Sam Roberts
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Forced Vortex-InducedVibrations on a Cylinder
with Two Axes of Motion
Samuel L. Roberts
Boston University Academy
Class of 2008
MIT Center for Ocean Engineering
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2008
Samuel L. Roberts
This work is licensed under the Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License. To view a copy ofthis license, visithttp://creativecommons.org/licenses/by-nc-nd/3.0/or send aletter to Creative Commons, 171 Second Street, Suite 300, San Francisco,California, 94105, USA.
This copyright excludes all parts specifically credited to other authors.
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Dedication
For Grandpa,
Be Happy, Have Fun, Keep Smiling.
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Table of ContentsAcknowledgements .....................................................................................1Abstract ...................................................................................................... 3Introduction ............................................................................................... 5
VORTICES .................................................................................................................... 5
DAMAGE FROMVIV.................................................................................................. 6IMPORTANT NUMBERS ............................................................................................. 7
The Strouhal Number ................................................................................................ 7The Reynolds Number ................................................................................................ 7The Reduced Velocity ................................................................................................. 8
WHYFORCEDVIBRATIONS? ................................................................................... 9WHY2-AXES OF MOTION? ...................................................................................... 9
Methods .................................................................................................... 11Results and Analysis ................................................................................. 13REDUCEDVELOCITY4.5 ........................................................................................ 14
REDUCEDVELOCITY5 ........................................................................................... 14REDUCEDVELOCITY5.5 ........................................................................................ 15REDUCEDVELOCITY6 ........................................................................................... 16REDUCEDVELOCITY6.5 ........................................................................................ 16REDUCEDVELOCITY7 ........................................................................................... 17REDUCEDVELOCITY7.5 ........................................................................................ 17REDUCEDVELOCITY8 ........................................................................................... 17
Conclusions ............................................................................................... 19SECOND SURFACES IN REDUCEDVELOCITIES 5 AND 5.5 ............................... 19PROTRUSION DOWNWARD OF THE 0NET POWERSURFACE ......................... 20ELONGATION OF THE NEGATIVE NET POWERSURFACES ............................ 21SHRINKING OF THE 0.3NET POWERSURFACE AT REDUCEDVELOCITY8 . 22GENERAL CONCLUSIONS....................................................................................... 23
Appendix 1Tables and Figures ............................................................. 25TABLES....................................................................................................................... 25GRAPHS...................................................................................................................... 27
APPARATUS ............................................................................................................... 35Appendix 2MATLab Code ................................................................... 39
Code for Generating Graphs. .................................................................................... 39Preprocessing code ...................................................................................................... 47Code for Detecting Peaks in Graphs ......................................................................... 54
Appendix 3 Coupled Damped Harmonic Oscillators: A Primitive lookat the larger effects of Vortex-Induced Vibrations ................................... 57
MODEL....................................................................................................................... 57NUMERICALANALYSIS ........................................................................................... 58
CONCLUSIONS .......................................................................................................... 64
Bibliography ............................................................................................. 66ARTICLES ................................................................................................................... 66PICTURES ................................................................................................................... 66
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Table of FiguresFigure 1: Representation of VIV on an Oil Rig ........................................................ 5Figure 2: Top View of Vortices Shedding Off of a Cylinder .................................. 6Figure 3: Examples of Turbulent and Laminar Flow ............................................... 7Figure 4: Side Representation of Vortices Shedding Off of a Cylinder................. 8Figure 5: Normalized Average Power for Reduced Velocity 7 ............................. 14Figure 6: Normalized Total Average Power for Reduced Velocity 4.5 ............... 27Figure 7: Normalized Total Average Power for Reduced Velocity 5 .................. 28Figure 8: Normalized Total Average Power for Reduced Velocity 5.5 ............... 29Figure 9: Normalized Total Average Power for Reduced Velocity 6 .................. 30Figure 10: Normalized Total Average Power for Reduced Velocity 6.5 ............. 31Figure 11: Normalized Total Average Power for Reduced Velocity 7 ................ 32Figure 12: Normalized Total Average Power for Reduced Velocity 7.5 ............. 33Figure 13: Normalized Total Average Power for Reduced Velocity 8 ................ 34Figure 14: Representation of a Side view of the Apparatus .................................. 35Figure 15: 3D Visualization of the Apparatus ......................................................... 36Figure 16: Picture of the Actual Apparatus .............................................................. 37
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Samuel Roberts 1
Acknowledgements
This research was conducted and this paper was written with the helpof Dr. Jason Dahl, Professor Michael Triantafyllou and Gary Garber.
