Simulation of Motion Responses of Spar Type
Oil Platform
by
Whitney Alan Cornforth
B. S., Ocean Engineering, Massachusetts Institute of Technology
(2001)
Submitted to the Department of Ocean Engineering in partial
fulfillment of the requirements for the degrees of
Master of Science in Naval Architecture and Marine Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2001
2001 Whitney Cornforth. All Rights Reserved.
The author hereby grants permission to reproduce and to distribute publicly paper and
electronic copies of this thesis document in whole or in part.
Signature of Author:
Department of Ocean Engineering
u91 4 , 2001
Certified by:
Profao
Pohl Sclavounos
al Architecture
Thesis Supervisor
Accepted by:
Chairman, Dep
omi
e
rar
pde
OF TECHNOLOGY
NOV 2 7 2001
LIBRARIES
Simulation of Motion Responses of Spar Type Oil Platform
by
Whitney Alan Cornforth
Submitted to the Department of Ocean Engineering on June 14, 2001, in
partial fulfillment of the requirements for the degrees of
Bachelor of Ocean Engineering
and
Master of Naval Architecture and Marine Engineering
Abstract
Computer based simulations of the motion responses of spar type oil
platforms were carried out. Theses simulations examined both the linear
and nonlinear coupled responses in surge, heave, and pitch to plane
progressive wave trains as well as random waves. A twelve line mooring
These
system was also incorporated for more accurate modeling.
largeplatforms
only
the
limit
to
simulations showed the mooring system
scale motions. Studies were made both at and far from the buoy's natural
frequencies in both the linear and nonlinear cases. A pitch instability due
to coupling between the pitch restoring term and the displaced heave
position was also examined. The pitch instability was present only in the
nonlinear simulation with all governing quantities evaluated at the local
wave elevation. The magnitude of this instability was limited, but not
eliminated, by the addition of the mooring system.
Thesis supervisor: Paul Sclavounos
Title: Professor of Naval Architecture
Acknowledgements
I
would
like
to
express my
to
thanks
the
all
members
of
the
Laboratory for Ship and Platform Flow in the Department of Ocean
In
Engineering.
Paul
Prof.
particular
Sclavounos
relentless questions,
for
I would
his
like
thank my
to
guidance
and
advisor
answering
my
and Peter Sabin for his sense of humor and
perspective when things looked bleak.
I
would
Institute,
also
like
to
my
classmates
and
faculty
at
Webb
without them I never would have made it to or made it
through M.I.T.
I would like to thank my mother for all her support throughout
my
education.
parents,
This work
is
dedicated
in
loving memory
to her
Papa and Granny Boats, without whom I would have never
have discovered my love for the sea.
5
Contents
Chapter 1 Introduction ................................
.........
0 .9
Chapter 2 Mathematical Formulation....................
12
2.1
Coordinate System........................... ......... 12
2.2
Linearized Free Surface Condition .......... ......... 12
2.3
Plane Progressive Waves..................... ......... 14
2.4
Body Response in Plane Progressive Waves ...
......... 16
2.4 . 1
Added Mass .............................. ......... 17
2.4 .2
Damping Forces........................... ......... 20
2.4 .3
Restoring Forces ........................ ......... 21
2.4 .4
Exciting Forces.......................... ......... 23
2.4 .5
Natural Frequency........................ ......... 24
Chapter 3 Linear Simulation........................... ........ .........25
3.1
Linear Simulation of a Freely Floating Buoy ......... 25
3.2
Linear Simulation Coupled To Mooring System ......... 28
3.3
Linear Simulation With A Random Excitation Wave and a
Moor ing System.............................................. 32
Chapter 4 Nonlinear Simulation.................................35
4.1
Nonlinear Simulation of A Freely Floating Buoy ...... 35
Chapter 5 Conclusions..........................................45
6
List of Figures
Figure 1 Illustration of the two pitch restoring force
terms ..................................................... 22
Figure 2 Spar buoy..............................................25
Figure 3 Linear simulation output...............................26
Figure 4 Responses near heave natural frequency ................ 27
Figure 5 Responses near pitch natural frequency ................ 28
Figure 6 Comparison of maximum responses with and without
mooring systems with excitation of w=0.15 Hz...............30
Figure 7 Responses with and without mooring system and a
fairlead location of % the draft...........................32
Figure 8 Linear responses to random wave with mooring
system.....................................................34
Figure 9 Nonlinear responses of freely floating buoy ........... 36
Figure 10 Pitch restoring term..................................37
Figure 11 Nonlinear responses w = 0.2700Hz ..................... 38
Figure 12 Responses with pitch restoring term evaluated at
local wave elevation to excitation at w = 0.2200 Hz........39
Figure 13 Nonlinear responses with all quantities evaluated
at local wave elevation and excitation with w =
0 .2700Hz .................................................. 40
7
Figure 14 Responses with all quantities evaluated at local
free surface elevation and excitation with w =
0.2200Hz...................................................41
Figure 15 Nonlinear response to excitation of w=0.2200 Hz
with a mooring system......................................42
Figure 16 Nonlinear simulation with random wave excitation ..... 44
List of Tables
Table 1 Mooring system restoring forces ....................... 29
8
Chapter 1 Introduction
Spar buoy type oil platforms are being considered for many of the
current
generation
particular
of
the
oilrigs.
interest for deployment
conditions and depth.
from
offshore
standpoint
in
Spar
buoys
are
of
seas with extreme weather
Motion on an oil platform is not desirable
of
production where
its
effects
can
range
from making the crew seasick to breaking risers and other vital
pumping
equipment.
