Preliminary design of differentiated compilance anchoring systems

0. Garza-Rios

Assoc. Mern. ASME

M. Bernitsas'

Mem.ASME .

0

micnaelb@engrnumich.edu

0

w Ianent 01 Naval Architecture and O CL _{Marine Engineering,}

w2

O.$ - university of Michigan, 2600 Draper Road, - . Ann Arbor, MI 48109-2145

K. Nishimoto

rtment of Naval Architecture and Ocean Engineering,

,,JUniversity of São Paulo,

- São Paulo. Brazil

I. Q. Masetti

Petrobrás Research Center/CENPES, Rio de Jäneiro, Brazil

1

Introduction

Petrobrás. the Brazilian oil company, is scheduled to install several floating production storage and offload (FPSO) units using conventional turret mooring systems (TMS) and single-point mooring systems (SPM) near the coast of Brazil. Due to the high cost of installation, PetrobrIs is searching for alterna-tive means to devise more economically and technologically feasible FPSO mooring systems. The environmental conditions of the Brazilian offshore are not as severe as those in the North Sea or the Gulf of Mexico. These relatively mild conditions allow the designer to implement successfully offshore produc-tion systems that could not be utilized under more severe envi-ronmental conditions [5].

The first floating storage and offload (FSO) system was in-stalled in 1993 in Brazil, in the Caravela Field. Santos basin, offshore Santa Catalina. at a water depth of 195 m. The mooring system consisted of eight lines. After intensive analyses and redesign, the FSO system became the first DICAS (differenti-ated compliance anchoring system). a type of spread mooring system (SMS). and is still in operation. DICAS is a system in which different mooring line pretensions, and therefore stiff-nesses, are used at the bow and at the stern of the ship. allowing the vessel to partially weathervane under different environmen-tal conditions. The lines moored forward of the center of gravity of the vessel have higher pretensions than the lines moored aft. This somewhat restricts the bow of the vessel from moving. while allowing the stem of the vessel to move relatively freely. Risers can then be installed near the bow of the vessel, where the horizontal motion is limited. A properly designed and imple-mented DICAS provides an alternative to swivels and turrets. thus reducing the cost of production systems based on existing tankers [13].

Corresponding author

Contributed by the OMAE Division and presented at the 16th lnrentatiônal Symposium and Exhibit on Offshore Mechanics and Arctic Engineering. Yoko-hasna. japan. April 13 17. 1997. of Tiix AMERIcAN SoctaivofMEC)IANICAL

ENGINEERS. Manuscript received by the OMAE Division. September 17. 1997: resised manuscript received September II. 1998. Associate Technical Editor. S. Causal.

Preliminary Design of

Differentiated Compliance

Anchoring Systems

Journal of Offshore Mechanics and Arctic Engineering

Copyright © 1999 byASME

The preliminary design of a differentiated compliance anchoring system (DIC'AS) is assessed based on stability of its slow-motion nonlinear dynamics using bifurcation theory. The system is to be installed in the ('ampos Basin. Brazil, for a freed waler depth under predominant current directions. Catastrophe sets are constructed in a two-dimensional parametric design space. separating regions of qualitatively different dvna,n,cs. Stability analyses define the morphogeneses occurring across bifurcation boundaries to find stable and limit cycle dynamical be/ia vior. These tools allow the designer to select appropriate values for the moonng parameters without resorting to trial and error, or extensive nonlinear time simulations. The vessel equilibrium and orientation, which are functions of the environmental excitation and their motion stability, define the location of the top of the production riser. This enables tire designer to s'errfv that the allowable limits of riser offset are satisfied. The matheniari-cal model consists of the nonlinear, horizontal plane fifth-order large-drift, low-speed maneuvering equations. Mooring lines are modeled by open-water catenary chains with touchdown effects and include nonlinear drag. External excitation consists of time-independent current, wind. and mean isave drift.

The success of the DICAS concept has prompted Petrohrás to consider this unique and relatively inexpensive type of mooring system as an alternative for deeper waters, such as the Campos basin (300 m). Due to the large number of design parameters related to this system (location of the fairlead. number. type and spread of mooring lines, pretension. etc.), the traditional methods of analysis by simulation prove to be ineffective, and sometimes inconclusive [6]. A design methodology for SMS based on nonlinear dynamics and bifurcation theories [2, 6] can be used to account for a thorough preliminary design .of a DICAS. This methodology provides in the design space the catastrophe sets which define regions of qualitatively similar dynamics. Thus, the designer can select appropriate system de-sign variables and virtually eliminate the number of nonlinear slow-motion simulations that must be performed.

