CI

Journal of Applied Ocean Research, in review

SLOW MOTION DYNAMICS OF TURR] T MOORING AND ITSAPPROXIMATION AS SINGLE POINT MOORING

ncMic UVRSTLTI.aboratoriurn voor

Scheepehydrom6ChaflcArchiet

Eekehveg 2.2628 CD Detft)d.Otg-78Th- ft Oi..18W3

by

Dr. Luis 0. Garza-RiosResearch Fellow

Dr. Michael M. BernitasProfessor

Submitted for publication inApplied Ocean Research

Department of Naval Architectuit and Marine EngineeringUniversity of Michigan

Ann Arbor, Michigan 48109-2145

SLOW MOTION DYNAMICS OF TURRET MOORING AND ITSAPPROXIMATION AS SINGLE POINT MOORING

ABSTRACT

The mathematical; thi$del for the nonlinear dynamics of slow motions in the horizontal plane ofTurret Mooring Sysienis (TMS) is presented. It is shown that. the TMS model differs from theclassical Single Point Mooring (SPM) model, which is used generally to study the TMS dynamics.

The friction moment exerted between the turret and the vessel, and the mooring line dampingmoment 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 identifiedusing nonlinear dynamics and bifurcation th&ny. In two and three!dimensional parametric designspaces, 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 ofa TMS can be located

approximately by the SPM static bifurcation. The dynamic loss of stability of TMS and theassociated morphogenesis may be affected strongly by the friction/damping moment, and oalesser extent, by the mooring line damping. Nonlinear time simulations are used to assess theeffects of these properties on TMS and compared to SPM systems. The TMS mathematical modelconsists of the nonlinear horizontal plane fifth-order, large drift, low speed maneuveringequations. Mooring line behavior is modeled quasistatically by submerged catenaries, includingnonlinear drag and touchdown. External excitation consists of time independent current, steadywind, 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

Yr

L= vessel length

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

addedmass in sway

a = number of mooring line

N= yaw hydrodynamic momentNv= 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.

2

co

xc

Presently, several TMS have been designed for and deployed in relatively deep waters under larshenvironmental conditions'6.'8. TMS design methods, however, depend mostly on experience andlimited data, and therefore TMS design can be studied only on a case by case basis. Due to thecomplexity 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 andintermediate water depths'7.26, show that SPM models have proven useful, especially indetermining 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 thehorizontal plane slow motion dynamics of TMS; (ii) to develop a methodology to understand TMSdynamical behavior as function of design parameters; (ili) to determine the validity of modelingTMS as SPM systems based on their slow motion nonlinear dynamics; and (iv) to assess theimpact 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 byBernitsas and Garza-Rios' for multiple lines. The latter is used in this paper for directcomparisons 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 indequilibria 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 themooring 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.

3

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 line model, 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. Thedirection 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') S the 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, wherer

VrV(+4hi. de (2)

,2.1 TMS equations of motion

(1)

The equations of motion of the TMS in the (X, 1, Z) reference frame in surge, sway, and yaw1l:

(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 thesystem (m=my + mT); 1 is the moment of mertia of the system about the Z-axis; m, rn, andJ,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 Wicatenary; 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 -

5

6

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' =7and 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, relativesize 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 theposition/velocity dependent mooring line forces is negligible; the damping moment created due to

these forces, however, differs considerably between SPM and TMS. This is because in 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 momentis non-zero, as the mooring lines are attached off the center of gravity of the turret. As the turretrotates, the mooring line damping moment becon non-zero.

The moment equation for the turret (10) shows that, even in the absence ofa moment exerted bythe vessel, the turret itself has a non-zero rotational acceleration. Exception to this presents thecase 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 ofrisers 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 are the transverse and longitudinal areas of the vesselprojected 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

8

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 isthe wave amplitude, and o, is the wave frequency The two-parameter Bretschnider 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 arenonlinearfunctionsof H113 id 7j134.

rd = 5S(0 xD(aO 11w0, (29)0 0.5pga J

Cy1j=JS(01){0

0' (30)FYD(0.)o)}fO.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 2 is 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 system yields 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 velocityvectors of the system. Thus, by selecting the following state variables: (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}

10

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.

(35)

11

In eqns (33)-(40), the quantities inside the summation signs T11, $ and fi' are functions of thestate position variables (x5, x6, x7, x8); and F, FN and F are functions of all statevariables. The expressIons for the external excitation Fue. F and vary with theorientation 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 continuouslydifferentiable functions'4. For the state space representation selected above, all variables, vectorfield 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. Allsystem equilibria, also called stationary solutions or singular points, can be found by setting thetime 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 principalequilibrium 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 theTMS 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, andthe linear velocities at equilibrium are exclusively functions of the relative angle of attack between

the heading of the vessel and the current Substitution of relations (43X46) into eqns (33)-{36)

yields the following set of four nonlinear algebraic equations that must be solved simultaneouslyfOr 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 particular TMS 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 andbifurcation 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.

