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Twenty-Fourth Symposium on Naval Hydrodynamics (2003)

Chapter: A Nonlinear Stability Analysis of Tandem Offloading System

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Suggested Citation:"A Nonlinear Stability Analysis of Tandem Offloading System." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"A Nonlinear Stability Analysis of Tandem Offloading System." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"A Nonlinear Stability Analysis of Tandem Offloading System." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"A Nonlinear Stability Analysis of Tandem Offloading System." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"A Nonlinear Stability Analysis of Tandem Offloading System." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"A Nonlinear Stability Analysis of Tandem Offloading System." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"A Nonlinear Stability Analysis of Tandem Offloading System." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"A Nonlinear Stability Analysis of Tandem Offloading System." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"A Nonlinear Stability Analysis of Tandem Offloading System." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"A Nonlinear Stability Analysis of Tandem Offloading System." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"A Nonlinear Stability Analysis of Tandem Offloading System." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"A Nonlinear Stability Analysis of Tandem Offloading System." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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24th Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 A Nonlinear Stability Analysis of Tandem Offloading System Dong H. Lee, Hang S. Choi (Seoul National University, Korea) ABSTRACT In this paper, we analyze the linear and nonlinear stability of a tandem offloading system in wind, current and waves. The wind and current forces are evaluated with the help of published experimental data, while the hydrodynamic coefficients and wave drift forces are carefully estimated by using a three-dimensional singularity distribution method based on potential theory. The bow-hawser and mooring lines are modeled quasi-statically by elastic catenary equations. In order to examine the static and dynamic stability of the system, equations for surge, sway and yaw are linearized. Based on the Hartman-Grobman Theorem, the Stable Manifold Theorem and the bifurcation theory, the effect of design parameters such as turret position, mooring stiffness, hawser length and stiffness on the stability is investigated. The stability diagram of the tandem offloading system was quite different from those of a single-point mooring system and a turret mooring system. The non-linearity in the strength of hawser and the environmental load are included. For consistency, the coupling effect of surge motion with sway and yaw motions is also included in simplified nonlinear equations of motions. The nonlinear stability analysis clarified the mechanism of limit cycle for the tandem offloading system, which is known as the fishtailing motion. INTRODUCTION Nowadays, turret-moored FPSOs (Floating Production Storage & Offloading) are increasingly deployed for oil exploitation in deep-water marginal fields. The produced crude oil is normally transported by shuttle tankers, which are connected in tandem to the FPSO through bow-hawsers during offloading process. Thus, an analysis on the relative motion between the FPSO and the shuttle tanker is critical for safe loading operation. It is well known that moored vessels undergo unstable large drift motions even in mild sea-states. This unstable phenomenon is known as a fishtailing motion in towed ship dynamics. It results from the interactions between flow and elastic structure. Galloping and flutter are similar phenomena in the field of aerodynamics. Petrobras reported that a tugboat was always required for shuttle tankers without Dynamic Positioning System (DPS) in Campos Basin (Sphaier et. al., 2001~. Therefore, a stability analysis is necessary for the design of a stable mooring system and the representative mooring parameters - mooring stiffness, turret position, hawser length, etc. - have to be carefully determined. Many studies have been conducted on the stability analysis and the motion simulation of moored vessels. Bernitsas at the University of Michigan has investigated design methodologies based on the stability analysis and bifurcation theory for several mooring systems (Bernitsas and Papoulias, 1990; Chung and Bernitsas, 1997; Bernitsas and Kim, 1998~. Another research group on this topic is Petrobras, the Federal University of Rio de Janeiro and the University of Sao Paulo (Fernades and Sphaier, 1997; Simos et. al., 2001~. Simos et. al. studied the fishtailing motion of a single-point moored tanker theoretically and experimentally. Recently, more researches on the tandem offloading system have been published. To name a few, Morishita and Cornet, 1998; Lee and Choi, 2000; Fucatu, et. al., 2001;Spahier, et. al., 2001. In this paper, we have studied a FPSO-shuttle tanker system with tandem configuration in current, wind and waves. The wind and current forces are evaluated with the help of experimental data. Bow-hawsers and mooring lines are modeled quasi-statically. The hydrodynamic coefficients for the moored vessels are rigorously estimated by using a singularity distribution method based on linear potential theory. Based on the Hartman-Grobman Theorem and the Stable Manifold Theorem, the stability analysis of the tandem offloading system is carried out. Bifurcation theory is used to understand the entire feature of the dynamic stability. The effects of environmental conditions and mooring parameters on stability have been examined in terms of the ratio between wind and current velocity, mooring stiffness, hawser length and its tension. The effect of the relative motion between vessel's motion and surrounding fluid is investigated. A simplified nonlinear equation of motion is also derived in order to include the nonlinear tension of the hawser.

