449
1 INTRODUCTION
For safe and efficient ship‐tug operation from the
viewpoint of a tugʹs master (towmaster) we need to
have exact knowledge and understanding of the
complexrelationshipbetweenmultipleinput control
variables and the output performance of a tug. The
output performance, ordered by a pilot and/or
captain, is mostly indicat
ed by the towing force (in
terms of direction and magnitude) applied on the
towed ship. This force is transferred by a hawser
(towingline)inpullingmodeoradirecthullcontact
in pushing mode. Especially in pulling mode, the
towing force can be decomposed, into the
steering/transverse and backing/
longitudinal
components–both directionsaretaken withrespect
to the assisted ship. On the other side, the required
tugʹscontrolparameters,primarilyconsistingofthree
variables:thehulldriftangle,thethrusterangleand
force,essentiallychangewiththespeedoftheassisted
ship.Inaddition,formediumandhighspeedsofthe
escort operation tugs a
pply the so‐called indirect
towing in that they can take advantage of the
hydrodynamic force developed on their underwater
hull. This way the effective towing force is much
higherthanthethrusterforce.
Since there are some specific, more precise
definitions within indust
ry, we simply consider the
focused ASD tug as a tug with the directional
propulsionlocatedaftandthetowingpointforward.
Bystatistics[Artyszuk,2013b],thiswillmostlybean
azimuthing (podded, z‐drive) propulsion tug, and
mainlywith dualpropulsors installedsymmetrically
versusatugʹscentrepla
neforindependentoperation.
However, this paper is essentially dealing with
indirect towing performed by a parallel/coupled
operationofbothpropulsors,sotheycanberegarded
as a single unit of twice increased power, which is
through the text uniquely called as the thruster.
Steady-state Manoeuvring of a Generic ASD Tug in
Escort Pull and Bow-rope Aided Push Operation
J
.Artyszuk
M
aritimeUniversityofSzczecin,Szczecin,Poland
ABSTRACT:Thispaperisdevotedtoexpandtheverypromisingresearchundertakenintheauthorʹsprevious
work,basicallydoneonsimplifiedmodellingtheescortpushoperation.Now,theothertwomodesofatugʹs
employment,asstatedinthetitle,arecovered.Thespecialfocusisagainsetontheindirecttowi
nginthatthe
towlineforceismuchhigherthanthethrusterforce.Theratioofthesetwoforces,referredtoastherelative
towingforce(oramplificationratio)isevaluatedtogetherwiththehulldriftangleandthethruster(‐s)anglefor
a given escort speed. This mutua
l relationship is known as the tug performance diagram. Although rather
generic (container‐type) formulas are derived, they are supplied for exemplification purposes with simple,
analytically given hull hydrodynamic forces. The aim is also here to provide a basis for further sensitivity
analysisofthemodelandpossiblei
mprovement/optimisationtothetugdesign.Theobtainedchartsalsocould
serveasroughandclearguidancefortowmasterswhileescorting.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 8
Number 3
September 2014
DOI:10.12716/1001.08.03.17
450
Because we are implementing rather general
hydrodynamicmodelof thethruster,the results can
be easily adopted to another types of directional
propeller, e.g. the Voith‐Schneider propeller (VSP),
evenwhentheseareinstalledintheforwardpartofa
tug (called then a tractor tug), as usual for this
propulsion.Inthelattercaseofatractortug,wehave
torememberthatthistugwillalsowork/assistbyits
end, which is free of the thrusters, i.e. by the stern,
quitesimilartoASDtugsactingthroughthebow.
Someresearchcentres,refere.g.to[Hensen,2003],
[Quadvlieg,
Kaul, 2006], [Renilson et al., 1992],
[Waclawek,Molyneux,2000],claimtheydevelopeda
software for computing tow forces, as well as the
necessarycontrolparametervaluesonatug,insteady
state situations. However, appropriate results and
discussionconcerningboththeappliedmathematical
model and the detailed, well documented output in
the form of charts are practically not published. If
any, such diagrams are sometimes very hard in
handling.
