UNDERWATER EXPERIMENTAL PLATFORM FOR STUDYING THE BEHAVIOR
AND PERFORMANCE OF A SUBMERSIBLE PROPELLED BY A
FLUID-CONVEYING FLUTTERING TAIL
By
Paul Christian Streﬂing

A THESIS
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
MASTER OF SCIENCE
Mechanical Engineering
2011

ABSTRACT
UNDERWATER EXPERIMENTAL PLATFORM FOR STUDYING THE
BEHAVIOR AND PERFORMANCE OF A SUBMERSIBLE PROPELLED
BY A FLUID-CONVEYING FLUTTERING TAIL
By
Paul Christian Streﬂing
We have successfully designed and developed an underwater experimental platform for
studying the dynamic behavior and performance of a novel submersible, which is propelled by
the synergistic combination of jet action and oscillatory motion of a ﬂuid-conveying ﬂexible
tail. The submersible, which we have named the Synergistically Propelled Ichthyoid, or
SPI, ingests water through an oriﬁce in its fore-body and expels it through a jet located at
the rear end of the tail. The ﬂuid ﬂow through the ﬂexible tail induces ﬂutter instability
and generates a traveling wave along the length of the tail. A single prime mover in the
form of a pump is used for propulsion but the thrust is generated by the combination of jet
action and jet-induced tail oscillations. The submersible has been designed to accommodate
prime movers for propulsion and buoyancy control, sensors for feedback control, and active
and passive ﬁn assortments for maneuvering and stability. In addition, the submersible
has a data acquisition system and on-board computing platform for autonomous and semiautonomous operation. With over 30 hours of underwater testing we have converged on
a design that is robust and dependable for further experimental research. This platform
is ideally suited for investigating thrust and eﬃciency characteristics of the tail waveform
for diﬀerent tail designs and is expected to provide clues for design of rapid tail-assisted
maneuvers.

if graduate_student(Ben_LeVesque,2015)1
echo "This thesis is dedicated to Ben LeVesque in hopes that it
will persuade him into perusing graduate school in the near future."
else
echo "This thesis is dedicated to my loving family."
end

1 Clearly

graduate student(name, year) is a function that can read into the future.

iii

ACKNOWLEDGMENTS

I would like to thank all of those people who helped make this thesis possible.
First, I wish to thank Dr. Ranjan Mukherjee for the design freedom he allowed and
encourage in regards to SPI. I am grateful that I was fortunate enough to work with him
and look forward to working with him in the future. I would also like to thank Aren Hellum
for his exceptional understanding of carangiform dynamics and overall ﬂuids as a whole. He
is a ﬂuid mechanician on a Fossian scale.
I would like to thank my family for their continued support and their curiosity in regards
to my research. Many thanks to Penniless Heiress and Rooftop Sy as well. Their lack of
interest in engineering and overall common sense never fails to amuse me.

iv

TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

List of Figures

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Nomenclature

xi

1 Introduction
1.1 Motivation and Background of Submersibles . . . . . . . . . . . . . . . . . .
1.2 Propulsion by Fluid Conveying Fluttering Tails . . . . . . . . . . . . . . . .

1
1
3

2 Dynamic Model

7

3 Mechanical Design
3.1 Requirements . . . . . . . . . . . .
3.2 Hull Construction . . . . . . . . . .
3.2.1 Nose Cone . . . . . . . . . .
3.2.2 Tail Clamp . . . . . . . . .
3.2.3 Buoyancy Control . . . . . .
3.2.4 Hydroplanes/Diving Planes
3.3 Prime Mover . . . . . . . . . . . .
3.4 Tail Construction . . . . . . . . . .
3.4.1 Material Selection . . . . . .
3.4.2 Fabrication . . . . . . . . .

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4 Computing Platform, Actuators, Sensors and Circuits
4.1 Computing Platform Requirements . . . . . . . . . . . .
4.2 Pumps and Servos . . . . . . . . . . . . . . . . . . . . .
4.3 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Microcontroller and Digital to Analog Converter . . . . .
4.5 Wireless Interface . . . . . . . . . . . . . . . . . . . . . .
4.6 Power Supply . . . . . . . . . . . . . . . . . . . . . . . .
5 Software
5.1 Introduction . . . . . . . . . . . .
5.2 Client Software . . . . . . . . . .
5.3 Wireless Encoding and Decoding
5.4 Microcontroller Software . . . . .

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41
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45

6 Experimental Methods
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Free Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Guided Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47
47
49
51

7 Experimental Results

53

8 Future Improvements

57

9 Conclusions and Future Research

60

A Dynamic Model Literature Review
A.1 Uniform Velocity Proﬁle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Non-Uniform Velocity Proﬁle . . . . . . . . . . . . . . . . . . . . . . . . . .

63
63
70

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

vi

LIST OF TABLES

3.1

Sample of acceptable materials for ﬂexible tails. Note that current ﬂexible
tails have been constructed out of latex. . . . . . . . . . . . . . . . . . . . .

23

4.1

Speciﬁcation of the HiTec HS-5056MG . . . . . . . . . . . . . . . . . . . . .

30

4.2

Gyro Electrical Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . .

34

4.3

Accelerometer Electrical Characteristics

. . . . . . . . . . . . . . . . . . . .

34

5.1

Encoding and checksum values for the wireless algorithm. . . . . . . . . . . .

45

7.1

Mean speeds of the three tested conﬁgurations in meters and body lengths
per second for all runs performed to date. . . . . . . . . . . . . . . . . . . .

54

Mean speeds of the three tested conﬁgurations in meters and body lengths
per second for all submerged runs. . . . . . . . . . . . . . . . . . . . . . . . .

55

7.2

vii

LIST OF FIGURES

1.1

2.1

3.1

3.2

3.3

3.4

3.5

3.6

The RoboTuna platform, on display at the Science Museum in London, was
built by a team of researchers at the Massachusetts Institute of Technology
(MIT) led by Michael Triantafyllou. David Barrett built the ﬁrst RoboTuna
known as Charlie I in 1995 for his PhD thesis. David Beal and Michael
Sachinis, who introduced modiﬁcations including a cable-pulley system to
produce RoboTuna II. Picture Credit: Science Museum/Science & Society
Picture Library. For interpretation of the references to color in this and all
other ﬁgures, the reader is referred to the electronic version of this thesis. . .

5

Neutral stability curves for diﬀerent values on rigid body mass fraction µ from
[23]. Regions to the left of the dotted line on each cure represent a region of
negative thrust produced from the tail oscillations. . . . . . . . . . . . . . .

10

(a)-Two piece monocoque hull. (b)-Modular nose cone. (c)-20 AA NiMh
Batteries. (d)-Rule Inline 500 GPH bilge pump. (e)-Dorsal Fin. (f)-Anal
Fin. (g)-Tail Clamp. (h)-Charging and data port. (i)-External location of
the charging and data port. . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

SPI’s custom designed seal geometry includes a male/female interface that
compresses a bead of silicon RTV as seen in the explosion in this ﬁgure. The
silicon RTV is user replaceable and easy to inspect. . . . . . . . . . . . . . .

14

A cross section view of SPI’s current nose cone. External sealing to the hull is
accomplished with two silicone o-rings that live in the nose cone. The wetted
surfaces of the nose cone are a continuation of the hull itself creating a smooth
transaction into the oriﬁce. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

Exploded view of nose cone installation showing how the hull seal is routed
around the centerline of the hull to avoid a three part interface resulting in a
less complicated sealing interface. . . . . . . . . . . . . . . . . . . . . . . . .

17

An exploded view of the tail clamp showing the nozzle, hose barb and sealing
features of the hull. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

The SPI tail clamping mechanism. The set screws are used to clamp the sheet
metal around the latex tail to ensure the clamped boundary condition. . . .

18

viii

3.7

Two 120ml buoyancy tanks in fore (a) and aft (b) locations in the hull. These
two buoyancy tanks can be used for static and dynamic depth control as well
as static and dynamic rake. . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Hydrofoils installed on SPI incorporating a symmetric design which has been
chosen such that lift is not created at zero camber angles. This reduces unnecessary drag as static depth is controlled by the buoyancy chambers. (a)
Isometric view of the hydrofoil location. (b) Cross section of the hydrofoil.
(c) Plan view of the hydrofoil. . . . . . . . . . . . . . . . . . . . . . . . . . .

21

A swift and energetic Blacktip shark with its pectoral ﬁns. These large, developed pectoral ﬁns are used for dynamic depth control and aggressive trajectory
control; the Blacktip shark has been known to launch itself out of the water
and spin three to four times about its axis before returning to the water.
Speed attained by the shark during these jumps has been estimated to average 6.3 m/s [39, 8]. This image was originally post to Flickr by StormyDog.
Shark. 10 May 2011. http://ﬂickr.com/photos/32184789@N00/134610871. .

21

3.10 Two manufacturing methods for creating latex ﬂexible tails. (Top) a laminated tail that is held together with silicon RTV and placed in a two piece
compression mold. (Bottom) a butt constructed tail that is also held together
with silicon RTV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.11 The tail mold for ﬂexible tails that are made using the laminated approach.
Pins are used to ensure alignment between the two halves. . . . . . . . . . .

25

3.8

3.9

4.1

Circuit diagram of SPI’s single direction motor driver. Where Q1 is the NChannel MOSFET, D1 is the ﬂyback diode, C1 acts as a ﬁltering capacitor,
R1 is a pull-down resistor and AGND is the circuits analog ground. . . . . .

29

Circuit diagram of SPI’s switching regulator for producing a clean 5 volts for
the hydroplane servos. NOTE: Need to deﬁne F1,C2,R2,D3 and L1(or refer
to spec sheet perhaps?). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

4.3

Switch output source saturation versus source current for the MC34167 . . .

32

4.4

The SPI antenna which passes through the hull and extends 22.6 cm above
the waterline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

An exploded NiMh cell as a result of a short in the system can be seen here.
The chemical resistant heat shrink completely contained the failure protecting,
SPI, the environment and personnel. . . . . . . . . . . . . . . . . . . . . . .

40

4.2

4.5

ix

5.1

A high level ﬂow chart that lays out the ﬂow of communications from the user
to SPI including the communications protocols and media that are used. . .

42

Three diﬀerent tail geometries have been testing with the experimental platform. (Top) A rigid tail with identical geometry and buoyancy of a ﬂexible
tail. (Middle) A rigid tube that has the same length and ﬂuid jet exit conditions as a corresponding rigid of ﬂexible tail. (Bottom) The ﬂexible tail, made
of neutrally buoyant latex a constructed using the butt-welded method. . .

49

SPI testing under guide string conditions with a rigid tail. The guide strings
only purpose is to provide a straight trajectory for SPI; power and communications are identical between free and guided locomotion. . . . . . . . . . . .

52

High-speed images of the SPI acquired over a period of 1 sec, soon after it
started from rest in the MSU swimming pool. The SPI was moving with an
average speed of ≈ 0.4BL/s, less than the maximum speed of ≈1.15BL/s; note
the large deﬂections of the tail which occur during acceleration. . . . . . . .

54

Average speeds of the conﬁgurations listed in Table 7.1 which include 175
runs, with error bars denoting the limits of the 95% conﬁdence intervals.
Points denoted by a indicate runs which traveled North in direction while
points denoted with a ♦ indicate runs in the opposite direction. . . . . . . .

55

Average speeds of the conﬁgurations listed in Table 7.2 which were performed
completely under the surface of the water totaling 60 runs. Error bars denoting
the limits of the 95% conﬁdence intervals, points denoted by a indicate runs
which traveled North in direction while points denoted with a ♦ indicate runs
in the opposite direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

The proposed calibrated elbow that will provide SPI with the means necessary
to provide a consistent ﬂow rate to the tail regardless of the state of the power
supply. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

A.1 A ﬂuid-conveying pipe and a magniﬁed view of a small element along its length
including the cross sectional dimensions. . . . . . . . . . . . . . . . . . . . .

