... 5
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1
COLUMBIAR.
R.
LIBRARY
Marine Science Laboratory
Oregon State University
DEPARTMENT of OCEANOGRAPHY
rILLAhvoot
11.4Y
SCHOOL of SCIENCE
OREGON STATE UNIVERSITY
ANALYSIS OF A TWO-POINT
MOORING
FOR A SPAR BUOY
Richard F. Dominguez, John H. Nath,
Steve Neshyba and David A. Young
Sponsor
Standard Oil of California
Reference 69-34
Data Report No. 38
December 1969
Analysis of a Two-Point Mooring for a Spar Buoy
By
Richard F. Dominguez, John H. Nath,
Steve Neshyba and David A. Young
Data Report No. 38
December 1969
Reference 69-34
Sponsor : Standard Oil of California
TABLE OF CONTENTS
Acknowledgements
List of Symbols
List of Tables
List of Figures
ii
iv
Introduction
Theoretical Considerations
7
Design Criteria
Motion of the Buoy . ....... .
A Numerical Model
An Analogue for Mooring Statics
Design and Operation Curves
7
10
21
23
26
Experimental Considerations
Froude Modeling Law
Couchy Modeling Law
Testing Facilities
Testing Procedure
Buoy Models
Frequency Response in Heave and Pitch.
39
.
39
40
41
42
42
46
Summary
51
Recommendations
53
References
82
Appendix
83
Hydraulic Model Data
84
A CKNOWLEDGEMENTS
We express appreciation to Standard Oil of California for supporting this study on the two-point mooring of a spar buoy. We express
appreciation to the Office of Naval Research for their general support
of research in which the spar buoy plays an important role. We thank
the National Science Foundation, Atmospheric Science Branch, for their
support of the construction of a Marine Atmospheric Research Facility
based on the TOTEM buoy.
Bill Fischer of Standard Oil has given valuable suggestions on
engineering techniques. G. B. Burdwell of ESSA has provided information leading to a design wave spectrum for the Oregon Coastal waters.
Ray Fajardo has contributed to the numerical model of the two
point mooring. Charles Nath, Linda and Cathy Neshyba assisted in
reduction of hydraulic model data.
We appreciate the cooperation of General Dynamics, Convair San
Diego Division, in the lease of their wave basin for testing of the model
buoy and mooring.
LIST OF SYMBOLS
Definition
Symbol
Acceleration
a
Area
Damping coefficient
Drag coefficient
d
d
Diameter
s
Depth from the surface to the spring buoy line
and chain junction
Wave frequency
Froude number
Buoyant force
F
F
F
F
F
F
d
np
nh
rh
rp
Drag force
Natural frequency of spar buoy in pitch
Natural frequency of spar buoy in heave
Resonant frequency of spar buoy in heave
Resonant frequency of spar buoy in pitch
g
Gravitational constant
H
Wave height
Spar buoy heave
Moment of inertia
k
Spring constant
Horizontal distance from the center line of spring
buoy to the lowest point on the anchor chain
L
Characteristic length
Wave length
Mass
Kinematic viscosity
Pitch
N Reynolds number
R
Mass density
Wave slope H/X
t
Time
Wave period
Unit weight of sea water
The angle the anchor line forms with the
horizontal
Velocity
W
Weight
wn
Natural radian frequency
x
Cartesian coordinate
y
Cartesian coordinate
Cartesian coordinate
Damping ratio
Line mass density per foot of length
LIST OF TABLES
Table 1.
Physical Properties of Prototype TOTEM Buoy
Table 2.
Static Mooring Line Analogue Model Results
Table 3.
Summary of Determinations of Natural Period in Seconds
Table 4.
Natural and Resonant Frequency Response of Spar Buoy
Models
Table 5.
Hydraulic Model Data - Frequency Response of Buoy 1, 3
and S
Table 6.
Hydraulic Model Data - Two-Point Mooring with Straight
Buoy Connecting Chain
Table 7.
Hydraulic Model Data - Two-Point Mooring with Weighted
Lines Connecting Spring Buoys to TOTEM
Table 8.
Frequency Response of Buoy S'
iv
LIST OF FIGURES
Fig. 1.
Prototype Spar Buoy, TOTEM
Fig. 2.
TOTEM I under construction at the Marine Science Center,
Newport, Oregon
Fig. 3.
Two-Point Mooring
Fig. 4.
Two-Point Mooring System and Nomenclature
Fig. 5.
Design Wave Spectrum
Fig. 6.
Forces Acting on Spar Buoy - Case I
Fig. 7.
Forces Acting on Spar Buoy - Case II
Fig. 8.
Mass-Spring-Damper System
Fig. 9.
Forces Acting on Spar Buoy during Pitch
Fig. 10.
Flow Chart of the Numerical Model
Fig. 11.
Static Mooring Line Model
Fig. 12.
Wind Velocity vs. Drag Force on Spar Buoy
Fig. 13.
Horizontal Displacement in X Direction
Fig. 14.
Vertical Displacement of Spar Buoy vs X
Fig. 15.
Connecting Chain Span and Spring Buoy Depth vs. Horizontal
(X) Displacement of TOTEM
Fig. 16.
Line and Chain Tension vs. Displacement
Fig. 17.
Line Tension vs. Anchor Line Bottom Angle.
Fig. 18.
Displacement vs. Horizontal and Vertical Force Acting on
Spar Buoy
Fig. 19.
Y vs. X' Displacement
Fig. 20.
Vertical Displacement of Spar Buoy vs. X'
Fig. 21.
Model Tests of TOTEM buoy and its two-point mooring
Fig. 22.
TOTEM buoy models, scale 1:100
Fig. 23.
General Dynamics Model towing basin with TOTEM model
and two-point mooring subjected to wind drag
Fig. 24.
Underwater photo of 2-point moor model in General Dynamics
wave basin under 100 kt wind force.
Fig. 25.
Frequency Response of Buoy 1
Fig. 26.
Frequency Response of Buoy 1
Fig. 27.
Frequency Response of Buoy 3
Fig. 27a.
Frequency Response of Buoy 3 (in pitch)
Fig. 28.
Frequency Response of Buoy S
Fig. 29.
Frequency Response of Buoy S
Fig. 30.
Frequency Response of Buoy 3
Fig. 31.
Frequency Response of Buoy 3
Fig. 32.
Frequency Response of Buoy 3
Fig. 33.
Frequency Response of Buoy 3
Fig. 34.
Frequency Response of Buoy S'
Fig. 35.
Frequency Response of Buoy S'
Fig. 36.
Frequency Response of Buoy 3
Fig. 37.
Frequency Response of Buoy 3
Fig. 38.
Frequency Response of Buoy 3
Fig. 39.
Frequency Response of Buoy 3
Fig. 40.
Frequency Response of Buoy 3
Fig. 41.
Frequency Response of Buoy 3
vi
Fig. 42.
Frequency Response of Buoy 3
Fig. 43.
Frequency Response of Buoy 3
Fig. 44.
Normalized Frequency Response - Buoy 1
Fig. 45.
Normalized Frequency Response - Buoy 3
Fig. 46.
Normalized Frequency Response - Buoy S
Fig. 47.
Pitch and Heave vs. Wave Height of Buoy 1
Fig. 48.
Pitch and Heave vs. Wave Height of Buoy 3
Fig. 49.
Pitch and Heave vs. Wave Height of Buoy S
Fig. 50.
Normalized Frequency vs. Pitch and Heave for Various Chain
Collar Positions
Fig. 51.
Normalized Frequency vs. Pitch and Heave for Various wind
Directions
vii
INTRODUCTION
Recent activity in the exploration and development of the ocean's
resources has created a need for scientific and technical knowledge in
many areas. One example is the need for rapid and reliable techniques
for ascertaining and for the predictions of sea state and the atmospheric
conditions at particular ocean locations. The Department of Oceanography at Oregon State University will attempt to make such atmospheric
and oceanographic measurements from a Spar buoy in the near future.
In September 1969, the buoy was moored on a two-point mooring some 30
miles off the Oregon coast, in 1800 ft of water. This is a report on the
engineering analysis of the mooring structure for the buoy. The continued development of ocean resources will depend heavily on the installation of moored structures and it is hoped that this report will contribute to the engineering knowledge of such structures and moorings.
The TOTEM spar buoy was built by the Department of Oceanography, Oregon State University. Figure 1 shows the shape and dimensions of the buoy. Figure 2 is a photograph of the prototype under construction. Listed in Table 1 are the physical properties of the buoy. It
will be used to monitor atmospheric and ocean conditions by means of
unmanned data collecting devices. Data 'will be telemetered directly to
the central digital computer at Oregon State University. The advantage
of such a system is that it can obtain measurements continuously at a
fixed location over extended periods of time and transmit this information
HO/ST
INSTRUMENT HOUSE - RATED
FOR NOMINAL 5000 LB. PAYLOAD
SPAR
d„..--ST/LL WATER LEVEL
TANK NO. / - FLOTATION
TANK NO. 2 - RESERVE TRIM / FLOTATION
TANK NO. 3 - TRIM
181'
73'
STEM
BALLAST
A
- --- TANK NO. 4 - BALLAST FLOTATION
12'
BALLAST
Fig. 1 Prototype Spar Buoy, TOTEM.
