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R-1066
MOORING MECHANICSA COMPREHENSIVE COMPUTER STUDY
I Volume 1IIThree Dimensional' Dynamic Analysis of4 Moored mnd Drifting Buoy Systems
byNarender K. Chhabra
December 1976
C>~ 0
The CAbiarles Stark Draper Laboratory, Inc.Cambridgea, WassachulefsO2439
buoy iReproduced FromBest Available Copy
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MOORING MEC kiv;:1c'5o- A CCJntr-htzrQ tt( j -- •r Au CO. 'u". "
6tudy - Th ree 1',4r.•,..rk1. Dynatnic An ilysi s U -
of *$cor.;:d jr.6 Tli _1 ~I- irr iuoy Systems. d104-5Cl6
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The Charles Stark Dýao.'r Laboratory, Inc.
Cambridge, Mass,3chusct:s 02139 --
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IS SUPPLEMENTARY NOTES
19. KEtY WOROS (Cstinm on reverse V odo Id nlcesr.w•p sen D•dntIIl Ay block a,.ber)
1. Oceanographic Systems Simulations 5. Drogued Buoys
2. Mooring Systems 6. Mathematical Modelling3. Cable Dynamics 7. Computer Simulations4. Buoy Dynamics
a40 Tr•Ac T (Colfno on, revWe*e do It necoee*y and Ide.ntfy by block nsmw *)
Ocean currents and surface waves may induce serious errors in
oceanographic measurements obtained from moored and drifting buoy systems.
A general, computationally efficient solution for the dynamics of moored
buoy systems, free drifting buoys, and drogued buoy systems in three-
dimensional space is described in this report. Time-domain computer
simulations of four specific configurations in various environments are
presented. The mathematical model of one of these configurations, a ..
DD A 1473 EDITION O0 1 NOV S S OBSOLETEy
CURITY CLASSIFICATION OF TNLS PAGE (Whle Does Enteted)
I- Cto*?Y CLAMF4ICA'VtOM OF V13Sg 0WAOKEM440 V. 'f#D)0
".:vhsurface mooritig, was evaluated and improved using full scale ocean test,�ta The modal of the surface mooring configuration studied will soon alqooý oilivahted using recent test data.
C
I
gaIcuniTY CLASIIIIgCATIOW OFr THIS PAOI('tal Dr~al 1Kmte*.
R-1066
MOORING MECHANICSA COMPREHENSIVE COMPUTER STUDY
Volume II
Three Dimensional Dynamic Analysis ofMoored and Drifting Buoy Systems
by
Narend K. Chhabra -- 1 -
December 1976
Approved:' /It
Philip N. BowditchHead, ScientificResearch Department
The Charles Stark Draper Laboratory, Inc.Cambridge, Massachusetts 02139
i 4!
ACK:JQrW1.EGMELNT
T're aut4&r ,•.Q!onnthis op•o7.tmicty to thank all ijtse at the
L~a-les.M1gar).. Ui-avon IADre.tm:r.y, ic who had vail-,able acntriixiticris to
"v. %rk d•crihkd. SpciJ. tan"k:;• arc- due to ýt. John Dahlen for his
!pfiL" gui,:cv k:0 c-i yimmnts dur:,nc, i;rcapraticn of thuis report. Tb
'. Jeffrev Lczc,. fcl his Ihtlp d.r.ino $To.hnmatjcral nodel formulations;
?t. Jinns SchY1t'_n for hi3 help duxing c mputer simulations; and
9v* Willimn Vachmn for Iis comt•nts during the preparation of this report.
Finally, the authc-W w_ ,hces Lo t] lnk Miss Cheryl Gibson and
mrs. Catherine- M!l for aoing such an excellent job in typing this report.
This repoit was preparcd under CSDL Project 53-68800, sponsored by
the ocean Scien-e and T-c•nologýy Division of the Office of Naval Research,
Departme.t of the Navy, through cntract N00014-75-C-1065.
7tve publication of this report does not constitute approvalby the U.S. Navy of the findings or the conclusions herein. It ispublished only for the exchange and stimulation of ideas.
AI ;TIPACT
Ocean currents ind surface waves may induce serious
,rrors in oceanoqriphic mraý3urements obtained from moored
and driftin; buoy :;v;t,.ms. A general, computatioaally
efficicnt solution for the dynamics of moored buoy systems,
free driftinq buoys, and drocrued buoy systems in three-
dimensional space is described in this report. Time-domain
computer siimulations of four specific configurations in
various environments are presented. The mathematical model
of one of these configurations, a subsurface mooring, was
evaluated and improved using full scale ocean test data.
The model of the surface n:oorinq configuration studied will
soon also be evaluated using recent test data.
-V-
TABLE OF CONTENTS
Section Page
].0 INTRODUCTION ................................ 1
1.1 Background .... ................. 3
1.2 Assumptions, Capabilities, andLimitations ......................... 5
2.0 THEORETICAL ANALYSIS - MATHEMATICALMODELS ................................... 9
2.1 Surface Buoys ...................... 9
2.1.1 Spar Buoy ....... ................... 22
2.1.2 Other Shapes ...................... 38
2.2 Mo.)ring Line ................. 39
2.2.1 Continuous Line Formulation ...... 41
2.2.2 Lumped Parameter Formulation ..... 51
2.3 Attachment Between Surface Buoyand a Mooring Line ............... 61
2.4 Window Shade Drogue ......... 65
3.0 METHOD OF SOLUTION ...................... 6r
3.1 Moored System Analysis ........... 68
3.2 Initial Conditions for the Steady-State Analysis of a Spar Buoy .... 75
3.3 Drifting Drogued Spar Buoy ....... 79
-vi-
TA3IE OF CONTENTS (Cont.)
Section
4.0 COMPUTER PROGRAM DETAILS ................. 82
4.1 Surface Moored/DriftingSystems ............................. 82
4.2 Subsurface Moored Systems ........ 91
3.0 CASE STUDIES/SIMULATIONS .................. 97
5.1 Spar Buoy .......... ...... ............. 97
3.1.1 Cylindrical Spar ................. 97
5.1.2 Tuned Spar ....................... 121
5.2 Subsurface Moored System ......... 143
5.3 Surface Moored System .............. 162
5.4 Drifting Drogued SparBuoy ......... 184
6.0 SUMMARY ................................... 193
Appendix
A COMPUTER PROGRAI LISTINGS ................ 196
REFERENCES ......................................... 271
-vii-
NOMENCLATURE
A Acceleration vector.
AD Area used in drag calculations.
AEc Acceleration of point c in the E frame.
A NoA T Normal and Tangential areas for acylindrical body.
AR Relative acceleration vector.
a Point of attachment between the surfacebuoy and the mooring line.
AM Added mass force vector.
c,x',y' ,z Body coordinate system, Frame B.
c Mid point of the mooring line element.
{CF} Array of constant nodal forces.
CD Drag coefficient.
CN,CT Added mass coefficients, normal andtangential directions, surface buoy.
C DNCDT Normal and tangential drag coefficientsfor a cylindrical body.
C DP Pressure drag coefficient for the endplate of a spar buoy.
CDIN,CDIT, CDIA Appropriate dra__constants correspondingto DN, DT, and DA.
Viscous drag force on a body inserted ina mooring line.
DF Viscous drag forcL vector.
DM Moment vector due to viscous drag force.
DN,D-T Normal and tangential drag forces per unitstretched length of the mooring line.
-vi. i-
thDN ,DT ,DA Viscous drag forces acting on the n
n n node of the moorinq line.
dB Mass of water displaced by a differentialdisk of the surface buoy, K" dz'
dM Mass of a differential dish of the surfacebuoy.
F Total force vector acting on the body.
{F A Array of additional forccs and moments.
{F D Array of wave exciting forces and moments.
FF Froude Krylov exciti.ng iorce vcctor.
{FG} Array of hydrostatic and gravitatioralforces and moments.
FH Hydrostatic pressure force acting on themooring line, per unit stretched length.
g Acceleration due to gravity.
1! Length of the surface buoy.
ho Instantaneous draft of the surface buoy,measured up to the mean free surface.
hSpar buoy dru height.
h2 Spar buoy mast height.
hi ho plus the wave elevation component.
Inertia force per unit stretched lengthof the moorinq line.
IFb Inertia force vector of the inserted body.
'xxIIyy Moments of inertia of the spar buoy inroll and pitch about c.
ix ly, z Unit vectors of o,x,y,z.
i,, ,,i Unit vectors of c,x',y' ,z'x y z
-ix-
Wiv, number v,.,ctcr.
K Wave number componrýnt, , cos
K Wave number component, iK' sin .
Y
3tiffness coefficient o" the nth segment.n
" Ul'U•U, 3 Integration limits as defined.
M• Total mass of the spar buoy.
Matrix of tensors of inertia and addedinertia.
ma Added mass of the inserted body, shapeother than cylindrical.
m Mass of the inser;ed body.
mdn Mass of water displaced by the n mass.
mn ,mt Added mass components (normal andtangential) due to cylindrical bodies.
N Number of nodes or segm~ents of a mooring
line.
1.-1B Net bouyancy of the inserted body.
n Arbitrary line segment, or lumped m.ns.
o,x,y,z Fixed coordinate system, frame E.
P Hydrostatic pressure of the fluid.
P- 1u2 dB (z')
p Location of the cuntroid of the differentialdisk of a surface buoy.
"Q= 1U2 dBeKzw(z)i
R Displacement vector.
Roc Displacement vector from point o to point c.
r o f the sppar buoy or the reducedf the moorinr line.
ri oI th 0i: , spa r hucy.
I :*iil 'I tor in the., x-y (horizontal). J,,scribinq; propagatior of the
"r•;•- 't irmnaI area of the surface buoy.
i. :;--7 :t ional ateas oF the tuked spar
s sIat -al coordInate of the rooring line.
-Y.h4 tension vector.",7e E!fec'tiv'e tension variablc along the
!oor:n'1 linlg.
"cme, of the ttension force ;.
TF Te,:sion force vector on a mooring line nor~e.
t Time variahle.
T FluiU velocity vector.
UR Relative fluiO velocity vector.
U4R,UTP, "Normal and tangential components of UR.
V Velocity vector.
Unit vector along the rnoorinq line.
4V, Array of tianslation and rotation rates.
"VBw Velocity, of water particle w in frame B.
"Vb Tangential relative flmitj velocity atbottom of the tuned spar.
"V Ox,V oy,V Components of surface current.
Vs Tangential relative fluid velocity at stepof the tuned spar, also static velocityof the drogued drifting buoy system.
w ~~Lc~caticn o'l -jo 0i
Y U:~mI 21 t: t 1t2.~ r
.,.. t
Xf t. *.(, ) i
Pt c hv -14 fl'. +
-1-
1.0 INTRODUCTION
Oceanographic measurements from moored buoy
systems are contaminated by mooring motion. Numerous
articles have been published in the last three years
pointing out quantitatively the errors introduced by the
surface wave field. Large vertical excursions have been
experienced at great depths by instruments located on
the synthe'ic rope portions of surface-buoyed mooring
lines (WUNSCH and DAHLEN, 1974). Observations during the
POLYMODE experiment showed subsurface mooring lines to
have undergone hundreds of meters of vertical excursions.
For any operational moored instrument system we need two
models whose accuracy can be established with known
confidence: a mooring system model from which the motion
environment of instruments can be predicted, and a model
of the instrument motion response characteristics from
which the zýeasurement error can be estimated. Possepsion
of these models is required for more effective mooring
systems design and for optimwq interpretation of
oceanographic measurements obtained from moored systems.
The first of these two models is the main subject of
this report.
An alternative to the moored approach is the
use of surface-trackable drogued drifting buoys. Such a
buoy system employs a high drag device (or drogue) at
some depth, tethered to a trackable buoy at the surface.
The major impediments in the widespread use of this approach
have been inadequate component and system design. Both
of these impediments can be removed by the development of
dynamic modeling of drogued buoys. A part of this report
deals with the dynamic modeling of drogued buoys.
Dynamic models of drogued buoys do not exist
except in primitive form, while such models of moored buoys
and free-drifting buoys are further along. This difference
stems from the fact that drogued buoy systems are dynamically
very complex, and they have only recently been considered
essential to major programs. Even though many dynairic
mathematical models for moored buoys and free-drifting
buoys are available, not much has been done toward the
evaluation of these models using full-scale ocean test
data. It is the purpose of this report to present a
general, computationally efficient approach for analyzing
moored buoy systems, free drifting buoys, and drogued
buoy systems and to simulat-e this analysis on the computer
so that these models can then be readily evaluated with
full scale ocean test data.
This report constiAutes parts 3 and 4 of a
four part repcrt. Parts 1 and 2 were published in
"-3--
volume 1 and dealt with the "Three Dimensional Static
Analy. is and Desiga of Single Point Taut and Slack Moored
Buoy SystemE." (CIIHABRA, 1973).
1.1 Background
We at Charles Stark Draper Laboratory Inc. have
evaluated one of the mathematical models presented in
this report. The mathematical model of a subsurface
mooring system (Section 2.2.1 and simulation 5.2 in this
report) has been shown previously to predict the mooring
motion forced by ocean currents of periods greater than
15 min.(CHHABRA, DAHLEN and rROIDEVAUX, 1974). The model
was evaluated in a full scale ocean test that provided
experimental data on mooring response and ocean current
forces. The ocean test was conducted jointly with Woods
Hole Oceanographic Institution on R. V. Chain cruise
107. We obtained a record of the motion of an acoustic
transceiver near the top of the subsurface mooring line
at the 500-m depth. The transceiver sent out a sound
pulse every minute and recorded the four return times
of replies from four near-bottom acoustic transponders
at 5460m. By comparison with this and other data from
precision pressure recorders, tensiometers, and
inclinometers, the mooring model had been found to
predict well the observed motions. The r.m.s. difference
betweer the ex!nerlrmntal and predicted trajectory of the
acoustic trans:eiver was 1l.8m, about 10% of the mean
excursion. Tiis mathematical model was evaluated a
second time by data from the central mooring (Mooring
No. 1, Station 431) of the Mid-Ocean Dynamics Experiment
(MODE). In that studiy (CIHIABRA, 1976), the current record
from the topmost vector-averaging current meter on the
central mooring of the MODE experiment was corrected fur the
effects of mooring motion, and rower spectra of the uncorrected
and corrected signals were compared. The correction was
1 cm s-I for that mooring line (vertical excursion
<12 m). Creep in the synthetic portion of the mooring
line was also identified.
In addition, we are planning to evaluate a
second mathematical model presented in this report. The
mathematical model is-of a tuned spar buoy (35 ft. long,
1 ft. dia. at W.L.) tethe red to a subsurface mooring line
by a stiff buoyant line. An instrument line is hanging
from the base cf the spar (Sections 2.1.1, 2.2.2, 2.3
and simulation 3.3). This configuration was recently
tested in the ocean durinc the October 1976 OUR/NDBO
Mooring Dynamics Experiment. A total of fifteen motion
sensing instruments (4 Force Vector Recorders, 6
Temperature/Pressure Recorders, I POPMIP, and 4 Acoustic
-5-
Beacons) were attached along this mooring system. The
evaluation and improvement of this mathematical model
would be done by comparing measured responses with
those computed by the computer simulation for the
measured/observcd environment.
The above mentioned evaluation and improvement
is planned to be completed in CY77. If such an
evaluation and improvement is done, the only mathematical
model to be evaluated and improved from the analysis
presented in this report would be the driftinq drogued
buoys. We plan to do that task in the near future.
1.2 Assumptions, Capabilities, and Limitations
Surface gravity waves treated in this report
have a single frequency and amplitude, propagating in a
single direction. It is assumed that the 4urface buoy
is oscillating in the path of small (i.e. amplitude
of wave train much less than its wavelength) incident
surface waves which are long in relation to the body
dimension in the direction of wave propagation. Wave
direction has been generalized to include any arbitrary
direction in the horizontal plane. For the three-
dimensional analysis of surface buoys, we also -ssume small
displacements of buoy when compared to vertical
dimension of the buoy. Hence only the first powers ofA-
small quantities are retained for the three-dimersionil
surface buoy analysis. Moments of inertia about all
axes in a horizontal plane are taken equal. Even though
moment of inertia about the longitudinal axis of the
spar buoy is negligible, it is included for ntunerical
comp.utational purposes.
In all mathematical models an earth fixed
frame is assumed to he the valid inertial frame. Viscous
drag forces are computed based on the square drag law.
For cylindrical shapes these forces are assumed to act
in directions normal and tangential to the longitudinal
axis; and for any other shape they act in the direction
of the relative flow. For the case of a tuned spar
viscous drag forces and 'idled mass' due to the bottom
b:,se and the step are also included. The ideal fluid
damping (wave-damping) as derived by Newman (1963) for a
spar buoy was found to 1e negligible and hence is neglected.
All mooring lines are ccnsidered elastic.