In
shallow
solutions to this problem.
the platform is
waters
there
are
a
of
One is the tension leg platform where
held with high tension mooring lines
mounted in the bottom;
number
another
to
anchors
is the jack-up platform where
the
platform is physically raised above the water's surface on rigid
legs
extending to the sea
floor.
These shallow water solutions
are not feasible in water deep water or harsh sea states.
As
shallow water
oil
the offshore
industry is
10,000m
farther
and
reserves
are
looking to move
offshore
where
less predictable and more punishing.
competitive alternative to
the
being
rapidly
depleted
to water depths up to
weather
conditions
are
The spar type platform is a
classical design
for
these difficult
environments.
The
vertical
classic
cylinder.
shape
of
a
spar
buoy platform
Traditionally they have a
that
constant
of roughly 30m and a draft of approximately 200m.
9
is
of
a
diameter
The motivation
for the use of such an unusual shape with its large volume is to
The large volume to
minimize the buoy's motion response in seas.
waterplane area ratio of the spar makes
to changes
summary
it
the
is
is
that
system
the mooring
than
rather
platform,
the
of
shape
In
slow.
(waves) quite
elevation
in water
its response
mainly responsible for the platform motions remaining small.
In the depth range for which spars are being considered the
steel risers are extremely flexible, so horizontal
(surge) motion
The risers cannot, however,
withstand huge
is of little concern.
and compressive
stretching
motions
(pitch)
rotations
can
could
or buckling,
be
minimized.
Rotational
lead
to
to
need
problems;
cause
also
small
capsizing
and
equipment
makes
its
natural
are
motions
extreme
while
conditions,
working
adverse
with
associated
rupturing
motions
(heave)
so vertical
loads without
and
breakage
loss.
The
the
of
shape
spar
With low natural frequencies the
heave and pitch extremely low.
threat
spar
to
Waves
wavelengths.
1
approximately
-,
comes
platforms
so
high
of
waves
from
mainly
break
to
begin
in
frequencies
of
waves
amplitude
also
is
slope
their
when
long
have
long
7
wavelengths,
a dangerous combination for spar platforms.
This research was aimed
type
oil
platforms.
This
computer-based simulation.
linear theory.
These
at predicting the motions
was
done
in
a
number
of
of
spar
steps
of
The first simulations were based upon
simulations predict
10
the
surge,
heave,
and
pitch responses of a freely floating spar to one wave train of a
specified amplitude and
responses
natural
under
frequency.
specific,
This was done to
known conditions,
This simulation
frequencies.
such
so
the
next
step
was
to
mooring system to the linear simulation.
generator
was
added
to
examine
the
around
the
effects that
known as coupling.
Real oil platforms are not freely floating,
and
as
included the
one mode of motion may have on another,
in place,
study the
add
they are moored
the
Finally,
to
responses
effects
of
a
a random wave
a
wave
more
representative of a true ocean wave.
Next
evaluated
elevation,
neglected
a
nonlinear
governing
thus
by
simulation
quantities
accounting
linear
theory.
for
was
at
a
The
developed.
the
number
same
local
of
free
effects
progression
mooring system and random wave was also carried out.
11
This
of
model
surface
that
are
adding
a
Chapter 2
Mathematical Formulation
2.1 Coordinate System
Throughout
this
discussion
a
Cartesian
coordinate
system
(x,y,z)
will be used with the origin coinciding with the undisturbed free
It will be located at the centerline of
surface.
the positive
The elevation of the free
z-axis pointing upwards.
Thus gravity
surface will be defined by 77(x,y,t).
the buoy with
(g)
acts in the
negative z-direction.
A body freely floating on the surface of a fluid is able to
move in six modes of motion.
and
heave,
coincide
with
x-y-z
axes respectively.
pitch,
represent
cylindrical,
rotation
there are
that must be studied,
Three modes, designated surge,
translation
motion
parallel
The other three modes,
about
only three
those
roll,
Since
axes.
a
sway
to
the
yaw,
and
spar
is
fundamentally different modes
surge, heave, and pitch.
2.2 Linearized Free Surface Condition
Assuming
an
ideal
and
irrotational
fluid,
there
will
exist
a
velocity potential 0 such that the fluid velocity is expressed as
the gradient of this velocity potential
12
Equation 2.1
Due
to
conservation
of
mass,
the
divergence
of
this
velocity
dynamic
boundary
potential must be zero,
Equation 2.2
V 20=0
it
must
also
satisfy
both
a
kinematic
and
condition on the free surface.
The
kinematic
boundary condition requires
the
velocity of
the free surface to be equal that to that of the fluid particles
of which
it
is
comprised.
found by requiring that
the free surface.
The kinematic
boundary condition
the substantial derivative
of
is
(z-)=Oon
The result of which is
Equation 2.3
O=kD(Z - 77
Dt
The last
two
?
