Course Stability of a Ship Towing System

Course Stability of a Ship Towing System
Ahmad Fitriadhy, Graduate student1
Hironori Yasukawa, Professor2
Abstract
This paper proposed a numerical modelling of course stability of a towed ship coupled to tow ship
through towline using both nonlinear and linear approaches. Captive model tests were conducted at
the towing tank to obtain hydrodynamic derivatives both of tow and towed ships. Eﬀects of unstable
towed ship (2B) and stable towed ship (2Bs), towline, tow points, dimension of tow ship associated
with an autopilot rudder at tow ship were treated on the numerical analyses. The simulation results
obtained that 2Bs showed better course stability than 2B. A longer towline and shifting the tow point
towards the bow both for 2B and 2Bs made the towing system more course stable indicated through
attenuation in slewing motion. Lateral motion of the tow ship increased gradually due to shifting
the tow point forward of the ship’s centre of gravity. This degrades the course stability of the towing
system. The ensuing rudder action demanded by the autopilot improved the course stability. In
general, the increase of the tow ship’s dimension was still insuﬃcient to improve the course stability
of the ship towing system.
1
Introduction
Ship towing systems are widely engaged for towing unpropelled barges or disabled ships. An
improper towing system may introduce serious towing instability and lead to accidents (e.g. collisions).
An extensive investigation with regard to course stability and turning ability of tow and towed ships
is therefore required. This paper deals with course stability of a towing system.
Several researchers have presented theoretical approaches to model towing systems. Using a single
towline model, Strandhagen et al.(1950) assumed constant towline tension in analyzing course stability
of the towed vessel. Kijima and Wada (1983) and Kijima and Varyani (1985) used an approximate
formula to estimate the towline tension, where the towline was assumed as rigid and straight. Bernitsas
(1985), Bernitsas (1986), Lee (1989) and Jiang (1998) employed a single elastic tow rope model. A
diﬀerent method was adopted by Nonaka (1990) and Yukawa et al.(2002) in modelling the towline,
estimating the towline tension using catenary model equations. Yasukawa et al.(2006) proposed a
new mathematical model for the towline motion equations, applying a 2D lumped mass approach in
which the towline was divided into a number of nodes and the equations of motion were written for
each node. Basically, the equation of towline tension was derived by considering the dynamic towline
associated with the towed ship motions and the external forces, which experienced on the towed ship
and towline. The results, Yasukawa et al. (2006), Yasukawa et al. (2007), agreed well with model
basin tests. In these studies, however, the towed vessel was decoupled to a tow ship, i.e. the tow ship
motion was assumed to be given.
In this paper, we present a reliable numerical tool, capable of predicting the course stability of
a towing system in the calm water. The model couples tow and towed ship via of a towline. An
autopilot system is applied for the tow ship to keep a desired track. A 2D lumped mass method
models the towline motion. This approach improves the accuracy of calculating normal and axial forces
experienced in dynamic towline motion. The linearized motion equations were derived to conﬁrm the
validity of the nonlinear analysis in which the boundary of towing stability and instability regions were
determined. Several towing parameters were treated to examine their eﬀects on course stability. We
expect that the presented numerical approach will reduce expensive experimental costs, even though
a validation in these investigations is highly recommended.
1
2
Hiroshima University, Kagamiyama 1-4-1, Higashi-Hiroshima, 739-8527, Hiroshima, Japan, [email protected]
Hiroshima University, Kagamiyama 1-4-1, Higashi-Hiroshima, 739-8527, Hiroshima, Japan
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1
2
Mathematical formulation
The mathematical model describes the tow and towed ship motions along with the towline motion,
expressed in three degrees-of-freedom (surge, sway, yaw). The components of hydrodynamic forces
and moment (due to the tow and towed ships hull form, rudder, and propeller) are modelled in the
numerical analysis. Finally, the numerical program is conﬁrmed against maneuvering sea trial data of
R.V. ”Discovery”.
2.1
Coordinate systems
Figure 1: Coordinate systems of tow and towed ships (left) and lumped mass model for towline (right)
In deriving the basic equations of motion of the tow and towed ships, three coordinate systems are
used, Fig. 1. One set of axes is ﬁxed to the earth coordinate system and denoted as O − XY, and
two set of axes G1 − x1 y1 and G2 − x2 y2 are ﬁxed relative to each ship’s moving coordinate system
aligned with its origin at centre of gravity. In the moving reference, the xi −axis points forward and
the yi −axis to starboard. i = 1 designates the tow ship, i = 2 the towed ship. The heading angle
ψi refers to the direction of the ship’s local longitudinal axis xi with respect to the ﬁxed X−axis.
The instantaneous speed of ship Ui can be decomposed into an advance velocity ui and a transverse
velocity vi . The angle between Ui and the xi −axis is the drift angle βi ≡ − tan−1 (vi /ui ).
The towline is represented as the sum of lumped mass of ﬁnite number (N ). Each mass is connected
by a segmented element over the entire truss element. The lumped mass particulars describe the
towline characteristics, such as the mass, the density and the drag. The coordinates of the i-th
lumped mass is labelled by (Xi , Yi ), where (i = 1, 2, 3, . . . , N + 1). The angle between the X−axis and
the length of i-th segmented towline ℓi is denoted as θi . Their connection points through a towline
in the earth ﬁxed coordinate systems are denoted as (X0 , Y0 ) and (XN +1 , YN +1 ) with respect to each
centre of gravity of the local ship coordinate systems are denoted as (ℓT , 0) and (ℓB , 0). The coordinate
system (Xi , Yi ), which deﬁnes θi and ℓi can be expressed as follow:
Xi = X0 −
i
∑
ℓj cos θj , Yi = Y0 −
j=1
i
∑
ℓj sin θj
(1)
j=1
where θN +2 = ψ2 and ℓN +1 = ℓB .
2.2
Motion equations of towed ship and towline
The motion equations of the towed ship are written in Eqs.(2) and (3), (Yasukawa et al., 2006).
(2)
= (m2 + mx2 ) and My = (m2 + my2 ) represent the virtual mass components in x2 and y2 ,
(2)
respectively. Iz = (I2 + J2 ) is the virtual moment of inertia. These are expressed as sum of mass
(2)
Mx
2
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(2)
(2)
(2)
(moment of inertia) and added mass (added moment of inertia) components. Fx , Fy and Mz
are the surge force, the sway force, and the yaw moment acting on the towed ship, respectively. The
superscripts (1) and (2) in the notations mean the tow and the towed ship, respectively.
−
N
+2
∑
ℓj (Mx1 sin θj − My1 cos θj ) θ¨j
j=1
=
N
+2
∑
¨ 0 + My1 Y¨0
ℓj (Mx1 cos θj + My1 sin θj ) θ˙j2 + TV 1 + Mx1 X
(2)
j=1
Iz(2) θ¨N +2 +
N
+2
∑
ℓj ℓB sin γ (My2 cos θj − Mx2 sin θj ) θ¨j
j=1
=
N
+2
∑
¨ 0 − My2 Y¨0 ) + M (2)
ℓj ℓB sin γ (My2 sin θj + Mx2 cos θj ) θ˙j2 − ℓB sin γ(TV 2 − Mx2 X
z
(3)
j=1
where
Mx1 = Mx(2) sin γ cos ψ2 + My(2) cos γ sin ψ2 ,
= Mx(2) sin γ sin ψ2 − My(2) cos γ cos ψ2
My1
Mx2 = Mx(2) cos γ cos ψ2 − My(2) sin γ sin ψ2 , My2 = Mx(2) cos γ sin ψ2 + My(2) sin γ cos ψ2
TV 1 = (M (2) v2 sin γ + M (2) u2 cos γ)ψ˙ 2 − (F (2) − M (2) u2 ψ˙ 2 ) sin γ + (F (2) − M (2) u2 ψ˙ 2 ) cos γ
x
TV 2 =
y
(−Mx(2) v2 cos γ
+
y
My(2) u2 sin γ)ψ˙2
+
(Fx(2)
x
+
y
My(2) v2 ψ˙ 2 ) cos γ
+
x
(Fy(2)
− Mx(2) u2 ψ˙ 2 ) sin γ
γ = θN +1 − ψ2
The Lagrange’s motion equations were used to describe the dynamic motion of the towline as derived
in Eq.(4). mi and kF i are the mass and the added mass coeﬃcients of the i-th lumped masses,
respectively.

