course stability of a ship towing system in wind oceanengineering03022013
TRANSCRIPT
-
8/10/2019 Course Stability of a Ship Towing System in Wind OceanEngineering03022013
1/11
Course stability of a ship towing system in wind
A. Fitriadhy a,n, H. Yasukawa b, K.K. Koh c
a Department of Maritime Technology, Faculty of Maritime Studies and Marine Science, Universiti Malaysia Terengganu, Malaysiab Department of Transportation and Environmental Systems, Hiroshima University, Japanc Department of Marine Technology, Universiti Teknologi Malaysia, Malaysia
a r t i c l e i n f o
Article history:
Received 7 June 2012
Accepted 3 February 2013Available online 3 April 2013
Keywords:
Stable barge
Unstable barge
Course stability
Wind angle
Wind velocity
Towline tension
a b s t r a c t
This paper proposes a numerical model for analyzing the course stability of a towed ship in uniform and
constant wind. The effects of an unstable towed ship and a stable towed ship were recorded using
numerical analysis at various angles and velocities of wind. The stability investigation of the ship towingsystem was discussed using the linear analysis, where a tugs motion was assumed to be given. When the
tug and the towed ships motions were coupled through a towline as a proper model of the ship towing
system, their dynamic interactions during towing was then captured using towing trajectories and
analyzed using nonlinear time-domain simulation. With increasing wind velocity, the simulation results
revealed that the towing instability of the unstable towed ship was recovered in the range of beam to
quartering winds; however, the towing stability of the stable towed ship in head and following winds
gradually degraded. It should be noted that this towing instability might have resulted in the impulsive
towline tension and could led to serious towing accident e.g. towline breakage or collisions.
& 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Course stability of a ship towing system is vital in still water andstill air conditions. In reality, tug and towed ship are always exposed
to some degrees of wind at different directions. A reliable investiga-
tion either using a theoretical or experimental approach is required to
obtain a deeper understanding of the course stability of the ship
towing system with such external disturbance.
In recent years, several studies regarding course stability of ship
towing systems in wind discussed investigating the motion char-
acteristics of a towed ship in various velocities and angles of wind.
Kijima and Varyani (1986) carried out a linear analysis and found
that when the wind angle changed from the head to the following
winds, the course stability of the two towed ships tended to become
unstable. In addition, Kijima and Wada (1983) presented that the
course stability of the towed barge would generally be unstable in
the range of beam to quartering wind conditions. Using an experi-
mental model in a towing tank, Yasukawa and Nakamura (2007a)
found that the course stability of an unstable towed barge was
recovered in the range of beam to quartering winds. In this work,
however, the towed barge was decoupled from the tug, i.e. the tugs
motion was assumed to be given.
This paper presents linear and nonlinear model analyses of
course stability for a ship towing system in uniform and constant
wind conditions as an extension study from the previous work by
Fitriadhy and Yasukawa (2011). In the nonlinear analysis, a
proper model of the ship towing system was modeled, where
the tug and towed barge are coupled by a towline. This is quitereasonable given the fact that wind forces will be exerted on
windage areas both of the towed ship and also the tug. Thus, the
analytical model of predicting course stability of a ship towing
system is deemed more reliable. The effect of wind velocities ( Uw)
and absolute wind angles (yw) were taken into account in the
models. A 2D lumped mass method was applied to model towline
motion incorporated with dynamic towline tension; and an
autopilot system was employed to reduce heading and deviation
of the tug from its desired track. The presented numerical
approach is expected to reduce experimental costs, even though
the model test validation is still recommended.
2. Mathematical formulation
The mathematical model of maneuvering motions equations
for a tug and towed ship associated with dynamic towline tension
relates to nonlinear three degrees of freedom in the time-domain,
i.e. surge, sway and yaw motions.
2.1. Coordinate systems
In deriving the basic equations of motion of the tug and towed
ships, three coordinate systems are used,Fig. 1. One set of axes is
fixed to the earths coordinate system that is used to specify
absolute wind velocity Uw and angle yw denoted as OXY, and
Contents lists available at SciVerse ScienceDirect
journal homepage: www.elsevier.com/locate/oceaneng
Ocean Engineering
0029-8018/$ - see front matter& 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.oceaneng.2013.02.001
n Corresponding author. Tel.: 60 1 9155 590; fax: 60 9 668 3719.
E-mail addresses: [email protected] (A. Fitriadhy),
[email protected] (H. Yasukawa),[email protected] (K.K. Koh).
Ocean Engineering 64 (2013) 135145
http://www.elsevier.com/locate/oceanenghttp://www.elsevier.com/locate/oceanenghttp://dx.doi.org/10.1016/j.oceaneng.2013.02.001mailto:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.oceaneng.2013.02.001http://dx.doi.org/10.1016/j.oceaneng.2013.02.001mailto:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.oceaneng.2013.02.001http://dx.doi.org/10.1016/j.oceaneng.2013.02.001http://dx.doi.org/10.1016/j.oceaneng.2013.02.001http://www.elsevier.com/locate/oceanenghttp://www.elsevier.com/locate/oceaneng -
8/10/2019 Course Stability of a Ship Towing System in Wind OceanEngineering03022013
2/11
two sets of axes G1
x1y
1 and G
2x
2y
2are fixed relative to each
ships moving coordinate system aligned with its origin at the
center of gravity. In the moving reference, the xi-axis points
forward and the yi-axis to starboard. i 1 designates the tug,
i 2 the towed ship. The heading anglecirefers to the direction of
the ships local longitudinal axis xiwith respect to the fixed x-axis.
