Chapter 4 –Hydrostatics

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

Chapter 4 – Hydrostatics

4.1RestoringForcesforUnderwaterVehicles4.2RestoringForcesforSurfaceVessels4.3LoadConditionsandNaturalPeriods4.4BallastSystems

Archimedes(287-212BC)derivedthebasiclawsoffluidstaticswhicharethefundamentalsofhydrostaticstoday.

Inhydrostaticterminology,thegravitationalandbuoyancyforcesarecalledrestoringforces,andtheyareequivalenttothespringforcesinamass-damper-springsystem.

M!! " C!!"! " D!!"! " g!"" " go # # " #wind " #wave

M ! MRB "MA - system inertia matrix (including added mass)C!!" ! CRB!!" " CA!!" - Coriolis-centripetal matrix (including added mass)D!!" - damping matrixg!"" - vector of gravitational/buoyancy forces and momentsgo - vector used for pretrimming (ballast control)# - vector of control inputs#wind - vector of wind loads#wave - vector of wave loads

6-DOFequationsofmotion

Chapter 4 – Hydrostatics

BallastcontrolGravitational/buoyancyterms

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

Inthederivationoftherestoringforcesandmomentswewilldistinguishbetweentwocases:

• Section4.1Underwatervehicles(ROVs,AUVsandsubmarines)• Section4.2Surfacevessels(ships,semi-submersiblesandhigh-speedcraft)

UnderwaterVehicles:AccordingtotheSNAME(1950)itisstandardtoexpressthesubmergedweight ofthebodyandbuoyancyforce as:

=waterdensity=volumeoffluiddisplacedbythevehicle

m =massofthevesselincludingwaterinfreefloodingspace

g =accelerationofgravity

4.1 Restoring Forces for Underwater Vehicles

W ! mg, B ! !g!

fbn ! !

TheweightandbuoyancyforcecanbetransformedfromNEDtoBODYby:

fgb ! Rbn!!"!1fg

n, fbb ! Rbn!!"!1fb

Thesignoftherestoringforcesandmomentsandmustbechangedwhenmovingthesetermstotheleft-handsideofNewton’s2nd law,e.g.ma=f⟹ma-f=0:

Wedenotethegeneralizedrestoringforces . Noticethattheforceandmomentvectorsaremultipliedwith-1.

Consequently,thegeneralizedrestoringforceinBODYwithcoordinateoriginCObecomes:

4.1.1 Hydrostatics of Submerged Vehicles

g!!" ! !fgb " fb

rgb ! fgb " rbb ! fb

! !Rbn!""!1!fg

n " fbn"

rgb ! Rbn!""!1fgn " rbb ! Rbn!""!1fb

mib ! rib!f ibfi

rbb ! !xb, yb, zb"! centerofbuoyancywithrespecttoCOcenterofgravitywithrespecttoCOrgb ! !xg, yg,zg"!

M!! " C!!"! " D!!"! " g!"" " go # # " #wind " #wave #

MainResult:UnderwaterVehicles:

The6-DOFgravityandbuoyancyforcesandmomentsaboutCOaregivenby:

g!!" !

!W ! B" sin!! !W ! B" cos! sin"! !W ! B" cos! cos"! !ygW ! ybB" cos ! cos" " !zgW ! zbB" cos! sin"

!zgW ! zbB" sin ! " !xgW ! xbB" cos! cos"

! !xgW ! xbB" cos ! sin" ! !ygW ! ybB" sin!

Example4.1:NeutrallyBuoyantUnderwaterVehicles:LetthedistancebetweenthecenterofgravityCGandthecenterofbuoyancyCB bedefinedbythevector:

ForneutrallybuoyantvehiclesW=B,andthissimplifiesto:

AnevensimplerrepresentationisobtainedforvehicleswheretheCGandCB arelocatedverticallyonthez-axis,thatisxb =xg andyg =yb.Thisyields:

BG ! !BGx, BGy, BGz"! ! !xg ! xb, yg ! yb, zg ! zb"!

g!!" !

