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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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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3

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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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

6

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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8

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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10

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.


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.


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


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:


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:


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:


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