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Stability of a Flapping Wing
Micro Air Vehicle
Marc Evan MacMaster
A thesis submitted in conformity with the requirements for the degree of Masters of Applied Science
Graduate Department of Aerospace Science and Engineering University of Toronto
0 Copynpynght by Marc MacMaster 2001
Acquisitions and Acquisitions et Bibbgraphic Se~ices seMeas bibliographiques
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Stability of a Flapping Wing Micro Air Vehicle
Masters of Appiied Science, 2001
Marc Evan MacMaster
Graduate Department of Aerospace Saence and Engineering
University of Toronto
Abstract
An experimental investigation into the stabiiii of a flappiug wing micro air
vehicle was perfomied at the University of Toronto Institute for Aerospace Studies. A
thtee-degree of W o m force balance was designed and constnicted to measure the
forces and moments exhibiteci by a set of flapping wings through 180' of rotation at
varied fiee-stream velocities. The same apparatus was a h used to test two tail
configurations.
A two-dimensional simulation program was wtitten using MA'IZAB software to
ident* stable whicle configurations at or near the hoveiliig condition. A total of four
case studies were performed, and each revealed the vehicIe had inherent stabiiity. The
presence of a tail on the vehicle produced only margmal effects. Of crucial importance
was the placement of the vehicle center of gravity with respect to the wings. A preferred
distance of 3.5 cm h m the c.g. to the leading edge of the whgs allowed for stable flight
under ali cases studied.
Acknowledgements
Many individuals need to be thanked for their time and guidance during the
course of this research. W i u t them I do not beiieve that 1 could ever have completed
this research on my own, First, I would like te *hdc Mr. Dave Loewea W i i u t his input
and experience, the work completed herein could w t even have been started His
patience in keeping me fiom h b l i n g about the workshop was much appreciated.
Dr. James DeLaurier also deserves much credit for adding his wealth of
knowledge and experience in supervishg my work. His office door always seemed to be
open, and he was ever prepared to m e r my questions and provide solutions to my
pro blems h s t instantly.
Patrick Zdunich and Derek Bilyk were two ikiends and members of the MAV
project for whom i couid always rely on for advice and judgement- They never seemed to
be bothered by my occasional questions, and were remarkably patient. A special thanks
goes to Patrick for the use of his wind tunnel for my testing.
I am especially gratetlll to the M i n g h m both UTIAS and the MAV pmject.
Wiiout their contniutions I could never have afEorded to pursue my Master's degree,
and in tum wouid have lost the great experience C had during my studies.
Finally, 1 wouid like to acknowledge my fbkh m God 1 do not think the stniggks
that arose both inside and outside my studies over the past 18 months couid ever have
been overcome without a steadfàst devotion to Him.
Table of Contents
.. .......... Absrnet ................... H ."....-... U
**- A~knowledgememts ....... .. ......................................... ............... ...... ..................... UI Table of Cmtenîs ... ...... .. ...... .. ............................. . ................................................. N
... List of Figures ............................................................................................................. VIII
List of T a b h ........................................................................................................ .. ...... xi
Chapter 1: Introduction ................................................................................................ 1
................. 1.1 MAV Project at UTIAS ....-..... ............. .. ............ .. .... ..... i 1.1.1 Roject Background ......... ., ..... ,.. .......................................................... 1
................................................................................... 1.1.2 ResearchObjectives 3
1.1.3 Year 3 hject Metamorphosis ................................................................... 4
1.1.4 About the Vehicle Components Used ......................................................... 5
.................................................................................. Chapter 2: Force BaIance Design 9
2.1 Rationale for Sekcted Design ........ .. ........ .. ......... ... .................. .. ................ 9
22 Design Sperifiertioas ...,.,, ......,.,.,,... .. .................. ....... . . 10
.......................................................................................... 72.1 How it Works I O
2.2.2 Axes System ............................................................................................ 12
. . 22.3 Design Adjusbbd ity. ................................................................................ 13
2.2.4 m e r Details .......................................................................................... 14
23 BaianceCalibrati6n ........................................ ......... ........... 17
23.1 Mependent Gauge Caliition ................................................................ 17
2.32 Complete System Caliaration ................................................................... 18
2.3.3 Performance Verification ......................................................................... 19
Chcrpter 3: Wnd Tunnel Calibmtion ........................................................................... 23
3.1 Wiid Tunnel Detaib ................................................................................. 23
.......................................................................... 3.2 Wind Tunnel Caübration 24
3.2.1 Initial Resuhs ........................................................................................... 24
........................................................................................ 3 22 Revised Design 24
3.2.3 Calibration Procedure .............................................................................. 26
Chapter 4: Ekperimenls ............................................................................................... 29
.................. 4.1 Wing Testing Procedure .., .. .............................................. 29
4.1.1 Methodology ............................................................................................ 29
4.1.2 Taring ...................................................................................................... 34
4.2 Tai1 Testing Procedure ........................................................................... 39
............................................................................................ 4.2.1 Tai1 Design 3 9
4.2.2 Methodology ........................................................................................... 41
Chapter 5: Experimental Results ............................................................................... 43
5.1 Wings .......................................................................................................... 43
. . 5.1.1 Repeatabriity ............................................................................................ 43
.................................................................... 5.1 2 Longitudinal (2-axis) Forces 44
5.1.3 Lateral (X-mis) Forces ............................................................................ 48
5.1.4 Moments (about Y-axis) ........................................................................... 49
5 2 Taib ............................................................................................................ 51
5.2.1 Results ..................................................................................................... 51
.............. .......................... 5 3 Ampliticition of ûah " . 52
5.3.1 Z Forces ................................................................................................... 52
5.3.2 X Forces ad Y Moments ......................................................................... 56
5.4 Cornparison ta hsumed Vahes ..........m.......m .... .................................. 58
Chopter 6: 2-0 Simulation .......................................................................................... 62
6.1 Numerid Mode1 ..................W.......... .... ............... .................................... 62
6.1.1 Application of Newton's Laws ........................................................ 6 2
6.1.2 Lookup Tables ................... .. ........................................................... 6 4
6.1.3 Numerical Procedure .............................................................................. 6 4
6.2 Initial Results ............................. .................. Do ...... ........................... 71 . . 6.2.1 Simple Hovering Condition ............ .... ........................................ 71
......................................................................... 6.2.2 Rotational Disc Damping 72
6 3 Disc Damping Experimeob ................ .. ....... .. ............................................ 72
6.3.1 Experimental Setup .................................................................................. 72
.................................................................................. 6.3.2 Dynamic Equations 73
6.3.3 Experiment ................................................................................... 75
6.3.4 Results ................................................................................................... 77
6.4 Case Studies ....,...... .. .......... .. .............. .. .......... 77 . .................................. 6.4.1 Test Cases .......................................................................................... 77
6.42 Case I - Hovering Condiiion with Tiltmg Disturbance ............................. 79
6.4.3 Case ii - Slight Ascent with T W g Disturbance ...................................... 82
................................... 6.4.4 Case iü - SLight Descent with T i Disturbance 85
6.4.5 Case IV - Lateral Gust ............................................................................. 88
Chapter 7: Conclmkns ................... .............. ....-...... ................................. . ........ 90 7.1 Case Study Analyses ................................... " ....-.........- .... -.....!Hl
..................................... .............................. Chapter 8: Refemnces and Bibibpphy ,. 92
8.1 References .................... .. ...................... ................... .......................... 92
8.2 Bibliognpby ................................. ....... .............. .,.... ................................... 92
Appendices:
Appendix A: Force Balance Design Specifications
Appendu lk Force Bahnce Caiibration Data
Appendix C: Wiad Tunnel Velocity Profiles
Appendix D: Experimental Resu hs
Appendix E: Dific Damping Experimentrl Data
List of Figures
Chapfer I Figures
................ Figure 1.1 : Flapping-Wig MAV Conceptual Drawing (by Dave Loewen) 2
............................................................................... Figure 1.2. BAT-12 Wig [l] 6
......................................................................................... Figure 1.3. ProtoSo uth. 7
Chapter 2 Figures
Figure 2.1. Side View of Force Balance Design .................................................... 1 1
....................................................... Figure 2.2. Top View of Force Balance Design 1 1
Figure 2.3. Wi-Hub Axes System .................................................................. 12
Figure 2.4. Clamping of the W i s to Fixed Upper Plate .......................................... 14
Figure 2.5.1. Cantilever Beam Configuration (by AC Sensor [4]) ............................ 15
Figure 2 . 5 2 Parallel Beam Configuration (by AC Sensor [43) ................................ 15
Figure 2.6. Final Constructeci Force Balance ........................................................ 16
Figure 2-7: Force Balance with ProtoSouth .............................................................. 17
.......................... Figure 2.8.1 : Pure Applied Moment (Top View) ............... ,... 2 0
Figure 2-82: Combined X and Z Forces (Top View) ............................................... 20
Figure 2.8.3. Combined X, 2 Forces with Moment (Top View) ............................... 21
Chripier 3 Figures
Figure 3.1 . 1. Open End Wmd Tunnel at UTIAS ..................................................... 23
Figure 3.1 2: Open End Wmd Tunnel at UTlAS ..................................................... 23
Figure 3.2. Sample VeIocity Field (with Cone) ..................................................... 2 5
........................................................... Figure 33: Pitot Tube and Manometer Setup 26
Chapter 4 Figures
Figure 4.1 : Wing Testing Procedure ........................................................................ 30
.................................................................... Figure 4.2. Original Mounting Bracket 35
.............................................. Figure 4.3. Original Mounting Bracket (Top View) 36
............................................................ Figure 4.4. "Goosenec kn Mounting Bracket 37
Figure 4.5. Foam Shroud and Mounting Bracket .................................................... 38
Figure 4.6. Exaggerated Mounting Misalignment ................................................... 39
.......................................................................................... Figure 4.7. Tail Designs 40
Figure 4.8. Tail Dimensions .................................................................................... 40
................................................................................ Figure 4.9. Tai1 Testing Mount 41
Chapter 5 Figures
. ..........................-*.***............ Figure 5.1 : Lateral (X-axis) Force vs Angle, J = 0.735 43
Figure 5.2. Longitudinal (Z-axis) Discontinuity at 90" ................ .. ....................... 44
Figure 5.3: Longitudinal (Z-axis) Force vs . Angle with Liaear
Trend Line, J = 0.735 ......................................................................................... 46
Figure 5.4: Longitudinal (Z-axis) Force vs . Angle with Linear
Trend Line. J = 0.735, (ûutiying Anomalies Removed) ..................................... 46
. Figure 5.5. Lateral (X-axis) Force vs Angle, AI1 Advance Ratios ........................... 48
. Figure 5.6. Moment (about Y-axis) vs Angle, Ail Advance Ratios .......................... 50
................................................................ Figure 5.7. Cr Curves for Tails #l and #2 51
............................................................... Figure 5.8. Co Curves br Tails #l and #2 52
........... Figure 5.9. Thrust Ratio vs . Free-Stream - Frequency Ratio (Origina[ Data) 54
Figure 5.10: Th- Ratio vs- Free-Stream . Frequency Ratio
(Orig . and Extra Data) .................................................................................. 5 5
Figure 5.1 1 : Extrapolated Z Force Data for 40 Hz ...................................... ,. ......... 56
Figure 5.12. Effect of Flapping Frequency on X Force ........................................... 57
Figure 5.13. Effect of FIapping Frequency on Y Moment ........................................ 58
Figure 5.14. X Force Cornparison to initiaiiy Assumed Values ................................ 59
Figure 5.15. Y Moment Comparison to Initially Assumed Values .................... ,... 60
................................ Figure 5.16. Z Force Cornparison to Initially Assumed Values 61
Chapter 6 Figures
Figure 6.1 : Mode1 Representation ......................................................................... 62
. . Figure 6.2. Disturbed Condition ............................................................................ 63
........................... Figure 6.3. Exarnple of Wigs' Tme Free-Stream Velocity Angle 67
............... Figure 6.4. Second Example of Wmgs' True Free-Stream Velocity Angle 68
......................... Figure 6.5. Force and Moment Summatioa Exampk (Wigs Only) 69
.......................... Figure 6.6. Force and Moment Summation Example (Tail Only) ... 70
Figure 6.7. Initiai Test Case Wahout Tai1 ................................................................ 71
..................................................... Figure 6.8.1 : Disc Dampuig Experimental Setup 73
.................... Figure 6.8.2. Dise Damping Experimental Setup, Perturbeci Condition 73
Figure 6.9. Disc Dampmg Apparanis ....................................................................... 75
Figure 6.10. Example Plot of Osdiatory Decay ................................................ 76
Figure6.11.1.CaseI-NoTa Il =75 cm ......................................................... 79
................................... . Figure 6.1 1.2. Case 1 -No Ta EfEct in the Reduction of 11 80
Figure 6.1 1.3. Case II -No Tail, EEct m the Reduction of 11 ................................. 83
Figure6.11.4: Case II - WithTail, h=-12.5 cm, Il = 7.5cm. ................. ., .............. 84
Figure 6.1 1.5: Case II - With Ta& h = -12.5 cm, 1, = 2 cm ..................................... 85
Figure 6.1 1.6: Case iII -No Tail, Effect in the Reduction of 1, .............................. 86
Figure 6.1 1.7: Case III - With Tail, h = 12.5 cm and -12.5 cm, II = 7.5 cm .........,... 87
Figure 6.1 1.8: Case HI - With and Without Tail, i2 = 12.5 cm, II = 2 cm ................. 88
Figure 6.1 1-9: Case IV -No Ta& Effect in the Reduction of 1, ...................... ..... 89
List of Tables
Chupter 2
Table 2.1: K-Value Summary .................................................................... . 2 2
*Note: Figures and fables in the Appendices are nos listed here. however the Appendix title and introductory poragraph should ahw the reader to determine w h t rypes of figures me contained therein,
Chapter 1 : INTRODUCTION
1.1 MAVPmjectatUTlAS
1.1.1 Projecf Background
In 1997, an initiative to develop a Micro Air Vehicle (MAV) was brought forward
by the U.S. Defense Advancd R e m h Projects Agency (DAWA), in Light of the
current and irnpeodiag developments in rnicroelectronics t ahg place at the the. The
intent of the project was to create a small airborne platform capable of perforrning
various surveillance missions to be used both in müitary and civil applications. in
outlining its objectives, DAWA required that the maximum dimension of the aimaft
should not exceed 15 cm, and have a total vehicle mass between 30 and 50 gram. Such a
vehicle would be expected to carry a variety of sensors, yet remai. portable and durable
enough so that it could be easily transponed inside a sofdier's pack Hence, a priority was
placed upon the devetopers to mate a tightweight, robust and efficient design that wodd
satisfL the demands of the agency.
DAWA awarded several research contracts to various hitutions and k m
across the United States. iacluded amongst these was a contractai partnership between
SRI International of Menlo Park, California and the University of Toronto Institute for
Aerospace Studies (üTiAS). Together, this team sought to evoive a vehicle design that
would combine the technologies of fiapping-wing propulsion and artif id muscle
actuation. This particular f o d a would stand apart h m other proposais in that it would
be directly aimed at producing a MAV capable of bvering ftight. F i 1.1 depicts an
early conceptuai mode1 of the anticipated design.
Figure 1.1: Flapping-Wmg MA V Coriceptuaf Druwing (by Dave Loewen)
This marriage of expertise between SRI and LmAS began in May of 1998, with
the total contract duration encompassing 3 years, With its strong background, knowledge
and expex-ience in hpping-wing flight, üTiAS wouid focus on developiug a successtùl
design for wing propulsion as well as the vehicle aerodyaamics. Alternatively, SRI would
direct its work toward perfectiog its technologies in Electrostrictive Polymer Arti f id
Muscles (EPAMs) as the wing actuating mecbanism, in addition to incorporahg the
vehicle's necessary electronics.
The unique flapping-wing concept was expected to yield distinct advantages m
the MAV context: better stability and conbol in slow translationai flight, improved
energy efficiency, and more steaitblike capabilities. Under the guidance and direction of
Dr. J m s DeLaurier, the OTlAS approach was io mode1 Mother Nature's s u c c d
design of the hummingbird. Due to the smaii d e s invoived, much research was
performed in order to investigate this untzxpiored and mysterious region of flight. At
UTIAS, the 6rst year of the project succeeded in investigating and producing a practicai
wing design tbat would provide sufficient thrust to cary a mass budget of approximately
50 grams. The second year continued with the wing research, addressing such areas as
flow visualization and developing numerical tools for analysis of this flight regime. At
the outset of the contract's finai year, there was a reorganization of the UTIASfSRI
position in the DAWA administration. What soon foiiowed was a subsequent
reclassification of the üTiASlSRI effort to Edl under the direction of the Micro-Adaptive
Flow Control (MAFC) branch of DARPA, rather than the original MAV group.
Also in the early stages of the final year, SRI completed an anaiyticai model
"flight simulator" of the MAV for the purpose of allowing rapid evaluation of stability
and control under different vehicle configurations. This would become an invaluable tool
for facilitating prototype design. The analytical model used by the simulator required
experimental data (i.e., forces and moments) for the MAV wings and t d under diffierent
flight conditions. The initial data that was King used were simply "best estimates" of
what performance could be expected
1.1.2 Research Objectives
The objective of this thesis was to identiQ and evaluate possible configurations
that will permit stable and controllable flight of the MAV. This encompassed the
evaluation of the forces and moments associated with the current generation MAV whgs
under different angles to the k-stream veiocity. Tai1 lift and drag data were aIso
determined, Together, these were to be used wiîh the aforementioned simuiation code to
coduct case studies of possiile tail-wing coufiguratr*ons that would lead to saîishtory
controI and stability when a fiynig modei is realkd Udbrtunately due to Iogistical
problems, the author muid not personaüy conduct such case studies with the SRI
simulation program. As an alternative, a 2-D program was produced so that these studies
couid still be performed, albeit at a somewhat less sophisticated level of programming.
1.1.3 Year 3 Project Metemorphosis
As previously mentioued, tiinding for the work performed between SRI and
UTlAS was switched to Mi under the jurisdiction of DARPA's MAFC branch of
research. With this change came tbe aiieviation of some of the restrictions piaced upon
the project in terms of size limitations. No longer did the vehicle need to conform to a 15
cm maximum dimension; however the pmject would stiü rernain hcused on producing a
platfom useful to the military. One of the main issues impeding progress of the initial
MAV prototypes was the lack of energy density available with even the latest generation
of batteries and capacitors. Free flyers powered in this mamer were very limited in their
tiight duration Thus, much of the finai year of contract work hcused on developing a 30
cm span flyer that would achieve successfùl flight. It was thought that by going to a
larger span, more thrust wouid be produced anci therefore the ability to use heavier, gas-
powered forms of propulsion wouid be made possible (which in tuni wouid extend flight
duration times). Indeed, at the time of this writhg, a gas-powered R/C flyer designeci at
UTIAS was repeatedly show to be s u c c d m achieving hovering flight (albeit
tethered to a pole). It should also be stated that the method of wing actuation was stiii
king performed through mechanid means, as SRi's EPAM techwlogy had yet to
mature to the appropriate level as to be brporated mto the existing design.
That being saki, the reader shouid be reminded that di the tests, experitnental
results and data, as weU as the 2-D simulation code, aü revert back to the original 15 cm
MAV flight modeL The idea being that if the contract were to continue beyond 3 years,
the initiai 15 cm platform may be revisited In order to meet DARPA1s original criteria.
