Riding and Handling Qualities of Light Aircraft

NASACONTRACTOR REPORT

N COPY: RETURN YO AFWL (DOUL) KBWTEAND AFB, N. MI.

RIDINGAND HANDLING QUALITIESOF LIGHT AIRCRAFT - A REVIEW AND ANALYSIS

by Frederick 0. Smetum, Delbert C. Summey, und W. Dondld Johnson

Prepared by

C AR O LINA STATE UNIVERSITYNORTH STATE CAROLINA ,, Raleigh, N. C. 27607 for LalzgleyResearch Center

NATIONALAERONAUTICS AND SPACE ADMINISTRATION WASHINGTON,D. C. MARCH 1972 TECH LIBRARY KAFB, NM

1. Report No. AccessionGovernment 2. No. 3. Recipient's Catalog No. - NASA CR - 1975 4. Title and7 5. Report Date RIDING AND HANDLING QUALIII?ES OF LIGHT AIRCRAFT - A REVIEW March 1972 AND ANALYSIS

"- 7. Author(s1 Organization 0. Performing Report No. Frederick 0. Eknetana, Delbert C. flunmey, andW. Donald Johnson None 10.Work Unit No. 9. MamingOrganization Name and Address 736-05-10-01-00 North Carolina State University at Raleigh Department of Mechanical and Aerospace Engineering 11.Contract or Grant No. Raleigh, North, Carolina 27607 NASL-9603 13. Type of Report andPeriod Covered 12. SponsoringAgency Name and Address Contractor Report National Aeronautics and Space Administration Washington, DC 20546 14. SponsoringAgency Code

I 15. SupplementaryNotes

16. Abstract Design procedures and supporting data necessary for configuring light aircraft to obtain desired responsesto pilot commands and gusts are presented. The procedures employ specializations of modern military and jet .transport practice where these provide an improvement over earlier practice.

General criteria for riding and handling qualities are discussed in terms of the airframe dynamics. Methods available in the literature for calculating the coefficients required for a linearized analysis of the airframe dynamics are reviewed in detail. The review also treats the relation of spin and stallto airframe geometry.

Root locus analysis is usedto indicate the sensitivity of airframe dynamics to variations in individual stability derivatives and to variations in geometric parameters. Computer programs are given for finding the frequencies, damping ratios, and time constants of all rigid body modes and for generating time histories of aircraft motions in responseto control inputs.

To the end that the work serveas a self-contained riding and handling characteristics design handbook, appendices are included presenting the derivation of the linearized equations of motion; the stability derivatives; the transfer functions; approximate solutions for the frequency, damping ratio, and time constants; an indication of methods to be used when linear analysis is inadequate; sample calculations; andan explanation of the use of root locus diagrams and Bode plots .

117. Key'(Suggested Words by Authoris) 1 10. Distribution Statement Stability derivatives Bode plots Unclassified - Unlimited Sample calculations Computer programs

I 19. Security Classif. (of this report) 20. Security Classif. (of this page) 21.No. of Pages 22. Rice* Unclassified . Unclassified 409 $6 .oo For sale by the National Technical InformationService, Springfield, Virginia 22151 " " TABLE OF CONTENTS

Page GENERAL I NTRODUCT I ON 1

Background 2

Objectives and Organization 4 Factors Contributing to the Pilot's Opinion of Aircraft Handling Qualities 6

Flight Motions of Light Aircraft-General Considerations 11

Control Forces and Deflect ions-General Considerations 14

I: I TERATURE REV I EW 15

Scope 16

Syrnbo I s 18

Stability Derivatives 27

Power Effects 110

Contro I Forces and Def I ect i ons 122

Inertial Characteristics 152

Stal I 156 \ 'i Spin Entry 165

Human Factors 179

Design for DesirableRiding Qualities 187

THE EFFECT OF VARIATIONSIN STABILITY DERIVATIVES ON THE MOTIONS OF A TYPICAL LIGHT AIRCRAFT 193

194

iii TABLE OF CONTENTS (continued)

Page

LongitudinalVariations 198

LateralVariations 203

REFERENCES 30 1

APPENDICES 31 1

A - Derivationof Equations of Mot ion 31 1

B - Definition of Stability Deriva ti ves 333

C - TransferFunctions 337

D - Simplified Response Characteristics 347

E - Use of theNon-Linear Forms of the Equations of Motion 353

F - Some Noteson the Construction and Interpretation of Bode Plots andRoot Locus Diagrams 355

G - Sample Calculations 363

Longitudinal Sample Calculations

Latera I Samp le Calculations

H - Computer Programs 379

Longitudinal Program

Latera I Program

Longitudinal and Lateral Sample Output

T ime Response Program

I - Bib1iography 393

iv GENERAL INTRODUCTION BACKGROUND

Recentengine advances, the upsurge Inbusiness flying, and generalaffluence have greatlyaccelerated the development of new and alteredmodels of generalavia- tionaircraft. Thesedevelopments follow thetraditional industry pattern of ev- olutionary change ratherthan revolutionary change.For example, a particular model may notbe.selling as well as expected, so it is determinedthat offering theaircraft with additional powerwould increase itssales appeal. It isthen up to the company'sengineering department to accomplish this refitting with a minimum of. expense.There are the obvious structural modifications to make, the weight and. balance to check,and then a check flight to determinewhether the ride and handlingcharacteristics remain, according to thesubjective opinion of the company pilot,satisfactory. It is, of course,not possible to explore,in this type of flight test program, all possible flight modes and conditions so that one may omit tests at a conditionwhich formerly was notuncomfortable or unsafebut has, because of thechanges become so. Thisseldom happens of course, butplace an aircraft with a largerengine in the hands of aninexperienced pilot, andhe will probably be surprisedthat the periods of thelongitudinal short period and dutch rol I modes increaseand their dampingdecreases; similarly, it is seldomexpected by the novice that moving the engine to thewing and adding a secondengine may, because ofthe nacelle design, give a seriouspitch-up condition at high ang!es of attack or thatproviding adequate static longitudinal stability or weathercock stability at some particular flight condition does not guaranteeacceptable dynamic characteristics. Yet these things can, givensuf- ficient and reliableanalytical andexperimental data, be forecast quite accurately andcan be included in the pilot's handbook.

To expectthe manufacturer of light aircraft to carryout the comprehensive preliminarydesign analyses, the extensive wind tunnel tests, and theexhaustive flight tests that wereperformed for the C-5A, for example, for each of his new aircraftis not being economically realistic. If onecould, however, supply the designer of lightaircraft with simple-to-use, accurate means of predicting all aspects of theflight performance of his airplane, this would go a long way to- wardenabling these manufacturers to providesafer, more enjoyable aircraft more economically.

Despitemore than 50 years of scientific study, much of the preliminary designis still an art. Desirable handling qualities, for example, arestill largelyunquantified. Theusual procedure is to have anexperienced pilotdecide whetheran airplane handles "satisfactorily." Little effort hasbeen made until recentlyto determine--by a series of the best available quantitative tests-- thosecontrol forces andresponse rates with which a human isconfortable and thelimiting aircraft accelerations, oscillatory amplitudes, and frequencies for acceptableriding qualities andthen incorporating this information into a determinationof the geometric, inertial, andpower characteristicsnecessary to achieve them.

Anothermajor problem in light aircraft design is the diversity and frag- mentation of reliableexperimental and analytica,l information. In an attempt to contribute to a solution of this problem,the National Aeronautics and Space

2 Administrationsupported a reviewof its published research since 1940 (Refs. I, 2, 3) with a viewtoward identifying those items of pertinence to light aircraft design. The presentstudy is anin-depth review, selection, and extensionof that portion of thisinformation and certainother information specifically related tothe handling and ridingqualities of lightaircraft. It isintended to serve as a compilation of applicable,available experimental data as well asmodern analysisprocedures for those interested in these aspects of light aircraft design

3 OBJECTIVESAND ORGANIZATION

Thiswork was undertakenwith the intention of providing a recentengineer- inggraduate with sufficient understanding, information, and procedures to permit him to predict,with reasonable accuracy, certain riding and handlingcharacter- isticsof projected light aircraft. To facilitatethe presentation, the discus- sionis limited, for the most part, tothose facets of the riding andhandling qualities which have a common basis:they trace their origin to the general dynamicalbehavior of the airframe and its components.This approach takes ad- vantageof the fact that few recentstudents have not at some time during their collegiatecareers seen derivedthe equations of motion of a rigid body in space and thesolution of a systemof linear differential equations by the method of LaplaceTransforms. Recent graduates are also generally familiar with some of thevocabulary of feedback control systems analysis. The presentwork, therefore, seeks to build upon thisfoundation in elaborating modern techniques for predict- ingessential features of the riding andhandling qualities of light aircraft. As a consequence of this choice, however, many perfectlyrespectable problem areasand effective analysis techniques are not discussed. For these omissions theauthors request the reader's forebearance.

Thework begins with a discussion of the origin of and basis for riding and handling qualities criteria and how these are distinguished from other fam- iliarcriteria, termed here control capability. This section outlines the more familiarhandling qualities design and evaluation procedure, that basedon control forcesand deflections. The relation betweensuch methods and those de- rivedfrom dynamical considerations is indicated.Next a sectionreviews briefly, in qualitative terms, the derivation of thelinearlzed equations of motion and theusual methods for analyzing the stability of their solutlons. Appendices detailthese procedures for tha,seunfamiliar with them.

These sections are intented to provide a basis for appreciating the use which may be made of +he data presented in the literature review and thatgener- atedduring the present investigation. The first portlm of thereview is basi- cally a compilation of thosetechniques reported in the literature which the authorsregard as most suitable for the calculation of aircraft stability derivatives if giventhe aircraft's geometry. In preparation is a computerprogram forperforming these calculations. One need thensupply only the airframe geo- metri c and inertia I parametersand the program wi I I provide numerical values of allthe derivatives. This program,along with typical results, will beavailable aspart of a follow-onreport which will discusstest and data reduction tech- niquesfor extracting the values of stability derivatives from flight data. A shortsection discusses what is known generally about the effect of applications ofpower tothe values of the individual stability derivatives.

The nextportion is concerned with the relation among the stability derivatives,the airframe geometry and mass distribution, the dynamicpressure, and thecontrol forces and controldeflections. Included in this discussion of handlingqualities are current views of desirable values asrepresented by the FAR and military specifications.

4 Informationon typical values of lightaircraft inertial characteristics andmethods ofcomputing them are also included as a section of the literaiure review. The ratio of aerodynamiccontrol moment to aircraft moment of inertia, of course, isthe primary factor in determining the responsiveness of an aircraft to controlsurface deflections; hence, it seemed desirableto provide suitable information to permit one to make estimates of these characteristics.

Only two conditionsinvolving significant departures from small pertubations aretreated: stall andspin. These are included to indicatethe state of knowledgeregarding some of the limits of thesmall pertubation analysis.

Finally, a sectionreviews the information available on the anthopological basis for riding andhandling evaluations by human pilots.Included is a new procedure for estimating the riding qualities in terms of theairframe dynamics.

Followingthe literature review are results of computer studies measuring thesensitivity of aircraft motions to variations in individual stability derivatives.It is, of course, not possible to varythese derivatives in this fashionphysically but these studies do indicatethose derivatives which influ- encethe motion significantly and must, therefore, be determined with great accuracy.More approximate values are quite acceptable for derivativeswhich do not strongly affect the motion.

A limited number of more elaboratecomputer studies are also presented. Forthese studies all the derivatives which dependupon a givengeometric vari- ationare varied appropriately. One canthen see the changes in motion which resultfrom a givenchange in airframe geometry.

Forthe sake of completeness, Bode plots of theprincipal longitudinal and lateraltransfer functions are then provided, using the original stability de- rivativevalues for the exampleairplane. Some personsfind such plots very helpfulin visualizing aircraft dynamicresponses.

The appendicespresent derivations of the equations of motion, transforma- tionof these equations to the frequency domain, a computerprogram for evaluatingthe constants in the transfer functions given the aircraft geometry and inertia, a computerprogram for extracting the poles and zerosof a 4thorder polynomialtransfer function and a computerprogram to calculate the time histories of the linear velocities andangular displacements given the poles and zeros of thetransfer functions; also included are discussions of oneand two degree of freedomsimplifications of theequations of motionas ameans of ob- tainingapproximate predictions of thefrequency anddamping of the principal modes, exampledeterminations of flight motion of a particularairplane using the methodspresented in the text, and theuse of unexpanded forceterms in generalizednon-linear equations of motion to investigate departures from the resultsof small pertubation theory.

A discussion of the correlation and interpretation of Bode plots and Root Locusdiagrams isincluded. The finalappendix presents a bibliography of pertinent documents not specifically cited in the text.

5 FACTORS CONTRIBUTING TO THE PI LOT 'S OPINION OF AIRCRAFT RIDING HANDLING QUALITIES

Pilots,like other humans, aresensitive to theinertial forces imposed on theirbodies by changes intheir motion. The relativeacceptability of these inertial forces gives rise to the pilot's subjective opinion of the riding qualities of theaircraft. To controlthe motions of theaircraft, the pilot must exert certain forces on the controls and displacethem through certain distances. The magnitudes of thesecontrol forces and controldisplacements and the magnitude andphase relationships between controlforce anddisplacement on the one handand the aircraft's motion onthe other hand combine to form theDsycholoqi- cal and anthropologicalbasis of the pi lot's opinion of the hand1 ing qualities of the ai rcraft.

Inaddition to the riding andhandling qualities defined in this fashion, a Dilot's satisfaction with his aircraft also deDends uDon the aircraft's control capabi lli ty, whichincludes such factors as how qhickl y 'it wi I I attain a given bankangle, how tight a turn it will make, how great a variationin c.g. location it-wi l l tolerate, how oftenretrimming is required, the minimum airspeed for ruddereffectiveness, whether it is possible to fly unyawed withasymmetric power,whether the stall is gentle and the recovery rapid, and whetherspin recoveryis simple and rapid. The logicfor separating what is commonly treated as a singlestudy, aircraft flying qualities, into three divisions as in the presentwork stems from a number of considerations:handling qualities can, in modern aircraft, be altered to achievedesired values without affecting the ridingcharacteristics or the control capabilities; many of thecontrol capabil- itylimits involve very large motions andaerodynamic non-linearities no+ normallyencountered in the motions associated in the pilot's mind with the ridingqualities; the control capabilities are primarily a direct result of the aircraft'sgeometric configuration; riding qualities are strongly affected by inertialcharacteristics aswell as geometric configuration; the pilot makes a directassociation between the "ride" and the force applied to his body; for many of thecontrol capabilities, his role is more like that of anobserver in that his gratification is dependentnot so much upon theforces applied to his bodyas to the psychologically satisfying condition of beingable to will a motion and observethat it is carried out or to beingable to refrain from performing a tedioustask. It is, of course,true that if oneconfigures the aircraft geometry, mass distribution, and controlsystem to obtain a givenset of riding and handlingqualities he has at the same timedetermined, even if unknowingly, much of theaircraft's control capability. Nevertheless, for the present,an aircraft's geometric and inertialparameters will be examined pri- marilyin the light of their influence on riding and handlingqualities. The exceptionto this is the discussion in the literature review of the control- labilityduring stall and spinentry.

As inother matters of opinion, it hasproven difficult to identify and to quantifythose necessary parameters and theirdesirable values which are significantin determining the riding andhandling qualities of aircraft. Military

6 flying quality specification, for example, are the result of the combined efforts of many trainedengineer-pilots over the years; yet until the release of MIL-F- 87858 (ASG) in 1969, mostrequirements were stillstated in qualitative terms. The new specificaticmand the accompanying 688 pagebackground report, AFFDL-TR- 69-72, August 1969 (Ref. 41, are the result of more thanthree years work by a largegroup of specialists, who consulted more than 660 references, some of them obscure or classifiedmilitary reports. The new specificationsets some quantitative limits onboth the riding and handlingqualities as well as continuing some qualitativerequirements from earliereditions of thespec.ification. The quantitativelimits are derived from modern stabilityanalysis, some limited anthropologicaltest data, and pilot comments on the characteristics of variable, stab i I ity ai rcraft. The select ion of parameters upon which to set requ i rements appears to havebeen based upon experience rather than upona rigorousdemonstra- tionthat these parameter values doindeed represent the necessary and suffucient conditionsfor adequate riding and handling qualities.

Whilethe new specification is a greatstep forward, it is to behoped that one will soon findstandards of acceptableaircraft motions, control forces and displacements,and aircraft response to control displacements which trace their claimfrom systematic anthropological studies of forces and limbextensions, accelerations,response rates, etc. found to be comfortable bya large number of pi lots.

Certainlyprior to the issuance of the new specification, and probablyeven today,the most widely used concept of aircraft flying qualities (handling qualities in the present context) was thatdeveloped more than 25 years agoand detailedin the standard textbook of Perkins and Hage and in NACA Report 927, "Appreciation and Predictionof Flying Qualities," by W. H. Phillips. Toappre- ciatethe reason that the group of parameters upon whichthe criteria of these authorsare based came into common use, it iswell to recall that an aircraft spendsmost of its flight time in a quasi-equilibriumcondition; thus, the forces and, to some extent,the amplitude andphase relations betweenforce application and aircraftresponse are the traditional basis for the pilot's opinion of the handlivgqualities.

Indeveloping criteria for the desirability of these forces, deflections, andresponses, theattempt wasmade to expressthem in termsmeaningful to the designer,readily measuredand applicable to a widerange of aircraft types. For example, the change inelevator angle with change in trimmed lift coefficient can beshown to be related to thelocation of the center of gravity in chord lengthsfrom the aerodynamic center, the ratio of horizontal tail area to wing area,the location of the horizontal tail plane aft of the wing aerodynamic ten- terin wing chord lengths, the tail lift curveslope, the relative dynamic pressure and flow direction at the tail compared to that at the wing,and the effec- tivenessof the elevator in changing the apparent tail angle of attack. Note thatthese quantities are all non-dimensional and, therefore,applicable to aircraftof varying size and configuration.If one iswilling to assume that the tail contribution to aircraft lift coefficientis small, then bymeasuring thedynamic'pressure and aircraft weight during flight the lift coefficient is readilyobtained. The elevatorangle can bemeasured easily by installingthe movableelement of a potentiometer or other position transducer on the elevator shaft.For a givenaircraft d6e/dCLcan be alteredreadily only bychanging center-of-gravitylocation. From thepilot's viewpoint, d6e/dCL issignificant 7 because a zerovalue of this parameter is indicative of therearmost permissible loadingwhich an aircraft may havebefore no motion of the stick is required to changespeed. Obvious I y, a pi lot w i I I preGr a load i ng such that at I east a smalldisplacement of the stick isrequired to changespeed. Usually, therefore, the aft c.g. limit will beset about .05c ahead of thepoint for d6e/dC =O L Unfortunately, a pilot is less able to discernvariations in dbe/dCL than he is in changes inforce with speed. When writtenas d(FS/q)/dCL theparameter dependson the same quantitiesas dGe/dCL with the addition of thecontrol sys- temgearing and theaerodynamic moments of theelevator about its hinge. De- pendingupon the elevator design, d(FS/q)/dCL can be zero forward or aft of the c.g.location for which dde/dCL = 0. It is desirable to attempt to designthe elevatorsuch that d(Fs/q)dCL = 0 occurs at or slightly aft of the c.g.location forwhich dGe/dCL = 0. One wouldnot wish the forces required to change speed to disappear while there is yet some stickmotion required nor would one wish theaircraft to continue to changespeed if thepilot released the stick. The conditions may bothoccur if the aircraft hasd(Fs/q)/dCL = 0 forward of the c.g.location for which dGe/dCL = 0 and is loadedsuch thatthe c.g. is between thetwo points. Herce, it isusually felt necessary to,check both parameters duringflight test. The aft c.g. limit isthen chosen to insurethat both parametershave atleast perceptable values.

Although an airplane may beloaded withthe c.g. so far aft that the control forces and e I evator def lections requ i red to change speed are opposite to those normallyrequired, it isnot necessarily an uncontrollableairplane. Only if the forces or def lections requi red for control are too large or requi re too rapid an application for the pilot to manage can the aircraft really becalled uncontrollable. The factthat the controls require actuation in a manner contrary tothe usual experience would not of itself be so serious if it werenot also forthe fact that at these c.g. locations the aircraft will spontaneouslyrotate to largeangles of attackrather rapidly unless countering controls are promptly applied and continuouslymodulated. If the rotation takes place more rapidly thanthe speed decreases, large load factors will bedeveloped and the aircraft willquickly destroy itself. Because of themotion damping effectsof the horizontaltail, however, thec.g. location at which the elevator angle or stick forceper unit of normal acceleration is zero is further aft than whendGe/dCL=O or d(FS/q)/dCL = 0; thus,even when the aircraft is loadedsuch that d(Fs/q)/dCL=O, it will retain a finitevalue for dFS/dn.Hence vehiclemotions--as contrasted to speedchanges--will require conventional, although light, forces for control. As thec.g. moves aftfrom the dFs/dn = 0 location,control becomes progressively more difficult.Determining the c.g. location at which dFs/dn = 0 from flight testprovides the designer with a checkon the calculated damping effectiveness of the horizontal tail andindicates to thepilot the control margin he has for maneuver if heshould inadvertantly load the aircraft such that d(Fs/q)dCL = 0.

Inaddition to providing a measure of therearmost permissible c.g. location theparameters dGe/dCL, d(Fs/q)/dCL,dde/dn, and dFs/dn are often used todefine desirablehandling at normal loadings. The presentlyaccepted standards for the values of these or similarparameters are discussed in the literature review. At thispoint it is sufficient to notethat for psychologicalreasons increasing parametervalues seem to bepreferred as aircraft size and grossweight increase although human physicalcapabilities provide the overall upper and lower bounds forall classes of aircraft. 8 In add tion to the s tick force arising from aerodynamichinge moments, a significant contribution isusually supplied by control system friction. This has a mater albearing on the pilot's evaluation of.the aircraft's handling characteristics.In Tact, customer feedback to themanufacture indicates that the average light aircraft pilot seems to prefer a controlsystem which requires a consciousapplication of force to changespeed and which is tolerant of small, inadvertantstick movements. Forthis reason most lightplane manufacturers evaluatethe handling characteristics of their products only in terms of the forcesrequired to changespeed and normal acceleration as functions of loading, speed, configuration,and power setting.

In addition to the handling about the y-axis of the aircraft, the pilot will also beconcerned with handling about the other two axes.Traditionally suchthings as the sideslip angle per unit of rudder deflection or rudderforce, the amount ofrudder needed to compensate for theyawing due to aileron deflection and rolling, andthe non-dimensional wing tiphelix angle (pb/2U)which is a measure of the rolling capability of the aircraft havebeen measured to pro- videquantitative support to the pilot handling impressions. As will beseen from theliterature review desirable values are still being specified for most of theseparameters.

The readerwill note that all of the parameters mentioned thus far are intended to bemeasured under steady flightconditions. One willgeneral ly find little in the way ofquantitative requirements on the dynamic behavior of aircraft.This situation arose because older aircraft usually were not very "clean" aerodynamically,had a low ratio of weightto volume, operatedat low altitudes, andhad relativelylarge horizontal tailplanes to provide adequate control at low speeds. It canbe shown that aerodynamicdrag isthe principal contributor to the damping of thelongitudinal phugoid oscillation; hence a "dirty"airplane--particularlyone with relatively low powerloading--seldom has unsatis- factoryphugoid characteristics. Damping of thedutch roll and longitudinal shortperiod modes isstrongly dependent upon the ratio of therestoring force generatedby rotation to the moment ofinertia about that axis. Thus low den- sityaircraft with large control surfaces operating in dense air usually exper- iencewell-damped longitudinal short period anddutch rolloscillations. However, as light aircraft become moredense, operateat higher altitudes, become more "clean",and utilizehigher powerloadings, it is nolonger reasonable to assume that if oneassures satisfactory static handling characteristics, as outlined above, thenthe dynamic flight characteristics, will naturally be satisfactory.

Two additionalconcerns of recentvintage have also served to focusatten- tion ondynamic characteristics:ride and handlingin turbulence. Now thatthe novelty of flight hasdisappeared for a significant fraction of thepopulation, standardsfoi- what is anacceptable ,level of ride are continuously being upgraded. Travelin current jet transports has convinced many that(a) flying can be very smoothand (b) an uncomfortable ride can be very fatiguing for passengers and canserve todiminish the effectiveness of a pilot. As a resultof these comfort and safetyconsiderations and the growing realization among designersthat aircraft riding characteristics can be altered through suitable design just as they can inother vehicles, increasing attention is being directed toward establishing thevehicle behavior identified with good ridingqualities. Since the human body associatesonly changes in motion with vehicle ride, it is apparentthat riding qualities canbe defined only in terms of thevehicle's dynamic characteristics. 9 In turbulent weather, the aircraft is struck continually by gusts of more or less random magnitude and orientation.Most of thecharacteristic motions ofthe aircraft are excited simultaneously. As a result, pilot workload may be veryhigh depending almost entirely on the vehicle's dynamic characteristics. Well damped aircraft, for example, materiallyreduce the amount ofcontrol input required of the pilot when compared with aircraft having lightly damped longitu- dinalshort period and dutchroll oscillations.

There is evenless justification for neglectingconsiderati9n of the air- framedynamics when oneobserves that many of the conventional static stability parametersare merely the zero frequency values of more generaltransfer functions. They are,therefore, still available, having been generatedas a part ofthe process of evaluating the general dynamical behavior. This point is 3laborated somewhat inthe next section.

10 FLIGHT MOTIONS OF LIGHTAIRCRAFT- GENERAL CONSIDERATIONS

The motion of a rigid body in space isdescribed completely by a system of six equations, three representing the translation of the center of mass (center of gravity) andthree representing the angular motion of the bodyabout its center of gravity.General solutions of this system of differentialequations are unknown becauseproducts and squares of thedependent variables appear in some terms andbecause the unbalancedaerodynamic forces and moments whichpro- ducethe motion are not known explicitly.Since these forces and moments are known to depend, among otherthings, upon the vehicle's inclination relative to the stream,the time rate of change of theinclination, the vehicle velocities, and thevehicle accelerations, it hasbeen usual to representthe forces and moments by Taylorseries expansions in these variables about the equilibrium condition. Even if thesevariables are all related to thesix dependentvariables and theirderivatives, one isfaced with the task of evaluating either empir- ically or analytically the first andhigher order partial derivatives of all theforces and moments with respect to each ofthe dependent variables and their derivatives in order to make possiblesolutions for particular sets of initial conditions--theonly possible solutions because of the presence of the non- linearproduct andsquare terms. The inverseproblem, finding the values of -all the partial derivatives if the motion is known, clearly has no unique solution.

If it werenecessary tofollow this procedure in all its detail every time someone wanted to know themotion of an aircraft following a momentary control surfacedeflection, the motion would probably remain unknown until a seriesof expensive flighttests established the safety of the particular control surface deflection and thecharacter of theresponse. Fortunately, most departures fromequilibrium are small and, also becausethere is a planeof symmetry, thereis little lateral-longitudinal cross-coupling. As a result,for most flight maneuversone can neglect products andsquares ofthe dependent variables becausethey are very small compared with the value of the variables themselves. One canalso consider the aircraft's motion to berepresented by two independent systems ofthree differential equations; further, the departures from equilibrium aresmall enough thatthe aerodynamic forces and moments canbe described with sufficientaccuracy by retainingonly the linear term of the Taylor expansion. The equationsare thus made linear andamenable to a vast literature of solution techniques.

The usualrange of values of the partial derivatives in the Taylor expan- sionsof the aerodynamic forces and moments, the so-called stability derivatives,* are such that most aircraft exhibit very characteristic motions in response to momentary controlsurface deflections. In response to an elevator deflection

* For a detailedderivation of theequations of motion, their linearization, and thenormalization of the partial derivatives to obtainthe conventional stabilityderivatives, the reader is referred to Appendix A.

11 the aircraft will exhibit a well-damped oscillationin the neighborhood of 6 radiandsec.,the so-called short period mode; at the same time it will also exhibit a verylong period (Phugoid) oscillation which may evenbe slightly divergent. The motionsabout the other two axesare always coupled. There is an oscillation,called the dutch roll, whichusually is not well damped andhas a frequencynear 6 radiandsec.There is an aperiodic motion, which is usual ly SI ight ly divergent, cat I ed the spira I mode,' and a second aperiodic motion,called the rolling mode or roll subsidence,which isheavily damped.

Dur i ng the past twenty years, it has become common practice to study flightmotions by transforming the lineardifferential equations into algebraic equations through the use of theLaplace transform. One canthen solve for theresponse to a specif i c controlsurface deflection readily by .. matrixtechniques'. The resulting ratio of aircraft response to control surfacedeflection is called a transferfunction, eg., thefunction which describesthe manner in whichthe aircraft converts a controlsurface deflection into a motion.If one then obta ins the inverse Lap I acetransform of thesetransfer functions, he thenhas the time history of the aircraft's linear or angularvelocity components resulting from a specified control surfacedeflection. The threedependent variables excited byan elevator motionare usually taken to be u, 8, and a, although w or a, may sometimes beused instead of a.

Findingthe inverse transform is not always straightforward and is really unnecessary if onedesires only to determine the frequency anddamping of oscillations or thesubsidence of aperiodicmotions. The transferfunction consistsof polynomials in the numerator and denominator. By factoring thesepolynomials and writing themas a product of factors,one can find the valuesof the Laplace variable, s, for whichthe numerator or denominator- iszero. The valueof s forwhich the numerator vanishes is called a zero, and thevalue for which the denominator vanishes, a pole. A firstorder polerepresents an aperiodicmotion in the time domain.The numerical valuedescribes the rate at which the motion subsides (or diverges). A pairof complex poles represent a periodicmotion. The realpart expresses the dampingand theimaginary part, the natural frequency. Understanding ofthe significance of thisprocedure is enhanced,perhaps, if one plots thepoles on a plane(the s-plane) with the abscissa the axis of reals and theordinate the axis of theimaginaries. If a polelies on the right halfplane, the motion is unstable. A pairof complex poles, or roots, on theimaginary axis represent an undamped sinusoid. The distance from the realaxis represents the frequency of the sinusoid. The distancefrom the imaginaryaxis represents the degree of damping or divergence.

The location of theroots (poles) on the s-planewill, of course,change asthe values of thestability derivatives change. By calculatingthe locus of rootsfor representative values of stability derivatives from negative infinity to positive infinity it ispossible to determinethe conditions underwhich instabilities will exist. This form of analysis, as well as the transferfunction approach, also facilitates study of the effect of addingan automaticcontrol system to theaircraft. If one plotsthe poles and the zerosof the aircraft-control system combination on the s-plane, hecan employ

12 thewell-developed methodology of the now classicallocus-of-roots analysis to sketchrather accurately the path which the roots must follow as the gain of thefeedback element in the combined system is changedfrom zero to infinity. Theconcept of thelocus of roots of the "openloop" transfer function* is used inI'ater sections of this work to illustratethe allowable range of values for singleaircraft stability parameters with all other parameters held constant. Details of the construction of the transfer functions for the airframe and the numericalprocedure for evaluatingthe constants in the transfer functions andextracting the roots may befound in the Appendices.

AS notedearlier, the study of the stability of acceleratedmotions (dynamic stability) has often beentreated as separate and apartfrom a study of the tendencies of the aircraft at equilibrium to return to equilibrium if disturbed (staticstability). It will berecognized, however, thatthe steady state, or infinite time portion of thesolutions of the general equations of motion arerepresentative of the"static" stability andhandling. Onemay cite as examples thelongitudinal derivatives dbe/daZ,dFs/daz, dFs/du, and dGe/dCL whichare often used to describe the handling and stability characteristics. The firstis simply the inverse of the zero frequency value of the transfer function 2 . Transferfunctions written in terms of theLaplace variable E 6e can be transformedinto the frequency domainby setting s = jw. The notation -a, is used to designatethe amplitude ratio of thetwo frequency dependent 6e quantitieswhich form thetransfer function. It isalso a convenientshort- hand way torepresent a specifictransfer function, particularly its zero frequencyvalue or gain. The lastderivative dGe/dCL, which is a measure of thestick-fixed neutral point location** can be obtainedfrom E by multiply- 6e ing by C LC%. The other two derivatives require that the transfer function - be obtained and multiplied by 6 or - . An a 1 ternate form of dF,/du, FS 'e dF / 6e e a 4,whichcan be found from - ,- , and CL,, is frequent I y used to dCL Fs/q 6, locatethe stick-free neutral point. Thus, whilethere are occasions when it is sufficient simolv to eauate the forces or moments actina on the aircraft and set the derivative wiih respect to an aerodynamic ang I e equa I to zero to determine the static stability tendenc es, thefact that the same information is available as a part of the solution for thedynamic stability should encour- ageone to examine the vehicle motions and tendenciescompletely.

* Closed loop (aircraftplus automat c controlsystem) transfer function with the gain of the feedbackelement equal to zero.

** The neutralpoint is that longitudinal location of thecenter of gravity for whichthe aircraft hasno tendencyeither to return to its initial condition or tocontinue to moveaway from it, if disturbed. The proximity of the c.g. to theneutral point is the conventional measure of the aircraft's static longitudinal stability. 13 CONTROL FORCES DEFLECTIONS- GENERAL CONSIDERATIONS

A pilot's evaluation of the handling qualities of his aircraft has its origin, of course, inhis anthropological capacity. He is able to move the control whee I fore and aft through bit a I im'i ted distance before the reach becomes uncomfortable;he is able to exert only a given maximum force with comfort; a controlforce below a certainvalue is disturbing because the pilot canapply it withoutconsciously willing it andhe has difficulty modulating it; if thetime between control force application and theresponse of the aircraftis longer than a particularvalue, the pilot will tend to become con- fusedand may applyerroneous contro! inputs; if theaircraft responds too rapidly,he cannot control potentially dangerous motions.

In addition to theseconstraints, the designer must provide a comfortable relationshipbetween wheel fore-and-afttravel andspeed changes and between wheel force and aircraft normalacceleration. If these gradients are too small,the aircraft is said to betoo sensitive; if theyare too large,the aircraft is said to be too unresponsive. The designermust also take care to provideproper control centering for trimmed flight. It is undesirable to permit the stick to move through an appreciabledistance without requiring theapplication of force, nor should the force required to set the stick into motionbe larger than the smallest force gradient for speed change or acceleration.

Somewhat similarconstraints govern the application of thelateral controls.

Toaccomplish these objectives, the light aircraft designer will usually manipulate(for control of thelongitudinal flight mode) theelevator deflection to wheel deflectionratio, provide aerodynamic balance on the elevator or a flyingtab, provide centering springs, control the linkage elasticity and friction, and limitthe aircraft's center of gravity travel; he will also select a suitableelevator area, horizontal tail aspect ratio, airfoil section, and horizontaltail area andhe willlocate the horizontal tail area suitably faraft. The designerwill recognize that pitch responsiveness is related to the aircraft's moment of inertiaabout its y-axis as well as the moment the elevatorcan produce and that there is a structurallimitation as well as a human toleranceto normal acceleration which the pilot should not be able to exceedinadvertently. The designerwill also endeavor to provide a reasonably linear relation betweencontrol force or deflection and thedesired result, suchas speed change or normalacceleration.

From theforegoing, it isevident that, to carryout this task in a direct fashion,considerable knowledge of desirable anthropological limits must be availableas well as a substantialknowledge of diverseaerodynamic and mechan- icalcharacteristics. The stateof understanding of this problem is discussed inthe literature review which follows.

14 LITERATUREREVIEW

15 SCOPE

During a review(Refs. 1, 2, 3) of all NACA/NASA literaturepublished between 1940 and1968, those reports dea I i ng with aircraft stab i I ity and controlanalyses and tests were setaside as a group.The documents in this groupwere then screened and the content of those regarded as most applicable tolight aircraft design was notedin the report. For the present study, theentire group was reexaminedand rereviewed. In addition, the review of NACA documents was extendedbackward to 1930, and a computersearch of Scientific and TechnicalAerospace Reports, International Aerospace Abstracts, and Department of Defensearchives wasmade for current (1962 to late 1969) stability and controlinformation from world-wide sources. Included in this were the USAF FlyingQuality Specification and Stability and Control Datcom.

As onewould suspect, much ofthe earlier (pre-19441information has beensuperceded; more refinedanalyses andmore accurate and reliable test dataare now available. Most ofthe information is also uncorrelated: flight testresults are not compared withresults from wind tunnel tests; wind tunneltest results are not compared withtheoretical analyses. To attempt such a correlation and evaluationhere is not possible with the information availableto the authors. Thereason for this is that the wind tunnel or flight test reports available in the open literature usually donot give sufficientinformation on the airplane's geometric or inertial parameters to permitsatisfactory correlations to be made. The courseadopted, therefore, was to list all the stability derivatives likely to be significantin thedesign of future light aircraft, detail the method of calculationregarded asmost reasonable for such aircraft,report the values of thesederivatives in such aircraft, and notepertinent references for eachderivative. This approach is anexpansion (in that it details methods ofcalculation and typicalresults) and specialization (to lightaircraft) of the approach adoptedby Ellison andHoak (Ref. 5).

Values for thesederivatives are required, of course, to calculate the ridingqualities of a lightaircraft. The handlingcharacteristics* will, forthe purposes of the present study, be definedas

(1) The longitudinalcontrol force and control surfaceposition variations with speedand load factor--their maximum andminimum values, phaserelationships, and controltravel required to achieve them

* thereader will recognizethat the definition of handlingis somewhat broadenedhere to includeitems previously identified with aircraft control- ability.This hasbeen done because many specificationrequirements state responsiveness to controlapplication only in terms of angular displacement reachedin a giventime. In suchinstances it ishelpful to correlate the initial responses to controlapplication--here considered as the handling characteristics--withthe final capabilities of the aircraft. 16 (2) The angulardisplacement and angular rate variationsproduced by application of rudder and aileron forces--their maximum andminimum values,phase relationships, degree of linearity, and controltravel required

(3) Controllability of theaircraft under special circumstances.

The reviewwhich follows cites some ofthe literature available on the forces, displacements,and application rates found comfortable by most humans. This apparentlyhas been considered in developing the force and rate limits set in Mil F-87858. Forthis reason, the discussion below follows the specifi- cationfairly closely, yet pointing out those areas not treated and describing methodsavailable in the literature for insuring that the aircraft complies withthe specification.

Forthe present compilation, only that literature containing data applicable to aircraftdesigned to operate within the following limits was cons i dered :

Maxi mum gross we i ght IO ,OOO#

Maximum wingloading 40#/f t2

Maximum indicatedairspeed 300 mph

M aximum MachMaximum number 0.4

Aspect ratio 2 5.0

Little or nowing sweep

Rigidstructure

Those portionsof the specifications and other documents referring to aircraft controllability asdefined above and to handlingin turbulent air orwith external stores are not discussed. Also, inexamining the literature upon whichthe following was based, many documents notcited but dealing with thesubject were examined. Some wereregarded as perhaps less complete than thereports cited, as confirming results found in cited documents, presenting datain a form not so wellsuited to comparisonwith theory, or presenting lessaccurate results than are now available. The readerinterested in perusing the more interesting of these documents isreferred to Appendix I for a biblio- graphicallisting.

17 SYMBOLS

Ae effective aspect ratio

AR aspect ratio

a commonly usedarea for a componentwhich is thereference area for CD A, 7.r a.c. aerodynamiccenter of wing a0 lift-curveslope at zero lift a, lift-curveslope of vertical tail

BH P eng i ne brake horsepower b wingspan ba ai leron span bV verticaltail span

C wingforce parallel to theairplane reference line C coefficien+ of wingforce parallel to theairplane reference line 2 CC apu s D drag coefficient - CD $pU2S

zero lift drag,found in CD section cDf

three-dimensionaldrag coefficient found in lateral stability cDO derivativessection

two-dimensionalparasite drag coefficient cdO

u 8% " cDU 2 au

18 component drag coe fficient based on the area AT

elevator hi nge-moment coefficient when =at Oo and 6, = Oo

(-IrndCL wing lift curve slope with infinite aspect ratio da

c!2 section I ift coefficient or rol I ing moment coefficient(L/fpU2Sb) maximum section liftcoefficient '&ma x

c%x section lift-curve slope of the horizontal tail t

19 change in rolling moment coefficient due to changein aileron deflection (found in control forces section)

section lift-curve slope due to elevator deflection (this is not to be confused with rolling moment coefficient)

Xi - a6R

pitch i ng-moment coeff i cient (M/ipU2Sc)

pitch ing-moment coefficient due to thrust force

ai rfo iI section Ditching moment about the aerodynamic center Lma.c.

coefficient of wing force normal to the airplane reference line yawing-moment coefficient (N/ipU2Sb)

20 mean aerodynamicchord a i leron chord e I evator chord flap chord airplane center of grav,i ty dragforce base of naturalsystem of logarithms or Oswald'sspan efficiencyfactor induced-angle span efficiency factor

21 stick-force due to aileron actuation

stick force

theequivalent parasite area discussed in the CD section

ratio of elevator displacement to the product of stick length and stick angulardisplacement

acceleration due to gravity (32.2 ft/sec2)

horsepower

heightof the horizontal tail a.c.above the c.g. (positivefor a.c.above c.g. 1

moment of inertiaabout the x-axis

moment ofinertia about the y-axis

moment ofinertia about the z-axis

productof inertia

incidence ang I e

radius of gyrationabout the x-axis

radius of gyration about the y-axis

radiusof gyration about the z-axis

lift or rolling moment

lengthof fuselage or body

lengthfrom c.g. to tail quarter chord

lengthfrom wing quarter chord to tail quarterchord

tail volume factor

lengthfrom c.g. to vertical tail aerodynamiccenter

p'itching-moment about the c.g.

