Mathematical model for the slow motion dynamics of turret mooring systems

Scheepshydromechartjca Archiof

Mekelweg 2,2628 CD De!ft

TeL Ot-?da7 F 018.181838

MATHEMATICAL MODEL FOR THE SLOW MOTION DYNAMICS OF TURRET MOORING SYSTEMS

by

0. Garza-Rios

M. Bernitsas

Prepared for the Uthversit of Mkhigan/Industry Consortiumin Offshore Engineering Februaiy 1998

Depaitmnt of Naval Anthitecture

and Marine Engineering College of Engineering The Umversity of Michigan Ann Arbor, Michigan 48 1O92 145

ABSTRACT

A mathematical model for the slow motion horizontal plane nonlinear dynamics of Turret Mooring Systems (TMS) is derived. The equations of motion consistOf the three equations of

motion of the system in surge, sway and yaw, and an independent equation for the rotation of the turret. The model is completed by establishing the fOur kinematic relations between the velocity vector of the system and the absolute velocity vector as defined by an earth-fixed reference frame. Two different nonlinear hydrodynaniic models of the TMS areincluded, Tie first is the traditional, third-order, small drift angle model; and the second is a fifth-order, low

speed, large drift angle model. Mooring line behavior is modeled quasistatically by submerged

catenaries, which include nonlinear drag and touchdown. External excitationconsists of time independent current, steady wind, and second order mean drift forces. Several models for the hydrodynamic excitation, mooring lines, and other external excitation can be incorporated in

the derived mathematical model.

This repOrt is a product of research Sponsored by the University of Michigan/Industry Consortium in Offshore Enginecring. Industry participants include Amoco, Inc.; Conoco,

Inc.; Exxon Production Research; Mobil Research and Development; Shell CompanIes

TABLE OF CONTENTS

ABSTRACT

-. ii ACKNOWLEDGMENTS -. iii

LIST OF FIGURES

v

LISTOF APPENDICES

=

vi

NOMENCLATURE Vii

CHAPTER

I. INTRODUCTON

1

IL GEOMETRY OF TMS

3

2.1. Geometiy of the Turret 5

2 2 Transformation Relations between the Three Coordmate Systems 6

2.3. Transformations of Velocity Components 8

III

DERIVATION OF THE TMS EQUATIONS OF MOTION

11

3.1. Equations of Motion fOr the Vessel 11

3 2 Equations of Motion for the Turret 12

3.3. Forces and Moments Transmitted between Turret and Vessel 13

IV. TMS EQUATIONS OF

17

4.1. Equations of Motion of the System 17

42. Kinematic Relations 19

4.3 Hydrodynamic Models 19

The Abkowitz Model

20

The Takashina Model 21

4.4. External Excitation 22

APPENDICES

26

BIBLIOGRAPHY

=

34

Figure

1, Geometry of TMS - 4

Intermediate reference frames in TMS

- 4

Geometry oftret

5

4 Forces and moments acting on TMS - 12

LIST OF APPENDIXES

Appendix

pJ

MOoring Line Tension Relations 27

a,b,c,d,e

dummy independent variables representing u, v, r and 5 in functions XHD,

HD and NHD

CG center of gravity of TMS CGv center of gravity of vessel CGT center of gravity of turret

DCG distance between CG and CGT

turret diameter

fa attachment point of ith mooring line at DT fa

FN horizontal drag force in the direction normal to the motion of the catenary in the

(X, Y, Z) reference frame

hOrizontal drag force in the direction normal to the motion of the catenary in the (X', Y', Z') reference frame

F

horizontal drag force in the direction parallel to the motion of the catenaly in the (X, Y, Z) reference frame

F

horizontal drag force in the direction parallel to the motion of the catenary in the

(X', Y', Z') reference framç

FXD damping force in the direction of motion of the vessel due to the mooring lines FYD damping force perpendicular to the direction of motion of the vessel due to the

mooring lines

turret moment of inertia about the Z' -axis

Iv vessel moment of inertia aboUt the Z"-axis

moment of inertia of system in yaw (vessel + turret) added moment of inertia in yaw (Takashina model)

horizontal distmce between mooring and turret attachment points m mass of system(vessel + turret)

my mass of vessel

rn7 mass of turret

m added mass in surge (Takashina model) m added mass in sway (Takashina model)

n number of mooring lines

Na derivative of yaw hydrodynamic momentwith repect to a

N21, N17j, NHD N1. N Nv NT

r

r1 rT TH TMS U V (x,y,z) (xm,ym,O) (xT,yT) (X,.Y, Z) (X', Y', Z') (K", Y", Z") Xa Xab Xabc XHD xa 'ab 'abc 'abcde HD Yr

derivative of yaw hydtQdynamic rnomep.witJ repect tO a and b derivative of yaw hydrodynamic moment with repect to a, b and c derivative of yaw hydrodynamic moment with repectto a,

b, c and d

yaw hydrodynamic moment

added moment of inertia in yaw (Abkowitz model)

added moment in yaw due to a unit SWa acceleration(Abkowi model) moment exerted by the vessel on the turret

moment exerted by the turret on the vessel yaw angular velocity of system w.r.t. water relative angular velocity between turret and vessel yaw angular velocity of turret w.r.t. water

horizontal tension in ith mcoring line Turret Mooring System(s)

forward velocity of system with respect to water lateral velocity of system with respect to water

inertial reference frame with origin on sea bed atmooring terminal 1 horizontal plane coordinates of the mooring point at the sea floor

horizontal plane coordinates of the attachment point of the catenary on the turret body fixed reference frame with origin at CG

