Ocean Engineering 64 (2013) 135–145

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Ocean Engineering

0029-80

http://d

n Corr

E-m

yasukaw

journal homepage: www.elsevier.com/locate/oceaneng

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

18/$ - see front matter & 2013 Elsevier Ltd. A

x.doi.org/10.1016/j.oceaneng.2013.02.001

esponding author. Tel.: þ60 1 9155 590; fax:

ail addresses: naoe.afit@gmail.com (A. Fitriad

a@naoe.hiroshima-u.ac.jp (H. Yasukawa), ko

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 towing

system was discussed using the linear analysis, where a tug’s motion was assumed to be given. When the

tug and the towed ship’s 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 exposedto some degrees of wind at different directions. A reliable investiga-tion either using a theoretical or experimental approach is required toobtain a deeper understanding of the course stability of the shiptowing system with such external disturbance.

In recent years, several studies regarding course stability of shiptowing 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 foundthat when the wind angle changed from the head to the followingwinds, the course stability of the two towed ships tended to becomeunstable. In addition, Kijima and Wada (1983) presented that thecourse stability of the towed barge would generally be unstable inthe 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 wasrecovered in the range of beam to quartering winds. In this work,however, the towed barge was decoupled from the tug, i.e. the tug’smotion was assumed to be given.

This paper presents linear and nonlinear model analyses ofcourse stability for a ship towing system in uniform and constantwind conditions as an extension study from the previous work by

ll rights reserved.

þ60 9 668 3719.

hy),

h@fkm.utm.my (K.K. Koh).

Fitriadhy and Yasukawa (2011). In the nonlinear analysis, aproper model of the ship towing system was modeled, wherethe tug and towed barge are coupled by a towline. This is quitereasonable given the fact that wind forces will be exerted onwindage areas both of the towed ship and also the tug. Thus, theanalytical model of predicting course stability of a ship towingsystem is deemed more reliable. The effect of wind velocities (Uw)and absolute wind angles (yw) were taken into account in themodels. A 2D lumped mass method was applied to model towlinemotion incorporated with dynamic towline tension; and anautopilot system was employed to reduce heading and deviationof the tug from its desired track. The presented numericalapproach is expected to reduce experimental costs, even thoughthe model test validation is still recommended.

2. Mathematical formulation

The mathematical model of maneuvering motions equationsfor a tug and towed ship associated with dynamic towline tensionrelates 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 towedships, three coordinate systems are used, Fig. 1. One set of axes isfixed to the earth’s coordinate system that is used to specifyabsolute wind velocity Uw and angle yw denoted as O�XY , and

Fig. 1. Coordinate systems of tug and towed ships (left) and lumped mass model for towline (right).

A. Fitriadhy et al. / Ocean Engineering 64 (2013) 135–145136

two sets of axes G1�x1y1 and G2�x2y2 are fixed relative to eachship’s moving coordinate system aligned with its origin at thecenter of gravity. In the moving reference, the xi-axis pointsforward and the yi-axis to starboard. i¼1 designates the tug,i¼2 the towed ship. The heading angle ci refers to the direction ofthe ship’s local longitudinal axis xi with respect to the fixed x-axis.The instantaneous speed of ship Ui can be decomposed into aforward velocity ui and a lateral velocity vi. The angle between Ui

and the xi-axis is the drift angle bi ��tan�1ðvi=uiÞ. Here, yw ¼ 01and yw ¼ 1801 are the head and following winds, respectively, andcoincide with the earth’s fixed system X; yw ¼ 901 is the beamwind, which coincides with the earth’s fixed system Y.

The towline is composed of a finite number N of lumpedmasses; the masses are connected by segments into the entiretruss element. The lumped mass particulars describe the towlinecharacteristics, such as the mass, the density and the drag. Thecoordinates of the ith lumped mass is labeled by ðXi,YiÞ, whereði¼ 1,2,3, . . . ,Nþ2Þ. The angle between the x-axis and the lengthof ith segmented towline ‘i is denoted as yi. Here, ‘Nþ2 is thedistance of the connection point at the towed ship with respect toher center of gravity and yNþ2 ¼c2 is the heading angle of thetowed ship. Their connection points with respect to the earth’sfixed coordinate systems ðX0,Y0Þ and ðXNþ1,YNþ1Þ, respectively,have the coordinates ð‘T ,0Þ and ð‘B,0Þ in the respective local shipcoordinate systems. Then, the coordinates of lumped massesðXi,YiÞ through yi and ‘i can be written as

Xi ¼ X0�Xi

j ¼ 1

‘j cos yj, Yi ¼ Y0�Xi

j ¼ 1

‘j sin yj ð1Þ

where yNþ2 ¼c2 and ‘Nþ1 ¼ ‘B.

2.2. Motion equations of towed ship and towline

The motion equations of the towed ship are written inEqs. (2) and (3) as follows:

