manouvering notes

8/4/2019 Manouvering Notes

1/68

CONTENTS

l.r. INT1~'ODUCTILO'.N"i'.''''''''''III''''''''II'''''''';fi'''iI!llf! '!! ! I I ! ' , . "!!!!l!.'I'''!lItI'!''1 i & i oI iili ,.. , ii ii r.,!!!!! 'I ! I!"'_" ii ,. 12~ GENERAL EQ. UATIONS 'OF M'OT I'OI, . .. ' . .. .. .~. ,, ,.. ' .. 0 ,0 .,0' 0, ,., ,.' .~ .3.3. S 'YS1"Er)JIS OFAXE .S . iifli i1,.. ""...... "'.''I!'.'''!' ,., ' ! ! ! ! . . . . .. I ji.i.i,ii. i ii~ . _ i i ii , ""~ ., ,,,, ..... " ! I I ' '!' ! '! I~ ii . oIi iit i .. oi, _",44 - . DEVELOPMENT OF~{A.NOEUVRING EQUA TION S., ..................... ,.7

4.1 Relating Inert ial ,Ffaroe of R e fe re nce .............................. "'~ """" ...14.2 E - q ' I. H ! .t i, o n s o f Mo ti,o n , ". , , '... 114 .3 Linearisation o f TI 'lC E.qu l iUtQ l lSof Mouon ,., 205. ,i s OF THE !\.UNOEUVR!N'G EQUATION ,.,. _. n , 35

5 .1 lntrcduction "" - , ~ , ,,'.. , , 35.5.2 Stability o f lV fo tions ,,~ n ~."." " ,.. 365.3 Non D im en si,o naU sa tto n o f Eq ua tio ru :, and C C e f : f ic i e _ n t s . . . . . .. .. .. .. .. 48: S o . 4 Repr es en ta tie n of CD .nD '\l~ .u rf ac es ,& . f ixOO:F in s , .. , , .525.5 Control of t - . 1 G t iO D S . . . . . . " ,. . , , , . , , . , , , " 57

RE EHENCE 6f 'i : ' I I ', '_ I' .. r l 4 - I .. 'F . . , , ' , . . S , t . Ii i ii . , . .. .. ' l ! ! ! 1 1 !'!I!'''. i ..... , . ~ . . " .. ~ ~II! I! !oJ ! I I . ' . I ! ! ll rI I ,I,iillilillll,i,ti.", ~~!!!!I ! II!!!!ItI! Ii." ,.,.,., """ " ! ! -II!! !! !II""! ! !Ill' IF. I,. iI! - ,7


2/68

\\

1. INTRODUCTION~I'. 1 ,~i '

/"

"Manoeuvr ing" , "Seakeeping". hStabil i ty" '~Re:sista_ncen and "propulsion": fourdistinct subiects within the field of Nava[ Architecture, which ere generally consideredin depen den tly o f each ather, AJj -our, o f co urse a re aspects o f the gen era l dynamics o f m a rin evehicles and w hilst the q ua lita tive d istin ctlo n be tw ee n the ca te go rie s is elear, 5:0 to o is 1 .1obvious that there exists a s l.ro n g quanritat lve in te rdependence .

The "Dynamics of nu~dsystems" is n sub.icc't wbieh is still, in its infancy even 1l1.0llghits been talked abou t C O ' t several hu nd red yearB . or tZr'ler~is srI! I n o reliable ~nd accuratem eans o f predlcl ing rh e performance of any but the sim plest ,(and usually ideallsed) fluidsy stem s, H i stc ric all y t h e onl,}, way fOrYl l' ard i 'n the i : l i n a J y s i s ;O I f a very c o m p l e x f l u i d system(such I : 1 S our m arine vehicle dynam ics ) was 1 1 1 1 ) reduee the problem to m ore manageab lepro por i ion s. In naval a rc hite ctu re ~ 1 'I;cevelopmeeu o f four l ar ge ly i nd e pe o de n t seriallsa:lionsgiven above was necessary, expedient and ~n "icw of I,bi'll qualitative distinction among them o tio ns co nce rn ed , lo gica l.

W ithin each of m e ' four topics nUlilhemfllk.a[ reL atio nships ha ve be e! '! d evelo pe d w hichmodel the physical system as srmplified). It would be nice if\vc. could just add 'thepredictions from eaeh cR legmy together t . I i ) define th e genera! dynamics of marine vehicle.Justificenon of this would demand thet the responses ineach category were truly ~inear(in thernathematieal sense), However, e ' \ ! ' , e n [hough it t is net trtlly' l b - e case it doesn't stop us fromapplying the laws of superposi t ient Such superposition, of course, is pe rfo rm ed , under __.engi f leer ing J ieense" an d de-m anus careful engineering judgement.

In 11113 course ~we will be eoncerned witbmanoeuvring and seakeeping,MANOEUVRING concerns the controlled changeof direction of motion of a marine vehiclea nd itssp ee d in Lhe sam e d irection , It in vo lves a n anaJ:Y Slso f all fo rce and momen t influencesand resulting motion parameters relevant to IDe manoeuvring space, For surface ships thismanoeuvring space is in the plane of the (assumed) stili water Interface, For submarines anddeeply submerged vessels it is t hr ee . .dimensional space.

SEAKEEPING concerns the respcnse of marine vehicle I to 'waves. Here the f orces andmoruent influences considered are those caused by the distortions of the fluid interface acting

j


3/68

-em th e hull o f the vesse l (w hl,eh is a ssum ed rig id . I. No mtez:act ion o f com ro l surfaces isco nsid ered in the pureseakeeplag topic. -

The need to have som e ,an a ly ticaW c a r p . a J b m t y 1[1 the f ie ld ,o f manoeuvring andseakeeping 1 . : ; ; ~8.thel"0bv i ons ,

Manoeuvr ing theory " J " , m allow us Ko:-~ A ss ess d ire cu on al stabil' ity o f c ra ft."'----- D e te rm i ne th e se ries o f contro] s ur fa ce s r eq u ir ed tel prov i de a certain manoeuvr ing

.capability and th e paw,er required to drive these with in the vessel's com 01 system.- D e sig n ship m o tio n c cn tr el ~ 1 's lc lD ! su ch a s a ura pilo t),

Ileakeeping theory will allow us 10: -- Pred ic t the un desirab le m o tio n responses ( i f iC lud ing ra t es o t' heave , p i tch, ro l l and

o ccu rre nce s o f p artia l hull emergence/submergence ~ead i ng '[0 s lamm ing I sl il i,p p in g uf water) .- M ake e stim a te s o f sh ip strue tu ra l lo ad in g.Sowe conclude thal both msnoeuvnng and seakeeping win involve the development

and use of equations gcverning the metlons of a rigid body_ These arc easy to deve lop withoutsimplifications or approximations but the oolul~on techniques employed still requireslmpl..ficarions and approximat ions . -

--

1


4/68

l A GENERAL EQUATIONS OF MOTIONAppHcBl t iOI1 of Ne:mon 'cS l.aw30f m otion results in the' l inked g en era l re la tio ns fo r

rig id b od )' mo tio n g iv,e n. by , eqwu ions (2. I) and 2 .2 ).In words: the total orce acting on, a body is l e q U a l to the {ate of change of its linear. -- -~-om en tum fin d th e l"eSullanl rnem e nt o f the appH e d f ( l ' r o r : . a ctin g o n the body equa l s the rate ofF = d {mIT )- d t- d hG=-dt

(2.q- I J.~t 'N~'" ,...h ...1 1 1 - " " ' I D I . \ i ' . . . . "'0"

~'-t C .tv1 (2.2)T h is is 1 : 1 5 f !l!.r as w e can go in ~hl;!gent.-ral s t a ~ . e m e n E o f th e equa t ions o f mot ion . T o

usefully expand upon the eornpeneat jerrns we need to ~dOpi~.nsxes system suited to th e type(1J ma t i 0 11 0 r !he type (1f rigid bod y we are it e r e st e d In.


5/68

3,.SYSTEM OF AXES -A s,hip at s e a , er a bod.y m O ' Y . m g in , S I .fluid, i s : ailowedte move and m a n y times doesm ove, in a ll six d eg rees o f fre ed om i.e, tra nsla tio n a lo ng 'th re e onhogona l axe s a nd ro ta tio nabo ut eaeh o f'the L lU "CCxes. [t is therefore necessary t cheese an a xe s s ys tem to describe si xmo t jo n fre ed om s and theehoiee sho uld be ODC~ wh i,c h i s mest eenvenicnr fo r th e d eve lo pm en tof ' the l11011on ana lysi s ,

. . . .

,_ ,

Con sid er a rig ht-ha nd ed inertlsl frrul1e ;QLu:&r nn ce OXYZ and a typical (r:igid) marinec--- .vehicle in 're la tio n to tb is, rig 3.1 illu stra te s tha t a ltho ug h [h e lin ea r : ln d th e angular veloci t iesQ f the vehicle could be described with respect to the axes system. there is no means ofs p e c i f y i n g th e orientation o r ~he vehicle a t aI!Y L im e' d l l l r . . m s its m o tio n. Furthermore, th ecOITI ]1 {H 'l l; ;n lSo f l inear and aagu lar velocities \ ' 1 . r i t h referen ce to th e inertial It me wil l g (.l n~ raUyno t bea r direct re l evance to Id en tifiable m otlcn om ponem s onb n rd u ie vesse l (i.c, surge,swaYI heave '; r o ll , p it ch 8Jld yaw) .

The em ploym en t o f a system o f body axes (l.e , axes llxed w ilbin the rig id v eh ic le ) ispro bably the mos t obvio us m eans o f overcom ing th e above probl.enl. In fig ,,3.2 we see a righthanded body axes system CXyz.\'I.~th origin a t the centre o f ~ass 0f tile vehicle. II ha s ex- ~ . . . . _ . . .ly in g in aft-to -fo re d ire clio n-;C y lyiTlg in the port 10 starboard ditectionvand Cz ' is , p a ss in gdown tl ll'O ll gh t J1 C vessel. The identification of motion eosnponents of the vehicle may now bedirectly related to the directions ef the body axes system I'surge along ex SW:lY ,a long Cy"heave along Cz, rull .abcut Cx~ pitch about Cy and Y3\ about Cz) and the' orientation angles(as we-shall see) are those between the axes in 'OXYZ' 3:rtd"Cxyz ' ,

~A less, obvious way of defining the motion o r a marine. vehicle in 'certain cases is bymeans of an intermediate set of equiiibrium axes, These could be defined where tho vehicle is\....--- . . . . ._performing some steady known reference morien and where the perturbations form this steadymotion arc of interest, Fig 33 illustrates ~ right-handed equilibrium axes system. Ax ~y'z.U sua lly th is is parallel to O XY Z , 'w ith A x"''! and OXY in the s ame plane. The "equilibrium_vehicle" (and axes) performs a steady linear motion h'l the Ax~ direction, if only smallperturbation of vehi cle is considered, then the kinematics, m ay be considered by the three~iuear a nd three angllJ le r m o tien s referred to ! .he l e q u H . i b r 1 u m a ?:e s: sy ste m w ith ou t e xp licitreference to the body ,~xessystem.


