Slow motion dynamics of turret mooring and its approximation as single point mooring

CI

Journal of Applied Ocean Research, in review

SLOW MOTION DYNAMICS OF TURR] T MOORING AND ITS

APPROXIMATION AS SINGLE POINT MOORING

ncMic UVRSTLT

I.aboratoriurn voor Scheepehydrom6Chaflc Archiet Eekehveg 2.2628 CD Detft )d.Otg-78Th- ft Oi..18W3 by Dr. Luis 0. Garza-Rios Research Fellow Dr. Michael M. Bernitas Professor

Submitted for publication in Applied Ocean Research

SLOW MOTION DYNAMICS OF TURRET MOORING AND ITS

APPROXIMATION AS SINGLE POINT MOORING

ABSTRACT

The mathematical; thi$del for the nonlinear dynamics of slow motions in the horizontal plane of Turret Mooring Sysienis (TMS) is presented. It is shown that. the TMS model differs from the

classical Single Point Mooring (SPM) model, which is used generallyto study the TMS dynamics. The friction moment exerted between the turret and the vessel, and the mooring line damping moment resulting from the turret rotation are the sources of difference between TMS and SPM.

Qualitative differences in the dynarnicaj behavior between these two mooring systems are identified using nonlinear dynamics and bifurcation th&ny. In two and three!dimensional_{parametric design} spaces, the dependence of stability boundaries and singularities of bifurcations for given TMS and SPM configurations is revealed. It is shown tb.t the static loss of stability of a TMS can be located approximately by the SPM static bifurcation. The dynamic loss of stability of TMS and the

associated morphogenesis may be affected strongly by the friction/damping

_{moment, and oa}

lesser extent, by the mooring line damping. Nonlinear time simulations _{are used to assess the}

effects of these properties on TMS and compared to SPM systems. The TMS mathematical_model consists of the nonlinear horizontal plane fifth-order, large drift, low speed maneuvering

equations. Mooring line behavior is modeled quasistatically by submerged catenaries, including nonlinear drag and touchdown. External excitation consists of time independent current, steady

wind, and second order mean drift forces.

1: NOMENCLATURE

CV, CG= centers of gravity of system nd turreJ1

D= distance between CG and CGT Dr turret thameter

f0=attachmentpointofithtnooringlineat Ifa(O

'T=turret moment of inertia

I7 =momenl of inertia of system (vessel -I- turret)' J= added moment ofineatia in yaw

1

L= vessel length

m = mass of system (vessel + turret) = added mass in surge

addedmass in sway

a = number of mooring line

N= yaw hydrodynamic moment

Nv= moment exerted by the vessel On the turret N.= moment exerted by the turret on the vessel SPM = Single Point Mooring

TH'= horizontal tension in ith mooring line TMS Turret Mooring System

(x, y)= inertial reference frame on sea bed with origin at mooring terminal 1 (X, Y,Z)= body fixed reference frame with origin at CG

XH= hydrodynamic force in surge = hydrodyiiamc force in sway a= current angle wit. (x,y)

()=

_{angle between x-axis and ith mooring line}

fixed angle of ith catenary attachment w.ri turret reference frame

r= rn nt friction coefficient between turret and vessel

0= angle of external excitation w.r.t (xy) i=dri1t angle

'VT= absolute yaw angle of turret

INTRODUCTION

f r

Moàrirg Systems (TMS) are used widely in stationkeeping of ships and floating roductiOiP

stem&.9'lñ general, TMS consists of a vessel and a turret, to which several rnooringlanchoring°' 1egs aie"aUached (see Figure 1). Bearings constitute the interface between turret and vessef

eâHowli the latter to rotate around the anchored tuffit in.iesponse to the changes in enyimnment

cöncIitiöüS. Turrets can be mounted externally ahead of the bow or aft of the stern, or internailr'

withi the vessel, and may also be disconnectable or permanent depending on the type of mooring,

projected time of operation, and the çnvironmental conditions. The weathervaning capability of TMS, along with its functions of production, storage and offloading in one facility, make this

concepta flexible and effective solution for a wide range of applications'5.

co

Presently, several TMS have been designed for and deployed in relatively deep waters under larsh environmental conditions'6.'8. TMS design methods, however, depend_{mostly on experience and} limited data, and therefore TMS design can be studied only on a case by case basis. Due to the complexity of these types of systems, presently TMS_{are modeled mathematically as Single Point} Mooring (SPM) systems5.6.7'7.21. Limited experimental and field data for TMS in shallow and

intermediate water depths'7.26, show that SPM models have

_{proven useful, especially in} determining the turret locations for which the system is statically stable.

The main purposes of this work are: (i) to provide a complete mathematical model for the

horizontal plane slow motion dynamics of TMS; (ii) to develop a methodology_{to understand TMS} dynamical behavior as function of design parameters; (ili) to determine the validity of modeling

TMS as SPM systems based on their slow motion nonlinear dynamics; and (iv) _{to assess the}

impact of approximating TMS as SPM systems in design. The slow motion nonlinear dynamics of SPM systems have been extensively studied by Bernitsas and Papoulias3 for_{a single line, an by}

Bernitsas and Garza-Rios' for multiple lines. The latter is used in this

_{paper for direct} comparisons to the TMS model developed in Section 2.