�XNþ2

j ¼ 1

‘jðMx1 sin yj�My1 cos yjÞ€yj

¼XNþ2

j ¼ 1

‘jðMx1 cos yjþMy1 sin yjÞ_y

2

j þTV1þMx1€X 0þMy1

€Y 0 ð2Þ

Ið2Þz€yNþ2þ

XNþ2

j ¼ 1

‘j‘B sin gðMy2 cos yj�Mx2 sin yjÞ€yj

¼XNþ2

j ¼ 1

‘j‘B sin gðMy2 sin yjþMx2 cos yjÞ_y

2

j �‘B sin g

�ðTV2�Mx2€X 0�My2

€Y 0ÞþMð2Þz ð3Þ

where

Mx1 ¼Mð2Þx sin g cos c2þMð2Þy cos g sin c2

My1 ¼Mð2Þx sin g sin c2�Mð2Þy cos g cos c2

Mx2 ¼Mð2Þx cos g cos c2�Mð2Þy sin g sin c2

My2 ¼Mð2Þx cos g sin c2þMð2Þy sin g cos c2

TV1 ¼ ðMð2Þx v2 sin gþMð2Þy u2 cos gÞ _c2�ðF

ð2Þx �Mð2Þy v2

_c2Þsin g

þðFð2Þy �Mð2Þx u2_c2Þcos g

TV2 ¼ ð�Mð2Þx v2 cos gþMð2Þy u2 sin gÞ _c2þðFð2Þx þMð2Þy v2

_c2Þcos g

þðFð2Þy �Mð2Þx u2_c2Þsin g

g¼ yNþ1�c2

The notations of Mð2Þx ¼ ðm2þmx2Þ and Mð2Þy ¼ ðm2þmy2Þ representthe virtual mass components in the direction x2 and y2, respec-tively; and Ið2Þz ¼ ðI2þ J2Þ is the virtual moment of inertia, which isexpressed as the sum of mass (moment of inertia) and addedmass (added moment of inertia) components. Fð2Þx , Fð2Þy and Mð2Þz arethe surge force, the sway force, and the yaw moment acting onthe towed ship, respectively. The superscripts (1) and (2) denotethe tug and the towed ship, respectively.

Lagrange’s motion equations are applied to describe thedynamic motion of the towline and are derived in Eq. (4). mi

and kFi are the mass and the added mass coefficients of the ithlumped masses, respectively.

XN

i ¼ k

Xi

j ¼ 1

ðmsi sin yk sin yjþmci cos yk cos yjÞ‘k‘j€yj

8<:

9=;

þ‘k sinðyk�yNþ1ÞXNþ2

j ¼ 1

‘jðMx2 sin yj�My2 cos yjÞ€yj

¼Q0k�Q1k�‘kTV3 sinðyk�yNþ1Þ ðk¼ 1,2,3, . . . ,NÞ ð4Þ

where

msi ¼mið1þkFi sin2 yiÞ

A. Fitriadhy et al. / Ocean Engineering 64 (2013) 135–145 137

mci ¼mið1þkFi cos2 yiÞ

TV3 ¼XNþ2

j ¼ 1

‘jðMx2 cos yjþMy2 sin yjÞ_y

2

j þMx2€X 0þMy2

€Y 0�TV1�TV2

Q0k ¼�‘k sin yk

XN

i ¼ k

ðRCi sin yi�FCi cos yiÞ

�‘k cos yk

XN

i ¼ k

ðRCi cos yiþFCi sin yiÞ

Q1k ¼ ‘k sin yk

XN

i ¼ k

€X 0þXi

j ¼ 1

_y2

j ‘j cos yj

0@

1Amsi

�‘k cos yk

XN

i ¼ k

€Y 0þXi

j ¼ 1

_y2

j ‘j sin yj

0@

1Amci

þ2‘k

XN

i ¼ k

ð _X i sin ykþ_Y i cos ykÞmikFi

_yi sin yi cos yi

�ð _X2

k�_Y

2

k ÞmkkFk cos yk sin yk

Two different external forces experienced on the segmentedtowline, Fig. 1 (right). These forces are decomposed into normaland axial force components:

RCi ¼�12 rSiCDi9VCi9VCi, FCi ¼�

12 rSiCFi9UCi9UCi ð5Þ

where VCi ¼�_X i sin yiþ

_Y i cos yi and UCi ¼_X i cos yiþ

_Y i sin yi. r isthe water density, Si the profile area of the segmented towline, CDi

the coefficient of the normal force, and CFi the coefficients of theaxial force.

2.3. Motion equation of tug

The equation of the tug motion was derived adequately asfollows:

Mð1Þx_u1�Mð1Þy v1

_c1 ¼ Fð1Þx þFTx ð6Þ

Mð1Þy_v1þMð1Þx u1

_c1 ¼ Fð1Þy þFTy ð7Þ

Ið1Þz€c1 ¼Mð1Þz þMTz ð8Þ

where Mð1Þx ¼ ðm1þmx1Þ and Mð1Þy ¼ ðm1þmy1Þ represent the vir-tual mass components in the direction x1 and y1, respectively,Ið1Þz ¼ ðI1þ J1Þ the virtual moment of inertia. Fð1Þx , Fð1Þy and Mð1Þz arethe surge force, the sway force and the yaw moment acting on thetug, respectively. FTx, FTy and MTz denote the surge force, the swayforce, and the yaw moment due to the towline tension acting atthe connection point of the tug, respectively, which can beexpressed as

FTx ¼�TX cos c1�TY sin c1 ð9Þ

FTy ¼ TX sin c1�TY cos c1 ð10Þ

MTz ¼ ‘T ðTX sin c1�TY cos c1Þ ð11Þ

The towline tension components TX and TY are expressedfollowing Yasukawa et al. (2006):

TX ¼XN

i ¼ 1

msi

Xi

j ¼ 1

‘j sin yj€yj

0@

1A

þXNþ2

j ¼ 1

‘jðMx2 sin yj�My2 cos yjÞ€yj cos yNþ1þTð1ÞX ð12Þ

TY ¼�XN

i ¼ 1

mci

Xi

j ¼ 1

‘j cos yj€yj

0@

1A

þXNþ2

j ¼ 1

‘jðMx2 sin yj�My2 cos yjÞ€yj sin yNþ1þTð1ÞY ð13Þ

where

Tð1ÞX ¼XN

i ¼ 1

€X0þXi

j ¼ 1

‘j cos yj_yj

2

0@

1Amsiþ2mi

_Xi kFi sin yi cos yi_yi

24

þRCi sin yi�FCi cos yi

�þTV3 cos yNþ1

Tð1ÞY ¼�XN

i ¼ 1

€Y0þXi

j ¼ 1

‘j sin yj_yj

2

0@

1Amci�2mi

_Yi kFi sin yi cos yi_yi

24

þRCi cos yiþFCi sin yi

�þTV3 sin yNþ1

The resultant towline tension at the tow point of the tug can

be expressed as TC ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT2

XþT2Y

q.

