607
1 INTRODUCTION
The concept of e‐navigation tends to comprise and
address the range of objectives related to the wide
spectrum of shipping challenges. According to the
proposed e‐navigation strategy „e‐navigation is
expectedtoprovidedigitalinformationandinfrastructure
forthebenefitofmaritimesafety,securityandprotectionof
the marine environment, reducing the administrative
burdenandincreasingtheefficiencyofmaritimetradeand
transport” (IMO 2005). The declared effort
s in the
fieldsofsafety,securityandincreaseintheefficiency
of maritime transport are inherently associated to
many issues and ship stability among them. The
stability matters need to be ta
ken into account both
duringportoperationsandseavoyageaswell.Even
though large amplitudes of ship roll are highly
unlikelyduringcargohandlinginaport(Krataetal.
2013, Krata 2015), dangerous excessive rolling may
happenatseaway(Kobylinski&Kastner2003).Thus,
the ship st
ability, among others, ought to be
addressedbye‐navigationsolutions.
In 2014 IMO has prepared e‐navigation Strategy
Implementation Plan‐SIP (IMO 2014). The SIP
includes tasks defined for five prioritized solutions
requiredforasuccessfule‐navigationimplementation
and deployment and the task no. S 4.1.5 was
formulated as “Routing and filtering of information on
board (weather, intend
ed route, etc.)”. This shows that
weather routing solutionstaking into account safety
issuesresultingfromdynamicalstabilityphenomena
are considered by IMO as e‐navigation tools
benefiting in increasing safety and security of
shipping. Numerous works are also recently
publishedwhichaimatcombinat
ionofshipstability
performanceandweatherrouting(Decó&Frangopol
2013;Dongetal.2016).
Prediction of Ship Resonant Rolling - Related
Dangerous Zones with Regard to the Equivalent
Metacentric Height Governing Natural Frequency of
Roll
P.Krata&W.Wawrzyński
GdyniaMaritimeUniversity,Gdynia,Poland
ABSTRACT: Potentially dangerous zones corresponding to dynamical stability phenomena, possibly
encounteredbyshipssailinginroughsea,areestimatednowadayswiththeuseofthemethodrecommended
by IMO in the guidance coded MSC.1/Circ.1228. In this IMO method the parameter governing the natural
periodofrollistheinit
ialmetacentricheight.Someearlierstudiesrevealedthattheinitialmetacentricheight
whichiscommonlyinuseon‐boardshipsforthepurposeofperformingtheMSC.1/Circ.1228‐recommended
calculations, may significantly vary from the so called equivalent metacentric height obtained for large
amplitudesofship’sroll.Inthelightofsuchascert
ainment,thepaperdealswithresultantresonancerollzones
locationswithregardtotheequivalentmetacentricheightconceptremainingappropriateforlargeamplitudes
of roll. The noteworthy transfer of the resonance zones location is disclosed which reflects the distinct
configurationsofpotentiallydangerousship’scourseandspeedconfigurationstha
ncouldbepredictedonthe
basistheinitialmetacentricheight.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 11
Number 4
December 2017
DOI:10.12716/1001.11.04.05
608
Weatherroutingapplicationscommonlytakeinto
accountsome hazards resulting fromthe ineluctable
rollingofshipswhichmayhappenatseaway(Krata
&Szlapczyńska2011,Amiseaware,Amarcon).They
aredesignedtohelpnavigatorsto calculate theship
coursespronetoarousethesynchronousrollandthe
parametric resonance,
since under the resonance
conditions the ship rolling may develop excessively
causingthedirectthreattotheship,cargoandcrew
(Kobylinski & Kastner 2003). One could expect that
the commonly accepted ship stability standards are
effective solution for stability‐related accidents.
Unfortunately, the decades of experience revealed
numerous incidents
resulted from dynamic
phenomenatakingplace inroughsea whichare not
covered by the contemporary stability standards
describedintheIMOISCode(IMO2008).Oneofthe
significant respondwas publishing in 1995 the IMO
guidanceMSC/Circ.707andinlateryearsdeveloped
and revised to MSC.1/Circ.1228, published in 2007.
