Basic FO
ODELLERS SECOND EDITION
FOR MODELLERS
BY ALASDAIR SUTHERLAND BSc
© 2002 Traplet Publi cations Limited
All rights re serv ed . All trad em ark s and register ed nam es ac kno w ledge d . No part o f th is book ma y be co pie d, reprodu ced or tran smitt ed in any fo rm w ithout the wri tte n co nsen t o f the Publish e rs. The informati on in this book is tru e to the best o f o ur kn owl ed ge at the time o f co mpilatio n. Recommendati ons are made without any gua rantee, impli ed or o the rwise , o n the part o f the a utho r o r publish er, wh o also discl aim any liability incurred in co nnec tion with the use o f d ata o r specific informatio n co ntaine d within th is publicat ion .
First ed ition publish ed by Trapl et Publi cat ions Limited in 1995 Publi sh ed by Tra plet Publi cation s Limited 2002 Traplet House , Severn Drive , Up to n-up o n-Seve rn,
Wo rces te rsh ire . WR8 OJ L United Kingdom .
ISBN 1 9003 7 1 41 3
Front Couer. Stefan If/u rlll seen bere exercising some ofb is considerableflying skills ioitb b is 1:2 scale Pitts 51. Stefa n brought tbe Pitts backwards, balancing the thrust oftbe engine against tbe stlffbrecze, until tbe rudder tou ched b im! (Photo: Peter Dauison)
Tecbnical D ra uiings by Lee \\7isedale Ca rtoons by Simo n Bates
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Acknowledgements o n ve n tio n a lly th is is a p a ge o f syco p ha n tic ramblings wh er ein I thank everyo ne in my life from th e midw ife wh o d elivered m e to my dent ist's rece ptio nist. Well , thank yo u o ne and all. I o we my parents a small a po logy , as I rem ember bu ying a mod el ae ro plane a nd then promi sin g that it would be my last ; not o nce but thr ee o r four times. I made no suc h rash pr omi ses to my wife Ann e who unwittingly made th e mistake o f marr yin g a dormant Ae rom od elle r, who ever since then has been e rupting with increasing magnitude and frequen cy, sprinkling the hou se with successive layer s o f styrene bead s, wood shav ings, balsa dust, glass fibre stra nds and Solarfilm fragmen ts. Sorry Anne . As for my daughters Ron a and Shee na , if the y ever live in Ame rica th e ir a nalysts w ill ma ke mu ch of the socia l a nd paternal deprivat ion the y have e ndured by being the offspring o f a fervent aero mode ller. Passi ng q uic kly over m y educat io n a t Le n zie Acade my, Glasgow Universi ty a nd the Hambl e College of Air Training, the grea t mileston e in my modellin g life was when Jo hn Mich ie had the time a nd p at ien ce to teach me to fly proporti on al R/C aeroplanes. And it was Brian Davies who introduced me to aeroba tics and wo rd p rocessing, which is whe n this book ge rmina ted . I have learn ed a grea t deal from my frie nds in the Alde rsho t club and W'indsor Park, a nd co ntinue to learn from my present circle of friends in Scotland . It was du e to one of th ese , Bob McGill , th a t I became imme rsed in wa ter plan es. Finally , th ank yo u to Dr. Fra n k Cot on of th e Depa rtment of Ae ro s pace Eng inee ring at Glas g o w University wh o read throu gh th e manuscript to check th at I wo u ld not e m ba rr ass th e Dep artm ent to o e xte ns ive ly b y preach ing fund am enta l ae ro dy na mic fallacies.
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Foreword ne of the first technical qu estions my son ever aske d me was "How do plan es fly?" Well, we all know how plan es fly .. . don't w e? Th ink again! If you were asked that simpl e qu estion , co uld yo u give a co ncise comprehensible a ns wer? If yo u co u ld, how would yo u deal with the retor t, delivered by the son of on e of my colleagues . . . "How do plan es fly upside down?". On e of the most fascinating as pec ts of th e modern w orld is th e science of flight. Wh ether it be a bird , heli co pter , fighter aircr aft or e ve n th e marvellous bumble bee, people ha ve always been intrigued by the same basic qu estion - "How does it fly?". Unfortunately, th e a nswe r is n ot a lways s traig h tfo rward a n d is co mp lica te d b y the w id e varie ty of mechanisms a nd physical ph enomena which interact to produce flight. Man 's interest in model aircra ft is a lon g stand ing one. Over the yea rs, the mot ivation for this has largely be en recreation al altho ugh since scientific studies ha ve been co nduc ted, most notabl y those in Ge rmany between the Wor ld Wars. As a res ult, tod ay's aeromo de ller is a fairly well info rmed ind ivid ua l w ho , inste ad of ask ing th e bas ic nature of flight qu estion, is more inter est ed in how to improve the performance of a n ai rcraft o r how to avoid problems during fligh t. The answers to most of these question s can be found in Basic Aeronautics for Mod ellers. Thi s book skillfully guides th e reade r through th e bas ics o f a irc raft flight a n d p erform anc e before addressing issues specific to model aircraft. Alasdair Su the rl a n d draws on his p e rs onal e xperience as a stude nt, a pilot, and most imp ortantly a n aeromodeller, to pr esent fundamental informati on in a friendly and eas ily accessib le form . He does so b y building th e kn owled ge bas e of the read er in a steady progressive m ann er, h ighlightin g a numb er o f co m m o n miscon ception s along the way. In this wa y, he en sures that the rea de r is prepared for each new sectio n of th e book as it is reache d. Thankfully, the use of complicated equa tions or tedi ou s derivation s wh ich, if excessive, can ofte n det er th e laym an , is either avoided o r they a re provided in appendices . Th rou gh ou t th e book, use is mad e of observat ions from flow visua lisation ex peri me nts to illustrate asp ects of fluid be hav iour. Over the years, flow visua lisation has been o ne of the mo st p owerful too ls in the development of our current understanding of fluid dynam ics. Ind eed , smo ke flow visua lisatio n w ind tunnels are still used in ma ny un ive rsitie s for resea rc h a n d s tu d e n t dem on str ations. It is o bvio us th at the demonst rations given to Alasdair Sutherland in his stude nt days had a co nsiderable impact; after all seeing is believing! Whether you consider yoursel f to be a novice or a
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well-season ed ae rornode ller, there is so meth ing in this b ook fo r yo u . Beginner s ca n le arn a bou t th e ba sic mech ani sms of lift generation and the manner in which for ces act on an aircraft. The more ex perience d , o n the othe r hand, can contemplate the detailed influ ence of model sca le and the role of the Re ynold s number. The book may even encou rage so me to raid the library for mor e informatio n or carry o ut so me res earch of their own. Most importantly though , this book was written by an e nthus iast for its readers to enjoy. I hope yo u do! Dr. Frank Cotton Department of Aerospace Engineering University of Glasgow. Alasd a ir Sutherl and w as b orn a n d e d ucated in th e Glasgow area , progressing from Lenzie Aca demy to Glasgow Univers ity wh ere he ea rned a B.Sc. w ith Honours in Aero nautical Engineering. Afte r training for a career as an airlin e pilot at Ham ble , near Southampton, he joine d BEA in 1973 to fly Trident aircraft arou nd Euro pe and Lockheed LlD11 aircraft wo rldwide. An aerorno delle r sinc e th e age of eleven, he flies most types of radio co ntrolled airc raft especi ally spo rts and aerobatic, and particul arly enjoys designin g models of va rio us typ e s. After man y years as a member o f Alder sh ot Mod el Club he mov ed back to Scotland as Captain o f British Airw ays turboprop aircra ft, first th e H.S. 748 a nd latt erly th e British Aerospace ATP. He is now a member of both the Clyde Valley Fliers and the Garn ock Valley !vIAe.
Tbe Author: Alasdair Sutherland
Contents Page Introduction
11 13
Chapter 1
The Aeroplane's Environment Tbe air. Mass toeigbt and grauity. Newton
Chapter 2
Requ ireme nt for Flight - Lift \fiatcbing tbe a irfloto. Pressure variation . Pressure exerts a force . Wind tunnel testi ng.
16
Chapter 3
The Stall's the Limit The lift cu rve. 17Je stall, tbe reason . Variation in sta lling cbaracteristtcs.
20
Chapter 4
The Drawback Drag 17Je boundary lay er. Wing drag; drag polar , effect of tbickness and ca m ber, la m tn a rfloui sections. Fuselage drag , strea mlin ing . A bit for golfers.
23
Cha pter 5
Have you a Moment? 17Je m om ent on tbe wing . Centre ofpressure. Aero dyna mic centre. A erofoil section su m m a ry, tbe effect oftbickness and ca m ber. Section classification and use.
26
Chapter 6
The Vortex Syste m The uortex around tb e wing. Seeing tb e cortices. Even m ore drag, tbe reason . Complications. Simp lifica tio ns. 17Je importa nce ofAspect Ratio. Lessons forpra ctical modellers . Ground effect,
30
Chapter 7
Planform and Twist Elliptical lift distribution. Local angle ofattack. Different planform shapes. Tipstalling . Wasbout, aerodynamic ioasbout. Sweep ba ck. Mean cho rd . Horses for courses.
35
Chap ter 8
CG and Stability 17Je CG. Stability in gen eral. Motio n ofan aeroplane. Stability ofaerop lan es in Pitcb , CG Position . Complica tion s. We can work it out? Simpler equations. Va riations on tbe formu la .
.41
Chapter 9
Directional and Late ral Stability Directional sta bility , the fin. Lateral sta bility, sideslip. Fin sideforce, wing p osition , d ihedral, sweep back . Aspects ofdesign . Directional and lat eral interaction, spiral divergen ce, dutch ro ll.
.49
Chapter 10
Control Rudder. Elevators. A ilerons, a ileron drag, aileron alternatives. Control su rface balances. Control effective ness, rotational inertia, sta bility, a erodynamic damping. Otberflying con trols, throttle, air brakes, flaps, sla ts. Control combinations, ta ilerons, flaper ons, eleuons, V-ta il.
.56
Cha pter 11
Turning Flight Mecbanics of turning . Turning aeroplan es, loa d f a ctor in a turn, refinem ent, stdeslipp tng and skidd ing, drag in a turn, stalling speed. Higb aspect ratio. Turning using rudder. Specia l effects. Wben is a rudder an eleva tor?
63
Chapt er 12
A Delicate Balan ce Equilibrium. Tail lift to trim. Elevator ang le to trim . Ta il Setting angle . The effec t of thru st o n trim.
67
s Laws. Inertia . Vectors. Moments.
Cha p te r 13
Glid er Performance Lift/Drag rati o. Speed ra nge. Ae rodyna m ic da ta . Optimising performance, strea mlin ing , toeigbt. Iiffect ofto ind on perfo rmance, down trim , ballast.
72
Cha p te r 14
Power ed Performance Propeller thrust, slipstrea m effects. Level flight, top speed, sta lling speed, effect on toeigbt. Take oJ(. Clim b. Descent and landing .
76
Cha p ter 15
Th e Ae ro dyn am ics of Aeroba tics 77Je sta ll. Sp in . Snap . Loop . In oerted. Roll. Yatu. Aero bat ic trim set up.
80
Chap te r 16
Special Cases Low asp ect ratio, handling, CG position . Ca na rd . sta bility, CG Position , Tail-less aeroplane, sta bility, trim, co n trol. Multitoing, performance, CG p osition .
85
Cha p te r 17
Reyn olds Number Definition, importance, n ontogra nt. In tbe bou ndary lay er, situation normal, laminar separation , separation bu bble, tbe underside. Re-eff ect on a erodynamic da ta . 77Je p roblem area . Hysteresis loop . 77Je effec t 0 11 m odel desig n a nd performance, wing tips, class rules, optim u m weigbt. Tu rbu lator strips. surface fi nish , Using publisbed data .
90
Chapte r 18
Aeroelasticity Effect on stability , ta il bend, wing twist. A ileron reversal. Wi ng divergen ce. Aileron flutter , tbe ca use, tbe cure. 117ingflutter. Tail Flutter .
96
Cha p te r 19
Tu ck Under Description . 77Je villa in unmasked. Wing twist. ta il bending.flexible con trols. 77Je elevator trim g rap h . Critica l speed . Tuck under speed . Getting away witb it. Tailplane insta bility. Rem edies/or tu ck under. Conclusions .
102
Cha p te r 20
Th e Air o n th e Move Navigatio n . Slop e lift . Tbennal lift , Windsbear a nd Win d Grad ien t. Gusts. Mytbs a nd miscon ceptions. Momentum . Kin etic energy . A nalogies. 77Je meaning ofl ife?
109
Cha p te r 21
Mod el Aircraft Structures Defining some words, composite structures, tobat a ir does to wings, bending m om ents, stru tted wings, torsional stiff ness, fuselages, tailplanes.
114
Cha p te r 22
Centre of Grav ity Pos ition Rigbt and wrong CGs, Fligb t testing, p opula r m isu nderstand ings, tobat m a tters, m ean cbo rds, tbe flying toing , biplanes, tb e neutral point , adjustm ents, p u tting it togeth er, sta bility m arg in .
.123
Appendices
131 A Bemoulli 's equation B Boundary Laye r C vortices D Dib edral and sweep E Usefu l Nomogra ms
143
Glos sary Symbols, Abb rev iations a nd Co mmo n Aero dy na m ic Terms Ind ex
145
No tes
147
Introduction hen the cold raw wind howls down from the North bringing grey fragme nted clouds which sc ud low o ver th e d amp d a rk fo rbidd in g landscap e like a demon arm y. When sheets of icy rain deluge incessantly from a leaden sky and the puddles join fo rces to threaten us with an oth er great flood . When the gre at oak trees bow down to the un seen forces of th e w ind like frightened peasants befor e th eir Gods . When ever the outside environment beco mes hostile to man and his aeroplane, I curl up in a cha ir by the fire with so me books and magazines, to absorb all the fact, fiction and folklore of o ur fascinating hobby. It is o n nights like the se as I lie in be d listening to the w ind howlin g or the rain lash ing o r the deathl y silence of th e s nowfa ll that I h e ar voices , vo ices from my pas t. They are the vo ices of aerodyna mics lecturers and au thors a n d the y remi nd m e h o w littl e acc u ra te knowledge of aerodyna mics is ava ilable to the ave rage modeller, and they tell me w hose fault it is. Mine! My fault for not writing this book soone r! I have three main aims in wri ting this boo k. The first is to disp el the half-truths and old wives tales passed on , usu ally in go od faith , into the folklore of the hobby. I o nce ha d a very pu zzling conversa tio n w ith a modell e r a bout th e u se of "flap s", until he cla rifi ed matt e rs by explai ni ng that he me ant th e "bac k flaps" (e leva tors) . So the se cond aim is to ge t us all speaking th e sa me langu age a s fa r as p o ssible so that our in e vit able di s cussions and a rg u me nts can b e more meaningful. The third aim of my book is an introduction to aerodynam ics so that you ca n understand how to make use of th e data available elsewhere wh en designing your own mod els . Und erstanding some simp le theory will not turn you o vern ight into the design e r of the most elegant and super-efficient models (that still requires experience, in s piration a n d talent) , but yo u ca n le arn what is p o ss ibl e und er the laws of Ph ys ic s , a n d w h a t is impossible - unlike the alche mists of o ld w ho was ted their lives trying to turn lead into go ld . Now let me plea for pati ence es pecially fro m th e more knowledgeable readers. I have started off with a simple , rosy , idealised view of the wo rld and I introduce the rea l co mplications little by little.
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Basic Aeronautics for Modellers
11
Chapter I
The Aeroplane's Environment The Air Please try this simple experiment. Take a can of beer, open it, and drink the contents. Now w ha t are you left with? Most peopl e say "an e mpty can " but th at is wrong . If you answered "a can full of air" give yo urself a pat o n the back . We ae ro mode llers mu st be co nscious of the air. We are depending o n it to su pp ly the lift for our aerop lanes. Next time yo u see a Jumbo jet lumberin g off th e runw a y reme m ber t ha t th e air is provi di ng th e upward force of up to 400 to ns. So how heavy is, say a room full of air, 4 met res by 3 an d 2.36 metres high? Wo uld yo u believe 35 kg or 77 Ib? At abou t 1.22 o u nces p e r c ubic foot ai r is not very d ens e , b u t yo u w ou ldn 't ca ll a roo m empty if it co nta ine d 77 Ib of balsa wood! Now, how stro ng is the air? In a school ex periment the halves of a four inch (lOO mm) diamet er hollow stee l sphere were pla ced togeth er and as mu ch as possible of the air ins ide wa s remo ved. The air hel d th e halv es togethe r. It too k a lot of effort from the four strongest lad s in the class to pull the tw o halves apa rt. Pres sure is defined as a force per un it area . The force which the air pressu re exerts on a surface wi th a vacu um o n th e ot he r side is 14.7 pounds per square inch or nearly a ton pe r sq u are foo t! Th e pull n e ed ed to se pa rate the h emispheres in sc hool was almost 180 Ib (8 00 N). Natura lly the air exerts its fo rce on a surface w he ther the re is a vac uum on the other side or not. Hold up a sq uare foot of paper and there is a ton of force on ea ch side , but so wh at? Th e two forces cancel o ut. Pressure is not direction al, or rather it is omnidirection al; it acts in all direction s at on ce . And it acts perpendicular to the surface at every point. So w hichever way up yo u hold the paper th e re is exactly th e sa me o ne ton for ce o n each side . You can see the air pressure vary ing slightly from day to day o n yo ur barometer. Both density and p ressure reduce w ith altitude but we aeromo de lle rs can igno re these small differences. The reduction in air pressure is about a tenth of one p er ce nt for every 30 feet climb ed . Incid entally it is by measuring that reduction in pressure that an aeropl an e's altim eter works. Low sp eed airflow is called "incompressible " because, although the pressure wiII vary, density does not. We all kno w air ca n be co mpressed, and its den sity change d, but only in a co ntainer. Aeroplanes in free air do no t co mpress it unl ess they travel at ne ar so nic speeds .
Mass, Weight Gravity An o bject's mass is the amo unt of mate rial which it contain s. Becau se we live o n the earth's surface we tend
Basic AeronauticsforModellers
to use the word weight instead and to us there is no differ ence. Whe re a n object's mass (as opposed to its weig h t) s hows itse lf is in its resistan ce to b e ing accelerate d . Take an iron ca nno nba ll into space and it wiII be "weightless" but try kicking the ca nno nball and yo u will bre a k yo u r fo ot. It s resistan ce to being accelerated , its mass, has not changed . Th e weight of th e ball is just th e for ce o f th e earth 's gra vi tatio n al a ttractio n o n its ma ss. To ge t th e w eight o f a body, multiply its mass times "g", the "gravitatio nal co nsta nt" which on th e earth' s surface is 32.2 It/ sec/ sec or 9.8 1 m/ sec/ se c. The we ight of a "kilogram" of mass is a force of 9.81 Newtons an d the we ight of a "slug " (yes really) of mass is a force of 32.2 pounds. (But you don 't ne ed to remember all th at) .
Newton's Laws If a body is in "equilibrium" it is e ither at res t or moving at co nstant speed in a straight line (tha t is, not acc elerating). Man y years ago Sir Isaac Newto n put into wo rds three funda mental Law s of Motion . • 1. The first says that a bo dy wiII be in eq uilibri um if a ll the fo rces o n it cancel ou t, Le . if there is n o resultant force . • 2. The second says th at the force nee ded to cause an acceleration equals the mass times the acce leration . • 3. Th e th ird is the old favo uri te ab o u t each fo rce ha ving an eq ua l and opposite rea ction.
Inertia When yo u kicked the canno nba ll in sp ace , it app lied an eq ua l and opposite fo rce to yo ur foot. Tha t kind of for ce is ca lled an "inertia force", and is the for ce w ith which a body res ists being accelera ted . Similarly, w hen yo u catc h a ba ll yo u appl y a force to s low it do wn , overcoming its "ine rtia" which makes it wa nt to carry on the way it was go ing .
Vectors A riddl e! The re was a car sitting on a level roa d with th e brakes off and three men pu shing it but it wasn't mo ving! Why not? One w as pu shing the front, o ne the back , and o ne was pushing the side. An important little de tail! Any quantity w hos e direct ion must be specified as well as its a mou n t, for exa m p le for ces, is ca lle d a "Vector". O the r examp les of vec tors are distance moved, acceleratio n and velocity. I prefer the word velocity to speed because it is a rem ind e r that it is a ve ctor.
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_....
..
. .: . .......
Vec to rs ca n b e added to g eth er b y a d d ing th eir am ounts o n ly if th ey are in th e sam e d irec tion . If two Fig u re 1.1
B
10
c
force s are in o p pos ite dir ections, like tw o men pu s hin g a t e ithe r e n d o f a c a r, th e y w ill ca nce l each o ther o ut. If ve ct ors a re at a n a ngl e to ea c h o the r th e y ca n b e added by drawing a "vecto r di agram" using a ru ler a n d protract or. A vecto r dia gr am is a scale drawin g in whi ch th e len g th o f th e lin e s re p rese n ts th e a mo u n t, and th e direction represe nts the d irection o f th e vectors . Figure 1.1 co u ld re prese n t a trea sure map . "Starting at A wa lk ten metres n o rth to B, th en go ten metres ea st to c." The e q uivale nt, or re s u lt ant , o f the tw o vec to rs AB and BC a d de d to get h er is the vecto r AC which is 14.14 metres to the northeast. Figure 1.1 co u ld just as ea sily hav e represented t h e addit ion of two for ces or veloci ties . Ve ctors ca n al s o be s p lit up , or "reso lve d" , int o two o r more "co mpon ents " whi c h wil l h a v e th e s a m e e ffec t (F ig u re 1. 2) . The tr e a sure is in a ca v e , "C ". The ins c rip tio n o n th e Azt e c Temp le , "A" says ; Go five kilometres on a bearing 037 0 East of No rth (b ut beware of th e Dragon at "0 "). Preferring an ea sy life to he ctic adve ntur e, o ur hero "Tri gon o metry" ]ones instead goes 4 km du e No rth, sto ps for a few beers at "B", and th en goes 3 km du e East w here he finds th e cave , treasure et c . e tc . Very precise and sc ie n tific but no u se for a mo vie script. From th e vector d iagram in Figure 1.2, vec to r AC can b e s p lit in to its tw o co m p o ne n ts , AB th e No rthe rly co mpo ne nt and BC the Easte rly co mp onent. The bigger a ng le A is, th e smaller AB become s as a proportion of AC and th e bigger BC become s as a p roportion of AC. The ratio of BC to AC is called th e sine of th e angl e , the ra tio of ve ctor AB to AC is ca lle d the co s in e of the a ngle , and the rati o of BC to AB is called th e tange nt of th e angle A. These ratios are usually sho rte ne d to sin, cos a nd tan a n d ca n be lo o ke d up in table s for an y angle. Us ing his ma themat ica l tabl e s "Tr ig " jones co u ld work out t he c omponents for a n y a n gl e with ou t reso rting to sca le drawi ng . Th e sine of 37 d egrees is 0.6 a nd cos 37 0 = 0.8. Of co urse th e same go es fo r other vecto rs like force s or ve locities e tc.
Moment A
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Th e "mo ment " of a forc e abo ut a point is the size of the force times the di sta nce of the for ce from the p oint.
eastc Aero nauticsfor Mode/!el :~
Figure 1.3
Figure 1.2 B
3
100
C
~
Easterly Co mponent
Nortbernly Component
50
5
10
~
J:
Figure 1.4
5 4
10
~- - - - --- -- -- --- - - --- --- ------- - ---- - --~ P ivo t
Mo ment = 5 .'\" 10 = 50
A
5
groundsp eed vec tor. Wind ha s no o ther e ffec t (b ut se e the cha pte r o n wind near the e nd anyway). To save any argument I shall I ass ume still air conditions in all th e cha pte rs until th en.
Figure 1.3 represents a seesaw th e pl ank o f w h ich is exactly balan ced . Th er e is a ch ild w eighing 100 lb 5 feet from th e pivot and a ch ild w eighing 50 Ib 10 feet from th e pi vot. Th e ch ild o n the right has a moment o f 500 ft. Ib clock wise abo ut th e pivot , and th e ch ild o n th e left has a mom ent of 500 ft. lb anticloc kw ise ab out th e piv ot. Th e tw o moments are equa l but in opposite d irections a nd so th e y c a nce l o u t whi ch le a v e s th e seesa w balan ced . It is in eq u ilibr ium as th er e is zero resu ltant mom en t. In Figure 1.4 two eq ua l but opposite fo rces act o n a b od y. Th e two fo rce vec tors ca nc el out, th ey h ave no resultant but th ey will o bvious ly tend to turn th e body. Th e turning effect , o r moment, o f the pair o f for ces is th e sa me about any p o int yo u care to choose. The tot al moment is Force tim es th e d istan ce b etween th em . Th is kind o f syste m is called a co up le and its moment is th e sa me 5 x 10 = 50 about a ny pi vot point. In Cha p ter 5 I'll remi nd yo u th at you ca n hav e a for ce sys te m w ith no res u lta nt excep t a mom ent wh ich is the sa me about an y point. You will ofte n see so me qu anti ty lik e a irspeed (V) w ith a number s u persc rip t. Fo r exam p le V3 me an s V "c u bed " o r V "to th e p owe r 3" or s peed x s peed x speed. Similarly th e "cu be root" of V (w ritte n 3jV) is th e numb er which , when mu ltiplied together th re e tim es , . gives V.
Win d I co uld have used th e w ind as another exam p le on vec tors. To find the e ffec t of th e wind , just ad d th e wi nd vector to th e ae ro p la ne 's ve loci ty ve ct or to ge t th e
Basic Ae rona utics f or Modellers
15
Chapter 2
Requirementfor Flight - Lift hat makes a n aer oplane s pecia l is its wing. The qu estion is, ho w does it produce lift? I wish I co uld tak e yo u to a wind tunnel with a p prop riate mode ls and mea surement eq u ip me n t. I co uld then dem on strate ho w lift is produced just as it wa s shown to me . Instead I sha ll have to att empt to describ e it in words and diagrams.
W
Wa tching th e Airflow It is interesting to watch the flow in a smo ke tunnel , wh ich is a specia l low speed wind tunnel in w hich many s ma ll s tre a ms o f s mo ke a re fe d in to t he ai rstream up wind o f th e wi ng. T h e t h in s t re a ms o f s m o ke travellin g wit h the air as it flows over the wi ng help to visualise the airflow . Figure 2.2 is a dia gram sho wing a typ ical flow pa ttern aro und a win g. Th e lines sho w the position of the smo ke streams . Th is is a co mmo n way o f s ho w ing a n airflow a nd th e lin e s drawn a re ca lle d "streamlines" . Strea mlines are lines drawn in the direction of th e airflow suc h th at no wh er e does th e air flo w across a line. As the airflow approaches the Lead ing Edge (L.E.) of th e w ing it s p lits in two, part going a bove a nd p a rt below. The strea mline which d ivides the air w hich w ill go over the w ing fro m the air whic h w ill flow unde r it meets the w ing at poi nt A. Air molecules flo win g exactly alo ng th is line will me et th e wi ng hea d on a n d be b rou ght to a d e ad s top a t A. Po int A is ca lled th e "stagnatio n poi nt" becau se the air's ve locity is red uced to ze ro. . Wa tching th e s mo ke st rea ms over th e top surface very closely, it ca n be seen that the air speeds up as it
Definitions Figure 2.1 sho ws th e cross-sectio n o f a wing. Th e straig ht line from the ce ntre of the leadi ng edge (L.E.) the trailing edge (T.E.) is the chord line . The len gth of the chord line is the cho rd of the w ing (the w ing tip to wi ng tip distan ce is the spa n) . Th e maximum distan ce b et w e en th e to p a n d b ottom su rfaces is th e win g th ickness , usu all y ex p ressed as a percen tage o f th e cho rd. The line drawn midway be tween top an d botto m surfaces is ca lle d th e mean line or ca mber lin e . Th e maximu m distan ce between the mean line and the cho rd line is the ca mbe r of the sec tio n and it too is give n as a pe rce ntage of th e chord. Th e leading edge is always smoothly ro unded and the trailin g edge is always sha rp. A typica l test wing fo r a w ind tunnel has a uniform chord and aerofoil sectio n from o ne e nd to the o the r and fits e xactly in th e width of th e tunn el F ig ure 2.2 wh ich do es a wa y with th e co mp licat ion of tip effec ts w hich w e don 't need at this stage. I s ha ll give you fair w arning w he n I come to a win g w ith tips. For the mom ent the flow is ass ume d to be the same a t an y p o sit ion a lo ng th e s pa n ( two dimensional flow).
Figure 2.1 Cam ber L ine
• Ca ll/be"
.• L E.
Chord Lbw
T.E.••
•
•
~--------------- ----- -------- --------- - - ---- ------- ----- - -~ CIJOI'd
16
Basic Ae rona uticsfar Modellers
pa sses over th e thick Figure 2 ,3 pa rt of the w ing a nd resumes its p re vi o u s speed by th e Trai ling Edge (T.E .) . Under th e wing the smoke bu nches u p as it slows down , an d then it accelerates to its or iginal speed at the T.E. If the smoke strea ms are pu lsed, Le. re leased in s ho rt burs ts , it ca n be seen that the start of the smoke pulse above the w ing re ach es the trailing edge before the smo ke Figure 2.4 be low th e w ing a s illustrated in Figure 2.3. Obv io us ly the air over the top surface has had to speed u p to cover a longer pat h in the same ti m e . No tice a lso t h a t w here th e f low h a s speede d up the streamlin e s a re close r a nd w he re t he flo w is slo we r th e streamlines are furthe r apart. As th e a ng le o f a ttack is in cre ased th e stag nat io n p o int A mov es down around the cu rve of the leading edge increasing the dis tance the air travels over the to p , a nd re duci ng the dis tance alo ng the unde rside . On a w ing w ith a symme trica l sectio n a t a n ang le to th e airflow, the stag na tio n poi nt is be low the ce nt re of the le ad ing e dge (as in Figure 2.4) so jus t as wi th th e cambered sect ion the air flowi ng over the to p surface has fu rther to go in the same time , and must therefore speed up .
~ ---------------
Pressure Variation You can 't get a change in velocity wi thou t a pp lying a force (Newton's First Law). The on ly force ava ilable to t h e free air is its press u re so th e p re s su re mu st be changing as speed cha nges across the chord of the wing (See App en dix A, Bern oull i's equation) . If we wish to measur e accurately the pressure cha nges we have dedu ced mu st b e occurri ng o ve r o ur Figure 2.5 aerofo il, we ca n drill a row of tiny holes in the top and bo ttom surfaces and connec t eac h one to a p re ss u re measur ing d e v ice . Eac h pres su re meas ured ac ts at right ang les to th e surface at th e po int w he re it was measured . The pressur e is, as ex pected , less on th e upper surface th a n on th e und e r s u rf ace a n d th e re is a h igh pre s su re p e ak a t t he stag na tio n p oint w here
Basie Aero nalilies fo r Modellers
---
the ai r me ets the wing head o n. See Figure 2.5 in whic h the len gth of each arrow represents the pr essure at tha t po int.
Pressure Exerts a Force Pressure is de fine d as force per uni t area . Imagine in Figure 2.5 that th ese pr essure arrows , o ne inch apart, each represent the fo rce o n the o ne squa re inch around eac h hole . If all those force vecto rs are added togeth er, the resu ltan t will be the total force on a o ne inc h wide strip of w ing . Its size and di rec tion de pend upon th e aerofoil section , the ang le to the air flow , the speed of the a irflow , ete. See Figure 2.6 in wh ich the res ultant force is sho w n as force F. Th e p o int where th is force crosses the chord line of the section is ca lled the Centre of Pressur e (or C P.) . It is the poi nt th rough w hich the total pressure effec t on the w ing ca n be repl aced by a sing le force .
17
Figu re 2.6
Ail flow ~
Figure 2. 7
Ailflow ~
L
,, ,,
, ,, , ,, , ,
c.r.
Figure 2.8
lV
It is in c on veni ent to hav e a fo rc e ac ting in an arbitrary direction like that and so it is split up into two co mpone nts at right an gles to each othe r. Th e d ire cti ons chose n a re th e obvio us on es fo r a w ind tunnel. Th e co mpone nt in th e dire cti on o f th e airflow is called Drag, and the co mpo ne nt at right ang les to the a irflow is called Lift (See Figure 2.7) . No te that I d id not sa y ve rtical and hor izontal! It is tru e if the w ind tunne l is built horizontal , but lift w ill not b e ve rtical wh en we co me to an aeroplane climbing or desce ndi ng o r ban kin g. Figures 2.8 and 2.9 show what I mean . No te that it is a mat hematical co nven ience to sho w forces like F, or L and D at the ce ntre of pressure . They are merel y representing the tru e situa tion of Figure 2.5. So me p re ssure m ea suring d e v ic e s m ea sure the diffe rence in p ressur e between the desired point and the static pr essure of the a ir in the room . Or if you like the pressure differen ce between the insid e and outside of a hollow win g. Figure 2.10 is simila r to Figure 2.5 but this tim e sho w ing th e pr essure difference betwe en inside and outside . Th e reduction in pressure whe re the air is speeded up ca uses an upward force over the top surface and w he re the air is slowed down ther e is an upward force on the lo wer surface . Thi s is a co mmo n meth od of sho w ing the lift d istribution whi ch yo u may have co me across before (so me times o nly the line joining the tops
18
of the arrows is shown) . The resultant of all thes e force s (o r pressur es ) is exac tly the sa me as in Figure 2.6. Ju st to get all this in perspective , co nsider how mu ch pressure cha nge is needed to su pport the w eight of a m od e l with a typ ical win g lo a d in g of 20 o z./ft -. Atmospheric pressure is ab out 14.7 pounds per sq uare inc h . An ave rage press ur e rise on th e und ers id e of 0.02%, and an averag e pr essure reduction of 0.04% on the top surface will suffice. We are n ot asking mu ch a re w e ? To ca ll t hi s a "va c u u m" w ou ld b e mi sleadin g. I ex ag gera te d enormously the arrows o n my diagram s 2.5 and 2.10 to mak e them mean ingful.
Wind Tu nnel Testing Of c o urse w e don 't real ly g o th rou gh a ll thi s r ig m a ro le o f m e a surin g pre ssure s a n d in vol v ed ca lc u la tio n to w ork out th e lift and dra g in a wind tunn e l. Beside s th e co mp lica tio n in volved , th e s kin friction drag has been igno red . The w ing co uld simply be mounted o n a bal an ce to mea sur e the forces directly. The force mu st be mea sured through the attac hme nt point (e .g. the L.E. or qu arte r cho rd point) together with the mom ent abo ut this poin t. This mom ent is ca lled the
Basic Ae rona uticsfor Modellers
Pitch in g Mom ent. As m om ent e q ua ls fo rce tim e s di st a nce , if th e lift a nd moment a re kn o wn th en th e position whe re the lift ac ts (the Centre of Press ure) can be calculated . Th e w ind tunnel sho uld be eq uipp ed w ith a ba la nce ca pa b le o f me asu rin g h o ri zontal force s , vertica l forces , an d p itch ing moments a ll at th e sa me time . Thi s eq uipment can be used to test a wing, adjus ting o ne variable at a time and kee ping everythi ng else the same to find out the effect of each variable. For instan ce test ing the sa me wing in the sa me position at d ifferent airs peeds s hows th at Lift, Dr ag a nd Moment are a ll prop ort ional to the sp eed sq uared. In o ther words at twic e the speed yo u ge t four times the force , and at three times the speed, nine times the force etc. By s imilar means it is found th at Lift a nd Drag are also proport ion al to the air den sity p and the wing area. The mom ent is proportional to the sp eed squared, the air den sity and the wing are a times the chord . To turn these relationships into use ful eq ua tions for es tima ting the lift from a wing , a co ns ta nt has to be intro duced and its valu e mu st be found ex pe rime ntally. 50 for exa mple • L
=
A d ifferent co ns tant is ne ed ed in each case but to sa ve running o ut of suitable lett ers, the letter C is used in all three equations w ith a d ifferent subscri pt. The people w ho mad e up the eq ua tions put in a !1 as we ll becau se the te rm !1 p V2 had turn ed up in Berno ulli's equ ation (see Appendix A agai n). We e nd up with these three familiar eq ua tions • L = !1 P V2 5 CL • D = !1 pV2 5 CD • M = !1 P V2 5 C C~ I
Wh ere CL is th e lift coe ffic ie n t a nd CD is th e dra g coefficie nt an d CM is th e pitching mom ent coefficient. Th ey all vary with ang le of attack as you w ill see.
P V2 5 x co nst.
Figure 2.9
Figure 2.10
! 1I t Basic Aeronautics forModellers
t t t t
+
~ 19
Chapter 3
The Stall's the Limit n w ind tunn els the win g is stationa ry and the air is drawn over it, so that is how it is usu ally describ ed in th e ory. It is just as valid to th ink of the a ir as stat ionary and the wing moving. Its directio n of motion is exact ly opposite to th e arrow marked "a irflo w ". The dir ection of the a irflow must be measured far enough ahead o f the wing so that it is not affected by the wing's approach.
I
Definitions Figure 3.1 s ho ws a wing se ctio n in an airflow. Th e angl e between the chord line and and the airflow is called the angle of attack . It is usually represent ed by the greek lett er a (alpha). Occa sionally a different datum line is used instead of the cho rd line. It may be a straight line o n the und erside of a flat bottom ed or und ercarnbered Wing, or the wing 's zero lift line. As the nam e suggests, if the airflow is parallel to the ze ro lift line, the lift is zero (usefu l in mathem atical formulae). Th e inciden ce of the wi ng is the a ng le betwe en its cho rd line (or oth er datum line) and the fuselage datum line . It bears no relation to the airflow and angle of attack at all. It is just a riggin g angle. It may be measured o n the aeroplane with an incidence meter or on the plan with a protractor. Those are the usu al definition s and I shall stick to t he m , but it is not uncommon to see th e w ord incide nce used mea ning angl e of atta ck .
Notice the shape of the graph! It is straight from A to C and then curves up to a maximum at D th en down to E a nd be yond . At point B the an gle of attack is ze ro as the wing has be en arranged as in Figure 3.3 suc h that the chord line is paralle l to the airflow. Although the an gle of attack is ze ro , the wing is still producing lift. At point A the wing has been tilted further le ad ing edge down as in Figure 3.4 and is now producing no lift. Th e ze ro lift angle of attack is written as a o (the s u bsc rip t 0 den oting n o lift ) . T he normal wa y of measuring a ng le o f atta ck is to mea sure UP from the direction of moti on to the cho rd line . Because the chord
E
The Lift Curve Testing a wing at man y different ang les of attack and worki ng out th e Cl. e ac h time ( fro m the for mu la in Chapter 2) enables a graph of lift coefficient again st angl e of attack to be drawn for that particular section . For most normal sectio ns the graph loo ks like Fig ur e 3.2. Th is g ra p h is true for this sectio n regardless of the size or s pee d a nd c a n b e u s ed to es tim a te th e lift in an y co nd ition.
0< =0
Figure 3.1
C
--- - -Cb';;"';-;' - - - - - - Airflo w
~"~ _ _ ~
Zero lift l .
Direction of Motion
""
A ng le ofAttack (measured from z ero lift U1Ie)
20
, , A ngle ofA tta ck (measur ed f rom c h o r d U1Ie)
Basic Aerol/l/ /Ifics/or Modellers
li n e is DOW N in t h is case the ang le of attack is a negative ang le (for e x a m p le th e a ng le o f attack for ze ro lift on an Eppl e r 19 5 s ectio n is g ive n as - 3 d e gre es) . The zero lift line (ZLL) d rawn on the w ing is by definitio n parallel to the airflow . At p o int D th e lift c oe ffic ie n t is C Lm a x w hich is the maximum lift coe fficient wh ich the section ca n prod uce and oc curs at as the stalling angle of attac k.
The Stall
Figu re 3.3
Figu re 3.4
----:----c-_ _ _.~~
ZLL
-0<;[ - - - - - - - - - - - - - - -
~
Figure 3.5
At points C, D and E the w ing is mounte d as in Figure 3. 5 wi t h a la rge p osit ive an gle o f a ttack but s o m e t h in g stra nge happe ns to the lift in this a rea. As a has be e n incre ase d , th e lift Di rection Of Mo tioll h a s bee n increa si ng s te a d ily in proportion but now it su ddenly reaches a peak and drops off again . towa rds the lead ing edge , Figure 3.7. At this poi nt the The p henomenon whereby lift d rop s beyond a certain wing is fully sta lled (p o int E on Figure 3.2) . The a ir angle of attack, rather than incre asing as before, is called makes no atte mp t to fo llow the w ing's top surface but the "STALL". The wing is said to have sta lled be cause it breaks up into tur bu lence. The result is a reductio n in c a n n o t be p e rsu a d e d to pro d u ce a ny grea te r lift lift coefficient. Note that there is still q uit e a lot of lift, b ut less th an the re was when the ang le of attack was coe fficient. just less than the sta lling angle.
The Reason To find th e reasons in the ai rflow fo r th e stall it is back to the smo ke tunne l. At sma ll ang les of attack the airflow over the wing is smooth but as an g le of atta ck is incre ased there co mes a po int wh en the flow starts to break away be fore it gets to the trailing edg e , Figure 3.6. The air ca n't quite mak e it down the back of the ae ro foil so th e smooth flow e nds as the st rea m lines a bru p tly br e a k away , o r "se p a ra te ", fro m th e s urface a t th e "separatio n point". If th e angle o f att ac k is increased e ve n mo re the se pa ration point mo ves progressive ly fu rthe r for wa rd
Variations Different sec tio ns have different sta lling characteristics d ep endin g u p on th e ir th ick n e s s , ca mber an d th e s harpness o r b lun tness o f th ei r le ad ing edges . So me s e ctio ns mi ss o u t th e Figu re 3.6 stage a nd th e flo w se pa ration starts suddenly at the le ading edge giving a ve ry abr up t s ta ll as in Fig u re 3.8 (NAC A 230 12 for ex a mple). Ot hers hav e a mor e progressive sta ll as in Figure 3.9 (fo r exa mple NACA 4415). In th e specia l case of a n un c a mb ered ( i .e . sy mme trical) wing sectio n, the graph of lift coefficie nt
Figure 3. 6
:~ Basic Aero na uticsfo r Modellers
21
Figure 3. 7
---== is just w hat yo u expect, and of co urse it perform s just as we ll inverted . Any sec tion will have a graph like Figure 3.10 if the ang le of attack is measured from the sectio n's ze ro lift line. It is merely a case of mov ing the vertical axis alo ng to where the lift is zero . Then, for the straight bit of the graph below the stall, the lift coefficie nt equals the slope of the line times the angle of atta ck . Co nve nie ntly it is found that CL = 0.1 per degree (a pprox) for all aerofoil sections. I sha ll use this idea in th e cha pte r on Pitch Stability.
F igure 3 .8
Notice To Airmen
0< against ang le of atta ck will look like Figure 3.10. Th at is, the lift coefficient is zero at ze ro angle of attack , which F igure 3.9
Figure 3 .10
0<
22
I hate to lab our the po int but not ice w ha t is o n the graph o n Figure 3.2, not speed bu t a ng le of attac k. A win g does not have a stalling speed . It has a stalling a ng le o f att ac k a t w h ic h it will s ta ll m ore o r le ss re gardl ess of th e speed. Tha t is o ne reaso n why lift coefficie nt is plotted , to ge t rid of airspeed and den sity variables which are unimportant to the prop ert ies of a sec tion . It is true that an aeroplan e has a stalling speed, but it is o nly a little true . When I come to mention th e stalling spe eds of an aeroplane I sha ll remind you that it is the stalling an gle of the win g which matters.
0<
Basic Aero nautics for Modellers
Chapter 4
The Drawback • • • Drag n my book Drag is no thin g to do with d ressing up . It is a force resisting mo tio n. To be mo re exact, DRAG is a force exerted by th e a ir o n a moving aeroplane, and it ac ts in exactly the opposite d irection to the di rec tion of motion of the aeroplane . Drag as measu red in the w ind tunnel is made up of tw o pa rts . First the re is the drag from the p ressu re d ist ributio n me n tio ned in Chapte r 2. If th e pressure d ist ributi on d ep ict e d in Figure 2.4 is a dde d up to produce a single resul tant force o n th e wi ng (Fig u re 2.5), then the co mponen t in the direction of the airflow is the Pressure Drag . That is one part, the ot her is good o ld friction. \V'he n o ne object s lides over a no ther , th e re is a friction force resisting mo tion. A friction force Cal; ex ist even witho ut mo tion whic h is w hy the ha nd brake ca n hold the car o n a hill. In fluids (e .g. helium, air, water, .o il, treacle) the friction effect is called "viscosity" an d the d ifference in th is case is that the visco us forces can not exis t withou t mot io n. The visco us dr ag o n an aeroplane is, for tunately, sma ll due to the air's low viscosity an d it occurs in the "boundary laye r". The bo unda ry layer is a ve ry thi n layer of ai r, th e bott o m of which is stuc k to the aeroplane 's surface, and th e to p of w hich is m o v in g wi th the airst rea m (See appe ndix B) . T he flow in th is re g io n may be smooth o r ro ug h ( larn ina r or tu rb ul e n t in te c h n ical jargo n) or more usually a b it of each. It s tarts off la m in ar a nd th e n u su al ly ch anges into a tu rb ule nt bound a ry la yer fu rt h e r dow nstream. A lami nar bo undary layer has less drag bu t is more prone to separate from the surface.
I
ca lcu lated. In the case of a test on a w ing section, the d rag is divided by Y, pVl and th e w ing area, and th e result ing Drag Coefficient , CD is a p roperty of th e section, inde pendent of speed and size , and can be used to es timate th e drag of a ny o the r wing usi ng th a t section . It w ill vary wi th the angle o f attack ho wever, so it is normal to tes t it at a w ide range of angles of attack and then p lot a grap h of drag coefficient against ang le of attack for that section . The typica l shape of such a graph is shown in Figure 4.1. Drag coefficient turns ou t to be a very sma ll nu mbe r which at small angles of a ttack does not vary muc h . There is a min im u m drag ang le of attack (point A) which is no t necessarily where (J. is zero. Approaching the stalling angle of attack (point B) the drag increase is more rapid wh ile above the sta lling angle the d ra g increases ve ry rapi dly indeed . Whe n the wing stalls at poi nt B, the drag increase is pro ba b ly mo re significa nt than the reduc tion in lift coefficient.
Drag Polar Knowi ng the drag of a wing at a ce rtain angle o f
Wing Drag In th e w ind tun ne l eac h aerofo il sec tion ca n be tested to find its drag by s im p ly me asu rin g it o n a ba la nce . Using the formu la at the e nd of Chapter 2 the drag coefficient can be
Basic Aero na utics for Mode llers
23
the resulting graph would look rather squashed so the drag is always show n greatly ex aggerated. From the drag polar you can read off the value of CLrnax and CDrnin • Notice that the minimum drag does not ne cessarily occur where lift is zero. The Lift/ Drag ratio is often taken as a measure of the "efficiency" of a section, and it can easily be worked out from the polar diagram. At any point on the graph div ide the lift coefficient by the drag coefficient. The best VD ratio occurs at the point C wh ere the straight line just touches the graph.
Figure 4.1
B
Thickness and Camber 01.
01.
S
Figure 4.2
..- - - _.. _.... -- -- - .. .. ---:..;--..,---....
The amount of the minimum drag depends mainly upon the section thickness. The less the thickness, the less the minimum drag, but thin wings are not strong so a compromise has to be reached. In addition, a very thin wing has a sharp leading edge, and that is one of the things which can cause an abrupt leading edge stall as on Figure 3.8, in the previous chapter. The angle of attack, or lift coefficient, at which the minimum drag occurs varies with the section's camber. The more the camber, the higher the angle of attack at which the minimum drag occurs. Therefore the drag on an aeroplane which usu ally flies slowly can be minimised by using a section with quite a lot of camber. There is however a large increase in drag if the aeroplane is flown fast. In other words it does not penetrate well. Highly cambered sections are often called "low speed sections".