The TMS nonlinear mathematical model, including vessel hydrodynamics, mooring lines and

external excitation is presented in Section 2. The concepts of state space representation ind

equilibria of the TMS model are covered in Section 3. These concepts are used to establish the TMS eq!lations to determine its stability properties around an equilibrium position. Section 4 introduces the concepts of nonlinear stability analysis and bifurcations of equilibria for TMS, and are used to develop a design methodology for TMS by performing bifurcation sequences aroind each equilibrium position. Such a methodology makes it possible to assess the dynamics of the

system by studying qualitative changes in behavior as function of design parameters. These include the number, length, fairlead coordinates, orientation, pretension and material of the

mooring lines; water depth; environmental condition, etc. flie design methodology is applied in Section 5 to compare SFM and TMSdynanncs. Design cha that separate regions of qualitatively

different dynamics in the parametricdesign space are onstnjcted in the two and three-dimensioal

spaces to evaluate the dynrnical behavior of the syms. fle location of the tuñet and direction of external excitation are used to detennine the effeCs of theSe parariietezs on the system dynamis.

A third parameter, tl friction moment exe ted between the turret and the vessel, which arises frém

the relative rotation between them, is studied as welL Nonlinear time simulations show how the friction momefl1 along with the mooring line damping, affect ThIS behavior. Finally, a discussion on the results obtained is presented along with some closing remarks.

2 TMS MATHEMATICAL MODEL

The slOw motion dynamics of TMS in the horizontal plane (surge, sway and yaw) is modeled in terms of the vessel/tuiret equations of motion without memory, mooring linemodel, and external excitation. Figure 1 depicts the geometry of TMS, with two principal reference frames: (x,y) = inertial reference frame With its origin located on the sea bed at mooring terminal 1; (X,Y, Z) = body fixed reference frame with its origin located at the center of gravity of the system CG, i.e. of the vessel and turret combined. In addition, n is the number of mooring lines; (x,y) are the horizontal plane coor4inates of the ith mooring terminal with respect to the

(x,y) framc; t'

is the honzOntal projection of the Ith mooring line; y@is the angle between the

x-axis and the ith mooring line, measured counterclockwise; and iy is the drift angle. The

direction of the external excitation (current, wind, waves) is measured with respect to the (x,y) frame as shown in Figure 1.

Figure 2 shows the intermediate reference frames of the vessel/turret system from which the TMS mathematical model is derived: (X', Y', Z') Sthe turret reference frathe with its origin located at the

center of gravity of the turret (CGr); and (X", Y", Z") is the vessel reference frame with its origin located at the center of gravity of the vessel (CGv). Moreover, DCG is the distance between the centers of gravity of the system (CG) and the turret; and 4 is the distance between the centers of

gravity of the system and the vesseL These are re1ateda.follows:

A rn7,.

"CG '-'CG' my

wheremvisthemassofthevesselandmTisthemasSOfthetUrret. Inaddition,'1istherelative

yaw anglebetweffie turret and the vessel measured countetclockwise as shown in Figure 2, and

is the absoluteyaw angle of the turret with respect to tbe (x,y) frane, where

r

VrV(+4hi.

de (2)

,2.1 TMS equations of motion

(1)

(in + m1)ü - (m + m)vr_{= XH(u,v,r) +} {[i

-

Ji)Jcosp(&)

+ F!)sinp(0}

_'surge'

(m + m)i +(m + m)ur

Y11(u,v,r) +

{[ii)

-

i]sinp(0 F/)cos$}+

(J + J )r

= NH (u, v, r) + DCG

-

F} sin

_fl

-

_{+Nya., + Nr,}

where, (u, v, r) and (ü, v, r) are the relative velocities and accelerations of the center of gravity of the system with respect to water in surge, sway,and yaw, respectively; m is the total mass of the

system (m=my + mT); 1 is the moment of mertia of the system about the Z-axis;

_{m, rn, and}

J,2 are the added mass and moment terms; Fsu,e,

and N

are external forces and moment acting on the vessel, such as wind and second order drift forcesL NT is the moment

exerted on the vessel by the turret; and XH _H and NH are the velocity dependent hydrodynainic

forces and moment expressed in terms of the fifth-order, large drift, slow motion derivatives23

following the non-dimensionalization by Takashin2Z4

XH

=Xo+Xuu+!Xuuu2+!Xuuuu3+Xvrvr,

(6)

=

Yv+ Yv3 +

+ Yur+ Y,1,1urIi1+ Y*1. (7)

NH Nv + Nuv + Nv3 + Nui3 + Nrr + N,j,jrJrJ + N,ju4ii + Nv2r.

(8)

In eqn (6), the first four terms represent the third order approximation of the resistance of the vessel. R, such that1:

=

_(x0

+

xu

+xu3).

i-iii

ed

9)

tat -ti

In addition, the terms inside the Summation in eqns (3)-(5) apply to each of the mooring lines

(i

= 1, ..., ii), and are described as follows: TH is the horizontal tension component of the Wi

catenary; F, and FN are the drag forces on the Wi catenaiy in th

directipns parallel and perpendicular to the motion of the mooring line12. They depend on the position of the turret with respect to the mooring points and the velocities of the mooring lines; and

_fi

is the angle between the X-axis and the ith mooring line, measured counterclockwise, _{$ =V}

A fourth equation of motion describes the relative rotation of the turret with respect to the

carth-fixed reference frame, and consists of a moment equation about an axis parallel to the Z-aAis (i.e.