•
---:011)=.11111101:1111#11111WRSINIMM--,,-;ffivt.
1PLAWSWS FrYir!"!
Imminammuld.
- mixtuir
",1=1
'
Fig. 2. TOTEM I under construction at the Marine Science Center, Newport, Oregon
.
(-4
4
TABLE 1
Physical Properties of Prototype Totem Buoy
Item
1. Weight
2. Center of gravity
from bottom of
buoy
Flotation and
ballast tanks
empty
Flotation and ballast
tanks flooded to normal design elevation
55, 460 lbs
141, 820 lbs
81. 0 ft
62. 7 ft
74.5 ft
3. Center of buoyancy
from bottom to buoy
4. Centroidal moment
of inertia
5, 329, 000 lbssec2-ft
8, 796, 000 lbs-
sec2 ft
ocean would allow for continuous measurement of physical oceanographic
and meteorological data simultaneously from widely separated locations.
This report covers the investigation of the static and dynamic
behavior of the particular two-point mooring shown in Figure 3. Shown
in Figure 4 is the nomenclature and coordinate system adopted in this
report. The primary objective of this study was to develop a reliable
and economical mooring. The influence of wind, current and surface
waves upon the motion and the internal forces within the system's components was considered.
It was intended initially to completely analyze the problem from
three viewpoints -- (1) an analytical approach where the equations of
motion were linearized in order to obtain a closed solution for the buoy
motion, (2) a numerical analysis which included all non-linearities and,
(3) a hydraulic model study where the motions and forces were modeled
in accordance with Froude law of similitude. However, due to a shortage of both time and available personnel the total analysis included the
following: (1) an analysis of the natural frequencies of the buoy
in heave and pitch; (2) an analysis, with the aid of a structural analogue,
of the static forces and displacements due to steady state wind and current; (3) a nearly completed numerical model of the motion of the unmoored buoy; and (4) a hydraulic model study which established the
sufficiency of the mooring for sinusoidal waves of frequencies that
Fig.
Two-Point Mooring
6
Spring buoy
PL AN
ds °
-i- 1
I
Spring buoy
epth
q
Chain sag
0.,,,--Totem buoy
I
ELEV.
O
Fig. 4. Two-Point Mooring System and Nomenclature.
covered completely the design wave spectrum shown in Figure 5.
THEORETICAL CONSIDERATIONS
First established were the design criteria, or probable environmental conditions which the mooring would be required to resist. Following this, internal forces and motion of the system components were
determined for given loading conditions.
Design Criteria
Two sets of design criteria were developed. A 100 knot wind with
a 2 knot surface current was used to establish the response of the system
to static loads. In addition, a 10-year storm was assumed in order to
develop a wave spectrum to design for dynamic action of the buoy and its
mooring. The first criterion is self-explanatory. The second will be
discussed briefly.
Watts and Faulkner (5) have described a set of recorded storms off
Vancouver Island which they used to develop curves for wind speed vs
average interval between storm occurrences in years. They did not
state what statistical techniques were used as a basis for the curves.
However, the curves appear to be reasonable. It was decided to use
these curves as it was felt that the information would be conservative
since the storms off Vancouver are notoriously worse than off the coast
of Oregon 240 miles south. The curves show that a 10-year wind can be
defined as:
8
(f) .404
0 .05
.32 .35
.123
Frequency in cps'
Fig. 5. Design Wave Spectrum
38'
.*N
143'
w
w= Fb
Fig. 6.
Forces Acting on Spar Buoy - Case I
w
Fb > w
zo at t=0
I
Fb
Fig.7. Forces Acting on Spar Buoy - Case II.
Fig. 8. Mass-Spring-Damper System
10
Magnitude of Wind
(Knots)
(MPH)
45
65
85
105
39.1
56.5
74.0
91.2
Duration
(Hours)
24
2
1
Gust
It was then necessary to establish a significant wave height to be
used. Both the S-M-B method for an 80% fully-developed sea and the
P-N-J method for a fully-developed sea were utilized as explained in
Ippen (1). It was also decided to assume that the durations were somewhat longer, and the most severe condition was determined to be that
where the wind duration is 44 hours. The resulting significant period is
14 seconds. A Gaussian sea surface was then assumed and an equilibrium spectrum where the high frequency spectral density values vary
with the frequency to the negative fifth power, as explained in Plate and
Nath (3), was then obtained. The resulting design spectrum is shown in
Figure 5.
Motion of the Buoy
Since the primary mission of the spar buoy is to serve as an instrumented floating platform from which oceanographic and atmospheric
measurements can be made, it is extremely important that the buoy be
sufficiently stable under most sea conditions. The important forces which
the buoy must withstand are those due to wind and current drag, and to
wave interaction with the buoy.
11
Structures which are subject to constraint, which provides a restoration force when displaced, will exhibit oscillations that will occur at the
natural frequency of the structure. Should the structure be subjected to
disturbing forces composed of many frequency components such as that
produced by waves, the possibility of resonance must be considered.
Resonance between a structure and the disturbing force can occur when
the frequency of the applied force matches the natural frequency of the
structure. This results in a condition where the displacement amplitude
of the structure may be large, depending on the amount of damping present. The occurrence of excessive displacements can be both destructive
to the system and detrimental to its intended use.
Consideration of the spar buoy's restraints show it to possess six
degrees of freedom of motion. Therefore analyses of its motion must
consider the possibility of both translation and rotation with respect to all
three coordinate axes. The important forms of oscillatory motion, in
terms of buoy stability, that can be expected to take place, are vertical
motion in the z direction (a condition known as heave) and rotation about
the horizontal x and y axes, termed pitch. Since the buoy is symmetrical
about the z axis it is necessary to investigate pitch for one direction only.
To obtain analytical solutions for the motion of the spar buoy with
respect to heave, two cases of motion were considered. The first case
neglected the effects of form drag on the buoy to obtain a first approximation of the natural period of the buoy. The second case made use of this
12
result to obtain an improved solution by including form drag effects due
to fluid viscosity in order to get a first approximation of the equivalent
linear relative damping coefficient.
Case I - Viscous Effects Neglected
If the buoy shown in Figure
6 is given a small initial vertical dis-
placement z from its static equilibrium position, the effect is to change
o
the buoyant force acting on the buoy by an amount equal to the additional
positive or negative displacement weight of the sea water displaced by
the buoy. The only two forces acting on the buoy are thus the weight of
the buoy which is constant, and the buoyant force which is dependent on
the vertical position of the buoy. Upon release from the initial displacement position, without the effects of viscosity, the buoy will oscillate
about its equilibrium position indefinitely. The equation of motion for
the buoy is arrived at by the application of Newton's second law.
E F = ma
w-F
b
= m
2
2
z/dt . .
(1)
where
m = mass of the buoy
a = vertical acceleration of the buoy
F = buoyant force
b
t = time
w = weight of the buoy.
The left-hand portion of this equation can be expressed in terms of
the displacement z and the diameter d of the buoy.
13
-
Tr
2
md2z
ydz2
4
dt
. . .
(2)
where
'y= unit weight of seawater.
3
Substitution of the following values, d = 3. 5 ft, y= 64 lbs/ft , and
m = 4, 400 slugs, reduces the above equation to the following form:
2
d z
2
dt
1
7. 15 z = 0
. . .
(3)
This differential equation has the following solution:
z = A sin t
1/7. 15 + B cost t V 1/7. 15 .
(4)
where V1/7. 15 is the radian frequency.
Evaluation of the constants A and B is obtained by applying the
boundary conditions:
51) t
= 0,
(d)t = 0,
Z=Z
dz
dt
0
n
thus
=0 and B = z
o
and the solution therefore becomes
z = z cos 0/1/7.15 . .
o
.". T = 16.8 sec or f = 1/T = 0.059 cycles/sec.
(5)
14
Case II - Viscous Effects Included
Formulation of the equation of motion in this case is identical to
that of Case I except for the inclusion of the drag force on the buoy which
can be expressed in terms of the following equation:
= 1/2pAC d 1 V IV ..
(6)
where:
F = drag force
D
p = specific density of the fluid
A = projected area or surface area in contact
with fluid for the case of pure shear drag
C d = drag coefficient
V = velocity
The above form of the equation is used to maintain the directional
sense of the drag force, which is always opposite to the direction of
motion of the buoy. The drag coefficient C d in this equation is dependent
upon the ratio of viscous forces to inertial forces. This can be expressed
VL
in terms of a Reynolds number NR
= v
where V = velocity, L =
characteristic length, and v= kinematic viscosity of the fluid. Drag is
also dependent upon the relative roughness of the surface and the shape
of the object. Values of experimentally determined drag coefficients are
presented in any number of standard fluid mechanics references and are
normally expressed in terms of the Reynolds number, determined from a
steady-state velocity condition. Since oscillatory flow is not steady, a
15
single representative drag coefficient, and hence solution based upon this
concept, can only be an approximate one. An additional added complication is the fact that what is the leading edge of the flow during half the
cycle becomes the trailing edge during the next half cycle. The degree
of influence this has upon the drag is not precisely known. From Eq. 6
it can be seen that the drag force is dependent upon the square of the
velocity. When the drag term is incorporated in the equation of motion
the resulting differential equation becomes non-linear. An approximate
solution to the equation can however be obtained by linearization of the
equation. By application of Newton's second law, the equation of motion
now becomes,
+
= ma
2
d z
w - F - F =m
b
D
2
dt
(7)
The drag experienced by the buoy can be considered to be made up
of two parts - that produced by the lower end which is comparable to a
circular disc oriented perpendicular to the flow; and that part due to the
shear drag developed by the cylinder sides equivalent to that produced by
a flat plate of equal area (see Fig. 7).
