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Analysis of a Tensegrity System for Ocean Wave Energy Harvesting

Permanent Link: http://ufdc.ufl.edu/UFE0043743/00001

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Title: Analysis of a Tensegrity System for Ocean Wave Energy Harvesting
Physical Description: 1 online resource (76 p.)
Language: english
Creator: Vasquez, Rafael E
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: energy -- ocean -- tensegrity
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Tensegrity systems have been used in several disciplines such as architecture, biology, aerospace, mechanics and robotics during the last fifty years. However, just a few references in literature have stated the possibility of using tensegrity systems in ocean or energy-related applications. This research addresses the analysis of a tensegrity mechanism for ocean wave energy harvesting. The mechanics of ocean waves is described using the linear theory developed by George B. Airy. The kinematic properties of the waves, the motion of particles and wave power calculation are addressed. Then, the fluid structure interaction is reviewed making emphasis on the concepts of radiation damping and viscous damping. A planar tensegrity mechanism is proposed based on a planar morphology known as X-frame that was developed by Kenneth Snelson in 1960s. A geometric approach is used to solve the forward and reverse displacement problems. The theory of screws is used to perform the forward and reverse velocity analyses of the device. The Lagrangian approach is used to deduce the equations of motion considering the interaction between the mechanism and ocean waves. The tensegrity configuration is compared to a purely heaving body that is commonly used in ocean wave energy harvesting. The result shows that tensegrity systems could play an important roll in the expansion of clean energy technologies that help the world's sustainable development.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Rafael E Vasquez.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Crane, Carl D.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-12-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043743:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043743/00001

Material Information

Title: Analysis of a Tensegrity System for Ocean Wave Energy Harvesting
Physical Description: 1 online resource (76 p.)
Language: english
Creator: Vasquez, Rafael E
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: energy -- ocean -- tensegrity
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Tensegrity systems have been used in several disciplines such as architecture, biology, aerospace, mechanics and robotics during the last fifty years. However, just a few references in literature have stated the possibility of using tensegrity systems in ocean or energy-related applications. This research addresses the analysis of a tensegrity mechanism for ocean wave energy harvesting. The mechanics of ocean waves is described using the linear theory developed by George B. Airy. The kinematic properties of the waves, the motion of particles and wave power calculation are addressed. Then, the fluid structure interaction is reviewed making emphasis on the concepts of radiation damping and viscous damping. A planar tensegrity mechanism is proposed based on a planar morphology known as X-frame that was developed by Kenneth Snelson in 1960s. A geometric approach is used to solve the forward and reverse displacement problems. The theory of screws is used to perform the forward and reverse velocity analyses of the device. The Lagrangian approach is used to deduce the equations of motion considering the interaction between the mechanism and ocean waves. The tensegrity configuration is compared to a purely heaving body that is commonly used in ocean wave energy harvesting. The result shows that tensegrity systems could play an important roll in the expansion of clean energy technologies that help the world's sustainable development.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Rafael E Vasquez.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Crane, Carl D.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043743:00001


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ANALYSISOFATENSEGRITYSYSTEMFOR OCEANWAVEENERGYHARVESTING By RAFAELESTEBANV ASQUEZMONCAYO ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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c r 2011RafaelEstebanVasquezMoncayo 2

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Idedicatethisworktomylove(G),myfamilyandmyfriends 3

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ACKNOWLEDGMENTS Iexpressmygratitudetomysupervisorycommittee:Dr.CarlCra ne,Dr.Jacob Hammer,Dr.JohnSchueller,Dr.WarrenDixonandDr.JulioCor reafortheirtime, eort,andinvaluablecontributionstomyacademicgrowthd uringmytimeatthe UniversityofFlorida. ThisresearchwasdonewithnancialsupportfromtheU.S.Depar tmentofState, throughtheFulbrightProgram;theUniversityofFlorida,thr oughtheCenterforLatin AmericanStudiesandtheDepartmentofMechanicalandAerospac eEngineering; theColombianAdministrativeDepartmentofScience,Technol ogyandInnovation: Colciencias;andtheUniversidadPonticiaBolivariana,Mede llin. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 11 1.1OceanEnergy .................................. 12 1.1.1WaveEnergy ............................... 13 1.1.2WaveEnergyResource ......................... 14 1.1.3WaveEnergyTechnology ........................ 15 1.2TensegritySystems ............................... 16 1.2.1AdvantagesofTensegritySystems ................... 16 1.2.2MotionApplicationsofTensegritySystems .............. 18 1.2.3OceanApplicationsofTensegritySystems ............... 19 2OCEANWAVEMECHANICS ........................... 21 2.1Airy'sLinearWaveTheory ........................... 21 2.2TravelingWaves ................................. 24 2.3WaterParticleMotions ............................. 27 2.4WaveEnergyandPower ............................ 28 2.5WaveMechanicsNumericalExample ..................... 30 2.5.1KinematicProperties .......................... 30 2.5.2WaterParticlesMotion ......................... 31 2.5.3WaveEnergy ............................... 31 2.6Wave-StructureInteraction ........................... 32 2.6.1HeavingandPitchingBodyMotions .................. 32 2.6.2HeavingEquationofMotion ...................... 34 2.6.2.1MassandAddedMass .................... 35 2.6.2.2RadiationandViscousDamping ............... 36 2.6.3Wave-InducedForces .......................... 37 2.6.4SteadyStateSolutionoftheHeavingEquation ............ 38 3TENSEGRITYMECHANISM ........................... 39 3.1MorphologyDenition ............................. 39 3.2PositionAnalysis ................................ 40 3.2.1ForwardPositionAnalysis ....................... 40 3.2.2ReversePositionAnalysis ........................ 42 5

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3.2.3NumericalExample ........................... 43 3.3VelocityAnalysis ................................ 44 3.3.1ForwardVelocityAnalysis ....................... 45 3.3.2ReverseVelocityAnalysis ........................ 46 3.3.3NumericalExample ........................... 47 3.4EquationofMotion ............................... 49 3.4.1Assumptions ............................... 49 3.4.2KineticEnergy ............................. 50 3.4.3PotentialEnergy ............................ 51 3.4.4FrictionVector .............................. 53 3.4.5GeneralizedForces ........................... 53 4OCEANWAVEENERGYHARVESTING ..................... 54 4.1ElectricalGeneratorsforWaveEnergyHarvesting .............. 54 4.2SeaStateSelection ............................... 55 4.2.1WavePowerandWaveKinematicProperties ............. 56 4.3DirectDriveHeavingSystem ......................... 57 4.4TensegritySystem ................................ 59 4.5DiscussionofResults .............................. 62 5CONCLUSIONSANDFUTUREWORK ...................... 64 5.1Conclusions ................................... 64 5.2FutureWork ................................... 65 REFERENCES ....................................... 66 BIOGRAPHICALSKETCH ................................ 76 6

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LISTOFTABLES Table page 1-1Oceanenergyglobalresource ............................ 12 3-1Mechanismparametersforkinematicchain ..................... 40 3-2Positionanalysisnumericalexampleresults ..................... 43 4-1Directdriveheavingroatcoecients ........................ 57 4-2Tensegrityharvestingsystemcoecients ...................... 60 7

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LISTOFFIGURES Figure page 1-1Oceanenergyconversiondevelopment ........................ 13 1-2Globalcoastalwavepowerestimates ........................ 14 1-3Tensegritymorphologies ............................... 16 2-1Notationforthelinearwaveanalysis ........................ 22 2-2ParticlepathspredictedbyAiry'slinearwavetheory ............... 28 2-3Notationforthewaveenergyanalysis ........................ 28 2-4Examplewaterparticlesmotion ........................... 31 2-5Rigidbodywithsixdegreesoffreedom ....................... 32 2-6Floatingbodiesundergoingheavingorpitchingmotion .............. 33 2-7Floatingbodies:pureheavingandpitchingconditions ............... 33 2-8Addedmassandadded-massmomentofinertiacoecientsfora rectangularbody 35 2-9Non-dimensionalradiationdampingcoecientforaheavin grectangularsection 36 3-1Conceptofawaveenergyharvesterbasedontensegritysystems ......... 39 3-2Kinematicdiagramofthemechanism ........................ 40 3-3Vectordiagramofthemechanism .......................... 41 3-4Positionanalysisnumericalexample ......................... 43 3-5Velocityanalysisnumericalexample ......................... 48 3-6Velocityanalysisnumericalexampleresults ..................... 48 4-1Locationandbathymetryof IslaFuerte ,Colombia ................. 56 4-2Wavepowervariationfortheseastateat9.408 N,76.180 W .......... 57 4-3Heavingbodysimulation:wave-inducedforce ................... 58 4-4Heavingbodysimulationresponse .......................... 58 4-5Tensegritysimulation:wave-inducedforces ..................... 60 4-6Tensegritysimulation:surging,heavingandpitchingmoti ons ........... 61 4-7Tensegritysimulation:instantdissipatedpower .................. 62 8

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4-8Tensegritysimulation:motioningeneratorsandsprings ............. 62 4-9Variationofpowerdissipationwithmechanismparameters ............ 63 9

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ANALYSISOFATENSEGRITYSYSTEMFOR OCEANWAVEENERGYHARVESTING By RafaelEstebanVasquezMoncayo December2011 Chair:CarlD.CraneIIIMajor:MechanicalEngineering Tensegritysystemshavebeenusedinseveraldisciplinessuchasarc hitecture, biology,aerospace,mechanicsandroboticsduringthelastf tyyears.However,just afewreferencesinliteraturehavestatedthepossibilityofusi ngtensegritysystemsin oceanorenergy-relatedapplications.Thisresearchaddresses theanalysisofatensegrity mechanismforoceanwaveenergyharvesting. Themechanicsofoceanwavesisdescribedusingthelineartheo rydevelopedby GeorgeB.Airy.Thekinematicpropertiesofthewaves,themoti onofparticlesandwave powercalculationareaddressed.Then,theruidstructureinte ractionisreviewedmaking emphasisontheconceptsofradiationdampingandviscousdamp ing. Aplanartensegritymechanismisproposedbasedonaplanarmorph ologyknown as\X-frame"thatwasdevelopedbyKennethSnelsonin1960s.Age ometricapproach isusedtosolvetheforwardandreversedisplacementproblems.Th etheoryofscrewsis usedtoperformtheforwardandreversevelocityanalysesofthe device.TheLagrangian approachisusedtodeducetheequationsofmotionconsidering theinteractionbetween themechanismandoceanwaves. Thetensegritycongurationiscomparedtoapurelyheavingb odythatiscommonly usedinoceanwaveenergyharvesting.Theresultshowsthattenseg ritysystemscould playanimportantrollintheexpansionofcleanenergytechno logiesthathelptheworld's sustainabledevelopment. 10

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CHAPTER1 INTRODUCTION Duetosustainabilityconcerns,aworldracestartedseveralyear sagotoincentivize theresearch,developmentandutilizationofrenewableener gysources[ 1 { 3 ].Theocean representsanenormouspotentialenergysource[ 4 5 ];however,itsexploitationisstill incipientcomparedtootherwell-establishedpowerharvestin gtechnologiessuchaswind andsolarenergies[ 6 ]. Oceanenergycanbeobtainedfromwaves,tides,currents,therm algradientsand salinitygradients.Duringthelast30yearstheR&Dworksinall theseresourceshave increasedconsiderably[ 7 { 17 ].Severalassessmentstudiestoevaluatetheamountofocean energywhichisavailableataparticularregion,andtodete rminethetechnologythatis convenientforthelocalconditionshavebeenconductedind ierentcountries[ 17 { 26 ]. Waveenergyconstitutesthemostnoticeableformofoceanener gy,maybebecause ofits(oftenimpressive)destructivecapabilities[ 27 ].TheU.S.DepartmentofEnergy (DoE)developedtheMarineandHydrokineticTechnologyData base[ 28 ],asashared resourceforthemarine/hydrokineticindustryandgovernmen t.Therearemorethan160 dierentdevicesforoceanenergyharvestingregisteredinth edatabase,withabout40% correspondingtowaveenergy. Thewordtensegrityisacombinationofthewordstensionandin tegrity[ 29 ]. Tensegritysystemswereintroducedinthe20 th centurybyFuller[ 29 ],Emmerich[ 30 ] andSnelson[ 31 ].Thesesystemsareformedbyacombinationofrigidelements(st ruts) undercompression,andelasticelements(ties)undertension[ 32 33 ]. Tensegritysystemshavebeenusedinseveraldisciplinessuchasarc hitecture,biology, aerospace,mechanicsandroboticsduringthelastftyyears[ 34 ].Applicationsinsciences andengineeringinclude,amongothers,developmentofstruct uraldomesandbridges [ 35 { 40 ],deployablesystemsforspaceapplications[ 41 { 46 ],descriptionandmodellingof livingorganismsandbiologicalsystems[ 47 { 51 ],andapplicationsinrobotics[ 52 { 60 ]. 11

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Justafewreferencesinliteraturehavestatedthepossibilityo fusingtensegrity systemsinoceanapplications.ScruggsandSkelton[ 61 ]madeapreliminaryinvestigation inthepotentialuseofcontrolledtensegritystructurestohar vestenergy,andsuggested theirsuitabilitytoharvestenergyfromoceanwaves.Theyshowe dhowatensegrity structure,withoneactivebar,canbeusedtoeectivelyharve stenergywhenitisexcited atasinglefrequency.Jensenetal.[ 62 ]proposedtensegritystructuresinthedesignof wavecompliantstructuresforoshoreaquaculture;theystudi eddierentcombinationsof pre-stressanddeterminedhowitinruencesthestinessofthewh olestructure.Wroldsen [ 63 ]developedanalysistoolsbasedondierential-algebraiceq uations(DAEs)ofmotion, andextendedtheformulationtoincludethedynamicsofrela tivelylongandheavycables withincreasedcomputationaleciency.Tensegritysystemscou ldplayanimportantroll intheexpansionofcleanenergytechnologiesthatwouldcont ributetoworld'ssustainable development.Therefore,thiscanbeapromisingeldofdevel opmentthatisstillona conceptuallevelandneedstobeexplored. 1.1OceanEnergy Theoceanscontainalargeamountofrenewableenergy.Repor tsoftheestimated globalresource,presentedbyAEAEnergy&Environment[ 6 ]andtheU.S.Departmentof Energy[ 64 ],aresummarizedinTable 1-1 Table1-1.Oceanenergyglobalresource. EnergytypeEstimatedglobalresource Waveenergy80000TWh/yTidalenergy300+TWh/yMarinecurrent800+TWh/yThermalenergy10000TWh/ySalinitygradient2000TWh/y TheInternationalEnergyAgency(IEA)[ 14 ]reportedtheglobalstatusoftechnology developmentforoceanenergysystems,Figure 1-1 A,highlightingtheUnitedKingdom, theUnitedStates,CanadaandNorway,asleadersindevelopment .Figure 1-1 Bshowsthe emphasisplacedonthedevelopmentofwaveenergytechnologi es. 12

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0 5 10 15 20 25 30 35 40 CountryFinland Germany Israel Italy NewZealand Korea Brazil Greece Portugal Russia Mexico France India Ireland Netherlands Sweden Australia China Denmark Japan NorwayCanada US UK Numberofsystems Salinity Thermal Wave Current Tidal ACountryparticipation. 0 5 10 15 20 25 30 TechnologySalinity Thermal Tidal Current Wave Number of systems Conceptdesign Part-scale(Tank) Part-scale(Sea) Full-scale Preandcommercialproduction BTechnologystatus. Figure1-1.Oceanenergyconversiondevelopment[ 14 ]. Althoughenvironmentallegislationforthedeploymentofoce anenergyprojectsis notclearyetinplentyofcountries,severalstudiesarebeingc arriedoutaroundtheworld todemonstratesustainabilityandcommercializationpotenti alsofoceanenergy,andto fullllegislativerequirementsthataimtoprotecttheenvi ronmentnearexploitationareas [ 65 { 67 ].Inthisdirection,theDepartmentoftheInterioroftheUni tedStates[ 68 ]stated policiesabouttheuseofrenewableenergyandalternateuseso fexistingfacilitieson theoutercontinentalshelf.Additionally,DetNorskeVeritas( DNV)[ 69 ]developedthe OshoreServiceSpecicationDNV-OSS-312,whichpresentsthep rinciplesandprocedures withrespecttocerticationoftidalandwaveenergyconvert ers. 1.1.1WaveEnergy Waveenergyisanindirectformofsolarenergy.Temperatured ierencesproduced bythesolarradiationaroundtheworldcreatewindsthatblow overtheoceansurface, generatingwaves.Suchwavescantravelthousandsofkilomete rsthroughdeepwaterswith minimallossofenergy,representingasourcewithhigherpower densitythanwindorsolar power[ 70 ]. 13