Laminar Flow Sections
attack is only part of the story. The "drag polar" (as in Figure 4. 2) is useful in showing how much drag the wing produces when generating a certain amount of lift. If lift and drag coefficients were shown to the same scale
Figure 4,3
Cl)
Certain sections have a drag curve like Figure 4.3, with a region of particularly low drag from point A to B. This is known as the "drag bucket", and it takes little imagination to see why . The drag coefficient is virtually constant in the drag bucket and rises stee p ly on either side. By careful design , and keeping the surfa ce very smooth, the designers of the sections have managed to keep the boundary layers laminar (se e Appendix B) as long as possible to take maximum advantage of the lower drag. If the section is not built accurately, or if it is not kept smooth and clean, the drag bucket will disappear. As with other sections the more the camber the larger the angl e of attack where the minimum drag occurs, and the mor e the thickness the more the minimum drag will be. Curiously also, the thicker the section, the wider the drag bucket will be.
Fuselage Drag
A
24
B
A wind tunnel can be used to me asure the drag of a fusel ag e (o r undercarriage or any other part of an aeroplane) . It too will consist of two parts . Surface friction drag will depend on the surface roughness, and on the surface area. The mo re surface area exposed to the airflow (the "w e tt e d area ") , and the greater the proportion of turbulent boundary layer, the more the surface friction drag, but more important is the pressure drag which will depend on the shape of the body.
Basic AeronClutics/orModellers
Streamlining Pressur e drag ca n be m in im is ed b y c a re fu l "s tre a m lin in g " of th e b o d y , that is sh apin g th e bod y s uc h that th e stre amline s in th e a irflo w foll ow th e sha pe of the bod y rath er than break ing away from the s u rface to leav e a turbulent wa ke. For ex ample su ppose the dra g o f a flat disc at right an gles to th e airflow is 100 unit s. Th e drag o f a sphere of the sa me d iam eter w ould be o nly 45 units whil e Scale a ircraft like this SkJ1walker often use wheel spats which significantly th e dr ag of a ca re fu lly reduce profile drag. streamline d body, agai n o f th e sa me d iameter , co u ld be reduced to o n ly four size of the turbul ent wak e whi ch reduces pressure dr ag by a subs tantial amount. It mor e than co mpe nsates for unit s. Yes, the profi le drag of a strea mline d body ca n be redu ced to onl y fou r p er cen t of th at of th e sa me the slight increas e in sk in friction drag. Hen ce the ball dia me ter of flat disc. goes further for a given clout. See Figure 4.4. Th e drag d ue to t h e wake c a use d b y th e flo w / se pa rating from the surface is so mu ch more important than th e sur face fricti on dr ag in th e boundary laye r, wh eth er laminar or turbulen t.
A Bit fo r Golfer s Why , yo u are w ond erin g , d oes a go lf b all h av e dimples? Well as it flies through the air at grea t speed, it has a boundary layer. The d impl es are there to e ns ure that it is a turbu lent boundary laye r, as turbu lent boundary layers cling to the surfa ce lon ger before they se parate . \'(Thich red uces the Figure 4.4
Smooth Ball
Turbulent Wake
..
Dimpled Ball
Basic Aeronautics for Modellers
25
Chapter 5
Have You A Moment lmost certainly! It does not ma tter at w hic h point on the wing yo u choose to attac h the ba lance, yo u w ill almost cer tainly be ab le to measur e a mo me nt abou t th at point. The le ad ing edge might be chosen as a co nvenient po int as it simplifies the ensu ing ca lculations . As w ith lift an d drag the moment coefficient CM is worked o ut from the for mu la at a wide range of different ang les of attack and then plo tte d o n a gra p h . For mat he matical reasons it was decided that nose up moments would be defined as positive, but of course the mo ment abou t the leading edge w ill be nose down , Le. nega tive .
A
Figure 5,1 C~ILE
Nose Up
0( 0 0(
A
B
Nose Doum D
E
Th e g ra ph wi ll lo o k li ke Figure 5.1 in w hich the points A, B, C, D and E correspond to those on Figure 3.2. Th e line is straight from poi nt A, the angle of attack for no lift, to point C, w here the wi ng starts to stall, and then curves down to D and E as the wing stalls. In o ther words, the mo ment ge ts progressively more nose down as angle of attack is incre ased and then at the stalling angle there is a furt he r increase in the nose down moment. Please notice also that at point A, w here lift is zero and a ng le of attack is a o (t he 0 meaning "no lift"), there is still a nose down moment. The corresponding momen t coefficient is ca lled CMo (where the 0 again me ans "no lift ") and it is a lways ne ga tive , Le . nose down, for normal sections. BUT, But , but! I he ar you say. The moment is the turning effect of the lift force so how can no lift have a moment? We ll remember th a t all th is stuff about Lift forces, Drag forces, Moments a nd the Cen tre of Pressure is just for ad ministra tive convenience. \Vh a t we are trying to describe is a pressure distribution around the wing , so let us go back to tha t; look at Figure 2.10 again. At the angle of attack at which the ba lance says there is no lift , th e pressure dist ribution wi ll h a ve cha nged to something like that in Figure 5.2. Th e re w ill be a sma ll downward pressure on the front part of the wing an d a small upwa rd pressure o n the re ar part of the wing, bu t th e angle of attack has been carefully adjusted so that these cancel o ut. However they w ill still have a moment about the lead ing edge, or any other point yo u care to name (see Figure 1.4) ,
Centre ofPressure If the lift and drag and the moment about a known point like the le ad ing edge are known, then the position of the Cen tre of Pressure (CP) ca n be calcu lated . As yo u
Figure 5.2
26
Basic AeronauticsJar Modellers
kn ow , th e CP m o ve s around and Figure 5.3 shows th e trend of the movem ent. Th e Centre of Pressure moves forw ard o n th e wing as a ng le o f a tt ac k is increased . It nearly get s to th e qu arte r c h o rd p o sit ion but then the s ta ll m ove s it ba ck ag ain . At th e other end a c u rio us thing happ ens . Wh en CL is ve ry small the Ce n tre of Pressure dis appears off th e ba ck of th e wing . Th at ca n happen be cau s e it is a mathema ti ca l con ve nience , no t tied to the wing ' with a piece of string . The distance of th e ce ntre of p ressure behind the leading edge is ca lculate d by dividing the momen t ab out the leading edge by th e lift coeffic ie nt. Wh en th e lift coeffic ie nt becom es very very sm all, the answer becomes very very large . When the lift is zero, the answe r is infin ity! You ca n imagine that the ide a of a min iscu le force a gigantic distan ce be hind the wing wou ld have the sa me effec t as the pressure distribution in Figure 5.2. Lift is defined as the co mpone nt of the resultant force at right angles to the a irflow so in Figure 5.2 there is ze ro lift.
Aerodynamic Centre In the spe cial case of a symmetrical aerofoil there is no mom ent at zero lift, and wh en th e CP position is calculated it turns out to be at abo ut the quarter ch ord point at all angles of atta ck right up to the stall, w here it moves back a bit as before. A fixed poi nt like this is so much more sa tisfying . It can be marked on diagrams , and you can take moments abo ut various points an d do little calculations (if that is what turns yo u on) . Wou ldn 't it be just Figure 5.3 thri llin g if we cou ld d o th a t for cambered _ ___ __ __ J_ I Stall_ sec tions as well? C Lll ltu : We ll , Figure 5 .4 is just like 5.1 exc ept that in ad d itio n to th e moment about the leading ed ge , it also shows th e g ra p h of the moment a bout th e trailing edge as w ell. This lin e al s o passes through point A s ho w ing th at th e zero lift mom ent is the sam e no matter abo ut which point it is measured. The onl y difference is t h e slope, wh ich is now the
Basic Aeronautics for Mode llers
"$0, TO LO$ETilE IIUN, t DIVED IIEK $TKAIGIITDOWN• • • • ANDTilE COFP $LlPPED klGIITOFFTilE WING, ZIPPED PA$THY EAK, ANDGOT TANGLED IN TilE TAIL KIGGING WIKE$I"
: .1 1
other way. You are perhaps wondering if it is possible to choose a point in between su ch that the graph will be in betw een the others, de ad le vel like the dotted line in fact? Yes, it's possible! Ba ck in t he da y s wh en Ca me ls fou ght a ga inst Albat rosses , th e Centre of Pressure was th e phrase on eve ryo ne 's lip s, in aerodynamic circl es that is. But in later years whe n aerodyn am icists found that there wa s a po int o n the ae ro foil about which the moment did not vary with a ng le of ,a ttack , they were so ple ased that they gave it a spe cial name, the ae rod yn amic ce ntre of the section (so me times shortened to ae rocentre or just A.C.). Here at last was a point at which th ey co uld pla ce the lift o n their dia grams and in their little calculations and all they had to do was add a mom ent o n the aerop lane which varied on ly with airspeed , not angle of attack . This new mat hematical conce pt described the pressure distribu tion (reme mber Chap ter 2) just as well as the old Centre of Pressure ma thematical co nce pt. The beauty of
_
I am a mathemati cal ~ co ncep t y ou knoui
CP "'"'"
~ Ch o rd
CP Position
27
the e ffect of the sam e co mple x ai rflow and pr e ss u re d istributio n , b ut do n ot fo rge t th a t it is the p ressure d ist rib u tio n which creates th e lift, no t the arrows or for m ulae w hich are just co nven ien t ways o f at tempting to describe it.
Figure 5.4
Aerofoil Section Summary / / / /
/ / /
z /
/
_
A
, \
III Bettoeen
it is th at for a pa rticular section, th e co efficie nt C'\ln is a
constan t, jus t a sma ll nega tive number, lik e -0 .05 for NACA 2415 for ex ample (but it is co nstant only below th e section's sta lling angle). Now the pressure distribu tion may be represented by forces in fou r d iffe rent ways. They are shown in Figure 5.5 . The first is the resu ltant force th rou gh the CP. O r one co u ld show th e tw o se parate compone n ts, Lift and Drag , at the CP. Bu t since t h a t is im practica l w he n yo u come to measure it in a wi nd tu nnel, th e for ces can be measured as Lift and Drag at a fixed point like the lead ing edge toget her with a momen t abou t th e lea di ng e dge , and finally the Lift and Drag at the aero dyn am ic ce n tre and a momen t )'1'10 . This last metho d is m o st convenient for calculations . These are all equally va lid ways of showing
Now th at I have ment ioned al l th e s e c tio n characteristics, I would like to de scr ib e , wit h the hel p o f Figure 5.6 , how an aerofoil sec tion may be made u p , an d ho w we can influence its aero dy namic coefficien ts. First d raw a straight line w hich wi ll be the cho rd line of th e section. Next d raw in the ca mber lin e . Th e maxim um ga p be tween it a n d th e chord lin e is th e camb er of th e section, w hic h may be fro m zero to 6% or possib ly 8% of th e chord . The ma x camber can occur b et w een 15% and 60% of the chord fro m the le ad ing edge. Th en a thickness dis tribution is w rap ped around the ca m ber line . This may b e do n e b y d rawi n g lines o f appro pria te le ngth across th e ca mber line an d jo ining thei r ends, or drawing a series of circles w ith centres o n the ca mber line an d jo in ing th e ir ta ngents as shown. The maximu m th ickness is usu ally between 6% and 18% and occ urs from 15% to 50% of th e chord from the lea di ng edge.
Thickness and Camber Increasing the thickness will • 1. increase the m inimum d rag , CDmin • 2. wi den the d rag bu cket o n lami nar flow sec tions • 3. increa se streng th/weight ratio . Increasing the camber will • 1. incre ase CU11:IX (very th ick or very th in sectio ns have a re duced Cl.llm due to an ea rly sta ll) . • 2. make th e ze ro li ft ang le of a ttack , 0: 0 more negative. • 3. inc rease the lift coefficient at w hich min imum drag occurs.
Figure 5.5
L
L
( 28
L
D
AC
Basic Aerona 11lies for 1110dellers
Figure
5.6 Cambe,. Line Ma.'\: Camber
Cb o r d Line
t Il(/If
Th ic k ness
Distributton
Tbickne ss
Half Thick ness added eacb side of ca m bel'
• 4. increase the negative va lue of C~ IO ' which will be be tween -0.02 and -0.03 for each 1% of cambe r. • 5. reduce the negative (inverted flight ) Clm a,
Section Classification and Use
the p ercentage thi ckness is chose n to b e a bou t 3.33 tim es th e ca m be r then the re a r 70 % o r 80% o f th e aerofo il underside often turns o ut flat. That makes it easy to build, it has a good upright pe rfo rma nce but is poor invert ed.
SYMl'l'JET RICAL se ctions hav e ze ro ca m be r and therefor e ao and CMo are also ze ro. \Vitho ut ca mber they have rath er a lo w Cu lla, but at lea st it is as good inverted as uprigh t. The least drag occurs at ze ro lift. Symm etri cal sectio ns a re th us best fo r h igh speed a nd a e ro ba tic aeroplane s. Their th ickness is a co mpro m ise be tween st rength and dr ag , typically 10% to 18% fo r wi ngs and 6% to 10% for tailpl an es.
All other sections are cambered sections. An UNDERCAMBERED sec tion IS Just a thin highly cambe red sectio n. Sometimes the camber is just enough to mak e th e underside slig h tly co ncave as o n WW I ae ro planes . On some free flight floaters the underside is very co ncave because the pe rce ntage cambe r is as mu ch as the th ickn ess. Such sections are very good at large lift coefficients (low speed) but poor at sma ll lift coefficients (high speed) , whi ch me an s they do not pen etr ate well. They are also useless inverted . The BICO NVEX (or SEMI-SYMMETRICAL) sectio n is so ca lle d because both top a nd bott om surfaces ar e co nve x, bu t the top o ne is mo re so. That is be cause the ca mbe r is small co mpa re d to the thickn ess, and the faster o r more aeroba tic th e aeroplan e will be , the smaller the cambe r sho uld be. Th e FLAT-BOTTOMED sectio n , like the Clark Y or Gottingen 796, is a specia l case of a cambe red sec tio n. If
Basic Aeronautics for Modellers
29
Chapter 6
The Vortex System ith no wing in the wi nd tun nel the strea mlines in the flow would be straig ht and paralle l as in Figure 6.1. Putti ng in a wing cha nges the airflow so mewha t as sho wn in Figure 6.2. The changes imposed o n the air's ve locity by the wi ng are an upwash just in front of the wi ng , a speed inc rea se abo ve and a decrease below the win g , and a downwash behin d the wing . Figure 6.3 shows these velocity changes (shown as dV) in isolation . The effec t of the wing seems to be to
W
F igure 6.1
indu ce a kind of sw irling mot ion to the air, around itself. A rota ting flow is c a ll e d a VO RTEX (Ap pe n d ix C ex plains vort ices in mo re de tail) . \'V'hene ve r a wing is produci ng lift it tends to indu ce this ci rcu la ting flo w around itself , and the more lift the more circulation . This vort ex is ca lled the "bo und vor tex " as it is fixed aro und the w ing . Vortices canno t end abru pt ly in mid air. In the win d tunne l th ey end o n the wi nd tu nne l wa ll w hich is fine. But w hat happen s if the wing d o e s no t exte n d from th e w all to w a ll? Wh at h app en s if t h e w ing has . . . (wait for it) . . . TIPS! YES fo lks we are now into THREE DIME NSIONAL FLOW . d id p romise to wa rn you) . We ll yo u know very w ell wha t happe ns , the
er
Figure 6.2
Fig u re
~
6.3
dV
dV
I
dV
.. 30
!
dV
Basic AeronauticsJar Modellers
Figu re 6.4
~ Loto Pressure
( ~Wi"gTiP
High Pres sure
~
LE
I
I I
I
Bottom Surface Flow Slightly Outward
I
, I
Top Surface Flow Slightly Inuiard
TE vortices do not just end, they trail off in the flow behind th e wing tips . Th ese vortices a re ca lled the Trailing Vortices . They would go on for ever if the air 's viscosity did not dissipate them an d absorb their energy. I have watched th e condensa tion trails of a Boeing 747 st ill gently rotating when fo llowing 2000 fee t be low and n ine teen miles be hind . There is another eq ua lly valid way of loo king at these trailing vor tices . At o ur newly acquired w ing tips , the air p ressure is lower above the w ing than below. The air inevitably tries to go from high pressure to low , arou nd the tips, whic h gives rise to a degree of spanwise flow , outward on the lo wer surface and inward on th e top. The trend co ntinues to a decreasing extent some way in from the tip . Wh en the to p and bottom flows re unite at the trailing edge , th ey are mo vin g in slight ly di fferent directions, sligh tly outward on the un derside and slight ly inward on top. In Figure 6.4 I have tried to show th e result of all th is. Along the trailing edge, especia lly near the tips, vor tices are formed which all roll up togeth er to form one la rge vortex trailing beh ind each wing tip .
Seeing The Vortices T here is a n easy way yo u ca n see your mod el 's trailing vor tices . Attach three streamers twe lve to fifteen feet lo ng to each wing tip of a suitab le model used a Ga ngs ter 63) . Lay them out straig ht o n the gro und for take off. O n climb out it w ill be seen that the strea mers
er
Basic Aeronautics/or Modellers
are bei ng w hirled rou nd by the airflow, clockwise o n the left an d an ticlockwise o n the right. TIY a slow flypast. The stre ame rs will be swi rled in large slow sw irls. Now try a high speed beat u p . Notice th e d iffe re n ce in th e way th e strea me rs a re swirling s ugges ting a less stro ng vortex . The lift equa ls th e weigh t in both cases sugges ting tha t a t low speed a stronger vortex is needed to ge nerate the sa me lift. O n the ne xt high speed pass try pulling a tight loop as the model passes. The swir ling noticeabl y increases as soon as yo u pull the u p elevator to inc rease the lift. Now do a low inverted pass. From your p~i nt of view nothing is d iffere nt. The lift is still up and the vortices still go clockwise o n your left, and an ticlockwise on your right (from the aeroplane 's po int of view the d irec tions of rota tion and lift have all reversed). The lift is related to the vortices in dire ction as well as strength. Check that by coming in slow and high and do ing a bunt, o r ou tside loo p. Watch the st rea mers ca refully as yo u ap p ly d o wn e leva tor. You will see the m stop rotating and then star t rotating again the o the r way round . Th is w ill contin ue all th e way round the b un t until when yo u re lease th e do wn e leva tor to co ntinue no rmal fligh t, th e rotations reverse aga in . The lesson to learn from this is that the stre ngth of th e vor tices increases wit h the lift coefficient of the wi ng . After abo u t five mi n u tes of this th e s treamers h ad flap ped themselves to pie ces and were down to two or thre e feet long.
31
Figure 6.5
AR = Infinite
AR = Infinite
,.-,6
.>: "\ //
I
,'
1/ /
. . --, 3
,/
1
I,'
1//
1:/ '
,:
1,/ '
~,r
:~
Even More Drag Th e ASPECT RATIO of a 3-D win g is defin ed as the spa n d ivided by the ave rage chord. It is found tha t when a rea l wing with tip s is tested in a w ind tunne l its dra g is more than if it fitted perfectly fro m wa ll to wa ll, and the lift is less. Th e loss in performance depends on its as pect ratio as illustrated in Figure 6.5 . The high e r the Aspect Ratio of the wing, the ne are r is its pe rformance to that of the ideal tw o di mensional wing (infinite aspect ratio) .
The Reason This sho rtfall in p erformance is caused by the trailing vortices whic h create a reg ion of descending air be hind the w ing, a fter all the energy to crea te the m mu st be p aid for so me ho w . Th at th es e vo r tices a ls o c a use d ownwash in the airflo w as it ap proaches the wing ca n be proven by the ory , or dem on strated at hom e by filling a tall glass w ith wa ter and placin g a few g rains of rice at the bottom . \Vith a spoo n , stir the wa ter in the glass near the top and yo u w ill soo n see th e rice g ra ins begin to
sw irl. Bec ause of the fluid 's viscosity a SWirling motion is ind uce d rig ht to the bo ttom of the glass. If the spoo n is the wingtip vo rtex stirring the air be hind the wing , the rice is be ing sw irle d ro und a head of th e wing, in the sa me d irection , bu t to a lesse r ex te nt. Fig ure 6.6 s hows th e airflow aro und a re a l thre e d imen sion al wi ng in more deta il. A long wa y ahead of the wi ng the airflow is undisturbed by its presence. As the air approac hes , it is angled down s ligh tly by the d ownwash ahead of the wi ng ind uced by the tra iling vo rtices and th e n jus t in fro nt of th e wi ng the air is swept up and over by the boun d vort ex as in 2-D flow . We started by defining the ang le of attack as the angle be twee n th e w ing and th e und isturbed a irflo w aw ay ahe ad of the w ing . No w we ca n see that th e "real" angle of attac k of the air meeting the wing has been red uced by the downwash. And the lift re lates well to the lift p redic ted from 2-D tests at this reduced an g le of attack. So t h e lo s s o f li ft is e xp la in e d b y t h e d ownwa s h red u ci n g th e a n g le of a tt a ck . But w h a t abo u t t he increase in d rag? Look ing back at Figure 6.6 agai n, the ae roplane mu st
'~OIlIlY, TIIAT ~1I0ULD BE 'NOW TIlYA BUNTCAIlEFULLY, WATCIIING rse~TIlEAMEIl~"
..
"..~
.. • .•
32
~. .
. C'"
Basic Aeronautics/or Modellers
Figure 6.6
Measw'ed
Cl.
Real
Real Lift
Cl.
---
---
Doumuiasb All
-------=======:----Undisturbed Ab'
Doumuiasb III Front Of lVi1lg
think it is constantly flying uphill, or rather flyin g le vel throu gh the sinking air of its o wn do w nwash . The lift for ce has been tilted back a little , by the amou nt of the dow n was h a ngle . Th at means th e lift ha s a s ma ll component in a d irec tio n opposite to the directio n of motio n . If it opposes motion it is Drag , isn't it? This compone nt of the total d rag is called "ind uce d drag" because it is caused by the tilting back of the lift ca use d by th e dow nwas h ind uced b y th e tra iling vo rtices. You may also see indu ced d rag referred to as "active drag". No amoun t of strea mlining or fiddling w ith th e section will red uce it. It is inevitable as it co mes from the lift.
Complications Figure 6.6 shows the airflow at o ne particul ar place on th e wing . Near the tip s w here th e vortices a re centred the downwash is grea ter than o n the ce ntreline, as in Figure 6.7. Th us o n th is recta ng u lar wi ng each point from the roo t to tip has a d ifferent "real" o r "local"
Doumuiasb Behind Wing
ang le of attack and th erefore a d iffere nt "loc a l lift coeffic ient ", u su all y d e n ot ed Cl (w it h a s ma ll I su bscript). The local lift coefficient red uces towards the wing tips. Th e situa tion even on a straight w ing is not as simple as the p icture I have pai nted up to now. On a ta p e red t w isted swep t wi ng the si t ua tio n is as co mplicate d as yo u ca n imagine , if not m o re ! Th e variatio n of the "local" ang le of attack de pends on the variatio n of the downwash w hich depends on . . . well just about everything , includ ing the variatio n of the local ang le of attack . So becau se the downwas h angle, lift coefficient, drag coefficien t, lo cal ang le of attack , ce n tre of pressure positio n, etc, all vary wi th po sition along the spa n , that makes it very difficult to use the section characteristics me asured in two dime nsional flow . It is just too muc h for the hu man bra in to cope wi th and is be st le ft to co mputers with time on the ir han ds. \'\fe co uld give u p the who le messy bu siness here and no w , o r we co uld just step back and look at it from a d istance.
Figure 6.7
~
Direction OfMotion Wi1lg's Apparent Angle OfAttack
Root Doumuias
Tip
Cl.
Direction OfMotiotl WhIg's Apparent Angle OfAttack
Basic Aero na ut ics fo r Mode llers
Tip Doumuiasb
33
F igu re 6.8
L
~ -- -- ~Airfloto
Simpiifications
Lessons f o r P r a ctica l Modellers
Thi s looks like a cla ssi c case for the bla ck box system . I shall draw an imaginary black bo x around the wing and care not a wh it for wha t is happening insid e . Air e nte rs th e fron t of the bo x a nd co mes o ut o f the back an gled d own slightly by the downwash, Th e angle of attac k is mea sured between the direction of motion and a referen ce line d rawn o n the outside of the box. Th e refer en ce line may be e ither the cho rd line at the ro ot sectio n o r th e ze ro lift lin e of th e who le w ing . Th er e is a Lift for ce perpendicu lar to the d ire ction o f motion and a total Drag for ce , including induced d rag, op posi te to the d ire ction of motion. Th e re w ill be an overall Centre o f Pressur e but I wouldn't care to guess at its p ositi on so I prefer to p ut th e lift a t th e w in g 's ae rody na mic ce ntre (25% mean chord) and apply a ze ro lift pit ch ing mom ent. See Figure 6.8. \Xrhe n I refe r to lift coefficient , or drag coefficient, I mean an ave rage for the wh o le win g wor ke d o ut fro m tests and th e Cha pter 2 formula e . The lift co efficient Cl. is the overall average for th e wh ole w ing and has a ca p ital L su bsc ript. Its grap h will still be a familiar sha pe but need not be the sa me as the section's curve . The wing would hav e to be tested to get acc ura te graphs but a re asonable guess could be mad e by making "allowances" for as pect ratio, tap e r and twi st. Th e slo pe of the straight bit will dep end o n the as pect ratio (as in Figure 6.5). And the position of aD and the stall will dep end o n the wing's planforrn a nd twi st as mu ch as its sec tio n.
Reducin g induced d rag is important fo r aeroplanes which cru ise a t a la rge lift coefficie nt (Le low speed) therefore glid ers mu st have as high a n Asp ect Ratio as is pra cticab le . Also a lig hte r ae ropl ane has less induced dr ag than an identical he avy o ne a t th e sa me spe ed be cau se of the lower lift co efficient. Any lift, even downward or s ide ways lift, will cause vor tices whi ch cause down wash which will tilt the lift back wh ich co ntributes to induced drag . If a tailp lan e is carrying a download , not o nly d oes the downward lift on th e tail produce indu ced dr ag but th e Wing must produce ex tra lift to co unterac t the download and that means ex tra indu ced dra g o n the wing as well. Most wind tunn el sectio n test s ar e d on e o n tw odim en sional models. If these resu lts a re used to es tima te th e p e rform an ce of a re al ae ro p la ne th e y will g ive opt im isti c a nswe rs becau se the y d o not include the induced drag. It is possib le to es tima te the charac teristics of a w ing from se ctio n da ta usin g var iou s fidd le factors but that is o utside the sco pe of this bo ok.
Th e Importa nce ofAspect R a tio A sho rt span broad win g mu st have a stro nger vort ex to g ive the sa me lift as a lo ng narrow w ing, a nd w ill co nseq ue ntly produce mor e downwash . Th er efor e the induced drag is greate r for a win g of low Asp ect Ratio than for o ne of high Asp ect Ratio (AR) becau se of the g re a te r d o wnwa s h , An d as th e a n g le of a ttac k is inc reased , bo th Lift a nd d ownwash a ng le in cr e as e (because the vo rtex stre ng th increa ses ). Thus induced dr ag increases rap id ly. In math ematical shorthand th e induced dra g coefficie nt (C n) is given by;
34
Grou nd Effect Lo oking at Figure 6.6 you probably thought that the wind tunnel wall s would co nstrain the downwash effect, and yo u we re right. Th e ind uced drag of a finite Wing will be underestimat ed in a wind tunnel becau se th e do wnwash is reduced . Th e sa me thing will happen if an aeroplane is flown very low over the gro und . Th e proxim ity of the gro und will reduce th e downw ash and therefore th e ind uced dra g will be reduced . Th e effec t is to prolon g the flare . For this rea son , flight testin g mod els with in a few win g cho rds of the grou nd will give misleading result s.
Basic Aeronauticsfor Modellers
Chapter 7
Planform and Twist s yo u sa w in the Figure 7. 1 previo us chapter, the Aspect Ratio Elliptical Lift Distrilnttton Loading - Lift/Unit Span o f th e wi ng lar g el y _-.,-.....-rTTT"'T"TTT"TT"T...........--,-.">----d et e rm in es ho w mu ch in d u c e d dra g it wi ll c a use . Bu t th e littl e e q ua tio n for in d u ce d drag coefficient w hich I ju st s lip pe d in th e re co ntaine d a co nstant, K, as we ll. Thi s K (ca lle d the Indu ce d Drag Factor) depen ds upo n how the load is sha red alo ng the wing . T he lift ma y Figure 7.2 be eve n ly sp read , o r mostly near the roo t, o r what eve r. Fig u re 7. 1 s hows w ha t theor y says is the Rectangular Area Dtstribution id e a l lift d istribu ti on w hich gives K its min imu m value o f 1. Th is is a d iagram of the Lift Per Unit Spa n . Each arrow rep re s ent s th e li ft o n a o ne inch wide strip of win g , and th e lin e joi ning the to p s of a ll the arro ws is an e llipse . Loading
A
Local Angle of Attack We m u st a fte r a ll p e e k ins ide th e b la ck b o x d e si gn e d to o b sc u re a ll thi s co m p licati on . O n o ur usu al recta ngular wing wh ose c ho rd, sectio n a nd in ci de nce a re th e sa me fro m ro ot to tip t h e load ing , show n o n th e rig ht of Fig u re 7 .2 , is ne ithe r rectang ular like th e pla nfo rm no r e llipt ica l as s h o w n b y th e
Figure 7.3
Elliptical Area Distribution
J
Local Cl
Basic Aerona utics/or Modellers
1
~
Loading
-s
Ltfr/Untt Sp a n
35
It is goo d m an a ge ment pr a ct ice to move work ers from wh ere they are id le to w he re th e y are more productive and so we do with win g a rea. Taper R a tio OS An o bvio us id e a mig ht be to use an elliptica l p la n forrn (Fig u re 7 .3) wit h n o twi s t. I Ta k ing a close lo o k a t I suc h a wi ng und e r test, I a n e lli ptica l lo ad in g is Local Cl I possibl e and the downI wash is co nsta nt ac ross Max the s pa n, which means th at th e lo ca l angle of a tta ck is a lso co ns ta n t. T he re fo re th e lo ca l lift coeffic ie n t is the same a ll ove r the win g. Every sq ua re inch is doing its Taper R a tio 0.33 fa ir s ha re of lifting , h en c e m a ximum e ffi ciency . It co uld be tricky to bu ild with acc u ra te sectio ns throu gh out. A reaso na b le co mp rom is e is to b u il d a L oading - wi n g wit h a st raig h t ta per fro m root to tip as in Fig ure 7 .4 in w h ic h yo u ca n see that the lift dist ribution ca n be ma de q uite clo se to the idea l e llip tica l shape. It is nea rly as attack towards the tip. The res ult ing local lift coefficie nt, o r lift per un it area , is shown o n the left side of Figure easy to build as the rectan gular wi ng but is sig nificantly 7.2. Loadin g and local Cl are the sa me becau se chord is mo re efficien t. Tapering the wing helps to improve co nstant in this case . e fficie ncy b y eve n ing out th e d o wn w a s h a n d so increasing the local lift coefficient towards the tip . Thus th e lift is more even ly sha re d th an o n a Re ctan gul ar Differe nt Planform Shapes w ing . Th e o p tim um Taper Rati o (defined as the tip To reduce the indu ced dra g factor the planform of the cho rd over the root chor d) is abo ut 0.4 . If the tap er ratio wing may be changed . is redu ced bel ow th e opt imum 0 .4 , th e lo cal angle
Fig ure 7.4
J
(
~
Fig u re 705
Double Taper Taper R a tio OS
--- - - - ---- --~-~-~-~~~
----
--
Loading
Local Cl
M ax
36
Basic Aeronautics/or Modellers
Figure 7.6 Root
Washollt
_-=- -=- -::. :. -=- I) -=- -:. r::::===--~__
of a tt ac k near th e tips b ecomes greater than ov e r the inboa rd s ect io n s, the tips beco me overloaded , and efficiency is again lost. Another slight decr ease in the K fac tor may be had by building a wing with a do uble taper , wh ich is very littl e e xtr a trouble esp ecially if the wing is too long to build in o ne piece anyw ay . See Figure 7.5. Th e double tap er ed planform gives a n a re a dist ribution even closer to tha t of an ellips e a nd so is more efficient still.
Fig ure 7.7 C,
Lift/Spa"
: LOllJ 1 Speed 1 I 1 1 1 1 1- _ _
Local C,
~
1
Load ing ".. - - -
High Speed
.... ....
.... ....
Ma~:
Tip
Tip
Tips talling If the lift distribution is of the ide al elliptical sh ape then the local angle of attack will be co nstant along the spa n . Theoretically the w ing sho uld st all all the wa y along the wing simultaneously. But the inevitable slight imp erfection or dirt or even a twitch of aile ron means that o ne side will stall before the o the r. Eve n if just the outboard portion of one wing stalls first the effect of the diffe ren ce in lift and drag between one wing an d the other is a vio lent ro ll, a "w ing drop". Because of its very efficiency , an elliptica l wing is lik ely to drop a wing when it stalls, and pilots just don't like aeroplanes w hich flick themselves upside down un expectedly. On a constant chord wing the local an gle of attack is least near the tip. The airflow approaching the wing is "tw isted" down at the tips ensuring that in stea dy flight it is most unlike ly to tipstall. But e ith e r surface irregularities or turbulence or a rudder input, if seve re en ou gh , can produce a tipst all on mo st wings . Of course avo iding tipstalling does not mean that a win g will not drop at all. Even a root stall o ne sid e at a time will give a ro ll, bu t nothing too violent. A tap ered wing's characteristics will be between the two , dep ending on the tap er ratio. The more taper, the more lik el y is a tipsta ll. If th e taper on a wing is ' ex ce ssive (ta p e r ra tio 0 .4 or le ss) then a tipstall is in e vit abl e as the local angl e o f a ttac k is greatest o utboard. In Fig ur e 7.4 I have shown the spa nw ise position wh er e local Cl is a maximum, and this is where the wing should stall first (assuming co nstant sections).
Basic Aeronautics for Modellers
Washo u t If a wing is twisted such that the incidence decreases toward s the tip , that is called "washo ut". Cl deliberately use the word incidence because th e angle of attack will depend also up on the variatio n of the downwash .) See Figure 7.6. Th e o pp os ite twist is, logically, called wash in. We mod ellers use washout as a cure for tipstalling . Twisting the leading edges down at the tips allow s them to fly at a lower angle of attack so hopefully the roots will stall first. Another effect of washout is to shift some of the lift inboard o n the wing (Figure 7.7) . Th e effec t of that will obviously be to increase the indu ced drag factor, K. The tips m ay e ve n be lifting downward a t high speed . Adding washout to an efficiently tapered planform takes aw ay so me of its inherent efficien cy. Consid er the effect of wa shout o n the gra ph of lift against ang le of a ttack for the whol e wing. In Figure 7.8, if line A is for an untwisted wing, then line B sho ws the effec t of the sa me w ing of con sid erable washout. The an gle of attack has been mea sured aga inst the chord line at the ro ot. The root has to be rotated to a higher angl e of atta ck to achieve the same overall lift becau se the tips start off lifting downwards. The stall is very mild and prolonged bec ause it starts at the root and works its way gradua lly towards the tip as angle of atta ck is increased. As th e lift on the tips is increasing , th e lift on th e roots is reducing, and the maximum lift de veloped by the wing is much less in case B. That means that , although th e stall is gentler, the stallin g spee d will be high er.
37
--
parts are do ing their fa ir s ha re of lifting . Inste ad of unl oa di ng the tips to p re vent th em fr om stalling (and sp oilin g the e fficie n cy), w hy not tr a in them to w o r k h arder? Ch oose a tip section w hich is capa ble of a hi gh er Clm"x than the roo t se ction , becau se it h as mo re , or d iffe re n tly s ha pe d, ca mber. Th en apply so me washo ut to the wi ng so that b oth ro ot and tip w ill have th e sam e Cl from zero lift to th e stall. But th e ro ot w ill stall first! It so u nds like a Uto p ia n id e a but tak e c are se lecti ng the sec tio ns, and read the cha pte r o n Re ynold s Nu m be r carefully.
-------=---_.
~t;;(J_
<
.
...
Sweepback
Aerodynamic Washout A similar effec t to wa sh out can be obtained by using a tip sectio n wh ich stalls at a high er geomet ric a ng le of attac k than the roo t, keep ing their ch o rd lines parallel as in Figure 7.9. Th is effec t is called "aerodynamic washo ut" since, as a is increased the roo t section w ill re ach stalling an gle first. Figure 7.10 is a d ifferent app roa ch to the pr obl em. We ha ve design ed a wi ng wh ich is efficient becau se all Figure 7.8
A .. ...
Tip
~
.
Washout Angle
38
The a ng le of sweep o ug ht to be me asured a t th e qu arter chord line of the w ing (altho ug h so me peo ple me asure it at th e leading e dge). Swee p ing th e w ings ba ck (o r forward for that matt e r) is a device for dela yin g the o nset of the tran son ic d rag rise, which is nothing to do w ith us mod elle rs. We d o not ca re th at a straig ht wing ex perie nces a sharp pr ofil e d ra g in cre ase as s peed ris es throu gh a Ma ch Num b e r o f 0.6 o r so ( 0.6 times th e s pee d o f sound) . Nor does it matter th at th is d rag rise ca n be del ayed to Mach 0.8 o r 0.9 by sweeping the wings ba ck . But models do ge t bu ilt wi th swept wi ngs so I had bette r mention it. Ap art from its ma in pu rpos e me n t io n e d above, it ca n be used to ad jus t the position of the mean ch o rd o f the wing back or forw ard with out moving the roo t fixings. Su p pos e yo u d es ign a model and it turns o u t tail he a v y . Yo u co u ld C build a new w ing with a little sweep ba ck and get it c o rrec tly b al an c ed Root without ball ast a n d w ith o u t a lte ri ng th e fuse lage . Swee pbac k a lso has various side effects, like the e ff ec t o n lat eral stability (see Cha p ter 9), and in large doses it has 0( a detrimental e ffec t on th e efficie ncy o f th e Measw'ed At Root wi ng. T he a ir h a s a
1:---
Basic Aerona utics f or Modellers
Figure 7.9
Root Tip
I I I
Ti
I
I
I I I I I I I
Root
Aerodynamic Waskrout
I
~ I I
~
I I
a tenden cy to flow out alon g a swep tbac k win g towards th e tip s which reduces th e lift. Th e tenda ncy ca n be counterac ted by using win g fen ces o r notch es etc.
Th e line from A to B crosses the 50% chord line at point C, the ce ntro id of the sha pe . The chord through C is the win g's M.A.C.
Mean Chord
Geometric Mean Chord
What does one use as the win g chord mea surem ent if the w ing is tapered and swe pt? The cor rec t va lue to us e is w hat ae ro dy namicists ca ll th e "Me an Aero dy na mic Cho rd " CM.A.C. fo r sho rt) o f th e wing. It need s rather co m p lic a te d math emat ics to d efine a nd ca lc u la te it. However it may be fo und graphically.
Th e j'd .A.C. is unn e c e ssaril y c o m p lica te d for us mod ell e rs . Th e Ge om etr ic Me an Cho rd is jus t th e a ve rage o f the root and tip cho rds, or the wing a re a di vid ed by the span . The G.M.C. is slightly smaller than the M.A.C. but with a tap er ratio of 0.65 th e e rro r is onl y 1.5%. I suggest that the geome tric me an chord is acc ura te e no ug h for o u r purpo ses and s ha ll us e it in fu tu re , dropping the "geo metric" and calling it just the "me an cho rd ".
Graphical Methods Figure 7.11 shows a g ra p hical w ay of find ing th e ce ntro id of a swept tap ered wing. Draw the w ing o ut to scale and square off the wing tip . Now join th e midp oint o f th e tip c ho rd to th e midpo int of the root cho rd . Extend the tip cho rd forward by the am ount of the root chord to po int A. Extend the root cho rd aft by the am ount of the tip cho rd to point B.
Horses for Courses Th e us e of tap er a nd was ho u t is a co mp ro m ise b etwe en perfo rman ce a nd handling and a different co mpromise is ne cessar y for different typ es of mod el.
Figure 7,10
Tip
" ""
"
""
" ""
" ""
"
""
,,- .... ,
""
-- -,, ,Tip
I I
I :
Tip
Add Some Was bout
I I
..
~ I
Tip
d:=
I
I
I I
I
I
I
I
Root
+=+:::,..
C-R-oo-t-~
0oot:~
Basic Aeronauticsfor Modellers
I I I I
Root
I I 0<
0<
39
Gliders Figure 7.11 Willg Centre Line
Half Span ~---------------------------------------~ I
,
C,.'
A \
\
,,
\
I I I
Indu c ed dra g is ve ry s ig n ifica nt, th erefor e gliders use taper ed w ings , if possible double ta pere d . Th e y a lso use as high a n aspe ct rati o as po s sibl e to re d uce induced drag. Howe ver th ere ma y be occasions wh en co ntest rule s ma ke the high as pec t rati o le s s n e cessa ry (see c ha p te r o n Re ynolds Number).
I I I
J
I I I
,, ,, \
,,, I
I
,c"
, Cl I
I I I \ \
\
\ I I I
ic,
B
Trainers It is imp ort ant that a train er sho uld not tipst all eve n when p rov ok ed by rough handling , and it mu st be easy to build . For these reasons a co ns tant cho rd un swept win g of mod est Aspect Ratio is usuall y employe d . The loss o f aerody na mic efficiency is unimportant.
Scale Scale mod els leave no cho ice of taper ratio , as pect ratio , o r sweep . If the tip chord is less than two third s of th e roo t c ho rd , or if it is e llip tica l, th en a co u p le of degrees of was ho ut would be prudent, more if the w ing is stee ply tap ered . The scale mod eller ad justs the amo unt of washout to achie ve the kind of handling he prefers.
Aerobatic The "Pattern ship" mu st above all be pr edi cta ble . It mu st not tipstall accide ntally a nd ye t it mu st sp in reliabl y wh en co mma nded with rudder co ntrol. It mu st fly as well invert ed as upri ght and so was ho ut is undesirable . For these re ason s a tap er ratio o f 0.55 to 0.65 is usually employe d. Any sweepba ck would o nly be for aes the tic reasons, altho ug h I have heard it sa id that it help s the rolls and disgui ses aerobatic er rors.
40
Basic Aerona uticsfor Modellers
Chapter 8
CG and Stability T
he cen tre of Gravity (or CG) of an aeroplane is the po int throu gh w hich its weight acts , o r where it ca n be su ppo rted with out falling o ver. Its fore and aft position , w hich is wh at matters most, is shown o n pl an s by the sy mbols ~ or
0
Most mod ellers have a general idea of w ha t stabi lity is and how it is related to Cen tre of Gra vity pos ition, bu t let us go a bit further for a better understa nd ing .
the p oo l table . Disturb it sligh tly and it w ill remai n in its new position (Figure 8.3). Ther e is no res tor ing force or d iverging force . It is neutrally stab le. Th is ex pe rime nt shows the sta tic sta bili ty an d is co ncerned on ly with the initial reac tio n to a disturban ce. Figure 8.1
Defining Stability in General Th e s ta bi lity o f a b ody is its tend e n cy to ret urn towards its original state if d isturbe d slightly. The re are tw o types of s ta b ili ty , Stat ic Sta b ili ty a nd Dyn ami c Stability. Dyna mic Stability is about what happens over a lo ng period of time following a d istur ba nce. Th at is, a bod y is dynam ically stab le if some time after a distu rban ce it has se ttled down into its o rigina l state . Dyn amic Stability is impossible unl ess you first achieve Static Stability. Sta tic Sta b ili ty is co nce rned o n ly with th e in itia l reaction of the body follo w ing a disturbance . Th at is, if the bod y is dis turbe d slightly from its equilibrium state , will forces automatica lly arise whic h will ten d to mak e it return back towards its eq uilibr ium state again? If so it is statica lly sta ble. Th ere a re thre e poss ible co ndit io ns : Uns ta b le, Ne u tra lly St able , or St abl e . Fo r a d e m o n stration of each of these s tates ge t a snoo ke r b all , a Ch inese Wo k , and an o ld British army steel helmet o n a pool tab le . Let the ball rest in equilibrium right in th e cen t re o f th e Wok. Distu rbing it slightly will pr oduce a fo rce w hic h will ma ke it wa nt to roll back towards the ce ntre . See Figure 8.1. Thi s is sta b le . No w ca refu lly b alan c e th e ball at th e ve ry to p o f th e steel helmet. Disturb it slightly and it will tend to c on tin ue in th at d ire cti on and fall right off (Figure 8.2) . That is unstable or "Divergent". No w pl ace th e ball o n
Basic Aeronauttcs f or Modellers
Figure 8.2
1
2
41
Figure 8.3 2
1
00 Wh at the final o utco me will be afte r a time has e lapsed will depend upon the dynamic stab ility. If a bod y (ball or aero plane) is statica lly stable like the ball inside the \'{ro k, we may investigate its dyn am ic sta b ility . If the ball is hel d four inch es up th e inside surface of th e Wok and re leased , it w ill return to the ce nt re bu t will overshoot and go up the other side say thre e and a half inch es , then down and u p three inc hes e te. until afte r a while it w ill settle in the centre . It is thu s dynamically sta ble d ue to the damp ing force ca used by the rolling resistan ce . If the Wok wer e full of wa ter it would have grea ter da mping (wa ter, damping, o h neve r mind) w hich wo uld increase the dynamic stab ility and the ball wou ld se ttle in the ce ntre more q uick ly. Figure 8.4 shows the kinds of stability graphically. T h e dynamic stabilit y o f a eropla n e s is a very co mp lica te d subject which I do not want to d iscuss. Suffice it to say that to a large ex te nt it depends upon the a ircraft havin g sufficie nt aerodynamic damping, or resistance to pitch ro ll and yaw motion s, which I sh all me nti on lat er, in Cha pte r 10. Th e rema inder of this cha pter is about Static Stabili ty.
Motion of an Aeroplane An ae ro plane is free to move in thre e d imen sions. In Figure 8.5 I ha ve d rawn the thr e e axes o n an ae roplane and ca lled them forwa rd side ways and down . Th e axe s
are fixed in the ae rop lane w heth er it is climbing, divi ng , ro llin g o r whatever. Sidewa ys movement is ca lled s ides li p , and th e ra te o f forwa rd mo vem ent is the airsp eed. The aero plane is free to ro tate ab out these axes as wel l. Rota tion about the lo ngitudinal axis is called roll, rot ation abo ut the lateral axis is ca lled pit ch and rotatio n abo ut the ve rtical axis is called yaw . See Figure 8.5. The re m a ind e r o f th is c ha p te r is co nce rned w ith s ta tic stability of pitching motion around the latera l axis which is neverthel ess ca lled "lo ngitu d ina l" stab ility. Ro ll a nd yaw are dealt wit h late r.
Stability ofAeroplanes in Pitch This is a n area full of misconception s and arg ume nt for mod e llers , therefore let me defi ne the subjec t clo se ly. Firstly the name is Lo ngitud inal Stat ic Stability. Th is is ab ou t an aeroplane flyin g alo ng in eq uilibrium (in trim) and sudden ly so me thing , like a gust o r a glitch or the pilot sneezing . ca uses it to p itch u p (or dow n) changing s ligh tly t h e a ngle of a ttack, b u t not th e s p e e d o r a nything e lse . I know th at a lift in crea se wi ll start a climb but I am no t co nce rned with that. I o nly wan t to kn ow "w ill th e c h a nge of fo rc e s te n d to make th e aeroplan e pitch further nose up , o r nose down , o r have no pitching mom ent at all abou t the CG?" First, le t us loo k at the fo rces o n a wing alone . Figure 8.6 s ho ws a wing flying along in equilib rium . All the forces and mom ents on it are exactly ba lan ced . Thrust and dra g are e qua l and are bo th throu gh the CG (and o m itted fo r s im p lic ity) and Mo exac tly bala nces the mom ent of L about the CG (Mo=L.x). Notice tha t I have sho wn the zero lift pitching mom ent with a nose Do\'{rN arrow (Le. the way it reall y acts) to mak e my dia grams cleare r (altho ugh it is normally defin ed as positive nos e up) in spi te of Chapte r 5. Now if the angle of attac k is increased , L increases by a sma ll amo unt, say XL (fo r Xtra Lift). Mo still balances
Figure 8.4
/
// Statically Unstable [diuergent)
/ /
/ /
Dynamically Stable
.... - ....
