' .+ ,Na1~libraryoiCariada i

.', '

'. , : . ' "

TH~~~'SCANADIENNES SURMICROFICHE

"Ie- :-LS.B.N,

' .+ ,Na1~libraryoiCariada i

^Blblool;n;uenallOOillel1lCanilda

..Col\tl:lie"!o OtvtlGJ>':"'fn18l~ch .IOill ction 6udelllleppomenI 6 O$c<llllC1ionl C.nod;.n Th".lon ~n-;oid;llh~lclnldiM'n"

, Mlcl ol iche_Sll~;a· . lu< ", ;,:,eloche 011lwa . ~t\idl.

•K1AON4·

'AVIS

FI~pr~I,I~[io~

^in'jl,lU

~r' i~ 'pari

ot this filmisgo.... , '.La.rel?roduc.t io n,I"(ll!me partielle,de cemitwl,ilm ernedbV the~nadian-CoPvri9htACI,.R.S./;:.1970, est's.01Jmiseala l oi l:lIhadiennesurJe dro it'd'aute ur, c.C·30:Please read theauthonz~ti onfo rms which'•.SRC19 70, c. C·30.Velullez prendreccrmarssencedes 'a~ompanvthislhl!$iS:·. . -.·fo rmu.les.d'aut.otisatj~:ql,li.acc(lmpaQnent cet~e Ih~se,

,I'

.Nt-J39.! ',

" L ; T~E S'E

ETE"

MtCfiOFtL MEE.TELLE QUE NOUS,L'AVONS,RECUE'

~~'

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':'!'

.:"v.

. .

EFFECT OF^~THE^.LOSS^, OF TE'j."HERSTI F FNESS^'

^.

ONTH.~^BEHAVIOR

OF

A'TENSI ON LOO'PLATro~

BY

CNitindraR•.Joqlek ar;B.Tech.(Honlll)

.AThedesubmit.ted to th e

.

SchooLofGradua te

.

st.udies·

·in.pa r't..ial f·ulf.ill'lIent'···o t th e ceq~i renierit:.B .

;~ ,

^f9rt.he,

d~gr~~'

^of

"!-a,. ter.oftngi~ee r in"q'.

, . .

St.Joh n: .

. -.

'

. .

^{' .Fll~~lt.Yof} En91~e ering andAppliedSci ence"

Memoria l Uni verdt.y of Newfoundland

.'

..:". l'U9ll~tt lC;li4

^{' ." ,}

.'-"

Ne wfl?und l and

)

,. ^.

r

i i

' . ~STAACT

. ' . ,-

.

Anu meri c a l JlIOdel'tor thehydr o dynami c llnaly els of a.Ten d onLe g'Platform hae heen",i? Pl eme nt edusing strip theory -".A compute ralgorithmhae beendev e lopedusing-beam ..

e;e~nt8 fO~.

^a

larged'i~placemen~ non;li~e~r ' anal Y8i8"o~ .t;.h~

cable.:n~t~or1t. , .

~A Paramet. r ic"s t ud y~fthe-piat f ormrootio n ree po n ee

• , . -. J . ' . '_.'

~~nd ~en8ionvariati~n8has,b.,eenrna~e ~o.~.~ll.e: elof..1088 in t.h e cableIttilfness. Ithaa .been shownth a t if't~etotaltens i on

: - - - - • •: • ,; - .- ,: .!j'

remlli~8 con8~nt . th,s,urge,motio n will not be!a f fec:edbut the hAve and pitch motions will beincrealled~'ndtether'

, : ^"

.tend o ne will vary.signif i c a ntl y.,The tether'tensionl,are

", " .'

>'affec te dby'th~'illtera ~.~able.dynamic8.

".

\

I.,'.

. ... ...

ACKNOWLEDG,EMENT

'.

.

t~.Dr :

.if:

aoOton~for his superv1don,suppo r tan d

'.-,

ellcouragemen!-during the entire cou r s e of thiswo ,rit. s1ncer-e"

than1t8" ~re.

^{due,toDr.G..'R".Peters, D,ean}

Of · E~~ine·er1.~g ~nd

-Appl i ed.Sc i e nce s~nd,Dr . T.R.~ari.AssociateDe'An

of

.E.ngln,ee.dng·and~pplied'seteoeee.,f o r their 'enco~ragem,ent; ~....

The author 1a indeb t e dt~Dr. ,F. A.Ald~£ch.'Dean ofGr ad uat e 8tUd~.~S"•..,f.or.aWArd,i ng'.11.,~ti.1Yerl'-ityfei~o";ehlPfor

the-pa~t't~o xeil,ra . . ,'_ .

,Theauthor.~'ishesto~x~.et.d,hb deepapp'r~ciatlon to Dr.,A. S._. ,J .s....-am1das;·viait.1ng'Iteaearch_. fellow,Dr.

K .

Munaswamy, po8t':d~ctor~lfel low-an~Mr.,A. Allievi,fe llow

.gradu ate,

~tlident ~or

^their

:val u;bl~

^{'sugge s ti ona"}

~n~

!"elp'ful

d~Bc:u,asio?s,

ana

~o Levln:i~vatcher' ,for:,the ~yping

of the ..

-manlls'cdpt.

"

:'"

'

I

.~,i

\

i'

f··

i \

\ \

I to

, ' I ⁱ '

T I !

I

,

"L"

.

,

'

. r

!.

'" ~

^.

ABSTRAcT"

... .

^{"~11 .}

;'11 1_"

'-,,\

. . .

CkAP'rER.5.AH-'hYSIS OFTHERESULTS

5.1 'I n troduct i on .

".;"5..2.Valida t Ion

5.3'Parametri c Studi••

S.4'IH.cu••ion

CKAP:~~i 1 -" 8;~:~~~~JON . - . .' . .j.. '

.. L'2 :Objjllc t i vn [. ..

C~P~fR -~

.LI TERATURE'

iu:vI~

^{. ,"}

~"

2.1'Enviroma entalLoadsand,Reapona.·

.2.2 Ho~ellln9 o~the':Te~her .

CHAPTER j THETENS ION LEGPLAT FORM 3.1 TheDe signPhilosophy J.2'Ganeral,-O••er i p tlon 3_~3 The.T~~h er sy.te ll

;. 'CHAPTER 4 PROBLEMFORMULATIO N

",4.1 Intr oduction

.4.2 Line ar Model

4..2.1 TheConst ituent Porc e s .. 4.2. 2 TheEquati onof'Mo t io n -: . 4.2:3 Teth er St iff n. . .Modal\oJ .4. 3 Nonllne arModel

4.3 .1 ",••ulIlpt. i ona", 4. 3 . 2 TheFiniteEIe'man t Modal"- 4.3.3.TheTIme'In t e g ration'SCh e me 4.4 Nu_r~calAnaly~b' ' -.

!,,'

.•j ,j" .;

I 'J ~ . I i-

I

",

^.

1· a vii

" 10

,10

,11-

1~

re

16.

·17

17 19,

'0 ""

, .

as .29 33 33 33 -"35

J'

'44_,

. . .

'91':

••

. . ..

. -' - -

" .:.

.'\ '.

:/

- I.

-- -. - .

-:---' _._-~:--:- '-'~""

CHAPT ER 6'CONCLUS ION-, NOMENCLATU.~

BI BLIOGRAPHY'.

,-,APPEN DI X'A HYDRODYNAMICANALf t IS .'..APPENDIX B'THESTiFFNESS COMPUTATION

LISTOPTABLES

I '-- .

./ LISTOP"FI GURES

"<? . '

!!2 APP ENDI XC,I NPUT ANDOUTPUTOF·THE~ROGRAMME:S' 100:'

A~PENDIXp'LISTINGO~TLPI 107

APPENDIX. E LISTI NGOF,TLP2.., 127

1 j"

.~i

.J j

p•••

,

^.

.•' i

V _!

; ...".!

vi

LIST'OF'TABLES.

~^.

"

' \

' .1"

.·conip'~ri.cn~f'~Olllputedresult.'with data

.presentedby Kirltand'Etak(1979)

~p~~~:~~;~iC^{study of.Iu r g e.reeE?naeamplitUde} A

,parametr{c"8t'~d~

^:,0£'heav e responss'

amplltud~-

^{.'c .}

operat ors . -. '

'.-Aparometric~tudy,o f.,t hepitch-~!Jlp"anBe ll.m.~l1t.u:de,.~perato-:~ . -

...,i. .

'p ara me tric"studyof,.t heten~ion:.reltponu

&mplitude 'ope r a tor s onth e,aft side "

,50:,--•

vii

._~LISTOF FI GURES

. !'.!.9.!.

5.' 51'.

51 58 59 .0 61 '2

,

63

.

ee

"

_'8'

69'• 70 71

72 73 7.

7S.•,

7.

" ~

.

^{"~ .' ,}

Tend onforwa~dve rs us'ti~l',pa.rame~ dcst,udY Ef fe ct'ofwa vil:he i g-h'tOn,t e'nll'lo n'hla t ory,

~arametr1c, "study " . ,' , .

SU! lJe.di.p-i.a~ellM!ntve-r·.u~~l/Q.u··Pa~~lI'le~ric

study.. . " .

.

". . . .

,

Heave dilpl~c.inen~:v~~su~^_.time;·p~ra~etric,

"study" ".