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1.1.2WaveEnergyResource Wavestransportbothkineticandpotentialenergy.Thetotal energyofawave dependsmainlyonitsheight( H )anditsperiod( T ),andisusuallymeasuredinWatt permeter(W/m)ofwavefront.Theglobalwavepowerpotentia lisofthesameorderof magnitudeastheworld'selectricityconsumption,around110TW[ 4 ].Cruz[ 71 ]states thatitispossibletoextract10-25%ofthisenergy,suggestingt hatwavepowercould makeasignicantcontributiontotherenewableenergyindust ry.Thebestwaveclimates, whoseannualaveragepowerlevelsarebetween20and70kW/mor higher,arefoundin zonesfrom30to60degreeslatitude,Figure 1-2 .Nonetheless,attractivewaveclimatesare foundalsowithin 30degreeslatitude,wherethelowerpowerleveliscompensat edby smallerpowervariability[ 4 ]. Figure1-2.GlobalcoastalwavepowerestimatesfromtheTopex altimeter[ 72 ]. Asdeep-waterwavesapproachshallowwaters,theyslowdown,the irwavelength decreaseandtheirheightgrows,whichleadstobreaking.Them ajorlossesofenergyare duetobreakingandtofrictionwiththeseabed;therefore,on lyafractionoftheresource reachestheshore[ 71 ].Notallsitesaresuitablefordeploymentofharvestingdevice sfor severalreasons,includingunsuitablegeomorphologicconditi onsattheshoreline,excessive tidalrangeandenvironmentalimpact[ 4 ].Nonetheless,shorelinedevicescouldprovidea substantialcontributiontotheelectricenergydemandsinsma llislandsorisolatedcoastal regionswheretheenergyconsumptionissmall. 14

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1.1.3WaveEnergyTechnology Thereareseveraltechnologiesthatresultfromdierentways inwhichtheenergycan beharnessedfromwaves,dependingonthewaterdepthandtheloc ation(i.e.shoreline, near-shore,oshore)[ 27 ].Severalwaystoclassifywaveenergydeviceshavebeenpropo sed. Falc~ao[ 27 ]recentlyproposedaclassicationbasedonthetype,thedeploy mentstyle,and thedeploymentplaceasfollows: Oscillatingwatercolumns(OWC)aredevicesthathaveaparti allysubmerged structure,openbelowthewatersurfacewhichholdsairinside. Theincidentwaves thatgothroughtheOWC,createoscillatingmotionoftheairi nsidethechamber, forcingtheairtorowthroughaturbinewhichdrivesanelect ricalgenerator. Self-rectifyingturbineswhichprovideaunidirectionalr otationforanalternatingair row,suchastheWellsturbine[ 73 74 ],areoftenusedinsuchdevices. Oscillatingbodysystemsaregenerallylocatedoshore(waterd epth > 40m),and canbeeitherroatingorfullysubmerged[ 27 75 ].Themainelementofthesesystems isanoscillatingbodythateitherroatsorissubmergednearth esurface[ 4 ].Several congurationssuchassingle-bodyheavingbuoys,two-bodyhea vingsystems,fully submergedheavingsystems,pitchingdevicesandbottom-hinged systems,have beenusedtobuildthistypeofenergyconverters[ 27 ].Severaltechniqueshave beenproposedtoharnesstheenergywithoscillatingbodies,usin geitherlinear (translational)orrotationalelectricalgenerators. Overtoppingconvertersconstituteanothertypeofwaveener gyconverters.The waterinthewavecrestisintroducedbyovertoppingthrougha slopingwallorramp intoareservoirlocatedatalevelhigherthanthesurrounding watersurface.The potentialenergyofthewateristhenconvertedintousefulen ergythroughanarray oflow-headhydraulicturbines(e.g.Kaplanturbines)[ 27 76 ]. DunnettandWallace[ 77 ]proposedanalternativeclassicationforwaveenergy converters.Thisclassicationmethodisbasedonhowthedevice sgetthemechanical energyfromthewaves: Pointabsorbersaredeviceswhosesurfaceareaisverysmallinco mparisontothe wavelengthofoceanwaves. Attenuatorsarerelativelylongdevicesthatareplacedpar alleltothegeneral directionofwavetravel. Terminatorsareplacedperpendicularlytothewavesandare intendedtoabsorba largeproportionoftheenergyofthewave. 15

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1.2TensegritySystems SkeltonanddeOliveira[ 34 ]recentlydenedatensegritycongurationasfollows: \ Intheabsenceofexternalforces,letasetofrigidbodiesin aspeciccongurationhave torquelessconnections.Thenthiscongurationformsaten segritycongurationifthe givencongurationcanbestabilizedbysomesetofinternal tensilemembers,i.e.connected betweentherigidbodies. ". Figure. 1-3 showsthreedierenttensegritymorphologiesthathavebeene xtensively addressedinliteraturefordierentapplications. AX-frame[ 31 ]. BOctahedron. C6-barprism. Figure1-3.Tensegritymorphologies. 1.2.1AdvantagesofTensegritySystems TensegritysystemsoeranumberofadvantagesasdescribedbySk eltonetal.[ 78 ] andWroldsen[ 63 ].Suchcharacteristicsaresummarizedinthefollowingparag raphs. Stabilizingtension.Atensegritystructuregetsastablestatic equilibriumconguration whenalltiesareintensionandallstrutsareincompression(int heabsenceofexternal forcesortorques)[ 63 ].Compressivememberslosestinessastheyareloaded,whereas tensilemembers,possessinglessweight,gainstinessastheyareloa ded[ 78 ]. Geometryandstructuraleciency.Traditionally,structure stendtobemadewith orthogonallyarrangedelements[ 78 ].However,thistypeofarchitecturedoesnotusually yieldminimalmassdesignsforgivensetsofstinessproperties[ 79 ].Tensegritysystems, 16

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ontheotherhand,uselongitudinalmembersarrangedinunusua lpatterns,toachieve strengthconditionswithlessmass[ 78 ]. Deployabilityandsmallstoragevolume.Sincethemembersinc ompressionareeither disjointorconnectedbysimplejoints,largedisplacement,dep loyability,andstowageina compactvolumeappearasadvantagesoftensegritysystems[ 46 78 ]. Tunablestiness.Oneremarkablepropertyoftensegritysystemsi sthepossibility tochangeshapewithoutchangingstinessandviceversa[ 80 ].Thesestructuresusually havelowstructuraldamping,leadingtochallenges/opportun itieswithrespecttovibration insomeapplications[ 63 ].Skeltonetal.[ 78 ]addressedthatstructuresdesignedtoallow tuningwouldplayanimportantroleinthedevelopmentofnex tgenerationmechanical systems. Reliabilityinmodelling.Sincemostmembersinatensegritysy stemareaxially loaded,theequationsneededtorepresentthestaticanddynam icbehaviorrequires,in general,lesssimplications,resultingintomorereliablemod els[ 63 78 ]. Activecontrol.Asinglememberofatensegritysystemcanservemul tiplefunctions: asaload-carryingmember,asensor,anactuator,etc.Thisrex ibilityprovidesan encouragingopportunityforintegratingstructuralandcon troldesignprocesses[ 78 81 ]. MotivationfromBiology.Ingber[ 47 49 ]andHuangetal.[ 50 ]statedthattensegrity isafundamentalbuildingarchitectureoflife.Hence,iften segrityispartofnature's buildingarchitecture,thecapabilitiesoftensegritycoul dmaketheeciencypresentin naturalsystemstransferabletoman-madesystems[ 78 ]. Modularitythroughcells.Tensegritysystemsareoftenmadeusin galargenumber ofidenticalbuildingblocks,orcells.Themodularityfacili tateslarge-scaleproductionof identicalunitsthatcanbelaterassembled[ 63 ]. Robustnessthroughredundancy.Usingmoretiesthanstrictlynee ded(redundancy) increasesthesystemrobustnessandhelpstoavoidinnitesimalme chanisms[ 63 ]. 17

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1.2.2MotionApplicationsofTensegritySystems Theinvolvementoftensegritysystemsinmotionapplicationsi srelativelynew, andseveralworkshavebeenincreasinglyappearingduringthe lasttwentyyears.At thebeginningofthe1990's,Pellegrino[ 82 ]presentedatheoryforthematrixanalysis ofkinematicallyindeterminateprestressedassembliesmadefro mpin-jointedbarsbased onalinearapproach.CalladineandPellegrino[ 83 ]discussedtheanalyticalconditions underwhichapin-jointedassembly,whichhasindependentstat esofself-stressand m independentmechanisms,tightensupwhenitsmechanismsareex cited.Djouadietal. [ 84 ]describedanumericalschemeofactivenonlinearcontroloft ensegritysystemsfor spaceapplications.Stern[ 41 ]developedgenericdesignequationstondthelengthsof thestrutsandelastictiesneededtocreateadesiredgeometry. Sultan[ 85 ]workedon modelling,design,andcontroloftensegritystructuresforsev eralapplications.Duyetal. [ 42 ]presentedareviewofafamilyoftensegritystructuresthatsel fdeployfromastowed orpackedconguration.Knight[ 43 ]addressedtheproblemofstabilityoftensegrity structuresforthedesignofadeployableantenna.Pellegrino etal.[ 44 ]studieddeployable structuresforsmallsatellitemissions.Sultanetal.[ 52 ]proposedthedevelopmentofa rightsimulatorbasedonatensegritystructure. Startingthedecadeof2000s,OppenheimandWilliams[ 86 ]examinedthedynamic behaviorofasimpleelastictensegritystructure.Skeltonetal .[ 87 ]developedanexplicit analyticalmodelofthenonlineardynamicsofalargeclassof tensegritystructures.Sultan etal.[ 88 ]formulatedthegeneralprestressabilityconditionsfortense gritystructures, expressedasasetofnonlinearequationsandinequalitiesonth etendontensions. KanchanasaratoolandWilliamson[ 54 ]developedapassivenonlinearconstrainedparticle dynamicmodelforaclassoftensegrityplatformstructures.Sul tanetal.[ 89 ]derived thelinearizedequationsofmotionfortensegritystructures aroundarbitraryequilibrium congurations.Tibert[ 45 ]workedonthedevelopmentofdeployabletensegritystructur es forspaceapplications.Tranetal.[ 53 ]performedthereversedisplacementandcompliance 18

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analysisofatensegritybasedparallelmechanism.SultanandSk elton[ 46 ]presenteda strategyfortensegritystructuresdeploymentusingsetsofequi libria.Bossensetal.[ 90 ] analyzedthedynamicbehaviorofatensegritystructurebycom paringaniteelement modelwithanexperimentalmodel.Chanetal.[ 91 ]usedsimplecontrolstrategiesfor theactivevibrationcontrolofathree-stagetensegritystruc ture.MarshallandCrane[ 56 ] proposedasix-degree-of-freedomtensegrity-basedparallelp latformwhichcombinesrigid andelasticelements.SultanandSkelton[ 55 ]usedtheintrinsicpropertiesoftensegrity structurestoconstructasmartforce/torquesensor. Morerecently,Craneetal.[ 92 ]obtainedtheequilibriumpositionforageneral skew-prismaticstructurewithavarietyofexternalloadsandm omentsactingonthe structure,usingthevirtualworkprinciple.ArsenaultandGossel in[ 57 { 59 ]addressedthe possibilityofthetensegritysystemstobeusedinthedevelopment ofnewlightweight mechanismsformotionapplicationswhereonlyrigid-linkro botshavebeenconsidered. BayatandCrane[ 93 ]presentedaclosed-formanalysisofaseriesofplanartensegrit y structurestodetermineallpossibleequilibriumconguratio nsforeachdevicewhen noexternalforcesormomentsareapplied.ArsenaultandGosseli n[ 94 ]usedgeneral staticbalancingconditionsadaptedforthecaseoftensegrity mechanisms.Vasquez andCorrea[ 60 ]presentedthekinematicandthedynamicanalyses,andanonlin ear controlstrategyforaplanarthree-degree-of-freedomtense grityrobotmanipulator. Craneetal.[ 95 ]performedtheequilibriumanalysisofaplanartensegrityme chanism showingthecomplexitythatresultsfromnon-zerofreelength sinthecompliantelements. ArsenaultandGosselin[ 96 ]presentedthedirectandinversestaticanalysesforanew spatialtensegritymechanismminimizingitspotentialenergy .Wroldsenetal.[ 97 ]useda non-linearfeedbackcontrollawfornon-minimalrealizati onsoftensegritysystems. 1.2.3OceanApplicationsofTensegritySystems ScruggsandSkelton[ 61 ]madeapreliminaryinvestigationinthepotentialuseof controlledtensegritystructurestoharvestenergy.Theyprese ntedanapproachtouse 19

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linearregenerativeactuatorsasactivebarsintothestruct ure.Theyillustratedthe approachinasimulationexampleforasmallscaleone-actuator system,andsuggestedthe suitabilityoftensegritysystemstoharvestenergyfromoceanwa ves. Jensenetal.[ 62 ]andWroldsen[ 63 ]proposedtensegritystructuresinthedesign ofwavecompliantstructuresforoshoreaquaculture.Theyad dressedthepromising propertieswithrespecttocontrolofgeometry,stinessandvi bration,thatcouldmake tensegrityanenablingtechnologyforfuturedevelopmentsi nopenoceanaquaculture constructionsystemsforhighenergyenvironments. 20

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CHAPTER2 OCEANWAVEMECHANICS Oceanwavescanbegeneratedbyseveralphenomena,suchasmoti onsofcelestial bodies(sunandmoon),seismicdisturbances(earthquakes),movi ngbodies(ships)and winds.Thewindproduceswavesofdierenttypes,fromtheshort capillarywavetothe longswell,thatcanbeclassiedasfollows[ 98 ]: Linearwaves(sinusoidalproles). Nonlinearwaves(non-symmetricalproles). Randomseas(predictableinthefrequencydomainundercerta inassumptions). Forthisresearch,alinearmodelofwind-generatedwaves,dev elopedbyGeorgeB. Airyin1841[ 99 ],isused. 2.1Airy'sLinearWaveTheory GeorgeB.Airydevelopedtherstmeaningfulanalysisofoceanw aves[ 100 ].It involvesthesolutionofthelinearequationofcontinuity(c onservationofmass)foran irrotationalrowsubjecttolinearizedboundaryconditions. Thewavepropertiesderived withthistheoryaregoodapproximationsforsmallvaluesoft hewavesteepness(dened astheratiobetweenthewaveheight H andthewavelength ,Figure 2-1 )[ 98 101 ],i.e. waveswithrelativelysmallamplitudes. Thefollowingparametersaredenedforthelinearwaveanal ysis,Figure 2-1 : SWL:stillwaterlevel.h :waterdepth,measuredfromtheroortoSWL. H :waveheight,measuredfromthetroughtothecrest. :wavelength,measuredfromcresttocrest. :verticalfree-surfacedisplacement,measuredfromtheSWL.T hisparameterisa functionof x and t c :wavecelerity(phasevelocity). 21