Even if a contract renewai were not to materiaiize, or if the 15 cm flyer is completely
abandoned, the research into the stability of a haif scde mode1 of the existmg gas-
powered prototype would most certainly be beneficial as a fkt approximation in
evaluating the its stability.
i . l .4 About the Vehicle Components Used
As stated previously, the primary requirement for the simulation code was to
obtain true qualitative data on the latest MAV wing design. Due to the unsteady nature of
the lifi mechanisrns involved, an analyticd method of ideu tmg the forces and moments
of such a wing configuration was yet to be fiilly devebped Thus, an entireiy empirical
approach was taken in detennining this information
The latest wing design showing the most promise had the ability to produce 50
gram of thnist when flapping at appximately 40 Hz [l]. Caiied the BAT-12, this
design is depicted in figure 1.2. It was this type of wing was used to evaiuate the test data
m this author's research The wings are constnicted using a unidirectional carbon fibre
(PEEK) fiamework together with a light myiar coverhg.
Figure 1.2: BAT-12 Wing [ I l
The total span of two mounted wings was 15 cm, thus conforming to DAWA'S
size restrictions. For more details on the evolution of this wing's design, the reader is
directed to Mr. Derek Bilyk's Masters thesis [ I l , a previous student at UTIAS during the
fist year of the project.
Since EPAMs remained unavailable for use in the tests, the wings were actuated
mechanically by incorporating two wncentric shafts, each having a pair of wings
attached. Driving these shafts were two wnnecting rods attached to a d DC eIectric
motor, Such a rnechanism was designed ad fabricated by SRI during the early stages of
the project. Figure t .3 shows this device (named ProtoSouth).
Figure L3: ProtoSourh
The details of the mectianism are as fobws: two concentric brass tubes comprise
the "mast" of the structure, and are supported by an aluminum brace. Connecting rods
attach to small tabs extending ftom these tubes, and aIlow for the linear motions of the
rods to be b.ansfomd into tube rotation. The rods extend 10.2 cm to a crank extending
h m the DC electric motor. When actuated, the motion of the tubes is nearly sinusoidal.
This motion is transfemd to the wings momted on hubs attached to the tubes. With two
wings per hub (m an opposing orieatation), they are able to tlap and rotate against one
another. Such motion produces the cIapfling effect - one of the prime aerodynamic
mechanisms sought to produce the required iift. For f k h r msight into this and other
hi& lifi mechanisms, the d e r is directeci to Ms. Jasmine El-Khatiis Masters thesis
that was also completed at UTIAS in relation to the MAV project [2].
Flapping amplitude is d e W as the magnitude of the angle one wing sweeps
through in one cycle of craak rotation, It is governeci by varying the Iengths of the
vertical links in the four bar mechanism. Unfortunately, the abiiity to vary the amplitude
was not a feature made available in the construction of ProtoSolnh, The flappmg
amplitude of this mechanisrn was fixed at a value of 60 degrees. Previous research fiom
[II revealed that 72 degrees of amplitude was a more desirable value. However, since the
existing prototype was both readily available for testing in addition to beiag more durable
than other existing mechanisrns, it was thought that it would be sufficient to evaiuate the
desired data.
No previous research had been done to mvestigate an optimum taii design, nor in
the placement of a tail with respect to the fklage of the MAV, except for some
conceptual drawings and sketches. The simulation code requkd only the coefficients of
tifi and drag of the tail through 360 degrees of rotation m a flow field, It was believed at
the outset of rhis research ihat the orientation of the tail (Le., above or below the wings)
would be the most important factor in govwning vehicle stability. Therefore more
emphasis was directed to investigating tail positionhg rather than on exhaustive testing
of various taii designs.
In order to evaluate the wings and tail, a balance was required tbat would be
sensitive enough to measure the inherentiy small forces to be encotmtered. Such a force
balance was buiit at the UTIAS Lab specifidy for these tasks, and its design is
descriibed in the foiiowing chapter.
Chapter 2: FORCE BALANCE DESIGN
2.2 Design Specifications
2.2.1 How if Workrs
First and hremost, the fundamental design had to be scaled down m order For it to
adapt to the anticipated forces encountered with the MAV wings. Essentiaiiy, the revised
concept consisted of an aluminum tray suspended h m a fixed upper pIate via thin wires.
This my could translate in two directions as weli as twist. A munting piece attached to
the Iower tray extended up through a hole in the centre of the 6 x 4 upper plate. It was to
this mounting piece that the ProtoSouth iiapping mecbanism attached and was able to
transfer loads. Siraïn gauges were rnounted to the h e d upper plate and reacted to aay
translations of the suspendeci lower tray. Figures 2.1 and 2.2 are simple depictions b t
more cleariy illustrate how loads were transferred to the strain gauges, as weU as their
layout. Three gauges were useci, each labeiieci #1, #2 and #3 as m figure 2.2.
The beauty of the design was ttiat it permitteci the sim-us measurwient of
two forces and a moment, which was preciseiy what was desired h r the planned testhg
to follow.
Fixed Plate Btacket -.
F- I Sbain
Gauge
Figure 2.1: Side Yiew of Force Bulance Design
Longitudinal
Lateni
th Gauge Gauge #1 #2
Fignre 2.2: Top Vicw of Force Bulance Dcsr'gn
Retérring to figure 22, the center mark represents the point of Ioad application on
the fk lower piate. Using the show11 force-labeling s c b , it is observed tbat iaîerai
loads were resisted by gauge #3. Longitudinal loads were determineci through a
summation of the readings h m gauges #I and #2. Any appiied moment manihted itself
as a diierence m these two gauge readings and was determined knowing the distance "8'
between them, using the simple formula:
2.2.2 Axes System
A wind-hub axes system was used in the simulation code and was adhered to m
actual testing and reporthg of data. The z-axis (called the longitudinal axis) extends
through the centre of the MAV dong the tlapping axis. The x-axis (labeled the laterai
axis) was aiways oriented so that its cosine component was pomted downstream. Hence,
when the simulation depicted the vehicle rotating past 180 degrees in a crossfiow, the x-
axk instantaneousiy changed to maintain its direction inro the wind. The y-axis
completed the orthogonal triad m the right-handeci seme. Figure 2.3 superimposes ttiis
system over a simple sketch of the MAV.
F&re 23: Wind-Hu6 Aus System
Adapting this system to the gauge Iayout in figure 22, tfme forces dong the z-axis
can now be referred to as longitudinal wùüe tbose dong the x-axis c m ww be d e W
as lateral Ioads.
2.2.3 Design Adjustabilify
Choosing the overall dimensions of the halance was relativeiy arbitmy, What was
most important howeva, was enabhg the device to be sensitive emugh to m u r e
minute forces yet still retain some durability so as not to be easiiy damaged. Thus, during
constmction. an effort was made to allow 6 r adjustment m order to make the device
more rigid or relaxed. With a fieroile desigu, it was believed that if the completed
bahce perfomied u~wtisfactorüy, it wuId be easily modined without scrapping the
entire device and sbrting over. One level of adjustnsent was the abiiity to alter the
distance separating the two plates. Taken in the extreme sense, a very short distance
wouid d e the balance very "SM" with respect to applied moments anà forces, whereas
as too Iong a separation would becorne impracticd Therefore, a degree of adjustability
was aiiowed for by ciamping the wires to the 6xed u p p plate of the balance rather tban
rigidly ancho~g them into position. Leaving the wires long p d e d the bwer plate to
descend h h e r shouid the mecl arise. Figure 2.4 ilbistrates how this was done.
Figure 2.4: îlamping of îhe W- io F d L rper Plate
Another pararneter that codd be changed, albeit somewhat less conveniently, was
the distance "d" separating gauges #1 mi #2 9i figure 2.2. A larger distance wodd d o w
for greater s e n s i t i i to appiied moments. CompIetely dmiensioned CAD 3rawings of
the force balance are mcluded m Appendix A.
2.2.4 Other Details
The gauges used were corrmietcially purchased AC Sensor Mode1 6000 Planar-
Beam Force Sensors [4]. Each sensor contameci a fuü bridge s tnm gauge mtegrated ont0
a thin-film s t ades steel element of 0.004 in thiclrness, This particular mode1 sensor was
the Iowest capacity (114 pound) avaiiable h m AC Sensor. It was decided that such a
commerciaüy manufactured product wouM tte more reliat,le and accurate than design&
and sizing appropriate flexures m-house. indeed, h m the detaiIs that foiiow m this
chapter and those ahead, this assirmptioa proved to be ûue. I-, the gauges were
mounted cantilevered as shown m fîgwe 2.5.1. It was discovered however that this type
of orientation performed quite poorly. Excessive drift in the gauge readings d e
&%ration nearly impossiile. It was befieved the gauges were fiexhg out of plane a d
succumbmg to Ioad misalignment, To recti& the problem, the gauges were mouutai m a
pardel beam fashion as a means of cornpeasating any applied moment and reducing
errors in off-centre loading. In other words, the gauges were consûained to react to pure
forces ody. Figure 2.52 illustrates this type of gauge set-up.
Figure 2.5.1: Cantilever Beam Configuration (by AC Sensor (41)
Figure 2.5.2: Parallei Beam C o n f i t i o n (by AC Sensor [4v
To transmit the appiied loads to the strain gauges, angle brackets were mounted to
the Fiee lower plate. Each bracket was aligned with one gauge (refèr to figures 2.1 and
2.6). Extendhg fkom the bracket to the gauge was a piece of heavy piano wk, which
acted as a rigid rod between them. Altogether, one could imagine the load path as
folIows: an applied force h m the ûapping rnechanism is passed through its mount d o m
to the k e Iower plate, which m tum is transmitted through the angIe brackets, through
the piaao wire, and finallv is resisted by the gauge. A photo of the fmished balance is
shown m figure 2.6. Figure 2.7 shows the balance together with ProtoSouth, attacheci to a
lripod as it was during actual testing.
Figure 2.6: Final Constructed Force Balance
Figure 2.7: Force Balance with PnrtoSouth
2.3 Balance Calibradion
As rnentioned previousiy, a cantilevered strain gauge design was scrapped in
hvor of the paralle1 bearn configuration. What foiiows focuses on the calibration of the
gauges m their latter form.
2.3. i independent Gauge Calibralion
Prior to 6nai attachment of the angle brackets to the paralle1 beam gauges, it was
d& to c a i i i each of the beams iedependently. Two reasons provideci the ration&
for this effort. First, s k e the *es attaching the angIe brackets to the gauges were glued
mto p k , these was m, way of "umloingy this ûuai step. Secondly, a cumpke gstm
caiiition (Le., with angle brackets attachai) wodd require the assumption that the
appüed longitudinai loads were equally shared 50150 between gauges #I and #2. By
perforrniug caliitions separateiy, any change m the hi system d f f k i s couid be
O bserved.
The resuits fiom these idependent triais revealed that there was no appreciable
change m the gauge slopes before and d e r the fird attachment of the angle brackets.
These tests are mciuded under case 1 of Appendix B.
2.3.2 Complete System Cdibmtion
Since the balance was expected to perform under a variety of appiied loads and
moment (both pure and combiued), exhaustive caüition tests were performed m order
to evaluate its performance, repeatabiüty, and level of crosstalk among strain gauges.
The complete system calhtion tests were performed by M y clamping the
balance to a level desktop. A handheld muitimeter together with a power suppiy was used
to take readings of each of the gauge outputs separately. A simple cyündrical pillar
(attactied to the lower plate of the balance) served as the attachent point for appiying
test loads. By using a pdey system, a series of known masses providing the forces were
appiied in both lateral and longitudinal directions. Output voltage readings were
recorded, dowing the determination of each gauge's slope, dehed as:
k = AV/m (2-2)
where AV represents the change m the gauge output volîage between the loaded and
unloaded condition (measuted in millivolts), and m is d e W as the applied m a s
providing the force (in gram)-
The initial tests sougbt to determine these k-values of by simple application of
niasses m one direction or@. For example, gauge #3's k-vahie was evaiuaîed by a p p m
a series of loads in the x direction (both in the positive and negative sense), ensuring no
force component emerged dong any 0 t h axk Similar tests were repeated for gauges #l
and #2 in the z direction. As expected, each gauge exhiiited ünear behavior in response
to loading, as weii as the remarkable virtue of zero crosstaIk among the gauges. With
zero crosstalk, a gauge's output was wt comrpted h m loadings outside of its intended
axis of measurement. This particular test's resutts are detailed m Appendix B under case
2.3.3 Pedomance Verifcation
Mer completion of the above tests, it was decided to perform t'urther cali'brations
m which combinations of known forces and moments were appiied to the balance. This
banage of tests would serve two purposes. F i , by using the h M l y derived k-values,
an estimate of the percent error incurred under ciifferent force conditions could be
evaluated. Second, new k-values codd be derived h m these extra test cases, allowing
the abity to assess any gross change th& magnitudes. This rather elaborate procedure
wouid gamer a deeper insight into the o v d perform~tnce of the balance, and aid m
determiniag the final k-values to use during actual experbentation. Complete data for
each test case are included in Appendix B. What wül fbllow will be a brief description of
each subsequent case and summarize its d t s .
Pute Applied Moment
The 6rst case entailed the application of a pure positive moment about the y-axïs.
This was achieved by bolting a d aluminum a m to the ercisting mouut, as ilhistrated
in figure 2.8.1. The distance "P' between the applied force T and the center of the plate
couid be varied aiong the arm, which allowed the magnitude of the applied moment to be
adjusted. A mass of 40.3 gram was applied at 1 cm increments outward aiong the arxn
Figure 2.8.1: Pure Applied Moment (Top View)
Results for this scenario were exemplary. Using the mitialiy derived k-values, al
erros for mass and moment were on or about 5%. No crosstak was observeci in gauge
#3.
Cmbined X and Z Forces
A mass of 17.6 gram was applied dong a diagonal, such that it allowed a
component of its force to appear m both the x and z directions. Figure 2.8.2 depicts this
scenh .
Figure 2.8.2: C d i n e d Xand Z Fumes (Top Vuw)
Choosing a diagonal travelling exact& through the corner of the reçtangulac plate
fàcilitated proper alignment. Simple geometry determined the angle 0 to be 53.04". Again
the balance performed admirabiy, save for mstances of small loads (below 7 grams).
Combined X and Z Forces wiîh Moment
This final test case was compteted by using the previous a m attachment aligned
dong a similar diagonal as shown m figure 2-8.3. Agaiu, the appiied test m a s was 40.3
granis-
Figure 2.8.3: Combined X 2 Forces wah Moment (Top Yiew)
As with the previous two cases, the re& were excellent. Error in the force
rneasurements remained m the 5% range, with some as b w as 0.2%.
As mentioued, with each test case came the ability to reevaiuate the gauges' k-
values, as one could view each scenario m its& as a calibration method includmg the
initial caiiiration @ e r f o d m both positive and oegative directions), there were seven
different conditions for which k-dues were detemimi, and they are summarized m
table 2.1 below,
+ Z Force 0.0541 0.087 1
+ X Fort4
- Z Force
- X Force
+ Y Momsnt
CornMnad X, Z Forcer
The variation in k-values amongst ali conditions was quite small, with the widest
Combinsd X, Z and Y Moment
margin of dEerence no greater that 10%. Et was aiso kk that the finai k's chosen stiould
nia
0.0563
nla
0.0546
0.0524
greater reflect test cases which involveci multiple forces. After much thought, it was
Table 2. I : K- Value Summary
0.0554
decided that the t e d s deriwd kom the combined x, z forces and y moment condition
nia
0.0820
nla
0.0897
0.081 5
would be the best representation of the overaü system's calibration cuefiients.
da
d a
0.0528
nia
0.0537
0.0829
J
0.0537
Chapter 3: WlND TUNNEL CALIBRATION
3.1 Wind Tunnel Details
A mail, open test-section wind tunnel was used to coqlete ail experimental
tests. This tunnel was built by Mr. Patrick Zdunich (a Master's student at üTTAS
involved with the MAV project) to mvestigate flow visualization aspects of his research.
Mr. Zdunich later decided to pursue other flow visualization methods, thus leaving his
wind tunnel available for use. This was a fortunate circumstance, as the larger wind
tunnet in the subsonic lab would likely produce velocities much higher than those desired
for testing a small MAV.
The tunnel measured some 49.5 in long, and was consmted fiom medium
density fibre (MDF). Air was accelerated by a small 18 m diameter indusiriai fàn initially
through a circular cross-section, which then passed through a section of flow
sûaighteners, and 6naüy convergeci to a rectangular shape nieasuring 20 in high by 10 m
wide. Figures 3.1.1 and 3.1.2 are photos of this tunnel at UTIAS.
Figures 3.1.1,jY.I.t: Open Wnd Tunnel at üThU
3.2 Wind Tunnel Calibrafion
3.2.1 M i a l Resulfs
Idedy, a wind tunnel should create a b w field tbat is entirely d o m in its
cross-section, Such an ideai is never t d y realized due to boundary Iayer effects dong the
wak of the tunnel as weU as turbulence in the flow. Early cali'bration tests were
performed by Mr. Darcy AUison, an undergraduate student who worked during the
summer of 2000 on MAV related tasks. Unfortunately, his ce& revealed a somewhat
disappointhg "weli" or "dip in the center of the velocity protile. Some steps were
required in order to rectify this problem.
3.2.2 Revised Design
in an attempt to dirninish this chanrcteristic, a cone was built by Mr. Allison that
f i e d to the fàn of the tunnel. This was expected to accelerate the air more uniformiy, as
the rather gewrically designed fiin was by no means coostnicted with îbe purpose of
wind tunnel testing in nhd. The cone addition yielded somewhat better results, however
the undesirable velocity dip was stiü apparent, as depicted in figue 3.2 (in three
dimensions).
Sampk Velacity Profih (dth Cone)
I l
1 I I I
I I
*Euch station height is separated by 2.54 cm. , and width by 3.8 cm.
I 1
I 1
-
Figure 3.2: Sample Velociiy Field (with Cone)
Mer much consideration of these eariy results, it was decided that this trait of the
flow field might not be as great a hindrance as initially expected. Although the tlow field
as a whole was decidediy non-dom, the velocities in the central "pocket" of the flow
were in iàct fàiriy consistent. The shallow dip measured roughiy 15 cm in height by about
20 cm in width. Recalling the span of the MAV wings were 15 cm, it was decided that
provided di of the tests were con6neà to this "sweet spot", respectable resuhs would be
attainable. As is d e m i m later chapters, this assumption proved to be accurate. For
the overail mean velocity for a setting, a weighted average of the sampled
velocities in the 15 cm square were caicuiated
3.2.3 Calibrafion Procedure
A pitot tube together with a nianometer was used to meme the flow field
velocities. The pitot tube was anchored to a rod supported by a U-shaped fiame situateci
in front of the tunnel exit, as shown in figure 3.3. The probe was positioned to take
sample readiigs at 1 in hcrements verticaiiy and 1.5 in horizontaüy. There was also the
ability to position the probe at various distances away fiom the tunnel exit. This aiiowed
the degree of velocity decay away from the tunnel exit to be observed.
The standard method for detennining velocity using a manometer was used,
whereby a change m the nianometer reading was translated mto a dynarnic pressure,
which in turn was used to calculate the air velocity at that pomt. The pressure P exerted
by a manometer fluid with a density puu& at a depth h is given by:
P=p&&h (3-1)
where g is the acceleration due to gravity. The change in the manometer reading h m the
zero vetocity condition constituted the value k Thetefore! P would quai the dynamic
pressure exerted by the air. The dynamic pressure q of the air is d e W as
q~ = sPurv2 (3-2)
The density of the air during testing was evaiuated by knowing the ambient temperature
and pressure recorded fiom a digital barometerlthemmeter situated in the lab.