Z'-axis). Such equation need not be transferred to the (X,Y,Z) reference frame. In the

(X', Y',Z') frame it is given by":

JTrT =

_fa{[7P

-

i)}sin($(I)

- F'

cos(p'(i) = i))} Nv, (10)

where tT is the rotational acceleration of the turret; Jr is the turret moment of inertia about the Z'-axis; B1. is the turret thameter Nv is the moment exerted on the turret by the vessel. The terms

'inside the summation in eqn (10) are defmed as follows: ía is the fraction of the turret diameter at which the ith catenary is attached (0 1);

F7 and F

are the dtag forces on the ith catenary in the directions parallel and perpendicular to the mooring line motion, respectively'2;

fi'

is the angle between the X'-axis and the mooring line measured counterclockwise

(fi'

=7

and y is the angle at which the ith catenaiy is attached to the turret with respect to the X'-axis

(fixed to the turret). In equations of motion (3}-(5) and (10), it has been assumed that the turret is

tternal to the vessel and that no hydrodynaniic or external forces or moments act directly on the turret.

The exerted moments between the turret and the vessel are a result of the relative rotation between

them. These moments are generally negligible. They incorporate the damping between the turret and the vessel, as well as the friction exerted as the turret rotates with respect to the vessel. The

excited moments axe modeled as follows11:

NV=NT=r(rTr),

(11)

where r is a smIl fridin fadthat depends on the geometry, móom g line damping, relative

size of the turret, separation betéën the turret and the vessel, and external excitation; and rT ' the rotatiorial velocity of th'strret

Eqns of motion (5) and (10) show the coupling that exists between the turret and the system via the transmitted friction moment (11). This coupling becomes stronger with increasing friction between

the turretand the vessel. in the absence of such a moment, eqns of motion (3)-(5) and (10) show that the TMS mathematical model resembles that of SPM systems in the absence of the turret1'3, provided that both types of systems have the same inertial propertiCs. In such a case, one major difference reminc between eqns of motion (3)-(5) and their SPM counterparts: the mooring lines

are all attached on the turret and move with it. This results in differences in mooring line damping

forces and moments as the turret rotates with respect to the vessel.

_{The diffrence in the}

position/velocity dependent mooring line forces is negligible; the damping _{moment created due to} these forces, however, differs considerably between SPM and TMS. This is becausein the SPM

case, the damping moment due to the mooring lines is negligible as the vessel rotates because all mooring lines are attached at the centerline of the vessel. In TMS, however, the damping _moment is non-zero, as the mooring lines are attached off the center of gravity of the turret. As the turret rotates, the mooring line damping moment becon _non-zero.

The moment equation for the turret (10) shows that, even in the absence of a moment exerted by the vessel, the turret itself has a non-zero rotational acceleration. Exception to this presents the

case in which all the mooring lines are attached at the center of the turret; in that case a SPM model

would be appropriate. In practice, however, this scenario is not possible, because a number of risers and other production/recoveiy devices pass through the center of the turret. In conclusion, the oscillatory motion of the turret affects the dynamical behavior of the system as the mooring lines attached to the turret oscillate with it, transmitting turret motion to the system. It is worth

pointing out that, in the limit, as the mass and diameter of the turret go to zero in equation (10), the

SPM model is recovered.

The kinematics of the TMS are governed by eqns (l2)-(15)1:

I =ucosyl' vsinyi+Ucosa,

$'=usiny+vcosij,+Usina,

f1 = = TT,

Where U = is the absOlute value of the relative velocity of the vessel with respecto water, and

a is the current angle as shon in Figure 1. The first three kinematic conditions pp1y a1soto

SPM systems"23. In SPM systems, however, the last: kinematic condition does not exist due to the absence of the turret

2.2

MoorIng line model

The mooring lines of the system are modeled quasistatically as submerged catenaiy chains with nonlinear drag and touchdowfl The total tension T in each of the catenaries is given by'2

WhCPa isthcairdensity; (4, stlewinspeed atastandardheightoflometersabovewater,

ar Is the relative angle of attack, 4 and AL arethe transverse and longitudinal areas of the vessel

projected to the wind,1 respectively; L is the length of the vessel; C,,N(aT), C(a,) and C are

the wind forces and moment coefficients, épresed in Fourier Series as follows:

C(ar)=

o +

1k cos(k4),

(23)

C(i )=

lSin(ka),

(24)

k1

urge

_Fj,.4+Fzve,

(17)

F5, =

_F;

,yj' +

_F;wave' (18)

Nywjp.i.+Fwave.

(19)

Wind forces du to time independent wind velocity and direction are modeled as19:

,vind = PaUw2Cxw(ar)4, (20) F = :pcUw2Cyw(ar)AL,

= paUw2C(ar)ML,

(21) (22) 4TH2 + TV2 = TH

cosh(..).

(16)

where T is the vertical tension in the mooring line, P is the weight of the line per unit length, and

1 is tie horizontally projected length of the suspended part of the lin&2.

2.3

External excitation

The model for external excitation in surge, sway and yaw in this work consists of time independent current, steady wind, and mean second order drift forces. Their directions of excitation are defined

in Figure 1. The current forces are modeled in equations of motion (3)-(5) by introducing the

relative velocities of the system with respect to water. For the other twO sources Of excitation, we have

C,(ar)=

sin(kar). k=1

Coefficients and _{in eqns (23)-(25) depend on the type of vessel, location of the} superstructure, and loading conditions.