P itchmoti~n:versus'time: Parame tri cat,udy Tendonar~venus't.i~;"l?ar'amet ric: st udy'

',~' r~e fi.rJ~;~:, ,~ ~~~;~t·:,~~~~

Definition'IIiltet ches

9- : , ~ethe~ Mo~~i/ ::

'D~~~il~: ~'i:" the:,~r~'n9 ~Y'9~elll

Mo~:'-rincit:ompa r t.l\l.ent..:;

.2,.'.Sch~tlMti'~'r'epreuntatioo..

ot

^i'Pla na<~t,iO;U;

"3 Nilltur al,'i;~u~'l'Icies~f ~.T-l!ruJio~ ~eg'^{P1atfo n}

". ", u t c . tif fitess lDatr:Lltof a'bf! aJII.e lelll~ni.', .

1;1";:.

~ur~

^force

' ~~f~';

^•.

f,~equen~ '

.I!!=.. ';•:', .. ,: :T~tle.,

.f:..'_

:.o ~er.a li ,.\d,-"':' ~f~LP:.With

^li;ey:

d~_ndOn..

',"·10

.·~,'·-,; F,i~ ~~~··.o~ "'linear\tonalYS,i~

^aJ.9?rithJll.

'11 F100:,'chart.o~.~ol)iiner'a"~alYsi8alg-~r1thm

, '. . ' ". - " .-

14· He a veforcevenue frequency.

. ~t;e~i~~~~-~v=~:=.~~~~r== ",,= '" :: .::-ncy=-~-~~-~

I.

17 18

'1')

20 21 22

~:' .!

.. ..

vi.!!

~. ~

23 Effectofwavefrequencyontenll.on nhtorY!

Paramet ri c study , ^{'7 7}

"'~~~"

c ,.

24

" ~u'rge reapona~' am~l~tude

operator ven ua t illle!

'Aca s e'atud y , 78

25'

...

Heav e reap o n s e ampli t ude ope ra tor,venua t illlel

Aca s'e'atud y . . 79

' .2"

2B 29 30

31 / 32

J4 .35

Surge dhp1aceme n t ver s ustime t,A.case st udy

-~---.. .

aeeve diap 1 a c eme n t vs r s u s ti mel A."ceee study"

B1

B2 B3 B4

BS

B6

B8 B9

I.

!(Xb.;.yb,tb) K

.9

-:

~--:

. . . . .. .

"

.',"NOMEN C LATURE

Th~':g lobal'f ral!leof're f e rence.fi :u;d ,the i.n i ti a l posit.ion0.£ cente r of'

gra vi t y

DiBp 1.sc ellle n t vece.o r (10:.-1, 2, .,•• 6 refe r to urg e,'lrfay.heave,·r ol 1. 'pitCh,and,yalrf

r~i1pective l y) ' .

. .... .

Wave,number

Amp~ itudeof,lnc .i d eJ:\t'';;aYe~

.'

.

-:~~

wave.s'with.,respec t to poaitive X-ax ~,~. .- :-'

'AceeTIr rti-o n,d a e't ogravi.ty

,

'~l

compla~V~.10~ity^{·poten ti ai}

cOlIlp 1er ' ve locit y potential•.functi on·o~' sp,,:cecO-:-ord~~ate9only

loca i frameof reference for an element as defin e din Fi g.B(b )' den s ityof ....ater·

VoJ.umeof ifllllle rse d body

.Gl o b alc~r'dina~e6

o f

^{tMbot tomof the}

ceo t.e at.an Y,ti me·t

Q(xO, yO•..zO)

" "

'.

Glo balco-ordii\ai~s'o£th~-'center bf gra ,,:i y~t anyt imet

Gl o b al'co-o r dinat esof thebottompf.

the.cable atany.~t~me t " .

f

[m..i.j]

'.'.[aij]:.

., .~'[brj]

[Qi j ] '{R }

{RaJ ,"

{Ra} 61C-j . S

.p

·L

"'IllaSS mat rb:

.added ~9Smatrix

damp1.n gco- e f fici e nt~at r~x hydr o stati c re~tor1.n9eo'ef~,ici ent,matrh ,~ ... ~nertiavec t or

dR/dt dr/d t d2r/dt~

.e·i.ci t. ~n9'forceYec~r Res t ori ngforceve ctor

surfa~earea of the h~l~

nor:nal v~ctor ,C<:>Sh (x+d )/ c o sh {kz}

Gable sti ff nessmatrix Equili b r i ulIltensi on in thec'a:b le length of the'cabl e/ e l ement ] Yo ung 's modu l us

)

J

VI,'~12

A.r eaof.cro~~ B ectionof the ca b le Elatt:ic stif f ne ss~at rix of the cable/ e l emen t ,', . • Geometri c st, i £f nes s' mat r ixof the cable /element.

Axial disp l acement a'of'a plane t;llB9.

elemen t , ."" ,.

' ,' - ' ^- ,, -

Rotati ons of a'planetrl!as:el e ment

I

t

.,' -'

';

i

! I :

t<.,

.,: .

dl, d2 '0.

,tG_Jcj]

•F~

"S j Kl. ~2'

Qua~.raticdrag C?efficient

Amplit"udeof the period(c flu id vel'bcity

· "Total' .re s t o r i ng" force'

b~t.r.iX

'·.A~p litu deof the·fo rc ing-function

, . : , ~

Amplitudeofth e jth mode,of moti9n

•St if fn e s s e ll atendt:and2 ..hef le c t i onsat end1and

. f

·Totai zcec e of,bu oyancy

• Totalweightof~the.st r uctur e j

1'(':tatf'ret,ersipn. l<',X..

::·~.~~tan~~B der~ned

ⁱⁿ

.Fig.~.' dQ.·(di::;.~y,

^dZ)

,D ~SPlacem.e.n~j

^{of point Q.}

t-/','dT Ch~ngein'cablet;.ension·

cha~g~

in'leng th"'o f

c~b~e'

.

' .

i l

l I

\.

I i

'I.

Number of degreesof frei'!dol11 .\

.1 I

I ·~·

·1 i .... .

^'

"

' , ' ..

. .

f,rom, la nd

,and' ~·ea,~

^__

,s,hor.~· dt~',· ~~ ,st ~u~tu r~o

ⁱⁿ

'~a~~f. ~I:p~ti~

'

. .

, '.,... ," "" ,

-

"

inexce,Sf;I,o~ l~,OO,~ter.$~'.Md;,into'p,u s h er en'''i',ronmimts ..fU,l1

,o.~ .,wa;~~~;" cu~r e~:s:,: ~,in~;" .:s'eaqUakes. ,~nd' ~~e,~·~9S.,.

^":,":.

Centralto th is dev elop me nt,iothede sign of'anine xp ens,ive,' ,

Insha~.low,";"aters.',th.echo iceis,ge ne rlllly a :f:l~ed'pla t f?X:m. whose,fund amen talperiod is'~~li'belo.,:th~

p~ak ~erJ.od

^of

~e exc~tizig 'sea-Ift~t~·s . W~th ,~.~ -incre~:s,~

'.ofwat~rdepth . thest r uc t u r e ' s'-na tu ral peri od'1nc,r e,a ses and 'i t'is

suscePt1~e

^to'reecnen c e ; resul t-io g'in

:la~ge-

^'

.d~sPlaceme'rits :

^:nd

9Ue8ll~S,: Th~

^{co s t of}

'fi ~'d s'tr~ctureB"

,alSo'i ncrease s

'expone~t~;'(;y

^{...it h the}

i~cr~ase~f 'w-ate~

^.

.'.d ep th.'

'r:,

b.n ~rder'

^to,

-ov,er~o~e t~e,s e "prOble~S,,' ,c6mPli;.'~t ,

^":,,,

structure s.havebe e n devel op ed• .Th ese str,:,ctures elth ib it.. IIlOtl on'under'load ing. wh! ;h'

~ener~ llY

le ad s to red\.lc ed

l~ad'tng :

^{'The reare'ma ny'}

vari~tle~ ^{, iJ f .}

co mp liant

'8truct~r~8'

but'th enost wi d e l y favoured typ e s'are,theSemisubmer81bl~.

the' G~~ed,T~e~,

^{. .}

·JTt.~I.

- z '

,.'~: ^{' .}

:, .. ,

^..'

' .

^" ,

..

^,

. .:' :., '

^"

, ,' . '

;h~ 'T'enS'i~ Le<j",P I~tf~~m i'~ 'de~igned

^0,":

'~~~" -.

^{,' .}

, .' : ,', '

..

',./ " .,', ' .' , :", ^{" , '} . ',

Pri nc i p l e of E,x ~essBuOyancy. It'seeke'toachieve'a.ee a e ure

, "

' .

e.vpenduLu m hung",ups idedown.,

The"

fi ~'~t,

^of,such"P l a ttor'rns;'

.the !rit~n.

^{wa s} pr~ucedallllOst '20 ye~rs ago'-',The d~ve'l"opmerit''o f'the'c;r{c~pt '

~~s ,~~~ ,

^'"

,1~n9

^way

~in~e :

^{then. ee e the}

fi'r's~co~~rci'~ l "

:~'r';ductio~

faei lity:

of:

^{;,h ia} type,

' th~ ,~,~t~o'~.',

^TL.P;

shortly ~ Cp?rat'iona l','~ri,~e ~ort~sea:.:,"

1:'2" O ;~i~Ct'~ v~s .

