Catenary mooring lines with nonlinear drag and touchdown

No. 333

January 1997

CATENARY MOORING UNES

WITH NONLINEAR DRAG

AND TOUCHDOWN

L.0. Gan^'Rios

M^iM. Beriiitsas

k. Nishimoto

.Ät'sä-THr ÜWVERSiïY OF MIM^^^^^^^

COUEGE OF ENGINEERING

No. 333 Jannsuy 1997

CATENARY MOORING UNES WUH NONUNEAR DRAG AND TOUCHDOWN

by

L. O. Gaiza-Rios M- M. Bemitsas K. Nishimoto

Prepared for the University of Midugan/Industiy Consortium in Offshore Engineering

Dqjaitmoit of Naval Ardiitecture and Marine Engineering CoUege of Engine^ing Tbe Uxiiv^ty of Mchigan Ann Aitxff, ïiÊdügan 48109*2145

ACKNOWLEDGMENTS

This report is a result of research sponsored by Üie University of Michigan/Sea Grant/Industry Consc^um in OfEshore Engineering under KGdiigan Sea Grant O^ege Program, project number R/T-35 under grant numb^ DOCrNA36RG0506 from tfae Office of Sea Griant, National Oceanic and Atmospheric Administration (NOAA)» Ü.S. D^artment of Commerce, and funds ftom the Stale of Michigan, Industry participants include Amoco, Inc.; Conoco, Inc.; Exxon Production Research; Mobil Research and Development; and Shell Conpanies Foundation.

ABSTRACT

Analytical expressions modeling the tension dbtribution ahd geomietric deformation along catenary mooring line, including touchdown effect and nonlinear drag, are developed. The motions of a catenary mooring line in the horizontal plane are studied thoroughly. Due to the complexity of the modeling equations, solution can be achieved only by iterative methods. Approximate values for recursion in the sc^ution ctf thé horizontal tensicm in the catenary are derived based on tibe first order approximation of the geometric properties of the line. Esqsesäohs for tfae nonliniear drag in the faorizontal plane of tfae catenary are dmyed based on energy dissq>ation principles. The resulting analytical expsiBssions serve to calculate the forces in die catenary due to drag In the horizontal plane. The nonlinear drag forces are then recast into a form ssàtalcàe for niKxmng plications.

TABLE OF CONTENTS p.# ABSTRACT i i A C I ^ O W L E D G M E N T S ™ . ^ . ^ . . . y ^ , . . . v . v » » . . . » y ^ .>,v...,.,, i i i LIST O F FIGURES . . v L I S T O F APPENDICES ..w...ww. w . . . v v v . . . ™ . . . ™ . . . v y ™ ^ v l NOMENCLATURE ...^ vU CHAPTER I . INTRODUCTON™ 1 I I . CATENARY MOORING LINES . 2

2.1. Catenary Equiations « « 2 2.2. Sohition fOT the Hcnrizprital Teasion in the (jc,y) Plane 5 2.3. Approxirnation of tiie Iterati(»t Values for the Hcmzontal tension — 9

2-4. Scdutimi to tibe Catenary Nfodel U I I L MOORING LINE DAMPING 15

3,1. DanyatipnofCate^i^Daiiqnn^ 15 34' Nonlinear pariçingR)rc^ o n & C a i e ^ 18

3 3 . NonUnear D a t i n g Forces on the Vessel — . — 23.

A P P E N D I C E S ...27

B I B L I O G R A P H Y . . . v . . . , . . . . , . w . . . 41

U S T OF FIGURES

BSUIX UsÈ

1. Geometry of catenary

2. Displacenient of the cateàoaiy in the 2-D plane » 16 3. Geomeaoyofthenaoormg^s^^ . . . i . . 24

LIST OF APPENDIXES

Appendix a J 1. I%st Order Variation ofthe Horizontal Tensi<H) m Tems of te

Projected Lengtii of the Sûspehded Catenary ...v....y....v..>>...,..>> 28

2. First Order Variation of the Hpiizcmtal Projected

Catenary in Terms of its total Horizontal Length 30 3. Order Expansion ofthe Drag Terms with Respect to die Hcrâ

Fronted Lragth of the Suspended Catenary . y . . . ™ ™ « . . . . , . . . . , . 35

4. Relaüäon ]Mwee3i tiie Effective Length of tiie C ^ ^ 40

NOMENCLATURE

A referoice point denoting the upper point of tiie catenary

B reference point denoting tfae mooring point of tfae catenary at the sea flocu* Cp drag coefScient for the catenary

CG center of gravity of the vessel

PICAS Differentiated Cbiiq)liance Anchoring Sysitem(s)

^ horizcHXtal lengtii ofthe undefcnm

Dtff effective diameter of the catenary

horizcHital drag foiœ in the direction paralld to tte^

Fit Fl horizcmtd drag foitie in the direction perpradicular to ^ ^ dancing force in the direction of motion of the vessel

danying forte peajjendiailar to the dijEection of rocfdon of tte vessd ^ water depth (or vertical projection of die catenary)

^ horizontally projected kingtii of the suspended portion of tiie catena ^' horizraital distance between noooring and attachment pdnts

^ length ofthe suspended catenary tccal loigtfa of tbe catenary

Mjo dampiiig moment about die Z-axiis

P v ^ c a l foroe per unit catenary length

^itT horizontal pretcaisioQ in tfae catenary line

r yaw angular vdodty of tiie vessd

horizontal rigiiËty (stiffiiess) of the cateaq^

s azclengtii coordinate of the sus^pended portioxi ofthe catenary SMS Spread Mooring System(s)

T{s) tendon (fistnbution of tiie catenary farQ^s^t^

TMS Turret Mooring System(s) Tq horizontal tenaon of tiie catenary Ty vertical tension of tiie catenary

û fcvward velocity of vessd with lespect to water

V lat^vel(Kity of vessd with respect to wator

{x{s\y(s)) horizontal geometric (fistribution of cat^iaxy ha&forO<^s<^l^

(x,y,z) reference &ame of catenary line at tiie poirtt of contact with sea floor

(X,y.Z) refnenœ coordinate fraroB of the v^sd

(xc-,yc) famzonM geotostric distribution of tbe catenary fii^ (-^0*yo>^) coordinates of zeiro skipe of the catenary

(x^,y„,Zm) àK^diiiates of the niooting point at die seaflopr

{Xpyyp) body fixed coordinates of the itfa fîadrïead

(xj^y-f^ZT) coordinates of the ^pper flttarhmftni point of tfae catenary

z{s) vertical geome^ distributicm of c a t ^ ^ line for 0 ^ 5 ^ igg

Greek Symbols

^ angle between tiie X-axis of tfae vessel and the catenary, measured ccMHiterdockwise

y angle betwie^ the §-axis and tbe catenary, measured counterclockwise 7(x,y) an^ between catenary endpoutts ineasured with respect to tfae (x,y) plane

energy dissq)ationfinx:ticn of the nKx>ring line

^(^) an^e of the cateaiary witii respect to tiie horizontiEd plane at point 5

^/ angle between the upper eod^xmt of the mocmng Dne and tfae horizontal plane (1,77) earth-fixed referuice fetme

P water dmsity ^ yaw {or drät) angle

Spedal Svmbols

(*) value of (•) before displaoem^t of tfae catei^ny 5(») first order variation of (•)

A{*) displaoement of (•) with respect to its iititial position

L INTRODUCTION

In deep water operations, several types of lines can be used for the purpose öf towing, mooring and andionng. Tfaese include nonlinear elastic strings (liylon, polyester), catenary chains, and steel cables. A nümber of mooring and anchoriiig systems, sucfa as Timet Mooring Systenis (TMS), Differentiated Compliance Anchoring Systems (DICAS), and general Spread Moprmg Systems (SMS), use a hybrid combination of rhooring lines dming operations, espedally in de^ waters. A cornlnnation of difîerent types of mooring lines is soibetimes required to decrease tfae overall weigfat of tfae lines, tfaus reducing the verticd force on tfae moored ves^..