Time
Statically Stable, but Dynamically Unstable
42
Basic Aeronautics/or Modellers
the moment of the Figure 8.5 original lift L, but the moment of XL about the CG will pitch the wing further nose up which is unstable (Figure 8.7) . Notice the difference between equilibrium and stability. This wing was originally in equilibrium because all the forces were balanced , like the ball on top of the helmet, but its reaction to a disturFortuard bance was unstable. Flying wings do fly (in Chapter 16) but the normal way to achieve stability is to add a tailplane , or what in America is more aptly called a "stabiliser". Figure 8.8 shows an aeroplane flying in trim. All the forces and moments balance out Le . T = D, L = Wand Figure 8.6 Mo balances the moment of L about the CG. L What happens when the aeroplane gets a slight nose up disturbance? Only the wing lift and tail lift change. In Figure 8.9 only the changes, the Xtra Lift Aerocentre forces, are shown because all the other forces cancel each other out. (If you like, all the Mo (=L.x) forces in Figure 8 .7 have been combined in one resultant force, which is Zero). Let us say that for a one degree pitch up the wing's Xtra Lift is 5 and the tail 's Xtra Lift is 1 (because it is smaller). These two are separated by a distance of 36 units . Some simple mechanics show that they have a resultant force of 6 at a distance of 6 units behind the wing's aerocentre. This force determines the stability. If it acts aft of the CG as shown in Figure 8.9 it will pitch the aeroplane nose down which is stable , and the further behind the CG the greater its moment and so the more the stability. Conversely if this resultant were forward of the CG it would be destabilising. In the special case where this resultant acts exactly through the CG then the static stability is neutral because it will tend to pitch the aeroplane neither up ' nor down, because it has no moment about the CG. The name Neutral Point is given to the point of action of the resultant force due to a pitch disturbance because if the CG is at this point then the stability is neutral. Notice what went into finding the Neutral Point: the size of the Xtra lift on the tail relative to the wing and
Basic Aeronalilies for Modellers
YllW
Vertical Axis
DOW1l
x
CG
w the distance between their aerocentres. Notice also that the further the CG is ahead of the NP the greater the stabilising moment.
CGPosition Achieving a good balance between the stability and controllability of an aeroplane is partly achieved by correctly positioning the CG. A compromise has to be reached whereby the aeroplane is safely stable and is also easily controllable. The Correct CG Position is what feels good during flight testing, and will vary according to the purpose of the model, and the ind ividua l preferences of the pilot. You will usually get a CG position from a plan , or perhaps a formula, but that does not make it exactly right. It is just a starting point. If an aeroplane's CG is too far aft it is very sensitive to small elevator movements and small elevator trim changes . It needs little or no down elevator to hold inverted. There is a dangerous tendency to over-control
43
Simplicity Its elf? Figure 8.7 L
If th e tail are a is a fifth of th e win g a rea th en its Xtra Lift should b e a fifth th at o f the wing and like in Figure 8.8 the Neutral Point will be a sixth of the tail arm a ft of th e wing' s aeroc e n tre ( ta il a r m is th e di stan ce b etwe en th e aeroce n tre s , or quarte r c h o rd p oints , of w ing and tail) .
+XL
x
CG
Aerocentre
Mo (=L.x)
Complications the aerop lane and a real risk of sp inning o ut of contro l o r overstressing the wing. If its CG is too far forw ard the model's response to e le vato r co ntro l will be sluggish , it will hold inve rted o nly with large amo unts of down elevator - if at all and it will be d ifficu lt or imp ossible to stall and spin. It will also pitch ex cessive ly with speed changes. The CG is cor re ct when the aircraft handles just the way you like it. But yo u ha ve to start somewhe re , and the nearer yo u can get it in the first pla ce the better. Th e parameter wh ich governs the stab ility is denoted Kn a nd in our simplified theory may be ca lled the CG Ma rgin , St ati c Margin , Stability Fa ctor or Stability Margin, which is th e one I sha ll us e as it se em s the most descript ive . The ph ysical meaning given to the stability margin is the distance betwee n the aeroplane 's CG and its Neutral Point (N P) . Kn is g ive n a s a d e c im al fra ction , or percentage, of the mean chord of the wing. A suitable figure for Kn is fro m 5% to 25% of wing mean chord with an optimum in the region of 10% to 15%. That is all we have to do then. Find the NI' and put the CG 15% of the chord ahead of it!
If onl y it were that e asy . I kept my ex ample nice and simp le to illustrate the principles involved but, as you are probably aware , wh at a lo t I left out! A larg e number of factors co mbine to reduce th at expected Xtra Lift on th e tail , so the Neutral Point is further forwa rd than I calculated . The most o bvious problem is the downwash. \'1!hen the wing 's a ng le of attac k increased by on e degree , the down wash over the tailpl ane a lso increas ed , reducing the angle of atta ck increase experien ced by the tailpl an e to s o me w ha t les s th an on e d e gr ee. \'Ve m ay lo s e a qu arter , a h alf or th re e quarte rs of that one degree depending on the wing's Aspect Ratio , and also on the tail ga p, the w ing lift distributi on , and the tail heigh t. Norma lly the tail ha s a lower asp ect ratio than th e wing. Going ba ck to Figure 6.5 , the tail's lowe r aspect ratio means that its lift increase will be less per degree th an the wing's . No t only th at but using a very thin se ctio n , especially a flat plate , may impl y a still lower lift curve slo pe . Th e w in g and the fu se lage ha ve boundary laye rs wh ich flow off behind the aeroplan e as a wake , movin g slowe r than the rest of th e a irflow . If the tail is flying in
Figure 8.8
.L
(=LII'
=
LrJ
I
D
lV(=L)
Figure 8.9 A XL = 6 I
Distance 36
-1 -- - - - --- I
30
44
Basic Aeronautics/or Modellers
this wake the n its lift increase will be less tha n expected because XL is p ro po rtio nal to airspeed sq uared . The effect is neglig ible for a T-tail but quite significan t for a "draggy " aeroplane.
We Can Work it Out? I would lik e to be as o pt imistic as the o ld Bea tles so ng but to be ho nest anyo ne w ho cla ims to be ab le to calcu la te accura tely th e co rrec t CG for a mode l aeroplane is kidd ing h imse lf Th e re a re just to o many va riab les and no t enough sources of informatio n . So wha t do we do? \'{1e gu ess . But the more accurately we ca n gues s the better. \'{1e make up a fo rmu la in co r po ra ting a ll the "allo wances" and the 15% stabil ity ma rgin to give a CG positio n directly. It can be as simp le as "Put it at a th ird chord mat e!" o r as compl ica ted as lo o k ing up a textbook and findi ng a for mul a for neutr al po int distance aft o f the aerodynamic centre . Then yo u add the .25 chord whic h the aerocentre is b e hi nd th e le ading edge, sub trac t the .15 chord stabil ity margin and end u p wit h something like CG pos = 0.1+ q T aT (I- DWF) V har q a w he re
The tail area should be measured excluding the part covered b y the fuse lage or cu t away to cle ar the rudd er, bu t the wing area should incl ude the a rea wi thi n the fuselage . In deriving this for mula it was assume d tha t the tail is beh ind the w ing and is appreciably sma lle r tha n it. It is possible to loo k up obscure tables to es tima te the information to pu t into the eq ua tion bu t is it worth all that tro ub le for each mo de l? I do ubt it. The answer w ill still o nly be as acc ura te as the "estimates" yo u p ut in.
Simpler Equations What we mod eller s need is an equa tio n which is easy to work ou t ju s t fro m t h e meas ure me nts of the aeroplane so let us make some assumptions and pu t some estimates in the above eq uation . If t he aerop la ne is reas o nab ly clea n , th e t a il un o bstru cted , and its area calculated as before , then we can make an a llowance for the airspeed fac tor. The lift curve slope factor is always about the same if the tail Aspect Ratio lo oks "no rmal" (say a third tha t of a h igh aspe ct rat io wi ng (20) , or half that of a med iu m aspect ra tio wi ng ( LO), or two thirds tha t of a lo w aspect ratio wi ng (5)) . If the tail gap is in th e usual ra nge of 1.5 to 3 chords an d the tail is about level wi th the wing then th e down was h a llowa nce w ill d e p e nd o nly o n th e wi ng 's aspect ra tio. I therefore prop os e this purel y em pirica l equa tion
.9.I is the ratio of the (a irspeed)! at ta il and w ing q
a T is the ratio of lift curve slopes of tail and wi ng , and a ( I -Down wash Frac tio n) is wh at is le ft of the angle of attack increase by the time it reaches the tail and Vim is tail area tail arm IT - - - - - X -------'wing area wing chord and is ca lle d the tail vo lume ratio . (see Nomogram in Appendix E)
My CG Formula This eq ua tion gives the CG posi tion , as a fraction of the wing me an chord , of gliders or conventional, engine at the fron t, tail at the back, mono pl anes with a wi ng of Aspect Ratio fro m abo u t 4 to 25. (See Chapter 22 fo r anyt h ing e lse .) CG posn = 0.1 + 0.25*(AR)i\0.25*V-bar (where (AR)i\0 .25 is the fourth root of the wing Aspect Ratio)
Locb Insb makes a lovely setting for Dr. feremy Sbaui's Stranraer. Tbe drag of all those struts and wires must considerably reduce tbe airfloto velocity over tbe tail,
Basic Aerona lilies for Mode llers
45
Figure 8.10
1 90
.9
80
.8
70 60
.7 50
.6 25
40
20
15
.5
10
30
8 6
.4
25
4 .35
3 20
.3
18
Wing Aspect Ratio
16
.25
15 14
.2
CG Position as a % of wing mean cbordfrom lea diug edge
Tail Volu m e Ratio Vsba r
It is dea d simple to wo rk o ut o n a calculato r with square root s. Enter the wing Aspect Ratio (span/chord), take the sq uare root , and pr ess sq ua re roo t aga in. Multiply by 0.25 an d th e V-bar, a nd a dd 0.1. T ha t g ives the CG positio n as a frac tion of the mean cho rd . Multip ly by 100 fo r a pe rce ntage. For examp le , if th e wi ngs pa n is 99" a nd th e mean cho rd is 11" the Aspect Rat io is 9. The sq ua re roo t of 9 is 3 and the sq ua re roo t o f 3 is 1.73. If the tail volume is 0.6 this gives a CG position of
The CG shou ld be at 36% of the mean cho rd, or 0.36 x 11" = 3.96" aft of the LE of the MEAN cho rd, for the first flight. Eve n easier to use in the nomogram in Figure 8.10. Yo u wi ll find nomo g ram s to help work o ut th e wi ng Aspect Ratio and Tail Volume Ratio in Append ix E. Most powered train ers and sport models have a win g AR of between 5 and 8 so if those are the o nly mod e ls yo u wa nt to co nsider yo u co uld simplify the formula to
0.1 + (0.25 x 0.6 x 1. 73)
CG posn
46
=
0.36
=
0.1 + 0.4*V-bar
Basic Aerona utics/or Modellers
Th e CG Formula bas to be adjusted to allow for tbe ve ' J' long nose
0/1
tbis model.
If yo u then ass u me that yo ur model has a V-bar of about 0.6, whi ch is co mmon, yo u ge t a CG position of one third cho rd . It is a standard , oft-use d, balan ce point, but app lies o nly to models whi ch fit the standard mod el proportions.
Variations on the Formula
Fifties Formula
Flexibility
Ano the r useful a nd po p u lar CG formula , fo u nd in Go rdo n Whitehead's sca le book, was mad e up by an ae ro e ng ineer in the nin eteen fifties (w he n mo de l tails wer e bigger).
The wh ol e book until Cha pter 18 ass u mes that the struc ture o f the ae roplane is rigid . O nly a slight flexib ility has been allowe d for in the formulae . If yo u r ae ro plane is not iceabl y flexible see Chapters 18 and 19 and mov e the CG forward a little. This is es pecially import ant on hig h Aspec t Ratio glide rs.
CG distance aft of mean cho rd LE area x tail arrn j/ S x wing area
=
chord/ri + 3x(Ta il
This can be simplified by dividing all the terms by the mea n cho rd to get the CG po sition as a fraction of the cho rd (as in my form ula). CG posn
=
1/6 + 3/ 8 V-bar
=
0.17 + .375 V-ba r
You can see that the formu la is very like min e, and on the abo ve ex a mp le aircr aft giv e s a CG po sition of 39 .5% mean cho rd, 3.5% furthe r aft than my formu la. Many fu ll size aircraft , lik e th e Hurrican e a nd the Turbulent, have a sma ll tail vo lum e o f 0.4 o r less. Man y scale mod el plan s use a CG position wh ich co nfor ms to the Fifties formula, but experien ce has sho w n that they fly better w ith the mo re forward CG given by my fo rmul a. Now, perh ap s yo u ca n see that all the above formulae , in fact any se rious CG formu la, will try to do the same thing . . . es tima te the NP position and place the CG a sa fe d ista nce ahead o f it. (T ha t d istan ce is ca lle d th e Stability Margin , and is usu ally about 15% of mean chord.) A more co mprehensive treatm ent of Centre o f Gravity is . detailed in Cha pter 22 which covers man y un conventional layo uts. Ther e are probab ly other formulae aro und whi ch you co uld use . To be realistic they sho uld allow for the tail area and tail arm, and possibly eve n the wing aspe ct ratio . Any formu la w hich does not ment io n the tail, or w hic h
Basic Aeronautics/or Mode llers
messes ab out with Centre of Pressur e , sho uld be viewed with sus picion.
Fuselage Influence O n con ve n tio na l aeroplan es th e extra lift on th e fuse lage , ca use d by a pitch disturbance , will co incide w ith the wi ng's aerocen tre. Tha t has been allow ed for by using the gross wi ng area . Howeve r if the fuse lage has an unusual proportion of its area ahead o f the w ing (as in Figu re 8.11) the n the NP will be mov ed forw ard and a forw ard ad justment of the CG is advisable . This applies also to eng ine nace lles on mu lti-en gined aeroplanes. See Cha pter 22 for wo rking out adjustments .
Personal Preferences So me p e ople have a preference for a part icu larly sensitive mod el, o r a part icularl y stable one . Fine! Adjust the formula to suit yourself.
Willg Section Influence Th ere isn't any! Th e sec tion thickness might have the tiniest effect, but not the ca mber. Cha nge the sec tion from symme trical to Clark Y and the formu lae all give the same answer be cause th e NP has no t mov ed . You ha ve to change the rigging ang le b ut not the CG position . Let us bury tha t myth forev er.
47
Figure 8.11
c__
~_---:_
C ,
Destablistng Nose A,"ea
, -,
\ ,
I
Q lI lI r ter ll1ell"
Cbortt Net Tail Area C
Mean
CIJortt
Gross lVillg Area
Tail Setting Angle Th e ang le w hich the tail is se t (leading edge do wn) from the wing's da tu m lin e is o fte n referre d to by mod ell er s as "lo ngitud ina l d ihe d ral", a n un fortu nat el y ina ppropr iate nam e as it lead s modellers to conclude , wro ngly, that it produces stability. That is like saying that ca rts are for pu shing horses. As you have seen, the Tail Setting Angle is not involved in determining the stability at all. Howeve r there is a link. In a later chapter you will see that the Stability Margin partly determines the Tail Setting Angle need ed for trim.
48
Basic Ae ronautics f or Modellers
Chapter 9
Directional and Lateral Stability ---
Directional Stability Th e cave me n knew the importance of directi onal s ta b ili ty ( o r yaw stability, or weathercock stab ility) . A plain stra ight piece of w ood in flight is un st abl e sinc e the centre o f press u re is ah ead of th e midpoint wh ere the CG lies as in Figure 9.l. I n ord er to m a ke their arrows fly stra ight th e y added w e ights a t the front and feath ers at th e ba ck . Thi s mo ved the CG forw ard and the ce ntre of pr essur e back and so the arr ow lined itsel f up w ith the airflow as in Figu re 9.2. O ur ae roplanes , with th eir CG w ell forward and a ve rtic a l s u rface ca lled a fin at the rear, use the sa me technique to e ns u re dire ct ion al stability. See Figure 9.3, which sho ws a n ae roplan e wh ich wh ile flying along happ ily in trim has been disturbed b y a s ma ll ya w o ffse t a nticloc kw ise. It is no lo nge r lined up with the a irflo w so th e fin w ill ha ve an an gle of attac k c a us ing a s idew ay s "lift", F. Thi s fo rce will tend to re al ign th e aero p la ne with th e a irflow. The co rrec ting mom ent dep ends up on th e fin a rea a n d its distan ce behind the CG. A co rrectly trimmed
-:. ' ~.p.
$r:~
1J1I'If~F
"
W
.
?~~
.-
~?).,.
.,.
-q,£f
~
c-
..... . .....
~
. _.. ,
~.... . .
-=~:J '.7
' iI. ~
Figure 9.1
Airflow
P ress ure
Increase
F
Figure 9.2
~
.. Airflow
Basic Aerona utics/o r Modellers
~=-+-nm-+ Resultant Force
49
Figure 93
aeroplane' will always fly straight relative to the air. Even if an aeroplane is flying North in a Westerly wind the air will flow straight from nose to tail. Were it not so, DIRECTIONAL STABILITY would soon line up the aeroplane with the airflow, eliminating any sideways component.
Lateral Stability An aeroplane which is statically stable in roll will, when disturbed slightly in roll, initially tend to roll back to a wings level condition , So we need to design in some mechanism to provide a rolling moment when the aeroplane is upset in roll,
Lift forces are caused by the air pressure on the wings' surfaces, Air pressure can only provide a force perpendicular to a surface. Therefore there will be' two lift forces, one perpendicular to each wing. Or they may be combined as a single resultant force up the centreline of the aircraft, because I am assuming it is completely symmetrical. Also, because of symmetry, the CG lies on the centreline. Figure 9.4 represents the rear view of an aeroplane which while flying happily along in trim has been upset and now has a slight bank to the right , angle B. I can see no force producing a moment which will tend to correct this bank. I happened to draw a high winged aeroplane with the wing above the CG. It may seem obvious that this in itself will provide stability, but is there a moment about the centre of gravity? The weight cannot have a moment about the CG (by definition) and since the lift acts on the centreline of the aeroplane, it cannot provide a moment either. Even if you split the lift in two to have half perpendicular to each wing, by symmetry their moments will cancel each other out. Even if you resolve each half into its vertical and horizontal components, the net moment is zero so there can be no restoring rolling effect. Let us see what will happen though. First I shall exercise my right to resolve (split up) any force into two components. In Figure 9.5 the weight has been split into two forces: one (W cos B) opposite and nearly equal to the lift, because B is a small angle, and the other (W sin B) towards the low, right, wing. Now you don't need me to tell you that this component will cause a sideslip .. . a sideways velocity to the right. When the aeroplane is sideslipping, the air is approaching at velocity V slightly from one side, at a small angle Y say to the centreline of the aeroplane. As shown in Figure 9.6 the velocity V may be split into two components, V.cosY along the centreline of the aeroplane and V.sinY called the sideslip velocity, at right angles to the centreline. It is convenient, and quite legitimate, to examine the effects of each part of the velocity in isolation. It is as if the aeroplane has two velocities simultaneously which cause separate effects which can be looked at separately. Now let us look at the design features which affect lateral stability.
1. Fin Sideforce Figure 9.4 L
w
50
Back to our aeroplane which ' was sideslipping to its right. I shall redraw it in Figure 9.7 as a mid winged aeroplane to avoid ambiguity, and I am sure you can see that a sideforce will arise due to the sideways velocity, or sideslip. The sideforce on the fuselage is unlikely to have much moment about the CG but the sideforce on the fin will. As the fin is normally mainly above the fuselage it will
Basic Aeronautics for Modellers
usu all y g ive a n a n ti clockwise , stable rolling mom ent . Yes, this sideforce on th e fin, F, is th e sa me s idefo rce w hic h p rovi ded th e di rection al sta b ility earlier (Figure 9.3).
Figure 9.5
L
2A. Wing Position (relative to CG) Fig u re 9.8 s h ows a "p a raso l wi ng" type aeroplan e, with its wing mounted away a b ove th e fu sel age , and th e four fo rces of Thrust , Weight , Drag and Lift. Most of the weight is in the fuselage . So be cause the wing co ntrib utes a large p rop ortion of the drag, the total drag will act somewhat above the CG posit ion. (The tot al lift w ill ac t s lightly be hi n d the CG to counteract the p itch ing effect o f th e Thru st / Drag co uple.) If this aero p la ne is g ive n a slig h t bank to th e right , none of th e for ce s ca n p rodu ce a ro lli ng moment as A low toing and sligbt dihedral gives little or 110 roU effect witb rudder. before , but as b efore the y will cause a sid eslip tow ards the low wing. Now Airfloto look at Figure 9.9. The drag acts, by defin ition , in the Figure 9.6 dire ction of the resultant airflow and may be split up into two components, D.sinY and D.cosY (where Y is the sideslip angl e) , as shown. The co mpone nt D.cosY can ex e rt no rolling mom ent. Transferring th e compon ent D.sinY onto a view from the tail , Figure 9.10 , shows that thi s component will hav e a rolling V Velocity Vcm'Y mom ent abo ut the CG w hich will depend on the ver tical Vector Forward distance between the CG and the line of action of the Component drag for ce . For this high winged aero plane the rollin g moment resulting from the Sideslip is stabilising, Le. it w ill pick up the low wing. Th is kind of sta bility is so meti mes referre d to as "pe ndulum" stability for a reason which I fail to see VSINY since aerodyna mic drag has little effect on pendulums. Sideways Component Not e tha t the w o rking o f th is source o f s ta b ility dep ends upon the sideslip , as without the sideslip the drag wo uld not have a sideways co mpo ne nt. /
IIIIIIII
2B. Wing Position (relative to fuselage) Now co nsider the flow of air rou nd a cylinde r. As the air ap proaches the cylinder it w ill eithe r be forced to flow up o ver an d down , or d own under an d up to resume its quiet steady flow (see Figure 9.11).
Basic Aeronautics for Modellers
51
Figure 9.7 Resultant MOlne"t L
Sideslip Velocity
w The extra upward flow (upwash) on the right gives an increase in angle of attack, and hence lift, and the downward flow on the left gives a decrease in lift. The net result is a stabilising (anticlockwise) rolling moment which tends to lift the right wing. On a low winged aeroplane you find that a Sideslip to the right causes a clockwise rolling moment which is destabilising. That is why a high winged aeroplane with no dihedral will turn on rudder but a low winger with no dihedral or sweep is more likely to bank the opposite way. More of that later.
Figure 9.8 L '1==r=~"D
T
w Now in Figure 9.12 the cylinder is the fuselage of an aeroplane with a w ing stuck on top and it is sideslipping to its rig ht. As the air approaches the fuselage so me of it is deflected up the right side, over the top, and down the left side . Figure 9.9
/ /
Airflow
A.gkY/
Sideslip VEL (VSIN Y)
1
3. Dihedral Now we come to the famous Hedral sisters, Di, Ann and Polly and their role in lateral stability. Figure 9.13 shows an aeroplane with a slight bank to the right which has started sideslipping to the right. As you can see , the air coming from the right will have an upward component on the right wing, tending to "get under" the right wing and lift it (at the same time pushing down the left wing) which gives a stable, anticlockwise, rolling moment. (A rather unscientific explanation but I hope you get my meaning.) In m uch the same way, Di 's sister Anhedral (a downward tilt of the wing tips), would give a clockwise (destabilising) rolling moment tending to push the low wing further down. If you want a more scientific explanation. complete with trigonometry, see appendix
D. The effect of dihedral was first explained to me by the "Pro jected Area Theory" many years ago (Anyone remember the ]etex powered Keil Kraft Cub , my first model aeroplane?). When the dihedralled aeroplane was banked, right say, the area of the right wing projected onto a horizontal surface was greater than that of the left wing. The theory was that the lift would therefore be greater on the right which would pick up the low wing. Well I eventually realised that the theory is wrong! Because the lift on each wing is still equal and at right
52
Basic Aeronautics/or Modellers
a n gle s to t h e w ing 's s u rface a nd so ha s a n eq ual mom ent abo ut the
F ig u re 9.10
cc. The refo re th e re ca n be n o m om en t unt il it s ta rts to s id es lip ( w h ic h t h e p roje ct ed a re a th eo ry o mi tted to ment io n ).
4. Sweepback Sw e e pi ng b a c k th e wi ng s o f a n ae ro p lan e co ntrib utes to its late ra l s ta b ility , Th a t is . if a sw eptba ck w ing ex pe rie nces a ro ll u pset to the rig ht (clockw ise) . it w ill s ta rt to Side s lip to its right. As a result o f the s id es lip it wil l e x pe rte nce a ro lling mo me nt to the le ft (a nticlo ckw ise) w h ich w ill tend to cor rect the initial ro ll distu rb ance . O ne ex p lanatio n o f why it works is show n in f igure 9 . 14. in w h ich an ae ro p lane wi th a swe p thack w ing is s ides li p p ing to its rig ht. T h e re sult in g ai rflo w is as sho w n. As far as the a p proac h ing a ir is co nce rned the tw o w ings h a ve th e sa me a rea but th e r igh t w in g appears to be o f a grea te r as pect ratio th an th e left wi ng . Th e r igh t wi n g w ill th e refo re have more lift Figu r e 9. 12 than th e left ""ing w h ich g ives sta b ilising ro llin g mom ent to lift th e rig ht win g . 11 is a s imp le matt er 10 p ro\'e b y T r ig o n o me try that the sta b ility ef fec t o f sw ce p bac k is re a lly du e to th e righ t w ing h a vi n g an in crease d a ng le of a tta ck caus ing the incre ase in lift. w ith a co rres po n d in g d e cre a s e o n th e le ft w ing , I show in Ap pe nd ix D th at t h e c h a n ge in
S id es lip Vel
w a ng le o f a tta c k du e to s ides lip. d e rive d from swee p hack . is p ro porti on a l to the s ideslip a ng le . the orig inal a n g le o f a t tac k . a n d th e tan g e nt o f t h e a n g le o f swee p hac k (or yo u co u ld jus t b e lie ve th e p re vio us pa rag rap h) . I have a lso seen ca mbe r bro ug ht into the a rgu me nt as fo llows . Th e ca mbe r o n bo th wi ngs is the sa me. bu t as a p ercentage o f chord in the d irection o f the a irflow th e
Figure 9.11
: -----0-------~:
_______
-
-
-
-__"""1111---
-------~------"""IIIf---
Basic Aerona u ticsfor Modellers
53
s ides l ip. Th e pri mar y effec t o f th e roll d is tu rb anc e is s ides lip to w ar ds the lo w w ing . Fo llow ing th a t ma y co me th e ro llin g mo men t du e to sideslip de rived fro m the above factors .
Figure 9.13 XL
M~ XL
D
t-
Aspects of Design
V SIN D
......
Th ese desig n charac ter ist ics ma y be com bi n e d to o b tai n th e require d de gr e e o f late ra l st a b ility . Yo u ma y see a ero p la nes wi th a high stra ig ht wi ng a nd a littl e d ihedral (Ces sna 150), o r a low straig ht win g and plenty of d ihed ral (Pipe r Cherokee), o r a swept high w ing w ith anhe dra l (B.A. 146), or a swept lo w wi ng with a little d ihed ral (Boe ings) . The wing position is no rmally chosen for mech ani cal re a so ns , th e s wee p b a c k to s u it th e c ru is ing Ma c h nu mbe r, and fina lly the di hed ral is chosen to ach ieve the req uired late ral stability. The same a p p lies to mod e ls except that sweep is used for appe a rance mainly. O n pattern models we try to ac hieve ne utral late ral s tab il ity b y u sing jus t e no ug h d ih edra l to ca nce l out th e Ai1j10W d e sta bili s in g effec t o f t h e lo w w ing . I h a ve sawn th rou gh the glass fibre ba ndage o n the top s urface o f a wi ng and rej o in e d wit h s lig h tly m o re di h ed ral. T h u s mod ified the model did n ot roll a t a ll w h e n yawed with rudde r. Yo u w o u ld ge t a s im ila r e ffe c t u si n g a h igh w ing and a littl e anhe d ral. If yo u lo o k a t t he form u la e d e ri v e d in appendix D, it is ev ide nt that, becau se the swee pb a c k e ffec t d ep e n d s up on an gle of attac k but the dih edra l effect d oes not , yo u can no t d irectly eq ua te o ne d e gree o f d ihedra l w ith so ma ny degrees of swee p . T he relat ive effects vary w ith the speed of the aircraft. T he effec t of t h e d ihedra l becomes mo re p rom in e n t as s pee d rises. Also, not e th at th e sweepba ck e ffect gives a V c:mD
ca mbe r is greater o n the right win g than o n the left and so w ill give a g rea te r lift. Thus providing a stab ilising rollin g mom e nt. So unds good , bu t that doesn 't explain why it still wor ks o n win gs w ith a symme trical se ction!
Summary To re ca p the n, the lateral stability is influe nc e d , in a p proxi ma te o rder of im porta nce, b y the foll o wi ng : d ihedral, sweepback , wi ng positio n and fin area above the CG. ALL of the m are co mplete ly dependant o n the
Figure 9.14
I
54
Basic Aeronautics/o r Modellers
sta ble ro lling mom ent w he n invert ed (w hile di hedral does not).
Figure 9.15 Rotation
Directional and Lateral Interaction Spirall)ivergence While the aeroplan e in Figure 9.15 is ac tually yaw ing , i.e. rotating clockwi se ab ou t its vertica l axis, the win g o n th e left is advan cin g into th e airflo w a nd is th er efo re faster than the one on the right so a ro ll to the right may de velop (the same wa y as the o rigina l yaw) , So wh at? You design a high w ing train er with a g rea t big fin for d irectional stability. Th e stude nt lets the right win g drop a n d it s ta rts to yaw to th e right du e to th e d irectional stability from the fin. Th e rate o f yaw to the right mean s that the left w ing is travelling faster than the right wing as in Figure 9.1 5 so th e aero p la ne banks furth er to the right wh ich increases the sideslip, so the fin produ ces mo re rate of yaw which ca uses more right bank. By th is tim e th e bank ha s d e veloped suc h th at right yaw is d ropping the nose as well. You kn ow wha t this is becoming .. . a sp iral d ive! When it is ban ked right over a nd pointing down , all up elevator will do is tighten the spiral. Sp iral di vergenc e is an ins ta b ili ty ca use d by the imbal an c e b etw e en we ak lat eral s ta b ility a n d a n excessive am ount of dir ection al stability. Th e tenden cy ca n be correc ted by reducing the fin size or increasing dihedral.
Forward Airspeed Slower Wi"g
Faster Wi"g
(
o
J
I ,..------J,\\ 1/1------....
1
IJutch Roll An imbalan c e th e o p pos ite w a y b et w e en w e ak directional and strong lateral stability, too mu ch dih edral, manifests itself as "Dutch Roll " whi ch is a rollin g/ yaw ing osc illation which is ve ry difficult to sto p. On radio co ntrolled mod els it is often prefe rable to put up with a slight tenden cy to wards sp iral d ivergen ce by go ing for the bigger fin and less d ihed ral. For free flight an y d ivergen ce is una cceptabl e so fins are smaller, d ihedral grea te r, and a slight Dut ch Rollin g tenden cy is acce pted if it is well damped o ut.
Basic Aerona utics f or Modellers
55
Chapter 10
Control 10.1. No te the di ffe re nce betwe e n yaw an d sid es lip . Yawi ng the nose to the right crea tes a sideslip to the le ft.
Rudder The rud de r ro ta tes the aero p lane a bo ut its verti cal ax is , or yaw ax is, and the farthe r the rudde r is beh ind th e CG th e mo re le vera ge it wi ll ha ve . Wh en rig ht rudder is ap plied , a force to the left is ge nerated over th e fin a n d rudd er w hi ch will y a w th e a e ro p la ne clo ck wise , i.e . nose to the right. Or to look at it from a pilot 's point of view, wh en he has righ t rudde r ap plied the a irflow is co ming from ahe ad and slightly fro m his left so the aero plane is sides lipp ing to its left. See Figure
Fi.~ I/ I'I'
10. I
Elevators The purpose and effect of moving the ele vators is to rotate the aero plane about its lateral or pitch axis. Moving the e levato rs up creates a downforce o n the tail wh ich te nds to rotat e the ae rop lane nose up. See Figure 10.2. Becau se of its ine rtia, its ce ntre of gravity initially tends to keep go ing in the sa me direction a nd so the net result is an increase in the angle of a ttac k . \\;rhat th e e lev a tors ha ve co n tro l ove r then is the angle of atta ck. Up e le vator will increase the wing's ang le of a ttac k an d the re fo re in crea s e th e wing lift , unless the angle of attack is in cre a s ed from ju st b el ow to a bo ve th e s ta lling an gl e . In th at }'a 1/ ' case u p e lev ato r is just a no the r down co n tro l. An All Mo v ing Ta il (AMT) wo rks in exactly the sa me way.
+ + + + + +
Ailerons
Ytt u :
56
T h e ae ro p la ne is ro lled a bo u t its lon g itudinal axis by mo vin g o ne a ile ro n d own and the o the r o ne up as in Fig ur e 10 .3. Moveme nt o f the a ileron c ha nges th e ai rflo w a ro und the w ho le sectio n , not just around the aileron itself. Unless the win g starts off near its stalling ang le of a tta c k, th e do wn-goi ng ail ero n p ro d u c es a n incre a s e in lift , a n inc rease in drag and an inc rease in t h e no s e do w n p itch ing mo men t coefficie n t CMo (o r a n aft mo vem ent of the CP if you prefe r). The e ffects a re re ve rs ed o n th e upgoin g a ileron.
Basic Aeronautics/o r Modellers
Aileron Drag
Figure 10.2 Th e unwanted d rag cha nge can be tro ubl eso me . Say yo u appl y A l rflo u i rig h t a ileron , th e drag increases on the left and Pilc!J decreases o n th e right giving a yaw to the left, not the way yo u wa nt to ~ ~ c.~~~~ go . Th is is ca lle d ad ve rse ailero n dr ag (a nd ~ is d iffe ren t from t h e "a ilero n reve rs a l" in PilclJ Cha pter 18). The co mbi na tio n of the late ral sta bility a nd the left yaw/ rig ht sideFigure 10.] s lip w ill g ive a ro lli ng moment to the left (see XL Roll Ch a p te r 9) . O n aeropl an e s w ith s tro n g lateral stability and large ailero n mo vem e nts th is latte r mome nt can overco me t h e ro lli ng moment to the right from t h e a ile ro ns a n d th e Roll X L aero p la ne co u ld eve n roll the wro ng way . O ne so lu tion is to a rra nge fo r d iffe rential move me n t o n th e Transmitter ai lero ns , t he up o ne co mes up more than the do wn o ne goes do wn , to eq ualise the d rag. Ano the r so lutio n is to fit "Frise ailero ns" in wh ich the Figure 10.4 upgoing ailero n has its lead ing edge stick ing down into the airstream (Figure 10.4), to eq ua lise the d rag. Anoth er is to o pe n the airbrake o n the side w ith the up goin g ailero n. The loss of lift and increase in d rag are just wha t is need ed to ass ist the ailero ns in a turn . The othe r o ptio n is to apply rudd er toget her w ith the a ilero n, and in th e same d ire c tio n , to o p p ose th e ad verse yaw from the aileron d rag. To mak e this eas ier man y mode rn ra d io s ha ve a co up ling sw itc h o n th e tran sm itt e r (c alle d a "C.A.R. fu n c tio n " for Coup led Aile ro n & Rud de r) wh ich a llows th e a ileron stick to Prise Aileron move the aileron s and rudder simultaneously.
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c
Aileron Alternatiue There a re alt e rn ati ves to ailerons fo r ro ll co n tro l. Either the wh ole w ing ca n be rotated about its qu arte r chord line , or th e w ing ca n be tw isted at the tips to produ ce a rolling mom en t (w ing warping) . Th e theo ry is good bu t the re ar e mech anical co mp lexi ties . Ano ther al te rn a tive it to u s e tail e ron s (s e e un de r Co ntro l Combina tions).
Figure 1005
Aerodynamic Balance
Control Surface Balances The re a re tw o kind s o f bal an ce fitt ed to co n tro l s u rfaces , a n d th e y p erform different fun cti on s . Aerody na m ic ba la nces a re a p a rt o f th e ar e a of the co ntro l surface ahead of the hinge line. The air p ressure
Basic Aerona uticsfor Modellers
57
a contro l surface already has an ae rodynami c bal an ce , it is a convenient place to put the mass balance also.
Figure 10.6
Control Eff ecu venes s Mass Balance
Figu re 10. 7 Damping Mo ment
- - --
R o tatio n
I~ -
, X --
~ CG
on this part counteracts to some ext ent the pressure on the rest of the surface and so reduces the effort needed from the pilot (or the servo) to mo ve the surface. An example of an ae ro d yna m ic ba lance on a rudder is shown in Figure 10.5. This technique can be applied to all three types of control surface . A mass bal ance on the other hand is simp ly a weight rigidly attac hed to th e co ntro l sur face in front of the hinge line . See Figure 10 .6 . Its purpose is to avoid control surface flutter (of which more in Chapter 18). If F igure 10.8
Rotation
CG
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R otation
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58
Wander along to a gymnasium and pi ck up a weight training bar, complete wi th weights. Now turn it sm art ly through 180 degrees and stop it. It takes quite a bit of effort to start it turning and to stop it. Now try the same thing with a shot putter's shot. Th at is much easier because it has less rotational inertia since the weight is concentrated near the centre . An aeroplane with long he avy wings and a long hea vy fusela ge is more d ifficult to rotate, and stop rotating, which mak es it le s s manoeuvrable, but it flies more smoothly. Th at is one reason why the Sopwith Camel, for examp le , whose main mass es of engine pilot and guns we re all placed together, was so agile .
Next there is stability which affects on ly pit ch and yaw (sin ce a roll offset in itself produces no opposing moment) . For example when the elevators change the angle of attack of the wing, th e longitudina l s ta tic stability of the aeroplane sees it as just another pit ch disturban ce and produces a moment tending to change it back. Th e control is fight ing against the stability. The greater the sta b ility , the les s th e effectiveness of the control. In a sens e Stability is the opposite of Control. Too much stability can leave yo u with too little control.
I
[
Rota tional Inertia
Stability
I I
Sideways Ve locity -----r---
The control forces , like all aerodynamic forces , increase with the sq ua re of the airspe ed . At low airspeed the co ntrols are less effective . That applies particularly to the ailerons o n a propeller d riven a e ro p lane w hose rudder and ele vato rs are usually in the slipstream. Co n ve rs e ly , at high speed th e controls can b e dange rously effe ctive. Ailerons will produce a very rapid roll as will a rudder/dihed ra l combination . So the higher the spee d , the more "g" will res ult from a given elevator deflection . So beware of a pplying full up elevator in a high sp eed div e . That is a D o umuiard rea l wing folder. Ve locity The rate at which the Rotatio n aeroplane rotates in response to a contro l input obviously depends upon the relative size of the co n tro l surfaces . It depends also upon three design features of the aeroplane itself, inertia , stability and damping. The control input ha s to overcome all three.
----- -
~
Damping
~ " M ome1lt
A e ro dynam ic Damping The th ird de sign fea tur e which affe cts the control
Basi e Aeronautics for Mode llers
Compact a nd ag ile ( 110, 1I0t Dounie, his Pills Special) due to tbe mini mal inertia a nd damping of tbe sbort uiings a ndfuselage. effec tiveness is aerody na mic damping. A damping force lift, XL-r which ha s a n ant icloc kw ise mom ent about the is a force wh ich is the res ult of a rate o f movem ent and CG, the dam pin g mom ent. Since the upwash velocity is whi ch o p p oses th at rat e o f mo vem ent . Drag is a n prop orti on al to x, so is XLr and so the damping mom ent exa mple. However here I want to look at forces whi ch is p roportio nal to x 2 a nd tail a rea , Th us while sta tic arise from rot ati on o f th e aerop lan e a bout th e thre e stability depends o n the tail area times moment arm, the axes. The greater the damping mom ent , the slowe r the damping (an d he nce dy namic stability) depends o n the of th e re sp on s e aerop lan e to a control input , again making it fly more s moo thly but less man oeu vrabl e . AEIZODY,.w..«c..OAMPIIIC;, Damping is al so an \.IOorul1E1 4 -3 important ingred ient in DYNAt'vIIC STABILITY.
P itch Da mping Figure 10 .7 repre se n ts a n aero p la ne w hic h has a clo ckwise (nose up) pitchi ng rate abo ut its CG. As a resu lt th e tai lp la ne h as a downward velocity p rop orti onal to x, its d istan ce back from the CG. Th at has the sa me effect as a n u p w as h o n th e ta ilplan e w ould ha ve . .. an increa se in tai l
Basic Aerona uticsfor Modellers
59
clo ckwi se , Le . rig ht win g down . Without labouring the point , yo u ca n see that , Rolling as for p it ch damping , a lon g tail a r m impro ve s ya w d ampin g XL a nd hi gh as pec t ra tio wings wil l hav e b etter ro ll damping. Thu s th e bi gg er th e ae roplane the g rea ter its XL ro ll d ampin g , a nd the grea te r its Asp ect Ratio th e g rea ter th e ro ll d amping . Th at is wh y large mod els , and large glid er s in particu lar , fly .... --... so much more smoothly . .,.,. As yo u ca n probab ly Damping Mome1lt gues s from Figure 10.9, the tailplane and fin add to the roll damping, but not much . tail area times distance sq ua red . So a small tail o n a lo ng fuselage w ill have better da mping than a large tail o n a sho rt fuselage even th ou gh the y are de sign ed to have Other Flying Controls the sa me tail volume ratio and static stability. There is s o me p itch d amping from th e wing itself but it is The Th rottl e is a flying co ntrol: it is the UP co ntro l. Ask an y glid er pilot , wh o has to do with out o ne . Glide rs relatively unimportant unl ess it is swe pt. can o nly co me down throu gh the air. AIRBRAKES w ill redu ce th e speed o f an aeropla ne Yaw and Roll Damping but are no t reall y a speed co ntro l, they ar e a "co me Figure 10.8 shows an ae roplan e wit h a rate of yaw do wn q uic ke r" co ntro l. They inc rease the profile d rag clockwise about its CG (nose to th e right) an d Figure wh ich stee pens the an gl e , a nd increases the rat e o f 10.9 shows a view from the rear of an ae roplane rollin g descent. They are mu ch mor e effec tive at high speed . The y also, incid entally, in crease th e s ta lli ng speed. They may ha ve a Fig ure 10.10 s ma ll pit ching e ffec t , Plain d ep ending o n th e d e s ig n . Airb ra kes c o mmonl y cause turbulence in th ei r wake and a re norma lly p lac ed outboard of the tailplane o n gliders. FLAPS in cr ea se the dra g to ste ep en t h e d escent , in addi tion to in cr ea sing th e liftin g ca pa b ility o f th e w ing , Clmax w hich red uces the s ta lli ng s p ee d , a n d allo ws the aeroplane to fly mor e slowly. Th re e ty pes o f fla p a re s h o w n in Fig u re 10 .1 0 , namel y a p la in flap , a sp lit flap , and a slo tted flap . T he effect o f a ll of these o n the lift curve of - ......the ae rofo il is sho w n o n Figure 10.11. The whole c u rv e is moved to the left du e to the increa se d c a m be r , a n d Cl ll",x is Figure 10.9
t
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60
Basic Aerona utics/or Modellers
increased. The ang le of Figure 10.11 att a c k a is m ea sur e d from the o rigina l cho rd line. Lower ing flap s mo re than a few deg rees ruins the Lift/ Drag ratio which reduces the climb angle, but a little flap may be u s ed fo r tak e- o ff to redu ce the tak e-off run . On land in g , a d efl e ct ion o f up to 40 d e gr e e s ma y b e u s ed (eve n m ore for s p li t flaps) . O ne p robl em I found usin g la rge flaps o n a model is that at the low sp e ed achi e ved th e a ilero ns b e cam e ineffe ctive with out co upled rud der. Fla ps u suall y a ffec t th e pit ch trim o f th e aeroplane. Be cau se o f th e ext ra ca mbe r, the y ca use an inc re ase in the nos e down mo ment o n the w ing , but they also ca use ex tra do wnwash o v er th e tail g iv in g a nose up moment cha nge from the extra download o n the tailpl an e . Th e overall e ffec t dep ends o n w hich of the above is th e mo re powe rful , which depends on th e layou t. SLATS o n the leading edge which yo u can select in and o ut are also a flyin g contro l. They are selected out, as in Figure 10.12, to let the aeroplane fly more slow ly. When th e s la t is out , ai r ru shing throu gh th e s lo t pr events th e flow from se parating at th e le ad ing ed ge wh ich prevents the wing from stalling until it reaches a high e r ang le of attack than usual , ena bling it to ac hieve a large r ma ximum lift coefficie n t th an be fore (Fig ur e 10.13). Like flaps they are bad for the Lift/Drag ratio and are best retracted for high speed flight although so me ae ro planes have fixed slats to save co mplexi ty and just put up w ith the inefficien cy. Th ere is also the velY clever "auto matic slat " w hic h is arrange d to o pen a t high angles of atta ck when th e stagna tion point (re member him?) mo ves down towards the e ntra nce of the slot. Their disadvantage is that the y can ope n when the pilot pull s high "g".
Control Combinations Wh en the sa me bit of co ntro l surface is used for two
wu» Flap Standard Aerofoil
0< diffe rent purposes it is oft en give n a sp e cial nam e . It usu all y al so involves u sing ex tra se rvo s a nd e ither me chan ical or ele ctronic mixers. TAILERONS is the nam e g ive n to e leva to rs, or th e hal v e s of a n a ll moving tail , wh ich ca n m o v e indepe nde ntly. The tw o sides ca n be moved tog eth er as n ormal for p itch co n tro l, or th e y can be mo ve d in opposite d irections to give a ro lling mom ent on the tail. It w ill be a bit s h ort on lev era ge b u t it w ill b e a de q ua te for lo w as pect ra ti o a eropl an es (e .g . jet fighters). FLAPERONS a re stri p a ilerons which ca n b oth b e moved down together to ac t as plain flap s, but which ca n at the sa me time still be mo ved inde pe ndently for roll co ntrol. Adve rse yaw can become a probl em . ELEVONS a re just like flap erons but they are used on tailless aeropl anes to co ntro l roll and pit ch . Th e e levons move tog ether for p itch co n tro l an d in opposition for roll co ntro l. Th e e le vo ns both move up for nose up pit ch an d down for nose down. A V-TAIL is jus t like a tailplane with extr em e d ihe d ral. It ac ts as a stabilise r in both pit ch and yaw, and the tw o co ntrol surfaces ca n be moved inde pende ntly. As show n in Figure 10.14, w he n mov ed togeth er they
Figure 10.12
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Basic Ae ronautics f or Modellers
- - - -
61
Figure 10.1 4
Figure 10.13
c,,"X
t
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V-Tail
Witb
Sla t
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CJ.MA X
Standard
Left Rudder
ac t like eleva tors , and when moved in opposition they have the e ffect of a rud der. The un wanted parts of the vectors ca ncel out.
up Elevator
Like most Delta toinged models, tbis one uses eleuous for pttcb and roll, tbe canard is fixed.
62
Basic Aero na uticsfor Mode llers
Chapter
,,
Turning Flight T urning flight is one of those areas wh ere a great d e al o f mi sinformati on and ha lf truth s h a ve becom e accepted into the fo lklore of the hobby. I have even seen an article in whi ch it was claimed that in turning flight the "curve d a irflow " reduced the risk of stalling. Wha t utter no nsense! In this chapter I want to put the record straig ht, starting with the basic mechanics of turning. I shall assume still air condition s.
