Basic Aeronautics for Modellers

Basic

FO ODELLERSSECOND EDITION

FOR MODELLERS

BY ALASDAIR SUTHERLAND BSc

© 2002 Traplet Publi cations Limited

All rights reserv ed . All trad em ark s and register ed nam es ac kno w ledge d . No part of th is book may be co pie d,reprodu ced or tran smitt ed in any fo rm w ithout the wri tte n co nsent of the Publish e rs.

The informati on in this book is true to the best of our knowl ed ge at the time of co mpilatio n. Recommendationsare made without any gua rantee, impli ed or o the rwise , o n the part of the autho r or publish er, wh o also disclaim any

liability incurred in co nnec tion with the use of data or specific information co ntained within th is publicat ion .

First ed ition publish ed by Trapl et Publi cat ions Limited in 1995Publi sh ed by Tra plet Publi cation s Limited 2002

Traplet House,Severn Drive ,

Upto n-upon-Severn,Wo rces tershire . WR8 OJL

United Kingdom .

ISBN 1 900371 41 3

Front Couer. Stefan If/u rlll seen bere exercising some ofb is considerableflying skills ioitb b is 1:2 scale Pitts 51.Stefan brought tbe Pitts backwards, balancing the thrust oftbe engine against tbe stlffbrecze, until tbe rudder

tou ched bim! (Photo: Peter Dauison)

Tecbnical D ra uiings by Lee \\7isedaleCartoons by Simo n Bates

TRAPLET~v;=---"=:P

r U lJl l C AT I O N S

Printed and bo und by Stephen s & George Limited ,Merrh yr Industrial Estate , Dowlais, Merthyr Tydfil , Mid Glamorga n CF48 31'D

AcknowledgementsC o nve ntio na lly th is is a page o f sycop ha n tic

ramblings wh erein I thank everyo ne in my lifefrom th e midw ife wh o d elivered me to my

dent ist's rece ptionist. Well , thank you one and all.I owe my parents a small apo logy , as I rem ember

bu ying a mod el ae ro plane and then promi sin g that itwould be my last ; not o nce but three or four times. Imade no suc h rash pr omi ses to my wife Ann e whounwittingly made the mistake of marr ying a dormantAerom od eller, who ever since then has been e ruptingwith increasing magnitude and frequen cy, sprinkling thehou se with successive layers o f styrene bead s, woodshavings, balsa dust, glass fibre strands and Solarfilmfragmen ts. Sorry Anne .

As for my daughters Ron a and Sheena , if the y everlive in Ame rica the ir analysts w ill ma ke mu ch of thesocia l and paternal deprivat ion they have endured bybeing the offspring of a fervent aeromode ller.

Passi ng q uic kly over my educat io n a t Le n zieAcade my, Glasgow Universi ty and the Hambl e Collegeof Air Training, the grea t mileston e in my modelling lifewas when Jo hn Mich ie had the time a nd pat ien ce toteach me to fly proportion al R/C aeroplanes. And it wasBrian Davies who introduced me to aeroba tics and wo rdprocessing, which is whe n this book ge rminated . I havelearn ed a grea t deal from my friends in the Alde rsho tclub and W'indsor Park, and co ntinue to learn from mypresent circle of friends in Scotland . It was du e to one ofth ese , Bob McGill , that I became imme rsed in wa terplanes.

Finally , th ank yo u to Dr. Fra n k Cot on of th eDepa rtment of Ae ro space Eng inee ring at Glas go wUniversity wh o read throu gh the manuscript to checkth at I wo u ld not e m ba rr ass th e Dep artm ent to oexte ns ive ly by preach ing fund am enta l ae ro dy na micfallacies.

Foreword

O ne of the first technical qu estions my son everaske d me was "How do plan es fly?" Well , we allknow how plan es fly .. . don't we? Th ink again!

If you were asked that simple qu estion , could you give aco ncise comprehensible a ns wer? If yo u co uld, howwould you deal with the retor t, delivered by the son ofon e of my colleagues . . . "How do plan es fly upsidedown?".

On e of the most fascinating aspec ts of the modernworld is th e science of flight. Wh ether it be a bird ,heli co pter , fighter aircr aft or e ven th e marvellousbumble bee, people have always been intrigued by thesame basic qu estion - "How does it fly?". Unfortunately,the answer is not a lways s traigh tfo rward an d isco mplica ted by the w ide varie ty of mechanisms andphysical 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 enrecreational altho ugh since scientific studies have beenconducted, most notably those in Germany between theWor ld Wars. As a result, tod ay's aeromode ller is a fairlywell info rmed ind ivid ua l who , instead of ask ing th ebas ic nature of flight qu estion, is more interested in howto improve the performance of an ai rcraft o r how toavoid problems during fligh t. The answers to most ofthese question s can be found in Basic Aeronautics forMod ellers.

Thi s book skillfully guides the reader through th ebas ics o f a irc raft flight a nd p erformance beforeaddressing issues specific to model aircraft. AlasdairSu the rland draws on his p e rsonal experience as astude nt, a pilot, and most imp ortantly an aeromodeller,to pr esent fundamental informati on in a friendly andeas ily accessib le form . He does so by building th eknowled ge bas e of the read er in a steady progressivemanner, h ighlightin g a number of co m m o nmiscon ception s along the way. In this wa y, he ensuresthat the reader is prepared for each new section of thebook as it is reache d. Thankfully, the use of complicatedequa tions or tedi ou s derivation s wh ich, if excessive, canoften deter the layman , is ei ther avoided or they areprovided in appendices .

Th rou gh ou t the book, use is mad e of observat ionsfrom flow visua lisation experi me nts to illustrate asp ectsof fluid be hav iour. Over the years, flow visua lisation hasbeen one of the most powerful too ls in the developmentof our current understanding of fluid dynam ics. Ind eed ,smo ke flow visua lisation wind tunnels are still used inma ny un ive rsities for resea rc h a nd s tu d e n tdem on strations. It is obvio us that the demonstrationsgiven to Alasdair Sutherland in his stude nt days had aconsiderable impact; after all seeing is believing!

Whether you consider yoursel f to be a novice or a

well-season ed ae rornode ller, there is someth ing in thisbook fo r yo u . Beginner s can learn about th e ba sicmechani sms of lift generation and the manner in whichforces act on an aircraft. The more experience d , on theother hand, can contemplate the detailed influ ence ofmodel sca le and the role of the Reynolds number. Thebook may even encourage some to raid the library formor e informatio n or carry out so me res earch of theirown. Most importantly though , this book was written byan enthus iast for its readers to enjoy. I hope you do!

Dr. Frank CottonDepartment of Aerospace Engineering

University of Glasgow.

Alasd a ir Sutherland w as born a nd ed ucated in th eGlasgow area , progressing from Lenzie Aca demy toGlasgow Univers ity where he earned a B.Sc. w ithHonours in Aero nautical Engineering. After training for acareer as an airlin e pilot at Hamble , near Southampton,he joine d BEA in 1973 to fly Trident aircraft aroundEuro pe and Lockheed LlD11 aircraft wo rldwide.

An aerornodelle r since th e age of eleven, he fliesmost types of radio controlled aircraft especi ally sportsand aerobatic, and particul arly enjoys designing modelsof va rio us typ es. After many years as a member ofAlder sh ot Mod el Club he mov ed back to Scotland asCaptain of British Airways turboprop aircra ft, first theH.S. 748 and latterly the British Aerospace ATP. He isnow a member of both the Clyde Valley Fliers and theGarnock Valley !vIAe.

Tbe Author: Alasdair Sutherland

ContentsPage

Introduction 11

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chap ter 8

Chapter 9

Chapter 10

Chapter 11

Chapt er 12

The Aeroplane's Environment 13Tbe air. Mass toeigbt and grauity. Newton sLaws. Inertia . Vectors. Moments.

Requ irement for Flight - Lift 16\fiatcbing tbe a irfloto. Pressure variation . Pressure exerts a force. Wind tunnel testing.

The Stall's the Limit 20The lift curve. 17Je stall, tbe reason. Variation in stalling cbaracteristtcs.

The Drawback Drag 2317Je boundary layer. Wing drag; drag polar , effect of tbickness and ca mber,la m tnarfloui sections. Fuselage drag , strea mlin ing . A bit for golfers.

Have you a Moment? 2617Je mom ent on tbe wing . Centre ofpressure. Aero dyna mic centre.Aerofoil section su m ma ry, tbe effect oftbickness and cam ber. Section classification and use.

The Vortex System 30The uortex around tbe wing. Seeing tbe cortices. Even more drag, tbe reason .Complications. Simp lifica tio ns. 17Je importance ofAspect Ratio.Lessons forpra ctical modellers . Ground effect,

Planform and Twist 35Elliptical lift distribution. Local angle ofattack. Different planform shapes.Tipstalling . Wasbout, aerodynamic ioasbout. Sweep back. Mean cho rd . Horses for courses.

CG and Stability .4117Je CG. Stability in general. Motio n ofan aeroplane. Stability ofaerop lan es in Pitcb ,CG Position . Complica tions. We can work it out? Simpler equations.Variations on tbe formu la .

Directional and Lateral Stability .49Directional stability , the fin. Lateral stability, sideslip. Fin sideforce, wing p osition ,d ihedral, sweep back . Aspects ofdesign . Directional and lateral interaction,spiral divergen ce, dutch roll.

Control .56Rudder. Elevators. A ilerons, aileron drag, aileron alternatives. Control surface balances.Control effectiveness, rotational inertia, stability, aerodynamic damping.Otberflying controls, throttle, air brakes, flaps, slats. Control combinations, ta ilerons,flaperons, eleuons, V-tail.

Turning Flight 63Mecbanics of turning . Turning aeroplan es, load f actor in a turn, refinem ent,stdeslipp tng and skidd ing, drag in a turn, stalling speed. Higb aspect ratio.Turning using rudder. Special effects. Wben is a rudder an elevator?

A Delicate Balance 67Equilibrium. Tail lift to trim. Elevator ang le to trim . Tail Setting angle .The effec t of thru st on trim.

Cha pte r 13

Cha pte r 14

Cha pter 15

Chapte r 16

Cha pte r 17

Chapte r 18

Cha pte r 19

Cha pte r 20

Cha pte r 21

Cha pte r 22

Glider Performance 72Lift/Drag rati o. Speed range. Ae rodyna mic da ta . Optimising performance, strea mlin ing ,toeigbt. Iiffect oftoind on perfo rmance, down trim , ballast .

Powered Performance 76Propeller thrust, slipstrea m effects. Levelflight, top speed, stalling speed,effect on toeigbt. Take oJ(. Clim b. Descent and landing .

Th e Aerodynam ics of Aerobatics 8077Je stall. Sp in . Snap . Loop. In oerted. Roll. Yatu. Aero bat ic trim set up.

Special Cases 85Low aspect ratio, handling, CG position . Ca na rd . sta bility, CG Position ,Tail-less aeroplane, stability, trim, control. Multitoing, performance, CG position .

Reynolds Number 90Definition, importance, nontogra nt. In tbe bou ndary lay er, situation normal,laminar separation , separation bu bble, tbe underside. Re-effect on aerodynamic da ta .77Je p roblem area . Hysteresis loop . 77Je effec t 0 11 model design a ndperformance,wing tips, class rules, optim u m weigbt. Tu rbulator strips. surface fi nish ,Using publisbed data .

Aeroelasticity 96Effect on stability , ta il bend, wing twist. A ileron reversal. Wing divergen ce.Aileron flutter , tbe cause, tbe cure. 117ingflutter. Tail Flutter .

Tuck Under 102Description . 77Je villa in unmasked. Wing twist. ta il bending.flexible controls.77Je elevator trim graph . Critical speed . Tuck under speed . Getting away witb it.Tailplane instability. Rem edies/or tu ck under. Conclusions .

Th e Air o n the Move 109Navigation . Slope lift . Tbennal lift , Windsbear a nd Win d Grad ient. Gusts.Mytbs a nd miscon ceptions. Momentum. Kin etic energy . Analogies. 77Je meaning ofl ife?

Mod el Aircraft Structures 114Defining some words, composite structures, tobat a ir does to wings,bending mom ents, stru tted wings, torsional stiffness, fuselages, tailplanes.

Centre of Grav ity Pos ition .123Rigbt and wrong CGs, Fligb t testing, p opular m isu nderstand ings, tobat matters,mean cbo rds, tbe flying toing , biplanes, tbe neutral point , adjustm ents,p utting it together, stability margin .

Appendices 131A Bemoulli 's equationB Boundary LayerC vorticesD Dib edral and sweepE Usefu l Nomogra ms

Glos sary 143Symbols, Abbrev iations and Commo n Aero dy na mic Terms

Index 145

Notes 147

IntroductionW

hen the cold raw wind howls down from theNorth bringing grey fragme nted clouds whichsc ud low o ver th e damp da rk fo rbidd in g

landscap e like a demon arm y. When sheets of icy raindeluge incessantly from a leaden sky and the puddlesjoin forces to threaten us with an oth er great flood . Whenthe great oak trees bow down to the unseen forces ofthe wind like frightened peasants befor e their Gods .When ever the outside environment becomes hostile toman and his aeroplane, I curl up in a cha ir by the firewith some books and magazines, to absorb all the fact,fiction and folklore of our fascinating hobby.

It is on nights like the se as I lie in bed listening to thewind howling or the rain lash ing or the deathl y silenceof the snowfa ll that I hear voices , vo ices from mypas t. They are the vo ices of aerodynamics lecturers andau thors a n d the y remind me h o w littl e acc u ra teknowledge of aerodyna mics is ava ilable to the averagemodeller, and they tell me whose fault it is. Mine! Myfault for not writing this book soone r!

I have three main aims in wri ting this book. The firstis to disp el the half-truths and old wives tales passed on,usually in go od faith , into the folklore of the hobby.

I once ha d a very puzzling conversa tio n with amodell er about th e use of "flaps", until he clarifi edmatters by explaining that he me ant th e "back flaps"(e leva tors) . So the second aim is to ge t us all speakingth e sa me langu age as fa r as p ossible so that ourinevit able di scussions and arg ume nts can b e moremeaningful.

The third aim of my book is an introduction to aerodynam ics so that you can understand how to make useof the data available elsewhere wh en designing yourown mod els . Understanding some simple theory will notturn you overn ight into the design er of the most elegantand super-efficient models (that still requires experience,in spiration a nd talent) , but yo u ca n le arn what isp oss ibl e under the laws of Phys ic s , a n d w h a t isimpossible - unlike the alche mists of o ld who was tedtheir lives trying to turn lead into go ld.

Now let me plea for pati ence es pecially fro m th emore knowledgeable readers. I have started off with asimple , rosy , idealised view of the wo rld and I introducethe rea l complications little by little.

Basic Aeronautics for Modellers 11

Chapter I

The Aeroplane's EnvironmentThe Air

Please try this simple experiment. Take a can of beer,open it, and drink the contents. Now what are you leftwith? Most peopl e say "an empty can " but that is wrong .If you answered "a can full of air" give yo urself a pat onthe back . We aeromode llers must be co nscious of theair. We are depending o n it to su pply the lift for ouraeroplanes. Next time you see a Jumbo jet lumberin g offth e runw ay remember tha t th e air is providi ng th eupward force of up to 400 tons.

So how heavy is, say a room full of air, 4 met res by 3an d 2.36 metres high? Would you believe 35 kg or 77 Ib?At about 1.22 ounces pe r cubic foot ai r is not veryd ens e , b u t yo u w ouldn 't ca ll a roo m empty if itconta ine d 77 Ib of balsa wood!

Now, how strong is the air? In a school experimentthe halves of a four inch (lOO mm) diameter hollow stee lsphere were placed togeth er and as mu ch as possible ofthe air ins ide wa s removed. The air hel d th e halv estogether. It took a lot of effort from the four strongestlads in the class to pull the two halves apa rt. Pres sure isdefined as a force per un it area . The force which the airpressu re exerts on a surface with a vacuum on the otherside is 14.7 pounds per square inch or nearly a ton persqu are foo t! The pull n eeded to sepa rate thehemispheres in sc hool was almost 180 Ib (8 00 N).Natura lly the air exerts its fo rce on a surface whe therthe re is a vacuum on the other side or not. Hold up asquare foot of paper and there is a ton of force on ea chside , but so wh at? The two forces cancel out. Pressure isnot direction al, or rather it is omnidirection al; it acts inall direction s at on ce . And it acts perpendicular to thesurface at every point. So whichever way up yo u holdthe paper there is exactly the same one ton force oneach side .

You can see the air pressure varying slightly from dayto day on yo ur barometer. Both density and pressurereduce with altitude but we aeromo de lle rs can igno rethese small differences. The reduction in air pressure isabout a tenth of one per cent for every 30 feet climb ed .Incid entally it is by measuring that reduction in pressurethat an aeropl an e's altimeter works.

Low sp eed airflow is called "incompressible" because,although the pressure wiII vary, density does not. We allkno w air can be compressed, and its den sity changed,but only in a container. Aeroplanes in free air do no tcompress it unless they travel at ne ar sonic speeds .

Mass, Weight Gravity

An object's mass is the amo unt of material which itcontain 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 nodifference. Where an object's mass (as opposed to itsweigh t) s hows itse lf is in its resistan ce to b e ingaccelerate d . Take an iron canno nba ll into space and itwiII be "weightless" but try kicking the canno nball andyo u will bre a k yo u r foot. It s resistance to beingaccelerated , its mass, has not changed . The weight ofth e ball is just th e for ce of th e earth 's gravi tation ala ttractio n on its ma ss. To ge t th e weight o f a body,multiply its mass times "g", the "gravitatio nal constant"which on the earth' s surface is 32.2 It/ sec/ sec or 9.81m/ sec/ sec. The we ight of a "kilogram" of mass is a forceof 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 needto remember all that) .

Newton's Laws

If a body is in "equilibrium" it is e ither at res t ormoving at co nstant speed in a straight line (tha t is, notaccelerating). Many years ago Sir Isaac Newton put intowords three funda mental Laws of Motion.

• 1. The first says that a bo dy wiII be in eq uilibrium ifa ll the fo rces on it cancel ou t, Le . if there is noresultant force .

• 2. The second says that the force nee ded to cause anacceleration equals the mass times the acce leration.

• 3. Th e th ird is the old favouri te ab ou t each fo rcehaving an eq ua l and opposite reaction.

Inertia

When yo u kicked the canno nba ll in space , it appliedan equa l and opposite force to your foot. Tha t kind offorce is ca lled an "inertia force", and is the force withwhich a body res ists being accelera ted . Similarly, whenyo u catch a ba ll yo u apply a force to slow it do wn,overcoming its "inertia" which makes it wa nt to carry onthe way it was go ing .

Vectors

A riddl e! The re was a car sitting on a level road withthe brakes off and three men pu shing it but it wasn'tmoving! Why not? One was pu shing the front, one theback , and o ne was pushing the side. An important littlede tail!

Any quantity whos e direct ion must be specified aswell as its amount, for example for ces, is ca lle d a"Vector". O the r examples of vec tors are distance moved,acceleration and velocity. I prefer the word velocity tospeed because it is a rem inder that it is a ve ctor.

13

Vec to rs ca n b e added to gether b y a d d ing th eiram ounts only if th ey are in the sam e d irec tion . If two

. .: . .......

Th e "mo ment " of a forc e abo ut a point is the size ofthe force times the dista nce of the force from the point.

force s are in o p pos itedir ections, like two menpu s hing a t e ithe r e n do f a c a r, th e y w illca nce l each o ther out. Ifvectors a re at an angleto ea ch o the r th ey ca nbe added by drawing a"vecto r diagram" using aru ler an d protract or. Avecto r dia gram is a scaledrawing in whi ch thelen g th o f th e lin e sre p resen ts th e amo unt,and the direction represe nts the d irection o f thevectors . Figure 1.1 co uldre prese n t a trea suremap. "Starting at A wa lkten metres no rth to B,th en go ten metres ea stto c." The equivalent, orre s u lt ant , o f the tw ovec to rs AB and BCa d ded toget her is thevecto r AC which is 14.14metres to the northeast.Figure 1.1 could just asea sily have representedt h e addit ion of twofor ces or veloci ties .

Ve ctors ca n al so besp lit up , or "reso lved" ,into two o r more "co mponents " whi c h wil lh a ve th e s a m e e ffec t

(F ig u re 1. 2) . The tre a sure is in a ca ve , "C". Theinsc rip tio n o n th e Azt ec Temple , "A" says ; Go fivekilometres on a bearing 0370 East of North (but bewareof the Dragon at "0 "). Preferring an ea sy life to hecticadve nture, our hero "Tri gonometry" ]ones instead goes 4km du e North, s tops for a few beers at "B", and thengoes 3 km du e East w here he finds the cave , treasureet c . e tc . Very precise and sc ien tific but no use for amo vie script.

From the vector d iagram in Figure 1.2, vec to r AC canbe s p lit in to its tw o co mp o ne n ts , AB th e No rtherlycompone nt and BC the Easte rly co mponent. The biggerangle A is, th e smaller AB becomes as a proportion ofAC and the bigger BC becomes as a p roportion of AC.The ratio of BC to AC is called th e sine of the angl e , thera tio of vector AB to AC is ca lle d the co s ine of theangle , and the rati o of BC to AB is called the tangent ofthe angle A. These ratios are usually sho rtened to sin,cos a nd tan and ca n be lo o ked up in tables for an yangle.

Us ing his mathemat ica l tabl es "Tr ig " jones couldwork out the c omponents for a ny a n gl e with ou treso rting to sca le drawing . Th e sine of 37 degrees is 0.6and cos 370 = 0.8. Of co urse the same go es fo r othervecto rs like force s or ve locities e tc .

Moment

c10

.._....

B

A

Fig u re 1.1

14 eastc Aero nauticsforMode/!el :~

5

Pivot

Moment =5 .'\" 10 =50

510

~- - - - --- -- -- --- - - --- --- ------- - ---- - --~

Figure 1.4

groundsp eed vector. Wind ha s no o ther e ffec t (but se ethe cha pte r on wind near the end anyway). To save anyargument I shall I ass ume still air conditions in all thecha pte rs until then.

Figure 1.2 Figure 1.3

B 3 C 100 50

Easterly Component ~ 5 10 J:~

4

A

NortbernlyComponent

Figure 1.3 represents a seesaw th e pl ank of w hich isexactly balanced . Th ere is a child weighing 100 lb 5 feetfrom the pivot and a child weighing 50 Ib 10 feet fromthe pivot. Th e child on the right has a moment of 500 ft.Ib clockwise about the pivot , and the child on the lefthas a mom ent of 500 ft. lb anticloc kw ise about the pivot.Th e two moments are equa l but in opposite d irectionsa nd so th e y c ance l o u t which le a ve s th e seesa wbalanced . It is in equilibr ium as there is zero resu ltantmom en t.

In Figure 1.4 two equa l but opposite fo rces act on abod y. Th e two fo rce vec tors ca ncel out, they have noresultant but they will obvious ly tend to turn the body.The turning effect , o r moment, of the pair of forces isthe sa me about any point yo u care to choose. The totalmoment is Force times the d istan ce between them . Th iskind of system is called a co up le and its moment is thesa me 5 x 10 = 50 about any pivot point. In Cha pter 5 I'llremind yo u that you ca n have a force sys tem w ith nores ultant except a mom ent wh ich is the sa me about an ypoint.

You will ofte n see so me qu anti ty like a irspeed (V)w ith a number supersc rip t. Fo r example V3 mean s V"c ubed " o r V "to th e powe r 3" or speed x speed xspeed. Similarly the "cube root" of V (w ritte n 3jV) is thenumber which , when mu ltiplied together th ree tim es , .gives V.

Wind

I co uld have used the w ind as another example onvec tors. To find the effec t of the wind , just add the wi ndvector to th e aero p lane 's ve loci ty vector to get th e

Basic Aeronautics f or Modellers 15

Chapter 2

Requirementfor Flight - Lift

W hat makes an aeroplane specia l is its wing.The qu estion is, how does it produce lift? Iwish I could tak e yo u to a wind tunnel with

ap prop riate models and mea surement eq u ip me n t. Ico uld then dem on strate how lift is produced just as itwa s shown to me . Instead I sha ll have to att empt todescrib e it in words and diagrams.

Definitions

Figure 2.1 sho ws th e cross-section of a wing. Th estraig ht line from the centre of the leadi ng edge (L.E.)the trailing edge (T.E.) is the chord line . The len gth ofthe chord line is the cho rd of the wing (the wing tip towing tip distan ce is the spa n) . The maximum distan cebet w e en th e to p a nd bottom su rfaces is th e wingth ickness , usu all y ex p ressed as a percen tage of th echo rd. The line drawn midway be tween top an d bottomsurfaces is ca lle d the mean line or ca mber lin e . Th emaximum distan ce between the mean line and the cho rdline is the cambe r of the sec tion and it too is given as ape rce ntage of the chord.

Th e leading edge is always smoothly rounded andthe trailing edge is always sharp.

A typical test wing for a wind tunnel has a uniformchord and aerofoil section from o ne end to the o the rand fits exactly in th ewidth of th e tunn elwh ich does awa y with F ig ure 2.2th e co mplicat ion of tipeffec ts w hich we don 'tneed at this stage.

I sha ll give you fairwarning w hen I cometo a win g with tips. Forthe moment the flow isass umed to be the samea t an y p osit ion a lo ngth e s pa n ( two dimensional flow).

Figure 2.1

Watching 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 which manys ma ll s t re a ms o f smo ke a re fe d in to the ai rstreamupwind o f th e wi ng. Th e th in s tre a ms o f s mo ketravellin g with the air as it flows over the wing help tovisualise the airflow . Figure 2.2 is a dia gram sho wing atyp ical flow pa ttern aro und a win g. The lines sho w theposition of the smo ke streams. Th is is a co mmo n way ofsho w ing an airflow a nd th e lin es drawn a re ca lle d"streamlines" .

Strea mlines are lines drawn in the direction of theairflow suc h that nowh er e does the air flow across aline.

As the airflow approaches the Lead ing Edge (L.E.) ofth e w ing it sp lits in two, part going above and partbelow. The strea mline which d ivides the air which willgo over the wing from the air whic h will flow under itmeets the wing at poi nt A. Air molecules flowin g exactlyalo ng th is line will meet th e wi ng hea d on and bebrought to a dead stop a t A. Po int A is ca lled th e"stagnation poi nt" becau se the air's velocity is red ucedto zero. .

Watching the smo ke st rea ms over th e top surfacevery closely, it can be seen that the air speeds up as it

16

Camber L ine •Ca ll/be"

• Chord Lbw •. L E. T.E.•• •~--------------- ----- -------- --------- - - ---- ------- ----- - -~

CIJOI'd

Basic Aerona uticsfar Modellers

the ai r meets the wing head on. See Figure 2.5 in whic hthe len gth of each arrow represents the pressure at tha tpo int.

Pressure is de fine d as force per uni t area . Imagine inFigure 2.5 that these pr essure arrows , one inch apart,each represent the force on the one square inch aroundeac h hole . If all those force vectors are added togeth er,the resultan t will be the total force on a one inch widestrip of wing . Its size and di rec tion de pend upon th eaerofoil section , the angle to the air flow , the speed ofthe airflow , ete. See Figure 2.6 in which the res ultantforce is sho wn as force F. Th e po int where th is forcecrosses the chord line of the section is ca lled the Centreof Pressure (or C P.) . It is the poi nt through which thetotal pressure effec t on the wing ca n be repl aced by asing le force .

Pressure Exerts a Force

~-------------------

Figure 2.4

Figure 2,3

Figure 2.5

pa sses over th e thickpa rt of the w ing a ndresumes its p re vi o usspeed by th e Trai lingEdge (T.E .) . Under thewing the smoke bu nchesup as it slows down , andthen it accelerates to itsor iginal speed at the T.E.If the smoke strea ms arepu lsed, Le. re leased insho rt burs ts , it ca n beseen that the start of thesmoke pulse above thewing reaches the trailingedge before the smo kebe low th e w ing asillustrated in Figure 2.3.Obvious ly the air overthe top surface has hadto speed up to cover alonger pat h in the sameti me . No tice also tha tw here th e f low hasspeede d up the streamline s a re close r a ndw he re the flo w isslo wer th e streamlinesare furthe r apart.

As th e ang le o f a ttack is increased th e stag nat io npoint A moves down around the cu rve of the leadingedge increasing the dis tance the air travels over the top ,and reducing the dis tance along the underside . On aw ing w ith a symmetrica l section a t an ang le to th eairflow, the stag na tion poi nt is be low the ce nt re of thele ad ing edge (as in Figure 2.4) so jus t as wi th th ecambered sect ion the air flowi ng over the top surfacehas further to go in the same time , and must thereforespeed up.

You can 't get a change in velocity wi thou t app lyinga force (Newton's First Law). The on ly force ava ilable tothe free air is its press u re so th e p re ssu re mu stbe changing as speed cha nges across the chord of thewing (See Appen dix A, Bern oull i's equation) .

If we wish to measure accurately the pressure changeswe have dedu ced mu stbe occurring over o uraerofo il, we ca n drill arow of tiny holes in thetop and bo ttom surfacesand connec t each one toa p re ssu re measur ingd ev ice . Eac h pressu remeasured ac ts at rightangles to the surface atthe po int w he re it wasmeasured . The pressureis, as ex pected , less onthe upper surface thanon th e und e r s u rfacea nd th e re is a h ighpres su re p e ak a t thestag na tio n point where

Pressure Variation

Basie Aero nalilies for Modellers 1 7

Figure 2.6

Ail flow~

Figure 2. 7

Ailflow~

Figure 2.8

L

,,,,,,,,,,,,

c.r.

lV

It is in con venient to have a fo rc e ac t ing in anarbitrary direction like that and so it is split up into twoco mponents at right angles to each othe r.

Th e d irections chosen are the obvio us on es fo r aw ind tunnel. Th e co mpone nt in th e direction o f th eairflow is called Drag, and the compone nt at right anglesto the airflow is called Lift (See Figure 2.7) . Note that Id id not say ve rtical and hor izontal! It is true if the windtunne l is built horizontal , but lift w ill not be ve rticalwh en we come to an aeroplane climbing or descendi ngor ban king. Figures 2.8 and 2.9 show what I mean. Notethat it is a mathematical co nven ience to show forces likeF, or L and D at the ce ntre of pressure . They are merelyrepresenting the true situa tion of Figure 2.5.

So me p re ssure measuring d e v ice s measure thediffe rence in p ressure between the desired point and thestatic pressure of the air in the room. Or if you like thepressure differen ce between the inside and outside of ahollow win g. Figure 2.10 is simila r to Figure 2.5 but thistim e show ing the pr essure difference between insideand outside . The reduction in pressure whe re the air isspeeded up causes an upward force over the top surfaceand whe re the air is slowed down there is an upwardforce on the lower surface . This is a co mmon meth od ofsho wing the lift d istribution whi ch you may have co meacross before (so me times only the line joining the tops

18

of the arrows is shown) . The resultant of all these forces(o r pressures ) is exac tly the sa me as in Figure 2.6.

Ju st to get all this in perspective , co nsider how mu chpressure cha nge is needed to support the weight of amod e l with a typ ical win g lo a d ing of 20 o z./ft - .Atmospheric pressure is about 14.7 pounds per sq uareinc h . An ave rage pressur e rise on th e unders ide of0.02%, and an average pressure reduction of 0.04% onthe top surface will suffice.

We are not asking mu ch a re we ? To ca ll thi s a"va c u um" w ou ld b e mi sleading. I exaggera te denormously the arrows o n my diagrams 2.5 and 2.10 tomak e them mean ingful.

Wind Tunnel Testing

Of co urse w e don 't real ly g o th rou gh a ll thi sr ig m a ro le o f me asuring pressure s a n d in vol vedcalc u la tio n to work out th e lift and drag in a windtunne l. Besides th e co mplica tio n in volved , th e skinfriction drag has been igno red .

The wing co uld simply be mounted on a balan ce tomeasure the forces directly.

The force must be measured through the attachme ntpoint (e .g. the L.E. or qu arte r chord point) together withthe mom ent abo ut this poin t. This mom ent is ca lled the

Basic Aeronauticsfor Modellers

Pitch in g Moment. As m oment e q ua ls fo rce timesdi stance , if th e lift and moment are kn own th en th eposition whe re the lift ac ts (the Centre of Press ure) canbe calculated . The wind tunnel sho uld be eq uipped w itha ba la nce ca pab le o f me asu ring ho ri zontal force s ,vertica l forces , an d p itch ing moments all at the sa metime .

This equipment can be used to test a wing, adjus tingone variable at a time and keeping everything else thesame to find out the effect of each variable. For instan cetest ing the same wing in the sa me position at differentairspeeds shows th at Lift, Drag and Moment are allprop ort ional to the sp eed sq uared.

In other words at twice the speed you ge t four timesthe force , and at three times the speed, nine times theforce etc.

By similar means it is found that Lift and Drag arealso proport ion al to the air den sity p and the wing area.The mom ent is proportional to the sp eed squared, theair density and the wing are a times the chord .

To turn these relationships into useful equa tions fores tima ting the lift from a wing , a co ns tant has to beintroduced and its valu e must be found ex pe rimentally.

50 for example

• L = P V2 5 x co nst.

Figure 2.9

Figure 2.10

A different co ns tant is need ed in each case but to saverunning out of suitable lett ers , the letter C is used in allthree equations with a different subscript. The peoplewho mad e up the equa tions put in a !1 as we ll becausethe te rm !1 p V2 had turn ed up in Berno ulli's equ ation(see Appendix A agai n).

We end 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 coeffic ie n t a nd CD is th e dra gcoefficie nt an d CM is the pitching mom ent coefficient.They all vary with angle of attack as you will see.

! 1 I t tBasic Aeronautics forModellers

t t t + ~

19

Chapter 3

The Stall's the LimitNotice the shape of the graph! It is straight from A to

C and then curves up to a maximum at D then down toE and be yond .

At point B the angle of attack is zero as the wing hasbe en arranged as in Figure 3.3 such that the chord line isparalle l to the airflow. Although the angle of attack isze ro , the wing is still producing lift.

At point A the wing has been tilted further lead ingedge down as in Figure 3.4 and is now producing nolift. The ze ro lift angle of attack is written as a o (thesu bsc rip t 0 denoting no lift ) . T he normal wa y ofmeasuring angle o f atta ck is to mea sure UP from thedirection of motion to the cho rd line . Because the chord

E

0< =0

Figure 3.1 shows a wing se ction in an airflow. Theangl e between the chord line and and the airflow is calledthe angle of attack . It is usually represent ed by the greeklett er a (alpha). Occa sionally a different datum line isused instead of the chord line. It may be a straight line onthe und erside of a flat bottom ed or und ercarnbered Wing,or the wing 's zero lift line. As the nam e suggests, if theairflow is parallel to the zero lift line, the lift is zero (usefu lin mathematical formulae).

Th e inciden ce of the wing is the ang le between itscho rd line (or oth er datum line) and the fuselage datumline . It bears no relation to the airflow and angle of attackat all. It is just a riggin g angle. It may be measured on theaeroplane with an incidence meter or on the plan with aprotractor. Those are the usual definition s and I shall stickto the m , but it is not uncommon to see th e wordincide nce used mea ning angle of attack .

Testing a wing at man y different ang les of attack andworki ng out th e Cl. eac h t ime (fro m the for mula inChapter 2) enables a graph of lift coefficient again st angl eof attack to be drawn for that particular section . For mostnormal sectio ns the graph loo ks like Figure 3.2. Th isgraph is true for this sectio n regardless of the size ors peed a nd ca n be used to es t ima te th e lift in an yco ndition.

The Lift Curve

Definitions

I n wind tunnels the win g is stationa ry and the air isdrawn over it, so that is how it is usu ally described inth 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 isexact ly opposite to the arrow marked "airflow ". Thedir ection of the airflow must be measured far enoughahead o f the wing so that it is not affected by the wing'sapproach.

Figure 3.1

Angle ofAtta ck(measuredf rom

ch o rd U1Ie)Ang le ofAttack

(measuredfrom z ero lift U1Ie)

C Zero lift l .

--- - -Cb';;"';-;' - - - - - - - ~"~ _ _~Direction ofMotion " " , ,Airflo w

20 Basic Aerol/l/ /Ifics/or Modellers

li n e is DOW N in th iscase the angle of attackis a negative ang le (fore xa m p le th e a ng le o fattack for ze ro lift on anEppl e r 195 sectio n isg ive n as - 3 d egrees) .The zero lift line (ZLL)drawn on the wing is bydefinition parallel to theairflow .

At p o int D th e liftcoeffic ie n t is CLm a x

which is the maximumlift coe fficient wh ich thesection can prod uce andoccurs at as the stallingangle of attack.

The Stall

Figu re 3.3

Figure 3.4

___.~~ ZLL----:----c-- -0<;[ - - - - - - - - - - - - - - -

~

Figure 3.5

Direction OfMo tioll

At points C, D and Ethe wing is mounted asin Figure 3. 5 wi th ala rge posit ive an gle o fa ttack but s o me th ingstrange happens to thelift in this area. As a hasbeen increased , the liftha s been increa si ngs te ad ily in proportionbut now it suddenly reaches a peak and drops off again .The phenomenon whereby lift drop s beyond a certainangle of attack, rather than increasing as before, is calledthe "STALL". The wing is said to have sta lled be cause itc a n n o t be p e rsua d e d to prod uce a ny grea te r liftcoe fficient.

towards the lead ing edge , Figure 3.7. At this poi nt thewing is fully sta lled (po int E on Figure 3.2) . The a irmakes no attemp t to follow the w ing's top surface butbreaks up into tur bu lence. The result is a reductio n inlift coefficient. Note that there is still quite a lot of lift,b ut less than there was when the angle of attack wasjust less than the sta lling angle.

The Reason

To find the reasons in the ai rflow for the stall it isback to the smo ke tunne l. At small ang les of attack theairflow over the wing is smooth but as angle of atta ck isincreased there comes a point wh en the flow starts tobreak away be fore it gets to the trailing edge , Figure 3.6.The air can't quite make it down the back of the ae rofoilso the smooth flow ends as the st rea mlines abru p tlybr e a k away , o r "se p ara te ", fro m th e s urface at th e"separation point".

If th e angle o f att ack is increased eve n mo re these pa ration point moves progressive ly fu rther for ward

Figure 3.6

~:

Basic Aeronauticsfo r Modellers

Variations

Different sec tions have different sta lling characteristicsd epending u pon th e ir th ick nes s , ca mber an d th esharpness o r b luntness o f th ei r le ad ing edges. Somesectio ns miss o u t th e Figu re 3.6 stage and th e flo wse pa ration starts suddenly at the leading edge giving ave ry abrup t s ta ll as in Fig u re 3.8 (NAC A 230 12 forex ample). Ot hers have a more progressive sta ll as inFigure 3.9 (for exa mple NACA4415).

In th e specia l case of a n un ca mbered ( i .e .symme trical) wing section, the graph of lift coefficie nt

21

Figure 3.7

---==Figure 3.8

0<

against ang le of attack will look like Figure 3.10. That is,the lift coefficient is zero at ze ro angle of attack , which

Figure 3.9

is just what yo u expect, and of course it perform s just aswe ll inverted .

Any sec tion will have a graph like Figure 3.10 if theang le of attack is measured from the sectio n's zero liftline. It is merely a case of mov ing the vertical axis alongto where the lift is zero . Then, for the straight bit of thegraph below the stall, the lift coefficient equals the slopeof the line times the angle of atta ck . Conveniently it isfound that CL = 0.1 per degree (a pprox) for all aerofoilsections. I sha ll use this idea in the cha pte r on PitchStability.

Notice To Airmen

I hate to lab our the po int but not ice wha t is on thegraph on Figure 3.2, not speed bu t ang le of attac k. Awin g does not have a stalling speed . It has a stallinga ng le o f att ac k a t w h ic h it will s ta ll more o r le ssregardl ess of the speed. Tha t is o ne reason why liftcoefficie nt is plotted , to ge t rid of airspeed and den sityvariables which are unimportant to the prop ert ies of asec tion. It is true that an aeroplane has a stalling speed,but it is only a little true .

When I come to mention the stalling spe eds of anaeroplane I sha ll remind you that it is the stalling angleof the win g which matters.

Figure 3.10

0< 0<

22 Basic Aero nautics for Modellers

Chapter 4

The Drawback • • • DragI

n my book Drag is nothin g to do with dressing up .It is a force resisting mo tio n. To be mo re exact,DRAG is a force exerted by th e air o n a moving

aeroplane, and it ac ts in exactly the opposite direction tothe direc tion of motion of the aeroplane .

Drag as measu red in the wind tunnel is made up oftwo pa rts . First the re is the drag from the p ressu redist ributio n me n tio ned in Chapte r 2. If th e pressured ist ribution dep ict e d in Figure 2.4 is added up toproduce a single resul tant force o n the wing (Fig u re2.5) , then the component in the direction of the airflowis the Pressure Drag . That is one part, the ot her is goodold friction.

\V'he n o ne object s lides over another , th e re is afriction force resisting mo tion. A friction force Cal; ex isteven witho ut mo tion which is why the hand brake ca nhold the car o n a hill. In fluids (e .g. helium, air, water,.o il, treacle) the friction effect is called "viscosity" and thedifference in th is case is that the visco us forces can notexis t without mot ion. The visco us drag on an aeroplaneis, for tunately, sma ll due to the air's low viscosity an d itoccurs in the "boundary layer".

The bo unda ry layer is a ve ry thin layer of ai r, thebottom of which is stuc k to the aeroplane 's surface, andth e to p of w hich ismo v in g wi th the air-st rea m (See appe ndixB) . T he flow in th isregio n may be smootho r ro ug h ( larn ina r ortu rb ul e n t in tech n icaljargo n) or more usuallya b it of each. It s tartsoff la m in ar a nd th e nusu al ly changes intoa tu rb ule nt bounda ryla yer fu rt h e r dow nstream.

A lami nar boundarylayer has less drag bu t ismore prone to separatefrom the surface.

Wing Drag

In th e w ind tunne leach aerofo il sec tionca n be tested to find itsdrag by s im p ly me asu ring it o n a ba la nce .Using the formu la at thee nd of Chapter 2 thedrag coefficient can be

Basic Aero na utics for Modellers

ca lcu lated. In the case of a test on a wing section, thed rag is divided by Y, pVl and the wing area, and theresult ing Drag Coefficient , CD is a p roperty of thesection, independent of speed and size , and can be usedto es timate th e drag of a ny o the r wing usi ng tha tsection . It w ill vary wi th the angle o f attack ho wever, soit is normal to tes t it at a wide range of angles of attackand then p lot a grap h of drag coefficient against ang le ofattack for that section .

The typica l shape of such a graph is shown in Figure4.1. Drag coefficient turns ou t to be a very sma ll nu mbe rwhich at small angles of a ttack does not vary much.There is a min imum drag ang le of attack (point A)which is no t necessarily where (J. is zero. Approachingthe stalling angle of attack (point B) the drag increase ismore rapid while above the sta lling angle the d ra gincreases ve ry rapi dly indeed .

Whe n the wing stalls at poi nt B, the drag increase isproba b ly mo re significant than the reduction in liftcoefficient.

Drag Polar

Knowing the drag of a wing at a ce rtain angle o f

23

Figure 4.1

B

the resulting graph would look rather squashed so thedrag is always show n greatly ex aggerated. From thedrag polar you can read off the value of CLrnax and CDrnin •

Notice that the minimum drag does not ne cessarilyoccur 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 outfrom the polar diagram. At any point on the graphdiv ide the lift coefficient by the drag coefficient. Thebest VD ratio occurs at the point C wh ere the straightline just touches the graph.

Thickness and Camber

attack is only part of the story. The "drag polar" (as inFigure 4.2) is useful in showing how much drag thewing produces when generating a certain amount of lift.If lift and drag coefficients were shown to the same scale

The amount of the minimum drag depends mainlyupon the section thickness. The less the thickness, theless the minimum drag, but thin wings are not strong soa compromise has to be reached. In addition, a very thinwing has a sharp leading edge, and that is one of thethings which can cause an abrupt leading edge stall ason Figure 3.8, in the previous chapter.

The angle of attack, or lift coefficient, at which theminimum drag occurs varies with the section's camber.The more the camber, the higher the angle of attack atwhich the minimum drag occurs. Therefore the drag onan aeroplane which usu ally flies slowly can beminimised by using a section with quite a lot of camber.There is however a large increase in drag if theaeroplane is flown fast. In other words it does notpenetrate well. Highly cambered sections are oftencalled "low speed sections".

A wind tunnel can be used to measure the drag of afusel ag e (o r undercarriage or any other part of anaeroplane) . It too will consist of two parts . Surfacefriction drag will depend on the surface roughness, andon the surface area.