Oa7aOa7a
az
at
ax a
ay ay
terms may be neglected because
they are of
order and therefore much smaller than the first two,
second
this results
in the linearized kinematic boundary condition
Equation 2.4
at
The dynamic
acting on
the
boundary
free
surface
(-y
condition
requires
from above be
13
that
equal
the pressure
to the pressure
acting
from
dynamic
The
below.
is
condition
boundary
found
through the use of Bernoulli's equation,
Equation 2.5
1
a# 1
p
at
-(p-p.)=-+-V~eVb+gz=0O
Substituting
?
for
condition
boundary
z
linearizing
and
in
results
2
the
as
in
linearized
kinematic
the
boundary
dynamic
condition
Equation 2.6
1 ao
g at
These
surface
two
boundary
can
conditions
be
combined
on
the
z=0 resulting in one boundary condition for the velocity
potential
Equation 2.7
-+g--0
at2
"y
2.3 Plane Progressive Waves
Plane progressive waves are
free-surface
this
single
and will be
condition,
discussion.
amplitude
These
(A)
the simplest waves
and
waves
are
frequency
sinusoidal motion in one direction.
14
that satisfy the
used extensively
two-dimensional
(w)
and
throughout
and
propagate
have
a
with
a
Equation 2.8
i7(x,t) = Acos(kx -wt+6)
describes
direction,
zero.
a
plane
a phase
The value
progressive
wave
moving
in
the
positive
x-
6 has been included but will be assumed to be
k is known as the wave number and is
defined as
Equation 2.9
where
V,
is
travels) and
the
phase
(t)
27r
w2
Vg
V,
)
g
velocity
(the velocity that
a wave peak
A is the wavelength.
The velocity potential
4
that
satisfies both Equation 2.2
and Equation 2.7 and through the use of Equation 2.6 will result
in
15
Equation 2.8 is given by
Equation 2.10
=gA e' sin(kx -t)
0)
In the ocean, waves are not two-dimensional,
sinusoidal,
by
and their amplitude and frequency cannot be described
extremely useful
plane
However,
parameters.
single
are not purely
progressive
are
waves
and
for examining both simplified problems,
it
will be discussed later how "real" ocean waves can be modeled as
a
superposition
many
of
progressive
plane
waves
with
each
differing amplitudes and frequencies.
2.4 Body Response in Plane Progressive Waves
As mentioned, a body freely floating on the surface of a fluid is
free
to move
in
all
six modes
section will
This
of motion.
explore the response of a body in the three modes that have been
identified as being of particular interest
spar
platform,
oil
surge,
namely
heave
equations of motion will be presented,
of each
of
the
to
the problem of a
and pitch.
First
the
followed by an examination
of the terms contained within them including discussion
coupling
effects
between
surge
and
out
over
pitch.
In
these
surface
equations
integrals
will
be
carried
the
j'dz
interval
-draft
the
actual
z
value
represented by
discussed later.
The equation of motion given by
16
the
term
'surface'
will
be
Equation 2.11
[-w 2 (Mij + aj )+imbj + c ]{j = AX
j=1,3,5
where
the
indices
are
ij
the modes
corresponds with surge, heave,
coupling
potential
represented.
motion
and pitch.
1,
3
or
5
that
When the indices i A j,
between
of
modes
motion
are
Some of the coupling terms represented in Equation
2.11 are equal
to zero,
possible,
it
will
further
be
effects
of
does
indicating that even though coupling is
not actually occur.
Non-zero
coupling terms
addressed
j
are
later.
terms
the
body
displacement in complex form.
In
inertial
Equation
2.11
the
M,
force upon the body.
mass, while
M
55
is
terms
M 1
are
and
the
M3
the moment of inertia
'
components
are
of
the
simply the body
defined by
Equation 2.12
M 55 =1 22 = 'fJpYdV
V
where
pb
There
are
is
the density of the body,
no
other
non-zero
mass
and V
terms
is
the body volume.
associated
with
this
problem.
2.4.1
The
Added Mass
afterms
in Equation
2.11 are
known as the added mass
since they are proportional to the body acceleration.
is the case,
terms,
Since this
they are obviously dependent on the frequency of the
17
motion.
found
Through
a method
analytically
for
a
of
images,
given
added
geometry;
mass
terms
however,
be
to
the
complexity of this method a combination of empirical values
and
strip
low
theory
were
employed
to
estimate
the
due
can
added mass
at
frequencies.
Empirical formulas exist determining the added mass of many
common geometric
shapes.
Of
importance
to
this discussion are
those for a circle moving in a two dimensional fluid.
Equation 2.13
"a
12D
1
_ ,,2
=7p..(a
a 22D = rpr2..........(b)
where
represents
a1
movement
in
cylindrical
and
movement
out
of
within
the
plane.
the
The
spar buoy in heave is due to the
and falling;
thus,
Equation 2.13
plane,
added
and
mass
a2 2
of
is
a
flat bottom rising
(b) can be used directly to find
the added mass of the buoy in heave.
Equation 2.13
surge,
pitch,
allows a
and
(a) was used to find both the added mass in
the
coupling between the
three-dimensional value to
be
two.
found by
Strip
theory
integration
of
two-dimension values over a body, provided that the change in the
two-dimensional values
is
small.
The
following
this was employed to find the surge added mass
Equation 2.14
surface
=i
f
-draft
18
,(z)dz
illustrates
how
In
this
diameter
discussion
removing
the
buoys
were
the dependence
the problem further.
of
assumed
2D
a,
on
to
z,
be
thus
of
constant
simplifying
The following two equations show how this
method was used for determining the added mass in pitch,
coupled added mass term between pitch and surge
Equation 2.15
surfiace
-d f zrafd
-draft
19
and the
Equation 2.16
a51
a3D
=a15
surface
r2Dd
3D
za dz
=
-draft
There are no other non-zero added mass terms in this problem.
