The Flight Dynamics of a Full-Scale Ornithopter
Tahir Rashid
A thesis submitted in confomity with the requiiements for the degree of Master of Applied Science Graduate Department of Aerospace Science and Engineering in the University of Toronto
O Copyright by Tahir Rashid, 1995 National Library Bibliothèque nationale du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 395 Wdliion Street 395. nie Wellington OnewaûN K1AW Ottawa ON K1A ON4 Canaada Canada
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The author retauis ownership of the L'auteur conserve la propriéte du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial exûacts ffom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. The Flight ûynamics of a FulCScale Ornithopter Master of Applied Science, 1995 Tahir Rashid Aerospace Science and Engineering University of Toronto
This t hesis investigates the non-lineaif light dynamics of a full-scale flapping-wing aircraft (ornithopter). Using simplifying assumptions, the equations of motion were developed for a 2-wing-panel and 3-wing-panel rnodel.
A cornputer program was written to examine the longitudinal and lateral stability of the omithopter. The pmgram was first tested using inputs for a mode1 ornithopter known as "MI. BilP and then using the inputs for a full-scale omithopter. The results indicate that both 'Mr. Billaand the fulkscale ornithopter are longitudinally and laterally stable. The accelerations and displacements are much less for the 3-panel model than for the 2-panel case because the 3-panel design serves to balance the time-varying forces seen by the fuselage. I wouid like to thank my supervisor, Di. J.O. Delaurier, for his assistance and guidance during this research, and as well as his patience. Also, thanks to all the mernôen of the ornithopter team for their help and the Natural Sciences and Engineering Council for financially sopporüng this projed. I am also very grateful to my father(AWu1 Rashid) and my biothers Akhtar, Zafar, Nasir, and
Qasir for their advice and continual encouragement. Their hard work and support has helped to make my dreams a reali. Finally, I wouîd like to dedicate this thesis to my mother, Mn. Seema Rashid, who passed away during the compilation of this research. Her love and wisdorn have benthe key reasons for my success in life. May she live on in our hearts and may God bless her. TABLE OF CONTENTS
ABSTRACT
7ABLE OF CONTENTS
LIST OF FIGURES
UST OF SYMBOLS vtii
3. LITERATURE SEARCH
4. DEVELOPMENT OF THE EQUATlONS OF MOTION
4.1 %PanelMode1 4.1.1 Dynamic Mode1 4.12 Body EquationsûfM~ai 4.13 Wmg Equaiions ûfhaotion 4.1.4 Kincmatic Analysis 4.1 5 Complctc Equarions Of Motion (2-Pncl)
433-P-1 M*l 42.1 Bady Equakms Of Mdim 422Ccnter Psael E41mticmsOf Motion 423 Pm Wmg Equalions Of Me 42.4 SIIlbard Wmg Equations Of Motion 4.25 KineniPtic Analysis 4.2.6 Campiece EquPtim Of MoQa (3--1) 5.3 Body contributhm 53.1 Body Dmg 5.32 Body Lin 53.3 Body Pitching Momcnt
SA TU)Contributho 5.4.1 Total Angle of Auck of Tail 5.42 Taü Lift 5.4.3 Tai1 Drag
S.!! Canbined Body and Tdi Coiitributioa
5.4 outer wm Panel coitribotim 5.6.1 Wig Lift 5.6.2 Wing Drag 5.6.3 Wing Pitching Moment
5.7 Cmm blCOObhthll 5.7.1 Centcr Panel Lift 5.7.2 &ter Panel Drag 5.7.3 Cawr Panel Pitching Moment
6.1 My~~~tributioP 6.1.1 Body Si- FaFe 6.12 Body Yawing Mormcnt 6.13 Body Rolling Moment
6.3 Fim coatmutia 6.3.1 Fin Sideslip Angle 63.2 Fi Side Lift 6.3.3 Fi Drag
8. DISCUSSION OF RESULTS
8.1 Imgitudid Stabiüty 8.1 .l Pipr Comanche PASI-?SO 8.12 2--1 "hi.. Billa 8.13 3-Puiel "Mr. Bill" 8.1.4 2-Pncl Fd-MOrnirhagca @SO) 8.1 s 3-PnJ FiiII-Scrlt Oniihaper (FSO) 8.2 LATERAL STABILll'Y 82.1 Pipa ComiPche PA24-W) 8.2.2 2-Ponel "Mr.Bill" 8.23 3-Puicl "M.Bill" 8.2.4 2-l'ad Full-SC& Oniiw(FSO) 8.25 3-Plael FU-S~CMtbapter (FSO)
9. CONCLUSION
APPENDIX A - GRAM APPENDlX B - DERIVATlON OF EQUATIONS OF MOTION FOR A RlGlD BODY
APPENDIX C - 2-PANEL COMPLETE NON-UNEAR EQUATIONS OF MOTION
APPENDlX D - 3-9 ANEL COMPLETE NON=UNEAREQUATIONS OF MOTION
APPENDIX E - MOMENTS AND PRODUCTS OF lNERllA Of WlNGS APPENDIX F - ORDER OF MAGNITUDE ANALYSS FOR TWlSTlNG
APPENDlX H - COMPUTER PROGRAMS LIST OF FIGURES
Figure 1: 3-Pricl madel anithopcr. .M r. BU ...... 3 Figm 2: Conventid notation...... 5 Fi- 3: 2-pluukl ...... 8
Figure 4: 3-panel m&i ...... *...... -9 Figure 5: Schemstic 2-panel mode1 with reaction farces and moments ...... 11
Fiipiie 6: Ax* sysiem ofwhg and body ...... e...... 12 Figure 7: Axes sysmns ...... 15 Figure 8: 3-plm&l with mction faa~and manaus ...... Figure 9: Cenm panel ...... 22 Figure 10: Pϔ wing ...... 23 Figure 11: Sirboud wing ...... a
Figure 12: Axes fa body ad whg ...... 25 Figure 13: Axcs fa body and ccnia pl...... 27 Figure 14: Lift ~IMIârag trmsfdon ...... Figure 15: Tdtd lift vscuiru, producc i)w9...... e...... 38 Figure 16: Changes in angle of aasdr due to a roll p ...... 4 Figure IR Chrigcs in angle of ritrk due to a yaw r...... 47 Figure 18: 'Ihiide-view drawing of the Piper ComPndie PA24-250...... 51 LIST OF SYMBOLS
f - fin cp - centei panel wl - port wing w2 - starboard wing
a - angle of attack fl - sideslipangic e - downwash angle Q - mUmgie 0 - pitchangle Y -yawmgle Y - WPing angle P - dcnsity of air - aspect ratio - "ng span - mean wing chord - drsg coefficient - drag coefficient at zero angle of attack - lia coefficient - lifl - CUNC slope - pit~hing- m~mc~~t- cuive dope - side - force stability dcrivative - yawing - moment stabiiity denvative - rohg- moment stability denvative - dragforce - x distance from body cg. a centa panel cg (3 - panel) x distance fiwi body cg. to point on mtof wing alignai with wing cg (2 - pand) - y distance frwi body cg. a center pand cg (3 - panel) y distance from body cg. 1~ point on root of wing aiigned with wing cg (2 - pand) - z distana hmbody cg. to centa pand cg (3 - panel) z distance hmbody cg. to point on mot of wing aligneci with wing cg (2 - pand) - x cunponent of œnta panel vdocity W. r. t. body (3 - panel) - y cmponmt of cuiter pand vdocity W. r. t. body (3 - peî) - z canponent of center panel vclocity w.r. t. body (3 - pd) - x canponent of anter pand acœleratim w.r. t body (3 - panel) - y component of anter panel occcleration W. r. t. body (3 - pand) - z cornpaient of anter pand ~c#xIeretionW. r. t body (3 - pand) - x distance fiom wing cg to pivot point - y distance frwi wing cg to pivot point - z clistana fiom wing cg to pivot point - x distana frwi mot of wing to cg of wing pand - y distance from root of wing O cg of wing pmd - z distana fnw mot of wing O cg of wing panel - Osdddfich~y factor - x dinction rcaction fcnce at wing pivot point - y direction rwction farce at wing pivot point - z cikation r#iction farce at wing pivot point - x distance frwi body cg to pivot point on wing - y distance hmbody cg to pivot point on wing - z distance ninn body cg to pivot point on whg I,I,I,I,I - momaitofinertias
~&,&Jyz - derivatives of moment of incrtias JL - xdiisoionlWCtimmnnitatwingpiwtpant J, -ydirccti011donmamtatwingpi~t~ JN - zdirixtiond011mmmtat~g~vapant L - liftfolice L, L, -idlingmnrnt L- - tdeaodynanicmb(lltlltsinthexdircctim m -IIliiSS M - pitchingmniwu ML -x~011rmctiaimmntbawmitheaioawingpsndandmiapsnel(3-pand) xdiribctiailCaCtionmmrntbaws#ithcwingandbody(2-pand) MM - ydinctiairitactiai~tbaweaidieaitadgpaadandmi~p~l(3-panci) y~onrc8ctimmonientbetweaithcwingandbody(2-panel) MN - z diiccticm dmmmnt bawcen the wing panel ami anter panel (3 - panci) zdiribctioniacrionmnmnbetwknthewingaadbody(2-pand) M, -taPlamdynamicrrmmtntsintheydiribctim N -yawingmnmt N- - totalasodynsmcrmnenffindrz~ P - roll an@r vdodty rate P - roll anpuiar accxkmion rate Q - pirh angular velocity rate Q - pitch anguiar aocderation nite R - yaw anguiar velocity rate R - yaw angular acœleration me r - distance hmwing mot to wing segmmt Re - Reynold's number Rx - x dllecriai action force betwcen the outa wing panel and centcr panel (3 - panel) x diriection maion force between the wing wd body (2 - pand) RY - y didonmion force between the outa wing padand center pend (3 - pand) y dimion naaion force betweai the wing adbody (2 - panel) Rz - z diriection rieaction foice betwem the outa wiag panel and cenoer panel (3 - panel) z didonnaaion force between the wing and body (2 - panel) S - reference area Vol - voheof body u - SU@^ vdadty u - smgingocaleratim v - sideslip velocity v - sideslip accderatiaa w - plunging velWty w - plunging axeleration X, - taPl aemdynamic forces in the x diicction Y - sideslip force y, - tapl aerodynaniic farces m the y dbtion 2- - totaï scrodynaniic foiccs in the z dixection 1. INTRODUCTION
The study of nature has often led to surprising breakfhroughs in science
and technology. One such example is the flapping-wing flight of birds. For
centuries, humanity has been fascinated by the ftight of birds and has often
made numerous attempts at imitating them. One of the first known accounts of
human flight cornes from Greek mythology in which Daedulus and Icanis escape
from their prison on the Mediterranean island of Crete using wings constnicted
of wax. Unfortunately, lcarus flew too close to the Sun which melted his wings
and he fell to his death. According to Chinese legend, Shun, the emperor of
China around 2000 B.C., was taught to fiy using wings. Among the Norsemen, a
blacksmith named llmienen fashioned metal wings to rise from the Earth. lt was considered for a long time that the only way humans could fly would be to imitate the birds. However, this notion was shattered in 1799 when Sir George Cayley
introduced the fixed-wing aircraft concept. Cayley's idea was the first of its kind which abandoned bird flig ht and introduced a new concept in which the lifting
surfaces, the wings, were separated from the mechanism of thnist. This
revolutionary idea eventually lead to the first successful engine-powered flight at
Kitty Hawk in 1903 by the Wright brothen. Cayley's principle is still used today in al1 modern aircraft. With the success of fixed-wing aircraft, flapping-wing flight was al1 but abandoned as a viable field of research. While fixed-wing aircraft have dominated the aeronautical research field, it is becoming apparent that a flapping-wing aircraft (ornithopter) holds great promise as an efficient means of generating lift and thrust simultaneously[Ref. 171. Also, the boundary layer does
not develop to its full potential in a non-stationary flow about an oscillating airfoil, which means that separation is delayed and the fom drag is reduced. The dificulües of designing of such a flapping-wing device would have to considei the aerodynamic, mechanical, and structural aspects. Many attempts have been
made to design ornithopters, the first of which was by Leonardo da Vinci.