In this work, the formulation of the design problem is pre-sented in Section II with a brief discussion on the mathematical model, the mooring line model, and external excitation. Particu-lars of DICAS and environment are defined for numerical appli-cations. In Section III. the preliminary design methodology of a proposed DICAS system based on stability analysis and bifur-cation theory is presented along with several numerical applica-tions. Finally, conclusions regarding the design methodology and guidelines for further research are discussed.

II

Formulation of the Design Problem

The slow-motion dynamics of mooring systems in the hori-zontal plane (surge. s''. ay and aw) are modeled in terms of the vessel equations of motion, mooring line model, and external excitation. The mathematical model presented in this work is based on the large drift angle. slow horizontal plane motion, fifthorder maneuvering equations [16. 17. 181.

11.1 Equations of Motion. The horizontal plane geome-try of a DICAS is shown in Fig. I. with two reference frames: (x. v) = inertial reference frame with its origin located at moor-ing terminal I; (X. Y. Z) = body fixed reference frame with its origin located at the center of gravity of the vessel (CG). In addition. n

is the number of mooring lines; (x, y') are

the mooring coordinates of the ith mooring line in the (x. )

ABS average breaking strength of catenary line(s)

B = beam of vessel

C9= block coefficient of vessel CG = center of gravity of vessel

D = draft of vessel

DICAS differentiated compliance an-cboring system

frame; (x, y)are the body-fixed coordinates of the ith moor-ing line fairlead is the horizontal length of the z th moonng line between the attachment point of the vessel and the moonng point; and cli is the diift angle.

The equations of motion in the horizontal plane are

(m + m1)ti - (m + m,)rv

= XH +

{ Tjge - Fij

+ Fjge (1)

(m +

+ (m + m,),'u

= YH + {T2,y

-

FJ) + Fay

(2)

(I_ + J)i =

NH

{x,(T2 - Fj)

-

y(T.jg. -

Fj)} +

(3)

where in is the mass of the vessel; I. 'is the moment of inertia

about the Zaxis; rn, m. and J- are the added masses and

moment of inertia in surge, sway, and yaw, respectively; u, v. and

r

are the relative vessel velocities with respect to water in surge. sway, and yaw, respectively; XH, YR. and Nfl are the velocity-dependent horizontal plane hull hydrodynamic forcçs and moment expressed in terms of the large drift angle, slow-motion derivatives s follows:

Fig. I Geometry of DICAS

LOA overall length of vessel-L, LWL = length of vessel

= yaw angular acceleratiOn of vessel With respect to water SMS = spread mooring system

a = surge acceleration of vessel with respect to water

(4)

YH = Y,v + Yv3 + Y,,v5

+ Y,jir + Y,jurIrJ

+ Yr:,VITI (5)

Nfl

Nv + Nuv + Nv3 + Nuv3 + N

+

N,rIrf + N,uufrI +

N,v2r

(6) Expressions (4)-(6) have been obtained using the method developed in [12. 20]. The first three terms- in (4) are related to the vessel resistance, R. by [2]

R

= -(Xu +

+ (7)

In addition, and represent the tension components in the horizontal plane of the ith moonng line in the surge and sway directions;

_FJ

and

F)

correspond to the mooring line nonlinear damping components in surge and sway, respectively; and

Fe,

and Nyaw are the external forces and moment acting on the vessel due to steady wind and second-order wave drift forces.

The kinematics of the system aze governed by Eqs. (8)- (10) i = U Coscl' - t sin i/i - U COs a

y = Ucoscl'-+ usin& + Usina

Nomenclature

-

-U, = current velocity

-= sway acceleration of vessel with re-spect to water

a = current angle with respect to (-x. v) frame

= displacement of vessel

(x,,.y,,,.z)

where U = IU is the absolute value of the relative velocity of the vessel with respect to water, and a is the current angle 'measured with respect to the (x, v) frame as defined in Fig. 1.