If a total vertical displacement of 10 ft is assumed, and the period
obtained in Case I is used, a representative average velocity, over a full
cycle, can be obtained for the buoy.
2 x 10
V = avg. cyclic velocity - 16. 8
1.2 ft/sec.
16
Assuming a kinematic viscosity we can now obtain drag force terms
as a linear function of the velocity as shown below.
Disk:
NR
Vd
Disk
.. C
D
1. 2 x 4. 5
-5
1. 5 x 10
- 3.6 x 10 5
= 1.2
F D= 1/2 pAC D VIVI= 22.9 V
Disk
Plate:
1 x 143
VL
x 10 7
N R- 1. 05 -5
Plate
1.5 x 10
C D= O. 035
F
FD
= 80. 5 v
Substitution for all terms in Eq. 7 yields:
2
d
-615 z - (22. 9 V + 80.5 V) = 4400 z
2
dt
dz
V dt
2
. d z
2
dt
+ 0.0236 dz + 0. 104z
dt
=0
• •
(8)
Examination of Eq. 8 will show that it is equivalent in form to that
of the equation for a mass-spring-damper system such as shown in Fig. 8.
The equation for this system can be written as
17
m2 + cz + kz = F(t) .
\X
inertia damping restoration
term
term
term
(9)
In this equation c is the damping coefficient, k is the linear spring
coefficient, and F(t) is the forcing function. The solution to this important second-order linear differential equation is well known and will be
used to obtain a solution to Eq. 8.
The equivalent linear relative damping coefficient is therefore obtained directly from the first form of Eq. 8.
c = 22. 9 + 80. 5 = 103. 4
The damping ratio
is thereby found to be
-
103.44
c
2I
rciT
21/615x4400
- O. 0315
This now allows for calculation of the natural frequency and period
fh -
w
-12
2 Tr
- 0.0594 cycles/sec
and
T
1/f h = 16. 8 sec.
The linearization procedure used to obtain the natural frequency of
oscillation in heave will again be used to obtain the natural frequency in
pitch. Summing moments about the center of gravity of the buoy shown
18
in Fig. 9 will obtain the following form of Newtons' second law.
=i
cg
cg
(10)
.
Here e = the angular displacement, e = the angular velocity, and & =
angular acceleration. We now seek to write Eq. 10 in the following form
which is equivalent to Eq. 9
Ie
.,._.
Inertia
term
ke
=
\--,.--/
Restoration
term
cê
...y___,
Damping
term
M(t) .
■-„-Forcing
moment
The damping term is obtained by integrating the drag force over the length
L of the buoy.
Moment of drag =
5
1/2 C D
e V 2 z dA
(12)
L
Now
V = ze
dA = D dz where D = ave buoy diameter = 4. 0 ft
P = 2. 0 slugs/ft3
Since the drag coefficient is dependent on the Reynolds number, hence the
velocity which will vary from zero at the center of rotation, to a maximum
near the free surface, an average weighted drag coefficient of 0. 5 is found
and will be used. This was arrived at in a similar manner as that used
in the heave solution. The natural period in pitch was calculated without
consideration of viscous damping, then an initial displacement of 10 degrees from the vertical was assumed which enabled the calculation of an
average angular cyclic velocity CZ.) = 0. 04 rad/sec. Substitution of the
appropriate values into Eq. 12 and integration from r l to r 2 yields the
following:
19
Fig.9 Forces Acting on Spar Buoy During Pitch.
20
4
4•
Moment of drag = Caw (rz + r
= 0. 5 (0. 04)(80 4 = 62. 74) 6
6
= 1. 15 x 10 8.
(13)
The restoration force is provided by the weight w acting through
the center of gravity and the buoyant force acting vertically through the
center of buoyancy. By summing moments about the center of gravity
the following relation is obtained
(14)
8 F sins .
E M cg = 11
b
•
If we assume small angular displacement, Sin 8 C. We further
assume that any accompanying heave is small therefore
ZF z = ma 0 and F b = w = 141, 820 lbs
Hence Eq. 14 becomes
= 1. 67 x 10 6
F Mcg
(15)
e .
Now by substitution of Eq. 13 and 15 into Eq. 10 we obtain
Ie+ 1. 15 x 10 6 6 + 1. 67 x 10 6 e = 0
.
(16)
which is the exact form of Eq. 11 which we seek. Substitution of a value
of I = 8. 79 x 10 6 gives
Damping ratio S =
c- 0. 152
2JkI
21
Natural undamped radian frequency in pitch
L') np = —
k = 0. 435 radians /sec
I
Natural frequency in pitch
_2
F =co
np
np
= 0. 070 cycles sec
21r
and
T = 14.3 sec
A Numerical Model
Our original intention was to develop a numerical model that would
predict the response of the system to sinusoidal waves, wind and surface
current. Drag and added mass coefficients were to be determined from
hydraulic model studies. It was hoped that the model could be used to
determine the optimum amount of chain for use in the buoy connecting
catenaries and to investigate other design factors such as the best attachment point for the chain on the Totem buoy. The model was based on
Newton's second law of motion. However, this portion of the study was
drastically limited because of time;
and we, therefore, attempted to
model just the motion of the Totem, untethered. This numerical model is
about 90% completed. The program is running but obviously contains an
error. A flow chart of the program is given in Figure 10.
22
PROGRAM SPAR BUOY
(Program to Determine Motion of TOTEM Buoy)
STOP
START
DA TA
Input buoy dimensions and other
parameters.
Input environmental data such as
wave height and length, wind, etc.
Print out all input.
t=t+At
No
PRINT 2
V
DEFINE
Print all accelerations
Define wave constants and other
constants.
Set all velocities = 0
Establish At
Set t = 0
I
Subroutine FRCA NOIVI
Establish various functions such
as water particle velocity, etc.
I
J
- -
Calculate moments about center of
gravity of buoy.
Calculate forces in longitudinal
direction.
Solve for all accelerations with
simultaneous equations.
Convert to primary coordinate system.
Find displacements and velocities for
next At with recurrence formulae
Subroutine STRIPFC
PRINT 1
Print time, buoy coordinates and
velocities
Calculate forces acting on cylindrical
strips of the buoy in the secondary
coordinate system.
Fig. 10. Flow Chart of the Numerical Model
23
An Analogue for Mooring Statics
It was originally intended that tension forces in the anchor line and
chain would be measured dynamically in the hydraulic model under representative wind, wave and current conditions. However, the experimental
difficulties to be discussed prohibited this.
The two-point mooring configuration can be considered to be a
cable system whose supports are free to move, except at the anchor
position. Therefore, any change in the external forces acting on the
system, such as a change in wind drag or surface current on the spar
buoy must result in movement of the mooring system components to new
equilibrium positions and cable configurations. Consideration of the chain
alone indicates a number of geometric possibilities depending on relative
position of each of the supporting buoys.
For a cable supporting its own weight, uniformly distributed along
its length, the resulting shape of the cable is that of a catenary. Unlike
most other structural members, a cable cannot develop resistance to
bending and compression forces but acts as a tension member only.
Determination of internal tension forces in the cable can be evaluated provided the locations of the supports are known, which for a generalized
loading of the mooring system is not the case. The result is a force
system which is not amenable to direct solution. Two alternative ways
of obtaining anchor line and chain tensions for design purposes were considered.
1.
Solution of the resulting set of equilibrium and displacement
equations for each of the component cables, buoy and the mooring
system as a whole by a reiteration scheme using a digital computer.
24
2.
Solution by means of a static-mooring line analogue model.
Due to a limitation on time, the second method of solution was
adopted.
Since the two-point mooring is symmetrical, it was only necessary
to construct an analogue model of one-half of the mooring system. The
external forces considered to be acting on the system were 1) the submerged weight of the anchor line, and the connecting chain, 2) the net
buoyant force produced by the spring buoy, and 3) the reactions at the
anchor and spar buoy end of the chain. Since the cable geometry is d e
pendent only on the external loads applied to the system at the spar buoy,
it was possible to displace the spar buoy end of the chain as shown in
Figure 11 to various selected positions and then measure the resulting
cable configuration.
The model was constructed to two scale ratios:
Length ratio: 1 inch = 19. 7 feet
Weight ratio: 1 gm 56.0 lbs.