In the continuous line formulation of mooring lines,
nonlinear elasticity depcndent on the prior loading history
as explained in volume 1 (CHHABRA, 1973) is considered.
For the lumped parameter formul'tion; a linear stress-strain
curve is derived from the nonli car curves for the range
of stresses in study. Dynamic effects on these stress-
-7-
strain curves are neglected. Internal damping forces in
the mooring lines are assumed negligible when. compared
with viscous drag ard stiffness forces. The hydrostatic
pressure forces are treated more rigorously than they
are in the traditional method employed in most previous
studies. As it turned out this new methou changed only
the stress distribution along the mooring system, while
the configuration of the mooring system remained
essentially unchanged. Instruments or buoyancy packages
attached along the mooring lines are treated as
concentrated forces which have length and give rise to
forces as explained in Section 2.2. A time varyinq
current profile of any nature and shape can be inputted
to the computer programs. A list of current profiles
used in various simulations is given in volume 1
(CHHABRA, 1973). In its present form no allowance for
shrinkage or creep of the synthetic ropes is taken into
account, but creep was identified in C1I1iABRA (1976).
Elongations due to rotation of non-torque balanced cables
is also not considered. In the mooring line models
viscous drag forces due to the velocity field generated
by the surface waves are included. Wave-damping forces
are neglected. In the continuous line formulation of the
mooring line, exciting forces exerted on the line by
the wave Eystem are neglected; whereas in the lumped
parameter formulation, exciting forces are taken equal
to the 'Froude-Krylov' forces as it is assumed that the
presence of the mooring line does not disturb the wave
particle motion. For added mass purposes, it is assum-d
for the continuous line formulation that the body
(continuous cylindrical line) motion accelerates fluid
only in the direction normal to its longitudinal axis.
2.0 THEORETICAL ANALYSIS - MATHEMATICAL MODELS
This section derives all the equdtions of motion
pertinent to the analysis presented in this report.
Surface floats are analyzed in Subsection 2.1, mooring line
in 2.2, their attachment in Section 2.3, and a window
shade drogue in Section 2.4. The mooring line analysis
includej all subsurface floats, instruments etc. attached
to the mooring line.
2.1 Surface Buoys
As is well known, the analysis of the wave
induced response of floating *hodies is, in general, a
most formidable task. Initially the action of the fluid
must be decomposed into real (viscous) and ideal (inviscid)
effects. Each effect then must be modeled as to its
interaction with the floating body. Further, it is
convenient to assume that the ocean waves are "gentle"
enough so as to permit first order linear surface wave
theory to seLve as a foundation for the calculation of the
wave exciting forces.
The ideal fluid problem is still so difficult
in general that only the simplest body shapes are
amenable to rigorous analytical solution (potential
flow theory). This solution involves the determination
of the (velocity) potential function P(),t) which rust
-10-
satisfy: 1) Laplace's equation, 2) the kinematic boundary
condition on the moving body suiface, 3) the free
surface (and bottom) boundary condition(s) and 4) the
radiation condition at great distance from the body.
Assuming this potential, 1S,t), has been determined
by some means, the next step is to substitute it into the
2ernoulli's expression for fluid pressure which in turn
4s integrated over the immersed portion of the body to
yield the instantaneous 'or. e and moment vectors. In
principle this is a striight-.orward procedure. In practice
it is hardly ever possible to find a tractable potential
function which satisfies the above four conditions. In a
few notable cases, however, for simple geometries and
small body motions, rigorous solutions have been worked
out. In particular Newman (1963) has derived the linearized
equations of motion for a vertical, cylindrical - ar buoy
responding to tMe influence of a unidirectional wave
train. The axisy'mm.etry of the buoy as well as its
postulated slenderness was greatly ezxploited in the work
to yield manaqable resu]ts. In Newman's work as well as
in othersof co'iparable rigor, the computed ,otential
is the result of an intricate distribution of singularities
(sources, sinks, dipoles, etc.) withir or over the wetted
body surface.
From these studies, it turns out that the linearized
-11-
hydrodynamic forces and moments in general body geometries
may be decomposed into constituents proportional to
body acceleration, body velocity, displaced fluid
acceleration, and the displaced fluid velocity. In
gesieral, rigorous integration of the fluid pressure, both
hydrostatic and wave, over the body surface yields:
1) the 'Froude-Krylov" force from the pressure distribution
due to the undisturbed wave system; 2) the "diffraction"
force from disturbance of the waves by the presence of
the body; 3) the force due to the motion of the body;
and 4) the hydrostatic pressure force. The Froude-Krylov
force and the diffraction force in combination are also
known as the exciting forces exerted on the body by the
wave system. The Froude-Krylov force equals in value
to the product of the mass of the displaced fluid times
the acceleration cf the local undisturbed fluid particles.
The diffraction force and the force due to the motion
of the body both yield "added mass" coefficients
proportional to accelerations and wave-damping
coefficients proportional to velocities of the fluid
and the body respectively. As shown in Chung (1976),
the added mass coefficients are same (opposite signs) for
both the fluid and body accelerations. Also the two wave-
damping coefficients have the same (opposite signs) value.
- i? -
Hence, these two forces may be combined to represent
an "added mass" force proportional to the relative
acceleration (fluid acceleration minus the bo0y acceleration),
and a wave-damping foxce proportional to the relative
velocity. Other forces (apart fron. the fluid pressure
forces) acting on the body are: 1) the weight of the body;
and 2) the actual unbalanced force which accelerates the
body. In the real (viscous) fluid another force, the
viscous drag force, acts on the submerged portion of the
body.
Analysis in this section includes all the above
mentioned forces, as derived from the potential flow theory,
except the following deviations: 1) the ideal fluid
damping (wave-damping) is omitted; 2) the added mass
force is treated slightly differently, and 3) the viscous
drag forces are added on to the general equations of motion.
The ideal fluid damping is omitted here for
expediency. Later analysis of specific problen addressed
in this report showed the wave damping force, as derived
by Newman (1963) for a spar buoy, to he negligible
compared to the viscous drag force on the submerged body.
The added mass force, which is proportional. to the
relative acceleration is separated into two terms; an
added mass force proportiornal to the lonclitudinal component
-13-
of the ielative acceleration, and an added mass force
proportional to the transverse component of the relative
acceleration. Each of these two terms is multiplied by
a different constant coefficient, depending on the shape
of che floating body. These coefficients and the
co-fficient.s used to derive the viscous drag force are
experimentally determineu hydrodynamic coefficients.
As will be shown below, our general equations of
motion reduce to thc ones given in Rudnick (1967) for a
particular value of these coefficients. Rudnick Zn his
analysis used the reasoning of Lamb (1945) who treats
a uniform two-dimcnsional flow across a long circular
cylinder.
Formulation of the General Equations of Motion:
Derivation of the equations of motion for surface piercing
buoys in a train of regular harmonic ocean waves is
given. In addition a surface current (not due to surface
waves) is present. Surface buoys are considered as rigid
bodies with six degrees of freedom. The problem under
consideration is represented schematically in Figure 2.1.
An earth fixed cartesian coordinate system (z positive
upwards) is situated at the undisturbed level of the free
surface. Call it the frame E with origin at o;
z- iziy, ix
w-',• • x,ix
Suriace c /wBuoy
S~x'
Figure 2.1 Coordinate System for SurfacePiercing Buoys
and ix,•y, and i. the unit vectors along x,y, and z
respectively. Frame D is a body (surface buoy) fixed
cartesian coordinate system with its origin at the centroid
(c) of the surface buoy, and x,' vz' being parallel to
.,y,z respectively when the vjoy is in the upright and
non-rotating position. In this analysis R represents
a displacement vector; 7, a velocity vector, and A, an
acceleration vector. It is assumed that the surface buoy
is oscillating in the path of small (i.e., amplitude
of wave train much less than its wavelength) incident
surface waves which are long in relation to the body
dimensions in the direction of wave propagation. Point
p is the location of the centroid of a differential
disk (height = dz') of tho surface buoy (Figure 2.1).
First, the wave direction will be generalized
to include any arbitrary direction in the x-y plane. To
this end let the wavenumber vector K specify the direction
of propagation (Figure 2.2) where:
2A
K =- (cosS ix + sinS i )g
z
y
(wave direction)
x
Figure 2.2 - Generalized Wave Direction
The generalized velocity potential field of the
incident wave system may now be written:
= g ! 'Iz=- e' cos (A - t)
where; S = x + i y; and the wave surface elevation E is
-16 -
given by:
r 0o sin(T.3 - w t)
Here, q is the acceleration due to gravity, and
0 is the amplitude of the incident wave of frequency w.
In the eatth fixed frame (frame E), assumed to be a valid
inertial fra&me for this problem, we define the fluid
particle acceleration vector:
A =i +. PEw - x y y z z
where w is the particle of water next to the differential
disk of centroid p. It is assmed that AEw is constant
over the entire differential disk. Components of AEw at
point p are given by:
X, =dt = g:xo e Cos(R.. - Wt)
= dt 9y = gy0 e cos(T,.' - ut)
d 3 IRjZWand 'Z == glKIlo e sin(i.?J - wt)
where; Kx= IKI cos3; Ky IKI sinR, and the higher order
terms in 0z have been neglected.
z_
-17-
11w; R R + RN~ow; Rop oc cp
Ep R•-op E R -cIE + PEB x Rp
and A Ep = OgC]E + EW'EB]E X Wp+ W E Bxx
where; WEB is the angular velocity of the buoy.
Let us also define the relative acceleration vector
of the fluid particles with respect to the buoy (AR), at
the location of point p as:
ARM AEw A AEP
.i±SO let, ART = (XR " ,)i,
and A ARN -A R T
The vector force equations of motion may now
be written as:
ut : M AEp = 2dB AEw C dB ARN +
(2.1)
CT 1dB ART u f dB +
£ £ 7
where the viscous drag forces and the tension forces (due /to the attached mooring lines) are left out for later
introduction.
Here; 2= -Zc u = H- 7c, u2=ho - :,c and
u 3 = hi - Zc Zc is the distance between c and the bottom
of the buoy. H is the length of the buoy, h0 is the
instantaneous draft of the buoy measured up to the mean
free surface, and hi is h0 plus the wave elevation component.
Distances 7c, il, h and h. are measured along i 1 ,. dMc 0 1
is the mass of the differential disk (Figure 2.1) of
height dz' and dB is the mass of water displaced by this
disk. CN AND CT are the appropriate hydrodyna.mic constants.
In equation (2.1) the integrals in order of their
appearance represent: 1) the actual unbalanced force which
accelerates the buoy; 2) the Froude-Krylov force;
3) the normal component of the added mass force; 4) the
tangential component of the added mass force; 5) the
hydrostatic pressure force; and 6) the weight oi the
buoy. This equation reduces to the equation in Rudnick (1967),
which was derived for a spar buoy, for C., = 1.0, and
CT =0.0.
The vector moment equation can now be written
similarly as: (moments about C.G.)/
-A
-lq-
JdM(Rcp x AE) J IdB(Rcp x TEW) +
+ CN jdB(Rcp x AR/]) +
(2.2)f
C+ 1dB(Rcp x AR) -
- idB(Rcp x g) + eM x
it is also necessary to determine an appropriate
transformation scheme to relate vectors in the E frame
to vectors in the B frame. A sequence of rotations
(Ficiure 2.3) about the body x',y',z' axes (roll =
pitch = '•, and yaw ',) respectively yield the following
transformation.
2 Fcosj cosp sin4, cos: sine, sine - i1X . I+ cosy sino sine -coslp sinO cost
iy -siný co,30 cosil, cos€ý cosý, sin¢ i '-sino sinO siný +sinw sinO cos€ Y
iz, j sin' -cosO siný Cos0 cos' izjL_
The inverse of this transformation matrix is given
by its transpose. The force and moment equations (2.1)
and (2.2) may be combined and written in matrix form as:
Yi,
-'0- ,
/
Ster 1I Roll (;:Rotate y,z about x to obtain X11 y1 , z1
z 1 z1 2
I _ _ _" II j
4 *ylY 2
'1 1
x. X, 21
Step 2 -Pitch (9): Rotate xizIabout yto obtain x'2z
1y y
x
2z 2,z 2b'
Step 3 - Yaw (8: Rotate x2 'y2 about z to obtain x,y1, z'
Figure 2.3 Transformation Matrix Rotations
-2 1-
[M] •- {V = {FD} + {FG: + F (2.3)
Here the matrix [M] is composed of the tensors
of inertia and added inertia. The {V} column vector has
three translation rates and three angular rates. The vector
{FD contains the wave exciting forces and moments. The
vector {F G comprises the hydrostatic pressure and weight
restoring forces and moments (i.e. gravitational and
buoyancy effects); and iFA) is any additional forces
and moments including viscous drag forces and moments
introduced next and tension forces/moments from attached
mooring lines introduced in Section 2.3.
Viscous drag forces and moments- A square
drag law, whcre viscous drag forces are proportional to
the square of the relative fluid veloicty is used. In
Figure 2.1
ow Roc +Rcp + pw
"'"VEw VEc + WEB x Rcp + [fpw]E
If point p and w are overlapping then:
[Rpw]E R pw]B VBw VEw Ec - EB x Rcp
-2.2-
Here; VEw i (V +t + i (0 + ) +*w x ox x y 1, y
1i (Voz + Cz) and VBW is the relative velocity of water
seen by the buoy at the local point p. Her!ce ti,ý
viscous d:ag force on the submerged pcrtion of L-:ie buoy is
given by:
Y= P CD 2CDiVBwAV 3 . (2.4)
where, p is the water density, CD the dreg coefficient, and
dAD the appropriate differential irea. The viscous drag
moment can similarly 1e written as:
5M- ½ CD J ~dDJVBwl R X VBw) (2.5)
Limits of integration for both (2.4) and (2,.5)
are the entire submerged depth of the buoy.
In the remainder of Section 2.1 the equations
(2.1), (2.2),- (2.4), and (2.5) presented above will be
specialized to the specific problems at hand. The
matrices of equation (2.3) will then be derived and
presented for these specific problems.
I
2.1.1 Spar Buoy
A tuned spar buoy (Figure 2.4) will be analyzed
in this section to obtain general spar buoy equations.
Note: Shown for zero rotation and translation.
z
"TT/
[Ih2
w 0h
dz
r 2
X h
hlz y
Figure 2.4 - A Tuned Spar Buoy
mA
C'ilindrical spar buoy equations car then be .,,tained as a
spiecial caso of the tuned spar buny. Loth two-dimensional
ankl three-c1:-i n:;tonal a will be presented. We
will find tKhi ,,uatxJn. <r": on of its response for a
tra'in of surfacc qravity waves h,%viinq a sing!e frequency
an- arnplitude, propaqatinq in any si.gle direction.
V'rjation in wave acc'leration ard velocity over the
-ori-ontal buoy dimensicns are noclected. In addition in
thtŽ three-dimensional analysis, all rotational angles
(roll, pitch, and yaw) are restricted to he snall.
T"w,'o-D-.mensional Analysis: Ir tli. analysis K
is a scaler wit.h 0= ; and . is replaced ',y the scaler
x which is the 6irection of propagation. A Jo, Ew =+ . .
x x z z
RN= (A r ix, )i x1
and the appropriate transformation with 0 = 0, and 0' = 0
becomes
ýci s, i n,-x ,COS9 -sin'2F X 4:
n= C
S\.
-25 -
Now let, [R OCE = ix c + iz zc
EB y
and Rcp z iz.
... A .
"A Ep ix [x= 4 z' 10 cos".-2 sin'I)]
+ i z Lzc + z; (-0 sin('- 2 cose)]
-- 2 eKz•Also, Aw 2. e' [i cos(Kx - tL + i sin(Kx - t%]
I u2Define; Pi i dBlz')' = 0,1,2
u2 Kz
and Qi = dB e (')w i = Ol
Integrals of equation (2.1) can now be written as:
1. dM AEp Mix xc+ z ze] ,
Where M is the total mass of the buoy.
