Stability of iceberg towing
Aleksey Marchenko
The University Centre in Svalbard
Norway
Barents Sea iceberg near FJL and 3 field researchers, 2005
Imaginary iceberg and ice resistant platforms in the Barents Sea
RV ”Mikhail Somov” and small icebergs near North Novaya Zemlia, 2006
RV ”Mikhail Somov” and small icebergs near North Novaya Zemlia, 2006
Collision with iceberg creates high stress concentration
5 m/s
Iceberg mass
4000 - 7500 t
Scheme of collision of oil tanker
with small iceberg on January 2, 1994
Alaska
F
σ=
S
Bulb of oil tanker Overseas Ohio
with mass ~75000 t
σ → ∞ by S → 0
Canadian method of iceberg towing in ice free waters
(20% of unsuccesful towings due to different reasons)
Heavy steel rope
Method of iceberg towing used in the expeditions by RV “M.Somov” (2004, 2005)
(in 2005 the rope was broken)
Floating poly-steel rope
Governing equations for 1D iceberg towing
Rope tension: T = T ( X , lr )
rope length
T
0
lr
X
dvi
⎧
⎪ M i dt = − Fwi (vi ) + T ( X )
⎪⎪
dvs
= P − Fws (vs ) − T ( X )
⎨M s
dt
⎪
dX
⎪
= vs − vi
⎪⎩
dt
Fwi = ρ wCwi Si vi2 , Cwi ≈ 0.5
Fws = ρ wCws S s vs2 , Cws ≈ 0.003
Steady towing (P=const)
⎧ ρ wCwi Si vi2 = T ( X )
vs = vi = v0
⎪
2
2
ρ
C
S
v
+
T
(
X
)
=
P
⇒
ρ
v
⎨ w ws s s
w 0 (C ws S s + C wi S i ) = P
2
⎪
v
=
v
ρ
C
S
v
s
i
w wi i 0 = T ( X 0 )
⎩
P = ρ wCws S s v∗2 = const
Ship propulsion versus v* (a ) and dimensionless iceberg velocity versus Si
(b) calculated with Ss=1000 m2, Cwi=0.6 and Cws=0.003.
Iceberg towing in 2004
RV ”Mikhail Somov”, Barents Sea
30 m
Rope length between the ship and
the iceberg is about 300 m
10 m
V,m/s
1.6
1.4
1.2
1
P ≈ 50 T, Si ≈ 300 m 2 , S s ≈ 1000 m 2
0.8
0.6
Duration of towing
0.6 h
0.4
Mean velocity of FD
0.17 m/s
0.2
Maximal velocity of FD
0.4 m/s
Mean towing velocity
0.8 m/s
Maximal towing velocity 1.46 m/s
Time
14:46:10
14:44:10
14:42:10
14:40:10
14:38:10
14:36:10
14:34:10
14:32:10
14:30:10
14:28:10
14:26:10
14:24:10
14:22:10
14:20:10
14:18:10
14:16:10
14:14:10
14:12:10
14:10:10
0
Global stability of the towing
⎞
vi2
vs2
d ⎛
⎜⎜ M i + M s + τ ⎟⎟ = Pvs − ρ wCwi Si vi3 − ρ wCws S s vs3
dt ⎝
2
2
⎠
τ = ∫ TdX ≥ 0
⇓
⎞
vi2
vs2
d ⎛
⎜⎜ M i + M s + τ ⎟⎟ < 0 when vi → ∞ or vs → ∞
2
2
dt ⎝
⎠
Velocities of the ship and the iceberg are limited
Main steps for the study of steady towing stability
1. Small perturbations of steady solutions
vi = v0 + δvi , vs = v0 + δvs , X = X 0 + δX
2. Linearization of governing equations in the vicinity of the steady solution
dT
⎧ dδvi
⎪M i dt = −ki v0δvi + T '0 δX , T '0 = dX ( X = X 0 )
⎪⎪
dδvs
= − k s v0δvs − T '0 δX , ki = 2 ρ wCwi Si , k s = 2 ρ wCws S s
⎨M s
dt
⎪
⎪ dδX = δv − δv
s
i
⎪⎩ dt
3. Finding of characteristic equation
δvi = vi 0 e λt , δvs = vs 0 e λt , δX = X 0 si e λt
a3λ3 + a2 λ2 + a1λ + a0 = 0
If some root of the characteristic equation has positive real part (Re λ>0)
then the steady solution is unstable
Characteristic equation
F (λ ) ≡ F0 (λ ) + F1 (λ ) = 0
F0 (λ ) ≡ λ ( M i λ + ki v0 )( M s λ + k s v0 ), F1 (λ ) ≡ T '0 (( M i + M s )λ + (ki + k s )v0 )
Increasing of T’0
F0 (λ )
F1 (λ )
λ2
λ0
λ1
0
λ
Graphs of F0(λ) and F1(λ) with different T’0
Roots of the equation
F0 (λ ) = 0
k s v0
ki v0
λ2 = −
, λ1 = −
, λ3 = 0
Ms
Mi
Root of the equation
λ0 = −
F1 (λ ) = 0
ki + k s
v0
Mi + Ms
When T’0 becomes sufficiently big two roots of the equation F(λ)=0 become complex.