Diffexent types of quasistatic and dynamic models of towmg and mooring lines faave been extensive^ smdied in recem years [1,3,5-8,10,11], with varicHis degrees of variations in the coniplexity of the naodels. (^uasistatic models can be adapted to study tiie slow motion dynamic of towing, moormg, and anchoring systems [10]. In tins work, an analytical modd for the quasistatic analysis of deep-water moodng/anchoriag catenary cfaains is devdoped. TMs model include^ touchdown effects and nonlinear drag. The modd developed is two-(Ëmensioiial due to the nature of the catenary deformation. Ihus, bottom friction due to off-plane motion of tiie cat^iary is npt taken into account

Cateiiary chains are heavy compared to other types of mppring lines, faave in-plahe deformation, are neariy fully submerged arid tfaixs faave faigfa faydrodynaimc resistance. They are commonly used in most moormg ffltd ancfaoring situatiims.

The equations for the cat^iary are deyél(^»ed in Chapter H with a special emphasis on deriving analytical expressions for the faorizontal tension at tiie tc^ of the cateiiary. These «qjréssiôns reqitire an iterative solution. Two methods for solution are develpped in tfais work. Expressions for the tension distribution dong the catenary as wdl as tfae geometry pf the cateoiary are also derived. In Chapter JE, analytical expressioi^ for tiie drag force con^nents on the catenary are developed using enexgy disdpation fun^ons. The horizraitd plane riimlmear drag forces due to the cat^iary Sterted on tfae vessd are obtained based ott tfae slow oQotion dynaoiics of tfae mooring/ancfaoring vessel. BnaDy, sjocfa expressions are recast in a form suitable for implementation in a dynamical nK>oriiig/ancfaoring modeL

n . CATENARY MOORING LINES

The typit^ ^iôatioas for inextensible catenariies in towing ^plications cannot be applied in deep water mooring operations because they do not take into account toucfa(k)wn efiects md iipiijOaiear drag. The mam focus of tfais di!q)(^ is to dev^^

for the horizontd tostsion at tfae top of tfae catenary using the catenary relations for mooring applications. Additiond expressions for tfae tension distribution dong tfae catenary, and tiie catenary geometry, are also doived.

2,1. OtwuiiyP/ptatiAfi«

Hgure 1 sfapws the geometry of a deq> water catenary line; The origfai of its reference firame (jcy,z) is located at tiie pomt of ccmtact of the catenary with tfae ground (x^.y^.z^), wfaicfa conespcmds to the point of z m dope. In this figare, x=^(x,y)Tepres^it5 the horizontd plane of the catenary; (x„,ymam) ^ ^ mooring point on the sea floor; {Xf,yf,ZT) is tfae attachm^ pdnt of tiie catenary on tte vessd; is the lengtii of tfae suspended catenary; t is tfae horizontd düsitance between the ihooring point and tbe attadbment pcnnt; ^ is tfae faprizisitally projected lengdi of die suspended catenary; d is tiie faorizontally projected leng^ of tbeandeforin6dcatEmaiy; and A iswaterdeptfa. hi ad<£tion, 5 is the arclengtii coordinate of die »ispended catenary, wiftt its CHÎjgjn at tfae poirit of zero slope (x0,y^,Zo)=(O,O,O); and d(s) is tiie a n ^ of tfae catenary with respect tp the (x,y) plane at point s. The geometric configuration of tiie cataiary is giv^ by x(5), y{s), z(s)and0(.s),fcjrO^s^igff,

The totd lengtii of tfae catraiary s ^ven by die geometric relation:

^ 7 = % + ^ . (1)

and fee totd horizontally projected length of the catenary is given by

/ ' = / + r f . (2)

2i (?)

wfaere P is the verticd force per unit catenary lengtii ahd To is tfae faorizontd tension in tfae (Jc,y) |dane, tfae dimerisioidess deqpi sea ihooring catenary equations are:

Xi=sinfa"^(L) ,

Hi=cosh(Xi)'l .

(4) (5)

Hgure 1: Geometry d deep vatat catenaiy

Expressions (3), (4) and (5), dong witfa the geometric constraints (1) and (2), are the governing expresdcms for deep water catejiaries The totd faorizontally projected lengdi of die catenary / ' is known, provided tfaat the positicni pf the upp^ enc^int of the cat^iary in the hocizcMtd plane (xx^yr) ts known witfa reispect to tfae mooring point (jc„,,y„) by tihe rdation:

In addition, tfae water depth h, and the totd length of the mooring line ij- are ccmstant and known, w h ^ i ^ the faCHizontd tension 7^ , tension distribution T(s), and configuration ctf tfae catenary (x(s)t y(j), z(5), and ^ i ) ) are unknown.

To find tfae facnrizontd tension 3^, catenary equatipns (4) and (5) are^ squared as follows:

Z? =rsinh2(Xi), = codi^(Jfi)-2cosh(>ri)+1, (7)

and tfaen substracted fiom one another to &id an ei^nession relating L and H:

- f f ^ =smh^(Xi)-cösh^(iri)+2cosh(Xi)-l=2[cosh(Xi)-l]==2ff. (8)

Thus,

Z.=Vi^+2ff . (9)

In dimeiKsiond form, equation (9) provideis a relation between the suspended lengtfa of tfae catenary and the horizontd tension 2^ :

(10)

A second dimeosipnd and T^^ is found from eqpi^on (4):

V = f s i n h ^ . (11)

By combming equations (10) and (11), an expression relating Tp to the horizontally projected loigtiioftfaesu^isaded catenary / is obtair^ed:

2.2. Solution fbr tiie Hmizrortd Tension in Ûie(x. v\ Plane

An exact analyticd solution for tiie faorizcxitd tension T^ can be dù^ctiy obtained provided botfa ^ and i are known by cp^>ining the geometric constraints (I) and (2) óf the system sucfa tfaat:

(13)

Then, rdatitm (13) cain be combined with (10) to find an expression m tent^ pf known quantities iy, i\ iand h wfaich can be readily solved for TqI

If Y\

^ \ PJ (14)

In the equation above, the only unlmown is To, and tiius tiie andyticd expressidi for tiie horizontal mooring line tension is given by:

In generd, however, the <HstaQce t is known (depends sdleiy on tfae position of tfae endpoints of tfae catenary as shown in (6)), but ^ is unknown. In such a case, Tq must be obtained iter^vdy. Ei^ression (12) done does not suffice for calculating Tg becaiKe it involves two unknowns m thiis case (7*^ and / ) .