This Revised guidance to the master for avoiding
dangeroussituationsinadverseweatherandseaconditions
isintended to givesome help toship masters when
sailing in adverse sea conditions. This publication
containsastraightforwardsetofremarksandadvices
regarding the avoidance of following dangerous
dynamical phenomena at
sea like surf‐riding and
broaching‐to,reductionofintactstabilitywhenriding
ona wavecrestamidships, synchronousrolling and
parametricrollmotions(IMO2007).Thelasttwoare
strictlyrelatedtothenaturalperiodofship’srolland
itspossibletuningwiththeencounteredwaveperiod.
The notion of
the natural period of roll plays a
vital role in the resonance determination procedure.
According to the guidance IMO MSC.1/Circ.1228
navigatorsareinstructedtoobtainthenaturalperiod
of roll by use of stop watch in calm seas at each
departure, which is usually hardly feasible, or
alternatively on the
basis of IMO recommended
simplified formula, referred later in this paper as
formula(3).Unfortunately,bothmethodsarenotvery
reliable. The natural period of roll for large
amplitudesmayconsiderablydifferfromobservedin
calm seas(small roll amplitudes) and the calculated
onewhentheformula(3)isapplied(amplitudes
close
to zero). Therefore the conditions allowing for
building up an excessive roll motion due to
parametric or synchronous rolling may be
significantly different than expected by navigators
nowadays(Wawrzynski&Krata2016a,2016b).Thus,
a new reliable method for the resonant rolling
conditions prediction was proposed (Wawrzynski &
Krata
2016a). The considered problem refers to the
scope of influence of the equivalent metacentric
height application on the locations of resonance
zones.
The rest of the paper is organized as follows.
Section2 introducesthe background ofthe ship roll
motion.Section3discussesresultsoftheresearch,e.g.
locations of
the resonance zones with regard to the
equivalent metacentric height. Finally, Section 4
presentsconcisesummaryandconclusions.
2 ROLLMOTIONOFTHESHIPMODELEDAS
THENONLINEARDYNAMICALSYSTEM
The ship rolling under external excitations is
commonly described as a nonlinear dynamical
system, for instance in the form of
formula (1). All
crucial components of this equation, e.g. restoring,
damping and excitation, may reveal nonlinearities,
however the most essential parameter determining
the nonlinearity of rolling is the restoring moment.
This moment is strictly related to the ship stability
which is generally described by the righting arm
curve(theGZcurve),
althoughsomeauthorsusethe
initialmetacentricheightasthestabilitycharacteristic.
The earlier studies show that the GZ curve and it
nonlinearities plays vital role regarding to the
resonancemodeofroll (Wawrzynski&Krata2016a,
2016b).
Therollequationappliedtosolvetheshiprolling
problemisformulated
asfollows:
2
2cos
we
x
g
GZ t
r
(1)
where μ is the damping coefficient, g is the gravity
acceleration, r
x is the gyration radius of a ship and
added masses (which is assumed to be constant for
the sake of simplicity), GZ is the righting arm, ξ
w
meanstheexcitingmomentcoefficientandω
edenotes
externalmomentfrequency.
The typical phenomenonrelated to oscillations is
theresonance.Whennonlinearoscillationstakeplace,
resonancefrequencyisalmostconstantforverysmall
amplitudes of motion while it depends on the
amplitude of the oscillations for large amplitudes
which are typical for ship rolling in rough sea.
The
exemplarycasescanbeseeninfigure1wheretheroll
spectraaredeterminedonthebasisoftheequation(1)
with the use of two approaches: first, the linear
righting arm which is written in the form
GZ(ϕ)=GM
0∙ϕ (left plot) and second, the real
nonlinear characteristic of the righting arm (right
plot).Theresonancefrequenciestracethemaximaof
the roll spectra obtained for the increasing value of
theexcitationmoment.Itcanbeclearlyseenthatthe
simplified approach(left plot) producesthe maxima
appearing for one
constant value of the roll
frequency,whilethemoreaccurateapproach,taking
intoaccounttheactualnonlinearityoftheGZcurve,
results in more complex pattern of the roll spectra
maximatrack(rightplot).Thevalueofthenaturalroll
frequency strictly depends on the amplitude of roll
which is
omitted nowadays in the IMO
recommendations provided in the MSC.1/Circ.1228
guidanceforshipmasters.