65

6.1

6.2

7.1

7.2

7.3

8.1

A.2 Free-body diagram of (a) a ﬂuid element and (b) its corresponding pipe element. 68
A.3 Cross sectional view of three concentric ﬂuid volumes described by triple plug
ﬂow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

71

Nomenclature
βe

nondimensional mass ratio for the external ﬂuid

βi

nondimensional mass ratio for the internal ﬂuid

λ

nondimensional parametrization coeﬃcient of rigid body’s length

µ

nondimensional parametrization coeﬃcient of rigid body’s mass

ψB

nondimensional parametrization coeﬃcient of rigid body’s shape

E

Young’s Modulus (MP a)

I

Area Moment of Inertia (m4 )

U

Internal ﬂuid velocity relative to the pipe (m/s)

ue

nondimensional external ﬂuid velocity

ui

nondimensional internal ﬂuid velocity

xi

Chapter 1
Introduction

1.1

Motivation and Background of Submersibles

It is a well-known fact that 71% of the Earth’s surface is covered by water; the majority
of this water, approximately 97% is contained in the oceans. Oceans are the life blood
of Earth, they drive the weather, regulate temperature and ultimately support life as we
currently know it. It is imperative that we understand this aquatic ecosystem to ensure
that it remains strong and resilient tomorrow as it does today. The oceans are a home to
numerous life-forms, as it currently stands only 230,000 marine species have been identiﬁed,
but it is estimated that over 2 million species exist. This is not surprising considering that
only 5% of the ocean has been explored. Clearly there is a need for greater underwater
exploration to obtain a better understanding of the most important ecosystem on earth.
It is not a trivial task to explore these vast bodies of water though. An eﬃcient submersible platform that can withstand extreme pressures and cover great distances is needed.
Ocean exploration from the surface of the water is signiﬁcantly easier than doing so from be1

low the surface; surface vessels existed for more than 10,000 years before the ﬁrst submersible
know as the Drebbel which was built in 1620. However, there are signiﬁcant beneﬁts from
exploring the oceans at depth. The ability for real time control and quick maneuverability provide beneﬁts to underwater exploration that can not be obtained from large surface
vessels.
Underwater exploration has been greatly accelerated with the use of remotely operated
underwater vehicles (ROVs) since the 1950s as the required technology to do so has been
developed and perfected. Unmanned ROVs have several beneﬁts compared to a manned
platform such as “the relative economy of development in time and equipment costs, unlimited operational endurance on site by virtue of the cable link to the surface, surface control
and coordination of project eﬀorts, ability to perform in hazardous areas without endangering personnel, ability to change or modify all system components to meet individual tasks
range needs without aﬀecting system safety or certiﬁcation status, and ease of changing
crews without disrupting the mission.” [45].
These beneﬁts inspired Navy funded ROV’s in the 1950s and 1960s for recovery of lost
objects such as the missing hydrogen bomb from the 1966 Palomares B-52 crash. The
hydrogen bomb, located by the manned DSV ALVIN submarine in 880 meters of water in
the Mediterranean sea was recovered by an unmanned torpedo recovery vehicle known as
the Cable-Controlled Underwater Recovery Vehicle or CURV-I [21]. Using technology for
the CURV and CURV-like ROVs the oil and gas industry began deploying similar ROVs in
oﬀshore drilling platforms and pipelines in the 1980s as the depths exceeded the reach of
human divers [10].
The academic and scientiﬁc communities have made substantial advancement in ROVs
2

since the 1980s producing autonomous platforms, highly accurate underwater positioning
systems, and an assortment of specialized tools for deep-sea tasks. Several key research
institutions have pushed the design of submersibles with such accomplishments as the Woods
Hole Oceanographic Institution’s (WHOI) hybrid autonomous under water vehicle (HROV)
Nereus1 , Jason, and Jason II. The Monterey Bay Aquarium Research Institute’s (MBARI)
ﬂeet of ROVs including the Tiburon and Ventana and the University of Rhode Island’s
(URI) Hercules, all capable of advanced underwater functions that have made signiﬁcant
contributions to advancement of ROVs [14, 32].
Modern technology such as microcontrollers and high capacity batteries has enabled variations and improvements in submersible platforms including miniaturizations and platforms
that have been developed for reasons other than object recovery or sea ﬂoor exploration. Such
examples include alternative propulsion methods and autonomous algorithms for schools of
robotic submersibles [9, 46]. These advanced platforms have gained the freedom of tether-less
operation with technologies common in consumer electronics as well as complete underwater
communications with technologies such as the WHOI micro-modem [15].

1.2

Propulsion by Fluid Conveying Fluttering Tails

Alternate methods of propulsion are not a new ﬁeld of study even though the technology to
support such endeavors is fairly young. The promise of highly eﬃcient, ﬁsh-like propulsion
has been a matter of academic interest since Gray’s pioneering work in the 1930s. In a 1936
paper [16], Gray determined that the speed attained by a dolphin should require approx1 In

May 2009 the Nereus dove to the bottom of the deepest surveyed point in the global ocean
known as the Challenger Deep, a small depression at the bottom of the Mariana Trench named in
honor of the original Challenger expedition of 1872 [14]

3

imately seven times the power available to the animal, a situation that has been dubbed
Gray’s Paradox, albeit a paradox that has been resolved by later workers [13].
In his seminal 1960 paper, Lighthill [28] used slender body theory to approximate the
eﬀect of the pressure ﬁeld surrounding the ﬁsh and was able to derive expressions approximating the thrust generated and power requirements of an idealized ﬁsh. Lighthill found that
a traveling waveform with a phase velocity in excess of the ﬁsh’s forward speed is required
to produce eﬃcient thrust and estimated that 90% of the power expended by a swimming
ﬁsh is available (before consideration of viscous losses) to propel the ﬁsh; the remainder is
wasted, serving only to raise the kinetic energy in the wake.
This predicted eﬃciency of this locomotion strategy provides a sound reason for robotic
investigations into ﬁshlike motion. A well-known biomimetic platform is MITs RoboTuna
[4, 47], which uses an articulated tail covered by a tail-shaped sheath. The links which make
up this tail are individually actuated in an approximation of a ﬁsh’s motion. A diﬀerent
approach was taken by Alvarado and Youcef-Toumi [48], who described a system which
used a wholly ﬂexible tail, excited with a single actuator at the base. The fundamental
diﬀerence between these approaches is that the RoboTuna’s need to independently actuate
the links of its articulated tail means that the system must essentially be driven against its
natural dynamics, whereas Alvarado’s ﬂexible tail can be induced to oscillate near its natural
frequency.
Compared to the literature produced on articulated mechanisms, work on ﬂexible tails is
relatively sparse. In addition to the aforementioned paper by Alvarado and Youcef- Toumi,
an earlier attempt to design the type of propulsor we describe here was made by Pa¨
ıdoussis
and coworkers in the mid-1970s [38]. Harper and coworkers [20] constructed an articulated
4

Figure 1.1: The RoboTuna platform, on display at the Science Museum in London, was built
by a team of researchers at the Massachusetts Institute of Technology (MIT) led by Michael
Triantafyllou. David Barrett built the ﬁrst RoboTuna known as Charlie I in 1995 for his
PhD thesis. David Beal and Michael Sachinis, who introduced modiﬁcations including a
cable-pulley system to produce RoboTuna II. Picture Credit: Science Museum/Science &
Society Picture Library. For interpretation of the references to color in this and all other
ﬁgures, the reader is referred to the electronic version of this thesis.

tail, which was actuated in series with lateral and torsional springs so that the motion of
the tail is under-actuated; this system represents a kind of intermediate form between the
inﬁnitely continuous beam-like tails and discrete systems. In this work, we propose a sort of
ﬂexible tail in which the desired traveling waveform is produced by ﬂutter instability induced
in a slender tail by conveying ﬂuid down its center rather than by an actuator at the base.
This instability is sometimes referred to as the garden hose instability, after the tendency of
such a hose to whip about when the faucet is opened too far.
The Synergistically Propelled Ichthyoid (SPI) is our implementation of an underwater experimental platform to test propulsion by means of a ﬂuttering tail. It is the third generation
of devices that have been created for this purpose. However, this iteration of SPI presented
in this paper diﬀers considerably from previous designs as it includes numerous features that
are unique to it and cannot be found on previous versions created here at Michigan State
University or anywhere else in the academic community. In addition to its rich features,
many aspects of SPI are based upon nature itself including an overall ﬁsh-like appearance
with ﬁn anatomy and curved aerodynamic surfaces often found in many species of carangi5

forms. Not only does SPI have features that are used in our testing today, provisions have
been set forth for numerous features and additions for future needs. Equipped with data
acquisition and a modular hull design this experimental platform has many opportunities for
use in the development of ﬂuttering tails as well as other submersible experimental testing.

6

Chapter 2
Dynamic Model
The concept behind propulsion by a ﬂuttering tail is based on the natural locomotion of a
carangiform and carangiform-like animals that swim by oscillatory movement of their posterior end. This movement, most notably the side-to-side motion of the caudal ﬁn produces
a thrust which propels the animal in the forward direction. In the case of SPI, ﬂutter instability is induced by conveying ﬂuid through a tube that forces the tail to oscillate and
generate a traveling wave, which provides a forward thrust in a similar means to that of a
ﬁsh. Forward thrust is provided by the oscillating tail as well as the ﬂuid jet that exits the
end of the ﬂuid-conveying pipe. The analysis of thrust from a ﬂuid-conveying pipe will begin
by considering the pipe and the pipe only. Consider a pipe that is suﬃciently ﬂexible such
that it begins to ﬂutter as the ﬂow rate of the ﬂuid increases. Whether the pipe is ﬂuttering
or not, the exit conditions of the pipe are equivalent to that of a jet or nozzle that results in
thrust from the ﬂuid jet.
For the SPI, the jet thrust will cause an acceleration in the direction opposite of the jet.
This jet will drive SPI to a velocity in which the drag forces match that of the jet force
7

and the platform reaches an equilibrium velocity. As the ﬂow rate increases and the ﬂexible
pipe begins to ﬂutter, the forward trust provided from the jet will no longer be parallel to
the center-line of the hull for all instances of time. As the tail ﬂutters back and forth, the
exit angle of the jet will pass through a range of angles that will still contribute to thrust
in the forward direction. The vectoring of the jet due to ﬂutter will result in a loss of
thrust in comparison to a straight jet. However, the oscillatory motion of the tail mimics the
traveling wave found in powerful carangiforms suggesting that further investigation may ﬁnd
that exceeding the eﬃciency of a standalone jet is possible. The SPI experimental platform
has been developed to study this phenomenon and help answer questions of feasibility and
eﬃciency, and investigate the possibility of developing control strategies that can provide
turning capability.
A complete review of the literature pertaining to the theory and experimentation of
ﬂuid conveying tubes beginning in 1878 is included in Appendix A. Which also details the
development of the equations presented in this chapter.
Theoretical development of the dynamic model that captures the instabilities of a ﬂuttering tail and the thrust produced by the oscillations has been investigated by Hellum
and Mukherjee [22, 23]. Based on the work presented in [23] the equation of motion for a
ﬂuid-conveying ﬂuid immersed pipe is even by

Y ′′′′ + u2 + u2 Y ′′ + 2 ui
e
i

βi + ue

˙
¨
βe Y + Y = 0

(2.1)

where ui and ue are the non-dimensional internal and external velocities respectively, and βi
and βe are the nondimensional mass ratios for the internal and external ﬂuid respectively.
The non-dimensional linearized boundary conditions for the tail with a clamped interface
8

with the rigid body represented at (X = 0) are given by [23]

¨
˙
[Y ′′′ + µ(Y − λY ′ )]X=0 = 0

(2.2)

¨
¨
[Y ′′ − µ(ψB + λ2 )Y ′ − λY ]X=0 = 0

(2.3)

Y ′′ (1) = 0

(2.4)

Y ′′′ (1) = 0

(2.5)

where the nondimensional coeﬃcients ψB , λ, and µ parametrize the shape, length and mass
of the rigid body. Eqs.(2.4) and (2.5) provid the boundary conditions of the free end.
Eqs.(2.1), (2.2), (2.3), (2.4) and (2.5) can be solved to obtain an expression for Y (X, T ),
4

4

An eℜ[Zn ]X ei(ℑ[Zn ]X+ℜ[ω]T ) e−ℑ[ω]T .

An ezn X eiωT =

Y (X, T ) =
n=1

n=1

(i)

(ii)

(2.6)

(iii)

which captures the the transverse deﬂection of the tail. Inspecting Eq.(2.6) indicates that
Y (X, T ) is a product of three exponential terms, (i), (ii) and (iii). The ﬁrst term, eℜ[Zn ]X
is bounded since X is bounded. The second term, ei(ℑ[Zn ]X+ℜ[ω]T ) is oscillatory from its
imaginary exponent. The third term, e−ℑ[ω]T can grow unbounded with time if ℑ[ω] < 0
which represents the unstable dynamics of the pipe. The exact mode and the velocity at
which the ﬂuid-conveying pipe becomes unstable depends on the ﬂuid mass fraction β which
is explained in detail in [23].
The stability boundaries for a ﬂuid-conveying ﬂuttering tail can be seen in Fig.2.1 which
is an Argand diagram for various values of µ, the mass of the rigid head. The value of
µ = inf represents a cantilever while µ = 0 represents a free end. The dotted line through
9

15
2.5
2.25

10

5

25

ue

2.0

∞

1.0

0.33

5
ui ≈ 5.7, ue ≈ 2.9

0

0

2

4

6

8

ui
Figure 2.1: Neutral stability curves for diﬀerent values on rigid body mass fraction µ from
[23]. Regions to the left of the dotted line on each cure represent a region of negative thrust
produced from the tail oscillations.
each curve represents the transition point at which a ﬂuttering tail provides negative and
positive thrust. Left of the dotted line, a curve represented in bold is an area in which it
is predicted that the tail oscillations will produce a negative thrust. Curves to the right
of the dotted line represent areas in which it is predicted that tail oscillations will provide
positive thrust. However, even in the event that tail oscillations produce negative thrust,
the synergistic action of the tail allows the ﬂuid jet to maintain bias towards positive thrust.
Based on the work in [23] and the inspiration of ﬁsh-like eﬃciency it was proposed that
if we were to implement a submersible with a ﬂuid-conveying ﬂuttering tail that it would
potentially outperform a ﬂuid jet of the same geometry. With this goal, we set forth to
design an experimental platform to quantify the performance of a ﬂuid-conveying ﬂuttering
tail.
10

Chapter 3
Mechanical Design

3.1

Requirements

We have developed a modular design to develop the SPI experimental platform to satisfy as
many current and future requirements as possible. The general requirements for the mechanical design are to maintain buoyancy, provide housing for both mechanical and electronic
components, as well as to remain dry under all testing conditions including situations resulting in impacts and extended periods of time on the bottom of the pool. In addition to these
general requirements, the modular design allows most components to be easily updated or
replaced to accommodate design or experimental changes and/or requirements without the
need for a complete redesign of the platform. For example, ﬁn geometry including the tail
ﬁn, pectoral ﬁns, dorsal ﬁn and anal ﬁn are independent of the main hull and can all be
modiﬁed and installed without the need for hull modiﬁcations.
The primary mechanical components of SPI, both internal and external are introduced
and labelled in Fig.3.1. With the exception of the Rule inline 500 gallon per hour (GPH)
11

(e)
(g)

(b)

(i)

(a) (c)

(d)

(f)

(h)

Figure 3.1: (a)-Two piece monocoque hull. (b)-Modular nose cone. (c)-20 AA NiMh Batteries. (d)-Rule Inline 500 GPH bilge pump. (e)-Dorsal Fin. (f)-Anal Fin. (g)-Tail Clamp.
(h)-Charging and data port. (i)-External location of the charging and data port.
bilge pump and the array of NiMh batteries, all of the components represented in the ﬁgure
have been custom-designed and manufactured for the SPI experimental platform. These
components and, speciﬁcally, the hull have been designed to be as universal as possible to
reduce the need for design iterations and additional manufacturing.

