1Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

Chapter 5 – Seakeeping Theory

5.1HydrodynamicConceptsandPotentialTheory5.2SeakeepingandManeuveringKinematics5.3TheClassicalFrequency-DomainModel5.4Time-DomainModelsincludingFluidMemoryEffects

2Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

Chapter 5 - Seakeeping TheoryEquationsofMotionSeakeepingtheoryisformulatedinequilibrium(SEAKEEPING)axes{s}butitcanbe

transformedtoBODYaxes{b}byincludingfluidmemoryeffectsrepresentedbyimpulseresponsefunctions.

Thetransformationisisdonewithinalinearframeworksuchthatadditionalnonlinearviscousdampingmustbeaddedinthetime-domainundertheassumptionoflinearsuperposition.

μisanadditionaltermrepresentingthefluidmemoryeffects.

Inertia forces: !MRB !MA" "! ! CRB#!$! ! CA#!r$!r

Damping forces: !#Dp ! DV$!r ! Dn#!r$!r ! "

Restoring forces: !g##$ ! goWind and wave forces: # $wind ! $wave

Propulsion forces: !$

3Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

StripTheory(2-DPotentialTheory)Forslenderbodies,themotionofthefluidcanbeformulatedasa2-Dproblem.Anaccurateestimateofthehydrodynamicforcescanbeobtainedbyapplyingstriptheory(Newman,1977;Faltinsen,1990;Journee andMassie,2001).

The2-Dtheorytakesintoaccountthatvariationoftheflowinthecross-directionalplaneismuchlargerthanthevariationinthelongitudinaldirectionoftheship.

Theprincipleofstriptheoryinvolvesdividingthesubmergedpartofthecraftintoafinitenumberofstrips.Hence,2-Dhydrodynamiccoefficientsforaddedmasscanbecomputedforeachstripandthensummedoverthelengthofthebodytoyieldthe3-Dcoefficients.

CommercialCodes:MARINTEK(ShipX-Veres)andAmarcon (OctopusOffice)

5.1 Hydrodynamic Concepts and Potential Theory

4Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

ShipX (VERES) by MARINTEK

VERES- VEssel RESponse program isaStripTheoryProgram whichcalculateswave-inducedloadsonandmotionsofmono-hullsandbargesindeeptoveryshallowwater.TheprogramisbasedonthefamouspaperbySalvesen,TuckandFaltinsen (1970).ShipMotionsandSeaLoads.Trans.SNAME.

MARINTEK- theNorwegianMarineTechnologyResearchInstitute- doesresearchanddevelopmentinthemaritimesectorforindustryandthepublicsector.TheInstitutedevelopsandverifiestechnologicalsolutionsfortheshippingandmaritimeequipmentindustriesandforoffshorepetroleumproduction.

5Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

ShipX (Veres)

ShipX (VERES) by MARINTEK

6Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

OCTOPUS SEAWAY by Amarcon

andAMARCONcooperateinfurtherdevelopmentofSEAWAY

SEAWAY isdevelopedbyProfessorJ.M.J.Journée attheDelftUniviversity ofTechnology

SEAWAY isaStripTheoryProgram tocalculatewave-inducedloadsonandmotionsofmono-hullsandbargesindeeptoveryshallowwater.Whennotaccountingforinteractioneffectsbetweenthehulls,alsocatamaranscanbeanalyzed.Workofveryacknowledgedhydromechanicscientists(suchas Ursell,Tasai,Frank,Keil,Newman,Faltinsen,Ikeda,etc.)hasbeenused,whendevelopingthiscode.

SEAWAY hasextensivelybeenverifiedandvalidatedusingothercomputercodesandexperimentaldata.

TheMaritimeResearchInstituteNetherlands(MARIN)andAMARCONagreetocooperateinfurtherdevelopmentofSEAWAY.MARIN isaninternationallyrecognizedauthorityonhydrodynamics,involvedinfrontierbreakingresearchprogramsforthemaritimeandoffshoreindustriesandnavies.

7Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) Copyright © 2005 Marine Cybernetics AS7

8Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

5.1 Hydrodynamic Concepts and Potential TheoryPanelMethods(3-DPotentialTheory)Forpotentialflows,theintegralsoverthefluiddomaincanbetransformedtointegralsovertheboundariesofthefluiddomain.Thisallowstheapplicationofpanelorboundaryelementmethodstosolvethe3-Dpotentialtheoryproblem.

Panelmethodsdividethesurfaceoftheshipandthesurroundingwaterintodiscreteelements(panels).Oneachoftheseelements,adistributionofsourcesandsinksisdefinedwhichfulfilltheLaplaceequation.

Commercialcode:WAMIT(www.wamit.com)

-40

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-10

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10

20

30

40 -30

-20

-10

0

10

20

30

-12

-10

-8

-6

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-2

0

2

4

Y-axis (m)

3D Visualization of the Wamit file: supply.gdf

X-axis (m)

Z-ax

is (m

)

3D Panelization ofa Supply Vessel

9Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

WAMIT

WAMIT® is the most advanced set of tools available for analyzing wave interactions with offshore platforms and other structures or vessels. WAMIT® was developed by Professor Newman and coworkers at MIT in 1987, and it has gained widespread recognition for its ability to analyze the complex structures with a high degree of accuracy and efficiency.

Over the past 20 years WAMIT has been licensed to more than80 industrial and research organizations worldwide.

Panelization of semi-submersible using WAMIT user supplied tools

10Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

5.1 Hydrodynamic Concepts and Potential TheoryPotentialtheoryprogramstypicallycompute:

• Frequency-dependentaddedmass,A(w)• Potentialdampingcoefficients,B(w)• Restoringterms,C• 1st- and2nd-orderwave-inducedforcesandmotions

(amplitudesandphases)forgivenwavedirectionsandfrequencies• …andmuchmore

OnespecialfeatureofWAMITisthattheprogramsolvesaboundaryvalueproblemforzeroandinfiniteaddedmass.Theseboundaryvaluesareparticularusefulwhencomputingtheretardationfunctionsdescribingthefluidmemoryeffects.

ProcessingofHydrodynamicDatausingMSSHYDRO– www.marinecontrol.orgThetoolboxreadsoutputdatafilesgeneratedbythehydrodynamicprograms:

• ShipX (Veres)byMARINTEKAS• WAMITbyWAMITInc.

andprocessesthedataforuseinMatlab/Simulink.

11Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

5.2 Seakeeping and Maneuvering KinematicsSeakeepingTheory(PerturbationCoordinates)TheSEAKEEPINGreferenceframe{s}isnotfixedtothecraft;itisfixedtotheequilibriumstate:

e1 ! !1,0,0,0,0,0"!

! ! !"

L :!

0 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 !1 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

!" ! !#" ! !$

! ! !!1,!2,!3,!4,!5,!6"T #

!! ! ! !U!L!" " e1"

!!# ! !# !UL! # #

Transformationbetween{b}and{s}

!

"

#

!

00#"

#

$!

$"

$#

#

vnsn ! !Ucos!, Usin!, 0"!

!nsn ! !0,0,0"!

"ns ! !0,0,!" "!

# # #

-Intheabsenceofwaveexcitation,{s}coincideswith{b}.- Undertheactionofthewaves,thehullisdisturbedfromitsequilibriumand{s}oscillates,withrespecttoitsequilibriumposition.

!sb ! !!4,!5,!6"! ! !"#,"#,"$"! #

12Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

SeakeepingAnalysisTheseakeepingequationsofmotionareconsideredtobeinertial:

5.3 The Classical Frequency-Domain Model

! ! !" ! !!x,!y,!z,!",!#,!$"! #

EquationsofMotion

MRB!" ! #hyd " #hs " #exc #

Cummins(1962)showedthattheradiation-inducedhydrodynamicforcesinanidealfluidcanberelatedtofrequency-dependentaddedmassA(ω) andpotentialdampingB(ω)accordingto:

!hyd ! !Ā"# ! "0

tK$ !t ! !""%!!"d! #

K!t" ! 2! !0

"B!""cos!"t"d" #

Ā ! A!!"

13Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

5.3 The Classical Frequency-Domain Model

Ā ! A!!"

Frequency-dependentaddedmassA22(ω)andpotentialdampingB22(ω)insway

14Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

CumminsModel

5.3 The Classical Frequency-Domain Model

K!t" ! 2! !0

"B!""cos!"t"d" #

!MRB ! A!!""!" ! "0

tK# !t # !"!$!!"d! ! C! " %exc #

Iflinearrestoringforcesτhs = -Cξ areincludedinthemodel,thisresultsinthetime-domainmodel:

Matrixofretardationfunctionsgivenby

!hyd ! !Ā"# ! "0

tK$ !t ! !""#!!"d! #

Thefluidmemoryeffectscanbereplacedbyastate-spacemodeltoavoidtheintegral

15Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

5.3 The Classical Frequency-Domain Model

Longitudinaladdedmasscoefficientsasafunctionoffrequency.

16Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

5.3 The Classical Frequency-Domain Model

Lateraladdedmasscoefficientsasafunctionoffrequency.

17Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

5.3 The Classical Frequency-Domain Model

Longitudinalpotentialdampingcoefficientsasafunctionoffrequency.ExponentialdecayingviscousdampingisincludedforB11.

18Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

5.3 The Classical Frequency-Domain Model

Lateralpotentialdampingcoefficientsasafunctionoffrequency.ExponentialdecayingviscousdampingisincludedforB22 andB66 whileviscousIKEDAdampingisincludedinB44

19Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

5.3.1 Potential Coefficients and the Concept of Forced Oscillations

foreachfrequencyω.

Thematrices A(ω),B(ω)andC representsa"hydrodynamicmass-damper-springsystem"whichvarieswiththefrequencyoftheforcedoscillation.

Thismodelisrooteddeeplyintheliteratureofhydrodynamicsandtheabuseofnotationofthisfalsetime-domainmodelhasbeendiscussedeloquentlyintheliterature(incorrectmixtureoftimeandfrequencyinanODE).Consequently,wewilluseCumminstime-domainmodelandtransformthismodeltothefrequencydomain– nomixtureoftimeandfrequency!

Inanexperimentalsetupwitharestrainedscalemodel,itispossibletovarythewaveexcitationfrequencyω andtheamplitudesfi oftheexcitationforce.Hence,bymeasuringthepositionandattitudevectorη,theresponseofthe2nd-orderordersystemcanbefittedtoalinearmodel:

MRB!" ! #hyd " #hs " fcos!!t" #

!MRB ! A"!#$!" ! B"!#!# ! C! " fcos"!t# #

harmonicexcitation

20Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

!MRB ! A!!""!" ! "0

tK# !t # !"!$!!"d! ! C! " %exc #

5.3.2 Frequency-Domain SeakeepingModels

Cumminsequationcanbetransformedtothefrequencydomain(Newman,1977;Faltinsen 1990)byassumingthatthevesselcarriesoutharmonicoscillationsin6DOF(seeSection5.4.1)::

ThepotentialcoefficientsA(ω)andB(ω)areusuallycomputedusingaseakeepingprogrambutthefrequencyresponsewillnot beaccurateunlessviscousdampingisincluded.

TheoptionalviscousdampingmatrixBV(ω) canbeusedtomodelviscousdampingsuchasskinfriction,surgeresistanceandviscousrolldamping(forinstanceIKEDArolldamping).

21Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

BV!!" !

"1e!#!"NITTC!A1" 0 0 0 0 0

0 "2e!#! 0 0 0 0

0 0 0 0 0 0

0 0 0 "IKEDA!!" 0 0

0 0 0 0 0 0

0 0 0 0 0 "6e!#!

