Design and Performance Evaluation of a
Biomimetic Flapping Foil
by
Craig Brian Martin
Submitted to the Department of Ocean Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Ocean Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2001
@ Massachusetts Institute of Technology 2001. All rights reserved.
A uthor ..............
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...
.......
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Department of Ocean Engineering
May 11, 2001
C ertified by ....... V.
Accepted by.......
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Michael S. Triantafyllou
Professor of Ocean Engineering
Thesis Supervisor
.....
Henrik Schmidt
Chairman, Department Committee on Graduate Students
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
BARKER
NOV 2 7 2001
LIBRARIES
C
Design and Performance Evaluation of a Biomimetic
Flapping Foil
by
Craig B. Martin
Submitted to the Department of Ocean Engineering
on May 11, 2001, in partial fulfillment of the
requirements for the degree of
Master of Science in Ocean Engineering
Abstract
In this thesis I designed and constructed an apparatus for the study of force produced
by a roll/pitch flapping wing and its associated three-dimensional flow. Provision was
made for advanced real-time feedback control using RTLinux and an integrated motion control and data acquisition system. Tests were conducted at Reynolds number
20,000 to explore the maneuvering capabilities of a NACA 0012 foil by changing mean
pitch angle under harmonic roll/pitch motion, and by non-harmonically driving the
roll and pitch through change of a symmetry parameter. It was found that mean lift
coefficients in excess of three are available at mean pitch angles of 30 degrees at low
Strouhal number and low maximum angle of attack. It was also found that generation
of maneuvering forces through change of mean pitch angle is superior to the use of
asymmetric motion throughout the range tested.
Thesis Supervisor: Michael S. Triantafyllou
Title: Professor of Ocean Engineering
3
4
Acknowledgments
This thesis would not have been possible without a number of people who supported
me through wise advise, a necessary third hand, and timely distraction from the
doldrums of "working in a basement".
I would like to thank Professor Triantafyllou for supporting my work here for two
years, and for putting me on this great project in the first place. Franz Hover was an
invaluable resource to me through all stages of this project. Without him, the project
would have spelled much more frustration and much less success.
I spent a great amount of time learning, laughing, exploring, and creating with
my fellow graduate students in the tow tank. Thanks so much for making me look
forward to coming to "work" everyday: Dave, Alex, Josh, Muriel, Anna, Michael,
Albert, Jennifer, and John. A number of undergraduates helped make the tow tank
fun and exciting. Among them Marisa and Karl directly contributed to this project
my making the physical apparatus, thanks!
A number of visiting students came and went during my time in the tow tank.
Their fresh perspectives, dedication to research, and thirst for a good time helped
make my time more enjoyable. Thank you: Oyvind, Harald, Martin, Andrin, and
Frederick.
Finally, I would like to thank my wife Becky, for standing by me, for paying the
bills, and for believing I could accomplish something like this. She sacrificed at least
as much as I did for this thesis, I couldn't have done it without her.
5
6
Contents
13
1.1
14
Hull-Form and Principal Dimensions
. . . . . .
22
2.2
General Arrangement . . . . . . . .
. . . . . .
23
2.3
Flapping Mechanism Design.....
. . . . . .
25
2.4
Sensor Arrangement
. . . . . .
29
2.5
Data Acquisition and Motion Control
. . . . . .
30
2.5.1
Data Acquisition
. . . . . .
. . . . . .
31
2.5.2
Motion Control . . . . . . .
. . . . . .
32
Real-Time Software . . . . . . . . .
. . . . . .
34
.
2.1
.
21
.
. . . . . . . .
37
3.1
Position-Dependent Offset Correction . . . . . . . . . . . . . . . . .
37
3.2
Coupling Matrix Solution
39
.
Sensor System Calibration
.
. . . . . . . . . . . . . . . . . . . . . . .
45
4.1
Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
46
4.2
Bias Pitch Angle Experiments . . . . . . . . . . . . . . . . . . . . .
48
4.2.1
Mean Performance Results . . . . . . . . . . . . . . . . . . .
49
4.2.2
Time-Domain Results
52
4.3
.
.
.
Maneuvering Experiments
.
. . . . . . . . . . . . . . . . . . . . .
Non-Harmonic Roll Motion Experiments
7
. . . . . . . . . . . . . . .
.
4
.
Design and Construction
2.6
3
Literature Review . . . . . . . . . .
.
2
Introduction
.
1
54
4.3.1
5
Mean Performance Results . . . . . . . . . . . . . . . . . . . .
55
59
Conclusions
A Pitch Bias Experiments Time-Domain Plots
61
B Asymmetric Roll Motion Experiments Time-Domain Plots
69
8
List of Figures
2-1
Hull Arrangement
2-2
Motor Arrangement
2-3
Waterproofing Details
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
. . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2-4
Flapping Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2-5
Bevel Gear Detail
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2-6
Roll/Pitch Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2-7
Sensor Arrangement
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2-8
Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2-9
Motion Control System. . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2-10 Host Computer Software Setup. . . . . . . . . . . . . . . . . . . . . . .
36
3-1
Position Dependent Offsets . . . . . . . . . . . . . . . . . . . . . . . . .
38
3-2
Force/Moment Coordinates
39
4-1
Kinematics at Various Pitch Bias Angles (Style #5)
4-2
Mean Performance Results for Pitch Bias Experiments
4-3
Lift Coefficient vs. Time for Various Pitch Bias Angles (Style #5)
4-4
Roll Moments vs. Time for Various Pitch Bias Angles (Style #5)
4-5
Pitch Moments vs. Time for Various Pitch Bias Angles (Style #5)
4-6
Kinematics at Values of s (Style #15)
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . .
48
51
. . . .
53
. . . . .
54
. . . .
55
. . . . . . . . . . . . . . . . . . .
56
4-7
Mean Performance Results for Non-Harmonic Roll Experiments . . . . . .
57
A-1
Lift Coefficent vs. Time for Various Pitch Bias Angles (Style #5)
. . . . .
61
A-2
Roll Moments vs. Time for Various Pitch Bias Angles (Style #5)
. . . . .
62
9
A-3
Pitch Moments vs. Time for Various Pitch Bias Angles (Style #5)
62
A-4
Lift Coefficent vs. Time for Various Pitch Bias Angles (Style #15)
63
A-5
Roll Moments vs. Time for Various Pitch Bias Angles (Style #15)
64
A-6
Pitch Moments vs. Time for Various Pitch Bias Angles (Style #15)
64
A-7 Lift Coefficent vs. Time for Various Pitch Bias Angles (Style #118)
65
A-8
66
Roll Moments vs. Time for Various Pitch Bias Angles (Style #118)
A-9 Pitch Moments vs. Time for Various Pitch Bias Angles (Style #118)
66
A-10 Lift Coefficent vs. Time for Various Pitch Bias Angles (Style #113)
67
A-11 Roll Moments vs. Time for Various Pitch Bias Angles (Style #113)
68
A-12 Pitch Moments vs. Time for Various Pitch Bias Angles (Style #113)
68
B-1
Lift Coefficent vs. Time for Various Symmetry Parameter (Style #5)
69
B-2 Roll Moments vs. Time for Various Symmetry Parameter (Style #5)
70
B-3 Pitch Moments vs Time for Various Symmetry Parameter (Style #5)
71
B-4 Lift Coefficent vs. Time for Various Symmetry Parameter (Style #15)
72
B-5 Roll Moments vs. Time for Various Symmetry Parameter Style #15)
73
B-6 Pitch Moments vs Time for Various Symmetry Parameter (Style #15)
73
B-7 Lift Coefficent vs. Time for Various Symmetry Parameter (Style #118)
74
B-8 Roll Moments vs. Time for Various Symmetry Parameter Style #118)
75
B-9 Pitch Moments vs Time for Various Symmetry Parameter (Style #118)
75
B-10 Lift Coefficent vs. Time for Various Symmetry Parameter (Style #113)
76
B-11 Roll Moments vs. Time for Various Symmetry Parameter Style #113)
77
B-12 Pitch Moments vs. Time for Various Symmetry Parameter (Style #113)
77
10
List of Tables
2.1
Principal Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.1
Applied Calibration Loads, L . . . . . . . . . . . . . . . . . . . . . .
41
3.2
Calibration Test Measurements, 1
(bits) . . . . . . . . . . . . . . .
42
3.3
Calibration Matrix, S . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.4
Percent Error of Maximum Load
. . . . . . . . . . . . . . . . . . . .
43
4.1
Flapping Styles Tested . . . . . . . . . . . . . . . . . . . . . . . . . .
47
11
12
Chapter 1
Introduction
Over two thirds of the earth's surface is covered by water, yet human exploration and
utilization of the oceans' vast resources have been limited by our ability to create
machines that can survive harsh conditions while remaining efficient and adaptable
agents for human aspirations. While the challenges of operating in the ocean environment are substantial, we find hope and inspiration in the variety of hugely successful
species of animals whose existence has proven the possibility of efficient and adaptable
systems suited to the ocean environment.
Clearly, nature's machines are not particularly suited for human directed missions such as transportation, resource exploration and production, warfare, or other
missions required of a man-made machine. Furthermore, biological systems such as
marine animals are incredibly complex, and likely impractical to emulate to any fine
level of detail. These systems involve intricate neural, chemical, and mechanical interactions that are well beyond human capabilities for design and construction. However, these are the most elegant and well optimized marine systems, and they show
a mastery of the marine environment beyond, perhaps, the hope of human marine
system designers. The approach behind this research is to emulate these beautiful
evolved machines in the hope that in the future we might apply our findings to the
improvement of our own designs.
One type of marine design that is particularly important to human exploration
of the oceans is that of the swimming and maneuvering vehicle. Examples of these
13
include ships, submarines, AUV's, and ROV's. Biological analogs of these designs
include whales, dolphins, various fishes, and the penguin. One nearly universal characteristic of these biological systems is the use of thrust vortex generation for propulsion and maneuvering. This characteristic stands in contrast to human designs which
are largely based on the spinning propeller and rudder combination.
The focus of this particular work is to study the interactions between a mechanical
swimming system and the surrounding fluid. The flapping penguin wing was chosen
as the inspiration for this work due to the impressive observed maneuverability of penguins "flying" through the water. This maneuverability is astonishing considering the
fact that penguin bodies are largely rigid while all maneuvering forces are generated
by the flapping of its wings, rotation of the head and beak, and small adjustments
of the tail. Such maneuverability of a rigid hull swimming in water is encouraging
to engineers seeking to incrementally (or drastically) improve the maneuvering capabilities of man-made rigid-hull vehicles. This thesis explores the possibility that
flapping wings inspired by the penguin could be used to improve the maneuverability
of existing designs.