3.2

Hull Construction

The hull of the SPI is machined from Delrin (Polyoxymethylene) to form two major monocoque halves to create a water tight enclosure. A monocoque design was chosen to simplify
the design and mounting of the various components that live inside the hull. Mounting
features that have been designed into the hull include the necessary geometry and mounting
features to accurately and easily incorporate 20 AA batteries; a 500 gallon per hour pump;
12

two large, 60ml ballast tanks with independent pressure pumps; as well as two hydroplanes
including their servo motors. The hull also has signiﬁcant room for electronics and sensors,
which currently include a custom designed circuit board equipped with a microcontroller,
an Attitude Heading Reference System/Inertial Measuring Unit (AHRS/IMU), data logging
capabilities, and other specialty sensors. The hull has also been machined with provisions for
static semi-permanent ballast in the form of highly dense materials such as lead or tungsten.
To ensure a robust and dependable water tight enclosure the hull incorporates a custom
designed sealing ﬂange that is easily to inspect, maintain, and replace if damaged. The hull
is split down the middle longitudinally to create two halves, speciﬁcally a left and a right.
The halves have four 3/16” stainless steel dowels to ensure proper alignment and is fastened
together with 18 6-32 stainless steel cap screws at approximately every 2 inches. The large
population of fastening screws ensure that there is a consistent and uniform compression on
the sealing surface removing the possibilities of local hull deformation, which may result in a
breach of the hull at the seal interface. The seal geometry includes a male/female interface as
seen in Fig. 3.2 that compresses a bead of silicon RTV (Room Temperature Vulcanization)
to create a water tight seal in a similar method to an o-ring design.
The monocoque design incorporates bolting locations that have been designed into the
hull to allow for various components to be fastened to the hull without the need for nuts,
nut plates, or through hole designs. To add thread strength and robustness, all threaded
holes have marine grade brass inserts to distribute the load over a larger shear area to reduce
the chance of fastener pull out from accidental over torquing or highly stressed components.
These inserts are available in three diﬀerent diametrical sizes, each size increasing the shear
area and permissible fastener tension. In the event of insert pull out the next size insert
13

Figure 3.2: SPI’s custom designed seal geometry includes a male/female interface that compresses a bead of silicon RTV as seen in the explosion in this ﬁgure. The silicon RTV is user
replaceable and easy to inspect.

can be installed in the hull without the need to repair the damages with sub-par methods
such as glues or epoxies that can not provide a sound and secure fastening method for an
extended period of time.
The monocoque has been designed to be as modular as possible. Many design features
have been included into this hull in hopes that it will be in service for many years because
it can be adopted to ﬁt many diﬀerent testing scenarios. Not only is the hull designed for
the evaluation of propulsion by ﬂuttering tail, it can easily be updated to use other forms of
locomotion without a complete redesign of the hull itself. Included in the modular design is
the possibility to change nose cones, tail mounting, buoyancy tanks, as well as hydroplanes
and the entire assortment of ﬁns. The hull is not built for a speciﬁc internal pump either; it
has been designed to accommodate a variety of pump geometries and sizes.
14

Figure 3.3: A cross section view of SPI’s current nose cone. External sealing to the hull is
accomplished with two silicone o-rings that live in the nose cone. The wetted surfaces of the
nose cone are a continuation of the hull itself creating a smooth transaction into the oriﬁce.

3.2.1

Nose Cone

The nose cone of the SPI is designed for two purposes, the ﬁrst is to maintain a smooth
transition from itself to the body. It has been modeled as a continuation of the hull itself
and maintains tangency in all three dimensions. The second purpose of the nose cone is to
provide a smooth inlet to the pump reducing pump losses. A cross section of the nose cone
can be seen in Fig.3.3, which shows the tangency that has been maintained between the
hull and the inlet. In addition to these surface features the nose cone also provides sealing
geometry in the form of o-ring grooves for the external seal as well as a barbed ﬁtting for
the internal plumbing to the prime mover. The internal plumbing is composed of a silicon
hose that bridges the connection between the nose cone and the pump. Internal seals are
maintained by compressing the silicon hose onto the barb with a low proﬁle pinch or “ear”
stainless steel hose clamp.
To avoid a three part interface between the two sides of the hull and the nose cone,
15

the sealing surface has been routed around the nose cone as seen in Fig.3.4, rather than
maintained at the centerline of the hull as it does around the remainder of the perimeter,
excluding the nozzle exit. This design decision has been made because it would be impossible
to incorporate a compression seal between three surfaces without administering uncured
silicon sealant during the assembly procedure. Although applying silicon sealant during
assembly would most likely maintain a water tight seal, it is not, however, a feasible solution
for an experimental platform because curing times are on the order of days when quick
adjustments need to be made during testing sessions that have a duration of 1-2 hours. The
current solution, as presented in Fig. 3.4 has proven to be robust and leak free with over
30 hours of underwater testing since the initial swim of SPI . Notice in the ﬁgure that the
additional component shown in gold could have been incorporated into the design of the
hull itself. However, by designing it as a separate component, it reduced the price of the
hull material by a signiﬁcant amount without greatly complicating the design. It can also
be changed to allow for a larger diameter entrance if desired.

3.2.2

Tail Clamp

The tail clamp has identical sealing features as seen in the nose cone with the exception that
it is a two-piece design that gives the user the ability to change the nozzle geometry or the
tail clamp mechanism without having to redesign both pieces. The tail clamp itself does not
provide any sealing features and is strictly used to provide the boundary conditions for the
ﬂexible tail and to remain as aerodynamic as possible. The tail clamp has been designed
to enable tails to be quickly changed with minimal eﬀort and minimal strain on the tail
while providing features to reproduce tail boundary conditions during testing sessions and
16

Figure 3.4: Exploded view of nose cone installation showing how the hull seal is routed around
the centerline of the hull to avoid a three part interface resulting in a less complicated sealing
interface.

throughout the lifetime of SPI.

Tails are “clamped” into the tail clamp by two stainless steel set screws that can be seen
in Fig. 3.6, which provide a compressive force on the two stainless steel sheets that pinch the
latex tail into the clamp and provide the desired boundary condition. The 5/16” ID latex
tube extends towards the hull from the stainless steel pinch clamps and interfaces with the
hose barb on the nozzle creating a leak proof seal to the high ﬂow-rate nozzle. The pinch
clamps not only provide the boundary condition for the tail, they also provide the necessary
restraint to keep the tail hose from slipping oﬀ the barbed nozzle.
17

Figure 3.5: An exploded view of the tail clamp showing the nozzle, hose barb and sealing
features of the hull.

Figure 3.6: The SPI tail clamping mechanism. The set screws are used to clamp the sheet
metal around the latex tail to ensure the clamped boundary condition.
18

3.2.3

Buoyancy Control

The SPI has been designed to incorporate buoyancy tanks to maintain depth without a
forward velocity. Two tanks as seen in Fig. 3.7 totaling 120 ml of capacity are strategically
placed not to interfere with future pump changes or upgrades. The buoyancy system, which
has not yet been installed, will be implemented such that each tank is independently operated
to statically adjust the fore/aft CG without having to open the hull and manually place the
mass. Independent buoyancy tank control is an important feature when tails are often
switched as current tails are not neutrally buoyant to the degree that small changes in
tail size do not eﬀect the static rake of the hull. It is also worth mentioning that tailless
performance and rigid tail performance measurements currently require the movement of
the CG to maintain proper rake for both hydrodynamic and stability reasons. Externally
controlling the hull’s static rake is a beneﬁcial feature that once implemented will greatly
reduce experimental setup time, accuracy, and understanding. The details of the hydraulics
of the buoyancy system have not been ﬁnalized at the time of this writing.

3.2.4

Hydroplanes/Diving Planes

The SPI has been designed to incorporate two hydroplanes or diving planes, one on each
side of the hull biased towards the nose as seen in Fig. 3.8. Hydroplanes are typically found
on submarines to assist the vehicle’s pitch requirements and depth control at speed. When
compared to the autonomy of a carangiform, hydroplanes are most similar to the pectoral
ﬁns as seen on the Blacktip Reef shark in Fig. 3.9. Highly developed pectoral ﬁns, such as
those found on sharks, are used for dynamic lifting for depth control as well as aggressive
trajectory control [39, 8]. Dynamic lift is necessary for species that do not have the means
19

(b)

(a)

Figure 3.7: Two 120ml buoyancy tanks in fore (a) and aft (b) locations in the hull. These
two buoyancy tanks can be used for static and dynamic depth control as well as static and
dynamic rake.

of changing their buoyancy, or changing their buoyancy quickly. Many deep sea sharks have
an oil rich liver to maintain the desired buoyancy, but to change depth quickly they rely on
the dynamic lift associated with their pectoral ﬁns.

SPI has incorporated hydroplanes for both depth control and yaw control similar to that
of species of ﬁsh with highly developed pectoral ﬁns. Unlike ﬁsh that depend solely on
dynamic lift to maintain depth, SPI’s primary depth control is achieved from the ballast
tanks while the hydroplanes or pectoral ﬁns are used for quick navigation changes. This
allows SPI to maintain a depth without a forward velocity that would be required if depth
control was maintained by the hydroplanes alone.
20

(a)

(b)

(c)

Figure 3.8: Hydrofoils installed on SPI incorporating a symmetric design which has been
chosen such that lift is not created at zero camber angles. This reduces unnecessary drag
as static depth is controlled by the buoyancy chambers. (a) Isometric view of the hydrofoil
location. (b) Cross section of the hydrofoil. (c) Plan view of the hydrofoil.

Figure 3.9: A swift and energetic Blacktip shark with its pectoral ﬁns. These large, developed pectoral ﬁns are used for dynamic depth control and aggressive trajectory control; the
Blacktip shark has been known to launch itself out of the water and spin three to four times
about its axis before returning to the water. Speed attained by the shark during these jumps
has been estimated to average 6.3 m/s [39, 8]. This image was originally post to Flickr by
StormyDog. Shark. 10 May 2011. http://ﬂickr.com/photos/32184789@N00/134610871.
21

3.3

Prime Mover

SPI’s current prime mover for propulsion is the Rule iL500PK, a 500 Gallon Per Hour (GPH)
marine inline submersible pump designed to run on a 12 volt system. It is an oﬀ-the-shelf part
that is unmodiﬁed except for the addition of a ﬂy-back diode and a large 1000 µF capacitor
on the leads to reduce unwanted dynamics and noise from the motor. It is designed for
continuous operation and has proven to be robust even when exceeding the manufacturer’s
recommended voltages for extended periods of time.
The Rule iL500PK is a centrifugal type pump that drives a plastic impeller at the inlet
of the housing and exhausts the pumped water over the electric motor providing necessary
and suﬃcient cooling for continuous operation. Current ﬂow restrictions limit the pump to
a maximum of 200 GPH.

3.4

Tail Construction

SPI’s ﬂexible tails needs to be suﬃciently ﬂexible to allow for ﬂuttering behavior given the
conditions imposed from the pump and its external velocity. Tail ﬂexibility can easily be
controlled with the combination of tail thickness and material properties. Tail knowledge is
currently heavily based on experimentation that has found tails with a second moment of
area of 5.6497e − 10m4 with a Youngs Modulus of 900KP ato be suﬃcient to sustain ﬂutter.
Tail buoyancy is also a factor that needs to be controlled to ensure proper hull dynamics.
Tails deviating from neutrally buoyant will cause static rake in the hull; which results in
surfacing, diving, and bobbing instabilities when under power. For example, a heavier-thanwater tail can cause the tail to droop behind the hull and the entire platform may behave
22

Table 3.1: Sample of acceptable materials for ﬂexible tails. Note that current ﬂexible tails
have been constructed out of latex.
Material
Latex
Evoprene
C-Flex
Softﬂex
Silastic
Dynaﬂex
Elastamax XL
Nexprene
Dryﬂex
Geniomer

Speciﬁc Gravity 100% Modulus psi
0.935
130
1
174-247
0.9
100-290
.947-.972
102-290
.930-.960
145-247
.910-1.10
110-360
.9-1.1
98-487
.93-.94
131-139
1.04-1.08
102-290
.98-1.0
87-290

Shore A Durometer Hardness
40
35
30-48
31-50
37-48
30-37
34-48
36-45
30-50
40-55

in an unpredictable manner.

3.4.1

Material Selection

Several materials have been found that ﬁt both the stiﬀness and buoyancy requirements for
tails with a nominal thickness of 1/20 of an inch. Acceptable materials can be found in
Table 3.1. As seen in the table, several materials are suﬃcient candidates for feasible tails.
However, current tails have been chosen to be fabricated from natural latex; it is highly
available in a variety of thicknesses and can easily be fabricated into composite shapes as
detailed in the following section.

