Polynomial representation and damping of series 60 hull forms

o

?OL1KflAL sENrATION AIW DL'INC OF SRIES 60 mILL FO1MS

By S. Gerritsm, ., . .Kerwin, and 3. N, Newman3

Hay 1962

t.1

1.. _{Thchnologjcaj University, Deift}

Nassachusett Institute of Technology, Cambridge, Mass. flavid Ta7lor Hodel Basin, W4shiugton, _D.C.

ABSTRACT

Polynomial representations and damping coefficients for the Series. 60 Block .60, .70, and .80 hull forms _{are presented.}

Several polynomials are given, ranging from 48 to 140 terms and providing varying degrees of accuracy as analytic representa-tions of the, hull forms, _{Damping coefficients based upon these}

polynomials are presented, for various frequencies and Froude numbers, and compared with experimeital data, _{The agreement}

between experimental and theoretical coefficients _{is generally}

i

INTRODUCTION

in

the

_{past few years, due to the establishment of modern digital}

computers, it has been possible in the field of theoretical naval

archi-tecture to study problems which were previously impractical in view of

their cOmplexity. in particular, refined hydrodynamical theories can.

now be applied to fairly realistic ship hulls. As two inter-related examples.of such problems, this paper presents several polynomial

ap-proximat ions to the Series 60 family of hull forms, together with

-corn-putations.of the damping coefficients of the Series 60 hulls, based upon the polynomials, and an experimental comparison of these damping coefficients.

The importance of polynomial hulls to facilitate hydrodynamic research

on ships has long been realized, and a practical method for fitting a polynomial to a given hull form has been developed by Kerwin [1]. This

method consists of approximating the given hull shape by a

two'-dimensio-nal polynomial wfth coefficients which are determined by a least-squares technique. For details of this technique and further examples of actual

ships with their polynomial approximation, we refer to (11. In the

pre-sent paper practical polynomials of -various degrees of, accuracy are given

for the analytical representation of the three Series .60 hulls, with

block coefficients .60, .70, and .80, which were developed by Thdd _(2], and upon which detailed experiments of seakeeping characteristics have been made by Gerritsma (3].,

Although a comprehensive theory of ship motLons is still lacking _even for regular head wa?v-es, one important aspect of this problem, the damping

0'

and a 'three-dimensional theory including the effects o forward speed has

been given by Newman [4] based upon the Michell or "thin_sip" approxima-tion. _{We present here computations of the damping coefficients from this} theory, utilizing the above obtaied polynomials. The resulting comparison with Gerritsma'a experiments [31 provides an opportunity to evaluate the

complete research cycle of polynomial representation, damping theory, and experiments.

TH POLYNOMIAL HULL FORMS

Although a fairly detailed description of the procedure used to obtain polynomial approximations appears in [11, it is possibly worthwhile to sum-marie some of the more important features of the method..

A general expression for a polynomial hull surface is

m xi

h(x,z) = B/2

E E

_{am x}

z

(1)

n U

where x and z are non-dimensional coordinates in the longitudiflal and

vertical directions, h is the half-breadth of the hull at a point (x,z) and B is the maximum beam. _{This notation is illustrated in Fig. 1.}

It is fair.y evident that most hull shapes are not easily

ap-proximated by. a single polynomial of the form given in equatio&I1.. This

is particularly true for single-screw merchant ship forms such as Series .60 where extremely complex curvatures are present and where there are

relatively abrupt transitions from flat to curved regions on the hull. Consequently if the principal objective is to obtain the most .ac*irate

representation of the hull, it is better not to attempt to find an

ex-pression such as. (1),. .but to divide the hull up into _{a number of regions}

which can be fitted more easily. However, the objective in the present

case is to compute the damping coefficients of these hull forms, so that the -visual quality of the fit is important only to the extent that the

accuracy of the hydrodynamic coefficients is affected. _{Since a single} polynomial expression is much more suitable for hydrodynamic calculations.,

it is obviously .better to express the hull shape in this way provided

S

that sufficient accuracy in the final result can be obtained. It can be concluded from the results given in the next _{section that sufficient} ac-curacy can be obtained with the hull form expressed as a single polynomial.

One of the principal difficulties in obtaining _{a good polync*nl.al fit} to a merchant ship hull form is the fact that the flat _{bottom requires the} hull functidn h(xz) to have _{an infinite slope when z = 1. While a} poly-nomial can never have an infinite ølope, _{a close approximation can be} ob-tamed by including extremely high powers of z such as _{In order to} do this without incitiding all of the lower p0wrs of , it is convenient

to

define a special daae 'of (1.)

N

C1-i

_-1

_C3-1

h(x,z)

= B

jm

{ '

az"

+

a,

nl+Di+D2

n=l

n=C1 n=C

+

'Z amn

₍₂₎

in

which

up

to three arbitrary blocks

_Of

_{powers of z}

_{haye been}

_deleted..

_The eight integers N, N, C1 and D may then be elected arbitrarily to define polynomials in which a wide range of possible powers of zcan be

_included

witiot an unreasonable increase in the total

number

of

terms. Once

these

constants, have

.been

selected,

the coefficients a

_{can be determined for}

mp

any arbitrary hull form by a modified least-squares technique described in

(I].

The results of six different polynomial approximations to the Series

60 hull forms with block coefficients of .60, .70, and .80 are shown in

Pigs. 2, 3, and 4. The eight parameters characterizing each of these potynornials are given in Table 1, ad the coefficients a for three of

the

polynomials

are tabulated in the Appendix. The figures were re-produced directly from .a cathode ray tube recordar connected to the IBM

709 computer which was used to perform the computations. In these figures

the solid 'lines are ,a plot of the polynomial and the spots are the input

data. The

polynomial. is

_{plotted only up to the waterline (z = 0) while} 'the input 'data is given

for a

èhort distance above

the

waterline mi order

toprovide sufficient constraint to the slope of the polynomial at z = 0.

It may be of some interest 'to

note

_{tha.t there are a few incorrect data} points evident in rig. .4 which are completely "smoothed out" by _he polynomial.

While 'there are a large number of possible combinations of the con-stants M, N, C and D1 for any given total number of terms., it is fortunate tba.t the choice is not very critical. Por the Series 60 fOrms, it appears that it is essential to include one high power of z such a dud sb

least. one

intermediate 'pomer such

as

_{or z°. It also seems. to be best}

to have

'the

'ratio M/N somewhere around 1.4. It appears

that

the choice of cone tants for the 48, 70 and 96 term polynomials given here is fairly close

to the optimum.

tt can be observed from Figures 4 - 6 that the 140 term polynomials

are nOt 58 good .a _{the 96 term polynomials due to oscillations in the stern}

sections. _{These oscillations are present whenever the number of data points}

is not sufficient. As the number

of

terms in the polynomial is increased,

the 'minimum number 'of dat pints naturafly ncreasee. In 'these example S

offsets were giyen at 21 stations and 51 waterlines, which seemed _{to be} sufficient for 96 terms but not for 140 terms. ]n a previous

cdmputa-tion using half as many waterlines, the oscillacdmputa-tions _{were mu1a worse,} and were present in both the 96 and 140 term polynomials.