For aii caiiition trials (save for the last), a manometer using decane as the
manometer fluid was used. This particular manometer was speciîidy designed for slow
speed use. As directed by its coostnictor, the dynamic pressure (q, in units w@)
m e a s d by the device was caiiited to be
q = 0.244 L (3-3)
where L was the change m manometer reading (mches), with the manometer tluid being
kerosene. This formula was easily modified for use with decane, as the onIy property that
changed was the fluid density. Thus, the equation becarne
q = 0.2199 L (3-4)
Of course for consistency, the results were convened and reported in metric units (Pa).
For the 6nai caiiition test, the mawrneter normally used with the large wind tume1
was empbyed, as it was fomd a more convenient apparatus. It read m mches of water, so
no speciai forrnuia for q was required. Equations (3-1) and (3-2) were set equal and
inimediateiy solved for the air velocity V.
The wind tunnel set* was governed by controlling the applied voitage to the
Ws AC motor by way of a variable voitage source. The mtor was capable of handling
voltages up to 110 volts, which therefore dictaid the maximum attamable wind velocity-
in each case, a specific voltage setting was correlated to a certain calibrateci wind speed.
Complete velocity protiles for the tunnel settings used in the experhmtal tests
are included in Appendix C. It shouid be noted that h m the prelEnioary tests performed
by Mr. Allison, it was discovered t h there was mniimal decay in the velocity field as
one moved away Eom the tunnel exit (i.e., m the order of 15 cm or less). Smce it would
be quite simple to constrain aii testing to within this distance, it was decided to take
cali'bration readings for a 15 cm square region centered only at the tunnel exit. No
additional profles were sampled at dhances away fiom the exit.
Chapter 4: EXPERIMENTS
4. i WSng Testing Procedure
4.1.1 Mefhodology
It would be wise for the reader to ce-- themselves with figure 2.3 in
Chapter 2, whiçh iilustrated the wind-hub axis system used by the simulation code. It was
decided that the best testing procedure wouid measme these forces directly, Le., have the
force bahce continually aligned with this body-tixed axes system This wouid elaninate
the need to convert the results with tngonornetry into the desired axiai components. Such
an added step may bave produced undue error.
The SRI simulation program required data for the MAV wings' lateral and
longinidinai forces and moments for 180" rotation in various ke-strem velocities.
These measurements wouid be perftorrned m the static sense, meanhg the wings wouid be
positioned at a îked angle of incidence to the crosdow, and then the forces would be
recorded whiie flapping at a steady state. The tests wouid not address the dynamic
scenario, whereby the mechanism wouid be rotated through the crossîlow at a constant
anguiar velocity while simuitaneously taking readïngs.
Due to the nature of wbg actuation in ProtoSouth (see figure 1.3), it was
irmnediately apparent there wouId be problem when the whgs were oriented past 90° m a
crodow. In the extreme sense, with the wings positioned at the 180° mark, the flapping
mechanhm (as wel as the mount attachai to it) wouid be upstream of the wiags. This
sort of flow blockage would be totaiiy unacceptabie. R d that the objective was to
obtain data for the wings alone (Le., mïau.s any driwig mecbanism). Since it was
mipossible to completeiy isolate the whgs h m the main body of ProtoSouth, some
alternative method of testing was necessary m order to record data at angles beyond 90".
A simple solution emerged whereby the wings were mounted backwards (Le., inverted)
on the mast of ProtoSouth. By dohg this, it was possible to accumulate ùiformation for
the extreme angies of crossfiow. Figure 4.1 illustrates this wing testing procedure.
O" 45O 90' (Reverse Wing Mounting)
This figure ûiustrates the two basic steps in the testmg ptocess. Step 1 depicts the
wing mounting used during the first 90" of rotatioa At the 90" point, the wings were
detached a d remounted as shown in step 2, suçh that the i d h g edge of the wing was
now upstream of the l d m g edge. This allowed the remahhg angIes to be tested.
One remaining drawback of the procedure was observeci during the mitialOO - 9û0
rotation phase. During these angles the wiugs were orienteci such that they were tbnistmg
d o m upon the flapping mechanism and mount, which acted to b k k the thnist. It would
have been more desirable if the mast of ProtoSouth were much longer than its current 5.5
cm Length, Such au elongated uwt would have acted as a shg, thereby dowing fess
downwash on the main body of RotoSouth. Effort was taken however in design& a
momt that would not add coderabiy to this bw impedance. Short of rebuilding
ProtoSouth, this was aü that could k done. Uniess the ensuing resuits appeared
completeiy out of sorts, no such recoastnrction would be atternpted.
Communication with SRI'S Tom Low, the progmmmr who developed the
simulation revealed that the code worked using a series of lookup tables. The computer
wodd evaiuate the MAV vehicle's flight condition based on the advance ratio J and the
vehicle's orientation in the fiee-çtre;un. Mr. Low defined J of the vehicle as:
where V is the fiee-stream velocity, b is the span of the wings (15 cm), o is the hpping
kequency (in Hertz), and 8 is the flappuig amplitude (in radians). Once the computer
determined the vahe of J, it would reference the tables and interpohte where necessary
to acquire the forces and moments acting upon the MAV.
Wbat beçame mimediateiy apparent was that J had dimensions of revolutiom?,
which meant that it was a kqueracy dependent variabIe. S k e the wuigspan and Eiapping
amplitude were fixed, the ody parameters that coukl be varied were the fiee-stream
velocity and the flapping kquency. Mr. Low had mitialiy programmeci his code with
advance ratios of 0.5, 1.0, 1.5 and 2.0. Matheniatidy speakin& there was an infinae
number of V and o combiions that couid produce these desired Ps. However, one
must te- this k t with hgic in that the Vlo ratio shouId ilhistrate a realista Bi@
condition, For example, knowing the top speed of the ninnieI to be about 7 mis, it can be
detennmed that an a d m e ratio of 2.0 codd be actiieved by flapphg at approximately
I 1 Hz However, would this be a realistic fIapping fkquency? In the context of the MAV
vehicie, the m e r was absoluteiy not. Reférring to the research performed both by Mr.
Bilyk [Il and Ms. EEKhatib [2], such a low hquency would not produce suflicient
thrust, nor wouid the wiugs twist in order to perform m th "clap-hg" region so mveted
in this scde of ûight, Therefore, the set of experiments wouid have to be performed in
such a way as to be meaaingfiil and approximaâe the tme &ght conditions.
As an initial approach, it was decided to perform tests at 40 Hz (a value
corresponding to rougiùy 50 gram of thrust), which was a reaüstic hqueracy to alIow
hovering of the anticipated MAV, Unforturaately, it was dikcovered thai this was a
padcularly demanding hquency in ternis of wing and motor durabiiity. in hct, once the
crossfiow cornponent was appiied to such fiapping, the wings were found to disintepte
only after a few trials - much too short for meaningful data to be recorded. A
compromise thus carrie by reducing the test fkquency to 30 Hz. The wings performed
much better at this value in t e m of durabi , aibeit at reduced sbtic thnist vaiues
(approlcmiateiy 22 gram). The conclusion was therefore to @nn ail testing at 30 Hz,
with uniy the variations in the fk-stream velocity king the method of aIîering the
advance ratio.
Reférring to the wnmd tunnel caii'bration resdts [Appendix C), the maximum
attaiuabIe vebcity was 7.0 mls. With the other parameters d o n e d above, this limited
the rmxbmm advance ratio to 0.743. Given this due , and wah fkther disckon with
SM, it was decided to acquire data for three othet ratios of about 020, 0.50 and 0.65,
Each of these would require a specinc velocity. Fiow kIcî vetocities wete based on an
average value of several manometer readiogs. Hem, it was extremeiy difiicult to make
these average values match to hose speciûed by the advance ratios above. An
effort was made to approach these as best as possible, and as a result, the &g J
values were 0.19,0.55,0.66 and 0.74, which were deemed acceptable by Mr. Low.
The force balance was mounted to a tripod for ail trials perfonned. This greatiy
fàcilitated Ieveliing of the system, as the tripod had numerous adjustments for this
purpose. In addition, the tripod aiiowed the tialance to be raised or lowered m the flow,
such that the wings wouid aiways be pked m the optimum "sweet spot" m the tunnel's
flow field.
The balance was wired to a Fluke NetDAQ data acquisition system attached to a
laptop cornputer. The NetDAQ monitored four channeis, aameîy the three gauge outputs
as weU as the applied excitation voItage. The NetDAQ proved to be a very convenient
apparatus, as its accompanying Windows software provided many options with regards to
sample times and output formats. With some advice fiom Mr. Dave Loewen, a 5 second
sample t h would be recoded to a data tile at 0.006 second intervais. Thus, a typical
test nin wouid begin with a m o reading immediately foiiowed by another reading with
the wings in motion. The gauge outputs w h k the wings were flappbg were quite
oscillatory, as can be expected by the nature of the motion. In order to determine the
mean change in voitage h m the zero cornlition, these oscinating outputs were sîmply
averaged over the 5 second tirne i n t d This was proven to be a valid assumption as a
graphical plot showed these aUctuathg outputs îakhg place about a lïxed mean due.
A h , simple thrust tests (with no crusdow) produced tbrusts m close approximation to
the numbers generated by ML Biiyk [1] on a compkteiy separate appamtm. This type of
cornparison inspiml much confidence m the accuracy of the baki6ce m addition to
veriS.ing that the caîculation method was a s o d approach. in d cases, the data was
reduced using Mimsofi Exce1 97 softwareftware
Each test run consisted of positioning the wings and tripod together at îhe desired
angles in the crodow. The kremental change in aogle was chosen to be IO0, whkh
proved to be of acceptable resolution. As illustrated in step 1 of figure 4.1, the wings
were swept MaIy to 9û0, wiih an added test doue at 100' prior to inverting the e s .
This was done to provide some overiap in the results. As descriid earlier, the wings
were inverted and cepositioned (or swept back) to compIete the fidi 1 %O0 rotation. Each of
these weeps was performed three times for each advance ratio to detennine spread of
data and he! degree of repeatability.
4.f.2 Taring
One of the greater (and unexpected) challenges during che course of the testing
involveci the tare values of the force baiance mount. Tare dues are the force and
moment conttriutions mide by components other than the wiags during testing. It was of
great importance to keep the tare to a minimum percentage of the total reading, as iarger
values tend to contaminate the r e d s . in this case, the muunhg bracket and hpping
mechanism w m susceptible to the crossflow and thus transmitted drag forces to the
balance. Fortunately, the fieestream did not affect the force bahnce kif as it was
psitioned below the Ievel of the tunnel exit. [a otder to obtain the truc redts (Le., tùr
the wings m isolation), these unwamed contriitions had to be suboracted h m tk test
data
The method for calculating the tare of the mount and flapping mecbanism was to
simply record their longitudiuai and laterd forces and moments (without the Wmgs
attacheci) under the same crossfiow conditions as those to be tested wiih the wings. As
can be seen lÏom figure 4.1, the tare values fiom O" to 90" wodd be completely
analogous to those fiom 90" to 180'.
The 6rst mounting bracket used was a disappomtment. Sketched m figure 4.2
below, it consisted of a simple post with gussets extendhg outward to support
ProtoSouth.
Figure 4.2: Original Mounting Brackei
in this cantilevered position, difnculty was encountered m transf'erring the
measured moments (about the centre of the baiance) to a position on the MAV wings.
The problem was with the laterai force's contriiution, which had a large lever arm, which
m turu increased the mgnitude of the readmgs. This is depicted more clearly in @re
43.
Lever A Darire Moments a lei ad in^ Edge
c Figure 4.3: Original Mouniing Bmcket (Top View)
These lateral contniions to the o v d moment essentiaiiy masked the true
wing moments, resuiting m data that was greatly scattered and erratic. Fortunately, the
force readiigs met no such problems m taring, and th& data couid be obsewed.
A lesson was leamed h m this rather dispieashg start, and much greater thought
went into the design of the second mount. Two issimes were addressed. Fi, the size of
the bracket's fiontal area was minimised at d e s near 90' to reduce lateral tares.
Secondly, there was the need to have a h e d reference point by whiçh moments would be
calculated about, rather than attempting to transfèr the moment to a selected point on the
MAV wings. The chosen reference point was taken to be the wings' leading edge.
Discussion with Mr. Low supported this decision, and revealed that his code couid be
adapted to d o w the moment to be r e h e d anywhere on the MAV body. No removaI
of the laterai force's moment contriion wodd be perfonned, With these issues in
mirad. a "gooseneck" type muat was constructeci (shown in ligure 4.4), which enableci
the leading edge of the MAV wiags (in either a f o d or inverteci attachment) to be
aligned with the centre of the force baiance.
The irnprovement was still far h m perfèct. At least in this instance the values
were les scattered and a trend was beginnnip to emerge. Apparently, the moments of the
MAV wings were either exceptiody small, the tare was d l too Large, or both
Determinhg their values with confidence continued ta be a challenge.
Two nnai options emerged, The ht was a redesign of the force bahce to d e
it les s t 8 in an effort to attenuate its sensitivitynsitivity Alternatively, an attempt to shud the
muutmg bracket wàh some type of shieki wouid reduce the tares even finrther. Tbe
decision feu to the latter, as it would be îhe quickest and e!zisiest to impIement.
A two-piece barn "cocoon" (seen m figure 4.5) was cut and mounted about the
braçket. The reduction m tare values was astonishing, reducing their dues by 64%. The
moment data (reportecl in more detail m the next chapter) became immediately more
clear, and an identifiable and repeatable trend was observed. The tare reduction effort had
One hi note on taring should be d e m regards to what this author labeiied
"thw taren. Because of minor misalignment, the flappmg mecbanism would sometimes
be pointhg off centre, which registered a moment on the balancebahnce This is shown (quite
exaggerated) m figure 4.6. The root of this error was due to the nature of the three-piece
attachment of the gooseraeck mount. Each piece had the ab%ty to rotate with respect to
one another, allowing slight alignmmt mors to emefge.
Figure 4.6: Exaggerated Mounting Mkalignment
This type of misaiignment was practicaiiy mipercephile to the naked eye,
however it was certainly perceptiile to the sttam gauges. Therefore, prior to performing
actual tests, a series of trials were performed without a crossflow to determine the
magnitude of this misalignment. Until this tare was minimised (through numerous
aàjustrnents), the test would not proceed. In addition, at certain pomts in the test
procedure these trials were repeated to ensure that the th- tare had not changed. If it
had, the test was either repeated or modified to reflect the new tare.
4.2 Tail TeMing Procedure
4.2. f Tail Design
Mer consulting the MAV team members, no preferzed tail design had yet to be
established. Thus, the tested designs were rather arbitrary in their dimensions. As stated
previously, more emphasis was to be d e on their placement with respect to the MAV
body in the simulation program. An m-depth and exhaustive study to create an optimum
tail consgUration was not the intention oLthis research,
Two taü designs were iuvestigated, both of crucifom c o ~ o n . These are
show m figure 4.7, wiîh their dimeasions ilhistrateci m figure 4.8.
Figure 4.7: Tai1 Designs
Figure 4.8: Tai1 Dimensions
The hst design was a basic rectanguIar sbape, wbile the secoiad was a simple
half-mon. Both were constructed of 1/16" batsa and glued to a metal rod wbich attacbed
to the rnounthg p s t (see figure 4.9).
4.2.2 Methodoiogy
The taiIs were tested in a similar fàshîon to the methods above; save in this
instance the gooserteck mount was mit use& In its place was a verticai post with a r d
attachent as iu figure 4.9.
Mr. Low's simulation required ody CL and CD c w e s for the tails d e r a 180'
rotation. Thus, rnoment meaSurements were w t requned duriug the procedure. This kt,
in combination with the smaller motrnting bracket and absence of a fhppîng meçbaniJm
meant there was no need for a shroud to duce the tare. T a dues were f o u d to pose
no difiicuity whatsoevet.
It was decided to test the taiIs at a wirmd velocity correspondmg to the typicai
downwash velocities detecmined from Ms. El-Khatb's [2] research with hot-wire
anemometry. However, given the probable s k an MAV size tail (iess than 7 cm), the
issue of taring problems ernerged once more. Thus it was decided to double the scak of
the tail dimensions, but test at haLf the velocity, This was to ensure that the Reynolds
nurnber remahed m a smiilar regime. Ms. El-Khati'b's research showed air velocities of
about 4 mis at positions 15 to 18 cm below the wings. Unfortunateiy, the enlarged wings
tested at 2 d s yielded very minute forces in the order of 3 grams or less. This greatly
pushed the sen~itivity iimits of the baIance, causing unrealistic drag and lift curves. Mer
much consideration, it was conçeded that the only option was to test at a higher velocity
of 5.24 d s . Although effectively more than doubling the Reynolds nutnber, it was stiü of
low value (below 25,000) such that there would be minimal error m the n~n-~onal
Lift and drag curves. Auy discrepancy wuld likeiy manifest itseif m the üf& curve's
stalling angle and drag would becorne decmsed slightly.
Mer these changes, the drag and lift curves became much mure reaüstic, and
their tùn results are descriid m the next chapter.
Chapter 5: EXPERIMENTAL RESULTS
5.1 Wings
5.1. 1 Repeafabiiity
As descn'bed in Chapter 4, a totai of four advame ratios were investigated in the
experimental anaiyses. Also mentioned was that for each advance ratio, a total of three
180° weeps were performed in order to establish the repeatability of the measwements
and the size of kir mor bands. In al cases, the degree of scatter in the recordeci data
was low, especialIy with respect to the lateral (x-axis) forces. Recaiiing the tahg
challenge that was encountered with the moment measurementç, it was a pleasant
experience to h d y iden@ clear and repeatable trends for this data. Figure 5.1 shows a
sample plot of the lateral (x-axis) force vs. crossflow angle for 3 triai rum.
X Force (9) vs. Angle
40.0 -[ 3 Trials
l '
S. 1.2 Longitudinal (2-mis) Forces
Upon inspection of the longitudinal (2-axis) data, it was readily observai that a
sharp discommuity o c 4 at the 90" mark. The uiitiai conclusion was that the
inversion of the w b g s (recd section 4.1.1) was the source of this abrupt "jump" in the
rneasurements. Why the force decreased m magnitude however, was somewhat
mysterious. One would intuitively expect that with the wings mverted, they would be k e
l+om the b w blockage caused by the Dapping mechanism a d shroud and subsequentiy
produce mure thnist. Yet it appeared the oppsite was me. Of particular interest m the
k t that this dikcontinuity was absent w k n the a d m e ratio eqded 0.19. Figure 5.2
illustrates the characteristic, and its non-appearance when J = 0.19.
Avg. Z Force vs. Angle
Figure 5.2: Longiktdinal ( Z h ) Dkcontinuitp a! 90 *
Due to the absence of the discontuiuity when J equaited 0.t9, it was believed
some aerodynamic condition at hi& velocities was attenuating this phenonmon. A
plausible explanaiion may have been the tbaî because the air was being acceIerated
about the fiont of the shroud (Le., the region immediately afi of the e s ) , it may have
interacted differentiy wiîh the complex vortex shedding of the wings, serving to ampli@
their thrust. Another reason may have been that the shroud and flapping mechanism
served to block the mcomhg air to the inverted wings, demashg their thrust. Or perhaps
the higher velocities disnipted the mtake of the whgs m their inverted condition in aii
iikelihood, it may be an elaborate combination of al1 of ttiese tactors that contriïuted to
the problem. No clear solution was apparent, and no remedy seemed to remove or lessen
the trait unless ProtoSouth codd be refitted with an elongated mast. Overlapping
readings were recordeci at 100" d e r the conventional aaâ mverted attacbments, and
reveaied the discontinuity continumg past the 90" mark. CompIete raw data for these tests
are mcluded m Appendix D.