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Figure2-1.Notationforthelinearwaveanalysis[ 98 ]. Theconservationofmasstheoremisexpressedbytheequationofc ontinuity,whose dierentialformisgivenby @ @t = r ( V ) ; (2{1) wherethedeloperatorisgiveninCartesiancoordinatesby r = @ @x i + @ @y j + @ @z k ; (2{2) andtheruidvelocityvectorisgivenby V = u i + v j + w k : (2{3) Therowbeneaththefreesurfaceisassumedtobeirrotational,i .e. r V =0.Then, thevelocityofwaterparticlesisaconservativevectoreld thatcanberepresentedbya potentialfunction,inthiscase,thevelocitypotential ,asfollows V = r : (2{4) Substituting( 2{4 )into( 2{1 ),andassumingasteadyandincompressiblerow,gives r 2 =0 : (2{5) 22

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Equation( 2{5 )isanellipticpartialdierentialequation(Laplace'seq uation), thatcanbesolved,subjecttoasetofboundaryconditions,todet erminethevelocity potential .Theboundaryconditionsaregivenby: Kinematicfree-surfacecondition:thevelocityofaparticl eonthefreesurfacemust equalthevelocityofthefreesurfaceitself. V j z = @ @t k @ @z z =0 k : (2{6) Sea-roorcondition:adjacentparticlestothesearoorcanno tcrossthesolid boundary. V N j z = h = @ @z z = h =0 : (2{7) Dynamicfree-surfacecondition:thepressureonthefreesurfac eiszero. = 1 g @ @t z = : (2{8) Combining( 2{6 )and( 2{8 ),andeliminating yieldsthelinearizedfree-surface boundarycondition,whichisgivenby 1 g @ 2 @t 2 + @ @z z = 0 =0 : (2{9) Thesolutionof( 2{5 )canbefoundbyseparationofvariablesforatravelingwavei n theform = X ( x ct ) Z ( z )= X ( ) Z ( z ) : (2{10) Substituting( 2{10 )into( 2{5 )yields 1 X d 2 X ( ) d 2 = 1 Z d 2 Z ( z ) dz 2 = k 2 ; (2{11) where k isaconstant. Thegeneralsolutionofthetwoordinarydierentialequatio nsisgivenby X ( )= C sin( k + ) ; (2{12) 23

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Z ( z )= C z cosh( kz + ) ; (2{13) where C C z and arearbitraryconstants. Sincetheoriginofthe x (and )coordinateisarbitrary, canbeassignedazero valuewithoutlossofgenerality.Therefore,( 2{12 )becomes X ( )= C sin( k ) : (2{14) Applyingthesea-roorboundarycondition( 2{7 )to( 2{13 ),weget = kh ,whichgives Z ( z )= C z cosh[ k ( z + h )] : (2{15) Substituing( 2{14 )and( 2{15 )backinto( 2{10 )yields = C cosh[ k ( z + h )]sin( k ) ; (2{16) where C = C C z Now,usingthelinearizedfree-surfaceboundarycondition( 2{9 ), C canberemoved from( 2{16 ).Hence,thevelocitypotentialforatravelingwaveisgiven by = H 2 g kc cosh[ k ( z + h )] cosh( kh ) sin[ k ( x ct )] ; (2{17) wheretheupper(+)signin representsright-travelingwaves,andthelower(-)sign representsleft-travelingwaves. 2.2TravelingWaves Two-dimensionalproblemsinwavemechanicsgeneralyconside rright-traveling waves[ 70 98 101 ].Fromthecombitationofthelinearizedfree-surfacecondi tion( 2{9 ) andthegeneralsolutionforthevelocitypotential( 2{16 )(eliminating ,andusingthe denitionofthehorizontalcoordinate ),itispossibletodemonstratethatthefree-surface displacementcausedbythewaveisgivenby = H 2 cos[ k ( x ct )] : (2{18) 24

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Sincethefree-surfacedisplacementfromtheSWLissinusoidali nbothspaceand time,themaximumvaluefor (thecrest),occurswhen k ( x ct )=0 ; 2 ; 4 ;::: (2{19) For t =0,thedistancebetweentwosuccessivecrestsisthewavelength .Then,from ( 2{19 )wecancomputethewavenumber k ,asfollows k = 2 : (2{20) Now,for x =0,thetimelapsebetweentwosuccesivecrestsisthewaveperiod T From( 2{19 ) kc = 2 T =2 f !; (2{21) where f isthewavefrequencyinHzand isthecircularwavefrequencyinrad/s. From( 2{20 )and( 2{21 ),wecancomputethecelerityorphasevelocityas c = T : (2{22) Thevelocitypotentialcanbewrittenintermsofthecircula rwavefrequencyas = H 2 g cosh[ k ( z + h )] cosh( kh ) sin( kx !t ) : (2{23) Eliminatingthevelocitypotentialfrom( 2{9 )and( 2{23 ),thecircularwavefrequency canbecomputedas = p gk tanh( kh ) : (2{24) Combining( 2{20 )and( 2{24 ),resultsinanexpressionforthewavelengthgivenby = 2 k = 2 g 2 tanh( kh )= gT 2 2 tanh 2 h = cT: (2{25) Equation( 2{25 )istranscendentalbecause cannotbeisolatedandsolved analytically.Therefore,anumericaltechniqueisrequire dforitssolution. 25

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Becauseofthebehaviorofthehyperbolictangent(tanh( kh ) 1as kh !1 )in ( 2{25 ),itisacommonpracticeinoceanengineeringtodividethei nnitespaceof h= intothreeregionswhichdenetherelativeseadepthasfollo ws[ 98 ]: 1. Shallowwater: h= 1 = 20 2. Intermediatewater:1 = 20 h= 1 = 2 3. Deepwater: h= 1 = 2 Asaresult,approximationsareusedinthecomputationofthewa velengthfor shallowanddeepwaterconditions,becausethedierencebetwe enthevaluesof kh and tanh( kh )isabout3%orlessfor h= 1 = 20,whereasfor h= 1 = 2thedierencebetween tanh( kh )andunityislessthan0.4%[ 98 ]. Thedeep-waterapproximationof( 2{25 )isgivenby 0 gT 2 2 c 0 T; (2{26) wherethesubscript0isusedtoidentifythedeep-waterpropert ies. Theshallow-waterapproximationof( 2{25 )isgivenby gT 2 2 kh = gT 2 2 2 h = gT 2 h cT; (2{27) then = p ghT cT: (2{28) Theresultin( 2{26 )showsthatthedeep-waterwavelengthandcelerityarefunct ions ofthewaveperiodonly.Ontheotherhand,theresultin( 2{28 )showsthattheshallow-water wavelengthisafunctionofbothdepthandperiod,andthatth ecelerityisafunctionof depthandindependentofperiod.Consequently,weseethatwav esshortenandslowdown astheyapproachtheshoreline. 26

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2.3WaterParticleMotions TheCartesianvelocitycomponentsinatwo-dimensionalirrot ationalrowaregiven,in termsofthevelocitypotential,bytheCauchy-Riemannequa tions u = @ @x = @ @z ; (2{29) w = @ @z = @ @x ; (2{30) wherethescalarfunction iscalledthestreamfunction. Then,thevelocitycomponentsintravelingwavescanbecomp utedsubstituting ( 2{23 )into( 2{29 )and( 2{30 ),whichyields u = H! cosh[ k ( z + h )] 2sinh( kh ) cos( kx !t ) ; (2{31) w = H! cosh[ k ( z + h )] 2sinh( kh ) sin( kx !t ) : (2{32) Thehorizontalandverticaldisplacements( ; )ofaparticleaboutaxedmeanpoint ( x o ;z o ),arefoundsubstituting( x;y )by( x o + ;z o + ),expandingtheresultsinMaclaurin seriesin and ,andnally,integratingtheCartesianvelocitycomponents ( 2{31 )and ( 2{32 )overtime. Theresultexpressionsare,totherstorder,givenby = Z udt j x o ;z o = H cosh[ k ( z o + h )] 2sinh( kh ) sin( kx o !t ) ; (2{33) = Z wdt j x o ;z o = H sinh[ k ( z o + h )] 2sinh( kh ) cos( kx o !t ) : (2{34) Then,thepositionofaparticlemeasuredfromthemeanposition ( x o ;z o )isgivenby thepositionvector r = i + k : (2{35) Substituting( 2{33 )and( 2{34 )into( 2{35 )yields r = H= 2 sinh( kh ) f cosh[ k ( z o + h )]sin( kx o !t ) i +sinh[ k ( z o + h )]cos( kx o !t ) k g : (2{36) 27

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Figure 2-2 showstheparticlepathspredictedbybyAiry'slinearwavethe ory,using appropriateapproximationsforthehyperbolicfunctionsi n( 2{36 ). ADeepwater. BIntermediatewater. CShallowwater. Figure2-2.ParticlepathspredictedbyAiry'slinearwaveth eory[ 98 ]. 2.4WaveEnergyandPower ThemassofanelementdisplacedabovetheSWL,Figure 2-3 ,isgivenby m = ( x ) b; (2{37) where b isthewidthofthewavecrest. Figure2-3.Notationforthewaveenergyanalysis[ 98 ]. 28

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Thecenterofmassislocatedatadistance = 2abovetheSWL.So,thepotential energyoftheelementisgivenby E p = g ( m ) 2 = 1 2 g 2 ( x ) b: (2{38) ThetotalpotentialenergyoftheelementsabovetheSWLisgi venby E p = gH 2 8 b Z 0 cos 2 ( kx !t ) dx = gH 2 b 16 : (2{39) Thekineticenergyofasubmergedelementisgivenby E k = 1 2 ( u 2 + w 2 ) bxz; (2{40) where( u;w )aretheparticleCartesianvelocitycomponentsgivenby( 2{31 )and( 2{32 ). Thetotalkineticenergyofthesubmergedelementsisgivenby E k = 1 2 b Z 0 h Z 0 ( u 2 + w 2 ) dxdz = gH 2 b 16 : (2{41) Finally,thetotalenergyofalinearwave,whichisequallyd ividedbetweenpotential andkineticenergy,isgivenby E = E p + E k = gH 2 b 8 : (2{42) Substituting( 2{26 )and( 2{28 )into( 2{42 )resultsintoapproximationsfordeep-and shallow-water,whicharerespectivelygivenby E 0 = g 2 H 2 T 2 b 16 ; (2{43) E = g 3 = 2 H 2 h 1 = 2 Tb 8 : (2{44) Thetime-rateofchangeofenergytransmissionperunitareanor maltothe rowdirection,i.e.theenergyrux,canbecomputedbyusingth eequationofenergy conservationforanirrotationalrow,i.e.asimpliedformof Bernoulli'sequation.The followingassumptionsweremadetousethesimpliedexpression:t hepressureatthe 29

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freesurfaceiszero,andthekineticenergyterm( V 2 = 2)isnegligibleforwaveswithsmall steepness( H=<< 1).ThesimpliedBernoulli'sequationisgivenby @ @t + gz + p =0 : (2{45) Theonlytime-dependentpartin( 2{45 )istherstterm,thatrepresentsboththe unsteadykineticenergyandthedynamicpressure.Therefore,t heenergyruxisgivenby theproductofthedynamicpressureandtheruidveloctiy,asfo llows @ @t V = @ @t r : (2{46) Thetotalenergyruxcanbecomputedsubstituting( 2{23 )into( 2{46 )andintegrating overboththenormalarea( bh )andthewaveperiod( T ).Theresultingexpressionis consideredthewavepower[ 98 ],andisgivenby P = b 1 T Z T 0 Z 0 h @ @t r dzdt = gH 2 cb 16 2 kh sinh(2 kh ) +1 i : (2{47) Deep-andshallow-waterapproximationsarerespectivelygiv enby P 0 = g 2 H 2 Tb 32 i ; (2{48) P = g 3 = 2 H 2 h 1 = 2 b 8 i : (2{49) 2.5WaveMechanicsNumericalExample Letussupposethatawaveistravelinginwaterthatis10mdeep( h =10m). Measurementsindicatethatthewavehasaperiodof3seconds( T =3s)andan amplitudeof2meters( H =2m). 2.5.1KinematicProperties IterativelysolvingEquation( 2{25 )yields =14 : 0367m. Then,equations( 2{20 ),( 2{22 )and( 2{21 ),canbeusedtocomputetheother kinematicspropertiesofthewave,whichgives 30

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k =0 : 4476rad/m, c =4 : 6789m/s, =2 : 0944rad/s. Theratio h= =0 : 7124,indicatesthatwehavedeepwaterconditions. 2.5.2WaterParticlesMotion ThewavecanberepresentedgraphicallyusingEquation( 2{36 ).Figure 2-4 shows adiscreteplotofwaterparticlesforthegivenwaveataparti cularinstantoftime. Thesmallclosedpathisdescribedbythemotionofonesurfacepar ticle.Aspredicted byAiry'swavetheory,fordeepwaterconditionssuchpathisci rcular(compareto Figure 2-2 A). 0 5 10 15 20 25 30 -10 -5 0 Distance, x (m)Depth, z (m) Figure2-4.Examplewaterparticlesmotion. 2.5.3WaveEnergy Forthegivenwave,thetotalenergyandpowerperunitofwave front,i.e. b =1m, canbecomputedusingequations( 2{43 )and( 2{48 ).Substitutingthecorrespondingvalues insuchequationsyields E =70 : 8471kJ, P =11 : 8078 i kW. 31

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2.6Wave-StructureInteraction 2.6.1HeavingandPitchingBodyMotions Themotionofarigidbodyinthethree-dimensionalspaceischa racterizedbysix componentscorrespondingtosixdegreesoffreedom(6-dof).I noceanengineeringitis commontodenesuchmotionsasdepictedinFigure 2-5 andnamedasfollows: 1. Surge:displacementalong x axis. 2. Sway:displacementalong y axis. 3. Heave:displacementalong z axis. 4. Roll:rotationabout x axis. 5. Pitch:rotationabout y axis. 6. Yaw:rotationabout z axis. Foranaxisymmetric(oranothernot-elongated)body,e.g.an sphere,thenumbers1 and2(and4and5)areambiguous(i.e.theycanbearbitrarily interchanged).However, thisambiguityisremovedbydeningthe x directionasthewavepropagationdirection. Figure2-5.Rigidbodywithsixdegreesoffreedom. Sincethisworkconsidersatwo-dimensionallinearmodelforo ceanwaves,onlythree degreesoffreedom(3-dof)areofinterest.Themotionisanal yzedinthe x z plane, hence,onlysurge,heaveandpitchmotionsareconsidered.Fig ure 2-6 showstwoofsuch congurationsinwhicharoatingbodycanbeexcitedbyocean waves. 32

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APurelyheavingroat. BPurelypitchingroat. Figure2-6.Floatingbodiesundergoingheavingorpitching motion[ 70 ]. Assumingasinusoidalwaveprole,wecanseethatwhen = L ,noheavingmotion willoccursinceacrestandatroughofawaveactsimultaneouslyo verthebodyasin Figure 2-7 B.Therefore,thereisnotverticalforceactingontheroata ndnoheaving motionswhen L = N;N =1 ; 2 ; 3 ; (2{50) Conversely,whenanextracrest(ortrough)actoverthebody,t hereisanetvertical forcethatcreatesheavingmotions.Heavingcanbeexpectedwh en L = N 2 ;N =1 ; 3 ; 5 ; (2{51) ThebodydepictedinFigure 2-6 Bisallowedtorotate(pitch)aboutitscenterof gravity.Aminimumpitchingmomentisexperiencedbythebod yunderthecondition describedby( 2{51 )when N =3.Ontheotherhand,amaximumpitchingmomentoccurs when 2{50 issatised. APureheavingcondition. BPurepitchingcondition. Figure2-7.Floatingbodies:pureheavingandpitchingcondi tions[ 70 ]. 33