Damping Forces
2.4.2
The
in Equation
terms
b.ii
the waves
present
do
to
radiate
outwards
O, ,
components
the six modes of motion.
generated
0
is
07
condition
2.7.
into
the
eight
for each of
is called the diffraction potential
These waves must
Equation
In
energy.
broken
that
body
the
representing the potential
#0 is the incident wave
and represents the body-generated waves.
potential.
by
dissipating
terms
damping
with 1, 2. .6
are
body
the
from
the
of
derivation
that
and
These forces are
to the velocity of the body.
are proportional
terms
damping
the
are called
2.11
satisfy the
From
this
fact
free
surface boundary
can
be
derived
the
Haskind relations'
Equation 2.17
X,=-pJJ#
is
the body surface and
where
SB
body
surface.
expressions
exciting
PijS
for
force.
The
the
It
Haskind
n
relations
damping
terms,
will
shown
be
is the normal vector of
20
which
in
are
used
are
2.4.4
to
related
that
the
the
derive
to
the
exciting
forces
are
therefore,
At
related
to
the
frequency
this point
was
the
exciting
wave;
the damping terms are dependant on the wave frequency.
in the
simulation a
in determining damping coefficients.
SML
of
used
to
determine
an
simplification was made
The general-purpose program
appropriate
constant
value
for
damping coefficients.
2.4.3
Restoring Forces
The C.ii
terms in Equation 2.11 are the restoring terms,
which are
responsible for trying to return the body back to it's
state.
original
When examining the freely floating body these are all due
to hydrostatics.
The restoring
original
force in heave is
displaced
volume
and
the difference between the
the
heaved
displaced
volume
multiplied by the density of the fluid and the force of gravity
Equation 2.18
C33 = pgA
This
term
is
multiplied
by
the
heaved
position
to
give
the
parts,
the
resultant force.
The
first
pitch
is
creates
due
a
restoring
to
moment.
the
term
gravity
Another moment
located within
is
located
comprised
redistribution
because the center of gravity
longer
is
the
below
body was initially stable),
(ZG)
of
is
two
submerged
created
by
volume
pitch
and center buoyancy
same vertical
the
of
center
of
plane.
buoyancy
The
(ZB)
that
motion
are no
center
(provided
of
the
and the force acting at the center of
21
gravity acts down while the force on the center of buoyancy acts
upwards,
since these two
forces must be equal
from the origin they create another
are now different distances
restoring
moment
in magnitude but
to
proportional
the distance
between the
two.
Figure 1 illustrates how these two terms are physically produced.
t
n
V
f
ZG
V
ZB
W
Center of gravity and
buoyancy term
Redistribution of
submerged volume
Figure 1 Illustration of the two pitch restoring force terms
These two terms can be seen in the following equation for C55
Equation 2.19
4
_pgnrd
C 55
64
+ pgV(YB
22
G
where
ZG
d
is
the buoy diameter,
V is
the buoy volume,
and ZB
are the centers of buoyancy and gravity respectively.
and
There
are no other non-zero restoring forces.
2.4.4
Exciting Forces
Exciting forces
and G.
I.
for
Taylor's
this
problem were
formula for
the
found using strip
force
of a
theory
two dimensional
section
Equation 2.20
dX, =(V+-)--(p-)
p ax,
where
xi
is
the
Cartesian
[(1,2,3)
axis
parallel to the desired exciting
employed
to
integrate
these
at
corresponds
force.
slices
over
Equation 2.21
X1
=
ao
a,
(V+a
-draft
(P
Pax
t
)dzx_,
Y=
Equation 2.22
X3
=(V+_a3)
XY=
a_(Pa)
p az
=
a3t IYO
Z=-draft
Equation 2.23
surface
a
a
-draft
P
ax
23
(x,y,z)]
Again strip theory was
follows for surge, heave, and pitch
suoface
with
)d
at
the
entire
buoy
as
2.4.5
If
Natural Frequency
no
exist
there
damping
and
coupling
or
external
forces
Equation 2.11 reduces to
Equation 2.24
-O
The
solution
of
this
2
(M i+ai)X+cii =0
equation
natural frequency for the t'
produces
what
is
known
as
the
mode of motion
Equation 2.25
ii+ a,
Since negative frequencies are physically impossible we are only
concerned with the positive value of Equation 2.25.
of a body excited at a
exactly
equal
Therefore,
it
to
is
its
frequency that
natural
important
that
frequency
a
excited at one of these frequencies.
24
is extremely
physical
will
The response
close to or
be
structure
extreme.
not
be
Chapter 3 Linear Simulation
3.1 Linear Simulation of a Freely Floating Buoy
A
time domain
was
created.
linear simulation of
This
a
freely floating
simulation was based upon
the
spar buoy
linear theory
that has been presented.
The spar buoy that will be used for all
these
draft
simulations
diameter
of 30m,
has
a
of
200m,
with
30m
freeboard,
a
and a center of gravity located 104m below the
undisturbed free surface.
With the exception of random waves the
wave amplitude will always be 10m.
A3 0
200
830
Measurements
in meters
Figure 2 Spar buoy
25
Sample output from the simulation is presented in Figure 3
Linear Simulation with amplitude
=
0.11 hz
10.00 m frequency =
5
E
0
(D
-3
-
-0
20
40
60
80
100
120
140
160
180
200
20
40
60
80
100
120
140
160
180
2 00
1
I
I
140
160
180
2 00
50
E
0
(I
Cn
C
-50
.01,
-U-
-
I
-
*0
TO
Cu
0
-0.01
0
I
I~
20
40
m
p
80
60
120
100
Time - sec
Figure 3 Linear simulation output
3
surge,
and pitch.
the
shows
the
Figure
response near
sinusoidal
This
motions
over
at
the buoy's natural
wo=0.22Hz.