22 Mfus pitching-momentabout the c.g. due to the fuselage and nacelles nac

aerodynamic moments aboutthe y-axis (pitching moment) m mass inslugs

N wingforce normal to theairplane reference line or yawing moment n normal accelerationin g's P. rol ling velocity P rate of change of rol ling velocity

helixroll angle (e)2U

4 dynamicpressure ($pU2) orpitching velocity 4 rate of change of pitching velocity qt dynamicpressure at the horizontal tail qv dynamic pressure at the vertical tail RPM propellerrevolutions per minute

RN Reynolds Number r yawingvelocity r. rate of change of yawingvelocity

S wi ng area

SRP seat reference poi nt

rudderarea sR

Se e I evator area

Sf flap area

SS bodyside area

'h hor i zonta I ta i I area

23 S distancein half chords which is usedas a non-dimensionalizing factor

T thrust

TDPF tail dampingpower factor,see figure 72

TDR tail damping ratio, see figure 72

-1 -aT (see Appendix A) TU m au t time or ai rfoiI thickness

t time

At I ag timeof the downwash at the tail plane

U airp lanevelocity

sta I I speed

UTri m trim speed w airplane weight

maximum width of thefuselage or nacelles Wf

X’ distancefrom c.g. to wing quarter chord (pos itive for c.g.ahead ofquarter chord) - X longitudinaldistance rearward from c.g. to w ng aerodynami c center

distanceparallel to relativewind from the w ng a.c. to the C.Q. (positive for a.c.ahead ofthe c.g.1

Yi distancefrom body centerlineto inboard edge of ai leron

ZT perpendiculardistance from thrust line to c.g. (positive for thrust line belowthe c.g.1

‘a verticaldistance from wing a.c. to c.g. (positive for a.c. above c.g.1

=V distancefrom the center of pressureof the vertical tail to the aircraft centerline(positive for vertical tail above thex-axis)

ZW distancefrom body centerline to quarter-chord point of exposedwing rootchord (positive for thequarter-chord point below the body center I i ne)

r dihedralangle

A wing sweep angle 24 resultant angular velocity

arlgle of attack

rate of change of angle of attack

inducedangle of attack

angle of attack for zero lift

sta I I angle of attack

e I evator eff iciency factor I (dCL/dBE1/(dCL/dat) I

sides1 ip ang le

flight path angle from thehorizontal

correctionfactor for induceddrag

a i I eron def I ect ion

elevator deflection

rudder def I ect ion

aileron deflection

e I evator def I ect ion

elevator angle at zero aircraft lift

flapdeflection

trim tab deflection

downwash angle

change in downwash angledue to change inangle of attack.

Dutch Roll damping ratio

%.p. shortperiod damping ratio rl efficiencyfactor for tail, qL./q, or thepropeller efficiency

25 pitch angle

rate of change of pitch ang le

taperratio (tip chord/root chord 1

airplane relative density f actor (m/pSb)

3.1416

density

s i dewash ang I e dat correctionfactor for induced angle or elevatoreffectiveness factor (-1 d6 E timeconstant for roll mode

rot I angle

second derivativewith respect to time of the roll angle

a measure of the ratio of the oscillatory cohponent of bankangle to the averagecomponent of bank anglefollowing a rudder-pedal-freeimpulse a i I eron contro 1 command yaw angle phaseang le of Dutch Roll component of sideslip

Dutch Rol I naturalfrequency

shortper iodnatura 1 frequency

Subscript:

a.c. aerodynamiccenter

c.g. center of gravity

c/4 wingquarter-chord

h horizontal tail

0 tai I

V vertical tail

W wing

26 STABILITY OERIVATIVES

It is customary,as noted earlier, to represent the aerodynamic and thrust forces on an aircraft by a Taylor expansion about the conditions for steady,level flight, Because of theassumption that perturbations from equilibriumare small, only the linear terns are retalnedJ squares, productterms, and higher order derivatives are neglected, The remaining derivatives,called stability derivatives, belng morenumerous thanthe equations of motion,can therefore beevaluated only through special tests orthrough analyses which limit responses tothose resulting from a single variable. The dlscussionbelow isolates each of thesederivatives, the methodfrom the literature regarded as most suitable for its evaluation, and typicalvalues for light aircraft,

27 Usually, by thetime an airplane is subjected to a stability analysis, thedesign has progressed far enough that the lift coefficients for the flight conditionsof interest are known forthe complete airplane. Since 2-D (section)data are available for many airfoil and airfoilf!ap configurations, it is desirable to have a means ofconverting section data to three-dimensional datafor a particularairfoil. In Theory of Flight,Richard Von Mises(Ref. 6) givesan estimate of lift coefficient,

CL = CQ 1 + 2/AR

An example 2-D plot of Cg versusangle of attack is shown below for the 2412 airfoilsection.

2.01

1.61 3.1 RNxL06 0 5.7

1-21 o 8.9

-4l-

-.8

-1.2 -32 -24 -16 -8 0 8 16 24 32

Section Angle of Attack, Q , deg

Figure 1. 2-0 plotof section lift coefficient versusangle of attack for 2412 airfoi I section.

The approximation does notinclude such factors as taper effects and tip effects.

28 .-

It should be noted that 2-D section data areavailable for a large number of airfoils in Theory of Wing Sections (Ref. 7); and the same data are included in TR-824 (Ref. 8). A refined analysis to determinelift coefficient should include the lift contribution of the tail and,if possible, the lift contribution of the fuselage. The interference effects between thewing and the fuselage may be significant and, for this reason,wind tunnel tests or actual flight tests contribute to determining lift coefficient. The tail coefficient,

can be included in that for the entire aircraft where = ‘Itq+/q: - St + c (CL)Airp lane - ‘LW + ‘Lt S, ‘It Lfuselage’ nacel les

The tail contribution to the totalairplane CL at cruise can be approximated by using the moment equation:

or CLt = c -Xa -Sw * Lw Rt St ‘It -

The fuselage lift coefficientmay be quite difficult to estimate, unless some simplification, such as slender body theory,is applied. NACA TR-540 (Ref. 9) discussed the interference effects between thewing and the fuselage and gives some example lift coefficients for the fuselage at various angles of attack. The table below gives an example of some of trlese experimental data.

a=OO a=40 a=8O a= 1 Z0

Fuse I ag6 Eng i ne CL CD CD CL cL cD CL CD

Round None Round .OOO .0041 .OOl .0042 .005 ,0049 .011 .0062

RoundUncow I ed .OOO .0189 .001.0191 .004 .0200 .008.0216

RoundCow I ed .OOO .0069 .008.0073 .017.0088 .028.0115

RectangularNone .OOO .0049 ,005 .0054 .014.0068 .026 .0097 ; Table 1. Example lift coefficients for the fuselage at various angles of attack.

* A method for calculating the elevator deflection required for equilibrium cruise is given in the sample calculationsin Appendix G. 29 The fuselagecoefficients in the table are based on wing area of thewing fuse- lagearrangement tested. Thus, byexamining the table with a knowledge of the airplaneangle of attack,one can estimate the fuselage contribution to the totalairplane lift coefficient.For small angles of attack, 00 to 8O, the fuselage lift canprobably be neglected without a significant change in the total airplane lift coefficient,unless the fuselage is highly cambered.

Datcom (Ref., 10) gives a method forcalculating CL, of a bodybased on potentialflow theory (see CL,). Thus, CL canbe approximated by multiplying CL, by a of the body, whichis known.

Forlight airplanes in the cruising mode of flight,the lift coefficient willprobably fall between 0.25 and 0.45. Inclimbing flight, CL will probably belarger”0.5 to 0.9.For the landing approach, the usual CL will probably rangefrom .95 to 1.18. Forthe Cessna 182, typicalvalues are 0.309 atcruise, 0.719 inclimbing flight, and 1.12 inthe landing approach.

30 Thedrag coefficient, CD, is anaerodynamic force coefficient which can bethought of as a damping coefficient. Theequilibrium airplane drag effectively damps the thrust by acting in the opposite direction of the relative windand is alwayspositive in sign. When consideringperformance of an alr- craft,the smallest possible value of CD is desired;however, in airframe dynamics, CD is the main contributor to.the damping of thephugoid mode. Thus, thelarger the value of CD, thebetter the damping.Since phugoid damping is notconsidered of majorimportance in the flying qualities of the airplane,the performance rather than the flying qualities of the aircraft should dictate the design value of CD.

A preliminarystage of dragestimation of an airplane in anincompress- ibleflow may beaccomplished by adding the individual drags of several components ofthe airplane. This method iscal led "drag breakdown."The majority of airplanedrag curves can be expressed as yr. 2

where CDf = zero lift orparasite drag,

Mathematically, CDf is'the intercept on the CD axis of a graph of CD versus CL~,and l/TeAR isthe slope. Only at very low or highvalues of lift coefficients does theparabolic approximation deviate from the actual drag polar.

The followingprocedure is commonly used toobtain the value of CD~for a particularairplane. The dragcoefficient of eachcomponent is basedon an area, A,, "proper"to that component. For example,one usually bases the drag coefficientfor the fuselage on the maximum fuselagearea. The equivalent parasitedrag area, f, is thenexpressed as

The total f for anairplane is approximately the sum of CD A, for theindivid- ua I components, p I us five or ten per cent for mutua I i nteryerence between the components.Another percentage error (3% to 5%) may a Is0 beadded for sma I I protuberancessuch as handles, hinges, antennas, and coverplates. Fifteen per cent of the total CD,A~ was used to account for protuberancesand interferences onthe Cessna 182 discussedlater in the present study. Once thetotal f is found, CDf for theairplane can bedetermined from

"- f where cDf s S '= wing area.

CL2 Total CD isthen .- cDf -t TeAR

* e canbe estimated inthe CL sect ion. a 31 Values of CD, and theareas on which they are based are given in the tables below.This information was selected from References 11, 12, 13, and 14.

Thewing drag CD, forairfoils with standard roughness can be obtained from a tablegiven in Reference (15) and shown below.The comparison is based onstandard roughness at RN = 6 x lo6; thewing CD, is basedon wing area and istabulated as Cd,. Wing flapdeflection may generatean additional CD, which should beaccounted for.

Airfoil Section

632-41 5 .15 .0098 .20 1 .33 12. -.070 23016.5 ,165 .0102 .15 1 .20 12. -.005 23012.0 .12 .0099 .18 1 .22 12. -.01 5 241 2 .12 .0098 .20 1 .22 14. -.048 2301 8 .18 .0105 .10 1 .05 11. -.005 241 0 .10 .0095 .20 1 .22 14. -.050 632-21 5 .15 .0097 .16 1 .26 14. -.030 641-412 .12 .0097 .31 1 .34 12. -.070 642-AZ 1 5 .15 .0102 .20 1.20 14. -. 035 652-4 15 .15 ,0100 .22 1.25 14. -.065 641 -21 2 .12 .0088 .18 1.18 11. -.025

Table 2. Airfoil sections.

The fuselage CD, canbe obtained from the table below by choosing the fuselage mostnearly like the one investigated. Twenty per cent of CD, canbe added to accountfor the canopyand windshield. For the Cessna 182, thefaurth fuselage was chosen. Ref.. Area

SC 0.266

0.062

SC 0.071

0.063

SC 0.116

Table 3. Fuselagedrag where Sc = maximum crosssectional area and L = Fuselagelength. 32 .- I II

The empennage CD valuesfor undeflected control surfaces can be found from Tab I e 4 based on'the tai I p I anform area where is is the hori zonta I tai I surfaceincidence.

Tai I Lrrangement "-Description \rea 'D, Tapered f i I lets, vertical and horizontaltapered surfaces is= OO St .0043 i5= -40 .0063 Tapered fillets,tail surfaces dith endplates is = OO St ,0058 is = -40 .0063

Symmetr ica I tapered f i I I ets is = 00 St .0059

dertical and horizontaltail surfaces is = OO St .0070 .'s = -40 .0058 T.a i I surfaces with end 3 I ates is= 0' St .0058 rapered f i I I ets, hor i zonta I ta i I surfaces is = 0' Sb .0039 is = -40 .0083 is = 4O -006 1 4

Table 4. Empennage drag.

The informationrequired for landing gear drag prediction is given in Table 5. Forthe nosegear, CD~isgiven; however, for theother landing gear, the valueof f is given. The tabulatedvalue of f isthe value for bothwheels of thelanding gear assembly.

33 Remarks

8.50-10 wheels, not faired ...... 1.67 8.50-10 wheels,faired ...... 1.50 8.50-10 wheels, no streamline members . . 3.83

8.50-10 wheels, faired...... 0.74 27-in. streamlined wheels, not faired ...... 0.98

27-in. streamlined wheels, not faired . . 0.84 8.50-10 wheels, faired...... 0.68 21-in. streamlined wheels, not faired . . 0.53

8.50-10 wheels ...... 0.51

8.50-10 wheels, not faired...... 1.52

8.50-10 wheels, faired ...... 1.02

8.50-10 wheels, not faired...... 1.60

24-in. streamlined wheels, 8, intersections filleted ...... 0.86 8.50-10 wheels, no fillets...... 1.13

8.50-10 wheels ...... 1.05

Low pressure wheels, intersections filleted 0.31 Low pressure wheels, no wheel fairing . . 0.47 Streamlined wheels, round strut,half fork no fairing ...... 1.25

For the nose gearCg = .%.8 based on Nose Gear 7T A, A, = (wheel diameterl(whee1 width) I

Table 5, Landing gear drag. 34 .-

The CD, valuesfor other components of interestcan be estimated by usingthe tablebelow:

AREAFOR DRAG COMPONENTS CA LCU LAT I ON cD7T

Nace I les 1. abovewing, smal I ai rplane areaion Crosssect .250 2. largeleading edge nacelle,small airplane Crosssection area . 1 20 3. smal I leadingedge nacelle, large airplane .080area sectionCross 4. improvednacelle, no cool ing flow areasectionCross .050 5. improvednacelle, typical cooling air flow areaion Crosssect .loo

Wing Tanks 1. centeredon tip Cross sectionarea .05-. 07 2. belowwing tip Cross sectionarea .07-. 10 3. inboardbelow wing Cross sect ion area .15-. 30

Wi res and Struts 1. smoothround wires and struts(per foot) Fronta I a rea 1.2-1.3 2. standard aircraft cab le (perfoot) Fronta I area 1.4-1.7 3. smooth ellipticalwire (perfoot) Fronta I area finenessratio 2:l 0.6-0.4 finenessratio 4:l .35 finenessratio 8:l .3-.2 4. standardstream1 ined wire (perfoot) Frontal area .45-. 20 5. squarewire (per foot) Fronta I area .16-. 20 6. streamlinedstruts (perfoot) Fronta I area .075-0.10

Table 6. Airplane components.

Forsmooth round wire of diameter less than 4 inch, assume twoend fittings equivalent to three feet of wire.

Forsmooth round struts of diametergreater than 5/16 inch, assume twoend fittings equivalent to one foot of strut. Forsmooth elliptical wire, assume twoend f ittings equiva I ent to 10 to 15 feet of wire.

Forsquare wire, assume two end fittings equ ivalent to two feet of wire.

Forstreamlined struts, assume two end fittings equivalent to fivefeet of strut if faired,ten feet if unfaired.

The total drag coefficient for the airplane is thus

The dragbreakdown method was used to estimate a drag coefficient of 0.0311 for the Cessna 182. Dragcoefficients for other light airplanes can be found usingthe table below taken from Reference (14).

YEAR A I RPLANE TYPE b S W "POWER" u cD (ft) (ft') (Ibs) (hpl (kts) I I 1903 Wright-Brofhersbiplane 40: 5.10 750 12 26 0.074 1945 Piper "Cub" persona I 35 5 79 1500 1 35 110 0.032 1950 Cessna rr170rf persona I 36 175 2200 140 122 0.032 1942 Messe,rschmitf-I09fighter 32 172 6700 1200 330 0.036 1943 N. A. P-51 "Mustang" f ighter 37 23.5 10000 1380 38,O 0.020

Table 7. Dragcoefficients for selected light airplanes

It shouldbe noted that Reference (15) givesanother method ofestimating dragcoefficient which sums friction drag and theprofile drag for each of the components.

* e = 1/(1 + 61 and a graphfor estimating (1 + 6) canbe found inthe discussionof the CL, section.

36 Theaerodynamic pitching moment aboutthe c.g. isdefined as Cm. In equilibrium flight, the aerodynamic pitching moment mustbalance the moment coefficientresulting from thrust.Although Cm appears in the list of dimensionalstability derivatives in Appendix B of thepresent study, it is not usuallyreferred to as a stabilityderivative. The principaleffect of Cm is to contribute to the period of the phugoid mode.

Severalforces and moments, suchas tail, wing,and fuselage lift and drag, and moments aboutthe aerodynamic centers of the wing and taii, contribute tothe pitching moment coefficient; however, for equilibrium,power-off flight, theelevator is positioned so thqt Cm iszero. For powered flight, Cm isthe negative of the Cm required to balancethe thrust force moment, CmT.CmT canbe written as T ZT -" ',T ',T qSi where ZT = perpendiculardistance from thrust line to c.g., posltlve for thrust line below thec.g. T = thrust

The particularairplane discussed later was examined withpower-off; thus, Cm = 0. From theequation above, it isalso apparent that C, iszero when the thrustline passedthrough the center of gravity (ZT = 0).

An equation for Cr? isalso very helpful in designing or sizingan airplane. The equationcan be writtensimply by summing theaerodynamic moments of the various ai rcraftcomponents about the c.g. The equationgiven below is a modification of that containedin TR-927 (Ref. 16). Defining "nose-up"as a positive moment, the moments canbe summed usingFigure 2. The followingequations arebased on the assumptionthat cos a = 1 and sin a = 0.

L \ i'

Figure 2. Forcesand moments inthe plane of symmetry. 37 Mc.g. = Lxa + Dza + Ma.c. - LtRt + Dtht+ where

I x, is positive f0r.a.c. ofwing ahead of c.g. Za is positive for a.c. ofwing above c.g. ht is positive for tail a.c.above c.g. = distance from c.g. to tai I quarterchord

Thus, the equation for the moment coefficient can be written as

Sincethe tail is usually a symmetricalsection,

where at = (aw - iw- E + it)

Thus,

It shouldbe noted that the above equation does notconsider the fuselage or propellerpitching moment. The fuselagepitching moment canbe approximated by

(Cma)f seage isevaluated in the discussion of the Cma inthe present study. The efyec4 s of the prope I ler are discussed in the sect Ion dea I i ng with power effects on stability derivatives.

38 The thrustcoefficient,.C~, like CL is not strictly a stability derivative butis.necessary in a dynamicana,lysis. . The coefficient is non-dimensional .. .. , .. and isdefined ,as .

Forthe,Cessna 182 discussedlater, CT = 0.0 sincethe analysis wasmade with power-off;tiowever, for power-on cruise CT = .04.

39 C C and Cm L"' D" U

The stabllity derivatives CL~, CD~, and Cmu arethe changes in lift, drag, and pitching moment coefficients,respectively, with airspeed. Thesechanges at low speeds arereally Reynolds number effects andcan usually beconsidered to bezero. The factthat they are very small for low Mach numbers ispointed outby the figure below taken from Reference (18).

0.3 0.5 0.7 0.9 0.3 0.5 0.7 Mach number Mach number

Figure 3. Variationof Cg and Cdowith Mach number.

Sincefor the cruise condition CT CD and CT~N CD, andi'n the same manner asabove CT isneglected for cruisingflight. It should benoted that for approach an8 I and i ng maneuvers, CT, and CD" may become significant and therefore shouldnot beneglected.

CL~,CDu, Cmu, and CT~were chosen to be zero for the Cessna 182 investigated laterin this report.

40 The stability derivative Cb is the change in lift coefficientwith angle of attack and is commonlyknown asthe lift curveslope. This derivative is alwayspositive for angles of attackbelow the stall. Ordinarily, the wing accounts for 85% to 90% of thetotal Cb. Thisderivative is very important inequilibrium flight and in dynamicconditions. Since an airplane with a highervalue of CLC, usually has a lowerdrag, a highvalue of Cb is desirable for optimumperformance. Higher values of C arenecessarily associated with high aspect ratio, unswept wings. The deriva % ive also makes an important contribution to the damping of thelongitudinal short period mode.

Thatportion of the effective wing angle of attack induced by wing lift canbe written from Reference (19):

where el = l+.r'I = induced-anglespan efficiency factor T = correctionfactor for induced angle (see Figure 4) AR = wingaspect ratio

Sincethe angle of attack for zero lift does notvary with aspect ratio, the geometricangles of attack for the same airfoil wing operating at the same CL butwith two different aspect ratios are related by

Thus, theexpression for the lift curveslope of a wingwith one aspect ratio interms of the lift curve slope at anotheraspect ratio becomes

where dCL/da = change in 'lift coefficlentper degree

The equationabove implies that the lift curveslope is a constant,which is a fairlyaccurate assumption for angles of attack up to 10 or 12 degrees. However, oncethe linear range of the 2-D lift curveslope is exceeded, the lift curveslope can only be evaluated for a particularvalue of a. If condition "2" is that of infinite aspect ratio, the lift curveslope per degree of a finite aspect ratio canbe written in terms of the 2-D airfoil lift curve slope, (dCL/da), as

41 Typicalvalues for CCk), perdegree range from a,bout 0.110 for thin airfoils to about 0.115 for thick airfoils at ReynoldsNumbers greater than about lo6. The slopeis somewhat lessat lower Reynolds Numbersand atthe higher (near 1.0) lift coefficients.For exacting use, referenceshould be made to careful windtunnel tests of theairfoil being examined.

A rapidapproximation to the exact value of el may beachieved by using eitheror both of thefigures below. Figure 4, takenfrom Reference (19) hasboth 'I and 6 plottedfor various taper ratios at an aspect ratio of 6.28 (6 is used inthe discussion of Coal. Figure 5 is a plotof 1 + T and 1 + 6 forvarious aspect ratios at a taperratio of 1.0 fromReference (20). Thus, if a givendesign hasan aspect ratio 6, Figure 4 may beused; if the design has a taperratio of 1.0, Figure 5 may beused. For most lightairplanes, Figure 4 shouldgive acceptable results.

Tip chod'Root chord

Figure 4. Variationof T and 6 with taper ratios at aspectratio of 6.28.

42 AR

Figure 5. Variationof 1 + T and 1 + 6 withaspect ratios at taper ratio of 1.0.

The lift curveslope of partial or full span flappedwings can often be approximatedsatisfactorily or directly found by consultingthe NACA literature. Many reportshave been writtenconcerning various flap configurations on several airfoilsor wings.Thus, NASA CR-1485 (Ref. 2) should be veryvaluable, because it lists NACA reportsdealing with specific flap configurations. In theusual case, CL islarger for flaps deflected than for flaps retracted. For the cruise codit ion, an undef lected flap can usual ly be considered part ofthe airfoil section, thus requiring no additionalcalculation. For other flight modes, however, it may benecessary to find CL~for a givenflap deflection. The liftcurve slope of the wing is probably known forall flight conditionsby the time a dynamicanalysis is made, but if not, NASA CR-1485 should be usedas a majorreference for informationon prediction of aerodynamic characteristicsof flaps. If the airplane analyzed has partial span flaps instead of full spanflaps, the approximate lift curveslope can be obtained from

where

S = wingarea

f = subscriptdenoting f lap EL - = mean sectionvalue of the lift curve da s I opebetween the root and tip section 43

L. Although ~LC,for the complete airplane, in many cases, is assumed equal to the Cb. of thewing alone, amethod of predicting the bodyand tail Cb shouldbe available. Themethod presented here for approximating the body contributionis taken from Reference (10).

where k2 - k, = apparent mass factorwhich is a functionof finenessratio (length/maximum thickness)

vb = total bodyvolume

So = crosssectional area at x.

x. = body station where flowceases to be potential,this is a functionof xi, the body station where theparameter dSx/dx first reaches its minimum value. (Thisstation where the change inarea withrespect to x first reaches its lowest valuecan be estimatedfrom a sketchof the body. 1

Sx = body crosssectional area at anybody station

Rb = bodylength.

Figures 6 and 7 representgraphs for estimating both k2 - kl and xo/Rb.

FINENESS RATIO

Figure 6. Reduced mass factor.

44 Figure 7. Body station where flow becomes viscous.

Foran estimate of some valuesof (C~,)Body, Table 1 from TR-540 (Ref. 9) inthe discussion of CL canbe considered. The table is a tabularform of CL versus a forvarious fuselage, wing combinations. The valueof CL is basedon wingarea, and although the fuselages given do not really represent the fuselage of a lightairplane, they serve as an indicatorfor the degree of fuselage contributionto (Cb)Total.

The tail aswell as the fuselage may makea significant contribution to CL,. Forthe usual aerodynamic analysis, the wing CL~isusually considered to be thetotal C of theairframe; however, for a dynamic stabilityanalysis, the total CLO, is t% e change in total CL resulting from a change in angle of attackonly. An airplanein flight mustbe trimmed afterthe angle of attack is changed if the new angleof attack is to bemaintained. Thus, the trim forceschange over the angle of attack range, making it impossibleto measure CL, inflight with everything else (including trim forces)constant.* Wind tunneltests can be used toobtain the total C , sincethe modelcan be restrained. Since the tail contribution is sma9 I compared to thewing contribution(usually less than lo%), in many cases, CL ofthe wing alone can beused. For the ai rpI ane to beana lyzed later i n tais study, the w i ng (4.61 I is used. cLa The tail contribution to Cbcan be estimatedas,

Total Lift = Liftwing + Liftfuselage + Liftta; I

Thus,

The derivative of the above expressionis taken to yield

* Althoughimpossible to measure exactly, CL~can be approximated with a gooddegree of accuracy from flight test records. 45 S i nce i, + it E , at =a, - -

thus,

The tailcontribution can therefore be writtenas

Typicalvalues of CL for lightairplanes fall in the range of 4.0 to 7.0per radian,depending chyef ly on the type of wingused.

46 The stability derivative CD~is the.changein drag Coefficient with varying angle of 'attack.: -Above .the an'gle of attack for minimum drag -the.drag coefficient

increases as the,angIe of attackincreases; thus,'Cd ispo'sittive in sign. '8 CDai,isua I I y Has .I ittle effect on the short period mo%e and has only a. sma I I ',effecton the phugoid mode in that a decrease'in CD~usually increases stability.

Co, is made up ofcontributions from the wing,fuselage, and tailsection, withthe wing being by far the largest. The wingdrag coefficient can be writtenas

CD = cdo + (CL*/.rreAR)

where Cdo = profile drag coefficient

e = 1/(1 + 6) = Oswald'sspan efficiency factor *

Thus, CIJ~ is wing - dCdo ~CL "+- CDawing da TeAR 'h

Forsmall angles of attack,(dCdo/da) isusually small; however, for flight modes otherthan cruise, (dCdo/da) could have a significantvalue. The wing CD~, inmost cases, serves as a goodapproximation of thetotal CD~.

Forfuselage angles of att-ackless than loo, thefuselage CIJ~can be ignored,as can the tail CD~,with goodaccuracy. Wind tunneltests should actually beused to measure airplane CD~,but because the wing CD dominates and sincethe equations are relatively insensitive to changes in it is felt that CD~can be approximatedby the wing contribution. Thus,

The value of CD~calculated for the Cessna 182 is 0.126 perradian.

* A graphfor estimating (1 + 8) can be foundin the discussion of CL,. 47

" is perhapsthe most important derivative related to longitudinal yand control,since it primarilyestablishes the natural frequency of theshort period mode and is a majorfactor in determining the response of theairframe to elevator motions and gusts.Usually, a largenegative value of C% isdesired (-0.5 to -1.0 forlight airplanes), but if C istoo large, the required elevator effectiveness may become unreasonably hig? .

Figure 8, takenfrom Reference (1 11, shows that

N = L cos(a - i,) + D sin(a - i,)

C = D cos(a - i,) - L sin(a - iw) where i = incidenceangle t = subscriptwhich refers to the tail w = subscriptwhich refers to the wing

-Airplanereference line $&t it Wind -. Direc

Figure 8. Forcesand moments inplane of symmetry.

Neglecting Cct, the derivative of the pitching moment withrespect to CL can be written as

ContributionwingofContribution Contribution of fuselage of horizontal & nacelles tai I

48 -

where CmaaC, = moment coefficient about aerodyanmi c center Rt = length from c.g. to hor i zonta I tai I quarter chord rlt = st/sw Eva I uat ing some ofthe derivatives shows that, if a - i, issmall then sin(a - i,) y (a - i,) andcos(a - i,) 1.0, CD(~- iwl/CLais small compared with (a, - iwl,and dCdo/dCL = 0 then

where e = span efficiency factor obtained from CL~ AR = wingaspect ratio

Usingthe above equations,

where CL~isper radian. If the c.g. is ahead ofthe aerodynamic center, X, ispegatlve (Xa = distancefrom c.g. to a.c.1. Ifthe c.g. position is under thewing a.c., Za ispositive (za = verticaldistance from c.g. to a.c.1.

The angleof attack of the tail is a functionof the wing angle of attack, aownwash, tailincidence, andwing incidence. Thus,

at =aw-~+it-iW.

The first term of a Taylorexpansion of CNt gives

Therefore,

49 Theabove equation can be used to write (dCm/dCL)tas:

Reference (19) givesthe change in downwash withrespect to a as

where X = tipchord/root chord (use 0.67for elliptical wing) AR = wingaspect ratio R’ = distancefrom wing quarter chord to horizontal tai I quarterchord.

Ifthe horizontal tail is located k0.5~ or more verticallyfrom the centerline ofthe wake,a constant of 18 insteadof 20 shouldbe used.

Two methods, explainedbelow, may beused to estimate the effects of the fuselageon Cb. The latter method is consideredthe more accurateof the two butrequires more calculations. The first method requiresthe evaluation of theequation

Nacel les where kf = empiricalfactor shown inFigure 9 wf = maximum width of thefuselage or nacelle E,, = lengthof fuselage or nacelle

50 3.2

2.8

2 A

2-0

I.., kf 1-6 .. 1.2

-8

-4

0 20 40 60 Position of wing 114 root chord on body,

Figure 9. Empiricalfactor for fuselage or nacelle contributions to C%.

Theabove formula, taken from TR-711 (Ref.21), is a simple method forestimating the effect of the fuselage or nacelles on CYa.

Thesecond method, taken from Perkins and Hage (Ref.111,considers the fact thatthe variation of the fuselage longitudinal pitching moment withangle of attackis greatly affected by the upwash in front of the wing and the downwash behindthe wing. The wing's induced flow has a heavy destabilizinginfluence onthe fuselage or nacellesections ahead of thewing and a stabilizinginfluence behindthe wing. Thus, thelocation of the wing with respect to the longitudinal axisis of considerable importance. Multhopp (Ref. 22) proposes the following formula to account for this phenomenon:

where B = angle of localflow (freestream angle of attackplus the angle of the induced flow ahead of or behindthe wing) wf = fuselagewidth = length of fuselage 51 Theequation can be integratednurnerica Ily in the manner illustratedin Figure 10.

Segments 1-5 da/da from curve a " Segment 6 dB/da from curve b

b-4 I

Figure 10. Typicallayout for computingfuselage moments.

The integrationrequires that the average value of wf2 (dB/da)Axbe found for eachsegment and thatthe individual segment parts be summed. The curvefor dB/da versuspositions ahead ofthe wing leading edge inpercent wing chord are givenin curve (a) of thefollowing figure. For thesection immediately ahead ofthe wing leading edge,dB/da rises so abruptlythat integrated values are given basedon thelength of this segment aft of thewing chord (curve b). Forthe segment aftof the wing, it is assumed that dB/da riseslinearly from zeroat the root trailing edge to (1 - de/da) atthe horizontal tail a.c. In theregion between the wing leading and trailing edge,dB/da isconsidered zero. It should beremembered thatusing the procedure above gives cm per rad i an. a.

52 0 -4 .8 - 1.2 1.6 2.0 X1 X1 X1 - and - C C

Figure 11. Fuselagepitching moment contribution to Cm,.

It may be desirableto include a factorwhich takes into account the interferenceeffect of the fuselage or nacelle onthe wing Cm. Thisaddition- alcontribution can beapproximated by the formula below andadded to the Ch of thewing. dCm - " C da (290) (SI (W~.~.+ WMid. - WT.E.) where WL.~., WM~~.,and WT.E. = widthsof the fuselage at thewing leading edge, mid- chord,and trailing edge, respectively.

Thus, the total airplane Cm can be written as the sum of the wing contribution,the fuselage contribution, and the tail contribution:

A typicalvalue for C,, is the oneassociated with the Cessna182. For the c.g. at 26% of the mean aerodynamicchord Cmcl = -0.885.Moving the c.g. forwardproduces higher negative values for Cb; moving it aft achievesless negativevalues. For most light airplanes, Cm probablyfalls in the range of -0.5 to -1.0.

53 C La

The stabilityderivative C isthe change in lift coefficientwith the rate of change of angle of attac? . This derivative arises from a type of "plunging"motion along the z-axis, during which the angle of pitch, 8, remains zero. For low speed flight,the derivative results primarily from the aerodynamic timelag effect at the horizontal tail, and itssign is positive. For theconventional light airplane, the horizontal tail is immersed inthe downwash field of the wing some distancebehind the wing. When thewing angle of attackis changed, the downwash fieldis also altered; however, it takes a finite length of time for the downwash alterations to reachthe tail, resulting in a tail lift whichlags the motion of the aircraft. C@ canalso arise fromaeroelastic effects at high speed, butthese aeroelastic contributions arenegligible for light aircraft.

Thisderivative is unimportant in a dynamic longitudinalstability analysisfor light airplanes, since its effect is essentially the same as if theairplane's mass orinertia werechanged about the z-axis. Many studies eitherneglect CG by callingIts effects small or fail to mention it at all because of the negligible effects.

An empiricalmethod for calculating Cbcan be found using the same approachthat is used for C . Cm& ismerely the moment resulting from the time lag of the tail lift; t2 us, Cucan be thought of as CG divided by the non-dimensional moment arm *. Thus, -C% -c C~ cG=Rt/c= Rt

The negativesign is includedbecause a positive pitching moment from the tai I requires a negative lift (nose-up is a positivepitching moment). Us i ng the equationderived for C~,CG can be written as,

CG = where kt' = distancebetween the wing quarter chord and thehorizontal tail quarter chord.

A typicalrange of valuesof C forlight airplanes is 1.5 to 3.0. For the Cessna 182, CG was found to beq.74.

* thelength from thec.g. to the tail quarter chord divided by the mean aerodynamicwing chord, (Q/c). 54 C Da

The stability derivative CDG is the change in drag coefficient with rate of change of angle of attack. , (3% arisesfrom the aerodynamic lag effect and various"dead-weight" aeroe L'ke astic effects. For airplanes in the speedand weightrange of light airplanes, however, the drag variation due to boththese effects is negligible. Consequently, CD~is taken to bezero.

" C ma

The derivative Cm6 is the change in pitching moment coefficient with respect to b, the t ime rate of change of theangle of attack. It is quite importantin longitudinal dynamics,since it isinvolved in the damping of theshort period mode. A negativevalue of Cm& increasesshort period damping; thus,high negative values are desirable. This derivative is actua!ly causedby a lageffect of the downwash at the horizontal tail of the aircraft.

War Report WR L-430 (Ref. 23) isconsidered to give the best derivation of an empiricalformula for Cabased on thelag of the downwash between the trailing edge of thewing and the horizontal tail. The downwash at the tail at the time t dependson the angle of attack of the wing at the time

t" a; U

where Ri = length from thequarter chord of thewing c tothe quarter chord of thehorizontal tail -" = lagtime of the downwash U

The angle of attack of the "ai I, at, canbe written as

For a given maneuver, theangle of attack can be writtenas a functionof time, a = f(t). Thus,expanding the downwash in a Taylorseries and ignoring second orderterms, the downwash canbe writtenas a functionof (t - At) as E=-" f(t - At) aa where At = - U

As a simp1 ifyingtechnique, f(t - At) canbe expanded in a Taylorseries as (At 1 f At f'(t) + -f"(t) (At)3 f"'(t) 2 - -6 + ... Thus,

A quantity s is now introduced to non-dimensionalize 6, where Therefore,can be expressed as a="=-* 2U da 2U da c ds c 2u d(- t) C andcan be expressed as

The downwash can now be written in terms of ,s as

da + (2 g)' E = Ea {a - 2- d2a - ...3 c 2u 2u d(- t) d(- C C

The equationfor CY+ can now be written,with the above equation substituted for the downwash, as 2% da 2q d2a at - a, - i, + it - {a - -- + (-1 z-... 1 c ds C

Since a and a, refer to the same angleof attack,

2a-i 2 da (TI at = a(l - + it - iw+ -- - 9- ... c ds Ea 2

Since Cm- results from the lag of the downwash at the tail, only the tail contr i buy ion to the pitchi ng moment must be considered when Cm& is derived . The partof the pitching moment coefficientcontributed by the tail canbe written as

where fit = distance from c.g. to & chord of ta i I

Since Cm is a function of tailangle of attack, the expression derived above for % can be substitutedinto the Cm formulato yield

2% 2 fit st da Cm = - "{a(l - Ea) + Ea - - - (T-) d2a + ...} cbt Ot c sw c tis Ea 2 z

The .partial derivative of Cm with respect to da/dscan now beexpressed as St 2ai - (71rlt SW

57 or

It shouldbe noted that the above formula agrees with the one developed in Perkins and Hage (Ref. 111, except for the manner inwhich & is non- dimensionalized. It shouldalso be pointedout that Cmi canbe found from the aboveequation by merely differentiating with respect to d2a/ds2. In general,however, angular accelerations are considered small when linearized equations of motionare used and thereare only small disturbances from equilibrium. Thus, for I ightairplanes, CmE would not be of greatimportance.

A typicalrange of Cm& for a lightairplane is -3.0 to -7.0. Forthe Cessna 182, Cm& was calculatedat -5.24.

58 The stability derivative CL,q representsthe change in airplane lift with varying pitching velocity while The angle of attack of the airplane as a whole remainsconstant. Contributions are made byboth the wing and horizontal tail,but the tail Is by farthe moreimportant. The generalconcensus is that CL playsonly a minorpart in estimating the longitudinal response of the a i rcraf?.

Volume V of AerodynamicTheory by Durand (Ref. 24) explainsthe physical phenomena associatedwith CL~. Figure 1'2 shows that an airplaneflying with velocity U in a circular flight path of radius R and center 0 hasan angular velocity, de,such that U = R(de/dtland a is constant.

Figure 12. Pitchingat constant a.

Forsmall perturbations, (dO/dt) = q (Appendix A). Ifthe airplane c.g. is traveling with velocity u while the airplane is rotating with angular velocity q, the direction of motion of any point on the tail, distance Rt behindthe c.g., makes anangle tan-l(qRt/U) with the direction of motion of the c.g. Provided qkt isnot too large compared to u, theeffective incidence of the tai I isincreased byapproximately qRJu radians.

The wingcontribution to CL can be explainedin much the same way. The change in angle of attackof the wing 9 (measured at the aerodynamiccenter) is xc.g. - Xa.c. ACX = - q( I U where xcag. = distance to the c.g. xaac. = distance to the a.c.

Thisexpression indicates that there is a reductionin lift if thec.g. is behindthe a.c. and an increase if thec.g. is in front of the a.c.For a c.g.very near the a.c., thewing contribution is negligible; however, the wingcontribution increases as the distance between the c.g. and a.c. increases. It isfelt that, for light airplanes, the fuselage contribution to CL~is smaller than that of the wing;thus, it isneglected here. I . , -.. .. .

A theoreticalderivation of CLq canbe obtained from WR L-430 (Ref. 231 bymodifying the Cmq derivation to be disGussedlater. At thehorizontal tail,the angle of attackis increased Rte/U bypitching. The total lift coefficientis, therefore, increased by the amount

where = tail I ift curveslope

The followingexpression is usedas a non-dimensionalizingtechnique.

Thus,

Differentiatingwith respect to (ci)/2U)

The wi ng ccmt ributio n can be obtainedsimilarly by sub‘st i tut i ng the distance fromthe c.g. to the wing quarter chord * for thedistance Rt. Thus,

where x’ = distancefrom c.g. to wingquarter chord (positivefor c.g. ahead ofquarter chord, negativefor c.g.behind quarter chord) CLC, = wing lift curveslope

* At thispoint it is assumed thatthe a.c. of the wing is very near the wingquarter chord; therefore, the quarter chord is usedas the reference.

60 The airplane C becomes Lq --- 8CL cLq a(c9-j 2u

For the Cessna 182 aircraft investigated later in the paper, the c.g. is located very near the quarter chord and the wing contribution is therefore of neglected. The value aircraft CL4 was calculated to be 3.9.

61 The stabilityderivative CD is the change indrag of theairplane with varying pitching velocity while ?he angle of attack of the airplane as a whole remainsconstant. This derivative has contributions from boththe wing and thefuselage but both contributions are very small. In all of the literature for subsonicflight, CD isignored because it isreally unimportant in analyzingflight dynami2sand very small inmagnitude. Qq for the Cessna 182 is taken to be 0.

62 The change in pitching moment coefficient due to a change in pitching velocityconstitutes the stability derivative Cm . If anairplane has a positive pitch rate with a constant angle of attack ?flying a curved f I-,igh't path) , the angleof attack at the tail is increased, thereby adding more positive lift to the tail and creating a moment to oppose thepitching motion. For this reason, the derivative is sometimes referred to as the "pitch damping" derivative and isusually negative. The wingcontribution to Cm either opposes orincreases the pitching motion depending on the c.g. lo2ation (see discussion of CL~); however, this is relatively insignificant compared to the tail contribution. The fuselagecontribution is always neglected for light airplanes.

Thisparticular derivative is very important in longitudinal dynamics because it plays a majorrole in the damping ofthe short period mode anda minorrole in phugoid damping.High negative values of Cm (-10.0 to -15.0) 9 usuallyinsure good shortperiod damping forlight airplanes.

' Cmq can be considered a sum of momentsdue to the component partsof CL~. Consequently, fit ac, -" a cm - -" I a(%, tai I a(:) tai I

(The negativesign appears because Cmqtail is alwaysnegative, while CL 9ta i I isalways positive) and 2% - lxcl acL " " 'mqw i ng a a(%) w ing wing LU wing where

1x.l = distance from c.g . to wingquarter chord(a I ways pos itive).

The signof Cmqwing isopposite that of CLqwing; thus, /xc/ is used instead of X*. When the a.c. infront of the c.g. CL~~~~~isnegative but Cmqwing is positive. The total Cmq is - 2x * St -" "r) c"q C2 where xc = distancefrom c.g. to wing quarter chord (positivefor c.g. ahead ofquarter chord, negativefor c.g.behind quarter chord. /xc/ = magnitudeof x*

The valueof Cm calculatedfor the Cessna 182 is -12.43. 4 63 Thechange in lift coefficient due to elevator deflection is the stability derivative CL~~.Since downward deflection of theelevator is defined as positive,produclng a positive lift, CL~is normally positive in sign. Many erroneously consider this derivative to Ee the same asthe change in equilibrium lift coefficientwith respect to elevator deflection, in which the elevator deflectioncauses a change inthe angle of attack,resulting in wing and tail lift changes.For the derivative CL~~,the elevator angle is the only quantity whichcan change; thus, the angle of attack as we I I as a I I other angles must remainthe same. CL6E doesnot appear in the characteristic equation of the aircraft,but it doesappear inthe numerator of thetransfer functions; it therefore affects the gain of a particular transfer function.