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2.6.2HeavingEquationofMotion Thelawofconservationoflinearmomentumstatesthattherate ofchangeofthe linearmomentumofthesysteminaninertialreferenceframe,i sequaltothetotal externalforceactingonthesystem.Forapurelyheavingroati ngbodytheequationof motionisgivenby[ 98 ] m d 2 z dt 2 = a wz d 2 z dt 2 b r z dz dt b v z dz dt dz dt N b p z dz dt gA wp z Nk s z + F zo cos( !t + z ) : (2{52) McCormick[ 98 ]deneseachtermintherighthandsideof( 2{52 )asfollows: 1. Inertialreactionforceofthewater,where a wz istheaddedmass. 2. Radiationdampingforce,whereb r z isthedampingcoecient. 3. Viscousdampingforce,where b v z istheviscousdampingcoecient. 4. Dampingduetopowertakeo,whereb p z isthepowertake-ocoecient. 5. Hydrostaticrestoringforce,where A wp isthewaterplaneareawhenthebodyisat rest. 6. Mooringrestoringforce,where k s istheeectivemooringspringconstantofeach line,and N isthenumberoflines. 7. Waveinducedverticalforce,where F zo istheforceamplitude, =2 =T isthe circularwavefrequency( T isthewaveperiod),and z isthephaseanglebetween thewaveandthewave-inducedforce. ThepowerNontheviscousdampingtermdependsontherowregim e,N=0for laminarrowandN=1forturbulentrow.WhenN=1theequationo fmotionis nonlinear.Inthiswork,weassumethattherowislaminarandN= 0.Therefore, b v z is replacedbyalineardampingcoecientb v z .Reorganizingtermsin( 2{52 ),thelinearized equationofmotionisgivenby ( m + a wz ) d 2 z dt 2 +(b r z +b v z +b p z ) dz dt +( gA wp + Nk s ) z = F zo cos( !t + z ) : (2{53) isthedensityofseawater(1030kg/m 3 ), g istheaccelerationduetogravity(9.8m/s 2 ). 34

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2.6.2.1MassandAddedMass Themassofaroatmustbeequaltothemassofdisplacedwater.Hence ,themassfor arectangularroatingbodycanbecomputedas m = dA wp ; (2{54) wherethewaterplaneareaisgivenby A wp = LB: (2{55) Theaddedmassistheinertiaaddedtoasystemduetothepresenceo fastructurein amovingruidorduetothemotionofastructureinastationaryr uid.Themagnitudeof suchmassisproportionaltotheinertialreactionforceonthe body[ 98 ].Forrectangular solidssuchtheonesshowninFigure 2-6 ,theadded-massandtheadded-massmomentof inertiaaregivenrespectivelyby[ 70 ] a wz = K m LB 2 4 ; (2{56) A w = K I L 4 B 16 : (2{57) Thevaluesforthecoecients K m and K I canbefoundgraphicallyfromFigure 2-8 0 5 10 15 20 0.5 0.75 1 K mB=d AAddedmasscoecient. 2 4 6 8 10 0.05 0.075 0.1 0.15 K IL=d BAdded-massmomentofinertiacoecient. Figure2-8.Addedmassandadded-massmomentofinertiacoecie ntsforarectangular body[ 70 ]. 35

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2.6.2.2RadiationandViscousDamping Duetobodymotionsneartotheruidfreesurface,wavesarealso created.Such wavestakeenergyawayfromthebody.Thisenergylossiscalle dradiationdamping.For rectangularsolidssuchtheonesshowninFigure 2-6 ,theradiationdampingcoecientis givenby[ 98 ] b r z =b 0r L = g 2 3 R 2Z L; (2{58) whereb 0r istheradiationdampingcoecientperunitlength,andR Z istheratioofthe radiated-waveamplitudeandthebodymotionamplitude,and isgivenby R Z =2 e 2 g d sin w 2 g B 2 : (2{59) Boththeaddedmassandtheradiationdampingcoecientsvary withfrequency. However,McCormick[ 98 ]statesthatafrequency-invariantvaluefortheaddedmassgi ven by( 2{56 )andtheexpressionfortheradiationdampingcoecientgiven in( 2{58 )match measurementsdonebyVugts[ 102 ]forabodywitharectangularsection.Figure 2-9 shows thevariationofradiationdampingcoecientforaheavingr ectangularsectionwhenthe draftisthehalfofthelength,i.e. d = L= 2. 0 0.5 1 1.5 0 1 b rz ( d p 2 Lg ) s L 2 g Figure2-9.Non-dimensionalradiationdampingcoecientfor aheavingrectangular section. Thenonlinearviscousdampingcoecientin( 2{52 )isgivenby b v z 1 2 C d A d ; (2{60) 36

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where A d istheprojectedarea,i.e. A wp forarectangularbody,and C d isthedrag coecient.Forarectangularbodyinheavingmotioninlamin arrow,thedragcoecient is C d =1 : 2[ 103 ]. McCormick[ 98 ]statesthatthenonlinearandthelinearviscousdampingcoe cients in( 2{52 )and( 2{53 )arerelatedbythefollowingexpression b v z = 8 3 Z o b v z ; (2{61) where Z o istheheavingamplitudewhichiscomputedasinSection 2.6.4 .Since Z o dependsonb v z ,anonlinearequationmustbesolvedfor Z o tocomputethelinearviscous dampingcoecient.2.6.3Wave-InducedForces Thewave-inducedheavingforceonarectangularroatinalin earwaveisgivenby[ 70 ] F z ( t )= F zo cos( !t ) ; (2{62) where F zo = gHB 2 e 2 d +1 sin L : (2{63) Thewave-inducedmomentonarectangularroat(assumingthedr aft d tobe constant)inalinearwaveisgivenby[ 70 ] M ( t )= M o sin( !t ) ; (2{64) where M o = gHB 4 e 2 d +1 sin L L cos L : (2{65) Thephaseanglebetweenthewaveandthewave-inducedforce/m oment, z ,isequal tozero( z =0)forabodythatissymmetricaboutthe x z and y z planes[ 70 ]. 37

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2.6.4SteadyStateSolutionoftheHeavingEquation Equation( 2{53 )isasecond-orderordinarydierentialequationwhosesteady -state solutioncanbefoundeasilyandisgivenby z ss ( t )= Z o cos( !t + z z ) ; (2{66) where Z o = F zo ( gA wp + Nk s ) s 1 2 2 n 2 + 2 n (b r z +b v z +b p z ) b cz 2 cos( !t + z z ) : (2{67) In( 2{66 )thenaturalheavingfrequencyisgivenby w nz = 2 T nz = r gA wp + Nk s m + a wz ; (2{68) where T nz isthenaturalheavingperiod;thecriticaldampingcoecie ntisgivenby b cz =2 q ( m + a wz )( gA wp + Nk s );(2{69) andthephaseanglebetweentheforceandmotionisgivenby z =arctan 2664 2 n (b r z +b v z +b p z ) b cz 1 2 2 n 3775 : (2{70) 38

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CHAPTER3 TENSEGRITYMECHANISM Salteretal.[ 104 ]statedthatmostwaveenergyharvestersmusthave,amongothers, thefollowingsubsystems: Slowprimarydisplacingelement,suchasaroatorbuoy. Connectinglinkagetotransmitthewave-generatedforces(a nalyzedinthiswork). Electricalgenerators,electricalnetwork,transformersan dswitchgear. 3.1MorphologyDenition Theproposedtensegritymechanismisbasedonthetwo-dimensiona l\X-frame" morphologyproposedbyKennethSnelsonin[ 31 ],Figure 1-3 A.Themechanismcomprises fourmembersintensionandtwomembersincompression.Thememb ersincompression canbereplacedbytwobarsconnectedbyprismaticjointswhic hrepresenttheelectrical generators.Twoofthetieshaveaveryhighmodulusofelastici ty(themechanismbase andtheelementexcitedbyoceanwaves)withrespecttotheothe rtwoties.Therefore, thedeformationsofthebaseandthebuoyantelementarenegli gibleandthelateralties arethetwodeformablemembersundertensionthatarenecessary tokeepthetensegrity conguration.Figure 3-1 depictstheconceptofatensegrity-basedwaveenergyharveste r. Figure3-1.Conceptofawaveenergyharvesterbasedontensegri tysystems. 39

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Thekinematicdiagramoftheproposedtensegritymechanismissh owninFigures 3-2 and 3-3 .Themechanismparametersfortheselectedkinematicchainar elistedin Table 3-1 0 1 1 2 2 3 3 4 4 5 5 6 Figure3-2.Kinematicdiagramofthemechanism.Table3-1.Mechanismparametersforkinematicchain. Linklength,mTwistangle,degJointoset,mJointangle,deg a 1 =0 1 =90 0 S 1 =0 0 1 = variable a 2 =0 2 =90 1 S 2 = variable 1 2 =180 a 3 = L 3 =0 2 S 3 =0 2 3 = variable a 4 =0 4 =90 3 S 4 =0 3 4 = variable a 5 =0 5 =90 4 S 5 = variable 4 5 =180 a 6 = L 0 6 =0 5 S 6 =0 5 6 = variable 3.2PositionAnalysis 3.2.1ForwardPositionAnalysis Theforwardpositionanalysisallowsonetodeterminetheposit ionandorientationof oneofthelinksforaspeciedsetofjointvariables.Theproble misstatedasfollows: Given:theconstantmechanismparameters a 3 a 6 and L pxz (positionofpoint p on thetopplatform);andthesetofjointvariables, 1 S 2 4 S 5 and 0 1 Find:thepositionandtheorientationofthetopplatform, x m z m and 40

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n Figure3-3.Vectordiagramofthemechanism. Letusdenethefollowingvectors S 2 = 1 S 2 cos( 0 1 ) i + 1 S 2 sin( 0 1 ) k ; (3{1) a 6 = a 6 i : (3{2) FromFigure 3-3 ,thefollowingvectorloopequationcanbewritten L B = S 2 a 6 : (3{3) Equation( 3{3 )permitstoevaluate L B and B .Theangle 3 canbecomputedusing thecosinelaw 3 =arccos L 2B + 4 S 2 5 a 23 2 L B 4 S 5 : (3{4) Hence, 5 6 = B + 3 + : (3{5) Then,thevector S 5 isgivenby S 5 = 4 S 5 cos( 5 6 ) i + 4 S 5 sin( 5 6 ) k : (3{6) 41

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FromFigure 3-3 ,othertwovectorloopequationscanbewrittenas L A = a 6 S 5 ; (3{7) a 3 = L A S 2 : (3{8) Equation( 3{8 )permitstoevaluatetheorientation .Therearetwosolutionsfor 3 however,onlytheoneintherstquadrantistakenforthismec hanism'sconguration. Finally,thepositionofthepoint p =[ x m 0 z m ] T canbecomputedas p = S 2 + L pxz L a 3 : (3{9) Thelengthofthespringsdenotedbyvectors L A and L B canbecomputedintermsof thepositionandorientation( = + )ofthetopplatform,when L pxz = L= 2,asfollows L A = ( x m L= 2cos ) 2 +( z m L= 2sin ) 2 1 2 ; (3{10) L B = ( x m + L= 2cos L 0 ) 2 +( z m + L= 2sin ) 2 1 2 : (3{11) 3.2.2ReversePositionAnalysis Thereversepositionanalysisallowsonetodeterminethepositi onandorientationof oneofthelinksforaspeciedsetofjointvariables.Theproble misstatedasfollows: Given:theconstantmechanismparameters a 3 a 6 and L pxz ;andthepositionandthe orientationofthetopplatform, x m z m and Find:thesetofjointvariables, 1 S 2 4 S 5 and 0 1 Letusdenethefollowingvector a 3 = a 3 cos( ) i + a 3 sin( ) k : (3{12) FromFigure 3-3 ,thefollowingvectorloopequationscanbewritten S 2 = p L pxz a 3 ; (3{13) L A = S 2 + a 3 ; (3{14) 42

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S 5 = a 6 L A : (3{15) Equation( 3{13 )permitstoevaluate 1 S 2 and 0 1 ,and( 3{15 )permitstoevaluate 4 S 5 3.2.3NumericalExample Lettheconstantmechanismparametersbe L =3m, L 0 =4 : 5mand L pxz =1 : 5 m.Toverifythepositionanalysis,adesiredpathcanbedenedin theCartesianspace. Then,thevaluesofthejointvariablesthatsatisfythepathca nbefoundusingthereverse kinematics.Finally,theforwardkinematicscanbeusedtocom putethepointsinthe Cartesianspacethataregeneratedwitheachsetofjointvariab les.Figure 3-4 showsthe mechanisminthenalpositionandthepath.Table 3-2 showsthenumericalresultswhich indicatethatthereverseandforwardkinematicsfunctionsw orkasexpected. -1 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Positionin x (m)Positionin z (m) Figure3-4.Positionanalysisnumericalexample.Table3-2.Positionanalysisnumericalexampleresults. OriginalpathReversekinematicsForwardkinematics x md (m) z md (m) d (deg) 1 S 2 (m) 4 S 5 (m) 0 1 (deg) x m (m) z m (m) (deg) 1.502.00170 3.44825.015530.2969 1.50002.0000170.0000 1.752.25175 3.87514.866433.1537 1.75002.2500175.0000 2.002.50180 4.30124.717035.5377 2.00002.5000180.0000 2.252.75185 4.72424.569537.5735 2.25002.7500185.0000 2.503.00190 5.14284.426739.3445 2.50003.0000190.0000 43

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3.3VelocityAnalysis Thevelocitystateisdenedasasetofparametersfromwhichth evelocityofany body/pointofthelinkagecanbedeterminedrelativetoaref erencebody[ 105 ].Then, usingthisconcept,thevelocityanalysisallowsonetoobtain therelationshipsbetweenthe velocityofanybody/pointofthemechanismandthevelocitie softhejointvariables.Rico etal.[ 106 ]andCraneetal.[ 105 ]presentedtheconceptofvelocitystateusingthetheory ofscrews,developedbySirRobertStawellBall[ 107 ]. Sincethepositionanalysisiscomplete,thedirectionsofthe unitvectors i s i +1 along eachaxisaswellthecoordinatesofonepoint r i oneachjointaxisareknown.Then, Pluckercoordinatesofthelinesalongtherevolutejointa xesaregivenby f i s i +1 ; i s i +1 OL g = f i s i +1 ; r i i s i +1 g : (3{16) Thecoordinatesofthelineatinnityassociatedwiththeprism aticjointsaregiven by f i s i +1 ; i s i +1 OL g = f 0 ; r i i s i +1 g : (3{17) Then,using( 3{16 )and( 3{17 ),thePluckercoordinatesofthelinesalongthejoint axesaregivenby 0 $ 1 = f 0 s 1 ; 0 s 1OL g = f s 1 ; 0 s 1 g ; (3{18) 1 $ 2 = f 0 ; 1 s 2OL g = f 0 ; s 2 g ; (3{19) 2 $ 3 = f 2 s 3 ; 2 s 3OL g = f s 3 ; S 2 s 3 g ; (3{20) 3 $ 4 = f 3 s 4 ; 3 s 4OL g = f s 4 ; L A s 4 g ; (3{21) 4 $ 5 = f 0 ; 4 s 5OL g = f 0 ; s 5 g ; (3{22) 5 $ 6 = f 5 s 6 ; 5 s 6OL g = f s 6 ; a 6 s 6 g : (3{23) Alltheunitvectorsin( 3{18 )to( 3{23 )areknownfromthepositionanalysisandare givenby: 44

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s 1 = s 3 = s 4 = s 6 =[ 0 10 ] T s 2 = S 2 j S 2 j s 5 = S 5 j S 5 j Sincethemechanismisaclosed-loopkinematicchain,bodies0 and6arethesame (i.e.thegroundofthemechanism).3.3.1ForwardVelocityAnalysis Theforwardvelocityanalysisallowsonetodeterminethevel ocitystateofoneofthe linksforagivensetofjointrates.Theproblemisstatedasfoll ows: Given:theconstantmechanismparameters a 3 a 6 and L pxz ;thesetofjointvariables, 1 S 2 4 S 5 and 0 1 ;andthesetofvelocitiesofthejointvariables 1 v 2 4 v 5 and 0 1 Find:thevelocitystateofthetopplatform 0 30 v 3 O T and_ x m ,_ z m and Thevelocityfortheclosedkinematicchaincanbewritteninsc rewformasfollows, [ 105 106 ]: 0 1 0 $ 1 + 1 v 2 1 $ 2 + 2 3 2 $ 3 + 3 4 3 $ 4 + 4 v 5 4 $ 5 + 5 6 5 $ 6 =0 : (3{24) Substituting( 3{18 )to( 3{23 )into( 3{24 )yields 0 1 2666666666666664 0 1 0000 3777777777777775 + 1 v 2 2666666666666664 000 s 2 x 0 s 2 z 3777777777777775 + 2 3 2666666666666664 0 1 0 s 2 z 0 s 2 x 3777777777777775 + 3 4 2666666666666664 0 1 0 L A z 0 L A x 3777777777777775 + 4 v 5 2666666666666664 000 s 5 x 0 s 5 z 3777777777777775 + 5 6 2666666666666664 0 1 000 a 6 x 3777777777777775 =0 (3{25) Equation( 3{25 )canbewrittenasa3x3systeminmatrixformasfollows 45