400m from a
the
simulation can also be
frequency of the buoy in heave is
responses
of
The
10m wave,
in
heave,
used to examine
frequencies.
The natural
w =0.2209Hz, Figure 4 shows the
magnitude
illustrating
the buoy near its natural frequency.
26
buoy
of
the
the
heave motion is
effect
of exciting
Linear Simulation with amplitude
10.00 m frequency
=
0.22 hz
=
500
E
0-
-5001
0
50
20
40
60
80
100
120
140
160
180
200
20
40
60
80
100
120
140
160
180
200
20
40
60
80
100
120
140
160
180
200
E
2)
50
0
00
0
Time - sec
Figure 4 Responses near heave natural frequency
In
Figure
4
the
heave
motion
completely leaves the water.
is
not
included in this
indicative
frequency,
of
the
is
so
great
that
the
buoy
The effect of this event occurring
simulation.
response
to
it will be seen in 3.2
Although
excitation
this behavior
near
the
is
natural
that the presence of a mooring
system prevents this occurrence.
The natural
Figure 5
frequency of
the buoy
in pitch
shows the responses at (O=0.05Hz.
pitch response
is
between
and
surge
extremely small.
pitch
has
In
overcome
is
ow=0.0491Hz,
The magnitude of the
this
the
case
the
coupling
effects
natural
frequency excitation as illustrated by Equation 3.1, which is the
27
expanded
form
of
Equation
2.11
for
pitch
including
coupling
effects
Equation 3.1
- 02(M + a55)
- ( 2 (MIJ + al,)+(ib 5 * +(ib1
Linear Simulation with amplitude
=
1 +c 554 +c&
= AX5
10.00 m frequency
0.05 hz
=
1
E
0
.
-1 41
0
20
40
60
80
100
120
140
160
180
200
0 10-320
40
60
80
100
120
140
160
180
200
160
180
200
onr
E
0
'
0)
5.
1
-o
-
L
0
0-
-5
0
20
I
I
I
40
60
80
I
100
120
Time - sec
I
140
Figure 5 Responses near pitch natural frequency
3.2 Linear Simulation Coupled To Mooring System
Oil
platforms
are not
freely floating bodies,
place with mooring systems.
ten to
buoy.
most
twenty mooring
they
are held in
Spar buoys are typically moored by
lines,
spaced
around
the
diameter
These lines typically consist of three segments,
that
connects
to
the
anchor is
28
made
of
chain,
of
the
the lower
the
middle
section
is
is made
also made
of steel
of chain.
or synthetic rope,
and the
top section
As mentioned previously the purpose of
the mooring system is to prevent large-scale motions,
it would be
unfeasible to hold the platform completely stationary.
For this
simulation the restoring effects of a 12-line mooring system were
added.
These
effects
were
found
through
simulation
using
the
LinesT' portion of SML.
The
buoy was
moored
in
830m
of
water
spaced lines that were each 1850m in length.
was a 150m length of chain,
followed by
moored
to
150m of
the
chain.
buoy is
by
twelve
equally
The lowest section
followed by 1500m of polyester rope,
The
defined as
point
the
at which
fairlead;
the
lines
this
are
point was
varied from % the draft of the buoy to the bottom to examine the
effects of fairlead location of platform motions.
Table 1 Mooring system restoring forces
Fairlead
110
120
130
140
150
160
170
180
190
200
3.59E+05
2.65E+05
2.27E+05
1.88E+05
1.54E+05
1.26E+05
1.05E+05
8.81 E+04
7.50E+04
6.46E+04
C33 -
C55 -
C51 -
C15
k /s^2
kg.m/sA2
ks^2
kg/s^2
1.96E+05
6.42E+09
-4.77E+07
-4.85E+07
1.32E+05
9.60E+04
8.26E+04
6.76E+04
5.47E+04
4.43E+04
3.61 E+04
2.97E+04
2.47E+04
2.08E+04
5.75E+09
5.39E+09
5.32E+09
5.26E+09
5.13E+09
5.OOE+09
4.88E+09
4.78E+09
4.71 E+09
4.66E+09
-4.44E+07
-2.97E+07
-2.77E+07
-2.48E+07
-2.19E+07
-1.93E+07
-1.71 E+07
-1.52E+07
-1.37E+07
-1.25E+07
-3.75E+07
-3.1OE+07
-2.78E+07
-2.51 E+07
-2.23E+07
-1.97E+07
-1.75E+07
-1.56E+07
-1.40E+07
-1.28E+07
29
-
location below
C11 free surface (m) kg/s^2
100
5.21 E+05
26.86
.