N ∑
i
∑
i=k

(msi sin θk sin θj + mci cos θk cos θj ) ℓk ℓj θ¨j
j=1
+ ℓk sin (θk − θN +1 )
N
+2
∑



ℓj (Mx2 sin θj − My2 cos θj )θ¨j
j=1
= Q0k − Q1k − ℓk TV 3 sin (θk − θN +1 ) (k = 1, 2, 3, ..., N )
(4)
where
(
)
mci = mi 1 + kF i sin2 θi , mci
TV 3 =
N
+2
∑
(
= mi 1 + kF i cos2 θi
)
¨ 0 + My2 Y¨0 − TV 1 − TV 2
ℓj (Mx2 cos θj + My2 sin θj ) θ˙j2 + Mx2 X
j=1
Q0k = −ℓk sin θk
N
∑
(RCi sin θi − FCi cos θi ) − ℓk cos θk
i=k
Q1k = ℓk sin θk
N
∑
i=k
+ 2ℓk
N (
∑

¨0 +
X
i
∑

N
∑
(RCi cos θi + FCi sin θi )
i=k
θ˙j2 ℓj cos θj  msi − ℓk cos θk
j=1
)
N
∑

Y¨0 +
i
∑
i=k
j=1
(
)

θ˙j2 ℓj sin θj  mci
X˙ i sin θk + Y˙ i cos θk mi kF i θ˙i sin θi cos θi − X˙ k2 − Y˙ k2 mk kF k cos θk sin θk
i=k
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Two diﬀerent external forces experienced on the segmented towline, Fig. 1 (right). These forces are
resolved into normal and axial forces components:
1
1
RCi = − ρSi CDi |VCi |VCi , FCi = − ρSi CF i |UCi |UCi
2
2
(5)
where VCi = −X˙ i sin θi + Y˙ i cos θi and UCi = X˙ i cos θi + Y˙ i sin θi . ρ is the water density, Si the proﬁle
area of the segmented towline, CDi the coeﬃcient of the normal force, and CF i the coeﬃcients of the
axial force.
2.3
Motion equation of tow ship
The equation of the tow ship motion was derived adequately as follows:
Mx(1) u˙1 − My(1) v1 ψ˙ 1 = Fx(1) + FT x
(6)
My(1) v˙1 + Mx(1) u1 ψ˙ 1 = Fy(1) + FT y
(7)
Iz(1) ψ¨1 = Mz(1) + MT z
(1)
(8)
(1)
Mx = (m1 + mx1 ) and My = (m1 + my1 ) represent the virtual mass components in the direction
(1)
(1)
(1)
(1)
x1 and y1 , respectively, Iz = (I1 + J1 ) the virtual moment of inertia. Fx , Fy and Mz denote
surge force, sway force and yaw moment around the centre of gravity of the tow ship, respectively.
FT x , FT y and MT z denote surge force, sway force, and yaw moment due to the towline tension acting
at the connection point of the tow ship:
FT x = −TX cos ψ1 − TY sin ψ1
(9)
FT y = TX sin ψ1 − TY cos ψ1
(10)
MT z = ℓT (TX sin ψ1 − TY cos ψ1 )
(11)
The towline tension components TX and TY are expressed as, (Yasukawa et al., 2006):
TX
N
∑
=

msi 
i=1
= −
TY
i
∑

ℓj sin θj θ¨j  +
j=1
N
∑

mci 
i=1
i
∑
N
+2
∑

)
(12)
j=1
ℓj cos θj θ¨j  +
j=1
(
(1)
ℓj Mx2 sin θj − My2 cos θj θ¨j cos θN +1 + TX
N
+2
∑
(
)
(1)
ℓj Mx2 sin θj − My2 cos θj θ¨j sin θN +1 + TY
(13)
j=1
where
(1)
TX
=


N [
i
]
∑
∑
2
X¨0 +
ℓj cos θj θ˙j  msi + 2mi X˙ i kF i sin θi cos θi θ˙i + RCi sin θi − FCi cos θi
i=1
j=1
+TV 3 cos θN +1
(1)
TY
= −
N [
∑
i=1

Y¨0 +
i
∑

2
ℓj sin θj θ˙j  mci − 2mi Y˙i kF i sin θi cos θi θ˙i + RCi cos θi + FCi sin θi
]
j=1
+TV 3 sin θN +1
√
The resultant towline tension at the tow point of the tow ship can be expressed as T =
2.4
2 + T2.
TX
Y
Forces and moments acting on the ships
The hydrodynamic forces and moments on the ships for constant speed and course can be expressed
(i)
as functions of the velocities, rudder angles and propeller rpm. Here, the forces and moments Fx ,
(i)
(i)
(i)
(i)
(i)
Fy and Mz are expressed according to the MMG model, i.e. hull forces (XH , YH , NH ), propeller
4
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(i)
(i)
(i)
(i)
forces (XP ), and rudder forces (XR , YR , NR ):
(i)
(i)
(i)
= XH + XP + XR 


(i)
= YH + YR
Fy
(i)
Mz
2.4.1

(i)
Fx
(i)

(i)
(i)