^{, ' " ,}

Thisthesis isconcern edwith'-the loss.ofstiffn~6s

' . .

in any tether'" and the study'o f:ite e ff ects'o nthe platform

, . . ', ", ':" ,"" ".' . . " ,•• .",'0'

reesceee- Hydr:odynam,ic intet"a ~t,ionwi,Il ,J::;e.,t:aken'~nto:

ac"count'ua1ng

', t he

,d r a g"a n'd,iri~'~th:'ef'~e:cta' 'an~.th'~'ad,d&'d'm~s li,

" ", , , ,', :" " .. , '

concept. The tethersare firstII"Ode ll e d aS,line arsptirigs~'

,at~~ctl~d t~

^a

IUlnped:',~'~el"

^cs."

~he ~~~8i~~~: w'~~ , .~i'~-'!d~'9~~~~ '

::b:::::::;Y~:t:·:~:;:::.:·::::::/:.tf~eq;::C:~,:::.;n, . .

displacements,wi l l be,st u d iedus in g 11Fin i t e Ele ment

"

', .. .

idea li zati on of:the 5Y6t~m"i nth e tirns'domain.

. . ' 1 . . .... , . ' .

•Theai m'

of

,thit,study is.to ascert~in'the'·

" " '.- ': '

!

the loa s "of tether',8t1f fn~ l!8.

,.

I

',,:

.j: '

, .

.CtiAPT~R ² LITERATURE'RE·VIEW.

·2.i Environmentai,Lo'a,d S and the'R e8pon~eof ..a '.eNllo"Leg P latfonn "': '. . ' :

. .' . .

.

Methode of erJ.virorJrnent.""l response.-an d,analysis.

,which~h.avebe e n,ap p lie d·i n thedesi~of floa:ting~~iesand. oth_ercomp lia n t plat forll'l!l

.

" and:.

s·e~isub~rsibl~S . P~OVid~

.

.

.. ^a. 'so\-ll1d:basis'forthe' ~al?alysisof .the motf on ~esponseof a.

·j'ena don·1.egPlatform.

. .

. The'

hYdr~dYfar.l~Cs

.;f th e;

pi~tf~l:m.

can

~ de~~rlllin~d,

.

·'bi:the followingtwo'lI'IO~e1s.

The first model· isbasedon,'theclas8ical

hY~~odyn~~,C

't n e o r y ,

w~ere.in

^{asolution.'t o}

~he ~:PIL:e

·eq ua t.ion is'·s9U9ht

_.

. in

th~

^fluid

dom~ij,.

^{SUbjec t}

'

^to

. ~e

'p r e va i li ng'boun d a ry

condit1on~''''

These.co"nd i tions

includ~:the

·

~ine~~ic bol,l~da~' ~~dition.'oo ~e _;ree. su;~ace' ~nd

^·.on'th.e.

·

~

'sur f a c e,iu e H "a

eon8ta~t 'p;~88ure

dyn,3mi c,eond,!tion

.:de'rived'from'

the :

rigid bOdy equations of nption.and'Ute.far

.

.

.

.

' ,

lJU9·9~8t.'ed·8JtlO1t9oth~rs;

'bi

Faltin,uinand,Michelson (1974),by'

t~e

^{-eo - c eiree}

thr~e i:Ul!lension~i ~ingu laritY diet.r!bu~ibn .. ·~'~·~~~iq~.~-, . .

, ::,'. . .

~h.e·

;s e c o rJd

ap'~rOa.~h- i~

^..\e.as exact. but·19 nadi r '

adaptable f'or:etructures', suchas space frames. The

·

.h~:~~Odynami·c

..prop.ert..f:e8 of'

ea~j, ~kb~~ ' ~r~

^.

~8ti~at ~~ ei~h'

^J;""': '

.'_".

experiment a 1.ly.orbyusi ngst. rip th e o r y. This formula tion is

·in

effect

;id~nt ~Cll.l to

^theanaiysia,

9Uq'~~'~'eied

^{byMoris onst.}

d•.(1951)_.. Thebas'is of assembly. termed'as

h~drC;;dynamiC

, syn t h es h . istheaSBu mp t.ion that,\h ehYdtodyn'amic.forces on

" _' .'. • " /Jo. •

th e asseillbledstn~ cture"e q ua l ~e,llIuJllllllltlonof th~'for,ce:e acting onthe in div idu a lmemben. SOllie.of the earU estwork,,

·

,.-.

-

' . :

such as the mo t i on analysis of8em i 8u~me r 8 i b.lesby Bur k e

'(1170 ) andHooft (1971), follow this'appro a ch . _.

. The

hYdrOdy~arni~

^synthe'eh

t~chniq~e

^{forthe' .}

~re;Ucti~n

of the IlOtion re s pons e.and

t~-nS ion

^{va:r iat i on'}'l ,n th,S tethers was fir s t prop o s edby Paul-ling

:~nd

^{J:lor t?n (i ?70).}

fo~a lin!'lar izedIlOd elofa Tens"t on.t-esPlatfor~,.lnre9~ lar, and ir r e g u l a r·waves, and thec;:omputedre s ult u we r e,'Dh own to be'in-_good'

ag~eement

.'"i t hec me

·eJ(pe drne~~a. l ,fip.di~~s,

Nonlinear ~ft:ect9 ' wh ich;wer e'n'~giecte",in the.' -,pre'v;io U9ana lys i s. were consi dered by:P~Ul ling 1l97 4 )'In'

a '

·maybeeeuee dby"th"in~s , ~l1ch 11.8'

ofmotron. if.· finit."e·ampli t u d e.J;v e·s ii i..Jlonlinear"drag'f~'rcell

.

^.

Th e re'e.ult s i~~,luded.~th tr~nllient andsteady stat e

~esponges wh'lc~, wa r~

coinp ut1ed.,UB.i Qg

~hS R-u'':'~e~~utta: te~hnique .

~

... ..

:'

..

~,

I '

,[

'" , J

Th e possibility of'd y na mic ins tab i li t i e s-"at . critipalw~v.e.'fr,:q\ole~c:ie6^and ~ubharmo:ni~^OGcil'lati'on6.at'. Cros6_, ,seas'ha~>^{bc en·}

.

,de lllOns't r 1l.ted

' by

Rai neY'i'ii n s )·. The~e

-

.

"h a ve·. 'ge:ne r ateclg~ea.t^..intereat~'_ea~eC_ia.llY,^{amOng'at.!:;he.Bri ti sh} research e rs,.whonev e car rLed outanumber of expe rimental

'·9iudi~a. . ' ,- ' '..'

. '. " -.

'~he re.o~ance

_{. ', ! .}

·~roblem

^{for tethe r :}_{". "}^,_.ⁱⁿ

:~d~t.ers

^'.,

has been discu lilSed byRo ~e~an dStein~ 1k(1977 ). ,They have

u'~ed ·th~

_,

. :

three dimenlion a'1

. ~~ng~1ari'ty

"

4'istrib~ti6n 'te~hniq'uq'

" -.

:

fo r,th e h,y d r o dy nami c:: analysis•

•

:~

^{T.h e'}

~n,line.llr a~,!,lYa.i~"ca.~ried ~t ~'Na;w;g-:

^{;an d'}

'·P e nc\e-r ed.(1977 )ueea theNe;.,~rk1I.awel l allth e Ne-et.crr-

'. " . '

.

^{, ,}

.RlIph e.on.Heth~,forthe time,' ,inte g r a t i o n. They.ho!l:ve,,l;'O~c l~ded:

.that the

U~~ar

·met J:'t.ods give.

r~l~able' :eBUl~~ .if

^{.: h e·}

me,a~

dr i ft forces are.in.c:~uded.

' hibI'eeh~. e~ ,~'l:' (197~ 1 hav~ ~~elle~ted : al90;i~hmll

^.

fo~, "l.in~~r ",~nd n.6.n~i~e.~r anali~:e~:

^:The:'

h,a~e

^.

~l.S;

^..

~iscu~~e~'

^,

tet. her<:tina~ics""andc::on~i~.er;e~'.the,c as es.

at

inc 1il'\e~.

.

-

'.a.cornp er-Leon of,li ~ea~'an4 I)OnHitear'a naJ.r sio. hili ' been.pr~~ e nted,?y^{'uenf ee and,}He~f.^{(1,9 ? 9).}'T h~Y" :have'

6,~~nted c:n

th&rang&of

v.aUdit~. of

each.anaiys is,a nd,

•' .,br~ugl;1t.f;>U~

t. he

(act~h~tthe·mOe t.:significan t,non li near '.

c:f7ect·' ·i~. ~u~ed'.

by"'the

l.5z:~e ·

displa,ce ment's inthe

geomet.ry ~­

-. ,of t.he,~ethel.'s: pt~e.reEfe'cta, such,a ll~~!.'buoyancyIn,.tl: e'

\.

'v-:

. .

•

.•.

Pla~~h

^..

zon~· and _ ~i.cbu. 'IO~Cn';

^{.ar e'}

~f

^{L.. .er}'c o naeo:iuenc:::er

".

The.~~~Z::lcnonl.ln-;'&rltY

^{.a lBo}

IXIndder~blY ~.l~n~~ne~~ ,

^the.

·t.n.lQ~in'the~.th·.r.:

. . -

..

