8 t h I n t e r n a t i o n a l C o n f e r e n c e on t h e B e h a v i o u r o f O f f s h o r e S t r u c t u r e s (BOSS) Délft, t h e N e t h e r l a n d s , J u l y 1 9 9 7 , p p . 177-188
NONLINEAR SLOW MOTION DYNANflCS OF
TURRET MOORING S Y S T E M S IN DEEP WATER
L. O. Garza-Rios and M . M . Bemitsas
Department of Naval Architecture and Marine Engineering. University of Michigan. Ann Arbor. Michigan 48I09-2I45. USA
ABSTRACT
A design methodology is developed to reveal the dependence of nonlinear slow motion dynamics of Turret Mooring Systems (TMS) on several design parameters, such as water depth, turret location and mooring line pretension. For a given TMS configuration, catastrophe sets are developed in the parametric design space, showing the dependence of subility boundaries and singularities of bifurcations on design variables. This paper provides fundamental understanding of the effects of design variables on the horizontal plane nonlinear dynamics of TMS. Thus, the Hrst guidelines are developed lo understand the qualitative properties of the system and r^hice trial and error in TMS design. Further, this methodology eliminates the need for extensive nonlinear simulations. The mathematical model consists of the nonlinear, horizontal plane fifth-order large drift, low speed maneuvering model. Mooring lines are modeled by deep water catenaries with touchdown effects and include nonlinear drag. A tanker TMS is used to illustrate the richness of the iK>nIinear system dynamics under external exciution consisting of lime independent current
KEYWORDS
Tunet Mooring Systems, Nonlinear Dynamics, Bifurcation Theory, Catastrophe Sets. Mooring Systems Design, Mooring Systems Dynamics.
INTRODUCTION
Station keeping of ships and floating production systems can be achieved by Single Point Mooring (SPM), Two-Point Mooring (TPM) or Spread Mooring Systems (SMS). Mooring systems intend to reduce die slow motion nonlinear horizontal plane^ikMipits^ejNhe moored vessel. InnplenKntations include floating production/drilling systems configured for o i l ^ ^ recovery'. - Oite such oonc^fis a concept of Turret Mooring System (TMS), where a iiumber o f ' ^ n a r y mooring legs are attached to a turret that is essentially part of the vessel to be moored. The birret inclfides bearings to allow the vessel to rotate fteely around tbe anchor legs, and can be mounted externaUy .iM$m tb^^ssel's b&w or stem, or internally widün tbe vessel. TMS allows the ship to weathervane by changing^t^headin|^ielative;to the actual environmental conditions without requiring a power drive roechanisin for rotatiSi. Thitjability ito take any orienution gives the vessel relatively good motion characteristics for production and/or^lling operations. The capability of combining the functions of productiofu storage and offloading in one facili^, a)ong with lower installation costs, make this concept a fiexibie and effective solution for a wide range of af^ications (Henery and Inglis, 199S).
Tbe first TMS was installed in 1986 in the Jaribo 09 field, offshore Australia, at a water depth of 120 meters (de Boom. 1989). Such ^stem was designed with a disconnectable Risisr T m e t Mooring (RTM). to allow the tanker to leave the site of operation io case of inclement weather conditions. With tbe recent advaocenteoi of techntdogy, sophisticated types of disconnectable and permanent TMS have been deployed in deeper waters under harsher eovinMunental conditions (Laures and de Boom. 1992; Mack, 1993). The need to venture into deeper waters has prompted die industiy, particularly oil companies, lo develop suitable TMS designs. The dynamics of
178 L. O. Garza-Rios and M . M . Bemitsas
TMS. however, arc not understood well, since the knowledge for this type of systems depends móstly on experience, limited experimental data, and numerous numerical simulations based on SPM models (Derclcsen and Wichers, 1992, Nishimoto et al. 1996),
The purpose of ihis paper is to develop a design methodology for TMS based on nonlinear dynamics and bifurcation theory that eliminates nonlinear simulations of the horizontal plane slow motion dynamics of the system, and reduces trial and error in TMS design. This methodology provides fundamental understaiiding of the effects of design variables (such as location of nirret, water depth, length and pretension of the mooring lines, etc.) on the nonlinear slow motion dynamics of TMS. Thus, qualiutive conclusions regarding the dynamical behavior ofTMS can be drawn.