The spar buoy end was displaced in either the positive or negative
x direction (see Figure 11). Vertical displacement of the spar buoy in
the z direction was disregarded as this would be small and would not significantly affect the cable configurations or tensions, even for extreme
conditions. Since the spring buoy exerts a constant vertical force owing
to its net buoyancy, a spring buoy location corresponding to this condition
was then obtained by moving the spring buoy end (mechanical spring) to
a position, by trial and error, which corresponded to the required vertical
force. Once this position was obtained, the resultant cable geometry.
25
EQUILIBRIUM POSITION
FOR ZERO DISPLACEMENT
OF SPAR BUOY
80'
d
SPAR BUOY END
OF CHAIN
CALIBRATED SPRING
AT SPRING BUOY
POSITION
SPAN
'Ng
Physical Description
Chain Length =
in Equilibrium position
600
ft.
9,300
es
Span =
465
ft.
Sag =
168
ft.
Chain Weight =
in air
Spring Buoy ÷ net Buoyancy = 13,400 #'s
1800
d =
bot
=
75°
ft.
degrees
Fig. 11 Static Mooring Line Model.
26
was measured. We then calculated the external force at the spar buoy
required to displace it, the line and chain tensions, and the reactions at
the anchorage and at the spring buoys. Cable reactions and tensions were
calculated using the equations for a catenary to locate the center of gravity
of the cable and its segments. Table 2 contains a summary of the results
obtained for this part of the model investigation. All dimensions and
forces have been converted and are presented in terms of the prototype.
Design and Operation Curves
In order to summarize and present the results of this study in a
convenient and readily usable manner, a series of curves were developed,
as given in Figures 12 through 20. It should be noted that these curves
were developed for the mooring configuration, including chain weights,
buoy size, etc. , shown in Figure 11 and, therefore, apply to these conditions only. The use of these curves is illustrated by means of an
example.
Example 1. A 60 knot wind accompanied by a 2 knot surface current
from the direction, act on the buoy and mooring system in the positive
x direction, parallel to the anchorage points. To determine the geometric
response of the mooring system and the forces in the anchor line and
connecting chain the curves are used in the following manner:
Step 1. Enter Figure 12 with the value of 80 knots for the wind,
hence the total drag force on the spar buoy is found to be 5, 500 lbs.
Note that this curve incorporates the affect of a 2 knot surface
current. For some other condition, the drag equation, Eq.
should be used instead.
6
Line tension
at top (lbs)
X(ft)
d s (ft)
l(ft)
m(ft)
Sag
spq(ft)
atop°
ebot°
Chain tension
at spar buoy
end (lbs)
-400
22. 0
139. 0
136. 0
248. 0
85. 1
83. 7
4, 040
7, 460
-320
22.0
173.0
149.0
222.0
82.5
80.5
3,900
7, 500
-200
23. 6
234. 5
231. 5
167. 5
, 75. 0
71. 3
4, 660
7, 825
-160
33.5
236.0
238. 0
157.5
74.3
70. 2
4, 820
8,905
-1.20
43. 2
237. 5
249. 0
148. 5
73. 4
69. 4
5, 050
8, 030
-80
56. 1
235. 0
251. 5
138. 0
73. 0
69. 2
5, 130
8,190
-40
66. 0
234. 5
260. 0
132. 0
72. 5
68.4
5, 280
8, 300
0
80. 0
234. 5
270. 0
120. 0
71. 5
67. 3
5, 500
8, 490
+40
93. 6
226. 0
283. 0
106. 5
71. 5
66. 8
5, 800
8, 740
+80
108. 3
221. 5
300. 0
90. 6
69. 6
65, 4
6,180
8, 960
+120
124.0
211.0
311.0
80. 8
69.2
64.8
6, 400
9, 250
+160
140. 0
202. 5
325. 0
69. 0
68. 8
64. 5
6, 730
9, 460
+200
152. 0
190. 0
345. 0
53. 2
67. 6
63.5
7, 250
9, 910
+240
169.0
164.5
357.0
39.4
67.8
63.4
7, 650
10, 250
+320
214.0
108.5
435.0
11.8
63.0
58.8
9,500
11, 700
+400
267. 0
6. 5
531. 0
0. 5
56. 2
51.6
11, 800
14, 200
j
i
Table 2,
Static Mooring Line Analogue Model Results
28
Step 2. With the total horizontal drag force now determined, we
enter Figure 13 and obtain from the lower graph the horizontal displacement in the x direction of the spar buoy to be 320 ft.
Step 3. From Figure 14 the vertical displacement of the spar buoy
is 2. 45 feet.
Step 4.
From Figure 15 (upper graph) the attachment point of the
left hand spring buoy is found to be located at a depth of 217. 3 feet
below the surface while that of the right hand buoy is 22. 5 feet
below the surface.
Step 5.
Figure 15 (lower graph) shows the horizontal span between
the spar and spring buoy of the left hand side of the mooring to be
550 feet while the right hand side span is 350 feet.
Step 6. In Figure 16 (upper graph) chain tension at the Totem end,
left hand side, is found to be 9700 lbs; on the right side, 4200 lbs.
Step 7. Anchor line tension at the spring buoy end is found from
Figure 16 (lower graph) to be 12,100 lbs. on the left hand side,
and 7, 600 lbs. on the right.
Step 8. Knowing the line tension at the spring buoy end enables us
to determine the angle that the anchor line makes with the bottom
by means of Figure 17. On the left hand side the bottom angle is
58. 5°; on the right hand side 75.4°.
29
Example 2. We can now consider the case where the same wind and
current as in Example 1 are now acting in the positive y direction, or perpendicular to the mooring line anchorage. With the addition of the Figures
18 through 20 the same previously used curves may be utilized to obtain
a solution for this situation. The spar buoy will be displaced in the
positive y direction, which results in a condition equivalent to having two
left hand moorings acting under a condition of positive x displacement,
as considered in Example 1.
Step 1. The drag force is determined exactly as in the previous
example.
Step 2. Utilizing the drag force of 5, 500 lbs. we enter Figure 18
and obtain the y displacement of the spar buoy as 770 feet.
Step 3. Using this value of y, we now obtain from Figure 19 the
equivalent x distance, x', equal to 222 feet.
Step 4. From Figure 20 the vertical displacement of the spar buoy
is 3. 5 ft. From this point on, determination of buoy spans, sub
surface depth, and line tensions, is identical to that used above
for displacement of the spar buoy in the positive x direction. Only
in this case we utilize the equivalent x' dimension for x and take
values for the left hand portion of the mooring only.
30
200
100
90
so
(f)
0
C
70
60
50
40
a)
30
20
10
1
2
3
4
5
6 7 8 9 10
Total Drag Force Including 2 Knot Current x 10 3 lbs.
Fig. 12 Wind Velocity vs. Drag Force on Spar Buoy.
20
31
300
Fig. 13. Horizontal Displacement in X Direction
400
32
5.0
4l
C
4.0
0
CO
s
-CO
CL
3.0
0
yC
(1)
U
2.0
co
a
_
co
U
1.0
L
0
X Displacement in ft.
Fig. 14 Vertical Displacement
of Spar Buoy vs X
33
400
30 0
200
100
100
200
300
400
X Displacement in ft.
600
L
0
0
4-
0
0
0
500
C
400
3 00
200
0
100
200
300
X Displacement in ft.
400
Fig. 15. Connecting Chain Span and Spring Buoy Depth vs. Horizontal
(X) Displacement of TOTEM
34
CD
C C
0
C
(I)
w
0
I— co
C s(I)
_c
N
X Displacement in Feet
15
14
13
12
11
10
9
8
04c:10
0____0
O
O
O
O
O
7
6
0
100
200
300
400
X Displacement in Feet
Fig. 16. Line and Chain Tension vs. Displacement
90
80
70
°
goT
60
50
7
9
10
11
Line-Tension x 10 3 lbs.
at spring buoy end
Fig. 17. Line Tension vs. Anchor Line Bottom Angle
12
13
36
60
56
52
48
44
40
36
32
c•-■
O
o
28
24
20
16
12
8
4
200
400
600
Y
Displacement in ft.
Fig. 18. Displacement vs. Horizontal and Vertical. Force Acting on
Spar Buoy.
37
1200
1100
1000
900
800
700
600
500
400
300
200
100
100
200
300
400
500
= Equivalent X Displacement of Spar Buoy in Feet
Fig.lq Y vs. X' Displacement
38
10
9
0
0
Wind
8
7
6
5
4
3
2
1
0
100
200
300
400
X' = Equivalent X Displacement of Spar Buoy in Feet.
Fig.20 Vertical Displacement of Spar Buoy vs X'
39
EXPERIMENTAL CONSIDERATIONS
Hydraulic model tests were conducted on the spar buoy and its
mooring. The objective of these tests was to determine the dynamic behavior of both the spar buoy and mooring system over a wide range of
wind, wave, and current conditions. Specifically sought were the frequency response of the buoy to waves, tension forces in the anchorage
system under dynamic conditions, and the drag characteristics of the
spar buoy. Model tests provided a means for evaluating several alternate
two-point mooring line configurations. The relative ease with which
models can be altered allowed for evaluation of such items as the opti
mum point of attachment for the connecting chains to the spar buoy.
