SLOW MOTION DYNAMICS OF DICAS MOORING STEADY …Offshore Engineering 51 (ID where T^ is ±e...

SLOW MOTION DYNAMICS OF DICAS MOORINGSYSTEMS UNDER STEADY CURRENT, WIND, ANDSTEADY DRIFT EXCITATION

Luis O. Garza-Rios, Research Fellow, and Michael M. Bernitsas, ProfessorDepartment of Naval Architecture and Marine EngineeringUniversity of Michigan, Ann Arbor, Michigan 48109-2145, U.S.A.

Kazuo Nishimoto, ProfessorDepartment of Naval Architecture and Ocean EngineeringUniversity of Sao Paulo, Sao Paulo, Brazil

Abstract - The slow motion dynamical behavior of a Differentiated ComplianceAnchoring System (DICAS), subject to a range of directions of externalexcitation of constant magnitude, located in the Marlin Field, Campos Basin,Brazil, is assessed based on nonlinear stability analysis and bifurcation theory.Excitation consists of steady current, wind, and second order mean wave driftforces. Catastrophe sets are constructed in a two-dimensional parametric designspace, separating regions of qualitatively different dynamics. Stability analysisdefines the morphogeneses occurring across bifurcation boundaries to findstable and limit cycle response near the principal equilibrium position. Theresulting design graphs allow the designer to select an appropriate orientationand other design variables for DICAS without resorting to trial and error, orextensive nonlinear time simulations. The position of the vessel at equilibriumdefines the mean horizontal displacement of the system with respect to aprescribed initial orientation. This position depends on the magnitude anddirection of the external excitation. Limited simulations or further nonlinearanalysis enable the designer to investigate whether or not DICAS slow motionscomply with the allowable limits of motions for safe operations. The effect ofwater depth variation on the stability of DICAS is considered, and it is shownthat the dynamics of the system change considerably with relatively smallvariation in water depth. The DICAS mathematical model consists of thenonlinear, horizontal plane fifth-order, large drift, low-speed maneuveringequations. Mooring lines are modeled by catenaries with touchdown andnonlinear drag. External excitation consists of time independent current, steadywind, and second order mean wave drift forces.

Transactions on the Built Environment vol 29, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509

48 Offshore Engineering

INTRODUCTION

In offshore operations, several types of mooring and anchoring systems arebeing designed and deployed, depending on the type and projected time ofoperation, and the environmental conditions. The search for more efficientmeans of production and recovery, has led to innovative ideas in the area ofmooring system design. One such concept is the Differentiated ComplianceAnchoring System (DICAS), a type of Spread Mooring System (SMS) that canbe effectively used, if properly designed, in relatively moderate weatherconditions, such as those encountered offshore Brazil. The first DICAS wasinstalled in the Caravela field, Santos basin, Brazil, in 1993 at water depth of195 meters, and is still in operation. DICAS is a type of spread mooringsystem with different stiffnesses in the mooring lines at the bow and stern of thevessel. Such differences in stiffnesses, somewhat restrict the bow of the vesselfrom moving due to their higher pretensions while allowing the stern of thevessel to move relatively freely. This property allows the vessel to partiallyweathervane as the environmental conditions vary. Due to the characteristics ofthis system, the DICAS design must take into account the slowly changingdirections of the environmental excitations as well as the best layout for theproduction risers.

The design of a DICAS, however, is tedious and time consuming due to thehigh number of parameters that must be considered, such as the number,length, orientation and pretension of the catenaries, fairlead position, etc., aswell as the environmental conditions in which the system is to operate. Adesign methodology for SMS based on nonlinear dynamics and bifurcationtheory l, can be used to analyze the dynamics of the system while eliminatingintense computations in SMS design. A previous introductory study of theslow motions of DICAS for the Campos Basin% based on these principles,shows the richness of the system nonlinear dynamics under varying currentdirections, fairlead positions of the mooring lines, and mooring linepretensions. In this paper, the horizontal plane slow motion dynamics of aDICAS under varying directions of the external excitations are analyzed usingnonlinear dynamics and bifurcation theory. The effect of water depth on thedynamical behavior of the system is considered as well.