2. 2.= :,Qo-[iQ cos(Kxc -. t) + i sin(Kxc - t)]
2. dB,~ A U . -....~-'~ \
-26-
3 C. dB KX, C 2 f Q [cos(Kx - Ot) Cos
-sin (Kx~. - it) sinOI cosO] +
2"+ p O(z csin9 cosCo - X Cos 0) - P1 cosoTJ} +
"+ CN i 2r (W [Oo (Kx - k.t)sin0 -cos(Kxc t)
2sine cosO] + P 0 ( (: sine cosO - z csin 0) +
"+ P sinfiV9}
4. CT!j dB KIT =CIL OQO 2 csx -ý.'t) sn9+
"+ sin(Kx~ - c't)sine cosO] +
2"+ P0 (-x c sin 0 - 7csinO cosP) +
+ sinOO I T £z( 2Eo 0 cos(Kx - t)sinCcosr +
2"+ sin(Kxc - ,t)cos C] + P 0 (-x csinecosQ -
- o 2 + p .S12
u3 cosK ,A/os
(U g U2 0 1o c
5. .jdB g =gJdB i z + g JdB iz
Also; dB =s pSdz', where S 0 is the cross-sectional
area of the disk of height dz'. Then;
-27-
(U3
dB g iz [p 0g -rS cjtr sin(Kxc -,It)/cose]
2.where; S o2 '
6. JdMg = Mg i z
Integrals of equation (2.2) can also be written
as:
(21. dMcp xEp J yM i yy 0 y
2. x ~) =i~, w rQl[cos(Kxc-wt)cose
-sin (Kx A~) si n9j
3.C4d(cp x~ =C {W 0Q1 [cos(Kxc-!Jt)coso-
-sin(Kx c-,'t)s-inel Picoso xc +
+ P sinu z~ p2 o
4. C ýdB(R xART =0
(U 3 x ) -f2d( x
- i(-Plg sine)
yI
-. -------:7
-28-
6. JdM(Rcp x )= 0
Viscous Drag Forces and Moments: Por a tuned
spar buoy the viscous drag forces are assumed to be
acting in normal and tangential directions proportional to
their respective drag coefficients and areas.
VBw =ix[Vox + 4x -x - Z'cose] +
+i z[Voz + (D - Zc + z'Osinf]
= x [ (Vox + Ox - C)cose
A- (Voz + 0z - zo )Sin6 - z'g]
+ iz'[(Vox + ox " )sinO
+ (Voz + Oz - zc)cose]
AA
Sx VBwx z + z' VBwz'
Drag force components can now be written:
DF, = PCDNjrIVBwxIVBw, dz'
DFz = PCDT TrIVBwzIVBwz, dz'
+ 1 PCTr {r2 IV 2VPDp 1rIVbb + (r 1 r 2 )IVsIVs,
Here; CDN and CDT are normal and tangential drag
coefficients. CDp is the pressure drag coefficient due
to the bottom, and the step, of the tuned spar buoy.
Vs =Bwz, (z'= hI - Zc)
and Vb = VBwz, (z' =-Zc
Also; DM = PCDNIrVBwxV Bwx' dzDNJBWdBwx'i
Using the transformation matrix:
DFx cosO sine DF x1
[DFZ j -sine cos8 [DF z'
Added Mass Force Due to the Step and Bottom Base
of the Tuned Spar: In addition to the forces mentioned
above, an added mass force proportional to the tangential
body acceleration (AEt) and acting perpendicular to the
step and the bottom end of the tuned spar is considered.
Lamb (1945) in his analysis of a cylindrical spar moving
in still water, gives an added mass coefficient equal
3to 4/3pr3. Following representation is used in this
analysis for this force (AM).
S.*.' I.-; , . 'x.:,
I A
-3C-
3 L 1rAM = C ip 1 AEt{ z'=-. .. r-
L 3 (- (2.6)
1-2 Etlz'=hl-: /
where; AEt (AEp'iz,)iz,, and a is a constant.
AM is added to the left hand side of equation. (2.1).
Rcp x AM would be equal to zero.
Combining all these forces and moments; the matrix
coefficients of equation (2.3) can be written as:
M 1 = M + P o(CN Cos 2 + CTsin 2) +
4 yP[:l3+ (r-r 2 3 ]sin2 e
12= PoSin~cosO (CT - CN) +
4 cpLrl3+ (r 1-r 2 ) 3 sinscos'
M1 3 = CNPlCoOe
M = Posin0cosO(CT - C ) +
4 ((;rI + lr,-r 2 ) 3 ]sinecoso
122 2
22= M + Po(CNsin20 + CTCOS 0) +
Qp3[r3 + (r,-r ) 3 ]cos2 s
12
-31-
M 2 3 C CN p sine
M 31 CNp 1cose
1432 -CNP 1sinO
M 33 1 = + C NP 2
{V} =Lx f,z ,O];which is a column vector.
2 CSK, o 2 6 Csn2 e
FDi w Coo osKx-Lt) (l+C~csecsn~
sin(Kxc-wt) sinecose(CT-CN))
F~~ 0 D2 w Q snK -t)(i+C Nsin 2e+c TCos 20) +
Cos (Ixc-,.,t) sinbcosO (C T-C N)] /
F D3 =w 2 OQ fcos(Kxc-'Jt)cosO-sin(Kx - 1t) sine (1+C1 4N
rG1 =0.
F G2 = P0 9 p S02 gCO sin(Kx C-.t)/cose-Mg
FG3 T~gsine
-32-
FAI = CTPlsinO 52 + DFx 4 3 +(rl-r23]7 sine62
Al T I r1 (rr 2 ) ]zic
FA2 = CTPICoseo + DF + a[r3+(rr
(h 1 -Zc) cosOO2
and FA3 = DM
Three-Dimensional Analysis: For small angles the
appropriate transformation matrix reduces to:
* -- "%x x
iz- lJ iz
In this analysis we let CN = 1.0, CT 0.0, and
define:
E c]E ix c + YC + izc
y z
and R =zp z PZ
A i(~+ :' + (v -z:+i/A + z CXC c
and; A E 2 ' eK w [i cosucos(k<'- t) +
sifl:cos (K-- t) + i s in (K2- t)3
Integrals of equation (2.1) can now bc written
a s:
1 d.M Al. (1x = + 1y C+ izC)
2. JdB AEW Q 2 Q 0 iCOS.COS(K< t
i ysin cos(KI- t) + iz sin(k---, t)]
ARN C~: C
x x o
+ *.z +. i )
o-R C p +
V +P -p :7 1 i p '0OC 1 0 C z 0' -
- -. sin (K-S- t
0
i [P 0 - ( F.0Sill - t
Similarly integrals of equation (2.2) can be
written as:
1. IdM(R5c x A ix I 0 + i YI 0Y
where; IY IY = dM7"
- 22. 'dB (Rc Xc AE) ix {-W F 0sin~cos(R~--E.'t) I
2+ I [W CQ costpcos(~.~ )
3. JdB(F~ x ARI ix I- i~o(-`tt+
2+ P 1 7 c + i w {W 0 cos~cosfk*--.q-,-)
- Pl x C- 20 +P 1 Oz ' zP3 {0',' + x
4. J BTc x 'g J Kc x g
i x (-P 1og) + i Y(-P log)
5. jdM(R cp x g) =0
-35-
Viscous drag Fcrces and Moitronts:
VBw x IVox + ýx c p
i[IV +~ *D + zisy oy y ic p
i z Lit oz + 4
x x x- c* oy C. oz C
"+ i y L-v ox +',x +V Y+,p Y-y c+z ý+V p-z -ZJ
"+ i [V e-x- 0-V 04,+ 4+V +(D~ZI OX C Oy .c oz Z- CI
=ix ,V Bwx' + iy V BwyS + iz ,V Bwz'
Hence; DFi = I)DDNJrI Bwx' +VBwy' ( Bwx')z
DFi =CDfr IV 2X + VB 2 (V ~)dz'
PCfrIVBwZ Bwz w
and, DFZ, = DTJ ,TjrVBz VBz dz' +
1 P rr2 +(,2 _r2)vls'I DC1 P 1 IV bIVb 2 r 1V1 5
Using the transformation matrix:
-3•6-
DFX1 1 -y ] fDF ,
DFy , . DFy
DFzj e DFz'
Also DM =f(Rcp xdDF)
- 'x, (DM,) + iy, (DMy,)
Therefore:IDMXJ= e
DM• fl 1M-,
DMz) -j 0
Combining all these forces and equation (2.6) of additional
added mass, we can write the non-zero elements of matrices
in equation (2.3) as:
M11 = M + Po M 22
= 3
M 1 3 4/3 ap[r 1 + (rI-r 2 ) 3] - P C = M3 1
M15 = 1 M 51
M2 3 = Po• - 4/3 ap[r 1 3 + (rl-r 2 )] = M32
M 24 -- - 1 M 42
M3 3 =M + 4/3 ap~r1 +(r-)
45 3 P10
M +=1
M55 1yy + p2
M 6 1 =Ppl M 62 Ple
IV) = [, ,, ,i,)is a column vector.
FD1 = 2 w 2 0 c~s~ncos(Ri-§-wt)
2
FD2 = 2w 2 0o~ sin~cos(R.T-wt)
FD5 2w 2 C Q1 cosacos(KP.,-wt)
S- =
-30-
FG3 = P0g - o So 2 grosin(K'.- t) - Mg
G4 V
FG5 = 'g
and {FA I [DFx, DFy, DFz, DMx, DMy, DMz] is a column
vector.
As can be seen from these matrix elements, the
sixth degree of freedom corresponding to '. does not drcop
out; but M6 6 equals zero. M6 6 is introduced in the equations
for computational purposes. "66 is given by:
M L1T[ 4 h + r 4(1-M6 6 = 2[rl hl + r 2 (1-hI)]
which represents the moment of inertia of a tuned spar
about its longitudinal axis.
2.1.2 Other Shapes
To obtain the velocity potential function
for any shapes other than for simple geometries and small
bedy motions is hardly ever possible. Hence to obtain
any reasonable solution to this problem one has to depend
on emperical approaches. Solution to these problems has
thus been left out of this report. In order to use the
/
-39--•
analysis and computer programs of this report, the readers
will have to substitute their own elements for [MW, [FD},
(FG), and {F A} matrices of equation 2.3; pertinent to the
particular surface buoy in question.
2.k Mooring Line
A mooring line connects a surface or a subsurface
buoy to the anchor. In this report, a line attached to a
buoy but not to an anchor is also considered a mooring line.
Such a line could be connecting a drogue with the surface
buoy or be an instrument line hanging from a moored surface
buoy. In general a mooring line is made of any type or
number of materials (steel, nylon, dacron, etc.) and has
any type or number of instruments (including subsurface
floats) inserted along its length.
The mathematical model of mooring line dynamics
will be formulated in two different approaches. The "first"
approach is called the "continuous line formulation". In
this approach the mooring line differential equations,
with respect to the spatial coordinate s, are integrated
incrementally down the mooring line, to obtain its
dynamic equilibrium at any instant of time. Velocities
and accelerationsof the mooring line differential elements
and the instruments (including subsurface floats) in3erted
in the mooring line are computed by differentiating
I I I I/
/
/
positions found by the dynamic equilibriurms. This approach
was used to model the low frequency motion of a subsurface
mooring system, as presented later in the report. Exciting
forces exerted on the mooring line by the wave system are
neglected and so are the wave-damping forces. Viscous
drag effects due to the velocity field generated by the
surface waves is included. This approach was found to be
computationally inefficient, and hard to solve numerically
(due to the differentiation of positions to find velocities
and accelerations) for the high frequency motion of a
surface moored system.
For the high frequency motion of a surface moored
system a "second" approach called the "lumped parme-ter
formulation" is presented. Here exciting forces exerted V
on the mooring line by the wave system are taken equal to the
"Froude-Krylov" forces, as it is assumed that the presence
of the mooring line does not disturb the wave particle
rn.tion. Again, viscous draa effects due to the velocity
field generated by the surface waves are included, and the
wave-damping forces are neglected. 1.. both approa'hes, a
velocity profile (could-be time varying) can be present
along with the surface wave. Both formulations are presented
in three-dimensions and can be reduced to two-dimensions
when needed.
•. < ... , / .'., -V' ...
-4 1-
2.2.1 Continuous Line Formulation
On a differential element of a continuous moorirlg
line the forces acting are: (a) the constant force due
to gravitational attraction, (bl the variable tensile
forces transmitted from Lhe adjoining elements, and
(c) the variable pressure (normal) and shear (tangential)
forces applied by the fluid. The fluid forces can be
broken down into (1) the hydrostatic pressure force,
(2) the pressure fozce due to the acceleration of the fluid
by the element (the so-called added mass force), and
(3) the viscous drag forces due to fluid relative velocity,
which have both pressure (normal) and shear (tangential)
components. Internal damping forces are neglected in
this analysis as these are assumed small compared to
tensilz and viscous drag forces. Exciting and damping
forces due to the wave system are also neglected. From
Newton's second law of mechanics the above forces should
equal the mass of the differential element multiplied
by its acceleration. By computing these forces the
differential equations for the mooring line dyramic
equilibrium are derived as follcws:
The mooring line is considered to be a cylindrical
slender body. A free body diagram of a differential
element of length ds of the mooring line is shown in
Figure 2.5.
- 12-
+TY ds (T+ -ds +v Ls(+ s2 ( (v + s;2 -s _T
,/
/C
a z
'#T0 3 dsT 12L (vO- LV A-
"-( -s 2 s- 2,s s s
Figure 2.5 Mooring Line DifferentialElement
The internal tension and the inclination of the
line segment change to keep all the above-mentioned forces
in equilibrium. In Figure 2.5 v is a unit vector along
the mooring line given by:
v = iCOSl + iycoS" 2 + zcoS" 3
-43-
'is a tension vector, and U is a fluid velocity vector
(obtained from the current profile and the velocity field
generated by the surface wave). The relative velocity
vector UrR is given -y:
-I
U-R U- Y
where; U = U + iyU + izU'
xx y y zz
* * A A
and, Yc = i Xc +c + i + c is the velocity of point
c on the differential segment.Tl-,e viscous drag forces are computed according
to the square drag law and are assumed to act in normal
and tangential directions to the cylindrical line,
proportional to their respective drag coefficients and
areas.
Let: UR =UTR + UNR
where; UT--R = (v)v
and, UNR =UR- UTR
From UFR and UTR, normal (DN) and tangential
(FT) drag forces can be calcuJated using the pertinent
drag coefficients (CDN and CDT). Representative areas of
the differential element 'ds' can be calculated using the
/'
* * ~
-44-
reduced diameter and stretched length. These are given by
dAN and dAT. The standard formulation is given by:
D-Nds = p/ 2 (CDN ) (dAN) IU-IRIU-FR
and D-Tds = r/2 (CDT) (dAT) IU--IUTR
The constant force due to gravitational attraction
is W = - a z Here, Wa is the weight in air per unit
stretched length of the mooring line. W is resolved into
normal (N) and tangential (WT) components as:
i'j = (W'v)v = - WadScos¢ 3 v
and TN R - T
= - Wads(i 7 - v cos13 )
The hydrostatic pressure force on the differential -
element by the surrounding fluid is given by the weight of
the fluid displaced minus the hydrostatic pressure forces
on the end cross-sections of the element. Or,
A 2*
FHds = (W - W,)ds i - 2r2( - -
a v1 j-v ds (Pb + P01
~ J
I>
where, Ww is the weight in water per unit stretched length,
and r is the reduced radius due to stretch of the mooring
line. Pb and Pt are the hydrostatic pressures at the
bottom and top ends of the differential element. Assume;
(P- P = t g ds cost 3
P b + P t
2
2and Wa -Ww = g r
FH ds = (Wa - Ww)ds (iz - v cos 3) +
pig asj
Here, Pc is the hydrostatic pressure at the
midpoint c of the differential element.
For a continuous cylindrical body it is also
assumed that the body motion accelerates water only in
the direction normal to its longitudinal axis. The
inertia forces due to body's (differential element)
own acceleration and the so called "added mass" term can
now be written as:
------------
"W' (Ila - ww)IF ds =-_adsY- a ds
g g
where; YN = Y - YT
YT (Y.v)v; and the added mass of the differential
element is assumed to be the mass of water displaced by
this element.
Combining all these forces, the force equilibrium
is written as:
AA
3aTds, Dvdsdss F DT ds
as 2 as as 2 as 2
Db•N ds + D-T ds -Wa ds cos" 3 v -
Wa ds(i - v cosý 3) +
(2.7)A A
(Wa - Wl)ds{ (i - v cos 3) +
I-I ds ..14 + Wa - t , . .Pc l ds ( +VfY) - ds YN=O
rg Dsjg g
raTJ + Waor, hi v = (-DT + Wa cosc 3 + - YT)v (2.8)
g
-47-
P (W~ WW,)j -OVand T+ .g DJ + -w DN + -'of4 +
.9 s
(2Wa - W .
YN (2.9)g
Also we can write:
. v (2.10)as
I'Bre T, the tension, v, the unit vector along
the mooring line, and 7, the geometric displacervent vector
are the dependent variables of interest. Equations (2.8)
and (2.9) can be simplified if we define a new variable
called the effective tension (Te) similar to the one
described in GOODMAN and BRESLIN (1976).
Te = T + Pc (aWw) (2.11)P9
Differentiating Te with respect to s
3Te aT Wa - Ww aPc
as as 9 as
aT- ( - Ww)cos3 (2.12)as
Substituting (2.11) and (2.12) in (2.8) and (2.9) we
obtain:
aTe W- = - DT + Ww cos 3 + a (2.13)as g
and Te - = - DN + 1w(iz - v cost 4 ) +3s
2Wag Ww (2.14)
Equations (2.13) and (2.14) are derived for a line
of an arbitrary stretched length and of conseauent reduced
diameter. Also we assume, as reasoned in GOODMAN and
BRESLIN (1976) for materials obeying 1,ooke's law,that the
effective tension and not the actual tension controls the
extensibility of the mooring line. The extension of the
mooring line is caused by (1) pulling on the line due to
tension and (2) squeezing of the line due to hydrostatic
pressure. Volume I (CHHABRA, 1973) plotted non-linear
curves between tension and elongation of various mooring
cables and ropes. Using the effective tension instead
of the actual tension, mooring line stretch and the
consequent reduced diameter from some initial reference
state are found as discussed in Volume I.