!BGyW cos! cos" "BGzW cos! sin"BGzW sin! "BGxW cos! cos"!BGxW cos ! sin" !BGyW sin !

g!!" ! 0, 0, 0, BGzW cos!sin", BGzW sin!, 0!

Forsurfacevessels,therestoringforceswilldependonthecraft'smetacentricheight,thelocationoftheCG andtheCB aswellastheshapeandsizeofthewaterplane.LetAwp denotethewaterplaneareaand:

GMT =transversemetacentricheight(m)GML =longitudinalmetacentricheight(m)

ThemetacentricheightGMi wherei={T,L}isthedistancebetweenthemetacenterMiandCG.

4.2 Restoring Forces for Surface Vessels

Definition4.1(Metacenter):ThetheoreticalpointMi atwhichanimaginaryverticallinethroughtheCBintersectsanotherimaginaryverticallinethroughanewCBcreatedwhenthebodyisdisplaced,ortilted,inthewater.

isthewaterplaneareaofthevesselasafunctionoftheheaveposition

Forafloatingvesselatrest,buoyancyandweightareinbalancesuchthat:

z=displacementinheavez=0 istheequilibriumposition

Thehydrostaticforceinheaveisrecognizedasthedifferenceofthegravitationalandbuoyancyforces:

4.2.1 Hydrostatics of Floating Vessels

mg ! !g!

Z ! mg ! !g!" " ""!z""! !!g""!z" #

wherethechangeindisplacedwateris:

!!!z" ! "0

z Awp!""d"

Awp!!"

B moves to B1 when the hullis rotated a roll angle f.G is fixed (rigid body).

Forconventionalrigsandshipswithbox-shapedwallsitcanbeassumedthat:

Thisexpressionisconstantforsmallperturbationsinz.Hence,therestoringforceZ willbelinear inz,thatis:

Z ! "!gAwp!0"Zzz

Awp!!" ! Awp!0"

zAwp!z"

This is physically equivalent to a spring with stiffness Zz ! !!gAwp!0" and position z.

!frb ! Rbn!!"!1

!"g "0

z Awp!#"d#

TherestoringforcesandmomentsdecomposedinBODY:

Themomentarmsinrollandpitchare and,respectively.Weightandbuoyancyactinthe z-directionandtheyformaforcepair.Hence,

4.2.1 Hydrostatics of Floating VesselsGMT sin! GML sin !

W ! B ! !g!

!GML sin !GMT sin"

frb ! Rbn!!"!1

!#g"! !#g"

! sin!cos ! sin"cos ! cos"

Neglecting the moment contribution due to !frb (only considering fr

b!implies that the restoring moment becomes:

mrb ! rrb ! fr

! !"g"GMT sin#cos $cos#GML sin$ cos$ cos#

!!GML cos $ "GMT"sin#sin $ #

g!!" ! !!frb

g!!" !

!!g "0

zAwp!""d" sin #

z Awp!""d" cos # sin$

z Awp!""d" cos #cos$

!g#GMT sin$cos #cos$!g#GML sin# cos# cos$

!g#!!GML cos # "GMT" sin$ sin#

MainResult:SurfaceVessels:

6-DOFgeneralizedgravityandbuoyancyforces:

Linear(SmallAngle)TheoryforBoxed-ShapedVesselsAssumesthat aresmallsuchthat:

4.2.2 Linear (Small Angle) Theory for Boxed-Shaped Vessels

g!!" ! G!

g!!" !

"!gAwp!0" z"

!gAwp!0" z#

!gAwp!0" z

!g#GMT #

!g#GML "

!g#!"GML!GMT" #"

!gAwp!0"z

!g#GMT#

!g#GML"

sin! ! !, cos! ! 1sin" ! ", cos" ! 1

z Awp!!"d! " Awp!0"z

!, ", z

G ! diag!0, 0, !gAwp"0#, !g!GMT, !g!GML, 0$

M!" #N! #G$ % & # go #w

Linearkinetics:

4.2.2 Linear (Small Angle) Theory for Boxed-Shaped VesselsThediagonalGmatrixisbasedontheassumptionofyz-symmetry(fore-aftsymmetry).IntheasymmetricalcaseG takestheform(noticethetwoadditionalcouplingtermsG35 =G53):

G ! G! !