For a conventionalaircraft with the horizontal tail mountedan appreciable distanceaft of the center of gravity, CL6E issmall, approximately 0.1 to 0.2 perradian. For light airplanes with relatively large tails, CL~~may takeon values between 0.3 and 0.5 perradian.

NACA TR-791 (.Ref. 25) mentionsthat CL~is usua I ly obtainedfrom wind tunneldata; however, an empirical relation from Reference (11) canbe used to approximate CL~~: dCLtdat CLQ = -- dat d6E where (dCLt)/(datl = lift curveslope of the tail and dat/d6E isplot-tedas a functionof elevator area divided by tail area (Figure 131, where SE/S~= 1.0 for al I-movable tai I.

0 .l .2 .3 .4 .S .6 .7

‘E”t

Figure 13. Elevatoreffectiveness.

The derivativecan beestimated more exactly by using the two-dimensional dataavailable in Reference (19) andmerely correcting for aspect ratio. The datapresented are for the 0009 airfoil and areuseful, therefore, for most lightairplanes (Figures 14 and 15). Dommash (Ref. 19) mentionsthat the data 64 given for the NACA 0009 airfoil with a plain flap illustrate the principles involved and arenot intended as exact engineering design data; however, the datashould give results well within the satisfactory requirements.

L V Y

-Test data V 0) r Extrapolated r 0 .02 e+ m

0 -2 .4 .6 .8 1.0 Flap-airfoil chord ratio, cf/c Figure 14. Flapeffectiveness parameter versus flap-airfoil chordratio for NACA 0009 airfoil with plain, unsealedflaD and O..OO5c gap.

0- 1.0 a- .w '0 -8 .-V r.- r v -6 0 0 c1 z- -4 .-c 0 Mc .2 a a 0 0 0 5 10 1s 20 25 30 Flapdeflection cf, deg Figure 15 Change in lift coefficient(with constant angleof attack) versus flap deflection forseveral values of flap-airfoilchord ratios. NACA 0009 airfoilwith plain, unsealedflap and 0.005~ gap. 65 The two-dimensionalvalues of CL~found above can be changed to CQ corrected for aspectratio using the correcFion procedure given in Reference (f0) for flappedwings:

sectionlift curve increment due to flap(elevator) deflection(this derivative should not beconfused with the rolling moment coefficientappearing in the I atera I dynamics 1. lift curveslope of tail without flap deflected (3-D1 sectionlift curve slope of basic airfoil

ratio of 3-D flap effective parameter to the 2-D flapeffectiveness parameter obtained from the figurebelow as a functionof wing (tail) aspect ratio and the theoretica I va I ue of (a6 )c2. The theoreticalvalue is also given as a function of flapchord to airfoil chord. flap-spanfactor which is = 1.0 forelevator horizontaltail surface for light airplanes.

66 For most lightairplanes, (a&)~/(a&) N 1.0 to 1.1; CL~~and Ckt canbe estimatedusing the procedures knumer%'ed for Ck, and Cg6 canbe estimated using thegraphs for the 0009 airfoi I. The formulafor CLgE based on wing areacan then be wri "ten as

or,since (a&) /(cl&)cRis approximately 1.05 for most lightaircraft, CL

The value of calculated for the Cessna 182 is 0.427.

"""__"..__._~."."""..._.I .._.".."_... *""-..."""- ---...... --..-.------. Thechange in drag coefficient due to a change in elevator angle is the controlsurface stability derivative CD~. Foran elevator of reasonable size, the total airplane drag does not c Fi ange appreciably with elevator deflection.For this reason, CD~~isoften neglected. The characteristic equationof the aircraft is not a functionof CD~,-; thus,affects only the gain of the particular transfer suchas u/~E.

Probably +he best method of estimating CD~, is to perform wind tunnel tests on a particularaircraft model;however, if windtunnel testing is not feasible, CD6 can be approximatedby using data from wind tunnel tests which havea I ready Eeen performed. NACA TR-688 (Ref. 26) discussedthe aerodynamic characteristics of severalhorizontal tails. Using the figures beluw and the appropriategraphs, can be estimated. Five tail shapes from NACA TR-688 werechosen forinclusion in this analysis. To estimate CD”, the tail surfacewhich is mostlike the one in question should 6.e use . Th,e numerical value of CD6 canbe taken from plots of CD versus CY for different elevator deflections For each tai I surface;however, the va I ues of CQ must be mulfiplied by St/S so thatwill be basedon wing area.

(5) Figure 17. Fivetail surfaces selected from NACA TR-688 (Ref. 26). 68 Tai I Span St Se Ct Test u Test Surface AR (in.) (sq. (sq. (fps) (in.) RN in.) in. 1

1 3.488.0 45.25 2450 155 5 701 1 ,960,000 2 3.1 23.668 181 7.68 110.0 448,000 3 4.3 3 81 361 39.4 9.15 ----- """"- 4 4.3 361 39.4 117 9.15 ----- """"_ 5 4.3 356 39.3 9.06 1 62 ----- """"-

I I

Table 8. Dimensionalcharacteristics for horizontal tails.

.5 2

.40

.3 2 4- c

b 0" .24 M .20 Ipe

.16

.12

.O 8

.04

-4 0 4 8 16 12 20 24 -4 0 4 8 16 12 20 24 Angle of attack (deg) Angle of attack (deg) Figure 19. Drag coefficient against angle of Figure 18. Drag coefficient attack at various agai nst angle of elevator deflec- attack at vari,ous tions for tai t elevator deflec- surface 2. tions for tai I 69 surface 1.

I -4 0 4 8 12 16 20 -4 0 4 8 12 16 20 Angle of attack (deg) Angle of attack (deg) Figure 20. Drag coefficient Figure 21. Drag coefficient against angle of aga i nst angle of attack at various attack at various elevatordeflec- elevatordeflec- tions for tai I tions for tai I surface 3. surface 4.

-4 0 4 8 16 12 20 24 Angle of attack (deg) Figure 22. Drag coefficient against angle of attack at various elevator def lections for tail surface 5. 7a The stability derivative Cm6E is the change in pitching moment coefficient with changes in elevator deflection, usually referred to as"elevator power", or "elevatoreffectiveness." If Cm6E and the maximum deflection of theelevator are known, the maximum rotation moment whichthe tail can exert can be estimated. It mustbe remembered that Cm6 isevaluated without allowing the airplane to rotate or any other parameF ers to change; thus, Cm6E is really p roduce d bythe moment produced CL~~. \

The primaryfunction of the elevator is to control the angle of attack of theairplane in equilibrium flight or in maneuvering flight. Depending onthe maximum allowableforward center of gravitytravel, the horizontal tail is designed to give enough tail power in all flight conditions to control the aircraft. If the needed tail or elevatorsize is entirely unreasonable, the desired c.g. travel may have to be limited.For all practical purposes the maximum practical Cm6E determinesthe maximum forwardcenter of gravity travel.

Since a positiveelevator deflection is defined as down,a positiveelevator deflectiongives a negativepitching moment contribution,making the sign of C!sE negative. A desirablevalue of Cm6E cannot be statedin general for all atrcraft becauseeach case must be analyzedseparately. For most light aircraft, however,should have a valuebetween-0.75 and-2.0.

The numericalvalue of CmBE can be obtained by multiplying CL~~bythe distancefrom the c.g. to thetail quarter chord divided by the wing mean aerodynamic chord :

The negativesign appears because CL~is positive and Cm6E must be negative, asdiscussed above. For the Cessna 162, Cm6E was calculatedto be-1.26. In theexperimental case used for this study, there is a contributionto Cm6E due to the moments aboutthe a.c. of the horizontal tail; however, this contributionis so small it isusually neglected.

71 The stability derivative C B is the change in side force causedby a variationin sides1 ip angle. den the airframe has a positivesideslip, f3, therelative wind strikes the wing,fuselage, and vertical tail obliquely from theright, resulting in a negativeside force. The majorcontribution to C comes fromthe vertical tail, with a smallercontribution from the fuseyageand a nearlynegligible contribution from the wing. This derivative contributesto the damping of theDutch Roll mode; thus,large negative values of CY@ might seem desirable.Large negative values of CYB, however, may create a largetime lag in the airplane's response and cause it to react sluggishly to the pilot's commands.

Inestimating values for CyB, force-testdata for thedesign in question shouldbe used, if possible.According to TR-1098 (Ref. 271, interference effectsare so largethat a generalizedformula would not be completely satisfactory.Instead, a method ofcorrecting data of a similardesign for use inanalyzing the design in question is recommended. A more recent publication, Datcom (Ref. 101, probablygives the most accurate method, sinceinterference effects basedon experimental results are included.

The wing contribution to CyB is sma I I, on the order of a2 (angle of attackin radians), so itsaccurate estimation is notvital to the total Cyb. Forswept wings, TN-1581 (Ref. 28) givesthe following formula for CYB of thewing: - 6 tan A sin A . (perradian) (Cyg)wing - cL2T AR(AR + 4 cos A)

For zero sweep (A = 0'1, (C )wingis equal to zero. TR-l098(Ref. 27) states thatthe aboveformula is n8 satlsfactory in practice and that no correction of (Cyglwing is necessaryfor similar designs since this contribution is so small. Themethod in Datcom (Ref. 10) givesthe effect of wingdihedral on 6 as fol lows: (CygIwing = -.0001 II'l perdegree, where r isin degrees.

The fuselagecontribution is greater than that of the wing and may be estimatedby a methodadapted from Datcom (Ref. 10). The basicrelation is

where Body ReferenceArea = (fuselage ~olume)~'~,

Ki = wing-fuselageinterference factor obtainedfrom the graph below,

72 zw = distancefrom body centerline to quarter-chord poi nt of exposed wingroot chord (positive for the quarter-chord point be I ow the body centerline),

d = maximum body height at wing-body intersection.

0 -1.0

Figure 23. Values for wing-fuselageinterference factor.

TR-540 (Ref. 9) givesvalues of (CbIfusas .0525 perradian for roundfuse- lagesand .1243 perradian for rectangular fuselages. Observation of these

73 dataindicates that, for mostconventional light aircraft, the value of (CyB)fus is small.

The vertical tail is the mostimportant contributor to CyB, and (Cygltail is used inthe calculation of tail effects on many ofthe other lateral stability derivatives. TR-1098 (Ref. 27) givesthe following formula for adapting i lar model to thedesign under consideration:

i-qn" refers to the new Configuration, and thesubscript "data" refersto the configuration for which (Cyg)tai; isalready known. Datcom (Ref.10) presentsprobably the most comprehensive method of obtaining(CyB)tai I yet developed.In this method, the following formula is shown:

The value of (CL~)~must be determined, using the effective aspect ratio ofthe vertical tail, to convert the two-dimensional lift-curve slope to the three-dimensionalvalue. The importanceof this approach is stressed in WR L-487 (Ref. 291, in whichdata of free-flight tests show that, for vertical tails of the same area,with the aspect ratio increased from 1.0 to 2.28, effectivenessincreases 67%. The valueof k,an empiricalfactor, may be obtainedfrom the following graph as a function of the ratio of vertical tail span to fuselagediameter in the tail region, (bv/Zrl).

0 1 2 3 4 5 6

Figure 2'4. Values for k as a functionof the ratio of vertical tail span tofuselage diameter in the tail region.

74 The value of thecombination sidewash and dynamic pressure ratio parameter for zero sweep is theempirically-derived expression, SV (1 +"I" qv - .724 + 1.53 + .4 a-ZW + .009 (AR), 9 W where zw = distance,parallel to z-axis, from wing rootquarter-chord point to fuselage centerline, d = maximum diameter of fuselage.

A comDarison of calculatedvalues of this parameter (using the aboveformula) with tested va I ues indicates thatthe average error is less than five percent. Putting together each of the components, thefollowing formula from Datcom (Ref. 10) fortotal $6 resu Its: Body ReferenceArea (cy6 )tots I = - Ki (CLJfus ( 1 - .OOOI Irl S W

Incorporatingthe aboveformula and thecharacteristics of a typical light airplaneyields CyB = -.31 perradian, which seems to be a typicalvalue.

75 The stability derivative Cg normally referred to as the "effective dihedral derivative," is the cha E'ge in rolling moment coefficient caused by variation in sideslip angle. In popular usage, a "positive dihedral effect" means a negative value of CQ . During aircraft sideslipping, the rolling moment produced is the resulf of wing dihedral effect and the moment resulting from the vertical tail center of pressure located above the equilibrium x-axis. For most conventional configurations, the value of CAB is negative; however, this value can easily be adjusted by changing the amount of built-in wing d i hedra 1 .

CgB is quite important to lateral stability, since it aids in damping both the Dutch RoI I mode and the spiral mode. For favorable Dutch Roll damping characteristics, small negative values of CQB are desired, but for improved spiral stability, large negative values are necessary. A compromise is thus in order, as indicated in TN-1094 (Ref. 301, which shows that best general flight behavior is obtained when the effective dihedral angle is approximately

2O. *

In actual practice, CQ~is usually not determined analytically because of the large errors involved as compared with force-test data on the design in question. If possible, force-test data should be used to determine the amount of wing dihedral, thus determining the value of CgB. An analytical approach for straight and level flight (B = -9) is given in Perkins and Hage (Ref. 11) for calculating CQ~of the total aircraft. This formula is given below, followed by a discussion of methods for obtaining each component:

is From Perkins and Hage, fhe wing dihedral contribution fo CQP

(cQB)w = (-1CQB r -4- (ACRg)tip shape r The effect of wing tip shape on the value of ACQ B is shown below. Maximum Ordinates On Upper Surface

~~~

AC"= --0°02 Maximum Ordinates

*%=O-O Maximum Ordinates On Lower Surfaces 1-p 1-p Aclp= -0002

Figure 25. Effect of winq tip shape on CgB per radian. 76 - ~~

The value of ((2%B /r) for a particular aspect ratio and taper ratio may be obtained from thefollowing graph.

0 2 4 6 8 10 AspectRatio, AR

Flgure 26. Values for (Cg /r) forvarious aspect and taperratios. 6 A theoreticalstudy for unswept, elliptical wings with zero dihedral is presentedin TR-1269 (Ref. 311, producingthe following equation for wing contr i but ion toCgB :

(c~~)~,=CL[- 16 + .05] per radian r=o -- IT AR

Alsopresented is a revisedformula, which considers changes in taper ratio:

(CRg) w, = CL [- k('71 x + + .05] perradian r=o AR A where k = 1.0 for straight wing tips k = 1.5 forround wing tips

Ifthe vertical tail is located abovethe longitudinal axis, the vertical tailcontribution is easily calculated from computation of the normal force caused by sideslip. Thus,

where zv = distancefrom the center of pressure of the verticaltail to the airplane's x-axis (positive for vertical tail above thex-axis). 77 Most studies on this subject considerinterference two effects on the vertical tail--wing-fuselage and wing-vertical tail interference--both influencedby the wing's location. These effects are tabulated in Perki nsand Hage, as shown in Table 9.

Wing-fuselageWing-vertical Tail ( mg1 ( ACRg 12 High Wing -.0006 .00016 Mid Wing 0 0 Low Wing .0008 -.0001 6

Table 9. Values for interference effects on the verticalfail per rad i an.

Typi.cal yaluesof CR8 range from -.03 to -.12 per radian.

78 The stability derivative Cn , oftencalled the "weathercock" or static directional derivative, is the c Eange in yawing moment coefficient resulting from a change insideslip angle. Physlcally, CnB isthe result of theairframe sideslipping,with the relative wind striking it obliquely,causing a yawing moment aboutthe center of gravity. The verticaltail, fuselage, andwing contribute to CnB, withthe vertical tail the dominant factor. For positive sideslip,the vertical tail causes a positiveyawing moment; thus, CnB is usuallypositive, even though the fuselage contribution is normally negative. The wingcontribution is usually positive, but quite small compared to that of the vertical tail andfuselage.

The value of CnB determinesprimarily the Dutch Roll naturalfrequency and affectsthe spiral stability of theaircraft. It isgenerally agreed that values of Cn ashigh as practically possible are desirable for good flying qualities. Iatues for CnB shouldbe obtained from force-test data for the model inquestion wherepossible.

At present, two analytical methods ofcalculating total Cne appearmost complete. The firstis that presented in Perkins and Hage (Ref. 11) for straight,level flight (6 = -$I. The value of CnB forthe total planecan be obta inedfrom the composite formula,

Perk ins and Hage alsogive a value for (CnglWas

(CngIw = ,00006 (A01 1'2 perdegree

Forunswept wings, TM-906 (Ref. 32) gives

C,. (CnBB)W = - per radian TAR

Thisformula is modified in TN-1581 (Ref.28) for sweep, by thefollowing relation: 1 tan A AR AR~ (CngIw = CL~ [" (COS A - 2 - 4rAR T AR(AR + 4 cos A) 8 cos A

+6CTIx sin per radian where -x = long itud i na I distance. rearward from c.g. to wingaerodynamic center

79 /

A value of (CngIfuscan be obtained from Perkins and Hage by use of

The valueof KB, anempirical constant, can be obtained from thefollowing graphas a function of finenessratio and C.Q. location. The distance from the nose to the c.g. is d; S, isthe body sidearea. The othervariables are shown below.

Figure 27. Empiricalconstant KB as a function offineness ratio and C.Q. location.

Perkinsand Hage alsopresent the following formula for vertical tail contribution to CnB:

The value of a,, lift-curveslope of vertical+ail, is determined by usingthe effectiveaspect ratio, which is calculated

Ae = 1.55 -bV SV

The value of a, can now be obtainedfrom the following graph for a particular valueof Ae. Thisvalue is basedon a conventional, low hor i zon ta I tail configuration;therefore, for designs significantly differen t, one i s ref erred to Datcom (Ref. 10) for an alternate method of finding Ae.

80 01 2 34 561 Effective aspect ratio vertical tail, Ae

Figure 28. Values ofav as a function ofvertical tail aspect ratio.

The vert ica I ta iIefficiency factor, vv, isalso needed for thedesign in quest ion. Two interferencefactors must bedetermined to completethe calculation of total Cn . The firstof these, A1Cn , isthe wing-fuselage inter- f erence factor, a punctionof wing location. 7 he followingchart is used to determine itsvalue.

WING POSITION AICnB PER DEGREE

High Wing .0002

Mid Wing ,000 1

Low Wing 0

Table 10. Values for A1CnB as a functionof wing location.

Thesecond interferencefactor, A2CnB, is theresult of sidewash at the vertical tail causedby wing-fuselage interference. Its value may bedetermined from thefollowing graph as a functionof wing position and maximum fuselageheight.

81 Figure 29. Valuesfor A2CnB as a function of wingposition and maximum fuselageheight.

Now that each of the componentshas been determined, the value for total CnB, usingthe method ofPerkins and Hage, canbe evaluated. They indicate that a CnB 1 .0005 tJ"/b) is necessary for adequate d i rect iona I stab i I ity.

The secondmethod of obtainingtotal C ispresented in Datcom (Ref. 10) nB where a wing-fuselagecorrection is added to the vertical tail contribution, resultingin

The interferencefactor, K, (perdegree), may bedetermined from the following graphas a functionof aircraft geometry andReynolds Number. SB, = Body Side Area w = Maximum Body Width

0.001 0.002 0.003 0-004

Figure 30. Empiricalinterference factor, Kn, as a function of aircraft geometryand Reynolds Number. 83 The value of (C ltaiI may be obtained from thediscussion of the stability derivative Cvs thepresent study.

Forlight aircraft, typical values of C seem to rangefrom 0.03 to 0.12 perradian. "6

84 The stability derivative Cyp is the change in side force resulting from rollingvelocity, with the vertlcal tail the main contributor, eventhough, for some configurations,the wing may make a significant contribution. The rolling velocity, p, createsan effective angle of attack on the tail, which, inturn, produces a side-force. The sign of C may be eitherpositive or negative. It is relativelyinsignificant and%nmonly neglected.

In TN-1581 (Ref. 281, a completelytheorefical approach, the following formula for C of thewing is presented: YP AR + cos A tan A 'YP = cL AR + 4 cos A

€orzero sweepback, thisformula gives C = 0.0, whichagrees with other theoreticaltreatments. From wind tunnel tes Yps on wings alone, TR-968 (Ref. 33) adds CL/AR to the above formulato account for wing tip suction, resulting in

Thisreduces to Cy = CL/AR forzero sweepback,which does notappear sma I I enough to be cons i Hered neg I igi b I e.

The wingcontribution is usually minor compared to the vertical tail contribution. From NASA MEMO 4-1-59t(Ref. 341, using a theoretical method of discrete-horseshoe-vortices, with thehorizontal tail in the mid-position, a valueof (C )tailis (-.8)[bv/(bw/2)] for the vertical tai 1 contribution. Also, in TR-1&6 (Ref. 351, a formulafor tai I contributionis

where zv = height of vertical tail center of pressureabove the longitudinal axis

(J = sidewash angle at the vertical tail

The ratio (acs/[a(pb/2U)]),, is the averageeffect of sidewash on the vertical tail andcan be obtained from the following graph as a function of angle of attack and vertical tail to semispan ratio for a wingwith aspect ratio in the vicinity of 6. 4 8 12

(Y , deg

Figure 31. istimationof average sidewash angle at thevertical tail with wing aspect ratio equal to 6.

In TR-1098 (Ref. 271, therelation for tail contribution is Z (Cyp)ta i 1 = 2 e - (~)a= 01 (CyB)ta i I w bw which seems quitesmall for cruise at an angleof attack near zero.

Datcom (Ref. 101 a I so presents a method for calculating Cyp, but,for conventionallight aircraft with zero wing sweep, lt appearsthat C isvery nearzero. The followingformula is taken from this work: YP

The followinggraphs, which have been adapted from Datcom forlight aircraft, show (C /CL) as a functionof wing sweep and taperratio, and [(ACyp)r/(C~p)r=o] as a funyg tion of dihedral angle.

86 AC14 30 0 -30

0 X -5 1.0"'"""'""'""!'"~ -.4 "2 0 -2 A

Figure 32. Values for (C /CL) as a functionof wing sweep andYP taper ratio.

r, Dihedral Angle [deg]

TN-40.66 (Ref. 361, whichuses flight measurements to determinestability derivatives,states that, in practice, Cyp may lie between 0.3 and -0.3. Using this range of Cyp, the other stability derivatives werecalculated, with CY showing a smalleffect only on CYB and C Thus, it appears that Cyp = 0.6 "P' is probablyas accurate an estimateas is necessary.

87 The stabilityderivative Cg, theroll damping derivative,is the change P’ inrolling moment coefficient due tovariation in rolling velocity. For a positive rol I, isthe result, primarily, of anincrease in lift on the down movingwing Cg%nd a decreasein lift on the up movingwing, thus creating a moment whichopposes the motion of the roll. This moment is negative, making CfiP negativein sign. Thewing, horizontaltail, and verticaltail contribute to CgP, withthe wing the dominant factor for airframes with conventional-sizetails. Cgp isthe principal determinant of thedamping-in- rollcharacteristics of the aircraft.

The basicwing contribution to Cgp may be foundfrom the graph below as a function of aspectand taper ratio for a wingwith zero or small sweep, a liftcoefficient of zero, and lift curveslope of 2~.For swept wings, the readeris referred to TR-1098 (Ref. 271, fromwhich thisfigure was adapted.

0 2 4 6 8 10

Aspect Ratio, AR

Figure 34. Wing contribution to Cg forwing with zero or sma I I sweeg.

The presentstudy assumes a linear lift curveslope with no correction for non-linearitiesin lift coefficient. In order to useFigure 34 to calculate wing Cgp forlift curve slopes other than ZIT, several means ofcorrecting the datamust be considered. TN-1839 (Ref.37) presents the following method:

which, forzero sweep, reduces to

88 The effectof wing dihedral on Cg is cons ideredby TN-1732 (Ref.38) in a correction for wi ng Cg ; however, sike most light aircraft havedihedral of sevendegrees or I ess, tR is correction appears quitesmall. The dragcontri- bution to wing Cg isgiven in TN-1924 (Ref. 39 1, for sweptwings with elliptic- chorddistributioR, by thefollowing formula:

whichreduces, forzero sweep, to thefollowing relation:

where CD = experimentallydetermined drag coefficient.

Thus, for an aircraftwith a linear lift curveslope, small dihedral, and zero sweep, thefollowing formula for wing Cg results: P AR + 4 (cEp)w i ng = [(Cgp)ao = 27r1 [( 27r - (ao)w)AR + A By consideringthe horizontal tail as an isolated airfoil, the basic value for (Cg )h may also bedetermined from Figure 34 as a functionof horizontal tai I asFectratio, taper ratio, andamount of sweep. Thisprocedure is the same as thatfor calculating the wing contribution; thus, (Cg )hmust be correctedfor lift curve slope other than 27r. The valueobtalned P is scaled down by the method of TR-1098 (Ref. 27):

The factor 0.5 isincluded to accountfor the flow rotation at the tail caused by thewinq; A isthe amount of sweep ofthe horizontal tail. The horizontal tail is assumed to have negligib le dihedral.

The vertica I tai I contribut ionto Ct isalso given in TR-1098 (Ref. 27): -p (CRp)V

Ot-

where zv = height of centerof pressure of vertical tail above thex-axis (different for each angle of attack).

89 . . . , ...... _...... , ... .I 11..111 1.1 I.I. I"..," _I. rn 11-11 , , . ...."...... - .,.", II ""- ...".. , _.,,.

From these formulas, it is obvious that the vertical tail contribution is negligible at low angles of attack. A much more elaborate formula for estimating vertical tail contribution is given in TN-2587 (Ref. 401, but because of the relative unimportance of this contribution to Cgp,total that method is not included.

In summary, foran aircraft with zero sweepa va I ue for tota I Cg may be obtained by simply adding the various contributions as follows:P AR + 4 1 (Ctp)tota I = [(Cgp)ao = ZT] - 8 CD [(-AR +

An adaptation from TN-1309 (Ref. 41) for various sideslip angles is

(CRp)tota I = [(cgp)+ota I J B = 00 cos2 B * Typical val ues of Cg range from -.25 per radian to -.60 per radian. P

90 C" P

_The stability derivativeC?p is the changein yawing moment caused by rol I ing, with the wing and vertlca I tai I the main contributors. For a positive roll, the producedyawing moment is a result of the unsymmetrical lift distribution causing increaseddrag on the left wing and decreased drag on the right wing and is, therefore, negative. The vertical tail contribution may be either positiveor negative, depending ontail geometry, angle of attack, and sidewash from the wing. Dutch Roll damping is influenced by Cnp in that the larger its negative value, the less DutchRoll damping. therefore, a positive C is desired. "P Using an elliptical lift distribution, Perkins and Hage (Ref. 11) give the following formula forCnp:

The results ofwind tunnel testing on a wing with AR = 5.16, h = 1.0, and 00,wing sweep is reported in TR-968 (Ref. 33). The resulting formula for (Cn Iw, by a curve fit, is P (CnpIw = -.043 CL - .0044

for 0 < CL < 1.05. This report shows that, forlarge CL (in the stall region), Cnp reverses signs, and large positive values are obtained. This same trend occurs at smaller CL when sweepback is encountered. Below is a formula for use with sweepback, from TN-1581 (Ref. 28). AR = + 4 1 + 6(1 +- (Cnp)w CL AR + 4 cos A AR 12

The ratio (Cnp/CL)A= 00 as a function of aspect ratio and taper ratio may be obtained from the following figure.

- 2 4 6 8 10 1612 14

AspectRatio, AR Figure 35. Values of (Cn /CL) for zero sweep. P 91 TN-2587 (Ref. 401 explains, through the following formula, how the vertical tail contribution to C"P is influenced by sidewash variations:

Values for ao7/[3(pb/2U)], effect of wing sidewash, as a function of (h /b 1, may be obtained from the fol lowing graph, reproduced TN-2332 from (Ref.t42r!2for wings with aspect ratiosin the vicinity of six, wherehe is the distance from the wing centerline to the center of pressure of the vertical tail (positive above the wing centerline).

"0 .1 .2 .3 .4 .5

Figure 36. Effect of wing sidewash on vertical tail. 92 The effect of fuselagesidewash, ao2/[a(pb/2U)], may be determinedfrom the followingformula adapted from TN-2587 (Ref. 40):

where Ah zv - (zvcos a - R, sin a) ”- bW bW

For zero angle of attack,this s idewash factorreduces to zero and is quite sma I I for lowang les of attack.With these factors, a value of (Cn Ivcan be determined and added to the wing contribution to yield total Cn P P’

(Cnp)tot- 1 = (Cn )W + (Cn )V P P Typicalvalues for Cn rangefrom -.01 perradian to -.lo perradian. P The stability derivative Cyr is the change inside force resulting from a change in yawingvelocity. As theairframe undergoes a positive yaw, an effectivepositive side force develops on the vertical tail, which is the dominantcontributor to Cyr. Sincethis force is normally small, Cyr usually has a smallpositlve value.

The wingcontribution is normally negligible, as indicated by TN-1669 (Ref. 431, which isthe result of windtunnel tests on rectangular wings with zero sweep and aspectratio of 5.16. The followingformula is produced by curve-fittingthe data of this report, which gives (CyrIwingas a function of lift coefficient for an angleof attack below the stall region:

(C 1 = ,143 CL - .05. Yr The tail contribution may be estimatedby the formula below from TR-1098 (Ref. 27) whichgives (Cyr)tai1 as a functionof the values for (CYb)tail or (Cng)tai I discussedin connection with the stability derivative Cyg In thepresent study:

Forwind tunnel tests of a model oscillatingin yaw, TR-1130 (Ref. 44) indicatesthat much largervalues for Cyr areobtained than from other testing methods.These resultsare presented graphically in Figure 37, including fuselage,wing, and taileffects. The fusefageeffects are relatively large and negativein sign.

94 -0- Fuselage "C+ Fuselage +Tail "0- frstlalt + Wing +Tail

0 2 4 6 8

Angle ofAttack, a , deg

Figure 37. Values forfuselage, wing, and tail contributions to Cy,.

TN-4066 (Ref.361, which utilizes flight test data to calculate stab i I ity derivatives,indicates that Cy, couldrange from -0.3 to 0.3 without show i ng any significanteffects on the other sfability derivatives. This helps demonstratethe insignificance of Cy, to the lateral stability of light a ircraft. ,

95 The change in rolling moment due to variation in yawingvelocity consti- tutesthe stability derivative Cgr. The wingprovides the major contribution, withthe vertical tail having a mlnoreffect. When there is a positive yaw rate,the left wing moves fasterthan the right wing,producing more lift on theleft wing and, consequently, a positiverolling moment. The tailcontribution may be eitherpositive or negative, depending on tail geometry and angle of attack of theairplane. Although Ctr has littleeffect on Dutch Roll damping, it isquite important to thespiral mode. Forspiral stability, it Is desirab le that Ckr beas small a positive number aspossible.

The w i ng contribution to Ckr ispresented in TR-589 (Ref. 45) as a function of lift coeff icient bythe following formulas:

(Ckr’w i ng = CL/~for rectangular lift distribution, or (Cgr)wing = CL/4 for an elliptical lift distribution.

In TN-1669 (Ref. 431, the results of windtunnel tests of a NACA 0012 airfoil of aspectratio 5.16 and zero sweep indicatethat (Ckr),,,ing = CL/~agrees with experimentaldata for lift coefficients below the stall regime. The wing contribution to Cgr as a functionof aspect ratio, taper ratio, and sweep angleis presented in Figure 38, adapted from Datcom (Ref. 10).

.1 .2 .3 .4

Figure 38. Wing contributionto Cgr as a functionof aspect ratio,taper ratio, and sweep angle. 96

...... , . Theminor contribution of the vertical tail to Cgr may be calculated from the formu la below from TN-I 984 (Ref.46) :

By addingthe wing and vertical tail contributions, total Cgr may be determined.

Typ.icalvalues of Ck, range from .04 perradian to .12 perradian.

97 Cnr, commonlyknown asthe yaw damping derivative, is thechange in yawing moment due to variationin yawing velocity. As theairframe undergoes a positive r, a yawing moment whichopposes the motion is produced. This moment consistsof contributions from thewing, fuselage, and vertical tail, all of whichare negative in sign. TN-1080 (Ref. 47) statesthat the vertical tail contributesabout 70% to 90% of Cnr forconventional designs. The derivative Cnr is the main contributor to the damping ofthe Dutch Roll mode and also plays a significantrole in determining spiral stability, making it vital to lateralstability. For best effects in each of these modes, largenegative values of Cnr aredesired.

Thewing contribution to Cnras a function of lift and dragcoefficients isgiven in TR-1098 (Ref.27) by the following relation:

Originallypresented in TN-1581 (Ref. 281, thisformula is the result of simple sweep theorywith strip integration; consequently, [(ACnr),/CL2]and [(AC~,)~/CD~] arefunctions of sweep, taperratio, and aspectratio. For wings withzero sweep andaspect ratio greater than five, however, theseterms are constantat the following approximate values:

(ACrlr) 1 = -0.020 , CL2 and (AC, 12 r = -0.30 . co, Withthese restrictions, the aboveformula reduces to

The graphthat follows from TN-1669 (Ref. 43) shows, for a rectangular wing, theclose agreement between the above theory and experimentaldata for wing contribution to Cn,-.

98 ---Theory

43- Experimental

0 .2 .4 1.2 .8 1.0

LiftCoefficient, CL

Figure 39. Resultsof experimental data on wing contribution to Cnr.

Thisgraph points out the relative insignificance of wing contribution to Cnr.

The tailcontribution to Cnr ispresented in TR-1098 (Ref. 271, as fol lows:

Blakelock(Ref. 18) treatsthis contribution as,

The verticaltail efficiency factor, qv, compensates forthe interference between thefuselage and thevertical tail. When thereis no interference, nv is equal to one.Total Cnr isthen determined by adding the wing and verticaltail contributions, since the fuselage contribution is negligible.

Anotherapproach to calculating total Cnr isfree-oscillation tests, as presented in WR L-387 (Ref. 48). These tests on a mid-wingairplane include effects of wing,fuselage, and vertical tail on Cnr. The wingcontribution is calculated as

which, for an aspect ratio of six and taper ratio of ope, seems to agreeclosely withthe previous formula for wing contribution. WR L-387 (Ref. 48) also

99 L .. . - ...... -...... ". -. . .. - .. . "- ..

statesthat the fuselage contribution often ranges frorrc -.003 to -.006, small enough to be neglected. The verticaltail contribution from Reference (48) is

Combiningthese two formulas, the result is the empirical expression below, showing Cnr for a conventional,mid-wing airplane:

Typicalvalues of Cnr for general aviation aircraft range from-.05 per radian to -.14 perradian.

100 C

The stability derivative CY^^ is the change in side force coefficlent with variationin aileron deflection. For most conventional light aircraft, this derivative is zero;however, for an airframewith low aspectratio and highly sweptwings, it may have a Jalueother than zero.

101

I The stabilityderivative CQ~ known asthe aileron effectiveness or A' "aileron power", is the variation in rolling moment coefficient with change inaileron deflection. Since left aileron down isdefined as positive, a positivedeflection produces a rolling moment to theright, which is also positive,making Cg6 positive.Desirable values of CQ~~aregiven in terms ofthe wing tip he1 18angle (pb/2U) for a ful I ailerondeflection. For I ight aircraft,this range is normally from 0.07 to 0.08 radians.In Perkins and Hage (Ref. 111, stripintegration can beused toevaluate CQ~~as,

b 4 i2 i2

Figure 40. Illustrationof strip integration.

Inorder to integrate,the chord, c, as a functionof the spanwise distance, y, mustbe known. Forstraight, tapered wings, this relationship is

where CR = rootchord.

The value of T may be obta inedfrom the following graph as a funct ion of aileron chord to wingchord ratio.

102 Figure 41. Valuesfor T as a functionof aileron chord to w i ng chord rat io.

Themethod ofstrip integration presented above is seldomused inpractice because of largeerrors incurred in assuming a discontinuous lift distribution. Inreality, the lift distribution adjustsb the aileron deflection quickly but smoothly.This being the case, the value of CQ is normallyfound from the spanwiseload distributiondata as presented in ?R-635 (Ref. 49). These dataare reproduced below for (CJ?,~~/T)as a functionof the extent of the unit antisymmetricalangle of attack.

.2 .4 . -6 1.0 Extent of unitantisymmetrical angle of attack; %b/2 Figure 42. Valuesfor (CQ~/T) as a function of the extent of unitantisymmetrical angle of attack. 103

"" ~ ...... - . . . .. A valueof (CQ~/T) isobtained from Figure 42 by firstusing the distance fromthe body centerline to the outboard edge of the aileron divided by the wingsemispan to get a value,from which is subtracted a valueobtained by usingthe distance from the body centerline to the inboard edge of the aileron dividedby the wing semispan. The value of (CR~~/T)from thegraph is multiplied by a valueof T fromthe preceding graph to give C!?,6A perradian.

Typicalvalues of CJ?,~~range from 0.1 to 0.25 perradian.

104 The stability derivative Cn6A, the change in yawing moment coefficient with variation in aileron deflection, results from the difference betweendrag onthe upand down ailerons.Since a positivedeflection is withthe aileron onthe. left wing down,Cn6A isusually negative, even though it isheavily dependenton the position and size of the ailerons and the angle of attack of theairframe. A negativevalue for Cn6A is known as"adverse yaw coefficient due to ailerons" because it is the result of initial yawing of the airframe in a directionopposite that desired for a turn. Thus, thedesired value of C is eitherzero or a verysmall positive value. "A Tocompute a valuefor Cn6A,a formula,adapted partially from Datcom (Ref. 101, is

Here, 6~ isagain positive for left aileron down and rightaileron up, with K being an empiricalfactor obtained from the following graph. spanwisedistance from centerline to the inboard ="- 'i - edge ofthe control surface bw/2 semi span

-25 -75 1.0

Figure 43. Empiricalfactor K as a function of TI for taper ratio = 0.5. 105 "3

-.1

.25 .5 1.o

Figure 44. Empiricalfactor K as a functionof Q for taper ratio = 1.0.

Typicalvalues of Cn6A range from -0.004 to -0.09 perradian.

106 The stability derivative CysR is the change inside force resulting from rudder def I ect ion. For a posit I ve rudder def I ect ion, or rudder toward the left wing, a positiveside force results; hence, thevalue of CysR ispositive.

Foran airplane.withoutautopilo-f, the effect of CYsR is relatively unimportant to lateral stability and often is assumed equal to zero.

InEtk in (Ref. 501, thefollowing formula for estimating is set forth:

where a, = lift curveslope of the vertical tail (calculatedas shown indiscussion of Cng), T = a functionof rudder area to vertical tailarea ratio as found from the graph below.

Figure45. Values for T as a function ofrudder area to vertical tail arearatio.

Typicalvalues of CYsR rangefrom .12 perradian to .24 perradian.

107 The stability derivative Cg,6R is the variation in rolling moment coefficient with change inrudder deflection. Because therudder is normally located above thex-axis, a positiverudder deflection (rudder to the left) causes a positive rolling moment, making CQ~positive. This value may possibly be negativefor an unusualairframe configuration or anabnormal angle of attack.For conventionallight aircraft, this derivative is of only minor importance and is usuallyneglected.

The followingformula adapted from Etkin (Ref. 50) may be used to determine CQR :

where zv = distancefrom the x-axis to aerodyanmic center of the vertica I ta i I

The valueof T may be obtainedfrom the graph below as a functionof rudder area to vertica I tai I area ratio.

I- -4

0 .1 .2 .3 .4 .5 .6 .7

Figure 46 Valuesfor T as a function of rudder area to vertica I tai I arearatio.

108 The stability derivative Cn6R is the variation in yawing moment coefficient with a change inrudder deflection. Also known asthe "rudder power," this derivativeis negative, since a positiverudder deflection toward the left wingcreates a negativeyawing moment.

Perkins and Hage (Ref. 111 give sv Rv C =-av~-- "6R sw bw nV

The value of T may be obtainedfrom the following graph for a particularrudder area to vertical tai I arearatio.

Figure 47. Values for T as a functionof rudderarea to vertical tai I arearatio.

The value of Cn6R isnormally on the order of -.06 perradian but may varygreatly, depending on the airframe configuration. Powerhas a greateffect on Cn6R, asseen inthe following graph from TR-781 (Ref. 51) for a single engineplane with propeller rotating to the right, flying at a speed of 111 milesper hour. -04

-0 2 Cnlfr: -.0012 Normal Power

Cn 0

"02 :"0023 Full Power '"6 r Flap+Gear Down

' II -.04- both per degree -2010 - 10 bo 20 Figure 48 . Effect of aircraft poweron Cn6R. 109 POWER EFFECTS

Althoughthe primary function of a propulsionsystem is to overcomeairplane drag,the location of thesystem with respect to aerodynamicsurfaces may influence some aerodynamicparameters and, ultimately,aircraft stability. Thus, aswing loadings increase, more pronounced power effects are to beexpected. The literatureindicates that analytical determination of power effects for lightaircraft is, at best, anapproximation. Many referencessuggest only the origin of a particulareffect and estimateonly its order of magnitude.The followingkill discuss the significant reports dealing with power, thecharacteristic changes to beexpected in longitudinal and lateral stability due to powerapplication, and theanalytical procedure given in Datcom(Ref. IO) for estimatingthe effects of poweron longitudinal stability.

Power effects for light aircraft canusually be classified as I) directpro- peller effects and 2) slipstreameffects, with more accurateestimation possible for propellerthan for slipstreameffects. Since the engine is directly in frontof the horizontal and verticaltails, the slipstream effects on single engine aircraft may bemore difficult to predict than those on twin or multi- engineaircraft, especially in the lateral mode. The lack of documentationon ways to predict analytically the effects of poweron lateralstability bears out this conclusion.

Theneed for higher-poweredaircraft during World War II led to consider- ableinterest in effects of poweron aircraftstability. WR L-710 (Ref. 102) indicatesthat powerhas a significantinfluence on both the longitudinal and lateralstability and controlcharacteristics. Using this report, oneconcludes that power appliedto single-engine, low-wing models decreases longitudinal stability and effectivedihedral. Directional stability and rudder and elevatoreffectiveness wereusually increased by applications of power. At the same time,the report warns that power-onwind tunnel tests should be used to predict actual flight stability andcontrol, lest the researcher be misledby theoreti ca I data.

NACA TR-690 (Ref.103) is a report of theeffects of propelleroperation onthe flow about eight wind tunnel models. A tabulationof downwash angles, thedynamic pressures at the tail, and thepitching-moment contribution of the propeller and thewing is presented. TR-941 (Ref.104) correlates some perti- nentexperimental data on power effects obtained during the War Years. It givessemiempirical procedures to predict power-on longitudinalstability char- acteristicswith flaps undeflected, andagrees well with experimental data.

In two TechnicalNotes, 1339 (Ref. 105) and 1379 (Ref. 1061,Hagerman discussesthe effects of power on the longitudinal and lateralstability of a single-engine,high-wing airplane model. He foundthat, for longitudinalstability, power greatlyincreased the lift increments and thetail-off lift curveslope while, in general, it decreasedthe stability of the model for all threeflap configurations tested. For lateral stability, application of power hadno effecton the effective dihedral, with the flap neutral; however, with bothsingle and double slotted flaps deflected, power increased the effective dihedral. The directionalstability of theentire model was increasedexcept withflaps neutral at low lift coefficients.Rudder effectiveness was decreased 110 withflaps neutral and double slotted flaps deflected and increasedwith single slottedflaps deflected. Trim changes caused by power were small, indicating good control. TN 1327 (Ref. 107) onlateral stability, written about the same timeas Hagerman's work, revealsthat power decreased the dihedral effect re- gardless of theflap condition, increased the directional stability, and in- creasedoverall lateral stability as lift coefficient was increased.

The problem of power effects also justified the investigation undertaken in TN I474 (Ref. 1081, whichsought to off-set power effects 'by usingan unsymmetrical tail onthe single-engine airplane. Although the tests and analyses showed that extreme asymmetry i n the hori zonta I tai I i nd i cated a reduction i n power effectson the longitudinal stability, the "practical" arrangement tested didnot show markedimprovement. Three years after the asymmetric investigation, a dynamicfree-flight study of dynamiclongitudinal stability as influencedby static stability measured inwind-tunnel force tests under conditions of constantthrust and constant power was undertaken.(Ref. 109) The results agreedwith previous studies that the longitudinal "steadiness" of airplanesis affected to amuch greater extent bychanges in constant-thrust static margin than bychanges inconstant-power static margin.

TN D-3726 (Ref. 1101, a recentreport directly related to light aircraft, discussesthe effects of power on thelanding configuration stick-fixed and sti'ck-freestatic longitudinal stabi Iity for the aircraft (including both twin and singleengine aircraft) with the most pronounced power effects. When the power was cycled from approach to maxi mum at an ai rspeed of 80 knots the pi lot had to push with a forceof approximately eight pounds to counter the resulting nose-up pitch.This characteristic also presented a problem when thepower was beingreduced in the landing phase. For light aircraft, power effects caused bypropel ler sI ipstream can become quitelarge. At a speed of I IO knots, approximately IO degrees of rudder and90 pounds offorce was requiredto maintain heading when changingpower from maximum to idle.This was consideredexcessive by the pi lot.

Anotherrecent paper (Ref. I I I) on light aircraft gives methods foranalyzing power-on staticlongitudinal stability. The methodsare simi lar tothose givenin Datcom (Ref. IO); however, thepaper also describes how power effects canbe analyzed using airplane stick force, elevator deflection, and neutral points. The method utilizes a point-to-pointcalculation technique for each speed or loadfactor variation from trim.

In 1969 and 1970, NASA investigatedlongitudinal and lateral stabi lity characteristics of both a lightsingle engine and a lighttwin engine aircraft in a ful I scaletunnel (Refs. 112 and113). These investigationswere carried out for severa I conditions of power,as i ndicated by the extensive data in each report. These two reports may beused eitherto obtain a roughestimate or to helpverify the exactness of analytical procedures for estimating power effects on ai rcraft simi lar to those investigated.

Althoughthe I i teraturementioned above may givethe general trends for power effects.,analytical techniques are often hard to fi,nd, cumbersome to use,and inaccurate.For the present, Datcom(Ref. IO> probablygives the best analyti- calprocedures for predicting powereffects. They areextensions from those of Perkins and Hage (Ref. Ill, among others(Refs. 114 and 1151, with improvements addedwhere possible.Datcom's methods of estimating power effects on lift and 111

I pitching moment variationwith angle of attack are summarized below. Curves of both ACL and ACm versus a canbe plotted for several flight conditions using theseprocedures; thus, CL~and Cma due to powercan be evaluated. Scrutiny of the present discussion points out that the estimation of power effects deals onlywith static longitudinal stability. It would seem that an approximation of power effects for thedynamic derivatives could beachieved by estimating the change in TI&with power,computed analytically in a simi lar manner to that given below,and usingthis corrected value. Some ofthe methods given can also beused toestimate lateral stabi l i tyderivative changes with power. For example,(C~,)p is analagous to (Cy Ipor (-CygIPfor cruising flight and could beused toestimate or (Cy6 Y p. When approachedfrom the engineering viewpoint,similar methods couldalso beused indetermining slipstream effects on the vertical tai 1.

Againthe reader should remember that,at best, analytical prediction of power effectsis crude.

Lift Increment Due to Propellet- Thrust

(ACLIT = nCT sin a T

where n = number of eng i nes = thrustaxis angle of attack to free stream in degrees ct T Lift Increment Due to ProDe I fer NormalForce

nN S (acLlN = cos a = nf(c 1 (ap)( $j cos a (perradian) 2 T T P qsW N" p W

where aP = anglebetween local ai rstream and thrust in degrees Sp = propel Ier disc area f = propel ler inflow factor (C 1 = propellernormal-force derivative at T ' = 0 perradian Na p C

where a0 = wingangle of attack for zero I i ft in degrees a€ A=a" wing upwash derivativegiven in Figure 49. The factor f accounts for the increase in ve locity at the prope I ler p I ane due tothe induced flow of the propeller and isgiven in Figure 50 asa function of

'w 'T 8R P wherr R = propeller radius in feet. P

112 (C 1 = [(C 13 [ 1 +0.8(- KN - Na p Na KN=80.7 80.7 11 (perradian)

b b b b (per where K,,, = 262 3Rp + 262 + b I ade) (8). (8).6Rp 35 (8).9R P P PP with b = bladewidth' in feet ' P C(C 3 = propellernormal-force derivative at CT Na KN =80.7 andgiven in Figure 51 as a function of B, thenominal b ladeangle at 0.75 radians; this blade angle shou Id be obtainedfrom a performanceengineer or from a power p I ant engineer.

113 0 2 4 6 8 10 12 14 16 18 20 22 SWC~/8R2 Figure 50. Propellerinflow factor.

0 Id 20- 30. 40. sd 60' Nominal Blade Angle-@ at.75 radius

Figure 5 I. Propeller normal force parameter.

Lift'Increment Due to a Change in SI i pstrearn'Dynarni c Pressure on the Wing

114 where K1 = correctionparameter for additional wing lift due to powerand givenin Figure 52, where ARi = effectiveaspect ratio of wing immersed inthe slipstream, Ai = 2Rp/ci , c i = average chord of w i ng immersed i n s I i pstream. - Ans - g = ratio of change in dynamicpressure in propeller slipstream to freestream dynamicpressure and given by SwcT Ans = 7(per engine). *RP

Si = wingarea immersed inthe slipstream in square feet .

115 Figure 52. Correlationparameter for additionalwing lift due to prope I ler power.

I16 Ljft Due -fothe Change in Ang le of _Attack -l.ndu-qed by. the Prope I ler Flow Fie1 d

“E where Aa = 1- - aa

aE U a = aT - -(a -a 1 P aa w o The propeller downwash derivative is given by

where C 1 and C2 arepresented in the figure below and (C 1 IS given in one ofthe previous sections. Na p

0 2 4 6 8 10 12 14 16 18 20

sw or/8 + Figure 53. Factorsfor determining downwash due topropellers. Lift Increment Due to the HorizontalTail

= (ACLt -(ACmIt (~tC

where (ACm), = total change in pitching moment of the horizontal tail due to powerand can be calculated using

(ACmIt = (AC, 1 + (ACm IE tq t given in the next section--pitching moment variations with power. Total Lift Increment Due to Power

Pitching Moment Increment Due to the Offset of the Thrust Axis from Oriain of Axis ZT (ACmIT = C - Tc Pitching Moment Due to the Propeller Normai Force Ihl (Acm)N = (*‘L)N c cos a P P T where x = distance from the intersection of the propel ler plane with the thrust axis to the wing quarter chord. (ACLIN is evaluated in a previous section. D Pitching Moment Due to the Changein the Lift of theWing Caused bv Power Effect X (ACmL = - [,A, L Ans + (ACLlaw] $-

where x = distance parallel to x-axis fromwing quarter- W chord to aerodynamic centerof wing area immersed in the sli~stream.in feet (AC 1 and (AC ) are eva’luated in a previous section. L Ans aw

118 Pitching Moment". ~ ~ Due to Change in Dynamic Pressure Acting on the Hori zontaI Ta i I sa (AC I = -C - -t -t Lt 9 sw = where 7is presentedin the figure below.

Figure 54. Effect of propeller poweron dynamic pressureratio at the horizontal tail. 119 Pitching Moment Due to the Change in Angle of Attack of theHorizontal Tai I

St qt (ACm 1 E = CL AE - -(-) Sw c q power a t t where AE isgiven in Figure 55 forsingle engine ai rp lanes and Figure 56 for mu1 ti-engine airplanes. off obtainedbecan from a formu I a i n the CL section.

Aq where -is t eval uated i n Figure 54 above. q

14O

12O

10"

y. 8' f t 6O

Y) 4"

e = Downwash 20

0 .4 .8 12

Figure 55. Incrementin downwash due topropeller power forsingle-engine airplanes.

120 Figure 56. Increment in downwash due to propeller power for multi-engine airplanes. Total Change in Pitching Moment Dueto the Propeller Power Effects

(ACm)power = (ACm.)T + (AC I + (ACmIL f (ACmtIq NP + (ACmt)E CONTROL FORCES AND DEFLECTIONS

PitchControl

As indicatedpreviously, the state of knowledgeregarding desirable air- crafthandling qualities up to 1948 is codifiedin NACA TR-927 (Ref.16) and in the text by Perkins and Hage (Ref. 11). Theseworks have served those of the presentgeneration of aeronautical engineers without access to largeresearch anddevelopment budgets virtually as holy writ, and the handling qualities of mostaircraft of less than 10,000 pounds grossweight now flying reflect this technology. It istherefore desirable to examine theparameters cited in these works, theirvalues, and thefactors which produce them in some detailas a foundationfor recent advances in understanding and new results.

Foran aircraftwhich employs a trim tab to set longitudinal flight velocity, it can be shown that at anygiven trimmed speed the incrementalstick forcevariation with speed isgiven by Perkins and Hage (Ref. 11) as

where G = ratio elevator displacement to the product ofstick length and stickangular displacement,

and

Since

and

then

A pullforce isnegative. It is seen thatfor a givenaircraft, the stick force gradient at t rim dependsupon the trim speed, theweight, and the c.g.location. Short of majorgeometric modifications, the force gradients for a given speed, weight, andc.g. locationcan bechanged by modifying G and thehinge moment parameters, Ch, and Chg. The effectof elevator balance point, elevator nose

122 shape, elevator trailing-edge shape, mass balancing and sealingon Cha and Ch6 aretreated extensivel'y In TR-868 (Ref, 521, Figures 57, 58, 59, 60, and 61 illustratingthese effects are taken from this work, A guide for uslngthese chartsfol.lows Fi.qure 61,

.36

.32

.2a

.24 Fl 4b 4, .2a

.16

.12

.oa

Figure 57. Chartsfor determining numerical values ofoverharg factor F1 and of nose-shape factor F2 fromgeometric constants of balancedailerons.

123

'I # Nose Section showing nose shape Nose shape factor, F2 type I

kcb -I

Figure58a. Various nose shapes considered in correlation of pIain,overhang,and Fr i se ba I ances and correspond i ng express ions for nose-shape factor. 1 24 I?-

.60

.so

- .4 0 -'b Ca .30

.2 0

.lo

.6 0

.s0

- .4 0 --'b 30

.20

.10

0 .04 .O 8 12 .16 0 .04 .OS .12 .16 4 F2 4 F2

Figure 58b. Charts for estimatingthe required lengths of overhangshaving various noseshapes. Letters A,B,C, and D refer to the correspond ingnose shapes of Figure 58a. 125 0 4 8 12 16 2820 24 32 36 40 Trailing-edge angle, 4, deg. Figure 59. Hinge moment parameters of plain 2-0 aileronswith various chords and values of 9. Gaps sealed;ca/c=aileron to wing chord ratio. 1 26 .OO!

.004

.003

.OOi ch, .001

,001

-001

-.007 -.005 -.003 -.001 .001 .003 .005

9.6'

A I RPLANE CONSTANTS

Wing span, b, ft...... 43 Wing area, S, sq ft ...... 308 Aspectratio, AR...... 6.0 Taperratio, X ...... 5 Root airfoilsection...... NACA 23015 Tipairfoil section ...... NACA 23009 Airplaneweight, Ibs...... 12,000 Stick length, ft...... 2.33 Figure 60, Effects of aerodynamicbalances on aileron hinge-moment para- metersestimated from correlations. c- = 0.25. 127 I

0 .002 .004 .008 .006 .018.016 .014 .012 .010 .020

0 .o 2 .04 .06 .OS .12 .16 .14 .lS .20

A+2

Figure61a. Effect of sealed internal balances onthe hinge-momentparameters of controlsurfaces. M=0.2 or less.

128 4 8 12 16 20 24 28 32 Trailing-edgeangle (deg) Figure 61b. Effectof gap onhinge-moment variationwith trailing edgeangle. NACA 0009 airfoil; 2-D model, 2 =0.30. C The generaiexpressions for changes in the hinge moment characteristics which resu It f rom geometric changes a re

AC = & [O.O17F, + 5.05 x A@] 9 ha AR+2 and

- -AR [o. loF1F2 + 5.7 X A@] . Ach6 AR+2 The trailing edge angledata are for balanceswith a gap of .005E.

One first selects a noseshape from Figure 58 and securestherefrom a valueof F2'. This involves, in addition, a selectionof cb/cq. Thesymbols Mg, MB, Mc, MD, ME, and MF refer to moments aboutthe hinge axts of the profile areasof exposedoverhang balances of typescorresponding to the sub- scripts 0, 6, C, and so forth. Thebalance profile area is defined as the totalprofile area of theairfoil ahead ofthe hinge axis. F2=F2' for nose shapes 0, A, 6, D, and G. F2can also befound from Figure 57. The same figureis used to determine F1. One thencalculates F1F2'and uses Figure 61b or theequations above to checkwhether the initial selection of nose shape andnose overhang will givethe desired modification in hinge moment characteristics.Additional modifications in ch and Cb canthen be obtained from trailing edgeangle variations as indicatgd in Ftaures 59 and61a or the equationsabove.

Forsealed balances, the equations are

129 and

Theincrements represented by these equations are to be added to the hinge moment coefficientsproduced by controlsurfaces with noarea forward of thehinge line. The results of calculations for thebasic hinge moment coefficients of thin,simple shapes shown in TR-868 (Ref. 52) canbe representedby C a Ca Ca Cha - ,0003 - < 0.4 , c C and

It shouldbe noted, however, thatthese values will be altered to some extent by thecondition of the flow overthe basic wing (condition of boundary layer,presence of tipvortices, etc.) andby the particular shape of the controlsurface. For this reason, precise values are usually obtained ex- perimentally.Further details may befound inReference 52.

One finalconsideration should bementioned here: To reducedrag, inertially-inducedcontrol surface deflection, the possibility of flutter, and effects of controlsurface droop, and thecontrol surface actuation force,the control surface is also mass balancedabout the hinge line. This will oftenrequire that lead weights beplaced in the nose of theaero- dynamicbalance or attached bylong arms to thehinge axis.

