605
1 INTRODUCTION
Shiphandlinghasbeenviewedbymanyasanʹart‘,
meaning that it cannot be performed by scientific
calculationsalone,butmustalsobereliedupononeʹs
own experience and intuition. One of the factors
contributed to this view was the concept of ‘pivot
point’. It has been the central and import
ant tool in
shiphandling,unfortunately,however, ithasbeena
ratherambiguousentity,resultinginsomeconfusion
andmisuse amongstshiphandlers.Yetpractitioners
hasbeentryingtounderstandships’motioninterms
ofit.
In recent yearsa number ofauthors gave clearer
expositionsofit‐
TsengC‐Y(1998),ArtyszukJ(2010),
Seo S‐G (2011‐6), for example‐and demonstrated
whatcanbeachievedwiththecorrectunderstanding.
These enabled the practitioners to have an
unambiguouspictureoftheconcept.
2 THECONCEPTOFPIVOTPOINT
Theconceptofpivotpointhasbeenanessentialtool
inshiphandling.Theknowledgeab
outthepositionof
the pivotpoint ina manoeuvring situationprovides
the ship handler with the information on the
geometry of motion of the ship. Rowe R W (2000),
ClarkI(2005),CauvierH(2008)andBauduH(2014).
Hence,ithasbeenarequirementfortheshiphandler
tounderstand howandwhytheshipbeha
vesineach
condition.
Ship’smotioninaconfinedareacanbemodelled
as a planar rigid body motion assuming no vertical
movementofanypointoftheship.Thisisjustifiedfor
therelativelycalmfreesurfaceinsuchanarea.
Whenswayandyawoccursimult
aneously,aship
handler can only perceive the combined effect of drift
andturn,whichgiveshimafalseimpressionthatonly
arotationalmotionhappenedaboutacertainpointon
theship’scentreline.Thisapparentcentreiscalledthe
Pivot Point of the ship. This is a simp
lification of
Safer and More Efficient Ship Handling with the Pivot
Point Concept
S.G.Seo
SouthamptonSolentUniversity,Southampton,UK
ABSTRACT:Theconceptofthepivotpointofaturningshiphasbeeninexistenceformorethantwocenturies.
Itwasnot,however,properlyunderstoodfromthebeginning,andthussomemisconceptionsdeveloped.This
inturncausedittobeviewedassomethingmystical,thuspreventingshiphandlingfromscient
ificapproach.
Theconceptisexpoundedinafreshlight,derivinganequationforthedefinitionandothersforthecalculation
ofthepivotpointlocationbothingeneralandforspecificexamplesinanidealizedcondition.Theimplications
ofthederivedequationsarediscussed.Theresultsofaverifica
tionexperimentarepresented,whichproved
centuries’ of teachings and learnings to have been incorrect. A number of exercises for both steady and
unsteadycaseshavebeensuggestedforthetrainingofthepractitionersinthelightofthesenewfindings.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 10
Number 4
December 2016
DOI:10.12716/1001.10.04.09
606
perceptionfromtwomotionsdowntoonemotion,whichis
the very reason why the pivot point concept is so
usefultoshiphandlers.
2.1 HowisthePivotPointbroughtabout?
Water being a yielding material, any active force
turningashipwillcausedriftingmotionatthe
same
time. In the case when ‘drift’ happens before ‘yaw’,
thetwomotionsinsequence areshown in Figure1.
The thick red arrow indicates the drift and the thin
arrowsdepictstheyawmotion.
Figure1.ActualMotion(SwayandYaw)
AfewnotesonFigure1:
Theshipisdepictedasturningaboutthecentreof
gravity,G2.
ThedistanceofG1fromPisnotthesameasthat
ofG2fromP.
Pisnotafixedpointintheshipbutdependenton
thesway
andtheyawmotion.
Forabriefmoment,G1andG2canbetakentobe
approximately the same distance from P. The two
motionsoftheshipinFigure1canthenbeimaginedas
aturningmotionaboutP,asshowninFigure2.This
imaginarymotion,when
takenoveraninfinitesimally
smalltimeinterval,iscorrectasfarasthekinematics
oftheshipisconcerned.Thisisnormallythewaythat
aship’smotionisperceivedbyshiphandlers.
Point“P”istraditionallycalledthe‘pivotpoint’of
theship.Forpracticalpurposes,thisisan extremely
usefulconceptbecauseitreducestwomotionsdown
toone.
Figure2:ApparentMotionforaSmallTimeInterval(Yaw
only)
Figure2actuallydepictsapureturningmotionof
a rigid body in which every point in the ship is
moving along a concentric circle centred on P.