MATHEMATICAL MODEL

The slow motion dynamics of the DICAS in the horizontal plane (surge, swayand yaw) are formulated in terms of the vessel equations of motion, mooringline model, and external excitation. The mathematical model is based on thelarge drift angle, slow horizontal plane motion, fifth-order maneuveringequations .

Equations of MotionThe horizontal plane geometry of a DICAS is shown in Figure 1, with tworeference frames: (x, y) = inertial reference frame with its origin located atmooring terminal 1; (X,y,Z) =body fixed reference frame with its originlocated at the center of gravity of the vessel (CG). In addition, n is the number


Offshore Engineering 49

of mooring lines; (x$,y$) are the mooring coordinates of the iih mooring line

in the (jc,y) frame; (x \y$) are the body-fixed coordinates of the iih mooring

line fairlead: Xp is positive forward of the CG, and yp is positive on the port

side of the vessel; t is the horizontal distance of the ith mooring linebetween the attachment point on the vessel and the mooring point at the oceanfloor; and y is the drift angle. The mooring lines are numberedcounterclockwise starting from the line moored connected to terminal 1.

DIRECTION OF CURRENT,WIND AND WAVES

270'

Figure 1. Geometry of DICAS

The equations of motion in the horizontal plane are given by2

(m + m )u - )rv =

r= +zW")(7 -*•

(1)

(2)

(3)

where m is the mass of the vessel; 7%z is the moment of inertia about the Z-axis;nix* my and J& are the added masses and moment of inertia in surge, sway andyaw, respectively; w, v, and r are the relative vessel velocities with respect to



water in surge, sway, and yaw, respectively; X#, 7# and JV# are the velocitydependent horizontal plane hull hydrodynamic forces and moment expressed interms of the large drift angle, slow motion derivatives

-n ^, (4)L o

(5)

Vr- (6)

The first three terms in (4) represent the third order approximation of the vesselresistance, tf:

In addition, in Equations (l)-(3), 7 and Ty represent the tension

components in the horizontal plane of the ith mooring line in the surge and

sway directions; F and F$ correspond to the mooring line nonlinear

damping components in surge and sway, respectively; and / , Fsway* and

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

The kinematics of the system are governed by Equations (8)-(10):

% = wcosy - vsinyf + (7cos# , (8)y = vcosy + wsin y + [/sina , (9)

V/ = r , (10)

where U = \U\ is the absolute value of the relative velocity of the vessel withrespect to water, and a is the current angle measured with respect to the (a,y)frame as defined in Figure 1.

Mooring Line ModelThe mooring lines of the DICAS are modeled quasistatically by open-watercatenary chains, which include touchdown and nonlinear drag*. Mooring linesare classified as Single (S) or Double (D). Single mooring lines have anAverage Breaking Strength (ABS) of 5159 KN; Double lines have an ABS of9715 KN*.

The total tension of the catenary, 7, is*



(ID

where T is ±e horizontal tension component in the catenary, 2J, is the vertical

tension component, P is the weight of the catenary per unit length (1510 kg/s%for Single lines; 3094 kg/s% for Double lines)*, and t is the horizontallyprojected length of the suspended portion of the catenary.

Mooring Line Drag Forces: The drag forces in the surge and swaydirections F and Fyj on each of the mooring lines are of the form :

(12)

(13)

where /J is the angle from the X-axis to the direction of the mooring linemeasured counterclockwise, and F and FI are the mooring line drag forces inthe directions parallel and perpendicular to the catenary motion, respectively.

