Turret mooring design based on analytical expressions of catastrophes of slow motion dynamics

17th International Conference on Offshore Mechanics and Arctic Engineering (OMAE) Lisbon, Portugal, July 1998.

Journal of Offshore Mechanics and Arctic Engineering, in press, August 1998

TURRET MOORING DESIGN BASED ON

ANALYTICAL EXPRESSIONS OF CATASTROPHES

OF SLOW MOTION DYNAMICS

MIchael M. Bernitsas and Luis O. Garza-Rios

Department of Naval Architecture and Manne Engineenng

The University of Michigan Ann Arbor, Michigan

ThCRN!SCHE UVERSTET

Laboratorium 'icor

Scheepshydromethai*a

4,rchIof

Mekeiweg 2.2828 CD Deft

IL Ol

- - Fmc O1 - 181C

ABSTRACT

Analytical expressions of the bifurcation boundaries exhibited by Turret Mooring Systems (TMS), and expressions that define the morphogeneses occurring across boundaries are developed. These expressions provide the necessary means for evaluating

the stability of a TMS around an equilibrium position, and

constructing catastrophe sets in two or three dimensional

parametric design spaces. Sensitivity analyses of the bifurcation

boundaries define the effect of any parameter or group of

parameters on the dynamical behavior of the system.

l'bese expressions allow the designer to select appropriate values for TMS design parameters without resorting to trial and error. A four-line TMS is used to demonstrate this design methodology. The mathematical model consists of the nonlinear, fifth-order, low speed, large drift maneuvering equations. Mooring lines are modeled with submerged catenaries, and include nonlinear drag. External excitation consists of time independent current, wind, and mean wave diifi

NOMENCLATURE

ABS = Average Breaking Strength CG= cenxm of gravny of ThIS CG center of gravity of turret

D= digce between CG and CGT

L= vessel length

in = ss of system

n= nwnber of mooring lines N= ynw bydrodynamic moment PE = Principal Equilibrium position SMS = Spread Mooring System SPM = Single Point Mooring

TH= horizontal component of the mooring line tension TMS = Turret Mooring System

(x,y)= inertial reference frame with originat mooring terminal L

(X,Y.Z)= body fixed reference frame with origin at CG Xj,= hydrodynamic force in surge

Y,,= hydrodynamic force in sway

current angle w.r.L (x,y) wave angle w.r.L (x,y) a,= wind angle WS.L (x,y)

r = moment friction coefficient between turret and vessel = angle of external excitation w.r.t. (x,y)

y= drift angle

'T= absolute yaw angle of turret

ith eigenvalue

4(.)= first order perturbation operator P= time derivative operator d/dt

5= equilibrium of (.) I. INTRODUCTION

Turret Mooring Systems (TMS) have been developed recently for stationkeeping of ships and floating/production systems for oil and gas production and recovery. ThLS consists, in general, of a vessel and a turret to which several mooring/anchoring legs are attached. The turret includes bearings which constitute the interface between turret and vessel, allowing the latter to rotate freely around ita

anchor legs

in response to changes in

environmental eitation and system dynamics. Thus, TMS can

weathervane b

changing

its

heading relative

to the

environmental excitation.

TMS may be mounted externally

ahead of the bo* or aft of the stern, or internally within the

vessel. Further, turrets may be disconnectable or permanent depending on th type of mooring, projected time of operation. and environmental conditions [19. 21).

The first TMS was deployed in 1986 offshore Australia, at a water depth of 120 Ineters (6). Such system was designed with a disconnectable Riser Turret Mooring (RTM) to allow the tanker to leave the site of operation in case of harsh environmental conditions. Recent adveatment in technology made possible deployment of more sophisticated types of ThIS in deeper waters (211. As the need to venture into even deeper waters increases.

dynamics. TMS nonlinear dynamics, however, are complex to

understand because TMS have a high number of design variables (parameters). such as number, length, pretension and orientation

of mooring lines; size and position of the turret; environmental conditions; etc.

To understand the qualitative properties of TMS. a design

methodology based on nonlinear dynamics and bifurcation

theory has been developed [13. 15). This methodology consists

of evaluating the stability of a particular TMS equilibrium, or group of equilibria, using eigenvalue analysis and numerical

search for the structural changes of eigenvalues (bifurcations) as parameters are varied. These can be represented graphically in design charts (catastrophe sets) that help visualize the effects of

design variables in design spaces up to three dimensions.

Previously. Garza-Rios and Rernitsas [2.. 10-12) devised a design methodology for general Spread Mooring Systems (SMS) based

on analytical expressions of catastrophes of slow motion

dynamics. These expressions make it possible for the designer co study the nonlinear slow motion dynamics of SMS around all

equilibria in any parametric design space, and identify the

associated morphogeneses. The designer is no more limited to

three-dimensional design spaces [2) due to graphic representation. Analytical expressions for the stability criteria and bifurcation boundaries have been derived for TMS, and successfully compared to numerically obtained system eigenvalues and its bifurcations [16). These expressions are physics-based. The aim of this work is to use those analytical

expressions to develop a TMS design methodology by deriving

analytical expressions of sensitivity of bifurcations on parameters, and applying them

in TMS design.

This

methodology completely eliminates trial and error. Further, it reduces dramatically the required nonlinear simulations since the morphogeneses occurring across bifurcation boundaries reveal

the qualitative behavior of TMS dynamics. Existence of

bifurcations in TMS nonlinear dynamics has been demonstrated by [7. 8. 20, 23] by modeling TMS as SPM systems.

The formulation of the TMS design problem is presented in Section Ii with a brief description of the mathematical model. mooring line model, and model for the external excitation. In Section

_(II.

a design methodology based on analytical

expressions

for the TMS stability

criteria and bifurcation

boundaries is developed. In Section IV, a particular application

of this

theory is presented. This includes construction of bifurcation boundaries in a two-dimensional parametric design space and the effects of a parameter on these boundaries (sensitivity analysis). Conclusions are summarized in Section V.

IL FORMULATION OF THE DESIGN PROBLEM

The slow motion dynamics of TMS in the horizontal plane (surge, sway and yaw) is modeled in terms of the vessel/turret equations of motion without memory. catenary mooring line model, and external excitation. The system considered in this paper corresponds to an internal turret configuration, and thus it is assumed that no hydrodynamic forces/moments act directly on

the wrret.

ILi. EquatIons of motloq

The general geometry of a ThIS is shown in Fig. I, with two principal reference frames: (x,y) = inertial (non-moving)

refee frame with its origin located at mooring terminai I;

(lIZ)

body-fixed reference frame with its origin located at

the center of gravity of the system (CG). i.e. vessel and turret combined. In addition, n is the number of mooring lines; (x.y) are the coordinates of the lilt mooring terminal with

respect to the (x,y) frame;

'

is the horizontal projection of

the ith mooring line; (s1

is the angle between the x-axis and

theish mooring line, measured counterclockwise; and r is the drift angle of the vessel. The direction of the external excitation (i.e. current, wind, waves) is measured with respect to the (x.y) frame as shown in Fig. I.

The intermediate reference frames of the vessel/turret system from which the TMS mathematical model is derived are shown in

Fig. 2. (X',Y'.Z') is the turret reference frame with its origin located at

the center of gravity of the

turret _(CGt); and

(X.Y.Z) is the vessel reference frame with its origin located ai the center of gravity of the vessel (CG). Moreover, D is

the distance between the centers of gravity of the system (CG)

and the turret; and d is the distance between the centers of

gravity of the system and the vessel. These are related as follows:

(I)

where m is the mass of the vessel and mr is the mass of the turret. In addition. y is the relative yaw angle between the turret and the vessel measured counterclockwise as shown in Fig. P'T is the absolute yaw angle of the turret with respect to

the x axis. Thus,

P'rYPti

(2)

The horizontal plane equations of motion of the TMS consist of che three vessel equations in surge, sway, and yaw, and the corresponding turret equations. Since the turret is fixed to the vessel, the linear translations of the turret depend on the motion of the vessel. 'Thus, the system has four independent degrees of freedom. The dynamics of the TMS can therefore be recast in

terms of four equations of motion [14).

three equations describing the motion of the system in surge, sway and yaw, and one equation pertaining to the rotation of the turret.

The equations of motion of the system in the (X.Y,Z)

reference frame in surge, sway, and yaw are [14]:

(m+m-(m+m,)vr=Xa(u.v.r)+F4,,

- F°Jcosß

+,op«},

(3) (m+m7)+(m+m1)irr= Y,(ig,v,r)+F

cosß}.

