at th

An Expe ri ment a l Study of the Dyn amics of Caten ary Mooring Lines

by

©

John F.CrossBSe.

A thesis submitt ed tothe School of Graduate Studies inpartlnlfullillmcntoftherequirements forthedegreeof

Masterof Engineering

Faculty ofEngineeringand Applied Science Memorial UniversityofNewfoundland

June,1992

St.John's Newfoundland Canada

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Abstra ct

Thiswork examinesthedynamic fortesthat aregenerated in catenary mooring lines whenthe lop partof thelineis subjected to aforcing function.Theforcing functionconsists ofa sinusoidalmotion of fixed amplitude but varying frequency and varying angle in theverticalplane.Theelfect ofchanges in the pretensionin thecha in is also examined.

Fourdifferentareas ofresponse arcclassified by theresponse of thelineand the characteristicsof the force in the line observed overtime.Theforceilleach areaisexaminedand the mechanismthatgeneratesthesefo rcCI! is discussed. The change of theseforceswithchanges inthe forcing Cunctionat the top end ofthe lineis alsoexamine d.

Themodelparameters for the catenarymooringsystem arc identifiedand modeltests performed .From these modeltests it is possibletoconcludethatlarge struct uressuch as oil rigsanularge veaeelawillcause mooringlinestorespondin a rangewhere dynamic forcesneed not be takeninto accou nt. However, smaller objectssuchas buoyswillsuffersignificant dynamicforces.

Dedica tion

Tomymoth ~rJessiu MargaretEyreFordCros8(192&1090) .From who m[learned themostaboutengineering.

iii

Ac knowledgements

Althoughmy name is the sole namealltheCO\"Crof thistllt'si9 1haveincurred theaid of many in bringing it to a successfulconclusion. First andforemostI would liketo thankmy supervisorDr.1\1.Bootonforhisexcellentguida nceand ad vice.lIis calmapproach to problemsolvinghelped memany times.

Iamgratefulfor financial supportfromtheCenter forCold OceanResources Engineeringin thefor m of a C-CORE fellowshipand alsoLa the NaturalScience and Engineering Research Council forsupportintile Ionn ofallNSERC poet- graduate scholarship.

I amalsoindebted to many others such asMr. Hussein andDr. Marshall who helped me perform my experiments. Iam abo gratefulto thestalf ofthe wavetank, the machine ahcp andthe CenterforComputer Ah.k-dEngineeringfor theirhelp and guidance. Ginny as she usually docs sortedme outwhenI became entangledin mathematicalexpressions.And also,my than ks to the Engineering Graduate Student Body Ior their adviceand support.

Contents

ListofFigures List ofTables List ofSymbols 1 Introduction

2 The CatenaryMooring System 3 Review ofPrevious Wor k

:1.1 The Static Problem .

vii ix

,8

3.2 ExperimentalWork .• .... 10

3.3 NumericalCalculationsof Mooring Line Dynamics.... 16 3.4 StatisticalAnalysis Techniques ,.. . . . 19

.. Dimension lessAnalysisofa Catenary 21

4.1 Definition of Physical Parameters.. ...••.. ...• ... 21 4,2 Derivation of the DimensionlessEquation .• •.•. .• •. .•.. 23 4.2.1 DimensionalAnalysisUsing a VelocityAmplitude... .• 27 4.3 Validationof theModelUsing the Static Case .•.• • •.. ... 29

4.4 Select ionor the Experimental Parameters....• .

5 Exp er iment alWor k 5.1 Equipment... ...

32 .••• • • • •.•••..•• J:!

5.2 Instrumentation ... ... 38

5.3 TheTest Chain .. . . .. . ... .•. .... -15

~P~~. .. ... ... . 46

5.5 Da.ta AnAlysis...• •..• • • • ...•...•. -17

6 Discussion 6.1 ExperimentalWork. .. .. 6.2 Scalingthe Model... 7Conclusions 8 Rec o m mend ations App end ix A App endixB Appendix C AppendixD

"

. 51 ..• . . . • •. .•.. . . . 57

6.

66

76

7 .

168

List of Figures

2.1 TheCatenary Mooring System .,.

2.2 A Cate na ry Mooring SystemUnder Horizontal Loads.

3.1 Force-time tracesfor the fourconditions . .. . . . ... . • 13

4.1 Pa rameters forthe catenarymooring system.. 23

5.1 Overview of equipment for imposing aforcing functionon a cate- nary mooring system.. ... •. . .•.. ... .. . . .. .•• 33

5.2 Side view of the equipment•.. . •••••• ••••• ••• 34

5.3 Theequipment mountedin placebothlevelandtilted. .•. ... 36

5.4 Photograph of theequipment .. .... . •... 37

5.5 Closeup oftheforce tra nsducer. 38 5.6 The equipmentmountedin place ••.•.. ... . .. ... . 39

5.7 Informati ongathering system . .. • . .. . .. . .... .•..• 40

5.8 Thelocat ionorthe instru mentation•. •... .. .. . • . •. 40

5.9 Thetwo wayforcetransducer . . .. .•.. •... 42

5.10The circu it diagramror thestraingauges intheforce tra nsducer . 42 5.11 The equipmenttiltedat 90 .. ... .. .. .... •• .•..•.. 44

5.12 Thetest chain.. .•.. . . .•.. ...•.•.. ... .. 45 vii

5.13A samplerunwith andwithoutfillt-Ting . . . .. . . ...•. 49 5.14The fillers used for the dat a.•. . ....•...•...••••. 5(1 6.1 TheOFR valuesplcuedagainstfretlueney fat:iONpre-teuslou

..

52 6.2 The OrRvalues ploUedagainstfreque ncyfat:roN pre-tensi on

..

'3 6.3 TheOFRval uesplcuedagainstfrequency for 15Npre-te nsion.. '·1 6.4 Resultsfrom thespeeduptrials... .. .. . .. .... . . '8 6.' Non-dimensiona.lgraphsofthe resultsfat30N pre-tension. GO 6.6 Nen-dhne nsione lgra phsorthe resultsfor20Npre-tension. . . GI 6.7 Non-dimensionalgraphs oftheresultsfor15Npre-tension. 62

viii

List of Tables

S.I Parametersfor the accelerometer .. ..•. . . ... . .... 41 5.2 Parametersforthe st rain gages

5.3 Properties of thechain..

ix

41 45

List of Symbo ls

T" horizontal componen tofline tens ion Til vertical component of line tension H waterdepth

S scope ofthe mooring line w weight per unit length of line T/J tangent angle at thetop endof the line

e

angle of indination of the lineofact ion R the amplitude ofthe motion ofthe forcingfunction w the frequencyof the forcingfunction Jl water dynamicviscosity PI waterdensity me modified mass ofthe chain E effectiveYoung's modulus D parameterfor chain diameter 9 gravitationallLccclcrat ion T the total tensionin the line

water kinemati cviscosity V velocity amplitud eof the forcing{unction

Chapter 1 Introduction

In the ocean envircnmcnt there arc manyforcesthat ad on amooredobject. Be- cause it is difficult to design a system to maintain an objectcompletelystat ionary when facing theseforces,most mooting systems allowa degreeofmovement. The catenarymooring systemis the most common system ofthis type.

Thetechniques for the staticanalysisofa catenarymooring are wellesta.b- lished. Howeverthe dynamic analysis can be quite complicated because of the contributionsfrom prete nsion,hyd rodynamic drag(andrelatedfactorssuch as vortexshedding),iner tia lefi"edsand addedmass. Althoughsome analyticalmod- elsexist, there isashortage ofexperimentaldata.This thesiswillprovidesome data011the propertiesofthe dynamics ofmooring chains.

Someof theprevious studies indicate that there arcvery large dynamic loads generatedin mooringlines.Publis heddata indica te thattheseloads can rea ch a magnitude of tentimes that of the staticload. At thepresenttime, moorings arcdesignedbased011expectedstatictension and on previous experience.The accurate predict ionofdynamic loadswould improve the selectio nprocess.

This thesisdescribes experimentsthatwill examine the dynamics ofcatenar y

mooring systems. Experimentstook placeinthe MemorialUnivcrllity of New- foundland wave tank.Asupport structure011whicha motorwas mounted^WM placedjust abovethewater surface.The motor oscillated a block ina st raight line. Oneend of thechainwas a.ltached to this block througha. tension measuring device.