11.2 Mooring Line Model. The moonng lines of the DI-CAS system are modeled with open-water catenary chains, which include touchdown effect and nonlinear drag [7]. Cate-nary chains are heavy, have high hydrodynamic resistance, are fully submerged, only slighiJy extensible. and have 2-D catenary deformation. For this analysis, mooring lines are classified as single (S) or double (D) [9]. Single mooring lines have an average breaking strength (ABS) of 5159 KN. The ABS for double mooring lines is approximately 9715 KN [11].

Figure 2 shows the geometry of a single open-water catenaiy, with the origin of its reference frame (, i) at the point of contact of the catenary with the ground (x0. yz,), which corresponds to the point of zero slope. The vector = (c.,. .) contains the coordinates of the catenary in the horizontal plane. In addition, (x,,. y. zr,) and (x7, Yr. Zr) are the endpoints of the catenaiy; is the length of the suspended catenary: is

the horizontal projected distance between the endpoints of the catenary. also shown in Fig. 1: f is the horizontal projected length of th suspended catenary; d is the horizontal length of the undeformed catenary; h is the water depth; and 9, is the slopeof the upperen4point of the catenary with respect to the horizontal. The total length of the mooring line . isgivenfrom

Fig. 2 as follows:

(II)

The horizontal and vertical components of the mooring line tension T0 and T, at the attachment point on the vessel can be found by solving the following relations [7]:

"inh()fh(h+2°)

(12)

T,=

Tosinh()

(13) The total tension Tin the catenary is calculated as

(x0.y.z_)

Journal of Offshore Mechanics and Arctic Engineering

FIg. 2 Geometry of open-water catenary

yjstr)

T = + T.: = T., cosh (14)

In Eqs. (l2)( 14). P is the weight of the catenary per unit length (1510 kg/s for single lines: 3094 kgls for double lines [II]). The horizontal mooring line pretension. PRT, is defined in terms of the vessel resistance, R. with respect to the zero current angle as

P1=NR

(15)

where N is a factor greater than zero that defines the horizontal pretension in terms of the vessel resistance. Mathematically, PRT defines the position of the mooring coordinaies of the system in the horizontal plane (i.e.. (x,Y,

v),

I = I...n) by

defining the horizontal projected length of the suspended cate-nary as follows:

( =

sinh

(-f- ih(h

+

2-'fl

(16)

P \PRT

\

P/I

The total pretension T in the catenary is given by [7]

TP=PRI-+Ph

(17) Mooring Line Drag Forces. The drag forZes in the surge and sway directiOns F and Fd on each of the mooring lines

are of the form [7]

F=FAcosFLsin

(18)

Fd = FA sin B + F, cos (19)

where

fi

is the horizontal angle from the X-axi to the direction of the mooring line, and F. and FL are the drag forces in the directions parallel and perpendicular to the motion of the moor-ing line, respectively. These are given by the relations

= PCDxDeff17'fDL'Ii' (20)

I 5mB,.

FL =

pC'01D,ah

k

4

lcos8

(21)

Property LOA LWL(L) B D Ca m, m, I.. J.

-In expressions (20) and (21). p is the water density, D(f is the effective diameter of the mooring line [9],_{CDXand C0. are the} drag coefficients in the directions of the horizontal motion of the mooring line: i' and )"_{are the horizontal velocities of the} mooring line measured at the point of_{attachment to the vessel;} and the coefficient Vo is a function of the energy dissipation due to an in-plane motion caused by the slow-di-ift motion of the catenary [14].

11,3. _{Particulars of DICAS. The DICAS under study}

consi-ts of a converted IPSO [10] with _{nine mooring lines (Six} moored forward of the CO. and three moored_{aft of the CO).} each with nominal catenarv length _{T = 1910 m. The system} is geometrcallv symmetric with_{respect to a prescribed current} direction (zero degree angle), has three different points of line attachment, with three mooring lines attached at each point as shown in Fig. _{I. The geometric properties of the vessel are} shown in Table I [10].

The system is to be designed for operation in a 300-rn water depth environment under _{a current velocity of 2 knots. The} predominant current direction is from the southeast, with a range of excitation of approximately ±45 deg _{[4, 5]. The flexible} risers of the system are allowed to have a maximum relative horizontal displacement of 20-25 percent of the water depth (i.e.. 60-75 m) with respect to their _{initial position (at zero} current angle) [1]. This can be achieved by placing _{the riser} tops as forward as possible on the vessel, where_{the system has} the least horizontal displacement regardless of the changes in environmental conditions. The type of mooring lines (single. double) in the system, and their orientation with respect to the X-axis are shown in Table 2 [101.

ifi

Preliminary Design of the DICAS Using

Bifurca-tion Theory

Nonlinear dynamical systems with several design _parameters require in general trial and error with numerous and lengthy simulations in each trial. Aspects of the methodology developed to design SMS based on their slow-motion, horizontal plane. nonlinear dynamics [2, 3, 6] are summarized in the following section and adopted for DICAS.

