459
1 INTRODUCTION
When a transport ship is sailing on a given sailing
routeinrealweatherconditions,anumberoffactors–
planerudderaction,amongothers–influenceactual
servicespeedofa ship.Whenashipisacteduponby
wind and wave of oblique directions, this results in
additional resistance, drift force and ship rota
ting
moment(inordertomaintainapresetshipcoursethis
momenthastobecounterbalancedbythemomentof
deflected rudder blade). When plane rudder is
deflected, apart from additional resistance, a side
forceemergeson a rudder causing the ship to drift.
Various methods for calculat
ing the forces and
momenton plane rudderlocated behind apropeller
areknownfromliterature[1],[2].Suchmethodsare
usefulforexaminingshipmaneuverabilityproperties.
Theyare notgood,however,toforecast shipservice
speedin realweatherconditionsatpreliminaryship
design. At this stage, ship hull shape, propeller
charact
eristics and propelling engine parameters are
not yet known. Therefore forecasting ship’s service
speed, models based only on basic geometrical
parameters of a designed ship are highly useful,
among other to calculate additional resistance from
deflectedplanerudder.
The article presents an approximate method of
calculating forces on pla
ne rudder located behind a
propeller, useful for forecasting service speed of a
shipatinitialstageofitsdesign.
2 FORCESANDMOMENTONARUDDER
LOCATEDBEHINDAPROPELLER
When a ship is sailing on wavy water, in particula r
whentheshipisacteduponbywindandwavefrom
oblique directions, side forces and moments emerge
whichenforcechangesonsailingrouteandresultina
drift.Inordertomaint
ainconstantcourse,rudderhas
to be deflected (Fig. 1) which causes additional
resistanceX
R.
In literature on ship maneuvering, there are
numerous algorithms to calculate hydrodynamic
forces on rudder, e.g. [3], [4], including also
additional resistance [5]. According to [4] forces on
aruddercanbecalculatedfromtheequationsbelow:
Approximate Method of Calculating Forces on
Rudder During Ship Sailing on a Shipping Route
K.Żelazny
WestPomeranianUniversityofTechnology,Szczecin,Poland
ABSTRACT:Servicespeedofashipinrealweatherconditionsisabasicdesignparameter.Forecastingofthis
speedatpreliminarydesignstageismadedifficultbythelackofsimplebutatthesameaccuratemodelsof
forcesactinguponashipsailingonapresetshippingroute.Theart
iclepresentsamodelforcalculatingforces
andmomentonplanerudder,usefulforforecastingofshipservicespeedatpreliminarystagesofshipdesign.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 8
Number 3
September 2014
DOI:10.12716/1001.08.03.18
460
sin ,
cos ,
cos ,
RN R
RyN R
RzN R
XF
YaF
MaF
(1)
where:
R rudder angle (Fig. 1 – rudder angle at port
side
R>0,rudderangleatstarboard
R<0),
a
y influencecoefficientofthehulloverYRforce
onrudder,
a
z influence coefficient of the hull over MR
momentonrudder,
zyR
aax
, (2)
x
R distance of rudder from the ship center of
gravityG(x
R<0)
F
N normalforceonrudder,
RRRwN
VAF
sin
25.2
13.6
2
1
2
, (3)
aspectratioofrudder,
A
R rudderarea,
V
R velocityofwater inflowtorudder,
R effectiveangleofattack.
Figure1.Forcesonarudder
As a result of rudder deflection, a moment MR
fromforceY
Remerges–therefore inordertokeepthe
presetcourseofaship,themomentonruddershould
have such value as to counterbalance the resultant
enforcing moment from wind, wave and resistance
moment (together with current action) in case of
movementwithdriftangle.
Effective angle of attack of
rudder
R occurring
alsoin equation (3) depends mainly on trajectory of
shipmovementanddirectionofinflowtorudder,i.e.
driftangleinrudderarea.
RR R


, (4)
where:
R driftangleatrudder,
coefficient accounting for
straightening/correctingactionofaship’spropeller
andhull(flowstraighteningcoefficient)).
ThevelocityofwaterinflowtorudderV
Rdepends
on ship’s speed V and the wake and slipstream
behindapropeller:
2
(1 ) 1
RR S
VV w KG

, (5)
where:
w
R wakefractioninrudderarea,
K
2 influencecoefficientofpropelleroverrudder,
G
S loadcoefficientofapropeller.
Taking into account prior assumptions to the
presented algorithm according [4] i.e. that the ship
sailswithconstantspeedanditsrudder is deflected
onlyinasmuchastocounterbalanceexternalmoments
tomaintainthepresetcourseofaship,then:
TR
ww
, (6)
2
)1(
)4.12(6.0
s
ss
b
D
G
R
p
S
, (7)
D
p propellerdiameter,
b
R rudderheight,
s slipratioofpropeller,
(1 )
1
T
p
Vw
s
Pn