TIlE PITCH AND }tEAVg DAJPING C0EFFICIEN!I'

The polynomials presented in the preceding section

_{have been}

used for thecmnputation of the Series 60 pitch and _{heave damping coefficients,} based upon the theory of iewman [4]. _{This theory Consists essentially of} replacing the ship hull by a distribution of translating, _{pulsating sources}

and dip1es which satisfy the 1inearied free surface _{condition; the}

damping ny then be obtained either by integrating the energy radiation at itlfinity,.or by integrating the pressure over the hull surface. Computa-tiOns were presented in. [4] for the Weinbium polynomial model and compared

with Golovat&s experiments _{The qualitative dependence on forward speed} was similar between the theory and experiments, but the theoretical coef-ftcients were substantially higher than the _{experimental results.}

Figures 5 _{6, and 7 show the computed Series 60 pitch and heave} damping coefficients for the three block coefficient8 .60., .70, and .80, as funetins.of the non.dimensional frequency parameter..,/L/g and the Froude number. AlsO showa in the same figure _{are the experimental}

re-sults .of Gertitsixza [31. _{The agreement between these theoretical and}

ex-b4

ho&4 L.t. i*Iitrpv-FiJ 'ijtt caIeoP

perimeutal results is n

oodin view

_{Of the earlier experience}

in comparing the sane theory with other _{experiments [41. As in [4] we note}

that the damping coefficients are infiiite at '1 = .Wc/g = . This mathe-matical singularity, is also suggested by the experiments except in the

case of the .60 block coefficient,

The theory appears to exaggerate the dependence on forward speed by comparison with the experinnts }Ioweyer it should be noted that the ex-perimental results cover a relatively narrow speed range, from '.15 to .30

Froucte numbers. Considerably more speed dependence was foufld by GOlOvato with different model [4], The effects of very high speed are shown in Figure 6 for the .70 block coefficient, including Froude numbers'up to 1.0.

We note in particular that the pitch damping becomes negative for high speeds at intermediate frequencies,

_{in much}

_{the same manner as wds noted in} the damping of a submerged ellipsoid [5]. At very low frequencies, with non-zero Froude numbers, the damping curves become quite erratic, while at high frequencies the results are essentially independent of forward speed.

The negative pitch damping implies,

if physically

_{realistic, that at} high speeds floating (and submerged) bodies will be unstable in pitch, unless

the positive damping introduced by 'ri8cosity can prevent this. The physical

sOüre of negative damping Is briefly, discussed in [5].

The calculat:ions shown are based upon the 70 term

polynomials

(Figures

2, 3, and 4). In order to determine the influence on the

_.damptn,

:ealcula

-Lions of modifying the polynomial accuracy, calculations were also made with polynomials of 48, 56, 96, nd 140 terms. _{Typical results.are shown in}

b1e 2. The maximum variation between the results with different.po1ynomtls is seen to be only a few percent. Thus we may conclude that for practical.

purposes even the 48 term polynomial is sufficient for damping computations,

and presumably also for many other hydiodynamical computatiOns sxch as the wave resistance. This is probably a consequence of the smoothing operation of pressure integration er

the hull surface

since, as was shocm above, the 48 and 56 term polynomials do not accurately.describe the

hull Shape

in the Stern.

CONCLUSIONS

Several polynomial representatiøns of Series 60 hill forms have been given,, with yaryin.g degrees 'of accuracy. It

has

been Shown that the simplest polynomial, with 48 terms., is sufficiently accurate or certain

hydrodynainical applications, but

that 70

to 96 terms are greatly superior for a accurate "visusl" fit to the entire hull surface

The damping coefficients obtained from thase polynomials show fairly good -agreement with experimental results, suggesting that the

thin-ship

theory of ship motions is potentially a valuable analytic approach to the theory of seaworthiness The negative 'pitch damping at high roude numbers

indicates that experimental investigations of the motions and stability of

very high-speed Vessels

re of vital importance, and should not await the

future 'develOpmeOt of these essels

£CKOWGNZNT

Mr. M.R. Dabcovich Of the Department of Naval Architecture and Marine Engineering at LIT. converted the polynomial program for use with IBM 709/7090 cOmputers and also ran many of the results showi

here The authors wish to express their thark to him for his eforts in what turned out to be very time consuming task.

The computations presented here were carried out on the IBM 704,

7O9 Sand 7090 digital computers at the LI.T,. Computation Center, LI,T, Cooperative Computer Laboratory, aid the Dayid Taylor Model Basin Applied Mathematics Laboratory.

The part of the work done at LI. was performed under Contracts NOnr 1841 (64) and Nonr 1841 (67) as part of the Bureau of Ships undamental Hydromechardes Research Program, Project :S-R009-0l-0l, administered by

the David Taylor Model Basin.

IEURENCS

j, _{"Po1yi,omial Surface Representation of Arbitrary Ship Forms,"}

by .T. E... I(erwin, Journal of Ship Research, Volume 4, No. 1, 1960, pp 12-21.

"3p

lurthsr xperiments Ott Single cr 'Merchant Ship Forms-Series Sixty," byF. L Todd, Transactions, .SNA, Volume 61, 1953, pp. 516-574.

"Ship Motions In Longitudinal Wayes," by S. .Gerritsma,

Inter-pat:iona.l Shipbuilding

Progreès, Volume 7, No. 66,

1960, pp. 49-7l.

"The flpiiig and Wave Resistance of a Pitching

and Ileasing Ship,"

by S. N. Newman, Journal of Ship Research., Vol. 3, No. 1, Stifle 1959, pp.

1-19.

"The Dampirtg of ap Oscillating Nilipsoid Near A Free Surface,"

by S. N. Newman, Journal of Ship

Research, VoL 5, No, 3, December 1961, pp. 44-58.

T4e Farce:

.,d

/'foe,,tt

o,

f4eavii;

S'h;,

f

j/ova.'o

ZflurJ

_.j'

gtse1tfr

/ N

n11..:i

/9c7

/9.2

ThBLE 1

TNPtJT DATA AR NGMN IN ALL CASES

21 stations x = -L00 (0.10)1.00 51 waterlines z -0.250 (0.025) 1.000

11

POLYNOMIAL PA1S FOR NUMERICAL W1PIS

140

Total

number Of, terms 48 56 70 84 96

Number of terms in x

M

8 8 10 12 12 14

Ntunbr

of

terms in N 6 7 7 '7 8 10

First

.gap position

C 0 0 0 0 0 8

2nd

gap position

5 6 6 6 7 9

3rd gap position C 6 7 7 7 8 10

First.

gap value D 0 0 0 0 0 13

2nd _gap value D 16 15 15 15 24 19

COMPARISON. OF DAMPING COEFFICIENTS. FOR C1, . ..70 4D...VARI0US. POLYNOMIALS

PrtCH

HEAE T&BLE 2 12

NO. of

Terms :

w7.=.4o

Pr=O

Fr= 25

Fr=0

= .2.0.

Fr= 25

= 20

Fr=0

Fr= 25

Fr=0

Fr= 25

.48

.56.

70 84 96 140

.0915

.0935

.0904

.. O905

.0906

.

.0905

.1282

.1321

.1240

.1250

.1248

.1254

.1245 .1283

.1181.

.1182

.1183 .1I85

.0900

..

.0937 . .0858 .0875 p.0873 .0881

3.809

3.179

3.848.

3.220

3.7453.403

3..4'5

.3.166

3.745

3.142

.3.741

3163

.2.204

.2.080

2.242

2.123

2.030

:1.929

2.031... 1.933

.

2030

1.932

.

