The control of heave and pitch of a semi-submarine in regular astern waves

"T CONTROL OF AVE AND PITCH (P A SISUBMARI1VE IN REGULAR ASTERN WAVES"

by

L M. Jonckheere

Contract No.: Nonr i84i (5Z.)

MIT DSR Project No. 8073

s research was carried out under the Bureau of Ships Fundsnental Hyd.rnechsnica Research Program

SR..009.O1-Ol administered by the David Taylor de1

Basin.

NASSACHUSES INSTITUTE OF TECHNOLOGY

Department of Naval Architecture and Marine Engineer1rg

Cambridge 39, Zssachusetts

J1STRACT

2Ii COW1OL QE EEAWE AD PITCH ('

A UBMARflE IN ASTER1 REGULAR1VES

The purpose of thIs Investigation vas twofold. First, mathematical expressions were sought to represent the coupled heaving and pitching motions of a semisubiusrine in astern regular waves. For the sake. of

simplicity, the linear theory was used; but corrections were made for some rionilnear effects. Since a complete analytical setup of the motions did not give results that corresponded well with experimental values, some correction factors, consistent with the theory, were

Introduced. In this way reasorble correlation was achieved in the

range of experimental results. Once the analytical erTreasIou of the motions was found, a mathematical model was set up with a control

system; end the motions in waves were siilatz'.d on the computer.

Different types of controls were studied, which led to a selection of

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A

i

-LZST Lg?

ftd.z

¿

&ft

L12

i

8oft

&O

1it

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90 91.

I!CAE

1y eyxnbola ueed zae t once in

tt 'ø included in thie table;

'oth o1g.i

defined dere ud. In

'eetiUy eli ceee dicmsl

ut1tiee re

lied in fOOtfr pound, end ec

u.te.

tu eZ4Ve O%tiou O?

j1±de t

_{fE'ee fG1}

at contrai. fina

O

1i& %t ¿ept

bOlow

2, 3

edpel 6on of body of

A

vtoci

t of

in

A (in cjtxe on atol)

of eont'oi fine.

ie vju nnt

of

itia 6ue to

th een

of te

Oonta'OI folle

tneee

etionel

wee. t 6itence z fron

of

b

¿epin coficient

&p

coeffielent & to peace

of folle

only

B

th den ccfioint

pitth

zin coefficient

th to

eenoe of folle only

bedtb

of

e

buocy coefficient in

twe ytlon

o

ve ceit

C coefficient of

i

iz8inei

%in nnt

foil th'

coefficient

coefficient of lo

itudinel ig1tlng

nt

only to t pW4eD

of

oentol

fine

foil lift coefficient

C3 nc'

coefficient in pit

etion

6

eonpled iael nrnt of intta coefficient in

nve ejation

oon2û vgtuel 'nt of inetia coefficient

e only to

of

cotol folle

Cge

D = coupled

D coupled control

-V-virtual mass coefficient in pitch equation

virtual mass coefficient due only to the presence of fins

e coupled béave damping coefficient coupled heave damping coefficient due control fins

E _{coupled pitch damping coefficient}

= coupled pitch damping coefficient due

control fins

function of time lag in expression

angle

f function of time lag in expression

angle

only to the presence of

only to the presence of

f2 _{còefficienta of cos wt and sin}

wt in expression of exciting force

of varying anti-heaving toil of varying antI-pitching foil

Fr Fraude number

g _{coupled spring coefficient In heave equation} g gravity acceleration

G coupled spring term in pItch equation

h _{depth coordinate below surface}

I _{longitudinal moment of zaterp1ane}

J maximum moment of inertia of senil _{ubinarine around the y-axis}

k1 ₂

3 ¿ control parameters in e:pressions of varying angles of

'

' contro]. fins

ratio of added mass, a, to actual mass, m

Iç ratio of A to J

L ship length

m mass of the ship

m1, n2 _{coefficients of cos wt and sin wt in expression of exciting} pitching moment

ri0 = maximum exciting pitching _{moment on semisubmarine due to strut and}

.

pitcin3 mtt poducsd on senigubine

tirou the contsol

fina

to1ec

b

odmic devivative cC pitdzg nsnt

attack

nl

hjc

i

to vee

ezciting pitckii aut fr

y iiat2on lu buoyancy due to the

1113

ezeiting pitthing "nt directly due to strut

n, 1(,

nt,, &2, hydrodynuiC derivatives defined in Ref. 21.

1(z) heave dszping f"i

of a strip of untt length at diatnce z

f

iddahip

O origin of ie syateu, o&uter of bo6v at 'evoiuticn

(also apprc1ate

position o center

.

iy8rodazuic PØU'

of gravity)

R rediue of body of revolution at distance fron ¡dduhip

t

tine coordinate

t1ß

tine leg of control

ratene

nataz'al heaving period

= naturat pitciI)ß period

T period at encomter

U absolute value of opsed of 'ava perticle vith respect to the

control foils

V ship velocity

VP iave particle velocity

L.

iave particle velocity at depth h y heave velocity

4z) =

.. local width of strut

W

york d.ne by a coutrI fil during a ue.rtcr

cycle

z ciordinate along longitudinal e2zifl of with origin

- vil

cf

teaving

fin f îp

itee of

tipit4

fin fz'

idttp

oiont1 dita beten enter of

bobj anti tr of

etut

X3

byro,nie surge foros on ctut

to avae

1yoattie a'

f o'ce on tiut due to

'es

y

h'lontal oon

_{iate in dieeiion of pt 'iith origin}

O

eeiti siarin focs

= maximum heave motion amplitude

s

heave z eoa of

the vessel

Z0 _{iaxiimim total exciting heavinz force on the vesse].}

hydrodynic exciting heaving force on submerged body of revolution due to the waves

Z2 exciting heaving force from vary-lag buoyancy force due to the waves

Z8

ljrodmic

ivtive cf the heave foice due to an

_{g3 of}

attach of a control tin

5q' 5q' zD

hyO

uic d ivati due to tî otions of

the Vt351 (see Ref 21)

W of attack of the

ve pstieles 4th

_{ect to the}

hozontal

(Ip

ee1e of attack on a ifitch$amg fin of tae partic2ts tith

1saj9ect to the borisontal

of ìttack on antihaaving fin t4th respect to the hoiaontm1

= angle beteeu control fin and 1oitudine1 exis of ship

also phase angle between heave motion and wave motion

8

varying eagle betieee antiahaaving foil and 1gitudinal xie of

ship

vøyiag

la beteen anti

thiig toil end lozigitndinel ante of

- e

total en1e cf attack on the

ontihes.ving to2.1

f- e total angle of attack on

ti.piting

foil

= maximum rotation nplitude of anti-pitching fin maximum rotation amplitude of anti-heaving fin

7! &i]aòenent in eu ft of EexniEubmri1Th

g1 btn $t

te -_ otiozì

oi1i bt

ciU

pati

mit

mtic

batn eoit

kve fca»

oti

e

:!t

»c

at t

vø1

e0-

______

'3 13Zt

the otic at 1ti-28viz2g

foU a

the ee

p=

pt

1e betn zotia at

ti-pitci

fin

-avpotati1

Lu

a a)

f

jzency at

cntea', cyclee/sec

ÂCQWL1MEwrE

The ithor w18he to acknovlede the gu1dnce and

enccurgent of Professor Ent Fkel nd Proeeaor ftlllp

)ndel of the Department of 1aval Arth1tcture and _rine Engineering at the

.aaachuetta Institute of Tecnoloy

througjiout the preparation and correction of thi report. Without the cooperation and facilities of the Cciiputetion Center at M.I.T., this work could never h3ve been accomplished.

.1

I.

XDUcT!OI

The ao..eAlled

ieubi

(Fig. 1) consista of a fully

subnsed body of revolution vit a relutively large nra strut

voae top rises Bt of t&s atew.

Interest in this type of vessel we aroused beceuse it seed

sa

of

to holdgthe edvantee of both s sbarine and a surface ship.