The instantaneous speed of ship Ui can be decomposed into a
forward velocityuiand a lateral velocity vi. The angle between Uiand the xi-axis is the drift angle b i tan
1vi=ui. Here, yw 01andyw 1801are the head and following winds, respectively, and
coincide with the earths fixed system X; yw 901 is the beam
wind, which coincides with the earths fixed system Y.
The towline is composed of a finite number N of lumped
masses; the masses are connected by segments into 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 ith lumped mass is labeled by Xi,Yi, where
i 1,2,3, . . . , N2. The angle between the x-axis and the length
of ith segmented towline i is denoted as yi. Here, N 2 is thedistance of the connection point at the towed ship with respect to
her center of gravity and yN 2 c2 is the heading angle of the
towed ship. Their connection points with respect to the earths
fixed coordinate systems X0 ,Y0 and XN 1,YN 1, respectively,
have the coordinates T,0 and B,0 in the respective local shipcoordinate systems. Then, the coordinates of lumped masses
Xi ,Yi throughy i andi can be written as
XiX0 Xi
j 1
jcos yj, Yi Y0 Xi
j 1
jsin yj 1
whereyN 2 c2 andN 1B.
2.2. Motion equations of towed ship and towline
The motion equations of the towed ship are written in
Eqs.(2) and (3) as follows:
XN 2
j 1
jMx1 sin yjMy1 cos yj yj
XN 2
j 1
jMx1cos yjMy1 sin yj _y2
j TV1 Mx1X0 My1 Y0 2
I2
zyN 2
XN 2j 1
jBsin gMy2cos yjMx2 sin yjyj
XN 2
j 1
jBsin gMy2sin yjMx2cos yj _y2
jBsin g
TV2Mx2 X0My2 Y0 M2
z 3
where
Mx1 M2
x singcosc2 M2
y cosgsinc2
My1 M2
x singsinc2M2
y cosgcosc2
Mx2 M2
x cosgcosc2M2
y singsinc2
My2 M2
x cosgsinc2 M2
y singcosc2
TV1 M2
x v2 sin gM2
y u2cos g_c2F
2x M
2y v2
_c2sing
F2y M2
x u2 _c2cosg
TV2 M2
x v2 cosg M2
y u2sin g _c2 F
2x M
2y v2
_c2cosg
F2y M2
x u2_c2sing
g yN 1c2
The notations ofM2x m2 mx2and M2
y m2 my2 represent
the virtual mass components in the direction x2 and y2, respec-
tively; andI2z I2 J2is the virtual moment of inertia, which is
expressed as the sum of mass (moment of inertia) and added
mass (added moment of inertia) components. F2x ,F2
y andM2
z are
the surge force, the sway force, and the yaw moment acting on
the towed ship, respectively. The superscripts (1) and (2) denote
the tug and the towed ship, respectively.Lagranges motion equations are applied to describe the
dynamic motion of the towline and are derived in Eq. (4). miand kFi are the mass and the added mass coefficients of the ith
lumped masses, respectively.
XNi k
Xij 1
msisin yk sin yjmcicos ykcos yjkjyj
8>>=>>>; 14
Estimation of wind forces on exposed windage areas of the tug
and the towed ship are modeled in various velocities and angles
of wind. Based onIsherwood (1972), the equation of wind forces
and moments are
XiA 1=2raAi
XVi2
A Ci
XAyi
A
YiA 1=2raAiYV
i2A C
iYAy
iA
NiA 1=2raAiYV
i2A LiC
iNAy
iA
9>>>=>>>; 15
where
yiA tan1viA=u
iA 16
ViA
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiui2A v
i2A
q 17
uiA uiUwcosywci 18
viA viUwsinywci 19
The notations of CiXA, CiYA and C
iNA are the force and moment
coefficients as a function ofyi
A (relative wind angle); ra is thedensity of air;AiX andA
iY are the front and lateral projected areas.
Here, Uw and yw are the absolute velocity and angle of winds,
respectively.
A. Fitriadhy et al. / Ocean Engineering 64 (2013) 135145 137
-
8/10/2019 Course Stability of a Ship Towing System in Wind OceanEngineering03022013
4/11
3. Linearization of motion equations for course stability
investigation in wind
Study on course stability of the ship towing system in wind
involves stronger nonlinearities than in calm water conditions. To
understand the basic mechanism of the ship towing system in the
wind condition, a course stability model using piecewise linear
system is utterly essential. This approach leads to provide a
threshold for identifying stable and unstable towing conditionsin the various angles and velocities of wind. Based on Fig. 2,
several simplifications have been considered:
1. Motions are considered in the horizontal plane only (surge,
sway, yaw).
2. The motion of the tug (X0,Y0) is assumed to be given with
Y0 0.
3. The towline is treated as non-extensible catenary model
(N0).