Themotor speedwas variedto provide specifiedexcitationfrequencies.The supportstructurecould be tiltedallowingthe trans lationaloscillationstooccur at different orient ationsin the vertical plane. The statictension could alsobe varied.

The work hw severalobjectiveswhichwere compose d oftwoprima.ryobjec- tive and severalsecondaryobjectives.These objectives arelisted belowwiththe prima ryobjectivesgiven first.Insumma ry, the objectivesoftheexperimentwere to determ ine:

1the magnitude of thedynamic forces present in an oscillating mooringchain (primaryobjective),

2whetherthe dynam icforces were more prominentinoneparticular orientation (primary objective),

3how the static tension affectsthedynamic tension (secondaryobject ive),

"if the four conditions that Suharaet ai,(1981)identifiedcould be roundillthe slackcondition(secondary object ive),

5whatthemodelscale effectsarc; iehowtheexper iment relates tothe ocean envi ronment(secondaryobjective).

Tointrodu cethetopic, a brief descriptionofthecatenary mooringsystem is givenillChapter 2.Paetwork in thisfield andrelate d fieldsis examinedin Chapter3, followedby the derivationof a dimensionless equation describingan oscillatingmooringchainin Chapter 4.Theapparatu sand procedurewill be examined illChapter5. Chapter6 will discuss theresults fromthis workand Chapter7willdraw conclusionsabouttheexperiments.Some recommendations regardingfut ureworkwill be made inChapter8.

Chapter 2

The Catenary Mooring System

The catenary mooring system takesits namefrom the shapeofthe cur ve for medhy themooring line(the word catena ry is from thelatin catenaorcha in). This curve is characteristic to anyline which is suspended from two poiutaand which hes a uniform weigh tperunitlength and negligiblebendingstilfnc8s.Thederivation ofthebeslc equationsforthe catenary line isgiven inAppendix A.The length oftheline from thetop towhereittouches tile bottomis called thescopeofthe line. Thescope andother ter minology rclalingtothe catenary mooring systemis shown in Figure2.1.

The catenarymooringsystem hasbeenlraditionallyused by shipsand other seagoing vesselsfor severalreasons. The systemcallbeeasilyandcompactly stowed when not inuse,itrequireslit tlemaintenance, ituses simpleeasilycb- tainablemateria ls,andin spite of its sim plicity,itprovides a. remarkaLly effective way topositionafloating object. If thetop endofthe chain ismoved,arestoring force is create d .Therestoring forceis smallatfirst but increasesvc~yrap idly. A descr iptive analysis of themooringsys te mwouldbeasfollows.

Ifthereare no displacingforces onthemoo red object , the mooringchainwill

Lineis JlOl t~"!1tUJ botton(tOllt{ondi tiMJ

MooredObject

---+

Line hJngsinthe --SMpeolOCQteruy

Mooring

"'---Line

Figure2.1:TheCatenaryMooringSystem

hang ina vertical[inc as shownin Figure2.2. As soon as the mooredobject is displaced,twoeventswilllake place. First,someof the cha inwillbepicked up fromthe bottom.Second,thechainwillstartto make an anglewiththe moored object awayfromthe vertical.Itis a combinatio n of these effedsthatcause the chainto applya restoringforceon the moored object.

As chainis picked upoff thebottommoreweight needsto be support edat the top.Thiswillshow up as anincreaseinthe tension.Also ,since theline rna'..es an angle with the verti cal,therewillnow beacompone ntof forcegenera ted in the horizonta ldirection.This horizontalforceis whatcounteractsthe displacing force,To boldthe chain againsthorizontalforcesan anchorisused,

The effectivenessof the mooring systemcan be seenin Figure 2.2.The diagram isdrawn to scale and itshows thatas the displacing forceincreasesfrom0.2 units

to20unit s, the .1c1u1I1 displncemcut Increasesbyatnctcr

o r

uboru.,I.TIll'fnr t'p un itin thiscaseisthe weig ht ofa ],'Ilgthor1Il0l)ri ll~lim-equa ltol1\1'cl"lll hof the wate r,

F is the disp loc ing force

Fig ure2,2:A Cat ena ry Moori ugSyste m IJwler [lorixoutnl I.oa ds

l1JClower end ofthechain is langellttt>;HId lOlldJing 1.11<'],,,UlIllJ.AtlIti.~[II/iut ther earconlyhorizontalforcesill thechainwhich must Ilt'n~isl.l,,1bytlwilnd rllr and the frictio noftheseabed.IIIthetalltC,L~C,tlu-chainis neverl:lrrW~lltto thebottomandstopsaltheanchorform ingan anglnwit l.tlu:houom.Inthis circumstance,theanchormust resistaverticalforceas wellIL~aIrorh'./lntalonc:.

Inthis exp e riment,thechainismnlntnincrlillthuslal:kI'(m.lil,iuJrItt alltim.!s.

isnoexplicitequation [orlc:rtsilllI!lIt,1sn anil.r:rilli v,~It:dllri'I'J<: Jltust bl'IIs,:II.

Theseeq ua ti ons so lveforTzand'[~(t ire horizcutnlilllliwrt icalwlJl jJl/JlI!nt sof

the tensioninterms of:wat er depth,H;scope, S; andwei ght perunit length,w.

The derivationfor these equationscan be found inAppend ixA.Otherequations using differentinput variables can be deriveddepe ndingon what valuesyou wish to useas input.

T.. Tsin!/J==wS (2.1)

II == TCiJs!/J cosh

(ainh-l ~)

^{_Tcos1/J (2.2)}

w Tcoa1/J w

Chapter 3

Review of Previous Work

The studyof the dynamics of mooringchainshasreceiveda great deAlofattention, at least intheoret ical developments.Parnelland Hicks (1976)slate thatover500 papers have been publishedthat dealwith cable dynamics,Howeverthey go on to say that experimental data arelimited .Patt ison (1974)agreeswiththis and statesthat although analyticalmodels exist, dataare scarce.

Thisreviewwillbedivided iota foursections.The firstwilldiseusethe simple static problem asitappliesin the marineenvironment.The n,experimental work willbereviewed,followed by an examinationorvarious numericaland computer techniques for finding dynamicforces in mooringlines. Finally,abrierreview of a relat ivelyDewtopic,statis t icaltreatmentofdyn amicforcesinmoor ing Iinca, will be undertaken.

3.1 The Static Problem

A good placeto startlookingatthe dyna micsof mooringlinesiswitb the static problem.The sta ticanalysisblUlbeen Cairlywell studiedandcan be {ound in severalbooks on differentia lcalculus(Fox (1950)and EdwardsandPenney(1982)

forexample.) The bailie approachisto solvethe geometri cut~."l.ryequat ions.

One of theearliestandbest analysisofcatena ryprob lemsasthey apply to theocean environment ill given by Collipp et al( 1962).Theyusethegeometr ic approachto derivethebasic catena ryequationswhich are thenusedto draw up tables thatgive propert iessuch astension, anglethe cablemakesin thevertical plane and potent ial energy in thecable. Howeverthepara metertheyuse for entry intothe tables may, in some cesee,beawkward.Their param eter requires knowingthe depth ofwa ter,the scopeofthe cable andthehorizontaldistan ce thlltthescopecovers.The equationsshown in Chapter2 givethe same result, but requirejust the scope anddept h of water.

Becauseof the non-linea rrelationbetweenthe displaceme ntand restor ing forcethe usc of tables and chartsfigureprominentlyin th eanalysi sofmooring systems. Adams(1969)andRadwanetal(1986) usc tablesISallaid tothe design ofmooring systems.Eventodaywhencomputerscan giveexact solu tions the speedandcase of useoftablesstill makethempopular.

Along with tab les,there wassome attempttolinearizethe equa t ions,Ogawa (1984) triedthis andused alinear ization coefficientmatrixto represent linetea- slcusasalinear function ofdisplacement.However,aswithall suchmethods, it isineccurnte outsidecerta in limits .

