595
1 INTRODUCTION
The linear dynamic models of first‐or higher‐order,
notonlyinthefieldofshipsteeringormanoeuvring,
havebeeninspiringresearchersfordecades,andstill
draw our attention nowadays. Despite some
drawbacks, they are simple, can often provide an
efficientanalyticalsolutionthatcanbeeasilystudied
forexactanddirectinherentrelat
ionshipswithinthe
investigateddynamics,mostlyconstitutingamoreor
lessnonlinearproblem.Thedynamicmodelsofship
manoeuvring can be of hydrodynamic type (with
parameters as hydrodynamic derivatives) or the
equivalentinput‐output(transferfunction) type.The
parameters of the latter type cover various ti
me
constantsandamplificationratios.
Withregard tothe coupledshipsway (drift)and
yaw motions in the linear formulation, they can be
well either described by a single two‐dimensional
linear model of first‐order (as set of two coupled
linearODEsoffirst‐order)orbytwouncoupledone‐
dimensionalmodelsofsecond‐orderforeachmotion.
Wecautiouslyomithereadiscussiononthevalidit
y
rangeofthislinearity.
Over the years, various identification techniques
(includingsystemidentification)forparametersofthe
twohydrodynamicandinput‐outputtypesofmodels,
especially intheir linear form and for the combined
sway‐yawmotions,asofconcerninthepresentpa
per,
were developed and are still under improvement
efforts – e.g. [Kallstrom, 1979],[Holzhuter, 1990],
[Terada, 2015]. The ship motion phenomenon and
measurement experiments are actually complicated.
The last word has not been said yet. Although the
conversion of hydrodynamic description to tra
nsfer
functiondescription,andanalysisofdynamicsystems
inthelatter,convenientform,isfirmlyestablishedin
literature,e.g.[Nomotoetal.,1957],[Lisowski,1981],
[Dudziak, 2008], the inverse transformation is
practicallymissing.
Within the full‐mission ship handling simulator
mathematical models, very sophisticated and
nonlinear,the so‐calledfour‐quadrant operationand
lookup‐ta
bledatastorageisstandardrequirementfor
Inherent Properties of Ship Manoeuvring Linear Models
in View of the Full-mission Model Adjustment
J
.Artyszuk
M
aritimeUniversityinSzczecin,Szczecin,Poland
ABSTRACT:Thepaperpresentsnewresultsontheinherentpropertiesofshiplineardynamics.Thefocusis
madeonthe second‐orderformulation forthe uncoupledequations ofswayandyaw,and on theirunique,
unknown performance within the zigzag test. From the standpoint ofapplicat
ion to full‐mission model
tuning,averyimportantloopinthedrift‐yawdomainofthezigzagbehaviour,asgovernedbytherudderrate
dependent time constants (of T3‐type), is brought to the light. This and some other dependent effects, like
overshoot angle performance, are likely to be lost, if the well‐known, rather am
biguous, first‐order
approximationsaredeployed.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 10
Number 4
December 2016
DOI:10.12716/1001.10.04.08
596
modelling the hull, propeller, and rudder
hydrodynamics – e.g. [Lebedeva et al., 2006],
[Artyszuk,2013], [Sutulo,GuedesSoares,2014].With
regard to hull and rudder forces in particular, we
focus on arbitrary combinations of drift angle and
dimensionlessyaw velocity, astheir arguments,and
consider appropriate plots/curves in the
drift‐yaw
plane.Basedonrecordedmotions,anattemptismade
to efficiently fix the values of hydrodynamic
coefficients – the nodes of lookup‐tables. In this
context,aspecialinterestisbeingplacedondesigning
highqualitymanoeuvringtrials,suchthatbringalot
of information for comprehensive and
unique
calibrationofthemathmodel.Inthisprocess,weare
alsolookingfor analytical techniques of thosetrials,
similar in type to that of [Nomoto, 1960], to
effortlessly and quickly arrive at some parameters,
thatcanbenexttransformedtothebackgroundʹfull‐
missionʹhydrodynamicmodel.
The existing zigzag test
also seems to provide
necessary data. However, the most frequently used
first‐order Nomoto approximation for uncoupled
motions,thoughoriginallyintroducedanddiscussed
for yaw motion [Nomoto et al., 1957], proves to be
inadequatewhenwewanttorevertbacktothebasic
hydrodynamics.Inthelatteraspect,atleast
forzigzag
test,wearethusforcedtofullymaintaintheoriginal
second‐orderformulationofuncoupledmotions.Very
crucial parameters of this representation are the so‐
called T
3‐time constants, derived from and
responsiblefortheessentialinteractionbetweensway
andyaw. Theseconstants surprisinglylack aproper
appreciationinthepastresearch.Atributeshallhere
bepassedto[Norrbin,1996],whoasoneofnotmany
tried to consider some aspects of T
3 problem in
respectofshiphydrodynamics.
Of course, a big challenge is here to develop a
deterministic,curvefittingmethodofzigzagdatafor
thisdual(sway&yaw)second‐ordermodel,butitis
outofthescope ofthepresentstudy.Insteadof,some
new facts on sensitivity
effects of T3 are revealed,
whichshallbehelpfulindesigningandimplementing
suchanidentificationmethod.