The mean second order drift excitation in the horizontal plane is given by4:

''xwave =pgW1

cos3 (9

_-u'),

₍₂₆₎

1ywwe

p),

(27)

=

pgL2 Cj

sin 2(9 - (28)

where P is the water density, 8 is the absolute angle of attack g is the gravitational constant; and

C, and C,

are the drift excitation coefficients in surge, sway, and yaw, respectively:

9

(25)

In the expressions above, the quantities in square brackets are the drift excitation operators; a is the wave amplitude, and o, is the wave frequency Thetwo-parameterBretschnider spectrum i.s

Used to express S(io,) in terms of the significant Wave height R and significant wave period(

as:

S(w0)=A _5e_80304, ₍₃₂₎

where A and B arenonlinearfunctionsofH113 id 7j134.

rd =

5S(0

xD(aO 11w0, (29) 0

0.5pga J

Cy1j=JS(01){ 0 0' (30) FYD(0.)o)}f O.5pga2

C =

JS(wo)[ 0 Fw(W0) 1*00. (31)

0.5pLga2

3 STATE SPACE REPRESENTATION AND EQUILIBRIA OF TMS

The nonlinear TMS mathematical model presented in Section 2is autonomous, and consists of

equations of motion (3)-(5) and (10), kinematic relations (12)-(15), and auxiliary equations for the

system geometry, the mooring lines, and the external excitations. These relations are grouped

below into a compact form that allows us to analyze the TMS dynamics by representing the

strongly nonlinear flow as a vector field.

3.1

State space representatIon

The TMS mathematical model can be recast as a set. of eight firstOrder nonlinear coupled differential equations by eliminating all but eight variables (the equivalent SPM systemyields six first-order differential equations' 2). FOr general TMS, where the number of mooring lines can be unspecified, it is convenient to represent the state space flow in terms of the position and velocity

vectors of the system. Thus, by selecting the following statevariables: (x1= u, x2= v, x3= r,

X4=J1, X5X, X6=y, X7/', x8=V(r),

the system nonlinear model can be wntten in Cauchy

standatd.form14 as:

=

(m+mx)[XH 1,x2,x3)( + m)x2x3

-

F0]cosØ') + FjPsinfl(0}

4,

(33)

(m+y)[YH1x2x3)(m+mx

x1x3

_+±{[7'

]sinp(°

i!)cosp(0}

+ (34)

X3

-

J)[NH(x1x2x3)+ DcGY{[2/

-

F0]sinp(O

-

Fcosp(0}

X4 =

fa{[7P -

F'(°]sin($'°

-x

=xeosx7 x2sinx7

+Ucosa,

6 =x1sinx7 +x2cosx7 +Usifla,

= '3,

18X4.

11

In eqns (33)-(40), the quantities inside the summation_{signs T11, $ and fi' are functions of the}

state position variables (x5, x6, x7,x8);

and F,

_{FN and F are functions of all state}

variables. The expressIons for the external excitation

_{Fue. F}

_{and vary with the} orientation of the vessel and, thus, are explicit functions of the vessel drift angle(x7). State space evolution eqns (33X40) are denoted by the vector field:

i=f(x),

fEC1, f:98_,.98,

₄₁₎

where

98

Is the eightdimensional Eucidean

_{space and C1 is the class of continuously}

differentiable functions'4. For the state space representation selected above, all variables, vector field

f,

_{and the consponding JacObian Df(x) are continuous functions of}

_x.

32 EquilibrIa of TMS

The equilibria of the nonlinear TMS model can be computed as intersections of null dines23. All system equilibria, also called stationary solutions or singular points, can be found by setting the

time derivatives of the state vector (41) equal to zero, i.e.

f(i)=O,

₍₄₂₎

where the overbar on the state vector x represents equilibrium. All TMS exhibit _{a principal}

equilibrium position (hereafter denoted as eqtlilibriuxn A") in which the vessel weathervanes. Other equilibria may exist as a result of static bifurcations, dcpending on the configuration of the

TMS and external excitation.

Eqns (3-(40) show that, at equilibrium:

11

=Ucii7a),

₍₄₃₎

X2 =Usin(17a),

(46)

1

The above conditions to the flow indicate that the rotational velocities at equilibrium are zero, and

the linear velocities at equilibrium are exclusively functions of the relative angle of attack between

yields the following set of four nonlinear algebraic equations that must be solved simultaneously

fOr x5' x6, 1 and x8:

o

=UXcoS(i - a)U2Xuucos2(

a)

-

i)}cos(i) +P)sin°}

+1'surge(7) (47)

o

=U,sin(i7 - a)+ U3Ysin3(7 - a)+ U5Y

sin5(17 a)

-

i)]sin(i)

_T$)

_F,(i7),

₍₄₈₎

o = tJN sm(17 a) U2 NUV cos(17 - a)sn(7 - a) + U3NV,V sin3(7 - a)

-

U4 NUWV cos(7

- a)

-

(OJsin (j) F$1)COsfl(t)

Q= (:){[7(i)

P(i)]sjn((i)

-

_riP)

(i) co(1)

i=1

Eqns (47)-(50) can be solved numerically using a Newton-Rapson22 or any other suitable

algorithm to fmd all equilibria of particularTMS configuration. SPMZ and weakly coupled TMS

exhibit three solutions to the equilibrium equations.

The concepts of state space representation in Section 3J and equilibria of TMS in 3.2 are used in the next section toy the stability properties of the TMS around an equilibrium position.