The use ofnumerical techniques tosolve the equat ionsbecamepopular with the wide spreaduse of computers.Nielsen (1976)useda Newton-Repheon root solving algorithmtomat chrestoringforcesto external loads. Theshapeof theforce versus displacementcurvelends itself well to solutionby theNewton- Raphsontechnique.

\0

Aprogramwhichsolves thestatic problemofa catenarysyste m hasbeen writtenand is includedin AppendixD.It uses the equationsderivedby Collipp eral(1962)along witha Newton-Raphsoualgorithm.Theres ults generatedby thisprogram areidentlcelto thosepublishedbyCollipp, Bergm an and Miller.

3.2 Experimental Work

Someof the earliestthoughts givento designingexperiments wer e direct ed towar ds scale model tests.Collier(1972) attemptedtoderive scalelawsfor amoored buoy in anocea n currentby usingthe governingequations.Const ants were introduced torelate the modelandthe prototypepropertiessuchasmas s,forceandlengt h.

Subst itut ionofthese constantsintothe governingequaticnaallowarelationship between the constantsto hedetermined. Dimensionlessnumbers were then drawn up. Collieruseda distortedscaleto achieve dynamicsimi larity in his model laws. While this does havethe advantageofavoidi ngproblems that are typicalin hydr oclasticmodels (ietheinabilityto scale alltherelevant paramet ers),itdocs havethe disadvantagethat the cableoscillatoryvelocitymust besma llcompared to thecurrent speedwhich inthe majorityofeasesis an un realisticassumption.

Unfortunately, Collierdid notdo testeusingthisideaand norecord of anytests witha distortedscalecan be found.

ParnellandHicks(1976)alsouse the governingcqueuone to derivetheirdi- mensionlessnumbers.Howevertheir&1lalysis differsfromCollier in thatthey do not useadistortedscale.They ignore thehydrodynamicfor cetangent to the cableby st at ing that itis small comparedto thenor malforce. Thenormalforce is accounted for by a 'hydrodyna mic force paramete r'which is the dragcoefficient

11

for a cylinder multipliedbythe ratioof cable lengt h todiameter.Pa rnelland Ilicks goon to statethat it isdifficultloscale all the relevantparame ters.Thei r solutionwas to dothe testsinafluidwithamuchhigher viscosity than water . Thissolut ionis wellknowntohydra ulicmod elers,but itappearsthatDo onehas usedthisidea to modelaca te nary.

Experimentalexaminatio nofthedynam ics of mooring cab lesseems tohave stnrtcd intile mid1970'swiththework of two projects.Pat t iso n(1974)seems to havebeen the first to publishexperimentalwork followedby vanSluijisand Blok (1977)severalyearslater.

Pattison'spurposewas to perfo rmexpe rimentsonsevera l differentkinds of mooringlinemater ialand develop theseinto adat abase for analyti cal model validation.The method wastooscilla te aslidersinuso idallyinthe ver t ical plane by means of a drive cableauached toa rotating cra nk arm. Themooringline wasjoined totheslidervia a twoway forcedynamometer.Fivedifferentmooring line materials weretestedwith scopetodepthratiOli between 1.1 and 1.6 anda frequencyofmotion between 0.1and 2.2 Hz.Dyna mictension s up to 1.2times thatof theslalic tension wererecorded. Higherloa dswererecorded butwere ignoredi1Sbeingunrealisti c ofcondit.ionsat sea.

Unfort u nately Pattison hadsome problemswithhisexperi ment.Hereported problemswith the two wayforce tr ans ducer thathe usedin hisexperiments.It app earsthat thetransducerswerecoupledand hespent quite a bit ofhisanalysis trying to decoupletheforces.Also,although be report edsinu soidal cecil letic ne, fromthe way his equipment was setuphewouldactuallypro duce motionwith twosinusoidalcomponents. U would helmpceeibletosaywhat theeffectoCthe

12

seco ndcom p cneut wouldbe without knowingmore about his equipment, vallSluljs and Blok(1!l7i)also prefor med similar experiments except ther tbey oscilla tedthecable in a horizontal plane. TheyreportedlUIincrease ofdynamic tension with an increase of frequencyupto a maximum. Afterthismaximum.

increasingthefrequencycaused a decrease inthe tension.They also reported that there was no significantscaleerror in thedynamiceffect.(Morewillbesaid about this inChapter 6).

In themid 1980's there werealso two published reports of experimental work.

Thes e weredonebyvan denBoom (1985)and Suhafaet al (1981).

van den Boomconductedexperimentsinvolvingthe harmonic oscillation of mooring lines. Hedocs notsayiftheoscilleucns were horizontal or vertical but theimpressionistbatthey werehorizontal. Noscaling parameters for the expe riment weregivenbut it is statedthat thesealingWa.!lcarriedout according to Froude'slaworsim ilitude.

Itishardtosee how Froude' slaw would be applicable inIt.CMelikethisand ether analysis donot have the Froude number as an important consideration.Thus it would seem that van den Boom'sattempts at modellingwere in error. However , the experimentalres ultsdo show theexpectedrise and drop off in dynamicteueion with increasing frequencyof the rortingfunction.

Suharaet aldid theirtests with bothhorizontalandvcr~icaJoscillationsand usedscopetodept h ratiosbetween 2.8end 8.3.They also used a widerangeof frequenciesfor the forcing oscilla ticnaand several different emplitudee.

One importantoutcome from their workwas theclassificationof thereeulteby the meansof severaldimensionlessparameters. Thestatic conditionis dete rmined

13

! l1flsi-S t .lJtic

~"..e",="..e"",::"..e",-,,-,:n~.~\I v.'l

"

:.-

...

iir.r

lin e

free foil

line

Figure3.1:Force-time traces for thefour conditions

bytheratiowXolTllowherew is the weight per unit length,XoiJtheamplit ude of motion of the top part of thechain, andTno istheItatichorizont al tension.

The dyn a mic characterietlcaare determinedby two ratioa: 1)Z...ID~whereZ...i.

theverticaldispla cement

or

thecenter of gravity cl tbecetenerypart andD~illthe diameter ofa.cylinderwit hthelamevolume as thechain,and 2)the ratioZ",w¹

/g

whereINisthe frequency

o r

the oscilla tionand 9 isgravitationalaccelera tion.

UsingtheseplLlameter.they iden tified four condition•.Thefirst is&quae..

14 stat iccond ition wherethe dynami ctensions can beca lculatedbythepositionof the topend and the static formula.Second is&harmonicoscillationcondition.

Thethirdis calledasnap cond itionandshowswhathappenswhen the ca.ble goes ta u t . Th e last is a free-fallconditionandillitpartsofthe chain actuallyfall untilbroughtupbythe restofthech ain generatingimpll.dforces. Thefinttwo conditionsseemto bewhatPatti son(1974)recorded.

Sincepart of this work wastoidentifythesefoursta ges,some timeshould hetaken hereto examine what happens in the fourstages.Thefiflltst age was des cribedaboveandnothing moreneedstobe said exceptthat the primarycontri- butionto the line tensionis fromthestaticpre- tension.In the harmonicoscillation region, th eforcetrace inthe time domainappearsdose to einuecidal.Inthis area theinert ialforcesofthe chainandth efluidpropert ies or added mass anddrag are theprimarycontribu tor stothe line tens ion.

Inthesnap region , a.new mechani s m com einlo play.The minimumtension in the chain, whichuntilthistimehas been grea terthan zero,becomeszero(since a chain cannotsupp ortcompressive leads,the tension can neverbe lessthan zero ). This meansthatfor a brieramountoftime, the links ofthe chainare not su pp ort ingany load. This takell placewhilethetop of thechain is moving in a directlontowards the anchor(in a direction that slackensthe chain). When the directionreverses(sothatthe top part isnow movingin adirectionthat tncreesce thetensionof the chain) , thesleekenin gsud denly stops. Thissuddenstopping ofthelin ks astheyfall generatesimpactload ing. The magnitudeof the impact loadsare dependantupon thetimethatthelinkshad notensionODthem.It is th ese impactloads thatcausethe ext remelyhigb tensionsthat are associated with

15

dynamic forces. Because oftheshape ofthe catenary and thelength of theline, notall the Jinks take part inthispheno mena.However, asthe speedof the top Pdf tincreases, moreand morclinkscomeintoplayand this cau sestheincreasing impa ct forces.