turret fixed reference frame with origin at CGT vessel fixed reference frame with origin at CGv

derivative of surge hydroclynarnic moment with repect t. a derivative of surge hydrodynamic moment with repect to a and b derivative of surge hydrodypamic moment with repect to a, b and c hydrodynarnic force in sl.irge

added mass iii surge (Abkowitz model)

derivative of sway hydrodynamic moment with repect to a derivative of sway hydrodyuamic moment with repect to a and b derivative of sway hydrodynaznic moment with repect to a, b and c

derivative of sway hydrodyami moment with repect to a,

b, c, d and e

hydrodynamic force in sway

added mass in sway (Abkowitz model)

added mass in sway due to a unft yaw acceleration (Abkowitz model)

current angle w.r.t. (x,y,z) distance between CGv and CG

angle between x-axis and it/i mooring line

fixed angle of it/i catenary aachme!1 w.r.t. turret reference frame

moment friction cOefficient between turret and vessel drift angle

relative yaw angle between turret and vessel absolute yaw angle of turret

I.

INTRODUCTION

Stationkeeping of ships and floating production systems can be achieved by several types of

mooring systems, such as Single Point Mooring (SPM), Two Point Mooring _{(TPM), and}

Spread Mooring Systems (SMS). Mooring systems, in general, are designed to restrain the

slow nonlinear horizontal plane motions of the moored vessel. Several specific types of

mooring systems can be used depending on the projected time of operation, environmental

conditions, and type of Operation.

One such type of mooring system is the Turret Mooring System (TMS), where several catenary mooring legs are attached to a turret Which is essentially part of the moored vessel. TMSs are used widely due to their capacity to allow the vessel to wèathervane_{by changing itS} heading relative to the actual environmental conditions, without requiring_{a power mechanism} for rotation. The turret includes bearings that allow the vessel_{to rotate freely around its anchor} legs. A turret can be mounted externally at the vessel's bow or stern or internally within the

vessel [11]. This ability to change the orientation provides the vessel_{with relatively good}

motion characteristics for production and/or drilling operations, thus making this concept a

flexible and effective solution for a wide range of applications [9].

The slow motion nonlinear dynamics of TMS is not well understood, however, because

most of the available knowledge about TMS depends on limited experimental data and

observations. Numerous mathematical models [4, 5, 6, 10, 14] have been developed as well studying TMS as SPM systems.

The pm-pose of this report is to derive the mathematical equations of motion and the

associated kinematic relations for the horizontal plane, slow motion nonlinear dynamics

_of

TMS. These equations incorporate the turret assuming that it is located inside the vessel,_thus

providing a complete model for the slow motion horizontal plane TMS dynamics.

The geometry of the turret, the coordinate transformations between the vessel, the turret, and the vessel/turret system as well as the transformations between velocity components are studied.

on Chapter H. In Chapter ifi, the equations of motion of the turret and the vessel are developed. These equations are coupled via the forces/moment transmitted between the vessel

and the turret and the damping in the mooring lines. These relations are incorporated in

Chapter IV to provide the rnthematica1 model for the. slow motion horizontal plane TMS dynamics This model. incorporates the hydrodynamic forces and moment acting on the

system, mooring line equations and external excitation. Two different nonlinear hydrodynamic

models of the TMS are included. The first is the traditional, third-order, small _{drift angle}

model; and the second isa fifth-order, low speed, large drift angle, model. Mooring lines are modeled quasistatically, consisting of submerged catenaries and include nonlinear drag and touchdown External excitation consists of steady current, wind and secOnd order mean drift forces.

II. GEOMETRY OF TMS

The TMS considered in this report consists of two interconnected rigid body components: a vessel (nominally a tanker), and a turret, which is located inside the vessel

Figure 1

depicts the geometry of TMS, with tWO principal reference

_frames: (x, y, z) = inertial reference frame with its origin located on the sea bed at mooring terminal 1;

(X, Y, Z) = body flxed reference frame with its origin located at the center of gravity of the system (CG), i.e. vessel and turret combined. In addition, n is the number of mooring _lines;

(x,y,O) are the coordinates of the ith mooring terminal located_{at the sea bed with respect}

to the (x, y, z) frame; £'' is the horizontal projection of the ith mooring line length; y() is_the angle between the x-axis and the ith mooring line, measured counterclockwise; and i,j, is. the

drift angle. The direction of the external excitation (current, wind, waves) is measured with

respect tO the (x, y, z) frame as shown in Figure 1

The TMS model shown in Figure 1 can be obtained by incorporating _{the intermediate}

reference frames. of the vessel/turret system as shown in Figure 2 In Figure 2: (X', Y', Z') is

the turret reference frame with its origin located at the center of gravity of the_{turret (CGT); and}

(X", Y", 2") is the vessel reference frame with its origin located at the center of gravity of the

vessel (CG). Moreover, DCG is the distance between the centers of gravity ofthe system

(CG) and the turret; and is the distance between the centers of gravity of the system and the vessel. These are related as follows:

A

mT,.

'-'CG '-'CG' my

-.

where my is the mass Of the vessel and is the mass of the turret. Further, &y is the

relative yaw angle bçtween the turret and the vessel measured counterclockwise as shown in

Figure 2, and lJ' is the absolute yaw angle of the turret with respect to the (x,y,z) frame,

where

(2.1)

Figure 1 GeOmetry of TMS

2.1. Geometry of the Turret

As shown in Figure 2, the center of gravity of the turret (CGT) is located at a distance DCG

from the center of gravity of the system (CG), and its geometry is described in the (X', Y', Z') reference frame. Figure 3 shows the geometry of. the turret, where: a0 is the angle between the X' -axis and the point of attachment of mooring line 1; 70(j) is the angle between the X' -axis and the point of attachment of mooring line i (fixed to the turret); and j3 is the angle

between two consecutive mooring line attachment points For equally spaced moonng lines,

we have /3 = 360/n.