NONLINEAR MOTION SIMULATION equation: Equations of Motion A FPSO-shuttle tanker system is considered as shown in Figure 1. To describe the motion of the FPSO-shuttle tanker system, two coordinates systems are introduced. Let o-xy be the body-fixed coordinate system with its origin located at the mid-ship of moored vessel. The x-axis points the bow. The O-XY denotes the inertial coordinates fixed to the earth. For simplicity, only the horizontal plane motions (surge, sway and yaw) are considered herein. y x 2 x 1 11 ~ ~—r— .- \ o ~ of an a O BOX Figure 1: Coordinate systems The FPSO is turret-moored, while the shuttle tanker is connected to the FPSO through the bow hawser. In this figure, a denotes the distance of the turret position, and ~x, ,8 and I denote the distances of the attached positions and the length of the hawser, respectively. The mathematical model is derived from the Newton's conservation law of linear and angular momentums. The non-linear coupled equation of motion for each vessel is formulated in the corresponding body-f~xed coordinate system (Fossen, 1994~. Mvi + C(vi jVi = Fi (t~memo~ + Fi nonviscous + F i (tidied + Fi (t)Curren' + Fi (trade + Fi (t~mooring + Fi (t~hawser , i = 1, 2. where M is the mass matrix, which includes added mass and added moment of inertia; Vi = [ui,vi, ri]T is the velocity vector of the i-th vessel; C(vi~vi are the Coriolis and the centripetal force and moment. Each vessel's velocity is expressed in terms of the corresponding body-fixed coordinates and it is related with the inertial coordinates by the following (Xi: Hi rli = Hi = sin Hi Nisi ~ O -Sinai O hi cosyri O vi . (2) O 1 Pi External forces consist of the wave radiation force, the viscous force, the wind force, the current force, the wave exciting force, the mooring force and the bow-hawser force. The wave radiation force contains the memory effect, which is represented by a convolution integral of the time-memory function. It is known that the time-memory function can be obtained effectively using the hydrodynamic damping coefficients. Fi (tremor = .L~ L`(t—r) Vi (~)a7r <3y The turret mooring system for the FPSO consists of anchored chains, which are modeled by catenary. In this study, we considered 12 catenary-chain lines, which are spread axisymmetrically. The dynamic effect of the mooring line is ignored and the restoring force is evaluated quasi-statically by using the catenary equation. For time simulations, the relation between the vessel's offset and the restoring force is established, in which the instantaneous touchdown points of mooring lines are considered in order to describe the geometric nonlinearity in mooring forces. The bow-hawser force is generated as a result of the relative distance between the FPSO and the shuttle tanker. The elastic catenary equation has been used for the bow-hawser force. The horizontal distance is related with the tension as given by (Irvine, 1981~: EA w Sit ( H ) (4) where d is the horizontal distance between the FPSO and the shuttle tanker; H is the horizontal component of the tension; w is the hawser weight per unit length; I is the hawser length; E and A are the Young's modulus and the cross-section area of the bow hawser, respectively. Environmental Loads Environmental loading exerted on moored vessels are the results of current, wind and wave. Wave forces consist of first and second order components. Slow drift forces can invoke resonated horizontal responses of the moored vessel. Wave forces are calculated by

using the singularity distribution method based on linear potential theory. For tandem offloading systems, hydrodynamic interactions between the FPSO and the shuttle tanker are included. The force and moment acting on bodies are calculated by integrating the pressure inferred from the Bernoulli's equation. Furthermore, wave drift forces are obtained by the direct integration method (Pinkster & Oortmerssen, 1977). In maneuvering models, current loads are included implicitly in the equation of motions by using the relative velocities between the vessel and the surrounding fluid. This approach is known to be more accurate than the method based on projected area and drag coefficients of the vessel. However, maneuvering models require many hydrodynamic coefficients that have to be measured by PMM test. Thus we use the projected area and drag coefficients method in this work. The current and wind forces are expressed in the following form: F = 2 CpVr2A, USA where p is the water or air density; C the drag coefficient; A the projected area exposed to current or wind; Or the relative velocity. In this study, the experimental data for VLCCs recommended by OCIMF are used for evaluating the current and wind loads. It is noted that the shuttle tanker behind the FPSO experiences the sheltering effect (Lee and Choi, 1998~. Some experimental results show that the presence