The mo re surface area exposed to the airflow (the"we tt e d area ") , and the greater the proportion ofturbulent boundary layer, the more the surface frictiondrag, but more important is the pressure drag which willdepend on the shape of the body.

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 littleimagination to see why . The drag coefficient is virtuallyconstant in the drag bucket and rises steeply on eitherside.

By careful design , and keeping the surface verysmooth, the designers of the sections have managed tokeep the boundary layers laminar (see Appendix B) aslong as possible to take maximum advantage of thelower drag. If the section is not built accurately, or if it isnot kept smooth and clean, the drag bucket willdisappear.

As with other sections the more the camber the largerthe angl e of attack where the minimum drag occurs, andthe more the thickness the more the minimum drag willbe. Curiously also, the thicker the section, the wider thedrag bucket will be.

Fuselage Drag

Laminar Flow Sections

01.01.S

B

..- - - _.. _.... -- -- - .. .. ---:..;--..,---....

A

Figure 4,3 Cl)

Figure 4.2

24 Basic AeronClutics/orModellers

Scale a ircraft like this SkJ1walker often use wheel spats which significantlyreduce profile drag.

StreamliningPressur e drag can be

m in im ised by c a re fu l"s tre a m lin ing " of th eb o d y , that is sh apingth e bod y suc h that th estre amlines in th e a irflo w foll ow th e sha peof the bod y rath er thanbreak ing away from thes u rface to leave aturbulent wa ke.

For example supposethe dra g o f a flat disc atright an gles to th e airflow is 100 unit s. Th edrag o f a sphere of thesa me d iameter w ouldbe o nly 45 units whil eth e dr ag of a ca re fu llystreamline d body, agai no f the sa me diameter , co u ld be reduced to o n ly fourunit s. Yes, the profi le drag of a strea mline d body ca n beredu ced to only fou r per cen t of th at of th e sa mediame ter of flat disc.

The drag d ue to the wake ca used b y th e flo wse pa rating from the surface is so mu ch more importantthan the sur face friction dr ag in th e boundary layer,wh eth er laminar or turbulen t.

A Bit for Golfer s

Why, yo u are w onderin g , does a go lf ball hav edimples? Well as it flies through the air at grea t speed, ithas a boundary layer.

The dimpl es are there to ensure that it is a turbu lentboundary layer, as turbu lent boundary layers cling to thesurfa ce lon ger before they se parate . \'(Thich red uces the

Figure 4.4

Smooth Ball

..

Dimpled Ball

Basic Aeronautics for Modellers

size of the turbul ent wak e whi ch reduces pressure dragby a subs tantial amount. It mor e than compe nsates forthe slight increas e in sk in friction drag. Hen ce the ballgoes further for a given clout. See Figure 4.4.

/

Turbulent Wake

25

Chapter 5

Have You A MomentA lmost certainly! It does not matter at which point

on the wing yo u choose to attac h the ba lance,yo u will almost cer tainly be ab le to measure a

moment about that point. The leading edge might bechosen as a co nvenient point as it simplifies the ensu ingca lculations . As w ith lift an d drag the moment coefficientCM is worked out from the formu la at a wide range ofdifferent ang les of attack and then plotted on a gra ph.For mat he matical reasons it was decided that nose upmoments would be defined as positive, but of coursethe moment about the leading edge w ill be nose down,Le. negative .

Figure 5,1

C~ILE

Nose Up

0( 0

0(

BA

NoseDoum

D

E

Figure 5.2

The gra ph wi ll lo ok like Figure 5.1 in w hich thepoints A, B, C, D and E correspond to those on Figure3.2. The line is straight from point A, the angle of attackfor no lift, to point C, where the wing starts to stall, andthen curves down to D and E as the wing stalls. In otherwords, the moment ge ts progressively more nose downas angle of attack is increased and then at the stallingangle there is a further increase in the nose downmoment. Please notice also that at point A, where lift iszero and angle of attack is ao (the 0 meaning "no lift"),there is still a nose down moment. The correspondingmomen t coefficient is ca lled CMo (where the 0 againmeans "no lift ") and it is a lways negative , Le . nosedown, for normal sections.

BUT, But , but! I hear you say. The moment is theturning effect of the lift force so how can no lift have amoment? We ll remember th a t all th is stuff about Liftforces, Drag forces, Moments and the Cen tre of Pressureis just for ad ministra tive convenience. \Vhat we aretrying to describe is a pressure distribution around thewing , so let us go back to that; look at Figure 2.10again. At the angle of attack at which the ba lance saysthere is no lift , the pressure dist ribution wi ll havecha nged to something like that in Figure 5.2. There w illbe a sma ll downward pressure on the front part of thewing an d a small upward pressure on the rear part ofthe wing, bu t th e angle of attack has been carefullyadjusted so that these cancel out. However they will stillhave a moment about the lead ing edge, or any otherpoint you care to name (see Figure 1.4) ,

Centre ofPressure

If the lift and drag and the moment about a knownpoint like the leading edge are known, then the positionof the Cen tre of Pressure (CP) can be calcu lated . As you

26 Basic AeronauticsJar Modellers

CP Position

"$0, TO LO$ETilE IIUN, t DIVEDIIEK $TKAIGIITDOWN• • • • ANDTilE

COFP $LlPPED klGIITOFFTilE WING,ZIPPED PA$THY EAK, ANDGOT TANGLED

IN TilE TAIL KIGGING WIKE$I"

: .1 1

I am a mathematical~ co ncep t y ou knoui

CP "'"'"

other way. You are perhaps wondering if it is possibleto choose a point in between such that the graph will bein between the others, de ad level like the dotted line infact? Yes, it's possible!

Ba ck in the da ys when Ca me ls fought a ga instAlbat rosses , the Centre of Pressure was the phrase oneve ryo ne 's lips , in aerodynamic circl es that is . But inlater years whe n aerodynamicists found that there was apoint o n the ae rofoil about which the moment did notvary with angle of ,attack , they were so pleased that theygave it a spe cial name, the aerod yn amic ce ntre of thesection (so me times shortened to ae rocentre or just A.C.).Here at last was a point at which they co uld pla ce thelift on their dia grams and in their little calculations andall they had to do was add a mom ent on the aerop lanewhich varied on ly with airspeed , not angle of attack .This new mathematical conce pt described the pressuredistribu tion (remember Chap ter 2) just as well as the oldCentre of Pressure mathematical conce pt. The beauty of

~ Chord

I Stall_ ___ __ __ J_ _ _

CLll ltu :

Figure 5.3

kn ow , th e CP m o ve saround and Figure 5.3shows the trend of themovem ent. Th e Centreof Pressure moves forw ard o n th e wing asa ng le o f a tt ac k isincreased . It nearly getsto th e qu arter ch o rdp osit ion but then thes ta ll moves it ba ckagain .

At th e other end ac u rio us thing happens. Wh en CL is ve rysmall the Ce n tre ofPressure dis appears offthe back of the wing.Th at ca n happenbecau se it is amathema tica l con-venience , no t tied to thewing ' with a piece ofstring . The distance ofth e ce ntre of p ressurebehind the leading edgeis ca lculated by dividing the momen t ab out the leadingedge by the lift coeffic ie nt. Wh en the lift coeffic ientbecom es very very small, the answer becomes very verylarge . When the lift is zero, the answe r is infinity! Youcan imagine that the ide a of a min iscu le force a giganticdistance be hind the wing wou ld have the sa me effec t asthe pressure distribution in Figure 5.2. Lift is defined asthe component of the resultant force at right angles tothe airflow so in Figure 5.2 there is zero lift.

In the spe cial case of a symmetrical aerofoil there isno mom ent at zero lift, and wh en the CP position iscalculated it turns out to be at abo ut the quarter chordpoint at all angles of atta ck right up to the stall, where itmoves back a bit as before. A fixed point like this is somuch more sa tisfying . It can be marked on diagrams ,and you can take moments about various points an d dolittle calculations (if thatis what turns you on) .

Wou ldn 't it be justthri lling if we cou ldd o th a t for camberedsec tions as well?

We ll , Figure 5 .4 isjust like 5.1 exc ept thatin ad d itio n to th e moment about the leadinged ge , it also shows theg ra p h of the momentabout th e trailing edgeas well. This lin e al sopasses through point Ashow ing th at th e zerolift mom ent is the sam eno matter about whichpoint it is measured. Theonl y difference is theslope, wh ich is now the

Aerodynamic Centre

Basic Aeronautics for Mode llers 27

Figure 5.4the effect of the same co mplex ai rflow and pr essured istribution , b ut do not fo rget th a t it is the p ressured ist ribu tio n which creates the lift , not the arrows orfor mulae which are just conven ien t ways of at temptingto describe it.

Aerofoil Section Summary/

/

/

Increasing the thickness will• 1. increase the minimum drag, CDmin• 2. widen the drag bucket on lami nar flow sec tions• 3. increa se streng th/weight ratio .

Increasing the camber will• 1. increase CU11:IX (very th ick or very th in sections 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.

Now th at I have ment ioned al l th e s e c tio ncharacteristics, I would like to descr ibe , wit h the hel p ofFigure 5.6 , how an aerofoil sec tion may be made up ,an d how we can influence its aero dy namic coefficien ts.

First d raw a straight line which wi ll be the chord lineof the section.

Next d raw in the ca mber lin e . The maximum ga pbe tween it a nd the chord lin e is the camber of th esection, w hich may be fro m zero to 6% or possib ly 8%of the chord . The ma x camber can occur between 15%and 60% of the chord fro m the lead ing edge.

Th en a thickness dis tribution is wrapped around theca mber line . This may b e do ne by d rawi ng lines o fappropria te length across th e ca mber line an d jo iningthei r ends, or drawing a series of circles w ith centres onthe camber line an d jo in ing the ir tangents as shown. Themaximum th ickness is usu ally between 6% and 18% andocc urs from 15% to 50% of th e chord from the lea dingedge.

Thickness and Camber

,\

III Bettoeen

z _/

A

/

//

//

it is that for a pa rticular section, the coefficient C'\ln is aconstant, jus t a small negative number, lik e -0 .05 forNACA 2415 for ex ample (but it is constant only belowthe section's sta lling angle) .

Now the pressure distribu tion may be represented byforces in fou r d iffe rent ways. They are shown in Figure5.5 . The first is the resu ltant force th rou gh the CP. Orone co uld show the tw o se parate compone n ts, Lift andDrag , at the CP.

Bu t since tha t is im practica l w he n you come tomeasure it in a wind tu nnel, the for ces can be measuredas Lift and Drag at a fixed point like the lead ing edgetogether with a momen t about th e lea di ng edge , andfinally the Lift and Drag at the aero dynamic ce n tre and amoment )'1'10 . This last method is most convenient forcalculations . These are all equally va lid ways of showing

Figure 5.5

L

AC

L

D

L

(

28 Basic Aerona 11lies for 1110dellers

Figure 5.6

Ma.'\: Camber

t

Cambe,. Line

Cb o r d Line

Il(/If

Th ick ness Distributton

Tbickness

Half Thick ness added eacb side ofca m bel'

• 4. increase the negative va lue of C~ IO ' which will bebe tween -0.02 and -0.03 for each 1% of cambe r.

• 5. reduce the negative (inverted flight ) Clm a,

Section Classification and Use

SYMl'l'JETRICAL sections hav e ze ro ca mber andtherefore ao and CMo are also zero. \Vithout camber theyhave rath er a low Cu lla, but at least it is as good invertedas uprigh t. The least drag occurs at ze ro lift. Symm etricalsectio ns are th us best fo r h igh speed a nd ae ro ba ticaeroplane s. Their th ickness is a co mpromise betweenst rength and drag , typically 10% to 18% for wings and6% to 10% for tailplanes.

All other sections are camberedsections.

An UNDERCAMBERED sec tion IS Just a thin highlycambe red section. Sometimes the camber is just enoughto mak e th e underside slig h tly co ncave as o n WW Iae roplanes . On some free flight floaters the underside isvery co ncave because the pe rcentage camber is as mu chas the th ickn ess. Such sections are very good at large liftcoefficients (low speed) but poor at small lift coefficients(high speed) , whi ch means they do not pen etrate well.They are also useless inverted .

The BICONVEX (or SEMI-SYMMETRICAL) section isso called because both top and bott om surfaces areco nvex, bu t the top o ne is more so. That is becausethe cambe r is small compa red to the thickn ess, and thefaster o r more aeroba tic th e aeroplan e will be , thesmaller the cambe r should be.

Th e FLAT-BOTTOMED section , like the Clark Y orGottingen 796, is a specia l case of a cambe red sec tion. If


the percentage thi ckness is chose n to be abou t 3.33tim es th e ca mber then the re a r 70% o r 80% o f theaerofo il underside often turns out flat. That makes iteasy to build, it has a good upright pe rformance but ispoor invert ed.

29

Chapter 6

The Vortex SystemW ith no wing in the wind tun nel the strea mlines

in the flow would be straig ht and paralle l asin Figure 6.1. Putti ng in a wing cha nges the

airflow so mewha t as sho wn in Figure 6.2. The changesimposed on the air's ve locity by the wing are an upwashjust in front of the wing , a speed inc rea se abo ve and adecrease below the win g , and a downwash behin d thewing . Figure 6.3 shows these velocity changes (shownas dV) in isolation . The effec t of the wing seems to be to

Figure 6.1

Figure 6.2

indu ce a kind of swirling mot ion to the air, around itself.A rota ting flow is ca ll e d a VO RTEX (Ap pe nd ix Cexplains vort ices in more detail) . \'V'heneve r a wing isproduci ng lift it tends to indu ce this ci rcu lating flowaround itself , and the more lift the more circulation . Thisvort ex is ca lled the "bound vor tex " as it is fixed aro undthe wing .

Vortices canno t end abrupt ly in mid air. In the win dtunnel they end on the wi nd tunnel wa ll which is fine.

But what happen s if thewing d o es no t exte ndfrom th e wall to w a ll?What happens if thewing has . . . (wait forit) . . . TIPS! YES fo lkswe are now into THREEDIME NSIONAL FLOW .er d id p romise to warnyou) .

We ll yo u know verywell what happens , the

Fig ure 6.3 ~

dV

dVIdV!

dV..30 Basic AeronauticsJar Modellers

~ Loto Pressure

~Wi"gTiPHigh Pressure

~

Top Surface FlowSlightly Inuiard

Bottom Surface FlowSlightly OutwardI

I

LE

TE

II

I

I,

(

Figure 6.4

vortices do not just end, they trail off in the flow behindthe wing tips . Th ese vortices are ca lled the TrailingVortices . They would go on for ever if the air 's viscositydid not dissipate them an d absorb their energy. I havewatched the condensation trails of a Boeing 747 st illgently rotating when fo llowing 2000 fee t be low andn ine teen miles be hind .

There is another eq ually valid way of looking at thesetrailing vor tices . At our newly acquired wing tips , the airp ressure is lower above the w ing than below. The airinevitably tries to go from high pressure to low, aroundthe tips, which gives rise to a degree of spanwise flow ,outward on the lower surface and inward on the top.The trend co ntinues to a decreasing extent some way infrom the tip . When the top and bottom flows reunite atthe trailing edge, they are moving in slight ly di fferentdirections, sligh tly outward on the un derside and slight lyinward on top.

In Figure 6.4 I have tried to show the result of all th is .Along the trailing edge, especia lly near the tips , vor ticesare formed which all roll up together to form one largevortex trailing beh ind each wing tip .

Seeing The Vortices

There is an easy way yo u ca n see your model 'strailing vor tices . Attach three streamers twe lve to fifteenfeet long to each wing tip of a suitab le model er used aGa ngs ter 63) . Lay them out straig ht on the gro und fortake off. On climb out it will be seen that the strea mers

are bei ng w hirled rou nd by the airflow, clockwise o nthe left an d an ticlockwise on the right.

TIY a slow flypast. The streamers will be swi rled inlarge slow sw irls. Now try a high speed beat up . Noticeth e diffe rence in the way th e streame rs are swirlingsuggesting a less strong vortex . The lift equa ls th eweight in both cases suggesting that a t low speed astronger vortex is needed to ge nerate the sa me lift. Onthe ne xt high speed pass try pulling a tight loop as themodel passes. The swir ling noticeably increases as soonas yo u pull the up elevator to inc rease the lift.

Now do a low inverted pass. From your p~int of viewnothing is d iffere nt. The lift is still up and the vorticesstill go clockwise on your left, and an ticlockwise on yourright (from the aeroplane 's point of view the direc tionsof rota tion and lift have all reversed). The lift is related tothe vortices in direction as well as strength. Check thatby coming in slow and high and do ing a bunt, or ou tsideloop.

Watch the st reamers ca refully as yo u ap p ly downeleva tor. You will see them stop rotating and then star trotat ing again the othe r way round . Th is will continue allth e way round the b un t until when yo u re lease th edown e leva tor to co ntinue no rmal fligh t, the rotationsreverse aga in .

The lesson to learn from this is that the stre ngth of thevor tices increases wit h the lift coefficient of the wing .After abo u t five mi n u tes of this th e s treamers hadflap ped themselves to pieces and were down to two orthree feet long.

Basic Aeronautics/or Modellers 31

Figure 6.5

AR = Infinite AR = Infinite

1

,.-,6/ / . . --, 3.>: "\

I , ' , /

1/ /I,'

1//1:/',:

1,/'~,r

:~

Even More Drag

The ASPECT RATIO of a 3-D win g is defin ed as thespa n divided by the ave rage chord. It is found tha t whena rea l wing with tips is tested in a wind tunne l its dra g ismore than if it fitted perfectly from wa ll to wa ll, and thelift is less. The loss in performance depends on its aspectrat io as illustrated in Figure 6.5 . The higher the AspectRatio of the wing, the nearer is its performance to that ofthe ideal two dimensional wing (infinite aspect ratio) .

The Reason

This sho rtfall in performance is caused by the trailingvortices which create a reg ion of descending air behindthe wing, after all the energy to crea te them mu st bep aid for somehow . Th at th es e vo rtices a lso causedownwash in the airflow as it approaches the wing canbe proven by theory , or dem on strated at hom e by fillinga tall glass with wa ter and placing a few grains of rice atthe bottom . \Vith a spoo n , stir the wa ter in the glass nearthe top and you will soon see the rice grains begin to

sw irl. Because of the fluid 's viscosity a SWirling motion isind uced right to the bo ttom of the glass. If the spoo n isthe wingtip vortex stirring the air be hind the wing , therice is being sw irled round ahead of the wing, in thesame direct ion , bu t to a lesser ex tent.

Figure 6.6 shows th e airflow around a re al threedimen sion al wing in more deta il. A long way ahead ofthe wing the airflow is undisturbed by its presence. Asthe air approaches , it is angled down sligh tly by thedownwash ahead of the wi ng ind uced by the tra ilingvo rtices and th en jus t in fro nt of th e wi ng the air isswept 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 anglebe tween the w ing and the und isturbed a irflo w awayahe ad of the wing . Now we can see that the "real" angleof attac k of the air meeting the wing has been red ucedby the downwash. And the lift re lates well to the liftp redic ted from 2-D tests at this reduced angle of attack.So the lo ss o f li ft is exp la in e d by the d ownwas hred uci ng the a ng le of a tt a ck . But w ha t abo u t theincrease in drag?

Look ing back at Figure 6.6 again, the ae roplane mu st

.."..~..

'~OIlIlY, TIIAT ~1I0ULD BE'NOW TIlYA BUNTCAIlEFULLY,WATCIIING rse~TIlEAMEIl~"

• . •~. . . C'"

32 Basic Aeronautics/or Modellers

Figure 6.6

Real Lift

---

Real Cl.

- - -

Measw'ed Cl.

Doumuiasb All

-------=======:------Undisturbed Ab' Doumuiasb III Front Of lVi1lg Doumuiasb Behind Wing

think it is constantly flying uphill , or rather flying levelthrou gh the sinking air of its own do wnwash. The liftforce has been tilted back a little , by the amount of thedow nwas h a ngle . Th at means th e lift ha s a sma llcomponent in a direc tion opposite to the direction ofmotion . If it opposes motion it is Drag , isn't it?

This compone nt of the total drag is called "induceddrag" because it is caused by the tilting back of the liftca used by th e downwas h ind uced b y th e tra ilingvortices. You may also see indu ced drag referred to as"active drag". No amoun t of strea mlining or fiddling withthe section will red uce it. It is inevitable as it co mesfrom the lift.

Complications

Figure 6.6 shows the airflow at one particular placeon th e wing . Near the tip s w here th e vortices arecentred the downwash is grea ter than on the ce ntreline,as in Figure 6.7. Thus o n th is recta ngu lar wi ng eachpoint from the roo t to tip has a different "real" or "local"

ang le of attack and therefore a d iffere nt "local liftcoeffic ient ", usu all y d e noted Cl (w ith a s ma ll Isubscript). The local lift coefficient reduces towards thewing tips. The situa tion even on a straight wing is not assimple as the p icture I have pai nted up to now. On ata p e red tw isted swep t wi ng the si tua tio n is asco mplicated as yo u ca n imagine , if not m o re ! Th evariation of the "local" angle of attack de pends on thevariation of the downwash which depends on . . . welljust about everything, including the variatio n of the localang le of attack .

So because the downwas h angle, lift coefficient, dragcoefficien t, lo cal ang le of attack , ce n tre of pressureposition, etc, all vary wi th po sition along the span , thatmakes it very difficult to use the section characteristicsmeasured in two dime nsional flow. It is just too muc hfor the hu man bra in to cope wi th and is be st left toco mputers with time on the ir hands.

\'\fe co uld give up the who le messy bu siness here andnow , or we co uld just step back and look at it from adistance.

Figure 6.7

~Direction OfMotion

Wi1lg's ApparentAngle OfAttack

Root Doumuias

Direction OfMotiotl

WhIg's ApparentAngle OfAttack

Tip Cl.

Tip Doumuiasb

Basic Aero naut ics fo r Modellers 33

Figu re 6.8

~-- -- ~-

Airfloto

Simpiifications

L

Lessons f o r P r a ctica l Modellers

This looks like a cla ssi c case for the bla ck boxsystem . I shall draw an imaginary black bo x around thewing and care not a wh it for what is happening insid e .Air ente rs the fron t of the box and co mes o ut o f theback angled down slightly by the downwash, The angleof attack is measured between the direction of motionand a referen ce line drawn on the outside of the box.The refer en ce line may be e ither the cho rd line at thero ot sectio n or the ze ro lift line of th e who le w ing .Th ere is a Lift force perpendicu lar to the d irection o fmotion and a total Drag force , including induced drag,op posi te to the direction of motion. There will be anoverall Centre o f Pressur e but I wouldn't care to guess atits position so I prefer to p ut the lift a t th e w ing 'sae rody namic ce ntre (25% mean chord) and apply a ze rolift pitch ing mom ent. See Figure 6.8. \Xrhen I refe r to liftcoefficient , or drag coefficient, I mean an average for thewh o le win g wor ke d o ut from tests and the Cha pter 2formula e .

The lift co efficient Cl. is the overall average for thewh ole wing and has a cap ital L su bsc ript. Its graph willstill be a familiar sha pe but need not be the sa me as thesection's curve . The wing would hav e to be tested to getacc ura te graphs but a reasonable guess could be madeby making "allowances" for as pect rat io, tap er and twist.The slope of the straight bit will dep end on the aspectrat io (as in Figure 6.5). And the posit ion of aD and thestall will dep end on the wing's planforrn and twist asmu ch as its sec tion.

The Importance ofAspect Ra tio

A sho rt span broad win g mu st have a stronger vortexto give the sa me lift as a lo ng narrow w ing, and willconseque ntly produce more downwash . Therefore theinduced drag is greate r for a win g of low Asp ect Ratiothan for one of high Asp ect Ratio (AR) because of theg re a te r d o wnwash , An d as th e a ng le of a ttac k isinc reased , bo th Lift a nd downwash a ng le in cr e ase(because the vortex stre ng th increa ses ). Thus induceddrag increases rap id ly. In math ematical shorthand theinduced dra g coefficient (Cn) is given by;

34

Reducin g induced drag is important for aeroplaneswhich cru ise a t a la rge lift coefficie nt (Le low speed)therefore glid ers must have as high an Asp ect Ratio as ispra cticab le . Also a lighte r aeroplane has less induceddr ag than an identical heavy o ne at th e same speedbe cau se of the lower lift coefficient.

Any lift, even downward or s ide ways lift, will causevor tices whi ch cause down wash which will tilt the liftback wh ich co ntributes to induced drag . If a tailp lan e iscarrying a download, not only does the downward lifton the tail produce induced dr ag but the Wing mustproduce ex tra lift to counterac t the download and thatmeans ex tra indu ced dra g on the wing as well.

Most wind tunnel 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 timateth e pe rforman ce of a re al ae ro p la ne th ey will g iveopt im isti c answe rs becau se the y do not include theinduced drag. It is possib le to es timate the characteristicsof a wing from section da ta using var iou s fidd le factorsbut that is outside the scope of this bo ok.

Grou nd Effect

Looking at Figure 6.6 you probably thought that thewind tunnel wall s would co nstrain the downwash effect,and you were right. Th e induced drag of a finite Wingwill be underestimated in a wind tunnel because thedo wnwash is reduced.

The same thing will happen if an aeroplane is flownvery low over the ground . The proxim ity of the groundwill reduce the downwash and therefore the ind uceddra g will be reduced. The effec t is to prolon g the flare .For this reason , flight testin g mod els with in a few win gcho rds of the grou nd will give misleading result s .


Chapter 7

Planform and Twist

Loading <Lift/Unit Span

dott ed line. As seen in the p revio us cha pte r induceddown wash ahead of the wing redu ces the local angle of

Rectangular Area Dtstribution

Figure 7.2

Figure 7.1

Elliptical Lift Distrilnttton Loading - Lift/Unit Span_-.,-.....-rTTT"'T"TTT"TT"T...........--,-.">-----

Local Cl -sLtft/Unit Area

A s you saw in theprevious chapter,the Aspect Ratio

o f th e wi ng lar gel yd et e rm ines ho w mu chind uce d drag it wi llca use . Bu t th e littl ee q ua t io n for ind u ce ddrag coefficient which Iju st s lip ped in th e reco ntaine d a co nstant, K,as we ll. Thi s K (ca lle dthe Indu ced Drag Factor) depen ds upon howthe load is sha red alongthe wing . The lift maybe eve n ly sp read , o rmostly near the roo t, orwhat ever.

Fig u re 7. 1 s howswha t theor y says is theid e a l lift d istribu ti onwhich gives K its min i-mu m value o f 1. Th is isa d iagram of the Lift PerUnit Spa n. Each arrowrep re sents th e li ft o na o ne inch wide strip ofwin g , and th e lin ejoi ning the to ps of allthe arrows is an ellipse .(I n future I s hall onl yd raw the line along thetops of all the arrows as is more usual). Fo r all o the r liftd istributions K will be a little grea ter than 1.

Local Angle ofAttack

Figure 7.3

We m ust a fte r a llpe e k ins ide the b lackbo x d esi gn e d to o bsc u re a ll thi s co mp lication . O n our usu alrectangular wing wh osecho rd, sectio n and in ci de nce a re the sa mefro m ro ot to tip theload ing , shown o n therig ht of Fig u re 7 .2 , isneithe r rectangular likethe planfo rm nor ellipt ica l as sho w n b y th e

Elliptical Area Distribution

J 1 ~Local Cl <Ltft/Unit Area Loading -s Ltfr/Untt Sp a n


To reduce the indu ced drag factor the planform of thewing may be changed .

It is good man age ment pr act ice to movework ers from wh ere theyare id le to w he re th e yare more productive andso we do with win g area.

An o bvio us id e amig ht be to use an elliptica l p la nforrn (Figu re7 .3) wit h n o twis t.Tak ing a close lo o k atsuc h a wing und er test ,an e lliptica l lo ad ing ispossibl e and the downwash is co nsta nt ac rossthe span, which meansth at th e lo ca l angle ofa tta ck is also co ns tan t.The re fo re th e lo cal liftcoeffic ie n t is the sameall ove r the win g. Everysq ua re inch is doing itsfa ir s ha re of lifting ,hen c e ma ximum e ffi ciency . It co uld be trickyto bu ild with accu ra tesections throu ghout.

A reaso na b le comp rom is e is to b u il d awi ng wit h a st raig h ttaper from root to tip asin Fig ure 7 .4 in w hic hyo u ca n see that the liftdist ribution ca n be made

quite close to the idea l e llip tica l shape. It is nea rly aseasy to build as the rectangular wi ng but is sig nificantlymo re efficien t. Tapering the wing helps to improvee fficie ncy b y eve n ing out th e d o wn w as h a n d soincreasing the local lift coefficient towards the tip . Thusth e lift is more even ly sha re d th an o n a Rectan gul arw ing . Th e o p timum Taper Rati o (defined as the tipcho rd over the root chord) is abo ut 0.4 . If the tap er ratiois redu ced bel ow th e opt imum 0.4 , th e lo cal angle

Loading - -~

Taper R atio 0.33

Taper Ra tio OS

(I

JI

Local Cl I

I

I

Max

Figure 7.4

attack towards the tip. The result ing local lift coefficient,or lift per un it area , is shown on the left side of Figure7.2. Loadin g and local Cl are the sa me because chord isco nstant in this case .

Different Planform Shapes

Fig u re 705

Double TaperTaper Ra tio OS

Loading

--- - - - ---- --~-~-~-~~~

Local Cl

- - - - --

M ax


Figure 7.6

RootWashollt

_-=- -=- -::. :. -=- I) -=- -:. r ::::===--~__

If the lift distribution is of the ideal elliptical shapethen the local angle of attack will be co nstant along thespa n. Theoretically the w ing sho uld st all all the wa yalong the wing simultaneously. But the inevitable slightimperfection or dirt or even a twitch of aileron meansthat o ne side will stall before the othe r. Even if just theoutboard portion of one wing stalls first the effect of thedifferen ce in lift and drag between one wing an d theother is a vio lent roll, a "wing drop". Because of its veryefficiency , an elliptica l wing is likely to drop a wingwhen it stalls, and pilots just don't like aeroplanes whichflick themselves upside down unexpectedly.

On a constant chord wing the local angle of attack isleast near the tip . The airflow approaching the wing is"twisted" down at the tips ensuring that in steady flight itis most unlike ly to tipstall. But e ithe r surfaceirregularities or turbulence or a rudder input, if severeen ou gh , can produce a tipstall on most wings . Of courseavo iding tipstalling does not mean that a wing will notdrop at all. Even a root stall one side at a time will givea roll, bu t nothing too violent.

A tap ered wing's characteristics will be between thetwo , dep ending on the tap er ratio. The more taper, themore lik el y is a tipsta ll. If th e taper on a wing is 'ex ce ssive (ta pe r ra tio 0 .4 or le ss) then a tipstall isin e vit able as the local angl e o f a ttack is greatestoutboard.

In Figure 7.4 I have shown the spa nw ise positionwh ere local Cl is a maximum, and this is where the wingshould stall first (assuming constant sections).

of a tt ack near thetips becomes greaterthan ov e r the inboardsect io ns, the tips beco me overloaded , andefficiency is again lost.

Another slight decr ease in the K fac tormay be had by buildinga wing with a doubletaper , wh ich is verylittl e extra trouble esp ecially if the wing istoo long to build ino ne piece anyway . SeeFigure 7.5. The doubletap ered planform givesa n a re a dist ributioneven closer to tha t of anellipse and so is moreefficient still.

Tips talling

Figure 7.7

Tip

C, Lift/Spa"

: LOllJ1Speed1I

Local C, 1 Load ing11 ".. ---1 High ....

~

1- _ _Speed ....

1....

....

Ma~:

Tip

WashoutIf a wing is twisted such that the incidence decreases

toward s the tip , that is called "washout". Cl deliberatelyuse the word incidence because the angle of attack willdepend also up on the variation of the downwash .) SeeFigure 7.6.

Th e oppos ite twist is, logically, called wash in. Wemod ellers use washout as a cure for tipstalling . Twistingthe leading edges down at the tips allows them to fly ata lower angle of attack so hopefully the roots will stallfirst.

Another effect of washout is to shift some of the liftinboard on the wing (Figure 7.7) . Th e effec t of that willobviously be to increase the indu ced drag factor, K. Thetips may e ve n be lifting downward at high speed .Adding washout to an efficiently tapered planform takesaw ay some of its inherent efficien cy.

Consider the effect of wa shout o n the graph of liftagainst ang le of attack for the whol e wing. In Figure 7.8,if line A is for an untwisted wing, then line B shows theeffec t of the same wing of con siderable washout. Theangle of attack has been measured aga inst the chord lineat the root. The root has to be rotated to a higher angl eof atta ck to achieve the same overall lift because the tipsstart off lifting downwards.

The stall is very mild and prolonged because it startsat the root and works its way gradua lly towards the tipas angle of atta ck is increased. As the lift on the tips isincreasing , the lift on the roots is reducing, and themaximum lift de veloped by the wing is much less incase B. That means that , although the stall is gentler, thestalling spee d will be high er.

Basic Aeronauticsfor Modellers 37

parts are do ing their fa irsha re of lifting . Insteadof unl oa ding the tips top re vent th em fr omstalling (and spoilin g thee fficie n cy), w hy nottr a in them to w o rkh arder? Ch oose a tipsection which is capa bleof a hi gher Clm"x thanthe roo t se ction , becauseit has mo re , or d iffe re n tly sha ped, ca mber.Then apply so me washout to the wing so thatboth ro ot and tip w il lhave th e sam e Cl fromzero lift to the stall. Butthe root will stall first! Itsounds like a Uto p ia nid e a but tak e carese lecting the sec tio ns,and read the cha pte r o nRe ynolds Num be rcarefully.

Sweepback

< .

~t;;(J_ ...

-------=---_.--

0(

Measw'ed At Root

Root

The ang le of sweep o ug ht to be me asured at th equ arter chord line of the wing (altho ugh so me peoplemeasure it at the leading edge). Sweep ing the wingsba ck (o r forward for that matt e r) is a device for delayingthe onset of the transon ic drag rise, which is nothing todo with us mod elle rs.

We do not ca re that a straig ht wing ex periences asharp pr ofil e d rag increase as speed rises through aMach Numbe r o f 0.6 o r so (0.6 times th e speed o fsound) . Nor does it matter that th is d rag rise ca n bedel ayed to Mach 0.8 or 0.9 by sweeping the wings back .

But models do ge t bu iltwi th swept wi ngs so Ihad bette r mention it.

Ap art from its ma inpu rpose me n t io ne dabove, it can be used toadjus t the position of themean chord o f the wingback or forw ard with outmoving the roo t fixings.Su ppose yo u d es ign amodel and it turns o u ttail heavy . Yo u cou ldbuild a new wing with alittle sweep back and getit co rrec t ly b al ancedwithout ball ast a n dw ith o u t a lte ri ng th efuse lage .

Sweepback also hasvarious side effects, likethe e ffec t o n lat eralstability (see Chap ter 9),and in large doses it hasa detrimental effec t onth e efficie ncy o f th ewi ng. T he a ir h a s a

C

Washout Angle 1:---

A Tip.. ...~ .

Figure 7.8

A similar effec t to wa shout can be obtained by usinga tip section wh ich stalls at a high er geomet ric angle ofattac k than the root, keep ing their chord lines parallel asin Figure 7.9.

Th is effec t is called "aerodynamic washo ut" since, asa is increased the roo t section will reach stalling anglefirst.

Figure 7.10 is a different app roa ch to the pr obl em.We have design ed a wing wh ich is efficient because all

Aerodynamic Washout

38 Basic Aeronautics f or Modellers

Figure 7.9

Root

Ti

Root

~II

I TipIII

IIIIIIII

I Aerodynamic~ WaskroutII

a

tenden cy to flow out alon g a swep tbac k win g towardsthe tip s which reduces the lift. Th e tendancy ca n becounterac ted by using win g fen ces or notches etc.

The line from A to B crosses the 50% chord line at pointC, the ce ntro id of the shape . The chord through C is thewin g's M.A.C.

Mean Chord Geometric Mean Chord

What does one use as the win g chord measurem ent ifthe wing is tapered and swept? The cor rec t va lue to useis what ae rody namicists ca ll the "Mean Aero dy na micCho rd" CM.A.C. for sho rt) o f the wing. It need s ratherco mplica ted mathemat ics to define a nd ca lcu la te it.However it may be found graphically.

Graphical Methods

Figure 7.11 shows a g ra phical way of find ing th ece ntro id of a swept tap ered wing. Draw the wing out toscale and square off the wing tip .

Now join th e midpoint o f th e tip c ho rd to th emidpoint of the root cho rd . Extend the tip cho rd forwardby the amount of the root chord to po int A. Extend theroot cho rd aft by the amount of the tip chord to point B.

Th e j'd .A.C. is unne cessaril y co m p lica te d for usmodelle rs . Th e Ge om etr ic Mean Cho rd is jus t th eave rage o f the root and tip cho rds, or the wing areadivided by the span . The G.M.C. is slightly smaller thanthe M.A.C. but with a tap er ratio of 0.65 the erro r is onl y1.5%.

I suggest that the geometric mean chord is acc ura tee no ug h for o u r purposes and shall us e it in fu tu re ,dropping the "geometric" and calling it just the "meancho rd".

Horses for Courses

Th e use of taper a nd was ho u t is a co mp ro misebetwe en perfo rmance and handling and a differentcompromise is ne cessary for different types of mod el.

Figure 7,10

0<

I I

-- -,, Tip,I

III RootI

I I+=+:::,..

Tip

d:=I I

C-R-oo-t-~

0<

I

I Root

I : Add Some WasboutTip I I ..

~I I

0oot:~I I

Tip ,,- .... ,

"""""""""""""""""


Figure 7.11

Willg Centre Line

I

C,.',I

II

J

I

II,,,,

\

,Cl

II

II

\\

B

HalfSpan~---------------------------------------~

A

\

\

\,,I

II,,,

I

I

,c"

\

\I

II

ic,

GlidersIndu ced dra g is

ve ry s ign ifica nt, th erefore gliders use taperedwings , if possible doubletapered . Th ey a lso useas high an aspect rati oas possible to re d uceinduced drag.

However th ere maybe occasions wh en co ntest rule s make the highas pec t rati o le ss n e cessa ry (see cha p te r o nReynolds Number).

TrainersIt is imp ortant that a train er should not tipst all eve n

when provok ed by rough handling , and it must be easyto build . For these reasons a co ns tant cho rd un sweptwin g of mod est Aspect Ratio is usuall y employe d . Theloss o f aerody namic efficiency is unimportant.

Scale

Scale mod els leave no cho ice of taper ratio , as pectratio , or sweep. If the tip chord is less than two third s ofth e root cho rd , or if it is e llip tica l, th en a co up le ofdegrees of washout would be prudent, more if the wingis stee ply tap ered . The scale mod eller adjusts the amo untof washout to achieve the kind of handling he prefers.

Aerobatic

The "Pattern ship" must above all be predi cta ble . Itmu st not tipstall accide ntally and ye t it mu st sp in reliablywh en co mmanded with rudder co ntrol.

It mu st fly as well invert ed as upri ght and so washoutis undesirable . For these reason s a tap er ratio o f 0.55 to0.65 is usually employe d. Any sweepback would only befor aes the tic reasons, altho ug h I have heard it sa id that ithelp s the rolls and disguises aerobatic er rors.

40 Basic Aerona uticsfor Modellers

Chapter 8

CG and StabilityThe cen tre of Gravity (or CG) of an aeroplane is

the point throu gh which its weight acts , or whereit can be suppo rted with out falling over. Its fore

and aft position , which is wh at matters most , is shownon plan s by the symbols~ or 0Most mod ellers have a general idea of wha t stabi lity isand how it is related to Cen tre of Gravity pos ition, bu tlet us go a bit further for a better understand ing .

Defining Stability in General

Th e s ta bi lity o f a body is its tend e ncy to ret urntowards its original state if disturbe d slightly. The re aretw o types of s ta b ili ty , Stat ic Sta b ili ty a nd Dyn amicStability.

Dyna mic Stability is about what happens over a longperiod of time following a disturba nce. That is, a bod y isdynam ically stab le if some time after a distu rban ce it hasse ttled down into its origina l state . Dyn amic Stability isimpossible unl ess you first achieve Static Stability.

Sta tic Sta b ili ty is co nce rned o n ly with the in itia lreaction of the body follo wing a disturbance . Th at is, ifthe bod y is dis turbe d slightly from its equilibrium state ,will forces automatica lly arise which will ten d to make itreturn back towards its equilibr ium state again? If so it isstatica lly sta ble.

Th ere a re thre e poss ible condit io ns : Uns ta b le,Ne u tra lly Stable , orStabl e . Fo r a d e mo nstration of each of theses tates ge t a snoo ke rb all , a Ch inese Wo k ,and an old British armysteel helmet o n a pooltab le . Let the ball rest inequilibrium right in thecen tre o f th e Wok.Distu rbing it slightly willpr oduce a force whic hwill make it wa nt to rollback towards the ce ntre .See Figure 8.1. Thi s issta b le . No w ca refu llyb alanc e th e ball atth e ve ry to p o f th esteel helmet. Disturb itslightly and it will tendto con tin ue in th atdirecti on and fall rightoff (Figure 8.2) . That isunstable or "Divergent".Now place th e ball o n

Basic Aeronauttcs f or Modellers

the pool table . Disturb it sligh tly and it will remai n in itsnew position (Figure 8.3). There is no res tor ing force ordiverging force . It is neutrally stable.

Th is ex perime nt shows the sta tic sta bili ty an d isco ncerned on ly with the initial reac tion to a disturban ce.

Figure 8.1

Figure 8.2

1 2

41

Motion ofan Aeroplane

An ae roplane is free to move in thre e dimen sions. InFigure 8.5 I have drawn the three axes on an aeroplaneand ca lled them forwa rd side ways and down. The axe s

00Wh at the final outco me will be afte r a time has elapsedwill depend upon the dynamic stab ility.

If a bod y (ball or aeroplane) is statica lly stable likethe ball inside the \'{rok, we may investigate its dynam icsta b ility . If the ball is held four inch es up the insidesurface of the Wok and re leased, it will return to thece nt re bu t will overshoot and go up the other side saythre e and a half inches , then down and up three inchese te. until afte r a while it will settle in the centre . It isthu s dynamically sta ble due to the damping force causedby the rol ling resistance . If the Wok were full of wa ter itwould have grea ter da mping (wa ter, damping, oh nevermind) which would increase the dynamic stab ility andthe ball wou ld se ttle in the ce ntre more quick ly.

Figure 8.4 shows the kinds of stability graphically.Th e dynamic stabilit y o f aeropla ne s is a very

co mplica ted subject which I do not want to d iscuss.Suffice it to say that to a large ex tent it depends uponthe aircraft having sufficie nt aerodynamic damping, orresistance to pitch ro ll and yaw motion s, which I sh allme nti on lat er, in Cha pte r 10. Th e rema inder of thischapter is about Static Stabili ty.

are fixed in the ae rop lane wheth er it is climbing, diving ,ro lling o r whatever. Sideways movement is ca lleds ides li p , and th e ra te o f forwa rd mo vement is theairspeed.

The aeroplane is free to ro tate ab out these axes aswel l. Rota tion about the lo ngitudinal axis is called roll,rotation abo ut the lateral axis is ca lled pitch and rotationabo ut the ve rtical axis is called yaw . See Figure 8.5. There ma ind e r o f th is cha p te r is conce rned w ith s ta ticstability of pitching motion around the latera l axis whichis neverthel ess ca lled "lo ngitud inal" stab ility. Ro ll andyaw are dealt wit h late r.

This is an area full of misconception s and arg ume ntfor mod ellers , therefore let me defi ne the subjec t close ly.First ly the name is Longitud inal Stat ic Stability. Th is isabou t an aeroplane flying along in equilibrium (in trim)and sudden ly so me thing , like a gust or a glitch or thepilot sneezing . causes it to p itch u p (or dow n) changings ligh tly the a ngle of a ttack, b u t not the sp e e d o ranything else . I know th at a lift increa se wi ll start aclimb but I am not co ncerned with that. I only wan t tokn ow "w ill th e change of fo rce s te nd to make th eaeroplane pitch further nose up, or nose down, or haveno pitching mom ent at all about the CG?"

First , le t us look at the forces on a wing alone . Figure8.6 sho ws a wing flying along in equilib rium. All theforces and mom ents on it are exactly balanced . Thrustand dra g are equa l and are bo th throu gh the CG (ando mitted fo r s implic ity) and Mo exactly balances themom ent of L about the CG (Mo=L.x). Notice tha t I havesho wn the zero lift pitching mom ent with a nose Do\'{rNarrow (Le. the way it reall y acts) to make my diagramsclearer (altho ugh it is normally defin ed as positive nos eup) in spi te of Chapte r 5.