6/68

From Our description of ' 1 : l J 1 edis inc .~opics ofMrulOeuvring and Seakeeping in s e c t i o n1,' yo u must expectthat body axes would be empioy,ed for manoeuvring (where ]31];e'changesin m otions and d i rections are .im plied ],. W hereas equim~riun axes waufd be m ore, suited fo rseakeeping s tud ie s : ( since : we me .i -n te :r es~ed .in ]Jertu;r;1:tallon to motion and loading. caused byw av es), T Ills in d ee d. is .b e ca se , a s you \wU see in ' m e li te ra tu re . H o w ev,e r, w e sha ll a lso seethat ill order to be able [0 do anyth ing : u s e r u r m th genera l m anoeuvring eq uations. it isn ecessa ry to red uce them Z ,O pertu rba tio n eq ua tio ns; 1 0 v iew o f th is the cho ice betw een bodyand equilibrium axes should bea matter ofL~I.as[ei", aJf.boughtradi t ion usually d icta te s theusage desc ribe d.

lit,


7/68

0'.r-~-----_,._ y. I

J'

I

Zo

~ . . . . . , . _ xc

y I1z

1--

z ,,, I / . . . . .L I . ,I

z.oi

~ - - ~ _ . x/:Y

z

x

Fig 4 .1 Seq uen ce o f ro ta tio n through"0 RU~NTAn.ON ANGLES" derjrdog[he orientation of the body axes w.r.t,th e in ertia l fra m e

6


8/68

~.DEVELOP"MENT OFMANOEUVRING EQUATION4.. 1 RELATING INITIAL FRAlIE OF REFERENCE

H avin g decid ed upon the body axes :sys~em in the con text o f m a no euvd ng it will beuseful 'to examine 'Lhe I 'C la ti cmsh~p , between menons as viewed by:. -

(i) a stationary observer fi.e. w.r.t, an inertial firame:)(ii) an observer m oving with t he v eh ic le (i.e, W.,F' .t b od y fram e o fre fe re nc e)

Figure 3.2 H Iu strales 'I,be t r ; r u d iliona I etnangen:J lcnt of [he body a nd in ertia l r~.a:m e s.nzpo in ts vertica lly down \,Y ilh O X and OV horizon~ .M, a n d o fte n s up po se d lying in ca lm water"

v. surface Cxz eclneides with the plane 0 symmelry. ex}, is ho ri zon ll J.l w it hi n the vessel inequilibrium with vectors U &. K . ill'ld i .1 & . ~ being unit vectors in the directors 'OX, OY &Q Z and ex. Cy & Cz respectively, Let l I. IS I Ll OW determine the relat ionship between th edeseripi io ,1 1o f a genera l vector in te rms or body Wild inert ial f rame cc-erd ina tes as defined inI!quilt~on (4.]) { tbis g en eral vecto r could W'eprrcS'Cnt~sp,~a,cerncnt1.velocity or acce le rat ion) .

- . . - - . .A;: [l!) I 4- b J +cnK "" i+b j+c k (4.1)The relat ionsh i between m e compon en t s a o , b e IC o &. a, b, c is a funetiOIl of

"orientaticn.angles" which describe the .o dematia :n o ftJl: ve body W.1.t. inertial axes,

These angular displacements when stricdy appl i ied in ilie fo l lowing order to the bodyaxes orientations, initially parallel to the meni a l . frame will result in . the actual orientation ofthe body axes.

L2 .

- a. " swing" to (he ' ac t 'l l la~ 'azimuth- a . "lilt'; [0 the ac tua l. e l eva t ion- a "heel" 'tQ the acnsal orienlation

The swing. lilt and heel are applied about. the intermediatepositiun ofthe Cz, Cy, andCx axis respect ively i n . accordance with ll1 e rig lu h an d corkscrew rule, The 'order is importantsince finite rotation In). diraensions is not cumulative and Qn'[ be vectorially described.


9/68

\~"

c a u Xua . ~= 3( 1 c o s \ - J ? + bf!1 s j n i . . IJb~:: ~,iI'ilsin"P + ba cos~F

a = (Ii co S'S - ell sinec, = II I si liB . e ll casE:)

( ' 1 . 1 , VI.~ ...-... b~P .. . ~ '. v-._ _ . - . . ' -CI ~.~ b= b b c os< l) .;. C I 5 ~ n ( ! : >c ~ ~br sincb - + C J cos$

Fig -1.1 II1l:remen!u! fr,wsHmnfltrnn nflt.-'ecwr ~\

-

-

01;,. "..


10/68

Figure 4.1 illustrates the nature cferientatien angles and effect and identifies theinitial, intermediate and f u J i ' ! l J I . o r i e n t 1 L U c l n of the body axes a s C X Q j 'o ,Z O , C X 1 Y l z o , l CXlYIZI &.C x yz r es pe ctiv ely .. Fig 4.2 illustrates 'IJheincremental development in the description o f v ec to rwhere the three stages are defined. by the equation 1(4.2). The geome t ry of Hg .;L2,indicateshow thesethree sta ge s m a y be described by the series of three' transfer matrices 1 '1 . 1 '2 T~which are applied as in equation (4.3).

.vector description in th e bod}1 and m en ia l axes systems as given in equation (4.4) (note t luueach o f UU! mat r i ces TI t 2, T J and the ir p rod ucts are I M agana l w uh the usua l i rr rp l l ca tions asfol l , loW8).

J) Ih c i nv ers e o r an or thogona l matrix J IS it s Lrn.1 l5POSC:2 ) th e ii i IIm s ofthe squares 0 r e lem en ts in any f10w .o r co ium n is un iy3)1 the sum. of (b e products of correspunding elements in any pair of .rDWS or columns

is zero

(4.2)

(4.3)

1 9


11/68

m T) Tl T r a ' l- ~II (4..4) ('I CQI ,s:in7oos@

sin' si i\(!h:i~4' ~osW ewelltiri~rilin8roli..,-,oo~q;ii,in([l'

-s,in ' ] ' 1COShi~,l(l)' coscos@

(l I .S)

T, = r T l',

[OO~lP c, 0 , S ,8..- sin lI.~cos ~-:f;lnB

cos t]J s in 9, s irn~) - si til q .t cos~'~ r l ' : l ' " P s . i I ' l l '9 sin q~+ co s q : l ! co s 4; ,COS,S sin ,d )

. " " " i" 9 C , 0 , " ," " , ' , " " ,i, n tFSin ( b ]sin "rfsin eH~O:Sd) - co s tF sin c ncos B cos ,d )

4.1.T :rOSl'f,fON ANI) ORI.ENTATION OF ,1 \ VEf--nCl~E-~

The orientation of a,marine vebic le js uniquely described by the orientation angles lj.e & (])a ppH e d in t.h e s tr ic t o rd e r p re vio 'U s l,y described, T he p ositio n o f a , ve hicle in sp ac e .m aybe def ined by the vector ,~ (see figure 3.2) given by equation (4.6). .

(4,6)

Consider now the velocity ofavebicle, 0 - mleID1:S of it s ccmponsats, w.r.t, each ofth.. axes systt:!1iTiSas given in e q ua tio n (4.1). W i tl 1 i: re fe rence to equa t ion (4..S) w e can s-ee tha tthe two sets of veloc]Ly componeats arerelated by equation. (4.8)

~ '~ _ I iiu~+J+wk (4.7)

(4 ..2)

\ 0 -


12/68

LOA,DING AND'KINEMATICS DESCRIPTION-W.,R~T,.BODYAXES

!"',. 'i.i~,i.

y. '

z


13/68

"LOADING AND KINEMA'TICS D'ESCRIPTIONW.R~T,.'8 ,'ODY AXES --. . . .

z


14/68

LOADING AND KINEMATICS DESCRIPTION.W R " T - ' B " ' O - " D " ' Y " " A ' X E " " S "iii I!!II ! I I " I_I,. I _ : _ ' / " , . ' , ' , ' I . -, _.

i r.~_ Q ~' ','~ ~ . . . . . - . . :: . ct....."HI- '"'6t..J '-"". , 0 1 , . , 1 ' : r"....i :' , . . T " _ . ; . " " " ! . W~U o S \ , p : .4~;: ' ,tt, Z ca-y~ (:?~~,k "'..... a,t(..o)


15/68

- D ~ ' I S T -: -U R B A N 'C '-E F ' O - R C E ' S ' A 'N D M - O M E N - T ' S -,....... , _ '~_ _ - :.i _ ...1 '. .. " . ' I , . , : ,I ._' 1_ _ ", ' - ' _ _ _ - , I ',._:. ... ". _ I r . I ~ ' . I _ ' , _ I ;' _;" _', ", _- _", ~(-

! ! !L. I, . '

y A M

z


16/68

. "

.,~, . .

y

u. ~

p ~.~~~.

I r

z.,

~:..1-~ t . < J - d


17/68

,4.1.2 ANGULAR VELOCITl(TIle in stan taneous m otion . o f a. mar rnne veh..k~e is specified completely by the linear

velocity U Iequation (4.7)] of NLscentre of mass and its angularvelocity fl., that may bedescribed in component form \,Y.rJ_ bodyaxes b o ) ' P Q and R - angularveloci t ies of ron, pi~chand yaw respectively as in equetion (4,.9). R d ah ng an gu j,a r velocity vector to the rates ofchange of orientation angles however . is mlbeJ a,l, l I lk"'1,wd because of ' their r a U l I : : r eumbersomedefin ition . A s we hlJ'ile said lh~ fin ite angl,es q /. e .& < D ' d e mo t ccn st it ut e 8 veetor qucmtity.The ra tes o f change o f these a n gle s d o . h,Ol,VCVCI. with the eempllcatien that the re l evan taxesof rotation arc not however mutually perpendicular,

-,~

n = Pi+ Q j; It k (4.9')(4.10)

It can be; seen that Q can be descr ibed , s in (4.10) Wi1CJ1ethe subscripted vectors relateto the eppropriate axes in figure 4.1. Employing our transformarion matrices we get therelationship as in equation (4.l [), between P l~Q .& R ~..Ildq.l, e& ([:I. ifwe invert the matrix of(4 . I 1 ) (which is not ortbcgooal) we gel ~. e & $~n terms of P, Q & R as in equation (4T12),At least in theory, then. the variation ofrhe orientation eo-ordinates can be traced 'in terms ofth e angular velocities of ron, pitch and yaw. However, 'P'.0 & 4 have no easily identifiablephysical interpretation. -

._

, 4 " 1 0

( 4.1 I)


18/68

sin 1 ( j ' I ) , ta n Bco s 4J

SiD .q) sec 8' (4.12)

4. :Z EQUAl ' '10Neo O F MOT IO ;_

'W ith the awareness o flhe xes system we are now in a po sitio n W expand the te rm s o rthe gen era l eq ua tio ns o f m o tio n (2 ..] ) & (2.2 ) in to a m o re cx.pH c it fo rm .

Let IJm to ta l fo rce e xerted by the fluid on the huU of marine ve hicle ha ve com po ne ntsX , y & : Z inthe direct ions ,0 1th e bo dy ax es, (~ olc lih aL these will, i nc lude the e f fec ts of con lmJsurfseea etc. but not th e weight). he resultant fosee F, then i nc ]uding the vchichts weightmay be gi ven by Icqua~ i lQI l(~t]]). The wcigh~~ef course. always acts dcwnwards whtch lS wbyit m ust be referred to 'the in cl1 'ia~ fdm.e . U sin g o ur lm l:I:sfo rm a lio n m a trice s ,~ vcca n e xp re ssmgK in terms f 'thc - d y axis as in 4 . . 1 . .such lhal f. m ay be rewri t ten as in equation (4.l5).

F = X i ,j- yj + Z k + rng K (4.13)(4.14)

F '"' (x - mgSin e il- (Y + mgCos eSin ( ! ) ) J + (z TmgCos ecos c D } k (4.15)4.2.2 MOlVfli:N'f

Let the resuhant moment G (about C) applied (0 the hull have components K, M & Nin the directions of body axes as given b equa t ion (4.16). Note that since weight ,acts th roughC. (j is entirely due to fluid forces,

4.2..3 MOMENT OF MOIVIENTUM


19/68

The m om en t ,o f m om en tum ;~ canbe IC)tPlressetl in componen t form a s in equa t ion(4,17) where the components are Cl(mvenie:nt~ye;t;;~ressedinterars of'rucments and prodects ofinertia as inequation (4. ~8). Note that ifCxz is a plane of symmetry of [he marine vehicle(which is usual ly the case), T):'Y~lvz=O.

---

(4,16) -(4.t7) -

[ 1 1 ' J 1 : I ~1= by - = -Iqh. -[., = r,] [ P ]1 ( 1 . QI R -( 4 .1 B)4.2.4 EQUATIONS Q :

Provided th e marine vehicle has (can be assumed to have) a constant mass; m, whichincludes I : ' I J I fluids etc, car r ied . ooboerd (hen equ-ati.on (2. l) suupljfies slightly and m a : y beexpanded E lS in equat ion (4.19). Expa ll tr l.i Jl .l g e q u a ti on ( 2.2 ) iu a s imi l a r way yields equat ion(4.20). We can now write down the expanded force' equation as in (4.21). Separating this intoscalar equations along free bodyases directions we gel equation (4.22). These are.jlre forceequations in surge, s\.'ay (or drift, and heave respectively. On expanding (4.20) wifh [he useof (4U 8) and separating out the components we obtain the moment equa t i ons in roll, pitch andya\\I as given in (4.23).

