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