A bine is not subject to the large forces sad mcnts that

ere caused by s change iñ buoyancy due to ves sad by the ve

orbitel velocity. Aio, the resietsace due to avezaking by the

vessel is. less for

vard body,

ila its icresse in

at high speede

frictional resistance ie not as

'eatas the dacx'eas-e .n vaya

eaiet&ncs (Rei. i).

the other hand, the erfs ship has

perznent accese to the air '.abich io an advantage fron a psycho.

loical viewpoint and 4th respect to installation of relatively

low.cot arabreathing mach.tnery.

perisents have been carried ont at the ¡4.1. T. and other of the sejsubzrarjne

ta4ng tanks to investigate the rfozusnce,.in. cain vater and in regular The motions in the vertical plane (pitchi

heaving, eM surging) received special attention. The tests

(Ref. 3) ehed that the semisubrged body behaved

very eU in

ahead but poorly in astern eaves. This can easily be understood

since the aatral frequency of both heave :nd pitch are eme.0

(see App. I) so. that synchronism yith waves occurs only in asterñ seas, while in ahead seas the disperity between the

natural. and the encounter fz'quencieo beca 1arar

1arr with

incraaoing f 'vard apeada.

It ,no the euthor a intention to find

theetie1 foundation

in a2pZaeOiflg thooe notions themtice11y, ouch tha it vould bacone poaaible to predict the motions of the oomioiiz in regular avea and, by ouperposition, in co used se.

eover, ,4th the

lticel mothoda, it i pooaiblc to find

the proportica nsooary to control eceooivo otiom.

Control of heave end pitch, vith thie type of veseol, dooa not oeem to be os

difficult a tack as it le vith eurfaca thip

Insad, as

s ntioued

th f'ces and

.itfltO acting n

¶;lill

as large as on the faos ship vhile the 1.reusncy of notion .t btch

control is neosesary io ieich loez then with surface seals. This io

due to the lo nstorel trequsnciee of t

miubrged ship.

Tho co iderations pronotod the ü

irability of Investigating

the notions anel4ticaU,y. Eovar, l torature on the motions of near surface veaselo is quite end. that

ic

io avoUable (Ref. 22)

proved. inadenjanto for the purposes cf thia

.vertheloas, theories

bave been developed in the past yasra for surface ship end deeply eub morged ouarine motions that bear zm.ich re to the osaisubmarmne.

In this ork sa attempt bas been nade to conbins, modify, and. ndapt

these theories in ouch a vej that en &e3.yti1 ezpraoeion for heaving

and pitching motion cf semioubtarmno is obt4*d..

large eurge motion observed e srlsntel1.y (Ref. 3) zeast have a

sail establishes e. relatively long moment arm. Thus, even the n&ll

surge force induced by the waves on the strut, or by the motion of the body itself, must have considerable effect on pitching and, by coupling, on heaving. The surge exciting force due to the wave s on

the strut has been taken into account in this work but not the effect of the self-induced force by sarging on either the strut or body. This

effect would cause the motion to become an involved, nonlinear vibration problem (see App. U) in which many properties have, as yet, not been

The tntione iii the vertical. pleiie of a body In

fluid ci be

se a eat

of the di

ntial aqaticae i îi

_the vae

pitb and

cp1ed %Fjth es other0

If the

ie nitted, tvo equatioe rcan.

!Ihey cf the

fn

+ c. z -i-4 ₊

w(heavforce eq.)

-f c.o

s)

E GZ

t-(pitch-nioment eq.)

¡4 E

vith

eeot to the uLin of twa.

no TIOI

basic tor equations by 'eans of Tay1ors

aymbois are thoee of Ref. li):

(2

-01 vn,) -(

+v )ft .-M

-tteein y

, q O, end

J.

1

eofficjente

"bdrodyxzedc divetie" (eee Ref

le).

Th e eyatan

icì

iee ?ith

the bty, i.e

¶be

i.e at conetant pth equal to j

le in

the di.etion of the lo it

inel

of tb body of *evolutia

at

reet,

oeitLe to the bows,

lle y le to port ad

u1vz(Le

trectice of

(note that these directions are opposite to those _{specified in Ref 21).}

this noviv coodinate tan do not

ana during t

motion,

eqtione (1)

reseoneble if the aotioni ae

t ta1l,

se se

aisn by Aba4ts (Ref0 ¡)

o e lced the

equaticee frca

eipansions (the

.-

Vz)8

4(w

-ft1z t

divntl

tu b f

ttw it

t2

iii

& t fcu

ata. nt

2.oe on t

eiibz'1ne

The e1t1ng Forcea nd nta

7]ßV, in

in

tntiaa

p

tic

QU C

t

t

ns

(t

oocnUs Fra.y1av

ìota)

t the oi ntlon ct t

flov in n

le ¡sot

iuenea6 by ta

a body in it eo thnt ùssion for

the

'cntn

t the

a'i

rts on

via

lp

le

iwd

fron bueacy eonnisntlone.

ainr, thin

setlon in not

tlre.y rtt einen it in

fron ezier2nnnte

orntica1

oicn tbnt n

ea7ln n

th otnetinI

the thai in itu nei b

nt1r.

tha. te nn intion beten

tnr

i body aule

it. To

oe9e to' 'ob1 the

t

' 1e

theory o' °Strip" tory

e

se bj

eU tez'e n in n a

nt&cn fnetorn in tii n

oeit

ontie1

i1e the bonor coniticen

e npeee

tcthnw iith

the oc

tin thnt the bath ntio bn

U. Xt te b1t

n

aeineneicn1

Xn the n'ip theory t

bc&y in ta3ßo

raee. tobe e1ongìte, bat aleo

te ape to be

11.

n thst

ne the f lo ora& thin bc' in not cone

e thn' bat

cn

in ivle into ertil oroen

For aurace ehips the Froude-Krylov hypothesis is satisfactory since buoyancy 18 the main exciting force. In the semlsubnar1ne the

etia1 ti %L

tto4

t1y frca e

aaz

nMct±

t 1itd1

oeLt.

ida

G1Ita

tati*

_

v

t

totol f oe

nt

t

8vet

c' t

owx t

thinp te'y

u

pottL1 thy.

ortute1yD

iric1 oceetion

to

to b u

to

tay copa Vt

mt1 eou1to.

2b

t2oy

'e&pectß U ys±o1 baun&'y caitionc but

1 atstioo

dint

i

ot yt b

fLcit1r dev'1opd to et

o

t

t1on c

zoD

t

to17

cioa to

th

ccc opt1eii

tetio

cn b

(

V) to

ve 1itt1, etftt o

tb iotio vLt tiv1,y cs Fwou

b

io

± th

ttt.

foe' tb bwe ec.tt

foe

t

pitth

olt

c

ub'

body been by ¡pLt (Ref. 5).

keivo foec in

t'n

(V-c)]4

i th

Utud9

t t

fre

faoe A(z)

ii

'

etiçn1

t

ditc x fc

foe'

body tttt

ott. To

tt into aoontD

Z9 ¿ : & _[it:

4W)t7j{s

r

Ar

-fI+j4

-J

the ion k

to b nolified..

on the body

ie

tven by foeu1 [1 'Ref 5:

then giver

1

Therefore1

wM iti

f roni the

The total force in the vertical piene expressed in. coeplex form is

by:

kj(co)d9

beget:

_J14&1

t

baavtg force for a strip of

it 3ai& mt 4itence

center of the ecordi.nat? 1-' system. .

t"

-

r _{?[ ('} O)sî.&. _{L9 +Jú}

ft)

ctzi ed8J

+ 2O _d&

f 2Tk

dj.

7L

(r

_°s-ia

--'e

W2t

f

z

0 en withcat

strat (i.e.,

₎

0), this eves:

Lii-eP

_LA.

_¡Q&1rr

iJ.

+(V-C)3

¡t

a ifforen beten

.