Figure 3. Geometry of turret

As shoWn in Figures 1 and 3, the mooring lines are attached o the turret. The body-fixed

fairlead coordinates of each of the mooring lines measured With respect to the turret center of

gravity are denoted by

y,(i),

i = 1, ..., n), and are given by:

y,(i)

_fa(')qtmro()

_(2.4)

where ía is the fraction of the turret diameter at which the mooring line

_{is attached}

(0

1); and D7. is the diameter of the turret detern med by the strength considerations

of the vessel's hull and the geometry of the vessel [13].

2.2. Transformation Relations between the Three Coordinate Systems

The reference frames of the system, the turret and the vessel are related through translations

and rotations. Translatioial transformations are straightforward, while the rotational _ones

require some algebra. From Figure 2, the rotational transformation that exists between the

reference frames of the turret and the vessel is given by:

X' cos ji1 siniy

0 X"

_{cos V'i}

sin

ip 0 X'

= -

Sin cOs 0 Y" , = _{sin Vi Cos Vi} 0 Y' (2.5)

z'

0 0 1 Z" Z" 0 0 1

Z'

Since the reference frame of the vessel is aligned to that: of the system as shown in Figure 2,

the same rotational transformation exists between the turret and the system, i.e.

X' cos vi sin I'i

0 X

X cos - sin V'i 0 X'

=

sinjt1

_COWl 0 Y , Y =

sin1

cos 0 Y' (2.6)

0

O1Z

Z 0 0

1Z'

Coordinate transformations (2.5) and (26) serve to recast the various terms pertaining to the

geometry of the turret (including mooring line reactions and damping) in terms of the vessel

and systeiii refereice frames.

The fairlead coordinates of the mooring lines can be measured with respect to CGv in the

body-fixed vessel reference frame as follows:

= DCG

-

'CG + cos y,(0 sin

XT' =

x

cos i

_y()

sin y

= x + Dca COS V + fa Pz. cos(y0(') +

= x+ DCG COSL( + fa Lcos(y(1) +

yrW_{= y+}

_{sin ' +}

y(i)

_cos

y+ DCG sin

iy+f(1) .L[sin

_k/i'cos(y0(0 ₊ + cos k/I Sin (i)

=y + DCG Sill k/I + fa° _{+ (2.12)}

In expressions (2.11) and (2.12), x and y denote the position of the center of gravity of the

systcm (CG), yi is the drift angle of the vessel, and k/IT denotes the rotation angle of the turret with respect to the fixed reference frame.

2

)=sinisin

(2.11)

y;(i)_{= + COS}

= fa0 Lsin(70(i) _(2.8)

The fairlead coordinates relative tO CG,, are not constant, since the mooring lines are not

attached at CG and change as the turret rotates with respect to the vessel.

The fairlead coordinates of the mooring lines measured from the center of gravity of the

system (CG) in the body-fixed reference frame of the systemare:

DCG + laO Lcos(y(O

_(2.9)

y(l)

_{= fa sin(yo(i) + (2.10)}

The values for

x(i)

and (i) change as the turret rotates with respect to the system. The

spatial positions of the attachment points of the mooring lines in the horizontalplane are given in terms of the attachment coordinates _{(XT'), yT'} i 1, ..., n). These are measUred with

2.3. Transformations of Velocity Components:

The horizOntal plane velocities in surge, sway and yaw for the vessel in the (X",Y",Z") reference frame, and those for the turret in the (X', Y', Z') reference frame can be recast in the

(X, Y, Z) reference frame with respect to the center of gravity of the system (CG )i Let _{uv, V}

and rv be the translational velocities of the vessel's center of gravity and its rotational velocity

in the (X", Y", Z") reference frame; u7., VT and rr be the translational velocities of the turret's

center of gravity and its rotational velocity in the (X', Y', Z') reference frame; u, v and r be the translational velocities of the center of gravity of the system and its angular velocity in the

(X, Y, Z) reference frame

The velocity vectors Of the center of gravity of the vessel (7) and the center of gravity of

the turret ( are obtained in terms of the velocity vector of the center of gravity of the system

(V) as follows:

Vv=+XAv,sys,

(2.13)

VTxRT/sys+o.il,

(2.14)

where ãì is the angular velocity of the vessel; ö)1 is the relative angular velocity between the

turret and the vessel; and R is the distance between the vessel (Rv) or turret (Rr) and the

system center of gravity.

The absolute vessel velocity vector of CG%, in terms of the system velocities is

The relative velocity vector of CG in the (X, Y, Z) reference frame is given by:

U"

_U

U i

j

k

v" =

v, ry V

r

+LICG o os

Or

sin 0 (2.15) which yields

u"=uv=ukGrsinhlf,

(2.16) v"=vv=v+/icarCOSII(

(2.17)'

r"

rV =r (2.18)

Similarly, the absolute velocity vector of the turret CGr in _{terms of the velocities of the} system is given by U' U V1 = V

r'

r

to yield

u'=uDrsin/F

v'=v+DrCOSJI

r'=r+r1

I

3 Q 0 0

r

coslv

sfl, 0

0

+0,

r1

where r1 is the relative rotational velocity between the turret and the vessel.

The rotatioiial transformation of the turret velocities to the (X, Y, Z) _{coOrdinate system can be}

expressed as:

(2.25)

(2.26) (2.27) (2.28)

U _{UTCOS IV1VT Sin (2.29)}

V'UTS1fl/F1+VTCO5l/f1

₍₂₃₀₎

= rr.

