|
Items in cart [1] |
Questions? Call 888-624-8373 |
| Copyright © 2009. National Academy of Sciences. All rights reserved. Terms of Use and Privacy Statement |
Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 348
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.
OCR for page 349
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
OCR for page 350
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
OCR for page 351
OCR for page 353
OCR for page 354
OCR for page 355
OCR for page 356
OCR for page 357
OCR for page 358
OCR for page 359
Representative terms from entire chapter:
stability analysis
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)
. . . . . .
~ . .. .... ~ . ... .. .. ~ ~ ~ ~
;~; ;~ ;
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.