Leonardo designed ornithopters in which the pilot sat in an inclined position and used pulleys which required both hands and feet to flap the wings. [Ref.l9] This concept was more sophisticated than the ones that preceded it because it involved more than strapping wings to arms. No one is certain if Leonardo's designs were actually constructed. Most ornit hopters were small tubber- powered models which derived from the 1874 model of Alphonse Penaud
[Ref.5]. More sophisticated rubber-powered omithopters used to study bird flight were built by von Holst [Ref.S]. Alexander Lippisch designed and built a
human-powered ornithopter in 1929 which achieved powered glides from catapult launches [Ref.l7]. In 1959, Emil Hartman built a human-powered ornithopter which was capable of extended glides but not sustained flight
[Ref.l'lj. One successful ewrnple of a human-powered aircraft which was not an ornithopter was the flight of the Gossarner Condor in 197ï [Ref.l9]. The 70
Ib. airplane with a 29 m wingspan had a large propeller in the back and was
pedaled by a pilot weighing approximately 140 Ibs. Although nibber-powered
and human-powered ornithopters have existed, successful examples of
motorized flapping-wing aircraft are few with the notable exception of the 18 ft
span robot pterosaur built by AeroVironment Inc. of Monrovia California [Ref.S].
However, this model was not able to sustain flight.
The first successful flight of a motorized radio-controlled flapping-wing
aircraft, known as 'MI. Bill", was made on September 4,1991 by Dr. James
DeLaurier and Mr. Jeremy Harris at the University of Toronto. This model aircraft was able to sustain flight for about 3 minutes and was landed
successfully. This omithoptefs wing design is different from that for birds in that it does not have the cornplex motions such as feather spreading, fore-and-aft swinging, semispan variation, etc. (See Appendix G). The wing consists of 3 panels: 2 outer panels and a center panel (See Figure 1).
Fgun 1: 3-Panel modal ornithopter, 'Mr. Bill ' The motion is such that the center panel moves. in a direction which is opposite to the flapping of the outer panels. This three-panel design serves to balance the time-varying lift seen by the fuselage and evens out the power required from the engine during the flapping cycle. With the two-panel design, the power required for the downstroke is greater than for the upstroke. The flapping is a simple harmonic motion driven by a lightweig ht transmission which reduces the high rotational speed of the engine down to the low flapping frequencies which are required. A linear twist is experienced by the wing and is 90 degrees out of phase with the flapping. No ailerons are present, thus turning is accomplished by yawlroll coupling produced by the rudder deflection in conjunction with the wing's average dihedral angle. The dihedral angle is accomplished by making the upstroke flapping angle larger than the downstroke angle.
Although the major motivation behind such an undertaking has been an interest in flapping-wing flight, the relative aeroacoustic silence of flapping wings can eventually be used foi surveillance applications.
Research is now continuing into constnicting a full-scale motonzed ornithopter capable of carrying a human being. This enomous task can be divided into several different sub-areas some of which include: wing design, landing and take-off simulations, drive mechanism design, and flight dynamic analysis. 2. PROJECT DEFINiTION
The purpose of this study is to investigate the complete non-linear flight dynamics of a full-scale ornithopter during cwising fligM. In conventional fixe& wing aircraft. the equations of motion can be linearized and then uncoupled into longitudinal and lateral equations. The longitudinal equations examine folward velocity(U), plunging velocity(W), pitch(8), and pitch rate@), while the lateral equations look at the roll(@),yaw(Y), and sideslip velocity(V) (See Figure 2). However. for the ornithopter, it is more difficult to linearize the equations of motion and thus the non-linearity was maintained. Also, if the lateral variables
(@,Y,V) are set to zero, the cornplete set of equations do break down to the longitudinal case. However, the longitudinal variables(B,U,W,Q) cannot be set to zero to oMain the lateral case. For the lateral case, the cornplete set of equations must be used. The equations of motion provide valuable insight into unstable flig ht regimes.
3. LITERATURE SEARCH
In dealing with the problem of exarnining the non-linear flight dynamics of the ornithopter, the first step was to become familiar with the field of flight dynamics, along with stability and control. Several excellent books have been written on this subject. The most recognized and widely used is written by Etkin,
[Ref.6] a pioneer in this field. This book contains al1 the basics of flight dynamics including a detailed derivation of the non-linear equations of motion, which are applicable to any aircraft. The equations are linearized and decoupled into longitudinal and lateral equations. The stability denvatives are presented along with an examination of the control aspects of the aircraft.
Lineanzation was not in the sape of this thesis and was only studied for informational purposes. A previous edition by Etkin [Ref.ï] was useful, but it dealt with the equations using a wind-axes system. Two other books written by Ashley [Ref.2] and Perkins and Hage [Ref.16] were excellent complements to
Etkin and proved to be valuable reference sources.
Once the topic of flight dynamics was mastered, the next step was to find literature that dealt with the ornithopter. The best resource for this was Dr.
James DeLaurier of the University of Toronto and Jeremy Harris of Battelle
Mernorial Institute. A joint paper by DeLaurier and Hams [Ref.5] proved extremely valuable because it showed the feasibility of motorized flapping-wing flight. The work concentrated on a model ornithopter, named "Mr. Bill", showing various aspects such as wing design, drive mechanism, stability and control, and flight tests. The importance of this work is that it showed the analytical and structural design considerations are applicable in constructing a full-scale ornithopter capable of carrying a human being. Two papers written by DeLaurier
[Ref.3&4] showed the complexities involved in the aerodynamics and structural dynarnics for an ornithopter.
Other omithopter designs worth mentioning are by Fowler [Ref.8] and
Skarsgard [Ref. 17] because t hey also dealt with the possibility of piloted flig ht.
Fowler dealt with the more traditional approach of a human-powered ornithopter whereas Skarsgard examined a motorized omithopter. Both show the complexities in the aerodynamic, structural, and mechanical design, looking at such aspects as wing design. powerplant, stability and control, landing gears, and weight considerations. With a good grasp of flight dynamics and a sound knowledge of omithopten, the next step was to search for literature which combined the two, i.e. the fligM dynamics of an ornithopter. Although, no such article was found, projects written by McGuire [Ref.I 31 and Grant [Ref.9] were the closest and &est resource. Both projeds developed the equations of motion for the Canada
Goose. Grant used the traditional formulation while McGuire used a multibody approach nomally considered for bodies which have parts moving relative to one another. Both projects examined only the longitudinal stability, looking at the perturbed and unpenurbed motions. The model consisted of three separate bodies: the fuselage and the two flapping wing panels. (See Figure 3)
wing / panel
fuselage :ire 3: 2-panel mode1 Since the vehicle was syrnmetnc about the xz plane, only han of the goose was considered. Grant used the traditional body-fixed axis system at the
body center, and an axis system at the wing which was aligned with the body axis even though the wing was moving relative to the body. In each case linear aerodynamics was assumed, twisting of the wing was considered, and sinusoidal flapping was implemented. This two-panel formulation was an excellent prelude to the three-panel case shown in Figure 4.
fuselage -ire4: 3-panel mockl
On a final note, a unique and interesting concept was presented by
Pantham [Ref.l5]. In his thesis on the dynamics of variable swept-wing aircraft,
Pantham develops the non-tinear equations of motion by using a body-fixed axis system a an arbitrary point ratheithan at the body's center of gravity. This formulation is powerful because it can be used even if the center of gravity changes, which is often the case in most aircraft. However, due to the complexities of this model, it was decided to remain with the conventional center-of-gravity based non-linear dynamic equations of motion. Another point of interest was the fact that this was the only other study, to the author's knowledge, where the equations were developed for an aircraft where the wings moved(sweep) relative to the body. This proved interesting because the center panel and the outer flapping panels of the ornithopter also move relative to the body.
4. DEVELOPMENT OF THE EQUATIONS OF MOTION
After completing the intensive literature search on aircraft dynamics and flapping-wing flight, the complete equations of motion were developed. The equations were first developed for the 2-panel design similar to Grant and then for the &panel design. Grant's 2-panel model provided a reference base from which further work could be carried on. Since Grant only developed the longitudinal case, the first step was to rework those equations to indude the complete set of equations of motion. As a check, the lateral variables O,P,V,R were set to zero and the resulüng equations were exactly the same as Grant's longitudinal case. After the fundamental equations were developed, the longitudinal and lateral aerodynamic terrns were deal with. Next, a cornputer program was written to solve the equations and, finally, al1 the required data for
"Mr. Bill" and the full-scale ornithopter(FS0) was input into the program to
determine the stability. For a detailed development of the equations of motion,
see Appendices B,C,D,E.
The 2-panel design consists of the body, which includes the tail+fin and the two wing panels. The wings are coupled to the body with the reaction forces
and moments at the joints (Figure 5).
Figure 5: SchematE 2-panel mode1 wlh readion forces and moments
11 The equations of motion were developed for the body and each of the wing halves separately, and then matched through their kinematics. This resulted in a set of 18 linear equations with 18 unknowns, which were the state variables (U,V,W,P,Q,R) and the reaction forces and moments at the interface of the body and wings. A body-fixed axis system was implemented for the fuselage while an axis systern which is continually aligned with the body's axis systern was used for the wing (Figure 6). This unique axis system for the wing was chosen because it proves to be convenient in the kinematic analysis.
I Figure 6: Axis systern of wing and body
4.1.1 Dynamic Model
Dynamically, the wing and the body were both assumed to be rigid, with the wing chosen to be a thin flat plate. The flapping was considered to be sinusoidal and the effects of the twisting of the wing were ignored in the dynamic analysis (Se8 Appendix F). However, in the aerodynamic analysis, twisting had to be considered and the wing was given a sinusoidally-varying linear twist. Twisting occurred at the same frequency as the.flapping but was 90 degrees out of phase to achieve optimum thrust.
4.1.2 Body Equations Of Motion
The body includes everything but the wing panels. As mentioned before, a body-fixed axis is used at the center of gravity. The distance Ob is the distance from the body center of gravity to the root of the wing. The following are the
4.1.3 Wing Equations Of Motion
The axis systern which was attached to the wing center of gravity is a pseudo-stability-axis frame. This means that this ais system is aligned with the body-fixed system of the body. The equations of motion for the port (left) wing are:
The equations for the starboard (right) wing are: 4.1.4 Kinematic Analysis
After obtaining the equations of motion for the body and wing, a relationship was required which would link the two different sets together. This couid only be accomplished by relating the kinematics of the body and wing, i.e., the velocities, accelerations, angular velocities, and angular accelerations had to be found (See Appendix C for complete development). If the axes systems shown in Figure 7 are aligned at al1 times, then il can be shown that for the port wing , the velocities and accelerations are: port wing velocities:
port wing accelerations:
The starboard wing velocities and accelerations are: starboard wing velocities: starboard wing accelerations:
4.1.5 Compkte Equations Of Motion (2-Panei)
Substituting the above kinematic relationships and noti ng t hat the ieaction forces and moments on the wing are equal and opposite to the body, the equations of motion now becorne:
These are a set of 18 simultaneous differential equations with 18 unknowns(U,V,W,P,Q,R, R~I,R*t, Ra,MM, MWI, MW, &, RM, MW, b,Mm). Given the initial conditions, these equations were solved numerically using a Gauss-Jordan elirnination routine to obtain the denvatives of the state variables. These were integrated using a second-order RungsKutta
routine to obtain the desired variables.