3.4.2

Fabrication

Tail fabrication consists of a composite design where latex components are adhered to other
latex components with Silicone Room Temperature Vulcanization (RTV). Current manufacturing methods include a laminated approach that can be seen in the top of Fig. 3.10 and
a butt construction as seen in the bottom of Fig. 3.10. Both methods take a considerable
23

Figure 3.10: Two manufacturing methods for creating latex ﬂexible tails. (Top) a laminated
tail that is held together with silicon RTV and placed in a two piece compression mold.
(Bottom) a butt constructed tail that is also held together with silicon RTV.
amount of time and a skilled and patient individual.
The butt construction is the simplest of the two and requires only a small ﬁxture to
assure that the tail sheet is in the center of the tail tube. Once centered a small bead of
silicon is introduced to the seam and allowed to dry. This silicon can be seen in Fig. 3.10
with a salmon color. The laminated construction is similar to that of a composite wet layup
used for ﬁberglass, carbon ﬁber, and Kevlar fabrication. Each internal surface of the tail
is coated with a thin layer of low viscosity silicon, placed in a mold, and then clamped as
seen in Fig. 3.11. Currently it has been found that laminated construction methods induces
strain into the latex during the mold clamping period resulting in a permanent set. The
permanent set is detrimental to tail performance resulting in current tails being produced
with the butt construction method.

24

Figure 3.11: The tail mold for ﬂexible tails that are made using the laminated approach.
Pins are used to ensure alignment between the two halves.

25

Chapter 4

Computing Platform, Actuators,
Sensors and Circuits

Acronyms and Abbreviations

ADC
AHRS
AVR
GCC
IMU
MOSFET
PCB
RDS
RISC
SD
SPIB
UART
USART
WHOI

Analog to Digital Converter
Attitude Heading Reference System
Atmel AVR, a family of microcontrollers
GNU Compiler Collection, and open source compiler
Inertial Measuring Unit
MetalOxideSemiconductor Field-Eﬀect Transistor
Printed Circuit Board
Resistance Drain-Source for MOSFETs
Reduced Instruction Set Computing
Secure Digital Card, used for data storage
Serial Peripheral Interface Bus, known as SPI outside of this document
Universal asynchronous receiver/transmitter
Universal synchronous/asynchronous receiver/transmitter
Woods Hole Oceanographic Institution
26

4.1

Computing Platform Requirements

The requirements for the on-board computing platform are primarily to facilitate communication between the platform and the shore or dock and to execute multiple control algorithms
in real time. In addition to these requirements, a design was sought to provide data acquisition and semi-autonomous decisions. Furthermore, it was crucial for the system to be inexpensive. Using an oﬀ-the-shelf miniature computer or other expensive all-in-one solutions
were not feasible for a device that is deployed int a highly conductive chlorine environment.
Even though many provisions have been set forth to minimize the possibility of a breach in
the hull, it is still a signiﬁcant threat and not worth the risk for a prototype device that has
not been ﬁeld tested for extended periods of time. To maintain a reasonable cost for possible
replacement, a microcontroller was sought for its abundant set of features, versatility, and
real-time computing environment. In addition to being inexpensive, the computing platform
needed to have the ability to drive several brushed DC motors, control servo motors, and
have the ability to run an Analog to Digital Converter (ADC).

4.2

Pumps and Servos

The Rule iL500PK is an centrifugal type pump that drives a plastic impeller pumping water
in the same direction, regardless of the polarity of the terminals1 . This characteristic, which
is common with all the pumps we have currently considered does not require the need for
the complex circuitry of an H-Bridge to reverse the polarity of the pump’s driving circuit.
To simplify the driving circuitry an N-Channel MOSFET was selected as a robust, and
1 The

intended pumping direction is more eﬃcient than the opposite direction given the biased
impeller design for centrifugal type pumps

27

inexpensive solution for the motor driver. This solution provides precise throttling ability and
virtually no rejected heat with an internal Resistance Drain-Source (RDS) in the MOSFET
of only 0.0095 Ω.
The Logic Level N-Channel MOSFET operates at battery voltage with a pulse width
modulation (PWM) signal from the microcontroller operating on the gate of the MOSFET.
The PWM signal operating on the oxide-insulated gate of the MOSFET induces a conducting
channel between the source and the drain located on the MOSFET. In our application
the source is connected to the system ground and the drain is connected on the negative
terminal of the driving pump. This setup can be seen in Fig. 4.1, which shows the NChannel MOSFET (Q1) and its supporting components including the PWM signal from the
microcontroller as OC0. In this conﬁguration the 100 kΩ pull-down resistor (R1) provides
a path to ground for the gate in the event that the microcontroller fails to do so, such as
a microcontroller restarting event. Without the pull-down resistor tying the gate to ground
the MOSFET has the potential to drift creating a conducting channel between the source
and the drain turning the pump on when it is not desired to do so. The diode (D1) in
this circuit, often referred to as a ﬂyback diode provides a path to eﬀectively eliminate the
sudden voltage spike from the inductive load of the pump when its supply voltage is reduced
or removed. Capacitance (C1) in this circuit provides noise ﬁltration from motor operation,
C1 is on the order of 1000 µF. The Rule iL500PK, as purchased, has capacitance between
its terminals inside the motor housing on the order of 0.1 µF, which removes a suﬃcient
amount of higher frequency noise.
SPI’s two hydroplanes are controlled with standard hobby-grade digital servos. Manufactured by HiTec, California, the 5056MG is a “mini” sized servo with a full metal gear train
28

Figure 4.1: Circuit diagram of SPI’s single direction motor driver. Where Q1 is the NChannel MOSFET, D1 is the ﬂyback diode, C1 acts as a ﬁltering capacitor, R1 is a pull-down
resistor and AGND is the circuits analog ground.

with ball bearing support. Its speciﬁcations can be seen in Table 4.1. The 5056MG’s small
size and relatively high torque output is an aﬀordable and robust solution for the pectoral
ﬁns, which do not require high rotational resolution or excessive torque that can be found
in more expensive alternatives to the hobby servo.
The 5056MG, like all HiTec servos, operates between 4.8 and 6 volts, which is not directly
available from SPI’s 24 volt battery pack. To provide a stable and robust voltage source
for the servos, a switching voltage regulator was chosen to provide a clean and consistent
voltage for the servos while maintaining a high eﬃciency that would otherwise be wasted as
excess heat by a linear regulator. The power circuitry can be seen in Fig.4.2, which is based
around ON Semiconductor’s MC34167 5.0 Amp step-down switching regulator.
As speciﬁed from ON Semiconductor, the diode (D2) on the output of the switching
regulator is a 1N5825 high frequency Schottky barrier rectiﬁer speciﬁcally designed for low
voltage high frequency circuits. The inductor (L1) on the output is speciﬁc to the application
29

Table 4.1: Speciﬁcation of the HiTec HS-5056MG
Motor Type:
3 Pole
Bearing Type:
Top Ball Bearing
Speed (4.8V/6.0V):
0.11 / 0.09
Torque oz./in. (4.8V/6.0V):
21/25
Torque kg./cm. (4.8V/6.0V):
1.5 / 1.8
Size in Inches:
0.88 x 0.45 x 0.94
Size in Millimeters: 22.35 x 11.43 x 23.88
Weight ounces:
0.45
Weight grams:
12.76
Voltage V:
4.8 - 6
Pulse Voltage V:
3-5
Pulse Duration mS: 0.9 - 2.1, 1.5 Center

Figure 4.2: Circuit diagram of SPI’s switching regulator for producing a clean 5 volts for
the hydroplane servos. NOTE: Need to deﬁne F1,C2,R2,D3 and L1(or refer to spec sheet
perhaps?).

30

and can be determined using step-down regulator design equations [41]

L=

Vin − Vsat − Vout
∆IL

ton

(4.1)

where Vin is approximately 24 volts, Vsat is determined from Fig.4.3, resulting in approximately −1 volts under heavy load, Vout is currently set at 5.0 volts to satisfy the input
requirements of the servos, ∆IL is the desired peak-to-peak inductor ripple current and can
be set at twice the output current and ∆IL = 2(Iout ). In Eq.(4.1), ton is given by the
expression
ton =

α
fosc (α + 1)

(4.2)

where fosc is the oscillator frequency of 72 kHz and α is deﬁned as

α = ton /tof f =

Vout + VF
Vin − Vsat − Vout

(4.3)

where VF is the output rectiﬁer forward voltage drop which is 0.35 V for the 1N5822.
Setting R2 and RF at 6.8 kΩ and 68 kΩ respectively the MC34167 outputs 5.05 volts,
which satisﬁes the input voltage requirements of the servos. However, initial testing of the
servos was performed at 6.0 volts by computing R3 from the relation

Vout = Vref

R2
+1
R3

(4.4)

which equated to an R3 value of 36 kΩ using 5.05 volts as Vref . Running the servos at 6.0
volts was however abandoned when the switching regulator failed. As a result of chip failure,
Vin shorted to Vout and supplied both servos with an excessive 24 volts resulting in servo
31

Figure 4.3: Switch output source saturation versus source current for the MC34167

failure. The shorting of the MC34167 is still an unknown mystery; however, provisions have
been implemented to reduce the chance of servo failure in the event of a regulator short. To
limit the voltage delivered to the servos, a transient voltage suppression (TVS) diode has
been placed on the output of the regulator. Seen as D3 in Fig 4.2 this TVS diode clamps
at just over 5 volts by shunting excess current, at approximately 6 volts the induced voltage
exceeds the avalanche breakdown potential and the diode absorbs much of the transient
energy. In the event that Vout approaches 6.0 volts for an extended period of time the fuse
F1 will blow, protecting the circuit and the servos attached to it.

Servo position is dictated by a PWM signal from the microcontroller, which is produced
from commands by the user from the dock, which are detailed in the wireless algorithm
section. As presented in Table 4.1, pulse durations are speciﬁed to be at 50Hz and between
0.9 ms-2.1 ms with 1.5 ms as the center.
32

4.3

Sensors

SPI has been implemented with an array of sensors for both real-time control and post
processing of their output data to quantify SPI’s performance under predetermined circumstances and to gain understanding of speciﬁc events such as instabilities and high drag events.
An Inertia Measuring Unit (IMU) and an Attitude and Heading Reference System (AHRS)
has been installed in SPI with a single sensor array from CH Robotics known as the CHR6dm AHRS. The CHR-6dm uses a 32-bit ARM Cortex processor to evaluate an Extended
Kalman Filter (EKF) that combines data from onboard accelerometers, rate gyros, and magnetic sensors to produce yaw, pitch, and roll angle estimates. It is relayed to the system
microcontroller over a Transistortransistor logic (TTL) 3.3V UART at 115200 Baud. In our
implementation we have conﬁgured it to transmit raw sensor data in addition to Euler angle
estimates for a packet that is formatted as [yaw, pitch, roll, yaw rate, pitch rate, roll rate,
mag z, mag y, mag x, gyro z, gyro y, gyro x, accel z, accel y, accel x].
The 3.3V UART is not directly compatible with the system microcontroller; an AVR
ATMEGA1284 expects a 5.0V UART. Communication between these two system requires
a bi-directional level shifter; the MAX3372E from Maxim Integrated Products has been
implemented to provide this functionality.
The CHR-6dm’s sensor characteristics can been seen in Tables 4.2 and 4.3, which include
their sensitivities [40]. In addition to acceleration measurements provided from the AHRS,
a second more sensitive accelerometer has been installed in the longitudinal direction to
achieve a higher accuracy of tail performance with the option to perform integration to
obtain velocity data. This additional accelerometer, the MMA1260KEG from Freescale
Semiconductor has a sensitivity of 1200mV /g which is 4 times higher than that of the CHR33

Table 4.2: Gyro Electrical Characteristics
Symbol
FSA
FS
SoA
So
SoDr

Voﬀ
Vref
OﬀDr

NL
BW
Rn

Parameter
Measurement
Range
Measurement
Range
Sensitivity
Sensitivity
Sensitivity
change
vs
temperature
Zero-rate
level
Reference
voltage
Zero-rate
level change
vs. temperature
Non linearity
Bandwidth
Rate
noise
density

Test Condition
4x OUT (ampliﬁed)

Min.

Typ.
Max.
+/- 100

Unit
/s

OUT (not ampliﬁed)

+/- 400

/s

4x OUT (ampliﬁed)
OUT (not ampliﬁed)
Delta from 25 C

10
2.5
0.03

mV//s
mV//s
%//s

1.23

V

1.23

V

0.02

/s/C

+/- 1
140
0.017

% FS
Hz
/s/rt Hz

Delta from 25 C

Best ﬁt straight line

Table 4.3: Accelerometer Electrical Characteristics
Symbol
aFS
SoXYZ
Stemp
nXY
nZ
oﬀXY
oﬀZ
aNL
aBW

Parameter
Measurement Range
Sensitivity
SCDTT
X,Y noise density
Z noise density
X,Y 0 g voltage
Z 0 g voltage
Non linearity

Test Condition
Vdd = 3 V
Vs = 3 V

Vs = 3 V
Vs = 3 V

Min.
+/- 3
270

1.35
1.2

With external ﬁlter

34

Typ.
+/- 3.6
300
0.01
150
300
1.5
1.5
+- .03
140

Max.
330

1.65
1.8

Unit
g
mV/g
%/C
ug/rt Hz
ug/rt Hz
V
V
% FS
Hz

6dm accelerometers.

