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 Copynpynghtby Marc MacMaster 2001 Acquisitions and Acquisitions et Bibbgraphic Se~ices seMeas bibliographiques
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The author retains ownershfp of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auîeur qai protège cette thèse. thesis nor substantid extracts tiom it Ni la thèse ni des extraits substantiels may be printed or othewise de celle-ci ne doivent être imprimés reproduced without the anthor's ou autrement reproduits sans son permission. autorisation. 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 Wom 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 ah 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 hmthe c.g. to the leading edge ofthe 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. Wiut them I do not beiieve that 1 could ever have completed this research on my own, First, I would like te *hdcMr. Dave Loewea Wiiut his input and experience, the work completed herein could wt even have been started His patience in keeping me fiom hbling 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 mermy questions and provide solutions to my pro blems hstinstantly.
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 Ming hmboth 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 Tabh...... 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 ...... IO
2.2.2 Axes System ...... 12 . . 22.3 Design Adjusbbd ity...... 13
2.2.4 merDetails ...... 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 ...... 39
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 52 Taib ...... 51 5.2.1 Results ...... 51
53 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 ...... 62
6.1.2 Lookup Tables ...... 64 6.1.3 Numerical Procedure...... 64
6.2 Initial Results ...... Do ...... 71 . . 6.2.1 Simple Hovering Condition...... 71
6.2.2 Rotational Disc Damping ...... 72
63 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 TWg Disturbance ...... 82 6.4.4 Case iü - SLight Descent with TiDisturbance ...... 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-12Wig [l]...... 6
Figure 1.3. ProtoSo uth...... 7
Chapter 2 Figures
Figure 2.1. Side View of Force Balance Design ...... 11
Figure 2.2. Top View of Force Balance Design ...... 11
Figure 2.3. Wi-Hub Axes System ...... 12
Figure 2.4. Clamping of the Wis to Fixed Upper Plate ...... 14
Figure 2.5.1. Cantilever Beam Configuration (by AC Sensor [4])...... 15
Figure 2.52 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) ...... ,...... 20 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) ...... 25
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. "Gooseneckn 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) ...... 55
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 ...... 22
*Note: Figures and fables in the Appendices are nos listed here. however the Appendix title and introductory poragraph should ahw the reader to determine wht 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 Remh Projects Agency (DAWA), in Light of the current and irnpeodiag developments in rnicroelectronics tahg 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 km 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 artifid muscle actuation. This particular fodawould stand apart hmother proposais in that it would be directly aimed at producing a MAV capable of bvering ftight. Fi1.1 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 Artifid
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. Jms DeLaurier, the OTlAS approach was io mode1 Mother Nature's succd design of the hummingbird. Due to the smaii des 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 tdunder 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 succd 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 ideutmg 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 [Il
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 [Il, 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 hmthe 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 fkhr msight into this and other hi& lifi mechanisms, the deris directeci to Ms. Jasmine El-Khatiis Masters thesis that was also completed at UTIAS in relation to the MAV project [2]. Flapping amplitude is deWas 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 hma 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 6x4 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 hed upper plate and reacted to aay
translations of the suspendeci lower tray. Figures 2.1 and 2.2 are simple depictions bt
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 hr the planned testhg
to follow. Fixed Plate Btacket -.