Report 927 (Ref.16) illustrates suitable values for Cha and Ch6 for a typical example. Thisis reproduced as Figure 62. Ingeneral, one desires to keep thevalues of Cha and Ch6 small.This will also keep thestick forces to reasonablevalues for high speeds or heavyweights.

Some of the means for al.tering the effective G mechanically(springs and bob weights)are discussed by Perkins and Hage (Ref. 11). Fully poweredcon- trol systems, of course,can use other means if necessary to producethe desiredfeel. (See, for example, TN D-632 (Ref. 531.1

Formerly,considerable effort was expendedtoward specifying acceptable values for dFs/dU, butrecently specification writers (both in FAR part 23 (Ref.54) and in MIL-F-8785B (Ref. 4) 1 haveasked only that dFs/dU bestable

130 P

at all flight conditions, e.g., thatincreasing pull forces and aftmotion ofthe elevator control be requiredto,,maintain lower than trim airspeed. There is no requirementthat dF,/dU be linear.*

In addition to calling for stab1.eforce-speed gradients, both FAR part 23 and MIL-F-8785B specify The maximum out-of-trim forces an aircraft should requirein a varietyof flight conditions. The tablebelow summarizes these maximum forces and conditions.

Stable ngim

Neutral stick-free stability

0 C hat Positive values of Chi. not used because of unstable shortperiod -.w5 oscillations with stick free stability for static margin of .05c I Unstable region -. 010 -.015- -.010 -. 005 0 .005

Figure 62. Boundarybetween stable and unstablevalues of ch and C for the example givenin TR-927. hb, Unstableside of boundaries i nd icated by cross-hatching.

* Indeed,dFg/dU isdirectly proportional to U away from trim. Other causesof non-linear force variations include the fact that all the derivativesin the equation are evaluated at specific points only. At high values of CL, for example, is lessthan for CL near 0. Thus,one shouldtake care that the vative values used forcalculations are thosevalid for the range of angles being considered.

131 1.5 Ustallwith power offandgear and flaps 10 Ibs FAR 23.145 down at most forwardc.g.

Any cruise speedbetween 1.3Us and Umax; Approachspeeds 40 Ibs FAR 23.175 between 1.1Us and 1.8Us

Maximum forcefor pro- longe d application;longed 10 Ibs FAR 23.143 Temporary application Stick 60 Ibs Whee I 75 Ibs

Ta ke-of f 20 Ibs pull to MI L-F-8785B 10push Ibs § 3.2.3.3.2

Land ing be Must a pull MI L-F-8785B forceof not more § 3.2.3.4.1 than 35 I bs

Diveswithaircraft < 50 pushIbs trimmedfor level flight < 10 Ibspull 1 stick < 75 Ibs push < 15, I bs pu I I] M t L-F-87855 5 3.2.3.5 with trim atdive entry < 10 fbs stick < 20 fbs wheel

Table 11. Specificationrequirements for maximum out-of-trimforces.

Thegeneral expression for stick force is given in Perkins and Hage (Ref. 11) as

132 where a, = aircraftangle of attack for zero lift, iw = wingincidence angle, ie = tailplane incidence angle, Cho = residualhinge moment coefficient (hinge moment notcaused by deflection or angle of attack), &eo = elevator angle at zero aircraft lift, 8, = trim tabdeflection.

Notethat for trimmed flight, 6t is such that Fs = 0. Foraircraft with irreversiblelongitudinal control systems, Fs is the force which the control systemmust apply to the elevator or stabilator, but the force which the pilot feelsis determined by controlsystem design.

The onlymention found in the literature for desirablecontrol deflection gradientsis the requirement in MIL-F-8785B93.2.2.2.2 (Ref. 4) thatthe pilot applynot less than 5 Ibsforce for eachinch of stick travel.

Anythingwhich changes the flow field in the neighborhood of the horizontal controlsurface can have an effect on its actuating forces. Such changescan be theresult of applications of power(which result in increases in nt and contributions to the downwash fieldfrom the propeller slipstream) or alterations inthe lift configuration, aswith deflection of flaps (which also result in altered downwash fields). Thechange in wingpitching moment accompanying the deflectionof flaps varies the trimming moment whichthe tail is required to generateand thus changes the stick force or the trim tab setting required. Additional moments associatedwith the application of powerdevelop when the thrust does notact through the c.9. andbecause propellersgenerate in-plane forcesin upwash fields. Some ofthese effects tend to reduce stick forces and gradients;others tend to increase them;none of themare easy to estimate accuratelyfrom theoretical considerations alone. Wind tunnelor flight testing isnecessary for accurate evaluation. For most aircraft, the application of power resultsin a more positivevalue of Cm, and a reductionin control forces. The discussionin Perkins and Hage (Ref. 11) enablesone to arrive at some veryapproximate quantitative values for theseeffects. The problem of undesirable trim changeswith application of power or extensionof gear and/or flaps is apparently a fairly common onesince several of the light aircraft tested for TN D-3726 (Ref. 55) evidencedsubstantial shifts in stick force with the application of power or theextension of flaps or gear.

Inaddition to theforces required to change speed, theforces needed to change flight direction contribute significantly to the pilot's impression of thevehicle's handling qualities. In the longitudinal mode, theseare usually

133 statedas the stick force per "g". The forcesrequired to change directionare higher than f-hose required to initiah a change in speedalone at the same dynamicpressure because an additional trim moment is required to pitch the aircraft to anangle of attack sufficient to generatethe additional lift needed to curve the flight path and because rotation(up-pitching) induces a posltiveangle of attack component in theflow about the tail plane, which one must counter with a compensatory elevator deflection.

To findthe elevator stick force for anarbitrary maneuver, it is necessary tosolve the general equations for the normal acceleration produced by that elevatormotion which results from the specified application of stick force. Forsimplicity and standardizationin analysis and inflight tests, however, maneuversdesigned toevaluate stick force per g arelimited to steadypull-ups inthe xz plane of inertial spaceand to steadyturns in the xy plane of inertial space.The general expressions for stickper g inthese circumstances a re - GntSeCe(W/S)Chg +"" (>Pu I I -ups chg C Sw '6 d6e cm6

Notethat the stick force gradients will be linearin pull-ups so longas G, Cma, Cb, Cha, Chg, (dat/d6,), (de/da), Ckt, and nt retainthe same values asfor n=l. Note also that as turns tighten, the stick force per g approaches the same valueas for pull-ups.

On thequestion of linearity, MIL-F-8785B(Ref. 4) specifiesthat the localvalue of dFs/dn shall not differ bymore than 50% from its averagevalue, The forcelimits in maneuvers arestated differently for stick andwheel controls, the rationale being that stick controllers are easier to manipulate precisely at low forces andwheel controllerspermit larger forces to be applied. Not unexpectedly,the force limits are stated in terms of the limit loadfactor whichthe structure can sustain. One wouldnot wish to be able to exceed this loadfactor with a verylight force; nor, on the other hand, wouldone wish the stick forces to be so great that the maneuver capabilities of the craft could

* ch6 isoften increased by IO$ to accountroughly for thefuselage damping, 134 notbe achieved. For the limit loadfactor of 3.8 typicalof most light aircraft, the specification stipulates that for stick controllers the maximum stickforce per g shall beno more than 28 Ibs per g norless than 20 Ibsper g.The minimum gradientis 7.5 Ibs perg.

With wheel con--rollers,the maximum values are at least 42.8 Ibsper g butnot more than 120 Ibsper g. Theminimum valueis 16 Ibsper g.

llThe'-.term gradient does not include that portion of theforce versus n curvewithin the preloaded breakout force or friction band."These, accord.ing to paragraph 3.5.2.1 (Ref.4) should be between 1/2 and 3 Ibs for a stick and 4 I bs for a wheel.

FAR part 23 (Ref.54) places no numerical limits on the elevator stick force in maneuvering flight other than to say(23.143) thatthe force on the pitchcontrol should never exceed 60 Ibs (stick) or 75 Ibs(wheel) during tem- poraryapplications or 10 Ibsduring prolonged applications. The popular flyingjournals generally do notreport such data perhaps because itsacqui- sitionrequires the installation of considerable instrumentation or because theomission of quantitative standards in the FAR'S suggests to some either a lack of significance or a lackof popular appreciation for the meaning of quantitativevalues in describing handling. Since there are few reports dealingwith light aircraft available elsewhere, it is difficult to determine theforce gradients nowcommon in light aircraft.

Two sources,however, are helpful. TND-3726 (Ref.55) reports that, for some ofthe light aircraft investigated, stick force gradients varied between 8and 17 Ibsper g for speeds of lessthan 100 knots. When compared withthe requirementsof MIL-F-8785B (Ref. 41, theseaircraft have gradientslower than desired. On theother hand, tests on a lightaircraft adapted for military use (Ref.56) showed compliancewith the specification at all flight conditions and aircraftloadings. Based onthese limited samplings, one would expect to find a widevariation in the handling during maneuvers exhibitedby contempor- ary I ight aircraft.

One additionalarea given prominance byMIL-F-87858 butnot referred to elsewhere inquantitative terms is the phase relation betweenthe application ofcontrol force and themotion of the cockpit control on the aerodynamic controlsurface. Paragraph 3.5.3.1 requires that control deflection should not leadthe application of control force. Paragraph 3.5.3 specifies that the controlsurface deflection shall not lag the cockpit control forces by mre than 30° in phaseangle for application frequencies equal or less than Wn SP ' Incontrol systems where theaerodynamic surfaces are actuated by rigid linkagesfrom the wheel orstick these requirements of course are always met. If, however, thesystem contains elastic elements, linkage force boosters, or remotelycontrolled actuators this may not be thecase. The characteristics ofsuch systems will require investigation and possiblealteration to insure compliancewith the specification. Reference 56 presentsresults taken with typicalforce and angularposition transducers which could beused alongwith oscillographicor tape recorders to obtain data suitable for analysis of the controlsystem phase responses. 1 35 Rol I Control

If onereviews the literature dealing with the handling qualities of light aircraft in the roll mode chronologically,he becomes aware of a subtle shift of emphasisfrom concern primarily with achievement of a given rolling rate at low speedsand limitation of aileronforces at high speeds to concern with total pilot workload (including rudder and elevator coordination required)during rolls and to theangle attained in a specifiedperiod of time. Thisis not surprising when one recallsthat during the era of World War II good rollingperformance in fighter aircraft (weighing generally less than 10,000 Ibs, atleast at the beginning of thewar) was essentialto survival in dog fights. As knowledge of how to achievethis performance aerodynamically grewand thecontrol forces at high speedwere reduced by power boost systems, attentioncould be devoted to factors affecting safety and precision ratherthan simple survival. Hence, the 1969 revisionof MIL-F-8785B (Ref.4) devotes considerable attention to limiting the forces the pilot must apply to rudder and elevatorduring aileron application. It also changed therequire- ment for the attainment of a giver; pb/2U to theattainment of a given bank anglein a fixedtime on the premise that such a requirement was more mean- ingfulin establishing collision avoidance capabilities and was equally suited to establishing the desirability of other areas of rolling performance.

The shift in areas of concernwith time also had its analog in the ana- lyticaltechniques employed and theparameter values identified. Whereas one wouldbegin a discussionof roll handling with consideration of thosecontrol forces,control deflections, and their yradients which affect the pilot's opinion of an aircraft'sroll handling qualities, it will berecognized that suchthings as the phase relationships between control application and aircraft response,the presence of spuriousresponses which require control input to counter,and the precision with which desired maneuvers can be executed also contributesignificantly to hisoveral! impression of handlingcharacteristics, Thesefactors plus the inevitable coupling of lateral and directional modes and thefact that aerodynamically the aircraft has no inherentbank orientation and thus no staticroll stability in the conventional sense, make it necessary to employ a moregeneral approach toroll handling analysis than thesimp!er view suitable for the treatment of pitch handling. The discussion whichfollows begins, therefore, with the simple, one-dimensional view and moves on to detail methodswhich the specification writers have used in an effort to quantify other phasesof acceptable roll handling.

The general,linear treatment of steady, one-dimensional roll has been availablesince the mid-1940Is. (See References II and 16, forexample.) With theassumption that all derivatives remain constant irrespective of aileron deflection, speed, altitude, and rollingrate, and thatthe time required to attain the maximum rolling rate for a givenaileron deflection can be considered

1 36 zero*,one can write

for the bankangle as a function of aileron deflection and

* One-dimensional rollingmotion is described by the equation ..

whose solution for a stepaileron input is

2 Inthis equation, q = 1/2 pU . Sincethe denominator of the second term ** (equal to (1/-cR) 1 generallyhas a valueof -10 or more (negative), it is obviousthat for times in excess of about 1.0 secthe second termcontributes verylittle to the bankangle attained. MIL-F-8785B, §3.3.1.2 requiresthat the maximum valueof TR forlight aircraft is 1.0. §3.3.1.4 furtherstates thatthe rolling mode and thespiral mode shallnot couple, ie. the time constants have the same value, to producean oscillatory mode. For a simulator study of this case, thereader is referred to TND-5466 (Ref. 57).

Forthe complete three-dimensional treatment of rolling response to ailerondeflection, the appendices to thepresent study should beexamined.

** See Appendix D for a discussion of the stability derivatives most importantin determing the value of the rolling mode timeconstant, TR.

137 pub SacaG

for the bank angleas a functionof control force. is the total (up plus down, assumed to be equaland equally effective) aileron deflection. y’ is thespanwise location of theaileron centroid, It is seen thatat low speed therolling performance increases with increasing speed untilthe force limits arereached; at this point the bankangle possible per unit time decreases withincreasing speed. The ratio (Cg6/CEp) is a function of taperratio, aileron-to-wingchord ratio, and per cent of spandenoted toailerons. In addition to making this ratio as large as practicable (maximum possiblevalue isabout 1.51, onedesires to adjust

to be approximately 0.5 so as to increase the roll capability at high speeds. Carefulattention to ailerongeometry is therefore necessary to obtain the properratio of cha to chg. TR-868 (Ref. 52) is an excellent source of experimentaldata on geometric effects and designmethods.

It is apparentfrom the foregoing that if oneminimizes friction and elasticityin the aileron linkage, the factors contributing to wheel orstick force and to rollingperformance are well known.Even non-linearitiesin the derivativeschg, cha, etc.,while requiring that tedious computai-ions be made to obtain +(t),are well documented for a large number ofconfigurations. Thus it is a fairlystraightforward matter to translate control force, control deflection, and rolling performancelimitations into hardware specifications. It remainsthen to state the appropriate values for these qualities.

The maximum forcegiven in FAR part 23 (Ref. 54) and MIL-F-8785B (Ref. 4) seem, for the most, to havebeen selected by experience from many years of pilot comments; nevertheless,comparison of theseresults with available anthropologicaldata (See discussion of Human Factorsin this work) for the com- fortableapplication of lateralforces showsgood agreement. The tablebelow gives asummary of thespecification requirements.

138 -__-" RE MEN T REFERENCE REQU I REMENT

.. - __T_" "~ , Temporary: Stick 601 Wheel FAR73# (Max) I43 23. ProI onged I O# I Stick Whee I Theminimum force for maximum M 1 L-F-87858 ro I I i ng performance sha I I not 9 3.3.4.2 beless than the breakout force plus 1/4 of the abovevalues.

Controlcentering and breakout forcesshall not bemore than 53.5.2. I 2# (stick) or 3# (wheel)nor less than I /2 #.

Rollingperformance: aircraft mustreach a 60' bankangle in 53.3.4.14 1.7 sec in cruise and a 30° bank in 1.3 sec forapproach

Notmore than 5# (stick) or IO# (wheel)should be required to achieve a 45O bank withrudder 53.3.2.6 free and theailerons trimmed for wingsin level flight. "_ Not morethan 60° of wheel +80° for a rotation,in either direction completely 53.3.4.4 -._C rnechan i ca I system Thereshall be no objectionable non-linearitiesin variation of 53.3.4.3 rollingresponse to wheel motion.

i Controlsurface response shall i notlag cockpit control force input bymore than 30° phase 53.5.3 angle for frequenciesless than 1/-rR or Wnd, whichever is I arger .

Cockpitcontrol deflection shallnot lead cockpit control 53.5.3. I force.

Table 12. Specificationrequirements for comfortable application of I atera I forces . 139 Therequirements quoted in the preceding table are those restricted to Class I (smalllight airplanes such as light utility, primary trainer, and lightobservation up to about 12,000 Ibsgross weight); category B whichin- cludesclimb, cruise, and descent and category C whichincludes take-off, approach,and landing; and Level I, ie,having flying qualities clearly adequate for theMission Flight Phase. Thisapproach was selected for the present-workbecause it was desired to show the present state of understanding of what isrequired to obtainsatisfactory handling qualities.

The requirement on the time constant of the rolling modewas givenpre- viouslyas not less than 1.0 sec.Examination of theequation of motion citedpreviously will also show that the aircraft will respondwell to sinu- soidalaileron inputs up to frequencies of

P rad/sec, 2u Txx where 2 q = 1/2 pu .

Thisis generally above therange at which the pilot can track. Thus, if the phaselag inthe control system is less than 30' up to thisfrequency, +he pilot will find the aircraft able to generate roll rates corresponding to wheel positionvirtually as rapidly as hecan turn the wheel or move the stick.

isnoteworthy that in preparing this specification, its writers knave quantitativelimits on

The rol I ingperformance of which the aircraft must be capable,

The maximum andminimum forces needed to produce maximum rolling velocity,

Permissiblebreakout and centeringforces,

The maximum controldeflection.

These, coupledwith the frequency response requirements mentioned earlier and the banon objectionable non-linearities, present a complete and quantitative descriptionof the factors contributing to the pilot's satisfaction with roll mode hand I i ng.

The usualmethods of producing a rollingmotion is the differential deflectionof outboard portions of trailing edge ofthe wing, or ailerons, If onewishes to roll to the right, he deflectsthe right aileron up to reduce lift on that wing and deflectsthe left aileron down to increase lift on that wing.Usually, such operation is a portion of a coordinatedturn to the right. The dragon the right wing is decreased and thedrag on the left wing increased by thisaileron deflection. The resultingyawing moment (calledadverse yaw),

140 if unbalancedby a rudderdeflection, reduces the effective rolling velocity bymoving the right wing at a higherforward velocity than the left wing. The lift on theright wing is therefore greater (and less on the left) than wouldbe the case if theadverse yaw werenot present.

Adverse yaw is also undesirablebecause if unopposed it excitesthe lightly damped Dutch Roll oscillation. The pilotthus finds himself pro- ducingoften disconcerting yawing motions unintentionally. The specification requirementslimiting its extent include 53.3.2.5 (Ref.4) which provides that nomore than 50 Ibs of rudderforce be required to makea coordinated (B=O) turn at cruise speeds or 100 Ibsin the approach configuration for highper- formance fighteraircraft. There is also 53.3.2.4, whichprovides that ad- versesideslip shall not exceed IOo and thatproverse sideslip shall not exceed 3O during f I ight phasecategories B and C (cruise, cI imb, descent,and take-off and landing). TND-3726 (Ref.55) suggests that pilots find a sideslip angle of 10' in response to a rudder-lockedaileron deflection the maximum acceptable. Some ofthe aircraft tested for this report exhibited sideslipangles in excess of 13'. Reference 56 reportsthat a convertedlight aircraftcarrying military stores under the wing reached sideslip angles in excess of 10' andrequired 125 Ibsof rudder force to perform a coordinated turnat low speed. The militarystores wereresponsible for a 5O increasein sidesl ip angle.

Inthe absence of a rudderforce to counter the adverse yawing moment, the aircraft will develop sufficient sideslip to permit a balancingyawing moment due tosideslip to beproduced, Since alllight aircraft employ some dihedralto aid in maintaining the wings' level during normal flight* and since this stabilizing rolling moment due to gideslip acts to reduce the roll rateachieved by a givenaileron deflection, it isimportant for good rolling responsethat adverse yaw be heldto a minimum. 53.3.6.3.2 of MIL-F-8785B (Ref. 4) statesthat the positive dihedral effect shall not be so greatthat morethan 75% of theroll control power availableto the pilot, andno more than 10 Ibsof aileron stick force or 20 Ibs of wheel forceare required for sideslipangles which might beexperienced inservice deployment,

Thecomments of AFFDL-TR-69-72 (Ref. 4) relative to sidesliprequirements developedfor the specification seem particularlypertinent: I' theprimary sourceof data from which the sideslip requirement evolved (See AFFDL-TR-~~-~~)~~ are those tests for which the ratio of bank angle to sidesl ip angle during Dutch Rol I equaledabout 1.5. "The pilot comments associatedwith these configurations indicated that pilot's difficulties werealmost exclusively associatedwith sidesl ip, rather than with bank angle tracking as was thecase for l($/f31dE6 configurations.Analysis of the data revealed that the amount ofsideslip a pilot will accept or tolerate is a strongfunction of the phase

* FAR part 23 (Ref.54) states that ''the static lateral stability, as shown bythe tendency to raise the lowwing in a slip,must be positive for anylanding gear and flapposition."

141 angleof the Dutch Roll component of sideslip, When thephase angle is such that B is primarily adverse, the pilot can tolerate quite a bit of sideslip. On theother hand, when thephasing is primarily proverse, the pilot can only tolerate a small amount of sideslip because of difficulty of coordination.

"There is more tocoordination, however, thanwhether the sideslip is adverse or proverse;the source andphasing of thedisturbing yawing moment alsosignificantly affect the coordination problem. If theyawing moment is causedby aileron and is inthe adverse sense, thenin order to coordinate the pilot mustphase either right rudder with right aileron or left rudder withleft aileron. Since pilots find this technique natural, they can gener- allycoordinate well even if theyawing moment islarge. If on the other hand, theyawing moment isin the proverse sense or is causedby rollrate, coordin- ationis far more difficult. Forproverse yaw-due-to-aileron the pilot must crosscontrol; and foreither adverse or proverse yaw-due-to-roll-rate, re- quiredrudder inputs must be proportionalto roll rate. Pilots find these techniquesunnatural and difficultto perform. Since yawing moments may also beintroduced by yaw rate, it can beseen that,depending on the magnitude andsense ofthe various yawing moments, coordination may either be easy or extremelydifficult. If coordination is sufficiently difficult that pilots cannot be expected tocoordinate routinely, the flying quality requirements mustrestrict rudder-pedals free unwanted motions to a sizeacceptable to pi lots."

"Analysisfurther revealed that it was not so much theabsolute magnitude of the sideslip that bothered the pilot, but rather the maximum changeoccuring insideslip. The latter wasa better measure ofthe amount ofcoordination required. Thus thedata...were plotted ...as the maximum change in sideslip occuringduring a rudder-pedals-fixedrolling maneuver, ABmax, versusthe phase angle of theDutch Roll component ofsideslip $6, Thephase angle, I)@, is a measure ofthe sense of the initial sideslip response,whether adverse orpro- verse,while ABmax is a measure of theamplitude of thesideslip generated. Boththe sense and amplitude affect the coordination problem."

Forsmall aileron control commands (93.3.2.4.11, the amount ofallowable sideslipis given in the figure below.

142 Figure 63. Allowablesideslip for small aileron control commands,

Thisrequirement applied for step aileron control commandsup to the magnitudewhich causes a60' bankangle change in two seconds or

2lT seconds,

whichever i s greater.

The differencein allowable ABmax with Jlg "is almost totally due to the differencein ability to coordinate during turn entries and exits, $6 is a directindicator of the difficulty a pilotwill experience in coordinating a turnentry, For -180°2 $6 2 -270°, normalcoordination may be effected .... As $B varies from -270' to -360' coordination becomes increasingly difficult, and inthe range -360' I $6 I -90' crosscontrolling is required to effect coordination.Since pilots do notnormally cross control and, if theymust, havegreat difficulty in doing so for -360' 5 Jlg I -go', oscillations in sideslip either go unchecked or are amplified by the pilot's efforts to co- ordinatewith rudder pedals."

To extendroll-sideslip coupling requirements to largercontrol deflections,§3.3.2.3 states that the value of the parameter

9, + 93 - 292 - 9osc"average 9, + 93 + 293 '

143 where$1, $2, $3 representsuccessive peaks in the bankangle time history ($2 is aminimum peak), shall be withinthe limits shown onthe figure below.

1.0

0 -40' -80' -120'-160' -200'-240' -28W-320"360. when p leads @ by 45' to 225' I I I I I I I I I -220'-260' -300' -340' -20' -60. -100. -140" -180' fie when p leads Q by 225" through 360'to 45"

Figure 64. Limits for extending roll-sideslip coupling requirements to larger control deflections.

Similar requirements designed to insure control precisionin the roll mode are given in §3.3.2.2.1, which requires that PoSc/P~vbe as shown in the figure below.

144 0.6 -

05 -

OA -

n!* 43- \ v,8 92- n a1 L L I I I I I I I 1 I -40' -80' -1zo' 160. -200' -240' -280' - 320'-360' #@ (see Fig. 64) I 1 I I I I I I I -220' -260' -300' -340' -20' -60' -100' - 140' -180' $p (see Fig. 64)

Figure 65. Requirements for insuringcontrol precision in the roll mode.

Thisrequirement applied for step aileron control commandsup to the magnitudewhich causes a 60° bank angle change in

3.4Tr seconds, w n r,-- 2 d - 'd

Forlarger roll rates, §3.3.2.2 statesthat following a rudder-pedals-free stepaileron control command, the roll rate at the first minimum following the first peakshall be of the same sign andno lessthan 60% (categories A and C) or 25% (categoryB) of the roll rate of the first peak.

Additionalevidence of thebearing which rolling moment due to sideslip hason pilotts opinion of anaircraft's handling qualities is provided by .:43.3.2.1 whichstates that roll acceleration, rate, anddisplacement responses toside gusts shall be investigated for airplanes with large rolling moments due to sideslip,

As mentioned at the beginning of this section, modernconcern with aircraft handling characteristics hascentered on precision of responseand totalpilot workload. The paragraphs of MIL-F-8785B (Ref. 4) cited above demonstratethis concern clearly. TND-3726 (Ref. 551, citingthe fact Nthat the stability and controlcharacteristics of the airplanes (tested) are

145 . , . .. . . - - ...... -

general!ysatisfactory but deteriorate with reduced speed, increased power, aftcenter-f-gravity location and changes to thelanding configuration,.. and are critically degraded in turbulent air" and that"in general, each stability and controldegradation is relatively small" suggest that "reduced pilot rating results from the combined effects of all deteriorations" and that it isV-herefore necessary to considerthe combined effect of all the stability and control characteristics in order to properlyassess the pilot ratings.t' The authors of TND-3726 (Ref. 55) suggestthat this be doneby evaluating a work loadfactor defined as

workload factor =

The integralsare the areas under thecurves of the three pilot-applied force time histories taken over a 30 sec period with the aircraft oriqin- al ly in a trimmedstate. "The satisfactoryairplane presents a workload factorof approximately 300; whereas theunsatisfactory airplane approaches a factor of 500."Clearly, a requirement of thistype will ultimatelybe made part of the flying quality specifications,

Many of the quantitative tests from whichthe specification writers gainedguidance were performed in simulators or in variable stability airplanes.Results reported in TND-746 (Ref.58) and TND-779 (Ref.59) for example, showed theimportance of theDutch Roll frequency anddamping in establishingthe limiting behavior which the pilot could control. TND-221 (Ref.60) reports pilots require 0.23 sec to begin to move the stick later- ally and0.33 sec longitudinally. TND-173 (Ref.61) reports results of a studyto provide artificial longitudinal stability. TND-1782 (Ref.62) providesdata on thetransfer function of human pilots.It should be noted, however, that suchdata cannot be accepted without some qualification: human pilots display a veryhigh degree of adaptability to devices designed to measure theirresponse capabilities and thusexhibit different transfer functionswith increasing time or asthe control task is varied.

Severalvery recent full-scale wind tunnel studies on the stability and controlcharacteristics of representativelight aircraft are reported in References 63,64, and 65.Results of a programof artificial stability augmentation andwork load reduction on a popularlight twin is presented inconsiderable engineering detail in Reference 66, This program is notable inbeginning with a competentanalytical attack on the problem (and in de- tailingthe numerical calculations), in devising an ingenious roll-yaw couplerespecially suited to this type aircraft, and inpresenting details, includingtransfer functions, of the flight hardware,

Finally, for a veryinteresting review of thechronology of manrs understanding of aircraft stability and control,the reader is referred to the 1970 von Ka"rrnSn Lecturegiven by Dean Perkins(Ref, 67).

146 6 ~~

Yaw Contro I

Theuses of the yaw controlin aircraft include, according to AFFDL-TR-69r72 (Ref. 41,

(a 1 To perform a cross-windIandingYTeither employ a steadyrudder-pedal-induced sideslip or else a decrab maneuver,

To augment roll rate anywhere within +he flight enve I ope,

To raise a wing when the pilot is busy with his hands,such as when taking a clearance.

For tracking . For wing-overs ...to obtain a rapid change in headingor bank angle.

For c I ose-format ion f I y i ng .

To losealtitude as in a forwardsideslip or to improve visibi I ity.

To counteryawing moments frompropeller torques, speedchange, asymmetric thrust, stores, etc.

To taxi.

The reader wi I I note that from a hand I ing standpoint these uses are of two types:primary control of motion about the yaw axis andprecision of control incoupled lateral-directional motions. Primary control about the yaw axis is much likeprimary control about the pitch axis: the aircraft can bepro- videdwith inherent static stability, the forces experienced by the pilot are a goodmeasure ofthe out-of-trim condition, andmost of thelongitudinal stabi I ity parametershave d i rectiona I ana logs. Thus, one can write

-dFr = -G(g)qvSrcr [.I + Ch , dB [$)I a "6 r

and

147 Wil-h theusual assumption of constancyin the values of the stability derivatives,these relations provide ameans to compare the primary directional handlingparameters of proposed aircraft with the specification requirements quotedbelow. Note that to keep thevariation in force required with changes in speedreasonable and thus minimize the need for retrimming it isnecessary to make Chcl andchg as small as possible. Such a stepwill also aid in keepingthe force required to producegiven sideslips (and thus aid in rolling maneuvers) withinreason.

Primary yaw controlin the present connotation consists of thosefunctions whichcan be provided only by rudder deflection. Included in this list would beitems such as (a), (e), (g), (h), and (i). The precision-of-controlitems inthe list indicate that for these circumstances the rudder is used princi- pallyto complement aileroncontrol, to improve its precision by countering adverseyawing moments (Cngr or Cn 1 orto provide small additional favorable P rolling moments (CEr).The table below shows how thedesire to permit these uses was translatedinto quantitative requirements.

REM ENT REFERENCE REQU I REMENT

Maximum rudder force temporary 150# FAR 23.143 steady 20# Rudder force sideslipcounter<50# to MIL-F-8785B in rolls. §3.3.2.5

Coordinatedturn which reaches 45O of b an k should require less thanless require should bank §3.3.2.540# of rudder.

50# ofrudder force must induce a roll 93.3.4.5 rate of 3O/sec.

Ifthe aircraft is trimmed with symmetric power it must beab le to changespeed ?30% §3.3,5.1 withoutrequiring more than 100# ofrudder force .

No more than 100# rudderforce must be of 93.3.5.1.1 necessaryfor asymmetric loading.

(continuedon next page)

148 . " REQUIREMENT REFERENCE

~ .- ~~ ~~ The ratio B/6, mustbe essentially linear to +15O withthe slope permitted to be smallerfor 8>15Opositive. stillbut 53.3.6.The I ratio B/Fr mustbe essentialty linear to +IOo. Fr may be lessfor larger values butnever zero.

~- -~ ~~ For the aircraft trimmed for @=O flight theserequirements apply at sideslip angles produced orlimited by (a ) full rudder pedal deflectionpedal rudder full (a) 53.3.6 (b 1 250H rudderpedal force (c) maximum aileroncontrol or surface deflection. - Controlcentering and breakoutforces shall §3.5.2. I be between I# and 7#.

Controlsurface response shall not lag cockpitcontrol force input by more §3.5.3 than 30° (phaseangle) for frequencies equal toor less than I/TR.

~~ .. - . .- ~. ~ " ~~ ~ Cockpitcontrol deflection shall not §3.5.3. I leadcontrol force.

It shall be possibleto take off and land with normal pilotskill in 90' crosswinds 93.3.7 fromeither side with velocities up to 20 knots.Rudder forces shall not exceed loo#.

Rudder control power shall beadequate to maintainwings level and sideslipzero w ithou t retrimmingthroughoutwithoutdives 93.3.8 and pullups. In theservice flight envelope, shallnot exceed 180#.

It shall be possible to taxi at anyangle 53.3.7.3 to a 35-knotwind.

Rudder forces shall notexceed #I80 to maintaineventthe a straight inpath 53.3.9. I of sudden loss ofthrust during take-off,

~ " " ~~~

'able 13. Yaw controlrequi rernents.

149 Currentlight aircraft often require fairly large ("185 Ibs)rudder forces in 15O steadysideslips (Ref. 56) oras a result of theapplication of power--up to 90 Ibs changebetween idle powerand maximum power at a givenairspeed (Ref, 55). Pilotsconsidered the latter unsatisfactory although it would satisfythe specification. Note that brake forces in auto- mobilesin excess of 100 Ibsare generally regarded as undesirable while forcesin excess of 200 Ibsare regarded as unacceptable, In this regard, the FAR maximum rudderforce requirement appears to be far morereasonable thanthat of MIL-F-87858 (Ref. 41, particularlyconsidering that diminuative women shouldbe able to operate a lightairplane comfortably. The specifi- cationsare also notable in omitting any quantitative limitation on rudder pedaltravel. While this is another aspect of thepresent unsatisfactory state of directionalcontrol requirements, the condition likely stemsfrom thefact that most directional control during flight is now accomplished throughaileron manipulation which the pilot can perform very precisely and forwhich the force-deflection-response relationsare prescribed in great deta i I.

It is felt by severalpilots consulted by theauthors tha+ the force limitsquoted in the military specifications should be appliedonly to the faster and heavieraircraft of thegeneral aviation class. It was their contentionthat one should reduce these limits progressively as maximum speedand weightare reduced so that for the lightest andslowest aircraft inthe class the maximum forces will beno greaterthan 1/3 or 1/2 the specificationvalues. The reasoning apparently follows the usual human expectationthat larger vehicles require larger forces to control them. Thatthis is not necessary, however, isevident from thepopular acceptance of power steering andpower brakes on automobiles. With these devices the forcesare made more or lessindependent of car size. It is to be expected,therefore, that as "fly-by-wire" control systems are installed more widelyin aircraft a trendtoward standardization of controlfeel, ridingqualities, and handlingqualities for aircraft of allsizes, speeds,and weights will develop.

150 The discussionabove has assumed that the air mass throughwhich an aircraftis flying is uniformly stationary with respect to inertial space. Nature,however, is seldom so accomodating,and many aircraftexperience severe deterioration in handling qualiti'es during flight in turbulent air (Ref. 55). Analysisreported in Reference 4 Indicatesthat a first order treatment of theproblem can be performed. Gusts are considered to be iso- tropic away fromthe ground, Gust velocity variation with spatial frequency fortwo models is used to obtain gust components ofu(x), v(x), andw(x,y). From these,one can calculate gust components for a, (3, p, q, and r and, subsequently,the altered values of the stability derivatives. It is seen thereforethat all airframe dynamic modes canbe excited in gusts. If these areinsufficiently damped, if thelateral-directional coupling is excessive, or if thestatic stability is marginal, the pilot's workload will increase markedly if he attemptsto maintain a reasonablyprecise course or comfortableride. Outstanding handling qualities in still air are consequently a necessaryprerequisite to acceptable handling in turbulent air.

151 INERTIALCHARACTERISTICS

Theairplane designer is directed to thesection of thisstudy dealing with spinentry for a discussion of dynamicresponse resulting from changes in moments of inertia;techniques for estimating moments of inertiaare detailed here.

When anairplane is rotated about its center of gravity,the resulting torque, I?, is equal tothe product of the moment of inertiaabout the c.g. and theangular acceleration (r = I dddt).Since the torque is applied by a controlsurface, the angular acceleration can befound for a particular controldeflection by dividingthe torque by the moment ofinertia. The threeangular degrees of freedom for the airplane are pitch, roll, and yaw; thus, it is necessary to know the moments ofinertia about the x,y, and z airplane bodyaxes. One product-of-inertiaterm, Ixz,alsoappears in the equationsof motion (see Appendix A), butthis term is usually small, and, for many analyses,is considered zero, indicating that all the axes are principalaxes.

Moments of inertia canbe obtained by experimental measurements orcan be estimatedfrom airplane mass and geometriccharacteristics. Theusual method for determiningexperimentally the moments of inertia is a pendulum method.

The applicationof the pendulummethod tothe experimental determination of moments ofinertia of airplanes is discussed in TR-467 (Ref.68). The moments ofinertia about the x and y axesare found by swinging the airplane as a compound pendulum, whereasthe moment ofinertia about the z axis is determinedby oscillating the airplane as a bifilar-torsional pendulum.The differentialequation which describes the pendulummotion is of the form,

d28 + be = 0 where I = measured moment ofinertia IF b = constantdepending on the weight and dimensions of thependulum 8 = angulardisplacement.

The periodcan be written as T = Z'IT//~, and the moment of i nertia canbe solved by solvingfor I. Foreach of thecases mentioned above, the true moments ofinertia are determined by correctingthe measured moments ofinertia for (1) thebuoyancy of the structure, (2) theair entrapped within the structure, and (3) theadditional mass effect. Thesethree factors cause an apparentadditional moment ofinertia, which is evaluated on the basis of (1) theairplane size andshape normal to thedirection of motion and (2) the resultsof tests of the additional mass effectof flat plates. Reference 68 shouldbe used to evaluatethe required corrections. The additional mass effect (moment ofinertia influenced by thesurrounding medium) results from thefact thatthe period of the pendulum's vibrating in air is to some extentdependent onthe momentum impartedby its motion through the air. The momentum imparted to the body isproportional to the momentum ofthe body;thus, the equivalent additional mass may beused.

The precisionof the pendulum method to estimatetrue moments of inertia isapproximately k2.5 percent, t1.3 percent,and kO.8 percent for thex, y, and z axesrespectively. Several types of airplanes of grossweight less than 152 10,000 poundshave been tested at the NACA laboratories,the results of which werecompiled and published as NACA TN-780 (Ref. 69) whichsupersedes TN-375. Only a few of theairplanes tested are characteristic of the light airplane designs of today.This report, however, is valuablein obtaining an approxi- matenumber for the moments of inertia and theradii of gyration.Agard Report 224(Ref. 101) also discusses methods of obtaining the moments of inertia by thespring oscillation method.Schematic representation of typical methods for determining(within 5% or better)the rolling and pitching moments of inertia aregiven as well as other references which may be helpfulin using these methods.

Because of theequipment required to measure the moment of inertia by the pendulummethod, probably the most popular method utilizes the weight and lo- cationof component partsoutlined in TN-575 (Ref.70). This is a step-by-step, tabulated methodwhich yields the moments of inertiaabout the three axes, the products of inertia, and thecenter of gravitylocations. It is believedthat the moments of inertia canbe estimated within 10% by this methodand can be applied by completingTable 14, whichis explained below.

The first step in the procedure is to define a set of threemutually perpendicularreference planes, x’z:y’z: and x’y’, as shown inthe figure below.

\ -y 1 2‘ \Referenceplane /

kXI 21 Reference plane

\ \ 1-yl Reference axis Figure 66. Set of threemutually perpendicular reference planes. 153 The plane of symmetry is chosenas the x*z* reference plane, since it contains thec.g. The y*z*reference plane is usually set at the nose of theairplane, whi.lethe x*y* reference plane can be established at the top or the bottom. These reference planes define the origin of the coordinate axes.

The tablewhich must becompleted is shown belowand explained column bycolumn.

(2) (31 (5) (6) (7 1 (8 1 (9) - X N X N N N 3 3 3 X r u 3 5 t 3

Tota Is

Table14. Sample tablefor obtaining inertial characteristics.

Column (1) Elements ofthe airplane which are considered (flaps, wheels,baggage doors, engine, seats, pilot,fuel, etc. 1.

Column (2) Weight of each individualelement.

Columns(31, (41, & (5) Distancefrom the x*, y* and z* axes to theelement. It isimportant to note that some of these distances may be either pos’i tive or negative.

Columns (6) 8, (7) Moments contributed byeach element. Since the airplane is symmetricabout the x*z* plane, wy = 0; therefore, it does notappear.

Columns (81, (91, & (10) The productof the squares of the indi- vidualdistances and theitem weight.

Columns(111, (121, & (13) The estimationof the moments ofinertia of the larger items about their own center of grav i ty.

Column (14) The totalproduct of ‘inertia, Ixz.

The informationin the abovecolumns can be used to find

(1) c.g.Location--The andx z c.g. locations (xc.g. and zc. 1 canbe found by dividing the totals of columns (6) and (?j respectively by thetotal weight.

154 (2) Moments ofInertia--The total moments of inertia of theairplane about the three reference axes are

Ix*x*= Cwy2 + CWZ~+ ZAIxx

Iy’y ’ = Cwx2 + Cwy2 + ZAIyy = Cwx2 Cwy2 CAIzz I,’z + + The moments of inertia about the c.g. canbe written as

- 2 Iyy - Iy’y’ - W(Xc.g. + zc.g. 2, I,, I,, = Ixy- w(zc,g.2)

I,, = - W(XC. g .2,

where W = total airplane weight

(3) Product of Inertia--Theproduct of inertia, Ixz,aboutthe c.g.can be found by the formula,

I,, = cwxz - W(Xcag. + zc.g.

Even for thelarge items, the products of inertiaabout their own c.g.can usuallybe neglected. Also, for lightaircraft, the products of inertia for thecomplete aircraft are usually small or negligible. Thus, for a first approximation,assuming the products of inertia to be zerois fairly accurate. If greateraccuracy is required, the formula above can beused.