Therefore,the planarmotionofa shipin aconfined
area could be represented by the composition of a
surge motion and a
yaw motion about P only, by
usingthepivotpointconcept.
2.2 TheMathematicalDefinitionofthePivotPoint
Among all the points in the ship in planar motion,
thereisonlyonepointonthecentrelineatwhichthe
sway and yaw completely cancel each other, thus
making this
point seem to be stationary. All other
points appear to be turning about this point. This
pointisthePivotPointasexplainedin2.1.Ifthesway
speedandyawspeedareknown,thepositionofthe
pivotpointcanbeobtainedasthedistancefromthe
centreofmass
(GP)usingEquation(1).
v+(GPxr)=0 (1)
where, v(m/s) = sway speed of G; G = Centre of
Gravity;P=PivotPoint;GP(m)=distance
toPfrom
G;r(rad/s)=yawSpeed.
2.3 ThePivotPointPositionduetoaTurningForce
Since the pivot point is defined on the centerline of
theship,onlyonedimensionalcoordinatesystemwill
sufficeforourpurpose.Theverticallinethroughthe
centre of gravity is taken as
the origin, one side of
which is taken as positive direction, the other side
negativedirection.SeeFigure3.
Figure3.1‐DCoordinateSystem
Whena forcecauses ashiptodriftand turn,the
centreofgravitywillmoveduetothedriftmotion.
2
1
1F
GG t
2
(2)
where,Fistheforce;∆isthemassdisplacementofthe
ship;tisthetimetaken.
The arcdrawn byG inan imaginary yawmotion
withPasthepivotalpointis:
2
1
1F (GF)
(arc)GG GP t
2
(3)
where, GF is the distance from G to F, the negative
signindicatingtheothersideofG(origin)fromP;Iis
the second moment of mass of the ship about the
origin.
For a minuscule change of heading, GG
1 can be
equatedto(arc)GG
1giving,
GP
(GF)
(4)
607
Equation(4)givesthepositionofthepivotalpoint
in terms of GP. This pivotal point (P) is naturally
calledthe‘Pivot Point’of the shipeven thoughit is
imaginary. Under the assumption of a solid ship of
uniformdensitywithmultiplenumberofcontrolling
forces,Equation(4)
becomes:
2
C
1
GP r dV
VGF
(5)
where, V is the volume of the ship; GF
C is the
longitudinaldistancealongthecentrelinebetweenG
and Fc, the position of the resultant of all applied
controlling forces; r in this equation is the radial
distanceoftheinfinitesimalvolumefromtheorigin.
Equation(5)reduces,foraboxbarge(C
B=Clp=1.0),
toanelegantlysimpleequation.
22
c
LB
GP
12GF
(6)
Awall‐sidedhullcouldbedefinedby:
2
B2x
y1
2L
(7)
where, L is the length of ship; x is the longitudinal
position of a point along L with the origin at
midships;yisthehalfbeam.
A Wigley Hull, of which the lines are shown in
Figure4,isdefinedby:
22
B2x z
y1 1
2L D
(8)
where,Disthedepthofship;zistheverticalposition
ofapointassumingtheshipiscompletelyimmersed
inwater.
Figure4.WigleyHullForm
AssumingFCatrudderstock(GFc=‐0.5L),B=L/7,
D = L/7, some calculations are carried out using
Equation5,andtheresultsareshowninTable1.
Table1.RelativePositionofthePivotPoints
_______________________________________________
Box Wall‐Wigley
BargeSidedHull
_______________________________________________
Cb1.02/34/9
I0.0017L
5
0.0007L
5
0.0005L
5
GP0.1701L 0.1023L 0.1016L
FromBow 0.3299L 0.3977L 0.3984L
toP
_______________________________________________
By comparing the results, one can deduce that a
bigger block coefficient will cause the pivot point to be
closer to the bow. The calculation was carried out
assuming an unresisting medium. The numerical
values are not the same as reality. However, the
qualitativedeductionsmadewouldstillbecorrect.
Anyapplied forceon aship inreal fluidwillset
her into motion. The gradually increasing motion
changes the aerodynamic and hydrodynamic
environment. The reactive forces increase until they
balance withtheactive forces. Bythen the shipwill
havegainedsomemomentum.Thismomentumadds
furthermovementinthe
pivotpoint position,which
settlesdownasthemotionbecomessteady.Inreality,
therefore,dealingthiswaywiththeunsteadyprocess
accurately is very difficult, if not impossible,
particularlywhenvariousforcesareinvolved.