4.4

Microcontroller and Digital to Analog Converter

Atmel’s AVR microcontroller series were investigated as a suitable platform for SPI and
was found to be more than adequate for the required goals of the project. There are many
beneﬁts to the AVR platform compared to similar platforms such as the PIC. The open
source nature of the AVR platform has not only made them extremely popular but has
fueled an assortment of development tools of which the majority are free. The ability to write
programming code in the C language and compile with the GNU Compiler Collection (GCC)
reduced a signiﬁcant amount of overhead and cost without reducing the performance of the
product. Currently, SPI is running the ATMEGA1284 microcontroller for its assortment
of features and its large 40 pin dual in-line package (DIP), which facilitates quick and easy
prototype development. The ATMEGA1284 is a high-performance 8-bit Reduced Instruction
Set Computing (RISC) microcontroller with 128 Kilo Bytes (KB) of ﬂash memory with readwhile-write capabilities. This microcontroller included all the requirements sought for the
project, which included a 10-bit analog to digital converter (ADC), three timers, several
pulse width modulation (PWM) channels, Serial Peripheral Interface Bus (SPIB), as well
as two programmable serial universal asynchronous receiver/transmitters (USARTs) . This
single microcontroller provides the majority of processing power for the entire platform and
performs many tasks. Its responsibilities include receiving wireless messages over the ﬁrst
USART and decoding them in real time to provide pulse widths for the motor driver and
pectoral ﬁns, operating the 8-channel ADC at 100Hz to obtain data from various sensors,
acquiring data over the second USART from the Attitude Heading Reference System/Inertial
35

Measuring Unit (AHRS/IMU), logging all relevant information to a Secure Digital (SD) card
on the Serial Peripheral Interface Bus, as well as making decisions for semi autonomous
behavior in the event of communications failure.
Data acquisition was implemented on SPI to increase the productivity of current experiments as well as future experiments by designing the system with enough bandwidth for
signiﬁcant expansion. Data are collected from various sources on the AVR and written to a
Secure Digital (SD) Card for post processing via the Serial Peripheral Interface Bus (SPIB)
in raw ASCII in 512 byte blocks. It was determined that implementing a ﬁle system on the
microcontroller was unnecessary. A ﬁle system required signiﬁcant processing overhead and
considerable programming space only for the luxury of being able to easily mount the SD
card using conventional ﬁle system techniques. Writing to the SD Card without a ﬁle system greatly improves logging performance despite the cost of imposing technical challenges
when trying to extract data from the card. This can easily be accomplished by the following
GNU/Linux commands. Data are ﬁrst extracted from the SD Card block by block into a
log ﬁle using the dd command:

dd i f =/dev /mmcblk0 o f=Data . l o g bs =512 count =10000
where /dev/mmcblk0 is the SD device and Data.log is the destination for the data. Zeros
are then trimmed from the log ﬁle using the tr command

t r −d ’ ’ \ 0 0 0 ’ ’ < Data . l o g > Data Trimmed . l o g .
Those two commands yield a ﬁle that can be interpreted with software of the user’s
choice. It is important to note here that the SD Card can easily be cleared or formatted
with the dd command as well by writing zeros to each block of the card.
36

dd i f =/dev / z e r o o f =/dev /mmcblk0 bs =512 count =20000
Data is acquired not only from the CHR-6dm AHRS over a USART on the AVR but
from the AVR’s Analog to Digital Converter (ADC). The Atmega1284’s ADC has 8 channels
with 10-bit resolution currently sampling the output from the longitudinal accelerometer at
100Hz; the remaining 7 channels are available for future use. The speciﬁcations for the
Atmega1284’s ADC are [11]

• 10-bit Resolution
• 0.5 LSB Integral Non-linearity
• ±2 LSB Absolute Accuracy
• 13 µs - 260 µs Conversion Time
• Up to 15 kSPS at Maximum Resolution
• 8 Multiplexed Single Ended Input Channels
• Diﬀerential mode with selectable gain at 1x, 10x or 200x
• Optional Left adjustment for ADC Result Readout
• 0 - VCC ADC Input Voltage Range
• 2.7 - VCC Diﬀerential ADC Voltage Range
• Selectable 2.56V or 1.1V ADC Reference Voltage
• Free Running or Single Conversion Mode
• ADC Start Conversion by Auto Triggering on Interrupt Sources
• Interrupt on ADC Conversion Complete
• Sleep Mode Noise Canceler

4.5

Wireless Interface

Single direction wireless communication from the dock to SPI is accomplished with a 315
MHz link over the air. This one-way communication is based around Holy Stone Enterprise’s
MO-SAWR-A transmitter on the dock and their MO-RXLC-A receiver installed inside of
37

Figure 4.4: The SPI antenna which passes through the hull and extends 22.6 cm above the
waterline.
SPI. Receiver installation inside the hull requires an antenna to pass through the hall and
extend past the surface of the water to receive the wireless signal over-the-air. This was
implemented by passing the 1/16” diameter antenna through the hull and sealing it with
an o-ring under heavy compression provided by a custom designed sealing fastener that has
provided a robust solution without the need for a sealant. The antenna, protected with
Raychem heat shrink wrap as seen in Fig. 4.4, is a 1/4 wavelength antenna that measures
22.6 cm out of the water.
Initial bench top implementation of this communications package was very promising
with distances over 100 feet and a fairly high baud rate of 2400bps, both of which were
more than suﬃcient for the requirements of the wireless link. However, when installed
inside the hull, signiﬁcant noise from the pump and other circuitry completely disabled
communications. After inspecting the circuitry and software, a solution of both hardware
38

and software was implemented to restore the wireless link. The hardware modiﬁcations
included isolating the receiver behind an LC (Inductor-Capacitor) circuit on both the power
and ground side as well as implementing a high pass ﬁlter on the antenna itself. Increased
encoding was implemented in the software which, with the combination of the hardware
modiﬁcations restored wireless communications with a slightly lower throughput from the
additional overhead of the checksums for the data packets. Details of the wireless algorithm
can be found in chapter 5.

4.6

Power Supply

SPI’s monocoque hull was designed to accommodate 20 AA batteries with room to spare for
additional, unanticipated components. The AA battery size was chosen for its availability
and ease of creating custom battery packs. AA batteries are plentiful in a variety of diﬀerent
chemistries and are relatively inexpensive compared to new battery technologies such as
lithium polymers. SPI’s current battery chemistry is Nickel-Metal Hydride (NiMh), it was
primarily chosen for its high ampere-hour rating and its density. Battery placement in
the hull has been designed to be as low as possible; the high density of NiMh batteries is
beneﬁcial for the CG location. Another important feature of NiMh batteries is their ability
to be charged with relative ease and safety compared to lithium based battery technologies.
SPI’s current batteries are Panasonic’s HHR-210AAC4B with a minimum rating of
2000mAh with a nominal voltage of 1.2V. These 20 AA batteries are installed in series
for a nominal battery pack voltage of 24V. The batteries are assembled into 4 packs of 5
batteries each and are enclosed in chemical resistant heat shrink wrap from Raychem to
contain the possibility of explosion as seen in Fig.4.5. In addition to providing safety from
39

Figure 4.5: An exploded NiMh cell as a result of a short in the system can be seen here.
The chemical resistant heat shrink completely contained the failure protecting, SPI, the
environment and personnel.
battery explosions, the Raychem shrink wrap provides an ultra durable protective coat that
not only protects the batteries from mishandling and abrasions but completely water proofs
them so that in the event of hull failure the in-line fuse will have the opportunity to blow
and completely isolate the battery system avoiding the possibility of contaminating the environment.
The batteries were chosen to be installed in series for the ability to drive the pump at
higher voltages for increased ﬂow for short periods of time it is found necessary. An added
beneﬁt from having the batteries in series is the simpliﬁed charging circuit as well as the
safety associated with the battery pack. SPI has been designed to enable charging through
the charging port rather than opening the entire hull, which greatly reduces the time and
eﬀort to charge SPI.

40

Chapter 5
Software

5.1

Introduction

The SPI project has included several software programs to provide the platform with an
assortment of tools and features to streamline testing procedures and get the full beneﬁt of
the data acquisition system. In general, there are two primary programs that are required to
facilitate a testing session: software running on the AVR microcontroller inside SPI as well
as supporting software running on a laptop computer on the shore, or dock. In addition to
these two primary programs, software has been written to post process sensor data including
ﬁltering and plotting capabilities.
Communications between the user and SPI are outlined in Fig. 5.1, which shows an
overview of the communications including the various protocols and media that are used
between devices. The wireless algorithm used for encoding and decoding the wireless signal as
outlined in the ﬁgure is included in this chapter. In addition to providing a communications
link between the user and SPI the programs running on both the laptop and on SPI perform
41

Figure 5.1: A high level ﬂow chart that lays out the ﬂow of communications from the user
to SPI including the communications protocols and media that are used.
many other tasks that are outlined in the following sections.

5.2

Client Software

Client software written in Python, is currently executed on an IBM Thinkpad running Debian
Linux and is responsible for capturing the user inputs through a game controller, assembling
the requests in a predeﬁned manner and broadcasting the user requests over the wireless link.
User commands are interpreted from their inputs on a Logitec Dual Action USB controller
plugged into the laptop. The user has two analog joy sticks, two triggers, and an array of
buttons that are custom programed to perform a variety of tasks. The analog joy sticks are
primarily used for turning and diving commands while the triggers are used for the throttle.
42

The various buttons on the controller have been programmed to perform predeﬁned sets of
instructions, such as issuing ramp inputs to the throttle or turning sequences. These user
commands are recorded to a local timestamped log ﬁle as well as supplying the user with
useful outputs to the terminal such as the throttle ON/OFF events.
The ability to program a previously determined set of instructions ensures that a given
test can be executed multiple times without concern that the user’s inputs have changed.
Signiﬁcant work has been put forth to minimize the amount of time SPI needs to be removed
from testing to perform software maintenance; soft limits and calibration are present in the
client software that give comlpete control to the user without the need to open SPI in order
to reprogramming the microcontroller. This is beneﬁcial for two reasons: ﬁrst, reducing
the programing overhead to adjust parameters saves signiﬁcant time during testing sessions
by adding ﬂexibility to testing procedures, and, modifying parameters and conﬁguration
changes in Python is signiﬁcantly easier and faster than performing the same task on the
microcontroller, which is written in C with very speciﬁc requirements to remain compatible
with the AVR microcontroller.

5.3

Wireless Encoding and Decoding

As mentioned in the Wireless Interface section noise in the electronics, primary from the
pump degraded the performance of the wireless link to a state that was no longer suﬃcient
for transmitting data. The excess noise picked up by both the antenna and the receiver
circuity overwhelmed the USART buﬀer resulting in the polluting of the data packets with
garbage. After designing and implementing analog hardware ﬁlters it was also required
to provide an encoding and decoding algorithm for the wireless signal to ensure that only
43

valid packets were processed by the microcontroller. Manchester style encoding was ﬁrst
evaluated but was found not to be suﬃcient for the desired data packet and the environment
to which it was exposed. Manchester encoding operates on the bit level; for simplicity of
packet transmission and programming in both SPI and in the client software it was desired to
directly send American Standard Code for Information Interchange (ASCII), which operates
on top of the bit level. It was found that including a simple checksum to each command
resulted in an acceptable and suﬃcient transmission of desired commands.
Currently, SPI operates on three independent commands that are sent to it from the
user: throttle, left pectoral position, and right pectoral potion. Each of these commands has
been compressed from its initial values of 000 − 255 to 000 − 099 to accommodate the packet
size and algorithm. For example, the throttle command has been allocated numbers between
000 − 099 with check sum values between 100 − 199. The algorithm is fairly straight forward:
consider a desired throttle value of 050, the client software ﬁrst sends that value, then adds
100 to the desired value and send the second value of 150. SPI receives the ﬁrst message,
because the value is between 000 and 099 it is assumed that this is a throttle command. If
the next message SPI receives is 150, SPI then processes the command as valid and applies
the desired throttle position. To maintain a high throughput it has been decided that all
commands will be 3 digits in length resulting in 500 unique commands between 000 and 999
which include their checksum values. These allocations have been outlined in table 5.1 with
the addition of reserved space for future commands.
The wireless system runs at a baud of 2400bps and has the ability to adjust its gain on
the receiver side to help increase distance. While this can be quite beneﬁcial, it requires a
continuous stream of data to be received for the gain to maintain an acceptable value. In
44

Table 5.1: Encoding and checksum values for the wireless algorithm.
Command Name
Throttle
Left Hydroplane
Right Hydroplane
Left Turn
Right Turn
Depth
Forward CG Buoyancy
Rearward CG Buoyancy
Dive Buoyancy
Surface Buoyancy

Command
0-99
200-299
400-499
600-649
700-749
800-849
900
902
904
906

Checksum
100
100
100
50
50
50
1
1
1
1

Checksum Value
100-199
300-399
500-599
650-699
750-799
850-899
901
903
905
907

the event that there is a pause in the data, gain in the receiver increases and a considerable
amount of garbage is send into the USART RX buﬀer when underlying noise is ampliﬁed.
As the buﬀer becomes overwhelmed from the increased noise, the actual data messages get
lost in the translation and it become erroneous. In addition to the wireless algorithm, zero
padding has been included in the header of each packet to help set the gain in the receiver
before critical data are received.