--f = 111U

= 1 1 1 i_(tJi + \ f j oj- W k )d l=m(Di+vJ+\Vk+ut V J + W k )= m ( u i + VJ+\ifk +Q x u )= I n [ ui + vl- W k . ~ ~ : 1 ]

I ~...

(4.19)

_ . . . .


20/68

l (4.20)! I I l r I i' I< _=h i+hyj+hzk+Qxhd (MV)t3iF( X - m g S i n '8)i 1" ( Y + 1 I T Il g C o s , " '. ' i n (E l ) ), ( Z + - m_gCo5 eCos I b rf j k I

= rn (C d + v j " * W Ik )+ In P 0 I : l .U V \V

[ 4.2 1 )

X-mgSin e =m (,' oj Q\ - RV)Y -I mg C os e S~n41 , = m (V 'RU - PWZ+ m B Co 's e C os (V ;;;;;; (\~ , j'V -QW)

(4 .22))r c ;

l( '" :1k I) - I'~~Q - I ... R + b ' . R . - I ~) - I 'ti Q )Q - ( I r Q - I" R - I.y P) R :.M "" - I o,y r + I Q - 1,1 It + (L P - I .~Q I ...F.)R - (I. R - t . . . r - I,. Q)p (4 .23)N - - I. ? - I 0 I R (I Q - 1 ,,' R - I. P I~ - I P - [ Q - I~. R )0

T he o nly sim p H fy in g as sumpt i ons in (4 .2 2) & (4 .23) a re cha it the vehicle is rig id andtha t its m ass is co nstan t an d is t:ra 'fe lH ng a t a speed le ss t ita n J.:.: H)lIm f s~t::!

Th ese e qu atio ns the n lJ,ilm app'!y to an~ constant mass rig id b od y (sh ip, sub m arin e e tc .)which are affected by external forces. Theequations can be sIi.gh'tly s impl i f i ed if the vehiclepossesses symmetry, (e.g, symmetry in Cxz causes h~Y = I V 2 = 0) but they still conta inproducts of velocity etc, which make them lOG complicated fo r analytical purposes. Notsu rp risin gly , (b e. e qu atio ns w illb ee om e ra th er m o re c om p lic ate d if w e o rig in o f H ie body axesdoes not coincide with th e centre ofmass, Such smtemems of sys t em . m ay be more convenientin c-ases where the vehicle load ing changes appreciably Or where the descr ipt ion of certainforce or moment compcnents can be ; made simpler. \~e shan continue the deve lopment ofmanoeuvring equations using the forms l.2l) and I. 4-23),


21/68

,4.) L~NEAR1SATION OF THE EQUATIONS OF 1\10TIONThe obj ectives of manoeuvring th eo ry ~ outliaed in merion I. require t h a t we area b J e

to sufficiently aocurateiy.predietthe motions of a vessel from knowledge of the deflection ofthe control surfaces (rudders, fills etc.). Our l equa t ions (4.22)1 and (4.21)' are clearly not y~t it ,S lJ C }I 8. U sa b l e , [onn s i n e e :

(i) the existence of products . f v,e:loci.t)'e mponents make tJ~e fonn of t he equa ti on s1 1 0 n - U n e . 8 1 l ' and therefore, ico dimCl' Ih f o r direct solution;

(ii) the force and m om ent term s X, Y. Z,' K. M &. N ind1.lde both th e applied surfacete rm s/m om en ts and the fOi 'Ccs/mQll1en~s 'On the hun caused by its mov,eUB,ent ~n w ~te r:. e ach o fwhic h a rt: v er y d if fic ult ' 1 , 0 Sl)ccify.

l n o rd e r 10 proceed it ~ s necessary t o S'i~np~irJ [he , e : q u a [ i . o n s by "llucarislng ' ~hQ m. F(1rli ncarisatiou we w ill co nsid er o nly m otio ns involv ing sm all p er tu rb atio ns f rom 8 s t eadyreference motion. \Vc must be c-Ilreful in our applicarion of tbt! result ing perturbaricn,eq u at i on s. W e '''''ill be II Ibl e to m odel those m otiens in vol vi ng g en t! e changes o r course: n aviolent manoeuvres cou ld he considered with jusnficat ion.

4.3.1 STEADY SV1\1"METRIC REFI:RE .,oCt: McOTION

The steady re fe rence mot i on normalIy considered IS on e o f 'tile Uprigh ttranslations, For 81 submarine, this could involve climbing 'or descent, but not banking o rsideways motion, A general reference motion Cor at vessel ill a 3-D manoeuvring space: isgiven by equation (4..24). Under these conditiens the equations. of motion (4.22) &. (4.23)reduce to (4.25) where the "bar' denotes steady value,

(4.24) .~

X ~mgSjn0 = 1Y = - IIZ+ n~gCo~A=0 Jl ( =M=N=O

-(4.25)


22/68

If tbe vessel is rel?trided to a 2-D manoeuvring space Le, a sur face vessel;then the additional constraint 19 = 0 simplifies the reference motion equations to (4.26).

lf a V C ! S ! , H : : : I wert . : [0 perform th e reference mot i on d ~s ej'jb ed , s m a ll deviat icua [TOIlI thatp os itio n w o uld inevi tably ( ICCu r m a re~d is~ ic cnvironJ'ucl1It If the se d evia tio n s grow o r ~lI!.ISC~~'O\Vlh o~ d evi .tions:n ether nKltiQ~s~h~n w e 'lJ~ uld sa y (h lilJ l[he vesse l is "l.mt-:t,nbjC~': 1 r theg iven re fe rence m onon . rf thc: deviat ions S:UbSI,de: then the ' vesse l cou ld be ' descr ibed a s"stable", Cases of Hncutn' i lsfabih[y" ' can else be 'envisaged c.g, head ing c1t~ng(!S d L k todeviations In yu w may be mainta ined e ve n [h o1 1J gh lh e yaw:f,iil:(C is stable. We shall disCLISS '[histopic of stsbiliry o f 1 l1 0UOnSin dC1:1i ] hucr.

Let the force and moments components be describe I by thesum of the "steady" y luesand th e d i!:tu .rb ed va lu es as in eq uat in (4 . 2?);, and le t the ' d isturbe d m otion 0 f vesse besimilf,lrly described as in equation (4.28). hl .! ipect ion of the' equa t ions of motion (4 .22) and( I .~3) indi .cates that .we mus t . a lso Lake aCC('liUD~ oflhe devtaJi.olllsil l th e o rie nta tio n a n,1 e): !.,which we call describe by < D . , e & l J J _ And although small displacements from the std,ldymotion position do not appear explicitly in the equations as yet. they may radically affect l thefluid forces and the moments (aswe shall seewhen considering the particularcase of surface~

(4.26)

EADV SYM.METl:UC RJ~FI1RIi;NCrr:MO')I"rON

uraft),

Y - Y-."I"Z=Z+~K - K-AKM = M-i ANIN=\1 +ll.N

(4.27)

~I


23/68

~:~~+u)~fV _ J + r w + W ) k }O-PI+qJ+,rk

. . . . .(4.28)

Substituting (4.27) & (4.28) iu (422) & (4.23), we: g et the disturbed motion equationsin {4.29). If we now invoke

(i) t he s imp l if yi ng symmetry o,f typi .ea l marine v eh ic le s s uc h that [:KiY= 0 = fy z(ii) the assumption of small perturbations ,-which means we can eliminate all products

o f sm a ll pCl1urhl'dions} an d(jii) th e ~ lcfld y slate re la tio ns hip >! fo r X . " 1 , . Z , K , M & N as, g ive n ~n eq ua Uo J1 (4 .2 S),

then we obtain the disturbance cq\lltl'ions ora 3-D manoeuvring space as in (4.jO) ..Note that(I symmetrical: surface vessel, w i l l have S U g J i l d ) f s impler force equat ions a s in (4.3,]) since W =

-

U co 9.

x ~ AX 1 11 gSin (@ ~- 0) - III f u . + q e W w )' - nP~Y -i 6 .Y 4 - mgCo .s (@ f 9}S jj'l.~ - m ] \. + - i r e u +u ) ~p(\V ..w}]Z ~ .aZ I mCos{6 ...acos~ '" mfw + I'H'q tU . ; .u nK ....At


24/68

In equatioJls (4.30) and (4.31 the perturbed force' and. moment terms still relate 10 th ecombination of the fluid loading em the' hu n due to it s movement and the loading caused bythe cen tro I surfaces and o therextem al effects,. The perl;Jtn :l"bedluid load ing on the hull, a reobviously dependent upon rhe inslanh'JIneous peruebed motions of tile vessel and theirp re vio us v alu es such lnat \:\',ecould IPO'sLulate a description (o r each of the components as in(4.32) If thi s w ere the case then all six. equ l ions of 14.30) Or 14.3 ] ) 0 would be interdependentHowever by applying physical arguments b sed n symmetry considerations we shall sec thalthe Junction al rn rm so [ ax . etc, arc simpler than (4 .3 2) fin d lhal th e OVC~11n coupl iI:II:!. ruq ua tiun sis le ss se ve re ,

AXb,YAZ f U i n l io l1s oQ ip ,9,y ,X.Y.2,U, \I,'W, p."I.r,U, v 1 w . , p . q . r .. .A K pre' l l ] (!IUS \I)! Itu :.s of. h es c a r i. J b . l c . s , t)1AMA N

(4.32)

The twelve perturba tion vs riahles x, Y. z, r p , . EI~I~; 1I.,V . w. II" q & I' fall. .naturally lnIO 2 : glit'HJPS

0) x , z ,u w, q. e wliieh 'VIe a ss ee ia te with sym .m e ~ric disturbances and(i.i) y, v, p, r. e, $ which ' l iveassociate with anti-symmetric disturbances,

XVhCi e: ": SY I n li l1 e tr ic " r ef er s to ] 1 1 1 [he Cxz plane of symmetry" of th e marinevehicle. It should be clear that Mull any symmetric disturbance on. a vessel the' fluid load ingI 1 Y . AK & ':A Nw ill a ll be zero (assum in g that the f low around th e' h ull re m ain s symmetrical,alth oug h pro pe lle r w ash es can cause ant i - symmetric f low). Thus AY .t'o.K& AN will dependonly upon the anti-symmetric dis.'luibance components. Although physical reasoning does no tlikewise suggest that the symmetric loading bX" liZ &, aM will all be zero with a purely anti-symmetric disturbance, II does indicate that [hey wiU be relatively small (second order)com pared with their values arising from symmetric. disturbances. Assuming" that Lo.X.62&.6M depend only on the symmetric disturbance eompcncms th e functional form of thelO.81dingsimp!'iry from (4.32) LO f4.:Ba &: bj,


25/68

-(4.33b)

Insp ectio n o f ihc equations (4.3'0) with (4J3Jll'D o~Yincie ares thal they separate into tw o'in Gc pcm ie n~ , u ne eu plc d g ro up s g ov ern lJ'lg sy;mmetdc m o tio ns (4 .3 4) a nd a nti-syn nn ctricm otio ns (4.35). A s "Io:fo re, lh,e equa t ions s irn plif y f or v es se ls e en strsin ed to th e interface m l i l l(4.36) and (iL37)

Syrmnclri,c: Per turbed Mnlirm in 3 -D Malloeu ,ring Space

t 1 , ' , X , - " , ,gG c o ~ e ' " 'm(il - ~ y ) }t'J.Z- m g9 SinS"" m (w ~qU } ,t.\M -- I~.i~

(4 .34)

Am i-sy mm c ulc P ertu rb ed M a lio n in 3 -D Ma nOtlliw in g S pa ce .

AV,+m,g$ CosB = 1 ' 1 ' 1 ( ' 1 ' 1;- r U - PW)}('.\K= L.p-I.iAN= l l r l .. f l (4.35) -

Symmetric Perturbed Motion in 2~DMaIl;lJeU\'ring Space.

(4 .36)

Ami-symmetric Perturbed Motio-n in Z~D Manoeuy r i ng Space.