5.45
26.84
5.4
Without mooring
26.82
Without mooring
5.35
26.8
E
>
5.3
26.78
+
With
mooring
5.25 -Z
26.76
With mooring
26.74
5.2
26.72
5 5
26 7
100
120
140
160
180
100
200
120
140
160
180
200
Fairlead location below undisturbed
Fairlead location below undisturbed free
free surface - m
surface - m
0.009225
0.00922
-Without
0.009215
0.00921
,I:
0I
mooring
0.009205
0.0092
0.009195
With mooring
0.00919
+
-
0.009185
100
120
140
160
180
200
Fairlead location below undisturbed free
surface - m
Figure 6 Comparison of maximum responses with and without mooring
systems with excitation of w=0.15 Hz
Figure
6
amplitude
shows
10m
the
and
responses
frequency
of
the
spar
buoy
to
a
wave
of
w=0.15Hz both with and without the
30
twelve
line
because
mooring
this
frequencies,
be
system
frequency
in
is
yet not close
spaced
between
to either,
in a realistic domain and natural
minimized.
For all
w=0.15Hz
place.
was
the
so that the
selected
two
natural
results would
frequency effects would be
three modes of motion the maximum responses
are diminished by the presence of the mooring system.
also
shows the
Figure 6
effect that the fairlead location has upon these
motions.
In heave as the
fairlead location approaches the bottom of
the buoy the motion approaches that of the unmoored buoy.
does take into account the elasticity,
of
the mooring
lines,
so
as
the
Lines"'
as well as the curvature,
lines are
connected
lower
the
heave restoring force they provide is lowered, as shown in Figure
6.
The results for surge and pitch do not show much dependence
upon
small
the
fairlead location.
relative magnitude
floating
buoy
moves
In both
of
only
cases
the motions.
26.84m,
this
In
while
the
is
due
to
surge
the
freely
anchors
the
for
the
mooring lines were positioned in a ring 1700m from the platform.
As mentioned,
positioning
a
the
large amount
anchors far
of
away
surge motion is
acceptable,
the mooring will prevent
by
very
large surge motion but allow motions on the order seen.
The
effect
of
the
mooring
system
in
all
three
modes
motion is extremely small,
as can be seen in Figure 7.
heave
with
plot
the
distinguishably
motions
lower
than
that
31
the
of
the
mooring
line
unmoored buoy.
of
In the
are
just
In
the
surge and pitch plots the two motions are indistinguishable from
one another.
Linear Simulation with amplitude = 10.00 frequency = 0.1500
--
5 E
-
X 0
5
10
15
20
25
30
35
5
10
15
20
25
30
35
5
10
15
20
Time - sec
25
30
35
20E
a)
0-
CO -20x 10-3
5'0
0
Figure 7 Responses with and without mooring system and a fairlead
location of % the draft
3.3
Linear Simulation With A Random Excitation Wave
and a Mooring System
True ocean waves
frequency.
This
cannot be described by a
due mainly to the
is
which was presented in Equation 2.9.
states
that waves
velocities,
of
as does
different
the
single
The dispersion relationship
frequencies
they are random.
32
and
dispersion relationship,
travel
energy associated with
are ocean waves dispersive,
amplitude
at
them.
different
Not
only
These random waves
can
be
waves,
modeled
each
as
a
randomly
superposition
selected
of
many
amplitude,
plane
progressive
frequency
and
phase
difference.
A
random wave
amplitude,
generator was
randomly
amplitudes
were
from 0-Im,
allowed
to
range
of 200 waves.
The first plot
used
the
same
Figure 8 shows the
using the
larger
superposition
twelve
line
and pitch responses.
mooring
before with a fairlead location of % the draft.
responses
0.05-0.45Hz,
shows the wave elevation and the
following three show the heave, surge,
simulation
from
and phase from 0-27.
output from the random wave simulation,
exhibits
selects
frequency and phase, each over a range making physical
Frequencies
sense.
created that
than
the wave
system
This
employed
The heave motion
elevation due
to
high
amounts of wave energy that happen to be focused around the heave
natural
that
frequency.
shown
in Figure
Note that
4 where
this response
the
buoy was
is
still
excited at
frequency that was very close to the natural frequency.
33
far below
a
single
Linear Simulation with random wave
20
E
10
c
0
0
(D
-10
-20
-3C
0
100
200
300
400
500
600
700
800
900
1000
0
100
200
300
400
500
600
700
800
900
1000
100
200
300
400
500
600
700
800
900
a
I
700
800
900
4C
2C
E
c
-2C
-4C
5C
E
0
co
0.01
*0
Cu
0
C,
0~
-0.01r
a
100
200
a
300
400
500
600
ime - sec
Figure 8 Linear responses to random wave with mooring system
34
Chapter 4 Nonlinear Simulation
4.1 Nonlinear Simulation of A Freely Floating Buoy
A nonlinear simulation was developed for more realistic modeling
of the motions of a spar buoy.
H.
A. Haslum and 0. M. Faltinsen
presented a paper titled Alternative Shapes of Spar Platforms for
Use in
Hostile
within
which
AreaS2 at the 1999 Offshore Technology Conference;
an
instability
due
to
indirect
coupling
between
heave and pitch was studied.
The
term
the
instability
contains a
centers
of
term
was
present
that
is
buoyancy
and
because
dependent
the
on
gravity.
the
The
pitch
restoring
distance between
center
remains basically fixed to a point within the buoy;
of
gravity
however,
the
center of buoyancy moves with the buoy's motion so it is always
in the center of the submerged volume.
In cases of large heave
it is possible for the center of gravity to rise above the center
of
buoyancy
negative.
to
become
making
this
part
of
the
pitch
restoring
force
In extreme cases of heave it is possible for this term
so
negative
as
to
make
the
pitch
negative, making the buoy unstable in pitch.