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Samuel L. Roberts
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Abstract
In the past, forced vibration experiments, for analyzing vortices, have
dealt extensively with only one axis of motion of a cylinder. It is much easier to
experiment with only one axis of motion than it is with two and in the past it
was assumed that the other axis of motion would have some impact but not a
large amount. More recent research, however, has shown that the second axis,
in some circumstances, can have a major impact on the vortices formed and,
therefore, the response the cylinder has.
This research investigated two axis motion of a cylinder, varying the
lateral axis amplitude as well as the transverse axis amplitude, the phase between
them, and the reduced velocity. Investigation into the matrix of obtained force
readings reveals that it is consistent, with respect to power transfer, with
previously conducted free vibration experiments with similar parameters.
Analysis reveals that there are many regions within the force matrix which
ought to be investigated more deeply, as the parameter values do not yield acomplete picture of the behavior of the system. These regions are described and
analyzed individually.
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Samuel L. Roberts
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Introduction
Vortices
Vortices occur everywhere in nature. Many types of movement of an
object through fluid can shed vortices. A vortex can be of many different sizes,
and can have many different shapes, but they have common properties. Hans
Lugt asserts that one way to define a vortex is as the rotating motion of a
multitude of material particles around a common center (Lugt 18). This
definition is able to cover vortices of all magnitudes, from quantized vortices in
liquid helium, to the rotation of galaxies (Lugt 26). One of the most common
examples of vortices,
however, is when an
object moves through
water, or when water
flows past an object.
The actual
rotation and
composition of a vortex
is not the only
interesting element. The
actual creation of the
vortices can effect a
force on the object the
vortex shed from.
Potentially, vortices can
shed at a certain
frequency and if that
frequency is close to the
Figure 1: Representation of VIV on an Oil Rig
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Forced Vortex-Induced Vibrations on a Cylinder with Two Axes of Motion
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Figure 2: Top View of Vortices Shedding Off of a Cylinder
natural frequency of the object, it can cause large oscillations through
resonance. This is called vortex-induced vibration, or VIV.
It is critical to understand vortices within the topic of ocean
engineering; VIV can play a major factor in the life span of various oceanicstructures. For example, any long, flexible, cylindrical object that is surrounded
by water is susceptible to VIV.
Damage from VIV
VIV can cause damage to any object in a fluid. This includes buildings
in air. Most concern with VIV, however, is centered on structures located in the
ocean. This includes cables and riser tubes on oil rigs as well as undersea pipes.
If the current of the water moves at the right velocity, various parts of the total
length can be excited by resonance. This resonance can cause energy to travel
along the structures in the form of more vibration. These vibrations, over time,
can severely damage the structural integrity of the structure. This is an
important consideration in the design and construction of ocean structures. In
order to account as best as possible for the results of VIV, it must be
understood as best as possible.
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Important Numbers
The Strouhal Number
What dictates the relationship between the frequency of vortex
shedding and the velocity of water? The relationship is represented by the
Strouhal number, given by the equation,
U
DfSt s
Equation 1: Strouhal Number
Where St is the Strouhal number, fs is the frequency of the vortex shedding
directly behind the
cylinder, D is the
diameter of the
cylinder, and Uis the
velocity of the
stream. (Belvins 15).
The units on the
right side cancel out
completely, making
the Strouhal number
non-dimensional.
The Strouhal
number is very useful, because for the general realm of Reynolds number of the
cylinders in the experiments, the Strouhal number is about 0.2 (Belvins 15 fig 3-
3).
The Reynolds Number
The Reynolds number is a dimensionless number used, in the case of
these experiments, to group together similar regimes of fluid flow around the
Figure 3: Examples of Turbulent and Laminar Flow
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cylinder (Belvins 14 fig 3-2). The
Reynolds number dictates when the flow
is turbulent, chaotic, versus laminar,
smooth. Flow separation can occur when
the boundary layer of fluid separates from
the cylinder. This layer becomes the
vortices that shed off the cylinder. As the
Reynolds number increases, the flow
separates off of the cylinder sooner and
produces larger vortices. This is importantbecause the Strouhal number is determined by the Reynolds number and
geometry for the circumstances of these experiments. (Belvins 15).
The Reduced Velocity
Another very important number is the reduced velocity. The equation
for the reduced velocity is very similar to Strouhal number. The reducedvelocity is given by:
Df
UV
s
r .