Some typicalvalues of moments ofinertia can be seen for light aircraft byexamining the table below:

Airplane Number Ixx IYY Izz Weight of (ibs. 1 Eng i nes (s 1 ug-f t2 (sI ug-ft2 1 (s I ug-ft2 1

2200 1335 902 1922

2650 94 1 1479 21 10

3350 1495 2207 2878

4650 1939 8884 11,001

5,000 64,811 17,300 64,543

21 00 766 1275 1805

Table 15. Typicallight aircraft moments of inertia. 155 STALL

The flow over a body is said to stall when thepressure gradient becomes so unfavorable that the velocity at the surface is zero and themainstream is nolonger attached tothe body. When stalloccurs on airfoils, there is both a loss of lift and an increasein drag. In general, stall can be definedas that flow conditionwhich follows the first lift-curve peak. TN-2502 (Ref. 71) indicatesthree classifications for stall at low speeds: 1) trailing-edge stall, 2) leading-edge stall, and 3) thinairfoil stall.

Trailing-edgestall is preceded by the movement of the point of turbulent boundary-layerseparation forward from the trailing edge withincreasing angle of attack(usually characteristic of airfoils of 15% thickness or more).This typeof stall is distinguished by a gradual,continuous force and moment variationwith a well-roundedlift-curve peak.

The leading-edge stall is an abruptflow separation of the laminar bound- arylayer near the leading edge, generallywithout subsequent flow reattachment (usuallytypical of airfoils 9% to 15% thick).Little or nochange in lift- curve>lope should beexpected prior to maximum lift andan abrupt, often substantial,decrease in lift should occur after maximum liftis attained. A combinedleading-and trailing-edge stall is possible.

The thinairfoil stall is precededby flow separation from the leading edge, withthe reattachment point moving progressively downstream with increasing angleof attack. This thin airfoil stall has a rounded lift-curve peak,generally preceded by a discontinuousforce and moment variation for airfoils with a roundedleading edge.

The stallsmentioned above are shown inthe figure below.

Angle of Attack -

Figure67. Characteristic stall types.

156 Notevery airfoil can be classifieduniquely into a givenstall category, nor is each type of sta I I I imited to a certain range of thickness ratios.

Sincestall characteristics play a majorrole in the design of any airplane, it is important to know at the outset at whatangle of attack the stall occurs,depending on the weight, speed,and maximum lift coefficient of the airplane. The abruptstall hasproven to be quite dangerousbecause the pilot receiveslittle or nowarning before attaining a criticalattitude. Also, if stalloccurs during land,ing or take-off, the pilot has little or no altitude inwhich to recover. Many parametersaffect the stall of a body or wing: taper,aspect, and thickness ratios; Reynolds number;camber; washout;and leading-edgeshape. Before considering the parameters vhkhaffect stall, some mentionshould be made of therequirements for good stallwarning andrecovery. TechnicalReport AFFDL-TR-69-72 (Ref. 4) definesstall warning requirements, stallcharacteristics, and stallrecovery requirements for light military a i rcraft.

Sta I I Warning

The sta I aproachshall beaccompanied by an easily percept blewarning.

Acceptab I e warn i ng forall types of stalls consists of shaking of the cockpitcontrols, bu f fetingor shaking of theairplane, or a combinationof these. The onset of thiswarning should occur within the ranges specified by thetwo tables below

Warningspeed for stallsat 1g normal to theflight path. Warning onset for stalls at lg normal tothe flight path shall occur be- I tween thefollowing limits:

FI ight Minimum Stall Warning Maximum Sta I I Warning Phase Speed Speed t ApproachHigher of 1.05UsHigher of or 1.1OUs or Us + 5 knots Us + 10 knots

AI I OtherHigher of 1.05Us or Higherof 1.15Us or Us + 5 knots Us + 15 knots

Table 16. Warningspeed for stalls at lg normal tothe flight path.

157 -

Warningrange for acceleratedstalls. Onset of stall warning shall occuroutside the Operational Flight Envelope associated with the Airplane Normal State and withinthe following angle-of-attack ranges:

FI ight FI Minimum Stall Warning Maximum Stall Warning Ph ase Angle Phase of Attack Angle of Attack

Approach a0 + 0.82 (a, - a,) a. + 0.90 (a, - a,) AI I Other a. + 0.75 (as - a,) a0 + 0.90 (a, - ao)

where as is the stall angle of attack and a, isthe angle of attack for zero I ift.

Table 17. Warningrange for acceleratedstat is.

Stall angle of attack is the angle of attack at constant speed forthe configuration,weight, andc.g. position which is the lowest of the following:

(a> The angle of attack for thehighest steady loadfactor, normal to theflight path, that canbe obtained at a given Mach number;

(b) The angleof attack for a given speed or Mach number at whichuncontrollable pitching,rolling, or yawing occurs (i.e., loss ofcontrol about a singleaxis);

(cl Angleof attack for a given speed or Mach number, at whichintolerable buffeting is encountered.

The increasein buffeting intensity with further increase in angle of attack shouldbe sufficiently marked to be notedby the pilot. This warning may be providedartificially only if it canbe shown thatnatural stall warning is not feasible.

StallCharacteristics

Inunaccelerated stalls, +he airplaneshal not exhibituncontrollable rolling, yawing, or downward pitchingat the stall in excess of 200.

It isdesired that no pitch-uptendencies occur in unaccelerated or acceleratedstalls. In unaccelerated stalls, mild nose-up p tch may be acceptable 158 if noelevator control force reversal occurs and if nodangerous, unrecoverable, or objectionableflight conditions result. A mild nose-uptendency may be acceptablein accelerated stalls if theoperational effectiveness of the airplane is not compromisedand

(a) The airplane hasadequate stall warning;

(b)Elevator effectiveness is such that it is possibleto stop the pitch-up promptly andreduce the angle of attack;

(c) At nopoint during the stall, stall approach, orrecovery doesany portion of theairplane exceed structural limit loads.

Theserequirements apply to all stalls resulting from rates of speed reduction up to fourknots per second (4.6 milesper hour, per second). Stall characteristicsare unacceptable if a spinis likely to result.

StallPrevention andRecovery

It shall be possible to preventthe complete stall bymoderate use of the controls at the onset of the stallwarning. It shall be possible to recoverfrom a completestall byuse ofthe elevator, ailerons, and ruddercontrols with reasonable forces, and to regainlevel flight without excessive loss of altitude or build-up of airspeed.Throttles shall remain fixeduntil speedhas begun to increase when anangle of attackbelow the stall has beenregained. In the straight-flight stalls with t he airplanetrimmed at a speed notgreater than 1.4 Us and with a speed re- ductionrate of at least 4.0 knotsper second, elevator control powersha I I be sufficient to recover fromany attainable angle of attack. On multiengine aircraft, it shall be possibl e to recover safe I y from stal Is withthe critical engineinoperative.

Thisrequirement applies with the remaining engines up to thrust for level flight at 1.4 US, butthese engines may be throttledback during recovery.

Although some light aircraft may havespecial stalling problems, such as nacelles stalling at high speedswhere theycan produce a significant lift, causingthe airplane to attain dangerous attitudes, the stall problem is usuallysolved if thewing hasgood stallcharacteristics. NACA TR-703 (Ref. 72) is a set of designcharts prepared to show theeffects of wing geometry on the stalling characteristics of taperedwings; a summary of these effects is presentedin Table 18.

159 I

I I GEOMETR IC PROPERTY/W ING EFFECTS ON STALL

Taper Increasing taper tends to move the stalling pointprogressively outboard and to decrease the stalling margin of most of theremaining wing.

AspectRatio An increasein aspect ratio tends to flatten thesection lift distribution; up to an aspectratio of 18, the effect onthe stalling pointis relatively small.

Th i ckness An increasein root thickness ratio beyond 0.15 causes the stalling point to move inboard,except for the lowest va I ues of Reynolds numbers tested, and tends to reduce therate at which the section lift and maximum section lift divergeinboard ofthe initial stalling point.

Camber Increasing camber linearly from root to tip (4%) appears to be useful to good stall characteristicsonly for low taperratios.

Washout Althoughwashout isexpensive in regard to dragconsiderations, it offers an effective means of improvingstalling characteristics. Washout becomes more effective asReynolds number is increased.

Reyno I ds Number An increaseReynoldsin number tendsto move theinitial stalling point inboard. I SharpLeading Edge A sharpleading edge reduces the maximum section lift coefficient so that stalling takesplace inboard where the sharp edge islocated.

Table 18. Effectsof wing geometry on stall characteristicsof tapered wings.

160 Thus, for good stallcharacteristics, wings with low taperratios, moderate thickness(10% to 15%),low to moderateReynolds numbers, loto 3O washout, and a sharpleading edge at the root seem the mostdesirable.

WR L-145 (Ref. 73) pointsout that surface roughness and propeller effects shouldalso beconsidered as parameters affecting airplane stall. The air- planestested for thereport were militaryaircraft of the mid 1940's. Because of the armament required,the airplanes wereequipped with numerousaccess doors,inspection plates, andnumerous otherfeatures that tend to make the wingextremely rough and allow air leakagethrough it. It was found,that a wingwhich was faired and sealedgave a higher CL~~~thenan unfaired wing, eventhough both wings stalled at approximately the same angleof attack. Thestudy also indicated that propeller operation generally increases the severity o.f thestall, especially on single-engine airplanes. The rotation within the slipstream increases the effective angle of attack of thewing sectionbehind the up-going propeller blades and decreasesthe effective angle of attackof the wing section behind the down-going propellerblades. An asymmetricalstall pattern is thus produced, leading to a sometlmesevere rot I.

WR L-296 (Ref. 74) indicatesthat airplanes of the 1930's solvedthe problemof stall by providing a definitewarning of theapproaching stall through backward movement, position, and forces on thecontrol column. Monoplanes usually had little or no taperwith "inefficient" wing-fuselage junctures, causing a graduallydeveloping stall, beginning at midspan. Thus, thestalled conditiondeveloped progressively after a reasonablydefinite warning; also, lateralcontrol was oftenmaintained up to or beyond thestall. Many ofthe airplanesdesigned in the 1940's depictedtrends toward higher wing loadings andlanding speeds; less emphasis was placedon stalling tendencies, thus leadingto airplanes with poor stalling characteristics. WR L-296 also indicatesthat sharp leading edges at the wingroot may improvepoor stall tendencies.Another proposed solution is limitation of longitudinalcontrol to preventthe wing from reaching maximum lift. Tobe effective, a warning mustoccur at an angle of attack considerably below that of maximum lift becausegusts or inertia effects may momentarilycarry the airplane beyond thewarning attitude. An investigation was made in WR L-296 (Ref.74) of a "stall-controlflap." The basicwing tested wasa 23012 sectionwlth a 60% chordflap. The idea was todeflect the stall-control flap so thatthe modifiedairfoil would have a shape similar to the 4412 airfoil, whichhas good stallcharacteristics because ofits relatively flat lift-curve peak. T'he flap was consideredaerodynamically satisfactory with or withouthigh lift devices,even though it may be veryexpensive toinstall. Thegraph belowindicates the effect of the flap deflection on Iift-curve shape.

161 3.0 23 25

-.2

Figure 68. Investigationof a stallcontrol flapin WR L-296 (Ref. 74).

NACA TN-1868 (Ref. 75) is a study made tocorrelate pilots' opinions of thestall warning properties of 16 airplanes,ranging from single-enginefighters to four-engine bombers, with a number of quantitativefactors obtained from time,history flight records for speedsnear stall. The altitudetest range was 4,000 to 12,000 feet, with stalls attained in straight flight by gradually approachingthe stall with the normal accelerationfactor as close to unity as possible. The testsindicate that, in general, the stal! warning is considered satisfactory by the pilots when characterizedby any of the following qualities:

162 (a)Airplane buffeting at speeds 3 to 15 mph above stalling speedand of a magnitude to produceincremental indicated values of normal accelerationfactor from 0.04 to 0.22;

(blPreliminary controllable rolling motion from 0.04 to 0.06 radiansper second oc- curing anywhere within a range of 2 to 12 mph above the stalling speed;

(c) At least 2.75 inchesrearward travel of thecontrol stick during the 15 mph speedrange immediately preceding the stal 1.

The twographs below indicate the satisfactory and unsatisfactoryranges of bothnormal acceleration increments, Figure 69, and rearward movement of the controlstick, Figure 70, versusspeed above the stall. Each linerepresents a separatetest of one ofthe 16 airplanes. A, isthe normal acceleration increment.

0 4 16 8 12 20 24 28 Speed above stall (rnph)

Figure 69. Correlation of pilotopinion of stall warning with airplane buffet at various speedsabove stall.

163 Speed above stall, mph

Figure 70. Correlation of pilot opinionof stall warning with rearward movement of control stick in the speed rangeimmediately above stall.

In a summary of variousstal I-warn,ing devices, TN-2676 (Ref. 76) points to the fact that a warninglight, stal [-warning dial,rudder shaker, ur.stick shaker may benecessary for any aircraft with very poor stall-warning characteristics. Such indicatorscan possibly be actuated bysuch devices as a leading-edgeorifice, leading-edge tab, spoiler and pitot-static tube, trailing- edge pitotstatic tube, or traiiing-edge vane. Lightairplanes without hydrau- lic controlsystems will probablynever need to usethese devices.

TN-2923 (Ref. 77) describesthe testing of a low-wing, lightairplane model duringthe stall and Intothe incipient spin. A more detailedreview of this reportis presented below in the discussion of spinentry.

It should be notedthat in seeking to analyze the motions of aircraft near stall,the usual assumption that the longitudinal stability derivatives are constantsis nolonger true. The derivativesare very strong functions of a. Also, downwash andsidewash fieldsare very strong and unstableunder these conditions. Dynamic pressure may varyconsiderably along the span. Finally, a deep stall is usually accompaniedby sufficientlylarge motions that onecan nolonger assume withaccuracy that products and squares of pertubationvelocities canbecneglected. For these reasons, extreme care must be used in attacking this problem analytically. 164 SPIN ENTRY

The capacity for attaining a spinning attitude while performing ordinary flight maneuversand the fact that many light aircraft accidents have been attributedin the last few years to stalI/spinproblems has led to extensive researchby NACA/NASA concerningspin entry and recovery. More than 25 NACA/NASA reportsdeal with spin applicable to lightaircraft design. Much of thisliterature was writtenin the late 1940’s and early 1950’s because thehigher wing and fuselageloadings of moremodern aircraft are not characteristic of the majority of the general aviation aircraft.

A developedspin is usually considered to be a motionin which an airplane in flight, at some angle of attackbetween the stall and 90°, descends rapidlytoward the earth while rotating, with the wings nearly perpendicular to a vertical or near-verticalaxis. Special consideration must be given to spinsin the design stage, since controls effective in normal flight might be inadequatefor recovery from the spin. The same factorswhich may cause difficulty in attaining a spinalso may impede recoveryfrom it, onceattained. Besidesthe problem of providingadequate controls for recovery,there is alsothe problem of pilot disorientation resulting from thedeveloped spin; thus,preventing the spin or recovering during the incipient phase (the motion between theinitial stall and thedeveloped spin) is essential. Civil Air Regulations require that the pilot of a personal-ownerairplane recover from a one-turnspin upon releaseof controls by the pilot; this may mean that controls bedesigned to float against the spin.

Of the numerous significantreports reviewed for thisstudy, two are consideredof Drimarv interest--AirDlane SDinnina bv James S. Bowman (Ref. 78) and Statusof Spin Research for RecentAirplane 6esigns by A. I.Neihouse, etal (Ref. 79). The latterdiscussed sDin-tunnel testina and itscorrelation ” u with actua I f I i ght tests, the i nf I uence of ai rp lane geometry on spinning, and asummary of thespinning characteristics of 21 model airplanes,including model and fullscale correlation. Airplane Spinning is a state-of-the-art summary ofspin prediction and alleviation.

Experiencehas indicated that spins and spinrecoveries of airplanes canbe investigated safely and at a comparativelymoderate cost using small dynamicmodels in a spintunnel.* Itis important that the modelbe designed so thatgeometrically similar paths of motionbetween the modeland the airplaneare attained in the spin; this is accomplished by holding the force, mass, andtime ratio constant as well as the ratio of lineardimensions in the model design. It is hand-launchedwith a spinningmotion into the spin tunnel, and thevertical speed of the column of air is adjusted to maintain a spinningattitude of the model at a particularheight in the tunnel. It is assumed thai,for most spins, the pilot would probably have the airplane

* a verticalwind tunnel controlled by a propellerwith very fast response time

165 controlsset approximately at "normal spinning control configuration"--that is,stick full back,and laterallyneutra1,rudder full with the spin. After thespin is established, the mode! controlsurfaces (rudder, elevator, and ailerons)are deflected to attemptrecovery. Experience has shown thatthe criterion for satisfactoryrecovery for model tests is recoverywithin 28 turns of the model afterthe control surface deflections, Basedon this analysis, when therecovery in the spin tunnel requires more than this number of turns, the controls are not sufficiently effective and thecorresponding airplaneprobably would have unsatisfactory recovery characteristics. One poorrecovery out of several recovery attempts is usually considered as undesirableas consistently poor recoveries. The philosophyis to assume that a proposeddesign is inadequatefor spin recovery unless it canbe provensatisfactory.

A developedspin involves a balanceof aerodynamic and inertial moments and forces;thus, the effectiveness of any controlin promoting or inter- minatingthe spin depends notonly on the aerodynamic moments and forces producedby the control but also on the inertial characteristics of the airplane. A spinabout any axis in space can be consideredas a rotational motionabout any axis through the center of gravity. The equationsfor the moments actingin a spin*are . u2z c, + Iyy - Izz = 2pkX qr IXX . U2 Izz - Ixx 4=- 2 cm,b + 'P 21-1 ky =Y Y

The airplanespinning attitude (steep or flat) and its rate of rotation depend primarilyon the yawing and pitching moment characteristics of the airplane. Low damping in yaw atspinning attitudes or highautorotative yawing moments lead toflat (high a), fastrotating (high S2) spins. Some interesting facts canbe pointedout by approximating the pitching-moment equation obtained inequating the aerodynamic and inertialpitching moments:

Q2 = - Myaero $(Izz - Ixx)sin 2a

Remembering that a positivepitching moment is noseup, it canbe seen that a nose-down (negative)pitching moment may nose theairplane down butlead to a higherrate of rotation and may, infact, flatten the spin. For given directional and lateralcharacteristics, the pitching moment caninfluence the motion so that it may varyfrom a highrotative spin to a low rotativespin.

The effect of any controlin bringing about spin recovery dependsupon the momi3nts that the control provides andupon theeffectiveness of those

* assuming the x, y, and z axesare principal axes and that engine effect can be i-gnored. 166 moments inproducing a change inangular velocity and thusin upsetting the spinequilibrium. Experience has shown thatthe best way to alleviatethe spinningmotion is provide a yawing moment aboutthe z body axis to oppose thespin rotation; -thus, therudder is usually considered the most important control,especially for lightairplanes. It alsoappears that elevator effectiveness and aileroneffectiveness, in the final analysis, dependupon their ability to alterthe yawing moment of thespin.* Thus, it would seem that the most effective way to influence the spin and to bringabout recovery is to obtain a yawing moment by applying a moment aboutan axis which experiences theleast resistance 't0.a change inangular velocity.** For example, themost proficient way to obtain an antispinyawing moment for recovery may be to roll the ai rp lane in such a d i rection that a gyroscopic yawing moment to oppose thespin is obtained. Similarly, if mass isheavily concentrated in the wings, movement of elevators downward may provide the most effective means of applying an antispinyawing moment.Thus, because of inertialcoupli-ng of a rotating body, if the moment aboutone axis is changed, the moments aboutthe other axesare also changed.

By inspectingthe equation given earlier for P, it canbe seen that the rudderis the most important control when IXX- Iyy = 0 because defined as m/pSb)and kzare relatively small, and Cn is a function of rudderdeflection. For modern, high speed fighters and researchairplanes, large negative values of Ixx- Iyy predominate, since the mass isheavily concentrated in the fuselage.For these airplanes, it wouldbe extremely important to make theinertia termsantispin (negative for right spin) for recovery.This can be accomplished by controllingthe algebraic sign of the pitching velocity, e.g.,by tiltingthe inner wing (right wing in a rightspin) down relative to thespin axis.This tilting of the wing downward makes pitchingvelocity positive (q = 52 sin + for low valuesof IXx) and givesrise to a cross-coupledeffect, whichacts in a directionthat terminates the spinning. Light airplanes, however, fall,for the mostpart, into the category of airplanesdesigned in thelate 1930's and early 1940's. These lightairplanes have relativelysmall changes inthe inertia terms which contribute to i-, indicatingthat the rudder shouldbe the most important control.

The principalfactors in spinning are mass distribution, by farthe most importantsingle parameter, and tail design,particularly important for conditions of zero or near-zeroloading (small I,, - IYY).By knowing the mass distribution and taildesign, it ispossible, In many cases, to predict whetheran airplane has satisfactory spin-recovery characteristics. The mass distribution of airplanescan be grouped into three general loading categories, as shown on the following page.

* It shouldbe pointed out that if spoilersare usedinstead of ailerons, thespoilers are generally ineffective in the developed spin because of the areashielded in the spinning attitude.

** a moment aboutthe axis with the least amount ofinertia.

167 Ailerons With RudderAgainst ElevatorsDown Plus Followed B y Plus RudderAgainst ElevatorsDown RudderAgainst

FuselageHeavy Zeroloading WingsHeavy loading Loading

Figure 71. Primaryrecovery controls as determinedby mass distribution.

The typeof loading shown onthe right is classified as wing-heavyloading, in whichthe roll moment ofinertia is greater than the pitch moment ofinertia. The airplaneon the left is an example offuselage-heavy loading, in which the roll moment ofinertia is less than the pitch moment ofinertia. The airplanein the center has roll and pitch moments of inertiawhich are about equal; thiscondition is referred to as zero loading.

The loadingof the airplane can dictate what controls are required for recovery,as explained in WR L-168, A Mass-DistributionCriterion for Predicting theEffect of Control Manipulation on the Recovery from a Spin(Ref. 80). Deflectingthe rudder against the spin is always recommended, but,for satisfactoryrecovery, deflection of othercontrols is sometimesrequired. For the caseof wing-heavy loading, down elevatoris the primary recovery control, whileailerons against should also be beneficial; for fuselage-heavy loading, theaileron is the primary recovery control. In the latter case,the aileron needs to be deflectedwith the spin--for example, stick right for a spin to theright. Predicting what the effects will be forthe zero loading is difficult;but, almost invariably, the proper recovery procedure is to move +he rudderagainst the spin and, a shorttime later, move theelevator down. The aboverecovery techniques seem toindicate that spin recovery is simple, but it is sometimeshard to class a givenairplane in oneof the three categoriesabove and,once classed, the controls still may or may notproduce enough antispin moment toachieve recovery.

Taildesign is also an important factor in designing an airplane to recover from spins.Since most light planes fall into the zero loading category, the rudderis of primary importance to a good design.In a spin,there is a dead airregion over much of the vertical tail causedby the wake of the stalled horizontaltail. For optimumrudder effectiveness, part of the rudder must be outsidethis stalled wake. Anotherfactor which affects tail design from thespin standpoint is that there should be a substantial amount offixed areabeneath the horizontal tail to provide damping ofthe spinning motion. A criterion for good taildesign was determinedin the middle 1940's 168 (TN 1045, Ref.81 and TN 1329, Ref.82) on the bas isof spin-tunne I tests withabout 100 differentdesigns. This criterion is cat led the ta i I damping Dower factor (TDPFI. a measure of the damping, -. prov i ded by the f ixed area beneaf-h the horizontal tail and thecontrol power provided by theunshielded part of therudder. The tail-damping power factorcan be calculated as shown below in parts a, b, and c of Figure 72.

Relative Wind /

Full-lengthrudder; a assumedto be 45’

Part a

-To c.g. of Airplane -+ll I

-To Air Relative Wind 1 Partial-lengthrudder; a assumed to be 45. TDR c 0.019 Part b

169 To c.g. of u” -Airplane Relative Wind / Partial-lengthrudder; a assumed to be 30‘ TDR 3 0 019

Part c

Figure 72. Calculationof tail-damping power factor.

The tail dampingpower factorrequired to insuresatisfactory recovery isgiven in the figure below.

AirplaneDensity mlSb ’’ DensityAir P

1600”x10-6- 1200 - Tai I - Damping Power 800 - Factor - - 400 - 0- I -2000 - 800 -400 0 400 n ld4 Fuselage Wing Heavy Heavy Mass Distribution, Clxx-lyy)/mb2

Figure 73. Taildesign requirements. 1 70 All the data are in the area of zero or near-zeroloading, where the rudder is a primaryrecovery control and thetail design is of particular importance. As canbe seen from thegraph in Figure 68, only a limitedpart of the total existingrange of mass distributionsapplies. The plot shows boundaries indicating the minimum valuesof the tail-damping power factor required to insuresati,stactory recovery. The hatchedside of the boundaries is the unsatisfactoryside. The solidlines are for recovery by rudder al-one, and thebroken line shows theboundary for recovery by rudder and elevator. The boundariesare presented in terms of the relative density factor p. The valueof 1.1 = 6 isrepresentative of light, single-engine, personal-owner airplanes,while p = 35 is representative of that for executive jets.

TN-1329, TaiI-Design Requirements for Satisfactory Spin Recovery for Personal-Owner-TypeLight Airplanes, (Ref. 82) is a reportdealing specifically withlight airplane design. Its conclusions are basedon tests of 60 models inthe Langley spin tunnels. Results of this'investigation are shown inthe plot below.

Regionfor satisfactory recoveryby rudder reversalalone

versa1of hoth rudderand elevator

Inertia Yawing- moment Parameter,-Ixx-lYY mb2

Figure 74. Vertical-taildesign requirements for personal-owner-typeairplanes.

171 An importantplot which indicates other influences of mass distribution on optimumcontrol movement for recovery from spin is presentedbelow.

"Weight alongWeight along- fus elage wing fuselage

Figure 75. Influence of mass distribution onoptimum control movement for recovery from spin.

The unsatisfactoryregion exists because thedifference in inertia is low. As thedifference increases either way thecontrol surface (elevator in the case ofpositive increase, aileron in the case of negative increase) gains effectiveness. Based onthe inertias, the reversal of aileroneffect should occurat I,, - Iyy.= 0; however,due to theaerodynamic effects, the reversal of aileroneffect IS shifted from 0 to[(Ixx-Iyy)/mb21 X IO4 = -50. Thus, in thisvicinity, ailerons with the spin (stick right in a rightspin) generally loosetheir favorable effect and become adverse; for aileronsagainst the spin, theconverse is true. This result, it isbelieved, is primarily a resultof a secondary effectassociated with positive Cng ofthe airplane and a resulting relativeprospin increment in yawing moment because ofthe increment in inward sideslipthat invariably occurs when aileronsare set with the spin. Another importantgraph, shown below, is a summary of themost important factors in spinning and indicatesthe present state-of-the-art.

172 I RECOVERYCONTROL 'RudderElevator AileronsWith\Against 1 Down PIus dFollowed#Plus RudderAgainst ;r' BY f Rudder #ElevatorAgainst $Down 1. 1600 '%- 146 TAIL 1200 - DAMPING - POWER 800 - FACTOR - p=35 400 - - 15 0-4 I I I I -2000 -800 -400 0 400 X Fuselage Wing Heavy Heavy

MASS DISTRIBUTION, ( Ixx-lyy)/mb2

Figure 76. Summary of themost important factors in spinning.

The figureabove agrees with current literature; however, it was previously consideredin rudder-elevator recovery procedures that rudder application shouldlead elevator application. For an erect spin down elevatorincreases theshielding of the rudder more than up elevator;thus, maintaining elevator up provides maximum ruddereffectiveness during rudder reversal. It should benoted that, for theextreme loadings on both ends of the scale, no criteria havebeen developed for predicting the effectiveness of the controls for satisfactoryrecovery. The safest way to insure good spinrecovery is by spintunnel or flight testing.

The prescribedmethods for spinrecovery have been givenabove for the particular mass configurationdesired, but a recovery may still behard to achieve.In flight, an airplaneenters a spinfollowing roll-off just above the stalling angle of attack after being brought up fromlower angles of attack. Itusually takes an airplane two to fiveturns to attain a fully developedspin after starting the incipient-spin motion, with the number of turnsdepending upon configuration and control technique. One important factshould be remembered; recoveriesare generally achieved much more readily when attemptedduring the incipient phase of the spin than when attempted afterthe spin becomes fully developed.Thus, some considerationshould be given to thetechniques of noticing a spin entry attitude and to ways of avoidingthe fully developed spin.

TN-2352 (Ref. 83) is a spin-tunnelinvestigation of a low-wing personal- owner aircraftwhich was conducted to providedesign information for proportion- ingpersonal-owner or liaisonairplanes for satisfactoryrecovery from spins 173 and for spinproofing. The investigation was intended to beextensive enough to determinethe configurations most likely to meet thespin-recovery require- mentsgiven in Part 3 of The Civil Air Regulation(Ref. 54) and summarized below.

For an airplanelicensed in the normal category:

(1) A IS-turnrecovery after a I-turnspin by releasing controls (controls assisted to theextent necessary to overcome friction) (2)"Uncontrollable spin" check--airplane capable of recovering from a I-turn spin with ailerons at neutral by first completely reversing elevator andthen, if necessary, fullyreversing the rudder.

Forairplanes licensed in the acrobatic category:

(1) A 4-turnrecovery after 6 turns of thespin by releasingcontrols (2) Recovery from a 6-turnspin in 1% additionalturns after neutral- ization of rudderand elevator, ailerons at neutral (3) "Uncontrollablespin" check--airplane capable of recovering from a 6-turnspin with ailerons at neutral by first completely reversing elevator andthen, if necessary,fully reversing the rudder (4) Recovery from "abnormalspins"--a 2-turn recovery after 6 turns of the spin with ailerons initially either full with or full againstthe spin by neutralizingailerons and fully reversing rudder and e levator (5) A IS-turnrecovery from a I-turnspin by neutralizationof rudder and elevatorwith flaps and landinggear extended.

An extensive amount oftesting of this light airplane indicated that satisfactoryrecovery can be readily obtained even if thetail-damping power factor isnot very great, provided the recovery technique used is full rapid rudder reversalfollowed approximately % turnlater by forward movement of thestick. The results also indicated that for recoveryby merely neutralizing both controls,especially for rearwardc.g. positions, high values of tail-damping power factor may havean adverse effect upon recoveries. It was foundthat differentwing planforms had little effect on the model spin and recovery characteristics. It was concludedthat unless the rudder can be designed to floatagainst the spin, recovery from a spin by releasingcontrols might be difficult unless the elevator can be made to float at deflections farther down thanneutral. !t was alsoconcluded that other requirements for recoveryby various movements ofthe controls as specified in the aforementioned regula- tionscould probably be met for the various model configurations and mass distributionsinvestigated by maintainingthe center of gravity at a forward position and utilizing a hightail-damping power factor.For fuselage heavy loading andlow TDPF a prematureforward stick movement may retardrecovery. Mass changeswere significantat low TDPF butnot at high TDPF. It may be of interestto mention that the modelspun with a totalangular velocity of approximately0.35 to 0.5revolutions per second.

An investigationof the effect of center of gravity location on the spinning characteristics of a low-wingmonoplane model isgiven in TR-672 (Ref. 84).

174 Movingthe c.g. forward steepens the spin, increases Qb/2U, andimproves recovery;similarly, moving the c.g. back tends to flatten the spin, decrease Qb/2U, and retardrecovery. A reportin the same series of investigations, TR-691 (Ref. 851, givesthe effects of airplanerelative density. The findings in this report are that, in most cases,an increase in relative density producesflatter spins, higher velocities, lower values of Qb/2U, andslower recover ies.

NACA TN-570 (Ref. 86) isdirectly applicable to light airplane design. This particular report investigates the effect of different tail arrangements onthe spinning characteristics of a low-wingmonoplane model. Results of this ihvestigationindicate that a reductionin tail length results in spins with higherangles of attack, higher values of Qb/2U, andslower rates of descent. Recoveries from thespin seem to depend critically upon theexact location ofthe vertical surfaces. It is also concluded that, bymaking certain reasonablysmall changes in the tail arrangement, allthe spinning characteristics,except the amount ofsideslip, can bechanged through wide ranges. Thisagain points out,the importance of tail design in light airplane spin recovery .

TN-608 (Ref.87) is another investigation in which a seriesof models were testedin the spin tunnel. It was found thatrectangular and fairedtips give thesteepest spins and flaps tend to retard recovery; for controls with the spin,tail B (below)gives steeper spins than tail A, withgenerally satisfactory recovery for either tail, while tail C generallygives slower recoveries.

EmpennageArrangements Investigated

Figure 77. Effectsof various aircraft tails on spin recovery.

175 The effect of such devices as antispin fillets anddorsal fins on spin recoverycharacteristics can befound in TN-1779 (Ref. 88). Data from 21 different modelswere used to determinethe action of fillets in damping of spinrotation, and 30 modelswere investigated for the effect of dorsalfins; The effectiveness of antispin fillets for spinrecovery appear to depend primarily upon the fact that the fuselage area below the fillets becomes effectivein damping thespin rotation. Whether or notthe fillets improve therecovery characteristics of a given design is still a function of the tail-damping power factor of thedesign and the mass distribution.Dorsal finsgenerally have little effect on spin and recoverycharacteristics.

TN-1801 (Ref.89) should be mentioned because it is a ;pin investigati& of a twin-taillight airplane model withlinked and unlinkedaileron controls. It was foundthat when therudders and aileronsare linked for two-control operation,the model generally does notspin. The spinsobtained in this studywere steep, and the tests resulted in satisfactory recovery.

TN-2923 (Ref. 77) isanother report in which a lightairplane was tested. Themotion of a personal-owneror liaison airplane through the incipientspin was analyzed. It was foundthat, after the initial stall and immedlatelyafter the model becomes unstalled,the rates of yaw and pitch arerelatively small, and therates of roll begin to decrease. There also is little loss of altitude up to this time and theresults indicate that, even thoughthe airplane may be inverted, a timesoon after the roll-off has started appears to be a desirable time to attempt to terminate the motion. It isfelt that the motions attained in this investigation are indicative of themotions of a low-wing, lightairplane in the incipient spin. Although evaluation of theparameters in incipient-spin motion is not treated specificallyin the equations of motion presented in Appendix A ofthe present study,these parameters could be obtained by proper application of thetech- niquesused in that derivation. Below are some example plotsof angular displacements and angularvelocities versus time for the incipient spin caused by a rightroll-off shown inFigures 78 and 79.

176 450 11111\1 Ill1 0 1 2 3 4 5 6 7 8 Time, sec.

c 0 -2ob Q) Repeat -- \ I - -25 - - M - \I P \/ 4 -30 I Ill1 111111111II 0 1 2 3 4 5 6 7 8 Time, sec. 30r

-5 I 11111111111111~ 0 1 2 3 4 5 6 7 8 Time, scc. Figure 78. Angular displacements versus time for the 'Encipient spin caused by a right roll-off. 177 0' 0

Original-- =3 '1 m L ReDeat- - - - - n

0 1 2 3 4 5 6 8 Time, sec.

2- - Original Repeat - - - - 1-

-1 I I I I I I I i I I I I I I I I 0 1 2 3 4 5 6 7 8 Time, sec.

3.0- Original 2.5 - - Repeat -- --- 2.0 -

1.5 -

1.0 -

.5 -

0 C'l I I I I 1 I I "5' I I I I I I I 0 1 2 3 4 5 6 7 8 lime, sec. Figure 79. Angularvelocities versus time for the incipient spin caused by a right rot I-off . 178 HUMAN FACTORS

For maximum safety to an aircraft and its occupants,the pilot must be ableto control aircraft motion during all flight conditions. One possible means of attaining this objective is to design the aircraft controls for the "average" man; however, AFSC DH 1-3 (Ref. 90) indicatesthat less than one percent of the population is "average" in the five dimensions considered. Thus, designingfor the "average" man appearsunsound. The concept of"design I imits" offers amore reali.sticapproach. With this idea in mind, thefollowing discussioncenters on the pilotls comfort from a "design limit" viewpoint. Usinganthropological data, comfort limits for pilots of general aviation aircraft are examined.

Settingthe proper minimum forcereduces the likelihood of accidental activation of a control,especially those controls on which the pilot must continuously keep his hands orfeet. An upper limitis needed toinsure that requiredcontrol forces do notexceed the pilotls capabilities. Several generallzationsconcerning force applications to control devices are given in AFSC DH 1-3 (Ref.90):

a)Force application is equally accurate for hands a nd feet;

b) Contro I s centeredin front of the operator permit maximum forceapplication;

c) Controc) I forcegreater than 30 to 40 Ibs applied byhand or greater than 60 I bs by footis fatiguing;

d) The preferred handand arm aregenerally 10% strongerthan the non-preferred.

The capabilitywhich 95$, orthe fifth percentile, of a populationcan be expected toexert is considered the standard.

From AFSC DH 1-3, thetable below shows arm strengthfor different angles of elbowflexion.

179 Arm Strength

4 in

SEATED

ELBOW FLEX I ON PUSH PULL DOWN UP IN OUT

R LR LRL LRLRLR

”””

180° 50 42 50 52 14 139 17 20 13 14 8 1 50° 42 30 42 56 1520 18 18 20 15 15 8 1200 26 36 34 42 24 17 2126 20 22 15 10 90032 37 22 36 20 17 26 21 16 18 16 10 60’ 22 34 26 24 15 20 20 18 20 17 17 12

Table 19. Arm strengthfor different angles of elbow flexion.

The maximum force exerted on an aircraft control stick by the right arm of male Air Forcepersonnel in the sitting position, according to Morgan(Ref. 911, is reproducedin Table 20.

180 RIGHT ARMON AIRCRAFT CONTROL STICK (POUNDS)

ISTANCE IN INCHES FROM

SRP* MIDPLANE PUSH PULL LEFT R I GHT

8 (left) 12 26 24 34 44 (left) 18 28 31 31 9 0 26 34 30 23 44 (right) 34 39 26 15 8 (right) 37 39 26 12

121/2 8 (left) 18 33 23 31 8 (right) 43 49 22 16

8 (left) 23 39 20 25 151/2 0 43 54 24 20 8 (right) 53 55 24 13

8 (left) 36 45 16 22 18 3/4 0 64 56 8 15 8 (right) 70 58 22 14

8 (left) 29 51 11 19 23 3/4 0 54 62 14 13 8 (right) 56 58 20 12

*Forward; controlis 13 1/2 inchesabove seat reference po i nt.

Table 20. Maximum forceexerted on an aircraft control stick by the right arm.

181 From this same reference,similar information for amount of forceon an aircraftcontrol wheel is givenin the table below.

13:

RIGHT ARM ON AIRCRAFTCONTROL WHEEL (POUNDS)

D I STANCE IN INCHES CONTROL FORWARD POSITION PUSH PULL LEFT R I GHT FROMSRP*

900 (left) 32 23 23 27 45O ( I eft) 48 40 21 24 10 3/4 0 52 44 26 20 450 (right) 40 39 31 24 80° (right) 19 18 21 15

13 1/4 90' (left) 32 33 26 21 90' (right) 25 31 25 19

goo ( I eft) 32 42 27 19 15 3/4 0 61 66 27 27 90° (right) 32 49 29 20

900 (left) 37 60 22 27 19 0 64 73 25 30 90' (right) 33 61 33 22

900 (left) 82 73 21 26 231/4 0 105 77 20 35 90° (right) 49 74 26 22

*Wheel grips 18 inchesabove SRP and 15 inchesapart.

Table 21. Amount offorce exerted on an aircraft control wheel.

1 82 Sinceeach of these tests gives the maximum force for the right arm of amale, it must beremembered thatthe standard left arm, which isusually weaker, and the possibility of female pilots for general aviation aircraft would dictatelower maximum forces. These limits onwheel forces need not be so stringent if a worstcase analysis shows them to be impractical,since the pilotcould useboth hands if necessary. When possible,use of twohands on the wheel shouldbe avoided on landing, as the pilot may need his right hand to performother tasks. Damon (Ref. 92) givesthe left rotation of the wheel asapproximately 25 Ibs and right rotation asabout 30 Ibs.

The maximum forcethat can be exertedin extension of the leg at the hip andknee for 17 testconditions on male British civ i I ians in the sitt i ng positionis given in Morgan(Ref. 91).

TestConditions Avg . force A BC D (Ib)

I 000 90 63 """""""" "_ 000 113 89 000 135 156 050 164 559 060 94 73 080 93 87 0 10 0 80 77 0 10 0 90 59 0 10 0 135 27 0 0 10 0 165 346 0 15 0 149 227 0 15 0 160 845 0 15 0 169 530 0 16 0 129 31 9 0 17 0 117 272 0 17 0 151 684 0 33 0 106 184

Table 22. Maximum forceexerted in extension ofthe leg at the hip andknee.

Morgan (Ref. 91) alsoconsiders the maximum forcethat canbe exertedin extensionof the ankle, corresponding to footpedal operation (Table 231, by male Air Forcepersonnel for 18 testconditions.

1 83 I TestConditions Percent i I es ( I b 1

A E F G 5th

13 10 13 14 35t 37 13 10 384 37 54 13 30 13 25 358 37 13 30 13 64 38; 37 13 2450 35$ 37 13 4850 384 37 13 10 13 35; 39 15 13 10 13 37 384 39 13 30 13 26 35; 39 13 30 13 60 38; 39 13 2250 358 39 38; 39 5013 3950 38; 13 10 13 35; 41 18 13 35 10 384 41 13 32 30 35; 41 13 30 13 50 384 41 13 50 13 23 35; 41 13 50 13 50 384 41

Tab le 23. Maximum forceexerted in extension of theankle.

The following table,adapted from MIL-F-8785B (Ref. 41, givesforce limits for the e I evator, ailerons, andrudder.

CONTROL MAXIMUMCONTROL (Ibs) MINIMUM (Ibs)

E I evator Stick controllers 28.0 3.0 Wheel controllers 120.0 6.0

Ai I erons Stick controllers 20.0 5.5 Wheel controllers 40.0 10.5

Rudder Peda I s for short duration 100.0 forsteady coordinated turns 40.0

Table 24. Force limits forthe elevator, ailerons, andrudder. 1 84 Bureau ofAeronautics Report AE-61-4-11 (Ref. 93) discussesfriction forcesin the control system as they affect the pilot's tracking accuracy and recommends that friction forces for hand controlsin excess of three pounds beavoided, since they do not improve performance but do increasepilot fatigue. At the same time,this report also recommends that nohand controlrequire lessthan two pounds of force andno pedal movement requireless than seven pounds of force for increasedpilot tracking accuracy.

