A linearized theory for the motion of a thin ship in regular waves

7d3iHD

Ace

 t ART7Rfl ThEO FOR

.)tON

OP A TWi

Lk

fl

REGTJL&R WAVES

by

JO NiEOLAS NEMA'T

S0B., Msanhusettz Iflatitu:te of Techno1ò

(1956)

LS., Masaahusetta Ins.tute. c TechnoTh (1957)

I ?APPTAL PULFTTIT!gNT OP.

_RUÏREMEIS

F

DE

OF DOC OP

at the

MASSAUSELTS NSij.xu OP TOIilOL Pebrual7, 1960

Si-.NavteotuxaMarjn;

Certified. y

j Stervisor

ted. by

Deparnt1 Coiunittee on

Gatite

_Students

Lab. y.

.Schpbowkte

Techgische Hogeschool

Delit

Ac1aowledement

This investigation

_{as carried out at}

Cambridge Univerai1,

Cambridge, England, by special permission

_{frOEn the Conunittée n Graduate}

School Policy of the Mshusetts

_{Institute of Technolo'-.}

_{The author}

is

inRbted to ma

people on both sides. of t

_{Atlantic for}

_mAlri1g

this arrangement possible, but special

thavik8 are due to Pxfeaaor

Martin À. Abkwitz of M. I. T. and Dr Fritz

UrseU of Cambridge, Who

acted as joint seviaors,

_and

to the

_{Analytical Ship-Wave Relations}

Panel

of the.

Society of Naval

_{Architects and Märine Eiigineers}

_for spoflaoring this work.

Bl,2

Nitation

unit vect'a

¡I

Denoten integration over the length

of

_the ship' a lead waterline

L,M,N

Coonenta of the 'ont veCtor

M

External nt

vector acting on ship

n

Unit.noi'vnl Vector

p

Pzeasure; Pourier cent index (aú a mpe

script).

R

Polar coordinate

of

radius

=

/(x-W.)"(y.-IJ.)' (.)1

r

Pbaitioñveotor in (x,y,z) system S Surface Of inte"4ton

Denotes integration over one side of the ship' a surface below the load waterline

a (u) Green's function si defined foliawing

n

_{quation (3.7)}

t

Ti

u

Duny variable of integratiOn

Dauqing oceffioienta

C

_{Contour of integration}

o

_{Ïsn forward}

_{ye.00ity it ahip}

Extezal foro. vector acting

on ship

G

Green'

s ftmction

g

_{Gravitational acceleration.}

h(x,z)

_{Kuli function defining the local been}

Velocity potential

Frequency of incideflt vaTes relative to space

=

fl

+ Ko = frequency

of

encounter

A dot denotes differentiation with respect to time.

A bar denotes the time average

A aeXscript index referÉ to the ooionente in.

the

Poier e,anaiona (2.9) to (2.13)

Multiple subscripts refer tO caoneiits of the

'perubaticn

ean3ions (2.9) to (2.13), except for

the matrix el.mmts

defined bet cre equation (4.22),

where double subscripts

Introduction .

Generai Ponilstion

of the Problem

fI

Derivation of the Velocity Potential

IV

The. Forcés and Moment

V

Diussion

à

At,pendices

Â.

Derivation of the Velocity Potentials

and

B

Derivation of the Velocity Potential 1101 56

Derivation of

the Forces and Moment 60

Figura

1

The CoortiinAte

Systms.

7

2 Diffraction Wave Crest for cosu> O

3 Diffraction Wave Crest for cosu<O and.T27/52 26

1,..

Diffraction Wave Crest

for oosucc0 and 27/32

L'<1

5 Diffraction Wave Crest for cosu <0 and

> i

27 6 Plot of

the

Parameters T and.

Táble of Contents 1 6 13 29 '9

u

Green'i funótio

huntof integration defined

following equation (3.7)

Velocity vector

X,T,Z Co*onenta

of

the force vector

Xy, z Rectangulax coordinate system tid in ship

107

_{° O}

,z Reotangula coordinate system

translating in

space with constant forward velocity o

XIDZG CoOrdinates of ship' s center

of

gravity Surge motion displacement

Heave motion displacement 3,

Free surface elevation:

Beam-Length ratio of ship

ÇMotion perturbation parameter: the order of

the asplitude of ship' s motions due to wavws

or other excitation

ÇTuident wave perti'bation parameter: the

ratio of hAlf-asplitude to length at the me dent wave syst

Du coordinate system rx'espovMug to

(x,y,z)

Diwr coordinate system corresponding to

POlar coordinate of angle

fo4o,;

e

A Linearized Theoiy for the Motion of a Thifl Ship

in RegUlat'Waves

by 30m N Chola Nensn

Subeiitted to the Dsparflt of Nav&i Ar iteituir aM Marine

Engineering on October 1, 1959, in partial fu].filimst of

the requirenents foe the degree of Doctor of Sòi1,ee.

The iresent investigation considere the lineaxized forces acting

on a thin ship in regular noxI. waves, on

the basis of a yetenatic

pertui-bation eion In tóX

of the beam-length

ratio aM the

alitndo of the oscillations. . It is

that the first-order forces

one4t solely of inertial ¡M hydrostatic restoring forces,

together with

the exciting forces Lue tO the incident 'wave

tion in the absence of

the ship, as Obtaiñed by the Pronde-Kr1oY 'hypothesis. In

present

psper the ascend- rder equations arC

derived, representing the

hydrod1ynamical diatuztance due to the presence of the ship, and inel"g

both the rsstóring fores Lue 'to the

11ations and. the exciting

forces due to diffraction of the incident waves.

At non-reømant

frequencies the displacenents of pitch, heave, ÓM surge are ôbte1nd

to first Order fron thé first-order forces, and' to second order fz

ccnsideraticn 'of the second-order forces,' but at resnìapàe the, solution

for the first-order 'diaplaconnt8 depende on the force8 cf both orders

of sgnitud.

'The present""Ayais ¿iffera fx earlier investigations

of the second-order foaee by thà uae«cf perturbation eaflsions

to oarxy

out the linearization dn á rstematio 'er. As a result ail

of

the

-forces cf second order are inoluded in the resultj*g eqUat4.

It is

suggested that a combined theoretical and eerinental program of

research

should be ocndted to detorwin 'the validity cf th. present result8

and those .of the seni-espirical

strIp-theoz!..

Thesis Siorvisorss

Nartil A. Abkowitz,

Associate' Professor cf' Naval Arohitécture,

Maéaaehusötts Institut. cf Technc1o.

Fritz Ursell,

I

Introd.uctio

In recent years considerable effort has been directed.

towards, the

linearized' a-"alysis of ship tions in regular

head

seas. These

investigations have

generaUr

fallen

into

two categories, being based either On the linearized. equations of motion with. send.-empiricai

co-efficients,. or on the complete h

,odaxiica]. analysis

of a floating body in' the' absence of viscosity. The f oxmer approach is exemplified

in the recent work of Korvin-Krotkovs1cy and JacobS [i) and. HaSlçthd [2). It remains, however for thé

.hdxoãnaniica]. approaoh

to be developed. so as to. pláce the solutiOn of the problem On a rationpl 'basis,

and it

is twards this goal that the present work ta directed4

The only significant difference between the

above

methods of approach lies in the imd.er]%ying assumption of each. In the

seni-empirio4: nLysis the coefficients of the equations of motion are de'

rived. f rn the "strip..tbeozy" whiob. neglects th ee-imenaio-1- effects

,,,S

he

effecs

of the forward velocity. On the other band. if the bdrodynaniical approach is to be practicable the ship. must be thin, or flat, or a

cozñbination

of the two, in order that the disturbance caused by its forward motion is sufficiently

5mRll

to linearize. Although the flat (iaithigtpé) hull offers scne advantages over the thin (Michell) hul].,

the solution of the foraer is

oonsi4erably more difficult and in the present Work we sh]l

concentrate

on the

study

of the thin case.