2.028

1.931

48 TERMS

70 TERMS

96 TERMS

INPUT DATA

56 TERMS

84 TERMS

POLYNOMIAL

140 TERMS

FIG. 2 POLYNOMIAL

_{APPROXIMATIONS TO THE SERIES}

48ITERMS

70 TERMS

96 TERMS

INPUT DATA

56 TERMS

84 TERMS

140 TERMS

POLYNOMIAL

48 TERMS

70 TERMS

96 TERMS

IN PUT DATA

56 TERMS

POLYNOMIAL

84 TERMS

140 TERMS

6

FIGURE 5 - DAMPING COEFFICIENTS FOR 0.60 BLOCK COEFFICIENT

KEY TO EXPERIMENTAL POINTS FOR VARIOUS FROUDE NUMBERS

0: 0.15 p:O.25 0.3 F:O PITCH Ch 0.60 5 F: 0

KEY TO EXPERIMENTAL POINTS FOR VARIOUS FROUDE NUMBERS

0:0.15 0.25 0.3 2 3 5 3 2 0.3 0.25 0.2 c.j 0.15 0.1 0.5

5_

4

3

2

0

KEY TO EXPERIMENTAL POINTS FOR VARIOUS FROUDE NUMBERS

0:0.15 A :0.2 0.25 xEQ.3 HEAVE Cb :0.70 N -J

FIGURE 6 - OAMPNG COEFFICIENTS FOR 0.10 BLOCX COEFFICIENT

KEY TO EXPERIMENTAL POINTS FOR VARIOUS FROUDE NUMBERS

0 = 0.15

A:0.2

b:O.25

03 0.2

0.5"

KEY TO EXPERIMENTAL POINTS FOR VARIOUS FROUDE NUMBERS

0:0.15 D0.25 . HEAVE Cb;O.BO 0 x I ID 6 0.3

0.25-0.2 0.l5 0.1 0.05 0.5

FIGURE 7 - DAMPING COEFFICITS FOR 0.80 310CK COEFFICIENT

0.2 O.i5 03

F: 0

KEY TO EXPERIMENTAL POINTS FOR VARIOUS FROUDE NUMBERS

0= _O.15 = 0.2 o: 0.25 X: 0.3 L PITCH Cb 0.80 5 6

T/BLE A.i

P0LYiOAL 00]

]C11NTS )48

1

.

.z

2 z

terms

3 .20 .z z2001

1

_1.00119

0.007611.

_0.27152

_-0.39583

_-0.15139

_-0.73.310

x

_0.00079

_-0.09079

_-0.25980

_0.358011.

_-0.10572

_0.09750

-0.38312

-0.37393.

-2.4663o.

0.75357

0.20608

_2.26362

3 _1.11.7238

0.62350

6.710511. -6.6111.05

0.31490

0.43711.3 x1I

-1.56638

.8773

6.8o8i

_-0.48576

_-0.60236.

_-2.27828.

x5 _2.62211.7 1.11.9665

_-16.37150

_13.73882

_0.07986

_-1,56656

0.98811.3

.1.94102

_-3.93306

_-0.32349

_.0.595.32

₀₄₃₁₄₉

-1.18908.

1. 731111.7

_9.28078

_-7.07054

_-0.33616

_.1.04978

1

TABE1E A.2

P0LIN01AL C0EPFICiNTS a

CB = 0.60

70 termE

2 z z

20

' z

200

1.00380

-0.00691

-0.06580

0.52208

-0.61459

-0.08320

-0.75537

0.00029

-0.03750

-0.08134

-o.i6o4i

0.26.530

-0.01757

0.03122

.O.31?62

-0.23778

1.35346

-

_.31900

3.07211.1

0.17163

2.27186

-1.39521

-0.29552

0.75880

11.11.9658 -1]... 68415

0.06858

1. 05097

-1.96611.0.

-2.52054

-11.81909

32.13318

-12.79666

-o.84888

-2.18137

2.24992

5.15154

io,o76

.47 08386

53.28766

-0.16262.

-3.49856

1.71801

2.8511.7

23.386311.

-'i.3.42'rrr

13. ?2127

2.00996

0.211.107 -0.715142

6.67850

-25.19531

98.19963

.68 .93 511.5 -p0.57225

3.89775

-0.40615

-0.38011.7 -12.133611.

16.41796

.2.68157

-1.23376

0.11.1806

-0.17596

2.15293

13.711.015'. -11.1.78005

26.87033

0.66699

1 1.O0339

O.oi8

-0.03350

x

o.004o6 -0.06262

_-0.09421

2

-0.32690

-0.13366 0.86728 -1.47891

0.758O

1.07995 -1.81642

_-3.98459

_-8.77268 2.75002 . 1.81791 8.31171 1.20829 _{8.51596 17.16052} -1.89679 0.77714 -22.39593 0.28430 _.85o79 8.16439 1.00263

-4.42681

.13.38316 -0.31433 4.11469 _-O.40995

XII

-0.41926 1.94884

-0.93173

TABLE A. 3

POL0AL C0ETIwJaS a

0D

0.60

96 terms

2

3.

z z.

-0.21250

1.06895 7.21636 -16.05129 -67.06824 95.88598 209.96434 -279.00048. -234.. 5783 318.07514 84.77609 -120.39790 -4 z 0.96116 .,2.32078 -25.21228 46.30231 195.59237 -289.94523 -.523.13014 .728.52983 533.05326 -738. 4088k -183.25503 ?57.83540 z5

-0.93120

1.38150 15.52977 -31.10522 -113 44058 183.85718 289.86147 J4.26 .39365 -285.96935 406.66397 96.20157 -135.65668

-0.03651

-0.05893 .0.38604 0.96476 2.82639

-5.23907

-6.29178 10.50876 5,24549

-8.53613

-1.41209

2.41516 200

-9.75903

0.08204 2.44.556 0. 81218

-3.33680

2.56396 2.71240

-10.13427

-1.34139

12.25187 0.29916

-4.79551

1 O,70 2 z 48terins 20 200 z. 1 0.98778 000683 0.15239 _-0.15500 -0.15267 -0.82563

x

o.6o

o.o3i79 -0.56969 0.611.967 0.02121

.-c:.i6ioi

x2 0.1'.310'.i _{0.118525 -1.92859 -0.31619}

-o.c586

1.35069 -0.20693 -o689 6.71933 -5.45107

-0.e8261

0.110911. -2.7816 -3.9992 5.597]A

1.21140

_0.320011.. 0.29399

x5

0.07059 4 30391'. -1k'. 811295 9 10695 1 18797 0 17264 3.3911.07 _{-3)12728 -1.03337} 0.54333 -0.84104 X7 0.10685 -3.56591 8.1'.5694 :J1..12816 -0.75506

-0.U1ioi

TABLE .A05

P0L0AL c0ETICNTS

1

x

H x3 X

x5

6

'1

z . z

0.70

.3

z

70 texs

z z

.20

z

200

0.99762

0.011487 -0.01i.625

_0.3379).

-0.33311.0 _0.11311.11.

-0.82757

-0.01928

-0.014511.

_0.03391

.

-0.35802

. 0.11.3294

0.03340

-0.11835

0.20100

O.L81

1.26810

_.2.45589

O.32686

_o.36i65:

o.11Ji82

_-o.o114r9

_i.94576

9.277b2

_-66o345

_o,6j36i

-0.51152

-1.60888

-2.711693

-9.35507

4.96689

7.67277

..3.21l.911.7 3.3221-i

-2.26447

1.17388

13.924311.