4

DYNAMIC ANALYSIS OF TMS

.:

?ifl(i

The slow motion dynamics of TMS can be analyzed usingnlihear stability analysis and

bifurcation theory around an equilibrium position as shown in the following subsections To

facilitate discussion, and for the purposes of comparison to SPMiynamics, the TMS dynamics in

this work focuses around three equilibria: its principal equili rium position A, which is the preferred orientation; and the two alternate equilibria that appear aS a result of static bifUrcation of the principal equilibrium A. Dynamic analysis of SPM. was thoroughly studied in reference3.

Equilibria of nonlinear systems are special degenerate trajectories. Local analysis near an

equilibrium solution reveals the behavior of trajectories in its vicinity. Local analysis can be

performed by studying the linear system

ig=[A]q,

qe98,

[AlE98X8 (51)

where itt) = x(t)

is the deviation from equilibrium, and [A] is the Jacobian matrix of

f

evaluated at the equilibrium condition. If all eigenvalues of [A] have negative real parts,

equilibrium position j is asymptotically stable, and all trajectories initiated _{near i will converge} toward it in forward time. If at least one eigenvalue of [A] has _{a positive real part, 2 is unstale,}

and all trajectories initiated near such an equilibrium will deviate from it in forward time'4. Local stability analysis can be performed near each and every equilibrium position of a particular system configuration to determine its global behavior.

4.2

Bifurcations of equilibria in TMS

The principle of nonlinear stability analysis applies to a specific TMS configuration unler prescribed external excitation. It can be used to determine the dynamical behavior of the system in forward time around an equilibrium position without the need to perform euensive nonlinear time

simulations. _{This principle is used to develop a TMS design methodology by performing}

bifurcation analysis as one or more system parameters (or design variables) change. l'his is achieved by studying the bifurcation sequences around an equilibrium position to find the qualitative behavior of the TMS as such system parameters are varied.

In TMS design, several design variables must be considered. These include the vessel

hydrodynamicproperties; the number, length, pretension, orientation and material of the moormg

lines; the size and poSition of the turret with respect to the vessel the fractiofl of the turret diameter

where the mooring lies are attached; the gap between the turret and the vessel, etc. In addition,

other parameters that must be taken into account, such as variation in water depth, and the

magnitude añddirectn of the external excitatiofl (current, Wind, waves), make the design process evi more complicated.

The dytuimic behavior of TMS around an equilibrium position may change significantly with systematic variations of the system parameters In order to petform bifurcation analysis on the

TMS, evolution eqns (41) are written as

i=f(x,p),

XE98,

PE9NP,

(52) where p is the design parameter vector, and N is the number of parameters in the system. For

the purpose of numerical applications in this paper, we study the behavior of the system in terms of

the position of the turret under different directions of external excitation. Thus, the principal design variables considered in the numerical applications are limited to the position of the turret with respect to the system (DCG), and the direction of external excitation. For the numerical

applications it is assumed that current, waves and wind are collinear, i.e. act in the same direction,

hereafter denoted as _0. The value of DCG for the corresponding SPM model is the distance of attachment of the mooring lines from the center of gravity of the system. A third parameter, the

damping/friction factor between the turret and the vessel, r, is considered to illustrate the dynamical differences between TMS and SPM systems.

Thus, a total of three design variables are used in the numerical applications. That is, N= 3 and

,i=[ DCG, 0! r 1T, while the other design variables are fixed. The behavior of the solutions to the dynamical system (described by eqn (52)) in the Ne-dimensional space is graphically illustrated with stability or design charts. The set of lines shown in a stability chart is called a "catastrophe set," which constitutes the boundaries between regions of qualitatively different dynamics'a. These boundaries, also known as "bifurcation" or "stability" boundaries, are a result of loss of stability in the system. Elementary catasttophes such as fold or cUsp'4'23, which correspond to

the bifurcatiOns mentioned above, play an important role in the nonlinear dynamical behavior of the

system.

5 TMS NUMERICAL APPLICATIONS AND COMPARISONS WITH SPM

As mentioned previously, the concepts of nonlinear dynamics and bifurcation theory serve to constnict design charts (or catastrophe sets) in 2 or 3-dimensional parametricdesign spaces. These

sets can be used to select appropriate values for the mooring system ameers in TMS (and SPM) design without resorting to trial and error.

The geometric properties of the system used in the numerical applications are shown in Table 1.

These properties correspond to a TMS tanker13 for which the complete set of slow motion

derivatives are available. Table 2 shows the values of the TMS design paranters that remain fixed in this analysis. The turret diameteriS not taken into account as a design variable, as its size is set

generally by the strength of the ship hull and the beam limitations of the vessel20. These parameters are used in the numencal applications for purposes of demonstration only.

Figure 3 shows a set of catastrophe sets about the principal system equilibrium, A, in the (DCGIL, Ø)planesuchthat

0.35

_DcG/L0.5,

_{0° 0}

600

for both the SPM and the TMS. For the TMS

_t=

_{0.001 (very high) and 0.000001 (low). To}

establish an appropriate basis of comparison between SPM and TMS, both systems have the same

vessel hydrodynamics, mooring line model, center of gravity, added and stnictural mass, added

and structural second moment of inertia.

In the catastrophe sets of Figure 3, there exist three regions (numbered I to ifi) of qualitatively

different dynamical behavior.