A few mteresting points were made atler a k trend h e was passed through
the data for each of the advance ratios. in ati cases (except for J = 0.19), it was
immediatety apparent that divergence h m the trend line began at 70" and ceased at 90"
(see figure 5.3). AU data points afier 90" were very near the trend Zinc, with those below
70" conforming as weU. With these outlymg points temoved, and the trend line reapplied,
it was discovered that the equation of the trend 1Eie changed oaly slightiy (figure 5.4).
Hence, the culprit causing this jump would most Likely be found by focusing an
mvestigation m the region between 70" - 90". Evadently, inversion of the wings was not
the contriïuting fàctor, but rather some intefaction ernerging near the 70" pomt.
! Z Forca vs. Angle, J = 0.735
Figure 5.3: Longitudinal (Zab) Force vs. Angle with Linear Trend Line, J = 0.735
Z Forca vs. Angle, J = 0.735 (Anomalies Removed)
Figure 5.4: Longitudinal (Zd) fiire vs Angle wak Liuear Trend Line, J = 0.735, (Odijdng Anornulits Runo@
The data was reported to Mr. Low at SRI m its origiual form, Le., without any
adjustments or alteration. Of course, it wodd te ükeiy that such changes would (and
certainly should) occur, however the author felt it best to report the accumuiated data m
its purest sense, without any manipulatioa
hother observation made kom the addition of the trend lines was the remarkable
linearity in their slopes, as indicated by their R' vahies (again with the exception of the
case where J = 0.19). This lead to the conchision îhat for advance ratios above 0.5, the
relationship between longitudinal Grce and the angle of incidence to the crossflow was of
a nearly linear nature. This conchision was Limited to these higher advance ratios (which
of course corresponded to higher velocities), as indicated by the marked difference in
bebaviour when J = 0.19. Dirring the accumulation of data, discussion with the Mr. Low
and his coiieague Bruce Knoth showed they had a preference for data at the higher
advance ratios (i-e., above OS), aad thus no f i r tests were performed to determine if a
similar dope relationship couid be made for the Iower region of advance ratios. It is of
importance to remind the reader that the simulation code worked on the principle of
lookup tables, rather than c o m t e mathematicd fornulas, to determine the forces on the
MAV. Thus, the above dope observation was an experimd conclusion reiated to this
thesis, but not reported w r desired by the SRI software developers.
Figure 5.2 shows a clear correlation between the force magnitude ami advance
ratio (Le., a logicai progression of highex advance ratios correspoedmg to higher forces).
However, again there was a contrase wben J = 0.19 which, as already mentioned, iacked
the discontmuity and simiiar dope of the other ratios.
What was certainly conmion to aii four ratios was their magnitude at 90°, which
was approximately 22 grams. This corresponded to the static th& value the BAT-12
wings produce at 30 Hz. This made sense because, when at right angles to the oncoming
tlow, the wings (at least m the longitudinai sense) were not "seeing" any component fiom
the crosstlow, and thus produced theh conventional thnist values the regardless of the
magnitude of the Eee-stream.
5.1.3 Lateral (X-axis) Forces
Figure 5.5 shows the iaterd forces vs. angle for the four advance ratios tested.
Again, the sensible trend of larger advauce ratios corresponding to larger forces is readiiy
apparent. The close symmetry of ali plots about the 90" pomt also foiiows as one might
expect.
Avg. X force vs. Angle
- --
Figure 5.5: Laferaï (Xd) Forte vs Angle, A11 Advonce Ratios
The addition of second-order poiymmiai trend lines (as was done to the
longitudinal data) data did not show similar rehtionships for any advance ratios. A 6nai
observation can be made on the disruption in the curve for J = 0.19, while the curves for
the other ratios remained srnooh Again, one mut assume that the source of the error is
ükely due to the change in whg aitachment, as the dip m measurement appeared near
90". A surnmary of the raw data used to generate these figures is mcluded m Appendix D.
5.1.4 Moments (about Y-axis)
The moment data obtained Eom the experirnents was by Far the most interesting,
and revealed a striking contrast ktween the higher advance ratios and the lower J = 0.19
condition. For the high advance ratios (0.55, O656 and 0.735), the similanty in trends
were obvious, with the lowest magninade of moment fonning an apex about the 60' - 70"
mark (see figure 5.6).
Avg. Y Moment vs. Angle
20.0 1
-70.0 J I
Angle m g )
Figure 5.6: Moment (about Y-k) vs. Angle, Al1 Advance Ratios
With regards to when J = 0.735, it was observeci that the moment did not return to
zero at 180° as one might expect. Since this was the highest advance ratio (and thus the
highest ûee-strearn velocity), it was probable that any minor misalignments of the
apparatus were amplüied at 1 80°, conmbuting to the zero o&t. Even through repetition,
this data point rernained an outlier fiom the zero pomt. Of course, one couid d y
recommend the data be altered artinciaiiy to "maice sense". Howevet, as mentionai
above, this author has decided to report aii data as it was recordeci, with w such
modi6cations inchdeci. Please refer to Appendix D for the coqlete moment data,
ïhe difkrence m the moment trend for when J = 0.19 was extraordinarydinary
Paruarucuiariy interesting was the pronouuced positive moment at aogles above 110°,
whereas the oher ratios for the most part were entirely negative. When J = 0.55, there
was a simiIar positive moment region, aithough here it was tightIy cordird between 160°
and 180". One can speculate b t were the advance ratio lowered fiom 0.55 to 0.19, this
positive zone would expand to encompass a wider berth of angles. At the hi@ a d m e
ratios of 0.656 and 0.735, there were no positive regions present.
5.2 lails
5.2.1 Results
The CL and CD curves for the two tails descriid in Chapter 4 are mcluded below.
In cornparison to the whg tests, ihese experimenîs were relativeiy simple, and therefore
only two 180" sweeps were perfonned, Very W e scatter was encountered between
readings. Unlike the wings, the simukition code required curves for 360° rotation. Thus,
the data was simply mirrored about the 180" mark to satisfj. these requirements. Figures
5.7 and 5.8 depict a s u . of the resuits.
Tail li1 and #2 CI vs. Angk Re = 22,000
F i g m S. 7: CL C m for Ta& #l and #2
i 1 tail #1 and lit Cd vs. Angle ! Re = 22,000
Figure 5.8: CD Cuwes for Tai& #I and #2
As is evident, the dierence between the two tail desigus was relatively minor,
especiaiiy wiîh regards to the CL cuves, which were nearly on top of one another. Any
performance difference would most likeiy d e s t itself in the slightty higher Co values
producd by the taii #2 design, As was discussed m Chapter 4, it was expected that
placement of the taii (and not some extraordinary tail stiape) would be the more miportant
fktor in determining a stable coafiguration.
5.3 Amplification of Daia
5.3.i Z Forces
For reasons aimdy meatioaed, al1 data was taken at a fiqping f k q u e q of 30
HL However, the projected mass of the actuaI MAV was expected to be close to 50 g, a
thrust value tbat could ody be reached by flappïrag at a 40 Hz fkqwncy- Thus came the
issue of scaIing the data mteiügently so as to represent the hrces kurred at 40 H z A
simple mdtipiication &or would be the poorest scaling as ihe nature of the
forces was anything but iiuear- Consuitation with Mr. Bilyk and Dr. DeLaurier raised the
bypothesis that the thnist produced by the MAV was directiy rekted to the axial
component of the fiee-stream velocity impmging upon it. This was taken fiom the fact
that when the MAV was oriented 90" to the flow (i.e., with no axial k-stream
comportent), it produced a thrust values neariy identical to those when the ke-stream
was absent. A nondiinsionai relationship was devised to properly test this theory, and
it included the above variables together with the f.lapping fkquency and span of the MAV
wings. The Girst nondimensional group compriseci the ratio of measureà hmst (Le., that
recorded during testing) to the static thrust (ie., the thnist pmduced when îbe crossflow
was absent), and is herein referred to as the tisrust ratio. The secorad group was a ratio
between the axiaI component of the k-stream velocity to the product of the flapping
tiequency and wnig span, and was labeiied the k-stream - kquency ratio.
Aii four advaace ratios were reduced to this format, and plotted as shown m
figure 5.9. The scatter plot showed a remarkable lineanty in this telationship. However,
this did not entireiy validate the hypothesis. AU the advance ratios were produced h m
30 Hz data. Some fkther investigation was required at other fkquewies to better
estabiish the theory.
t T h t u i Ratio v a W a ü W
2.50 1 I
Figure 5.9: Thrwî R& vs. FreeSîream - Freqnency Ratio (Original Data)
In lieu of completely re-evaluating the data at other fkquencies, it was decided to
perform a few tests at the extreme ends and at the centre of the anguhr sweep, for
Werent îlapping fkquencies. Tbnist data for 25 Hz and 35 Hz was obtaiued in a 524
ds crosdow, and is shown with the previous data m figure 5.10.
l
Thrust Ra* vs. VaxiaüWb
4.30 -020 4.10 0,W 0.10 0.20 0.30 / Vaxh W b l 1
Figure 5.1 O: Tkrrrst Ratio us, Free-Streum - Frequency Ratio (Orig. and Ertrp Daia)
From the d t s , it was apparent that this relationshrp extendeci to other
keqwncies, with a tolerable degree of scatter m the plots. The next task was thus to
extrapolate h m this to anah the &ta for 40 Hz A simple addition of a trend iine
through tbe data ( h w n m figure 5.10) allowed for this, remit& in the eqation
where b is the span of the wings and o is the ûapping ikqueclcy. The q u e of mterest
was now L at 40 Hz for each tested advance ratio. Thus, by hwing the Vd
components for each advance ratio, ami knowing the sîatk thnrst at 40 Hz to be 50
grams, it was a relatively minor task to obtain the required curves. These are sbwn m
figure 5.1 1.
Extrapolated Z Foree vs. Angle (40 Hz)
Figure 5.1 1: ExtrapoIuted Z Force Data for 40 Hz
Of course, due to the chaage m Dapphg fiequency to 40 Hz, the advance ratios
were altered accordiigiy. It is this step that emphasizes how J is a iÏequency dependent
variable, as the 40 Hz extrapoIated d e for J = 0.55 do not at ail match the origmal30
Hz values for J = 0.55.
5.3.2 X Forces and Y Moments
A similar methodology was pursueci to investigate the effect of flappiug
fiequency on the x-axk forces and y-axk moments of the MAV. From testing
experience, it was mtuitiveiy beLieved that the f h p m q effèct wodd be niargmal as the
x forces were feh to be largely due to a sectional area drag, and y moments about ttie
Ieading edge of the wing were essentiaiiy a by-product of these drag forces. Intuition of
course, was not enough to s a w this hypothesis, and so fiapping tests at kquencies
above and below 30 Hz in a moderate crossflow of 524 mis were performed.
The results were supportive of the above bekf, and are shown m figures 5.12 and
5.13. Unfoctunately, a complete 180° sweep at 33.3 Hz was unavaiiable as one of the
strain gauges on the balance was damaged (ükely due to fatigue Mure). Enough
evidence was present however to conclude that the effect of ûequency on the lateral
forces and Y moments was not significant.
1 I
i X Force rit Various f lapping Fmquencies V = 5.24 mls
Figure 5.12: Effea of Rapphg Freq~ency on X Fome
Y Moment at Various Flapping Fmquencies v = 5.24 mls
- - - -
Figure 5.13: Effect of Flapping Frequency on Y Moment
5.4 Compn'son to Assumed Values
The simutatKia was initiauy p r o g r a d with force aod moment data that were
essentiaiiy educated guesses as to what type of aerodynamic perfbrmance could be
expected fiom the MAV. Early test nms with this estimateci data showed the aircrail to be
unstable, and therefore it was important to determine (br better of for worse) what
degree of instabiiity truiy existed. This section briefiy compares the differences between
the estimated and experimeatal data.
The most convenieat comparisons can be made between the eqerhm&d resuiîs
recordeci at an advance ratio of 0.55, and the a s s d vahses for J = 0.50. The ciifferences
encouniered bmamn ihem were astonishing. Both lateral forces and monients dïfkxd by
neariy two orders of mgnitude. One couid infér h m these substantiai increases,
parti'cuiarty in the shaiiow @es of cro~ow, that there would likely be greater righthg
forces to the MAV if it were disturbed fiom a steady hoverhg position. Figures 5.14 and
5.15 illustrate the moment and lateral force cornparison to the initiay. a s s u d values.
X Force Comparison of Measured (J = 0.55) v a humed (J = 0.50)
1 40.0 1-ksuedXForceJ=0.55 1
35.0 -. Assumd X Force J = 0.50
I 1 1 I I
I
I I
1 I I
3.0 b la0 150 290 1 I
Angle (de91
Figure 5.14: X Force Comparison to InitiaIiy Assumed VaIues
Y Moment Comparison of Measured (J = 0.55) vs. Pasumed (J = 0.50)
l 1 -Assumed Y Moment J = 0.50 j 1
An@k (deQI Figure 5. 15: Y Moment Comparison to Initiafiy Assumed Values
With respect to the Z forces, the assumed values feii more m h e with the actuai
(albeit amplined) data Yet a ciifference between them was stdi readiîy apparent, Both
were nearly linear in shape, but thei. dopes dBièred simcantiy. This simply meant a
less steep thrust degradationkittenuation with angle of crodow (see figure 5.16). It is
also important to note that the assumed values showed a considerable thnist surplus (to a
value of roughiy 80 grams), which also gave a discrepancy.
Z Force Cornparison of Extnpolrted (J = 0.55) vs.
-Ewtrapdated 40Hz Z Force J = 0.551
- - --
Figure 5.16: Z Force Cornparison to InitiaIiy Assumed Values
Chapter 6: 2 0 SIMULATDN
6.1 Numerical Model
6.1.1 Application of Newton's laws
in order to simulate the MAV dynamically, Newton's iaws were appiied directly,
using a mathematid mode1 coded m the MATLAB version 5.0 programming language.
This model is depicted in figure 6.1 below.
Figure 6.1: Mdel Representdion
The two dimensional model was governeci by the Mamental equations F = ma
in horizoutaüvertical translational motion, and M = la in the rotationai sense about the y-
&. The distance 11 represented the Iength betwea the vehicle centre of gravity and the
leading edge of the wings. This parameter was d d k d by the fact thai al1 forces recocded
during the experimentai testing were resolved about the wings' leadimg edge. In the same
vein, h represented the dktance fbm the W s quarter chord to the vehicle centre of
Pvi ty-
In order to gauge the stability of the MAV d e r perturbeci comütions. it was
decided to descrii the aircraft's motion in a globaI coordinate system, Le., how a
stationary observer would withess the flight trajectory. The ongin of this system was
centred at the c.g. of the vehicle when time equalled zero. The ensuing caicuiated
motions would indicate the vehicle's path 6om this initial state. For rotations in 0, the
vertical 2-axis was designated as the 0" reference point with clockwise rotations king
positive. An example of the MAV in a d i i b e d state is shown in figure 62.
z Flight Path from Origin
.,--..,
t \,
Figure 42: DrSIurbed Condition
As indicated m the figure, the thrust of the wings was onented almg the
longitudinal axis to coincide wi th how the data h m Chapter 5 was recorded. The same
can be said of the drag force h m the vehicle's wings b and Dm of course reptesented
iifl and drag contriutions h m the tail,
6. f.2 Lookup Tables
Values for thnist and drag of the MAV were necessary to properly calculate the
vehicle acceIeration at each tirne step. This was performed by using Iookup tables
generated h m the experimental results of Chapter 5. Smce these results depended upon
both the angle to the k-stream and the fke-stream velocity, a double interpoiation
scheme was required in the cornputer pro- With these values in han& it then became
a matter of resoiving them appropriately into the global coordinate system duriag the
summation of forces and moments upon the vehicle.
6.1.3 Numerical Procedure
The simulation could be broken down into five steps. ïhe tüst step was to specify
the initial conditions of the vehicle. This would include both x and z velocities, aagular
displacement i?om the vertical, and any 0 t h accelerations or velocities of interest.
(Smce the focus of this chapter is primarily upon the hovering condition, the mode1 was
tyjically only displaceci h m the vertical with ail other conditions zero). The second step
was to use the lookup tables to evaiuate the! wing thnist and drag contniutions. The
forces iiom the tail were determined fiom a mathematical equation taken h m a trend
Sie placed tfrough its CL VS. a and CD vs. a graphs (see Appendix D). These coefficients
were then simply muitipiied by the tail area and dynamic pressure to give totd Iift and
drag h m the ta i l Step three mvoived evaiuatiog the angular acceleration of the MAV
body through a summation of moments, which was then mtegrated twice over the time
step to get the new anguiar displacement, The dynamic equations goveming this step are
Iya = D* II (6-1)
anCui = a*dt + (6-2)
en, = uacw*dt + 00ld (6-3)
where a represents the ringular acceImtion about the vehicIe over the tirne step dt, o is
the angular velocity, and 8 is the angular âisphcement 6om the vertical. 1, represents the
moment of inertia of the a i r c d about its y-axis (see figure 6.2), D denotes drag force
and 1, is dehed as in figure 6.1. The "old" subscript refers to a variablets integrated value
at the end of the previous tirne step, whereas the "new" subscript indicates the updated
value of the variable at the end of the current time step.
Similarly with steps four and tive, the z and x acceleraîions were calculateci and
integrated to yield the new velocities and positions at the end of the time interval.
Referring to the global coordinate system iliustrated in figure 62, the dynamic equations
goveming this step were
ma, = T*cos& - mg + D*sinûdd + aooid*vX d,j (6-4)
Vzn, = a,*dt + Vzold (6-5)
Zn, = v* ,*dt + &Id (6-6)
max = T*sinûdd - D*COS~~M - u,id*vZdd (6-7)
v, , = a,*dt + v, (6-8)
X m = vx ncw*dt + &id (6-9)
where a, and a, are the hear acceierations of the vehicle, v, and v, are the hear
velocities, and z and x are the total displaced positions of the vehicle fiom its initial
position. Again, subscripts "OH" and "newn refér to the vaiue of the variable at either the
end of the previous time step or the newly updated vdue at the end of the iatest time step
during the integration process.
At this point di the variables had k e n updated and the process was repeated for
the next increment in the. A complete iisting of the code is included (with conments) in
Appendix E.
One procedure m the program that required carehl hught and planning deserves
some elaboration here. This was conceniing the method for evafuating the magnitude of
the fie-stream velocity and the angle of attack, which had to be assesseci with respect to
both the leading edge of the wiags and the quarter chord of the tail. The fk-stream
velocity was simply the magnitude of the resultant vector generated by the x and z
velocities at the current time step. Note that due to rotation, the h-siream velocity at the
tail would not be qua1 to that at the leading edge of the wings. A more cornplicated
scheme was required however in determinhg the angle these vectors made to either the
wings or tail. Taking the inverse tan of the ratio of the velocity components was not
suficient to defme the angle corredy under al1 vehicle conditions.
For example, since the experiments m Chapter 5 were perfomred with al1
crosdow angks measured with respect to the longitudinal axis of the vehicle, it was
necessary to define the ûee-stream angle for the MAV wings m the same way. Hence, the
"me" angle to the b s t r e a m incorporateci mt only the inverse tan of the velocity
components, but aIso the current tirne step's tilt angle h m the vertical (in the global
coordiite system). Figure 6.3 better iiiustrates the situation.