of the FPSO stabilized the motion of the shuttle tanker (Fucatu, et. al., 20014. This effect can be significant for the dynamic behavior of tandem offloading system. However, we did not include the sheltering effect here because published data on this topic are not yet available. The vessel's motion also induces viscous damping forces, which can be readily included into the current and wind loads by using the relative velocity concept for surge and sway motions. The drag moment resulting from pure yaw motion cannot be considered in eq. (4), but it can be calculated with cross-flow model, as follows: Fmv 2 PWaierT ICD(X~)X |x|cabc rare, (6) where CD(xj is the transverse drag coefficient for two-dimensional cross-flow; r is the yaw angular velocity and x the longitudinal coordinate measured from the mid-ship. Numerical Results and Discussion For numerical simulations, we considered a typical FPSO-shuttle tanker system. The principal dimensions of the FPSO are 277m(1ength) 45.5m~breadth) 20m~draft). The dimensions of the shuttle tanker are assumed to be the same as the FPSO. The submerged weight of a catenary chain-line is 2943 N/m. The stiffness of the turret mooring system is approximately 235 kN/m. The position of the turret system is variable as a design parameter. The elasticity of bow-hawser EA is 1.0E7 N. Its length is also variable as a design parameter. The restoring force and the horizontal tension are given in Figure 2 and Figure 3, respectively. As one can see in these figures, the hawser tension has strong nonlinear characteristics, while the mooring stiffness is nearly constant in the region of small excursion. 2( -~.2 -0.15 -0.1 -o.oS o o.OS 0.1 0.15 0.2 excursion: xIL Figure 2: Restoring force of turret mooring system 2C. t.4 -o ~ ~2 ~ · ~ ~ 3 horizontal distance: (HIM Figure 3: Horizontal tension of bow-hawser The incident angles of current, wind and waves are all taken as 180° and the current velocity is 0.5 m/sec. The wind velocity is variable and the ratio

between wind and current velocity ~ ~ = Vw / Vc ~ is taken as a design parameter. As mentioned in the A, previous section, the sheltering effect between the = FPSO and the shuttle tanker is not considered in this study, although it may affect the dynamic behavior of the FPSO-shuttle tanker system. Figure 4 shows the time history of yaw motions of the FPSO and the shuttle tanker. In this case, the turret is located at 0.2L and the hawser length is 0.6L, where L is the length of FPSO, and the wind velocity is 0.0 m/s. The steady-state response corresponds to a large periodic oscillation, which is called the fishtailing motion or limit cycle motion. The nonlinearity of hawser tension and the viscous damping force prevent the eventual blowup of motion. The trajectories of the mid-ships are plotted in Figure 5, which shows a typically nonlinear behavior. The phase diagram is given in Figure 6. FPSO 90 6( 3( ~ ( >~-3( -6( -9( 90 60 30 O -30 -60 -90 Figure 4: Yaw motions of FPSO-shuttle tanker system (a=0.2L, 1=0.6L, Vw-Om/s) shuttle tanker ~_A 200 1SO Inn an nn -1 50 Inn FPSO 100 80 .. 60 .. 40 .. 20 .. ~ O .. -20 .. 40 .. -60 .. : : : 1 1 -250 -1 00 -410 -400 -390 -380 -370 -20 -10 0 10 20 Am) xLm) Figure 5: X-Y trajectory of FPSO-shuttle tanker system (a=0.2L, 1=0.6L, Vw-Om/s) -0.1 . . . . -02 , , . . . . . -40 -30 -20 -10 0 10 20 30 40 yaw~deg) Figure 6: Phase diagram of yaw motion (a=0.2L, 1=0.6L, Vw-Om/s) . . . . . . ~ . .. .... ~ . ... .. .. ~ ~ ~ ~ ;~; ;~ ;<j ;~\ ~ ~ ;j,:; - ~~~~--~~~ 7--~--r~~;7---r---~-r-~~~~~--~~~~/----~~-~-~' , , , , , , . FPSO ~ . ; , ; ; 20 . ~ ~ . , 0 50 100 150 200 250 300 350 SHUTTLE ~ 10 _ . ... ..... ~ r ~~~~~~~~~ ~ (U ~ : : : : : : ~ . ~ O t~ ~ , ~ t ~ ~ ~ t ~ 1 t t ~ 1 `-10 _ .... . .. , . . .. a. .. .. . . ~ _ . ~ ~ ~ if :7 ~ ~ _ ; , ; 1 1 1 1 1 1 0 50 100 150 200 250 300 350 time(min) A typical stable motion is given in Figure 7. In this case, the turret position is 0.2L, the hawser length is 0.2L and the wind velocity is 30.0 m/s. It is noted that the yaw motion is very small but still oscillating. _ 10 O `-10 -20 O 50 100 150 200 250 300 350 10 O ^-10 2OA 50 100 150 200 25G bme(min) Figure 7: Yaw motions of FPSO-shuttle tanker system (a=0.2L, 1=0.2L, Vw-30m/s) From the parameter study, the yaw motion amplitudes of the tandem offloading system are summarized in Table 1 and 2. It is found that the velocity ratio is most significant parameter affecting the yaw motion and there may exist a critical velocity ratio between 20 and 40, where the amplitude dramatically decreases. In order to investigate the effect of the drift force on the motion of the FPSO-shuttle tanker system, an irregular sea with the significant wave height of 6.0 m and its peak period of 8.6 sec is considered. The result is given in Figure 5. It is found that the drift force negatively contributes to the stability of the tandem offloading system.