:

-Tble

Design parameters of the system

Mooring Line Propeities

number of mooring lines =4

average breakigtrength 5159 KN/Line

length of moonng lines =2625 rn

orientation of mooring line I =

0,

mooring line spacing =90

pretension = 1444 KN/line fiaction of attachment=O.75 not applicable to 5PM

Environmental Conditions

waterdepth=750rn current speed=3.3 knots

significant wave height=3.66 rn significant wave period=8.5 sec

windspeed= 15 knots

15

Table 1.

_{Geometric properties of the system}

Propert

length overall (LOA) _{272.8 m}

length of the waterliie (L) _{259.4 m}

beam(B)

_43.lOm

draft(D)

_{16.15 m}

turret diameter(Dr)° 22.50 m

block coefficient (CB) _0.83

total dispincement (4) _{1.5374x 10 tons}

r

Region I (R-l): The principal equilibrium A. is.stable. All eight TMS eigenvalues (six in the case of

SPM systems) have negative real parts and therefore equilibrium A is a stable focus. Since no

other equilibria are present to attract the trajectories, all flow trajectories converge toward eqUilibrium A in forward tune. The resulting relative angle of attack between the orientation of the system at equilibrium and the environmental excitation is zero.

Region II (R-ll): The principal equilibrium A is unstable with a one-dimensional unstable

manifold. One real eigenvalue has a positive real part. Equilibrium A is an unstable node and therefore all trajectories initiated in the vicinity of A deviate from it in forward time. A static bifurcation of the supercritical pitchfork type'4 occurs, when crossing the bifurcation boundary

from R-I and R-ll, resulting in static loss of stability with the appearance of two additional

statically stable equilibria The flow trajectories will converge to one of these alternate equilibria depending on the initial conditions of the system. These flew equilibria have a non-zero relative

angle of attack between the orientation of the system and the environmental excitation. This angle increases as the system parameter DCG moves to the left deeper in R-l1

Region Ill (R-ffl): The principal equilibrium A is Unstable with a two-dimensional unstable manifold (i.e. a complex conjugate pair of eigenvalues with a positive real part). A dynamic bifurcatiOn of the Hopf type occurs when crossing from R-I to R-ffl'4 resulting in dynamic loss of

stability. The corresponding rnorphogenesis;is the appca ice of a limit cycle around equilibrium

A.

The catastrophe sets in FigUre 3 show that qualitatively the dynamics of the TMS and SPM

systems are similar in the (P

IL,

0)plane. As noted in Figure 3,tbe static stability of both

systems depends mostly on the value of L). Obviously, they differ quantitatively. The location

of the turret (mooring lines in the SPM case) largely determines whether or not thëprincipal

equilibrium will bifUrcate create two additional equilibria irrespective of the direction ofihe

external excitation Both SPM and TMS tend to achieve their principal weathervamng equilibrium as the turret (or location of the mooring lines) is moved forward, Le. withincreasing D vahies, as shown by stable region R-I in Figure 3. Alternative stable equilibria (regio4 R-U)4ipear

á a

result of moving the turret toward the center of gravtty of the system. These results are in

accordance with the conclusions reached by Yashima e al for large scale experiments on TMS.

Notice that the static bifurcalions that occur in both SPM and TMS in Figure 3 are very similar.

turret has important design considerations, since higher hydrodynarnic forces act on the vessel

when achieving a non-weathervaning configuration.

Figure 3 also shows that the dynamic stability for both systems is affected by the direction of external excitation. Both SPM and TMS dynamics may fall in R-Ill achieving_{a stable limit cycle} around equilibrium A for large values of DCG. This occurs for excitation angles _{for which the} total projected area of the catenaries to the inflow is smaller, resulting in less damping_{due to the}

mooring lines.

The third parameter, which corresponds to the magnitude of the moment exerted between the turret

and the vessel, is shown to alter significantly the region where dynamic loss ofstability occurs. For non-zero friction coefficient r, the TMS dynamic stability increases with increasing value _of

r. As the friction coefficient decreases, the region of limit cycles (R-ffl) exhibited by the TMS

approaches that of SPM systems.

It is significant to point out that the static bifurcation boundaries (i.e. the boundary between R-I and RH) for TMS and SPM systems do not coincide as shown in Figure 3. As theturret diameter decreases (and therefore its mass decreases), the TMS catastrophe set approaches that of the SPM system, and their bifurcation boundaries between regions R-I and R-ll, and R-I and R-ffl, converge in the limit of zero turret mass.

Figures 4-6 show nonlinear time simulations of the drift angle ' for the TMS and SPM system in

each of the three regions depicted in Figure 3. To additionally demonstrate some of the differenèes between these types of systems, the non-dimensional friction coefficient for the TMS in Figures 4-6 is taken as 0.001.

Fig re4sh

the simulation for geometiint S1 (D/L = 0.45,

₀= 180°), which falls in

R-mmFigure3forboththesPMandtheTMs. AsanticipatedfromFigure3,itshows

oscillatory behavior about the priixipalqquilibrium. Notice that the TMS takes longer time to

achieve its stable limit cycle as cQp1pazedip its SPM counterpart, and exhibits smaller amplitude

osillations. This difference in behavior isIue both to the exerted moment that exists between he turret and vessel, and the damping çoment created by the mooring lines as the turret rotates. This

also results in lower frequency of oscillations for the TMS, as shown in Figure 4.

An increasein the angle of external excitation for both systems (from 180 to 2200) yields stable

configurations around equilibrium A. as proven by the catastrophe sets of Figure 3. This is shown

in Figure 5 for geometry point S2 (DCG I.L = 0.45, 0 = 220°).