1.1
Literature Review
Understanding of the hydrodynamics of flapping wings is limited by the fact that the
vast majority of previous aerodynamic and hydrodynamic research has been focused
on steady problems such as rigid wings at constant angles of attack. The flapping
wing is an inherently unsteady problem where the instantaneous flow patterns about
the wing are never the same over the course of a flapping cycle.
A number of researchers have looked at the dynamics of unsteady flow about
wings. A selection of these inquiries are described here with particular attention paid
to those touching on vortex wakes and the generation of unsteady lift.
A number of parameters are refered to throughout the literature on flapping foils.
A few of these are listed here for convenience:
* Free stream velocity: U
14
9
Flapping radial frequency: w
* Vortex-wake width: A
" Chord length: C
" Reynolds number: Re
* Reduced velocity: k =
"
Strouhal number: St =
Freymuth
U
Cavg
2c
wA
[6] performed experiments on a NACA 0015 profile airfoil in separate
heaving and pitching modes. These experiments were conducted in a low-speed wind
tunnel, with foil-chord Reynolds numbers between 5200 and 12,000. Freymuth identified the "propulsive signature" as a staggered vortex street which forms a jet-like
velocity component away from the foil.
Flow visualizations are presented for pure plunging motion at k = 2.7, Re = 5200,
h/c
=
0.2, and mean angle of attack of 5 degrees. This combination of parameters
resulted in the described propulsive signature.
Similar results are shown for pure
pitching about the quarter chord axis for k = 2.9, Re = 12000, pitch amplitude 20
degrees, and mean angle of attack 5 degrees.
Careful examination of the figures reveals asymmetry in the vortex wake, suggesting a mean lift as would be expected with a non-zero mean angle of attack. The
asymmetry in these figures is characterized by the shedding of vortices in distinct
pairs that tend to create a mushroom-shaped jet at an angle to the incoming flow.
The deduced mean lift is most likely the result of this transverse acceleration of mass.
Koochesfahani
[12] presents a variety of visualization and laser doppler velocime-
try (LDV) data for a NACA 0012 wing section pitching about its quarter chord point
at Reynolds number about 12,000. These experiments were performed in a low speed
water channel using a feedback control system to control the pitch of the wing for
sinusoidal and non-sinusoidal profiles at amplitudes of 2 degrees and 4 degrees.
15
For sinusoidal oscillating pitch angle the author identifies a thrust/drag dependence on reduced frequency where above a certain threshold the oscillating wing
produces thrust. This threshold value was shown to be different for the 2 degree and
4 degree cases.
Flow visualizations for these cases showed that the result of drag
or thrust production could be deduced by the arrangement of down-stream vortices.
In particular, it was shown that the case of all vortices in line corresponds to zero
thrust/drag, and that any staggering results in non-zero thrust/drag with a sign determined by the direction of rotation of the staggered vortices. Furthermore, a unique
wake mode was identified with four vortices shed per cycle at high frequencies for 4
degree pitch amplitude.
Non-sinusoidal experiments were conducted by accelerating one of the strokes
(half of the cycle) while decelerating the stroke moving in the opposite direction.
This motion is described by a symmetry parameter, S, where S=50% corresponds to
sinusoidal motion.
For these experiments, Koochesfahani shows that typically one
large vortex is shed during the fast portion of the cycle, while the slow portion of the
cycle sheds multiple vortices of the same sign. Figures show that this often results in
three vortices shed per cycle, in a clearly asymmetrical arrangement. Measured mean
wake velocity profiles confirm this asymmetry, and suggest mean lift on the foil.
Gursul and Ho
[8] performed experiments on a stationary two-dimensional NACA
0012 wing section at an angle of attack of 20 degrees to an unsteady stream. These
experiments were performed in a vertical gravity-fed water channel equipped with
variable position output slats for controlling the velocity of the flow in the test section.
The velocity of the inflow is described as a steady state component plus an oscillating
component where the amplitude of the oscillating component (R) is varied between
30% and 70% of the mean flow velocity.
Results show a peak in mean lift coefficient for all values of R at a reduced velocity,
k, of about 0.8. It is also shown that mean lift coefficient increases with increasing
R. Flow visualizations suggest that mean lift is primarily a function of the amount
of the cycle for which the unsteady leading edge stall vortex remains attached to
16
the wing before shedding into the wake.
They demonstrate that this behavior is
maximized at around k = 0.8. The maximum mean lift coefficient shown is about 2.7
for R = 0.70 and k = 0.7 (tests with higher values of k are not presented presumably
due to limitations of the apparatus).
Gursul and Ho also present phase-averaged instantaneous lift-coefficients for three
values of k. These curves show a strong peak midway through the cycle with maximum values approaching 14 for k = 0.70. It is suggested that these transient forces
are a function of the intensity of the separation (leading edge) vortex as it passes the
mid-chord of the wing.
Bandyopadhyay et al.
[3] investigated the morphology of various fish species, and
identified common characteristics of both fast and maneuverable fishes. Maneuvering
capabilities of fast and maneuverable fish (mackerel and bluefish) are quantified using
video analysis.
Comparison of this maneuverablity to that of man-made vehicles
clearly shows the superiority of live-animal maneuvering to that of man-made vehicles.
Experiments on an unsteady maneuvering device consisting of a controllable-camber
dorsal fin are also discussed.
Experiments show unsteady forces not predicted by
airfoil lifting theory.
Anderson et al.
[1] conducted extensive experiments with heaving and pitching
NACA 0012 foil sections at zero mean angle of attack for a variety of harmonic
heave/pitch motions. The authors' experiments focused on a search for maximum
propulsive efficiency using both force measurement and Particle Image Velocimetry (PIV). The force measurement experiments were conducted at Reynolds number
40,000, while the visualization was at Reynolds number 1100.
The search resulted in propulsive efficiency as high as 87%. Visualization shows
that high efficiency corresponds to the synchronization of the shedding of same-sign
leading edge and trailing edge vorticity resulting in a reverse Karman street. Furthermore, it was shown that efficiency is highly dependent on the phase angle of motion
between heave and pitch which effects the synchronization of leading and trailing
17
vorticity shedding.
Read
[13] performed a number of experiments on an apparatus similar to that of
Anderson et al. [1]. Read's experiments were in four categories: Harmonic Propulsion,
non-Harmonic Propulsion, Harmonic Maneuvering, and Starting Maneuvers.
Harmonic Propulsion experiments showed an efficiency plateau for high thrust
coefficients on order of unity at about 45% efficiency. A drop-off in efficiency was
identified for high Strouhal numbers and attributed to the kinematic breakdown of
the angle of attack profile that results from harmonically driven heave and pitch
motions.
Non-harmonic propulsion experiments corrected the deteriorating angle of attack
profile by modulating the heave velocity in order to produce a nearly sinusoidal angle of attack profile.
Read was able to apply this approach to a number of high
Strouhal number runs, and showed that trends of increasing thrust-coefficient at low
Strouhal numbers could be continued into higher Strouhal number ranges by using
the corrected motion.
Harmonic maneuvering experiments consisted of harmonic heave/pitch motion
with a non-zero mean pitch position. Read showed that this motion could be used
to generate large mean lift coefficients and results in a general vectoring capability of
flapping foils for possible application to surface and underwater vehicles. Interpolated
results show a maximum mean lift coefficient with zero thrust of 4.34 at 100 degree
phase angle and 30 degree maximum angle of attack, and bias angle of 24 degrees.
Haugsdal
[9]
conducted experiments on the same apparatus as Read [13].
The
experiments of interest were those visualizing the flow in the wake of the flapping foil.
Haugsdal did a number of experiments contrasting the wake behind harmonically
oscillating foils, and non-harmonically oscillating foils. The most interesting of these
were cases similar to those conducted by Read.
For these cases, Haugsdal shows
that the wake behind a harmonically driven foil at high Strouhal number is relatively
unorganized with multiple "drag" vortices shed per cycle. In contrast, visualizations
18
show that, for the same Strouhal number and heave-to-chord ratio, fixing the angle of
attack with a sinusoidal profile results in an ordered wake with vortices staggered in
a standard two vortex-per-cycle configuration resulting in markedly better efficiency
and thrust production.
19
20
Chapter 2
Design and Construction
Most any design project requires the effective balance of competing design objectives.
Typically the process of design involves an iteration through the objectives, where
a decision made to improve the meeting of one objective affects the design's ability
to satisfy the others. The design of the Sea-Lion was achieved through just such a
process. As such, the discussion presented in this chapter shows the role that each
of the primary objectives played in the various design decisions. The discussion does
not attempt to narrate the actual iterative design process complete with all of the
dead-ends that are revealed through such a process. The design and construction of
the Sea-Lion apparatus were driven by the following objectives:
" testing in the MIT Towing Tank
" measure all six forces and moment generated in time by foil motion during
a constant speed run as well as foil position in order to calculate propulsive
efficiency and force and moment coefficients
* two degree-of-freedom flapping motion (roll/pitch)
* maximize motion controllability
" minimize free surface effects and non-uniform inflow effects
* forces large enough to measure accurately
21
. provide for the possibility of real-time feedback control
* maximize ease of testing by a single operator
* construction by a student using MIT facilities with moderate outside assistance
2.1
Hull-Form and Principal Dimensions
More than any other design component, the hull-form best shows the general concept
of a flapping wing attached to a vehicle. The final design of this hull-form resembles
the typical tubular underwater vehicle. However, the hull-form design required significant balance of the design objectives before a workable final design was complete
A streamlined submarine-like hull form was chosen as the platform to attach a
flapping wing for testing.
This decision was reached as a result of the balance of
previously stated design objectives.
In order to minimize free surface effects and
maximize the uniformity of the input flow to the foil, the apparatus is located in
the vertical middle of the tow-tank. An underwater body is required to house the
mechanism required to create the flapping motion of an external wing and to simulate
the existence of the vehicle or animal that the wing is typically attached to.
The principal dimensions of the hull-form were chosen to be the maximum size
that would remain workable for one person to move and lift alone.
Proximity of
the flapping wing to the free-surface and tank floor were also considerations, but the
limiting factor was the single operator limitation.