Table 1 Yaw amplitude of FPSO cr. IIL O 20 40 60 0.2 43.0 16.4 4.3 1.6 0.4 38.0 13.3 4.0 1.8 Table 2 Yaw amplitude of shuttle tanker 0.4 60.0 40.5 10.8 3.9 60 or an 0.6 62.1 41.5 11.1 4.1 nr 1 50 200 time(min) 2SC 350 Figure 8: Yaw motions of FPSO-shuttle tanker system M1l 0 (a=0.2L, l=0.2L, Vw-30m/s, mean wave drift) 0 M22 In order to clarify the entire dynamic response of the mooring system, a stability analysis is required where in terms of design parameters. The next section will describe a static and dynamic stability of the tandem offloading system. STABILITY ANALYSIS Linearized Equations of Motion In order to thoroughly understand the dynamic behavior of mooring systems, a stability analysis must be carried out. In order to do so, we must first find out the equilibrium points of nonlinear equations of motion and the equations are linearized near these equilibrium points. The equilibrium points can be obtained from the static force and moment balance between steady environmental loads and mooring forces. The equilibrium point of the shuttle tanker is determined from the following relations: .- 0.6 35.7 12.6 3.4 1.8 ~ where y and Fo are the hawser angle and tension, respectively; Arc and GO are the incident angles of current and wind, respectively. (X, Y. N) denote the steady forces and moment of wind, current and waves. The above equations can be rearranged as follows: Fo cos(`v2 - y) = -X2 (§UC, {VW, {V2 ~ ' (7a) Fo sin(`v2 - Y) = Y2 (~c, tow, ivy ), (7b) ,/3Fo sin(~y'2 - y) = N2 (arc, Yew' )/2 ~ ' (7c) Y2 (§UC, tow, 5V2 ~ = N2 ( {/c ~ yew ~ ~2 ~ ' (8) Fo2 = X22 + y22, (9) The heading angle of the shuttle tanker and the hawser tension is calculated from equations (8) and (9~. Then, the equilibrium point of the FPSO satisfies the following equations: (a + lox jFo sin(`v, - y) + at (arc, {VW, {v, ~ = N. (by, {VW, tu, ~ (10) Herein, the equilibrium point with zero heading is considered. By assuming the mooring stiffness and the hawser tension are constant and ignoring the quadratic viscous damping force, the linearized equations of motion for surge, sway and yaw are derived as follows (Lee and Choi, 2000~: x2~+[S2; S22~ X2~ = 0~ (11) m,, O Mii= O m22 O mi S., =~ -Kit sI2= O - o o my , m66 K+KH O K Fo Fo -a I O O _ Fo _p Fo 1 1 ax Fo ap Fo 1 1 O aK o o aK -—(cx + 1) - ~ I dy'' a K + or—(or + 1) -— I dye,_

--KH O O S21 = 0 Fo ax Fo , O _p Fo Up Fo KH O O ~ O ~ ~ (A + l) - d 2 . O ~ Fo ~ Fo (p + 1) _ dN2 S22 = where scripts 1 and 2 stand for the FPSO and the shuttle tanker, respectively. M is the mass matrix including added mass and S is the restoring stiffness matrix. K is the turret-mooring stiffness. Fo and KH are the hawser tension and the axial stiffness, respectively. The definitions of ~x, ,d, a and l are indicated in Figure 1. Static and Dynamic Stability According to the Hartman-Grobman Theorem and the Stable Manifold Theorem, the local behavior of the nonlinear behavior of the nonlinear equation near an equilibrium point is qualitatively determined by the linear equation. The condition for the static stability is that the restoring moment induced by environmental loads is positive at static equilibrium points. That is, the determinant of the stiffness matrix, dettS], should be positive. The stable condition for the FPSO-shuttle tanker system requires the following inequality: FO | _ dN2 + dY22 K dv2 dN1 _ + deal dY, a + dial aFO + F Lao. (12) Notice that the static stability is not affected by the presence of the FPSO. Generally, the shuttle tanker with single point mooring system satisfies the static stability criteria and has zero heading angle in this state. 1°(-d 2+{dY2)>0. (~13) But the turret-moored FPSO may have a bifurcation point. For the static stability of the FPSO-shuttle tanker system, the turret position, a, must be located forward of the critical turret position: dNI _ OF do,, a = or dYl +F d Y/} (14) The above equation indicates that the shuttle tanker, which is connected to the FPSO, invokes the critical point to move toward the mid-ship or a stern and enhances the static stability. The dynamic stability is determined in terms of design parameters such as a, K, l and FO, which govern the restoring stiffness matrix, S. In addition, stability is affected by the intensities of current and wind. The stability can be examined by checking the eigen-values ofthe equation (11~. I) = Xe2~. (15) The sign of eigen-values gives us the complete feature of the nonlinear dynamics. Particularly, if there exists a complex conjugate pair of eigen-values with positive real part, the equilibrium point is unstable with two-dimensional unstable manifold. Then, motions asymptotically reach a periodic oscillation, which is called a limit cycle. It is noted that the restoring stiffness matrix S is not symmetric unlike a spring-mass system. Such asymmetry of the restoring stiffness matrix is caused by the interaction between fluid loading and mooring stiffness. Thus, it may invite the unstable motion of moored vessels such as fishtailing. This kind of unstable phenomenon is prominent in aerodynamics. Numerical