External ExcitationExternal excitation corresponds to time independent current, wind, and meansecond-order wave drift forces, with direction of excitation with respect to the(jc,y) reference frame as shown in Figure 1. The current force is formulated inthe maneuvering equations by introducing the relative velocities of the vesselwith respect to water. Thus,

~ Vx wind "*" ** wave » (14)

~ **y wind "*" y wave » (15)

~ M£ wind "*" * z wave •

Wind forces and moment corresponding to steady wind action exerted on theship can be modeled as :

(18)

(19)

where p^ is the density; U^ is wind velocity at a standard height of 10 m

above water; ay is the relative angle of attack between the wind direction andthe ship heading; Aj and AL are the transverse and longitudinal projected areasof the vessel, respectively; L is the length of the vessel; C(a,.), C (a ),



and Cy#(&r) are wind longitudinal and lateral force and moment coefficients,expressed in Fourier Series as follows:

, (20)n=l

, (21)n=l00

(22)

Coefficients , £„, #„, and £„ in Equations (20)-(22) depend on the type ofvessel, superstructure location, and loading conditions. These coefficients canbe obtained from a suitable set of data.

The mean wave drift excitation in the horizontal plane is* * :

FX wave = PwfLCxl oosty* - f) , (23)

Fy wave = P yd ™?Wo ~ V) . (24)

MZ wave = PwgtfCtf sin2(6 - y), (25)

where p^ is water density; QQ is the absolute angle of attack; g is the

gravitational constant; and C%y, Cy , and C f are the drift excitationcoefficients in surge, sway and yaw, respectively:

(26)

In the expressions above, the quantities in square brackets are the driftexcitation operators; a is the wave amplitude; &)<, is the wave frequency. The

two parameter Bretschnider spectrum is used to relate 5(fi)) in terms of the

significant wave height HI / 3 and significant wave period 7J / 3 as:

V*"""\ (29)



where A and B are nonlinear functions of #1/3 and 75/3.

PARTICULARS OF THE SYSTEM

DICAS GeometryThe DICAS under consideration consists of a converted FPSO system, whosegeometry is shown in Table 1%. The system consists of nine mooring lines (sixmoored forward of the CG\ three moored aft of the CG), with nominalmooring line length If* 1910 meters each. The system is geometricallysymmetric with respect to its X-axis, with three different points of attachment,with three mooring lines attached to each of them as shown in Figure 1. Thetypes of catenaries (single, double) in the system, their orientation with respectto the X-axis, the body-fixed coordinates of the mooring line fairlead, and themooring line pretensions as percentages of ABS, are shown on Table 2. Thisconfiguration has been shown to provide highly stable slow motion dynamicsand adequate displacement properties for operation in the Campos Basin,Brazil, in the absence of wind and wave excitation under a current speed of 2knots%.

Table 1: Geometric properties of the FPSO

Length overall (LOA) 272.80 mLength between perpendiculars (LWL, L) 259.40 mBeam (5) 43.10mDraft(D) 16.15mBlock Coefficient (Q) 0.83Displacement (A) 1.5374 x 10 tonm, 9.110x106 kgmy 1.360x108 kg

7 7.180xlOUkg.m2J 5.430x10* lkg.m2

The production risers in the system are placed near the bow of the vessel, wherethe system has the least horizontal displacement as the environmental excitationschange direction, due to the relatively high pretensions in the forward mooringlines. These are flexible risers, with a maximum allowable horizontaldisplacement of 60-75 meters (20-25% of the water depth), with respect to aprescribed initial orientation .

Environmental ConditionsThe DICAS configuration described in the previous sub-section is to bedeployed in the Marlin field, Campos basin, Brazil, under a water depth of 300meters. The predominant direction of each environmental force at that site,along with the percentage of occurrences for each, are as follows :



predominant direction occurrencesurface current: S-N 54.56%wind: NE-SW 33.50%waves: NE-SW 38.60%

The values for the magnitudes of surface current, wind, and mean second orderwave drift forces acting on the system, are based on a 10-year return period,and are shown on Table 3.