(2.31)

Equaling expressions (2.26)-(2.28) With (2.29)-(2.31) we fmd

UT UCOSWI +Vsiflu/,l +Dc&sinOVi

j'),

_(2.32)

Uv = U _(2.19)

vv=v+ficGr.,

rv=r,

arid the accelerations of the vessel are

uv=a

(2.20) (2.21) (2.22)

vV+ziCGr

(2.23) rv = r _(2.24)

VT =US1flIV1 vcOs V!i +DcGrcosOI'l V!) (2.33)

rT=r+?1

(2.34)

Expressions (232)-(2.34) can be recast in terms of the (X, Y, Z) reference frame as follows:

UT =UCOS V!i +VSifl V!i +DcGrsinl _(2.35)

VT =US1fl + vcosJJ1 + DCGrcos W1 _(2.36)

TTT+T1.

(2.37)

The accelerations of the turret in the (X, Y, Z) reference frame can be derived by taking time

derivatives of the velcxities as follows:

UT =UcosV!i+VsiflIVi +Dccisin V!i +DCGrr1 cos ur1 sin yi ±vri COS (2.38) VT =UsinV!l+VcosV!1+DCGTcosV!lDCGrrlsinV!l_u7co5I/1I_vrl5jflV!l (2.39)

rT=r+rI

(2.40)

Expressions (2 19)-(2.24) and (2.35)-(2.40) express the velocities and accelerations of the

vessel and the turret in terms of the velocities and accelerations Of the system. These

expressions are used in the following chapter to derive the mathematical, fOrms of the equations

III. DERIVATION OF

THE TMS EQUATIONS OF MOTION

The equations of motion of the system are derived from the equations of motion _{of the}

vessel in surge, sway, and yaw; and the corresponding turret equations. In this Chapter, the equations of motion of the vessel with respect to the (X", Y", Z") and (X', Y', Z') reference

frames, nd the equations of motion of the turret withrespect to the turret reference frame

(X', Y',Z') are shown. These equations will be combined to yield three equations of motion of

the system in the (X, Y,Z) reference frame and a turret rotational equation about the Z'-axis.

3.1. Equations of Motion for the Vessel:

The equations of motion of the vessel in surge, sway and yaw in the vessel reference frame (X", Y", Z") with respect to (x,y, z) are:

where !, is the moment of inertia of the vessel; _{YflTh and ND are the velocity/}

acceleration dependent hydrodynamic forces and moments acting on the vessel in surge, sway

and yaw, respectively; Frge. F

and N are the external forces and moment acting on the vessel; Fffr and Ff4. are the forces in surge and sway exetted by the turret on the vessel, and are assumed to act at the center of gravity of the turret; and N is the moment exerted by

the turret on the vessel.

In general, however, the hydrodynamic forces and moment as well as the external excitation

are measured on the system, rather than on the vessel itself, as shown in Figure 4. Therefore, it is convenient to recast the equations of motion of the vessel with respect to the center of gravity of the system. Using the velocities and accelerations of the system, the_{e uations of} motion in surge, sway and yaw become:

mv1 - myr(v+ CGr) =XHD + Fsurge + (3.4) 11

mv(1v rvvv)=Xb+Fs'rge+F/fr

,

rnV(vVtVuV)=Y/.4DPY±F}'4...,

(3.1) (3.2)

Jvtv=ND+NW+(bcG-11cG)tj'4+N,

(3.3)

mV(+Gi)+mVru=YHD+Fway+F,,

(3.5)

Ivr+mvG(v+ru)=NHD+Nyaw+DCGF}Pr+NT

(3.6)

The various terms on, the right hand side of equations (3.4)-(3.6) have the same definitions as those on the right hand side of equations (3.1)-(3.3). The former are measured, however, with respect to CG.

Figure 4. Forces and moments acting on TMS

Several different models for the hydrodynamic forces/moment and external excitation can be

implemented in the equations of motion (3.4)-(3.6). In the next chapter,. two classical

hydrodynamic models, which take into account the excitation due to current, are implemented in the equations of motion A model for external excitation due to steady wind anda model for second order mean drift forces are presented in Chapter IV as well.

3.2. Equations of Motionfor the Turret:

The eqUations of motion for the turret in surge, sway and yaw are given in the (X', Y', Z') reference frame with respect to (x,y,z) as:

mT(uT-rTvT)=TX-FXD+FXV, (3.7)

rnT(vT+rTuT)=Ty-FYD+Ffl,, (3.8)

where _'T is the turret moment of inertia; _Tk, T and T are the horizontal tension components of the mooring lines in the X', Y', and Z' directionS, respectively; _{FkD, FD and Nj are} the mooring line damping forces and moment; F$, and F are the vessel forces acting on the turret; and NC, is the moment on the turret exerted by the vessel.

Expressions for several models of mooring line tension/damping can be implemented in equations(3.7)-(3.9). Appendices 1 and 2 show expressions for the mooring line tension and

mooring line damping components, respectively, for submerged catenariçs. A complete

derivation of these models is provided in [8].

3.3. Forces and Moments Transmitted between Turret and Vessel

The vessel forces acting on the turret _Ffv

and F, from equations (3.7) and (3.8) can be

recast in the (X, Y, Z) coordinate frame by a simple rotational transformation as follows:

F F,cos,1 -

_Fh,sin1

, (3.10)

FFsiifly1+F,cos(1

. _(3.11)

Since the Z and the Z' axes are parallel to each other, the moment exerted on the turret by the vessel NC,, remains the same with respect to the (X,Y,Z) frame, i.e.