The Mechanics ofTurning In General By turning I mean chang ing the direction of motion of an object. For example, the ball in Figure 11.1 is going at co nstant speed round in a hori zont al circle radiu s R. On e minut e its velocity is V ft/ sec to the north and a short time late r its velocity is V ft/sec to the eas t. Vectors VI and V2 are not ident ical. In the time the ball has taken to mo ve from A to B its velocity has cha nge d becau se its dire ction has change d. The shorter the time interval, the more rapid the change. The rate of cha nge of velocity is defined as acce lera tio n . A turning body is not in e q uili briu m, therefore it is acce lerating. The rate of turn N is the rate at whi ch the radius OA is turn ing clockw ise . It may be given in re vo luti ons pe r minut e , o r d e grees pe r seco n d o r, to s im p li fy th e equations , in radians per sec ond (a rad ian is an angle of abo ut 57.3 degrees). Then the simple equation V = N.R. relates the velocity to the rate of turn and radiu s of turn . The velocity V is always tangential to the circle, Le. at right angles to the radiu s from O . The accelera tion of the ba ll in Figure 11.1 is given by • a=V2/R o r in ter ms of rate of turn , a=N2 R In simple language the high er the speed, or the tighter th e turn , th e more accele ratio n . T he direction of th e acceleration is towards the ce ntre of the circle at 0 , at right angles to the dire ction of motion. From Newto n's go od o ld laws of motion an o bject will not turn unl ess a force is applied in the d irection of the required acce leration, Le. towards the ce ntre of the turn . If you want to turn right , then yo u mu st app ly a force to the right. This force is called the centripe tal fo rce becaus e it is towa rds the ce ntre (from the Latin for "centre seeking"). The mo re co mmonly kno wn ce ntrifuga l force (from the Latin fo r "flee ing the ce nt re") is its eq ua l and o ppos ite ' reaction . (Newto n's 3rd law). When you whirl a weight roun d o n a piece of string yo u su pply the ce ntripetal force to the weight via the string to keep it moving in a circle. The weight applies a ce ntrifug al for ce to yo ur hand , pullin g it towards th e outside of the circle .
Basic Aeronautics fo r Modellers
Back to Aeroplanes The force required to turn th e aero p la ne must be produced from the air. The fin and rudder do not provide it. Left rudder give s an aero dy na mic for ce to the right whi c h is the oppos ite o f w h a t we w ant. T he yaw produced by the left rudder will pro d uce a side force on the fuselage to the left. This force co uld, given time , produce a left turn but at the cos t of a co nsiderable drag increa se. The only surface which can produce a large aerodyna mic force, and do it efficie ntly, is the w ing . It is th e w ing w hich turns th e aeroplane. Th e lift fo rce alwa ys acts perpendicular to the wing and so a horizonta l co mpo ne nt ca n be pro vided b y b ank ing th e ae ro p la ne towards the d ire cti on of th e desired turn . Figure 11.2 shows an ae roplane banked to the right at ang le B. The horizontal co mpo ne nt of the lift will be L.sinB caus ing th e turn (ce nt ripe ta l force) . The ve rtica l co mpone nt, L.cosB, will now be less th an th e weight if the aeroplane started off in trim in level flight. The pilot has to apply en ou gh up eleva tor to increase the angle of attack to increase the lift eno ugh so that L.c osB = \Vl. That will ensure tha t the ae roplane will perform a leve l turn . The lift is now g reater than the we ight, and the ratio of lift to w ei ght is c alle d the lo ad factor n , w h ich o nl y depends o n th e ba nk angle (n = L!W = I!cosB) . The banked win g turns the aeroplan e while the elevator keeps the nose up .
Figure 11.1 B
v
A
63
usual not to bother with the rudder in a turn .
Figure 11.2
Side slipping and Sk id d ing Tur ns Figure 11.4 is a view of an aeroplane in a level banked turn to the left as seen from the centre of the turn. The aeroplane has just the corre ct amount of rudder applied to hold its fuselage in line with the airflow. This is called a balanced turn . The aeroplane in Figure 11.5 h as in sufficient left rudder applied and so is sideslipping to its left. A little more left rudder will bring the tail up into line as in 11.4. The aeroplane in Figure 11.6 has too much left rudder applied and so is skidding in the turn . Less left rudder will allow the tail to drop down into alignment with the airflow . In practice, the slip or skid is so slight it is not noticeable and no t worth correcting, except perhaps for some scale models, or high aspect ratio models.
w Load Factor in a Turn That equation tells us how many "g" to pull in the turn to keep the turn level, for a given bank angle. Using less up elevator will decrease the turn rate and let the nose drop. More up elevator will tighten the turn and make the aeroplane climb. Figure 11.3 gives an idea of the relationship between the bank angle (in degrees) and the up elevator needed to keep the turn level. Not much for small bank angles, but quite a lot for steeply banked turns. For example for a 60 degree banked level turn you need to "pu ll 2g " (i .e . double the wing lift) to prevent the nose from dropping. However if the bank angle is 20 degrees, the wing lift need be increased by only six per-cent.
Drag in A Turn
Refinement There is more to turning an aeroplane than just having a centripetal force turn its CG to move in a different direction. Its fuselage must be realigned with the new direction of motion . If you think about it, rotating a banked aeroplane about a vertical axis is a combination of nose up pitch, and yaw in the direction of the turn . You have already applied up elevator which takes care of the nose up pitch component, and a little rudder in the d irection of the turn will line up the fuselage centreline with the airflow. The bank turns the plane, the elevator keeps the nose up, and the rudder keeps the tail in line . If you don't put on a little rudder in the direction of the turn , then the aeroplane 's weathercock stability will supply the necessary yaw once the aeroplane starts to sideslip. It is normal practice to coordinate the turn on light aircraft , but on most models (and jet transports) it is
All parts of an aeroplane cause drag but the wing in particular causes drag for two reasons. There is the profile drag which depends on its sectio n , and ind uce d drag which depends on the lift. In a turn the wing's angle of attack is increased and so its profile drag is likely to increase . More significant though is the increase in induced drag which increases as the square of the load Fig u r e 11.4
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5
10
15
20
25
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11.5
Basie Aeronauticsfo r Modellers
factor. Which mea ns that if you pu ll 2g you quad rup le the induced drag, and if yo u pull 6g yo u get th irty-six times as much induced drag . So do not be surprised when a sharp turn kills some speed, especia lly on low aspec t ratio aeroplanes. The control surfaces being offset from their ne utra l pos ition is likely to cause a little extra drag in a turn . The drag of the fuselage, and things like th e undercarriage wi ll ca use significant extra drag if there is significant skidding or sideslipping.
Figure 11.5
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Stalling Speed in a Turn An aeroplane does no t always stall at a certain speed. When you see a "Stalling Speed" referred to it is short for "the speed at which the wing is at the stalling angle of a ttack in wings level 19 fligh t at its maximum all up weigh t a nd the recommended CG position" . The significant bit is the stalling ang le of attack because that is always the same. Whe n an aeroplane is put into a banked turn th e angle of attack has to be increased . The aeroplane mus t therefore start off flying faster than its "stalling speed" by a reasonable margin .
Because the wing will always stall at the same ang le of attack, when you increa se the ang le of attack (pull "g") to keep the nose level, the stalling speed in a tu rn will rise as the square root of the load factor n. In othe r words if you pull 9g the stalling speed treb les. Pull 4g and the stalling speed doubles, and so on. Remember that the drag increases when you pu ll g in a turn so un less yo u add extra power the actual flying speed will decrease. Beware of stalling . If the aeroplane is docile it w ill merely refuse to pu ll the turn as tightly as
F igu re 11.6
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L Basic Aeronautics/or Modellers
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65
you want , but so me mod els co uld tipstall and flick out of the turn into a spin!
High Aspect Ratio Whe n an aeroplane is turning left say , the win g o n the right , on the o utside of the turn , is go ing faste r than the left win g. This has two seco nda ry effects wh ich are not usuall y not iceabl e unless the win g has a high aspect ratio . (I) The righ t wing will develop more lift tha n the left o ne . The aeroplane will tend to increase its ba nk ang le all by itself, even with the ailero ns held at neu tral. It may eve n be necessar y to hold a little o p posite (Le . right) aileron to maintain the desired left bank angle. (2) The right wing will ca use mor e dra g than the left wing, giving a yaw to the right. A sideslipping turn like in Fig u re 11.5 wi ll re sul t , w hic h w o uld be fu rth e r co mplica ted by strong lateral stab ility. The so lution is to apply left rudder. So to ge t a good acc urate left turn you may need a trifle of right aileron to hold the bank ang le, so me left rudder to yaw the tail into line, and some up eleva tor of co urse . Coupled Ailero n and Rudder would be a disad vant age here and differential o r Frise ailero ns won 't help during the turn (b ut they would still be effec tive as ailero n is bein g applied to roll into the turn ).
Turning Using Rudder I do not mean flat turns like those wh ich boats pe rform and wh ich are possi ble on ce rtain aeroplanes . If rud der is ap plied to produce co nside rable yaw then the sidefor ce on the fuselage may provide eno ugh ce ntripetal force for a wide gradual turn with the wings level but it look s a bit odd , crea tes a lot of drag, and is very inefficien t. What 1 me an is the co mmo n pr actice o n mod els of using rudde r to create yaw an d then late ral stab ility to produce the banked turn. The wing turns the ae roplane , not the rudder , becau se the w ing provides the ce ntripe tal force to crea te the turn . Up elevato r has to be applied to maintain level flight in the turn . Th e turn so produced will be a s kiddi ng turn as in Figure 11.6. There will be no adverse yaw pro blem, eve n w ith high as pect ratios, a nd o nc e th e turn has starte d so me of the rudd er ca n be taken off. Tak ing off all of the rud de r will allow a sideslip to devel op. Lateral stab ility will usually then roll the aeroplane out of the turn (a few ae ro pla nes would e nter a sp iral d ive and so o p posite rudder must be applied to leve l the w ings) . It is a perfectly satisfactory way to turn a simple mod el (never used o n full size that I kn ow of) but the entry and ex it ca n look rather un tidy. The rea ction time is slow as we are using a secondary effect. Rudd er -> yaw + lateral stab -> roll.
Summary To turn an aeroplane a force mu st be provided tow ards the ce ntre of the turn . The tight er the turn req uired, the bigger th e fo rce has to be . Th e force is provided by banking the aeroplane so that the wing lift has a sideways co mpone nt. The total lift has to be increased so that the ve rtical co mpone nt ca n still suppo rt the weight , w hich mean s up el ev ator is n e ce ssary . O n som e types o f aeroplane a little rudder in the direction of the turn may ne ed to be a pp lie d to co u nte rac t adve rse yaw from a ileron d rag, o r asymme tric dr ag o n high as pect rat io
66
ae ro planes . On ce the turn has started the a ilero n often needs to be returned to near neutral to stop the bank angle increasing. To achi e ve a ve ry ra pi d turn , a "pylo n race turn ", dem an ds a very large ce ntripetal force . The aeroplane mu st be flying at a speed well above its stalling sp eed. Point the win g's lift in the direction of the desired turn (80 to 8 5 d e gr ees o f bank ) , and in crea se it as mu ch as possible using up ele vator , but bew are! Pull too hard and the win g will stall in spite of the high speed and it may flick out of the turn. And make sure it is strong e nough to take the load s. Beginners! Avoid stee p ly banked turns, the y lead to trouble. The nose drops if you don't apply a lot of up , yo u risk stalling if yo u do . Start off us ing sma ll bank ang les of 20 to 30 degrees which means that very little up eleva tor is necessary. Then any failure to apply the small amo unt of up results in a very gradua l height loss, noth ing d ramati c. I hav e noticed when instru cting beginners that when they roll o ut of a turn their mod el often zooms upward s and so far I ha ve spotted two causes. If they let the nose drop in a turn and let the aeropla ne pick up speed (sa fer th an lett ing the speed d rop off) th e excess speed w ill ca use a zoom up ward s in level flight. Or so me times they do a goo d level turn with correc t up elevator applied and th en on rolling le vel forget to release the u p e levato r. Again a zoom results.
Special Effects There are so me very special pilots wh o can fly the ir speci al aeroplan es (mode l or full size) in a manner which seems to defy all the norm al rules of flying and ph ysics. O ne ca n but ma rvel a t the lik es o f Hann o Prettner performing his rolling circles with suc h precision . I kn ow that there is a ce ntripe tal force fo rming the circle and a force supportin g the weight , but how does he keep the resultants so con stant wh en the surfa ces producing them are co nstantly chang ing? Eve n those pilots ca nno t defy the laws of phys ics however. The ce ntripe tal force causing the turn , and the force supporting the weig ht must be there if yo u take the trouble to look.
When is a Rudder an Elevator? Whe n is a cow a horse? Whe n it is pu lling a plough pe rha ps? To a Vet a cow in harness is still a cow, and to me a rud der is always a rudder and always ca uses yaw w h ile a n e leva tor is always an e leva to r and a lways co ntro ls a ngle o f att a ck (p itch). Wh en I hear a cha p ren amin g the co ntrols during a manoeu vre like a turn or knife edge flight it suggests he has a dee p miscon cepti on of how the controls wor k and it crea tes so me co nfusio n, in me at least. I have visions of so me co mplicated device swa pping ove r the pushrods wh en his aeroplane banks. For example, in a stee p turn old Bloggs says that his e levator has becom e the rudder , but it seems still to be d oing its o ld jo b of cha ngi ng the a ng le o f a ttac k to increase the wing lift w hich ca uses th e tight turn . The rudder never did that in the first place. If you tell him to put on mo re up eleva to r which stick will he move? Which co ntrol will mov e, the o ld e levator or the new on e?
Basic Aeronautics/ or Modellers
Chapte r 12
A Delicate Balance T
his cha pter is not abo ut stability. Th at has already been dealt with. We have given the ae ropl ane a po sitive stability margin , Le. made Kn pos itive, by putting the CG ahead of the NP position, and that is an end of it. The next task is to trim it out to fly "hands oft" in eq uilibrium, wh ich is acco mplished by adjusting the force on the tailplane. As a schoolboy I ha d a flight in a Bolk ow 2 seater , wh ich had a trim leve r marked "nose heavy " and "tail heavy". I imagined that it moved a big we ight back and forward in the fuse lage . Wro ng! The term "nose heavy" in this context is nothing to do w ith CG position . Saying the aeroplane is "nose heavy" just mean s that it is o ut of trim such that left to itself it would pitch nose down. Moving the trim lever towards "tail heavy" (o r nose up ) adjusted the trim tab to correct the tenden cy.
Note that, in the interests of clarity, I have gone aga inst co nve ntio n by showi ng the no lift pitchin g moment Mo as a nose down arrow , because that is the way it really acts. Othe r books co nve ntionally show 1'1'10 as a nose up arro w but wi th a ne gati ve va lue . In my formulae mak e Mo positive for norm al wings. It is ass umed that thru st and dra g have no mom ent about the CG. Therefor e, taking mom ents ab out the CG, yo u can see that the direction of the tail force mu st be partly down to balan ce Mo and partly up to ba lance the wing lift. Tail lift may therefore be up or down but I sha ll define posi tive lift as up ward s. Skipping ove r the algebra , and ass uming that the tail is small co mpared to the wing , this Simple equa tion for tail lift coe fficient can be found . CI.T;(X,CL - C,' IO)N"'Ir ' . . eq uatio n 12.1 where Cl. is the lift coefficient of the "total lift".
Tail Lift Requiredfor Trim
Note that there is no ment ion of ang le A so the e quat ion is true whe n A is zero (in level flight) o r nin ety degrees (a vertical dive at terminal ve locity) or anyw here in between . Nor d oes stability matter to the tail lift. The ac tua l lift force on the tail will of course depend on airspeed squa red. The re la tio ns hi p is best ex plaine d o n the grap h in Figure 12.2. The tail lift will ofte n be up war ds at lo w speeds nea r the stall. At so me interme diate speed it will be zero , while at high speed the tail lift must be down ward. In a vertical d ive the total lift coefficient Cl. is zero a nd the tail lift is well and truly down, how mu ch dep ends o n the wing ca mber. To go from low speed trim to high speed
The aeroplane in Figure 12.1, which may be a glider or powered, has been trimmed o ut to fly in a dive of ang le A, whi ch can be any angle. From stability calculations the CG has turn ed out distan ce x behind the wing's aerodynamic ce n tre and I hav e show n the we ig h t s p lit int o tw o co mpo ne nts, W.sin A along the flight path , and W.cos A pe rpe nd icular to it. For the aeroplane to be in equilibrium, th e fo rces mu st ca ncel eac h o the r o u t a n d th e tot al mom ent o n th e ae ro plane mu st be ze ro . Th e total lift balances W.cos A, and the Drag is balan ced by W.sin A plus the Thru st T (if any). Figure 12.1
D
I I I
I I I
..1w
COS
A
W \..W sINA
Bas ic Aeronautics f or Modellers
67
the CG is further aft than inB. In the special case where the wing has a symmetrical section the graph will look like Figure 12.3. No tail load is needed in a vertical dive but otherwise tail lift coefficient is always up and increases as speed is reduced.
Figure 12.2
CG Well , Behind AC v"
~ (--'
Paradox Stalled
In the previous paragraphs I exposed a paradox. I stated that to Vtrim for a higher speed I the change in the tail lift Vcoefficient is downwards, but you know that it V requires down elevator trim. You may be tempted to suppose that down trim would cause a I tail lift increase, but this is V no place for intuition. You must stick to believing the equation which must be true for equilibrium and try to find a reason. Suppose an aeroplane has been speeded up such that its angle of attack is decreased from 10 degrees to 2 degrees. Because of the downwash the tail's angle of attack changes from 5 degrees to 1 degree, a reduction of 4 degrees which gives the reduction in tail lift to satisfy the equation . It is then retrimmed with down trim. So as long as the amount of the downtrim is less than 4 degrees (say it is 2 degrees) then both the conditions which you know must he true can be satisfied. Down trim has been applied and yet the tail's angle of attack (and hence its lift coefficient) has been reduced.
,
, ,
(--'
trim you need to move down the line to reduce the tail lift coefficient. The slope of the line is proportional to the distance of the CG behind the wing's aerocentre. In line A Figure 12.3
(" (" (" (" ~ Stalled
Low Speed
(" ("
,, 68
Elevator Angle to Trim As you know, to trim for a faster flying speed you need to apply down trim . Figure 12.4 shows how the elevator angle to trim varies with the lift coefficient. I have shown three lines for three CG
Basic Aeronautics/or Modellers
pOSItiOnS, i.e . th ree d ifferen t va lue s o f K, on the sa me aerop lane. The steepe r the lines, th e mo re the stab ility. Line A is for an aerop lan e set up w ith an aft CG to mak e it agile . Line I3 is for an aero p lane w ith a n ave rage CG position, and line C is fo r an aeroplane se t up with a ve ry far forward CG so that it will not stall. (The re is just not e noug h up elevator). Th e trim change betw een sp eed s 1 a nd 2 ca n cle arl y be se en to d epend o n th e sta b ility ma rgin . The less th e sta bi lity margin th e less th e trim cha ng e need ed. If there wer e no stability, then the grap h would be horizont al and no trim cha nge w oul d be need ed to cha ng e the aerop lane 's speed .
Tail Setting Ang le In Cha pter 3 I menti on ed several lines on an aerofoil from whi ch ang le of attac k ca n be measured . In this and subsequent cha pters the ang le of attack will be measur ed from the zero lift line of the wh ole win g. O ne reason for this is that at ze ro lift there is no downwash and as the an gle o f attack is increased , the downwash is a co nstant p roporti on o f it. Double th e an gle of attac k (measu re d from the zero lift line of the w ing) and yo u double the downwash . Tail Setting Ang le is the nam e for the ang le between the zero lift line of the whole w ing and that of the tailplan e with the eleva tor neutral. It therefore ap plies
to o nly o ne spe ed , or rather o ne lift co efficie nt. It does not co nfer lo ngitu din al stab ility as its mo dell e rs' nickname "lo ngitud inal dihedral" unfo rtun ately implies . By ta kin g e q ua tio n 12.1 a bove a n d d oing a littl e al ge bra ic magi c o n it a n e q ua tio n ca n b e fo u n d fo r wor kin g o ut the Tail Setting Ang le o f an aeroplane at a ce rt ain CL (or speed). Equation 12.2
Figure 12.5 illustrates th e variatio n of Tail Setting Ang le w ith w ing ca mber, stability ma rgi n , and als o w ith the chose n spe ed at which th e el evator is requi red to be neutral. \'('hat the eq ua tio n and the grap h bo th sho w is that the Tail Setting Ang le is in tw o pa rts, both negative (tail lead ing edge down). One part depends on the wing's ca mbe r, the mor e the cambe r the more the Ta il Setting Angle . Th e ot he r part d epends on th e sta bility and th e se lec te d lift coefficie nt. Th e mor e th e sta b ility and the slowe r the trimm ed speed , the more the Tail Setting Angle . The bottom line o n the eq ua tion is the tail's effectiveness, its tail volume rat io Vha r times aT th e slo pe o f its CL (J. cu rve. Th e more the tail's effect iveness the less the angle has
Figu re 12.4
Ele vator a ng le to trim
Fil II Doum Elev
Speed 1 1 1 I 1 1 I
forwa rd
Speed 2 I I 1 1 1 1
Stalled
-I I 1 1 I 1 1 I I 1 - 1-
cg
FilII Up Elev
Basic Aeronautics/or Modellers
1 1
69
Figure 12.5 Cbosen Speed for Neut ral Ele vator
Stall
1 I Higb Speed
1
2
tI ----,----Part due
to Camber
3
4
Low Speed
I I I I 1 1 1 -I I I I I 1
5
6
7
8
Tail Setting Angle ill degress LE DowII
to be to get th e desired effect. Even aeroplanes with symmetrica l wings need some Tail Setting Ang le, be it ever so small, unless they also hav e zero stability. That's no t an aeroplane , it's a guided missile!
All Moving Tail In the above d iscussio n I referred to the co nventiona l prac tice o f using a fixed tailplane with an eleva tor hinged o n the ba ck . The functio n s a re comb ined on a n All Moving Tail (AMT). The Tail Setting Ang le diagram 12.5 above becomes also the "tailp lane angle to trim" d iagram for the AMT w he re the tailp lane's angle is measured down from the Wing 's zero lift angle . If some middl e position is co nsi de red th e ne utra l position, then all the followi ng diag rams are still va lid , replacing "e levato r a ng le " by "tailplane displace ment from neutral".
The Effect ofThrust on Trim Un til now I have assu med tha t the th rust line was in
70
the d irection of motion and passed through the CG, but that is not alway s the case. The eng ine may be in the nose hut angled down at an angle ca lled the downthrust angle (the angle between the thrustline and the fuselage da tum line) such that the thrustline passes above the CG. Or the engine may be mou nted h igh above th e fuse lage o n a py lo n w h ich wi ll have the same effect b ut to a more extreme ex tent. See Figure 12.6. Line A of Figure 12.7 shows the normally sha ped trim line of an aeroplane like that in Figure 12.6, w hen it is gliding with power o ff. Line B of Figure 12.7 is th e tr im lin e of the sa me aeroplane in level powered flight. The more thrus t that is applied the faster the aeroplane flies and the more dow n tr im effect its mo me nt h a s (represe n ted by th e gap betw een lines A and B). Th erefor e less down e levato r trim is need ed . An aeroplane wi th excessive pit ch stab ility w ill ne ed large changes of eleva tor trim as power is changed, bu t it is so me times "fixed" by using excessive downthrust. The ste ep ness of the power off trim curve (l ine A) w ill be
Basic Aeronautics for Modellers
Figure 12.6 L T~----1"
w flatten ed by the downthrust as you ca n see (line B) and if the e ng ine cuts at high speed (po int 1 o n line 8) , the aeroplane will reve rt to its power off trim line and glide with the sa me trim setting but at a low er spee d (po int 2 on line A). Safe eno ug h. Line B is representative of a po werfu l eng ine with a little downthrust or a small eng ine high o n a powe r pod.
A Dangerous Situation
stall and persistentl y tries to do so until you wind in so me down trim. Can't happen yo u think? Once after a repair I glued the firewall back in at slightly the wron g angle , and I did not notice that in add ition the cast alloy mount had becom e bent in the cras h. I did notice that after take off it need ed mu ch more up trim than I ex pected, but thin gs do change when yo u re move a nd replace radi o ge ar. Wh en th e e ng ine qu it o n me it was trimmed to stall and I was in for o ne of my mor e ex citing deadstick landings'! (I know, write out o ne hundred times "After rep airs I must chec k rigging and downthrust angles").
If a powerful engin e has excessive downthrust , o r worse still is mounted high in a pow er pod, thrust will cause a large trim change . T h e trim c urve for straig ht and le vel flight Figure 12. 7 will look like line C on Figure 12.7. Line A is still Elevator a ng le to t rim th e trim lin e fo r zero thru st , but putting on D ow n p ow er add s an Elev e n o rmo us n o se d own mom ent. The aeroplane is in trim at full pow er at p oint 3 on th e g ra p h with a grea t deal of up 1 tr im . If th e p ower is turn ed o ff it tries to B rev ert to trim line A ( point 4 on the graph) but with that mu ch up trim it is sta lle d wh en th ere is no p ow e r to ho ld th e no s e down . T h a t is a p ot en ti all y I CI dangerous situa tio n if it I ca tc hes yo u un awa res. I O n clo sing th e thrott le I the mod el zoo ms up and tri es to s ta ll o r e ven up loo p. It may flick roll or Elev spin if it is so inclined. At best yo u ha ve a mode l wh ich on its ve ry first un e xpect ed d ead stick landing is trimm ed to
~
I
Y Y
- ----
3t
Basic Aeronauticsfor Modellers
I
V I
--
V
_
CL
4 I
~
I
V
71
Chapter 13
Glider Performance F
igure 13.1 shows a glider in a steady glide at angle A to the horizontal. The total Lift is at right ang les to the d irection of mo tion and is the result ant of all
• Lift = We ight x cos A • and Drag = Weight x sin A • o r Lift/Drag Ratio (L D) = l/t an A
Table 13.1
A1lg1eA
L/DRatio
Load Factor
1 2 3 4 5 6 7 8 9 10 15 25 35 45 55 65 75 85 90
58.8 28.6 19.2 14.3 11.5 9.5 8.1 7.1 6.3 5.7 3.7 2.1 1.4 1 0.7 0.5 0.3 0.1 0
0.999 0.999 0.999 0.998 0.996 0.995 0.993 0.990 0.988 0.985 0.966 0.906 0.819 0.707 0.574 0.423 0.259 0.087 0
th e lift for ces on th e wi ng , ta il and fuselage a t their centres of p ressure . The to ta l Drag is opposite to the direc tion of the mo tion. By splitting the weight into two co mponents as shown it can be seen that.
Th e Lift/ Dr ag ratio o f the ae ro p la ne is im p orta nt becau se it is directly co nnected to the glide ang le. If yo u know the g lide ang le yo u ca n div ide 1 by the tan gent ratio (loo ked up in your tabl es) and Bingo! . .. You have the Lift/Drag ratio . You w ill see (from the equations at the end of Cha pte r 2) tha t the Lift/ Drag ratio LID is just the sa me as th e rati o o f Lift coefficie n t CL ove r D rag coefficient Co becau se eve rything else ca ncels o ut. In a div e Lift is alway s less than Weigh t. The "load factor ", LlW, dep en ds on the glide ang le and is in fact eq ua l to cos A wh ich is always less than 1. Tabl e 13.1 shows the var iation of bot h LID ratio and load factor in a d ive of ang le A (in degrees).
Speed Range In a vertical dive, the angle A is 90 degre es an d the ratio sin A is 1, so that Drag = Weight (and Lift is zero) and the aeroplane is at the maximum speed at which it is aerodyna mically capa ble of flying, its "te rminal velocity".
Figure 13.1
Horizontat \
lV
~
... lV cu s A lV . IIIA
72
Basic Aerona utics/or Mod ellers
Its min imum limiting speed is its "sta lling speed ", at which the overall lift coefficient is a maximum. Th at w ill occ ur whe n th e an gle of attac k so me whe re alo ng the wing rea ch es its local stalling angle. Whe n the aeroplane is turning or "p ulling g" the w ing w ill stall at th e sa me a ng le o f a ttack , b ut th e sta lling speed in a man oeuvre is high er than the stalling speed in a stea dy glide .
Looking at Aerodynamic Data For a particular ae roplane Cl. and CD will vary wi th the angle of attack and ther efore so will the ir ratio . Each e levator trim positio n gives a part icul ar flying speed at a par ticular ang le o f attac k and a pa rticul ar glide ang le . Therefore a graph can be d raw n showi ng how all these th ings vary fo r ea ch trim position from the stall to the vertical di ve . The re are severa l w ays to present th e info rma tio n . I hav e dr awn u p a se t of graphs which , althoug h ma de up for an imaginary glide r, are co nsis tent
Figure 13.2
within them selves. They are pr esented as Figures 13.2 to 13.7. The first two, 13.2 and 13.3 are the assumed lift and d rag coe fficients graphe d aga inst angle of attack (alp ha a ) . Figure 13.4 is the most ob vio us presen tation of glidi ng performance , just a graph of LID ratio against angle of attack. Whe n L is zero at a the LID must ob viously be zero . As a is increas ed the LID increases to a maximum an d then red uces wh en d rag starts to increase rapid ly. An alte rna tive pr esent ation is to draw a graph of Cl. ag ai ns t CD which is ca lled a "po la r cu rve " or "p o lar d iag ra m ". An exa m p le is g ive n in Figure 13.5. The max imum valu e of LID is the slo pe of the line which just to uches the curve and the Cl. at whi ch it occur s can be read off. Becau se the bes t glide ang le is associa ted with a parti cula r lift coefficient like this yo u ca n see that o nly one ang le of attack , and therefore o ne trim se tting, will give the flattest glide . Anoth er alternative prese ntation is to draw a "Hodograph", Figure 13.6. Each po int on the ho dograph (e .g. po int X) represe nts a velocity vec tor (e .g. V) for on e pa rticula r trim setting. Th e di stan ce of th e point from the origin is the glider's airspeed , and the angle down from the hor izontal is its glide ang le. Thu s the a irspeed and its ho rizont al and vertical co mponents
Figure 13.3
a a
Figure 13.4
LID Figure 13.5
I I I
Basic Aeronautics f or Modellers
73
Figure 13.6 Horiz
Comp
Speed for Best
---
Glide Angle I I I I I I I
Mi" Si"k Speed
_________ J
Figure 13.6 Mag"ified
and the glide angle may all be seen together. You can also see straight away what glide angle will give the greatest horizontal velocity for penetration, what is the minimum glide angle and what is the minimum sinking speed. It is also evident from the magnified portion that the speed for minimum sink is less than the speed for the shallowest glide angle. It is important for a glider to have not only a shallow glide but also a low minimum sink rate so that it can remain airborne on the slightest whiff of lift.
Optimising Performance There are basically two aims when flying a glider, either to go as far as possible from a given height, or to stay airborne for as much time as possible from a given height, both of which require a fairly flat glide. The best of the full size soarers can achieve a glide angle of about one degree, but the best models would do well to achieve a two degree glide angle. I shall now assume that in flight the Lift is always equal to the Weight. Look at table 13.1 again and you will see that even at a glide angle of 8 degrees there is less than 1% Figure 13.7 Drag
Profile......- -. / ~
74
Induced
Vel
error in this assumption. If the lift is constant the LID ratio is greatest when the total Drag is a minimum . As you know, the total drag is made up of the induced drag mainly from the wing, plus the profile drag of the wing, tail, fuselage and struts etc combined. Figure 13.7 shows how the total drag, and its two components, vary with airspeed. Profile Drag is proportional to speed squared, whereas Induced Drag is inversely proportional to speed squared. Odd as it may seem, the induced drag of an aeroplane is greatest at low speed. At very high speeds the induced drag all but disappears but the profile drag is very large, and eventually equals the weight. (Note that this is actual drag force for once, not the coefficients). You can see from Figures 13.6 and 13.7 that the min. drag speed, which gives the flattest glide, is quite near stalling speed and the speed for minimum sink is very near stalling speed. It can be shown (trust me) that the airspeed for minimum sink rate is theoretically about 75% of the airspeed for minimum glide angle.
Effect ofStreamlining I have already mentioned how and why to reduce induced drag, but to be efficient a glider must be a clean machine . Tailplanes with an up or down load have induced drag like wings. Tails, fins , fuselages, struts and undercarriages have skin friction drag from their surface area and form drag which depends upon their streamlining. The drag of the whole aeroplane is more than the sum of the parts. The extra drag is called INTERFERENCE DRAG and is caused by the airflows around the various parts interfering with each other. As much of this profile drag as possible must be eliminated by careful streamlining and fairing adjoining shapes into each other. The reduction in total drag obviously gives a better LID ratio . What is equally important is that this better LID ratio is achieved at a higher airspeed, which brings three further benefits. The aeroplane has a greater speed margin above its stalling speed. The controls will work more effectively and as you will see later a further increase in efficiency is possible because of the higher Reynolds Number at the higher airspeed.
Basic Aeronautics for Modellers
All this su pe r e fficie nt strea mline d e lega nce mak es landing a nightmare at an angle of two degrees or less whi ch is wh y it is necessary to add handfuls of add itional profile drag in the form of a irbrakes to ruin the UD ratio and give a stee per descen t.
Figure 13.8
..
Wind Vector,
HorizComp
Effect of Weight on Performance Non dimensional constants like CD and CLare just that . . constant. They do not vary with weight or speed or anything else but angle of attack. Therefore the U D ratio at a particular ang le of atta ck will not va ry with weight. Exactly the same trim will give yo u the same o ptimum glide angle whatever the weight. How ever the speed at whi ch it is achieved will rise. The speed increase will be prop ortional to the square root of the we ight increase. So that for example if the weight is doubled , all referen ce speeds will rise by 41.4% and that includ es Stalling spee d, terminal Diving speed , Minimum Sink speed, and optimum Glide Angle speed . Th e glide r ca n still glide th e same distanc e from a given height but will do it more qui ckly. Therefore its rate of descent will be (41.4%) greater. Although the coefficients have rema ine d the sa me , the actua l Drag has doubled and so has the Lift. Even at the ne w minimum s inking s peed the rat e of descent has increased . Ballast, or a heavily built mod el, will not hinder your mod el from getting about the sky, but it will reduce its endurance .
Effect of Wind on Performance The most supe r efficient sailplane with a still air glide a ngle o f one d egre e at 40 mph w ill s till h av e ze ro gro undspeed in a 40 mph headwind and so will descen d vert ically. Its nice low sinkrate is not affected by the w ind , but it will still not get ba ck to its field from downwind . A glide r needs to have a good speed range to get abo ut the co untrys ide in windy w e athe r. Th ere a re usu all y two choice s if you are not ge tting enough pen etration . Either trim to a high er spe ed, or ballast up.
Down Trim Figure 13.8 sho ws the magn ified top part of an oth er hod ograph. A glide r is trimmed to fly at point B o n the graph and has a great glide angle relative to the air. Now I shall draw in a wind vector from po int A to the origin (the op posi te direction to the aeroplane 's velocity to represent a headwind). Vectors from A to point s on the hod ograph sho w the speed and glide angle of the glide r relative to the ground. The line from A to B shows that the glide angle relative to the ground is ab ysmal. It would be best to trim the ae roplane to point C o n th e graph . At thi s high e r airspe ed th e glide a ng le rel ative to the gro u n d is optimised. Don 't ask me how you find this po int witho ut instruments - that tak es ex pe rience. I'm alright with a pencil but rubbish on the sticks.
B
Doum Comp same pen etration as speed C (w ith down trim) but with a low er sink rate and also a shallower glide relative to the ground . The mod el will ge t back from do wnwind with more height rem ainin g. Of co urse once you have returned from downwind yo u a re s till stuc k with th e b all ast , wh ereas do wntrim you can take off. I have ex plained the options, the choice is up to you.
A Third Option As I menti on ed previou sly you ca n try reducing the profile drag. Change the win g sec tio n, sharpen the trailing edge, polish th e surfaces , blend the win g and fuse lage sha pes togeth er carefully etc. The result is an eve n better glide angle ·and an eve n slowe r sink rate at an eve n high er speed . Who says you can't have your cake and ea t it? Figure 13.9 Wi"d Horiz Comp
C\ \ \
I
I I
I
1,I "
Weight W/ ' ,
Ballast Figur e 13.9 is yet ano the r hod og raph, this time with two lines. Line 2 is for the same aeroplane as line 1 but at double the weig ht. The bes t glide angle relative to the air ha s not cha nged but the minimum sink rate ha s. It is worse. How ever look carefully. The second optio n is to add ballast and fly trinun ed at speed D, which gives the
Basic Aerona uticsfor Modellers
Dow" Comp
2
Weight2W
75
Chapter 14
Powered Performance f yo ur main int erest is in p owered mode ls th en p erhaps yo u have skip ped th e cha p te r o n g lide r p e rformance . Well go b a ck a nd read it a ny way becau se a pow er ed mod el becom es a glider wh en the engi ne stops. Bear in mind that power ed mod els tend to use lowe r aspect ratio and so have mo re indu ced dr ag, a nd less careful str eamlining and so mor e p rofil e drag . The power so urce co uld be a gas turbine o r d uct ed fan wh ich fo r pract ical purpo ses produ c e co ns ta n t thrust , but usually we use a prope lle r. Prop s ize is given as diamet er x pit ch e .g . 10 x 6, th e pit ch b eing th e forw ard m o v em ent in on e re vo luti on , in th e dir ection of th e aerofoil cho rd line . A "fine " pit ch propeller has a sma ll pit ch and a "coarse " pi tch p ropell er has a large pit ch . A p rop e ller bl ad e is, u su a lly , a fl at b ott om ed a e ro fo il fi tted with th e c urved s id e o f th e aerofo il fo rward, to w ards the dir e ction of mot io n of th e ai rcraft (o bvio us perhaps , but I have seen pu sh er props back to fron t) .
I
P ropelle r Th rust Th e velocity of the air re lative to the pr opeller , as sho w n in Figure 14.1, is a co mbina tion of its rotatio na l ve lo c ity and its forw ard ve loci ty . Th e fast er th e aeroplane is flyin g the less th e b lad e a ng le of a ttack. Figure 14.2 shows how th e thrust of pr ope llers vari es wi th th e forward spe e d of th e aeroplan e for different co mbina tions of diame ter and pitch . When the a irspeed rises and the blad e's ang le of attack reduces , the to rqu e ne e de d to dri ve the prope lle r a t a g ive n speed a lso reduces and so the engine ca n speed up . It is sa id to "unload in the air". Th e pit ch of the p rop eller is chose n to suit the flying s pee d of th e model and th e c o a rs e r th e p itch , th e sma ller the d iam eter mu st be to avoi d overloadi ng the e ng ine .
Slipstream Effects The wake of a propeller is a co lumn of air moving aft
F igu re 14.1 L
......
~
R esultant •• • •• •••• ' Fo,~e
It
~---_._._---_ ._--\-------
..• , ,
.. , ,
••
Thrust
\
,,
\
.. .. , , ,, .. ,, .. , .. \
\
\
\
\
Torqlle Component
76
\ \
Rota tion
Basic Aerona utics/or Modellers
rela tive to the aerop lane at a speed greater tha n that of th e s urro undi ng air. If th e tail con trol s urfaces ar e mo un te d in th e sli ps tream the ir effec tiveness will be inc reased as the pro p ell e r thru st is increased . For exam ple rudder gives ve ry good d irectio nal co ntro l on take off but is mu ch less effective on lan din g. In Figure 14.3 (in which the effect is exaggerated) an aerop la ne has a pr opell e r ro ta ting clockwise (fro m b eh ind ) and the sli ps tream wi ll strike th e fin slightly fro m th e le ft. A fo rce to the right w ill be generated yawing the nose to the left. If in an attemp t to avoid this problem the fin is turned to line up with the slips trea m, then when the throttle is closed the aerop lane has some right rudder ap plied and will now yaw to the right. A better so lu tion is to mou nt the e ngine suc h that the thru stline is angled as shown to the righ t, called "right sidet hrust". The slipstream still causes left yaw , bu t the thru st has an o pposing mo me nt ab out the eG. The fin is straight whic h lets the model fly straigh t power off, an d th e si det hrus t can b e a djus te d to give straight flig ht power on . One to three degrees is a typic al sidethrust angle.
Figure 14.2 Thrust
-,
" "\ Fine """\
, ,\
\ , \ ,
\
", ,
\
\ \
Coarse
,
,
,
,
, Airspeed
Figure 14.]
Level Flight Top Speed
~~~= T
--- -
-_
A powered aeroplane ca n fly level , regardless of the - --=ac-~ drag. Just bo lt a big enough vibra to r o n the front and away yo u go . The thrust eq uals the total dr ag, and the to tal lift eq ua ls the weight. Figure 14.4 illustrates how the thrus t an d drag vary with the forw ard speed of the aeroplane . Where the two cross Th ru st eq ua ls Dra g an d th e aeroplane is in from th e surface increasi ng th e drag coefficient a nd eq uilibrium at its maximum speed in level fligh t. A mo re reducing the lift coefficient. When th e lift coefficient is at p owerful eng ine will o bviously produce more thrust and its ma xim um, the airspeed is at its min imu m, whic h is so th ru st w ill eq ual d rag at a highe r speed . Using a called the "stalling speed in level flight" (for the exis ting coa rser pitch prop on the same engi ne the thru st ca n be weight and co nfiguration) . It is an oft quot ed statis tic of sustained to a hig her speed allowing a high er to p speed an aeroplane and is a usefu l reference speed. for the aeroplane , at the expense of lo w s peed Figure 14.4 acceleration . Th e s peed a t w hic h th ru st and drag are equal is an airspeed of co urse . Thrust will equal d ra g a t this ai rspee d regardless of whe ther a Fi"e Prop wi n d is b lo wing . If some thing makes th e a e ropla ne s low dow n, Drag Thrust then t hr ust will b e grea ter th an drag an d Coarse Prop w ill accelerate the a e ropl ane back to the a irs peed at which they ,, were eq ua l. _
m
_
_
m_
,,
Stalling Speed
,, -,
As fo r a glider it is no t th e s p eed bu t th e sta lling angle of attack which is fixed. The wing sta lls a t th e ang le of at tack a t w hich the airflow ove r the to p s u rface breaks away
Basic Aero na utics for Modellers
,, \
\ \
\
Pine Top
Co
Speed
se Speed
77
Figure 14.5
Thrust
Drag
I I I
, I I I I
I
) Stall I ,Speed)
I
Top I
ISpeed
I
)
I
High Weight
Speed
Low We ight Speed Range
Effect of Weight
Climb Performance
Obviously from the above the sta lling speed in leve l flight w ill increase as th e weigh t is inc reased . T he stalling speed increases as the sq ua re root of the weight , d o ubling the weigh t gives a 41.4% increase in stalling speed. Thrust from the propeller is independent of weight. How ever the greater the weight, the greater the drag w ill be, partly because the induced d rag is greater and partly because the associated inc rease in angle of attack w ill increa se th e profi le drag . Figure 14.5 shows th e thrust from a particu lar engine/propeller combination an d also the drag at two weigh ts. As yo u can see the speed range of the aeroplane reduces as weight is increased, stalling speed increasing and maxi mum speed red uc ing .
Figur e 14.6 shows an aeroplane in a climb at ang le A to the ho rizontal wit h its thrust , weight, overall drag and overall lift all th rough the CG. The weight has been sp lit into two components. You ca n see that the Lift is eq ua l to the component of the weight W cos A, w hich mean s tha t lift is less than weig ht in a climb . Yo u can also see fro m th e diagra m th at th rust has to balance not only drag but also a component of the we igh t, W.sin A. The steeper the climb the more thrust will be needed. In a ver tica l climb the th rust must e qua l the weight plus the drag at the particu lar airspeed, but lift will be zero. Figure 14.7 shows the maximum th rust an d tota l drag plo tted against speed. This time howeve r the aeroplane is climbing at speed V. The thrust at this speed is grea ter than the drag, and the excess thru st is use d to overco me the weight co mponent W.sin A. The lower the climbing speed the more excess thr ust and so the steeper th e climb . Th e heavier the aeroplane th e more its drag w ill be at th e same speed . But it w ill have to be flown faster to keep th e same margin above sta lling speed , w h ich incre ases the d rag even mo re . Th e re is th e re fo re less spare thrust ava ilable, but the "dow nhill" co mponent of weight, W.sin A, is g rea ter. These factors all combine to re du ce the climb angle ac hievable at the high er weight.
Take off Performance For accelera tion yo u need low weight and plenty of th ru st. Whe n the speed is safely above sta lling speed yo u need enough eleva tor power and clearance under the tail for rotation. When the a ng le of attack is sufficient the lift w ill exceed the weigh t and the aeropla ne will take to the air. The faster the model is travelling th e less pitch up needed to "unstick". The heavier the aeroplane the greater the take off speed and the slower the accelera tion . Weight w ill have a ma rke d e ffect on runway length requ ired, as will the surface. Smooth concrete is great, long wet grass really holds the model back. A larger diame ter fine p itch prop will give mo re thrust and hence a quicker accelera tion at the expense of to p speed .
78
Descent and Landing Figure 14.8 shows an aeroplane in a descent, with a sma ll amo unt of thrust hel ping the weight component \X7.sin A to overcome the dr ag . The mo re thrust ther e is the sha llower the descent. In ot he r words o pe ning the throttle a little will g ive a sha llower descent at the sa me
Basic Aero na lilies fo r Modellers
Figure 14.6
Hortzontal I I
, I
I
I
lVcosA W
speed, whil e closing th e th rottl e w ill g ive a ste e per descent. The throttl e se tting co ntrols rate of descent. The d ra gg ie r th e ae ropl an e , th e more th e po w e r need ed , especially not iceabl e in the case of low aspect ra tio biplan es clutter ed w ith rigg ing w ires a nd stru ts. So me cl e an p o w e red mod el s ha ve to o s ha llow a n approach ang le for safe te rrain cleara nce . The best cure for too sha llow a n ap proach pa th is more d rag from airhrakes , flap s, slats or so me thing. Diving the aeroplane over the obstructions on the approach w ill o nly increase th e s peed a t w hic h it reac hes the run w a y a nd th e aeroplane will "float" right down the run w ay and land in the roug h. The fina l a p proach se ts up th e aero p la ne fo r the landing, low over the run way thresh old w ith the speed safely above stalling speed . In th e "flare " th e nose is pit ch ed up to ar res t th e rat e o f d e s cent until th e ae roplane land s almos t at stalling speed.
Figu r e 14. 7
Th rust
Drag
WsI"A
D
Speed
Figure 14.8
W
Basic Aeronautics/or Modellers
WcosA
79
Chapter 15
The Aerodynamics of Aerobatics have already de alt with straight night a nd turning. Any t h ing e lse is not us e ful in tr an s p orting th e aero plane from A to B a nd is therefore co ns ide red to be an acroba tic man oeu vre , don e jus t for the sa ke of doin g it. All th e acroba tic man oeu vres w hic h I ca n think o f co ns is t o f a few basi c e leme n ts s tr u ng to geth er in var io us wa ys. I do not claim to be any kind o f hot sho t aerobatic p ilot and therefore do not feel q ua lified to go into the re fine me nts of fl y in g ae roba tics. Ne ithe r do I have the eq uipme nt ava ilable to test o ut the su btleties of acrobatic model design . I sha ll leave the refinements to the expe rts and stick to the basic e leme nts. It is wort h rem em ber ing w h en per for ming or watching aerobatics that the law s of ph ysics canno t be brok en , by a nyo ne.
I
The Stall The pu p il p ilo t is ofte n tau gh t the sta ll as his first "ma n o e uv re ", so th at he ca n le a rn ho w to avo id a dan ge rou s un intenti on al stall. The aeroplane is slo wed down by gra dually applying up e levato r in level night , increasing the ang le of attack , until so mew here on the w ing th e flow s ta rts to s e p a ra te . T here are seve ra l possible o utco mes . • 1. If the se paratio n occ urs ove r both win g roots more
o r less simultaneo us ly then the nose do wn pitching mom e nt coe fficie nt C ~ I O whi ch had b e en co ns ta nt sudde nly inc reases . Or if yo u are a Centre of Pres sur e fan, the Ce ntre of Press ure moves a ft aga in . The result is a ge ntle nose down p itch , a nice safe re acti on . Applying more up e leva tor ca n co unteract th e nose down mom ent an d kee p the nose up . The ae ro plane w ill th en descend in a le vel a ttitude a nd w ill still respond to ailero ns , rudd er and elevato r. • 2. If se pa ratio n occu rs over th e in board regio n on o nly o ne side, the wi ng on th at side will drop. DO NOT try to pick it up wit h aileron . That is like ly to stall the outboard pa rt of the wi ng as we ll and make the situation worse. Th e co rrec t rec ov ery is to apply down e levator and opposite RUDDER. • 3. If the o utbo ard part of the w ing stalls o n o ne s ide , a "tips tall", the n a viole nt wi ng d rop occurs usua lly le ading stra ig ht into a sp in. T he re is no tim e to recov er from the stall. See below fo r spin reco very. • 4. As I sa id in Cha pter 12, if the ele vator is small and the stability ma rgin is large even fu ll up e levator may not be enoug h to stall the w ing . It may just fly slo wly in trim. See Figure 12.4 agai n .