Eventually a time comes where thetop is moving50fastthat itis able toreach the endofitsrunand startback beforethe remainder of the chaincan respond.

Now,fewerand fewerlinks are involvedandthetimethe links areallowedtofall, Ircc offorcesin the chain,becomessmallersothedynamicforces starttodecrease.

This is the Free-fullregion characterisedby thedecreaseinthe dynamicforces.

Suhara ct al(1981)also comparedtheir resultstoa theoret ical predict ion methodcalled the lumpedmass metho d.Agreementwas goodin conditions 1 and2butWiUIelgniflcantlyoffforcondi tions3and4.

Anot herpaperdiscussing thederiva tionof thedimensionlessparametersrel- evanttooscillatingmooring lines isbyPapazoglouet a1(1990).Theyusedthe methodof governingequat ionstoderivetheimpor ta ntparameters. They also rec- ognized thatfor deep water moorings the elast icityoC thecable is animp ortant consideratio nin the calculatio nofdynamicforces.

Theyhave proposeda modeling scheme whereall parametersexcept for the elast icity paramet er Aremetas wellaspossible and thentbe elast icityismodelled withsprings.

The ideaofusingspringstocompe nsat e for the inabilityto correctlymodel the scaling parametersisalso presented inareportby Faure (1989).Faureused a set of springs that bestsimulatedthecate naryeffect and the elasticit y of the missingsection aswell as compensated for theelasticity of the rema.iningmodel.

16

Unfortu natelyno mention of the modellingpara metersis madeexcept toSAythat line lengt hs andweightswere scaledcorrectly.

3.3 N umerical Calculations of M oo ring Li n e Dy- namics

Numerica lsimulationsofthe dynam icsof mooring lineshave beenebundantill the past. Due to non-linear materials,non-lineardrag coefficients,non-linear forcingfunctions andnon-linearities due tothe nature ofthe catenary,apurely analytica lapproachwas impossible. Therewas somethought givento linearizing the equations of motionandusingothertypes of analysissuchIIJIperturbation expansions, howeverthese method s were inaccura te outs idecertainlimits . Com- pute rrun numericalsimulatio nsseemed to providethebest tool loranalyzingthe problem.

Compu terprogramsto solvethe quasl-stetieproblem are relati velyeasy to createand veryinexpensiveinCP U time to run and thesewere the fiut attempt to solvethe problem.However,usingthequasi-staticapproachmeant thatthe mostinteresting phenomena were ignored. Sometimesthequasi-staticmethodis still usedbut this is usually fora special application (Nakajima (1986» and,even so, it canbe mislead ing.

To handlethenon-llnearitiesofthesystem several numerical techniques were derived.Themost powerfu l of theseprovedtobeBOrnesortof discreteelement technique. Therearetwomaintypal of discrete element algorithmaueed La an- alyze the motionsof mooring lines. They arethe knownas the finiteelement method and thelumped massmethod and a good reviewof the two methods is

17

given by van denBoom(1985). There have beenalargenumberof schemes for either methodproposed.

In the finite elementmet hodthe cable is broken into a number ofsegments witha set ofassumedbeha viourfuncti ons. These equati on sateappliedto the kinematic anddynamicequ ilibriumequatio ns as wellas the equati ons descri bing themater ialproper ties.

Johansson (1978) proposedafiniteelement modelwithlinearlyelasticmaterial proper ties and aconstant dragcoefficient. Webster(19Bl )developeda program calledSEADYNbasedonthefiniteelement method.SEADYN brokethe problem into twopath, one dealingwiththequasi-s tatic solut ion,the other handlingthe dynamicaspects.Theprogram could handle non-linear materials anditattempted to modelfhridIorceeas well as strununi ngphenomena. One or thebeetfinite elementmodelsisthat proposed byHwang(1986). Aswell aa an improvedmethod ofaccountingror drag, itwaaabletohan dle comp ositelines (lines wheretwo or moremateriels arejoinedtogether) andclumpedweights. Italsouseda 3- dimensio na lanalys isor the line.

Theother mainmeth od used was the lumped mass method .Thelum ped massmeth odwas fin t proposed by Walto n and Polacheck(&3reported in van den Boom(1985) ).Itinvolvesthe lumping ofthe mass andtheexternal and internal forces ofthe line at a number ofdiscretepoints.Equatio nsof equilibrium and continuity at thesepoints ornodescan be derivedand then solved usingnumerical methods.Severalauthorsnotethat thelumpedmass methodseemstobe more computationallyefficientthanthe finiteelementmeth od(van den Boom (1985), Faure(1989)).

18

Thesimplestimplernt'nl&lionof thelumpedm&Qmethodwouldbe to work wit bjustone node.Suharaet&1(1983.19S7) didthis andcbtel ned feiUODioble resultsfotthe smaller frequencies

tie

where the dynamicforceswererelatively ,mall).Liu (1973)hadmoreelementsinhismodelbutthenumericaJ modelling of the 8uid dragandaddedmesswas poorand theI1lAteriaiVllUIassumedto follow Hooke',Ia.w (exceptwhen the chain wentinto compression).

vanden Boom(1985) proposedalumpedmMe modelwhich beincorporated into a progr amcalled DYNLI N.Heproposed~hattheftuid loadingof a lineill dueto the orbitalmoti on ofthewaves,the currentand themotion,of theline(it sho uld benot edtha this modelneglectedthewa ve motions).Hi.2·d imensiona.1 modelalso includedseafloorreactionforces .Hereported good agreement with experimentaldata.

Recently,Faureproposed amodelbasedon vallden Boom',model(Faure (1989».Thismodelhad a betterrepresentat ionoCbottom reactionforGei and bottomfrictio n.It ab oused Morisoo 'sequationto calculatethe fluidloadin& on thelines.

There havealsobeen seve ral attemptstode velop com puterpackagetthat"&lid inthe design ofwholemoori ngsystems,eg Owen and Linfoot(1976).

All the numericalmeth odsmentioned above needed some values for thevar- ious hydrodynamiccoefficients. Someexperimenten usedpre vjoualypublished da.tawhereethers suchallFaur e (1989)cameup withtheirown.Itshouldbe mentionedherethat thecoefficientsneeded arethesetha toccurin 08ciUatin ll:flu- ids. While these values are neededfor the harmonicOlIdllati ngconditio n,thecue isdifferentforthe soa p andfree-Call cendltlona,Inthese caaeathechain motion is

19

not ha.rmonie110coefficient. measured£rom random osciUationlest.IUch".tba.e reportedbyLongor ia d .l (1991)shouldbeused.

3.4 St atistical Analysi s Tech niques

wahtherealizationof theexistenceandmagnitudeofoscillato ryIorcee and the increased knowledgeofmetalwear andfat igue, att entionwasnat urally turned totheseeffectsin mooringlines.Shaw(1989) analyzedthe problemusingwave stat istics and a mooringline tensiontransferfunct ion whichbe obtainedfrom matching model testa to anumericalsimulation.Howeverbewas interestedin obtaining amethodofcalculati ngthecyclicloadingthat a mooringline would experienceandnet indeter mining themooring line',maximumload,

SincockandLalani(1990)wereinterested indevelopingguiddinesforfaUgue analysisof anchorchains,Theystatethatthere are twoway.toc&Icu1~eload.

inthe mooringline:thequasi-static approachand&dynamic procedure.They ahago ontotalk aboutlOmeolthe designcodes in effectfor semi-submersibles and float ingprod uct ionunill .

TheAmericanBureauof ShippingandLloyd'smake nomentionof guide- linesformooringsystemandleavethespecificationupt.otheowner. TheUK's guide onoffshoreinstallat ionsmakes no mentionolany specificdesign method- ology bywhichtocalcula teloads. The Norwegiansalloweither aquasi-sta tic or dynamic analysis.ThenewlydraftedAmericanPetroleum Instit ute's"Rec- ommended Prac ticesforDesign,AnalysisandMaintenance or Catenary Mooring lor Floa.tingProd uctionSystems~does acknowledgethe significance ordynamic Io..'\dingandrecommendsa time domaindynamic mooring load anal}'llie.