(4)

(I J=

NN(u,v,r)+N+NT

+

_-

"]sinß«)

- F'

cosß(°}. (5) where (ig,v,r) and

(i.p,i)

are the relative velocities and .ekrations of the center of gravity of the system with respect

Io water in surge. sway, and yaw, respectively; m is the total

mass of the system (m = mv + mT); ! is the moment of inertia of the system about the Z-axis; m, ms,, and J are the added mass and moment terms:

F5,. and N,

are externai forces and moment acting on the vesseL such as wind and second

order wave drift forces [I, 3); N is the moment exerted on the

vessel by the turret; and XH. Y and NH are the velocity dependent hydrodynamic forces and moment. These are expressed below in terms of the fifth-order, large drift slow motion derivatives (27] following the non-dimerisionalization by Takashina [26].

Ifi

=X.uX,u2+Xu3+Xvr,

(6)

Yu=Y,Ywv3+Y5r+lw1114'i+l+1.

(7)

NH

=N,v+Nv+Nv3+Ns4v3+N,r+N«jrj

+N4,1+Nv2r.

(8)

The terms inside the summation in equations of motion (3)-(5)

apply to each of the mooring lines (i = I...n).

TH is the horizontal tension component of the catenary; F, and FN are the drag forces on the catenary in the directions parallel and

perpendicular to the motion of the mooring line (17]. and depend

on the position of the turret with respect to the mooring points and the velocities of the mooring Lines; and ß is the angle

between

the X -axis

and the mooring line, measured

counterclockwise (ß=y-p).

The fourth equation of motion describes the relative rotation

of the turret with respect to the earth-fixed reference frame, and

consists of a moment equation about an axis parallel to the Z-axis (i.e. Z' -Z-axis). Such an equation needs not be transferred to

the (X.Y.Z) reference frame.

In the (X',Y',Z') frame, it

is

given by [14]:

ÎTrT = (O{[i4i)

-

F;(')]sin(ß'"

-_pj(os(p

_-rP)}

_+N,

(9)

where ,'. is the rotational acceleration of the turret; 'T iS the

turret moment of inertia about the Z' -axis; DT is the turret diameter N,. is the moment exerted on the turret by the vessel The terms inside the summation in equation (9) are defined as follows:

f

is the fraction of the turret diameter at which the

casenary is attached (Ofgl);

F and F. are the drag

forces on the catenary in the directions parallel and perpendicular

co the mooring line motion, respectively (17]; 5' is the angle between the X'-axis and the mooring line measured

counterclockwise (ß'=y-ç7);and y

is the angle at which the

catenary is attached to the turret with respect to the X'-axis (flied to the turret). In equations of motion (3)-(5) and (9), it

has been assumed that the turret is internal to the vessel and that

no hydrodynamic or external forces or moments act directly on

the turret.

The moments exerted on the turret and the vessel (N and N,.) are a result of the relative rotation that exists between

them. These moments arc of equal magnitude, and 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. in this paper, the exerted moments are modeled as follows:

NV=NT-T(rT-r),

(IO)

where is a small positive friction factor that depends on the

geometry, mooring line damping, relative size of the turret, separation between the turret and the vessel, and external excitation; and r. is the rotational velocity of the turret.

The kinematics of the TMS are governed by equations

(Il)-(14) [14]:

where U = )U is the absolute value of the relative velocity of

the vessel with respect to the water, and a, is the current angle whose direction of excitation is shown in Fig. I.

11.2. MoorIng line model

The mooring lines of the system are modeled quasistatically

by catenary chains with nonlinear hydrodynamic drag and

touchdown [17). The total tension T in a catenary is given by:

T=4TN2 _{= TNCOSh(TJ.} (15)

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

projected length of the suspended part of the line [17].

11.3.

gxternai

excitation

The model for external excitation consists of cime independent current, steady wind, and mean second order wave drift forces.

Each component has an independent direction of excitation defined in Fig. I. The current forces which act on the vessel are

modeled in equations of motion (3)-(5) by introducing the

relative velocities of the system with respect to waxer. Thus.

Fm,,, F1.,ù1i+FI1In.e, (16)

Fm., _F; +F, (17)

N,, z N wñid + N (IS)

Wind forces due to time independent wind velocity and

direction are modeled as

[fi:

rwiid (19) Fyw =.!p.Uw2C,,(ar)At, (20) upmUw2Czw(ar)LAa, (21) i = us ç - vsin ç + Ucosa3,

j=usinç+vcosç+Usina,

p' = r, = r7,

(Il)

(12) (13) (14)

where p is the air density; U, s the time independent wind

speed at a standard height of tO meters above water, a, is the

relative angle of attack; AT and AL are the transverse and

longitudinal areas of the

vessel projected to the wind,

respectively; L is the length of the vessel; C,.(a,), C(a,)

and C.(a,) are the wind forces and moment coefficients,

expressed in Fourier Senes as follows:

C(a,)= ¿, +

cos(ka,), (22)

k=I

C(a,)=

Öksin(ka,). (23)

kI

C.w(or)= Ctsin(ka,). (24)

Coefficients

Ç

t and Ç in equations (22)-(24) depend on the type of vessel, location of the superstructure, and loading

conditions.

The mean second order wave drift excitation in the horizontal

plane is given by [4]:

pgLC1d

cos3(cç.

- y).

(25)

F,,

=pgLC1sin3(a

_-y),

(26)

=pgL2Ç,sin2(a.., -y),

(27)

where p,, is the water density, a,,, is the absolute angle of attack; g is the gravitational constant; and C, C,, and C3, are the drift excitation coefficients in surge, sway, and yaw, respectively:

C1 =

JS(W.

F(o,0)

i

1,0.5p11,ga jdWo

C_

=j1,I F(Û0)

ldw O.5p,,ga _J ]dw..

Cj

_°'4O7a2

In the expressions above, the quantities in square brackets are

the drift excitation operators; a is the wave amplitude; and ,

is wave frequency. The two-parameter Breischnider specmim ta used to express S(w0) in terms of the significant wave height

H1 and significant wave period 7 as:

S(tD,) = Aøe'°',

(31)

whse A and B are nonlinear functions of H,,3and T,,3. Other autonomous models of wind and drift forces/moments can be

incorporated in che formulation as welL

Ill.

TUS DESIGN METHODOLOGY

The design methodology for TMS developed in this paper is based on analytical expressions for the stability criteria and bifurcation boundaries derived in this section. Physics-based

analytical expressions for the bifurcation boundaries and the associated morphogeneses for TMS nonlinear dynamics are

derived and then applied to TMS design. This is achieved by constructing design charts (catastrophe sets) in two or

three-dimensional parametric design spaces, and then performing

sensitivity analyses of the bifurcation boundaries with respect to one or more design variables.

In brief, the equations of motion (3)-(5) and (9) and kinematic relations (11H14) can be perturbed near an equilibrium position to study the stability characteristics around that equilibrium [18). The concept of TMS equilibrium is studied in [151. To find TMS equilibria requires solution of a system of eight coupled

nonlinear first order differential equations. After perturbation of the system. the equations of motion (3)-(5) and (9) can be recast in the following generic form:

f = + +

i')Lif(')a1 +

b,dx +c,dy + d14y +e,4y7)}

_{((i)

_+P"a1

_{+be4x+c14v+djiy+e14p'r)}}

+4TL'1)Xa21 + b24x +c214y + d,4y +e214Y)}. (32)

The overbar in each term in (32) indicates the value of each

variable evaluated at equilibrium, where f is an autonomous function that depends exclusively of terms containing 41. 4k,

4j. 4y 4' 4

4y.

_{4T' 4y.,}

and 4p'.; h? is the sum

of the hydrodynamic force/moment and external excitation

evaluated at equilibrium (zero in the turret rotation equation (9));

i

is the first derivative of the horizontal tension of the mooring line with respect to the horizontal projection of the catenary; 4t' is the first order perturbation of the horizontal

length of the catenary; and are the mooring line drag

forces in the perpendicular and normai directions respectively to the motion of che catenary evaluated at equilibrium; and _4FA and

¿ are the first order perturbations of these forces.

A?, f.

and all coefficients in (32) for surge, sway, yaw, and turret

rotation equations are defined in the Appendix.