This conceptual, theoretic paper, though
supportedbyanumericalanalysis,issubdividedinto
several chapters. We start from recalling and
discussing the basic linear system of differential
equations for sway and yaw manoeuvring motions,
its hydrodynamic
structure and the second‐order
uncoupled version. The most innovative yet very
important and meaningful results, though simple in
methodology, are presented in the next three
chapters. Therein starting from deriving the inverse
formulas forthesecond‐order models, by which the
transfer function parameters are converted to
hydrodynamic coefficients. Based
on them, some
investigationsarenextconductedonthegreatroleof
thementionedso‐calledT
3timeconstantsintransfer
function description. Finally, a rational proposal
follows on how to fix the detailed hydrodynamic
coefficients, if the aggregate hydrodynamic
coefficients, as obtained fromthementionedinverse
formulas,areknown.
2 2DLINEARMODELOFSHIPMANOEUVRING
Thecoupledlinearordinarydifferentialequationsof
thefirst‐order
withconstantcoefficientsforswayand
yawvelocitiesofa shipareworldwideknowninthe
field of ship manoeuvring and ship control
engineering. Theyconstituteabasis for deriving the
very famous direct (uncoupled, or independent of
sway)the2
nd
orderlineardifferentialequationofyaw
motion, traditionally referred to as the 2
nd
order
Nomotomodel.Thiscanbenextapproximatedtothe
first‐order linear equation of yaw, the so‐called 1
st
orderNomotomodel[Nomotoetal.,1957],[Dudziak,
2008].
The stated above models can be formulated in
dimensional,i.e.absoluteunitsofvelocitiesandtime,
orjustmadedimensionless.Thedimensionlesstime,
byrule,isexpressingtheadvancingtimecountedin
units of time that a ship requires to cover
its own
length and is fully equivalent to dimensionless
distance,i.e.thedistancetravelledbyashipasrated
inherownlengthunits.
The dimensionless quantities are much better in
analysis, since they provide universal steering
characteristics, as independent of the shipʹs
size/length and forward (surge) velocity. One of
essential assumptions underlying the linear model,
muchstrongerinthefullydimensionlesscase,isthe
constantsurgevelocity.
Theadoptednotationofcoefficientsinthecoupled
equations for sway and yaw varies from author to
author, where we may generally distinguish two
styles‐the western (international) and the eastern
(Russian)
one.Forthe purposeof thepresent study,
however,thefollowingisapplied:
222
111
'
'
'
'
'
cba
ds
d
cba
ds
d
z
z
z
(1)
where:
‐driftangle[rad]atshipʹsorigin, positivewhen
laterallymovingtoport,
xy
vv arctan
,vxand
v
y‐shipʹssurgeandswayvelocities,
z
'
‐relativeyawvelocity[‐],positiveforturningto
starboard,
vL
zz
/'
,where:
z‐yawvelocity, L‐
shipʹslength,v‐totallinearvelocity(asresultedfrom
v
xandvy),
's ‐dimensionless time/distance [‐], LvtLss
' ,
inwhich:t‐timeors‐distance,
‐ rudder angle in [rad] as input control, positive
whentoport.
Additionally,onecanwrite
dtd
z
,where
istheheadingangle.
3 STRUCTURINGTHEMODEL
Except for c
1 and c2, as solely connected with the
rudder hydrodynamic force, all other coefficients in
(1) combine the effects from both a shipʹs hull (ʹHʹ)
andrudder(ʹRʹ). In addition,the coefficient b
1 hasa
veryimportantcontributionfromthecentrifugalforce
597
(ʹCʹ) involved in the development of drift angle. All
the coefficients can easily be derived (or
approximated) from a detailed description of
hydrodynamic forces laid down at a core of the
mentioned full mission models. The core mainly
consists instoring relevant relationshipsin the form
of lookup
‐tables. Around a certain point, those can
makeuptheusualanalyticalform,knownfromother
simplermodels,andevenbereducedtolinearmodel.
