chapter 4 –hydrostatics - · pdf file1 lecturenotes ttk 4190 guidance and control of...

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.

2

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)

3

UnderwaterVehicles:AccordingtotheSNAME(1950)itisstandardtoexpressthesubmergedweight ofthebodyandbuoyancyforce as:

=waterdensity=volumeoffluiddisplacedbythevehicle

m =massofthevesselincludingwaterinfreefloodingspace

g =accelerationofgravity

z

fg

CGCB

fb

n

n

4.1 Restoring Forces for Underwater Vehicles

W ! mg, B ! !g!

fgn !

00W

fbn ! !

00B

TheweightandbuoyancyforcecanbetransformedfromNEDtoBODYby:

fgb ! Rbn!!"!1fg

n, fbb ! Rbn!!"!1fb

n

!

!


4

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

Wedenotethegeneralizedrestoringforces . Noticethattheforceandmomentvectorsaremultipliedwith-1.

Consequently,thegeneralizedrestoringforceinBODYwithcoordinateoriginCObecomes:

where

4.1.1 Hydrostatics of Submerged Vehicles

g!!" ! !fgb " fb

b

rgb ! fgb " rbb ! fb

b

! !Rbn!""!1!fg

n " fbn"

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

n #

mib ! rib!f ibfi

b

g!!"

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

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


5

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!

z

fg

CGCB

fb

n

n


Copyright © Bjarne Stenberg/NTNU

6

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!!" !

000

!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!


7

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.


8

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).

9

Forconventionalrigsandshipswithbox-shapedwallsitcanbeassumedthat:

Thisexpressionisconstantforsmallperturbationsinz.Hence,therestoringforceZ willbelinear inz,thatis:


Z ! "!gAwp!0"Zzz

Awp!!" ! Awp!0"

Z

zAwp!z"

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

!frb ! Rbn!!"!1

00

!"g "0

z Awp!#"d#

TherestoringforcesandmomentsdecomposedinBODY:


10

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

4.2.1 Hydrostatics of Floating VesselsGMT sin! GML sin !

W ! B ! !g!

rrb !

!GML sin !GMT sin"

0

frb ! Rbn!!"!1

00

!#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

b

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

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

g!!" ! !!frb

mrb



11


g!!" !

!!g "0

zAwp!""d" sin #

!g "0

z Awp!""d" cos # sin$

!g "0

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:


12

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" #"

!

00

!gAwp!0"z

!g#GMT#

!g#GML"

0

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

!0

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

!, ", z

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

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

Linearkinetics:


13

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

where

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

" 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

14

4.2.3 Computation of Metacenter Height for Surface Vessels

MetacenterM,centerofgravityG andcenterofbuoyancyB forasubmergedandafloatingvessel.ThereferenceisthekeellineK.


15

Forsmallrollandpitchanglesthetransverseandlongitudinalradiusofcurvaturecanbeapproximatedby:

wherethemomentsofareaaboutthewaterplanearedefinedas:


GMT ! BMT ! BG, GML ! BML ! BG

K

M

B

G

BMT ! IT! , BML ! IL

!

IL ! ! !Awpx2dA, IT ! ! !

Awpy2dA

ForconventionalshipsanupperboundontheseintegralscanbefoundbyconsideringarectangularwaterplaneareaAwp=BL whereB andL arethebeamandlengthofthehullupperboundedby:

IL ! 112 L

3B, IT ! 112 B

3L


ThemetacenterheightM canbecomputedbyusingbasichydrostatics:

16

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).



17

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"

!roll

!pitch !C55

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

!pitch

#

#

#

Mustbesolvedbyiterationsinceaddedmassisafunctionoffrequency.Thisgivesanimplicitequationforfrequency.


18

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)


19

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

Assumethatthefloatingvesselcarriesoutharmonicoscillations

Then

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 #

20


MSStoolbox:6-DOFDecoupledAnalysisfortheTankerModel(WAMITdata)


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

load tanker

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


CoupledAnalysisDecoupledAnalysis


21


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?

22

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

23

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:

24

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

12 #

ir isthemomentofinertiaofthewatersurface


25

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

p


26

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

#

Afloatingorsubmergedvesselcanbepretrimmed bypumpingwaterbetweentheballasttanksofthevessel.Thisimpliesthatthevesselcanbetrimmedinheave,pitchandroll:

4.4 Ballast Systems

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

Steady-statesolution:

XX X

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

Theballastvectorgo iscomputedbyusinghydrostaticanalyses(steady-statecondition).

mainequationforballastcomputationsg!!d" ! go " w #


27

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

n Vi

ThegravitationalforceinheaveduetoWi is:

Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)Copyright © Bjarne Stenberg/NTNU

28

4.4 Ballast Systems

RestoringmomentsduetotheheaveforceZballast

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

Kballast ! !g!i!1

n

y iVi

Mballast ! "!g!i!1

n

x iVi

#

#

m ! r ! f

!

xyz

!

00

Zballast

!

yZballast

!xZballast

0

#

go !

00

ZballastKballastMballast

0

! !g

00

! i!1n Vi

!i!1n yiVi

"!i!1n xiVi0

Resultingballastmodel:


29

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

! #g

!"i"1n Vi

!"i"1n yiVi

"i"1n xiVi

" 0

#

Trimmingisusuallydoneundertheassumptionsthatandaresmallsuch:

4.4.1 Conditions for Manual Pretrimming

Reduced-ordersystem(heave,roll,andpitch):

!d !d

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$!


30

Assumethatthedisturbancesinheave,roll,andpitchhavezeromeans.Consequently:

and


!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

V1V2"

Vn

!

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

H! ! y"

#


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

Courtesy to SeaLaunch

31

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


P P

PP

P P

V1 V2

V4 V3

xb

yb

O

p1

p2

p3+

+

+

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

y !

!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

y1x

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

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

Inputs: zd ,!d," d


32

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!


33

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

, p "

p1p2p3

, L "

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



34

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$ #


35

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

36

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

37


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

38

SMMarine Segment




39

0 187.5 375 562.5 750 937.5 1125 1312.5 150021.5

10.5

00.5

11.5

22.5

33.5

44.5

55.5

6

Roll and pitch during launch time (secs)

roll

and

pitc

h (d

eg)

420 430 440 450 460 4702

0

2

4

6

Measured pitch during launch

time (secs)

Pitc

h ang

le (d

eg)

4.21

0.95

A 1< >jp

470420 jp20 10 0 10 20 30

2

0

2

4

6

Calculated pitch motionstime (secs)

pitch

angl

e (de

g)

4.326

0.202

Z 4< >l

180p

.

29.77515 Z 1< >l

roll

pitch

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

CNN10th October 1999



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