Results and Discussion As a case study, we considered the same FPSO-shuttle tanker in the previous section. Figure 9 shows the stability diagram of the FPSO-shuttle tanker. The hawser tension(FO) is normalized by the longitudinal drag force (Fe) due to current and wind. The hawser length is normalized by vessel's length. ~ denotes the ratio between the wind velocity and the current velocity. Figure 10 is the stability diagram of a single-point- moored shuttle tanker, which verifies that a relatively short bow-hawser suppresses the unstable motion of SPM system. However, the tandem-offloading system is quite different from the SPM. For tandem offloading system, the short hawser does not guarantee stability, as shown in Figure 9. It is easily observed in Figure 9 and Figure 10 that the wind velocity is the most important parameter for the stability of mooring system. As the wind velocity increases, the unstable region decreases. The

1 8 ,_1 6 a, O 1 4 IL ~ 12 o tin 1 a, ~ 08 ~7 3 0 6 04 no 0 0.2 0.4 0.6 0.8 1 1.2 1.4 hawser length (IJL) Figure 9: Stability diagram for TANDEM system as function of hawser tension and length 1 81 1 6 a, TO 1 4 ~ 12 o ._ 1 a, a, ° ~ ~ 0.6 s o4 no 8C 1r 0 005 01 016 02 025 03 0.35 04 045 06 Turret Location(alL) Figure 11: Stability diagram for TANDEM system as function of mooring stiffness and turret location 0 02 0.4 06 0.8 1 12 14 hawser length (IIL) Figure 10: Stability diagram for SPM system wind shows such a trend to stabilize the system because the selected model in OCIMF data has a deckhouse at stern, which induces a positive restoring moment. Figures 11 and 12 represent the effect of turret mooring stiffness and location on the stability. In this case, the diagram is illustrated for only one wind velocity to avoid confusion between lines because the stability boundary lines are more complicated. Wind loads tend to enlarge the unstable region in the case of the turret mooring system, as opposed to the SPM. In the case of the tandem offloading system, there exists an unstable island for relatively strong mooring stiffness. This result comes from the interaction between FPSO and shuttle tanker. Thus, the mooring stiffness and the location of turret must be carefully determined. Figure 13 shows the critical wind velocity for different hawser lengths. For the tandem offloading system, the stability boundary is not sensitive to the hawser length. It is also found that the tandem fir 1n 0 0.05 01 015 02 025 03 0.35 04 045 05 turret location~aJL) Figure 12: Stability diagram for TURRET system 35 30 ~,26 a 20 ·, ~ .?16 o a, 10 _ 02 04 06 08 1 1.2 14 hawser length (AL) Figure 13: Stability diagram for Tandem and SPM systems as function of velocity ratio and hawser length offloading system considered herein is always unstable in week wind conditions. In order to examine the occurrence of limit

cycle, eigen-values depending on the velocity ratio are plotted in Figure 14. It can be seen that there exists a complex conjugate pair of eigen-values with positive real part, when the velocity ratio is less than 24. This value of the velocity ratio is a Hopf bifurcation point. Fishtailing motions will occur below the critical velocity ratio. This is readily expected from nonlinear simulations, which are summarized in Table 1 and 2. U.U 1, n non - ~c n non O,OOc n non 3 ~ U ~ 3 ZU 23 dU 63 MU velocity ratio (\fw IVc) Figure 14: Variation of eigen-values depending on velocity ratio NONLINEAR DYNAMICS In the previous section, the stability is analyzed based on a linearized model. The behavior of the nonlinear system is hereby qualitatively investigated. As a result it is found that a nonlinear stability analysis predicts the occurrence of the limit cycle in terms of the Hopf bifurcation, but the quantities of the limit cycle such as motion amplitude cannot be obtained. In this section, we derived simplified nonlinear equations of motion for the tandem offloading system in order to understand the fishtailing phenomena. First, we consider the damping force resulting from the relative motion between vessel and environment. It is assumed that the wind or current velocities are greater than vessel's surge motion. In the linear model, the following damping forces and moment are included. ( al< AN ) (16) where V is wind or current velocity. Figure 15 shows the stability diagram with including the above damping forces and moment. Compared with Figure 13, the unstable region is enlarged when the damping forces are introduced. The reason for it is thought caused by the asymmetric stiffness matrix and the phase difference between the sway and the yaw motions, as mentioned in the previous section. It can be interpreted as a galloping phenomenon. ~ ~ . ~ , UNSTABLE, , . . . 0 2 0 4 0.6 0.8 1 1.2 1 4 hawser length (ILL) Figure 15: Effect of damping force on stability The nonlinear drag moment for yaw motions and the nonlinear tension of the hawser are considered. By assuming the constant drag coefficient in equation (6), the drag moment for each vessel is simply written by D6 = 64 CDPTL4 l Hi | Vi (17) The hawser tension shows a strong nonlinearity as observed in Figure 3. Thus, it is worthy to investigate the effect of the hawser tension on vessel's motion. Fo ~ Fo + K~/\l . ^1 ~ Xi + ~ y2 + ~ all + a'v2 _ ~ a~ - X2 + 2l Y2 + 2l p(1 + p)~2 + `' {!V2Y2 - l Y~Y2 + l Acme - / {Y,IV2 + / Yang (18) (19) Substituting equations (16), (17) and (18) into equation (11), the simplified nonlinear equations of motion can be written in the following matrix form: Mx+Dx+Sx+r~x) = 0, (20) where Rex) denotes a cubic restoring force and moment in sway and yaw, while it has a quadratic restoring force in surge. The detailed expressions have been omitted herein. Based on numerical simulations, it is