The 3-panel design consists of the body (which includes the tail and fin), the center panel, and the two outer wing panels. The outer wings are coupled to the center panel with the reaction forces and moments at the joints (Figure 8).
center Y;a,X panel 2 pivot pivot point point
Outer panel F- FLP
Jw Jnt
Figure 8: 3-panel delwith reaction forces and moments
Note that the x moment (Mc, ) is non-existent at the interface of the outer and center panel because there is a hinge connection, which means that the outer panel is free to rotate about the x-ais. The center panel only plunges up and down while the outer panels are pivoted to operate in a rocking "seesaw" manner. There are reaction forces and moments present at the pivot point as well. Hawever, there no moments in the x and y directions (i.e. no Ju ,Jm ). The x direction moment, Ju , is non-existent because the outer wing panel is free to turn about the x-axis and thus no moment is resisting its motion. The same can approximately be said for the y direction moment, Jm because the chordwise location of the pivot point is such that it provides no pitching constraint. This is not completely tnie because there is a moment, MW, at the hinge connection between the center panel and outer panel, which is resisting the motion. However, an engineering decision was made to ignore Jim to reduce the numbef of unknowns.
The equations of motion were developed for the body, center panel, and both wing halves separately and then related through their kinematics. This resutted in a set of 24 linear equations with the 24 unknowns being the date variables (U,V,W,P,Q,R) and the reaction forces and moments at the hinge connecting the center panel and outer panels as well as at the pivot point. As was the case with the 2-panel model, a body-fixed axis system was used for the fuselage. The center panel and outer panel both had an axis system which is continually aligned with the body's axis system. 43.1 Body Equations Of Motion
The body equations of motion are:
4.2.2 Centet Panel Equations Of Motion
Tgure 9: Center panel The center panel equations of motion are:
4.2.3 Port Wing Equations Of Motion
Figure 10: Porî wing 4mX4 Starboard Wing Equations Of Motion 4.2.5 Kinematic Analysis The kinernatic relationship between the body and the wing is similar to th8
2-panel model. However, an extra relationship is necessary between the body and the center panel.
Using the axis system shown in Figure 12, the following equations can be developed (See Appendix D for a complete development):
- Fmwe 12: Axas for body and wing
25 port wing velocities:
port wing accelerations:
starboard wing velocities: starboard wing accelerations:
Using the axis system shown in Figure 13 ,the velocities and accelerations of the center panel are:
Figure 13: Axes for body and center panel The x and y components @, . D,) of the distance from the center panel
c.g. to the body c.g., along with the derivatives @,, , D*) are equal to zero, but
are added to the equations for the sake of completeness.
4.2.6 Complete Equations Of Motion (3-Panel)
Making all the appropriate substitutions yields the complete non-linear equations of motion for a 3-panel ornithopter:
These are a set of 24 simuttaneous differential equations with 24
unknowns(U,V,W,P,Q,R, Li,Rwi, Rzw, Mm,, Mwi, FM, Fvri, Fmi, JNW
Rm, Rnn, MW, MM, FM, FW, FnR, Jm). Given the initial conditions, these equations were solved numerically using a Gauss-Jordan dimination routine to obtain the derivatives of the state variables. These variables were integrated using a second-order Runge-Kutta routine to obtain the desired variables.
Since the flapping is relatively slow, a quasi-steady aerodynarnic model was used where unsieady wake effects are ignored. Also, linear aerodynamics was assumed in the analysis. Unlike conventional aircraft where the angles of attack used must be small to be considered in the linear range, the ornithopter can have larger angles of attack (approximately 15 degrees). This is because the flapping causes separation to occur at larger-than-usual angles of attack
(Ref.9). This condition, known as dynamic-stall delay, allows higher instantaneous angles of attack to be reached without stalling.
To examine the aerodynamics of the wing, the segment method was used.
This involves dividing the wing into a finite number of segments, analyzing the forces and moments of each segment, and finally summing al1 the segments to
obtain the total forces and moments on the wing.
The aerodynamic terms for the body and tait were analyzed separately
and then summed together.
5.1 Flapping Angle
The flapping is considered to be sinusoidal.
Port winq:
The flapping angle and its derivatives for the port wing are:
y, = y, cos(ot)+dihedral angle P,, = y, = -yoocos(ot)
tarboard winq:
The flapping angle and its derivatives for the starboard wing are: 5.2 Twist Angle
The twist varies linearly along the wing span and is 90 degrees out of
phase with the flapping to achieve optimum thnist.
5.3 Body Contribution
53.1 Body Drag The drag of the body consists of two separate effects: the drag at zero angle of attack, and drag due to a finite angle of attack. Because the body is fairly streamlined, it was assumed that the drag at zero angle of attack was due to skin friction only.
The drag at a finite angle of attack can be obtained frorn USAF Stability and Control Methods Datcom (Ref. 18).
The drag can now be found by summing up the two drag coefficients and using the drag equation: 5.3.2 Body UR Due to symmetry, the lift of the body at zero angle of attack was assumed to be zero. The lift at non-zero angles of attack was obtained from USAF
Datcom. The equation for lift-curve dope is given by :
The lift can ihen be found from:
53.3 Body Pitching Moment Due to symmetry, the pitching moment of the body at zero angle of attack was assumed to be zero. The pitching moment at non-zero angles of attack was obtained from USAF Datcom. The equation for pitching-moment-CUwe slope is given by:
- CM. - Vol
The pitching moment is given by: 5.4 Tail Contribution
5.4.1 Total Angk of Attack of Tai1 The total angle of attack of the tail is composed of three components:
1) angle of incidence of tail
2) induced angle of attack due to plunging(Wb) and pitch rate effects
3) induced downwash from the wing
1) Angle of incidence is given by an equilibrium angle of attack
2) plunging:
Plunging causes an induced angle of attack which, after assuming small
angles, is
pitch rate: The pitch rate effect consists of two parts -
i)rotation of the body about the c.g. causes an induced plunging velocity at
the tail
ii) rotation about the aerodynamic center of the tail has an effect similar to
adding chamber to the lifting surface
3) downwash e Using an average lift coefficient the downwash was estimated from Etkin,
(Ref.3, Figures 6.6.1 ,B.6.2) which provide the ratio of downwash over lift
coefficient.
The downwash was obtained by multiplying this ratio by the estimated lift
coefficient. The time it takes for the downwash to reach the tail is estirnated
by :
Thus the total angle of attack of the tail is:
5.4.2 Tai1 Lift The lift at the tail can be found using:
5.43 Tai1 Dng The drag of the tail can be estirnated using the conventional lift-drag polar for a wing:
Thus the drag of the tail is: 5.5 Comblned Body and Tall Contribution .
Considering drag to be parallel to the local flow and lift to be perpendicular, the angle of attack is taken into account to transforrn the equations into the body-fixed axis (See Figure 14).
I Figure 14: Lift and drag transformation
In the body-fixed axis system, the lift and drag can now be calculated as follows:
Applying a small angle approximation means that: sin6d cos6 = 1 The equations simplify to:
Using this information and summing both the contributions from the body and tail for üft, drag, and pitching moment gives the following aerodynamic terms: 5.6 Outer Wing Panel Contrlbutfon
The twisting of the wing which was neglected in the dynamic analysis
must be taken into consideration in the aerodynamic analysis. The twist varies
linearly along the wing and is 90 degrees out of phase with the flapping angle.
The thrust is assumed to be generated entirely by tilt of the lift vector(Figure 15).
c Figure 15: Tiit of lift vector to produce thrust
The discrete-element method is used to divide up the wing into a finite
nurnber of segments of width Ar. The distance from the wing root to the segment
is *P. Each segment is analyzed separately and the forces and moments are
summed together to obtain the total for the wing. The velocities are found for
each segment. However, these velocities are parallel to the body-fixed ais system of the body. It is useful to obtain these velocities parallel to the wing (i.0. a body fixed ais of the wing) and the following transformation accomplishes this:
O port wing [?:]= clefy,] [?il wwci 0 -siny, cosy, W,I,
starboard wing
5.6.1 Wing Lift The angle of attack of the wing is similar to that developed foi the tail except no downwash is required, pitch rate about c.g. is included in W, ,and a linear twist ra is added.
With this, the lift of the segment of width Ar can be calculated.
1 starboard wing L~~~~~= -p(uI,t + WL)cd( r, W~.~~wlrz 2 Sm63Wing Drag The drag for a segment is given by the conventional lift-drag polar:
starboard wing
5.63 Wing Pitching Moment The pitching moment at zero angle of attack has to be taken into account and was estimated using Etkin(Ref. 6, Fig 88.2)
Using the srnall angle approximations, the aerodynamic ternis for one segment are:
Port wing: H - Starboard wing :
To get these aerodynamic terms back into the proper coordinat8 system
(i.8. axis system aligned with the axis system of the body), the following transformation matrix is used:
port wing
starboard wing
5.7 Center Panel Contribution
The center panel is considered rigid which means that twisting is neglected. Like the wing, thrust is assumed to be generated entirely by tilt of the lift vector(Figure 1 5).
Since the center panel only plunges in the veaical diredion, A is easier to analyze aerodynamically and the segment rnethod is not necessary* 5.7.1 Center Panel Lift The angle of attack of the center panel is similar to that developed for the outer wing panel except no twisting is taken into account.
With this, the lift of the center panel can be calculated.
5.7.2 Center Panel Drag The drag for a panel is given by the conventional lift-drag polar:
5.73 Center Panel Pitching Moment The pitching moment was estimated by:
Using the small angle approximations, the aerodynamic terms for the center panel are: 6. LATERAL AERODYNAMIC ANALYSE
6.1 Body contribution Stability derivatives were used to estimate the contribution from the body.
6.i.l Body Sideslip Force The stability derivative used to measure the sideslip force can be found by the following formula:
The sideslip force can then be calculated:
6.13 Body Yawing Moment The equation for the stability derivative is:
The yawing moment is given by: 6.U Body Rolling Moment The formula for estimating the rolling moment stability denvative is:
Cu, = -îcpCm from which the rolling moment can be calculated.
6.2 Hoilzontal Tail Contribution
The horizontal tail contribution to the lateral stability is minimal, especially since there is no dihedral for the tail. The only lateral characteristic modeled was the change in angle of attack during a roll. When the aircraft rolls, the angle of attack varies linearly with the right wing tip having a value of bh/2uband the left wing tip (Figure 16). This change must be taken into account for the spanwise distribution of angle of attack for the tail developed in Section 5.4.1.
Figure 16: Changes in angle of attack due to a dl p
44 6.3 Fin Contribution
6.3.1 An Sideslip Angle The sideslip angle of the fin is composed of two components:
1) control-surface angle of fin
2) induced angle of attack due to siddip(Vb) and yaw rate effects
1) control-surface angle Ba
2) sideslip: pi = -Vb Ub
yaw rate The yaw rate effect consists of two parts - i) rotation of the body about the c.g. causes an induced velocity at
the fin
ii) rotation about the aerodynarnic center of the fin
There is also an effect from the sidewash which is analogous to the downwash for the tail. However, due the complexity in estimatirtg it, the sidewash contribution was negleded.
Thus the total sideslip of the fin is: 6.3.2 Fin Side Lift The side lift can be found using:
6.3.3 Fin Drag The drag of the fin can be estimated using the conventional lift-drag polar for a wing:
Thus the drag of the fin is:
The fin aerodynamic forces are:
nie Xd tenn can be added to the total body drag term Xrm developed in the longitudinal aerodynamics (Section 5.5).