When the linear righting arm characteristic is
considered the maxima of roll spectra reflect the so
called natural frequency of ship roll which can be
easycalculatedaccordingtothewellknownformula
providedintheIntact
StabilityCode(IMO2008):
609
Figure1.RollspectrafordifferentvaluesofanexcitingmomentfortheLNGcarrier(T =12.00m,GM0=5.00m)calculated
withthelinearrestoringarm(leftplot)andactualnonlinearshapeoftheGZcurve(rightplot)
Figure2.Rollspectraandresonancecurvesforthe5000TEUPanamaxcontainership(T=7.50m,GM0=1.50m)(leftplot)and
forthegeneralcargoship(T=6.00m,GM
0=1.00m)(rightplot)
2
r
r
where
2
o
cB
GM
withthevalueofccoefficient:
0.373 0.023 0.00043
B
cL
T
(3)
where
isthenaturalperiodofroll,GM0istheinitial
metacentric height, c is the coefficient describing
transversegyrationradiusr
x(rx=c∙B),Bisthebreadth,
ListhelengthatthewaterlineandTisthemeandraft
oftheship.
Generally, the formula (3) is in common use on‐
board for the purpose of IMO MSC.1/Circ.1228
guidance application. In fig. 1 the line of resonance
frequencycalculatedusing
theGM0ismarked„GM0”
and itcan be seen that the maxima of all presented
curvescoincidewiththe„GM0”lineinleftplotwhile
theysignificantlydiffersinrightplotoffigure1.
The roll period and the resonance frequency,
taking the nonlinearityof the GZcurve and the roll
amplitude into account, can be calculated according
to the method proposed in (Wawrzyński & Krata
2016a).ThemethodisbasedontheareaundertheGZ
curveandtheaverageinclinationofthetangentline
totheGZcurve,bothcalculatedfromzerouptothe
rollamplitude.The main
formula looksvery similar
totheIMO’sone:
2
A
A
cB
GMeq
(4)
wherethevalueof c coefficientremains thesameas
given in the formula (3) and GM
eq(ϕA) denotes the
equivalent metacentric height for a specified roll
amplitudeϕ
A.
Theformulafortheequivalentmetacentricheight
calculationisthefollowing:
0
2
2
A
A
A
A
A
GZ d
GZ
GMeq
(5)
According to formula (5), the value of the
equivalent metacentric height GM
eq and as a
consequencealso therollperiod, dependon theroll
amplitude.Thisismakesupanoteworthydifference
comparing to the original IMO recommended
formula.
Theverificationofthismethodwasperformedfor
seven ships in different loading conditions (total
number of analyzed cases add up to 30),
and it
revealed very good consistency of the roll period
predictedbytheformulae(4)and(5)withtheresults
of numerical simulations for a wide range of roll
amplitudes(Wawrzyński&Krata,2016b).
Duetothecombinationwithformulas(4)and(5),
the roll resonance frequency depends on the
amplitude:
610
2
rA
A
(6)
Formula (6) combined with formulas (4) and (5)
was extensively analyzed (Wawrzyński & Krata
2016b).Inthecourseofthoseinvestigations,thefinal
formoftheformula(6)wasdetermined:
2
22
2
2
rA
A
(7)
Theelaboratedformulasallowtocalculatetheroll
resonance frequencyas a function ofamplitude and
further they are called the “GM
eq method”. The
influence of the GZ curve nonlinearity affecting the
rollcharacteristicswiththeincreaseinrollamplitude
can be conveniently presented in the form of the
resonance curve which is also called the backbone
curve‐inthefollowingfiguresthiscurveismarked
“GMeq”. In the research (Wawrzyński
& Krata,
2016b) the accuracy of the GM
eq method was
examined for different stability characteristics. In
mostcases,themethodrevealsverygoodagreement
with the results of rolling simulations. In Fig. 2, the
roll spectra for different values of an exciting
moment,calculatedfortwoships,arepresented.Each
curveshows therollamplitudeversusrollfrequency
withthemaximumreflectingtheresonancefrequency
ofoscillations.Itcanbeseen,thattheresonancecurve
calculated by the GM
eq method accurately traces the
maxima of all curves. However, for the loading
conditions where the simulation results reveal roll
amplitudebifurcations theresonancecurve obtained
fromGM
eqmethodisnotveryaccuratewhereasitstill
performs far better than the IMO recommended
formula(3).Although,thebifurcationisknownasthe
relatively difficult problem to modeling and
analyzing (Francescutto & Contento 1999). The
observed inaccuracy was found in regions with
significantjumpsoftheamplitude.