Apracticalexampleofsuchrelationships,whichcan
besuited toany existingapproach waspresented in
[Artyszuk, 2013]. Those arequoted,rearranged,and
simplified
tomeetthedefinitionofthesixcoefficients
in(1)‐a
i,bi,ci,wherei=1,2‐asfollows:
RH
aaa
111
,
CRH
bbbb
1111
(2a)
RH
aaa
222
,
RH
bbb
222
(2b)
180
'
1
1
5.0
22
1
b
B
H
Y
k
cB
L
a ,where 0'
b
Y (3)
Th
Ry
R
cw
c
k
R
a
11
1
22
1
,where 0
Ry
c and (4)
H
ThL
ThR
B
a
cc
cwA
cB
L
R
1
180
,
11'
1
5.0
0
2
,
where
0
H
a
(5)
w
B
H
Y
k
cB
L
b '
1
1
5.0
22
1
,where
0'
w
Y
(mostlyfor
modernships,ofskeg‐shapedstern), (6)
ReffR
Th
ReffRy
R
xa
cw
xc
k
R
b '
11
'
1
1
22
1
,where 0'
Reff
x (7)
22
11
1
1
1
k
k
b
C
(8)
22
1
1 k
R
c
(9)
180
'
''
1
5.0
2
66
2
2
b
z
B
H
N
rr
cB
L
a
,where 0'
b
N (10)
ReffR
z
Reff
Th
Ry
z
R
xa
rr
k
x
cw
c
rr
R
a '
''
1
'
11
''
1
2
66
2
22
2
66
2
2
(11)
w
z
B
H
N
rr
cB
L
b '
''
1
5.0
2
66
2
2
,where
0'
w
N
(12)
2
1
2
66
2
22
2
66
2
2
'
''
1
'
11
'
''
ReffR
z
Reff
Th
ReffRy
z
R
xa
rr
k
x
cw
xc
rr
R
b
(13)
ff
z
ff
z
xc
rr
k
x
rr
R
c
Re1
2
66
2
22
Re
2
66
2
2
'
''
1
'
''
(14)
Particular dimensionless elements of the above
expressions(3)to(14)canbeexplainedasbelow:
hull‐relateditems:
BL ‐ shipʹshulllength‐to‐beamratio,
B
c ‐blockcoefficient,
11
k ‐surge added mass coefficient, mmk
1111
,
where
11
m
‐ surge added mass, m‐shipʹs
displacement(mass),
22
k ‐sway added mass coefficient, mmk
2222
,
where
22
m ‐swayaddedmass,
z
r' ‐shipʹs gyration dimensionless radius,
2
' mLJr
zz
, where
z
J
‐ shipʹs mass moment of
inertia,
66
'r ‐added gyration dimensionless radius, that is
2
6666
' mLmr
, where
66
m
‐ added moment of
inertia,
b
Y'
,
w
Y'
,
b
N'
,
w
N'
‐hull hydrodynamic
(dimensionless) derivatives;
b
Y' and
b
N' , in view
of the conversion factor
180 in (3) and (10), are
computedwithreferenceto
in[];
b
N' isassumed
to include/integrate the Munk moment contribution,
asusuallyseenwhilereportingexperimentalresults.
rudder‐relateditems:
R
A' ‐rudderarearatio,
LTAA
RR
' ,where:AR‐
rudderarea,LT
‐shipʹslength‐draftproduct,
w‐ propellerwakefraction,
Th
c ‐propeller thrust loading coefficient,
2
8
J
Jk
c
T
Th
, where J‐advance ratio, kT‐thrust
coefficient,
0
,
ThL
cc
‐rudder lift coefficient derivative
vs.flowincidenceangle
[]foragivencth,takenat
;cLisdefinedhereinwithregardtopropellerrace
velocity–see(5),
H
a ‐empirical amplification factor of (effective)
rudderforceduetohull‐rudderinteraction,
Ry
c ‐empirical multiplier (≥1 or <1) to the rudder
geometric local drift angle to arrive at its effective
local drift angle; c
Ry=1 means equality of both;
furthermore,itisindirectlyassumedthatashipʹsdrift
andyawhaveequaleffectsonthiseffectivelocaldrift,
Reff
x' ‐effective rudder longitudinal position (the
effectivelocationoftherudder force),dimensionless
in shipʹs length units;for xʹ
Reff=0.5 we get the
598
nominal/physical position of the rudder force at aft
perpendicular; as supplementing theʹactionʹ of a
H,
this coefficient also arises from hull‐rudder
interactionbutintermsof theeffectiverudderforce
arm,xʹ
Reff≥0.5orevenxʹReff<0.5areallowed.
All the terms in (3) to (14), except for the seven
mostly uncertain and empirically determined
coefficients‐4relatedtohull(Yʹ
b,Yʹw,Nʹb,Nʹw)and3
associatedwithrudder(aH,cRy,xʹReff)‐canbereferred
toastheformal(reference,nominal)quantities.Their
values are to be established by means of usually
available geometric or hydrodynamic prediction
methods. Any uncertainty/bias within them is
allowed since theʹfinalʹ accurateness of forces and
moments is to be reached through tuning of the
aforementioned 7 dimensionless empirical
coefficients.At
thisstageofresearch,thethreerudder
parameters are considered constants, however,
accordingtothisauthorʹspastinvestigations,acertain
functional relationship with motion and control
variablesseemsquitelikely.
4 DECOUPLEDCLASSICALDRIFTANDYAW
EQUATIONS
The basic hydrodynamic equations (1) impose
problems when someone wants to relate
a shipʹs
kinematicresponseforagivencontrolinput(interms
of rudder angle) to their coefficients. The
improvement goals using such efforts may be
multiple – from ship design, through ship steering
control, to full mission simulator performance in
nauticalstudies,likeinourcase.Inthiscontext,and
in view of transformations proposed in the next
section, it seems necessary to recall and briefly
discuss the well‐known classical relationships
relevanttotheuncoupledequations.