1t tvo e

esioas

late

ttoa

about 15

pr

ot 1ass for tbs boj of

reo1atioa vit a

Ej&eesioa (3) lategated ivs

I

=

-

49r

p À

ILR

Jsf(V-J&]dL

4

This exp!ession is nwneric&Uy evaluated for tvo wave lengths In App.

_III

(pg.48)

'sci.

[z

_¿v-cJJ{

(a)

-£(z) in ti

aio

ea t

'e of _{cta' CACC &t z,}

Fft'11y9 Zbcome a - of tvo t:

cee' oit +

fein

wt.

2b fo'ce ee'1u1st4 tow _{of four e}

e'it ft0

a a positive is (u

f c) an

_{ve tr3u paaaes}

e wi1e ce t

t

_{t viu}

courter-act t

buccy fc

due to ta pt that pier

faee.

equation of the tve

_{cUa te md to be eeoida1:}

a gj

=

(z - et)

zo z is

bto1uta abscissa. Ie1atj to tw eovin &p,

ta

tt

sin ozxÍz(cV)tJ

2W

c

[z

(V

c)tJ

.s áiftwoe ta dLep1c nt in the

ve in osxz eater 2.e'

k

Yofw()s$

_{+(v»j

1 L Cc5

_fr-

t/)

_5Lt,

r

+ a.

5

F'

vJ

8i

t

tat i

triI,

e t

y

_{tø o4}

ite

O. t&e b&oyìcy f 'ce becu g

()

=

[ef*idx}t(v-cJ't

total Pvig fo i. tTa

Z Z + Z2 ₁ + f2

This is also numerically evaluated ix?App. III, pg. 98.

4citiDg mnt thz to t

%svee cie.tgts beielly

fo'

i.

e

t

te body of

_{oloio de to t}

orbithi vave

velocity e

to te se.body ictiom.

ct izd by tis th

£ bucy elo

th. loth

of t

sut

_{to t}

ve elo

effect of

foros os t

stat.

force nut

be *'g into scc*gìt

ts oit of aplicatic of this

te relotive3y ter fros

'ii of cctes. £

iO

tic 'isostal

ss

os the eat ih

to the peitios of the

vith

_{e»ect to the st.}

LVs

for the snt os

_{near surface body}

_rvolutjon

in astern uaves (Ref. 5):

L

Ji

= -

¿p

4

f(i -i:)

A (xJ

r

Jfr)

2.

£1ta

tia szessios

ve1ope d t colsteIy g1 m'JD45

baiee,

without apendeges . _et

of t

eut oi te ¡it

_{ecittor of}

_body,

_{, io}

nelig.b1e eio

t

_{etr p3rt of that nt}

_i

_cot'iatd

_{by th:}

on

egtc,tieo

of t

iran

bo6y it1S.

direct efct of

e

ic

le celled

Le disesed abent1y.

Kaplen°s exprelon for L 0

1it

M1

4Tf

c(v-JL

JL(

= coo wt + eLe

(e

. XX). _¡n coo

t io fowd to be te ddt te.

ce

eiai Z, t

to in aLe

eo 6it9 the

¿1e bee

TVLe f

LtLe

¿za oued 90°.

'e ffet of

ozz baoy

be egeooed oet

I

'

5d.h ..

(V c)E} o(%

L

'j

c

ir(Vc» ï:

Edz;

4

aince the $at Le ayintiicel.

flt9 tO

£flto cot ±8

_6üSOt

_{fect tb9t t}

yUrc-dde ee ct

ve 3rtB o

e 3tat.

e ¡i o _{f t}

awt8 of tø tzat L not 1v

_{80 that izojjc}

_f

nont a

t

ta

_{?9 nov}

_{cvn. t}

_{tzt to b8}

80 Lp1ce '

i v1i.

/ + 1octy

-

cf

-.

lo±t poetL

k X.

--

_'

o± et depth h:

ir

-4ur

vjj

M'j to

li'8

- .=.

_4r

a _% t& _{1t7 83.}

e

M18h t the tuz

in t i £aco

cz,

absolute

4Eivgt _{14th respect to t in}

anj. rtéi.L

_y

V

, ( +V

4&si

inen

bsoaae

t

tt io

t intÇ

b

ïie

tb jotoia of

f10 o. noo

in - Îoui toe'

_e

y

t

₌

-= .. z o px ¿ _2g =

[x(v-c)E]

N

-

fr

The eLlt1D surge forr

i

th2 L

X

2irrfc2

f

A

£

@fi fi.- a..

Jccs

j7'v-cJ L

5j4

A

(5 App. XXX, pg. 19)

L

X times the oiaet arm.

3 3

To find the mnt srm, the center c

eure bee to be given. Since this pessure decreaeea ezpou ntiaU..y dth d.epth, the center

2iv

cen eaeily be fmd (App.

iT.

We get cos (V

c)t

The hrostetic

Ont tel dl.rect4 o will bave a

resultant. different fron zero d.ie to the vave of ile. Thin

esere (prendicular

to the bull) will be p g 11 end le a function.

of x.

In the direction of the iotudins1 saLts the

essure over

1p1 _{; p}

pg 1 ... d

dfl.

-1

ad

dv

pp---2--Th tot1

f 00D

a t

po3.t i

t tbe t

bo'

e tbmns

P

t

t

ue foeco. X2 ti

= Th t& fc

vili ae

ctceUy ao

t

fine ft

oirions:

-

°r2 [

±

?E(V)&

'V

= CoS

of tc

that.

f

to be

qm]-1

ttty

ce be jog

ie toti G

ti

the

aitw

iU then be:

+ ₊

oo w +

wt +

ti

tut. (See p. 50.)

a1tc

of

kOVe' (f. 6

i) hae

Ue t

e'ip theory

to gj

the oeffLdeate of

'.

ottii of tht fiov

'o

qritnw øvi

pzpLci

to £te

ie

y Gos

r is the

iw of

cyiIi&r,

nd R i a vector to any point in the f 10v.

-]JL

vr co

..

ei

i1ectd p

Jp

o

vticL v1ccity c

t

tp

y + V$ zyt out the

1ffnt2t10

ud tto t&e

_{t11 f as a}

_{bo8 induced}

Is

y the body s own motionp.found by Korvin to be:

=

_{- e}

f'tL

_(.

-te

= ... F

this

'_' ip-Vj

4

'= rr

1dz

_

L pv- / J

_-

prrY J' x dx

= -

rr VL J t

Lat, 4

inxt

s crkd by

Lt4p1yi

_{evej Qeei% wer}

th9 it'e1

La vß.th .

&

ß

j4

-LpVJL

d.

*

2 script 1. eeLte t&? h

_{cdyn4c pe-t}

_{e erivtivee c,}

15

Zu ti zpc

attGmpt i3

to coTi*t tb fo offteiGnta

nia offaot not f11y oci

£n oviu'o

'1

incn

te cefrot ce t jency of notion

ce tED fzee

Zn OdeZ

to

t

eooffiaiont8 o cocted frz tb

'in

cct eficiont

ao

vithat ta bipt 1.

i.

a -I

of Itia a

A

Zt

a been t oun by etprixnt (Raf. lo

_U)

t'y that t

i

not a ecnatnt,

coløa1ate by

ip tby, but

pMo

on fOzÑ

v1ooit1, fe ency of nocia, eM

ezy at1y on

e1ati bn foe'

boie. At w

O th edd

táetioaUy ae to iinit.

'eU, en

e1te th

fron that fouM by

3t'i»

tory

to both uz'fcie effecte

feqncr

oticn. eleo ca1on]ate th

*ango fo?

bo8y

ee difgLa fron

crfln'.

foaM a o

ecti

facto? of

finity foe w= O. _{ad on LQeUa ioe'& aM on t u1ta}

of

f. 10 eM 11, vion vaina of

eM ute

of in'tia -_

ta8.

ae ve eM

cf the otb coefficiente of equation (1)

1ote ee &ao?ibe in the ta11oaiu

egee$e

tn utilieM in n it

_{tive caate?}

_{'e'ea in}

:effcot to

èe

_{1min cc'e1atLon vith the pata of ref. 3.}

beet eepon iae

taieM 4th

ua cze3. to abtt

60 per cent f the actual noac at

V c O up to 200 per cent.. at the

w = o (V

e).