By introducing the differential operator V = did,, the equations of motion can be written as:

+ A,2V+A,3)Ax+(A21V2 +AV+A,)4y

+(A31V2 +AD+A33)4y+(A41V2 +A2V+A,)4. = O, (33)

(B1V2

+ßD+B3)4x+(B21D2 +B,P+B)A;

4(B31D2 + BD+B&4p' +(B41D2

+ 842Vf Q)r = O,

(34) (C11f12 + C,2V + C13)4x +(C21D2 +D+ f,)4y +(C31D2 +CD+C33)4y+(C41V2 +C42V+C)A'T = 0. (35) (11V2+Ll2V+D3)4x+(D11V2 +D,2D+L)Ay

(D1D2 +DV+D)Ayr+(D41f72 +DV1'D,3)4y

= 0. (36)

The expansions of coefficients A,1-D43 are shown in _[16]. The determinant of these coefficients gives the characteristic equation for any one of Ax, Ay,

4y, or 4

as

Acy8+Bc7+Ccr6+Da5+Ea4+Fa3+Ga2+Ha+I =0.

(37) Coefficients A - I are the Routh-Hurwitz coefficients of the

eighth order TMS nonlinear dynamics. The expansions of these

coefficients are too long to be included in this paper. Suffice to

say that these can be easily coded even in a small PC. For guidance in deriving these expressions, see Lili for the sixth

order nonlinear dynamics of a spread mooring system.

¡11.1.

nalytIcaI eWressions for

identification

of

morphogeneses In TM

The characteristic equation (37) in a can be written in the form:

A[(a-a1Xa-a2Xc-a3)(a-a4)(a -a5Xa-a6)

(a-a7Xa-ag)]=0.

(38)

where o, i =1...8 are the eigenvalues of the eighth order system, and the roots of equation (37). An equilibrium position

is stable if all the eigenvalues have negative real parts [28]. Comparison between equations (37) and (38) produces the following relations between the coefficients and the roots of the eighth order system:

*

B/A

₌

(8 terms).

C/A=Xc7aj.

sj,

(28 terms). i=tj=2

678

D/A=->a,a1ø, ijk, (56

terms).

i=Ij=2k=3

5678

_ijkJ.

(70 terms), i=Ij=2k=3=4

45678

F/A

= -

aøJakala_. i

j

k ¡ m, i1 j=2ka.314m5 (56 terms).

345678

cTapkaa,.

iIfr2*'.3b'4=5a=6

i*jktlm*n,

(28 terms).

2345678

H/A =

-îIj.Y4.5=6o.7

i*j*k1*mno,

(8 terms).

I/A =

Expressions (39)-(46) are functions of TMS particulars, and

thus ase real numbers. These equations provide eight necessary

conditions for the stability of a TMS equilibrium as follows:

B/A>0. C/A>O, D/A>0, E/A>O, F/A>O, G/A>0,

H/A >0 aixl I/A > O. For the eight-order system, the necessary and sufficient conditions for TMS stability are compiled into the following eight criteria [241:

Caii>0.

(47) C12

= - - -

C D

_AB

>0.

D B(BE-AF)0

CR3_A _A(BC-AD) C

E(BG-CF-AHXBC-AD)AF(BE-AF)0

14 - A A[D(BC-AD)-B(BE-AF)J F (BC -

ADBF- D)2 +

H- BflJ B(BJ+ DG)(BE- A + A[(BC_ADXDE_CF+BG_AH)_(BE_AF)2j H(A2DF-A2BH+2k82G-B2CE)- B3G2

(BC-AD{HI(BC-AD)(EH

- D!XCF- DE)-(CH Bfl2] Den6

(BC-AD(GH(CD- BE)+ H(2AEW- BC!)]

+ Dm6

E_.H_BrXE_AF)+81(BG_2AJ1)+ACH21

Der6 HfA(BG_AH)2 +G(A2DF_B2CE)] +

>0,

(52) Dm6 H ¡ (BC_AD4&(DI_FJI_FG)_H2(BC_AD)] Dm7

j

(BC_ADF2(2BG_AH_CF+DE)} A Dm7 ¡ (BC_AD42H(B21+CDF)_3BDFI1 Dm7

¡(BE- AF((FG+DI-EHXRE-AF)-2B1(BG- AI!)J

A Deal ¡ (BE_AF4(ACH2 BDG2)J A Deal ¡ BfBDI(BG_AR)_F(BG_AR)2]

"A

Dm7

I

B2E(DE2h_B2I2_CEFH)Lo , A Dm7

C13f>O.

where Dm6= A[(BC-ADXDFJ+CDH+2BGF)J _Af(BC- ADXBDI+AFH +

&G

+CF2)J

+AL(BE_AFXB2!_BEF+ BDG+AF2)+A2DFHJ

_A4(BE_Afl+(AH_

BG)2+ BCE!!]. C,5 + ;:a

&

>0.

(51) J

C,,. =[(a1 +a2Xa1 +a,Xa1 +a4Xa1 +a5Xa1 +a6Xa1 +a7) (a1 +a3Xa2+a3Xa2+a4XcT2+a5Xa2 +a6Xa2 +a7) (a2 +a5xa3 +a4Xa3 +a5xa3 +a6Xa3 +a7Xa3 +a8)

(ty, _{+t75X0'4 +a6Xa4 +a7Xa4 +tlgXa5 +a6Xa5 +a7)}

(a5 +a8Xa6 +a7Xa6 +a8Xa7 +a5)J. (59)

This statement was proven for a sixth order SMS in

[9].

Expression (59) consists of the product of twenty eight terms, each containing the sum of two roots, or cigenvalues. of the eighth order system.

When the system loses

its stability

dynamically, the real part of a complex conjugate pair of

eigenvalues becomes zero. Expression (59) vanishes when one or more complex pair of eigenvalues crosses the real axis, thus establishing a boundary when dynamic loss of stability occurs. In terms of the Routh-Hurwitz coefficients,

_Cc

can be written as

C =(BC-AD)f (FI -GH)(02G+ BD!- DEF+CF2)J

.4-(BC_AD{H2I(BC_ AD)+(BG- AHXFGH- F2!- EH2)J +(BC- ADH(CF+AH)(EH- Di)+2AB1(DF-BH)J ....(BC...AD42CDHBFG+D(FJIDf)2 +H(B1.-CH)2J +(BE-AF)2[(F21_ FGH+E112)]

«BE - AF'{2F1(ADH - 821)+H(BDG2 +ACH2

«BE -AF)I[(BG - AJIX3BH - DF)+8D21]

+B4F(BG- AH)2+FJi(F(BC- AD) - 2BDE)J +82!2[82!_ D(BG_AH)J

+HL2GH(A2DF_ B2CE)_(BG_AH)3j. (60)

Derivation of the counterpart expressions for a sixth order SMS

is provided in

[91. The boundary determined by BDL is necessary for determining the dynamic loss of stability of the system with respect to any equilibrium position. Setting C,,

equal to zero, however, is not a sufficient condition for the

system to undergo dynamic loss of stability.

A pair of real

eigenvaJues with the same magnitude and opposite signs, or two or more real zero eigenvalues, also would make C,r equal to

zezo. Such exceptions are treated in detail in [16].

The bifurcation boundaries thai define the quabtatively.,

different dynamics experienced by the TMS as parameters are varied can be obtained analytically by setting expressions

(Sl)-and (60) to zero (i.e. making C,,1 and_Cac active). _.

111.3

Outline of the design procedurt

The Thf S design procedure developed in this work consists of the following steps: First, the expressions for the bifurcation

boundaries (57X58) aan be used to construct stability charts in the two or three-dimensional parametric design spaces. These

charts, also known as catastrophe sets, constitute the loci of bifurcations defined by setting C,, and C equal to zero.

Bifurcation boundaries separate regions of qualitatively different

dynamics in a parametric design sp.

These sets can either Den7=(BC_AD)((CH_DG)2+(CF- DE)(DI- EH+ FG)J

(BC - AD)H1(AFG - BC! -2AEH) -l(BC - AD)J

+(BC-AD)[BG(EH + DI- 2FG)+B212J +(BE-AF)2[(FG+D!- EH)]

_(BE_AF42BI(BG_AH)+ACH2+BDG2}

+2GH(B2CE- A2DF) +(BG - AH)3. (56)

The TMS

is stable if all relations (47)-(54) hold. Thus, the stability of the system around an equilibrium position can be assessed without explicitly solving for the system eigenvalues. The first stability criterion (B/A>0) 's always satisfied for ship

hulls

1161. and thus, the number of eigenvalues with positive

real pails (and the corresponding dimension of the unstable

manifold) in the system is equal to the number of sign changes between the values of the stability criteria [5].