Summing both the contributions from the body(Section 6.1) and fin for side force, yawing moment and rolling moment gives the following aerodynarnic terms: 6.4 Outer Wing Panel Contribution
Unlike the horizontal tail, the wing plays a major role in the lateral stability of the aircraft. As was the case with the tail, when the aircraft rolls, the angle of attack varies linearly with the right wing tip having a value of pbbrJ2uband the left wing tip -gbw/2ub(See Figure 16). Also there is a component of angle of attack equivalent to (V&)siny, which arises when the plane yaws (Figure 17).
These changes must be taken into account for the total angle of attack of the wing developed in Section 5.6.1.
Fiure 17: Changes in angle of attack due to a yaw r Again the segment method is employed to calculate the lateral aerodynamics tems. These aerodynamic ternis for one segment are:
Port wing:
Starboard wing:
Although the values for Y, and N, are equal to zero in this coordinate system, this will not necessarily be the case when the aerodynamic tenns are transformed back to the body-fixed mis system using the coordinate transformation in Section 5.6.3.
6.5 Center Panel Contribution
The lateral aerodynamics of the center panel is much like that of the outer wing panel with the angle of attack component. pcpb(d2uep.king present when the aircmft rolls. 7. COMPUTER PROGRAM
Solving the non-linear equations of motion analytically is a diflicult task; and thus a numerical technique was used. This took the fom of a cornputer program written in FORTRAN. Initial conditions of the state variables, flapping information, ornithopter geometric characteristics, and aerodynamic information was input into the amputer program. This information was read from four data files: BODY.DAT, WING.DAT, TAILDAT, INIT.DAT. The latter file contains al1 the initial information required while the other files contain information for the body, wing, and tail, respedively. The program solves the equations sirnultaneously at tirne t using a Gaussian Elimination routine. This method was used because it is straightfonuard, easy to use, and is efficient as any other method. The major drawback is that 1 is computationally slower than the other routines but, for this case, computational time was not a crucial factor. The answers retumed by the equation wlver were the derivatives of the state variables. To obtain the state variables, a second-order Runge-Kutta technique was used to integrate the derivatives. Nonnally , a fourth-order Runge-Kutta is used, but in this case fast amputational time and high accuracy were not required. The output from the program was written to a file (0UTPUT.DAT). A very powerful feature of the program is that any perîuibaüons of the state variables or changes in the control systems et time t can be specified in the input. Solving these equations by using a computer program, as opposed to finding an analytical solution, is much easier; however, the major diffiartty arises when predicting how one variable depends upon another. The approach for solving the linear equations of motion for conventional aircraft (shown in Etkin) produces root locus plots which provide valuable information of how certain parameters, such as C, , may affect the aircraft's dynamic stability. Valuable insight is provided into the stability of the aircraft from such plots. Such insight cannot be gained from the computer analysis because of its cornplexity.
Determining whether the value of a certain parameter will cause any instability is a trial and error approach.
8. DISCUSSION OF RESULTS
8.1 Longitudinal Stability
The longitudinal stability was determined from the computer program by setting the lateral variables (P,R,V,Q,y)equal to zero. The first test to determine the accuracy of the program was performed on the Canada goose to examine its longitudinal stability with the inputs being provided by Grant. The resuts were exactly those of Grant's, which meant no recoding flaws had been made and this instilled confidence in the veracity of the program.
The next step was to input data from Fowler's human-powered ornithopter for the 2-panel model. Unfoiainately, many of the inputs, such as the aerodynamic coefficients and the geometric values, were not provided and had to be estimated. This led to erroneous results foc the longitudinal stabiiity
variables.
8.1.1 Piper Comanche PA24-tSO
An aircraft for which there was an abundance of information is the Piper
Comanche PA24-250 (Figure 18). The longitudinal and lateral stability of the
Comanche had ben studied by Zingg (Ref. 20) using the ünearized equations of motion. 8y setting th8 flapping frequency and flapping angle equal to zero, and using the information provided by Zingg, the 2-panel and 3-panel models were compared to Zingg's linearized model for the Comanche.
Fguie 18: Three-view drawing of the Piper Comanche PA24-2% This aircraft is longitudinally stable with two modes: phugoid and short- period. The phugoid mode (Graph 1) is characterized by lig ht damping and a long penod. Graph 1 indicates that the Bpanel model and 3-panel model are also stable with a phugoid motion closely following Zingg's linear model. The 2- panel model is more damped and has a lower penod, while the 3-panel model is less damped with a higher period. However, it is clear that al1 three models converge to a z' increase of about 5 m from the reference level.
The short-penod mode (Graph 2) is characterized by heavy damping and a short period (hence the name). From Graph 2 it is evident that the Bpanel model and &panel model are stable and comparable to the theoretical short- period motion. Again, the Ppanel model is more damped and has a lower period, while the &panel model is less darnped with a higher period. The convergence to about 1 m above the reference point is apparent for al1 three rnodels.
The similarity of the resub between the linear model with that of the 2- panel and 3-panel models confirms that the Bpanel and 3-panel models have been developed correctly for longitudinal stability calculations. However, the correctness of the lateral stability modeling will be examined later in Section
8.2.1 .
8.1.2 2-Panel "Mr. Bill" With the success of the Piper Cornanche comparisons, it was time to begin inputting the values for 'Mr. Bill" because al1 the relevant data was readily available. The initial conditions given were: forwardvelocity[U]: 13.7mls(45fVs) -
plunging velocity[W] : O m/s
pitch rate[Q]: O degls
pitch [O]: O deg
flapping frequency[w): 3 Hz
The results indicate that after a transient stage, 'MI. Bilî" reaches an equilibrium Mate. Due to the f lapping motion, sinusoidal-type variations of the longitudinal variables occur about the equilibrium value. The resutts are as fol lows:
forward velocity[U]: 14.5 f 0.1 mls (48 Ws) (Graph 3)
forward acceleration[U '1: 1 f 3 rn/s2 (Graph 3)
plunging velocityw: O f 1 mis (Graph 5)
plunging acceleration[~I]:O f 20 m/s2 (2g) (Graph 6)
pitch rate[Q]: O î 15 deg/s (Graph 7)
pitch [O]: O k0.5 deg/s (Graphs 8 & 9)
A plot of the flight-path trajectory shows that after an initial vertical decline, the ornithopter achieves level flig ht (Graph 10). The peak-to-peak amplitude of the vertical displacernent is 10 cm (Graph 1l)which is higher than what was observed in the actual flights (Ref. 5). Another point of concem is that the plunging acceleration is varying between f 20 m/s2 (2g). 60th the high vertical displacement and plunging acceleration are a resut of the simple Bpanel model. The use of the 3-panel model dramatically decreases these values because it serves to balance the inertial and aerodynamic loads(Ref. 5).
1.$Panel Mr. Bill" Like the 2-panel model, the results from the 3-panel model also indicate sinusoidal variations about the equilibrium value. lncorporating the 3-panel feature greatly improves the results as shown below:
forwardvelocity[U]: 14.7f 0.05 m/s (48Ws) (Graph 12)
forward acceleration[~'1: 0.5 i 1.25 m/s2 (Graph 13) plunging velocityM: O * 0.25 mfs (Graph 14)
plunging acceleration[w '1: O I5 rn/s2 (0.59) (Graph 15)
pitch rate[Q] : O I4 deg/s (Graph 16)
pitch [el: O f 0.25 deg/s (Graphs 17 & 18)
The trajectory shows that the &panel ornithopter is able to achieve level flight after an initial drop (Graph 19). The two points of concern, high plunging acceleration and peak-to-peak plunging amplitude, have been rectified by the 3- panel model. The plunging acceleration variation is reduced to f 5rn/s2 (OSg), while the peak-to-peak amplitude of the vertical displacement is 2 cm (Graph
20). These parameters indicate ver' tolerable levels if a pilot were seated inside. û.1.4 2-Panel Full-Sale Onithopter (FSO) The initial conditions given for the FSO were the same as for "Mr. Bill" except for the following two variables:
forward velocity[U]: 23.77mls (78 Ws)
flapping frequency: 1.O5 Hz
The FSO goes through an initial transient stage before reaching an equilibrium state. It also experiences sinusoidal-type variations of the longitudinal variables about the equilibriurn value. The resuts aie:
fomrard velocity[U]: 22.75 10.375 m/s (75 ïüs) (Graph 21)
forward acceleration[~Il: 0.5 f 2 m/s2 (Graph 22)
plunging velocity(\l\lj: O f 1.25 m/s (Graph 23)
plunging acceleration[~'1: O I8 m/s2 (1 g) (Graph 24)
pitch rate[Q]: O f 8 deg/s (Graph 25)
pitch [el: O Ildeg/s (Graphs 26 & 27)
A plot of the flight-path trajectory shows that aftei an initial vertical rise, the ornithopter achieves level flight (Graph 28). The peak-to-peak amplitude of the vertical displacement is 40 cm over a range of 10 m (Graph 29). The plunging acceleration is varying between f 8 rn/s2 (1g). 60th the high vertical displacernent and plunging acceleration aie, again. the result of th8 simple 2- panel model. 8.1.5 1PuieI Full-Sale Ornithopter (FSO) The same trends that were noticed for "Mr. Billn are also present for the
FSO. Namely, the plunging acceleration is reduced to f 2.0 rn/s2 (0.2g) and the peak-to-peak vertical displacement is reduced to 15 cm (Graphs 37 and 38).
The program yields the following results:
foiward velocity[U]: 24.6 f 0.1 mls (80 Ws) (Graph 30)
fotward acceleration[lll]: O f 1.25 m/s2 (Graph 31)
plunging velocitym: -0.2 f 0.2 mls (Graph 32)
plunging acceleration[~I]: O f 2.0 m/s2 (0.2g) (Graph 33)
pitch rate[Q] : O f 1.5 deg/s (Graph 34)
pitch [el: -0.3 f 0.3 degls (Graphs 35 & 36)
8.2 LATERAL STABlLlTY The lateral stability of al1 four cases was examined using the complete equations of motion, with the output variables of interest being : roll angle, yaw angle, sideslip velocity, and sideslip acceleration. The longitudinal variables could not be set to zero because al1 the variables are coupled and thus the complete equations of motion were used. The lateral stability analysis for the
Canada goose and Fowlets human-powered ornithopter was ignored because very limited information was available. 8.2.1 Piper Cornanche PA24-ZSO The validity of the lateral stability predictions of the Ppanel and 3-panel models was assessed by setting the flapping frequency and flapping angle equal to zero, and comparing it to Zingg's (Ref. 20) lineanzed model for the
Comanche.
The Comanche has two lateral modes: an unstable spiral mode, and a stable dutch-roll mode. The spiral mode (Graph 39) is characterized by heavy damping and a short period. Graph 39 indicates that the 2-panel mode1 and 3- panel model are also unstable, rnatching the linear model at the beginning but diverging from the theoretical results as the instability grows. The divergence is such that the developed models are spiraling at a faster rate, which means greater instability. This implies that the models are consewative in nature, showing more instability than there actually is. From a design point of view, this is preferred in cornparison to a model that shows less instability.
The dutch-roll mode (Graph 40) is characterked by light damping and a long perîod. From Graph 40 it is evident that the Bpanel model and 3-panel model are stable and comparable to the theoretical short-period motion. The 2- panel model is more darnped and has a lower period, while the 3-panel model is less damped with a higher period. All three models converge close to the reference mark of O m.
The similanty of the results with the linear model confirm that the Ppanel and Spanel models have been developed carrectly for lateral stability. This, combined with the successful companson of the longitudinal stability (See Section 8.1.1), instills a high degree of confidence in the correctness of the models.