2.4 InterpretationoftheformulaforthePivotPoint
Position
Twoimportantaspectsarenotedfrom
Equation(5).
Firstly,the minus(‐)sign indicatesthatthepivot
pointappearsontheothersideofGfromF
c.Secondly,a
biggerGF
cyieldsasmallerabsoluteGP,whichmeans
thatanexternalforcefartherawayfromGcausesthepivot
pointtobeclosertoG.Thesetwofindingsareessential
knowledgeforthepractitionerstoproactivelycontrol
thepivotpoint.
3 TRADITIONALLYHELDVIEWPOINTSONTHE
PIVOTPOINTANDDISCUSSION
3.1 TraditionalViewofthePivotPoint
A few examples of traditional views found in the
literaturearethatthepivotpoint:
moves towards the bow or towards the stern
dependingonthesenseofsurgemotion
isthecentreofrotation(yaw)
hasinstantaneousmovement
isthefulcrumoftheturningmoment
Alloftheabovefourviewsareincorrect.Thefacts
arethatthepivotpoint:
isindependentofsurgemotion,nottomentionthe
sense
isonlyanimaginarypoint
movesgradually
isnotaphysicalentity.
However,shiphandling
professionals,particularly
the seasoned practitioners, find it very difficult to
acceptthesefindings.Forexample,drawingslikethe
onebelowarecommonlyfoundintheliterature.
608
Figure5.MovingAsternandthePivotPoint
The situation is described as (i) two equally‐
powered tugsare pulling the shiplaterally withthe
same turning moment (ii) the engine starts making
sternway (iii)thenthe shipturnsbow tostarboard.
Supposedly,this isbecausethepivotpointisonthe
aftsideofthecentreof
gravity,thusgivinga longer
moment armtothe tug at bow. In other words, the
drawing is used for two fold purposes. One, as the
proof of the pivot point being near the stern when
moving astern, and two, as the explanation of the
headingchangeintermsof
thepivotpoint.
Theaccountismistakenintwoaspects.Oneisthat
thepivotpoint istreatedasa physicalentity(asthe
fulcrum).Theotheristhatthepivotpointistreatedas
acause.Thecorrectexplanationisthatwhentheship
starts moving astern, the centre of
lateral resistance
moves sternwards from midship. Thus the reactive
hydrodynamicforcesproviding anextraturningforce
aboutthecentreofgravity,whichisthenetforcethat
actuallyturnstheshipaboutitscentreofgravity.The
tugsarepullingtheshiptostarboard.Thestarboard
swaycausingthe pivot
pointtoappear between the
centreofgravityandthesternastheresult.
3.2 AVerificationExperiment
Historically,itwassaidthatthepivotpointislocated
near the stern when a ship is moving astern. An
example from a ship handling book says, “When
making sternway, the pivot point
moves aft and
establishesitselfapproximately1/4Lfromthestern.”
The derived Equation 5, however, implies that the
sense of surge motion is irrelevant with the pivot
pointlocation,butrather,ifthepropellerandrudder
combinationatthesternisusedastheonlypropulsor
system, the pivot point
will always appear near the
bow.
A verification experiment was conducted at
Warsash Maritime Academy Ship Handling Centre
on11
th
February2016.Theship,“Progress”,isa1:25
scalemodelofPanamax,LOA225m,beam32m.
Figure 6 shows the starting point. The ship’s
turningforcewasprovidedbysettingtheenginehalf
astern.Thepropellerisrighthandedwithfixedpitch.
Figure7showsthefinalposition.
Figure6.TheStartingPosition
Figure7.TheFinalPosition
Forthepurposeofanalysis,thewholeexperiment
wasdividedinto8timeintervals.Ineachinterval,the
result was analyzed calculating the position of the
pivot point as the average in the interval. The
positionsaregivenaspercentagelengthsbetweenthe
bow and the pivot point, to the length
of the ship.
TheyareshowninTable2.
Table2.PositionsofthePivotPoint
_______________________________________________
Interval 1 2 3 4 5 6 7 8
Position 16.7 16.7 16.7 17.0 17.4 17.0 15.4 14.8
_______________________________________________
Thetableshowsplainlythatthepivotpointwasat
around17%oftheshiplengthfromthebow.Nearthe
end of the experiment, it is obvious that the pier is
interferingwith thewater flow beingcreated bythe
propeller. This experiment conclusively proves that
thetraditionalteachingsand
learningsaboutthepivot
pointforcenturiesareincorrect.