The

bydrod'nazical 'study

of this model was initiated by the

classical papr of J.H. Miohe].l in 1898. Subsequent treatments of the' problem of motions in the plane of

Bynet17

have essentially been

1Numbers in brackets refer to the references on pages ¿

2

generalizati of this analysis. _{In particular the}

wo± of

M.D. HAnk-Sd

[j)

_{may be mantioned. as the first}

attezit at the oon].ete

solixj

of the

wiateady problem. _{More recently, however,}

Peters and Stoker [ii.] have ei1oyed _{a aystematio perturbation}

eansion in

powers of the beam-length, ratio to ShoW _{that the a. priori}

means-atiene of

T1ri4

_{ad. others are not ooflaistent.}

It is shown by POtere and StÖIr that the lowest order restoring forces _{ñaist oxily} of those due to hydrostatic _{pressure and ri.d-.bÖdy inertia, while} what are usually _{as danping and. added}

ss forces are of higher

be

order with respect\to the beam-olength

ratio, and should ofllyinóluded in the course of investigating _{SU forces of this òrd.er.}

These results have been critically received by naval architects,

who frn eerience

have felt that the daning _{and added mass 'fóroes} are

of

the same order as

the

_{hydrostatic and. inertial terms.}

However

the

argisnsnt of Peters SM Stoker is. based _{on mathematical results,} and. in fact may be verified by _{elementary physical considerations.} If a thin body of beam-length ratio _{is given a n1l displacement} in the

_{].oflgitiiMiui1}

plane, the bydrodynainjca], _{jaturbance will be of} order in the a'bsence of viscosity. _{The rei1ting force acting on} the body will equal the integra], of _{this pressure over a projected} area which itself is of ozer _{,. whence the hydro&ynamic force will be} .of order

2

On the other hnniI _{the hydtostatic and inertial forces}

are easily

seen

to be of order _{, and. will therefore be the d.nitnt}

forces.

The tendency is to resolve this apparent conflict between theory ant]. practice by concluding that the thin ship _{is an imaatisfaotoxy}

3

model for investigations of ship itiOns. This appears to be thS

conclusion

of Peters and Stoker, who for this reason have oonside'ed the flat type

of

hull. How ver the mathematical n*y

sia at the fiat

model involves

the solutión o

singular integral equations. Purther-more the usual surface ah is of

mi 1

beam-l8ngth ratio, and

reason-able sucoess has

been attaiñed in the study of steady-state wave

reeistance using the Mioheil model. Moreover an

e,rminatiou of the

coefficients

tabulated by Korvin.4Croukovs]Qr aM Jacobs shows that the

hydrostatic and

ioertal fârces are indeed &ì4m't except near

resonant frequencies, where their determ4nsitt

vanishes.

It is just

this tact

which makes neoeasazy

the study of the

second order forces,

vonj

for at resonant frequencies the first order restoring forcesleaving those

at

second order as the leading terms.

It is apparent therefore that an nálraìs which is válid at

all

frequencies must include forces of both ordeza of

mgnitude.

HOweVer' it jó equal]y apparent from

the investigation of

Petera

and Stoker that

axq second

order

study

must be done .in a systentio 'nner so as to

include all

of the appropriate forces. The present york is therefore

a study Of the

first and

second order' oscillating forces, aMlysed

by the methOd Of

perturbation eanSiofla.

U4le the work of

Peters

and

Stoké', three separate perturbation parameters az'e employed to represent the orders of. the beam-length ratio, the slope of the cn.

emth'g waves, and the osoil].ations of the ship. The relations between these parameters may be deterjixLned from the

&ynamioal

conditions that, for exaile, no external f croes except a given propeller thrust be

acting on the

Ship. The use of three paraitera offers, át the

4.

general theory which by

appropriate àhoice of the. re].ationcihip. between

parameters can be ide to oòrrespond. to the problems of reso'nt

_Or

non.-reaona.t treciuenoies,. free or res

.ined

rnoti.On in. waves,' or foroed oaoj11Atiofla

in

Cala water.

The aflalytioal methods enloyed az

.ssexitlafly

those used by Peters Stker; that is,. Gree& s theorem. is

used

to derive the

velocity potential and the

forces are foimd by integration of pressure on the hull surface, although it has bee shown

(5]

that the simpler

approach of ener flux at infinity leads to the same rèsulta for the dissipative forceS. Aside from the naidaratiön Of

higher order

tez, the

present

rk differs from that of

'Peters and Stoker

by

.Lh

use of three pertuitation parameters ànd by the

development of the

surface integrals on

the actual,

hull surface rather

than

its' mean position 'with respect to time. The

_in

ortanoe of

thia. ist point i

not obvious 'but must not

be underestimated, for in thé M].5is'

of

the

tmatead bydrodynamio forces n error results frofli

the use of the

mean-position surface integrals, arising from the effect Of the f

orb'-ward ve].óoity.

In general the pre sónt

ial7sis

retains forces of first and second order in

the beam-lenth

ratio ,

and

of first order in the amplitudes

of the onO%mi

waves and

the oscillations of the ship. An exception to this ja. the stu&y of the mean increased

wave

resstauce, which requiros the oonsid.eration of second order teris

in

the oscillation

amplitudé. Throughout this investigation it is assumed that the fluid motion is

incompressible, mvi

acid, and. irrotational, and. that

of the relevant physical vaiiÄble i. aytoticaUy convergent power

series with respect to the se parainters.

Section Ii formu1iitos the general boun4ai7-øvalue problain of a

thin skip nv1ng into infiniteas1l nozmal plane waves and. performing

periodic oscillations in pitCh, heave, and suge. In Section III an

outline is given of the solution of this problem for the velocity

potential.

In Section IV this potential fiiqtion is used to derive

ressicns for the forces and

ment aáting on the ship, followed

b

a discussion of theas.

ults in Section V

The derivations; are

II Generai. Pox,milaticn of

the Problea

We conaider

_{a hip of}

₁₁

beaa.length ratio _{, cont*iiing a} vertical plane of tranayeree _txy

_and

prooeeding with a

_an

for" ward Velocity _{on th. surface of an infinite}

heavy fluid. Regular plane waves of _{1l a1ita.=1ength ratio t,,}

p°'1,

rests of hioh

are peiendjc,gar to the ship' s longitn4n*I azis. It is athzned that th. _{re ulting nation. is periodic and}

tric about the ship' a

pi-ini,

_{of etz7, and that the alituda Of the} oaOiUatox7 part of the ship's wotion is

1i

of order

Two rectangular coord4nate

_st

_are

_1oy.d

(Pigurs i). The (z, y, z) _{stem is fi,d in the abip while the}

(za,

y0, z)

8tea

vea in apace 111th aonatirt

velocity o such that the two ayatems Coincide when the motion 0f the ship

_va1

ahea.