Undersuchbackgroundtheauthorʹsconducteda
researchonthemechanismofequilibriumforatugin
the escort operation, i.e. the towing assistance
renderedundersignificantspeedoftheassisted
ship.
Thestudypreliminarilyinvolvedthecaseofpushing
operation [Artyszuk, 2013a]. The reader is
encouragedtorefertothisworkwhichisavailablein
open‐access through the web‐site of the authorʹs
affiliation.Now,inthepresentpaper,thoseanalytical
solutions are being generalised to cover a more
sophisticated case, namely the pulling mode. Under
appropriateparametervalues,thepresentedhereafter
solution converges to the previous pure pushing
mode. At the end of paper, however, some
considerationsarealsomadewithregardtoapplying
theresultstopushingmodewithbow‐linesupportor
thefrictioneffectincluded.
2 MATHEMATICALMODEL
The ship‐tug arrangement during the so‐called
indirect pulling operation, together with forces and
conventionsforangles,ispresentedinFigure1.The
indirecttowinginvolvestaking advantageofatugʹs
underwaterhullhydrodynamicforcewhilerendering
assistance at significant escort speed. The tug‐fixed
coordinate
systemMxyispositionedforconvenience
at the intersection of her centre plane and midship
section, with x axis pointing forward and y axis to
starboardside.
Theequilibriumconditionsfora tugbetween the
hull(H),thruster/propeller(P),andtowing(T)forces
intugʹscoordinates
taketheform:
0
0
0
xH xP xT
yH yP yT
zH zP zT
FFF
FFF
MMM
(1)
where:
F
x,Fy–longitudinal and lateral components of each
force[N],
M
z –momentdevelopedbyparticularforce[Nm].
towing force
F
T
as acting on ship
SHIP
at speed
v
inflow speed
v
thruster
force
F
P
hull
force
F
H
F
yH
F
xH
X
Y
O
x
y
M
drift angle
M
tug
body axes
ship
body axes
TUG
towing (reaction) force
F
T
as acting on tug
F
T
>0 (always for pulling)
F
T
>0
x
T
x
P
F
P
>0 (always)
hawser
angle
<0
Figure 1. Definition sketch of forces and angles in tugʹs
dynamics
The hull moment MxH is specific in that it is
directlymeasuredorcomputed,andpublished,andis
beingrarelybasedonconstructingtheproductofthe
hulllateralforceF
yHandanabscissaofitsapplication
point,whichisalsosometimescalledanarmorlever.
Thelatterisnamely generally outofinterestin hull
hydrodynamics.
The tughullhydrodynamic forces are commonly
writtenasfollows:
2
0.5
xH fxh
yH fyh
zH mzh
Fc
FLTvc
MLc
(2)
where:
–waterdensity[kg/m
3
],
L,T –tugʹs length (between perpendiculars) and
draft(extreme)[m],
v –absoluteinflowspeed(equaltotheescortspeed)
[m/s],
c
fxh,cfyh,cmzh –nondimensional hydrodynamic
coefficients[],
–drift angle (equal to tugʹs inclination angle vs.
shipʹshull)[].
Thehullhydrodynamiccoefficientsforrectilinear,
obliquemotion,asincaseofourstaticconditions,are
functionsofthedriftangle andusuallylookup‐table
stored.Thelookup‐tableapproachisalsoanessential
451
part of the developed and hereafter presented
algorithm for solving the equilibrium condition.
However, in view of the undertaken preliminary
research some simple, analytical, and qualitative
relationshipsareintroducedtosuchtables:
0.03cos
0.5sin
0.1sin 2
fxh
fyh
mzh
c
c
c
(3)
where
180 , 180
.
Due to symmetry and for some practical reasons
connected with physically justified equilibrium
conditions, we will seek the equilibrium solution in
the range of negative drift angles (
<0), strictly
for
0,90
.Thiscorrespondstoa tugsecured
on the shipʹs port quarter and facing its starboard
bowtowardstheinflow,asshowninFigure1.
