International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 6
Number 4
December 2012
1 INTRODUCTION
Icebergs may cause a threat to installations, vessels
and operations in a number of Arctic and Antarctic
regions. If icebergs are detected and considered to be
a threat, it has been documented that they can be
deflected into a safe direction in approximately 75%
of the events (Rudkin et al., 2005). The preferred
method for iceberg deflection is single vessel tow
rope (Fig 1a). Experimental towing of the iceberg in
the Barents Sea was realized with a rope taken
around the iceberg by a loop in 2005, and the rope
was broken during the towing. Photograph of the
towing is shown in Fig. 1.
The majority of the unsuccessful tows ended
because the tow slipped over the iceberg while
ruptures of tow line or iceberg rolling over are other
common explanations (Rudkin et al., 2005). There
are also examples of towing where towing in the
planned direction was not possible. In order to
increase the understanding of what happens when an
iceberg tow is started Marchenko and Gudoshnikov
(2005) and Marchenko and Ulrich (2008) developed
a numerical model for iceberg towing. Eik and
Marchenko (2010) analysed results of HSVA
experiments on the towing of model iceberg in open
water and when broken ice was floating on the water
surface. Stability of iceberg towing with floating
rope was discussed in (Marchenko, 2010).
Figure 1. Towing of 200000 t iceberg in the Barents Sea.
In the present paper we compare two different
methods of iceberg towing, formulate governing
Methods of Iceberg Towing
A. Marchenko & K. Eik
The University Centre in Svalbard, Longyearbyen
STATOIL
ABSTRACT: Mathematical models of iceberg towing by a ship connected to the iceberg by mooring lines are
considered. Governing equations describing the towing and the tension of mooring lines in two different
schemes of the towing are formulated. Stability of steady solutions describing the towing with constant speed
is studied. Numerical simulations are realized to compare results of the modeling with experimental results of
model towing in HSVA ice tank.
507
equations, analyse the stability of steady towing and
perform numerical simulations of the HSVA
experiments on iceberg towing in open water taking
into account the influence of natural oscillations of
water in the tank.
2 METHODS OF ICEBERGS TOWING
In the practice of iceberg management two methods
(I and II) of icebergs towing were performed. In
both methods towed iceberg was trapped by a loop
of floating synthetics rope ringed around the iceberg.
The ends of the rope are fastened on the boat stern in
the method I (Fig. 1a) and connected to heavy steel
hawser fastened at the boat stern in the method II
(Fig. 1b). The rope floats or hangs above the water
surface in the method I. In the method II the hawser
is submerged and the rope is floating near the
iceberg and submerged near its connection to the
hawser. Typically the hawser is shorter the rope. In
Fig. 2 the line FTR shows floating tow rope and the
line W shows the steel hawser. Thin line SS shows
the water line around the iceberg and between the
iceberg and the boat. In Fig. 2a) the rope segments
D
1
O
1
and D
2
O
2
are floating, and the segments O
1
C
and O
2
C are hanging above water surface. In Fig.
2b) the rope segments O
1
B and O
2
B are submerged.
The realization of the method I is simpler on the
practice, but in case of the rope break up the boat
stern can be damaged by the rope. In the method II
the water takes buffer role in case of the rope break
up.
Figure 2. Schemes of iceberg towing with floating (a) (method
I) a
nd submerged (b) (method II) tow lines.
The forces applied to the iceberg (
Ir
F
) and to the
boat (
Br
F
) by the rope in the method I are calculated
with formulas
21 DDIr
TTF +=
,
21 CCBr
TTF +=
. (1)
The rope tension forces
,
2D
T
,
1C
T
and
2C
T
are shown in Fig. 3. The forces applied to the
iceberg (
Ir
F
) by the rope and to the boat (
Bh
F
) by the
hawser in the method II are calculated with formula
21 DDIr
TTF +=
,
CBh
TF =
. (2)
The rope tension forces
1D
T
,
2D
T
and
C
T
are
shown in Fig. 4.
Safety requires avoid the approaching of iceberg
to the boat. Therefore in both methods of icebergs
towing distances
1
X
and
2
X
shown in Fig. 3 and
Fig. 4 should be much greater distances
1
R
and
2
R
.
It can be expressed by the inequality
1<<
χ
, where
XR /=
χ
and
R
represents iceberg radius. Quantity
X
represents distance between the boat stern and
the points D
1
and D
2
, where the rope approaches to
the iceberg. Assuming
m 500≈X
and
representative radius of the iceberg 50 m we find
that
1.0=
χ
. Distances
C
z
and
B
z
are typically
smaller 10 m. Therefore parameter
r
lz /
BC,
=
ε
is
smaller 0.02.