'.-~·Qheme ~8be.8t.·8u,it."ed·fo'r the·~olut ion::Therefor e, t:ht:;

c:~o~ce ?~.the",87he .ne ShOll~dbe".made:ac c or'cUng~oth e~_at~ r~..

' ~he

^{lJt.at e.of the artve e.}

qu~te

^....

i~.

e.tllb liehe,d

~ .

·~he'':late-8e}'ent~e8 .·.-,.~OSt...Ul~ubI18hed^{....or k}

1 n

^therece.~t.'Ie ll.n

ea~ ~ ~te90riud'~. : : ^{' . .}

. _.'U_-.l

"

_.

L1t~rature

_{,' ,}

~eait..n9

_{.-.' ..,}

with

_{", '}^the_.^dedq"n_{", . I}^{eepe ce.e}_{. ', '}^, _.

-s uch

as

-the work ofCapano91~ (1979). Li u,e t"er. (1?8 0)-,"Ch oo -et·ai.-(1980). Mer~i.r'~t^{al . (1 9 80),Te tlow} ~~^at.'"

(198 2'). ,.

e;lil

~t

^{d .'}

' (~982 ). 'L l~c:i

^-n'

~l.

^{}19831.;.: . ':, ' " . .}

. .(11) Liten.s-ure·d~al1it9wit.b^lthe~lat.f~r"'..R)li.on analY8:is.".lIla jo~ity.^aJ-ting t.~eae^{report'uperi ..}~ntil~--

·_··inv~.tfgat ion,!"~ " , a....ch.. 'all.the work.of.Yo s h Ula

.. . . .

et d.

.

(~;981.

.

) ;

·F ~ldn.se~

^{et .11.} (19 8 21.

K atlloYllm4. · {~98~ )

^•.

~YO~

^{.e t.'}

~1.

^(,1993),

a~d OI:Jn~l re' N"id :~en . (~98~) : '

:.' The Clu';lmt"trend in:tho,ana'1Y~h··..e~ma,:to^be'.

"

t~t;~8'ed:'

^on

~e~~u~, in~eatigaHon

^o'f·.•

~ecip~ ,.prO~le~ .

^",'

comb inat io not.variou s'hy'd~ody'namicnlode18 have~en,.i~Ud hd.

If ·th e.plat for;,has large r.pon tpone.cloae'ee, the.wete r.

. ....

'.".

•<.":

....:..::.;!.

" > :

.. ...

~U:·~i'on

^•.wui,'besignificant . _

~he vi~c;~~~ 'e'~ects

^aremore

significa n t in~e"sle nd e r member s. and .they"Cannotbe

"ac~ount~fo r inth epotent ial floW' ·-a·na lysls'•.-~'synt h e sis of

.'the two

'-~del9 "g~Vea the -~et' ~esults. as r.~p~rted ~

Standin~U'979l ;- ^.

,

" ~ ,~n

_.^{• recene.}" 'c l e

_{. .}

Dillingham

_'

(19B 4 )b "

d"ee".'

the modeiecate simu iat10n. of

a

^TLP. the teeearch.'\ .areas,_. .of- particula r int e r e stin the curre nt~e~·-·-'ts.t~-.and.- 'made

r~connenda£~~n8 far 'fu~ure

^{teste .}

Mo s t of'the'ava ila ble literat u r e dealswit h 'IlDd e ls.

r .'

2. 2.Modelling'o f-t heTether SYs tem

".'

,.-'

';, .... . .

the .te thers bei ngr.ep r e s.en t edby,linear apri n~.s., The ana'lyticaltra ns mis sio ri line solu t ionwas,

' ·"F'

. ,

,emp l o y edby~ong.(1914) -topred~ctthe·behavior of the

lightl yt;enslone}1'~aut ropes at t a chedto~eanograPhic bucye•.;

"lbre~~~

^{'e t}

a.~.

^:

C1.~:8j\ave:,pc~s'en,-e~

^a

d~t.aile~

tr'~atmentof~thenon li~ea ri t i e8.c!l',:"a ~d ~lar,gedeforma~ions in

a :

qu aa j st a t i ci~tud'y,_of-a,Fin ite,EI~ment'll"'?de~,of"the,. '",?o ring .ystem~

'?-'he-analyai~ofRichardaon andPi~to (197 9 )"is

"'-bss e d ,?" tetherp;'!'Cideiied as,-heavyceeeee eree, which .9~vea

~imp~s.^{clos ed},-f~rm^{solu t io n.}

/

/ ' .

. .

-~-I-- -_.'_

. ._ ,:.

investigat e eth e effe c ts of water dep th , eurre ce platform Th.eres.e~,rch gr~pat UniversityCOlle~e'.,Londo n has deve lope dthe ",naly t ic",l solu tions in a sy s tematic

.manne r. Jeftr iesand

P at.~l ( l';BI)

have carri ed

~ut

^a

.comparison of various IlIIlthema t i c llleoa e re of the tether system., They",havs us e d -all analyti Clllmodel, the linearmodal

}'"

".

,

:-,.

ana l ysis.~!1<tthe Finit eEl e me ht IlDdel forthi s compariao n"

Onl y,the Fini t eElement lI'Ode ,l.~ancater'for;the nonlinearit ies•.Pat e l;"~dLy nch (1963')'hll'~~presentedII.

mathematic al mod el:of th ecoup l eddy na mi c sof aTensionLeg

,-P lat for m and'the·l~te ral.dynami cs ofit s tethe·rs. Th is stu<!y.'

--- .-"",

_~s s; and the tethermass per unit length on.th e tethe r disp lace~entand the bending strength; 11.8well ason the re8 u lting,Pl~,t formnotion. Ithas bee n,ah own thllt, whilethe tet h erdy na mics,do nOt affect the pl a t.f o rm mo t io na, the platf~'rlnITOtioll'!l af {e c t the t.ethe r'dynamics Con sidera b l y.

Keysorand Mott'(1 9 7 1)'repo r t_t.h at the of f shore II industty has in'th e pas t use d fr eCtu e ncy'reeeeur ement; t,echniq~e

for't heesti~~tionot''t h e cable tens,l'on s .

Stateof-th:e'a~:of thev~b~a t ionana l y sia of fixe doff s h ore plat-forina:-'wher e shifts_inth~readnant frequ en c i es.:are usedto IIt ud y the loss oftll~ersti f f ness, ,forthepu~pos,eofInt,egri~ymo!1~tOr1ng:~~,w:~lestablished.

MOIl. of the reviewed ll.iterature con s i dere d the tethers'f.a~aveequal stiffn es,s. The investiga tion carri ed

I;

.t\. I

f·'

/

- I ·

1- 0 1"

^bY.YOlJhi~et

d. - ;. .. . ; : . '

^I

lr~8on atau.~ ~r~.d.·p~!I;.to.r~.,.·

A _

~rre~t_inveat{9a~.i.on·at th e,Har riot - Wat t Un ~ve r8 ity .-UK 18 aimedatacomprehe nsive

Jd;'

I.

. '..'a

.

·

..

progr.';';.

.

;",'=v,,"

.

the "~dY · 0;.

.

the

.

'effe r ta·

C?~\tether,_fai~li.re. i.~Clu~in9're:dhtr-i but i on-.o f.loads and.

r~r i n ::h:::::::::' :n::::::~::e~dtllin9'he roee of . . .

.te t.:t'tel'IIt i f f n e as and lts effect s on

the

platfo !"mbehavior,ha s.

.!

· . n ot\ beenteportedin · ,he ~pen · li'rr.tu r ., ... .

\

^..

^{.. ..}

1

\.:

!

. \ I

I

\"

-!

(/ .)

··. 1'

I L ·

l. r .

I i

r I~^.

..:;;;"

THE TENSION

LEG PLATFORM '

, .

.thatha sbeen connected to theanchor/fou~dat i'ori'tiltedat the

, " ' . :

sea-bedby yerti e:alimoring.'linea• .~h,eae,·tl!.the~s virtua l:lY e:liminatet'hevertical',pianemotio~^{of heave:.pitdl'and}

', '. '

:.',"

^- "':' ' .

'ro ll•.wh ile,>the;~t~ ral'IlOti o n a.or fhlay. surge,an.:l.,yaw,~;e

.

^.

system. Th~8 e6!lOdes~re'de8c~ibedas.the~rincip~I"llOdea .todisti~gui8~'t'hemfromt.h!!!secondary.:modlll,:w'hic:h-a x e

clo 8e l yre l a t ed'to the eo -eeiteat..eth er dynamics, d..at'is.

thevibr ati on'o f t.e t'h erswithi n't 'he msel v e s .

' "

"

,. -

- .,

to.:tninimizethe wei g'ht"ari~lnaltimiie'the~yloa,d.

to

,achi~ ve,

....atertight

:p~rti:tfon8 \O' withsta~d :~artb l"

^'.

oii.fi eld:

.t.CH~,b~~;'.n~er.lIte";"'~~e;':,~in~:^and~urr.e:~t,for.c.es_~~ls·

.re~pon8e.:

-

\. "

'compa'ri s onwi t h,ell c i t i n g'sea- state energ'Y. andit. sh o u ld

TJ:le

pri~r:Y ,d~sig'n ?b?e'Cti~es

^{·a r e.:'·"•.}

:~.

,.( 1)..t.oprovide the:spaceand~uPv<:' rt.,.forall tl\eJ

.

^{.' ".-.. . ..}

.