111.2

Analytical

_expressions

for

_the

_TMS

Dlfurcatlon boundarlea

The types of bifurcations present

in

a system can be

determined numerically by finding the system eigenvalues and detennining their changes (performing bifurcation search) as one o( more parameters are varied. The boundaries that correspond to

loas of stability

in the TMS are presented in closed-form

analytical expressions in this sub-section. These expressions eliminare the needs to obtain the eigenvalues numerically and to perform bifurcation search. Further, they are physics-based expressions which reveal the exact dependence of bifurcations on design variables. Bifurcation boundaries define regions where TMS exhibit qualitatively similar dynamics in the parametric

design space.

TMS exhibit both static and dynamic loss of stability 113. 15, 16]. both of which are described below.

Static Loss of Stability. Static loss of stability arises when a real cigenvalue crosses from the negative to the positive real axis as cee parameter is varied [1g, 28], and may appear in the form of z pitchfork, saddle-node, or transcritical bifurcation.

The boundary where static loss of stability occurs

(BSL) is

obtained by setting expression (46) equal to zero. i.e..

8SL

=

C, i/A=

=0.

(57)

The term

I/A is the product of the eight eigenvalues of the

TM& As cee real eigenvalue crosses zero along the real axis as

a parameter is varied, equation (57) becomes active, thus

tabIishing the boundary where static loss of stability occurs. This bowidary repi'esents the necessary and sufficient conditions

stic toss of stability (9].

Dynaaiic

Loss of Stability. Dynamic loss of stability corresponds so Hopf bifurcation, and is realized when a pair of complex conjugate cigenvalues crosses the complex axis with

aoa-zero speed (18,

28]. The boundary where dynamic loss

(BDL) of stability occurs in the system can be determined

a1yticafly by multiplying the first seven stability criteria and setting this new expression to zero, i.e.

IDI.

Car C,1C,C,jC,4CC,C4 =0, (58)

wIze

it ena be shown that iS

be obtained numerically with eigenvalue analysis and bifurcation

theory 113. 15), or analytically as defined by expressions (57) and (58). In each region of a catastrophe set, the behavior of

the system exhibits qualitatively different dynamics, such as a

stable node, unstable node, stable limit cycle. etc. The stability criteria (47)-04) are used then to determine the morphogeneses across bifurcation boundaries. This is achieved by finding the

dimension of the unstable manifold of a

particular TMS configuration based on the number of sign changes in the stability criteria. This is illustrated in the next section in Table 3. Neat, since plotting in design spaces of more than three dimensions is not possible, sensitivity analyses

of the

bifurcation boundaries with respect to one or more parameters are

performul. This is achieved by taking partial derivatives of

expressions (57) and (58) with respect

to one or more

parameters, and evaluating the resulting changes in the bifurcation boundaries. Thus,

it

is possible to determine

analytically the dependence of the TMS dynamics on any design parameter.

In practice, the designer starts

with a

specific TMS

configuration that satisfies the design requirements, the location

where the system is to be deployed, time of operation,

environmental conditions. etc. Then, the designer uses simulations to assess the behavior of the system, and trial and

error to improve the limited picture of the

TMS dynamics revealed by simulations. The design methodology developed in

this pagr makes it possible to select appropriate TMS design parameters without resorting to thai and error and performing nonlinear time simulations. In the following section, we present a specific application of the TMS design methodology outlined above. This example reveals the effect of certain

parameters on the TMS dynamics as well.

IV. APPLICATION

In this section, we first construct a senes of catastrophe sets

of a T'MS by finding analytically the bifurcation boundaries and their associated morphogeneses. The direction of external excitation, the turret position, and the mooring line pretension are used as design parameters. A simple example of sensitivity analysis of the bifurcation boundaries with respect to the turret position is used then to understand the dynamics exhibited by the ThlS

To facilitate discussion, this example is focused on the design

of a TMS based on its dynamics near its principal equilibrium (PE) position. PE is the weathervaning position that the ThiS achieves in response to the environmental excitation. The geometric opesties of the tanker T'MS used in this example are

shown inbk i (131.

Table 2 shows the values of the TMS

design psqne*ers relating to the mooring lines and the external cxcisation..ihat are fixed throughout this application. The

momea* fnc*ioo coefficient r is taken as 1540 Nm.

Figure 3 shows a series of catastrophe sets of the tanker TMS in the two.dimensional design space for the principal equilibrium position (PE). The variables shown in Figure 3 are the mooring

line pretension (set equal in each mooring line, and ranging from 1250 KN to 1450 KN) and the direction of external excitation (cwyen*, wind, and waves which are assumed to be cotIln ie. act ¡n the same direction, hereafter denoted by )

ranging troia 0 to i80.

DÇ/L (0.39, 0.40, 0.42 and 0.45), the locMioo of the turret with respect to the center of gravity of

the systesa. is used as the third parameter. The catastrophe sets

of Figure 3 have been constructed analytically by setting the

expressions for the bifurcation boundaries (57) and (58) equal to

zero. Only dynamic loss of stability occurs in the ranges of the

specified parameters, and thus only the boundary for which BDL is zero is shown in the figure.

Table 2: Fixed design parameters of l'MS

Mooring Line Propertie number of lines = 4 ABS = 5159 KN/line length of lines = 2625 m

orientation of line I = 0

mooring line spacing = 90 fraction of attachment = 0.75

Environmental Conditions

water depth = 750 m current speed = 3.2 knots sig. wave height = 3.66 m sig. wave period = 8.5 sec wind speed =0 knots

In Figure 3, there are four regions (numbered I to IV) of

qualitatively different dynamical behavior around PE. Regions I and Ill correspond to the dynamics exhibited by the system for

D/L values of 0.42 and 0.45: this dynamic behavior is

denoted by A in the figure. Regions Il and 1V correspond co the

dynamics exhibited by the system for D/L values of 0.39 and

0.40; this dynamic behavior is denoted by B.

Region I (R-I): The principal equilibrium (PE) is stable. All

eigenvalues of the system have negative real parts and therefore this equilibrium is a stable focus.

Region II (R-Il): The principal equilibrium is statically unstable

with a one-dimensional unstable manifold. One real eigenvalue

has a positive real part (unstable node). Two statically stable equilibria exist in this region as a result of the static pitchfork

bifurcation [28) of PE.

Region ill(R-III): The principal equilibrium is unstable with a two-dimensional unstable manifold (i.e. a complex conjugate pair of cigenvalues with a positive real part).

A 4namic

bifurcatico (Hopf) occurs when crossing from R-1 to R-W [25]

resulting

in dynamic

loss

of

stability. The ensuing

morphogenesis is the development of a limit cycle around PE. Region IV (R-1V): The pnncipal equilibrium is unszabk with a

three-dimensional unstable manifold (i.e. a real positive eigenvalue and a complex conjugate pair of eigenvalucs with a positive real part). A dynamic bifurcation occurs when crossing

from R-U' to R-IV, thus resulkag in a statically a well as

dynamically unstable principal equilibrium The dimension of

the unstable manifold in this region around PE corresponds to the minimum dimension for the onset of chaos (25).

Regions I and 111 (bebabior A) are located to the right and to the left of the bifurcation boundaries for the values

of D/L z

0.42 and 0.45. That is, the TMS is stable to the right of the

Table I: Geometric properties of TMS

fropert'(

length overall (LOA) 272.8 m

length of the waterline

(L)

259.4 m

beam (B) 43.10 m

draft (D) 16.15 m

turret diameter (Di) 22.50 m

block coefficient (C,) 0.83

bifurcation boundaries and becomes dynamically unstable (i.e. undergoes a Flopf bifurcation) as the bifurcation boundary is crossed from right to left in Figure 3. Since the system loses its stability dynamically, the TMS oscillates around its principal equilibrium position, and no other equilibria that may attract the

trajectories exist.

Regions H and IV (behavior B) correspond to the right and to

the left of the bifurcation boundaries for che _DCG/L values of 0.39 and 0.40. The PE is statically unstable for all ranges of

parameters considered in Figure 3. As the bifurcation boundary

is crossed from right (11) to left (IV), dynamic loss of stability

of the PE occurs; that is. the PE undergoes a Hopf bifurcation.

In R-IV, the system PE is statically as well as dynamically

unstable, thus turning into a strange attractor [25J. Since the

system has undergone static loss of stability, two additional equilibria exist [15) as a result of a supercritical pitchfork bifurcation [15, 28]. These equilibria may be dynamically

unstable (13J.