4. This virtual tug moves in a straight-course withU _X0 , where
c1, X0 , _X0 and Y0 are equal to zero, while y2 c2.
Here,y1
and c2
are defined in steady and unsteady motions as
y1 y0 Dy1,_y1 D
_y1 20
c2 c0 Dc2,_c2 D
_c2 21
where Dy1 and Dc2 are negligibly small (Oe); g0 c0g0 andDg Dy1Dc2.
3.1. Linearized motion equation of forces and moments acting on a
towed ship
The basic linearization of external force under wind condition
is based on the relative wind angle y2
A , which is exerted
forcefully on superstructure of the towed ship as written in
Eq. (16). Using Taylor series expansion with respect to D _
y1 ,D _c2 , Dc2, theny
2A is solved as
y2
A tan1v2A =u
2A
Cy2
A0y2
AqD _y1y
2ArD
_c2y2
AcDc2 22
where
y2
A0 tan1v2A0=u
2A0
y2
Aq1fUcosy0Uwcosywy0g
u22A0 v22
A0
y2
Ar Bu
2A0
u2A0 v2
A0
y2
A0c 1
u2A0 Ucosc0 Uwcosywc0
v
2
A0 Usin c0 Uwsinywc0
The square term of relative wind velocity V2A in Eq.(17)can
be recast into the linearized form:
V2A V2
A0 V2
AqD _y1 V
2ArD
_c2 23
where
V2A0 U2 Uw2 2UUwcos yw
V2Aq 2sing0fUwcosywc0 Ucosc0g
cosg0fUwsinywc0Usin c0g
V2Ar 2BUwsinywc0Usin c0
The equations of forces and yaw moment X2A ,Y2A ,N2A underwind condition are denoted as FkA (k 1,2 and 3, respectively), as
follows:
FkA Fk
A0Fk
AqD _y1 F
kArD
_c2 Fk
Ac2Dc 24
where
FkA0 1=2raAkCkA yA0VA0
FkAq 1=2raAk CkA yA0VAq
@CkA yA0
@yAyAqVA0
" #
FkAr 1=2raAk CkA yA0VAr
@CkA yA0
@yAyArVA0
" #
FkAc
1=2ra Ak@CkA yA0
@yAyAcVA0
and A1 A2X , A2
A2Y and A3
A2Y L2.
Then, the hydrodynamic forces and moment acting on the hull
X2
H ,Y2H ,N
2H are denoted as F
kH (k 1,2 and 3, respectively) and
expressed as
FkH FkH0 F
kHqD
_y1 FkHrD
_c2 FkHcDc2 25
where
F1H0X0U2 cos2 c0 XvvU
2 sin2 c0
F1
Hq 2UXvvsin c0 cosg0 X0 cos c0sin g0F1Hr Usin c02BXvvXvr
F1Hc2
2U2 sinc0 cos c0XvvX0
F2H0 YvUsin c0YvvvU3 sin
3c0
F2Hq cosg0Yv3YvvvU2 sin
2c0
F2Hr Yr YvvrU2 sin2 c0BYv3YvvvU
2 sin2 c0
F2Hc2
Ucos c0Yv3YvvvU2 sin2 c0
F3H0 NvUsin c0NvvvU3 sin3 c0xGF
2H0
F3Hq cosg0Nv3NvvvU2 sin2 c0xGF
2Hq
F3Hr NrNvvrU2 sin2 c0BNv3NvvvU
2 sin2 c0xGF2Hr
F3
Hc2 Ucos c
0Nv3NvvvU
2 sin2 c0
xG
F2
Hc2Fig. 2. Coordinate systems for linear model of a towed ship in wind.
A. Fitriadhy et al. / Ocean Engineering 64 (2013) 135 145138
-
8/10/2019 Course Stability of a Ship Towing System in Wind OceanEngineering03022013
5/11
Referring to Eq. (14), the linearized equation of the total
external forces and moments Fkx ,Fk
y ,Mk
z is denoted as Fk
(k 1,2 and 3, respectively) and take the following form:
Fk Fk0 Fkq D
_y1 Fkr D
_c2 Fkc2Dc2 26
where
F
k
0 F
k
H0 F
k
A0,
F
k
q F
k
HqF
k
Aq
Fkr FkHr F
kAr, F
kc2
FkHc2
FkAc2
The notation ofFk0 is the steady component of the lateral forces
and yaw moments; Fkq , Fkr and F
kc2
are the unsteady derivative
values of lateral forces and yaw moments with respect to D _y1 ,
D _c2 and Dc2, respectively.