26/68

"'.y...m~.'~=~1(V -+ r u ) }.6.K = I x p -l,ar ..~ N = ,]i - ~ > ; l P ' . (4.37)

The equations (4.34) to ' (4.37) are Hneu- (products of variablds have been removed),A nd providing '''''C re strict the expansion ,0 th e lo ad in g t e rms 10 l inear fOtiTIS (which is heonly logical th ing 10 do ) [hen b)' th e principle o f superpo sition w e can ~pl,il any perturbedmot ion into hs sym m etric and snri-symmeme com po nts an d analyse t hem s ep ar at el y. l4.3 .4 fORCE AND M OME'NT DISTURBANCES FOR A DEIlPL Y SU IJM6RGI' DVESSEL

Fo r a deeply submerged vesse l the fo rce& 'm om cn Js w ill be in dependen t of the lin '~\!'d isp lacem en ts Jrom refe rence m o tion . hus (4 .33 ) w ill red uce to eq ua tio n (4 .38 ). ~hC"previous va lues" o t" the m otio n va riable s m ay be de te rm ined by the ir hig he r im ~ la num c ousder ivat ives and provid .ed they a re ~ fC ~smw b !y behaved", die r U D c d , o n s , ~ay be C X p ' l " C ~ ! - I e d ! i t lthe fo rm of Tay lo r series. The sw ay t,on;e, fo r example , the re fo re m uy be re prese nted (10 '~lielint order) as in (4.39) where the coet l ic : iems of IIIhe variable arc refer red 10 as th ir

l l . X }!:iZ :: t {S, ll,W, q , u ~~",~,q.revio us 'V alues o f ~hese ' 'M ' < l r . i ab l es , . t)LlM

(4.38a)

~~}.= ( $ , v.p.r, v.p.r ,previolLlsvallJesa.f these var iables , E )l'iN (4.38b)

(4.39)


27/68

-

Y o}~1 = -1~!g~1K~~h ,= ~H1yh$NI}~ 0

= -rug= -mgh= 0

'('II [ ! mgH ~ . " < 1 1 lngZd8 = (J ::::> Z II = 0 T lIe = -1J)~lle ~ M n = -rngh

rig.


28/68

S im ila r e xpressio ns can be deve~op.ed,or AX , AZ 1' t lM, aK & :AN but def in~ng theinfinite number of derivatives is clearly imprn


29/68

t c ) [ c i n . g term X(t) Y(t) etc, allow fur the effe,cts of control surfaces and/or other forcingin n uences o n the vesse l.

Substituting (4A] )hno equations (4.34) & 1(,4.35) yields the lirlcarlsocl (perturbation)equations of motion for 3.deeply submerged craft given in (4.42). These may e lr er na ii ve ly bewritten in Ihe..matrix form as in eq uation . 4.43) bu r only \ hen 8-0 (i.c. when the referencemotion is symmetricand M C ' i fe l ) . Note that a few of the fo rce /momen t influences can be(Breclly anributed ttl, hydrostatic Cffr.!'C1S (assum ing , these m e [he sam e as their values WhenU t::: 0 ) and the r el ev a nt d e ri va ti ve s tan be simpJ) ' ,q~uOloo(sec' figul'IC ' ' 1 . 3 ) ,

x u t' , + X w u + X , .. \ \ / ~ - X , ., w + X ~ q + X qq . X ~ I - rngCeti,eZy,j - + z~u .. Z W ' ; , ' 'If" Z...!I V 4 ,Z qC I1-Z q + 2 . 0 ' e - m,gS in9M~t!I+ - M II H;' M w \v +M ...w + 'Mqq t-Mqlq +Mil a

=m{~Jrll W) - X t t } ): : : ; :m (w -l.IU) - t.:(t) { . l ~ , 4 2 S l );;:;;)'I-M(O

y , , : + Y,' ,y,P,+,Y'. P+Y;'; Y,,~ Y _ , .. g~_ :Ol(U+,jj ~ pWI- VlIJ}K" v T ~ < .~ v ,. K ~P TK\,,'P -iI' Krr + K.f -t K , . Q 1 - = !~ -l>gr~ I


30/68

[m-XiJ

A- -Z...;-'iI-M.u

[-x. u

B = -Z...1. 0. -MI.I-x ..._ ,Z . . .-M..

[ Y ( Q ] [ Y JO C t ) = = .K ( I . ) ; q .= 4 1 N(t) w [m-y

A = _K t- ~-N~

[ -"B~ -K'- \-N -(mW -I- Yp) rmU- Y ,-Kll -Kr-Np -Nr(note: j ., ; : . J udt; Y = J'"dt; "l= f wdl)

-x,m~Zw-Mwo m g - X I l ]o mg-ZQo -Ml

-v i1[ ~KI1-(f._ F Np)

-y I- ' " > 1 , . K1 )

N" ,-(mg f Yo)-K(!

-N ~~ ]o

For a vessel operating at t he in te rf ac e: the 101m force and mom e n t disturbances wiltdepend on the Lineardisplacerneats of [he craft Irem the steady re fe re n ce ! Ilt llk ln as given il lequation (4.33). The nature of dependence upon the body axes displacemenrs x,'l L r q _ z,

-however, is rather Iconfusing since the er i emat inn keeps chang ing , lf we postu late an"equilibrium ship" which ah\paysmoves in the steady reference motion, [he picture becomescle are r. W ith a sso cia te d axes A x,~>y'. z ' a s i l l . 1 ! J S t t a t e d in figu re 3 ,.3 the lin e ar d isp la cem en ts o fthe craft mayahernatively be descr ibed . by x', y. & z and immediately we ca n say that theforces and moments will not be affected by ;{ '.&y" which simply describe the position of thecra f t o n the intejface, The dependence 0]1z " however is equa l ly obv ious in . that changes in '[h epar ameter win al ter the displacement o f the bu ll a nd the buoyancy dis t r ibut ion ,

Before s implyadopl img z~(and its: time history) as an. extra variable in force andmoment equa t ions we mustexamine how the general d i sp l acemen t IX'., y~. z') relates to the


31/68

rather abstract (x, Y.z), This can be approached. via the descriptions of veloclty of the centerofthe vehicle as given and developed in equatlon(4.44.2).

ACd ( .; l; i y.' j+ z ' K)

= d t=:fi v,'j+ zi- (4.44)

0= UI +AC=(TI + 1 i ; . ' ) j +fj ~" K1 3 L it a l so,

I.C.

to give 10 the first order .:0 ; =it . ' =.:7 - =

(W =0)_ o : ! ! ! ! ! . .

" ' l ~j-0 o 1 [ U + l . l j -~ 'IfI ,

Integration with respect (Otime givesI

x " (l) = JtL{l ') dl' - x{'tjn

--y ., (r) = y(11+ VJ (t ')d~'~,

IZ ~(I) = Z(l} ~UJ (11tH'

IJ

At this point we are sill] leftwith the rather abstract x, y and z bm remember that we .are only concerned with the- z' component a nd ' (4.44) show's t I m E . this is a tunc t ion of only th esymmetric variables 1 :. and o . Thus z'"maybe considered as a symmetric disturbance variable


32/68

Prom equation (4.44) we can gel the reletienship between zt. IN-and w as givell in 4,46 whichwin facilitat'e the matrix description of tile symmetric equations o f motion. Using the equation(4 .4 -6) .~ th e m om e n t'f orc e d e sc rip tio n s m a y b e c xp re s,s ed 3:5 in (4 .4 ( 7 ) .

AX = X~ lI . KIIll + X;j, ,O t +X", w + Xq q Xllq + x~le+x.:z+X ,.z + X(O 1tiz = Z il z, + Z .. \i .r+ z ..w+ Z q q Lq,q... Ze9+Z1 .i".+ Zl'~~~.Z{t).. '. . (4.45)11 M =MyLl+M~u M .. "' M "w I-M 'lq , M 'I'I1 "Mof.l f -M , . ,7 . , M .. z. ~M{t)

w = f~GV I - : ( + U D }

W h ere fo r a symm e iric d is tu rb a n ee .

[ X ( I ) J r ' ) ] [ m - x . -X;; - X ' j(t) = I z~t) " 9 = z'u) , A= -z m - z . . . -2.-. ~( -~e t) 9(t) -Mu -t...w (4.48.a)I Ii- 1 mg-X. - i l x . l- x . -x"' -(X


33/68

_ -From equation (4.44) we can gettbe re.iationslilip between ~" w and o W as given in4.46 whichwi 11fa cilita te the m atri x description of t h e s :YWMemcequa t i o n s o f m o tio n, USi l lg th e equa t ion(4.46), the momern/force d e : s m ,p ti o ;f ] s m a y he , C ' x p r e s S ! O O ,asm(4,.41).

I.\X " " X " u +X ~IJ+ x . . . . , ~ oj. x i ! ! W + X qq X qq ....K i l l e+Xl.] : + X Z' " .1 - X(t)6 .Z ~ Z.:.lH Z " tl+ z . , . , ~ tI- Z "" w + Z ~q+t; q + Z e E l I + Z", .Z + - z~." ' Z(t)I ' J , M""M~u Mwu'i-M.;,. ,v+M~~ +Mqlq+Mq!q+Ml)e+M~.f Mr.'z" } (4.45)Met) --,\I ;;;;;i : + u a. , ._ . U 0 'V i'=Z ..J ' } (4 .A6) .t . \ X " , X il X"IJ+X


34/68

_N :o U a ~Igl e: ! f a

. . ... Y,PI "= ' ~m~~1K j . d l - -mghN1~1 ~ I},

y ,_

"" ~ll1g= -mgh = - IfIgCM,= 0 (Oneil as sumed )

X n SM u S m gS :::> ~.:: -mghS ~ Mu = m g= ~mgb

ACZ .z :,

z

z, -jJ_gA ',\'1'


35/68

q ( t ) ~ r n - -r t ' ] r ~ Y - -y - Y o 1= -K," p -(Itt ;K,)( 'i ) " " ket) ,. r -K.- . 'If ~ "M(t) -N IQ+N p t, - N. (404Sb)"[ ~ y -y m u - v ' l ~ = [ : -(lm_g+Y.) : 1= - K : l"-K -Kr ; -K~p-N. -N -N. 0 -N, 0'To SUJru1HUi 'l .c~TIle vertical posi'uoning 0' a surface eraft will rulTect only Ute sym nrietric

fo rce and on ly symm:l;tlrlc 'brcc ;and m om en t disturbances, The equa t ions o f m oticn tO l i 'ent i symmet r i c disturbances a re th ere fo re ~ he !S1llll1,e as fo r a f u J . l y s.ubm.ergcdw:.hich:, a s B iven i n(4 .43) except tha t N. is no longe r ze ro . The l !q lu~Hion.so f sym metric m otion m ay be w ritten ina sim ila r fo rm ~ '" i n (4.43), but 11lC d isp lacemen t r n a e r i x con sists now o f x, 'l."~0 and thcocffle ients matriees At B &. C axe iu acco rd ance w ith the fo rce .and m om en t descrip tio ns of(4 ,47), 'l he st m aufx eq ua tion s o fd~s[lD rbed mot ion o r a surface cra ft a re g iven by (4 .4 8), A sfo r th e slJbm erg ecl v~bit.lc so m e o f ~hc U h y d m s t a r i c d etiv ativ cs " c an sim ply be quo ted asindicated in . f ig t l rc4.4~il : ssuming again that their values arc unchanged from the:case whereD = o . . . . .

, . . . .

-


36/68

5. USES OF THE MANOEUVRING EOUATION,5.1IN"faODucnON

In the last section we succeeded in eKpf iCu ingEhe equa t i ons of (disturbed) motion inthe fo rm '(5.1 ) { see eq ua tio n s 4 .4 3 and 4 .4 ,8 ), The se a re lin ea l' m a trix second o rd e r d iffe ren tia lequations, which we can hand l e : , 1 I i f l : I J y 1 i c a l . l ' I up 10 a point). W ben using t h e s e e qu a t i o n s we'must 8 1 ways rem em ber the asS:UI11pUOnS ;;lJnd,flpproxima:tions" which have been m ade in theirderlva tion an d avo id an cJn pti:ng .0 use ILhem or m odona: where these approximat ions arecle ' ' F l y inval id

In this course we V/UJ~ l n cmp~ ~o C o ve r some 0 th e P in ti ' ~ I use s o f these equa t i onswhiehare

"'CciJ1sidennioll! .l o r hRTABiUl 'Y '1 0' m tion where Q(t is zero and WIJ won t to examinew ha t haJ pen s ""hell Ihc 'V esse l u em p ts ~'O execu te ~ steady re fe ren ce rn o tlo nw lth no controlsapplied,"'CONTROL purposes WhCfii:: Q(t I will b s om e function of ~. ntrol surface (rudders,

stsbilizers, hydroplanes) settings (note that [he concepts of STABI1..ITY wil) apply tocontrolled motions also),

'tIM AN OEU VR 1N G sim "!Jla~jl)n an d pred ic tio n where such equations (o r usually morecom plica ted on es) a re num erica lly in teg ra ted us ing a oom puu lr.