Now if the angle of attac k is increased , L increases bya sma ll amo unt, say XL (for Xtra Lift) . Mo still balances

Stability ofAeroplanes in Pitch

21

Figure 8.3

Figure 8.4

StaticallyStable, but

//Statically Unstable/ [diuergent)

//

//

Dynamically Stable

Dynamically Unstable

.... -....Time


w

CG

YllW

DOW1l

x

Vertical Axis

the distance between their aerocentres. Notice also thatthe further the CG is ahead of the NP the greater thestabilising moment.

Achieving a good balance between the stability andcontrollability of an aeroplane is partly achieved bycorrectly positioning the CG. A compromise has to bereached whereby the aeroplane is safely stable and isalso easily controllable.

The Correct CG Position is what feels good duringflight testing, and will vary according to the purpose ofthe model, and the individua l preferences of the pilot.You will usually get a CG position from a plan , orperhaps a formula, but that does not make it exactlyright. It is just a starting point.

If an aeroplane's CG is too far aft it is very sensitiveto small elevator movements and small elevator trimchanges. It needs little or no down elevator to holdinverted. There is a dangerous tendency to over-control

CGPosition

L

Aerocentre

Mo (=L.x)

Fortuard

Figure 8.6

Figure 8.5the moment of theoriginal lift L, but themoment of XL aboutthe CG will pitch thewing further nose upwhich is unstable (Figure 8.7) .

Notice the differencebetween equilibriumand stability. This wingwas originally in equilibrium because all theforces were balanced ,like the ball on top ofthe helmet, but itsreaction to a disturbance was unstable.

Flying wings do fly(in Chapter 16) but thenormal way to achievestability is to add atailplane , or what inAmerica is more aptlycalled a "stabiliser".

Figure 8.8 shows anaeroplane flying in trim.All the forces andmoments balance outLe. T = D, L = WandMo balances the moment of L about the CG.What happens whenthe aeroplane gets aslight nose up disturbance? Only the winglift and tail lift change.In Figure 8.9 only thechanges, the Xtra Liftforces, are shownbecause all the otherforces cancel each otherout. (If you like, all theforces in Figure 8 .7have been combined inone resultant force,which is Zero).

Let us say that for a one degree pitch up the wing'sXtra Lift is 5 and the tail 's Xtra Lift is 1 (because it issmaller). These two are separated by a distance of 36units . Some simple mechanics show that they have aresultant force of 6 at a distance of 6 units behind thewing's aerocentre. This force determines the stability. Ifit acts aft of the CG as shown in Figure 8.9 it will pitchthe aeroplane nose down which is stable , and thefurther behind the CG the greater its moment and so themore the stability. Conversely if this resultant wereforward of the CG it would be destabilising.

In the special case where this resultant acts exactlythrough the CG then the static stability is neutralbecause it will tend to pitch the aeroplane neither up 'nor down, because it has no moment about the CG. Thename Neutral Point is given to the point of action ofthe resultant force due to a pitch disturbance because ifthe CG is at this point then the stability is neutral.

Notice what went into finding the Neutral Point: thesize of the Xtra lift on the tail relative to the wing and

Basic Aeronalilies for Modellers 43

Figure 8.7

Mo (=L.x)

L +XL

Aerocentre

x CG

Simplicity Itself?If th e tail are a is a

fifth of th e wing a reathen its Xtra Lift shouldb e a fifth that o f thewing and like in Figure8.8 the Neutral Point willbe a sixth of the tail armaft of th e wing' s aeroce n tre ( ta il a rm is th edi stance b etween th eaerocen tre s , or quarte rc ho rd p oints , of w ingand tail) .

Complications

the aerop lane and a real risk of sp inning out of contro lor overstressing the wing.

If its CG is too far forw ard the model's response toelevato r contro l will be sluggish , it will hold inve rtedonly with large amounts of down elevator - if at all and it will be d ifficu lt or imp ossible to stall and spin. Itwill also pitch excessively with speed changes.

The CG is cor rect when the aircraft handles just theway you like it. But yo u have to start somewhere , andthe nearer you can get it in the first pla ce the better.

The parameter wh ich governs the stab ility is denotedKn and in our simplified theory may be ca lled the CGMa rgin , Stati c Margin , Stability Factor or StabilityMargin, which is th e one I sha ll use as it seem s themost descript ive .

The ph ysical meaning given to the stability margin isthe distance betwee n the aeroplane 's CG and its NeutralPoint (N P) . Kn is g iven as a d e c im al fraction , orpercentage, of the mean chord of the wing. A suitablefigure for Kn is from 5% to 25% of wing mean chordwith an optimum in the region of 10% to 15%. That is allwe have to do then. Find the NI' and put the CG 15% ofthe chord ahead of it!

If onl y it were that easy . I kept my example nice andsimple to illustrate the principles involved but, as youare probably aware , wh at a lo t I left out! A larg e numberof factors co mbine to reduce that expected Xtra Lift onthe tail , so the Neutral Point is further forwa rd than Icalculated .

The most obvious problem is the downwash. \'1!henthe wing 's angle of attac k increased by on e degree , thedownwash over the tailpl ane also increased , reducingthe angle of atta ck increase experien ced by the tailpl an eto somew hat les s th an one d e gr ee. \'Ve may lo se aqu arter , a half or th re e quarte rs of that one degreedepending on the wing's Aspect Ratio , and also on thetail gap, the wing lift distr ibuti on , and the tail heigh t.

Norma lly the tail ha s a lower asp ect ratio than thewing. Going ba ck to Figure 6.5 , the tail 's lower aspectrat io means that its lift increase will be less per degreeth an the wing's . Not only th at but using a very thinse ction , especially a flat plate , may impl y a still lower liftcurve slope .

Th e wing and the fuselage have boundary layerswh ich flow off behind the aeroplan e as a wake , movingslower than the rest of the airflow . If the tail is flying in

Figure 8.8

.L(=LII' = LrJ

I

D

lV(=L)

30

A X L = 6

I Distance 36-1 -- - - - --- -I

Figure 8.9


this wake the n its lift increase will be less tha n expectedbecause XL is pro po rt io nal to airspeed sq uared . Theeffect is neglig ible for a T-tail but quite significan t for a"draggy " aeroplane.

We Can Work it Out?

I would like to be as opt imistic as the o ld Bea tlesso ng but to be honest anyone w ho cla ims to be ab le tocalcu la te accura tely th e co rrec t CG for a mode laeroplane is kidd ing himself Th ere a re just too manyva riables and not enough sources of informatio n . Sowhat do we do? \'{1e gu ess . But the more accurately wecan gues s the better.

\'{1e make up a fo rmu la in co rpo ra ting a ll the"allowances" and the 15% stabil ity ma rgin to give a CGposition directly. It can be as simp le as "Put it at a th irdchord mate!" or as compl ica ted as looking up a textbook and finding a formul a for neutr al po int distance aftof the aerodynamic centre . Then yo u add the .25 chordwhic h the aerocentre is be hi nd th e le ading edge,sub trac t the .15 chord stabil ity margin and end up wit hsomething like

CG pos = 0.1+ q T aT (I- DWF) Vharq a

w he re

.9.I is the ratio of the (a irspeed)! at ta il and w ingq

a T is the ratio of lift curve slopes of tail and wing , anda

( I -Down wash Frac tion) is wh at is left of the angle ofattack increase by the time it reaches the tail and Vim is

tail area tail arm IT----- X -------'-wing area wing chord

and is ca lled the tail vo lume ratio . (see Nomogram inAppendix E)

The tail area should be measured excluding the partcovered by the fuse lage or cu t away to clear the rudder,bu t the wing area should incl ude the a rea wi thin thefuselage . In deriving this formula it was assumed tha t thetail is behind the w ing and is appreciably sma lle r tha n it.

It is possible to look up obscure tables to es tima te theinformation to pu t into the eq uation bu t is it worth allthat tro ub le for each mo de l? I do ubt it. The answer w illstill only be as acc ura te as the "estimates" yo u put in.

Simpler Equations

What we modellers need is an equa tion which is easyto work ou t ju s t fro m t he measure me nts of theaeroplane so let us make some assumptions and pu tsome estimates in the above eq uation .

If the aerop la ne is reas o nab ly clea n , the ta ilun obstru cted, and its area calculated as before , then wecan make an allowance for the airspeed fac tor.

The lift curve slope factor is always about the same ifthe tail Aspect Ratio looks "normal" (say a third tha t of ahigh aspect rat io wing (20) , or half that of a med iumaspect ra tio wi ng ( LO), or two thirds tha t of a low aspectratio wi ng (5)) . If the tail gap is in the usual range of 1.5to 3 chords an d the tail is about level with the wing thenth e downwas h a llowa nce w ill d e p e nd o nly o n th ewi ng 's aspect ra tio. I therefore propose this purel yem pirica l equa tion

My CG Formula

This eq ua tion gives the CG posi tion , as a fraction ofthe wing me an chord , of gliders or conventional, engineat the fron t, tail at the back, monoplanes with a wi ng ofAspect Ratio fro m about 4 to 25. (See Chapter 22 fo ranyt hing 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 AspectRatio)

45

Locb Insb makes a lovely setting for Dr. feremy Sbaui's Stranraer. Tbe drag ofall those struts and wiresmust considerably reduce tbe airfloto velocity over tbe tail,

Basic Aerona lilies for Mode llers

Figure 8.10

1

.9 90

80

.8 70

60. 7

50

.625

40 20

15.5

30 10

8

.46

25

4

.353

20

.318 Wing

Aspect Ratio

.25 16

15

.214

Tail Volum e CG PositionRatio Vsba r as a % ofwing mean

cbordfrom leadiug edge

It is dea d simple to work out o n a calculator with squareroot s . Enter the wing Aspect Ratio (span/chord), take thesq uare root , and press sq ua re root aga in. Mult iply by0.25 an d th e V-bar, and add 0.1. Tha t gives the CGposition as a frac tion of the mean cho rd . Multip ly by 100for a pe rce ntage. For examp le , if the wings pa n is 99"and the mean cho rd is 11" the Aspect Rat io is 9. Thesqua re roo t of 9 is 3 and the squa re roo t o f 3 is 1.73. Ifthe tail volume is 0.6 this gives a CG position of

0.1 + (0.25 x 0.6 x 1. 73) = 0.36

46

The CG shou ld be at 36% of the mean chord, or 0.36 x11" = 3.96" aft of the LE of the MEAN cho rd, for the firstflight.

Even easier to use in the nomogram in Figure 8.10.You wi ll find nomogram s to help work o ut the wingAspect Ratio and Tail Volume Ratio in Appendix E.

Most powered trainers and sport models have a win gAR of between 5 and 8 so if those are the o nly mod elsyou wa nt to consider you co uld simplify the formula to

CG posn = 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 you then assume that your model has a V-bar of about0.6, whi ch is common, you ge t a CG position of one thirdcho rd . It is a standard , oft-use d, balance point, but app liesonly to models whi ch fit the standard mod el proportions.

Fifties Formula

Ano the r useful and popu lar CG formula, fo und inGo rdo n Whitehead's sca le book, was mad e up by an ae roeng ineer in the nin eteen fifties (w he n mo de l tails werebigger) .

CG distance aft of mean cho rd LE = chord/ri + 3x(Ta ilarea x tail arrnj/ S x wing area

This can be simplified by dividing all the terms by themea n chord to get the CG position as a fraction of thecho 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 theabove ex ample aircraft giv es a CG position of 39 .5%mean cho rd, 3.5% furthe r aft than my formula.

Many full size aircraft , lik e th e Hurrican e and theTurbulent, have a sma ll tail volume of 0.4 or less. Man yscale mod el plan s use a CG position wh ich co nfor ms tothe Fifties formula, but experien ce has sho wn that they flybetter w ith the more forward CG given by my formul a.

Now, perh ap s you can see that all the above formulae ,in fact any se rious CG formula, will try to do the samething . . . es tima te the NP position and place the CG asa fe d ista nce ahead o f it. (Tha t d istan ce is ca lle d theStability Margin , and is usually about 15% of mean chord.)A more comprehensive treatm ent of Centre of Gravity is .detailed in Chapter 22 which covers man y unconventionallayouts.

There are probab ly other formulae aro und whi ch youco uld use . To be realistic they sho uld allow for the tailarea and tail arm, and possibly even the wing aspe ct ratio .Any formu la which does not ment ion the tail, or which

Basic Aeronautics/or Mode llers

messes about with Centre of Pressure , sho uld be viewedwith suspicion.

Variations on the FormulaFlexibility

The wh ol e book until Cha pter 18 ass umes that thestructure of the ae roplane is rigid . Only a slight flexib ilityhas been allowe d for in the formulae . If your aeroplane isnot iceably flexible see Chapters 18 and 19 and move theCG forward a little . This is especially import ant on hig hAspec t Ratio glide rs.

Fuselage Influence

O n conve n tio nal aeroplanes th e extra lift on th efuse lage, caused by a pitch disturbance, will co incide w iththe wing's aerocentre. Tha t has been allowed for by usingthe gross wing area .

However if the fuse lage has an unusual proportion ofits area ahead of the w ing (as in Figure 8.11) then the NPwill be moved forw ard and a forw ard ad justment of theCG is advisable . This applies also to eng ine nacelles onmulti-en gined aeroplanes. See Cha pter 22 for wo rking outadjustments .

Personal Preferences

So me people have a preference for a part icu larlysensitive mod el, o r a part icularl y stable one . Fine! Adjustthe formula to suit yourself.

Willg Section Influence

There isn 't any! The sec tion thickness might have thetiniest effect, but not the camber. Cha nge the sec tion fromsymme trical to Clark Y and the formulae all give the sameanswer because the NP has no t mov ed . You ha ve tochange the rigging ang le but not the CG position . Let usbury that myth forev er.

47

Figure 8.11

-, , I

, , \

Net Tail Area

Gross lVillg Area

C

MeanCIJortt

QlI lI r ter ll1ell"Cbortt

c__~_---:_

C

Destablistng Nose A,"ea

Tail Setting AngleThe angle which 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 bymod ell er s as "lo ngitudinal d ihedral", an un fortu nat elyina ppropr iate nam e as it lead s modellers to conclude ,wro ngly, that it produces stability. That is like saying thatcarts are for pushing horses. As you have seen, the TailSetting Angle is not involved in determining the stability atall. However there is a link. In a later chapter you will seethat the Stability Margin partly determines the Tail SettingAngle needed for trim.


Chapter 9

Directional and LateralStability

F

ResultantForce

P ressureIncrease

~=-+-nm-+

--- -:. '

~.p.

1J 1I'If~F$r:~

W~ -q,£f.,.

"~

c -

~

..

?~~ . _.. ,~?).,. ~.... . .

.- -=~:J' . 7 ' iI.

~

Airflow

Figure 9.2

Figure 9.1

Airflow

.. ... . ..... .

Th e caveme n knewthe importance of directional s ta b ili ty (o ryaw stability, or weathercock stab ility) . A plainstra ight piece of woodin flight is unst abl esince the centre o f press u re is ah ead of th emidpoint wh ere the CGlies as in Figure 9.l.

In order to m a ketheir arrows fly stra ightth ey added w e ights a tthe front and feath ers atth e back . This mo vedthe CG forw ard and thece ntre of pressur e backand so the arr ow lineditsel f up with the airflowas in Figu re 9.2.

Our ae roplanes , withth eir CG well forwardand a ve rtica l su rfaceca lled a fin at the rear,use the same techniqueto e ns u re dire ct ion alstability. See Figure 9.3,which sho ws an ae roplan e wh ich wh ileflying along happ ily intrim has been disturbedby a s ma ll yaw o ffse tanticlockw ise. It is nolonger lined up with thea irflo w so th e fin w illhave an angle of attac kc a us ing a s ideway s"lift", F. Thi s fo rce willtend to re al ign th eaero pla ne with th eairflow. The co rrec tingmom ent dep ends up onth e fin a rea a nd itsdistan ce behind the CG.

A co rrectly trimmed

DirectionalStability


Figure 93

aeroplane' will always fly straight relative to the air. Evenif an aeroplane is flying North in a Westerly wind the airwill flow straight from nose to tail. Were it not so,DIRECTIONAL STABILITY would soon line up theaeroplane with the airflow, eliminating any sidewayscomponent.

Lateral Stability

An aeroplane which is statically stable in roll will,when disturbed slightly in roll, initially tend to roll backto a wings level condition, So we need to design insome mechanism to provide a rolling moment when theaeroplane is upset in roll,

Figure 9.4

L

w

50

Lift forces are caused by the air pressure on thewings' surfaces, Air pressure can only provide a forceperpendicular to a surface. Therefore there will be' twolift forces, one perpendicular to each wing. Or they maybe combined as a single resultant force up the centrelineof the aircraft, because I am assuming it is completelysymmetrical. Also, because of symmetry, the CG lies onthe centreline.

Figure 9.4 represents the rear view of an aeroplanewhich while flying happily along in trim has been upsetand now has a slight bank to the right , angle B. I cansee no force producing a moment which will tend tocorrect this bank. I happened to draw a high wingedaeroplane with the wing above the CG. It may seemobvious that this in itself will provide stability, but isthere a moment about the centre of gravity? The weightcannot have a moment about the CG (by definition) andsince the lift acts on the centreline of the aeroplane, itcannot provide a moment either. Even if you split the liftin two to have half perpendicular to each wing, bysymmetry their moments will cancel each other out.Even if you resolve each half into its vertical andhorizontal components, the net moment is zero so therecan be no restoring rolling effect.

Let us see what will happen though. First I shallexercise my right to resolve (split up) any force into twocomponents. In Figure 9.5 the weight has been split intotwo forces: one (W cos B) opposite and nearly equal tothe lift, because B is a small angle, and the other (W sinB) towards the low, right, wing. Now you don't needme to tell you that this component will cause asideslip .. . a sideways velocity to the right.

When the aeroplane is sideslipping, the air isapproaching at velocity V slightly from one side, at asmall angle Y say to the centreline of the aeroplane. Asshown in Figure 9.6 the velocity V may be split into twocomponents, V.cosY along the centreline of theaeroplane and V.sinY called the sideslip velocity, at rightangles to the centreline. It is convenient, and quitelegitimate, to examine the effects of each part of thevelocity in isolation. It is as if the aeroplane has twovelocities simultaneously which cause separate effectswhich can be looked at separately.

Now let us look at the design features which affectlateral stability.

1. Fin Sideforce

Back to our aero-plane which ' was sideslipping to its right. Ishall redraw it in Figure9.7 as a mid wingedaeroplane to avoid ambiguity, and I am sureyou can see that asideforce will arise dueto the sideways velocity,or sideslip. The sideforceon the fuselage isunlikely to have muchmoment about the CGbut the sideforce on thefin will. As the fin isnormally mainly abovethe fuselage it will


A low toing and sligbt dihedral gives little or 110 roU effect witb rudder.

Airfloto

/

V VelocityVector

VSINYSideways Component

L

Vcm'YForwardComponent

IIIIIIIIFigure 9.6

Figure 9.5usu all y g ive an a n ti clockwise , stable rollingmoment .

Yes, this sideforce onth e fin, F, is the sa mes idefo rce w hich p rovi ded th e di rectionalstability earlier (Figure9.3).

2A. WingPosition(relative to CG)

Fig u re 9.8 shows a"p a raso l wi ng" typeaeroplane, with its wingmounted away ab oveth e fu sel age , and th efour fo rces of Thrust ,Weight , Drag and Lift.Most of the weight is inthe fuselage . So becausethe wing co ntributes alarge prop ortion of thedrag, the total drag willact somewhat above theCG posit ion. (The totallift w ill ac t s lightlybe hi nd the CG tocounteract the pitch ingeffect o f th e Thrust /Drag couple.)

If this aerop la ne isg ive n a sligh t bank toth e right , none of th eforces ca n p rodu ce aro lli ng moment asbefore , but as beforethey will cause a sideslip tow ards the low wing. Nowlook at Figure 9.9. The drag acts, by definition , in thedire ction of the resultant airflow and may be split upinto two components, D.sinY and D.cosY (where Y isthe sideslip angle) , as shown. The component D.cosYcan exe rt no rolling moment. Transferring th ecompon ent D.sinY onto a view from the tail , Figure9.10 , shows that thi s component will have a rollingmoment abo ut the CG which will depend on the ver ticaldistance between the CG and the line of action of thedrag force . For this high winged aeroplane the rollin gmoment resulting from the Sideslip is stabilising, Le. itw ill pick up the low wing.

Th is kind of sta bility is so metimes referred to as"pendulum" stability for a reason which I fail to seesince aerodynamic drag has little effect on pendulums.

Not e tha t the working o f th is source of s ta b ilitydep ends upon the sideslip , as without the sideslip thedrag wo uld not have a sideways compo ne nt.

2B. Wing Position(relative to fuselage)

Now consider the flow of air round a cylinde r. As theair approaches the cylinder it w ill eithe r be forced toflow up over and down, or down under and up toresume its quiet steady flow (see Figure 9.11).


Figure 9.7

Resultant MOlne"tL

SideslipVelocity

w

Figure 9.8

L

'1==r=~"D

T

w

The extra upward flow (upwash) on the right givesan increase in angle of attack, and hence lift, and thedownward flow on the left gives a decrease in lift. Thenet result is a stabilising (anticlockwise) rolling momentwhich tends to lift the right wing.

On a low winged aeroplane you find that a Sideslipto the right causes a clockwise rolling moment which isdestabilising. That is why a high winged aeroplane withno dihedral will turn on rudder but a low winger withno dihedral or sweep is more likely to bank theopposite way. More of that later.

Now in Figure 9.12 the cylinder is the fuselage of anaeroplane with a wing stuck on top and it is sideslippingto its right. As the air approaches the fuselage some of itis deflected up the right side, over the top, and downthe left side .

Now we come to the famous Hedral sisters, Di, Annand Polly and their role in lateral stability.

Figure 9.13 shows an aeroplane with a slight bank tothe right which has started sideslipping to the right. Asyou can see , the air coming from the right will have anupward component on the right wing, tending to "getunder" the right wing and lift it (at the same timepushing down the left wing) which gives a stable,anticlockwise, rolling moment. (A rather unscientificexplanation but I hope you get my meaning.)

In much the same way, Di 's sister Anhedral (adownward tilt of the wing tips), would give a clockwise(destabilising) rolling moment tending to push the lowwing further down. If you want a more scientificexplanation. complete with trigonometry, see appendixD.

The effect of dihedral was first explained to me bythe "Pro jected Area Theory" many years ago (Anyoneremember the ]etex powered Keil Kraft Cub , my firstmodel aeroplane?). When the dihedralled aeroplane wasbanked, right say, the area of the right wing projectedonto a horizontal surface was greater than that of the leftwing. The theory was that the lift would therefore begreater 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

3. Dihedral

Airflow Sideslip VEL(VSIN Y)

A.gkY/ 1/ /Figure 9.9

52 Basic Aeronautics/orModellers

Sid es lip Vel

w

a ng le o f a ttack du e to s ides lip. d e rived from swee phack . is proporti on a l to the s ideslip ang le . the orig inala ng le o f a t tac k . a nd th e tan g e nt o f the a ng le o fswee p hac k (or yo u co u ld jus t be lie ve th e p re vio uspa rag rap h) .

I have a lso seen ca mbe r broug ht into the a rgume nt asfo llows . Th e ca mbe r on bo th wi ngs is the sa me. bu t as apercentage of chord in the d irection of the a irflow the

Figu re 9.10a ngles to t he w ing 'ss u rface a nd so ha s a nequal mom ent about thecc.

The refo re the re ca nbe n o mom en t unt ilit s ta rts to s id es lip( w h ic h th e p roje ct eda re a th eo ry o mi tted toment ion ).

Sw e e pi ng b ac k th ewi ngs o f a n ae ro p lan eco ntributes to its late ra ls ta b ility , Th a t is . if asweptba ck wing ex pe riences a ro ll upset to theright (clockw ise) . it wills ta rt to Sides lip to itsright. As a result o f thes id es lip it wil l ex pe rt-e nce a ro lling mo me ntto the le ft (a nticlo ckwise) which will tend to cor rect theinitial ro ll distu rbance .

O ne ex planatio n o f why it works is show n in f igure9. 14. in which an ae roplane wi th a swepthack wing iss ides li p p ing to its rig ht. The result in g ai rflo w is assho wn. As far as the a pproac hing a ir is co ncerned thetw o w ings h ave th e sa me a rea but th e r igh t w in gappears to be of a greate r as pect ratio th an the left wing .Th e r igh t wi n g w illthe refo re have more liftthan the left ""ing w hich Figu r e 9.12gives stab ilising ro llingmom ent to lift the rightwing.

11 is a s imple matt er10 p ro\'e b y T r igo n ome try that the sta bilityef fec t o f swcep back isre a lly du e to the righ tw ing h a vi n g an in creased ang le of a tta ckcaus ing the increase inlift . w ith a co rres pond ing d e cre a se o n th ele ft wing ,

I show in Ap pe nd ixD th at th e c ha nge in

4. Sweepback

Figure 9.11

::

_______ - - - -__"""1111---

-----0-------~--------~------"""IIIf---


Figure 9.14

s ides l ip. Th e pri maryeffec t o f th e rolld is tu rb ance is s ides lipto war ds the lo w w ing .Fo llowing th a t ma ycome th e ro llingmomen t du e to sideslipderived from the abovefactors .

Aspects ofDesign

I

Ai1j10W

V SIND

These design character ist ics ma y be combi ne d to o b tai n th erequired degr ee o f lat-e ra l st a b ility . Yo u may

see aerop lanes wi th a high stra ight wi ng and a littl edihedral (Cessna 150), or a low straight win g and plentyof d ihed ral (Piper Cherokee), or a swept high wing withanhe dra l (B.A. 146), or a swept low wing with a littledihed ral (Boe ings) .

The wing position is normally chosen for mechani calre a so ns , th e s wee p b ac k to s u it th e c ru is ing Ma chnu mber, and fina lly the dihed ral is chosen to ach ieve thereq uired late ral stability.

The same ap p lies to mod e ls except that sweep isused for appe arance mainly. On pattern models we try

to ac hieve ne utral late rals tab il ity by using jus te noug h d ih edra l toca nce l o u t th ed esta bili s in g effec t o fthe lo w w ing . I ha vesawn th rou gh the glassfibre bandage on the topsurface o f a wi ng andrej o ine d wit h s ligh t lymo re di h ed ral. T h usmod ified the model didnot roll a t a ll w h e nyawed with rudder.

Yo u w o u ld ge t as im ila r e ffe c t usi ng ah igh w ing and a littl eanhedral.

If yo u lo o k a t theform u la e d e rive d inappendix D, it is ev identthat, becau se the sweepb a c k e ffec t d epe n dsup on angle of attack butthe dih edra l effect doesnot , you can no t directlyeq ua te o ne d egree o fd ihedra l w ith so ma nydegrees of sweep . Therelat ive effects vary w iththe speed of the aircraft.

T he effec t of thed ihedral becomes morep rom in e n t as s peedrises.

Also, not e th at th esweepback effect gives a

-......V c:mD

XL

tD

M~XL

Figure 9.13

To reca p the n, the lateral stability is influe nced , ina pproxi ma te o rder of importa nce, by the foll o wi ng:d ihedral, sweepback , wing posit ion and fin area abovethe CG. ALL of them are co mplete ly dependant o n the

ca mbe r is greater on the right win g than o n the left andso will give a grea te r lift. Thus providing a stab ilisingrollin g mom ent. Sounds good , bu t that doesn 't explainwhy it still wor ks on win gs with a symme trical se ction!

Summary


stable ro lling mom ent whe n invert ed (w hile di hedraldoes not). Figure 9.15

Rotation

1

J

ForwardAirspeed

Slower Wi"g

o

\ I,..------J,\ 1/1------....

Faster Wi"g

(

Directional and Lateral InteractionSpirall)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 onthe left is advan cin g into the airflo w and is therefo refaster than the one on the right so a ro ll to the right mayde velop (the same wa y as the origina l yaw) , So wh at?

You design a high wing train er with a grea t big finfor directional stability. The stude nt lets the right win gdrop a nd it s ta rts to yaw to th e right du e to th ed irectional stability from the fin. Th e rate o f yaw to theright means that the left w ing is travelling faster than theright wing as in Figure 9.15 so th e aerop lane banksfurth er to the right wh ich increases the sideslip, so thefin produ ces more rate of yaw which causes more rightbank. By th is time the bank ha s d eveloped such thatright yaw is dropping the nose as well.

You know wha t this is becoming .. . a sp iral d ive!When it is ban ked right over and pointing down, all upelevator will do is tighten the spiral.

Sp iral di vergence is an ins ta b ili ty ca used by theimbal an ce b etwe en we ak lat eral s ta b ility a nd a nexcessive amount of direction al stability. The tenden cycan be correc ted by reducing the fin size or increasingdihedral.

IJutch Roll

An imbalance th e o p pos ite w ay betw een w eakdirectional and strong lateral stability, too mu ch dihedral,manifests itself as "Dutch Roll" whi ch is a rollin g/ yawingosc illation which is ve ry difficult to sto p.

On radio co ntrolled mod els it is often preferable toput up with a slight tenden cy towards sp iral d ivergen ceby go ing for the bigger fin and less d ihed ral.

For free flight an y divergen ce is una cceptabl e so finsare smaller, d ihedral grea te r, and a slight Dutch Rollin gtenden cy is accepted if it is well damped out.


Chapter 10

Control10.1. No te the di ffe re nce betwee n yaw an d sid es lip .Yawi ng the nose to the right crea tes a sideslip to the left.

T he ae ro p la ne isro lled a bou t its long itudinal axis by movin go ne a ile ro n down andthe o ther o ne up as inFigur e 10 .3. Moveme nto f the a ileron changesth e ai rflo w aro und thew ho le sectio n , not justaround the aileron itself.Unless the win g starts offnear its stalling angle ofa ttack, th e down-goingail ero n p ro d uc es a nincre a se in lift , a ninc rease in drag and aninc rease in the nosedow n pitch ing mo men tcoefficie n t CMo (o r anaft movem ent of the CPif you prefer). The effectsa re re ve rsed o n th eupgoing aileron.

Ailerons

The purpose and effect of moving the ele vators is torotate the aeroplane about its lateral or pitch axis. Movingthe elevato rs up creates a downforce o n the tail wh ichtends to rotat e the ae rop lane nose up. See Figure 10.2.Because of its ine rtia, its ce ntre of gravity initially tends tokeep go ing in the sa me direction and so the net result is

an increase in the angleof a ttac k . \\;rhat th ee lev a tors have co n tro love r then is the angle ofattack. Up elevator willincrease the wing's angleof a ttack an d the refo rein crease the wing lift ,unless the angle of attackis in cre a sed from justb el ow to a bove th es ta lling angl e . In th atcase u p e lev ato r is justa no the r down co n tro l.An All Mo ving Ta il(AMT) works in exactlythe same way.

}'a 1/ '

Elevators

+ + + + + +

Ytt u :

Fi.~ I/ I'I' 10. I

The rud de r ro ta tes the aerop lane abo ut its verti calax is , or yaw ax is, and the farthe r the rudder is beh indthe CG th e mo re leverage it wi ll have . Wh en rig htrudder is ap plied , a force to the left is generated overth e fin a nd rudder w hi ch will ya w th e a e ro p la neclockwise , i.e . nose to the right. Or to look at it from apilot 's point of view, wh en he has righ t rudder ap pliedthe airflow is coming from ahe ad and slightly from hisleft so the aeroplane is sides lipp ing to its left. See Figure

Rudder


)PilclJ

XL Roll

Prise Aileron

Figure 10.4

c

Transmitter

Roll X L

Figure 10.]

Figure 10.2

A l rflo u i..Pilc!J

"(~IOliO" ~-_..~ ~ .. c.~~~~--..~

Aileron DragTh e unwanted d rag

cha nge can be troubl e-some . Say yo u appl yrig h t a ileron , th e dragincreases on the left anddecreases o n th e rightgiving a yaw to the left,not the way you wa nt togo. Th is is ca lle d adverse ailero n drag (andis d iffe ren t from the"a ilero n reve rs a l" inChapter 18).

The co mbi na tio n ofthe late ral stability andthe left yaw/ right sides lip w ill g ive a ro llingmoment to the left (seeCha p te r 9) . O n aeropl an es w ith s tro nglateral stability and largeailero n movem ents th islatter moment can overcome the ro lli ng moment to the right fromthe a ile ro ns a nd th eaero pla ne could evenroll the wro ng way .

O ne so lu tion is toarra nge fo r d ifferentialmove me n t o n th eai lerons , the up o necomes up more than thedo wn o ne goes do wn ,to eq ualise the d rag.

Ano the r so lution is to fit "Frise ailero ns" in wh ich theupgoing ailero n has its lead ing edge stick ing down intothe airstream (Figure 10.4), to equa lise the drag.

Anoth er is to ope n the airbrake o n the side with theupgoing ailero n. The loss of lift and increase in drag arejust 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 together with thea ileron, and in the same d irectio n , to o p pose theadverse yaw from the aileron drag. To make this eas ierman y mode rn rad ios have a co upling sw itch o n thetran sm itt e r (c alle d a "C.A.R. fu nctio n" for CoupledAile ro n & Rud de r) wh ich a llows th e a ileron stick tomove the aileron s and rudder simultaneously.

Aileron Alternatiue

There a re alt e rnati ves to ailerons fo r ro ll co n tro l.Either the whole wing can be rotated about its qu arterchord line , or the wing ca n be tw isted at the tips toprodu ce a rolling momen t (w ing warping) . The theo ry isgood bu t the re are mech anical co mp lexi ties . Ano theral te rn a tive it to us e tail e rons (s ee unde r Co ntro lCombinations).

Control Surface Balances

The re a re tw o kind s of bal an ce fitt ed to co n tro lsu rfaces , a nd th e y p erform different functi ons .Aerody namic ba lances are a part o f the ar ea of theco ntro l surface ahead of the hinge line. The air p ressure

Figure 1005

Aerodynamic Balance

Basic Aeronauticsfor Modellers 5 7

Figure 10.6a contro l surface already has an ae rodynami c bal ance , itis a convenient place to put the mass balance also.

Rotation

D o umuiardVelocity

Wander along to a gymnasium and pick up a weighttraining bar, complete wi th weights. Now turn it smart lythrough 180 degrees and stop it. It takes quite a bit ofeffort to start it turning and to stop it. Now try the samething with a shot putter's shot. Th at is much easierbecause it has less rotational inertia since the weight isconcentrated near the centre . An aeroplane with longhe avy wings and a long heavy fuselage is more d ifficultto rotate, and stop rotating, which makes it le s smanoeuvrable, but it flies more smoothly. That is onereason why the Sopwith Camel, for example , whosemain masses of engine pilot and guns we re all placedtogether, was so agile .

The control forces , like all aerodynamic forces ,increase with the square of the airspe ed . At low airspeedthe controls are less effective. That applies particularly tothe ailerons o n a propeller d riven aero p lane w hose

rudder and ele vato rs areusually in the slipstream.

Co n ve rse ly , at highspeed th e controls canb e dangerously effective.

Ailerons will producea very rapid roll as willa rudder/dihedra l combination . So the higherthe speed , the more "g"will res ult from a givenelevator deflection . Sobeware of applying fullup elevator in a highspeed dive . That is area l wing folder.

The rate at which theaeroplane rotates inresponse to a contro l

input obviously depends upon the relative size of theco n tro l surfaces . It depends also upon three designfeatures of the aeroplane itself, inertia , stability anddamping. The control input has to overcome all three.

Next there is stability which affects on ly pit ch andyaw (sin ce a roll offset in itself produces no opposingmoment) . For example when the elevators change theangle of attack of the wing, th e longitudina l s ta ticstability of the aeroplane sees it as just another pit chdisturban ce and produces a moment tending to changeit back. The control is fight ing against the stability. Thegreater the stab ility , the les s the effectiveness of thecontrol. In a sense Stability is the opposite of Control.Too much stability can leave you with too little control.

Control Eff ecu veness

Stability

Rota tional Inertia

Aerodynam ic Damping

,X

I~ - --

- - --

~CG

Rotation

CG•,III

II

X,I

SidewaysI,

Velocity -----r---

[ \l -~F]

Figure 10.8

Damping Mo ment

R otation

Mass Balance

R otation~ Damping

-, ~ " Mome1lt... ~

Figure 10. 7

on this part counteracts to some extent the pressure onthe rest of the surface and so reduces the effort neededfrom the pilot (or the servo) to mo ve the surface. Anexample of an ae ro dynamic ba lance on a rudder isshown in Figure 10.5. This technique can be applied toall three types of control surface .

A mass bal ance on the other hand is simply a weightrigidly attached to the co ntro l sur face in front of thehinge line . See Figure 10 .6 . Its purpose is to avoidcontrol surface flutter (of which more in Chapter 18) . If

----- -The th ird de sign fea tur e which affe cts the control

58 Basie Aeronautics for Modellers

Compact and ag ile ( 110, 1I0t Dounie, his Pills Special) due to tbe minimal inertia and damping oftbe sbortuiings andfuselage.

lift, XL-r which has an ant icloc kw ise mom ent about theCG, the dam pin g mom ent. Since the upwash velocity isprop orti on al to x, so is XLr and so the damping mom entis p roportio nal to x2 and tail a rea , Th us while sta ticstability depends o n the tail area times moment arm, thedamping (an d hence dy namic stability) depends on the

AEIZODY,.w..«c..OAMPIIIC;,\.IOorul1E14-3

effec tiveness is aerody namic damping. A damping forceis a force wh ich is the res ult of a rate o f movem ent andwhich o p poses th at rat e o f mo vement . Drag is anexa mple. However here I want to look at forces whi charise from rot ati on o f th e aeroplan e about th e threeaxes. The greater the damping mom ent , the slowe r there sponse o f th eaeroplan e to a controlinput , again making itfly more smoothly butless manoeu vrable .

Damping is al so animportant ingred ient inDYNAt'vIIC STABILITY.

Figure 10 .7 repre se n ts a n aero p la new hic h has a clockwise(nose up) pitchi ng rateabo ut its CG. As a resu ltth e tai lp la ne h as adownward velocity p roportional to x, its d istan ce back from the CG.That has the sa me effectas a n u p w as h o n th eta ilplan e w ould ha ve. .. an increa se in tai l

Pitch Da mping


)

Plain

- _......-

--

XL

clockwise , Le . rig htwin g down.

Without labouring thepoint , yo u can see that ,as for p it ch damping ,a long tail a rm impro ves ya w d ampinga nd hi gh as pec t ra t iowings wil l hav e betterroll damping.

Thus th e bigg er th eaeroplane the grea ter itsro ll d amping , a nd thegrea te r its Asp ect Ratioth e g rea ter th e ro lldamping . Th at is wh ylarge mod els , and largeglid ers in particu lar , flyso much more smoothly .As yo u ca n probablyguess from Figure 10.9,the tailplane and fin add

to the roll damping, but not much .

The Th rottle is a flying co ntrol: it is the UP co ntrol.Ask any glider pilot , wh o has to do with out one . Glide rscan o nly come down throu gh the air.

AIRBRAKES w ill reduce the speed of an aeroplanebut are no t really a speed co ntro l, they are a "co medown quicke r" co ntrol. They inc rease the profile d ragwhich stee pens the an gle , a nd increases the rat e o fdescent. They are mu ch more effec tive at high speed .

The y also, incid entally,in crease th e s ta lli ngspeed. They may have as ma ll pitching e ffec t ,depending o n th e d e s ig n . Airb ra kes co mmonl y cause turbulencein th ei r wake and a renorma lly p lac ed outboard of the tailplane ongliders.

FLAPS in cr ea se thedra g to steepen thedescent , in addi tion toin cr ea sing th e liftingcapa bility o f th e w ing,Clmax which reduces thes ta lli ng s peed , a n dallows the aeroplane tofly more slowly.

Th ree ty pes o f fla pa re s ho w n in Figu re10 .10 , namely a p la inflap , a sp lit flap , and aslotted flap .

The effect o f a ll ofthese on the lift curve ofthe ae rofo il is sho wn onFigure 10.11. The wholecu rve is moved to theleft du e to the increasedca m be r , a n d Cl ll",x is

Other Flying Controls

Rolling

\)::=========

--... - - .,.,.....

Damping Mome1lt

"

XL

,t

Fig ure 10.10

Figure 10.9

Figure 10.8 shows an aeroplan e with a rate of yawclockwise about its CG (nose to the right) an d Figure10.9 shows a view from the rear of an ae roplane rolling

tail area times distance squa red . So a small tail on a longfuselage will have better da mping than a large tail on asho rt fuselage even thou gh the y are de sign ed to havethe same tail volume ratio and static stability. There isso me p itch d amping from th e wing itself but it isrelatively unimportant unless it is swe pt.

Yaw and Roll Damping

60 Basic Aerona utics/or Modellers

Figure 10.11increased. The angle ofatt a c k a is measure dfrom the origina l cho rdline.

Lower ing flap s morethan a few degrees ruinsthe Lift/ Drag ratio whichreduces the climb angle,but a little flap may beu sed fo r tak e- o ff toredu ce the take-off run .

On land in g , a d efl e ct ion o f up to 40d e gr e e s ma y be used(eve n more for sp li tflaps) . O ne p roblem Ifound usin g large flapson a model is that at thelow speed achie vedth e a ilerons b ecam eineffective with outcoupled rud der.

Flaps usually a ffec tthe pitch trim o f th eaeroplane. Be cau se o fth e ext ra ca mbe r, theycause an inc rease in thenose down mo ment onthe w ing , but they alsoca use ex tra downwashover th e tail g iv ing anose up moment cha nge from the extra download onthe tailpl ane . The overall effec t dep ends on which of theabove is th e mo re powerful , which depends on th elayou t.

SLATS on the leading edge which you can select inand out are also a flying contro l. They are selected out,as in Figure 10.12, to let the aeroplane fly more slowly.When th e s la t is out , ai r ru shing through th e s lo tprevents the flow from se parat ing at the lead ing ed gewh ich prevents the wing from stalling until it reaches ahigh er angle of attack than usual , enabling it to achievea larger maximum lift coefficie n t th an before (Fig ure10.13). Like flaps they are bad for the Lift/Drag ratio andare best retracted for high speed flight although so meaeroplanes have fixed slats to save complexi ty and justput up with the inefficien cy.

There is also the velY clever "automatic slat " which isarrange d to o pen at high angles of attack when thestagna tion point (re member him?) moves down towardsthe entra nce of the slot. Their disadvantage is that the ycan ope n when the pilot pull s high "g".

Control Combinations

When the sa me bit of contro l surface is used for two

Figure 10.12

wu» Flap

Standard Aerofoil

0<

diffe rent purposes it is often give n a sp ecial nam e . Itusuall y al so involves using ex tra servo s and e itherme chan ical or electronic mixers.

TAILERONS is the nam e given to e leva to rs, or th ehalve s of an a ll moving tail , wh ich ca n mo veindepe nde ntly. The two sides can be moved togeth er asnormal for p itch co n tro l, or th ey can be mo ved inopposite directions to give a ro lling mom ent on the tail.

It w ill be a bit s hort on lev erage b u t it w ill beadeq ua te for lo w as pect ra tio aeroplanes (e .g . jetfighters).

FLAPERONS are strip a ilerons which ca n both bemoved down together to act as plain flap s, but whichcan at the same time still be moved inde pe ndently forroll co ntrol. Adverse yaw can become a probl em .

ELEVONS are just like flap erons but they are used ontailless aeropl anes to co ntro l roll and pitch . The elevonsmove tog ether for p itch co n tro l and in opposition forroll co ntro l.

Th e elevons both move up for nose up pit ch an ddown for nose down.

A V-TAILis jus t like a tailplane with extreme dihedral.It ac ts as a stabilise r in both pit ch and yaw, and the twocontrol surfaces ca n be moved inde pende ntly.

As shown in Figure 10.14, whe n moved togeth er they

~~C--__S_(ot ----


Figure 10.13 Figure 10.1 4 V-Tail

ac t like eleva tors , and when moved in opposition theyhave the effect of a rud der. The un wanted parts of thevectors ca ncel out.

Left Rudder

up Elevator

WitbSla t

Standard

- - -"- ,/

//

c,,"X tCJ.MAX

Like most Delta toinged models, tbis one uses eleuous for pttcb and roll, tbe canard is fixed.

62 Basic Aero na uticsfor Mode llers

Turning Flight

Chapter ,,

Turning flight is one of those areas wh ere a greatd e al o f mi sinformation and ha lf truths ha vebecom e accepted into the folklore of the hobby. I

have even seen an article in whi ch it was claimed that inturning flight the "curved airflow" reduced the risk ofstalling. Wha t utter no nsense! In this chapter I want to putthe record straig ht, starting with the basic mechanics ofturning. I shall assume still air condition s.