Thesetoflinearequationsofthefirst‐order(1)can
easily be transformed to fully equivalent time
responses of drift and yaw,
being the second‐order
linearequationsofparticularmotions:
''
'
321
2
2
21
ds
d
TK
ds
d
TT
ds
d
TT
bb
(15)
'
'
'
'
'
'
321
2
2
21
ds
d
TK
ds
d
TT
ds
d
TT
wwz
zz
(16)
Although the drift equation (15) is rather of less
interest and seldom challenged in literature, it is
obviously very crucial for keeping uniqueness and
identificationofthebasicset(1).Particulardefinitions
of time constants (marked withʹTʹ symbols) and
amplificationconstants(ʹKʹnotation),bothofpractical
response
interpretation,aresummarisedbelow:
1221
2
2121
1
45.0
1
babababa
T
(17)
1221
2
2121
2
45.0
1
babababa
T
(18)
1221
2
3
caca
c
T
w
(19)
2112
1
3
cbcb
c
T
b
(20)
1221
1221
baba
caca
K
w
(21)
1221
2112
baba
cbcb
K
b
(22)
The time constants T
1 and T2, given above
explicitlyandappearingidenticallyinbothequations
(15) and (16), are sometimes quoted in a more
convenient,equivalentway,namelyimplicitlyinthe
formoftheirproductandsum:
1221
21
1
baba
TT
(23)
2121
1221
21
21
TTba
baba
ba
TT
(24)
Alltheexpressions(17)to(22),particularlywhen
applying (23) and (24), have a direct, practical
meaningwhilestudyingthetimeresponseofashipto
certainrudderactions.
T
3b and T3w, called hereafter as T3‐type constants,
areconnectedwithrudder(deflection)rate–itssign
and magnitude. They can oppose or magnify the
effectofrudderangle.
Foradynamically(directionally)stableshipthere
holds a practical dual condition (see e.g. [Dudziak,
2008]):
0
21
TT (25)
0
21
21
TT
TT
or 0
21
TT inviewof(25) (26)
whichleadtothebasicstabilitycriterion,inwhichT
1
andT
2shouldbebothpositive.Theseinequalitiesare
satisfiedwhen:
0
1221
baba (27)
0
21
ba
(28)
599
butwithregardto(3),(4)‐ensuringa
1<0‐and(12),
(13)‐leading to b2<0‐the stability condition for
marinevesselscanbesolelyreducedtoequation(27).
However,themagnitudesforbotha
1andb2havetheir
directinfluenceonthefirsttermin(27)andthuson
thestability.
The time‐domain simulation of response to any
rudderactionissymmetricalversusT
1andT2inthat
ifweinterchangetheirvaluesinplaceofoneanother
there will be no change in response. In addition, T
1
and T
2 calculated by the expression (17) and (18)
accordingly always provide the case T
1>T2 (even
T
1>>T2)forastableship.
Insummary,wehaveasetof6newcoefficients(of
T‐, and K‐class) instead of the original set (of a‐, b‐,
andc‐class)inequations(1).Bothsetsareinvertibleto
each other as being shown next. However, as
mentioned
before,theinverseproblemofgettingthe
original coefficients of (1) is seldom undertaken in
research. Moreover, the identification procedures of
T‐ and K‐class constants based on ship motion
responsedonotexistforthefullsecond‐orderlinear
equations. This is even true in case of the single
equationforyawmotion(16).
Suchalgorithmsmostlydealwiththereduced(of
loweramountofinformation),first‐orderequationsof
two unknown parameters, and stable ships. The
widelyusedherezigzagof10/10typeorofanother
type,butwithfiniteyawresponse,oftenseemstobe
excessive
to establish a linear yaw model for an
unstable ship. In that, the identification procedure
itself (of a certain integral approximation/fitting
towards a linear model), as redefined in [Nomoto,
1960],andthe usedactuallyʹoverlinearizedʹzigzag
responseduetotheassumedrelativelylargevariation
of nominal rudder and heading (even of
only 10
magnitude),nearly alwaysleads toresponse models
ofmoreorlessbutstableships.
5 DERIVATIONOFINVERSEFORMULAS
Thementioned inverseconversion ofthe sixT‐,and
K‐classconstants,ifsuchareknownforbothdriftand
yaw,tothebasicsixhydrodynamiccoefficients(of
a‐,
b‐,andc‐class)in(1)ispresentedbelowincondense,
naturalorder:
2133
321321
1
TTTT
TTTTTT
a
wb
wb
(29)
2133
321321
1
TTTT
TTTTTT
K
K
b
wb
bb
w
b
(30)
2133
321321
2
TTTT
TTTTTT
K
K
a
wb
ww
b
w
(31)
2133
321321
2
TTTT
TTTTTT
b
wb
bw
(32)
21
3
1
TT
KT
c
bb
(33)
21
3
2
TT
KT
c
ww
(34)
wherea
1andb2aresolelybasedonthetimeconstants.
Thedetailsofthosederivationsareasfollows:
Step1
After combining the four equations (19) to (22)
withtherelationship(23)wehave:
w
w
K
TTc
T
212
3
(35)
b
b
K
TTc
T
211
3
(36)
whichleadstraightto(33)and(34).
Step2
Substitutingthejustreceiveddefinitionsofc1and
c
2,storedin(27)and(28),toequations(21)and(22),
andagaindeploying(23),wearriveat:
wbbww
KaKTaKT
2313
(37)
bbbww
KbKTbKT
2313
(38)
whichshallbenextcoupledwith(23)and(24),as
uniquely representing (17) and (18), but written in
suchaform:
21
1221
1
TT
baba
(39)
21
21
21
TT
TT
ba
(40)
Henceasetoffour,apparentlynonlinearalgebraic
equations–(37) to (40) – is being received, that
shallbesolvedagainstthemissingunknowns:a
1,a2,
b
1, b2. Speaking precisely, equation (39) is the only
ʹnonlinearʹ within this set, but thisʹnonlinearityʹ can
beresolvedintoelementary,linearrelationshipsafter
takingadvantageoftheotherthreeequations.Atfirst
glance,however,therequiredtransformationsforthis
taskarenotsoclear.