I
MATHEMATICAL MODEL \
The slow motion dynamics of TMS in the horizontal plar»e (surge, sway and yaw) are modeled in terinslof the vessel/turret equations of motion without menwry, nworing line model, and external exciution. Figure I depicts the general arrangement of the system, with two principal reference frames: (x,y) = inertia! reference frame with its origin located at mooring temunal 1; (X.Y.Z) = 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^m .yi^) are the mooring coordinates of the iih mooring terminal with respect to the (x.y) frame; is the horizontal projection of the ith mooring line; y^'^ is the angle between the x-axis and the ith mooring line, measured counterclockwise: yr is the drift angle; Ug is the current speed; and g is the current angle measured with respect lo die (x.y) frame as shown in Figure I .
Figure l::^ige«i]eirj|(;pf Turret M ó ^ System (TMS)
'•.iHf ' ^-W ' ' ' • The intennediate reference frames of the vesset^unct system used in die derivation of die madtematical model are
shown in Figure 2. where iX', Y'. Z') is;d>e turret reference frame widi its origin located at the center of gravity jof \ht turret (CGr); and ( X ' . K ' . Z * ) is flte rcfefinice fraim o f die vessel widi i u origin located at die center of gravity of die ship (€0$). Moreover. ,pcc is ti^distance between die centers of gravity of die system (CG) and die turret; and is the distance from the cent^ of gravity of die system and the vessel. These are related as in Eq.(l): •
4a; = - ^ 0 a ; . (i)
where is tbe mass of the vessel and mf hSiè ihass of tbe turret, lo adt&tion, Yi relative yaw angle between die turret and tbe vessel measured counterclockwise as shown in Figure 2. and Yf is die absolute! yaw angle of dK turret widi respect to die (x.y) frame:
Nonlinear Slow Motion Djnamics o f Turret Mooring Systems
Figure 2: intermediate reference frames in TMS
Equaüons of Motion
Tbe equations of motion of the vessel in surge, sway and yaw can be written witfi respect to the iX',Y',Z'') frame It is convenient, however, to recast them with respect to the (X.K.Z) frame as the hydrodynamic forces and moments, and the external exciuüons acüng on the system, are measured with respect to the center of gravity of the system CG. The vessel equations of motion in the (X.K.Z) frame art given by.
(ms + my)v + m^Gsr + (m, + in,)iK = r«(M, v.r) + F^wy + ^ÏT' m^CGsv + (l, + + nts^Gsnt = v.r) + DajFrr + ^ww +
(3) (4) (5)
I n die equations above. («. v.r) and i;r) are the relative velocities and accelerations of die center of gravity of tbe system widi respect to water in surge, sway and yaw. risspectively. addition, I, is die moment of inertia of die vessel widi respect to die Z-axis; . /n^ and ƒ ^ are die added mass and moment terms; F„, Fyr and Nj are die forces and moment diat die turret exerts on die vessel assumed to act at CGj; F^gg. Fsway ^ the external forces and moments acting on die vessel, such as wind and second order drift forces (Bemitsas and Garza-Rios. 1996); X», JV, aiid are die velocity dependent hydrodynamic forces and moment expressed in Getms of die fifth-order, large drift slow motion derivatives (Tanaka, 1995) following die non-dimensionalization l»yTakashina(1986). r i l ^ ^
The horizontal plane equations of m o t i o i ^ f the t i i j ; ^ in die (X'. T.ZO frame are: .ith ÏCtit. mrür - " r ' ^ ' Y ^ l S f l l ^ ' ^ - /JJ<«)c0s/r<'T + f l ^ ' ^ s i n ^ ' ^ } + f^s-„ , v , * mrurrr = / 5 < ' ^ s i n ^ « - fï^^cos^-^'^} * Ff^s-„. (6) (7) 1=1 / , ^ = 5 t f F / 0 { [ 7 ^ ' > - / ^ ' ^ j s i o ( / r < ' ^ - / ; ^ ) ^ (8)
b l die expiessiohs above. (i<r.v>-.»l-) and («r-'^fV'ï) *rc die velocities and accelerations of die turret center of « a v i t y measured in die turret local coordinate system; I j is die turret moment of inertia about die Z'-axis; Dr i$
the timet diameter, f x , , f i , and ^ , are die forces and momertdiat die ves^^ Tbetemis msde dK summations in Eqs, (6)-<8) apply to each of die mooring lines ( i « 1,..., n). and are defined as follows.