^{,,; .}

tobeil:omeamod ljil for the futur egenerat.ion ofplatfJ)rme• .

Thh',d~8'iqn,

shOWn\J.n ',Fi;.1','

is',deSCri~ed 'as f:i·~~w.!i' ·'

-.

.~~ ~rov~'d e ~<lIBY ~cce~II' ~or

^--:ark'<lind

'inspec.~ion.

J ." G.n",~rD;. c;rp"o. ~ ~- . ^-:-. -,,--,----,--,---~

... ~n~;:c.db~:t:·:::::::~~::::c::L:·f~:::9~.:: . :::;::~.'

All the.studies'priprt~ ,r.he,design0,£theHJJ£t~:m.TI.,P.'

.nec~BBlIrllyincluded"e~eral,a~t;~r~a~~'-.;te.861~t·i:n~.,.:P~Ct.:O·~.1

' "

liIuch asenvir,onmentalcond i t.io ns'jwate~,dep~hand'~.Oi l·

properti u'he,ve'le d to a configuration-clioice :wh i c h'is likeiy".' .

12

,

:.~-

. .--- '- - - - - -:-- - - ';-..,.- -.,-

The hull structure of'the'Hilt tQ n TLP.,consists of

·

~Ylindr1.ca{COl~~S·

^{. ,}

r~ctan91l1ar ~on.t.oon.~.

^{-; ji ng}

.·tiffene·d~ ·cQl\lmn~ hav~

^a'

Q:er~e8 of wate~~'19ht

^{flats to.}

li;;;it

" .

f.lO~d·1'n9: · The~·.hOU8e" a -.jy~f;~ -f~r ~nc~,?rin~ .t~~ Ten8~on. Le~~

^"

~nd~tr4n~ferrlng.their',l o a d s~t.O.the8hell~-~f'thec~lu"mnsa,nd

.... .. ~

^..

Placed"at the~inJ^.:....

'i:>ulkh,elldll, int owh i chmo~u larpallet sar

...

·,.'

.. .

^{" :}

'~the'-~hecolumn-pont~nno de•.,"Ats~a:lev e l,.wh e r e.th e- ,

~~l1bl1;i~y^'of'coH iIl',i o rt'_~~ma9~' :i!'_:!"'?~t^{.'s ,ever e.}~^ildOU.bi.e~'Bki~

con'8t~~ction,is',used't~pro~ld'e^.e'secondaryp~th""tOr:^the'

mezzanine'andwe a th e r deckle ve leto for.ani~te'grated

~tru'ct~~e ~'

~ehull.and thedecksare,f a b icatedseparately.

..a~d ~t"ea ,in.ehall~water, aft erwhich.he~'aretowedtothe

,8 i~e a~d

.a r e,

In~tailed

with the helP.'6f he rooring'

By.~te.fD "

, Thol1jJhth e etructureis si;"Ua .inconfi g urat ipn'

ee.

.

:lI~:m~eubmerB ibl.e

drilling

~

rigB, th e t:Qr"lOW'ng featllre_s'are

...,.. " '.

.

'1~ad8-on't h e,coj. umne,

· .The'

~ntOO~I.; 'ract"'~9u-l;r

^in.eec

lad.

^are

;:.::::~:::d1:9::::::::::1:n:t::r:::: :':: ::;::::~ .•:

. "."j o i n't·is Bubjec.te dto;"xb, l a~~ibend'i~?10dsana'·~'lso.'

. ... . ::::::::~:~~):~?n., ~.Y 0' . ,e: ,n;~9 t· e Ten"nnLeg

. . ---.<--Th edecjc provides the wo r1cspac r.for'th eto.l?sid e

~

. . . . • . . . . .

:.f a c i l i t i e sand c.onsists ofa.grillllg&<of 2.S..metet's deep'

< :

13-

..

~-" (1) disti nc t ive:

. .-' ',: .

-. ,

. .'." " :' .,"

~heportJ.on of buoyancy ,provided-by the ccmrene,

inducedheav e'motions.

jo~nts.

t~~

^{'o v e r alldeptllis.much}

9reate~.·:~h~~· :~tlat '~<-a

se misubme r lli bl e .·Th is,isd:n~_':~,o.:a.V9~d-t~jhi~h::s'~am in thecase Of"a highl.d,de

.tog~ther,

^{wi thill'hi g h}

~ave

^.•

. -

.

. . .

-"- '-"

..

co mp a r e d...it.l'\.thlltprovided.by'the.

po ntoons-L e

much greater'in:a,

Te~S'ic;n .~e~ .:pi~J;()j~. · Th'~9

^,ls,d o n"e.ec

~i~imize,the tensionle g'.l'oads~·^.,rat h e r. t h a n'·the.,- (11)

TheMo orinq sveea

T"Ile."'~utton^{TLP1.8,a.pe rmane nt} ins t a ll at i o n·in:~he'~^.:',

.B,ense:that

it ....

il l··

be .

operat ionalin all but most'adve rse'

env Lz-onment.a LC?nd i ~ iona.andisfu~!<tiomsllyequivai e nt to a

.:(i i i )

0'

,',

. '

lO!,dtoth ec~OB s"loadbearing (CL B) . Below'~heCLB-the re

are"1 1,el eme b t s each approx imately'9. 5

~eters', lo~g.

^a,nd

"'i~~ '

• a

mad'm~~

^{'y1e 1 Q}

-1It.~en9tl1.

^{of 795 N/1MI2(F i g.}

',:, < d l.

^{,Thl!' ,:}

• bottom o'f t.he_{' , I} .-tag h_,. as_,."the.'anchor

con~~·ct~r.:(·F~. I • ,

^4(0.)I,.

. Wl\~Ch

isadeviceforat t a chi n g thelegto the.tem p l a t e on.th:' ae~~ed. .Thifl ccnnec ccrisactivat~dby th efO,r ce'o fgravity andre ex e .itselffn, ,It cenbe re mov e d from the.'d.e c k remo t ely_wi th a hydraulic device.

Both en{ls of

_th-~ te~8ion

^{leg have}

elastom~-~ic

^.

••'. '._f.l e itj <:,i,n.t 8, wh i challo.... fre e~otation-~Pt?

is.s

deg ~ee~:• . The_. depr~ymentof th e leg_\.is

.

under taken -fr o m

.

': ...:'

the

. l1I00ring compartillent'(Fig.5), whichis loc~tedintheuppe r

,

.

^,

partof the column. The leg eremencs are storedhere pr i o r

, ' t ', ...

toinsta llation, The compartmen thouses speciallydeveloped.

~ ' - '.<

mechanical'h andling equipmenta. Thesear !'us ed'f o r insta l la t i on'a.ndremoval ofthe1'eg.e 1E;meot s.

,:'~_;~

^{Th e}

tethltSy~tem ins~~i.l~·tion

^is

c~~ried

out"'in '

calm"'weather. ,The'

plat~o:m

^istempora rilymooredwi t h the

"h e Ip of an~ightpoint catenarymoo~ingsystem, .A.'aiogle leg element ~neachcorn er isas s e m~ ledand low e~ed , alongwi th theconn,ectingan em1;JJ.y. The "succeeding slem,ent sar e adde~

.',. on tilljt h e lower endrea ches~-meters abovethe se abed. Th e motioh,compens~toroneach le g isthens~trokedful l y to raise

'';~Ch le~

^{by aboutiOmeters.}

F'in~ltY. th~

'anChor connecto rs

~restahbed intothe"templates, At thisstage', the

. ,

. i

I.,

,-.~,

,I

.

.-'

t

!

C:~mpen8a.tc;n\GC~~ri'£: f6r'. th~ 'y~~s'el,'

^e"

,mot.'i~n8." ^.

^{A nominal}

·.:--t·~~;!·o~

^{;8ap plie d}

t~:,eac~ :i~9 " to cl:l~ck

^{:for.iOCki ng}

.in~

^the

ch~Ck'

veIvee on._t h e"

c~m~~riutOJ:'6'

^{"at e}

~108ed.

^{;Whe n}

tl\ e" he ave

mo"tion

i8~·f!!.ilY ' ~upprea8e~ ;: t~e'

pl'at f or m'is

~uile.(3. -d~r:

^to

".",i t-iiiope r~tin9draf t.wi tht'he.hel p of.h ydrau l-ic,ja c ks. 'Th e-.

lo ad-?lat~8aren~'lo'cke~'~~dthe pl a t form Lepartia~lY

deballu't~d. ~h~ pi~tfotm

;1;'novin'the TLl"

mod~ .,

:The remaini ng12

e l~me~t8 ar~

. -

'now

^{Lowe r-ed}

~ith'~th~

. . .^helpof.a pol a rcr an e...once.~11the-eleme n t sar e:in pos .i:tion , th e

. ,

.platformisdeba lla sted.to:itsoperationaldraft .,

•.• ". i- " ". . . :,

Asforthe!r e d.undanc y?fthe,.sys t em,~n ytwo

\ethers'~4achcor ner can take.thedes ign'lo adif the remai nin g' ee ee r ein.t he clus terare notfu ncti onal."The r eis

" .aprovi sionfor;et!aCl i ngany of the

tet~er9

te mpot'll.;UY

~or"

inspe~tion. ~venthe,unth i nkable fai lu re , of allthetet.hers

. '

~ . "

in'a'corn e r wou ld no tbe'p r ogres si ve, al t h o u ghth eplat fo~m.