3.2. Course stability of a towed ship
Referring Eq. (2), the linearized equations of the towed ship
can be written in the following form:
Mx0 sin y0My0 cos y0D
y1BMx0 sin c0My0 cos c0D
c2
F1q sing0 F2q cosgD
_y1F10 cosg0 F
20 sing0Dy1
Ucos c0 cos g0MyMx Usin c0sin g0MyMx
F2r cosg0F1r sing0D
_c2
F10 F2c cosg0 F
20 F
1c2
sing0Dc2
F10 sing0 F20 cosg0 27
Similarly, Eq.(3) becomes:
Iy0 cos y0Ix0 sin y0D y1 IzBIy0cos c0Ix0sin c0D c2
BF1q sing0cos g0 F
2q sin
2 g0F3q =BD
_y1
BF10 sin
2 g0cos2 g02Fy0sin g0 cosg0Dy1
BUsin c0 sin g0 cos g0MxMy Usin2
g0 cos c0MyMx
F3r =BF1r sing0 cosg0 F
2r sin
2 g0D _c2
Bsing0F1c cosg0 F
2c sing02F
20 cosg0
F10 sin2 g0cos
2 g0F3c2
=BDc2
Bsin g0F10 cosg0 F
20 sing0 Mz0 28
where
Mx0 Mxsin g0 cos c0 Mycos g0sin c0My0 Mxsin g0sin c0My cosg0 cos c0Ix0Bsin g0Mxcosg0 cos c0Mysin g0 sin c0
Iy0Bsin g0Mxcos g0 sin c0 Mysin g0 cos c0
Eqs. (27) and (28) are non-dimensionalized with respect to
1=2rL2d2U2 and1=2rL22d2U
2, respectively.L2,d2and U denote
the length and the draft of the towed ship and the tows speed,
respectively. Through separating these equations into the non-
dimensional steady and unsteady motion terms, the following
equations are expressed:
Steady components:
F010 sing0F020 cosg0 0 29
0Bsin g0F010 cosg0 F
020 sing0M
0z0 0 30
Unsteady components:
a1D
y
0
b1D
c
0
c1D_
y
0
d1D_
c
0
e1Dyf1Dc 0 31
a2Dy
0b2D
c0c2D
_y0d2D
_c0e2Dyf2Dc 0 32
where
a1 0M0x0sin y0M
0y1cos y0
b1 0BM
0x0 sin c0M
0y1cos c0
c1 F01q sing0F
02q cosg0
d1 cosc0 cosg0M0
yM0
x sinc0sin g0M0
yM0
x
F02r cosg0F01r sing0
e1 F010 cosg0 F
020 sing0
f1 F010 F
01c cosg0F
020 F
01c sing0
a20I0y0cos y0I
0x0sin y0
b2 I0
z0BI
0y0cos c0I
0x0 sin c0
c20Bsin g0F
01q cosg0 F
02q sing0F
03q
d20Bsinc0 sin g0 cos g0M
0xM
0y sin
2 g0 cos c0M0
yM0
x
F01r sing0 cosg0 F02r sin
2 g0F03r
e2 0BF010 sin
2 g0cos2 g02F
020 sing0 cosg0
f20Bsing0F
01c cosg0 F
02c sing02F
020 cosg0
F010 sin2 g0cos
2 g0F03c
From Eqs.(29) and (30), the value for the variables ofy0and c0is obtained. By substituting those values accordingly into Eqs. (31)
and (32), the unsteady motion equations of the towed ship are
then solved. When the wind coefficient is equal to zero, this work
follows essentially the approach of Peters (1950) and Shigehiro
et al. (1997).
The non-dimensional motion are
M0xM0
yM0
x0M0
y0MxMyMx0My0
1=2rL22d2
I0x0,I0
y0 Ix0,Iy0
1=2rL3
2d2
I0z Iz
1=2rL42d2
F010 ,F020 ,F
01c ,F
02c
F10 ,F20 ,F
1c ,F
2c
1=2rL2d2U2
F01q ,F02q ,F
01r ,F
02r
F1q ,F2q ,F
1r ,F
2r
1=2rL22d2U
F030 ,F03c
F30 ,F3c
1=2rL22d2U2
F03q ,F03r
F3q ,F3r
1=2rL32d2U
0,
0
B
,B
L2
D y 01,Dc02
D y1,Dc2
U=L22
D _y 01,D_c02
D _y1,D_c2
U=L2
where0 and0Bdenote the ratios of towline length and tow pointto length of the towed ship, respectively, where 0 =L2 and0BB=L2 (B40.
3.3. Course stability criterion
Simultaneous solution of equations can be used for the
assessment of the stability of the straight-line motion in steady
A. Fitriadhy et al. / Ocean Engineering 64 (2013) 135145 139
-
8/10/2019 Course Stability of a Ship Towing System in Wind OceanEngineering03022013
6/11
wind, i.e. motion with Y00 _Y00 0. The values ofDy1 and Dc2
are described by
Dy1 C1elt
, Dc2 C2elt 33
By substituting Eq. (33) into Eqs. (31) and (32), a fourth-order
characteristic equation with respect to l should satisfy the
following conditions:
D0l4 D1l
3 D2l2 D3l D4 0 34
where the values of D0,D1,D2,D3 and D0 are obtained (see
Appendix A). By applying the Hurwits method in Eq. (34), the
basic solution of stability criteria is written in Eqs. (35) and (36).