In this case steady solution can be instable.
A model of floating towing rope
Momentum balance equation for the rope
W
dσ
a=
τ − σkn + W
g
ds
k=
η xx
(1 + η )
2 3
x
curvature
Boundary conditions
(2)
(3)
Steady rope:
η = Hs, x = 0
∂η
= 0, x = A
η=
∂x
d 2η
σ0 2 =W
dx
WA2
K
σ0 =
=
,
2
2 H s (lr − X )
(1)
, W = (0,−W )
weight of unit
length rope
a=0
(1) • τ
σ = σ 0 + Wη
(1) • n
d 2η
⎛ dη ⎞
σ 2 = W 1+ ⎜ ⎟
dx
⎝ dx ⎠
2 H s2
2
A=
, K = WH s3
3 lr − X
9
rope length
2
Analysis of instability with W=45 Nm-1 , Ss=1000 m2, Ms=20000 t, Cwi=0.6 and Cws=0.003.
Locations of the roots of characteristic equation Representative time of the instability
Example of numerical simulations
Si = 3200 m 2 , lr = 450 m, P = ρ wCws S s v∗2 tanh 2 [t / t s ], v∗ = 14 m/s, t s = 360 s
Rotational mode of oscillations of vertical cylinder connected
by 2 branches rope to the fixing point
d 2ϕ
I i 2 = R (σ − − σ + )
dt
C
C
σ+ =
σ
,
=
−
(l+ − c) 2
(l− − c) 2
dϕ
dl+
dϕ dl−
= −R
,
=R
dt
dt dt
dt
l+ + l− = 2 L = const
(
)
+ H ), 0 < κ < 1
max l± = c 2 + H s2 + γ c + H s − c 2 + H s2 , 0 < γ < 1
(
min l± = c 2 + H s2 + γκ c + H s − c 2
2
s
Period of rotational oscillations T0
versus γ and κ
Excitation of rotational oscillations of the iceberg under
the change of towing direction
Towing in constant direction
Towing with change of direction
The townig of iceberg in the Barents Sea, 2005
80 m
40 m
Rope length between the ship and the iceberg was about 450 m
The situation of the rope break up in 2005
Spatial locations of 2 GPS sensors installed on the iceberg in 2005
Rope tension in iceberg towing in 2005
WA2
2σ 0 =
Hs
W = 25 N, H s = 7 m
A ≈ 450 m
Conclusions
Stability of steady towing is determined by ship propulsion,
weight of the rope and towing method
The instability of iceberg towing is related to the excitation of
- Iceberg oscillations in the towing direction
- Rotational oscillations of the iceberg
The instability influences
- Hitches of the rope tension
- The lost of mean towing speed
Rotational oscillations are excited under changes of towing direction
The rotation of iceberg created the breakup of the towing rope in
experimental iceberg towing in 2005
Thank you for your attention