"would be

in

je,op ardy,unles s itsce n t e rofgra vityismov ed

towards t.h e unt ethered cor n er .

LOad cella in.ea chcorner.a r e ueed tocon'tinuously mp"ni t.orthetensionin the t;e t he r and thesevalue~^{are.us e d}

fo r.onboa r dlO~lldistri bution .

.~.

4 •.1 In t r od uct i on

Theobje.ctiveofth i s

fO,rm~lation'·· ,il'.'~O d~velO~'

.~-th.ell\li..ti~o!I1models toatudy the effect'

of

^{·"t h.}'l~lJaof tethe r .stiffnesson the behavior ofaTensionLI!9 Platform.

,

I t is appa ren t that a dalllagedstr"uc:ture, 'will have: _ differentrnotiLoncharacterist icsfrom ~o8 eof an intact struc t4re'. The.,usual'a~sumption8pertdningto the,res tor~n9 ecececomput~tions. ~U 8t'^thereforebemodified·to'C:llt~'r.f~~'

.

^.,'.

.

- ,' ^' . . "

the un eventether stiffneBQea. Thestudy,ha s beencarFied out, in

t....

os~ages.

Fi rs t II. frequ encydomain analysis'iscarrie~<,o ut' .with

~he aIl8u~Pti~n ~hat

the

systembehav~s lin~arly.

Strictly8pe~krng:_:~hb~anitiysis',has a limte?va l i d i t y;'~ut,.

itservessaa anindicatorof.the'natureof·th'echange R."The.

8tt"uctur~,is'll'lOde l l e d asII lump e d masa~ith ~degrees'of freedom...Asch ematicdia9ram'of t.hismodel'isshownin

F,~9.

6.' ERSe?tlally-, this

f~rrnuiation'

foilow8'a 8trip

. th~O~Y

.

,apprC?aCh" ,~~ S~~dy

^'.

the te~Bion V~~h~~ci~,

^{,a.n J},l:h e"...

·llI.~t~.ori

^'..

~

^.

.reapottaee , .,

I. ", ." .

~thasbeens~own'th~t't hegeome t, ricnonline~rity' cauaed by.the'lar gedefOl-lIlal:ionainf l uenees'the'plat f o r m

beh~vior ~a , we~l

^as'

th~ .te~~ion ' hi9t~ry

slg:n!f!cantl.y

{Denise '~

"1,'

I

. .. J

'..~.

.

·· ·~ .~<!:af.· l'7~ 1 .

^1\

'la'r~<!:' dlaplaee.~nt

n'Otll1nea r inode l 18

de velopedfor Il time d::lma.1n",n~ lYd~^•.Thehy~rod.;n:"rli,c .'

prope:r~ie;'

^{are.det-lvell frOll}• ^{th e}j

pre~io~.

.' ^{';4el; The}•'.^{'p{a tf orm}•

·18'~e lled,as I.'J':'diMns'ional n~!-vorkof.bel..element s wi t h

·4S

n~es

^{{Ind.249 degrees of}

freedOll.~ T bi. ·

model 11

lI~pwn

ⁱⁿ

Fig.7~

. t ,

4.?,LinearAnalysis'.~. .4'.2.1,T.heconsti t u t ive

f~~ee~

.' . J--:' ':i ~'

,'co ns ider a

'~igi4 'f1Qa~ln9 ~dY os'~i llat.·i~g\ ~~ut

^'Il

.,~~ =.

. . . .

·st l.te---of re s t in re sponse to,e xci t llt ionby.longcrested

...

·;~'n.~~r .~;~f.~~' .. A~:1~ert,ia~ syst~m.

^of'.

~din~t~s

^{.OXYZ. · •}

i'..'a ttached to t.h eequilibr i u';lpos itionofthe c.~t;·eror-gravity

· ' . • _ " . t.h '"

.,i.-s.defi n~ l1.nd.:~e ~tionis ~.enoted^by.r )c.for't.~~"It ,'.degree offr e edo.. as shown-I n·Fi g.8 (a) .,Th'e'prob l empolled

.

here~eai s

^{wit,,! the'fluid':rotlo n}

:indu ced' ~

^....U

a.rI;'ii t~de·

o8cli latio~

^{of t.he.objectaI...}

ell· ~· th~ ·t-~U·id.

^{ll'IO·tion·}

a8~~~~ edwit hthe int e raction'ofthe.~je ~;t ~ith ~tr.ai~ o~

~nCOrai~gv~..~es. Th~ theoret~e~ l~sisof.th e f~~nUllltilon along....i th de ta qe d derivation s.h.a ~beenprese nte d.t:rt.H ~~ft

(1970.19.71) forIl

selllil~hmenihle .is 8u~in9 ·pot~ nt.ial fl~~.

Thl~f.orm~lationhas'bee n adoptedh~re.wi th 'ce~t';"irmi~~r .·

',

mo'difica.~ions. The " u~ ~ent i~'~t~ree ~!

^thiu

fOrmUla,t~iori

^..·

~10J;l g ~i ththeimplicit lllS8ul)'lptions are d1seulled In"

... .

'. '

. .

'." ,

.

^~'".

'. ~ )

. . '

.

.~.--~- - .

.. ..'

^,'

'8

_"

.

... ...

\ . e n

The

undiltU:':~d.pre

^{. .are}:force"

·(.~so

^kne,.,n

~,

.thePre u d e -« r y lov fo.rc e l. whi~la t he.te r ce tha~arises, frcn Ap pendixA•.Themodifi cation

~ropo.~..

^{1n regardto'}

~;'her . a~i~f~e.ss ,~·~el1in9'·

^b -presented 1n'th e MJI:'t

~e.ct.i:on.

_!J'n

.the .·.~ a i.· of·

the.·:afon -"Illentioned

~o~ll latiorl.

-itI

\s~ .i~f~r.;1fJ

.that.'th e

Pl~tf~t"~.~~ion

^{In ...}

a~':.s.

i.

~auud ~'

^.•

· ·. ~:~:~~:E::;?: "nr,:=i:~· ····

H,) .:o",.j~ . ^,'th tho ^{,.,,,ity IDam p,:, 'fo ,,,, )}

( i1il~ ' t/,~elJ,

prop6rtiOna 1

.:~

^the.:

~~.piac~,:ne.nt·" (R·e~tofing

",'forc u')

/~:.;ma,:':~::~::~::~io. ,~~~~. ~. th . . rod, . a r.

"',:

..

^'~

~^"

't hep~~asur.ov~r^{hul l' 1;':4}

W a y;

that,bas'~t~~n'disturbed'

.'by:-he.h u 11•

.

li~·) Th~: ~ner~(a fcir~e~:whic~

',ris es'fro... the acceie rati onof.l he

add~

^masa'of tbe

" h'~ll

init.wa v e that' haa

,,', .

.n?t:been'.dhtur':tadby..~e'huH•

.The',potenth~~ld'.,-8oiuti~n'neq ~e~ts.th ~ viscou s.,.

effec ts. wh i ch"~are^of.ai~nificllnt^{.mll gni tude::} 'Th~'~iSCOU8 'f o r c e'lI"ilr~lh'ere fo re'in:Clu4e~^additiona

11Y .

liS'.propose'aby

'

.. ..

^{, ' ,}

. ....

^{' " '.} ".

."

:,. '

Pau ll i nq (1911) .':Th e viscouspa rth'u8 uslly'aseu med'to bea QUadnt1c".fUnct.l0nO.f:the

velC?d~y ·bt. a~~1~9Y ;tO '~e ~ca8e o~ .,

... ..

-;

rs

(.

8t~a~ :n~"' , ~,~r.- t~e ~r~e~t an~lY~is. th~

^{qu adraticdrag.is}

rep l a c ed~,.¥i.M~qU~ Ya1~ n,::.11n,e ar.dra g-.force,.... The , '" . equiv}~,lent'"li ne a,t' drag,~oef fiCi ent CDI.i~giyen.:by.

CDL •

~.uo ~DQ

where coO.isthe qUadrat,it:,dra9coef~i<:~ent ^and.uo'ill.t.tt e. amplitu.~0.£-the,peri<?di~.f~uid'\relacitl"

The

~8t

of'th e

6:,n9~it~ent

forc es'is

'~n~ re8t6~ing'

~. " ". . " .

fo r ce introd.uced.'r:ty,th~t;ether sy a t e m.,"'ITh is_.f orc e i~_

~roportional i~'

^the

diJla~ernent

^and

ig'·llot.l~ibuiedt:

^the,

.compon.nto

of'·.'n~,,~r lih. t';~OM t.n.in9 :t~. i.•

^toro·the

.at~ucture:foil~ing.~ ,smal i'~i8Place~ent.

4; 2.2 Th~i'guadono"fH~tion'.

"The equationo~lIOtion

in'oUte'

8iinplifi~d form-

c .

res ultsfrom equatir:t9th~ :e~~tivefor ce swi th the e:lI:~t'tation. - f.~rceB,'.'.::-

·....her e

' 6 " " .,') . _

..•j: 1J~j'+~kj)rt ~j^rj..~^{Clc j r}

r"

Fk

, . ..' .: , ,I .

)1;"1.2•• •.•• 6,j·

exp(iwtl

(4.1:)

I'~kexp (i..,~ )~~YdrOdy.~ami~.excita:~~o~.:~C;rce [~'j'j~. ....Has ll:/iner t:l ama~:ri'x

. '

. } a)l;jl ..