_-30.44687

_12.15857

₃₅₆₅₅₅

_1.88897

-047131

.0.92717

i7.8ob6o

6.73350

-23.76374

759

-6.38683

3.12984

1.92009

21l..13.9852

_23.24867

_4.81744

_7.20811.7

.4.40795

0.88614

_2.47209

-9.20258

-10.35328

.17.16653

4.03673

3.06913

-1.29088

-2.97083

12.19425

-1.24941

-11.06455

4.23463

0.14591

TABLE A.6

2.

3 z z

z-1 0.99925 0.0121].. _-0,07915 -0.27111.11. 1.22970 1.O1677

-o.b4ili.4

-0.83225

x

-0.00388 0.06769

-0.20927

1.O52k1 2.86917 -.1.628811. 0.1017].

-0.14413

0.11120 0.L5582 1.08215 3.52818 _-13.29296 7.9i.Q83

-.220].

0.69488 O.1C2Q9

_-1.3189

2.84857 i4..86726

-35.81971

20.70053

-0.82463

-0.56174

-0.83146

..3.26492

-6.718(2

..24.60516 65.82087 -35.23973

-0.80072

5.639112 -0.220112 8.89193

-14.18847

.!65.43905

i89.6o89

-124.911221 4.36663 2.52470 .2.77827 5.84552 7.78356 52.43337 -91.77592 38.74701 4.57839

-14.83454

xl

-1.81676. -18.7287? 146.61076 142.609411.

-508.188k7

353.72791 . -12.14313

-2.07477

3.64067 .8.05329 5.25234 -19.87137 -14.88192 25.611.933

_-6.26797

_14.53338 3.88478 21.48633 -64.92600 -176.65012 650.52440 -448.17633 14.49375

-0.63384

-1.13728

5.21319

-6.86637

-.12.083911. 53.59393

-36.26641

2.74359

-5.19723

x11 -1.911897 -10.34313 30.17866 86.17354 -299.35741 200.39751

-5.98809

0.887(2

P0LO)AL CO

iC.usrriS a Inn

CB=O.TO

96term

11_

TAE

A.7

P0LYIIOMIAL .00ETiC]INTS

a__

20

200

z I 0.99511.9

.0.03088

_0.011.076 0.006.511.

_-0.12898

-0.88289

z

-o.022i

0.11279.

-0.28977

0.10219

0.03995

Ô.05903

x2

o..iu6

1.0311.18

-0.32987

-1. 33123

0 07220

.0.12285

-1.60577

2.11.0825. 1 £2706 -0.51006 -2..(11206 -0.28513,

,Q59l 39

-1i..68623 6.02oo9

-0.096k5

-3.9k375

11.. 23302

-.112858

-1.75422

1. 358111 59020

.3 39]

b828

3.11.4038 1.01311.9 -3. 337311 1.118925

-1.79735

xT

_-o.7o516

_11..,38502 1.11.5189

5.96885

_-0.92653

-1.110336

48 terms

CB

O.8o

1

z CB =

2

z

0.80

3 z

70 terms

4 z z

20

z

200

1.

1.00115

0.00553

-0.00523

-0.03549

0.03415

-0.11242

-0.88767

0.02598 _{_O.079911.}

_-0.09008

_0.38'i-18

_-0.40399

_0.97888

_0.08499

2

-003999

-0.48987

-0.26992

6.31288

_-6.09833

_0.58824

_-0.00323

-0.64856

1.47289

1.49067

-10.08535

11.79959

-1.88888

_-2.14034

0.43297

3.800811.

2.23265

_-46.85543

40.81862

_-3.87310.

3.iiiii

3.65938

-6.53419

-5.74234

58.21311.8

_-61.58751

_9.05186

_2.938ô9

-1.85356

-11.42216

_-7.66868

_10048151

_-83.20669

_7.38591

_-.4.01798

-5.13990

14.21261

11.29255

_-114.49785

_108.78478

_-15.07299

0.42369

0.47065

7.91311

6.46734

-61.29307

48.97929

-4.00562

1.46921

x9

_2.09584

_-8.951422

_-7.4110O

_66.65480

_-58.91880

_7.83997

_-1.30831

TA&E A.8

L0AL

C.uthrxs

a

1.

0.99930

0.00203

-0.03356

-0.09558

0.46785

-0.38286

-0.08136

-0.87582

0.02048

0.04.939 0.0011.911. 0.570311.

1.03328

-O.6121i.1

-o.oi4i4

0.08883

0.06579

o.563?7

-0.17834.

.5.95334

14.17331

-9406t

1.0114.16

_-0.5719$

-0.52592

_1.14.3005

-0.61816

12.1211.19

-22.18625

14.70202

0.15956

-2.22533

0.11.7029

_-5.87423

_2.20697

_70.08940

_-163,37639

_97.63599

_-9.05112

_8.83966

x

2.91425

_11.73400

6.24.786

-89.42125

171.70397

-110.51909

-2.06798

3.11.0657

0.80907

18.311.911.9

-9.15365

_266.11.4460

77.65bb5

.3?6.2?4911. 25.142237

-20.4.0739

-3.31551

-32.96618

-14.88312

294.611142

-600.62596

350.1i.0423 6.11.11.379

0.33683

2.6937i

..28a6'r538

9.85905

396.91272

-7911.50901

426.00732

-28.17851

21.27671

0.16627

11.3.80342

16.54893

-1420.751422

833.82379

-462.84.530

-8.15208

-2.59718

x1°

1.30177

15.42911.7

_-1.93469

..195..33502

365.48222

-187.53618

10.83115

-823825

0.73310

..21.06408

-.7678o

204.49269

-389.63491

208.60542

3.6584.14.

0.97962

-

2

1

z

TALE A. 9

P0LYN0)AL C0J1ICJ..NT8

-3

4. 5 30

200

z -z

-z

z

POLYNOMIAL REPRESENTATION AND

DAMPING OF SERIES 60 HULL

FORMS.

by

J. GERRITSMA, J. E. KERWIN, J. N. NEWMAN

Reprinted from

INTERNATIONAL SHIPBUILDING PROGRESS

SHIPBUILDING AND MARINE ENGINEERING MONTHLY

ROTFERDAM

Volume 9 - No. 95 - July 1962

POLYNOMIAL REPRESENTATION AND DAMPING

OF SERIES 60 HULL FORMS

by

J. GERRITSMA*), J. E. KERWIN**) and J. N. NEWMAN**)

Abstract

Polynomial representations and damping coefficients for the Series60 Block .60,. .70 and .80 hull forms are pre-sented. Several polynomials are given, ranging from 4.8 to 140 terms and providing varying degrees of accuracy as analytic representations of the hull forms. Damping coefficients based upon these polynomials are presented, for various frequencies and Froude numbers, and comparedwith.experimentaLdata..Jheagreement between experimental

and theoretical coefficients is generally good. introduction

in the past few years; due to the establishment of modern digital computers, it has been possible

in the field of theoretical naval architecfure

to

study problems which were previously impractical

in view of their complexity. in particular, refined hydrodynamical theories can now Ibeappliedto fairly

realistic ship hulls. As two inter-related examples

of such problems, this paper presents several

poly-nomial approximations to the Series 60 family of

hull forms, together with computations of the damp-ing coefficients of the Series 60 hulls, based upon the

polynomials, and an experimental comparison of

these damping coefficients.