Figure 3. Schemes of towing line in method I in lateral (a) and
upward (b) projections.
Figure 4. Schemes of towing line in method II in lateral (a) and
upward (b) projections.
Angles
2,1
α
and
2,1
β
shown in Fig. 3 and Fig. 4
are of the orders
ε
and
χ
respectively. Projections
of the rope tensions on the
x
-direction are
508
proportional to
2,1
cos
α
and
2,1
cos
β
. Therefore their
difference has the order
),(
22
χε
O
, and formulas (1)
and (2) can be written with accuracy
),(
χε
O
as
follows
TF
Ir
=
,
TF
Br
−=
, (1a)
TF
Ir
=
,
TTF
CBh
−=−=
, (2a)
where
2/T
is the rope tension,
Ir
F
,
Br
F
and
Bh
F
are
the absolute values of the forces applied to the
iceberg and to the boat along the
x
-axis.
Further it is assumed that the difference between
x
-coordinates of the points D
1
and D
2
is much
smaller than the iceberg radius
R
, and
XXX ==
21
with accuracy
)
,(
χ
ε
O
. In this case the rope tension
is expressed as a function
)
(X
T
T =
,
0/ >dXdT
(3)
where
X
is the distance between points of the rope
fastening on the boat stern and on the iceberg
surface. Inequality in (3) means that the increase of
X
is accompanied by the increase of the rope
tension.
3 GOVERNING EQUATIONS
Equations of momentum balance for boat and
iceberg connected by tow line are written as follows
xhgMP
TR
dt
dv
M
ss
s
s
∂∂−+−−= /
, (4)
xhgMTR
dt
dv
M
ii
i
i
∂∂−+−= /
, (5)
where
s
M
and
i
M
are the masses of the boat and
the iceberg including the added mass,
P
is the boat
propulsion, and
xh ∂
∂ /
is the water surface gradient.
It is assumed that horizontal scale of water surface
elevation is much greater than the iceberg diameter
and ship length. Water resistances to boat motion (
s
R
) and to iceberg motion (
i
R
) are described by
square low as follows
)( uvuvSCR
ssswsws
−−=
ρ
,
)( u
vuvSCR
iii
wiwi
−−=
ρ
, (6)
where
s
S
is wet surface of the boat hull, and
i
S
is
representative area of vertical cross-section of
submerged surface of the iceberg in perpendicular
direction to the tow direction, and
u
is water
velocity. Drag coefficients are equal to
003.0=
ws
C
(Voitkunsky, 1988) and
)1,5.0(∈
wi
C
(Robe, 1980).
Equations (4) and (5) are completed by the
definition of relative velocity as follows
is
vv
dt
dX
−=
. (7)
Equations (4), (5) and (7) perform closed system
of ordinary differential equations relatively unknown
functions of the time
)
(t
v
s
,
)
(t
v
i
and
)(tX
. They
have steady solution describing steady towing with
constant speed
0
v
, constant propulsion
0
P
, constant
water velocity
u
and
0/ =∂∂ xh
. For the steady
towing it follows
0
vvv
is
==
,
i
RT =
0
,
00
PTR
s
=+
, (8)
)(
0
0
iwiswsw
SCSC
P
uv
+
=−
ρ
,
iwiw
SCuvT
ρ
2
00
)( −=
(9)
Distance
0
XX =
is determined from the first
formula (3) when
0
TT =
.
Dimensionless variables: time
't
, velocities
s
v'
and
i
v'
, rope tension
τ
, boat propulsion
π
and
distance
Ζ
between points of the rope fastening are
introduced as follows
1,
'
r
t
t
t =
,
0
'
v
v
v
s
s
=
,
0
'
v
v
v
i
i
=
,
0
'
v
u
u
=
,
0
T
T
=
τ
,
0
P
P
=
π
,
r
l
X
=Ζ
, (10)
where
001
,
/TvM
t
sr
=
is the representative time, and
r
l
is the rope length. Equations (4), (5) and (7) are
written in dimensionless variables as follows
(primes near dimensionless variables are omitted)
xuvuv
dt
dv
wss
s
∂∂−++−−−−= /)1()(
11
ηεπετε
, (11)
xuvuv
dt
dv
wii
i
∂∂−+−−−= /)(
22
ηετεε
,
is
vv
dt
d
−=
Ζ
γ
,
where
)/(
1
iwi
sws
SCSC
=
ε
,
is
M
M /
2
=
ε
,
)/(
2
00
v
MT
l
sr
=
γ
,
rw
la /=
ε
, and
a
is the amplitude
of water surface elevation. Simple estimates show
that
1
2,
1
<<
ε
when the iceberg is not very small.