Table 3 shows the values of the stability criteria for geometry

points S1 and S2 depicted in Fig. 3 (corresponding to

= 90'

and pretension values of 1300 KN and 142.5 KN respectively) about PE for the values

D/L=

0.40 and 0.45. As shown in the table, the number of sign changes between the stability criteria(NSC) are three and one for

D/L=

0.44) and two and

zero for

D/L=

0.45 respectively. This number indicates the

number of cigenvalues with positive real parts in the system.

and shows chat a dynamic loss of stability in the system arises when moving from the higher

₍₂₎

to the lower (Si) pretension in both cases, as denoted by two additional eigenvalues with positive real parts (a complex conjugate pair).

Table 3 also shows that the system is stable for point S2 and

D/L=

0.45. As the location of the turret decreases, geometry 2 becomes statically unstable, as denoted by one sign change between the higher (0.45) and lower (0.40) values of

Dc/L

in

the table.

Table 3: Stability criteria for geometry points S and Sa about PE

The catastrophe sets of Figure 3 cannot fully explain the drastically different behavior exhibited by the system as the turret position,

_DrIL,

changes. To understand the behavior

of

the system as a function of _DCG. the values of

_{Cas and C.}

are

plotted in Figures 4 and 5 for pretension values of 1300, 1350 and 1400 KN for = 90 in the range 0.30

Dcc/L

0.50.

Due to the symmetry of the

system about its principal

equilibrium with respect to the environmental conditions, and

the SPM-like behavior of the TMS [15], the values of

C,

and

C,,

are sufficient for determining the dynamics of TMS near the principal equilibrium [IO],

C,>

O and

C>

O imply static

and dynamic stability respectively.

As shown in Figure 4, the PE of the TMS is statically unstable (C45<0) for _Dcc/L< 0.40274. A static bifurcation (Cas = 0) occurs ai Dcc/L = 0.40274. and is denoted by BP in the figure. The system becomes statically more stable as the turret is moved further forward. The static bifurcation of PE occurs at the point for which

D NUsin

-

N1.U2 sin cos + NU3 sin3

NEr.,U4 sin3 cos-

,

Y,Usin+ YVU3 sin3 +_YYW,U5

_{sin5 Ô+,,}

where = - a,,, and the overbar indicates the equilibrium position. In this particular case, where the external excitation is collinear to the current direction, this expression is reduced to

N,, +

N,,U+2pgL2C/U

-Y',

Expression (62) shows that the static bifurcation point is the same with respect to the turret position independently of

mooring line configuration. pretension, and direction of external excitation (i.e. a SPM-Iike behavior).

Figure 5 shows the effect of turret location on C,. for three

different values of mooring line pretension. As shown in che figure. the dynamic stability of the system is drastically affected by changes in the pretension of the mooring lines and the turret location. The bifurcation points (BP), defined by C,= 0. strongly depend on turret position and pretension. The dynamically stable range around PE (C,> 0) increases with

increasing pretension, as shown in the figure. Notice in Figure

S that the bifurcation point (BP) for the smallest pretension (1300 1(N) occurs at the point Dr/Lz 0.38738. To the right of this value, C is negative and the system is dynamically

unstable (oscillatory). In the range 0.38738 D$.YJ/L

0.40274. the system is both statically and dynamically unstable (refer to Figure 4) as determined by negative values of _Car aDd

C,.

It is worth noting that the counter intuitive observation

that placing the turret toward the bow has a destabilizing effect The sensitivity of cxpressipnz Car

aiI

Cac with respect to changes in Dr (i.e. _{s= C'arItDcc aral C7,c}

_dCacIcv)

is shown in Figures 6 and 7, respectively, for several values of mooring line pretension (O = 90).

Figure 6 shows that C is constant, and drastically increases with increasing moonng line pretension. For statically stable Drc/L= 0.40 Si Sa Dcc/L Si 0.45 Sa Ca 28081 2.8383k 2.8083 2.8384

C,

13.052 22.64 13.144 22.776 C,3 199.42 333.71 197.53 331.10 C,,

c,

14.371 108.97 44.465 274.4Ò 23.720 9.0732 66.398 149.56 C'1,5 -0.6749 -1.1363 -12.185 24.135

C,,

2.8616 -89.284 19.230 6.3385

C,

-0.3570 -1.3587 6.0915 23.180 NSC 3 1 2 0

systems, an increase in pretension increases the static stability of the syszein around PE. Notice, however, that an increase in

pretension for a statically

unstable system yields a more

statically unstable system around PE.

FIgure 7 shows that C vanes widely with turret position and moonng line pretension. As the pretension in the system increases, the value of C becomes more sensitive to small

changes in turret location. The relatively large values of C, and C. in Figures 5 and 7 imply that for two dynamically unstable systems, the one with higher pretension will have larger amplitude oscillations.

An alternative way to understand the behavior of the TMS as

the location of the turret varies, is to construct catastrophe sets as functions of D. Figure 8 shows a series of catastrophe sets

in the (D/L, O) plane about PE for several values of

pretension, ranging from ¡250 KN to 1400 KN. The regions that &oc the dynamics of the system are the same as those in FIgure 3. In Figure 8. however, it is easier to understand that a

pitchfork bifurcation occurs when moving from R-1 to R-IL, and Hop( bifurcatsons our when moving from R-I to R-III and R-II

to R-tv. These sets show how different values of pretension and

direction of excitation do not affect the static stability of the system, whose bifurcation is determined by equation (62). Increasing the mooring line pretension increases the stable region R-I. while environmental excitation at angles in the

oeighlxrbood of

= 90 increases the unstable domain. This is chae to the fact that the drag due to the mooring lines at these

positions is smaller, thus, destabilizing the system.

The values of C,5. C. C,

and C have been used co

visualize the effects of the turret

location, mooring line pretension., and direction of excitation on the dynamics of TMS and the bifurcation boundaries. The analytical expressions for

C4,. C. C,. and C. can be used to determine the relative strength of degree of stability or instability as a function of its

position in the catastrophe set.

This measure cannot be

visualized in any o the catastrophe sets shown in Figures 3 and

g-V. CONCLUSIONS

The paeliminasy design stages of Turret Mooring Systems can

be greatly simplified uing the analytical expressions for the stability altena and bifurcation boundaries presented in Section 1H. The numerical applications in Section IV illustrate how t eaptessions. derived from first physics-based principles, can be used in practice to analyze the slow motion nonlinear dyn__=rs of different TMS configurations and to design TMS

Lsed on is atabiy characteristics.

The design methodology

peesend in dus paper yields qualitatively the same results as

the TMS design methodology based on nonlinear dynamics and

mnieal computation of the bifurcation boundaries developed in 113. 15J. Bc*h lediniques virtually eliminate trial and enot' in pnraicn selection, thus minimizing the number of nonlinear tinte simulations the mu be peormed. The theory derived in this pen can also be used to determine the sensitivity of the stability criteria and bifurcation boundaries on TMS design variables, and thus the effect of each design variable on the system dynamics, as shown in Section (V. This method

elii'

the ocied for graphical representation, which is

restnicand as three-dimensional spaces. The complexity of the

analytical expressions for the stability criteria and bifurcation

boundaries, proves that it is unlikely that any rules of thumb can be derived in TMS design. A thorough study of TMS nonlinear

dynamics is mandatory. The similarity of TMS with SPM

systems, however, can help simplify and comprehend the TMS nonlinear dynamics [15).

In this work, only the principal equilibrium of the system has been studied. A complete preliminary design consists of finding

all TMS equilibria, and studying their stability characteristics

using the methodology presented in this paper. This methodology is capable of handling several parameters at once

without the need to resort to visualization of several

three-dimensional stability charts.

ACKNOWLEDGMENTS

This work is sponsored by the University of Michigan/Industry Consortium in Offshore Engineering. Industry participants include Amoco, Inc.; Conoco, Inc.; Exxon Production Research; Mobil Research and Development; Shell Companies Foundation; and Petrobrãs. Rio de Janeiro, Brazil

REFERENCES

[I J Bernitsas M.M. and Garza-Rios, L.O., 'Effect of Mooring Line Arrangement on the Dynamics of Spread Mooring Systems,

Joirna1 of Offshore Mechanics and Arctic

En,gineering, VoL I 18, No. 1, February 1996, pp. 7-20.

[2] Bernitsas M.M. and Garza-Rios, L.O., Mooring System Design Based on Analytical Expressions of Catastrophes of

Slow-Motion Dynamics,' Journal of Offshore Mechanics and Arctic E.ngirseeri.ng, Vol. I ¡9, No. 2, May 1997. pp. 86-95.