Figure 63: &le of Wings' True Free-*am VeIoci@ Angie
The figure shows the vehicIe in a typicai displaced codiion with horizontal and
vertical velocities together with a tilt angle of 0. The angle of interest is Ob, which is tlme
angle the resuitant velucity vector makes with the IongitudinaI axis of the MAV. The
angle 0, is calcuIated knowing the magnitudes of the! vebcity componeots V, and V,
Thus in this instance, On is simpIy defmd as 90 - 0 + degrees. However this would
ody hoid true for the above situation It was neçessary to wosider the various
combinations of angies and velocities (in both the positive and negative sense) such that
the correct Oa wouid be calculated every time. A mire complicated example is illustrateci
in figure 6.4.
Figure 6.4: Second Ewmple of Wings' True Free-stream Veldty Angle
In this example, the vehicle is tilted with a negative tilt angle (8) but with positive
values of x and z velocities. The angle the total velocity vector (Vd) makes with the
longitudinal axis O f the vehicle is Qf, = 90 - 9, + 8.
Sirnilar cases can be made with x and z velocities king negative together with
positive and negative tilt angles. Seven general cases were necessary to encompass al1
these possibiiities, a d are included in the program in Appendix E. In the case of the tail,
ody six such cases were necessary.
More effort was still required in order to resolve the forces into the global
coordinate system used by the program. To correctiy d e t h the appropriate signs for the
drag, thnist and Lift forces, one must know the tilt angle and the direction of the k-
stream velocity. For example, with regards to the wings the direction of the drag force
wouid aiways be oriented in the same direction as the ke-stream. Also, the tilt of the
vehicIe to the left or nght of the vertical wouid determine which direction the wings'
thnist component wouid be orienteci m the x direction. Figure 6.5 depicts an example of
such a tlïght condition.
Figure 65: Force and Moment Summation Example (Wings On&)
The three equations of motion goveraing îhk particular condition would be
CF, = ~*sin0 + D*sine (6- 1 O)
ZFz = T*cosû - D*sinû - mg (6- 1 1)
CM = D*li (6- 12)
Again, the reader is reminded of the use of lookup tables in the simulation
procedure. AU drag and thrust values were recordeci as positive, and there was no
mathematicai formula to reIy upon to take care of the sign convention. Thus the onus fell
upon the author to ensure any possible combination of velocity and tih angles
encountered would always sum the forces wiîh the correct signs.
The tail forces relied on an angle of attack caldation, and thus WK orientation
relied on a cornbition of x and z velucities aad Iüt angle. This necessitatecl eight
separate cases, each involving a specinc mmbiaation of x and z velocities and tilt angles
that in turn yielded a unique force or moment summation. See figure 6.6 for an example.
Y Tail t in
Figure 6.6: Force and Moment Summation Example (Tail Only)
In thi example al1 forces are resolved through the quarter chord of the fin. The
summation of forces and moments for this particular scenario wouki become
XF, = L*sinû' + D*sinût (6-13)
CFz = L*cose' - D*sinû' (6-14)
XM = (L*cosû 6 + Dssinû6)*12 (6-1s)
where l2 is the distance h m the quater chord to the vehicie c.g. as depicted in figure 6.1
and 0' in this Înstance is dehed as 90 - 0 - Os degrees. Again, caution had to be taken to
ensure the correct sign convention was obtained d e r ail possible combinations of 0 and
linear velocities. The reader is referenced to the code in Appendix E For firrther details to
gain insight on this procedure.
On a final note, it shouid also be mentioned that the code used Euler inîegration,
and the program's results were checked for convergence by coritinually haiving the tirne
step util no perceptiile changes in the output couid be obsaved.
6.2 Initial Resulfs
6.2. i Simple Hovenng Condition
The code was ûrst used to analyse the MAV under a disturbed condition with and
without a tail. In both cases, the vehicle was placed into a hovering state (where thnist
equaiied the vehicle mass), but with a 2 O initial disturbance fiom the vertical, The ensuing
motion was found to be a steady oscillation between * 18", which neither grew nor
decayed sigdlcantly (ie., neutral stabiiii). This occurred regardless of the presence of
the tan. Apparently, the force of the MAV wings were much larger than those of the tail,
mainly because the k-stream velocities at the tail were low which, in turn, reduced the
amount of dynamic pressure. Oniy if the tail was made ridiculously large did one begin to
see its influence on the system. An example of the oscillation without the tail is show in
figure 6.7.
7
i
-301 1 t 1 l 1 1 I 1 1 1
O 1 2 3 4 5 6 7 8 9 1 0 Tirne (sec)
Rgvre 6 7: lilitial Test Glse W d h t T d
AIthough the motion was by no nieans chaotic, it lacked a certain degree of
reaiism, as one wouid expect some sort of convergence or divergence if the vehicle was
to be truly placed under such conditions. Hence, it became a matter of h d h g a way to
incorporate such an element of reality.
6.2.2 Rotational Dise Damping
Mer discussion with Dr, DeLaurier, it was felt that the code lacked a disc
damping tem. This damping would be due to the physicai act of ''tibg" the hpping-
wing disc plane about the y-axis, which in turn would become a sink for retnoving energy
from the system It made sense that such an effect would be absent m the static tests
morrned in Chapter 5, as it was a dynamic property of the vehicle. The term wouid
appear during the summation of the moments on the body as sorne yet-unknown
coefficient multiplied by the angular veiocity of t&e vehicle. The task then was to
determine experimentalIy the value of this unknown coefficient.
6.3 Disc Damping Ekperiments
6.3.1 Experimentrl Seîup
Dr. DeLaurier devised a procedm h m which the unknown disc damping term
could be detamined It was a relatively simple concept by which a penduium was set up
such that its rate of decay (influenced by the arnot.int of dampmg containecl within the
system, whether by bearùig friction, aerodynamic drag or an actuathg d i s ) was readily
monitored and calcuiated. Figure 6.8.1 iilustrates the ide. while figure 6.8.2 shows the
system in a perturbed state (again with positive 0 in the cIockwise direction).
P A
12
countemeight v
pendulum mass
Figure 6.8. I: D k Dumping Experimentul Setup
Figure 68.2: Disc Dumping ExperUnentd Sptirp, Perturbed Condition
6.3.2 Dynrmic Equations
The oscillaîocy equations of motion for a peoduium are WU documenied in any
dynamics or vii ions text. As seen h m figure 6.82, three components serve to
dampem the motion, Bearing fliction and aerodynamic drag were combined h o ooe
prameter (labelled F in the d i ) which, a s a first approximation, was multiplieci by
the anguiar velocity of the apparatus to caiculate tbe resistance. More important,
however, are the two remaining damping parameters. Daqing due to the bbsurging" of
the wing disc area in the z direction was labelIed quai to Ci2 The damping due to
%itingW of the wing disc a m about the y-axis was, m turn, labeiied Czû. By sumrning the
moments about the pivot point in the perturbed condition, it is hund that
r,b@ = - mg*h*sin 0 - C& - ~ 2 8 - F 6 (6-16)
with 1, being the mass moment of inertia about the y-ais, m king the mas of the
pendulurn weight, and 11 and h as defineci in figure 6.8.1. Simpli@ing through the use of e
the small angle approximation and letting 'z = It 0,
where
In the most general case of such an oscillatory system, fiom reference [q the
solution to the dflèrential equation written in (6- 17) is given as
The bracketed terms are responsble for the osdations m tbe system, while the
exponentiai term outside gives the damping. Et is eady m u that in the absence of the
damping terms Ci, CZ and F, then D would @ zero and t6ere would be no exponentiai
decay in the soiution. Thus it becornes a matter of empmcaily determinhg the exponent
in the patameter e4"' and solving for the unknown coefficients of interest.
6.3.3 Experimenf
The experiment was set up sunilady to figure 6.8.2, and a photo of the appamtus
is shown in figure 6.9.
Figure 6 9: D k Domping Apparatus
Bail bearings were used at the pivot point. A thin aluminum rectaaguiat arm
(representing h) was attacheci to a mounting bracket on the pivot. The actud distance of
h was caicuiated by detefmiLUng the centre of gravity of both the aiuminum arm and
penduium mas together with respect to the pivot point. Attached at 90" to this ann was a
slender steel rod PmtoSordh was attachai to tk end of the rod in such a way as to allow
it to siide up or down the length, thus allowing variance m 1,. A compass and nede were
mounted above the pivot point so that angular displacements could be accurately
measured.
In order to monitor the osciilations precisely, each test was videotaped using a
Canon digital video camera. Knowing thaî there were 30 fiames per second and replaying
the video in a h e by h e advance alIowed for a plot of theta versus tirne to be
produced. An example of such a plot is sbwn in figure 6.10 for the tare damping of the
system As mentioned, the envelope of decay in the oscillations is govemed by the term
e4DR''. Plotting the upper peaks of oscillation alone and adding au exponential trend line
in Microsofi Excel determineci this value of interest. Hence, it becarne a simple matter of
taking the exponent, equating it to D/2, and solving for the unknown parameters.
Theta v s fime, TARE MMPlNG
, 30.0
Prior to testing, it was wcessary to calculate the system moment of mertia (I,,),
which can be found in Appendix F. The f b t test then hvolved the caiculation of the loss
parameter F of the system, This parameter wuid be viewed essentially as the ~ ' '
damping in the system, and was simply measured by obseMng the pendulu. motions in
the absence ofany wing flapping. W i a solution for F, the next s tqs were to observe
the decaying oscillations while the wings were flapping. Since there were two remaining
unknowns (Ci and C2), it was necessary to generate two separate equations. Thus the
expriment was performed twice but with a different It during each test.
6.3.4 Resulfs
Complete raw data for the disc damping tests are included in Appendix F. It was
discovered that F had a value of 0.00 1688 N*m-s/rad, and the ensuing tests revealed Ci to
equal0.0002673 N.s/rad/m and C2 to be 0.001734 N-mdrad. In terms of the simulation
code, the effects of surging (represented by CI) should have aiready manifested
themselves in the lookup tables, as this was simply the motion of the wings translating
through a flow fieId. Therefore, the newly implemented parameter was C2, which would
make its appearance in the summation of moments about the vehicle cg. at each time
step. Its effects were substantiai, and discussed in the sections to foUow.
6.4 Case Studies
6.4.1 lest Cases
These sections invoIve ninning the simulation under different condiions of
mterest, and then varying the geometry of the aircrafl in hopes of establishing a stable
configuration, as weU as gainhg insight into the behaviour of the vehicle in tlight. Smce
the number of possiile variations was nearly idhite, it was decided to focus on
situations where the MAV was at or near the hoverhg state. This was the flight reghe
for which the MAV was most intended, and so it seemed a nahuai flight condition m
which to investigate.
A total of four scenarios were devised, and each was evaluated both with and
without the presence of a tail. The geometry of the vehicle was initially taken directly
fiom the early 15 cm span k e flyer whose body moment of inertia and centre of gravity
location were determineci fiom a Solidworks mode1 of the MAV. The first scenario
consisted of a typical hovering condition with a 2" initial disturbance h m the vertical,
with ail other initial conditions zero. The second case would involve the MAV in a slight
ascent (by simply by lowering the mass) and disturbhg the MAV by 2' fiom the vertical.
Similarly, a third case would put the vehicle into a slight descent together with a 2" tilt
disturbance. F i y , îhe fourth case wodd simulate a lateral gust of 2 m/s with the MAV
initially unperturbeci Eom the vertical. In al1 cases the motion of the vehicle would be
observed to determîue if d converged to a 0" vertical displacement. Divergence was
certainly a possibiiity, and indeed it was hoped that if such situations were encountered
that they could be remedied with appropriate modification to the vehicle geornetry aod
configuration.
Parameters that could be modified included the vehicle's mas, the tail geometry,
the taii's position above or below the wings, and the cg. position with respect to both the
wings and tail. Each case wouki have the MAV begin m what the author defines as the
"standard configuration". This meant that the Ieading edge of the wings would be 7.5 cm
above the c.g. of the vehicle (labeiled 1, m figure 6.1), which geometrically msitched the
prototype drawn m Solidworks. From this initial configuration, each case study wodd be
run and the above parameters wouM be mditïed to observe their impact on stability. in
al1 cases, the parameters were never increased beyond the 15 cm maximum dimension
estabüshed in the MAV project requirements.
6.4.2 Case I - Hovering Condition with lilting Disturbance
As descriid above, this case involved the MAV beginning in a hover, but
disturbed by an angle of 2" and observing its ability to right itseif to a steady hovering
state. Ail other initial conditions were kept at zero. The resuits for the vehicle without a
taii in the standard configuration are shown in figure 6.1 1 .l.
Theta w. Time
Time, sec
Figure 6 I l . l : Case 1- No Tail, = M c m
It was immediately apparent that the vehicle had the abiiity to converge to a zero
vertid disphcement. It then becarne a question of determining the effect of changing the
79
c.g. location with respect to the leading edge of the wings (11). Reduçing 11 to 3.5 cm
revealed improved convergence and indeed with a value of 2 cm there were even better
results. Figure 6.1 12 shows this trend caused by the reduction o f 11.
Theta W. Time
1 5 10 1s Time. sec
Figure 411.2 Case 1- No Taii, Effect in the Reduction of lr
Values of Il below 2 cm were w t mvestigated, as they wodd put into guestion
which side of the c g the wing forces would truly act on. R e d that the forces were
experimentally recorded as acting h u g h the Ieading edge of the wingq and tbat the
chords of the wings are roughiy 3.5 cm, Thus7 11 values less than 3.5 cm meant t4at some
of the projeçted wing area was below the c.g. Anaiysis of the moment and faîeral force
data obtained h m the wind tunnel experiments maleci on average that the wings'
center of pressure was siightiy less than 2 cm below the kadhg edge. Values of Il
s d e r tban this would yield misleading results. For example, the simulation would
resolve the drag force acting through the leading edge of the wings 0.5 cm above the c.g.,
even though the center of pressure was obviously below. Hence caution would be needed
to take in the interpretation of such results. Instances in which 1, extended beyond 7.5 cm
only proved to be less satisfactory than those in figure 6.1 1.2
Continuhg one step M e r and truiy placing the wings below the c.g. added
nothhg to aid stability. In k t , the vehicle immediately began to diverge catastrophically.
The next step was to evaluate the effects of the tail. Under such conditions it
intuitively made sense to place the tail above the wings in hopes of enhancing
convergence. Since the properties of the two tail designs tested m Chapter 5 were closely
matched, either wodd suEce in conjunction with the code. For al1 the case studies, tail
#2 was chosen to cornpiete the analyses. The tail's area was made 0.007 mZ and placed at
an initial distance of 12.5 cm (h) above the c.g., but its effects on the performance of the
MAV were truiy negiigible. The results were almost an exact dupiicate of the
performance without a tail (note that Il was kept at 7.5 cm in thk case). Similar tests at
distances of 15 cm, 1 cm, and even below the c.g. remained ineffective. This was because
the velocities encountered by the tail were very small and hence theh abiüty to produce
aerodynamic forces were limited by a lack of dynamic pressure. Further investigation
showed the taîl drag forces to be four orders of magnitude greater than those of the
wings.
Doubling the tail area (to 0.014 m2) and combmmg this with the above dues of
12 stiU proved fiuitless m enhancing stability- Further mcreases in area were not
investigated, as this wouId mvolve a tail d c e area more than double the size of the
wings, a tnily ridiculous notion (To aid tbe reader conceptually, the tail area of 0.007 m2
would be an area siightly d e r than that of the wings). Thus it was conciuded that m
this instance, the presence of the tail was optionai.
6.4.3 Case II - Slight Ascent wifh Tilting Disturbance
in this study the niass of the MAV was teduceci to a value of 48 granis which,
compared to the th- value of 50 grams, would aiiow the vehicle to slowly gain altitude.
A tihing disturbance of 2' was imposed, with aü other initial conditions set to zero,
Under the standard configuration without a ta& the vehicle began to diverge noticeably
der 6 seconds to a peak displacement of roughly 9.5'. after which the oscilIation neither
grew nor decayed. However, lowering the distauce 1, produced profound stabilising
effects, as evinced in figure 6.1 1.3. At lùrther distaitces of 3.5 cm and 2 cm, the abiiity to
retum to a 0° tilt angle was even more effective.
Figure 6.1 1.3: Case II - No Tait, Egeet in the Redudion of 1,
As was encountered in Case 1, placement of the wings below the c.g. only
produced immediate divergence.
The addition of the tail to the standard configuration (11 = 7.5 cm) only made the
response worse. In this instance, the tail area was again set to 0.007 m2 and h made 12.5
cm. Altering 12 to values of 1 cm and -1 cm stiii proved ineffective. However, when l2
was extended to -12.5 cm (below the cg.), convergence was achieved. This is shown in
figure 6.1 1.4.
5 10 Tirne, sec
Figure 6.11.4: Case Ll- Wuh Tail, III = -12.5 cm, 1, = 7.5 cm
This noticeable improvement comes with a caveat however, as the tail would be
situated in the region of downwash of the wings. This phenonnon was not modeUd in
the code and m y or may not have si@cant effects un the behaviour of the vehicle.
C e M v if there were some form of active contml mrfkes on the tail, then this type of
configursttion could be beneficial. But due to the its passive nature, however, conciusions
based on this type of set up must be taken with some caution.
Taking the best co&gurations fiom both sceuarios (Le., with and without a tail)
and combining hem reveald an overd bene& in performance. That is, with Ir set to 2
cm and l2 set to -12.5 cm, the vehicle dispiayed the best convergence trajectory of a i i as
depicted in figure 6-1 t S.
ïïme, sec
6.4.4 Case Ill - Slight Descent with Tiltiiig Distur6snce
Along the sirnilar ünes as in Case II, the MAV mas was inçreased to a value of
52 grams in order to impose a ttinist deficit and thus mate a siight descent. In addition, a
3" vertical disturbance was added, with di otber initial conditions remaining at zero. in
the absence of a tail under the standard configuration, the vehicle showed a converging
oscillation. As was seen m Cases I aml II, redmtion m the value of 11 produced better
results. Likewise, placement of the wings bdow the c.g. caused diergence. Figure 6.1 1.6
displays îhe effects of demashg II.
Figure 6.11.6: Cose iX! - No Tail, Eflect in the Reduction of 1,
Theta us. Time 2
1.5
1 d Q g CD
4 0.s ai- - Q
c O
Wiih the addition o f the taii (hawig the same geometry as that in Cases 1 and iI),
0 . 5
-1
the motion was found to be only siightly better for values o f h king eitber 12.5 cm or
- k!"'"- L ,
- 12.5 cm. The results for both cases are shown in figure 6.1 1.7.
O 5 10 15 Time, sec
- W i Taii, 12 = t2.5cm - W i i Tail, 12 = -1 2.5cm - No Tail. H = 7.5cm 1
Tme, sec
The same caution must be taken here as in Case II with respect to downwasti
effects. Upon wmbining the best fiom bot& scenafios (with and without a tail), one fhds
the configuration where 11 = 2 cm and h = 12.5 cm. This revealed convergence, but
certainly not as great as tbat having the tail absent. See fÏgure! 6.1 1.8.
