Course Stability of a Ship Towing System in Wind OceanEngineering03022013

8/10/2019 Course Stability of a Ship Towing System in Wind OceanEngineering03022013

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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

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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


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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


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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.



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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



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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



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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.



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



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



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



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


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


2 g0gF03r

F010 cosg0 F020 sing0

F01

0 F01c cosg0F

020 F

01c sing0

0Bsin g0F01q cosg0 F

02q sing0F

03q


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


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).


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