Both systems align to the

direction of the external excitation in forward time (an xcitàtion angle of 220° corresponds to a

drift angle V1 of 40°). The TMS oscillations are slower with smaller amplitude. Notice that convergence to the principal equilibrium exhibited by the TMS is not slower than that for the SPM

system.

The simulations in Figure 6 show the effect of reduced Dca for both systems for geometry point

S3 (D/L = 0.40, 0 = 180°). This results in an unstable principal equilibrium, as inferred

from Figure 3, with the trajectories converging to stable alternative equilibria in forward time. Figure. 6 also shows slower oscillations by the TMS due to the damping moment created by the mooring lines and the friction between the turret and the vessel. Notice that in this case the TMS

converges faster to its stable equilibrium.

These differences in response between SPM and TMS can be explained in terms Of the Degree Of Stability (DOS) inherent in the system. The DOS is defined as the maximum value of the real parts of the system eigenvalues, and represents a certain measure of minimum damping (if negative) or maximum growth rate (if positive), and therefore a measure of stability or instability of a specific

equilibrium. In Figure 4, where the principal equilibrium has two eigenvalues with positive real

parts, the DOS of the SPM system is higher

In Figure 5, the DOS for both systems is

appmximately the same, while in Figure 6 the TMS has a higher DOS.

CONCLUSIONS

The mathematical model for the nonlinear slow motion dynamics of TMS presented in Section 2

includes the vessel hydrodynamics, mooring line dynamics and external excitation. This model is e general, and it is not restricted to any specific application. The model takes into account theexact

1otion of the mooring lines, which are attached to the turret, and the friction moment exerl as L th turret rotates with respect to the vessel. The former results in mooring line damping ment .er i

due to the hydrodynamic drag created as the turret rotates In analyzing TMS dynamics, theswo

°

cómponerits are often neglected, and SPM models are used instead. As shown in this papcthe ' effect of theses components may not be negligible. Each of these components results in c*upling

of the equations of motion of the system and the turret

The TMS dynamics was analyzed with the concepts of nonlinear dynamics and bifurcation theory

shown in Section 5. These charts, also known_{as catastrophe sets, give qualitative information}

regarding the TMS dynamics, and constitute the basis for a design methodology used to select appropriate design parameters without trial and _{error and lengthy nonlinear time simulations. This}

design methodology has been used to illustrate the differences in qualitative behavior that TMS and

SPM systems exhibit as a parameter or group ofparameters are varied.

It has been shown that, if the friction moment between the turret and the vessel is_{ignored, both}

SPM and TMS exhibit qualitatively the same dynamics in the parametric design space described in

this work. The TMS mooring line damping moment (not present in SPM systems) does_{not affect}

significantly the qualitative behavior of the system as shown in Figure 3, and TMS can be

appiox mated by a SPM system. Inclusion of the friction moment in the TMS equations of motion

could result in major differences with respect to the SPM model, particularly in the dynamic behavior of the system. This is shown in Figure 3 by the expansion of unstable region R-ffl, corresponding to stable limit cycle behavior. Notice that the friction moment has little effect on the

static loss of stability of the system, which is marked in Figure 3 by the boundary between regions

R-I and R-ll.

The nonlinear time simulations in Figures 4-6 help visualize the effects of the friction_{moment and} the mooring line damping moment on the TMS. For purposes of comparison to SPM_systems, these simulations have been performed using the same geometries. As shown in all simulations, the frequency and amplitude of oscillation of the TMS are lower than those exhibited by the SPM

systems in each case. The frequency of oscillations may interact with excitation such as that due to

slowly-varying drift forces. Accordingly, the fact that the SPM approximation results in a limit cycle of frequency different than that of the corresponding TMS, renders the SPM approximation

invalid. In addition, Figure 4 shows that the TMS takes longer to achieve a stable limit cyde, while Figures 5 and 6 show rapid convergence to stable equilibria with respect to the SPM system.

vst. es!g e

The friction moment between the turret and the veI becomes less significant as the turret

diameter decreases. Mooring line damping becomes less important as h mooring lines e moved doser to the center of gravity of the turret As the tundlsmeter decreases, the TMS tends toward

a SPM system, provided that the inertia pmpettià of $&h types1 of systems are equaL

While SPM models can predict the static properties of the TMS and its asso fated morphogeness,

a complete TMS model is needed to study its complete dynamical behavior, particularly the

ciated bifurcations. This is important for design d implementation of TMS in deeper waters, where dynamic instabilities, not captured with SPM models, become increasingly important'°.

REFERENCES

Bernitsas MM. & Garza-Rios, L.O., Effect of mooring line arrangement on the dynain cs of

Spread mooring systems. Journal of Offshore Mechanics and Arctic Engineering 118:1

(1996) 7-20.

Beritsas 4.M. & Garza-Rios, LO.k Moonng system design based on analytical expressions

of catasphes of slow-motion dynamics Journal of Offshore Mechanics and Arctic

Engineen.g 1i;2 (1997) 86-95.

Bern tsasM.M. & Papoulias, F.A., Nonlinear stability and maneuvering simulation of single

pint mooring systems

Proceedings of Offshore Station Keeping Symposium, SNAME, Houston (1990) 1-19.