-49-
The analysis also allows for an arbitrary
number of intermediate bodies such as sensor packages
and subsurface floats inserted in the mooring line. For
this case equations (2.13) and (2.14) are replaced by:
Ten = Tel + F (2.15)
Here, F is the summatior of gravitational
attraction, fluid, and inertia forces of the body inserted
between n, the differential element above the body, and
n+l, below it.
For cylindrical packages, the viscous drag
representation remains similar to the one for cylindrical
mooring line. For a spherical or any other shaped package
this viscous drag force is computed as:
DA = p/2 CDb ADb jURIU-R
where; CDb and ADb are the appropriate drag coefficient
and area.
The gravitational force and the hydrostatic
pressure force are combined to give:
NB =- Wwb iz
where; W w is the weight in water of the inserted body.Wb
Inertia forces for cylindrical packages are
changed slightly from that of a continuous cylindrical
mooring line. Uere the assumption of the body motion
accelerating water only in the direction normal to its
longitudinal axis is dropped. Instead two added mass
terms; one normal and the other tangential to the
longitudinal axis are used. For cylindrical packages
inertia force is gi',cn by:
IF = (Mrb Y ÷ mnb YN + mtb YT)
and for any other shape this force is:
IFh = (mb + m ab) Y
Here; m. is the mass, fhb and mtb are the normal and
tangential components of added mass for cylindrical bodies,
and mab is the added mass for any other shaped body.
Summation of gravitational attraction, inertia, and fluid
forces gives F. Integration of equations (2.10), (2.13),
(2.14) and (2.15) along the spatial coordinate, s, of
the mooring line gives the dynamic equilibrium of the
-51-
mooring system at discrete times. At any time these four
equations can be solved for positions, W (t), of the
mooring system. Positions Y are calculated at short
enough time intervals, Ats, so that velocities Y and
accelerations Y during these time intervals can be
computed as piecewise constant. Accelerations are neglected
at t 0 and tI, velocities are neglected at to.
2.2.2 Lumped Parameter Formulation
In this formulation, the mooring line is reduced
to a discretized dynamic system which is solved by a lumped
parameter approach. All forces acting on the mooring line
and the inserted packages are transferred to a fixed
number of nodes (say N). If each node has q degrees of
freedom then N x q simultaneous differential equations, q
for each node, describing these nodes can be solved
simultaneously by using any of the numerical integration
techniques. Once again the forces transferred at each
node (lumped mass) are the same as listed in 2.2.1, with
the exception of Froude-Krylov forces whiich are not
neglected here. The forces for the nth mass for a N node
system are derived next.
Consider the discretized dynamic system as
shown in Figure 2.6, where an N-mass system is shown. In
this figure v n is a unit vector along the nth segment
., \
-32-
"AY N-1
p n+i
nfln
in 1 n
m1 m
z2
I
Figure~~~ 2.2-asSse
-53-
(below the nt" mass). This can be written as:
V = i coSi + i cost2 + i cos3n x In y 2n z ýý3n
Tn' T!+lare the tension vectors, and Un is the fluid
velocity vector at the location of the nth mass. A
typical mass could consist of cylindrical components,
spherical components, and any other shaped components.
The transfer of all adjacent forces to a typical nth mass
is done as explained below.
Viscous drag forces acting on the mooring system
have to be transferred at '11' nodes. Any scheme devised
for this transfer should take into account the different 7
behavior of cylindrical and spherical cor.mponents of the
mooring system. Let DNn be the drag force transferred
to nth node which is equivalent to contribution of normal
drag forces on cylindrical components lumped at the nth
node. Similarly, D-Tn is the drag force on nt" node
equivalent to tangential. drag forces on cylindrical
components lumped at the nth node." DA n is the drag force
acting on the nth node in the direction of the relative
velocity URn' and is a contribution from spherical or
any other shaped components lumped t the ntb node. These
forces are given by:
-54-
D-Nn (CDIN)nIU-RnI U-n
D-T= (CDIT) nTU-R I UTRn n, n n
DA = (CDIA)nJU-RnI iU-Rn
and f 1- + D +T (2 16)n n n n
Here; n is computed from the current profile and tne
velocity of water due to surface waves at the location
of the nth mass which is exponentially attcnua$--d
with depth. CDIN, CDIT, and CDIA are the appropriate
drag coefficients multiplied by respective areas.
The gravitational forces and the hydrostatic
pressure forces are combined for all the transferred
components to give:
N-•n = wn iz (2.17)
Inertia forces for the components, in general, are given
by:
IF n ( n + man) Y n - nn Yn (2.18)
-mtn YTn
-55-
This representation is similar to the one used
for inserted bodies in the continuous line formulation.
The tensile forces acting on the nth mass change
in direction as well as magnitude with the motion of the
nodes. The direction can be obtained by Yn :nd Yn-1
(refer Figure 2.6); but as the present problem is one
of large displacemcnts the magniturle is determined at each
integration step by updating its value at the beginning
of the time step by the stiffness force due to the
incrementalduforrmation during ths timr step. Let the
incremertalchange in length of the nth segment ALn be:
Ln = n - Yn-1
For small AYn and !Yn-1;
ALn L -n.V =
n n-
-cosi in- cos 2n- COS 3n COS lncosP 2ncoso 3n] JT1-1
If k n (stiffness coefficient) is defined as the
force required at node n for a unit displacement along Vn;
then the change in tension magnitude ATn is:
ATn = k ALn
*/ 4
/,-56 -
or; Tn =(Tn -A•Tn)vn + ATnVn
Also, as tension vectors Tn and Tn+1 are actingth
at the n mass, we have:
TF =T - Tn (2.19)
Froude-Krylov force on the submerged 'mass n' is
assumed to be given by the product of mass of water
displaced by the 'nth mass' and the acceleration of water
at that location which is exponentially attenuated with
depth. Or,
IKizn _ _-FF = mdn gý e Ii xK xcos(K-rwt) +
A A
i yK ycos(K-wt) + izIjfsin(K.&S-wt) (2.20)
where, Mdn is the mass of water dispiaced by the n~h mass.
Combining equations (2.16) through (2.20) for all
nodes and taking into account all the boundary conditions,
the force balance reduces in matrix notation to:
LM]{Y} = {DFI+{CFI+{TF}+[K]{AY}+{FF} (2.21)
,7
- 37-
where; [M] is a matrix of appropriate inertia terms.
[K] is a matrix of stiffness terms.
{TF} is an array of tension components at the nodes.
{DF} is an array of appropriate viscous drag terms.
{CF} is an array of constant nodal forces(gravitational and hydrostatic).
{FF} is an array of Froude-Krylov forces.
{Y1 is an array of accelerations at the nodes.
and {AY} is an array of incremental deformations ofthe nodes.
derivation of Matrices: In equation (2.21), the
order of each matrix and array is equal to the number of
nodes (N) multiplied by degrees of freedom (q = 3) per
node. For simplicity the masses were assumed to be lumped
at 'N' nodes, in which case the off-diagonal terms in the
mass matrix are zero, i.e., force at node 'i' due to an
acceleration at node 'j' equals zero. A consistent mass
matrix analysis can be done to allow for distributel mass.
Mass Matrix - [MJ
14 3
[M] =rn
MNl1TMNJ
- Z8-
where; MI, M2 . . . .. MN are 3x3 matrices given by:
+nn i2in (fltn-mnnnCOSinCOS2n j (itn-m )coS~inCOS43nnfmain 3n2112
M +M tn C os 2 In"
M (n (Mt-i n)cosý 2 cos n 'n +M a+rnnsin ~2n (M~ -M IOn (tn-nn S2n°in, nman n 22n tnm nn)C°S 2nCoSý 3 n
+M Costn 2n
I , i 2
(Mn Mi )cosý COS (M Cssi2tn- nn 3nSin tn- nn )COS3nCos2n mn+man+mnn 3n
S2MI tncS 3n
th
m = mass lumped at the nth node.
man = added mass at nth node due to non-cylindrical components
and mnn'mtn = Normal and tangential components of
added mass at nth node due tocylindrical components
Stiffness Matrix ] - 4 Mass System
IlK1 + K2 K2 0 I --- K 0
.K2 K2 + K I -K 3 0L':] = \ LF--K3 K3 +K -K4
4 4T -i
-39-
where; KS are 3Y3 matrices whose elements are given by
SK -k cs. Cs
13 s IS j,s
ks = Stiffness coefficient for segment s.
and csi's = cos~i of segment s.
{TF}, {DF}, {CF}, {FF}, {Y} and {AYI are column
matrices. These matrices can be written as
-T- AT ) CS1 , + (T2 AT2 ) CS 1 2I 1F-(TI - AT1 ) CS2 , 1 + (T 2 - AT2 ) CS2 , 2 FF2, 1
-(T 1 - AT1 ) CS3, 1 + (T2, - AT2 ) CS3, 2 F3, 1
{TF} [FF=
-(TN I- ATN,_I)Cs3,M 1 + (TN- ATU)C-3,N
- (TN - ATN)CS 1,N FF,1,
- (TN - ATN)CS2,N FF2,N
- (TN - ATN)CS3,N FF3,E 4
/o,
-60-
fDF CF Y AYDFI, CFI,I iYl~Y,
DF2,1 CF2,1 Y2,1 AY2,1
DF3, 1 CF3,1 Y3,1 AY3,1
{DFI = . ; {CF} ; {Y} = . ; {fY} =
DF CFI, Y AY1,N1, . 114 1,11 ,
DF CF., Y A2,N 41,. 2,N2,
DF2N CFY AYDF3,N CF3, N 3,N 3,N
Equation (2.21) can be re-written as
{y} = [M- {TF} + [M]- [K] {AY} +
[M]-I {DF} + EM]-I {FF} + [M]-' {CF}
These 3N second-order differential equations can be
reduced to a set of 6N first-order differential equations
which can be solved by any of the numerical methods.
-61-
2.3 Attachment Between Surface Buoy and a MooringLine
Figure 2.7 shows the attachment between a surface
buoy and a mooring line. Let the attachment point be 'a',
which is fixed rigid1ly to the buoy.
z
y
x
-T~uyadaMoigLn
Figure 2.7 Attachment Between SurfaceBuoy and a Mooring Line
In addition to all the other forces acting on the
surface buoy as outlined in Section 2.1; a tension vector
T is acting at point a, which should be included in the force
balance.
T= Tv =Txix + Tyiy + Tziz (2.22)
S / "
-.62-
also let vector Rca be given by:
AA A
+- 'I I + I
Rca = 1 xa y a Iz' Zaa
Taking moment of T about point c we obtain:
TM =-Rca x T (2.23)
for inclusion in the moment balance.
Two Dimensional Analysis:
A A
Rca ix, xaI + iz, za
A A
= ix (xacos + Za'sine) + i z (za 'cos - Xa 'sine)
and; M = i [T (Xa 'cos + zisinO) - Tx(za 'cosO - Xa 'sin)]
Three Dimensional Analysis:-
Rca ix(xa- ýy' +eza) + iy(x' + Y! Oz!) +
iZ(-Ox + "'+z!)
-63-
And; T-M i iT (-Sxa+ -y• +za) Tz(ýxa+ya-"a)] +
i[Tz(Xa-ya+Oz') - Tx( eXa÷"yaza] +
iz[Tx(Xa+ Ya- za) - T y(xa- -Ya+ 6Za)]
Change in mooring line tension magnitude due to buoy rotation
(coefficients in the stiffness matrix).
Two Dimensional Analysis:
Rotate Rca through AO to obtain Rca1
c La 1] L-sinO Cose (za
For small rotations change in tension magnitude AT is given
by
AT= k(R Rca).v(ca 1 c
where k is the stiffness coefficient for the line segment
attached to the surface buoy.
If v= ixcos4 + iz sins
AT = k[cosj,(za cosO-xa' sinO) + siný(-za sinO -a a a
x! cose)]Ae
N'
-64-
Three Dimensional Analysis:
' 1ca
1 =- i K I a)
and;
°s2( a-Ya-z')ýc~s'al(-t-Xa+!*Ya'+Zal°Sl-•x-'+ a •a SaAT=k L
(- l+ZaY' Ozcosco" (Xa'-'Ya+Oza )L 3 a+cos2a a)
L_
If more than one mooring lines are attached to the
surface buoy, as is the case of simulation study in Section 5.3,
force and moment of both these lines are to be included
in the appropriate equations. It is important to note that
within the definition of the coordinate frame (fixed at
the anchor) if a mooring line hangs from the buoy (as is the
case in 5.3), - T of this section would be replaced by + T.
Also in this case .T would be given by
AT =-k(Rca -R ca)'v
ca 1
-65-
2.4 Window Shade Drogue
Sometimes an alternative approach to mooring
surface buoys is the use of surface trackable drogued
drifting buoys. Such a buoy system employs a high drag
device (or drogue) tethered to a trackable buoy at the
surface. A drogue is subjected to the same forces
(including the wave effects) as a differential element
of tne mooring line. A special case of drogues, the
window shade drogue, as shown in Figure 2.3 ic considered
in this analysis. The drogue, modeled dynamically as
a lumped parameter system (Figure 2.8), is reduced to
strips. The mass, the acting forces, anO the dynamic
behavior of each strip is lumped at an assigned nodal
point. Nodal masses are linked by suitably elastic lines.
The window shade drogue is a compliant sheet.
Close observation of scale model drogues (Va:hon, 1973)
has revealed that it seems nearly locked to the local
water mass insofar as motion normal to the -heeL is
concerned. However, in the tangential directionpit seems
to slip through the local water mass rathei easily. The
forces were therefore described as follows: the wave
forces (Froude-Krylov force; as it is assumed that there
is no diffracted wave) acting on a material strip are
estimated by calculating the force normal to the axis
/ / . /
-66-
XV
Surface Buoy
Tether Line Window Shade
ruDrogue
Attachment Point
l /
Ballast Weight
Figure 2.8 Drifting Drogued Buoy System
-67-
of revolution of a hypothetical cylindrical element, the
diameter and length of which is equal to the breadth and
length of the strip respectively. This force is given
by the mass of the hypothetical water cylinder multiplied
by the component of water acceleration normal to the
cylinder'slongitudinal axis. Or;
FF mn[AEw Ew Vn )vn] (2.24)
where w is the particle of water next to the nth node,
and mdn is the mass of the hypothetical water cylinder.
For inertia forces man and mtn equal zero; whereas mnn
equals mrdn. The viscous drag of this strip in the normal
direction is calculated using the frontal area of the strip
and the drag coefficient measured in water-filled quarry
tests (Vachon, 1975). The elemeptal strips of the drogue
are assumed subject to tangential viscous drag calculated
using the area of the strip and a tangential drag
coefficient measured during drop tests in the ocean
(Vachon, 1975).
Gravitational, hydrostatic pressure, and tensile
forces on a drogue strip are similar to the ones outlined
in Section 2.2 for a discretized dynamic system.
I•.L,,"
-68-
3.0 METHOD OF SOLUTION
Section 2 outlined all the necessary ingredients
of the mathematictal models used in this report. In this
section some details of how to use these ingredients, to
obtain a solution for a specific problem on hand, are
given. For illustrative purposes a specific case study,
of a spar buoy at the surface and a mooring line connecting
this buoy with the anchor, will be discussed. Method
of solution for both the two-dimensional and the tbree-
dimensional analysis of the moored system will be outlined in
3.1. Section 3.2 outlines the approximate initial
conditions for the steady state analysis of a spar buoy
freely floating in a surface wave.
Another case study of a spar buoy at the surface
and connected to a window shade drogue at a given depth
is discussed in section 3.3.
3.1 Moored System Analysis
Two-Dimensional:
In this case, the current profile is in the same
plane as the train of surface gravity waves having a
single frequency and amplitude; anQ its direction parallel
or anti-parallel to the direction of surface wave
propagation.
A
-69-
Static Solution:
For the static solution, the analysis is started
at the spar buoy. The pertinent static equations for the
soar buoy (Section 2.1) with an attached mooring line
(Section 2.3) are:
DFx - Tx = 0 (3.1)
POg - Mg + DFz -Tz =0 (3.2)
and -Pig sin" + DM + T (x' coso + za sinO) -a
- Tx (za' cosO -j sinO) =0 (3.3)
IPere; we hove three equations and four
(Tx, T, e, hs) unknowns,where h. is the static draft of
the spar buoy. Now if we assume a value for hs, the
other three unknowns can be solved by equations(3.1)
through (3.3). The solution for the three unknowns T
Tz, and 0 is found by iterating on the pitch angle 0.
