On stability of ships in irregular following sea

acJ, ceeh

On Stability of Ships ¡n Irregular Following Sea

by

Tor Virije, Associate Professor, Division of Ship Hydrodynamics,

University of Trondheim, Norwegian Institute of Technology,

Trondheim.

Abstract

'l'he probkiii of cap.iiIiig of ships in u regular sea has during recent years

beculiic a inure and inure iiiiportaiit protkiii. Ibis paper gives a re%iew uf

previous papers, published by authors on electrical engineering iii analogus

fields. Their results are here applied to the ship-rolling problem.

In (he paper the author caciids the results for lightly damped systems to:

The ship is stable if: (J <

Where p is the linear damping coerncient in rolling and u2 is the variance of the time-dependent part of the metacentric height. Al the end of' the paper the author refers to some of the stability analyses of the non-linear and stochastically,

para-metric excilated equation of rolling.

1. Introduction

I lie iut hors intention ah t his paper is noi to prov ide a

coniplete soltitioii to the coinplictted stability pi ohicin ut

ships iii follos iiig sea. hut iatiuet to ',huw stone soiuuiions tu the siiiiplutie&l lineariied piohleiiu. I lue pitubleiti .is such is cLissical ulkt has pieviuuisly been iieaicd by niiuiy authors.

.ifl utitiodiictioii tiuc eu.iei

is ieteiicd to (Jill)) Ill.

li

is easil' seen that the dyniiiiic cquu.itioil ut

ailing iii

lulltusiiìg sea s gi' cui [he luim.

f 2PI) )

g1 i (Pt g2t i t '»

q (

I i)

(i)

Iueic q())is a juitu

11h11 tl.iITilulig lei Ill. g1ii is,) iioui

litic,ii

'esitiiiulg ii.! ill .iuitl Cj i) is J_itIc it) Ilk: chl Linissii fiLJ( i'/tall L t)uIilhiiii1. .111th is 1iCIi .i .1 t.iiut,ii.ti, t ,.ulissi.iii

sit)Lhi.istiL piaLu.ss hi iliis Lise tille IS hoi SpCL .uiI

iliicic

sted ti tiat tite stuliutioii ol l.q I t looks like. i:itiiei I ii is stable ui

noi atid

ii it is. iii what sense t is stable

I his papel ssill tiiii be &le.iling with the hi,ieait,ed lrm

Il Iqi ft

2pw

f

iltg,(t))P-()

intl will gite sutil Lient orudiiians ttìai [tie StIlilililil ti) ibis

ut1ui,iiititl is

stable. iogether with a

LhisCLISSI))ui iii he LIIIILtCIui stability LI lelia

'

tui et disctussioiu al tue ieinu g 4f i' )v ill he given iii idduilin. Note thai il itie Saliutut)Fu al is iiiust,ihlc hic sahiiuii ol the equiivaleuit Iqi lus riot necessarily irustible. On the other hand: il the soluuiion ut I-q)2) is si,ihle does riot grow so rituich thai the non-linear lei ifl5 ut Iqi I) tiaVe Lu be taken into account, arid hcnee the

slnliun ¡s said lu he stable. So: a sufficient condiiuon loi stability uf Ihe solution tif Fq(2) is also a suilicicitt cuiidi-lion loi stability of the sulul ion uf EqI I).

I ti siinphfy lIte discussion in the lollowing chaptet s. I-q(2) is icwi liten:

f2pb+(lfl(r))çb

1) (3) 15

TCH UTE

Laboratorium voor

Scheepshydromechanlca

kohle?

Mekeweg 2, 2628 CD Deft

1

O1-78Th--Fc 01G -81C

where r=i0t í(r)=g2(r/w0) and the difkrentìation is

taken with respect to T.

2. Definition of stability

To give a proper delunittun of stability of ships

in waves has been tinund tobe raiherdiffienit: the lit-st session of the t I ruernational Conference on Stability ut Ships and Ocean Vehieles in Glasgow in March 1975 uRefI2/) dealt with t his suhject . ,uiid

the conclusion ol

lIte papel s presented and of the discussion was that one has nut yet

ioutid a delunition ilkut is both sutisIaetur us such md

usehul tor practical purposes. lt is not the author's

inten-tion with this paper to introduce such a definiinten-tion, hut

rather a brief discussion of the suhjecl.