20 A verygood review of general techniques fortensionrespon se statis t icscanbe foundin Lua(1990) .Luoshows that a Gaussiandistr ibutioncannotaccount for the nonlinearities in the system.He goes onto stalethat significantimprovement in thetension ranges can be obtainedby using the Weibulldistrib ution.

Computerapplicationsof~hismethod are usually writtentosolve thisspeci fic problemand thusallow more versatility on thepa.rt of the person derivingthe equations.

Chapter 4

Dimensionless Analysis of a Catenary

Thefirst stepin developing anexperimentisto determ ine whatis to be measured andtheparamete rsthat affect the experiment. Dimensionalanalysis can show ways inwhichthepar ameters can begroupedtogether .Thedimensionlessequa- tion is derivedhereforan oscillatingcatenary system andthen thegrou psinit arccompared to thedimensionlessgrollpSfoundbyotherexperimenters.

4.1 Definition of Physical Paramet ers

The parametersnecessary todescribe the systemcan be divided intothree groups.

The first groupconsistsofthe quantitytobemeasuredintheexperiment.The second group contains the input parametersfor the~xperiment.Thelast group holds the parametersdescribingprop ertiesofthesyste m.

Inthis experimentthe quantitythat is to be measuredistheten sion,T,in thecable.Sincethetension is chan ging alongthelengthor the chain,aposition for thetension measureme ntmustbechosen. The twologicalplacestomeasure tensionare ateither endortl-echai n. Thetensionsatthebottom areor interest

21

22

tothosedesigning anchorsystems. Inthisexperiment theforces that the chain would exertonanobjectarcimportant so tensionis measuredat thetop.

Theinput parameter sdescribethe initial condit ionsofthe systemand the forcing function thatwill move the topend of the chain.The initial conditions consistsofthescope of thechain S,the depthof water/Iandthe angle of inclination thelineof actionmakeswith the wat er,

e.

Although the stalic tension orpre-tensionis animportant considera tionit is notincluded here becauseitis setoncethe scopeanddepth of water are set.

To describethe forcingfunction that actson thecableend, two parameters are needed. First ,thefrequencyatwhich theoscillationoccurs,w,isimportant.

Secondthe amplitudeofthe oscillation,fl,is important.Sincethe o9cHlation is assumed to beapure sinusoid,the amplitudecanbeeithera velocity amplitude ora displacement amplitude. Once oneisfixed, theother is set. In thiscase displacementis chosen because itgives a betterphysicalunderstandingofthe model.

Thesystemparam eters describevarious propertiesofthe physical set up. For the fluid,the importantproperti esarethe viscosity,p,andthe density,P" For the cablethe descriptiveparameters are a measure of the mass of thechain,m~, andthe effective Young'smodulus,E.Themeasure of themass ofthe chainis defined113thedensity of the chain materialminus thefluid density.Thusm,is seento have thesameunitsas densitybut ismodified to accountfor buoyancy.

Eis called theeffecti ve Young' s modulusbecauseit is not the Young'smodulus ofthe material but the slope ofthe stressstraincurveforthewholechain.

Afinal proper tyof thechain that must bedescribed isits drag, Subara.et al

forci ng Fundion:

fI Yi~i~~~::~~: ~

Cabl eDensity:I:

Diameler:J

Frequency:__

Amplilude: : ...-(.

-.-,1-1.~

, j /

23

Figure ·1.1:Parametersforthe catenarymooring system

(HlSl)IIsl~dfortheir pnrnmeter the diameterof acylinderD,havingthe sa me volume asthechniupnrnmcrer .Thisquantityls just tot,ivea characteristic length to thechnin.A lastphr~k.,lparameter that mustbe included isthe gravitational acceleration.9.Figurl'·1.1showsthe~(:variables.

4.2 Derivation of the Dimension less Equation

Note thatunder till'pl'Opl'l"til"~oftheca ble , 110 accountwasmadeof thebending sti lrl1l~~s. Thisisfon~islt·lll.witIItlu-catenary analysiswhichassumestllll.tthe IWlIllingswrn('Ssoftill'cableis7.1'1"0.

TIll'SCpnrnructcrsformthe functionul relattonfO Tthe system. To startthe derivation of thefunct ion al relation. the tensionis assumedto be a function of thl'other properties:

T

=

¢J{P/ll-l,

m.,

E, D,S ,ll,9,R,w.O}.

2.

(U) Thequantity0isalready dimensionless and thusmay beleft outofthe Malysis fornow.IncludingTin the expression on the right handside ofEquation4.1and reordering this expressionthefollowing is obtained.

4J{Pf .m.,/l,E,T,9,w,

n,

D,II, S}

=

0 (4.2)

To allow the greatest amount of control in derivingthe fun ct ional expression, the method of synthesiswit h linear proporlionaliticsisused by Sharp(1981).

Eitherthe fluid density or the modifiedcabledensity is usedto cancel the mMs dimensions wherethey appear.Theparameters that have units ofmassin them are P,EandT./Iis dividedbyPIto get/IandEandTarc dividedbym., Also, m. is dividedbypt-

'{II, !l, !...,g'r..J,S,D. ll. R,~}

_{me m. PI} ⁼

^o.

^(4.3)

Linear proportionsare createdfromthefirst five parameters/I,E/m~,

T/m.,l.J

and g,the rest already havingdimensions of length except for theterm m./Plwhich is dimensionless and,like0, willbeleft outFrom the analysisIornow.Iti.noted thatEhas the same unitsas pressureandTis a force. Allthe possible linear proportions are listed below.

{. 'I' ,.!..., (L)'I' ,.!L, (".)'n, (!.)'I',

gl/3m.g m.g w2 w E

2S

The fiveparameters should give10 linear proport ions(4

+

3

+

²

+

1)butwe onlyhave9 because linearpmport ions can notheformedfromthe parametersI'

andT, Fromthe 9 group' only4areneeded,howevereachofthe variablesmust bechosen atleastonce and theymust allbe interrelated.

Alittleart andinstinctisneeded here to select appropriat egroups although thegroupsselectedcan be compoundedto form any of the non-selectedgroups.

However a goodplacetostartistoselectthe term withTinit.SinceTis the quantity wearcinterestedin,only oneter mwithit shouldbeselected.We have a choice of threebut the term withTas a ratio of thetensionto the weightof thecablcfigures promine ntly inthe stati c analysis. Thus the group{Tl mcg)t/3 is chosen.

Also,itwouldbe mostusefulto haveEin termsof propertiesofthe cableso Ejmc9ischosen.

The forcingfrequency is an impcTtantconsiderationso it should appear in only one factor.Thisgivesa choice of selecting oneof thefollowing:9/1.lJ',(E/m.w)I/2, (vjl.lJ)l/2or(TjmcllJ,)I/~.ThetermswithvandEare eliminated because they arcnot as relevanttothe experiment as the othertwoterms.

It ispossible thatthetermTjm.,4JJlcould prove tobe a veryusefulparameter, howeverthe other termis selected as9/1.lJ2becauseit is desirabletohaveTin onlyoneterm.

The la.stter m selectedis v2/"Jj9^1/3and itis chosen to providethe missing parameter,v.Thusthe listof linearproportiona Utiesbecomes:

26

jv'" (T )' " ( E)'" , }

19 1/ 3' ;f~

•

meg '-;;;i.Il,S,JI,D . Now each termis divided by one of thefour lengths to make the expression dimensionless. Again,suitable selectionsmustbemadethatwillallowuseful parameters to be formed.Thetermv2/3/g1/3is the only term that hasboth the fluidviscosity anddensity,90itwou ld make sensethatitshouldbedividedby the drag paramete r,D. The term (T/mcg)ncodatobedividedbythree lengths b makeitdimensionless. Choosing the termsD²11willgive the ratio of tension tothe massofalengthofchainequalto the depthofthe wat er . (Act ually, the term ditTersCrom this value by a factor of1I'/4,however constants ,orlack of tbem do not harm the validity of a dimensional analysis.)The termElme9needs to be dividedbyonly one lengthto ma keit dimensionless andSis chosen.Itwould seem to make sense to haveaparameterthatcontainsboth theamplitudeof motion and thefrequency, soglw~is dividedbyR.Finall y, the lengt htermsare made dimensionless to giveRIH ,SIJIandDIS.Also, the 8term endthe m./PI arereintroduced into theequationtogive:

T {V"' (E),RSD m, }

m.gD2J1

=

¢Jgl/3D' m.gS 'w2R ' D 'H' S' p;,9 . (U ) Several of the parametersin this equationarefamili ar.The teaelcn term and the ratio51Hare used in the static analysis.Thetermv2/3/g1/3Dis afor m of the Fronde-Reynoldsnumber.AlsothetermRIDillsimilar to the Keulegan- Carpenternumber.