Theta us. Tme
5 10 Tirne, sec
-- - - - - - - -- - - - - -
Figure 6.11.8: Case Lü - With and Without Taiï, lz = 12.5 cm, 1, = 2 cm
To rationalise the minor effects of the tail, it was again determineci that the taii
forces were many orders of magnitude smaller than those of the wings. It was concluded
that under this condition the best performance would be achieved without a tail; however
a tail's presence would do nothhg to M e r an e v d convergence.
6.4.5 Case W- Latemi Gus!
In this 6nai case shdy, the MAV was distrrrbed fiom a steady hovering condition
by a laterd gust of 2 mis. Ttiere was M> initial tihmg d i i e and aii other initial
conditions were set to zero. in the absence of the tail, the standard con.fïguration showed
a remarkable ability to right itseif der the disturbance. It is noticed h m figure 6.11.9
that the Iliaximum deflected amplitude reaches roughly 28". Reducing Ir t'urther to 2 cm
damped this maximum deflection to ody 18".
Theta m. Time
Time, sec
Figure 6.11.9: Case IV- No Td, Efled i~ the Redrcdion of 11
The addition of a tail (of same geometry as the previous cases) above or below the
wings had negligible effécts, again due to minute taii forces. The vehicle retained its
abirity to converge. in this case, the best performance occurred in îhe absence ofa tail.
Chapter 7: CONCLUSIONS
7.1 Case Study Analyses
In each case study perfonned in Chapter 6, it was determined that through a
judicious placement of the vehicle's c.g. position a stable design was entirely possible
without the presence of a taiL Hence, future MAV prototypes should stress component
layout such that the cg. falls much cbser to the wuigs' leading edges, preferably at
distances near 3.5 cm. The addition of a tail thetefore may be viewed as optiod, and
indeed its presence for the most paa did üttle to augment the inherent stability of the
system. In the interest of saving weight, this appears to be a t d y bemficial
characteristic. However caution must be taken with any simulation, and real worId
experimental analyses would certainly be required to estabtish the vdidity of this
statement.
The final MAV prototype will obviously need a method of flight controi, and thus
some sort of active control surfaces will have to be incorporated. It is cornforthg to kmw
that the presence of a tail in the conditions descn'bed in Chapter 6 does not hinder the
vehicle's abiiity to stabilise ilself d e r a disturbance. The case studies reveaI that the best
taiI placement would be below tfme c.g. at a distançe of 12.5 cm to the quarter chord of the
k. This would be coupIed with the wings' leadmg edges placeci 2 cm above the c.g.
This Iayout, under ali case studies, was a weU balanceci configuration with respect to
overall performance. Also, with the taiI piaced in the downwash of the wing thrusr, it
could be suggested that better pedonnance of the control surfices wouid be encotmtered.
Improvements that could be made in this study e w g e primarily in the area of the
wind tunnel velocity profiIe. It is believed tht the industrial type fàn used in the wind
tunnel is the Likely culprit in the non-uniform flow field. Another recommendation could
be made towards increasing the sensitivity of the s W gauges, as some of the data was
obtained in their lower range. At the tirne of the experiments these were the most
sensitive gauges available commerciaily, but newer versions may have ernerged since
that time.
This provides the necessary closure for this body of work, and lends optimism to
the realisation of a stable and controüable fiapping-wing MAV. Future research should
focus on the development of an actual flight vehicle testbed fiom which observations of
stability can be made. Appropriate modifications should then be applied based upon both
the observed flight characteristics of the vehicle and the conclusions dram Eom this
thesis. Work should also continue with the 3 dimensional simulation code developed at
SRI using a sirnilar case snidy analysis perforrned in this document. Its conclusions
should be compared to those of the 2 dimensional code to see what descrepancies may
exist between them Outputs of both programs under similar initial conditions should in
the very least be cotnplimentary.
In closing, it is sincerely hoped that the hdings herein wiii benefit the îùture
research and development of such an extraordmuy airçraft,
Chapter 8: REFERENCES AND BIBLIOGRAPHY
8.1 References
111 Bilyk, Derek. The Developnient of Flapping Wings for a Hovering Micro Air Vehicle. University of Toronto Institute for Aerospace Studies; Master of Applied Science Degree, 2000.
121 El-Khati'b, Jasmine. Flow Msu(11isation for a Micro Air Vehicle. University of Toronto Institute for Aerospace Studies; Master of Applied Science Degree, 2000.
P l Loewen, Dave C. An Experimenfd Investigation of Closely Spaced Membrane Aifloils. University of Toronto; Bachelor of Applied Science Degree, 199 1.
141 'Mode1 6000: Planar-Beam Force Seasor." Advanceci Custom Sensors Inc. 19 July 2000. < h t t p ~ I ~ ~ ~ . a c s e n s o r . w m l P a g ~ o d e l - 6 0 0 .
FI Thompson, William T. and Dahleh, Marie Diiion Throry of Vibration ivith Applications. 5' ed. Upper Saddle River, NJ: Prentice Hali, 1998, p. 27 - 3 1.
8.2 Bibliography
SRI International, UTIAS. FIapping Wing Proplsion Udng EIectrosiriclive Polymer Artifcial Muscle Actuators: Semi-And Report. 14 December 2000.
Fiuke NeetDAQ Data Logger User's Marnral, Fidce Corporation, 1995.
Anderson, John D., Ir. Introduction ro Fïighr. 3d ed, New Yotk: McGraw-Hill, 1989.
Appendix A: FORCE BALANCE DESIGN SPECIFICATIONS
As mentioned in the main body of the thesis, the balance dimensions were
selected rather arbitrarily, with the provision for adjustment should the need arise. The
essential design was a scaled dom version derived fiom an existing cantilevered balance
residing in the UnAS subsonic aerodynamics lab. The CAD drawings that depict tbe
overail dimensions and layout of the design are included on the foiiowing page. Al1
measurements show are in mm and the drawings are not to scale.
Appendix B: FORCE BALANCE CALIBRATION DATA
in order to acquire an o v e d assessrnent of the force balance performance, a
series of test cases were perfonned. These wodd determine if there were any adverse
effects on the gauges, or their behavior would change under d i i e n t load combinations.
The foilowing appendix is divided into sections correspondhg to each different test case.
Each section wiii include a description and summary of the caiiiration results.
CASE 1 : lndependent Gauge Calibration
Recalling the balance design, it is worth repeating here that there was an
assumption that the loadhg dong the longitudinal (2-axis) direction was divided 50/50
between the gauges #I and #2. This particular test case investigates this hypothesis by
ailowing a comparison between the gauge k-values (or dopes) before and d e r they were
permanently attached.
Two AC Sensor Mode1 6000 pianar beam sensors were used to create one @el
beam balance. As mentioned m the main body of the text, this particular orientation
compensated for load misaiignments, allowhg for the measutement of pure forces ody.
For simplicity, the author will refer to a parailei beam configuration as a single unit and
cal1 it a gauge. Hence, three of these "gauges" were mounted on the balance. Prier to
their final attachment to the lower plate, these gauge units were individually dihatecl
through the appiication of k w w n masses and the recordmg their voltage outputs. This
was performed by using a Keithley 177 Microvott DMM (digital multimeter) m
conjunction with a Sorensen DC power supply. As per the mamhamds specincations,
the appiied input voltage was 10 VOL. This value was monitored both before and afler
the ca i i i i on tests to ensure that it did w t drift appreciably during the readings. Proper
strain relief of the leads extending fiom the gauges was crucial in order to obtain
repeatabie and accurate resuits. This was done by clamping them M y to a rigid surface
so that they could not move during caliibration.
Known masses were applied to each gauge by using an attached looped thread.
The thread passed over a pulley and ended in a hook onto which these masses were hung
in the positive axis direction ody. Aii gauges exhibiteci Iinear behavior, with the dope
curves shown below. The k-values correspond to the siope of the trend lime equations
show on the graphs.
CASE 2: Complete System Calibration - Pure Fortes
Once the gauges were permanently attached to the lower plate of the balance, the
next step was to determine if the k-values of the gauges had changed sisniflcantly. The
fist load condition to determine these new k-values was an application of a pure load
dong a single axis only (in both positive and negative directions). Loads applied dong
the z-axis were considered to be shared equally between gauges #1 and #2. Under these
conditions another set of k-values were determined. Linearity with respect to Ioading was
tetained, with gauge #1 yielding a k-value of 0.0541 mVlg in the positive z direction and
0.0563 mV/g in the aegative z direction. Gauge #2 produceci a value of 0.0871 mVlg and
0.0820 in the positive and negative z directions respectively. Finally, Gauge #3 gave k-
values of 0.516 mV/g (positive x direction) and 0.0528 mV/g (negative x direction).
Upon cornparison between these and the previous independent tests, one can immediately
deduce that there was üttle effect on the gauge slopes due to their final attachment. Thus,
the initial assumption of equaiiy shared Ioading between gauges #1 and #2 was
considered vaiid. It is also worth noting that tbere was no crosstalk observed. The
hiIo wing graph iIlustrates the results of this test.
CASE 3: Complete System Calibratian - Pure Moment
Wi the initial set of k-values determineci h m case 2 above, it was then desired
to apply a variety of load conditions to see if any appreciable change occurred in the
gauge slopes. Ideally, these values should not change- These extra tests however, would
shed light on the overd behavior of the balance. This was important, as the loading
conditions expected durhg actual testing would be quite variable.
By means of a small arm attached to a vertical post extending fiom the lower tray,
a pure moment was applied to the balance. Loads were attached at various positions
dong the length of the arm to aIlow variation in the moment's magnitude. The results of
the test are shown in table B-1 below. An error anaIysis was performed using the k-dues
iiom case 2, and revealed most of the mors did not stray fat. kom the 5% value. In
addition, k-values could be derived fiom the test condition using a system of equations as
follows: let AVl = change in voltage of gauge #1 between loaded and unloaded
conditions, and similarly let AV2 = change in voltage of gauge #2. ktine xl and xt as
AVIIm and AV2/m respectively, with m king the apptied mas. The distance separatiag
gauges #1 and #2 is labeled d. Together, this data reduces to a system of two equations
(using the previously d e k d sign conventions):
AVi xi + AV2 xz = -m (1)
-d/2 * (AVI xi - AVz x2} = m (2)
These are easily solved hr simuhaneously for the imknowns XI and x2, which in
turu are the inverse of the gauge k-vahies. These equations were used for each moment
apptied in the test. Hence, since seven different moments were used, seven different
dopes could be deduced. An average of these values reveaied that the k-value for gauge
#1 was 0.0546 mV/g and gauge #2 was 0.0897 mV/g. On account of the load orientation,
no component of the force occurred in the x direction, and therefore no k-due could be
deduced for gauge #3. One final comment should be made on the caiculated slope of
gauge #2 for the instance where the applied moment was 322.4 g c m This number m e d
out to be 0.2238 mV/g, which was decidedly out of sync with the rest of the calculated
slopes for gauge #2. It was therefore not included in calculating the overail average and
considered an anomaly due to the hi& moment Ioading on the gauge. It was t d y
unlikely that such a large moment would be encomtemi in practice.
Table RI: Case 3 Resu1t.s / Emr Ana&sis
CASE 4: Complete System Calibration - Corn bined X and Z Forces
A combination of x a d z forces was obtained by aliping an applied load dong
an angle to the center of the lower plate. The appIied force was simpIy reduced mto its
component vectors m order to detemine the forces aiong these orthogonal axes. As
before, use of the initiai k-dues h m case 2 gave percent mors m the m g e of 5%, with
poorer performance occurring only in the extremely tight loading condition. The test
resuits are depicted below.
1 62.2 1 46.8 -21.7 -23.3 64.3 3.3 -45.0 4.0
Table û-2: Case 4 Resu1i.v / Error Ana&sis
Simply plotting the gauge outputs vs. mass showed the k-values to be 0.0524
mV/g for gauge #1,O.O8 15 mVlg for gauge #2 and 0.0537 mV1g for gauge #3. These are
shown in the fo Uowing graphs.
OZ- 92-
CASE 4 Combined X & Z Forces Callbration - Gauge #3
CASE 5: Complete System Calibration - Combined X, Z Forces and Moment
This test was a dupliçation of case 4 except the load was off center of the plate by
use of an arm attachment. Thus a twisthg force was added to the x and z force
components. The results are summarized below, and reveal errors (using k-values fiom
case 2) no greater than 5%. Again, average k-values were bund to be 0.0554 mV/g for
gauge #1,0.0829 mV/g for gauge #2 and 0.0539 mVIg for gauge #3.
appked moment. Table B-3: Case 5 Results / E m r Ana&sis
120.9
161.2
201.5
241 -8
282.8
322.4
' l n al1 cases the applied mass was -40.4 g. Placement along the bar uttachment uilowed variation in the
-24.2
-24.2
-24.2
-24.2
-24.2
-24.2
32.2
32.2
32.2
32.2
32.2
32.2
33.8
33.6
33.4
33.6
33.8
34.0
4.9
4.3
3.7
4.3
4.9
5.5
-24.2
-24.7
-24.8
-24.4
-24.5
-24.4
0.0
1.8
2.3
0.7
1.0
0.7
119.2
159.0
195.8
234.0
268.4
298.0
-1 -4
-1.4
-2.9
-3-4
-5.4
8.2
Table B-4: Case 5 CuIcuItated Slopesfor Gauges #1, #2 and #3
Average:
*In cuch i m n c e rhe applied Z Mas WPT -242g und the applied X M m was 32.2g. as indicared in Table B-3. T h e vaiues 0.0688 and -L?l.(1 were not includedin culculution of the average.
0.0562
0.0655
0.0549
0.0538
0.0554
-0.2144
0.0681
0.0700
0.0688
0.08#1**
0.534
0.537
0.0540
0.0543
0.0639
Appendix C: WlND TUNNEL MLOCITY PROFILES
A s d l open-section wind tunnel was used to conduct al1 tests for this research
work. As mentioned in the main body of the text, Mr. Darcy Allison performed
prelirninary calidnation tests, with the remainder performed by the author. A drawiug of
the tunnel is included below (not to d e ) .
Fan lnlet with Circular / crosssection
Side Yimt Cross-Secîion Wind Tunnel
Wind Tunnel Exiî
Tunnel Fan Voltage = 51 V, 1.5 inches from Tunnel Exit
*Each station helght was separated by a distance of 2, M om (7 in.) and each station wldth by 3.81 cm (1.5 in.).
*Esch station a distance of station width
Tunnel Fan Voltage = 56 V, 2 inches from Tunnel Exit
rht was separated by cm (1 in.) and each 81 cm (1.5 In.).
Appendix D: EXPERIMEHTAL RESULTS
This appendix contains a numerid summary of the resuits from the wind tunnel
tests performed on both the BAT- 12 w i q s and tails, as descri'bed in the main body of this
document. Each table gives the data accumulateci for one advance ratio. Each table is
Further divided into sections for IateraVlongitudinal forces and y moments. As each
advance ratio was repeated h e e times, a final column correspondhg to the test average
was used to determine the overd trend of each force or moment vs. angle to the
crossfiow. Following these tables is another table giving the amplified data deduced for
the z forces. Finaily, the 1st table contains the CJCD data obtained for the two tail
designs.
Table D-1: J = 0.19 Wing Test Data
Table D-2: J = 0.55 Wing Test Data
Table D-4: J = 0.735 Wing Test Data
Table D-7: Tail 2 Lift and Drag Data
Appendix E: 2-0 SIMULATION PROGRAM
The foiiowing pages contain the simulation codes used to produce the resuits
descri i in Chapter 6. The fkt code, titied "SlMMAV3", models the MAV in the
absence of any tail surfaces. The second program, named "SiMMAVTAIL3", is an
extension of the former code, but with the presence of a tail surface hcorporated into the
routine. Both programs were completed using the MATLAB version 5.2 propmming
language. The author bas doue his best to provide as much commenting as possible in
order to reflect the logic used w i t h each program.
-A Simple 2D MAV S i r n i a c i o n , HO TAIL -Yarz EiacMaster, Yarch 2001
- D - lookup t a b l e f o r wing d r a g valries: eacn column corresponds CO J's - cf O,~I .LC ,9 .55 ,0 . i 5€ and 0 . 7 3 5 and - éash row f o r ang le s O t h r u t o 180 degrees - 7 - lookug t a n l e as above, cxcept f o r t h e t h r u s t vaiues of t h e wings
- :heta - angu ia r d i sp iacement from Che vertical, degrees - - - a l i i t u d e (metzes!, x - i a t e r a l d i sp lacement ime t r e s i , a lpha -
- angu la r a c c e l . i r a d / s a 2 i cmeqa - anguhar v e l o c i t y Ired/si , xao t - l a t e r a l v e l o c i t y !rn/s!, xdd . - l a ~ e r a : accel (n /s"Z ' i
- zd - v e r t i c a l v e l o c i t y ! m / s ) , x!d - v e c t i c a l accel. [rn/sa2) - t h e t a v - a n g l e t h e f r e e strearn makss w i th t h e horz . due :O boch z r x - v e i c c t l e s ac l .e. of r i n g s - L - d i s r a n c e fzom che c.g. t o t h e 1.e. of Che wings im:
in - nass of Che v e h i c l s : k g ] , ï y - inass rnomen: of i n c e r i a anout t h e y - a x i s ,kg m"2i rho - air i s n s i q : k g / m " 3 ) , 5 = t a i l s u r f a c e a r e a (mA2) , cl - d i s c - damping c c e f f i c i e ~ t :!cg n A 2 / s )
- x v t - t c t a l l a t e r a l v e l c c i t y a t 1 . e . o f t h e wings due t o : rans la t ion - and r o t a t i o n :m/si - z v t - t o t a l v e r t i c a i v e i c c i t ÿ a t 1 .e . o f t h e wings due CO ::anslation - and r o t a t i o n Im/s I c h e t a f s - maqnitude o f iree Stream a n g l e t o t h e l e a c i n g edge of t h e - wlngs x r t t h e v e r t i c a l !deqreesi
+ Thr - t h r u s t from wings :gi , Drag - d r a g frcm wings (g i - . . . - . . . . . . . . . . . -* . - -* . . **" .* ' . - -*----- - . . -~.--*--- . - - - . - . . . - -* . .~-*----- .**------ - . - . .