III.! _{Theoretical Considerations. The slow-motion dv} namics of DICAS are studied based on nonlinear dynamics and bifurcation theory. In this methodology, the system is modeled by a set of six first-order nonlinear coupled differential equa-tions by combining the equaequa-tions of motion_{(I ) -(3) and the} kinematic relations for thesystem (8) -(10). The stability of a particular DICAS configuration is then obtained by performing eigenvalue analysis around an equilibrium position [2]. If all six eigenvalues have negative real parts. such equilibrium is stable and all trajectories initiated near that equilibrium position will converge to it. If at least_{one eigenvalue has a positive real} part. such equilibrium is unstable and_{a small disturbance from} equilibrium will cause the system trajectories to diverge from ii (19]. Once the stability properties of_{a particular DICAS}

12 I Vol. 121, FEBRUARY 1999 272.8 m 259.4 in 43.1 m 16.15 m 0.83 1.5374 x 1O tons 9.lIOx 10kgs 1.360 x IO kgs 7.180 x 10" kg'm 5.430 x 10" kg'm

under a defined set of environmental conditions_{have been} ana-lyzed, the qualitative behavior of the_{system can be discerned.} Bifurcation sequences are then studied to find qualitative changes in the dynamic behavior of the_{system by varying one} or several design parameters/excitations for _{a specific mooring} configuration [2]. Such sequences are used to determine regions in two or three-dimensional parametric _{spaces of qualitatively} different system dynamics (i.e., stable, unstable, periodic, quasi-periodic, chaotic), and are graphically_{represented with stability} charts known as "catastrophe sets" [61._{The designer can select} appropriate system parameters, thus virtually eliminating the number of nonlinear time simulations in the design process. Nonlinear simulations are used to calculate the dynamic ten-sions, maximum displacement of the vessel, _{relative motion of} the vessel's endpoints, etc.

The number of possible equilibria_{present in the system} de-pends on the system geometry and hydrodynamics,_the orienta-tion and pretension of the mooring lines, _{as well as on the} external excitations [61. The stability analysis for the DICAS under consideration is limited to the principal _equilibrium posi-tion. Due to the relatively high pretensions_{of the forward} moor-ing lines in the system, other equilibrium positions may exist, but most likely will not be attainable [6].

111.2 _{Analysis of the DICAS Mooring System. The}

DI-CAS design process can be studied _{as a function of several} parameters based on the methodology developed in_{[3. 6]. In} this work, catastrophe sets are constructed_{as a function of two} parameters: the fairlead position of the forward mooring lines (rn. ",,), and the current angle Cr. The fairlead position of the aft mooring lines is fixed to the aft_{most part of the vessel (x,/}

L = -0.5.

,,, = _{to allow the vessel to achieve larger} displacements at the stern by reducing the moments created with the rotation of the vessel. The forward mooring lines are attached at two points with the same x,, values, but different y,, values. Since the objective of DICAS is _{to reduce the bow} displacement only, the designer tends _{to place the forward} mooring lines as far forward as possible. _{This geometry, in} general, reduces the horizontal bow displacement _{of the vessel} and the moments created with the rotation _{of the vessel. It} results, however, in higher vertical mooring line tensions due to the first-order vertical motions of the vessel [15].

Figure 3 .shows a series of catastrophe_{sets in the parametric} space (a. xIL) for the following values:

l80dega

l8Odeg

0

(.t,/L)d '

0.5

for three different horizontal forward_{mooring line pretensions} (PRr)d ranging from 10 to 20 percent ABS The _horizontal pretension in each of the aft mooring lines _{has been set to 5} percent ABS to allow the aft part of the vessel to move relatively freely in the horizontal plane. These pretension cases correspond to _{of 10, 15, and 20 percent ABS. and are denoted as}

Pg:.