Parnell andHicks (1976)gave as their dimensionlesspar ame ten:PI IP. ,TlmgL , CDLID, Elp.gLandU:lgL.All the termsexceptforCDLI DandU:/ gLere

21

foundillequatlcn404. CoLI Disnottherebecause of thewaythatthedraa i~defined(Cuis.ameasure ofthedrag)andU:/gLisDOt.there becausethe experimentisbasedonthedilpluementamplitude not the velocityamplitude,

SuharactaJ (1981) haveMtheir par&mclersUlX.ITHO.Z..ID~andZ...w²

/g.

The firstparameterlistedisjusta measureoftheItal ictension.ThetermsZ...

!

Dc am!Z...wllgcanbefoundin equation4.4except. tha t the amplit udeofthe forcing Iuuctic nis usedinsteadofZ...which isthemotionof the center ofgravity of the chain.glw²Risverysimilartothe a term usedby Suhafa.

4.2.1 Dimension al Analysis Using a Velocity Amplitude

AsWa.:Imentioned in thepre vioussectiontheforcingfunct ion canbegivenin terms of avelocityampliLudeu wellasoildisplacementamplitu de.Itis interestingto repeatthe dimensionalanalysisofu~aryusingthis.Theequation forteDlioo is now asshown below.

(U) TIn:initialpa rtoftheanalysisiscarriedoutexactlyas before.ThisgiVe:! the equat ionas sllownbelow.

• {., .§.., I..,g. ". V.S. D, H,~} =

O.

m~m, PI (4.6)

There arcnowsixvariablesfromwhich linearproportions can beder ived.The six variab lesshouldgive15groupsof linearproportio ns,however nogroupscan bemade fromthecombinationsIIandTlindE andV&0thereereactuallyonly 13groups.Thesegroupsarelistedbelow.

28

{v,

_gJ/3'm.g'

" .s. (L)'

_meg

"

_'

J... ("-)'"

_{w" w '}

(!.)'" "-

_{E 'V'}

(...I-)'

_m.V1

!2,!::,

_{9 w}

(...£)"

_meW'

',(...I-)"',

_mew'

(v'm,)"'}.

_E

Thesamerulesare used to selectthe paramet ersMbefore except this time five grou ps areneeded.Some of the groups are the same as in the previous methodAnd arestill significant paramete rs sothey areselectedagain.These are the groups (Tlm.g) ln, (Elmcg) andg/",P. There arestill two more groups to select, at least oneof whichmustcontain1/and oneV.There arctwo groupsthatfulfill this conditionandarefamiliar.TheseI\.I'Cthe groupII/V(which can be developed into a Reynoldsnumber)and the groupV'j9(whichcanbe developedinto a Froude number).

Tocomplete the analysisall lerms arc dividedbrone of thelengthsin the syste mtoobtain the dimensionlessequationbelow.

T {VD ( E) 9 V'

S

D

m, }

m.gD2H

= ,p 7 '

m.gS

'w2R'DU'H,s,;?D .

(4.7)

There arcot her groupsthatcan be found here thatare familiar. Compounding thegroup

g lw

^{2wit h}V²

/g

and taking thesquarerootwillgive the Keulegen-Car- pente r numb er.Combiningthe Reynolds numberwiththe Keulegen-Carpent er numberwill produce thefrequency parameter,1IIw~.

Theanalysisbasedon the velocity amplitudegivesmore of the numbers Iami- larto hydrodyna mics.Howeverthis is just becauseof theway that the forcing Cunction is defined. It would beequallypossible andvalid to repeat the analy.

sisusing an acceleration amplitude instead of the velocity amplitude but no new

29

iwi,!;huwould be gainedfromthis exerci5e.

4.3 Validation of the M o del U sing th e Stat ic C ase

Aswith many problemainhydrodynamics,itis difficul t toprovethat the dimen- sicnlcssequationderiveditindeed valid.However,a.nexpression for thedynamics DC a catenarymooring system muslincludethe parametersforthe slaticsyste m.

Thus,ifallthetcr methat containinformationaboutthedyna micsofthe system arcset cqucltczero, theremaining terms must describethesta ticsituation.

To start,examine equation4.4.Themostobvious termsthat containdynamic informationarethe termsinvolvingfrequen cy ofthe Cordng funct ion and the amplitudeDCthe mot ions.Thus theterm 9/W1Risdroppedfrom the equation.

Also since thereisno motion , theviscosity ofthewateriscot relevant.Theangle 0,the anglethaLthe Corcingfunc~ionactsa~is&Isoirrelevan t new.The diameter orthecablewas imporLa.nt .u&dra& parameteraliiwastheratiom,:!PIso they can beremoved.The removalortheseexpression,leavestherollowingequMion tha~descelbcs thetensio nina.,~&l.iccatenary,

('.8)

Usually the catenaryie assumedtohave infiniteaxia.l,tiffned so theterm Elm,gSisignoredbutinsomecase! (as inCollippetal(1962))it is included. Thus thest atic tensionisdependentuponthe weigh tperunitlengthofcheiu and therati oSIll.

Lookingatthec.1tena ryequat ionas shown inChapter2.wefind thatthisi.

30 the case.Thisisnotsurprising,since west arted with this equation,howeverif this had notworkedit would have pointed alitthat there wassomething wrong withthetheory.

4.4 Sel ec t ion of the Experimental Parameters

Fromthefu nctional equation (equati on4.4) three parameterswerechosen to be the variables inthe experiment. These parametersarc:T/m c9D'H,9 /W'R,and

o.

Severalnotesshould be made atthistime explaining whytheseparameters werechosenandwhy otherswerenot.

Itmightseeminappropriate to includethetension termillthis listsince that is whatisto bemeasured,howeverthis isduetothefactthat the forcethat we actuallymeasure isa combina tionofthe staticanddynamic force andone ofthe parameters that isvaried is the stat ic tension.ThesetwoIorceecould be separ ate d,howeverthisservesno usefulpurpose sincethedynamic tension that we are int ereste d in isa combinationof both.

Sincethe systemwillbe operat ing in wate randsincethe constant9will not change significantly,there isno need tocha nge the fluidpropertyII. Also,the materialusedtomakethe model chain issteelwhichis alsothe most common material forcommercialchains, thus thematerial properlym~does not needto be changed .Thisisalsotrueto a certainexten tfor the Young'smodulusof the chainE.However ,asEis definedas the Young'smodulusforthewhole chain, the geometry of the chainbecomesimportant . Nonetheless, thisW&8consideredII.

smalleffect in shallow water mooring and thus noat te mptwas made to modelit.

This left the followingparametersthatwereim port ant to the experiment:

31

1'/rnt:9D' 1l

,g/w'

ft,and0.Todoa completeanalysisof the system, these pa- rameters wouldneedtobe changed, oneat a time ,whilethe others remained constan t.However,this soon means that totalnumber of tests needed is too large tobeaccommodated.Thussome limits hadto beimposed.

Frompreviousworkitwasfound that frequencyisthe singlemost important paramet er,thusitmustbeva ried throughoutthetests. Also,theexperiment wasilliliallydesigned toexamine responseas the forcingfunction was applied at variousangles. Thus the anglethat tbe linc of action makeswit hthe water is alsoimportan t. Finally,Il8ment ionedbefore,thepre-tensionin the chain was changed.

Withtheexperimental parameters selected,theexperiment and the necessary equip mentcould be designed.

Chapter 5

Experimental Work

The experimentalport ionofthe work had118its goal themeasurement of tension at the topend ofthe chain, whilethe cha inunderwent different for cingfunct ions withdifferentpre-t ensions.Experiment slookplacebetween Sept ember26and October31990,in thewave/towingtank at Memorial Universityof Newfoundland.