Iterative expressions for the solution to the horizontd tension can be obtained as a fimcticnt of a dngle unknown (i.e. To) by combining expresdons (1) and (11) as foUows:

T f Pl

ij.-^mh " - r + / = 0 .

P ^ T o ) (16)

The expression above, dong witfa an additional expresdon that relates To and / are iised to obtain a dngle equation in terms of To. There are two methods to cdculate T^ if / is unknown:

MâhQâl

An expresdön relating r^, and ^ is denved fixim equation (12)

^ = %dnfa-* 4 - j / J A + 2 ^

p [TO^X PJ)

(17)

wfaicfa, combined witfa tfae constraint relation (16), yields a single equation that can be solved

forr^:

wah — ih Ä + 2 ^ -^t-^tr ~ J h =0

[Tp[i\ PJ ^)) Toi { PJ (18)

Letting a = ^,ejq)resdon (18) can be recast as follows:

( ( I ( 2\ ^ if 2\ sinfa a Jh A+— +t-lj - a , A A+— =0 .

_I

U ^ aJ )) \ V aJ (19)

A non-trivid solution to equation (19) is obtained by iteratiiig for a , and can be computed numeriü^y using a Newton^R^ison algoritinh [2] of the fornr

ÛE/+l=ÛEjî-fiaj) ƒ'(«;•) ' (20) where /(ay)-dnh V V a JJ \ 2 ' _{A + —} V ^3) (21) and

La]

A

+.1

l « i j

Y

In tfaismetfaod, equatipn (5) is c^mendpnalized

PA

= cosh PT - 1 ,

to obtdh tfae following eiqprësdôn of ^ in terms of the unknown T^:

^ = ^codi"

P _{\JO\ P J )}

(23)

(24)

Then, equation (24) is combmed with tiie constramt (16) to arrive to an equation witii a single unknown, 7^, of the fontc

T ( ^( P f T\^ T a( P ( T ^^--^dnfa cosh-^ ^ A + - ^ - r + ^ o o d i - ' ^ A + ^ =0 .

P \ \JO ^ P J J ) P \JO V P J,

(25)

A direct itmtive solution to tiie equiation above is mâtiiematically tedious, but it can be sni^lified accordingly tp obtain a single fayperboUc furrction in terns of tfae unknown

To-P ( T \ I

Letting x = — , and using tfae fayperbolic rdation cosfa"^(a:) = sinfa~Wx^+l [2],

\ P J

8

/ p I

f f pvf r ^^ T \( p f T \

U ï i j v P>' P {ToK P) = 0 , (26)

and tiien simplified in tfae fonn.

\JO\ PJJ TO \\^OJ

^Ph «

+ 2 ^ =0 (27)

Equation (27) can be fiirtfa^ reciast as

codi - a - i = o .

3i

(28)

Again, letting a = , the expresdon above becomes:

cosh

^V

(o*)^

Equaticm (29) can be solved for a using a Newton-Ripson algorithm of the form (20) witfa

f(aj) = cosh ^[ajkf + 2ayA - OJUT - O - a ^ A - l , (30)

and

gjÀ+1

J{ajh)' +2ajh

( \ ï

dnh|^^{ayA) +2a^A-a;(^r-r)J-A . (31)

Expresdpns (19) and (29). wfaich have beoi obtained witii different metiiods, are a function of a d i ^ e untaiown (a), and can be solved to find the tensi<m in the (x,y) ptoe To. Once the vdue for a has beniobiamed, 7^ can be cdculated fiom die seiatum:

2 i = ^ . (32)

2.3. Approximation of the Iteration Vdues fortiie Horizontal Tension

Equations (19) or (29) can be solved readily by iteration if an ^)propriate initid vdue fpr a in sequence (20) is provided. In this secticm, an ^^xhnation for a based on tfae first order expandons of tfae mooring line geometric i^c^rties is derived.

Ccnddér an initid known faorizontd pretendon Pj^ of the catenary line in the {x,y) plane. The vdues of tfae geometric propres of die catenary can be found readily as foltows:

03)

(34)

(35) (36)

Tfae overbar in expresdons (33)-(36) indicates tfae vdue of the geometric properties when ^<f -

PRT-As the catenary line is displaced a small amount in the (x,y) plaine, the geometry of the catenary changes, thus producing a cfaahge in T^. To find tfae faorizontd tendon after tfae £s{dacment of the catenary takes place, an spfscadxaate initial value for a must be ^ven for eidhser equatipn (19) or (29) to convey in tiie hexatipn process. Sûdi vahie can be foimd from the equations of eitfa^ of the two methods dè\élpped in Secticm 22. Letting

a ' = % = i , (37) P a

and using the equations from Method 2 of Section 2.2, relations (16) and (23) are recast in termsof a ' as follows:

10 f+i = 0, (38) f i \ \a'J f £\ h cosh — = - 7 + 1 . (39) \a'J a'

By squaring the two expressions above and substiacting (38) from (39), we obtain an expression of the form:

a'^ codi^f-^1 - a'^ dnh^f-^)=A^ +2a%+a'^ ~{tj -r-¥t? , (40)

Kd J \<X J

which can be sinplified to yield

0 - = - ^ — . (41)

Tfae vdue of or' can be solved for readily if botfa V and I are known (i.e. t~T and t=l). Wfaen tfae upper end of tfae mooring line is offset by a small faorizcmtd ainpunt, bowevor, tfae values ^ V and t change nonproportionatdy, depending <m tfae arnount of the displacement Ccmsequentiy, equatipn (41) can be solved cGrectiy ody if € is known.

Letting: t~l'-¥dt and l—l-k-M, where At and M are the changes of the geometric prop^ties after the displac^oi^ of the upper end of the line, die relative di^tlaceincnt betweëa

M and M' can be emulated to find an equation that can be integrated into (41) to solve for tibe single unknown a f .

The rdaticm between and i l f ' is derived in Appendix 2 as fdlows:

M

^ ' 1- cosh 'pt\ (idT\. j:pt\\' _{+ — s i n f a -^r} Jo) \Pdi)\To)^

(42)

where ^ is ti» derivative of tiie iKmzontd tendon To witii r^pect to t evduated at die ûdtid

di _

11

P,WA+2^codifP

Jhh+^Yp Ç c o d i ^ - d n f a a +A

ï V ^IK^oJ \^oJ \*oJ,

(43)

Fürtto,substimtihgrelations i~£+-^A£' and t-l'-^di' into (41), we find ei^qiresdon for a' ofthe form:

an

f - - - Ai ^ ij'tAi'+i+'^Ae -- i Ai J

2h

E^qxesdon (42) is substkuted into eqpratiiôn (44) to obtain

(44) 2Â<1- 1 - = ^ cosh — + ~ — smh — \ Ll ^oàir^Tp)\Pdi)'^To codi = + ^-77 smh ==- ïir-i T^dir^To) \:pdir\To)[^

Tl ^ 3 f V Jp?^ f i ^ . Jp?^l

In tiie exinesdon above, a' is the only unknown and con be cdculated readily. An appropriate initid ^^xhnaticm for the vahie pf a ineq^udpns(l9)or(29)isthusgiVenby a=^.

2A S to ti» Catenary Model

To conqdete tfae study of catenaries, the geometry and tension distrtbutions pf the catenary line are doived in this section.