Leee ast10 corz'ectioua e foaM foe the

adM

nt of inertia.

e wvievieal valvee cf a, A, n, aM J c

be foaM in A. IV (N, 51).

2!.

i

Co

fi eo (bed

In a »efet

fluid.

i

iatiCa of

Y

(htc ie

g) n cnly occ

by th pÖ3 of er beft

ari

by gitr

vea at &vur

- th

er

ßiati

b1c ta azf. Since

cft cf t,be fzee

3 ot ccned in th e

eeion of b1

thy

e not v1id.

Th

att to t

account of ftoe

mfaoe

orvn-!ouLovey 1ai then ctive1y by j 1(E) &

j' I(x) x2 + 2VD _{2Vpî J}

' tan

: l«2C)

the denpL for'ca pe unit 1ingth. o1atein (193e) ecuateÛ N(iz) to pg A2/w3 vi A le the et10 Qf the e

1itue of the

tavee rtate by the etrip over

the np1itut1.e cf eava of this

strip eeetion.

irceU (i9Z) a

Glni (193) ve f ou vaIaie

that coweì.t.e to

certain derae ith G

i3ent3a 3ove?,

tbe

thiee cennot be usen in tio case

since they crp1y to

curface ipe.

A sj to f

the eping wu4 be to

coridsr the body of

revolution aa a bydzcfoil on

ich a lift is serate due to

the heaving pltthing*

i

acL

dL

r

cc y dS.

of. a eroi; ;ction

aaL

zp -

_òc'

V 17 Z + Ve -L

ththt

ß

_,jxdZ

=

_t

o1e difficulty lie

aever, tu finding -

ain it viea

over the ]n8th

t

body for _{peculiar abape vith ita very 10}

aepect ratio.

zcae, ttii

thod iae aleo diaded.

a atcd aelected in tbla Eeport foe calculating tìa d&ing

coafficiante ae given in a raoent pper by

_{(Raso 8) pizig} coefficients

re ecnuted for en oscillating ellipsoid vith f oeard

iotion near the free aurfaoe _{ellipsoid Le r e'eeentea by a} diatributl.cn cf Gi

tiee, neaely, at

.atate and oaeiuing

dipoles sad uedripo1ee.

The diing cofficient

ere then fcur.(

fron the errz'

radiation at i'inity.

_{Using the so}

_{!ej' (8).}

tae dieanaion dating Coefficient,

, of

tiie report i expressed er

foilova:

2À ¿

re

-

fcte&3Z

,L .

_x

f

()j

L

()

d

, . 8). =

I-

-q

i

are Green s inteals.

a1, e2,

reapective1y the h f..1enth, hbo,

sd k .t'hett of the esroid.

j(q)

= _{sphericel Bessel Function}

ellipsoidal coordinate u heave xelocity .i

q

)[(e

a3')co a

+ (2 -

2)8i2uf/2

r

r

--iI

° t.

(..L.)

or

t > IlI

A &iiiai lar e eeei as found for

355

'where B55 is none 'sional pitch amping coefficient0

M XD14 7O catr

eist to find these velues s a

function of 'r,

end a, e

s. be have

that the

of our body of zevolution &e not differ ajpreciably

» i'oid, euch that the cs

f 'aulse are

plicable Fig. 2 zhoe the apìcd end freueur dependence of b b foe'

different

ve legtha

It te seen on Fig. 2 thet uofOrnate13J

xininun dp2j is sacociateû 4th the rezonant condition.

For

vmraee1

--.

Mio. IV (. 52 -

53)

.

i*id I$ng Cos

icientQ, e and E

In en attt to account foe the free

rface, orvin. oukover also

dif ted the eeeicms of e

ad E.

e

_(,)

3bere end are coewection fectars dje to abaps of the

body, end the free surface (the average k2k1 .uounts to about

75 per cent foe' xZaos ebipa). Ravelock (Ref. 9) found

*

19 0

e'1 the t et f

a flta 23I eoi:

pnv, e .pìv vith p

0.515 fc a 1enth .ter

ratio of 8. bkawitß (ReZA) fia& e = o(2. + VZj.)

e t

eeioa for e io jh1

n forni to that auete

b ,ia. We 1eo e that

eaú thi bee eleo b,

1F"

by eriatG

Tie ot by .Grrit

(nef. 13). e

ies f' to bave a

iffeat eiz

a 1r vabze then E.

To f2

ezoeiaa foe'

o

eee 1a (io)

faz

-

Vf'3

I-/1

_(

_"L

j

-j-te fcad in the fr1.

'J

=

íj

L

jte beve been cueted at

to fiM

for iffent

'

of eeiy

ib mee. The fo11ovi

iice1

a=-j

eM k1 virtoal, 3$ coofficint ©

be fiM in

Th2 Ve1U

foe e s

f2neUy ea cted _{a b}

ip

ezeetly tk'

se that 1t1ie

. 15 f o

thsj s

L

asee veaies

e given t ¿. U! (. 53)

If

1e

fcn

jg

this

aot

ah ifferient fri

thoe ccu1stet is3.ng the thcd

decibe in the

ovi

pgrt

_{¡n e' svont}

the vs.bs cf e

b'e

body very

&U cce to

_{s. end fOU foe}

the

eubmi

.eWzippe

vit cctro1 f azis

e

eitie c

he

vaseel (see

ts

Iv).

40 Contante o sndc

lbs coefficient, e, exists becense of the

_{presence of the} surface-piercing strut Qid, C, bécuse _{the metacentric sthbility of the}

semi-submrjne. Fröz quatjc,

_{(1)oid(2)t is}

_{Been thatc,Z aid CN0+ V24. The}

heaving fce &ae to e zbenoe

is Z _-p Z ₌

thez'efoe'e, o _p &,. ₌ A

A + p gZ hee X in the.

1cngituì1 ¡nt of irtis cf t

tere.

At

LT. TQdng Tnh teets

_{iod out to detgne the}

od&de

nt c

sdeubi

it

' t: rt ee

_{(est. 3)}

At ev'y seed it

s fawi

that the

e1 ouoe iren _{U initß.el}

a1e, took oa a eeteia 1, stebis

in

'. i

tiok

s

twiable vith V se

in Fig. 4.

At tha etab

t0I

tbe 1oegi

taius1 oti1i

cnt is eas1 to the bong mmt. For

11er

e the IÑdo cnt ou1d be

tar,

for

r 0 the eti1it

mnt vaalß dntQ.

ài12 the sti thìt the stsbtlity te...

1$.

ii

_{with e in reoe it in uîess1s to asswe}

that the bcync wwnt is siso Uzr A

o1ic fuactioe

Q

1J

U etitioe

I,,

5

e C1

Vi't%g

tQt

C ZZt& D

'o

iciete

et&'lp thecy to be e.a1 to

p j' x

f

to

ß

eEez't3 c

2J4 ì .

the itute

e

ot es itve

to ve

aìa1t.

Xt iW O32a thet tb.eee co1i

teze e10

eab2.y

et vy 1oî f

c. be

iie ef

co

dente,

1ct

D 55.

6.

_

Like t

cfficito o e. C, tee e

icLet eict becee a

t

th eat. Frai eqztic

(1) e

_(e),

e fi3 Z

Z9 + a G

=.

i

OV8f

(Ref_{Q 7) tha e ei} øeioe £ _the

, $

+ Vb G

'p $ E

i

t

e otut. Beoaee

eiiat i

tit

ect to the

GC.1TG' lixe, Z9

O. L.c1 e1at1ipe f

eve1ce fit

e1

aAbt'øe i'e the f

. + -C---.- u

ie th

8L21ieE lift.