-popu ia t e lookup =abLes D = [1.98 0.3 -.11 .89 0;7.85 6.98 4.93 3.40 0;15.88 12.94 10.10 6.02 0;23.81 18.4 14.46 7.7 0;29.58 21.79 17.36 9.79 0;33.23 24.6 19.33 10.67 0;35.6 26.4 19.33 10.67 0;35.6 26.4 20.96 12.01 0;37.03 27.01 21.41 12.56 O; ... 35.59 25.81 21.1 11.13 0;34.74 25.28 20.64 9.21 0;34.02 24.66 20.41 10.08 0;30.92 22.52 17.92 10.27 0;27.35 18.07 16.08 8.61 0;24.24 15.25 12.53 5.41 0;20.63 12.23 9.43 3.12 0;16.06 8.63 5.14 0.74 0;10.22 5.07 2.39 -.O6 0;4.99 3.29 1.06 -.6 0;-1.56 -95 1.43 -.66 O];
-specify initial conditions t h e t a = ( 2 / 1 8 0 * p i ) ; z = 0 ; x = O ; x d o t = O . O ; a l p h a = O ; xdd=0.O;thetaplot(1)=theta/pif18O;xdotplot(1)=xdot;alphaplot(1)=alpha; dragplot (l)=O;thrplot (lI=SO;tplot (l)=O;xpIot (l)=O;zplot (l)=O; omegaplot(l)=0;velplot(l)=0;thetav(l)=0; cl = -0.0017345:
-orner variables l=O.OiS; dt=0 -001; m=O -050; Iy=0.000031;
-üse loo~üp tobles to evaluatè the curreat D, T, using xdot as -reférence veiocit? and reference rheta and assuming al1 flapping done -at 4gRr
-designate Che velocities "heaaings" for eacn marrix column in lookup - raDles vl=7.04;v2=6,28;v3=5.24;v4=l.87;v5=0; thetav(1) =O; -beu:n los-ing chrough tirne steps tplot (1) =O;i=l;q=O; -Iccç beçins 3t dt, and al1 variables ger updated based on avg. -acce le ra t ions -thr=ugh rbat Fncrenentai rime step. The values a t the end of 3 stec Srepresênt Lhe vaiues sf zhat v ü r i a b l e at ~ n d t tirne at. fsr t=dt:dt: 15,
i=i+l; tplot (il =t;
- t c za i x velocity at leaaing edge of wings xv t (il =xdoc+omegaf 1-0s Cabs (thetal 1 ; -cotai = -~el=city at leading edge of wings zvt (il =zd+omega*l*sin (abs ( theta) ) ;
-deremne thetav, cne angie zhe free Stream makes with tne ho:z. -due to 50th vert & korz translations ( x v t and zvt; F f zvt(i)==O & xvt(i)==O,
-prevent undefined 0 / 0 when using arctan function! thetav(i) =O;
eise thetav(i) =atan (zvt (il /xvt (i) 1 ;
end;
if z v t (i) <=O if .wt(i)<O,
if theta<=O, thetafs=pi/Z+theta+thetav(i};
eise thetafs=pi/2+thetav(i) +theta;
enci; end;
end; if zvt(i)C=O
if xvt(i)>O, if theta>=O,
thetafs=pi/2-theta-thetav(i);
end; end;
and; if zvt(il>=O
if xvt(i)>O, if theta>=O,
sise thetafs=pi/2-thetav(i1 -theta;
end; end;
end;
-case wnerr wings are risinq purel? v e r t i c a i i ÿ
etc; if rvt(i)<Q,
-case of purs descent of wings
end; end;
~f zvt(i)< 0.00001, ~f zvtti) >-0.00001,
if xvt (i)<0.00001, if xut(i) >-O ,00001,
thetafs=O; q=q+ 1 ;
ena; end;
ènc ; ena; tfs (i) =thetafs*lEO/pi; -thetafs is the H?GtITUDE of the angle between the free stream -and the vertical wrt 1.e. of wings,
*determine uhat range the velocity falls under - columnl is always -the higher veLocity calumn Vel=(xvt(i) "2+~vt(i)^2)~.S; velplot (il =Vel; if Vel<vl h Vel>=v2,
columnl=l;column2=2;vlow=v2;vhigh=vl; end;
if Vel<v2 & Vel>=v3, co1-1=2 ; column2=3 ; v l o m 3 ; vhigh-2;
end; if Vel<v3 & Vel>-4,
col~l=3;column2=4;vlow=v4;vhigh=v3; end; if Vel<v4 & Veb-5,
columnl=4;column2=5;vlow=v5;vhigh=v4; end;
-determirie w n i c n :ou the angle fa115 under - rowi is always the -higher angie row angle = abs(theta£s/pi+l801; ang ( i) =angle; :f anglecl0 & anqle>=O
rowl=2; row2=l; end; if angle<20 & angle>-10
rowl-3; row2=2; end; rf anglec30 6 angle>=20
rowl=4 ; row2=3; eria; if anglec40 & anqle>=30
rowl=5; row2=4 ; ènd; :f angle<50 & angle>=40
rowl=6; r0~2=5 ; md; 15 anglec60 & angle>=50
rowl=7; row2-6; ena; . +
:: anqle<70 & anqle>=60 rowl=8 ; row2=7 ;
end; if angleC80 & angle>=70
rowl=9 ; row2=8 ; end; i f angle<90 & angle>=80
rowl=lO;row2=9; end; if angle<100 & angle>=90
rowl=ll; row2=l0; end; if angle<llO h anqle>=lOO
rowl=12; row2=ll; end; if angle<120 & angle>=llO
rowl=l3;row2=12; end; if anglecl30 & angle>=l20
rowl=L4;row2=13; end; i f anglecl40 & angle>=130
rowl=15;rou2=14; end; if anglecl50 & angle>=L40
rowl=16; row2=lS ; end; if angle<l60 6 angle>=150
rowl=lf;row2=16; end; if anglecl70 h anqle>=l60
rowl-18 ; row2=17 ; and; if anqle<l80 & angle>=170
rawl=19;row2=18; end; anglelo- (row2-1) *l0; - s c a r ~ double i n t e r p o l a c l a n f o r t h r u s t - ' J ~ ~ O C ~ L Y :
-ve l sc icy pair ac l o w e r angle VLowl=T (row2, columnl) ;VHighl=T (row2, column2) ; -velocity- pair for higher a n q l e vLow2=T(rowl,columnl);VHighS=T(roi~l,~0I~mn2); - i n t a r p o i a c i wich t h e s e p o i n t s a t t h e currenc u e l c c i t y ThrLow~ngle=~~owl-~Highl)/(vhigh-vlaw)~(Ve~-vlow~+VHigh~; ThrHiqhAnqle=(VLow2-VHiqh2}/(vhigh-vlow)*(VeL-v10w)+VHigh2;
- inte:coiate =o ?et a: the desired angle Thr = (ThrHiqhAngle-ThrLUwAnqle] /IO' (angle-anglelow) +ThrLowAngle; thrplot [ i l =Thr; -F.epeat rhe same cCinq fo r getting Che ara9 - v e i o c i t : ~ ? a i r at i auer angit VLowT=D (row2, coLumn1) ;VHighl=D (row2, c o l u 2 ) ; - v e l o c i t y pair for higner ang le VLow2-D ( rowL, columnl);uHigh2=D ( rowl, c o l u 2 1 ; -icrerpoLace -&th these poincs a t t h e current v d o c i t y ~ r a g ~ o w ~ n g l e - (VLowl-VHighl) / (vhigh-vlow) (Vel-vlow) +VHighl; DragHighAngle= (VLow2-VHighZ] / (vhigh-vlow) el-vlow] tVHigh2;
- i n t e r p o l a t e t a get Drag a t :ne desired angle Drag =(DragKigh~nqLe-Drag~ow~ngle]/10*(angle-anglelow)+DragLowRngle;
-evaiüate angu la r a c r e l e r a t i o n , ensuring Drag f o r c e always points - 0 p p o s i t e to Che a i r e c t i o n of zhe velocity thetaold=theta; ornegaold=amega; draqpLot (i 1 =Drag;
- t h e or igif ia l alpha at the beginning of the current rime interval alphaold=alpha; discdampl = cI*omegaold;
if x u t ( i ) > = O , a lpha = -Drag/L000t9.81+L/Iy+discdampl/Iy;
tlse alpha = ~rag /1000*9 .81*1 / Iy+d i scd~ l / Iy ;
enà;
alphaplot ( i l =alpha;
- in tegra te alpha t w i c e t a get angular p o s i t i o n -once:
-total omega at the END of current time interval
omega = alphafdt+omegaold; omegaplot ( i =amega; - twice: -total theta at the END of current time interval theta = omegafdt+thetaold; thetaplot(i)=theta/piW; if thetaplot (il >80,
fprintf ( ' 1 AM OUT OF CONTROL ! ! ! ' ) end; -evaluate vertrcal accieration zcid, and incegrace twice for -"aLticüde" postition zddold=zdd; if xvt (i) >=O,
if thetaold>=O, zdd=Thr/1000*9.81/m*cos (abs (thetaold)
9.81+Drag/1000*9.8i/mfsin(abs(thetaold) l g a else
zdd=Thr/1000*9.81/m*cos(abs(thetaold) Draq/1000*9.81/nfsin(abs(thetaold))+omeqa+xdot
m u ; ei se
r f thetaold>=O, zdd=~hr/lO00*9.81/m*cos (abs (thetaoid) ) -9 -81-
Draq/1000*9.81/mfsin(abs(thetao1d) )+omega*xdot; else
zdd=~hr/1000*9.8i/m*cos(abs (thetaold) ) - 9.81+~rag/1000*9.81/m+sin(abs[thetaold))+omega+xdot;
and; end; -integrate twice zdold=zd; rd = zdd*dt+zdold;
zold=z; z = zd*dt+zold; zplot (il =z; zdplot (i) =zd; -evaluate horizontal accieration xdd, integrate once for x velocity, -then twice for horizontal position xddold=xdd; -use appropriate equation aepending upon vhich side of the theta -equals O nark. -mut use theta at s t a r t of this interval to update -xdot if thetaold>=O,
if xvt(i)>O, xdd=Thr*9.81/1000/m*sin tabs (thetaold) ) -
Drag/l000*9.81/m*cos(abs(thetaold~~-omeg*zd; else
xdd~hr*9.81/1000/mfsin[abs (thetaold) 1 +D~ag/1000*9.8l/m*cos (ab s (thetaold) 1 -omega*zd;
end; else
if xvt(i) >O, xdd=-Drag/1000*9.81/m~os(abs(thetaold))-
Thr/1000*9.81/rn*sin(abs~thetao~d))-O-;
else xdd=Drag/1000*9.Bl/m*cos(abs(thetaold))-
Thr/1000+9,8l/m*sin(abs (thetaold) )-omega*zd; end;
end; xdotold=xdot; xdot=xddfdt+xdotold;~the xdot at the END cf this interval xdotplot (il =xdot; xold=x;
x = xdotcdt+xold; xplot (i) =x;
-ail variablès have now Seen updated. Laop.
end;
figure ( 1 ) ; plot(tplot,thetaploti;title('Theta vs. Time');
-A Simple ZD MnV Simula t ion , WITX TAIL :Marc NacMascer, March 2001
clear a l l ; . . . . . . . , '. 3F K R I - U L E S -;iL;--L;;;;;;;;-;;i-iL-i-L------ .....*.*..**....*-. . . , . . . . . . . .
- D - Lockup t a b l e f o r wing d r a g v a l u e s : each column cor responds CO S's - cf f i l ,Q . l9 ,~~ .55 ,O.E56 and 0 . 7 3 5 and each row f o r a n g l e s 0 th== t o l e 0
- - - : - ,ockup t a b l e a s &ove, except for ~ h e t h r u s t va lues of che wings - r n e t a - angu la r d i sp lacement Ercm t h e vertical, degrees . - - - al~itudc : m e t r e s ) , x - l a t e r a l dispiacernent irneczesl, alpha -
- anqu la r acce i . [ rad/ sa 2 1 - zmega - anqu ia r , r e l a c r t y ; r a a / s ! , xdot - L a t e r a l v e l a c i t y ! d s i , xdd . - l a c e r a l a c c e i !m/sA2 : - =c - . f e r z i ca l v e l o c i r y ;rn/si , zCd - v e r t i c a l acce l . im/SmSi - xddwing - L a t e r s i a c c e l . cont r ibu tec i by~ wings ; m / s A 2 : - zadwing - v e r ~ l c a l a c c e l . c o n t r i b u t e d by wings (m/s'2i - :<ad:ail - L a t e r a l a c c e i . z o n t r i b u t e a By c a i l im/sA2i - z d d t a i l - verticai accei. c o n t r i b u t e d by t a i i Irn/sAS> - awing - u i q zon t r ibuc t an t o t o t a i anqu'ar a c c e l e r a t i o n about rhe
- a c a i l - r a i l r onc r ibuc ion co t o c a l anqu ia r a c c e l e r a t i o n about che . -.q. - ' rad/s"2:
- zk.ecaq; - a q i t :kit free s t r e m ;riakts witb =ha nort . due Co bcth h ; ::i - v e l o c t i e s ac 1.e. î f winqs - I - a i s t a n c e fzom the c.g. to zhe l . e . of che wings i m ) - in - zass cf =he v e h l c l e [ k g ! , Iy - mass marnent of i n t e r i a about t h e y - azis i kg m'2 - L? - d i s t a n c e from the r .g . t o :he q u a r t e r chord o f Che t a i l ;rit) - rhc - al : c e n s i t y tkg/mA3), 5 = tail s ü r f a c e area ( n " Z l , c l - cüsc - darging =aef f i c i e n t kq m A Z / s - xvc - t o t a i l a t e r a l v e l o c i t y a t 1.e. of che wings due t o :ransla:ion - and r o t a c i o n ids i - z t c ~ - t o t a l v e r t i c a l v e l o c i t y a t i , e . of t h e wings due to t r a n s l a t i o n - and rocaticn ' d s i - xvtail - z u t a l l a t e r a l velocity a t c a i l q u a r t e r chord due to - z r a n s l a c i o n and rotaticn ! m / s ) - s v t a i L - c o r a i v e r t i c a l v e l o c i t y a t t a i l q u a r t e r chord due t o - translation and r o t a t i o n I m / s ) - r h e t a f s - rnagniïude o f f r e e Stream angle t o the i e a d i n g edge of t h e - w i n g s wrc the v e r t i c a l ( d e g r e e s ; - the tavta l l - angle t h e f r e e stream makes wi th the horz. due :O b o t h z - S x v e l o c = i e s a t - c /4 af t a i i Idegrees) - c h e t a f s t a i i - .magnitude of free stream a n g l e t o zhe parter cnoca o f - c a i l w r t t h e v e r t i c a l idegreesi - L t a i l - i i f t due t a tail (NI, D t a i i - drag due ta tail !Ni , Thr -
t h r u s t f rom winqs ! gj - 3raq - draq from winqs (g) ~ ~ ~ ; ; ; ; ; ; ~ ; ~ i ; ; ~ ; L ~ ~ ~ i L ~ i ~ i > 2 > ~ ~ i ~ ~ G G + . G ~ ~ ~ + ~ ~ ~ ~ ~ ~ & A ~ ~ 2 ~ ~ & ~ ~ ~ ~ & ~ ~ ~ ~ < ~ ~ ~ ~ . . . . . . . . , . . . . . -~ . .
-pcpu la t a Lookup tables f o r L i f t and Drag a a c a D = [1,98 0.3 -.Il -89 0;7.85 6.98 4-93 3 - 4 0 0;15.88 12.94 10.18 6.02 0;23.81 18.4 14.46 7.7 0;29.58 21.79 17.36 9.79 0~33.23 24.6 19-33
10.67 0;35.6 26.4 19.33 10.67 0;35.6 26.4 20.96 12-01 0;37.03 27.01 21.41 12.56 O; ... 35.59 25.81 21.1 11.13 0;34.74 25.28 20.64 9-21 0;34.02 24.66 20.41 10.08 0;30.92 22.52 17.92 10.27 0;27,35 18.07 16.08 8.61 0;24.24 15-25 12.53 5.41 0;20.63 12.23 9.43 3.12 0;16.06 8.63 5.14 0.74 0;10,22 5.07 2.39 -.O6 0;4,99 3.29 1.06 -.6 0;-1-56 -95 1.43 -.66 O];
-spec: f y ~nit;ai tondicions theta=(2/180+pi);z=0;x=O;xdot=O.O;alpha=O.OO;omega=O;zdd=O.OOOOO;zd=- 0.00;xdd=0.0;thetaplot(l)=theta/pi*l8O;xdotplot(l~=xdot; alphaplot (1) =alpha;dragplot (1) =O; thrplot (1) =SO; tplot (1) =O; xplot(l)=0;zplot(l)=O;omegaplot(l~=O;velplot~l)=O;thetav~1~=O; xddwing(1) =O; zddwing (1) =O;xddtail(l) =O;zddtail(l) =O;awing(l) =O; atail(l)=O; -othsr v a r i a 8 i . é ~ l=O.O75; dt=0.001; m=0.050; Iy=0.000031; 12=0.125; -distance t o tail . c i 4 rho=l,225;S=0,007; cl=-0.0017345; -use Lockup tables ta -valuate the surrent E, T, using xdot as .refsrence velocity and reference theta -and assurmng al1 flapping aone at 40Hz
.designate che velocities "headings" for each matrix ealumn in lookup -tables v1=7.04;~2=6.28;~3=5.24;~4=1.87;~5=0; thetav ( 11 =O; -begin Looping through ~ i m e ateps tplot (1) =O;i=l; -lccp ~egins at dt, ana aii variables get updated based on avg. -acceiera tions -thzough chat inczemental time step. The values at the end of a step -repzesent the -values of that variable at that time dt. for t=dt:dt:l5,
i=i+l;tplot (il =t; ;total x velocity at leading edge of wings xvt(i)=xdot+omega*l*cos(abs(theta)); rtotal z velocity at leading edqe of wings zvt (il =zd+omega*l*sin (abs (theta) ) ; 'determine thetav, the angle the free strevn makes with the horz. -due to both vert h horz translations (xvt and zv t ) if zvt(i)=O,
if xvt(i)=O, thetaw(i)=O;iprevent undefined 0/0 when using arctan function!
end; eise
thetav(i1 =atan(zvt (il /xvt(i) ) ; end ;
ena; ena; if zvt (i) <O
~f xvt (il > O I rf theta>=O,
anc; snc;
end; 15 zvt (i) >O
if xvt ( i l >O, ;f theta>=O,
t h e t a f s = p i / Z - t h e t a - t h e t a v 0 ; eise
thetafs=pi/2-thetav(î1-theta; ana;
end; if zvt(i) >O
if xvt(i)<O, F E thetat=O,
thetafs=pi/S+theta+thetav(i); else
thetafs=pi/2+theta+thetav(i) ; end;
-case where wings are rising purely vertically
end; if zvt (il <O,
-case of pure descent of wings
end; end; if zvt(i) =r O,
if xvt(i)-=O, thetafs=pi/2+theta;
end; end; if zvt(i)=O,
E-II
rowl=12; row2=ll; enà; if angle<l20 & angle>=llO
rowl=l3; row2=l2; end; if angle<l30 & angLe>=120
rowl=14;row2=13; end; if anglecl40 & angle>=130
rowl=1S;row2=14; end; ~f anqle<150 & angie>=140
rowl=16;row2=15; enà; if angle<l60 & angle>=150
rowl=l7; row2=16; end ; if angle<170 & anglo>=160
rowl=18; row2=i7; end; . - iZ angle<l80 & angie>=l70
rowl=19;row2=18; rnà; anglelow= (row2-LI +IO ;
-star: i o c b l e interpolation for thrus: ... , a ? - A L c i t y : - -veiccity p a r at Lower angle VLowl=T ( raw2, columnll ;VHighl=T (row2, column2) ; - v e l c c i t ; ~ p u r for higher angle Vtow2=T (rowl ,columnl) ;VHigh2=T (rowl, colrimn2) ; -interpolate with chese points a t the current velocicy ThrLowAnqle= (VLowl-VHighl) / (vhigh-vlow) + (Vel-vlow) +VHighl; ThrHighAngle= (VLow2-VHigh2) / (vhigh-vlow) *(Vel-vlow) +Vkiigh2;
-interpoiate =a get at the desired angle Thr = (ThrKighAngle-ThrLowAngle)/l0*(angle-anglelow)+ThrLowAngle; thrplot (i ) =Thr; -Repeat the same thing for getting the drag -veioci=ÿ gai1 at lower angle VLowl=D(row2,columl);VEIighl=D(row2,column2); -veiocity ?ai= fcr higher angle VLow2=D ( rowl , c o l m l ) ;VHigh2=D (rowl, colu.1~2 1 ; -interpoiate with these points at the eurent velocity DragLowAngle= (VLuwl-VHighl ) / (vhigh-vlow) Y Vel-vlow) +VHighl ; DragHighAngle=(VLow2-VHigh2) /(vhigh-vlowl *(Vel-vlow)+Wigh2;
-inteqaiate to get Drag at the desired angle Drag = (DragHighAngle-DragLowAngle)/lO*(angle-
anglelow) +DraglowAngle;
.~ ~ . . . . . . . . . . . . . . . . . ......... ---..-.*.---**.-.*--*- TAIL FORCES . . . . - . . . . . . . . . . . .