2.4. Forces and moments acting on the ships

The equation of external forces and moments on hull, propel-ler, rudder, and wind acting on the tug and towed ship areexpressed in Eq. (14). The first three of the aforementioned forcesand moments were described in the previous study (Fitriadhy andYasukawa, 2011).

FðiÞx ¼ XðiÞH þXðiÞP þXðiÞR þXðiÞA

FðiÞy ¼ Y ðiÞH þY ðiÞR þY ðiÞA

MðiÞz ¼NðiÞH þNðiÞR þNðiÞA �FðiÞy xGi

9>>>=>>>;

ð14Þ

Estimation of wind forces on exposed windage areas of the tugand the towed ship are modeled in various velocities and anglesof wind. Based on Isherwood (1972), the equation of wind forcesand moments are

XðiÞA ¼ ð1=2ÞraAðiÞX V ðiÞ2A CðiÞXAðyðiÞA Þ

Y ðiÞA ¼ ð1=2ÞraAðiÞY V ðiÞ2A CðiÞYAðyðiÞA Þ

NðiÞA ¼ ð1=2ÞraAðiÞY V ðiÞ2A LiCðiÞNAðy

ðiÞA Þ

9>>>=>>>;

ð15Þ

where

yðiÞA ¼ tan�1ðvðiÞA =uðiÞA Þ ð16Þ

V ðiÞA ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuðiÞ2A þvðiÞ2A

qð17Þ

uðiÞA ¼ uiþUw cosðyw�ciÞ ð18Þ

vðiÞA ¼ viþUw sinðyw�ciÞ ð19Þ

The notations of CðiÞXA, CðiÞYA and CðiÞNA are the force and momentcoefficients as a function of yðiÞA (relative wind angle); ra is thedensity of air; AðiÞX and AðiÞY 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) 135–145138

3. Linearization of motion equations for course stabilityinvestigation in wind

Study on course stability of the ship towing system in windinvolves stronger nonlinearities than in calm water conditions. Tounderstand the basic mechanism of the ship towing system in thewind condition, a course stability model using piecewise linearsystem is utterly essential. This approach leads to provide athreshold 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 withY0 ¼ 0.

3.

The towline is treated as non-extensible catenary model(N¼0).

4.

This virtual tug moves in a straight-course with U ¼ _X0 , wherec1, €X0 , _X0 and €Y0 are equal to zero, while y2 ¼c2.

Here, y1 and c2 are defined in steady and unsteady motions as

y1 ¼ y0þDy1, _y1 ¼D _y1 ð20Þ

c2 ¼c0þDc2, _c2 ¼D _c2 ð21Þ

where Dy1 and Dc2 are negligibly small (OðeÞ); g0 ¼c0�g0 andDg¼Dy1�Dc2.

3.1. Linearized motion equation of forces and moments acting on a

towed ship

The basic linearization of external force under wind conditionis based on the relative wind angle ðyð2ÞA Þ, which is exertedforcefully on superstructure of the towed ship as written inEq. (16). Using Taylor series expansion with respect to D _y1 ,D _c2 , Dc2, then yð2ÞA is solved as

yð2ÞA ¼ tan�1ðvð2ÞA =uð2ÞA Þ

Cyð2ÞA0þyð2ÞAqD

_y1þyð2ÞAr D

_c2þyð2ÞAcDc2 ð22Þ

where

yð2ÞA0 ¼ tan�1ðvð2ÞA0=uð2ÞA0 Þ

Fig. 2. Coordinate systems for linear model of a towed ship in wind.

yð2ÞAq1 ¼‘fU cos y0�Uw cosðyw�y0Þg

uð2Þ2A0 þvð2Þ2A0

yð2ÞAr ¼�‘Buð2ÞA0

uð2ÞA0þvð2ÞA0

yð2ÞA0c ¼�1

uð2ÞA0 ¼U cos c0þUw cosðyw�c0Þ

vð2ÞA0 ¼�U sin c0þUw sinðyw�c0Þ

The square term of relative wind velocity ðV2AÞ in Eq. (17) can

be recast into the linearized form:

V ð2ÞA ¼ V ð2ÞA0þV ð2ÞAqD_y1þV ð2ÞAr D

_c2 ð23Þ

where

V ð2ÞA0 ¼U2þUw2þ2UUw cos yw

V ð2ÞAq ¼ 2‘½sin g0fUw cosðyw�c0ÞþU cos c0g

�cos g0fUw sinðyw�c0Þ�U sin c0g�

V ð2ÞAr ¼�2‘B½Uw sinðyw�c0Þ�U sin c0�

The equations of forces and yaw moment ðXð2ÞA ,Y ð2ÞA ,Nð2ÞA Þ underwind condition are denoted as FðkÞA (k¼1,2 and 3, respectively), asfollows:

FðkÞA ¼ FðkÞA0þFðkÞAqD_y1þFðkÞAr D

_c2þFðkÞAc2Dc ð24Þ

where

FðkÞA0 ¼ ð1=2ÞraAðkÞCðkÞA ðyA0ÞVA0

FðkÞAq ¼ ð1=2ÞraAðkÞ CðkÞA ðyA0ÞVAqþ@CðkÞA ðyA0Þ

@yAyAqVA0

" #

FðkÞAr ¼ ð1=2ÞraAðkÞ CðkÞA ðyA0ÞVArþ@CðkÞA ðyA0Þ

@yAyArVA0

" #

FðkÞAc ¼ ð1=2Þra AðkÞ@CðkÞA ðyA0Þ

@yAyAcVA0

and Að1Þ ¼ Að2ÞX , Að2Þ ¼ Að2ÞY and Að3Þ ¼ Að2ÞY L2.Then, the hydrodynamic forces and moment acting on the hull

ðXð2ÞH ,Y ð2ÞH ,Nð2ÞH Þ are denoted as FðkÞH (k¼1,2 and 3, respectively) and

expressed as

FðkÞH ¼ FðkÞH0þFðkÞHqD_y1þFðkÞHrD

_c2þFðkÞHcDc2 ð25Þ

where

Fð1ÞH0 ¼ X0U2 cos2 c0þXvvU2 sin2 c0

Fð1ÞHq ¼ 2‘UðXvv sin c0 cos g0þX0 cos c0 sin g0Þ

Fð1ÞHr ¼U sin c0ð2‘BXvv�XvrÞ

Fð1ÞHc2¼ 2U2 sin c0 cos c0ðXvv�X0Þ

Fð2ÞH0 ¼�YvU sin c0�YvvvU3 sin3 c0

Fð2ÞHq ¼�‘ cos g0ðYvþ3YvvvU2 sin2 c0Þ

Fð2ÞHr ¼ YrþYvvrU2 sin2 c0�‘BðYvþ3YvvvU2 sin2 c0Þ

Fð2ÞHc2¼�U cos c0ðYvþ3YvvvU2 sin2 c0Þ

Fð3ÞH0 ¼�NvU sin c0�NvvvU3 sin3 c0�xGFð2ÞH0

Fð3ÞHq ¼�‘ cos g0ðNvþ3NvvvU2 sin2 c0Þ�xGFð2ÞHq

Fð3ÞHr ¼NrþNvvrU2 sin2 c0�‘BðNvþ3NvvvU2 sin2 c0Þ�xGFð2ÞHr

Fð3ÞHc2¼�U cos c0ðNvþ3NvvvU2 sin2 c0Þ�xGFð2ÞHc2

A. Fitriadhy et al. / Ocean Engineering 64 (2013) 135–145 139

Referring to Eq. (14), the linearized equation of the totalexternal forces and moments ðFðkÞx ,FðkÞy ,MðkÞz Þ is denoted as FðkÞ

(k¼1,2 and 3, respectively) and take the following form:

FðkÞ ¼ FðkÞ0 þFðkÞq D _y1þFðkÞr D _c2þFðkÞc2Dc2 ð26Þ

where

FðkÞ0 ¼ FðkÞH0þFðkÞA0 , FðkÞq ¼ FðkÞHqþFðkÞAq

FðkÞr ¼ FðkÞHrþFðkÞAr , FðkÞc2¼ FðkÞHc2

þFðkÞAc2

The notation of FðkÞ0 is the steady component of the lateral forcesand yaw moments; FðkÞq , FðkÞr and FðkÞc2

are the unsteady derivativevalues 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 shipcan be written in the following form:

�‘ðMx0 sin y0�My0 cos y0ÞD €y1�‘BðMx0 sin c0�My0 cos c0ÞD €c2

¼ ½�Fð1Þq sin g0þFð2Þq cos g�D _y1�½Fð1Þ0 cos g0þFð2Þ0 sin g0�Dy1

þ½U cos c0 cos g0ðMy�MxÞþU sin c0 sin g0ðMy�MxÞ

þFð2Þr cos g0�Fð1Þr sin g0�D _c2

þ½ðFð1Þ0 þFð2Þc Þcos g0þðFð2Þ0 �Fð1Þc2

Þsin g0�Dc2

�Fð1Þ0 sin g0þFð2Þ0 cos g0 ð27Þ

Similarly, Eq. (3) becomes:

‘ðIy0 cos y0�Ix0 sin y0ÞD €y1þ½Izþ‘BðIy0 cos c0�Ix0 sin c0Þ�D €c2

¼�‘BðFð1Þq sin g0 cos g0þFð2Þq sin2 g0�Fð3Þq =‘BÞD _y1

þ‘B½Fð1Þ0 ðsin2 g0�cos2 g0Þ�2Fy0 sin g0 cos g0�Dy1

�‘B½U sin c0 sin g0 cos g0ðMx�MyÞþU sin2 g0 cos c0ðMy�MxÞ

�Fð3Þr =‘BþFð1Þr sin g0 cos g0þFð2Þr sin2 g0�D _c2

�‘B½sin g0ðFð1Þc cos g0þFð2Þc sin g0�2Fð2Þ0 cos g0Þ

þFð1Þ0 ðsin2 g0�cos2 g0Þ�Fð3Þc2=‘B�Dc2

�‘B sin g0½Fð1Þ0 cos g0þFð2Þ0 sin g0�þMz0 ð28Þ

where

Mx0 ¼Mx sin g0 cos c0þMy cos g0 sin c0

My0 ¼Mx sin g0 sin c0�My cos g0 cos c0

Ix0 ¼ ‘B sin g0ðMx cos g0 cos c0�My sin g0 sin c0Þ

Iy0 ¼ ‘B sin g0ðMx cos g0 sin c0þMy sin g0 cos c0Þ

Eqs. (27) and (28) are non-dimensionalized with respect toð1=2ÞrL2d2U2 and ð1=2ÞrL2

2d2U2, respectively. L2, d2 and U denotethe length and the draft of the towed ship and the tow’s speed,respectively. Through separating these equations into the non-dimensional steady and unsteady motion terms, the followingequations are expressed:

Steady components:

F 0ð1Þ0 sin g0�F 0ð2Þ0 cos g0 ¼ 0 ð29Þ

‘0B sin g0ðF0ð1Þ0 cos g0þF 0ð2Þ0 sin g0Þ�M0z0 ¼ 0 ð30Þ

Unsteady components:

a1D €y0

þb1D €c0

þc1D _y0þd1D _c

0þe1Dyþ f 1Dc¼ 0 ð31Þ

a2D €y0

þb2D €c0

þc2D _y0þd2D _c

0þe2Dyþ f 2Dc¼ 0 ð32Þ

where

a1 ¼�‘0ðM0x0 sin y0�M0y1 cos y0Þ

b1 ¼�‘0BðM0x0 sin c0�M0y1 cos c0Þ

c1 ¼ F 0ð1Þq sin g0�F 0ð2Þq cos g0

d1 ¼�½cos c0 cos g0ðM0y�M0xÞþsin c0 sin g0ðM

0y�M0xÞ

þF 0ð2Þr cos g0�F 0ð1Þr sin g0�

e1 ¼ F 0ð1Þ0 cos g0þF 0ð2Þ0 sin g0

f 1 ¼�ðF0ð1Þ0 þF 0ð1Þc Þcos g0�ðF

0ð2Þ0 �F 0ð1Þc Þsin g0

a2 ¼ ‘0ðI0y0 cos y0�I0x0 sin y0Þ

b2 ¼ I0zþ‘0BðI0y0 cos c0�I0x0 sin c0Þ

c2 ¼ ‘0B sin g0ðF0ð1Þq cos g0þF 0ð2Þq sin g0Þ�F 0ð3Þq

d2 ¼ ‘0B½sin c0 sin g0 cos g0ðM0x�M0yÞþsin2 g0 cos c0ðM

0y�M0xÞ

þF 0ð1Þr sin g0 cos g0þF 0ð2Þr sin2 g0��F 0ð3Þr

e2 ¼�‘0B½F0ð1Þ0 ðsin2 g0�cos2 g0Þ�2F 0ð2Þ0 sin g0 cos g0�

f 2 ¼ ‘0B½sin g0ðF

0ð1Þc cos g0þF 0ð2Þc sin g0�2F 0ð2Þ0 cos g0Þ

þF 0ð1Þ0 ðsin2 g0�cos2 g0Þ��F 0ð3Þc

From Eqs. (29) and (30), the value for the variables of y0 and c0

is obtained. By substituting those values accordingly into Eqs. (31)and (32), the unsteady motion equations of the towed ship arethen solved. When the wind coefficient is equal to zero, this workfollows essentially the approach of Peters (1950) and Shigehiroet al. (1997).

The non-dimensional motion are

M0xM0yM0x0M0y0 ¼MxMyMx0My0

1=2rL22d2

I0x0,I0y0 ¼Ix0,Iy0

ð1=2ÞrL32d2

I0z ¼Iz

ð1=2ÞrL42d2

F 0ð1Þ0 ,F 0ð2Þ0 ,F 0ð1Þc ,F 0ð2Þc ¼Fð1Þ0 ,Fð2Þ0 ,Fð1Þc ,Fð2Þc

ð1=2ÞrL2d2U2

F 0ð1Þq ,F 0ð2Þq ,F 0ð1Þr ,F 0ð2Þr ¼Fð1Þq ,Fð2Þq ,Fð1Þr ,Fð2Þr

ð1=2ÞrL22d2U

F 0ð3Þ0 ,F 0ð3Þc ¼Fð3Þ0 ,Fð3Þc

ð1=2ÞrL22d2U2

F 0ð3Þq ,F 0ð3Þr ¼Fð3Þq ,Fð3Þr

ð1=2ÞrL32d2U

‘0,‘0B ¼‘,‘B

L2

D €y 01,D €c02 ¼D €y1,D €c2

ðU=L2Þ2

D _y 01,D _c02 ¼D _y1,D _c2

U=L2

where ‘0 and ‘0B denote the ratios of towline length and tow pointto length of the towed ship, respectively, where ‘0 ¼ ‘=L2 and‘0B ¼ ‘B=L2 (‘B40Þ.

3.3. Course stability criterion

Simultaneous solution of equations can be used for theassessment of the stability of the straight-line motion in steady

Table 2Resistance 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 �0.13110

A. Fitriadhy et al. / Ocean Engineering 64 (2013) 135–145140

wind, i.e. motion with ð €Y 00 ¼_Y 00 ¼ 0Þ. The values of Dy1 and Dc2

are described by

Dy1 ¼ C1elt , Dc2 ¼ C2elt ð33Þ

By substituting Eq. (33) into Eqs. (31) and (32), a fourth-ordercharacteristic equation with respect to l should satisfy thefollowing conditions:

D0l4þD1l

3þD2l

2þD3lþD4 ¼ 0 ð34Þ

where the values of D0,D1,D2,D3 and D0 are obtained (seeAppendix A). By applying the Hurwit’s method in Eq. (34), thebasic solution of stability criteria is written in Eqs. (35) and (36).

D0,D1,D2,D3,D440 ð35Þ

D�D1D2D3�D21D4�D0D2

340 ð36Þ

4. Simulation condition

4.1. Ships

The principal dimensions of tug and barge including theirlateral and longitudinal windage areas used in the simulation arepresented in Table 1. The length of the tug and the barge aredenoted as L1 and L2, respectively. The towing point at the tug isdenoted as ‘T and non-dimensionalized as ‘0T ¼ ‘T=L1. Negative ‘0Tmeans that the tow point is located behind the center of gravity ofthe tug. Two conditions of the barge, namely with and withoutattached skegs, are denoted as ‘‘barge 2B’’ and ‘‘barge 2Bs’’,respectively, hereafter named the stable and unstable barge. Thetug has twin CPP propellers and twin rudders. Each CPP Propellerhas a diameter of 1.8 m, revolution of 300 rpm and a total enginepower of 1050 kW, used in the simulations for maintaining aconstant speed of 7.0 knots on the tug alone. The rudder design is

Table 1Principal dimensions of tug and barge.