The importance of polynomial hulls to facilitate hydrodynamic research on ships has long been

real-ized, and a practical method for fitting

_a poly-nomial to .a given hull form. has been developed by Kerwin [1]. This method consists of approxi-mating the given hull shape by a two-dimensional polynomial with coefficients which are determined by a least-squares technique. For details of this

tech-nique and further examples of. actual ships with their polynomial approximation,

_{we refer to [1].}

In the

present paper practical polynomials of

variOus degrees of accuracy are given for the

analy-tical representation of the three Series

60 hulls,. with block coefficients .60, 70, and .80, whichwere

developed b.y Todd [2], and upon which detailed

experiments of seakeeping characteristics have been

made by Gerritsma [3].

Although a comprehensive theory of ship motions

is still lacking even for regular head

waves, one

important aspect of this problem, the damping

coefficients in the equations of motion, has received extensive attention, and a three dimensional theory

including the effects of forward speed has been

given by Newman.

[4]

based upon the Michell or

"thin-ship" approximation. We present here

com-putations of the damping coefficients from .this

theory, utilizing the above obtained polynomials.

*) Technological University, Deift.

* *) Massachusetts Institute of Technology, Cambridge, Mass. ***) David Taylor Model Basin, washington, D.C.

Reprinted from mt. Shipbuilding Progress_{- Vol. 9, No. 95 - July 1962}

The resulting comparison with Gerritsma's

experi-ments [3] provides an opportunity to evaluate the

complete research cycle of polynomial

representa-tion, damping theory, and experiments.

The polynomial hull forms

Although a

fairly detailed description of the

procedure used to obtain polynomial approxima-tions appears in [1], it is possibly worthwhile to

summarize some of the more important features of

the method. .

-A general expression for a polynominal hull

surface is

h (x, z) = B/2 E L' a5515 xm z (1)

m n

where x and z are non-dimensional coordinates in

the longitudinal and vertical directions, h is the

half-breadth of the hull at a point (x, z) and B is the maximum beam. This notation is illustrated in

Fig. 1.

It is fairly evident that most hull shapes are not easily approximated by a single polynomial of the form given in equation1. _{This is particularly true for}

single-screw merchant ship forms such as Series 60

where extremely complex curvatures are present and

where there are relatively abrupt transitions from

3

4

flat to curved regions on the hull. Consequently if

the principal objective is to obtain the most accurate representation of the hull, it is better not to attempt to find an expression such as (1), but to divide the hull up into a number of regions which can be fitted more easily. However, the objective in the present case is to compute the damping coefficients of these

hull forms, so that the visual quality of the fit is

important only to the extent that the accuracy of the hydrodynamic coefficients is affected. Since a

single polynomial expression is much more suitable for hydrodynamic calculations, it is obviously better

to express the hull shape in this way provided that

sufficient accuracy in the final result can be obtain-ed. It can be concluded from the results given in the next section that sufficient accuracy can be obtained with the hull form expressed as a single polynomial.

One of the principal difficulties in obtaining a good polynomial fit to a merchant ship hull form

is the fact that the flat bottom requires the hull

function h (x, z) to have an infinite slope when

z = 1. While a polynominal can never have an

infinite slope, a close approximation can be obtained by including extremely high powers of z such aszCOO.

In order to do this without including all of the lower powers of z, it is convenient to define a special case

of (1)

B M C1i

h (x, z) = -

x"1 {

a,,z°' +

2 m=l

,i

c2T C)!

--

a,,,,, Zn 1+/)1 _j_. a,,,,, z° 1+D1+D2 _.4. n=C2 N

+

a,,,,,z_l+D1+l)2+l)3} (2) n = C3

in which up to three arbitrary blocks of powers of

z have been deleted. The eight integers M, N, Ci and

D may then be selected arbitrarily to define

poly-nomials in which a wide range of possible powers

of z can be included without an unreasonable

in-crease in the total number of terms. Once these

constants have been selected, the coefficients a,,,,, can

be determined for any arbitrary hull form by a

modified least-squares technique described in [1].

The results of six different polynomial approxi-mations to the Series 60 hull forms with block

coeffi-cients of .60, .70 and .80 are shown in Figs. 2, 3 and 4. The eight parameters characterizing each of these

polynomials are given in Table 1, and the coeffi-cients a,,,,, for three of the polynominals are tabulated in the Appendix 1. The figures were reproduced

directly from a cathode ray tube recorder connected

to the IBM 709 computer which was used to per-form the computations. In these figures, the solid

lines are a plot of the polynomial and the spots

are the input data. The polynomial is plotted only

up to the waterline (z = 0) while the input data

is given for a short distance above the waterline

in order to provide sufficient constraint to the slope

70 TERMS

Fig. 2. Polynomial approximalions lo the series

60__C 0.60 hull for,,, 48 TERMS 70 TERMS 84 TERMS 56 TERMS 84 TERMS

Fig.4. Poly,,on:iaI apjiroxi,iiahio,Is toI/,c series

6O_Ca = .80 hull form

70 TERMS 84 TERMS

96 TERMS 140 TERMS INPUT DATA POLYNOMIAL

Fig. 3. Poly,,o,,,ial ajsproxi,natio,,s In the series

60C1 = .70 hull form

96 TERMS 140 TERMS INPUT DATA POLYNOMIAL

48 TERMS 56 TERMS 48 TERMS §6 TERMS

96 TERMS

INPUT DATA POLYNOMIAL 40 TERMS

TABLE 1. Polynomial parameters for numerical examples

of the polynomial at z = 0. It may be of some

interest to note that there are a few incorrect data

points evident in Fig. 4 which are completely

"smooted out" by the polynomial.

While there are a large number of possible

com-binations of the constants M, N, C and D2 for any

given total number of terms, it is fortunate that the choice is not very critical. For the Series 60 forms,

it appears that it is essential to include one high

power of z such as z200, and at least one intermediate

power such as or z40. It also seems to be best to

have the ratio M/N somewhere around

1.4. It appears that the choice of constants for the 48, 70 and 96 term polynomials given here is fairly close

to. the optimum.

It can be observed from Figures 4-6 that the

140 term polynomials are not as good as the 96

term polynomials due to oscillations in the stern

sections. These oscillations are present whenever the

number of data points is not sufficient. As the

number of terms in the polynomial is increased, the minimum number of data points naturally

in-creases. In these examples offsets were given at 21 stations and 51 waterlines, which seemed to be

suffi-cient for 96 terms but no,t for 140 terms In

a

previous.computation using half as many waterlines, the oscillations were much worse, and were present

in both the 96 and 140 term polynominals.

The pitch and heave damping coefficients

The polynomials presented in the preceding

section have been used for the computation of the

Series 60 pitch and heave damping coefficients,

based upon the theory of Newman [4]. This theory

consists essentially of replacing the ship hull by a

distribution of translating, pulsating sources and

dipoles which satisfy the linearized free surface

condition; the damping may then be obtained either

by integrating the energy radiation. at infinity, or by integrating the pressure over thq hull surface.

Input data arrangement in all cases

21 stations

x = 1.00 (OJO) 1.00

51 waterlinesz = 0.250 (0.025.) 1.000

Reprinted from mt. Shipbuilding Progress - Vol. 9, No. 95 - July 1962

Computations were presented in [4] for the

Wein-blum polynomial model and compared with

Gob-vato's experiments [4], [6]. The qualitative

de-pendence on forward speed was similar between the theory and experiments, but the theoretical

coeffi-cients were substantially higher than the

experi-mental results.

Figures 5, 6 and 7 show the computed Series 60 pitch and heave damping coefficients for the three

block coefficients .60, .70 and .80, as functions of

the non-dimensional frequency parameter v'w v'L/g

and the Froude number. Also shown in the same

figures are the experimental results of Gerritsma [3]. A short description of the experimental

determina-tion of the damping coefficients for heave and pitch is given in Appendix 2.