5.4

Microcontroller Software

Software that resides on the Atmega1284 has been written in the C programming language
and compiled with AVR-GCC. The software is quite complex and includes many features
that are speciﬁc to the family of AVR microcontrollers. This software is the heart of SPI
and is responsible for all of the Input/Output (I/O) operations that happen during standard operation. In addition to providing and maintaining I/O operations this code has been
implemented with autonomous operations to surface the platform in the event the communications have been lost for a duration of 10 seconds.
45

Hardware interrupts have been implemented in the software to guarantee log rates and to
ensure that all packets received over the USART are accounted for. Log rates are guaranteed
with the use of a timer and an interrupt. A timer is analogous to a simple counter; however,
its advantage is that it is independent of program execution and relies only on the input
clock of the microcontroller, which results in a very stable frequency. Hardware interrupts
are issued when the programed timer reaches a designated value. At that instant the required
logging sequence is initiated and data are logged are a consistent rate for the duration of
experiment. When packets are received into one of the USART buﬀers, either from the
wireless transmitter or the AHRS, a hardware interrupt is issued and the packet is processed
immediately, clearing the USART buﬀer for future packets.
There are several timers on the Atmega1284 that are used for guaranteeing log rates as
well as controlling PWM outputs. Using a timer to control a PWM output removes the need
manually apply a modulated signal to an output pin of the microcontroller. By associating
a PWM output pin with a hardware timer the PWM signal is generated automatically,
without the need to for software intervention. In addition to these advanced features of
AVR microcontrollers, it is important to note that they are not limited to 8 bit math, using
the AVR-GCC compiler, these microcontrollers can easily perform ﬂoating point math and
even handle 64 bit numbers. These features of the microcontroller reduce programming time
and eﬀort and result in readable and eﬃcient code that is not restricted by their 8 bit core.

46

Chapter 6
Experimental Methods

6.1

Introduction

Quantifying the performance of ﬂuttering tails has proven to be time consuming as well
as challenging. Testing a submersible adds signiﬁcant complications that are not present
with platforms that exist outside of water. Not only does the waterprooﬁng of the platform
require signiﬁcant design work and maintenance, testing sessions have an extended duration
while waiting for pool currents to reside and conditions to be acceptable for testing. The
testing conditions SPI is exposed to have helped develop the primary goals of the platform.
These goals were set forth to minimize the down time during testing by designing a robust
platform that could sustain the forces exposed to it from the testing environment and to
ensure that maintenance and routine changes were straightforward and required minimal
tools and knowledge of the platform.
Numerous experiments have been conducted to characterize and test SPI for durability
and longevity as a whole, including that the hull is water tight, wireless algorithms are suﬃ47

cient, data acquisition is in working order, and that SPI is robust enough handle unforeseen
circumstances such as collisions and extreme depths. It is crucial that the platform performs as expected and carries out its required tasks such that testing sessions are dedicated
to quantifying the performance of ﬂuttering tails rather than debugging the experimental
platform. Initial testing of the hull was performed in a 50 gallon ﬁsh tank where it was
submerged for a total of 24 hours. This test proved that the hull was indeed a water tight
enclosure and it was then populated with electronics and the actual testing of the platform
began. It was proven to be a feasible platform in several tests in the diving well at Michigan
State University’s Intramural West Facility including an extended swim for the duration
of several minutes at a depth of 17 feet in the bottom of the diving well during which SPI
collided with the walls at approximately 1m/s. The results of this particular test (which was
not intended to happen) proved that SPI was suﬃciently durable and that semi-autonomous
behavior to return to the surface was needed when communications were lost.
Since the initial validation of the experimental platform, signiﬁcant time has been spent
characterizing both tail and hydroplane performance. Fluttering tail performance has been
compared to a rigid tail as well as a rigid tube to quantify its performance and eﬃciency,
these tails can be seen in Fig.6.1 . The rigid tail is geometrically similar to that of the
ﬂuttering tail, it maintains the same aspect ratio, area, and density; it only diﬀers from the
ﬂuttering tail by its thickness and material choice which make it suﬃciently stiﬀ. The rigid
tube maintains the same cross sectional area of the ﬂuid-conveying pipe as the ﬂexible and
rigid tails and is designed such that the exit conditions of the jet are the same as the ﬂexible
and rigid tails.
Open water testing has been performed exclusively in the indoor diving well at Michigan
48

Figure 6.1: Three diﬀerent tail geometries have been testing with the experimental platform.
(Top) A rigid tail with identical geometry and buoyancy of a ﬂexible tail. (Middle) A rigid
tube that has the same length and ﬂuid jet exit conditions as a corresponding rigid of ﬂexible
tail. (Bottom) The ﬂexible tail, made of neutrally buoyant latex a constructed using the
butt-welded method.
State University’s Intramural West Facility. The diving well is approximately 2500 square
feet on the surface and is suﬃciently deep for our current needs. At this time of writing
SPI has been in service for a total of 5 months and has swam in the diving well for a total
of 21.0 hours, 20.7 of those hours have been logged as time when the pump has been on.
There have only been two notable incidents during these test events, both incidents were of
human error, which resulted in a collection of moisture in the access port on the left side of
SPI as a result of not applying enough torque to the sealing fastener. In both cases, the leak
was unnoticed during the testing session and was only discovered when the access port was
opened to turn oﬀ the electronics.

6.2

Free Locomotion

Initial testing of the SPI was completely untethered. We began by giving SPI a throttle
command and then studied the stability and robustness of the platform. In regards to the
49

stability of SPI we focused on its behaviors in roll, pitch, and yaw, as well as its ability track a
straight line trajectory. Using our observations we made small adjustments to the location of
the center of gravity with respect to the center of buoyancy to improve stability in roll, pitch,
and yaw. Straight line tracking was improved with the addition of control surfaces including
keels and rudders with the latter greatly improving the straight line tracking ability.
It is important to note here that SPI was not designed to swim in a straight line, it
was designed to be quick and agile and accepting of a control strategy that would provide
the means necessary to track trajectory. Furthermore, it was designed to investigate the
possibility of induced turning events by modulating the ﬂow rate of the ﬂuid conveying pipe.
The control strategy is a topic of future research and important steps need to be taken to
support these future goals. First and foremost, quantifying the performance of ﬂuttering
tails is the ﬁrst step to continuing this research. Our motivation behind improving SPI’s
straight line tracking ability was to minimize the error when collecting data to quantify tail
performance.
Even though SPI was designed with quickness and agility in mind, its modular design
has allowed it to accommodate a selection of ﬁns to modify its behavior in order to provide
satisfactory tracking performance. The ﬂexibility of this platform has proven extremely
beneﬁcial. Fin and rudder changes can be performed pool side with a single tool without
the need to visit a machine shop or carry an entire box of tools.
Unfortunately, currents in the Intramural swimming pool have proven to be considerably
strong. Their eﬀects on SPI’s straight line trajectory have been so severe that we have opted
to design a guide-string system to maintain a relatively straight path to minimize erroneous
data.
50

6.3

Guided Locomotion

As previously mentioned, experimental testing with the guide-string system was implemented
to reduce erroneous data and increase the eﬃciency of our test sessions. However, the
tethering of SPI with a guide string is not to be confused with tethering systems that
incorporate data and power to mobile robots. Our guide-string system has a single purpose
which is to provide straight line tracking for SPI. It does not provide SPI with power or
communications, SPI is still operated over the 315MHz wireless link and provides its own
source for power with its 20 NiMh AA batteries. Guided locomotion has been achieved over a
50 foot span in the diving well by implementing a taunt nylon guide string over the distance.
SPI has an attachment that mounts to its dorsal ﬁn to accommodate the guide string, which
can be seen in Fig 6.2. The guide string forces SPI to maintain a straight trajectory so a
valid comparison between the performance of various tails can easily be achieved.
To quickly and eﬃciently test SPI on a guide string, a custom tram has been developed
as seen in Fig 6.2. The tram has an internally polished stainless steel tube that provides
guidance that is connected to the dorsal ﬁn with a rotary locking mechanism that allows SPI
to be quickly turned around on the tram without removing it from the guide string. The
rotary locking mechanism is equipped with two detents to lock SPIs position in either 0 or
180 degrees ensuring proper alignment. The stainless steel tube only provides a suggested
guidance for SPI. SPI is still free to roll about the string as well as pitch forward and rearward
at the location of the dorsal ﬁn/tram junction. This combination of constraints allows SPI
to operate with a minimum amount of inﬂuence from the guide string, ensuring that stability
and control of the platform is still a requirement.

51

Figure 6.2: SPI testing under guide string conditions with a rigid tail. The guide strings only
purpose is to provide a straight trajectory for SPI; power and communications are identical
between free and guided locomotion.

52

Chapter 7
Experimental Results
As mentioned in Experimental Methods, to quantify the performance of SPI, its speed was
determined using both ﬂexible tails and a rigid tail of identical dimensions and buoyancy.
In addition to these tails, a rigid tube of equal length and internal diameter aﬃxed in place
of the tails has also been included. Measurements conducted with the rigid tube represent a
baseline performance which a viable implementation that SPI must exceed. The presence of
the tube, as opposed to simply removing all attachments from the tail barb, was required in
order to stabilize the vehicle; it was found that removing the tube led to a pitch instability of
the hull. Figure 7.1 is a sequence of images taken over 1 sec. of operation; these images were
acquired while SPI was accelerating from rest, during which time we could take pictures more
normal to the free surface to reduce glare. The reduced forward speed during acceleration
also leads to a longer period of oscillation of the tail; Fig 7.1 indicates a period of 0.8 seconds,
compared to a period of 0.33 seconds observed when SPI is at top speed.
The average speed of SPI has been measured by analyzing 30 frame/second video taken
during operation; the guide string has been marked at two locations 9.14m apart, and the
53

Figure 7.1: High-speed images of the SPI acquired over a period of 1 sec, soon after it
started from rest in the MSU swimming pool. The SPI was moving with an average speed
of ≈ 0.4BL/s, less than the maximum speed of ≈1.15BL/s; note the large deﬂections of the
tail which occur during acceleration.
Table 7.1: Mean speeds of the three tested conﬁgurations in meters and body lengths per
second for all runs performed to date.
Conﬁguration
Rigid Tube
Rigid Tail
Flexible Tail

Trails Speed [m/s](N)
42
0.725
45
0.759
88
0.648

Speed [BL/s](N)
0.929
0.973
0.831

Speed [m/s](S)
0.795
0.814
0.707

Speed [BL/s](S)
1.019
1.044
0.906

times at which the SPI crosses these marks can be determined, and the speed calculated.
To date we have performed 175 runs over the 50’ span in the swimming pool, the calculated
speeds are given in Table 7.1 as well as in Fig.7.2 which includes error bars denoting the
limits of 95% conﬁdence intervals. The majority of these tests have been performed near the
surface of the pool to maintain a wireless connection. Through iterations of our testing we
have learned that the loss of energy due to free surface eﬀects has a substantial inﬂuence on
the speed of the platform.
To limit loss of energy due to free surface eﬀects, the mass and center of mass of SPI
were adjusted such that the measured portion of the run was entirely below the surface. The
results of these runs can be seen in Table 7.2 as well as in Fig.7.3 which includes error bars.
54

1.1
1.05
1
Speed [m/s]

0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6

Flexible

Rigid

Tube

Figure 7.2: Average speeds of the conﬁgurations listed in Table 7.1 which include 175 runs,
with error bars denoting the limits of the 95% conﬁdence intervals. Points denoted by a
indicate runs which traveled North in direction while points denoted with a ♦ indicate runs
in the opposite direction.
Table 7.2: Mean speeds of the three tested conﬁgurations in meters and body lengths per
second for all submerged runs.
Conﬁguration
Rigid Tube
Rigid Tail
Flexible Tail

Trails Speed [m/s](N)
14
0.806
32
0.794
14
0.744

Speed [BL/s](N)
1.033
1.018
0.954

Speed [m/s](S)
0.872
0.845
0.811

Speed [BL/s](S)
1.118
1.083
1.040

The diﬀerence between these selected runs is signiﬁcant and shows that the performance of
the ﬂutter tail is on par with the both the rigid tail and the rigid tube. We believe that
future improvements in tail design which are currently being pursued will allow the rigid
tube baseline to be surpassed.
As seen in Figs.7.2 and 7.3 there is a signiﬁcant diﬀerence between runs which travel in
the North direction compared to ones which travel in the South direction. This diﬀerence has
been contributed to the various currents that are present in the pool. These pool currents are
55

1.1
1.05
1
Speed [m/s]

0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6

Flexible

Rigid

Tube

Figure 7.3: Average speeds of the conﬁgurations listed in Table 7.2 which were performed
completely under the surface of the water totaling 60 runs. Error bars denoting the limits
of the 95% conﬁdence intervals, points denoted by a indicate runs which traveled North
in direction while points denoted with a ♦ indicate runs in the opposite direction.
artifacts of the circulation and ﬁltration system as well as the disruptions from lap swimmers
and occasional divers. It will be desirable in future test sessions to perform our testing in
the absence of the circulation and ﬁltration system as well as the pools patrons.