1/2pL D e on1ee ¡ °

O.9

2 L2V2 & = av

efe,

1/2pV2L2( l + D').

ez'e vilI be e eeU coefficient

Ibe el1e e

9eSiofl

Le uee here i lieu cf the

ecUetios1 ezeaeloe for b on

17 - 18, beci

tbe aoup1et flvìg

coetent

le a e1ativey iOinifict o.

e, t

ee no

22

to tb ftet tt the

_{tat does zot stead in}

_te

_{d1s of th body.}

Yit e

g vili be:

3

s the oriitl

_{teee btuej, th}

_{oezter of z4}

of strut.

¡ a

2g

g = O G = _{Ze Atr}

- 23

Lt1 SOW'I 3F

Th2 equati.na of aotin ee of the type

i

(mtpt».bi4c.z A.ü

(J4i

+ü

+ Eq.)

21V

(D V e)

The coefficienta ere hither tPate ç

f

tone of V. inoe the

equation of pitch le not 1r, the

iap32 iithod of eolvin a set

of lineer Oifferentie1 aatione cennot be ueed ier to

aeliminery setinate of boy the tly cte4 espo

of heave

d pitch d.0

rsso

%rth the

ar1tsI ¡eeult,

the pit Ch eation %s

livieer.

In this

y

it te

possible to

tito a eptor

'

that eolvee ths euaticse in a very chct

t1,

such that co

ic2nts that hñ to be f oun( by triai. end error

reatbod could easily be verieO.

The e perirsntsl pitching responses tiere _{used to 1in}

for CB +

for different seede.

n the

voee ere ivi6ed

by B, a cofficlent C' is fcun

id vîU be the ecefficint of

e

in the lineeri2ed pitchi

equation. For apseds above the

elntel ren

s

_aße e e end later oea corrected

by trial end error.

solatica of the oat of tvo meer euatlons is .ven by

Q -da + lew + g

S:

J +

+ jbw + C o

Z0 ez

is

o

ee elE.

E and e _{thee phase ng1ea between re8pectively}

torce and plteMng mcent and the wave motion.

ar4 .

e ta

_{anglee betmen}

plt

z. =

o

° 24 °

exciting heaving

a + ib.

1a2 +

+.

_în

_1oow

mek goes c' Z

A coezte

_v4

t

viue

e0

1e. w

in f oL.oin ,ee 8 t0 F 3 cmgr

tb reeulta of two rthoe of ciipating the

_{reeponze curves in coarison with}

e

pit

iWIrL1 re ou1t can be in ¡V lineerlzed

p. ziethod of eo1vixj thQ o _{atloe yieided satisfactory résui t&}

f o te pitci, a'ew the avee.

¶ oUo in

anpez1zanta1 curves, oueve, are

ot recuce im te

theeti1 soltic

eve, te

aintt!1 &ta of be

pttc

foe ees cf 1.5 ft/sc

up ecv g'eat fi

tmetion,

it

y be

ty preent t

ccect

Ues

25

-Por the heavini, Eotiono it in f oun1 the eol3ztion of the lin.arined

ithi

oií too 1o7

lituds for eped3 bz1

_{i ft/soc.}

_he

6iearity le fron 50 to 30 er ooat in

t.

end a littls

better in 84t

_{n in posib1 that either the etrut, the}

or a t

_{di rsicaaL factor be a}

_{ort&nt ith12zn on}

This inoa io aich that it inereaeso the heave

reeponae

cf the

_{At a ed ce bt L8 ft/oiie}

the entai

- the tical datì .lo:tn

_{for biber ojiae«o theorr iveo -reee}

larger then the teeto. This is not a eriaias deficioxey becuae

_it

s that ezy controls deeißzzed to 1ila the theareticthUy _predicted

ezcitaticn wiU overcontrol in reality.

1s heve no

_{of cheadv heu}

_{the they r}

_{.eeute the} actoal notione .t

_{apseo obovs 2 ft/c (P}

_{0.18). n ie econ that}

tJ notione inci'ease cat1.y o

U beyca t

_{liir z@nge}

_{so that}

the 1ried

zaticne do not reaont the actuel aoti

_{j cre.}

opedo of eyndironji occur £eod at FE'zde

_{nbez'o etarti}

_at

0.25 for

heave 2n 4..ft eveo en at 0.29 t

the pitcì in

2t

o . go

ually up with ta

tIaV

1the. In the

solation of

_{liarisad euatione, the aako at ayncronin}

_ze

arent;

t they are only çielitatively cect. 1evethe1esa,

the ccoptatlons eboi that, dthat controls

_{the sicubino cannot}

be opate8 in speed rs of fr

0.2 and up in atarn eeee.

'o test the y ildity of th

of solving the est of

26

-nothcd9 vea used Ti±dt gives itndss u

eeie end pitc sa s

function tino. A set

_1itts1

liidss is given nd efter a

certain treneition airiod

'elstive

tndr

onic raotion is

found. For evezj aped the litudes ro taken to re'eesnt tb2 noti One w..

no siso dono with tb eerizntal

results.

M con be eson in Aa).

fr

F.3, the

in k.ft. yaw foUs vet'y c.ose1

the ot

solution, ii1e in 8..

ft.

vse the tiesaist'y

thod corre po

betts with the

riitsl &esve.

nonlinear tern in tbe pltdi equation icb

is tikn accmt of in t

Tafl _{tbo i account f cy&' t?ÌKS}

M for the pitci.ng the

rves eta' close to

in te

ai'1jntai rss (V fran O to 2 't/noc)

it

the f 2'st

s

1itts better thsn the sod in I4.Ift. vaves, vi1e it is te

oosite oses in 8..ft.

: (Fig. 3B).

ier then 2 ft/sec (

o.i8), thse is csieeb1s

issity beteen te

o

ys o eoivt t e sties eseoinii in

the io

ives.

GOV52'fr

1itas go

11 beyond t

iino'

ronge the %astiose su

tbst ty do not o

spând vith reelity.

A1tha

qaentitstheiy these results re not velid, quslitetively

tby give

iiction ttsre

bi

lthdos vifl be

e

5O3 0L found at the roe ce speeds9 i.e., re

tuniflg factor A. = .L1. = .

or bave the rosce speeds go fron

M iffnti1 ejtto

oc

E3 io fr

to i.O fs/c. lt io 000n in

both lso

ite

tico tt

o not coineio ctl.y

wtt to t

taUr

. o ¡r

b &e to tv

foot tt or acnli

og

tioo

M itin

ot tb tLst

tho io tt it

o.ct1y

o4 t&o va;1

in

¡t

3, Vai' inotaoo

taiO tt

Voy 1G?a

ioy

onnt

tb vìo1 b33

tnc to scU13to at ito zto

icio

ic

g,

pit.

& f aa1ato

c not o _{= O.25,} y ottU aa

bo we if

cy

to coito1 t}o

tio

¡g tbi

y

028

!!_1

coøL

' AVE A

rii

ezpre salons for beavi.ng

id pitcbin motion derid in

the rvioue chapter form a v33uQblC tool to investigate if control of these motions La poealblß snd how this can be done. We know that

the equations do not give the rlt pictere at apede at aa

hier then

3yUCh'Cn16 sinon gows bey

t

coátrol hoier it

y be poosible to deease the motions in a

r that they rin in the 1.ine' r

CO that

cn 55

thea equations even c' speeds clo. to eynchani1

siiesi&

io5t econ4cal sj to control a

notion is by a

hydrofoil that can rotate tg

ie

ionl'

to the stream.

To control the pitching it le aventecue to put the foil in

the stern of the Eip sinca thin gives a etclo

such that %ihe

there is pitch

lift worts in the direction ooslte to the

pitching an31 To control the heave the vot 1voreb1 plaon for

foil is ne

the center of

rnitrj cf t

oip.

¡n thie ve4y the

pitching mrt

by this foil vili be

flero. This pwsvsnta the

efl1ct of the

t1wpitching fin to be amuUe by the ent..hsaving

fint ¡t fòllows that in the equation for pitch the effect of

ánttching tin alone

is tiam into aaconnt, 1i32 both

foils

effect t

equation heave.

is the distttin force.

control elsrsnt bzssvor,

worts in this aistiance aich,

¡' tvru, has en effect cm the control.