ISO L . O. Garza-Rios and M . M . Bemitsas
in the catenary: f ; and are the drag forces on the catenary in the directions parallel and perpendicijlar to the mooring line motion, respectively (Garza-Rios tt al, 1997); fi' is the angle between the X'-axis and the mooring line measured counterclockwise ( ^ ' = y - VT)' and YO ' f * angle « which the catenary is atuched in the turret with respect to the X'-axis (fixed to the turrei). Eqs. (6)-(8) above assume that the turret is internal ito the system and that no hydrodynamic or external forces or moments act directly on the turret.
The forces that the vessel exert on the turret ( f ^ j and a « of equal magnitude to those exerted' by the njrret on the vessel ( % and F^). They are related as follows:
fxT = -^xscos(v'r-V') + fwsin(yr^M'). , (9) ^w = - ' j w s i n ( V r - V ) - ' > 5 C o s ( V r - r ) - ! (10)
The exerted moments and Nj are small compared to the other terms in Eqs, (5) and (8). and are generally neglected. Modeling these terms is difficult, as there are almost no existing experimental data. These are ijf equal magnitude as well, and incorporate the damping between the vessel and the turret, as well as the frictionjexened as the turret rotates with respect to the vessel. These moments are thus a function of their relative rotational velocities. A proposed model for these would be:
Ns=-Nr='-nrr-r), (11) where r is a small factor that depends on geometry, line damping, and external system exciutions.
The linear velocities and accelerations of the turret center of gravity (CGj) depend on the motion of the vessel, and thus the turret linear Eqs. (6) and (7) are not independent from the vessel Eqs. (3) and (4). These can be combined through relations (9) and (10). and the system has therefore four independem degrees of fnéóotii. The djrnamics of the system can be recast in terms of four equations of motion. OHresponding to the three equatibns of njotion for the system (i.e. vessel and nirret combined) in surge, sway and yaw. arxl one equation involving the roution of the turret. The equations of motion of the system with respect to tbe (X,Y.Z) reference franic. are given by:
(m + m^)u- (m + m,)vr = X^iu.v.r) + ^{[T^'^ - ^^'^jcos^'^ + FI'^ sin^'>} + F,^^,. (12) i=l
(m + in,)v + (m + m,)i«- = Y^iu.v.r) + ^^{[T^'^ - /^'^Jsin^'^ - /^''^cos^''^} + F^^, (13)
( / « I - y „ ) r = NH(u,v,r)^ DCCXJT^'^ - i^'^lsin^'^ -f^')cos^''>} + t(rr - r ) . '(14)
•vC.aU,. "
where m is the total mass of the system (ffl-fn^tfemrX is tfie moment of inertia of the system^about the Z-axis. In addition. and Fi, are tbe drag foriMs on dié^Ëitenary in the directions parallel and perpendicular to die mooring line motion nwasured widi respect to the system velocities; and /3 is the angle between die X-axis and die mooring line, measured coonterclockwiscLC^ » y ).
The fourth equation (i.e. rotation of die turret) is^ètnK>oiait^6quatiori*^iOut aii axis parallel to the Zaxis (i.e. Z '
-axis) and need not be transferred. In die ( X ' . r . Z O frame it is given by: i
fyft" 1'^/.^'^{['^'^ - /5^'^]sin(^'<'^ - y « ) - f^'^co,(/r<'^ - y « > ) } - r(o. - r ) . (15)
Eqs. of motion (12)-(14) show diat. i f die exerted momeim between die vessel and die turret are ignored, the systemtendstobebaveasif all mooring lines were attached to die saiiK point at a (fistance Deo from die center of gravity of die system in die absence of a torreL This is equivalent to mod^ng a SPM system, provided diat the ineitia of this t y ^ of system is equivaloit to diat for die TMS. Eq. (IS) above, however, shows diat even in the
Nonlinear Slow Motion Djiiarhics of Turret Mooring S)'stenis 18!
absence of an external moment exerted by the vessel, the turret itself has a non zero rotational acceleration, unless all the mooring lines are attached at the center of the turret, in which case a SPM model given by Eqs. (12>-( 14) would be appropriate. In real mooring applications, however, this scenario is not possible, since a number of risers or other production/recovery devises pass through the center of the nirret. Then, the oscillatory motion of the turret affects the dynamical behavior of the system, as the mooring lines attKhed to the nirret. oscillate w ith it transmining turret motion to the system.