(i)
= NH + NR
(14)
Hull
The hull forces (surge, sway, yaw) are expressed as follows:
(i)
=
(i)
=
(i)
=
XH
YH
NH
(
)
1
′
′
′2
′
′ ′
′
′2
ρLi di Ui2 Xuui
u′2
+
X
v
+
X
v
r
+
X
r
i
vvi i
vri i i
rri i
2
(
)
1
′
′
′
′
ρLi di Ui2 Yvi′ vi′ + Yri′ ri′ + Yvvvi
vi′3 + Yvvri
vi′2 ri′ + Yvrri vi ri′2 + Yrrri ri′3
2
(
)
1 2
′ ′
′ ′
′
′
′
′
ρLi di Ui2 Nvi
vi + Nri
ri + Nvvvi
vi′3 + Nvvri
vi′2 ri′ + Nvrri
vi′ ri′2 + Nrrri
ri′3
2
(15)
(16)
(17)
√
u2i + vi2 its speed. u′i = ui /Ui and vi′ = vi /Ui are
the non-dimensional surge and sway velocities, respectively. ri′ = ψ˙ i Li /Ui is the non-dimensional yaw
′ , N ′ ... etc. are the hydrodynamic manoeuvring derivatives of ship i. −X
rate. Yvi′ , Yri′ , Nvi
uui is the
ri
resistance coeﬃcient.
Li denotes the length of ship i, di its draft, Ui =
2.4.2
Propeller
In the simulation program, a twin-propeller coupled with twin-rudder was employed. The twinpropeller thrust was described in terms of the surge propulsive force:
(1)
XP = (1 − t1 )
∑
TP 1 = ρ(1 − t1 )
∑
DP4 1 n21 KT 1
(18)
∑
DP 1 denotes the propeller diameter, n1 the propeller revolutions, TP 1 the total thrust, and t1 the
thrust deduction factor. KT 1 is the open-water characteristics of propeller thrust.
2.4.3
Rudder
(1)
(1)
The summation of twin-rudder forces XR and YR in x1 and y1 , respectively, and the yaw moment
(1)
NR , while considering the rudder-to-hull interaction, are calculated as follows:
(1)
∑
XR
= − (1 − tR )
(1)
YR
(1)
NR
= − (1 + aH )
FN 1 sin δ1
∑
FN 1 cos δ1
= − (xR + aH xH )
∑
(19)
FN 1 cos δ1
δ1 denotes the rudder angle, tR , aH and xH the rudder and hull interaction parameters. xR is the
x-coordinate point on which the rudder force YR acts. The rudder normal force FN 1 is:
1
FN 1 = ρAR1 UR1 2 fα1 sin (αR1 )
2
(20)
AR1 denotes the rudder area, fα1 the lift coeﬃcient gradient of rudder, UR1 the (average) inﬂow speed
to the rudder, and αR1 the eﬀective rudder in-ﬂow angle:
√
UR1 =
2
u2R + vR
αR1 = δ1 − tan−1
(
vR
uR
)
(21)
(22)
uR and vR denote the inﬂow velocity components with:
vR = U γR βR
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(23)
5
vR is inﬂuenced by the ﬂow entrance angle to the rudder βR (≡ β1 −ℓ′R r1′ ). γR is the ﬂow straightening
coeﬃcient to the rudder. ℓ′R is the non-dimensional eﬀective inﬂow coordinate to the rudder of the
tow ship. uR is calculated as, (Yoshimura et al., 1978):
uR =
εu1 (1 − wp ) √
1 − 2 (1 − ηκ) s + 1 − ηκ (2 − κ) s2
1−s
(24)
wp denotes the eﬀective wake fraction at propeller position, s the propeller slip ratio, η the ratio of the
propeller diameter to the rudder height, κ the propeller ﬂow correction factor (κ = 0.6/ε is normally
used), and ε is the ﬂow coeﬃcient of the rudder with respect to its location.
In order to preserve a desired straight-line course of the tow ship, an autopilot system controls the
rudder angle as follows:
δ1 = −G1 (ψ1 − ψT ) − G2 ψ˙ 1
(25)
ψ1 denotes the actual heading angle and ψT the target heading angle (ψT = 0). G1 and G2 are yaw
and yaw rate gains, respectively.
3
Linearization of motion equations for course stability investigation
This section describes course stability of tow and towed ships motion using a linear approach. Here,
the following assumptions are employed:
• Tow and towed ships move to X-axis direction.
• Ship motions (θ1 , ψ1 , ψ2 ) are suﬃciently small.
• The towline is treated as non-extensible catenary model, Fig. 2.
The towline length is denoted by ℓ.
Figure 2: Coordinate systems for linear model of a towed ship in straight-path motion
3.1
Course stability of towed ship
First, the motion of tow ship (X0 , Y0 ) is assumed to be given. Eqs.(2) and (3), the linearized motion
equations of the towed ship incorporated with the towline motions are derived in the following:
My(2) θ¨1 ℓ + ℓB My(2) ψ¨2
[
(
= Yv2 ℓθ˙1 + Yv2 ℓB − Yr2 − X˙ 0 My(2) − Mx(2)
[
(
)
]
)]
[
¨ 0 ψ2 − Yv2 Y˙ 0 + M (2) Y¨0
− X˙ 02 Xuu2 − X˙ 0 Yv2 + My(2) − Mx(2) X
y
6
]
¨ 0 θ1
ψ˙ 2 + X˙ 02 Xuu2 − Mx(2) X
(26)
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[
]
(2)
¨ 0 Mx(2) θ1
IZ ψ¨2 = −Nv2 ℓθ˙1 + (−Nv2 ℓB + Nr2 ) ψ˙ 2 − ℓB X˙ 02 Xuu2 − X
[
]
¨ 0 M (2) ψ2 + Y˙ 0 Nv2
+ ℓB X˙ 02 Xuu2 − X˙ 0 Nv2 − ℓB X
x
(27)
In the simpliﬁed derivative notation to non-dimensionalize the linear equations of motion, Eq.(26) is
divided by (1/2)ρL2 d2 U 2 and Eq.(27) by (1/2)ρL22 d2 U 2 , where L2 , d2 and U = X˙ 0 are the length of
towed ship, the draft of towed ship, and the tow speed, respectively. The non-dimensional equations
of motion are then:
ℓ′ My′(2) θ¨1′ + ℓ′B My′(2) ψ¨2′
)
(
(
)
′
′ ′ ˙′
′ ′
′
¨ ′ θ1
− Mx′(2) X
= Yv2
ℓ θ1 + Yv2
ℓB − Yr2
+ Mx′(2) − My′(2) ψ˙ 2′ + Xuu2
0
(
[
)
]
′
′
¨ ′ ψ2 − Y ′ Y˙ ′ + M ′(2) Y¨ ′
− Xuu2
− Yv2
+ My′(2) − Mx′(2) X
0
y
0
v2 0
Iz′(2) ψ¨2′
(28)
)
(
(
′
′ ′ ˙′
′ ′
′ ) ˙′
¨ ′ M ′(2) θ1
ψ2 − ℓ′B Xuu2
−X
= −Nv2
ℓ θ1 + −Nv2
ℓB + Nr2
0 x
)
(
′
¨ ′ M ′(2) ψ2 + Y˙ ′ N ′
+ ℓ′B Xuu2 − Nv2
− ℓ′B X
0 x
0 v2
(29)
where
′(2)
′(2)
Mx , My
′(2)
Iz
(2)
=
=
′ =
Yv2
(2)
Mx , My
(1/2)ρL22 d2
(1)
(2)
I z , Iz
(1/2)ρL42 d2
Yv2
(1/2)ρL2 d2 U
θ¨1′ , ψ¨2′ =
¨′, Y
¨′ =
X
0
0
′ , N′
Yr2
v2 =
θ¨1 , ψ¨2
(U/L2 )2
¨ 0 Y¨0
X
U 2 /L2
Yr2 , Nv2
(1/2)ρL22 d2 U
θ˙1′ , ψ˙ 2′
=
′
Xuu2
=
′
Nr2
=
θ˙i , ψ˙2
U/L2
Xuu2
(1/2)ρL2 d2 U 2
Nr2
(1/2)ρL32 d2 U
The primes in the Eqs.(28) and (29) indicate that all the terms included in the equation of motions
are non-dimensional. ℓ′ and ℓ′B denote the ratios of towline length and tow point to length of the
towed ship, respectively, where ℓ′ = ℓ/L2 and ℓ′B = ℓB /L2 .
(
)
The simultaneous solution of equations yields the concept of straight-line stability Y¨0′ = Y˙ 0′ = 0 ,
where the values of θ1 and ψ2 can be written as:
θ1 = C1∗ eλt ,
ψ2 = C2∗ eλt
(30)
Substituting Eq.(30) into Eqs.(28) and (29), and elminating C1∗ and C2∗ , a fourth-order characteristic
equation in λ can be obtained as:
D0 λ4 + D1 λ3 + D2 λ2 + D3 λ + D4 = 0
D0 = ℓ′ My′(2) Iz′(2)
(32)
(
′
′
D1 = ℓ′ My′(2) Nr2
+ Iz′(2) Yv2
D2 = ℓ′
(
D3 =
−
D4 =
[
[(
)
)
]
(
′
′
′
′
′
¨ 0′
Mx′(2) − Yr2
Nv2
+ Yv2
Nr2
− Xuu2
− Mx′(2) X
′
¨ 0′
Xuu2
− Mx′(2) X
(
)[
(
(
)
(
)[
′(2) (
ℓ′B My
)
ℓ′ + ℓ′B + Iz′(2)
′
′
′
′
Nr2
+ ℓ′ + ℓ′B ℓ′B Yv2
− ℓ′B Yr2
− Mx′(2) + My′(2) + Nv2
)
)
′
¨′
Xuu2
− m′2 − m′x2 X
0
][
)]
]
(33)
(34)
]
′
¨ ′ ℓ′ N ′
Xuu2
+ My′(2) − Mx′(2) X
0
v2
[
(31)
(
′
′
¨ ′ m′ − m′
ℓ′B Yv2
− Nv2
− ℓ′B X
0
2
y2
)]
(35)
(36)
The conditions for the Hurwitz stability criterion are:
D0 , D1 , D2 , D3 , D4 > 0
D ≡ D1 D2 D3 −
D12 D4
−
D0 D32
Schiﬀstechnik Bd... - ..../Ship Technology Research Vol. .. - ....
> 0
(37)
(38)
7
Eqs.(37) and (38) provide the basic criterion for dynamic course stability. This work follows essentially
the analysis approach of Peters (1950) and Shigehiro (1998).
The stability of the towed ship in the linear system has the following features:
• D0 in Eq.(32) is always positive.
′ and N ′ are normally negative.
• D1 in Eq.(33) is normally positive because Yv2
v2
¨ ′ = 0, D2 in Eq.(34) can be written as
• For X
0
D2 = ℓ′
[(
)
]
[
(
)
′
′
′
′
′
Mx′(2) − Yr2
Nv2
+ Yv2
Nr2
− Xuu2
ℓ′B My′(2) ℓ′ + ℓ′B + Iz′(2)
]
(39)
The ﬁrst term deﬁnes the stability criterion of the towed ship itself. When the ﬁrst term is
positive, the course stability of the towed ship will be achieved since the second term is always
positive. Higher towed ship resistance and towline length lead to more stable towing conditions,
i.e. larger D2 . If the ﬁrst term is negative, the increase of ℓ′ may tend to degrade the towing
stability, which results in increasing transverse oscillation of the towed ship. Shifting the tow
point ℓB forward on the towed ship increases the second term and increases towing stability
again.
¨ ′ = 0, D3 in Eq.(35) can be written as:
• For X
0
[
(
)
(
′
′
′
D3 = Xuu2
Nr2
− ℓ′B Yr2
− Mx′(2) + My′(2) + ℓ′ + ℓ′B
)(
′
′
ℓ′B Yv2
− Nv2
)]
(40)
′
Since the resistance coeﬃcient Xuu2
is negative, for D3 > 0, we need to satisfy:
(
ℓ′B −
(
′(2)
′ )
Nv2
′
Yv2
′(2)
(
>
′(2)
′ −M
ℓ′B Yr2
x
′(2)
+ My
′ (ℓ′ + ℓ′ )
Yv2
B
)
′
− Nr2
(41)
)
′ < 0, Y ′ − M
′ < 0, the sign of the r.h.s. of Eq.(41) is negative.
For Nr2
+ My
> 0 and Yv2
x
r2
In this case, the condition of Eq.(41) coincides with the condition of D4 . Therefore, for D4 > 0
we have D3 > 0. Thus the sign of D3 depends on the sign of D4 .
¨ ′ = 0, D4 in Eq.(36) becomes:
• For X
0
(
′
′
′
D4 = Xuu2
ℓ′B Yv2
− Nv2
)
(42)
′
′ −N ′ ) < 0.
Since the sign of Xuu2
is negative, the condition of D4 > 0 can be achieved for (ℓ′B Yv2
v2
′ /Y ′ ), this indicates that an acting tow point ℓ on the oblique motion of the towed
For (ℓ′B > Nv2
B
v2
ship should be located in front of the acting point of hydrodynamic force. Physically, it implies
the tow point ℓB should be moved towards the bow of the towed ship.
• The sign of D in Eq.(38) could not be identiﬁed easily due to the complex problem. Thus, the
value of D should be checked in the numerical calculation.
Based on the above considerations, it can be concluded that the linear analysis of course stability
for the single towed ship involves mainly three parameters, namely D2 , D4 and D. For course stable
towed ship, longer towline and tow point ℓB nearer the bow of the towed ship increase D2 and hence
D, which indicates as a more stable condition of towing.
¨ ′ ) of the towed ship aﬀects the towing stability. The acceleration
In addition, surge acceleration (X
0
or the deceleration of the towed ship during towing normally occurs due to the loosening and the
tightening of dynamic tension in the towline. In acceleration, the resistance of the towed ship increases
and directly enhances the towing stability and vice-versa.
8
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3.2
Course stability of tow and towed ships
The course stability of a ship towing system entirely should be assessed as an integrated system
(tow ship, towline and towed ship). A course instability of the tow ship will lead to oscillatory yaw
motions which will aﬀect the course stability of the towed ship itself. Dynamic ship towing systems
involve complex nonlinearities. Here, a linearized motion equation of the tow and towed ships is
derived ﬁrst.
In the presence of the tow ship, we determine (X0 , Y0 ) as its towing reference and assume X˙ 0 = U ,
X¨0 = 0, Y˙0 = U ψ1 + v1 + ℓT ψ˙1 and Y¨0 = U ψ˙1 + v˙1 + ℓT ψ¨1 . Referring to Eqs.(26) and (27), the motion
equations of the towed ship is expressed as follows
(
My(2) −θ¨1 ℓ − ψ¨2 ℓB + ψ¨1 ℓT + v˙ 1
)
[
(
= −Yv2 θ˙1 ℓ − Yv2 ℓB − Yr2 − U My(2) − Mx(2)
(
)]
)
ψ˙ 2
+ Yv2 ℓT − My(2) U ψ˙ 1 − Xuu2 U 2 (θ1 − ψ2 ) + Yv2 [v1 + U (ψ1 − ψ2 )]
(2)
IZ ψ¨2 = −Nv2 θ˙1 ℓ − (Nv2 ℓB − Nr2 ) ψ˙ 2
+Nv2 ℓT ψ˙ 1 − Xuu2 U 2 ℓB (θ1 − ψ2 ) + Nv2 [v1 + U (ψ1 − ψ2 )]
(43)
(44)
As deﬁned in Eqs.(7) and (8), the linearized equations of the tow ship motion can be written as
(
)
(
)
My(1) v˙ 1 = Yv1 v1 + Yr1 − Mx(1) U ψ˙ 1 − Yδ1 G1 ψ1 + G2 ψ˙ 1 + FT y
(
)
(1)
IZ ψ¨1 = Nv1 v1 + Nr1 ψ˙ 1 − Nδ1 G1 ψ1 + G2 ψ˙ 1 + FT y ℓT
(45)
(46)
Referring to Eqs.(45) and (46), the dynamic response of the tow ship motion was computed including
an autopilot system, where Nδ1 and Yδ1 are sway external force and yaw external moment of the
rudder. FT y is the lateral towline tension in y1 -direction.
Using the linearized equation, the simpliﬁed formula of FT y in Eq.(10) can be re-written as
FT y = Xuu2 U 2 (θ1 − ψ1 ) + O(ϵ2 )
(47)
Substituting FT y into Eqs.(45) and (46) yields a set of equation:
(
)
(
)
My(1) v˙ 1 = Yv1 v1 + Yr1 − Mx(1) U − Yδ1 G2 ψ˙ 1 − Yδ1 G1 + Xuu2 U 2 ψ1 + Xuu2 U 2 θ1
(
)
(1)
IZ ψ¨1 = Nv1 v1 − (Nr1 Nδ1 G2 ) ψ˙ 1 − Nδ1 G1 + Xuu2 U 2 ℓT ψ1 + Xuu2 U 2 ℓT θ1
(48)
(49)
Finally, Eqs.(43), (44), (48) and (49) can be expressed in matrix form as























