August; 1978 .
.
Joint Program in Ocean Engineering, ~ssachu-
setts Institute of Technology/Woods Hole Ocean-
ographic Institution, and Department of Ocean
Engineering, Massachusetts Institute of Tech-
n0i-Y' August, 1978 .
RESPONSE OF A PENDULUM SPAR TO 2-DIMENSIONAL
RANDOM WAVES AND A UNIFORM CURNT
by
GEORGE RODENBUSCH
B. S., RICE UNIVERSITY
1974
M.M.E., RICE UNIVERSITY
1974
SUBMITTED IN PARTIAL FULFILLMNT OF TH
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
a t the
MASSACHUSETTS INSTITUT OF TECHOLOGY
and the
WOODS HOLE OCEANOGRAPHIC INSTITUTION .
Signature of Author
Certified by
Accepted by
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MARINE
BIOLOGICAL
lABORATOF?Y
-~
LIBRi,RY
WOODS por E ~"A-:
. .- i ..'1 0.)
W, f-. O. I. .
_.
Chairmn, Joint Comm' tee on Ocean Engineering,
Massachusetts Insti te of Technology/Woods
Hole Oceanographic Institution
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R. (p;¿
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-2-
RESPONSE OF A PENDULUM SPAR TO 2-DIMENSIONAL
RANDOM WAVES AND A UNIFORM CURNT
by
GEORGE RODENBUSCH
Submitted to the Department of Ocean Engineering
in August 1978 in partial fulfillment of the requirements
for the Degree of Doctor of Philosophy.
ABSTRACT
A linearized theory for the response of a circular pen-
du1um spar in .2-dimensional waves and a uniform current is deve1-
oped. The linear forces on the cylinder are predicted using an
approximate potential flow theory for slender bodies. The dynamic
equations are then amended to account for the wake effects of
viscous bluff body flow by including
a quadratic drag law and
neglecting wave damping. A spectral model for the forces on a
cylinder due to an oscillating wake, modeling the force as a
frequency modulation process, is proposed. The non-linear
equations of motion which result are then solved, assuming con-
stant force coefficients, by linearization
for use
with a Gaussian
random sea. The method of equivalent linearization is extended to
include mean flow effects and a spatially distributed process.
Some numerical experiments are then used to test the performnce
;. .
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-3-
of the linearization. For a variety of environments, the
linearization predicts the standard deviation of the simulation
response to within lO% and the mean angle of inclination to
within 30%. Results of the numerical experiments indicate that
there is significant variation (order of magnitude changes) in
both response and mean angle of inclination. Thus, significant
changes are followed by the linearization.
A laboratory experiment was carried out to test the
linearized spar model in a realistic fluid environment. Only
the low Keu1egan Carpenter number regime was investigated.
With some minimal mani~ulations, good agreement is obtained
between the experiment and the linearized estimates. It appears
that the drag coefficients for vortex induced in-line forces may
be an order of magnitude larger than those reported in the
literature, .5 instead of .06, and that the shedding of vortices
due to stea~y flow may reduce the added mass coefficient signif-
icantly, as observed in oscillating flows with significant vortex
shedding.
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ACKNOWLDGEMENTS
James W. Mavor encouraged me to investigate the dynamics
of the spar current meter which was the original stimulus for this
work. He was also the driving force in obtaining Sea Grant Con-
tract #04-6-L58-44L06, under which the instrumentation for the tow
tank experiment was purchased. The National Science Foundation
provided tuition and stipend support under an NSF Graduate Fellow-
ship for three years. I was fortunate to have been selected by
the Board of Trustees of the Naval Postgraduate School Foundation
as the first recipient of the Carl E. Menneken Fellowship for
Scientific Research, which provided partial support during 1976-77.
My final year's support was provided by the Woods Hole Oceanographic
Institution. I am indebted to the Acoustics Group at Woods Hole for
unlimited access to their mini-computer and the loan of numerous
instruments, and especially to Stan Rosenblad who, on numerous
occasions, came in at odd hours to revive the computer. Susan
Smith and Karen Pires typed the draft and final copy of the thesis.
Special thanks go to W.D. Grant for his careful reading of the
draft and his assistance in making the final copy readable and
consistent. I would also like to recognize my wife Nancy, who
gave me the encouragement I needed to begin my graduate education,
and my daughter Megan, who made me realize that it was time to
finish.
-5-
TABLE OF CONTENTS
Ti tle Page
Abstràct
Acknow1edgemen ts
Table of Contents
List of Figures
List of Tables
List of Symbols
1
2
2.1
2.2
3.
3.1
3.1.1
3.1.2
3.1.2.1
3.1.2.2
3.1.3
3.2
3.3
3.3.1
3.3.2
3.3.2.1
3.3.2.2
3.3.3
4
4.1
4.2
4.2.1
4.2.2
4.3
5
5.1
5.1.1
5.1.2
5.1.3
5.2
Introduction
Linearized Potential Solution
Zero Mean Curren t
The Effect of a Uniform Current
Viscous Effects
Steady Flow
Fixed Cyl inder
Forced Cylinder Oscillation
Transverse Oscillations
In-Line Oscillations
Elastic Vibrations
Oscillatory Flow
Equations of Motion
"Steady" Drag
Vortex Forces
Sinusoidal Flow
Lift Moment Spectrum
Eaua tions of Motion
Equivalent Linearization of Equations
Simple Analog
Linearized Analysis of Pendulum Spar
Linearized Spa tia 1 Process
Lineariza tion A1gori thm
Numerical Experiment
Experiment
Spar and Instrumentation
Spar
Ra te Gyros
Accelerometers
Experimental Configuration