Fr oude Modeling Law
Since complete similarity between model and prototype is rarely
ever possible, the investigator utilizing a model must design it with the
view that it will reproduce those effects which are essential to his investigation. The primary interest was to determine the dynamic response of
the system to wave action, which is a gravity phenomena. The model
was therefore constructed and operated on the basis of Froude number
criteria. The Froude number is a dimensionless quantity which expresses
the ratio of gravity forces to inertia forces. Consequently, if the Froude
number in both the model and the prototype are identical then similarity
is achieved with respect to gravity forces. The Froude number is defined
as
N
F=
gL
(17)
where
v = velocity
g = gravity
L= characteristic length
40
The model-to-prototype relationships derived from considering both the
Froude law of similitude and Newton's second law are:
F
r
I
=
=
V
=
f
=
r
t
r
r
Sr
=
=
3
r
5
Lr
1/2
L
r
-1/2
L
r
1/2
L
r
5/2
L
r
L
(18)
(19)
(20)
(21)
(22)
(23)
Where the subscript, r, refers to the ratio between model-toprototype, F is force, L is length, I is moment of inertia, V is velocity,
f is frequency, t is time, and S is the wave spectrum ordinate.
Using this criteria, in a model having a length scale ratio of 1: 100,
a model wave with a period of 2. 0 sec. would correspond to a period of
20. 0 sec. in the prototype. A 6 inch model wave height would correspond
to a 50 ft. wave in the prototype as it would scale directly in terms of
the length ratio.
Cauchy Modeling Law
Plate and Nath (3) have shown that it is important to establish the
natural frequencies of motion of the model in the same relative position
of the wave spectrum. If structural vibrations are important, then the
Cauchy number must be the same in both the model and the prototype.
For mooring lines, this requires that the following relation he satisfied
(24)
41
where A is the cross-sectional area of the line, E is its modulus of
elasticity and
p,
is the per foot mass density of the line. The above ex-
pression represents the speed of propagation of a disturbance in the line.
However, because of the flexible arrangement of the two-point mooring
and the fact that the lines are relatively short, it was not necessary to
model the modulus of elasticity, E. The area, A, was modeled using
copper wire and mass density, p., was thus approximated.
Testing Facilities
All of the model tests were conducted at the General Dynamics Electric Boat Towing Basin located at San Diego, California. The Towing
Basin facility utilizes a channel which is 320 feet long by 12 feet wide
and 6 feet deep. It is provided with an overhead model towing carriage.
Periodic surface waves are produced by a hydraulically operated compound wedge wave generator. To simulate wind conditions, the basin
has a mobile air-flow generator which can be moved along with the model
when towed, or located in a stationary position. The facility is also
equipped to measure and record both force and moment by means of
strain gauge dynamometers. The mooring lines for the model were
equipped with such instruments in order to determine the line tension in
the anchorage when subjected to directional wave conditions. However,
instrumentation difficulty occurred during the initial phase of the testing
program and this part of the model investigation had to be abandoned.
42
Testing Procedure
The model testing program consisted of 79 separate runs. The
dynamic behavior of buoys and alternate mooring systems were recorded
on 16 mm film, at a speed of 24 frames per sec. Fig. 21 shows a typical
movie frame sequence. Wave height, length, buoy pitch and heave as
well as other quantities of interest were then later obtained by analyzing
the films. This was possible by utilizing a 1 inch square background grid
which was located on the far side of the towing tank as a reference.
During all runs, the position of the buoy and mooring system components
relative to the camera and background grid was carefully noted since an
optical correction had to be applied to convert the apparent motion recorded on the film to true motion. Variation in wave amplitude and
period was achieved by altering the speed and amplitude of the wave
maker. Wave period was checked by comparison of the recorded period
with that observable in the movies.
Buoy Models
For the hydraulic model experiments, four buoy models were constructed at a scale of 1:100. Three of the TOTEM models, buoys 1, 2,
and 3 are shown in Fig. 22.
The fourth buoy, not shown,was a simple cylindrical spar buoy,
referred to as buoy S. It was tested both as a matter of interest and to
provide a basis for comparison of the dynamic behavior of the TOTEM
buoy models.
43
6
I0
Fig. 21 Model tests of TOTEM buoy and its two-point mooring.
44
Buoy 1: This model was constructed to copy exactly, the center
of gravity and moment of inertia characteristics of the
prototype buoy. The model was entirely solid. During
the initial testing sequence, the model warped, resulting
in the longitudinal axis becoming slightly curved which
in turn caused the model to rotate in an oscillatory manner
about this axis when subjected to wave motion.
Buoy 2:
This model was constructed as a spare in the event it
was needed. However, this model was never utilized in
any of the mooring testing programs.
Buoy 3:
This model was also constructed to model the center of
gravity and moment of inertia of the prototype but was
built with a hollow stem section, provided with connected
openings at the top and bottom to test their possible effect
in reducing buoy heave created by imbalance of dynamic
wave pressure with depth acting on the horizcntal surfaces
of the buoy.
Buoy S: This model was a solid, cylindrical spar buoy of constant
cross section and was not designed to model a particular
prototype.
Buoy S': This model was a modification of Buoy S through the
addition of added ballast at the lower end plus concentrated
buoyancy in the form of buoyancy tanks attached to the
cylindrical spar. The net result was a spar buoy with
characteristics similar to that of the TOTEM.
45
2
Fig. 22. TOTEM buoy models, scale 1:100. From left to right are
buoy models 2, 1 and 3 respectively.
46
All TOTEM models were carefully weighed and checked for balance
to determine the degree to which they modeled center of gravity, center
of buoyancy and the moment of inertia characteristics of the prototype.
Adjustments were made when necessary. Each of the buoys were then
placed in a tank of water and subjected to small vertical and lateral dis placements to determine their natural period of oscillation in both heave
and pitch. The degree of damping exhibited by each of the models was
observed by noting the time required to damp out all oscillations.
Figures 23 and 24 (lower photo) show the TOTEM model and its mooring
during testing at General Dynamics wave basin.
Frequency Response in Heave and Pitch
The data obtained for the hydraulic model phase of this investigation
is presented in Tables 4 through 8 in the Appendix. Frequency response
curves for both heave and pitch are plotted in Figures 25 through 51.
The experimental setup is described below by run numbers to aid in interpreting frequency response curves.
Runs 1-26
Buoys 1, 3, and S were allowed to float freely
without restraint and then subjected to wave action
to determine their frequency response heave and
pitch.
Runs 27-39
These run numbers were reserved for the dynamic
evaluation of mooring line tensions. Because of
instrumentation difficulties no appreciable data
was obtained.
47
Fig. 23. General Dynamics Model towing basin with TOTEM model and
two-point mooring subjected to wind drag.
48
Summer installation of TOTEM I off coast of Oregon. Sensors
gear atop platform is the AN-GMT-5 Automatic Meterology Station
on loan from NR L.
Fig. 24
Underwater photo of 2-point moor model in General Dynamics
wave basin under 100 kt wind force. Spring buoys plus energyabsorbing catenary lines keep surge loads off main anchor lines.
49
Runs 40-42
This series of tests were made on Buoy 2. The
model was towed at a constant rate by means of the
overhead carriage while waves generated in the
basin impinged upon it. In all three runs a Karmanvortex street was noted to form behind the buoy.
The towing velocities were; Run 40 - 0. 6 ft/sec;
Run 41 - 1. 0 ft/sec;
Run 42 - 0.3 ft/sec.
Runs 43-47
A two-point mooring incorporating a weightless
line connecting the spring buoys to the TOTEM
buoy was tested. Buoy 3 was used for this series.
The mooring system was oriented such that waves
approached from the positive y direction or perpendicular to the anchorage point.
Runs 48-49
The buoy mooring and orientation was the same
as for Runs 43-47 except the wind was now allowed
to blow against the buoy from the same direction
as that of wave propagation.
Runs 50-57
The buoy and mooring of the previous runs was
rotated through an angle of 45 degrees so that
wave approach was at an angle of 45 degrees to a
vertical plane through the anchorage points. No
wind was utilized.
Runs 58-63
The mooring of the previous runs 50-57 was again
rotated such that the wave advance direction was
now parallel to the anchorage points (in the positive
50
x direction). In Runs 61-63 extremely violent
jerking of both spring buoys was noted.
Runs 64-69
This series of tests was conducted using Buoy 3
again, but now with weighted lines to simulate
buoy connecting chains, as shown in Figure 4.
Here the chain attachment collar was located at
0.12 feet below the top of the flotation tank.
Wave advance was in the positive x direction
parallel to the anchorage.
Runs 70-73
All conditions were identical to that of runs 64-69
except that the chain, attachment collar was now
placed at the top of the flotation tank.
Runs 74-77
Same as previous Run 70-73, except that the chain
attachment collar was placed at the bottom of the
flotation tank.