In view of getting an equilibrium solution, the
ratiosoftughullhydrodynamiccoefficients turntobe
veryuseful:
''
,
fxh
mzh
fxh mzh
fyh fyh
c
c
cc
cc
(4)
The graphical image and detailed discussion of
theserelationshipsiscontainedin[Artyszuk,2013a].
Thethrusterforcesandmomentin(1)read:
cos
sin
sin
xP
yP P
zP P
F
FF
Mx
(5)
where:
F
P–absolutevalueofthrust(alwayspositive)[N],
–thrusterangle(equaltothethrustangle)[],
x
P–thrusterposition(negativeinaftdirection)[m].
Though most harbour tugs have dual,
independent propulsors to enhance their
manoeuvrability, itisassumedin the present study,
as mentioned before, that both thrusters rotate
parallelandworkequally.Thismeanswecanadopta
singlethrusteroftwiceincreasedforce.
Additionally,
theadvancespeedeffectonthethrusterperformance
loss and the influence of local drift angle on
producingthelateralcomponentofthethrusterforce
aredisregardedinthispreliminaryinvestigations.So,
both symbols F
P and
are denoting the effective
thrusterforceanditsdirectionangle.
The towing (pull) force, as the reaction force
excited on a tug, according to the conventions
adoptedinFigure 1,i.e.withfullsupportofsign, is
describedby:
cos
sin
sin
xT
yT T
zT T
F
FF
Mx
(6)
where:
F
T–absolutevalue of towing force (always positive)
[N],
–hawserangle(negativewhenleadingtoportside
ofthetowedship)[],
x
T–towing point position (positive in forward
direction)[m].
3 ANALYTICALSOLUTIONOFTHE
EQUILIBRIUM
Identically to [Artyszuk, 2013a], one can easily find
that:
'
'
'
cos cos
sin sin
T
fxh
T
F
c
F
(7)
'''
'
'
sin sin
sin sin
PTT
mzh
T
xxF
c
F
(8)
wherewehavedefinedtherelativetowingforce
'
T
F
,
asbeingastheratioofthethrusterforce:
'
T
T
P
F
F
F
(9)
and the other nondimensional quantities connected
with the geometrical positions of towing point and
thethruster:
'
T
T
x
x
L
,
'
P
P
x
x
L
(10)
Intheexemplarycalculationspresentedinthenext
chapterweareassuming:
'
0.5
T
x
L
,
'
0.5
P
x
L
(11)
Theformulas(7)and(8)canbeconvertedinto:
'
'
'
cos sin
cos sin
fxh
T
fxh
c
F
c
(12)
''
'
''
sin
sin
Pmzh
T
Tmzh
xc
F
xc
(13)
Making both them equal, we are arriving at the
first (starting) fundamental relationship
=
(
,
),
where
istheparameter:
1
''
1' 1 ' '
''
tan tan
Pmzh
Tfxhfxh
Tmzh
xc
Fcc
xc
(14)
Thedirectequation(14),explicitvs.thrusterangle
,shallbesolvedinthedriftangledomain.So,fora
452
series of discrete values of the drift angle
we are
computingthecorrespondingvaluesofthebalancing
thrusterangle
The second fundamental equation is one of the
two equivalent formulas: (12 or (13). Both make a
dependence of the relative towing force
'
T
F
on the
just determined thruster angle
. Below the latter
formulaisbeingchosen:
2
''
''
'
sin
sin
mzhT
mzhP
T
cx
cx
F
(13)
Thethirdfundamentalexpressioninthesequence
ofourcomputationsconsistsofthebalanceequation
forlateralforces,see(1):
3
sinsin
''
TyH
FF
(15)
wherewehavedefinedtherelativehulllateralforce
'
yH
F
inthesimilarwayto
'
T
F
in(9):
'
yH
yH
P
F
F
F
(16)
Therelationship(15)takesoninputthepreviously
establishedvaluesof
and
'
T
F
.