A fourth use is in I inear i sed seakeeping analys is where Q(t) would rep resen t 'W aveloading on the- hull, ,Af; discussed in. sect ion '3 . t radi t i .onaUy a rather different app roach isused. The seakeeping '~heol"Yleads ~'a equations of same: form as (5..1). but where"hydrodynamic 'coefficients" rather than fluid "de:rh ; ,anves" describe the components of thematrices A, B &C. Ul l sw:p r iS iDg~y there is di rect ooue]atkm among these parameters.

In. theequations we shall deal with which are o. [be form (5.0 the matrices may besaid to be of the order 11.;,,,'here n is the number of "degrees or freedom" needed to describeadeqmlieiy the motion being studied, For (II rigid ship with control surfaces undeflected, Ute


37/68

gen era l m o tio n m ay be d e S ; C ; n b e d by n ;- 1 5 , ; , we b.a \le , shown however tha t the componen teq ua tio ns fo rm two in d epend en t g ro ups w ,her,e n = 3 in sym m etric an d a nti-sym m e tricm o tio ns. If co n tro l su rfaces a re e ffec tive (~..c. operataoMi) . tben we 'w ou ld expect n ~ 3 .

5.:2 .ST A Un .l[TY O F M OTIO NS

5.2.1 SOLUTION OF ROLL MOTIO- D , 1 i ' A smrFor the purpose o f :in llod ucU on to m heoo licep~s of stabHity ef m otion le t us consider

rh e o n e- de g r~ e of f r eedom s y s t C n 1 o f u nc oup le d r o U i n g 0 ,( if vesse l, le re w e a x , a ~o ing ' Laassume that neither 5,\\'3y nor yaw mon o n s influtsnce in an y \\'tI.y th e vessel's be ha vio ur in roll(i .e , K v = K " = K , = K . " " 0 ;: ; ; ; J !! , o z J i' ) a n d th er ef or e linn th e equa t ionof mot ion m ay be g iven bye qua tio n (5.2 , (T he va !'id 'hy o f this a ppro xim a tio n is v,e rr d oubtful a nd th e re sults w e o bta inuH1.SIb e ' h e a te d wuh a pp ropr ia te c au ti on .)

Aq(l) I 1 3 q ( 1 . ) + Cq(t) = Q(O . . . .(5.]) . . .. . .(5,2)

The gen era l so lu tio n ~()WI equas ion such as 1(5.2) m ay be expre ssed a s the sum of twoparts which are separate: the COMPlEMENTA1:{Y FUNCTION and PARTICULARlNTEGRAL. The complementary function is [he gener a l solurlon 0 th e e q ua ti on (5.2) whereK( t l =O..It represents the "tran sien t respon se " o r "fre e rn o tio n" of the dyn am ic system whenresponding initially to an applied external input K(t) (or change in K(t). The parti~ularin teg ra l so lu tio n is the solution o f eq ua tio a (52 ) when K (Q 7# C . W hen K (t) is a co n stan t va lueor a periodic function then the particular integral represents the "steady state r esponse" orsteady periodic response of the system to K(t).

For the r e fe r ence motien of the vessel K t) = o . If we cons ider a small dis turoenceaffecting roll such lhat K(t is something fora. short timeand thenzero, we Call. see tlrnt therespon se to sucha d istu rb an ce is , g ive n by the' ccm ple rn en ta ry fun ctio n a lon e. " the pa rucu la rin teg ra l be ing ze ro , W e w ill be in terested E ! D se e whethe r the eom plem en ta ry func tio nd i m in ishe s (stab ility ) o r grows (instability)


38/68

~Hl) 0;& P . . - :0 (S~able} a or , P :> 0 (-Unstable)

. ,. .

GEN ERA!. CON [)IT,ION F'-ORSTA n,IIIUT\'::

REAL ROOTS OR REAL P.A;RTSOF COMPLEX. ROOTS MUST BE.NEGATIVE


39/68

Far our rolling examples the Ic-ompJemenlary fimction is the solution to (5.3). As th ecoefficients are independent of time thea a t r i a J : sclutian would be of the forzn (5.4).Substituting fo r (5. . '1) in (5,3) yieMse 'qu ;a~ion (5.S).f'or a . non-trivial snluticn q~o . , A . l i: - 0 andtherefore 2 '- is determined by the 'ch~rac.ter:istic equation" given in (5.6). This quadrsticequation 1 1 1 A . has the roots AJ. 7 1 , 2 which are either both real or complex conjugates. togene ra l terms, these may b e g iv en by ( 5.7 )., T h e c omp J.em e n twy f un c ti on therefore win be o fthe fo rm (Jf 5.8. -

(5.3)

S.4)

(5.5)

(5.6)

I.l'"0: )/..,=;, 1er '" = ~ . . . ~ \. . .t c , ~ J .I -IW

(5,?'J

wher e ..c z , ~11 ! - t&w arc rea l

o r (t) =,eiJI [ $ 1 e i U > L +$ ~e-'!ilt J=~el't sin(roH E)

where] r,q ) ~ ,$ , e;are constants determined from initial conditions

(5.8)

The roots of the ~baracteris1ic equation (:5.6 .are given by (5_9')& now we shall arguethe conditions for stabi lity in b:-:rms of the d riva ives,


40/68

(5.9)

And the two roots are real. Ho\ ever, one rent wUI be positive and the other negativeand th erefo re 'O ne co od h]Q U fo r s[8bjliit),! is that KiI>'([~-K~) < O. If-the h itte r h o ld s Du l stili

W be I' the term in ~} is posi Iiv c. ~ J~ re ~he m o [jon is slr~b]ei f

Al terna t ive ly if

But

Then the [0018 are complex conjugaro- pair for which

,C ! J _ . = [K p - ,.4KII (1...- Kp)J4 [, - Kp) ' : !


41/68

In a nutshell; for a stable mUing response (to srnaU disturbances) we , re qu ir e ' t ha tequat ion (5.]9) ho ld s. In g en era l al31U fo rw ard speeds, U I 1 < ' 1 1 and K p a re both negative so tha ith e condi t ion fo r st: 'tb iU ty red uces to (5.1 I],

It is worth po in tin g ou t Lhat re i ted to genera l (CimlS we have shown that if th ed~ ID ' a c't eJ ii st ic e qua ti o n o f ,3 one degree o f freedcm s,ystem m ay be given by ( 5. ] 2 )~ then th en ece ssa ry a nd su fficie nt condi t ion f O l r sUi,bmly ,Me fha1 81>0, t!2,>O.

( 5.1 0 )

-= > - pgVGM < U for a su rf a ce shi pl.e GM>O=:) -mgCD - < 0 For iii 5 u b ' 1 ' n e r s , i b r eI.e CB > 0

(5.1 1 )

(5.12) . . . .5.2.2 STABILITY OF eOlfFLEDEOUA nONS

Suppose our system's free motion is governed by the eq~;ation (5.13). Consider a.solution of th e form 0[(5.14) where o k ' is a celomn vector of [ he , s ame order '11" as IAl B &.C'. Subsntution of equation (5..14) Into (5,,])) yieldsequetioa (5.15), which constitutes ahomogenous system of 'n' equations in On; unknowns and bas a non-trivial solution for K (i.e ..K~O) ifal'1d only ifirs co-efficient determinant is zero. Thus the complementary function Ke"l. is valid only for values of ~}I.which satisfy (5,J6). Expanding (5,16) we obtain thecharac te ristic eq ua tio n fo r the system as in the equation ( 5. L1 ) .

-, r r ' < -

-


42/68

(5.13)

(5,'14)

(5.15)

(5.16)

., m 111 '1 - In -1 _ r,U!){I,. + a j / J . +


43/68

f

HI a~ a Ia D '-I,T ~I TJ = 3) a all2 : l;3~ a ,a at a~ , al(5.18)------9

A nd where al.,an: tahn to be ze re where k~O ( 'm' k>m .(NO'lI;~ fo r !hc last lest f tmc t ion rM= aMTM.I) -

BXAMPLE: S ay . 6 . ( 1 1 . ) = 80/\." ;. all' + all. 211A + 34I:;; 0 where . fl:> 0"he necessa ry and suf'~ciellt co nd itio ns fo ,r,s tab ility a re th at:

(i)

(ii)

(5.19)III ltD 0

T~::, 3] 3.1 HI =tltll.:I_3,':la~al: -aai.l!/ = a T: -a,~r!l!ll '> fl(iii)o a~ al

Note that as ~ > 0 from (iv) then 3.3> 0 from (ii) &(iii) fo r any possibility of Ty> O.(This makes (ii) redundant],

. -- .Similarly, ifa], aJ, ~> 0 then ell:;:' o for any chanceofTs > 0 , Thus the condi t ion for

stab ility red uces 'he re to ,a l > a.a:>:> Q " a.> .0 &T 3:> I[) (3.2 0' is sa t isf ied in p li ci t ly ).The final sratement in the example of I 5. ~9) reflects an important point as fOUO\!lS. it

is not necessary to use the R:OUTH-HUR WITZ criterion if any of the co-efficients of th e- characteristic equa t ion are negative because when this is , so it cab be shown that there is atleast one rout with a positive real part.

. -. . . .

. . .


44/68

Th e c o n sid e ra tio n of . s t abHi ty ,ofmot ioJ lm this s~on ],5 only a part o f o ur mteres t ind esi g n in g a ga in st undesirable f ,e~. turesof sk~plmad,a ' l l respcnses. For exam ple we could designi3 I ship which is rat ional ly stable ~.DI roll ye t . 1 1 3 3 a I iJa ,t I.u3 ]f:requenc:.yright in the' midd le o f th ewave freq uency range fo r its a re a o f lo pe ra tii.c n. .cD m ay be no ted ma t lh eROUTH~ ,F IUR W1TZcondi t ion if s:atis.fied. doe s :DOl U ~U 'U s hOI'ItV s~3.ble 8 s : y s r e m is . Once we know n system isr ati on a ll y s ta b le it Inay be n ece ssary to e xa m in e the so hn ie n co m po ne nts , in mere detai ls) . TIledam ping o f ro ll m otion m ay be v er y li gh t, which ,coU[dproduce unacceptably la rge roll anglesand velocities. PJ'ed ictionof such foroes: m otion ancircoonances is : tile dom elns ofSEA KEEl' iN G analysis.

5 .2 .3 STA JU

C on sid er th e: co up Jo d e qu atlc ns of sy m m en ic m o tio n 0 a surface sh ip a sg iven mequation (4.48) . SUbs'li111tilngfor "A, B '& C ' in equatien (5 . 16) y ie ld s th e d e te rm f nam fo rm . o fth e cha racte ristic eq ua tio n as 5.10 ) w hich we cou ld expand to th e po Iynotni.al (Oil11 o fequa t ion (5.2])t where the loo-dfldeu~s "at w]u b e cembina r ions of the: der ivat ives.Inspect icn indicatl~s thst a u = 0 g iv .i.llg n I~o(}l "] ::::;O J w h k:h corresponds to .[ 1 C(1nstantd lsp la ee rn e ru com ponen t in the com ph ::m cn ta ry functio n . This a rise s because o f a "sp rin g 'in fluence in I J 1 C fore-an mo t i on I f surg e and m ean s tha t a , s m all di s tu rbance to the vesse lco u ld re su lt in irs trave llin g sligbU y ahe ad o r behind the "eq u ilibrium sh ip",

(5.21)

(ru - XIJ )~~ - Xu60.) "" 1 0o

o[m- z.",)l.! - Z,.,l-l.z-M i!-M X- yf ~ -w Z

g en era li z in g roR OU TH - H U R W ITZ conditicn f ur s ta bi l ity

(5.22)

Note: b}> 0 is a redundant condition.

wuq


45/68

Xq = X q = Z u = Z u .= M iJ = Mu= 0 ,Wldwe baveX a= m g (see .6g.4.44), th e de te rminant (S.20.)facte rises in tQ two p o f . y n o m i , a 1 : s as. . s o o ' l ' , : v n in e qu atio n ( 5 . 2 2 ) . f i l l p l.) "'" 1 0 is o f course th eeharacteristlcequauon Ior uncoupled surge and h a s , 1 I " 0 m s A I : . 0 (constant displacementcomponent) and A 2 = XJ(m-X 0 1 ) . Since (or s l i l i p s X l i i i - c 0 1 it follows that A .J is negative(cor respond ing to an exponen tia I d ecay Icomponent caused by decay ill sm all speedperlurbatiens) and thus th e free motion .in surge is stab]e . 4 1 C i l ) = 0 is the cheraeterist icequation for coupl ing a lo ne in he illlV e..ndpitch. and cxp imd ing th e co-eft icients "b " we obtainequal iou (5.23) (c he ck th isl), ReIerring back 1 ,0 1 our general c xw np l.c ;n e qu atio n (5.19)~ ~henecessary and Sum d~n1 cond ition s fo r ::;I!iil ibmty.ar:c given in e q ea tio n (5 .2 4) ..