4.1 TMS stability analysis

12

Equilibria of nonlinear systems are special degenerate trajectories. Local analysis near anequilibrium solution reveals the behavior of trajectories in its vicinity. Local analysis can beperformed 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 fevaluated 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 convergetoward 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 systemconfiguration to determine its global behavior.

4.2 Bifurcations of equilibria in TMS

The principle of nonlinear stability analysis applies to a specific TMS configuration unlerprescribed 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 performingbifurcation 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 thequalitative behavior of the TMS as such system parameters are varied.

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

13

14

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 turretwith 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. Theseparameters are used in the numencal applications forpurposes 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). Toestablish an appropriate basis of comparison between SPM and TMS, both systems have the samevessel hydrodynamics, mooring line model, center of gravity, added and stnictural mass, addedand structural second moment of inertia.

In the catastrophe sets of Figure 3, there exist three regions (numbered I to ifi) of qualitativelydifferent dynamical behavior.

:

-Tble Design parameters of the system

Mooring Line Propeities

number of mooring lines =4average breakigtrength 5159 KN/Linelength of moonng lines =2625 rnorientation of mooring line I = 0,mooring line spacing =90pretension = 1444 KN/linefiaction of attachment = O.75

not applicable to 5PM

Environmental Conditions

waterdepth=750rncurrent speed = 3.3 knotssignificant wave height = 3.66 rnsignificant wave period = 8.5 secwindspeed= 15 knots

15

Table 1. Geometric properties of the system

Propert

length overall (LOA) 272.8 mlength of the waterliie (L) 259.4 m

beam(B) 43.lOmdraft(D) 16.15 m

turret diameter(Dr)° 22.50 mblock coefficient (CB) 0.83total dispincement (4) 1.5374x 10 tons

not applicable to SPM

r

16

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 towardeqUilibrium 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 unstablemanifold. 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 additionalstatically 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 angleincreases 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 unstablemanifold (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 SPMsystems 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. with increasing D vahies,

as shown by stable region R-I in Figure 3. Alternative stable equilibria (regio4 R-U)4ipear á aresult of moving the turret toward the center of gravtty of the system. These results are inaccordance 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.

This charactenstic of TMS's is well captured with SPM models3,6.7J7a'. The position of the

turret has important design considerations, since higher hydrodynarnic forces act on the vesselwhen achieving a non-weathervaning configuration.

Figure 3 also shows that the dynamic stability for both systems is affected by the direction ofexternal excitation. Both SPM and TMS dynamics may fall in R-Ill achieving a stable limit cyclearound 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 themooring lines.

The third parameter, which corresponds to the magnitude of the moment exerted between the turretand the vessel, is shown to alter significantly the region where dynamic loss of stability occurs.For non-zero friction coefficient r, the TMS dynamic stability increases with increasing value ofr. 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 the turret 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 Figures4-6 is taken as 0.001.

Fig re4sh the simulation for geometiint S1 (D/L = 0.45, 0 = 180°), which falls in

R-mmFigure3forboththesPMandtheTMs. AsanticipatedfromFigure3,itshowsoscillatory 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

17

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 isappmximately 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 asL 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

presented in Section 4 to constrict stability or design graphs in the parametric design space as

18

shown in Section 5. These charts, also known as catastrophe sets, give qualitative informationregarding the TMS dynamics, and constitute the basis for a design methodology used to selectappropriate 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 andSPM 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, bothSPM and TMS exhibit qualitatively the same dynamics in the parametric design space described inthis work. The TMS mooring line damping moment (not present in SPM systems) does not affectsignificantly the qualitative behavior of the system as shown in Figure 3, and TMS can beappiox 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 thestatic 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 limitcycle 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 turretdiameter 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'°.

19

REFERENCES

Bernitsas MM. & Garza-Rios, L.O., Effect of mooring line arrangement on the dynain cs ofSpread 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 expressionsof catasphes of slow-motion dynamics Journal of Offshore Mechanics and ArcticEngineen.g 1i;2 (1997) 86-95.

Bern tsasM.M. & Papoulias, F.A., Nonlinear stability and maneuvering simulation of singlepint 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-A119979, Naval Civil Engineering Laboratoiy, 1982.

Dercksen, A. & Wichers, LEW., A discrete element method on chain turret tanker exposedto survial conditions. Proceedings of the Sixth International Conference on the BehavioUr ofOffshore Structures (BOSS), 1, London, U.K. (1992) 238-250.