Thesolutionoftheset(37)
to(40)canbeobtained
analytically.Forexample,letʹsdeterminea
2from(37)
andb
2from(38)andthensubstitutebothto(39)and
(40).Thelattertwoequationsshallnowbesolvedfor
theunknownsa
1andb1.Usingthesevalues,thefinal
valuesofa
2andb2areprovidedafterreturningback
to(37)and(38).
600
Not only the final relationships (29) to (34) are
useful, but such are also the intermediate equations
(37) to (40), especially while seeking for mutual
relationships between the hydrodynamic coefficients
inarbitrarygroups,whensomeofthemhavealready
beenfixed.
Ofmajor importancealsoappears asensitivityof
the
resultsfora‐andb‐classcoefficients–see(29)to
(32)–totheaccuracyofestimatingthetimeconstants
relatedtorudderrate:T
3bandT3w,particularlytotheir
difference.Thevalueclosetozerointhedenominator
oftheseexpressionsimpliesveryhighvaluesofthe
parameters:a
1,a2,b1,b2.
6 ROLEOFT
3BANDT3WINSHIP
HYDRODYNAMICS–NUMERICALEXAMPLE
Exemplaryvaluesofparticulardatainformulas(3)to
(14)fora hypotheticalship,withoutanyclaim tobe
exact,arepresentedinTab.1,thoughtosomeextent
they originate in the authorʹs previous full‐scale
identificationstudies and lookup‐table modelling
on
a small chemical tanker [Artyszuk, 2013]. The
dimensionalvaluesofshipʹslength(constant)andher
forward speed (variable as specific to a given
manoeuvre)arenecessarytoconverttherudderrate
fromthe absoluted
dtin[/s] tod
dsʹ[/‐].Except
asexplicitlystated,therudderrateof2.5/shasbeen
chosen that is slightly above the minimum
internationalrequirementforsteeringgear(2.3/s).
Theindividualcontributionstothecoefficientsin
(1), classified according to the source of forces, see
(2a) and (2b), are
collected in Tab. 2. Herein, the
rudder has up to 30% significant contribution in all
terms, which is sometimes forgotten, when using a
simple rudder effect as coupled only with the helm
angle
. Practically, the sign of the rudder
contribution,ascomparedtohull,isonlyoppositefor
thedrift‐relatedyawmoment– refertoa
2Handa2Rin
Tab.2.
The final values of the direct (a‐, b‐, and c‐)
constants in(1),in parallel with T‐ and K‐constants,
are demonstrated in the upper part of Tab. 3. This
condition of the ship is referred to as the reference
case.Thelowerpart
ofTab.3containstheinfluenceof
thevariationofconstantsT
3bandT3w(rathersmallin
magnitude, at least, as compared to T
1) on the
computationofa‐andb‐,c‐constantswhile keeping
thevaluesoftheotherT‐andK‐typeparameters.The
hugesensitivityofthesteeringdynamicmodeltothe
considered T
3‐type constants is here evident. They
almostaffectallbasicparametersofthemodelin(1),
sometimesevenchangingthesign.
Table1. Elementary input data (dimensionless by default,
exceptasgivenexplicitly)
_______________________________________________
hullrudder
_______________________________________________
L[m]97.4AʹR0.0177
v[m/s]7.272w0.326
L/v[s]13.39c
Th2.127
cL/
@cTh0.0385
L/B5.867
c
B0.761aH0.6
rʹ
z0.247cRy1.0
k
110.056xʹReff‐0.5
k
221.004
rʹ
660.225
Yʹ
b0.0043
Yʹ
w0.0260
Nʹ
b0.0024
Nʹ
w‐0.0630
_______________________________________________
Table2. Final contributions to constants of equations by
theirnature(thereferencecase)
_______________________________________________
a1Hb1H a2Hb2H
a1Rb1Ra2Rb2R
b1C
_______________________________________________
‐0.479 ‐0.0504.843‐2.182
‐0.144‐0.072‐1.291 ‐0.646
0.527
_______________________________________________
Table3.Constantsofequations
_______________________________________________
CASET1T3b Kb
‐reference T
2T3w Kw
_______________________________________________
10.491 0.154‐3.464
0.298 0.983‐4.896
_______________________________________________
a1 b1 c1
a
2 b2 c2
_______________________________________________
‐0.6220.405‐0.171
3.552‐2.827‐1.539
_______________________________________________
Belowgiventhea‐,b‐& c‐constants
insameorderasabove
_______________________________________________
CASE‐T3b ‐0.046‐0.037 ‐0.342
variation 4.366
‐3.403 ‐1.539
T
3b=0.309
_______________________________________________
CASE‐T3w ‐1.4580.996 ‐0.171
variation 2.594 ‐1.992‐0.770
T
3w=0.492
_______________________________________________
The simulation of standard 10/10 zigzag
manoeuvreisperformedinthesubsequentFigs.1to
9.TheʹREFʹcurveshere correspondtothe reference
case,seeTab.3.Thisrathersimpletest,ascompared
toothers, bringscomprehensiveinformationon ship
behaviour, especially if we consider both drift and
yaw together of varying signs All the computations
havebeenmadebydirectintegrationof(1),including
thedifferentialequationforheadingangle,usingthe
Eulermethod(stillpowerfulforthisspecificproblem)
withdimensionlesstimestep
sʹ=0.05.