NC, . (3.12)

Figure 5 shows the transmission of the forces between the turret and the vessel, which act on the turret through its center of gravity. In this figure, the (X, Y) axes are shown positioned at the center of gravity Of the turret to illustrate the direction of the inteini forces

_{F,, F,}

F, and

Assuming that all vessel forces and moment acting on the turret are transmitted through its

center of gravity, the forces and moment exerted on the turret by the vessel are of equal

magnitude and opposite direction to those exerted on the vessel by the turret, i.e.

Fxv =F (3.13)

Fw=F}-,., (3.14)

Since the turret is located inside the vessel and mOves with it, we can eliminate two of the six equations of motion. The surge and sway translational velocities of the center of gravity of the turret (CGT), depend on the motIon of the system. Thus, the system has only four degrees of freedom. These correspond to the three equations of motion of the system (i.e. vessel and turret combined) in surge, sway and yaw, and one equation of motion for the turret, that of the

turret rotation.

Figure 5. Forces transmitted to/from the turret through it center of gravity

To eliminate two of the three. turret degrees of freedom, we take the following steps:

First we solve equations (3.7) and (3.8) for Fkv and F,. The resulting expressions are

substituted in the coordinate transformation expressions

(3.10)

and (3.1 1) to obtain

expressions for _{and Fw.}

Next, we change the signs of Fr, and F to derive the expressions forF, and F as dictatedby equations (3.13) and (3.14) These expressions are introduced in equations of

15

To obtain an expression for Fkv expressions (3.10) -and (3.11) are combined by applying transformation (2.6) and eliminating Fw. This yields

Fxy = mT{(uT - TTVT)COS_{lu! - (} + rruT)sm!III} +(T

-

Fft)sin

(TkFD)cosI

. (3.16)

The expressions for T and T

_{re dei ved in AppendiX 1, and those fOr FD and are}

shown in, Appendix 2. By combining the expressions of the velocities derived in Chapter II and the tension and mooring line drag forces in the (X, Y, Z) reference frame, we can recast expression (3.16) as

Fx

where Tsurge and FXD are the mooring line horizontal tension forces and the damping forces measured in the (X,Y, Z) frame in surge. These _{are given in Appendices 1 and 2, respectively.}

Similarly, equations (3.10) and (3.11), can be combined with transformation (2.6) to find the expression for F as follows:

Fyv =mT{(urrTvT)srnIIl

(TFD)sinv1l.

(3.18)

By combining the expressions for the velocities of the tUrret in terms of system velocities

(Chapter II), and the tension and mooring line drag forces in the (X, Y, Z) reference frame, we recast equation (3.18) as follows:

FYv=mT(v+ru'+DcGr)Tay+FyD,

' (3.19)

where

and F

are the mooring line, horizontal tension forces and the damping forces

With respect to the (X, Y, Z) frame in sway, as given in Appendices 1 and 2, respectively.

The exerted moments Nv and N- are a result of the relative rotation that_{exists between th'e} turret and the vessel. These are of equal magnitude, are small compared to other_{terms in the} rotation equations (3.6) and (3.9), and generally are neglected. These incorporate_{the damping} (3.17)

bçtween the vessel and the turret as well as the friction exerted as the turret rotates with respect to the vessel. In this report, these terms are modeled as f011ows:

Nv =NT =- r(r. - r)

, (3.20)

where r is a small positive factor that depends on geometry, relative size of the turret, and

IV. TMS EQUATIONS OF MOTION

The generic equations of motion of the vessel and the turret derived in Chapter ifi can be combined to derive the e flations of motion of the TMS. This is accomplished by eliminating the equations of motion of the turret in the X' d Y' directions, and implementing these in the

surge and sway equations of motion of the vessel in the (X, Y, Z) reference frame. This is

achieved by using relations (3.13)-(3.15) to determine the rclationship between the turret/vessel

internal forces and moment, and then substituting expressions (3.17), (3.19) and (3.20) into the equations. of motion of the vessel. A fourth equation of motion, modeling the rotation of

the turret, is independent and needs to be included in the mathematical model.

4 1. Equations of Motion of the System

After substitution of the terms mentioned above, the cquations of motion for the system (i.e. vessel and turret combined) in surge, sway and yaw can be writtefl as follows

m(ãrv)=XHD

+l'çurgeFxD+Furge, (4.1)

m(+ru)=Y,D+TswFyD+Fay,

(4.2)

NHD +DCG(TSWaY FYD)±NY +'r(rT

where m is the total mass of the system (m =my + mr), and I is the moment of inertia of

the system (I

=IVrnTDCG2). The tension forces Tsurge and are derived in

Appendix 1 as

1curge = (4.4)

i=1

= TH(i)sinJ3(1), (45)

i=1

where TH' is the horizontal tension cornponent of the ith mooring line and

$I)

is the angle given by

7 =fa(0T11(05

(4.6)

(4.10)

The mooring line damping forces FXD and FYD are derived in Appendix 2 as

FXD ={Fpcosf3(') - FN(')sifl/3}

, (4.7)

FYD

={F(0 sinJ3

+FN()cosI3}

, (4.8)

where F1J') and FN' are the damping forces on the mooring lines parallel and normal to the direction of mooring line motion, respectively [8]. These depend on the position and velocity

vectors of the mooring lines, which are attached on the turret and move with it.