56

Chapter 8
Future Improvements
Although SPI has proven to be a successful and robust platform we believe we can increase
our productivity by improving a few designs and adding additional features to the platform.
There are several features, which will be detailed in this chapter, that are currently in the
design stages that have not been completed at the time of this writing. In addition to these
new features, there are some improvements that can be made to the platform to increase its
ease of use.
To fully quantify the performance of the platform, it is desirable to know the characteristics of the prime mover. We have proposed to include a calibrated elbow inside the hull as
seen in Fig.8.1, which connects the exit of the pump to the nozzle of the platform in two 90◦
opposing bends. The addition to this calibrated elbow, we will require a single diﬀerential
pressure sensor as well as a means for measuring the voltage and current delivered to the
prime mover. After performing a bench-top calibration of the ﬂow system we will have a
characteristic curve of our prime mover for the range in which it operates. The characteristic
curve will provide the means necessary to have the ability to close the loop for ﬂow control,
57

Figure 8.1: The proposed calibrated elbow that will provide SPI with the means necessary
to provide a consistent ﬂow rate to the tail regardless of the state of the power supply.

enabling us to provide consistent ﬂow rates regardless of the state of the power supply.
As the circuit designs have approached their ﬁnal iterations we have started to lay out
a ﬁnal Printed Circuit Board (PCB) design to shrink the current package from its axially
installed components to smaller surface mount devices. The move to surface mount devices
will decrease the size and complexity of the circuits providing more real estate inside the
hull in addition to providing us with a quick method to produce multiple circuit boards.
Current plans are to include a surface mount pressure sensor for the calibrated elbow as well
as the required current and voltage sensors for the prime mover. The PCB will also include
the necessary components to control ballast tanks that are currently in the design process
as well.
We have found a single drawback with SPI’s current design, the method in which the tail
is clamped to the hull has proved to be somewhat time consuming and slightly disruptive
to our tails when removing them from the tail clamping component. Although the current
58

solution is acceptable and works in the manner envisioned, removal of the tail is not as
easy and forgiving as it should be and has caused de-lamination on some of our tail specimens. With slight modiﬁcations to the current tail clamp, we will be able to overcome its
current limitations and provide a solution to quickly change tails without fear of possible
de-lamination.
At some point in the future it will be desirable to develop a tail manufacturing process
that will enable us to manufacture tails that are not restricted to 2D shapes as they currently
are by our laminating and butt adhering methods. The ability to construct tails that vary
in cross sectional widths and heights will be beneﬁcial as we push the envelope in the design
of ﬂuttering tails. Tail shapes that are similar to ﬁsh may be feasible by developing casting
compounds that are neutrally buoyant and creating 3D molds that represent desirable shapes.
The most anticipated improvement to the SPI platform is providing a solution for underwater communications. This has proven not to be a trivial task with solutions such as
the WHOI micro-modem exceeding the cost of SPI by a signiﬁcant margin. To add to the
complications, it is not guaranteed that the WHOI micro-modem will even package inside of
SPI. These constraints have restricted us to perform our experiments at the surface of the
water where we can maintain over-the-air wireless communications.

59

Chapter 9
Conclusions and Future Research
We have successfully developed an underwater experimental platform for studying the behavior and performance of a submersible propelled by a ﬂuid-conveying ﬂuttering tail. We
have proved its robustness with over 20 hours of pool testing without a single substantial
breech in the hull. Not only have we designed a robust and dependable platform, we have
designed it to be as modular as possible to accommodate a variety of prime movers and ﬁn
assortments, which include both active and passive arrangements. We have included unique
features in the design that have streamlined testing procedures and provided a platform that
will sustain further research and development by enabling adjustments and modiﬁcations
without the need for a complete redesign of the major hull, reducing design time and complexity. We have included a data acquisition system running at 100 Hz to obtain data from
an Attitude Heading Reference System/Inertial Measuring Unit (AHRS/IMU) along with
an 8 channel 10 bit Analog to Digital Converter to provide the tools necessary to completely
understand the dynamics of the platform. We’ve designed the electronics system including
the circuitry, computing platform, actuators, and drivers to be as simple as possible with60

out reducing SPI’s performance or limiting its potential. It has been designed built using
oﬀ-the-shelf components that are easily and inexpensively replaced in the event of a breech
in the hull.
SPI has already proven its eﬀectiveness by producing a candidate for the IEEE/ASME
Transactions on Mechatronics journal [42] as well as two conference papers: one for 2011
Dynamic Systems and Control Conference [43] and another for the IEEE/RSJ International
Conference on Intelligent Robots and Systems [44]. These initial papers are based on the
investigation of straight line performance of ﬂuttering tails, which represents the beginning of
our research into this topic. It has been proposed that further research on the topic of ﬂuidconveying ﬂuttering tails is to investigate the possibility of induced turning by modulating
the ﬂow and pressure through the ﬂuid conveying tail. In addition to this investigation it
will be necessary to add additional supporting features to the platform including depth and
stability control. These additional features will help minimize the losses and eﬀects of the
wake and to maintain a stable platform that is acceptable to our control inputs.

61

APPENDICIES

62

Appendix A
Dynamic Model Literature Review

A.1

Uniform Velocity Proﬁle

The earliest work pertinent to the study of dynamics of ﬂuid-conveying ﬂexible pipes can be
traced back to a series of experiments by John Aitken F.R.S.E. in the Philosophical Magazine
and Journal of Science in 1878. Aitken, (a Scottish meteorologist, physicist, and engineer)
stated that these experiments did not contain much that was new but helped to ﬁll an evident
gap in the experimental dynamics and hoped that they would enable the general reader to
form a clearer idea of the action of the so-called centrifugal force [1]. The phenomenon
of spontaneous motions from the free end of a ﬂexible hose was ﬁrst recognized as a selfexcited oscillation by the French physicist and mathematician Louis Marcel Brillouin in 1885;
however, his work was never published and existed only as dans une Note de laboratoire or
laboratory notes. Bourri`res, a student of Brillouin embarked on the study of the oscillatory
e
instability of cantilevered ﬂuid-conveying pipes, which was published in 1939 including both
theoretical and experimental results [7] that included the correct equation of motion and
63

experimental results outlining the critical ﬂow velocity for the onset of oscillation.
Vibration in ﬂuid-conveying pipes was revisited in 1950 in connection with the study of
the Trans-Arabian pipeline vibrations between simply supported ends by Ashley and Haviland [2]. Feodosev [12] and Housner [27] studied the problem independently and came to
the conclusion that for suﬃciently high ﬂow velocities, pipes may buckle in a similar manner
to column buckling from suﬃciently high compressive loads in the axial direction. Using a
more general approach, Niordson [33] obtained the same equation of motion and the same
conclusions of buckling and stability as Feodosev and Housner. In 1955, Long [29] revisited,
among other boundary conditions, the dynamics of cantilevered ﬂuid-conveying pipes previously studied by Bourri`res to obtain methods of solution for relatively low ﬂuid velocities
e
which were below the critical threshold of the self-excited oscillation. He successfully perceived and experimentally veriﬁed that forced motions of cantilevered ﬂuid-conveying pipes
were dampened by the internal ﬂow in the pipe for all ﬂow vibrations he considered.
Oscillatory instability or ﬂutter was anticipated by Benjamin [5, 6] in 1961, when he
considering the dynamics of articulated pipes conveying ﬂuid. Benjamin noted that the
dynamics of cantilevered ﬂuid-conveying pipes were independent of the ﬂuid friction and
analytically predicted the existence of ﬂutter instability. In 1966 Gregory and Pa¨
ıdoussis [17,
18] predicted theoretically and proved experimentally that at suﬃciently high ﬂow velocities
cantilevered pipes were subject to ﬂutter rather than buckling. Benjamin found that buckling
instability was present in vertical cantilevered articulated pipes, Pa¨
ıdoussis found that a
continuously ﬂexible pipe subject to the same scenario would never be subject to buckling
[34], which was clariﬁed in 1970 by Pa¨
ıdoussis and Diksnis [36].
Signiﬁcant research was performed in the 1960s through 1980s extending the analysis
64

y
U

r

U

y(x,t)
x

R

x
dx
dx

Figure A.1: A ﬂuid-conveying pipe and a magniﬁed view of a small element along its length
including the cross sectional dimensions.

of cantilevered ﬂuid-conveying pipes to include stability analysis by Nemat-Nasser, Prasad,
and Herrmann [31, 24, 25], Timoshenko beam models by Pa¨
ıdoussis and Laither [38], nonlinear formulations by Baja et al., Holmes, and Lundgren et al. [3, 26, 30], external ﬂows by
Hannoyer and Pa¨
ıdoussis [19] as well as unsteady ﬂows by Pa¨
ıdoussis and Issid [37] of which
details the analysis of uniform velocity proﬁles in ﬂexible pipes.
Consider the ﬂuid element in Fig.A.2 of length dx with an enclosed volume of δV, which
is a section of the ﬂuid-conveying pipe in Fig.A.1. Assume that ﬂuttering motions of the
pipe are constrained to a single plane and that motions in this plane are relatively small in
comparison to the length of the pipe. The rate of change of momentum over the ﬂuid volume
δV can be expressed as

dM
=
dt

∂Vf
+ (Vf · ∇)Vf ρdV
∂t
δV

(A.1)

where dV is a small element found within the ﬂuid volume ∂V and Vf is an approximation
of the ﬂuid velocity inside the pipe. Assume that the ﬂuid inside the pipe travels through
the pipe as a single inﬁnitely ﬂexible rod at the same velocity U relative to the pipe itself
at all points. This approximation is known as plug ﬂow and is a reasonable approximation
for a fully developed turbulent ﬂow proﬁle. Ignoring secondary ﬂow eﬀects as well as radial
65

variations [37], the velocity of the pipe itself, Vp can be expressed as

Vp =

∂r
= xi + yj
˙
˙
∂t

(A.2)

where r is the position vector from the origin to an arbitrary point along the ﬂuid-conveying
pipe in Fig.A.1. x and y are the velocities of a particular point in the x and y directions
˙
˙
respectively. The velocity, Vf of the center point of the ﬂuid element in Fig.A.2 can be
expressed as a function of the velocity, Vp of the pipe as well as the ﬂuid velocity U relative
to the pipe
Vf = Vp + Uτ

(A.3)

where τ is the unit vector tangent to the pipe at the point of interest

τ =

∂y
∂x
i+
j
∂s
∂s

(A.4)

where s is the curvilinear coordinate along the length of the pipe. Expanding Eq.(A.3) and
collecting terms
Vf =

x+U
˙

∂x
∂s

i+ y+U
˙

∂y
∂s

j.

(A.5)

Vf in the above equation is known as the material derivative, Dr/Dt, for the ﬂuid element
in Fig.A.2. Eq.(A.5) can be reduced further based on previous assumptions; relatively small
oscillations suggests that ∂(dx/ds) ≃ 1 as well as ∂(dx/dt) ∼ O(ǫ2 ). These assumptions
yield the ﬂuid velocity as found in Pa¨
ıdoussis’ Fluid-Structure Interactions [35]

Vf = Ui +

∂y
∂y
j
+U
∂t
∂s

66

(A.6)

Taking the derivative of Vf in Eq.(A.6) with respect to time to obtain the ﬁrst term found
in the integrand of Eq.(A.1) yields
∂Vf
dU
=
i+
∂t
dt

∂2y
∂2y
dU ∂y
+U
+
2
∂s∂t
dt ∂s
∂t

j.

(A.7)

The second term in the integrand, (Vf · ∇)Vf , known as the convective operator, is evaluated
as follows:






0
0
❃
✚
❃
∂Vs
∂V✚
s
✚  i + V ∂Vy + V ∂Vy  j
✚
(Vf · ∇)Vf = Vs
+ Vy✚
s
y
✚
∂s
∂s
✚ ∂y
✚ ∂y

(A.8)

where Vy ∂Vs /∂y and Vy ∂Vy /∂y are exactly zero. Evaluating the remaining partial derivatives
and recalling that ∂(dx/ds) ≃ 1 yields

(Vf · ∇)Vf = U

∂
Ui +
∂x

∂y
∂y
+U
∂t
∂s

j =

U

∂2y
∂2y
+ U2 2
∂x∂t
∂x

j.