0 0 0 0 0 00 0 0 0 0 00 0 !Zz 0 !Z! 00 0 0 !K" 0 00 0 !Mz 0 !M! 00 0 0 0 0 0

! Zz ! !gAwp!0"

! Z" ! !g " "AwpxdA

! Mz ! !Z"

! K# ! !g#!zg ! zg" " !g " "Awpy2dA ! !g#GMT

! M" ! !g#!zg ! zb" " !g " "Awpx2dA ! !g#GML

ThecouplingtermsdependonthelocationofCO

4.2.3 Computation of Metacenter Height for Surface Vessels

MetacenterM,centerofgravityG andcenterofbuoyancyB forasubmergedandafloatingvessel.ThereferenceisthekeellineK.

Forsmallrollandpitchanglesthetransverseandlongitudinalradiusofcurvaturecanbeapproximatedby:

wherethemomentsofareaaboutthewaterplanearedefinedas:

GMT ! BMT ! BG, GML ! BML ! BG

BMT ! IT! , BML ! IL

IL ! ! !Awpx2dA, IT ! ! !

Awpy2dA

ForconventionalshipsanupperboundontheseintegralscanbefoundbyconsideringarectangularwaterplaneareaAwp=BL whereB andL arethebeamandlengthofthehullupperboundedby:

IL ! 112 L

3B, IT ! 112 B

ThemetacenterheightM canbecomputedbyusingbasichydrostatics:

Definition4.2(MetacenterStability):Afloatingvesselissaidtobe:

Transversemetacentrically stable ifGMT≥GMT,min >0

Longitudinalmetacentrically stable if GML≥GML,min >0

Thelongitudinalstabilityrequirementiseasytosatisfyforshipssincethepitchingmotionisquitelimited.ThiscorrespondstoalargeGMLvalue.

Thelateralrequirement,however,isanimportantdesigncriterionusedtoprescribesufficientstabilityinrolltoavoidthatthevesseldoesnotrollaround. Thevesselmustalsohavedamagestability(stabilitymargins)incaseofaccidents.

Typically,inroll GMT,min >0.5m whileinpitch GML,min ismuchlarger(morethan100.0m)

Atrade-offbetweenstabilityandcomfortshouldbemadesincealargestabilitymarginwillresultinlargerestoringforceswhichcanbequiteuncomfortableforpassengers(themechanicalequivalentisastiffspring).

Theloadcondition willdeterminetheheave,rollandpitchperiodsofamarinecraft.Theloadconditionvariesovertime(duetoloading,offloading,fuelburning,watertanks,etc.)

Inalinearsystem,thenaturalperiodswillbeindependentonthecoordinateoriginiftheyarecomputedusingthe6-DOFcoupledequationsofmotion.Thisisduetothefactthattheeigenvaluesofalinearsystemdonotchangewhenapplyingasimilaritytransformation!

1-DOFDecoupledAnalysis(NaturalPeriods)ThedecouplednaturalperiodsshouldbecomputedinCFusingthedecoupledequationsofmotion.Ifnot,theresultscanbeverywrongsincetheeigenvaluesofthedecoupledequationsdependonthecoordinateoriginasopposedtothe6-DOFcoupledsystem

4.3 Load Conditions and Natural Periods

!heave !C33

m " A33!!heave", Theave ! 2"

!heave

!roll !C44

Ix " A44!!roll", Troll ! 2"

!pitch !C55

Iy " A55!!pitch", Tpitch ! 2"

!pitch

Mustbesolvedbyiterationsinceaddedmassisafunctionoffrequency.Thisgivesanimplicitequationforfrequency.