Thewave/ towingtan kwas selected becauseitprovidedthelarges tdepth of water whilehaving sufficient lengthfor thechain tobe st retc hedcut .

5.1 E q u ip ment

The main itemof equipment needed for thetests was ade vicethat couldapplya forcing functionto the top end of the chainatdifferentIrcq ucncice and atdifferent orienta tionsin the verticalplane. The forcingfunction hadto besinusoida l and thedevice also had to becapable of supportingthechain ncar thewetc r and attach ingtothe catwalk.Aschematic oftheequipmcntis shown in Figure5.1.

Since this was a very specialized itemofequip ment,thereWallnothing suitable readily available.Thustheapparatushad to bedesignedand built.Thedevice designed was a variation of a scotch yoke. A variable speed electric molorrotalcd

32

support

yok e _ J

n oto r

'--- pin

33

Fi/tllfC5.1:OvervieworequipmentforimposingII.forcingIuuctlonona catena ry tuooringsystem

:1-1

Rotating P in Y oke Meehan i sn

W hee ls

R aiI s

The notor and th e g ears h ave b een anitted in this viev.

Figureij.2:Side\'icworUtel.'l:llIipUlI!lIt

adiskonwhichwasmountedalleccentricpin.TheyukllIl1ld11VI:l'tim.l~llItill whichthe pinwouldlit.Thl!pinWAS rll'l'toIIlIlVll illtIll'vl·rtil·al"I..tI.lItIVI'lll,1 forcethe yoketofollowib horizontalmotion s(SI'CFil;lln!;;.1).!Iuri7.11lltiJ;111.1

verticalinthis descriptionmeanraralldand perpcndiculurtutIll!I.mw plat", Themotor sekxtedIc r1I1(~ I!l(PI~rill\{:l1twas1II

n

lipvMiah!l!SlO':!"!"l'!drk

orthegearreducerWilS10cII.1l.,,1tbllroliltillgdisk.TIll!111I1J1111shaflW;L"sllJ>J", rL.:<1 by bearingsat each end.ThediskWitSmadoorsteel whiletbeposl(,flIUl1L'I.'it'IIII.ly

35

Wallmadeofaluminum. The reasonforusingsteel wasto add inertia.tothe system.

Since thepretensioninthecha.in would add energytothesystem over halt the strokeand subt rac t energy forthe other half, theinert iawasneededto smooththe motionout. The calculation for theforcerequiredtomovethe chainwasbased 011theclassic mechanica ldesignproblemof a continuouslyoperatingpunch.

The blockwasatt achedto thefront ofthe yokeand ranon rails.Itprovided a.suppo rt for the twowayforce transd ucerand, throughit,the chain. Under the block, aslotwas cut in thesupportplate through whichthechain hung.

Thesuppor tplat ewas mounted underthe catwalkandwasfree to tilt relative tothewa ter as showninFigure 5.3.Once at thedesiredangle itwasheld in place . bybolts fromthe sidefram e to themou nti ngframe.Itcouldbeplacedin one of six angles: 0, 15, 30, 45, 60,75, and90°.

The pivotpointofthe base platewas locatedat thecenterof theline ofmotion experiencedbythe toplink of the chain .Thereasonforthis was to make sure that the motionofthechain for all teststookplace abouta common cent er.

Most of the dev icewasmadefrom3{8 inch aluminu m plate althoughsteelwas usedinsomeplacesliketheshaltinthegearredu cer and the diskasment ioned above.

Frequencywaschanged throughthe contro ller onthevariablespeedmotor.

The speedcont rol for thevariablespeedmotor wasasimpledialwith 10grad- uationsonit.After the runs, theact ualspeed wascalculatedby examiningthe accelerometertrace,however duringtheteststhespeedwassettoapproximat ely the desiredvaluethroughthe dialwhichwascalibra tedwith a strobelight.

Since thedesign of most of the devicewasunique, completetechnicaldraw ings

:\li

3;

Figure 5.4:Photograph of the equipment

andassembly drawingsarc given in Appendix C. Overallthe performance of the equipment was very good .Wear was noticed on the pin/yokeinterface,however that was to be expected and the wear during the testswas negligible.The only

improvement that could be madewould be to allow eitherthe toprail orthe bottom rail to move and be set in place after the wheels were mounted. This wouldallow a betterfitin between the rails for the wheels on theyokeandthe block.

Figure 5.4 shows a photograph of the side view ofthedevice.A close upview of the two way forcetransducer (described in the nextsect ion) is shownin Figure 5.5.The equipment mountedin place is shown in Figure 5.6.

Figure ."i..S: Closeup of the forceIrnnsducer

5.2 Instrument a tion

Figure 5.7 shows theinstrument ation set up and theflowof information. ,.\t the startis the physical apparatus(ie the accele rom eterand two way force trans- ducer). at the end is a \'.\X~."i;JO.and in between were amplifiers. filters. a Keith- ley data acquisition system and a:!S6PC computer. Figure.j.r-shows the actual loca t ion of theinst ru ment at ion .

The data gatheringappa rat usfor the experiment consisted of an accelerometer anda twoway force rrausduccr. The accelerometer was aBrulcr andKoch l:l7S accelerometerwith the characteristicsshown in Table ,j.1.

The two way force transducer was designed and huilt atXle mo rie l University.

What was needed was au instrument that could rm-asure the force thatthechain

Figure.').6: The equipment mounted in place

.ru

I

^{AID ,.'}

r-=-I ,

^{Converte r}

---1-_\

Filter

r -C _ _ .

Lowpass Filterand Moving Average

End o f Yoke

Force T ronsducer

Figure,). 7:lufonnation~nlherill,tt SYSll~1Il

Figure .':i.8:Theloca t io n oftheinstrumeutntion

41

Serialnumber 1426469 Chargesens itivity 315Pc/g VoltngeSensitivity 258 mV/g NaturalFrequency 13 kHz Table5.1:Parametersforthe accelerometer

Configuration WheatstoneBridge Type StudentEA-06·120LZ.120 Resistance 120 n±.03%

Gagefactor 2.05±0.5%

Strainlimits 5%

Table 5.2:Param ete rsfo r the strain gages

exerted onthe drivingmechanism(theblock)whileallowing thetop link of the ch ainto pivotfreely. Since the angleof thetop link wasnot knownan d always chang ed.force had tobe measured in twodirections which couldthenbeadded vcctorinllyto get thetotal force.

The twowayforce tra nsducermeasuredforcebot h paralleland perpendicular to the plate.Itwas madeoftwo sets of thin platesinstrumentedwithst rain gauges.Oue setwas parallel to the supportplateandthe other was perpend icular to it(see Figure5.9).Each setwasmade oftwoidenticalplates witheachplate huvlng two straingaugesfixed in perpe ndiculardirections .Propertiesof theItrain gauges arcshownin Table5.2.The fourstraingaugesin eachset were connec ted illil.wbeatetcneBridgeconfiguration.SeeFigure5.10 forthe circuitdiagram.

Each platewasfixed bymachine screwsto a supportat eithe rend.At one end, the support was fixedto theblock throu ghapivot and at the other end the

\ B IOCk (wheels oM itted) :0 :

Pin C onnectors

Th in S teel P lates ~

Ip orallelt abose plo tel

.---.,~-

Thi n S teel Plotes

(perpend icula r to bose plc te)

T op Iin k free to rotate an pi n.

Figure 5.!):Thetwo wayrorcC!Lrnusducer

Figure5.10:The circuitdiagramforthest raingallg($inthl!rorClJlranlldul:l!f

43

aupporta {rom the twotransducerswerejoined together.Thepj votconnections allowedtheaxial forces tobe measuredwithoutintroducing the problemsthat IwucJiug forces would cause.

The tran sducer was calibratedbyclamping it to arigid structure and loading it with known weights . The parallelandthe perp endiculartransd ucers were cali- brated independently,and checks forone set interferingwiththeother weremade.

There was no interferenceobserved and thusitcan be concluded that the forces were decouplcd.The data acquisitionsystemwas usedtocheck onthe linearity of lhe voltage-forcecurve.In general,thecorrelationcoefficient,r,tor thetrials Willi.999 indicatinga good linearfit.