The kinematics of the system are governed by Eqs. \ I 6 H I9):
x = ucosv-vs\niif-Ucosa. ( 1 6 ) y = «sinv + vc()sV' + t/sina. ( 1 7 )
V=r, ( 1 8 )
> r = 0-- ( 1 9 )
where {/ = is the absolute value of the relative velocity of the vessel with respect to water.
Moonng Une Model
The mooring lines in the system arc modeled quasistatically with catenary chaii». and include nonlinear drag and touchdown effects (Garza-Rios et al. 1997). For the numerical applicatioiis in this paper, the mooring lines have an Average Breaking Strength (ABS) of 5 1 5 9 KN (Nippon Kaiji Kyokai. 1983). The total tension T in the cateiiary is given by:
7" = V ^ ^ ^ = T « c o s h (20)
If
where 7",. is the vertical comporient of the rnooring line tension; F is the weight of the Catenary per unit length (1510 kg/s2). and / is the horizontally projected length of the suspended part of the catenary . Tbe horizontal pretension of the mooring line is defined in terms of the vessel resistance R, with respect to the zero current angle as:
P„=cR. ( 2 1 )
where c is a factor greater than zero. defines the position of the mooring coordinates in the horizontal plane. The total pretension imposed on the catenary Tp is given by (Garza-Rios et al, 1997):
T,, = P„*Ph. (22)
where A is the water depth, or vertical length of ihe^i||spe.rid^.^atenary, In this analysis, all mooring lines are pretensioned by the same amount with a nnaximum hor|zootal,preiension Pgj- of 2.5% A B S (128.975 KN). Table
I shows the values of the total tension T in the catenary (in' KN) at various water depths for several values of Pgr. ranging from I % to 10% A B S . i j : : j L
TOTAL TENSION T IN CATENARY (KN) AT VAJÉföUS W A % t DEPTHS POR SEVERAL VALUES OF Pgr
water depth (m) P/rr^ I % ABS PUT=2% P C T = 3 ^ A B S F g r = I O % A B S
250 500 750 1000 15ÓÖ 2000 429.09 806J9 18 84.09 I S 6 l i 9 2316.39 3 0 7 I J 9 i 635.45 TI012.95 1390.45 • 1767.95 £2522.95 ? 3277.95 893.40 1270.90 1648.40 2023.90 2780.90 3333.90
As shown in Table 1, the total tension in the catenary the weight of the suspended catenary, irrespective of
dramatically with increasing water depth due to amount of horizonul tension or pretension In the
182 L. Ö. Garza-Rios and M . M . Bemitsas
mooring line. In deep water mooring siniations. it is commoii to use a hybrid combination of catenary /polyester strings to reduce the weight of the mooring line, or to place buoys along the catenary .
THEORETICAL CONSTOERATIONS: NONLINEAR STABILITY ANALYSIS
For a given TMS configuration, there is a number of design variables (system parameters) that the designer needs to select, such as the hydrodynamic propeaies of the hull and the size and location of the turret; the number of mooring lines in the system; length, orientation and pretension of the lines, etc. The hydrodynamids of the hull can only be modified during the hull design process, and the turret size is set by the strength of the ship hull and the beam limitations of the vessel (McClure et al, 1989). The selection process for the other parameters is, in general, very tedious and numerous nonlinear time simulations are needed at each cycle in a trial and error approach. In addition, variations in the ocean environment, such as current, water depth and other external excitations, make the design process even more difficult. In this section, the principle of nonlinear stability analysis is applied to the TMS mathematical model. This principle is used to obuin qualitative information on the slow motion dynamics of a particular TMS configuration to avoid trial and error and virtually eliminate nonlinear simulations.
State Space Representation
The nonlinear mathematical model presented in the previous section is autonomous, and can be recast as a set of eight first-order nonlinear coupled differential equations by combining equations of motion (12:)-(IS) and kinematic relations (I6)-(I9). By selecting the following sute variables: (X|=H,ar2= v.jr3=r,x4=r;-.