/ confirmed that equation (20) reflect the nonlinear dynamics of the FPSO-shuttle tanker system quite well. In order to estimate the fishtailing motion accurately and easily, the closed form solution for the limit cycle motion is required in near future. CONCLUSIONS A nonlinear stability of a tandem offloading system is analyzed. To do it, nonlinear motions are simulated in order to investigate the motion response more realistically. Wind, current and waves are considered. Numerical simulations show that the tandem offloading system experiences large fishtailing motions, when the wind is mild. It is also found that the velocity ratio between wind and current is the most important parameter, while the stability is not much sensitive to the hawser length. Wave drift force and moment contribute negatively to the stability of the system. According to the Hartman-Grobman Theorem and the Stable Manifold Theorem, the stability analysis of the tandem offloading system is carried out based on the linearized equations of motion. Bifurcation theory was used to figure out the entire feature of the dynamic stability. Parameter space consists of the hawser length and tension, the turret mooring stiffness and location, and the velocity ratio between wind and current. The stability diagram of the tandem offloading system is quite different from those of a single point moored shuttle tanker and a turret moored FPSO. In the case of the tandem offloading system, the hawser length does not affect the stability significantly and the turret mooring may worsen the stability depending on its stiffness. For a more precise analysis, the effects of relative motion and nonlinear hawser tension are examined. It is found that the relative motion brings damping effects. The damping, however, enlarges the unstable region, which may be understood as a galloping phenomenon. It is thought that it is caused by the phase difference between sway and yaw motions. In order to estimate the fishtailing motion accurately and easily, a further study is necessary to derive the solution of the simplified nonlinear equations of motion. REFERENCES Bernitsas M.M. and Kim B.K. "Effect of Slow-Drift Loads on Nonlinear Dynamics of Spread Mooring System", Journal of Offshore Mechanics and Arctic Engineering, Vol. 120, No. 4, 1998. Bernitsas M.M. and Papoulias F.A. " Nonlinear Stability and Maneuvering Simulation of Single Point Mooring Systems", Proceedings of the 1St Offshore Station Keeping Symposium, Houston, Texas, 1990. Chung J.S. and Bernitsas M.M. "Hydrodynamic Memory Effect on Stability, Bifurcation, and Chaos of Two-Point Mooring Systems", Journal of Ship Research, Vol. 41, No. 1, 1997. Fernandes, A.C. and Sphaier, S. "Dynamic Analysis of FPSO System", Proceedings of the 11th International Offshore and Polar Engineering Conference 1997. Fossen, T.I. Guidance and Control of Ocean Vehicles, John Wiley & Sons Ltd., 1994. Fucatu C.H., Nishimoto K., Maeda H. and Masetti I.Q. "The Shadow Effect on the Dynamics of a Shuttle Tanker" Proceedings of the 20th International Conference on Offshore Mechanics and Artic Engineering. Rio de Janeiro. 2001. Irvine M. Cable Structures, Dover Publication, Inc., New York, 1981. Lee D.H. and Choi H.S. "The Motion Behavior of the Shuttle Tanker Connected to a Turret-Moored FPSO", Proceedings of the 3rd Internatinal Conference on Hydrodynamics, 1998. Lee D.H. and Choi H.S. "A Dynamic Analysis of FPSO-Shuttle Tanker System", Proceedings of the 10th Int Offshore and Polar Engineering Conference Vol. 1.,2000. Morishita H.M. and Cornet B.J.J. `'Dynamics of a Turret-FPSO and Shuttle Vessel due to Current", IFAC Conference CAMS '98, 1998. Pinkster,J.A. and van Oortmerssen G "Computation of the first and second order wave forces on oscillating bodies in regular waves", Proceedings of the 2 International Conference on Numerical Ship Hydrodynamics, 1977. Simos A.N., Tannuri E.A., Aranha J.A.P. and Leite A.J.P. "Theoretical Analysis and Experimental Evaluation of the Fishtailing Phenomenon in a Single-Point Moored Tanker", Proceedings of the 11th International Offshore and Polar Engineering Conference, Vol. 1, 2001. Sphaier, S.H., Fernandes A.C., Correa S.H.S. and Castro G.A.V. "Maneuvering Model for FPSOs and Offloading Analysis", Proceedings of the 20th International Conference on Offshore Mechanics and Artic Engineering Rio de Janeiro 2001.