The Spin A spin is e nte red from low spe e d night. The w ing is a t o r ne ar its s ta lling a ng le of a ttack a nd o ne w ing d ro p s , du e to a g us t, ai le ro n in put , o r w hateve r.
Figure 15.1 Figure 15.2
Rising Wing Dropping Wi"g
a Of Rising Wi"g
a Of Dropping Willg
0< Original a
80
0<
Basic Aeronautics/o r Mode llers
F igure 15.4
Fig"re 15.3
-. W
V
W
L
Consta nt V W
L
W
L
W
2V - - -
w Appli cation of rudd e r in the appropriate dir ection a lso helps. Becau se the air is now co ming up towards the downgoing win g it ge ts an an gle of attack increa se , and the u pgo ing wing gets a n a ng le of a ttac k de c re ase . Th es e would n ormall y ca use "ro ll d amp in g ", but becau se the win g is alre ad y at its stalling ang le of attac k, the an gle of attac k incr ease pa st the stalling a ng le le ads to a re d uctio n in the lift coefficient. A ro lling mom ent in the direction o f the ex isting roll result s. The d ropping w ing w ill also ha ve th e g re a te r dra g p rodu c ing yaw tow ard s the ce ntre of the spin. Figure 15.1 sho ws that the dropping wing has a sma ller lift coe fficient than the rising wing and so the situatio n co ntinues . This pro cess is kn own as "auto rotatio n". The a irspeed rem ain s low in a sp in du e to the ve ry high dra g o f s ta lle d win gs . Wh ile th e a e roplan e is sp inn ing the angle of attack o n th e w ing o n the insid e o f the s p in is g re ate r th an that of th e o utsi de Wing but bec au se o ne o r pr obab ly both , are g rea te r th an th e sta lling an gle , th e s p in co ntin ues (Fig ure 15. 2) . Th e aeroplane is both rolling and yaw ing toward s th e mo re stalled wing. The rate of rot ation is faster than in a spi ral di ve a nd th e hei ght los s per turn is mu ch less. Most mod e ls have to be held in the sp in with up e levator a nd usu ally rudder and aileron in the same d irection . Recov er y action from a spin is ve ry dependant o n the aeroplan e. Most models will reco ver if all the co ntro ls a re ce ntralised . If no t then positive recove ry action mu st be tak en . Appl y full o p posite rudd er to co unteract the rotati on (the ailerons are ineffective), and down e levator to un sta ll the win g. It is impo rtant that aeroplan es design ed for ae rob atics will s p in re liably whe n required . The d esign fact ors whic h assist in re liabl e s p in ning a re taper ratio , CG position , co ntrol throws and LE sharpness . A taper ratio of 0.65 or less ens ures that the an gle of attac k at the tips is o nly slig htly less than near th e root. A modest CG mar gin and a de q ua te e leva tor throw e nsure th at th e win g can be pitch ed to its stalling a ng le of attac k. And adeq uate rudder movem en t ens ures that it w ill sp in in the direction yo u want. Havin g the lead ing e dge sha rpe r at the tip than the root also helps spinning .
Basic Aero nauticsfor Modellers
L
L
L
L
:= _~t=::==n
lV
In o rde r to avo id spinning , trainers go th e othe r way ; co ns tant chord, fo rwa rd CG, reduced e leva tor autho rity and blunt lead ing ed ge. Scale mod els have no cho ice of tap er rat io but ca n use wash out and a forward CG to avo id pr obl em s.
The Snap (or flick) Roll Exce pt that it ma y be don e travelling in any d irection at a ny sp eed , a sna p roll is just like a spin and the sa me design factors are involved. If full up ele vator is applied , the ang le of attac k wil l be incr ea sed , probably up to the stalling a ng le o r beyond . If one wing stalls but the oth er does not , the differ en ce in lift wi ll produce a ver y rapid ro ll rat e . On e Wing ca n be mad e to stall by ap plying rudder and aileron in th e directi on of the intende d ro ll. Rele asing the up e lev ator and/ or the rudd er and ailero n will sto p th e snap ro ll. A sna p roll may be performed in ho rizo n ta l fli ght , o r on a 4 5 d e gre e d ownlin e , o r vert ically upward s o r what ever. You sho uld be awa re of the structural loads imposed by a snap ro ll at high speed . If the ae ro plane is flying at a s p eed fa ctor of three times its lev e l flight sta lling sp eed , then the win g will be su bjecte d to a brief load ing of nin e "g", if at four times stalling speed , sixtee n "g". The load factor is the sq ua re of the sp eed factor.
Th e Lo op To make a loop round a ce ntripe tal force towards the ce nt re of the loop is necessary and is provid ed by th e win g lift. The tight er the loo p the mor e up e levator is need ed . If th e airspe ed ca n be kept co ns ta nt, th en ke eping th e ce n tripe tal for ce co ns tant w ill produ ce a round loo p , Th e ce ntripe tal for ce is the result ant of all th e force s a cting p erp endicular to th e a eropl an e' s dir ecti on of mo tion , the re fo re the lift force mu st va ry round the loo p . For ex ample, in Figure 15.3 lift is five times weight at the bott om , thr ee times we ight at the top, and four times the we ight at the ve rtical positions. \V'ithou t ve ry ca refu l power co ntro l the s peed w ill var y round th e loop . Th e ce n tripet a l force requ ired
81
Figure 15.5
varies as speed sq ua red therefor e the lift variation wi ll be mu ch greater. Taking the arbitrary ex a mp le w her e spe e d at the to p w ill be half wha t it was at the bott o m , the lift variation w ill be as in Figure 15.4. Th is time , if the lift is five "g" at the bott om, it must be zero at the to p . Lo o ps a re easy, but pe rfectly round loo ps are ve ry d ifficult ind eed .
Inuerted Flight Figu re 15.5 shows a n ae ropla ne wi th a cambere d wi ng in stead y le vel flight. T he tail lift LT mus t b e u pward so that its mom ent abo ut the CG balan ces bot h Mo a nd the win g lift. Yes I kn o w that yo u have ap plied do wn e levator but the who le aeroplane has been tilted Figure 15.6 Elevator Angle To Trim
- - - - - - - - - - - Full Doum Elevator
Inuerted Flight
Stall
Full Up Elevator
82
so mu ch to ge t the wing to lift up sid e do wn that th e tail is at a liftin g a ngle . Were it not for th e down e lev a to r LT wou ld be so g reat that it wou ld h a lf loop th e a ero p la n e into le vel flight. Figure 15.6 is a trim g ra p h ju st like Figu re 12.4 e xcept that it has been ex tende d to s how negative lift coefficie nts, for invert ed fli ght. It shows the e levator ang le to trim for an ae ro plane, like a traine r o r g lider, w ith a ca m be re d wing w h ic h has been set u p suc h that the ele vator is neutral in upr ight fligh t. Th ree lines for thre e d iffe ren t va lues of Stability Mar gin are sho wn. The trim change between uprigh t and inverted fligh t at the same spe ed is sho w n fo r a mid CG positio n. You ca n see th at the furth er forward th e CG, Le . the bigger the Stab ility Margin , the more trim cha nge there will be . Inve rte d flight is not possib le a t low spee ds becau se the wing is not as good at liftin g invert ed as it is upright. Figure 15 .7 is another tri m g ra p h, thi s tim e for a pattern ship with symme trica l wings and ta il bo th se t u p a t zero inciden ce . Fo r u pri gh t flight a little up trim is needed , th e more Stability Margin the more up trim . And fo r in ve rte d flight the same a mo un t of d own trim is req uired . The trim ch ange betwe en upright an d inverted fli gh t ca n b e see n to dep e nd o n the Stab ility Margin just like o n Figure 15.6. In fact for the sa me Stab ili ty Ma rg in , the same trim cha nge is need ed . It se ems stran ge but it is true that a ca mbere d wi ng d o es not requi re any more d ow n trim to fly inverted . It does have o the r e ffec ts ho w e ve r. T h e p ro fil e dra g of a ca mbered section flyin g invert ed is mu ch grea ter whi ch reduces the speed ca pa b ili ty o f the a erop la ne . And th e w ing ca nno t provid e nearly as muc h n e ga ti ve lift as p ositive lift be cause of th e s ta ll. Bo t h th e s e e ffec ts co ns p ire to Stall redu ce the negat ive "s" ca pa b ility o f a n ae ro plan e with a ca mbe re d wing . It is but a sma ll ste p from doing le ve l flight in vert ed to d o in g
Basic Aeronautics/o r Modellers
man o eu vr e s inv erted . To s pi n inve rte d from le vel in ve rt e d fli ght , g ra d u a lly a p p ly fu ll d o wn e levato r to sta ll a nd th en r u dde r and aile ro n in o p pos ite direc tions to put it into a spin. T he sa me co nt ro ls s ma rt ly a p p l ie d ca n produ ce a negative sna p roll and simply applying e no ug h down e lev a to r will pe rform a negative loo p, or "o utside" loo p, o r "b u n t" . Be caus e o f t h e lim ita ti on s in t h e p revi o us para gr ap h an aer opla ne wit h a ca mb e red win g will not b unt as ti g h tly as it loo ps (if at all).
Rolling Manoeuvres
Figu re 15, 7 Eleu ato r A ng le To Trim
FilII Dow1I
I I I lnuerted I I I I I I
I I
FilII Up
Yawing Manoeuvres
Basic Aeronautics/or Modellers
Sta ll
Stall
The re is little to say that was no t sa id in the c ha pter o n Co ntro ls. I sha ll jus t remind you that the rate of roll p roduced by the ailero ns depends o n their size obv iously, a nd on the aeroplane's roll damping. The bigger the win gsp an the mor e roll damping and so the slower th e rate of ro ll. Also , be care ful of using ailero ns near stalling speed .
Th e o nly co nt ro l left is th e ru d de r, whi c h causes the ae ro plane to y a w . T h is is u s e d in "k n ife e dge " fli g ht as de picted in Figure 15.8. The aeroplane has been ro lled thro u g h n in e ty d e gre e s a n d h eld . Rud d er h a s been a p p lied to h o ld th e fuselage at a n ang le o f att ack to th e o nc o ming air. Th e press u re d is tri bu tio n over the fuselage sides can give e no ug h lift to support the aeroplan e's weight , aided by the vertica l co mpo ne nt of th e thru st. There is o nly a sma ll amo unt of surface area involved so th is is a hi gh s pee d game. Rudd e r is a lso used in th e "Sta ll Turn " ma no e uvre . The a e ro-
Up r ig h t
plane is pointed ve rtically up and allowed to slow down to a sto p. Ju st befo re it stops , rudde r is used to rotate it 180 d e gre es a bout its "ve rt ica l" axis, w hic h is now po int ing to the far hor izon. T here foll o w s a ve rt ica l d es cen t. No te th a t th e aeroplane stops in this man oeu vre but it does not stall! Th e a ng le o f a tt ac k is ze ro in th e cli mb a n d ge ts no wh ere near the stalling angle of attack .
Figure 15. 8
M o tio 1l
83
Aerobatic Trim SetUp Acro ba tic mod els are se t up wi th th e CG in suc h a position that the mod el is smoothly stable but w il l s pi n re lia b ly w he n requ ir ed . Th e e leva tor th ro w s ho u ld b e s uch th at a s u ffic ie n tly ti ght lo o pin g rad ius ca n be a tta ine d , but witho ut flicking out. T he a ile ro n throws a re adj us te d to give a roll rate of thr ee rolls in four to fi v e sec o n d s for p ow er p at tern c o m p etiti on s . An d th e rudde r th row sho uld be e no ug h to give cris p ly c o n tro ll e d sta ll turn s . The d ihe d ral is ad justed as d e s crib ed und er Lat eral Sta b ility to g ive n o roll effec t w it h rudd er in put a n d th e w ing a nd tail ma y both b e rig g ed a t ze ro in ci d en c e usin g trim offse t to a ch ie ve le vel flight , or th e w ing may be rigged a t a pos itive inciden ce of ab out half a degree.
A typical pattern model has a low wing, sltm fuselage, generous tail area and large rudder. This one is floum by tbe late "Wee f obu" Robertson, then (1995) cliairntan of tbe Scottisb Aeromodellers Association.
Malcolm Harris's model sboios tbe typical pattern model's ptauform; a medium aspect ratio tapered wing and a long tail arm toblcb assists smooth j/ying toitb generous damping.
84
Basic Aero na utics/o r Modellers
Chapter 16
Special Cases Low Aspect Ratio Aeroplanes De ltas and novelt ies like flying discs or playing cards often have aspect ratios of 3 o r less. Th e lo w aspect ratio wing is often co mbined with a Cana rd or Tailless layo ut therefore that section must be rea d as wel l.
Handling Peculiarities One obvio us res ult of the low aspect ratio is the lack of roll d am pi ng. Th e y can be mad e to ro ll incred ibly q uic kly and tend to be twit ch y in roll. Use sma ll a ilerons wit h little movement , perhaps inse t fro m the tips, or even taileron s. Another character istic of low aspect ratio wings is the stro ng vo rtex th ey genera te , pro du cin g la rge ind uced drag at hig h angles of attack. Th ey tend to lose speed q uick ly in very tight turns or loops. And w he reas most powered mod els have suc h a lo w minimum d rag speed that it is not no ticeable in practice , the low aspect ratio aeroplane has a mar ke d minim um d ra g speed as in Figure 16.1. O n a powered model , whe n you gradually reduce the power a nd fee d in up trim to fly slower a nd slower the re co mes a po int whe re th e model wi ll no lo nger ma intain le vel flight unl ess yo u act ua lly increase the power aga in . It is very difficult to fly it at low speeds becau se its speed is un stabl e . . . a red uction in speed increases the dr ag ca us ing a further speed red uction and vice ve rsa . Th is is known in common parlance as flying "o n the back of th e d rag curve". At moderate speeds they will glide, but try to stretch the glide and they fall o u t of th e s ky, not beca use th e y h a ve s ta lled b ut becaus e the Lift/ Drag ratio has been assassina ted . Deltas and o the r lo w as pect ratio aeroplanes are best su ited to hig h speed fligh t whe re they ha ndle most co mfo rtably. And Figure 16.2 th ey are best sui te d to sma ll d ia met er coarse p itched propellers . The app roac h s ho u ld b e LI' flown at a speed a little above m in imu m dra g speed.
form u lae , pri n cip all y because of th e very strong d ow nw ash over the tailplane . For models without tails see la ter. O n de lta w ings you mus t u se the l'vJ.A.C. worked out graphically as in Figure 7. 11.
Canard Layout In my ancie nt French d ictiona ry, CANARD is a word meaning "d uck" or "hoax" or a "lump of suga r di pped in coffee" . I suppose a du ck's win g is so far back tha t it d o e s rese mb le a tail firs t aerop la ne . Or maybe the Fre nch tho ught it was a hoax! Figure 16.2 shows the forces in trimm ed flight on a canard aeroplane, LF the lift on the fo rep lane, L.\" the lift on the wi ng , Mo the zero lift pitc hing moment du e to wing camber and \ '\1 the we ight. Taking mo ments about Figure 16.1 D
V~TAJ.L
V,\/INDRA G
v
I "
I
I
, I
CGPosition Low aspect ratio aeropl a ne s wi th ta ilpl an es will be ne fit from a mo re forwa rd CG posi tio n than give n by the usu al
Basic Aeronautics/or Modellers
lV(=L)
85
Figure 16.3
. _~_
:"...-t--_
A
t he CG yo u ca n see th at th e foreplane mu st lift to co unteract the momen t o f the wing lift a nd lift so me more to co unteract the Mo of the wing. Th e foreplane is always lifting and its lift coefficient will be greater than that of the wing. It mu st the refore be set at a more positive angle of inciden ce th an th e w ing an d si nce th e forep lane will always have to pr ovid e upward lift it would make se nse to give it a cambe red section .
My formu la for ca na rds. CG position = 0.15 - Vhar gives a CG position aft of the leadi ng edge of the mea n cho rd , as a fractio n of the mea n cho rd. I have red uced the p lanned Stab ility Margin to 10% chord and th e Vhar is the fore plane vo lume ratio , w hich if la rge w ill g ive a NEGATIVE a nswe r wh ic h mean s the CG is in FRO NT of the lead ing edge of the mea n chord . 1 \
Stability No w thi s a ero p lan e ge ts a disturb an ce w hich incr eases its ang le of attac k. See Figure 16.3 in which the lift increa ses o n wing and foreplane, XLw and XLF are sho wn alo ng w ith the resu ltant AL. I have omi tted all the forces o n Figure 16. 2 w hich cance l o ut. The res ulta nt ex tra lift, AL, acts at po int A which is the Neutral Po int a nd w hich is d istan ce x fro m the CG. The aeroplane is stab le if the NI' is beh ind the CG as before and d istance x is the Stabi lity Margin.
CG Position I have been unable to find a fo rmula in textbooks fo r the Neutral Point of a cana rd whi ch le ad s me back to Tbe formula e for CG doesn't toork
text.
Oil
first prin cip le s a n d co m m o n s e nse . I a ssume that the foreplane is sma ll re lative to the w ing. I therefore believe that th e factors w hic h we co ns id e red in Cha pter 8, Le. the effec ts o f wake an d downwash a n d fl e x ibilit y and aspect ratio differences, w ill be sma ll and partly self cancelling .
har
.
IS
foreplane area . w mg area
fo replan e armlj, - - ----''-win g mean chord
X ----'----
BUT, the are a of the forep lan e must include the area in planform, of part of the forward fusel age (see Chapter 22). The CG obvious ly e nds up q uite far ba ck on th e aeroplane. The fore plan e ar m IF is the distance of the forep la ne 's q ua rter chord po int a head of th e w ing 's q ua rter chord point.
Control Eleva tors on the for ep lane o r an all moving foreplane ma y be us ed for pitch co ntrol. O bvious ly th e y mu st mo ve trailing edge down for no se up pi tch . On can ard deltas it may be more effe ctive to use ele von s.
tbis so it 's back to basic principles. See
Tailless Aeroplane - To Achieve Stability Th e re a re tw o w ays to m ak e taill e s s ae rop lanes stable . So rry , I'll rep hrase th at. T here is o n e w a y to m ake an a eropl an e s ta b le and that is to p lace the CG ahe ad of t h e Ne u tr a l Poi nt. Having done that there a re tw o wa ys to ma k e a ta illes s ae rop lane fly in trim . Th e Ne utral Poi nt of a n ae ro p la n e is th e p o int th rou g h w hic h th e resu ltant extra lift caused by a small pitch c h a nge w ill ac t. For a flying wing this po int is the w ing 's aerodynamic ce n t re, b y d efini tion , whic h is at ab o ut 25% of
86
Basic Aeronaut ics/or Modellers
its mean chord which can be found using the gra phical method in Chapter 7. T he CG shou ld be pl aced by tre at in g th e aero p la ne as a can ard without a fore p la ne using th e canard formul a abo ve an d treat in g fus elage area ahead of the lead ing edge if any , as foreplane a rea . If there is no fuselage then Vha , is zero and the CG is at 15% mea n chord.
Figure 16.4
r
NP
To Fly in Trim
w
On e so lu tio n is to use a special aerofoil section called a "reflex sec tion" as depicted in Figure 16.4. As yo u see it is turne d up at the trailing edge which gives it a nose u p zero lift pitchi ng mo me nt Mo (o r if yo u like the Centre of Pressure moves aft as ang le of attack is increased). A normal th in slightly cambered sectio n w ith th e control surface re flexed up w ards works as we ll. The further fo rward the CG is placed the more reflex is needed to co mpensa te, and the slower the flying speed the mor e reflex need ed to stay in trim. Th e second method is to sw ee p the w ings back and bu ild in a hea p of wash out at th e tips as in Figure 16.5. Conve ntional sections with CMO acting nos e down can be used. In my d iag ra m th e inboard p arts of the wi ng a re lifting up wards and th e p art s nea r th e tips are lifting downwards an d th e w ho le thing ad justed so that th e total lift is zero. Th e nose up mo ment from the lift forces mo re than balances the nose down moment from th e wi ng sections, a nd so the overa ll Ze ro Lift Pitchin g Moment is Mo as shown, acti ng nose up. The wing tips act just like a tailpl an e . The more ca m bered th e sectio ns u sed , and the fur ther fo rward the CG, the more washo ut is needed. Both solution s can be used together of co urse to end up w ith a swept flying wing with a little wa sh out and a little reflex on the sec tions.
Control Control is exercised by move ment of co ntrol surfaces at the trailing edg e. The co ntro l surfaces are a bit sh o rt of leverage in pitch co ntrol, but on the other hand, th e
flying wi ng has very little pitch damping (especially if unsw ep t) so not mu ch co ntrol is necessary. It does mean that suc h models can be a little se nsitive in p itch , a nd ca n be a little short of dynamic stab ility. Because th e fin will be so close beh ind th e CG, it must b e ve ry la rge to ac hieve e noug h d irectio na l sta bility. Even then, Yaw Damp ing will be q uite sma ll.
Multiwing Bipl anes, Tripl an es, Qu adruplanes etc. mean drag . Although structurally efficient, all those rigging wires and struts give ex tra profil e drag. All those wingtips give ex tra induced drag. Yo u ge t all the extr a drag from the bits interfer ing wi th each oth er. They even ten d to have big bul ky draggy fuselages, bu t they do have chara cter and th ou gh Boei ng haven't built o ne for man y a yea r, they are a firm favouri te with mod ellers.
Performance Becau se of the high d rag , the Lift/D rag ratio is poor, leading to a fairly stee p glide angle, which in itself is no great problem wh en the engine is running . It just means that approach es are best car ried o ut w ith a little power on, and yo u have to be wary of deadstick landings. It also ex p lai ns the re lative scarcity of b ipl an e th ermal soare rs. Becau se of the low flying speed (us ua lly), larg e fuse lage , and high d rag, most multiwin gs are bes t suited to larg e diameter fine pitched prop ellers. The compactness (re lative to the area) o f multiwinged aeroplanes gives them less pitch and ro ll dam ping. Th at
Figure 16.5
11
Mo
I
I I
-------
I
I I
---
--~
--
--
---------------------
\ \ \
,,
Basic Aeronautics for Modellers
87
a n d th e lightn e s s o f th e w ings ma ke s t h e m q uit e man oeu vrab le if the y have an a ilero n on each wing.
CG Position The fo rmula tor CG p o sit i o n in Chapte r 8 does not ap ply to biplan es becau se o ne w ing has bee n ass ume d in its deri vat ion . But if so me fiddl e fa ctors ar e used perh ap s it can be mad e to give acce p table result s. O ne pro blem is th at the tailpl an e is o pe rating in a ir w hich has bee n slowe d down by its flow aro und the fuselage a nd riggin g , a n d th e o the r is t ha t eac h w ing tip is cre a ti ng vo rt ices w hic h c rea te d o wn w a sh o n th e tailplane therefore so me account must be taken of the numbe r o f w ings . I th e re fore ma k e the fo llo w in g s ugges tio n . Fo r biplanes wit h two identical w ings, let the mean cho rd be the actual w ing cho rd and place it mid way be twe en the tw o wings (and rem ember to allow for swee pbac k). As befo re calculate the total gross w ing area a nd net ta il are a. Mea sure the tail a nn betw een the q uarter cho rd po ints o f the w ing mean cho rd and the tail , and wo rk o ut th e tail vo lume ratio . Now use o nly HALF the ta il vo lume ratio an d HALF the Asp ect Ratio of each w ing in the usua l fo rmula
CG posn
=
0.1 + 0.25 x VI"" x 4j AI{
For tripl an es the ave rage cho rd w ill be the midd le o ne and the factor ed Asp ect Ratio a nd tail vo lume ratios will eac h be a thi rd of the ac tua l o nes. The no mog ram in Figure 16.6 is the sa me as Figure 8.10 but ex te nde d to sma ller values of Vhar to cope w ith this fac toring . Nose le ng ths mor e than o ne cho rd a hea d of the mean cho rd le ad ing e dge , or pa rticul arl y w ide co w lings , o r eng ine nacelles on multien g ined aeroplan es will all have a d es tab ilis in g e ffec t a nd th e CG s ho u ld b e mo ve d forward a few pe rce nt to compe nsa te . For furth er details, including ho w to handle un equ a l wings a nd a wo rke d exa mple, see Chapter 22.
N u m ero u s toingtips, plenty induced drag, steep glide (pretty though isn't it).
88
Basic Aerona nticsfar Modellers
Note: 'factored' means actual ualue divided by number of ioings
.~
30
.35
25
8
.25
.2
------- -----
20
6
18
4
_-..:.-- - - - - - - - - - 16
3 2
15 1
14
.15 factored toing
13
aspect radio
12
.1
CG posttion as a % of toing
.09
Leading Edge
mean chordfrom
.08
Figure 16.6 .07 factored tail
oolume ratio
Basic Aeron a lilies f or Modellers
89
Chapter 17
Reynolds Number M
y ass u mpt io n th a t th e lift dr ag a nd momen t coefficien ts are ind ep endent of airspeed is not exactly tru e (ve ry littl e in life seems to be
exactly true). Th e cri terion w hich de termines wh ether the results of o ne test can be used in ano ther applicatio n is that they mu st be at the sa me Reynold s Number.
Figure 1 7.1
25 0 150 3 0 00
2 00
25
70 60
6 20 00
50
20
50
150 40
40
15
35
1000
35 8 00 30
30
25
25
20 20
15 15 100 80 10
10
60
30
40
25
9 8
8
3 7
15
30 20
2.5
6
6
20
5
15
15
em s. Willg cb o r d
90
5
2
10
15
ins. Re te
iooo»
fP s/mp b kps/mpb Airsp ee d
Basie Aero na lilies f or Mode llers
Os borne Re y n ol d s Figure 17.2 (1842 to 191 2) was a Brit is h scie n tis t w ho discovered t h a t fo r geome t rica lly si m ila r te st s th e flow p att ern w ill be id ent ical if a pa rtic u la r co mb ina tio n of the d imen sion s of the tes t piece and the speed of th e flo w , an d th e viscosi ty and density of th e fl ui d is kept co nstant. For aerodyna mic purposes th e m ag ic combi na tio n ca lle d th e Reyn old s Num ber (usually ab breviated to Re ) is give n by
p c V
Re = - - Jl • • • •
whe re p is the air's den sity c is the wi ng chord V is the a irspeed and Jl is the air's viscosity
R,:
Th e go ing has to ge t reall y tou gh be fore it succumbs to the ev ils of se paration . Lam inar on the o the r hand will give yo u a n e asy time w ith little d rag but , me et a little adve rsity o r obsta cles, and it will skip off an d leave yo u an d yo u know wh at suc h se pa ration me an s - less lift and ve ry high d rag, the stall.
Situation Normal
Tha t eq uatio n ca n be simplifie d to Re = 536.v .c w ith speed in It/ sec and chord in inc hes or Re = 70.V.c. wi th speed in m/ sec and cho rd in mm . The answer has no unit s. It is "d irn e nsio nless". It is just a number wh ose only purpose is to co mpare it with o ther Reyno lds Numbers. An a nswer to the nearest few th ou sand is acc ura te e no ug h. The simplest way of wo rking o ut a Re is to use my no mog ram in Figure 17.1. The influen ce of Reyn olds Number o n aerody na mic properties is irregular. A very simple illustration of th is is the var iation w ith Re of the d rag of a smo oth sphere. As sho wn o n Figure 17.2 the drag coefficient of a sp here is relatively large at ve ry low Re . As Re is inc reased the d rag coefficient grad ua lly reduces and then ove r quite a ra nge of Re re mai ns co nsta n t. Sudde n ly, at o ne pa rticular value of Re , the drag coefficient d rops to less tha n half its previou s steady value. Fur the r inc reasi ng the Re produ ces no mor e cha nge in dr ag coefficie nt. Rou gh en ing the surface of the sp he re reduces the Re of the sudde n drag reductio n . The step moves left.
In The Boundary Layer Th e scene of the ac tio n in th is part of the story is the laye r of air right next to t he aero foi l s urface . Figure 17.] Whe n air mo ves over a s u rface th e b ou nd a ry layer may be o ne of two ki n ds, lam in ar o r tu rb ul ent. The turbul e nt b ound a ry la yer g ives mo re d rag but is a mo re d e p en d abl e so r t of bo undary la ye r w hic h w ill s tic k b y yo u thr ou gh th ick and thin .
~
The nor mal situa tio n is for the boundary layer to start o ff lam in ar and the n some di st an ce b a ck fro m th e le ading edge it be com es a turbulent bo undary layer and rem a ins so to the traili ng edge . Th e point at which it c ha nges fro m lam in a r to tu rbul ent is ca lle d th e "Transition Po int ". Figure 17.3 shows a normal tran sition from a lam in a r to a turbul e nt bou nda ry la yer . T he turbul ent bound ar y layer is thi ck er th an th e Iarnina r bo undary layer w hich it replaces .
Laminar Separation Wh en the air flows over the aerofoil, its pr essu re on th e u pper su rface re d uces to a m inimu m , and th e n incre ases aga in to no rmal a t the tra iling edge . Figure 17.4 shows that as the larn ina r bo unda ry layer flows into this area of increasi ng pr essure its alrea dy slow progress is broug ht to a halt by the steadily incre asing pr essure w hic h it is me eting. Air co n tin ues to flow in to th is region from th e lead ing edge and so of co urse a "lump" of stationary air builds up , the strea mlines of the main airflo w are forced to se para te from the w ing surface . and th e wi ng has sta lle d . As I sa id in Cha p te r 3 th e sta ll usu ally starts near the trailin g edge but as an gle of attac k is increased the se pa ration point mo ves rapi dly forward . Th e drag increase due to se paration is very grea t.
/
Transition Point
~e -. ...........--.
"
-------------~~--\ Transition P--;;;llt
Basic Aero na utics fo r Modellers
91
Figure 17.4 Flow Getting Slower
Streamlines Separate from Surface
~~~~ 11 \\
Minimum Pressure Point
-'/
\\
~
11
l'
'11
~ 11 \\ ~
-'/
\\
~ /I -'/' 11 1\ 1' -;:::-
If -;:::- \\ ~ II ~ II -;:::- \\ II
Area of
/ Turbulence fl ""
~ ""~ ~ 1' -, If
\\ II .;
"" 111\
bubbl e ma y fo rm o n th e unde rs ide a t low a ng les o f attac k.
Separation Bubble At mod el Re it ofte n happen s th at shortly after the laminar boundary la ye r sepa ra tes fro m th e aero fo il surface, it tran siti on s into a turbulent bounda ry layer. The con sequent thickening allows it to re-attach itself to th e s u rface le a vin g a small p o ck et of s tag n a n t a ir tr app ed against th e ae ro foi l ca lle d a "Se p a ra tio n Bubble ". Figure 17.5 sho ws an aerofoil w ith a se pa ratio n bubble . Within the se para tio n bubble a ge ntle rotational flow develops as shown, du e to the pressur e distribution and the visco us forces. As ang le of atta ck is increa sed , th e minimum pr essure point will move towards th e leading edge and the se paration bubble will go w ith it. T he p o int w ill co me when ei the r trans ition will not occur , o r o ccurs to o lat e for th e tu rbulent bounda ry layer to re-att ach itself. Th e flow se parates co mpletely, the win g stalls, and the se pa ration bubble is said to have burst. The flow will then look like that in Figure 17.4.
The Underside On the lower surface of the wing the boundary layer starts off, as always, lamin ar. At high Re it will tran sition at so me stag e into a tur bul ent boundary layer with the atte ndant high er drag, but at low Re there is no good re ason why it sho uld be com e turbulent and wh en the pr es sur e is reducing over the rear porti on of the section, ther e is no reas on fo r it to se parate e ither. It is ho we ver possibl e that o n underc ambe red sectio ns a se pa ratio n
The Influence of Reynolds Number on Aerodynamic Data The e ffec t o f redu cin g th e Re o n th e aerodyn am ic prop erti es of w ings is usu ally to mak e them worse . From a ve ry high Re of 10 million down to a Re of 0.5 million nearly all sectio ns work pr ett y well. As Re is red uced w ithin thi s ran ge , as a ge ne ra l rul e , profil e d rag incre ases slowly but ste ad ily , a nd th e sectio n 's CLm " , red uces gradually. At very low Re , say 10 or 20 thou sand , most sections will hardl y w o rk a t a ll, givi ng a di sappoint ingl y lo w Cl m " , a nd very high d rag sugges ting laminar se paration o n the top surface.
The Problem Area Some w he re between thes e two ex tremes eac h section seems to have what yo u might ca ll a "Critica l Re Band " a bove which it o pe ra tes quit e normall y and be lo w whi ch it is virtuall y useless. W ithin th e c ri t ica l Re b and th e aero dy na m ic prop erti e s o f th e sec tio n c a n var y drasti c all y a nd sudde nly as illustrate in Figure 17.6 whi ch is fo r a typ ical mod ern section . Not all sectio ns show the same patte rn of variation. At a Re of 200,000 the lift curve and the dr ag polar
Figure 17.5 Laminar B.L.
Separatton Poln1 /
92
Basic Aero na uticsfor Modellers
Figure 17.6
-- . ---....
...
I
~
1 00,000 -
/ :' •
• ;(.::.• 100,000 •
-.... ...........
//.
<,
~
• :85,OOO! I
....
i 80,000,.....
80,000
--
•
,e -:..---- --
T D
:
z :
85,000
.
A. --- • <,B
200,000
200,000
/~_ . ~
/'
60,000
./
..... r ::
.... -.:
..:/" , , I
\
, . I
\ I
\
....
....
are bo th qui te normal. At this Re an d above almos t "full size" performan ce sho uld be achievab le. At 100,000 the d rag ha s more than doubled over most of the ran ge and the lift has reduced at eve ry a ngle of attack . Th e zero lift a ng le of attac k has reduced by a degree or so , but the curves are both a normal shap e (apa rt fro m that odd bit around the sta lling a ng les of a ttack). Th e pe rfo rmance ava ilable in terms o f g lide an gle and d u ra tio n will be reduced , but at le ast th e handling will be fairly norm al. At 85,000 the curves are fairly normal for low ang les of attack. At interm ed iate an gles o f attack lift produ ction falt ers associa te d w ith a sha rp rise in d rag indicating trou ble in the bounda ry layer , and prob abl y produci ng
handling peculia rities , but at high er angles of attac k the d rag red uces to acce ptable lev els and the lift ge ts back o n the job suggesting a turbulent boundary laye r ge tting thin gs stuck back down over the top surface again. However at 80,0 00 there is no suc h d ramatic rescu e . Lift produ ction star ts off ba d and ge ts stea d ily wor se . Dr a g is e normo us and th e Lift/Drag ra tio has be en assassina ted . A revolution has tak en place and it too k o nly a tiny change in Re to effec t it. Performan ce is ve ry poor ind eed and as for handling , who ca res anyw ay? At a Re of 60 ,000 th e det erioration is even worse . Thi s is bel o w the critical band whi ch I menti on ed and th e sec tio n is us el e ss for g liders. Flight w o u ld b e possibl e with e no ugh e ng ine pow e r. Figure 17.8
Figure 17. 7
v
•
+ + +
,- - -,
J---1-e D
Low lVeigbt Higb lVeight
Basic Aerona utics/o r Modellers
93
The Hysteris Loop Figure 17.7 shows the lift curve fo r a Re of 100,000 repeat ed from Figure 17.6. No tice th e "ex tra" po rtio n und erne ath . As a ng le o f a ttack is in c re a se d th e lift coefficien t fo llows th e top cu rve all the way u p past point A and o n to B. If, at any poi nt be fore B is reach e d , a is red uce d agai n, then CL follows the sa me line back d o wn . Ho we ver if a is increased past B then th e CL d rops abr up tly, do wn to the lower line at C. At the sa me time the d rag increases enormo us ly, so muc h that it goes rig ht o ff the gra p h. So mewhere in th e bound ary layer th e flow has se para ted dr astically. If a is now decreased the flow stays se pa ra ted and Cl. foll ows the lowe r line to D. Th e n so me how the flow re- att ach es itse lf, Cl. zooms back up to point A o n the origina l line , the associa ted dr a g retu rn s to a m o re reasona b le a mo un t, an d everything is back to no rmal. Thi s "o ne way sys te m" o n a grap h is ca lle d a "Hyste resis Loop" an d o ften occurs in the critica l Re ba nd . It is likely to have stra nge effec ts o n the handling o f the mod el aro u nd the sta ll. O n a ro ug h, d irty o r inaccurate ae rofo il the lo op will disappear and revert to the lowe r line , or even the line fo r Re o f 80,0 00.
Effect on Model Design and Performance Th e Re ban d within whic h a sec tion's performan ce will be se rious ly affected depends up on its thickn ess, and its ca mbe r. Ge ne rally, the more the thickn ess o r the mor e the ca mber, the h ighe r the Re at wh ich the sec tion sho uld be ope rated . For example, the \X1or tmann section FX60126 (ca mbe r 3.9%, thickness 12.6%) has a similar dr ag polar at a Re o f 60 ,00 0 as d o e s th e FX63 - 137 (ca m ber 6% , thickn ess 13.7%) at 100,000.
Wing Tips Th e wi ng tip o n a tap ered wi ng is flying at the sma llest Re. Th e bes t taper rat io ma y be mor e than the theor et ical optimum of abou t 0.45 becau se reducing the tip Re co uld increase profile drag more than the redu ction in induced dra g ac hieved b y a p p roxi ma ti ng an e llip tica l wi ng loadi ng. In add ition , the tip sec tion will ge t into its critical Re ba nd first as the aerop lane is slowed down, and Re is redu ced . Ind eed , because the lift loss an d dr ag rise can occur qu ite sud de n ly over a small cha nge in Re , it may be that it w ill hap pen on one side before the other showing all the signs of a tips tall. We cou ld cha nge the tip sec tion to o ne wi th less ca mber and thickn ess to raise the critical Re o f the tips. However that goes agai nst the adv ice in Chap ter 7 (F ig 7 .10) to increase tip ca mber to avo id tipstall. It depends a great deal on Re , h ut read o n a few paragraph s.
Class Rules Competition rul es can radi call y influe nce d esign . A large aspect ratio is usu ally good for performa nce , b ut if fo r exa mple the rules limit the wings pa n, the n a h igh AR wi ng wo uld have a very sma ll chord an d a sma ll w ing area and therefore a high wi ng loadin g. Th e problem with a tiny chord is a very low Re , at w hic h most sec tions w ill n ot perfor m a t a ll. T he o u tco me is th at g lide rs compromise o n a lowe r AR giving a large w ing area, and
94
hen ce light load ing , cou pled with a reasonable ope rating Re.
Optimum Weight In Cha p te r 13 I sai d that ball asting a glide r wou ld a lways give th e sa me best glide ang le, a nd a h ig her mi n imu m s ink rate , a nd I illu str ate d it in Figure 13.9, w hic h is rep rodu ced as Figure 17.8. Howeve r now I have ass umed th at th e aerofoi l w ill not work bel ow a Re corres pondi ng to speed "V". Co nseq ue ntly the p oints w he re the theor etical best glide ang le and the minimu m sink a re not ava ila ble . No w th e best glide an gl e a nd minimum sink occur at the high er weight. I am no t sayi ng that ballastin g is likely to improve a glide r's pe rforma nce , just tha t it is possible in certain circ ums tances.
Turbulator Strips Turb ulato r strips are very sma ll steps o r irregul arities on the surface of the wing , usu ally between 5% and 25% of the chord back from the leading edge on the up pe r surface. The ir purpose is to make the lami nar boundary laye r become tu rb ule n t be fore th e no rm al se pa ra tio n poi nt, in the hope that it w ill th en not separa te at all. Thei r use is no rma lly restr icted to g lide rs or free fligh t dura tion mod els, and es pecially o n o uter panels. Th e effec tive ness of the te chnique is illu strate d in Figure 17.9. The effect o f o ne strip of adhesive tap e 2 mm w ide and 0.5 mm thi ck at 20% chord is shown by the lin e s wit h cross-ba rs . At a Re o f 60,000 the sectio n 's performance has been tran sformed . It has a res pectable Lift/D rag rat io agai n , rat her th an a n ab ru p t sta ll, but normal han dlin g othe rw ise . Befor e yo u rush off to stick tape all over your wings, rem ember that this is jus t a br ief mention o f a sub ject w hic h co uld take up a book in itse lf. The tur bul ator sho uld not be regarde d as a pan acea to be used indiscrimina tely o n all w ings in all circu ms tances. It can have an amazing effect o n so me sec tions at ce rtain Re but tur bulators ca n have an adverse effect. Notice that at a Re of 200,000 the tur bulator strip has a det rimental effect, causing an earlier stall and mor e drag . Also ther e is a lower limit to the Re at w hic h they wi ll wor k.
Surface Finish It is noti ced in practice that a w ing with a ro ugh or un even surface finish so me times has a bett er pe rformance tha n a perfectly smooth wing . The rou gh texture , o r irregul arities in the surface, will turbulate the boundary layer just like a turbul ator strip . However it is indiscrim ina te in its ac tion . Altho ug h bu ildi ng in tur bu lence co uld so me times help, yo u may find that so me of it, or even all o f it, is having an adve rse effect. I suggest that it would be better to buil d the wing with as acc ura te an aerofoil, an d as smooth a surface , as possib le. Th e n yo u can ex pe r ime n t wi th s tick o n turbulator strips w hic h ca n be moved or rem oved at w ill.
Using P ublished Data Th ere is little point in loo king at aerodynamic da ta on w ing sectio ns unl e ss it is for th e co rrect Re , and yo u ca nno t interp olate grap hs . If yo u wa nt data for a Re of 150,000 bu t have gra p hs o n ly fo r 100,000 and 200,000 ,
Ba sic AeronauticsforMode llers
Fig ure 1 7.9
/
.-
/
/ /
.-
/
.-
/ /
/
/
/
a
R e 2 0 0, 000 Clean
I· ···H··+· ··I
R e 2 00, 000 Turbulated Re 60, 000 Clean Re 60, 000 Turbulated
yo u cannot just draw a line halfwa y between the two. The graph for 150,000 cou ld be anyw he re between th e ot he r two.
Theoretical Data In the world of full size aircraft, pe rformance can be p redicted with amazing accuracy by computers, though the test p ilot a lways has th e fina l say. Drag po lars for mo de l aerofo il se ctions at specific Re can also be worked ou t by computers . How accura te the res ults are is open to deba te . Th ere is often a good correlati on w ith wind tunnel tests at Re above the critica l band but wit hin it th e comp uters seem to get a bit op timistic.
Further Reading "Model Aircraft Aerodyn amics" by Martin Simons gives a m u ch more detailed a c cou nt of th e prob lems of aero fo ils at low Re . "Pro filp o laren fur den Modellflug " by Die te r Altha us and "Airfo ils at Low Speeds" by Selig Donovan & Fraser contain resu lts of ma ny tests of useful section at relevant Re .
Basic Aerona utics/or Modellers
95
Chapter 18
Aeroelasticity
s
o far I have been working on the assumption that the struc tur e of the aeroplane is co mple te ly rigid and will not d eform at all un d er the ae ro dy na mic load s. Th is as su mp tion is pr etty un iver sal in that nearly all the books and articl es o n ae rody na mics w h ich you are l ik el y to co me ac ross w ork u n de r thi s sa me a ssumpt ion , even th ough th e y d on 't a lw ays say so . However it has to be ad mitte d that no structure is to tally rigid so sh ould we th rowaway ae rodynamic theory as we know it? Have yo u co m p le te ly w ast ed yo ur time re ad ing abou t aero dy na mics u p to thi s stage? I don 't thin k so. Most of the time this assumpt ion is perfectly jus tified and ca uses no me asu ra ble error. As long as yo u are aware o f it, yo u ca n ign ore the flexibility o f the struc ture until it causes p roblems . The re are ce rtai n a reas in which struc tura l flexibility is kn own to ca use undesirabl e effec ts and I int end to me ntion them briefly.
1. The Effect on Stability Wh en I ex p lai ned Long itudin a l Sta tic Sta b ili ty , I d escrib ed a n ae ropl a ne meeting a d istur ba nce w hic h increases the ang le of atta ck, a nd so also the lift, of its wing an d its tail. The lift in creases both ac t at ab ou t the q uart er c ho rd p oints of th eir resp e ct ive s u rfaces , w ha tever th e sect io n . The res u ltan t o f th e tw o lift incre ases ac ts thro ug h a po int ca lled th e NEUTRAL PO INT a nd its mome nt a b out th e CG p ro v id e s th e sta bilising mo me nt on the aeroplane. Th e distance of the CG ah ead o f the NP is a measure o f the stab ility of the aeroplane. Now imagine w ha t th e effect wou ld be if th e ta il we re mo u nted at the end of a lo ng flex ible tail boom. Th e lift increase on the tail would bend th e tailb o om u p (as in Figure 18 .1). The tail 's angle of att ack is reduced so th e lift inc rease w ill no t now be quite as b ig as if the tail had been rigid . Co nseq ue nt ly th e resu ltant fo rce
thr ou g h the NP w hic h stabilises th e aerop lane w ill be further forward , i.e . nearer the CG, than was calc ulated using rigid theory . The ae roplane has a little less stab ility than was calc ulated.
The Torsional Axis of the Wing Wing flexibility can also have an effect. As I am sure you ca n imagine, an upward force o n the lead ing edge of a w ing will be nd it upwards AND tw ist it trailin g edge down . Similarly an u pward force at the trailin g e dge w ill twi st it lead ing edge down . So mewhere be twee n th e le adi ng an d tra iling edges there is a line ru nning from root to tip ca lled the "tor sio nal axis" (o r flexura l axis or e las tic axis) throu gh w h ich a force w ill jus t ca use bending, wi th no tor sion . It is the line about w h ich the structure tw ists . Th e po sitio n of this line will depend u p on th e wing 's s tr uc tu re, b ut w ill b e somewhe re between say 15% and 50% of th e chord fro m th e leadi ng edge.
How Tw is t Affects Stability The lift increases du e to a d isturbance w ill ac t at the qu a rt e r chord p o int a n d if the tors io na l axis of the struc ture is we ll aft, it w ill tend to twis t the w ing lead ing edge u p (see Figure 18.2). The w ing 's lift inc rease w ill therefore be a little larger than forecast and the NP w ill be fur ther forw ard than calculated for a rig id wing. The w ing's flexibili ty in torsion has tak en away so me o f the aeroplane 's sta bility. Th e proble m is at its worst if the tor sional axis of the wi ng 's structure is well aft and the wi ng is ve ry flexibl e in torsio n. The wi ng sectio n d oesn 't matt er. The wors t cas e is p ro ba b ly a comple te ly ope n struc ture wi th just le ad ing and trailing e dges, main spars at 30%, and aft spa rs at 70%, covered in p las tic film. A symmetrical sectio n is jus t as affec ted as any o ther.
Figure 18.1 XL
XL
(fIe.~) ( l"ig ld)
- -- -- -- --
Datum - line NP
96
{flex]
Basic Aeronautics/or Modellers
The Solution The CG for mulae in Cha p te r 8 ass u me a fairly rigid structure. The forward shifts of the NP mention ed above (not usu all y s ig n ifica n t ex cept in the case of h igh aspect ratio w ings) ca n be co mpe nsa ted fo r o n fle xi bl e mo d el s b y starting off with the CG slightly furth er forward th an yo u h a ve ca lc u la te d. How mu ch d e p end s up on yo u r assessment of the flexibil it y o f yo ur mod el 's win gs and tail (say 2 to 5 per-cent ").