In statistical terms the best way of detiiiing stability oía ship is by introducing the probability, P. that the ship will

capoLe within a predicted time interval and define the ship It) he stable when P becomes less than a given probability level. e. The time interval should in this case he at least tite

estimated lifetime of the ship and P calculated according to the long-term distribution of rolling. Ihis definition 01 stability is very tractable fiom any point at view. escept

It-onu the user's: ills easily seen that this leads Io gteat

CollI-putiunal efforts due to the fact that both the probability

dislributioiì of

(t) (see Eq( I» and of the individual

maxima arc unknown.

KoLin /3/ has in a paper from l96 made a surey uf the

different stability criteria applied to Eq(3). The one we aie

going to deal with here:

Almost sure asymptotic stability'

is not bused on the variable as such. hut on a

I.yaptunov-tunetion in the form:

V

x'-A-x

₍₄₎

where XT = I,

I and A isa positive definite matrix. 'lhe

Norwegian Maritime Research No. 2/1976

variable 0 is said to be almost surely asymptotically stable

il Prob (V0) las

T

In chapter 4 it will he shown tlìa the deviation Irtun a exponential growth or decay ot V will he:

exp(

-

j l-í(rHdt-E(Il-fI))

which rapidly approaches I, and hence stability according

io this criterion will he moie restrictive than the one given previously, and sufficient condii ions of stability according to this criterion will he of great interest.

Before leaving this subject it should he mentioned t hai other stability criteria have been regarded in the liter'aiuie without the same positive results as for the one pointed i)lit

hei e. For an introduction, the reader is referred to Ko,ui or Arraiatnanr /4/.

3. Previous work on stability of Eq.(3).

'['he discussion of stahrlity of what is known

as the Ito

eqIi

ilion

4

F(r) ' X

_(f

(5)

into which Fq.( I) - (';q.( 3) can he ti'anstormed. started in

1959 by Saniuels & Eringen /5/ with F(r)

given as a

white-noise process. '['his work was continued by Sairitiels

in a papei _{¡(r/ where he presented: a genei cal (hein y} l

stability ol r indorir lrrìcar systems. In I%2 ('aiighey &

I )ines /7/ showed

t hat when F (r) of one single

time-dependent term. sshich is a white- noise process, the pinihleni can 1w treated by the Fokker - Planck equation (st'C

Ref. //f.

I lic work on F (r) whit e e citat ion was e xtended tin noii while-noise by S.inìizels /9/ arid Ko/ifl /10/ who. in-tl epc irdciit I y. de s- eloped a sii lilt. lent condition tif -almost

sure asymptotic stability of I ql t) I heir results arc given in I-rg. 1. where for oui pii i pose, orn y the region for P< I

rs of any interest

Kiniin 's result s were gr eatly extended by Airai atn.int /4/ in 1967 arid his results ;ime gis-en in I-rg (t) together with

Norwegian Maritime Re.cean-h No. 2/1976

Tr

Fig i

_{The results of KOZIN, ARIARATNAM}

and

CAUGHEY & al.

The area between the

curve and theq -axes indicates the

stabi-ity domain

I (r

6

l.0

cAiicirr-;Y &

ii, 196e.

(Ref/Il/i

XNF/\NTF,,.1R68

(Ref./12/)

Present

csu1t.n Irons the

inethci qiven

by KOZIN 8 si, 1973

(Ref./14/)

Fig. 2, The results of CAUGHEY & al., INFANTE

and from the present paper. The

area

be-tween the curve and the q-axes indicates

the stability-domain

4. Contribution by Kozin & Wu

I he work in this field seems tempoi'arrly _{to have been} brought to an. end by Koimn & Wur /14/in_{a paper form 1973,}

t'herr work will he referred rn _{stime detail, due to the fact}

that the author of this paper will, by the use of their

method, extend their results for slightly damped _systenm,

Koirn & Wiu hase their work on Fq(3). written in the lint rin:

X' -

(A +

F(r))- X

(9)

61

lu

1.0 KQ7,IN, 1963 _(Ref./1O/) AP IArtATPAJI 1967

_(Ref./4/)