D0,D1,D2,D3,D440 35
D D1D2D3D21D4D0D
2340 36
4. Simulation condition
4.1. Ships
The principal dimensions of tug and barge including theirlateral and longitudinal windage areas used in the simulation are
presented in Table 1. The length of the tug and the barge are
denoted as L1 and L2, respectively. The towing point at the tug is
denoted asTand non-dimensionalized as0TT=L1. Negative0T
means that the tow point is located behind the center of gravity of
the tug. Two conditions of the barge, namely with and without
attached skegs, are denoted as barge 2B and barge 2Bs,
respectively, hereafter named the stable and unstable barge. The
tug has twin CPP propellers and twin rudders. Each CPP Propeller
has a diameter of 1.8 m, revolution of 300 rpm and a total engine
power of 1050 kW, used in the simulations for maintaining a
constant speed of 7.0 knots on the tug alone. The rudder design is
of square shape with both span and chord lengths of 2.0 m. The
steering speed of the rudder was set to 2.0 1/s.
4.2. Hydrodynamic derivatives
Hydrodynamic derivatives for the tug and barges 2B and 2Bs,
including their resistance coefficients, were obtained from captive
model test in the towing tank (see Fig. 3), which are completely
summarized inTable 2. Based on the stability index C, barges 2Band 2Bs are considered respectively unstable and stable motions
in course-keeping. In addition, added mass coefficients m0x,m0
y,J0
z
were calculated using singular distribution method under the
rigid free-surface condition.
4.3. Wind coefficients
Referring to Eq. (15), the wind coefficients for the tug and
barges were obtained using the linear multiple regression tech-
nique,Fujiwara et al. (1998) and are shown inFig. 4.
4.4. Autopilot of the tug
During ship towing operation, the autopilot is often employed.
The rudder of the tug as an actuator automatically adjusts the
backlash of the controller according to the heading angle and
lateral position of the tug. The control law of the tug is given in
Table 1
Principal dimensions of tug and barge.
Symbol Tug Barge
Ship length L (m) 40.0 60.96
BreadthB (m) 9.0 21.34
Draftd (m) 2.2 2.74
VolumeV(m3) 494.7 3292.4
Lateral wind area AX (m2) 57.35 77.5
Longitudinal wind area AY (m2) 28.91 250.5
LCB position xG(m) 2.23 1.04
Block coefficient Cb 0.63 0.92
kyy=L 0.25 0.252
L/B 4.44 2.86
Fig. 3. Model of tug (left) and barge (right).
Table 2
Resistance coefficient, hydrodynamic derivatives on maneuvering and added mass
coefficients.
Symbol Tug 2B 2Bs
X0uu 0.0330 0.0635 0.0641
X0vv 0.0491 0.0188 0.1152
X0vr 0.1201 0.0085 0.1086
X0rr 0.0509 0.0272 0.1311
Y0v 0.3579 0.4027 0.4373
Y0R 0.127 0.0568 0.1355
Y0vvv 0.2509 0.2159 0.7265
Y0vvr 0.1352 0.4840 0.3263
Y0vrr 0.000 0.495 0.2424
Y0rrr 0.000 0.8469 0.4167
N0v 0.0698 0.1160 0.0491
N0R 0.0435 0.0237 0.0742
N0vvv 0.0588 0.0458 0.0067
N0vvr 0.0367 0.0578 0.2486
N0vrr 0.000 0.2099 0.0360
N0rrr 0.000 0.0982 0.000
Y0d 0.05
N0d 0.025
m0x 0.0187 0.0391 0.0391
m0y 0.1554 0.2180 0.2180
J0z 0.0056 0.0124 0.0124
C 0.0509 0.251 0.023
A. Fitriadhy et al. / Ocean Engineering 64 (2013) 135 145140
-
8/10/2019 Course Stability of a Ship Towing System in Wind OceanEngineering03022013
7/11
the form of
d1 GPcTc1GD _c1 GYPYTY1GYD
_Y1 37
The notations ofc1 andY1are the actual heading angle and lateral
motion, respectively;cT
andYT
are the targeted heading angle and
lateral motion, respectively, (cT,YT 0).GPand GDare the propor-
tional and derivative gains with respect to the heading angle; GYPand GYD are the proportional and derivative gains with respect to the
lateral motion. Here, the constant controller gains ofGP,GD,GYPand
GYDare applied, i.e. 9, 10, 10 and 3.5, respectively.
5. Results
Course stability of the towing system at different wind velocities
and wind angles are numerically simulated using linear and nonlinear
approaches. In these simulations, the authors employed the towing
parameters of0T 0:44,0B 0:5 and different
0from 1:0 to 5:0;
whereas0
2:0 was only used for the nonlinear analysis.
5.1. Course stability of the ship towing system in wind: linear
analysis
Following the work of Yasukawa and Nakamura (2007a), the
stability conditions of the linearized system are determined by the
signs of the real part of its eigen values from Eq. (34): negative andpositive values represent stable and unstable motion responses,
respectively. The analysis was discussed in course diagram stability
designating stable (white color) and unstable (black color) zones, as
shown inFigs. 5 and 6. In this analysis, the tug motion was assumed
to be given as explained earlier in Section 3.
For barge 2B, the course stability diagrams of the ship towing
system using linear approach vs. the angle of wind are plotted in
Fig. 5. Based on the diagrams, the increase ofUw=Ufrom 0.0 (no
wind) to 4.0 took place in the unstable towing regions although0
was lengthened from 1.0 to 5.0. Using the linear theory from
Fitriadhy and Yasukawa (2011), the towing stability was dom-
inantly determined by the inherent stability criterion of the
towed ship itself: therefore the increase of 0 on the towing of
the unstable barge (negative course stability index) was
Fig. 4. Wind coefficients for tug and barge 2B/s in various angles of wind.