The Mechanics ofTurning In General

By turning I mean changing the direction of motion ofan object. For example, the ball in Figure 11.1 is going atco nstant speed round in a hori zont al circle radiu s R. On eminut e its velocity is V ft/ sec to the north and a short timelate r its velocity is V ft/sec to the eas t. Vectors VI and V2are not ident ical. In the time the ball has taken to movefrom A to B its velocity has cha nge d because its directionhas change d. The shorter the time interval, the more rapidthe change. The rate of change of velocity is defined asacce lera tio n . A turning body is not in eq uilibriu m,therefore it is acce lerating.

The rate of turn N is the rate at whi ch the radius OA isturning clockw ise . It may be given in revo luti ons perminute , o r d egrees per second o r, to s imp li fy th eequations , in radians per second (a rad ian is an angle ofabout 57.3 degrees).

Then the simple equation V = N.R. relates the velocityto the rate of turn and radiu s of turn . The velocity V isalways tangential to the circle, Le. at right angles to theradius from O.

The accelera tion of the ba ll in Figure 11.1 is given by

• a=V2/R or in terms of rate of turn , a=N2 R

In simple language the higher the speed, or the tighterthe turn , th e more accele ratio n . The direction of th eacceleration is towards the ce ntre of the circle at 0 , atright angles to the direction of motion.

From Newton's good old laws of motion an object willnot turn unless a force is applied in the direction of therequired acce leration, Le. towards the centre of the turn. Ifyou want to turn right , then you must app ly a force to theright. This force is called the centripe tal force becaus e it istowards the centre (from the Latin for "centre seeking").The more co mmonly known ce ntrifuga l force (from theLatin for "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 stringyo u supply the centripetal force to the weight via thestring to keep it moving in a circle. The weight applies ace ntrifug al force to your hand , pulling it towards theoutside of the circle .

Basic Aeronautics fo r Modellers

Back to Aeroplanes

The force required to turn the aerop lane must beproduced from the air. The fin and rudder do not provideit. Left rudder gives an aerody na mic force to the rightwhi c h is the oppos ite o f w hat we w ant. T he yawproduced by the left rudder will pro duce a side force onthe fuselage to the left.

This force co uld, given time , produce a left turn but atthe cos t of a considerable drag increase. The only surfacewhich can produce a large aerodynamic force, and do itefficie ntly, is the wing . It is the wing which turns theaeroplane.

The lift force always acts perpendicular to the wingand so a horizonta l co mpo ne nt ca n be pro vided bybank ing th e ae rop la ne towards the d ire ction of th edesired turn . Figure 11.2 shows an ae roplane banked tothe right at angle B. The horizontal co mpo ne nt of the liftwill be L.sinB caus ing the turn (ce nt ripe ta l force) . Theve rtica l co mpone nt, L.cosB, will now be less than theweight if the aeroplane started off in trim in level flight.The pilot has to apply enough up eleva tor to increase theangle of attack to increase the lift eno ugh so that L.cosB =\Vl. That will ensure tha t the ae roplane will perform a levelturn .

The lift is now greater than the we ight, and the ratio oflift to weight is calle d the lo ad factor n , wh ich o nlydepends o n th e ba nk angle (n = L!W = I!cosB). Thebanked wing turns the aeroplane while the elevator keepsthe nose up .

Figure 11.1B

v

A

63

Figure 11.2

w

Load Factor in a Turn

That equation tells us how many "g" to pull in the turnto keep the turn level, for a given bank angle. Using lessup elevator will decrease the turn rate and let the nosedrop. More up elevator will tighten the turn and make theaeroplane climb.

Figure 11.3 gives an idea of the relationship betweenthe bank angle (in degrees) and the up elevator needed tokeep the turn level. Not much for small bank angles, butquite a lot for steeply banked turns. For example for a 60degree 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 liftneed be increased by only six per-cent.

Refinement

There is more to turning an aeroplane than just havinga centripetal force turn its CG to move in a differentdirection. Its fuselage must be realigned with the newdirection of motion. If you think about it, rotating abanked aeroplane about a vertical axis is a combination ofnose up pitch, and yaw in the direction of the turn . Youhave already applied up elevator which takes care of thenose up pitch component, and a little rudder in thedirection of the turn will line up the fuselage centrelinewith the airflow. The bank turns the plane, the elevatorkeeps the nose up, and the rudder keeps the tail in line .

If you don't put on a little rudder in the direction of theturn , then the aeroplane 's weathercock stability willsupply the necessary yaw once the aeroplane starts tosideslip. It is normal practice to coordinate the turn onlight aircraft , but on most models (and jet transports) it is

Figu re 11.]

usual not to bother withthe rudder in a turn .

Sideslipping andSkid d ing Tur ns

Figure 11.4 is a viewof an aeroplane in a levelbanked turn to the left asseen from the centre ofthe turn. The aeroplanehas just the correctamount of rudder appliedto hold its fuselage inline with the airflow. Thisis called a balanced turn .

The aeroplane inFigure 11.5 has in suf

ficient left rudder applied and so is sideslipping to its left.A little more left rudder will bring the tail up into line asin 11.4.

The aeroplane in Figure 11.6 has too much left rudderapplied and so is skidding in the turn . Less left rudder willallow the tail to drop down into alignment with theairflow . In practice, the slip or skid is so slight it is notnoticeable and no t worth correcting, except perhaps forsome scale models, or high aspect ratio models.

Drag in A Turn

All parts of an aeroplane cause drag but the wing inparticular causes drag for two reasons. There is the profiledrag which depends on its sectio n , and induced dragwhich depends on the lift. In a turn the wing's angle ofattack is increased and so its profile drag is likely toincrease. More significant though is the increase ininduced drag which increases as the square of the load

Fig ur e 11.4

l~]-~------== - L-J-

IBala nced rum

1 5

J.()002 1.004

10

1.015

15

1.035

2 0

1.06

25

1.10

30

1.15

35

1.22

40

1.]1

45

1.4 1

»:

64

60

2

65

2.]7

70

2.92

75

3.86

80

5.76

85

11.5

Basie Aeronauticsfo r Modellers

factor. Which mea ns thatif you pu ll 2g youquad ruple the induceddrag, and if you pull 6gyou get th irty-six timesas much induced drag .So do not be surprisedwhen a sharp turn killssome speed, especia llyon low aspec t ratioaeroplanes. The controlsurfaces being offsetfrom their ne utra lpos ition is likely to causea little extra drag in aturn . The drag of thefuselage, and things liketh e undercarriage wi llca use significant extradrag if there is significantskidding or sideslipping.

Stalling Speed ina Turn

Figure 11.5

( -,

,<; I -------- ~ <,, ...

.....Sidesltpping rum

'- I

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 ofattack in wings level 19 fligh t at its maximum all upweigh t and the recommended CG position" . Thesignificant bit is the stalling ang le of attack because that isalways the same. When an aeroplane is put into a bankedturn the angle of attack has to be increased . Theaeroplane 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 ofattack, when you increase the ang le of attack (pull "g") tokeep the nose level, the stalling speed in a tu rn will rise asthe square root of the load factor n. In othe r words if youpull 9g the stalling speed treb les. Pull 4g and the stallingspeed doubles, and so on.

Remember that the drag increases when you pu ll g in aturn so un less yo u add extra power the actual flyingspeed will decrease. Beware of stalling . If the aeroplane isdocile it will merely refuse to pu ll the turn as tightly as

( \

-I \ \

'..- \.....£ ~

~cr----r ~ \

""""'- L U

Sk iddillg Turr

L ..J

Figure 11.6


you want , but so me models co uld tipstall and flick out ofthe turn into a spin!

High Aspect Ratio

When an aeroplane is turning left say , the win g on theright , on the outside of the turn , is go ing faste r than theleft win g. This has two seconda ry effects wh ich are notusually not iceable unless the wing has a high aspect ratio .

(I) The righ t wing will develop more lift than the leftone . The aeroplane will tend to increase its bank angle allby itself, even with the ailero ns held at neu tral. It mayeve n be necessar y to hold a little op posite (Le . right)aileron to maintain the desired left bank angle.

(2) The right wing will cause more drag than the leftwing, giving a yaw to the right. A sideslipping turn like inFig u re 11.5 wi ll re sul t , w hic h w o uld be fu rth e rco mplica ted by strong lateral stab ility. The so lution is toapply left rudder. So to ge t a good accurate left turn youmay need a trifle of right aileron to hold the bank angle,so me left rudder to yaw the tail into line, and some upeleva tor of co urse . Coupled Aileron and Rudder would bea disadvantage here and differential or Frise ailerons won 'thelp during the turn (b ut they would still be effec tive asaileron is being applied to roll into the turn ).

Turning Using Rudder

I do not mean flat turns like those wh ich boats pe rformand wh ich are possi ble on certain aeroplanes . If rudder isap plied to produce conside rable yaw then the sideforceon the fuselage may provide enough centripetal force fora wide gradual turn with the wings level but it looks a bitodd , crea tes a lot of drag, and is very inefficien t.

What 1 mean is the co mmo n pract ice on mod els ofusing rudder to create yaw an d then late ral stab ility toproduce the banked turn. The wing turns the aeroplane ,not the rudder , becau se the wing provides the centripe talforce to create the turn . Up elevator has to be applied tomaintain level flight in the turn .

The turn so produced will be a skiddi ng turn as inFigure 11.6. There will be no adverse yaw pro blem, eve nwith high as pect ratios , and once the turn has starte dso me of the rudder can be taken off. Tak ing off all of therudde r will allow a sideslip to devel op. Lateral stab ilitywill usually then roll the aeroplane out of the turn (a fewae roplanes would enter a sp iral dive and so op positerudder must be applied to leve l the wings) .

It is a perfectly satisfactory way to turn a simple mod el(never used on full size that I know of) but the entry andex it ca n look rather un tidy. The reaction time is slow aswe are using a secondary effect. Rudder -> yaw + lateralstab -> roll.

Summary

To turn an aeroplane a force must be provided tow ardsthe ce ntre of the turn . The tighter the turn required, thebigger the fo rce has to be . Th e force is provided bybanking the aeroplane so that the wing lift has a sidewayscomponent. The total lift has to be increased so that theve rtical co mpone nt ca n still support the weight , whichmeans up el ev ator is ne cessary . O n some types o faeroplane a little rudder in the direction of the turn mayneed to be a pp lie d to co u nte ract adverse yaw fromaileron drag, o r asymme tric drag o n high as pect rat io

66

ae roplanes . On ce the turn has started the ailero n oftenneeds to be returned to near neutral to stop the bankangle increasing.

To achi eve a ve ry ra pid turn , a "pylo n race turn",dem an ds a very large ce ntripetal force . The aeroplanemust be flying at a speed well above its stalling sp eed.Point the wing's lift in the direction of the desired turn (80to 85 d egr ees o f bank) , and increase it as mu ch aspossible using up elevator , but bew are! Pull too hard andthe wing will stall in spite of the high speed and it mayflick out of the turn. And make sure it is strong enough totake the loads.

Beginners! Avoid stee p ly banked turns , they lead totrouble. The nose drops if you don't apply a lot of up ,yo u risk stalling if you do . Start off us ing sma ll bankangles of 20 to 30 degrees which means that very little upeleva tor is necessary. Then any failure to apply the smallamount of up results in a very gradual height loss, noth ingdramati c.

I have noticed when instru cting beginners that whenthey roll o ut of a turn their mod el often zooms upwardsand so far I have spotted two causes. If they let the nosedrop in a turn and let the aeropla ne pick up speed (sa ferthan lett ing the speed drop off) the excess speed willcause a zoom up ward s in level flight. Or some times theydo a goo d level turn with correc t up elevator applied andthen on rolling level forget to release the up elevato r.Again a zoom results.

Special Effects

There are so me very special pilots wh o can fly theirspeci al aeroplanes (mode l or full size) in a manner whichseems to defy all the norm al rules of flying and ph ysics.O ne ca n but marvel a t the lik es o f Hanno Prettnerperforming his rolling circles with suc h precision . I knowthat there is a ce ntripe tal force forming the circle and aforce supportin g the weight , but how does he keep theresultants so con stant wh en the surfaces producing themare co nstantly changing? Even those pilots canno t defy thelaws of physics however. The centripe tal force causing theturn , and the force supporting the weig ht must be there ifyou 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 ploughpe rhaps? To a Vet a cow in harness is still a cow, and tome a rud der is always a rudder and always causes yaww hile an e leva tor is always an e leva to r and a lwaysco ntro ls angle o f att ack (p itch). Wh en I hear a cha pren amin g the co ntrols during a manoeu vre like a turn orknife edge flight it suggests he has a deep miscon cepti onof 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 deviceswa pping ove r the pushrods wh en his aeroplane banks.

For example, in a stee p turn old Bloggs says that hise levator has become the rudder , but it seems still to bedoing its o ld jo b of cha ngi ng the ang le o f a ttack toincrease the wing lift which ca uses the tight turn . Therudder never did that in the first place. If you tell him toput on more up eleva tor which stick will he move? Whichco ntrol will move, the old elevator or the new on e?

Basic Aeronautics/ or Modellers

Chapte r

ADelicate Balance12

T his chapter is not about stability. That has alreadybeen dealt with. We have given the aeropl ane apositive stability margin , Le. made Kn pos itive, by

putting the CG ahead of the NP position, and that is anend of it. The next task is to trim it out to fly "hands oft" ineq uilibrium, wh ich is acco mplished by adjusting the forceon the tailplane.

As a schoolboy I had a flight in a Bolk ow 2 seater ,wh ich had a trim lever marked "nose heavy" and "tailheavy". I imagined that it moved a big we ight back andforward in the fuse lage . Wrong! The term "nose heavy" inthis context is nothing to do with CG position . Saying theaeroplane is "nose heavy" just means that it is out of trimsuch that left to itself it would pitch nose down. Movingthe trim lever towards "tail heavy" (o r nose up) adjustedthe trim tab to correct the tendency.

Tail Lift Requiredfor Trim

The aeroplane in Figure 12.1, which may be a glider orpowered, has been trimmed out to fly in a dive of angle A,which can be any angle. From stability calculations the CGhas turn ed out distance x behind the wing's aerodynamiccen tre and I hav e show n the weig h t s p lit int o twocompo nents, W.sin A along the flight path , and W.cos Ape rpe ndicular to it. For the aeroplane to be in equilibrium,th e fo rces mu st ca ncel each o ther o u t and th e tot almom ent on the ae ro plane mu st be zero . The total liftbalances W.cos A, and the Drag is balanced by W.sin Aplus the Thrust T (if any).

Figure 12.1

D

III

III

..1w COS A

W \..W sINA

Basic Aeronautics f or Modellers

Note that, in the interests of clarity, I have gone aga instconvention by showi ng the no lift pitching moment Mo asa nose down arrow , because that is the way it really acts.Othe r books co nventionally show 1'1'10 as a nose up arro wbut wi th a negati ve va lue . In my formulae mak e Mopositive for norm al wings.

It is ass umed that thru st and dra g have no mom entabout the CG. Therefore, taking mom ents about the CG,you can see that the direction of the tail force must bepartly down to balan ce Mo and partly up to ba lance thewing lift. Tail lift may therefore be up or down but I sha lldefine posi tive lift as up wards. Skipping ove r the algebra ,and ass uming that the tail is small compared to the wing ,this Simple equa tion for tail lift coe fficient can be found.

CI.T;(X,CL - C,' IO)N"'Ir ' . . equation 12.1where Cl. is the lift coefficient of the "total lift".

Note that there is no ment ion of angle A so the equat ion istrue when A is zero (in level flight) or ninety degrees (avertical dive at terminal ve locity) or anywhere in between .Nor does stability matter to the tail lift. The actua l lift forceon the tail will of course depend on airspeed squared.

The re la tio nship is best ex plaine d o n the grap h inFigure 12.2. The tail lift will often be up wards at lowspeeds nea r the stall. At so me intermediate speed it will bezero , while at high speed the tail lift must be down ward.In a vertical dive the total lift coefficient Cl. is zero and thetail lift is well and truly down, how much dep ends on thewing camber. To go from low speed trim to high speed

67

the CG is further aft thaninB.

In the special casewhere the wing has asymmetrical section thegraph will look likeFigure 12.3. No tail loadis needed in a verticaldive but otherwise tail liftcoefficient is always upand increases as speed isreduced.

As you know, to trimfor a faster flying speedyou need to apply downtrim . Figure 12.4 showshow the elevator angle totrim varies with the liftcoefficient. I have shownthree lines for three CG

Paradox

Elevator Angle toTrim

Stalled

("("("("

~Stalled

("(",,

,V-IV-,V,(--'IV

In the previousparagraphs I exposed aparadox. I stated that totrim for a higher speedthe change in the tail liftcoefficient is downwards,but you know that itrequires down elevatortrim. You may betempted to suppose thatdown trim would cause atail lift increase, but this isno place for intuition.You must stick tobelieving the equation

which must be true for equilibrium and try to find areason. Suppose an aeroplane has been speeded up suchthat its angle of attack is decreased from 10 degrees to 2

degrees. Because of thedownwash the tail's angleof attack changes from 5degrees to 1 degree, areduction of 4 degreeswhich gives the reductionin tail lift to satisfy theequation . It is thenretrimmed with downtrim. So as long as theamount of the downtrimis less than 4 degrees (sayit is 2 degrees) then boththe conditions which youknow must he true canbe satisfied. Down trimhas been applied and yetthe tail 's angle of attack(and hence its liftcoefficient) has beenreduced.

CG Well ,Behind AC v"

~(--'

Low Speed

Figure 12.2

Figure 12.3

trim you need to move down the line to reduce the tail liftcoefficient. The slope of the line is proportional to thedistance of the CG behind the wing's aerocentre. In line A


pOSItiOnS, i.e . th ree d ifferen t va lue s of K, on the sa meaeroplane. The steepe r the lines, the more the stability.Line A is for an aeroplan e set up with an aft CG to mak e itagile . Line I3 is for an aero plane w ith an average CGposition, and line C is fo r an aeroplane se t up with a veryfar forward CG so that it will not stall. (The re is just notenough up elevator). The trim change between speed s 1and 2 ca n clearl y be se en to depend o n th e stab ilitymargin . The less th e stabi lity margin the less th e trimchange need ed. If there were no stability, then the graphwould be horizont al and no trim change woul d be need edto change the aeroplane 's speed .

Tail Setting Ang le

In Cha pter 3 I menti on ed several lines on an aerofoilfrom which angle of attack can be measured . In this andsubsequent chapters the angle of attack will be measuredfrom the zero lift line of the wh ole win g. One reason forthis is that at ze ro lift there is no downwash and as theangle of attack is increased , the downwash is a constantp roporti on of it. Double the an gle of attac k (measuredfrom the zero lift line of the w ing) and you double thedownwash . Tail Setting Ang le is the nam e for the ang lebetween the zero lift line of the whole w ing and that ofthe tailplane with the eleva tor neutral. It therefore applies

Figure 12.4

to only one spe ed , or rather one lift coefficient. It does notconfer longitu dinal stab ility as its modell e rs' nickname"longitud inal dihedral" unfo rtun ately implies .

By ta king e q ua tion 12.1 above and doing a littl eal ge bra ic magi c o n it a n eq ua tio n ca n be fo und fo rworking o ut the Tail Setting Angle of an aeroplane at ace rt ain CL (or speed). Equation 12.2

Figure 12.5 illustrates the variatio n of Tail Setting Ang lew ith w ing camber, stability ma rgin , and als o w ith thechosen speed at which th e elevator is requi red to beneutral. \'('hat the eq ua tion and the grap h bo th sho w isthat the Tail Setting Angle is in two parts, both negative(tail lead ing edge down). One part depends on the wing'scambe r, the more the cambe r the more the Ta il SettingAngle . Th e ot he r part depends on the stability and these lec te d lift coefficie nt. Th e more the stability and theslowe r the trimm ed speed, the more the Tail Setting Angle .The bottom line on the equa tion is the tail's effectiveness,its tail volume rat io Vhar times aT th e slope o f its CL (J.

cu rve.The more the tail's effect iveness the less the angle has

Ele vator a ng leto trim

Fil II Doum Elev

forwa rd

cg

FilII Up Elev

Speed 1

1

1

I

11

I

-II

11

I

1

1

I

I

1- 1-

1

1

Speed 2

II

1

1

1

1 Stalled


Figure 12.5

Cbosen SpeedforNeutral Ele vator

1I

Higb Speed Low Speed

Stall

II

1

2

3

4

5

6

7

8

tPart dueIto Camber

----,-----

Tail Setting Angle illdegress LE DowII

II

111

-IIII

I1

to be to get th e desired effect. Even aeroplanes withsymmetrica l wings need some Tail Setting Angle, be it everso small, unless they also have zero stability. That's no t anaeroplane , it's a guided missile!

All Moving Tail

In the above d iscussio n I referred to the conventiona lprac tice of using a fixed tailplane with an eleva tor hingedo n the ba ck . The functio ns a re combined on a n AllMoving Tail (AMT). The Tail Setting Angle diagram 12.5above becomes also the "tailp lane angle to trim" d iagramfor the AMT whe re the tailp lane's angle is measured downfrom the Wing 's zero lift angle . If some middl e position isco nsi dered the ne utra l position, then all the followingdiagrams are still va lid , replacing "e levato r ang le " by"tailplane displace ment from neutral".

The Effect ofThrust on Trim

Until now I have assu med tha t the th rust line was in

70

the d irection of motion and passed through the CG, butthat is not always the case. The eng ine may be in the nosehut angled down at an angle ca lled the downthrust angle(the angle between the thrustline and the fuselage da tumline) such that the thrustline passes above the CG. Or theengine may be mou nted high above the fuse lage o n apy lon w hich wi ll have the same effect b ut to a moreextreme ex tent. See Figure 12.6.

Line A of Figure 12.7 shows the normally sha ped trimline of an aeroplane like that in Figure 12.6, when it isgliding with power off.

Line B of Figure 12.7 is the tr im lin e of the sa meaeroplane in level powered flight. The more thrus t that isapplied the faster the aeroplane flies and the more downtr im effect its moment has (represen ted by th e gapbetween lines A and B). Therefore less down elevato r trimis need ed .

An aeroplane wi th excessive pitch stability will ne edlarge changes of eleva tor trim as power is changed, bu t itis so me times "fixed" by using excessive downthrust. Thesteep ness of the power off trim curve (l ine A) w ill be


Figure 12.6

L

T~----1"

w

flattened by the downthrust as you can see (line B) and ifthe engine cuts at high speed (po int 1 on line 8) , theaeroplane will reve rt to its power off trim line and glidewith the same trim setting but at a lower speed (po int 2 online A). Safe enough.

Line B is representat ive of a po werfu l eng ine with alittle downthrust or a small engine high on a power pod.

4

I

~I

V

~I

YYI

VI

V CL- - -

stall and persistently tries to do so until you wind in so medown trim.

Can't happen you think? Once after a repair I glued thefirewall back in at slightly the wrong angle , and I did notnotice that in addition the cast alloy mount had becom ebent in the crash. I did notice that after take off it need edmuch more up trim than I expected, but things do changewhen yo u re move and replace radi o ge ar. Wh en th eengine qu it on me it was trimmed to stall and I was in foro ne of my more exciting deadstick landings'! (I know,write out one hundred times "After rep airs I must chec krigging and downthrust angles").

B

C II

II

I

1- ----

3t _

Elevator a ngle to trim

upElev

Figure 12. 7

D ow nElev

If a powerful engin e has excessive downthrust , o rworse still is mounted high in a pow er pod, thrust willcause a large trim change .

The trim c urve forstraig ht and level flightwill look like line C onFigure 12.7. Line A is stillth e trim line fo r zerothrust , but putting onpower adds ane no rmo us nose downmom ent. The aeroplaneis in trim at full power atpoint 3 on th e gra p hwith a grea t deal of uptr im . If th e power isturned o ff it tries torev ert to trim line A(point 4 on the graph)but with that mu ch uptrim it is sta lle d whenth ere is no powe r toho ld th e nose down .That is a p ot en ti all ydangerous situa tion if itca tches yo u un awares.O n clo sing the thrott lethe mod el zoo ms up andtri es to s ta ll o r e venloop. It may flick roll orspin if it is so inclined. Atbest you ha ve a modelwh ich on its ve ry firstun e xpect ed deadsticklanding is trimmed to

A Dangerous Situation


Chapter

Glider Performance

13

F igure 13.1 shows a glider in a steady glide at angleA to the horizontal. The total Lift is at right anglesto the direction of motion and is the resultant of all

Table 13.1

A1lg1eA L/DRatio Load Factor

1 58.8 0.9992 28.6 0.9993 19.2 0.9994 14.3 0.9985 11.5 0.9966 9.5 0.9957 8.1 0.9938 7.1 0.9909 6.3 0.98810 5.7 0.98515 3.7 0.96625 2.1 0.90635 1.4 0.81945 1 0.70755 0.7 0.57465 0.5 0.42375 0.3 0.25985 0.1 0.08790 0 0

Figure 13.1

th e lift for ces on th e wi ng , ta il and fuselage at theircentres of p ressure . The tota l Drag is opposite to thedirec tion of the motion. By splitting the weight into twocomponents as shown it can be seen that.

• Lift = We ight x cos A• and Drag = Weight x sin A• or Lift/Drag Ratio (L D) = l/t an A

Th e Lift/ Drag ratio o f the ae ro p lane is im portantbecause it is directly co nnected to the glide angle. If youknow the glide ang le you ca n div ide 1 by the tangentratio (loo ked up in your tables) and Bingo! . .. You havethe Lift/Drag ratio . You will see (from the equations at theend of Chapter 2) tha t the Lift/ Drag ratio LID is just thesa me as th e rati o o f Lift coefficien t CL ove r Dragcoefficient Co because eve rything else cancels out. In adive Lift is alway s less than Weigh t. The "load factor ",LlW, dep en ds on the glide angle and is in fact equa l tocos A wh ich is always less than 1.

Tabl e 13.1 shows the var iation of bot h LID ratio andload factor in a dive of angle A (in degrees).

Speed Range

In a vertical dive, the angle A is 90 degrees an d theratio sin A is 1, so that Drag = Weight (and Lift is zero)and the aeroplane is at the maximum speed at which it isaerodynamically capa ble of flying, its "te rminal velocity".

Horizontat

72

\

lV ...~ lV cu s A

lV . IIIA

Basic Aerona utics/or Mod ellers

Its min imum limiting speed is its "sta lling speed", atwhich the overall lift coefficient is a maximum. That willocc ur whe n the an gle of attac k so me whe re alo ng thewing reaches its local stalling angle.

When the aeroplane is turning or "p ulling g" the wingwill stall at the sa me ang le o f a ttack , b ut the sta llingspeed in a man oeuvre is high er than the stalling speedin a stea dy glide .

Looking at Aerodynamic Data

For a particular ae roplane Cl. and CD will vary wi ththe angle of attack and therefore so will their ratio . Eachelevator trim position gives a part icular flying speed at apar ticular angle o f attac k and a particul ar glide ang le .Therefore a graph can be draw n showing how all theseth ings vary fo r each trim position from the stall to thevertical di ve . There are severa l ways to present th einformatio n . I have drawn up a se t of graphs which ,althoug h made up for an imaginary glide r, are consis tent

Figure 13.2

a

Figure 13.4

LID

Basic Aeronautics f or Modellers

within them selves. They are presented as Figures 13.2 to13.7.

The first two, 13.2 and 13.3 are the assumed lift anddrag coe fficients graphe d aga inst angle of attack (alphaa ) . Figure 13.4 is the most ob vious presen tat ion of glidingperformance , just a graph of LID ratio against angle ofattack. Whe n L is zero at a the LID must ob viously bezero . As a is increas ed the LID increases to a maximumand then red uces wh en drag starts to increase rapidly.

An alte rna tive present ation is to draw a graph of Cl.ag ai ns t CD which is ca lled a "po la r cu rve " or "po lard iag ra m ". An exa mple is g ive n in Figure 13.5. Themax imum value of LID is the slope of the line which justtouches the curve and the Cl. at whi ch it occur s can beread off. Becau se the best glide ang le is associated with aparti cula r lift coefficient like this you can see that onlyone angle of attack , and therefore one trim se tting, willgive the flattest glide . Anoth er alternative prese ntation isto draw a "Hodograph", Figure 13.6. Each po int on thehodograph (e .g. po int X) represe nts a velocity vec tor (e .g.V) for on e pa rticula r trim setting. The di stan ce of th epoint from the origin is the glider's airspeed , and theangle down from the hor izontal is its glide angle. Thu sthe airspeed and its horizont al and vertical components

Figure 13.3

a

Figure 13.5

II

I

73

Figure 13.6

Horiz Comp

---

Mi" Si"kSpeed

Speedfor Best

Glide Angle I

IIIIII

_________ J

Figure 13.6 Mag"ified

Optimising Performance

and the glide angle may all be seen together. You canalso see straight away what glide angle will give thegreatest horizontal velocity for penetration, what is theminimum glide angle and what is the minimum sinkingspeed. It is also evident from the magnified portion thatthe speed for minimum sink is less than the speed for theshallowest glide angle. It is important for a glider to havenot only a shallow glide but also a low minimum sink rateso that it can remain airborne on the slightest whiff of lift.

There are basically two aims when flying a glider, eitherto go as far as possible from a given height, or to stayairborne for as much time as possible from a given height,both of which require a fairly flat glide. The best of the fullsize soarers can achieve a glide angle of about one degree,but the best models would do well to achieve a twodegree glide angle.

I shall now assume that in flight the Lift is always equalto the Weight. Look at table 13.1 again and you will seethat even at a glide angle of 8 degrees there is less than 1%

error in this assumption. If the lift is constant the LID ratiois greatest when the total Drag is a minimum. As youknow, the total drag is made up of the induced dragmainly from the wing, plus the profile drag of the wing,tail, fuselage and struts etc combined. Figure 13.7 showshow the total drag, and its two components, vary withairspeed. Profile Drag is proportional to speed squared,whereas Induced Drag is inversely proportional to speedsquared. Odd as it may seem, the induced drag of anaeroplane is greatest at low speed. At very high speeds theinduced drag all but disappears but the profile drag is verylarge, and eventually equals the weight. (Note that this isactual 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 nearstalling speed and the speed for minimum sink is very nearstalling speed. It can be shown (trust me) that the airspeedfor minimum sink rate is theoretically about 75% of theairspeed for minimum glide angle.

I have already mentioned how and why to reduceinduced drag, but to be efficient a glider must be a cleanmachine . Tailplanes with an up or down load haveinduced drag like wings. Tails, fins , fuselages, struts andundercarriages have skin friction drag from their surfacearea and form drag which depends upon theirstreamlining. The drag of the whole aeroplane is morethan the sum of the parts. The extra drag is calledINTERFERENCE DRAG and is caused by the airflowsaround the various parts interfering with each other. Asmuch of this profile drag as possible must be eliminatedby careful streamlining and fairing adjoining shapes intoeach other. The reduction in total drag obviously gives abetter LID ratio . What is equally important is that thisbetter LID ratio is achieved at a higher airspeed, whichbrings three further benefits.

The aeroplane has a greater speed margin above itsstalling speed. The controls will work more effectively andas you will see later a further increase in efficiency ispossible because of the higher Reynolds Number at thehigher airspeed.

Effect ofStreamlining

VelProfile......--./~ Induced

Figure 13.7

Drag

74 Basic Aeronautics for Modellers

A Third Option

As I menti on ed previou sly you ca n try reducing theprofile drag. Change the wing sec tion, sharpen the trailingedge, polish the surfaces , blend the win g and fuse lageshapes together carefully etc. The result is an eve n betterglide angle ·and an eve n slowe r sink rate at an eve n higherspeed . Who says you can't have your cake and ea t it?

same penetration as speed C (w ith down trim) but with alower sink rate and also a shallower glide relative to theground . The mod el will ge t back from do wnwind withmore height remaining. Of course once you have returnedfrom downwind yo u are s till stuc k with th e ball ast ,whereas do wntrim you can take off. I have explained theoptions, the choice is up to you.

HorizComp

B

Wind Vector,..

Doum Comp

Figure 13.8All this supe r efficie nt strea mline d elegance mak es

landing a nightmare at an angle of two degrees or lesswhich is why it is necessary to add handfuls of additionalprofile drag in the form of airbrakes to ruin the UD ratioand give a steeper descen t.

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 ata particular ang le of atta ck will not va ry with weight.Exactly the same trim will give you the same optimumglide angle whatever the weight. How ever the speed atwhich it is achieved will rise. The speed increase will beprop ortional to the square root of the weight increase. Sothat for example if the weight is doubled , all referen cespeeds will rise by 41.4% and that includ es Stalling speed,terminal Diving speed , Minimum Sink speed, and optimumGlide Angle speed . Th e glide r ca n still glide the samedistanc e from a given height but will do it more quickly.Therefore its rate of descent will be (41.4%) greater.

Although the coefficients have remained the same, theactua l Drag has doubled and so has the Lift. Even at thenew minimum sinking speed the rat e of descent hasincreased . Ballast, or a heavily built model, will not hinderyour mod el from getting about the sky, but it will reduceits endurance .

The most super efficient sailplane with a still air glideangle o f one degre e at 40 mph w ill s till hav e ze rogro undspeed in a 40 mph headwind and so will descendvert ically. Its nice low sinkrate is not affected by the wind,but it will still not get back to its field from downwind . Aglide r needs to have a good speed range to get abo ut theco untrys ide in windy weather. Th ere are usu all y twochoice s if you are not ge tting enough pen etration . Eithertrim to a higher speed, or ballast up.

Effect ofWeight on Performance

Effect ofWind on Performance

Down TrimFigure 13.9

Figure 13.8 shows the magn ified top part of anoth erhodograph. A glide r is trimmed to fly at point B on thegraph and has a great glide angle relative to the air. Now Ishall draw in a wind vector from po int A to the origin (theopposi te direction to the aeroplane 's velocity to represent aheadwind). Vectors from A to point s on the hod ographshow the speed and glide angle of the glide r relative to theground. The line from A to B shows that the glide anglerelative to the ground is abysmal. It would be best to trimthe aeroplane to point C on the graph. At this high erairspeed th e glide ang le rel ative to the grou nd isoptimised. Don 't ask me how you find this point withoutinstruments - that tak es ex pe rience. I'm alright with apencil but rubbish on the sticks.

Ballast

Wi"dHoriz Comp

C\\

\

I

II

I

1 "I,Weight W/ '

,

Weight2W

Figure 13.9 is yet ano the r hodog raph, this time with twolines. Line 2 is for the same aeroplane as line 1 but atdouble the weig ht. The bes t glide angle relative to the airhas not cha nged but the minimum sink rate has. It isworse. However look carefully. The second optio n is toadd ballast and fly trinun ed at speed D, which gives the

Dow"Comp 2


Chapter 14

Powered Performance

If yo ur main int erest is in powered mode ls th enperhaps yo u have skip ped th e cha p te r o n g liderpe rformance . Well go back a nd read it any way

becau se a pow ered mod el becom es a glider wh en theengine stops.

Bear in mind that powered mod els tend to use loweraspect ratio and so have mo re induced drag, and lesscareful str eamlining and so more p rofil e drag .

The power source could be a gas turbine or d uct edfan which fo r pract ical purposes produce co ns tan tthrust , but usually we use a propelle 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 forward mo vement in on erevo luti on, in the dir ection of the aerofoil cho rdline . 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 rope ller bl ad e is, usu a lly , a fl at b ottom edae ro fo il fi tted with th e curved s id e o f th e aerofo ilfo rward, towards the direct ion of mot io n of the ai rcraft(obvio us perhaps, but I have seen pu sh er props backto fron t) .

Propeller Thrust

Th e velocity of the air re lative to the pr opeller , assho wn in Figure 14.1, is a co mbina tion of its rotationa lve lo c ity and its forw ard ve loci ty . Th e fast er th eaeroplane is flyin g the less the b lad e ang le of a ttack.Figure 14.2 shows how the thrust of propellers vari eswi th the forward speed of the aeroplan e for differentco mbina tions of diameter and pitch . When the airspeedrises and the blade's ang le of attack reduces , the torqu enee de d to dri ve the prope lle r a t a g ive n speed alsoreduces and so the engine ca n speed up . It is sa id to"unload in the air".

The pitch of the p rop eller is chosen to suit the flyingspeed of th e model and th e co a rse r th e p itch , th esmaller the diam eter must be to avoi d overloadi ng theengine .

Slipstream Effects

The wake of a propeller is a column of air moving aft

Figure 14.1

Rota tion

Thrust

\

\

\

\

\

\\\

\

, , , , , , , , , , , ,

L......~

Resultant •• • •• •••• ' ItFo,~e ~---_._._---_._--\-------. ,.