Figure 18.2
2. Aileron Reversal
XL Flexible
XL If'
Increased by twist
---------
XL Rigid
- -- - -- ------
\
Figure 18.3
'
t
XL (Aileron) Wh en a n a ilero n is d efl e ct ed d o wn , the ex tra lift ac ts we ll back =-=-=--=--==----=--=-=-::-:-~ o n the wing. It is likely To r si o nal to act aft of the torsion al / Axis axis of the struc ture in --which case it will tw ist the wing le ading edge down reducing its ang le of a ttac k and hen ce its lift (see Fig ure 18 .3) . The redu ctio n in the lift XL (Twist) mi ght turn out to b e mor e th an th e incre ase in lift from the a ilero n deflection so the w ing as a whole spectac ular whe n it happen s. Figure 18.4 represe nts an may lose lift causi ng a rolling mom en t in the o p posite aeroplane in normal flight. Th e torsion al axis of its w ing is qu ite fa r a ft, say 40% o r 45% o f the cho rd . So me dire ct ion to th at intend ed . For si m ilar re as ons th e d isturbance ca uses the wing to twist very slightly lead ing upgoing a ile ro n ma y ca use a n incr ease in lift o n its w ing . The ailero ns seems to have o pe rated in reverse, edge u p whi ch will incre ase the win g lift at the q ua rte r hen ce th e ex p ressio n "Aile ro n Rever sal ". The furth er cho rd po int. Th e wing 's torsional stiffness w ill try to tw ist it ba ck to its o rigina l position , an d no rmally does, forwa rd the torsional ax is, an d the weaker the struc ture is in to rsi on , th e m ore prone it will b e to a ilero n but the ex tra lift, XL, is trying to tw ist it eve n furth er reve rsa l. Th e wing section used mak es no differ en ce at leading edge up . The restorin g moment from th e wi ng 's stiffness is all. co nstant, reg ardl ess of speed , but the ex tra lift increases Becau se the twistin g mom ent o n the w ing rises as the sq uare of th e a irspeed , but th e stiffness do esn 't , th e as the sq ua re of the airsp eed . If the aeroplan e is flown a ilero n reversal will ge t wo rse as speed rises. In fact at fast enoug h th is latte r mom ent will w in. It will mak e the o ne p art icu lar s peed th e a ile ro ns will have no e ffec t wing twist leading edge up inc reas ing its ang le of attac k eve n furth er , wh ich inc reases the ex tra lift even more , whateve r. As speed rises to wards this critica l speed the ailerons gradually lose the ir effectiveness , whil e ab ov e it whi ch in c rea ses th e twi st even more a nd th e w ing they will ac t in the opposite se nse to that inten ded . This rap id ly twi sts le ad ing e dge up until it e ithe r sta lls or critical speed is called the "AILERO N REVERSAL SPEED". br eak s in bend ing o r twists right o ff. Th e initial slight Ailero n Reve rsa l is a speed dependent pr obl em . All d isturban ce rapidly be com es worse and w orse whic h I sa id in the cha p ter on Stability is called DIVERGENCE. aero planes w ith ailerons wi ll have an aileron reversa l speed , the trick is to mak e it faster then the ae rop lane Wing divergence usu ally o nly happen s in a stee p d ive ca n fly. Th is ca n be d on e by incre asing the to rsion al and ca n a ffec t w ings w ith ca mbered, sy mme trica l, or stiffness of the win g and/or by having the tor siona l ax is reflexed sectio ns. It is a speed dep endent probl em , i.e. not too far forward . there is a critica l sp eed called the WING DIVERGENCE SPEED above whi ch it will happen . The struc ture most at risk is a high as pect ratio open structu red wing w ith 3. Wing Divergence just a main spar, aft spa r, and lead ing and trailing edges, cove red in pla stic film. Win g di vergen ce is a n un common p ro bl em , but
-- -
- ----
Basic Aerona utics/or Modellers
97
Wing di vergen ce ca n be avo ide d by mak ing the wing stiff in torsion , mo ving the to rsio nal ax is forward nearer the aerodyna mic ce ntre o r sweep ing back the wings . If the w ings are swe pt back the lift increase near the tips ten ds to twi st them leadi ng edge down whic h helps to avoid the problem .
div e, co ming in for a hig h speed pass wh e n yo u hea r it. BRRRRRRRRRR BANG! a nd o ff co mes a p ie c e , an ailero n, an elevator, the tailp lane , o r the rudder. I have even heard of a case where the vibration was so severe that the w ho le wing dis integra ted in a shower o f w hite foa m beads . Not a p retty sight! Fl u tter is a viole n t osci llation of p a r t o f t h e aeropla ne 's str ucture, often, but not a lways, lead ing to 4. Flutter failure of pa rt o f that str uc ture . There a re many b its Your mod el is really goi ng we ll, full pow er, a bit of a whi ch can start to flutter and I sha ll br iefly me ntio n the common o nes . \X'hate ver it is that is flutt erin g, it is Figure 18.4 a s peed d ep enden t problem . That is there is a c ritica l s pee d (c u nXL From twist due to n in gl y ca lle d th e FLUTTER SPEED) above which that bit w ill start to flu tte r. If yo u do XL From twist due to hap pe n to ge t nutter at an y time, cut the throttle XL From initial twist an d climb im med iat el y ....... ............. ............ Original L (if not sooner) . ' Le t m e s ta r t wit h '. a ile ro n fl utt er w h ic h ". '. w rot e o ff o ne o f m y - - - <, ...... m o d e ls so me ye a rs ~. ba ck , and caused me to ............. lo ok u p th e th e o r y . Th e re are tw o mod es of ailero n flutt er, be nd ing ......7"..: -:-.. ::::.:..: :: ....... "'0:::-:.-- ....... and torsion . ..... ......... <. --Th e y have the same ' .. ........ : .. re su lt a nd th e same remedy an d m a y eve n occ u r to g eth e r but I s ha ll de scr ibe th e w a y they happen separate ly, Figure 18.5 starting with bending. So me thi ng, ma ybe a g us t, s ta rts bot h w ings be nd ing up . See Figur e A 18.5. The bending stiffc ness of the wings stops their u pwa rd mo tio n at Inertia p o s iti on A b u t th e a ile ron s, bein g attache d B b y s lig h tly fle xi b le li n k a g e s , d on 't s to p . c The ir CG is behind the hinge lin e a nd so the ir in erti a ca rr ies th e m a litt le furth er u p to positio n B. In th is pos itio n t he ai lero ns c ca use a loss of lift w hic h let s th e wi ng 's stiffness bri ng the m down , past thei r o rigina l position, to po si tion C w he re th e ir D in ert ia m a k e s th em d roop a little to position D. In thi s position they inc rease th e lift of th e w ing a n d it s ta rts Inertia co ming up aga in. When
t
t
t --.
--
.....
..
-- - --
--=--.."
--.
98
Basic Aeronautics/o r Modellers
Mass balance bonis visible under the uiings oftbis typical trainer style model 011 floats. the win g a t last s to ps go ing u p aga in, th e a ilero ns overshoot to positi o n B agai n a nd th e who le p rocess repea ts itself.
The to rsion al mod e of flutter sta rts whe n an a ilero n's position is d isturbed . In Figure 18.6 the ailero n has been d isturbed do wn ward slightly in position A. In Cha pter 10
Figure 18.6
A--+-----~m--=--:-----=--:---;--------
TOI'sIOlUlI Axis
Drooping aileron makes toing twist about its _______________ tors1011a I a xis.
Inertia
-----~
Tbe aileron's inertia carries it further lip.
Aileron's in ertia carries it
_....I..:::-E
~¥t_----------=:::..--===------~II!.:'..:.·t.!"b~e~I_: · d~owll tobicb will make it twist 1I0se doum again . . .
Inertia
Basic Aeronauticsfor Modellers
99
Figure 18,7
Mass ba la nce --...-:: insid e Aero dyna mic
3
balance
I
1
]
2
8
Set back binge line
-:
Set back binge
-...SlI·ipof solder o r p iano uiire
o L----'
--~
5
9
elm be u se d 01'
OIl
elenator,
ailel'OIl L
J.----.l
rndder.
I sai d th at a d o wn g o in g a ilero n in crea s e s li ft and increases Mo th e no se down mo ment. This nose down moment twi st s th e w ing tr ailin g e dge up a b out its tor sional axis. \\! hen th e wing 's tor sional stiffness sto ps th e twi sting, p o siti o n B , th e a ilero n ov e rsh oot s t o a n up w ard d efl e ct ion , p o siti o n C, b e cau se its CG is b eh in d th e hinge lin e . In th is p ositi on the moment o n th e w ing is reduced , it untwists , a nd w he n it sto ps in positio n D , th e a ile ro n 's inertia again ca rries it th rou gh to p osition E. We are back where w e started a nd th e whole cy cle be gins again.
Damaging aile ro n flutt er is not inevitab le though , the disturbance u sually d yin g out du e to damping. Ho wever, as I hav e alr ead y sa id, th e angular deflections depend on st iffn es s (whic h is the sa me at a ny spe ed) but th e for ces th ey produce inc reas e with speed sq ua red . If the ae ro p la n e is fly in g fa st enough , a bov e its AILERO N FLU TT ER SP EED , the wh ol e pro ce s s wi ll b e se lf p er p etuating and ve ry rapi d ly de vel op into a po tentially da maging vibration . Th e pilot mu st immed iate ly kill the s p ee d by closing th e throttle , increa sing the dr ag a n d pointi ng th e ae ro p la ne up . This is th e most co mmo n fo rm of flutter o n model aeroplan es .
Curing the Flutter Fro m the above two descriptions yo u ca n see th at s tiffe n ing the w ing in b ending and torsion will he lp b y redu cin g the amount that th ey flex up a nd down or twist and s tiffe n ing th e aile rons and their linkages will help by
100
reducing the amo u n t th at th ey will flap up a nd down, but both th e se soluti ons w ill just inc rease the flutter s peed , n o t remo ve it. All aerop la n es, h owe ve r s tif f, howe ver pe rfe ct their co ntro l linkage s, will have a flutter speed and th e pilot must ensure that he ne ver, ev er, flies above th at sp eed . There is an o ther sim p le so lution, used o n pra cticall y all full size aircraft since a bo ut the th irties . Th is so lu tio n , known as MASS BALANCING , is ac hieved by moving th e h inge line of th e aile ro n back or fixing w eights to the a ile ro n a h ea d of it s hing e line or a co m b in a tio n o f the two as in the exa mp les in Figure 18. 7. If the CG o f th e aileron is ex ac tly on its hinge line, its inertia ca n no t m ake it o vershoot th e position h eld b y it s co n tro l link ages a nd flutt er is eliminated . Su ch perf e ct b al an cing is u nn e c e s sary , a p artial ba lan ce being e noug h to raise th e flutt er s peed out of rea ch . The mass b al an ces ma y b e hi dd en in side th e hollow w ing tip as on th e Piper Ch erokee . If th e co ntro l surface has a n ae ro d yna mic balance (s o me a rea a head of the hi ng e line ) to reduce th e contro l for ce s , thi s is oft en a co nve nie n t lo cation for some ma ss balancin g as w ell. Beware though . If the mass is a tta che d to th e co n tro l sur face by a flimsy pi ece of wir e , the wire will bend a nd the m a ss will re ma in stationary a nd be co m p le te ly ineffective as in Figure 18.8.
Wing Flutter I b e t someone has thought of a nea t wa y to e lim inate
Basic Aeronautics/or Modellers
Figure 18.8
c
---~~-- • • • • • . . .. . .. :- # ..
a ile ro n flutt er , lea vin g o ff th e a ile ro ns . So rry to d isappoint yo u but yo u ca n still ge t w ing flutt e r. It is unco mm on I th ink, but it happens w hen a wing has its torsio nal ax is we ll ahead of its CG. If the wi ng in Figure 18.9, position A, is disturbed to give it a sma ll ang le of attack increase, the ex tra lift will make it bend upwards. Wh en its b e nd in g s tiffness s to ps it s up w ard mo vement the CG , be ing behind th e torsio na l a xis , overshoots to positio n B ca using a reduction in ang le of attac k es pecially towards the tip . Th e loss of lift allows th e wi ng to dro p to position C where its inertia has twisted it lead ing edge up aga in. The cycle repea ts e tc., e tc . and if th e s peed is h igh e n o ug h beco mes se lf sustaining flutte r. Th e mot ion is a co mbina tion of be nd ing and torsion. This wing flutte r ca n happen to wings wi tho ut a ilero ns, a nd it ca n also ha ppen to wings wi th aile rons , even perfe ctly mass bal an ced ail e rons. Inde ed th e a ile ro n mass bal ances can ma ke wi ng flutt e r w o rse as th e y mo ve the CG of the who le wi ng aft. Wing flutt e r ca n be cured by mo ving the torsio nal ax is of the w ing aft and the CG o f its struc ture forward.
co ntrols, hin ge d at its q ua rter chord poi nt but with its CG co mmo nly nearer 45 per ce nt ch ord . Cut light en ing holes aft of the hinge an d add lead to the leading edge to mo ve the CG fo rwa rd to the hin ge line.
Footnote With regard to flutter, there are man y report ed cures, a few of wh ich run co ntra ry to the theory. Perh aps so me of what is d iagn osed as flutter is re ally so mething e lse, like a sy mpa the tic vibra tion. By a ll mea ns try on e o f the o ld wives rem ed ies, it just might work, bu t first ask yourself o ne qu estion , "Do I feel lucky?".
Tail Flutter I ex pec t yo u h a ve guessed that if the tail is flexible o r is mounte d o n a fle xibl e fuse lage then flutter of the other co n tro l s u rface s , th e rudder and eleva to rs, is li ke ly at hi g h spee ds . You are right. Altho ug h eleva tor o r rudder flutter is le s s c o m m o n o n mod el s th an ail e ron flutt e r be aw are th at it ca n ha pp en . Secu re fi x in g of th e fin a nd tailpl an e to the fuse lage is essential. Mass balan cing of the elevator and ru d de r wi ll c ure th e problem and is standard pra ct ic e o n full s ize ae ro p la nes . Th e sa me thing ca n happen to an All Mo ving Ta il (or fore plane) , with flexible
Figure 18.9
Basic Aeronautics f or Modellers
XL starts to bend unng up View from front A
Tip
Root Inertia
B
Force from bending stiffness stops upward motion
Loss of lift due to twist lets uiing descend
Force from bending stifness stops doumuiard motion
c Inertia
~ Lift increase from twist starts wing going up again
101
Chapter 19
Tuck Under o me years ago, I was test flying a high as pect ratio pow ered mod el of my ow n design . Th e low speed hand ling was fine so I open ed the throttle fully to see how fast it would go . As th e aer oplan e began to acce lerate the nose started to rise so I app lied four clicks o f down tr im to maintain le ve l flight. As th e sp e e d increa sed furth er it appeared that I had o ve rdo ne the down trim a nd I had to apply four clicks of up trim. Even that wasn 't eno ugh as the nose co ntinue d to drop. I was back to the trim position for level flight at low speed and here it wa s still putting its no se d own. It co uld no t be ca use d by excessive downthrust as the initial tendency had been nose Up. Then I realised . This was it! This wa s the dread ed Tu ck Unde r. I found that , starting from trimmed flight at whatever speed, if I applied down e leva tor to start a dive , there ca me a po int whe re the model just ke pt stee pe ning the d ive of its own acco rd , eve n wh e n I re turn ed th e elevator to its or igina l position. The slower the aeroplane was !lying to begin with , the more di ve it ne e ded to make it tuck un de r. It w ou ld even do it fro m gliding fligh t, bu t each time it re co vere d w he n I closed th e throttle and applied full up eleva tor. Until that is, on e day I mad e it tuck under with the rate swi tches ina dve rte ntly se t at low . This time the d ive was te rminal.
S
Fig u re 19.1 Tail Angle to Trim
The Villa i« Unmasked Th is incid ent made me think in so me depth about the probl em . Figu re 19.1 shows the tailp lan e an gle to trim for the full ran ge of lift coefficients, from Chapter 12. If the aeroplan e is s peed e d up a n d th e lif t coe fficie n t is reduced , to keep lift equal to wei ght , then basic theory says that the aeroplan e will be trimm ed tail heavy and d own e le va to r trim m ust b e a p p li e d to k eep the aerop lane in trim . Failure to put on down elevator means tha t the aeroplane will zoom nose up . Th e greater the Stability Margin, the ste eper the graph and greater the nose up mom ent will be . There is no way that the usua l Sim p lifie d th eory o f ae ro dyna mics co u ld a llo w tu ck und er to happen . Since I had just pr o ved that it does happen th e a ns w e r has to b e th at o ne o f th e three standa rd ass ump tio ns of simp le aerodyna mic theory has le t us down. It ca nnot be Compressibility, it is un likely to be Reyno lds Numbe r, so it must be Flexibility. If dis tor tio n of th e aerop la ne 's struct ure e ffect ive ly red uced the ang le be tween the w ing a nd tail just like whe n yo u apply down elevator, then like down eleva tor it would ca use a nose do wn moment , a nd if it w ere severe eno ug h it co uld overcome the restori ng mom ent from the stability. I ha ve co me to th e co nclusion that the re are several ways in w h ic h stru ctural fle xib ilit y w ould c a use a c h a nge in tai l setting an g le le a d in g to tu ck und er.
1. Willg Twist
H igb Speed Tri m P O SIl
up Elev
102
Low Speed
As I sa id in Chapter 5, the forces on a normally ca mbe re d wing can b e repre s ent ed b y a lift fo rce th rou gh the ae ro dyn ami c ce ntre (q uarte r c ho rd p o int ) to gethe r w it h a m om e nt Mo w hich acts lead ing edge down . This mo me nt will tend to twis t th e w in g lead ing edge down. Th e structure of the w ing will therefore acquire a twist suc h th at th e w ing tip s w ill de ve lop "was ho u t" relative to the roo ts. The twi s t wil l b e w orst on high as pect ratio wings.
Basic Aerona utics fo r Modellers
Figure 19.2
Tip
-----_T_-_- itlJist-~---
True Situation
I
Bquiualent t»
A
----------==~
~E::::q......iualent Ta~
_____
---------
----------
Figure 19.4
Figure 19.3
C=--?Sf~¥J LT As the aeroplane flies faster the twisting moment will increase as the square of the speed. That is, when you double the speed you get four times the twist. The effect of Mo pushing the nose of the aeroplane down is not the problem. That ha s been taken care of because the restoring force from the tail also increases as the square of the speed. The problem is that the wing twist gets worse as the speed increases. This speed dependent washout can be considered equivalent to the entire wing twisting as a rigid unit relative to the fuselage by an angle A in a leading edge down direction. See Figure 19.2. The greater the moment Mo and the less the wing's torsional stiffness, the greater will be the angle of twist A. The effect on the aeroplane's trim of the wing twisting leading edge down will be the same as if the tailplane were rotated leading edge up by angle A. The aeroplane has put on its own down trim, angle A, without the pilot moving the controls.
2. Tail Bending Figure 19.3 shows a model with a slender tail boom. As you have seen in Chapter 12, an aeroplane trimmed for a fairly high speed will have a download on the tail. The download increases greatly as the trimmed speed is
Basic Aeronautics/or Modellers
increased. If the tailboom which carries this download to the rest of the aeroplane is flexible , it will deform downwards tilting the tailplane through angle B which will be proportional to the tail load Lr and inversely proportional to the bending stiffness of the tailboom. It is just as if an angle of down trim B had been applied but without moving the controls.
3. Flexible Controls Most models have a tail a rra n ge d such that the pushrod, or Bowden cable (snake), has to "push for up". That can cause problems when it is connected to an all moving tail with its pivot well ahead of its aerodynamic centre (25% of its Mean Chord), as in Figure 19.4. In high speed flight there will be a downforce on a symmetrical tail at the 25% chord point. This force will put the pushrod or snake into compression. Unless it is very stiff, a pushrod in compression acts a bit like a spring and it will bow, or buckle, under the load . (Figure 19.5) The problem is made worse by using long wire ends with Z-bends to clear obstacles or exit the fuselage. The greater the compressive load , the more it will buckle and so effectively shorten. Unless it is well supported over its full length, a Bowden cable will also bow, and so
103
shorten, under a compressive load. Because of the layout, shortening the pushrod will reduce the tail setting angle by an angle of downtrim C, shown in Figure 19.5. The faster the aeroplane is flown, the more the downforce on the tail will be and the more the shortening of the pushrod . Therefore the angle C will increase with speed.
The Elevator Trim Graph The twisting moment on the wing and the download on the tail increase roughly as the square of the airspeed. Therefore the angle of downtrim due to the wing twist, the bending of the tailboom and shortening of the pushrod all added together, increases as speed squared.
The pilot will have to compensate by applying a small amount of up trim at low speeds, and more and more up trim as speed increases . At the aeroplane 's terminal diving speed the trim change due to flexibility will be very large indeed. The trim graph will become distorted as shown in Figure 19.6. The theoretical trim line from the formula is shown as a broken line. To this, an uptrim correction will have to be added as shown to produce the trim line of a flexible aeroplane. The aeroplane still behaves quite normally at lift coefficients from A to B on Figure 19.6. To avoid having up trim on all the time it would be best to change the tailplane setting, or the reference position of the tailplane, as shown. This adds a third part to the tail setting angle equation, 12.2.
Figure 19.5
-------- -----
---
Pushrod Bowed Pushrod Compression
Figure 19.6 Tail angle to trim
Trim Line <, (Rigid Theory)
.......
/
Angle
Neutral Elevator
ofFlex
Low Speed
High Speed
New Neutral
-------
.... CL
Trim Line (Flexible)
) A
up
104
Basic Aeronautics/or Modellers
The Critical Speed The airs peed correspondi ng to the lift coefficie nt at p o in t B o n Fig ure 19 .7 is th e Critica l Speed of the aeroplane. If the aeroplane is in trim at its Critical Speed, ANY slight speed increase beyon d this speed will lead to a tuck und er. In Figure 19.8 I have drawn another graph of elevato r trim position but against airspeed this time. Po int B is aga in the Critical Speed .
Tail Area Influence It may seem strange to you, it did at first to me , but the Critical Spee d dep end s o n the size of the tailplan e. The Tuck Unde r is ca used by structura l flexibility chang ing the angle be tween the wing and the tail. For a given ang le of flex, the bigger the are a of the tail, the mor e moment it will have abo ut the CG and the lo wer the speed at which it will overcome the stab ility. Tha t means the bigger the tail, the low er the Critical Speed w ill be. The tail has no t ca used the tuck under, in the same way that you cannot blam e the gro und for caus ing a cras h jus t becau se it is there.
Stability Influence As I said in the beginn ing, a rigid aeroplane ca nnot tuck und er because its stability will not let it. Similarly the stab ility of a flexible aeroplane will try to prevent it from tucki ng unde r. The mo re the speed devia tes fro m the trimmed speed the more the nose up pitching mo me nt from the stab ility increases, BUT the nose down mom ent fro m th e fle xibility inc reases m uc h more rapidl y as illustrated on Figure 19.9. Where the two lines cross ove r the aeroplane will aga in be in trim, but the slightes t furthe r inc rease in speed will leave a nose down net mo ment whi ch will cause a tuck und er. However the mor e Stab ility Ma rg in an aero p la ne has , th e stronger th e re co ve ry
mo ment will be and so the faster it mu st be going to tuck unde r. In ot her words the further forward the CG, the more Stab ility Margin, and the higher the Critical Spee d.
Tuck Under Speed Fro m Figures 19.7 an d 19.8 yo u ca n see th at if a n ae roplane is in trim at so me speed, po int C, speed ing it up a little w ill lea ve it trimmed tail heavy . If down trim is no t ad de d it w ill pitch nose up . However if it is speeded up beyond point D it wo uld beco me nose heavy agai n. If e nough up trim is not adde d it will tuck und e r. I therefore call the speed at D the "Tuck Under Speed". It is not a fixed speed, it depends on the initial trimmed speed of the aeroplane . The further the trimmed spee d is be low the critical speed (B), the furthe r the tuck under speed will be above the Critical Speed, bu t eve n if it is trimmed to fly a t its m inimum s peed, o r eve n w ith fu ll up e levator app lied, it will Tuck Unde r if speeded up eno ugh .
Getting Away With It If the speed of the model is a little above its tuck un der speed it will try to stee pe n the dive but ap plying full u p elevator will pull it o ut. However if the model accelerates away above the tuck un de r speed , the nose d own net mo ment may be so grea t that even full up eleva tor (on high rates) is not eno ug h to pull it out (see Figur e 19.8 an d 19.9). Tha t may be beca use the servo is not strong e no ug h, or the pu sh rod is be nding. Or there isn't e no ugh trave l on the tailpl an e or elevator. Or if the tailpl ane or elevator is too small, it may be incapa b le of develo p ing e no ug h do wn ward lift. Increasing th e up movem en t on an a ll moving tailplane too much will just let a small tailplan e stall at a nega tive angle. See Figure 19.10. Just before it re ach es its stalling an gle the tail will be developing its
Figure 19.7
B
DOWII
-~:~:~)~~
Elev
I I
Neutral
Critical Speed CL
A
up Elev. Vertical Dive
Basic Aero na uticsfo r Modellers
105
Figure 19.8
Full Dowll
,., ,.,
B
- --
Rigid -I I I I V Max I I I I Flexible I
Full Up
I I I I I
A
Figure 19.9
Tailplane Instability
Su p pose something, like a gust, deflects the All Moving Tail in Figure 19.11 leading edge up by a small angle. The pushrod will be compressed and it will try to spring back and return the tail to its original position. This restoring force from the push rod will be independent of airspeed. However the addi tio na l lift XLT caused by th e disturbance will tend to ro tate the tail the other way, leading edge up , and this desta bilising force will increase as the square of the speed. If I the speed is high enough I the d es ta b ilis in g force I will win. The tail will flip I lead ing edge up until it R ecover I Tu ck Ullder reaches the fu ll "dow n " V III I - -......~ travel stop. The result is obviously a tuck un der, TrlmSpeed Tuck Under Speed or even an outside loop. All th e up elevator you can apply on the servo maximum downward lift coefficient. If that is not sufficient will disappear into further bending of the pushrod. Th is to pull it out of the dive , it is doomed. Its only chance Tailplane Instability Speed, above w hich the pushrod is would be to bunt right round into climbing flight inverted incapable of preventing the tail runaway, can be increased until the speed red uces. by stiffening the pushrod, reducing the tail area, increasing Then . .. get out of that! distance y, or best of all placing the tailplane 's pivot on, or Pitching Momellt
10 6
Basic Aerona utics for Modellers
just a head of, its aeroFigure 19.11 dyna mic centre. The rest of the aeroplane doesn 't co me into it. If th e aero plane is be ing flown fast, above the tailplane's P ivo t (well aft) ins ta bility speed, there will be a download on th e tail h olding the pu sh rod in ten sion and __ I so there is no probl em , =- x rega rdless of the speed. However w ha t happen s if a glitch, or the pilot for I so me reason, put s o n a I larg e ja b of down e leI y vator, eno ug h to give the I tail an upl o ad? If th e I aeroplane is flying faster I I th an th e tailplane 's "'t'!,J\~1 cr itica l s peed , that is Pusbrod Ben t a nd Sho rtened -..: :~ enough to let it flip into its full d own p o sition and there is noth ing you can do abo ut it. It will seem as if the servo has run away. Fig ure 19.10 I have just had a vision of glide r designers crowding the window ledges twenty floors up. I had better put away the doom and gloom and come up with some answers. Airflow
....
_.~
- - - --
Remedies for Tuck Under Now that we have a better idea of how Tuck Under is ca used w e ca n thi n k of w ays to design it o ut of o ur aero planes. Many of the remedies which I sha ll suggest are well known fro m experi en ce of course, wh ich is as it sho uld be, but I believe it is impo rtant to know why these design featu res alleviate tuck under. • 1. The struc ture of the aeroplane sho uld be as stiff as possible . The less the w ings twist and the less the fuselage be nds, the higher the critical speed will be. • 2. Attach the wing rigidly to the fuselage using dowels and wing bo lts, substantial plug in joiners or plen ty of elastic bands. • 3. Move th e pivot of a n all movin g tail n ear to its aerodynamic centre (25% of its MA e. ). The less load the pushrod is carrying the less it will deform. This will also eliminate the tailplane instability problem. • 4. Use a large diameter stiff pushrod and keep the w ire en ds as sho rt and straight as possible. • 5. Leng then ing the tailplane or elevator horn and using a longer servo arm to achieve the required movement also reduces the load in the pushrod. • 6. Use of close d loop controls or a pull for up pu shrod or snake will eliminate my third cause of Tuck Under. See Figure 19.12 for possible layou ts. I have show n the tailplane's mean chord , not its root chord . • 7. If using a Bowden cab le make sure it is completely sup porte d alo ng its le ngth w ith as litt le bare inne r showing as poss ible. • 8. If you are choosing from a range of wing aerofoils all' of w hich are su itable, use the one with least camber. • 9. Move th e e.G. fo rw ar d to increase the stability margin, bu t remember that control responsiven ess will be re duce d . Yo u m a y h a ve to increase contro l movement. • 10. A smaller tailplane will increase the Critical Speed
Basic Aeronautics fo r Modellers
Tail Sta lled
an d make Tuck Unde r less likely to occ ur, bu t it may make recovery less likely if a tuck under does occ ur . The e.G. will have to be move d forward to maintain the same stability ma rgin. Too small a tail may reduce the controllability of the aero plane. • 11. Try a tail with negative camber. I haven't me ntione d this before for a good reason. It do esn 't help preven t tuck under! It do esn't raise the critical speed at all, but remember Figure 19.10 where the tailplane with full up applied was jus t stalled? A tail w ith negative cambe r wouldn't be stalled at that angle. It would be good for ano the r degree or two if it doesn 't have too sha rp a lead ing edge. Its maximum downward lift coefficient will be slightly grea ter than for a symme trical tail. Th is extra downward lift migh t make the difference between just pulling ou t of the dive and just failing to pull ou t, but I wouldn 't like to depend on it! The tail's d rag will be fraction ally less w hile carrying a do wn load as an adde d bo nus. It is possible to des ign a model whose win gs and tail are stiff enough, and whose C011U"ol runs are effective enough, that its critical speed is ab ove its termin al velocity, which mea ns that its trim CUIve will be like Figure 19.13. No way will this aeroplane tuck under.
Conclusion Tuck Unde r is ano ther aeroelastic problem caused by the flexibility of the structure, and not, as the uninformed may try to misle ad yo u into be lievi ng, ca used by the
107
Figure 19.12
•
•
w in g 's cen tre of pres sure mo vem ent or "ta ilp la ne tak eover". Although it is clo sely ass ociated with static stability, an ae roplane w hich nicks under is not "unstable" in the context of Cha pter 8 in that it would respond in a stabl e fash ion to an ang le of atta ck cha nge at co nstant speed . Becau se all ae roplanes are flexible to some extent , a ll a e ro p la nes wi th ca m bered se ctions will have a Figure 19.13
Theory Do w " Elev
Ac tual
up Elev
108
CRITICAL TUCK UNDER SPEED. We have to try to m ak e s u re that th e structure and controls are stiff eno ugh to mak e the critical sp eed so high that th e aeroplan e c a n n o t reach it. All the potential c a us es of tu c k un d er must be removed. Curing only on e may in cr e as e the Critica l speed a bit , but n ot e no u g h. Th e unfortunat e m od ell er h aving cure d on e faul t and still ex perience d tuck under may start to think th at w ha t he ha s d on e was not a cure at all. The prima ry cure is to stiffen u p th e structure, piv ot all mo ving tails at 25% m e an c ho rd , a n d use a pull for up control system . I would cha nge the tailpla ne to a smaller, negat ive ca mbe re d o ne only if the o riginal w as exceptio nally big. If the ae ro plane is flown with a very s ma ll Stabilit y Ma rgin to mak e it very sensitive, only a very little fle x will ca use a tu ck und er. In th is cas e a forw a rd CG movement m a y b e th e b e st cu re . Th ose a re all the thin gs wh ich I ca n p ro ve wi ll lead to tuck und er. There m a y b e mo re d esi gn faults which will cause tu ck u nde r w h ic h I do know about. If you find o ne , please let me kno w .
Footnote In the past ther e hav e been mod els whose performanc e has seeme d a little odd , unu sual , disappoi nting . Fo r exa mple, I brou ght myoid tra iner out of retire me nt in order to run in th e eng ine for my first "hot" aerobatic sp o rts mod el. It wa s a ve ry good tra in er w ith docile predictable handling characteristics but wh en I installed an engine with two and a half times the power of its intende d engine, it changed . At low spee ds, w ith the engine at idle , it still flew velY nicely, but at high sp eed it just felt wrong, perh ap s a bit sen sitive to elevator trim, but difficult to put into words at the time . Looking ba ck I wonder if I wa s o pe rating it at aro und its Critical tuck under speed, and if that could account for its odd handling. I wonder if oth er people have encountered unu sual handling cha racteristics on mod els which could be related to inadequate structura l stiffnes s. I wonder is it wise to operat e an aeroplane so far outside its intended design envelope? It is worth giving it so me thought.
Basi c Aero na III ics for Modellers
Chapter 20
The Air on the Move could read half a dozen aerodynamic textbooks from cover to cover and not see wind mentioned once, which indicates that it has no effect whatever on aerodynamics, only on navigation. You see by "wind " I mean a block of air whose molecules are not moving relative to each other but which are all moving at the same speed over smooth level ground.
Y
OU
Navigation The velocity of the aeroplane relative to the air is its "airspeed" , and its "groundspeed" is just the vector addition of the airspeed and the windspeed as in Figure 20.1. The direction of the groundspeed is called the "track" of the aeroplane. The airflow over the aeroplane will still be straight from nose to tail. If you are really interested (or mistrustful) you will find proof of this vector addition idea in a mechanics textbook, for example "MECHANICS" by Den Hartog. You will also find proof that the acceleration of the aeroplane relative to the ground is the same as its acceleration relative to the air, because we are assuming that the air has no acceleration. If a steady wind does not affect the acceleration of the aeroplane then Newton would agree that it does not affect the forces on the aeroplane or its behaviour.
Slope Lift
generally reduce with height as in Figure 20.2, there will therefore be a "ce iling" above which a particular glider cannot climb , and the more efficient the glider the higher its ceiling will be . You will notice in Figure 20.2 that the streamlines are closer together over the brow of the hill, indicating an increase in wind speed as you would expect. The other requirement is that the wind must be strong enough to have its vertical component greater than the glider's minimum sink rate, but the wind must not be so strong that it blows the glider backwards over the hill. In a strong wind it is better to ballast the glider than add down trim to maintain its good glide at a higher speed.
Thermal Lift Hot patches develop on the surface of the Earth due to solar heating, or buildings or whatever. A hot patch of ground shares its heat with the surrounding air which, when heated, expands. The warmer air is less dense than its surroundings so when a big enough "p ile" of it has developed it breaks away from the surface and begins to rise as illustrated in Figure 20 .3. The surrounding air flows in to replace it, and the hot patch of ground starts to warm up another batch.
Figure 20.1
When this solid block of air moving over a smooth level plain (Le. the wind) comes to a hill, something has got to give, and it isn't the hill. The air will be forced up, and this "slo pe lift" is an ideal way of keeping a glider airborne indefinitely. A particular glider will obviously require to fly in a "wind slope" steeper than its minimum glide angle. The more efficient the glider the less steep the hill needs to be. Also, because the "slo pe" of the wind will
Basic Aeronautics/or Modellers
Wi"d
Vector
109
Figure 20.2
Right down amongst the heather wh ere the midgies hide there is very little wind. At a height of a few inches there is more, while at tree top height it could be quite strong. It is just like a boundary layer on a giant scale. Surface friction reduces the wind speed the closer you are to the surface . Not only that but the direction of the wind changes with height also . If you are facing the wind on the surface, the wind higher up is usually from slightly to your right. For example the wind could be»
-~---
-~~-
-~---------------/_---------------------As the bubble of warm air (the "thermal") rises, it cools at a fixed rate, so how far it will rise depends upon the weather. If the temperature of the surrounding air is constant all the way up, the "thermal" will soon cool to the same temperature as its surroundings and so stop rising. But if the temperature of the surrounding air drops rapidly with height, as when a cold air mass moves over a warm surface, it is unstable and once a thermal starts to rise it just keeps going and going. The big ones build up into towering Cumulo Nimbus clouds several miles high containing strong updraughts. And if there is an updraught in one place there must be a downdraught somewhere else to compensate. Even a little slope lift can trigger thermals in unstable air. Now that this idea of thermals has broken my idealisation of the w ind as a mass of air molecules moving uniformly together, I might as well come clean and mention other "real" variations on the ideal wind.
Windshear and Wind Gradient The term "w in d s hear" applies to any significant change in wind speed or direction for a small change in height or position and it can sometimes be so severe that it causes jet transport aircraft to crash. The wind cannot be the same at every height.
• 3 ft/sec at north-east. • 10 ft/sec at • 30 ft/sec at • 50 ft/sec at
a height of 3 inches, blowing from the 3 feet 30 feet 300 feet now coming from the east.
Now suppose you take off from your runway heading north . On the runway there is a small headwind component which is fine , and a small crosswind from the right which is no problem. The model is kept straight using rudder. On lift off the aeroplane is suddenly exposed to a crosswind which would make an aeroplane with strong lateral stability tend to roll to its left and as the model gains height the headwind component increases (no problem) and the crosswind from the right increases. That will continue any tendancy to roll to the left. The solution is to apply a little aileron into wind on take off to prevent the crosswind from lifting one wing. See Figure 20.4. As your aeroplane seems so keen to turn left it is a shame to disappoint it so you climb your model in a gentle left turn onto a westerly heading. You are careful to keep the speed constant, by applying up elevator as you have not had time to adjust the trims yet , but the speed apparent to you is the groundspeed, because you
Figure 20.3
Hot Patch
110
Basic Aeronauticsfor Modellers
are stand ing with yo ur we llies se t firmly in the mud . So, climbing into that incre asing tailwind with yo u kee ping a co nsta nt grou nds peed , yo ur mo d el h as a s te ad ily decrea sing airspeed , w hic h is all it cares abo ut, and the resu lt co u ld we ll b e a n une x pe ct e d s ta ll. Th e o n ly an swer to this o ne is to be aware of the wind var iation and make a rule never to take off downwind , an d neve r to tu rn down w ind after take off. Yes, I bro ke a model o nce doi ng that. Ano the r good crash scena rio, illustra ted in Fig ur e 20.5, is an ap proach with a good head w ind at he igh t dropping off to o nly a little head wind ne ar the ground . Th e m od el p ilot w ith hi s mind g ro u n d referenced maint a in s a cons ta n t g ro u ndspee d wh ich gives th e aeroplane a decreasing airspee d leadi ng to a stall. Th e p ro b lem is mad e w orse if a p owered ae ro p la ne is trimmed with too mu ch downthru st such that red ucing power redu ces speed . The th rottle sho uld be a rate of descent co ntro l and the e le vato r trim shou ld be the speed co ntro l. If the speed is set a go od margin abo ve stalling speed a t a sa fe he igh t with the elevator trim, and le ft the re , a co rrec tly trimm ed stable aero pla ne w ill mainta in th at airspeed . T he p ilo t th e n w ill be a d jus ting p o w e r to ke ep th e mo de l o n the desired de scen t path.
Gusts Re m embe r w he n flying b ecame avai lable to th e ge n era l p ub li c o n cha rte r fligh ts to Palrna Figure 20. 4 and Be n idorm wh ich were c h ea p e r th a n staying at hom e? People came back wit h stories of new ho te ls in slee py Spanis h fishing villages, Sa ng r ia a n d "Cu b a Libre " che aper than tea Gro""dspeed 100 ( a n d mu ch s tro n ge r) , Airspeed 50 a n d o n t h e fl igh ts of Stalling Speed 51 h ittin g "Air Po ck e ts " Oops! whi ch so unde d like the holes in Sw iss che ese or th e lu m p s in sch o o l cus tard . We ll of co ur se th ere w ere no t re all y hol e s in th e a ir, w ha t th e y h ad e nc o u n te re d we re GUSTS. It wo u ld h ave be en n ic e if th e air m o v ed smoo thly and unifor mly over th e sur face but it d o e s n ot. It is fu ll o f tur bu len ce , what w ith th e ro u ghn e s s o f th e ground sur face a nd th e th e rm a ls a n d so o n . Win ds hear is fi x ed in p la c e , ie: - a fte r on e ae ro p lane flies throu gh it the winds hea r w ill still be the re w he n the next ae ro plane comes alo ng. However tur bul e nt gust s
Basic Aerona utics/or Modellers
are a pure ly random va riatio n in th e local windspe e d and ca n be in any di rection . Whe n flying into w ind , say, a sudde n increase in w inds peed is a gust fro m in front , a redu ction in w indspeed is a tailgu st, a temporary shift in wi nd d irection is a side gust , a the rmal is an up gu st, and a do wnd rau ght is a down gus t. Any cha nge in the w ind means that the wind now has an acc e leratio n whi ch w ill affect the aeroplane. Gu sts from front o r re ar, above o r below , w ill cause some spe e d variation, so me pitchin g up a nd down and prob ably so me wing flexing but a stable aerop lane will be ab le to ride it o ut w ith no probl em s. The da nger is that, if the w ing 's a ng le of attack is just be lo w its stalling ang le , a sudde n increase from a gust ca n stall the wing, whi ch mak es it prude nt to fly at a speed a safe margin above stalling speed, and the mo re the tur bulence the bigger the margin. Also bewa re of pu lling "g", tig ht turn s e tc., near the ground . Gus ts from the side can disturb the head ing because of di re ctio na l stability , o r bank the aerop lane due to late ral stability. Ve rtical gusts affecting o ne wing more than the o the r ca n also produce unw anted banks. They are just a nui san ce which has to be co rrec te d , un less the aerop lane is near the gro und and th e wing's ang le of attack is ne ar its sta lling angle . O n the ap proach a gust dropping one wing can be a li ttl e e m ba rr ass ing . Pic ki ng it up w ith rudde r, o r a co mbina tio n or rudder and ailero n is safe r than relying jus t o n ailero n .
Groundspeed 100 Airspeed 100
~
/ Wi"d
Right aileron to prevent left roll =T"1I"""=
111
Figure 20.5
»: ~8'$'45
~THUMP u..J
LJ ·g.s.45 a.s.48 Stalling Speed 51
«ess
g.s.45 a.s.75
Groundspeed (g.s) 45 Airspeed (a.s) 95
Myths and Misconceptions I was watching a magic show on television w ith my little da ug hter. Paul Dan iels stoo d on its edge a b ig sheet of pl ate glass w hic h he dem on strated to be so lid all over. Then he se t up his a ppa ratus and pu shed his lad y assis tant straight thro ug h the middl e of the sheet of glass with out a scra tc h . My d au ghter was a mazed . I w as a maze d. She belie ved that the lad y had passed th rou gh th e g lass by magi c . I, b ecau se of my grea t age and ex pe rie nce and knowledge of ph ysics, know that it was a trick, an illusion , but I do not ye t kn ow how it was don e. I d o kn o w that what a p pea re d to happ en is imp ossible and I a m sticking to that beli ef. I know there is an oth er logical explana tion whi ch fits in with the laws of scie nce . I just can't find it. There a re man y mod elle rs wh o believe so me popular unscientific myths eithe r becau se they are misinform ed o r hav e w ron gly interp ret ed th e b eha viou r of th eir model. You , dear re ad er, ca n no t be o ne of them o f co urse having had the se nse to bu y this book, but yo u have prob abl y heard them and perh ap s wondered how to break it to them ge nt ly. First of all, wh en flying in a stea dy wi nd the airspeed o f the aeroplane is unaffected by the wi nd . How co uld it be? Sure if yo u sudde nly app ly a tailw ind the airspe ed will reduce . But d rag w ill redu ce and the aerop lane w ill accele ra te back to its origi na l airspeed at wh ich dr ag eq ua ls th ru s t. Dra g d epend s o n a irs pee d , n ot gro unds peed . In a crossw ind the track of the aeroplane is affected but the air still flows along the ce ntre line from front to back. It must do. If it we re offset, the ae rop lane wo uld h a ve a s ides li p velo cit y , and from Cha p te r 9 its d irecti on al sta bility w ould lin e it up with the airflow ag ain . Wh en flying th e mod el to wards yo urself in a
112
crosswind, you must cock the nose of the aeroplane into wi nd suc h tha t it loo ks as if it is flying sideways towards yo u , but th e airflow over th e aerop lane w ill s till b e straig ht from nose to tail an d a flag o n the nose wo uld always fly stra ight back, ass um ing the aeroplane has no t been yawed delibe ratel y with the rudder. Th e o the r th orn y p robl em is th at of tu rnin g a n aero plane in a wind . Some people swear tha t wh en the y turn the ir mod el fro m flyin g downw ind to flyin g int o win d it zooms upwards, and the y ca n demonstrate to prove it. Howe ver w he n I turn their aerop lane it does not happen , and they canno t tell me wh er e I am go ing wron g. How do yo u prove that so me thing doesn't (o r at least sho uld n't) happen? Perhaps a clarifica tio n o f the probl em will help. By "tu rn ing " the ae ro p la ne I m e an s u p p ly ing a ce ntripe tal force to ge t its CG mo ving in a new direction as we ll as aligning the fuselage w ith the new direction o f motion . Ju st rot ating it a bou t its yaw ax is is not turning. I a lso me an a bal anced turn as d es cr ib ed in th e cha pter o n turning wh ere the lift is increased so that its ve rtical co mpone nt still sup por ts the weight , and thru st eq ua ls d rag . I am also disregardin g gusts and wi nds hear. Given those co nditio ns then , a turn , ho wever tigh t or slow, w ill ca use no ve rtica l effect. I have flown full size aero p lanes a nd kn ow th at th ere is no way th at th e aeroplane ca n se nse the wind directio n . The re is no way th e p ilo t ca n kn o w th e wind di rec tio n except fro m gro und based instrume nts . All the forces o n a n aerop lane o ther than grav ity arise fro m the air and so its eq uat ions of motion are related o nly to th e air. Wh en yo u a re sitting up th er e in an aeroplane looking at the clo uds , the surface of the earth co u ld b e m o vin g in a ny dire cti on a t an y s pee d undern eath the air mass and it makes not o ne wh it of d iffere nce to the flying of the aerop lane .
Basic Aeronautics/o r Modellers
What about Momentum? O f co urse so meo ne w ill say "An aero p la ne flyin g north into a stro ng w ind has very little mom entum , so if it tu rns so uth it mu st los e airsp eed!", o r "\X1he re does the ex tra momentum co me from?". I'm gla d yo u ask ed . Everyo ne has a roug h idea of wha t mom entum is but to be a bit mor e specific, I loo ked up a boo k. Mom entum is a vec tor quantity, mass times ve locity, and the change in mom entum (w hic h is what we are afte r) is defined math em atically as the integ ral of the for ce produ cing it w ith resp ect to time . I w ill not bor e yo u with the de tails but th e force pr oducing th e mom entum cha ng e o n a turn ing aeroplane is the o nly unbalan ced forc e in Sight, the ce ntripe tal force . I integrated it round a 180 degre e turn and found that the mom entum cha nge produ ced by the ce ntripe tal force is 2mV, in a d irection o pposite to the o rigina l ve locity. If the initial airsp eed is V to the north into a northerl y w ind of V!2 say , the northbound momentum is mV!2. Th e cha nge is -2mV so the final mom entum is -3mV!2 (minus becau se it is so uthbo und) . Th e grounds peed is the refo re 3V!2, makin g the airsp eed V. Th e ce ntripe tal force supplies the d ifferen ce in mom e ntum with out loss of airsp eed , see?
What About K inetic Ene rgy Unde te rred o ur doubting Th om as says "But whe re d oes the ex tra kin et ic energy co me from?". I loo ked tha t up as w ell. Kin eti c e ne rgy is Y, mV2 a n d is a Sca la r qu antit y ( i.e . no t a vec to r) . Th e in cr e ase in kine tic e ne rgy is defin ed as the wor k don e by th e result ant of a ll th e fo rces ac ting o n th e bod y. Work do ne is th e resu ltant force times the d istan ce moved in the direction o f t ha t force . I in te grat ed th e co m po ne n t o f th e centri pe ta l fo rce in th e dir e cti on o f the grou nds peed round a 180 degree turn , from so uth to north th is time . The cha nge in kin et ic energy is -2mVU where U is the wi nds peed . Th e K.E. cha nge turn s o ut negat ive if the wind is in the sa me di rection as the initial airsp eed . Su p pose th e w inds peed U is h alf the a irs peed V aga in . Go ing so uth, with the wind , the K.E. relat ive to the grou nd is Y,.m(3V/ 2)2 or % m V2 The cha nge is -mV2 giv ing a final K.E. of ~ mV2 or Y, m (V/ 2)2, whic h means that the final ground speed, goi ng Nor th again is V/ 2, which mean s the a irspeed is V. Th e centripetal force has rem o ved the excess ICE. with no increase in airsp e ed , see?