Runs 78-79
Here the attachment collar was placed at 0.12 feet
above the bottom of the flotation tank. All conditions were the same as in previous runs except
that wind was now allowed to blow on the model.
51
SUMMARY
The natural frequencies of the buoy were determined by several
means and are summarized below:
Table 3. Summary of Determinations of Natural Period in Seconds
Analysis
Mode
Experimental
Displacement
Curves
Experimental
Frequency
Response Curves
Numerical
Model
Heave
16.8
18
22
16.6
Pitch
14.3
22-28
37
19.6
Comparison of frequencies will show that the prototype frequencies
in heave and pitch will not occur within the design wave spectrum.
Therefore, if the design wave spectrum is a good representation of wave
energy and frequency distribution for a ten-year storm, the mooring
should have no resonance problems.
If one compares Figures 38 through 43 (the frequency response
curves for pitch and heave of Buoy 3 while moored) with Figures 27
and 27a
(the same curves for the buoy unmoored), it can be seen that
the mooring has little influence on the motion of the buoy. The static
analogue study indicated that the mooring line and chain forces will be
well within the allowable limits even at maximum conditions considered
of 100 kt winds accompanied by 2 kt surface current. Reduction in freeboard of the TOTEM buoy will be within tolerable limits.
Design information that was not obtained was the dynamic forces
in the mooring line and chain. However, close study of movies taken
52
of the buoy and mooring models indicated no severe dynamic response of
the system. Therefore, considering the sizable factor of safety, based
on the static analysis, it was decided to deptoy the two-point mooring
considered. The prototype installation will be instrumented to measure
dynamic loads within its mooring chains.
53
RECOMMENDATIONS
It is intended that the TOTEM buoy be moored in 1800 feet of water.
At a date subsequent to the completion of the study contained herein it
was decided to add 75 feet of 7/8" diameter cable from the spar buoy to
the chain in order to facilitate installation procedures. It is felt that this
will not influence the motion of this mooring in a detrimental way. The
spring buoys will be submerged to a depth of 80. feet in the absence of
wind or surface currents. They will be located at a distance of 550 feet
to 590 feet from the spar buoy. Anchor weights of 25,000 pounds, submerged, should be adequate to prevent dragging of the anchor. If
possible, an accelerometer should be installed on board the spar buoy
to measure heave motion so that this motion can be filtered from the
wave pressure measurement to be taken.
Future analytical work should be directed toward determining the
optimum size and lengths of chain for this type of mooring, development
of a means to calculate the dynamic line and chain forces, the development of transfer functions to describe the response spectrum of any part
of the system when the wave spectrum is specified, the determination of
all drag and added mass coefficients, etc. The foregoing items are presented as examples. It is not the intention of the authors that future work
be limited to them only. Indeed, future developments may well point out
that other areas of research on this type of mooring will be most beneficial.
54
30
25
Runs 1- 26
20
15
10
•
0
0
0.5
1. 0
1. 5
f in cps
Fig. 25 Frequency Response of Buoy 1
2.0
55
2:0
Runs 1-26
1. 5
A
•
MO
•
•
Hv
1. 0
•
•
•
.5
S
o
.5
1. 0
1. 5
f in cps
Fig. 26 Frequency Response of Buoy 1
2. 0
56
2.0 _
Runs 1-26
1.5
Hv
H
•
1.0
•
•
•
•
0.5
•
•
•
0
0
0.5
1.0
2.0
1.5
f in cps
Fig. 27 Frequency Response of Buoy
3
57
30
25
20
7
15
10
5
•
1.0
1.5
f in cps
Fig. 27 (a) Frequency Response of Buoy 3 (in pitch)
2, 0
58
30
Runs 1- 26
S
25
20
15
10
5
•
•
0
0
0.5
1.0
1.5
f in cps
Fig. 28 Frequency Response of Buoy S
2. 0
59
3.0_
•
2.5
Runs 1-26
•
•
2.0
• •
1.5
•
•
•
1.0
•
••
• •
•
•
•
0.5
0
0
0.5
1.0
1.5
f in cps
Fig. 29 Frequency Response of Buoy S
2.0
60
25
Runs 43-47
20
15
10
5
0
0
.5
1.0
1.5
f in cps
Fig. 30 Frequency Response of Buoy 3
1
2.0
61
2 .0 _
Runs 43-47
1.5
Hv
H
1.0
•
0.5
•
•
0
0.5
1.0
1.5
f in cps
Fig. 31 Frequency Response of Buoy
3
2.0
62
Runs 50-57
25
20
S
15
10
5
•
•
0
0
0.5
1.0
1.5
f in cps
Fig. 32 Frequency Response of Buoy 3
2.0
63
2.0
Runs 50 - 57
1.5
Hv
H
•
1.0
•
•
0.5
•
•
0.5
1.0
1.5
f in cps
Fig. 33 Frequency Response of Buoy 3
2.0
64
25
Runs 51-54, 56, 57, 60, 63
20
15
10 -
w •
•
0
•
0
0.5
1.0
1.5
f in cps
Fig. 34 Frequency Response of Buoy S
2. 0
65
2.0—
Runs 51-54, 56, 57, 60, 63
1.5
Hv
H
1.0
0 .5
•
•
•
•
•
•
•
•
1
0.5
1.0
1.5
f in cps
Fig. 35 Frequency Response of Buoy ST
2.0
66
25
Runs 58-63
20
15
10
0
0
0.5
1.0
1.5
f in cps
Fig. 36 Frequency Response of Buoy 3
2.0
67
2.0
1.5
Runs 58-63
iv
1.0
•
•
0.5
•
•
•
•
0
0
0.5
1.0
1.5
f in cps
Fig. 37 Frequency Response of Buoy 3
2.0
68
25
Runs 64-69
20
15
10
5
0
0.5
1.0
1.5
f in cps
Fig. 38 Frequency Response of Buoy 3
2.0
69
2.0
Runs 64-69
1.5
Hv
H
1.0
•
•
0 .5
•
•
0
0.5
1.0
1.5
f in cps
Fig. 39 Frequency Response of Buoy 3
2.0
70
25
Runs 70-73
20
15
10
5
0
0
0.5
1.0
1.5
f in cps
Fig. 40 Frequency Response of Buoy 3
2.0
71
2 .0 _
1 .5
Runs 70-73
1.0
0.5
•
•
0
0.5
1.0
1.5
f in cps
Fig. 41 Frequency Response of Buoy 3
2.0
72
25
Runs 74-77
20
15
10
5
•
0.5
1.0
1.5
f in cps
Fig. 42 Frequency Response of Buoy 3
2. 0
73
2.0
1.5
Runs
74-77
Hv
H
1 .0
0.5
•
•
•
0
0
0.5
1.0
.1
2.0
1.5
f in cps
Fig. 43
Frequency Response of Buoy
3
3. 0
2. 8
2. 4
2. 0
f
1.6
Hv
rh
= 0.46
•
H
•
•S
1. 2
•
0. 8
i•
•
•.
0.4
•
•
•
•
•
fifrh 0. 46
30
f
rp
=
0. 27
20
•
10
•
'0 0
•
•
•
•
•
0
•
2
3
•
4
5
I
6
f/f
rp
Fig. 44 Normalized Frequency Response - Buoy 1
75
2.5
2.4
2.0
Hv
1.6
H
j=.0.45
rh
•
•
1.2
•
.8
•
.4
•
••
•
4.o
3.0
2.0
1.0
5.0
3.0
•
•
2.0
P
S
•
•
f
•
rp
=
0.26
•
1 .0
•
•
0
1
2
3
•
•
4
5
fifrp
Fig.45
Normalized Frequency Response - Buoy 3
76
2. 8
2.4
2. 0
F
Hy
rh
= 0.56
1.6
1. 2
.8
•
•
•
1.0
0
3. 0
2.0
f/f
4.0
rh
30
f
rp
= 0.30
20
•
10
• •
•
•
0
••
1
•
2
3
4
5
6
f/f
7
8
rp
Fig. 46 Normalized Frequency Response - Buoy S
5.0
77
H
v
(In.)
H (In)
Pitch
0
(In.)
Fig. 47 Pitch and Heave vs. Wave Height of Buoy 1.
78
H
v
(in)
12
10
Pitch°
8
6
4
2
00
5
3
6
(In)
Fig.
48 Pitch and Heave vs. Wave Height of Buoy 3
79
Hv
(In.)
12
10
Pitch°
8
6
4
2
1
2
5
3
6
7
8
H
( In.)
Fig. 49. Pitch and Heave vs. Wave Height of Buoy S.
80
LEGEND - Chain collar attachment
location from top of floatation tank.
8
0
.12 =
7
0
0
.36 =
0
6
0) 0
0
O
5
0
Pitch
0
0
O
3
2
1
2
f/frp
4.0
0
3.0
0
Heave
(In.)
0
2.0
1.0
O
1
2
3
5
f/frh
Fig. 50. Normalized Frequency vs. Pitch and Heave for Various Chain
Collar Positions.