The agreement between these theoretical and ex-perimental results is good, but should be interpreted

with caution, in view of the earlier experience in

comparing the same theory with other experiments [4].. As in [4] we note that thedamping coefficients

are infinite at

= wclg = ¼. This mathematical

singularity is also suggested by the experiments

except in the case of the .60 block coefficient. The theory appears to exaggerate the dependence

on forward speed by comparison with the

ments. However it should be noted that the experi-mental results cover a relatively narrow speed range, from .15 to .30 Froude numbers. Considerably more

speed dependence. was found by Golovato with a

different model [4]. The effects of very high speed are shown in Figure 6 for the .70 block coefficient,

including Froude numbers up to 1.0. We note in

particular that the pitch damping becomes negative for high speeds at intermediate frequencies in much the. same manner as was noted in the damping of a

submerged ellipsoid [5]. At very iow frequencies,

with non-zero Froude numbers, the damping curves become quite erratic, while at high frequencies the results are essentially independent of forward speed. The negacive pitch damping implies, if physically

Total number of terms 48 56 70 84 96 140

Number of terms in x M 8 8 10 12 12 14

Number of terms in z N 6 7 7 7 8 10

First gap position C1 0 0 0 0 8

2nd gap position C2 5 6 6 6 7 9

3rd gap position C:i 6 7 7 7 8 10

First gap value D1 0 0 0 0 0 13

2nd gap value D2 16 15 15 24 19

6 .5 5 4 3 5 .3 2 .0.15 0.2

KEY TO EXPERIMENTAL POINTS FOR VARIOUS FROUDE NUMBERS

o 0.15. FO A0.2 o 0.25 x 0.3 HEAVE CbO.6O 0 025 0.2 0.3 . 015 F 0

Fig. 5. Damping coefficients for 0.60 bloc/c coefficient

KEY TO EXPERIMENTAL POINTS FOR VARIOUS-FROUDE NUMBERS

o 0.15 A 0.2 o 0.25 x 0.3 HEAVE Cb O.7O Cd H m 0.3

lig. 6Dam ping coefficients for 070blok coef1ient

KEY TOEXPERIMENIAL POINTS FOR VARIOUS FROUDE NUMBERS

O 0.15

A 0.2

o 025

x 0.3

PITCH Cb 0.60

KEY TOEXPERIMENTAL POINTS FOR VARIOUS FROUDE NUMBERS

.0 0.15 A O.2 o 025 0.3 7 2 5 6 4 _0.25

4 3

r

0.3 0.25/02 O5-..' 0.15 FO KEY TO EXPERIMENTALP0!NTS

FOR VARIOUS FROUDE NUMBERS

OO.I5 A 0.2 o 0.25 )( 0.3 HEAVE CbO.BO X 0 X a 0 A X I I I sL0 2 3 4 5 6

realistic, that at high speeds floating (and

sub-merged) bodies will be unstable in pitch, unless the

positive damping introduced by viscosity can

pre-vent this. The physical source of negative damping

is briefly discussed in [5].

The calculations shown. are based upon the 70 term polynomials (Figures 2, 3 and 4). In order to.

determine the influence on the damping calculations of modifying the polynomial accuracy, calculations were also made with polynomials of 48, 56, 96 and.

140 terms. Typical results are shown in Table 2. The maximum variation between the results with

different polynomials is

seen to be only a few

percent. Thus we made conclude that for practical

purposes even tlie 48 term polynomial is sufficient

Reprinted from mt. Shipbuilding Progress - Vol. 9, No. 95 - July 1962

0.3 0I25 0.2 0.15 0.I 0.05 0.5 çO.2 0.15 0.3-s 1 2

Fig. 7. Damping coefficients for 0.80 block coefficient

F0

for damping computations, and presumably also for many other hydrodynamical computations such as the wave resistance. This is probably a consequence of the smoothing Operation of pressure integration over the hull surface since, as was shown above,, the

48 and 56 term polynomials do not accurately

describe the hull shape in the stern.

Concius ions

Several polynomial representations of Series 60 hull forms have been given, with varying degrees of accuracy. It has been shown that the simplest

poly-nomial,, with 48 terms, is sufficiently accurate for

certain hydrodynamical applications, but that 70 to

TABLE 2. Comparison, of damping coefficients for Gb = .70 and various polynomials

KEY TO EXPERIMENTAL POINTS' FOR VARIOUS FROUDE NUMBERS

0 0.15 A 0.2 o 0.25 x 0.3 X

A0

_OX

-3 4 5 6 7 PITCH _{Cb 0.80} No. of Terms PITCH HEAVE w.VL/g = 2.0

Fr0

Fr=.2 w/L/g=4.0

Fr0

Fr=.2S w.I/L/g = 2.0

Fr0

Fr=.2 wVL/g = 4.0 Fr=0 Fr=.25 48 .0915 .1282 .124.5 .0900 3.809 3.179 2204 2.080 56 .0935 .1321 .1283 .093 7 3.848 3.220 2.242 - 2.123 70 .0904 .1240 .1181 :08.58 3.745 3.103 2.030 1.929 84 .0905 .1250 .1182 .0875 3.745 3.166 2.031 1.93 3 96 .09.06 .1248 .1183 0 873 3.745 3.142 2.030 . 1.932 140 .090 5 .1254 .1185 .088 1 3.741 3.163 2.028 1.931

.8

96 terms are greatly superior for an accurate "visual" fit to the entire hull surface.

The damping coefficients obtained from these

polynomials show fairly good agreement with expe-rimental results, suggesting that the thin-ship theory

of ship motions is potentially a valuable analytic

approach to the .theory of seaworthiness. The nega-tive pitch damping at high Froude numbers indicates that experimental investigations of the motions and

stability of very high-speed vessels are of vital

impor-tance, and should not await the future development

of these vessels.

Acknowledgement

Mr. M. R. Dabcovich of the Department of Naval Architecture and Marine Engineering at M.I.T. con-verted the polynomial program for use with IBM

709/7090 computers and also ran many of the

results shown here. The authors wish to express their

thanks to him for his efforts in what turned out to

be a very time consuming task.

The computations presented here were carried out on the IBM 704, 709 and 7090 digital computers at

APPENDIX

1.-TABLE A. 1. Polynonüal coefficients a,,171

TABLE A. 2. Polynomial coefficients a,,,,,

the M.I.T. Computation Center, M.I.T. Cooperative Computer Laboratory, and the David Taylor Model Basin Applied Mathematics Laboratory.

The part of the work done at M.I.T. was perform-ed under Contracts Nonr 1841 (64) and Nonr 1841

(67) as part of the Bureau of Ships Fundamental

Hydromechanics Research Program, Project

S-R009-0-1-01., administered by the David Taylor

Model Basin.

References

I. Kerwin, f. E.: "Polynomial Surface Representation of Arbitrary

Ship Forms." Journal of Ship Research, Volume 4, No. I,

1960, p. 12-21.

Todd, F. H.: "Sonic Further Experiments on Single Screw

Mer-chant Ship Forms-Series Sixty." Transactions, SNAME, Volume 61, 1953, p. 516-174.

Gerritsma, I.: "Ship Motions in Longitudinal Waves." Internatio-nal Shipbuilding Progress, Volume 7, No. 66, 1960, p. 49-71. Newman, J. N.: "The Damping and Wave Resistance-of a Pitching and Heaving Ship." Journal of Ship Research, Vol. 3, No. 1,

June 3959,p. 1-19.