Steady solution in dimensionless variables is
written as
1====
πτ
is
vv
. (12)
Value
0
Ζ
=Ζ
for steady towing is constructed as
solution of the equation
1)(
0
=Z
τ
.
4 STABILITY OF STEADY TOWING
Solution of equations (11) in the vicinity of the
steady point is written in the form
509
ss
vv
δ
+=1
,
ii
vv
δ
+=1
,
Ζ+Ζ=Ζ
δ
0
, (13)
where new independent variables
)(tv
s
δ
,
)(tv
i
δ
and
)(tΣ
δ
describe small fluctuations in the vicinity of
the steady solution. Substitution of formulas (13) in
equations (11) leads to the following equations
Ζ−−=
δτδε
δ
'2
01 s
s
v
dt
vd
,
Zv
dt
vd
i
i
δτεδε
δ
'2
022
+−=
,
is
vv
dt
d
δδ
δ
γ
−=
Ζ
, (14)
where
dZd /'
0
ττ
=
by
0
ZZ =
.
Substituting exponential solution
t
ss
evv
ψ
δδ
0,
=
,
t
ii
evv
ψ
δδ
0,
=
,
t
e
ψ
δδ
0
Ζ=Ζ
, in equations (14) we
find that eigenvalues
ψ
are the roots of the cubic
equation
0))(2)1(()2)(2()(
212221
=++++++≡
εεεεψµεψεψψψ
F
, (15)
where
0/'
0
>=
γτµ
. Roots
0,i
ψ
of equation (15) by
0
21
==
εε
are chosen as the first approximation of
the roots when
1
2,1
<<
ε
. One finds
0
0,1
=
ψ
,
µψ
i=
0,2
,
µψ
i−=
0,3
. (16)
Next approximation is expressed by the formulas
21
2
εψ
−=
,
212
2
ε
µ
εµψ
i
i +−=
,
213
2
ε
µ
εµψ
i
i −−−=
. (17)
From formulas (17) follows that real parts of
eigen-values
ψ
are negative. Therefore the solution
describing steady towing is stable. At the same time
absolute values the of real parts of eigen-values
2
ψ
and
3
ψ
are much smaller than absolute values of
their imaginary parts:
)Im()Re(
3,23,2
ψψ
<<
.
Therefore damping of perturbations of the steady
solution will be accompanied by oscillations with
dimensionless frequency
µ
. Dimensional period
of the oscillations is calculated as follows
µπ
/2
1,1, rp
tt =
. Oscillations of mooring system
with floating submerged sensors were observed and
studied by Hamilton (2000).
5 MODEL TESTS OF ICEBERGS TOWING
Model tests on the iceberg towing were performed at
Hamburg Ship Model Basin (HSVA), Germany.
Two model icebergs with cylindrical and rectangular
shapes were made from water ice and towed using
scheme shown in Fig. 1b) in the tank with different
concentration of ice floes on the water surface. The
description and results of the experimental studies
are performed in the paper (Eik and Marchenko,
2010). In the present paper we consider only results
of the experiment when cylindrical iceberg was
towed in the water with free surface. Dimensions of
model iceberg and fragment of the towing are shown
in Fig. 5.
Figure 5. Dimensions of model iceberg (a) and fragment of the
HSVA experiment (b).
The movement and rotation of towed icebergs
were recorded in all six degrees of freedom with a
Qualisys-Motion Capture System. The platform with
sensors was installed on the iceberg surface to
monitor the iceberg motion as it is visible in Fig.
5b). Three degrees of the movement are
characterized by the horizontal distance between the
carriage and the sensors (surge), sideways
movement between fixed point at the carriage and
the sensors (sway), and by the vertical displacement
of the sensors (heave). Three degrees of the rotation
are performed by the pitch, the roll and the yaw of
the platform with sensors with respect to the
direction of the tank extension.
The tow line consisted of floating rope Dyneema
and steel wire. Characteristics of the towing rope
Dyneema and the wire used in the tests are
performed in Tables 1 and 2 for model scale. The
tension in the tow line was recorded in three
locations: on the end of the wire on the carriage and
on the rope ends near the point of their connection
with the wire (point B in Fig. 5). Some of the loads
cells in some of the tests were “drifting” and
manually corrected. This causes some unfortunate
uncertainties in the load results. Average tow loads
were varied in the range from 0.78 N to 4 N in the
tests with open water.