PRT2. andPR73. _{respectively. Table 3 shows the valuesof}

the total pretensions T,. in each of the mooring_lines

correspond-Table 2 Mooring line arrangement of DICAS_{(S = single;} D = double) Mooring line 3 4 5 6 7 8 9

Transaction5 of the ASME

Line type _{Mooring angle (deg)}

S ₂₆₈ D ₂₇₁ D ₂₈₄ D 66 D S 89₉₂ D ₁₇₅ S D 180 185

0.50 0.45 - 0.40-0.35 0.300.25 - 0.20- 0.15- 0.10-0.05 10% ABS (Rr)ji.d= 15% ABS (RT)fr.s. 20% ABS 0.00 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140 160 180 current engje (deecs)

ing to these cases as a percentage of the average breaking strength (ABS).

The fairlead offsets (yp) of the forward moorng lines are given by the relation

Fig. 3 Catastrophe set, DICAS (xIL).0 -0.5, (yIL). 0.0, = 0.05 ABS U0 = 2 knob

The catastrophe sets in Fig. 3 present two different regions of qualitatively similar dynamics, numbered I and H and de-noted by R-1 and R-fl. respectively. These regions have the following characteristics:

Region I (R-I): Stable equilibrium. All eigenvalues in the system have negative real parts. A small random disturbance from a stable equilibrium initiates trajectories converging to it 'in forward time.

Region 11 (R-ll): Unstable equilibrium with a two-dimen-sional unstable manifold (i.e,, a complex conjugate pair of ci-genvalues with positive real part). A small disturbance from equilibrium results in a limit cycle around such equilibrium [81. Dynamic loss of stability occurs when crossing from R-I to R-H, and is due to a change in the structure of the limit sets of solutions of the vector field as a parameter is varied [2]. At the boundary between R-1 and R-ll. a Hopf bifurcation, which

Table 3 Total pretension in terms of the horizontal preten-sion as a percentage of ABS

Il-2(

'V

\LJ

(22)

is of dynañiic nature, occurs and the system oscillates around the equilibrium position if there are no other nearby equilibria that may attract such trajectories. The limit cycle that the system exhibits may be stable or unstable depending on the interactions between the mooring lines as the system oscillates around its eqwlibrium position [6]. A DICAS may fall in R-II as the direction of the current changes. To design a safe DICAS in region R-ll. further analysis is required based on the Center Manifold Theorem [19] and nonlinear time simulations to as-sure safe limits on the amplitude of oscillations in terms of mooring line tensions and vessel displacement.

Figure 3 shows that the stable doma.in (R-l) increases as the pretension in the forward mooring lines increases. Notice, however, that the upper unstable region in Fig. 3 moves towards smaller current angles with increasing pretension for large val-ues of (xP/L)d, thus destabilizing a stable system operating under such environmental conditions. Asshown in Fig. 3. there is a range of (xIL values for which the system is stable under all current directions for the three pretension cases

consid-ered.

A DICAS can be designed with a suitable choice of x/L values based on the catastrophe sets of Fig. 3 fOr a prescribed pretension. As mentioned previously, the designer can select (xPIL)d values as close to the bow as possible to ensure lesser horizontal displacements of the vessel's bow as well as lower horizontal mooring line tensions. As shown in Fig. 3. however. these values for (xP/L)d are unstable for a large range of current angles. In addition, such fairlead positions result in higher impact on the first-order vertical motions due to waves, causing higher dynamic vertical tensionsin the forward mooring liries. It is then reasonable to select (./L)fWd values closer to the vessel's CG. where the system is stable for a larger range of current angles. at the expense of larger relative displacements of the bow and higher horizontal mooring line tensions. Interme-diate values of (x/L ) have the advantage of rendering highly stable equilibria without compromising large horizontal dis-placements/mooring line tensions. For the remainder of this analysis, the values (xP/L)fwd = 0.25 are selected.

Table 4 shows the horizontal distance, in meters. between the equilibrium position of the vessel's bow for several current

Journal of Offsho,e Mechanics and Arctic Engineering FEBRUARY 1999, Vol. 121 / 13

Mooring line PR71 PRI7 PRTh

I (S) 18.73 23.71 28.69 2 (D) 19.49 24.48 29.46 3 (D) 19.49 24.48 29.46 4 CD) 19.49 24.48 29.46 5 (D) 19.49 24.48 29.46 6 (S) 18.73 23.71 28.69 7 (D) 14.51 14.51 14.51 8 (S) 13.74 13.74 13.74 9 (D) 14.51 14.51 14.51 B =

-2

angles. with respect to the equilibrium position of the vessel's bow for zero current angle ((xp/L)?Wd = 0.25) for the three pretension cases considered. At the bottom of Table 4, the maxi-mum relative horizontal displacement at equilibrium and the current angles at which it occurs for each case are listed.