Regardlessthe
validationoftheproposedmethod
presented in (Wawrzynski & Krata, 2016b) this
methodwas applied in irregularwave conditionsto
theC11classcontainership(Acanforaetal.2017).The
advanced nonlinear numerical model called LaiDyn
was utilized. A set of ship motion simulations was
carriedouttocollecthistories of
rolloftheC11vessel.
TheirregularseamodeledwiththeuseofJONSWAP
spectrum was applied and numerous sea states and
the ship courses were analyzed. The study revealed
that the maxima of recorded roll spectra are much
closer to the natural frequency of the ship roll
obtainedaccording
totheproposedmethod(formula
4 and 5) than the natural roll frequency predicted
accordingtothecontemporaryformula(3)(Acanfora
et al. 2017). This study seems to be the strong point
speakinginfavoroftheproposedmethod.
3 EQUIVALENTMETACENTRICHEIGHT
VARIATIONANDTHERESULTANT
RESONANCEZONESLOCATIONS
As
the roll amplitude depended equivalent
metacentric height governs the natural frequency of
shiproll,theaccuracyofsynchronousrollprediction
isaffected.Thus,thewelljustifiedquestionisrelated
to the realistic extent of the equivalent metacentric
height variation. To assess this crucial matter the
series of calculations was carried
out aiming at the
typicalloadingconditionsofnumerouscargovessels.
Sixdifferentshipsweretakenintoaccounttocovera
varietyofshippurpose,theirdifferentparticularsand
loading condition from ballast up to fully loaded
ones. The main particulars of considered ships are
givenintable1.
Theequivalent
metacentricheightwascalculated
according to formula (5) for all the vessels listed in
Table1.Thewiderangeofpossibleamplitudesofroll
was taken into account. The performed series of
calculationsrevealedthattheequivalentvalueofthe
metacentric height may vary significantly according
totheamplitudeof
rollinmanycasesbutnotinallof
them. Generally, the range of variations depends on
theshapeoftheGZcurve.
The research reveals that in almost half of
examined cases the equivalent metacentric height
may significantly vary comparing to the initial GM
evenby50%whichispresented
inFig.3.Moreover,
inoneeighthofcasestheGM
eqiseventwiceaslarge
as the initial GM. The GM
eq variations ranging in
extreme cases from 169% to 10% which reflects
different shapes of GZ curves remaining close to
linear in low or wide range of angles of heel,
respectively. In the light of the preformed analysis
onemaystatethattheformula(3)shouldbereplaced
by the formula
(4) or (7) to address the problem of
accuratepredictionofpossiblesynchronousrollofthe
ship.
Table1.Mainparticularsofshipsconsideredinthecourse
oftheresearch.