The length and the water depth in the tank are
equal
m 72=
t
L
and
m 5.2=
t
H
respectively (Fig.
5). Natural oscillations of the water in the tank can
influence the towing. Horizontal water velocity
u
and water surface elevation
h
of the first natural
mode are described by formulas
xktuu
x110
sincos
ω
=
,
xkthh
x110
cossin
ω
=
, (18)
510
where
t
is the time,
x
is the horizontal coordinate
directed along the tank,
0
h
is the amplitude of water
surface elevation and
)/(
1010 tx
Hkhu
ω
−=
is the
amplitude of water velocity oscillations in the first
mode,
tx
Lk /
1
π
=
is the wave number, and
11
kgH
t
=
ω
is the wave frequency. The period
1
T
of the first natural mode is calculated with formula
s 29/2
11
≈=
ωπ
T
. (19)
Water surface deformed by the first natural mode
of the tank is shown in Fig. 5 by blue dashed and
continuous lines at the different phases of the
oscillation. Fig. 5 also explains that the distance
X
between the center of the iceberg and the carriage
can be different from the distance
cs
X
between the
sensor platform and the carriage because of the
natural oscillation of the water in the tank and the
iceberg pitch and roll. Difference
cs
XX −
can
depend on the sway and the yaw of the iceberg if the
position of the sensor platform relatively the iceberg
center is determined with insufficient accuracy.
Figure 6. Scheme of the towing in the HSVA tank.
Period of heave oscillations of floating ice
cylinder is estimated with the formula
( )
2/1
)/(2 ghT
iwcwc
ρρρπ
−=
, (20)
where
3
kg/m 1000=
w
ρ
and
3
kg/m 887=
i
ρ
are
water and ice densities, and
c
h
is the cylinder height.
Assuming
m 645.0=
c
h
we find the period of heave
oscillation
s 8.4=
c
T
for the iceberg model.
The speed of the carriage in the experiment is
shown in Fig. 7a) versus the time Mean tow speed is
equal to 0.11 m/s when
s 250s 125 << t
, and it is
0.13 m/s when
s 400s 702 << t
.
t
. Fig. 8a) shows
the difference between the carriage speed and its
moving average calculated over 6 s. The mean
deviation of the carriage speed is about 0.005 m/s.
The carriage speed doesn’t include oscillations with
well recognized periodicity. Fig. 7b,c,d) show the
surge, sway and heave of the iceberg versus the
time. The characteristics of iceberg rotation
performed by the pitch, roll and yaw are shown in
Fig. 8b,c,d). The surge, sway, roll and yaw have
visible correlation due to long term trend changing
its direction when
s 200≈t
. The surge and heave
have oscillations with period about 30 s closed to the
period
1
T
of the first natural mode of the tank.
The amplitude of the heave oscillations varied
within 0.5 - 1 cm when the iceberg was near the
edge of the tank and decreases to smaller values
when the iceberg was towed to the center of the
tank. From the first formula (18) it follows that the
amplitude of water velocity oscillations in the first
mode
0
u
is varied from 2 cm/s to 4 cm/s in the
middle part of the tank when
cm 15.0
0
−=h
. The
decrease of the heave amplitude in Fig. 6d) with the
time can relate to the decrease of the amplitude of
water surface elevation in the middle part of tank
according to the second formula (18).
Figure 7. The speed of the carriage (a), horizontal distance
between the carriage and the sensors installed on the iceberg
(b), sideways movement relative between fixed points at
carriage and the sensors (c) and iceberg heave (d) versus the
time.
Figure 8. Difference between the carriage speed and its moving
average (a), pitch (b), roll (c) and yaw (d) of the iceberg versus
the time.
6 RESULTS OF NUMERICAL SIMULATIONS
Characteristics of towing lines in steady towing.
Numerical simulations are performed in model and
full scales. Properties of the towing rope and the
hawser are performed in Tables 1 and 2. Towing
511
rope Dyneema was used in the HSVA tests.
Characteristics of the towing rope Dyneema
performed in Table 1 for the full scale are taken
from the web-address www.dynamica-ropes.dk. It is
assumed that the diameter of the hawser should be
the same as the rope diameter since their strengths
are almost the same.