As shown in Table 4. the maximum relative horizontal motion of the bow at equilibrium occurs in all pretension cases for angles near 142 deg. and is less than the allowable value (60-75 m) under which flexible risers can operate properly. Ariser placed near the bow of the ship would be appropriate for all pretension cases accordin2 to Table 4. Nonlinear simulations. however, are required to determine the dynamic displacement in each case [15].

Once the values for (x,/L ) have been selected, the next step in the DICAS design stage is to select appropriate values for the moorng line pretension. As shown in Fig. 3. thesystem is stable for (PRT)fWd = 0.15 and 0.20 ABS under all current directions. These values of pretensirin also satisfy the require-ments for the maximum relative horizorital displacerequire-ments (see Table 4), and thus, any of these (or intermediate values of these) pretensions could be selected. The lowest of these preten-sions (i.e.,(PRT )fwd =0.15 ABS1,would be the preferred choice

since it results in lower moorng line tensions. In general, it is expected that the resulting dynamical tensions due to slow and fast motions will be about the same, resulting in a lower total tension for the lowest pretensions. This may not always be the case, depending on the orientation of the vessel around equilibrium. An illustration to scale of the final equilibrium position of the vessel for several current angles for(xp/L)rd =

0.25 and(PRT)f,,,d 0.15 ABS is shown in Fig. 4. The bottom part of the figure shows in detail the pOsiiion of the system at equilibrium for several current angles.

Table 5 shows the values of the total tension T in each of the mooring lines at equilibrium(xP/L)...d =0.25. (PRT)d =

0.15 ABS) for selected current angles (ranging from 0to 103.67 deg). as well as the sum of the tensions in the mooring lines (SUM), and the relative angle between the vessel orientation and the current at equilibrium (RAVC) for each of these angles.

In Table 5. the a = 0 deg case corresponds to the symmetric equilibrium position of the vessel, while a= 103.67 deg corre-sponds to a relative angle of 90 deg between the vessel and the current at equilibrium. At this angle. the highest hydrodynamic forces act on the ship. The maximum tension at equilibrium of a specific mooring line, however, does not occur at this current angle. This is because of the mooring line arrangement of the system (i.e., the relatively small values of(x/L)1,.. the orienta-tion of the mooring lines, and their differences in pretension), and the fact that the aft mooring lines, which have a smaller pretension. share some of the h drodynamic load. Instead, the maximum mooring line tension occurs at current angles of

a

= _{73.14 deg for a double (D) line (mooring line 3) and}

_a

₌

79.24 deg fur a single S) line Imooring line I). as shown in

italics in Table 5. These correspond to 30.19 and 28.02 percent 0

ire

0= 180'-,. _{-s-- a=O'}

Fig 4 Equilibria of DICAS in differentcurrent _{directions: (x0IL) =}

0.25, (Pj074,.. = 0.15 ABS, U, = 2knots

of the breaking strength of the mooring lines, respectively. Due to the symmetry of the system, the maximuin tension also occurs for a =- _{-73.14 deg for a double (D) line (mooring line 4)}

and a = -79.24 deg for a single (S) line (mooring line 6). Table 5 also shows that the total sum (SUM) of the mooring line tensions at equilibrium is higher for

a

= 103.67 deg. The current angles for which the total sum of all mooring line ten-sions is highest in the (- 180 deg 180 deg) range. however, does not occur at this angle. even though the hydrody-narnic forces are maximum. The maximum sum of mooring line tensions occurs for a = ±137.41 deg (RAVC = 117.88

deg, SUM = 16154.69 KN). due to an increase in tension in

14 / Vol. 121, FEBRUARY 1999

. Transactions of the ASME

Table 5 _{Equilibrium tension (KN) of DICAS for various} current angles: (Xp/L)r.d . 0.25. (PRT)r,,d 0.15 ABS (S = single; D = doUble)