0.5 xxi c'AucaiEy 8 1,

_{1965 tRer./1l)}

Koiin's. All these results are based inn the

Gronwall-Hell-man appioxr mat ion, which gives an nneq rial ii y foi the

solution of the integral-equation, replacing Eq .( 3). I-or P <T t) Artai'atnarlr's result becomes: 0 (r) rs almost surely

'asymptotically stable it:

i <'.Jiri2'p

16)

where o=virt(r))

In _{I 9M ('arighey & Gray / li / introduced i .yapunov's}

direct method to the problem, introducing the I .yapunov-fu n ciii m

V-(p+ 102)fl,b-0

ji2

(7)

.-\pplyurig a _{t (nenni erri on h(nundness tnt qiialiaric fini ms.} pros-ed ru lie paper-. ihe _{ciniuld extend Ariaiatnam's tlateri} results. In Fig. tI) then tesnilts arc shown in addition tin the pie Vitilts Ieslilts.

In 196$ Inlante /12/ applied a clever approach based upon properties of quadratic hrms (introduced by Cesan /13/) and extended the results to:

( r) is almost surely asymptotically stable if:

(3<2p (s)

which is shown in ['ig. (2). together with ('aughey & Grey's

i estilt s-.

o I.0

110111 this _tile) pi

ocd that:

¿(I-2P)

_{kf2I)klkII). 2p}

where

I?

uid

P 1-.

_T

I I

_{C)p(_f2/2iJ2)J)}

k k i)

CS)) )-k/12)

li

iii

oduc ing this into uiiequality

_{(lO) one gel s.}

III

-

21y

i +

C)(f) (- i2/2u)J

together

with the reslrictioll

k <2p

₍₁₉₎

tJnequality 419) includes thatk issmall, and hence (18) can he written:

L-°L

r

-i

o

_J

(20)

where

the first

_terms

_{in a series-expansion of unequality}

(18) ai

taken into

account.

lt is then easily seen

thatliii:

(J

(21) X is almost_{asymptotically stable. 1 his is found by puiting k} eqii;il toLCI) in (20).

_{A careful calculation horn unequalily}

(lt) shows that for p< I

hasgot its maximum fork=Ù,und

henceíp isa

_{good approximation for the stability}

cillerion also t'or slightly non-lightly

_{damped systems.}

5. Additional Topics

Soute work has

_{been completed}

_{on non-linear} equa-louis; withotut bringing

_{any interesting results....he}

etitid-tion

2 ø'+( 1 +f(r))

-4- ) -(4

(224

s liete g4 4 _{) is strictlymonotonicallyincreasing noii-liiteii}

function in ,

_{has been examined by both Inlante}

_{/12/ and}

1<

o,'in & Wu

_{/14/, gi ing as result that the condiI}

ion of

stahilit) ofq. according t) this_{equat ton. IS the same asloi}

I-q(3), which should

_{have been expected beforehand.}

lxcept tor this, flO work

seems lo have

_{been done in this}

held.

Au _{equation of still more interest has}

been treated by

Koiin

/10/, Infante /12/ and

Mehr- & Wang 11.6/:

(A±F(TftX+h(r)

₍₂₃₎

dr

hich, except forthe h (r)_{term, coincides with Eqt9(. [his} equation describes in practice the

_{roltiiig of a ship iii}

qtiaiteriflg sea, having an Input

_{( h r ))}

_together

with the

_{heave/toll coupling (F( T )).}

_{Unfortunately,}

this equation is not_{easy to handle, and hence the results} ate not very interesting. lfboth fi h(r) fi

_{and II F (r)}

I are hounded then there can he found some stability conditions

for this equation which_{are given by the authors quoted, to} shonì the reader is referred for _{more details.}

A recent paper by Price /16/ deals with this problem and one of the conclusions is that the stability of the solution of

Eq(23) does not depend

_{on h(r), and hence that the}

criterion of stability of the solution_{of Eq(3) is valid in this}

case too, which should have been

_{expected from the}

classical theory of stability and _{front Rosenberg's work}

/17/ on

the

Mathieu equation.