Symbol Tug Barge

Ship length L (m) 40.0 60.96

Breadth B (m) 9.0 21.34

Draft d (m) 2.2 2.74

Volume V (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 (lef

of square shape with both span and chord lengths of 2.0 m. Thesteering 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 captivemodel test in the towing tank (see Fig. 3), which are completelysummarized in Table 2. Based on the stability index C, barges 2Band 2Bs are considered respectively unstable and stable motionsin course-keeping. In addition, added mass coefficients ðm0x,m0y,J0zÞwere calculated using singular distribution method under therigid free-surface condition.

4.3. Wind coefficients

Referring to Eq. (15), the wind coefficients for the tug andbarges were obtained using the linear multiple regression tech-nique, Fujiwara et al. (1998) and are shown in Fig. 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 thebacklash of the controller according to the heading angle andlateral position of the tug. The control law of the tug is given in

t) and barge (right).

Yv �0.3579 �0.4027 �0.4373

Y 0R 0.127 0.0568 0.1355

Y 0vvv �0.2509 �0.2159 �0.7265

Y 0vvr 0.1352 0.4840 0.3263

Y 0vrr 0.000 0.495 �0.2424

Y 0rrr 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

Y 0d �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

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) 135–145 141

the form of

d1 ¼ GPðcT�c1Þ�GD_c1þGYPðYT�Y1Þ�GYD

_Y 1 ð37Þ

The notations of c1 and Y1 are the actual heading angle and lateralmotion, respectively; cT and YT are the targeted heading angle andlateral motion, respectively, (cT ,YT ¼ 0). GP and GD are the propor-tional and derivative gains with respect to the heading angle; GYP

and GYD are the proportional and derivative gains with respect to thelateral motion. Here, the constant controller gains of GP, GD, GYP andGYD are applied, i.e. 9, 10, 10 and 3.5, respectively.

5. Results

Course stability of the towing system at different wind velocitiesand wind angles are numerically simulated using linear and nonlinearapproaches. In these simulations, the authors employed the towingparameters of ‘0T ¼�0:44, ‘0B ¼ 0:5 and different ‘0 from 1:0 to 5:0;whereas ‘0 ¼ 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), thestability conditions of the linearized system are determined by thesigns 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 stabilitydesignating stable (white color) and unstable (black color) zones, asshown in Figs. 5 and 6. In this analysis, the tug motion was assumedto be given as explained earlier in Section 3.

For barge 2B, the course stability diagrams of the ship towingsystem using linear approach vs. the angle of wind are plotted inFig. 5. Based on the diagrams, the increase of Uw=U from 0.0 (nowind) to 4.0 took place in the unstable towing regions although ‘0

was lengthened from 1.0 to 5.0. Using the linear theory fromFitriadhy and Yasukawa (2011), the towing stability was dom-inantly determined by the inherent stability criterion of thetowed ship itself: therefore the increase of ‘0 on the towing ofthe unstable barge (negative course stability index) was

A. Fitriadhy et al. / Ocean Engineering 64 (2013) 135–145142

unnecessary and even prone to degrade the towing stability.However, the stable region then appeared in the range of beam

Fig. 8. Time histories and trajectories of towing for

Fig. 7. Time histories and trajectories of towing fo

Table 3Case of 2B, effect of wind velocity on motion amplitude of ship towing system

with yw ¼ 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

and following winds as a further increase of Uw=U up to 8.0. Thiscould possibly be explained by the wind forces exerted on theexposed windage of barge 2B, which would increase of the yawdamping on her hull and result in significant reduction ofamplitude of the lateral motion. The results agreed well withmodel basin tests conducted by Yasukawa et al. (2007b), wherebarge 2B was towed in uniform and constant wind conditions.

For barge 2Bs, the course stability analysis is plotted in Fig. 6.For the no wind case (Uw=U ¼ 0:0), the towing of barge 2Bs wasabsolutely stable. When Uw=U increased up to 4.0, the towinginstability appeared in the range of 1541rywr1801 at0:4r‘0r5:0. The same tendencies showed that the towingcondition took place in the unstable region in the range of

2B in various wind velocities with yw ¼ 1201.

r 2B in various wind velocities with yw ¼ 01.

A. Fitriadhy et al. / Ocean Engineering 64 (2013) 135–145 143

01rywr261 and 1671rywr1801 at 0:4r‘0r2:65 and0:2r‘0r5:0, respectively, as Uw=U increased from 4.0 to 8.0.Similar to what is noted by Yasukawa et al. (2012), the instabilitytowing regions in the head and following winds occurred mainlydue to the effect of the positive sign for N0Ac2 (the restoringmoment derivative with respect to yaw angle). As discussed inSection 5.2, this towing instability was presented in the form ofincreasing oscillation of the lateral motion for barge 2B (seeFigs. 9 and 10). However, the towing instability regions in thehead wind case with Uw=U ¼ 8:0 vanished by lengthening thetowline (‘042:65). For this reason, the higher resistance of thestable barge (positive stability index) associated with the longertowline led to more stable towing conditions, similar to thefinding by Fitriadhy and Yasukawa (2011). In general, thetowing stability of barge 2Bs was found to be more stable thanbarge 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 atug and towed ship coupled through a towline, has revealed theenormous complexities involving two ships motions associatedwith dynamic tension in a towline. Therefore, nonlinear analysisis required to capture this phenomenon, which would be efficientto obtain a more reliable prediction for the course stability of theship towing system.