Newman, J N.: "The Damping and Oscillating Ellipsoid Near A Free Surface." Journal of Ship Research, Vol. 5, No. 3,

December 1961, p 44-58.

Golovalo, P.:: "The Forces and Moments on a Heaving Surface Ship." journal of Ship Research, Vol. 1, No. 1, April 1957,

p. -19-26. C,3 = 0.60 70 terms a z2 z4 z20 a200 1 - 1.00380 -0.00691 -0.06580 -0.52208 -0.61459 -0.08320 -0.75537 x 0.00029 -0.03750 -0.08134 _0.16041: 0.26530 -0.01757 0.03122 x2 -0.31262 -0.23778 1.35346 -6.31900 3.07241 0.17163 2.27186 -1.39521 -0.29552 0.75880 11.49658 -11.68415 0.06858 1.05097 -1.96640 -2.52054 -11.81909 32.13318 -12.79666 -0.848-88 -2.18137 xa 2.24992 5.15154 10.05576 -67.08386 53.28766 -0.16262 -3.49856 x 1.71801 2.85047 23.38634 -43.42777 13.22127 2.00996 0.24107 -0.71542 -6.67850 -25.19531 98.19963 -68.93545 -0.57225 3.89775 x8 -0.40615 -0.38047 -12.13364 16.41796 -2.68157 -1.23376 0.41806 x" -0.17596 215293 13..7401i -41.7-8005 26.87033 0.66699 -1.47471 C,3 = 0.60 48 terms i . z z200 1.00119 -0.00764 0.27152 -0.39583 -0.15139 -0.733 10 x 0.00079 -0.09079 . -0.25980 0.35804 -0.10572 0.09750 x2 -0.3 83.12 -0.373:93 -2.46630 0.75357 0.20608 2.263 62 -1.47238 0.62350 6.71054 -6.61-405 0.31490 0.43 743 x4 -1.5:6638 -1.87273 6.80581 -0.48576 -0.60236 -2 .27 828 x5 2.62247 1.49665 -16.37150 13.73882 0.07986 -1.56656 x 0.98843 1.941-02 -3.93306 -0.32349 0.59532 0.73 149 x7 -1.18908 -1.73447 9.28078 -7.07054 -0.33616 1-. 0497 8

TABLE A. 3. Polynomial coefficients aiim

TABLE A. 4., Polynomial coefficients a,,11

TABLE A 5. Polynomial coefficients afl1fl

Reprinted from mt. Shipbuilding Progress- Vol. 9, No. 95 - July 1982

C1 = 0.60 '96 terms I Z 72 z3 z31) 1 1.00339 0.00818 -0.03350 _0.21250 0.9611.6 -0.93120 -0.03'6S1 -.0.75903 x 0.00406 -0.06262 -0.09421 L06895 -2.32078 1.38.150 -0.05893 O'.08204 x2 -0.32690 -0.13366 0.86728: 7.2.1636' -25.21228 15.52977 0.38604 2.44556 x3 -1.47891 0.27580 1.07995 -16M5.i29 46.30231 -31.10522 0.96476 0.0121.8 x4 x5 -1.81642 2.75002 -3.984.59 1.81791

-8.7728

831171 -67.06824, 9..885,98 195.59237 -289.94523 -113.44058 183.85718 2.82639 -5.23907 -3.33680 2.56396 x 1.20829 8.51596 17.1:6.052 209.96434 -523.13014 289.86147 -6.29178 2.71240 xT -1.89679 0.77714 -22.39593 -279.00048 728.52983 -426.39365 10.50.876 -10.13427 0.28430 -8.85079 -8.16439 -234.25783 533.05326 -285.9.6935 5.24:549 -1.34139 x9 1.00263 -4.42681 13.38316 318.07514 -738.40884 406.66397 -8.53613 12.25187 x10 -0.31433 4A1469 -0.40995 84.77.609 -183.25503 9.6.20157 -1.41209 0.29916

xU

_{-0.41926 1.94884 -0.93173 -120.39790 257.83540 -135.65668 2.41516 T479551} GB = 0.70 . 70 terms z z2. 1 0.99762. -0.01487 -0.04.625 0.33794' -0.33340 -0.11344 -0.82757 x -0.0:1928 -O.O'144 0.o3391 -0.35:802' , 0.44294 0.03340- -0.11835 0.20100 0.11814 1.2681.0 ' -2.455:89

-0.2686

0.36465 0.83054 x 0.44182 -0.04479 ' -1.94576 9.27702 -6.60345 -0.613-61 -0.51152 x4 -1.60888 -1.74693

-9.3507

4.96689 . 7.6727.7 -3.24947 3.3.2211 -2.26447 1.17388 13,92434 -30.44687' 1215857 ' 3i.6'555 1.88897 x6 -0.47137 -0.92717 17.80060 6.73350 -23.76374 7;01259 6.38683 x7 3.12984 1.92009 -4.4985.2 23.248,67 4.81744 -7.20847 -1.40795 0.88614 2.47209 -9.20258 -10.35328 17.16653 -4.03673 3.06913 -1.29088 -2.97083 12.19425 -1.24941 -11.06455 4.23463 . 0.14591 C,1 -. 0.70 48 terms z20 z20° 1' - 0.98778 -0.00683 0.15239 -0.15500 -0.15267 -0.82563 x . 0.0260.5 0.03 179 -0.56769 .0.64967 0.02 121 -0. 16 101 x2 0.43441 0.4852.5 -1.92859 -0.31619 -0.02586 1.35.069 x3 -0.20693 -0.68947 6.7193.3 TS.45107 -0.48261 0.11094 x4 -2.78176 -3.99992 5.59714 1.21140 -0.32004 0.293,99 x5 0.07059 4.30394 -14.84295 9.10695 1.18797 0. 172 64 x 1.36368 3.3940.7 - -3.42728 -1.03337 0.54333 -0.84104 xT 0.1068,5 -3.56591 8.45694 -4.12816 -0.75506 -0..1 1401

10

TABLE A. 6. Polynomial coefficients aiim

TABLE A. 7. Polynomial coefficients aiim

TABLE A. 8. ' Polynomial coefficients 'aiim

= 070

96 terms z2 Z' z$o z200 1 0.99925 0.01211 -0.07915 -0.27144: 1.2 2970 -1.01677 -0.04 144 -0.83225 x -0.00388 0.06769 -0.20927 -1.0 5 241 2.8 69 17 -1.628 84 0.10:171 -0. 144 13 x2 0.11120 0.155 82 1.08215 3.52818 -13.29296 7.94083 -0.22 016 0.6948 8 x3 0.10209 -1.3.12 89 2.848 5,7 14.86726 -35.81971 20.7005 3 -0. 82463 -0.5 6174 x -0.83146 -3 .26492 -6.71802 -24.6.05 16 65.82087 -35.23973 -0.80072 5.63942 x -0.22042 8.89193 -14.78 847 -65.43905 189.60899 -124.94221 4.3 6663 2.5 2470 x6 -2.77827