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266664 1 1 1 s 2 z L A z 0 s 2 x L A x a 6 x 377775 266664 2 3 3 4 5 6 377775 = 0 1 266664 1 00 377775 1 v 2 266664 0 s 2 x s 2 z 377775 4 v 5 266664 0 s 5 x s 5 z 377775 (3{26) Thesolutionof( 3{26 )givesthemagnitudesoftheangularvelocitiesbetween consecutivebodies.Then,thevelocitystateofthetopplatfor mcanbecomputedas 264 0 3 0 v 3 O 375 = 0 1 0 $ 1 + 1 v 2 1 $ 2 + 2 3 2 $ 3 : (3{27) Nowthevelocityofanypoint p onthetopplatform,whosepositionisrepresentedby r O P ,isgivenintermsofthevelocitystateby 0 v 3 p = 0 v 3 O + 0 3 r O P : (3{28) Equations( 3{26 ),( 3{27 )and( 3{28 )completetheforwardvelocityanalysisandallows onetocomputethevelocityofanypointinthetopplatformof themechanismforagiven setofvelocitiesofthejointvariables.3.3.2ReverseVelocityAnalysis Thereversevelocityanalysisallowsonetodeterminethesetof jointratesforagiven velocitystateofoneofthelinks.Theproblemisstatedasfollo ws: Given:theconstantmechanismparameters a 3 a 6 and L pxz ;thesetofjointvariables, 1 S 2 4 S 5 and 0 1 ;andthevelocitystateofthetopplatform 0 30 v 3 O T interms of_ x m ,_ z m and Find:thesetofvelocitiesofthejointvariables 1 v 2 4 v 5 and 0 1 Sincethethevelocityofthepoint p ( 0 v 3 p =_ x m i +_ z m k ),andtheangularvelocityof thetopplatform( 0 3 = j )areknown,theelementassociatedwithlinearvelocitiesint he velocitystatecanbecomputedusing( 3{28 )as 0 v 3 O = 0 v 3 p 0 3 r O P : (3{29) 46

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Equation( 3{27 )denesthevelocitystateofthetopplatform.Substituting( 3{27 ) into( 3{24 )yields 264 0 3 0 v 3 O 375 = 3 4 3 $ 4 4 v 5 4 $ 5 5 0 5 $ 0 : (3{30) Sincethevelocitystateofthetopplatformisknown,thejoin tvelocitiesofthe mechanismcanbecomputedfrom( 3{27 )and( 3{30 )asfollows 266664 10 1 0 s 2 x s 2 z 0 s 2 z s 2 x 377775 266664 0 1 1 v 2 2 3 377775 = 266664 0 3 0 v 3 x 0 v 3 z 377775 ; (3{31) 266664 10 1 L A z s 5 x 0 L A x s 5 z 0 377775 266664 3 4 4 v 5 5 6 377775 = 266664 0 3 0 v 3 x 0 v 3 z 377775 : (3{32) Equations( 3{31 )and( 3{32 )completethereversevelocityanalysisandallowsone tocomputethevelocitiesofthejointvariables(andallthe jointvelocities)foragiven velocitystateofthetopplatform.3.3.3NumericalExample Lettheconstantmechanismparametersbe L =3m, L 0 =4 : 5mand L pxz =1 : 5m. Then,wedeneacircularpathintheCartesianspace,centered inthepoint( x m ;z m )= 2 : 5 i +3 k witharadiusof0.5meters,Fig 3-5 A.Wewantthemechanismtofollowsuch pathin5secondswithconstantcelerityandxedorientation( =190deg).Fig 3-5 B showsthecorrespondingdesiredpositionandvelocityofthepoi nt p (centeredinthetop platformofthemechanism),whicharecomputedusingthefollo wingparametricequations (parameter t 0 )ofthecircle ( x m ;z m )=[ x 0 + r cos( t 0 )] i +[ z 0 + r sin( t 0 )] k : (3{33) Thevelocityof p canbecomputedusingthederivativewithrespecttothetimeof ( 3{33 ). 47

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Finally,thereversepositionandthereversevelocityanalysis equationscanbeused tocomputethecorrespondingvaluesofthejointvariablesth atwouldmakethepoint p followthedescribedCartesianpath. Figure 3-6 showstheresultswiththejointvariablesvaluesandtheircor responding velocities. -1 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Positionin x (m)Positionin z (m) AMechanismandCartesianpath. 0 1 2 3 4 5 0 2 4 Time(s)Position(m) x z 0 1 2 3 4 5 -1 0 1 Time(s)Velocity(m/s) x z BPositionandvelocityofpoint p Figure3-5.Velocityanalysisnumericalexample. 0 1 2 3 4 5 3.5 4 4.5 5 5.5 Time(s)Position(m) 1 S 2 4 S 5 0 1 2 3 4 5 -2 -1 0 1 Time(s)Velocity(m/s) 1 v 2 4 v 5 APositionandvelocityfor 1 S 2 4 S 5 0 1 2 3 4 5 35 40 45 50 Time(s)0 1 (deg) 0 1 2 3 4 5 -0.2 0 0.2 Time (s)0 1 (rad/s) BPositionandvelocityfor 0 1 Figure3-6.Velocityanalysisnumericalexampleresults. 48

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3.4EquationofMotion ThedierentialequationofmotioncanbeobtainedusingtheL agrangianapproach. Thisequationrepresentsthedynamicbehaviorofthemechani sm,andisgiven,intermsof thegeneralizedcoordinatesanditsderivatives( q ; q ; q ),by M ( q ) q + N ( q ; q )+ G ( q )= Q ; (3{34) where: M ( q )isdenedastheinertiamatrix, N ( q ; q )= V ( q ; q )+ F ( q )accountsforCoriolis/centripetaleectsandfriction, G ( q )iscalledthegravityvector,and Q isthegeneralizedforcevector. TheLagrange'sequationofmotionisgiven,intermsofthege neralizedterms,by [ 108 109 ] d dt @E K @ q @E K @ q + @E P @ q = Q ; (3{35) where: E K isthekineticenergy, E P isthepotentialenergy, Q isthegeneralizedforcevector, q isthevectorofgeneralizedcoordinates, q isthevectorofgeneralizedcoordinatesderivatives(velo cities). 3.4.1Assumptions Thefollowingassumptionsaremadeforthetensegritymechanism Thelinksofthemechanism,exceptforthetopplatform,arema ssless.Itisassumed thattheycanbedesignedtohaveneutralbuoyancy. Theelastictiesaremassless. Thestinessofeachtieisconstant,i.e.theybehaveaslinearsp rings. 49

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Thevectorofgeneralizedcoordinatesisdenedas q = 24 q 1 q 2 q 3 35 = 24 x m z m 35 : (3{36) 3.4.2KineticEnergy Sincethelinksofthemechanismareconsideredmassless,thekinet icenergyofthe mechanismisallconcentratedatthetopplatform.Thekineti cenergyofthetopplatform duetosurge,heaveandpitchmotionsisgiven,intermsoftheg eneralizedcoordinates,by E K = 1 2 ( m + a wx )_ q 1 2 + 1 2 ( m + a wz )_ q 2 2 + 1 2 ( I y + A w )_ q 3 2 ; (3{37) where m isthemass, I y = m ( L 2 + Z 2 ) = 12isthemassmomentofinertia, a wx =1 = 4 d 2 B and a wz aretheaddedmassesduetosurgingandheaving,and A w istheadded-mass momentofinertiaduetopitching. Therstandtermof( 3{35 )isobtainedbytakingthederivativeof( 3{37 )with respecttothegeneralizedcoordinatesas @E K @ q = 266664 ( m + a wx )_ q 1 ( m + a wz )_ q 2 ( I y + A w )_ q 3 377775 ; (3{38) andthen,takingthederivativeof( 3{38 )withrespecttothetimegives d dt @E K @ q = 266664 ( m + a wx ) q 1 ( m + a wz ) q 2 ( I y + A w ) q 3 377775 : (3{39) Equation( 3{39 )canbewritteninmatrixformas d dt @E K @ q = M q = 266664 ( m + a wx )00 0( m + a wz )0 00( I y + A w ) 377775 q : (3{40) 50

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From( 3{37 )weseethatthekineticenergyofthetopplatformdoesnotdep endon theposition.Hence,thesecondtermof( 3{35 )isgivenby @E K @ q = 0 : (3{41) 3.4.3PotentialEnergy Thepotentialenergyduetoheavingandpitchingmotionsoft hetopplatformare describedbyMcCormick[ 70 ]as E Pz = 1 2 gA wp q 2 2 ; (3{42) E P = 1 2 Cq 2 3 ; (3{43) where A wp isthewaterplaneareaand C istherestoringmomentconstantwhichis dened,forabottom-ratbodyintermsofthedraft,as C = gI y d : (3{44) Thepotentialenergyoftheelastictiesisgivenby E Pk = 1 2 k A ( L A L A 0 ) 2 + 1 2 k B ( L B L B 0 ) 2 ; (3{45) where L A 0 L B 0 ;and k A k B ;arethefreelengthsandthestinessesofthetiesrespectively Substituting( 3{10 )and( 3{11 )into( 3{45 )gives E Pk = 1 2 k A ( q 1 L= 2cos q 3 ) 2 +( q 2 L= 2sin q 3 ) 2 1 2 L A 0 2 + 1 2 k B ( q 1 + L= 2cos q 3 L 0 ) 2 +( q 2 + L= 2sin q 3 ) 2 1 2 L B 0 2 : (3{46) Thetotalpotentialenergyofthemechanism, E P = E Pz + E P + E Pk ,isgivenby E P = 1 2 gA wp q 2 2 + 1 2 Cq 2 3 + 1 2 k A ( q 1 L= 2cos q 3 ) 2 +( q 2 L= 2sin q 3 ) 2 1 2 L A 0 2 + 1 2 k B ( q 1 + L= 2cos q 3 L 0 ) 2 +( q 2 + L= 2sin q 3 ) 2 1 2 L B 0 2 : (3{47) 51

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Thelasttermof( 3{35 )iscomputedtakingthederivativeof( 3{47 )withrespectto eachgeneralizedcoordinateasfollows: @E P @q 1 = k A ( q 1 L= 2cos q 3 ) 2 +( q 2 L= 2sin q 3 ) 2 1 2 L A 0 ( q 1 L= 2cos q 3 ) ( q 1 L= 2cos q 3 ) 2 +( q 2 L= 2sin q 3 ) 2 1 2 + k B ( q 1 + L= 2cos q 3 L 0 ) 2 +( q 2 + L= 2sin q 3 ) 2 1 2 L B 0 ( q 1 + L= 2cos q 3 L 0 ) ( q 1 + L= 2cos q 3 L 0 ) 2 +( q 2 + L= 2sin q 3 ) 2 1 2 ; (3{48) @E P @q 2 = gA wp q 2 + k A ( q 1 L= 2cos q 3 ) 2 +( q 2 L= 2sin q 3 ) 2 1 2 L A 0 ( q 2 L= 2sin q 3 ) ( q 1 L= 2cos q 3 ) 2 +( q 2 L= 2sin q 3 ) 2 1 2 + k B ( q 1 + L= 2cos q 3 L 0 ) 2 +( q 2 + L= 2sin q 3 ) 2 1 2 L B 0 ( q 2 + L= 2sin q 3 ) ( q 1 + L= 2cos q 3 L 0 ) 2 +( q 2 + L= 2sin q 3 ) 2 1 2 ; (3{49) @E P @q 3 = Cq 3 + k A ( q 1 L= 2cos q 3 ) 2 +( q 2 L= 2sin q 3 ) 2 1 2 L A 0 [( q 1 L= 2cos q 3 ) L= 2sin q 3 ( q 2 L= 2sin q 3 ) L= 2cos q 3 ] ( q 1 L= 2cos q 3 ) 2 +( q 2 L= 2sin q 3 ) 2 1 2 + k B ( q 1 + L= 2cos q 3 L 0 ) 2 +( q 2 + L= 2sin q 3 ) 2 1 2 L B 0 [ ( q 1 + L= 2cos q 3 L 0 ) L= 2sin q 3 +( q 2 + L= 2sin q 3 ) L= 2cos q 3 ] ( q 1 + L= 2cos q 3 L 0 ) 2 +( q 2 + L= 2sin q 3 ) 2 1 2 : (3{50) Equations( 3{48 ),( 3{49 )and( 3{50 )canbewritteninmatrixformas @E P @ q = G ( q )= @E P @q 1 @E P @q 2 @E P @q 3 T : (3{51) Equations( 3{40 ),( 3{41 )and( 3{51 )representtherstandthirdtermsof( 3{34 ). 52

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3.4.4FrictionVector AsdescribedinSection( 2.6.2.2 ),themostimportantcharacteristicintheruid-structure interactionphenomenonisthepresenceofdampingduetovisco useectsandradiationof waves.Viscousdampingisconsideredinbothsurgingandheavingm otions,andradiation dampingisconsideredforheavingandpitchingmotions.Thefr ictionvectoristhengiven by F ( q )= 266664 b v x q 1 (b r z +b v z )_ q 2 b r q 3 377775 ; (3{52) wherethedampingcoecientsintheheave/surgedirectionsa regivenby( 2{58 )and ( 2{61 ),andtheradiationdampingcoecientduetopitchingmotio nisgivenby b r =b 0r L 3 12 ; (3{53) whereb 0r istheradiationdampingcoecientperunitlengthgivenin( 2{58 ). 3.4.5GeneralizedForces Thegeneralizedforcevectorisformedbythewave-inducedf orcesthatactoverthe generalizedcoordinates.Sincethegeneralizedcoordinate swerechosentodescribedirectly themotionoftheroat,thegeneralizedforcevectorisgiven by Q ( t )= 266664 0 F zo cos( !t ) M o sin( !t ) 377775 : (3{54) Thecompleteequationofmotionisthengivenby 266664 m + a wx 00 0 m + a wz 0 00 I y + A w 377775 q + 266664 b v x q 1 (b r z +b v z )_ q 2 b r q 3 377775 + 2666664 @E P @q 1 @E P @q 2 @E P @q 3 3777775 = 266664 0 F zo cos( !t ) M o sin( !t ) 377775 : (3{55) 53

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CHAPTER4 OCEANWAVEENERGYHARVESTING 4.1ElectricalGeneratorsforWaveEnergyHarvesting Severalworkshavebeendonewithelectricalgeneratorsfor waveenergyharvesting. BakerandMuller[ 110 ]andMuller[ 111 ]investigatedtheconceptofdirectdrivewave energyconverters.Salteretal.[ 104 ]describedarangeofdierentcontrolstrategies forwaveenergypowerconversionmechanisms.Danielsson[ 112 ]designedathree-phase permanentmagnetlineargeneratorfordirectcouplingtoar oatingbuoy.Bakeretal. [ 113 ]outlinedtheperformanceandmodelingofaprototypelinea rtubularpermanent magnetmachinewithanaircoredstator.Polinderetal.[ 114 ]describedthemain characteristicsoftheArchimedesWaveSwing.Thorburnetal. [ 115 ]presenteddierent topologiesfortheelectricalsystemtransmittingpowertothe grid.MuellerandBaker [ 116 ]investigatedtheissuesassociatedwithconvertingtheenergyp roducedbymarine renewableenergyconvertersusingdirectdriveelectricalp owertake-o.Leijonetal. [ 117 ]presentedanovelapproachforelectricpowerconversiondisc ussingalsothe economicalandsomeenvironmentalconsiderations.Rhinefran ketal.[ 118 ]described theresearch,design,constructionandprototypetestingproce ssofanoveloceanenergy directdrivepermanentmagnetlineargeneratorbuoy.Danie lsson[ 119 ]studiedthe electromagneticproperties,builtalaboratoryprototypea ndanalyzedtheperformanceof linearsynchronouspermanentmagnetgenerators.Mulleretal .[ 120 ]describeddierent powertakeomechanismsanddescribedhowsomedisadvantagessho wnbyconventional rotarygeneratorscanbeovercomewithdirectdrivesystems.Sz aboetal.[ 121 ]proposeda novelmodularpermanentmagnettubularlineargenerator,a nalyzedbymeansofnumeric eldcomputations.Trapanese[ 122 ]performedtheoptimizationofapermanentmagnet lineargeneratordirectlycoupledtoseawaves.Elwoodetal.[ 123 ]developedahybrid numerical/physicalmodelingapproachforthedesignofa10kW energyconversionsystem. Liuetal.[ 124 ]presentedananalyticalmodelforpredictingtheelectroma gneticand 54