Fig. 5. Course stability diagram of 2B in various velocities and angles of wind.
Fig. 6. Course stability diagram of 2Bs in various velocities and angles of wind.
A. Fitriadhy et al. / Ocean Engineering 64 (2013) 135145 141
-
8/10/2019 Course Stability of a Ship Towing System in Wind OceanEngineering03022013
8/11
unnecessary and even prone to degrade the towing stability.
However, the stable region then appeared in the range of beam
and following winds as a further increase ofUw=Uup to 8.0. This
could possibly be explained by the wind forces exerted on the
exposed windage of barge 2B, which would increase of the yaw
damping on her hull and result in significant reduction of
amplitude of the lateral motion. The results agreed well with
model basin tests conducted byYasukawa et al. (2007b), where
barge 2B was towed in uniform and constant wind conditions.
For barge 2Bs, the course stability analysis is plotted inFig. 6.
For the no wind case (Uw=U 0:0), the towing of barge 2Bs wasabsolutely stable. When Uw=U increased up to 4.0, the towing
instability appeared in the range of 1541rywr1801 at
0:4r0r5:0. The same tendencies showed that the towingcondition took place in the unstable region in the range of
Fig. 8. Time histories and trajectories of towing for 2B in various wind velocities with yw 1201
.
Fig. 7. Time histories and trajectories of towing for 2B in various wind velocities with yw 01.
Table 3
Case of 2B, effect of wind velocity on motion amplitude of ship towing system
withyw 01.
Uw=U u1 (m/s) c1 (1) c2 (
1) d1 (
1)
0 2.67 1.01 51.1 5.0
4 2.34 1.19 50.3 5.3
8 1.91 1.24 51.6 6.4
A. Fitriadhy et al. / Ocean Engineering 64 (2013) 135 145142
-
8/10/2019 Course Stability of a Ship Towing System in Wind OceanEngineering03022013
9/11
01rywr261 and 1671rywr1801 at 0:4r0r2:65 and
0:2r0r5:0, respectively, as Uw=U increased from 4.0 to 8.0.Similar to what is noted byYasukawa et al. (2012), the instability
towing regions in the head and following winds occurred mainly
due to the effect of the positive sign for N0Ac2 (the restoring
moment derivative with respect to yaw angle). As discussed in
Section 5.2, this towing instability was presented in the form of
increasing oscillation of the lateral motion for barge 2B (see
Figs. 9 and 10). However, the towing instability regions in thehead wind case with Uw=U 8:0 vanished by lengthening the
towline (042:65). For this reason, the higher resistance of thestable barge (positive stability index) associated with the longer
towline led to more stable towing conditions, similar to the
finding by Fitriadhy and Yasukawa (2011). In general, the
towing stability of barge 2Bs was found to be more stable than
barge 2B.
5.2. Course stability of the ship towing system in wind: Nonlinear
analysis
In the presence of wind, the ship towing model, composed of a
tug and towed ship coupled through a towline, has revealed theenormous complexities involving two ships motions associated
with dynamic tension in a towline. Therefore, nonlinear analysis
is required to capture this phenomenon, which would be efficient
to obtain a more reliable prediction for the course stability of the
ship towing system.
As seen, the entire towing performance of barge 2B at yw 01with the various wind velocities was still directionally unstable as
indicated by the sufficient large lateral motions (Y2) and ampli-
tude ofc2 (see Fig. 7). The results are presented in Table 3. In
head wind condition, u1 decreased adequately by 28% as Uw=U
increased from 0.0 to 8.0. This occurred since the quadratic
function of Uw was proportional to the total ships resistances.
Meanwhile, the yaw motion of barge 2B oscillated more fre-
quently by 65%; and the period ofY2 became faster by 41% withrespect to the horizontal trajectories (X2). However, the increase
of head wind velocity in general had a relatively small effect on
the mean magnitude ofTC; and the motion performance of barge
2B as indicated by the insignificant influence to the amplitude of
c2, Y2, c1 and d1. This can be explained as the behavior of the
towing independently correlates to the inherent course stability
index of the barge itself as well-noted in Table 2.
The changing of wind angle from beam to quartering remarkably
affects the course towing stability as illustrated in Fig. 8. These
towing trajectories were captured at yw 1201. With the subse-
quent increase of Uw=U from 0.0 to 8.0, the simulation results
showed that the motion of barge 2B veered off to the starboard side
from the initial course and then settled then in relatively steady
course withc2
35:51, Table 4. This can be explained (Section 5.1 at
Paragraph 2) as the sway forces in the towing of barge 2B were more
dominant than her yaw moment induced by the wind forces, which
acted alongside the windage. In addition, the mean amplitude ofc2was reduced by 32%, which revealed less fluctuating of TC and
implied a towing to speed up u1 by 23%. At the same time, to
Table 4
Case of 2B, effect of wind velocity on motion amplitude of ship towing system
withyw 1201.