I Sbain F- Gauge
Figure 2.1: Side Yiew of Force Bulance Design
Longitudinal
Lateni
Gauge th 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 scb, it is observed tbat iaîerai loads were resisted by gauge #3. Longitudinal loads were determineci through a summation of the readings hmgauges #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 cm ww be deW 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 mure
minute forces yet still retain some durability so as not to be easiiy damaged. Thus, during
constmction. an effort was made to allow 6r 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 dethe 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 uppplate of the balance rather tban
rigidly ancho~gthem into position. Leaving the wires long pdedthe bwer plate to
descend hher shouid the mecl arise. Figure 2.4 ilbistrates how this was done. Figure 2.4: îlamping of îhe W- io FdL 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 dow for greater sensitii 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 stnm gauge mtegrated ont0 a thin-film stades steel element of 0.004 in thiclrness, This particular mode1 sensor was the Iowest capacity (114 pound) avaiiable hm 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, hm 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 de
&%ration nearly impossiile. It was befieved the gauges were fiexhg out of plane ad 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 Confition (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 hmthe ûapping rnechanism is passed through its mount dom
to the keIower 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 caiiieach of the beams iedependently. Two reasons provideci the ration& for this effort. First, ske the *es attaching the angIe brackets to the gauges were glued mto pk,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 dffkis couid be
Observed.
The resuits fiom these idependent triais revealed that there was no appreciable change m the gauge slopes before and der 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 Myclamping 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 deW as the applied mas 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 appm a series of loads in the x direction (both in the positive and negative sense), ensuring no force component emerged dong any 0thaxk 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 hmloadings 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. Fi, by using the hMly derived k-values, an estimate of the percent error incurred under ciifferent force conditions could be evaluated. Second, new k-values codd be derived hmthese extra test cases, allowing the abity to assess any gross change th& magnitudes. This rather elaborate procedure wouid gamer a deeper insight into the ovdperform~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 dts.
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 amto 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: CdinedXand 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 amattachment aligned dong a similar diagonal as shown m figure 2-8.3. Agaiu, the appiied test mas 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 bwas 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 @erfodm 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 nia nia da
- Z Force 0.0563 0.0820 da
- X Force nla nla 0.0528
+ Y Momsnt 0.0546 0.0897 nia CornMnad X, Z 0.0524 0.081 5 0.0537 Forcer J Combinsd X, Z and 0.0554 0.0829 0.0537 Y Moment Table 2. I: K-Value Summary
The variation in k-values amongst ali conditions was quite small, with the widest margin of dEerence no greater that 10%. Et was aiso kk that the finai k's chosen stiould greater reflect test cases which involveci multiple forces. After much thought, it was decided that the teds deriwd kom the combined x, z forces and y moment condition would be the best representation of the overaü system's calibration cuefiients. 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 Mial Resulfs
Idedy, a wind tunnel should create a bw field tbat is entirely domin its cross-section, Such an ideai is never tdy 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 fiedto 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 demim 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 hmthe 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 deWas
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@) measdby 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 hmthe prelEnioary tests performed by Mr. Allison, it was discovered ththere 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. Rdthat 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 hm 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 ad remounted as shown in step 2, suçh that the idhg edge of the wing was
now upstream of the ldmg 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
dom upon the flapping mechanism and mount, which acted to bkkthe 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. Skethe 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 ktwith 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 adme 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 merwas 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 tem 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 donedabove, 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 thwould be recoded to a data tile at 0.006 second intervais. Thus, a typical test nin wouid begin with a mo reading immediately foiiowed by another reading with the wings in motion. The gauge outputs whk 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 hmthe zero cornlition, these oscinating outputs were sîmply averaged over the 5 second tirne intdThis was proven to be a valid assumption as a graphical plot showed these aUctuathg outputs îakhg place about a lïxed mean due.
Ah, 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 sodapproach. 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 es.
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 reds. in this case, the muunhg bracket and hpping
mechanism wm 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 hmtk 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. Darire Moments a lei ad in^ Edge Lever A
c Figure 4.3: Original Mouniing Bmcket (Top View)
These lateral contniions to the ovdmoment 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 hmthis 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 desnear 90' to reduce lateral tares.
Secondly, there was the need to have a hed 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 dow the moment to be rehedanywhere 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 fodor inverteci attachment) to be aligned with the centre of the force baiance.
The irnprovement was still far hmperfèct. 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 dl 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 de
it les st8in an effort to attenuate its sensitivitynsitivityAlternatively, 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!zisiestto 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 dem 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 deon 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 co~on.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 pst (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 rd
attachent as iu figure 4.9.