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electromechanicalcharacteristicsofaslotlesstubularline argenerator.Ruellanetal.[ 125 ] presentedadesignmethodologyfortheall-electricsolutiona daptedtoSEAREV.Chenget al.[ 126 ]presentedastudyofamulti-polemagneticgeneratorforener gyharvestingatlow frequencies. Sincethisresearchinvolvestherstanalysisofatensegritysyst emforenergy harvestingundertheinruenceofseawaves,theworkisconcentr atedintothebehavior oftheconnectinglinkagewhichtransmitsthewave-generate dforcestotheelectrical generators.Therefore,theelectricalmodelisnotincluded andisconsideredasdamping duetopowertakeo,assuggestedin[ 123 ]anddescribedin( 2{52 ),whereb p z isthe powertake-odampingcoecient.Theeciencyofpermanent magnetlineargenerators usedinoceanwaveenergyconversionisabout75-85%.[ 70 112 112 ]. 4.2SeaStateSelection Althoughthebestwaveclimatesarefoundinzonesfrom30to60d egreeslatitude asstatedinSection 1.1.2 ,attractivewaveclimatesarefoundalsowithin 30degrees latitude,wherethelowerpowerleveliscompensatedbysmalle rpowervariability. Additionally,thereareseveralisolatedplacesthatcannotbe connectedtocontinental powergridsinalotofcountriesaroundtheworld,makingoce anenergyafeasible renewableenergysource.Therefore,waveenergycanbeespeci allyusefulforsmall communitieslivingnearshoreorinislands,avoidingthetransp ortationandutilizationof fossilfuels. Osorioetal.[ 127 ]developedaroadmapforharnessingmarinerenewableenergy forColombia,pointingoutthataprofoundknowledgeofthea vailableresourceand oceanographicconditionsisrequired.Inthatsense,Ortegae tal.[ 128 ]designeda methodologyforestimatingwavepowerpotentialinplacesla ckinginstrumentationby usingreanalysiswindsandwavegenerationmodels. Ortega[ 129 ]performedastudyontheexploitationofwaveenergyfor IslaFuerte ,a smallColombianisland,locatedintheCaribbeanSeathatdoes nothaveaccesstoelectric 55

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powerfromthenationalgridandgeneratesitspowerusingfossi lfuels.Thewaveavailable resourceanditstimebehaviorwasidentiedusingthirdgener ationwavegeneration models,withbathymetriesandreanalysiswindsasinputs.Orte gaselectedaplacewith coordinates9.408 Nand76.180 W,whosedepthisaroundthirtymeters,i.e. h =30m. Suchplacewaschosentakinginaccountenvironmentalissuesin ordertoprotectthecoral reef,whichisfoundevennearplaceswithhigherwavepowera vailability,Figure 4-1 A IslaFuerte .TakenfromGoogleEarth TM LATLON 25303 54 04 55 04 54 03 53 02 5105101 520102 01 55 -76.22 -76.21 -76.2 -76.19 -76.18 -76.17 -76.16 -76.15 9.34 9.35 9.36 9.37 9.38 9.39 9.4 9.41 9.42 9.43 BBathymetryof IslaFuerte [ 129 ]. Figure4-1.Locationandbathymetryof IslaFuerte. 4.2.1WavePowerandWaveKinematicProperties Attheselectedlocationthewaveheightandperiodvaryfrom0 .2to1.2meters andfrom3to10secondsrespectively;andthejointprobabilit yfor H and T showsthat H 2 [0 : 40 : 6]mand T 2 [46]s[ 129 ].Using( 2{48 )with h =30m,theavailablepower permeterofwavefrontcanbecomputedovertherangeof T and H .FromFigure 4-2 ,the maximumavailablepower, P =2 : 125kW/m,isfoundat H =0 : 6m, T =6s. Thewavepropertiesfor h =30m, H =0 : 6m, T =6scanbefoundbysolving ( 2{25 ),andsubstitutingtheresultinto( 2{20 ),( 2{22 )and( 2{21 ),yields =56 : 1081m, k =0 : 1120rad/m, c =9 : 3514m/s, =1 : 0472rad/s. Theratio h= =0 : 5347 1 = 2,indicatesthatwehavedeepwaterconditions. 56

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4 5 6 0.4 0.5 0.6 1 1.5 2 T (s) H (m) P (kW/m) Figure4-2.Wavepowervariationfortheseastateat9.408 N,76.180 W. 4.3DirectDriveHeavingSystem Severaldirectdrivewaveenergyconvertersbehaveasapure lyheavingbody, connectedtoelectricalgenerators,see[ 110 111 117 118 121 122 ]forreference.The dimensionsoftheheavingroat,astheoneshowninFigure 2-6 A,are L =2m, Z =0 : 5m, B =1m,and d =0 : 25m. Thecoecientsof( 2{52 )canbecalculatedusingequationsdescribedinSection 2.6.2 andarelistedinTable 4-1 Table4-1.Directdriveheavingroatcoecients. CoecientValueUnitsEquation m =515kg( 2{54 ) a wz =1.09 10 3 kg( 2{56 ) b r z =2.04 10 3 N-s/m( 2{58 ) b v z =695.09N-s/m( 2{61 ) b p z =0N-s/mAssumed A wp =2m 2 ( 2{55 ) N =0 Assumed k s =0N/mAssumed F zo =1.19 10 4 N( 2{63 ) z =0radSymmetricbodyinlongwaves w nz =3.55rad/s( 2{68 ) b cz =1.14 10 4 p N( 2{69 ) z =0.2090rad( 2{70 ) Z o =0.6327m( 2{67 )&( 2{61 ) 57

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Figure 4-3 showstheforcethatthewaveinducesoverthebody,andFigure 4-4 shows theposition,velocityandinstantpowerdissipatedbytheroat. 0 2 4 6 8 10 12 -1.5 -1 -0.5 0 0.5 1 1.5 x 10 4 t (s)F z ( t )(N) Figure4-3.Heavingbodysimulation:wave-inducedforce. 0 2 4 6 8 10 12 -1.5 -1 -0.5 0 0.5 1 1.5 t (s)z ( t )(m),_ z ( t )(m/s) z ( t ) z ( t ) APositionandvelocity. 0 2 4 6 8 10 12 -5 0 5 10 15 t (s)P ( t )(kW) BInstantdissipatedpower. Figure4-4.Heavingbodysimulationresponse. Thepoweroftheheavingbodyisgivenby P z ( t )= F z ( t ) dz ( t ) dt : (4{1) Theaveragepoweroveroneperiodoftimeisgivenby 58

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P ave = 1 T Z T P z ( t ) dt: (4{2) Applying( 4{2 )overonewaveperiod(insteady-state)ofthefunctionshownin Figure 4-4 B,givesanaveragepower P ave =0 : 82kW.Sincetheroat'sbreadthis1m,then wecancomparethisresultwiththepowercontainedinonemete rofwavefront. Theavailablepowerfortake-o, P pz ,isgivenbythedierencebetweenthewave powerandthepowerdissipatedduetoradiation( P r z )andviscouseects( P v z ) P p z = P w z P r z P v z = 1 2 2 b p z Z 2 o ; (4{3) where P w z istheavailablewavepower. FromFigure 4-2 theaveragepowerpermeterofwavefrontis P =2 : 125kW. Therefore,38.6%ofthewaveenergyisdissipatedasradiation andviscousdamping;hence 1.3kWareavailabletobeharvestedwithelectricalgenerato rs(61.4%ofthewavepower). From( 4{3 ),themaximumdampingcoecientduetopowertake-oisgive nby b p z = 2 P p z 2 Z 2 o : (4{4) Equation( 4{4 )givesaroughestimateofthemaximumadditionaldampingtha tcan beaddedtothesystemusinganelectricalgeneratortoharvesten ergy.Thisequation becomesusefulindesignstageswherethephysicalparametersof thesystemhavetobe determinedinordertodevelopawaveenergyharvestingdevic e. 4.4TensegritySystem LetthedimensionsofthetopplatformbethesameoftheroatinS ection 4.3 ,i.e. L =2m, Z =0 : 5m, B =1m,and d =0 : 25m.Theadditionalconstantmechanism parametersare L 0 =6mand L pxz =1 = 2 L m.Thebaseofthemechanismislocatedat adepth h m =6m.Thecoecientsinallthetermsof( 3{55 )canbecalculatedusing equationsdescribedinChapters 2 and 3 .Table 4-2 containsthevaluesofthecoecients. 59

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Table4-2.Tensegrityharvestingsystemcoecients. CoecientValueUnitsEquation m =515kg( 2{54 ) a wx =50.56kg( 3{37 ) a wz =1.09 10 3 kg( 2{56 ) A w =240.45kg( 2{57 ) I y =182.4kg-m 2 ( 3{37 ) C =7.15 10 3 N-m/rad( 3{44 ) L A 0 =4.32mReversekinematics L B 0 =4.32mReversekinematics k A =200N/mAssumed k B =200N/mAssumed b r z =2.04 10 3 N-s/m( 2{58 ) b r =679.24N-m-s/rad( 3{53 ) b v x =6.95N-s/m( 2{61 ) b v z =6.95N-s/m( 2{61 ) b p z =0N-s/mAssumed A wp =2m 2 ( 2{55 ) F zo =1.19 10 4 N( 2{63 ) M o =445N-m( 2{65 ) z =0radSymmetricbodyinlongwaves Figure 4-5 showstheforcesthatthewaveinducesoverthebody.Figure 4-6 showsthe positionandvelocityresponseofthetopplatformthethreedir ectionsofmotion:surge, heaveandpitch. 0 2 4 6 8 10 12 -2 -1 0 1 2 x 10 4 t (s)F z ( t )(N) 0 2 4 6 8 10 12 -500 0 500 t (s)M ( t )(N-m) Figure4-5.Tensegritysimulation:wave-inducedforces. 60

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0 5 10 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 t (s)x ( t )(m),_ x ( t )(m/s) x ( t ) x ( t ) ASurgingmotion. 0 5 10 -1.5 -1 -0.5 0 0.5 1 1.5 t (s)z ( t )(m),_ z ( t )(m/s) z ( t ) z ( t ) BHeavingmotion. 0 2 4 6 8 10 12 -6 -4 -2 0 2 4 6 8 10 t (s) ( t )(deg), ( t )(deg/s) ( t ) ( t ) CPitchingmotion. Figure4-6.Tensegritysimulation:surging,heavingandpitch ingmotions. Figure 4-7 showstheinstantaneouspowerofthebody.Thispowerisameasur eof theenergythatisbeingdissipatedduetoradiationandviscous eects.Applying( 4{2 ) overonewaveperiod(insteady-state)givesanaveragepower P ave =0 : 6231kW.Since theroat'sbreadthis1m,thenwecancomparethisresultwitht hepowercontained inonemeterofwavefront.Theavailablewavepowerfortakeowascomputedas P =2 : 125kW.Therefore,29.31%oftheavailableenergyisbeingdi ssipateddueto radiationandviscouseects;hence,1.5kWareavailabletobe harvestedbyelectrical generators(70.69%ofthewaveenergy). Thesolutionofthedierentialequationofmotionofthetense gritymechanismwas performedforasetofgeneralizedcoordinatesdenedintheC artesianspace.Hence,the reversekinematicanalysisdevelopedinChapter 3 allowsonetocomputethevaluesofany variableinthejointspace. Figure 4-8 Ashowsthebehaviorofthehypotheticalelectricalgenerato rs;therequired displacementofsuchgeneratorsislessthan2m,avaluethatcan beachievedformodern linearpermanentmagnetgenerators.Thepositionandvelocit ydatacanbeusedasinputs intheanalysisoftheelectricaldevicesinfuturework. 61

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0 2 4 6 8 10 12 -10 -5 0 5 10 15 20 t (s)P ( t )(kW) Figure4-7.Tensegritysimulation:instantdissipatedpower. Figure 4-8 Bshowsthelengthsofthesprings,whicharealwaysgreaterthan thefree length,arequirementtomaintainthetensegritycongurati onofthemechanism. 0 2 4 6 8 10 12 6 7 8 9 10 11 t (s)Position(m) 1 S 2 4 S 5 0 2 4 6 8 10 12 -2 0 2 4 t (s)Velocity(m/s) 1 v 2 4 v 5 AMotionofgenerators. 0 2 4 6 8 10 12 2 4 6 8 t (s)Length(m) L A L A 0 0 2 4 6 8 10 12 2 4 6 8 t (s)Length(m) L B L B 0 BLengthofelasticties. Figure4-8.Tensegritysimulation:motioningeneratorsandsp rings. 4.5DiscussionofResults Fromsections 4.3 and 4.4 ,weseethatthetensegritycongurationdissipatesless energythanthesameroatinpureheavingmotion.Thedirectdr iveheavingsystemallows toharvest1.3kW(61.4%ofthewavepower)whilethetensegrity mechanismallowsto harvest1.5kW(70.69%ofthewaveenergy).Thisvalueisabout 10%morethanthe 62

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valuegivenbythecongurationthatisusedinmostdirectdriv esystemssuchtheonesin [ 110 111 117 118 121 122 ]. Additionally,noparameters,otherthanthedimensionsofther oat,canbemodied inthepurelyheavingdevice.Ontheotherhand,severalparam eterscanbechangedin thetensegritycongurationtomodifytheenergydissipationd uetoradiationandviscous eects.Forinstance,Figure 4-9 showsthevariationofthepercentageoftheavailable powerthatisdissipatedduetoradiationandviscousdamping,a safunctionofthebase length(Figure 4-9 A)andthestinessoftheelasticties(Figure 4-9 B). 2 4 6 8 10 28 28.5 29 29.5 30 30.5 31 31.5 32 L 0 (m)P ave (%) AVariationof P ave with L 0 500 1000 1500 2000 29 30 31 32 33 34 35 36 k A k B (N/m)P ave (%) BVariationof P ave with k A k B Figure4-9.Variationofpowerdissipationwithmechanismpara meters. 63

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CHAPTER5 CONCLUSIONSANDFUTUREWORK 5.1Conclusions Theideaofgettingenergyfromoceanwavesisnotnew,andonl yafewreferences inliteraturehavestatedthepossibilityofharvestingenergyf romexternaldisturbances inatensegritystructure.Thisresearchpresentedtherstappro ximationtotheenergy harvestingpotentialofatensegritysysteminteractingwithoc eanwaves. Thisworkaddressedtheanalysisofatensegritymechanismforoce anwaveenergy harvesting.Aplanartensegritymorphologywasselectedforth esystem,andthedynamic analysiswasperformedconsideringatwo-dimensionallinearm odeloftheoceansurface waves,consideringtheaddedmass,radiationdampingandviscous dampingphenomena. Theforwardandreversepositionanalyseswereperformedusinga geometric approach.Theforwardandreversevelocityanalyseswereperf ormedusingtheoryof screws.Numericalexamplesareprovidedinbothcases.TheLagran gianapproach wasusedtodeducetheequationsofmotionofthemechanismsubje cttotheactionof wave-inducedforcesandmoments. Thetensegritypotentialforwaveenergyharvestingwasdemon stratedbycomparison withapurelyheavingcongurationthatiscommonlyusedinha rvestingdevices.It wasshownthatthetensegritycongurationallowstoharvestab out70%oftheenergy containedinalinearwave,against60%thatwasallowedbythe conventionalsystemusing thesameroatingbodyasinputintothesystem.Itwasshownhowthe changeinsome recongurableparametersofthetensegritysystemaectthepo werdissipationdueto radiationandviscousdamping. Theinteractionbetweenoceanwaves,amulti-degree-of-fre edomlinkageandelectrical generatorsposeschallengingproblemsintermsofmathemati calmodelingandsimulation. Nonetheless,theideaspresentedinthisdocumentwillbeusefulf ortheanalysisand testingofmoreadvancedandcomplexenergyharvestingdevice s. 64