T
able 1. Tow rope properties (model and full scales)
___________________________________________________
Property Unit Model scale Full scale
___________________________________________________
Total length [m] 23 920
Length CD (Fig. 2) [m] 10 400
Length BD (Fig. 3) [m] 10 400
Weight [Kg/m] 0.0069 6.22
Diameter [m] 0.004 0.12
E-Module [GPa] 95 95
Ultimate Load [T] 1.3 1000
___________________________________________________
Table 2. Tow hawser properties (model and full scales)
___________________________________________________
Property Unit Model scale Full scale
___________________________________________________
Total length [m] 2.05 82
Weight [Kg/m] 0.049 79.6
Diameter [m] 0.003 0.12
E-Module [GPa] 200 200
Ultimate Load [T] - 1000
___________________________________________________
From formulas (1a) and (2a) it follows that the
towing with rope loop around the iceberg can be
performed as the towing with one rope having
double weight
r
W2
and double buoyancy force
b
W2
.
Models of towing lines related to towing schemes
shown in Fig. 1 are performed in the Appendix. Fig.
9 perform the dependence of the rope tension
T
from the distance
XlX
TL
−=∆
, where
TL
l
is total
length of the towing line between the points C and
D, in model and full scales. Curves 1 and 2 are
related to towing method I and II shown in Fig. 1a)
and Fig. 1b) respectively,
m 05.12=
TL
l
in model
scale and
m 482=
TL
l
in full scale. One can see that
the slope of the tension curve for the first towing
scheme is higher than for the second towing scheme
when the tension is relatively high.
Figure 9. Rope tension
T
versus distance decreasing
X∆
b
etween the boat and the iceberg in model (a) and full (b)
scales.
The shape of towing lines in steady towing
performed by schemes shown in Fig.1a) and Fig. 1b)
are performed in Fig. 10a,b) and Fig. 10c,d)
respectively in model and full scales. One can see
that all curves in Fig. 10 have small slopes. Thus
assumptions made in the Appendix for the
construction of the towing lines models are satisfied.
It is also visible that towing rope always has floating
part. In this case variations of the rope tension
influence significantly vertical displacement of the
towing lines and have insignificant influence on
horizontal displacements of points C, D and B in
comparison with the rope length. These
displacements are invisible in Fig. 10 and therefore
are shown by squares.
Figure10. The shape of towing lines in steady towing in model
(a,c) and full scales (b,d) calculated with different values of the
rope tension
T
.
Oscillations in unsteady towing. Dimensional
period of these oscillations is estimated by the
formula
µπ
/2
∗
= tt
p
, where
002,
/TvMt
ir
=
.
Periods
p
t
are shown in Fig. 11 versus the tow load
T
in model and full scales. Curves 1 and 2 are
related to the towing schemes shown respectively in
Fig. 1a) and Fig. 1b). The periods are decreased with
the increasing of the tow load. Representative value
of the period is few tens of seconds when the towing
is occurred according to Fig. 1b). Periods of
oscillations in the towing scheme in Fig. 1a) are
smaller 10 sec in model scale and 20 sec in full scale
when tow load is relatively high. Periods of
oscillations in the towing scheme in Fig. 1b) are
higher then in the towing scheme in Fig. 1a) . From
Fig. 11a) follows that period
p
t
is closed to period
1
T
of the first natural mode of the tank, when the
rope tension is about 2 N. It can create resonance
effect. In full scale period
p
t
can be closed to swell
period. For the conditions of the Barents Sea swell
period is about 12 s. From Fig. 10b) it follows that
the swell period can be closed to the period
1
T
when
the towing is performed according to the scheme in
Fig. 1a).
512
Figure 11. Periods of oscillations in model (a) and full (b)
scales.
Fig. 12a,c) show the data measured in the
experiment on the towing of model iceberg with
cylindrical shape in the HSVA tank (Eik and
Marchenko, 2010). Iceberg diameter and height are
1.909 m and 0.645 m. The towing was realized as it
is shown in Fig. 1b). Characteristics of tow line are
shown in Tables 1, 2. The rate of the distance
between the carriage and iceberg (
dtdX /
) is
calculated using the experimental data. The iceberg
motion was calculated using equations (11), where
s
v
was substituted equaling to the carriage velocity
)(tv
c
. Initial conditions were
0=
i
v
and
19.1=Z
by
0=t
. It is assumed that
m/s 11.0
0
=v
,
kg 1643=
i
M
and
N 34.2
0
=T
. Figure 13a) shows
carriage velocity used in numerical simulations.