Line

123456789

Type

SDDDDSDSD

Table 2: Moon

Angle

268°271°284°66°89°92°175°180°185°

ing line arranger

fairlead coc

Xp

64.8564.8564.8564.8564.8564.85

-129.70-129.70-129.70

nentofDICAS

>rdinates (m)

yp

-15.24-15.24-15.2415.2415.2415.240.000.000.00

Pretension(% ABS)

23.7124.4824.4824.4824.4823.7114.5113.7414.51

Table 3: 10-year period return values, Marlin field

Force Predominant Direction Magnitude

surface current S current speed: 1.58 m/swind NE wind speed: 25.06 m/swaves NE max. wave height: 8.6 m

max. period: 11.7 ssignificant wave height: 4.7 msignificant period: 9.2 speak period: 12.9s

DYNAMIC ANALYSIS OF DICAS

Analyses of nonlinear dynamical systems undergoing changes in theenvironmental excitation over long periods of time require, in general, trial anderror with numerous and lengthy nonlinear time simulations in each trial. Themethodology for the design of SMS based on nonlinear dynamics andbifurcation theory as several parameters vary over the life of the moored vessel



(in this case direction of current, wave, and wind)*/? can be used to eliminatesuch a tedious process, and is adopted for DICAS.

Theoretical Background: Nonlinear Dynamics/Bifurcation TheoryThe slow motion dynamics of DICAS are studied based on nonlinear dynamicsand bifurcation theory. First, the system is modeled as a set of six first-ordernonlinear differential equations by combining equations of motion (l)-(3) andkinematic relations (8)-(10). Then, the stability properties of a specific DICASconfiguration are obtained around an equilibrium position by performingeigenvalue analysis for a given set of environmental forces. If all eigenvalueshave negative real parts, such equilibrium is locally stable and all trajectoriesinitiated near that equilibrium position will converge to it in forward time. If atleast one eigenvalue has a positive real part, such equilibrium is unstable, and asmall disturbance from equilibrium will cause the system trajectories to divergefrom itK

Once the local stability properties of DIG AS have been established, bifurcationsequences can be studied to find qualitative changes in the dynamic behavior ofthe system as one or several parameters (direction of current, wind, waves)vary with respect to a prescribed DICAS orientation. These sequences are usedto construct stability charts in the two or three-dimensional parametric designspace that separate regions of qualitatively different dynamical behavior (stable,unstable, periodic, quasiperiodic, chaotic)?. These charts, known as"catastrophe sets," can be used to virtually eliminate the number and length ofnonlinear time simulations in the DICAS design process. Nonlinear simulationsare used to calculate the dynamical tensions, maximum displacement of thevessel, relative motion of the vessel endpoints, etc.

Analysis of the System Under Variable External ExcitationTo study the dynamics of DICAS as the environmental excitation changesdirection over the lifetime of the system, catastrophe sets are constructed aroundthe principal equilibrium position of the system in the design space of: thecurrent direction (a), and the wave direction (64). The wind direction (6 ) isintroduced as a parameter as shown in Figure 2. The excitation directions aremeasured with respect to the (x,y) reference frame according to the conventionshown in Figure 1. The values for the magnitudes of the external excitationscorrespond to a 10-year return period (Table 3).

Figure 2 shows a series of catastrophe sets in the (a, Oj) parametric space forthe following ranges in the current and wave directions:

0°<<%< 360° , 0° < 0^ < 360%

for four different values of the wind direction 0 ,, ranging from 0" (or 360°) to270° in intervals of 90°. These sets present two regions that denote qualitativelydifferent dynamical behavior of the system. These regions, numbered I and II,and denoted as R-I and R-n, have the following characteristics:



(i) R-I: Stable equilibrium. All eigenvalues in the system have negative realparts. A small random disturbance from this equilibrium position initiatestrajectories converging toward it in forward time.