D ue to strong slJring action U1 heave and. pitch mOLiot' l ee nd ltio ns in (5.2 4) a r e generallysatisfied with case for surface vessels, Submar ines I when dived however lack the heaverestoring 81,,';11, and 1.hci' l 'p itc h re sto rin g spring is m uch sm rillcr than fo r a surface vesse l,S ubm a l'in ~s could f ' L ; f l S i ' b l y be unstable in c ou ple d h ea ve /p it ch . w h ic h in practice would beun acce pta bl e .

S.2A_STABILITY IN ANTISYMMEl'RJC ~OT10N

A JLho ,ugh our lin ear equation (4 .41 8 ) i l l l . l J S t : : a L t e s the interdependence between the ant i -symmetric motions of sway. roll and ya\" th~ usefidness of a stability analysis for this 3-degree of freedom motion is ! l / . s u b j e c t o f de1 b - : ; U e .It is usual and easier to consider twoqualitatively distinct phenomena .sepa.rn~e]y in an ti sy ri .l fn e lr ic s tab il it y analysis:

(0 Directional stability (s'way-yaw coup]ed motion)(ii) Righting srability,

I Note; that different equations apply but. ~he form of the characteristic equation , e re is th esame.


46/68

" . _ ,

ttl -- ........------0../--

IV

UIIIIV

- Directionally unstable- Directiruja Ily srah!e

( N o ,o o u [n ;d ~ p :p ll,ie d )(No control appHe.dI){Amopil:oi" control)(Suph isticated autopilot control)

,- S ta bH iIY il l head ing- Stabili r y i!'l track


47/68

Iaconventicnal treatment of directional stability Slwy and yaw are assumed to beuncoupled from roll 'Whi~ using qualitative arrgumenM: and indeed in practice, appears ~ohe areescnable approx im fi t ion.

' . . .5.2 ..;'UlRECTIO ~,A'LSTABIl.ITY

Directional stabili ty could be def ined at differe1fI :E l e v e l s " For directionally stablevesse ls a sm all perturbation from lJllc straight-line r ef eJ1 en .;e mo tio n could leave th e vesselexecuting, s a . , ) " a turn wit hou t t he rudd JC ta :pp~ i'l :: ld .1 1 , e . lo west level o f 5 w b i 1 i ' t y would be wherethe vessel cominues in , E ! I straight line after th e di.s 'lurbanc.c- - 81 though~ p rob ll bly with a s ll ghJ Iyd if fe re n t I u: ad in s. ]f" s ,im p ic a uto pi~Ol d riv en b)' 1 . 1 : ecm pass is ~ns;IBU ed then. a h ig hc ,F form ofstabiliry cou ld be ob ta ined where the vessel rctums to it's Qdgbrol ht lad lng after disturbance.A more sop l istica ted . ,u to piJo '. w ith acec le rom ' te rse te migh t re tu rn (he ship to it s originaltrack. These leve ls o r slability are illustra ted in fiB . S .2 . -

In n av al a rchite cture "D ire ctio na l stabHityU u1 tu aUy r e fe r s: 't o t he charncterisE!C o f theh ull w ith ou t control appl ied . ]L gives a n in dic atio n o r rh,~ Man3,gCilbilily" of the vesse l. Avesse l tha t is s lig htly d irec tio na lly un stable eou ld still po ssib ly be steered in a rea scnab lys tr aig h t lin e by a helmsman Of an ,au l.O-pi lm .aJlhou :gh h e - , 1 1 l l corrections would like ly n eed . tobe:continueuswith an associated penal ly In increased resistance and fuel consumption. A verydlrectionally unstable ship may well be uncaatrolla ble; and. atthe other extreme a vesselwhich is directionally stable 10 the beim and eonsequently very d.ifficul'~to manoeuvre .Vesse ls , such as warships, which req uire a high deg r ee of manoeu.vrabi l i ty ,willi b e d e si- gn e ddose to the margin of directional-stability (on th e stable sidel),

. . .

Here '1Ne shall c~msider this. conventional directional stability without control. Theeq ua tlo ns o f horizontal m onon" fur swayand yaw un coupled from rO:Ufrom (4 ..4 8 ) a re givenin equation (5.25). In our consideration ofthe lowesr level of s:I.~bmry weare not particularlyinterested in yaw velocity 'v' since we cannot measure it from the ship..Stability 'of yaw is ofdirect interest. Wifh 110 control applied Y r) = N(l =C and on differentiating the resulting~q1J8r_ ions with re spect to , tim e we obtain 4 " equa t ions altogether f rom which we caneliminate v , v & . \I through combination to leave one second order differential equation in 'r',the an gu la r ve lo city in yaw as shown in equa tion (5.26 ) . C om pasin g equa tion C ,"E ) o f (5.26) '[0

-


48/68

o ur g en era l conditions ofstabiUty fa r a 2nd o n : l : i e r system respon se d em and s tha t (BfA) > n and(CIA) > O. By con sid e rin g the llnure , a :ndreJa t iv,e mSlpludes o f co e ff ic ien ts in each of A" B& C we can , see th a t A & B w iU , r u w a r y s b e po s it iv e fu conventional vesse ls an d , tha tit 's then a tu re o f the ~c~e rm w hich g ove rn s the [o [ve~ l d irec tio n a ] s t ! l J b H f r y .

To con clude , fo r the ho rizcm al d i11 ectiom lll smbiHty the necessary a nd su ffic ie ntcondition is tha t (5.27) applies. Y1 r and Nt are both nega t ive and (mU - y~) is norma l lypos il ive , Nr. howeve r has a sig n dun depends o n th e oen t r e o f lateral resistance o f the hull inre le t icn {O the c~n1 re o f g rav ity . ,Bo d:! t he s~gn and relative magni tude of N r wU l I;l f fe c't th ed ir ec t io n al s ta b ili ty .

(m - VI,)\I - Y,' v , - YJ+{mU - Y,,Jr =Y 1 ) }-N \'-N v J -N,J)i-N r .. N(l)1" ~ I (5.25)(m - Y ~ )v - V " - V,i ' . .. (mU - V r ) r : ; ; ;, 0(m - Y ~ )V - Y , , t - V,r +' (m .U - y~).r > , i j ] i U )

(i ii)(iv)

-Nvv-N,,"'+ l ..-N,;:)r-N,i=O- N.\i - N v - i I . (I - N lr - Nj "" 01, I. I ' I

E! imina t ingv between (~i)and (iv) y~e lds~

E lim in a tin g v be tw een (i) a nd (iii) y ie Jd s:-

rNV (m ~ 't.) + N~ YvJv+[- N y Yr-Y~(II' -N . .) ]r+ [N,,(mU- Y r ) + Y y N r j r = _ O (vi)

El imina t ing v be tween (v) & (vi) yie lds :

[- Y N , , +(m - Y. )(1 ~ N - ) } r ;_ rN (J" 'U -Y ),- N (m-y,)- N Y _ - Y (1 _ N _ . ) ' ] j - - Li' ';' 1I..z : ~ r , [ ... . II~II I" " ' li 'I I -"II ~ r- + ,~.{IliU-Yr,,+y,N,lr=o


49/68

'- :;;;AI +Br +Cr = 0 (5.26}

(5.27)

.5.3,NUN nlMENSlONALISA TON OF EiQUA.TION'S AN1)COEFFlClENTS ,'"

11 1see t ion 4 - when ,d :criv ill lg e xp re ss io ns fo r ( he foree an d moment di s tu rbances actingon a marine vehicle we dealt explicit] on their disrerbaace upon the dis turbance kinematicso r the vesse l, W e derived expressions fo r 1 J J 1 C foroesl l ' l lOrmen~ acting in term s of c ompcnen t srelated 10 m otion s in each o f the degree f t r e edom by way o f "ceriva tlve s". It Illa)~ i etemp t inQ to th ink lhal~ v ess el sim p Jy "co m es w ith it s d eriv ativ es ", T his is [ru e 1 1 1 " to a polmbut nothing 'in flu id dynam ics is 1 h s L $wmph: t1nd m 'l shou'ld b e ' clear that our "derivat ives 'would be functions o r;

. . . ., _. . . .

D l 11~runn ing a ttitude of the vesse l (ill te rm s o f dm.ft i and t r i n ' l lii) the speed o f reference rno : io n o f (he e sse l,

Fh ~ d epen den ce o n i J is obvious; tba t 0 1 ii) is ru th er m o re o bvi.o us. C on sid er first thechange in rate o f C re la ti v el O'DW past [he hull alene, This ' ! I o ' i l l likely change th e pressuredistributicn over the hull and m ay cause a shift in the position of separadon points etc, I! bmwil l this a lo n e a lte r the v alu es o f L 1 : 1 C hydrodynamic d e riv a ti ve s ? U s ,in g arguments o f l in e a ri tyof the system the answer for the motion derivatives \\foaM be 1 1 0 . . The now disturbances dueto m0ti~~~pe~ur~alions (which refleet thevalnes of the disturbance forces and moments) aresimply superimposed upon whatever reference mot ion flow veloci t ies exist This argumen tdoes not, of course hold for the pcsitional deri ,varive;gl which are associated with theactualpressure distribution on the hull The speed variation in the reference motion, and theassociated changes in pressure distribution will however, affectall the derivatives in thef o ll ow i n g in te ra c ti ve \ va y .s ~-

-(a) the running attitude changes

lNeverthe less . the "hydrostatic values ' geJ1r6raHy apply Je as on ab ly w ell fo r m o s t opet~tingr a n g e : ' > ,


50/68

(b) su rfa ce w av e Ji'iH e m (fo r vessels operating on or near the interface) changes andthere fo re the bo dy of water to be m oved w iw th e p er tu rb atio n c ha n ge s.

Tn Ino r e q i ;J a n ti ~'a tie te rm s le t us ccnside r the n am re o f fo rce and moment influences'de pen den ce o n systlem va ria ble s b y w ay IQ f d :im e n sio n al a n aly sis. C o n sid e r th e component o fsw ay fo rce tJY due to S\:I/a)r vel city 'v', Per fo rming a d imens iona l analysis on a systemconsisting of !'I surface vessel , e ) ( ' C C l 1 : Iiug the steady reference motion with a c o n s t a n t driftsup erim p ose d yie ld s (5.2 ,S "~ ! and imrm:dhu .e ly \I io re can see tha t t i l i t e l ikely speed a n d wavem akinG dependence are confirm ed . B)' au r ,ar,gumcnl o f Iiuea rity and sm aU gen tledisturbances, however; which yielded (4.4 I) 'w e c-a n se eth at ( A Y )~ is g i.v cl\ by (5.29) whereY v is n ot d .t.:p u nd clu o n '~hl.!g en tle m o ucn v', \ ' I , I . e ; can there fore sim p lify (5 .:2 8) ,I~S.30) anddefine dimensionless forces, derivauvcs and motiens as in 5.31 . Note. el f course, that them om ents wonld have 10 be non-dimel1sil~)n81' iscd by II ra th er ihan L2.

(1\ Y) \' II - . m f X I ), - IJncncn P""C. n . . ~J.~Ut: 'l (5 ,28)W he re L = l eng th orthe vessel

Re =Reynold s numberF n =F ro nd e N umb er

(~Y) IJ = Y,,"I (5.29)i.e, (8Y)\.. =Y'vv' (5.30)

.where the prime indicates dimensionless quantity,. (AYL(6 V)' v = {pUL:i! - dimensiculess perturbed force component in sway

due to sway velocity,

j For deeply submerged submarine Fn dependence does not apply.