Computed displacement of the iceberg is shown in
Fig. 13b). Fig. 13a) shows surge rate
dtdX /
calculated using experimental data. Fig. 13b) shows
the surge rate without accounting of the natural
oscillations of the water in the tank. Figures 13c)
and Fig. 13d) shows the surge rate with accounting
of the natural oscillations with period 29 s and 16 s
respectively. One can see that periods and
amplitudes of oscillations of
dtdX /
in Fig. 13a) are
most close to those performed in Fig. 13c).
Figure 12. Carriage velocity versus the time used in
simulations (a). Calculated iceberg displacement versus the
time (b).
Figure 13. Surge rates calculated with experimental data (a),
with numerical simulations without accounting of natural
oscillations of the water in the tank (b) and with accounting of
natural oscillations of the water in the tank with periods 29 s
(c) and 16 s (d).
CONCLUSIONS
Stability of steady towing of iceberg is analyzed for
two methods of the towing (Fig. 2). It is shown that
steady towing is stable, but damping of steady
towing perturbations is accompanied by oscillations
in the system ship-tow line-iceberg. The resonance
of these oscillations with swell of 12 s period
observed in the Barents Sea is available when the
towing is realized with floating tow line (method I).
Model experiments on iceberg towing were
performed in HSVA ice tank. Surge and heave
oscillations of model icebergs were observed in the
experiments. Periods of these oscillations were close
to the period of the first natural mode of water
oscillations in the tank 29 s. Numerical simulations
confirm that the amplitudes of surge and heave
oscillations of iceberg models were influenced by
the natural oscillations of the water in the tank.
REFERENCES
Eik, K., Marchenko, A., 2010. Model tests of iceberg towing.
Cold Regions Science and Technology, 61, pp. 13-28.
Hamilton, J.M., 2000. Vibration-based technique for measuring
the elastic properties of ropes and the added masses of
submerged objects. J. of Atmospheric and Oceanic
Technology. Vol. 17, pp. 688-697.
Marchenko, A., Gudoshnikov, Yu., 2005. The Influence of
Surface Waves on Rope Tension by Iceberg Towing. Proc.
of 18
th
Int. Conference on Port and Ocean Engineering
under Arctic Conditions (POAC’05), Vol. 2, Clarkson
University, Potsdam, NY, pp.543-553.
Marchenko, A., and C., Ulrich, 2008. Iceberg towing: analysis
af field experiments and numerical simulations.
Proceedings of 19
th
IAHR International Symposium on Ice
“Using New Technology to Understand Water-Ice
Interaction”, Vancouver, BC, Canada, July 6-11, 2008,
Vol.2, 909-923.
513
Marchenko, A.V., 2010. Stability of icebergs
towing. Transactions of the Krylov Shipbuilding Research
Institute. Marine Ice Technology Issue. 51 (335), 69-82.
ISBN 0869-8422. (in Russian)
McClintock, J., McKenna, R. and Woodworth-Lynas, C.
(2007). Grand Banks Iceberg Management. PERD/CHC
Report 20-84, Report prepared by AMEC Earth &
Environmental, St. John’s, NL, R.F. McKenna &
Associates, Wakefield, QC, and PETRA International Ltd.,
Cupids, NL., 84 p.
Robe, R.Q., 1980. Iceberg drift and deterioration. In: Colbeck,
S. (Ed.), Dynamics of Snow and Ice Masses. Academic
Press, New York. pp. 211-259.
Rudkin, P., Boldrick, C. and Barron Jr., P., 2005. PERD
Iceberg Management Database. PERD/CHC Report 20-72,
Report prepared by Provincial Aerospace Environmental
Services (PAL), St. John’s, NL., 71 p.
Voitkunsky J.I., 1988. Resistance to ship motion. Leningrad:
Sudostroenie. 287 p. (in Russian)
APPENDIX
Model of the towing with floating rope
The scheme of icebergs towing with floating rope is
shown in Fig. 14a). The rope segment OC hangs in
the air, and the rope segment OD floats on the water
surface. The coordinate of the point O is equal to
O
x−
, and the coordinate of point D is equal to
X−
.
The rope is fastened to the boat stern in the point C
with coordinates
0=x
and
C
zz =
. The motion of
the rope is occurred under the influence of the
gravity force, the rope tension and the rope inertia.
Momentum balance of the rope segment hanging in
the air with length
ds
is written as follows
WTa +=
ds
d
g
W
, (П1)
where
),0(
r
W−=W
is the weight of the rope of unit
length,
T
is the rope tension,
a
is the acceleration
(Fig. 14c). Since the vector of the rope tension is
tangential to the rope it can be performed as
τT T=
,
where
τ
is the tangential vector of unit length to the
rope segment
ds
. Using Frenet formulas equation
(П1) is written in the form
Wnτa +
−= Tk
ds
dT
g
W
, (П2)
where
n
is unit normal vector to the rope segment
ds
and
k
is the rope curvature.