Bernitsas M.M. and Papoulias, F.A.. 'Nonlinear Stability and Maneuvering Simulation of Single Point Mooring Systems.' Proceedings of Offshore Station Keeping

Symposium, SHAME. Houston, Texas, February 2, pp. 1-19.

[4] Cox, J.V., Statmoor - A Single Point Mooring Static

Analysis Program.' Report No. AD-AI 19 979, Naval Civil Engineering Laboratory, June 1982.

[S] D'Azzo, LI. and Houpis, C.H., Linear Control System Analysis Design, McGraw-Hill, New York, 1915.

dc Boom, W.C., 'The Development of Turret Mooring

Systems for Floating Production Units.' Proceedings of the 21st Offshore Technology Conference, Paper OTC.5978. Vol. II, Houston, 1'X, 1989, pp. 201-212.

Fernandes, A.C. and Aratanha. M., 'Classical Assessment to the Single Point Mooring and Turret Dynamics Stability

Problema,'

Proceedings of the

15th International Conference on Offshore Mechanics and Arctic Engineering

(ØMAE), Voi I-A, Florence, Italy, June ¡996, pp. 423-430.

[8) Fernandes, A.C. and Sphaier, S., 'Dynamic Analysis of a

FPSO System,' Proceedings of the 7th International Offshore and Polar Engineering Conference (ISOPE), Honolulu, Voi 1. 1997, pp. 330-335.

[9]' Garza-Rios Eychenme, LO., Developmeiu of a Desigr Methodology for Mooring Systenu Based on Casastroph

Theory. Ph.D. Dissertation, Department of Naval Architecture and Manne Engineering, The University of

Michigan. Ann Arbor. May 1996. (3]

[IO) Garza-Rios. LO. and

Bernitsas. M.M.. "Analytical Expressions of the Bifurcation Boundaries for Symmetric

Spread Mooring Systems."

Journal of Applied Ocean

Research, Vol. 17. December 1995, pp. 325-341.

(II) Garza-Rios. L.O. and Bernitsas. MM., "Stability Criteria for the Slow Motion Nonlinear Dynamics of Towing and

Mooring Systems."

Report to the University

of

Michigan/Sea Grant/ Industry Consortium in Offshore

Engineering, and Department of Naval Architecture and Marine Engineering. University of Michigan. Ann Arbor. Publication No. 332, November 1996.

(12) Garza-Rios,

L.O. and

Bernitsas, MM.. "Analytical

Expressions of the Stability and Bifurcation Boundaries for

General Spread Mooring Systems,"

Journal

of

Ship

Research, SNAME. Vol. 40. No. 4, December 1996, pp.

337-350.

[13) Garza-Rios. L.O. and 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). Vol. 2. Deift. The Netherlands. July 1997. pp. 177-188.

[14] Garza-Rios, L.O. and Bernitsas, MM., "Mathematical

Model for the Slow Motion Dynamics of Turret Mooring Systems." Report to the Urnversity of Michigan/Industry Consortium in Offshore Engineering, and Department of Naval Architecture and Marine Engineering, Ann Arbor. Publication No. 336, February 1998.

(IS) Garza-Rios, L.O. and Bernitsas, M.M., "Slow Motion

Dynamics of Turret Mooring and Its Approximation as

Single Point

Mooring. in review, Applied Ocean

[24] Routh, EJ.. Dynamicsof a System

of

Rigid Bodies, Mac-Millan. New York, 1892.

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." Journal ofthe Society of Naval Architects ofJapan, Vol. 160. 1986, pp. 93-104. Tanaka, S.." On the Hydrodynamic Forces Acting on a Ship at Large Drift Angles," Journal

of

the West Society of Naval Architects ofJapan. _{Vol. 91. 1995, pp. 81-94 (in} Japanese).

[28] Wiggins, S.. Introduction to Applied Nonlinear Dynamical Systems and Chaos. _{Springer-Verlag. New York. 1990.}

APPENDIX

Expansion of generic equation (32): In surge, the left hand side takes the fonn given in equation (A-1).

f =

(ni + m )cos

(cos7_

sin

Ji

+(m + m )sin

{.Lsin

+ --cos Jd

+[(m +mi)_(m+mv_L}4P

(:

i;'

_wrL.

d)

(A-1)

Research.

[16] Garta-Rios, L.O.

and Bernitsas. M.M.. "Analytical

Expressions for

Stability and

Bifurcations of Turret

Mooring Systems." in review. JournalofShip Research.

[171 Garza-Rios, L.O., Bernitsas. M.M. and Nishimoto, K..

Catenary Mooring Lines With Nonlinear Drag and

The coefficients appearing on the right hand side are defined below.

M

=XN+F,

(A-2)

a =

(A-3)

Touchdown," Report to

the University of Michigan. _{b1 =c21 =}

_-cosi,

_(A-4)

Department of Naval Architecture and Marine Engineering.

Ann Arbor, Publication No. 333. January 1997.

C =-b =-sinj,

(A-5)

[181 Guckenheimer. J. and Holmes. P..Nonlinear Oscillations,

₄

=a

=r«)sinß«,

(A-6) Dynamical Systems, and Bifurcations ofVector Fields.

Springer-Verlag. New York. 1983. e1

=-f»sm(y.'

T"i).

_(A-7)

[191 Lanres, J.?. and de Boom. W.C., "Analysis of Turret

Moored Storage Vessel for the Alba Field," Froceeding of

=-<D+0cosßt),

(A-8)

the Sixth International Conference on the Behaviour of

Offshore Structures (BOSS), Vol. I. London, U.K.. July C2j (A-9)

1992. pp. 211-223. 1,.

[20) Leite. AJ.P.. Aranha, J.A.P.. Umedaj. and de Conti.

In sway, the left hand side of equation (32) becomes

M.B.. "Current Forces in Tankers and Bifurcation of

Equibbrium of Turret Systems: Hydrodynamic Model and

f=m+m,)sini._?sini?)4k

Experiments," fouinai ofApplied Ocean Research.in press.

[211 Mack. RC., Gruy, R.H. and Hall, R.A., "Turret Moorings

-..

(a)ay

for Extreme Design Conditions." roccedingi of the 27th

_{+(m+m)cvsrr4y -I--sinp'}

Offshore Technology Conference. Paper OTC-7696, Vol. II,

+_Lcos!jl1Y

Hoosson. TX, 1995. pp. 23-3 I.

(22) Maglie, 1.1..., "Ship Maneuvering and Coçtrol in Wind," _+[(m

_{+mz)_(m+ai,W_?1j4,,}

SNAME Transactions, Vol. 88. 1980, pp. 257-281.

[23) Nishimoto, K., Brinati, H.L and Fucatu C.H., "Dynamics

of Moored Tankers SPM and Turret," Proceedings of the 7th _(A-LO)

International Offshore and Polar Engineering Conference (ISOPE). Vol. I, Honolulu, 1997, pp. 370-378.

The coefficients appearing on the right hand side of equation

(32) are defined below.

M =Y,, +

In yaw, the left hand side of equation (32) becomes

f

=_(Lcosi_iLsin }*

(

Ç.

Th0

dp'

-V

(A-19)

The coefficients appearing on the right hand side of equation

(32) are

Ag I: Geonruy Moixrng Sy (TMS)

(A-Il)

In the turret rotation equation. = <' and =

Rgw 2:Iniee rekrenci frames in Th(S

a- =d ='sinß.

b1 =c24 =sin,

c- =-b, =-cosi.

d1 =-(D +7cosßW).

e1

-2 fdCOS(7Ot'+7T

(A-12) (A-13) (A-14) (À-15) (A- 16) (A-17) (A-IS)

a =''cosß,

e21

-i).

_{(32) are defined below.}

A?=o. (À-29)

a- = -e21

_Drf(i)7o

_-70e).

(A-30)

b = C21 =

Sifl(7» + ir).

(A31)

= -b21 =-'-f.cos(r0+1r).

(À-32)

4

=_Df,.(0cs(yY)+r_),

(À-33)

e1 =

_-t.f.((?'

(i(I)

-

ye»))+

.T_f(i)).

(À-34)

024 = _-f4 cos(1 -

y,0)), _(A-35)

d21 = D ..j(I) sin(70W +

- )

(A36)

M = N,, + N,

a = -d2 = Dc'sinß".

(À-20) (À-21)

b1 =c21 =Dsin.

(A-22)

c. = -b = -D cosi.

