10 5

Table 2

Pitch and Wave Characteristics Added Resistance MOdel 241 Fn. A/L _RAW (N) .za/ca

_6zç

(de.gr)

6eç

(de.gr) RAW*L, (cm) 21r.ca pgB2ç 0.2

L00

0.62 1.17 - 41 L2'O -194 16.0 H _1.15

:03

1.00 0.09 0.61 -116

057

-228 2.2 1.18 0.4 1.00 1.25 -0.85 0.45 0.30 0.86 -149 ., -. 2 0.27 1.65 -248 -.124 -21.5 11.4 .1.16 1.1.6 0.3. 1.25 0:8.1

238

- :58 H 1.34 -193 22.0 _1.12 0.4 0.2 .1.25. 1.50 -0.83 H 0.2:0 1.57 0.96 H -119 - 6. : 0.79 1,.i3. 2i9 -:104 -.25.6, 4.0 1.05 1.30 0.3 H1..50 0.8:6 1.56 0: 2. oO: -137 .1.6.3 _1.34 0.4 1.50 .: 0.4.1. 3.3L - 60 .' .1.28.. -179 8.3 1.30 0.2 1.75 0.0:2 0.97 1..:09 . -100 0.3 1.55 H 0.3 1.75 0.3.5 11.21 0 1.34 -110 . 5.6 : 1.46 0.4

175

0.26 2.50 - 7 1.35 -161 3.7 1.54 0.2 0.75 . 0.00 0.13 6.8 0.23 H -224 0.0 1.20 0.3. .0.75 -0.14 _0.09 - 5.4 0.11 _-239

- 35

_1.17 0.4 0.75 0.90.

006

- 0.07 23.2 _1.15 0.2 1.1.0 1.0.0 1.21 - 6 1.68 . -16.5 25..7 1.15 0.3 1.10 0.35 . 1.24 - 99 0.83 -214 9.3 . 1.13 0.4 1.10 -0.20 0.65 -135 0.42 -223- - 5.8 1.0:8 0.2 1.40 0.19 0.89 - 6 1.24 . -.106 3.8 1.3.0 0.3 1.40 1.29 :2.14 - 12 1.94 -163 27.6 1.26 0.4 1.40

062 .3.03

- .8,2 _1.26 . -189 14.9 1.19 0.2 1.6.0 -0.43. 0.96 0 0.95 - 9:9 H 7.5 1.;40 0 3 1 60 0 31 1 27 0 1 58 -116 5 4 1 40 0..4 1.60 -0.03 .3.07 - 36 1.16 -171 H _{- 0.5 1.37} 2.00 -0.03 .0.99 .0 0.68 - 95 0.4 1.54 0.3 2.0:0 0.08 .1.18 .0 '. 0.88 -101 1.2 1.52 0...4 .2.00 -0.65 1.53 + .2. . '-].14 -. .9.8 .1.50 0.2 1.0.5 2.09 .1.3.2 . - 27 . 1.44 -.18.2 26.1 1.65 '0.2 1.0'S .0.99 1.29 19 145 . -1.75 '25.5 .H .1.15 0.2 1.05 0.42 1.31 - 17 . 1.49 ;-.175 .. 31.8 0.67 0.3 1.25 , 1.74. 2.31 - .59 1.3:4 -19.3 2.3.1 1.60 0.3 0.4 1.25 1.50 0.14. 0.11 2.52 : 3.35 - '58 0 1.39. 1.20 I95 -.11.3 5.6 0.65' 0.82 0.4 1.50 -0.5.4 3.54 0 '1.2.8 - _-2:0.4 0.95