. . . . . . . . . . -1 ' - ' - - - - . - - . - " - - - - - . - 7 f f - . . . . . . . . . . . . . .
;total :< velocity at c/4 of tail xvtail (i) =xdot+omega*12*cos (abs (theta] ) ; -total z velocity at c / 4 of tail zvtail (i) =zd+omega*l2*sin (abs (theta) ) ;
determine cnetamaii, cne angie cne free scream maices witn the -horz. due to both vert h horz translations ixvttail and zvtail) if zvtail (il ==O & xvtarl (i) =O,
-prevent undefinea 0 / 0 when using a r c t a n function! thetavtail (il =O;
else thetavtail (i) =atan (zvtail ( i l /xvtail (i) ) ;
end:
if zvtail (il <=O rf xvtail(i)<=O,
if theta<=O, thetafstail=pi/2+theta+thetavtail(i);
alse thetafstail=pi/2+thetavtail(i)+theta;
nna; end;
and; if zvtail (i) <=O
~f xvtail (i) >=O, rf theta>=O,
thetafstail=pi/2-theta-thetavtailti); elsé
thetafstail=pi/2-thetavtaiL(i1-theta; end;
a d ; cna; F f zvrail( i) >=O
~f xvtail(i)>=O, r f theta>=O,
thetafstail=pi/2-theta-thetavtail(i); êISS
thetafstail=pi/2-thetavtailcil-theta; ma;
ena; ma; if zvtail( i) >=O
rf mail(i)c=O, if theta<=O,
thetafstail=pi/2+t~eta+thetavtaii(i); else
t h e t a f s t a i l = p i / 2 + t h e t a f t h ~ f t a v t a i l ( i ) ; end;
2r.d; end; rf xv-tail(i) == 0,
if zvtail (i) >=O, -case where wings are rising purely vertically
thetafstail=theta; end; if zvtail(i)<=O,
thetafstail=pi-abs(theta1;-case of pure descent of wings end;
end;
if xvtail (i) =O, thetafstail=O;
end; end; tfstail(i)=thetafstail; -thetahtail is the PlAGNITUDE of the tail's angle -ci arrack between the free stream velocity vector -and the ocdy-f ixed t-axis (longitudinal axis
-deré,nine magnitude of :ail free stream velocity Veltail=(xvtail (i) *2+zvtail(i) ̂2) ̂ .5; -calculate :he C L , cd of :ne tail at the free-stream angle anqletail=abs(thetafstail/pi+l80); cl = 3*10A (-6) +angletail^3-
,0008*angletai1''2+0.0487+angletai1+0.006l; cd = 2*lO"(-8)*angletailA4-7*lOn(-6)fangletailA3+O.OOO6*anqletailA2-
0.001*angletai1+.0984;
-caicula:e :oral drag and lift frcm :ail :in body-fixeci frame: Ltail=1/2*rho+Veltailn2*S*cl; Dtail=1/2*rho*VeLtailA2+S*cd; Ltl(i)=Ltail;Dtl(i)=Dtail:
-evalratt angular acreleration, ensuring 9raq farce alwzys pornts -opposiLè :O t n è direction o f the velocitÿ thetaold-theta; ûmegaold=omega; dragplot ( i ) =Drag;
-the original alpha a: the beginning of the current time interval alphaold=alpha; discdampl = cl*omegaold; -qet the zoncrinution of the wings to alpna if xvt (i)>=O,
awing(i1 = -Drag/1000+9.81+1/Iy+discdampl/Iy; eLse
awing(i1 = Drag/1000+9.81+1/Iy+discdampl/Iy; end;
-gec the contribution of the taii to alpha -CASES Wt?'ERE Thetaold < 0: - M E ia: ThetaCo, xvtail<O, zvtailC0, thetafstail<90deg. if thetaold<=O h xvtail(i)<=O h zvtail(i)<=O d thetafstail<=pi/2,
atail(i1 = (Dtailfsin (abs (thetafstail) ) +Ltail'cos (abs (thetafstail 1 ) 1 *12/1y;
end; -CASE Lb : ThetaCo, :rvtail<O, zvtail<O, thetafstail>=90deg. rf thetaold<=O h xvtail (il <=O h zvtail (i) <=O h thetafstail>pi/2,
atail (i) = (Dtail*cos (abs (thetafstail) -pi/2) - Ltail*sin ( a h (thetahtail) -pi/2) ) *12/Iy;
end; ;CASE 3: Theta<O, :wtail<O, zvtail>O, thetafstail<oOdeg. if thetaold<=O h xvtail (il <=O & zvtail (i) >O h thetaf stail<=pi/Z,
atail(i) = (Dtail*sin (abs (thetafstail) 1 +Ltailtcos (abs (thetafstail) ) 1 *12/1y;
enci; - Q U E Ja: Theta<Ü, .uvcaii>Ü, zmaii>ü, cnetafscaii<50aeg. if thetaold<=O & m a i l (i) >O h zvtail (il >=O & thetafstail<=pi/2,
atail (i) =(-Dtail*sin (abs (thetafstail) ) - Ltail*cos(abs(thetafstail)l )*l2/Iy;
end; -CASE 5B: Theta<O, xvcail>O, zvtaib0, thetafstail>=90deg. if thetaold<=O & xvtail (il >O & zvtail (il >=O h thetafstail>pi/2,
atail (i) =(-Dtail*sin (pi-abs (thetafstail) ) +Ltailfcos (pi- abs (thetafstail) ) ) *l2/Iy;
enci; -CASE 7 : Theta<O, xvtail>a, zvtail<O, thetaLstail>=90deg. if thetaold<=O & xvtail(i)>O & zvtail(i)<=O & thetafstail>pi/2,
atail(i1 =(-Dtailhsin(pi-abs(thetafstai1) )+Ltail*cos(pi- abs (thetafstail) ) ) * U / I y ;
end; -CASES WHERE TheCa > 0: -CASE 2: Theta>O, xv:ail<O, zlrtail<O, thetafstail>=90deg. L E thetaold>O & xvtail ( i l <=O & zvtail (i) <=O & thetafstail>pi/2,
atail(i) =(Dtailcsin(pi-absithetafstai1))-Ltail*cos(pi- abs(thetafstai1) l )*12/Iy;
end; -CASE 4a: The:a>O, :wtail<O, =vtaFf >O, thetafstail<90deq. if thetaold>O 6 xvtail (i) <=O & zvtail (i) >=O h thetafstail<pi/2,
atail(i) = (Dtail*sin(abs(thetafstail) l+Ltailtcos(abs(thetafstail)))*12/Iy;
-CAS= I b : The:a>O, xvtarl<O, z~rtaib0, :hetafstail>=90deg. L f thetaold>O & xvtail (i)<=O h rvtail(i) >O & thetafstail>pi/2,
atail (i) = (Dtail*cos (abs (thetafstail) -pi/2) - Ltail*sin(abs(thetafstail)-pi/2))*12/Iy;
end; - W E 5: Theta>O, :tmail>O, zviail>O, thetafstail<=?Odeg. ~f thetaold>O & xvtail(i) >O h zvtail(i) >O h thetafstailcpU2,
atail(i) = (Dtail*sin(abs(thetafstail) )+Ltail+cos(abs~thetafstail)))*l2/Iy;
ezd; -CASE Sa: Theta>O, xvtail>O, zvta i l<O, =hetafstail<90deg. if thetaold>O & xvtail (i) >O 6 zvtail (il <=O & thetafstail<pi/2,
atail (il = (-Dtail*sin(abs (thetafstaill 1 - Ltail*cos (abs (thetafstail) 1 *12/Iy;
enc; -CEE; ab: Theta>r), xvLail>O, zvtail<O, thetafstail>90deg. if thetaold>O h xvtail (i) >O & zvtail (i) <=O & thetafstail>pi/2,
atail(i1 = ( -Dta i l* s in (p i -abs ( the ta f s ta i l+cos (p i - itbs (thetafstail) ) ) *l2/Iy;
end;
alpha = awing(i) +atail (i) ;
-integrate alpha twice to get angular position -once : -total omega at the D I D of current time internai omega = (alpha *dt+omegaold) ; ; twice : -totai rheta at the END of current time inte,yal theta = omegafdt+thetaold;
thetaplot(i)=theta/pi*180; if ~hetaplot(i)MO,
fprintf('STOP!! - 1 AM OUT OF CONTROL!!!!'); end; -evaluate vertical accleration zdd, and integrate twice for ;"altitudew postitiûn zddold=zdd;xddold=xdd;
-get the wing =ontribution to the z acceleration if x v t (i) >=O,
rf thetaold>=O, zddwing (i) =Thr/l000+9.8l/m*cos (abs (thetaold) ) -
9.81+~raq/1000*9.81/m+sin(abs(thetaold))+omeqa*xdot; èise
zddwinq(i)=Thr/1000+9.8l/m~os(abs(thetaold))-9.81- ~raq/1000+9.81/m*sin(abs(thetaold)~+omeqa*xdot;
end; ?Ise
:f thetaold>=O, zddwinq (i) =Thr/Z000+9,8l/m+cos (abs (thetaold) ) -9.81-
~raq/1000+9.81/m+sin(abs(thetaold) l+omega*xdot; else
zddwing (i) =~hr/2000*9.8l/m*cos (abs (thetaold) 1 - 9.81+Drag/1000*9.81/mfsin(abs(thetaold))+omeqa*xdot;
end; end;
-aec cte winq toncributron t u the x acceleraticn -use appropriate equation depending upon which side of rhe theta -equals 11 ;nar~.
-must use theta at starc of r h i s mte~val co upaace xact 1: thetaold>=O,
:f xv t (il >O, xddwing(i) =~hr* 9.81/1000/m*sin(abs (thetaold) 1 -
Drag/l000*9.81/m*cos(abs(rhetao~d~ 1-omega*zd; eise
xddwinq(i)=~hr+9.81/1000/mfsin(abs(thetaold))+Draq/1000+9.8l/m +cos (abs (thetaold) 1 -omega*zd;
end; else
if xvt(i)>O, xddwinq (i) =-Drag/1000+9.81/mtcos (abs (thetaoid) ) -
Thr/1000*9.81/m*sin(abs(thetaoldH-omega*zd; else
xddwing(i) =Drag/1000*9.8l/m*cos (abs [thetaold) - Thr/1000+9,8l/m*sin (abs (thetaold) ) -omeqa+zd;
end; end;
-qet the tail's contribution to the z and x accelerations -CASES WHERE Theta < O: ;îASE ia: Theta<O, xvtail<O, zMail4, thetafstail<gOdeg. if thetaold<=O & xvtail(i)<=O d zvtail(i)<=O h thetafstail<pi/2,
zddtail (i) = (Dtailtsin (abs (thetaddl +abs (thetafstail) - pi/2) +Ltail+cos (abs (thetaold) +abs (thetafstaill -pi121 1 lm;
xddtail (i) = (Dtail*cos (abs (thetaoldl +ab9 (thetafstail) -pi/2)- Ltail'sin (abs (thetaold) +abs (thetahtail) -pi/2) ) /nt;
2nd; -O.SE Lb : ThétaCo, x-taii<O, rvtaii<O, thetafstail>=?0deg. if thetaold<=O & xvtail (i) <=O & zvtail(i) <=O & thetafstail>=pi/2,
zddtail(i)=(Ltail*sin(pi-abs(theEafstai1)- abs (thetaold) ) +Dtailfcas (pi-abs ( t h e t a f s t s (thetaold) 1 1 lm;
xddtail(i)=(Dtail*sin(pi-abs(thetafstail)-abs(thetaold))- Ltail'cos (pi-abs i thetafstail) -abs (thetaold) ) 1 lm;
end; -CASE 3: Theta<O, xvtail<O, zvtaib0, thetafstail<9Odeg. L£ thetaold<=O & xvtail (i) <=O & zvtail (il >=O & thetafstail<pi/2,
zddtail (i) = (Ltailfcos (pi/2-abs (thetaold) -ab9 [thetafstail) 1 - Dtail'sin (pi/2-abs (thetaold) -abs (thetafstail) 1 1 /m;
xddtail(i)=(Dtail*cos(pi/Z-abs(thetaold)- abs (thetafstail) ) +LtailCsin(pi/2-abs (thetaold) -abs (thetafstail) 1 1 lm;
end; - P W &-SE 5a: ThetacCl, xvtaib0, z7n;aii>0, chetafstail<9Odeg. if thetaold<=O & xvtail (i) >=O 6 zvtail (il >=O 6 thetafstail<=pi/2,
zddtail (i)=(ltail*sin(abs (thetafstail) -abs (thetaold) 1 - Dtai14cos (abs (thetafsrail) -abs (thetaold) 1 lm;
xddtail (i) =(-Dtailtsin (abs (thetafstail) -abs (thetaold) 1 - Lraîl*cns !abs (rhetafstail) -abs lthetaold) ) ) /in;
ena; -CASE 5 ~ : ThetaCo, :wtail>O, zvcail>O, :hetaf staii>=SOdeq. if thetaold<=O & xvtail (il >=O & zvtail (i) >=O & thetafstail>=pi/2,
zddtail (i) = (Ltailf cos (pi/2+abs (thetaoldl -abs (thetafstail) 1 - Dtail4sin(pi/2+abs(thetaold]-abs(theta£stail]))/m;
xddtail (i) =(-Dtail+cos (pi/2iabs(thetaoldl -abs(theta£stail) 1 - ~tail*cos(pi/2+abs(thetaold)-abs(thetafstai1)) )/m;
end; -CASE 7: Theta<[), xnaii>O, zvtail<O, thetafstaii>=?Odeg. if thetaold<=O & xvtail(i)>=O & zvtail(i)<=O & thetafstail>=pi/2,
zddtail (i) = (Ltail*cos (ab8 (thetafstailbpil2- abs(thetaold))+Dtail*sin(abs(thetafstail)-pi/2-abs(thetao~d)))/m;
xddtail (i) =(-Dtailfcos(abs (thetafstail) -pi/2- abs (thetaold) ) +Ltail'sin (abs {thetafstail) -pi/Z-abs (thetaold) ) ) /m;
end; -CASES -HERE Theta > 0: -CASE 7: ThetaiO, xvcail<O, zvtail<O, thetafstail>=gOdeg. if thetaoldHI & xvtail(i)€=O & zvtail(i)C=O & thetafstail>pi/2,
zddtail (i) = (Ltail'cos (abs (thetafstail) -abs (thetaold) - pi/2) +~tail+sin (abs (thetafstail) -abs (thetaold}-pi/2)) /m;
xddtail (i) = (Dtail*cos (abs (thetafstail) -abs (thetaold) -pi/2) - Ltail*sin (abs (thetafstail) -abs (thetaold) -pi/2) 1 /m;
tnci; -CASE 4a: TbetaiO, a v t a i i C 0 , Z~tail>0, thetafstail<gOaeg. if thetaold>O & xvtail(i)<=O & zvtail(iI>=O 6 thetafstail<pi/2,
zddtail (i) = (Ltaif *cos (abs (thetaold) +pi/2-ab5 (thetafstail) ) - Dtail*sin(abs (thetaold] +pi/2-abs (thetafstail} 1 ] /m;
xddtail (il =(DtaiL*cos Iabs (thetaddl +pi/2- abs(thetafstail))+Ltailfsin(abs(thetaold~+pi/2-abs(theta£stail) ) )lm;
ena; -CASE 4b: ?heta>O, xvtaild?, zvtaib0, thetafstail>=?Odeg. if thetaold>O & xvtail (i) <=O 6 &ail (i) >=O & thetafstaiDpU2,
zddtail (i) =(Ltail*cos (abs (thetaold) -abs (thetafstail) +pi/2) - Dtailesin (abs (thetaold) -abs (thetafstail) +pi121 ) /m;
xddtail (i) =( Dtail*cos (abs (thetaold) - abs (thetafstail) +pi/2)+Ltail%in iabs (thetaoldl- abs (thetafstail) +pi/2) 1 /m;
end; .CASE 6: Tbeta>O, xvtail>O, zvta i l>ù, thetafstai l<=oOdeq. if thetaold>O 5 &ail (i) >=O & zvtail(i) >=O & thetafstail<pi/2,
zddtail (i) = (Ltail'cos (pi/2-abs (thetaoldl -abs (thetafstail) 1 - Dtailtsin(pi/2-abs(thetaold)-abs(thetafstail)))/m;
xddtail ( i) = (-Dtail'cos (PX-abs (thetaold) -abs (thetahtail) - Leail*sin(pi/2-abs(thetao1d)-abs(thetafstai1)) ) / m i
enc ; -CASE 9a: T h e t a > Q , xvtail>O, zvtailC0, thetafstail<cOdeq. :f thetaolcb0 & m a i l (il >=O h zvtail (il <=O & thetaf stail<pi/S,
zddtail (il =(Ltail*cos (abs I t h e t a f s t a i l ) +abs (thetaold) - pi/2)+Dtail4sin(abs(thetafstail)+abs(thetaold)-pi/2))/m;
xddtail (i 1 = (-Dtail+cos (abs (thetdfstaill+abs (thetaoldl - pi/2)+Ltail*sin(abs(thetafstail)+abs(thetaold)-pi/2) )/m;
2nd; -CASE 3b: Theta>O, xvtailz0, zvtail<0, thetafs ta i l>?Odeg. ~f thetaold>O & xvtail(i)>=O b zVtail(i)<=O & thetafstail>pi/2,
zddtail(i) = (Ltail*sin(pi-abs (thetafstail) - abs(thetao1d) )+Dtail'cos(pi-abs(thetafstai1)-abs(theta01d)) )lm;
xddtail (i) = (-0tailfsin (pi-abs Ithetafstail) - abs (thetaold) ) +Ltail'cos (pi-abs ( t h e t a f s t s (thetaold) 1 1 /m;
ena;
zdd = zddwing ( i l +zddtail (il ; zddplot=zdd; -in:sqrate zwice zdo Ld=zd; zd = zdd'dt+zdold;
-evaLüace h o r i z o n ~ a l accierat ian xdd, ln tegrace once for x ve loc i ty , -then twice f o r hor izcn ta l p o s i t i o n xdd=xddwing(i)+xddtail(i);
xdotold=xdot; - the adot at tke END of this i n t e r v a l xdot=xdd*dt+xdotold; xold=x; x = xdottdt+xold;
-al1 var iables have now been updated. Locp.
êna; figure(l1 ; plot (tplot, thetaplot) ; title ( 'Theta vs. Time' ) ;
Appendix F: MSC DAMPINO EXPERIMENTAL DATA
Listed below are tables pertaining to the disc dampiog experiments descriid in
Chapter 6, Table F-1 details the moment of mertia of the apparatus used in testing when
the distance between ProtoSouth and the pivot point was 25 cm. The foiiowing figures
are plots of the oscillatory decay observed for the dBerent configurations descriid in
Chapter 6.
Pendulum Mass (Ath connectfon) 0.0814 0.187 0.0152
Table F-1: Summaiy of D k Damping Apparatus Moment of Inerti'ri About Piwt Point (If,)
Horizontal Steel Rad 0.00987 0.122 0.00 120
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