v˙ 1
ψ¨1
θ¨1
ψ¨2
ψ˙ 1
θ˙1
ψ˙ 2
























=











E1 E2
0
0
E3 E4
0
E5 E6
0
0
E7 E8
0
E9 E10 E11 E12 E13 E14 E15
E16 E17 E18 E19 E20 E21 E22
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
Schiﬀstechnik Bd... - ..../Ship Technology Research Vol. .. - ....
























v1
ψ˙1
θ˙1
ψ˙ 2
ψ1
θ1
ψ2























(50)
9
where
E1
=
E4
=
E8
=
(1)
Yv1
(1)
My
Xuu2 U 2
(1)
My
+
E5
=
(1)
My
E9
(1)
Iz
E10 =
=
Nv1
E6
(1)
Iz
(1)
Xuu2 U ℓT
(2)
E2
Yr1 − Mx U − Yδ1 G2
=
(
(1)
Iz
(1)
=
E7
=
(2)
−ℓB Nv2 Iz + ℓT Nv1 Iz
(2) (1)
)
−Yδ1 G1 − Xuu2 U 2
(1)
My
−Xuu2 U ℓT − Nδ1 G1
(1)
(2) (1)
+ Iz Iz
(1)
(2)
(2)
)}
(
Iz
(2)
(1)
Yv1 My − Yv2 My
)
ℓIz Iz My My
{
(1)
(2)
My My
Nr1 − Nδ1 G2
=
E3
(1)
}
(2)
My My ℓT −Iz ℓB Nv2 + Iz (Nr1 − Nδ1 G2 )
(2) (1)
Iz Iz
{
(2)
My
( ℓIz(2) Iz(1) My(2) My(1)
(1)
)
(1)
Yr1 − Mx U − Yδ1 G2 − My
(2) (1)
(2)
(
Yv2 ℓT − My U
(1)
ℓIz Iz My My
(2)
E11
=
E12
=
(2)
My ℓB Nv2 + Iz Yv2
(2)
(2)
Iz My
{
(
)
}
(2)
(2)
(2)
(2)
−My ℓB (Nr2 + ℓB Nv2 ) − Iz
Yr2 − My − Mx U − Yv2 ℓB
(2)
(2)
ℓIz My
(1)
E13
=
(1)
E14
=
E15
=
(2)
(1)
(2) (1)
(2) (1)
+
(2)
(2)
(
−Iz My My (ℓB Nv2 U ) + Iz My My ℓT −Nδ1 G1 − Xuu2 U 2 ℓT
−Iz Iz
{
(2)
(
My
(1)
(2)
ℓIz Iz My My
)
(1)
Xuu2 U 2 + My Yv2 U + Yδ1 G1
(2) (1)
(1)
)
}
(2)
ℓIz( Iz My My )
(
)
(1)
(2)
(1)
(2)
(2) (1)
(1)
(2)
My My Xuu2 U 2 Iz ℓ2B + Iz ℓ2T + Iz Iz Xuu2 U 2 My + My
(2) (1)
(1)
(2)
ℓIz Iz My My
(2)
(2)
−ℓB My U (ℓB Xuu2 U + Nv2 ) + Iz U (Yv2 − Xuu2 U )
E18 = −
(2)
(2)
ℓIz My
Nv2 ℓ
(2)
Iz
E19 =
Nr2 − Nv2 ℓB
(2)
Iz
E20
=
Nv2
(2)
Iz
E16
E21 =
=
ℓB Xuu2
(2)
Iz
Nv2 ℓT
(2)
Iz
E17
E22 =
Nv2 ℓT
=
(2)
Iz
ℓB Xuu2 − Nv2
(2)
Iz
Referring to Eq.(50), the ﬁrst term of matrices 7 × 7 clariﬁes that the stability conditions of a towing
system are determined by the signs of the real part of eigenvalues. Negative and positive values
represent stable and unstable motion responses, respectively.
4
Simulation condition
4.1
Ships
The principal dimension of tug (three diﬀerent lengths) and barge are presented in Table 1. The
length of the tug and the barge are denoted as L1 and L2 , respectively. The ratio of the increase of the
tug with respect to the length of the barge is denoted as SL ≡ L1 /L2 . The tug has twin propellers and
twin rudders. The CPP propellers have 1.8 m diameter, 300 rpm and a total engine power of 1050 kW
were used in the simulations for maintaining a constant speed of 7 knots on each of the three tugs. The
rudder is assumed as rectangular shape with 2.0 m × 2.0 m for span and cord length, respectively.
The steering speed of the rudder was set to 2.0◦ /s. The towing point at the tug is denoted as ℓT
and non-dimensionalized as ℓ′T = ℓT /L1 . Negative ℓ′T means that the tow point is located behind
the centre of gravity of the tug. Two conditions of the barge (with and without attached skegs) are
denoted as “2B” and “2Bs”, respectively. The skegs were attached at port and starboard side at the
aft end of the barge.
4.2
10
Hydrodynamic derivatives on manoeuvring
Schiﬀstechnik Bd... - ..../Ship Technology Research Vol. .. - ....
Figure 3: Model of tug (left) and barge (right)
The hydrodynamic derivatives of tug and barges (2B and 2Bs) including resistance coeﬃcients in
′ ) were obtained by captive model test in the towing tank, Fig. 3. They are summarized in
full scale (Xuu
Table 2. Based on the stability indices (C), 2B and 2Bs are considered as unstable and stable coursekeeping motions. Added mass coeﬃcients (m′x , m′y , Jz′ ) were calculated using singular distribution
method under the rigid free-surface condition. In addition, several data related to rudder coeﬃcients
are presented in Table 3.
Table 1: Principal dimensions of tug and barge
Description
SL
LBP (m)
Breadth (m)
draft (m)
Volume (m3 )
LCB (m)
Block coef.
kzz /L
L/B
1.0
60.96
13.72
3.35
1751.1
3.39
Tug
0.66
40.0
9.0
2.2
494.7
2.23
0.63
0.25
4.44
2B/s
0.5
30.0
6.75
1.65
208.7
1.672
60.96
21.34
2.74
3292.4
-1.04
0.92
0.252
2.86
The symbol of kzz is radius of gyration for yaw
Table 3: Related coeﬃcients of rudder
Symbol
tR
aH
x′H
ε
γR
ℓ′R
fα
Value
0.02
0.2
−0.45
1.25
0.4
−1.0
1.89
Table 2: Resistance coeﬃcient, hydrodynamic
derivatives on maneuvering and added mass
coeﬃcients
Symbol
′
Xuu
′
Xvv
′
Xvr
′
Xrr
Yv′
YR′
′
Yvvv
′
Yvvr
′
Yvrr
′
Yrrr
Nv′
NR′
′
Nvvv
′
Nvvr
′
Nvrr
′
Nrrr
Yδ′
Nδ′
m′x
m′y
Jz′
C
Tug
-0.0330
-0.0491
-0.1201
-0.0509
-0.3579
0.127
-0.2509
0.1352
0.000
0.000
-0.0698
-0.0435
-0.0588
-0.0367
0.000
0.000
-0.05
0.025
0.0187
0.1554
0.0056
0.0509
2B
-0.0635
-0.0188
-0.0085
-0.0272
-0.4027
0.0568
-0.2159
0.4840
0.495
-0.8469
-0.1160
-0.0237
0.0458
-0.0578
0.2099
-0.0982
0.0391
0.2180
0.0124
-0.251
2Bs
-0.0641
-0.1152
-0.1086
-0.1311
-0.4373
0.1355
-0.7265
0.3263
-0.2424
-0.4167
-0.0491
-0.0742
-0.0067
-0.2486
0.0360
0.000
0.0391
0.2180
0.0124
0.023
In the simulation program, we used a dimensionalized model for the tug and the barge motions as
written in Eq.(50). The dimensional coeﬃcient expressions are shown in Table 4.
Table 4: Equation of dimensional terms for Table 3
Schiﬀstechnik Bd... - ..../Ship Technology Research Vol. .. - ....
11
Xuu(1,2)
′
= (1/2) ρL(1,2) d(1,2) U 2 Xuu(1,2)
Nv(1,2)
′
= (1/2) ρL2(1,2) d(1,2) U Nv(1,2)
Xvv(1,2)
′
= (1/2) ρL(1,2) d(i) Xvv(1,2)
Nr(1,2)
′
= (1/2) ρL3(1,2) d(1,2) U Nr(1,2)
Xvr(1,2)
′
= (1/2) ρL2(1,2) d(1,2) Xvr(1,2)
Nvvv(1,2)
′
= (1/2) ρL2(1,2) d(1,2) Nvvv(1,2)
/U
Xrr(1,2)