_{The displace-i}

nnts of surge, heave,

and pitch are

dencted by x, and res-peotively. Thus

X0XOO5Ø+ZßiflO+xj

Z0

= z; coz

0M

x sin

eM

+ £lL

y0

= y

(2.1)

IrrOtatioii1 inooapressible flow

iB asammd, whence there oziata

a velocity potential

(xe,

z)

satisfying Laplace's equation, such that the fluid velocity vector relative

to space is

or

relative to the (z0,

y0,(

z) aysten,

i.0

Sinne

the

pbyaical

quantities of

pressne and velocity are

epressib1e in terms

of the velocity potential

it follows that the analysis of our problem

Figure 1 -. The Coordinate Systems

depe essentially on the solution of the boiara21Ie problem for the fmction (. !L'his problem is fomi1ated aa f ol].ows:

i..

The fuaction Ø' satisfies Laplace's equation

sL

#

V.r

_i;s1.

+

in the

&1n

occied by the flui

2. On the ship hull

the normal veloci

of the fluid equals that of

the ship. We

ereaS

the

equation of the hiil]. szf ace in the form

H(su,7,,.1t)

ÇLcc,) t7 20

er. h(x,z) is the local half-beam

at the point (x,z).

It is

aaainsed thzaghout that this

fmotiòn is twice continuously

differemtiable. The bounaax7 comlition on this suxace

w

be

urittem [61

in the fc

.DH

aH

i

A.

¡

V.,'

or since -

, the imit

normal vector into the hull, ve

ob-Iv.IJ

tain the

ao"iiitii

(c)jv.iii': f(cj

P

ae

als,

+(-.$)

1r,..jlv.11i

-ss;Om, Ó,,i +cc..sOm)

taJ'

1...k.

.t(i,,sss0,

sí, c.s9,

.Ò,,x

ic

(2.3)

ontheBnaoe,4h(i,t).y.o.

o

Derivatives of (2.6) are obtained as follow

P;.

'iii

_?#.

\i#

1#

3Lt2t

t

t_s P.

(!_'

.Q'+)

bt I dt

+v.4

ìí.LSt'

I

_a4

9

free surf ace axe equál and the pressure is constant.

Denoting the

eievatiofl of this surface by

3

(x0, y0;t) the equation of this

urZace is

j(iri1y.;t)

and the t

botirptexy

it4tions axe (6) then given by

,)

_{+!# 1)}

I.) ä'..)J1

r-

1J 79.P9$ *;j)

(2.5)

on

Fr

(2.5)

and.

" lìarly for

-

3

_w)'.

1 I in4 nlitjflg thee. deriva-'

tivea fr

(2..)

e obtain se

tI

exact free surface noMtion

8

°iat

#G

4.

?tÛr

L)

4

(2.7)

on

Since the position Of this saoe is un]ooWn it

s desirable to

obtain an

approrhnate bomda17

ooM tion which may be inosed on the

surface

=.

O.

Por thu pulposo

we

assw

that the Velocity potential

may be continued analytically

to

_z

₌

O, using Taylor' s

theorem, a.. we also ans

that derivatives of

_$

order sa the potent ial

f'w3otiòn

itself.

Thus

«Jc1,y.

_o;t)

and we obtain fron. (2.7)

ao

ã'

1?#.

#i,j:! ?3

2_!

_

d3óz

f 9a.?

?a.Jt

(2.8)

on z0

=

O.

4

t large distances fru the ship a suitable radiation condition

is iioaed, cor espeinMng physicafly to the fact that the only distuxi.

banoes present ars the regular

I 'w-dent wave system and the motion gen..

erate&bythahp.

5.

At large depth the disturbano.

Iñ order to linearize this bow4ary value

problem ail the physics].

variables are eipanded as

series in tetms óf the three peL-tubation

paraneters

_,

and

E,.

Since the motion ia periodic of frequenay

we may further eandi these

functions.

i Fourier series with

reeÒt

to tine.

Thim

_{we obtain aa the basic eansiona}

y., .ft)

2

Ç'

_{4e.,.. (x.}

_{.i t)}

4..,,

-Re dZ

e; z;

"4,,

¡(p)

iS

re..,.,

(x1y,.)e

36'.,y.;

t)

,/4,"e:JC,., (x.,

)'.;è)

Re'

,øS¡fUIS

t.."J1

(xi, y.

.,.,p

x

M') Z

"1js)

e,.,,

6'.'.

e

"SW,

n, Ji.'

(P)

0p'

"«e Afn

_'ml,

= Re Z

,41Ç

e'

(2.9)

Hers it ahou].L be noted

that the displacamentà are not eanda&:

in

pera- of ¡

by virÑe of the fact that t,is detned to be the

Order-of the

s»1aoent dUe to the fn»l-deùt wavea, or other distuibance,

The fmctiomal depet.enoe of

on f,, U fo110 fron the soligion

of the problem.

This fact also allows us to delete terms higher

than m=1 in the displacnt eansioxia, emd. the following term& may

be at to yM'ahS

u

(#)&'.'

'.'

X,,,,..

_s

(p)

(p)

-a ¿m

e.

(p.)

4')

rsdp)1.

₎

.

(by suitable definition of the coordinate

-system)

(for a 2 (by d8finition of

The probiem

nou be iiueaxizd by

aubstitutiug the ,.bovs

expanaiona in Lpieae' s equation aM. the boIary cm1tioflS

idL

collecting tenas. of the ease oeder.

The result ot these cperatioZ1S

i& a seQuence of linear boUnda17ValUO

px'oblz for the potential.

fmctiona

An:. outline of this

mlysia for the functions

i'

duo' 1ooi'

is presented in the next section,

aM the

details are given in 1endice A

and B.

III Derivation of the Velocity Potential

Thé baaic metho& of soLVi!g the bondazyval..Ue problems formuléted

in SeCtion U iS the use of Greent a theorem, in the f oem

-

Jis

(3.1).

h6'

S is ar cosed.' aurface nt1"1'g the poiflt

(x,y,z) an&. n is

the

unit nox*ll gut of S.

The Green' s function. G is. a harwonic

function exépt at the'

point B1 = O where it be

haves. lue the imit source

That is

The inteation az

normi

differentatiofl

in. (34)

'e with'repeot

).

It is shown in AppeMiX

A that for a periodic velocity

potential

satiying the linear free suif ace condition

#

.75;.. .

74

t

-C

_X.

in S.

:0

_on 13.

a boniy

condition of the

f oem (2.3) on the huI1 the

integral.

equation (.1)

for ' reduces to 'an integral.

eression of the fota

.1!

,

j:

JS

(3.3)

where S1

is that pù't of the hull

surface 1

below

This re»

duc'tion

is valid provid the

Green' s ftmction is properly chosen ax

texs of order

neglected..

Since

the appropriate 'Green' B

fmctiofl

is

own e

_açi/

n is

iown on S1, the böunIa7VelUe problem for

sia! ace cóndition (3.2) i.e in principle solved. However some

re

is

reqed since

is the instantaneous position

_Of

the ship' s hull i* space.

and

therefore in a moving surface. This is of urae the suztac on 'ehioh ?