["xj]

#,a.dded'M~S8·'m~·t r iir:•.~

If, ', , -'

..Dompin~ c~efflclentmatrix

:.

.!

Tho

l

'"

..

"

,

r..

,.

'.-

",

~C~j]' ^{.'"Restor.1ng.force}.a~r~l.

'..e irelH a r'freq u'ency,0 /. .cit~ti.on-

Ifl-.orde~

^to

redu~~_'

^the

co~ple~ ~e~ond'

^order

.diff erential

equll.t l~N1

^to

~' 8e~

^of

~inll;!lr .co~lex eqU~tiO~"

^.

.

1~.~8 nec~s~ary^to.a'lIu'~~.~arrtl?nic^{'lIOtion. :·Sub ati tu ti l'!S}

~

^,.I.j.

eIP.(i :t~

^).".-'

6 . . -./ /... . ". " '.

j:l· (~~' ('j .' -~~) " + '.'i~:'. ~j

^"+

Ckj.~ ,l j

^{-» ,}

F . t:

where 8jis.the.ll.m~}~tudeof !h~ j~.mode.of /!lOt ion:

'..

';qu~t:;i~n ~f.· ~.tiqn" .~.' .~:l~~.~: _ _

^for

8 j" . : . .,

.4.2. 3 Th~.fetherSt H'f nessHodel~':,,:.-'

Ca l culati onof-.theEquilibrium'hulo n_

Fig_9'(a) aha:.a the-pla~r'eO~fi~uritionofa". . 'T en'.i o nLeg

~btforlll

^{with the}

tet~ers ~el.led;'s ' ' lln~~r

springs of.

;"'- nequai .•

t.t:tt.?e8~.KlaM K2., _At eqllil~~iiw:..th~i J:' J:'e. pe ctiYedefl~cti~naae edl a.nd·Q. 'Th etotal force ~f

buoy~ncy

^{ie'0, ilcting}

il~

^{thece Mer'}

~f .b~oy~ n~

^{"(Xun!U from}

te t her:1I1th atU fne•• Kl,). Ttte tota ~:welgt; t",11"1;; acti z:tgilt th e.midpo int betwenthe twotettLen (or:L ',/2'~nit!ll fr::om~the

...teth er

wi~h a~Ufre

^{. ..Kl)•.}

I~ i~ _th~ ' tOta~

^undel'"ool'ilter volu me.

'. '.

.

Whi.ch,.reJna.in~_cons t ant , Th e. tbtd pretension'18P,10"that p...D-W'

TheequillbJ:'i ull'l equatto n.s

~

. :

" .,

....,

j' ,

-. _ _

.",:-.'-' ~~~-~--

! . .• .

i

Kl .dl +K2.d2-P

/ .

(4.2)

}l '

Kl. ei .x'=<K2. d2 . (L' - X}:..W.,(X-L,'/2 )

,•••••(4:3)

'Th is problem is~taticaliyindetermina 'teinX.,ai.a nd d2.',

From~the<;Jeometry depieted in 'Fig. 9(hl:the'deflectionat X is

.,dX:''".dI~ ^(d2-dl)" K/L'· (4.4)

I-

..Thedef le.ction ·at.x isc~u~edby t~etra nslat~onof ~he b~Y and rot:.a.tional !rOt.ien-about th.e·midpoint, so'th a t

' . / ." ' . "

.

•

~.

^{··dX ....·P(l/Kl';t}

~2) : ~

^{4' . (X"-L'/2 )2 /}IKl -K2 )L, 2 )

. '" ... .~(~',5)

".f r ctld 4.4)and (4' ,5 )

.

~IKl "- 'K2)(d1 .L '+ ( d2;f)- l5i~2 )"..;'~·~i~

^-.X}2,:,O

··•••:.(4.6)

f;quations4.2. 4.3,'and4.6are solveditera t ivelYjfor X.dl and·d2. This principle,is further app li ed to tl\eoth~r Plan~

to arrive at the 3-dirrie neienalequilibrium. CableStiffne aeMatrix \.

Ca b l e a areeeeumedto,be ....eightless. taut aridof linear stiffneas_. . Th e

~del i~c'~udee

^{th e}

ela!~ii. . stiffn~lJ8

due to the extension of the cable as wel l as ~thependUlum- ...~t iffne8sdue toth e pz-e'te ne Ion Inth,:, ca'ble 'under la t e r a l .displacement.

The.configuration of-a typ icalcableha s beenshown

. . . .

inFIg. 9(e)".Considera cablef~ xedatth e point~(xb, yb ,

. ' . " .

zb) ....it hitsothe r end attllc hedtoa'IlOving structure, at'the

" ~

. '

.

. /1'

,poi~t 2.(~o. ^{yo, z.o)" The.motion of the}.ot r u !5tu re isabo u t:

I:

the~~ nt~(xg.yg;"zg~~ ^{I f}

the

pointof atbchm_~n_tl1P....ee' .bya. smalldista~ce~I d x. dy•.dz)~,tlle forceen the

.~ t ru'ctureto·thefir s t'order is

whe r e

dT. " '"<J) +(~L;^{Tl.dL.;(a--o)}

..

r..-Length E',:,Yolmg' s".-m:xtulus A 2'

x

eee.of.cr~SS8ection.

.dL...,-Change in le ngt h

It

~an be8h~n. f~om

Eqn.4.7

t~at

't h e jtl'lcomp o ne nt.of the displacement t"j at ex;Wi ll,.ca u s efo~.c e ll.given_by"

(4.8)

wh e r e'Ki j

i.,

"th e-st.iffneElIl.dueth e,the,cable:

. The detailedstit"f n~8 8^matrixwa s derive(land the

. . . .. . . . \

final·equat i.ons are pre 8 e n t ed ~ i n.App e nd ix B•

.

,

4.3 Non l inea r Ana ly s is

. .

4.3. 1 Th e Equa tionof-Mot i o n

It has'been shownbyAlbrecht !!taI.'·(19 7 6l that.:ih

j I

23

~functi ons, has to,pe used. Gelle rally, theequ.at.Lortaof

• , -

~"t_

,.... ' . .

IIlQtionfo r a.lIUtt~e9:r~ e ~.ffreedom8~stemre a d .

N ~ ^{- ' .}

K~iCfl'Jct"Kj]{ rJ'+[~.jJ{r}+{Nes tori n9^(r)}=

(F(r,.r.r,_.tl ) where

(4.9 )

fm~jJ^,"Mliss/ Iner.tia"matrt.li:

[~Jcj]..-Added massmatr ix [bk j J.-',oa~Pin~^matrix

IFrestoring(r) } ..Res toring fo rcevec t or {F( r. r, r, tll ..E:xci t i ngforce ve ctor {d. LrIarethedi s p la cement , velocity and ac ce lera t ionvecto r s resp ec t ive l y. The numberof deqr eee of fr e edOlll'isH,....h~ c:h i~c lud es.boththe r~ 9 i dand'ela stic

89rt1es offreed~m^•

.T~ ere..or,:,anumb~ rof non li ne a r ter mswhi ch may

-lI"~iearinEqn . 4.9. Deniseand Heaf {197.9).ha ve shOlo'n tha t

.' 1ImO,z

^thevariO~8no n linea r eff,ecta.

the

variation in the restori ng for c e due to largedil!lplll.c~lllents is th e most.

inf\~.ntial

^{fa ctor}

90voroin9-<il~

^{t.ene Lcn reepo ne e}

o~

^.the

.

tethe~ 'a~ ~eil

^as

'the~'iooti'on

^Of',:

the p~~tform

^•

.,\\

~ e hy~r.odynamiC prOPE!'rths~, tha~ ,.iB, ,~ ~--:-

da\Pinq andth e wave in@c~exe±n~~aa~e ' . .. .

-1-~- - '-'\ '''- ''O;:~:,: " ".. ...

ⁱ. . . .. _ ,

.• ...

" "

.. , .

The' the or etical eeet e and"a $9umpt i ons,ma defor,this,

'.~-i: ;o:rmul~tion

^are_

d;~~us89d_ in A~pendix~. Il~~ver, t~e

^io'ad

....

~

...

^'

^.

vec:t or-Ln:th eno n li ne a r form ulat io nhas tobe comput ed,at ell~h^{ste p':'}b~a-u.e·^{it is}depe~dent^.on'thefnstantlineou~^.'

.

.. . ./

pee-Ltion ofth e'pl atf 0 t:I1. .• •

.The'restort:ng

forc~; vec~¢r

^has.cont ribu t i o n s fro m

.

, . . , "

t~ehydr o,stat i e - eorce ,.theelu.ti.Csti f f nes s",of~he .t et h e"ro as we ll,a,s.the geollletricst i ffnessofthe' tet hersY';Jt emo .:~ Thedhcr etizat i on..of the,con ti nuumia..car: ried'out.

usingth~ ^{fini teele me ntmOd el: Theconcep t cif,finit e} ele me nt intime di ioens i on

.

i s emp10yedfo r th e sol uti on o. f'\the. eClu ation_ofeoe.Icnby,anit e rative numerica l.scl1em!'!'.

4.3.2 The Fi ni teEle me nt MOClel

For th ela rge dis place mentdynamic respo nD e

c~leUlat.l0n

^of

~he , teth~r .