X3=x,jC(=y,jC7=yr,xg=y^7-), the system nonlinear nnxielcan be written in Cauchy standard form as:
I X2 = (ffl + ffij) 1 i3 = (m + ffiy) I surge 1 ' 4 = T XHUU'2''3) + ( " + 'nv)'2'3 + É J ^ ^ ' ^ " 'I'^jcos^'^ + i ^ ' \ i n ^ ' ^ } + F,
YH(xi,X2,Xi)-(m + mJx,X3 + f { [ 7 ^ ' ^ - /<'^]sin^'> - F^'^cos^'^}* F « ^
\ « ( x „ X 2 . X 3 ) +
Dect§7^'^
- /l'^]sin^'> - /^'^cos^'^} + + r ( x , - X 3 ) 1=1X5 = X| COSX7 - X2 s i n x 7 - f/cosa, X( = X| sin X7 + X2 C0SX7I;* Uüng),
•* >k- fokt (23) (24) (25) (26) (27) (28) (29) (30) In the equations above, 7^, ^ and ^ are functions of state,^i;iriable$j('X5,x^,X7.xg); F^, ^ , and Fi are fiuKtions of all state variables; aitd / ^ « . F f ^ . and are^functions of X7 exchisively. Hereafter, evolution Eqs. (23H30) will be denoted as:
* «
i = / ( x ) . f e d , f:^-^^, (31)
where 9 t ' is the ei^t-dimensional Euclidean ^kaoe and C ' is the dass of continuously differeotiable ftinctioas.TUSEquUiMa
Equilibria of die nonlinear TMS model can be computed as mtersectiODS of null ctines (Seydel. 1988). Thus, all equilibria can be found by setting die time derivatives of tbe state vector (31) equal to zero, i je..
Nonlinear Slow Motion Dyriamics of Turret Mooring Systems 183
where the overbar on the sute vector x represents equilibrium. The TMS may present more than one equilibriumu depending on the configuration of the system and the external exciutions under which the system operates (Garza-Rios. 1996). All TMS exhibit a principal equilibrium position (hereafter denoted as 'Equilibrium A'), in which the vessel orients itself to the direction of the vector of external exciution (i.e. weathervanes). Other equilibria may be present in the system as well, and are denoted as B. C. D, &c.
^abiliiy Star Equilibrium
Local analysis near equilibriuin points reveals the behavior of the trajectories in its vicinity. Local analysis is performed by studying the linear system:
«Hiere ^ represents deviation from equilibrium, and [A] is the Jacdbian matrix of ƒ evaluated at equilibrium ( x ) . I f all the eigenvalues of Eq. (33) have negative real parts, equilibrium x is asymptotically suble, and all trajectories originating near x will converge toward it in forward Ume. If at least one eigenvalue has a positi ve real part, J is unsuble, and all trajectories originating near such equilibrium will deviate from it in forward time (Guckenheimer and Holmes. 1983). The principle of nonlinear subility analysis based on nonlinear dynamics can be used to discern the dynamics of the system without performing itonlinear simulations. This principle can be ai^lied to develop a TMS design methodology by performing bifiircation theory, as shown in the following section. A more detailed description of these concepu applied specifically to mooring systems is given by Bemitsas and Garza-Rios (1996).
BIFURGATIONS OF EQUELIBRU AND CATASTROPHE SÉTS
Ths dynamics of the system may change significantly with variation óf (ksign variables such as water depth, position of the turret, number and length of the mooring lines, pretension, etc. In this Section, bifurcation sequences are snidied to find the qualiutive changes in the dynamical behavior of TMS as a function of several design parameters. In order lo perform bifurcation analysis, evolution Eqs. (31) are wriaen as:
x = /(x,fi). x € 9 l * , | i € 5 l ^ ' , (34) where is the design parameter vector, and A/p is the number of parameters in the system. In this analysis, only
the water depth (A), the turret location (Dec) a"<l horizontal pretension (Pgf) are considered as parameters, aitd thus M^lff'Dcc-Ffr]^- behavior of the solutions to the dynamical system described by (34) is gny>hically illustrated with a parameter or subility chart. The set of lines shown in a subility chart is called a 'catastrophe set,' (Bemitsas and Garza-Rios. 1996). Those lines constitute the boundaries between regions of qualitatively different dynamics. A catasuophe set is a very powerful tool which serves the purpose of illustrating ihequaliiaavebehaviorof iheTMS asanumberofparainetersvary., lo/a
-Catastn^heS^forTMSDynamia -ivBI I n this subsection, the nonlinear dynarhics of a 4-line tanker t j ^ ^ who^^ropertïes are shown in Table 2 are
discemed as a function of die parameters mentioned above.