(A-23)

d1 =-D(D

(A-24) ei

_{_Dcctfa(cos(yo(') +}

(A-23) 024 = (A-26)

e24 = Dc,

+ij

(A-21)

In the surge. sway and yaw equations of motion. TAW = (1) and

= FNt0.

In the turret rotation degree of freedom, the left hand side of equation (32) is

f =

+ r4p' - (A-28)

Rgur 3: Catasuopbe set. 4-line TMS: effect of pretension and diretioc of etanai excilaflon for v4ous twiet posiflocs

Agnee S Effect of 0oe °° Cc foc vanous valves of moenag line pretension;

4-line Th4S

Ag 7: Effect of D on Ck (or vous ,aIve* ci oonng liuc pretension;

4-live Th4S

Fgute 4: Effect of 0oe oc C, foc vous values of moonog line pretension;

4-line TMS

ligure 6: Effect of Doe os C for various vaines of mooring line pretensio.

4-hoc litS

Rgwe $: Catassrop ve. 4-line TMS: efkc* ai mires location a cive. ezcgthas 1e foc vso .sooclng line xesio

II

.j

4

...

.-.-.--.---A

III

.

D4L C DSMd JI

. -.I_w w

SM IM 537 SAS SM IA SAI SM 543 SM SM SAS $47 SM SM Si

t.m

4_.. JI u. \_

--

¡

i

I3UI .

\

'

SAS SM 535 54 141 IC 543 ¡55 50 SM SAS SM SM Si '4

p.iSw

-533 SiS 5.37 SM SM 5.5 5.41 S.0 SM 5.44 SC SC SAI SC 5.45 Si

I.

IM (III.

J:

-: . t I3 .

V.

n I__li

I

-f

5

. i_54,.

_/

/

/

/

1.541 /

...

_>'---

...

t

C_50) 535 SAS LO 535 lAS SA 5.1 5.43 50 5.. 543 5.45 547 545 545 S.J

CHNSCHE UPUVERSITEIf Zaboratorium voor Scheepshydromechanica I rchief Mekolweg 2, 2628 CD Deift

'ilL 016- 1U8?3 Fazt 015 -18183$

M. M. Bernitsas

P ro tesso r Mem. ASME michaelb©engin umich.edu L. O. Garza-Ríos Research eIow, Assoc. Mem ASME

luisinon@engin.um ¡ ch ed u

Department of Naval Architecture and Marine Engineering, The University ot Michigan. 2600 Draper Road, Ann Arbor, Michigan 48109-2145

I

Introduction

Turret mooring systems (TMS) have been developed recently

for stationkeeping of ships and floating/productionsystems for

oil and gas production and recover-v. TMS consists, in general. of a vessel and a turret to which several mooring/anchoring legs are attached. The turret includes bearings which constitute the interface between turret and vessel, allowing the latter to rotate freel _{around its anchor legs in response to changes in} environmental excitation and system dynamics. Thus. TMScan weathervane by changing its heading relative to the environ-mental excitation. TMS may be mounted externally ahead of the bow or aft of the stern, or internally within the vessel.

Further, turrets may be disconnectable or permanent. depending

on the type of mooring, projected time of operation, and envi-ronmental conditions [19, 21].

The first TMS was deployed in 1986 offshore Australia.at a

water depth of 120 m [6]. Such system was designed with a disconnectable riser turret mooring (RTM) to allow the tanker to leave the site of operation in case of harsh environmental

conditions. Recent advent in technology made possible deploy-ment of more sophisticated types of TMS in deeper waters [211.

As the need to venture into even deeper waters increases, the

design of TMS requires a thorough understanding of its

dynam-ics. TMS nonlinear dynamdynam-ics. however, are complex to under-stand because TMS have a high number of design variables (parameters). such as number. length. pretension. and orienta-tion of mooring lines: size and posiorienta-tion of the turret:

environ-mental conditions; etc.

To understand the qualitative properties of TMS. a design methodology based on nonlinear dynamics and bifurcation the-orv has been developed [13. ¡5]. This methodology consists of evaluating the stability of a particular TMS equilibrium, or

group of equilibria. using eigenvalue analysis and numerical search for the structural changes of eigenvalues (bifurcations) as parameters are varied. These can be represented graphically

Contnbuted b the OMAE Dj'.ision for publication in the iO1RJ. OF OFF.

SHOR.E MECHÀ'.:CS ND ARC-nc ENGL%EERING. Mariuscripi received b the OMAE

Disision. .tir.h 0. 998, revised m.Inuscnpi receised Apnl 7, IN8 Technicai

Editor: S L:u

154 I Vol. 120, AUGUST 1998

Turret Mooring Design Based

on Analytical Expressions of

Catastrophes of

Slow-Motion Dynamics

Analytical expressions of the bifurcation boundaries exhibited by turret mooring systems (TMS), and expressions that define the morphogeneses occurring across

boundaries are developed. These expressions provide the necessary means for evalu-ating the stability of a TMS around an equilibrium position, and constructing

catastro-phe sets in two or three-dimensional parametric design spaces. Sensitivii-v _analyses

of the bifurcation boundaries define the effect of any parameter or group of pa

rame-ters on the dynamical behavior of the system. These expressions allow the designer to select appropriate values for TMS design parameters without resorting to trial and error. A four-line TMS is used to demonstrate this design methodology. _The

mathematical model consists of the nonlinear, fifth-order, low-speed, large-drift

ma-neuvering equations. Mooring lines are modeled vi-ith submerged catenaries, and include nonlinear drag. External excitation consists of time-independent _current, wind, and mean wave drfl.

Copyright © 1998 by ASME

in design charts (catastrophe sets) that help visualize the effects

of design variables in design spaces up to three dimensions. Previously. Garza-Rios and Bernitsas [2. 10-12] devised a

design methodology for general spread mooring systems (SMS) based on analytical expressions of catastrophes of slow motion

dynamics. These expressions make it possible for the designer to study the nonlinear slow motion dynamics of SMS around all equilibria in any parametric design space. and identify the associated morphogeneses. The designer is no more limited to three-dimensional design spaces [2] due to graphic

representa-tion. Analytical expressions for the stability criteria and

bifurca-tion boundaries have been derived for TMS. and successfully compared to numerically obtained system eigenvalues and its

bifurcations [16]. These expressions are physics-based. The aim

of this work is to use those analytical expressions to deelop_a

TMS design methodology by deriving analytical expressions of sensitivity of bifurcations on parameters. and applying them in TMS design. This methodology completely eliminates trial and

error. Further, it reduces dramatically the required nonlinear

simulations since the morphogeneses occurring across bifurca-tion boundaries reveal the qualitative behavior of TMS

dynam-ics. Existence of bifurcations in TMS nonlinear dynamics has

been demonstrated by [7, 8. 20, 23] by modeling TMS as SPM systems.

The formulation of the TMS design problem is presented in Section II with a brief description of the mathematical model. mooring line model, and model for the external excitation. In Section Ill, a design methodology based on analytical

expres-sions for the TMS stability criteria and bifurcation boundaries is developed. In Section IV. a particular application of this theory is presented. This includes construction of bifurcation boundaries in a two-dimensional parametric design space and the effects of a parameter on these boundaries I sensitivity

analy-sis). Conclusions are summarized in Section V.

II

Formulation of the Design Problem

The slow motion dynamics of TMS in the horizontal plane (surge. sway and yaw) is modeled in terms of the vessel/turret equations of motion without memory. catenan mooring line model, and external excitation. The sstem considered in this

Dutjon (a) of Currnni (a.). Wind (a), and Wac,, (a,,,)

(a,,. a,, and a,, rndcpendcni) 270

Fig. i Geometry of turret mooring system (TMS)

paper corresponds to an interna! turret configuration. andthus

it is assumed that no hydrodynamic forces/moments act directly

on the turret.

11.1 Equations of Motion. The general geometry of a

TMS is shown in Fig. 1, with two principal reference frames: (X, y) = inertial (nonmoving) reference frame with its origin located at mooring terminal 1; (X, Y, Z) = both-fixed reference frame with its origin located at the center of gravity of the system (CG). i.e. vessel and turret combined. In addition, n is the number of mooring lines; (x4. v,Y) are the coordinates of the ith moonng terminal with respect to the (X. y) frame; f'' is the horizontal projection of the ith mooring line: y ' is the

angle between the x-axis and the ith mooring line, measured counterclockwise; and i is the drift angle of the vessel. The

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

Fig. 1.