_______________________________________________
Typeofvesselconsidered
metacentric
length breadthdraftT heightGM0
__________________
min max min max
[m] [m] [m] [m] [m] [m]
_______________________________________________
1 generalcargoship 140.00 22.006.00 9.00 0.40 1.00
2 bulkcarrier156.10 25.906.50 10.501.00 3.50
3 LNGcarrier 278.80 42.607.50 12.001.00 5.00
4 5000TEUPanamax283.20 32.207.50 13.500.50 3.00
(old)containership
5 7500TEUcontainer285.00 45.607.50 12.502.00
5.00
ship
6 motortanker 320.00 58.0010.0022.005.00 10.00
_______________________________________________
0 20 40 60 80 100 120 140 160 180
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Data
Cumulative probability
d_GM data
Log-Logistic
50
Figure3. Cumulative distribution function for the
equivalentmetacentricheightvariation
611
0 0.2 0.4 0.6 0.8 1
roll amplitude (rad)
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
extreme GM variation = 169%
GM equivalent
GM initial
0 0.2 0.4 0.6 0.8 1
angle of heel (rad)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
draft T= 6 m; GM= 0.4 m
GZ curve
tangential line (initial GM)
0 0.2 0.4 0.6 0.8 1
roll amplitude (rad)
16
18
20
22
24
26
28
30
natural period of roll T
N
T
N
based on GM
eq
T
N
based on initial GM
Figure4. General cargo ship, LBP = 140.00 m, B = 22.00 m. Ship stability characteristics and the natural period of roll
calculatedaccordingtotheformula(5)–upperplots.Locationofresonancezonedobtainedfortheinitial metacentricheight
(lowerleftplot)andfortheequivalentmetacentricheight(lower
rightplot)
Figure5.Bulkcarrier,LBP=156.10m,B=25.90m.Shipstabilitycharacteristicsandthenaturalperiodofrollcalculated
accordingtotheformula(5)–upperplots.Locationofresonancezonedobtainedfortheinitialmetacentricheight(lowerleft
plot)andfortheequivalentmetace ntricheight(lowerright
plot)
612
GM (m)
GZ (m)
natural period of roll (s)
Figure6.LNGcarrier,LBP=278.80m,B=42.60m.Shipstabilitycharacteristicsandthenaturalperiodofrollcalculated
accordingtotheformula(5)–upperplots.Locationofresonancezonedobtainedfortheinitialmetacentricheight(lowerleft
plot)andfortheequivalentmetace ntricheight(lowerright
plot)
00.20.40.60.81
roll amplitude (rad)
4
5
6
7
8
9
10
11
extreme GM variation = 56%
GM equivalent
GM initial
00.20.40.60.81
angle of heel (rad)
0
2
4
6
8
10
draft T= 10 m; GM= 10 m
GZ curve
tangential line (initial GM)
00.20.40.60.81
roll amplitude (rad)
14
16
18
20
22
natural period of roll T
N
T
N
based on GM
eq
T
N
based on initial GM
Figure7. Motor tanker, LBP=320.00 m, B=58.00 m. Ship stability characteristics and the natural period of roll calculated
accordingtotheformula(5)–upperplots.Locationofresonancezonedobtainedfortheinitialmetacentricheight(lowerleft
plot)andfortheequivalentmetace ntricheight(lowerrightplot)
613
The locations of resonance‐related dangerous
zones established according to the IMO Circ. 1228
recommendations,varyincorrespondingextend.The
calculations and graphical presentation of resonance
zones locations are performed with the use of a
software tool described in (Krata & Szlapczyńska
2011).TheshipsmentionedinTable1
areconsidered
and the dangerous zones are obtained for the
assumed wind force 8B and the sea state reflecting
fully developed wave system with the dominant
waveperiodequalto11seconds.Thesampleresults
ofcalculationsareshowninFig.4to7.
In case of relatively small difference
of the
metacentric height, comparing the initial and
equivalent value, the resonance zones are located
veryclose(seeFig.7).Thismeansthatregardlessthe
formulaenablingthenaturalperiodofrollcalculation
(formula3or4)theconfigurationofship’sspeedand
courses to be avoided by the officer of
the watch is
alike.Suchasituationtakesplace whenGZcurve is
close to linear for a wide range of angles of heel.
Otherwise, the reliable resonance zones obtained on
thebasisoftheequivalentmetacentricheightcanbe
far from the inaccurate expectations resulting from
theinitialGM
utilization(seeFig.4and5).