If 0 is known; DFx, DFz, and Dri can be computed explicitly.
P0 and P1 can be calculated if hs is known and thus Tx
anO Tz are given by equations (3.1) and (3.2) respectiqely.
Moment balance can now be performed by using equation (3.3)
-70-
to obtain the error. This eiror is minimized by iterating
on the pitch ang'.e 0.
Having solved the buoy force and moment balances
for a assumed hs the mooring line can be analyzed next.
For the continuous line formulation, a numerical
integration is carried down the mooring line by dividing
it into a finite number of segments. In this representation
equations (2.13), (2.14), and (2.10) for two-dimensions
can be written as:
ATe = (-DT + Ww sinf)AS (3.4)
TeA¢= [DN + Ww cost]AS (3.5)
and AY =ix's.S cosp + i AS siný (3.6)
Havi.ng found T,, and Tz (or T and •) at the top
of the mooring line equations (3.4) through (3.6) are (
used to find the. configuration and the tension distribution
along the rioori:ng lJne. At the enl of the mooring line
the bounda.iy condition of the ocean floor depth should
be met. if not, the uhole process is repeated with a new
hs; thus computing the correct value of hs for the given
bound&ry condition. Whenever the mooring line is
-71-
discontinued to insert an intermediate body, the
discontinuity is taken as a lumped mass and resolved as I7
done below for the lumped parameter approach.
For the lumped parameter formulation, the
configuration an,' the tension distribution along the
mooring line can be found by (Section 2.2):
Tn T n+1 - Wwn iz + DFn (3.7) o.nw\
where; n is the tension vectot below the nth mass and
1n+l abcve it.
Dynamic Solution:
Equations of Section 2.2 which area written for
three-dimensional analysis can be easily reduced for the
two dimensional problem. Using the lumped parameter
approach for the mooring line; eauations (2.21) are
combined with equations (2.3), (2.22), and (2.23) to
obtain a global matrix equation for the mooring/buoy
system. Stiffness coefficients for the change in mooring
line tension magnitude due to buoy rotation are obtained
from Section 2.3. This matrix equation can be solved
in the time domain using various numerical integration
techniques. Integrals of Pi and Qi, given in Section 2.1,1 1
-72-
are computed exactly; whereas the integralo for viscous
drag forces and moments DFx, DFz, and DM are computed
numerically by dividing the spar buoy in disks of height
dz'. The composit matrices are of order 2N + 3. Matrices
[M], {TF), {KW, [DF}, {FF1, {FD}, {FG), and {FA) are
computed at each time step of the integration.
For the continuous line formulation; the matrices
of equation (2.3) combined with equaticns (2.22) and (2.23)
can be solved for T in equation (2.22) if the buoy draft
h is known. Then integration of equations (2.13),
(2.14), (2.10), and (2.15) along the spatial coordinate, s,
of the mooring line gives the dynamic er-luilibriun of the
mooring system at discrete times. Iteration on the
unknown h0 may be necessary to meet the boundary condition
of the ocean floor depth.
Three Dimensional:
Here; the train of surface gravity waves having
a single frequency and amplitude, can propagate in a
general direction. The current profile is also three-
dimensional and can vary, in magnitude and direction, with
depth.
/
-73-
Static Solution:
The pertinent static equations for the spar buoy
(Section 2.1) with an attached mooring line (Section 2.3)
are:
-JdBg + [d~fg + - =0 (3.8)
and; -JdB(Rcp x g) + DM + TM = 0 (3.9)
Integrations for dB are over the static draft of the
buoy, and integration for DIA is over the length of the buoy.
Equations (3.8) and (3.9) can be written as:
Pog iz -Mg iz + DF- T =0 (3.10)
and { = [S]- {F} (3.11)
where; [S] is a 3 x 3 matrix with components qiven by:
S Fg PoZ 1 + DFz£2 - Ty - Tza
11 z-r y Tz
-74-
S =TX ; S -Tx'12 yXa S13 za
T';= T ' "'
$21 Txa ; S23 Tzya
$22 = Pogi 1 + DFz£ 2 - TxXa - TzZa
$31 - DFx2 + TxZa
$32 = - DF y2 + T yza
S33 = T Tx'a - Tyya
and {F} is a 3 x 1 array
F1 - DF y 2 + Ty za - Tzy
F2 = DF Z2 - Txza + TzXa
F 3 = Ty ' - TX
also Po = fdB; and Vi' 12 are the distances between
center of gravity and center of buoyancy and center of
N'
-75-
drag respectively. For the static case we also assume
(small angles)
DF X IVBwx VBwx
DFy XIVBwy IVBwy
and DFz QiVBwzIVBwz
.. T and ý, 0, and ý can be solved explicitly from
equations (3.10) and (3.11) if hs is known. hs is
determined by iteration on the boundary condition of the
ocean floor depth.
Mooring line solution is similar to the one
presented for the two-dimensional analysis.
Dynamic Solution:
This dynamic solution is similar to the one for
two-dimensional analysis except that the order of
matrices in this case is 3N + 6.
3.2 Initial Conditions for the Steady-State
Analysis of a Spar Buoy
Two-Dimensional Analysis:
Let CN = 1.0, CT = 0.0, and xc = 0.0. Also
assuming small pitch of the buoy and neglecting viscous
-76-
drag forces, the equations of motion for a spar buoy
become:
2Mkc + P1 = 2wý20oo coswt (3.12)
M*zc + OSo2gz = - 2 E sinwt
+ PSo 2 g~o sinwt (3.13)
and Plxc + (Iyy + P2 )e + PlgO =,2w2 &0Q coswt (3.14)
where, M* = M + 4/3 p[r 13 + (rI-r2 ) 3
Surge Equation 3.12:
Let P1 < < 2Mxc and x = C1 sinwt + C2 coswt + C3 .
Now differentiating x twice and substituting in (3.12) we
obtain
x = -roQo coswt/11 + C3
and x = wýoQo sinwt/M + C3 t
Hence x(o) = 0
and x(o) =- oQo/M
,'0.. 0
-77-
Heave equation (3.13)
Let 7 = C 1 sinwt + C 2 coswt + C 3
Differentiating twice and substituting in equation (3.13) we
obtain:
C2 = C3 = 0
and C 1 = Fo(pSo 2 g w 2Qo)/(PSo2g - M*w2
Hence, z(o) = 0
•(o) = 1
Pitch Equation 3.14:
Substituting Rc from equation (3.12) into
equation (3.14) we obtain: do
AO + Be = C coswt (3.15)
where, A = (Iyy + P2 - 2/2M)
B = P1g
-78-
and C (2,, 2 oQi- P1W oQo /M)
Now let (i C1 sinwt + C2 coswt + C3
Differentiating 0 twice and substituting in equation
(3.15) gives:
C= C3 0
and C, C/(B-Aw2)
COr, 0 2 coswtB- Awo
Hence, W2 0 (2QI-PIQo/M)
0(o) = 2
and 6 (o) = 0
Three Dimensional Analysis:
Similarly; for the three dimensional case with
wave direction at an angle 8 with the x-axis we have:
-79-
x(o) = - r,° coss Qo/M
y(o) = - ýo sin6 Qo/M
z(o) = 0.
X(o) = y(o) = 0
-(O) = •wo(PSo 2 g-w 2Qo)
PSo 2 g-M*w2
w' & sin$(2'Q-lomý(O) 2 2 2 1 P1 0 M
2
22
Plg-W2 (I yy+P2-Pl2 /2M)w2 o cosS(2Q1 -PlQo/M)e(o) - Pl-7 (y+ 2-2/M
1P (o) = 0
*(o) = (o) 0)= 0.
3.3 Drifting Drogued Spar Buoy
In this analysis the drogue and the mooring line
connecting it to the spa# buoy are modeled dynamically
as a lumped parameter sy tem.
-80-
/
Two-Dimensional Analysis
Static Solution:
The pertinent static equations for the spar buoy
(Section 2.1) with an attached mooring line (Section 2.3)
are:
DFx + Tx = 0 (3.16)
Pog - Mg + DF + Tz =0 (3.17)
and -Pig sine + DM - Tz(x' cose +a Z sine)
+ T (z! cosO - xa' sinS) = 0 (3.18)
Here; we have three equations and five (Vs, hs,
Tx, Tz, 8) unknowns where Vs is the static velocity of
the drogued drifting system. These equations are solved
by iterating on Vs and 8. Assuming a value for Vs, Tx
and Tz can be calculated by known weights, buoyancies,
viscous drag forces, and elastic characteristics of the
system beneath the spar. Now if a value for e is assumed
equation (3.17) is solved for hs and the error in
equation (3.18) is found. This error is minimized by
iterations on 8. Next Vs is iterated upon to satisfy
equation (3.16).
-81-
Dynamic Solution
This solution is similar to the lumped parameter
solution discussed in Section 3.1.
Three Dimensional Analysis
Static Solution:
Here the pertinent equationo (static) remain the
same as Section 3.1 expect for replacing -T by +Y and + TM
by -TM.
Also; following similar reasoning of small angles
T, o, e, and ' can be solved explicitly from equations (3.10)
and (3.11) with new T and TM if Vs is known. V is
determined by iteration on the equation similar to (3.10).
Dynamic Solution
This solution is similar to the one in Section 3.1
\
__ _ _ ... . ... __ _ __ _ _ __ _ _ __ _ _ __ _ __ _ _ __ _ _ __ _ _ / |
-- 2
-82-
4.0 COMPUTER PROGRAM DETAILS
This section will describe the various computer
programs written for the theoretical analysis presented in
Section 2 and using solution methods of Section 3.
Description of the prograrsis general and specific details
can be found in the program listings presented in Appendix A.
All computer programs are written in FORTRAN IV and have
been run for many case studies on the CSDL AMDAHL 470 V6
computer. Innut data required for these programs is also
explained in this section. Some of the simulations, along
with the accompanying input data, are presented in Section 5.
Computer programs written for tne surface moored
and drifting systems are tabulated in Table 4.1. Programs
for subsurface moored systems are tabulated in Table 4.2.
Program listings for some of these computer progra;ms are
presented in Appendiz: A. Listings not preserted are duplications
of what is presented and can he obtained after minor modifications.
4.1 Surface Moored/Drifting Systems
Computer program SD3.FORT is coded for the
three-dimensional analysis of surface moored and drifting
systems presented in Section 2. Four case studies can be
simulated by this program:
1. A freely floating spar buoy.
-83-
z . 0 40 4. 014 .
C C:0 0 IQ j( :
Z() 0. 0'4 0-4 0 -q 0-400 9 ( dU) 4-4 inU4-4 0tl4-) U) L M4-4 0Cfl4-
r_ 41 C4.)
t" w 4-4 1ý4 2 En.4 f" .4 w 44E
z- rl. :) 0 :3 :3 0 :3 11 0 : V 0 : Z 0 :
UU i ' 4 ) u u '
-4 0 -4 o -4 o C. 0 4 0
r_04.) 0 04 -r.- 0' r r00 0 no 0. )0 Q40)0 00
o 0 0 a)2 0 0 0) 0V)4 t&4E 4 ) tw (4 (4
-'- ý 0
U) 4 41 (r. 4-1 4)AI I~. I
4-4
ý4 0 00 4 50 0 0 0
C.) to
Ci)~U t 'i1 i)
r. t - v-H -0 It' 1.4 g- 1045
co 24 a ti a r C4 w iCi) ty fu-4 >1ý . , r
C) s: a 0CflQ40 (
:1o' o 00 4-
(n to P, § -4 %4 ( 4 En4-' $4
11 0 i4 , 4 0 " 0~- w .4 0 E4z $4 0 0 -4 >4-i ~0 -10-4 Q):5$qt 0 L4
-4 -4 "-A > ~ 4~ v -moka l -r 4 -41-0 U :,T
Ic '0 4 0 W tP 4J -A 0T3 C: nI) I- c .1- r
"a41) '-4 c H -4 -H E; 4-4 -4 -4*.0-4 -4 00 U4-4-40 r- 0C
L) LiU JU 44~i E_4 0 4
0-
00
clA
.1)'
u C'
-4 0.
'ELn
C) 1).4~
4J AA
o ~ U) 0
'-4 44 41
U) 1))
IxI00
0 54 0
U:U
C))I-. 1. ~ 4 I-
-85-
2. A spar buoy anchored with a lumped
parameter mooring line.
3. A spar buoy anchored with a lumped
parameter mooring line and a lumped
parameter mooring line hanqing from
the buoy.
4. A spar buoy attached to a window shade
drogue.
SD2.FORT does the same simulations as SD3.FORT,
but is coded for the two dimensional analysis of Section 2.
Computer programs BUOY.FORT, MDE.FORT, and DDB.FORT are
subsets of the computer program SD3.FORT. BUOY.FORT
simulates case study 1, MDE.FORT simulates case study 3,
and DDB.FORT simulates case study 4. All these programs
use the lumped parameter formulation for the mooring line.
Input data necessary for these programs is outlined below.
Input Data Required for Computer ProgramSD3.FORT (All units in F.P.S.)
General data:
NM: Number of nodes (masses). Equals
one for case study 1.
NB: Number of the surface buoy. Starting
from the anchor, the number of the node
(anchor is not counted as a node) where
-86-
the buoy is located. NM = NB for
case study 3. NB equals one for
case studies 1 and 4.
ND: Number of the first node on the drogue.
NST: Number of steps. This is used for
dividing the spar buoy in NST strips
for integration of viscous drag
forces and moments over the entire
submerged buoy.
NC: Number of cycles, i.e. number of
surface waves for which the simulations
are tc be done.
MOOR: Index for case study cnntrol. Equals
zero for case study 1; one for case
studies 2 and 3, and two for case
study 4.
DEPTH: Ocean depth.
DT: Integration time step AT.
TMAX: Maximum time of simulation.
T2: Time interval for simulation printout.
Buoy Data:
RDI,RD2: Two radii (r1 and r 2 ) of the tuned
spar buoy. RDl is for the drum and
equals RD2 for a cylindrical spar.
-87-
ZCG: Distance (Z CO) between centeroid of
the spar base and its C.G.
RGYR: Radius of gyration of the buoy about
any axis in the horizontal Plane.
HMAX: Length of the spar buoy (H).
CDL: Normal viscous drag coefficient (C DN)
used for the spar. 7!CDT is found by
riultinlying CDLxO.02
IIST: Length of the drum for the tuned spa7
buoy (h). For cylindrical spar IIST
is some arbitrary number between zero
and the submerged depth of the spar.
CON: Added mass coefficient (Cd~ for the
buoy. (Used & read in SD2. FORT only).
COT: Added mass coefficient (C T) for the
buoy. (Used & read in SD2.FORT only).
CDP: Viscous drag coefficient for the base
and step of the tuned spar (C DP).
ALPHA: Added mass coefficient for the base
and step of the tuned spar (A).
Mooring Line Data:
WM: Weight in water (w .) of components
lumped at the I th node. Equals weight
in air for the buoy (NBth node)
CM(): Mass of components lumped at the Ith
noce (m ).
CMS(I)M Added mass of spherical componentsth
lumped at the I node (mai).
CMD(I): Mass of water displaced by components
lumped at the Ith node (mdi).
CM(I), raormal and tangential components ofCMT ():
added mass of cylindrical components
lumped at the Ith node (mni and mti).
SL(I): Slaci. length of the moorinq line
proceeding the Ith node (Li).
EK(I): Elastic coefficient for SL(I) (ki).
CDIN(I), Normal and tangentialCDIT(I):
viscous drag constants (1/2 ,CDAD)
of cylindrical components lumped at
the I th node.
CDIA(I): Viscous drag constant of non-
cylindrical components lumped at the
I th node.
DEP(I): Depth (less than zero) of the Ith
node (approximate) hi).
-89-
Attachment Data:
XCI,YCI, Coordinates of the attachment pointZC1:
between spar and the anchoring line
in spar buoy coordinate frame (x',
y', z').
XC2,YC2, Coordinates of the attachment pointZC2:
between the spar and the instrument
line in x', y', z' frame.
Current Profile Data:
V(IJ,I): Absolute velocity of water in Jth
(x, y, and z) direction at the depth
of I th node.
Surface Wave Data:
WE: Frequency (w) of the surface wave.
AMP: Amplitude (0 ) of the surface wave.
BETA: Wave direction (a).
All of the input data are read in the main
program. The main program calls subroutines: (a) STATIC;
which calculates the mean configuration or steady state
(static) solution without any surface wave forcing.
(b) BUOYS; which calculates the mass matrix [M] for the
spar buoy. (c) DBDRG, which calculates the viscous drag
-90-
forces/moments on tVi spar buoy. (d) MIUVER, which inverts
a matrix. (e) ?VATR X, which calculates the wave exciting
plus hydrostatic forces on the spar buoy, the stiffress
matrix L[], the remaining mass matrix for the mocring
lines, viscous drag forces on the mooring lines, Froude
Krylov forces on the mooring lines, and the tension force
components at each node. It also in egrates the
differential ec'uations of motion. (f) NEXT, which updates
the geometry of the system.