In fact, though it is possible for
a single person to move the apparatus from stowed position to testing position and
vice versa, nearly any other movement of the device requires two people.
Another
consideration for the principal dimensions was the bulk of the internal sensors and
flapping mechanism. This bulk proves to be a limiting factor in maintaining small
overall length.
The wing is located on the bottom of the hull opposite the carriage attachment
mast. This placement minimizes any effect of the fixed mast on the fluid flow around
the wing, as well as minimizing any vertical accelerations of fluid which would be
22
Table 2.1: Principal Dimensions
Length
5.25 ft
Hull Diameter
Wing Span
8 in
30 cm
Wing Plan-form Area 1135m
Average Chord
5.51 cm
most likely to dissipate energy through the generation of surface wave energy. The
wing dimensions are limited by the torque capacity of the servo gear-motors.
The
wing-span to chord ratio is similar to that of a penguin, and the length was chosen
based on 2-D flapping foil experiments conducted in [13]. The final dimensions are
listed in table 2.1.
2.2
General Arrangement
With the general external hull form and wing placement determined, the internal
arrangement of sensors and actuators is possible.
The factors of primary impor-
tance during this design phase are the environmental requirements of various sensors
and actuators. Economical electric motors and sensors are rarely water-proof. Understandably, this creates difficulty when coupled with the fact that the vehicle is
required to operate primarily under water.
The design requires that at least a portion of the hull that contains the primary
motors and force sensors remain dry. The awkward attachment of the flapping wing
to the hull makes the possibility of a reliable water-proof seal at the point of wing
attachment unlikely. The final design does not attempt such a questionable arrangement. Rather, the final design employs the usee of a proven rotary shaft seal for the
primary moving seal (see figure 2-3). All other seals required in the final design are
static seals that can be constructed with ease.
The final arrangement consists of a horizontally split central hull machined from
a solid Delrin cylinder (see figure 2-1). This central Delrin portion is extended at the
23
Carriage Attachment
Strut
Upper Central
Section
Nose Cone
Tail Cone
Wing Attachment
Point
Lower Central
Section
Forward Acrylic
Section
Aft Acrylic
Section
Figure 2-1: Hull Arrangement
O-ring
O-ring
Seals
Seals
Pitch Motor
Beam Couplings
Roll Motor
Figure 2-2: Motor Arrangement
nose and tail by transparent acrylic tubes which are further capped by a wooden nose
and tail cones. The interfaces between the acrylic hulls and the central Delrin section
are sealed by a double o-ring seal. The carriage attachment mast attaches to the top
of the central Delrin section, while the wing attaches at the bottom of this section.
For reasons mentioned above, the bottom central hull is designed to be flooded
during normal operation. This section houses the flapping mechanism which is di-
rectly inside of the external wing attachment point. This lower central section also
houses two water-proof torque sensors. The top of this section is capped by a removable anodized aluminum sealing plate fitted with an o-ring gasket seal and easily
removable stainless steel fasteners. This sealing plate contains a cut-out for the forcedynamometer, and is fitted with a flexible membrane seal so that the dynamometer
remains dry while still able to sense forces. The ends of this section are fitted with
24
Flexible Watertight
Membrane Seal
Watertight Deck
Rotary Shaft Seals and
Watertight Bulkheads
Figure 2-3: Waterproofing Details
Teflon v-ring rotary shaft seals that allow shafts attached to the motors to actuate
the flooded flapping mechanism.
The two servo-motors are housed within the tubular acrylic sections, and are
mounted on two aluminum angles embedded in the central hull. The top sealing plate
and the rotary shaft seals of the lower central Delrin section, and the double o-ring
seals between the acrylic and Delrin sections allow the acrylic sections of the hull to
remain dry and maintain suitable environmental conditions for standard economical
servo-motors.
2.3
Flapping Mechanism Design
The design approach to the flapping mechanism is dominated by the required roll/pitch
wing kinematics and the objective of maximizing controllability. However, considerations of watertight sealing and sensor arrangement affected the design considerably,
resulting in design compromises that often hindered the satisfaction of the primary
objectives of controllability and proper kinematics.
The kinematic requirements of the design are to independently control the roll and
pitch positions and velocities in time to create the flapping-wing-like motion observed
25
in penguins and other marine animals.
The design requires that the mechanism
be capable of a wide range of motion amplitudes, frequencies, and motion shapes.
In particular, the design must be capable of phase shifted harmonic motion where
the roll angle leads the pitch angle by a specified phase lag. The mechanism must
also be capable of non-sinusoidal motion profiles within reasonable physical limits
(continuous, differentiable, without excessive velocity or acceleration, etc.).
In order to achieve the variety of different amplitudes, frequencies, and motion
profiles, computer controlled servo-motors were chosen to drive the flapping mechanism. Servo-motors are ideal for this type of application because nearly any motion
profile is achievable through computer control without any physical change such as
changing linkage points or changing out gears or cams. Changes in motion profiles
require only small software changes that can be accomplished in the middle of a single
run if required.
The controllability of the above kinematic requirements is the second primary design objective and can be described as the accuracy with which the mechanism can
create the required kinematics. For any physical system, one-hundred percent accuracy is likely impossible, therefore the goal is to minimize any error, while satisfying
all other design objectives.
The main obstacles to the controllability of this flapping mechanism design are
the torque capabilities of the servo-motors, mechanical backlash, transmission compliance, and proximity of feedback sensors to the objective output of the mechanism.
All of these factors lead to increased error in the desired roll and pitch motion of the
wing. The final design attempts to balance the sources of kinematic error:
* Torque capability of the servo motors: minimal error because of gear-boxes, but
increases backlash.
* Mechanical backlash: moderate error due to motor gear-box backlash and pitch
bevel-gear, reduces required motor torque.
" Transmission compliance: moderate error, steel shafts are stiff, but somewhat
long, and beam couplings are less stiff but are required to allow for small mis26
-J
alignments.
* Error coupling: coupling between the two motor inputs for the wing pitch
output results in the addition of roll error into the pitch error.
" Proximity of feedback sensors to output: moderate error because the above
sources of error are not measured at the motor encoder. Design is limited due
to lack of availability of small water-proof position sensors.
Pitch
Input
ShaftRoll
Input
Shaft
Bearing Housing
Bevel Gear
Wing Attachment
Housing
Figure 2-4: Flapping Mechanism
The final flapping mechanism design uses the unique bevel-gear arrangement illustrated in figures 2-4 and 2-5. This design allows for a direct connection from the roll
control motor to the roll input of the mechanism. This direct connection minimizes
backlash and compliance. The pitch transmission includes a bevel-gear arrangement
that allows the pitch control motor to be placed horizontally which is ideal considering the 8 inch hull diameter limitation. This does, however, introduce some backlash
into the pitch transmission. Furthermore, this arrangement couples the control of the
wing pitch. This coupling is a result of the fact that the bevel-gear housing actually
rolls with the same angle of the wing. The coupling between the roll and pitch motors
that results in the wing pitch position is described in equations 2.1 and 2.2. The roll
and pitch coordinates are shown in figure 2-6.
27
Bevel Gears
Pitch Input
Shaft
Wing Attachment
Point
Figure 2-5: Bevel Gear Detail
#
= wing roll position
0 = wing pitch position
4D = roll motor position
E
= pitch motor position
# = <D
(2.1)
0 =E) - 4D
(2.2)
pitch, 0
Fu 2oll,
(}
Figure 2-6: Roll/Pitch Coordinates
28
The coupling of input motion to pitch angle output is accounted for by adding the
commanded roll position to the desired pitch position when calculating the motion
profiles for a given run. This corrects for the coupling, giving the desired pitch profile.
Unfortunately, this method does not account for the roll positioning error which is
added to the pitch error due to the coupling.
2.4
Sensor Arrangement
In order to accomplish the objective of measuring forces, power input, and power
output, force and position sensors are a required element of the system design. Furthermore, the three dimensional nature of the flow about a rolling and pitching wing
all but guarantees large forces and moments in directions other than pure thrust,
side-force, roll, and pitch. In all likelihood the forces and moments generated by the
wing in the fluid will consist of a full combination of all six possible forces and moments. It is desirable to know the characteristics of these forces in order to describe
the generated forces, as well as to correct for force coupling in a typical force sensing
system.
The final sensor arrangement measures all of the applied forces and moments
through the use of a six degree-of-freedom force/moment dynamometer and two
torque sensors. This arrangement is illustrated in figure 2-7. The force dynamometer
is anchored to the mast attachment point, and absorbs the load carried by the main
bearing housing of the flapping mechanism. The torque sensors measure the torque
applied to the roll and pitch input shafts to the flapping mechanism. The axial and
transverse forces carried by the input shafts are expected to be small, yet affect the
overall sensitivity of the dynamometer measurement. These effects are to be taken
into account in the sensor system calibration described in chapter 3.
Roll and pitch position and velocity measurements are made using the servo-motor
encoder feedback sensors. Use of these sensors eliminates the need for specialized
water-proof position sensors, and needless sensor duplication.
29
Force/Moment
Dynamometer
Torque Sensors
Figure 2-7: Sensor Arrangement
2.5
Data Acquisition and Motion Control
The process of doing an experiment with the Sea-Lion apparatus involves both controlling the motion of the wing, and recording the physical measurements made by
the force and position sensors. In other similar robotic systems these two tasks are
typically delegated to totally separate systems. This approach of separating the two
systems is driven by a desire to modularize the process making it simpler to troubleshoot and easier to understand. This typical approach, however, limits the flexibility
of control feedback and requires unnecessary sensor duplication in order to maintain
full modularization.
The Sea-Lion Data Acquisition and Motion Control systems are integrated into
one single system. While this tends to make the single integrated system more complex than the two modular systems, it makes elaborate feedback control a possibility
and requires no duplication of sensors. There remain, however, two distinct tasks to
be accomplished by the system: record data for later analysis and actively control
the flapping of the wing.
30
2.5.1
Data Acquisition
Measuring and recording physical quantities requires measuring the physical quantity
with a sensor which outputs an electrical signal somehow related to the value of the
measured quantity.
Often this sensor must be powered by an external unit which
supplies power to the sensor, and converts its electrical output into another electrical
output which typically relates the value of the measured quantity proportionally to
the output voltage. This output voltage is then recorded using some type of logging
machine.