81
0
8
0
0
e
7
6
1?)o
0
Wave Advance
0 - 0 = 90°
4
0 - 0 = 45°
0°
e-0
9
O e
e
2
O
0
4
3
5
f/frp
4.0
0
3.0
2.0
O
9
1 .0
O
0
O
O 9
0
1
2
9
4
3
5
6
f?frli
Fig. 51. Normalized Frequency vs. Pitch and Heave for Various Wind
Directions.
82
REFERENCES
1.
Ippen, A. T. , "Estuary and Coastline Hydrodynamics", Chapter 3
by C. L. Bretschneider, McGraw Hill, 1966
2.
Kinsman, B. , "Wind Waves", Prentice-Hall, Englewood Cliffs,
1965
3.
Plate, E. J. and Nath, J. H. , "Modeling of Structures Subjected
to Wind Generated Waves", Proceedings of the ASCE and Society
of Civil Engineering, London, September, 1968
4.
Schlichting, H. , "Boundary Layer Theory", McGraw Hill, New
York, 1960
5.
Watts, J. S. and Faulkner, R. E. , "Designing a Drill Rig for
Severe Seas", Ocean Industry, November 1968
TABLE 4
Natural and Resonant Frequency Response of Spar Buoy Models
Buoy
1
Natural Frequency (Cycles/sec.))
Pitch - f
Heave - fnh
np
0.47
0.55
Resonant Frequency (Cycles/sec. )
Heave - f
Pitch - f rp
'
rh
0.35
0.46
ho
z
3
0.36
0,263 for
runs 70-73
0.57
0.26
0.45
S
0.29
0.59
0.30
0.56
TABLE 5
Hydraulic Model Data - Frequency Response of Buoy 1, 3, and S
Wave Conditions
Run No.
Buoy
T (sec)
f (sec)
H (ft)
x (ft)
Pitch Response
S
(radians)
5
1
3
S
2.15
2.15
2.15
0.465
0.465
0.465
0.0816
0.0816
0.0816
22.2
22.2
22.2
6
1
3
S
2.22
2.22
2.22
0.45
0.45
0.45
0.163
0.163
0.163
23.5
23.5
23.5
0.007
0.007
0.007
0.105
0.105
0.035
7
1
3
2.08
2.08
2.08
0.481
0.481
0.481
0.177
0.177
0.177
21.0
21.0
21.0
0.0084
0.0084
0.0084
0.096
0.070
0.070
P/S
0.0436
0.0436
0.0174
15.0
15.0
5.0
1 14
8.3
8.3
f/f
•
f/f
Heave Response
•
Hv (ft)
Hv/H
f/f
f f/f
f
r
0.99
1.28
1.61
1.32
1.77
1.53
0.1
0.1
0.125
1.22
1.23
1.48
0.83
0.80
0.77
1.00
1.02
0.82
0.97
1.25
1.57
1.29
1.73
1.50
0.1
0.183
0.166
1.17
1.12
1.39
0.81
0.78
0.76
0.97
1.00
0.80
1.03
1.34
1.68
1.37
1.85
1.60
0.216
0.175
0.274
1.22
1. 13
1.56
0.86
0.84
0.80
1.04
1.07
0.86
TABLE 5 - continued
Hydraulic Model Data - Frequence Response of Buoy 1, 3 and S
Run No.
Buoy
T (sec) f (sec)
H (ft)
Heave Response
Pitch Response
Wave Conditions
) (ft)
S
P
(radians)
P/S
f/f
np
f/f
rp
Hv (ft)
Hv/H
f/f
nh
f/f
rh
8
1
3
S
1.08
1. 08
1.08
0.93
0.93
0.93
0. 207
0. 207
0. 207
6.0
6. 0
6. 0
0. 0344
0. 0344
0. 0344
O. 113
0. 096
0.07
3. 28
2. 79
2.03
1.99
2. 58
3. 22
2.64
3. 56
3.08
0.066
0.066
0.0083
0.30
0.33
0.03
1.66
1.61
1. 56
2.01
2. 06
1.65
9
1
3
S
1.08
1.08
1.08
0.93
0.93
0.93
0.318
0.318
0.318
6.0
6.0
6.0
0.0530
0.0530
0.0530
0.131
O. 131
0.07
2.47
2.47
1.32
1.99
2.58
3.22
2. 64
3.56
3.08
0.083
0.075
0.025
0. 26
0.24
0.09
1.66
1.61
1.56
2.01
2.06
1.65
10
1
3
S
1.08
1.08
1.08
0.93
0.93
0.93
O. 126
0. 126
0. 126
6.0
6.0
6.0
0.021
0.021
0.021
0.0871
0.0785
0.0261
4. 15
3.74
1.24
1.99
2.58
3.22
2.64
3.56
3.08
0.05
0.05
0.00
0.36
0.41
0.00
1.66
1.61
1.56
2.01
2.06
1.65
11
1
3
S
1.08
1.08
1.08
0.93
0.93
0.93
0.0601
0.0601
0.0601
6.0
6.0
6.0
0.010
0.010
0.010
0.035
0.0436
0.00
3.5
4.36
0.00
1.99
2.58
3. 22
2.64
3.56
3.08
0.0166
0.0166
0.00
0.30
0.29
0.00
1.66
1.61
1. 56
2.01
2.06
1.65
12
1
3
S
0.80
0.80
0.80
1.25
1.25
1. 25
0.158
O. 158
0.158
3.3
3.3
3. 3
0.048
0.048
0. 048
0.0785
0.061
0. 035
1.63
1.27
0.73
2.69
3.48
4.36
3.56
4.82
4. 17
0.025
0.025
0.00
0.17
0.02
0.00
2.25
2. 18
2. 10
2.72
2.78
2.23
13
1
3
S
0.54
1.86
0.0645
1.47
0.044
0.035
0.80
4.00
5.33
0.0083
0.06
3.35
4.05
0.54
1.86
0.0645
1.47
0.044
0.0175
0.398
6.5
6.21
0.00
0.00
3.14
3.32
1
3
S
0.476
0.476
0.476
2.1
2.1
2. 1
0.0143
0.0143
0.0143
1.15
1.15
1. 15
0.0124
0.0124
0.0124
0.0175
0.0175
0.00
1.41
1.41
0.00
4.52
5.70
7.35
6.00
8.07
7.00
0.00
0.033
0.0083-
0.29
2.4
0.49
3.78
3.67
3.54
4.57
4.68
3.76
.
14
TABLE 5 - continued
Hydraulic Model Data - Frequency Response of Buoy 1, 3 and S
Pitch Response
Wave Conditions
Run No.
Buoy
T (sec)
f (sec)
H (ft)
X. (ft)
S
P
(radians)
P/S
np
f/f
Heave Response
rp
Hv (ft)
Hv/H
f/f
nh
f/f
rh
0.94
1.15
0.94
1.27
1.10
0.158
0.167
0.183
1. 03
1.09
1.23
0.59
0.57
0.56
0.72
0.73
0.59
1.76
0.71
0.94
1.15
0.94
1.27
1.10
0.158
0.167
0.183
0.76
0.80
0.90
0.59
0.57
0.56
0.72
0.73
0.59
0.0838
0.0785
0.0436
14.2
13.3
7.41
0.80
1.03
1.29
1.06
1.42
1.23
0.192
0.192
0.217
1.00
1.01
1.17
0.66
0.64
0.62
0.80
0.82
0.66
0.0088
0.0088
0.0088
0.105
0.096
0.061
11.9
10.9
6.93
0.94
1.26
1.54
1.26
1.69
1.47
0.217
0.217
0.217
1.14
1.10
1.31
0.96
0.77
0.74
0.96
0.60
0.79
47.0
47.0
47.0
0.00379
0.00379
0.00379
0.0662 17.4
0.0838 22.1
18.4
0.07
0.58
0.75
0.94
0.77
1.04
0.90
0.175
0.175
0.175
1.21
1.21
1. 29
0.48
0.47
0.46
0.59
0.60
0.48
0.243
0.243
0.243
47.0
47.0
47.0
0.00517
0.00517
0.00517
0.58
0.75
0.94
0.77
1.04
0.90
0.217
0.217
0.242
0.89
0.89
0.95
0.48
0.47
0.46
0.59
0.60
0.48
0.28
0.28
0.28
0.216
0. Z16
0.216
45.0
45.0
45.0
0.0048
0.0048
0.0048
0.0384
0.0523
0.0175
8.0
10.9
3.65
0.60
0.78
0.98
0.80
1.08
0.93
0.30
0.242
0.15
1.38
1.12
0.68
0.51
0.49
0.47
0.61
0.62
0.50
0.29
0.29
0.29
0.147
0.147
0.147
43.2
43.2
43.2
0.0034
0.0034
0.0034
0.035
0.061
0.07
10.3
18.0
20.6
0.62
0.81
1.01
1.12
0.97
0.158
0.158
0.117
1.09
1.12
0.79
0.52
0.50
0.49
0.63
0.64
0.52
15a
1
3
S
3.33
3.33
3.33
0.33
0.33
0.33
0.153
0.153
0.153
42.0
42.0
42.0
0.00364
0.00364
0.00364
0.0175
0.0436
0.0871
4.81
12.0
2.39
15b
1
3
S
3.33
3.33
3.33
0.33
0.33
0.33
0.208
0.208
0.208
42.0
42.0
42.0
0.00495
0.00495
0.00495
0.0175
3.54
0.0871
16
1
3
S
2.70
2.70
2.70
0.37
0.37
0.37
0.191
0.191
0.191
32.5
32.5
32.5
0.00588
0.00588
0.00588
17
1
3
S
2.28
2.28
2.28
0.44
0.44
0.44
0.220
0.220
0.220
25.0
25.0
25.0
18a
1
3
S
3.70
3.70
3.70
0.27
0.27
0.27
0.178
0.178
0.178
181)
1
3
S
3.70
3.70
3.70
0.27
0.27
0.27
1
3
3.58
3.58
3.58
1
3
S
3.44
3.44
3.44
20
f/f
0.71
e
TABLE 5 - continued
Hydraulic Model Data - Frequency Response of Buoy 1, 3 and S
Buoy T (sec)
f (sec)
H (ft)
Heave Response
Pitch Response
Wave Conditions
Run No.