Figs.1and2 presenttheheading variation,helm
angle and the both kinematical variables as directly
governedbyourdynamicequations– driftangleand
dimensionless rate ofturn.The manoeuvre itself for
our ship essentially lies within transient states
because the range of kinematics shown in Fig.
2 is
much lower than the steady‐state values –
34.6
for the drift angle (=K
b
) and
ʹz0=0.854 for the
relativeyawvelocity(=K
w
).
601
Oneofthemostnoticeablefeaturesofthesecond‐
order linear formulation of uncoupled steering
dynamicswithregardtodriftandyaw,see(15)and
(16), as equivalent to the full set of (1), is a partly
independentchangeofdriftangleanddimensionless
yawvelocity.Moreover,theincrease/decreasein
yaw
ismuchhigher(ʹoflowinertiaʹ)thanofthedriftangle
– Fig. 2 and 5. For zigzag manoeuvre, such a
behaviour produces– Fig. 3 and 4– a certain,
closedloopofthemutualrelationship
ʹz=
ʹz(
)inthe
plane of the domain of the dimensionless hull
hydrodynamic forces, being functions of just drift
angleanddimensionlessyawvelocity.Thewiderthe
loop,thebetterforthefittingorvalidationofthehull
forceresponsesurfaceasa3Drepresentationoftwo‐
variable relationship [Artyszuk, 2013]. The
stated
hereinperformancesarenotexhibitedatallbytheso‐
called first‐order uncoupled Nomoto models, for
certainreasons,muchmorefrequentlyusedthanthe
former, original ones. These first‐order
approximationsarebasedonthecriterionproposedin
[Nomotoetal.,1957]andquotedbelow:
bb
K
ds
d
T
'
,where
bb
TTTT
321
(41)
wz
z
w
K
ds
d
T '
'
'
,where
ww
TTTT
321
(42)
SinceT
3bandT3warerathersmall,Tbof(41)isquite
closeinmagnitudetoT
win(42).
Fordetailed comparison,theoutput oftheabove
1
st
ordermodelsisalsoincludedinouranalysisand
markedbyʹ1stORDʹinFigs.3to6,and8.However,
inthecaseofthese1
st
ordermodels,thederivativesof
driftangleandyawvelocityinFigs.5,6,and8,and
the resulting direct values of these two variables in
Figs. 3 and 4, are considered only for the most
representative,initialperiodofthezigzagmanoeuvre
with the first rudder execute. The rudder
is then
simplisticallykeptinthisposition(thecounter‐rudder
is no longer applied), that enables a very efficient
analyticalsolutionof(41)and(42),whichisadopted.
Twoversionsofruddercontrolarestudiedforthe1
st
ordermodels‐the infinitelyrapid (step)movement,
as the limiting case, denoted byʹ=constʹ and cyan‐
coloured,andthetrapezoidalsteering(ʹ=varʹ,brown
color), with the same rudder rate as used in
computing the correspondingʹsecond‐orderʹ
response.
The corresponding
‐
ʹz curve for the first‐order
modelsispracticallyanopen,straightlineinclinedan
anglearisingfromtheratioofsteadystatevalues of
drift angle and dimensionless yaw velocity, or just
directly from K
w/Kb. Figs. 3 and 4 present the 1st‐
quadrant section of this curve, which is quite
independentofthemodelversionused–withinfinite
orwithfiniterudderrate–andoftherudderalternate
control strategy like in zigzag test. Combining both
models(41)and(42),thiscurveis
definedby:
wb
TT
zz
0
0
11''
(43)
Itthusmeansthatthe
ʹz=
ʹz(
)relationshipofthe
1
st
orderuncoupledmodels,(41)and(42),losesalot
ofessentialinformationfromtheoriginalbackground
hydrodynamicsexpressedby(15)and(16),orjustby
(1).Moreover,inthelattercase,thederivativeofyaw
velocityinFig.5experiencesasignificantpeakthat
isdamped forthe1
st
order approximation, whichin
consequenceleadstoquitedifferentovershootangles
and oscillation periods in the heading diagram.
However,thisheading performancefor the 1
st
order
uncoupled equations, has not been shown in the
paper.
Whenreducingtherudderratefromthereference
2.5/sto theabstractvalueof 0.5/s,one canachieve
an efficient convergence of the second‐order
uncoupleddynamicstothefirst‐orderonebecauseof
the relatively low influence
of the T3b and T3w
constants,whichshallberatherobvious–seeFigs.6
and7.However,thederivativeofyawinthesecond‐
order response still displays the initial jump that is
responsible for the occurrence of a loop around the
straightlinesectionofthefirst‐ordermodelinthe
‐
ʹzdomain.