= (1/2) ρL3(1,2) d(1,2) Xrr(1,2)
Nvvr(1,2)
′
= (1/2) ρL3(1,2) d(1,2) Nvvr(1,2)
/U
Yv(1,2)
′
= (1/2) ρL(1,2) d(i) U Yv(1,2)
Nvrr(1,2)
′
= (1/2) ρL4(1,2) d(1,2) Nvrr(1,2)
/U
Yr(1,2)
′
= (1/2) ρL2(1,2) d(1,2) U Yr(1,2)
Nrrr(1,2)
′
= (1/2) ρL5(1,2) d(1,2) Nrrr(1,2)
/U
′
Yvvv(1,2) = (1/2)ρL(1,2) d(1,2) Yvvv(1,2)
/U
Yδ(1)
′
= (1/2) ρL(1) d(1) U 2 Yδ(1)
′
/U
Yvvr(1,2) = (1/2)ρL2(1,2) d(1,2) Yvvr(1,2)
Nδ(1)
′
= (1/2) ρL2(1) d(1) U 2 Nδ(1)
Yvrr(1,2)
′
/U
= (1/2)ρL3(1,2) d(1,2) Yvrr(1,2)
mx(1,2) , my(1,2) = (1/2) ρL2(1,2) d(1,2) m′x(1,2) , m′y(1,2)
Yrrr(1,2)
′
/U
= (1/2)ρL4(1,2) d(1,2) Yrrr(1,2)
Iz(1,2)
4.3
′
= (1/2) ρL4(i) d(1,2) Iz(1,2)
Autopilot
In actual ship towing operations, a scheme for an autopilot rudder is necessary to reduce heading
and lateral deviation of the tow ship from its desired track. This autopilot rudder should have a
feedback compensator to counteract external disturbances due to periodic slewing of a towed ship,
which may cause serious instabilities in seaways. In a case with no autopilot, the tug drifted away from
a straight course and featured for 2B slewing motions. With autopilot (with G1 = 5 and G2 = 10),
there was no severe veering motion and the tug kept its path once it reached the prescribed heading
angle. Correspondingly, 2B ﬁshtailed on both sides, Fig. 4(left). For no autopilot and 2Bs, both tug
and 2Bs deviated, then steadily veered and settled at 10◦ . Thus the inherent good course-keeping of
2Bs resulted in less deviation of heading angle even without autopilot. Using the autopilot, tug and
2Bs returned to the initial straight-line path, Fig. 4(right).
Figure 4: Eﬀect of autopilot for tug towing 2B (left) and 2Bs (right); tug = 0.66L2 ; ℓ′ = 2.0; ℓ′B = 0, 5;
ℓ′T = −0.44
The eﬀect of control gains on the dynamic performance of the ship towing system was also examined.
Three control gains were determined (G1 = 1, 5 and 10), With ﬁxed ratio G2 : G1 = 2 : 1. The results
provided useful guidance and insight into the tug’s motion. Increasing G1 from 1 to 5 increased the
rudder angle deﬂection and decreased its deﬂection period. Consequently, the ﬂuctuation of the tug’s
heading angle was reduced signiﬁcantly, Fig. 5. Increasing G1 further from 5 to 10, gave essentially
similar tow speed u1 , ψ1 , ψ2 and δ1 . Hence, G1 = 5 and G2 = 10 were selected as the appropriate
autopilot parameters for the numerical simulation. This seemed to be suﬃcient for maintaining stable
periodic motion of the tug.
12
Schiﬀstechnik Bd... - ..../Ship Technology Research Vol. .. - ....
Figure 5: Eﬀect of control gain on motion of tug and barge, barge = 2B, tug = 0.66L2 , ℓ′ = 2.0,
ℓ′B = 0, 5 and ℓ′T = −0.44
4.4
Towline cable
The dynamic tension in the towline cable was simulated using a lumped mass approach. Three
diﬀerent numbers of towline nodes (namely N = 20, 30 and 40) were investigated. The towline data
were: ℓ′ = 1.0, CD = 1.0, CF = 0.01, kF = 1.0, diameter of towline cable = 0.075 m, and mass of
towline = 0.225 t/m. The towline was assumed to be inclined initially to 10◦ with respect to the
X−axis, with most part of the towline submerged under water. Fig. 6 shows that the already N = 20
gave rather good results. For the subsequent simulations, N = 30 was chosen.
Figure 6: Time histories of towline tension in diﬀerent numbers of towline (barge = 2B, tug = 0.66L2 ,
ℓ′ = 1.0, ℓ′B = 0, 5 and ℓ′T = −0.44)
5
Nonlinear simulation of slewing motion of 2B
This section presents the analysis of course stability of a towing system. Snapshots of towing
motions trajectories were taken from t = 1 s until t = 650 s. From these snapshots, the lateral tug
motion (Y1 ) and barge motion (Y2 ) were measured. The autopilot for the tug was activated in the
simulations.
5.1
Eﬀect of towline length
The eﬀect of the towline length ℓ on the towing system is shown in Fig. 7. For small towline ratio
(ℓ′ = 1.0), the barge’s slewing motion strongly aﬀects the tug’s motions; the tug experienced vigorous
motions and occasionally veered well oﬀ course. Trying to keep the tug on the desired track inevitably
resulted in larger rudder angles δ1 . Extending ℓ′ from 1.0 to 2.0 and 3.0 decreased Y1 , ψ1 and δ1 ,
Table 5. Y1 was substantially reduced. Increasing ℓ′ from 1.0 to 3.0 decreased Y1 from 14 m to 2 m.
Schiﬀstechnik Bd... - ..../Ship Technology Research Vol. .. - ....
13
Figure 7: Eﬀect of towline length on slewing motion of 2B with autopilot rudder, 2B (ℓ′T = −0.44, ℓ′B =
0.5, and SL = 0.66)
Table 5: Eﬀect of ℓ′ on motion amplitudes of towing with autopilot
ℓ′
1.0
2.0
3.0
u
¯1
2.51 m/s
2.60 m/s
2.64 m/s
ψ1
1.68◦
0.95◦
0.81◦
ψ2
46.24◦
47.86◦
45.86◦
Y1 /L1
0.35
0.035
0.05
Y2 /L2
0.52
0.65
0.66
δ1
8.45◦
4.35◦
3.82◦
Period of ψ2
225.0 s
239.0 s
250.0 s
This was suﬃcient to improve the motion performance of the tug and tended to reduce its resistance.
Unfortunately, this did not result in a signiﬁcantly higher mean tow speed u
¯1 . A comparison for
diﬀerent ℓ′ was therefore deemed unnecessary. The results were assumed to be insigniﬁcant for the
general motion performance of 2B, particularly its heading angle ψ2 and the towline tension T . The
lateral motion of 2B (Y2 ) increased by 27% when increasing ℓ′ from 1.0 to 3.0. On the other hand, the
slewing period of 2B was lower by 11% as ℓ′ increased from 1.0 to 3.0. This is plausible, as a simple
swinging pendulum shows similar behaviour when its cable is lengthened.