*

By 'nti-heaving fin" it is understood that s of f0118 are. located

on either side of the ves8el In the vicinity of the longitudinal e.g. of the ship.

heaving notion is not depent

* _{th of' the centerline}

ca the ve

The en&m of catta

es s

function of tine is given byz

'T L .00S (w 1? + T)

PV

A

cosCLut+.v)

+ the

ve s1ita

-. li = .75 f t; b/L .1875.

In this vey the 'ie dietabs

'ees1 an the cccat'o1.

The

et

efficient conto1 ou1

in fcact, oec in cabn tez. Dy most

efficient it is iG1iQdZ efficient on the averge since,

Lth

certain vave 1enth eM with the vessel itiv

freqncy of

oncoìiater, the obitca1 velocity t the

fin vili be so

as to inssos tb eftót of the foiL1

At othe' gave 1ethe the

opposite vou2ß oecu

The'of'e, it is batte

to the

of the obitsi velocity sevli

'ais oea be &e by 1ath tha fin

io

tsn the ebip.

Tha ave cbite1 velocity is given by

ae

vve 1zgths.

The thge in ele cf cattcach on

foil

e to thin

is 2

end heS been cutcd for d 2erent eseds.

It

s

found that t V

2ft/c for s.

sve length I

8 ft.

the chenge in

of attcac cat depth of the centerline is 200,

j

at fi;t judged to be

1ae.

in of attec t L/10 bc,.oTi the centerline is

100

iz vld be scoeptablo for g f nctiiiog at' te ft foil.

entiaheee foil vouA rt cat the de»th of the

ztsrline

sinos the hese 1a betveen the

of eagle of atts, eM

30

T get en idea about the alza of the toile and the mg1ee they

- .*_'-

.-have to incline ta

eonìrl the hesvleg and pitcki iotlone

aucceeaful, o

¿en rüìn ea f QUOVe.

The viotione viii be coepletely annulled Si the force a on the hydrofoils act in aucth a vey that the pitda1.ig mxnt and heaving force ich they produce le, at ali tiase, euel to the ecitin

momenta end forces due to the waves. This would not be a cont-ol

have to

system in the regu1sr cense becaise it would)mo frì btoiehand

how to react. In regular waves thie rstevi could work but not in

a confuced 9ea.

7e absll first investigate the pitcM motion.

The lift

force on the anti-pitthing Lin cnn be ezssed ea 1/2 CL p A U2

where A

eo the foil.

produces la .M

i/ (--)dp

A U2L for a 2O. «1ie the total.

angle of attack.

If 6

6 sln(at ii') representa the rotation of

the angle of t fin aiamd its n axis, thaá the total. ongle of

attack becs:

o2=

_5&t+-7)

The pit.4'tvg

mut will be:

M

c(r)jpCìL

4

_.4.

U is the resultant of V and VP.

iJ2

_=Yt

..t4

ApL kL

4

o(Lu..

(2)

_CoSZ=

cupae to

U.

This is

iesible if the level of the foil i

at

ieat L/lO belo the center of bovgrnv

3

esc M' than

2 ft/eec.

r ssttin sil ed the terse first in sin wt and then in

cos t ei to

ro h

equation -

oes t - n2 su wt

O and

solviig f

and Lt is found that

- L1 4n2-31 -t i.-rC)

x.

V

r

x

[Vt

¿ +

me

unoari

a'e S, and 4.

Tht mont ie

to be 5qu91 to the

itin mcent

o3 wt + 2

To ilify the ßQluticn

the ter + 2V Vp _{( +}

4

a'e and to be negligible

ApL ')C1.

i1'

e

t

Â;sL y4rpcos1it2

ac

(For a diecusion on the

uidee of A and

ee App. V! and i' heve been calculated foe' an anti-pitdUng fin vith A 5%

of the longthzdinsl cross sectiossi eren end

3.4 in avea of

4 and 8 ft. end foe' speeds of 2 and 4 ft/eec.

1he resulta (A. VII,

pg. 69) show that it is vpoesib1s to control the pitshin at V

2

ft/see, ithile at V

4 ft/eec it i

theoe'eticsilr posib]ié to eHi1nate

32

(i)

1f4O the

iix rotation of

in order to control the pitch.

¡hese ez1e bet'e the motion en the origin of ti

To control

the

_{aie, the affct of both folle hc to be ten into eccont.}

o ont heaving toil,

ich to oltoatad

cipo at the

it of

the canter of bUOJanCy, io aeeuxied to have the ee _and

the ont pitchin fin.

So, the total heave exciting force fron bôth the pitch and heave toila ta:

F

(.L)y[

_{st(IAL_1O_}

°(

..ir)

L

f

particle velocity nt depth h

10

"p2 -

particle velocity at

th h

roxin rotation of center foil

heOO lag beteon center foil and origin of tine.

ie force io At equal to th aciting force:

F f1 coo wt + f, ein uit.

ìutting the teroa in cos wt end 0m rt on both oide of thin equation 2qual to ea

_{other, it io fcmd}

_{ter èo}

_{coleclatione:}

+

L

s(wL)_

cD5

-(:05

=

CD5

33

iii

'

()yL

after rep1acin 3 i br theiz' ve

in (i)

(2) (i. 31):

(3) Ç

(f2.

£,i

SLIt

It in f oud thit for eee

of 2 f t/e it le ipoaeXb1

to control

the heave idecl].y (Appt, VII, p 61&) Thie Le eiznil

to the conclueion f,

- xitch.

ee cattione,

iI

_{ae in t__lves on'y}

prit±one

of the rel. piani con Lt.ane,

.tht it Le

to take the

eeteet

_{ee for the oUa poneible end thet there vîU nivey'e}

be ee pitdin end efln et ¡o apede (F

_<.5).

A tore reeUetic

_{ttod of control in 'to une the ììotion of the}

veneel en' ipt to the euntro3. elent.

control fins vili

in aw

e t

_{an nivja to cmteract the notion the vencel pertoren}

at that ìont0

Th moticr

utic!e ci

then be e teased a follc,a:

(4

+

L+.

è)+z(_e)

i-Z

= ( (g

) Mt-)tN(

J) it

there Z8 and M5 are hydrodynamic der vativee of respectively heave force and pitch mount due to angle of attack on the foils.

8

e12 betien

foil ed

i

tudini

ie of the

ip

of tttck 2dtaced by the

s4th respect to the h'iaontal.

I IO

neUib1e çwtitiee f

ws

lg ee cr lin3r

S ft

thn

ebip 8 e 8 e

1l foe t

vessel in ate'n ene since

s O.

the motioe have 1I f eqviney

&teee

cc, S

considewed to be of cder end re nog1scted.

In tb

eJABtia of notion, the ocefficienta ,

b, e,

. do

not bze the sn velue ea in

the urnontzoLbd vesseL

The added mass of two foils placed at x _{and XH from the center}

of gravity of the ship is given by (Ref. 16):

Zr1

z

a

where _{8K bH are} reapQctively span _and chord of the anti-heaving foil.

Analogously for ap and b.

Zqf = i pri (Hx +

Px)

.pîr +

Px)

=

Mq = _4pri _(xH2 + x2).- -Af In our case xH = O.

These vthlvee re eded to t 'eepcttva coeffioteate c the voee. idthonte fine (An. XV.).

B,

o Ef

e cad b7 tb

3.Sit dreg the falle

due to the ship'e a

L=

l/2pA{V2+(ê+

l/2eAC4V2+(è+)2].

Frc tse to oate the

oofficients c

b fou:.

7 -b + l/2pa V ____ 4-

C)

35 21

Ij

l/4f'A V L + =

- l/8pA

L2 .t

c)

i

PAVL(!

+).

Cttio

Z6 i Z68 (z8) 6

eg1e bet'ieo longitudleel

_{le of ship end the pltivg foil.}

(z1 ) (z6

H en both toile equal to àach other.