Mr. Low's simulation required ody CLand CD cwes for the tails dera 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. Tadues were foud 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 skan 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 dsyielded very minute forces in the order of 3 grams or less. This greatly pushed the sen~itivityiimits 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 ds. 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 hdyiden@ 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 3 Trials 40.0 -[ l ' S. 1.2 Longitudinal (2-mis) Forces
Upon inspection of the longitudinal (2-axis) data, it was readily observai that a sharp discommuity oc4at the 90" mark. The uiitiai conclusion was that the inversion of the wbgs (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 ke
l+om the bwblockage caused by the Dapping mechanism ad shroud and subsequentiy produce mure thnist. Yet it appeared the oppsite was me. Of particular interest m the
ktthat this dikcontinuity was absent wkn the adme 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 (Zh)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 es),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" derthe 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 he 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 hmthe 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 firtests 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 comte 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 wr 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 mg)
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 180°, conmbuting to the zero o&t. Even through repetition, this data point rernained an outlier fiom the zero pomt. Of course, one couid dy 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
Paruarucuiariyinteresting 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 btwere 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@ adme ratios of 0.656 and 0.735, there were no positive regions present.
5.2 lails
5.2.1 Results
The CLand 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 Wescatter 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 su.of the resuits.
Tail li1 and #2 CI vs. Angk Re = 22,000
Figm S. 7: CLCm for Ta& #l and #2 i 1 tail #1 and lit Cd vs. Angle ! Re = 22,000
Figure 5.8: CDCuwes 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 destitself 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 fkqueq 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 Hz 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 hm
30 Hz data. Some fkther investigation was required at other fkquewies to better estabiish the theory. t T htuiRatio vaWaüW
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 / VaxhWb 1l
Figure 5.1 O: Tkrrrst Ratio us, Free-Streum - Frequency Ratio (Orig. and Ertrp Daia)
From the dts, 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 hm this to anah the &ta for 40 Hz A simple addition of a trend iine through tbe data (hwnm 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 queof mterest was now Lat 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 defor 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 fhpmq 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 saw 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 X Force rit Various f lapping Fmquencies i V = 5.24 mls
Figure 5.12: Effea of Rapphg Freq~encyon 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 prograd 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 assdvahses 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 hm 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. assudvalues.
X Force Comparison of Measured (J = 0.55) va humed(J = 0.50) 1 40.0 1-ksuedXForceJ=0.55 1 35.0 -. Assumd X Force J = 0.50
I 1 I I
I I I 1 I I b la0 150 290 1 3.0 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 he 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: 20SIMULATDN
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 ddkdby 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 Wsquarter chord to the vehicle centre of
Pvity- In order to gauge the stability of the MAV derperturbeci 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 diibed 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 with how the data hmChapter 5 was recorded. The same can be said of the drag force hmthe vehicle's wings b and Dm of course reptesented iifl and drag contriutions hmthe 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 hmthe 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 0th accelerations or velocities of interest.