TIll:treueduce rwas designed10withstanda.ma.ximum force of500N.However, in actualtests, theforce neverwentabove150N.

Althoughfor mostofthe tests theforcetransducer measuredtensile stress, there were cccaslons when compressive(orcei:l were important . This occurre d when the test apparatus was tiltedat 90^Das can be seenin Figure 5.11.The transducerfunctionedwell in compression,howeverin thisconfiguration tbetop link pUl abending stress on the perpendiculareetofplates and this causedan errorillthereadings.The errorwas eliminated.byusing a small pieceof wire to attachthe chain to the force transducer,howeverthissolutionwasimplemente d onlyaI~erthe firstrun had been made.

The accelerometerwaspowered byacharge amplifierandthesignalfromitwas Iedintoafilter.Thestraingaugeswere connectedto astrain gauge conditioner and amplifier and thenalso tothe samefilter.Thisfilterwasalcwpeeefilte r set tofilteroutfrequencies above 250hz. From thefilter thesignals,onefor

In this case the top I ink piacel l a b cn ding m maenl on lhe b ol lom f orrr l rau sduccr

0 :

Th is prob le m w as solved If-"'...l---~

hy iuserl ing a sma l l Icnglh of r i re 10 keep Ihc chain clear of 1.11(' l ransducer

" re

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- cha in

Figure;;.11:The'~IIU ipl1lfmttilll:,1III_~J(J

Typo:

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,25

Pignro~,12:The test chain

till';1('I:dl~ rnll1t'll'I'nud1\1'0forIIi{'force transducer,wentto aKeithley analog todigitalrcnverter.l-ronrlu-reitwentto the 2S6PCtOlT1pUII~ram] thenwas lrallsft'rrl'll to til('VAX~.'j:JOfor flll'lllt'rfilteringandaualyais.

5.3 The Test C hain

,],111~dlninsole-tedwasanopenlinksteelchain, Properties of thechainarc found in'Iulde5.:1andtlu-clinu-usiouso[the chainarcshownin Figure:,.12,

There II'il:'ll1o d fnrtmadeIIIl1lolld11pa rticularchain since many cliffcfcnlkinds ,In'used.1I0\\"1'1'l'ftill'modellinglaws dictatethatthe densityratio betweenthe

working fluidandthe mooring line mater ial mustbe thesame in bothmodel and prototype.Since water is usedinboth instances(thedensitydifferencebetween sailwate rasusedintheprototypeand fresh as usedillthe modelisnegligible for thisexperiment)themooringlinematerial density should alsobethe same in modeland prototype. Thisjustifies the usc ofa stee l link chain.

5.4 P r o cedure

As wasderived inChapte r<I,thevariablestobe changed in thisexperiment were thepre-tens ion,and thefrequency amiangle oftheforcingfunction.The frequency was veryeasy to changethroughthecontrollerofthe variablespeed . motor. Anglewasalittlemore difficult tochangebecause the baseplatehad to beliltedand then fastene dinplaceat the new angle.Thepre-tension wasthe mostdifficultto changebecause itinvolvedmoving a:lOkg anchor.

Withthisin mind, thefollowing proced urewas followed. The apparatus was initially setup ina configurati on withthe baseplatelevel to thewater and the anchor setto give thesmallest pre-tensionin thechain,Tile fil'llttest wasthen runwith the lowestfrequency,followedbytests withincrensiug frequenciesup toamaximum. At thispoint , theangleof thebase plateWallchangec! and thefrequenciesrunthrough again.Thiscont inued untilalltile angleswere run through and thenthepre-te nsion was changed.

Frequencies were startedinit ially at 0.5liz. and then Increased to a maximum ofabou t 4.0 Hz.Frequenciesbelow 0.5Hz.were not slguiflcantsince:they were well into thequasi-sta t icarea.Frequencies grealerthan 4.0liz.were not used becausetheequivalent scaledfrequenciesarc notencounte red in nature.

47

The stan. freq uency, thecutoff frequency andthe numberof frequenciesum·

pled underwentsmall cha.nSCllthrough outthe series ofexperiments.Forexample it was foundthatunderthelasSestpre-te nsion ronditionand ataninclinedaope, the motorcouldnot provideenoughtorque to producepuresinusoidal motion at aIrequcncyof0.5liz.sothetestsstarted atIHz.

The equipmentWIl.'JrunCorabout10secondsbeforedatarecordingbega.nto allowtransientstodieolr. Also, observation.weremadeduring thefirstaeries oflest!to make surethat neither the equipment northe catwalk vibratedto ...

lIignificnal amou nt

Therewerefouranglesused inthis experiment:0,3D,60 and 90".alt hough• asmentionedbefore,someproblemawerefound at 90·.Therewerealsothree pre-tensions used:ISN'.20NandJON.

5.5 Data Analysis

Asment ion ed beforethe,ignahfromthe accelerometer and the atrain &au&e3 were filtered at 2MHz.before theAIDconverter.Afterthedata.werecollected, theywere sent tot.heVAX8530computerforfurtheranalysis.

The analysisconsistedof twostages. First, the signalwasrun t.hrough a program whichapplied&movingaveragewithawindow of

to

tothedat.aset.

Thisremovedmuchof therandomnoisein thesignal.

However itwasobservedthat there was stillsomenoiseinthesystem.An examinat ion ofthesignalrevealed tha tmuch of t.hisnoisewasfound in the40to 90liz.range.This is possiblydueto the elect ricalnoisefro m the meter,orthe mechan icalnoise(vibrations) lromthema.cbineryoperating(egge&nl meshing) .

'8 Itis interestingto note that there was not/l.significantcontribution al the 60 liz.

freque ncy;a commonfrequencyfor interference.Thusitwas decided to filter the signal again toremove compo nentsof higherfrequency.

To performthis a Yule-Walker filterwas designed. Lookingatthe signalit was determined that thesignificantportion ofitwasCOl\l P OlIcOoffrequencies leas than 25Hz.(or the highfrequencytestsdecreasingto bethan10liz. forthe low frequency tests. Consequently, two fillersweredesigned,one thatcut out frequenciesof more than 25Hz. and one thatcut out frequenciesofmore thanl5 H,.

The signalfromthelower frequencytests appearedto benoisierthanthat . from the highfrequencytests.The reason fat this isthat the force inthe small frequency testswason the order of atenth of a Newton and thus thesignaloutput fromthe straingauges was quietweak.Forthe higherfrequen cytellts the forces involved wentaahigh as 100 Newtons andeffect ively drownedout the random noise.

A graphshowingtheresults from alow frequency test with and without1IJ.

teringisshown inFigure 5.13. Graphs of the signalvers usthe frequency for the two Butterwo rth fil te rs used are shown in Figure5. 11.

Hc---·~~-~~~~~-49

7 i 1

I

6 1 5 1 i

: ~1\1\f\NN

10 1 2 3 4 5 6 7 8 9

Figure5.13:Asample runwithandwithoutfiltering

-ac

·100

Resporuc or Yule· Walku Faler (forhighspeed \(IU)

su

FrequcllC)'(Hz)

ResponseotYuIe.WalkerFiller(forlow speed IClU)

·100

FrequcllC)'(Hz)

Figure 5.1-1:Tile filtersllSf~flfor tIll',llll.;l

Chapter 6 D iscussion

The discussion in thischapter willbebroken into two parts. Thefirstwill look atthe dat afrom thetests and discussand draw conclusion s from this data. The secondsectionwillexaminethemet hod and effects ofscalingthe modeltests to a prototype environment.

6.1 Ex perimental Work

Thedata from thetestswereanalyzedand graphswere drawn of forceagainst timefor each run.These graphsareshown in AppendixD.From these graph!

it was possible to lind thedynamic forceratio(OFR).Thedynamicforce ratio is theratio of the maximumchange in theforce(iefrom trough to crest) to the static value.Forexample ,ifthestatic pretension value was15 N and the force variedbetween10 and 40N,the DFRwould be2.

Once allthe OFRvalu es for therunswerecalculated. they were plotted against the frequency for the threedifferentvaluesof pretensionand are sbown in Figures 13.1•13.3.These graphsenableseveral conclusionsto be drawn.

The fourconditions firetdescribedby Suhera anddiscussed in Cbapter4 can

51

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