Before finding the steady state dynamic response
of a system to a sinusoidal surface wave, an equivalent
current profile has to be determined to compute the mean
configuration of the system. This current profile depends
on the original current profile plus the contribution due
to the surface wave. In effect it is a rectification of
the time constant current profile by the oscillatory
velocity field generated by the surface wave. From Section 2.1
we have:
VEw = x (Vox + 0x + iy(Voy + Py
+ iz(V oz + Z)
also; IVEw12 = (Vox + Px)2 + (Voy + y 2
+(V oz +z)
, |' | z
-91-
And as viscous draq forces determine the moored
or drifting system configurations for the static solution,
and these are proportional to VEw VEw, the square roct
of mean (iVwVw) over a wave period (T) would give theEw Ew
equivalent rectified current, vector at this location.
Let; E d V )dt)= -T ]o IEwl (Ew d(t
=ix EX + iy Ey + iz Ez
Now the equivalent current vector can be written as:
V F-qE/EE
where: EE = [TI
This calculation to find V , before a static
solution is performed, is done in the main program. The
numerical integration of differential equations in the
subroutine MATRIX was performed by the rectangular rule,
which was found adequate for all case studies simulated.
4.2 Subsurface Moored Systems
Computer programn SSD3.FORT is coded for the three
dimensional analysis of subsurface moored systems
-52-
(continuous line formulation) given in Soction 2. The
program simulates the response of a subsurface moored
system to the time varying current fields. SSD3.FORT
reads in the relative (relative to the mooring system)
.elocity profile in the three orthogonal x, y, and z
directions. From these three components the viscous drag
forces can be compulted directly. These components of
the relative velocity could be the ones measured by
current sensors on the mooring line. A]ternatively,
SSD31.FORT reads in the absolute (relative to the ean
floor) velocity profile in the three orthogonal (x, y,
and z) directions. Profile is read in as amplitudes of
the sinusoidally varying currents. All components (varying
with depth) are driven by a single frequency. In addition
a surface wave of given amplitude, frequency, and direction
generates velocity field exponentially attenuated in depth.
The program computes the total absolute velocity and then
substracts the mooring line velocity from this absolute
velocity to compute the visco'-b drag forces. SSS3.FORT is
the same as SSD3.FORT but it neglects all inertia forces
of the mooring system. All three programs use the
continuous line formulation of the mooring line, given in
Section 2, as the mathematical model. Innut data, except
-93-
for the input forcing function (velocity profiles and surface
wave) are the same for all three programs and aredetailed
below:
General Data:
NI: Number of intermediate bodies
(instrument clusters or inserted
floats) to be included in the analysis.
NP: Number of mooring parts (each mooring
part can have different properties).
NS: Number of segments the continuous
line is to be divided in.
NPT: Number of dynamic equilibriumisto be
performed.
IKK: Number of locations the velocity profile
changes along the ocean depth.
DEPTH: Depth of the ocean. (Meters)
DDT: Approximate depth of the mooring line
top. (Meters)
ER: Used for iteration on the boundary
condition of ocean depth. If DDT is
known with reasonable accuracy, ER
is not required. (Meters)
ERI: Used for error control in SSD31.FORT.
-94-
In.st u'rent Data:
P I.J Position of the J i nstrtuient in
slack length distance from the
top. (Meters)
SI (J): Length of the Jth instrument.
Equals zero for non-cylindrical
instruments. (Meters)
ZM(J) : Mass of the Tth instrument (m.). (Slugs)
ZMC(J): Normal. added mass component (m nj)
for cylindrical instruments. (Slugs)
ZMV(J): Added mass (maj) of the jth
instrument for a non-cylindrical
instrument. Tangential added mass
component (m j) for cylindricaltJ
instruments. (Slugs)
F(3,J): Weight in water (Wwj) of the Jth
instrument. (Pounds)
CDIN(J), Normal and tangential viscousCDIT(J):
drag constants (1/2 pCDAD) of the
instruments. CDIT is not used for
spherical instruments. (F.P.S. units)
Mooring Line Data:
DIAL(I): Nominaldiameter of the Ith mooring
part. (Inches)
-95-
SLL(W): Slack length of the Ith moorincg
part. (Meters)
AWL(I): Weight in air per unit slack I.:ngth
of the Ith mooring part (Wa). (.os/M)
•JL(I) : Weight in water per unit slacK
thlength of the I mooring part (Ww). (IDs/M)
TPL(I- Transient peak load on the Ith mooring
part. For materials which obey Hook's
law this is substituted with the Young's
modulus of elasticity. (Refer volume 1
CHHABRA, 1973). (ibs, or lbs/in )
COl(i), Constants for stress-strainC02 (I),P01(I), relationships of mooring lineP02 (i),AO(I): materials. (CHHABRA, 1973).Col is used for
jacket diameter (in.) in jacketed wire rope.
CDN(I), Normal and tangential drag coefficientsCDT (I): tof the Ith mooring part. (CDN and CDT)
Current Profile Data:
D(I): Depths, greater than zero, where the
current changes direction. I goes from
1 to IKK. (Meters)
V(I,J): For SSD3.FORT: Relative velocity in
the Ith (x, y, and z) direction for the
Jth zone. J goes from 1 to INK + 1.
(mm/sec)
For SSD31.FORT: Absolute ve]locity,
amplitude in the I h ditect iofl forthb
the J zonM. Following (]ata are
used only in SSD31.FORT.
WE: Fren7uoncy of current profile
ami,] iu-les V(I,J) . (Rad7:,c
T)M: Time step for calculation of the
mooring system dynamic eqcuilibrium.
T2: Time step for simulation output.
TMAýX: Maximum time for simulation.
WS: Surface wave frequency (). (Rad/sec)
AMP: Surface wave amplitude (0o). (,m)
BETA: Surface wave direction ([). (Ratiians)
All of input data are read in the main program.
The main program calls subroutine M'TION which calculates
the dynamic equilibrium of the subsurface mooring system at
discrete times. Subroutine MOTION calls subroutine FORCES,
whenever an instrument package or a subsurface float is
encountered in the mooring line incremental integration
scheme. Both MOTION and FORCES call subroutine SPEED to
find the current for viscous drag calculations.
-97-
5.0 CASE STUDIES/SIMULATIONS
This section presents computer simulations of
specific case studies; using the computer programs outlined
in Section 4. All simulations are done in the time-domain.,
Simulations of the freely floating spar buoy, as presented
in Section 5.1, were combined to obtain the response
amplitude operator (RAO) vs Kh (wave number multiplied by
the appropriate buoy draft). Section 5.2 presents the
subsurface moored system simulations. Section 5.3 gives
simulations of a surface moored system, and Section 5.4
presents simulations of a drifting drogued buoy.
5.1 Spar Buoy
Both the two-dimensional computer program
(SD2.FORT) and the three-dimensional computer program
(SD3.FORT) are used to simulate each of the two spar buoys;
one cylindrical, and the other tuned presented in
Sections 5.1.1 and 5.1.2 respectively. RAO vs Kh plots
a-e presented for the two-dimensional simulations of
cylindrical anA tuned spar buoys.
5.1.1 Cylindrical Spar
The spar buoy (Figure 5.1) used in these
simulations was tested in a series of wave tank tests to
-98-
derive empirical response data for comparison with the
computer simulations. This comparison has not been
presented in this report. The dimensions
0.75" O.D.
7TTotal Weight of Buoy =
W.L. 157.6 grams (0.35 pounds)
I Buoy Radius of Gyration =
L 27.1" 14g ICG 8.5",j21.4" F r 8.4
M
10. 7"81 7 '
T
Figure S.1 Scale Model Spar BuoyDescription
and characteristics of this scale model spar buoy are
shown in Figure 5.1.
Two Dimensional Simulations
Figures 5.2, 5.3, and 5.4 present the typical surge,
heave, and pitch response of the buoy to surface waves of
-99-
NCO- CIO
2 C4)
) '.
9)n
() -4
.444
0 0
41.0
L4
I4- J) x9- ý l xx( j)
-100-
Nl 44
-< C
06I S6I;0 ,I S0 L6S0 E''ý51:f~j) z fi) Z fIA) z (IJ)
-101-
44 _
II II I IIEiLn C-)
73 0
41
C.
[y)-.z HIJ 0H IfJH H I()HiC Y
-102-
amplitude 0.05 ft. and different frequencies. Table 5.1 is
the input data used in conjunction with the computer program
SD2.FORT to obtain these simulations. Noteworthy is the fact
that the viscous (CDL) and pressure (CDP) drag coefficients
in this-study eqxual zero. Surge drift evident in Fig. 5.2
is due to the bias or rectification of wave exciting forces,
which would be present for example if an oscillatory hydro-
dynamic force were to act on a heaving bod4 whose motions
were at varying with this force. Simulations for the same
data as Table 5.1 and viscous and pressure drag coefficients
equal to 1.0 are shown in Figures 5.5, 5.6, and 5.7.
Comparison between the surge motions of Figures 5.2 and 5.5
shows the effect of viscous drag on. surge drift. Surge drift
due to viscous drag occurs as the submerged area of the
spar and its inclination vary over a wave period, and is a
function of their phase relationships. This drift is most
predominant near resonance. In this case heave resonance
occurs at Kb = 1.0, and pitch resonance occurs at Kh = 0.4.
Heave motions (Figures 5.3 and 5.6) are not effected much
by viscous drag because the tengential drag due to CDL
and CDP is comparitively small. Comparison of Figures 5.4
and 5.7 shows that the system frequency present in the top
Sfour curves of Figure 5.4, due to approximate initial
conditions, is damped out in Figure 5.7 due to viscous
drag damrping. Otherwise, the amplitudes do not show
much change.
-10 3-
0
1-4
-4
* 4.40~ 1-0
'-4
0. 01
.44J
0
411
NJ 02n
N 00
000 *80
0 0
,-4 L 0 0 04-4
Cj 0 . N q"n-NM
.000000 *0o
-104-
CV t
-44 .-40
0
) ()
-4-
444
It!I
LA ~-- n
C) Lý 0Y i i' n 00Z n - n 0)D
(j x )- xC x x
-105-
ri riN'.0 .-4
-4 0 o 'U-
II It II II 0
-C
2S �zz�--� _____ -.---- IC r.
C
C-
__ U
-� 1�4
-� -, V
-� �- ( -4
C-s. -I,->,C-
- 7�. (C.1�0
L� 0
0 (CII
2 � -4
C)
-- C -.
4-, 2>
-.4 - 0)
-� (t� 4'-�
4-'-
0-'-I cn
'0 $-� -�------�
3 *1-44-'
0
71-
C)'CC)
----- C
-- S
� � -
�ti� z n z z z ui-� z
/ -I.
-10 6-
4 c-4
r- O *
* 0 n
-a-
4 1
-~C U
0-0---) n 0 n 01 n 0)
-n *0 )
Cýo
-107-
Response amplitudes of surge, heave, and pitch;
and surge drift as calculated from the steady state portions
of the times histories presented in Figures 5.2 through
5.7, and other similar simulations are tabulated in
Table 5.2. In this table column 1 tabulates the freauencies
(w) of waves studied. From these frequencies Kh is
calculated and tabulated in column 2. Columns 3 through 6
are four columns of surge amplitude divided by Eo" The
first of these four columns presents the results of the
undamped linear analytical solution as derived in Section 3.2.
The second pzesents the computer simulation results with ýo
equal to 0.05 ft. and no viscous drag. Third and fourth
present the computer simulat-on results with viscous drag
and o equal to 0.05 ft. and 0.25 ft. respectively. The
simulated plots are shown for two wave amplitudes to
emphasize the nonlinear response obtained by the nonlinear
theory. Similarly columns 7 through 10 tabulate four columns
of heave amplitude divided by &o' and columns 11 through 14
tabulate four columns of pitch amplitude divided by Kro. Kro
is the maximum wave slope. Analytical solution for surge
drift was not found hence only three columns 15 through 17
2 2are given for surge drift divided by Kw& . Kt 0o is the
mean "stok:es drift" velocity at the mean free surface. Data
points of Table 5.2 are plotted in Figures 5.8 through 5.11
~0 0o
- - j1 0
CO C'
C ; r- ` 4. L 0 0n- N (N 0
-4 -
N- In 0 - Ln 00
(InI
(N (N7 -j 1-4 L n r4In
In 00 .ý C . .ý 1 C-4 r cc 'i? (N>I N - n '0 0 0 N C I
u .1
-4 a; ~ ( ,-. 0 0 C 0 0 0 0 .-
-4I I I r-cc fo 'N C
-, *r 0ý 1. V U)'4 '40 0 nr 4 In 4 N ' C: Jn 0v (N >a;~ ~~ 4 0 N ( H -'- 0 0 0 u
-- 4 (C1 II I I
en .14 (N 1- 0 w -
CA
-4-4C
rqn
o0 00
0
o o o 0 N3 0 0 OD
r ' JI ' . 0 n C ) -
I c I InC)
-109-
Analytical Solution - 1
Computer Simulation - 2
* o * Computer Simulation - 3
t t t Computer Simulation - 4
2.0
4j
'-4o+
1.0
,...t
"0.51
m+
0 1.0 2.0
Figure 5.8 Surge Magnification Vs Kh
*
-110-
2.0
0.0
- -2.0 TS-4 0
-6.04.0
-6.0 xComputer Simulation - 2
- Computer Simulation - 3
,-,- •Computer Simulation - 4-8.0
-10.0 A 4-------4-------0 2.0
Kh
Figure 5.9 Surge Drift Vs Kh
-111-
I , 1
ILIi '
II I
4.0 I,4.0----Analytical Solution •i
IN - --- Computer Simulation 3 20
•. II --Computer Simulation-3S4 -4-Computer Simulation -_4I 3.0
2.0
1.0
0 1.0 2.0Kh
Figure 5.10 Heave Magnification Vs Kh
-112-
I
---- Analytical Solution - 1
--- Computer Simulation - 2
8.0 e-4--Computer Simulation - 3
I ,4,#--4 Computer Simulation - 4
V
o 6.0
4.0
2.0/\
0 1.0 2.0
Kh
Figure 5.11 Pitch Magnification Vs Kh
-113-
to present plots of RAO vs Kh. Figure 5.8 shows the buoy
surge per foot of wave amplitude as a function of Kh. For
zero frequency this ratio is one and for increasing
frequencies it decreases to zero. Figure 5.9 shows the
buoy drift (in surge direction) per Kwý 0 2as a function of
IKh. Figure 5.10 shows the buoy heave per foot of wave
amplitude as a function of Kh. This ratio is one for zero
frequency and exhibits resonance at Kh of approximately 1.0.
The ratio decreases to zero with increasing frequencies.
Figure 5.11 shows the buoy pitch per Ký as a function of Kh.
This ratio is one at zero frequency and, after exhibiting
resonance phenomenon at arcund 0.4 Kh, the ratio goes to
zero with increasing frequencies. Figures 5.8 through
5.11 are further discussed in Section 6.
Three Dimensional Simulations
Figures 5.12 through 5.17 present the surge., sway,
heave, roll, pitch and yaw responses of the buoy to
surface waves of the same amplitude and frequencies as in
Figures 5.5 and 5.7. Table 5.3 is the input data used in
conjunction with the computer program SD3.FORT to obtain
these simulations. Here the angle tý, giving the wave
propagation direction is zero. These simulations should
give the same response as Figures 5.5 through 5.7. Comparing
Figures 5.5 and 5.12, we see a lot of difference in the
surge drifts. Figure 5.12 looks more like Figure 5.2 which
-114-
P-40 0o
~. .0
CD)
0))
'4..J
40
0 CDL- 0
u) ý0
-,4 - 4.-44
L~0
>) 0I
14.c~0
Ln 0 10 L 00 0 Ln 00 1 Ln 0 10 n 0 10 0 00
x C; J-) 'C; (i ) X ) x C; AI
-115-
C'4 .-4
oo1- 0 -40
1-1 co
tn0
r-4
C"
41
0Ln 0-
"4-4
03
03rZ4
CD
0)m
Ln 00*0 n 000 Ln00,0 LO 0 *0 -00c; C; (41 1 c
-4 0 C-)
II >t
__ -4
0 C;
C r-4
IIte - op~
-117-
0 0 OCD
r-
tn U)
CD P-
41~431
00
0Y)
0
'44
:3:0
tC)
Lnf
00 0 'n 00,0 'rm 00 0 Ln 0010 Ln 00,0 n 0of d(YU) [Hd (Y) Hd 9 U) [d 9ý (Y) I-Id (Y) H d 9 W -I0l 0H 0 d
-118-(N (N q
- 144
-Jn o I;
4-4- -0
:>~
Ln 000 L
(ýJ)~~~J HIH 1H)H_H1_ H I
-119-
cc (4
$4
-4
ME
-404
00
> C )
4J 4)
'-0
CD
tp
'44
Ln 01 'o tn 0*0 n 0 *0 n 000 L 00 0 L 0., C)
(U d9 (H) sd 9 U)S 9 (H) sd 9 (H) Sd (U) Isdj
-120-
0
0.