#

5.3.2 Frequency-Domain SeakeepingModels

B total!!" ! B!!" " BV!!" #

u ! Asin!!t" #

y ! c1x " c2x|x|"c3x33 #

N!A" ! c1 " 8A3! c2 " 3A2

4 c3 # y ! N!A"u #

X ! !X |u|u|u|u" NITTC!A1"u #

Viscousfrequency-dependentdamping:

Quadraticdampingisapproximatedusingdescribingfunctions(similartotheequivalentlinearizationmethod):

!ie!"#

QuadraticITTCdrag:

Viscousskinfriction:

22Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

5.4 Time-Domain Models including Fluid Memory Effects

Unifiedmaneuveringandseakeeping model(nonlinearviscousdamping/maneuveringcoefficients

areaddedmanually)

Linearseakeeping equationsinBODYcoordinates(fluidmemoryeffectsareapproximatedasstate-spacemodels)

TransformfromSEAKEEPINGtoBODYcoordinates(linearizedkinematictransformation)

CumminsequationinSEAKEEPINGcoordinates(lineartheorywhichincludesfluidmemoryeffects)

23Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

!MRB ! A"!#$!" ! B total"!#!# ! "0

tK"t # !#!#"!#d! ! C! " $wind ! $wave ! "$ #

Fromanumericalpointofviewisitbettertointegratethedifference:

ThiscanbedonbyrewritingCumminsequationas:

5.4.1 Cummins Equation in SEAKEEPING Coordinates

!MRB ! Ā"!" ! !"#

tK# !t " !"!$!!"d! ! C# ! " %wind ! %wave ! "% #

Ā ! A!!" #

K! !t" ! 2! !0

"B total!""cos!"t"d" #

Cummins(1962)Equation

TheOgilvie(1964)Transformationgives

K!t" ! 2! !0

"#B total!"" # B total!""$ cos!"t"d" #

24Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

Itispossibletotransformthetime-domainrepresentationofCumminsequationfrom{s}to{b}usingthekinematicrelationships:

Thisgives:

Thesteady-statecontrolforceτ neededtoobtaintheforwardspeedU whenτwind =τwave=0 andδη =0 is:

Hence,

5.4.2 Linear Time-Domain SeakeepingEquations in BODY Coordinates!MRB ! A"!#$!" ! B total"!#!# ! "

0

tK"t # !#!#"!#d! ! C! " $wind ! $wave ! "$ #

!MRB ! A"!#$!!" !UL!$ ! B total"!#!! !U"L!# " e1#$ ! #0

tK"t " "#!!""#d" ! C!# " $wind ! $wave ! "$ " $%# #

!! ! ! !U!L!" " e1"

!!# ! !# !UL! # #

!" ! B total!!"Ue1 #

! ! !"

!" ! !#

!MRB ! A"!#$!!" !UL!$ ! B total"!#!! ! UL!#$ ! "0

tK"t # "#!!""#d" ! C!# " $wind ! $wave ! $ #

25Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

Whencomputingthedampingandretardationfunctions,itiscommontoneglecttheinfluenceofδη ontheforwardspeedsuchthat:

Finally,letusereplaceν bytherelativevelocityνr toincludeoceancurrentsanddefine:M =MRB+MA suchthat:

where

5.4.2 Linear Time-Domain SeakeepingEquations in BODY Coordinates!MRB ! A"!#$!!" !UL!$ ! B total"!#!! ! UL!#$ ! "

0

tK"t # "#!!""#d" ! C!# " $wind ! $wave ! $ #

!! ! v !U!L!" " e1" ! v "Ue1 #

M!" ! CRB! ! ! CA!!r ! D!r ! "0

tK!t # !"#!!!"#Ue1$d! ! G# " $wind ! $wave ! $ #

MA ! A!!"CA" ! UA!!"LCRB" ! UMRBLD ! B total!!"

G ! C

LinearCoriolis andcentripetalforcesduetoarotationof{b}about{s}

26Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

5.4.2 Linear Time-Domain SeakeepingEquations in BODY Coordinates

! :! !0

tK!t " !"#"!!""Ue1$

""d! #

FluidMemoryEffectsTheintegralinthefollowingequationrepresentsthefluidmemoryeffects:

! ! H!s"#" !Ue1$ #

!x " Arx # B r!!" " Crx

#

Approximatedbyastate-spacemodel

K!t" ! 2! !0

"#B!"" # B!""$ cos!"t"d" #

Impulseresponsefunction

0 5 10 15 20 25-1

-0.5

0

0.5

1

1.5

2

2.5x 107

time (s)