Equation 2: Reduced Velocity
The reduced velocity is a dimensionless way to relate the velocity of the fluid
flow, or the velocity of the cylinder, to the frequency of movement and the
diameter. As can clearly be seen, the reduced velocity is the inverse of the
Strouhal number. This would seem to imply that the reduced velocity that a
cylinder would have free vibrations at ought to be around 5, the inverse of the
Strouhal number 0.2. It is for this reason that the specific reduced velocities
used were chosen.
Figure 4: Side Representation ofVortices Shedding Off of a Cylinder
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Why Forced Vibrations?
There are several reasons to study the forces acting upon a cylinder
with forced oscillations. First among them is to understand under what
circumstances free vibrations can occur naturally. Due to the law of
conservation of energy, the net energy of a closed system cannot change. From
force and velocity data, it is possible to calculate the total average power
equation. Looking at the circumstances where the average power is zero,
different the types of motion can be inferred that are potentially possible in a
cylinder undergoing free vibrations.
While the zero power points are useful for free vibration prediction, the
positive and negative net power points are useful in the study of how vibrations
at one part of a long flexible cylinder interact with other parts. Because the
system is not the ideal closed system, energy is transferred along the cylinder in
wave form, and may be lost to structural damping. In situations with a positive
net power, something must be adding energy to the system: the fluid forces. Ifthe forces that result from the movement of the fluid around the cylinder add
power, something must be removing it from the system. In the case of the
apparatus, that would be the motors. In the case of the long, flexible cylinder,
the energy is lost to structural damping. This is how energy is transferred along
the cylinder. With the data collected from the experiments, one could
extrapolate potential reactions of various parts of the long cylinder, as forced bythe parts excited by VIV.
Why 2-Axes of Motion?
Historically, VIV experiments have been conducted with only the
transverse axis of motion. It was believed that the lateral motion would not
have much impact on the vortices and the forces that result. More recently,
however, it has been realized that there can be very different results depending
on the cylinder, particularly the mass ratio,
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f
s
m
m,
Equation 3: Mass Ratio
where ms is the mass of oscillating structure and mf is the mass of displaced fluid
(Jauvtis 2004, 24) and motion. Specifically, the periodic forces demonstrate that
there is a third harmonic component of the generally accepted and understood
forces. If this third harmonic can cause enough difference in the action of the
cylinders, it could have impacts on the design and implementation of structures
designed to function in environments where VIV is a major factor.
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Methods
These experiments were conducted on a small test tank of dimensions
2.4 meters long by 0.75 meters wide by 0.7 meters deep. The cylinder had a
span, length in the water, of 0.6858 meters with a diameter of 0.0381 meters and
was made out of aluminum. The X and Y motions of the cylinder were
controlled by two Parker Trilogy linear motors controlled by PMAC software.
Force measurements were gathered by a JR3 Model 20E12A-125 six-axis load
transducer. Images of this apparatus can be seen in the appendix.
In this tank, 2240 runs were conducted. Four degrees of freedom were
varied: the dimensionless lateral amplitude, between 0 and 0.75 in increments of
0.15; the dimensionless transverse amplitude, between 0.25 and 1.5 in
increments 0.25; the phase between, between -180 degrees and 135 degrees in
increments of 45 degrees; and the reduced velocity of the cylinder, between 4.5
and 8. The position was controlled by the two linear motors, one for each axis.
The apparatus was driven up the tank for each run and reset. Data was collected
for the position of each linear motor, the velocity of motor that ran the
apparatus up and down the tank, and the forces on the cylinder. This meter
collected data for each axis, lateral and transverse. This entire process, including
the running of the test, resetting of the apparatus, and collection of the force
readings, motor readings and velocity readings, was completely programmed
with LabVIEW. Processing of the data was done in MATLab.
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Results and Analysis
From all the data collected, many different values can be calculated.
The value considered here, however, is power. As previously mentioned, the
total power in the system can tell a lot about what is going on between the
cylinder and external factors. If the total power was zero, then all the energy
gained by the cylinder was also lost, representing a configuration for possible
free vibration. If the average total power is positive, the fluid forces are adding
energy to the system, and if the average total power is negative, the fluid forces
are taking energy out of the system: energy is transferred from the cylinder tothe fluid.
Since there are four degrees of freedom in this experiment, there are
several ways to graphically represent the total power in the system over many of
the experimental runs. Here, each of the eight reduced velocities is held
constant, making the axes the lateral amplitude, the transverse amplitude, and
the phase shift between them. All graphs can be found in the appendix.