, (8)
w
T wakefraction,
P propellerpitch,
n
p propellerrevolutions.
3 APPROXIMATIONOFFORCESANDMOMENT
ONPLANERUDDER
The above algorithm for calculating forces on plane
rudder contains a number of parameters regarding
ship hull, rudder as well as a propeller and engine
whicharenotknownatinitialstageofshipdesign.
In order to
work out a simple approximating
formula,thefollowingassumptionshavebeenmade:
From carried out calculations of the drift angle
value
Rinrudderarea[6]anddistributionofthis
angle performed during ship movement along a
shipping route (examples of obtained results –
Fig.2), it has been assumed that effective rudder
angle(4)equals:
RR
, (9)
Rudder area A
R and aspect ratio
will be made
dependantonbasicshipdimensions.
461
Figure2. HistogramofdriftangleforcontainershipK1on
ashipping route between WesternEurope– East Coast of
USA(meandriftangle
R=0.075)
Search for relationship between AR from ship
dimensionshasbeenperformedfor:
variousshiptypes(129shipsaltogether),
for specific ship types, eg. container ships, bulk
carriers,tankers.
During our research, it has turned out that the
division into specific ship types does not give us a
significant increase of approximation adjustment to
modelvalues.
Exemplary results of obtained approximations of
rudderareahavebeengiveninFig.3‐6.
Figure3.ApproximationofrudderareaARdependingona
product of the ship length between perpendiculars L and
draughtTforvariousshiptypes(129shipsaltogether)
Figure4.ApproximationofrudderareaARdependingona
product of the ship length between perpendiculars L and
draughtTforbulkcarriers(37ships)
Figure 5. Approximation of rudder area A
R
depending on a
product of the ship length between perpendiculars L and
draught T for container ships (42 ships)
Figure6.Approximationofrudderarea ARdependingona
product of the ship length between perpendiculars L and
draughtTfortankers(31ships)
Basedoncarriedoutcalculations,ithasturnedout
that rudder aspect ratio
depends on ship
dimensionsonlyinaverysmalldegree,therefore for
specified ship types constant mean values can be
adopted:
bulkcarriers
=1.695,
containerships
=1.795,
tanker
=1.826.
Functions approximating forces and moment on
rudderhavebeensearchedforintheform:
( , , basic ship dimensions)
R
RR
R
X
YfV
M
(10)
with the assumption that, rudder angle
R will only
change within a small range on a given shipping
route – example of calculation (to algorithm [7])
results for rudder angle distribution for a
containership on a given shipping route have been
presentedinFig.7.
462
Figure7.HistogramofrudderangleforacontainershipK1
onasailingroutebetweenWesternEuropeandEastCoast
ofUSA.(meanrudderangle
R=1.40)
Apartfromshipgeometricparametersandrudder
angle, ship speed V occurs in equation (3) – normal
forceonrudderaswellasinequation(8)–slipratio
of propeller. It has therefore been assumed, that
approximatingmodelforrudderinflowvelocitywill
taketheformof:
R
VabV
(11)
Takingintoaccountapproximationofrudderarea
A
R and aspect ratio
the final expressions of forces
andmomentonrudderarethefollowing:



,2sin)(1874.20194.06.014.1
4
1
,2sin)(1874.20194.06.014.1
2
1
,sin)(1874.20194.0
2
2
22
RBPPR
RBR
RR
VbacTLCLM
VbacTLCY
VbacTLX

(12)
where:
25.2
13.6
2
1
w
c
.
Table1.Coefficientvaluesa,b,c foranadoptedmodelof
variousshiptypes
_______________________________________________
a[m/s] b[‐] c[kg/m
3
]
_______________________________________________
bulkcarriers4.2520.262  1.3498
containerships 5.3330.329  1.3941
_______________________________________________
4 VERIFICATIONOFWORKEDOUT
APPROXIMATIONSANDFINAL
CONCLUSIONS
Comparison of calculations performed according to
algorithm described in section2 [7] with the results
obtained from approximating formulas (12) for
selected ships, whose basic parameters are given in
Table2,hasbeenpresentedinFig.811.
Figure8. Forces and moment on rudder calculated for
obtained approximations (12) and according to the
algorithm[7]foracontainershipK1andtwovaluesofship
463
Figure9.Forces and moment on rudder calculated for
obtained approximations (12) and according to the
algorithm[7]foracontainershipK2andtwovaluesofship
speed
Figure10. Forces and moment on rudder calculated for
obtained approximations (12) and according to the
algorithm[7]forabulkcarrierM2andtwovaluesofship
speed
Figure11. Forces and moment on rudder calculated for
obtained approximations (12) and according to the
algorithm[7]forabulkcarrierM3andtwovaluesofship
speed
Table 2. Basic exemplary ship parameters used for model
verification
_______________________________________________
K1K2M2 M3
container container bulk bulk
ship ship carrier carr.
_______________________________________________
shiplengthbetween 140.14 171.94 189.9 180.0
perpendicularsL
PP[m]
shipbreadthB[m] 22.3 25.3 25.3 32.2
draughtT[m]8.25 9.85 10.6 12.0
blockcoefficientC
B 0.641  0.698  0.820 0.805
waterplanearea 0.809  0.828 0.854 0.873
coefficientC
WP[‐]
displacementvolume 17290 29900 40831 56396
[m
3
]
shipspeedV[m/s] 9.31 10.08  7.51 8.69
_______________________________________________
Results presented in Fig.811 have been
calculatedforsmallrudderangles
R,sinceinorderto
keepashiponagivenshippingroute,therudderwill
onlybedeflectedtoasmalldegreeinmostcases,Fig.
7 [7]. For such small rudder angles, approximations
presentedhereareaccurateenough.Approximations
workedoutherearehighlyusefulto test,alreadyat
initialstageofshipdesign, servicespeedoftheship
inrealweatherconditionsonapresetshippingroute.
464
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