_W

?n is given, but the Green' s firnction

is defined in

t.r

at th

steady

coor&in,te rstema

(xo,y0,z,)

ath ( .,

The classical method at del('g. 'with auch probla in to ispos. the

boundary candition foe' ?1 /

n on the mean position of the hull 'with

respect to time, or if greEter socuracy is required,

expand the

riit hann side

of

(2.3)

ficIs instantaneous position

te the mean

position by expd1ig th. hull function in a Taylor series

lui(xì) :L(x.,t.) t(X-X.)j

_*

t

H0w6yvZ' this procedi. in. not justifiable in the

.

pres.ùt application

siEbe the function h is singular near the lower boiin

Of. the hufl

dL it is not possible to sapd the bounery condition on the hull

a point just 1nia4A

the luÖ

bOtnidaxy of' this surface to a point

just outsiñe, or vice-.versa.

To circumvent th1-

diffioû1y we shall adapt the apposite pro-s

cedere of expiiM

'tg the Green's function to tbe. instantaneous sUrface

-

sva]mting the integrals, thereon. Por this piapoae the surface

is livided into tso pers*

'where

S

is one side of that portion Of S1 lying below z=O (the

static undez.ater aurfaàe) and S is one

side of the

s4nder cf'

(the strip lying between the load waterline z=O ' the p]an z,zO).

The f actor 2 is. intioduced:. to reduce the integration to on1

one aide

of the hull, by virtue of the

etiy of the hull. srae aM the

integraM G f. The Green! a function is epmRAd

by Taylor' s

theorem

4(,

ÇAy.

LQ j) -

_(i-t)

...

a

the suztace integraL over S1 may then be evaluated

to the dasiX'ed

ord.er of appro.me.tii.

(Sse Apendi. A regaz'ding the jus ificstii

of oaning the Green' s function).

Wo tn nov to the &Lcuaion of

specific

in

the ex panaion (2.9). The functions pL

azidØ'

represent the

steady-stats Miohell potential

and: thó lowest order potenti&. due to the

ship' a oscillations, reepoctve1y. Both possess

the advantage that

they atify the, linear fre

surface oiM tian (3.2),

4 'ce there

no non.lifloar toras in the boundaXY comiton (2.8) of order

or Se,,.

The abóve reduction

of

Green' s theorem

may therefore be

lie4 to

direot]y.

The results, as

given equations (Ä40)

2lirE

# z,r

tt'.:::-'

)

41?

oXe the aerscrpt refore to the

corresponding Fourier cQ11eflt

in the

ension

(2.11.2.].3).

The function

is the Green' s

function hioh satiiss the free

surf äce cónR4tiOn (3.2) with

tine

'1.

epa

.rmcß Thua

L

ap

_{- 7II.0}

f 6'

on zumO.

Thia firnotion bas been

.ven in varioB peZ'5

[3,4,1)

4

correapoda pb'aic81]y to the

ve].00i1

potential of a soi'ee which

translates with velocity a and pu].satea with frecefl

ç.

The

solutions f ox poaitive c axe

=

-f(wr) s;.vf4.(X-r)C..&11J ca.iD.(y-)Siis']

la

£

:

e.4%(y-5,zisl3 J ¿

whóz

(x,y,z;,7,)=

g

c&

_c.saL

*

v-u.

i.;Zf

_{a -«e.}

(; Ì )

4(")

ep 1Lfi

#I(i-f) M

(j] J&L

.L(

L

.4U0

irj

Z

3.6)

Ic<

'Ill,

't

Çdönotes the CaucIy principal

va]aie.

The ereasion foi

b. obtained by substituting pc forc throughotit equation (3.7)

Rq.uation is the classical

Miobell potential of liuiearizsd

steady-state wave resistance

theozy, and. is pbyaioall.y

_{R?1ogou$ to} a source distribution on the centerline plan.

of density equal

to

_the

nol velocity. Distribution of

the

&ugularities on the

_{centerH ie}

plane follows as. a logto'1 ocmaequence of expw'ding the xeen'

_a

fmctin from one s%nace to another ai negLeoting temis of order.

P. It ¡bould

be

noted

that this devel,ment f011ows without assuming

aaal.ytj.c

oont1-uation of the

velocity potential into the hull.

quaticn (3.5)

for the oscillating distUitance differs salewbat from earlier woÈk, as a consequence of the .nloyment of a. steniatic pe-tui'bation procedure,

and

the deVelapnent of the. surface intOa1a on the instantaneous hr11 surface. The potential

₄

pbysjcall1y to

a

source distribution

of

density equal to thà

flonai

os'4

flnting

Velocity and a

longitnM-'ial

dipole distribution

of

dity

equal to the prodìiot

of

_{the forwardvelo4ty and the normal oscillating} dd.aplaCem.nt. Tid is to be eciarE with the result of satisfying the botmdazy conditions on the an surface

of

the hull, which yields

en eression of th. form

if

{(;tajo_c,)L

t

_ì15iiJ

(3.8)

t,)

o,

JdJ

From

Stokes' théorem the difference betWeen this d.ou1Je inte'a. a

that of (3.5) is a. line intepal ero

the boUndax

cf 3a or, sinCe

d

O on the waterline, aro

the lower boaiy

&(-)]

®

is the difference between the inatanteneoua aM

an position

It in not suzpriaing to find, that the .zor is in the f01*

Of a line intepal around. the boundaz7 .f

S,

for thin im. the re&.on

ihére àansian of the hull function h(x,z) is not valid. As m{ht

be e,ted. the error vanishes if ¿b,/

=Jh/

=0 On this bonTi7,

although it iculd still appear in treating higher order terms in

AB further evidence of the oozTeothss of (3.5) relative to the

alternative eression (3.8), it ml«kt be added. 'that the present WeX

was based in its ear]y stages on the

an position analysis, aM re

au].tant use of the

_{integr. (3.8)}

Uáing tbi& ereSsion the

danping aM

resistance forcei were obtained, both

by intepating presae at the hull aM by integrating enero- fl

at ñrififli ..With differing results. It will be shown that (3.5)

leads to consistent reSults in thiS repeet.

Next we eTa4ne, the effect Of the incident wave sy8ten. by cons»

sidring terms in (2.2) which arS linear

In

, ,

the alitUde..

length rtio of t. Incident wavBa.

Thus we Wih to obtain the

leeH 'ig terms in the

19

Ç.., #/2E

i.,

,4E, #2.,

m.

(3.9)

The tiret tex

ja the veils.bp.own potential of a p1ex

pro'eesive

wavo(6]

#0Ipg

-2K

e" eos(kJr.1'4t)

3.1O)

Wave frequ.noy l.a a ee.tionaz'y. cooz'ete aratem

K

wave number

...

+ Ko

frequancy of enøomte'

___

t

.a the 1esA4g

tez

of (3.9)

mit o,jider the

problea aolvóè, at isaat ia the linear id Bense.

inaarting

Beru1W a equation ax

integrating over the hiill aiacs vs obtain

the tiret Order eiting foraea due to the inC4A.tt eavo, uhioh when

squatet to. the ristoriiig torosa will give the tiret order

solutión tO

Q

problem...flsOeBity Of

seçond order forcs

at XrO

scmanee does not

here.

In effect then the linearized. theoxy

'leide

a logical procese to the

PrOude.'zylov by'potb.ate that the

Offset Of the ship on the incident wave yatem may be

gd

svsrthsl.s

there,. ax'. several roazefle for Midàing the

'second-'ts

.

_'

i) Th

study Of this potential poxud.te an evaluation Of

the Froude'

K.tlov bpothO.z

.

e have Olrsa

analysed 'iuo; t

developing the oØ

po1g i

allant wave potential. vs will obtain a cclete

second order theoxy

which may be used as a crude evaluation Of, and.

correction to,

first order resula.