•••......................TorqlleComponent


~~~=T --- -

_ m _ _ m_ ---=ac-~ -_

from th e surface increasing the drag coefficient andreducing the lift coefficient. When the lift coefficient is atits maximum, the airspeed is at its min imu m, which iscalled the "stalling speed in level flight" (for the exis tingweight and co nfiguration) . It is an oft quoted statis tic ofan aeroplane and is a usefu l reference speed.

"-, "\ Fine

" " " \, ,\

\ ,\ ,

\ "

Speed

Airspeed

Drag

se

, , , , , ,-, ,,

\\

\

\

Coarse

Co

Speed

,,,,,,,

Top

Pine

\

\

\

Thrust

Figure 14.2

Figure 14.]

Fi"e Prop

Thrust

Coarse Prop

Figure 14.4

A powered aeroplane can fly level , regardless of thedrag. Just bo lt a big enough vibra to r o n the front andaway you go . The thrust eq uals the total drag, and theto tal lift eq ua ls the weight.

Figure 14.4 illustrates how the thrus t an d drag varywith the forw ard speed of the aeroplane . Where the twocross Th ru st eq ua ls Dra g and th e aeroplane is ineq uilibrium at its maximum speed in level fligh t. A morepowerful engine will obviously produce more thrust andso th ru st w ill eq ual d rag at a higher speed . Using acoa rser pitch prop on the same engi ne the thru st ca n besustained to a higher speed allowing a high er top speedfor the aeroplane , at theexpense of lo w speedacceleration .

Th e speed at w hichth ru st and drag areequal is an airspeed ofcourse . Thrust will equald rag a t this ai rspeedregardless of whether awi n d is b lo wing . Ifsomething makes th eae ropla ne slow down,then thrust will b egrea ter than drag andw ill accelerate theae roplane back to theairspeed at which theywere eq ual.

rela tive to the aerop lane at a speed greater than that ofth e surro undi ng air. If th e tail con trol surfaces ar emo un ted in the sli ps tream the ir effec tiveness will beinc reased as the pro p elle r thrust is increased . Forexam ple rudder gives very good directional co ntro l ontake off but is mu ch less effective on lan din g.

In Figure 14.3 (in which the effect is exaggerated) anaerop la ne has a pr opelle r ro ta ting clockwise (fro mbeh ind ) and the slips tream will strike the fin slightlyfro m th e left. A fo rce to the right will be generatedyawing the nose to the left. If in an attemp t to avoid thisproblem the fin is turned to line up with the slips trea m,then when the throttle is closed the aerop lane has someright rudder ap plied and will now yaw to the right.

A better so lution is to mou nt the engine suc h that thethru stline is angled as shown to the righ t, called "rightsidet hrust". The slipstream still causes left yaw , bu t thethru st has an opposing mo me nt ab out the eG. The fin isstraight which lets the model fly straigh t power off, andth e sidethrust can be adjus te d to give straight flightpower on . One to three degrees is a typic al sidethrustangle.

As fo r a glider it isno t th e sp eed bu t thesta lling angle of attackwhich is fixed. The wingsta lls a t th e ang le ofat tack a t w hich theairflow ove r the to ps u rface breaks away

Stalling Speed

Level Flight Top Speed

Basic Aero na utics for Modellers 77

Figure 14.5

Thrust

I) Stall I,Speed)

)

High Weight

Low Weight

Speed Range

Effect of Weight

Obviously from the above the sta lling speed in leve lflight w ill increase as th e weight is inc reased . Thestalling speed increases as the sq ua re root of the weight ,doubling the weigh t gives a 41.4% increase in stallingspeed.

Thrust from the propeller is independent of weight.However the greater the weight, the greater the drag willbe, partly because the induced drag is greater and partlybecause the associated inc rease in angle of attack willincrease the profi le drag . Figure 14.5 shows the thrustfrom a particu lar engine/propeller combination and alsothe drag at two weights. As yo u can see the speed rangeof the aeroplane reduces as weight is increased, stallingspeed increasing and maxi mum speed red ucing .

Take offPerformance

For accelera tion you need low weight and plenty ofth rust. Whe n the speed is safely above sta lling speedyou need enough eleva tor power and clearance underthe tail for rotation.

When the ang le of attack is sufficient the lift willexceed the weight and the aeropla ne will take to the air.

The faster the model is travelling the less pitch upneeded to "unstick".

The heavier the aeroplane the greater the take offspeed and the slower the accelera tion . Weight will havea marke d effect on runway length required, as will thesurface. Smooth concrete is great, long wet grass reallyholds the model back. A larger diameter fine p itch propwill give more thrust and hence a quicker accelera tion atthe expense of top speed.

78

Drag

I

I

I,III

I ITop I

ISpeedI I

Speed

Climb Performance

Figure 14.6 shows an aeroplane in a climb at ang le Ato the horizontal wit h its thrust , weight, overall drag andoverall lift all th rough the CG. The weight has been sp litinto two components. You ca n see that the Lift is eq ua lto the component of the weight W cos A, which mean stha t lift is less than weight in a climb. You can also seefro m the diagram that th rust has to balance not onlydrag but also a component of the we igh t, W.sin A. Thesteeper the climb the more thrust will be needed. In aver tica l climb the th rust must equa l the weight plus thedrag at the particu lar airspeed, but lift will be zero.

Figure 14.7 shows the maximum th rust and tota l dragplotted against speed. This time however the aeroplaneis climbing at speed V. The thrust at this speed is grea terthan the drag, and the excess thru st is used to overco methe weight component W.sin A. The lower the climbingspeed the more excess thr ust and so the steeper theclimb .

The heavier the aeroplane the more its drag will be atthe same speed. But it will have to be flown faster tokeep th e same margin above sta lling speed , w h ichincreases the d rag even more . Th ere is therefo re lessspare thrust ava ilable, but the "downhill" component ofweight, W.sin A, is grea ter. These factors all combine toredu ce the climb angle achievable at the high er weight.

Descent and Landing

Figure 14.8 shows an aeroplane in a descent, with asma ll amount of thrust helping the weight component\X7.sin A to overcome the drag . The mo re thrust there isthe sha llower the descent. In ot he r words ope ning thethrottle a little will give a sha llower descent at the same

Basic Aero na lilies fo r Modellers

Figure 14.6

Hortzontal

II

I

I

I,lVcosA

speed, whil e closing th e th rottl e w ill g ive a stee perdescent. The throttle se tting co ntrols rate of descent.

The d ragg ie r th e ae ropl an e , th e more th e po we rneed ed , especially not iceabl e in the case of low aspectra tio biplan es clutter ed w ith rigg ing wires and stru ts .So me cl ean po w e red model s ha ve too s ha llow anapproach ang le for safe terrain clearance . The best curefor too sha llow an approach pa th is more d rag fromairhrakes , flaps, slats or some thing. Diving the aeroplaneover the obstructions on the approach will o nly increaseth e s peed a t w hic h it reaches the run w ay a nd th eaeroplane will "float" right down the runway and land inthe rough.

The fina l approach se ts up the aerop lane fo r thelanding, low over the run way threshold with the speedsafely above stalling speed . In the "flare " the nose ispitch ed up to arres t th e rat e o f des cent until th eae roplane land s almos t at stalling speed.

Figure 14.8

W

Figur e 14. 7

Thrust

WsI"A

D

Drag

Speed

Basic Aeronautics/or Modellers

W WcosA

79

Chapter

The Aerodynamics ofAerobatics

15

I have already de alt with straight night and turning.Any th ing e lse is not use ful in tr an s porting th eaeroplane from A to B and is therefore co ns ide red

to be an acroba tic man oeu vre , don e jus t for the sa ke ofdoin g it.

All the acroba tic man oeu vres w hic h I ca n think o fco ns is t o f a few basi c e leme n ts s tr u ng to gether invar ious wa ys. I do not claim to be any kind o f hot sho taerobatic p ilot and therefore do not feel qualified to gointo the re fine me nts of fl y in g aerobatics. Neither do Ihave the eq uipment available to test out the subtleties ofacrobatic model design . I sha ll leave the refinements tothe expe rts and stick to the basic eleme nts.

It is wort h remem ber ing w hen perfor ming orwatching aerobatics that the law s of ph ysics cannot bebrok en , by anyone.

The Stall

The pu p il p ilo t is often tau gh t the sta ll as his first"ma no e uvre ", so th at he ca n lea rn ho w to avo id adangerou s un intenti on al stall. The aeroplane is sloweddown by gra dually applying up elevator in level night ,increasing the angle of attack , until so mew here on thew ing th e flow sta rts to se pa ra te . There are seve ra lpossible outcomes .• 1. If the separation occ urs over both win g roots more

Figure 15.1

Rising Wing

DroppingWi"g

or less simultaneous ly then the nose down pitchingmom e nt coefficie nt C~ IO which had been co ns ta ntsudde nly inc reases . Or if you are a Centre of Pressur efan, the Centre of Press ure moves aft aga in . The resultis a ge ntle nose down p itch , a nice safe reacti on .Applying more up eleva tor ca n counteract the nosedown mom ent an d keep the nose up . The ae roplanew ill then descend in a level a tt itude and w ill stillrespond to ailero ns , rudd er and elevato r.

• 2. If se pa ratio n occu rs over the inboard region ononly one side, the wing on that side will drop. DONOT try to pick it up wit h aileron. That is like ly tostall the outboard pa rt of the wing as we ll and makethe situation worse. The correc t recovery is to applydown elevator and opposite RUDDER.

• 3. If the outbo ard part of the wing stalls o n o ne s ide ,a "tips tall", then a viole nt wi ng d rop occurs usua llyleading stra ig ht into a sp in. The re is no tim e torecover from the stall. See below fo r spin recovery.

• 4. As I said in Cha pter 12, if the elevator is small andthe stability margin is large even fu ll up e levator maynot be enough to stall the wing . It may just fly slowlyin trim. See Figure 12.4 agai n .

The Spin

A spin is ente red from low spe ed night. The w ing isa t o r near its s ta lling ang le of a ttack and o ne w ingd ro p s , du e to a g us t, ai le ro n in put , o r w hateve r.

Figure 15.2

Original a

a OfRisingWi"g

a OfDroppingWillg

0<0<

80 Basic Aeronautics/or Mode llers

Fig"re 15.3 Figure 15.4

-. V

WW

L

Consta nt V

L L L L

W W LW L

w2V - - - :=_~t=::==n

lV

Application of rudder in the appropriate direction alsohelps. Becau se the air is now co ming up towards thedowngoing win g it ge ts an angle of attack increase , andthe upgo ing wing gets a n a ng le of a ttack de crease .Th ese would normally ca use "ro ll d amp in g ", butbecause the win g is alre ad y at its stalling ang le of attac k,the angle of attack incr ease past the stalling angle leadsto a reductio n in the lift coefficient. A ro lling mom ent inthe direct ion o f the ex isting roll result s. The droppingw ing w ill also ha ve th e g rea te r dra g p rodu cing yawtow ard s the ce ntre of the spin. Figure 15.1 sho ws thatthe dropping wing has a smaller lift coe fficient than therising wing and so the situation co ntinues . This pro cessis known as "auto rotation".

The airspeed rem ain s low in a sp in du e to the ve ryhigh dra g o f s ta lle d wings . Wh ile th e ae roplane issp inn ing the angle of attack on the wing on the inside o fthe sp in is greate r than that of th e o utside Wing butbec au se o ne o r pr obab ly both , are grea te r th an th esta lling an gle , th e sp in co ntinues (Figure 15.2) . Th eaeroplane is both rolling and yawing toward s the morestalled wing. The rate of rotat ion is faster than in a spi raldi ve and the height los s per turn is mu ch less. Mostmod els have to be held in the sp in with up elevator andusu ally rudder and aileron in the same direction .

Recovery action from a spin is very dependant on theaeroplane. Most models will recover if all the contro lsare ce ntralised . If not then positive recovery action mustbe taken . Appl y full opposite rudd er to co unteract therotati on (the ailerons are ineffective), and down elevatorto unsta ll the win g.

It is impo rtant that aeroplan es designed for aerob aticswill sp in re liably whe n required . The d esign factorswhich assist in re liabl e sp in ning are taper ratio , CGposition , control throws and LE sharpness . A taper ratioof 0.65 or less ens ures that the angle of attac k at the tipsis only slightly less than near the root. A modest CGmargin and adeq ua te e leva tor throw e nsure th at thewin g can be pitched to its stalling angle of attack. Andadequate rudder movem en t ensures that it will sp in inthe direction you want. Havin g the lead ing edge sha rpe rat the tip than the root also helps spinning .

In o rder to avoid spinning, trainers go the other way ;co ns tant chord, forward CG, reduced eleva tor authorityand blunt lead ing ed ge. Scale mod els have no choice oftap er rat io but ca n use wash out and a forward CG toavo id probl ems.

The Snap (or flick) Roll

Exce pt that it may be don e travelling in any directionat any sp eed , a sna p roll is just like a spin and the sa medesign factors are involved. If full up elevator is applied ,the ang le of attack wil l be increased , probably up to thestalling angle o r beyond. If one wing stalls but the oth erdoes not , the differ en ce in lift wi ll produce a very rapidro ll rate . On e Wing ca n be mad e to stall by ap plyingrudder and aileron in the direction of the intende d ro ll.Rele asing the up elevator and/ or the rudder and aileronwill sto p the snap ro ll. A snap roll may be performed inho rizo n ta l fli ght , o r on a 45 d e gree d ownline , o rvert ically upward s or what ever.

You should be awa re of the structural loads imposedby a snap ro ll at high speed . If the ae roplane is flying ata speed factor of three times its leve l flight sta llingspeed, then the win g will be subjected to a brief load ingof nin e "g", if at four times stalling speed , sixtee n "g".The load factor is the squa re of the sp eed factor.

Th e Loop

To make a loop round a ce ntripe tal force towards thece nt re of the loop is necessary and is provid ed by thewin g lift. The tight er the loo p the mor e up elevator isneeded . If th e airspeed can be kept co ns ta nt, th enkeeping the ce n tripe tal force co ns tant will produce around loop, The ce ntripe tal force is the resultant of allth e force s acting perpendicular to th e aeroplane' sdir ection of mo tion , the refo re the lift force mu st varyround the loop. For ex ample, in Figure 15.3 lift is fivetimes weight at the bott om , three times we ight at thetop, and four times the weight at the ve rtical posit ions.

\V'ithou t ve ry ca refu l power co ntro l the speed w illvar y round th e loop . Th e ce n tripet a l force requ ired

Basic Aero nauticsfor Modellers 81

Stall

so mu ch to ge t the wingto lift up sid e down thatth e tail is at a lift inga ngle . Were it not forth e down e leva to r LTwou ld be so great that itwou ld ha lf loop th ea erop la n e into le velflight.

Figure 15.6 is a trimg ra p h just like Figu re12.4 e xcept that it hasbeen ex tende d to shownegative lift coefficients,for invert ed fli ght. Itshows the elevator ang leto trim for an ae roplane,like a traine r o r g lider,w ith a ca mbe re d wingwhich has been set up

such that the elevator is neutral in upr ight fligh t. Th reelines for three d ifferent va lues of Stability Margin aresho wn. The trim change between uprigh t and invertedfligh t at the same spe ed is sho wn fo r a mid CG position.You can see that the furth er forward the CG, Le . thebigger the Stab ility Margin , the more trim cha nge therewill be . Inve rted flight is not possib le a t low speedsbecau se the wing is not as good at lifting invert ed as it isupright.

Figure 15.7 is another tri m grap h, thi s tim e for apattern ship with symmetrical wings and ta il both se t upat zero inciden ce . For uprigh t flight a little up trim isneeded , the more Stability Margin the more up trim. Andfo r inve rted flight the same amo un t of down trim isrequired . The trim ch ange between upright and inverted

fli gh t ca n be see n todepend o n the Stab ilityMargin just like o n Figure 15.6. In fact for thesa me Stab ili ty Marg in ,the same trim change isneed ed .

It seems stran ge but itis true that a camberedwi ng d o es not requi reany more down trim tofly inverted . It does haveo the r e ffec ts how eve r.T he p ro fil e dra g of acambered section flyinginvert ed is much grea terwhi ch reduces the speedca pab ili ty o f the aerop la ne . And th e w ingcannot provide nearly asmuch ne ga ti ve lift asposit ive lift because ofth e s ta ll. Bo th th e see ffec ts co ns p ire toredu ce the negat ive "s"ca pab il ity o f a n ae ro plan e with a ca mbe redwing .

It is but a sma ll ste pfrom doing leve l flightin verted to d o in g

- - - - - - - - - - - Full Doum Elevator

Inuerted Flight

Stall

Full Up Elevator

Figure 15.5

Figure 15.6

Elevator Angle To Trim

Figu re 15.5 shows an ae roplane wi th a cambere dwi ng in steady level flight. The tail lift LT mus t beupward so that its mom ent about the CG balan ces bot hMo and the win g lift. Yes I know that you have ap plieddown elevator but the whole aeroplane has been tilted

varies as speed sq ua red therefore the lift variation wi llbe mu ch greater. Taking the arbitrary ex ample wherespe ed at the top will be half what it was at the bottom,the lift variation will be as in Figure 15.4. Th is time , ifthe lift is five "g" at the bottom, it must be zero at theto p. Loops are easy, but perfectly round loops are ve rydifficult ind eed .

Inuerted Flight

82 Basic Aeronautics/o r Modellers

Yawing Manoeuvres

The re is little to saythat was not sa id in thecha pter o n Co ntro ls. Isha ll jus t remind you that the rate of roll p roduced bythe ailero ns depends on their size obv iously, and on theaeroplane's roll damping. The bigger the wingsp an themore roll damping and so the slower the rate of roll.Also , be care ful of using ailero ns near stalling speed.

Sta ll

II

Up rig ht

FilII Dow1I

Eleuato r A ng le To Trim

FilII Up

plane is pointed vertically up and allowed to slow downto a sto p. Ju st before it stops , rudde r is used to rotate it180 degrees a bout its "ve rt ica l" axis, w hic h is nowpo int ing to the far hor izon.

There foll o w s a ve rt ica l d escen t. No te th at th eaeroplane stops in this manoeu vre but it does not stall!Th e a ng le o f att ac k is ze ro in th e cli mb a nd ge tsnowh ere near the stalling angle of attack .

II

III

I

Stall

III lnuerted

Figu re 15, 7

The o nly co nt rol leftis th e ru d de r, whi c hcauses the ae roplane to Figure 15. 8yaw . T his is us e d in"knife edge " fli g ht asdepicted in Figure 15.8.The aeroplane has beenro lled thro ug h n in e tyde gre es a nd held .Rud d er has been a p p lied to h o ld th efuselage at an ang le o fattack to the onc o mingair.

Th e pressu re d is tri bu tion over the fuselagesides can give e no ug h M otio1llift to support the aero-plane 's weight , aided bythe vertica l co mpone ntof th e thrust. There iso nly a sma ll amo unt ofsurface area involved soth is is a hi gh s peedgame.

Rudde r is a lso usedin th e "Sta ll Turn "ma noe uvre . The ae ro-

mano eu vr es inverted .To s pi n inve rte d fromle vel inve rt e d fli ght ,g ra d ua lly a p p ly fu l ldown e levato r to sta lla nd th en rudde r andaile ro n in o p pos itedirec tions to put it intoa spin.

The sa me co nt ro lss ma rt ly a p p l ie d ca nprodu ce a negative sna proll and simply applyingeno ug h down e lev ato rwill perform a negativeloop, or "outside" loop,o r "bu n t" . Because o fthe lim ita ti ons in theprevi o us para graph anaer oplane wit h a cambe red win g will notb unt as ti g h tly as itloops (if at all).

RollingManoeuvres


A typical pattern model has a low wing, sltm fuselage, generous tail area andlarge 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 mediumaspect ratio tapered wing and a long tail arm toblcb assists smooth j/ying toitbgenerous damping.

Aerobatic TrimSetUp

Acro ba tic mod els arese t up wi th th e CG insuc h a position that themod el is smoothly stablebut w il l s pi n re lia b lyw he n requ ired . Th ee leva tor th ro w s ho u ldbe s uch th at a s u ffic ie n tly ti ght lo o pingrad ius ca n be atta ined ,but witho ut flicking out.The a ile ro n throws areadj us te d to give a rollrate of three rolls in fourto five seco nd s forp ower p at tern c o mp etiti on s . An d th erudde r th row sho uld bee no ug h to give cris p lyco n tro ll e d sta ll turn s .The d ihedral is ad justedas d es cribed und erLateral Sta b ility to g iveno roll effec t w ithrudd er in put a n d th ewing and tail may bothbe rig g ed a t ze roin ci den c e usin g trimoffse t to ach ie ve le velflight , or th e wing maybe rigged at a pos itiveinciden ce of ab out half adegree.

84 Basic Aero nautics/or Modellers

Chapter 16

Special CasesLow Aspect Ratio Aeroplanes

Deltas and novelt ies like flying discs or playing cardsoften have aspect ratios of 3 or less. The low aspect ratiowing is often co mbined with a Canard or Tailless layo uttherefore that section must be rea d as wel l.

form u lae , pri ncipall y because of th e very strongdownw ash over the tailplane . For models without tailssee la ter. O n delta w ings you mus t use the l'vJ.A.C.worked out graphically as in Figure 7. 11.

Canard Layout

Figure 16.1

vV,\/INDRA GV~TAJ.L

I

" I

I

I,

D

In my ancie nt French dictiona ry, CANARD is a wordmeaning "d uck" or "hoax" or a "lump of suga r di pped incoffee" . I suppose a du ck's win g is so far back tha t itd o es rese mble a tail firs t aerop la ne . Or maybe theFrench thought it was a hoax!

Figure 16.2 shows the forces in trimm ed flight on acanard aeroplane, LF the lift on the forep lane, L.\" the lifton the wing , Mo the zero lift pitc hing moment du e towing camber and \ '\1 the we ight. Taking mo ments about

One obvious result of the low aspect ratio is the lackof roll dam ping. They can be mad e to ro ll incred iblyquickly and tend to be twit ch y in roll. Use small a ileronswit h little movement , perhaps inset fro m the tips , oreven taileron s.

Another character istic of low aspect ratio wings is thestro ng vo rtex they genera te , produ cin g large induceddrag at high angles of attack. They tend to lose speedquick ly in very tight turns or loops. And whe reas mostpowered mod els have such a low minimum drag speedthat it is not no ticeable in practice , the low aspect rat ioaeroplane has a mar ke d minimum d rag speed as inFigure 16.1.

On a powered model , whe n you gradually reduce thepower and fee d in up trim to fly slower and slowerthere co mes a po int where the model wi ll no longermaintain level flight unl ess yo u act ua lly increase thepower aga in . It is very difficult to fly it at low speedsbecause its speed is un stabl e . . . a red uction in speedincreases the drag caus ing a further speed reduction andvice ve rsa . Th is is known in common parlance as flying"on the back of the drag curve". At moderate speedsthey will glide, but try to stretch the glide and they fallo u t of th e sky, not beca use the y have s ta lled b utbecaus e the Lift/ Drag ratio has been assassinated .

Deltas and othe r low aspect ratio aeroplanes are bestsu ited to hig h speedfligh t whe re they ha ndlemost co mfo rtably. And Figure 16.2th ey are best sui ted tosma ll d ia met er coarsep itched propellers . Theapp roac h s ho u ld be LI'flown at a speed a littleabove m in imu m dragspeed.

Handling Peculiarities

CGPosition

Low aspect ratio aeropl a nes wi th ta ilplaneswill be nefit from a mo reforwa rd CG posi tio nthan give n by the usual

lV(=L)


first princip le s a ndcom m o n s e nse . I a ssume that the foreplaneis sma ll re lative to thewing. I therefore believethat th e factors w hic hwe cons id e red inCha pter 8, Le. the effec tso f wake an d downwasha n d fl e x ibility andaspect ratio differences,will be sma ll and partlyself cancelling .

\1 . foreplane area foreplane armlj,

har IS . X ----'---- - - ----''--wmg area wing mean chord

My formu la for cana rds.CG position = 0.15 - Vhar gives a CG position aft of

the leadi ng edge of the mean chord , as a fractio n of themea n chord. I have red uced the p lanned Stab ility Marginto 10% chord and the Vhar is the fore plane vo lume ratio ,which if la rge w ill g ive a NEGATIVE a nswe r whichmean s the CG is in FRONT of the lead ing edge of themea n chord .

._~_ :"...-t--_A

Figure 16.3

the CG yo u ca n see th at th e foreplane mu st lift toco unteract the moment o f the wing lift and lift somemore to co unteract the Mo of the wing. The foreplane isalways lifting and its lift coefficient will be greater thanthat of the wing.

It mu st therefore be set at a more positive angle ofinciden ce than th e wing an d since the forep lane willalways have to provide upward lift it would make se nseto give it a cambe red section .

Stability

No w thi s aero p lane ge ts a disturbance w hichincreases its angle of attac k. See Figure 16.3 in which thelift increa ses on wing and foreplane, XLw and XLF areshown along with the resu ltant AL. I have omi tted all theforces o n Figure 16.2 which cancel out. The res ultantex tra lift, AL, acts at point A which is the Neutral Pointand which is distan ce x fro m the CG. The aeroplane isstab le if the NI' is beh ind the CG as before and distancex is the Stabi lity Margin.

BUT, the area of the foreplan e must include the area inplanform, of part of the forward fusel age (see Chapter22).

The CG obvious ly e nds up quite far ba ck on th eaeroplane. The foreplan e ar m IF is the distance of theforep lane 's q ua rter chord po int ahead of the wing 'squarter chord point.

Control

CG Position

I have been unable to find a formula in textbooks forthe Neutral Point of a cana rd whi ch leads me back to

Eleva tors on the forep lane or an all moving foreplanema y be us ed for pitch co ntrol. O bvious ly th ey mustmove trail ing edge down for no se up pi tch . On canarddeltas it may be more effe ctive to use ele von s.

Tbe formula e for CG doesn't toork Oil tbis so it 's back to basic principles. Seetext.

TaillessAeroplane - ToAchieve Stability

Th ere a re two waysto m ak e taill e ss ae rop lanes stable . Sorry , I' llrep hrase th at. There iso ne w a y to m ake anaeroplane s ta b le andthat is to p lace the CGahead of th e Ne u tr a lPoint. Having done thatthere a re tw o wa ys toma k e a ta illess ae rop lane fly in trim .

Th e Neutral Poi nt ofa n ae ro p la n e is th ep o int th rou g h w hic hth e resu ltant extra liftcaused by a small pitchcha nge w ill act. For aflying wing this po int isthe wing 's aerodynamicce n tre, by d efini t ion ,whic h is at ab out 25% of

86 Basic Aeronaut ics/or Modellers

its mean chord whichcan be found using thegraphical method inChapter 7.

The CG shou ld bepl aced by treat ing th eaero p lane as a canardwithout a fore p la neusing the canard formula above an d treat ingfus elage area ahead ofthe lead ing edge if any ,as foreplane a rea . Ifthere is no fuselage thenVha, is zero and the CGis at 15% mea n chord.

To Fly in Trim

Figure 16.4

w

rNP

One solu tio n is touse a special aerofoil section called a "reflex sec tion" asdepicted in Figure 16.4. As you see it is turne d up at thetrailing edge which gives it a nose up zero lift pitchingmome nt Mo (or if you like the Centre of Pressure movesaft as ang le of attack is increased). A normal th in slightlycambered sectio n w ith th e control surface re flexedup wards works as we ll. The further forward the CG isplaced the more reflex is needed to co mpensa te, and theslower the flying speed the more reflex need ed to stayin trim. The second method is to sweep the wings backand bu ild in a hea p of washout at the tips as in Figure16.5. Conve ntional sections with CMO acting nos e downcan be used.

In my d iag ram th e inboard parts of the wi ng arelifting up wards and the parts nea r the tips are liftingdownwards an d the who le thing ad justed so that thetotal lift is zero. Th e nose up mo ment from the lift forcesmore than balances the nose down moment from thewi ng sections, and so the overa ll Ze ro Lift Pitchin gMoment is Mo as shown, acti ng nose up. The wing tipsact just like a tailpl ane .

The more ca mbered th e sections used , and thefur ther forward the CG, the more washo ut is needed.Both solution s can be used together of co urse to end upwith a swept flying wing with a little washout and a littlereflex on the sec tions.

Control

Control is exercised by movement of control surfacesat the trailing edge. The contro l surfaces are a bit shortof leverage in pitch control, but on the other hand, the

flying wing has very little pitch damping (especially ifunswept) so not much control is necessary. It does meanthat suc h models can be a little sensitive in pitch , andcan be a little short of dynamic stab ility.

Because the fin will be so close behind th e CG, itmust b e very la rge to ac hieve e nough d irectiona lstability. Even then, Yaw Damping will be quite sma ll.

Multiwing

Biplanes, Tripl an es, Qu adruplanes etc. mean drag .Although structurally efficient, all those rigging wiresand struts give ex tra profile drag. All those wingtips giveex tra induced drag. You ge t all the extra drag from thebits interfer ing with each oth er. They even ten d to havebig bul ky draggy fuselages, bu t they do have chara cterand though Boei ng haven't built one for man y a yea r,they are a firm favouri te with mod ellers.

Performance

Becau se of the high drag , the Lift/D rag ratio is poor,leading to a fairly steep glide angle, which in itself is nogreat problem wh en the engine is running. It just meansthat approaches are best car ried out w ith a little poweron, and yo u have to be wary of deadstick landings. Italso ex plai ns the re lative scarcity of b ipl an e thermalsoare rs. Because of the low flying speed (usua lly), largefuse lage , and high drag, most multiwings are bes t suitedto large diameter fine pitched propellers.

The compactness (re lative to the area) o f multiwingedaeroplanes gives them less pitch and roll damping. That

Figure 16.5

11 MoI

II

I

II\\

\ , ,

------- -----~ -- -----------------------


a nd the lightness o f th e w ings ma kes the m q uit eman oeu vrab le if they have an aileron on each wing.

CG Position

The formula tor CG posit ion in Chapte r 8 does notap ply to biplan es becau se one wing has been ass ume din its deri vat ion . But if so me fiddl e factors ar e usedperh ap s it can be mad e to give accep table result s. Oneproblem is that the tailpl an e is operating in air w hichhas been slowe d down by its flow around the fuselagea nd rigging , a nd the o the r is tha t eac h w ingtip iscrea ti ng vo rt ices w hic h c rea te d o wn w ash o n th etailplane therefore so me account must be taken of thenumber o f wings .

I th e re fore ma ke the fo llo w in g s ugges tio n . Fo rbiplanes with two identical wings, let the mean cho rd bethe actual wing cho rd and place it mid way between thetwo wings (and rem ember to allow for sweepback).

As befo re calculate the total gross wing area and netta il area. Measure the tail ann betw een the quarter cho rdpoints o f the wing mean cho rd and the tail , and workout the tail vo lume ratio . Now use only HALF the ta ilvo lume ratio an d HALF the Aspect Ratio of each wing inthe usua l formula

CG posn = 0.1 + 0.25 x VI"" x 4j AI{

For triplanes the ave rage cho rd will be the midd le oneand the factored Asp ect Ratio and tail vo lume ratios willeac h be a thi rd of the ac tua l o nes. The no mogram inFigure 16.6 is the same as Figure 8.10 but ex tende d tosmaller values of Vhar to cope with this factoring .

Nose lengths mor e than o ne cho rd ahea d of the meancho rd lead ing edge , or particul arl y wide co wlings , o reng ine nacelles on multien gined aeroplanes will all havea d es tab ilis ing e ffec t and th e CG sho u ld be mo vedforward a few percent to compe nsa te .

For further details, including how to handle unequ alwings and a wo rke d exa mple, see Chapter 22.

N u merou s toingtips, plenty induced drag, steep glide (pretty though isn't it).

88 Basic Aerona nticsfar Modellers

Note: 'factored' means actual ualue dividedby number of ioings

.~

30.35

25

8

.25

.2

.15

-------------

20

18

----------_-..:.--

16

15

14

6

4

3

2

1

.1

.09

.08

.07

factored tailoolume ratio

Basic Aeron alilies f or Modellers

13

12

CG posttionas a % of toingmean chordfromLeading Edge

Figure 16.6

factored toingaspect radio

89

Chapter 17

Reynolds NumberM y ass umpt io n tha t the lift dr ag and momen t

coefficien ts are indep endent of airspeed is notexactly tru e (ve ry littl e in life seems to be

exactly true). The cri terion which determines wh etherthe results of one test can be used in ano ther applicationis that they mu st be at the sa me Reynold s Number.

253 000

6

10

8

15

5

15

1015

fPs/mpbkps/mpb

Airsp ee d

15

20

25

3 0

25 0 70150

2 00 60

50

150

40

35

3 0

25

20

15

20

40

80

60

3 0

Re te iooo»

100

1000

800

2000

15

20

Willg cb o rd

Figure 17.1

15

10

9

83

7

2.56

5 2

em s. ins.

6

2 0

40

25

3 0

50

35

90 Basie Aero na lilies f or Mode llers

~e-- . ...........--.

"\ Transition P--;;;llt

Transition Point/

R,:

The nor mal situa tion is for the boundary layer to starto ff lam inar and the n some di st ance b ack fro m th eleading edge it becom es a turbulent bo undary layer andrem ains so to the traili ng edge . Th e point at which itc ha nges fro m lam in a r to tu rbul ent is ca lled th e"Transition Point". Figure 17.3 shows a normal transitionfrom a lamin a r to a turbule nt bounda ry layer . Theturbulent bound ar y layer is thick er th an the Iarnina rbo undary layer which it replaces .

The go ing has to ge t really tou gh before it succumbs tothe ev ils of se paration . Laminar on the othe r hand willgive you an easy time with little drag but , meet a littleadve rsity o r obsta cles, and it will skip off an d leave youan d yo u know wh at such se pa ration means - less liftand very high d rag, the stall.

When the air flows over the aerofoil, its pressu re onth e u pper su rface re d uces to a minimum, and th e nincre ases aga in to no rmal a t the tra iling edge . Figure17.4 shows that as the larn ina r bo unda ry layer flows intothis area of increasing pressure its already slow progressis brought to a halt by the steadily incre asing pressurew hich it is meeting. Air co n tin ues to flow in to th isregion from the lead ing edge and so of co urse a "lump"of stationary air builds up , the strea mlines of the mainairflow are forced to se para te from the wing surface . andth e wi ng has sta lled . As I sa id in Cha p te r 3 the sta llusually starts near the trailing edge but as angle of attac kis increased the se pa ration point moves rapi dly forward .The drag increase due to separation is very grea t.

Situation Normal

Laminar Separation

Figure 17.]

Figure 17.2

~-------------~~---

• whe re p is the air's den sity• c is the wing chord• V is the airspeed• and Jl is the air's viscosity

p c VRe=---

Jl

The scene of the action in th is part of the story is thelayer of air right next tothe aerofoi l s urface .Whe n air mo ves over asu rface th e b ou nda rylayer may be o ne of twoki nds, lam in ar o r tu rb ul ent. The turbul e ntbound a ry layer g ivesmo re drag but is a mored e p end abl e so rt ofbo undary laye r w hic hw ill s tick by yo uthr ou gh th ick and thin .

Os borne Re y nol ds(1842 to 191 2) was aBrit is h scien tis t w hodiscovered tha t fo rgeometrica lly si m ila rte st s th e flow patternw ill be id ent ical if apa rticu lar co mbina tio nof the dimen sion s of thetes t piece and the speedof th e flo w , an d th eviscosi ty and density ofth e flui d is kept co n-stant.

For aerodynamic purposes th e mag ic combi na tio n ca lle d th eReynold s Num ber (usu-ally ab breviated to Re) is give n by

Tha t equation ca n be simplifie d to Re = 536.v .c withspeed in It/ sec and chord in inches or Re = 70.V.c. wi thspeed in m/ sec and cho rd in mm .

The answer has no unit s. It is "d irnensionless". It isjust a number wh ose only purpose is to compare it withother Reyno lds Numbers.

An answer to the nearest few thou sand is acc ura teeno ug h. The simplest way of wo rking out a Re is to usemy no mog ram in Figure 17.1.

The influen ce of Reynolds Number on aerody na micproperties is irregular. A very simple illustrat ion of th is isthe var iation with Re of the drag of a smo oth sphere. Assho wn on Figure 17.2 the drag coefficient of a sphere isrelatively large at very low Re . As Re is inc reased thedrag coefficient grad ua lly reduces and then ove r quite ara nge of Re re mai ns consta n t. Sudden ly, at o nepa rticular value of Re , the drag coefficient drops to lesstha n half its previou s steady value. Fur the r inc reasi ngthe Re produces no more cha nge in dr ag coefficie nt.Rou ghen ing the surface of the sp he re reduces the Re ofthe sudde n drag reduction . The step moves left.

In The Boundary Layer

Basic Aeronautics fo r Modellers 91

Figure 17.4

Flow GettingSlower

~~~~MinimumPressure

Point

Separation Bubble

Streamlines Separate from Surface

11 \\ \\ ~ 11 ~ 11 \\ \\ ~ /I -'/' 11 1\-'/ l' ~ -'/ 1' -;:::-

'11 If -;:::- \\ ~ II~ II -;:::- \\ Area of

II / Turbulencefl ""

~ ""~ ~ 1'-, If \\ II .;

"" 111\

bubble ma y fo rm o n the unde rs ide a t low ang les o fattack.

At mod el Re it ofte n happen s that shortly after thelaminar boundary la ye r sepa ra tes fro m th e aero fo ilsurface, it tran sition s into a turbulent boundary layer.The con sequent thickening allows it to re-attach itself toth e su rface le a vin g a small p o ck et of s tag na n t a irtr apped against th e ae ro foi l ca lle d a "Se p a ra t io nBubble". Figure 17.5 shows an aerofoil with a se pa rationbubble . Within the se para tion bubble a ge ntle rotationalflow develops as shown, du e to the pressure distributionand the viscous forces. As ang le of atta ck is increased ,th e minimum pr essure point will move towards th eleading edge and the se paration bubble will go with it.The po int w ill co me when ei the r transit ion will notoccur , o r o ccurs to o lat e for th e tu rbulent boundarylayer to re-att ach itself. The flow se parates co mpletely,the win g stalls, and the se pa ration bubble is said to haveburst. The flow will then look like that in Figure 17.4.

The Underside

On the lower surface of the wing the boundary layerstarts off, as always, lamin ar. At high Re it will tran sitionat so me stage into a turbul ent boundary layer with theattendant high er drag, but at low Re there is no goodreason why it sho uld becom e turbulent and wh en thepressure is reducing over the rear portion of the section,there is no reas on for it to se parate e ither. It is howeverpossibl e that o n underc ambered sections a se pa ration

Figure 17.5

Laminar B.L.

Separatton Poln1 /

92

The Influence ofReynolds Number onAerodynamic Data

The effec t o f reducin g th e Re on the aerodyn am icprop erties of wings is usually to mak e them worse . Froma very high Re of 10 million down to a Re of 0.5 millionnearly all sections work prett y well. As Re is red ucedw ithin thi s ran ge , as a ge ne ra l rul e , profil e d ragincreases slowly but ste ad ily , and the sectio n 's CLm " ,

reduces gradually.At very low Re , say 10 or 20 thou sand , most sections

will hardl y w ork at a ll, givi ng a disappoint ingly lowCl m", and very high drag suggesting laminar se parationon the top surface.

The Problem Area

Somewhere between these two ex tremes each sectionseems to have what you might ca ll a "Critica l Re Band"above which it o pe ra tes quit e normally and belo wwhi ch it is virtuall y useless.

Within th e c ri t ica l Re b and th e aerody na m icpropertie s o f th e sec tio n ca n vary drasticall y a ndsudde nly as illustrate in Figure 17.6 whi ch is fo r a typ icalmod ern section . Not all sections show the same patte rnof variation.

At a Re of 200,000 the lift curve and the drag polar


Figure 17.6

85,000

200,000 A. --- • <,B

• ;(.::.• 100,000 •

//. .• :85,OOO! •

I : T D ,ez : -:..---- --i 80,000,.....

/'

200,000---....100,000 - -.... ...........

~

60,000--80,000

<,

-- .

.... ./

..... r ::.... -.:

:/" ,.. ,I

I\ ,\ I.\

/~_ . ~

. ..~/ :'

I •

........

are bo th qui te normal. At this Re and above almos t "fullsize" performan ce sho uld be achievab le.

At 100,000 the d rag has more than doubled over mostof the range and the lift has reduced at eve ry angle ofattack . The zero lift ang le of attac k has reduced by adegree or so , but the curves are both a normal shap e(apa rt fro m that odd bit around the sta lling ang les ofa ttack). Th e pe rfo rmance ava ilable in terms o f g lidean gle and d u ra tion will be reduced , but at le ast thehandling will be fairly norm al.

At 85,000 the curves are fairly normal for low ang lesof attack. At interm ed iate angles o f attack lift productionfalters associa ted with a sha rp rise in d rag indicatingtrou ble in the bounda ry layer , and prob ably producing

handling peculia rities , but at higher angles of attack thedrag red uces to acceptable levels and the lift ge ts backon the job suggesting a turbulent boundary layer ge ttingthin gs stuck back down over the top surface again.

However at 80,000 there is no such d ramatic rescu e .Lift production star ts off ba d and ge ts stea dily worse .Dr ag is e normous and th e Lift/Drag ra t io has beenassassina ted . A revolution has taken place and it too konly a tiny change in Re to effec t it. Performance is ve rypoor ind eed and as for handling , who cares anyw ay?

At a Re of 60 ,000 the deterioration is even worse .This is bel ow the critical band whi ch I menti on ed andth e sec tio n is usel e ss for g liders. Flight w o u ld bepossible with eno ugh engine pow er.

Figure 17. 7 Figure 17.8

D

Higb lVeight

v- - ,, -

LowlVeigbt

++

• +J---1--e


The Hysteris LoopFigure 17.7 shows the lift curve for a Re of 100,000

repeat ed from Figure 17.6. No tice th e "extra" po rt io nund erne ath . As ang le o f a ttack is in c rease d th e liftcoefficien t fo llows the top curve all the way up pastpoint A and on to B. If, at any poi nt be fore B is reach ed ,a is red uce d agai n, then CL follows the sa me line backdown . Ho wever if a is increased past B then th e CLd rops abr up tly, down to the lower line at C. At the sametime the drag increases enormo us ly, so much that it goesright off the grap h. Somewhere in th e boundary layerthe flow has se para ted drastically. If a is now decreasedthe flow stays se pa ra ted and Cl. follows the lower line toD. Th en so me how the flow re-attach es itse lf, Cl. zoomsback up to point A o n the origina l line , the associa teddra g retu rn s to a mo re reasona b le a moun t, an deverything is back to normal. Thi s "one way sys tem" o na grap h is ca lled a "Hyste resis Loop" an d often occurs inthe critica l Re ba nd . It is likely to have stra nge effec ts onthe handling of the mod el aro und the sta ll.

On a roug h, d irty or inaccurate ae rofo il the loop willdisappear and revert to the lower line , or even the linefo r Re of 80,000.

Effect on Model Design andPerformance

Th e Re ban d within which a sec tion's performan ce willbe se rious ly affected depends up on its thickn ess, and itsca mbe r. Ge ne rally, the more the thickness or the mor ethe camber, the highe r the Re at wh ich the sec tion sho uldbe ope rated . For example, the \X1or tmann section FX60126(ca mbe r 3.9%, thickness 12.6%) has a similar drag polar ata Re o f 60 ,00 0 as d o e s th e FX63 -137 (ca m ber 6% ,thickness 13.7%) at 100,000.

Wing Tips

Th e wing tip on a tap ered wing is flying at the sma llestRe. The bes t taper rat io may be more than the theoret icaloptimum of about 0.45 because reducing the tip Re co uldincrease profile drag more than the redu ction in induceddrag ac hieved b y ap p roxi ma ting an e llip tica l wi ngloading.

In add ition , the tip sec tion will ge t into its critical Reba nd first as the aerop lane is slowed down, and Re isredu ced . Indeed , because the lift loss an d drag rise canoccur qu ite sudde nly over a small cha nge in Re, it may bethat it w ill hap pen on one side before the other showingall the signs of a tips tall. We cou ld cha nge the tip sec tionto one with less camber and thickn ess to raise the criticalRe of the tips. However that goes agai nst the adv ice inChap ter 7 (Fig 7 .10) to increase tip ca mber to avoidtipstall. It depends a great deal on Re , h ut read on a fewparagraphs.

Class Rules

Competition rul es can radi cally influe nce design . Alarge aspect ratio is usually good for performa nce , but iffo r exa mple the rules limit the wings pa n, the n a high ARwi ng wo uld have a very sma ll chord an d a sma ll w ingarea and therefore a high wing loadin g. The problem witha tiny chord is a very low Re, at w hich most sec tions w illnot perfor m a t a ll. T he o u tco me is th at g lide rscompromise on a lower AR giving a large w ing area, and

94

hen ce light load ing, cou pled with a reasonable ope ratingRe.

Optimum Weight

In Cha pte r 13 I sai d that ballasting a glide r wou ldalways give th e sa me best glide ang le, and a h ighermin imu m sink rate , and I illu str ated it in Figure 13.9,which is reproduced as Figure 17.8. However now I haveassumed th at th e aerofoi l w ill not work bel ow a Recorres ponding to speed "V". Co nseq ue ntly the pointswhe re the theoretical best glide ang le and the minimu msink are not ava ila ble . No w th e best glide an gle andminimum sink occur at the high er weight. I am no t sayi ngthat ballasting is likely to improve a glide r's pe rforma nce ,just tha t it is possible in certain circums tances.

Turbulator Strips

Turbulator strips are very sma ll steps or irregul aritieson the surface of the wing , usu ally between 5% and 25%of the chord back from the leading edge on the uppe rsurface. Their purpose is to make the laminar boundarylayer become tu rbule nt before th e norm al se pa ra tio npoi nt, in the hope that it w ill then not separate at all.Thei r use is no rma lly restr icted to glide rs or free fligh tdura tion mod els, and especially on outer panels.

Th e effec tive ness of the te chnique is illu strated inFigure 17.9. The effect of one strip of adhesive tape 2 mmw ide and 0.5 mm thick at 20% chord is shown by thelines wit h cross-ba rs . At a Re o f 60,000 the sectio n 'sperformance has been tran sformed . It has a res pectableLift/Drag rat io agai n , rat her th an an ab rup t sta ll, butnormal han dlin g othe rwise . Before yo u rush off to sticktape all over your wings, rem ember that this is just a br iefmention of a sub ject w hich co uld take up a book in itse lf.The turbul ator sho uld not be regarde d as a pan acea to beused indiscrimina tely on all w ings in all circu mstances. Itcan have an amazing effect on so me sec tions at ce rtain Rebut tur bulators can have an adverse effect.

Notice that at a Re of 200,000 the turbulator strip has adet rimental effect, causing an earlier stall and more drag .Also there is a lower limit to the Re at which they wi llwor k.

Surface Finish

It is noticed in practice that a w ing with a rough oruneven surface finish so me times has a better pe rformancetha n a perfectly smooth wing .

The rough texture , o r irregul arities in the surface, willturbulate the boundary layer just like a turbul ator strip .

However it is indiscrim ina te in its ac tion . Altho ug hbu ilding in tur bu lence co uld so me times help, yo u mayfind that so me of it, or even all of it, is having an adve rseeffect. I suggest that it would be better to build the wingwith as acc ura te an aerofoil, an d as smooth a surface , aspossib le. The n you can expe rime n t wi th s tick o nturbulator strips w hich can be moved or rem oved at w ill.

Using Published Data

There is little point in looking at aerodynamic da ta onw ing sectio ns unl ess it is for th e co rrect Re , and yo ucanno t interpolate grap hs . If yo u wa nt data for a Re of150,000 bu t have gra p hs only for 100,000 and 200,000 ,


Fig ure 17.9

..-/

/

./

/

//

//

//

a

I· ···H··+· ··I

R e 200, 000 Clean

R e 200, 000 Turbulated

Re 60, 000 Clean

Re 60, 000 Turbulated

you cannot just draw a line halfwa y between the two.The graph for 150,000 cou ld be anyw he re between theothe r two.

Theoretical Data

In the world of full size aircraft, pe rformance can bepredicted with amazing accuracy by computers, thoughthe test p ilot a lways has the fina l say. Drag polars formo de l aerofo il sections at specific Re can also be workedou t by computers . How accura te the res ults are is open todeba te . Th ere is often a good correlati on w ith windtunnel tests at Re above the critica l band but within it thecomputers seem to get a bit optimistic.

Further Reading

"Model Aircraft Aerodynamics" by Martin Simons givesa m u ch more detailed account of th e problems ofaerofo ils at low Re.

"Profilpo laren fur den Modellflug " by Die te r Altha usand "Airfo ils at Low Speeds" by Selig Donovan & Frasercontain resu lts of many tests of useful section at relevantRe.

Basic Aerona utics/or Modellers 95

Aeroelasticity

Chapter 18

so far I have been working on the assumption thatthe struc ture of the aeroplane is co mple tely rigidand will not deform at all un der the ae rodyna mic

load s. Th is assu mption is pretty un iversal in that nearlyall the books and articl es on ae rody na mics w hich youare l ik el y to co me across w ork u n de r thi s sa meassumpt ion , even th ough th ey d on 't a lw ays say so .However it has to be admitted that no structure is to tallyrigid so sh ould we th rowaway ae rodynamic theory aswe know it? Have yo u co mple te ly w asted yo ur timereading abou t aero dy na mics up to thi s stage? I don 'tthink so.

Most of the time this assumpt ion is perfectly jus tifiedand ca uses no measu ra ble error. As long as yo u areaware of it, yo u can ign ore the flexibility of the struc tureuntil it causes p roblems. There are ce rtai n areas inwhich structura l flexibility is kn own to cause undesirabl eeffec ts and I intend to me ntion them briefly.

1. The Effect on Stability

Wh en I exp lai ned Longitudin a l Sta tic Sta b ili ty , Idescribed an aeroplane meeting a d istur ba nce w hic hincreases the ang le of atta ck, and so also the lift, of itswing an d its tail. The lift increases both ac t at ab ou t theq uarter cho rd p oints of th eir respe ct ive s u rfaces ,w ha tever the sect io n . The res u ltan t o f the tw o liftincre ases ac ts throug h a point ca lled th e NEUTRALPO INT and its mome nt about th e CG p ro vid es th esta bilising mo me nt on the aeroplane. The distance of theCG ah ead of the NP is a measure of the stab ility of theaeroplane.

Now imagine w ha t th e effect wou ld be if th e ta ilwe re mo unted at the end of a long flexible tail boom.The lift increase on the tail would bend the tailb oom u p(as in Figure 18.1). The tail 's angle of attack is reducedso the lift inc rease w ill no t now be quite as b ig as if thetail had been rigid . Co nseque nt ly th e resu ltant fo rce

Figure 18.1

throu gh the NP whic h stabilises the aerop lane w ill befurther forward , i.e . nearer the CG, than was calc ulatedusing rigid theory . The ae roplane has a little less stab ilitythan was calc ulated.

The Torsional Axis ofthe Wing

Wing flexibil ity can also have an effect. As I am sureyou can imagine, an upward force on the lead ing edgeof a w ing will be nd it upwards AND tw ist it trailin g edgedown. Similarly an upward force at the trailin g edge w illtwi st it lead ing edge down . Somewhere be tween th eleading an d tra iling edges there is a line ru nning fromroot to tip called the "torsional axis" (o r flexura l axis ore las tic axis) through w h ich a force w ill jus t ca usebending, wi th no torsion. It is the line about w hich thestructure tw ists . Th e positio n of this line will dependu pon the wing 's s tr uc tu re, b ut w ill be somewherebetween say 15% and 50% of the chord fro m the leadingedge.

How Tw ist Affects Stability

The lift increases du e to a disturbance w ill ac t at thequ art e r chord p o int an d if the torsio na l axis of thestruc ture is we ll aft, it w ill tend to twis t the w ing leadingedge up (see Figure 18.2). The w ing 's lift increase w illtherefore be a little larger than forecast and the NP w illbe fur ther forw ard than calculated for a rig id wing. Thew ing's flexibili ty in torsion has tak en away so me of theaeroplane 's stabili ty. Th e problem is at its worst if thetorsional axis of the wing 's structure is well aft and thewi ng is ve ry flexibl e in torsion. The wi ng sectio n doesn 'tmatt er.

The wors t case is p ro ba b ly a comple te ly ope nstruc ture with just lead ing and trailing edges, main sparsat 30%, and aft spa rs at 70%, covered in p las tic film. Asymmetrical sectio n is just as affec ted as any other.