35
restoring
term
Nonlinear Simulation with amplitude = 10.00 frequency =
0.15
10
E
0
CO
-10
0
20
40
60
80
100
120
140
160
180
200
0
20
40
60
80
100
120
140
160
180
200
20
40
60
80
120
100
Time - sec
140
160
180
200
50
E
0
C50
0.01
0
a.
-0.01
0
Figure 9 Nonlinear responses of freely floating buoy
Figure 9 shows the responses of the freely floating buoy with the
pitch restoring
instability is
pitch
The
even
term evaluated at the displaced heave position.
though
the
heave
not
evident in
response
is
due
equation
to
of
the
relative
motion,
each
for
the
pitch
as shown in Figure
10.
This
the
pitch
large
is
restoring term to become negative,
magnitude
of
simulated response
the
which
of
is
enough
the
terms
in
at
least
one
magnitude larger than the pitch restoring term.
36
order
of
x 1010
2
1.5-
S1(>
0)
-E 0.5
0
!0
a.
0
-0.5-
0
20
40
60
80
100
120
Time - sec
140
160
180
200
Figure 10 Pitch restoring term
Haslum
frequency,
and
Faltinsen
involving the natural
which produces the pitch
Where
4 1 Wh
and
r
heave;
respectively,
T
these
equation
for
the
excitation
frequencies in heave and pitch,
instability;
and T
TN, p tch
an
this is
given in Equation
refer to the natural periods in pitch
N,heave
natural
frequencies
are
resulting in a (critica0.2700Hz
.
4.1.
gave
Equation 4.1
Tcritical
TN,pitch
37
1
TN,heave
0.2209Hz and
0.0491Hz
11
Figure
shows
the
at
responses
nonlinear
enough
small
are
that
force
restoring
the pitch
as
In this case the
well as the value of the pitch restoring term.
motions
co=0.2700Hz
never
attains a negative value.
-
10
E
0
a:
-10
80
100
120
140
160
180
200
60
80
100
120
140
160
180
200
I
I
I
I
I
I
I
40
60
80
40
20
60
0.01.r
-o
LU
0
K...
a.
-0.01'
xc10
9 20
40
8
z
6
4
20
140
120
100
Time - sec
Figure 11 Nonlinear responses w
Figure
12
shows
the
responses
at
o
0.2200Hz .
response is very large because this value is
natural
frequency.
restoring
term
(c55)
These
heave
motions
the
heave
motion,
negative values for nearly 50% of the time.
restoring terms
heave
the
pitch
reaching
large
make
These large negative
are still not enough to overcome the
the other terms in the pitch equation of motion.
38
The
close to the heave
large
follow
180
0.2700Hz
=
=
160
effects of
500
A
IA IA /\ I IA
I II
E
9?
I,
0
0
-5000
20
40
60
80
100
120
140
160
180
200
-0.02'
0 1120
x 10
40
60
80
100
120
140
160
180
200
40
60
80
100
120
Time - sec
140
160
180
200
0.02
0
C,
5:
z
LO
0
20
Figure 12 Responses with pitch restoring term evaluated at local
wave elevation to excitation at w = 0.2200 Hz
As
in the
linear model
the buoy leaves the water;
the heave motion is
so large
that
again this will be prevented by the
mooring system.
Figure
evaluation
12
of
shows
only
the
that
in
pitch
restoring
the
coupled
term
simulation
at
the
the
displaced
heave position does not produce the pitch instability that Haslum
and Faltinsen studied.
The
quantities
final
nonlinear
at the local
simulation
free surface.
evaluated
all
governing
Again the simulation was
run with a wave excitation at ( = 0.2700Hz and o = 0.2200Hz .
results are shown in Figure 13 and Figure 14.
39
These
In Figure 13
the
nonlinear
be
can
behavior
both
in
seen
surge
the
pitch
and
the magnitude of these motions is not increased
motions; however,
over that of the linear simulation.
Nonlinear Simulation with frequency w= 0.2700
10
E
Ca
150.
0
100
200
300
400
500
600
700
800
900
1000
0
100
200
300
400
500
600
700
800
900
1000
0
100
200
300
400
600
500
Time - sec
700
800
900
1000
E
0.02
S0.01
.0
-0.01
Figure 13 Nonlinear responses with all quantities evaluated at
local wave elevation and excitation with w = 0.2700Hz
Figure
pitch
14
shows
motion magnitude
The
simulation.
of
all
more
a
only
radical
approximately double
large heave motions
large pitch motions
evaluation
not
that
of
for
are responsible
governing
quantities
at
the
a
linear
the
as Haslum and Faltinsen indicated.
the
but
behavior
these
Through
local
free
surface a pitch instability was produced.
One
item of
interest
is
that
the
surge
and pitch motions
shown in Figure 13 are not symmetric nor are they centered around
40
the origin,
for
this is even more noticeable in Figure 14. The reason
this behavior
is
unknown and warrants
further
study of
the
nonlinear responses.
Nonlinear Simulation with frequency w= 0.2200
500
E
0
-
-II
0
-500'
100
200
50
400
500
600
I
I
I
I
300
400
500
600
700
500
600
Time - sec
700
300
800
700
900
1i 00
I
0
2
C,,
-50
E -100
C
100
200
800
900
1C 00
800
900
1000
:. 0.04
'V
LO 0.02
0
0
-0.02'
0
I
I
I
I
100
200
300
400
I
I
Figure 14 Responses with all quantities evaluated at local free
surface elevation and excitation with w = 0.2200Hz
Nonlinear simulations were run with the twelve line mooring
system also.