The anàLy.ia Of '

ia essentially the linearized analysis Of''

plane wave diffra,aton b a maying .hlp,

which by itself is

'pu''. interesting.

. '

Thus it ae

apprqpz'iate to

pzrue

th4

anaiys the det4 i

_of

_iah

'e

pz'eented in âppsix B.

I)

PXIi

the 1a+htioa1 point of viav, the assential differ. b.

tesen th. potentials Ø'

i. thi feat that

eatisfies the

1inu fe. ufaae oavnjtjØn (3.2)

_{zwe. hoi}

en

i..igsneoiw conditien on the free suZ'face.

.

Inserting (2.9) in

(2.8) a. retaining t.r

order

f_fl

w fi

ea thi, bomAazy

f

af.

t4.

- 2c

1.

i

)*.

=

t.

+2k(s.r)-Gl

?4'saO

.2672

_!X.'

_ifZj

iû1

-

r eMff

_c21r

uk3

c

-ePj1'

;O.-V..«

¡y. sii.

ij ,JIa

'fJJÌ

(3.u)

ihers T =

iø the ratio of the forwardì weed of the. 8hip to the

phase e]aoity of the 4dent waves.

The aec& equali

f011ows by

.stitirU.ng equation (3.i,.) with the co reon6.ing Green! s fwction

introducing oo1ex integration

the ).91anA and. doti nlflg the contoiw

to pass Over ' un

)

according as cosu is positive or

negative.

The hull bonndazy ømi tion for

7p

(i(%.;K*...;et.) j

t'.

J

₍₃₄₂₎

in adition thin pOtentiòl m2st saticfy the r*A4ation condition afld.

the ai,tiMticai for s out]inedì in Section II.

In order to slVe this problem we adopt tL usl technique cf

deeling with inhosogeneous O'inaXy d

rential .qUations if à

c

findi a particular solution

of Lp1aós' a equation '.hii

satisfies the bonntRary condition (3.11), then the hoeigeneoua funótion

H .

P. pill satisfy the hcgeneoua ta of (3.11) acà. can

be dealt 'with

Green's. theorem ifl the 8a

neer sa ,L.

uordertof1the fmotion

lockterabarsonic

tiOn Thióh takes the sane fcii as the right hl aide of (3.11) 'uhen

O, or a ttOiction et the form

e

;frlc.tt)f

_:3r»r

-)

.iif

J;

.

4;y.s.)

+

cia.

IJaiJJJ

(.i3)

ere the function

F

,u,t) end

the Ooflt

C ars to be detexndned

from the boRaxy and radiation oónditj.onae

S8tituticfl of this

expxeasion in (3.11) gives. en ordinaz7 secondi order differential

equatiofl foe' the functida P. By diotatthg that the sotiOn starts from

a Btate of reat at taO and that the

transients vanish as. t

'

is

obtain the approprdate functiçn and the contoUr 0.

1* this

''er

the functiofl

may be found., and Is must net ocuaFler the

hoTIgeneoua fUnction

irr(343)

it is readily qparent that

ow

(31(;a)

_±y

and thu8 that the hrgeneous lolution

satiofies (3.12) without

mei.fioation. $ine the boundary omMt1ona aatiofed by

Yff.

axe of

the same fOx

as those f or

Green' s threm may. be uset in pze

.ae]y the same

ro

Prom e4uaion (38) the fina]. real]Lt f ór the

colete vsloàity potential

Ø'

is

e

(

;

e

#;

X

G

";}; o )

jjd--i.

SC e".')ff

f7

e[r .-

#.'r.-Vc.

s.

j

,,s..-=

'/;,>, =

id*J

}.

edcff

7

_{j(;':z). ei9'C.E*.}

ji

dtIa'I?dì

4 ;(*._r)GDs

..;,,s..w]

h/K

*..2

KAcasi. }

jJ.IFI

(3.]Ji)

ehere C0 and C

are the

Contours.

of inteation apprupriate to the

Green' s functions

and

¡

passea over or under the Singularity ?

₌

aócorting as

cosa is positive or negative.

C pas

over the singularity )..

=

and over or under the

SingU].ar'ity

)b = 7,

aocoxding as cosa is positive or negative.

f. (, .)

I$.4(2?(a,T)c,al]I(

.aka'

7c.51s1

..To.sY,4}

23

(2

p (I

_{_27(IT).sa,.3pc}

₊₂

(3.16)

zI(3-rc.42)J

(3.17)

Ç, (L,.1) 54 (y#aic

,

The ffret te

in 13.14.),; repreaent& the h,goous function

whil, the aeeond

_{d tbird texna corz'eepond tO the parti'1ar aolutii}

The firnotion

Y'

_{oorraaponda plvaical]y to à distribution c}

.(' .deity prpcitional to the normal veloai

_{of the ii1emt}

TaVS at the ConeBpc.tng point on the hull.

The Wv. pattezis at iufiui

due to diffraction by the ship

now be analymad to order

ß. Fron (2.6), Uang Taylor' s theorem,

we obtain

g

)Iei

34,.,

,g.,=

i-)

1D

2w

kß

e9'aP4)(

aL

-a ITa

y Ylos.

(3.l8

ThiA ez'esaicn will not be evaluatet e]i..citl7,

but we ith1 I atu4

the form of the wave patterns.

These

ay be divided into two gxo1s.

The first are the waves arespoa4g to the firet and third. multiple

integs.1]

_{of (3.ii) which ars of the sama form}

_{as the OErsefl' s function}

G$1)

iju thereí's be. identical with the TavUs generstedi b

_the

i:

ak ivr)

.

T1"

IC(('i,zr) T £a

3Tc.sg& ,Z

potential 4i1,

or withthó wavea generated b

a. tr Rlting pulsating

soce. These wave patterns have been atudie

in aeveral pere [811].

The. aeaondL po

of waves is rather different.

These ax

the wavea.

generatedi b the seoOth. nit1tip1e integral in (3.14) together with the

secondi

at téras in (3.18), all of .wh1h ere of the fo

I

_e.(Kx.#.rb)J

.#lyo sp

_{i }}

JiJj

(3.19

ihere P (7,u) is an im1-nortant might function.

In order to determ4ne

the wave patterna of this second gro

at

large distances

fzrn

the sh

we sb11 investigate the atotio foz

of (3.19), folla

nethos uae in (5).

Por the puxpose. we intro..

dioà polar coordinates

X0 = R cosO

Iy0IRO

let

On neglect of texs of order 1/E, it foliows that tb.

Wily contribution to (3.19) cunes from the resides at ) =

5

of stationary phase to the resulting single integral

we obtain, the clition

or

o

(3.20)

which defines u imp]4.citLy as a fction of O.

Thisis precisely the

relation obtained. in deriving the Kelvin wave pattern from

Çijs

the Waves y lie within the weli.]aiown sector

Ko ever their foxn wiU differ frn the Kelvin patteXi due to the presence of the fè.otor Inserting thia factor, it follows that

the wave height )! will be

of

the fOr

_

ßwLiR[A.cuJI(u_9) +SØJ

.*

where

r( e)

is a slowly val7ing function. Thus the locus of the crests is given by-'

R()0 ocs(n'e) + K. cose] = aatant

0*., uc'izig (3.20),

I 1+Tosw(2-c...s'L11)

¡

o('/R)

where is the locus of the Kelvin Wave for the sai pola.r aflgle

e.