The sensors used in the Sea-Lion are the force/moment dynamometer, the shaft
torque sensors, and the motor angular position encoders.
The force/moment dy-
namometer is based on strain-gage technology and was purchased from AMTI along
with the required external power supply and signal conditioning unit.
The shaft
torque sensors were purchased from Kistler Instruments and rely on the piezo-electric
properties of the Quartz crystal embedded inside the sensors.
This technology re-
quires the use of a specialized charge amplifier to convert the minute amounts of
charge produced by the Quartz into a reasonable voltage signal proportional to the
shaft torque. These charge amplifiers were also acquired from Kistler Instruments.
The motor angular position encoders are distinct from the other sensors because they
make direct digital measurements by emitting a pulse whenever a slot in a spinning
disk passes a stationary reading head.
These digital pulses are fed directly to a
computer and counted to determine angular position.
In order to record the voltage signals output by the sensor signal conditioners, a
data acquisition system is needed to maintain a log of all the data for the experiments. Digital sampling and recording is by far the simplest and easiest method for
logging finely spaced time-domain measurements for use in later analysis.
For this
purpose a Computer Boards DAS-16 analog-to-digital (A/D) sampling board was
used to record the voltage data output from the dedicated sensor signal conditioners.
The position measurement does not require A/D conversion because the encoders on
the servo motors make digital measurements directly. The position measurement is
31
DAS-16
Data
Acquisition
MEl
Motion
Control
Host Computer
Card
Card
Digital Pulses
Analog Voltages
AMTI Strain Gauge
Conditioner (6 channel)
AMTI
Kistler
Kistler
Charge
Charge
0
0
6-DOF
Dynaometer
Encoder
Physical
Forces and
Moments
Physical Moments
Encoder
Motor Angular Positions
Figure 2-8: Data Acquisition System
obtained directly from the motion control board circuitry which is described later in
this chapter.
The DAS-16 board used has 12 bit resolution (4096 increments) set to measure
between
5 Volts. The encoders used 500 line disks which give 1,000 increments of
measurement per revolution of the motor. Since the motor has a 5.9:1 reduction gearset, the position resolution at the motor output is 5,900 increments per revolution.
These measurements are recorded at 200 samples/second, allowing digital filtering of
frequencies below 100 Hz.
2.5.2
Motion Control
Servo control of the motors requires an active feedback controller that continuously
compares a commanded position signal to the actual motor position as measured
by the encoder.
The error between these two signals is then used to generate a
commanded torque sent to the motor in the form of electrical current. The system
32
used in the Sea-Lion includes a DC brushed servo-motor, a current control amplifier,
and a digital control card integrated into the data acquisition and motion control
computer. The motors, purchased from Pittman, are simple DC motors which output
torque roughly proportional to the amount of current applied to the motor coils, and
include integral encoders to measure the angular position of the rotor.
A Copley
Controls current amplifier is used to control the current applied to the motor. The
digital control card was purchased from Motion Engineering Inc. (MEI), and uses
digital circuitry to provide continuous control.
DC Powero
Supply
,
MEl
Voltage
Control Card
Copley Controlsly__
Current Amplifier
at
s
r
Copley Controls
LPCurrent Amplifier
ue
t DC Motor
Encoder
Torque
Output
==DC Motor
Enojder
Figure 2-9: Motion Control System
The MEI control card uses encoder interfaces, analog voltage outputs, and a dedicated digital signal processor (DSP) to calculate commanded positions, measure
position error, and execute a discrete PID control algorithm to generate an analog
torque command to send to the Copley Controls amplifier. Because the MEI board
uses a dedicated DSP, the host computer is not burdened with any of the PID computational load and is free to execute other tasks such as data acquisition or driving
the user interface.
33
2.6
Real-Time Software
In order to integrate the data acquisition and motion control tasks into one computer,
real-time software is required. The software must be able to make use of the various
sensor inputs for logging data from the experiment and also make use of the motor
control card for managing the motion of the wing. The primary challenge of this
integration is to create and use software that can collect and feed data to and from
the proper sensors and controllers in a timely and reliable manner. This challenge is
overcome by the use of real-time software.
By integrating the motion control and data acquisition systems into one system,
it is possible to log motor position information collected by the control card, or to
use sensor information collected by the data acquisition card as feedback for use
in the control system. In order for this integration to succeed, the host computer
and its software must be capable of shuffling the information where it is needed,
when it is needed.
The standard data acquisition rate for Sea-Lion experiments
is 200 samples/second, resulting in a time increment of 0.005 seconds. In order to
properly log synchronized data between the control card's encoder inputs and the data
acquisition card's A/D converters, the software must be able to collect the data in
intervals smaller than .005 seconds, and do so repeatedly without delaying or skipping
any of the scheduled increments. Good performance feedback control for this system
could require reliable scheduling increments an order of magnitude smaller than those
required for simple data logging.
Today's standard operating systems are not capable of providing this level of
scheduling guarantee. Most modern operating systems including Windows and Linux
are based on a preemptive multi-tasking scheduler that does not give any hard scheduling guarantees for any of the various tasks running on the system. This design is ideal
for applications where no guarantee is required, and the system is required to balance
a number of computing tasks simultaneously. Unfortunately, controlling an apparatus such as the Sea-Lion requires tight scheduling guarantees that standard systems
cannot offer.
34
Fortunately, many engineers in various industries have the need for such scheduling
capability, and a small but dedicated market exists for software that can perform
these non-standard tasks. Among these specialized alternative exists a free and open
extension to the equally free and open Linux operating system called Real-Time Linux
(RTLinux). This extension allows a few specially coded process threads to rely on a
real-time scheduler that can provide hard guarantees about when processes are started
and repeated. The scheduling accuracy for this system is limited by the computer
hardware rather than by the system software. In the case of the commodity Intel
hardware used in the Sea-Lion, scheduling accuracy is within an error of about 15pas.
An advantage to this approach is that the rest of the Linux operating system remains,
along with its capability for system maintenance, disk i/o, network communication,
etc.. These operating system functions cannot be used with the real-time scheduler,
but this is not necessary for application in the Sea-Lion.
The real-time software system design for the Sea-Lion uses RTLinux to control
the timing of the data acquisition samples. This design requires hardware drivers to
be modified for use with the real-time scheduler. Since the data acquisition requires
data from both the data acquisition and motor control hardware, drivers had to be
modified for both pieces of hardware. Fortunately, source code is available for the
drivers for both pieces of hardware, and only minor changes are required to allow
their compilation and use with the real-time extensions.
With these changes, the
integrated software and hardware system is fully capable of logging the data from the
A/D converters as well as the encoder feedbacks in regular increments. Figure 2-10
illustrates the real-time software design.
This system is also easily extensible to use custom control algorithms using any
of the sensors attached to the computer as feedback. While this is not yet implemented in the system, very few modifications are required to use the apparatus as
a platform for experimenting with force-feedback control in biomimetics and marine
vehicle maneuvering and propulsion.
35
-
-
- -
-
- ---
-
-
--
High Priority Tasks
(RTLinux)
Ir
-
i
1
I
-
-
-
-
U-
Periodic Data Acquisition
-
-
-
-
-
-
-
-
-
-
Data
Da Logging
Hardware
Safety Monitoring
-
Low Priority Tasks
(Linux User-Space)
Initialization
Network Services
Advanced Feedback
(SSHd, FTPd, HTTPd)
Control
User Interface
Custom
Drivers
Shared Resources
Non-Real-Time Access Only
DA-6User
Data
Disk
MEI
Control
Acq.
Storage
(Ethernet)
Card
mue
mose
C r
-
Interface
(keyboard,
I
Cad
Network
-
display)
- -- --
Figure 2-10: Host Computer Software Setup
36
-
-
-
Chapter 3
Sensor System Calibration
After an experiment is completed, and the outputs from the sensors have been
recorded, these outputs must be translated into real units of force and position.
This translation is accomplished through the use of a calibration which consists of
a series of experiments conducted under controlled conditions with known loads and
positions. If these calibration experiments are complete, their results, in combination
with the known loads, are used to create the sensor system calibration.
Often, for a simple apparatus with a single sensor which is known to have linear
response a single experiment can give a reliable calibration. For complex systems,
however, a number of factors can complicate this process. The factors complicating
the Sea-Lion sensing system are cross-coupling and position-dependent offsets. These
two complicating factors are handled independently and sequentially.
3.1
Position-Dependent Offset Correction
An unfortunate consequence of the design discussed in section 2.4 is that minor misalignments in the flapping mechanism and external shafts and bearings result in a
non-linear position-dependent offset at the dynamometer output channels. This is
best shown by removing the wing from the apparatus while in air and observing the
dynamometer measurements while rotating the motor shafts. The dynamometer measurements show significant changes in force, while it is known that there is negligible
37
change in external load while the vehicle is in air with no wing attached.
These forces measured by the dynamometer are real forces caused by the lack of
perfect alignment in the various shafts, seals, bearings, and couplings in the flapping
mechanism and external transmission. These forces, however, are not of interest in
the study of the forces generated by wing/fluid interactions. It is important, therefore,
to understand the nature of these internal forces so that they can be distinguished
from the forces of interest.
The internal forces can be eliminated from the dynamometer measurement by
describing the offset of each of the dynamometer channels as a two-parameter function
of the two motor positions. Because the motors are computer controlled, it is simple to
finely sample points on these two parameter functions, and later interpolate between
these points to calculate the offset corrections to subtract from the dynamometer
measurements for all of the combinations of motor positions encountered during the
run.
20200,
0,
-10W-20-
-200,
-40,
00
_400-
30-
9 (rad)
4 (rad)
0.- .5
9 (rad)
-0
0 (rad)
D (rad)
cD(rad)
d4
d6
0
-
-
40
200
0
Z100_
-~
-20
0101
5)05-,
0
-0
0
9 (rad)
(D (rad)
0 (rad)
5
-2
E) (rad)
Fig re 3-1: Position Dependent Offsets
Figure 3-1 shows the result of a typical set of zero external load experiments for
various combinations of motor positions. These samples are taken at 25 pitch motor
positions for each of 25 roll motor positions. All positions are evenly incremented
38
between the extremes of available motion. The result is the interpolation of a grid of
625 samples describing a surface that is a function of the two motor positions.
Once a run is completed, the recorded motor position profiles are used to calculate
the offset of each of the dynamometer channels. These offsets are then subtracted
from the recorded dynamometer signals. After the subtraction the remaining signals
are the result of external forces exclusively.