DISCUSSION Sa Y. Hong Korea Research Institute of Ships and Ocean Engineering (KRODI), Korea First of all, I'd like to congratulate you two for having successfully done one of current very interesting and complicated physical phenomena of floating body dynamics, stability analysis of tandem offloading system. I'd like to discuss about basic concept of the analysis method and interpretation of the analysis result. Your analysis model included the memory effect accounting for the frequency dependency of radiation damping. This paper mainly deals with the stability of slow planar motion in which radiation damping is negligible. Did you find any noticeable difference in the simulation results when you included the memory effect comparing to low frequency equation model neglecting the memory effect? And, have you check numerically Kramer- Kronig relation in very low frequency region where radiation damping is actually zero? Concerning the viscous damping effect on dynamic stability, you said galloping could explain broadening of unstable region when the damping force is accounted. As you are well aware, galloping is due to vortex induced vibration of bluff body where incidence of flow is insensitive. I think the incidence of flow is important in the case you considered and it seems that the damping force effect is accounted just like the way you used in the linear restoring coefficients, which explains the importance of acting direction of viscous forces as you have shown in equations (11) and (16~. So, I don't think galloping is mainly related with the broadening of unstable region. If I misunderstood, would you kindly give me the reason why you used galloping for explanation of increasing unstable region due to damping force? Congratulations again on your accomplishments. AUTHORS' REPLY Authors would like to appreciate the discusser's interest in our work. The time memory function was included in our numerical simulations in order to investigate its effect on the stability. From the numerical results, it could be found that the memory function affects drift motions of tandem-moored system, although the effect is not significant quantitatively. It is to mention that Kim (1999) also reported that the memory effect could destabilize the moored system to some degree. As for the accuracy of radiation damping, it was confirmed that the computed radiation damping approached to zero in very low frequency region, although authors did not check Kramer-Kronig relation. 'Galloping' is the term favored by civil engineers for one degree of freedom instability of bluff structures exposed to wind and current. It is known that the frequency of galloping is low relative to its natural vortex-shedding frequency. The galloping results from negative damping, which is caused by the relative motion between the bluff body and the uniform flow. In our analysis, we included the damping force resulting from the relative motion between the vessel and the environment. Although the damping force was not always negative, it tends to destabilize the system due to the asymmetric stiffness matrix. This is the reason why we used the term. 1. Kim, B.K. Stability Analysis and Design of Spread Mooring Systems, Ph.D. Thesis, The University of Michigan, 1999. DISCUSSION Michael Bernitsas and Joao Paulo J. Matsuura University of Michigan, USA Application of station keeping stability theory to a tandem system by the authors is original work. The authors may like to consider some of the modeling and analysis issues listed below in the order of appearance in the paper. 1. The memory effect in this paper is only used in simulations. It can be introduced in the stability analysis by use of the method of extended dynamics t63. 2. It is not clear from the text if the mooring line hydrodynamic drag/damping t5] is taken into account in the mathematical model for simulations. Also, the wave frequency dynamics of the mooring line affect the damping significantly - by a factor of up to two - and should be considered.

3. Slowly varying wave drift forces are non- autonomous. Do the authors include them in the simulations or the stability analysis? Are the mean wave drift forces, which are autonomous, included? We treat the spectrum of slowly varying drift forces as external excitation to an otherwise autonomous system. In reference t3], we have revealed seven interaction phenomena between mooring systems and such forces. Resonance, often quoted since the 1 980's, is only one of those and actually among the least important ones. We would anticipate that slowly varying wave drift would have similar effects on the application considered by the authors and it is worth investigating. 4. In our assessment of maneuvering models for mooring systems t7], the cross-flow models do not predict well dynamic bifurcation boundaries (Hopf bifurcations) due to the lack of linear hydrodynamic derivatives (Nr, Yr) with respect to the rotational speed. Does equation (6) account for the missing terms, Nr and Yr? Does it correct the stability of cross-flow models to predict Hopf bifurcations? 5. In the paper, it is mentioned that the motion in Figure 7 is stable. The transient indicates convergence, but it is actually followed by a limit cycle. This proves that the primary equilibrium is unstable and the system has undergone a Hopf bifurcation resulting in a limit cycle. 6. To obtain a global picture of the system's dynamics and be able to design a mooring system, all equilibria should be found and their local stability should be analyzed. Then, the global system behavior can be surmised from the local behavior near all equilibria. 