Still water resistance Model 241

Fn 0.2 0.3 0.4

RT (N) 3.79 9.80 25.96

Table 4

Wave force and moment Fn - 0.2 Model 241 Fn _ca cm L/A _{Fa Ma} 6(pç degr Mç degr

pgAc

pgIkç

* 1o3 * io3 0.2 2.80 1.00 140 282 38 -113 0.2 2.55 0.80 . 275 403 28 -106 0.2 2.45 0.50 540 .635 8 - 89 0.2 2.55 1.33 70 108 15.8 -122 0.2 4.60 1.00 ₁₄₉ 272 40 -104 0.2 4.45 0.80 297 420 33 -107 0.2 4.30 0.67 402 504 22 -100 0!2 4.05

057

506 611 18 81 0.2 3.90 0.50 545 660 14 - 70 0.2 4.70 1.3.3 71 103 160 -122 0.2 2.70 0.67 .397 512 2'2 . - 98 0.2 2.45 0.57 500 605 16 - 95

Table 5

Wave force and moment Fn = 0.3 Model 241

Table 6

Wave force and moment Fn 0.4 Model 241

-7-Fn _ca cm Fa Ma _Fç degr degr

pgAc

H

pgIkc

*' io H

*

0.3 2.75 1.00 163 296 43 -111 0.3 2.50 0.80 285 417 28 -119 0.3 2.55 0.57 480 576 17 - 90 0.3 2.40 0.50 572 665 10 - 86 0.3 2.5.0 1.33 98 126, 140 -112 0,3

251

2.00 130 .30 177 119 O.3 4.70 1.0.0 1.72 '292. .48 -119 0.3 4.40 0.80 316 429 27 -.100 0.. 3. 4...27 0.67 415 537 25 -132 0.3 4.03 0.5.7 530 606 21 - 95 0.3 3.88 0.5.0 563 658 1.6 - 92 0.3 4.37 1.33 96 127 153 -130 0.3 1.81 0.67 415 46.4 21 , - 95 Fn _a cm L/A Fa Ma '.. Fç de.gr . M,ç d'e.gr

pgAc

- pgi.kc. * io3 * 0.4 2.. 6.5 1.00 108 288 _{15 -120} 0.4 2.60 0.80 327 395 21 -101 0.4 2.42 0.57 533 556 20 - 96 0.4 2.40 0.50 579 623 16 - 94 0.4 2.45 1.33 - 117 - -117 0.4 4.65 1.00 206 ' 284 30 . -12,0 0.4 4,40 0.80 363 412' , 27 -109 0.4 4.25 0.67 46i 489 ' 22 - 95 0.4 4.10 0.57 54.5 577 20 -100 0.4 3.95 0.50 608 636' 13 - 91 0.4 4.45 1.33 66 122 - -165 0,4 2.85 0.67 438 477 23 -111

Added mass, damping- and cross coupling coefficients

Heave _{Model 241 Za '0.025 in} Fn

d9

_e0

0.2 1.639 1.310 - .21,3 .311 1.659 0.3 .744 1.697 - .1.20 .110 1.659 0.4 .626 1.698 - .099 .125 1.659 0.2 .530 1.560 - .028 .048 2.21.2

03

.548 1.665 - 044 .088 2.21.2 0.4 .492 1.597 -.042 .115 2.21:2 0.2 .5.11 1.5.00 -.01.2 .048 2.765 0.3 .481 1.439 .-..016 .090 2.76.5

04

.464 .. 1.474 .. -.01.4 _{.1.1.6 .2.765} 0.2 .450 1.2:89 . .002 .060 3.318 0.3 .465 1.262 .104 3.318 0.4 .454 ' 1.313 . .0:01 .125 .3.31.8 0.2 .452 . 1.0:54

.07

.085 ' 3.871 0.3 .456 1.0c85 _{..0,07 .128 3.871} 0.4 .4.53 1.1.57 .01.0

.44

_3.871 0.2 .472 .824 . .0:07 .107 4.424 0.3 .472 .888 .009 .152 . 4.4.24 0.4 .462 .991 .01,3 ' .167 4.4.24 0.2 .548 .511 .005 .129 5.5.30 0.3 .536 .570 .007 .192 5.530 0.4 .508 .699 . .015 .212 ' 5.530 0.2' : .608 .389 .004 ' .136 '.' 6.636 0.3 .6C1 .446' .004 .204 .6.636 0.4 . 5.74 523 . .013 .246 6.636