3.2
Coupling Matrix Solution
Due to the design of the force sensing system it is not possible to merely use the
output of the dynamometer as representative of the actual external forces, even when
applying the position dependent correction. This is a result of the fact that the dynamometer is integrated into a structure that includes two input shafts that carry
loads to the surrounding structure that cannot be measured by the dynamometer.
In fact, given the nature of the integrated structure, it is difficult to make any assumptions about what types and magnitudes of forces are absorbed by these input
shafts. Therefore, it is necessary to execute a complete set of calibration tests that
can identify any possible coupling between sensor measurements and applied external
loads. Furthermore, it is necessary to make these measurements at a number of load
magnitudes in order to show linearity of the coupled system.
x
Figure 3-2: Force/Moment Coordinates
For simple systems with minor coupling it is possible to approach the problem in
a manual fashion by identifying coupling parameters individually while isolating the
39
exact source of the coupling. For sensing systems as large as the Sea-Lion, however,
this painstaking process can spiral into a system of too many coupling parameters
to understand individually.
As a result of this complexity a more general method
was developed that increases the simplicity of the process, perhaps at the expense of
detailed understanding of the cause of each of the individual parameters.
The general system that was used was to identify a series of calibration tests
with known loads that exercise the sensing system in a complete series of linearly
independent modes. Each of these modes is exercised with varying load magnitudes.
The results of these tests are used as two matrices: the matrix of known loads in
each of the primary directions for each test (L), and the matrix of measured voltages
for each of the sensor outputs for each test (R). The relationship between these two
matrices is a matrix (S) that contains the calibration and coupling parameters needed
to interpret the sensor data recorded for any run. The relation is as follows:
R x S = L
(3.1)
S = R\L
(3.2)
where the backslash (\) operator solves the over-determined linear system in the leastsquares sense. The system is over-determined because tests were conducted with three
sets of weights, giving more information than is required for a unique solution. This
calculation gives the calibration matrix (S) such that the actual external loads (1) are
determined by right multiplying the sensor measurements (r) by S.
I= i x S
1i=
(3.3)
[Fx , Fy , Fz , Mx, My, Mz , Mpitchl
T=T
S= [di, d2, d3, d4, d5 , d6, 71, T21
Tables 3.1-3.3 show the calibration data and solution for a typical set of calibration tests.
Notice that identical modes are tested at different magnitudes, and
that the 24 tests in seven principal axes create a full-rank loading matrix. It is also
40
Test #
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Fx
Fy
Fz
Mx
My
Mz
Mpitch
(N)
0
0
0
0
0
0
0
0
0
(N)
-2.57
(N)
0
0
0
0
0
0
0
0
0
(N-m)
(N-m)
0
0
0
0
0
0
0
0
0
(N-m)
-0.00967
(N-m)
-0.00967
-3.1
-5.67
-8.09
-3.1
-5.67
-8.09
0
0
0
0
0
0
0
0
0
-2.57
-2.57
-2.57
-5.26
-5.26
-5.26
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.208
-0.0057
-0.0198
0.426
-0.0117
-0.0293
0.631
-0.0173
0.0158
0.0411
0.0158
0.208
-0.0057
-0.0198
0.426
-0.0117
-0.0293
0.631
-0.0173
0.0158
0.0288
0.0411
0.0158
0.849
0.0288
0.0288
1.21
-0.109
-0.198
0.0411
0.0411
0
0
0
0
0
0
-0.00804
0
0
0
0
0
0
0
0
0
-2.57
-2.57
-5.26
-5.26
-5.26
-7.79
-7.79
-7.79
-0.207
-0.204
-0.427
-0.423
-0.416
-0.873
-0.627
-0.617
-1.29
3.1
5.67
8.09
0
0
0
0
0
0
-0.213
-0.21
-0.396
-0.491
-0.485
-0.865
0.241
0.441
0.629
0.465
-0.283
0
0
0
0
0
0
0.0288
-0.00967
0.208
-0.0057
-0.0198
0.426
-0.0117
0.173
-0.00474
-0.0165
0.354
-0.00969
Table 3.1: Applied Calibration Loads, L
important to note that the resulting measurements also form a full-ranked matrix.
The final calibration matrix S shows that the system is very coupled. In order to
calculate a single external load, the measurement of more than one sensor is needed.
Furthermore, the existence of a dependence of the load calculation on the shaft torque
sensor data implies that the shafts carry non-torque loads that are proportional to
the torque measured by the sensor.
Table 3.4 shows the accuracy of the least squares solution. Numbers shown in
the table are percent error of R x S based on the maximum load applied for each
principle direction for all 24 calibration runs. A brief look at the errors found shows
41
Test #
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
di[
-0.200
-0.104
-0.192
-0.385
-0.287
-0.363
-0.542
-0.466
-0.519
-0.039
-0.058
-0.071
-0.038
-0.039
-0.067
-0.007
-0.010
-0.011
-0.165
-0.080
-0.128
-0.321
-0.242
-0.313
d2 I
d3 !
d4 I
0.181
0.011
0.213
0.158 0.024 0.167
0.366 0.010 0.427
0.365 0.017 0.433
0.340 0.031 0.345
0.741 0.014 0.861
0.533 0.019 0.626
0.501 0.042 0.531
1.091 0.020 1.268
0.182 -0.002 0.471
0.312 -0.001 0.812
0.435 0.002 1.137
0.147 -0.003 0.637
0.233 0.008 1.114
0.338 0.018 1.623
-0.008 0.066 -0.052
0.022 0.119 -0.056
0.038 0.172 -0.087
0.197 -0.001 0.219
0.174 -0.017 0.215
0.353 -0.008 0.423
0.419 -0.002 0.478
0.394 -0.017 0.427
0.865
0.742 -0.001
d5 I
d6 I
0.245 0.005
0.105 0.429
0.227 -0.013
0.029
0.471
0.318
0.787
0.428 -0.002
0.663 0.056
0.526 1.154
0.605 0.072
0.028 0.034
0.037 0.050
0.040 0.053
0.020 0.039
0.000 0.017
0.013 0.027
0.009 0.022
0.011 0.026
0.010 0.031
0.203 0.035
0.078 0.462
0.149 0.052
0.392 0.062
0.263 0.820
0.367 0.041
Table 3.2: Calibration Test Measurements,
T1[
0.161
0.001
0.323
0.326
-0.001
0.652
0.469
-0.011
0.950
0.004
-0.002
-0.006
-0.004
-0.026
-0.026
-0.002
-0.004
-0.003
0.178
0.039
0.320
0.378
0.095
0.666
(bits)
that the solution is very accurate, implying that the system is linear.
42
T2
0.008
-0.156
0.006
0.015
-0.324
0.009
0.016
-0.483
0.005
-0.009
-0.015
-0.019
-0.009
-0.012
-0.015
-0.000
0.000
-0.000
0.006
-0.138
0.005
0.012
-0.284
0.010
Ff
di
d2
d3
d4
d5
d6
T1
T2
-0.0192
-0.0136
0.0013
-0.0027
-0.0145
-0.0013
0.0180
-0.0177
FJ
-0.0256
0.0032
0.0001
-0.0015
-0.0297
-0.0072
-0.0045
-0.0068
Fz
Mx
0.1533 0.0040
0.0025 0.0002
0.0489 0.0000
0.0043 0.0001
0.1177 0.0031
0.0113 0.0002
-0.0019 -0.0015
0.0190 0.0016
MyI
-0.0023
-0.0012
-0.0009
0.0009
-0.0018
-0.0003
0.0000
-0.0007
Mz[
0.0001
-0.0000
-0.0000
0.0000
0.0000
0.0002
-0.0000
-0.0009
Mpitch
0.0007
0.0000
0.0000
0.0000
0.0006
-0.0001
-0.0000
-0.0015
Table 3.3: Calibration Matrix, S
Test #
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
F1 1
0.25
1.41
0.27
0.24
0.64
0.13
-0.20
-0.58
-0.61
-1.16
-0.13
1.02
-1.60
0.05
0.31
3.31
-0.76
-1.02
0.16
-0.08
-0.27
0.20
-0.43
0.64
FyJ
-1.79
4.13
-1.24
-1.58
2.83
-0.49
1.57
-1.71
-1.29
-1.87
-0.96
1.53
-3.89
2.05
0.04
-2.24
-0.76
0.65
-0.31
-0.56
1.05
2.92
-2.21
1.34
F2J Mx
-0.40 -0.55
7.74 -0.50
-2.12 -0.69
0.03 -0.53
1.64 -0.58
-1.98
0.02
-0.31
0.43
-0.26
0.12
0.74
-0.70
-1.65 -0.80
0.15 -0.08
0.29
1.43
-2.88 -0.64
-2.21
0.57
2.04 -0.08
1.06
0.17
-0.44
0.31
-1.57 -0.19
1.73 -0.85
-0.63 -0.42
3.33
-0.96
1.97
-5.26
1.85
0.69
0.91
-0.42
My
0.20
Mz [ Mpitch
-0.32
-1.59
-0.40
-0.60
-0.58
-0.05
0.60
0.23
1.01
0.09
0.01
-0.23
0.28
0.12
-0.07
-0.01
0.06
0.22
-0.27
-0.30
-0.51
-0.43
1.23
-0.92
-1.48
1.60
0.49 -1.62
0.49 -0.91
-1.22 0.63
0.46 -1.70
0.29 0.26
0.94 0.73
-0.12
1.23
1.42 0.49
0.20 0.33
-0.59 -0.33
1.11
0.00
-0.83 -0.15
0.43 -0.42
0.28 0.13
0.62 -0.18
-0.34 -0.32
-1.24 0.67
1.40 0.66
1.36
0.33
-1.31
1.27
-0.09 -3.21
-0.17 0.04
Table 3.4: Percent Error of Maximum Load
43
-0.83
44
Chapter 4
Maneuvering Experiments
With the apparatus designed built and calibrated, the system is ready to run and
collect data.
One of the primary goals in pursuing the study of flapping foils is
to improve the maneuvering capability of human vehicle designs. This thesis does
some preliminary exploration into the lift producing capability of three-dimensional
flapping foils that may eventually be useful in application to current designs.