Finally,weusethemiddleformulain(2)torelate
the escort speed to the absolute magnitude of the
thrusterforce
P
F
intheformof:
4a
fyh
PyH
LTc
FF
v
5.0
'
(17a)
or
4b
'
2
5.0
yH
fyh
P
F
cLTv
F
(17b)
Four fundamental equations (14), (13), (15), and
(17) constitute the basic mechanism of the wanted
tugʹsequilibrium.
4 NUMERICALRESULTS
For below computations we adopt the following
conditions of the environment and the tug: water
density1000kg/m
3
,L=30.5m,T=5m.
Figure2presentsthebasiccomputationresultsof
ourformulas.Twodifferent,ratherextremeandthus
meaningful thruster force values F
P have been here
selected, corresponding to 50t and 10t. The unit of
tonnehasbeenhereconsciouslytaken,sincethisstill
serves as the industry language of evaluating tug
capabilitiesandconductingtowingoperations.Figure
2iscomprisingfoursub‐chartsforeachcaseoftheF
P
magnitude.Theyshowaccordingly:thethrusterangle
,driftangle
(insomestudiesreferredtoastheyaw
orslipangle),therelativehulllateralforce
'
yH
F
,and
finallythemostimportantrelativetowingforce
'
T
F
.
Theyareplottedversustheescortspeed.Thetypical
rangeofspeedisincluded,i.e.uptoabout10knots.
Thehawserangle
istheparameterforallthecurves,
90,180
,thoughitsname onlyappearsfor
the top‐level subdiagrams. The value of
correspondingto180meansahawserinthecentre
planeandaftdirectionoftheassistedship,while90
marksthehawsersetabeamoftheship,alsoreferto
Figure1.
The excellent
indirect towing performance is
achievedforthelowerthrusterforce,sinceforthetug
hullsizeoforder30minlength(thetypicaldimension
ofaharbourtug)and theinvestigated escortspeeds
much of the equilibrium is relatively dominated by
the tug hull hydrodynamic force. It shall be
mentioned
that both columns of Figure 2 are
essentially similar to each other in that the adopted
thruster force is causing the horizontal scaling
(multiplyingtheʹx‐valuesʹ)ofthecharts.
In case of
=90 we are receiving the same
results as for pushing operation which were
publishedin[Artyszuk,2013a].
Forsomehawserdirectionswemayfindevenup
to three different equilibrium solutions in terms of
thruster angle and drift angle. Those are of course
accompanied by a different relative
towing force
contributing to an effective tugʹs pull force rated in
tonnes.
InFigure3thereisshowna mutualrelationshipof
thethrusteranddriftangles.AsclearfromEquation
(14),itis neither influencedbythethruster absolute
force,northe escortspeed.The curvefor
=90 in
the vicinity of zero drift angle slightly differs from
that in [Artyszuk, 2013a]. This small discrepancy is
due to a better discretization (lookup‐table based
interpolation) of the tug hull hydrodynamic
coefficientsforthepurposeofthepresentstudy.
Therelativetowingforce
'
T
F
canbedecomposed
for a practical application in ship towing operations
intothebackingandsteeringcomponents.Thisway,
theyarealsoexpressedastherelativequantities,i.e.
comparedversusthethrusterforce:
''
cos
back T
FF
,
''
sin
steer T
FF
(18)
BotharedemonstratedinFigure4.Forthehigher
thrusterforce50t,theyaregenerallyhardlyeffective
(notethevalueslessthanunity).
The subsequentFigure5 comprises the results of
calculation of the required thruster force (absolute
oneintonnes)foragivenescortspeed,seeEquation
(17b).
Ofcourse,Figure5repeats tosomeextentthe
data of Figure 2. Nevertheless, it provides data in a
differentformat,discretization,andisveryuseful to
directly study the thruster force under input escort
speed. Only three distinct hawser directions are
considered: 90 (steering action only) 135 (equal
backing and steering action), andsteering action
only,180(backingactiononly).