(17)

Norwegian Muriiu,ne Research

No. 2/197Ó

l:k '(f(rlt(r)J)

l'hts piool is based

_{on a}

_{I .yapunov-ttinct ion:}

V- xTL'x

(Il)

L is it this stage &iiiteiciiiìiiied. but icstl iLICI.lR) he posillc

definite. I)utu ng the c,ulc alioli L is detcuinutued logetlucu

titi j paiaiiielei k. o (uhu:

?''

124

(ici e k hecanies.

k .( ₂ t. _{- :1' )I}

I))

I

and k us chiuscit licei) uiIuuiu he consti hits

_{-O, I)}

_'o

SoI.iiig this

uuueujiiulity. unte gels:

(tI.

_{\(t)(ej)(( 2p}

luJlIi)IA S

1ul tl

t 00

1k

III ),dl

.j:( kf()J)

and hence

X is *luiuost sillely_{asy nÌjRulicaIl) st:ihle it;}

2p t DL ( 1k -f(î)j ):: t)

_(IS)

hich can he

piu

_{ved to}

_{coincide with}

unequal ut y (Il)).

liom

luis uiìequiality J is

_{I)piirnh/ed by varying k}

lhe uthous

havegiven an_{example with t(r) Gaussian}

distithuled

with teto

mean iiid one with a

_{narrow-banded}

process. Stiangely enough, hey have not

extended thcii

isuIts

lo

_{lightly damped systems,}

_{and this will he done}

heu e.

JIJI ( i.iUSSij(I variables:

'iI

exp(-12/202)dl

heie

A

Among addiiioniI

topics sorth itienioning, is an

ilteresting result given by (',raek / l/. I he present

piescii-tat iou _{is due io Aiiaratniim /4/. Assume i (r J u Fq( 3 Is}

given as:

e1

F (r)

f 74)

where i«l and

I

(r) is

normalized accoiding it) for

IflsIilhlCC E(E2) I Rewriting Eq(3):

where:

/ .exp( .pr)

(r)

(25)

o2 =(l-p2 )_I/2

= e

and solving this equation by the method of sloWly varying phase and amplitude (see Ref./ 19/), one finds thai the

solti-lion of I-q(3) is stable ¡n mean square if:

4P(l-p2i>sf(2VïT,

(26)

where S) is the spectrum of f.

Foi lightly damped systems this simplifies to:

>TfS1t2

(27)

tuch. as foi' the Mathieu equation. coirelaics the stability I 1f i he soliO ion to the amplitude of t he input ¿it t he double (Il

the characteristic frequency.

6. Concluding Remarks

'\'. his been shown iii t his paper. the siahility (uf_{a ship} iii following sea is dependent on both the roll-damping_¿irid on the spectrum of the mnetacentric height. when i'egaided

slit (mai y stochastic proCess

l'he al iliost suie asymptotic staliilit _{seems to depend on}

the total valiance oh' the metacenti ic height and that a

stil licicimt coud it ion of stability of lightly damped systems

can he written in the fornì:

P>ktJ

(28)

wheme K

_{seems to he the}

_{best value found.}

When trying to apply condition (28)to a ship in service. OflC

t

has to go through a rather complicated calculation: first the

characteristic frequency in rolling and the _{linear damping} ii this frequency has tobe found, either ('rom_experiments

or by calculation by a ship-motion

_{computer-program.} l'he next step is to calculate the statistical transfer-f'unction li orn the waves W the (dy mimic part of t hie nictacentric

height 'the main effects on the transfer_{-function are given}

by: the variation of the water-plane area, due to the

not-parallel form foie and aft,

ne st the variation tuf the

posmtiun uf the centre of buoyancy. which has a tendency io

cancel ou, the first ei'kct, and the _{efTcct of the dynìnumc} pressure, which works in the sante direction _{as (loes the}

change oh' water-plane_area

't he calctiliution of the transfer -functit 'n can he done by computing both the dynamic and the static forces acting on

the ship from the sea when

_{running in reguilar waves.}

together with their local points of

attack From this,

together with information ¿ihomit the centre of' gravity and the static montent tuf inertia of the actual waler-plane aiea,

uric is ¿ihie tti calcuilte the amplitude of _{(the vai mable pa; t}

Norwegian Marilime Research

No. 2/1Q76

18

(lithe) mnetacentniC height. and hence the traiust'ei fuuriction l'i'omtm t he svaves to the ritetacerut r ic height ( 'omiuhining i his

htiitetituit with the save-'pecti tira in the cniiuiiultrl way described in both Ref/2(t/ und Ret/2 I I. one is able to calculate t he spect muni uI f' (i' ) (in Fq.( 3)). which in titin

gives the vanianLe. O. when integrated uvei'alllreqtiericies. As far as has been hmoiight to the author's kruttwtedge. no