4 SOMEBASICEXERCISESTOACTIVELY
CONTROLTHEPIVOTPOINT
Some basic exercises to actively control the pivot
point are suggested below. Some of them require
additionalmeansofapplyingactiveforcesotherthan
the conventional propeller and rudder combination,
such
asabowthruster,asternthruster,apod,tugs,
etc.
4.1 KeepingPcoincidedwithGwhileyawing
This is the case when the ship is yawing about the
centre of gravity, which is taken as the centre of
rotational motion. The ship has no translational
motion (no surge, no
sway). The pivot point (P)
coincideswiththecentreofgravity(G).Thecentreof
609
turningcircle(C)wouldalsobeatthesamepoint,ifit
werenotasingularpoint–Figure8.
The radius of swept area will be the minimum.
This will allow ship’s heading turned in a smallest
possiblearea.
Figure8.Yawonly,GandPcoincide
4.2 KeepingPbetweenGandBow,whileswayingand
yawing
In the absence of any longitudinal movement (no
surge),iftheshipdriftsatthesametimeasturning,
andifthepivotpointisbetweenGandthebow,the
motion shown in Figure 9 will result. The centre
of
turning circle (C) could have appeared at the same
spotasP,ifitwerenotasingularpoint.
Figure9.YawandSway,PbetweenGandBow
4.3 KeepingPforwardofBow,whileswayingandyawing
Figure10 depictsthe situation.Again, Ccould have
appearedatthesamelocationasP,iftherehadbeen
anysurgemotion.
Figure10.SwayandYaw,PaheadofBow
4.4 KeepingPcoincidedwithG,whileturning
If aship makes headwaywhile yawing but without
any sway motion, the resulting movement will look
liketheoneshowninFigure9.Inthiscase,thepivot
point(P)willcoincidewiththecentreofgravity(G).
Thismanoeuvrecausesno
swingoutofthestern,
thus it may be a necessary manoeuvre in a tightly
restricted waters. The area of sweeping path is the
minimum possible among the manoeuvres with the
sameturningcircleradiusbyP–comparewithFigure
12and13.
Figure11.SurgeandYaw,GandPcoincide
4.5 KeepingPbetweenGandBow,whilemakinga
generalplanarmotion
When all the three motions (Surge, Sway, Yaw) are
present,allthethreedistinctivepoints(P,G,C)will
exist separately as shown in Figure 10. In this
particular case, the stern swings out sweeping a
bigger area,
as all skilled ship handlers are most
conscious of. Ship motions in general fall in this
610
category. The position of the pivot point is directly
relatedtotheamountofswingout.
Figure12.GeneralPlanarMotion, PbetweenGandBow
4.6 KeepingPaheadofBow,whilemakingageneral
planarmotion
Thesweepingareaforthismanoeuvreismuchbigger
now than in Figure 10. However, this may be a
necessary manoeuvre for some situations such as
preparingtoenteranarrowpassage.
Figure13.GeneralPlanarMotion, PivotPointaheadofBow
5 SOMEPRACTICALMANOEUVRES
Nowthetrainingcouldcontinue onpracticingmore
routines in which the pivot point positions are
continuously changing. Good candidates for the
routinesareturningaroundasharpcorner,enteringa
narrowcutandturningshortround.
5.1 TurningShortRound
Figure14.ShortRound
1 Start manoeuvre at a slow speed, then hard to
starboardwithakickahead
2 Stopengines–ruddermidships
3 Engineastern,transversethrust continuestoturn
vessel.
4 Vessel at a stop over the ground, continue with
engines astern. Transverse thrust still acting on
vessel.
5 Enginestill
runningastern
6 Enginestillrunningastern,abouttostopengine.
7 Rudderhardtostarboard,engineahead.
8 Vesselcompletedshortround.
Note:Thisisasimplifiedversionratherthanwhat
wouldberequiredinreality.
5.2 EnteringaCut
Figure15.EnteringaCut
While preparing for the manoeuvre shown in
Figure14, the clearance from the jetty and the
longitudinalpositionarecruciallyimportantsoasnot
to come into contact with any port structureduring
themanoeuvre..
Thepivotpointwillinitiallyappearnearthecentre
ofgravity,notnearertothebowas
normallyquoted
inshiphandlingliterature,andthengraduallymove
forwardastheshipgainsdriftingmomentum.