Uw=U u1 (m/s) c1 (1) c2 (1) d1 (1)
0 2.6 3.2 52.0 5.1
4 2.8 8.4 53.7 3.78 3.2 21.5 35.5 25.0
Fig. 9. Time histories and trajectories of towing for 2Bs in various wind velocities with yw 01.
Table 5
Case of 2Bs, effect of wind velocity on motion amplitude of ship towing system
withyw 01.
Uw=U u1 (m/s) c1 (1) c2 (1) d1 (1)
0 3.6 0.0 0.0 0.0
4 3.1 0.6 8.8 1.88 2.2 1.4 35.6 6.0
A. Fitriadhy et al. / Ocean Engineering 64 (2013) 135145 143
-
8/10/2019 Course Stability of a Ship Towing System in Wind OceanEngineering03022013
10/11
preserve the tug on the desired track inevitably resulted in a larger
deflection of rudder angles d1 by 251 to port. However, the
subsequent increase of Uw=U at yw 1201 had an insignificant
influence on the mean magnitude ofTC.
For barge 2Bs, the towing characteristics in the various head
wind velocities are illustrated inFig. 9. By increasing Uw=U from
0.0, 4.0 to 8.0, the motion of barge 2Bs is prone to be unstable as
indicated by the increase ofc2 up to 35.61, Table 5. The lateral
motion of barge 2Bs increased almost 5.5 times asUw=Uchanged
from 4.0 to 8.0. Similar to what was explained inSection 4.1, the
restoring force of the aerodynamic derivative N0Ac2 acting on barge
2Bs led to diverge her yaw motion. This vigorous manoeuvring
from barge 2Bs resulted in a considerable increase of maximum TCfrom 7.3 t to 9.2 t. This condition might pose structural concerns
and become even worse when the snatching frequency of the
towline coincides the with motion frequencies of the tug, Varyani
et al. (2007). From the trajectories, the resistance of barge 2Bs
seemed to increase as indicated by a decrease in the tows speed
ofu 1 by 14% and 39% asUw=Uincreased from 0.0 to 4.0 and 0.0 to
8.0, respectively. Even though the deflection of d1 increased to
stabilize the towing, an unwieldy slewing motion of barge 2Bs at
Uw=U 8:0 still occurred, which is absolutely unfavorable from
the towing stability point of view.
Fig. 10shows the effect of the following wind conditions on
the course stability of barge 2Bs. In general, the towing char-acteristics of barge 2Bs have been shown to bear qualitative
similarities to its characteristics in head wind condition. This
means that the increase of wind velocity gradually degrades the
entire towing performance as indicated by the excessive c2 up to
61.51 at Uw=U 8:0,Table 6. Similar to the head wind case, the
diverging motion of barge 2Bs in following wind condition
occurred due to the aerodynamic derivative value ofN0Ac2, which
was positive, Yasukawa et al. (2012). It was noted that Y2increased at almost nine times as Uw=U changed from 4.0 to
8.0. Because of the severity of barge 2Bss motion, this strongly
affects the tugs motions, where the tug experienced rigorous
motions indicated by the violent oscillation of c1, d1 and u1.
However, the increase of following wind velocity up to
Uw=U 8:0 is also detrimental to the tow by causing a very
impulsive towline tension with the maximum ofTCof 18.7 t. This
amount was almost twice the maximum ofTC in the head wind
case. The reason for this is that in the following seas the surge of
the tug increased the snatching of the towline due to rigorous
loosening and tightening of the towline with the violent motion
of barge 2Bs.
6. Conclusion
The course stability of the ship towing system in uniform and
constant wind conditions was solved by using theoretical
approaches. The agreement between linear and nonlinear analysis
was obtained. Using the linear analysis, the stability investigation
of the ship towing system showed that the course stability of theunstable barge was recovered in the range of beam to following
winds as the wind velocity increased. In addition, the towing
performance of the stable barge was prone to be unstable in head
and following winds as indicated by the large amplitude of her
headings angle and lateral motion. Employing a longer towline
for the towing of the unstable barge was ineffective in stabilizing
the towing system; conversely, for the towing of the stable barge,
the longer towline led to more stable towing conditions. In the
nonlinear analysis, the results revealed that the towing instability
of the unstable barge 2B at yw 1201 and Uw=U 8:0 was
recovered as indicated through attenuation in her fishtailing
motions. In general, the towing of the stable barge associated
with the longer towline led to more stable towing conditions than
the towing of the unstable barge. The increase of following wind
Fig. 10. Time histories and trajectories of towing for 2Bs in various wind velocities with yw 1801.
Table 6
Case of 2Bs, effect of wind velocity on motion amplitude of ship towing system
withyw 1801.
Uw=U u1 (m/s) c1 (1) c2 (1) d1 (1)
0 3.6 0.0 0.0 0.25
4 3.7 0.8 18.4 2.6
8 3.9 9.2 61.5 18.5
A. Fitriadhy et al. / Ocean Engineering 64 (2013) 135 145144
-
8/10/2019 Course Stability of a Ship Towing System in Wind OceanEngineering03022013
11/11
velocity resulted in a very impulsive towline tension, which is
almost twice the maximum ofTCin the head wind case.