1 — : — • -^^r^ S T T T T — 1
TABLE 2
GEOMETRIC PROPERTIES OF TTIE FFSO TANKER
PropfflY
Lengdi overall (LOA) 272.8m Lengdi of die wateiline ( L ) 2S9.4m
Bc»m(B) 4 i l 0 m
D n f t ( D ) 16.13 ra Tunet diameler ( D ^ ) 22J0m Block cocfTicieat (Cg) 0.83 IXsi^acefneat(iS) IJ374xlOStons
184 L. O. Garza-Rios and M . M . Bemitsas
•7i
0-Figure 3 shows a set of catastrophe sets in the (DcclL h) plane in the range (0.30 S Oc^/i S0.50; 400 m < A S 2000 m). for four different mooring line horizonul pretensions, ranging from I * ABS to 23*k ABS. under a current speed Uc of 2 knots. These correspond to 1.0.1.5.2.0 and 2 J% ABS (3.96 R. 5.94 R, 7.92 R. and 9.90 R). and are denoted as F ^ . F ^ . P„3 and '*jpr4. respectively. Such small values for the horizontal pretension are used mainly to show the different types of dynamical behavior that TMS may exhibit. In addition, the length of the catenaries is fixed to 5 times the water depth, and the mooring lines are spaced 90* apart, and oriented as shown in the figure. This orientation ( a = 0') provides the minimum mooring load from the drag on the lines, yielding the least amount of tension in the mooring lines, but also showing the richness of the TMS nonlinear dynamics.
Figure 3: catastrophe set. 4-line TMS: effect of turret location, waier depth and hcHizontal prtension
The catastrophe sets of Figure 3 show five regions of qualiutively different dynamical system behavior
Region I (R-I): Equilibrium A is suble; all eigenvalues have negative real parts, and all trajectories initiated near A will converge toward it in forward time. Figure 4 shows a time simulation of the TMS in this region (Z>cc/L= OJ75, h= 600 m, Pgn), showing convergence (a suble focus) to,^ui|ibriuj^A ( y = 0*).
Region n (R-U): Equilibrium A is unsuble, with a one-dimensional unsublè manifö^ Sutic loss of stability takes place when crossing from R-I to R-II; a transcriiical pitchfork bifurcation o6curs, and as a result, two additional suble equilibriai. which are mirror images of each other and denoted as B ahd B', appear in R-U. In dus region, the vessel diverges from equilibrium A and converges to equilit^un B orB' (Y* 0^ dqiending on die initial conditions. Figure 5 shows die behavior o f dié system.in R-D (D^^/t=0.3$ Ji= 600 m. F/m) for two different sets o f initial conditions, converging to B or B ' .
Region H I (R-m): Equilibrium A is unsubk. with a two-dimensional unstable manifold. Dynamic loss of stability occurs when crossing from R-I to R-III, wid) development of a Hopf bifurcation, tn diis regitMi. die system oscillates about equililmum A. achieving a suble limit cycle, widi no other suble equilibria diat may attract die system trajectories, as shows io die simulation of Figure 6 ( D c c f i ^ 0.40, 1000 m,
Pgn)-Region IV (R-IV): This region is characterized by die merging of R-II and R-IU. Equnibrium A is onsuUe widi a three-dimensiotial unsuble manifold, and die dynamics of the system are chaotic widi respect to das equilibrium position ( B m i t s a s and Garza-Rios. 1996). Equilibria B and B ' are unsuble widi iwo-dimenstonal unsuble maniftrfds; all tngectories will deviate from A and converge to a suble limit cycle about B or B', dqcnding on the initial conditions. Figure 7 shows diis type of behavior ( D ^ / L = 0.55, As 2000 m. Pgn), in wfaidi die system reaches a limit cycle about equilibrium B .
Nonlinear Slow Motion Dynamics o f Turret Mooring Systems iOS
Rgure 4: stable focus, equilibrium A (regim R-I)
-6H
-10
so 100 ISO 200 2S0
—rr
300 3S0 400 430 - SOO
Figure S: stable focus, equilibria B and B' (region R-D)
Region V (R-V): Equilibrium A is unsuble with a one-dimensional unsuble manifold; equilibria B and B' undergo dynamic loss o f stability, and art nnsuble with two-dimensional iuisuble manifolds. The dynamics in d i u r ^ i o n result in a stable limit cycle about either equilibrium B or B'.