8,2.2 2-Panel "Mr, Bill" The first step in performing the analysis was to determine which initial conditions to use. lt was decided that they would be the same as those used foi the longitudinal analysis with the added initial roll of 20 degrees.
forward velocity[U]: 13.7m/s (45 fVs)
sideslip velocity[V] : O m/s roll rate[P] : O deg/s
yaw rate[R]: O degls O 1:20 deg
yaw [y]:O deg
flapping frequency: 3 Hz
The ornithopter goes through a transient phase with oscillations, but these oscillations diminish with tirne to mach an equilibriurn stage where the state variables, sideslip velocity[Vl, sideslip acceleration[Vl, roll [$], yaw [y] are equal to zero (See Graphs 41 -44). Unlike the longitudinal case, there are no sinusoidal variations about the equilibnurn value.
This motion is a charactertic of a dutch-roll mode which is stable because of the diminishing oscillations with time. The period is 6.3 seconds while the time it takes to half amplitude(tta) is 3 seconds. The simulation shows that with a positive initial roll (right wing down, left wing up). the aircraft shifts to the right by about 2.1 4 m after the transient stage damps out (Graph 45). û.2.3 $Panel "Mr. Bill"
The cornputer simulation for the 3-panel mode1 shows that the equilibnum stage for the state variables (V,V',@)is zero (See Graphs 46-49). The motion is a stable dutch-roll mode with a period of 6.3 seconds and the time-to-half amplitude of 1.4 seconds. The period is essentially the same as the 2-panel "Mr. Bill"; however, the time-to-half amplitude has decreased by a factor of 2. Because the 3-panel design is more damped. this implies that it is more stable. The trajectory of the &panel ornithopter also shifts to the right but by only 1.4 m (Graph 50).
L2A 2-Panel FulCScale Ornithopter (FSO)
The initial conditions given for the FSO were the same as for "Mr. Bilr except for the following two variables: forward velocity[U]: 23.77rn/s (78 fVs) flapping fiequency: 1.O5 Hz Graphs 51 to 54 reveal that the 2-panel FSO goes through a stable dutch- roll mode with the state variables (V,V,+)damping out to an equilibrium value of zero. The period is 5.5 seconds while the time-to-half amplitude is 1.3 seconds. The trajectory shifts to the right by 101.35 rn (Graph 55).
Be23 3-Panel Full-Scale Omithopter (FSO)
Once again, the state variables of the 3-panel model damp out to zero (See Graphs 56-59) having a period of 5.3 seconds and a time-to-half amplitude of 0.74 seconds. The same decrease in ttn is apparent as it was for 'Mr. BilF. This indicates that the 3-panel design is inherently more stable than the 2-panel model. The trajectory shifts to the nght by 70 m (Graph 60). 9. CONCLUSION
The main objective of this study was to examine the complete non-linear flight dynamics of a full-sale ornithopter. This goal has been attained with the results indicating that the full-scale ornithopter is longitudinally and laterally stable. After an initial transient phase, the state variables reach equilibrium values with small sinusoidal variations about this value. The results show that a
3-panel ornithopter model experiences lower accelerations and lower peak-to- peak displacement amplitudes than the 2-panel model. This information will prove valuable because the pilot will normally be able to tolerate a 0.59 variation and small displacements.
The completion of this stage means that al1 the necessary equations for a complete dynamic analysis have been developed and it would be a simple matter to refine both the dynamic and aerodynamic models. For example, the wing can be modeled with a double taper instead of as a flat plate and aeroelastic effects can be taken into account (including the twisting of the wings in the dynamic analysis). More sophisticated models which use non-linear unsteady aerodynamics can also be incorporated. Further, the data inputs such as: moment of inertias, lift-curve slopes, and downwash angles can be refined.
The control systems may include feedback loops to add a dimension of piloting realism to the rnodel. At this stage, it is possible to determine responses to different control inputs, flight conditions. and flapping frequencies. It would also be a valuable exercise to see how the location of the center of gravity would affect the stability.
Atthough the simulation results are very promising, it will only be until the full-SC& ornithopter is constructecl and flown, with a pilot inside, that confirmation of th8 results can take place. REFERENCES . [l] Anderson, John D. Fundamentais of Aerodvnamics (2nd editionl,. New York: McGraw-Hill Inc., 1991. pp.247-374
[2] Ashley, Holt Enaineenna- Analysis of Fliaht Vehicles, Don Mills, Ontario: Addison-Wesley Publishing Company, 1974.
[3] DeLaurier, James An Aerodvnamic Model for Flawina-Wing FliahtLThe Aeronautical Journal, April 1993.
[4] Delaurier, James The Oevelo~mentof an Efficient Ornithabter Wina. The Aeronautical Journal, May 1993.
[5] DeLauner,J & J. Harris A Studv of Mechanical Flao~ina-WinaFlia ht, The Aeronautical Journal, October 1993. [6] Etkin, Bernard pvnamics of Fliaht - Stabilihr and Control, Toronto: John Wiley and Sons lnc., 1982. pp.13-84,85-1 55,167-21 0
[7] Etkin, Bernard Dvnamics of Atmosbheric Fliaht Toronto: John Wiley and Sons Inc., 1972.
[8] Fowler, Stuart A Feasibilitv Studv for the Desian of a Human- Powered Fiamina-Wina Aircraft, B.A.Sc. Thesis, Division of Engineering Science, Facuity of Applied Science and Engineering, University of Toronto, April 1993.
[9] Grant, Peter The Lonaitudinal Stabilitv of the Canada Goose. Project submitted to Dr. J. Delaurier for AER1137S, University of Toronto lnstitute for Aerospace Studies.
[1 O] Greenwood, Donald princides of Dvnamics, New Jersey: Prentice-Hall Inc., 1965. [Il] Hoerner, Sighard Fluid Dvnamic Draa Brick Town, New Jersey: Published by author, 1965.
(121 Hoemer, S & Henry Borst Fluid Dvnamic Lift Brick Town, New Jersey: Published by Mrs. Liselotte Hoemer, 1975.
[13] McGuire, Peter Multibodv Simulation of the Canas Goose, Project subrnitted to Dr. J. DeLaurier for AER502F, University of Toronto lnstitute foi Aerospace Studies.
[id] Meriam, James L. Enaineerina- - Mechanics: Statics and Dv namiCS. New York: John Wiley and Sons Inc., 1980.
[15] Pant ham , Satyaraj Classical Dvnarnics of Variable Swee~Wing Aircraft. PhD Thesis, Department of Aerospace Engineering, lndian lnstitute of Science, Bangalore-560012(lndia), September 1993.
[16] Perkins, C. & Robert Hage Airplane Performance. Stabilitv and Control, New York: John Wiley and Sons Inc., 1949.
[17] Skarsgard, Andrew The lm~lementationof Flamina Wing Pro~ulsionfor a Full-Scale Omitho~ter, Masten Thesis, Dept. of Aerospace Engineering, University of Toronto, June 1991.
[18] United States Air Force USAF Stabilitv and Control DATCOM. Wright-Patterson Air Force Base, Ohio: Revised August 1968.
[19] Wegener, Peter What Makes Aimlanes Flv?. New York: Springer-Verlag, 1991.
[20] Zingg, D.W. A Studv of the Stabilitv of a Sinale-Enaing Aircraft .Project submitted to Dr. J. Delaurier for AER2059F, University of Toronto, Jan. 1980 APPENDIX A - GRAPHS Vertic al Displacement vs. Horizontal Displac ement Cornparison of Piper Comanc he PA24-250 Phugoid Mode Using ~ifferehtModels (U=76.4 m/s,Theta=4 deg)
Theoretic al Lineor Model 2-Panel Model (f1op.f req=O)
C______3-Panel Model (flap.freq=O)
Graph 1
...... -.- ...... C...... *...*..-- ...... -*...... *-...... --...... i...i..*..,.. .., . , ...... a ...... *...... *.*-.." ...... " ...... - ...... CV ...... -*-...... P.. ... , ...... , ...... , ... .- ...... , ...... , ...... S...... S.., ...... a ...... S.*.. ..-*...... -.* ...... -.a .-...... -a-,- ...... *..+...... *.-...... *...... -0 ...... iIiliIiIil~I~IiI~. m7C\Lnr>-6>l'm ll54t464 - 7 7 7 7 7 - 7 7 ....m... L ...... C..
*~*.,.*.r.r..t*...*-..,-r
*..> ......
......
...... ,...... ,. . , ..., ...... ,. . , ...... f...... , .*-z... ..- . .
0 =- Ir,
Vertic al Displac ement vs. Horizontal Displac ement (3-Panel Mr. Bill, Flapping freq=3~z)
b,.., ...... ,......
"...... e...... * ...... * ...... ,,...... ,., " ...:...... *...... ,.;......
-,,.,* ....I ...... ,..,*,... 1...... I....'...... ,4.,..,...... I ...... *,.* ....* .,,. ,,.,,. ,,,. , - .,.,. .,...... , '<.,'+'...~...i v...... ,...... ,...... ,...... *,..,. ,.,., ,.
m...... ,...... <...... -t...,...... t...... ,...... :...... :...... :,.,.*..,,.,...
Graph 20 u') tTi > c
..... S....*-..* ...... -P...... S. ..-...... * ... ..-...... --.- S..
...
...... ; ...... a ..... - ...... ,...... a...... ** ...... -
P.. f .... .S...... - ...... -...... : ;;.. i- .: -- -7--c. , . . .-...... -..- ...... r'......
a.... i...... i...... l...... I.. .. rl... .. :...... I...... :...... I:: . . ...
f..
...... , ......
.... :...... :. ... C.. ..:....:....:......
l ...... -.*...... cno . Forward Displac ement vs. Sideslip Displac ement Cornparison of Piper Comanc he PA24-250 Spiral Mode Using Different Models (~=76.4m/s,Psi=-1 O deg)
Theoretic al lineor Model ,.. , , . . . . 2-Ponel Model (flop.freq=O) ------3- Panel Model (flop.freq=O)
Graph 39
'"Y
..-...... -...... fC,*.*.-.C.....-.....*.*...... *...... *.....-...... *...... If...... I ...... * .... *...... ++.-...... -...... *t-..--**...... +..... -...... *.*.. .-,.- ...... -. . . *. .-...... - ...... it...... f...... *...... - ...... -...... + ...:...... : ...-... . *...... - ...... * ...... * ...... - . . . . .-...... -...... - ...... - *. - ...... * ...... * - -. *...... -...... ,. . . . . *...... * ...... ,...... ,... , ...... , ...... - . . . , ...... - -...... * ...... *...... , ...... *...... -. . * ...... - ...... * ...... *...... f...... *...... *-...... *...... * .-..-. . . . - ...... , . . . . .-..S...... *.. . -. ~~~~~~~~~~~~~~~~~~~~~....-. -*.-. .-..-. - - -. a...... - ...... *.-i...... -...,...... -...a ......
...... a...... m...... -A.- ...... :...... !A
APPENDIX B - Derivation of Equations of Motion for a Rigid Body
For a rig id body:
Newton's 2nd Law: - F=ma
where - n=ffi,+Qa,+Rn,
- - n2 n3 ~,=&~+%+lh,+ P Q R ruvw =G1+Yn2+Wn3 + E,(QW-RV) X2(PW-RU) + E3(PV-QU) =@+QW-RV)E~ + (v-PW+RU)~,+ (W+PV-QU)% :. Newton's 2nd Law in cornponent form is: -
The torque equations are:
Evaluate each term 2nd terrn:
let ?=XE, +y% +ui,
= (b2-QX~-~Z+PZ~)~, + (-Pxy + QX~+ QZ~RZ~)Ï~~ + (RX* - Pxz - Qyt + R~~)Ï~~ The moment of inertia terms are defined as:
I,, = Jrydrn
Adding al1 three terms yields the torque equations. Rigid Body Dynamic Equations
Force Equations: x =~(u+Qw-RV)
Moment Equations: L = I,P-I,~Q- I,R +I,,PR+(I, - I,,)QR + I,(R~ -Q*)-I,PQ M = I~Q~I~~P~I~R-I~~QR+(I~~-1,)PR + 1,(p2 -R~)+I~PQ APPENDIX C - 2-Panel Complete Non-Linear Equations of Motion
Assumptions:
rigid body (no aeroelastic effects) mass is constant negligible angular momentum of rotating machinery(engines, etc) control systems fixed negligible buoyancy negligible twisting of wing in the dynamic analysis (however twisting is considered in the aerodynamic analysis) wings are thin rectangular plates
BQ& Use a body-fixed axis system at the center of gravity
Assume that the "wing slice" is forward and below the body c.g.