All of the maneuvering experiments were conducted in the MIT Towing Tank
during April, 2001. The towing speed for all tests is set at 0.40 meters per second,
resulting in about 10 to 20 steady-state cycles measured during a run. The Reynolds
number based on the average chord for all experiments is 20,000. Mean performance
data is calculated as the mean over the maximum number of steady state cycles, while
time-domain data is presented as phase-averaged values.
In order to reduce the effects of control backlash, a high-freqeuncy low-amplitude
dither was added to the motor command position.
This dither was intended to
be of the same amplitude of the backlash (about 3 degrees) in the transmission.
It was found through trial and error that a dither frequency of 50 hertz gave the
most apparent stiffness while manually handling the wing underwater. Small highfreqeuncy pressure waves could be felt radiating from the wing underwater due to the
induced vibration. It is possible that this vibration could have effected the large-scale,
low-frequency forces measured during the experiment. This result was not intended,
and was deemed to be unlikely.
45
4.1
Dimensional Analysis
All experiments conducted were based on a harmonic swimming model consisting of
a sinusoidal roll angle with sinusoidal wing pitch angle. The parameters used in the
definition of the wing motion are as follows:
U = carriage forward speed, m/sec
W = flapping frequency, rad/sec
0
= roll amplitude, rad
00 = pitch amplitude, rad
4= phase
t
=
angle between roll and pitch, rad
time, sec
r = distance from the roll center to a point on the span, m
These parameters are used to define the roll angle, q, pitch angle, 0, and angle of
attack, a, as functions of time:
0(t) = #o sin (wt)
0(t) = 0 sin (wt -
(4.1)
4')
(4.2)
a(t) = 0(t) - arctan 0 0w cos(wt)
U
(4.3)
In order to generalize the results from the experiments at this scale and speed,
non-dimensional parameters are chosen such that performance is expected to be dependent on the non-dimensional parameters rather than the dimensional parameters
for a particular scale. The non-dimensional parameters used to describe the style of
flapping motion are: phase angle between roll and pitch, Strouhal number based on
mid-span distance, maximum angle of attack achieved at the mid-span, and the roll
46
amplitude.
7r
2
St
Omax
midspan
#0
2
=(4.4)
(4.5)
U ormidspan
= Maximum angle of attack achieved mid-span
(4.6)
= Roll amplitude
(4.7)
All of the maneuvering experiments were conducted for four unique combinations
of the non-dimensional parameters.
These unique combinations are referred to as
"styles" because they define the harmonic flapping motion of any roll/pitch flapping
system. The styles were selected based on a preliminary sampling of the parameter
space.
Four styles were selected in two groups that were expected to show signifi-
cantly different results. The preliminary sampling of the parameter space was used to
calculate a rough estimate of the relative propulsive efficiencies between runs. Styles
#5 and #15 showed the largest efficiency based on this calculation. Styles #118 and
#113 showed very low efficiency, but relatively large side-force amplitudes.
The primary kinematic difference between the first style group and the second are
both higher strouhal number and lower maximum midspan angle of attack. Within
each group, the only difference is the roll amplitude. The actual physical frequency
and pitch amplitudes that achieve the flapping styles are shown in the last two lines
of Table 4.1.
Table 4.1: Flapping Styles Tested
1 #15
Style
St
#5
0.359
#0
midspan
0.639
0.262
1.012
0.262
1.198
0.873
1.012
0.873
w(rad/sec)
6o(rad)
9.425
1.184
5.953
1.184
4.183
0.209
4.953
0.209
amax
1#118 1#113
0.359 0.299 0.299
47
Bias Pitch Angle Experiments
4.2
One of the more obvious ways to create mean side-forces with a harmonically flapping
wing is to change the average pitch angle to a non-zero value. By flapping with a
non-zero pitch angle, the resulting angle of attack profile will have non-zero mean
presumably resulting in non-symmetric lift forces and thus average side-forces.
Ex-
periments were conducted for the for each of the styles listed in table 4.1 at eleven
increments of bias angle ranging evenly between +0.524 rad (+30 deg). Figure 4-1
shows the flapping kinematics for style #5
at various pitch bias angles.
The roll
profile does not change for the various pitch bias angles.
0.6
-
..
'
I-..
--e-
0.4
0.2
0
-0.2
-0.4
'_.-I ..
'
~n r,
i
-3pi/2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
1.5
0.524
-
0.419
1
-
-_-
0.5
0
2pi
.
-\
4\
....
...--
\-
\'-,*
....
-1
-0.105
-0.209
\--
-0.5
0.314
0.209
0.105
0.000
A
-0.419
-0.524
0 avg
-1.5
-3pi/2
-pi
-pi/2
pi/2
0
Phase During Cycle (rad)
pi
3pi/2
Figure 4-1: Kinematics at Various Pitch Bias Angles (Style #5)
48
2pi
4.2.1
Mean Performance Results
There are a number of different ways to evaluate the data collected for the bias pitch
angle experiments.
One of the important results from these experiments are the
mean lift coefficients and required power inputs. Of course, the flapping wing is an
oscillatory system, and none of the forces or power inputs are constant in time, but
the means of these time-varying values are critical components for the consideration
of one of these devices for use on a ship or underwater vehicle.
The primary performance parameter for evaluating suitability for use as a maneuvering enhancement device is the lift coefficient CL. The lift coefficient is the force in
the direction perpendicular to the incoming flow direction normalized by stagnation
pressure times the plan-form area of the wing.
CL=
Fy
pU2Apian
(4-8)
Another important performance parameter is the power input from the motors
required to generate lift forces. Power output for an angular actuation system is the
product of torque and angular velocity, and the average power input is the average
of the power input over a number of complete cycles. It is important to note that
calculating the average power in this way is not representative of the actual average
power applied to this particular apparatus. The reason for this discrepancy is the fact
that by averaging the power input over a cycle, any negative power input balances
a corresponding amount of positive power input. The consequence is that any real
mechanical system must take advantage of the negative power portions of the cycle
by absorbing that energy and releasing it in the positive power portion of the cycle.
A tuned spring system is one of many possible systems that accomplishes this.
Pin = MXO + MpitehO
f
Pinav
C
P
ndt
(4.10)
P
P
(4.11)
nT
SpU3 Aplan
49
(4.9)
Figure 4-2 shows the average results for lift coefficient and power input for all
tested swimming styles and pitch bias angles. As expected, larger pitch bias angles
result in larger lift coefficients. In fact, for the range of data collected, the dependence
of lift on pitch bias is nearly linear for all swimming styles over the range of pitch biases
tested. The results show a surprisingly large lift coefficients, particularly for style #5.
Typical zero-camber, infinite aspect-ratio wings without lift enhancement devices are
capable of lift coefficients of about one before stalling at around 15 degrees angle of
attack. Finite aspect-ratio wings begin stall at higher angles of attack, and very low
aspect-ratio wings can generate lift at very high angles of attack (>40 degrees), but
with lower maximum lift coefficients. Equation 4.12 gives the slope of a typical lift
vs. angle-of-attack (a) curve for varying effective aspect-ratio (AR) [10].
dCL
da
1
1
11
1
+ 7r(AR) + 2,r(AR) 2
(4.12)
It is important to note that the average lift plot reveals a weakness of this experimental apparatus. It is reasonable to expect that the average lift coefficient would be
zero for all swimming styles at zero pitch bias angle. Clearly, the plot reveals that this
is not the case. Furthermore, it is difficult to imagine that position error would result
in such force asymmetry. This, however, is precisely the source of this non-zero lift at
zero pitch bias angle. The key to understanding the resulting asymmetry is the fact
that the roll error is coupled into the pitch error because of the unique arrangement
of bevel gears in the flapping mechanism. The result is that the the pitch position is
amplified over half of the cycle and attenuated over the remaining half. The result is
an inherent asymmetric motion, and unintended bias toward non-zero lift coefficients.
The plot of average power input also reveals a lack of symmetry between positive
and negative pitch bias angles. This asymmetry is magnified by the plot presentation,
after noticing the scale of the power input plot it is apparent that the asymmetry
here is small compared to the total magnitude of the power input.
It is clear from analysis of the time-averaged results that the two groups of swim-
50
C
--
-j
0
.... ..
..... ..
.X
....0>. ...:. ...............
x-
.... Style St 0 a
X
#5
#15
x
* #118
.-.0 #113
--
--0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
o
0.359 0.639
0.359 1.012
0.299 1.198
0.299 1.012
0.2
0.3
max
0.262
0.262
0.873
0.873
0.4
0.5
5.8
5.6
5.4
.....................
..
5.2
01
........x.................
fl5
4.8 - -
x.
-
- -
-x
-
xx.
-0.5
-0.4
-0.3
-
- - -
-0.2
-
-
.. . . . . . . . . x..
. . .. . . . . . . . ...
. .
. ...3
.. . . .
.
...........
... . . . . .x
4.4
-
4.6
-0.1
0
0
(rad)
0.1
0.2
0.3
0.4
0.5
Figure 4-2: Mean Performance Results for Pitch Bias Experiments
ming styles are a group of styles #5 and #15 and styles #118
and #15
and #113.
Styles #5
appear to be most capable of producing large lift forces, while at the same
time using less motor power than the remaining two tested styles. There is, however,
a missing component of the true input power required to generate lift that is not
shown in these plots. This component is the thrust/drag produced by the flapping
wing. If at any of these bias angles the flapping wing creates a significant amount of
drag, then any fair comparison would have to take into account the power required
to drag the wing through the fluid. On the other hand, if at any of the bias angles
significant thrust is produced, this should also be taken into account. Attempts at
measuring the thrust/drag force with this apparatus were unreliable, and none of the
data is presented here. Therefore, it would be premature to conclude that the first
two styles are superior in average performance to the last two.
51
4.2.2
Time-Domain Results
The time-domain results show the actual forces and positions measured through the
course of the flapping cycle. The time-domain results give clues as to the fluidic source
of the forces, the role of angle of attack in producing the forces. These time-domain
values are also important for the design of a mechanical system that might undertake
to create these forces in an efficient manner. They are also of interest for dynamic
analysis for the possibility of integrating a flapping wing into a system that responds
at the flapping frequency.
All of the time-domain results are presented as a function of "phase during a
cycle" in radians. This phase is defined as zero at the up-crossing of the roll motion,
7r at the down crossing, and 27r or zero at the following up-crossing.
This allows
presentation on the same axes of data that are of different frequencies. The flapping
frequency (and cycle period) is a function of the flapping style.
Figure 4-3 shows the instantaneous lift coefficient over the range of tested pitch
bias angles.