Furthermore, besides the case of simultaneously
very low T
3b and T3w, the second‐order uncoupled
dynamics also converges to first‐order one for T
3b
closetoT
3w,independentoftheirabsolutevalues.Of
course,inviewof(29)to(34),thisgivesexaggerated,
almostenormousvaluesfora‐,b‐,c‐constants.
T
3bandT3whavegreatimpact(beingmuchhigher
forT
3w)onflatteningorspreadingofthe
‐
ʹzplotin
Figs.3and4.Thecorrespondingcoefficientsof(1)for
the tested variations in those time constants were
quotedinTab.3.TheconstantT
3bdoesnotaffectthe
yaw behaviour at all. The same should obviously
happenwithregardtotheT
3wvariationasexpectedto
completely preserve the drift angle image. This is
demonstrated in Fig. 8 and can even be proved
analytically.The heading accompanying the
reductionofT
3whasalreadybeenincorporatedinthe
initialFig.1.However,sincetheruddercontrolinthe
zigzag manoeuvre is essentially heading‐ or yaw‐
based, the preservation of the drift angle for the
varyingT
3wisbeingheldonlywithintheinitialperiod
of the test, i.e. up to the first counter‐rudder.
Thereafter, the drift angle differential equation is
being solved with the relative yaw velocities asʹnot
correspondingʹ to the actual drift angles and helm
angles. Fig. 9 shows this situation. In general,
T3b
impliesahorizontalexpansion/contractionintheloop
ofthe
‐
ʹz,while T3wactsmoreuniversally,namely
changingtheloopinbothdirection‐seeagainFigs.3
and4.IncreasingT
3w,whichishowevernotshownin
thechartofFig.4,leadstotheinversescalingofthe
ʹz(
)loop,suchthatwehaveasignificantcontraction
along
‐axisandalargeexpansioninthedirectionof
ʹz‐axis.
602
-40
-30
-20
-10
0
10
20
30
40
0 5 10 15 20 25
REF
s'-
REF
|T
3w
-50%
Figure1.Headingandhelm(incl.T3wvariation)
-15
-10
-5
0
5
10
15
0 5 10 15 20 25
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
s'
-
'
z
REF
REF
'
z
-
Fig.2.Driftangleandrelativeyawvelocity
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-15 -10 -5 0 5 10 15
'
z
-
1st ORD
REF
T
3b
+100%
Figure3.Dimensionlessdrift‐yawdomain‐T3beffect
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-15 -10 -5 0 5 10 15
'
z
-
1st ORD
REF
T
3w
-50%
Figure4.Dimensionlessdrift‐yawdomain‐T3weffect
0
0.05
0.1
0.15
0.2
012345
0
0.02
0.04
0.06
0.08
012345
d
/ds'
rad
-
s'
-
1st ORD
=const
REF
s'
-
d
'
z
/ds'
-
REF
1st ORD
=var
1st ORD
Figure5.Derivativesvs.1storderNomotomodel
603
0
0.05
0.1
0.15
0.2
012345
0
0.02
0.04
0.06
0.08
012345
d
/ds'
rad
-
s'
-
1st ORD
=const
0.5
/s
s'
-
d
'
z
/ds'
-
1st ORD
=var
1st ORD
0.5
/s
Figure6.Derivativesforreducedrudderrate
-60
-40
-20
0
20
40
60
0 5 10 15 20 25
0.5
/s
s'
-
0.5
/s
Figure7.Helmandheadingforreducedrudderrate
0
0.05
0.1
0.15
0.2
012345
0
0.02
0.04
0.06
0.08
012345
d
/ds'
rad
-
s'
-
s'
-
d
'
z
/ds'
-
1st ORD
Effect of
T
3
w
T
3w
= 0.5
T
3w
|
REF
Effect of
T
3
b
T
3b
= 2
T
3b
|
REF
- v.close to 1st ORD at =var
1st ORD
Figure8.EffectsofT3bandT3wonderivatives
-15
-10
-5
0
5
10
15
0 5 10 15 20 25
s'
-
REF
|T
3w
50%
Figure9.DriftangleresponsetoincreasingT3w
7 PROPOSEDDETERMINATIONOFDETAILED
HYDRODYNAMICPARAMETERS
Ifwemovefromthe T‐andK‐classconstants, sixin
total, determined through the analysis of kinematic
responsetoacertainruddercontrol,tothesixa‐toc‐
classconstants,bymeansoftheinverseformulas(29)
to (34), we can
attempt to obtain the detailed
parameters underlying the latter constants. In the
ʹStructuring the modelʹ section seven natural
unknowns–{Yʹ
b,Yʹw,Nʹb,Nʹw}and{aH,cRy,xʹReff}– were
specifiedinthiscontext.However,wenowarriveat
anindeterminate(overparameterised)setofalgebraic
equationsbecauseoftoomanyunknownsinrelation
to the number of equations. One of the parameters
should be thus fixed. Based on available model test
604
dataand/orothermethods,anyofthose7coefficients
could be selected for this purpose. In view of the
potential estimation uncertainty, such selection will
have an impact on the model validity while
simulatingspecificmanoeuvres.
In view of our derivations, the terms c
1 and c2
responsiblefortheeffectofhelmangle
,see(5),(9)
and(14),canberesolvedinto:
111
1 faacc
HH
(44)
222
'1', fxaxacc
ReffHReffH
(45)
wheref
1andf2havebeenintroducedtoreflecttherest
partofthecorrespondingexpressions.