944
244
0 V) M
o
00 .o *010
0000
00000
04
0 4J
m 0
V) -4C N,I. 4V n 000000 E4-
In 0 V 0I
09
000 000 00000ow09400oo0000oooo
-121-
was obtained by neglecting viscous drag. Hence we see
that the viscous drag effects on. surge drift which show
up in the two-dimensional non-linear simulations (Figure 5.5),
do not show up in the three-dimensional simulation
(Figure 5.12) where assumptions of small motions are made.
This assumption may be ini error the most near the
resonance frequency (Heave resonance; Kh = 1.0. Pitch
resonance; Kh =0.4), where the two Figures5.5 and 5'.12
differ the most. Comparison of Figures 5.14 and 5.16
to Figures 5.6 and 5.7 shows only slight differences.
Figures 5.13, 5.15, and 5.17 show no response as expected.
Simulations for the same data as Table 5.3 but e =r/
are presented in Figures 5.18 through 5.23. These figures
show the resolution of surge in Figure 5.12 to surge and
sway in Figures 5.18 and 5.19; and the resolution of pitch
in Figure 5.16 to roll and pitch in Figures 5.21 and 5.22.
Yaw response remains the sane as expected.
5.1.2 Tuned Spar
The spar buoy used in these simulations along
with its dimensions and characteristics is shown in
Figure 5.24.
Two-Dimensional Simulations
Figures 5.25, 5.26, and 5.27 show the typical surge,
heave, and pitch response of the buoy to surface waves of
amplitude 0.5 ft. and different frequencies. Table 5.4
is the input data used in conjunction with the computer
program SD2.FORT to obtain these simulations. Here again
-122-
04 -40
IIII Il UI;'
2 1 w-
7,A
4 2
> ,, -- -j
-,--
"II 4' ) '
A) " r,' ' i
k C)
\" 4-4_ I 2" 2I '/ 4 4.) /
, ~ 'Q U, , ? L I'-,0" fiJ~ X 4 fi• x 0 (jj/ X x JI.• / (40 '9 0 {i _
-123-
'0 w' C4. N
(11
1-4 0
D~) -4
CD
.4.4V)
4.7)
CLn
$4
Ci
004)A c 0 t 00 L 0, n 010L i d
-12 4-
ii Hn 0
Li
H~c H
C,
244
H0
7t.2
IIo
0 1) f--c4l 0 0 ,c - f
J)i Z (I ) zJ) Z fil iji z AI) Zsr*o
-125-
0 0
C3
_ _ _ _ .1
S 44
L±JzcJI-
'4-4c 0
C)
- "-4
00 0Y1) Ln 0010 Ln0 O[n 000 Col Oc;(H) H J (Y)HdA CH J- 9J (J)Rd H (U)L CIHd CHI Hd00
-126-
II~0 III!J" -
'0 NN - C
11,-
POEo
I" . _0
- ~ - Li,
Z44
00. CO' L 0010 Lno 0010 Coto010on° 1YU) HI (. (H) HI (H) HI (Y . ) Hi • (H) H () HI0 0 0 0
-12 7-
r- 00
00 N
If Li I'444
00
GC)
Ln
"-40
Ln~~~~~ Coo2 oo L oo-a (Y G C Ln 0 Ln 00 * in 0 c
c;V
-128-
1.0'
30'
Buoy Radius of Gyration = 5.11'
Total Weight of the Buoy = 3745.62 pounds
Figure 5.24 Tuned Spar Buoy Description
-129-
C.)
SL
C)
KC.)
(0
S CID5LOf) l- 9 f) I- f) 1- 0 'T-00 1 C2i( 1 X (I
DLL X IJ1 X r4 (J 1 ) X 4 (
-130-
/ ci)
(I Al z
44 I Z
-131-
0 C
C3
4-44
r 1
-~00.0 ' w OrY .0 01)O *0 0C -i Oa0 -
o UH 0 (Hd)k HI HI (H) Hi (H)1 Hi ()kH
-132-
0
0 E-4
0U
00 o0 0 ."4CN0 01 *4
4-40
0
0 - 00 >.0 *m 00
04
0
V4-
0 in 000 0 n Ifl In V*
0*0a. . 00000000000 L
%00
4c NTOO - - *0 0H* *% ** 000000
I.-00-I00000000o0oo-
flOO0000000000Ijj0 0000 000 oo00w
viscous and pressure drag coefficients equal zero.
Figure 5.28 through 5.30 use the same data as Table 5.4
except that viscous and pressure drag coefficients are
equal to 1.0. Once again comparisons between the two sets
of figures (5.25 through 5.30) show similar differences
as explained for the cylindrical spar buoy. In this case
the heave resonance occurs at Kh = 0.22 and the pitch
resonance at Kh = 0.4. Also in this case the viscous
damping in heave and pitch respons es is more pronounced.
Response amplitudes of surge, heave, and pitch;
and surge drift as calculated from the steady-state portions
of the time histories presented in Figures 5.25 through 5.30,
and other similar simulations are tabulated in Table 5.5
which is similar to Table 5.2. In this case the wave
amplitudes F, studied are 0.5 ft. and 2.5 fc. respectively.
Data points of Table 5.5 are plotted in Figures 5.31
through 5.34 to presents plots of RAO vs Kh. These
plots are similar to Figures 5.8 through 5.11. Figure 5.33b
is an expanded view of Figure 5.33a, near the heave
resonance frequency. The ratio between buoy heave and the
wave amplitude is one at zero frequency and exhibits resoance
at Kh of approximately 0.2. The ratio decreases to zero at Kh
equal to 0.32, after which it increases a little before
decreasing to zero with increasing frequencies. Such behavior,
where the heave response is minimized at a given frequency is
the main function of a tuned (as opposed to cylindrical) spar
buoy.
-134-
Ln ccLfl
C-ý
CDC:)
5LLJ~4.;
00
*11
C)
C) c 00 * - c O T - aa Oli')
(I Al X (I AlJ X U (Ii X A fIi) X A l (Ii A ) x
-135-
CLO
S C-)
-'-
LAn
C)
0,
Of) leg 01 ý29 01 f~~r, 0' -G9 00'be~ 00IhP
U i)z (iJ) z (I A z Ii)a LA
G*C;
-136-
tr01/ ( I
I)-
~5ji/l c0O T0,.o-4 0 1 14 o?*
(H) H (Y) HI ; f ) Hi f~ II
-137-
1 0I 0 0
0 C , '-4-1 11 -4 r- 0 0
C.,)
Iq r. .NV)
.INCN en~
-4N r 0 fn4 -4
14 -4 Q).
I) en r 0
r4 -4
2 .- 4( (. CN >4 - 0 0vi
L M
en0 -4C ý- N II ( '-00! C!00 00(l
-4 j1l l (1 C, C) )
iiI.
.r, CN LA m-44
L) 45 j 0 0 0 - 4
a) N o N1
U A N -4 r' 'I 0 u? in
UP o o o O O 0 0
--- - - -- - - - - - - - - -
cc V) c -4 NI w -4O 00'M ~ ~ ~ ~ ~ ~ ~ L CC) I iO O O ko %0n f
01 ~ ~0 0 0 0 0 CC)L
~j~o 0 0 00.
U) w) a% N N CD) r-1N0 '-
C; c. o 0 C; 0 0 -4 14 14
-138-
Analytical Solutin -1
SComputer Simulation -2
0 (kOmputer Simulation -3
2.0 t + ÷Omwputer Simulation -4
1.5
1.0
Xh
Figure 5.31 Surge Magnification Vs Xh
-139-
10.0
8.0
'--~ Corrputer Shimulation -2
a-ea-a Owputer Simu~lation -3
6.0 Cw--- puter Simul.aation -4
2.
-24.04
~2 01.2.
Figure 5.32 Surge Drift Vs M0
-140-
6.0-t
I --- -- Analytical Solution -1
5.0+ I- ruter Silation -2
,I " Qxp uter Simulation -3
1 I 4• C am"p- ter Sinulation -4
40
5 2.04
1.0
0 14.0 6
Figure 5.33a Heave Magnification Vs Rh
-141-
I I
5. 0T - Analytical Solution -1
I Caputer Simulation -2
I - Coaputer Simulation -3
4.0- I oputer Simation -4
* I
1.0.0'0 0.1. 0.2 0'3 0.405
Figure 5.33b Expanded View of Figure 5.33a
-142-
I~I.
1-. Analytical Solution -1
i0. - - o-M-uter Simulation -2
I Computer Simulation -3
_ +----- f xmputer Simulation -48:~ Ii1 T01'
6.0
4.0-P4
2.0
011.0 2.0
Figure 5.34 Pitch flagnific on Vs Mh
-143-
Three-Dimensional Simulations
Figures 5.35 through 5.40 present the surge,
sway, heave, roll, pitch, and yaw responses of the buoy
to surface waves of the same amplitude and frequencies as
in Figures 5.28 to 5.30. Table 5.6 is the input data
used in conjunction with the computer program SD3.FORT
to obtain these simulations. dere 3 equals zero.
Differencesbetween Figure 5.28 and 5.35 can be attributed
directly to the assumptions of three-dimensional
analysis. Simulations for the same data as Table 5.6
and 7 = 1/4 are presented in Figures 5.41 through 5.46.
5.2 Subsurface Moored System
Three dimensionai computer programs (S3S3.FORT,
SSD3.FORT and SSD31.FORT) described in Section 4.2 are
used for these simulatioi.s. Mooring system used as a
case study for these simulations is shown in Figure 5.47.
This mooring system was actually deployed during the MODE
experiment and was called mooring No. 1 (station 481).
Figures 5.48 and 5.49 display the three coordinates
(x, y, z) of the top of this mooring line in response to
the relative velocity data displayed in Figures 5.50 and
5.51. This simulation is a part of the stud~y (CHHABRA, 1976),
where the current record from the topmost vector-averaging
-144-
(N rI0 Ltn
Ci
ci
C3)
"--4 4-'
00
-- I)
-44
C))
Oý~ T f)00
J) xx P- A) J-)x J- X ý
CN tn 14-145-Ve
/L
cn00
0'
C'.-
* >1
C~) 4.n
C.)
0011- 00,1- 0011- 01)1- 00,1- 0Ol-0-9 (1•1)'9 IJ 9 1 ) 9 UIJ A) 9 fII Ik U AIi)'
-146-
CC
41
LAj
4-4
c0 - r-f
00 !'p~i 00' flO9 no, JO pY 0
tlJ Z 1~ Z [I)ZIi Z I 7I)
-147-
C44 0nGo ~~ 0 00D
CL,
C.)
C-2
44J
00IIn
LUn.1DC)
C:)
00 0 00 0 0010 0131 00,10 ono(ýJ) Nd ý (H) Hd f Y) Hd (H) Nd ] (H) dJ W HU Nd
-148
I'-4
~Ln0r
W4
IIjLIO 0 -0 *0r-4 11)0 00 a 0L" i p
C;I
-149-
m eq N CD
C)
0
0 Ln
CD
CD)
c2fY0 00, OfVO '-4 C(O'4 O) 0010~V -oy s(U d ý) (dGdd C(WJ cd fJ (nd~ (,Y)sd
-150-
0
0 -4*(
010
0C
0*
0000
0 .
0
0.0
0t~) 000 0
z :4zz * z Z
1010 000 0 0
0 " 0O 0""m 1
0000000000 4.0000000000000w
LN -4coA C)
00
CD
4-)
C)-
C-)
Ln
0 0-0 0-1 c- 00T1- 0 001 001- oA4 (1I Alx1 (1j) X A4 U(I x ) X 4 (1•1 X A (1i A' (i X
-15 2-
Cn co L
C:.5
Ii i II I i f- )
u-i
cn
C-)
) nci
oil -o o I)
r4 U. J'IP Ii Ir IJ
-153-
P-4 4 Lfn C)
cm)
oil t,9j 0' bl~cj00'fips 0) h9S 0 K-s 0 fncj0
U J) fi ) z I ý)z (iJ) z (I A z ( J)
-154-
Ln co 0n
gi II II IIit
lC)
F-4.)0010~~~~~ ~jL g- - 00, " 00.
IýJ H () d f ) d ; H)H (Y HJ ; Y) H
-155-
C14 Ll H4 coM' r(N H 0 C)
CDd
C)
( CJ
C.)
C)
.4U-)
C) 4.4-4 a) 0
C) L
9 4C)
00*0 00,0 0070 0070 j 10 0.0H H H H HH HI Hý)Hic H )H1 W
-156-
C..
0* -0
I4I
C)
Le)
.2
(U)~C ISC) C; (H s
-157-
Ocean Depth - 5400 Meters 4480
MLight and RadioRadio Float
1/2" Chain
Instruments 20 M 20 16" glass spheres on 3/8" chain
Vector Averaging Currentl Meter 96 M
O- Pressure/TemperatureRecorder 196 M
198 M
All lengths are 3/16", 199 N3 x 19, wire rope 9jacketed
280 M
15 M 12 16" glass spheres on 3/8" chain
500 M
476 M
470 M10 M 1_0 10 16" glass spheres on 3/8" chain
475 M
All lengths are 3/8" 5 M --- 5 16" gless spheres on 3/8" chaindacron _
376 M
898 M
56 m
15 M 15 16" glass spheres on 3/8" chain
Acoustic Release
20 M 3/4" Nylon3 M 1/2" Chain
2,500 pounds Stimson anchor
Figure 5.47 Subsurface moored System
""4
~17W
cnc
4-J
Ln
0. *16 S 0' fI6 0 m - 00 0i +..----- i--.--.---IU I - 00I- I I- o I IC
-159-
M C
0I
C.
0
cr)
0
Cý
"-210 .00o 170. 0o - 130. oo -90.00o -50.00o -10 .00o 30.00oX (METERSJ
Figure 5.49 S.'ajectory of the Subaxface Pbored System '-bo.
PV~CM-ERST COMPONENTS
U0
Lij
CD
c2
CD
CD .. ....
-- 4
LUcLL,,:.
cn.0 30 .0 .0 J.0 50 80
TIM .H.U....Fiue55 ltv eoiyItafrSbufc xoe yt3
VRCM-NORTH COMPONENTS
Ct)
Ljc;
CD)
LO
Lil
Li
LT.L!30 .090 10 ~ c01i0TIM 0H0B
Fiue55 LaieVlct aa crteSbufc br]Sj~
-162-
current meter of this mooring was corrected for the effects
of mooring motion, and power spectra of the uncorrected
and corrected signals were compared. The relative velocity
data as shown in Figures 5.50 and 5.51 was sampled every
fifteen minutes at eight locations on this moori line.
Computer program SSS3.FORT was used for this simulation
and no surface wave was assumed present. Figures 5.52
and 5.53 show this same response with the computer program
SSD3.FORT, which calculates and includes inertia forces.
Overlay of the two responses shows no noticeable difference.
Input data used with these two computer programs, describing
the system are shown in Tables 5.7 and 5.8.
Response of the top of this mooring line to
hypothetical absolute velocity data (Figures 5.54 and 5.55),
is shown in Figures 5.56 and 5.57. This simulation was
done with computer program SSD31.FORT and the input data
shown in Table 5.9. Surface wave was again assumed to be
absent in this simulation.
5.3 Surface L.oored System
The three-dimenýional computer program SD3.rORT
is used for this simulation. The surface moored system
simulated is shown in Figure 5.58. This system of spar
-16C-
/ t 4
) I 4
'S / 1- £
in
5> mg i 01(()J~iwl f~~lll
-164-
C-,
C.)
ri
CdCC
0
C-,/
tr.J
='-21 0. Cl- 170.00 - 130, 00 -91-1 DO -50.00 -1I0.00 30.010X METERS)
Figure 5.S3 'I~rajctor y of the subsurface Nbo~red Systelm Top
-S.
-165-
0,F4 OO 44 CIn m Cl* .. . ... .. .. .. .. .. .. 00 . . . . .
00 0 0 0 00 0 0 0 0 00000 * *000
000*
tol 0
%N 00
. N * * * 'I * * . W: * .0 * 4 * 4 . . .
mn InM
0~' (; C
0 0
000'.
t~000 00 00 00 00 0 0 0 00 0 0 .0CD . M4 .n .0 . . 0. 0 *4 -C ol . .n 'CNC .0 00Nt7 4I 0N D0
CD ~ ~ ~ ~ ~ ~ ~ ~ ~~~~0 00 00 0 0. 0000% - 4'4' 4-4 NNNNNNNN
0w00000 000000000000000000000000000 000 000 000 00 000 0000000 0000 0000
-16 6-
00000000000000000 .0* * * * C C C C C COC
In 4J
00000000040 . 0 00 0 MOMMv0 *, a C NC 0,o000 N~ 0N N V 'al V. o oN N 000 0W
In I N li N In Na N -4
V9%a49T00000000000000oo ao Z 4
000000000 000009499494 49 444440
00000000002.o In
* m n ONc * * C C C CCCC *
0ýMo l -I00 00 0 IW 0 In 00 .0 40 II0 vv 10 NM0 0N000 00 0
*ý . to V In LO In in In It'rT WWVo
c;0 MO n AO O *0 0 0 49 4P 4.