(z0)

1/2eAV2(.!_

+ cD). zo

M6

1/4?AV2L.L!+

L/2

The control angles en

e we

dependent on 6 and such that

relatively empio controlling devices can be tzed

'k1 +k2

+

here k1, k2, k3, k are four control parmìieterú to be determined.

¡f timo lags in the response of the control system are teken into

account, the anß1 becm z

£ .:=.jè

_-

-i-SN _.-

-tere t1 sie the timo lags

in the control system. }tsnoe the heave

equation beces:

+

L +

[e

+ eç

Z(L4Q3)J *f,iZs]

=

o+fvkW&+ ZÇ(,ø(H)

Vi

_/

L \.

- » _{coejwt 4- irj is the angla induced on the}

anti-V

1/

pitching foil by the sve. Aalogously, for the anti-heaving

fin,

- _eo , The total angle of attack on the anti-pitching

fin is then

+ - e.

*

3?

The pitch eqwtion becomee:

+

cL4P44-ic(

The different coefficiente of theee equti one re ev1uated in App. VIII A in the unconto11ed notion two metho& of aoiinz

the etie of rotiome

e ume6 c, the 1ineeid

tho de'ibe

pg. 2

the ot, Ms' 1rice1

tho6 (pg. 25 s 27).

In ¿&cf t uavee (VL L) it is seen (App.. VII!) that the

motions are f eirly aU eatroiled if the time lags srs teloen enU

enough. Large time lega f or high speeds cañ heve such en effect es

to 3faatiCelly diminLsh trae virt,ual mges and moment of inertia of

the ship. This effect is even stronger in the motions in 6..ft veves.

Thue, heaving end pitching can be eet1 increased. Even en increase

in foil area Cannot make. up for the lose in

_{. virttal inertia (Tables II' 8Th).}

The solution of thé linearized equations gives amallar _plittzdes

f or heave mcd pitch thera Adena Mthcd. This vas to be epacted,

Just as in the uncontrolled aotion, ein _nlinsar

rii inces

the pitching (App. 1V) Therefore,,

d a10

because of lack of computer time, this method vas discontinued.

(Tables IV end V)

Since the motions in 8..ft vairez were harder to controle túrther calculations

were confined to these waves.

_{Fis-Uy, the control in}

_t

waves was iwreetigated with the system that gave optimum control in 8.ft waves

(see pg. il).

The numerical resulta end details are given in App. VIII2 while

a description of different control systems th were investigated

*

A convenient stary of el]. the data of App. VIII is given in tabuler

follows below.

'.kbeaytem

6R k39 4- rI c'

fE(ti

leg)

did &ot give

_{tisfactory z.a1te.}

_{This cen be szplsined not only}

by the t4ix lag but also by t

_{fact that for speeds 4ger then}

3 ft/sac (Fr

.26)9 the phase le between heave end pitch

beces l8O0 which nesna that the

_nti

_{ving f011 is cmteracted}

by t

foil aft.

_{To e1'in.ta this effect the antipitcbing cont'1}

has to be such that onl.y a pitching mceaent is _{produced. The solution}

io:

two antipitching fins, one fore and _{one oft9 which rotate in}

opposite angles and whose resultant response dosa not produce any

vertical force but only a pitching ment.

Fz the

A.. VIII

Table VI)

be seen that the ant .heavs,ng fin is _{now nora affective; but the}

pitching ontro1 is worse, since the presence of

_{fin in the foed}

end of the veseel has a negative effect

_{on tciing tGbi1ity.}

This new system requires, cf course9 alteration of neny te'ina in the motion equations (App. VIII, pg. 70).

To mehe up for the loes of stability in

_{pitching, a different}

set of foils wee introduced, nane1j, fle

_flue.

_{¶Lbsy hawe a much}

higher CL than the conventional simple

fin.

The Sperry Corporation

manufactures anti..rolling gyro fins which have _{a CL of 1.5 at}

angle of tilt, with the flap at an edditiomel. angle cf 20°. lbs with this type of fin anoizute to

_{A fther inasse in}

control ability can be obtained by a 1'etepcontrol," _{which means that}

39.

the

le of the foil.

ovee etepvi

frca a

ez1 in o

nee to a iia

8nle in the oeito ss3:'

e F

*

See App. IV.

'

In this way the antlpitching foil asitchea over hen the pitching

motion

chas its

iystnia

aUtude end reverses its sene. Ts

entîbeaving foil works analogously.

sult vith thoa etrens

conditions of and control

thod ve4 en al,ost c'lste

eliainr4jon of mot4on.

In fact, the veel te

controlled., since the motions degenerated into almost a vibration.

accelerations oed, indse4, very high values,

tie the frs

noy

of the motions vea consideebly higher then w. (Tale VII, pg. 72)..

___

atepoontrol with the su foil c

aetertetics vea

uae in th nsat tryout but dth only ons antiitcbing fin ('t) (T1e

vin).

lbs zesult choed a fsily gco ditch control., but again the heaving

-becs too laie .'t epseda of reeonan

(V _{= 3}

ft/aac).

ce the

vas i is over this apses, the cont2 works afficiently.

.Eover, foe

.tsa bigh pae2e, the value o

i for La rather Ìigh _since

vitation nsj bec a cWan.

it is, therefore, safer to ta

a

lOwor. ¡1n lift officient t those dgh peeds, at srM 1.2v

ac7; 1»

%cbich gives a of p

itely,.

tce, it is reslUd

that the large strut needed to support the large pitch roil at the very

ste!fl Qf th? Si S'±Z1!!C

nay create Co _{6erable structural p'obleie.}

_{eoer, the}

increaae in dreg due to thin trut i&t be eo 1ar

at hi&i speeds that _{resietzoe Mventsga or tbe eedsub me ayer}

the e ventione3.

_{ip could be reersed.}

_{t is also trt}

_that

t

mer the ford speed, the less

_{tbe dieturbig i1nen}

of tbe vss _{the angle of attack.}

¶Leae coei4eatjone led to t

intoduetion of Wo euti-each

p2tchin fine of

_{U sise 9.8 per cent of the lotudjne1}

_croas

section erse) that are dirsct].r attacked

_{to tm fore a}

_{C t ende}

of t

bodj of revolj.ition at the level of the apis. The

antiving f o1 kaa a 2.5 per

_{nt area and eine !uidsbip0}

_A

5CL _6C1

of b and _{peontrol bere aaswed. :;r va.iae cf 1 fc}

8a.

io more realistic, since the lift coefficient cannot be taken ea lae

at higb speeds so as to avoid cavitation (seo App0 _{VI) ¶e resulte (Table IX)}

th thi

systenere as good s.s vith the przvioua

_{one, ibi1e t}

inm angle of attack on the foils io

Uer,

ick diriinigh the

danger of stalling. _{oavez', again the pitching plitudas have a}

terzdencr to oscillate and, thin t1,

_{not eround e}

_{= o bt e3rcwd a}

negative value.. This canses the veoael. to assuma larga pitching 'n1es (10° sud more), which is not allovsble0 _{a asma tiaa true, i:}

lesser deee, foe t

ve motion.

_{Pitch angle and beae dlaplaont}

eirls sre, tberatare,

died to

_{zt tryout:}

.

lt

f ouM tt this eyat.ez is very. effective to 1t the pitob

at sfl apsede,

i1

tZss heaving s 1ies

zaein eigrdficet at

t

epeed of oaoan

A better zelt et this speed is obtained

%n the heave .d 1spLament snn1is tehen aver (Ape. VIII, Table XI).

Jsj mo

syrtena of conto1 have been. tried, bit it asome that

the ysm

/6/

//

gives the best results.

pitch remains below 5° around the aeed

of pite rescnce, vhile for higher speeds it LS

Ch less.

heßve tehes vlueo of 2.5 times t wave height t the speed of heave

resonance, but outside this re it becomes negligible (see Fig. 5). ¶thie system applied to the vesse]. in waves ( L) shoved a'most a cpI.ete disappe ce of coy metionø, even with Systeme

that attU tshov'ed lera omplitides in &ft taves.