Table 4Case of 2B, effect of wind velocity on motion amplitude of ship towing system

with yw ¼ 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 3.7

8 3.2 21.5 35.5 25.0

Fig. 9. Time histories and trajectories of towing fo

As seen, the entire towing performance of barge 2B at yw ¼ 01with the various wind velocities was still directionally unstable asindicated by the sufficient large lateral motions (Y2) and ampli-tude of c2 (see Fig. 7). The results are presented in Table 3. Inhead wind condition, u1 decreased adequately by 28% as Uw=U

increased from 0.0 to 8.0. This occurred since the quadraticfunction of Uw was proportional to the total ship’s resistances.Meanwhile, the yaw motion of barge 2B oscillated more fre-quently by 65%; and the period of Y2 became faster by 41% withrespect to the horizontal trajectories (X2). However, the increaseof head wind velocity in general had a relatively small effect onthe mean magnitude of TC; and the motion performance of barge2B as indicated by the insignificant influence to the amplitude ofc2, Y2, c1 and d1. This can be explained as the behavior of thetowing independently correlates to the inherent course stabilityindex of the barge itself as well-noted in Table 2.

The changing of wind angle from beam to quartering remarkablyaffects the course towing stability as illustrated in Fig. 8. Thesetowing trajectories were captured at yw ¼ 1201. With the subse-quent increase of Uw=U from 0.0 to 8.0, the simulation resultsshowed that the motion of barge 2B veered off to the starboard sidefrom the initial course and then settled then in relatively steadycourse with c2 ¼ 35:51, Table 4. This can be explained (Section 5.1 atParagraph 2) as the sway forces in the towing of barge 2B were moredominant than her yaw moment induced by the wind forces, whichacted alongside the windage. In addition, the mean amplitude of c2

was reduced by 32%, which revealed less fluctuating of TC andimplied a towing to speed up u1 by 23%. At the same time, to

r 2Bs in various wind velocities with yw ¼ 01.

Table 5Case of 2Bs, effect of wind velocity on motion amplitude of ship towing system

with yw ¼ 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 1.8

8 2.2 1.4 35.6 6.0

Fig. 10. Time histories and trajectories of towing for 2Bs in various wind velocities with yw ¼ 1801.

Table 6Case of 2Bs, effect of wind velocity on motion amplitude of ship towing system

with yw ¼ 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

preserve the tug on the desired track inevitably resulted in a largerdeflection of rudder angles d1 by 251 to port. However, thesubsequent increase of Uw=U at yw ¼ 1201 had an insignificantinfluence on the mean magnitude of TC.

For barge 2Bs, the towing characteristics in the various headwind velocities are illustrated in Fig. 9. By increasing Uw=U from0.0, 4.0 to 8.0, the motion of barge 2Bs is prone to be unstable asindicated by the increase of c2 up to 35.61, Table 5. The lateralmotion of barge 2Bs increased almost 5.5 times as Uw=U changedfrom 4.0 to 8.0. Similar to what was explained in Section 4.1, therestoring force of the aerodynamic derivative N0Ac2

acting on barge2Bs led to diverge her yaw motion. This vigorous manoeuvringfrom barge 2Bs resulted in a considerable increase of maximum TC

from 7.3 t to 9.2 t. This condition might pose structural concernsand become even worse when the snatching frequency of thetowline coincides the with motion frequencies of the tug, Varyaniet al. (2007). From the trajectories, the resistance of barge 2Bsseemed to increase as indicated by a decrease in the tow’s speedof u1 by 14% and 39% as Uw=U increased from 0.0 to 4.0 and 0.0 to8.0, respectively. Even though the deflection of d1 increased tostabilize the towing, an unwieldy slewing motion of barge 2Bs atUw=U ¼ 8:0 still occurred, which is absolutely unfavorable fromthe towing stability point of view.

Fig. 10 shows the effect of the following wind conditions onthe course stability of barge 2Bs. In general, the towing char-acteristics of barge 2Bs have been shown to bear qualitativesimilarities to its characteristics in head wind condition. Thismeans that the increase of wind velocity gradually degrades theentire towing performance as indicated by the excessive c2 up to61.51 at Uw=U ¼ 8:0, Table 6. Similar to the head wind case, thediverging motion of barge 2Bs in following wind conditionoccurred due to the aerodynamic derivative value of N0Ac2

, whichwas positive, Yasukawa et al. (2012). It was noted that Y2

increased at almost nine times as Uw=U changed from 4.0 to8.0. Because of the severity of barge 2Bs’s motion, this stronglyaffects the tug’s motions, where the tug experienced rigorousmotions indicated by the violent oscillation of c1, d1 and u1.However, the increase of following wind velocity up toUw=U ¼ 8:0 is also detrimental to the tow by causing a very

impulsive towline tension with the maximum of TC of 18.7 t. Thisamount was almost twice the maximum of TC in the head windcase. The reason for this is that in the following seas the surge ofthe tug increased the snatching of the towline due to rigorousloosening and tightening of the towline with the violent motionof barge 2Bs.