453
0
2
4
6
0123 45
0
1
2
3
012345
-6
-4
-2
0
012345
-3
-2
-1
0
012345
-90
-75
-60
-45
-30
-15
0
012345
-90
-75
-60
-45
-30
-15
0
012345
0
30
60
90
120
150
180
012345
0
30
60
90
120
150
180
012345
thruster force F
P
= 50t
escort speed
v
[m/s]
thruster force F
P
= 10t
escort speed
v
[m/s]
thruster angle
[]
thruster angle
[]
-180
-150
-135
-120
-105
-90
-180
-165
-150
-120
-105
-90
-135
hawser
angle
[]
hawser
angle
[]
escort speed
v
[m/s]
escort speed
v
[m/s]
drift angle
[]
-180
-135
-120
-105
-90
-180
-165
-150
-120
-105
-90
-135
drift angle
[]
escort speed
v
[m/s]
escort speed
v
[m/s]
rel. hull lateral force F'
yH
[-]
-180
-135
-120
-105
-90
-180
-165
-150
-120
-105
-90
-135
rel. hull lateral force F'
yH
[-]
escort speed
v
[m/s] escort speed
v
[m/s]
relative towing force F'
T
[-]
-135
-90
-180
-165
-150
-120
-105
-90
-135
relative towing force F'
T
[-]
-120
-180
Figure2.Kinematicanddynamicparametersofindirect(pull)towingversusescortspeed
454
0
30
60
90
120
150
180
-90-75-60-45-30-150
thruster angle
[
]
-180
-165
-150
-120
-105
-90
-135
hawser
angle
[
]
drift angle
[
]
Figure3.Thedrift‐thrusteranglerelationship(asindependentofescortspeedandthrusterforce)
-3
-2
-1
0
012345
-3
-2
-1
0
012345
-3
-2
-1
0
012345
-3
-2
-1
0
012345
thruster force F
P
= 50t thruster force F
P
= 10t
hawser
angle
[]
hawser
angle
[]
-150
-105
escort speed
v
[m/s] escort speed
v
[m/s]
relative backing force F'
back
[-]
-135
-90
-180
-165
-150
-120
-105
-135
-120
-180
relative backing force F'
back
[-]
-90
-150
-105
escort speed
v
[m/s] escort speed
v
[m/s]
relative steering force F'
steer
[-]
-135
-180
-180
-165
-150
-120
-105
-135
-120
-90
relative steering force F'
steer
[-]
-90
Figure4.Distributionoftherelativetowingforcetobackingandsteeringcomponents
The parameter for all the curves in sub‐charts of
Figure 5 is the escort speed. The pattern of these
curves seems to be more interesting, rich in
informationandbeneficialforpracticalpurposesthan
thatinFigure2.Amongothers,Figure5isdivedinto
columns according to the
hawser angle, which
constitutesadirectorderfromapilot.
Both Figures 2 and 5 can predict the control
parametersofatugforgivenescortspeedandhawser
direction. However, a consequence in terms of the
absolute tension of the hawser (towing force in
tonnes)isnoteasilyseen.Namely,
thehigherthruster
forceisaccompaniedwithlowerrelativetowingforce,
whilethelower thrusterforceisincontrastassociated
withhigherindirecttowingeffectiveness.
Onemightwonderwhetherabsolutetowingforces
for high and low thruster forces are close to each
other. The plots of Figure 2 and 5
are thus
supplementedinFigure6withabsolutevaluesofthe
towingforce.