It seems that the contro]. of this oamiu'.zbime is the most

difficult in long waves which bave hX energy and causo frequency of encounter wound the naturel frequencies cd' the vessel.

Stepcozitrol is, bo4avsr, an idealized assumption; moreover, it

le not even desireble since it cisee vibrations and high accelerations.

Fron t1be C utatiomsi resulta, it oea be asen that a baonic

control fmctton je sufficient at high spds (Jr

0.35) but that step..

control or, ra, a 'rounded" stapoostrol is naceseary at

ede cf

It is advicab]a that time legs in the control èyeten be as ____ as possible.

vo CLU

Pzavioua

crt

pit1 ícr1z ohgaie the

_U

nottonì c

t

in hed ezìne

voz, 4t no

ion îe cnt

in a1 Oton

'Dhen

$znt

C flta

enW

_p in

the notion _{1itude ot} L oleo been found to be thze

in thin etndy, ich in entie2.y

1rticol in

Difteent

oiet1

_{theiee tAt yly to either eux'fece cip}

_{c' doepl.y}

bn

Ued to the oble

of t

ntabino. ne' the

afe

It Ia

' found that the notione o f beyond the

zee, ad the noctionn

for ebove the Unit of c't0

!ie report boe, hoenr tt 'ith ct.vtd

_controle incorportin

d n.ia4 control

_{xfeeee, it io poeibb to}

ro&& the notions to very ninel

vee

ino the f jaonoy of

encounters a eere notions oconr, e relatively enU, the

control _{ciry would not aave to be of a 2}

_{pied te}

neither need the voar roquirenent be loreL,

in the by&oyo

eiciting forone end nts

not lore

1bi fteibU.tt

of _{effective notion control ccu2d with}

the fact that the eniouine cot

_oricn

_eln

_{tende to}

ccjrn the zotation tt thin ty» of

veeeel hould be able to &astain ita cin epeed in rery heavy ea

'3

To

ive t & nø

oew&t

of t

not in

1D

tçve,

of __

of thtin

jpi'otios tht eau1ß. bs 2eettg&te

I,aaUtcU,f

low f«jmeio of naoter th8t oc

in

In taie

rpo't t

lø 1w

e ffiCtnt

-_

leebrti&l

. ocG3m,

i

s,

wr)1e8s,

coictent vit

mzietin theo'io e

nte1 ie&*lt.

o f

ecjg

_.in3

fient

ot fcwiú to cio vita eiit

of

E' fco bs not toen stLied

fully in tain woet.

problem vould, 1ideed, bee

th

eonlex (see App. III) ad forms a topic of research by itself.

The surge motiona may iave, _{lo effeCt}

_t

_±t

of t e19 ainee t

3j

atut.

t

ve&el.

¡t

b useful to iwestn

t

capiin of t te tiono

ptt

in

futo 141 of

eations uou1 ainly e'i

t

colutiena so tt th e1atiG

Lt

aS1ut&1 z4ta wcul8

e ¶1a

_{ou1â ,t eltow tb oeneinsiono yavioly}

n.

£ nntu1 e

eueni

to this

t

th rp1itiom

-AiDX L1

.? _{TS CW} 6m

Leatb14ft

2.14 m Vaterp1

₄₌

0.214

ç ft

Depaoaîit of body of

volution = o.Z8 rt3.

Dijìacnt of

il = 01214 &

Itr1 eavi

period = 2.3 sec; natura]. heaving frequency = 2.73 cycies/

Bec Nfttra1 pitthing period 142 sect natural pitching frequency = 1.83 " "

Body of Revolution

Sttim

o

D/D.

o

(D/)2

o

i

O.1e55, 0.207 2 0.702. 0.i93 3 _{0.872 0.762} 14 _{0.971 0.9142} 5 1.000 1.000 6 _0.976 0,952 7 0.910 0.828 8 _{0.778 0.605} 9 0.530 0.281 10 0 0 3tntt

Statiot

o o

i

0.253 2 0.523 3 _0.769 14 0.939 5 1.000 6 0.939 7 0.769 8 _0.523 9 0.253 10 0

-AflR1DU XI.

I LNCR

' QN J1QUTXOWS CF NOTION

1ile the effect of the

rge exciting toree cm t pitda exciting

mnt bee been partiall.y te4an into accnt in thie report (eee pg.

u-12), for the zet ¡art, it hae not been posaible to intro&ce other

effecte of eurge.

If -

call u the

uge velocity, have that the tiz dependent

velocity V V0 + u there u le a bernonic function of t in regul vee

and V0 le the en velocity of the ahip in vee.

The .freuency of encounti

()

Thta e

ìriinta (Ref0 3) have ehom that te

u con bec

:7:cT

coderab3e, up to 30 par cent of the foerd velocity.

In ad vee

er

c

t)

4. (V + e), thin &ge icu34 not have a

intbience but in ten vîvea the eiount V

e een bece quite

naU co that the frequency of encanter can chenge con iderebly

6urtzg the ti

that a

ve lonth

eaae the chip0

Iba ccple equnticna of ötion vuld beoea:

X.i a Xu a X.a L

a _{Xqa (Xq +}

a vxe

X0cos(ct

-a

+ (nia Z)Z

a a _{+ Vz1)é - VZ&} Z0c0s(wt _E)

(+Wt)éa(P+VÇ)O

-X, -X, Z.z, IÇ

e U ,antitlee and can be ¡g3ectede ¶1b97

e

eitbeT conpied eee or coupled momente of lrtia. Xq X,

inc'eead ze

ue to &r tu

tt,

eat

(y + u)2.

vili be t

_{ee tu iitt f}

_{ue te}

_{tu f}

_v1ocity

a functiou of V2 (Y0 + u)2. Froa

_{;vtor ZU it foilov.e that}

the coefficieuta Z» Z ZP LL,. -- d _{fuctLoie Of}

V.. V0+u

Solutjo

for the three

uatio Of IotioA

_{ou p. 6 eau}

only be fd by açiproxtute rtioda

_{vita igøpead coutere.}

_e

p'obLm te

_{tiaUy a n nlinear -ibrtiou probleì}

_{eny cd'}

t

_{eueffjcjents are difficult to define}

_mthmticaUy.

_{bre reeea'c}

on thie

_{te k1y d treble in this case cf}

- iB

AiaDxx III. EXCITIIG 1tcS t ¡OZT8

o ve lengths wer's ccsidered to

t en i6e

of the notion

of the eamisubnsr'iTì

in mae. They ere

scttV51& GJ1e1 to

end teice e length of the vessel They

vere çofl

cenes

th ntaz'e1 f'e.anei5

of the vesesi ere lo& eu t.et sckìz'onien v,th the encounter

freusncy is Uly to ocour

et

oerßtth

for these vsve lengths.

7

The integrels in f orenle (Z)were aolvcd by Siinpan°3 Ri1,s.

Ten etettone were considered sufficient since the body is @lsnder,

the dimensions do not very nnc

fron station to stetion.

The

rea1ts of the cutations for

A

4 ft:

z1 (0.072 0.0006 V)cos 1057(V 4.54)t (0.69 +

o.ci5 v)etu L57(V - h51)t. A

8 ft

Z1

(-

2813 + _0.15 V)ein o.185(V 6,)t + (Ol3 coi V)coe o.785(V

6Ji)t.

pg. 9

?ora2.e (5ivee for A

&

ft:

+

0.13 sin 1.57

(V

8ft:

Z2 1.17 sin 0.785 (V 6.4)t

the tota. heeving forces beo for' A = 4 ft

z = (0.13 + obole V)sin i.i(v &.54)t + (0.072

-o.0008 V)co5 l.57(V - 4.5Z)t (I.bs)

A8ft:

Z (.4.1 + 0.2 v)sin .785(v

6.l)t

+ (0.13 -0.01 Y)eoe 0.785(V -

6.4)t.

(lbs)