51/68

dimensionless sway velocity ~derivative of . sway force.-_ _

'It = v =U(5.31)

The deri v ~tj i v e val ues are aalai ned from O ' l l c 1 I 1 e 1 h!stmg (which we shall d iscuss ta teron ) sin ce com puta tio n a l U uid m echae ie s is . s m i J I .u nd er d e-v el, 'p m en L F rom k n o w le d g e o fd im e n sio n al a n aly sis and "physical similari lyn re qa ire m en es w e knew lhat ideally 'V i noe r, cx a lnp le . i fmea su red [roll:l IIImode l ~CSl.is independen t of E n e scaJ,e o f the m odel used. Thesam e valu e o f the d ',im e nsio nl ss d elii, a tive thete 'o r-c .::applies eq uaU y [0 th e fuU~saLevess I.T IN a ctu al d ir ne nsio n el derivative o f the vesse l w ,o uld be obtalned by redimensionalising Y';u sin g th e f ull-s ea le s)'!ilem parameters i[j. L & . p as illJu5lra(ed in e q ua tio n s (5 .3 2) . fo r true" dy n am i c s im i la rity " the relevant mod e l tesi m ust be run at th e arne R e & . tn as U 1C f ull s ea lesystem, lu practice of course ~hisis not p C 1 s ! d b J .e ( l ' e f c t to Ship l iydrodyncamics-I lecture notes)an d te sts a re 1 ' 1 1 1 ' \ sa .lisfy jn g F n a nd g eu ln g a s high a R c as possible (\~'ith sca le ) and s i m u J : t ; H i n gturbulent flow overthe mode! hull,

. . .

(V' l-Y' - {Y.}..v S - ,,)m - .L U L~\.(!p. " 1(5.32)

N01C~ s =e-shiprn=omodel

a turns om that the dependence I O f dimensionless motion derivatives on both Re andFn is not normally a sensitive Cineand [n at m o st of thedimensicnless derivative-sam m ore o rless independent of speed over a substantial range of , o p e : r < " t l m g conditions, This isuseful inthal the nnn-dimensionalised equations of motion can often be used to represent such a rangeof operating conditions asopposed to the dimensional equations we have so far been usingwhieh strictly apply to cnlyene operating conditi,on. For example, the: dimensionless f om1 o f-rile (horizontal) maneuve r i ng equations for a submerged or surface craft a re g iven illequa t ion(5.....) and the dimensionless derivative may often be taken as constant.


52/68

(5.:na)

(5.33b)

(. I_Y')" -V iV '_Y""+( '-Y' '=Y'(t)n II' ..~ ;rm r . '- N' v' - N' v' i (l' - N' I r j - N' r' =N.(.~v 'II .I:.! r IW he n we exam i ne the d im ~m ;jro 'n less form of the other i por tan t m aneu veri ng

cqualiorls namely s;ymnl,{~'!nc nUl 'l ions ' o l f subm erine (5.34) we m ay no te tha l ti ledi rnens ion leas derivatives l U " ' C n ot w lt.h om 'the ir d ra wb acks a nd that som e ca fe in ana lysisneeds to be taken. The little problem here lies \ : v i m the POsil,Ional, derivative MEl SUC l 1restoringinfluences for displacemeota are f und \0 be fai:dy in se nsitiv e to flu id dynamics, andhydrostatic effects dotniusnr. if we express Mo as 'mgh' (see fig-ure 4.3) and ]]011-dimensienalize it, we see that Me" Is di.reCJtly depe .nden t on re fe re n ce sp ee d and canno t be B ,constant [see (5.35)~. This illusaates UlaJ ' m e vertical stability and control of a submergedvehicle are markedly affected by the forward speed.

Or more neatly,

( I 2" J . , Z Z '" ( . Z, t, Z'()nl- ...." , : ! , . - . , ~ , , ' - . ~ q - m + .;P , I = I~ t M j I M' ( I ' 'M ') " &.~, ~ M'tl i!..1'('- " 'IN + ! , ! , 'w +' r - I. q 'I - I\'~IIqj - ,~g = l~ t j

S I '


53/68

mgh~ X,ptrl?,b gL=-m --=;L U- (5.35)

. 'l.gL:::::-.:n ~l~. U~

A s a fin al wo rd , the precess of non.~rumeMionaHsiFli ,gl ie .equations in the way iH u s tr ate d d o esn O ' l work if U = O! (Hovering submerines, ho v,e re ~ sta Uo nk.ee pin g vessels). In such c a s e ss o m e o th e r "reference force" descripl i o n m : J i J s (b e : u s ed c th .e r t h r u : : i !, 4 p ulL 1 ..

So fa r i t ap l't.: arn [ IUH we have beea ceesidering (vill. d~ri"aU.vcs) th e hydrodynmmicc ha ra c te rls tl es o f th e bare hun ofa vessel N 0 n~en: tkmbas. been made o f c on tro l sur facesandf ixed E ns. O bviou s] y the prese nce o f sueh appe :n d ag e :s (in [he ir un de flec te d st~ te ) o n @ . hul lwill i n e f feo lm odify 'Ihe d e ri va ti ve v al ue s: indcc.d t to a nex ten t i t is possible to "tune" certainderivatives to improve corresponding chsraceeristies -

Yo begin with we wi]l .illluslrnte the broa:d principles involved in ihe installation ofcennolsurfaces and fins b y means of a f a : i . E J y idealbed tb.oo.r)'. This considers the controlsurface or fin as a Hf"l.ingsurface with ronsmnt lifting cheraeteristics (CL versus af anduniform inflow characteristics (speedand c r . ) deEennm.edby atlgular defleetion S ofthe surface(if movable) and. the instantaneous body motien, p_clC;rs. T h e obvi.ou~ nmissiens in thetreatment are:

~. Flow deflect] OH s due to the 'sh~e IO f hull....2. Non uniform flow caused by b o: un d ai llJ la ye r, . .s ep an uio :n vortices and propeller

swirl.3 - , Increased ve i0 cities in . t he p ro p eH e .r W~s?tL

In the literature you may find some or all of these aeenunred for (probablyempirically) in various ~pplicadons- betthey ]'l,eq;uire careful eng inee . dn ,g judgemen ts ; indeeddoes-the use of the simple model. We willconsider,

-


54/68

5.4.1 CONTROL SURFACES

'8 ) To calcula te the fo rces (i:!n d mernents) o n any specif ied fin at an angle to ensure tha t,it is adequa te ly strong and powerf-I. I I ,enough servo-actuators etc, o r e installed in th e vesse l fo ri ts c en tr o I and D 1 : i o v cmem ,

1 T o c alcu la te ' Ihe resul t ing force on the vesse l which can be slo tted in io ourequa t ions ,

It shou ld be no ted tha t (a &. (b ) a rc n 01 .tfu e'S8Irw:since th e cont rol surfae wi l l . possessin ertia an d daJ't l l"jng eha rae reristlcs o f its O\llIJ1. Needless [0 S ') ', we are only co ncern ed w ith(0) 1 0 0 examine q u a n ' l i h l , 1 h i C I . y how th e vessel motions can be centrclled by cont rol surfacedeflec t ions.

L ei us consider the acuo n {I < :I rudde r on a surface ship e xe cu tin g h oriz on ta l motion,neglecting "I the momentangular orces due to angu la r velocity and accelera tion of therudder, Figure 5.3 illustrates the instantaneous kinernar ics an d geometry of the idealizedinc ident now to the rudder, yielding me ang le of at tack a as, given in equation (5.36 ). Fro mth e elementary theory o f tifting snrfaees, (h e h'l1emi force; FR exer ted o n ID ,erudder by water is .given by equation (5.31) where Cl is the lift coefficient (dimensionless). (Note that here el. isnon-d imens iena l i zed using the leng th and re feren ce speed of the vehicle ). (OC J8 ct.JIi= Q is rileslope of the l ift curve at zero angle of attack fo r m e particular control surface design. We cansee that F I{ cons iitutes a contribuucc to the side force tilstuibance &Y and that FR . i s : a

Expanding for le t in 5.37) yields .5.3821) where we can see the Linear contributions, ofunde flected rudde r to Y II, and Yr' g iven by RY\.'and RY( respectively and lhus convenientlyseparated effect of rudder deflection. The component contribution to l'l.Nr caused by the ruddermaysimilarly be descr ibed as, 5.3Sf b). 'h e re la t ionships between the [udder der ivat ives are


55/68

given by (5.39). The effects due toinstallation aad deflection TIof a set of hydroplanes on asubmarine may be s imi lady analyzed as given in (5.40')

(5.36)

(5.37)

-5.3Sa)(5.3gb) , -

where,

Contribution to !U.'

{OC ): = C . t _ = ' Q Q . " " II

= ( _ \ " ' , , '+ bq ' , f Gel. 1T J U t:A fu )0=(n~ wf + b q ( ~ - l= zr"+ Z w~ Z' ,q ''1'" FI.. H Q

(5.4030) -c-v;rhereb =d istan ce o f hydrodynalJ l lc cen tre o f hyd ro pla ne s a ft o f C)


56/68

acceleration of th e COJi]II"O~ surface wb ic l: l s ho u ld be considered, Generally the various"u nd efle cte d cc ntr lb utio ns" a re '~I,um'pedi" '" wfuth th e h ull fo rm d eriv ativ es 'With m o de l tes tsb ein g p erfo rm e d w ith fixe d su rfa ce s a tta che d. T 1 : H ' : c ou tm l su rf ac e d ef le ctio n d eriv ativ es winlikely be obtained viaempirical data, calculation and 8epa:m~e eXperiments on a larger scale.

In o ur lin ea r tb eo ry a rud de r m o ve m en t is co nsid ere d a s a n a nti-sym m e tric d is tu rba nce.and we would e}!pecl l an ant i-symmetr ic forting inf luence Q(t). see equat ions (4.43) &, (4.48)~given by 1(5.42) . S imHady , iii bytiropblne movemenr 0 0 1 ' 1 a subm arin e is cons idered as asymmet r ic disturbance and we ~ ocld expect syJml e tric f orc in g in :l: 1u .e nc eg iv en by (5.43),The eq ua tio n s o f nntisymn1clr ic m o tio n [p le ase re fer ba ck to (4.43) '& (4~.48)]1co n si d di l'l .g o n l yrudde r d e fl.e c tio ns a s, ex te rn a l fa ,rci.n g in lluen ees a re g iven by (5.44 ). A nd the eq ua tio n s o fsymmet r ic mot ions fo r a d e cp ly s ubm erg ed su bma rin e ecnsidering on ly h yd rop la n e def lec t iona s e xt er na l load ing . is g iven in (5.4 5).

(5.42)

where (I is the r u dd e r d e f le c ti o n ,15L ).

XII I f i i ]Z.'1 iIM 1'1i I J , (5.43)X T JZijM~'INhere 11 is th e hyd ro pla ne deflect ion 11 1t),

'_

(5.44)

(5..45)Whe r e llF i sth e d e fle c ti on of fw d h yd ro pla ne s a nd

l i l A is th e defleeticn o f a n hydrcplanes ,


57/68

COlltr Ibut ion to 6M! ~ ,

(S.40b)

whereZ 1 . , , ( acL)111=0-,-O { f . n; r " ( DCI )= - 1 ' 1 ' " ----'i;I..,Ii II Ba. 'n

5.4.2 SKEGS &: 'j:.IXE:tl [?JNSSkegs .& fixed [ins can be t rea ted in a s imi la r manner to rudder &hydroplanes. l- ' r

exam ple a , skeg m ay be tho ugh; o f as a fixed n Jd d~r and silnilat results m ay be o bta in ed lorccnt r ibu t i cns ' 1 , ( ' ) th e sway fo rce an d yaw m om en t a s given in eq ua tio n (5A l). T he d iffe ren ceis. o f c ou rs e 'that there is no deflecnoncontnbution,

,V : = _ I a ~ 1 )lCfJ. "I _ ' l e c ) . ). N " " -a .....c o : . 0

v: = 3 ' ( ~ )! 0 0 . D(5.41)N ' , ~ ( B C I . . )=-a --, r !l....~,I'i.'" Q

where

and 'a' I . S distance cf'hydrodynarmc centre ef skegaft of C,

5 . 0 4 . 3 REPRESEN'fA: ION OF CONTROL SURFACES IN EQlUATIONS OLMOTION

We have shewn how the installations of control s-urfaces may be represented byadditioualcentributious to certainderix atives phIS an active forcing term which is dependentuprm the control surface deflection .We discussed in the simple treatment on 'the contributionsto the veloci ty derivatives but clearly ,accene~'lIon dcrh~a~ive.'l,wil l also be affected. Similarlythere will be add ruo na l ac tive fo rc ing . re rm s proportional [0 th e angular ve lo city and

-


58/68

The signifiesnce of the centro I surface velocity and acceleretion terms is largely afunction of ship size and sensitivity,. For example in a large directionally (very) stable tanker,a helm alteration will need to beapplied for a long time to get response from the vessel, Inthis case the deflection term (acting fer a long time) dominates the: response, At El]eotherextreme, consider a smal] boat with a large rudder and' a very fast application capability - theresponse will be very quick.