Table 7b

Added mass, damping- and cross coupling coefficients

Heave Model 241

Za = O05O

Fn a d,9 e.9 0.2 1.633. 1.35.2 - .21.6 .306 1.659 0.3 .798' 1.6.45 -'.118 .113 1.659 0.4 .675 1.726 - .105 .133 1.659 .0.2 .554 1.5.57 - .029 .050 .2.212 0.3 . .595 1.614 - .044 .093 2.212 0.4 .493 1.579 - .044 .118 . 2.21,2 0.2 .512 1.487. .- 013 _{.. 04.3} 2.765 0.3 .4.89

1470

- .0'16 .089 . 2.765 0.4 .471 1.454 . -.015 ' .1.18 2.765 0.2 .454 .1. 302 :002 .059 . 3.318 0.3 .454 1.287

_.00l

102 ., 3.31:8 0.4 .446. 1.334 0 .126 3.31.8 0.2 .4.56 1.069 .007 .086 3.871 0.3 .460 1.C'94 .007 .127 3.871 0.4 .450 1.1,83 .009 .14.5 3.871 0.2 _.471 .84,6 .007 .108 4..424

03

.470 .904 .009 .158 4.424 0.4 .458 1,014 .013 .167 4.424 0.2 .568 . 54.8 003 . 179 5.530 0.3 . 561 ' . 595 .004 .196 5 5.30 0.4 .532 .728 .013 .220 5.530 0.2 .639 .437 , .0'Oi .188 6.636 0.3 .632 .483 .0.01 .207' 6.6.36 0.4 .597 .563 .010 .255 6.636.

Added mass moment of inertia, damping- and

cross coupling

coefficient

Pitch model 241 0.026 rad

Fn a,9 b.9 0.2 ;0209 .047 - .003 - .087 3.866 0.3 .0235 .039 - .001 - .128 3.871 0.4 .0275 .031 .009. -.205 3.871 0.2 .0200 f 141 24O - .271 1.665 0. 3 .02 50 .068 . 136 - . 1.20 _{i.. 6 59} 0.4 .0426 .044 .. 106 - .109 1.692 0.2 .0255 .056 .01.2 - 101 :2.21.2 0.3 .0236, .051 -' .0.53 - .076 2.212 0.4 .0333 .046 ...073 -.169 _2.212 0,2 .O2i5 .051 .01:8 ..0:46. 2.765 0.3 .0233 .044 .024 -'.096 2.765 0.4. .0297 .037 _.04.2 . -.164 2.760 0.2 .0214 .049 .00.1 -.062 3.324 0.3 .0236 .040 .007 - .103 3.32.4 0.4 .0:280 .032 . .020 - .181 3.324 0.2 .0206 .04.2 - .003 - .109 4.419

03

.0:234 .037 -0'03 -.154 4.424 0.4 .0271 .03.0 .0.04 - . 223' 4.424 0.2 .0217 .030 .000 - .126 5.530 0.3 .0;233 .03.2 - .001 - .188 5.530 0.4 .0364 .028 .005 - .263 5.536, 0.2. .0236 .024 0 - .131 . 6.647 0.3 .0246 .027 0 - .197 6.6:47 0.4 .0270 .025 .007 -.277 '6647

Table. 8b

Adde,d mass moment of inertia, damping- and cross coupling

coefficient

Pitch Model 241. 0a = 0.052. rad

Fn

a9

b0

0.2 .0333 .147 .232 - .330 1.659 0.3 .0301 .0!66 .132 - .116 1.659 0.4 .0484 .0:60 .160 - .155 1.659 0.2 .0274 .056 .039 - .015 2.212 0.3 .0260 .050 .049 - .093 .2.212 0.4 .0356 .045 .073 - 164