455
0
1
2
3
0 1020304050
0
2
4
6
0 1020304050
0
1
2
3
0 1020304050
-90
-75
-60
-45
-30
-15
0
0 1020304050
-90
-75
-60
-45
-30
-15
0
0 1020304050
-90
-75
-60
-45
-30
-15
0
0 1020304050
0
3
6
9
12
15
0 1020304050
speed
v
4m/s
3m/s
1m/s
2m/s
5m/s
0
30
60
90
120
150
180
0 102030 4050
0
30
60
90
120
150
180
01020304050
hawser angle
= -90 hawser angle
= -135 hawser angle
= -180
t
h
ruster ang
l
e
[
]
thruster angle
[
]
thruster angle
[]
thruster force
F
P
[t] thruster force
F
P
[t] thruster force
F
P
[t]
4m/s
3m/s
1m/s
2m/s
5m/s
speed
v
4m/s
3m/s1m/s
2m/s
5m/s
speed v
4m/s
3m/s
1m/s
2m/s
5m/s
d
r
i
f
t ang
l
e
[
]
drift angle
[
]
drift angle
[]
thruster force
F
P
[t] thruster force
F
P
[t] thruster force
F
P
[t]
4m/s
3m/s
1m/s
2m/s
5m/s
4m/s
3m/s
1m/s
2m/s
5m/s
4m/s
3m/s
1m/s
2m/s
5m/s
rel. towing force F'
T
[-]
rel. towing force F'
T
[-]
rel. towing force F'
T
[-]
thruster force
F
P
[t] thruster force
F
P
[t] thruster force
F
P
[t]
4m/s
3m/s
1m/s
2m/s
5m/s
4m/s
3m/s
1m/s
2m/s
5m/s
Figure5.Parametersoftugʹsequilibriumversusthrusterforce
0
10
20
30
40
50
60
70
012345
0
10
20
30
40
50
60
70
012345
0
10
20
30
40
50
60
70
012345
0
10
20
30
40
50
60
70
01020304050
0
10
20
30
40
50
60
70
0 1020304050
0
10
20
30
40
50
60
70
0 1020304050
hawser angle
= -90 hawser angle
= -135 hawser angle
= -180
4m/s
3m/s
1m/s
2m/s
5m/s
towing force F
T
[t]
towing force F
T
[t]
towing force F
T
[t]
thruster force
F
P
[t] thruster force
F
P
[t] thruster force
F
P
[t]
4m/s
3m/s
1m/s
2m/s
5m/s
4m/s
3m/s
1m/s
2m/s
5m/s
40t
30t
10t
20t
50t
towing force F
T
[t]
towing force F
T
[t]
towing force F
T
[t]
escort speed
v
[m/s] escort speed
v
[m/s] escort speed
v
[m/s]
40t
30t
10t
20t
50t
40t
30t
10t
20t
50t
speed
v
speed
v
speed
v
thruster
force
F
P
thruster
force
F
P
thruster
force
F
P
Figure6.Absolutetowingforce(intonnes)versusthrusterforceorescortspeed.
456
Relative (vs. the thruster force) and absolute
towingforcesbothhavetheirownindividualmerits.
The absolute force is anyhow more practical and
directly ordered by a pilot. It also leads to a tugʹs
capability in terms of the maximum effective pull
under given speed, if the maximum thruster
force
developed for such speed is input to the
computationalalgorithm.
Thesimpletughullhydrodynamicsadoptedinthe
presentpaperintermsofshape andsize ofthehull
nondimensional hydrodynamic coefficients, refer to
Equation (3), implies the following features, which
canbedrawnfromFigure6:
in
pure steering (
=90) and pure backing
(
=180) we are receiving the global indirect
towing effectiveness of approx. 50%, i.e. the
thrusterforceistransformedtoabout50%higher
values in the hawser, see the upper diagrams of
Figure6,
in case of
=135 the effectiveness is varying
between zero and even several hundred percent
dependent on the thruster force (the lower the
better)andescortspeed(thehigherthebetter);the
lowerdiagramforthishawserangle(showingthe
towing force versus escort speed) indicates the
towing force during the
equilibrium of a tug be
almost independent of the thruster force and be
influencedbytheescortspeedonly.
The raised abovepointsshallbevalidatedinthe
futureforactualhydrodynamiccharacteristicsoftug
hulls.
5 EQUIVALENCETOPUSHINGOPERATION
WITHFRICTIONEFFECTORMOORINGROPE
SUPPORT
The model and
results having beendescribedsofar
arealsovalidifweeitherconsiderthepushoperation
with the friction effect between a shipʹs and a tugʹs
hullor ifthepushactionofatugis supportedby a
longitudinal (with reference to the ship) mooring
rope. As
mentioned before, these effects were not
included in [Artyszuk, 2013a]. In Figure 7 only the
mooringropecaseisconsidered,exactlyconsisting of
a bow line. However, the friction force can be
modelled in the same way, it will also point up,
identically to F
M in Figure 7, since in both the
situationsatughasatendencytomovetowardsthe
sternofashipanddeceleratetheship.Anadditional
usefulsimplification,thoughquitereasonable,would
be if we assume the pushing point on a tug to
coincidewithitsmooringfairlead.