(ii) R-II: Unstable equilibrium with a two-dimensional unstable manifold (i.e. acomplex conjugate pair of eigenvalues with positive real part). A smalldisturbance from equilibrium results in a limit cycle around it*5. The systemloses its stability dynamically when crossing from R-I to R-H At the boundarybetween these two regions, a Hopf bifurcation occurs?, resulting in oscillationsaround the equilibrium position. The limit cycle exhibited by the system may bestable or unstable, depending on the interaction between the mooring lines asthe system oscillates *. A DICAS may fall in R-II as the environmentalconditions change. If operating in this region is deemed necessary, the limits ofthe amplitude of oscillations of the system in terms of mooring line tensions andvessel relative displacements must be studied. In such a case, further analysesbased on the center manifold theorem^ and nonlinear time simulations arerequired.

360 -i

330-

300-

270-

240-

210-

180-

150-

120-

90-

60-

30-

(

JKl\l

•

%../

wave

direction (deg)

M

1 1 1) 30 60 90

• j . .wind direction = 0

wind direction = 90°

wind direction =180

wind direction = 270*

I

"II ii

I j

current direction (deg)I I I I I

120 150 180 210 240

/' *-%

I n |

\j

$ ,

A

K/°* v I/1 1 1

270 300 330 3()0

Figure 2. Effect of external excitation on catastrophe set of DICAS

The catastrophe sets in Figure 2 show that the principal equilibrium of thesystem is stable under a large range of current, wave, and wind directions. Thesystem falls into oscillatory region R-n, undergoing a limit cycle behavior, forcurrent directions at or near right angles with respect to the vessel orientation.Waves destabilize the system in the range of following to beam seas, andstabilize the system in or near head seas. This is partly due to the fact that thisorientation of the drift forces provides added resistance to the already



pretensioned forward mooring lines, thus limiting the motions of the system.The same principles apply for wind forces, as shown in the figure.

The nonlinear time simulations in Figure 3 show the richness of the DICASdynamics depicted in the catastrophe sets in Figure 2 as functions of the driftangle if. Simulation (1) in Figure 3 corresponds to a stable principalequilibrium for the parameter values (#= 100% @ = 165% 0 ,= 0"). A smalldecrease in the current angle (15°) results in a stable limit cycle around theprincipal equilibrium, as shown by simulation (2) for the parameter values(#=85°, #d= 165% #w,=0°). As previously mentioned, if the DICAS is to bedesigned such that the environmental conditions of scenario (2) apply, furtheranalyses would be required to determine the plausibility of such a design.

-0.05-

-0.15-

-0.25 -

0 25 50 75 100 125 150 175 200

Figure 3. Drift angle, DICAS: stable and oscillatory principal equilibrium

DICAS OrientationAs shown in Figure 2, there are ranges of current directions with respect to the(*,y) reference frame (0° < a < 60% 100° < a < 260% 290° < a < 360°)for which the system yields a stable principal equilibrium irrespective of thedirection of wind and waves. A DICAS may be designed with a suitable choiceof vessel orientation based on the catastrophe sets of Figure 2. An appropriateorientation for the system would be that in which the predominant currentdirection (coming from the south, see Table 3) is aligned to the (jc, v) and(X,7) frames of the vessel (i.e. the relative angle between the current and thevessel is 180°). This would be achieved by orienting the system with its bowpointing into the principal direction of the current (i.e. south).



Figure 4 shows the relative horizontal displacement of the vessel's bow atequilibrium with respect to the position of the bow in the absence of externalexcitation, as a function of the current angle in the predominant range SW-SE.In this application, the water depth is 300 m; wind and waves are co-linear withdirections ranging from E-N. The vessel is oriented with its bow pointingsouth.

50

45-

40-

35-

30-

25-

20-

15-

10-

5-

Direction of wind and waves

.22

current direction

SW SE

Figure 4. Relative horizontal bow displacement; principal orientation: bow south

As shown in Figure 4, the maximum horizontal displacement of the bow inthese excitation ranges is under 30 meters (approx. 27.8 m), which is less thanthe allowable limits of motion for operation with flexible risers. Additionalanalyses are needed, however, to determine the dynamical tensions in themooring lines and the maximum displacement of the vessel.