Appendix A
D00
0
BM0
x0sin c0M0
y0 cos c0I0
y0 cos y0I0
x0sin y0I0z
0BI
0y0 cos c0I
0x0 sin c0
0M0x0sin y0M
0y0cos y0
D1 0BM
0x0 sin c0M
0y0 cos c0
0Bsin g0F
01q cosg0 F
02q sing0
F03q I0
z0BI
0y0 cos c0I
0x0sin c0
2F01q sing0F02q cosg0cosc0 cosg0M
0yM
0x
sinc0sin g0M0
yM0
x F02r cosg0F
01r sing0
0I0y0cos y0I0
x0sin y00Bsinc0 sin g0 cosg0M
0xM
0y
sin2 g0cos c0M
0yM
0x
0M0x0sin y0M
0y0cos y0
D2 0BM
0x0 sin c0M
0y0 cos c0
0BfF
010 sin
2 g0cos2 g0
2F020 sing0 cosg0gI0
z0BI
0y0 cos c0I
0x0 sin c0
F01
0
cosg0 F02
0
sing0cosc0 cosg0M0
yM0
x
sinc0sin g0M0
yM0
x F02r cosg0F
01r sing0
0Bsin g0F01q cosg0 F
02q sing0F
03q
0Bfsinc0 sin g0cos g0M0
xM0
y sin2 g0 cos c0M
0yM
0x
F01r sing0 cosg0 F02r sin
2 g0gF03r
F01q sing0F02q cosg0
F010 F01c cosg0F
020 F
01c sing0
0I0y0 cos y0I0
x0sin y0
0Bfsing0F01c cosg0 F
02c sing02F
020 cosg0
F010 sin2 g0cos
2 g0gF03c
0M0x0sin y0M
0y0cos y0
D3 cosc0 cosg0M0
yM0
x sinc0 sin g0M0
yM0
x F02r cosg0
F01r sing00BfF
010 sin
2 g0cos2 g02F
020 sing0cos g0g
0Bfsinc0 sin g0cos g0M0
xM0
y sin2 g0 cos c0M
0yM
0x
F01r sing0 cosg0 F02r sin
2 g0gF03r
F010 cosg0 F020 sing0
F01
0 F01c cosg0F
020 F
01c sing0
0Bsin g0F01q cosg0 F
02q sing0F
03q
0Bfsing0F01c cosg0 F
02c sing02F
020 cosg0
F010 sin2 g0cos
2 g0gF03c F
01q sing0F
02q cosg0
D4 F010 F
01c cosg0F
020 F
01c sing0
0BfF010 sin
2 g0cos2 g02F
020 sing0 cosg0g
0Bfsing0F01c cosg0 F
02c sing02F
020 cosg0
F01
0 sin2 g0cos
2 g0gF03c F
010 cosg0 F
020 sing0
References
Fitriadhy, A., Yasukawa, H., 2011. Course stability of a ship towing system. J. ShipTechnol. Res. 58, 424.
Fujiwara, T., Ueno, M., Nimura, T., 1998. Estimation of wind forces and momentsacting on ship. Japan Society of Naval Architects and Ocean Engineers 183,7790. (Japanese).
Isherwood, R.M., 1972. Wind resistance of merchant ships. RINA Trans. 115,327338.
Kijima, K., Wada, Y., 1983. Course stability of towed vessel with wind effect. Japan
Society of Naval Architects and Ocean Engineers 153, 117126. (Japanese).Kijima, K., Varyani, K., 1986. Wind effect on course stability of two towed vessels.
Japan Society of Naval Architects and Ocean Engineers 24, 103114.Peters, B.H., 1950. Discussion in the paper of Strandhagen, A.G. et al. Trans. Society
of Naval Architects and Marine Engineers 58, 4652.Shigehiro, R., Ueda, K., Arii, T., Nakayama, H., 1997. Course stability of the high-
speed-towed fish preserve with wind effect. J. Kansai Soc. Nav. Archit. 224,167174.
Varyani, K.S., Barltrop, N., Clelland, D., Day, A.H., Pham, X., Van Essen, K., Doyle, R.,Speller, L., 2007. Experimental investigation of the dynamics of a tug towing adisabled tanker in emergency salvage operation. In: International Conferenceon Towing and Salvage Disabled Tankers, pp. 117125.
Yasukawa, H., Hirata, N., Nakamura, N., Matsumoto, Y., 2006. Simulations ofslewing motion of a towed ship. Japan Society of Naval Architects and OceanEngineers 4, 137146. (Japanese).
Yasukawa, H., Hirata, N., Tanaka, K., Hashizume, Y., Yamada, R., 2007b. Circulationwater tunnel tests on slewing motion of a towed ship in wind. Japan Society ofNaval Architects and Ocean Engineers 6, 323329. (Japanese).
Yasukawa, H., Hirono, T., Nakayama, Y. and Koh, K.K., 2012. Course Stability andYaw Motion of a Ship in Steady Wind, J.Marine Science and Technology.Vol.17, No.3, 291304.
Yasukawa, H., Nakamura, N., 2007a. Analysis of course stability of a towed ship inwind. Japan Society of Naval Architects and Ocean Engineers 6, 313322.(Japanese).
A. Fitriadhy et al. / Ocean Engineering 64 (2013) 135145 145