Hence,atthefirststagewedirectlyreceivea
Hfrom
(44), and next xʹ
Reff from (45), and come to the
following4equationswith5unknowns:
ReffHbRyaw
ReffHbRyab
CReffHbRyaw
HbRyab
xafcfNb
xafcfNa
bxafcfYb
afcfYa
','
','
','
'
662
552
1441
331
(46)
where similarly the f‐symbols represent the
appropriaterelationships.
In the light of the state‐of‐the‐art in ship
manoeuvring hydrodynamics, it is suggested to fix
oneof thedrift‐dependenthulltermsin (46)‐Yʹ
bor
Nʹ
b‐as relativelywell workedout andpublished in
theliterature.
Of course, any uncertainty in estimating the
rudder parameters will beʹcorrectedʹ by relevant
recalibration of the hull parameters (and vice versa)
due to the same physics or dependence involved in
the background expressions. However, this will be
paidfor
byerrorsinsimulationwhen,forexample,a
shipissubjecttomanoeuvringwithoutrudderaction,
likeinwind,inwhichcasethehulleffectsdominate.
8 CONCLUSIONS
It seems that existing ground‐related shipʹs
positioning (satellite) systems and collected data
during full‐scale sea manoeuvring trials are still
insufficient
in order to conduct a reliable
identification of the uncoupled second‐order (ʹfullʹ)
linear dynamics, as aimed in the paper. This is also
true, when we apply a certain amount of post‐
processing,asconnectedwithfiltering/smoothingthe
measurements and eliminating the environmental
disturbances.Inparticular,theproblem
liesinalow
adequacy/reliabilityofthe indirectlyacquired runof
the drift angle, which is often subject to significant
water current effects. If the very sensitive T
3‐type
constantsarenotaccuratelyestablished,thenwecan
lose the physical sense of the final hydrodynamic
derivatives.Herein,acertainrolehereinisobviously
playedbyamanoeuvringtest(controlinput)selected.
Tosomeextent,aremedycanbe offeredbyfree‐
runningmodeltests.Nevertheless,thisisreally
abig
challenge for the future research to design special
kind, high quality manoeuvres, conduct their
measurements,andperformidentification.
It is believed that the zigzag test, served in the
present investigation as an example to demonstrate
peculiarities of the second‐order uncoupled linear
dynamics, couldalsobe used for
that purpose. This
manoeuvrehasaveryextensive recordofpublished
experimental and theoretical (simulation) data.
However, some systematic parametric studies are
required for zigzag performances being essentially
ʹgeneratedʹ by the concerned dynamics. This would
yield a benchmarking material in order to detect
hiddennonlinearitiesinactual(ofgivenship)
zigzag
behaviour and then to prevent from fitting the
consideredlinearpartofthefull‐missionmodel.
BIBLIOGRAPHY
[1]ArtyszukJ.: ModellingandSimulationinShipManoeuvring
Safety and Effectiveness Issues. ISBN 978‐83‐89901‐78‐1,
MaritimeUniversity,Szczecin,2013(inPolish).
[2]Dudziak J.: Ship Theory. 2nd Ed., Foundation of
Shipbuilding Industry and Maritime Economy
Promotion,Gdansk,2008(inPolish).
[3]HolzhuterT.:Aworkabledynamic
modelforthetrackcontrol
of ships. Ninth Ship Control Systems Symposium.
Proceedings,Sep24‐27,vol.4,Bethesda,1990.
[4]KallstromC.G.:IdentificationandAdaptiveControlApplied
to Ship Steering. Ph.D. Thesis, Institute of Technology,
Lund,1979.
[5]Lebedeva M.P. et al.: Development of Criteria for
Certification of the
Mathematical Models Used by Marine
Simulators, International Conference on Marine
SimulationandShipManoeuvrability–MARSIM2006,
Jun25–30,Terschelling,2006.
[6]Lisowski J.: Ship as Automatic Control Object. Wyd.
Morskie,Gdansk,1981(inPolish)
[7]NomotoK.:AnalysisofKempfʹsstandardmaneuvertestand
proposedsteeringquality
indices.FirstSymposiumonShip
Maneuverability, May 24‐25, DTMB Rep. 1461 (AD
442036),DTMB,Washington,1960.
[8]Nomoto K. et al.: On the Steering Qualities of Ships.
InternationalShipbuildingProgress(ISP),vol.4,no.35
(Jul),1957.
[9]Norrbin N.H.: Further Notes on the Dynamic Stability
Parameterandthe
PredictionofManoeuvringCharacteristics.
MARSIMʹ96 Proc., Marine Simulation and Ship
Manoeuvrability (A.A. Balkema, Rotterdam), Chislett
M.S.(ed.),Sep9‐13,DMI,Copenhagen,1996.
[10]Sutulo S., Guedes Soares C.: An algorithm for offline
identificationofshipmanoeuvringmathematicalmodelsfrom
free‐running tests. Ocean Engineering, v. 79, pp. 10‐
25,
2014.
[11]Terada D.: Novel identification method of maneuverability
indices concerning drift motion based on successive data
assimilation. Journal of Maritime Researches, Kobe
University,vol.5(Mar),pp.1‐13,2015.