An alternative DICAS orientation that renders a highly stable system accordingto the catastrophe sets in Figure 2, would be to place the vessel at 0" withrespect to the current principal direction (i.e. the bow of the vessel facingnorth). In such a case, the counterpart of Figure 4 is shown on Figure 5. Fromthis figure, it is shown that the maximum relative bow horizontal displacementfor the directions of wind and waves shown (approx. 23.2 m), is less than theallowable limit of motion. This orientation also yields smaller displacementsthan those shown in Figure 4, and it is therefore a better design configuration interms of vessel displacements.

It is important to point out, however, that in this particular case, such alternativeorientation yields smaller relative displacements due to the fact that the principal



direction of the surface current is opposite to the direction of wind and waves.If all forces were to act in the same principal direction, the alternative of placingthe vessel in following seas would result in an increase in the tensions of the aftcatenaries. This could possibly restrain the aft part of the vessel from moving,while allowing the fore part of the vessel to move relatively freely.

50

45-

40-

35-

30-

25-

20

15-

10-

5-

0

Direction of wind and waves

NEE

ME

NNEN

SW

current direction

SE

Figure 5. Relative horizontal bow displacement; principal orientation: bow north

EFFECT OF WATER DEPTH

The water depth at the Marlin field/Campos basin is not constant, and it istherefore important to consider the dynamics of the system under different waterdepths.

Figure 6 shows a series of catastrophe sets in the (a, 64) parametric space

(wind direction 9 is fixed at 180'), for three different values of the waterdepth, ranging from 250 meters to 350 meters. The qualitative behavior of thesystem about its principal equilibrium position in regions R-I and R-II in thisfigure, is the same as that in Figure 2.

As shown in Figure 6, the stable region R-I tends to increase with increasingwater depth. This is due to higher resistance (drag) on the system as a result ofincreasing the length of the suspended catenaries. Such an increase in thecatenary length, however, causes the horizontal displacement of the vessel, andthe tensions in the mooring lines, to increase as well.



360-i

330-

300-

270-

240-

210-

180-

150-

120-

90-

60-

30-

C

1 •*'

faV

1

1

/fc\

In\- \ |\\ \ }\\ S J\:k

i i i) 30 60 9C

}

X

)

depth = 250 m

depth = 300 m

depth = 350m

11

current direction (deg)l l t l l120 150 180 210 240

\

I

2

/'X\

i"l\z/

•o

H)\\-'n\/ i

>70 31

'

30 330 3()0

Figure 6. Effect of water depth on catastrophe set of DICAS

CONCLUDING REMARKS

Preliminary design of a DICAS based on its slow motion dynamics in thehorizontal plane can be developed using the methodology presented in thispaper. To illustrate some of the applications of the design methodology, it hasbeen shown that, for a prescribed DICAS configuration, it is possible to selectan appropriate system orientation that yields a stable configuration under a widerange of external excitations without resorting to trial and error, and lengthynonlinear time simulations. This orientation was selected based on catastrophesets with the predominant directions of the external excitations as parameters.An increase in water depth increases the stable domain of the system due tolonger suspended portions of the catenaries, and thereby induced higher drag.It results, however, in higher mooring line tensions and increased horizontaldisplacement of the system. The DICAS concept in deeper waters maytherefore not be feasible, unless the horizontal pretensions in the mooring linesare increased, and buoys are placed along the catenaries to reduce the verticaltensions on the vessel exerted by their weight.