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_ &

A

Fernandes, A.C. & Aratanha, M., Classical assessment to the single point mooring and turretdynamics stability problems. Proceedings of the 15th International Conference on OffshoreMechanics and Arctic Engineering (OMAE), I-A, Florence, Italy (1996) 423-430.

Fernandes, A.C. & Sphaier, S, Dynamic analysis of a FPSO system. Proceedings of the7th International Offshore and Polar Engineering Conference (ISOPE), 1, Honolulu, Hawaii(1997) 330-335.

Garza-Rios, L.O. & Bernitsas, M.M., Analytical expressions of the bifurcation boundariesfor 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 bifurcationboundaries 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 mooringsystems in deep water. Proceedings of the Eight International Conferenceon the Behaviour ofOffshore Structures (BOSS), 2, Ddft, The Netherlands (1997) 177-188.

Garza-Rios, L.O. & Bernitsas, M.M., Mathematical model for the slow motion dynamics ofturret mooring systems. Report to the University of Michigan/Sea Grant/Industry Consortiumin Offshore Engineering. Publication No. 336, Ann Arbor (1998).

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

Garza-Rios, L.O., Bernitsas, M.M., Nishinioto, K. & Masetti I.Q., Preliminary design of aDICAS mooring system for the brazilian Campos basin. Proceedings of the 16th InternationalConference on Offshore Mechanics and Arctic Engineering (OMAE), I-A, Yokohama. Japan(1997) 153-161.

Guckenheimer, J. & Holmes, P., Nonlinear Oscillations, Dynamical Systems, andBfurcations of Vector Fieldr, Springer-Verlag, New York (1983).

Henery, D. & Inglis, R.B., Prospects and challenges for the IPSO. Proceedings of the 27thOffshore 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 andbifurcation of equilibrium of tutret systems: hydrodynamic model and experiments. JourhalofApplied Ocean Research, in press.

Mack, R.C., Gruy, R.H. & Hall, R.A, Turret moorings for extreme design conditions.Proceedings of the 27th Offshore Technology Conference, Paper OTC-7696, 2 Houston(1995) 23-31.

Martin, LL., Ship maneuvering and control in wind. SNAJI$ Transactions, 88 (1980)257-281.

21

McClure, B., Gay, TA. & Slagsvold, L.; Design of a turret-moored production system(TUMOPS) Proceedings of the 21st Offshore Technol6gy Conference, Paper OTC-5979, 2,Houston (1989) 213-222.

Nishimoto, K., Brinati, H.L& Fucatu,C.H., Dynarnis of moored tankers SPM and turret.Proceedings of the 7th International Offshore and Polar Engineering Conference (JSOPE) 1,Honolulu, Hawaii (1997) 370-378.

Press, W.H, Flannery, BP., Teukoisky, S.A. & Vetterling, WiT., Numerical Recipes: TheAn of Scientific Computing. Cambridge University PreS, New York (1989).

Seydel, R., From Equilibrium to Chaos. Elsevier Science Publishing Co., Inc. New York(1988).

Takashina, J., Ship maneuvering motion due to tugboats and its mathematical mOdel. Journalof the Society of Naval Architects of Japan, 160(1986) 93-104.

25, Tanaka, S. On.the hydrodynamic forces acting on a ship at latge drift angles. Journal of theWest Society of Naval A rchitects.of Japan, 91 (1995) 81-94.

26. Yashima, N., Matsunaga, E., & Nakamuta, M. A large..scale model test of turret mooringsystem for floating production storage offloading (FPSO) Proceedings of the 21st OffshoreTechnology Conference, Paper OTC-5980, 2, Houston (1989) 223-232.

c)fr

thc

SLOW MOTION. NONLINEAR DYNAMICS OF TURRETMOORING 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 andfriction 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

Design parameters of the system

DIRECTIONS OF CURRENT, WIND, AND WAVES

360

340-

320-

300-

280-

260-

240

220-

200

180-

160-

140

120 -

100-

SPM

TMS 'r= 1x10

TMS T= ix103

i

1U

tt itiS3

HI

. S2

0.5

80

60-

40-'

20-

0DcdL

III

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

I

0.35 0.36I

0.37I

0.38I

0.39I I I I 1

0.4 O41 0,42' 0.43 0.441

0.45T

0.46I

0.47I

0.48 0.49

50-

40

30-

20-

-30 -

-50 -

-60,nondimensional time ((JilL)

0 5O 100 150 200 250 300

4

V

SPM

TMS

a

10- I

-10- 9S

I

-20-

nondimensional time ((JilL)

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 7500 50 100 150 200 250 300 350 400 450 500