Orte has yet carried ont such a laborious, hut stiaightfor.

wand ealcmilit ion , alt homigh it seems to he the most att rae-(ive way of' solving ube problem.

Another point rs that the mull-damping is often mIlic

non-linear than non-linear, If

_{(inc tries tul apply the method oh} Flquivalent t _ineani/atlon» to the non- linear equation

t see Ref/2O/ (Ir Rei/2 I/I, (une iifl5 irittu the tiuiuhtein t ti.tt t) is not Stationary and that its statistical distribution is quite

unknown, SO ¡t seems thai no adequate hincariialion is possible, one is then left with the possibility tif_{doing sorne}

numerical simulation on the non-Immicai equatitimi. tui get sorne information. I will just point out a way tut doing this: From a nunier'ical simuibitiuun tuf the variahleg2(l ) t in Fq( I J)

it _{is possible ('mont the Fq( I J to integrate (I J trout somiie}

properly chosen nit mal conditions. Ry es.inuining I (

2(i)

J

during i he i ntegma t mon one can find out mf' thieie is a n y t mend

to increase or decrease t: ( 2) and omit

then has iii

indication as to whcthen(t Jis stable inumtc,u'ì sqtJui u or riot

'l'his seems tobe the way lo search iftune wishes tui stul e the meal ship rolling pmohlerniii ttullostirig sea. cemt though suiuuuilition is a mather coinpuuremtiinc _{ctmisiiiiiuiig pIou}

Jume,

References

/1/ ( R I 1 . ( ). : Beitrag iii den l'rtthlcuu; Ici Sicher heut des Schitles unì Seegang

Schiff und I talen 1961, I-i .6

/2/ l'meprmnt. Iriteinatuonal ('onference tumi _.Siahmliiv

tif

Ships ¿utiil ocean Vehicles.

Univ. uf Stmathclytle, (dasgow March l75.

/3/ KO/.IN, l'.: A Survey tut' Stability ot Stuichustic

sy-stems, Autouulat ea . \' ul . S pp t)S II 2, 1969.

/4! ARIARA [NAM

. S. 1, : l)yiiuniic Stability ut' a (

'ol-u rn ii t'ol-unde r Ra ndom I Oath ing. I'ioc , t ,l'I nl , ('&uii f. uil

Dynamic Stability of Struuctmuues. _{pp 267-284.} Peu'gamon Press. NY, 1967.

/5/ SAMUELS. J.C. und [RINGEN. AC

: On Stoclìus-tic Linear' Systems.

MIr J. Math, 11hys. 38. 83 (1959).

/6/ SA M UEtS. J.C,: On the Srahìtity of

_Ramidoni Sy-stems and the Stahmlizatuon of t)eternimnmctic SySy-stems

with Random Noise.

J. Acouist. Soc. Am. 32, 594 1960).

/7/ CAIJ(IIJFY. T,K. and DIENES. J.K.: 'rhe

Hehiuvi-oui oil mear Systems with Random Pai'anietrmc Exci-tat ron.

J. Math, Phys.. vol 41. pp (W)-3 IO. 1*2.

/8/ ('OX, DR. and MIt I ER. HI).: The Theory

of'Siuc-hastie Processes.

Mcthiueti & ('o ltd. london. 1970,

/9/ SA M U ELS. J.C,: 'l'heony of Stochastic _Linear

Ss'-stemlis ss uth Gaussian l'ararnetrie Variations, J. Acou. Soc,, vol 33. pp I 782-1786. 1961,

I 10/ KOï,I N, l'a: On Almost Sure Stability of

_linear

Sy-sterns wmttu Random ( 'oetìit'ients.

J. Math. I'hys.. VIlI 42. pp S9-fi7, 1963,

.