6. Conclusion

The course stability of the ship towing system in uniform andconstant wind conditions was solved by using theoreticalapproaches. The agreement between linear and nonlinear analysiswas obtained. Using the linear analysis, the stability investigationof the ship towing system showed that the course stability of theunstable barge was recovered in the range of beam to followingwinds as the wind velocity increased. In addition, the towingperformance of the stable barge was prone to be unstable in headand following winds as indicated by the large amplitude of herheading’s angle and lateral motion. Employing a longer towlinefor the towing of the unstable barge was ineffective in stabilizingthe towing system; conversely, for the towing of the stable barge,the longer towline led to more stable towing conditions. In thenonlinear analysis, the results revealed that the towing instabilityof the unstable barge 2B at yw ¼ 1201 and Uw=U ¼ 8:0 wasrecovered as indicated through attenuation in her fishtailingmotions. In general, the towing of the stable barge associatedwith the longer towline led to more stable towing conditions thanthe towing of the unstable barge. The increase of following wind

A. Fitriadhy et al. / Ocean Engineering 64 (2013) 135–145 145

velocity resulted in a very impulsive towline tension, which isalmost twice the maximum of TC in the head wind case.

Appendix A

D0 ¼ ‘0½�‘0BðM

0x0 sin c0�M0y0 cos c0Þ�ðI

0y0 cos y0�I0x0 sin y0Þ

�½I0zþ‘0BðI0y0 cos c0�I0x0 sin c0Þ�½�‘

0ðM0x0 sin y0�M0y0 cos y0Þ�

D1 ¼�‘0BðM0x0 sin c0�M0y0 cos c0Þ½‘

0B sin g0ðF

0ð1Þq cos g0þF 0ð2Þq sin g0Þ

�F 0ð3Þq ��½I0zþ‘

0BðI0y0 cos c0�I0x0 sin c0Þ�

�ð2F 0ð1Þq sin g0�F 0ð2Þq cos g0Þ�½cos c0 cos g0ðM0y�M0xÞ

þsin c0 sin g0ðM0y�M0xÞþF 0ð2Þr cos g0�F 0ð1Þr sin g0�

�‘0ðI0y0 cos y0�I0x0 sin y0Þ�‘0B½sin c0 sin g0 cos g0ðM0x�M0yÞ

þsin2 g0 cos c0ðM0y�M0xÞ�½�‘

0ðM0x0 sin y0�M0y0 cos y0Þ�

D2 ¼�‘0BðM0x0 sin c0�M0y0 cos c0Þ½�‘

0BfF0ð1Þ0 ðsin2 g0�cos2 g0Þ

�2F 0ð2Þ0 sin g0 cos g0g��½I0zþ‘

0BðI0y0 cos c0�I0x0 sin c0Þ�

�ðF 0ð1Þ0 cos g0þF 0ð2Þ0 sin g0Þ�½cos c0 cos g0ðM0y�M0xÞ

þsin c0 sin g0ðM0y�M0xÞþF 0ð2Þr cos g0�F 0ð1Þr sin g0�

�½‘0B sin g0ðF0ð1Þq cos g0þF 0ð2Þq sin g0Þ�F 0ð3Þq �

�½‘0Bfsin c0 sin g0 cos g0ðM0x�M0yÞþsin2 g0 cos c0ðM

0y�M0xÞ

þF 0ð1Þr sin g0 cos g0þF 0ð2Þr sin2 g0g�F 0ð3Þr �

�ðF 0ð1Þq sin g0�F 0ð2Þq cos g0Þ

þ½�ðF 0ð1Þ0 þF 0ð1Þc Þcos g0�ðF0ð2Þ0 �F 0ð1Þc Þsin g0�

�½‘0ðI0y0 cos y0�I0x0 sin y0Þ�

�½‘0Bfsin g0ðF0ð1Þc cos g0þF 0ð2Þc sin g0�2F 0ð2Þ0 cos g0Þ

þF 0ð1Þ0 ðsin2 g0�cos2 g0Þg�F 0ð3Þc �½�‘0ðM0x0 sin y0�M0y0 cos y0Þ�

D3 ¼�½cos c0 cos g0ðM0y�M0xÞþsin c0 sin g0ðM

0y�M0xÞþF 0ð2Þr cos g0

�F 0ð1Þr sin g0�½�‘0BfF0ð1Þ0 ðsin2 g0�cos2 g0Þ�2F 0ð2Þ0 sin g0 cos g0g�

�½‘0Bfsin c0 sin g0 cos g0ðM0x�M0yÞþsin2 g0 cos c0ðM

0y�M0xÞ

þF 0ð1Þr sin g0 cos g0þF 0ð2Þr sin2 g0g�F 0ð3Þr �

�ðF 0ð1Þ0 cos g0þF 0ð2Þ0 sin g0Þ

þ½�ðF 0ð1Þ0 þF 0ð1Þc Þcos g0�ðF0ð2Þ0 �F 0ð1Þc Þsin g0�

�½‘0B sin g0ðF0ð1Þq cos g0þF 0ð2Þq sin g0Þ�F 0ð3Þq �

�½‘0Bfsin g0ðF0ð1Þc cos g0þF 0ð2Þc sin g0�2F 0ð2Þ0 cos g0Þ

þF 0ð1Þ0 ðsin2 g0�cos2 g0Þg�F 0ð3Þc �ðF0ð1Þq sin g0�F 0ð2Þq cos g0Þ

D4 ¼ ½�ðF0ð1Þ0 þF 0ð1Þc Þcos g0�ðF

0ð2Þ0 �F 0ð1Þc Þsin g0�

�½�‘0BfF0ð1Þ0 ðsin2 g0�cos2 g0Þ�2F 0ð2Þ0 sin g0 cos g0g�

�½‘0Bfsin g0ðF0ð1Þc cos g0þF 0ð2Þc sin g0�2F 0ð2Þ0 cos g0Þ

þF 0ð1Þ0 ðsin2 g0�cos2 g0Þg�F 0ð3Þc �ðF0ð1Þ0 cos g0þF 0ð2Þ0 sin g0Þ

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