The hodzcmtd geomûric disttibûtUHi ôf the cäitehary x{s) = {x(s),y(s)) - starting at the pcnnt of zero slope {Xg^yo^Zo) - is obtained by dimendtmaldng equation (4) as follows:

12

x(s) sï-^-smfa ' ,

P {To) (46)

wfaere O ^ s^ igff, Dénotfaig tiie angle between tfae endpoints of tbe catenary measined wxtfa respect to tiie (x,y) plane by y(x,y)y tfae faorizontd components Pf tfae catenary distrîbiition become: xis) = y(^) = T \( Ps\ - p ^ _ cosr(,,y), r \^oJ, "T -\( Ps\ f Sinr(x.y) • (47) (48)

Ihie above ^resdpns can be incorporated into expresdons for the horizontal distribution of die catenary (jc^.y^) as measured firmh die niocùing point at tiie sea floor (i.e. from the p ^

(j^,y„,Z|„))asfoUowK

Jfe = x(^/ + 5) = [Ei COSr(xiy) • (49) L

Pi

dny(jc,y). (50)

The verticd distribution z(*). of tiie cadtenary, is obtained from the dimendond form of equaticm (5):

V^/x(5)^+y(j)^

2(5)=^

and tiie verticd aingle between tiie catenary att any point 5 (j a 0^ plane (x^y) is given by:

0(5) = tan" • i f ^ l t

13

Fîndly, the tendon distributicm T{s) dong tfae catenary ( 0 ^ 5 ^ i^) is calculated as a function of die horizontd tâodon of the line To, and the verticd tension Ty, where

Ty(S)=^PS, ₍₅₃₎

as follows:

T(s) = 4T/-^Tyisf . (54)

At the upper erid of tfae catenary, wfaere s = ttie verticd tension in tfae mooring line is ^ven by

(55)

based ficmi equation (10), or by

2v(5 = V ) = ToSmh^ , (pi^ (56)

based on equation (11).

Using tiie expressicm forTy in e^qtriesdon (55), the totd tendon at the upper poûot of the catenary becomes:

(57)

wtoch can be reduced fuidier to the fonn:

ns = t ^ ) = To + Ph . ₍₅₈₎

Similariy, tfy combiiiiiig equations (54) and (56), an alternative form for tension at the top of the mooring line can be obtained:

14

Tis = i^)^ To^ + To^sinh^(^],

\ \*oJ (59)

oréquivalentiy:

T(j = %)=ToCO

By substituting s = if^ in rdations (49H52X the georhetric distribution àt tite txpp^ end of the catenary is ^ven by:

d(/^)=tan-l z ( ^ ) = ^ P 1+ =^[^l+tan2{e(^^))-l (61) (62) (63) (64)

Recognizing tiK$ -Jl+tan^(jr) = sec^(*)=l/cos(x), the expression abovecan also be written in tfae form:

i n . MOORING LINE DAMPING

Wfaen a dynandcd system moves hanncmically with a small period of oscillation (of the order of 15Ö seconds), tfae mcxHing fines act qitadstatically. The vUcons dissq>aticm of the mooring fine can bé estimated üdng esier^ {»inciples. As discussied in Chuter I, deep wateir çateiuay chains exhibit high faydro^namic drag, whicfa becomes increasingly inq>oitant witfa respect to the moored vessel redstance as the wat^ depth ihcreases. In this chapter, an andytkd modd for the ncmlinear catenary drag is derived based on energy prindples.

3.L Derii^on of CatenarV Damping Coefficients

Hgure2diowstfaegeonietryof a £ s p l a c ^ c ; a t t ^ ^ In this figure, xj^ is tfae faorizontd projected distance of the ciatenary; Ô(XA) is the faorizontd displacement of the top of the c^tt^xary ^ i n t A); S(v(ß)) is tfae dîsplacem^ ncnrmal to tfae catiâiary liiae; and is the faorizontd fence at tfae upper end^xnnt of tiie cat^iary ( ^=7^) .

The objective in tfais analysis is to relate the cfisplaceinent Ô(v(s)) to the horizonitd displacement ^(i:^ ) at the top of tfae mooring line. Tfae resulting expesdcm is incc»porateid into the formdation of the mcxiring line damping forces tbat act on the vesseL

Letimg:

^(/^)=viaiiati<HiofdiebcHizontdfcuce 7^ atpcùntA,

Âjï=-^^—horizontd rigidity (stifoess) of the catenary,

S{ifff)= variatipn of tfae catenary siispended kngth.

S[zji ) - variation of die Vóticd diqilacemrait of tfae cat^iäry at point A,

Ihe rdaticn ^(za)= ^ O-^- Verticd variation of tfae cateniary at tfae upper point pf aüOadmiem) ytelds an eâqpsiesdfHi for ^(^^) in teans of ^ ( ^ ) , wfaich can be used witfa the otpressicm for tfaetnrizantalvariaticni ^(jc^) to determine die faorizontd rigidity of die catenary Rji .

16

BgBré 2: DispaMimàst of tbe çafenary in die l-D plane Löting i i = 7 i . and :^=z(s=/^) we have, firom (51)

1+ (Pi \2

-1 =A

Takiiig tbe variation of expre^on (66) above àiid siettu^ h to zoo, we have:

(66)

1 . ^ fx

« a (67)

P^eff

Further, letting = e ( j = % ) , and observing from (61) tiiat - ^ = = t a n ( ö / ) , tiie expresdon a(bove can be rewritten as:

1 7 i - c ^ ( ö / ) ^ ^ dn^(Ö/)'

ö»(d/) J a»{Ö/) 5(F,)+[Pdn(0^)]5(%) = 0.

(68)

17

The expresdon above can be substituted m an expresdpn fen* S{xj^) (to be derived below) tp obtain a rdationsfaip that allows us to cdcdate tiie horizpntd rigidity of tfae catenary line. Assuming tfaat tfae mooring line moves in tfae two-dimendomd plane (i.e. s e ^ g 7(^x,y) - ^)* we find, fiorh âqnesdon (49):

^ A = ' r - % + § s i n f a - f

^ l

. (70)

Taking a first order variation of tiie expression above, tfae expressicm fór Ô{x/^) in terms of ô[i^) and S{Fj^) becomes;

S{^A)=-S{i^)'^^\\sh^^^^

V

which can be simplified further to yidd

5(^A)=^I-cos(d^)KV)+7 sidi-Htan(0^))- S(F^). (72)

Relation (69) can be substimted into expresdón (72) to find the following rdation between S(x^) and 5(F,):

Ex|He5sipn (73) is a singk equatipn recast in terms of the variations 0{xj^) and â(Fjg). The eqsesdpnfn-dte liprizcmtd rigidity /^^ is thus given by:

18

(74)

For values of 0^ ä 45*. e^qnession (74) above can be asynqitotically approximated by:

P\e,^4sr Bi (75)

From 63q)resd(ms (69) and (72), ah expresdon that rdates ^(ieff) in terms of tfae faorizontd displacement S(x/^ can also be fonujl:

The expressicm above can be siisplified for 0/ ^ 45* as:

(76)

(77)

3.2. Nonlinear Damping Foreeff ftn tiffî ^ffTfînPrV

In tiiis section, andyticd expresdons for tfae facmzontd plaiie daimping forces on the n«x)ring liiie in die clirections paialld and peq)^^

This mcticm is in the catenary defoi mation plane.