XL XL(fIe.~) ( l"ig ld)

Datum- line -

96

NP {flex]

- - - - - - - - -


The SolutionThe CG for mulae in

Cha p te r 8 ass u me afairly rigid structure. Theforward shifts of the NPmentioned above (notusu all y s ign ifica n t ex cept in the case of h ighaspect ratio wings) ca nbe compe nsa ted for o nfle xi bl e mo d el s b ystarting off with the CGslightly furth er forwardth an yo u h a ve ca lc u la te d. How mu chd e p end s upon yo u rassessment of the flexibil ity o f yo ur model 'swin gs and tail (say 2 to5 per-cent ").

Figure 18.2

XL If'

Increased by twist

------- - -- - -- ---

--- ----

XLFlexible

XLRigid

\ '

Figure 18.3

---

-- -

- - - - -

spectacular whe n it happen s. Figure 18.4 represe nts anaeroplane in normal flight. The torsion al axis of its wingis qu ite far aft , say 40% o r 45% o f the cho rd . So med isturbance causes the wing to twist very slightly leadingedge up whi ch will increase the win g lift at the qua rte rchord po int. Th e wing 's torsional stiffness w ill try totwist it back to its origina l position , and normally does,but the ex tra lift, XL, is trying to tw ist it eve n furth erleading edge up .

The restorin g moment from th e wi ng 's stiffness isco nstant, regardl ess of speed , but the ex tra lift increasesas the squa re of the airsp eed . If the aeroplan e is flownfast enough th is latte r mom ent will win. It will make thewing twist leading edge up inc reas ing its angle of attac keven furth er , wh ich inc reases the ex tra lift even more ,which increa ses th e twi st even more a nd th e w ingrap id ly twi sts lead ing edge up until it e ithe r sta lls orbreaks in bending o r twists right o ff. Th e initial slightdisturban ce rapidly be comes worse and worse which Isaid in the cha p ter on Stability is called DIVERGENCE.

Wing divergence usually only happens in a stee p d iveand ca n affec t w ings w ith ca mbered, sy mme trica l, orreflexed sections. It is a speed dep endent probl em , i.e.there is a critica l sp eed called the WING DIVERGENCESPEED above whi ch it will happen . The struc ture mostat risk is a high aspect ratio open structu red wing w ithjust a main spar, aft spa r, and leading and trailing edges,covered in pla stic film.

To r si o nalAxis

XL (Twist)

tXL (Aileron)

=-=-=--=--==----=--=-=-::-:-~/

Win g di vergence is an un common p ro bl em , but

Wh en a n a ileron isd efl e ct ed d o wn , theex tra lift ac ts we ll backo n the wing. It is likelyto act aft of the torsion alaxis of the structure inwhich case it will twistthe wing le ading edgedown reducing its angleof a ttac k and hen ce itslift (see Fig ure 18 .3) .The redu ction in the liftmi ght turn out to bemore than the increasein lift from the aileron deflection so the wing as a wholemay lose lift causi ng a rolling mom en t in the oppositedire ct ion to th at intend ed . For si milar reasons th eupgoing aile ro n may ca use a n increase in lift o n itswing . The ailero ns seems to have operated in reverse,hen ce the ex pression "Aile ro n Rever sal ". The furtherforwa rd the torsional ax is, and the weaker the struc tureis in to rsi on , th e more prone it will be to a ilero nreve rsa l. The wing section used makes no differ en ce atall.

Because the twistin g mom ent o n the w ing rises as thesq uare of the airspeed , but the stiffness do esn 't , theailero n reversal will ge t worse as speed rises. In fact ato ne part icu lar speed the aile ro ns will have no effec twhatever. As speed rises towards this critica l speed theailerons gradually lose their effectiveness , whil e above itthey will act in the opposite se nse to that inten ded . Thiscritical speed is called the "AILERON REVERSALSPEED".Aileron Reve rsa l is a speed dependent pr oblem. Allaeroplanes with ailerons wi ll have an aileron reversa lspeed , the trick is to mak e it faster then the ae rop laneca n fly. Th is ca n be don e by increasing the torsion alstiffness of the win g and/or by having the tor siona l ax isnot too far forward .

2. AileronReversal

3. Wing Divergence


Inertia

Inertia

c

dive, co ming in for a high speed pass wh en you hear it.BRRRRRRRRRR BANG! a nd o ff comes a p ie ce , anailero n, an elevator, the tailp lane , or the rudder. I haveeven heard of a case where the vibration was so severethat the who le wing dis integrated in a shower o f whitefoa m beads . Not a p retty sight!

Fl u tter is a viole n t osci llation of p a r t o f th eaeropla ne 's str ucture, often, but not always, lead ing tofailure of pa rt o f that struc ture . There a re many bitswhi ch can start to flutter and I sha ll br iefly mention the

common o nes . \X'hateverit is that is flutterin g, it isa s peed d ependen tproblem . That is there isa c rit ica l s peed (cu nn ingl y ca lle d th eFLUTTER SPEED) abovewhich that bit will startto flu tte r. If yo u dohap pen to ge t nutter atan y time, cut the throttlean d climb immediat el y(if not sooner) .

Le t m e s ta r t witha ile ro n fl utter w h ichw rote o ff o ne o f m ymo d e ls so me yea rsback , and caused me tolo ok u p th e the o ry .There are tw o mod es ofaileron flutt er, be nd ingand torsion .

Th ey have the sameresu lt a nd th e sameremedy an d may evenocc u r to geth e r but Isha ll de scr ibe th e waythey happen separate ly,starting with bending.

Some thing, ma ybe agus t, s ta rts bot h w ingsbend ing up . See Figure18.5. The bending stiffness of the wings stopstheir u pward motion atp o s iti on A b u t th eaile ron s, bein g attache db y s lig h t ly fle xi b leli n kag es , d on 't s to p .The ir CG is behind thehinge line and so the irin erti a ca rr ies th e m alitt le furth er u p topositio n B. In th ispos itio n t he ai lero nsca use a loss of lift whichlet s the wi ng 's stiffnessbri ng the m down, pastthei r origina l position, toposi tion C w he re th e irin ert ia m a kes th emdroop a little to positionD. In this position theyinc rease th e lif t of th ew ing a nd it s ta rtsco ming up aga in. When

c

.............

.... ..7"..:-:-.. ::::.:..::: --=--.."....... "'0:::-:.-- .......

..... ......... <. ----.' ..........: ..

".'.

--- <,- ......-- ~ ..

XL From twist due to

.....XL From initial twist

Original L

XL From twist due to

' .. ' .--

ttt

--

....... ............. ............ --.

Figure 18.4

Your mod el is really goi ng we ll, full pow er, a bit of a

Figure 18.5

B

A

c

D

Wing divergen ce ca n be avo ide d by mak ing the wingstiff in torsion , moving the to rsio nal ax is forward nearerthe aerodynamic ce ntre or sweep ing back the wings . Ifthe wings are swe pt back the lift increase near the tipsten ds to twi st them leading edge down whic h helps toavoid the problem .

4. Flutter


Mass balance bonis visible under the uiings oftbis typical trainer style model 011 floats.

the wing at last s to ps go ing u p aga in, th e a ilero nsovershoot to positio n B again and the who le p rocessrepea ts itself.

The torsion al mod e of flutter sta rts whe n an ailero n'sposition is d isturbed. In Figure 18.6 the ailero n has beend isturbed down ward slightly in position A. In Cha pter 10

Figure 18.6

Drooping aileron makestoing twist about its

_______________ tors1011a I a xis.TOI'sIOlUlI AxisA--+------~m--=--:-----=--:---;--------

Inertia

-----~Tbe aileron's

inertia carries itfurther lip.

Aileron's in ertia carries it_....I..:::-- ~¥t_----------=:::..--===------~II!.:'..:.·t.!"b~e~I_:· d~owll tobicb will make

E it twist 1I0se doum again . . .

Inertia


Figure 18,7

2

3

Mass ba la nce --...-::insid e Aerodynamicbalance

I1

5o

Set back binge

-:-...SlI·ipof

solder o rp iano uiire L----' --~

8Set backbinge line

9

elm be u sed OIl elenator, ailel'OIlL J.----.l01' rndder.

]

I sai d th at a d o wng o in g a ilero n increa se s li ft andincreases Mo the nose down mo ment. This nose downmoment twi st s th e w ing tr ailing e dge up a bout itstor sional axis.

\\! hen the wing 's tor sional stiffness sto ps the twi sting,p o siti o n B, th e a ileron ove rsh oots to a n up w ardd efl ect ion , po siti o n C, becau se its CG is beh ind th ehinge lin e . In th is position the moment o n the wing isreduced , it untwists , and when it sto ps in position D, thea ile ro n 's inertia again ca rries it th rough to position E. Weare back where we started and the whole cycle be ginsagain.

Damaging ailero n flutt er is not inevitab le though , thedisturbance usually dyin g out du e to damping. However,as I hav e alr ead y sa id, the angular deflections dependon st iffnes s (whic h is the same at any speed) but theforces they produce inc reas e with speed sq ua red . If theaero p la ne is fly ing fast enough , a bove its AILERO NFLUTT ER SPEED , the wh ol e pro ces s wi ll b e selfper petuating and ve ry rapid ly develop into a po tentiallydamaging vibration . Th e pilot mu st immediate ly kill thespeed by closing the throttle , increa sing the dr ag andpointi ng th e ae ro pla ne up . This is th e most co mmo nfo rm of flutter o n model aeroplan es .

Curing the Flutter

Fro m the above two descriptions yo u ca n see th ats tiffen ing the w ing in bending and torsion will he lp byreducin g the amount that th ey flex up and down or twistand stiffen ing the ailerons and their linkages will help by

reducing the amoun t that th ey will flap up and down,but both th ese solutions w ill just increase the flutterspeed , no t remo ve it. All aerop la nes, howe ve r s tif f,howe ver pe rfect their co ntro l linkage s, will have a flutterspeed and the pilot must ensure that he never, ever, fliesabove that speed .

There is an o ther simple so lution, used o n pra cticall yall full size aircraft since abo ut the th irties . Th is so lu tion ,known as MASS BALANCING, is ac hieved by moving th eh inge line of the aile ro n back or fixing w eights to thea ile ro n a head of it s hinge line or a com b ina tio n o fthe two as in the exa mp les in Figure 18. 7. If the CG ofthe aileron is ex ac tly on its hinge line, its inertia cannotm ake it o vershoot th e position held by it s co n tro llink ages and flutter is eliminated .

Su ch perfe ct b al ancing is u nne cessary , a p artialba lan ce being enoug h to raise the flutt er speed out ofreach . The mass bal ances ma y b e hi dden in side th ehollow w ing tip as on the Piper Cherokee. If the co ntro lsurface has an ae rodyna mic balance (s ome area aheadof the hi ng e line ) to reduce th e contro l for ce s , thi s isoft en a co nve nien t location for some ma ss balancin g aswell.

Beware though . If the mass is attache d to the con tro lsur face by a flimsy piece of wir e , the wire will bend andthe mass will re ma in stationary and be co m p le te lyineffective as in Figure 18.8.

Wing Flutter

I bet someone has thought of a nea t wa y to e liminate


Figure 18.8

--- ••••• . . .. . .. :- # ..

c --~~--

a ile ro n flutt er , lea vin g o ff th e a ile ro ns . So rry todisappoint yo u but yo u ca n still ge t wing flutte r. It isuncomm on I th ink, but it happens when a wing has itstorsional ax is we ll ahead of its CG. If the wing in Figure18.9, position A, is disturbed to give it a small ang le ofattack increase, the ex tra lift will make it bend upwards.

Wh en its be nd in g s t iffness s to ps it s up w ardmo vement the CG, be ing behind th e torsio na l axis ,overshoots to position B ca using a reduction in ang le ofattac k es pecially towards the tip . The loss of lift allowsth e wi ng to drop to position C where its inertia hastwisted it lead ing edge up again. The cycle repea ts e tc.,e tc . and if th e speed is h igh e no ug h becomes se lfsustaining flutte r.

The mot ion is a co mbina tion of be nding and torsion.This wing flutte r can happen to wings wi tho ut ailerons,and it ca n also happen to wings wi th aile rons , evenperfectly mass bal an ced ail e rons. Indeed th e aile ro nmass balances can make wi ng flutt e r wo rse as th eymove the CG of the who le wing aft.

Wing flutt e r ca n be cured by moving the torsio nalax is of the w ing aft and the CG o f its struc ture forward.

co ntrols, hin ged at its qua rter chord poi nt but with itsCG co mmo nly nearer 45 per ce nt ch ord . Cut light en ingholes aft of the hinge an d add lead to the leading edgeto move the CG forwa rd to the hinge line.

Footnote

With regard to flutter, there are man y reported cures,a few of wh ich run co ntra ry to the theory. Perh aps so meof what is diagnosed as flutter is really so mething else,like a sympa the tic vibra tion.

By all means try on e o f the o ld wives rem ed ies, it justmight work, bu t first ask yourself o ne qu estion , "Do Ifeel lucky?".

Tail Flutter

Loss of lift due to twistlets uiing descend

Tip

View from front

Root

c ~

B

A

Lift increase from twist startsInertia wing going up again

Inertia

Force from bending stiffnessstops upward motion

Force from bending stifness stops doumuiard motion

XL starts to bend unng up

Figure 18.9I ex pec t yo u h ave

guessed that if the tail isflexible o r is mountedo n a fle xibl e fuse lagethen flutter of the othercon tro l s u rfaces , th erudder and eleva to rs, islike ly at hi g h speeds .You are right. Althougheleva tor or rudder flutteris le ss co m mo n o nmod el s th an ail e ronflutt e r be aware that itca n ha ppen . Secu refi x in g of th e fin a ndtailpl an e to the fuse lageis essential. Mass balan cing of the elevator andru d de r wi ll c ure th eproblem and is standardpract ic e o n full s izeae ro p la nes . Th e sa mething ca n happen to anAll Mo ving Ta il (orfore plane) , with flexible


Chapter 19

Tuck Under

As I sa id in Chapter 5,the forces on a normallycambe re d wing can berepresented by a liftforce th rou gh the aerodynami c ce ntre (q uarte rc ho rd p o int ) to getherw ith a mome nt Mowhich acts lead ing edgedown . This mome nt willtend to twis t th e w inglead ing edge down. Thestructure of the wing willtherefore acquire a twistsuc h that the w ing tipsw ill develop "washou t"relative to the roo ts . Thetwi s t wil l be w orst onhigh as pect ratio wings.

1. Willg Twist

Th is incident made me think in so me depth about theprobl em . Figu re 19.1 shows the tailp lan e angle to trim forthe full range of lift coefficients , from Chapter 12. If theaeroplane is speed e d up and th e lif t coefficie n t isreduced , to keep lift equal to wei ght , then basic theorysays that the aeroplane will be trimm ed tail heavy andd own e le va to r trim m ust b e a p p li e d to keep theaerop lane in trim. Failure to put on down elevator meanstha t the aeroplane will zoom nose up . The greater theStability Margin, the ste eper the graph and greater thenose up mom ent will be . There is no way that the usua lSimplifie d th eory o f aero dyna mics co u ld allo w tu ckund er to happen . Since I had just proved that it doeshappen th e a ns w er has to be th at o ne o f th e threestandard ass ump tions of simp le aerodynamic theory hasle t us down. It cannot be Compressibility, it is un likely tobe Reynolds Numbe r, so it must be Flexibility.

If dis tor tio n of the aerop lane 's structure effect ive lyreduced the ang le be tween the wing and tail just likewhe n you apply down elevator, then like down eleva torit would ca use a nose down moment , and if it weresevere eno ugh it co uld overcome the restoring mom entfrom the stability. I ha ve co me to the co nclusion that

there are several ways inw hic h structural fle xib ilit y w ould ca use ac ha nge in tai l settingang le le a d ing to tu ckund er.

The Villa i« Unmasked

Low SpeedH igb Speed

up Elev

Tri mP O SIl

Tail Angle to Trim

Fig u re 19.1

Some years ago, I was test flying a high aspect ratiopow ered mod el of my own design . The low speedhand ling was fine so I open ed the throttle fully to

see how fast it would go . As the aeroplan e began toaccelerate the nose started to rise so I app lied four clickso f down tr im to maintain leve l flight. As th e spe edincrea sed furth er it appeared that I had overdone thedown trim and I had to apply four clicks of up trim. Eventhat wasn 't enough as the nose co ntinued to drop. I wasback to the trim position for level flight at low speed andhere it was still putting its nose down. It could no t beca use d by excessive downthrust as the initial tendencyhad been nose Up. Then I realised . This was it! This wa sthe dreaded Tuck Under.

I found that , starting from trimmed flight at whateverspeed, if I applied down eleva tor to start a dive , thereca me a point whe re the model just kept stee pe ning thed ive of its own acco rd , even whe n I re turned th eelevator to its or igina l position. The slower the aeroplanewas !lying to begin with , the more dive it needed tomake it tuck un der. It wou ld even do it fro m glidingfligh t, bu t each time it recovered w he n I closed th ethrottle and applied full up eleva tor. Until that is, on e dayI made it tuck under with the rate swi tches inadve rtentlyse t at low . This time the dive was te rminal.

102 Basic Aerona utics fo r Modellers

Figure 19.2

----------

A

I Bquiualent t»

----------==~

True Situation

---------

_____~E::::q......iualentTa~

Tip

-----_T_-_- itlJist-~---


C=--?Sf~¥JLT

As the aeroplane flies faster the twisting moment willincrease as the square of the speed. That is, when youdouble the speed you get four times the twist. The effectof Mo pushing the nose of the aeroplane down is not theproblem. That ha s been taken care of because therestoring force from the tail also increases as the squareof the speed. The problem is that the wing twist getsworse as the speed increases.

This speed dependent washout can be consideredequivalent to the entire wing twisting as a rigid unitrelative to the fuselage by an angle A in a leading edgedown direction. See Figure 19.2. The greater the momentMo and the less the wing's torsional stiffness, the greaterwill be the angle of twist A. The effect on the aeroplane'strim of the wing twisting leading edge down will be thesame as if the tailplane were rotated leading edge up byangle 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 trimmedfor a fairly high speed will have a download on the tail.The download increases greatly as the trimmed speed is

increased. If the tailboom which carries this download tothe rest of the aeroplane is flexible , it will deformdownwards tilting the tailplane through angle B whichwill be proportional to the tail load Lr and inverselyproportional to the bending stiffness of the tailboom. It isjust as if an angle of down trim B had been applied butwithout moving the controls.

3. Flexible Controls

Most models have a tail arra nged such that thepushrod, or Bowden cable (snake), has to "push for up".That can cause problems when it is connected to an allmoving tail with its pivot well ahead of its aerodynamiccentre (25% of its Mean Chord), as in Figure 19.4.

In high speed flight there will be a downforce on asymmetrical tail at the 25% chord point. This force willput the pushrod or snake into compression. Unless it isvery stiff, a pushrod in compression acts a bit like aspring and it will bow, or buckle, under the load . (Figure19.5) The problem is made worse by using long wireends with Z-bends to clear obstacles or exit the fuselage.The greater the compressive load , the more it will buckleand so effectively shorten. Unless it is well supportedover its full length, a Bowden cable will also bow, and so

Basic Aeronautics/orModellers 103

shorten, under a compressive load. Because of thelayout, shortening the pushrod will reduce the tail settingangle by an angle of downtrim C, shown in Figure 19.5.The faster the aeroplane is flown, the more thedownforce on the tail will be and the more theshortening of the pushrod. Therefore the angle C willincrease with speed.

The Elevator Trim Graph

The twisting moment on the wing and the downloadon 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 thepushrod all added together, increases as speed squared.

The pilot will have to compensate by applying a smallamount of up trim at low speeds, and more and more uptrim as speed increases. At the aeroplane 's terminaldiving speed the trim change due to flexibility will bevery large indeed. The trim graph will become distortedas shown in Figure 19.6. The theoretical trim line fromthe formula is shown as a broken line. To this, an uptrimcorrection will have to be added as shown to producethe trim line of a flexible aeroplane.

The aeroplane still behaves quite normally at liftcoefficients from A to B on Figure 19.6. To avoid havingup trim on all the time it would be best to change thetailplane setting, or the reference position of thetailplane, as shown. This adds a third part to the tailsetting angle equation, 12.2.

Figure 19.5

-------- ----- ---

Pushrod Bowed

Pushrod Compression

Figure 19.6

Tail angle to trim

Trim Line<, (Rigid Theory)

.......

Neutral

Elevator

NewNeutral

/AngleofFlex

HighSpeed

Trim Line (Flexible)

Low Speed

------- ....CL

)up

A


The Critical SpeedThe airs peed corresponding to the lift coefficient at

po in t B o n Fig ure 19 .7 is th e Critica l Speed of theaeroplane. If the aeroplane is in trim at its Critical Speed,ANYslight speed increase beyond this speed will lead to atuck und er. In Figure 19.8 I have drawn another graph ofelevato r trim position but against airspeed this time. PointB is aga in the Critical Speed .

Tail Area Influence

It may seem strange to you, it did at first to me , but theCritical Spee d dep end s on the size of the tailplane. TheTuck Unde r is caused by structural flexibility changing theangle be tween the wing and the tail. For a given ang le offlex, the bigger the area of the tail, the mor e moment itwill have abo ut the CG and the lower the speed at whichit will overcome the stability. Tha t means the bigger thetail, the lower the Critical Speed will be. The tail has no tcaused the tuck under, in the same way that you cannotblame the gro und for caus ing a crash jus t because it isthere.

Stability Influence

As I said in the beginn ing, a rigid aeroplane ca nnottuck und er because its stability will not let it. Similarly thestability of a flexible aeroplane will try to prevent it fromtucking under. The mo re the speed devia tes fro m thetrimmed speed the more the nose up pitching momentfrom the stab ility increases, BUT the nose down momentfro m th e fle xibility inc reases muc h more rapidl y asillustrated on Figure 19.9. Where the two lines cross ove rthe aeroplane will aga in be in trim, but the slightes t furthe rinc rease in speed will leave a nose down net mo mentwhich will cause a tuck und er. However the mor e Stab ilityMargin an aerop la ne has , th e stronger th e re co ve ry

Figure 19.7

moment will be and so the faster it must be going to tuckunder. In ot her words the further forward the CG, themore Stab ility Margin, and the higher the Critical Spee d.

Tuck Under Speed

Fro m Figures 19.7 an d 19.8 you ca n see that if anae roplane is in trim at so me speed, point C, speeding it upa little will leave it trimmed tail heavy. If down trim is no tadded it will pitch nose up . However if it is speeded upbeyond point D it would become nose heavy again.

If enough up trim is not adde d it will tuck und er. Itherefore call the speed at D the "Tuck Under Speed". It isnot a fixed speed, it depends on the initial trimmed speedof the aeroplane . The further the trimmed spee d is be lowthe critical speed (B), the further the tuck under speed willbe above the Critical Speed, bu t eve n if it is trimmed to flya t its minimum s peed, o r even w ith fu ll up elevatorapp 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 underspeed it will try to stee pe n the dive but ap plying full upelevator will pull it out. However if the model acceleratesaway above the tuck un der speed , the nose down netmoment may be so grea t that even full up eleva tor (onhigh rates) is not eno ugh to pull it out (see Figure 19.8and 19.9).

Tha t may be beca use the servo is not strong enough, orthe pu sh rod is be nding. Or there isn't enough trave l onthe tailplane or elevator. Or if the tailplane or elevator istoo small, it may be incapa b le of develo p ing eno ug hdown ward lift. Increasing the up movem en t on an allmoving tailplane too much will just let a small tailplanestall at a nega tive angle. See Figure 19.10. Just before itreaches its stalling angle the tail will be developing its

DOWII

Elev

Neutral

upElev.

B

-~:~:~)~~I

I

CriticalSpeed CL

Vertical Dive

A

Basic Aero na uticsfo r Modellers 105

Figure 19.8

FullDowll

Full Up A

,., ,., B- --

Rigid-I

II

I V

Max IIII

Flexible I

II

I

II

maximum downward lift coefficient. If that is not sufficientto pull it out of the dive , it is doomed. Its only chancewould be to bunt right round into climbing flight inverteduntil the speed red uces.

Then . .. get out of that!

V

TailplaneInstability

Suppose something,like a gust, deflects theAll Moving Tail in Figure19.11 leading edge up bya small angle. The pushrod will be compressedand it will try to springback and return the tailto its original position.This restoring force fromthe pushrod will beindependent of airspeed.However the addi tio na llift XLT caused by th edisturbance will tend toro tate the tail the otherway, leading edge up,and this destabilisingforce will increase as thesquare of the speed. Ifthe speed is high enoughthe des tab ilis ing forcewill win. The tail will fliplead ing edge up until itreaches the fu ll "down"travel stop. The result isobviously a tuck un der,or even an outside loop.All the up elevator youcan apply on the servo

will disappear into further bending of the pushrod. ThisTailplane Instability Speed, above which the pushrod isincapable of preventing the tail runaway, can be increasedby stiffening the pushrod, reducing the tail area, increasingdistance y, or best of all placing the tailplane 's pivot on, or

Tuck Ullder

Tuck Under Speed

II

II

Recover I

III I--......~

TrlmSpeed

Pitching Momellt

Figure 19.9

106 Basic Aerona utics for Modellers

y

Pivo t (well aft)

=- x

Fig ure 19.10

Airflow

- - - --

- - - - - - _.~__ I

....

IIIII

I I

- "'t'!,J\~1

Pusbrod Bent a nd Shortened -..: :~

Figure 19.11just ahead of, its aerodyna mic centre. The restof the aeroplane doesn 'tcome into it. If th eaero plane is be ing flownfast, above the tailplane 'sins tability speed, therewill be a download onth e tail h olding thepu sh rod in ten sion andso there is no probl em ,rega rdless of the speed.However what happen sif a glitch, or the pilot forso me reason, put s on alarg e jab of down e le-vator, enough to give thetail an upload? If th eaeroplane is flying fasterth an th e tailplane 'scr itica l speed , that isenough to let it flip intoits full down p ositionand there is noth ing you can do abo ut it. It will seem as ifthe servo has run away.

I have just had a vision of glide r designers crowding thewindow ledges twenty floors up. I had better put away thedoom and gloom and come up with some answers.

Remedies for Tuck Under

Now that we have a better idea of how Tuck Under isca used we ca n think of w ays to design it out of ouraeroplanes. Many of the remedies which I shall suggest arewell known from experi en ce of course, wh ich is as itshould be, but I believe it is impo rtant to know why thesedesign features alleviate tuck under.• 1. The struc ture of the aeroplane should be as stiff as

possible . The less the wings twist and the less thefuselage bends, the higher the critical speed will be.

• 2. Attach the wing rigidly to the fuselage using dowelsand wing bo lts, substantial plug in joiners or plen ty ofelastic bands.

• 3. Move the pivot of an all moving tail near to itsaerodynamic centre (25% of its MA e. ). The less loadthe pushrod is carrying the less it will deform. This willalso eliminate the tailplane instability problem.

• 4. Use a large diameter stiff pushrod and keep the wireends as sho rt and straight as possible.

• 5. Lengthen ing the tailplane or elevator horn and usinga longer servo arm to achieve the required movementalso reduces the load in the pushrod.

• 6. Use of close d loop controls or a pull for up pushrodor snake will eliminate my third cause of Tuck Under.See Figure 19.12 for possible layou ts. I have shown thetailplane 's mean chord , not its root chord .

• 7. If using a Bowden cab le make sure it is completelysupported along its length with as litt le bare inne rshowing as poss ible.

• 8. If you are choosing from a range of wing aerofoils all'of which are su itable, use the one with least camber.

• 9. Move th e e.G. fo rward to increase the stabilitymargin, but remember that control responsiven ess willbe re duced . Yo u may have to increase contro lmovement.

• 10. A smaller tailplane will increase the Critical Speed

Tail Sta lled

and make Tuck Unde r less likely to occur, bu t it maymake recovery less likely if a tuck under does occur .The e.G. will have to be move d forward to maintainthe same stability margin. Too small a tail may reducethe controllability of the aero plane.

• 11. Try a tail with negative camber. I haven't me ntione dthis before for a good reason. It doesn 't help preven ttuck under! It do esn't raise the critical speed at all, butremember Figure 19.10 where the tailplane with full upapplied was jus t stalled? A tail w ith negative camberwouldn't be stalled at that angle. It would be good forano ther degree or two if it doesn 't have too sharp alead ing edge. Its maximum downward lift coefficientwill be slightly grea ter than for a symme trical tail. Thisextra downward lift might make the difference betweenjust 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 drag willbe fraction ally less while carrying a down load as anadded bo nus.

It is possible to des ign a model whose wings and tail arestiff enough, and whose C011U"ol runs are effective enough,that its critical speed is above its terminal velocity, whichmea ns that its trim CUIve will be like Figure 19.13. No waywill this aeroplane tuck under.

Conclusion

Tuck Under is another aeroelastic problem caused bythe flexibility of the structure, and not, as the uninformedmay try to mislead yo u into be lievi ng, ca used by the

Basic Aeronautics fo r Modellers 107

CRITICAL TUCK UNDERSPEED. We have to try tomake su re that th estructure and controls arestiff enough to mak e thecritical speed so high thatth e aeroplane ca n no treach it. All the potentialca us es of tu ck un dermust be removed. Curingonly one may incr eas ethe Critica l speed a bit ,but not e no ug h. Th eunfortunat e modellerhaving cured one faul tand still experience d tuckunder may start to thinkth at w hat he ha s d on ewas not a cure at all.

The prima ry cure is tostiffen up the structure,piv ot all mo ving tails at25% mean cho rd , a nduse a pull for up controlsystem . I would cha ngethe tailplane to a smaller,negat ive cambe re d o neonly if the original wasexceptionally big. If theaeroplane is flown with avery s ma ll StabilityMargin to mak e it verysensitive, only a very littlefle x will ca use a tu ckunder. In th is case aforward CG movementmay be th e b est cu re .Th ose are all the thin gswhich I ca n p rove wi lllead to tuck und er. Theremay be mo re d esi gnfaults which will causetu ck u nde r w hich I do

know about. If you find one , please let me kno w.

In the past ther e have been mod els whose performancehas seeme d a little odd , unusual , disappoi nting . Forexa mple, I brou ght myoid tra iner out of retireme nt inorder to run in the eng ine for my first "hot" aerobaticsports mod el. It wa s a ve ry good tra iner w ith docilepredictable handling characteristics but wh en I installed anengine with two and a half times the power of its intendedengine, it changed . At low speeds, with the engine at idle ,it still flew velY nicely, but at high speed it just felt wrong,perhap s a bit sen sitive to elevator trim, but difficult to putinto words at the time . Looking back I wonder if I wasoperating it at aro und its Critical tuck under speed, and ifthat could account for its odd handling. I wonder if oth erpeople have encountered unu sual handling cha racteristicson mod els which could be related to inadequate structura lstiffnes s. I wonder is it wise to operate an aeroplane so faroutside its intended design envelope? It is worth giving itsome thought.

Footnote

•

•

Theory

Actual

Figure 19.12

Do w"Elev

upElev

Figure 19.13

w in g 's cen tre of pressure mo vement or "ta ilp la netak eover". Although it is clo sely ass ociated with staticstability, an aeroplane which nicks under is not "unstable"in the context of Chapter 8 in that it would respond in astabl e fash ion to an ang le of atta ck cha nge at co nstantspeed . Because all aeroplanes are flexible to some extent ,a ll a e ro p la nes wi th ca m bered sections will have a

108 Basic Aero na IIIics for Modellers

Chapter 20

The Air on the MoveY O U could read half a dozen aerodynamic

textbooks from cover to cover and not see windmentioned once, which indicates that it has no

effect whatever on aerodynamics, only on navigation.You see by "wind" I mean a block of air whosemolecules are not moving relative to each other butwhich are all moving at the same speed over smoothlevel ground.

Navigation

The velocity of the aeroplane relative to the air is its"airspeed" , and its "groundspeed" is just the vectoraddition of the airspeed and the windspeed as in Figure20.1. The direction of the groundspeed is called the"track" of the aeroplane. The airflow over the aeroplanewill still be straight from nose to tail.

If you are really interested (or mistrustful) you willfind proof of this vector addition idea in a mechanicstextbook, for example "MECHANICS" by Den Hartog.You will also find proof that the acceleration of theaeroplane relative to the ground is the same as itsacceleration relative to the air, because we are assumingthat the air has no acceleration. If a steady wind doesnot affect the acceleration of the aeroplane then Newtonwould agree that it does not affect the forces on theaeroplane or its be-haviour.

generally reduce with height as in Figure 20.2, there willtherefore be a "ceiling" above which a particular glidercannot climb , and the more efficient the glider thehigher its ceiling will be . You will notice in Figure 20.2that the streamlines are closer together over the brow ofthe hill, indicating an increase in wind speed as youwould expect.

The other requirement is that the wind must bestrong enough to have its vertical component greaterthan the glider's minimum sink rate, but the wind mustnot be so strong that it blows the glider backwards overthe hill. In a strong wind it is better to ballast the gliderthan add down trim to maintain its good glide at ahigher speed.

Thermal Lift

Hot patches develop on the surface of the Earth dueto solar heating, or buildings or whatever. A hot patch ofground shares its heat with the surrounding air which,when heated, expands. The warmer air is less densethan its surroundings so when a big enough "p ile" of ithas developed it breaks away from the surface andbegins to rise as illustrated in Figure 20 .3. Thesurrounding air flows in to replace it, and the hot patchof ground starts to warm up another batch.

Slope Lift

When this solidblock of air movingover a smooth levelplain (Le. the wind)comes to a hill, something has got to give,and it isn't the hill. Theair will be forced up,and this "slope lift" is anideal way of keeping aglider airborne indefinitely.

A particular gliderwill obviously require tofly in a "wind slope"steeper than its minimum glide angle. Themore efficient the gliderthe less steep the hillneeds to be.

Also, because the"slope" of the wind will

Figure 20.1

Wi"dVector

Basic Aeronautics/orModellers 109

---------------/_----------------------

Right down amongstthe heather where themidgies hide there isvery little wind.

At a height of a fewinches there is more,while at tree top heightit could be quite strong.It is just like a boundarylayer on a giant scale.Surface friction reducesthe wind speed thecloser you are to thesurface . Not only thatbut the direction of thewind changes withheight also . If you arefacing the wind on thesurface, the wind higherup is usually fromslightly to your right.

For example the wind could be»

• 3 ft/sec at a height of 3 inches, blowing from thenorth-east.

• 10 ft/sec at 3 feet• 30 ft/sec at 30 feet• 50 ft/sec at 300 feet now coming from the east.

Now suppose you take off from your runway headingnorth . On the runway there is a small headwindcomponent which is fine , and a small crosswind fromthe right which is no problem. The model is keptstraight using rudder. On lift off the aeroplane issuddenly exposed to a crosswind which would make anaeroplane with strong lateral stability tend to roll to itsleft and as the model gains height the headwindcomponent increases (no problem) and the crosswindfrom the right increases. That will continue any tendancyto roll to the left. The solution is to apply a little aileroninto wind on take off to prevent the crosswind fromlifting one wing. See Figure 20.4.

As your aeroplane seems so keen to turn left it is ashame to disappoint it so you climb your model in agentle left turn onto a westerly heading. You are carefulto keep the speed constant, by applying up elevator asyou have not had time to adjust the trims yet , but thespeed apparent to you is the groundspeed, because you

Figure 20.2

-~----~~--~-

The term "w inds hear" applies to any significantchange in wind speed or direction for a small change inheight or position and it can sometimes be so severethat it causes jet transport aircraft to crash.

The wind cannot be the same at every height.

As the bubble of warm air (the "thermal") rises, itcools at a fixed rate, so how far it will rise dependsupon the weather.

If the temperature of the surrounding air is constantall the way up, the "thermal" will soon cool to the sametemperature as its surroundings and so stop rising. But ifthe temperature of the surrounding air drops rapidlywith height, as when a cold air mass moves over a warmsurface, it is unstable and once a thermal starts to rise itjust keeps going and going. The big ones build up intotowering Cumulo Nimbus clouds several miles highcontaining strong updraughts. And if there is anupdraught in one place there must be a downdraughtsomewhere else to compensate. Even a little slope liftcan trigger thermals in unstable air.

Now that this idea of thermals has broken myidealisation of the wind as a mass of air moleculesmoving uniformly together, I might as well come cleanand mention other "real" variations on the ideal wind.

Windshear and Wind Gradient

Figure 20.3

Hot Patch

110 Basic Aeronauticsfor Modellers

are standing with your we llies se t firmly in the mud. So,climbing into that increasing tailwind with you keepinga co nsta nt grounds peed , yo ur mo d el has a s te ad ilydecreasing airspeed , which is all it cares about, and theresu lt co u ld we ll be a n une xpe ct ed s ta ll. Th e o n lyan swer to this o ne is to be aware of the wind var iationand make a rule never to take off downwind , an d neverto tu rn down wind after take off. Yes, I broke a modelonce doi ng that.

Ano the r good crash scena rio, illustra ted in Figur e20.5, is an approach with a good headwind at he igh tdropping off to only a little head wind near the ground .Th e model p ilot w ith hi s mind g ro u nd referencedmaintains a cons ta n t gro u ndspeed wh ich gives theaeroplane a decreasing airspee d leading to a stall. Thep ro b lem is mad e w orse if a powered ae ro p la ne istrimmed with too much downthru st such that red ucingpower redu ces speed .

The th rottle should be a rate of descent co ntro l andthe e levato r trim shou ld be the speed co ntro l. If thespeed is set a go od margin abo ve stalling speed at a sa feheigh t with the elevator trim, and left there , a co rrec tlytrimm ed stable aeroplane will mainta in that airspeed.The p ilo t th e n w ill be ad jus ting po we r to keep th emo de l on the desired de scen t path.

Right aileron toprevent left roll =T"1I"""=

are a pure ly random variation in the local windspeedand can be in any direction . When flying into w ind , say,a sudden increase in winds peed is a gust fro m in front , aredu ction in windspeed is a tailgust , a temporary shift inwind direction is a side gust , a the rmal is an up gu st , anda downdrau ght is a down gus t. Any change in the windmeans that the wind now has an acceleratio n whi ch willaffect the aeroplane.

Gu sts from front or rear, above o r below , w ill causesome spe ed variation, so me pitching up and down andprob ably so me wing flexing but a stable aerop lane willbe ab le to ride it out with no probl em s. The da nger isthat, if the wing 's angle of attack is just below its stallingang le , a sudde n increase from a gust can stall the wing,whi ch makes it prude nt to fly at a speed a safe marginabove stalling speed, and the more the tur bulence thebigger the margin. Also bewa re of pu lling "g", tight turnse tc., near the ground .

Gus ts from the side can disturb the head ing becauseof di rectio nal stability , o r bank the aerop lane due tolate ral stability. Vertical gusts affecting one wing morethan the othe r can also produce unwanted banks. Theyare just a nui san ce which has to be co rrec ted , un less theaerop lane is near the ground and the wing's angle ofattack is near its sta lling angle .

O n the ap proach a gust dropping one wing can be ali ttl e e mba rrass ing . Picking it up w ith rudde r, o r acombina tion or rudder and ailero n is safer than relyingjus t on aileron .

/~ Wi"d

Groundspeed 100Airspeed 100

Gro""dspeed 100Airspeed 50Stalling Speed 51Oops!

Figure 20. 4

became avai lable to th eRe membe r when flyingge nera l p ub li c o ncharte r fligh ts to Palrnaand Be n idorm wh ichwere c hea p e r th a nstaying at hom e? Peoplecame back wit h storiesof new hotels in slee pySpanish fishing villages,Sa ng r ia a nd "Cu b aLibre " cheaper than tea(a nd mu ch s tro nge r) ,a nd o n the fl igh ts ofh itt ing "Air Pocke ts "whi ch sounded like theholes in Swiss che ese orth e lu m ps in sch o o lcus tard . Well of co urseth ere w ere no t re all yhol e s in th e a ir, w ha tth e y had e nc o u n te re dwere GUSTS.

It would have beenn ice if th e air mo vedsmoothly and unifor mlyover th e sur face but itd o e s not. It is fu ll o fturbu lence , what w ithth e ro ughness o f th eground sur face and theth e rm a ls a nd so o n .Windshear is fixed inp lac e , ie: - a fte r on eae ro p lane flies throu ghit the winds hea r will stillbe there when the nextaeroplane comes along.However turbulent gust s

Gusts


Figure 20.5

»:~8'$'45

~THUMPu..J LJ ·g.s.45

a.s.48Stalling Speed 51

Groundspeed (g.s) 45Airspeed (a.s) 95

g.s.45a.s.75

«ess

Myths and Misconceptions

I was watching a magic show on television w ith mylittle da ug hter. Paul Dan iels stoo d on its edge a big sheetof plate glass whic h he dem on strated to be so lid allover. Then he se t up his appa ratus and pushed his lad yassis tant straight thro ugh the middl e of the sheet of glasswithout a scra tc h . My dau ghter was amazed . I wasamaze d. She believed that the lad y had passed throughth e g lass by magic . I, becau se of my grea t age andex pe rience and knowledge of ph ysics, know that it wasa trick, an illusion , but I do not yet know how it wasdon e.

I d o kn o w that what a p pea re d to happen isimp ossible and I am sticking to that belief. I know thereis an oth er logical explana tion whi ch fits in with the lawsof scie nce . I just can't find it.

There are many mod ellers who believe some popularunscientific myths eithe r because they are misinform edo r hav e w rongly interp ret ed th e behaviou r of th eirmodel. You , dear read er, canno t be o ne of them ofco urse having had the sense to bu y this book, but youhave prob abl y heard them and perh ap s wondered howto break it to them ge nt ly.

First of all, wh en flying in a steady wind the airspeedof the aeroplane is unaffected by the wind . How co uld itbe? Sure if you sudde nly apply a tailw ind the airspeedwill reduce . But d rag will redu ce and the aeroplane willaccelera te back to its origi na l airspeed at wh ich drageq ua ls th ru s t. Drag d epends o n a irs peed , notgro undspeed.

In a crosswind the track of the aeroplane is affectedbut the air still flows along the ce ntre line from front toback. It must do. If it were offset, the ae roplane wo uldha ve a s ides li p velocity , and from Cha p te r 9 itsd irection al stability would line it up with the airflowag ain . Wh en flying th e mod el to wards yo urself in a

112

crosswind, you must cock the nose of the aeroplane intowind suc h tha t it loo ks as if it is flying sideways towardsyo u, but th e airflow over th e aerop lane w ill s till bestraig ht from nose to tail an d a flag o n the nose wo uldalways fly stra ight back, assuming the aeroplane has no tbeen yawed deliberatel y with the rudder.

Th e o the r th orny p robl em is th at of tu rning anaeroplane in a wind. Some people swear tha t wh en the yturn the ir mod el fro m flying downw ind to flying intowin d it zooms upwards, and they ca n demonstrate toprove it. However whe n I turn their aeroplane it doesnot happen , and they canno t tell me where I am go ingwron g. How do yo u prove that some thing doesn't (o r atleast sho uldn't) happen? Perhaps a clarification of theproblem will help.

By "tu rn ing " the ae ro p la ne I mean su p p ly ing ace ntripe tal force to ge t its CG moving in a new directionas we ll as aligning the fuselage with the new directionof motion . Ju st rot ating it a bou t its yaw ax is is notturning.

I a lso mean a bal anced turn as descr ibed in th echa pter on turning wh ere the lift is increased so that itsvertical compone nt still suppor ts the weight , and thru steq ua ls drag. I am also disregardin g gusts and winds hear.

Given those conditions then , a turn , however tigh t orslow, will cause no vertica l effect. I have flown full sizeaeroplanes a nd kn ow th at th ere is no way th at th eaeroplane can sense the wind direction. The re is no wayth e p ilo t ca n kn o w th e wind di rectio n except fro mgro und based instrume nts .

All the forces on an aeroplane other than grav ity arisefrom the air and so its equat ions of motion are relatedo nly to th e air. Wh en yo u are si tting up ther e in anaeroplane looking at the clouds , the surface of the earthcou ld be mo ving in a ny dire ction a t an y s peedundern eath the air mass and it makes not o ne whit ofd iffere nce to the flying of the aeroplane .


What about Momentum?Of co urse so meone will say "An aerop la ne flyin g

north into a stro ng wind has very little mom entum, so ifit tu rns so uth it must lose airsp eed!", or "\X1here does theex tra momentum co me from?". I'm glad yo u ask ed .Everyo ne has a rough idea of wha t mom entum is but tobe a bit mor e specific, I looked up a boo k. Mom entumis a vec tor quantity, mass times velocity, and the changein mom entum (w hich is what we are afte r) is definedmath em atically as the integ ral of the force producing itwith resp ect to time . I will not bore you with the de tailsbut the force producing the mom entum change o n aturn ing aeroplane is the only unbalan ced force in Sight,the ce ntripe tal force . I integrated it round a 180 degreeturn and found that the mom entum change produced bythe ce ntripe tal force is 2mV, in a direction opposite tothe origina l ve locity.