The
oppositebutrathertheoreticaldirectionofthe
longitudinal force due to mooring of friction is also
possibleinthataspringlineisimplementedinstead.
Anyhow, such specific case is dealing with the
equivalent hawser direction angle
0,90
,
which is not examined in the present paper. Under
such conditions, as well as in other not discussed
situations, a necessity of independent operation of
boththrustersmightoccurtoachieveatugʹssteady‐
statemovement.Thisbringsanarbitrarycombination
of the balancing force and the moment excited
by
thrusters., while the coupled/parallel mode of
operation, widely used in the paper, limits (or
ʹstiffensʹ)themomenttotheproductofthe resulting
lateral force and the longitudinal location of both
thrusters.
SHIP
at speed
v
inflow speed
v
push
force
push reaction
force
< 0
M
TUG
mooring
rope
(bow line)
F
M
F
T push
mooring rope
pull force
towing force
F
T
> 0
Figure 7. Equivalent pushing operation supported with a
bowline
Using the numerical results of the previous
chapterforthementioned caseofapushingtugwith
bow line requires the following substitutions or
conversions(checkFigure7formeaningofsymbols):
90
90
(19)
22
TTpushM
FF F
cos
MT
FF
,
sin
Tpush T
FF
(20)
In case of friction, the force F
M in Figure 7 will
depictthefrictionforce,whileF
Tpushwillconstitutethe
normal reaction, as essential in computation of the
former one. The angle
represents thus the friction
angle, and its tangent denotes the standard friction
coefficient. Of course, the magnitude of friction‐
related force F
M is a small, fixed proportion of the
othercomponentF
Tpush.However,inthebowlinecase
thereisnopracticallimitforF
M.
6 CONCLUSIONS
The present study has revealed that both modes of
towing assistance at speed: pulling and pushing
operationsareofthesamephysicsandmathematical
model.Moreover,thepushingmodelisjustaportion
of the most general pulling model in that it derives
fromthelatter,referparticularly
toFigure6andthe
left diagrams pertaining tocase(
=90). However,
thetermʹindirecttowingʹisusuallyappliedtopulling
457
operation only. So, in situations when tugʹs.hull
hydrodynamic force is taken to advantage while
developing a towing force, the latter being much in
excess of the propeller force, either in pulling or
pushing,theexpressionʹindirectassistanceʹisequally
authorisedaccordingto its literal,primary meaning.
This involves of
course a certain drift angle of tugʹs
hull,butitsroleismuchmorethanpurelypreserving
appropriatekinematicfollowingofthe assistedship.
Suchcommitmentrelatingtotheproposedwideruse
ofthewordʹindirectʹtookalsoplaceintheprevious
work [Artyszuk, 2013a], but now has been
additionallyproved.
The presented algorithm is flexible enough to
accommodateanypatternoftugʹshullhydrodynamic
coefficients, and thus may result in the actual tug
behaviour.Theonlyitemsufferingsomedeficiencies
is the applied simple thruster (propeller) model, i.a.
notincludingtheadvancespeeddetrimentaleffect.It
is believed
however that the results of the present
report will still be valid to some extent, since the
constant values of the thruster force used in our
computations shall be considered as the effective
thrustforce,i.e.requiringhigherrpm/pitchsettingsif
strong speed effects exist. The other phenomenon
wortha
futureconcernisalsothetransverseforceon
thepropellerduetolocaldriftangle,ascontributing
tothetotalforceandthusalteringtheeffectivethrust
direction. The latter can be quite different than the
propelleraxis.
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[3]HensenH.:TugUseInPort,APracticalGuide.2ndEd.,
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[4]Quadvlieg F., Kaul S.: Development of a calculation
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