Line a =0 deg a =73.14deg a = 79.24deg a ₌ 03.67 deg

I (S) 1211.08 1443.17 /445.59 1433.69 2 (D) 2397.10 28IQ.68 2813.75 2789.32 3 (D) 2594.42 2932.85 2929.51 2892.32 4 (D 2594.42 2138.62 2131.76 2135.36 5 (DI 2397.10 2083.33 2082.42 2105.00 6 (S) 1211.08 1058.06 1058.27 1072.63 7 ID) 1230.91 315.69 1323.46 1356.88 8 iS) 6l3.95 673.15 678.57 703.32 9(D) 1230.91 1371CM 1383.95 1445.35 SUM 15480.97 15826.59 5847.28 15934.5 RAVC (derz) 0.000 64.57 69.86 9(1.00

with respect to zero current angle at equilibrium

'000-CUrrent angle (deg) PRfl PRr a-O' a=±45' a=±9O' a=i135' a=i180' 0.0 _-22.5 :45.0 :67.5 :90.0 :112.5 :135.0 l57.5 180.0 maximumvalues

currentangle (deg)

0.00 4.77 10.02 17.94 20.15 21.34 23.39 22.08 12.98 23.60 :141.7 0.00 3.06 6.45 12.08 1430 16.89 19.90 18.36 10.88 20.23 l42.7 0.00 2.25 4.76 9.19 11.41 14.55 17.76 15.56 9.66 18.02 :141.4

Pflae aft mooring lines caused by the arrangement of the system geometry. Further analyses with nonlinear simulations are re-quired to obtain the maximum values of the dynamic tension.

From the results shown in Fig. 3. and Tables 4 and 5. the

DICAS design with (xP/L)f,d = 0.25. (PRT)(Wd = 0.15 ABS

seems to be appropriate under a 2 knot current velocity (further simulations are required to verify the limits of vessel displace-rnent and mooring line tension). In the Campos basin, where the current direction is predominant in one direction with a

relatively small range of variation [51. larger values of (x,,/

L)

and lower values of (PRT )d can be considered, provided the system configuration satisfies the design requirements (i.e., limit of oscillations and mooring line tensions). Such arrange-ment would decrease, in general, both the horizontal displace-ment of the fore part of the vessel as well as the horizontal tensions in the mooring lines while rendering stable equilibria.

Concluding Remarks

The design methodology for DICAS, developed in this work, is based on nonlinear dynamics and bifurcation theory. Itwas shown that for the environmental conditions considered, it is possible to select different DICAS configurations for which the system satisfies the operational requirements. Based on this design methodology, it is possible to select appropriate design parameters without performing extensive nonlinear simulations. The following conclusions regarding DICAS dynamics_{can be}

drawn from this work: (a) intuitive means for stabilizing a

DICAS (i.e., increasing the mooring line pretension: moving the attachment points forward . may destabilize a stable system. as shown in Fig. 3. Intermediate values of (xp/L)4 for DICAS have the advantage of rendering stable configurations without compromising the allowable limits of riser offset, as shown in Fig. 3 and Table 4. (b) With this methodology, it is possible so select suitable DICAS designs without over-pretensióning the forward mooring lines, as shown in Fig. 3 and Table 5. (c) The direction of the external excitation plays an important role in the dynamics of the system. DICAS tend to be stable when the current is at large angles of attack with respect to the vessel

orientation, irrespective of the (x/L )d values, as shown in Fig. 3. (d) Due to the geometric constraints of the system, the largest tensions in the mooring lines do not occur for external excitation at right angles with the orientation of the vessel, as

shown in Table 5. These conclusions demonstrate that the

mhod developed in this paper is needed to design DICAS

efficiently, since it is much more powerful and efficient than systematic parametric search for a good design.

Adaiowledgments

l'his work is sponsored by the University of Michigan/Sea

Grant/Industry Consortium in Offshore Engineering under

Michigan Sea Grant College Program, projectno. R/T-35 under grant no. DOC-NA36RG0506 from the Office of Sea Grant, National Oceanic and Atmospheric Administration (NOAA). U.S. Department of Commerce, and ftnds from the State of Michigan Industry participants include Amoco. Inc.; Conoco, Inc.: Exxon Production Research: Mobil Research and

Develop-ment: and Shelr Companies Foundation. Special gratitude is extended to Petrobrás. Brazil, for providing the characteristics of the FPSO. The U.S. Government is authorized to produce and distribute reprints for governmental purposes notwithstand-ing any copyright notation appearnotwithstand-ing hereon.

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