Analogies For a n auti c al a na lo gy th ink of th e e ffec t o n a powered boat of a river curre nt, or better still an ocea n current miles from land . (Forget the e ffec t of wi nd o n a boat becau se that is a diffe rent case e ntire ly) . In th e middle of the ocea n there is no way of kn owing if the wat er is moving relati ve to the e arth unl ess yo u ca n see th e b ott om . If th ere is a c u r ren t it will m a ke n o differ en ce to the handling of a boat and I have ne ver hea rd of a boat zooming upwards when turn ed head o n ' to the Gulf Strea m. However back to ae roplanes in the wind . Whe n yo u sit back with a glass in yo ur hand and th ink ab out life, the wi nd is just a block of air movin g at so me speed relative to the ea rth's surface , and if a mo de l ae ropl an e stays w ithin that block of air, the speed of the a ir over
Basic Aero na uticsf or Modellers
the g round ma kes no d ifference. Th e block of air co uld be inside a Boe ing 747 for exa mple, moving at 600 miles per hour, but if yo u had a s uitab ly s ize d slo w flying mod el I bet that in spi te of the 600 miles p e r hour wind yo u cou ld fly qui te a resp ectable circle inside , if yo u were ins ide w ith it. There is yo ur probl em! I rate yo ur cha nces of flying a model in a circle as nil if yo u are o n the ground and it is in a passin g 747. The 600 miles a n hour "w ind" has no t mad e things d ifficult fo r the mod el , on ly for yo u. Inside the aeroplane you d o not fee l as thoug h you are moving, an d it does not matter wh eth er yo u are o r not. If yo ur brain is in the 747 then that is yo ur refe re nce. Th e p ro b le m is n ot on e o f aero dy n a m ics , or me ch an ics . It is a psych o lo g ical p rob lem . Yo u mu st "think air borne". Your we lly boots might be stuc k in the mud but yo u mu st deta ch yo ur brain an d put it up in th e air floatin g free like a balloon.
The Mea ning of Life Subtitled "Don 't as k q ues tions to wh ich ther e are no answers" I ca n no t tell yo u why yo ur aerop lan e zoo ms and d ives w he n yo u turn it , b e cau s e w he n I fl y yo u r aero plane it does not happen . I know it happen s wh en yo u fly it becau se I have see n , but yo u a re the o ne wh o mu st say why becau se it is yo u do ing it. Is it becau se w he n you turn it into wind yo u see its gro undspeed red uci ng so you let it speed up in case it sta lls? Wh en yo u lev e l th e w ings th e extra a irs peed wou ld ca use a zoom. Is it becau se yo u a pply u p elevator to hold the nose up in a turn and whe n you stop the turn yo u forget to rem o ve the up elevator? Th at u p eleva tor would ca use a zoo m. Do yo u turn th e ae ro p la ne wi tho u t app ly ing up e leva tor? If so the nose will d rop and the airspeed wi ll increa se althou gh yo u may not not ice as it turns into w ind and the gro undspeed reduces . The ex tra a irspeed would ca use a zoom. Maybe you speed the plane up in a turn , by add ing powe r. to increa se the lift. O n res uming le vel flight the excess sp eed (a nd ex cess po wer ) w ill ca use a zo o m upward . Do yo u turn th e ae ro plane by ba nk ing it ov er o n a wi ng tip and hea vin g o n the u p eleva tor? High "g" turns like th at dr ast icall y increa se the drag of an aer opl an e and so it w ill lose speed in the turn , but it will lose just as much speed turn ing downwind as turning into wind . It loo ks worse to yo u turn ing into w ind bu t it is just th e sa me to the aerop lane. The aeroplane is just as near its stalling speed after a turn downwind , but your mind is co mfo rte d b y th e a pp arent e x tr a s pee d fr om t he tailwind. I still ca nnot work o ut ho w Paul Dan iels does his lad y th rough the sheet o f glass trick , and I do not know how you d o yo ur zoo ming a nd d ivin g tric ks ei the r. When yo u have w ork ed o ut w ha t it is you are doin g to ca use the uninte ntio nal climbs and dives, please tell me . Te ll eve ryo ne!
113
Chapter 21
Model Aircraft Structures Figure 2 1.1
Fig u re 2 1.2 Support
~~
»>
~ Compression
j
~ Te nsio n
\~
I
Load
Support
I
n line with the res t of the book, this is not about how to design structures, it is just a lirtle bit of theory to explain how the structures used in model aeroplanes work. There are qui te a few words, most of which wil l be familia r, whose meanings in this context I wou ld like to clarify.
I
LOAD . The load carried by a part of the structure is the force in it. For example, a piece of balsa with a one pound weight hanging on it is carrying a load of one pound.
TENSION . A structure is in "tension" when a load is trying
to pu ll its two ends apart as in Figure 21.1. COMPRESSION. A st ructure is in compression when a load is trying to push its two en ds towards each other as in
Figure 21.2. SHEAR. The wing dowel in Figure 21.3 is in "shear" when
the wing is lifting it up and the fuselage is holding it down. TORSION. A p iece of structure is in "tor sion" when the load it is carrying is trying to twist one end relative to the
other. BENDING. The beam in
Figure 21.3
Figure 21.4 is carrying a "b e n d in g moment " when it is supported at both ends and a weight is p laced in the middle.
Fuselage holding dowel doum
114
STRESS is force per unit
of cross-sectional area, and tells how concentrated the load is . If a piece of quarter inch sq uare wood is canying a one pound load then the "stress" in it is 16 lb ./sq . in . . . If the same load is on eighth square balsa the stress is 64 lb ./sq . in . Each material has a limiting stress beyond which it will break.
Basic Aerona 11tics for Modellers
Figure21S
Figure 21 .4 Load
c~
1\
Support
Tensile stress 16
No
, J
load
Tensile stress 64
SUPPOl-t
STRAIN is the amount of deformation due to the stress, either stretch or compression . The s t ra in is approximately proportional to the stress applied so the eigh th square wood above will stretch four times as muc h as th e quarter square under the same lo ad . See Figure 21.5 . It is ass u me d that whe n the load is tak en o ff , th e s tretc h w ill d is ap p e ar , i.e. th e ma terial is "elastic".
llb
llb
load
load
Figure 21.6
STRENGTH, The strength of a piece of materia l or structure is the maximum load it can carry before some part of it reaches its limiting stres s and breaks. STIFFNESS . The stiffness of a ma terial is stress divid ed by stra in. The stiffe r a material is the less it will deform und er a give n stress. Figure 21.6 shows two d ifferen t ma teria ls each o f th e sa me cross-sectio n and unstre tched length carrying the same load . Th e str ess is the sa me in each but the stiffer materi al stre tche s less. Either m ateri al could break first, depen ding on th eir different strengths. STIFFNESS of a STRUCTURE is load over deformation, Le . the stiffer a struc ture is, the less it will deform under a given load . The stiffness of a structur e (e .g. a wi ng) depends u pon the stiffne ss of the materials used and th e ma nner in which it is co nstructed .
Stress
Stress
100
100
Unstretcbed -----" le1lgth
Stiff material Flexible naterial
Load
Load Figure 21 .7
LOAD PATH, The ro u te the lo ad takes th rou gh th e stru ctu re fro m where the force is made to where it is used. Just like a chain, any structure is o n ly as stro ng as its weakest link .
The Strength and Stiffness of Composites Imagine a steel wire and a le ngth of rubber bungee each of which w ill carry a load of 100 pounds before breaking. They have the same strength but th e steel wire is much stiffer. If a steel ring is su p ported by a foo t o f the wire and a lso a foot (u ns tre tch ed) of the ru b b er , w ha t is th e stre ng th of th e combina tion? Wha t is th e maximu m weight that ca n be hung on th e ring? Q uick as a flash yo u answer 100 pounds. Th e point is tha t th e stiffer member carries an unfa ir p roport ion of the lo ad because the stretch in each has to b e th e sa me, as in Figure 21.7 . The m inu scul e stretch of th e steel allows only a tiny lo ad to b e carried by th e ru b b er. When a . load o f 101 p ounds bre aks the steel wire th e ru bber w ill stretch until it a lso breaks. When ma ter ials of d iffe rent stiffness share a loa d in a model aerop lane structure , re member th at the stiffer material will carry an u nfair proportion of the load.
Basic Aeronautics fo r Modellers
Rubber
With this much stretch, the wire carries 100 load and the rubber has negligible load Unstretcbed length
115
Twist and Bend An airflow tends to do two things to a wing, and the effects are almost independent. The moving air tends to push the wing aside, and tends to twist it. The push force , or Lift, depends on the angle of the wing to the airflow, and not much else. By varying the angle you can make the Lift increase, or decrease to zero (but it doesn't change the twist) . The twisting effect, or Pitching Moment, depends only on how much camber the wing section has. Naturally both of these effects depend also on two properties of the airflow itself, the ve locity of its motion and the density of air. One end of the wing is anchored to the fuselage and the airflow tries to lift the w ing tip up above the w ing root (and occasionally it succeeds) so the wing bends. To resist this tendency the bottom surface of the wing will be pulled o ut in tension , and the top will be pushed towards the root in compression. (Sure, struts and wires Figure 21. 8 2
point
2
2
2
bending m o m ent
total =2
2xl
A B
2xl + 2x3
=8
c
2xl + 2x3 + 2x5
= 18
D
2xl + 2x3 + 2x5 + 2x7
= 32
E
2xl + 2x3 + 2x5 + 2x7 + 2x9
= 50
2xl + 2x3 + 2x5 + 2x7 + 2x9 + 2xll
=72
F
2xl + 2x3 + 2xS + 2x7 + 2x9 + 2xll + 2x13 = 98
G
Figure 2 1.9
Bending moment
can be used to hold the wing tips down, and we will come to them also.)
Bending Moments in Wings In Figure 21.8 I h ave divided up a wing into 2 inch wide strips , each of which has a lift force of 2 ounces . The "BENDING MOMENT" at any point in the wing is the moment of all the lift forces outboard of that point. Therefore the bending moment at the tip is zero. The bending moment at various points along the wing is shown in the table below. From this table and the graph of bending moment in Figure 21.9 you can see that by far the greatest bending load on the wing is at the root , and so the greatest strength must also be at th e root. The two wings cannot simply be attach ed directly to the fuselage sides, because the transfer of the bending moment would distort the fuselage as in Figure 21.10 . The tw o wings are best built in one 2 2 2 piece so that the stress is carried across the join. If the wings have to be detachable for transport then there must be some kind of wooden or metal beam running across the fuselage to carry the bending moment from one wing to the other. Two main typ es of wing structures are used . One consists of a foam core covered wi th wood veneer. The other is an assembly built u p with ribs , spars and usually some thin balsa sheeting, the whole thing being covered with doped fabri c or plastic film .
Built up Wings The usual structure for carrying a bending moment is the beam, shown bending in Figure 21.11. The strength of such a beam depends on its breadth (b), and its depth (d) cubed. The material nea r th e top surface is being compressed and along the bottom surface it is being stretched while in the middle the material is under no stress and carries no lo ad . It might as well not be there. The 'I beam' consists of a top spar and a bottom sp ar with a thin sheet web in between . For ease of construction the web may be glued to the rear faces of the spars as in Figure 21.12 in which a load is tending to bend one end upwards like the lift on a wing . When such a beam is canying a bending moment the top spar is in compression and the bottom spar is in tension. The web holds the two spars apart. The strength of the beam depends on the cross Figure 2 1.10
G
116
F
E
D
c
B
A
Tip
Wings a ttached to fuselage sides
Basic Aeronautics/or Modellers
Figure 21.12
Figure 21.11 Load
Load
Compression
Support
sectiona l areas of the spa rs (A), the distan ce bet we en them , and of co urse the strength of the materi al from whi ch they are mad e . To ge t the spars as far a part as p ossibl e th e y a re pl a ced a t th e th ic ke st p art of th e aerofoil section and of co urse the thicker the sectio n the stro nger the win g w ill be . Wing ribs, as we ll as giving an aerofoil section , hold th e spa rs in position w ith the sup po rt of leading an d tra iling edges . The leading and trailin g edges add little to th e w ing 's bending stre ng th but if th e wi ng is pa rtly sheered with medium de ns ity balsa shee t, the to p and bott om sheets may add to the ben d ing stre ng th (Figure 21. 13) , de pe ndi ng o n th e rel ati ve sti ffness o f t he mat erials. If the spa rs are mad e of spr uce , or es pecia lly car bo n fibre, then the spa rs w ill ca rry a ll th e bending for ces. The best way to streng the n th is typ e of structure is to in cre ase th e size o f th e s pars a t th e roo t end , preferabl y by wid ening rath er than dee pe ning them as shown in Figure 21.14.
so that it is ab le to car ry a co rn pressive load . Thus in a foa m w ing the to p sk in ca rries co mpressive stress an d th e bott om skin ca rries tensile stress, wh ile th e foa m preve nts th e s kins fro m bu c klin g a nd p rese rve s th e aero foi l s ha pe . T he b e ndi n g s t re ng th o f th e w ing depen ds o n th e sk ins ' th ickn ess a nd c hord , ho w fa r apa rt th e y are , th e stre ng th o f th e mater ial used , but ma inly on how well they are suppo rted . Obech i veneer 0.03" thick is usual , but o n w ings wi th a thick sect io n a nd /o r a lo w as pect rat io , thin ca rd or even pap er is adeq ua te . Becau se th e skins a re so thin th ey will just buck le up if they are not stuc k securely all over to the suppo rting foam . In fact [ sha ll stick my neck o ut and
Figure 24.14
Foam Wings A th in shee t o f mat eri al is ve ry go o d a t ca rryi ng ten sion . A strip of my printer pap e r o nly 1.5 inch es wid e wi ll ca rry a ten sile load o f 20 sma ll ca ns of beer (or 15 pounds to ge t technical), How mu ch load do yo u thin k that pap er co uld carry in co mpression? No ne, becau se it jus t bu ckles up, unl ess you ca n find a way of sup po rting it. If I ro ll up a strip 9 inc hes wid e into a cylinde r one inch in d iam eter and 1.5 inch es lon g, add a da b of paste to hold it in position and sta nd it o n e nd, I ca n ge t that sa me paper to ca rry a co mpressive load of 4.5 pou nd s before it co lla pses s ide ways . \'{Iith b etter s u p port it wo u ld proba bl y ca rry even more load , bu t it w ill a lways be wea ke r Figure 21.13 in co m press io n th an te nsio n . Gi ving th e Sbeet s hee t th e right su p port is vitally imp ort ant to its co m press ive s t re ng th and stiffness . T ha t is w here th e foa m in a fo am w ing co mes in . It s up ports Sheet th e thin s ki n covering
't"· ~ ...
Extra material more effective bere l' tban bere
.- ...
, ,
.___-::::::::::::::::::=======r=n--------
Basic Aeronautics/o r Modellers
Rib
11 7
Figure 21. 16
Figure 21.15
Bending str eng th
Bending lo a d
Root
Tip
predict that whe n a foam wing fails th e ve neer will fail on the co mpressio n side and it w ill be du e to a lack of support.
J o ining Wings The object is to transfer the stresses from o ne wi ng to the othe r and so a suitable load path through wh ich the stresses can flow is necessary. The best w ay to join spars is the usu al plywood dih edral brace glue d to the front or rea r faces of the s p ars (o r pre ferabl y b oth faces to increase the glue area) . When the stress is ca rried in a sheet mat e rial , as in a veneered fo am w ing , the two sheets of ve neer should be joined smo othly to gether with, say, a strip of glas scloth and epoxy resin . Joining fo am wings w ith a pl y dihedral brace is as king for trouble. You ca n no t ex pect th e foam to tran sfe r any stres s, and the joint between the ply and the skins is o nly 0.03" thick, and that is IF you get a good fit. It doe sn 't give a lot of gluing are a! When foam wings MUST b e joined by spars, as when they plug in for exa mple , great care is needed to ens ure that the stresses ca n tran sfer from the sk ins to the sp ars without leaving a weak link . Some designers incorporate a ca rb o n fibr e spa r to ca rry all the bending load in the first place which mak es win g joining ea sier.
Tapered Strength The bending moment o n a wing is g rea te st a t th e root. Th e stru ctur e is therefo re design ed to co pe with the load at the root, but for building co nve nie nce the co ns tructio n is often the sa me fro m roo t to tip , Figure 21.15. Wha t a waste of streng th! It wo uld make sense to tape r th e stre ngth a bit to save weig ht. For examp le, very ofte n the w ing sect ion at the roo t is much th icke r than the o ptimum to ga in ex tra streng th w ithout adding to o mu ch ex tr a weight , a n d th e sectio n thickness reduces toward s the tip (as in the Zlin SOL whose win g is 18% thick at the root and 12% thick at the tip). The spa rs can be tap ered in cross sectiona l area towards the tip , and with a plywood dih edral bra ce the result is a wing whose stre ngth varies as in Figure 21.16 .
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Tip
Root
A similar effect can be obtained on a foam wing by tap erin g it in thickness and chord and by using a broad glasscloth joining strip . I usually use several thin layers o f gl a sscl oth , a narrow o ne ov er th e join a n d progr essivel y broader ones ex te nd ing se ve ral inches each side of the centreline.
Bending Stiffness Althou gh the bending strength is our chief co ncern, if th e w ing ha s insufficient bending stiffness it may be p ron e to so me form of flutte r which is undesirable. To cure or avoi d the problem use a stiffer mate rial of the sam e stre ng th a nd weight, or just beef it up a bit to make it stro nge r and stiffe r at the sa me time .
Strength ofDamaged Wings The slightest interrup tion in th e load path will cause a structure to fail. A slight crack (a we ak link) across a spar carrying te nsio n , or a chordwise crack in th e obechi veneer, can allow a wing to fold. Beware of the fact tha t bec au se plast ic coverings a re more flex ible th an the wooden s truc tur es underneath , it is possible for the structur e to be fatally cracked with no mark on the covering. If you can feel any so ftness, or any movement, or are in any doubt, strip off to inves tigate . Figure 21.17 shows typ ical dam age caused to a foam wing. The ve ne er cracks on one sid e and creases on the other side. Both surfaces should be levelled with filler and covered w ith glasscloth and epoxy res in to replace the loadpath . Sp lits along the grain of obechi ve nee r will not seriou sly weake n the wing in bending. If there is a cav ity in the foam , ca used by heat or c he m ica l atta ck , the v e n e e r wi ll n ot b e p roperl y sup ported and when a compressiv e lo ad is a p plied it may fail. I always join my wi ngs w ith epoxy resin and/or white glue as the y do not attack foam .
Strutted Wings Figure 21. 18 represents a wing a ttac hed to t h e fuselage at X by a bolt running fore and aft, so tha t it
Basic Aeronauticsfor Modellers
ac ts li ke a h in ge . T he Fig u re 2 1.17 wing is also su pporte d hal fw a y bet w e en ro ot Crease a n d tip b y a s tr u t attac he d at bo th e nds in a similar manne r. I based it loosely o n ~ a full size aeroplane and rounded o ff th e numFoam core b ers , ju st to gi ve a n a p prec ia tio n o f th e rel a tiv e sizes of th e 't forc es in vo lve d . Th e Crack strut makes an ang le Y (27 d e gr e es ) w it h th e w ing . In th e lo w e r d rawing the par ts have been se parated an d the I~ forces s hown mo re Filler clea rly. The stru t is o bviously ca nying a ten sion force. ~ Taking mom ents abo ut Glassclo tb X, I worked o ut that the s tr u t ten si on e q ua ls 1.1W, 10% mo re th an th e w ho le aeropla ne 's weig ht , the fuse lage by a w ire wou ld be just as effective but yo u there is a co mp ressive for ce of W o n each side of the w ill need o ne o n the top (a so -called land ing wire) to top of the fuselage and a pu ll of 1.1W to each side of hold up the wi ng when on the gro und, o r in inverted the fuselage bott om . flight an d a ll the sa me arguments ap ply to fully rigged T ha t is in s tea dy le vel fli ght. Pull 6'g ' a n d yo u biplan es. Try working o ut the load path s yo urse lf to see whic h mult iply all these fo rces by 6. A stro ng strut and good strong jo ints between it and the w ing are neede d , and pa rts of the str uc ture and rigging are ca rrying the heavy th e fu sel age mu st be s tro ng e no ug h to resist be in g loads. sq ueezed to ge the r at th e to p and pulle d a pa rt at th e bottom . What of the be nd ing Figure 2 1.18 m om ent? We ll a t th e s t rut pos itio n noth in g Tota l lift = b a lf uieigbt of aircraft lV/2 h a s c ha nge d so t he bend ing mom ent here is jus t what it wo uld be if th ere were n o s tr ut. In bo ard of the strut, the be nding mo ment car ried by the wing red uces to zero at the roo t. Bend ing mo me n t is g rea tes t a t th e s tr u t fix ing (as in Fig u re 21.19), so that is w he re to stre ng the n th e wing lVillg roots co mp ress in b end in g , a n d yo u Fuse/age z certa in ly mu st not jus t Tcos Y X A Tcos YY ~ ~-=======~=-=pc=-==<;========== d rill thro ug h the spar to Top Tcos Y=W fix the stru t. Th e bo nus in us ing T T w ing s tru ts is tha t the T= 1.1W m a x im um b e nd in g m o m ent is , in th is insta nce , on ly about a Wi llg struts pull quarter of wha t it wou ld fu s e/ag e bo ttom b e a t th e roo t of a ca ntileve r w ing (Le. o ne w itho u t a s tr u t) pro du cing the sa me lift. T Attaching the wing to
J
D
r
~ TCOS Y
/
Basic Aerona utics/o r Modellers
119
th e c ho rd . a n d th e p o sition o r th e Ilexura l a xi s . but a n y twi st b etw e en t h e ro ot a n d th e tip is und e s ir able a nd must be reduce d as far as possibl e .
Figu re 2 1. 19 Bending moment (ill a.\· bending m oment 4 times g rea ter 0 11 a ca utileuer ioing )
Built up Wings
\ \ \ \
\ \ \ \
\
{Cautlle oer uiing} \ \
,,
,, "-
ili a ,\' moment
"-
Root
The Torsional Stiffness of Wings You wi ll h a ve no ti c ed th at in t h e c h a p te rs on Aeroe lasti city and Tu c k Unde r I w as co nc e rne d w ith st ru ctura l STI FF NESS . n o t s t re n g t h. Eit h e r ail e ron de flect ion o r. 1110re us uall y. th e p itching moment Mo , te nds to twi st th e \\'ing lead ing edge do wn . The amo unt o r tw isting mom ent in a w ing wil l va ry w ith th e sec tion.
Figure 2 1.20
T he s im p lest w ay to d e s ign a t ors io na llv stiffe r Wing is to reduce t h e asp e ct ra t io . T h at costs g lid ing p erform an c e so w ould app ly o nly to powe red models. Nex t yo u must co ns id e r a stiffer struc tu re . Figure 21 . 20 re pr e s e nt s tw o wi ng struc tu res . I h a ve idea lised them a bit and m is sed out th e trailing e d ge p ie c e a n d co ve r ing . Both u s e the SAIVIE am ount o f w o od but type A w ith th e Dbo x is IIIUCH (may be 23 tim e s ) st i ffe r in torsion th an typ e B. The reaso n for the dramatic increa se in sti ffness . and stre ngth. is th e closed box . Th e tor sion al stiffn ess o f struc tu re B is th e s u m Tip o f its part s . but the to rs ion al s t iffness of structu re A d epends on the skin th ickn ess and the AREA e nclosed . Fully shec ting the wh o le wing w ill enclo se mor e than twi ce the a rea and giv e more th an twi ce the torsi on a l stre ng th and stiffn es s. Any add itio nal stiffne ss fro m the co ve ring wi ll depend up on the co ve ring mat e rial used .
Foam Wings Th e typi cal veneered foam wing is pr obably as stiff as . ma yb e eve n s t iffe r th an . a bu ilt up fu ll y sh e ered win g , Aga in the s t ru ct u re d e riv c s it s Co mp risingrs tre n g t h a n d s t iffn e s s 1'1'0111 being in th e form 3 pieces 6 ,\' 6 "1/11 o f a c lo s e d b o x a n d 2 pieces lOO,\' 2 "1/11 in add itio n th e o hec h i 1 p iece 25 ,\' 2 111/11 ve n e e r is a s t iffe r ma te rial than ba lsa ,
Comprisingt3 pieces 6 .\' 6 mm 1 piece 100 .v 4 mm 1 p iece 25 ,\' 2 111/11
120
R educed Stiffness Th e tors ion b o x MUST b e co m p le te ly e nc lose d , Make a s lit or a c ra c k ri ght a lo ng a typ e A struc ture a nd its
Ba sic Aeronauticsfor , 1Iodel/el ~~
stre ng th an d stiffn e ss d isa ppear. Simi larly , s pa n wise cracks in the ven e er o f a fo a m w ing can serio us ly reduce its tor sion al stiffness, altho ug h th ey may have no effect o n th e bend ing streng th .
Figure 21.21
Increased Stiffness Either a fully shee te d w ing or a ve neere d foa m w ing can be fur ther stiffene d by co ve ring, either with dop ed tissu e or p re ferably g lass , Kevlar or ca rbo n fibr es. Fo r max imum resis ta nc e to ,--------------------------------------------, to rsion al load s the fibreFigure 21.22 glass clo th or strands o f Kev lar or ca rbo n fibr e s shou ld be ap p lie d wi th th e fibr e s r u n n ing di a g o n all y a cro s s th e w ing as in Figure 21.2 1. All th e s e m o dern mat er ial s are light er st ronger and stiffer th an steel w ith ca rbo n fibr e the stiffest and Kev lar th e stro ngest (jus t). wooden up right s carry compression . As long as ten sion is ma inta ine d in the wi res th e glue d wooden jo ints are in sim p le co mpressi on and are therefo re ve ry stro ng . When modell e rs first co pied thi s built up struc ture Aside from th e for ces applied to fuselages by stru tted th e y left o ut the di ag o n al w ires for s im p lic ity , bu t w ings a lrea dy me n tione d , aero dy na m ic load s a re covered it in d op ed tissue , silk or ny lo n. The co ve ring u n like ly to exceed th e stre ng th o f th e fu sel ag e o f a w as st uc k we ll a ll a long th e long e ron s a nd to the norm al mod el aircraft. Th ere are how ever tw o as pects I u pri ght s as we ll, so th e ta u t cove ring s up p lies the wo uld like to h ighli ght. d iago na l te nsion force, and th e stru ctu re is adequately The first is th e bendin g stiffness of slender tai lbooms stro ng in both di rec tions . The ability of just the woode n structure , dep icte d in so me ti mes u sed o n m ode l g lide rs . As des cr ibe d in Chapte r 19, too flexible a tailbo o m cou ld lead to tuck Figur e 21.2 4, to ca rry a lo ad de pends en tire ly o n th e und er. What is required is a boo m o f ad equ ate de pth be ndin g stre ng th of th e g lued joints , w hic h is hardly fair. made of a st iff mate rial. A th in wa lled ca rbo n fib re tub e No r is it ve ry stro ng. It is no t even stro ng e nough to mai nta in its own sha pe aga ins t the tensio n in a curved might be a good so lutio n . A suita ble fix for an existing ina de quate structure mig ht be to add a fin strake as in long er on , o nce it h as b e en lift ed fro m the b u ild ing Figure 21.22 to stiffen the ta ilboo rn. O r you co u ld add bo ard . If a built-up struc ture is to b e stro ng eno ug h (wi thou t ca rb on fibres to top and bottom. Fig ure 21.23 rep rese n ts a struc tu re o fte n used o n a d o p e d fa br ic covering) to main tai n its own shape, aircraft of the \Xf\V'l era . Side fra mes were construc te d ne ve r m ind ca rry a lo ad , it needs cross-b raci ng . Yo u usi ng wooden longerons and upr ights , a nd each bay mu st in sert a d iag on al mem b er into each bay to carry te n si on o r co mp ression force s , as in Fig u re 2 1.25 . had tw o w ires ru n ni ng di agonally an d tensio ned wi th turnbu ckles. Th ese wires were necessar y to maint ain the In st e ad o f a si ng le d iago na l, a pai r o f th inn e r ones shape und e r lo ad . In th e case shown, wi th a down lo ad (gl ue d togethe r w here th ey cross) would also d o the job . from the tail, the w ires shown with arrows ca rry tens ion Diagona l membe rs are essent ial, but th e u pri gh ts are (the o the r wi res ca ter for an u pwa rd ta il for ce) and th e n o t. You ma y see a s tr uc tu re usi ng only diago na l
Fuselage Stiffness
Figure 21.23
Doumload from ta il
Tension ill top longerons and wires
Compression ill bottom Iongerons and uprights
Basic Aero nautics/or Modelle rs
121
Figure 21.24 Load
members between the lon gerons, becau se the y will take either ten sion or compression, making the vertical struts redundant.
Tailplanes The lift for ce on the tail depends o n the ai rspeed sq ua red, the tail area Sr and the tail's lift coefficient el T . Just as in wings the greatest bending moment is at the root. From Figure 12 .2 , the tail 's lift coefficient is generally upward at very low airsp eeds, and becomes more and more negative as airsp eed inc rea se s. If it is going to break in bending, it will bre ak off downward s at the root in a high speed dive when the speed and lift coefficient a re both great est. It is quite poss ible fo r tailplane s to flutter just like w in gs . The c ur e is to combine stiffness w ith stre ngth, use inflexibl e controls, and mass balance at the leading edge .
Pushrods Th e stiffness of pu sh rods cropped up in the chapter on tu ck unde r, bu t even if that is not one of yo ur problem s it is a good idea to make the pu sh rod s as stiff as possible to prevent blowback o f the controls a nd maintain control effectiveness . They should sho rte n as little as po ssible when compressed. Balsa pu shrods are the norm, a nd the bigger in diamet e r the better. Th e recent innovation of using a carbon fibre tube seems an excellent idea . Piano wire ends for connections sho uld be as shor t and stra igh t as possib le .
Figure 21.25 Doumload from tail
Usu al m odel structure
Woode" diag o nals m aint ain shape
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Basic Aeronauticsfor Modellers
Chapter 22
Centre ofGravity Position ince the publication of the first edition I have been wri ting a co lumn called "Ae rodynamic For um " in th e monthly magazin e "R/C Mod e l World ", also publish ed by Traplet. In th e co lum n I try to a nsw er qu estion s from readers, and fully half of the probl em s I e ncounte r invol ve findin g a safe Ce ntre o f Gravity. I fo und it useful to co llect together into o ne place all the theo ry relevant to CG locatio n - an d I thou ght yo u ma y find it useful too , so h e re it is added to th e second ed ition. The Centre of Grav ity is the ba lance poi nt, the poi nt through which weigh t ac ts or, if yo u are a stickler for acc uracy, it is the poi nt about w hich the weight has no mom e nt (rotation effect) . It is a three d ime nsio nal poi nt , bu t it s hould be ne ar enoug h o n th e centreli ne of a sym metrical aircraft, the heigh t doesn 't much ma tter, so a ll we are co nce rned about he re is its fore a nd aft position . From th e above de finit ion , the CG is the p o int a t whi ch yo u ca n sup port the model sus pe nded fro m string or balan ced on finger tips, the edge of a ruler , the blunt e n ds of tw o p en c ils , or o ne o f th e p u rpose made devices for checking CG position s. T he pos itio n of th e CG d et ermi n e s how mu ch stability the ae ro plane w ill ha ve , and also ho w mu ch co ntro l over it yo u w ill have . Stability an d Contro l are o p posite s;- the more of one yo u have th e less of the o the r yo u ge t, so CG position is a co mpromise . You are try ing to ge t just the right mix of Stability and Contro l. We are co ncerned only w ith Eleva to r Control he re , the rud de r and ailerons do not concern us. \Ve w ant the elevator to pitch the nose of the aeroplane up and down in a reliabl e and pred ictabl e fashio n, so tha t we have the full range of co ntro l witho ut it be ing ove rse ns itive . If yo u wa n t to arg ue t ha t th e CG is a four di me nsiona l po int, because it cha nges w ith time as fue l is b urned , w he n bombs are dropped or t h e un derca rriage is raise d, the n we plan for the worst case. Pla n th e aft-most CG whic h will be e nco u nte red in flight, Le. with the fue l tank empty if it is at the fro nt, or full if it is at the back.
S
If the CG is wror« As yo u move the CG forward from its ideal position the aeroplane becomes mor e stab le. That 's OK, but yo u start to run out of CONTROL o n elevator. You gradua lly lose the ability to sp in, stall, fly invert ed , loop , and even
Basic Aeronauticsfo r Modellers
ge t the nose u p for a flared o ut land ing. The elevato r trim becom es less effe ctive , re q uiring mo re and more trim movem ent when you cha nge from high speed trim to lo w speed trim , a nd yo u ne ed more a nd mor e up trim. As yo u move the CG rearw ard th e aircraft becomes le ss a n d le ss STA BLE. Yo u h a ve to ma ke co ns ta n t co rrectio ns to the flight path, yo u need to add down trim a nd the elevato r trim becomes se ns itive to sma ll movements maki ng it difficult to trim the aeroplane in le vel flig ht , a nd a s ma ll m o ve ment of the e levator prod uces a large co ntro l resp on se . If the CG is too far back an aeroplane beco mes UNSTABLE, its flight path persistently d iverges, it cannot be trimmed, and a small up elevator movemen t produces a gut wrenching, wi ng folding , loo p.
Th e Correct CG The CORRECT position for the CG is tha t which suits YOU th e p ilot. Your CG mi ght differ fro m someo ne else 's , but if it is right for you , the n it is the right CG. T he CG fro m a form ul a or th e p la n is a sa fe recommended starting point, then with ex pe rience you can experime nt. You perform various Fligh t Tes ts.
Flight Testing The popular "Dip" test ca n be use d o n most mod els , bu t is pa rtic ularly rel evant to e lec tric soarers, glide rs, tr a in e rs or vi ntage typ e mod el s, w hic h do no t ta ke kind ly to aeroba tic man oeu vres. First yo u trim the model o ut in straig ht steady flight. If it is a p o wer model it is best do ne flying level a t cr uising power, one thi rd to two third s throttle . Once trimmed o ut you pus h fo rwa rd on the elevator stick a little to put the model into a shallow d ive a nd hold it for a co uple of seconds to pick up speed . The n le t the stick return to neutral. A stable model w ill gently pitch nose up into a sligh t climb and its speed w ill reduce to its origi nal trimmed speed . Thi s initia l reaction shows us the 'Static Stabili ty' of th e model. If th e mod el ra pidly se ttles dow n in to steady le vel flight at its orig ina l trimmed speed the n that shows tha t it has good aerody na mic da mping as we ll, and so it has good "Dynamic Stability". This is the usu al resul t. It may be that the mod el will pitch up q uickly from
123
the sha llow di ve into too steep a climb with reducin g speed. It is Stability which pull s it o ut of the div e , trying to ge t back to its o rigina l trimmed s peed . Too mu ch sta bility will mak e it rear up too qui ckly, whi ch me an s the CG is too far forward . Paradoxica lly (see belo w ), yo u ne ed to rem ove weight from the nose , o r add it to the tail, and re-trim. It may be th at the mod el w ill co ntinue in th e di ve , p o s sibl y b e cau s e th e e levato r has n ot return ed to neutral , w h ic h is b ad . Wi e rea lly ne ed to ha ve fre e co n tro l s u rfaces whi ch a lways fo llow t he co n t ro l co mmands ex ac tly with out binding. If the co ntro ls are free but the mod el remains in the dive then it may not hav e e no ug h stability so yo u ne ed to mo ve th e Ce nt re o f Gravity forward a littl e . So me mod e ls not o nly stay in the div e, bu t they low er their noses eve n mor e, into a steepe r dive , and acce lerat e . If so close the throttle and pu ll o ut of the div e, urg e nt ly. Yo u have a problem. Either the CG is mu ch too far aft, or yo u have Tuck Unde r ca use d by fle xibi lity of the structure - o r a co mbina tio n of the two . To find o ut, try th e opposit e o f the dip test. From stea dy trimm ed flight ho ld a little up eleva tor to raise the no se a nd le t the model s lo w d own a bit , a nd th en re lea se it. If the model co ntinues to pit ch up then it is u nsta b le and ne eds a more forw ard CG . (O r is the e leva to r s tick ing in th e up p o sit ionr) If it b eha ve s norm all y to th e pitch up b y lowering its n os e a nd speedi ng up ag ain then look for a Structura l Flexibili ty probl em as a ca use of the tuck under. The wings may be tw ist ing, the e levator co ntrol may be flexin g o r the fuselage bending o r so me thing . You wo uld then need to stiffen up the structure (see Cha pters 18, 19 and 21).
The Paradox Th ere may be sceptics wh o find it difficu lt to beli eve that adding lead to the nose w ill help a mod e l pull o ut of a dive , so let 's look at it an other way. When yo u add the lead to the nose to mov e the CG furth er forwa rd yo u will need to re-trim for stead y flight at the same speed . Som e u p e leva to r trim w ill red uc e the tail lift, o r eve n ap ply a little down force , to support that we ight up front. When you spe ed up the mod el , the mom en t of the nose we igh t does no t change , but the effect o f this up trim is e n ha nced by t he s peed sq ua re d effe c t so it prod uces an increa sed nos e up mom ent. So it is actually th e up trim you applied to trim o ut the forward CG which is lifting the nose.
More Plight Tests Most mod els will be ca pable of a loo p . If the CG is too far aft it w ill loop tightl y with very little up eleva tor and will be prone to sc rewi ng o ut of the loop. Roll ing (or flicking) o ut of tight loops and stee p turns is a sign o f a n aft CG (or too mu ch e leva to r mo vem ent) . With a forward CG the loop will be large, or it may not ha ve e no ug h up elevator autho rity to ge t around a loop at all. My favo ur ite check is inve rte d flight. Ha lf-loop (o r ha lf-roll) the model a nd appl y down e leva to r to hold in vert ed flight. If yo u ne ed less th an a bo ut 3 mm o f down mov em ent o n the stick then I suggest a furt her fo rward CG - yo u have plent y o f co ntro l bu t insufficient s ta b il ity . If yo u n e ed m o re than h al f th e d ow n
124
mo vem e nt to fly ste ady inve rte d th en th e mod el ha s more than e no ug h stab ility and the CG co uld be mov ed aft a little to imp rove co ntro l. Aeropl an es with ca mbe red win gs o r lo w powe r may not be capa ble of susta ine d invert ed flight , so co nsider the o the r tests. Che ck o ut how the e leva tor trim va ries w ith airspeed . Se tting a trim position sho uld se t a flying speed (if the th ru st line is right a nd th e st ruc ture stiff) . T rim yo ur mod el to fly fast , th e n retrim it to fly slow ly, and th e cha nge in trim indi cates ho w mu ch sta bility yo u have . With a very aft CG the mod e l is se ns itive to trim change; a very sma ll trim cha nge is need ed between high speed flight a nd lo w speed flight. It is easier to fly a mod e l wh ich is positivel y sp eed stable. A good se t up for train ers and sca le mod e ls is to ad just th e CG and trims so that w ith fu ll up trim th e model is flying as slow ly as it ca n, right o n the stall, a nd it will fly le vel at its maximum speed wit h the trim half wa y between the ce nt re and full down. \Vith the trim se t for a fast cruise , close the throttle (if you hav e o ne) and maint ain level flight o r a slight climb. Check that yo u have e no ugh u p e levato r mov em e nt to stall the mod el. If the CG is ve ry far forw ard the n full up ma y a llo w the a irc ra ft to fly nos e d own , a bo ve its sta lling speed, le aving yo u witho ut e no ug h co ntrol to land the mod e l properly. I am sur e there a re man y other useful l1ight tests for CG pos ition , but th e fina l test for me is th e s pi n . A mod el w ith a n a ft CG will s p in eas ily whil e w ith a for ward CG it will not sp in at a ll. Adjus t it ho w yo u like it.
Popular Misunderstandings If a new mod el "kee ps wanting to climb" , do yo u put lead in the nose to mo ve the CG forwa rd? No , yo u add down trim . You ad just the elevator trim to ge t the model flyin g stra ight and le ve l 'ha nds o ff' and then p e rform so me of the tests a bove. You find after lan di ng that the e leva tor need s to be d ow n for trimm ed fli g ht, d o yo u mo ve th e CG to re move the trim requ irem en t? No! If the flight tests were satisfacto ry, it just mea ns that the wi ng (o r tail) is o n at the wrong ang le. To co rrec t fo r down eleva tor pa ck the w ing 's TE up o r the LE dow n , o r a ng le the tailp lan e itse lf mo re lead ing e dge u p if that is eas ie r (and vice ve rsa fo r up e lev ato r). The riggin g an gles o f the wi ng and tail a re nothing to d o with stab ility, they o nly get the mod e l to fly in trim . "I've change d the win g sectio n from se mi-symme trical to a less stable, flat bottom ed , sectio n:- s ho uld I cha nge th e CG?" No! Th e re is no s uc h thing as a sta b le o r unst a bl e w ing sec tio n . Th e wing sectio n d o es no t s ign ificantly a ffec t th e s tabi lity , but yo u may have to change the w ing or tail rigging a ng le for trim . "If I c ha nge to a lifting sectio n ta ilplane sho u ld I move the CG to co mpe nsa te?" No! Again, cha ngi ng the tail's se ctio n doesn't affec t stab ility - a nd it does NOT mak e it lift! In the followi ng sections yo u will not ice that the wing o r tail sections o r rigg ing ang les do not matter significantly in stability a nd so are not invo lved in the rules and formu lae for CG positio n. "The CG s hou ld be at the thickest part of the win g." It o fte n is, but that is pure co inc ide nce . Th ere is no scientifi c co nnec tion so ignore suc h ad vice . "T he CG s ho uld be just in front o f th e Ce ntre o f
Basic Aeronauticsfor Modellers
Pr e s s u re . " NO ! Suc h a dv ice b e tra ys a la c k o f understanding, so igno re it. Th e Centre o f Pressure of the win g is a floatin g po int wh ich o n a ca mbered wing moves from 30% chor d at slow speed to infin itely far beh ind the wing in a d ive . The C of I' is ahead of the wi ng whe n inverted . O n a sy mmet rical section w ing the C o f I' is fixed at 25% cho rd w ith the CG usually be hin d it. "In the dip tes t the mod el keeps d iving , so the nose mu st be too heavy?" No! There is not e no ug h stability to reco ver from the d ive so mo ve the CG forward (a nd check for struc tura l flexibility). The new CG will need more up trim.
Where should the CG ofa new model be? For a kit o r p lan built model start where the design e r says, though personally I woul d check first having seen a few se rious ly wrong ones. Th e CG will be mark ed o n the plan by a sy mbol like those in Fig ure 22.1. Start w ith this positio n and do the flight tests described abo ve to see ho w th e d es igner's CG su its YO U, a nd a dj us t if ne c e s s a ry. No w, s u p pose t ha t yo u r m od e l co m e s witho ut a suggested CG positio n, o r yo u don 't trust the plan, o r yo u have design ed yo u r o w n, w here do yo u start? Whe re is a go od CG position ? Do yo u start w ith a fo rward CG? And wh at IS a forward CG, or an aft CG? You mus t wo nder when I men tion a forward o r aft CG, Forwa rd o r Aft relative to \XTHAT?
What Matters? You ofte n find the CG mar ke d o n the fuselage , but if yo u move the wing the CG mu st go with it becau se it's the wing that ma tters. Th e CG may be marked o n the roo t chord or tip chord of the wing, but if yo u change the sweep of the w ing then the CG has to cha nge. Th e CG is refere nced to the MEAN CHO RD, but it co uld be at 50%, o r 30% or 15% or even ah ead of the w ing mea n chor d because its position depends upo n the WHO LE AEROP LANE, not just the wing. I w ill no w b rin g th e Ce ntre o f Press u re in to th e d iscu ssio n, but only to throw it straig ht o ut aga in . It is a floating po int o f no relevance to anything, ce rta inly not the CG. Any ad vice to positio n the CG relative to th e Ce nt re of Pre ss ur e , o r th e thi ckest par t o f the w ing sho uld be treated w ith deepest sus p icio n . The NEUTRAL POI NT is the referen ce po int for the CG. The NP is defined as the CG position at w hic h yo u
Figure 22.1
ts ge t Zero, o r Ne u tra l, Sta bility (the ba ll o n the p o ol table). An aft CG means back close in front of the NP (a ball in a sha llow d ish), and a forward CG me ans well ah e ad of the NI' (a ba ll in a dee p d ish). See Figure 22.2. T he Ne u tra l Poi n t (N P) b e lon g s to th e WH O LE AEROPLANE and is w here the STABILITY FORCE on the who le ae rop lane acts. This force (marked XL o n Figure 8.9) is ca used by a p itc h ang le cha nge, a nd it is th is fo rc e w h ic h ro ta tes th e a irc ra ft a bo u t its CG ba c k to w ard s its trimmed p os ition . The ma in part of th e Stability Force co me s from the win g, but the tail makes a major co ntribution and all o ther parts of the a eropla ne shou ld be co ns ide red.
In a Nutshell To calc ulate a CG position we need a refe re nce o n the aircraft, and fo r this we use the ave rage win g cho rd. \'lI e need to find the wing's mean chord, me asur e it, and measure the model's tail and fuselage, and in fact any pa rts w ith s ignifica nt ho rizon ta l area . The n we es timate w he re the Ne utra l Poi n t wil l be. We w ill c hoose a Stabi lity Margin wh ich will g ive the kind of han d ling we req uire , mark that off as a frac tion of the me an chord ahead of the NP, a nd there we have it a sa fe CG for th e first test flight.
Mean chords The first ste p in any CG calculation must be to find the mean (ano the r word for average) cho rd and transfer it to a s ide vie w o f th e aero p lan e . If th e wi ng has round e d tip s, just sq ua re the m o ff w ith a ch o rd lin e pa ra lle l to the roo t c ho rd . Mak e th e ex tra a rea yo u create in the co rners equa l the are a yo u cut off at the tip as in Figure 22.3.
GMC It is custo ma ry to d raw the roo t cho rd of the wing o n
Figure 22.2 Stable
3. Reactioll
More Stable 2. Disturbed . . - position
1. Initial position
2. Disturbed position
Basic Aerona uticsf or Modellers
125
Figure 22.4
Figure 22.3
Ct
Semi span A
C,. DistanceD
AC
Taper T = Tip C RootC
the aircraft centreline when calculating its chord and Area (gross are a) . The "Geometric Mean Chord" (GMC), also called the "Standard Mean Chord" (SMC), on a tapered wing is l1 (root c + tip c), and it is found half way between the centreline and tip . For example on a wing with a 10" root and a 7" tip the GMC is l1 00 + 7) =
MAC
4
Root chord
Tip chord Ct
Cl-
Ct
Meatl
Aerodynamic chord MAC
8.5".
C,.
MAC The Mean Aerodynamic Chord (MAC from now on) is the technically correct refe rence length for calculating aerodynamic forces and moments , and hence CG po sitions, but the MAC is defined in terms of complicated mathematics. The MAC is always bigger than the GMC, but only very slightly bigger, unless the wing is sharply tapered. For a simply tapered wing we can use the simple equation MAC
= root
20+t+t 2 ) chord x ----=.----'30 + t )
where t is the taper ratio , tip chord/root chord. The distance of the MAC from the centreline ('d' in Figure 22.4) can be found fromd
=
aO +2t)/30 +t)
For example , for the same wing as before with a root chord of 10" and tip chord of 7" taper ratio t = 0.7 and the MAC is given by MAC = 10·2·0+0.7+0.49)/ (3·0 +0.7)) = 8.588" The MAC is very slightly bigger than the GMC, but the difference is hardly worth bothering with . However, if a wing is more sh arply tapered, say the root is 21" and the tip is 6.3", the taper t is 0.3. So the GMC is 13.65 but the MAC is 14.97" or IS" in practical terms. This time there is a significant difference, especially if the wing is swept as well. The formula for d w ill tell you where on the wing it lies, to find the fore and aft position (vital if swept). An easy way of finding the MAC of a tapered wing is the graphical method , illustrated in Figure 22.4. You extend the root chord forward (or aft) by the length of the tip chord, and extend the tip chord aft (or forwa rd) by the length of the root chord (whichever fits your paper) . Then you join the points just marked with a diagonal line . Where either of these diagonals crosses the 50% chord line marks the position of the mean
126
chord. You can of course draw both diagonals and miss out the 50% chord line . This method finds the location of the MAC, which you then measure , and project onto the centreline.