22l2

0.2 .0224 .050 018 -.046 2.771 0.3 .0244 .043 .025 - .09.3 2.765 0.4 .030.1 03.7 .038 - .162 2.771 0.2 .0220 .049 . ..O01 -.0:58 3.318 0.3 .024.1 .040 .00.6 - .104. 3.313 0.4 .0282 .03.3 . . .019 - .174 3.318 .0210 .046 - .003 - .088 3.876 0.3 .0239 .039. - .001 - .1.28 3.871 0.4 . .0278 .032 .009 - .1.93 3.866 0.2 .0209 .043 - .003 - .109 4.424 0.3 .0237 .038 -.003 -.152 4.430 0.4 .0273 .030 .006 - . 220 4,424 0.2 .022]. .032 0 - .126 5.536 0.3 .0239 .034 -.001 - .187 5.536 0.4 .0307 .005 .005 : -.259 5.536 0.2 . .0241 .026 .001 _- . 130 6.636 0.3 . .0250 .028 .001 - .196 6,636

04

. .0272 028 .006 - .274 6.631

Pitch and Heave characteristics, Added Resistance

Model 253 Fn A/L Raw (N) za/ca z.ç degr OaL H degr RawL ca (cm)

2ca

pg'B2c 0.2

O75

0.90 0.12 -13 0.08 -183 2.1 1.91 0.2 0.90 1.06 0.30 -43 0.22 -165 2.5 1.90

02

1.00 2.62 0.50 -49 0.35 -178 . 5.9 1.95 0.2 1.05 2.98 0.69 38 0.46 '-166 7.8 '1.80 0.2 1.15 3.55, 0.95 -26: 0.62 -154 9.6 1.77 .0.2, 1.20 3.02 1.03 -15 0.69 -147 .11.1 1.52 0.2, .1.25 4.15

L06'

- '8

0.76-14.3

'9:8 _.1.90

0.2.1.35

... :2.4.2.1.02 .0 .0.7.8. -1.25 7.6 1.65 0...2 1.5,0:: .1.79 .0.98: 6:..0,..71. -109 . .4.5 . 1.83 0'2H 1.75,, 1.0:2'. 0.93 '

0.0.53

- 99 .2.2 1.97 0.2 2.00 0.6:8 _0.98 ' . 9

.0.43..- 93

.1.7 1.84 0.2 0.75 0.86 0.10 - 4 0.07 175. 1.1 . 2.54 0.2 0.90 2.06 0.28 -40. 0.21 -165 2.6 2.62 0.2 1.00 4.01 _0.54 5'O 0.37 . 170 4.7 2.70 0.2 1.05. 5.34 0.71 -3.8 0.46 -160 5.8 2.80 0.2 1.15 6.59 0.96 -28 0.61 -150

76

2.72 0.2 1.20

759

1.05 -18 0.69 -145 ' 7.7 2.90 ' 0.2 1.2,5 7.18 1.09 -15 0.74 . -143 ' 7.6 2.83 0.2 1.35 5.67. 1.03 ' 0 0.77' -125 5.7 2.9'i 0.2 1.50 2.36. 0.97 '9 ' 0.70 -107 2.4 2.90 0.2 1.75 0.36 0.95 6 0.52 - 97 '0.4 2.8:3 0.2 2.00 -0.25 0.96 ' 4 0.45 ' -. 93. -0.3' .2.65