Let d,=d,,~+d,,~+d,Z Define Y: yaw angle (+ right) 8: pitch angle (+ nose up) O: roll angle (+ right wingdown)
The cornponents of weight are:
S#eYiew
X, = -mbgsin eb Y, = m,gcos 8, sin ab 2, = mbgcos8, cos@,,
Add al1 extemal forces and moments and substitut0 into the rigid body equations from Appendix B. Winas
OR~IeR1 Wng Use a stability axis system at c.g. of wing
Note tha! with this axis systern, the moments and products of ineitia are changing with time. Note from the wing moment. of inertia analysis (Appendix E) Ixywl= Ixrrl= Ixxwl = IlW= Ixzwl = 0 oaidlRightl Wing
Use a stability axis systern at c.g. of wing.
Note that with this axis systern, the moments and products of inertia are changing with time. Note from the .wing moment. of inertia analysis (Appendix E) Ixw2 =Im2 =Iuw2 =Ixp2 = O
Kinematic Analvsis
We need to relate the velocity and acceleration of the body and wing. velocity :- vc =&+ab x (-&)+%
port wing velocities:
starboard wing velocities:
port wing accelerations:
starboard wing accelerations: The following substitutions for the reaction forces and moments can be made:
The geometric properties of the two wing halves are identical, which means
Also, since the wing is attached rigidly to the body,
Substitut8 the above relationships along with the kinematic equations into the equations of motion for the wing. For the port wing equations of motion Simplifying these equations yields:
For the starboard wing equations of motion Simplifying the equations yields: Summarv - Eauations of Motion -
body equations:
port wing equations: starboard wing equations:
Reananging al1 the unknowns to one side gives:
Traiectonr From page 103 Etkin,
dx' -= UcosBcosY + V(sin@sin8cosY-cosOsinY)+ W(cosasinBcosY +sinOsinY) dt APPENDIX D - &Panel Complete on-Linear Equations of Motion
Assumptions:
1) rigid body (no aeroelastic effects) 2) mass is constant 3) negligible angular momentum of rotating machinery (engines, etc) 4) control systems fixed 5) negligible buoyancy 6) negligible twisting of wing in the dynamic analysis (however twisting is considered in the aerodynamic analysis) 7) wings are thin rectangular plates
centet panel TA pivot
Let the distance from the body cg. to the wing pivot point be:
Note that there no pivot-point moments in the x and y directions (Le. no Jlb ,JUb ). The x direction moment, JL~, is non-existent because the outer wing panel is free to tum about the x-axis with no moment resisting its motion. The same can be said for the y direction moment, Ju) because the chordwise location of the pivot point is such that it provides no pitching constraint. This is not completely tnie because the moment, MW, at the hinge connedion between the center panel and outer panel, resists the motion. However, an engineering decision was made to ignore Jls to reduce the number of unknowns. Define 'Y: yaw angle (+ right) 8: pitch angle (+ nose up) O: rollangle (+rightwingdown)
The components of weight are:
SideV[ew
X, = -m,gsin 8, Y, = m,gcos 8, sin ab 2, = m,gcosû, cos 0,
Add al1 extemal forces and moments and substitut0 into the ngid body equations Becauseof symmetty IN, Iw=O
Winq
With the 3 panel design, the wing is divided into two dynamic portions: 1) center panel 2) outer panel. The wing motion can be modeled as a superposition of the motion of both segments. Outer Panel
Use a stability axis system at c.g. of wing
Note that only the x moment (ML~) is non-existent at the interface of the outer and center panel because there is a hinge connection, which means that the outer panel is free to rotate about the x-axis. Also, with this axis system, the moments and products of inertia are changing with time. Note from the wing moment of inertia analysis Iaw = 1- - inW= iZF = 1- =O
tarboardlRiaht1 Winq
Use a stability axis system at c.g. of wing
Note that with this axis system, the moments and products of ineitia are changing with time.
Center Panel
Because of syrnmetry I*, 1- = O Also in this case, 1, = O because the center panel is treated as a flat plate.
The center of gravity is located in the middle of the plate with the reaction forces being located from the center of gravity a distance of f bd2 in the y direction only. There is no x and z distance components This implies that the reaction forces only create moments in the x and z direction as cm be seen from the equations below. Kinematic Analvsis
odv and Center Panel
Also- let the distance from the center panel c.g. to the body c.g. be: Db= ~~~i+ D~~j + D *k This is changing with so that 5, =KI ve locity: - vb +nbx (Db)+Vml - - ijk
Pb Qb Rb Dxb Dyb D~
(QbDlb-~b~~b$- (P~D* s+(pbDyb -Q~D,~)E acceleration: œ A, +Sbx ((Db)+nbx (nb xV, +ZRbxVd +Ani
O O - - ijk &XE,= P, Q, R~
Dxb Dyb Dzb odv and Wing
Since point a and b are rigidly attached: - - ab =na=P~~+Q,~+R~T; - - n, =P,i port wing velocities:
starboard wing velocities: - A, =ubi+vbj+Kk - - - ijk nbxHb = pb Q~ rjb H%b H,b H, port wing accelerations:
starboard wing accelerations: The following substitutions for the readion forces and moments can be made:
The geometnc properties of the two wing halves are identical, which means m,, = m, = m,
Also, since the wing is attached rigidly to the body,
One substitutes the above relationships, along with the kinematic equations, into the equations of motion for the wing. For the pott wing equations of motion Simplifying these equations yields:
For the starboard wing equations of motion, one has Simplifying these equations yields: Substitue the kinematic equations into the equations of motion for the center panel to obtain
These simplify to: Summarv - Eauations of Motion . body equations:
port wing equations: starboaid wing equations: center panel equations:
X,- X,- -m,gsin8-Rxwl -Rxwz = m,[ob + Q~D, -R~D~~-2~t~~~-zR~D~~ +2PbQbDyb +2PbRbDa +2QbWb-2RbVb +~Q~D*-~R~D~~ +r)xb] y=% +m,gcosûsin@- RyW1- RYw2 = m,[Yb - P~D*+ R~D~~-2~i~~~-2~$~~ + 2PbQbDxb +2QbRbDh -2PbWb + 2RbUb-3pbI)* + ~R,I),+SjYb] +m,gcosûcos4- R,, - R,, = m,[Wb + QIyb- Q~D,, - ~P,ZD* -2Q$* + 2PbRbDxb+2QbRbDyb +2PbVb-2QbUb +3pbI),, -~Q~D,~+D,]
L, L, +(b, /2)R,, -(b, / 2)R,z =I~~&+(IBp -Incp)QbRb - MMwl-MMu~ 'ryyqqb +(Ixx~-ItrpIPbRb
N, N, - (b- / 2)Rxwl+ (bq / 2)Rxw2 MNWI MNW~= 1-h + (Iyycp Ixxcp Reananging al1 the unknowns to one side gives:
O O 1 O 000 0 0 001 0 000 0 000 3:: : O O O 1000 0 0 000 1000 0 000 O O O O O 100 O O 00O 10O O00 E'? 6 O L O +%O0 O O 000~~00O O00 000 O r, O y, o*oo O O oort,oo O 000 000 O O r O O ooo O O ooo~~ok-q,,#oo %004r~'~LIC~-I)i) 0 -l O 004 O O 000 0 00O O00 Oip,OmJ&-~) O *&-qy) O -l O00 4 O O0 O O O0 O O O O0 0 oii ) O O -100 O -l 00 O O 000 O O 00 O00 *% O E$, +Oil+&O O40 O 000 O O00 O00 -r, 1, O -o, oq,'o-o,oD,-loo O 000 O 000 000 O O I, O D,-o,00~-EIC,000 0 O00 0 000 000 O O C, O O oo O o+,nooo O ooo otpo 000 O 4m O O O O00 O O 100 0000 O 010 O00 O O O O 00b/20 O O10 000+/2O O01 % 0 o~+&&b&-~ 0 O O 00 O O O 004 O 004 O O00 oig,oiSstp-4p) O **O O 00 O O O O0 O 4 O0 O 4 O00 00% O *-)tehsoliP-qdo O O0 O O O O0 O O 40 O O 400 000 t O + &O1 O O O OO~H,OlO O O00 %O0* ap, O O O 00 1 O O O00 0001 0 000 0-0 * O O O0001 00000000 1 O00 00% O aLqr O O 000 O i 000 O000 0 100 O00 + b O O O O0 O O O OO+OgO~ O k40 O00 h O O O O0 O O O 00~+044,+0 M +2QbWb -2RbVb +~Q,D* -3~,b,,+Osb] Y,, Y,, +m,gcos~sin~-m,[-2~~~yb-?R:D,~ +2PbQbDab+2QbRbD, APPENDIX E - Moments and Products of lnertia of Wings
Assume the wing is a t hin rectangular plate
1 2 2 Inni =-m,(c, + t,) 12 if we rotate the wing by -y
this is the same as rotating the coordinate axis by y:
We now find I w.r.t TJ,t direction cosine matnx
cos9Oo O cosy cos(9O0-y) oos(90'+y) cosy 1 O t= O cosy 4ny [O -,.y O t'=O cosy -&y [O &y "1 however since 7and Y=Y*=a
1 - 1, 1, = 1, sin 2 (-y)+I, cos2(-y)
9 1, 1, = 41, - I,)sin(-y)cos(-y)
dÏP = y,o sin ut (1, - 1, ) 2cos(-y)sin(-y) dt
"wv =y,osinut(sin 2 y-cos 2 y) (1, -1,) dt APPENDIX F - Order of Magnitude ~naiysisfor Twisting This order of magnitude analysis shows why the twisting of the wing can be ignored in the dynamic analysis.
Flapping occun about the mot of the wing. and thus the moment is M, = I,&" Since the wing flaps sinusoidally, P, =y0 cosait ,then Mx= 1,y0a2coso< If is the wing half-span then 1 2 =-rnw(b:)+mw(%) = 12
Therefore,
If the tip of the wing has a maximum twist angle Q, then
If G is the mean wing chorâ then 12 1, =pwcw Therefore,
Now taking the ratio of Y gives:
Plugging in the numben gives:
MW 1 APPENDIX F - Order of Magnitude Analysis for Twisting
This order of magnitude analysis shows why the twisting of the wing can be ignored in the dynamic analysis.
Flapping ocairs about the mot of the wing, and thus the moment is Mz= I,& Since the wing flaps sinusoidally , P, = y, cos an ,then Mx= ~~~y,o>~cusa If b, is the wing half-span then
Therefore,
If the tip of the wing has a maximum twist angle ab, then
If cr is the mean wing chorâ then
Therefore. 1 M~ = -m&a0d ~06a 24
Now taking the ratio of Md M, gives:
Plugging in the numbers gives: L=-M 1 Mx 50 APPENDIX G - BlRD FLIGHT
The main ofference between fixedwing aimafi and birds is the fad that birds use their wings to generate both lift and thnist. An efficient airfoil is composed to two forces: the normal force which is perpendiailar to the airfoil and the leading edge sucüon force which is parallel. The typical airfoil for birds and bats is thin and cambered, which means no leading edge suction is created.