The x and y axes of the graph, representing the X/D amplitude of the
cylinder and the Y amplitude of the cylinder respectively, have been
nondimensionalized by dividing them by the diameter of the cylinder. The
average power values have also been nondimensionalized.
The surfaces present are total average power surfaces. The colored
surfaces correspond to colors in the legend on the side. Here, the green surface
represents the values with 0 net power. These are the values with theoretically
could be observed in free vibration situations.
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Figure 5: Normalized Average Power for Reduced Velocity 7
Reduced Velocity 4.5
In this graph, the 0 net power surface is much smaller compared to the
other two negative net power surfaces. From this graph on its own we can tell
that we should not expect to see many free vibrations when the reduced velocity
is 4.5. It is clearly far more likely to see damped behavior around this reduced
velocity than free vibrations. It is difficult, however, to gather very much
information about the behavior of the system.
Reduced Velocity 5
Moving onto reduced velocity 5, we see a significantly larger 0 net
power surface. This seems reasonable because of the relationship between theStrouhal number and the Reynolds number. Again we have large negative net
power surfaces. In this graph, however, they are many more areas comparable
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to the 0 net power surface. In reduced velocity 5, we find something we did not
find in reduced velocity 4.5: a positive net power surface. Although this surface
is very small relative to the 0 and negative net power surfaces, it represents the
fluid forces acting upon the cylinder to amplify the motion rather than dampen
it. Another interesting note of reduced velocity 5 is the second smaller surface
for 0 net power just below the larger one, which is very likely an extension of
the larger one, simply without data points collected to confirm this.
Reduced Velocity 5.5
Each surface in reduced velocity 5.5 is, again, another step larger thanthey were in reduced velocity 5. Most noticeable are the growth of the 0.1 net
power and 0 net power surfaces. The positive net power surface, the yellow 0.1
surface, is much larger than the tiny area it was in reduced velocity 5. This
reveals that at reduced velocity 5.5, the cylinder is much more likely to respond
to the fluid forces from the faster fluid flow around it: this seems reasonable.
Again, there is another smaller surface of the 0 net power surface; this
one is very similar to the pervious, but for larger phase values. This one appears
to simply be an extension of the larger 0 net power surface, and some more data
collection in that area of the values would most likely confirm this.
What makes is of particular note is the small yellow surface found
around X/D amplitude of 0.002, Y/D amplitude of 0.5 and phase shift ofapproximately -135. In the other two reduced velocities the surfaces were
generally single surfaces, the only exception being the smaller 0 net power
surface that seemed to be an extension of the larger one. This smaller 0.1 net
power surface however, does not appear to be an extension of the larger 0.1 net
power surface. It seems more likely that it is an entirely different surface from
the larger one. It is impossible to tell from this graph and further investigation is
warranted.
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Reduced Velocity 6
Analyzing Reduced Velocity 6, we see that all four surfaces grew
considerably larger. The 0.1 net power surface grew perhaps the most of all four
of the surfaces. This makes some sense, because the faster the velocity, the
more the fluid forces can add energy to the system. The other, smaller 0.1 net
power surface disappeared completely, which makes its appearance in the
reduced velocity 5.5 graph all that more pertinent.
The 0 net power surface grew in much the same fashion as the 0.1 net
power surface. In fact, it is much the same shape, except for the sections that
seem almost broken off from the rest and folded downward. The two
negative net power surfaces continued to expand out as if they are making room
for the 0 net power surface to grow within.
Reduced Velocity 6.5
In the graph of reduced velocity 6.5, the biggest change from reducedvelocity 6 is the formation of the 0.3 net power surface within the 0.1 net power
surface. It is relatively small compared with even the 0.1 net power surface.
The rest of the surfaces continued to grow. They all seem to be
elongating along the Y/D amplitude axis. The two negative net power surfaces
have taken on much more of a partial cylindrical shape around the 0 and
positive net power surfaces. It would appear that they would reach points
similar to the 0 net power surface if we were able to see further down the Y/D
amplitude axis.
The 0 net power surface is longer along the Y/D amplitude axis as if it
has been stretched out. It should be noted that it continues to have the section
pointing downward instead of closed off as the rest do.
The 0.1 net power surface has grown in very much the same way as the
0 net power surface: elongated as if stretched. It is closer to being closed off
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than it has been in the previous graphs. This is in contrast with the 0 net power
surface, which is not nearly closed around low Y/D amplitudes.