Force in tiie Direction of tiie Mo<»inp Line Motion: Let be tiie drag force tiiat tiie line adds to tite vessd resistanœ or otii^ oceanic system ia its £rec^ Ihe disdpated power, doe tp tids phenomencm can be expressed as [9]:

19

where S{xj^ ) h a unit vdocity at tiie point of ^tachment in the direction to ttie mc^bn of tfae mooring line, and 5(v(5)) is tfaë variation ctf tfaie speed at eacfa point of the catenary. Thedrag coefficient Cp is purdy a function of tfae speed S{v(s)). The term in bradcets in tfae equation above rqsresents tiie energy dissô>at!on at each point of die mpcniz^ line.

The relation between tfae displacements S{v{s}) and 0{xj^ ) (as well as tfaeir time derivatives) can be obtained and tfaen substituted into (78) to c^)tain a rehttipn fo^ the mooring line danqnng forces in tfae phne pf mpticm of tt^ catenary,

Observing fimh Hgurè 2 that,

5(v(j))=4di^ô(^))]5(jc(</+5))+[cos(Ô(5))]5(z(j)), (79)

the variation of tfae displacment S{v{s)) of tfae catenary measured fiom tfae mooring point at tfaè sea floor cain be otoined in terms of 6^ and S{xji). To achieve tfais, the expresdons for

S{x(d-¥s)) and 0{zis)) must be expanded in t^ms of 9^ and S(xJ^). The ciiffeientid S{x(d t j)) is obtained fipm (62) as follows:

3{x(d-^s))^-d{i^)+l smh'

K'^xJ f PSYI 1+ "

(80)

Substimtmg ^presdons (52) and (74) hito tite equation above, the following relation is obtaiited:

5(jc(rf+5))=^[dnh-Vtan(Ö(5)))5(F^)-dn(0(j))]^(x4)+[^^^^ . (81)

Letting 0{s) - given by (76), the e;qaesdon above can he recast ais:

5(jc(rf+5))=%- s î d i - ï ( t a n ( Ô ( 5 ) ) ) - ( l . œ s ( d ( j ) ) ) i ! - ^  . (82) Jr smtpf)

20

Similarly, an expressicm for S(z(s)) in terms of S{xj^ ) is obtained by expanding equation (67) in tiie form:

fPsf

-1 Ps^ \FJ,P6{S)- (83)

the expressicm above <»n further be written as

5(z(*))=- (1-C0S(9(^))) „ U r n c r f l r c J ^ ^ f ^ * f ff Sir \

-^JÎ^(x^)+cos(e(.)) — - [ ^ J (84)

viluch can be as^Bfied fiirthn tp ^ d d

Sim = %[^^"^!fy^sin(g(.)) -1+cos(e(*))Vx^)

P L sin(a/) J (85)

Notice tiiat S[x{i^))=Ô{xj^); S{Z(1^))=Ô{ZA)^0; 5(x(0))=5(z(0))= 0, since tiie variation at tiie pcnnt of toiüfadown is of second coder m 5(x^). After sdbstimticm of expressicms (82) and (85) into (79), tfae first order variation ^(v(f )) can be written as:

5(v(j)) = ^ - ^^".'^^^^^^(l-c<M(e(5)))-dnfa"^(tan(g(f)))+dn(g(j)^ sm(0(5)) P sin(pf) + ^^^^^^mim)^i-^cos(eis)) cos{e(s)) ^ x ^ ) , sm(d/) J J (86) or S{v(s))^^ ii^:^^^^dn(Ô(5))+l-oos(0(j))-dnfa-Htan(ô(j))) r |_ sin(o/) (87)

21

(88)

where

ƒ(«*))=^î-r^^^sin(e(i))+1 -cos(«*)) - sinh-'(tan(«*))).

sm{0£) (89)

Urne derivatives of expression (88) can be derived in order to obtain an expression that Greedy rdates S{Xs)) to ^(x^)* Since ody tbe slow motions of the catenary are considered, tfae façnizontd stiffoess of &e catenary as well as die function in expression (89) above do not cfaange in time. Thus,

(90)

Expression (90) above can besubsätuted into (78) to obtain an expression for the d a n ^ g force in the duection of the mooräig Hne motion as follows:

(91)

In e]q)ressnn (91), p is the water density, D^ia tbe effective diameter of the cateoary, h is the depdi ofiiDmeisi<»i o f dKcatenaiy, and isaficnKtioaof tbeene^gy diss^ittiinu

(92)

By observing that dQ ~ -^cos^(0), expression (92) aibove can be written in terms of dg as

fdlows:

22

Fory9h)Gs,ofei^&,ex^Bs^<m(93)çm

^'^ks^^mf)'

Tbe first order esqiandonc^ tite energy dissipatK» function (93) v^ $l isshown inAppöidix3.

Force Perpendicular to die Horizontd Mooriny Line Morion: Let 5(y^) be tite Variation of tite velocity in tiie mooring line in tite directipn perpeodicnlar to tite mption of the catenary Une. The danqiing fence on tite catenary in tfais dtrecticHi is grven 1^ [9]:

Fl = |pCo(y^)%»|5(y^)|5(y^). (95)

wfam A is a fiinctionofdtediqdacenientoftfae catenary, defined as fc^ows:

If we condda diat dte comptete suspended catenary acquires a vdodty peipeiidiculartotbe direction of motion, l y ignoring the bottcnn fiiction, tiien

ft = V "

Equation (9Q can be givai interns of A and 9/(^q)eodix4)asfollows:

and, thus,

A=A ^ ¥ . (98)

l-cosiBi)

Substituting (98) into (95) wé find an expresdon for the dancing fince in tite nonnd directicm of motiion as follows;

23

If we consid», faowever, tfaat the catenary has no displacement at its bottom, tben tfae cixpressicm for A is given by:

A = f y ^^^0 dh'^^l 1, (100)

J A l-cos(Ö^) 2[l-cc^(Ô<)J ^ ^

and tfae «qnession fcn'tbe danqnng force becomes:

F, -\pCo(yA)t>^''j^^\S{y,^{y,) . (101)

33. Nptf i i m P^roping Fmes on ftte Vgssgl

Hgure 3 shows the faorizontd platte geometry of a moored vessel witii a deqp water catenary line. In this figure, (^,17) is die earth-fixed referälce frame; (x,y) is the catenary refiemice frain&asineasutedfiT»n the poiittofcomactofdtediainmdtegroim^ (xo^yo)\ ( X K , ^ is tite bodyreforenœfi:arneiinbedcMattliec^t|»'ofgraW^cyftfae ves^ (x',yO is dte coordinate system of the catenary witfa its origin at the attacfament point on the vessel, x' xheasurediärääd totfaefactiünitdpl^ y'nonnd to x ' . In addition, {Xp^yp) aretfaebc>d^-fixedcoc>rdinatescïftiie attaclinteixtppiiitondte v e s ^ yf is tte drift angle; y is tite angle betweea tite ^-axis and the cac^azy, mea^xred ccnnt^ is the distance between tbe ihoonhg pomt B and the attachment point A on the vessel^ also diown in Figure 1 ;

Fj^ and fi. are tite drag forces in the directkms paraltel and perpendtctilar to tite mooring fine modem respectivdy.