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Thisresearchconstitutesaninterestingapproachtoshowhowth eextensive knowledgeacquiredintheanalysisanddesignofmechanismcanb eusedinnew applicationsthatcontributetotheworld'ssustainabledeve lopment. 5.2FutureWork Oneimportantstageinthedevelopmentofwaveenergyconvert ersistheexperimental phase.Sincethisresearchincludedonlysimulationresults,itw ouldbeinterestingto developaphysicalscalemodeltobetestedinatank.Appropriate powerscalingmethods arewellreferencedinliteratureandmustbetakeninaccount Inthisresearchweusedalinearmodelfortheseawaves,andthean alysiswas performedforaplanarmechanism.However,morerealisticnonl inearandrandommodels forthebehaviorofoceanwavescanbeused.Suchmodelswoulda llowonetousenotonly aplanarlinkage,butspatialtensegritymechanismsandseveral dierentmorphologies. Optimizationtoolscanbeusedtoimprovetheperformanceoft heanalyzed mechanism,andtotondamoresuitable,ormaybeoptimal,tense gritymorphology foroceanwaveenergyharvesting. Becausethereareseveraloptionsforelectricalgenerators,f urtherstudiesmust becarriedoutinordertodeterminewhichcongurationoer betterconditionstobe usedwithdierenttensegritycongurations.Additionally,th econtrolmethodsandthe connectiontoapowergridorenergystorageoptionsshallbead dressedinordertodevelop acommercialdevice. Futureprojectscanincludethedevelopmentofmobileocean energyharvesting stations.Suchstationscanserveascomplementaryenergysource storelativelysmall oceandevicessuchasunderwaterremotelyoperatedvehicles. 65

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REFERENCES [1] Johansson,T.,McCormick,K.,Neij,L.,andTurkenburg,W.,200 4.\Thepotentials ofrenewableenergy".InProceedingsofthe2004Internatio nalConferencefor RenewableEnergies. [2] Favre{Perrod,P.,Geidl,M.,Klockl,B.,andKoeppel,G.,20 05.\Avisionoffuture energynetworks".InProceedingsofthe2005IEEEPowerEngin eeringSociety InauguralConferenceandExpositioninAfrica,IEEE. [3] Redondo{Gil,C.,Esquibel,L.,AlonsoSanchez,A.,andZapico., P.,2009.\European strategicenergytechnologyplan".InProceedingsofthe200 9International ConferenceonRenewableEnergyandPowerQualityICREPQ09, EA4EPQ. [4] Pontes,M.,andFalc~ao,A.,2001.\Oceanenergies:Resourcesan dutilisation". InProceedingsofthe18thCongressoftheWorldEnergyCounci l,BuenosAires, Argentina. [5] Scruggs,J.,andJacob,P.,2009.\ENGINEERING:HarvestingOceanW ave Energy". Science, 323 (5918),pp.1176{1178. [6] Murray,R.,2006.Reviewandanalysisofoceanenergysystems,de velopmentand supportingpolicies.Tech.rep.,PreparedbyAEAEnergy&Envir onmentonthe behalfofSustainableEnergyIrelandfortheIEAsImplementin gAgreementon OceanEnergySystems. [7] Clement,A.,McCullenc,P.,Falc~ao,A.,Fiorentino,A.,Gardn er,F.,Hammarlund, K.,Lemonis,G.,Lewish,T.,Nielseni,K.,Petroncini,S.,Ponte s,M.,Schild,P., Sjostromm,B.,Sorensen,H.,andThorpe,T.,2002.\Waveenergy inEurope: Currentstatusandperspectives". RenewableSustainableEnergyRev., 6 (5),Oct, pp.405{431. [8] TechnologyFutureEnergySolutionsAEA,2003.Waveandmarine currentenergy. Tech.rep.,DepartmentofTradeandIndustryUnitedKingdom,Ne wandRenewable EnergyProgramme. [9] Previsik,M.,Hagerman,G.,andBedard,R.,2004.Oshorewavee nergyconversion devices.reportE2IEPRIWP{004{US.Tech.rep.,ElectricPowe rResearchInstitute, PaloAlto,CA. [10] Pontes,M.,Aguiar,R.,andOliveiraPires,H.,2005.\Anearshore waveenergyatlas forPortugal". J.OshoreMech.Arct.Eng., 127 (3),pp.249{255. [11] Cornett,A.,andTabotton,M.,2006.InventoryofCanadianma rinerenewable energyresources.Tech.rep.,CanadianHydraulicsCentre,Nati onalResearch CouncilCanadaandTritonConsultantsLtd.,Canada. [12] Bedard,R.,Previsic,M.,Hagerman,G.,Polagye,B.,Musial,W. ,Klure,J., vonJouanne,A.,Mathur,U.,Partin,J.,Collar,C.,Hopper,C., andAmsden, 66

PAGE 67

S.,2007.NorthAmericanoceanenergystatus.Tech.rep.,Electr icPowerResearch Institute,PaloAlto,CA. [13] Musial,W.,2008.Statusofwaveandtidalpowertechnologies fortheUnitedStates. Tech.rep.,NationalRenewableEnergyLaboratoryNREL,U.S.De partmentof Energy,Golden,Colorado,Aug. [14] Khan,J.,andBhuyan,G.,2009.Oceanenergy:Globaltechnol ogydevelopment status.Tech.rep.,PreparedbyPowertechLabsfortheIEA-OES, [Online], Available:www.iea-oceans.org. [15] Hao,W.,Qiang,Y.,Donju,W.,Yiru,W.,Jianhui,Y.,andHao,Z.,20 09.\The statusandprospectofoceanenergygenerationinChina".InPr oceedingsof theInternationalConferenceonSustainablePowerGenerati onandSupply,2009. SUPERGEN'09.,IEEE,pp.1{6. [16] GeoscienceAustraliaandABARE,2010.Australianenergyresourceasse ssment. Tech.Rep.70142,PreparedbyGeoscienceAustraliaandAustralia nBureauof AgriculturalandResourceEconomics(ABARE)fortheAustralianGo vernment DepartmentofResources,EnergyandTourism(RET),Canberra,Au stralia. [17] Florez,D.A.,Correa,J.C.,Posada,N.L.,Valencia,R.A.,andZu luaga,C.A., 2010.\MarineenergydevicesforColombianseas".InProceedi ngsoftheASME 2010InternationalMechanicalEngineeringCongress&Exposi tionIMECE2010, ASME. [18] Blunden,L.,andBahaj,A.,2007.\Tidalenergyresourceassessmen tfortidalstream generators". P.I.Mech.Eng.A-J.Pow, 221 (2),pp.137{146. [19] Henfridsson,U.,Neimane,V.,Strand,K.,Kapper,R.,Bernho,H.,D anielsson, O.,Leijon,M.,Sundberg,J.,Thorburn,K.,Ericsson,E.,andB ergman,K.,2007. \WaveenergypotentialintheBalticSeaandtheDanishpartof theNorthSea,with rerectionsontheSkagerrak". RenewableEnergy, 32 (12),Oct,pp.2069{2084. [20] Defne,Z.,Haas,K.,andFritz,H.,2009.\Wavepowerpotentiala longtheAtlantic coastofthesoutheasternUSA". RenewableEnergy, 34 (10),Oct,pp.2197{2205. [21] Iglesias,G.,andCarballo,R.,2009.\Waveenergypotentiala longtheDeathCoast (Spain)". Energy, 34 (11),Nov,pp.1963{1975. [22] Lavrakas,J.,Smith,J.,andCorporation,A.R.,2009.Waveene rgyinfrastructure assessmentinOregon.Tech.rep.,OregonWaveEnergyTrust,Portl and. [23] Mackay,E.,Bahaj,A.,andChallenor,P.,2009.\Uncertaintyi nwaveenergy resourceassessment.part1:Historicdata". RenewableEnergy, 35 (8), pp.1792{1808. 67

PAGE 68

[24] Pontes,M.T.,Bruck,M.,andLehner,S.,2009.\Assessingthewavee nergyresource usingremotesenseddata".InProceedingsofthe8thEuropeanWa veandTidal EnergyConference,Uppsala,Sweden. [25] SengLim,Y.,andLeeKoh,S.,2010.\Analyticalassessmentsonthep otentialof harnessingtidalcurrentsforelectricitygenerationinMala ysia". RenewableEnergy, 35 (5),pp.1024{1032. [26] Iglesias,G.,andCarballo,R.,2010.\Waveenergyresourceint heEstacadeBares area(Spain)". RenewableEnergy, 35 (7),pp.1574{1584. [27] Falc~ao,A.,2010.\Waveenergyutilization:Areviewofthet echnologies". Renewable SustainableEnergyRev., 14 (3),pp.899{918. [28] U.S.DepartmentofEnergy.Marineandhydrokinetictechnologydatabase,[Online].Available: http://www1.eere.energy.gov/windandhydro/hydrokinet ic/default.aspx [29] Fuller,R.,1962.Tensile-integritystructures,USPatent3,0 63,521,Nov.13. [30] Emmerich,D.,1964.Constructionderseauxautotendants,Fren chpatentno. 1,377,290,Sep.28. [31] Snelson,K.,1965.Continuoustension,discontinuouscompressi onstructures,US Patent3,169,611,Feb.16. [32] Pugh,A.,1976. Anintroductiontotensegrity .UniversityofCaliforniaPress. [33] Motro,R.,1992.\Tensegritysystems:thestateoftheart". Int.J.SpaceStruct., 7 (2),pp.75{83. [34] Skelton,R.,anddeOliveira,M.C.,2009. TensegritySystems .Springer. [35] Hanaor,A.,1992.\Aspectsofdesignofdoublelayertensegritydom es". Int.J.Space Struct., 7 (2),pp.101{103. [36] Motro,R.,2003. Tensegrity:structuralsystemsforthefuture .KoganPageScience, Guildford,UK. [37] Fu,F.,2005.\Structuralbehavioranddesignmethodsoftense gritydomes". J. Constr.SteelRes., 61 (1),pp.23{35. [38] Gomez,V.,2010. TensegrityStructuresandtheirApplicationtoArchitectur e PUbliCan{EdicionesdelaUniversidaddeCantabria,Santander [39] Rhode-Barbarigos,L.,BelHadjAli,N.,Motro,R.,andSmith,I.F .,2010. \Designingtensegritymodulesforpedestrianbridges". Eng.Struct., 32 (4),pp.1158 {1167. 68

PAGE 69

[40] BelHadjAli,N.,Rhode-Barbarigos,L.,PascualAlbi,A.A.,andSmith ,I.F.,2010. \Designoptimizationanddynamicanalysisofatensegrity-base dfootbridge". Eng. Struct., 32 (11),pp.3650{3659. [41] Stern,I.,1999.\Developmentofdesignequationsforself-de ployableN-strut tensegritysystems".Master'sthesis,UniversityofFlorida,Gainesv ille,FL. [42] Duy,J.,Rooney,J.,Knight,B.,andCraneIII,C.D.,2000.\ Areviewofafamily ofself-deployingtensegritystructureswithelasticties". TheShockandVibration Digest, 32 (2),pp.100{106. [43] Knight,B.,2000.\Deployableantennakinematicsusingtense gritystructuredesign". PhDthesis,MechanicalEngineering,UniversityofFlorida,Gai nesville,FL. [44] Pellegrino,S.,Kukathasan,S.,Tibert,G.,andWatt,A.,2000 .Smallsatellite deploymentmechanisms.Tech.rep.,PreparedbytheDefenceEv aluationResearch AgencyandtheUniversityofCambridgeonbehalfoftheBritishNat ionalSpace Centre. [45] Tibert,G.,2002.\Deployabletensegritystructuresforspace applications".PhD thesis,DepartmentofMechanics,RoyalInstituteofTechnology ,Sweden. [46] Sultan,C.,andSkelton,R.,2003.\Deploymentoftensegrity structures". Int.J. SolidsStruct., 40 (18),pp.4637{4657. [47] Ingber,D.E.,1998.\Thearchitectureoflife". Sci.Am., 278 ,pp.48{57. [48] Caadas,P.,Laurent,V.M.,Oddou,C.,Isabey,D.,andWendling, S.,2002. \Acellulartensegritymodeltoanalysethestructuralviscoela sticityofthe cytoskeleton". J.Theor.Biol., 218 (2),pp.155{173. [49] Ingber,D.,2003.\TensegrityI.Cellstructureandhierarchi calsystemsbiology". J. CellSci., 116 ,Jan,pp.1157{1163. [50] Huang,S.,Sultan,C.,andIngber,D.,2007. ComplexSystemsSciencein Biomedicine .SpringerUS,ch.Tensegrity,DynamicNetworks,andComplex SystemsBiology:EmergenceinStructuralandInformationNet worksWithinLiving Cells,pp.283{310. [51] Cretu,S.,2009. Modeling,SimulationandControlofNonlinearEngineering DynamicalSystems .Springer,Netherlands,ch.TensegrityasaStructuralFramew ork inLifeSciencesandBioengineering,pp.301{311. [52] Sultan,C.,Corless,M.,andSkelton,R.,2000.\Tensegrityrig htsimulator". J.of Guid.ControlDynam., 23 (6),Nov,pp.1055{1064. [53] Tran,T.,Crane,C.,andDuy,J.,2002.\Thereversedisplacem entanalysisofa tensegritybasedparallelmechanism".InProceedingsofthe5t hBiannualWorld AutomationCongress,pp.637{643. 69

PAGE 70

[54] Kanchanasaratool,N.,andWilliamson,D.,2002.\Motioncontr olofatensegrity platform". CommunicationsinInformationandSystems, 2 (3),pp.299{324. [55] Sultan,C.,andSkelton,R.,2004.\Aforceandtorquetensegr itysensor". Sens. Actuators,A, 112 (2-3),pp.220{231. [56] Marshall,M.,andCrane,C.,2004.\Designandanalysisofahybr idparallel platformthatincorporatestensegrity".InProceedingsoft heASME2004 InternationalDesignEngineeringTechnicalConferencesan dComputersand InformationinEngineeringConference(IDETC/CIE2004),n o.46954,ASME, pp.535{540. [57] Arsenault,M.,andGosselin,C.,2005.\Kinematic,static,anddy namicanalysisofa planarone-degree-of-freedomtensegritymechanism". ASMEJ.Mech.Des., 127 (6), pp.1152{1160. [58] Arsenault,M.,andGosselin,C.,2006.\Kinematic,static,anddy namicanalysis ofaspatialthree-degree-of-freedomtensegritymechanism". ASMEJ.Mech.Des., 128 (5),pp.1061{1069. [59] Arsenault,M.,andGosselin,C.,2006.\Kinematic,staticanddyn amicanalysisofa planar2-DOFtensegritymechanism". Mech.Mach.Theory, 41 (9),pp.1072{1089. [60] Vasquez,R.,andCorrea,J.,2007.\Kinematics,dynamicsandc ontrolofa planar3-DOFtensegrityrobotmanipulator".InProceedings oftheASME2007 InternationalDesignEngineeringTechnicalConferencesan dComputersand InformationinEngineeringConference(IDETC/CIE2007),AS ME,pp.855{866. [61] Scruggs,J.,andSkelton,R.,2006.\Regenerativetensegrity structuresforenergy harvestingapplications".InProceedingsofthe45thIEEECon ferenceonDecision& Control,IEEE,pp.2282{2287. [62] Jensen,O.,Wroldsen,A.,Lader,P.,Fredheim,A.,andHeide,M.,2 007.\Finite elementanalysisoftensegritystructuresinoshoreaquacultu reinstallations". Aquacult.Eng., 36 (3),May,pp.272{284. [63] Wroldsen,A.,2007.\Modellingandcontroloftensegritystruct ures".PhDthesis, DepartmentofMarineTechnology,NorwegianUniversityofScie nceandTechnology. [64] NationalRenewableEnergyLaboratory,2009.Oceanenergyte chnologyoverview. Tech.Rep.DOE/GO-102009-2823,PreparedfortheU.S.Depart mentofEnergy, OceofEnergyEciencyandRenewableEnergyFederalEnergy Management Program,Golden,CO. [65] ElectricPowerResearchInstitute,2008.PrioritizedRDD&Dn eeds:Marineand otherhydrokineticenergy.Tech.rep.,ElectricPowerResea rchInstitute,Final Report2008,PaloAlto,CA. 70