III' ('AtI(iIlI\.

I k .iiìd ()R/\'

:.li

Ji.: ii ¡li..' '\)-lutist iiic Stal')ility III I in..'.ii I) tl,IIIIIL ysicIII'. tsiilu

St LII i)tl ( (leiht eill'.

J. AjifIl. MtL Ê.. \oI.

_{2. No. 2. Pute lUi)}

/121 INI

Nl I

I I ( )i _{iii).' Siahiht} III OIIlL I lit'.)

Non AlIOlIOlÌlti(l' R.kl.li Sysleiii'

J AppL

_sletli

_{. Vol t}

_Nt

_t.

1.i

l)f?

/ I

/ ('IS.\I.l

_I

_,iljitti..

Itt_ll.Il.)lII .tt1.t

tilitti!iI

lI t)lIIL'llì'. _{II (>1 tJIil.ii ) I)iIi11e1Ll,ll I}

-

_{ci Ltg l9.}

II-il K( )i.IN. }

,iuid

_{('.1,: (hi}

_lii'

_{i.ihilt oil men}

Stochisi i. l)lt,'eiiti.m(

_l.qtiilttii'

liS! t1i.tlR. (' I. iii

_{\VAN(. l'K.('}

_{i)i'.tti'.ioii liti d}

P.lPCi h) ('aiighc

_{liti (ii .i}

J. AppI. N4ch.

_{. Vil. H, NI) I. Miii. ltkit.}

Some considerations on shaft alignment

of marine

_shaftings

by

Odd C. Larsen, M.Sc., Del norske Ventas

Abtratt

reiiess et _{e ni the miliire d}

ant eterzimI factors that _{inh1unce the}

shalt alignmuucmmi in s'rs ice is

gi'.en. Ihe effects of draught alterations, thermal

in-Hucints...&teiutrit propeller thrust,_{thrust hearing Iiltiiig and bearing flesibulity}

are distussed. S

' humilier aspe&ts of shalt ulignnicmit are also dealt s,iih, such as the e , _{hctsseeii requircilients (ii aligiunicut sersus ss hirling vibrations,}

(lit' imnpor(alhe (it (olusl(lcIiiig

uiugular dcflel lumi between shalt and aft stern tube hearing, amid align1111111 iii duck

lt is tomnluded that Ihitsi' _{(actors miiust lie takemi into}

consideration ¡mu the design a iialysis il wie is tu cilsure

aim approach lo I he concept et opt ini nui alignment

-1. Introduction

i lit.

_{Itf)til'.IIII 'hut '.}

'.11_III ICIIL'.Clit'. titIc t! IL' Its)

IIIII)1)II.IIIT Itl,it liti . '.t.tit. cspccuauly

kim '.tiigle '.LiCVi

slIu1t'. _{LI lIt.1! ,i'. li i'. t)}

lIt lIlium .Itiml iii th .ilt1t I itt III'.

iCl',tliI

II I', _{C'.'tt..iliI,tl}

_litt)

I _IluglI

'tsttIII

_cli.11II!Ii I',

otit,uttit,tl i )itc 'Al'1 _{IiI CII'.IIIL' i!'.}

_{illiìetIllil}

s lt _c'.t,ihlistt (ht i sit,! _{u! lImilieiii}

I tti.

_{.uJ)7ihleitioli iti tite ulF LIII t C}

lilgItIttutil ilicui .

tu

tilt lC'.lgtl o! p 1titIlsI(Iii shit

s, .leimis h.'. guienully ht't'it .)tCCftti.'i.i I'. I _{', ,IIII,)hulC Ittil ¡ti Okt.11}

V. hut i'. tutelI

I Lili iCI! lii ,is _{.iit ulj)lttlitttli 511,111 aligililicimm}

Stmicily. uii lIpliillIutIt

stillt

_{.illgillttctlm itiu he}

deilmued t'. an .11 allgt.ItICIIt ill !hc.II 111g'._littititlict

hi_ai iig sputi'. ..Ii_i with spucliluui pl 1511101'. (t ills_i

_{t diluye lit}

I given mild dIlL e lift,') ti. tilt, li Ilililt.I tu 111)1 lit.11 'Ail kilig l_omi(!IiltumlS_ditsIlids