The drag force on die moormg line m the catenary coonrdin^ systm (ï\j^ can be expressedas:

F„ = F ^ r ' + F j ' . (102)

24

(103)

(104)

In tite êiqiresdons dwve. Qxpand Q>y aretitedragcoeffidoitsintbe and y'directions of motipn measored widi respect to tfae velodties at the top óf tfae catenary line.

Rgure 3. (joomttry pf Ibe nxxxing qntem

The coordinate transformation betweesi tte v^sel coordinate system (J.J) and tte mooring fine coonfinate system (ï'.JO in dte horizontd plane is given by:

r'=oosiff+sin)5t/. J ' = - d n j f + c o s ß J ,

25

Therefore, tte drag forces in tte (X, coordiiuite system are giyen by

Fxo = FJ^co&ß-Fi^smß , % - / > d h / J + f i c o s p .

(105) (106)

Tte expresdons for tte vdocity cotnpcntents in the moormg line also can te recast in tibe (J.J) fiaine witii respect to tte center of gravity of the vessd (CG). Notice from Bgore 3 that tte U { ^ enc^^int of tte mooring line is located at the pomt (x^»y^) as measured widi respect tptbe (X.y) coordinate firame.

Letting û and V te tite horizontd velocity components of tte vessd at its center of gravity, and r te its rotaticmd velod^, we can calculate tte vdodty ccmçonents at tte attacfament point ofttevessd »4 = (My(, v^, ) with reflect to tte (X.I^ firame as follow^

(107)

where îic; is tte velocity vector of tte vessel's center of gravity, = (x^.y^.O) is tte di^ance vectc» between tte vessd CG aiidpcünt A, and w^ r is tte rotationd velocày of tte ship. Equaticm (107) can te recast as:

(108)

Tte horizontd vdodty of tte upper end^xtint of tiie mocning fine fl^ also can te recast in ietms of tite (J, J) fi:ante as follows:

• 1 u i _{J i' ti-ypT} Va • — • V 0 0 r v+Xpr Ta r . yp <>. r ^=iA^'+yiJ-J^(c<»^+sinjaJ)+y;(-sin)^+ toyidd ^=(xicos^-y;dnJ^J+(xisMiP+yicosft^^

Equating expresdcms (108) aind (110) we find:

(109)

26

iÄcosß-yxmß^u-yj^,

i^dnP+yicosP= v+XpT.

(111) (112)

Thus, dte horizontal vdodty comppnents pf tte mcioring line at ppntt A are given in terms of v^seL vdodties and cocndiiiates as:

XA =(tt~ypr)cQS^+(v+Xpr)sinp , (113)

^A = ( v + y ) c o s ^ - ( « - y y ) d n j 8 . (114)

Tte dànçûng forces can te esqniessed in terms of viessd Vdodties as follows:

^A =jpCox^^rp\<ü'yi^)co&ßHv-i-Xpr)änß[{u-ypr)cosß+^^ ,(115)

[(v+Xpr)cos/î- (« - ypr)ànfi^ . (116)

Tte expresdons dK>ve can te iiisetted into equations (105) atùl (106) to obtdn tte

forces FxD and i^D respect tb tite vessd c<x>rdinate systeuL Tte moment Mjp on the veäd aibont tte Z-aus due to tite daäQmig force components is 9

^ZD = XpPyD~ypPxD •

which can te expanded further te yield:

^fm = PA{xp^ß--ypCOsß)+Fi[xpCosß-^ypänß)

(117)

(118)

Expresdons (105), (106) and (118) &note tte damping focces and momoot on eacfa of tte moQonglin^. Ttese must te soinnted up accordmg to tte nuniber of mocving lines: {vie^^ dtesysteuL Ncitiœ tiiat tfaey act in tite sante clirection as tte faydrpdyiiai^

exoted on tte vessd Q,e. resisting its motion). Tte expressions for tite mdcning line damping fcsces andnicnxienttfaer^ore nmst teiinplen^ any mathematicd model

A P P ^ D I C Ë S

APPENDIX 1; FIRST ORDER V A R I A T I O N OF T H E H O R I Z O N T A L TENSION IN TERMS O F THE HORIZONTAL PROJECTED LENGTH OF THE SUSPENDED CATENARY

Tte first order variaticni of tte hcnizontd rnooring line tehdcm (Tp = T), is obtaited fiom equation (12) in terms of tte first order variatiPh of tte terizonUd prt^ected length c>f tte si^tended cateoaxy (/). This e^qnessicm, in turn, can be subsequentiy expressed as a function c^ other gecKnetric pn^ierties of tte catenary. This is achieved by taking the first order variation of / in terms of tte variaUe of interest, such as tte borizcmtd ten^ tetween the GodpointsofiSae catenary ( O (Appencfix 2),.and ^ t y i n g tite chain rule of differentiation.

Consider equaticm (12), which is of tte form

T

fpi\

% = 4 d i i f a ^ = J A A + 2 - ^ . ( A M )

This ex{nession relates tte horizontd tension in tte mooring lizte (7), and tte horizcmtd siûpended length of ttecatenaiy (/), wfaere P and h aie known constants. Tte derivative of T witii respect to i cantederivedfiomequaticn(Al.l)by taking tte cienyative of all iiivolved tenns m tte equation witfa respect to / as follows:

ldT.JPi\J, idT\JP€\ ^di / A 1 , ^

.-^suah _ + 1 - - - J codi ^ = — . ( A O ) P di \T J \ TdiJ \T J B - / . . - T l A

V

hi this equation, — Utted£dvatrêeof ttefac>riz<)ntdtezidónrwitfarü|tecttotteh(>rizo

di

inôjècted lengtii of tte aisprad^ catraaiy i . Tte equation ateve can te furtfaer recast as follows:

29

dT

P^

T

di~(Pi\^JPi\ . JPi\ h

V r J ^ r J ^ r j

Rearranging further rdation (Al.3), tte find form for becomesl:

(AU)

/At jl\ (Al.4)

APPENDIX 2î FIRST ORDER VARIATION O F T H E H O R I Z O N T A L PROJECTED LENGTH OF THE SUSPENDED CATENARY IN TERMS OF ITS TOTAL HORIZONTAL LENGTH

Tte geometry of a deep water catienaty mocving lirte does not cfaange propc^onalty to a small hprizcmtd cfispkte^nent of its upper endpcnnt (i.e. a change in tte totd faorizontd length in tite catenary t does not equd to tte change in tte horopntd projected length of tte suspended part ofthe catenary/). Tte rdative cfaange tetween ttese geometric prc)perties as tte inooririg lixte is diqdaoed hoiizontalfy by a sni^

Hgure A2.1 shows tte gecntetty c^dte cateaiary tefore (1) and after (2) dte uppermost pofait of tte c^iain has been £splaoedby an amount Ax horizontally in tte plane of tite catenaxy. In FigureA2.1, {Ai^^Aii.

Frwn tte geoiiKtric relaticms in ^ u r e A2.1, tte hcnizootd oompöneiits of tte catenary teve tite foUowing relations:

r^T+At=T^Aii , (A2.1)

d^3+Ad^3-Ai2. (A2^)

/=?+iW = ?+4fi+4^2 - (A2.3)

Tte ov^bar hi tte ei^nesdons above denotes the vdue of tte geometric property before tte hormmtd displacement foaddto, tite foUowihg relations teld:

^j.=+AiT= COTStam, (A2.4)

wfaere

31

Aif^Ai^-hAd^O.