Reduced Velocity 7
There is minor growth in the graph of reduced velocity 7. It is,
however, extremely minor and the graph overall looks almost identical to the
graph of reduced velocity 6.5. No real conclusions can be made
Reduced Velocity 7.5
The graph of reduced velocity 7.5 is similar to the previous two. The
growth from the graph of reduced velocity 7 shows a larger change than the
minor growth between the graphs of reduced velocity 6.5 and 7. The difference
can really be seen in the 0 and 0.1 positive net power surfaces. The 0 and 0.1 net
power surfaces are wider than they previously were. Also, the protrusion
downward from the 0 net power surface is much less than it has been since the
graph of reduced velocity 5.5. The 0.1 net power surface is almost identical in
shape to the 0 net power surface, just a little smaller.
The 0.3 net power surface has grown a little longer than it was in
reduced velocities 6.5 and 7, but it does not seem any wider.
Reduced Velocity 8
Here we see significant change in all of the surfaces, except for the 0.3
net power surface, which changed slightly back to the smaller surface found in
reduced velocities 6.5 and 7. While this change is interesting, it does not reflect
the overall behavior of the surfaces in the graph of reduced velocity 8.
The negative net power surfaces are now much larger. In fact, much of
the -2 net power surface cannot be seen within the realm of these experiments.
The -1 net power surface is also much larger, but still very much contained in
the bounds of the X and Y/D amplitudes and the phase shift.
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The 0 net power surface is approximately as long as it was previously,
but is more of a closed surface towards the lower end of the Y/D amplitudes
then it had been in previous reduced velocities It appears to be multiple
surfaces, this time with two smaller surfaces, on at lower and one at higher
phase shifts. This, however, is similar to the graph of reduced velocity 5. They
are most likely just continuations of the larger surface but within the
experimental data obtained, it is not possible to know for certain.
The 0.1 net power surface is essentially identical to the 0.1 net power
surface in the previous graph, reduced velocity 7.5. The only difference is that
the part of the surface at the lower Y/D amplitudes is much closer to being a
closed surface than in pervious graphs.
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Conclusions
Within the graphs of reduced velocities, there are several
interesting characteristics. Unfortunately, most of these representations are
incomplete, and therefore it is not possible at this time to determine a precise
behavior of the energy and power of the system based on the data attained. It is
clear that more data ought to be collected: for example, more data points
between those already gathered and analyzed. Since general trends are
discernable from the current data, the specific parts of the graphs that would be
particularly interesting to study further are raised and discussed below.
Second Surfaces in Reduced Velocities 5 and 5.5
As mentioned in the Error! Not a valid link., there are 2 surfaces for 0
net power and 0.1 net power in the graphs of reduced velocities 5 and 5.5. In
the graph of reduced velocity 5, there is a small 0 net power surface. It appears
as if it is simply an extension of larger one, but the current graph it cannot be
discerned. This could be an area of further interest to collect more data:
VariableApprox. Lower Bound
of Area
Approx. Upper Bound
of Area
Phase () -190 -100
Y/D 0.3 0.6
X/D 0 0.02
Table 1: Area of Second Surface in Reduced Velocity 5
The graph of reduced velocity 5.5 has a far more interesting second
smaller surface for 0.1 net power. The smaller space does not appear to be a
continuation of the larger space as was the case for reduced velocity 5. It does
not seem reasonable that this smaller surface would be connected with the
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larger one, because the 0 net power surface appears to divide the two of them.
Because of this, it is recommended that more data be collected within the
following region:
Variable Approx. Lower Boundof Area
Approx. Upper Boundof Area
Phase () -150 -100
Y/D 0.2 0.6
X/D 0 0.017
Table 2: Area of One Second Surface in Reduced Velocity 5.5
This would give a clearer picture of the smaller surface.