These results are shown in Figure 15.
system has greatly diminished
maximum values
from nearly
100m in Figure 15.
the heave magnitude,
500m
in
Figure
14
to
The mooring
dropping the
approximately
The mooring system also limits the surge and
pitch motions as well as smoothing the motion in both.
41
Nonlinear Simulation with frequency w= 0.2200
200
E
0
)
-200'
50
100
200
300
400
500
600
700
800
900
1i 00
100
200
300
400
500
600
700
800
900
U 00
I
I
600
500
Time - sec
700
800
900
1000
r
E
0
CO
-50'
C
0.02
-I---------
0.01
a.
0
-0.01
I
I
0
I
--
400
300
200
100
Figure 15 Nonlinear response to excitation of w=0.2200 Hz with a
mooring system
Even with
the mooring system in place to prevent
fro heaving completely
The
present.
still
approximately 0.0125
radians
in
the
15
stability
is
motion
is
pitch
compared with approximately 0.0095
Like
model.
Figure
the
of
magnitude
radians,
linear
simulation results,
the water the pitch
out of
the buoy
shows
the
unmoored
nonlinear
the surge and pitch motions
to be centered around a non-zero position.
wave
random
simulation
progressive
frequencies
and
was
waves
from
generator
was
wave
added
with
a
each
with
amplitudes
0.5-.45Hz
and phases from
42
with
composed
run
the
to
ranging
0-2
.
nonlinear
200
from
plane
0-1m
,
The
The responses
with the mooring system in place are shown in Figure 16.
In this
trial
the
because
there
is not
pitch
instability
is
consistently large
not
evident.
heave motion.
and pitch motions do not appear to be biased
the origin.
this
is
just
This
Also,
is
the
surge
to either side of
The heave motion appears to diminish over the trial,
due
to
the random wave
used,
this
indicative of the behavior to all random waves.
43
behavior
is
not
20
E
10
0
0
Ca
-10
3:
-20
0
100
200
40
E
300
400
I
500
600
I
I
700
800
900
1000
1000
20
0
-20
-40
I
0
I
I
700
800
900
100
200
300
400
500
ime - sec
600
100
200
300
400
500
600
700
800
900
100
200
300
400
600
500
Time - sec
700
800
900
20
E
'0
a,
-40
0.01
-D
cc
0
0L
-0.01
Figure 16 Nonlinear simulation with random wave excitation
44
Chapter 5 Conclusions
Computer
based
simulation
of
the
motion
platforms has been created to examine
of motion,
surge,
heave,
spar
type
oil
the three important modes
and pitch.
carried out with the addition of a
of
Linear simulation has been
realistic mooring system and
the capabilities of examining the response to random waves.
Examples
that
examine
circumstances
have
been
the
responses
presented
for
under
a
response
variety
of
comparisons,
including response studies near the natural
frequencies in heave
and pitch.
heave motion of
These
examples
showed
that
the
the
buoy grows extremely large near the natural frequency.
Examples
negr
the
coupling between surge
pitch
natural
and pitch is
frequency
show
responsible for
that
the
there being
very little pitch motion at these frequencies.
The
addition
of
the mooring
effect on the motion of the buoy,
around the natural frequencies.
the motivation
for
motivation being
system
the
to have
little
except in extreme cases such as
This result was consistent with
using platforms
that
proved
shape of
of
the
spar
type
buoy is
design.
This
responsible for
limiting the motions, not the mooring system.
A
linear
simulation was
also
created with a
which is more representative of a true ocean wave.
45
random
wave,
This
showed
be
to
the motions
the
consistent with
that were
within ranges
linear single wave trials.
force.
restoring
pitch
the
on
position
heave
of
effects
the
taking into account
simulations were carried out
Additional
These simulations showed the effect of this indirect coupling to
be very small, not affecting the overall motions.
In these trials
quantities at the local free surface elevation.
of
magnitude
the
were
pitch motions
the
to
these
limit
to
proved
simulations
these
doubled
approximately
The addition of the mooring
over those of the linear simulation.
system
governing
all
evaluated
that
carried out
were
Simulations
responses,
resulting in over all motions of the magnitude seen in the linear
trials.
these
since
The
overall
extreme.
consistent
with
the
responses
the
of
and
these
responses
nonlinear,
moored
of
magnitudes
random
was
motion
heave
the
trials
mooring
the
with
The pitch instability was not present
system and a random wave.
in
made
also
were
simulations
Nonlinear
not
were
trials
with sinusoidal excitation.
Future work on this topic should be done in the examination
of
platform
responses
to
additional
These
effects.
nonlinear
include second order terms in governing equations.
Another
important
area
for
future
work
addition of more complex spar buoy geometries.
have
additional
structures
well
46
below
the
would
be
the
Many buoy designs
surface
to
further
lower the natural frequencies.
These are important to model as
are buoys with non-constant cross section.
Additional mooring system effects,
forces,
system
should
also
be
configuration
examined.
for
optimum
included in this work.
47
A
such as drag a vibratory
thorough
performance
study
of mooring
should
also
be
Bibliography
Newman, J. N., Marine Hydrodynamics, M.I.T. Press, Cambridge,
Massachusetts,
2
1977.
Haslum, H. A. and Faltinsen, 0. M., Alternative Shapes of Spar
Platforms for Use in Hostile Areas, Offshore Technology
Conference,
1999.
48