ACKNOWLEDGMENTS

This work was sponsored by the University of Michigan/Sea Grant/IndustryConsortium in Offshore Engineering under Michigan Sea Grant CollegeProgram, project number R/T-35 under grant number DOC-NA36RG0506from the Office of Sea GranJ, National Oceanic and Atmospheric Administration



(NOAA), U.S. Department of Commerce, and funds from the state ofMichigan. Industry participants include Amoco, Inc.; Conoco, Inc.; ExxonProduction Research; Mobil Research and Development; and Shell CompaniesFoundation. The U.S. Government is authorized to produce and distributereprints for governmental purposes notwithstanding any copyright notationappearing hereon.

REFERENCES

[1] Garza-Rios Eychenne, L.O., Development of a Design Methodology forMooring Systems Based on Catastrophe Theory. Ph.D. Dissertation,Department of Naval Architecture and Marine Engineering, University ofMichigan, Ann Arbor, 1996.

[2] Garza-Rios, L.O., Bernitsas, M.M., Nishimoto, K., and Masetti, L,"Preliminary Design of a DICAS Mooring System for the BrazilianCampos Basin," Proceedings of the 16th International Conference onOffshore Mechanics and Arctic Engineering, Yokohama, Japan, April1997.

[3] Takashina, J., "Ship Maneuvering Motion due to Tugboats and itsMathematical Model," Journal of the Society of Naval Architects of Japan,Vol. 160, December 1986, pp. 93-104.

[4] Tanaka, S.,"On the Hydrodynamic Forces Acting on a Ship at Large DriftAngles," Journal of the West Society of Naval Architects of Japan, Vol.91, 1995, pp. 81-94 (in Japanese).

[5] Nishimoto, K., Brinati, H.L., and Fucatu, C.H., "Analysis of SinglePoint Moored Tanker Using Manoeuvering Hydrodynamic Model,"Proceedings of the ASME 14th International Conference on OffshoreMechanics and Arctic Engineering (OMAE'95), Vol. I-B, Copenhagen,June 1995, pp. 253-261.

[6] Yumuro, A., "Some Experiments on Maneuvering Hydrodynamic Forcesin Low Speed Conditions," Journal ofKansai Zousen Kyoukai Shi, Vol.209, 1988, pp. 91-101 (in Japanese)

[7] Bernitsas, M.M. and Garza-Rios, L.O., "Effect of Mooring LineArrangement on the Dynamics of Spread Mooring Systems," Journal ofOffshore Mechanics and Arctic Engineering, ASME Transactions, Vol.118, No. 1, February 1996, pp. 7-20.

[8] Garza-Rios, L.O., Bernitsas, M.M. and Nishimoto, K., "CatenaryMooring Lines with Nonlinear Drag and Touchdown," Report to theUniversity of Michigan/Industry Consortium in Offshore Engineering,and Department of Naval Architecture and Marine Engineering, Universityof Michigan, Ann Arbor, Publication No. 333, January 1997.

[9] Nippon Kaiji Kyokai, Guide to Mooring Systems. N.K.K., June 1996(in Japanese).

[10] Martin, L.L., "Ship Maneuvering and Control in Wind," SNAMETransactions. Vol. 88, 1980, pp. 257-281.

[11] Cox, J.V., "Statmoor - A Single Point Mooring Static AnalysisProgram," Naval Civil Engineering Laboratory, Report No. AD-A119979, June 1982.



[12] API, "Draft Recommended Practice for Design, Analysis and Maintenanceof Mooring for Floating Production Systems," API RecommendedPractice 2FP1 (RP2FP1), First Edition, 1991.

[13] CENPES/DIPREX/SEPRON, "Meteocean Data, Soil Data andBathymmetry," Report to Petrobras Research Center, Brazil, 1996 (inPortuguese).

[14] Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems andChaos. Springer-Verlag, New York, Inc., 1990.

[15] Guckenheimer, J. and Holmes, P., Nonlinear Oscillations. DynamicalSystems, and Bifurcations of Vector Fields. Springer-Verlag, New York,Inc., 1983.


SLOW MOTION DYNAMICS OF DICAS MOORING STEADY …Offshore Engineering 51 (ID where T^ is ±e...

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