4.3.1 Decoupled Computation of Natural PeriodsMSStoolbox:1-DOFDecoupledAnalysisfortheTankerModel(WAMITdata)

Theave ! 9. 68 sTroll ! 12. 84 sTpitch ! 9. 14 s

w_n ! natfrequency(vessel,dof,w_0,speed,LCF)

vessel ! MSS vessel data (computed in CO)

dof ! degree of freedom (3,4,5), use -1 for 6 DOF analysis

w_0 ! initial natural frequency (typical 0.5)

speed ! speed index 1,2,3...

LCF ! (optionally) longitudinal distance to CF from CO

load tanker

T_heave ! 2*pi/natfrequency(vessel,3,0.5,1)

T_roll ! 2*pi/natfrequency(vessel,4,0.5,1)

T_pitch ! 2*pi/natfrequency(vessel,5,0.5,1)

6-DOFCoupledAnalysis(NaturalPeriods)Inthe6-DOFcoupledcaseafrequency-dependentmodalanalysiscanbeusedtocomputethenaturalfrequencies:

Assumethatthefloatingvesselcarriesoutharmonicoscillations

Theundamped system()representsafrequency-dependenteigenvalueproblem:

4.3.2 Computation of Natural Periods in a 6-DOF Coupled System

Theeigenvalues mustbecomputedforallfrequencies

! ! cos!!t"i, i ! #1,1,1,1,1,1$! #

! ! "2 #

M!!"!" ! D!!"!# ! G! " 0 # M!!" ! MRB " A!!"D!!" ! B!!" " BV!!" " Kd

G ! C " Kp

!G ! !2M"!# ! j!D"!#$a ! 0 #

D!!" ! 0

|G !! iM!""| ! 0 #

MSStoolbox:6-DOFDecoupledAnalysisfortheTankerModel(WAMITdata)

dof ! -1 % use -1 for 6 DOF analysis

load tanker

T ! 2*pi./natfrequency(vessel,dof,0.5,1)

CoupledAnalysisDecoupledAnalysis

MSStoolbox:

Theave ! 2! m " A33!"heave"#gAwp!0"

Troll ! 2! Ix " A44!"roll"#g!GMT

Tpitch ! 2! Ix " A55!"pitch"#g!GML

loadcond(vessel)

R55 ! R66 ! 0.25LppR44 ! 0.37B

Offshorevessels:

R55 ! R66 ! 0.27LppR44 ! 0.35BTankers:

Ix ! mR442

Iy ! mR552

Iz ! mR662

Therollperiodclearlydependontheloadcondition—thatis,addedmomentofinertiaA44,massm radiusofgyrationR44 andmetacentricheightGMT

Radiusofgyration

Whathappensifyouincreasethemass?

Damped Harmonic Oscillator (Linear 2nd-order ODE without Forcing)

Relative damping ratio

Solution:

1) Will a ship oscillate with ω0when excited by regular waves?

2) Will the ship oscillate at different frequencies in 6 DOF?

Natural frequency (undamped angular frequency)

For ships we add waves as forcing

Forced Harmonic Oscillator

Comments: Resonance at ω/ω0 = 1.0

Marin craft oscillates at ω and not ω0 in all DOF when sufficient excited (fully developed sea with peak frequency ω)

Before “fully excited” the craft can oscillate with different frequencies in all DOF

Steady-state variation of amplitude with relative frequency ω/ω0 and damping ζ of a forced harmonic oscillator.

Phase:

Impedance:

4.3.2 Computation of Natural Periods in a 6-DOF Coupled SystemManyshipsareequippedliquidtankslikeballast andanti-rolltanks.Apartiallyfilledtankisknownasaslacktankandinthesetankstheliquidcanmoveandendangertheshipstability.

Thereductionofmetacentric heightGMT causedbytheliquidsinslacktanksisknownasthefree-surface-effect.

Themassoftheliquidorthelocationofthetanksplaynorole,onlythemomentofinertiaofthesurfaceaffectsstability.