If the initial airsp eed is V to the north into a northerl ywind of V!2 say , the northbound momentum is mV!2.The cha nge is -2mV so the final mom entum is -3mV!2(minus because it is so uthbo und) . The grounds peed istherefore 3V!2, makin g the airsp eed V. The ce ntripe talforce supplies the differen ce in mom entum with out lossof airspeed , see?

What About K inetic Energy

Unde te rred o ur doubting Thom as says "But wheredoes the ex tra kin et ic energy co me from?". I looked tha tup as well. Kineti c energy is Y, mV2 and is a Sca la rquantity ( i.e . not a vecto r) . Th e incre ase in kineticene rgy is defined as the wor k don e by the result ant ofall th e forces ac ting o n the bod y. Work done is th eresu ltant force times the distan ce moved in the directiono f tha t force . I in tegrat ed th e co m po ne n t o f th ecentri pe ta l force in the dir ection o f the grou nds peedround a 180 degree turn , from so uth to north th is time .The cha nge in kinet ic energy is -2mVU where U is thewinds peed . The K.E. cha nge turn s out negat ive if thewind is in the same direction as the initial airsp eed .

Su ppose th e w indspeed U is half the a irspeed Vaga in . Go ing so uth, with the wind , the K.E. relat ive tothe grou nd is Y, .m(3V/ 2)2 or % m V2 The cha nge is -mV2giv ing a final K.E. of ~ mV2 or Y, m (V/ 2)2, which meansthat the final ground speed, goi ng North again is V/ 2,which means the airspeed is V. The centripetal force hasrem oved the excess ICE. with no increase in airspeed ,see?

Analogies

For a nauti cal a na lo gy th ink of th e e ffec t o n apowered boat of a river current, or better still an oceancurrent miles from land. (Forget the effec t of wind on aboat becau se that is a diffe rent case e ntire ly) . In themiddle of the ocean there is no way of kn owing if thewat er is moving relati ve to the earth unl ess you can seeth e b ottom . If th ere is a c u r ren t it will m a ke n odiffer en ce to the handling of a boat and I have neverhea rd of a boat zooming upwards when turn ed head on 'to the Gulf Strea m.

However back to ae roplanes in the wind . When yousit back with a glass in your hand and th ink ab out life ,the wind is just a block of air movin g at so me speedrelat ive to the ea rth 's surface , and if a mo de l aeropl an estays within that block of air, the speed of the air over

Basic Aeronauticsf or Modellers

the ground makes no difference. The block of air co uldbe inside a Boe ing 747 for exa mple, moving at 600 milesper hour, but if yo u had a suitab ly s ized slow flyingmod el I bet that in spi te of the 600 miles per hour windyo u cou ld fly qui te a resp ectable circle inside , if yo uwere ins ide w ith it. There is yo ur probl em! I rate yourchances of flying a model in a circle as nil if yo u are o nthe ground and it is in a passing 747. The 600 miles anhour "wind" has no t mad e things difficult for the mod el ,on ly for yo u. Inside the aeroplane you do not feel asthough you are moving, and it does not matter wh eth eryo u are o r not. If yo ur brain is in the 747 then that isyour reference.

Th e p ro b le m is not on e o f aerody na m ics , ormech an ics . It is a psych o log ical p rob lem . Yo u mu st"think airborne". Your we lly boots might be stuck in themud but you must deta ch your brain and put it up in theair floatin g free like a balloon.

The Mea ning ofLife

Subtitled "Don 't as k ques tions to wh ich there are noanswers"

I ca n no t tell yo u why yo ur aeroplan e zooms andd ives w he n yo u turn it , be cau se w he n I fly yo u raeroplane it does not happen . I know it happen s wh enyo u fly it becau se I have see n , but yo u are the one wh omu st say why because it is you do ing it.

Is it becau se whe n you turn it into wind yo u see itsgro undspeed reducing so you let it speed up in case itsta lls? When yo u lev e l th e w ings the extra airspeedwou ld cause a zoom.

Is it becau se yo u apply up elevator to hold the noseup in a turn and whe n you stop the turn you forget torem ove the up elevator? That up eleva tor would cause azoo m.

Do yo u turn th e ae ro p la ne wi thou t app ly ing upeleva tor? If so the nose will drop and the airspeed wi llincrea se althou gh yo u may not not ice as it turns intowind and the gro undspeed reduces . The ex tra airspeedwould cause a zoom.

Maybe you speed the plane up in a turn , by add ingpower. to increase the lift. On resuming level flight theexcess speed (a nd ex cess power ) will ca use a zo o mupward .

Do you turn the aeroplane by ba nk ing it over o n awing tip and hea ving on the up eleva tor? High "g" turnslike that dr ast icall y increase the drag of an aeropl an eand so it w ill lose speed in the turn , but it will lose justas 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 thesa me to the aerop lane. The aeroplane is just as near itsstalling speed after a turn downwind , but your mind isco mfo rted b y th e a pparent e x tr a s peed from thetailwind.

I still ca nnot work o ut how Paul Dan iels does hislady th rough the sheet o f glass trick , and I do not knowhow you d o yo ur zooming and d ivin g tricks ei the r.When you have work ed out wha t it is you are doin g toca use the unintentional climbs and dives , please tell me .Tell eve ryo ne!

113

Chapter 21

Model Aircraft StructuresFigure 2 1.1

Support

~~ »> ~

~j

Te nsio n

\~

I Load I

I n line with the res t of the book, this is not about howto design structures, it is just a lirtle bit of theory toexplain how the structures used in model aeroplanes

work. There are qui te a few words, most of which wil l befamilia r, whose meanings in this context I would like toclarify.

LOAD . The load carried by a part of the structure is theforce in it. For example, a piece of balsa with a one poundweight hanging on it is carrying a load of one pound.

Figure 21.3

Fuselage holdingdowel doum

114

Fig ure 2 1.2

Compression

Support

TENSION. A structure is in "tension" when a load is tryingto pu ll its two ends apart as in Figure 21.1.

COMPRESSION. A st ructure is in compression when aload is trying to push its two en ds towards each other as inFigure 21.2.

SHEAR. The wing dowel in Figure 21.3 is in "shear" whenthe wing is lifting it up and the fuselage is holding it down.

TORSION. A piece of structure is in "torsion" when theload it is carrying is trying to twist one end relative to theother.

BENDING. The beam inFigure 21.4 is carryinga "b e n d in g moment"when it is supported atboth ends and a weightis placed in the middle.

STRESS is force per unitof cross-sectional area,and tells how concentrated the load is . If apiece of quarter inchsq uare wood is canying aone pound load then the"stress" in it is 16 lb ./sq .in . . . If the same load ison eighth square balsathe stress is 64 lb ./sq . in .Each material has alimiting stress beyondwhich it will break.

Basic Aerona11tics for Modellers

Figure 21.4

Load

Figure21S

c~1\

Support

, J

SUPPOl-t

Noload

Tensilestress 16

Tensilestress 64

STRAIN is the amount of deformation due to the stress,either stretch or compression . The s t ra in isapproximately proportional to the stress applied so theeighth square wood above will stretch four times asmuc h as the quarter square under the same load . SeeFigure 21.5 . It is ass ume d that when the load is tak eno ff , th e s tretch w ill d isap p e ar , i .e. the ma terial is"elastic".

STRENGTH, The strength of a piece of materia l orstructure is the maximum load it can carry before somepart of it reaches its limiting stress and breaks.

Figure 21.6

llbload

llbload

Stress100

STIFFNESS . The stiffness of a ma terial is stress dividedby stra in. The stiffe r a material is the less it will deformunder a give n stress. Figure 21.6 shows two d ifferen tma teria ls each of the sa me cross-section and unstre tchedlength carrying the same load . Th e str ess is the sa me ineach but the stiffer materi al stre tche s less. Either materi alcould break first, depen ding on their different strengths.

STIFFNESS of a STRUCTURE is load over deformation,Le. the stiffer a struc ture is, the less it will deform undera given load . The stiffness of a structure (e .g. a wi ng)depends upon the stiffne ss of the materials used and thema nner in which it is co nstructed .

Unstretcbed-----" le1lgth

Load

Figure 21 .7

Stress100

Stiffmaterial

Flexiblenaterial

Load

LOAD PATH, The ro ute the lo ad takes th rough th estructure fro m where the force is made to where it isused. Just like a chain, any structure is only as stro ng asits weakest link .

The Strength and Stiffness ofComposites

Imagine a steel wire and a length of rubber bungeeeach of which will carry a load of 100 pounds beforebreaking. They have the same strength but the steel wireis much stiffer. If a steel ring is su p ported by a foo t ofthe wire and a lso a foot (uns tre tched) of the ru bb er ,w ha t is th e streng th of th e combina tion? Wha t is th emaximum weight that can be hung on the ring? Q uick asa flash yo u answer 100 pounds. Th e point is tha t thestiffer member carries an unfa ir proport ion of the loadbecause the stretch in each has to be th e sa me, as inFigure 21.7 . The minu scul e stretch of the steel allowsonly a tiny load to be carried by th e ru bber. When a .load of 101 pounds breaks the steel wire the rubber w illstretch until it a lso breaks.

When ma ter ials of d iffe rent stiffness share a loa d in amodel aeroplane structure , remember th at the stiffermaterial will carry an unfair proportion of the load.

Basic Aeronautics fo r Modellers

With this muchstretch, the wirecarries 100 load andthe rubber hasnegligible load

Rubber

Unstretcbed length

115

point bending m o ment total

A 2xl = 2

B 2xl + 2x3 =8

c 2xl + 2x3 + 2x5 = 18

D 2xl + 2x3 + 2x5 + 2x7 = 32

E 2xl + 2x3 + 2x5 + 2x7 + 2x9 = 50

F 2xl + 2x3 + 2x5 + 2x7 + 2x9 + 2xll =72

G 2xl + 2x3 + 2xS + 2x7 + 2x9 + 2xll + 2x13 = 98

Wings a ttachedto fuselage sides

22

In Figure 21.8 I have divided up a wing into 2 inchwide strips , each of which has a lift force of 2 ounces .The "BENDING MOMENT" at any point in the wing isthe moment of all the lift forces outboard of that point.Therefore the bending moment at the tip is zero. Thebending moment at various points along the wing isshown in the table below.From this table and the graph of bending moment inFigure 21.9 you can see that by far the greatest bendingload on the wing is at the root , and so the greateststrength must also be at the root.

The two wings cannot simply be attached directly tothe fuselage sides, because the transfer of the bending

moment would distortthe fuselage as in Figure21.10 . The tw o wingsare best built in onepiece so that the stress iscarried across the join. Ifthe wings have to bedetachable for transportthen there must be somekind of wooden ormetal beam runningacross 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. Theother is an assembly buil t up with ribs , spars and usuallysome thin balsa sheeting, the whole thing being coveredwith doped fabri c or plastic film.

can be used to hold the wing tips down, and we willcome to them also.)

Figure 21.10

The usual structure for carrying a bending moment isthe beam, shown bending in Figure 21.11. The strengthof such a beam depends on its breadth (b), and itsdepth (d) cubed. The material nea r the top surface isbeing compressed and along the bottom surface it isbeing stretched while in the middle the material is underno stress and carries no load . It might as well not bethere.

The 'I beam' consists of a top spar and a bottom sp arwith a thin sheet web in between . For ease ofconstruction the web may be glued to the rear faces ofthe spars as in Figure 21.12 in which a load is tending tobend one end upwards like the lift on a wing . Whensuch a beam is canying a bending moment the top sparis in compression and the bottom spar is in tension. Theweb holds the two spars apart.

The strength of the beam depends on the cross

Bending Moments in Wings

Built up Wings

22

TipA

2

B

2

cD

2

EFG

Figure 21. 8

Bending moment

Figure 21.9

Twist and BendAn airflow tends to do two things to a wing, and the

effects are almost independent. The moving air tends topush the wing aside, and tends to twist it. The pushforce , or Lift, depends on the angle of the wing to theairflow, and not much else. By varying the angle youcan make the Lift increase, or decrease to zero (but itdoesn't change the twist) . The twisting effect, or PitchingMoment, depends only on how much camber the wingsection has. Naturally both of these effects depend alsoon two properties of the airflow itself, the ve locity of itsmotion and the density of air.

One end of the wing is anchored to the fuselage andthe airflow tries to lift the w ing tip up above the wingroot (and occasionally it succeeds) so the wing bends.To resist this tendency the bottom surface of the wingwill be pulled o ut in tension , and the top will be pushedtowards the root in compression. (Sure, struts and wires



Load Load

Support

sectiona l areas of the spa rs (A), the distan ce bet weenthem , and of co urse the strength of the materi al fromwhi ch they are mad e . To ge t the spars as far apart aspossible th e y a re pl aced at th e th ic kest part of th eaerofoil section and of co urse the thicker the section thestro nger the win g will be .

Wing ribs, as we ll as giving an aerofoil section , holdthe spa rs in position w ith the suppo rt of leading an dtra iling edges . The leading and trailing edges add little tothe w ing 's bending streng th but if the wi ng is pa rtlysheered with medium de ns ity balsa sheet, the to p andbottom sheets may add to the ben ding stre ng th (Figure21. 13) , de pendi ng o n th e rel ati ve sti ffness o f t hematerials. If the spa rs are made of spr uce , or es pecia llycar bo n fibre, then the spa rs will ca rry all the bendingforces. The best way to streng the n th is typ e of structureis to incre ase th e size o f th e spars a t th e root end ,preferabl y by wid ening rath er than deepe ning them asshown in Figure 21.14.

Compression

so that it is ab le to car ry a co rn pressive load . Thus in afoa m wing the top sk in carries co mpressive stress an dth e bott om skin ca rries tensile stress, wh ile the foa mprevents th e skins fro m bu cklin g a nd p rese rve s th eaerofoi l sha pe . T he be ndi ng s t re ng th o f th e w ingdepen ds o n th e sk ins ' th ickn ess a nd chord , ho w farapa rt they are , the stre ng th o f the mater ial used , butmainly on how well they are suppo rted . Obech i veneer0.03" thick is usual , but on wings wi th a thick sect ionand /or a low as pect rat io , thin ca rd or even pap er isadequa te . Becau se the skins a re so thin they will justbuck le up if they are not stuck securely all over to thesupporting foam . In fact [ sha ll stick my neck out and

Foam WingsFigure 24.14

.- ...,,

Extra material 't"· ~ ...more effective bere l'

tban bere

Rib

A th in sheet o f mat eri al is ve ry go o d a t ca rryi ngten sion . A strip of my printer pap er only 1.5 inch es wid ewi ll ca rry a ten sile load o f 20 sma ll ca ns of beer (or 15pounds to ge t technical), How much load do yo u thin kthat pap er co uld carry in co mpression? None, because itjus t bu ckles up, unl ess you ca n find a way of sup po rtingit. If I ro ll up a strip 9 inches wid e into a cylinde r oneinch in diameter and 1.5 inches lon g, add a da b of pasteto hold it in posit ion and sta nd it on end, I ca n ge t thatsa me paper to carry a co mpressive load of 4.5 pou nd sbefore it co lla pses s ideways . \'{Iith better s up port itwo u ld proba bl y ca rryeven more load , bu t itw ill a lways be wea ke r Figure 21.13in co m press io n th ante nsio n . Gi ving th e Sbeetsheet the right su pport .___-::::::::::::::::::=======r=n--------is vitally imp ortant to itsco mpress ive s t re ng thand stiffness .

T ha t is w here th efoa m in a fo am w ingcomes in . It sup ports Sheetth e thin skin covering

Basic Aeronautics/or Modellers 11 7

Figure 21.15

Bending strength

Figure 21. 16

Bending loa d

Root Tip Root Tip

predict that whe n a foam wing fails the veneer will failon the co mpression side and it will be du e to a lack ofsupport.

J oining Wings

The object is to transfer the stresses from one wing tothe othe r and so a suitable load path through which thestresses can flow is necessary. The best way to join sparsis the usual plywood dih edral brace glue d to the front orrea r faces of the sp ars (o r pre ferably both faces toincrease the glue area) . When the stress is ca rried in asheet material , as in a veneered foam wing, the twosheets of ve neer should be joined smo othly togetherwith, say, a strip of glas scloth and epoxy resin .

Joining foam wings w ith a pl y dihedral brace isas king for trouble. You ca nnot ex pect th e foam totran sfer any stress, and the joint between the ply and theskins is only 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 wingsMUST be joined by spars, as when they plug in forexample , great care is needed to ensure that the stressescan tran sfer from the sk ins to the sp ars without leaving aweak link . Some designers incorporate a ca rbon fibrespa r to ca rry all the bending load in the first place whichmak es win g joining easier.

Tapered Strength

The bending moment o n a wing is grea test a t th eroot. Th e structure is therefore design ed to cope withthe load at the root, but for building co nve nience thecons truction is often the same from roo t to tip , Figure21.15. What a waste of strength! It wo uld make sense totaper the strength a bit to save weight. For example,very often the wing sect ion at the roo t is much thicke rthan the optimum to gain ex tra strength without addingto o mu ch ex tr a weight , a nd th e sectio n thicknessreduces towards the tip (as in the Zlin SOL whose win gis 18% thick at the root and 12% thick at the tip). Thespa rs can be tap ered in cross sectiona l area towards thetip , and with a plywood dih edral bra ce the result is awing whose strength varies as in Figure 21.16 .

118

A similar effect can be obtained on a foam wing bytap erin g it in thickness and chord and by using a broadglasscloth joining strip . I usually use several thin layerso f gl asscl oth , a narrow o ne over th e join a n dprogressivel y broader ones ex te nd ing se ve ral incheseach side of the centreline.

Bending Stiffness

Althou gh the bending strength is our chief concern, ifth e w ing has insufficient bending stiffness it may bepron e to so me form of flutte r which is undesirable. Tocure or avoid the problem use a stiffer material of thesam e stre ng th and weight, or just beef it up a bit tomake it stronge r and stiffer at the same time .

Strength ofDamaged Wings

The slightest interrup tion in the load path will cause astructure to fail. A slight crack (a we ak link) across aspar carrying tension , or a chordwise crack in the obechiveneer, can allow a wing to fold. Beware of the fact tha tbec au se plast ic coverings are more flex ible th an thewooden structur es underneath , it is possible for thestructur e to be fatally cracked with no mark on thecovering. If you can feel any softness, or any movement,or are in any doubt, strip off to inves tigate .

Figure 21.17 shows typ ical dam age caused to a foamwing. The veneer cracks on one side and creases on theother side. Both surfaces should be levelled with fillerand covered with glasscloth and epoxy res in to replacethe loadpath . Splits along the grain of obechi venee r willnot seriou sly weaken the wing in bending.

If there is a cavity in the foam , caused by heat orchem ica l attack , the ve n e e r wi ll not b e p roperlysupported and when a compressive load is applied itmay fail. I always join my wi ngs w ith epoxy resin and/orwhite glue as the y do not attack foam .

Strutted Wings

Figure 21. 18 represents a wing a ttached to thefuselage at X by a bolt running fore and aft , so tha t it


Fig u re 21.17

Crease

J~

DFoam core

'tCrack

r I~Filler

~Glassclo tb

zTcos YY

~TCOS YT T

A

Tota l lift = b alf uieigbt ofaircraft lV/2

the fuse lage by a wire wou ld be just as effective but yo uwill need one on the top (a so -called landing wire) tohold up the wing when on the gro und, o r in invertedflight an d all the same arguments ap ply to fully riggedbiplanes.

Try working out the load path s yourse lf to see whichpa rts of the struc ture and rigging are carrying the heavyloads.

T= 1.1W

/T

Tcos Y X~ ~-=======~=-=pc=-==<;==========

Tcos Y=WTop

lVillg roots co mp ress

Figure 2 1.18

Fuse/age

Wi llg struts pullfu se/ag e bo ttom

ac ts like a h in ge . Thewing is also su pportedhal fway bet ween ro ota nd tip by a s tr u tattac hed at bo th ends ina similar manner.

I based it loosely o na full size aeroplane androunded o ff th e numb ers , ju st to gi ve a na p precia t io n o f th erel a tive sizes of th eforc es in vo lve d . Thestrut makes an ang le Y(27 d egr e es) w ith th ew ing . In th e lo w e rdrawing the par ts havebeen se parated an d theforces s hown mo reclea rly.

The stru t is o bviouslycanying a ten sion force.Taking mom ents aboutX, I worked out that thes tr u t tensi on e q ua ls1.1W, 10% more th an th e who le aeroplane 's weight ,there is a co mpressive force of W on each side of thetop of the fuselage and a pu ll of 1.1W to each side ofthe fuselage bottom.

Tha t is in s teady le vel flight. Pull 6'g ' a n d yo umult iply all these forces by 6. A stro ng strut and goodstrong joints between it and the wing are neede d , andth e fusel age mu st be s tro ng e nough to resist be ingsq ueezed together at the to p and pulled apa rt at thebottom .

What of the be ndingmoment? We ll a t th es t rut position nothingh as c ha nge d so thebend ing mom ent here isjus t what it wo uld be ifth ere were no s tr ut.Inbo ard of the strut, thebe nding mo ment car riedby the wing red uces tozero at the roo t.

Bend ing mo me n t isg rea tes t a t th e s tr u tfix ing (as in Fig u re21.19), so that is whereto stre ng the n the wingin b end in g , a nd yo ucerta in ly mu st not jus td rill thro ugh the spar tofix the stru t.

Th e bonus in us ingw ing s tru ts is tha t themax im um be nd in gmo ment is , in th isinstance , on ly about aquarter of wha t it wou ldb e a t th e roo t of aca ntileve r wing (Le. onew itho u t a s tr u t) pro du cing the sa me lift.

Attaching the wing to


Figu re 2 1.19

Bending moment

(il l a.\· bending m oment 4 timesg rea ter 0 11 a ca utileuer ioing )

th e c ho rd . a n d th eposition o r th e Ilexura la xi s . but a n y twi stbetwe en the ro ot a n dth e tip is und e s ir ableand must be reduced asfar as possibl e .

Built up Wings

wing is probably as stiffas . ma ybe eve n s t iffe rth an . a bu ilt up fu ll ysh eered win g, Again thes t ru ct u re d e riv c s it ss tre n g t h a n d s t iffn e ss1'1'0 111 being in th e formo f a c lo s e d b o x a n din add itio n th e o hec h ive n e e r is a s t iffe rmate rial than ba lsa ,

The s imp lest w ay tod e s ign a t ors io na llvstiffe r Wing is to reducet he aspe ct ra t io . T h atcosts g lid ing perform an ce so w ould app lyo nly to powe red models.Nex t yo u must co ns ide ra stiffer struc ture . Figure21 .20 re pre s e nt s tw owi ng struc tures . I haveidea lised them a bit andm issed out th e trailinge d ge p ie ce a n d cove r ing . Both u se theSAIVIE am ount o f woodbut type A w ith th e Dbox is IIIUCH (may be 23tim e s ) st i ffe r in torsionthan typ e B. The reaso nfor the dramatic increa sein sti ffness . and strength.is the closed box.

The torsion al stiffnesso f struc ture B is the sum

Tip o f its part s . but theto rs ion al s t iffness ofstructu re A depends on

the skin th ickn ess and the AREA enclosed . Fully shectingthe wh ole wing w ill enclose mor e than twice the a reaand giv e more th an twi ce the torsion a l stre ng th andstiffness. Any add itional stiffne ss from the co ve ring wi lldepend up on the co vering mat e rial used .

Comp risingr-

Foam Wings

The typi cal veneered foam

3 pieces 6 ,\' 6 "1/11

2 pieces lOO,\' 2 "1/11

1 p iece 25 ,\' 2 111/11

Root

\\

\

\

\\

\\

\ {Cautlle oer uiing}\

\ , , , ,"-

ili a ,\' moment "-

Figure 2 1.20

The Torsional Stiffness ofWings

You wi ll h a ve no ti ced th at in t he c h a p te rs onAeroe lasti ci ty and Tu ck Unde r I was co nc e rned w ithst ru ctura l STI FF NESS. no t s t re ng th. Eithe r ail e ronde flect ion o r. 1110re usuall y. the p itching moment Mo,te nds to twist the \\'ing lead ing edge down . The amo unto r tw isting mom ent in a wing wil l va ry with the sec tion.

Comprisingt-

3 pieces 6 .\' 6 mm1 piece 100 .v 4 mm1 p iece 25 ,\' 2 111/11

R educedStiffness

Th e tors ion b o xMUST b e co m p le te lye nc losed , Make a s lit ora c ra c k ri ght a lo ng atyp e A struc ture and its

120 Basic Aeronauticsfor , 1Iodel/el ~~

Figure 21.21

wooden up right s carry compression . As long as ten sionis mainta ined in the wi res the glue d wooden jo ints arein simple co mpressi on and are therefo re ve ry stro ng .

When modelle rs first copied this built up struc tureth ey left o ut the di ago nal w ires for s im p lic ity , bu tcovered it in doped tissue , silk or nylon. The co veringw as st uck we ll a ll a long th e long e rons and to theu prights as we ll, so th e ta u t covering sup p lies thed iagona l te nsion force, and the structure is adequatelystro ng in both di rec tions .

The ability of just the wooden structure , dep icted inFigur e 21.24, to ca rry a load de pends en tire ly on th ebendin g stre ng th of the g lued joints , which is hardly fair.No r is it ve ry stro ng. It is not even stro ng enough tomai nta in its own shape aga ins t the tension in a curvedlonger on , o nce it has been lift ed fro m the b u ild ingbo ard .

If a built-up struc ture is to be stro ng enoug h (wi thou ta do pe d fa br ic covering) to main tai n its own shape,ne ve r mind ca rry a load , it needs cross-bracing . Yo umu st insert a d iagon al mem ber into each bay to carryte nsi on o r compression force s , as in Figu re 21.25 .In st ead of a si ngle d iagona l, a pai r o f th inne r ones(gl ue d togethe r w here they cross) would also do the job .Diagona l membe rs are essent ial, but the upri gh ts aren o t. You ma y see a s tr uc tu re usi ng only diagona l

Figure 21.22

stre ng th an d stiffness d isa ppear. Simi larly , spanwisecracks in the veneer o f a fo am w ing can serious lyreduce its torsion al stiffness, altho ug h they may have noeffect on the bend ing streng th .

Either a fully sheeted w ing or a ve neered foa m w ingcan be fur ther stiffene d by co vering, either with dop ed

Fortissu e or preferably g lass , Kevlar or ca rbon fibr es.maximum resis ta nc e to ,--------------------------------------------,torsion al load s the fibreglass clo th or strands ofKevlar or ca rbon fibr esshould be applied wi thth e fibr e s ru n n ingdi a g o n all y a cros s th ewing as in Figure 21.2 1.All th e s e m o dernmat er ial s are light erst ronger and stiffer thansteel w ith ca rbon fibrethe stiffest and Kev lar the stro ngest (jus t) .

Aside from the for ces applied to fuselages by stru ttedw ings a lrea dy me n tioned , aerody na m ic loads a reu nlikely to exceed the stre ng th o f th e fu sel age o f anormal mod el aircraft. There are however two as pects Iwould like to highli ght.

The first is the bendin g stiffness of slender tai lboomssometimes u sed o n m ode l g lide rs . As descr ibed inChapte r 19, too flexible a tailbo om could lead to tuckunder. What is required is a boom of ad equ ate depthmade of a st iff mate rial. A th in wa lled ca rbon fibre tub emight be a good so lutio n . A suita ble fix for an existinginade quate structure mig ht be to add a fin strake as inFigure 21.22 to stiffen the ta ilboorn. Or you co uld addca rbon fibres to top and bottom.

Fig ure 21.23 rep rese n ts a struc tu re o fte n used o naircraft of the \Xf\V'l era . Side frames were construc tedusi ng wooden longerons and uprights , and each bayhad two w ires running diagonally and tensioned wi thturnbuckles. These wires were necessary to maint ain theshape unde r load . In the case shown, wi th a downloadfrom the tail, the wires shown with arrows ca rry tens ion(the o the r wi res ca ter for an upwa rd ta il force) and the

Increased Stiffness

Fuselage Stiffness

Figure 21.23

Doumloadfrom tail

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 takeeither ten sion or compression, making the vertical strutsredundant.

Tailplanes

The lift for ce on the tail depends o n the ai rspeedsqua red, the tail area Sr and the tail 's lift coefficient el T.

Just as in wings the greatest bending moment is at theroot. From Figure 12 .2 , the tail 's lift coefficient isgenerally upward at very low airsp eeds, and becomesmore and more negative as airsp eed inc reases. If it isgoing to break in bending, it will bre ak off downward sat the root in a high speed dive when the speed and liftcoefficient are both greatest. It is quite poss ible fo rtailplanes to flutter just like w in gs . The cur e is tocombine stiffness with strength, use inflexibl e controls,and mass balance at the leading edge .

Pushrods

The stiffness of pu sh rods cropped up in the chapteron tu ck unde r, bu t even if that is not one of yo urproblem s it is a good idea to make the push rod s as stiffas possible to prevent blowback of the controls andmaintain control effectiveness . They should shorten aslittle as possible when compressed. Balsa pu shrods arethe norm, and the bigger in diamet er the better. Th erecent innovation of using a carbon fibre tube seems anexcellent idea . Piano wire ends for connections sho uldbe as shor t and stra ight as possib le .

Figure 21.25

Usu al model structure

Doumloadfrom tail

Woode" diagonals maintain shape


Chapter 22

Centre ofGravityPosition

Since the publication of the first edition I have beenwri ting a column called "Aerodynamic Forum " inthe monthly magazin e "R/C Mod el World", also

publish ed by Traplet. In the co lumn I try to answerqu estion s from readers , and fully half of the probl em s Iencounte r involve finding a safe Centre o f Gravity. Ifound it useful to co llect together into one place all thetheory relevant to CG location - and I thou ght you mayfind it useful too , so here it is added to th e seconded ition.

The Centre of Gravity is the ba lance poi nt, the poi ntthrough which weigh t ac ts or, if yo u are a stickler foracc uracy, it is the point about which the weight has nomom ent (rotation effect) . It is a three dimensional poi nt ,bu t it should be near enoug h o n the centreli ne of asym metrical aircraft, the heigh t doesn 't much ma tter, soa ll we are co nce rned about he re is its fore and aftposition .

From th e above definit ion , the CG is the po int a twhi ch you can support the model suspe nded from stringor balanced on finger tips, the edge of a ruler , the blunte nds of tw o pencils , or o ne o f th e p u rpose madedevices for checking CG position s.

The pos itio n of th e CG d etermin e s how mu chstability the ae roplane will ha ve , and also ho w mu chco ntro l over it you w ill have . Stability an d Contro l areop posites;- the more of one yo u have the less of theothe r you ge t, so CG position is a co mpromise . You aretry ing to ge t just the right mix of Stability and Contro l.

We are co ncerned only with Eleva to r Control he re ,the rud der and ailerons do not concern us. \Ve want theelevator to pitch the nose of the aeroplane up and downin a reliable and pred ictabl e fashio n, so tha t we have thefull 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 fourdime nsiona l po int, because it changes with time as fue lis b urned , w hen bombs are dropped or t heun derca rriage is raise d, the n we plan for the worst case.Pla n th e aft-most CG which will be e ncou nte red inflight, Le. with the fuel tank empty if it is at the fro nt, orfull if it is at the back.

If the CG is wror«

As you move the CG forward from its ideal positionthe aeroplane becomes mor e stab le. That 's OK, but yo ustart to run out of CONTROL on elevator. You gradua llylose the ability to sp in, stall, fly invert ed , loop , and even

Basic Aeronauticsfo r Modellers

ge t the nose up for a flared out landing. The elevato rtr im becom es less effe ctive , req uiring more and moretrim movem ent when you cha nge from high speed trimto low speed trim, and yo u need more and more uptrim.

As yo u move the CG rearward the aircraft becomesle ss a nd le ss STABLE. Yo u have to ma ke co ns ta n tco rrect io ns to the flight path, you need to add downtrim and the elevato r trim becomes se ns itive to sma llmovements maki ng it difficult to trim the aeroplane inlevel flig ht , a nd a sma ll movement of the e levatorproduces a large co ntro l respon se . If the CG is too farback an aeroplane becomes UNSTABLE, its flight pathpersistently d iverges, it cannot be trimmed, and a smallup elevator movemen t produces a gut wrenching, wingfolding, loop.

The Correct CG

The CORRECT position for the CG is tha t which suitsYOU the p ilot. Your CG might differ fro m someoneelse '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 ferecommended starting point, then with ex pe rience youcan experime nt. You perform various Fligh t Tes ts.

Flight Testing

The popular "Dip" test can be used on most mod els ,bu t is pa rtic ularly rel evant to elec tric soarers, glide rs,tr a in e rs or vintage typ e model s , which do not ta kekind ly to aerobatic manoeu vres.

First you trim the model out in straig ht steady flight.If it is a power model it is best done flying level a tcruising power, one thi rd to two third s throttle . Oncetrimmed o ut you push forwa rd on the elevator stick alittle to put the model into a shallow dive and hold it fora co uple of seconds to pick up speed. The n le t the stickreturn to neutral.

A stable model will gently pitch nose up into a sligh tclimb and its speed will reduce to its origi nal trimmedspeed . This initia l reaction shows us the 'Static Stabili ty'of th e model. If the mod el rapidly se ttles down in tosteady level flight at its orig ina l trimmed speed then thatshows tha t it has good aerody namic da mping as we ll,and so it has good "Dynamic Stability". This is the usualresul t.

It may be that the mod el will pitch up quickly from

123

the sha llow dive into too steep a climb with reducingspeed. It is Stability which pull s it out of the dive , tryingto ge t back to its o rigina l trimmed speed . Too mu chstability will make it rear up too qui ckly, whi ch meansthe CG is too far forward . Paradoxica lly (see below),yo u need to rem ove weight from the nose , or add it tothe tail , and re-tr im.

It may be that the mod el will co ntinue in the dive ,possibly because th e e levato r has not returned toneutral , w h ic h is bad . Wi e rea lly need to have fre eco n tro l s u rfaces which a lways fo llow the con tro lcommands exactly without binding.

If the co ntro ls are free but the mod el remains in thedive then it may not hav e eno ug h stability so you needto move the Ce nt re o f Gravity forward a little . Somemod els not o nly stay in the dive, bu t they low er theirnoses even more, into a steepe r dive , and accelerat e . Ifso close the throttle and pu ll out of the dive, urg ent ly.You have a problem. Either the CG is much too far aft,or yo u have Tuck Unde r ca used by flexibi lity of thestructure - o r a combina tion of the two .

To find o ut, try the opposit e o f the dip test. Fromstea dy trimm ed flight hold a little up eleva tor to raise thenose a nd le t the model s lo w down a bit , a nd th enre lea se it. If the model co ntinues to pitch up then it isu nsta b le and needs a more forward CG . (O r is thee leva to r s t ick ing in th e up posit ionr) If it behave snormall y to th e pitch up by lowering its nose a ndspeedi ng up again then look for a Structura l Flexibili typrobl em as a ca use of the tuck under. The wings maybe twist ing, the elevator co ntrol may be flexing or thefuselage bending or so mething . You would then need tostiffen up the structure (see Chapters 18, 19 and 21).

The Paradox

There may be sceptics wh o find it difficu lt to beli evethat adding lead to the nose w ill help a mod el pull o utof a dive , so let 's look at it an other way.

When you add the lead to the nose to move the CGfurth er forward you will need to re-tr im for steady flightat the same speed . Som e up e leva to r trim will red ucethe tail lift, o r even ap ply a little down force , to supportthat we ight up front.

When you spe ed up the mod el , the mom en t of thenoseweigh t does not change , but the effect o f this uptrim is e n hanced by the speed sq ua re d effe c t so itprod uces an increased nos e up mom ent. So it is actuallyth e up trim you applied to trim o ut the forward CGwhich is lifting the nose.

More Plight Tests

Most mod els will be ca pable of a loop. If the CG istoo far aft it will loop tightl y with very little up eleva torand will be prone to sc rewing out of the loop. Roll ing(or flicking) out of tight loops and stee p turns is a signo f an aft CG (or too much eleva to r movem ent) . With aforward CG the loop will be large, or it may not havee no ugh up elevator autho rity to ge t around a loop at all.

My favo ur ite check is inve rted flight. Half-loop (o rhalf-roll) the model and apply down eleva to r to holdinvert ed flight. If yo u need less than abo ut 3 mm o fdown movem ent o n the stick then I suggest a furt herfo rward CG - you have plent y o f co ntro l bu t insufficients ta b il ity . If yo u ne ed m o re than hal f th e d own

124

mo vem e nt to fly steady inve rte d th en th e mod el ha smore than eno ugh stab ility and the CG co uld be movedaft a little to improve co ntro l. Aeropl anes with cambe redwin gs or low power may not be capa ble of susta ine dinvert ed flight , so co nsider the othe r tests.

Che ck out how the eleva tor trim varies with airspeed .Se tting a trim position sho uld se t a flying speed (if theth ru st line is right and th e st ruc ture stiff) . Trim yo urmod el to fly fast , the n retrim it to fly slow ly, and thecha nge in trim indi cates how much sta bility you have .With a very aft CG the mod el is se ns itive to trim change;a very sma ll trim cha nge is need ed between high speedflight and low speed flight.

It is easier to fly a mod el wh ich is positively speedstable. A good se t up for train ers and scale mod els is toad just the CG and trims so that with fu ll up trim themodel is flying as slow ly as it can, right o n the stall, andit will fly level at its maximum speed with the trim halfway between the cent re and full down.

\Vith the trim se t for a fast cruise , close the throttle (ifyou hav e one) and maint ain level flight o r a slight climb.Check that yo u have enough up elevato r movem ent tostall the mod el. If the CG is ve ry far forw ard then full upma y a llo w the a irc ra ft to fly nos e d own , abo ve itssta lling speed, leaving yo u witho ut eno ug h co ntrol toland the mod el properly.

I am sure there are man y other useful l1ight tests forCG position , but th e fina l test for me is the spi n . Amodel w ith a n aft CG will sp in eas ily whil e w ith afor ward CG it will not sp in at a ll. Adjus t it how yo u likeit.

Popular Misunderstandings

If a new mod el "kee ps wanting to climb" , do you putlead in the nose to move the CG forward? No , you adddown trim . You adjust the elevator trim to ge t the modelflyin g stra ight and level 'hands o ff' and then performso me of the tests above.

You find after lan ding that the eleva tor need s to bed ow n for trimmed flig ht, d o yo u mo ve th e CG tore move the trim requ irem en t? No! If the flight tests weresatisfacto ry, it just means that the wing (o r tail) is on atthe wrong ang le. To co rrec t for down eleva tor pa ck thew ing 's TE up o r the LE down, o r angle the tailp lan eitse lf mo re leading edge up if that is eas ier (and viceve rsa for up elevato r). The riggin g angles o f the wingand tail a re nothing to do with stab ility, they only getthe mod e l to fly in trim .

"I've change d the win g section from se mi-symme tricalto a less stable, flat bottom ed , section:- sho uld I changeth e CG?" No! Th e re is no suc h thing as a sta b le o runst abl e w ing sec tio n . Th e wing sectio n d o es no ts ign ificantly affec t the s tabi lity , but yo u may have tochange the wing or tail rigging angle for trim .

"If I change to a lifting sectio n ta ilplane shou ld Imove the CG to co mpe nsa te?" No! Again, cha nging thetail 's se ction doesn't affec t stab ility - and it does NOTmake it lift! In the following sections yo u will not ice thatthe wing o r tail sections or rigg ing angles do not mattersignificantly in stability and so are not invo lved in therules and formu lae for CG positio n.

"The CG shou 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 noscientifi c co nnec tion so ignore such ad vice .

"The CG should be just in front o f th e Ce ntre o f


Pr e ss u re ." NO ! Suc h adv ice be tra ys a la c k o funderstanding, so igno re it. The Centre o f Pressure ofthe win g is a floatin g point wh ich on a ca mbered wingmoves from 30% chord at slow speed to infin itely farbeh ind the wing in a dive . The C of I' is ahead of thewing whe n inverted. On a symmet rical section wing theC o f I' is fixed at 25% cho rd with the CG usually be hin dit.

"In the dip tes t the mod el keeps diving , so the nosemust be too heavy?" No! There is not eno ug h stability torecover from the d ive so move the CG forward (a ndcheck for struc tura l flexibility). The new CG will needmore up trim.

Where should the CG ofa newmodel be?

For a kit or p lan built model start where the design ersays, though personally I woul d check first having seena few se rious ly wrong ones. The CG will be mark ed onthe plan by a symbol like those in Figure 22.1. Start w iththis positio n and do the flight tests described above tosee how th e d es igner's CG su its YO U, and adj us t ifne ce ssa ry. No w, s u p pose tha t yo u r mode l comeswithout a suggested CG positio n, or yo u don 't trust theplan, or yo u have design ed you r own, w here do yo ustart? Where is a good CG position ? Do yo u start with afo rward CG? And wh at IS a forward CG, or an aft CG?You mus t wonder when I men tion a forward or aft CG,Forwa rd or Aft relative to \XTHAT?

What Matters?

You ofte n find the CG marked on the fuselage , but ifyou move the wing the CG must go with it because it'sthe wing that matters. The CG may be marked on theroo t chord or tip chord of the wing, but if you changethe sweep of the wing then the CG has to cha nge. Th eCG is referenced to the MEAN CHORD, but it co uld beat 50%, or 30% or 15% or even ah ead of the wing meanchord because its position depends upon the WHO LEAEROP LANE, not just the wing.

I w ill now bring th e Ce ntre o f Pressu re in to th ediscussion, but only to throw it straight out aga in . It is afloating point o f no relevance to anything, ce rta inly notthe CG. Any advice to position the CG relat ive to theCe nt re of Press ur e , o r th e thi ckest par t o f the w ingsho uld be treated w ith deepest susp icion .

The NEUTRAL POINT is the referen ce point for theCG. The NP is defined as the CG position at which you

Figure 22.2

Stable

Figure 22.1

tsge t Zero, o r Ne u tra l, Sta bility (the ball o n the po oltable). An aft CG means back close in front of the NP (aball in a sha llow d ish), and a forward CG means wellahead of the NI' (a ball in a deep dish). See Figure 22.2.

T he Ne u tra l Poi n t (N P) b e longs to the WH O LEAEROPLANE and is where the STABILITY FORCE on thewho le ae rop lane acts . This force (marked XL o n Figure8.9) is ca used by a p itch ang le cha nge, and it is th isfo rce w hic h ro ta tes th e a irc ra ft abou t its CG ba cktowards its trimmed pos ition . The ma in part of th eStability Force co mes from the win g, but the tail makes amajor contribution and all o ther parts of the aeroplaneshou ld be co ns ide red.

In a Nutshell

To calc ulate a CG position we need a refe rence onthe aircraft, and for this we use the average win g cho rd.\'lI e need to find the wing's mean chord, measure it, andmeasure the model's tail and fuselage, and in fact anypa rts with s ignifica nt horizon tal area . The n we estimatewhere the Ne utra l Poi n t wil l be. We w ill choose aStabi lity Margin wh ich will give the kind of han dling wereq uire , mark that off as a frac tion of the mean chordahead of the NP, and there we have it a sa fe CG for thefirst test flight.

Mean chordsThe first ste p in any CG calculation must be to find

the mean (ano the r word for average) cho rd and transferit to a s ide view o f th e aerop lane . If the wi ng hasrounded tips , just squa re the m o ff with a chord lin epa ra lle l to the root cho rd . Mak e th e ex tra area youcreate in the corners equa l the are a you cut off at the tipas in Figure 22.3.

GMC

It is customary to draw the roo t cho rd of the wing on

More Stable

3. Reactioll

1. Initial position

Basic Aeronauticsf or Modellers

2. Disturbed..- position

2. Disturbed position

125


MAC

the aircraft centreline when calculating its chord andArea (gross area) . The "Geometric Mean Chord" (GMC),also called the "Standard Mean Chord" (SMC), on atapered wing is l1 (root c + tip c), and it is found halfway between the centreline and tip . For example on awing with a 10" root and a 7" tip the GMC is l1 00 + 7)= 8.5".

C,.

C,.

Tipchord

Ct

MeatlAerodynamicchord MAC

MAC4

Ct

AC

RootchordCl-

chord. You can of course draw both diagonals and missout the 50% chord line . This method finds the locationof the MAC, which you then measure , and project ontothe centreline.