I piI9h1'i lii'.) IltItiluttI o! ltc.ti_{tug lomd'. .t'. 'A ,11 I'.}

_itcptihlc

shah sim esses uit! uleilcu_tiittis

I lie lliii(l,ltulLuIi,II', III th l,mit LIII LC uligilitudlti tltm_{y WIll u.0 he tes icttud licue Is}

lIlly

p,l[ìCI lid iIi c.utl _{plIhtIslILih till}

_{III'. sllhJLd(. cg. /lj}

k liti lILI _{II lulls tu tIlfulliel}

it ugt .uiiIIiiI,''. _{lIli. L lICeI} iici_'Ii lutti Ill 1lit.' L.II¼

_{tii,iiittii tul liti}

_lll

It':icl lois ,Iilul slì.itt '.iit.s'uL''.

,iiiii

IIt.'lldCIiu.Il litt _{spcu_ihei.I sttullmttg II Ii1gL} imiu.'uiIs. u

_{g /2!}

i lful l'kl( i'

_{W.(i : A Stability Ai.ilysusoi ¡lue Roil Momuni}

nl .1 _{iIii) ill ail Im iegiuiui ciiway}

Sit. _{ilupsh. Ping.. VIII 22. Nu 247. i)7't}

1171 _{1<1 )Si.Niiil«i. R ri.i . Um the Stihuliiy tI}

u Nluiihimlcul

NOII ,\IIII.tIiI)Iilt)ii'. Syiu_'mti. 2 (J S. I nrlgm. oui Appi

M Ii Il

t ins _{oh Mmcii. I)54}

lUll (RAliF.. i1W.( I

: lug Aucliiv. ItU(u

119/ NAVI l'li,

_{Ail.: PdiliiIh,Itiulii Mclhods}

Jitiimi Wiley & Sons, N.Y. 1973.

/21)! liN, Y .I'

. Pii.Ii)iiI)lliStmL

i tId.ui), tui Stu tictiim.ui l)

na-umile'.. McGraw-hill, N.Y. l%7.

/211 l'R lCl, W.(

. amid IIISI lOP, R. Ei).: Puohahilislic

I heomy nl Ship_{i)ynamìmmcs.}

('hapinan and H all, I .ouidoii, i 974.

ii

_{l's ttuuJuy c&tuiinioni puai_lmce (t)}

L'iIIi)i (lilt Sohle kund oh

sfì,uit (IhIgmiunemIl cticlii(l(lOniS_{fuji at least insIill.ltul,uuus}

L'lullì-pmlsuulg mediuctmon geaIs. Ihe basIs _{and extent ai these} euldill,lI omis, however, change _{ensiderably fromm sh l[) Iti} shIp tiepentidmit on

experience, accepted practice (lid usai-table 1.1,11(1. A conveniitunìiI (iligillilemIt (Ilualysis (IstItill) comuipmmses only the evaluation of the _{statue.. shafting con} dit oit at an average dIauight. Ihus, possible changes in the

uimgmuniient _{emicotiritemed d&uming lunnmng condmtmon}

tom valions iluiiughts ate mint consmdemed. _{ihere (lFd a niimnhem nl} dmiieient exteuniil mulitienices that may cause changes In the

ilmgnmmuemìi mn semvmce. lt _{is essential that one is aware ut}

ihese iactot's and that they aie taken unit) consideration _in

the design analysis as carefully as possible if _{one is ti)}

approach the concept tut ooptmmulnu _almgnmeni

The scope of this

_{paper is to discuss some of}

thsc

lactois with special emphasis_{on geared installations,}

on

time hiisi'.

_{of experience gained}

_lrotiu

Iheometmciil _anti

e \pCIifllei)t.11 investigai inns cammmed _{OIt un recent seam}

s.

wmth pamticulam icleiciuce _{ti) large tiiihune tankers}

(2$-41.5(10111 .dw. ). _{i he Comliluent's will, hoss ever, un most cases}

.Iiso he relevant ton smnillen ships. ( )nl _{the alignment olthe} rmi.uitì shalting system ms eonsudeied

Al iriieglU?l if muruti,pie R('searl /i A'u. 2//971')