'Whether jl.l'ilified or not , the control surface (denecrion) acceleration and velocityd erivatives a re o fien ig nariSd in m an oeuvriag ,!IDalysis and the resulilng equa tions o f m otionare sim plified , Forexam pte , ihe di rnensionless {ann o f the ma n oe uv rin g e qu atio n rn ~f er backto (5,25)]1 arc g iven by 5.46 ). N Olc 1ha l ytr.8B1.ml N '6 0 re fe r E O changes o f fo rce and mome n tdue to the ' deflection o f co ntro l sur aces d no '! the flu id fo rce and moment acting on theru Id er,

( m l,- V : , )v~- V : v ' - v : r , ' ~ ( m ' - Y ; ) r ~ =Y ; , 5 }- N r , " /- N~ 1 / .I{f - N~ i t - N' r ' =N~s~ ~ d I !,

5.5 CONTROL 0" MOTlO S

Considering OL1requ ation of horizontal motion (5.46) when a steady state is reachedwith a constant rudder deflection B.c.yields (S. 7 ). EHmm a .t i'n g v' (which is aow a constant)from (5.47) relate to the constant y.aw Nne to rudder deflection as in (5.48). The steady rateof yaw per unit rudder deflection given by equation (5.48) is an indication of the effectiveness

"_of the rudder. By expanding Y\'1 and N'r in terms of rudder, skeg and hull contributions as in(5"4 . 9) and relating the rudder and :s,kegeentri bunonsro their respective (de r J B Q . ) a . = O valu es[sec equation ( 5 . 3 9 ) ,& (5.41)] " , 1 . 1 1 allow us to ~eied a rudder and skeg ccnfiguraticr; toprovide specified level of steering capacity.

'1.;" .'.j ,. y')'1 'VV .,. ~m - I r(5.41)


59/68

(5.48)N ' ( ' , .1 ) A . " l J 1 ' O N ,ro.~' tn~Y,r ,-.-~. l~ .. ' r

y'~,= R Y\) - i I - sY'~ - ' I F (Y"Ii f u r b a ;, r .e huH)N" ,= ...N ~,) + sN' '. + (1);J' .fer ~81;re hun)' y l~ ,. . iii I, ~ JU ~ 0 0 " (5.49)

5 . , 5 . 2 1"ONGITUD1NAl..POSJl'~.O'N -G i l ii ' RUOllEl't.. -

Fro m eq uatio n (5 .3 '1 ') w e have the re la t ionship b~Lwe-enyt6& N 1; e given by (5 .5 0); ands u b s U tut in g f O l ' N ' e i n ( 5 . 4 8 ) yields 1 ( 5 .: 5 ~ ) . N ' O i t , e lho'l U l : e direet ional st~bilily t o pr~v a l leq uat ;,o n (5.27) req uires U U l i 1 . the d en c'rn in a[c,['o C (5.,5 t) is pO : f ; i , i t i ve, For the g reatesteffectiveness oftbe rud der in te rr:ns o ,f its pesition, a ' weW8rU the rnagn itude of the numeratort C ' i be as l fl l'g ,e ~,S pos si bl e ..Forn,':IO'S~ShlPS 'N~! l is rnog.!1iHvc,and as Y' ~ is ciearly negative then a'should be es positively lerge as pessible, Hence JJ::Imlficnf~(Dn for the positioning (If rudder at ,..th ~ stern partly for pro jection , tatgc:~y [0 m ak.e use 'o f pro pe lle r s tream r.:ftcclS which areI" sJ "! .1cub lr ly use fu l f o r man~uvelring au ]o w speeds.

In the equatiou ( 5 . S ] ) a' could be regarded as th e polO[of appl lcat lon of a side force(i.e, a force in.rhc )I direction), It ean be seen lhal~f a~' is chosen such tbat the numerator iszero then r~=0 andensuasg motion!l:l,'iH be a pW'-ed.rift superimposed UP0'l"l the referencemotion. The position is called the ~'neutr:a~ poi['lC~for la tera] disturbance forces (andeerresponds to the center of lateral resisrance) and is,given by (5.52),

N'~ (5.50)

r' " y 1 . 1 f , e N ' . . +a~y l~ , )5 C NJ y (m'


60/68

L et us he re co .n sid er the steady state sttaiglu-line m o tio n o f su bm a rin es a s affected bythe deflect ion of hydroplanes . , For such m otion ,00:::::;' " " 0;;;,:q an d our dimensionless equa t ionsin (534) reduce [0 (5,.53) where Z'.Hand M" represent the resultant force and mome n teffected by the hydroplane combination {usually 2 sets - fore and aft} as given il l (5.54) ..fig ur e 5.4 H lYs t rn t c s the li .mmta~i'om o f the possibilities fo r Z Han d M\\ fo r just o n e o p er at iv eset of hydroplanes, andthe greater 'fl!cxibi~ily' when eoth sets are operative,

Ahl1C:l'llghwe co uld b e in te re ste d in W or specifically, m ore re levant , oael ' l , is rile rateo f c ha ng e of depth g ]ve n by,,,' (r e l er re d t o tCq u ]1b ri ur n tIi lU!S - se ctio n 4 .4 4) where V o l ~ i J : l gi \ lenby ' (5.5'-).

tlZ'11 = /jZ' Uf AZII!A: : 2 ;'1 n . 1 Z " f t 1 1 "140, -. 'otti ,- (S.S4a)

L \ M " i = . n . M 'al -AlI.,1''''A~Mi'l 1 1 . . . . . 'TM\l Ih110 f ' : ' " (S.S4b)

- - \'1 ' -w' ==w-1I0~ Ul=-G)= U(WI-O)u= u [ - Ai,' j:f 'T _.l_ { ~ . J f o . . l :u. ' + l \ I ' o v ; . } lZ' M' Z H HtIr D -.. .....-[ -I I {M' 1 1= u ~+~ __ W + x ' AZ',Z"",!\{'(, . Z'.. H J _ . .

(5.55)

- ~ l I 1 - I,.11 " " ' : ' 1 1 . ,- U .--,-. + -.,-, [x ~p-x , ~ } . C i Z HZ . . . ~ ' - a J{a) Mahd:enan.ce of Constant Trim:

In (5.55) it cal] be seenthat if AZ'M is applied! at X'H = i\'NP chen e = 0 and the boat willm a i n t a i n level t r im, x ' J ' o O ' ? : i Sthe "neutral palm" for applied vertical forces: a f o r c e applied llerewould effect a depth change but no change in trim,


61/68

-

. liZ I I

"-._ -I

iaUI'tl21 1 \ z --

X II - Pos i uon or resu lta nt h yd ro dynnm i"c fo rce in th e z di.recll,on -,!jM .m = - ililW AZ nr

W i'rh o nly (me operative set othydrophllles -~ X L i is lixed < 1 1 hydrodyuamic centre o r relevant fins

W lth tWIJ operati ve sets of hyd roplanes: - In theorv vsc ... x l i t < :x; I C I X I I I~ x: = = pure moment) - 821 1 and ~M Ij may be controlled separately.. M 0 : i . : , S ' : n U U 1 val ues of dZn an d 1l,Mu are approximately dQubL':; if_ hydroplane: sets

are located near bow and stern

H yd ro r Ianes d rng e ll III l ' I1lU1i1 ru has heen u rnule t! since for sm a ll a ng le s bei n g considered[ 1 1 1 $ is m ore l H [ess C L 11 1 st., m wu h 1 1


62/68

Ml-I I' . (I _ ,X H = X i" 'P' - -- =X . fil


63/68

, , 0 _ '

S - crin ca l poiluN - neuun] pm nl (ll\1dlUd~ nanuc 1,;1 :111e uf veri ical forces on h u II)

,ISIIr'm;.i r j~'II' i depends 11 1 I\vdspeed "alwa)'~; a n of N}. ~Si

~I!F ix ~d (a pp rn x .)(co liid lle fw d ur 'a f t uf C "}I .i

S inn] h ,::r'''__---I---~I''' La rSterU U

6ZH M P V1jl!d here Jll11dlU;~!inochange in UE.II~TU t..lu applied he re producesItO change in TR J M

Fi g. .5 .5 S ig ni ficam :p{ l~nts ~n su bma r in e conrml

-. -

-

6L


64/68

x '1 - 1 :Po si ti ,o n o f f,eswtmt : hyd ro p la n e f or ce in th e z d ir ec tio n

AMIIA. =-XUAZI-t,\; AMHF =-XHfAZ HFliZH=AZI-IA + AZFlF;6.MH = = ll,MHl1+AMl'IFf!.MII = -X'II filZIl

W ith o nly o ne opera tive set o f hyd ro plan es: - i;i i s f ii 'r !d at hydrodymlilIDc center of'relevan r fin s'!II AZI rand aM11 are in a f ix e d re la u cnsh ip dtrcdly I inked via x' u

Wi th two o pc :ra ti 'v e s et s o fh yd ro p Ja n es :-in theory -00 < :: x: . < :: (;Q I x ;~ I'~'00 ~ pure m om ent6ZH a nd f l.M J.! rnny be co n1 40 H ed $~para[dyMax imum values G f fiZ H snd A,! '! .' h< lrc approx , tmatel ,y doubled if hydroplane setsarc lo ca ted near bow and stern

fN UT E: H yd roplan es drag com.dbutlon has been .om in ed since for smal l angles beingoons idered these ant m o re o r le ss co nsta nt with l'l.)

BEE FIG 5 .4 .b SIGNIFICANT POINTS m S UBMA lU NE CON TROLC - cen te r o f-m ati'S -S - critical pointN - neutral pohn (liydrodynamic center of v nical forces on hull)

General arrangement

SPosrucn depends onfw-d speed(alwaysaft o fN)

c N.F ixed (approx.)(could l ie ' fwd oraft ofe)

'ixed


65/68

AZJ-lapplled here produces AZH applied here. producesno change in depth I } . O chan ge in trim

We have seen 111al We have the scope to adjust -al< x.H


66/68

K ';)~ ( ",'~.~...'.,u '[

FIg. S .6 Variation in : : n 1 .W and y a w rate wid, (ide.atised) IA orderresponse to stepped rudder detleerion ~.~. . . . . . .


67/68

-deflection. while T gives a measure of the "sluggishness" in responding to the helm. K' andT' in the context ofNaval Architecrnre are refbred to as "Steering Indices",

Ai~'+B:r'+Cr';;;;;ES + F B ,where A ;::= {m '- Y !. V r - 1 ' \ ' \ - Y ~ N '. " u,l\~ ] ~ 1I

13;;;;( m L _ Y ' )N ', -N' Y' (m '-Y ' )N ' . . . . . J ( t ' -N1 )y .r ... ,\ ., r r r '"C.: ;; ;(m" -Y" , )N '~ +Y '. N'r (5.58)E = = N' Y' -Y' N'- v a ~ ' ~F = tm ' -Y ', )N ',,-N . \ ' 1 >

(5.;i9)

E { f ( B F ) ( A F ~ " I " 'C 1+ C - E D- C -+ E: D- - - - - aif we let K"=Ere ;InaT = H/C -.F E aadwetruncare 1he:series 0 (Vl'I1':ri" t' = .,{)

Output : r'= K'a.-[I-f:H~)](5,61) -

~ .: "


68/68

REFEllENCES:

J . Lecture Noles. Uu,i'ller$ily Co/l 'egf! Londof,12 , 8 labiU lY and Mo ,tio n C on tr ol o /O ce cm Vehrc,It .U';Ab k l !HO \ - ! l t z M A3. M echanics of M ar ine Vc 1 1c ie s. C lo yln 1 '1 BR & B is ho p J rlED4. Basic Sl1lP TJ:eo,ry. Rawson KJ,and Tupper EC: 5 . Principles of Naval chltrecture. edited by Lewis t:W

manouvering notes

Documents