The intermediate reference frames of the vessel/turret system

from which the TMS mathematical model is derived are shown in Fig. 2. (X', Y', Z') is the turret reference frame with its origin located at the center of gravity of the turret (CGt); and

(X", Y", Z") is the vessel reference frame with its origin located

at the center of gravity of the vessel (CG). Moreover,

D0

is

the distance between the centers of gravity of the system (CG) and the turret: and AC0 is the distance between the centers

of gravity of the system and the vessel. These are related as

follows:

ACGDCG

(1)

Fflv

where m is the mass of the vessel and mT is the mass of the turret. In addition. is the relative yaw angle between the turret and the vesse! measured counterclockwise, as shown in

Nomenclature

CG = center of gravity of TMS CG1 = center of gravity of turret

distance between CG and CG L = vessel length

m = mass of system n = no. of mooring lines

NH = yaw hydrodynamic moment

PE = principal equilibrium position SMS = spread mooring system SPM = single-point mooring

T,,, = horizontal component of mooring line tension

TMS = turret mooring system (x, y) = inertial reference frame with

origin at mooring terminal I

(X. Y, Z) = body fixed reference frame

with origin at CG X,,, = hydrodynamic force in

surge

= hydrodynamic force in sway

= current angle w.r.t. (x. y) = wave angle w.r.t. (x, y) = wind angle w.r.t. (x, y)

tic0

CC

Fig. 2 Intermediate reference frames in TMS

Fig. 2, and ç!,,. is the absolute yaw angle of the turret with respect

to the x axis. Thus

(2)

The horizontal plane equations of motion of the TMS consist

of the three vessel equations in surge, sway, and yaw, and the corresponding turret equations. Since the turret is fixed to the

vessel, the linear translations of the turret depend on the motion

of the vessel. Thus, the system has four independent degrees of freedom. The dynamics of the TMS can therefore be recast in terms of four equations of motion [14], three equations

de-scribing the motion of the system in surge. sway. and yaw, and

one equation pertaining to the rotation of the turret.

The equations of motion of the system in the (X. Y, Z) reference frame in surge. sway, and yav are [14]

(m + m)ü - (m + m,)ur =

XH(U.u, r)

+ {[T' - F'P'] cos ß" + F'P» sin fiO)) + F,,,g. (3)

(m + m)U + (m + m)ur =

YH(U,u, r)

+

{[T' - F'] sin ß() - F' cos ß"} +

(4)

('a +

= NH(u,u, r) + DCG {

[T'i - FJ'] sin ß()

-

cos ß''} + N,,,,. + N,. (5) where (u, y, r) and (ii, u, ,) 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 = mv + mT): 1 is the moment of inertia ofthe system about the Z-axis: ,n, m,. and J.. are the added

mass and moment terms: F,,,,, F,,.,,. and N,,.,. are external forces and moment acting on the vessel, such as wind and second order wave drift forces [1, 3]; N is the moment exerted

r = moment friction coefficient

be-tween turret and vessel

= angle of external excitation wit. (x, y)

ib = drift angle

= absolute yaw angle of turret = ith eigenvalue

= first-order perturbation operator

= time derivative operator did:

() = equilibrium of ()

on the vessel by the turret; andXH. YRand NH are the

velocity-dependent hydrodynamic forces and moment. These are ex-pressed in the forthcoming in terms of the fifth-order, large-drift, slow-motion derivatives [27] following the nondimen-sionalization by Takashina [26]:

XH = Xu + Xu2 +

+ X,vr, (6)

Y,,, = }Çv + Y,,.t'3 +

Yv5

+

)Ç,,urjri + Y, ,t'jri,

(7)

NH = Nv

+ Nuv +

N,,v3

+ Nuv3

+ N,r

+ N,,.riri -

N,,,uvlrl

+

N.,ur

(8) The terms inside the summation in equations of motion (3)

-(5) apply to each of the mooring lines (i = I...n). TR is

the horizontal tension component of the catenary: F and F are the drag forces on the catenar. in the directions parallel and perpendicular to the motion of the mooring line [17]. and de-pend on the position of the turret with respect to the mooring

points and the velocities of the mooring lines: and j3 is the angle

between the X-axis and the mooring line, measured counter-clockwise

_{(/3 = y}

-The fourth equation of motion describes the relative rotation of the turret with respect to the earth-fixed reference frame, and consists of a moment equation about an axis parallel to the Z-axis (i.e.. Z '-Z-axis). Such an equation needs not be transferred to the (X, Y. Z) reference frame. In the (X', Y', Z') frame, it is given by [14]

= ..L

_{f{[T - F.] sin (ßU)}

y,i))

- FJ) cos

(ßi()

_{- y) }}

+ N (9) where _{is the rotational acceleration of the turret; I. is the} turret moment of inertia about the Z '-axis; D7 is the turret diameter; N is the moment exerted on the turret by the vessel. The terms inside the summation in Eq. (9) are defined as fol-lows: f is the fraction of the turret diameter at which the

cate-nary is attached (0

_f,

I): F and F. are the drag forces

on the catenary in the directions parallel and perpendicular to the mooring line motion, respectively [17]: /3' is the angle between the X '-axis and the mooring line measured

counter-clockwise (/3' = y - /'T); and y. is the angle at which the

catenary is attached to the turret with respect to the X '-axis (fixed to the turret). In equations of motion (3)(5) and (9). it has been assumed that the turret is internal to the vessel and

that no hydrodynamic or external forces or moments act directiy

on the turret.

The moments exerted on the turret and the vessel (N7 and N ) are a result of the relative rotation that exists between

them. These moments are of equal magnitude. and 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. In this paper, the exerted moments are modeled as follows:

N - N = -

(T- - r) (IO)

where r is a small positive friction factor that depends on the geometry, mooring line damping. relative size of the turret.

separation between the turret and the vessel, and external

excita-tion: and r7is the rotational velocity of the turret.

The kinematics of the TMS _{are governed byEqs. (11) (14)}

114]

= u cos¿)s - t Sin tb + U cos _,

= u sin¿ji + i: cos & +

U sin (,

¿ji = r.

where p is the air density: U, is the time-independent wind

speed at a standard height of 10m above water: a, is the relative

angle of attack: A7and AL are the transverse and longitudinal

areas of the vessel projected to the wind, respectively: L is the length of the vessel: c,,(a,). C,.(a,). and C...ja,)are the wind forces and moment coefficients, expressed in Fourier series as

follows:

Coefficients . 9k. and Z,

in Eqs. (22)(24) depend on

the type of vessel, location of the superstructure. and loading

conditions.

The mean second-order wave drift excitation in the horizontal

plane is given by [4]

F,,,,, = p,t'LC,dcos'(a,, - ¿b) (25) = p,gLC sin3 (aw, - ¿b) (26) N.,.,1, = p,gLC.1sin 2(aw, - _{¿ji) (27)}

where p, isthe water density:a,, is the absolute angle of attack;

g is the gravitational constant: and C,d. C,,,,.and Ca,, are the drift

excitation coefficients in surge. sway. and yaw. respectively.

156 / Vol. 120. AUGUST 1998 _{Transactions of the ASME}

C,,,(a,) = + _{cos (ka,) (22)}

1= I C,,. (a,) = k-I i9 sin (ka,) (23) C..ja,) = k, Ç sin (ka,) (24)

Wind forces due to time-independent wind velocity and

direc-tion are modeled as [221

= pnU,C=(a,)Ar

(19)

Fiwind = PU.CV.(a,)AL (20)

N-,fld = (21)

Fsurge = F,wind + FT,i&ase (16)

= F.wind + F, (17)

Nyaw = NWd + Nwn,. (18)

= r7 (14)

where U =

lui

is the absolute value of the relative velocity of the vessel with respect to the water, and , is the current angle whose direction of excitation is shown in Fig. 1.

11.2 Mooring Line Model. The mooring lines of the sys-

-tem are modeled quasistaucally by catenary chains with

nonlin-ear hydrodynamic drag and touchdown [17]. The total tension Tin a catenary is given b

T=T,,+ T= THcosh

(\)

(15) \ T1,, /

where T1. is the vertical tension in the mooring line. P is the

weight _{of the line per unit length, and t is the horizontally}

projected length of the suspended part of the line [17]. 11.3 External Excitation. The model forexternal

excita-tion consists of time-independent current, steady wind, and mean second-order wave drift forces. Each component has an

independent direction of excitation defined in Fig. I. The current

forces which act on the vessel are modeled in equations of motion (3)(5) by introducing the relative velocities of the system with respect to water. Thus.