Drop the Formality For most models, with little or no taper, it matters not the slightest which mean chord you use, because there is next to no difference. On deltas and sharply tapered wings I use the MAC, found as in Figure 22.4, but normally I just use whatever is ea siest, drop the formal capital letters, and call it the mean chord or average chord. When you find the location of the mean chord it is important to transfer it accurately to the side view of the fuselage, in the correct fore and aft position as indicated on Figure 22.4. While there, mark the Aerodynamic Centre (AC), or quarter chord point:- that 's a quarter of the mean chord from the front.
Elliptical Wings Sometimes the wing of an aeroplane has an elliptical planform (the Spitfire just springs to mind) or sometimes just the outer panel is elliptical. The MAC of an elliptical wing panel is 85% of its root chord, and you will find it 53% of the panel's span from its root chord, as show n in Figure 22.5. The panel Area = 0.785 x span x root chord. This also works for semi-circular panels, by the way , as they are just special ellipses.
Combining Panels Use the following method for wings with two panels,
Basic Aeronautics for Modellers
wh ich m a y ha ve differen t tapers a nd di fferent sweep back . Find th e MAC of eac h pan e l using th e formula e above or the g ra p hica l method in Figure 22 .4, an d then join the mean cho rds of th e 2 panels as in Figure 22.6. The d istance y in Fig 22.6 is found from
Figure 22.5 Sp m l2a
53% a
MAC
(Al · y1 + A2 · y2) y= (AI + A2) and th e length of the co mb ine d me an chord ca n be calculated from co mbined mean chord
(MAC1 - MAC2)(y 2- y ) (y 2 - y1)
= MAC2 + - - - - - - - , . .--'-------''-
Having found th e MAC o f th e wing, th e tail and th e foreplane , tran sfer them all acc urately to the ce ntreline of the aircra ft a nd ma rk them o n the side o r plan view, o r on the aeroplane itsel f.
Flying Wing If yo ur mod el is a flyin g wi ng yo u ne ed go no further. Put the CG at 15% of the mean cho rd to start with, and rig so me elevon reflex for trim . Having found the mean cho rd , and mea sured it at say 8.)", mark yo u r mean cho rd o n th e sid e view in th e co rrec t locati on , and mea sure hack 15% of 8.)" , that's 1.275" back fro m the front of the mean ch ord , and there is yo ur CG. However most aircraft have a tailpl an e or , as they say in Amer ica, a ho rizontal stabili ser and yo u will have to make allowa nces fo r it.
Tail Volume Ratio Ta il Volum e Ratio, V-bar, is the tail area , as a frac tion of the wing area , times the tail arm as a mu ltiple of the wing me an cho rd , and it is a m ea s ure o f the e ffec tive ness of th e Figure 22.6 tailplane , o r ho rizon tal s tab ilise r. A s mall tail ,,, w ith a lon g le vera g e ) '1 : ar m will have the sa me - - - -- -- i>;, e ffe c t a s a larg e tail ,, ne are r to the wing. Being dim en sionless, _______ _ : ,, it is th e s a me at any ,, s ca le , and u s in g any MA:C uni ts. You can measure ,, 1 th e fu ll s ize ai rcra ft in - - - - - - - - ,- ,, fe et , yo ur m odel in ,, inch es , o r a sma ll scale , dr awing in millimetres, a n d yo u get th e same answer. PallelI It is u su a l to u s e A rea A l g ro s s w ing a re a (i ,e ,
Basic Aeronauticsfor Mode llers
including the area inside the fuselage) and net tail area ( i,e . on ly th e ar ea o ut in th e a irflo w) . Tail a rm is mea sured between the q uarter chord po ints of the wing and tail me an cho rds. Wo rk it out o n the No mogram in App en dix E, Figure E.5 or fro m V-ba r
tail area win g area
tail arm wing MAC
- - - - x - - - --
Biplanes Use the following proced ure to find the mean chord of a biplane, o r a sesquiplane wit h unequa l wi ngs . For example the Fokker DVII has two unequa l wings with no rma l (positive) stagger, Le. the top wing is ahead of th e bo ttom wing . I work ed o u t tha t th e top wing su p p lie d 6 1% of th e to tal wing ar ea , a nd the bottom wing obviously 39%. Referr ing no w to Figure 22.7, I joined the mean cho rd lines of the top and bottom w ings. I th en divided th e ga p in th e rati o o f th e wing area s, 39 :61. Th e mean cho rd is 39% of the gap from the upper wing . (So me autho rs bia s the mean chord more toward s the top wing b ut I have k ept it s im p le r a n d mad e ge ne ro u s allow ances else whe re.) The CG formu lae in Cha p te r 8 do not a p p ly to bip lanes b e c a use t here is n earl y twi c e a s much
L _
----
Mea " c h o r d of co m bina tion
Panel Z
A rea A2
12 7
Th us we need to co nsi de r th e a rea o f ea ch co m po ne n t, th e point Top uiing a rea A l o n it wh ere th e lift c ha nge a ct s , a n d it s Ch o r d Cl re la t ive effic ie ncy in turnin g a o ne d e gre e pit ch u p into extra lift. O n a n aero fo il th e lift c h a nge ac ts a t th e Aero -d yn am ic Ce n t re, o r AC (q ua rte r ch ord) . M EA N CHORD O n o t he r o d d bits o f area, like the nose or na ce lles, I shall assum e , fo r co n-sistency, that the ex tra lift ac ts a q uarter o f its le n g th from th e Gap front. \ Al All parts beh ind the Gap x \ Al + A2 wing ar e le ss effe ctive ____ __ g.1.-O:..I~«- Q _ b e ca us e o f th e w ing 's wake and downwash. Air wil l spill around Lower toing area A2 th e fu s e la ge s id es, redu cin g it s lift c h a nge A I. Cl + A2.C2 co ns idera b ly , unl e s s Mean cbord c = A l +A 2 th e re is a ca na rd o n e it he r s ide . Let 's tak e o ne bit at a time , and as an exa mple I'll use my du cted fan Spectre , illustrated in downwash as the formul ae allow for , and tw ice as mu ch Figure 22.8. inte rfe ren ce to the airflow , Le . loss of a irspeed ov er the tail. Ho wever, I have fou nd that if yo u red uce both the Tail Volume and the win g Aspect Rat io by dividing them The Wing's the Boss by the number of w ings , th en putti ng th ese re du ced Most of the lift change we are chasing co mes from num be rs into the ori ginal formula gives a se ns ible NP th e wing and acts through th e Aeroce ntre , o r qua rter shift from the no rnograrn in Figure E.7. chord point (o f the MAC). If the a ircra ft is a flying wing In the case of the Fokke r, I ca lled the top wing 1 and th at is the e nd o f the story. The NP is at the wing 's the lowe r w ing is 0.64 of it, so the number o f wings is 1.64. That gave a reduced AR of 3.3 and a reduced V-bar q uarte r chord point, o r 0.25 MAC from the LE. So the of 0.24 which in my usual Cha pte r 8 formula gav e a CG eq uation for the NP of a flying wi ng is simply position at 18% of mean chor d, whi ch gave sat isfacto ry NP positio n = 0.25 handling. as a fraction of the MAC aft of its LE. As imp lied above, I advi se a Stability Margi n of 10% MAC for flying wings Finding the NP g iving a CG at 15% MAC. Fo r othe r a ircraft we can start from this neutral point Th e actu al lift on a co mpo ne nt does no t matter. \'\fe ne ed to wor k o ut the lift curve slo pe, or LIFT CHANGE and mak e a dj us tme n ts for a ll th e o the r pa rts o f the per degree of an gle change, a nd multiply by the area . ai rcraft whose litt le bits o f add itio na l e ffect are to be adde d or su btracted .
Fig ure 22.7
\
\
\
\
\
\
__
---=-~~--W
Figure 22.8
Adjustments The Tail V-bar
128
You could g uess from its o the r nam e , the hori zo ntal stabiliser, that th e ta ilpl an e is import ant. Following as it does a co u ple of chords beh ind th e w in g , it is imm ersed in th e w ing's w ak e a n d d o wn was h , and has a lo we r aspect ratio. The re are co mp lex
Basic Aerona utics/or Modellers
and acc urate ways of wor king out the tail's effec t, but my emp irical es timate ha s proven clo se e no ugh and is easy to appl y. NI' co rrec tion
=
0.25*(AR)t\0.25*V-bar
On a calculator with squa re roots enter the w ing Aspect Rat io (s pa n/c h or d), tak e th e sq ua re ro ot , a nd press squa re roo t again . Multipl y by 0.25 an d the V-b ar. For ex am ple, my Spec tre 's Asp ect Ratio (s pa n/cho rd) is 4.19, the squa re root is 2.047 an d the square root of 2.047 is 1.43. Ta il volume is 0.292 and th is gives NI' correc tio n = (0 .25 x1.43 x 0.292) = 0.104 (positive as it moves the NI' aft) Th e Tail moves the NI' aft 0.104 times the !vIAC, from 0.25 MAC to 0.354 MAe. If yo u do no t like the for mula, you ca n't go w rong with Figure E.7 in App end ix E, a nomogram w hic h is easier than "jo in the dots". Ju st mark the wing's Aspect Rati o (spa n /c hord rat io ) a n d the V-ba r , a n d th e suggeste d NI' co rrec tio n is on th e middle lin e . For a biplane use 'fac tore d' AR and V-ba r as d iscu ssed abo ve under "Biplanes".
The Fuselage We norm ally ass ume that there is on e chord le ngth of fuselage ahead of the w ing and 2 or 3 cho rds beh ind , so by using th e gross w ing area we hav e taken a suitable proportion of the fusel age area into account, and we hav e ass u med th at th e quarte r le n g th p o int of th e fuselage coi nc ides w ith the qu arter chord point of the wing. On most aircraft that ta kes care of the fuselage , but the Spectre has a very lo ng nose . The uns ha de d pa rt o n Fig ure 22 .8 wo u ld b e a "no rma l" fu s el a g e . Th e sha de d part is "Excess area" and will tend to de-st ab ilise the aircraft. Rou ghly sq ua re off th e curves to me asure the fuselage area w hich is more than o ne !vIAC len gth ahead of the wing Aero centre, and mak e it into a nose volum e rat io b y d iv idin g by t he wi ng a rea a n d multip lying by the d istan ce of its "q uarter cho rd " point from the wing Aeroce nt re, d ivide d by the !vIAe. So n ose vo lu me Vn = ( nose a rea x n o s e distance)/(wi ng area x w ing MAC) In the case of the Spectre this turned o ut to be (90 x 30)/(553 x 12.63) = 0.387 Th is a rea is n ot very e ffec tive so m ulti p ly by a co nstant 0.2 from Tabl e 22.1 to allow for th e air spillage aro und the nose. Th e no se moves the NI' (fo rwa rd) by 0.387 x 0.2 = 0.077 or 7.7% of MAe.
the air has less cha nce to "spill" arou nd the nose with the fore plane in the way.
LE Extensions O n the Spectre I have tre ated the wing LE extensio ns (co loure d dark in Fig ure 22.8) in a s imi lar way to "Io re p lane " a rea . Th eir ar ea times d ist an ce ah e ad o f win g Ae rocentr e gives a volume of o nly 0.025 , an d they are suc h narrow strips th at ai r will sp ill aro und th e m givi ng a Fac tor of only, let 's say, 0.4. They mov e the NI' fo rward by a vo lume rat io Vf = 0.025 x 0.4 . or o nly 1% of !vIAe. For any o the r od d b its of area on yo ur model mak e allowance in a sim ilar way and use your jud ge me nt and Table 22 .1 to c hoose a su ita b le vo lu me rati o and efficiency facto r.
Floats If an a irc ra ft is co nverte d to a sea p la ne b y fitting floats, th e NI' is likely to ch an ge . Floa ts o n a D/ F Spectre wou ld st re tch cre di bility too far, but my So na s sport ae roba tic model has a span of 73", w ing area of 993 sq . in ., a nd flo ats 40" long with 300 s q . in . of area. Th e float s are rigge d w ith th e ir ce ntres abou t o n th e CG which puts the ir qu art er chor d point 9" or 66% of the wing mean chord ahead of the w ing Ae. Float area is 300/993 or 30% of wing area so the ir volume coefficient is 0.2. Multiplying by the factor 0.2 for long slim floats from the table g ives a NI' shift (forward) of 0.04 or 4% of !vIAe.
V-Tails Th e to ta l a rea of a V-ta il is the se mi-s pa n of o ne panel, measured a long its surface, times two , times its ave rage chord . Its projecte d area o n a hor izont al surface is n ot , as yo u m ight s u ppose, its effective a rea as ho rizontal stabilise r.
Figure 22.9 Include shaded fuselage area with foreplane
Foreplane arm
The Poreplane
F
The for ep lan e , or ca na rd , sits right o u t in fro nt in und isturbed air, so it is as efficient as the w ing . Wor k o ut a Fo repl an e vo lu me u s in g th e n omo gra m in Ap pe nd ix E (Figure E.5) or the formula Vf == forep lane area x foreplane arm w ing area w ing mean cho rd Use the forepl an e 's gross ar ea (acting at its q uar ter chord) , a n d in cl u de th e fu s el a ge a rea ba c k to th e foreplane trailing e dge as shown in Figure 22.9. I use the gross area and a factor of 1 (or even 1.2 if the cana rd has less sweep and a high er AR than the wing) because
Basic Aerona utics/or Modellers
129
T he ta il, forep la ne a nd nose vo lu mes are each mult ip lied by a facto r chosen using the ta bl e as g u idance. A nose is slim if its le ngt h is, sa y, mo re than three times its average width , while a fat no se is as broad as it's lo ng . A fore p la ne 's AR is high if it is more than the wi ng 's .
Stability Margill Reme m be r ne a r th e beginning (Figure 22 .2) w he n I co m pare d th e Sta b ili ty of a n aircra ft flyin g a long in trim to a ball in a dish ? Well th e Fitting floats to Sonas, a 73" span ASP 108 powered sport model moved its NP Sta bi lity Marg in is th e forward all estimated 4 % of toing MAC. steepness of the d ish . In a very sha llow dish th e ball w ill just gra dua lly ro ll back For di he dra l o r a nhe dra l tail s , o r slo pi ng fins , th e into the ce ntre . In a deeper , steep side d , dish the ball effective a rea is the tot al area times the cosine of the will retu rn to the centre mo re q uick ly. d ihedra l a ngle , SQUA RED. Yo u h a ve to sq uare it becau se the area is redu ced by the slo pe, and so it its If a trimmed ai rcraft is pitched up slightly there w ill ang le of attack. be a STABILITY FORCE at the NI' w hic h w ill rotate it For example , if the included ang le of the Vee is 110 no se down about the CG. Th e further the CG is ahead of the NI' the more leve rage this Stab ility Forc e has so its degrees then the d ihed ral angle is 35 degrees, cos35= gre ater moment will restore stability more q uickly. That's 0.8192 , co s sq uared is 0.671, so the area effective as a tail is 67% of the to tal area . why a forward CG gives more Stability. How mu ch Stability Margin yo u use depends o n yo u a n d the mode ls you fly a n d your technique and Putting it all together experience. I always recomme nd a SM of 0.15 or 15% of On a complex aero pla ne wit h engine nacelles , a MAC. In the event of a 5% error in NI' position (about as ta ilplane a nd l or ca nard an d a lo ng nose the NI' form ula close as we ca n calc ulate ) we still have a flyab le aircraft. w ill use th e wing MAC as a re fe rence , sta rt w ith the Ho we ver I kn ow that co mpe tition g liders are flow n w ith less stability, and it seems that jet fight e rs and sport jets wing AC positio n at 25%, add the stab ilising e ffec t of use 10% (o r less) Stability . Cana rds a nd flyin g wings p arts aft of th e a ircraft like th e ta il, a nd sub tract the destab ilising effects of the for e pl an e and excess nose should use a SM of 10% of MAC. After test flying yo u area to e nd up as can adjus t it furth er aft as yo u wis h. O n my Spectre the NI' ca lculate d above is at 0.267 NI' = 0.25 + tail bit - foreplan e bit - nose bit chord o r 26.7% of MAC. A Stabilit y Margin o f 0.1 (or 10% MAC) gives a CG position of 16.7% MAC w hic h is 0.167 x 12.63" = 2.1" af t of th e MAC Lead ing Edge . Th at is In the case of the Spectre th is gives abo ut 0 .1" (o r 2.5 mm) a head o f th e manufacturer's recomme nde d point, and it flies just fine at that. NI' = 0.25 + 0.104 - 0.4 x 0.025 - 0.2 x 0.387 = .25 + .104 - .01 - .077 = 0.267
Table 22.1 Component Wing Tail slim nose fat no se Wing strakes ca nar d (foreplane) high AR canard LE Nace lles Aft Nace lles Floa ts
130
Factor 1 0.25 x ARwl\.25 0.2 0.4 0.4
Figure 22.10
1
1.2 0.4 0.2 0.2
Basic Aeronautics/or Modellers
Appendices Appendix A Bernoulli's Equation Thi s e q u a tio n , wh ich w as d e ri ve d th e oret icall y , describes how the pressure var ies w ith velocity in the flo w . It holds go od if th e flow is stea dy, e ne rgy is co nserve d, the flow speed is we ll below the speed of so und , and the fluid ha s no viscosi ty. It therefore a pplies ap p rox ima tely to air flowing around slo w aerop lanes, e xc e p t in th e bo u n dary la ye r wh ere v isc osi ty is imp ort an t. At any point alo ng a particul ar strea mline . • p + Y, P V2 = co ns t (ca lled th e "to tal pr essure " o r "stagnatio n pressure" of the flow ) • where P = stati c pr essur e at the point as measured by a barometer moving with the fluid. • P = air de nsity • V = air ve loci ty at the point. • y, p V2 is called the "dynamic pressure " of the flow .
B2. A go od demonstration of Lam inar and T ur bule nt flow is sho wn in Figure B3 wh erein a large tank of still water is run off through a glass tub e . At th e inlet of the tube a sma ll filam en t of liquid dye is introduced into the water strea m . At first the filam ent of dye is so steady as to appear statio nary. Furth er alo ng the tube it wave rs and th en breaks up and mixes wi th the water as the flow tran sition s into turbulent flow . Lik e th e w at e r in th e tub e , th e bound ar y la y e r comme nce s as laminar near the leading edge of a bod y and becom es turbulent at the "transitio n p oint " whose position depends upon spee d, viscosity and surface rou ghness.
Figure Bl H e ight
A "p ito t" tube (a n open ende d tu be fa cin g th e flow) me asures th e "to ta l pres su re ". An a irs peed ind icat o r s u b trac ts s ta tic pressure fro m tot al pressu re to ge t d ynamic p ressure from w h ic h ai rs peed ca n th en b e calculated.
Appendix B Boundary Layer It is a fa ct th at fo r all flu id s flo w in g p ast a solid sur face , wh ether water in p ipes o r air over wings, the m ol e cul es of flu id ne xt to th e s urface d o not mo ve relative to it. As th e relative ve loci ty at th e sur face is zero , it follows that there mu st be a regi on in th e flow wh ere the velocity rises gradua lly from ze ro to the free strea m ve locity. Th e region may be very thin but within it the velocity rises co ntinuo usly even if rap idly. There can be no abrupt step in veloci ty. Thi s region is called the "Boundary Layer ".
Two Types ofFlow Two types of flow can ex ist in the bo undary layer. If speed is low o r viscosity is high , the flow w ill be smoo th and laminar with no vertical movem ent, like man y thin sheets moving over eac h othe r, each slightly faster tha n the o ne be low . Th e ve loci ty p rofil e in th e bounda ry layer is show n in Figure Bl. But if spee d is high or viscosity is low, th e flow will . be turbulent. In tur bul en t flow there is greater mixing o f the particl es and so the boundary layer is not so slow but gives mor e dr ag. Th e slow mo vin g flu id part icles near the surface are continu ally re-en ergised by mixin g with the faster particles fro m furth e r out in the stre am giving great er drag. The veloc ity profil e is as in Figure
Basic Aerona uticsforModellers
Vel
Fig u r e B2 Height
Vel
131
FigureB3
Dye
Water
Laminar Flow
Turbulent Flow
~~ Transistion
Boundary Layer Thickness The thickness of the boundary layer is usually defined as the distance from the surface at which velocity of air reached 99% of the free steam velocity. The boundary layer thickness grows as it travel over the surface and a turbulent boundary level thickens more rapidly than a laminar one. For example a larninar boundary layer might reach a thickness of 1 mm after travelling 200 mm over a smooth flat plate. Whereas a turbulent boundary layer might be 5 mm thick 200 mm after becoming turbulent.
Appendix C Vortices There are two kinds of rotating flow. In the forced vortex the speed of the fluid is proportional to its FigureC2
Figure Cl
Forced Vortex
Free Vortex
distance from the centre and it is seen when water is stirred round in a drum for example. The surface forms as in Figure Cl. The other kind is the free vortex in which the speed of the particles of fluid reduces with increasing radius and the surface would be as in Figure C2. (P article velocity is inversely proportional to radius). This is the kind of vortex we see when we watch the bathwater go down the plug hole, blow a smoke ring or see a tornado approaching. All natural vortices are these free vortices. However when the fluid speed reaches nature's practical limit, e .g. at the centre of a tornado, the vortex breaks down at the centre into a forced vortex (forced by the fluid's viscosity) in which the speed reduces towards the centre (see Figure C3). Wing vortices are of course free vortices, therefore the speed of rotation is greatest near the centre and reduces with distance from the Figure C3 vortex core. Vortices can never end abruptly except against a boundary, e .g. a container side , wind tunnel wall, or the water surface. They may be in continuous loops as in a smoke ring or they may Free continue in the fluid Vortex until dissipated gradually by the viscosity .
Forced Core
Appendix D Dihedral and Sweep From Figure Dl, if an
132
Basic Aeronauticsfor Modellers
FigureD}
Pigure Dz
Airflow Velocity
Sideslip Component
-- _
Xc<
..
~ '
u
V
=
u (Upuiash Component)
• PQ = d/ cosL • XQ = d/cos(L-A) • YQ = d/ cos(L+A)
ae ro plane has a small sideslip veloci ty, then the an gle of sideslip A is given by • A ' = . v/V
Figure D2 is a vie w from the rea r o f a n ae roplan e w ith a d ih edral a ng le D. In Figure D2 th e si des lip velo city v has be en sp lit in to tw o co mpo nents - o ne p ar all el to th e wing w hi c h ha s no e ffe c t, and o ne perpendicul ar to th e wi ng w hich w ill be a n up wa sh ve loci ty u o n the right win g and a downwash o n the left wing. From Figure D2 th e upw ash co mpone nt u = v sin D From Figure D3 the ex tra ang le of attack du e to the up wash (Xa) is approx imately given by u Xa = V
D
at ang le L. The air flows in the dire ction PQ across the cho rd . But whe n the ae ro plane is sides lip ping , the air flow s alo ng YQ for left sides lip , and alo ng XQ for right sideslip . Both sideslip a ng les a re A. From the figure you ca n see that distan ce
FigureD3
Resultant Ait:flo,!:! _ r - -
V SIN
Th e le ading edg e is at a co ns tant height Y a bo ve th e trailin g e dge so a s we c a n see from Fig u re D .5 sin ( n -X«) = y/X Q ' =. (o -rXu) if a is fairly sma ll, so that ex + Xo ex - Xo
= =
cos (L-A).y/ d cos (L+A).y/ d
e q uatio n 1 equa tio n 2
now subtract e qua tio n 1 - e qua tion 2 to ge t 2Xa = cos (L-A).y/ d - cos (L+A).y/ d y/ d .(cosL.cosA + sinL.sinA - cos L.cosA + sinL.sinA) = 2y/ d .sin L.sin A equa tio n 3 =
FigureD4
and th ere is a co rre s pond ing d e creas e o n the left w ing .
I I I
Straight Airflow
1
Therefore v.sin D
Xa = -
-
V
- =A .
D
I
Sidesiipping Airflow
if both ang les are sm all. All an gles mu st be in radian me asure . ( O ne radi an is an a ng le of 180 ""it degrees = 57.29578 degrees). Th e change in an gle of attac k du e to sideslip derived from d ihedral is g ive n b y Xa = A.D whe re A = sideslip angle D = d ihed ral ang le . Fi gure D .4 s h ows a n aerop lane w ith parall el cho rd win gs swept ba ck
Basic Aeronautics/o r Modellers
133
Figure D5
~
Airflow
From Figure D5 a eq uatio n 3 gives
y.c os l./ d which when put in
2Xex = 2ex tan L.sinA so Xo = ex sinA.tanL o r Xn = ex A.tani if the sideslip angle is small. All angles mu st be in radian mea sure. Th e change in angle of attac k du e to sid eslip derived from swee pback is given by Xa = a A tanL wh er e • • • •
A = sideslip angle L = an gle of sweep-back a = an gle of atta ck (ave rage) Xo; = chang e in an gle of atta ck (average between 2 wings)
Th e follow ing books have be en my main referen ces. Th ey are listed as Auth or, TITLE, publisher (no tes).
Reference List 1.
A W Bab ist er , AIR CRAFT STA BILITY AND CO NTROL , Pe rgamon Pr ess , (for e q u a tio ns of a irc ra ft st abilit y a nd trim , tail loading , and tail setting angle) .
2.
F G Irving , AN INTRODUCTI ON TO THE LONGIT UDI NAL STATIC STABILITY OF LOWSPEED AIRCRAFT, Pergam on Pre ss, (fo r equation s of aircraft stability and trim , tail load ing, and tail setting an gle).
3.
4.
Ab b ott and Doe n h o ff, THEORY OF WI NG SECTI O NS, Do ver Publi cation s , ( Masses of infor mation on the NACA families of wing sec tions and famil y relationships, alb eit at high Re) . N A V Pie rcy , AERODYNAMIC S, English Un iversities Press Ltd, (w ing downwash and gen era l ba ckground information).
5.
A C Kermode, MECHANICS OF FLIGHT, Pitman , (general ba ckground).
6.
Den Hartog , MECHANI CS, Do ver Publicat ions , (re la tive motion , momentum and kinetic e ne rgy consid eration s for the cha pte r on wind).
7.
Martin Sim o n s , MODEL AIR CRAFT AERO DYNAMICS, Argu s Books, (sectio n data at low Re , and ge ne ral back ground .Re commended re ading for co mpetitio n glide r or free flight enthusiasts.).
8.
Dieter Althau s , PROFILPOLAREN FUR DEN
13 4
MOD ELLFLUG , Nec k ar-Ve rl a g, ( e xce lle n t information on model sectio ns at model Re with flap s and turbulators , but the text is in Ge rma n). 9.
Seli g Donovan & Fr a s er , AIRFOIiS AT LOW SPEEDS, Soartech/H . A. Stokely, (valuable test s o n sectio ns at model Re).
10.
Selig Gug lie lmo Broeren & Giguere, SUMMARY O F LOW-SPEED AIRFOIL DATA - VOL.1, Soartech/H. A. Stokely, (tes ts o n more sectio ns at model Re).
11.
Seli g Lyon G ig ue re N inh am & G u g l ie lm o, SUMMARY OF LOW -SPEED AIRFOIL DATA VOL.2, Soarte ch/ H. A. Stok ely , (te sts o n even more sections at model Re) . The whole se ries of Soartech book s is ava ilable from SoarTec h Publications, 1504 N. Ho rsesh oe Circle, Virginia Beach, Virginia 23451, USA
Basic Aeronautics f or Modellers
The exa mp le o n the nomo gram bel ow is o f a 10 " prop doin g 12000 rp m and its tip speed is just below the limi t. To chec k yo ur prop , mark a d ot a t yo ur prop d iam et e r o n line A, and a dot o n yo ur e ng ine rpm o n line C, join them with a ruler and read off o n line 13 yo ur pr op's tip spe e d in ft/s.
Appendix E This co llec tio n of nomogram s is meant to he lp you get an sw ers withou t even kn owing you a re de aling with math em atical e quatio ns . It is just a ga me o f join the dots. This first one , Figure El , is to hel p yo u ge t started us ing nom ogram s and also to save flying fie lds. O nce we bu y a mod e l engi ne it is too late to worry abo ut its design . \V'e co uld bu y an add-on ex tra silence r, but th e easies t way to make them qui eter is to red uce the tip speed of the prope llers. On ce the tip speed of the prop s exceeds 550 ft/s (ha lf the sp eed of so und) the prop noise seems to predominate and become un acceptable.
A
Figure El
C 30
35 30
25
25 B
M acb2
20
20
15 Macb 1
15
800
10
9 10
8
9 7 8
6 7
5
6 100
5
4 80
Tip speed
3.5 4 Prop diameter in ches
Basic Aeron autics for Modellers
Prop RPM ill 1000s
135
O r the a nswer co u ld be w ha t wi ng ar ea is needed to s u p port a g ive n w e ight fo r a d esired wi ng lo ading . O r yo u mig ht wan t to kn ow th e finish ed target we ight for a mo del g ive n its wing a rea and d esired wi ng loading.
Another simple nomogra rn. Figure E2, links th e wing ar ea o n line A via th e model weig ht o n line B to th e an sw e r, the model' s wi ng lo ad ing in po unds o r o un ce s per sq uare foot , o n line C.
Figure £2 C
A
B
6
90
30 150
40 00
2000
25 100 3 000
100
20
2500
15
1500
80 70 60
1000
50
800
40
2000 30
5
80 70
4 60
3.5 3
50
600 500
2.5
40
35
400
2
1500
30
1.5 1000
25
20
900 800
15
700
4
600
3
60 50 40
500
2
10 30
1.5
400
20
3.5 300
1 2
250
1.5
15
.8 10
.5
8
.4
.s 1
ft 2 ins 2 Willg A,'ea
136
.2 Ibs
7
5
6
5 4
150
8
6
.6
200
9
3
OZS Weigbt
4
3.5 3 Ibs/ft2 OZS/ft2 Willg Loading
Basic Aerona uticsfor Modellers
Yo u ca n use the wing lo ad ing fro m Figure E2 to find o ut from Figure E3 w hat speed your model will fly at a particular lift coefficie nt , or wh at lift coefficient it needs for a pa rticular speed. In particul ar yo u can estimate the stalling speed . Accord ing to Selig's tests in Soa rtech 8 the Clark Y section at mo de l Reynolds Numbers has a maximu m lift
coefficien t of 1.2. Becau se of uneven lift d istribution and lo s s e s aro u n d th e fuse lage th e ove ra ll wing li ft coefficient w he n it stalls may be a bo ut 1. Plott ing 1 o n line C and a w ing -loading of 20 oz /sq . ft. o n line A gives a likely sta lling speed in stea dy level flight of aro und 22 ftl s on line B.
15
4 1
0.1
138
CL
m 2/ft2
Lift Coefft
Willg Area S
N/lb Lift
m/sft/s Speed V
Basic Aerona utics f or Modellers
Tail Volume Ratio (or Coefficient) is conventionally written in textbooks as a capital V wi th a bar over it, but as that is tricky to type I shall write it as it is said, V-bar. It is a measure of the effectiveness of the tailplane . You will find it in all the best CG formulae and design criteria (though sometimes in disguise) . Basically it is the tail area, as a fraction of the wing area, times the tail arm as a multiple of the wing mean chord. It is usu al to use gross wing area (Le. including the area inside the fuselage) and net tail area (Le. on ly the area out in the airflow). Tail arm is measured between the quarter chord points of the wing and tail mean
chords. Start at the sides of Figure ES and work towards the centre. Mark the known values on lines A, B, F and G, join A to B a nd continue to C, join G to F and continue to E. Then join the points on C and E to get your answer on D in the middle. Use this nomogram for canard foreplanes as well. A typical value is 0.4 to 0.7 for normal RC models, maybe down to 0.3 for gliders while some free flight and vintage models, and the odd sca le model, can have Vbar over 1.
FigureE5 B
A
C
E
D
F
G
2000
15 00
30 V-bar
1000
= ST x.lI. S
25
c
150
800
1000 900 800 700
600 5 00
6
400
5
3 00
1.5
20
300 25 0 200
.3
10
20
9 8
7
6 10 8
.2
1.5
30 20
40
2 .15
15
60 50
15
.4
400 60 50 40
2.5
.5
.2
80
3
.6 100
100
30
.8
80 35 0
3.5
1
600 500
4
20
6
5 .1
5 4
4
3
15
3 2.5
15 0 Tail (H Stab) Willg Area S Area ST
Basic Aeronautics for Modellers
Sp'S
V-ba r
I p'c
Tail Arm IT
Willg Meall Chordc
139
10) use these nomograms in Figure E6 to ge t yo ur wing Aspect Ratio from its span, and chord or area .
Before getting stu ck into ca lculating Neutral Po int position s (o r even simple CG positions from Figure 8. Pigure Bti
B
A
C
30
30 600
20
400
20
300
15
15
200
10 8
100
10
60
8
40
6
6
30
5
5 20
4
4 10
3
3
6 2
1500
2
4 Chord
Wi"gSP""
AR
x
y
z
100 200
30
50
80 1000
800 600 500
150
40 20
60 50
30 15
100
40
80
30
400
20 10
60
15
8
50 300
20 40
10
30
8
5
15 200 150
6
4
6
10 20
3
5
8
100
15 ills
4 2
dm
AR
Wi"gSP""
140
Basic Aeronautics for Modellers
To save yo u wo rki ng o ut sq ua re ro ot s of squ are ro ots, this no mog ram (Figure E7) multiplies the fourth root o f wi ng Asp e ct Rat io (o n th e rig h t) tim es Ta il Volu me on the left to give the Neu tral Po int correc tion from the Tailplan e.
Th e exa mp le, my Spectre , shows that a tail vo lume of 0.29 behind a w ing of AR 4.2 will shift th e NP aft by about O. 105 times wing !'vIAe.
Figure £7 .8
.7
.40
35
.6
20
30
.25
10 8
.5
6 4
3
.4
2
35
1
.05
.25
.04 .2
Tail Volume Ratio V·bar
Basic Aero nautics for Modellers
NPSbift ·
due to tail as a fra ction of toing MAC
Wi1lgAspect Ratio
141
If you were put off trying to work out your tailplane angle to trim from equation 12.2 then try this nomogram (Figure E8), which comes in two parts. If you know the lift curve slope of your tailplane (unlikely) mark it, otherwise estimate it by marking the tail's aspect ratio on line A. Draw a line from this point to your V-bar on line C, then from where that line cros sed line B to Cmo (the moment coefficient of the wing) on line E. Where this line crosses line D gives the first part of TSA (that due to wing camber) 2 degrees in the example. It is negative (tail LE down) if Cmo is nega tive.
On line V bring down the value of al V-bar from line B. Draw a line across to the Stability Margin on line X, and then a line from the intersection on W to the lift coefficient on Z. The second part of TSA on line Y is another 2 degrees in this example. The total is 4 degrees. That is the angle between the tail and wi ng Zero Lift Lines, at th is lift coefficient. This is for trim , so of course it varies with speed, so choose the speed at which you want the elevator neutral.
Ptgure Ed A
C
B
E
D
.2 .00 6 .03
.008
1 .035
.2
30
.25
20
.3
10
.01
.1
.04 .015
6
.35
.08
4 .05
2
.06
3_
- - -.03
4
.07
.04 -
6 .08
.05 .06
8
12
.09
.4
.02
.1
":6-
.03
.6 .4 .3 .2
.9
.02
.1 .01 V-bar
a , V-bar p e r degree V
.04
1
.8
Tail Aspect ratio
.0 6 .05
_.5
.7
.08
Lift s lope p er degree
3 Z-
lV
X
.0 1
T.SA (camber) degrees
C. IO
y
Z
.3 2
.25 20
.015
.2 1
10 8 .02
.15
.025
,,~
.03 .035
,,-
,,-
.04
.05 .0 6
,,-
,,-
6
.8
4
.6 .5
3
- -
.:l-
.1
---
.4
.3
1 .8
.09
.6
.08
.4 .3
.07
.2
.06
- -
.2 .15 .1
.1
.08
.05
.06
.0 7 .08 a,V-bar' per degree
142
.04 .0 4 Stab illlJ' Margill K"
T.SA ( s tab) degrees
CL
Basic Aeronauticsfor Modellers
Glossary Symbols ~
0
Centre of Gravity Gre ek lett er alp ha , angle of attac k ze ro lift an gle of a ttac k stalling ang le o f attack
P J.L
Gree k lett er pi , ratio of circle circumferenc e over di am eter Gree k letter rho, air d en sity Greek letter mew , air viscosity
Abbreviations a ay AC AR c Co CD; Coo COmin CG CLor Cl CLmax CLy C~ I
Cl\lo C~ILE
CP D D\xrF
dV F
lift curve slope , dCL/ do tailplane lift curve slope, dCLy/ dnr Aerodynamic centre , or ae rocentre . Aspect Ratio wing chord Drag coefficien t induced drag coefficie nt profil e dr ag coefficient minimum drag coefficient Centre of Grav ity Lift co efficient section or wing 's ma ximum lift coefficie nt lift coefficient of tail Pitching moment co efficie nt. Pitching moment coe fficient, at zero lift Pitching moment coefficien t, about the lead ing edg e Centre of Pressure Drag DownWash Fraction, or in standard terminology, de/do a cha nge in velocity a general for ce
g GMC K KE Ko
If IN i, L LE MAC NP q qy Re S Sy TE V V-bar (o r Vhar) XL
gravitatio nal co nstan t, or (so metimes in " ") load factor geome tric mean cho rd a cons tant, in induced drag formula Kinetic energy Stab ility Facto r for epl an e moment ann fuselag e nose moment arm tail moment arm Lift Leading edge Mean Aerodynam ic Chord neutral p o int airflow dyn amic pr es sur e , V,pV2 dyn am ic pressure of airflow over tail Reyn olds number wing area Tailplane area Trailing edge vel ocity , a vec tor, speed in a particular direction Tail Volume Ratio Xtra Lift, an incre ase in lift
Co mmon Aerodynamic Terms If you cannot find what yo u want in the Glo ssary then p erhaps I cou ld not easily expl ain it without a d iagram . Look for it in th e main te xt via th e Index .
• Aerodynamic Centre The aerodyn amic cen tre of an aero fo il Section o r a wing is th e point ab out w hic h its pit ching moment do es not vary w ith angl e of atta ck . Th e po int is p ract icall y a lw ays within 2% of 25% cho rd a n d is often ca lled the quarter chor d point, c/4. • Angle of Attac k Th e angle between th e d ire ct ion of motion a n d a datum lin e on a w ing. The datum line ma y b e th e ze ro lift line , th e chor d line at th e root, or so me o the r easily defined line specified for th e purpose .
Basi c Aerona utics for Modellers
• Angle of Incidence Th e angle b etween th e fu selage datum line an d a d at um lin e on a wing . Th e datum lin e m ay be th e ze ro lift line , th e chord line at th e root , or so me other easil y defin ed line sp eci fied for th e purpose .
• Aspect Ratio Th e span of a wing divided by th e mean chord . O r it is some times ea sier to use span square d divid ed by wing area . (Fig ure E6)
• Ballast Weight carried e ithe r to ad jus t the ce ntre of gravity of th e aeroplan e , or to increase its w eig ht temporarily.
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• Boundary Layer The layer of air next to th e surface of a mo ving bod y.
as a d e ci mal frac tion or perce ntage o f th e mean chord.
• Canard An aeropla ne whose horizontal stabiliser is ahea d of the wi ng . The hori zontal stab iliser may be ca lled the "canard" or forepl ane.
• Stall Loss o f lift and increase in d rag beca use o f flo w separatio n o n the w ing.
• Centre of Gravity Th e balance p o int , or the poi nt throu g h w hic h the res ultant weight ac ts. It is the po int abo ut whic h the weight ha s no moment. \V'e norma lly ass ume it lies o n th e ce n tre line and q uot e only its fore an d aft position, ign or ing its height. • Centre of Pressure The im ag in a ry p oi nt th rou gh wh ich th e resul ta n t aerodyna mic fo rce ac ts. It is fo und by d ividin g the m om ent coefficie n t abo u t a p o int by th e lift coe fficie nt through that point. • Dihedral Th e a ng le by which each wi ng tip may be ra ise d above the height of the roo ts. • Drag Th e co mpo ne nt o f th e aerody na m ic fo rce in th e d irection exactly op pos ite to the dir ection of mot ion. • Drag Polar The Drag Po lar of an aerofoil o r w ing is a plo t of lift coefficient on the ve rtica l axis against drag coefficient on the hor izontal axis . • Flaps A mova ble port ion o n the trailing edge of the w ings which may be low er ed to incre ase lift and drag . • Incidence See "angle of inciden ce". • Leading Edge The lead ing edg e of an aerofoil or wi ng is the front, the pa rt whic h the airflow meets first. • Lift
The co mpo ne nt o f th e aerodyna mic force a t rig ht angles to the directio n of motion . • Longitudinal Dihedral See Tail Setting Angle. • Mean Chord The average cho rd of a tap ered w ing . (See index for Mea n Ae rody na mic Chord a nd Geo me tr ic Mea n Cho rd .) • Moment Th e mom ent of a force abo ut a point is the amo unt of the force times its distan ce from the poi nt. It is a mea su re of its twi stin g , or turning effect about th e poi nt. • Stability Margin (or Static Margin) A measure of the sta bility of an aeroplane . Physically the distance of the CG fo rward of the Neu tral Po int ,
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• Stalling Angle Th e stalling a ng le of attack of an aerofoil or w ing is the angle of attack at which it develops its maximum lift coefficient. • Streamlines Im ag ina ry lines d rawn to re prese nt th e flo w o f a flui d , su ch th a t th e re is no net flui d flow ac ross a strea mline . • Tail Setting Angle Th e angle betwee n th e zero lift line of the tail and the zero lift line of th e w ing . Always tail le ad ing edge down. Some times referred to as lo ngitud inal dihed ral. Taper Ratio Th e cho rd a t the w ing tip d ivide d by the chord at th e wing roo t, or at the fuselage centreli ne . (Express as a decimal fraction.) • Tipstall A viole nt wi ng d rop ca used by flo w se pa ration (a stall) on the o utboard port ion of o ne wi ng o nly. • Trailing Vortices A pai r of vortices trailing behind the wing tip s of a flying aeroplane . Th ey rot ate in opposite directio ns such tha t the air between them is descend ing . • Tuck Under Th e tenden cy of so me mod e ls to p itch nose d own w hen the ir airspeed is increased. • Turbulator Strip A stri p of ad hesive tape (usua lly) on th e top o f a w ing to improve its p e rforma nce a t lo w Re ynolds Num be r. • Vortex, Vortices A vortex is a ro tatio na l flow.
• 'Vashin A tw ist in a wi ng w hich increases the incid e nce at the wing tip . Not usu ally delib erate . • Washout A twis t in a wing whic h red uces the incidence at the wing tip . O ften done delib erately. • Wind Gradient T he vari ation of wi n d spee d and d ire ct ion w ith altitude . • Zero lift line If th e Zero Lift Line of a n ae rofoil sectio n or a w ing is aligne d with the airflow then the result ant lift will be zero.
Basic Ae rona 11ties for Modellers
Index aerobatics aerocentre aerodynamic balance aerodynamic centre aerodynamic damping aeroeIasticity aerofoil camber aerofo il section aerofoil thickn ess aile ron drag aileron flutter aileron reversa l aileron air air den sity airbrakes alpha an gle of attack as pe ct ratio autoro tatio n ax is, latera l, axis, longitudinal axis , vertical ballast bending moment BernoulIi biconvex section biplane boundary layer camber cambe r line cambered section can ard ce ntre of gravity centre of pre ssure centripetal force CG formula chord chord line climb coefficient , drag coefficient, lift coe fficient, pitching moment compone nt compression cosine couple damping descent dihedral dimpled balls directional stability divergence, wing
Basic Aeronautics for Modellers
80 27 57 27 58 96 29 28 29 57 98 97 56 13 13 60 20 20 32,34, 85, 140 81 42,43 42,43 42,43 75 116
131 29 87, 127 23,91,131 16,24 ,28 16 29 85, 129 41, 123 17, 26 63 45 16 16 78 19, 23 19, 20 19, 26 14 114
14 15 59 78 52,133 25 49 97
diverg ent double taper downthrust downwa sh drag drag bu cket drag coefficient drag polar dutch roll dynamic stability elevators eIevon elliptical load ing elliptical planform flap flap eron flat-bottom ed sectio n flexibility flick ro ll flight testin g floats flutter flutter, aileron flutter, tail flutter, wing forepl an e Frise aileron ge om etric mean chord graphical method gravity ground effe ct gusts hodograp h hysteresis loop incidence incompressible induced dra g interferen ce d rag invert ed flight knife ed ge laminar flow sections laminar se pa ration lateral axis lateral stability leading edge lift lift co efficient lift cu rve Lift/Drag ratio load facto r longitudinal axis lon gitudinal dihedral longitud inal static stability
41 36, 127 70 32 18, 23 24 19, 23 24 55 41, 42, 59 56 61 35 35,37, 126 60 61 29 47 81 123 129 98 98 101 100
129 57 39, 125 40 13 34 111
73 93 20 13 35 72
82 83 24
91 42,43 50 16 16, 18 19, 20 20 24,72 64
42,43 48, 69 41
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loop mass mass balan ce mean aerodyna mic chord mean cho rd mean line mom ent na vigatio n neutral point Newton 's Laws pendulum stability pitch damping pitching moment coefficient pressur e prop eller thru st Reynold s number rotational ine rtia rudder se mi-symmetrical section separation bubble se pa ratio n point shear sides lip sine skidding turn slat slipstrea m slope lift slot slug smo ke tunnel snap ro ll spin spiral dive spiral divergence stabiliser stability stab ility margin stagna tion point stall stalling angle stalling speed static marg in static stability stiffness strea mline strea mlining stress stru ctur es strutted win gs sweepbac k symme trical sec tion tail lift tail setting angle Tail volume ratio taile ron s tail-less tan gent taper ratio ten sion therm al lift th ickness throttl e tipstalling torsion trailing edge
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81 13 58, 100 39, 126 39, 125 16 14, 15, 26 109 43,44 , 128 13 51 59 19, 26 13,1 7 76 90 58 56 29 92 21 114 50, 64 14 64 61 76
trailing vo rtices trimm ed flight tuck under turbulator strips turning flight undercambered sec tion V-bar vector vec tor co mpone nt ve rtical ax is viscos ity Vortex, vortices V-tail washin , (see wash out) washout wash out, ae rodyna mic weight wind gradient wind tunnel wind shear wing divergence w ing flutter wing loading win g struts zero lift line
31 67 102 94 63 29 45, 139 13 14 42,43 23 30, 132 61,62 , 129 37 37 38 13 110 18 110 97 100 136, 137 118 20
109
61 13 16 81 80 55 55 43 41 44, 130 16 21 , 80 21 65, 77 44 41 115 16 25 114 114 118 38, 53, 133 29 67 48,6 9, 142 45 61 86 14 36,38 114 109 16, 24, 28 62 37 114, 120 16 Basie Aeronalilies for Modellers
Notes
Basic Aeronautics for Modellers
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BASIC ERO AUTICS FOR MODELLERS This second edition skillfully guides the reader through the basics of oircrolt flight and performance before addressing issues specific to model oircrolt Alasdair Sutherland draws on his personal experience as a student, a pilot, and most importantly, an aeromodeller to present fundamental information in a friendly and easily accessible form. He does so by building the knowledge base of the reader in a steady progressive manner, highlighting a number of common misconceptions along the way. In this way, he ensures that the reader is prepared for each new section of the book as it is reoched. Thankfully, the use of complicated equations or tedious derivations which, if excessive, can olten deter the layman, is either avoided or they are provided in appendices.
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ISBN 1-900371 -41-3
9 781900371414
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ISBN 7 900377 4 7 3