They obtain üft and thnist by twisting their wing fonnrard on the downstroke so that the nomlforce propels them forward and produces positive lit On the upstroke, the wing tilts backwards giving positive üft but could possibly give negative thrust (See Figure G.1) However, on the upstroke birds collapse their wings to produce neither lift nor thnist [Ref.3] (See Figure 0.2).
This variable-span model allows flapping flight to be attained with constant bound vorlicity through the flapping cycle. No transverse unsteady vortex wake is shed.
At kw Reynolcrs numbers, the boundary layer on the wing remains turbulent and thus delays staliing. Many other complicateâ effects are present such as featheting, up and down movements of m'ng tip vortices, downwash, and contribution of the tail feathers.
The major advantage of bird flight is the bird's ability to change speed, shape(camber) and the angle of different parts of the wing to achieve an optimum mix of lift and thrust. APPENDIX H - COMPUTER PROGRAMS * FlmR * LONGITUDINAL STABUTY (2-PANEL) 8
*-Red in wing chamtaistics-• opEN(UNIT=Lu,FJLE=mG1 .DAT) READ(LU,*)CLALFAW~AOW,Q)OW,CW,BW,GAMAO~,TW~,W WC, @CMOW cLc=&U) !Chdck if time interval is the time spa5ficd IF(I 4.PBTIM)THEN UB=UB+PBUB !Mdprmbtion values to the statc variables WB=WB+PBWB THETA=THETA+PBQB UBC=UBC+PBUB WBC=WIKl+PBWB THETAC=THETAC+PBQB ENDIF
*-Bbck to sdve fa wing faiccs 4i time T by integratingover BW-• 1IilltirfiPelOIPI wingraodynamicf~ XAEROC=O.o ZAEROCsOmO MAERûC=OmO LAEROC=O.O
LWtO.0 !1nii;si;ll lift over entire whg ALFA=ALFAO+cOS(WYT+PH1) !Calculate wing twist angle R=J8DELR !Colcuhc dis- fhm wing roai io wing panel
!CPIculote wing panel lift and drag Lwc=CLALFAWR.o*(WwC/UWC+ALFA+R+ALFAow+( @ *RoE*(UWC~UWc+WWc*WWC)*DEI,RLCW LWtLWC+LW !Record total lift over entire wing, to be psed for domiwash DW~CDOW+(&WC+2.0/~OE*(UWC*WC+WC*WWC)*DELR+CW))882~ @ ((@W(2.0)/CW)+PI~)8(S/2.*ROE*D~+CW8~C*UWC+WC @ +WwC))
rs CONTINUE !-utoie ltmaining wing ritro farces which required no pantl ~~~m~tion YAEROc=PW/BW NAEROC=YAEROC*XWc +-Bbck po colculaie taü forces Md moments-*
!Calah& tail aero forces XAEROT=LT*(AX-(QBTï)/(2.0*üB~ALFAOT)-DT ZAEROT=-LT-DT*(ATI(QBICI?/(2,O*UB)-ALFAUI') ~CYr=iCMOT+(I~.*ROE8(UB*UB+~*WB)~*BT)-.(ZAEROT*~XAEROT @+zT
ALFAOT=Gï.QBC+ALFAOT !Fecdback of gain GT on pitch-rate 125 formot(lUi0.2) 150 #)RMAT(XXW-~3X,W-VEL~3X,'PITCHRATE',3X, @WKHANCrF'4~'GAMA',7X((ALFAT) ~T(2X,F8.4~.4~04,!X~.4,!5X,F7.4~.4) 2000 PORMAT(2X,'SINGULAR MATRIX) 3000 n)RMAT(XWARMNG WROR IN IXTDECXMAL PLACE') 5000 FORMAT(2X,'SW ANGLE EXCEEDED !9 6000 #)RMAT(2X,TAL AT MAX DEFLECïiOIT) 8000 FORMAT(2X'LINEAR AERO EXCEEDED !') 8500 #)RMAT(12(F16.8)) 9000 FDRMAT('AV.U-m1,F8.4, 1X,'(m/s) AV.W-W,F8.4,lX, @'(m/s) AV.THETA=l,F8.4,1 X,'(&g)') STOP END A - kft band si& maarix fm eqn solver AI - col aigle of at(ack of uil inciuding ioisl incidencc angle of Uü. inddangle duc UBplmgiog and pitch ne effects. d inducrd downwash ALSA-wingtwistangkattimct MAO- maximum wing twist angle ALPAOT - totd incidaice angk of tail ALFAOW - mean total incidence angle of wing B - rifit hand si& fa eqn sdva BT-fulltailspan BW - wing haif-span CDOB - body -lift drag ~~~BTiciait Cf30T - tail zero-lift cirag coefficient CDOW - mliftdrag ooeffi#nt CLALFAB - body liftlcufve dope CLALFAT - iail liftumt sbpt CLALFAW - wing liftarue Jbpe CWUFAB - body pitching-momtntmc sbpe MOT- tail pitching-momeat coefficient CMOW - wing pitching-moment coefficient Ci' - mean tail ckd CW - mean chord of wing DELR - section Uurnments almg wing DWC-dragofmng DXB-~diançebanbodycg.~ppo9ntmmaoCwuigiligocdwiihwingcg DYB-ydirincefmcnbody~g.*>mot~fwing DZB - z distance hn body cg. mot of wing DYW-y~fromioaofwingtocgofwingpnel Dm-zdistancefromiooiafwing~,cgofwingpanel EL- &wnwash rstio(*l) EFFI' - OsWald efficiieiry foEtor EFFW - Oswsld dti~wCyfra~r ENG-alergy EW - &wnwash angk EW- initial Qwnwash angle 0 - graviioiionsi SEeCidon GAMA0 - muUnuin fhpping riyk Wl JE betdais GAMA - n9ppuig Mgle GT - fmkgain lDOT-input~uiradfœcqi~vertochccLdsimPlran~y END - total numbcr of rime inttrvals IRREC - opl numkof panels wing is sddinto iYYB-bdymonicatotYiertir IXXUtJYVW~JYZW- wing moment of matin WrWOJYYWO~O- wing moment of iDerrio at fiapping angle=O lyyDoTllv~,IYZDûTW- wing moinent of hath daivrtiM K - corinra ta 2nd- Rw-KutU Uittgrotiolt tAEROC - mal wing dihgmoment (&mg x) in body-fixed system UER0 - plVJue wing mihg inomcat (aicmg x) in body-fued sysom -0W - tail wing rdlhg moment (&mg x) in stPbility-axh sysom LW~raullift~vacntPe* LWC - lift over wing pmel
*-Crilculorie moments of indaof wing at flapPng an@e(GAMA)oO.* lXXWOcMW/12.~((BW+BW2)*(B W+BW)+'iWVW) IYYW~)5MW/l2.~(CWm6'+TW+TW) iZWû=MW/l2.0I(CWW+(B W+BW2)*@W+B W2)) & 5 i=1,100 ewi(i)=û.oOl 5 continue
ALFAOW=IW+ZLLW !CaMatc wing angk of sttack ALFAOCP=ICP+ZLLCP !Calcula& cairn panel angle of atta~k
!Cakulote flapping angk at thne T and its dcrivruivcs GAMA=eAMA(rcos(W"T) PW=-ûAMAO+W*SIN(W+T) PDOTW=GAMAO~W+W+COS(W~ *-Bbck to sdve fa wing foicc9 zu tùne T by integrating over BW-•
!InitUh td wing amdynamic farces XAEROC=O.O ZAEROC=O.O MAEROC50.0 LAEROCIO.0
LW=O.O !fnitiaiize Iift ova ench wing ALFA=#AO+COS(WT'+PHï) !Wulote wing twist angk
!Detamine am iwms fa the section of wing fnnn the pivot to the wing tip iRREC=INT(BW/DELR+.S) !Tdnumber of wing panels Do 1s J=l,mREc
R=J*DELR !CaiculPrc disinnce fiom wing mot to wing pantl
!Caîcutotc wing -1 lift anâ drag LW~AW/2.0+(WWC/UWC+~A8R+ALFAOW+(CW @ *ROE*(UWC*UWC+WWC*WWC)*DELR+CW LW=LWC+LW !Record GMal iift over aitire wing: to be used far &wnwash DW~CDOW~(ZWCrZq(ROE=(UWC8wC+wC8wC)*D~'CW))*(I)/ @ ((((BW+BW2+0.5*BCP)11.OyCW)*PI*EFFW))*(1/2.*ROE*DELR~* @ (uwc'uwc+wwc*wwc)) !Calculate distance fioin wing roat to wing panel
iCakuintewingp~nelLiftMddrag !Norebattwistingis ~~~gkcîcd~ihissectionisrigid LWWAW/2eW~C/UWC+UAOW+(cwllQWc/)) @ *ROE*~*UWC+WC1WWC)*DELR2'CW LW=LWC+LW !Rdtoral lih ovcr aith wing; to bc used for downwash DWC=(CDOW+(&WC.2.0/(ROE*(Ulfi'C*WC+WWC*WC)*DELR2+CW))*+2)/ @ (((@W+BW245*BCP)(ZeOyCW)*PI*EFFW))*(IB.* @ (uwc8UWC+WWclwwc))
!CJculPie nmPining wing rao fares which requircû no plsummation YAERW=PW/(BW+BW2) NAERûC=YAERûC*XWC
!Cakulooe disince fnm wing cg a wing ph DXPlO
A(Wk A(12,4)= -DZW A(12,6)= DXW A(l2.7)~-Dm A(12Qk DXP A(12,lO)L -1 A(12,12)= -1 A(1 l* -m A(l1.4)t DYW A(1lS)e -DXW A(11,7)= DYP A(11,8)= -DXP A(1 l,ll)= -1 B(i)= XAEROB - MB.G8SXN(THETA) - MB.QB8WB B(2)= ZAEROB + MB.G'COS~A)+ ZCB + MB.QB*üB B(3)= MAEROB + MCB
Bm XAEROW - MW.OeSINmA) + TXW - MW*(2.QBQB*(DXP-HXB) @ +2QB*WB - 3QB*W*DYP) B(8k YAEROW - MW8(PDOTW8DZ' + 2*PW'PW8DYP - PWWB @ -PWQBe(DXP-FIX8)) B@)i ZAEROW + MW---A) - MWe(2QBIQB'@P-HZB) @ -2.QB9UB - PDOTW8DYP + 2.PWW8DZP) B(+ tAWOW - DMW8PDOTW + IYZWQBQB B(12)c MAEROW - IYYDOïW.QB - IYZW*PWQB B(l1F NAEROW + IYZDOTWWB - (IYYW-IXXW)+PWQB ALFAOT=G'PQBC+ALFAOT IFeedbricL of gain GT on pi&-rsie