The time-domain plot shows that the maximum measured forces are
three to four times the average forces. It shows that for the high lift runs, there are
two lift peaks, one large and one small per cycle.
The asymmetry noticed in the average lift results can also be seen in these time
domain plots, yet it is clear that the force profiles are very similar between the positive
and negative pitch bias angle cases. This is evident by looking closely at the curves
marked
0.524. By flipping one of these profiles vertically about the zero force line
and phase shifting by a half cycle, it is evident that while the actual magnitudes of
the forces do not correspond exactly, the general form of the curves are very similar,
suggesting that any effect of asymmetric position error coupling is incremental.
Figures 4-4 and 4-5 show the moments in roll and pitch generated by the foil, and
balanced by the motor input. These plots are important because they show the input
torques required by any actuation system of such a device. These plots show the
maximum torques required, as well as the time varying nature of the applied torques
for use in the design of a tuned oscillatory system.
52
...
10-a-
5
.
,-
-
--
\
0.419
0.314
0.209
0.105
0.000
- 0.105
-
0.524
---
- -0.209
-0.314
00
0-
-5
-10--3pi/2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure 4-3: Lift Coefficient vs. Time for Various Pitch Bias Angles (Style #5)
Moments are presented in their non-dimensional coefficient form, where the normalization factor is stagnation pressure times plan-form area times wing-span.
CM
1
M
2
SpU Apianlspan
(4.13)
The measured roll torques shown in figure 4-4 contain, generally, the same frequency of the flapping motion, with a smaller amplitude higher frequency disturbance
seen most clearly at the torque peaks. This is suggestive of the effects of vortex shedding, but flow visualization is required to confirm such intuitions. The pattern caused
by changing the pitch bias angle in increments is striking in its regularity and apparent symmetry. Again, these profiles do not show perfect symmetry between positive
and negative bias angles, but continue to support the belief that any asymmetry is
having a small, incremental effect.
The measured pitch torques shown in figure 4-5 are much less clear than the other
measurements. The cause of this is the fact that the magnitude of these forces is much
less than that of the other forces. This is important in itself, suggesting that driving
53
the pitch of the foil requires much less power than the roll input. Consequently a much
smaller pitch motor is required than the one that was used for these experiments. It
is possible that such a motor could be small enough to be integrated into the bevel
gear housing, eliminating the need for bevel gears altogether.
Another interesting possibility for control of the pitch of the wing would be to
design a passive system that could approximate these force profiles.
Clearly, such
a system would be incapable of mean power input, but with such small forces and
power involved, perhaps passive control would be sufficient.
A complete set of time-domain plots for these pitch bias angle experiments is
provided in Appendix A.
-
-
3-
....
0.524
0.419
0.314
0.209
-0.105
2'-~
-
-
0.000
-0.105
-0.209
-0.314
-0.419
\
-
-
1
0-0.524
|!L
--
\
- . -avg
00 -.....-
0-
I
-3pi/2
-Pi
-pi/2
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure 4-4: Roll Moments vs. Time for Various Pitch Bias Angles (Style #5)
4.3
Non-Harmonic Roll Motion Experiments
Another way to induce non-zero mean angle of attack profiles is to drive the flapping
wing with a non-symmetric roll motion. The fact that non-harmonic roll motion can
result in non-zero mean angle of attack is evident in equation 4.3 which shows that
54
-
0.524
0.419
-
0.4
0.314
0.209
0.105
-f
0.3
*...
--
0.000
U0.105
0.209
-0.314
-0.419
-0.524
0.1
0
-0.3 --
-3pi2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
Pi
3pi/2
2pi
Figure 4-5: Pitch Moments vs. Time for Various Pitch Bias Angles (Style #5)
angle of attack is a function of both the pitch angle and the roll velocity. Theoretically,
an infinite number of non-harmonic roll motion profiles is possible. A roll profile was
selected that is very similar to the non-harmonic pitch motion selected for study by
Koochesfahani [12]. That is, the roll motion is divided into two portions: left stroke,
and right stroke. A symmetry parameter, s determines the percentage increase in the
velocity of one stroke, while also determining the percentage decrease in velocity of
the other stroke. This calculation ensures that the total cycle period is independent
of s. Figure 4-6 illustrates the roll and pitch profiles for eleven values of s evenly
spaced between +0.30.
4.3.1
Mean Performance Results
Mean performance results for the non-harmonic roll motion experiments are presented in the same way as the pitch bias experiments in section 4.2.1. Calculations of
mean lift and power coefficient are consistent with the calculations of the pitch bias
experiments as well.
Figure 4-7 shows the mean lift and power coefficient for all four tested styles, and
55
0.5-
0-
-0.5-
-1
-3pi/2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
1
pi
.
3pi/2
2pi
---
0.300
-
.\. -- 0.240
.- --0.180
0.5
-.
-
~'/
- .
00
-
\4.
-0.5
V
-
-1
---
0.120
0.060
0.000
--- - -\-0.060
- -0.120
---0.180
----0.240
--0.300
/1
-3pi/2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure 4-6: Kinematics at Values of s (Style #15)
all eleven tested values of symmetry parameter. The figure shows a roughly linear dependence of lift coefficient on symmetry parameter, while showing a roughly quadratic
dependence of power coefficient on symmetry parameter. Both these experiments and
the pitch bias experiments show that the higher Strouhal number, lower maximum
angle of attack styles are better at creating large lift forces. The precise reason for
this apparent trend is unknown, and requires further study to identify the nature of
fluid/wing interaction.
Again, as in the pitch bias experiments, this plot reveals a lack of precise control
of the flapping motion which results in non-zero lift coefficient for commanded motion
that is purely symmetric. The asymmetry is also evident by comparing the lift coefficients at the extremes of the tested symmetry parameter. Style #5,
a lift coefficient in excess of -2 at s
=
for example, has
0.3 and a lift coefficient of only 0.5 at s = -0.3.
56
The same trend is evident in style #15, while the opposite trend shows in the other
two styles.
0x
x
.
0 .
-.
...-...-..
.--.
...
-O_
0 .5 - . .. .-.............-.
-1
-0
#5
X #15
* #118
#113
-2 -0.25
0
0.359
0.359
0.299
0.299
0.639
1.012
1.198
1.012
-0.2
-0.15
a
maxx
-
Style St
-
0.262
0.262
0.873
0.873
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
6.5-
>0
C)
0'
5 -
0
050
4.5 -------
-0.25
--
-
x.-..
-.-.
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
S
Figure 4-7: Mean Performance Results for Non-Harmonic Roll Experiments
Taken in context of the power input data, it seems that style #5 shows the most
promise for efficiently producing large lift forces with non-zero symmetry parameter.
However, it must be recognized that these data are somewhat incomplete without
information about the mean thrust/drag produced by the foil in motion. Style #5
may produce large side forces for s = 0.30 but if this occurs at the cost of very large
drag then the suitability of the use of this type of motion may be reduced.
Time domain results were also collected for these tests as in the pitch bias experiments. These plots are presented in Appendix B.
57
58
Chapter 5
Conclusions
An apparatus is designed and constructed for the study of the forces generated by
the three-dimensional flow about a rolling and pitching wing. This apparatus has
the capability to run advanced feedback control programs for simulating a variety
of mechanical actuation systems and real-time dynamic control. The apparatus also
has the capability to measure the full complement of forces and moments generated
by the interaction of the wing in the fluid, as well as the applied motor torques and
power input.
The apparatus is used in preliminary testing to explore the side-force generating
capability of a roll/pitch flapping wing. Average lift coefficients in excess of 3 are
measured at a pitch bias angle of 30 degrees. It is also shown that maneuvering by
pitch bias is likely superior to introducing asymmetry in the roll motion.
This thesis confirms earlier studies showing that large lift coefficients are available
in flapping wings of symmetric section with out "lift enhancement" devices such a
flaps or boundary layer injection [8] [13]. Furthermore, tests show that these large lift
forces are not limited to two-dimensional sections, but can be produced by roll/pitch
kinematics.
The results of this thesis further support the idea that these devices may find
practical application as maneuvering devices because of their capability to produce
large side forces while in an ambient flow.
Of course, no optimization has been
achieved over the large parameter space of these types of devices, which suggests that
59
application in the field may only become more practical with further development.
A number of mechanical obstacles were encountered during the building and testing of the Sea-Lion apparatus. The primary consequence of these problems was the
lack of reliability of the thrust/drag measurement, and the lack of precise control of
wing pitch. Both of these results could be improved by integrating the pitch control
motor into the bevel-gear housing, eliminating the need for the bevel-gears.
This
design change would require a water-proof motor, but would eliminate much of the
backlash in the pitch transmission, and decouple the pitch error from the roll error.
This change would also eliminate the pitch input shaft as a source for force measurement corruption. As with any prototype, a number of other improvements could be
made, but this change would drastically improve the performance of the apparatus.
In addition to the rigid harmonic kinematic parameters explored in this thesis,
a number of parameters exist for exploration: span-wise distribution of wing area,
wing thickness profile, camber, wing flexibility (homogeneous and directional), forcefeedback control, etc. Steady-state wings and propellers have matured over the course
of a hundred years of study. Clearly, there is plenty of room for progress in the study
of unsteady devices.
60
Appendix A
Pitch Bias Experiments
Time-Domain Plots
- II
0.524
.
....
10-
-
-
0.419
0.314
-
0.209
--
-0.105
0.000
-0.105
--
0.209
\-0.314
-I
- \
-
-
'
-
-
-
-0.524
- .0
avg
0-
-5
.-
-3pi/2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure A-1: Lift Coefficent vs. Time for Various Pitch Bias Angles (Style #5)
61
-
-
-
-
-
0.524
0.419-
-- ....-
3 -
2 -
..
0.000
--
-0.105
-0.209
-0.314
-0.419
-
2-/
-
-\
\
1
--
0.524
- -\avg
-
\
-
0.314
0.209
0.105
00
0-
-3-
-4
3pi/2
-pi
-pi/2
pi/2
0
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure A-2: Roll Moments vs. Time for Various Pitch Bias Angles (Style #5)
.--
0.4 .-
-
-
0.524
0.419
0.314
0.209
0.105
0.000
....