5.2
Eﬀect of tug’s dimension
Fig. 8 shows time histories and trajectories of the towing system for various dimension (ratios) of
the tug SL . The performance of the towing is summarised in Table 6. Increasing SL = 0.5, 0.66 and
1.0 was expected to enhance the yaw damping moment. The simulation showed that the increase of
SL had a stronger eﬀect than the rudder angle δ1 in helping to reduce Y1 , which was proportional
with decreasing ψ1 . This can be explained by the fact that although the magnitude of δ1 gradually
decreased from 6.21◦ to 4.04◦ as SL increased from 0.5 to 1.0, it signiﬁcantly reduced Y1 by 81%, and
involuntarily increased u
¯1 by up to 30%. For SL = 1.0, there was no signiﬁcant eﬀect on Y2 or ψ2 . For
this reason, the motion interaction eﬀect between the tug and 2B was considered as negligible, due
14
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to large enough separation between them (twice the length of the barge). The increase of SL reduced
the ﬂuctuation of u
¯1 . This implied that 2B has a faster slewing period with respect to its ψ2 . This
could possibly lead to severely ﬂuctuating surge acceleration. Correspondingly, an impulsive towline
tension with amplitude of 2.05 t appeared due to rigorous loosening and tightening of the towline for
SL = 1.0.
Figure 8: Eﬀect of tow ship dimension on slewing motion of 2B with autopilot rudder, 2B (ℓ′T =
−0.44, ℓ′B = 0.5 and ℓ′ = 2.0)
Table 6: Eﬀect of SL on motion amplitudes of the towing system with autopilot rudder
SL
0.5
0.66
1.0
u
¯1
2.33 m/s
2.60 m/s
3.03 m/s
ψ1
1.21◦
0.95◦
1.03◦
ψ2
48.63◦
47.86◦
47.23◦
Y1 /L1
0.16
0.035
0.03
Y2 /L2
0.66
0.65
0.62
δ1
6.21
4.35
4.04
Period of ψ2 (s)
247.5 s
239.0 s
233.5 s
This may pose a serious threat to the ship’s structure at the tow point. It may become even worse
when the snatching frequency of the towline coincides with motion frequencies of the tug, Varyani et
al. (2007).
5.3
Eﬀect of tow point ℓB
The eﬀects of increasing ℓ′B on the towing characteristics is shown in Fig. 9. The rudder angle δ1
reduced, but its period also reduced in compensation, Table 7. As a result, u
¯1 was increased by up
to 25%. This is reasonable, as the slewing motion was improved (signiﬁcantly reduced ψ2 and Y2 ) as
ℓ′B increased. ψ2 was reduced by 63% and Y2 by 69% , as ℓ′B increased from 0.5 to 1.0. The slewing
period was faster by 40%. Although the decrement of towline tension T generally was insigniﬁcant,
the increase of ℓ′B would diminish the unwieldy slewing motion of 2B which leads directly to better
Schiﬀstechnik Bd... - ..../Ship Technology Research Vol. .. - ....
15
towing stability.
Figure 9: Eﬀect of tow point ℓB on slewing motion of 2B with autopilot rudder, 2B (ℓ′T = −0.44, ℓ′ =
2.0, and SL = 0.66
Table 7: Eﬀect of ℓ′B on motion amplitude of ship towing system with autopilot rudder
ℓ′B
0.5
0.75
1.0
5.4
u
¯1
2.60 m/s
2.97 m/s
3.26 m/s
ψ1
0.95◦
0.55◦
0.47◦
ψ2
47.86◦
36.75◦
23.53◦
Y1 /L1
0.035
0.08
0.08
Y2 /L2
0.65
0.35
0.20
δ1
4.35◦
2.46◦
1.75◦
Period of ψ2 (s)
239.0 s
184.0 s
142.5 s
Eﬀect of tow point ℓT
Fig. 10 shows time histories and trajectories of the towing performance varying the location of the
tow point ℓT on the tug, namely ℓ′T at −0.44, −0.25 and 0.25. The results are summarised in Table
8. The tug veered to port side initially and then settled to relatively stable heading. The maximum
Y1 was 2.80 m for ℓ′T = −0.44 and 7.56 m for ℓ′T = −0.25. Thus shifting ℓT towards the tugs centre
of gravity (COG) made the tug deviate more from its prescribed path. This is possibly due to the
fact that for ℓT near the tug’s COG, the sway force is more dominant than the yaw moment, due
to the slewing motion of 2B. ψ1 oscillated evenly to both sides when a steady motion of the tug was
achieved. For ℓ′T = 0.25, the towing was unstable with maximum Y1 24.80 m. Because of the large
slewing motion of 2B, this instability was highly nonlinear, i.e. sensitive to shifting the acting tow point
′ /Y ′ = 0.195).
ℓT near the acting point of the hydrodynamic force on the obliquely moving hull (Nv1
v1
′
Subsequent shifting of ℓT from −0.44 to 0.25 reduced ψ2 by 13.7% and Y2 by 15.4%. Although the
slewing motion of 2B was improved gradually associated with faster period of its heading angle, the
entire towing performance was still directionally unstable, as indicated by the increase of the lateral
motion of the tug.
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Figure 10: Eﬀect of tow point ℓT on slewing motion of 2B incorporated with rudder control system,
2B (ℓ′B = 0.5, ℓ′ = 2.0 and SL = 0.66)
Table 8: Eﬀect of ℓ′T on motion amplitude of ship towing system with autopilot rudder
ℓ′T
-0.44
-0.25
0.25
6
u
¯1
2.60 m/s
2.65 m/s
2.74 m/s
ψ1
0.95◦
0.57◦
0.62◦
ψ2
47.86◦
46.15◦
41.26◦
Y1 /L1
0.035
0.080
0.295
Y2 /L2
0.65
0.63
0.55
δ1
4.35◦
2.36◦
2.84◦
Period of ψ2
239.0 s
224.0 s
218.0 s
Eﬀect of towing parameters on course stability: linear analysis
Based on Eq.(50), the course stability conditions for the towing of 2B and 2Bs associated with
several towing parameters were solved numerically. The inherent eﬀect of stable and unstable barges
associated with the autopilot rudder played a major role for the towing stablity performance, as
presented in the course stability diagrams Figs. 11 and 12.
For 2B, the course stability diagrams of the towing system due to eﬀect of the tow point ℓB with