The turret rOtational equation is a moment equation about an axis parallel to the Z-axis, and needs not be transferred to the (X, Y, Z) reference frame. That equation is

Jr=TN!w-1(rr-

(4.9)

The expression for T is derived in Appendix 1, and is given by

where

7(i)

- V'i (4.11)

N is derived in Appendix 2 as

N = L

f(0{F'1(0 Sin(I3T(' + F1(') cos($r() (4.12) The definitions for FJ' and FJT' in the rotational equation ofthe turret (4.12) are the same as those for F(') and FN(s) appearing in the equations of motion of the system in surge, sway

19

turret reference frame, while the latter are measured With respect tO the center of gravity of the system in the system's reference frame.

4.2. Kinematic Relations

The kinematics of the system relate the independent telative velocities of the system with

respect to Water (u, v, r, rT) to the absolute velocities with respect to the earth- fixed reference

frame (1, ',

_,,

These are governed by equations (4.13)-(4.16) below:

In equations (4.13) and (4.14), U is the absolute value of the relative velocity of the vessel

with respect to water, and a is the current angle with its direction of _{excitation defined in}

Figure 1.

4.3. Hydrodynamic. Models

Several different hydrodynamic models for the horizontal plane slow motions of a vessel can

be incorporated in equations (3.4)-(3.6). In this report, we use two important hydrodynaniic models: the third-order model by Abkowitz [1], and the fifth-order, large _{cirift maneuvering} model by Takashina [18] Both models take mto account the_{current excitation by introducing}

the relative velocities of the system with respect to water. The hydrodynarnic forces (XHD and HD.) and moment (NHD) can be separated into their acceleration terms (A) and their velocity

terms (V), such that:

XHD = XHA + XHV, (4.17) 'Hb = tIA

liv'

(4.18) NHD=NHA+NHV. (4.19)

X=ucosJyvsin,,+Ucosa

_(4.13)

y=usiny+vcosupr+Usjna,

_(4.14)

= r

_(4.15) uVT=rT (4.16)

(a) The Abkowitz Model

The hydrodynamic velocity dependent forces and moments are expanded in Taylor seris to third-order following [1]. The following expressions are valid:

XHA

Xu,

_(4.20)

XHV = + X,U + Xvrvr+Xy5VÔ + Xrör&+XvruVTU + XvSu +

+!(Xuuu2+

Xv2

+

Xr2 +

Xv2u

+

Xr2u +

X5552 + Xô&82u)

(4.21)

HA = +

HV =

+!(y0u2

+ 1vrr Y,vvrv2 + Y11vU2 + + + ru2)

+ +l'äuuSt?)

+(içv3

+ +

NHA=N,v+Nr,

NHV N, + N0u + Nv + Nrr+ Nvu + Nruru + N58+ N&Su +NVTÔVrS

± NVr2 + N,,rv2 + Nvu2 +Nvö2 + NrrS2 + Nruuru2

!(Naov2

+ N&rSr2 + No&s2) ± Nrrrr3 +

N5o3),

where subscripts "o" and "S" represent propeller and rudder angle effects.

The complete e uatibns of motion using the Abkowitz model in surge, sway, and yaw for the

system become:

(m - X)a - mrv= XHV +

-

Fp(1)]cos 13(')_{+ FN') Sjfl I3'}± urge' (4.26)} i=1

(m - Xi'- r+

= HV

_±{[TH(°

-

Fp(')]sin/3

-

FNCOss} + Fwy

, (4:27)

(4.22)

(4.23)

N1,i) + (I

_{- Nr)t}

=NHV + DCG{[TH() - Fp(')]sm (i)

-

FNcosJ3}

NYr(rTr)

(4.28)

- (4.24)

)

21

Several other mathematical models for the hydrodynamic forces and _{mOments found in the}

literature are based on simplifications and modifications of the Abkowitz model.

(b) The Takashina Model

The slow speed, large drift Takashina mOdel is based on approximating the experimentally measured hydrodynamic forces and moment by _{a Fourier series. Expansion of this model}

yields [18]:

XHA

= mâ,

_(4.29)

XHV ₌XuU + (Xyp. + m.)vr, _(4.30)

YHA=m)v,

_(4.31)

HV =

Y,v+ Yv3 + Y17v

(1sr _{ur+Yurp,.jurIrI+Yi,,rivIrI (4.32)}

NHA =

Jr

_(4.33)

NHV =

Nv ± Nuv + Nv3 + Nuv3 + Nrr+ NMr I

r I +NUVI,.fuv I r I +Nrv2r . (4.34)

Letting

XH XUu + Xvrvr _(4.35)

= yvv + Yvvvv3 + + Y,.ur + ,rIrIU Ir i+iç,1v I

r

I (4.36)

NH=NJ,,

_(4.37)

the equations of motion using the Tàkashinahydrodynarnic_{model in surge, sway, and yaw can} be written as

(m + m)ã (m + m)rv

=_{XH +} {[TH(i)

FPJcosf$')

_{FN(')Sin18(')} + "surge (4.38)}

(m + m)'+(m m)ru

_{H + {[TH)}

-

_Fp()]Sin/3

_-

_{FN(')cQsI3}+ Fsway ' (4.39)}

(I + J)i

= NH + DCG{[TH(i) - Fp

jsin6(')

-

_{FN() cos/3}+}

Assuming that the turret is located inside the vessel, and that no hydrodynamic force or

moment acts on it, the equation of motion corresponding to the turret rotation is not affected by

the hydrodynamic model. Incorporating relations (4.10) and (4.12) into (4.9), the latter

becomes:

ITrT = _{'_fa(){[T11() - FIP)]sin(I3T(')} (i))

-

Fk(1) C0S(/3T(')

(rTr)

(4.41)

Equations (4. 1)-(4.3) are not restricted to the hydrodynarnic models presented above. Other

models, such as those presented in [7, 10] can be implemented as well into the equations of

motiOn.