0.3
-
-0.105
-0.209
-0.314
-0.419
0.2
J
-0.524
\r
avg
0.1
1
0
-0.1 --CLI
-
-0.2
-
-0.3
-3pi/2
-pi
-pi2
pi/2
0
Phase During Cycle (rad)
pi
3p/2
2pi
Figure A-3: Pitch Moments vs. Time for Various Pitch Bias Angles (Style #5)
62
-.
..
-
8
0.419
0.314
-
0.209
--
6
--
-
~
-
j
4-
-...
/..
2
.
0.105
.
0.000
-0.105
-0.209
-0.314
-0.419
-0.524
avg
-.
00
-
0
0.524
--
-j
-2-
-3pi/2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure A-4: Lift Coefficent vs. Time for Various Pitch Bias Angles (Style #15)
63
. / ..
-
3
.*~
0.524
.--
0.419
..
0.314
0.209
~
2
0.105
0.000
-A
-..
-- I
1
-.
-
- -
-0.419
-
-0.524
--
0
-0.105
-0.209
-0.314
-
-
0avg
--
-
IA
-1
/
\
j
"I I
-2
-3
-3pi/2
-p/2
-pi
pi
0
pi/2
Phase During Cycle (rad)
2pi
3pi/2
Figure A-5: Roll Moments vs. Time for Various Pitch Bias Angles (Style #15)
I
I
I
0.524
.
0.3
0.419
0.314
-
-
-
0.25
-
0.2
0.15
...
4
-
avg
0.1
0c
0.209
0.105
0.000
-0.105
-0.209
-0.314
-0.419
-0.524
0.05
-"AA
0
-0.05-0.1
-0.15
-0.2
-3pi/2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure A-6: Pitch Moments vs. Time for Various Pitch Bias Angles (Style #15)
64
I
I
I
I
a
10 -
I
.
--
0.524
....
0.419
'
"\
-
8-
-.- .
8
-~--
6-/
I
J-\~
\_.
f
. .. .
\\
-
.- - .....
-
4-\
0.314
0.209
0.105
0.000
-0.105
-0.209
0.314
-0.419
-0.524
avg
-
2--
~
0- . ~.
-4-
It
-/ I1/j
-
-6 ---
-3pi/2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure A-7: Lift Coefficent vs. Time for Various Pitch Bias Angles (Style #118)
65
.--3
- -I
2
0.524
0.419
0.314
-....
---
0.209
-- --
0.105
--.
-
~
-
-
...
0.000
____
-0.105
-\ --0.209
--
-
0.314
- -
1
-0.524
-
-'
0.419
-9
avg
:0
IA
-1
I
-2
A-
/
-3
-3pi/2
-pi
0
pi/2
Phase During Cycle (rad)
-pi/2
pi
-
1
-j\
3pi/2
2pi
Figure A-8: Roll Moments vs. Time for Various Pitch Bias Angles (Style #118)
0.3
0.524
0.419
0.314
0.25
0.209
f_
0.105
0.000
.
--
0.2
-.
0.15
-
-0.105
-0.209
-0.314
... -0.419
-0.524
avg
0.1
0
IV
I
0.05
1
0
I'It
-0.05
-0.1
-3pi/2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure A-9: Pitch Moments vs. Time for Various Pitch Bias Angles (Style #118)
66
-0.524
8-
8
~
r
0.419
S-
0.314
0.209
--
-
-
6-f~I- -I
....
.
--
0.105
-
--
I -
/-0.314
-
-
-0.209
-0.419
0 .5 2 4
-..
\-
avg
21 --
-
j- t
-
2
/ -I
-
-2-7/
-2
-6-3pi/2
-pi
-pi/2
pi/2
0
Phase During Cycle (rad)
pi
3pi/2
Figure A-10: Lift Coefficent vs. Time for Various Pitch Bias Angles (Style
67
~
--
4 -
0.105
0.000
2pi
#113)
-
10-
-
0.524
-
0.314
0.209
0.105
3
S -
-
.
-
6
2 -
0.000
....
-0.105
-0.209
-0.314
-0.419
-0.524
~-
-/
A\
....
\\
1-
-
'avg
-
:9 0
-1-
I
I
-2-
-3-
-3pi/2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure A-11: Roll Moments vs. Time for Various Pitch Bias Angles (Style #113)
.-
0.3
-
--
0.524
0.419
0.314
-
- -
0.209
-
0.105
0.000
0.25-
-0.209
-0.314
-0.419
-0.524
Oavg
-
-.-
-
-
....
0.15
0.1
-
-
-0.105
0.2-
0.050
-0.05
-0.15 -11-
-3pi/2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure A-12: Pitch Moments vs. Time for Various Pitch Bias Angles (Style #113)
68
Asymmetric Roll Motion
Experiments Time-Domain Plots
0.300
..-
6-
0.180
-
-
-- -
4
---
-----0.240
0.240
-2
2
0.120
0.060
0.000
-0.060
-0.120
-0.180
-
-
-
--
- .
-0.300
-0i
--
-4-
-
--
Appendix B
-6
1
-10
-3pi/2
-pi
-p/2
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure B-1: Lift Coefficent vs. Time for Various Symmetry Parameter (Style #5)
69
0.300
-
-.-.-
0.240
0.180
0.120
-'0.060
2-
0.000
1 --
-0.060
-0.120
-0.180
-0.240
-0.300
-
-
0
h
s
-2
-5
-
-4-
-3pi/2
-pi
-pi/2
pi/2
0
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure B-2: Roll Moments vs. Time for Various Symmetry Parameter (Style #5)
70
0.300
1 ---0.120
0.060
-0.060
0.6--1-
0.4
0.000
----
t
--
-0.120
---
-0.180
-0.240
--
-
-
0.2
- - --
.
-
.
0.8
0.240
--- 0.180
0.300
\
0--0.4 -
t
'
I
-0.6-0.8--
-3pi/2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure B-3: Pitch Moments vs. Time for Various Symmetry Parameter (Style #5)
71
0.300
-
-...
0.240
0.180
6 --
-%
/
\
0.060
0.000
-006
Y2-
-
--
...'
- --
-0.180
-0.240
0.300
-2
-4-
-6-
-8-II
-3pi/2
-pi
-pi/2
I
I/
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure B-4: Lift Coefficent vs. Time for Various Symmetry Parameter (Style #15)
72
-
4-
*.,
-0.120
0.300
-
3-
0.240
-
0.060
0.000
-0.060
-0.120
-0.180
-0.240
-
1
0
0.180
0.120
-
N
-
.
2-
-
0.300
-1,-
N
N
-2
-3-4 -I
-5-6 -I
-3pi/2
-pi
0
pi/2
Phase During Cycle (rad)
-pi/2
pi
3pi/2
2pi
0.4
-
-0.300 -L
- -.-.
-
Figure B-5: Roll Moments vs. Time for Various Symmetry Parameter (Style #15)
0.240
0.180
0.120
---
.--.
S -
0.2
0.1
--- --
--
I
QjI
0-
I
-
I
-
0.3
0.060
0.000
0.060
-0.120
-0.180
-0.240
-0.300
-
- Ij
Ii
-0.1
I
-
i
II
-
-0.2 -
-
-0.3
-0.4 -1-3pi/2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure B-6: Pitch Moments vs. Time for Various Symmetry Parameter (Style #15)
73
0.300
-
10
0.240
....
0.180
8-S-$:--1
---
-
- ----
6 -.-
K-
/
-
\
-
-
\
2
-s--
0
/--
-0.060
-0.120
-0.180
-0.240
-0.300
-
\./
4
0.120
0.060
0.000
-
s
-.
2/
.,
/-
0
-2-
-4-
-6-8-3pi/2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure B-7: Lift Coefficent vs. Time for Various Symmetry Parameter (Style #118)
74
0.300
-
4
0.240
0.180
-
0.120
0.060
-
3
0.000
.
-
-
2
-
~
--
-
1
-0.060
-0.120
-0.180
-0.240
-0.300
-
-.
-
s
-
0 //\-
\///
-2
-
-
-
-3
J \\!
-
-
-5
I
-3pi/2
-pi
-pi/2
a
-
11 1
0
pi/2
Phase During Cycle (rad)
pi
,
.
-
-
-4
1
3pi/2
2pi
Figure B-8: Roll Moments vs. Time for Various Symmetry Parameter (Style #118)
0.3
0.300
0.240
....
0.180
0.120
1
0.2
-.- 0.000
- - -0.120
0.060
- 4
0.1
-I
-1.
3. R.i
-.-
1---
-0.180
-0.240
--
L
6.
0.060
-
-0.300
-
i
s
0
0
-0.1'
-r.
-0.2
-0.3
-3pi/2
-pi
-pi/2
pi/2
0
Phase During Cycle (rad)
pi
3p/2
2pi
Figure B-9: Pitch Moments vs. Time for Various Symmetry Parameter (Style #118)
75
10
0.300
-
0.240
....
0.180
8
0.120
--
-
-
0.060
\
0.000
- --
6 -/--
/-
-/
-
-
-
-
-0.060
-0.120
-0.180
-0.240
-0.300
j4 --i/I
S
2IA
0Ii
A
-2-
\V~
-6-
-
-8I
-3pi/2
I
0
pi/2
Phase During Cycle (rad)
'/
-pi
-pi/2
I
pi
I
3pi/2
I.
I
2pi
Figure B-10: Lift Coefficent vs. Time for Various Symmetry Parameter (Style #113)
76
0.300
-
0.240
.
4 -.
0.180
-
0.120
0.060
-
3 -\-
0.000
-0.060
\\-
0.120
--
2-
- ----
1 -.
0
0.240
-0.300
-
-0.180
-
-\t
-
-2N.
-3-
4*
-5
I
-3pi/2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
pi
-
-
-4 - -
Ij
3pi/2
2pi
Figure B-11: Roll Moments vs. Time for Various Symmetry Parameter (Style
0.300
0.240
-
....
0.3 J.--
0.180
0.120
1
~--0.060
t-....
0.000
0.2 -V
-0.060
-0.120
- -
j:
- -
-0.180
0.240
--
0.1
-0.300
-
s
IV
0
H.
44
-
-0.1 I -
--:
#113)
I-
-l
**l
-
4
-0.2-
I
I
-0.3 -\-3pi/2
-pi
-pi/2
0
pi/2
Phase During Cycle (rad)
pi
3pi/2
2pi
Figure B-12: Pitch Moments vs. Time for Various Symmetry Parameter (Style #113)
77
78
Bibliography
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80