(Smce the focus of this chapter is primarily upon the hovering condition, the mode1 was
tyjically only displaceci hmthe 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 hma trend
Sie placed tfrough its CL VS. a and CDvs. a graphs (see Appendix D). These coefficients
were then simply muitipiied by the tail area and dynamic pressure to give totd Iift and
drag hmthe tail 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 aircd 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*vXd,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)
Xm= 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 "newnrefér 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 ken 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 bstream incorporateci mt only the inverse tan of the velocity components, but aIso the current tirne step's tilt angle hmthe 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, adare 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 deth 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 hmthe 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 derail 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 2O 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
1 I 1 1 1 -301 1 t 1 l 1 O12345678910 Tirne (sec) i Rgvre 6 7: lilitial Test Glse WdhtTd 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 hdhg 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 hmwhich 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 dis) 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). A P 12 countemeight v
pendulum mass
Figure 6.8. I: DkDumping 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 viiions text. As seen hm figure 6.82, three components serve to dampem the motion, Bearing fliction and aerodynamic drag were combined hoooe prameter (labelled F in the di)which, as 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
%itingWof the wing disc amabout 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& - ~28- 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 = It0,
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 muthat in the absence of the
damping terms Ci, CZand 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: DkDomping 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 heby headvance 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 vs 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 fbt 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. Wia solution for F, the next stqs 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 keflyer 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 hmthe 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. Fiy,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 6Il.l: Case 1- No Tail, = Mcm
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 of 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 wt mvestigated, as they wodd put into guestion
which side of the cgthe wing forces would truly act on. Redthat the forces were
experimentally recorded as acting hughthe 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 hmthe 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 sder 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 Merand truiy placing the wings below the c.g. added
nothhg to aid stability. In kt,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 dcearea 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 derthan 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 myor may not have si@cant effects un the behaviour of the vehicle.
CeMv 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 aii as depicted in figure 6-1t 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. Theta us. Time 2
1.5
1
d Q g CD 4 0.s -ai- cQ O
- 0.5 k!"'"-
-1 L , O 5 10 15 Time, sec
Figure 6.11.6: Cose iX! - No Tail, Eflect in the Reduction of 1,
Wiih the addition of the taii (hawig the same geometry as that in Cases 1 and iI), the motion was found to be only siightly better for values of h king eitber 12.5 cm or
- 12.5 cm. The results for both cases are shown in figure 6.1 1.7. - WiTaii, 12 = t2.5cm - Wii 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 Meran evdconvergence.
