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)
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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
#