0000000004 1 N000000 00000004 494004494049
N M V 100 14NM ý0 W 10W M V na M0 04q q . . . . 0 V NM
00000000000000000000000000944444499990000000 00000 0000000 0000 0000000 000
-167-
m)0O %a 4o k* W4 n
. . . 000Inv4000M
1flC4O .' 0~ Il~ll000w>
000400
DO* %o *4 IV4W
InmnoN .4400''00 WOOMOOON . OO.T V0.000
.: *t .0 CD "0 * . . 0. . . . . * 00
0o If I. ; N 0 0
%0000n . 40 0
0 0 0 '000'.LA*40a0I TI Vi nI TwMIn W0 N N *in 4 In 0 4TIV I
* N V * V * 0 * N *ý N W *** **. 0 a'4 0N 1ln In0000'40004000 NOO.4IN 4 1 qV
IT In40. .0
N 0 . 0 00U.0 ;44. * V * z 4 * * 4 4 U* 4 4 * z *; 4 0o. o. 0
00OOC 000 0qowo 0000000 00,000000000 000N W4 P; 04 In* V0 N ý 44 0. 0 4 rN 44 In %00 l r; 0 ; 0 ; '4 (NI 01 n cooN W 0 0 0'-
0noo 0 00 04 00 000 00 0 00 0 0 0
0P 000 0 NLIn00LO0 0 000w 0 000" 0000000 000N00 4I
-168-
00 00 000 000 000 000 '0N N N l N C N N i e N N N N 01
00000000000000000
Coo o* 4 084
0000000
/0
000 C0 U C
O000~000000O'0000000N•
0 0N 400Q000NOOOO0O0OO00000000000
94"4000000000000 %0aaa'0'0a00
04
mmmvl~v%0VV0VN0000000000m00
O0000 000000000 inI41..r
0 *0 no0W0W00~i
NNNUOOO ,Or%000 b7
*0 0 0ý0 *0 * * * * 0 * r N N N N N 0
00000000000000000000000000-00N~~~~~~~ .~ . N, N N00' N IN~ c U, ' N 0 0 Ntq U'0N 04.0 N CD
000000000000000000000000000000000 0000000000 0000 0000000 00 0000 00 00
.c) VRCrA-ERST COMPONENTS
LLJj
U~cUflc
LLJ~(F) LI
IT
(2H
UiC-)
27n 1 I K CIOII
TIME I(5ECONB3j XIO)Figure 5.54 Absolute Velocity Data for the Subsurface tijored Systemf
VRCM-NORTH COMPONENTS
Li
WCD
LL2C
rli
(n~
Li)
LU 0
L 0
bU.
27~o B61.00 1211.0j H8. 00 2L40.00 300. 00 3530.00TIME (5F-CONDSJ (Xl10'
Figure 5. 55 Absolute Velocity Data for the Subsurface Moorecl cystarn
C,)
co
C:,
L±J
4-40
CD
U-,
LO
C)
00'105 0,96r, o~gml 015al 00 NUS- 00*9- o Cn
OO'L6OEI~~(yl OfVY6O IOIIJ3i OOHIlJ xnmt~ OO
-172-
r-J
C)
0C)
u2
'_J
ci
1-r
7• (to -70.0'0 -55O0 -RO.0]O 0
x (M[:TEPS~J
Figure 5.57 Trajectory of the Subfsurface Mo~ored Syst~n ¶ibp
-17 3-
*OO 0 IV '4 Nn m0 iCIAN N M M W4.'. NO0
~~;0 0 co oc...... E
000000000000
W 000000In0NVIO *0 *4 I4I IfM0000000
NQV' ' *0T V0'44 40 to T - toN * -04T .i-4oý000000.4 .** .* .* 0 * *.** *.*.00.*.0 * 000000
00r-, 0 0. . . .4
InN %-4 .4 ON 0 4N N CD 94 CAN W 4 r4 CJN I I In N tj "M 0.0 0 In In In
.QI~~~~~4 ~00 00 a)40 ~ ~ 0~W 0 0 0 n0 I ~ -
w '4 M 0. IN Mv4 M "4 w.4 w.4 Mn N 4 w.4 "4 ( O'4 . (4I M v-4 N 4 0. O'O v-4 ". 0. v4M
0n N N N N NN
10 If) r 0 0000
0 *N '404000 -000MCW00ý0000In0 -0,0*CD *0 * * - * -* - *'. * - * - * * - - # * . . 4T . 3
N"-M'.4OI NOInN N N '.4N N N 0 '0CNN InN NM N '4W '4000 0
In IV0 0 In
0 *0 0 yr Go 00 0 0 0 0- Nv4 -4 I
N * .N * * . 4 . . *C . .t . . . .00 9 CO-r4 0 0N 1 0 0In 0 0 00-4 00 0 .0 00W N00 v4 IT
N0 0 .0.4 V V C) .) . 00. 0.
* * *NN~NNNMM00In044I0,.4TvnwNNwWWW.OOON'-4'.mmoN'q-4¶-4'NNNNNMMMM4444400000
000O000 000 0000000 000000000000 000000
0000000000000000000000000000000000
-174-
rl i rCl rti rCl 64J 1'.j 6l r'I cI Ci rCl 61 C. 6l 6l '0j '0 a00000000000000000 0a-
o
t.J.Ninor'0vmtýNO0000O0000O000000OO4O N 4- * 0 * 0 * * 0 * * . . * . * . *,vq.4 I
r .4."4."4."14. "1.q 4 ,qq4. q. -I."t4 .q 4.4
N N -O N N ýo N N %a N V) r * 00
00000000000 1O010010,010'00-.0 lco0 -0.0 0
in 4-)
qTdTov %v % vN000000000O0ml * 00
00 00 0 80 In InO0 00 .! . . .
CD -COO -NO qo* . . . .* . . . . .. . . *.4T40 I 4rMo I4T q t1 1 in rqm I ON ti '0 C'j 0 00 0 0 0.4"4.-4."4,44.4.4-4 V4,4
0In
N 0NN NNONN NfLO-40-010101040000000000 N*f In -00
00 000 00 0000 000 00 0000 000 4)00 00 00000in '0 N m 00 0. t4 "~ m in 0 %N m 0% 0 --4 04 m~ v 0n0 N w0 os 0 C-i N m in 0'0N M0 C)O0000000000000000000000000000000000
000000000 000000000OOOOOOOOOOOOOOoooo
-I75 !'l SPAR BUOY
BRDL FVR
S! POPMIP
10 M -- T/P
FLOAT PINGER
/ WIRE ROPE
67 M -FVR
TETHER L IINE i TIP
MASS TANK
75 M PINGERFVRCHAIN
" I100 M -- SPHERE
S~FVR
,,, PINGER
T/PWIRE ROPE
400 M -- T/P
SPINGER
GB'S
RELEASE
NYLON
1700 M -CLUMP W/DANFORTH
Figure 5.58 Surface 1!tored System
-176-
buoy/buoyant tether/subsurface mooring line with an
instrument line hanging beneath the spar was deployed
as part of the ONR/NDBO Mooring Dynamics Experiment
during October 1976. The system is reduced to seven
nodes for this simulation. These nodes are shown in
Figure 5.59. Figure 5.59 shows the mean configuration
of these nodes in the x-z plane. This configuration is
the result of the static solution calculated by SD3.FORT
using the equivalent velocity profile described in
Section 4.1. The equivalent velocity profile and the
actual velocity profile in the x-z plane is also shown in
Figure 5.59. Figures 5.60 through 5.64 display time responses
of these seven nodes to the input data presented in Table 5.10
which describes the-moored system and the environment forcing
this system. Forcing the system is a constant current
profile and a surface wave of amplitude 1 foot, a frequency
of 1 radian per second, and having a wave direction
inclined at 0.1 radians with the x-axis of the earth
fixed frame. Figures 5.60 through 5.62 are the plots of
surge, sway and heave responses of the seven nodes.
Figure 5.63 displays the roll, pitch and yaw of the tuned
spar buoy (node 6), and Figure 5.64 displays tensions in
the seven segments of mooring lines preceding the seven nodes.
-177-
r- -4 CIS (N4 r
Cý~~~~ cl o-
0~0
I ~41
.4J-4
*.44
LI
C',,
00 0 0D 0D0 0 0 0D 00n en 0H
cii- I-ole 7
Spar Buoy
LL- ri
Li~
LLp
£~ure 2CO~ZM ~kbde 102.10-
10. ~ ~ ai -1.03.0 00 00
TIME- (SEC
ppr: #1.
________ _____�---- -______ ---------
LL.
4-
I-- ..�.. j--- 4
I--
c���1�*-- , o.cI2 A-A
r �
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TIME (SECIFigure 5.63 Roll, Pitch, and Yaw fibtim of the Ntored
Spar Buoy
r -
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Fqre 5.64 7tsix K...itde inDifeen Sryens f6
Surface Ltborad System~
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-184-
5.4 Driftina DroguctBuoy Sy tem
A submerged window shade droque attached to a
cylindrical spar buoy by means of an elastic nylon line
is simulated on the computer by the three dimensional
computer program SD3.FORT. The system is subjected to
simple harmonic surface waves without any current profile.
A 47 feet long 3/3" nylon line is attached to the center
of the bottom of the buoy. The spar buoy is 30 feet long
and 0.66 foot in diameter. Its static draft with no lines
attached is 19.43 feet and weighs 426 pcunds in air.
The distance between the center of gravity and the bottom
end of the buoy is 8.69 feet. The window shade drogue
is made of nylon cloth (9 1/4 oz/yd 2) aid has the
dimensions of 7.5' x 32'. Its approximate thickness is
0.02". The droque has two metallic bars, one at each end.
The bottom bar also acts ar the dead weight, Yeeping the
drogue in tension. For computer simulation the droque
was divided into four strips and the parameters of each
strip lumped at a node. Tensions in three links between
these four nodes and ihe tension in the nylon tether line
were simulated on the computer. Figure 5.65 shows the
system. Figures 5.66 through 5.70 show the time histories
of the response of the system in a surface wave of
amplitude 1 foot and circular frequency of 1 radian per
second, and having a wave direction inclined at 0.25 radians
with the x-axis. No current profile is present. Figures 5.66
-135-
SparBuoy 30'long-0. 6' diaewter;
3/8" Nylon
Node 1o. 2rce Vector Recorder +
Tb bar + Drogue Strip
Figue Strips
Figure 5.65 Drifting Drge Buo System
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TIME (SEC.)1'igure 5.70 i!)enstioni Uirauitu&~ in j).iffteroiit Pr'polats of ti-k-Ž
Dritting Drr urxt I3ixy!-,yrtaT,
-191-
through 5.66 shz- the surge, sway and heave coordinates
of the spar buoy C.G. and the four nodes on the drogue.
Figure 5.69 shows the roll, pitch, and yaw angles rf the
luoy and Figure 5.70 shows the tension magnitudrs in
the nylon line Yelow the buoy and the three vegments cf
the drogue. The input data used for this simulation is
presented in Table 5.11.
O~Oi
o -e 0
00
03
90 0000 . ...
Nrr~ 000 . .. ..
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-193-
6.0 SUMMARY
Dynamics of moored and drifting buoy systems in
a three-dimensional space has been presented in this
report. Formulation of mathematical models, which are
programmed on the inhouse computer AMDAHL 470 V6, along
with time-domain computer simulations of these systems are
presented. Four case studies (freely floating spar buoy;
a single point subsurface moored system; a spar buoy plus
instrument line/buoyant tether/subsurface mooring; a
freely drifting spar buoy attached to a window shade drogue)
are simulated with forcing being supplied by a velocity
profile and a fully developed surface wave field. One
(a single point subsurface moored system) of these four
case studies was evaluated with full scale ocean test
data (CHHABRA, DAHLZN and FROIDEVAUX, 1974; CHILAERA, 1976).
Simulations of other three case studies can now be readily
evaluated with full scale ocean test data.
For the two-dimensional analysis oW a freely
floating spar buoy, where the assumption of small motions
is not made, three sets of computer simulations are
compared with the undamped linear analytical solution.
This comparison is presented in Figures 5.8 through 5.11
or a cylindrical spar and Figures 5.31 through 5.34
or a tuned spar buoy. This comparison shows the effects
of viscous drag and varying wave amplitude on the response
-194-
of a freely floating spar buoy. Plots for the undamped
linear analytical solution and the computer simulation
where wave amplitude is small and viscous drag is neglected
are almosT, identical. For the case of a cylindrical spar,
when viscous drag is added, the surge magnification vs
Kh plots show a scatter near the heave and pitch natural.
frequencies (Figure 5.8). This emphasizes the coupling
effects. Surge drift vs. Kh plots in Figure 5.9 show
a markedly different responses near these samc frequencies
and viscous dam.ping is evident in heave and pitch
magnification plots (Figures 5.10 and 5.11). The non-
linearity in the response with respect to wave amplitude
is also evident. These plots also show a heave-pitch
coupling near their resonance frequencies. Similar
observations can be made for the case of the tuned spar.
For the three-dimensional analysis oý a freely
floating spar buoy, comparison is made with the two-
dimensional analysis for tCe same input data. Th2 only
appreciable difference is observed in the surge drift,
which can be attrilbuted directly to the small motion assumption
of the three-dimensional analysis.
Simulation of the spar buoy plus instrument
line tethered to a subsurface mooring line shows a note-
worthy transmittal of motion from the surface buoy to the
subsurface line, by the tether. The tether line seems to
-195-
move mainly in the tangential direction due to the high
viscous drag force in the transverse direction. As a
result the combined surge, sway, and heave motions of the
spar are transmitted to the subsurface line. Hence, the
surge/sway motions of the spar (even if heave was
negligible) could be transmitted to the subsurface line as
heave. Therefore some modifications of the system may be
necessary to minimize longitudinal motions of the subsurface
line.
Simulation of the freely drifting spar buoy
attached to a window shade droque shows drift of the buoy
and the drogue even though no current (except surface wave)
is present. This drift is due to the same reasons as
postulated for the freely floating spar buoy.
Full scale ocean test data for the spar buoy plus
instrument line/buoyant tether/subsurface mooring is now
available (October 1976 ONR/NDBO Mooring Dynamics Experiment).
Using this and other data which might be available in the
future; the next logical step is the evaluation of the
mathematical models of the remaining three case studies.
-196-
APPENDIX A
Computer Program Listings Page
SD3.FORT. ........................................... 197
SD2.FORT .......................................... 222
SSD31.FORT ...... . ................................... 245
SSS3.FORT ........................................ 255
SSD3.FORT ............... .......................... 262
-197-
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REFERENCES
CHHABRA, N. K. (1973) Mooring Mechanics aComprehensive Computer Study, Vol. 1, C. S.Draper Laboratory, Inc., Report R-775.
CHHABRA, N. K., J. M. DAHLEN and M. R. FROIDEVAUX(1974) Mooring Dynamics Experiment-Determinationof a Verified Dynamic Model of the WHOI Inter-mediate Mooring, C. S. Draper Laboratory, Inc.,Report R-823.
CHHABRA, N. K. (1976) Correction of VectorAveraging Current Meter Records from the MODE-ICentral Mooring for the Effects of Low-FrequencyMooring Line Motion, Accepted Deep-Sea Research.
CHUNG, J. S. (1976) Motion of a FloatingStructure in Water of Uniform Depth, Journal ofHydronautics, Vol. 10, 65-73.
GOODM4AN, T. R., and J. P. BRESLIN (1976) Staticsand Dynamics of Anchoring Cables in Waves,Journal of IIydronzutics, Vol. 10, 113-120.
LAMB, If. (1945) Hydrodynamics, Sixth edition,Dover Publications, New York, N.Y., Article 68.
NEWMAN, J. N. (1963) The Motions of a Spar Buoyin Regular Waves, David Taylor Model Basin,Washington, D.C., Report 1499.
RUDNICK, P. (1967) Motion of a Large Spar Buoyin Sea Waves, Journal of Ship Research,December 1967, 257-267.
VACHON, W. A. (1973) Scale Model Testing ofDrogues for Free Drifting Buoys, C. S. DraperLaboratory, Inc., Report R-769.
VACHON, W. A. (1975) Instrumented Full-Scale Testsof a Drifting Buoy and Drogue, C.S. Draper Lab.,Inc., Report R-947.
WUNSCIH, C. and J. M. DAHLEN (1974) A MooredTemperature and Pressure Recorder, Deep-SeaResearch, 21, 145-154.
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