By combùimg expressions (A2.2) and (A2.6), tte following geometric relation is obtairwd:

^^"^2- (A2.7) Tte equation ateve sfaows dtat tite cfaange in tte lengtii of the siispended catenary Ai^ is

equd to tte itegative die change in leiigth of the horizontal ^

Ai2=-Ad,

figure A2.1: Geometiy of the ^splaced catenaiy

foprdtf to obtain an expresdcm rdating tite changes between tite gecmietric properties i and /'.first ccmsider equation (11)

i T . jPi\

32

Equation (A2.8) can te expanded reeadily in terms o f / . a n d tte horizpntd t ^ c m (7) asfdiows:

7 ^ Aé T^^T - J^±dQ\

V ^ 4 « < f l ^ = - ^ ^ ^ 7 ^ J • (A2.9)

^T

Letting X = (small), wd usnig tite binomial expandon rdation — - 1 - x + x ^ - x ^ + x * - . . . ,

l+x

tte rightmost term indde tte fayperbolic fonction of equaticm (A2.9) can te expanded as follows:

P ( 7 t A 0 f ( ?

(Axio)

Tte expresdon for tte derivative of tte facnizontd tendon with respea to the faorizontd

dT AT

SQspoKled lengdb of tte catenary &e. ^ = ~ 7 T ) (see Appendix 1). can te incorporated into

di Ai

(A2.10) by luting:

AT^^Ai^^Ai^T^Ai. (A2.11)

Ai oi

and tte right hand dde of expresdcm (A2.10) can te recast in terms ctf Ai such tiiat:

P{UAi)_Pl,p(,

Sub^itnticm (^(^12) into (A2.9) yields:

{T-¥TtAi\ (pl P( T,

/ ^ + 4 / ^ = i — ^ d n f a [ ^ ^ + £ [ l - . (A2.13)

Usirig tte hyperbolic rdation sidK«-*- v>==srnfa(ir)cosh(v)+cosh(M)smh(v). equation (A2.U) àteve can te wrätai in tiie form

33

^ ^ . ^ ^ . i î l ^ j s i n h f f l c o J ^ ^ ^

' p7\ fpr T \

+codi dnfa ^ 1 - ^ 7 Ai +... .

\ T J \T\ T J J

Tfae terms on tte rigfat faand side of equaticm (A2.14) can te expanded forther in Maclauren Stties[2]:

x^

x^

{T-¥TtAi\{ (pTi lfP\^( Ti^'S^ 9

, c a s h f « ï r f Y i 4 ? > 4 r j f f i 4 ? T ^ . . . . '

A r J[UA f J 6KT)K T )

Tfaen. tte rigfat faand dde of equation (A2.15) can te linearized:

T (

1>t\ f

P}\

' ^ = -^smfa . an expressicm for Ai^ can te foiind focnn (A2.16) as

^ - f f = 1—é-/ cosfa +^smfa — Ai .

^ [ l . r > / ^ r j p ^ T )p

In equaticm (2.17), ^ = ^ i - l - ^ 2 ^ c ^ = ^ 2 - Tte equaticm above can te recast in terms of diese displacentents às follows:

34

4<2 , [y T J \^Tj p V T ; J _ ^ ^

1- r i - S - 7 cosh ^ +^sinh ^

[\ T )^^TJ P \T)^

(A2.18)

To find Ai-Aii'^Ai2 in terms of At = Aii tte expresdcm ateve can te further recast to obtamr

(A2vl9)

Once tte relative cfaange between / and is cdculated fiom tte equaticm ateve, tte cfaange in other geometric properties (»i<^ as tte relative change in positicm pf tiie endpoints of tte catenary, for exanqile) can te obtained using tte chdn rule of dffferéntfaricHi foÙowng tte fötpresdcms doived in [4].

APPENDIX 3: FIRST ORDER EXPANSION OF T H E DRAG TERMS WITH RESPECT TO THE HORIZONTAL PROJECTED L ^ G T H OF T H E SUSPErâOED CATENARY

In order to perform staMtity anafysis of tte horizontd platte equations of motion in mooring and ^boring^ the derivatives of tte various terms involved in the drag equations witfa respect to fte velodties (u, v.r) and die podtion vector (x^y.V^) are needed Takiiig derivatives witfa réspea to tfae velodty t^ms is s t r a i ^ t f o m ^ Derivatives witfa respect to dte positieveen, are. fapwever more difficdt to derive.

Tfae derivatives of tte vairioas terms involved in tte drag equations witfa respect to the positipn vcytor can te obtamed frcnn relaticms derived based cm tiK

tte mooring and tte attadiment points on tte catenary ( as diown in [4]. Tte derivative with respect to tte viariable t cainte c^btained from i as stewn in Appendix 2. and the derivative with respect to / can te obteined fitom 0/ as is stewn telow.

In tfais Appäadix, tte first order expandcm of tte nonlinear terms in tte drag ccm^xment equations (91) and (95), specificdly tte hi»izontd rigidity R^ and tte energy dissipation function y ^ are derived with respect to d/.

The expresdon for 0/ is given from (61) as follows:

Ö ( / ^ ) - Ö / = t o n - f ^ l . (A3.1)

wfaere i^ is given by dtiœr equaticm (10) or (11) as:

V = (A3^)

or

36

T (Pi\

/ r f =-jdnh .

Expnssdioo (A3.1) above câs te recast in either of tte following two forms:

i( (PiW d/ = tan sinfa — , V \ T JJ (A3.3) (A3.4) or

M(P r7~~r\

Tte derivative of 9( with respea to / can te c^taiihed by tddhg tte ai^pidiate partid dttivatives of eitfaer esqnesdcm (A3.4) or (A3.5X Froth (A3.4) diis ytelds:

dei_

(m

1 - ^ ^ L (A3.6)

di cosh(W/r)l . TdiJ' (A3.6)

and from (A3.5):

dße (PhfT) 1 dT

(A3.7)

di l^Phß) _{P-jA(A+2r/P).} di' (A3.7)

dT

In ejqiresdons (A3.6) and (A3.7). is tte derivative of die tedzontd tendcnT with respect

di

to tfae horizontd projected lengtfa df tte suspended catenary / , given in Appendixl (equation (Al.4)). Bptii »pressicms (^.6) and (A3,7) are equilvdent

Once tte rdation between / and 0/ is caknhted from eititer (A3.6) or (A3.7), tte first cnder expauadons of tite rest of tte terms involved m tfae npnfinear danqnng equations can te taken directly witii reqieatp 0/ -, Ttese expaiisions, in turn, can te recast as functicms of / via tfae chainrdecfcfifSsrentzadott. Expandonsofätehtenns withrêqieate/'orany othtfpod^^ vector can te ^>tained following [4].

37

As mooned previously, tte first cnder expandons of tte horizontal rigufity (stiffoess) R^ and tite energy dlisdpatian function y^arederivedwithte^tectto^/.