ThedangerouszoneslocationsshowninFig.4to7
are established for the dominant wave period equal
11 s. This allows for presentation of the direct
dependence of the potentially dangerous courses
arrangement on the equivalent metacentric height
obtained for the realistic range of
roll amplitudes.
According to thesegraphs one may suspect that for
somevesselstheresonanceproblemactuallydoesnot
exist(seeFig.6forinstancewhere theshipspeedis
insufficient to enter the resonance zone). Such a
conclusionwouldbemisleadingsincethedangerous
zonesarecalculatedforone
valueofthewaveperiod
equal11seconds.Incaseofreallylongshipssucha
period is related to the wave length significantly
shorter than the ship’s length. Thus, the dangerous
resonancezoneswouldappearforlongerwaveswith
correspondinglylongerperiods.Suchcalculationsare
presentedin Fig.8obtained
for theincreasing wave
period. The largest considered vessel mentioned in
Table1istakenintoaccountwiththedraftequal22
meters and the equivalent metacentric height equal
5.4meter.
Discussingthesamplecomputingresultswecan
noticethat,evendespiteapplicationoftherelatively
simple model of resonance
zones locations, the
practical use of the IMO Circ. 1228 requires a
dedicatedsoftwareavailableon‐board.Thenumberof
input data is large enough to exclude the manual
calculations.Theequivalentmetacentricheightneeds
to be computed, the ship characteristics including
draftplayavital role and thewaveperiod
andasa
consequence the wave length varies significantly
depending on sea conditions. Thus, the proper
decision support tool would be helpful on‐board
every single vessel to enable a reliable and accurate
predictionofthedangerouszoneslocation.
waveperiod6swaveperiod9swaveperiod12
s
waveperiod15swaveperiod18swaveperiod21s
Figure8.Locationsofresonancezonesforthelargestconsideredmotortanker(LBP=320m,B=58m,T=22m,initialGM=5.4
m)forincreasingwaveperiodequalto:6s;9s;12s;15s;18s;21s
614
4 SUMMARYANDCONCLUSION
The problem of prediction the dangerous zones
locations related to the ship synchronous roll and
parametric resonance is discussed in the paper. Itis
emphasized that the nonlinearity of the GZ curve
plays a vital role in roll modeling while the
contemporary recommended simplified formula for
the natural roll frequency estimation omits this
nonlinearity at all. To address the problem the
equivalentmetacentricheightisutilizedinsteadofthe
initialGM.Itisrevealedthatsuchmodificationofthe
natural roll frequency evaluation method makes it
strictlydependedontheactualrollamplitudeofthe
ship.
This creates the important question related to
thesignificanceoftheGMeqintermsofpracticalshift
in dangerous resonance zones location. The study
comprisesnumerousdifferenttypesofshipssailingin
realisticloadingconditions.
The conducted research revealed that the
inaccuracy of the resonance zone location may be
very
significant when the GZ curve is strongly
nonlinear within the range of the considered roll
amplitudes.Thecontemporary methodbased onthe
initialGMsuggests thenavigatortoavoidspeedand
courseconfigurationsfarirrelevantincomparisonto
thereliablemodelingbasedontheequivalentGM.On
the other hand
when the GZ curve is linear up to
relatively large angle of ship’s heel, both analyzed
methodsproduceverysimilarresults.
Bearing in mind the wide scope of navigator’s
interestwhenonthewatch,wesuggesttoimplement
a dedicated software designed to support decisions
regarding avoidance of the potentially dangerous
zonesobtainedaccordingtotheIMOMSC.1/Circ.1228
Revised guidance to the master for avoiding dangerous
situations in adverse weather and sea conditions. The
equivalentmetacentricheightshallbeinuseinstead
of the previous initial metacentric height. Such an
approach would facilitate the application of
trustworthy set of dynamic constraints
to the route
planning which is an intention of the e‐navigation
conceptmakingthemarinenavigationsaferandmore
efficient.
ACKNOWLEDGEMENT
The computations for all polar plots presenting
resonancezonelocationsarecarriedoutwiththeuse
of the software tool provided by Joanna
Szlapczynska,forwhichbothauthors
aregratefulto
her.
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