Table 10

Wave. force and moment

Fn =0.2 Model 253 Fn _ca cm L/A _{Fa Ma EFç} H degr

Eç

.degr

pgAc

p'gIkc H H * * 0.2 2.2:3 '0. 50 494 572 H ₂₃ 9.7 0.2 2.35 0.57 412 53:4 2.5 96 0.2 H 2.27 0.67 349 460 3.5 105 0.2 2.5.5 0.80 . 2.47 .347 '4.2 . 111 0.2 2.15 1.0O. 199. H '280 74 116 0.2 0.2, , 2.50 . 2,. 28. 1.3.3 2 .0.0 98' 79 H. .106 . '4.0 ., .155 .220 144 273:. 0.2 3.83' H 0.50 501 &09 .22 .96 0.2 4.12 ' 0.57 44.6 538 32 10.5 0.2 4.40

067

3.55 4.56 38 10.8 0.2 4.50 . 0.80 . 263 365 50 113 0.,2 4.48 1.00 157 251 74 1.27 0...2 4.80 1.33 8'S 94 ' -184 -20.1 0.2 3.85

2O0

. 95 43 -101 -1.20

Added mass, damping- and cross coupling coefficients

He ave _{Model 253} Fn _d9.

_e9

Za 0.025 in 0.2 2.619 1.009 - .176 .619 1.646 0.2 1.041 2.820 - .032 .103 2.207 0.2 .989 2.256 - .019 .114 2.783 0.2 .907 1.949 .001 .137 3.341

O2

.887 1.596 .010 .173 3.869 0.2: .939 1.237 .007 .201 .4.434 0.2 1.042 H .758 .00.5 _H .229.' 5.492 .0.2 1i.73: .534 .002 .246 .6.636 Za = 0.050 m 0.. 2 2.791 _{1.261 - .213 .606 1.678} 0.2 1.048 2.560 - .033 .107 2.227

02

.996 2.264 - .020 .117 2.771 0.2 .912 1.986 0 .138 3.301 0.2 .908 1.588 .007 ,174 3.884 0.2 .949 1.237 .007 .206 4.445 0.2 1.08.2 .771 .004 .228 5.519

Table 12

Added mass moment of Inertia, damping- and cross coupling

coeffic ien.ts Pitch _{Model 253} Fn

ag

b9

0a = 0.026 rad 0.2 .0.841 .192

.190

- .631 1.658 0.2 .0305 .0.83 .042 . -.108 2.229 0.2 H .0194 :077 .027

:-I24

_.2766

0.2 .0160 .071 OO4 - .145 3:321 0.2 H. _{.0:1.36 .066 -.MO,3 -.179} 3.86.9 0.2 .0122 .059 -.00.3 .209. 4.421 0.2 .0134 .0.42. -.00,1 -.239 . 5.56.3 0.. 2. . .0I48.. .029 .0 - .253 . 6.674 = 0.052..r'ad 0.2 .0:831 .1.93 .198 - .601

i654

0 2 .0.331 . 080 .054 - 10;0 2 .2.11 0.2 .0.201 .077 .028 - .124 2.76.5 0.2 .0164 .071 .005 - .1.43 3.316 0.2 .0139 .066 - .003 - .179 3'.865

02

.0.124 .059 - .004 - .208 4.428 0.2 .0129 .041 - .002 - .245 5.531 0.2. .016.4 .O27 .002. - .261 6.65.8

0.1 0.2 0.3 0.4 0.5 0.6 0.. 7 0.8 0.9 1.0

F,))

Figure 1. Body Plan of Wigley Model.with midship-area coefficient 0.909 and beam-length ratio 0.1.

-90 180 2.0 Za/1a

;$

.1.0 0 MODEL 241 HEAVE

- OFn= 0.2

LFn = 0.3

-

DFn=0.4

1.0

Figure 2. Heave amplitude and phase model 241. 2.0 C, ec -180 AlL 1.0 1.5 MODEL 241 PITCH

.QFn.0.2

i?fl

0.3 OFti = 0.4 2.0

Figure 3. Pitch amplitude and phase model 241.

0 1.0 1.5

AlL

2.0 0 1.0 1.5 2.0

0.5

1.0

1.5

AlL

Figure 4. Added resistance model 241.