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 diieand 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 hmfigure 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 tdy 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 dera 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 ewgeprimarily 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 sWgauges, 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 Wingsfor 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. Pl 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. 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 oved 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 diient 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 der 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 kwwn 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 caiiiion tests to ensure that it did wt 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 Myto 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 hmcase 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 dekdsign 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 cm This number med 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 tdy 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 adz 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 hmcase 2 gave percent mors m the mgeof 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. 120.9 -24.2 32.2 33.8 4.9 -24.2 0.0 119.2 -1 -4 161.2 -24.2 32.2 33.6 4.3 -24.7 1.8 159.0 -1.4 201.5 -24.2 32.2 33.4 3.7 -24.8 2.3 195.8 -2.9 241 -8 -24.2 32.2 33.6 4.3 -24.4 0.7 234.0 -3-4 282.8 -24.2 32.2 33.8 4.9 -24.5 1.0 268.4 -5.4 322.4 -24.2 32.2 34.0 5.5 -24.4 0.7 298.0 8.2 'ln al1 cases the applied mass was -40.4 g. Placement along the bar uttachment uilowed variation in the appked moment. Table B-3: Case 5 Results / Emr Ana&sis 0.0562 -0.2144 0.534 0.0655 0.0681 0.537 0.0549 0.0700 0.0540 0.0538 0.0688 0.0543 Average: 0.0554 0.08#1** 0.0639 *In cuch imnce rhe applied Z Mas WPT -242g und the applied X Mm was 32.2g. as indicared in Table B-3. Thevaiues 0.0688 and -L?l.(1were not includedin culculution of the average. Table B-4: Case 5 CuIcuItated Slopesfor Gauges #1, #2 and #3 Appendix C: WlND TUNNEL MLOCITY PROFILES A sdlopen-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 de). 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.). Tunnel Fan Voltage = 56 V, 2 inches from Tunnel Exit *Esch station rht was separated by a distance of cm (1 in.) and each station width 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 wiqs 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 hee 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 descrii 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 witheach program. -A Simple 2D MAV Sirniacion, HO TAIL -Yarz EiacMaster, Yarch 2001 - D - lookup table for wing drag valries: eacn column corresponds CO J's - cf O,~I.LC,9.55,0.i5€ and 0.735 and - éash row for angles O thru to 180 degrees -7- lookug tanle as above, cxcept for the thrust vaiues of the wings - :heta - anguiar dispiacement from Che vertical, degrees ---aliitude (metzes!, x - iateral displacement imetresi, alpha - - angular accel.irad/sa2i cmeqa - anguhar velocity Ired/si , xaot - lateral velocity !rn/s!, xdd .- la~era: accel (n/s"Z'i - zd - vertical velocity !m/s), x!d - vectical accel. [rn/sa2) - thetav - angle the free strearn makss with the horz. due :O boch z r x - veicctles ac l .e. of rings - L - disrance fzom che c.g. to the 1.e. of Che wings im: in - nass of Che vehicls :kg], ïy - inass rnomen: of inceria anout the y - axis ,kg m"2i rho - air isnsiq :kg/m"3), 5 = tail surface area (mA2), cl - disc - damping ccefficie~t:!cg nA2/s) - xvt - tctal lateral velccity at 1.e. of the wings due to :ranslation - and rotation :m/si - zvt - total verticai veiccitÿ at 1.e. of the wings due CO ::anslation - and rotation Im/s I chetafs - maqnitude of iree Stream angle to the leacing edge of the - wlngs xrt the vertical !deqreesi + Thr - thrust from wings :gi , Drag - drag frcm wings (gi -. . . - ...... -*.--*..**".* '.--*------..-~.--*---.---.-...--*..~-*-----.**------.-. . -popuiate 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 theta=(2/180*pi);z=0;x=O;xdot=O.O;alpha=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~üptobles 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. -accelerations -thr=ugh rbat Fncrenentai rime step. The values at the end of 3 stec Srepresênt Lhe vaiues sf zhat vüriable at ~ndttirne at. fsr t=dt:dt: 15, i=i+l;tplot (il=t; -tczai x velocity at leaaing edge of wings xvt (il =xdoc+omegaf1-0s Cabs (thetal 1 ; -cotai = -~el=cityat 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 (xvt and zvt; Ff 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 zvt (i)<=O if .wt(i) sise thetafs=pi/2-thetav(i1-theta; end; end; end; -case wnerr wings are risinq purel? verticaiiÿ etc; if rvt(i) 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 -determirie wnicn :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; if angle<90 & angle>=80 rowl=lO;row2=9; end; if angle<100 & angle>=90 rowl=ll; row2=l0; end; if angle -inte:coiate =o ?et a: the desired angle Thr = (ThrHiqhAngle-ThrLUwAnqle]/IO' (angle-anglelow) +ThrLowAngle; thrplot [il=Thr; -F.epeat rhe same cCinq for getting Che ara9 -veiocit:~?air at iauer angit VLowT=D (row2,coLumn1) ;VHighl=D (row2,colu2) ; -velocity pair for higner angle VLow2-D ( rowL, columnl);uHigh2=D ( rowl, colu21; -icrerpoLace -&th these poincs at the current vdocity ~rag~ow~ngle-(VLowl-VHighl) / (vhigh-vlow) (Vel-vlow)+VHighl; DragHighAngle= (VLow2-VHighZ] / (vhigh-vlow) el-vlow] tVHigh2; -interpolate ta get Drag at :ne desired angle Drag =(DragKigh~nqLe-Drag~ow~ngle]/10*(angle-anglelow)+DragLowRngle; -evaiüate angular acreleration, ensuring Drag force always points -0pposite to Che airection of zhe velocity thetaold=theta; ornegaold=amega; draqpLot (i1 =Drag; -the origifial alpha at the beginning of the current rime interval alphaold=alpha; discdampl = cI*omegaold; if xut(i)>=O, alpha = -Drag/L000t9.81+L/Iy+discdampl/Iy; tlse alpha = ~rag/1000*9.81*1/Iy+discd~l/Iy; enà; alphaplot (il=alpha; -integrate alpha twice ta get angular position -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) lga else zdd=Thr/1000*9.81/m*cos(abs(thetaold) Draq/1000*9.81/nfsin(abs(thetaold))+omeqa+xdot mu; 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 start 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;~thexdot 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 Simulation, WITX TAIL :Marc NacMascer, March 2001 clear all; .. . .. , . . -;iL;--L;;;;;;;;-;;i-iL-i-L------.....*.*..**....*-. '. 3F KRI-ULES . . , ...... - D - Lockup table for wing drag values: each column corresponds CO S's - cf fil,Q.l9,~~.55,O.E56and 0.735 and each row for angles 0 th== to le0 - -: - ,ockup- table as &ove, except for ~hethrust values of che wings - rneta - angular displacement Ercm the vertical, degrees .--- al~itudc :metres), x - lateral dispiacernent irneczesl, alpha - - anqular accei. [ rad/ sa2 1 - zmega - anquiar ,relacrty ;raa/s!, xdot - Lateral velacity !dsi, xdd . - laceral accei !m/sA2: - =c - .ferzical velociry ;rn/si, zCd - vertical accel. im/SmSi - xddwing - Latersi accel. contributeci by~ wings ;m/sA2: - zadwing - ver~lcalaccel. contributed by wings (m/s'2i - : -pcpulata Lookup tables for Lift and Drag aaca D = [1,98 0.3 -.Il -89 0;7.85 6.98 4-93 3-40 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: fy ~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 varia8i.é~ l=O.O75; dt=0.001; m=0.050; Iy=0.000031; 12=0.125; -distance to tail .ci4 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 ~imeateps tplot (1)=O;i=l; -lccp ~eginsat dt, ana aii variables get updated based on avg. -acceierations -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 zvt) 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) anc; snc; end; 15 zvt (i)>O if xvt (il>O, ;f theta>=O, thetafs=pi/Z-theta-thetav0 ; eise thetafs=pi/2-thetav(î1-theta; ana; end; if zvt(i)>O if xvt(i) -case where wings are rising purely vertically end; if zvt (il 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 -star: iocble interpolation for thrus: ...,a?-ALcity: - -veiccity par at Lower angle VLowl=T ( raw2,columnll ;VHighl=T (row2,column2) ; -velccit;~pur for higher angle Vtow2=T (rowl,columnl) ;VHigh2=T (rowl, colrimn2) ; -interpolate with chese points at 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 ,colml ) ;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 '-'----.--.-"-----.-7ff-...... ;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 arctan function! thetavtail (il=O; else thetavtail (i)=atan (zvtail (il/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; ad; cna; F f zvrail( i)>=O ~f xvtail(i)>=O, rf 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 thetafstail=pi/2+thetafth~ftavtail(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 CL, 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 tnè direction of 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: -MEia: ThetaCo, xvtail -CAS= Ib: The:a>O, xvtarl alpha = awing(i) +atail (i); -integrate alpha twice to get angular position -once: -total omega at the DID 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 xvt (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 tu the x acceleraticn -use appropriate equation depending upon which side of rhe theta -equals 11 ;nar~. -must use theta at starc of rhis mte~valco upaace xact 1: thetaold>=O, :f xvt (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 zdd = zddwing (il+zddtail (il ; zddplot=zdd; -in:sqrate zwice zdoLd=zd; zd = zdd'dt+zdold; -evaLüace horizon~alaccieratian xdd, lntegrace once for x velocity, -then twice for horizcntal position xdd=xddwing(i)+xddtail(i); xdotold=xdot; -the adot at tke END of this interval xdot=xdd*dt+xdotold; xold=x; x = xdottdt+xold; -al1 variables 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 0.0814 0.187 0.0152 (Ath connectfon) Horizontal Steel Rad 0.00987 0.122 0.00 120 Table F-1: Summaiy of DkDamping Apparatus Moment of Inerti'ri About Piwt Point (If,)