chapter 5 –seakeeping theory · lecture notes ttk 4190 guidance and control of vehicles (t. i....

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

Chapter 5 – Seakeeping Theory

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


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: !$


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


ShipX (VERES) by MARINTEK

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

MARINTEK- theNorwegianMarineTechnologyResearchInstitute- doesresearchanddevelopmentinthemaritimesectorforindustryandthepublicsector.TheInstitutedevelopsandverifiestechnologicalsolutionsfortheshippingandmaritimeequipmentindustriesandforoffshorepetroleumproduction.


ShipX (Veres)

ShipX (VERES) by MARINTEK


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.


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

Panelmethodsdividethesurfaceoftheshipandthesurroundingwaterintodiscreteelements(panels).Oneachoftheseelements,adistributionofsourcesandsinksisdefinedwhichfulfilltheLaplaceequation.

Commercialcode:WAMIT(www.wamit.com)

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3D Visualization of the Wamit file: supply.gdf

X-axis (m)

Z-ax

is (m

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3D Panelization ofa Supply Vessel


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


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.


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"! ! !"#,"#,"$"! #


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!!"



Ā ! A!!"

Frequency-dependentaddedmassA22(ω)andpotentialdampingB22(ω)insway


CumminsModel


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



Longitudinaladdedmasscoefficientsasafunctionoffrequency.



Lateraladdedmasscoefficientsasafunctionoffrequency.



Longitudinalpotentialdampingcoefficientsasafunctionoffrequency.ExponentialdecayingviscousdampingisincludedforB11.



Lateralpotentialdampingcoefficientsasafunctionoffrequency.ExponentialdecayingviscousdampingisincludedforB22 andB66 whileviscousIKEDAdampingisincludedinB44


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


!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).


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:


5.4 Time-Domain Models including Fluid Memory Effects

Unifiedmaneuveringandseakeeping model(nonlinearviscousdamping/maneuveringcoefficients

areaddedmanually)

Linearseakeeping equationsinBODYcoordinates(fluidmemoryeffectsareapproximatedasstate-spacemodels)

TransformfromSEAKEEPINGtoBODYcoordinates(linearizedkinematictransformation)

CumminsequationinSEAKEEPINGcoordinates(lineartheorywhichincludesfluidmemoryeffects)


!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" #


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 ! $ #


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}


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

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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 ! $ #


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


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:


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)


5.5.1 Frequency-Domain Identification using the MSS FDI Toolbox

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


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

#

chapter 5 –seakeeping theory · lecture notes ttk 4190 guidance and control of vehicles (t. i....

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