Figure 14. Schemes of iceberg towing with floating (a) and
submerged (b) ropes. Schemes of forces applied to the rope
hanging in the air (c) and to submerged rope (b).
The shape of the hanging rope is described by
equation
),( txhz =
, where
x
and
z
are the
horizontal and vertical coordinates and
t
is the time.
The rope curvature is calculated by the formula
( )
( )
( )
2/3
2
22
/1/
−
∂∂+∂∂= xhxhk
. In static case the
integration of the projection of equation (П2) on
tangential vector
τ
leads to the formula
hWTT
rst
+=
, where
st
T
is a constant. The
projection of equation (П2) on normal vector
n
is
written as follows
2
2
2
1
+=
dx
dh
W
dx
hd
T
r
, (П3)
Dimensionless variables are introduced by
formulas
0,r
l
x
=
ς
,
0
T
T
st
=
τ
,
C
z
h
=
η
, (П4)
where
0,r
l
is the rope length and
0
T
is the rope
tension in steady towing. In dimensionless variables
equation (П3) has the following form
2
2
2
1)(
+=+
ς
η
ε
ς
η
υ
ε
υητ
d
d
d
d
, (П5)
where dimensionless coefficients
ε
and
υ
are
introduced as follows
r
l
z
C
=
ε
,
0
C
T
zW
r
=
υ
. (П6)
514
The rope tension
0
T
is estimated with formula
(8). Assuming
m 10
C
=z
,
-1
Nm 50=
r
W
,
23
0
m10=S
,
6.0=
wi
C
and
-1
0
ms 5.0=v
we find
that
kN 150
0
=T
and
033.0≈
υ
.
With accuracy up to high order terms equation
(П3) is written in dimensionless variables (П4) as
follows
ετς
η
r
k
d
d
=
2
2
,
,0)(-
O
ςς
∈
. (П7)
where
0
/TWlk
rrr
=
and
τ
is dimensionless rope
tension.
Boundary conditions for equation (П7) are
formulated as follows
1=
η
,
0=
ς
, (П8)
0=
η
,
0=
ς
η
d
d
,
O
ςς
−=
. (П9)
From (П8) and (П9) it follows
1
2
O
+
+=
ς
ς
ετ
ς
η
r
k
,
)0,(
O
ςς
−∈
, (П10)
ε
ς
τ
2
2
Or
k
=
. (П11)
Dimensionless length of the rope segment
hinging in the air is calculated from the formula
( )
4
0
2
2
0
2
2
OC
OO
2
11
ες
ς
ηε
ς
ς
η
ε
ςς
Od
d
d
d
d
d
l +
+=
+=
∫∫
−−
, (П12)
and total length of the rope is equal to
OOC
1
ς
−Ζ+= l
. (П13)
From formula (П12) it follows
+=
2
O
2
OOC
3
2
1
ς
ε
ς
l
. (П14)
From formulas (П13) and (П14) follows
)1(3
2
2
O
Ζ−
=
ε
ς
. (П15)
Substituting (П15) in formula (П11) we find the
expression of the rope tension
2
3
)1(9
2
Ζ−
=
ε
τ
r
k
. (П16)
In the steady towing
1=
τ
and
3/21
0 r
kZ
ε
−=
.
Dimensionless parameter
γτµ
/'
0
=
is calculated
with the formula
r
k
εγε
µ
23
=
. (П17)
Formula (П16) is applicable when
Z≤
O
ς
. This
condition imposes limitation for the tension
cr
ττ
≤
,
where
)2/(
2
ετ
rcr
k=
. The rope hangs above the
water when
cr
ττ
>
. Dimensional value of critical
rope tension
T 1632)2/(
2
≈=
ε
rrcr
WlT
, when
m 400=
r
l
,
m 10
C
=z
,
-1
Nm 50=
r
W
. This value is
much greater the strength of synthetic ropes.