»
The richness of die nonlinear TMS dynamics is clear ih the catastrophe sett of Figure 3, and verified by die nonlinear simulations of Figures 4-7. These sets also provid^ die necessary infonnation to understand die system nonlinear dynamics as a parameter or sets <rf parameters are i ^ e d .
As can be inferred from Figure 3, d » slow motioo dynamics pf the system about equilibriuin A are not altered by either water depth or horizontal pietensioa for small value^of Dec 0*ss dian 0.36). and dut die system may oscillate in this region about an altemaie equilibrium B or B', unless die horizontal pretensioo in die system is increased. For small and intermediate water depths (h 1 3 4 0 m), die system tends to align to die diredioa of the
186 L. O. Garza-Rios and M . M . Bemitsas
current as the tiirixi is moved forward (i e. Dec « increased). These results agree with the conclusions by Yashima et al (1989). and the nonlinear simulations by Nishimoto ei a/ (1996). using a SPM system model. Figure 3 also shows that the system may fall into an oscillatory behavior irrespective of the location of the turrei as the water depth increases. Operation in any of diese regions (III. IV. V) may be feasible, provided! that the limits of TMS nrotion and dynamical tensions are satisfied. In such a case, the designer can select an appropriate location for the turret, i.e. either close to amidships to reduce the turning moment for heading conut>l of the vessel, or further forward to attain higher weathervaning capabilities.
.0.
Figure 6: stable limit cycle about equilibrium A (region K-UD
Figure 7: stable limit cycle about equilibrium B ( r e ^ R-IV)
Stability of tbe system m d e ^ watert can be adiieved by increasing the horizontal pretension in the catenaries, as shown in Figure 3. Aoodier optit» for stability would be to increase the number of mooring lines in the^ystem. This alternative, which increases the drag in tftemooring lines, has the same effect as increasing the pretensions, and can be considered at the expense of higher mooring line tensions.
Nonlinear Slow M o t i o n Dynatriics of Turret Mooring Systems 187 Other design parameters, such as number and length of the rnooring lines, as well as varying environnwntai
conditions; can be incorporated readily in the catastrophe sets. The selection of TMS parameters ultimately depends on the type of operation and the ocean environment as well as the strength characteristics of the vesselAurm system
CONCLUSIONS
A mathematical model for the nonlinear slow motion dynamics of Turret Mooring Systems has been derived. It has been shown that SPM models for TMS are not appropriate even In the absence of moments transmitted between the turret and the vessel. This is because the rotation of the tunet affects the dynamical behavior of the system. It has been also shown that water depth does not affect the static stability of the system about its principal equilibrium, but that the system may fall into an oscillatory behavin^ in deep waters. Further analysis is required 10 determine the plausibility of operating in such regions. The relatively small horizontal pretensions selected in this work show the richness of the nonlinear dynamics that TMS may exhibit Such pretensions, however, may result in cases of 'slack' mooring lines (i.e. catenaries with no tension in die horizontal plane). The possibility of implementing additional mooring lines to the system will result generally in higher horizontal tensions, due mostly to the drag on die catenaries in deep waier widi changing current directions. This scenario, however, may be an attractive option if required by the environment Due to space limitations, only three design parameters have been considered. This methodology can be readily extended to several parameters by applying die analytical relations for the subility and bifurcation boundaries derived for SMS in (Garza-Rios and Bemitsas, 1996) following the design methodology proposed by Garza-Rios (1996).
ACKNOUXEDGMENTS
This work is spotöored by die University of Michigan/Sea Grant/Industiy Consortium in Offshore Engineering under Michigan Sea Grant College Program, project number R/T-35 under grant number DOCr;NA36RG0506 from the Office of Sea Grant National Oceanic and Atmospheric Administratioo (NOAA), VS. Depaitment of Conunerce. and funds Prom die Sute of Michigan. Industry participants inchide Amoco. Inc.; Conoco, Inc.;
ExxMi Production Research; Mobil Research and Development: and Shell Companies Foundation. The U.S.
Government is authorized to prodiice and distribute reprints for govemmenul purposes notwidistanding any copyright noution appearing hereon.
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