Fa/p gwCa Ma/p gI1kç

/

C.D w I I )

0.5

1.0

1.5

Figure 5. Wave force and moment Fn = 0.2, model 241.

Figure 6. Wave force and moment Fn 0.3, model 241. 100

I

0.5

100

Figure 7 Wave force and moment Fn = 0.4, model 241. 1.0 0.5

t

1.5 1.0 0.5 L/A

zz 1.0 2.0 1.0 Fn = 0.3 .1 II 00 Fn = 0.2 OZa =0.025 m HEAVE MODEL 241 Za 0.050 m Fn = 0.3 Fn = 0.4

Figure 9. Heave damping coefficient.

Fn = 0.4

HEAVE MODEL 241 = 0.050 m

5 5 5

WI_____

Figure 8. Added mass.

0 5 0 5 0 5 U)' U)' WI Fn = 0.2 1 a' zz 0.5

d'

F

O. 5

-l..0

Fu = 0.2 = 0.3 Fn = 0.4

HEAVE MODEL 241 0 Za= 0.025 m

L Z=

0.050 m Figure 10. Mass cross coupling coefficient.

0 Za= 0.025 in

HEAVE MODEL 241 L Za = 0.050 rn

Fn = 0.2 Fii = 0.3 Fn = 0.4

Figure 11. Damping cross coupling coefficient.

5 *10 -2.0 3.0 2.0 *10 1.0 ezO WI 0 5 5 0 5 5 0

* 100 6.0 4.0 '2. 0 Fn 0.2 J, 00 3

Figure .12. Addied mass moment o.f inertia..

Fn = 0.3 Pu = 0.4 PITCH MODEL 241 Oa 0.052 rad.

00

= 0.3

3

.5

Figure 11. PItch &ampingeoefflcicnt.

= 0.4 L

IA)'

OOa = 0.026 rad.

PITCH MODEL 2.1 Oa =O.O52 .rad.

15 10 * 100 5 b' 0 0 0. 5

5.

0 5 0 5 0 0 5

e'

Oz

2.0

Oz -.0.5

-1.0

-2.. 0

-3.0

= 0.2

0

Figure 14. Mass cross cOup1i.ng coefficient.

5;

OGa = 0.026 rad.

PITCH MODEL. 241 ,Oa 0.052 rad.

= 0.3

5

Figure 15. Damping cross coupling coeffIiWt.

= 0.4

5

w'I. 5

Fn 0.2. Fn = 0.3

PITCH MODEL 241. OGa = 0.026 rad., A'Oa = 0.052 rad. * 10

C Z 90 (degr) 180 20 0.5 1.0 0 0.5 A/L 1.5 2.0 Fn = 0.2

0a =

'i'0.018 m = '%O.028 in I - I -90 -180 (degr) OaL 20 10 0..5 0.5 1.0 1.5 I I -I

LO

1.5 2.0 A/L 2.0 -

FnO..2

OCa '10.018 I

= 4.08

in

Figure 16. Have amplitude and phase Figure 17. Pitch amplitude and phase

model 253. _{model 253.}

1.0 1.5 2.0

Za" a 10

RAW. L

pgB2

Figure 18. Added resistance model 253.

100

I

0

Figure 19. Wave force and moment Fn = 0.2, model 253. 1.0 1.5 0.5 AlL 1.0 1.5 A/L

0.5 *10.

d0

-2.0 0 *10

eO

Fn = 0.2, niodel 253. OZa = 0.025 m. Za = 0.050 m 2.0 P1.0 5 0 WI

Figure 21.. Cross coupling coefficients Heave, Fn .0..2.,

Figure 22. Added mass moment of inertia .and damping coe.fficientPiitch, En 0.2, mode:1 253. 00a 0:.026 rad.

'Oa

'0052rad

*10 -1.0 -2.0

Figure. 23. Cross àoup1ing coefficients. Pitch, En = 0.2, model 253.