It isnot considered necessary to set standards for switches and dials, sincethey will most likelynot require limit forces; however, Woodson and Conover(Ref. 94) suggest that, for increased efficiency, rotary knobdiameters rangefrom one-half to two inches andhave amaximum resistance of onepound or less.

Often,the force to be overcome is usedas a feedbackcue; thus, it i.c necessary to reproduce a previouslyexperienced force and associatewith it a certainreaction of theaircraft. The ability to reproduce a givenforce, as statedby McCormick(Ref. 951, varieswith type of control andamount offorce to be exerted,as shown inthe graph below. The controlstested were of the pressuretype, so various amounts ofpressure could be applied with little or nodisplacement, making amount ofdisplacement constant. The devicestested were a stick, an a ircraft-type wheel,and a rudder-l i ke peda I. The difference between theactual force reproduction and thedesired reproduction was expressed in I imens (thestandard deviation divided-by the standard pressure). This figureindicates that, for pressures of five pounds orless, the errors in reproducingthe desired forces are proportionally greater. For five to ten pounds, theerrors are somewhat less,but stillgreater than those of forces ten to forty pounds; pressuresgreater than forty pounds, overlong periods of time, are apt to cause pilot fatigue.

-stick control - - - - - wheel control ----o----pedal control

J 5 15 10 20 25 30 35 Pressure,pounds

Figure 80. Results of dataon reproduclng control forces.

1 85

L Of considerableimportance in the design of any controlsystem are the limitations on the pilot's response time, or the speed, consideringthe effect of load, at whichthe pilot can actuate the controls. Orlansky (Ref. 96) reportsan experiment to determinethe maximum rate at which pilots canpush or pull a controlstick as the load per unit displacement changes.From this and otherstudies, he concludes that, for a 35-lbload, the maximum rate of stick movement is about 50 in./sec,pushing a stick is nearly 25% fasterthan pulling,and the rate of controlstick motion decreases as the load on the stickincreases. TheHandbook of Human EngineeringData (Ref. 97) indicates that, as thedistance for a positioning movement increases,the operator increases his speed of movement; thetime required for the response does notincrease as much aswould be expected. For example, data from this handbook indicatethat thetime for total movement increases 15% when thedistance is doubled and 25% if thedistance is tripled. Also notedis the loss of timein making a position movement when the pilot mustchange directions with the control, when about 15% to 24% of the movement timeis involved in stopping the movement i n one direction and beginning it inanother direction. Continuous curved motions, therefore,are desired over motions with sharp directional changes.

The pilot's responsetime is the sum of hisreaction time and his movement time.Reaction time, as defined in AFSC DH 1-3 (Ref. 901, isthe period between theonset of the signal to respondand the beginning of theactual response. Among thefactors affecting reaction time are type of signal, motion unit responding,precision of the response, ageand sex of theresponder, preparation for theresponse, practice for complex responses, and ambientconditions. AFSC DH 1-3 reportsthat handresponse is 20% fasterthan foot response, and thepreferred limb is about three percent faster than the non-preferred. TheHandbook of Human EngineeringData (Ref. 97) gives mean reactiontimes for simple movements ranqinqfrom about 0.22 sec to 0.3 sec.The conclusion from these data is that the auditory system reacts faster than the vi sua I system.

The foregoinginformation has been includedto indicate the anthropological basisfor satisfactory handling: the actuation force levels, limb displacements,and phase relationshipswith which a pilotis comfortable. It is then thedesigner's task to providesatisfactory aircraft response using these anthropologicaldata to describethe input to the aircraft control system. The readerwill note that the primary emphasis of this report is on insuringsatis- factoryaircraft response. While many futurelight aircraft will retain entirely manual controlsystems which require the designer to make certain com- promisesbetween what forces, displacements, etc. he would like to presentthe pilot with the responses thatthese inputs can produce, some future light aircraft will employ artificial feel systemswhich can present the pilot with whathe would like while at the same timeproviding satisfactory responses. When thesystem is capable of optimizingboth these facets it then becomes importantto insure that anthropological requirements are considered in detail.

186 DESIGN FOR DESIRABLE RIDING QUALITIES

In the past the primary design and specification efforPs for aircraft have rightly beenconcerned withinsuring safe operation and successful com- pletion of themission. Progress in these areas now appears to havereached the point that some attention may be directed toward providing the pilot with a comfortableride as w.ell. A significantportion of what thepilot describes as riding qualities dependsupon thedesign of his seat and otherrestraints. Noise-inducedvibrations also contribute to thepilot's gross impression of theride. For the present discussion, however, considerationwill be restricted to thoseaspects of ride which can be altered by theaerodynamic de- signof the aircraft. Thus theconcern will be desirablevalues of linear and angularaccelerations associated with the airframe dynamics.

As statedearlier it isthese changes invelocity which the pilot feels asimposed forceson his bodyand which he interprets as major contributors to theriding qualities of hisaircraft. Unfortunately, no substantivedis- cussionof the relation between themagnitude and frequencyof these accelerations and theacceptability of the ride was foundin the literature. The followingarguments, however, lead tocriteria which may find utility.

Thereare five characteristic motions associated with rigid aircraft. The spiral mode has a verylong time constant and isaperiodic. It isthere- foreunlikely to inducesignificant accelerations in normaloperation or to occurat a ratewhich will beuncomfortable. The roll mode, alsoaperiodic, isvery heavily damped and issenerally not sensedby thepilot. This leaves the three oscillatory modes asthe source of ride discomforts.

Consider firstthe longitudinal case. From the second equationof A-34 it is evident that to a first order

a, - w - Uoq = ZUu + Zww + Z~G~G

The notation dG is usedhere toindicate an effectiveaerodynamic input, - similarto a flap onelevator deflection, due to a verticalgust, w. Thus,

Sincethis form is similar to thatresulting from a simplecontrol input it may beused to describe pi lot inducedoscillations as well as gust induced oscillations.Only the va lue of Z6G and 6~ mustbe changed. Following this tack,then

This may beevaluated through the use of Equations C-4 and C-7.

187 The resulting Bode plot has two well-definedpeaks corresponding to the Phugoid mode and theshort period mode. Sincethe peak accelerations are reallythe values of interest, it ishelpful to develop approximate forms for theamplitude of a,/bE correspondingto these two peaks. The numerator ofthe transfer function contains four zeros: one at the origin, one just to the right of it, one far to the right of the origin andone far to the leftof the origin. The lattertwo have no influence on the phugoid mode and very little onthe short period mode.

The denominatorhas the two second order factors corresponding to the phugoidand short period modes. It will be recalledthat the phugoid factor 25 s s2 +p+1 7w n n n P P reduces to 2Cp when w = wn andcan be approximated by Wn 2/wnp2 when w = wnsp.The shortperiofj factor is about I .O when w = wpSP and2cSp when w = wsp. Forthese two conditionsthe transfer function becomes

K Talwn 2 a "=- P &E 25P

When thegain and timeconstant, Tal, are evaluated as given byRef. 17 one has w2 a pu 4 'L n o! "- z- cm I+- 6E * o! where the damping- ratio refers to the mode beingconsidered. In terms of the gustvelocity, w, this canbe written

The quantitiesin parentheses are relatively constant with speed inthe rangeover which light aircraft operate. Since

ww = constant nn P SP and

w n - uo SP

1 88 then

aZ ... W u0/s z ssp isapproximately independent of speedwhile sp increaseswith Uo .

Notethat increasing the wing loading is favorablefor improved ride whilereducing oscillatory damping resultsin a poorerride. Two additional comments arein order regarding this relation: (1) w' hereis the amplitude of an oscillatorygust having a frequency of either Unp or wnssp. aZ will be less for oscillatorygusts of any otherfrequency. (2) Theequations from whichthe relation was derived assume nosteady pitching velocity. Hence onecannot find the value of az/6E in a steadypull-up from this relation.

Typically,the damping ofthe phugoid mode isabout 1/10 that of the shortperiod mode whileits frequency is about 1/20 thatof the short period mode. This means thatthe accelerations associated with the phugoid mode canbe ten times as large as those associated with the short period mode. On theother hand, sincedesirable values of short period mode frequencyrange betweenabout I and 6 radians/secfor effective handling, the period of time required for thephugoid acceleration to build up to its peak is onthe order of 5 to 30 seconds.During this period of time the pilot has the opportunity to takecorrective action. Autopilots also damp thismotion effectively. In anycase, a pilotis not likely to associate phugoid induced accelerations withride but rather with handling. It is evident, however, thatboth ride and handlingand therefore safety can beimproved significantly by increasing the damping.

The phugoid mode isalso accompaniedby a longitudinalacceleration which has a magnitudeabout 1/10 to 1/6 as largeas az. Since either the pilot or an autopilot will attemptto suppress az at wnp, it seems reasonable to ignore ax.Note thatvariations in ax areessentially zero at wnsp. Otherthan possible nausea resulting from thelong period swaying motion and the pilot fa- tigueincurred in controlling it, thephugoid oscillation can probably be ignored. It appearstherefore that one may takethe following as the ride criteria in the x-zplane:

between I and 6 rad/sec. w"sp A similarargument can be made for the lateral acceleration with the result that

* The anthropo logicalbas is for suitablevalues istreated later. 189 ...". . . ." ..

where ..

Interms of a lateral gust with osci I lation amp1 itude v, this becomes

- Psuo 2 Wd a =vKa '-- Y 4w

Psuo 2 Wd a =;K~'"--- Y 4w Sd Y w between I and 6 rad/sec. d withthe maximum valuefor ay to be specified. ay of courserefers only to per- tubationsfrom straight line flight.

Additional componentsmust be added to the accelerations to account for steadyrotation. For example, in a steadyturn with 0 = 0 theaccelerometer indicationsare

A =UR-gsit~@~ Y 00

A, = QoUo - COS @o .

Ifthe turn is coordinated,

R = 9 sin ou- 0, , 0 A =O Y 9

l-cos2@o g Qo = -uo cos@o ' and

A z-9 . z cos @ 0 The accelerations felt by the pilot are those values which differ from A, = g and Ay = 0. Thus in a coordinatedturn the comfort limit is determined by

AZ = g (cos 1@o - 1) .

190 2 McFarland(Ref. 98) suggests a comfort limit of IO ft/sec for I inear accelerations.According to thiscriterion coordinated turns nclear air shouldnot exceed bank angles of 40°. If the air is turbulent the al lowable bankangle apparently would be reduced.

While a pilot may tolerate these acceleration levels during turning flight becausethey are unavoidable and are imposed for relatively short periods of time, it does notappear reasonable that a 180 Ib.pilot would findnearly 60 Ibs. of forceapplied sinusoidally along any of his principal axes for an extendedperiod of timecomfortable. It isalso reasonable to expectthat because of hisconstruction, a pilot will bemore sensitive to lateralforces than to verticalforces. While no substantive information is available to supportthe author's qualitative experience, it is suggested that because of theseconsiderations 2 a = 4 ft/sec Z and 2 a = 2 ft/sec Y may bemore suitableacceleration limits for thedutch roll and shortperiod motionsthan that offered byMcFarland. Since such accelerations are substantially belowthose imposed by steady turns, the pilot probably will not be as sensitiveto themduring turns as at other times. The 40° bankangle limit may thereforebe acceptable during turns, provided the ayand aZlimits quoted above are met during straight-line flight.

Interestingly enough,McFarland suggests a limit of 5O bankangle at low altitudes and 25O bankangle at high altitudes for tilt angleswith which passengerswould be comfortable. Ifthe 25O tilt occurredin a steadyturn, it wou I d correspond to a 3.2 ft/sec2 acceleration i ncrement a I ong the z- d i rect ion.

McFarland was alsoconcerned with tolerable levels of angularacceleration. However, it canbe shown thatthe gain of

is aboutis times the gain of

It appears,therefore, that angular accelerations are of no significance if az and ay aremaintained at the levels indicated..

Otherdata of interest to theriding quality discussion are presented in Figure 81. This shows a summary of p ilot comments on the short period mode hand I i ng qua lities of a jet fighter. Notethat good handlingcharacteristics wi I I virtual lyinsure good riding qua lities according to the criteria developed here. 191 a

.l .2 3 .I 5 .6 .7 .8 .9 1.0 Damping Ratio

Figure 81. Results of pilotopinion ratings onthe handlingqualities of a jet fighter (Ref. 99).

One final comment regardingthe riding qualities of an aircraft in gusty air may be made. Rigidaircraft exhibit fairly rapid attenuation of normal accelerationresponse for w>wsP. Forexample, atfrequencies in the range where human internalorgan resonances are excited (-42 rad/sec), aZ isless than 10% as much as at wsp for a givenamplitude oscillatory gust. Only if the aircraft has a poorly damped fuselagebending mode or wingbending mode atthese frequencies would one expect there to be significant structural shake resultingin pilot discomfort.

192 THE EFFECT OF VARIATIONSIN STABILITY DERIVATIVES ON THE MOTIONS OF A TYPICAL LIGHTAIRCRAFT

193 INTRODUCTION

Specification of aircraft geometry, mass distribution, and controlsystem characteristics to yield given riding and handlingqualities is unfortunately an interativeprocedure and therefore a laboriousprocess. One canconstruct thenecessary transfer functions, but there is no unique method to assign numericalvalues to particular stability derivatives which assumes thatthese valueswill bear the often necessary interrelation with one another nor correspond tophysically realizable geometry or mass distributions. The approachemployed here proceeded through several phases. $he first was to takean existing light aircraft, in this case a Cessna182 , compute its stabilityderivatives, substitute in the transfer function, and extractthe roots. The derivatives werethen varied individually to determine the sensitivity of thelocus of roots to changes inthat particular parameter. Thisprocedure also permits one to determine approximately the range of values whichthe particular derivative may have for satisfactoryperformance. It is onlyan approximate range because some derivativescannot physically be variedindependently of others. This aspect of theprocedure and itssigni- ficancewill become clearin the subsequent discussion. Once reasonable valuesare obtained for the desirable values of the stability derivatives, particularlythose which have a strongeffect on movement of theroots, one thenproceeds to determinethe geometric and mass distributions which will producethese values and are atthe same timeself-consistent. This phase will be elaboratedlater.

Inpreparing the figures for the stability derivative variation, the derivativeschosen for examination were generally within !IUS or minusone orderof magnitude of those calculated for the Cessna 182 , atcruise. Includedwith the figures for thelocus of roots due to a variation of a single stability derivative are tables whIchindicate how the gain of each particular transfer function varies as a function of the stability derivative. Table 16 throughTable 31 andTable 35 throughTable 46 are tabulations of thenumerator roots for the longitudinal and lateralstability derivative variations,respectively. Included at the end ofthis section are a series of six Bode plots(Figures 90 through 95) illustratingthe motions of the aircraftin response to controlsurface step inputs. Thenumerical values used to preparethese graphs are those for a Cessna182 at cruise.

* The principalgeometric dimensions for the Cessna182 are shown in Figures 61a and61b. The longitudinal and lateral derivatives used for theanalysis are tabulated in Tables 14~and 14b.

** The valueswere calculated by the methods presented in earlier sections of thisreport. Theycompare favorablywith those calculated by Cessna according to a personalcommunication.

194 I8 80 LIS uwD.Io 2aOoLrn 12 IN NACA 2412 WINO AIRFOIL ROOT VERT. TAIL AIRFOIL ROOT NACA 0001.5 TIP NACA OW8 WRZ. TAIL AIRFUL ROOT NACA OOOO TIP MICA OW6 ANGLES OF INCIDENCE WHO-ROOT cwm tlo30' YIN --TIP CHORD -1. 30' NIN -3OtlS' NIN STABILIZER DIHEDRAL +lo 44' MIN WINO MOMENTSOF INERTIA 941 SLUO-FT~ IP I 346 SLUO-FT~ IY Y I967 SLUO-FT2 In

7-I329.37

THRUST 225.441

I 1\ I SJ.00

I- -1

Figure 82b. Three View DrawingContinued

196 -

......

CL 0.309

CD 0.031 1

Cm 0.0

CT 0.0

0.0 -0.3086 cLU GYB 0.0 -0.089 cDU c%3 0.0 0.06455 CmU c"B C -0.0373 CTU 0.0 YP 4.61/rad. -0.4708 cLa cgP 0.126/rad. -0.0292 cDct cnP -0.885/rad. 0.2103 cmct r 0.0958 CL& 1.74/rad. cgr- -0.09924 CD& 0.0 'nr

Cm& -5.24/rad. 'Y6R 0.187 3.9/rad. 0.0147 cLq cg6 R 0.0 C -0.0658 cDq n6R -12.43/rad. P 0.00205ugs/ft. SI cmq 0.427/rad. U 219.0 ft./sec.

CD6E 0.0596/rad. Y 00

Cm6 E - 1 .28/rad. Table 25b. LateralStability P 0.00205 s I ugs/f t . derivatives(per radian).

U ft./sec. 219.0

Y O0

Table 25a. Longitudinalstability der ivat ives.

197 LONGITUDINAL VARIATIONS

Figures 83aand 83b show the effect of CL variationson the longitudinal dynamlcs. It will beseen thatthere is little change inthe location of the shortperiod roots for all usualvalues of CL. Thephugoid mode, however, issignificantly altered bychanges in CL. At high CL'S, boththe frequency anddamping ratioare increased. At CL'S nearzero, the roots become real with onegoing unstable. Thus one would expect difficulty ip maintaining speed stabilityin a shallow,high-speed dive.

The effort to provide low drag for good performanceleads to a neutrally damped phugoid,as pointed out in Figures 84aand 84b. Stabilityaugmentation istherefore required if one is to obtainboth low (QO) dragand good riding and handling(low work load) qualities.

Cm and CT mustbe considered simultaneously because Cm providesthe aerodynamic moment tocounter the moment produced by thethrust. In gliding flight, Cm = 0. Figures 85aand 86a show thatthe short period mode isnot affected by changes ineither Cm or CT. Figures 85b and 86b show theeffect onthe phugoid mode ofaltering Cm and CT respectively.Adding power will make CT positive and Cm negative.Reference to the figures will show that making Cm negativewill cause the phugoid roots to split along the real axis with onegoing unstable. On theother handmaking CT positiveresults in an unstablephugoid oscillation. Thus while it isnot possible to conclude from thesefigures alone the detailed airplane behavior when power is added(because Cm and CT cannotbe varied independently in flight but only through design changessuch as thelocation of theengine thrust line), it isobvious that theapplication of power isdestabilizing.

Figures 87aand 87b indicatethe variations in longitudinal dynamics produced by changing CL~. Increasing CL is seen to reducethe frequency and toincrease the damping of theshort perlod zr. mode. A sufficientlylarge value of CL~suggest aperiodic short period roots. The time for the phugoid to damp to halfamplitude is little affected bychanging CL~(Figure 87b), but theoscillation frequency is a directfunction of CL~. Allusual values of CL~are therefore acceptable for satisfactory aircraft riding qualities.

From Figure 88a, it is seen thatthe short period mode isvirtuall:! insensitive to moderatechanges in CDa. Increasingvalues of CD~destabilizes thephugoid mode (Figure88b). Instability is most likelyto occur in the approachconfiguration where CD~isgreatest. The maximum acceptablevalue isabout twice the value calculated for the Cessna182.

Figures 89aand 89b show the movement ofthe short period andphugoid rootsrespectively due to a variationin Cma. Formost light airplanes as usuallyloaded, Cma will probablylie between -.3 and-1.5. Inthis range, the time to damp theshort period oscillation to half-amplitudeis independent of thevalue of Cma, whilethe frequency increases as Cma assumes greater negative v-al-ues. Cma withgreater negative values than -1.5 givesshort 198 period mode frequencieshigher than the 5-6 rad/seclimit desired. Values of Cma greaterthat about -0.23 causethe short period oscillation to disappear and become twoaperiodic modes.

Reasonablevariations in Cma (Figure89b) also have little effect on the time to damp thephugoid oscillation. The phugoid frequency increases as Cma becomes morenegative, while values more than about -0.05 causethe phugoid to degenerateinto aperiodic motions. It is seen thereforethat the propervalue of Cma is usuallydetermined by requirements for acceptableshort period mode characteristics. A notableexception may bementioned, however. A recentpaper (Ref. 116) discussedresults obtained with a light aircraft modified to evaluatethe pilot acceptance of variousshort period mode damping ratios. Thedamping ratio wasmade unity(i.e. both roots lie on the negativereal axis) by increasing Cma (making it lessnegative). One would expectthat the pilot would bepleased with this condition since it means that the aircraft wouldtrack rapid elevator commands without oscillating. Instead,the pilot found it quiteobjectionable, saying that it was difficultto maintain airspeed stability. Thepaper therefore recommends that damping ratiosapproaching unity not beused., The falacyin the argument isimmediately obvious when oneexamines theeffects of nearzero Cma valueson the phugoid mode. The roots become aperiodicwith one going unstableas Cma increases. Of coursethe pilot would find this undesirable. It is therefore important that unity damping ratio for theshort period mode not be achievedby increasing Cma alone. As discussedbelow the most satisfactorysingle means ofobtaining this behavior (unity damping ratio)is to make Cmq more negative.Increasing the tail length is the most effective geometricchange one can make to accomplishthis increase is damping. Because ofthe very significant lengthening required, however, it isdesirable to combine thiswith small rearward shift in operating c.g. (effectively a smallincrease in Cma).

Figures 90a, 90b,91a, and 91b show thatthe roots are relatively insensitiveto changes in CL& and CQ&. Changing CL& byan orderof magnitude fromits original value moves theroots only slightly for both the short period andphugoid modes. CD& isusually considered to be zerofor most lightaircraft. If a valuewere calculated, it wouldbe atleast an order ofmagnitude smaller than C&, atleast assmall as .177. Thus, forall normalvalues of CD&, bothphugoid and shortperiod roots remain in virtually the same location.

Figures 92aand 92b indicatethat Cm& isquite important in tlle short period mode butunimportant in the phugoid mode. Morenegative values of Cm& decreasethe frequencies and increasethe damping.High negative values caneven lead to twohighly damped aperiodic modes. Valuesgreater than zero shouldbe avoided. In the normal range of values,an error in Cm& of 20% couldcause a 5% errorin the calculated frequency of oscillation. Thephugoid mode rootsare relatively unaffected, even when Cm&, is changedby an orderof magnitude.

Figures 93a, 93b,94a, and94b indicatethat both the short period mode and thephugoid mode arerelatively insensitive to variations in CLq and CD~. It is noteworthythat a valueof CL~= 0 wouldgive almost the same characteristics asthe value calculated. 199

L Cmq is quite important to thefrequency and damping of theshort period mode butrelatively unimportant to those of thephugoid mode. The short period roots for variousvalues of Cmq aresimilar in behaviour to theroots for variousvalues of Cm&.(see above). As Cmq becomes morenegative, the frequencydecreases and the damping increases for theshort. period mode (Figure95a). For the normal range of Cmq values,the damping is moresensi- tive to changes in Cm thanthe frequency of oscillation.For very negative va I ues of Cmq, aperioaic motions are achieved. Variations of Cmq have a Imos-t no effecton the phugoid damping.The frequency ofthe phugoid mode decreases as Cmq becomes morenegative, as can be observed from Figure95b. Cmq, of course, isalways negative.

Theabove resultsare discussed below from theviewpoint of acceptable ridingqualities, since the proper range of the more importantstability derivativesis significant in achieving these qualities. Theimportant derivativeswhich affect the short period mode appear to be CL~,Cma, Cm&, and Cmq. It would be desirable to have theshort period damp to one-halfamplitude in onesecond orless, with a damping ratiogreater than 0.6 and to have the frequencyless than five radians per second for acceptableriding qualities and lessthan four radians per second for good ridingqualities. Any reasonablevalue of CL~will provide a damping ratiogreater than 0.6, a frequehcy less than 4.25rad/sec and a time for damping to one-halfamplitude of lessthan 0.25seconds. Cma valuesbetween -.23 and -1.0 giveacceptable shortperiod riding qualities with a frequencyof less than 5.0 rad/sec and a damping ratiogreater than 0.6. Itis desirable to have Cm& lie between -4.0and -17.0; the more negativethe value, the lower the frequency and higherthe damping. Cmq shouldhave values between -6.0 and-29.0 for good rid.ingqualities.

The frequencyand damping criteriamentioned above probab.ly should be regardedas applying only to aircraft without artificial stability augmentation. Thegeometric changes necessary to improvethe riding and handling qualitiesfurther (i.e. to obtainunity damping ratio for an undamped natural frequencyof 6 radiansper second) sufficiently degrade aircraft performance andpayload capacity as to make thesechanges unattractive. Improvements in riding and handlingqualities, however,can be obtainedwith stability augmentation without sacrificing either performance or payload.

Thephugoid mode mustalso be considered when discussingriding qualities. This mode shouldhave a damping ratio of at least 0.04, with a maximum frequency ofabout 0.3 rad/sec. Ifthe period is overthree to four seconds, it canbe tracked by the pilot although the pilot work load will behigh. Thephugoid frequency is really not very critical as long as it does not approachthe short period range. The derivatives Cm and CT contributeonly tothe phugoid mode andshould be considered together. More positive values of CT yieldunstable phugoid roots; for the particular airplane analyzed, Cm becomes more negativeas CT became more positivewhich leads to anincrease inphugoid frequency and a decreasein damping. CL has the most effect on thefrequency of the phugoid mode, while CD mainlyaffects damping.For the normalrange of CL values (.1-1.5), thefrequencies may take on values as highas 0.3 to 0.4rad/sec; however, as the frequency increases, the damping alsoincreases. The larger the value of CD, thebetter the riding qualities; however,performance requirements must dictate the value of CD. 200 Values of Cma between -0.01 and-2.5 appear to givesatisfactory phugoid responseby affecting the frequency while keeping the damping essentially constant. Cmq has a smalleffect on the phugoid frequency, but values between 0.0 and -30 givedesirable phugoid riding qualities.

In the abovediscussion, a range of desirable or acceptablevalues of theimportant stability derivatives hasbeen given for thephugoid and short periodriding qualities. It shouldbe emphasized thatthe derivatives have beendiscussed as if theywere independent of the other derivatives, while inreality they are not. In the table below, the mostimportant longitudinal stabilityderivatives are given with the range of valueswhich should result inthe desirable handling qualities indicated above. It shouldbe remembered that eventhough values of Cmq between 0.0 and -30.0 appearacceptable, if thevalues are very near zero then it willprobably beimpossible to obtain acceptablevalues for other stability derivatives.

StabilityDerivative Acceptable Ranqe

0.03" to 1 .O

-0.23 to -1.0

-4.0 to -17.0

0.0 to -30.0

Table 26. Acceptable range for longitudinalstability der i vat i ves.

Sincethe longitudinal stability derivatives cannot really be varied independently, it was felt that the movement of theroots due to the changes inthe airplane's geometry would be more informativethan just a variation of thelongitudinal stability derivatives. In the case of longitudinal dynamics, the movement ofthe roots was calculatedfor various c.g. locations,horizontal tail areas, tail lengths, and tailefficiencies. It should be notedthat these variations were made withoutchanging the original inertiacharacteristics of theairplane. As anexample of how inertia changes would effectthe geometric variations, the short period plot of Rt indicates theroots with and withoutthe inertia changes. It shouldalso be pointed outthat the tail area was variedin such a way thatthe tail aspect ratio was heldconstant. The figures and tables below can be used to track the roots.

Thephugoid and short period frequencies decreased as the c.g. was moved aft,until aperiodic modes wereobtained for both (Figure 96 andTable 44). The shortperiod dampingdecreased and thephugoid damping remained almost constantas the C.Q. moved aft UP to 42.5% m.a.c. The shortperiod roots were aperiodic with thec.g. at 42.5% m.a-.c., wh i le with the c.g . at 45%

*Lowervalues of CD will require artificial damping to meet the phugo i d damping criterion. 20 1 m.a.c., thephugoid roots wereaperiodic with one unstable root. For a c.g.location at 47.5%, there was a phugoid-shortperiod coupling as well asone very stable short period root andan unstablephugoid root.

The variationsin horizontal tail area with tail aspect ratio held constant showed that the short period dampingincreased and the frequency decreased fortail areas greater than that of the Cessna 182. Tailareas less thanthe original area gave lower frequencies and lower short period damping (Figure97a). Thephugoid damping was virtuallyunaffected by thevariations intail area while the natural frequency showed onlysmall changes.

Forthe variation in tail length without inertia effects, the short period dampingincreased and the frequency decreased for taillengths greaterthan that of the Cessna182 (Figure98a). Tail lengths less than the originallength gave lower frequencies with lower short period damping. Two aperiodicroots are obtained for tail lengths slightly below 45% ofthe originallength. The trajectoryof the short period roots considering inertia changes was similar to thatcalculated without considering inertia changes butexhibited higher damping ratiosfor longer tail lengths and lowerdamping ratios for shorter tail lengths. Thephugoid damping was almost unaffected by variation in tai I lengthfrom 200% to 45% of the origi na I value,as seen inFigure 98b.The frequency did decrease as taillength decreased,but no unreasonable frequencies were obtained for therange of taillength variations. As may beexpected, the variation of tail efficiency, qt, gave resultssimilar to thosefor tail length variation (Figures 99a and99b).

202 R

LATERAL VARIATIONS

Figure 100shows theeffect of CYB variationson lateral dynamics. In Figure 100a, theDutch Roll naturalfrequency appears nearly unaffected by changes in CY@; however, Dutch Roll dampingincreases slowly for more negative values of cy^. Forpractical airframe configurations, cy^ isnever positive. F.igures 100band 1OOc show the nearly negligible effect of cy^ onthe spiral and Poll modes, respectively.In general, the three modes seem relatively insensitive to changes in CY6.

The effect of CQ variation onthe Dutch Roll mode is seen in Figure lola, wherelarge negative values of CQ decreasethe dampingand increase the naturalfrequency, and positivevalues improve dampingand decrease the naturalfrequency; however, for typicalvalues of CRB, theDutch Roll mode isonly slightly affected. The spiral mode, aspointed out in Figure 101b, is more stable for largenegative values of Cj$ butless stable for both smallnegative values and any positive values. Thus an aircraft may be made more spirallystable by increasingthe wing dihedral; however, thereis a limit to the amount ofdihedral because of theadverse effect of large negativevalues of C@ onthe Dutch Roll mode. The roll mode (Figure101c) is also more stablefor larger negative values of CQ, but substantial changes arenecessary for the effect to be verynoticeable.

The Dutch Roll frequencyis highly sensitive to changes in CnB, Figure 102a, withlarge positive values causing very high frequencies; small negative values,which are possible for airframes with small vertical tails, produce an unstablesystem. Dutch Roll damping isrelatively unaffected by changes in Cng. The spiral mode (Figure 102b) ismoderately sensitive to changes in Cng; it becomes more stableas CnB growssmaller and becomes negative. The roll mode (Figure102~) is unaffected bychanges in CnB.

Variationin Cy has no effect onany ofthe lateral modes, asindicated inFigures 103a, 103&, and103c.

Figure 104ashows thatthe Dutch Roll mode isonly slightly affected by variationin CgP.The spiral mode inFigure 104b is somewhatmore sensitive; lessnegative values of Cgp increasethe stability; positive values of CgP arenot physically possible. Obviously, the roll mode inFigure 104c is highlysensitive to changes in CkP, thedamping-in-roll derivative, as it becomes more stablefor increasingly negative values of Cf, Forvery small P' negativevalues and for positive values, the roll and spiral modes couple to producethe roll-spiral oscillatory mode. MIL-F-8785B, (Ref. 41, specifically prohibits this.

Figure 105ashows theeffect on the Dutch Roll mode of variationsin Cnp. Largenegative values cause an increase in natural frequency, whereas positive valuesdecrease the frequency and damping until the system becomes unstable. The spiral mode (Figure 105b) isvirtually unaffected bychanges in Cnp, and the roll mode (Figure105~) becomes lessstable for large,-negative values of Cnp. 203 Variation in Cyr hasalmost no effecton any of thelateral modes, as Figures 106a, 106b, and106c indicate.

Largerpositive values of CRr (Figure 107a)increase the Dutch Roll damping, buthave little effect on the natural frequency. More positive values,though, seem to givean unstable spiral mode (Figure 107b)and a I ess stable rol I mode (Figure 107~).

Figure108 shows the effect of variations in Cnr on all three modes. Forlarger negative values of Cnr, theDutch Roll becomes two aperiodic modes, with one root movingtoward the roll mode root andanother root approaching thespiral root. Theseroots meet and, at one point, two oscillatory modes exist.This situation, however, may notbe physically obtainable. For less negativevalues of Cnr, theDutch Roll dampingdecreases, butthe roll and spiral modes arelittle affected.

Theprevious results are examinedbelow from theviewpoint of acceptable ridingqualities. Consideration is given specifically to the allowable range of severalimportant stability derivatives governing these qualities. Two stability derivatives, CnB andCnr, seem to have themost effect on the Dutch RoI.1 mode. Using a minimum

The stability derivative Cg,g seems to have the most effect on the spiral mode. Forbest riding qualities, atleast a neutrallystable spiral mode is desired. To attainthis, a value of Cg,g more negativethan -.05 is needed. This may be accomplished,physica Ily, byincreasing the amount of aircraft wingdihedral, but large negative valuesof Cgg adverselyaffect the Dutch Roll mode byincreasing the frequencyand decreasing the damping, thus, a Cg,g ofabout -0.4 perradian isconsidered the most negative value permissible.

The stabi!ity of the roll mode isdetermined primarily by Cg, , the dampi ng-in-rol I derivative. For a ~/TRless than about 0.7, wh icR seems desirablefor easyhandling, the value of CgP mustbe less than -.04 per radian;however, too large a negativevalue of Cg, may causethe aircraft P toreact sluggishly to the pilot's commands, andone of therequirements for easymaneuverability is that the aircraft be able to roll a certain number ofdegrees in a finite amount of time.

The followingtable lists the important lateral stability derivatives withtheir suggested range of values for desirable handling qualities.

204 - __ - ~~ Stability Derivatives Acceptable Range

" ~ ~"~ rad(per ian)

c"B .01 to .15

'nr -0.45 to -0.09

QB -0.40 to -0.05 lessthan -0.04 QP

Table 27. Acceptablerange for lateralstability derivatives.

Sincethe lateral stability derivatives cannot physically be varied independently, it seems necessary to discussthe movement ofthe roots of the characteristicequation due to a variation of theairplane's geometry. For thelateral dynamicscase, thelength to the vertical tail, Rv, thearea of thevertical tail, Sv, thewing dihedral angle, and a combinationof vertical tail areaand wing dihedral were varied.

The effecton the Dutch Roll (Figure 109a) ofincreasing Rv is to in- creaseboth frequency anddamping of this mode. Thereason for this can be gatheredby examining the effect on Dutch Roll characteristics of CnB and Cnr variations.Increasing Rv makes Cng more positive.This has a direct influenceon the frequency of theDutch Roll. Increasing Rv also makes Cnrmore negative. For small increases in the magnitude of Cnr, the damping of theDutch Roll is increasedbut the period is unaffected. Finally, for largenegative values of Cnr, theperiod of theDutch Roll increases and the damping alsoincreases. For Rv up to 150% of theoriginal, Cng seems to dominateas the frequency rises quite rapidly, but for Rv greaterthan 150% thefrequency levels off and thesystem becomes more highly damped as Cnr becomes dominant.For decreasing Rv, boththe frequency anddamping decrease until,at about 35% ofthe original tail length, the system becomes unstable. The spiral mode (Figure109b) becomes more stable for bothincreasing and decreasing Rv. Thisis possible since in the expression, CgP Cng/(CgB Cnr - Cng Cfir)CyP, whichgoverns the spiral root, Cnr, CnB, and Cgr all change with Rv variation. The roll mode (Figure109~) is unaffected by changes in verticaltail length since Cgp isnot dependent on Rv. The moment of inertia IZZwas heldconstant for one set of Rv variations and allowed to change for the second setof Rv variations. The resultsof both are shown inFigure 109 and inertia changesproduce only small deflections in the curves with the generaltrends remaining the same.

Figure 110ashows the effect of vertical tail area variation on the Dutch Roll mode. Increasing Sv givesboth a higherfrequency and a slightly more damped systemsince CnB and Cnr areagain substantially affected by changes invertical tail area. (Cnr becomes lessnegative as Sv decreases; for the same conditions CnBmay gothrough zero and become negative)with a value of about 48% of theoriginal tail area, the system becomes unstable. The spiral mode (Figure 110b) becomes more stablefor decreasing vertical tail area,since this produces small positive or evennegative values of Cne (Figure102b). The roll mode (Figure110~) remains unaffected since CR is P 20 5 only slightly affected by changes in vertical tai I area.

The effect of variations in wing dihedral on the DutchRoll mode is modest as shown in Figure llla. However, the spiral mode (Figure lllb) is heavily dependent on wing dihedral, since this determines the valueC~QB, of the "effective dihedral" derivative. Positive dihedral angle givesa more stable spiral mode and negative dihedral causes the mode to become unstable. The roll mode (Figure lllc) is essentially constant forwing dihedral variation.

The combination of changing both vertical tail areaand wing dihedral is shown in Figure 112. The Dutch Roll mode (Figure 112a) is the same as that for SV variation alone since dihedralangle has little effect on this mode. However, the spiral mode (Figure112b1, which previously became less stable for increasing SV, now can be made more stableby an increase in the dihedral angle. Thus the Dutch Roll and spiral modesmay both be improved by simultaneously increasing vertical tail area and dihedral angle. The roll mode (Figure 112c) remains unaffected.

206 Short Psriod Mode Roots for the CL Variation

Irn

"5 /rO.3093

" 3

" 2

" 1

I 1 1 I I I I I I I Re -5 -4 -3 -2 -1 1 2

" CLValues listed -1 beside roots -- -2

"-3

"-4

"-5

Figure 83a

207 Phugoid Mode Roots for theCLVariation

Irn

.3 .751

.2 .3093

.1 .1

1 c. " W 1 ;r Re .1 .2 .3

.I -. 1 CLValues I isted beside roots .3093( -.2

-3

-.4

-. 5

-.6

Figure 83b

208 NUMERATORROOTS

U I STAB I L I TY DERIVATIVE REAL I MAG I NARY REAL REAL

-0.5 - 0.71701 0.0 -26.9358 I 2 I .6049 -0. I - I .29165 0.0 -2 I .2794 I 15. I774 0. I - 2.12606 0.0 - I 7.28866 I I .34820 0.3093 - 7.68330 0.0 - 7.93156 6.84425 0.75 - 5.92058 k15.08719 I .58786 I .5 - 6.66843 k26.45906 0.56020 3.0 - 9.03239 k40.574 I7 0.24139 4.0 -10.68138 k47.65224 0. I7489 4.5 -I I .51 191 k50.80265 0.15371 5.0 - I 2.34472 k53.75583 0. I371 I

-0.5 -0.28084 0.0 0.24735 -195.51762 -0. I -0. I3526 0.0 0. I0320 -195.46607 0. I -0.0 I509 k0. 1 I668 -195.44030 0.3093 -0.0 I 472 k0.20640 -195.41333 0.75 -0.0 I 393 f0.32196 -195.35655 I .5 -0.0 I259 k0. 045569 -195.25991 3.0 -0.0099 1 k0. 64493 -195.06664 4.0 -0.008 I2 -10.74498 -194.93780 4.5 -0.00723 k0. 79032 -194.87339 5.0 -0.00633 k0. 83322 -194.80897

-0.5 -0.09750 0.0 -9.99285 -0. I -0.03463 0.0 -2.0562 I 0. I -0.03088 0.0 -2.06020 0.3093 -0.04605 0.0 -2.04529 0.75 -0. I4779 0.0 - I .94407 I .5 -0.768 I 7 0.0 -I .3246 I 3.0 - I .04730 k I .70249 4.0 - I ,04790 k2.45078 4.5 - I ,04820 k2.81025 5.0 -I .0485 I k3. I6440

Table 28 Numeratorroots for CL variations.

209 ShortPeriod Mode Roots for CDVariation

I I -5 -4 -3 -2 -1

-1 CD Values listed beside roots -2

-3

-4

-5

Figure 84a

210 Phugoid Mode Roots for the Cg Variation

CDValues listed beside roots

t --.20

Figure 84b

21 1 NUMERATORROOTS

STAB I L I TY DER I VAT I VE REAL I MAGREAL I NARY REAL

-0.03 -7. I7723 0.0 -8.39084 6.82668 0.031 I -7.68330 0.0 -7.93 I56 6.84425 0.3 -7.907 I9 51.21744 6.91800 0.35 -7.92566 kl .32588 6.93144 0.4 -7.94408 51.42569 6.94479 0.5 -7.98077 *I .60575 6.971 18

-0.03 0.01478 10.20640 -195.41334 0.031 I -0.0 I472 W. 20640 -195.41333 0.3 -0.14171 +O. I5078 -195.41328 0.35 -0. I6542 *O. I2430 -195.41327 0.4 -0.18914 k0.08393 -!95.4!326 0.5 -0.35238 0.0 -195.41324-0.12191

-0.03 0.01315 0.0 -2.0 I60 I 0.031 I -0.04605 0.0 -2.04529 0.3 -0.30099 0.0 -2.17126 0.35 -0.34860 0.0 -2.19476 0.4 -0.39622 0.0 -2.2 I826 0.5 -0,49148 0.0 -2.26524

Table 24 Numerator rootsfor C variations. D

212 Short Period Mode Roots for the Cm Variation

rn 5 0.0 -.5 0.5 4

I -e de -5 -4 -3 -2 -1 1 2

-1

C, Values listed beside roots -2

-3

-4

-5

Figure 8%

21 3 PhugoidMode Roots for the C, Variation

O.(

-.25 -.I I AI L I -. I Re -. 4 73 72 -.l

Cm Values listed besideroots

.I 5

21 4 NUMERATOR ROOTS

STAB1 LlTY DERIVATIVE REAL I MAG I NARY REAL REAL

-0.5 -7.68330 0.0 -7.93156 6.84425 -0.25 -7.68330 0.0 -7.93156 6.84425 -0. I -7.68330 0.0 -7.93 I56 6.84425 0.0 -7.68330 0.0 -7.93156 6.84425 9.15 -7.68330 0.0 -7.93156 6.84425 0.30 -7.68330 0.0 -7.93 I56 6.84425 0.50 -7.68330 0.0 -7.93156 6.84425

-0.5 -0.00400 +O. I4062 -195.43477 -0.25 -0,00936 +O. I7668 -195.42405 -0. I -0.0 I 257 *O. I9507 -195.41762 0.0 -0.0 I472 k0.20640 -195.41333 0.15 -0.0 I793 k0. 22227 -195.40690 0.30 -0.021 I5 k0.23704 -195.40047 0.50 -0.02544 k0. 25535 -195.39189

-0.5 -0.01557 0.0 -2.05353 -0.25 -0.03078 0.0 -2.04944 -0. I -0.03993 0.0 -2.04696 0.0 -0.04605 0.0 -2.04529 0.15 -0.05524 0.0 -2.04277 0.30 -0.06445 0.0 -2.04022 0.50 -0.07677 0.0 -2.03679

Table 30. Numerator roots for C, variations.

21 5 ShortPeriod Mode Roots for thehriation

I m

" 5 0.0 -5 0.5 4

I 1 I " I I .I I I Re -5 -4 -3 -2 -1 1 2

-1 CT Values listed beside roots -2

-3

.) -4 25 0.5 0.0 -5

Figure 86a

216 r

PhugoM Mode Roots for the CT Variatlon

4 D.20 0.0

-.2

-.lo

--.OS

.5 .S I 1 1I 1 I 1 . ".+ -25 720 -.is -.lo -.05 ,os .lo .ls a20 .25 .3O *35

"-a5 CT Valuer listed beside roots - -.lo

OD "20

Figure 86b NUMERATORROOTS

STAB I L I TY DERIVATIVE REAL I MAGREAL I NARY REAL

-0.5 -7.68330 0.0 -7.93 I56 6.84425 -0.2 -7.68330 0.0 -7.93 I56 6.84425, 0.0 -7.68330 0.0 -7.93 I 56 6.84425 0.2 -7.68330 0.0 -7.93 I56 6.84425 0.4 -7.68330 0.0 -7.93 ! 56 6.84425 0.5 -7.68330 0.0 -7.93 I56 6.84425

AI2

-0.5 -0.40952 0.0 -195.41718-0.09450 -0.2 -0. I0940 k0. I7076 -195.41487 0.0 -0.0 I472 +O. 20640 -195.41333 0.2 0.07997 k0. I9522 -195.41180 0.4 0. ! 7466 k0.12527 -195.41027 0.5 0. I7346 0.0 -195.409500.26937

e -0.5 -0.52 I96 0.0 -2.04256 -0.2 -0.23624 0.0 -2.04437 0.0 -0.04605 0.0 -2.04529 0.2 0. I4399 0.0 -2.04605 0.4 0.33390 0.0 -2.04669 0.5 0.42882 0.0 -2.04697