To find tite first ender«tpandpn for tite borizcnitd rigidity Rg, exfnessioo (74) is éxpanded witfa respect to 0^ Recall from (74):

1

sinh"'(tan(ô/)) J l - c o s ( g / ) \ dti(Ö/)

(A3.8)

Tte derivative of Rfg/P with tespect to is pyen by.

swrjBj) - <x^{9i)+cps^(g/) d?(Ô^) sec^(g/) Jl+tan^(ö/) dBi sidi' (A3^)

wfaich can te arranged to cA)taih:

4RH/P)^ 2cos(g/)-l-cos^(ft)

<^Bt pös(Ö^)[i^dn^(0^)-4t^dn(e/Xl-cos(Ö/))+4(l-cos(Ö/))2 (A3.10)

In tte equaticm above.

d=sinh~^(ten(Ö/)) . (A3.11)

To obtain tte derivative of tte Clergy dissipation function y^ witiire^teato 9f, consider equaticm (93), whicfa is repeated telow:

_(R„\^ cos(g,) f'\\f(Bif\B)]

KP) (i-cös(fli^))Jo [ cos\&) y ' (A3.12)

38

/(^) -fizf^§dWe)-dn(ô)fdnfa"VtaaW^^ . (A3.13) sin(0^) ; 1 1

Equation (A3.12) can te expanded by taldng dëriviatîves with respect to 9;.

Tte derivative of the integrd in equatipn (Aßvl2) can te derived usmg tfae generalized version öf Ldbnitz tiieoron [12], wbicfa states tfaat:

Ä = K a . « a ) ) ^ - K a . a ( « ) ) ^ . r ^

da da da J«(a) da dx . (A3J4)

Lettn^

IKa)=\ria,x)dx , (A3.15)

wfaere

ö(a)=0, Ha) = 9„ Ka.x) = i ^ ^ ? i 4 ^ . a n d cos'(x)

/(a.x)=(i-^^^|dn(x)-dn(x)fdnfa~'(tan(x))l+l-cos(^^.

and appl^ng (A3.15). dte dedvative pf (A3.14) witii reflect to 6/ beccmtes:

dR{9t) _\fiBii^HBi)^ f^c a (\mfhB)\Q

d9i cosHBi) Jo d9([ cos^(9) ™* (A3.16)

wfaere dBi '\f(.ffifH0)] (i-cos(fl/)süi(e))

c^(e)

J "

39

By mcoiporating cnpresdpns (A3.9), (A3.16) and (A3.17) into tiie ei^andon for equation (A3.12), tte derivative of jp with respect to Bi is obtained and given by:

d9 Pj

.IcQsjBt) d{Rff/P) am(g/) l-cos(Ö/) dBi I P J(l-cos(90

+ •

(l-cos(fl<)f

f f

cos(e/)

if

f».

[/^(fl)^/(e))

+

2i/(e)i/(e)

p ) {mHBè)) Jo ct^iB) cr^iB) ^9)dB

Ä f f

11

APPENDIX 4; RELATION BETWEEN THE EFFECTIVE L E N G T H OF T H E CATENARY AND WATER DEPTH

Tte geornetricrelationslup tetween the eâîBctiveleng^ i^ and tite water depth A (or verticd distance tetween tte catenary endpomts) can te obtained by sdistituting e;xinesdon (61) mto (66) as foilov/s:

' 4

^FJx-coäjBi)

" ? [ cos(ö/) (A4.1)

Tte expresdpn above can te recast as

1 ^ P r cosißi) '

A ^[l-cos(0/)J " (A4.2)

By multiplyfaig bodi ddes by / ^ into (A4.2) and performmg tte ^iinopriate trigonometric relations we find:

^ / ^ ' cos(d/)

Fx 1-008(0/) ; = t a n ( ö / { - . ^ ^ l = ^ _{^ ' ^ 1 C 0 S ( Ö / ) J 1} -sih(g/)

cos(Ô,) (A4.3)

Tte relation tetween tite suspended lengtii of tite catenary ( / ^ ) and die water d^tii (A) cain titen te dnp&fied m dte form:

BIBLIOGRAPHY

BIBLIOGRAPHY

[1] Ansari, K. A,, and Khan, N . U., T t e Effect of Cable Dynamics on tte Station Keeimig Response of a Moored Offshore Vessel," Proceedings of die ASME 5tii Ihtonationd conference on Offsfaore Mechanics and Arctic Engineering (OMAE), Vol, m, Tokyo. April 1986. pp. 514-521.

[2] Béyêr. W. H.. CRC Standard Mariamaticd Tables, 28tii Edition, CUC Pm_«. Ann

Arbor. 1990.

[3] Cox. J. v., "StatuKXH* - A Smgle Point Mooring Static Andysis Program." Navd Cî>il Éngiteâing Lateratory. Rqi^rt No. A D - A l 19 979, June 1981

[4] Garza-Rios, L. O., and Bemitsas. M . M., "Stability Qitieria for tte Slow Motion Kpnlhtear Dynamics of Towing and Mociring Syst^." R ^ r t to tite l^verdty of Michjgan/Sea Grant/Dadustry Consortium in Offiiiore Engineering, and Dqpartnmt of Naval Arclutectüre anid Mariite Engmeering. University of Mk^gan, Ann Arbor, Pd>lication No. 332, November 1996.

[5] Huang, S., "Dynamic Andysis of Tfaree-Dimendond Marine Caljles." Ocean Engineering, Vol. 21. No. 6. Elsevi^ Science LTD, 1994, pp. 587-605.

[6] Kwan, C. T., and Bruen, F. J., "Mooring Une Dynantics: Conq>aris(m of Tlnoe Domain, Frenuency Domain, arid (2uad-Static A n ^ ^ . " ^Proceedings of die 23rd Offehore Tec^olc^ Conference (OTC), Paper N a aiX>6657, Hcmston. May 1991, pp. 95-108.

[7] Mc Kenna, H. A., and Wong. R. K., "Syntfaetic Fiber Rope, Properties and Cahnilaticms Rdating te Moôqng Sy^^ns." Deepwater Mooring and DriUmg, A S M Ê Transactions. Ocean Engineering Dividon, Vol. 7. Dec. 1979. pp. 189-203.

[8] Nakajima. T-, Mptpra, S., and Fujino. M ^ "On tte Dynamic Analysis of Multi-Ccmmonoit Mooriiig lioes." Proceedings of tte 14tii Ofifehore Technology Conforenoe (OTC), Paper No. OtC-43q9. Houston. May 1982, pp. 105-120.

[9] Nishhsoto. K.. Kaster, F., and Aranba. J. A. P.. "Fdl Scde Decay Test of a Moored Tanker, ALAGÖAS-DICAS System." Report to Petrobräs, B r ^ , Fïïbruary 19^ (in Portuguese).

[10] P^ufias. F.A., and Bermtsas,M.M^"MOORLINE: A Program for Static Andysis of MOORing UDSEs," Report to the Üiüvmity of Middgan/Sea Grant/Industry Conscxtinm in Offetere Engmeering, and Department óf Navd AreUtectme and Marine Engineering, Tte Univeidty of Michigan, Arm Arbor, Pdilication No. 309. Mity

1988.

43

[11] van den Boom, H. L J.. "Dvnanuc Betevioùr of Mooring Lines." Proceedings of tte 4tii Intemationd Conference on the Betevioùr of Offshore Stmctures (BOSS). Amsterdam, 1985, pp. 359-368.

[12] Wyüe, C R. Jr.. Advanced Engineering Matfiémàtics. Third Edition. Mc C3raw-HiU, 196L