Thus there will be two steaa of "distorted Kölvin waves",

corres-to the cases Where cosu is positive or negative. POr

coau)O

we have

y

d thua a single crest will be of the fox diowa in Pigure 2.

Por cosu (O it is neoesaaX to distinguish between three cassa. Por 27/32 the

distortion

factor varies, between ene and a finite

mam vsLue, giving an

increased but finite radiva relative to th. Kelvin wave,. as shoi in. Figure 3. POr 27/32t T 'l

f actàr has 'two poles, one

on

either side of the cusp aigLs

0082U

= 2/3. Thus the transverse and diverging Wavea are bOth distorted to iinity

(Figure

ii).

In

z.tiining

oase when T 7 1 there is only one pale, at which

o082U

(2/3, Whence only the diverging wave is

distorted

to jut iñity (Figure 5).

Figure 2 - Diffraction Wave Crest for cos

>:0 Case shown is T -1/2.

Figure 3 Diffraction Wave Crest for cos u < O and T4 <27/32

Case Bhowfl is T = 1/2.

Figure 4 - Diffract&on Wave Crest for cos u <0 and 27/32 <T4

_<1

Case shown is T 0.98.

6.

5. 4.

ao

'.9

Figure 5

Diffraction Wave Crest for. cos u <0 and T4 >.1.

Case Shown is T .1.5.

27

T:-0.50 T;-0.25'-flÏ875 TrO.2510.3I25

TO.75

T0.I875 T0.50

'o.

_7:1.3125

ro.

. --. T: 1.88 :-0.0399 T:-l.0 . O . . r:2.0 - --

-.. -05 -04 -03 -0.2 -0 O 01 0.2 03 04 06 qL. J I I I I

IiI

I-

I I i i -05 -0.5 -oÄ

-a3-Q2-OJOJ2.a3

0.4 05 o Fraude Number

28

The parantera T and. t are shown in Figure 6 as fimotions of the

PrOth3.e .n=ber and the ratio of wave-length to ship length.

It ia seen

that the case TC 27/32 is pre&Óm(nnt for nonnal ship operatioi'al

oonditiona.

Therefore in the usual case there is c)nly a

11 disab

toz'tion of the diffraction waves J) .. relative to the Kelvin system.

The doiiii T 1 where oné of the diverging

syátems goes to infinie

corresponds to very

_14gh

spèeds, or very short inciderrt

laves, while

the di 27/32

i is. a vry narrow band between the first two.

To the

author' a

Iow].edge, .3istorte& waves of the above type have

not been observed .perimenta12y. Although they ars of r11 amplitude

relative to the

regular wave

system

and the steady-state Kelvin

system, they are of the sa

order as the

pusating.#rsr'tati.g-source type of waves, which are observable in eiperiinents. Possibly

this is due to

e relatively pit11

distortion at

nwi1 speeds audi

wave-lengths,

whence the diatortedi waves are not

distinguishable from

IV The Forces and MOment.

Raving obtainedi the vsioci

potentiel, tO arder

, ve m

pro..

oeed to find the forces se moment acting on tue ap, to otor

Té

a

ta as reference point the origin of the

(z,y,z)

'stea or

the iútórseation of

trhO ship'

s ioadter11iY, mj'&'ip section, az.

centerline pla1e. The directions of cojionenta of the farce

and

'.nt vili e retee&. to the ctsad

(x0,y0,z0) .

ystem. Thus

+J11t.VJ

¿II

PI L&,,M.1

$3.

-

_s

ffp(rJPk)SS ijJJrx(3k. i 4)

¿U

V

ers F. and l'i

ere -vectors '.preaenting the external force and.

moUUnt.

The surfaCe integrale Sie; taken over the submerged

portion

of the hnl1, thó vo.

inteals aver the entire mass of. the huit,

and ds is the difte.oitial elent of mass.

The vector 1% is the

unit noim. into S:

..

r is the position vector frdn

the origin:

The detailed rtduction of equations:

(4.1) and (4.2) is carxiedi

ont in kppe"

C, ana. vit on2y be; outlinedi hers.

The vaa.ne inte

'als aie treated. in the Usnal

ner of daai3-'g with rigid-bodY os

jliationa, .&. deacripticn of.

iioh is unnòcesaaz. The surfaCe int.i

aals are treated in a u"er

g41-r to that use& in the preoedi'g

section to derive the vs1ocit

potential. The .prsae is f oint fron

Bernoulli' s equation in terms att

the veloCity potential. Since this

is a function of the steady

oocrdinate

atem (x0,y0, za), Taylor1 s

(4..i)

30

theorem is used. to eand.. to the inatanteneous position

of the hull

in )reciseLy the mamer neodi with the

Creen' a. function. The auhea*gedi

surface

S ja

then

intc, three parts, 2S

₊

₊

S0

is one side of the hiil1

zfaco below the load

waterline znO,

Ç

is the atrip

between z=0

s,=O, end S" is the. atrii between

z0=o

sIz3.

The treatment of the sf ace 5

j3

straightforward. In order to treat the integrals over the strip

S'

aM 0" the integrenL is expende&

in a.pow

series and integrated

with reeot to a,

Thus these integrals az's

eresaed. ja

tezwa c

liiie integráis ¿ong the load

waterline z=O.

The resulting ereaaions az's

quite lengthy, even when ereased

in taras of

.ccty potentials. It is

convenient to present

them .

the tora of coefficients in the

epa1Jt1OflB2

¡2" X,.

X,.. efi ,c

₊

'fX,

_{+ / X,., #}

flZ.O

tf,ßZ0

_{rfZ,4. ç,14114,}

.ø...

M

_-

fi MaD.

₍₂

M £4 M,,,

t

c,/3$fl

s t,v,t"fl... t"

These coefficiente ax's given by

following . ex'sa8ienM

a...

za00

_2,cfl

rZ

(14.07)

2 It is aa8ued. in the definitian of

the oòordiflato qstem

that, in

accordance 'with

lzchimeded principle, the force aM nriAltt of

oder

.vaiah*

Z

-P;

JP

o

fr1,1

-7p.

ft«Dtf.4

Also in these

ereasiODß it is assuiid

that to the preøt

degree

of appioxLtiOfl

the Fourier

ooeffioita of order

p

I

yah a$

will follow in the case

of free osoillatiane

in waves.

Generali-zatii is straitforVaz'dm

. .

z,'.

Z,.,

#...

_J

dS =

.xv.

21a t(4Jt)4J.

+

a()

_¡(491

o.

(i', l

c

:

Jirda.

*9.,

!;°

_JiI*

-.

r491j+ 2p; f

(.'i&)I.,o) Jx

)zeje,,

+

4; ,9,(x )]

42pCX.,+I.a

+

2j

(z,_xe«)(;0_9.)

J

Za,.

3].

(.16)

2,.

vi cifir

(;;

(..8)

#2,.

14,,

_A,*(,V(X.s

_iq

ir4)

-

(s.s

D.w)JCL(a,.)

Jáir

-

()

4,

+2p;

e,

!"

!P es

f

eP«

(i..i2)

!iet

_"JJ

eK(

9JirJa (..13)

k

- 2p j(,,x4.) )cI.(ço) h

&()o,, _(,)2z9. #2,

_e

_if

2p

)(1ai*r.

-c

c/il

"8

.

(

x 0.,)(z,1_ i9..)

x Jx

2,6JTf.,+z.,;

+4Ø

+

32

9,0(z,M4 i. X,, X4.)