5.3 SouthamptonContainerPort
The following sequence of screen shots have been
taken from the PPU on the departure of the “CMA
611
CGM Marco Polo” from Southampton. The pilots
portable unit is an AD Navigation ADX‐XR which
includes RTK, giving a very precise position. The
performancecriteriaare:‐
PositionAccuracy:1‐2cm(RTKmode)
0.8mwithEGNOS/WAAS
2muncorrectedGPS/GLONASS
BowandSternSpeed:1cm/sec
(0.02knots)
Vertical/Squat:2‐3cm(RTKmode)
Heading: 0.01degree(20mPODseparation)
RateofTurn:0.1degree/min
Figure16.
The“CMACGMMarcoPolo”isclearoftheberth,
movingastern.Thevectorsforthebowandsternare
indicated by the black arrow, whilst the predicted
positionofthevesselisoutlinedfor4positions.
Figure17.
The gray fill outline is the actual position of the
vessel (400m LOA). The vessel has 3 tugs in
attendance,andneedsto swingwithinthe swinging
grounddepictedbythepurplecircle.
Figure18.
The ship outlines with no fill, are the predicted
positionsofthevesselafterasettimeduration.Thisis
setbythepilot/operator.Thevectorofthebowand
sterncanalsobeseen,indicatedonthechartwiththe
black arrow from the centre line, fore and
aft
respectively.
Figure19.
This is the sort of manoeuvre as the exercise
shown in Figure 8. The ship is using tugs, in
combination with own engine, rudder and bow
thruster so as to maintain a rotational movement
about the vessels midship position. Bow velocity is
depictedat theright handsideofthe screen,
in this
instance1.6knotstoport,anddepictsthatthevesselis
moving 0.06 knots astern. The stern velocity is
presently1.76knotstostarboard.
612
Figure20.
This depicts the vessel having completed her
swing 141m off 107 berth proceeding outwards at
3.7knots
ScreenshotscourtesyofABPSouthampton.
6 CONCLUSION
For two hundred years or more the pivot point
location in ship handling has been a rather
ambiguousentity. Yetpractitioners were taughtand
practicedtounderstand
ships’motionintermsofit.
Logical reasoning and mathematical derivations
brought the mystified pivot point into the light,
rectifyingafewmistakenconcepts.Thisbroughtship
handling closer to science than art. With correct
understandingoftheconcept,shiphandlerscannow
makemanoeuvresmoreefficientandsafer.
All
ship handling books, instruction manuals,
lecture notes, etc. should have a major rewrite
reflectingthecorrectunderstandingofthepivotpoint
concept.
All deck officers need to be re‐educated and
retrained to the new paradigm that the pivot point,
which is the most important theoretical element of
shiphandling,is
nowscienceratherthanart.
REFERENCES&BIBLIOGRAPHY
BlackburnI.1836NavalArchitectureLongman,Rees,ORME,
&Co.London
Hwang W‐Y. 1980 Application of System Identification to
ShipManeuveringPhDThesis,MIT
TsengC‐Y.1998 Analysis of thePivotPointfor a Turning
ShipJournalofMaritimeScienceandTechnology
Andy Chase G. 1999 Sailing Vessel Handling
and
Seamanship – The Moving Pivot Point The Northern
MarinerJuly1999:53‐59
Grassi C. R. 2000 A Task Analysis of Pier Side Ship‐
Handling for Virtual Environment Ship‐Handling
Simulator Scenario Development Master’s Thesis, Naval
PostgraduateSchool
Rowe R.W. 2000 The Shiphandler’s Guide The Nautical
Institute
CauvierH.
2008ThePivotPointThePilotOctober2008
ArtyszukJ.2010PivotPointinshipmanoeuvringScientific
Journals2010:13‐24
Seo, Seong‐Gi. 2011. The Use of Pivot Point in Ship
Handling for Safer and More Accurate Ship
Manoeuvring. Proceedings of International Conference
IMLA19September2011:271‐280
BauduH.
2014ShipHandlingDokmarMaritimePublishers
Seo,Seong‐Gi.2015.Rethinkingthe PivotPointII.Seaways
(TheInternationalJournalofTheNauticalInstitute),August
2015:21‐23
Seo,Seong‐Gi. 2015.ParadigmShiftin Ship Handlingand
its Training. Safety of Marine Transport, PP89‐95, 2015,
CRC Press/Balkema, ISBN: 978‐
1‐138‐02859‐3, ISBN: 978‐
1‐315‐67261‐8(eBookPDF)
Seo,Seong‐Gi. 2015.A Paradigm ShiftinShipHandling –
The Pivot Point. Proceedings of MARSIM 2015, 8‐11
TH
September2015
*Special thanks are due to my student Mr. David
Tracy who went through the tedious calculation
processtoverifythevaluesinTable1.