The graph of reduced velocity 5.5 also has a smaller surface for 0 net
power. This surface appears to just be a continuation of the larger one, and
more data in the following region would most likely confirm this:
VariableApprox. Lower Bound
of Area
Approx. Upper Bound
of Area
Phase () 50 150
Y/D 0.2 0.6
X/D 0 0.015
Table 3: Area of Another Second Surface in Reduced Velocity 5
Protrusion Downward of the 0 Net Power Surface
In the graphs of reduced velocities from 5.5 to 8, there is consistently a
portion of the 0 net power surface that protrudes downward towards lowerphases. The full table of regions in each reduced velocity containing this
protrusion can be found in the appendix. The following table represents the
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broad area of values which might potentially have further data collected for all
reduced velocities from 5.5 to 8:
VariableApprox. Lower Bound
of Area
Approx. Upper Bound
of Area
Phase () -200 0
Y/D 0.2 0.6
X/D 0 0.35
Table 4: Area of Protrusion Downward of the 0 Net Power Surface
Elongation of the Negative Net Power Surfaces
As the reduced velocity increased, so did the general size of the
surfaces. In particular, the negative net power surfaces grew to expand beyond
the scope of the data collected. While this may not be critical information, if we
desire to understand the full potential behavior of a cylinder, it would be a good
idea to analyze the possible net power beyond the maximum X/D and Y/D
amplitudes tested. This would involve further data collection in the following
region:
VariableApprox. Lower Bound
of Area
Approx. Upper Bound
of Area
Phase () -200 150
Y/D 1.5 Unknown
X/D 0.75 Unknown
Table 5: Area of Elongation of the Negative Net Power Surfaces
The unknown values would be enough to see the surfaces close off, as can be
seen on both the -2 and -1 net power surfaces in the graph of reduced velocity
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5, and in only the -1 net power surface in the graphs of reduced velocities 5.5
and 6. The upper boundaries would be determined by the surfaces.
Shrinking of the 0.3 Net Power Surface at Reduced
Velocity 8
One of the most interesting characteristics of the graph of reduced
velocity 8 is the way the 0.3 net power surface shrinks instead of grows as was
observed with all previous reduced velocities. This is the first and only time that
a surface significantly shrinks between a reduced velocity and the next. Because
there is limited data in the area of the values at which the 0.3 net surface exists,
more specific data ought to be collected. It would be advantageous to collect
more data points between the previously collected data points specifically within
the following region of the values:
Variable Approx. Lower Bound
of Area
Approx. Upper Bound
of Area
Phase () -100 -10
Y/D 0.35 1.1
X/D 0.12 0.33
Table 6: Area of 0.3 Net Power Surface at Reduced Velocity 8
For this characteristic, it would also be beneficial to test reduced velocities
between the previously measured ones, 7, 7.5, and 8. Measuring data between
reduced velocities would be helpful in many of the previous situations as well.
Here, however, it would be most useful because the main question raised here
by the current data is what happens to the 0.3 net power surfaces between
reduced velocity 7 and reduced velocity 8?
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General Conclusions
Clearly, the data collected with these experiments reveals a lot of useful
information, and has revealed several holes for further investigation has several
holes in it. On the whole, there are clear surfaces representing 0 net power in
the system. These surfaces should match up well with free vibration
experiments. The positive and negative net power surfaces correspond to
energy being added and removed by fluid forces respectively: the first being
forcing, the second, damping.
This report, however, focused on the areas where the system should be
investigated further. Since more data overall can only help the understanding of
the system, several specific regions to study further were outlined, described,
and analyzed.
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Appendix 1Tables and Figures
Tables
Protrusion Downward of the 0 Net Power Surface
Reduced Velocity Variable Approx. LowerBound of Area
Approx. UpperBound of Area
5 Phase () -200 0
Y/D 0.3 0.6
X/D 0 0.05
5.5 Phase () -200 0Y/D 0.2 0.2
X/D 0 0.2
6 Phase () -200 -40
Y/D 0.2 0.55
X/D 0 0.25
6.5 Phase () -200 -50
Y/D 0.2 0.6
X/D 0 0.3
7 Phase () -200 -50
Y/D 0.2 0.6
X/D 0 0.35
7.5 Phase () -200 -50
Y/D 0.2 0.55
X/D 0 0.2
8 Phase () -175 -50
Y/D 0.2 0.55
X/D 0 0.3
Table 7: Locations of the Protrusion
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Table 8: Locations of the 0.3 Net Power Surface
Shrinking of the 0.3 Net Power Surface
Reduced Velocity Variable Approx. LowerBound of Area
Approx. UpperBound of Area
7 Phase () -100 -40
Y/D 0.4 1.1
X/D 0.12 0.21
7.5 Phase () -50 -10
Y/D 0.35 0.8
X/D 0.12 0.2
8 Phase () -80 -35
Y/D 0.5 0.9
X/D 0.26 0.33
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Graphs
Figure 6: Normalized Total Average Power for Reduced Velocity 4.5
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Figure 7: Normalized Total Average Power for Reduced Velocity 5
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Figure 8: Normalized Total Average Power for Reduced Velocity 5.5
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Figure 9: Normalized Total Average Power for Reduced Velocity 6
ht