(.19)

#. 2,#c

jo,

- x

8.,)

X

(.17)

2..

2p. Q

(f

c

+ K/

(J#so

+ Ic i 4's,, (-;

+

eJj

L

J JKd

.K4lcJ'001(.?..+ Oi.+içO,.)J

¡.12 e'.., e,.

-

r;j3... (3e..

_,o)

,U;i.'c

'

n!',;

₀₀₎

+z.

;J(b !)

¿j,

(4.20)

in the above expressions (x,zG) is the location of the Éhip' s

center of gravity, n is its nass, ana I the iwmnt of inertia about

the

a.. L

denotes

integration along the waterline zd.) a

in all

casea the argtient Of the

eloàit' potential, is (x,0,z).

Substitution of the potential

ø

in (4.6) gives the Wel].-knoun

Michel]. integral for the steady-state wave resiStance

=

p;a.f

(r'#) ;.c«

¿s

;o.

ere

I.4.«3.

=Ç

j

Rcj.uations (4.7) ana (4.8) givs the steady-state is,'n,nt an

vertical

torce, or in' the absence of these, the trim 010

diicticn of these two equations to a toxin 54n11A' to (4.21)

ia

not

possib].e.

Equations (4.9-li) give the lowest ordSr oso4l1tO' r storing

forces, consisting only of h.ydrcstatio 6M inertial tenas1

whilé..

equations (4.15-17) give the second order forces, including the

dani.ug sudi adde

masa effects. The contributions involving the

osciiLting potential.

are. a' i'nl

r' (although not identical) to

the equations' derived. by

Hartd

[3]

'but the xemaii,4

tenas in

(4.15-17), involving the steady-state potential

afla.

coxespond-i.ig displacements, axe not

includad by Ijrhi,

In fact, it is flot

auxrising to find a ontrbUtion, from

the steady-state potefltial.

since the

hip

s oscillating within the

steady pressure field.D

s

(a)

aM z10. However re.

3' is to be eected, the _{cozesponding forces axe in phase with the} di1acemants. Thu8 th& on]y daing contribution

_{cca from the}

terms

in

In view of this siz1ifioation end

_{the particular}

iiortance

which is attached to dsnping we shall give

_elicit

ereaaicna for the daing coefficients, which _{may be obtained}

by inserting

sçuation (3.5) for _{in (4.15-17).}

To facilitate this presentation we adopt the matrix no-. tation3 of St. Denia and Craven

_[13]

_{wherein the daiiing âoeff} i-cienta axe given by the matriz elements _{The first index} reter. to the direction

_of

_{the foe-ce, with LiB, Z}

=

B, and

M = B3 _{and the second subscript refera to the direction of the} velocity giving rise to the ib roe with an na1ogous designation of indices.

We define the foUo'ing matrix.. elements:

f1(x,z) = t3(x,z)

f(x,z)

= z

1/?x.! x

aWz

P(Au) + i Q(X,u)

_f3

(at)

_{e*p( %(}

_{t;cr.s«)] .1K}

₁

= Pi(),u)

P(Xu)

= Qi()

= Qj,(Xu)

With this notation the diping coefficients are given by

3 It ahòul& be noted that the directions of our forces ax'e defined relative to space, rather than to the body añs as in [13].

-i1?'(.' 'V('s )

i,

2,

af1f

°

Ii

- !!j

_{'"] "}

."

j

(i=].3,5;

j=1,3,5

The first two muLtiple integrals are antinetric

matrice3 with

zero

hi

_{the last term is a}

_{mmetrio matriz with non-sero}

diagonal.

Thus only the last term contributes to the deeping

co-effiLenta B, B», and B55, with identical

resulta

to those ob..

tainéd

_{integration of energ flux at infinie [5].}

_Calculations

of the coéffioients B33 and 355 have been made [5]. for

a.

_____

h'11i and caie& with eeziir1nts. In view of the antiayminetry

of

the first two teziis it is now apparent *by the oroaa.coup1i

i

coefficients B35

LB53 could not be separated: iñ [5].

_{For a hull}

with fore-en&-eft ymaetzy we obtain the result8

2

B133l

;r(í

_(-a,)ti-3)

71'3 (),.)jL.

35

(.22)

Usual result.

B13 =

B31= B35

B53

= O

Howevór

there will still be coiling beteen pitch and. singez

15

=

51

(f )d

B135

= oß53

Çe

2(1f

(44,4)

B51

PS

a'?

(W_U.

LuJs,cs&,

i9+c.

Turning to the exciting forces, - equations (j..l2..l4.) give the

lowest order contributions, which are identical to those following

troni the roude.-KrylOv hypothesis. Equations (.ì8.'2Q) give the second-order exciting

forces. In the case Of zero

fo ; speed these seoond-órd.er equations oOrrespond..

to Ihe thiCe-dimensional

form of the exciting force correction used. by Korvin-Kroukovaky and

Jacobs [li, and obta11ie&. by

oonside'ing the s of the oscillatory

velocity of the ship

the otbital velocity 0f thà incident wave.

However for

non-zero

f orTard ßpeedì the steadystate solution

clioates the second-order exciting forces

nsidàably, as i8 the

case with the corresponding

restoring torces

(4.15-17).

In the case of free

oa&1ltiCis in waves, when there

are no

extermál forces acting on the Ship except a constant

thrust

the desiredi solution

s obtsiuie& b7 adM iig the

restoring afld excitiflg

förces

and

equating their s

to zero.

It f011ows that

t,.,

ç

'O(fw')

and. the first order solution is Obtsined fron

the three simultaneous

equations of ordèr C .

The aeoon& order displacements may then be

o tamed

br

solution of thè three simultaneous: equations

of order

At resonance, however, this prcøedure

:be,

down since

the.

tirEt order osoillAt7 system becomes degenerate, or

I

36

Ç4f 23)

I

where

cfl X.

LM,g

z4.

fM/

MSI.

s.

In oòr that the restoring ant eting

torces

hji11 be of the

sa

order, it

follows that

f

,=

_(i;îJ

That is

the

.amDlitudes of

the ow1l1ktions ire. of the order of: the

incident wave wnplituda divided by the

beam'lenzth

ratio.

The

]ineárized theoz7 re1nlna va Lit rOvided. £W4!/

.

This aM*ion

appears to be reasonable in practice, and in any event

s i].ied.

when we neglect teza of order

while retaiiing those of order

2

pSSoiapc

 solution Which is valid for all freuenoies i..nÓll!M?IgAmay be

obtained by forming the following system of

.nii1taneoua equations:

+

£Mfl

+f..fl&X

Mfl'Za,..,

m/2'a.,

i tar,

/1Sø

37

These may be solved for the three imnown dilacenta (x01 +

+

and. (o

+

At non. eaon1t frequenoieabóth the

first aflt seoofld order dilacemefltai re deteflainedì frun (4o21i.) by

equai4ri powers of the pertuxbation paramaters, but - at resonance oniy

thó first orde di

laoemants. may be Obtained.

I:

It ahoul&be noted. that we have made no assuntions iegarding the

foreand-aft .nmatx

of the 1'uU or the cowling of surge with pitch

and heaveo

Such- asauntions are cnn in the literatuj'e ant

would

o

11ify .oiu' foros equations awhat, but the baeic1exit(

u1d be .iged, and in the present age of electronic coiiuters it

558

_P

t,eas to .-

X5tX'iOtie

aastiOfl8: tiich only result

in giertleous si)1itioatiens.