Model of the towing with submerged rope
Scheme of iceberg towing with submerged rope is
shown in Fig. 14b). The rope is fastened on iceberg
surface behind the point D, it is floating between
points D and O and submerged between points O
and B. The rope BD is connected to the wire BC
fastened at the boat stern. The length of the rope
between the points C and D is equal
r
l
, and the
length of the wire between the points B and C is
equal to
w
l
. It is convenient to set up the origin in
the point O. The coordinates of the points D, O, B
and C are equal to
)0,(
D
x−
, (0,0),
),(
BB
zx −
and
)0,(
C
x
respectively. It is assumed that the weight of
the rope with unit length is equal to
r
W
, and it is
smaller than the buoyancy force
b
W
applied to the
rope. Therefore the resulting force
rbrb
WWW −=
is
upward directed (Fig. 14d). The weight of the wire
w
W
with unit length is much greater the buoyancy
force applied to the wire, therefore the influence of
the buoyancy force is ignored.
Analogically (П7) the shape of submerged rope
OB is described in dimensionless variables by the
equation
τς
η
rb
k
d
d
−=
2
2
,
)(0,
B
ςς
∈
, (П18)
where
0
/TWlk
rbrrb
=
and
τ
is dimensionless tension
of the rope and the wire. Since
B
z
is unknown
quantity depending on the solution dimensionless
variable
η
is determined by formula
r
lh /=
η
in
contrast with last formula (П4). Nevertheless further
it is assumed that
1/ <<
ςη
dd
.
Boundary conditions in the point O for the
construction of the solution of equation (П18)
describing the shape of the rope BD are formulated
as follows
515
0
=
=
η
ς
η
d
d
,
0
=
ς
(П19)
From (П18) and (П19) it follows
2
2
ς
τ
η
rb
k
−
=
,
)(0,
B
ςς
∈
. (П20)
The shape of the wire BC is described by the
equation
τς
η
w
k
d
d
=
2
2
,
),(
CB
ςςς
∈
, (П21)
where
0
/TWlk
wrw
=
. The solution of equation (П21)
satisfying to the condition
0=
η
by
C
ςς
=
is written
as follows
))((
2
C
B
k
w
+−=
ςςς
τ
η
. (П22)
Matching conditions for solutions (П20) and
(П22) have the form
ηη
ςςςς
00
BB
limlim
+→−→
=
,
ς
η
ς
η
ςςςς
d
d
d
d
00
BB
limlim
+→−→
=
. (П23)
Analogically (П12) dimensionless unit length of
the rope between the points D and B and the wire
length between the points B and C are expressed by
formulas
1
2
1
1
B
D
2
DB
=
+=
∫
−
ς
ς
ς
ς
η
d
d
d
l
,
wr
d
d
d
l
λς
ς
η
ς
ς
=
+=
∫
C
B
2
BC
2
1
1
, (П24)
Where dimensionless length of the wire in equal
to the ratio or dimensional lengths of the wire and
the rope (
w
l
and
r
l
):
rwwr
l
l /=
λ
.
Dimensionless distance between the boat and the
iceberg is introduced by the formula
CD
ςς
+=Ζ
. (П25)
Formulas (П20), ( П22) and conditions (П23) -
(П25) are used to perform implicit dependence
between dimensionless rope and wire tension
τ
and
distance
Ζ
. After simple algebra the implicit
dependence is reduced to algebraic equations
wr
λ
τ
ας
βς
=+
2
3
C
C
,
1
2
3
C
C
=+−
Ζ
τ
δς
βς
, (П26)
where coefficients
α
,
β
and
δ
are expressed by
formulas
−−
−+=
w
rb
w
rbw
k
k
k
kk
)1(3)1(3
6
2
22
2
ββββ
β
α
,
wrb
rb
kk
k
+
=
β
,
6
)1(
32
β
δ
−
=
rb
k
. (П27)
Excluding
C
ς
from equations (П26) we find
explicit formula for the rope tension
C
3
C
βς
ας
τ
−
=
wr
λ
,
)(
)1(
C
δαβ
αδ
ς
+
−−
=
Zλ
wr
. (П28)
Tension
∞→
τ
when
wr
Z
λ
+=1
or when
wr
llX +=
in dimensional variables.
In steady solutions
1=
τ
and rope characteristics
in steady towing
0
C
ς
and
0
Ζ
are determined as
follows
(
)
wr
λ
ςα
βς
=+
3
0
C
0
C
,
( )
3
0
C
0
C0
1
ςδβς
−+=Ζ
. (П29)
Taking differential from equations (П26) in the
vicinity of
1=
τ
we find
( )
( )
3
0
C
2
0
C
0
)(
2
3
'
ςδαβ
ς
αβ
τ
+
+
=
. (П30)
Dimensionless parameter
µ
characterizing the
stability of steady towing is calculated with formula
γτµ
/'
0
=
.
516