7. The simplification of the turret as a multidirectional, single stiffness system probably results in an incorrect assessment of the system's stability. The drag/damping due to the mooring lines is not taken into account in the paper. Damping has a strong influence on the location of the Hopf bifurcation sequences t73. The turret restoring force is modeled in the paper only as a function of the FPSO displacement, thus adding to the model inaccuracy. 8. It should be pointed out that the statement in the paper that "generally, the shuttle tanker with single point mooring systems satisfies the static stability criteria" is not true. Depending on parameters such as the hydrodynamic derivatives of the hull considered and the point of attachment of the hawser on the hull, the system may undergo a pitchfork bifurcation; that is, not satisfy the static stability criterion [13. 9. In the paragraph following equation (14), it is stated that the presence of the shuttle tanker stabilizes the FPSO. In equation (14), Fo is the hawser pretension, thus affects the FPSO stability. Indeed, pretension is often used as a means of stabilizing a mooring system. We have shown, however t1], that a general rule of thumb cannot be stated. Pretension may shrink or simply move unstable domains thus throwing a stable system into a limit cycle or even chaos. 10. The statement that "wave drift force and moment contribute negatively to the stability of the system" is not generally true. In reference t2], it is shown that depending on the operational parameters, the presence of mean wave drift forces alone may render an unstable system stable. Likewise, in reference t3], it is shown that the presence of slowly varying wave drift forces may reduce the amplitude of oscillation of a mooring system. 11. On the same issue of wave drift, it would be helpful if the authors would clarify how they study the effect of slowly varying drift forces, which are a non-autonomous excitation, on the system stability. t1] Bernitsas, M.M., Garza-Rios, L.O., and Kim, B.K., "Mooring Design Based on Catastrophes of Slow Dynamics." Proceedings of the 8th Offshore Symposium, Texas Section of the Society of Naval Architects and Marine Engineers, Houston, Texas, February 25-26, 1999, pp. 76-123. t2] Bernitsas, M.M. and Kim, B.K., Effect of Slow-Drift Loads on Nonlinear Dynamics of Spread Mooring Systems", Journal of Onshore Mechanics and Arctic Engineering, ASME Transactions, Vol. 120, No. 3, August 1998, pp. 154-164. t3] Bernitsas, M.M, Matsuura, J.P.J., and Andersen, T., "Mooring Dynamics Phenomena Due to Slowly-Varying Wave Drift", Proceedings, of the 21St OMAE, Oslo, Norway, OMAE2002-28162, June 2002. t4] Garza-Rios, L.O. and Bernitsas, M.M., "Slow Motion Dynamics of Turret Mooring and its Approximation as Single Point Mooring", Applied Ocean Research, Vol. 20, No. 6, December 1998. t5] Garza-Rios, L.O., Bernitsas, M.M., and Nishimoto, K., "Catena~ Mooring Lines with Nonlinear Drag and Touchdown", University of Michigan, Ann Arbor, Report No. 333, January 1997. [6] Kim, B.K. and Bernitsas, M.M, "Effect of Memory on the Stability of Spread Mooring

Systems", Journal of Ship Research, Vol. 43, No. 3, September 1999, pp. 157-169. t7] Matsuura, J.P.J., Nishimoto, K., Bernitsas, M.M., and Garza-Rios, L.O., "Comparative Assessment of Hydrodynamics Models in Slow Motion Mooring Dynamics", Journal of Offshore Mechanics and Arctic Engineering, ASME Transactions, Vol. 122, No. 2, May 2000, pp. 109-117. AUTHORS' REPLY First of all, let us sincerely thank Dr. Bernitsas and Dr. Matsuura for their detailed questions and comments. Authors agree with most of the comments they raised. Among these, let us respond to some main issues. In our mathematical model, the mooring line damping was not considered, because the typical unstable mode of tandem moored vessels under consideration is yaw and the turret-moored vessel rotates freely. In such a circumstance, it is thought that the mooring line damping may be less important. Since slowly varying wave drift forces are non- autonomous, it is difficult to include them in the stability analysis, as discussers pointed out. In this paper, drift forces were considered only in numerical simulations. We are not in a position to make a concrete conclusion on this topic, which is the area for further studies. It is well known that there are two approaches of modeling mooring systems: maneuvering model and cross-flow model. The cross-flow model uses fewer coefficients than the maneuvering model to describe hydrodynamic forces. It implies that the dynamic stability may be more sensitive to the coefficients in the cross-flow model. However, it is thought that the cross-flow model can be used to predict the stability boundaries at the initial stage because its coefficients can be easily obtained. To their questions (8) and (9), authors agree that shuttle tankers have the pitchfork bifurcation depending on the point of attachment of the hawser. Since the hawsers are generally installed at the bow of shuttle tankers, the static stability of shuttle tanker can be satisfied in most cases. In addition, hawser tension improves the static stability of the FPSO according to eq. (143. Accordingly, the hawser tension shifts the pitchfork bifurcation point to the stern. It is to note that the pretension of mooring line can trigger the dynamic instability of moored vessels, as Dr. Bernitsas rightly pointed out.

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