respect to the towline lengths was plotted in Fig. 11. Without autopilot, the increase of ℓ′ and ℓ′B
did not recover suﬃcient course stability, Fig. 11a. Both barge 2B and tug veered excessively oﬀ
the desired course. The behaviour of the towing correlates to the inherent course instability of 2B.
Referring to Eq.(39), the value of the ﬁrst term for the unstable barge was clearly negative. Therefore,
an attempt to increase ℓ′ was essentially unnecessary. On the other hand, the positive initial course
stability index of 2Bs could improve signiﬁcantly the stability of towing, Fig. 11b. The attached skegs
on 2Bs contribute a great deal to the towing characteristics, as also reported by Lee (1989). Moreover,
the increase of ℓ′B and ℓ′ eﬀectively allowed better stability of towing.
Shifting the tow point ℓB forward on the towed barge, aﬀecting the second term of Eq.(39), could
be achieved by employing a bridle towline conﬁguration, as commonly used in towing operations. This
is a practical solution, which enlarges D2 and D4 , and thus also D. In addition to achieving optimal
Schiﬀstechnik Bd... - ..../Ship Technology Research Vol. .. - ....
17
Figure 11: Eﬀect of tow point ℓB with respect to towline length (ℓ) on towing stability performance
with and without autopilot rudder (ℓ′T = −0.44)
towing direction of the tug, the autopilot was then applied appropriately. Here, both G1 and G2 were
taken as positive, Yδ1 as negative, and Nδ1 as positive. Referring to Eq.(48) and (49), the tuning of G1
enhanced the value of Yδ1 G1 + Xuu2 U 2 and reduced the value of Nδ1 G1 + Xuu2 U 2 ℓT . The tuning of
G2 generated a damping moment of the turning motion on the tug. By setting, G1 = 5 and G2 = 10,
the towing system was relatively stiﬀ, resulting in remarkably reduction of veering motion, Fig. 4,
which increased the towing stability. As seen in Fig. 11c, the increase of ℓ′B and ℓ′ associated with the
autopilot control system on the rudder was able to remove the region of towing instability. Similar to
the remarks on 2B, the autopilot enhanced the area of towing stability for 2Bs, Fig. 11d.
Fig. 12 shows course stability of the ship towing varying the tow point ℓT with respect to the
tow point ℓB . Shifting ℓT in either direction did not remove the towing instability areas for 2B
without autopilot, Fig. 11a. However, the instability regions (especially for ℓ′T from −0.5 to 0.0) were
stabilised with increasing ℓ′B , Fig. 12b. Thus the increase of ℓ′B played a signiﬁcant role in determining
the boundary of stable course in the towing case of 2B. Increasing ℓB is recommended as an eﬀective
way to recover this towing instability. For positive ℓ′T (from 0.0 to 0.5), increasing ℓ′B was found
unnecessary since the towing area of unstableness could not be diminished. By shifting ℓT aft of
the tugs COG resolves this unfavourable condition. As in Figs. 11c and 11d, autopilot improves the
stability of the towing system reducing the areas of instability, Figs. 12c and 12d.
For 2Bs, unstable regions were signiﬁcantly reduced, including in the areas of the tow point ℓ′T >
′ /Y ′ . This was very diﬀerent from the case for 2B. This was mainly due to the inherently stable
Nv1
v1
course of 2Bs with little slewing.
7
Conclusion
The main aim of this study was to develop a non-linear numerical simulation of course stability of the
ship towing system. The linearized motion equations was further solved to conﬁrm the validity of the
nonlinear analysis in which the boundary of towing stability and instability regions were determined.
In order to achieve this goal, the coupled manoeuvring motion equations of the tow and towed ships
associated with the towline motion was derived. A 2D lumped mass method was applied to express
the dynamic towline. Several towing parameters were taken into account to investigate each of their
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Schiﬀstechnik Bd... - ..../Ship Technology Research Vol. .. - ....
Figure 12: Eﬀect of tow point ℓT with respect to tow point ℓB on course stability of towing associated
with and without autopilot rudder, (ℓ′ = 2.0)
eﬀects on course stability of the ship towing system.
Several conclusions can be drawn:
• For both 2B and 2Bs, the non-linear and linear analysis results showed that the increase of ℓ′
and ℓ′B associated with the autopilot rudder at the tug assist to improve the entirely course
stability of the towing system.
• The inherent good course-keeping of 2Bs (with attached skegs) is an important factor for the
course stability of the towing system.
• Without autopilot, when towing 2Bs, the linear analysis results indicate that the instability of
the towing system in the range of ℓ′T from −0.5 to 0.0 can be recovered by increasing ℓ′B . For
0.0 ≤ ℓ′T ≤ 0.5, the area of unstable towing may be recovered also by shifting the tow point ℓT
aft of the tugs COG.
• Increasing tug’s dimension is insuﬃcient to enhance the course stability of the towing system.
The reason is that the motion interaction eﬀect between tug and 2B is negligible due to large
enough separation between them. Taking advantage of this, it is recommended to increase the
mean towing speed. However, increasing the tug’s dimension may cause the impulses in the
towline, which may threaten the safety of towing.
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