4.4. External Excitation

As in the cases of the mooring line and vessel hydrodynamic models, equations of motion (4.1)-(4.3) can incorporate several different extetnal excitation models. External excitation consists of time Independent current, steady wind, and second order mean drift forces Each

component has an independent direction of excitation, as shown in Figure 1, The current

forces acting on the system are incorporated in the equations of motion with the maneuvering model.s by using the relative velocity of the system with respect to the water flow. For the

other two sources of excitation we have

urge='xwind+Fxwave (4.42)

1sway = wind + _wave' (4.43)

= 1j wind + 'z wave (4.44)

Wind forces due to tithe independent wind velocity and direction are modeled as [12]:

wind = PaUw2Cxw('2r)4T, (4.45)

y wind

PaUw2v(1r)34L,

(4.46)

where p is the air density; U is the wind speed at a standard height of 10 meters above

water; a,. is the relative angle of attack; Ar and AL are the _{transverse and longitudinal areas of} the vessel projected to the wifld, respectively; L is the length ofthe vessel; C,

(a,.), C(a)

and C _{are the. wind forces and moment coefficients, expressed in Fourier Series as follows:}

C(a,.)

₌ ±

_'k

cos(ka,.) k=I

C(ar)=

ks1n(kar) k=1 Czw(ar) = Ck sin(ka,) k=1

Coefficients _{3i and Ck in equations (4.48)-(4.50) depend on the type of vessel,} location of the superstructure, aid loading conditions.

The mean second order wave drift excitation in the horizontal plane is given _{by [3]:}

, (4.51)

wave=PwgLydSin3(9oVf)

. (4.52)

wave =

pgL2 csin

2(O

- 'v)

, . (4.53)

where P _{is the water density, 9, is the absolute angle of attack; g is the gravitational constant;}

fld Cxj, Cyd, and cj are the drift excitation coefficients in

_{surge, sway, and yaw,} respectively:

Cd

=JS(o.0)[ IXD(Wo) Jdwo.

0

O.5pga

Cyd_{= 5}S(coo)[ FYD(O.0) 0

0.5pga

zd =

JS(wo)[ Fw(W0)

0

0.5pLga

j

23 (4.48) (4.49) (4.50) (4.54) (4.55) (4.56)

If we neglect the following three effects, we can approximate a TMS as a spread mooring

system [2] with all mooring lines attached at the centerof gravity of the turret in the absence of it, i.e. a SPM system provided that the inertia properties of the SPM system match those of the TMS:

The difference in the moorrng line damping forces, which is due to the distance of the

attach±nent points of the lines on the turret from its center of gravity, and to the

additional velocity components of the top of the mooring lines due to turret rotation. The moment on the turret due to mooring line damping.

The friction moment exerted between the turret and the vessel due to the relative rotation that exists between them.

Of the above, effects, the first one is negligible. The second and third affect the system dynamics. It is' worthy pointing out that as the turret mass (and therefore its size) and the

turret/vessel exerted moment go to zero, the turret rotational equation (4.9) is satisfied, until the turret size goes to zero. In the limit, the SPM model is recovered.

APPENDIX 1: MOORING LINE TENSION RELATIONS

The tension in the mooring lines can be recast in the reference frame of the _{system. The}

projection of the line tens on in the horizontal plane in the X' and Y' directions ar given by

T

=

T()

COSCOT

T =TH0sino.T()

where TH(' is the horiontal component of the ith mooring line tension and O)T( is the angle between the X' axis and the ith mooring line, measured counterclockwise (see Figures _{1 and}

2), i.e.

corW = _{V'T (A1.3)}

Substituting from equation (Al.3) into (A 1.1) and (Ai.2), yields

T TJJ(') cosfr(°

-= T1LJ(') sin(y(

-27

(Al.4)

(A1.5)

The tension components in surge and sway measured in the (X, Y, Z) reference frame can be

computed using the coordinate transformation given by equation (2.6) as follows:

1urge = cos

- T sin

-

41r)cosi

-

sinfr:o- _VIT)SiflV(l] =

T'

0(7:i)

=T,sini/1 +Tcosv1

=

T(')[cosfr(')

-

V'T)Sffl L11 + sin =

sin(y('

= TH()sin(7('

Letting 13(i) (t) -; ,j(, we have

Tsurge = , . (A1.8)

= (i) . (A 1.9)

The moment exerted by the mooring lines on the turret, T, can. be calculated as follows:

=

_{T()[x,(') sin(y}

_T)

y,() cos(y(°

-

_{T)] (A 1.10)}

This moment needs not be transferred to the Z-axis. Noticing that:

=

(x(i)

_{- DCG ) cos} + yp(0 Sfl

y,()

_=(DCC

_-

_xp(0)sini1+

i=1

cos

(AL6)

(A1.7)

we can expand (A1.10) as follows:

=

- DCG ) cos + y() sin iy ] sin

='vi)

i=i

i)[(x(i)

_-

_DcG)sinfr Ih(/z+2111_')

P

_{LTH) 'V}

_P.)