!Check fa maximum deflection of taü JF(ALFA0T .GE 2)T)IEN -6m) ALFAOT=.2 ELSE IF(ALFA0T LE. -2)THEN wRm616ooo) ALFAUb.2 END IF 125 format(l2fi0.3) 1SO FoRMAr(lXV-~3X,W-VELOCITT3X,'PIRATEt,3X, @'PITCHANGLE',4X,'ûAMA'.7X,'AW:Aï") 200 K)RMAT(2X,F8.4,SX,F0.4$1Y18.4,5X,F8.4,!XJV.4~.4) 2000 R)RMAT(2X,'SINGULAR MATRIX) 30 FORMAT(2XWARNING ERROR IN IDGT DECIMAL PLACE') SMKl FORMAT(W,SMAU ANûLE EXCEEDED !') 600 FORMAT(2X.TAIL AT MAX D-ON') 8000 PoRMAT(2XWNEAR AERO EXCEEDED !3 8500 #)RMAT(12(F17.8)) 9OoO EoRMEIT(2X,IAV.U-~(F9.5~AV.W-WJIaA~;AV.THETA=', @H.4) STOP END A - left hand si& matrix for eqn solver Aï - totai angle ofattack of EPÜ including tornl incidence ngk of W. induced angk duc to plunging and pitch rptt cffects, and inducai downwash ALFA - wing twist angle at time t ALFA0 - maximum wing twist angle ALFAOT - total incidence angle of tail ALFAOW - mean UWineidcna angle of wing B - right hand side ma& fa eqn solvu BT-fdlcailspen BW-dïstarmofwingfiomwingtiptopivot BW2 - distance afwing fim pivot to wing mat CDOB -body amblift c&ag cocffiiiait CDûT - taü zero-lift drag -cicnt CDOW - a#o-iift drag coefficient CLAI3AB - body liftlcurvc doge UALFAT - rPil lift-curve sbpt CLALFAW - wing liftsurve shpe CLDT - tail liftarve slop due to ekvator dcflection CMALFAB - body pitching-momaitlcurve sbpe CMOT - toil pir~bing-m~maiic~efficicnt CMOW - wing pitching-moment caffiicnt CT-mWchord CW-mcuichorddwing DE - elevauu &flacth (radians) DELR - ssction incranmis aOm wing tig to pivot DEtR2-~4~tioriincnmeirt9finmpivot~,rvingroo~ DWC-dragofwing DCP - drag of cenier panel DXB - x distancc fiom body cg. to centa panel cg DYB -ydhtancefnmi bodycg. toamicr~cg DZB-zdiswce~bodycg.toccnm~lcg DXSEQL - epuilibnum @am&) x diarre fiun body cg. ta cenîerpgncI cg DYEEQL - aquilibrium (givO) y distance fmn body cg. to antcr pniel cg DZBEQL - @hi~m w)z diwsnce Cmn body cg. tû p~nelcg DXBMAX - x nmi quilbn~mpositioir@XBEQL) tû mnim~n,posiiion (magnitude ûdy) DYBMAX - y dbtam! hmquiibrium position@XBEQL) to maximum posiÉi011 (magnitude dy) DZsW- z dUiMce fhm quilbnm position(DXBEQL) to maximum position (magnitude dy) DlDXB - x campentof cemer panel vcbcity wrt body (MMtive of DXB) DlDYB - y c~mp~nent~CCII~C~ -1 v~beüynt body (daiMtivt d DYB) DlDZB - z cmnpawatdcsnapiacl vebcity -body (derivative of DZB) D2DXB - x compmnt of cmte placœktotion wst body (deriMtivt of DIDXB) IXDXB - y-t d--1 rcsbuioa b~dy(daiMtiveof DIDYB) D2DXB -z~ot~~ll~trpncl~lcroEianntbôdy(~Mtiive~DI~) DXP-~~fnwn~~g~,piv~tpoint DYP-yaiP"^fninwhg~g~)pi~~pht DZP-zdisEPnce6amwi~cgtopi~olpoint DYW-ydiuiacebminocdwinglocgdwingpacl (mpitukonly) Dzw-z~fiOml~~~Of~~tocgofwiugpsnel(mÉudeonly) EiQI -&wnwa!b rorio(eEc1) EFm-~cffiyfocror
F3.FûR * LATERAL AND LONGITUDINAL STABILlTY (2-PANEL) iEND=IIUT~EL+.!i) !Total nurnbcr of timt inttrvals IRREC--INT(BW/DELR+S) !Total numbcr of wing panels !CaWaîc disPace fiun wing na< to wing cg DXWl=o DYWI=(BW/PCW(GAMAl))
LWkO.0 !IiDiYliire lilt ova caiirr pm wing ALFA=ALFA(rCOS(Wr]r+PHASE) !Wulate wing twist angle
!TIMSfœm to ccmvaùart-Ws(body-fiXCd)SYStCIIIBcolculote wing panci vebcities üWCdJB+QB*@ZB l+RbSIN(GAMAI)~B*GR~(GAMA1)-DYB 1) Wq+PB8GR8SIN(GAMA1)DZB I)+RB*@XB 1-DXWI) @ -PW1*R8SN(GAMA1))T~(GAMA1) @ -PB- l+R*CoS(GAMAl)~B*@XWl-DXB1) @ +PWPRTûS(GAUl))*SIN(GAMA1) WWC=(VB+PB'(-R8SIN(GAMA1)-DZB 1bRB8@XB 1-DXW 1) @ -PWl%8SN(GAMAl))8SIN(GAMA 1) @ +(WB+PB=(DYB l+RSCOS(GAMA1)~B*~XW1-DXBl) @ +PWl'R~(GAMAl))+COS(GAMAl) QWc3QB-(GAMAl)+W8SIN(GAMAl) PWC=PB+PWl !Cakuiptewingp~ellifrand~ CLW=CLALFAW8(WWC/UWC + UA*R+ ~U~AOW+ [email protected]) + R*PWc#Wc) LWC=CLW/2.0.ROE8(üWC*UWC+VWC8VWC+WWC*WWC)*D~~ LWl=LWC+LWl !R##rd tatai lifi ova entire wing, ta be usad fa dowbwash DWC=(CDOW+CLW*~W12.O/CW*PI*EFFW)) /2.(rROE*(UWc*UWC+wC*wc+wC*WWC)*DELR'CW
15 CONTINUE
!Tdmwing scio f- back to stabüity-axis systan XAEROWl=XAEROC YAEROW l=YAEROC.COS(GAMA 1)-ZAEiaûC*SlN(GAMA 1) ZAEROW l~~~S(GAMA1~Y~OC*S~(GAMA1) ~OWI=~Cx MAEROW l=MAER~(GAMA1)-NAEROC*SIN(GAMA 1) N~OW1=NAEROCICOS(GAMA1)+MAEROC*SIN(GAMAl)+YAEROWl*xwC !InitiPlilL total wmg raodymunic ~QCCJ XAERw.0 YAERûCd.0 ZAEROC=O*o LAEROC=O~O MAEROCW.0 NAERocd.0 LW2=0.0 !~~ lift over aiire sîarbaud wing
R=JmDELR !Caklote dist~ncehm wing nmt to wing -1
!Tdamio conveni«it-oxis(body-fixad)sysran Rr9ieulpce wing panel vebcities UWC=~B*~2+R.SIN(GAMA2)~RB*cR~GAMAS~DYB2) WmPB8(-R*SW(GAMA2)DZB2>,RB*@XB2-DXW2) -PW2*RLSiN(GAMA2))VOS(GM) w+PB*@YB2+R+COS(GAMA2))+QB*@XW2-DXB2) +PW2+R-(GAMA2))*SIN(GAMA2) WWC=(VB+PB~-R*SIN(GAMAS~DZB2~RB*~XB2-DXW2) -W2*R8S[N(GAMA2))*SIN(GAMA2) +(WB+PB~YB2+ReCOS(GAMAS))+QB*(DXw2-DXBZ) +PW2*R+Cos(GAMA2))TOS(GAMA2) QWwB*Cos(GAMA2)RB*SIN(GAMA2) PWC=PB+PW2
!Cakulalri wingpaneiüRanddmg awaaFAw*(wwc/uwc+ ALFA*R + ~AOW+ CW~WC/(~.~+UWC) - R*PWc/uwC) LW~W/2~0LROE8(UWC*WC+WC*WC+WWC'WWC)mDELR+CW LW2=LWC+L2 !R#3ord torol lift over entire wing; to be used fa downwash DW~CDOW+CLWL+Y(BW*2.~*PI*~) /2.0.ROE8(UWC*UWC+WC8WC+WC*Wc)*D~W !Transf'wing rso forces brk r> sribüity-axb sy-m
*-Sirnultaneuos solution of unLnrwn dabits at T+DELT -*
!Colciii.tr, tiaal values for the unbw,wn stafc variables UBT=UBC+UBT VBT=VBC+VBT WBT=WBC+WBT T)1ETAT=THETAC+THETAT PHIT=PHK+PHXT PSIT=PsIC+PSn'
!Mgn wlues to the tanpruary variables fa the next time inmmcnt UB=UBC VB=VBC WB=:WBC PB=PBC QB=QK RB== THETAtTHETAC PHI=PHIC PSI=PSIC
END iF 25 CONTINUE 10 CONTINUE STOP END * F4.FOR *LATERAL AND LONGITUDINAL STABILITY &PANEL) do 5 i=1,100 ewi(i)=û.OOl 5 continue
!CPlculste distance fiiom wing cg to whg pivot - DXPl=O DYPl=(@W-BWS)/2+Cos(GAMA1)) DZPl=((B W-BW2)/2*SIN(GWl))
8-Bkxkto sdve fa wing faces a timc T by mtegrPting ovet BW-*
I1nii;rii.t total wing wmdyirpmic fm XAEROCzO.0 YAERûC=û.O ZAERm=o*O LAEROC=O.O MAERCKXI.0 NAERoCa.0
LW 1=0.0 IInitiaiize lift ova entire wing ALFA=ALFAO+COS(WT+PHASE) !Calculate wing twist angle
!TrYlSf01111 io coavaii~~~t-axis(body-fued)systcm&cainibie wing pnel vebcities üWC=UB+QB*(FEB l+R8SW(GAMAl)~Re*(-R+COS(GAMAl)-HYB1) w~PB*(-Rrsm(GAMEI1~HZB1~m*(HXBl-DXPI) -PW 1*R*Sm(GAMAl))VOS(GAMAl) +(WB+PB8(HYB1+R~S(GAMAl))+QB8@XP1-HXB 1) -PW1 *RLCOS(GAMAl))*SIN(GAMAl) WW~+PB*'R8SIN(GAMA1)HZB l)+RB*(HX81-DXP1) -pWl'R*Sm(GAMAI))'SIN(GAMA 1) +(WB+PB*(HYBl+R.COS(GAMAl))+QB*@XPI-HXB1) -PW 1+RICOS(GAMA l))+COS(GAMA 1) QWWB-(GAMAl)+RB*SIN(GAMAl) PW&PB+PWl
16 CONTINUE
!Wulatedisrance fiun wing cg to whg moi DXVISZ=O DYW2=-((B W+BW2)/2TûS(GAMA2)) DZW2=-((B W+BW2)/2*SIN(GAMA2))
!CplCUrptE &stance ûun wing cg to wing pivot DXP2=O Dm(BW-BW2y2TOS(GAMA2)) ~-((sW-SW2)/2'SIN(GAMAS))
*-Bbck to sdve fa wing facce8 at timc T by intcgrating over BW-• !CIrlculate wing pallia and drag CLW=CULFAW*(WWC/UWC + ALFA*R + ALFAOW + CW~WC/(2.O*UWC) - R*PWC/UWC) LWCsCLW/2.O+R~*(UWC*UWC+WC*WC+WCWC)*D~+CW LW2=LWC+LW2 !Record total lift ovcr entire wing; to be uscd fœ bwnwash DWC=(CDOW+Q.W*lu((BW+BW2+OS*BCP)~.O/CW*PI*EFFW)) /2.0+ROE*(UWC*UWC+WC8WC+WC8WC)*D~+CW
17 CONTINUE !CPlculatt wing pancl lift and drag CLW=UALFAW*(WWC/UWC + ALFA8R + ALFAOW + CW$QWC/(2.WUWC) - R*PWC/UWC) LW~W/2.0+ROE*(UWC*WC+WC8VWC+WC*WWC)*DELR2ICW LW%LWC+LW2 !Rac#d totpl lift over entire wing; to be used for &wnwash DWC=(CDOW+CI,W8+u((BW+BW24.5*BCP)(Z.~'CPI*EFFW)) ~0+RoE*(UWc*UWC+YWC+YWC*wC+wC*wC)*D~~
18 CONTINUE
STOP END