J::\etwork. the.finite^. el eJlen t^. ecd ej,'is emp Lcyed, The.adva n t age of thi s model lie!!_ no ton1.y in the -.f.8 ci l1 tY.toin,c.orpor.at~·al~the paril.'meters .a~f~et~ng\h e

syst em',~ut^{al s o}

in

th'e'hi gh deg reeof t"eliaoili~Y

of

th e results.

Fig. i.shows,Hne dia g rall\of ~h~f in i t ee~ement mod e l. ,I

. .

^{. Thetethers aremo~e l1 edas b}eam e1.ementa wit~

12'

d~greeS ~f ' fr~ed~"'_{three ~tnn~ "

^{. :}

~.--- ^{. . , '}

.. .- -r ota ti on alat each-end ). , The end nodesof eachteth ~rare

.'~h ing ed .· Thfa model-in corporate s the va rh.ti o nin t;hetetller

... - -, J

/ '

25

'~'.

-'"

.pr~~~·~i:. ie B.·and tl!n~ionalon g.th e leng.~~

a s

well'as:t h e"

computat~on

Of'the contri buti ons"due'·t.o.

rH~~tic

and

·~eOinet.ri~

.. ~ti. ffnes s.T·helfOIie l'has'thecapab~lity

«:

ine or pOrating theexc i t ati oncaus edby'vort~x8headi. n9~t

tether~._ and.,exi!l.~t c~rrent.

^andwave fo rc eson,

the,:·t~~h·ers-.

Th e s eE!ffect~aie.not'inc~udedinthe 'fp r ea ent._ani!l.iys~s. The respon8~

'o r

the",'t e t h e rnet~~rk^isat if f nllBB"d omina t. ed,

Thetopsi de,18 modelled au"a lumpedrig id mass ha:-,:ing.si.xd~9r eeBof freedq rn•.The.lIIOti o.fl1iatthe top are lran"ferredto thenode s' 2 thr ough S. Si llcea~accur at e

, ^'

e~timat.ionof thefirst:d~r ivativeof theexc iti ng for ce h

",~~t availab le ,'th~^{moU onof·the'topside}1& solved us i~g^the

Wi.lso n-9 alg orithm,gi vllnbyBat h eand Wils on(1976). Thus the

t~pside co~fi9uratiO?

^{-i s a}

i{~id

^{bod yeod e lof tho}

plat f;ormhU~ l an ditsIfIOti onsa.0 i,ner t iadominated .

' l .

^{itha,sbeenbroug h t0,to-in the}dis c ussion of the nonl.!

ea~ Sti.f f_~e~s ' fOIll\Ulatioi ·{ see, APpendi~

^{B)thatth e.}

.-.\

~~;~er th~· n~~'b·er

^of

element~

^{in·a n,Yte t her, t}h il'greatll.r

i~

_ -,~he_acc u r acy of.t hefor~~£fne'5~er, ~

.,. _'. ... -:_,'the largernum'b ~-of-eTellle nt8le ad---t~la rge rrequi r eme nt s of

fO;-the'

an~,l.Y'!Jb.

^The

'5Y~teJ:l

^{"';as firs t '}

, mod e lled wi th large r nUllbElr ofelements (40) . and . su'bBequentl ythenumberof element s was reduce dto {25}•.

"Th ia help ed-i n a 6U-s a ving in thecomput er time wh'i1e

.

"affectin~

^{the.a.ccur ac y}

of

ther, s ultsmargina'1 IY .

26

.:

,

^{4.3.,] Thi! Ti.me}Integration^{.~ ".}Scheme

Argyris e t. al: (l913) have deri ved th e concept.'of , the

apPl!~atio~

^of"; t nit i;.

el~me:nt.m~t'htdol09Y, for·

^th.::.time

st~p .i;htegrati~R^'of ~onl i"ne.ar^{cab lene tworks: The"equa t.Lcn}

"

.o flQOt i o n ( 4.9) ·!s rewritte n \l'1s

(

,a nd

....here

{R} .. {R a )- {Ra}':(b]Ir)

u~.)^"·C~.a }.-{l~.s}^{-,(b ]} {r}

{~}" '" Iner tiascreevec'tor {RaJ .-ExcitingfO;C've c tor' JRs}.= Restorin",forceveoecr-.

[b] • Damping Matri x

(4. 10)

\

.--..,- - - '£r} = Dhplacement vector

The8uper~cri p t... ..de no t~V-.!....'.ol_ti;}L re s pec t tillle. ~e_ine rti a:---:fOr c eis assumed to bea cu b ic function

~e .

it hall1;Jeenshownby Argyris.that ;

. -1"' .~, , .

__

~t= ^(rO+m1 2"(6 RO+

'f Ro

+6R1-.TRIll.

"IrO+dr ) •.• •(4: 12)

!

and

I I

(4. 1)

""her,","'~he SUbscr~Pt

'''0''

,'~ ~di,ca tes t~~value~f ~lIevariable

. '\ ?<

~(?E!ginning of 110 -time'~te.pa.ndthe''SU? l c r.:i p t'~l ~ .~nd f~,ates,^the,'V~lue:o f,:t ll evariabl~^{'a f t era -}

tt.;e.

st.ep,(T)'and

" rn;)

is the,eq~~~t'~S8~tri.i 't;lle~umof-/nas s ' andadde~

~ss matr~.x:). An.'i te r a ti vesO,i."uti9~scheme is'adop ted for',

".the.s?ll1tion,,of theini t ia l valueproblemus i n g'sqne• 4. 12 ,

"· :·'<-~':i3::·'~.~·

_- ^{4.l4 ';}

_ :·:- F'~r t .~-.·.f·irst

_- ^..

'.9t~.mate 'the

^value.s_-^{(R, land·}_~...,;;.:i^;:;"...

iR1 J a r e'approxim"~tedas ,~

.Str,~eti~''~peaking" th"ek.i~em.&.ticlill:':.,~~uiva lent and damp'ing matrices are, changing with the geometry, but

. • _ , . ",: , -l '

,t h eY, a r e assumed to_00'',,?ons t a nt.

JThe:

~hoice

^of'

~~:,

^scheme

ee " in.tegra,ti~il'

^{is closely,}

linkeli to thenature.ofthe prpblemin terlllS of'numeric~l

~~abiii'ty^and'?~si~ed'^{accuracy. The}idea,:.;·alg~7.ithni~hou1d alsohllov~ th~"minimum,requ i r ement of'compu'ter',time and

.

.

Theal g or ith m'

of,' .

finit,eelements in'time .

in t e gra t l p n is beai:;su"ited·':forthe pre6ent"'analY8isbeca~se

. '

.. '- .

.of the-f0110....ing reasons,:.

.(i ) The equationof':nOtion in the p'rlil'sent,

'" . " ^.:,.

< :-. ;"

.'

. ...

)

( ;;~

, I !

-,

.

.

~h~ reaathe.t i"ffne"u mat r"1ces areposit i ondep ende? t'- -This. raet hoddo es not requir eaninve~. i onofth e st iffn ess ~tr i . and ju st.o ne lnver. lonof th e massmatri x."'''against.a l l .t he

:::::.::":! .~:~h:t:::·,",u<e"W.';iCh call ^{io :} r- ""

"(i l ) The

~et~er '·8tiffnesa ._trix

^{ha·s,alarge r}

.bandw id th. AIIchemehasbeen deve lop"ed,'ali de8crib~din

the ".

.ne x t " ": " •.'wh i chd?ey'.not.

re~~ir.e 910b~1 ,~8em~1;

^{of.t h}@.": :':":,,'_ .. st i f fne s s matr.ix andUJe s a col umn ve ctor prod u c t of-the '_

8.~iffne8S ' an~

^{"t h e}

dt.7~lac~me~t 'v~:to~', ,d~rec~iy

^{'; .Thh}

,. ~.c~em.~

^{.' -, :}

saves onthetota lfn-core mell\Ory reqrire lll.8nto ,. . . . ..(iii) .rehillSbeen

~h~n.

^{by Argyria.(1973')}

" th~t

^'""

th isl1Iethod 1nef fectalllOu nts to.",st

~rder' inte~~lat i~rt "af

.thedis p1a celloent•

fu~~tion

• and 18• IlI()r elJtab :!i·."and'5"'

..aeeura:t_t:"""~

.- • than

~.t

^{of thepn}vei ling nu••rica'

'nt~·,f" .ion

^.•

cho~'

^{': '...}

in nonlinear ..~ehanie8. Al brec h t.e t d ..(197 8)'h~ve ^also establishedthat method swhichwork dir.e~1Y·arenot.lJuit a b le· . '.:

fot:the s"alu tton~fnoniinearpr oble ma inv o lv i ng cabf~-' 'net";ork a and'have us ed this'"

~eheme

^{fo r'th ei ranalysi.lJ. ;n';}

theeOlllputation'of the~t~.onres pons e, t~8'eff e et of.the , s~ar,tin9transient halJ'be en'r e duced

!' _

~:'~owingtheeXd t in\}& . term to bu'"i ldup

gradU~llY"

^{Th 10is"}ae complli:hedby

7~i

• "i.I!I

mUlti plyi ng theforc e andmome nt ter ms by'"taperin g ,"- fun ction: The first 2 cycle s",r.e'·ign~~edWhil.~preli-e'n~i~9'

)