29 )

y() cos(yu

Using the expressions for and derived in Chapter II in terms of the body-fixed turret parameters (see equations (2.9) and (2.10)), becomes

=

Lf(i)Tfj(i)[jfl(y(i)

-

i)cosfr0(')₊

iv)

-

cos(y()

-

,y)sin(y0(0 +

'_fa(0TH(i) sin(y(i)

7(i)

_-

_{VT) (A 1.14)}

Letting $0

-

we can reWrite the expression above as

T =

faTH sin(I3r()

(A 1.15)

For submerged catenpy mooring lines, the horizontal component of the tension in each

mooring line can be obtained by solving the following relation [8]

(A1.13)

(Ai.16)

where P is the vertical force per unit catenary length, £ is the horizontally projected length of the suspended part of the submerged catenary, and h is the water depth (or vertical projection of the catenary). Other mooring line models, such as steel cables [16] can be incorporated_as

The mooring line drag forces acting on the turret carl be translated into the (X,Y,Z)

reference frame The drag forces in the X' and Y' directions are given by [8]

{F;(i)

cosflT

-

Sin 13T}

Fb

₌± {F1(i) sin + i=1

where F' and F11'

are the forces parallel and normal to the ith mooring Jine direction

measuied from the turret's center of gravity and f.4 is the angle such that

These forces can be transferred to the surge and sway directions using coordinate

transfOrmation (Z6) as follows:

= FkD cos iy - FD sin (A2.4)

FYD = FD Sin + F COS/F1. (A2.5)

Substituting (A2. 1 )-(AZ3) in equations (A2.4) and (A2.5), we obtain

FXD

j[F;(i)

COS$T

-

SiflI3T(')]CoSJ/Il

s + FJ1'

cos.(i)]sin1

cos(y(° ir)

-

F,,(1) 5(7(i)

-

_(A2.6)

FYD =

j[F;(')

_cos13i

-

sin $rW]sin p'

i=1

30

t[r()sin$T(1)

+

cosflW]cos,1

={Fi) sin(y(1)

) +

p(i) cos(y)

-Letting /3(1)₌

-

y, equations (A2.6) and (A2.7) can be recast as

FXD ={Fj&)cosf3(i) =Fk()sinØ')}

FYD₌

{i)

sin/3(1)+ 13(i)

}

31

We notice that and F1,1' are the drag forces on mooring line i measured in the turret coordinate system (X', Y', Z'). These can be transferred to the system reference frame hi the.

forms F,P) and _{by introducing the velocities of the mooring lines With respect to CC.}

The expressions for the forces and FN(1) are given by [8, 15]:

F')

= -

PCD Deff'hYp0 kAkA'

FN' P'.-D' "-'eff' 'si-' (i)ri (i')L sing,(i)

cOsO/') JYA(1)IYA(1)

where p is the water density and h is the water depth. For each mooring line, CD is the drag coefficient; D is its effective diameter; y is the energy dissipation functiOn of the mooring line; O is the angle between the upper end point of the mooring line and the horizontal _plane; XA is the velocity of the mooring line in the direction parallel to its motion; and _{'A is the}

velocity of the mooring line in the direction perpendicular to its motion [8].

The horizontal velocity components of the mooring line _A and YA are given in terms of the

system velocities as [8]:

XA = (u 9(1)r1 ) cos/3' + (v + DCGr + £(')ri)sin /3(1) _(A2..12)

YA =_(u _{9(1)rj )}sin$ + (v + DCGr + p0,l ) CS/30) (42.13) (A2.7) (42.8) (A2.9) (A2. 10) (A2.1l)

where r is the relative rotational velocity betWeen the turret and the vessel (r1 = rT = r), and

.(i) and

are the fairlead coordinates of the ith mooring line with respect to the center of gravity of the turret measured in the (X, Y, Z) coordinate frame:

(i)

= f(i) .Lc(y(i)

(A2. 1 4)

9(i)

= ..L sin(y (i) (A2. 15)

This yields:

XA(') = ucos13(i) + (v + DCGr) sn/3(1)

± (rT - r)-fa' sin

(y()

70(j) 1tfr) , (A2. 16)

YA

=usinf3±(v+DCGr)cos/3° +(rT

Using the expressions for XA and YA (A2. 12) and (A2.13), we can write expressions for

the forces FXD and FYD as

FXD

={FwCOS$i)

-

FN(')sin/3}

(A2.18)

FYD = sin (:0+FN0)cosfiO}. (A2. 19)

For the tUrret rotatk)nal equation, the mooring line damping is given by [8]

N

x,(i)[F,0

sin I3T + Fk(') cos flT]

_y,(i)[F(i)

_CO5/3T Ffr) sin (A2.20)

i=1

This expression can be rewritten in terms of fixed-body turret parameters by using expressions (A2.3) and (A2.4) to yield:

Prf(i)[F'(i)

sini3r' +

_{CoSf3T']COS7o}

at

2

33

(i)[F;(i)_{COS$T(' -}

p(I)

Siflf3i')]S1fl7o(''. (A2.21)

Expression (A2.21) above can be simplified to yield

9j(p(i)

(i))_{+ Fj' cos(J3r1)} y0(i))} _(A2.22)

where

=

!pCDDff(i)hy(1)IfAj.*'(t)

(A2.23)

Fk' = PCD(

j) 'IZ (A224)

In the equaUons above, the definitions for ± and are the same as those for A and $'A.

The former are measured, however, With respect to the turret refereflce frame, Followmgthe

defmitions in [8], these are given by:

=(u

-

_y,(

j) rT)cosJ3_T i)+ (v + DCGr + x,c0rr)siri (A225)

=_(u

-

0 rT)sinI3T0

+(v+

DCGr+4i)rT)cos/3TW (A2.26) where and

y()

are the fairlead coordinates of mooring line i measured from the center

of gravity of the turret. These are given by equations (2.3) and (2.4) rcspectiyely. Expanding

the terms of the equations above, and rearranging, we find:

= UCOS$T' +(v + DCGr)sin/37.W + rT..qr_fa(i) sin(IJr(')

-

(A2.27)

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