PAGE 71

[66] U.S.DepartmentofEnergy,EISAReporttoCongress,2008.Poten tial environmentaleectsofmarineandhydrokineticenergytec hnologies.Tech. rep.,PreparedinresponsetotheEnergyIndependenceandSecu rityActof2007, Section633(b). [67] Dalton,G.,andGallachir,B.,2010.\Buildingawaveenergy policyfocusingon innovation,manufacturinganddeployment". RenewableSustainableEnergyRev., 14 (8),pp.2339{2358. [68] U.S.DepartmentoftheInterior,MineralsManagementServic e,2009.\Renewable energyandalternateusesofexistingfacilitiesontheouterc ontinentalshelf;nal rule". Fed.Regist., 74 (81),Apr,pp.19637{19871. [69] DetNorskeVeritas,2008.Certicationoftidalandwaveenergy converters,Oshore ServiceSpecicationDNV-OSS-312. [70] McCormick,M.,2007. OceanWaveEnergyConversion .DoverPublications. OriginallypublishedbyWiley-Interscience,NewYorkin1981. [71] Cruz,J.,2008. OceanWaveEnergy:CurrentStatusandFuturePrespectives Springer. [72] FugroOCEANOR.Wavepower,[Online].Available: http://www.oceanor.no/index.htm [73] Gato,L.,andFalc~ao,A.,1988.\Aerodynamicsofthewellstur bine". Int.J.Mech. Sci., 30 (6),pp.383{395. [74] Falc~ao,A.,2004.\First-generationwavepowerplants:Curre ntstatusandR&D requirements". J.OshoreMech.Arct.Eng., 126 (4),pp.384{388. [75] ArupEnergy,2005.Oscillatingwatercolumnwaveenergyconve rterevaluation report.Tech.rep.,TheCarbonTrust. [76] Nielsen,K.,andPedersen,T.,2009.\Adynamicmodelforcontro lpurposesofa waveenergypowerplantbuoyancysystem".InProceedingofthe IEEEInternational ConferenceonControlandAutomation,ICCA2009.,IEEE,pp.8 25{830. [77] Dunnett,D.,andWallace,J.S.,2009.\Electricitygenerat ionfromwavepowerin Canada". RenewableEnergy, 34 (1),pp.179{195. [78] Skelton,R.,Helton,J.,Adhiraki,R.,Pinaud,J.,andChan,W. ,2001. Handbook ofmechanicalsystemsdesign .CRCPress,ch.AnIntroductiontotheMechanicsof TensegrityStructures,pp.316{449. [79] Michell,A.,1904.\Thelimitsofeconomyinframestructures". Philos.Mag., 8 (47), Nov,pp.589{597. 71

PAGE 72

[80] Skelton,R.,Adhikari,R.,Pinaud,J.,Chan,W.,andHelton,J. ,2001.\An introductiontothemechanicsoftensegritystructures".InPr oceedingsofthe 40 th IEEEConferenceonDecisionandControl,2001.,Vol.5,pp.42 54{4259. [81] Aldrich,J.,andSkelton,R.,2003.\Control/structureoptim izationapproachfor minimum-timerecongurationoftensegritysystems".InProcee dingsoftheSPIE: SmartStructuresandMaterials2003:Modeling,SignalProc essing,andControl, R.Smith,ed.,SPIE,pp.448{459. [82] Pellegrino,S.,1990.\Analysisofprestressedmechanisms". Int.J.SolidsStruct., 26 (12),pp.1329{1350. [83] Calladine,C.,andPellegrino,S.,1991.\First-orderinni tesimalmechanism". Int.J. SolidsStruct., 27 (4),pp.505{515. [84] Djouadi,S.,Motro,R.,Pons,J.,andCrosnier,B.,1998.\Activ econtrolof tensegritysystems". J.Aerosp.Eng., 11 (2),April,pp.37{44. [85] Sultan,C.,1999.\Modeling,design,andcontroloftensegrit ystructureswith applications".PhDthesis,AerospaceEngineering,PurdueUniver sity. [86] Oppenheim,I.J.,andWilliams,W.O.,2001.\Vibrationofanel astictensegrity structure". Eur.J.Mech.A.Solids, 20 (6),pp.1023{1031. [87] Skelton,R.E.,Pinaud,J.P.,andMingori,D.L.,2001.\Dyna micsoftheshellclass oftensegritystructures". J.FranklinInst., 338 (2-3),pp.255{320. [88] Sultan,C.,Corless,M.,andSkelton,R.,2001.\Theprestressabi lityproblemof tensegritystructures:someanalyticalsolutions". Int.J.SolidsStruct., 38 (30-31), pp.5223{5252. [89] Sultan,C.,Corless,M.,andSkelton,R.E.,2002.\Lineardyna micsoftensegrity structures". Eng.Struct., 24 (6),pp.671{685. [90] Bossens,F.,Callafon,R.,andSkelton,R.,2004.Modalanalysis ofatensegrity structure:anexperimentalstudy.Tech.rep.,DepartmentofM echanicaland AerospaceEngineering,UniversityofCalifornia,SanDiego. [91] Chan,W.L.,Arbelaez,D.,Bossens,F.,andSkelton,R.E.,2004.\ Activevibration controlofathree-stagetensegritystructure".InProceeding oftheSPIE11th AnnualInternationalSymposiumonSmartStructuresandMater ials,K.-W.Wang, ed.,no.1,SPIE,pp.340{346. [92] Crane,C.,Duy,J.,andCorrea,J.,2005.\Staticanalysisof tensegritystructures". ASMEJ.Mech.Des., 127 (2),pp.257{268. [93] Bayat,J.,andCrane,C.,2007.\Closed-formequilibriumana lysisofplanar tensegritystructures".InProceedingsoftheASME2007Interna tionalDesign 72

PAGE 73

EngineeringTechnicalConferencesandComputersandInfor mationinEngineering Conference(IDETC/CIE2007),ASME,pp.13{23. [94] Arsenault,M.,andGosselin,C.,2007.\Staticbalancingoftense gritymechanisms". ASMEJ.Mech.Des., 129 (3),pp.295{300. [95] Crane,C.,Bayat,J.,Vikas,V.,andRoberts,R.,2008. AdvancesinRobotKinematics:AnalysisandDesign .SpringerNetherlands,May,ch.KinematicAnalysisofa PlanarTensegrityMechanismwithPre-StressedSprings,pp.419{ 427. [96] Arsenault,M.,andGosselin,C.,2009.\Kinematicandstaticanal ysisofa3-PUPS spatialtensegritymechanism". Mech.Mach.Theory, 44 (1),pp.162{179. [97] Wroldsen,A.S.,deOliveira,M.C.,andSkelton,R.E.,2009.\M odellingand controlofnon-minimalnon-linearrealisationsoftensegrit ysystems". Int.J.Control, 82 (3),pp.389{407. [98] McCormick,M.,2009. OceanEngineeringMechanicswithApplications .Cambridge UniversityPress. [99] Craik,A.D.,2004.\Theoriginsofwaterwavetheory". Annu.Rev.FluidMech., 36 (1),pp.1{28. [100] Airy,G.B.,1841. EncyclopaediaMetropolitana(1817-1845),MixedSciences ,Vol.3. Partpublication,ch.TidesandWaves,p.396. [101] Falnes,J.,2002. OceanWavesandOscillatingSystems:LinearInteractions IncludingWave-EnergyExtraction .CambridgeUniversityPress. [102] Vugts,J.,1968.\Thehydrodynamicscoecientsforswaying,h eaving,androlling cylindersinafreesurface". Int.Shipbg.Prog., 5 (167),pp.251{276. [103] Szuladzinski,G.,2010. FormulasforMechanicalandStructuralShockandImpact CRCPress,ch.AerodynamicDragCoecients,pp.747{748. [104] Salter,S.,Taylor,J.,andCaldwell,N.,2002.\Powerconver sionmechanismsfor waveenergy". P.I.Mech.Eng.M-J.Eng., 216 (1),pp.1{27. [105] CraneIII,C.D.,Rico,J.M.,andDuy,J.,2009.Screwtheory anditsapplication tospatialrobotmanipulators.Tech.rep.,CenterforIntelli gentMachinesand Robotics,UniversityofFlorida. [106] Rico,J.M.,Gallardo,J.,andDuy,J.,1999.\Screwtheorya ndhigherorder kinematicanalysisofopenserialandclosedchains". Mech.Mach.Theory, 34 (4), pp.559{586. [107] Ball,S.R.S.,1900. Atreatiseonthetheoryofscrews .CambridgeUniversityPress. [108] Doughty,S.,1988. Mechanicsofmachines .JohnWiley&Sons,NewYork. 73

PAGE 74

[109] Rao,A.,2006. DynamicsofParticlesandRigidBodies:ASystematicApproach CambridgeUniversityPress. [110] Baker,N.J.,andMueller,M.A.,2001.\Directdrivewaveenerg yconverters". Rev. desEnerg.Ren., PowerEngineering ,pp.1{7. [111] Mueller,M.,2002.\Electricalgeneratorsfordirectdrive waveenergyconverters". IEEEProc-C, 149 (4),pp.446{456. [112] Danielsson,O.,2003.\Designofalineargeneratorforwaveene rgyplant".Master's thesis,UppsalaUniversity,SchoolofEngineering. [113] Baker,N.,Mueller,M.,andSpooner,E.,2004.\Permanentmag netair-cored tubularlineargeneratorformarineenergyconverters".InP roceedingsoftheSecond InternationalConferenceonPowerElectronics,Machinesan dDrivesPEMD2004, Vol.2,IEEE,pp.862{867. [114] Polinder,H.,Damen,M.,andGardner,F.,2004.\Linearpmgen eratorsystemfor waveenergyconversionintheaws". IEEET.Energy.Conver., 19 (3),pp.583{589. [115] Thorburn,K.,Bernho,H.,andLeijon,M.,2004.\Waveenergy transmissionsystem conceptsforlineargeneratorarrays". OceanEng., 31 (11-12),Aug,pp.1339{1349. [116] Mueller,M.,andBaker,N.,2005.\Directdriveelectricalpo wertake-oforoshore marineenergyconverters". P.I.Mech.Eng.A-J.Pow, 219 (3),pp.223{234. [117] Leijon,M.,Danielsson,O.,Eriksson,M.,Thorburn,K.,Bernho ,H.,Isberg,J., Sundberg,J.,Ivanova,I.,Sjstedt,E.,gren,O.,Karlsson,K., andWolfbrandt,A., 2006.\Anelectricalapproachtowaveenergyconversion". RenewableEnergy, 31 (9), Jul,pp.1309{1319. [118] Rhinefrank,K.,Agamloh,E.,vonJouanne,A.,Wallace,A.,Prud ell,J.,Kimble, K.,Aills,J.,Schmidt,E.,Chan,P.,Sweeny,B.,andSchacher, A.,2006.\Novel oceanenergypermanentmagnetlineargeneratorbuoy". RenewableEnergy, 31 (9), pp.1279{1298. [119] Danielsson,O.,2006.\Waveenergyconversion.linearsynchron ouspermanent magnetgenerator".PhDthesis,UppsalaUniversity,FacultyofSci enceand Technology. [120] Mueller,M.,Polinder,H.,andBaker,N.,2007.\Currentandno velelectrical generatortechnologyforwaveenergyconverters".InIEEEIn ternationalElectric MachinesDrivesConferenceIEMDC'07.,Vol.2,pp.1401{140 6. [121] Szabo,L.,Oprea,C.,Viorel,I.,andBiro,K.A.,2007.\Novel permanentmagnet tubularlineargeneratorforwaveenergyconverters".InPro ceedingoftheIEEE InternationalElectricMachinesDrivesConference,IEMDC '07,IEEE. 74

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[122] Trapanese,M.,2008.\Optimizationofaseawaveenergyharvest ingelectromagnetic device". IEEET.Magn., 44 (11),pp.4365{4368. [123] Elwood,D.,Schacher,A.,andRhinefrank,K.,2009.\Numerica lmodelingandocean testingofadirect-drivewaveenergydeviceutilizingaperm anentmagnetlinear generatorforpowertake-o".InProceedingsoftheASME2009 28thInternational ConferenceonOcean,OshoreandArcticEngineering(OMAE2009 ),ASME, pp.817{824. [124] Liu,C.-T.,Lin,C.-L.,Hwang,C.-C.,andTu,C.-H.,2010.\Com pactmodelofa slotlesstubularlineargeneratorforrenewableenergyperfo rmanceassessments". IEEET.Magn., 46 (6),pp.1467{1470. [125] Ruellan,M.,BenAhmed,H.,Multon,B.,Josset,C.,Babarit,A.,an dClement, A.,2010.\DesignmethodologyforaSEAREVwaveenergyconverte r". IEEET. Energy.Conver., 25 (3),pp.760{767. [126] Cheng,S.,andArnold,D.P.,2010.\Astudyofamulti-polemagn eticgenerator forlow-frequencyvibrationalenergyharvesting". J.Micromech.Microeng., 20 (2), 02/2010,p.025015. [127] Osorio,A.,Agudelo,P.,Correa,J.,Otero,L.,Ortega,S.,Herna ndez,J.,and Restrepo,J.,2011.\Buildingaroadmapfortheimplementati onofmarinerenewable energyinColombia".InProceedingsofOCEANS2011IEEE-Spain ,pp.1{5. [128] Ortega,S.,Osorio,A.,Agudelo-Restrepo,P.,andVelez,J.,201 1.\Methodology forestimatingwavepowerpotentialinplaceswithscarceinstr umentationinthe CaribbeanSea".InProceedingsofOCEANS2011IEEE-Spain,pp. 1{5. [129] Ortega,S.,2010.\Estudiodeaprovechamientodelaenerga deloleajeenIslaFuerte (Caribecolombiano)".Master'sthesis,SchoolofGeosciencesa ndEnvironment, NationalUniversityofColombia. 75

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BIOGRAPHICALSKETCH RafaelE.VasquezreceivedhisB.S.inmechanicalengineerin gin2002andhis M.Sc.inengineeringwithemphasisinautomationin2007,bot hfromtheUniversidad PonticiaBolivarianaUPB,Medellin,Colombia.Hejoinedthe facultyoftheUPB in2003whereheisassociateprofessorintheareaofdynamics,syste msandcontrol. Heiscurrentlycompletingadoctoraldegreeinmechanicalen gineering,undera Fulbright-Colciencias-DNPscholarship,attheCenterforInte lligentMachinesand Robotics(CIMAR)attheUniversityofFlorida,Gainesville,Flo rida.Hisresearch interestsaretheoryofmechanisms;design,analysisandcontrol ofdynamicsystems andnewtechnologiesforenergyharvesting.Heismemberofthe AmericanSocietyof MechanicalEngineers(ASME)since2005.AftercompletinghisP hD,Rafaelwillgoback toColombiatoconductresearchandteachintheDepartmentof MechanicalEngineering attheUniversidadPonticiaBolivarianaUPB,Medellin,Colom bia. 76