(A.9)

Combining Eq.(A.7) and Eq.(A.9) yields the rate of change of momentum with respect to
the free-body diagram found in Fig.A.2, which is
dM
∂2
dU
∂2
∂2
+ 2U
=M
δs i + M
+ U 2 2 yδs j.
dt
dt
∂t∂x
∂t2
∂x

(A.10)

To obtain an equation of motion for the ﬂuid-conveying pipe, the equations of motion for
the ﬂuid element and the corresponding ﬂuid-conveying pipe element need to be determined.
Once determined and combined with the momentum over the ﬂuid volume an Euler-Bernoulli
beam model can thenn be presented. For the ﬂuid element in Fig.A.2(a) balancing the forces
67

q S dx

1 ∂p
A(p + 2 ∂x dx)

F dx
∂p
A(p − 1 ∂x dx)
2

M − 1 ∂M dx
2 ∂x
1
Q − 2 ∂Q dx
∂x

q S dx

y

2 ∂x

T − 1 ∂T dx
2 ∂x

dx

x

Q + 1 ∂Q dx
2 ∂x
M + 1 ∂M dx
q S dx
2 ∂x
x T + 1 ∂T dx
F dx q S d

(a)

(b)

Figure A.2: Free-body diagram of (a) a ﬂuid element and (b) its corresponding pipe element.

in the x and y directions yields the following for the ﬂuid element:

−A

F −A

∂
∂x

p

∂p
∂y
− qS − F
=0
∂x
∂x

∂y
∂x

− qS

∂y
∂
∂ 2
y.
=M
+U
∂x
∂t
∂x

(A.11)

(A.12)

In a similar approach, balancing forces on the pipe element (see Fig.A.2(b)) yields
∂T
∂y
∂2y
+ qS + F
−Q 2 =0
∂x
∂x
∂x
∂Q
∂
−F +
∂x
∂x

T

∂y
∂x

+ qS

∂y
∂2y
=m 2.
∂x
∂t

(A.13)

(A.14)

In Eqs.(A.11)-(A.14), A denotes the internal cross sectional area of the pipe, S denotes
the internal surface area of the pipe per its unit length, p is the pressure of the ﬂuid, q the
shear stress in the ﬂuid, and F denotes the force per unit length of the pipe normal to the
pipe in the radial direction as presented in Fig.A.2(a). For the pipe element in Fig.A.2(b)
Q denotes the transverse shear force, M denotes the bending moment of the pipe, and T
denotes the tension of the pipe along its length.
In accordance with the Euler-Bernoulli beam approximation Eq.(A.13) can be simplifed
68

by dropping the second order term, Q
partial2 y/∂x2 to obtain
∂T
∂y
+ qS + F
= 0.
∂x
∂x

(A.15)

Summing the forces in the axial direction for the ﬂuid element in Eq.(A.11) and the pipe
element in Eq.(A.15) yields the relationship between the tension T and the product of the
pressure p and cross sectional area A

∂
(T − pA) = 0
∂x

(A.16)

which suggests that T − pA = 0 for x ∈ [0, L].
To obtain the Euler-Bernoulli beam model equation, the transverse shear force Q needs
to be related to the physical properties of the pipe. Starting from a general stress-stain
relationship of σ = Eǫ where σ denotes stress, E denotes the Young’s modulus, and ǫ
denotes strain, the strain in the xdirection of an Euler-Bernoulli beam can be expressed as

σx = −yE

d2 y
,
dx2

(A.17)

relating the axial stress to the bending moment of the pipe leads to

M = −EI

d2 y
.
dx2

(A.18)

The transverse shear force Q is deﬁned as

Q = dM/dx,
69

(A.19)

which leads to the relationship between the physical pipe properties and the transverse shear
force
d3 y
Q = −EI 3
dx

(A.20)

where E and I denotes the Young’s modulus and area moment of inertia of the pipe respectively.
Combining Eqs.(A.12), (A.14), and (A.20) yields the Euler-Bernoulli beam approximation for small lateral motions [35]

EI

A.2

∂4y
∂2y
∂2y
∂2y
+ MU 2 2 + 2MU
+ (M + m) 2 = 0.
∂x∂t
∂x4
∂x
∂t

(A.21)

Non-Uniform Velocity Proﬁle

In the previous section the Euler-Bernoulli beam equation was developed for a ﬂuid-conveying
pipe with a uniform velocity proﬁle known as plug ﬂow. Plug ﬂow is a simple model of a
velocity proﬁle for ﬂuid ﬂowing in a pipe that assumes the velocity of the ﬂuid is constant
across any cross section of the pipe. It assumes that there is an inﬁnitely thin boundary
layer at the inner wall of the pipe and this greatly simpliﬁes the analysis. When higher
accuracy is desired for pipe ﬂow analysis, a non-uniform velocity proﬁle must be developed.
The following development will ﬁrst assume a triple plug ﬂow model and then generalize it
to an N-plug ﬂow model as proposed by Hellum [23].
The triple plug ﬂow model assumes three volumes of ﬂuid being conveyed through the
pipe, these volumes are concentric with respect to each other as well as the pipe itself. The
cross section of the triple plug ﬂow can be seen in Fig.A.3; the ﬂuid velocity of each plug
70

y

z
1
2
3
Figure A.3: Cross sectional view of three concentric ﬂuid volumes described by triple plug
ﬂow
is constant and uniform throughout a given plug’s volume, but the velocity of each plug is
diﬀerent. Although the triple plug ﬂow is not physically possible, it develops the framework
to introduce more plugs using the same convention. As more plugs are introduced into the
model it approaches a velocity proﬁle similar to that of a parabolic shape. The three plug
ﬂows seen in Fig.A.3 are labeled from the inside out. Fluid volume 1 exhibits a single ﬂuidﬂuid interface with volume 2, volume 2 has two ﬂuid-ﬂuid interfaces between ﬂuid volumes
1 and 3, and ﬂuid volume 3 has a single ﬂuid-ﬂuid interface with volume 2 and a single
ﬂuid-pipe interface with the inside surface of the ﬂuid conveying pipe.
To proceed with an analysis similar to that of the previous section, parameters of the
triple plug ﬂow need to be deﬁned. The cross sectional areas of ﬂuid volumes 1, 2, and 3
will be A1 , A2 , and A3 , respectively; their ﬂow velocities will be known as U1 , U2 , and U3 ,
respectively. Following the same convention, their mass per unit length will be known as
M1 , M2 , and M3 . The radial force per unit length between ﬂuid volumes 1 and 2 will be
F12 ; between ﬂuid volumes 2 and 3 will be F23 ; and between ﬂuid volume 3 and the pipe
will F3p . Using the same convention, the shear forces at these interfaces will be denoted by
71

q12 , q23 , and q3p . The surface area per unit length at these interfaces will be denoted by S12 ,
S23 , and S3p . Balancing the forces in the x and y direction in Fig.A.2 yields the following
equations for the ﬂuid volumes which are similar in nature to Eqs.(A.11) and (A.12) for
single plug ﬂow.
Volume 1:

−A1

∂y
∂p
− q12 S12 − F12
=0
∂x
∂x

(A.22)

∂2y
∂y
∂
∂ 2
F12 − pA1 2 − q12 S12
y
= M1
+ U1
∂x
∂t
∂x
∂x

(A.23)

∂y
∂p
− q23 S23 + q12 S12 − (F23 − F12 )
=0
∂x
∂x

(A.24)

∂2y
∂y
∂
∂ 2
− (q23 S23 − q12 S12 )
= M2
+ U2
y
∂x
∂t
∂x
∂x2

(A.25)

∂y
∂p
− q3p S3p + q23 S23 − F3p − F23
=0
∂x
∂x

(A.26)

∂
∂2y
∂ 2
∂y
y
= M3
+ U3
− q3p S3p − q23 S23
∂x
∂t
∂x
∂x2

(A.27)

Volume 2:

−A2

(F23 − F12 ) − pA2

Volume 3:
−A3
F3p − F23 − pA3

Similar to Eqs.(A.13) and (A.14) for single plug ﬂow, balancing the forces for triple plug
72

ﬂow yields the equations for the pipe
∂2y
∂y
∂T
+ q3p S3p + F3p
−Q 2 =0
∂x
∂x
∂x
∂Q
∂
− F3p +
∂x
∂x

T

∂y
∂x

+ q3p S3p

(A.28)

∂y
∂2y
=m 2
∂x
∂t

(A.29)

Summing Eqs.(A.22), (A.24), (A.26), and (A.28) in the x direction yields

− (A1 + A2 + A3 )

∂p ∂T
+
=0
∂x ∂x

(A.30)

where (A1 + A2 + A3 ) = A the total inner cross sectional area of the pipe. Noting that A is
not a function of x, Eq.(A.30) can be expressed as

∂
(T − pA) = 0
∂x

(A.31)

which is identical to Eq.(A.16) developed for single plug ﬂow by Pa¨
ıdoussis [35]. Summing
the equations in the y direction, Eqs.(A.23), (A.25), (A.27), and (A.29) yields

−pA

∂2y
∂ 2 y ∂Q
+T 2 =
+
∂x
∂x2
∂x

3

Mn
n=1

∂2y
∂ 2
∂
y+m 2.
+ Un
∂t
∂x
∂t

(A.32)

Following the previous derivation for single plug ﬂow, the Euler-Bernoulli beam approximation for triple plug ﬂow can be obtained with the substitutions of (T − pA) = 0 and
Q = −EI(∂ 3 y/∂x3 ) into Eq.(A.32):
∂4y
EI 4 +
∂x

3
n=1

2
MnUn

∂2y
+2
∂x2

3

MnUn
n=1

73

∂2y
+
∂x∂t

3

m+

Mn
n=1

∂2y
= 0 (A.33)
∂t2

A continuous solution of the Euler-Bernoulli beam approximation can be found by increasing the number of ﬂuid volumes from 3 to N with additional volumes similar to that of
volume 2, which include two ﬂuid-ﬂuid interfaces. The additional volumes do not complicate
the analysis since the additional ﬂuid-ﬂuid interactions terms are canceled when summing
the equations in the x and y directions just as they did with triple plug ﬂow in Eqs.(A.30)
and (A.32). Thus the Euler-Bernoulli beam approximation for triple plug ﬂow, Eq.(A.33)
can be rewritten as

EI

∂4y
∂x4



+

N
n=1



2
MnUn 

∂2y
∂x2





N

+2

n=1

MnUn

∂2y
∂x∂t



+ m +

N
n=1



2

∂ y
Mn 2 = 0 (A.34)
∂t

where N is any integer greater than 3. As N approaches ∞, the thickness of each plug
approaches 0 and the summations of Eq.(A.34) can be replaced with integrals. The average
velocity of the ﬂuid can be expressed as

1
¯
U=
A

U(A)dA,

(A.35)

A

where the area A is the inside of the pipe. For a pipe with an internal radius R, the average
velocity of the ﬂuid inside the pipe can be expressed as
R
2
¯
U= 2
U(r)rdr.
R 0

Note that Mn is the mass per unit length for each volume, thus

74

(A.36)

N
n=1 Mn

= M. The terms

of the Euler-Bernoulli beam approximation for N plug ﬂow as N → ∞ can be expressed as



N

n=1



2
MnUn  = ρf



2

N

n=1

A

U 2 (A)dA = 2πρf



2
MnUn  = 2ρf



N

m +

n=1

A

R

¯
U 2 (r)rdr = µM U 2 ,

0

¯
¯
U(A)dA = 2ρf AU = 2M U ,


Mn = m + M,

(A.37)

where ρf is the density of the conveyed ﬂuid and µ is the non-dimensional momentum
correction factor. The momentum correction factor is an artifact of the conveying geometry
and the ﬂuid velocity proﬁle U(r), for the case of a ﬂuid conveying pipe, the momentum
correction factor is
µ=

R U(r) 2
2
rdr.
¯
U
R2 0

(A.38)

Substituting the algebraic simpliﬁcations found in Eq.(A.37) into Eq.(A.34) the EulerBernoulli beam approximation can be expressed as

EI

2
2
2
∂4y
¯ ∂ y
¯ ∂ y + (M + m) ∂ y = 0
+ µM U 2 2 + 2M U
∂x∂t
∂x4
∂x
∂t2

(A.39)

which shares the same form as the Euler-Bernoulli beam equation for single plug ﬂow as
expressed by Eq.(A.21). Notice that for a uniform velocity proﬁle, µ = 1, and Eq.(A.21) and
(A.39) are identical.

Hellum et al. [23] went on further to provide the solution of the diﬀerential equation to
analyze the behavior a ﬂuid-conveying pipe by introducing the familiar boundary conditions
75

for a cantilever

y(0, t) = 0, yx (0, t) = 0, yxx (L, t) = 0, yxxx (L, t) = 0,

(A.40)

where yx , yxx , and yxxx are the ﬁrst, second, and third partial derivatives of y with respect
to x respectively. With the following change of variables the equation of motion is nondimensionalized
Y =

x
y
, X = , T = tΩ.
L
L

The non-dimensional velocity of the ﬂuid can be expressed as

u=

M 1/2 ¯
UL,
EI

the mass fraction as
β=

M
,
m+M

and the frequency as follows:
ω=

M + m 1/2
ΩL2 .
EI

If the solution is assumed to be separable such that

Y (X, T ) = φ(X)e−iωT ,

(A.41)

it is then then possible to produce the non-dimensional equation of motion and the corre-

76

sponding boundary conditions
d4 φ
d2 φ
dφ
− ω 2 φ = 0,
+ µu2
+ 2β 1/2uiω
dX
dX 4
dX 2

(A.42)

φ(0) = 0, φX (0), φXX (1) = 0, φXXX (1) = 0,

(A.43)

where φx , φxx , and φxxx are the ﬁrst, second, and third partial derivatives of φ respectively.
Assuming the solution of φ to be of the form of

φ(X) = AezX

yields the characteristic polynomial

z 4 + µu2 z 2 + 2β 1/2uiωz − ω 2 = 0.

(A.44)

For speciﬁc values of µ, u, and β which are obtained by the physical parameters of the
ﬂuid-conveying pipe Eq.(A.44) provides four roots, zn , n = 1, 2, 3, 4, where zn = zn (ω). The
complete solution of φ(X) has the form of

φ(X) = A1 ez1 X + A2 ez2 X + A3 ez3 X + A4 ez4 X .

(A.45)

The solution of Eq.(A.45) based on the boundary conditions presented in Eq.(A.43) results
in the complete solution
4

Y (X, T ) =
n=1

An ezn X eiωT =

4

An eℜ[Zn ]X ei(ℑ[Zn ]X+ℜ[ω]T ) e−ℑ[ω]T .

n=1

(i)

77

(ii)

(iii)

(A.46)

Inspecting the above equation indicates that Y (X, T ) is a product of three exponential terms,
(i), (ii) and (iii). The ﬁrst term, eℜ[Zn ]X is bounded since X is bounded. The second term,
ei(ℑ[Zn ]X+ℜ[ω]T ) is oscillatory from its imaginary exponent. The third term, e−ℑ[ω]T can
grow unbounded with time if ℑ[ω] < 0 which represents the unstable dynamics of the pipe.
The exact mode and the velocity at which the ﬂuid-conveying pipe becomes unstable depends
on the ﬂuid mass fraction β which is explained in detail in [23].

78

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84