k (ft)
S
P
(radians)
P/S
f/f
np
f/f
rp
Hv (ft)
Hv/H
f/f
nh
f/f
rh
1
3
S
1.33
1.35
1.35
0.74
0.74
0.74
0.319
0. 319
0.319
9.3
9.3
9.3
0.0343
0. 0343
0.0343
O. 131
0. 122
0.07
3.82
3.56
2.04
1.54
2.06
2.58
2.84
2.46
0.075
0. 1
0. 133
0.24
0.32
0.42
1.33
1. 29
1. 24
1.61
1.64
1.32
22
1
3
S
1.79
1.79
1.79
0.56
0.56
0.56
0. 285
0. 285
0.285
15.8
15.8
15.8
0. 018
0. 018
0.018
0. 131
0. 122
0. 165
7. 27
6.78
9. 16
1. 20
1. 56
1.96
2. 16
1.87
0.308
0.300
0.675
1.08
1.06
2.36
1. 00
0.98
0.94
1. 22
1.21
1.00
23
S
1.69
0.59
0. 228
14.3
0.0159
0.201
12.6
2.07
1.97
0.582
2.56
0.99
1.05
24
S
1.89
0.53
0.215
17.6
0.0122
0.165
13.5
1.86
1.77
0.482
2. 22
0.89
0.94
25
S
1. 79
0.56
0. 227
15. 8
0.0144
0. 244
16.9
1.96
1. 87
0.608
2.68
0.94
1. 00
26
S
0.90
1. 11
O. 182
0.0438
0.0523
3.88
3.70
0.0166
0.78
1.86
1.98
21
4.15
1. 19
TABLE 6
Hydraulic Model Data
Two-Point Mooring with Weightless Line Connecting Spring Buoys to TOTEM
Wave Conditions
Run No.
0=90°
43
44
45
46
47
48
49
Buoy
T (sec) f (sec)
H (ft)
Pitch Response
X
(ft)
S
P
P/S
Heave Response
f/f
f f
1.03
1.39
2.07
3.37
5.60
-------
Hv (ft\
Hv/H
f/f
1. 06
1.43
2. 12
3.43
5.70
-------
0. 166
0.266
0.083
0.89
1.06
0.34
0.14
0.00
-------
1.54
1. 14
0.77
2. 10
3.50
-------
0.82
1.11
1.64
2.67
4.44
----
f/frh
Wind Direction
3
2.70
2.00
1. 35
0. 83
0.50
----
0.37
0.50
0.74
1. 20
2.00
----
0. 189
0. 250
0. 248
0. 162
0.0555
0.133
0.133
32.5
19.5
9.3
3.55
1.28
1. 28
1. 28
0. 0058
0.0128
0.0267
0.0457
0.0433
0.0104
0.0104
0. 0785
0. 131
0. 105
0.052
0.0174
13. 5
10.24
3.93
1. 15
0.402
0.37
0.37
0.57
0.74
0.96
0.96
1. 35
1.66
0. 201
0.336
0.350
0.312
0. 295
0. 303
0. 144
0.095
32.5
32.5
15.4
9.3
5.6
5.6
2.8
1.85
0.0062
0.0100
0.0227
0.0335
0.0527
0.0540
0.0514
0.0512
0.0436
O. 122
0. 157
O. 105
0. 139
0.096
0. 052
0.052
7.06
12.2
6.91
3. 13
2.64
1. 78
1.02
1.02
1.03
1.03
1.60
2.07
2.67
2. 67
3.77
4.64
1.06
1.06
1.63
2. 12
2.74
2. 74
3.85
4.71
0. 175
0.30
0.358
O. 108
0.075
0.075
0.017
0.008
0. 86
0.90
1.02
0.36
0.25
0. 26
0. 12
O. 12
1.54
1.54
1.00
0.77
1.68
1. 68
2. 37
2.90
0.82
0.82
1.27
1.64
Z. 12
2. 12
2.94
2.68
0.37
0.57
0.67
0. 248
0.362
0.347
32.5
15.4
11.5
0. 0076
0.0235
0.0302
0. 087
0. 139
0.096
11.40
5.92
3.18
1. 03
1. 60
1.86
1.06
1.63
1.91
0. 208
0.275
0.142
0.84
0.76
0.40
1. 54
1.00
1.16
0.82
1. 27
1.48
0. 80
0.93
0. 259
0. 212
8.0
6.0
0.0324
0.0353
0. 105
0.0698
3. 24
1.975
2. 24
2.58
1. 13
2.38
2.64
0.075
0.05
0. 198
0. 27
0. 24
4.0
1.40
1.62
1.78
2. 04
0.0495
0.0785
1.585
3. 16
3. 23
0.033
O. 18
1.98
2.51
--
0=45° Wind Direction
50
51
52
53
54
55
56
57
3
2. 70
2.70
1.74
1. 35
1.05
1. 05
0.74
0.60
0=0° Wind Direction
58
59
60
61
62
63
3
2.70
1.75
1.50
1. 25
1.08
0.88
TABLE 7
Hydraulic Model Data
Two-Point Mooring with Weighted Lines Connecting Spring Buoys to TOTEM
Run No. Buoy T (sec)
f (sec)
f/f
Hv (ft)
Hv/H
f/f
1.06
1.63
1.91
2.38
2.63
3. 23
0.216
0.308
O. 166
0.10
0.067
0.05
0.91
0.91
0.52
0.41
0.30
0. 26
1.54
1.00
1.16
1.40
1.62
1.98
0.82
1.27
1.48
1.78
2.04
2.51
1.40
3.04
3.51
4.30
1.06
2.38
2.63
3.23
0. 208
0.125
O. 092
0.042
0. 83
0.46
O. 35
0.20
1.54
1.40
1.62
1.98
0.82
1.78
2. 04
2.51
2. 24
3.49
3. 16
2. 58
2.38
3.58
3.23
2.63
0. 117
0.042
0.042
0.083
0.43
O. 13
0.20
0.31
1.40
2.20
1.98
1.62
1.78
2.78
2.51
2.04
f/f
), (ft)
S
P
(radians)
P/S
2.39
0.336
0.321
0. 246
0. 238
0. 199
32.5
15.4
11.5
8.0
6.0
4.0
0.00735
0.0218
0.0279
0.0308
0.0397
0.0498
0.113
0.113
0. 122
0.087
O. 105
0.086
15.4
5. 18
4.37
2.83
2.65
1.73
1.03
1.60
1.86
2.24
2.58
3. 16
0. 252
0.263
0. 259
0.208
32.5
8.0
6. 0
4.0
0. 00775
0.0328
O. 0432
0.0520
0.0785 10. 1
2.92
0.096
2. 22
O. 096
1.01
0.0524
0. 0334
0. 0864
0.0560
0.0467
0.096
0.096
0.0785
0. 113
2.87
1. 11
1.40
2.42
H (ft)
Heave Response
Pitch Response
Wave Conditions
np
f/f
rp
nh
rh
Chain attachment collaz located
64
65
66
67
68
69
3
2.70
1.75
1.50
1.25
1.08
0.88
0.37
0.57
0.67
0.80
0.93
1. 13
Chain attachment collar located
70
71
72
73
3
2.70
1.25
1.08
0.88
0.37
0.80
0.93
1. 13
Chain attachment collar located
74
75
76
77
3
1.25
0.80
0.88
1.08
0.80
1.25
1. 13
0.93
0. 267
0.285
0.224
O. 280
8. 0
3.3
4.0
6.0
90
Table'■8
Frequency Response of Buoy S'
Run No. f (sec)
51
52
53
54
56
57
60
63
0.37
0.57
0.74
0.995
1.35
1.66
0.666
1.13
S
P
0.0089
0.02
0.0334
0.0522
0.0512
0.0512
0.0302
0.0498
0.061
0.174
0.227
0.139
0.0697
0.017
0.21
0.096
P/S
Hv/H
6.85
8.72
6.80
2.66
1.36
0.33
6.95
1.93
0.28
0.39
0.31
0.22
0.08
0.05
0.37
0.24