At ±]/l' the potentiAls 1110 and havó iogarithsio ainguisritißs, as was pointed out by Kavelock

[71.

These

aingularitie5 fect the fbraes

ce

oiaJ'

2, anL woul& aft eat the higher order texia sa well. it is e*iily aho that as ¶

aII}

.

'

l

e'/ix$ +zjt

a,

,d'

e4'

'jj'dJ'

,

'(

aL!

-Li; 4'-t/e"tJj j'

aii*.

I

3*. Jip

dd

X

8k(2;a')(r ';ì)

38

i,. The notation

o(&)

will be uasd. to represent tszas ce.

order i

.an4/r

¿

. .

Pox auffiòient]y iil válnea

of

_I i ; these eresaiona will donl $e the lower order torees and a solution at the singulaity fox' the displacezta. r, and.. _{00j may} be obtainod.

by

solving

the eativel,y eile 'stem gLven by

(4.25)

a

(426). However

such a soLutjon is

of

doubtful

velue

since the an]itu4z of the

hydrodnamio. ¿i8turbónce becomes iofinite at

''

= d the linear

theoiy therefore bteaka down.

PinalLy w turn to the mean iEcrease& was resistance.

It is

e.-17 seen that the lowest order oontribaticn is of order

2,

it suat .po8aeaa;

a

finite time average over ano cycle.

We are there"

fore led to consider the

teXwa

C, Ew

X',,,

4

,rw'

g,

.+f

Xaa.

,

where.

a

ba denotes t time average. These coefficients y be obs taino& fr the analysis

of £pp,nM

C by ret1"4'g terme of order

In tia

neri

tht

0 8M

JQdrb L

V

Í'M "S*

2p

ff1

s..

e 39

)

If we restrict oursilvea to

ncn.re"t free osoiliatioI%a

in waves it folioTa that Xioi

X, etc., and thus that

= Ó since the ,fbz'cos of order do not co*t4n

a.inj teia.

An ecptation

84in41"tO

(4.27)

bas been given

by

Have3.Ook [12j,

ana d5ed.

Xt is thereby shoWn

thát

m

represents a positive t increase of resistance, which depends

critically on the diflg nd

a a j-nnn at xeao"e.

Häwuver or non-resonant frequencies the pre8ent eq.ustions

of

motion. are liyiRed. (to this order)

and.

thus the onclusian is not generally applicable. However for frequencies close

to resonance thére will be daing to

ordr C ¿w a)

and this

kqpotheaia. should. hold.

It fellows

that

will be positive

at

resonance,, but will tend. to mero óla8where.

Thus we ¡rrive at the oonolusioñ that the an increased, wave resistanCe is oforder ( for all frequencies, at

M C

'g,,

i view of this fact it is

förbmate that the forces

and.

X202 are ,no

at&

an&.74, for

at

,ofl.'resònant

freusfloies the se

Will

contribute to thé lOWest order increased

résistance. Howver we sb1

i

inveitigate the tera

presenta thé mean increase& resistanCe due to the o&rtTl*tiofla

th'Óles,

and. together with. this represents the lowest 'dei inoreaaedk resistance at resonane.

The

resistance coefficient X depends on the velooi potent1

_øoo'

1'

and The first

two

tanna have alre been derivedi ant the third follows

by retRl1g terms ofoxer

fiE4,'

-

G

7s2.

_4,,.

{

+ _{¡_}

+ : f

e,, ji Ls

-2i

g)

g')

-r

4r'5sJ

=

.

2iï. î(

..s.(G'(f ;)d,

2 y; 9;.

if

(r

;

2)

-J'

did.?-J

(4.28)

RetAlI1lIg teria of. ozar

f:,ea

in the

ta]. foece

anályaia of Appndix C wo thon obtain

-.íf ij

_iF'

q-j

a

;

=.p,

*2O _7!.

EI".

(e..;x c..zs)3 J,4

t,4*

D».

epL.(a.r.sJ

_dIJIdII

(i..29)

f

f"'.

r z )

d4'

P+:Q

ß.f(Oh

,o

9(Sfq

-t)].x

t),*cdsa

(;J.x4

2.

_,iLzrc.a«

4a. 4 dI.8!c.&'&)J

₂

2 ?. 9.. ;:_4)

t.cj4O(3.;JC.&.I)j

_411J%

ç11[

at

.2

Tha f

1!at

integral in (29) has also been derived [5] ty in..

tegratiEg .ncr flux &t irifini1,

and mm.rke1 reeult az

_.ven

for a po].yncial hull. The second

integre]., representing the c.

tributii fron the potential is not

oders&

r1 and

the

eaiatace given ' .s tez should be

added te that ooiuted in

[53

to find the total pean inorèased resistance tú àei

water.

It

shoUld be noted that this;

aMI ttonl

ji

ii

piit of the

freayency

and y therefore by derived . a qua8i..steadj.atate

analysis based on

Mithe1I' s

integral.

in ofexenoe [5] it was pointed out that there is a nftn..

tiEn .fi the first integral proportional to the product

l6"JIz/ s.;,I

where 5. ii: the phase lag of heave be14

pitch, and that this will

result in either a. reaistanoe or & prulsion dep,nHig on thi phase

lag S. SITÌe there is no

{4

i

_{tena in the second integre]. this.}

conclusion still hoi''s, and

in

fact (l,..29) shown that a _;:

I.:

Oziata between aahge

and pitch. As was noted in

_[5]

these

are the cnl.y nonvaniRhiT1g

oes when o=O, ahòwing that a floating

body may ho propelled by osoiU ting it either in pitch and nge, or

in pitch aM heave, prowided that the two nodes ere out of phaaó.

((n

phenomenon can be. dnstratedi for an axbitraXy body ai].y by oon

V DisCussion and Conclusiona

The resulta ot thi& inestigation axe priiari1 oont8ied.

in

eajiationa (4. 92(i), whiCh represent the restoring and eiting torees of first and. seoimd order With respect to the beaii.'length.

ratio . Unfortunately the second order equations 8re rather con-plox éven by ocearison pith the earlier wOrk of Ea4nd t)),

_as

siit

be eeote&. fron the use of a ystematia perturbation pro

cedlire.

Even mer. serious is the q2eati0fl of relative orders of

mag-ziitud.. f the rigid body and 1ydxodhaic roes, ehLoh Waa dis. Cussed. in the IfltXducticn.. In particular the oonlUaion of Section IV that the oscillation airplitudea at resonance are of oxer I/p times those at nonresonant frequencies seems vex

questionable imviec' .erimental results,

ihich show the

alitudea at reso"oe

to

be

of

thea

order as those at other frequencies. On the othir hand, the d.wing ooefficiénta oOiuted. on the basis ofthe present theoxy are

clearly of

the same order as

oàrimental results [5].

In a event the present inwetigation does not appear

to

help in p1-.ini hg the smei.'ueepirieal

(stri#heoxy)

approaoh

on

a rational. basis, for the tie methods of snalysis. seen to b. totally diverse.

The present method has e advantage of a rational £bundation, but thò disadvantage of oonplioated resulté, the Ya1idit of tich has not been established eerimentally. On the other hand the 8emi.' ëni,irioai approach baa the advafltages of relatively spie form

and

good

nt with eex'imenta]. measuznt

of

ship motions '3