K22(t)

M!" ! CRB! ! ! CA!!r ! D!r ! "0

tK!t # !"#!!!"#Ue1$d! ! G# " $wind ! $wave ! $ #

27Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

5.4.3 Nonlinear Unified Seakeeping and Maneuvering Model with Fluid Memory EffectsLinearSeakeeping Equations(BODYcoordinates)

UnifiedNonlinearSeakeeping andManeuveringModel

• Usenonlinearkinematics• ReplacelinearCoriolis andcentripetalforceswiththeirnonlinearcounterparts• Includemaneuveringcoefficientsinanonlineardampingmatrix(linearsuperposition)

!" ! J"!!"#M#" r # CRB!#"# # CA!#r"#r # D!#r"#r # $ # G! ! %wind # %wave # %

# #

M!" ! CRB! ! ! CA!!r ! D!r ! # ! G$ " %wind ! %wave ! % # Copyright © Bjarne Stenberg/NTNU

Copyright © The US Navy

28Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

5.5 Case Study: Identification of Fluid Memory EffectsThefluidmemoryeffectscanbeapproximatedusingfrequency-domainidentification.ThemaintoolforthisistheMSSFDItoolbox(PerezandFossen2009)- www.marinecontrol.org

Whenusingthefrequency-domainapproach,thepropertythatthemapping:hasrelativedegreeoneisexploited.Hence,thefluidmemoryeffectsμ canbeapproximatedbyamatrixH(s)containingrelativedegreeonetransferfunctions:

!! ! " #

! ! H!s"!" #

H!s" ! Cr!sI ! Ar"!1B r #

!x " Arx # B r!!" " Crx

#

hij!s" ! prsr"pr!1sr!1"..."p0sn"qn!1sn!1"..."q0

r ! n ! 1, n " 2

State-spacemodel:

29Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

5.5.1 Frequency-Domain Identification using the MSS FDI ToolboxConsidertheFPSOdatasetintheMSStoolbox(FDItool)andassumesthattheinfiniteaddedmassmatrixisunknown.Hence,wecanestimatethefluidtransferfunctionh33(s)byusingthefollowingMatlab code:

load fpsoDof = [3,3]; %Use coupling 3-3 heave-heaveNf = length(vessel.freqs);W = vessel.freqs(1:Nf-1)';Ainf = vessel.A(Dof(1),Dof(2),Nf); % Ainf computed by WAMIT

A = reshape(vessel.A(Dof(1),Dof(2),1:Nf-1),1,length(W))';B = reshape(vessel.B(Dof(1),Dof(2),1:Nf-1),1,length(W))';

FDIopt.OrdMax = 20;FDIopt.AinfFlag = 0;FDIopt.Method = 2;FDIopt.Iterations = 20;FDIopt.PlotFlag = 0;FDIopt.LogLin = 1;FDIopt.wsFactor = 0.1;FDIopt.wminFactor = 0.1;FDIopt.wmaxFactor = 5;

[KradNum,KradDen,Ainf] = FDIRadMod(W,A,0,B,FDIopt,Dof)

30Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

5.5.1 Frequency-Domain Identification using the MSS FDI Toolbox

FPSOidentificationresultsforh₃₃(s)withoutusingtheinfiniteaddedmassA₃₃(∞).Theleft-hand-sideplotsshowthecomplexcoefficientanditsestimatewhileaddedmassanddampingareplottedontheright-hand-side.

31Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

5.5.1 Frequency-Domain Identification using the MSS FDI Toolbox

h33!s" !1.672e007 s3 " 2.286e007 s2 " 2.06e006 s

s4 " 1.233 s3 " 0.7295 s2 " 0.1955 s " 0.01639

Ar !

!1.2335 !0.7295 !0.1955 !0.01641 0 0 00 1 0 00 0 1 0

B r !

1000

Cr ! 1.672e007 2.286e007 2.06e006 0

Dr ! 0

! ! H!s"#" !Ue1$ #

!x " Arx # B r!!" " Crx

#