Cox, J.V., Statmoor - a single point mooring static analysis program. Report No.AD-A119

979, Naval Civil Engineering Laboratoiy, 1982.

lii this work, only three design parametçrs have been swdied::.the location of the turret (the distance

of mooring line attachment from the center of gravity for SPM systems), the direction of external excitation, and the magnitude of the exerted moment. To gain a complete understanding of TMS

behavior, it is essential to determine the effect of each parameter on the system dynamics. This can

be achieved by constructing catastrophe sets around each equilibrium position and superposing them.. This task, however, becomes difficult in the sense that it is impossible to plot graphs in parametric spaces of dimensions higher than three. I future work, analytical expressions for the stability/bifurcation boundaries will be derived for TMS following Garza-Rios and Bernitsas8'9

that will help understand better the dynamics of TMS. These physics-based expressions can be used to develop a design methodology for TMS capable of revealing TMS dependence on any

number of parameters without the need to recur to plotting and superposing several three-dimensional stability charts2.

ACKNOWLEDGMENTS

This work is sponsored by the University of Mich.iganhlndustry Consortium in Offshore Engineering. Industry participants include Amoco, Inc.; Conoco, Inc.; Exxon Production Research; Mobil Research and Development; and Shell Oil Company.

s_j_ &

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Mechanics and Arctic Engineering (OMAE), I-A, Florence, Italy (1996) 423-430.

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Garza-Rios, L.O. & Bernitsas, M.M., Analytical expressions of the bifurcation boundaries for symmetric spread mooring systems. Journal of Applied Ocean _{Research 17(1995)} 325-341.

Garza-Rios, L.O. & Bernitsas, M.M., Analytical expressions of the stability and bifurcation boundaries for general spread mooring systems. Journal of Ship Research, SNAME 40:4 (1996) 337-350.

Garza-Rios, L.O. & Bernitsas, M.M., Nonlinear slow motion dynamics of turret mooring

systems in deep water. Proceedings of the Eight International Conference_{on the Behaviour of}

Offshore Structures (BOSS), 2, Ddft, The Netherlands (1997) 177-188.

Garza-Rios, L.O. & Bernitsas, M.M., Mathematical model for the slow motion dynamics of

turret mooring systems. Report to the University of Michigan/Sea Grant/Industry Consortium in Offshore Engineering. Publication No. 336, Ann Arbor (1998).

Garza-Rios, L.O., Bernitsas, MM. & Nishimoto, K., Catenary mooring lines with nonlinear drag and touchdown. Report to the University of Michigan/Industry Consortium_inOffshore Engineering. Publication No. 333, Ann Arbor (1997).

Garza-Rios, L.O., Bernitsas, M.M., Nishinioto, K. & Masetti I.Q., Preliminary design of a

DICAS mooring system for the brazilian Campos basin. Proceedings of the 16th_{International} Conference on Offshore Mechanics and Arctic Engineering (OMAE), I-A, Yokohama._Japan

(1997) 153-161.

Guckenheimer, J. & Holmes, P., Nonlinear Oscillations, Dynamical Systems, and

Bfurcations of Vector Fieldr, Springer-Verlag, New York (1983).

Henery, D. & Inglis, R.B., Prospects and challenges for the IPSO. Proceedings of the 27th

Offshore Technology Conference, Paper OTC-7695, 2, Houston (1995) 9-2 1.

Laures J.P. & de Boom, W.C., Analysis of turret moor d storage vessel for the Alba fi1d.

Proceedings of the Sixth International Conference on the Behaviour of Offshore Structures (BOSS), 1, London, U.K. (1992) 211-223.

Leite, A.J.P., Aranha, J.A.P., Umeda, C. & Dc Conti, M.B., Current forces in tankers and bifurcation of equilibrium of tutret systems: hydrodynamic model and experiments. Jourhal ofApplied Ocean Research, in press.

Mack, R.C., Gruy, R.H. & Hall, R.A, Turret moorings for extreme design conditions.

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257-281.

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West Society of Naval A rchitects.of Japan, 91 (1995) 81-94.

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Technology Conference, Paper OTC-5980, 2, Houston (1989) 223-232.

c)fr

SLOW MOTION. NONLINEAR DYNAMICS

_{OF TURRET}

MOORING AND ITS DISTINCTION FROM SINGLE

POINT MOORING

LIST OF FIGURES

Geometry of Turret Moormg System (TMS)

Intermediate reference frames in TMS

I Catastrophe set., 4-line TMS: effect of turaet location,_{external excitation angle and} friction factor

4. Stable limit cycle about equilibrium A: SPM and TMS (region R-ffl)

.5! Stable focus, equilibrium A: SPM and TMS (region R-I)

6. Stable focus, alternate equilibtium SPM and TMS (Region R-ll)

LIST OF TABLES

GeOmetric properties of the system

360 340- 320- 300- 280- 260-240

220-200 180- 160-140 120

-

100-SPM

TMS 'r=

1x10 TMS T= ix103

i

1U

tt iti

S3

HI

.

S2 80

60-40-'

20-DcdL

III

-.---.---.-.-.'-.

I

50-40

30-

2030 50

--60,

nondimensional time ((JilL)

0 5O 100 150 200 250 300 4 V SPM TMS

a

10- _I

-10-

9 S I

-20-2.5 -2.5 1.5 0.5

-

-0.5-

-1.5-nondimensional time (UtIL)

SPM TMS I I I I I I I . I I I 550 600 650 700 750 0 50 100 150 200 250 300 350 400 450 500