The effectivemetacentric heightcorrectedforslacktanksfilledwithseawateris

Free-surface-correction(FSC)forNtanks

Rectangulartankwithlengthl inthex-directionandwidthb inthey-direction

GMT,eff ! GMT ! FSC #

FSC ! !r!1

N!m ir #

ir ! lb3

ir isthemomentofinertiaofthewatersurface

4.3.2 Computation of Natural Periods in a 6-DOF Coupled SystemThemetacentric heightGMT isreducedonboardashipifapayloadwithmassmp isliftedupandsuspendedattheendofaropeoflengthh.

The effectivemetacentric heightis

wherem isthemassofthevessel.

GMT,eff ! GMT ! hmpm #

Thedestabilizingeffectappearsimmediatelyafterraisingtheloadsufficientlytoletitmovefreely

M !! " C!!"! " D!!"! " g!"" " go # # " #wind " #wavew

Afloatingorsubmergedvesselcanbepretrimmed bypumpingwaterbetweentheballasttanksofthevessel.Thisimpliesthatthevesselcanbetrimmedinheave,pitchandroll:

4.4 Ballast Systems

z ! zd, ! ! !d, " ! "d 3modeswithrestoringforceandmoments

Steady-statesolution:

!d ! !!,!,!,zd,!d, "d,!"!where

Theballastvectorgo iscomputedbyusinghydrostaticanalyses(steady-statecondition).

mainequationforballastcomputationsg!!d" ! go " w #

Consideramarinecraftwithn ballasttanksofvolumesVi≤Vi,max (i=1,…,n)

Foreachballasttankthewatervolumeisgivenbytheintegral

4.4 Ballast Systems

Vi!hi" ! !o

hi Ai!h"dh " Aihi, (Ai!h" ! constant)

Zballast ! !i!1n Wi ! !g!i!1

ThegravitationalforceinheaveduetoWi is:

4.4 Ballast Systems

RestoringmomentsduetotheheaveforceZballast

rib ! !xi, yi , zi"!, #i ! 1," ,n$BallasttanklocationswithrespecttoCO:

Kballast ! !g!i!1

Mballast ! "!g!i!1

m ! r ! f

Zballast

yZballast

!xZballast

ZballastKballastMballast

! i!1n Vi

!i!1n yiVi

"!i!1n xiVi0

Resultingballastmodel:

G!3,4,5"!d!3,4,5" ! go!3,4,5" " 0

!Zz 0 !Z!0 !K" 0

!Mz 0 !M!

zd"d!d

!"i"1n Vi

!"i"1n yiVi

"i"1n xiVi

Trimmingisusuallydoneundertheassumptionsthatandaresmallsuch:

4.4.1 Conditions for Manual Pretrimming

Reduced-ordersystem(heave,roll,andpitch):

g!!d" ! G!d

Steady-statecondition: Thisisasetoflinear

equationswherethevolumesVi canbefoundbyassumingthatw=0(zerodisturbances)

G!3,4,5" !

!Zz 0 !Z!0 !K" 0

!Mz 0 !M!

go!3,4,5" ! !g

!"i!1n Vi

!"i!1n yiVi

"i!1n xiVi

!d!3,4,5" ! #zd,!d,"d $!

w!3,4,5" ! #w3,w4,w5$!

Assumethatthedisturbancesinheave,roll,andpitchhavezeromeans.Consequently:

!Zz 0 !Z!0 !K" 0

!Mz 0 !M!

zd"d!d

#g"i!1n Vi " w3

#g"i!1n yiVi " w4

!#g"i!1n xiVi " w5

! ! H!y ! H"!HH""!1y

ThewatervolumesVi isfoundbyusingthepseudo-inverse:

!g1 ! 1 1y1 ! yn!1 yn!x1 ! !xn!1 !xn

!Zzzd ! Z""d!K##d!Mzzd ! M""d

H! ! y"

w!3,4,5" ! #w3,w4,w5$! ! 0

Courtesy to SeaLaunch

Example4.3(Semi-SubmersibleBallastControl) Considerasemi-submersiblewith4ballasttankslocatedatInaddition,yz-symmetryimpliesthat

r1b ! !!x,!y", r2b ! !x, !y",r3b ! !x, y",r4b ! !!x,y"Z! ! Mz ! 0

H ! !g1 1 1 1!y !y y yx !x !x x

!Zzzd!K""d!M##d

!gAwp!0"zd!g"GMT"d!g"GML#d

V1V2V3V4

! 14!g

1 ! 1y 1x

1 ! 1y ! 1x1 1

y ! 1x1 1

!gAwp!0"zd!g"GMT"d!g"GML#d

! ! H!y ! H"!HH""!1y

Inputs: zd ,!d," d

4.4.2 Automatic Pretrimming using Feedback from z, φ and θInthemanualpretrimming caseitwasassumedthatw{3,4,5}=0.Thisassumptioncanberelaxedbyusingfeedback.

Theclosed-loopdynamicsofaPIDcontrolledwaterpumpcanbedescribedbya1st-ordermodelwithamplitudesaturation:

Tj (s) isapositivetimeconstantpj (m³/s)isthevolumetricflowratepumpjpdj isthepumpset-point.

Thewaterpumpcapacityisdifferentforpositiveandnegativeflowdirections:

Tjp! j " pj # sat!pdj"

sat!pdj" !

pj,max" pj # pj,max"

pdj pj,max! " pdj " pj,max"

pj,max! pdj $ pj,max!

Example4.4(Semi-SubmersibleBallastControl,Continues):Thewaterflowmodelcorrespondingtothefigureis:

4.4.2 Automatic Pretrimming using Feedback from z, φ and θ

V! 1 " !p1V! 2 " !p3V! 3 " p2 # p3V! 4 " p1 ! p2

Tp! " p ! sat!pd"#! ! Lp

V1V2V3V4

p1p2p3

!1 0 00 0 !10 1 11 !1 0

4.4.2 Automatic Pretrimming using Feedback from z, φ and θFeedbackcontrolsystem:

Hpid!s" ! diag#h1,pid!s", h2,pid!s", . . . ,hm,pid!s"$

Tp! " p ! sat!pd"#! ! Lp

Equilibriumequation:

Dynamics:

G!3,4,5"!!3,4,5" ! go!3,4,5"#"$ " w!3,4,5" #

pd ! Hpid!s"G#3,4,5$ !d#3,4,5$ ! !#3,4,5$ #

Anexampleofahighlysophisticatedpretrimming systemistheSeaLaunch trimand heelcorrectionsystem (THCS):

4.4.2 Automatic Pretrimming using Feedback from z, φ and θSeaLaunch:

Thissystemisdesignedsuchthattheplatformmaintainsconstantrollandpitchanglesduringchangesinweight.Themostcriticaloperationiswhentherocketistransportedfromthegarageononesideoftheplatformtothelaunchpad.Duringthisoperationthewaterpumpsoperateattheirmaximumcapacitytocounteracttheshiftinweight.

Afeedbacksystemcontrolsthepumpstomaintainthecorrectwaterlevelineachofthelegsduringtransportationoftherocket

http://www.sea-launch.com

4.4.2 Automatic Pretrimming using Feedback from z, φ and θSeaLaunch Trim and Heel Correction System (THCS)

Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)Courtesy to SeaLaunch

SMMarine Segment

0 187.5 375 562.5 750 937.5 1125 1312.5 150021.5

Roll and pitch during launch time (secs)

420 430 440 450 460 4702

Measured pitch during launch

time (secs)

A 1< >jp

470420 jp20 10 0 10 20 30

Calculated pitch motionstime (secs)

Z 4< >l

29.77515 Z 1< >l

4.4.2 Automatic Pretrimming using Feedback from z, φ and θRollandpitchanglesduringlift-off

CNN10th October 1999

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