Ct

Taper T = Tip C

RootC

DistanceD

Semi span A

The Mean Aerodynamic Chord (MAC from now on) isthe technically correct refe rence length for calculatingaerodynamic forces and moments , and hence CGpositions, but the MAC is defined in terms ofcomplicated mathematics. The MAC is always biggerthan the GMC, but only very slightly bigger, unless thewing is sharply tapered. For a simply tapered wing wecan use the simple equation

20+t+t 2)

MAC = root 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 from-

d = aO +2t)/30+t)

For example , for the same wing as before with a rootchord of 10" and tip chord of 7" taper ratio t = 0.7 andthe MACis 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 thedifference is hardly worth bothering with . However, if awing is more sharply tapered, say the root is 21" and thetip is 6.3", the taper t is 0.3. So the GMC is 13.65 but theMACis 14.97" or IS" in practical terms. This time there isa significant difference, especially if the wing is swept aswell. The formula for d will tell you where on the wingit lies, to find the fore and aft position (vital if swept).

An easy way of finding the MAC of a tapered wing isthe graphical method, illustrated in Figure 22.4. Youextend the root chord forward (or aft) by the length ofthe tip chord, and extend the tip chord aft (or forwa rd)by the length of the root chord (whichever fits yourpaper) . Then you join the points just marked with adiagonal line . Where either of these diagonals crossesthe 50% chord line marks the position of the mean

Drop the Formality

For most models, with little or no taper, it matters notthe slightest which mean chord you use, because thereis next to no difference. On deltas and sharply taperedwings I use the MAC, found as in Figure 22.4, butnormally I just use whatever is easiest, drop the formalcapital letters, and call it the mean chord or averagechord.

When you find the location of the mean chord it isimportant to transfer it accurately to the side view of thefuselage, in the correct fore and aft position as indicatedon Figure 22.4. While there, mark the AerodynamicCentre (AC), or quarter chord point:- that 's a quarter ofthe mean chord from the front.

Elliptical Wings

Sometimes the wing of an aeroplane has an ellipticalplanform (the Spitfire just springs to mind) or sometimesjust the outer panel is elliptical. The MACof an ellipticalwing panel is 85% of its root chord, and you will find it53% of the panel's span from its root chord, as shown inFigure 22.5. The panel Area = 0.785 x span x root chord.This also works for semi-circular panels, by the way , asthey are just special ellipses.

Combining Panels

Use the following method for wings with two panels,


wh ich may ha ve differen t tapers a nd di fferent sweep back . Findthe MAC of eac h panelusing th e formula eabove or the graphicalmethod in Figure 22.4,and then join the meancho rds of th e 2 panelsas in Figure 22.6. Thedistance y in Fig 22.6 isfound from

(Al · y1 + A2 · y2)y =

(AI + A2)

Figure 22.5

Sp m l2a

53% a

MAC

and the length of the co mb ine d mean chord ca n becalculated from

combined mean chord(MAC1 - MAC2)(y 2- y)

= MAC2 + -------,..--'-------''-(y 2 - y1)

including the area inside the fuselage) and net tail area( i,e . only th e ar ea o ut in th e a irflo w) . Tail a rm ismeasured between the quarter chord po ints of the wingand tail mean cho rds.

Wo rk it out o n the Nomogram in App en dix E, FigureE.5 or from

,,- --- -- - - ,- - ----,,,,,

,,,_______ _ : L _

,,,,MA:C 1

Having found the MAC o f the wing, the tail and th eforeplane , transfer them all accurately to the ce ntrelineof the aircra ft and mark them on the side or plan view,or on the aeroplane itsel f.

Panel ZArea A2

tail area tail arm----x-----win g area wing MAC

V-bar

Use the following proced ure to find the mean chordof a biplane, o r a sesquiplane with unequa l wings . Forexample the Fokker DVII has two unequa l wings withnorma l (positive) stagger, Le. the top wing is ahead ofth e bo ttom wing . I worked o u t tha t th e top wingsup p lied 61% of the to tal wing area , and the bottomwing obviously 39%.

Referr ing now to Figure 22.7, I joined the mean cho rdlines of the top and bottom wings. I then divided thega p in th e rati o o f th e wing area s, 39:61. Th e meancho rd is 39% of the gap from the upper wing . (Someauthors bia s the mean chord more toward s the top wingb ut I have kept it s im p le r a nd mad e ge ne ro u sallow ances else whe re.)

The CG formu lae in Cha p te r 8 do not a p p ly tobip lanes b e ca use there is nearly twi c e a s much

Biplanes

Mea" ch o rd ofcom bina tion

,,,) '1 :

- - --- -- i>;

Figure 22.6

PallelIArea A l

Tail Volum e Ratio, V-bar, is the tail area , as a frac tionof the wing area , times the tail arm as a mu ltiple of thewing mean cho rd , and itis a mea s ure o f thee ffec t iveness of th etailplane , o r ho rizontals tab il ise r. A s mall tailw ith a long le veragearm will have the sa mee ffe c t a s a large tailneare r to the wing.

Being dim en sionless,it is th e s a me at anysca le , and us ing anyuni ts. You can measureth e fu ll s ize ai rcra ft infe et , yo ur model ininches , o r a sma ll scaledrawing in mill imetres ,and yo u get th e sameanswer.

It is u su a l to u seg ro ss w ing a re a (i ,e ,

If yo ur model is a flyin g wi ng yo u need go nofurther. Put the CG at 15% of the mean chord to startwith, and rig some elevon reflex for trim .

Having found the mean chord , and measured it at say8.)", mark yo u r mean chord o n the sid e view in thecorrec t locati on , and mea sure hack 15% of 8.)" , that's1.275" back from the front of the mean chord , and thereis your CG. However most aircraft have a tailpl an e or , asthey say in Amer ica, a horizontal stabili ser and you willhave to make allowa nces for it.

Tail Volume Ratio

Flying Wing

Basic Aeronauticsfor Mode llers 12 7

The actu al lift on a compo ne nt does not matter. \'\feneed to wor k out the lift curve slope, or LIFT CHANGEper degree of angle change, and multiply by the area .

downwash as the formulae allow for , and tw ice as muchinte rfe ren ce to the airflow , Le . loss of airspeed over thetail. However, I have fou nd that if yo u reduce both theTail Volume and the win g Aspect Rat io by dividing themby the number of w ings, then putti ng th ese redu cednumbers into the original formula gives a se ns ible NPshift from the nornograrn in Figure E.7.

In the case of the Fokker, I ca lled the top wing 1 andthe lowe r wing is 0.64 of it, so the number o f wings is1.64. That gave a reduced AR of 3.3 and a reduced V-barof 0.24 which in my usual Cha pte r 8 formula gav e a CGposition at 18% of mean chord, whi ch gave sat isfacto ryhandling.

Most of the lift change we are chasing co mes fromthe wing and acts through the Aerocentre , or quarterchord point (o f the MAC). If the aircra ft is a flying wingth at is the e nd o f the story. The NP is at the wing 'squarte r chord point, or 0.25 MAC from the LE. So theequation for the NP of a flying wing is simply

Th us we need to co nsi de r th e a rea o f eachcom po ne n t, th e pointo n it wh ere th e liftc ha nge a ct s , a nd it sre la t ive effic ie ncy inturning a o ne d egre epitch up into extra lift.

O n a n aerofo il th elift cha nge ac ts a t th eAero -d yn am ic Ce n tre,o r AC (q ua rte r ch ord) .O n o the r o d d bitso f area, like the nose ornace lles , I shall assum e ,for co n-sistency, that theex tra lift ac ts a q uartero f its le ng th from thefront.

All parts behind thewing ar e le ss effectiveb eca use o f th e w ing 'swake and downwash.

Air wil l spill aroundthe fu se lage s ides, redu cing it s lift c ha ngeco ns idera b ly , unle ssth e re is a ca na rd o ne ithe r s ide . Let 's takeone bit at a time , and as

an example I'll use my du cted fan Spectre , illustrated inFigure 22.8.

NP posit ion = 0.25as a fraction of the MAC aft of its LE. As imp lied above, Iadvi se a Stability Margin of 10% MAC for flying wingsgiving a CG at 15% MAC.

For other aircraft we can start from this neutral pointand mak e adj us tmen ts for a ll th e o the r pa rts o f theai rcraft whose litt le bits o f add itio na l effect are to be

adde d or subtracted .

The Wing's the Boss

A I. Cl + A2.C2Mean cbord c = A l +A2

Lower toing area A2

MEA N CHORD

\

\ Al

---=-~~--W__ Gap x \ Al + A2

Gap

____ __g.1.-O:..I~«- Q _

\

\

Ch o r d Cl

\

Top uiing area A l

\

\

Figure 22.7

Finding the NP

Figure 22.8

V-bar

AdjustmentsThe Tail

You could g uessfrom its o the r nam e , thehori zontal stabiliser, thatth e ta ilpl ane is import ant. Following as itdoes a co uple of chordsbeh ind th e w in g , it isimm ersed in the w ing'sw ak e a nd d o wnwash ,and has a lowe r aspectratio. The re are co mplex


and accurate ways of wor king out the tail 's effec t, butmy empirical es timate has proven close eno ugh and iseasy to appl y.

NI' co rrec tion = 0.25*(AR)t\0.25*V-bar

On a calculator with squa re roots enter the wing AspectRat io (s pa n/chor d), tak e the sq uare ro ot , and presssqua re roo t again . Multiply by 0.25 and the V-bar. Forexam ple, my Spectre 's Aspect Ratio (spa n/cho rd) is 4.19,the squa re root is 2.047 an d the square root of 2.047 is1.43. Ta il volume is 0.292 and th is gives

NI' correc tion = (0 .25 x1.43 x 0.292) = 0.104(positive as it moves the NI' aft)

The Tail moves the NI' aft 0.104 times the !vIAC, from0.25 MAC to 0.354 MAe.

If you do not like the formula, you ca n't go wrongwith Figure E.7 in App endix E, a nomogram w hich iseasier than "join the dots". Ju st mark the wing's AspectRatio (spa n/chord rat io ) a n d the V-ba r , a nd th esuggested NI' co rrec tio n is on the middle lin e . For abiplane use 'fac tore d' AR and V-ba r as discussed abo veunder "Biplanes".

The Fuselage

We norm ally assume that there is on e chord length offuselage ahead of the wing and 2 or 3 cho rds beh ind , soby using the gross wing area we hav e taken a suitableproportion of the fusel age area into account, and wehav e assumed that th e quarte r le ngth po int of th efuselage coinc ides with the qu arter chord point of thewing. On most aircraft that takes care of the fuselage ,but the Spectre has a very long nose . The unshade d pa rto n Fig ure 22 .8 would be a "no rmal" fu sel age . Th eshade d part is "Excess area" and will tend to de-st ab ilisethe aircraft. Rou ghly squa re off the curves to measurethe fuselage area which is more than one !vIAC len gthahead of the wing Aero centre, and make it into a nosevolum e rat io by d iv idin g by the wi ng a rea a ndmultip lying by the distance of its "quarter chord" pointfrom the wing Aeroce nt re, d ivide d by the !vIAe.

So n ose vo lu me Vn = ( nose a rea x no sedistance)/(wing area x wing MAC)

In the case of the Spectre this turned out to be (90 x30)/(553 x 12.63) = 0.387

Th is area is not very e ffec tive so m ulti p ly by aco nstant 0.2 from Tabl e 22.1 to allow for the air spillagearound the nose. The no se moves the NI' (forwa rd) by0.387 x 0.2 = 0.077 or 7.7% of MAe.

The Poreplane

the air has less cha nce to "spill" around the nose withthe fore plane in the way.

LE Extensions

On the Spectre I have treated the wing LE extensions(coloured dark in Figure 22.8) in a s imi lar way to"Io re p lane " area . Th eir ar ea times dist ance ah ead o fwin g Aerocentre gives a volume of o nly 0.025, and theyare suc h narrow strips that ai r will sp ill aro und themgiving a Fac tor of only, let 's say, 0.4. They move the NI'forward by a volume rat io Vf = 0.025 x 0.4 . or only 1%of !vIAe.

For any othe r od d b its of area on your model mak eallowance in a sim ilar way and use your judgement andTable 22 .1 to choose a su ita b le vo lume ratio andefficiency factor.

Floats

If an airc ra ft is co nverte d to a seap la ne by fittingfloats, the NI' is likely to change . Floa ts on a D/ F Spectrewould st re tch cre di bility too far, but my So na s sportae roba tic model has a span of 73", w ing area of 993 sq .in ., and flo ats 40" long with 300 sq . in . of area. Th efloat s are rigged with th eir ce ntres abou t on the CGwhich puts their qu art er chord point 9" or 66% of thewing mean chord ahead of the wing Ae. Float area is300/993 or 30% of wing area so the ir volume coefficientis 0.2. Multiplying by the factor 0.2 for long slim floatsfrom the table gives 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 onepanel, measured along its surface, times two , times itsave rage chord . Its projected area on a hor izont al surfaceis not , as yo u m ight su ppose, its effective area ashorizontal stabilise r.

Figure 22.9

Include shadedfuselage areawith foreplane

Foreplane armF

The forep lan e , or ca na rd , sits right ou t in fro nt inund isturbed air, so it is as efficient as the wing . Wor ko ut a Fo replane vo lu me u s in g th e n omogra m inAp pe ndix E (Figure E.5) or the formula -

Vf == forep lane area x foreplane armwing area wing mean cho rd

Use the forepl an e 's gross area (acting at its q uar terchord) , and incl ude th e fu sel age area ba ck to th eforeplane trailing edge as shown in Figure 22.9. I use thegross area and a factor of 1 (or even 1.2 if the cana rdhas less sweep and a high er AR than the wing) because


Fitting floats to Sonas, a 73" span ASP 108powered sport model moved its NPforward all estimated 4% oftoing MAC.

For di he dral o r anhe dra l tail s , or slo ping fins , th eeffective area is the tot al area times the cosine of thed ihedra l a ngle , SQUARED. Yo u h ave to square itbecau se the area is redu ced by the slope, and so it itsangle of attack.

For example , if the included ang le of the Vee is 110degrees then the dihedral angle is 35 degrees, cos35=0.8192 , cos squared is 0.671, so the area effective as atail is 67% of the to tal area .

Putting it all together

On a complex aeroplane with engine nacelles , ata ilplane and l or canard an d a long nose the NI' form ulaw ill use the wing MAC as a refe rence , sta rt with thewing AC position at 25%, add the stab ilising effec t ofparts aft of th e aircraft like the ta il, and sub tract thedestab ilising effects of the foreplan e and excess nosearea to end up as

NI' = 0.25 + tail bit - foreplan e bit - nose bit

In the case of the Spectre th is gives

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 FactorWing 1Tail 0.25 x ARwl\.25slim nose 0.2fat no se 0.4Wing strakes 0.4ca nard (foreplane) 1

high AR canard 1.2LENace lles 0.4Aft Nace lles 0.2Floa ts 0.2

130

The ta il, forep lane andnose vo lu mes are eachmult ip lied by a facto rchosen using the tabl eas gu idance. A nose isslim if its length is, sa y,mo re than three times itsaverage width , while afat nose is as broad asit 's lo ng . A fore p lane 'sAR is high if it is morethan the wing 's .

Stability Margill

Reme mbe r ne a r th ebeginning (Figure 22.2)w he n I compared th eSta b ili ty of a n aircra ftflying along in trim to aball in a dish? Well theSta bi lity Marg in is th esteepness of the dish . In

a very sha llow dish the ball will just gra dua lly ro ll backinto the ce ntre . In a deeper , steep side d , dish the ballwill retu rn to the centre more quick ly.

If a trimmed aircraft is pitched up slightly there willbe a STABILITY FORCE at the NI' which will rotate itno se down about the CG. The further the CG is ahead ofthe NI' the more leverage this Stab ility Force has so itsgre ater moment will restore stability more quickly. That 'swhy a forward CG gives more Stability.

How much Stability Margin you use depends on youa n d the mode ls you fly a n d your technique andexperience. I always recommend a SM of 0.15 or 15% ofMAC. In the event of a 5% error in NI' position (about asclose as we ca n calc ulate ) we still have a flyab le aircraft.However I know that co mpe tition gliders are flow n withless stability, and it seems that jet fight e rs and sport jetsuse 10% (o r less) Stability . Cana rds and flyin g wingsshould use a SM of 10% of MAC. After test flying yo ucan adjus t it further aft as you wis h.

On my Spectre the NI' ca lculate d above is at 0.267chord or 26.7% of MAC. A Stability Margin o f 0.1 (or 10%MAC) gives a CG position of 16.7% MAC which is 0.167x 12.63" = 2.1" af t of the MAC Leading Edge . Th at isabo ut 0 .1" (o r 2.5 mm) a head o f th e manufacturer'srecomme nde d point, and it flies just fine at that.

Figure 22.10


AppendicesAppendix A Bernoulli's Equation

This e q uatio n , which was d e rived th eoret icall y ,describes how the pressure var ies with velocity in theflo w . It holds go od if th e flow is stea dy, e ne rgy isco nserved, the flow speed is we ll below the speed ofso und , and the fluid has no viscosi ty. It therefore appliesapp rox ima tely to air flowing around slow aerop lanes,e xce p t in th e bou ndary la ye r wh ere v iscosi ty isimp ortan t. At any point along a particular strea mline .

• p + Y, P V2 = co ns t (ca lled th e "to tal pr essure" or"stagnation pressure" of the flow)

• where P = stati c pressur e at the point as measured bya barometer moving with the fluid.

• P = air density• V = air ve loci ty at the point.• y, p V2 is called the "dynamic pressure" of the flow .

A "p ito t" tube (a n open ende d tu be facin g th e flow)measures th e "to ta l pres su re ". An airspeed ind icat o rs ub trac ts s ta tic pressure fro m tot al pressu re to ge tdynamic p ressure from w hich ai rspeed ca n th en becalculated.

Appendix B Boundary Layer

It is a fact th at fo r all flu id s flo wing past a solidsur face , wh ether water in p ipes or air over wings, themolecul es of flu id next to th e surface d o not mo verelat ive to it. As the relat ive ve loci ty at the sur face iszero , it follows that there mu st be a region in the flowwh ere the velocity rises gradua lly from ze ro to the freestrea m velocity. The region may be very thin but withinit the velocity rises co ntinuo usly even if rap idly. Therecan be no abrupt step in veloci ty. This region is calledthe "Boundary Layer".

Two Types ofFlow

Two types of flow can ex ist in the bo undary layer. Ifspeed is low or viscosity is high , the flow will be smoo thand laminar with no vertical movem ent, like man y thinsheets moving over eac h othe r, each slightly faster tha nthe one be low . Th e ve loci ty p rofil e in the boundarylayer is show n in Figure Bl.

But if spee d is high or viscosity is low, the flow will .be turbulent. In turbul en t flow there is greater mixing o fthe particl es and so the boundary layer is not so slowbut gives more dr ag. The slow moving flu id part iclesnear the surface are continu ally re-en ergised by mixin gwith the faster particles from furth er out in the streamgiving great er drag. The veloc ity profile is as in Figure


B2. A go od demonstration of Lam inar and Turbule ntflow is shown in Figure B3 wh erein a large tank of stillwater is run off through a glass tub e . At the inlet of thetube a sma ll filamen t of liquid dye is introduced into thewater strea m. At first the filament of dye is so steady asto appear stationary. Further along the tube it wave rsand then breaks up and mixes wi th the water as theflow transitions into turbulent flow .

Lik e th e w at e r in th e tube , th e boundary laye rcomme nces as laminar near the leading edge of a bod yand becom es turbulent at the "transition point" whoseposition depends upon speed, viscosi ty and surfaceroughness.

Figure Bl

Height

Vel

Fig u re B2

Height

Vel

131

FigureB3

Dye

Water

Laminar Flow Turbulent Flow

~~Transistion

Appendix C Vortices

FreeVortex

distance from the centre and it is seen when water isstirred round in a drum for example. The surface formsas in Figure Cl.

The other kind is the free vortex in which the speedof the particles of fluid reduces with increasing radiusand the surface would be as in Figure C2. (Particlevelocity is inversely proportional to radius). This is thekind of vortex we see when we watch the bathwater godown the plug hole, blow a smoke ring or see a tornadoapproaching. All natural vortices are these free vortices.However when the fluid speed reaches nature 's practicallimit, e .g. at the centre of a tornado, the vortex breaksdown at the centre into a forced vortex (forced by thefluid's viscosity) in which the speed reduces towards thecentre (see Figure C3). Wing vortices are of course freevortices, therefore the speed of rotation is greatest near

the centre and reduceswith distance from thevortex core.

Vortices can neverend abruptly exceptagainst a boundary, e .g.a container side, windtunnel wall, or the watersurface.

They may be in continuous loops as in asmoke ring or they maycontinue in the fluiduntil dissipated gradually by the viscosity .

Figure C3

FreeVortex

FigureC2

ForcedVortex

Figure Cl

There are two kinds of rotating flow. In the forcedvortex the speed of the fluid is proportional to its

Boundary Layer ThicknessThe thickness of the boundary layer is usually

defined as the distance from the surface at whichvelocity of air reached 99% of the free steam velocity.The boundary layer thickness grows as it travel over thesurface and a turbulent boundary level thickens morerapidly than a laminar one.

For example a larninar boundary layer might reach athickness of 1 mm after travelling 200 mm over a smoothflat plate. Whereas a turbulent boundary layer might be5 mm thick 200 mm after becoming turbulent.

ForcedCore

Appendix DDihedral andSweep

From Figure Dl, if an


FigureD} Pigure Dz

Airflow VelocitySideslip

ComponentV SIN D = u (Upuiash Component)

now subtract equa tion 1 - equa tion 2 to ge t

• PQ = d/ cosL• XQ = d/cos(L-A)• YQ = d/ cos(L+A)

2Xa = cos (L-A).y/ d - cos (L+A).y/ d= y/ d .(cosL.cosA + sinL.sinA - cos L.cosA + sinL.sinA)= 2y/ d .sinL.sin A equa tion 3

equation 1equa tion 2

ex + Xo = cos (L-A).y/ dex - Xo = cos (L+A).y/ d

at ang le L. The air flows in the direction PQ across thecho rd . But whe n the ae roplane is sides lip ping , the airflow s along YQ for left sides lip , and along XQ for rightsideslip .

Both sideslip angles are A. From the figure you ca nsee that distance

Th e leading edg e is at a co ns tant height Y a bove thetrailing e dge so as we c a n see from Fig u re D.5sin (n -X«) = y/XQ ' =. (o -rXu) if a is fairly small, so that

FigureD3

ResultantAit:flo,!:! _ r - - ~ u-- _ .. '

Xc< V

ae roplane has a small sideslip veloci ty, then the angle ofsideslip A is given by• A ' = . v/V

Figure D2 is a view from the rea r o f an aeroplan ew ith a d ih edral a ng le D. In Figure D2 th e si des lipvelocity v has been sp lit in to two co mponents - o nepar all el to th e wing w hi c h ha s no e ffe c t , and o neperpendicul ar to the wi ng w hich w ill be a n up wa shve loci ty u on the right win g and a downwash on the leftwing. From Figure D2 the upwash co mpone nt u = vsin D

From Figure D3 the ex tra ang le of attack du e to theup wash (Xa) is approx imately given by

uXa =

V FigureD4

and th ere is a corre spond ing d ecreas e o nthe left wing .

Therefore

v.sin D DXa = - - - =A .

V

if both angles are small.All angles mu st be in

radian me asure . (O neradi an is an a ng le of180""it degrees =

57.29578 degrees).The change in angle

of attack du e to sideslipderived from dihedral isg ive n by Xa = A.Dwhe re

A = sideslip angleD = dihed ral ang le .

Figure D.4 s h ows a naerop lane with parall elcho rd win gs swept back

I

I

I

1Straight Airflow

ISidesiippingAirflow


Figure D5

~

Airflow

From Figure D5 a y.cos l./ d which when put ineq uation 3 gives

2Xex = 2ex tan L.sinAso Xo = ex sinA.tanLor Xn = ex A.tani

if the sideslip angle is small. All angles mu st be in radianmea sure. The change in angle of attack du e to sideslipderived from sweepback is given by

Xa = a A tanL wh ere

• A = sideslip angle• L = an gle of sweep-back• a = angle of atta ck (average)• Xo; = change in an gle of atta ck

(average between 2 wings)

Th e follow ing books have been my main referen ces.They are listed as Auth or, TITLE, publisher (notes).

Reference List

5. A C Kermode, MECHANICS OF FLIGHT, Pitman ,(general background).

1.

2.

3.

4.

A W Bab ist er , AIRCRAFT STA BILITY ANDCO NTROL , Pe rgamon Press , (for e q uatio ns ofairc ra ft st ability and trim , tail loading, and tailsetting angle) .

F G Irving , AN INTRODUCTI ON TO THELONGIT UDI NAL STATIC STABILITY OF LOWSPEED AIRCRAFT, Pergamon Press, (for equation sof aircraft stability and trim , tail load ing, and tailsetting angle).

Ab b ott and Doe n h o ff , THEORY OF WI NGSECTI O NS, Do ver Publi cations , (Masses ofinfor mation on the NACA families of wing sec tionsand family relationships, alb eit at high Re) .

N A V Pie rcy , AERODYNAMICS, English Universities Press Ltd, (wing downwash and gen era lbackground information).

MOD ELLFLUG , Neckar-Ve rl ag, ( exce lle n tinformation on model sections at model Re withflaps and turbulators , but the text is in German).

9. Selig Donovan & Fr a ser , AIRFOIiS AT LOWSPEEDS, Soartech/H. A. Stokely, (valuable test s onsections at model Re).

10. Selig Guglie lmo Broeren & Giguere, SUMMARY OFLOW-SPEED AIRFOIL DATA - VOL.1, Soartech/H.A. Stokely, (tes ts on more sections at model Re).

11. Selig Lyon Gig ue re Ninh am & Gu g lie lm o,SUMMARY OF LOW-SPEED AIRFOIL DATA VOL.2, Soartech/ H. A. Stok ely , (te sts on even moresections at model Re) .

The whole series of Soartech book s is availablefrom SoarTec h Publications, 1504 N. Ho rsesh oeCircle, Virginia Beach, Virginia 23451, USA

6. Den Hartog, MECHANI CS, Do ver Publicat ions ,(re la tive motion , momentum and kinetic ene rgyconsideration s for the cha pte r on wind).

7. Martin Simo ns , MODEL AIRCRAFT AERO DYNAMICS, Argus Books, (section data at low Re ,and ge ne ral background.Recommended readingfor co mpetition glider or free flight enthusiasts.).

8. Dieter Althau s, PROFILPOLAREN FUR DEN


Appendix E

This co llec tion of nomograms is meant to help you getanswers without even knowing you are dealing withmath em atical equations . It is just a ga me o f join the dots.This first one , Figure El , is to help you ge t started us ingnom ograms and also to save flying fie lds. O nce we bu ya mod el engi ne it is too late to worry abo ut its design .\V'e co uld bu y an add-on ex tra silencer, but the easies tway to make them qui eter is to reduce the tip speed ofthe prope llers. On ce the tip speed of the prop s exceeds550 ft/s (ha lf the sp eed of so und) the prop noise seemsto predominate and become un acceptable.

The exa mple o n the nomogram below is o f a 10"prop doin g 12000 rpm and its tip speed is just below thelimi t. To check yo ur prop , mark a dot a t yo ur propdiam eter o n line A, and a dot on yo ur eng ine rpm o nline C, join them with a ruler and read off o n line 13 yo urprop's tip spe ed in ft/s.

A Figure El C 30

35

3 0 25B

25 M acb2 20

20

15Macb 1

100

6

7

4

8

10

9

5

800

80Tip speed

10

9

7

8

6

5

15

3.54

Prop diameterin ches

Prop RPMill 1000s


Another simple nomogra rn. Figure E2, links the wingarea o n line A via the model weight o n line B to thean sw er, the model' s wing load ing in pounds o r oun ce sper sq uare foot , on line C.

O r the answer could be w ha t wi ng area is needed tosupport a g iven weight fo r a desired wing loading . O ryo u might want to kn ow the finish ed target we ight for amodel g iven its wing a rea and desired wi ng loading.

1500

154

60

3 50

40

2 1030

91.5

20 8

1 157

.8 6

.6 10

.5 8 5

.46

54

.s4 3.5

.2 3 3Ibs/ft2 OZS/ft2

Ibs OZS WillgWeigbt Loading

Basic Aerona uticsfor Modellers

C

B100

690

150

2000 5 80

100 701500

80 4

7060

60 1000 3.5

50 800 503

40 600

500 2.5 4030

400 352

30

251.5

20

1ins2 ft2

Willg A,'ea

150

1000

900

800

700

600

500

400

3.5

300 2

250

1.5200

Figure £2

A

30

4000

25

3 00020

2500

152000

136

You ca n use the wing load ing fro m Figure E2 to findout from Figure E3 what speed your model will fly at aparticular lift coefficient , or wh at lift coefficient it needsfor a pa rticular speed. In particul ar yo u can estimate thestalling speed.

Accord ing to Selig's tests in Soa rtech 8 the Clark Ysection at mo de l Reynolds Numbers has a maximu m lift

coefficient of 1.2. Becau se of uneven lift d istribution andlo sse s aro u n d th e fuse lage th e ove ra ll wing li ftcoefficient whe n it stalls may be about 1. Plott ing 1 online C and a w ing -loading of 20 oz /sq . ft. o n line A givesa likely stalling speed in stea dy level flight of aro und 22ftl s on line B.

15

138

0.1

CLLift Coefft

1

m2/ft2Willg Area S

N/lbLift

4

m/sft/sSpeed V

Basic Aerona utics f or Modellers

Tail Volume Ratio (or Coefficient) is conventionallywritten in textbooks as a capital V with a bar over it, butas 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 . Youwill find it in all the best CG formulae and design criteria(though sometimes in disguise) . Basically it is the tailarea, as a fraction of the wing area, times the tail arm asa multiple of the wing mean chord.

It is usu al to use gross wing area (Le. including thearea inside the fuselage) and net tail area (Le. on ly thearea out in the airflow). Tail arm is measured betweenthe quarter chord points of the wing and tail mean

chords. Start at the sides of Figure ES and work towardsthe centre. Mark the known values on lines A, B, F andG, join A to B a nd continue to C, join G to F andcontinue to E. Then join the points on C and E to getyour answer on D in the middle. Use this nomogram forcanard 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 andvintage models, and the odd scale model, can have Vbar over 1.

FigureE5

A B C D E F G

2000 30

V-bar = ST x.lI. 251500 1000 S c

800 15020

600 6 100500

1000 80 15400 5900 60800 300 1.5 504700 3.5

4020 1 10

600.8

3093

.6 8500 20

100 .5 2.5 7.2 1580 .4400 2 6

60 .3 1035050 .15 58

300 40 .2 1.56

30 4250 5

.1 420

2003 3

15

2.5150

Willg Area Tail (H Stab) Sp'S V-bar I p'c Tail Arm Willg MeallS Area ST IT Chordc


Before getting stu ck into ca lculating Neutral Po intposition s (o r even simple CG positions from Figure 8.

10) use these nomograms in Figure E6 to ge t your wingAspect Ratio from its span, and chord or area .

Pigure Bti A B C

30

20

15

10

8

6

5

4

3

2

600

400

300

200

100

60

40

30

20

10

6

4

30

20

15

10

8

6

5

4

3

2

Chord Wi"gSP"" AR

x y z

ills dmWi"gSP""

4

2

15

8

10

3

6

5

20

30

AR

4

8

10

15

6

5

20

50

40

30

15

20

40

80

60

50

30

150

100

200

8

15

10

100

20

40

80

30

60

50

100

150

200

400

800

600

500

300

1000

1500


To save yo u worki ng out sq ua re ro ot s of squareroots, this no mog ram (Figure E7) multiplies the fourthroot o f wi ng Aspect Rat io (o n the righ t) tim es Ta ilVolu me on the left to give the Neu tral Po int correc tionfrom the Tailplane.

The exa mp le, my Spectre , shows that a tail vo lume of0.29 behind a wing of AR 4.2 will shift the NP aft byabout O. 105 times wing !'vIAe.

Figure £7

.8

1

.7 .40

35 20

.6 30

.2510

8.5

6

4

.4 3

2

35

.25 .05

.04

.2

Tail VolumeRatio V·bar

NPSbift ·due to tail as a

fraction of toing MAC

Wi1lgAspectRatio

Basic Aero nautics for Modellers 141

If you were put off trying to work out your tailplaneangle to trim from equation 12.2 then try this nomogram(Figure E8), which comes in two parts. If you know thelift curve slope of your tailplane (unlikely) mark it,otherwise estimate it by marking the tail's aspect ratio online A. Draw a line from this point to your V-bar on lineC, then from where that line cros sed line B to Cmo (themoment coefficient of the wing) on line E. Where thisline crosses line D gives the first part of TSA (that due towing camber) 2 degrees in the example. It is negative(tail LEdown) if Cmo is negative.

On line V bring down the value of al V-bar from lineB. Draw a line across to the Stability Margin on line X,and then a line from the intersection on W to the liftcoefficient on Z.

The second part of TSA on line Y is another 2degrees in this example.

The total is 4 degrees. That is the angle between thetail and wi ng Zero Lift Lines, at th is lift coefficient. Thisis for trim , so of course it varies with speed, so choosethe speed at which you want the elevator neutral.

Ptgure Ed

A B C D E.2

.00 630 .2

.03.008 .25 2 01

.035 .01.3 10

.04 .1.015 .35 6

.084

.05 2 .02 .4 3 .06Z- .05

.06 3 _ .03 _.5 .04- - -4 - 1

.07 .04- .036 ":6- .6

.08 8 .0512 .06

.4.09 .7 .3 .02

.1 .08 .8 .2

Lift s lope Tail Aspect .9p er degree ratio .1

.01a , V-bar V-bar T.SA(camber) C.IO

p er degree degrees

V lV X y Z.01

.3

.25 2

.015 20.2

10 1

8 .8.02 6

.15 .64

.025 ,,- 3 .5- - -,,- .:l- - -~ - - .4

.03 ,,- .1 1 .3,,- .8.035 ,,- .09 .6

.08 .4 .2.04

.3.07 .15

.2.05 .06

.1 .1

.06 .08.05

.0 7 .06

.08.04

.0 4a,V-bar' StabilllJ' T.SA ( stab) CL

per degree Margill K" degrees


GlossarySymbols

~ 0

Abbreviations

Centre of GravityGre ek lett er alp ha , angle of attac kze ro lift an gle of a ttackstalling ang le of attack

PJ.L

Greek lett er pi , ratio of circlecircumference over diameterGreek letter rho, air den sityGreek letter mew , air viscosity

a lift curve slope , dCL/ do g gravitational constan t, oray tailplane lift curve slope, dCLy/ dnr (so metimes in " ") load factorAC Aerodynamic centre , or ae rocentre . GMC geome tric mean chordAR Aspect Ratio K a cons tant, in induced dragc wing chord formulaCo Drag coefficien t KE Kinetic energyCD; induced drag coefficie nt Ko Stability Facto rCoo profil e drag coefficient If forepl an e moment annCOmin minimum drag coefficient IN fuselage nose moment armCG Centre of Grav ity i, tail moment armCLor Cl Lift coefficient L LiftCLmax section or wing 's maximum lift LE Leading edge

coefficient MAC Mean Aerodynam ic ChordCLy lift coefficient of tail NP neutral po intC~I Pitching moment co efficient. q airflow dynamic pres sur e , V,pV2Cl\lo Pitching moment coe fficient, at qy dynam ic pressure of airflow over

zero lift tailC~ILE Pitching moment coefficien t, about Re Reynolds number

the lead ing edg e S wing areaCP Centre of Pressure Sy Tailplane areaD Drag TE Trailing edgeD\xrF DownWash Fraction, or in V velocity , a vec tor, speed in a

standard terminology, de/do particular directiondV a cha nge in velocity V-bar (o r Vhar) Tail Volume RatioF a general force XL Xtra Lift, an incre ase in lift

Common Aerodynamic Terms

If you cannot find what you want in the Glo ssary thenperhaps I could not easily expl ain it without a d iagram.Look for it in the main text via the Index .

• Aerodynamic CentreThe aerodyn amic cen tre of an aero fo il Section o r awing is the point about w hic h its pit ching momentdoes not vary w ith angl e of attack . Th e po int isp ract icall y a lw ays within 2% of 25% chord and isoften ca lled the quarter chor d point, c/4.

• Angle of Attac kTh e angle between th e d ire ct ion of motion and adatum line on a wing. The datum line ma y be th eze ro lift line , the chor d line at the root, or so me o the reasily defined line specified for the purpose .

Basic Aerona utics for Modellers

• Angle of IncidenceTh e angle between th e fuselage datum line and adatum lin e on a wing . Th e datum lin e may be th eze ro lift line , the chord line at th e root , or so me othereasil y defin ed line speci fied for th e purpose .

• Aspect RatioTh e span of a wing divided by th e mean chord . Or itis some times easier to use span squared divid ed bywing area . (Fig ure E6)

• BallastWeight carried e ithe r to ad jus t the ce ntre of gravity ofthe aeroplan e , or to increase its weig ht temporarily.

143

• Boundary LayerThe layer of air next to the surface of a mo ving bod y.

• CanardAn aeroplane whose horizontal stabiliser is ahea d ofthe wing . The hori zontal stab iliser may be ca lled the"canard" or forepl ane.

• Centre of GravityTh e balance point , or the point throu gh which theresultant weight acts. It is the point about which theweight has no moment. \V'e normally ass ume it lieso n th e ce n tre line and q uot e only its fore an d aftposition, ign or ing its height.

• Centre of PressureThe imag inary poi nt th rou gh which th e resul ta n taerodyna mic force ac ts. It is found by d ividin g them om ent coefficie n t abo u t a p o int by th e liftcoe fficient through that point.

• DihedralTh e ang le by which each wi ng tip may be raise dabove the height of the roo ts.

• DragTh e co mpone nt o f th e aerody na mic fo rce in th edirection exactly op pos ite to the direction of mot ion.

• Drag PolarThe Drag Polar of an aerofoil or wing is a plot of liftcoefficient on the ve rtica l axis against drag coefficienton the hor izontal axis .

• FlapsA movable port ion on the trailing edge of the w ingswhich may be lowered to increase lift and drag .

• IncidenceSee "angle of inciden ce".

• Leading EdgeThe leading edg e of an aerofoil or wing is the front,the pa rt whic h the airflow meets first.

• LiftThe co mpone nt o f th e aerodynamic force a t rightangles to the direction of motion .

• Longitudinal DihedralSee Tail Setting Angle.

• Mean ChordThe average cho rd of a tapered wing . (See index forMea n Ae rody na mic Chord a nd Geome tr ic Mea nChord.)

• MomentThe mom ent of a force abo ut a point is the amountof the force times its distan ce from the poi nt. It is amea su re of its twistin g , or turning effect about thepoi nt.

• Stability Margin (or Static Margin)A measure of the sta bility of an aeroplane . Physicallythe distance of the CG forward of the Neu tral Point ,

144

as a d eci mal frac tion or percentage o f th e meanchord.

• StallLoss o f lift and increase in drag because o f flo wseparation o n the wing.

• Stalling AngleThe stalling angle of attack of an aerofoil or wing isthe angle of attack at which it develops its maximumlift coefficient.

• StreamlinesImag inary lines d rawn to re prese nt th e flow o f aflui d , su ch tha t there is no net fluid flow ac ross astrea mline .

• Tail Setting AngleThe angle between the zero lift line of the tail and thezero lift line of the wing . Always tail lead ing edgedown. Some times referred to as longitud inal dihedral.

Taper RatioTh e cho rd at the wing tip divide d by the chord at thewing roo t, or at the fuselage centreline . (Express as adecimal fraction.)

• TipstallA violent wi ng drop ca used by flow se pa ration (astall) on the outboard port ion of one wing only.

• Trailing VorticesA pai r of vortices trailing behind the wing tips of aflying aeroplane . They rot ate in opposite directionssuch tha t the air between them is descend ing .

• Tuck UnderTh e tenden cy of so me mod e ls to p itch nose downwhen their airspeed is increased.

• Turbulator StripA stri p of ad hesive tape (usua lly) on th e top o f aw ing to improve its performance a t low ReynoldsNum be r.

• Vortex, VorticesA vortex is a ro tationa l flow.

• 'VashinA tw ist in a wi ng w hich increases the incidence at thewing tip . Not usually delib erate .

• WashoutA twis t in a wing whic h reduces the incidence at thewing tip . Often done delib erately.

• Wind GradientThe vari ation of wi nd speed and d ire ct ion w ithaltitude .

• Zero lift lineIf the Zero Lift Line of an ae rofoil section or a wing isaligne d with the airflow then the resultant lift will bezero.

Basic Ae rona 11ties for Modellers

Indexaerobaticsaerocentreaerodynamic balanceaerodynamic centreaerodynamic dampingaeroeIasticityaerofoil camberaerofoil sectionaerofoil thicknessaile ron dragaileron flutteraileron reversa laileronairair densityairbrakesalphaangle of attackaspe ct ratioautoro tationax is, latera l,axis, longitudinalaxis , verticalballastbending momentBernoulIibiconvex sectionbiplaneboundary layercambercambe r linecambered sectioncanardce ntre of gravitycentre of pre ssurecentripetal forceCG formulachordchord lineclimbcoefficient , dragcoefficient, liftcoe fficient, pitching momentcomponentcompressioncosinecoupledampingdescentdihedraldimpled ballsdirectional stabilitydivergence, wing


802757275896292829579897561313602020

32,34, 85, 14081

42,4342,4342,43

7511613129

87, 12723,91,13116,24 ,28

1629

85, 12941, 12317, 26

6345161678

19, 2319, 2019, 26

1411414155978

52,133254997

divergentdouble taperdownthrustdownwa shdragdrag bu cketdrag coefficientdrag polardutch rolldynamic stabilityelevatorseIevonelliptical loadingelliptical planformflapflap eronflat-bottom ed sectionflexibilityflick ro llflight testingfloatsflutterflutter, aileronflutter, tailflutter, wingforepl aneFrise ailerongeom etric mean chordgraphical methodgravityground effectgustshodograp hhysteresis loopincidenceincompressibleinduced dra ginterferen ce draginvert ed flightknife ed gelaminar flow sectionslaminar se pa rationlateral axislateral stabilityleading edgeliftlift coefficientlift cu rveLift/Drag ratioload factorlongitudinal axislongitudinal dihedrallongitudinal static stability

4136, 127

7032

18, 2324

19, 232455

41, 42, 59566135

35,37, 1266061294781

1231299898

10110012957

39, 125401334

11173932013357282832491

42,435016

16, 1819, 20

2024,72

6442,4348, 69

41

145

loopmassmass balancemean aerodyna mic chordmean cho rdmean linemomentnavigationneutral pointNewton 's Lawspendulum stabilitypitch dampingpitching moment coefficientpressureprop eller thru stReynold s numberrotational ine rtiaruddersemi-symmetrical sectionseparation bubblese pa ration pointshearsides lipsineskidding turnslatslipstrea mslope liftslotslugsmoke tunnelsnap rollspinspiral divespiral divergencestabiliserstabilitystab ility marginstagna tion pointstallstalling anglestalling speedstatic marg instatic stabilitystiffnessstrea mlinestrea mliningstressstructuresstrutted win gssweepbacksymme trical sec tiontail lifttail setting angleTail volume ratiotaileronstail-lesstangenttaper ratioten siontherm al liftth icknessthrottl etipstallingtorsiontrailing edge

146

8113

58, 10039, 12639, 125

1614, 15, 26

10943,44 , 128

135159

19, 2613,1 7

76905856299221

11450, 64

14646176

109611316818055554341

44, 13016

21 , 8021

65, 774441

1151625

114114118

38, 53, 1332967

48,69, 14245618614

36,38114109

16, 24, 286237

114, 12016

trailing vorticestrimm ed flighttuck underturbulator stripsturning flightundercambered sec tionV-barvectorvec tor compone ntvertical ax isviscosityVortex, vorticesV-tailwashin , (see washout)washoutwashout, ae rodynamicweightwind gradientwind tunnelwind shearwing divergencewing flutterwing loadingwin g strutszero lift line

3167

102946329

45, 1391314

42,4323

30, 13261,62 , 129

37373813

11018

11097

100136, 137

11820

Basie Aeronalilies for Modellers

Notes


BASIC ERO AUTICS FOR MODELLERSThis second edition skillfully guides the reader through the basics of oircroltflight and performance before addressing issues specific to model oircroltAlasdair Sutherland draws on his personal experience as a student, a pilot,and most importantly, an aeromodeller to present fundamental information ina friendly and easily accessible form. He does so by building the knowledgebase of the reader in a steady progressive manner, highlighting a number ofcommon misconceptions along the way. In this way, he ensures that thereader is prepared for each new section of the book as it is reoched.Thankfully, the use of complicated equations or tedious derivations which, ifexcessive, can olten deter the layman, is either avoided or they are providedin appendices.

T R AP L E TP U B __l _ I_C A T I ~O N

ISBN 1-900371 -41-3

9 781900371414 >

ISBN 7 900377 4 7 3

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Basic Aeronautics for Modellers