5.8452

7.78356 52.43337 -91.77 592 38.74701 4.57839 -14.83454 xT -1.81676 -18.72872 46.61076 142 .6 0944 -S 08.18 847 3 5 3 .7279 1 -12. 143 13 -2.07477 x8 3.64067 -8 .0 5 3 29 5.25234 -19.8 7.13 7 -14.88192 25.64933 -6.26797 14.53338-x' 3.88478 21.48633 -64.92600 -176.65 012 650.52440 -448.1763 3 14.49375 -0.63 384 X11 -1.13728 5.2 13 19 -6.86637 -12.083.94 53.59393 -36.26641 2.74359 -5.19723 x11 -1.94897 -10.343 13 30.17866 86.17354 -299.3 5741 200.39751 -5.98809 0.8 8702 C1. = 0.80 70 terms z4 !fflP 1.00115. .0.00553 -0.00523 -0.03549 0.03415 -0.11242 -0.88767 x 0.02598 L007994 -0.09008 0.38418 -0.40399 0.07888 0.08499 -0.03 999 -0.48987' -0.26992 6.3 1.2 88 -6.09833 0.58824 -0.00323 x3 -0.64856 1.47289 1.49067 -10.08535 11.79959 -1.88888 -2.14034 0.43297 ' 3.80084 2.23265 46.85543 40;81862 -3.87310 3.44441 x 3.65938 ' -6.5.3419 -5.74234 58.21348 -615'8751 9.05186 2.93809 x -1.85356 -11.42216 -7.66868 100.78151 -83.20669 7.38591 -4.01798 xT -5.13990 14.21261 11.29255 -114.49785 ' 108.78478 -15.07299 0.42369 x8 0.47065 7.91311 6.46734 -61.29307 4:8.97929 '-4.00.5 62 1.4692-1 x9 2.09584 -8.95422 -7.411.00 --66.6-5480 --5-8.9-1-8 80 7.83997 -1.30831

C, = 0.80

48 terms z2° 0.99549 -0.03088 0.04076 0.00654 -0.12 898

-0.8 829

x -0.02421 0.11279 -0.28977 0.10219 0.03995 0.05 903 x- 0.11126 1.0347:8 -0.3 2987 -1.33123 0.44247 0.07220 x3 0.1228 5 -1.60577 2.40825 1.62706- -0.5 1006 -2.04206 x4 -0.28513 -4.68623 -0.09645 4.2 3302 -1.7 5 422 2.5 9020 x') 0.5'9439 6.02009 -3.943,75 -7.42858' 1.35814 3 .3 98 8 1 -0. 8092 8 3.44038 '1.01349 -3.33734. 1.48925 -1.7973 5 x7 -0.70516 -4.38502. 1.45 189 5.96885 -0.92653 -1.40336

The experimental determination of the damping coefficients' for heave and pitch was carried out

with three eight-feet models of the Series-Sixty

[2]. The main dimensions of the ship models are

given in Table A 10.

The forced oscillation technique was used to

de-termine the' damping coefficients. In the case of

heaving a vertical sinusoidal force is applied to the

ship model by means of a Scotch Yoke and a 'soft

spring. The model is restrained for all motions

except for heave.

The vertical motion of ' the upper part of the

spring is given by:

r

r() sin ('J I,

where:

the circular frequency of the motion,

r0 _{half the stroke of the Scotch Yoke.}

The resulting heaving motion of the model can

TABLE A. 9. Polynomial, coefficients a,,,

APPENDIX 2.

'be described with sufficient acëuracy. by:

z = z0 sin

(co I- a)

where:

z0 = heaving amplitude

a = phase angle with respect to the motion of

the Scotch Yoke.

Assuming a linear damping term, the equation

of motion for this system'can be written as follows:

az + b + cz=.k. (r-z),

or:

az + bz+ (c+ k) z

krsinw/,

where:

a = total mass, including the hydroclynamic

mass

b damping coefficient

c _{waterplane area, multiplied by the specific}

'weight of the fluid.

TABLE A 10. Main dimensions of ship models.

Reprinted from _{mt. Shipbuilding Progress - Vol. 9 No. 95 - July 1982}

11. C11 = 0.80 96 terms z .z2 z4 z3° 1 0.99930 0.00203 -0.03356 -0.09558 0.46785 -0.38286 -0.08136 -0 . 8 7 5 8 2 x 0O2048 0.04939 0.00494 -0.57034 1.03328 -0.61241. -0.01414 0.08883 x2 '0.06579 0.56327 -0.17834 -5.95334 14.17331' -9.10691 1.01416 -0.57798 x:l _{-0.52592 -1.43:005 -0.61816 12.12419 -22.18625 l4.702O2 0.15956 -22253,3}

x4 -0.47029

-5.87423 2.20697 70.08940 -163.37639 97.63599 -9.05112 8. 8:3 9 66 x 2.91425 11.73400 6.24786 -89.4212'S 177.70397 -110.51909 -2.06798 3.40657 x6 0.80907 18.34949 -9.15365 -266.44460 577.65005 -326.22494 25.42237 -20.40739

x7 -3.31551

-32.96618 -14.88312 294.61142 -600.62596 35040423 6.44379 0.33683

x8 -2.69371

-28.67538 9.85905 396.91272 -794.50901 426.00732 -28.17851 21.27671 x9 0.16627 43.80342 16.54893 -420.75422 833.82319 -46284530 -8.15208 -2.59718 X1° 1.30177 15.42947 -1.93469 195;.33502 365.48222 187.53618 10.83115 -8.23825

X1 0.73310

-21.06408 -7.76780 20449269 -389.63491 208.60542 3.65844 0.97962

Length between perpendiculars _{'2.438 m 2.43 8 m 2.43 8 m}

Breadth 0.325 m _{0.348 m} ' 0.376 m

Draught _{0.130 m 0.139 m 0150 m}

Displacement _{61.9. kg 82.9 kg 109.9 kg}

Block-coefficient 0.600 0.700 0'800

Waterplane coefficient 0.706 0.785 0.871

Longitudinal radius of gyration

_{0.25 L}

_{0.25 L}

_{0.25 L}

Moment of inertia of waterplane 0.170 m4 0.229 m4 0.321 m2

Centre of flotation from _{A 0.094 m} ' A 0.041 m F 0.010 m

0

12

Substitution

of z'= z0 sin (oi I - a)

in the

equation of motion gives an expression for. the

damping coefficient:

kr0

b =

sin a

w z0

During the experiments, r and z were recorded

on a base of time and consequently r0, z0, w and a

could be determined from the recordings. A high

recording speed was used to increase the accuracy

of the phase readings.

The experiments covered a frequency range of

w = 3 to w= 14 and each model was tested at

four speeds namely:

V//gL = .15, .20, .25 and .30

The same program was carried out for the

pitch-ing motion ip. In this case the model is free to

pitch but is restrained for the other motions. The pitching axis goes through the centre of gravity of the ship model and the spring of the oscillator is, connected' to the 'model at a distance 1 from the

centre of gravity. Now the equation of motion

will be:

A.y -j-- By -j-Cy = It! (r

-

lip) or:

4';' ± By + (C + k12) 'ip = kir0 sin w t,

where:

A = total mass moment of inertia of the model

including the hydrodynamic mass moment of inertia.

B = damping coefficient.

C = moment of inertia of waterplane

multi-plied by the specific weight of the fluid.

The pitching motion can be described by:

p=yosin(wtfl),

where:

= pitch amplitude

fl

= phase angle with respect' to the motion of

the Scotch Yoke.

Substitution of this expression in the equation of

motions leads to:

B =Afro

1/)o W

Due to the finite tank width (4.3 meters) the

damping coefficients are influenced by wall effects

when w < 3 to 4.

Moreover the absolute values of a and fi in' this

region are too small to give a reliable phase