Table 31. Numeratorroots for 5 variations. Short Period Mode Roots for the CL2riation

In 5

4.608

4 -4.0

20.0

S I 1 S I I I I I I I I I I -8 -7 -6 -5 -4 -3 -2 -1

-1

-2

-3

-4

CL,Values listed beside roots

Figure 87a

21 9 Phugoid Mode Roots for the CL~Variation

20.0

Figure 87b

220 NUMERATORROOTS

STAB I L I TY DERIVATIVE REAL I MAG I NARY REAL REAL

- 4.0 3.8569 I Q. 24363 - 12.4401 0 0.0 0.48723 0.0 4.46793 - I I .56084 4.608 6.84425 0.0 -7.68330 - 7.93156 8.0 - 8.98685 k4. I0263 7.60946 12.0 - IO. 23230 k5.68619 8.22106 15.0 -I I. 10984 k6.45655 8.56664 20.0 - I 2.50753 k7.331 19 9.01289

- 4.0 -0.0 I472 k0.20640 -195.41333 0.0 -0.0 I 472 k0. 20640 -195.41333 4.608 -0.0 I472 k0. 20640 -195.41333 8.0 -0.0 I472 k0. 20640 -195.41333 12.0 -0.0 I472 k0.20640 -195.41333 15.0 -0.0 I 472 k0. 20640 -195.41333 20.0 -0.0 I472 k0.20640 -195.41333

- 4.0 -0.01518 0.0 2.00513 0.0 0.04672 a.I5901 4.608 -0.04605 0.0 -2.04529 8.0 -0.03914 0.0 -3.66043 12.0 -0.036 I 9 0.0 -5.55989 15.0 -0.03503 0.0 -6.98343 20.0 -0.03389 0.0 -9.35520

Table 32. Numerator roots for qCcvariations.

22 1 .

Short Period Mode Roots for the CgaVariation

“ 5 ao -.5 01.5

-0 4

“3

“ 2

“ 1

I I I I I 1 1 I I I 1 I*Re -5 -4 -3 -2 -; -1 2 “-1 CgaValueslisted beside roots ---2

“-3

““4 a0 -50 15

“-5

Figure 88a

222 Phugoid Mode Roots tor the Cpa Vartation

- m.20

-J256 -5 ) 1.5

' -.l5

- -.lo

--.OS

I a I a I I , rn Re .10 .O 5 .os .10 cDa Values listed - - -.os beside roots

-.5

-.20

p i.gu re 88b

223 NUMERATORROOTS

STAB1 LITY DERIVATIVE REAL I MAG I NARY REALREAL

-0.5 -5.98845 k18.51477 I .I0150 0. t256 -7.68330 0.0 6.84425- 7.93156 I .5 -0.43878 0.0 29.04580 -32.75372

ACI

-0.5 -0.01472 +O .20640 -195.41333 0. I256 -0.0 I 472 k0.20640 -195.41333 I .5 -0.01472 +O. 20640 -195.41333

-0.5 -0.09055 0.0 -2.00078 0. I256 -0.04605 0.0 -2.04529 I .5 0.04537 0.0 I367 -2. I

Table 33. Numeratorroots for C,, variations. c1

224 Short. &riod Mode Roots for the *Variation

- 25

- 1.5

-3

-. 3

CnG,Values listed beside roots -.6

-L5

p.i.gure. 8% PhugoidMode Roots for the Cm,Variation

~m,Values listed beside roots

p i\gu re 89b

226 NUMERATORROOTS

STAB I L ITY DERIVATIVE REAL I MAG I NARY REAL REAL

-2.5 -6.9 I423 -+4.94 608 5.05789 -I .5 -7.46700 k2.95479 6. I6344 -0.8852 -7.68330 0.0 - 7.93156 6.84425 -0.6 -5.99482 0.0 - 9.93315 7. I5734 -0.3 -5.330 I2 0.0 - I 0.92447 7.48395 -0.25 -5.24383 0.0 - I I .06490 7.53808 -0.2 -5. I6223 0.0 - I I .20053 7.59212 -0.08 -4.98270 0.0 - I I .50937 7.72143 -0.02 -4.90034 0.0 -I I .65618 7.78587 -0.01 -4.88704 0.0 -I 1.68021 7.79660 0.0 -4.87385 0.0 -I I .70412 7.80732 0. I -4.74795 0.0 -I I. 93702 7.91432

-2.5 -0.0 I 472 k0.20640 -195.41333 -I .5 -0.0 I 472 k0.20640 -195.41333 -0.8852 -0.0 I472 +O. 20640 -195.41333 -0.6 -0.0 I472 +O. 20640 -195.41333 -0.3 -0.0 I472 +O .20640 -195.41333 -0.25 -0.0 I472 +O. 20640 -195.41333 -0.2 -0.0 I472 +O. 20640 -195.41333 -0.08 -0.0 I472 -+O. 20640 -195.41333 -0.02 -0.0 I472 +O. 20640 -195.41333 -0.01 -0.0 I 472 -TO. 20640 -195.41333 0.0 -0.0 I 472 k0. 20640 -195.41333 0. I -0.0 I472 +O. 20640 -195.41333

-2.5 -0.05433 0.0 - I .78232 -I .5 -0.04893 0.0 - I .94544 -0.8852 -0.04605 0.0 -2.04529 -0.6 -0.04480 0.0 -2.09151 -0.3 -0.04356 0.0 -2. I4008 -0.25 -0.04335 0.0 -2.14816 -0.2 -0.04315 0.0 -2. I5625 -0.08 -0.04268 0.0 -2. I7565 -0.02 -0.04244 0.0 -2. I8535 -0.01 -0.04240 0.0 -2. I8697 0.0 -0.04237 0.0 -2. I8858 0. I -0.04 I98 0.0 -2.20474

Tab le 34. Numeratorroots for C, variations. a 227

- .. " Short Period Mode Roots for the CLfariation

Im 4

9 -5

-3

- -1

n 1 1 I i Re I = n I t I i r=Re -"5 -4 --3 -2 "f -1 2

""1. ChValues listed beside roots "-2

"-3

" -4

"-5

228 I'

Phugoid Mode Roots for the Cb%rlation

.20

.15

.io

.O 5

710 -d5

-.a Cdalues listed beside roots

-.lo

-.15

200 -2.00

".20

229 NUMERATORROOTS

STAB1 LIP/ DERIVATIVE REAL I MAGREAL I NARY REAL

2.0 -7.81212 - 2.0 fl .36416 6.7637 I I .7419 -7.68330 0.0 - 7.931566.84425 20.0 -4.97289 0.0 7.21373-10.61842

- 2.0 -0.0 I472 t-0.20640 -195.41333 I .7419 -0.0 I472 t-0.20640 -195.41333 20.0 -0.0 I 472 20.20640 -195.41333

- 2.0 -0.04604 0.0 -2.08659 I .7419 -0.04605 0.0 -2.04529 20.0 0.0 -0.04608 -I .86499

Table 35. Numeratorroots for variations. G.c1

230 n 5

1 I Re -5 -4 -3 -2 -1 -1 2 -I

CD& Values listed -2 beside roots

-3

-4 - 1-00 1.0 -5

Figure 91a

23 1 . . . Phugoid Mode Roots for tbc CM Variation

10 0 -10

.os

I I I -Re -30 -.os .os .10

-.os c Valueslisted DL besideroots

-u) 0 1.0 -.2 0

Figure 91b

232 NUMERATORROOTS

STAB I L I TY DERIVATIVE REAL I MAG I NARY REAL REAL

-I .o -4.34161 0.0 - 16.685 19 5.55105 0.0 -7.68330 0.0 - 7.931566.84425 I .o -5.29188 k4.46484 9.03719

ACi

-I .o -0.0 I 472 +O. 20640 -195.41333 0.0 -0.0 I472 *O .20640 -195.41333 I .o -0.0 I472 +O. 20640 -195.41333

-I .o -0.0460 I 0.0 -2.04686 0.0 -0.04605 0.0 -2.04529 I .o -0.04608 0.0 -2.0437 I

Table 36. Numeratorroots for C variations. D; D;

233 ShortPeriod Mode Roots for the CmVariation

I I Re -10 -9 -8 -7 -1 -12 -1

-2

-3

hd,Values listed -4 besideroots -5

-6

Figure 92a

234 m

.20

I I "= 110 -.OS 905 .lo

C,, Values listed -905 beside roots

-.lo .

-.15

-.20

p i:gu re 92h

235 NUMERATOR ROOTS

STAB I L ITY DER1 VAT IVE REAL I MAGREAL I NARY REAL

-50.0 -3.90188 0.0 -24. I I478 4.43293 -20.0 -4.97207 0.0 -14.47813 5.79415 -15.0 -5.34358 0.0 - I 2.77020 6. I I235 -10.0 -5.9201 5 0.0 -10.89391 6.46725 - 5.237 -7.68330 0.0 - 7.93156 6.84425 0.0 -7. I7309 k2.36959 7.30864 5.0 -6.59500 +3. I5130 7.80707

ACi

-50.0 -0.01472 +O. 20640 -195.41333 -20.0 -0.0 I472 k0.20640 -195.41333 -15.0 -0.0 I472 +O. 20640 -195.41333 -10.0 -0.0 I472 k0. 20640 -195.41333 - 5.237 -0.0 I 472 +O. 20640 -195.41333 0.0 -0.0 I472 +O. 20640 -195.41333 5.0 '-0.01472 k0.20640 -195.41333

-50.0 -0.04594 0.0 -2.22363 -20.0 -0.0460 I 0.0 -2. I0095 -!5.0 -0.04602 0.0 -2.08 I77 -10.0 -0.04603 0.0 -2.06293 - 5.237 -0.04605 0.0 -2.04529 0.0 -0.04606 0.0 -2.02623 5.0 -0.04607 0.0 -2.00835

Table 37. Numeratorroots for C,. variations. 0:

236 5

I I I I I I I I I I __t__tc Re -5. -4 -3 -2 -1 1 2

-1

C Values listed Lq -2 beside roots

-3

-4

-5

pi,gure 9.3d

237 Phugoid Mode Roots for the CLqVariation

Im -20

.10

.os

-7 10 -.O 5

CL Values listed 9 45 beside roots

720

F i:gu r.e 9.3h

238 NUMERATORROOTS

STAB1 LlTY DERIVATIVE REAL I MAG I NARY REAL REAL

- 5.0 -7.73020 6.60700 ? I .83643 3.9168 -7.68330 0.0 6.84425- 7.93156 20.0 -5.52374 0.0 7.27783- I 0.37532 50.0 I075 -4.2 0.0 8.09894- 12.23076

- 5.0 -0.0 I 475 *0.20171 -204.55258 3.9168 -0.0 1472 k0.20640 -195.41333 20.0 -0.0 I465 +O. 2 I575 -178.92893 50.0 -0.0 1449 k0.237 I8 -148,18066

- 5.0 -0.04605 0.0 -2.04529 3.9168 -0.04605 0.0 -2.04529 20.0 -0.04605 0.0 -2.04529 50.0 -0.04605 0.0 -2.04529

Table 38. Numerator roots for E, variations.

239 Short Period Mode Roots for the ChVariation

in 5

0 -1.0 11) 4

I I I 1 I I I I I Re -5 -4 -3 -2 -1 -.1 2 -1

C Values listed Dq -2 beside roots

-3

-4 0 - 1.0 1.0 - 5

p i,gure 94a

240 Phugoid Mode Roots for the CbVariation

' .a0

m.10

5 -A5 C Values listed Dq beside roots

-1 720

Figure 94b

24 1 . " ...... I., I,

NUMERATORROOTS

STAB I L I TY DER1 VAT I VE REAL I MAGREAL I NARY REAL

-I .o -4.88199 0.0 - I 6.378345.21643 0.0 -7.68330 0.0 - 7.931566.84425 I .o -5.50557 k3.67822 9.51381

-I .o -0.0 I395 %.20645 -195.41487 0.0 -0.0 I472 3,20640 -195.41333 I .o -0.0 I 549 %.20634 -195.41179

-I .o -0.04605 0.0 -2.04529 0.0 -0.04605 0.0 -2.04529 I .o -0.04605 0.0 -2.04529

Table 39. Numeratorroots for CD variations. 9

242 Short Period Mode Roots for the Cmq Variatkn

- 12.43/

Re -10 -9 -8 -6 -S -4 -3 -2 -1 I 1 2 I \ t -l

Cm, Values listed beside roots

t-5

Figure 95a Phugoid Mode Roots for the +Variation

- -a5

I 1 ,Re -.lo -.os .os ,10

% Values listed t -mi beside roots

Figure 95b

244 NUMERATORROOTS

STAB I L I TY DERIVATIVE REAL I MAG I NARY REAL REAL

-50.0 -4.67832 0.0 -21.12196 4.22104 -30.0 -5.26804 0.0 - 14.83064 5.33863 -29.0 -5.3 I 297 0.0 -14.51478 5.40867 -28.0 -5.36047 0.0 -14.19800 5.48035 -20.0 -5.88484 0.0 -I I .58404 6. I1845 - 12.4337 -7.68330 0.0 - 7.93156 6.84425 - 5.0 -6.96597 Q.38 1.45 7.69596 0.0 -6.44403 k2.89559 8.35687 5.0 -5.96067 k3.21405 9.09494 10.0 -5.5 I777 53.4096 I 9.91393

ACi

-50 .O -0.01350 +o. 20000 -208.22433 -30.0 -0.01413 +O .20333 -201.40389 -29.0 -0.01416 +O .20350 -201.06287 -28.0 -0.01 4 I9 +O .20368 -200.72185 -20 .o -0 .O t 446 +O. 20506 - I 97.99364 - I 2.4337 -0.01472 +O .20640 -195.41333 - 5.0 -0.0 I498 fO .20774 - 192.87823 0 .o -0.01516 +O -20866 -191.17308 5.0 -0.0 I 534 +O. 20958 - 189.46792 10.0 -0. o i 552 +0.21053 - 187.76275

-50.0 -0.04605 0.0 -2.04529 -30.0 -0.04605 0.0 -2.04529 -29.0 -0.04605 0.0 -2.04529 -28 .O -0.04605 0.0 -2.04529 -20.0 -0.04605 0.0 -2.04529 - 12.4337 -0.04605 0.0 -2.04529 - 5.0 -0.04605 0.0 -2.04529 0.0 -0.04605 0.0 -2 04529 5.0 -0.04605 0.0 -2 04529 10.0 -0.04605 0.0 -2 04529

Table 48 Numeratorroots for C variations .

245

L NUMERATORROOTS

STAB I L I TY DERIVATIVE REAL I MAGREAL I NARY REAL

-0.5 -7.585 I 1 k2.06610 7.74177 -0. I -7.67989 &I .57952 7.35208 0.0 -7.70387 +I .42618 7.25523 0.1 -7.72796 +I .25 I35 7. I5859 0.25 -7.76429 G.92196 7.01404 0.4268 -7.68330 0.0 - 7.93156 6.84425 0.75 -6.59867 0.0 - 9.17542 6.53544 I .o -6.220 I 5 0.0 - 9.67839 6.29786 2.0 -5.22593 0.0 -! I. 17913 5.35625 5.0 -3. I3507 0.0 - 14.73205 2.47395

Act

-0.5 -0.0 I 495 G.21 144 158.94513 -0. I -0.0 I 484 a.20920 81 I .6855l 0.0 -0.0 I482 S.20868 0. I -0.0 I479 S.20816 -820.16606 0.25 -0.0 I476 S.20734 -330.61069 0.4268 -0.0 I472 S.20640 -195.41333 0.75 -0.0 I465 %.20472 -113.03072 I .o -0.0 I460 a.20345 - 85.83329 2.0 -0.0 I442 3.I9859 - 45.03736 5.0 -0.0 I 406 S. I8585 - 20.56030

-0.5 -0.0440 I 0.0 -2.30446 -0. I -0.04482 0.0 -2. I939 I 0.0 -0.04504 0.0 -2. ! 6597 0. I -0.04526 0.0 -2. I3790 0.25 -0.0456 I 0.0 -2.09556 0.4268 -0.04605 0.0 -2.04529 0.75 -0.0469 I 0.0 - I .95233 I .o -0.04765 0.0 - I .87948 2.0 -0.05 I42 0.0 -I .57923 5.0 -0.09587 0.0 -0.56084

Table41.Numerator roots for variations. &E 246 NUMERATOR ROOTS

STAB I L I TY DER I VAT I VE REAL I MAG.1REALNARY REAL

-0. I -2.0983 I k 8.05539 -3.58753 -0.05 - I .62690 k IO. 80785 -4.16199 0.0 -4.9259 I I 36.99857 0.0596 -7.68330 0.0 -7.93 I56 6.84425 0. I -6.67433 k 2.63517 4.82778 0.5 -4.76 I77 k 3.95572 I .29735

-0. I -0.0 I 590 k0.20626 -195.52065 -0.05 -0.0 I553 k0. 20630 -195.48703 0.0 -0.01516 k0.20635 -195.45341 0.0596 -0.0 I 472 k0.20640 -195.41333 0. I -0.0 I442 k0.20644 -195.38617 0.5 -0.01 145 +O. 20676 -195.11722

-0. I -0.03843 0.0 -2.05 I90 -0.05 -0.0408 I 0.0 -2.04983 0.0 -0.04320 0.0 -2.04776 0.0596 -0.04605 0.0 -2.04529 0. I -0.04798 0.0 -2.0436 I 0.5 -0.0673 I 0.0 -2.02679

Table42. Numerator roots for C, variations. &E

247 NUMERATORROOTS

STAB I L 1 TY DERIVATIVE REAL I MAG I NARY REAL REAL

-15.0 -5,258I5 0.0 -33. I889I 29.6762 I -10.0 -5.30278 0.0 -27.27432 23.80632 - 5.0 -5.44662 0.0 -19.44515 l6.lZI06 - 3.0 -5.66806 0.0 -15.01042 I I .9078I - I .283 -7.68330 0.0 - 7.93156 6.84425 - 0.75 -6.66992 5.51054 4.56927 0.0 -3.963 I7 k4.21546 - 0.84424 0.5 -2.55070 k6.96412 - 3.66917 1.0 -2.25392 k9. I0942 - 4.26274

-15.0 -0.0 I 522 +O. 2085 I -2239.746 -10.0 -0.01519 *O. 20839 - 1494.563 - 5.0 -0.01512 +-0.20807 - 749.381 I3 - 3.0 -0.01503 +O. 20767 - 451.30835 - I .283 -0.0I472 k0. 20640 - 195.41333 - 0.75 -0.0 I 434 +-0.20486 - 115.97763 0.0 -0.00003 0.0 -4.23938 0.01042 0.5 -0.0 I 682 k0.21467 70.32293 I .o -0 .O I602 kO.21 158 I 44.83956

-15.0 -0.04267 0.0 - 2. I5785 -10.0 -0.04282 0.0 - 2. I 5263 - 5.0 -0.04327 0.0 - 2. I3696 - 3.0 -0.04388 0.0 - 2. I 1600 - I .283 -0.04605 0.0 - 2.04529 - 0.75 -0.04899 0.0 - 1.95623 0.0 0.01042 0.0 - I 5.26480 0.5 -0.03486 0.0 - 2.47223 I .o -0.03832 - 2.32224

Table 43. Numeratorroots for C variations. msE Sort hriod and mtgoid Roots kr c.g. Vanatii

-1

-2

-3

Figure 96 249 c.g. LOCATION (% m.a.c.1 SHORT PERIOD ROOTS SHORT PERIOD / PHUGOID PHUGO I D ROOTS

Rea I lmag i nary Rea I Rea I I magi nary Rea I I magi nary Rea I

0.0 - 4.6345k7.5094 -0,0136 k0. 1865

26.4 - 4.0848k4.3679 -0.0136 k0. I801

42.5 - 5.3421 -2. I568 -0.0182 +O. I I75

45.0 - 6.1725 -I. 1979 -0.0763 0.02000

47.0 - 6.6547 -0.4796 -0.3432 0. I3957

47. I - 6.6764 -0,40098k0.06724 0.14501

47.5 - 6.7615 -0.36069k0.19146 0. I6725

48.5 - 6.9626 -0.26908k0.30056 0.22966

50.0 - 7.2381 -0.15946k0.34295 0.3566 I

52.5 - 7.6453 -0.06470k0.31623 0.69487

55.0 - 8.0056 -0.03672k0.28517 I. I I858

100.0 - I I .5233 -0.01703k0.23765 7.3668 I

Table 44. Short Period and Phugoid Roots for c.g. Variations. Short Period Mode Roots for the St Variation

2 ARt =constant 1

__f_c Re 1

-1 St values listed beside roots as percent of -2 original value

-3

-4

-5

-6

Figure 97a

25 1 Phugoid Mode Roots for the St Variation

20 200

ARt = constant

St values listed beside roots as percent of original value

200fo

Figure 976

252 Short Period Mode Roots for the It Variation

1 I I I I Re -8 -7 -6 -5 -4 -3 -1 -1

-1 It values listed beside mots as percent of original value -2 25

-3

-4

-5

Figure 98a

253 Phugoid Mode Roots forthe It Variation

15 45

" -05

It values listed beside roots as percent of original value

200:J 100

720

Figure 98b

254 Short Psrlod Mode Roots for the vt Ylrlatlon

I I I .I I I I I I 7 6 5 4 3 2 1

vt values listed beside roots

Figure 99a

255 Phugold Mode Rootsfor the I+ Variation

9 values listed t beside roots -05

-t - 9.10

- - 45

-- 720

Figure 99b

256 % Values listed beside roots t -lp

t" Figure 1OOa 257 Spiral Mode Roots for theVariation % - -.oos

3.0 Q4 908 -2.0 -LO I L. I ~n a, " I ,Re I - I I I I -p20 w.015 70IO :005 ,005

C values listed ye beside roots Figure 100b

Roll Mode Roots for theVariation 93 Im

-10 3.0 a I I I I* I I 40 I I I Re - 14 -13 -12 -11 1

Figure 100c I

NUMERATORROOTS

STAB1 LlTY DERIVATIVE REAL I MAG I NARY REAL REAL

-5.0 .02272 0.0 - I 2.58832 -115.40198 -2.0 .02272 0.0 - I 2.58832 -115.40198 -I .o .02272 0.0 - 12.58832 -115.4Ol98 - ,308 .02272 0.0 - I 2.58832 -115.40198 .6 .02272 0.0 - I 2.58832 -115.40198 1.5 .02272 0.0 - 12.58832 -115.40198 2.5 .02272 0.0 - 12.58832 -115.40198 3.0 .02272 0.0 - I 2.58832 -115.40198

-5.0 0.0 0.0 -6.8 I839 9.22719 -2.0 0.0 0.0 -5.8646 I 9.69719 -I .o 0.0 0.0 -5.55962 9.86678 - .308 0.0 0.0 -5.29036 9.86759 0.6 0.0 0.0 -5.08575 IO. I5225 I .5 0.0 0.0 -4.82705 I 0.32069 2.5 0.0 0.0 -4.54640 10.51463 3.0 0.0 0.0 -4.40880 10.61431

-5.0 -2. I I867 0.0 - .I3344 - I 2.65862 -2.0 - .41643 f .33074 - I 2.65543 -I .o - .I7978 f .50051 -12.65501 - .308 - .01387 k .52896 - 1 2.65493 0.6 . I9895 f .49324 - 12.65086 I .5 .4 I 204 k ,33633 - I 2.64990 2.5 .27717 0.0 I .0204 I - 12.64890 3.0 .2 I422 0.0 I .32018 - 1 2.04842

Table45.Numerator roots for C variations. yB

259 ” 5

I a ’1 rnm I m 1 -7 4 -s -4

C, Values listed P beside mats

“-5

Figure 101a

260 Spiral Mode Roots for the C Variation lp

-1.0 +a9 --Q5

CI e values listed beside roots

Figure 101b

Roll Mode Roots for the C Variation t3 Im “1

1.0 7089 1.0 1.5 I - 0.5 I n I t- t rn I 1 I - I w I“ m- I L I *Re -14 - 13 -12 -11 -10 -9 1

-- -1 Figure 101c NUMERATORROOTS

STAB1 LlTY DERIVATIVE REAL I MAGREAL I NARY REAL

-I .o .02272 0.0 -12.588 -I 15.42 - .089 .02272 0.0 -12.588 -I 15.42 .5 .02272 0.0 -12.588 -I 15.42 I .o .02272 0.0 -12.588 -I 15.42 I .25 .02272 0.0 - 12.588 -I 15.42 I .5 .02272 0.0 -12.588 -I 15.42

-I .o -21.8216 0.0 31.9475 0.0 - .089 - 5.2904 0.0 9.8676 0.0 .5 .54328 k19.03747 0.0 I .o - .96334 k26.74 I 0.0 I .25 - I .71665 529.8279 0.0 I .5 - 2.4700 k32.6067 0.0

-I .o . I6499 +I .8909 -13.0132 - .089 - .01387 k .52896 - 12.65493 .5 - I .5533 0.0 I .268l I -12.3981 I .o -2.2648 0.0 -12.16725 I .74875 I .25 -2.57468 0.0 I .9370 - I 2.04559 I .5 -2.8686 I 2. 0.0 I0452 -I I .91919

Table 46. Numeratorroots for CQ variations. B Dutch Roll Mode Roots for the %Variation

-25

.I!

.064t

C C Values listed "P beside roots

.064 6

.2 5

Figure 102a 263 Spiral Mode Roots for the Variation Im c"p A -* 1

71 0 -.Os -.o 2 1 I 1 m I c. I c.1 I 0.0 AM6 I I 1 1 w 1 " 1 -1 I U I Re -7 -6 -5 -4 -3 -2 -1 *?bo 1

Cn, values listed t -l beside roots Figure 102b

Roll Mode Roots for the Cn Variation P Im A

I I I I I I 1 I #@ Re -5 -4 -3 -2 -1 1

- "1 Figure 102c NUMERATORROOTS

STAB1 LlTy DERIVATIVE REAL I MAG I NARY REAL REAL

- .05 .02272 0.0 - 12.588 -I 15.402 - .02 .02272 0.0 -12.588 -I 15.402 0.0 .02272 0.0 - 12.588 -I 15.402 ,06455 .02272 0.0 - 12.588 -I 15.402 .I5 .02272 0.0 -12.588 -I 15.402 .25 .02272 0.0 - 12.588 -I 15.402 .4 .02272 0.0 - 12.588 -I 15.402 .5 .02272 0.0 - 12.588 -I 15.402 .6 .02272 0.0 - 12.588 -I 15.402

- .05 I I .0918 0.0 -6.4556 0.0 - .02 10.81052 0.0 -6. I7427 0.0 0.0 10.61768 0.0 -5.98 I44 0.0 .06455 9.86759 0.0 -5.29036 0.0 .I5 8.9962 0.0 -4.3599 0.0 .25 7.6478 0.0 -3.01 16 0.0 .4 4.3478 0.0 .28833 0.0 .5 2.3181 I 3-3.47434 0.0 .6 2.3181 I 3-5.31620 0.0 0.0

- .05 - ,08494 3- .60700 - 12.66270 - .02 - .0664 I t- .5886 - 12.6598 0.0 - .05405 2 .5756 - 12.65785 0.06455 -.01387 3- .52896 - 12.6549 0. I5 .03877 3- .45654 - 12.64358 0.25 . I0076 3- ,34167 - I 2.63429 0.4 .00538 0.0 .38233 - 12.62066 0.5 -. 12719 0.0 .63929 -12.61 I77 0.6 -. I9755 0.0 .834 I9 - 12.60305

Table 47. Numeratorroots for Cn variations. B Dutch Roll Mode Roots for the CypVariation

Im 4 - -3.5

- -3.Q

- -25

- -m

"l.5

--LO

"0.5

CypValues listed -95 beside roots -lD

-15

-2B

I-25

-3.0 7.0w5 -35 Figure 103a 266 I Spiral Mode Roots for the Cyp Variation :" w.005

-z5 zo I I I I I 8 I "Re -.020 -,OlS -.a373 -.o 10 ,oos .005

Cyp values listed -405 beside roots F i gure 103b Roll Mode Roots for the CrpVariatlon C"

- 7.5 -.a373 7.0 1 Ir 1 T' - 1- I W - I- I I' Re - 14 - 13 -12 - 11 1

Figure 103c

I NUMERATORROOTS B STAB1 LIN DERIVATIVE REAL I MAG 1 NARY REAL REAL

-7.5 0.02177 0.0 -15.624 - 98.470 - .0373 0.02272 0.0 -12.588 -I 15.402 7.0 0.02373 0.0 -10.552 -133.746

-7.5 0.0 0.0 -5.2904 9.8676 - .0373 0.0 0.0 -5.2904 9.8676 7.0 0.0 0.0 -5.2904 9.8676

J, -7.5 0.25191 a.45598 -13.187 - .0373 -0.0 I387 S.52896 -12.655 7.0 -0.29057 a.45964 -12.102

Tab I e 48 . Numerator roots for Cy variations. P

268 Dutch Roll Mode Roots for the C Variation 'P

CI values listed P beside roots - "lD

- "ls -"

- "25

Figure 104a

269 0.0 -.os -. 2 -.4707 -4.0 I 1 I I a I I t- I I I- I I I I w I -vu c Re I - I 716 714 “12 710 ~08 -.06 -.04 -*02 -1.0 .02

t-,02 Clp values listed beside roots Figure 104b

Roll Mode Rootsfor the ClpVariation Im

“10

-3.0 -2.0 -1 .o -4707 -.2 -.os I I I L. I 1- I -I .P I I -w I I VI I -I I- I Re I *” .02 - 90 -80 -70 - 60 -50 =40 - 30 -20 -10 10

- “10 Figure 104c NUMERATORROOTS

STAB1 LlTY DERIVATIVE REAL IMAG I NARY REAL REAL

-4.0 .00270 0.0 -107.822 -I 15.135 -3.0 .00360 0.0 - 79.796 -I 16.713 -2.0 .00539 0.0 - 53.132 -I 16.934 -I .o .0 I 076 0.0 - 26.599 -I 17.025 - .4708 .02272 0.0 - 12.588 -I 17.032 - .2 .05256 0.0 - 5.4426 -I 17.066 - .05 . I7927 0.0 - I .5962 -I 17.0724 0.0 .49080 0.0 - .58346 -I 17.0743 .02 I. I5326 0.0 - .3527 I -I 17.0769

-4.0 -5.29036 0.0 9.8676 0.0 -3.0 -5.29036 0.0 9.8676 0.0 -2.0 -5.29036 0.0 9.8676 0.0 -I .o -5.29036 0.0 9.8676 0.0 - .4708 -5.29036 0.0 9.8676 0.0 - .2 -5.29036 0.0 9.8676 0.0 - .05 -5.29036 0.0 9.8676 0.0 0.0 -5.29036 0.0 9.8676 0.0 .02 -5.29036 0.0 9.8676 0.0

-4.0 - .03070 k .I8119 - 105.960 -3.0 - .03039 k .20996 - 79,514 -2.0 - .02969 k .25798 - 53.068 -I .o - .02673 + .36562 - 26.628 - .4708 - .01387 k .52896 - 12.6549 - .2 ,03660 k .79875 - 5.59728 - .05 .3534 I k I .20658 - 2.26388 0.0 .70033 +I .30297 - I .63539 .02 I .00127 ?I .I7834 - I .45731

Table 49. Numeratorroots for CA variations. P

27 1 Dutch Roll Mode Roots for the Cnp Variation m 5

Cnp values listed 2 beside roots

I I s 1 -2

-1

-2

-3

-4

-5 Figure 105a 272 Spiral Mods hotsfor tbo Cnp Varlrrtlelr

0.25 0.15 -.0292 -05 -1.0 I I I * I I n ~. I I -* I I I U I w I WW , ,Re -.06 -.OS -.04 - ,03 -802 -.Ol .01

Cnp values listed t -.O1 beside roots Figure 105b Roll Mode Roots for the CllpVarlatian t 1

.. NUMERATORROOTS

STAB1 LIP( DERIVATIVE REAL I MAG 1 NARY REAL REAL -I .o .0 I 584 0.0 -I 9.1345 -I 10.497 - .5 .O 1 877 0.0 -15.6501 -1 13.984 - .02923 .02272 0.0 - I 2.58832 -I 15.402 .I5 .02472 0.0 -I I .4622 I -I 18. I78 .25 .02599 0.0 -10.84717 -I 18.794 .30 .02667 0.0 - IO. 54233 -I 19.099 .4 I .0283 I 0.0 - 9.87778 -I 19.766

-I .o 9.86759 0.0 -5.29036 0.0 - .5 9.86759 0.0 -5.29036 0.0 - .02923 9.86759 0.0 -5.29036 0.0 .I5 9.86759 0.0 -5.29036 0.0 .25 9.86759 0.0 -5.29036 0.0 .30 9.86759 0.0 -5.29036 0.0 .4 I 9.86759 0.0 -5.29036 0.0

1cI

-I .o .04698 k. 43785 - 18.45365 - .5 .02029 k. 48044 - I 5.47606 - .02923 -.01387 k. 52896 -12.655 .I5 - .0337 I k.55521 - I I ,56659 .25 -.04523 k .56965 -10.95871 .30 -.05145 f .57728 - IO. 65385 .4 I - ,06638 k.59510 - 9.98065

Table 50. Numerator rootsfor Cn variations. P

273 Dutch Roll Mode Roots for the %Varktion

,2103

Cy, values listed beside roots

- 4.0 Figurc 27 5 Spiral Mode Roots forthe CyrVarlation Im

4.0 0.2103 -4.0 I ne I I "8 v r I ,I .L Re -.020 -.015 -.010 -.m ,005

Cyrvalues listed beside roots

Figure 106b

Roll Mode Roots for the Cy,Variatlon

I 'I I I'

i I t" Figure 1Q6c NUMERATOR ROOTS

STAB I L I TY DERIVATIVE REAL I MAGREAL I NARY REAL

-4.0 .0 I 953 0.0 -12.589-136. I67 .2103 .02272 0.0 - 12..588 -I 15.402 4.0 .02666 0.0 - 12.5767 - 99.875

-4.0 0.0 0.0 -5.8982 10.5344 .2103 0.0 0.0 -5.2904 9.8676 4.0 0.0 0.0 -4.8256 9.4618

-4.0 -.Of387 k. 52896 -12.655 .2 I03 -.01387 +.52896 -12.655 4.0 -.01387 +.52896 -12.655

Table 51. Numerator rootsfor Cy, variations. Dutch Roll Roots for the qrVariation

I . I I -4 -3 -2 -1

Clr values listed -1 beside roots

-2

-4

Figure 107a

278 Short Mode Roots for the C,rV.*latkm Ah

-3.0 - 2.0 0 .o .0959 2,o 40 I a rI I Re I 1- \. 1- - *I - 1- -15 - 1.0 -a5 Qs ip

Clr values listed beside roots

"-Qs

Figure 107b

Roll Mode Roots for the Clr Variation

Figure 107c

! NUMERATORROOTS

STAB1 LIN DERIVATIVE REAL I MAG INARY REAL REAL

-5.0 -I .8946 0.0 -10.261 I7 -I 16.459 -3.0 - I .03898 0.0 -I I .2754 -116.30101 -2.0 - .66903 0.0 -I I .72497 -I 16.0221 0.0 - .00695 0.0 - 12.5467 I -I 15.8325 0.0959 0.02272 0.0 -12.588 -I 15.402 2.0 0.57860 0.0 - 13.29262 -I 14.901 4.0 I . I0784 0.0 - 13,98292 -I 14.740 5.0 I .35548 0.0 -14.31 I36 -I 14.659 7.0 I .82262 0.0 - I 4.94063 -I 14.497

-5.0 -289.2 I 0.0 0. I2108 0.0 -3.0 -174.06 0.0 .24276 0.0 -2.0 -I 16.57 0.0 .39352 0.0 0.0 - 7.75308 0.0 6.85029 0.0 0.0959 - 5.29036 0.0 9.8676 0.0 2.0 - .52488 0.0 114.8989 0.0 4.0 - .29323 0.0 229.9438 0.0 5.0 - .24677 0.0 287.5357 0.0 7.0 - . I9355 0.0 402.759 0.0

-5.0 -.01387 2. 52896 - 12.65493 -3 .O -.01387 f ,52896 -12.65493 -2.0 -.01387 f .52896 - 12.65493 0.0 -.01387 2.52896 -12.65493 .0959 -.01387 2. 52896 - I 2.65493 2.0 -.01387 2. 52896 -12.65493 4.0 -.01387 f .52896 -12.65493 5.0 -.01387 f .52896 - I 2.65493 7.0 -.01387 f .52896 - 12.65493

Table 52. Numerator roots for Cg variations. r

280 Roots for the Cnr Variatkn

Figure 108 NUMERATORROOTS B

STAB1 LIT" DERIVATIVE REAL I MAG I NARY REAL REAL

-I .o -.03548 0.0 - I 2.4639 - 128.568 - .9 - .02947 0.0 - I 2.4762 - I 27.287 - .6 -.01077 0.0 -12.51447 -123.444 - .55 -.00755 0.0 -12.521 I1 - I 22.803 - .5 -.00430 0.0 - 12,5278 -122. I63 - .4 .00229 0.0 -12.5415 -120.881 - .2 .O I586 0.0 - 12.5697 -I 18.317 - .09924 .02272 0.0 -12.588 -I 15.402 - .07 .02496 0.0 -12.589 -I 14.650 - .05 .02638 0.0 -12.592 -1 14.394 0.0 ,02996 0.0 -12.599 -1 13.752

-I .o 4.93162 -I I .748 0.0 - -9 5.28858 -10.840 0.0 - .6 6.6698 - 8.3875 0.0 - -55 6.94212 - 8.0225 0.0 - .5 7.2278 - 7.6709 0.0 -. 4 7.8399 - 7.0084 0.0 - .2 9.2274 - 5.8466 0.0 - .09924 9.8676 - 5.2904 0.0 - .07 IO. 2420 - 5.2042 0.0 - .05 10.4055 - 5.1 128 0.0 0.0 10.823 - 4.893 0.0

-1 .o -.01387 f .52896 - I 2.65493 - .9 -.01387 f .52896 - 12.65493 - .6 -.01387 f ,52896 - 12.65493 - .55 -.01387 k. 52896 -12.65493 - .5 - .O I387 f .52896 - 12.65493 - .4 -.01387 k. 52896 - I 2.65493 - .2 -.01387 f .52896 - I 2.65493 - .09924 -.01387 2.52896 - 12.65493 - .07 -.01387 &. 52896 - 12.65493 - .05 -.01387 f .52896 - 12.65493 0.0 -.01387 k. 52896 - 12.65493

Tab le 53 . Numeratorroots for $, variations. r 282 NUMERATORROOTS

B STAB I L I TY DERIVATIVE REAL I MAG I NARY REAL REAL

-1 .o .02418 0.0 -12.55031 20.22192 - .75 .02377 0.0 - 12.55567 27.3650 - .5 .02333 0.0 -12.56214 4 I .64652 - .25 .02272 0.0 - 12.57005 84.47888 0.0 .02297 0.0 - 12.58048 .I874 .02272 0.0 - 12.59078 -115.40684 .25 .02264 0.0 -12.59431 - 86.7928 I .5 .02232 0.0 -12.61223 - 43.95034 -75 .02202 0.0 - 12.63853 - 29.64899 I .oo .02 172 0.0 -12.68051 - 22.46934

4 -I .o 0.0 0.0 -6.57089 7.74879 - .75 0.0 0.0 -6.26998 8. I6388 - .5 0.0 0.0 -5.98623 8.5961 I - .25 0.0 0.0 -5.7191 I 9.04498 0.0 0.0 0.0 -5.46802 9.50987 . 187 0.0 0.0 -5.2899 I 9.86846 .25 0.0 0.0 -5.23228 9.99012 .5 0.0 0.0 -5.01 I16 IO. 48499 .75 0.0 0.0 -4.80392 IO. 99373 I .oo 0.0 0.0 -4.60975 I I .51555

-1 .o -.31014 f .4297 -12.61691 - .75 -.24760 f .46833 - 12.62560 - .5 -. 18515 k.49616 -12.6341 I - .25 - . I2278 f.51497 - 12.64245 0.0 -.06049 f .52576 - 12.65062 .I874 -.01386 k. 52892 - 12.65664 .25 .00171 f .52906 - 12.65863 .5 .06384 f .52503 - I 2.66649 .75 . I2589 f.51353 - I 2.67420 I .oo . I8787 f .49407 - 12.68 I76

Table 54 . Numeratorroots for variations. 56.R 283 NUMERATOR ROOTS B STAB I L I TY DERIVATIVE REAL IMAG I NARY REAL REAL -3.0 - .48468 0.0 -168.74511 25.38497 -2.0 - .57868 0.0 -153.71383 15.71 175 -I .o - 1.17612 0.0 -136.61884 4.47829 0.000 I - I 2.32283 0.0 -115.75927 .03009 .o I475 - I 2.59078 0.0 -115.40684 .02272 .05 - 13.24437 0.0 -114.55102 .00605 .I5 -15.16965 0.0 -I 12.0589 - .03437 .50 -22.91592 0.0 -102.37878 - . 12576 I .o -39.3933 0.0 - 83.20187 - .I9315 2.0 -58.6349 '4 I .37965 - .25419 3.0 -55.98877 +62.391 19 - .28250

4 -3.0 - .72027 +3. I5419 0.0 -2.0 - .72763 +3. I7672 0.0 -I .o - .74972 k3.24328 0.0 0.0001 - IO. 29436 0.0 892.35499 0.0 .o I475 - 5.28991 0.0 9.86846 0.0 .05 - 2.69752 0.0 3.05337 0.0 ; 15 - .41 106 + 1 ,96457 0.0 .50 - .6172 k2.81787 0.0 I .o - .66137 k2.967 I2 0.0 2.0 - ,68346 k3.03875 0.0 3.0 - .69082 k3.06222 0.0 * -3.0 2.51434 0.0 22.92565 - 2.50023 -2.0 3.29649 0.0 10.61332 - 2.78665 -I .o I .70459 k3.08103 - 4. I0259 .000 I - .01042 + .58161 - I 2.49040 .o I475 - ,01386 * .52892 - I 2.65664 .05 - .02153 f .38078 - 13.05783 .I5 - .48245 .40343 - 14.20363 .50 - I .09987 .94997 - I 8.26858 I .o - I ,42295 I .23465-24. I3854 2.0 -I .68133 I .47382-35.93603 3.0 - I .79385 I .58414-47.75055

Table 55. Numeratorroots for C % variations.

284 NUMERATOR ROOTS

STAB I L I TY DERIVATIVE REAL I MAG I NARY REAL REAL

-. 12 - I 2.47042 0.0 .026 I7 -209.71058 -.IO - 12.49848 0.0 .02532 -174.92893 -.08 -12.54195 0.0 .02407 -140.13146 - .0658 - 12.59078 0.0 .02272 -115.40684 -.04 - I 2.78700 0.0 .O I796 - 70.374.82 -.02 - 13,49442 0.0 .00657 - 34.90327 -.005 -I I. 14076 f4.97905 -.04503 ,005 -10.91735 0.0 .2 1055 5.75643 .01 - I I .38982 0.0 .09625 15.03131 .02 -I I .761 14 0.0 .05942 32.81567

0 -. 12 -6.5992 I 0.0 15.66940 0.0 -. IO -6.20569 0.0 13.61845 0.0 -.08 -5.7 I878 0.0 I 1.4741 I 0.0 -.0658 -5.2899 I 0.0 9.86846 0.0 -.04 -4. I9447 0.0 6.63494 0.0 -.02 -2.62335 0.0 3.40639 0.0 -.005 - .23002 * 2.27629 0.0 .005 - .64437 f 3.77708 0.0 .o I - .85155 f4.32203 0.0 .02 - I .2659 I k5.22012 0.0

-. 12 -.0335 f .55252 - 12.57804 -.IO - .0287 I f .54692 -12.59717 - .08 -.02155 * .53837 -12.62581 - ,0658 -.01386 f .52892 - I 2.65664 - .04 .O I380 * .4925 I - 12.76809 -.02 ,08232 * .37954 - I 3.04833 - .005 -.28175 -14.63102I. I6989 .005 -.78637 f .85217 - 9.87907 .01 - .37960 = * .79577 -1 I 626536 .02 -.21060 * ,7084 I - I I .88973

Table 56. Numeratorroots for C;, variations. 6R 285 Dutch Roll Mode Roots for I,Variatkn

a- 0-

I 1 lo 1- I I Ru -3 -2 -1 1

I, V~I- 11~t.d beside mots as percent of original values

A-5 Figure 109a

286 A- with inertia effects 0 -without inertia effects I, values listed beside roots as percent of original value Figure 109b

Roll Mode Roots for Iv Variation

all values I . 1 I " I I >R8 -13 -1 2 -1 1 0

Figure 109c Dutch Roll Mode Rootsfor S,, Variation with Camtant AR

150

I i I I -3 -2 -1

50! -1 Sv Values listed beside roots as I percent of original Value

-2

150

Figure 110a 288 Spiral Mode Roots for S,Variation

50 75 100 In a a I I * Re I- F I I I I - -.30 - .2S - .20 -.Is -.lo - .05 0.0

S, vakes listed beside roots as percent of original values Figure 110b

Roll Mode Rootsfor S, Variation

Im t VV - 13 -11 lo Figure 110c Dutch Roll Mode Root for the Dihedral Variation !m

4 ‘3 -7 3

Dihedral angle in degrees 2 listed beside roots

I -2 -1

-1

-2

-7 -3 w35

-4 Figure 11 290 Splral Mode Roots forDihedral Varlrtion

7 5

Dihedral angk in degrm listed beside roots Figure 11lb

Roll ModeRoots for Dihedral Variation

Figure 11IC Dutch Roll Mode Roots for Dihedral nd S, Writtion

I I t I R -3 -2 -1

Values of dihedral in degreesand -1 Sv as percent of original value are givenbeside roots

-2

-3 5/ 1.73 100 -4 -5 Figure 112a

292 Spiral Mode Roots for Dihedral and S, Variation

Im t 11) 5 1.73 3 I -I* I ' -75 I I Re -.OM - .o;o '590;s - .O$- - ,008 - .ow 0.0

Values of dihedral in degrees and S, as percmt of original value are givenbeside roots Figure 112b

MI Mode Rootsfor Dihedral and SVVariation f 5 1.73s 1 I I Re I -"0 13 - 12 - 11 0

Figure 112c

" 294 Bcde Hot for -Aa 6E

.01 .02 .os A .2 10.5 2~ 50 10.0 2ao 50.0 lo~o200.0 SOQO 1000 Frequency G.ad/sec) Figure 114

295 Frequency (rad /sed Figure 115

296 P Bode Plot for - 6R

.001 .002 .005 .01 .02 .os .1 .2 .5 LO 21) 50 10.0 20.0 50.0 IM) Frequency (rad /sec ) Figure 116

297 @ 50- - Bode Plot for - 6R 40.-

30--

20- - a3 I g 10" 'E u 0" -0 z* 10" 0, 20" '0 3 +r 30.- n E a 40"

SO"

60- -

70- -

200- -

100- -

So--

Om- iL M 4, '0 -*- @m-- 4 41"-Q E-200. -

I I I I I I I I I I 1 I 1 I I I . I . I . .001 .002 .005 .a .02 .os .1 .2 .5 1.0 2.0 5.0 10.0 20.0 50.0 Loo

Frequency (rad kc) Figure 117 298 .001 .005.002 n1 .02 .os .l 9 5 Lo 2.0 sn l0D a.0 w.0 100 Frequency (rad/ sed Figure 118

299

REFERENCES

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21. Gilruth, R.;R. and White, M. D.: "Analysis and Predictionof Longitudinal Stabilityof Airplanes". NACA TR-711, 1941.

22. Multhopp, H.: "Aerodynamics ofthe Fuselage". NACA TM-1036, December 1 942.

23. Greenberg,Harry; and Sternfield, Leonard: "A TheoreticalInvestigation ofLongitudina! Stability of Airplanes with Free Controls Including Effectof Friction in Control System". WR L-430, February 1944.

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25. Greenberg, Harry; and Sternfield, Leonard: "A TheoreticalInvestigation ofLongitudinal Stability of Airplanes with Free Controls Including Effectof Friction in Control System". NACA TR-791, 1944.

26. Silverstein, Abe; and Katzoff, S.: "Aerodynamic Characteristicsof HorizontalTail Surfaces". NACA TR-688.

27. Campbell,John P.; andMcKinney, Marion 0.: "Summary of Methods for Calculating Dynamic LateralStability and Response and forEstimating LateralStability Derivatives". NACA TR-1098, 1952.

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302 29. Shortal,Joseph A,; andDraper, John W.: "Free-Flight-TunnelInvestigation of the Effect of the Fuselage Length and the AspectRatio and Size of the Vertica 1 Ta i I on Latera 1 Stabi I ity and Control". WR L-487, Apri I 1943.

30. McKinney, Marion O., Jr.:"Experimental Determination of theEffects of Dihedral,Vertical-Tail Area,and LiftCoefficient onLateral Stability and ControlCharacteristics". NACA TN-1094, July 1946.

31. Queijo, M. J. : "Theoretical SpanLoad Distributions andRoI I ing Moments for Sideslipping Wings of Arbitrary Plan Form in Incompressible Flow". NACA TR-1269, 1956.

32. Hoerner,Sighard: "Forces and Moments on a Yawed Airfoil". NACA TM-906, August 1939.

33. Goodman, Alex; and Fisher,Lewis R.: "Investigationat Low Speeds of the Effect of AspectRatio and Sweep on Rolling Stability Derivatives of Untapered Wi ngs". NACA TR-968, 1950.

34. Booth,Katherine W.: "Effectof Horizontal-tail Chord onthe Calculated Subsonic Span Loadsand StabilityDerivatives of Isolated Unswept TailAssemblies in Sideslip and Steady Roll". NASA MEMO 4-1-59L, March 1959.

35. Michael,William H., Jr.:"Analysis of the Effects of Wing Interference onthe Tail Contributions to the Rolling Derivatives". NACA TR-1086, 1952.

36. Klawans, Bernard B; and White,Jack A.: "A Method Utilizing Dataon the Spiral,Roll-Subsidence, and DutchRoll Modes forDetermining Lateral Stabi I ity Derivatives for FI ight Measurements". NACA TN-4066, August 1957.

37. Bird, John P.: "Some Theoretical Low-SpeedSpan LoadingCharacteristics of Swept Wings inRoll and Sideslip". NACA TN-1839, March 1949.

38. Queijo, M. J.; and Jaquet,Byron M.: "CalculatedEffects of Geometric Dihedralon the Low-Speed RollingDerivatives of Swept Wings. NACA TN-1732, October 1948.

39. Goodman, Alex; and Adair,Glenn H.: "Estimation of the Damping in Roll of Wings throughthe Normal Flight Range of Lift Coefficient". NACA TN-1924, July 1949.

40. Wolhart,Walter D.: "Influenceof Wing and Fuselageon theVertical Tail Contribution to the Low Speed RoIIi,ng Derivative of MidwingAirplane Models with 45O Sweptback Surfaces". NACA TN-2587, December 1951.

303 41. MacLachlan,Robert; and Letko, WI! I iam: "Correlation of Two Experimkntal Methods of Determining the RoI I ing Characteristics of UnsweptWings". NACA TN-1309, May 1947.

42. Michael,William H., Jr.:"Analysis of theEffects of Wing Interference onthe Tail Contributions to theRolling Derivatives. NACA TN-2332, Apri 1 1951.

43. Goodman, Alex;and Brewer, Jack D.: "Investigationat Low Speeds of the Effect of AspectRatio and Sweep onStatic and Yawing Stability Deriva- tives of UntaperedWings". NACA TN-1669, August 1948.

44. Bird,John D.; Fisher,Lewis R.; andHubbard, Sadie M.: "Some Effects of Frequencyon the Contribution of a VerticalTail to theFree Aerodynamic Damping of aModel Oscillatingin Yaw". NACA TR-1130, 1953.

45. Zimmerman, Charles H.: "An Analysis of LateralStability in Power-Off Flightwith Charts for Use in Design". NACA TR-589, 1937.

46. Campbell,John P.; and Goodman, Alex: "A SemiempiricalMethod for Esti- matingthe Rolling Moment Due to Yawing ofAirplanes". NACA TN-1984, December1949.

47. Cotter Wi I I iam E., Jr.: "Summary and Analysis of Dataon Damping in Yaw and Pitch for a Number of AirplaneModels". NACA TN-1080, May 1946.

48. Campbe I I, John P.; andMathews, Ward 0.: "ExperimentalDetermination of the Yawing Moment Due to YawingContributed by the Wing, Fuselage, and Ve rt icalTail of a MfdwingAirplane Model". WR L-387, June 1943.

49. Pearson,Henry A.; andJones, Robert T.: "TheoreticalStability and Control Characteristics of Wings withVarious Amounts of Taperand Twist". NACA TR-635, 1938.

50. Etkin,Bernard: Dynamics of Flight,Stability and Control.John Wiley andSons, Inc., 1959.

51. Goett,Harry J.; Jackson, Roy P.; andBelsley, Steven E.: "Wing-Tunnel Procedure for Determination of Critical Stability andControl Characteris- ticsof Airplanes". NACA TR-781, 1944.

52.Langley Research Staff (compiled by Toll, Thomas A.1: "Summary of Lateral- Contro I Research". NACA TR-868, 1947.

53. Faber,Stan!ey; and Crane, Harold L.: "A LongitudinalControl Feel System for In-flight Researchon Response Feel". NASA TN 0-632, January 1961.

304 54. FederalAviation Agency:"Federal Aviation Regulations, Part 23, AirworthinessStandards Normal, Utility, and AcrobaticCategory Air- planes". 1967.

55. Barber,Marvin R.; Jones,Charles K.; Sisk, Thomas R.; andHaise, Fred W.: "An Evaluation of theHandling Qualities of Seven General-Aviation Ai rcraft". NASA TN D-3726, November1966.

56. Lay, L. W.; andIkerd, L. R.: "FlyingQualities Tests Results". Cessna Rep. No. F-M337-12, January 1968.

57. Grantham, William D.; Moore, Frederick L.; Deal,Perry L.; andPatton, James M., Jr.:"Simulator Study of CoupledRoll-Spiral Mode Effects onLateral-Directional Handling Qualities". NASA TN D-5466, March 1970.

58. Taylor,Lawrence W., Jr.;and Day, Richard E.: "FI ightControl labi I ity Limits andRelated Human TransferFunctions as Determined from Simulator andFI lght Tests". NASA TN D-746, May 1961.

59. McFadden, Norman M.; Vomaske, Richard F.; andHeinle, Donovan R. : "FIight InvestigationUsing Variable-Stability Airplanes of Minimum Stability Requirements for High-speed,High-Altitude Vehicles". NASA TN 0-779, Apri I 1961.

60. Kuehnel,Helmut A.: "In-flight Measurement ofthe TimeRequired for a Pilot to Respond to an Aircraft Disturbance". NASA TN D-221, March 1 960.

61. Russell,Walter R.; Sjoberg, S. A.; and Alford, William L.: "Flight Investigationof Automatic Stabilization of an AirplaneHaving Static LongitudinalInstability". NASA TN D-173, December1959.

62. Adams, James J.: "A Simplified Method for Measuring Human Transfer Functions". NASA TN D-1782, April 1963.

63. Fink,Marvin P.; and Freeman, Delma C., Jr.:"Ful I-Scale Wind-Tunnel Investigation of StaticLongitudinal and LateralCharacteristics of a Twin-EngineAirplane. NASA TN D-4983, January 1969.

64.Fink, Marvin P.; Freeman, Delma C., Jr.;and Greer, H. Douglas:"Full- Scale Wind-Tunnel Investigationof the Static Longitudinal and Lateral Characteristics of a LightSingle-Engine Airplane". NASA TN D-5700, March1970.

65. Sh ivers, James P.; Fink,-Marvin P.; and Ware, George M.: l'FulI-Sca le Wind-Tunnel Investigation of theStatic Longitudinal and Lateral Characteristics of a LightSingle-Engine Low-Wing Airplane". NASA TN D-5857, June 1970.

305 66. Jarvis,Calvin R.; Loschke,Paul C.; andEnevoldson, Einar K.: "Evaluation of the Effect of a Yaw-RateDamper on the Flying Qualities of a Light Twin-EngineAirplane". NASA TN 0-5890, July 1970.

67. Perkins,Courtland D.: "Development of AirplaneStability and Control Technologyft.Journal of Aircraft,July-August 1970.

68. Soulg, Hartley A.; and Miller,Marvel P.:'!The ExperimentalDetermination ofthe Moments of Inertia of Airplanes". NACA TR-467, 1933.

69. Gracey, Wi I I iam: "Measured Moments of Inertia of 32Airplanes". NACA TN-780, October 1940.

70. Kirschbaum, H. W.: "Estimation of Moments of Inertia of Airplanesfrom Design Data". NACA TN-575, July 1936.

71. McCullough,George B.; and Gault,Donald E.: "Examples of ThreeRepresentative Types of Airfoil-SectionStall at Low Speed:. NACA TN-2502, September 1951.

72. Soule', H. A.; andAnderson, R. F.: "DesignCharts Relating tothe Stalling of Tapered W ings" . NACA TR-703, 1 940.

73. Sweberg, Harold H.; and Dingeldein,Richard C.: "Summary of Measurements in LangleyFull-scale Tunnel of Maximum Lift Coefficients and Stalling Characteristics of Airp lanes". .NACA WR L-145, Apri I 1945.

74. Jacobs, Eastman N.: "Tapered Wings, TipStalling, and Preliminary Results from Testsof the Stal I-Control F.lap". NACA WR L-296,November 1937.

75. Anderson, Seth B.: "Correlationof Pi lot Opinionof Stal I Warning with FI ight Measurements of VariousFactors which Produce the Warning". NACA TN-1868, April 1949.

76. Zalovcik,John A.: "Summary of StalI-Warning Devices". NACA TN-2676, May 1952.

77. Stone, Ralph W., Jr.; Garner, William G.; andGale, Lawrence J.: "Study of Motion of Model of Personal-Owner or Liaison .Airplane through the Stall and intothe Incipient Spin by Means of a Free-FlightTesting Technique". NACA TN-2923, April 1953.

78. bwman, James S.: "AirplaneSpinning". NASA SP-83,May 1965.

79. Neihouse,Anshal I.; Klinar,Walter J.;and Scher, Stanley H.: "Status of Spin Resea-rch for RecentAirplane Designs". NASATR R-57,1960.

80. Neihouse,Anshal I.: "A Mass-DistributionCriterion for Predicting the Effectof Control Manipulation on the Recovery from a Spin". NACA WR L-168, August 1942.

306 81. Nei house, Anshal I.; Lichtenstein,Jacob H.; and Pepoon, Phil ip w.: "Tail-DesignRequirements for Satisfactory Spin Recovery". NACA TN-1045, Apri I 1946.

82. Nei house, A. I.: "Tail-DesignRequirements for Satisfactory.. Sp in Recovery for Personal-Owner-Type LightAirplanes". NACA TN-1329, June 1947.

83. Klinar, Walter J.; andWilson, Jack H.: "Spin-Tunnel Investigation of the Effects of Mass and DimensionalVariations on the Spinning Characteristics of a Low-W ing S ing Ie-Vert Ica 1 -Ta i I Mode'l Typ i :a I of Personal-OwnerAirplanes". NACA TN-2352,May 1951.

84. Seidman, Oscar;and Neihouse, A. I.: "Free-Spinning Wind-Tunnel Tests of a Low-Wing Monoplane withSystematic Changes in Wingsand Tails - IV. Effectof Center-of-Gravity Location". NACA TR-672, 1939.

85. Siedman, Oscar;and Neihouse, A. I.: "Free-Spinning Wind-Tunnel Testsof a Low-Wing Monoplane withSystematic Changes in Wings and Tails - V. Effectof Airplane Relative Density". NACA TR-691, 1940.

86.Zimmerman, C. H.: "Effect of Changes inTail Arrangement upon theSpinning of a Low-Wing MonoplaneModel". NACA TN-570, June 1936.

87. Seidman, Oscar; andNeihouse, A. I.: "Free-Spinning Wind-Tunnel Tests of a Low-Wing Monoplane withSystematic Changes in Wings and Tai Is, 1. BasicLoading Condition". NACA TN-608, August 1937.

88. Gale,Lawrence J.; andJones, Ira P., Jr.:"Effects of AntispinFi I lets and DorsalFins on the Spin andRecovery Characteristics of Airplanes asDetermined from Free-Spinning-TunnelTests". NACA TN-1779,December 1 948.

89. Klinar,Walter J.; and Gale, Lawrence J.: "Wind-Tunnel Investigationof theSpinning Characteristics of a Model of a Twin-Tail Low-Wing Personal-Owner-Type Airplanewith Linked and Unlinked Rudderand AileronControls". NACA TN-1801, January 1945.

90. Headquarters, Air Force Systems Comnand: "Design Handbook, Series 1-0, General,Personnel Subsystems". AFSC DH 1-3, January 1969.

91. Morgan, C. T.; Cook, J. S.; Chapanis, A.; and Lund, M. W.: Human Engineering Gtiide to EquipmentDesign. McGraw-Hill, 1963.

92.Damon, Albert;Stroudt, Howard W.; andMcFarland, Ross A.: The Human Body in EquipmentDesign. Cambridge, Harvard UniversityPress, 1966.

93. Bureau ofAeronautics, Navy Department: !'TheHuman Pi lot". EL. AER Rep. AE-61-4111, August 1954.

307

. .. .- ...... - .- -. .. .. "..._.." ".. . -....._...."-...... ". .... 94. Woodson, Wesley E.; andConover, Donald W.: Human EngineeringGuide for EquipmentDesigners. University of CaliforniaPress, 1964.

95.McCormick, ErnestJ.: Human FactorsEngineering.. McGraw-Hill Book Company, 1964.

96. Orlansky,J.: "Psychological Aspects of Stick andRudder Controls in Aircraft",Aeronautical Engineering Review. Vol. 8, January 8, 1949.

97. Tufts CoII,ege, Institute of AppliedExperimental Psychology: "Handbook of Human Engineeri.ngDatal'. Special Devices Center, Technical Report No. SDC 199-1-2,November 1952.

98. McFarland,Ross A.: Human Factorsin Air Transportation.McGraw-Hill Book Company, Inc.,1953.

99.Harper , R. P.: "FlightEvaluations of VariousLongitudinal Handling Qua I tiesin a VariableStability Jet Fighter". WADC TR-55-299, July 1955.

300. Sevant, C. J.,Jr.: Control System Design. MCGraw-Hill Book Company, I nc. 1 964.

101. Wolowicz,Chester H.; andHolleman, Euclid C.: "StabilityDerivative Determination from Flight Data.'' AGARD Report 224,October 1958.

102. Recant, lsidore G.; and Swanson, Robert S.: "Determination of the Stability and ControlCharacteristics of Airplanes from Flight Tests of PoweredModels." NACA WR L-710, July 1942.

103. Katzoff, S.: "LongitudinalStability andControl with Special Reference to SlipstreamEffects," NACA TR-690, 1940.

104. Wei I, Joseph;and Sleeman, William C., Jr.:"Perdiction of the Effects of PropellerOperation on the Static Longitudinal Stability of Single- EngineTractor Monoplanes with Flaps Retracted." NACA TR-941, 1949.

105. Hagerman, John R.: "Wind-Tunnel Investigation of theEffect of Powerand Flapson the Static Longitudinal Stability andControl Characteristics of a Single-EngineHigh-Wing Airplane Model." NACA TN-1339, July 1947.

106. Hagerman, John R.: "Wind-Tunnel Investigationof the Effect of Powerand Flapson the Static Lateral Stability andControl Characteristics of a Single-EngineHigh-Wing Airplane Model." NACA TN-1379, July 1947.

107. Tamburel lo, Vito; andWeiI, Joseph: "Wind-Tunnel Investigation of the Effect of Powerand Flapson the Static Lateral Characteristics of a Single-Engine Low-Wing AirplaneModel." NACA TN-1327, June 1947.

308 108. Purser,Paul E.; andSpear, Margaret F.: "Wind-Tunnel Investigationof Effects of UnsymmetricalHorizontal-Tail Arrangements on Power-On Static Longitudinal Stability of a Single-EngineAirplane Model." NACA TN-1474, October 1947.

109.Schade, Robert .O.: "Free-FlightInvestigation of Dynamic Longitudinal Stability as Influenced by the Static Stability Measured in Wind- TunnelForce Tests under Conditions of ConstantThrust and Constant Power." NA€A TN-2075, April 1950.

110. Barber,Marvin R.; Jones, Charles K.; Sisk, Thomas R.; and Haise,Fred W.: "An Evaluation of theHandling Qualities of Seven GeneralAviation Aircraft." NASA TN D-3726, November 1966.

111. Marr,Roger L.: "A Method forAnalyzing Power-on StaticLongitudinal Stability.'! SAE Paper 700238, 1970.

112. Fink,Marvin P.; and Freeman, Delma C., Jr.:"Ful I-Scale Wind-Tunnel Investigation of StaticLongitudinal and LateralCharacteristics of a Light Twin-EngineAirplane." NASA TN D-4983, January 1969.

113. Fink,Marvin P.; Freeman, Delma C., Jr.; and Greer, H. Douglas:"Full- Scale Wind-Tunnel Investigation of theStatic Longitudinal and Lateral Characteristicsof a LightSingle-Engine Airplane." NASA TN D-5700, March 1970.

114. Ribner,Herbert S.: "Noteson the Propel ler and Slipstreamin Relation toStability." NACA WR L-25, October 1944.

115. Ribner,Herbert S.: "Formulas for Propel lers in Yaw and Charts of the Si de-Force Der i vat i ve .'I NACA WR L-2 I7, May 1943.

116. Ellis,David R.: "FlyingQualities Criteria for SmallGeneral Aviation Airplanes as Determined by In-FlightSimulation.'' SAE Paper 710373, March 24-26, 1971.

309

APPENDIX A

31 1 DERIVATION OF THE EQUATIONS OF MOTION

Derivation for theequations of motion,following Dynamics of theAirframe (Ref. 12, is basedon Newton's laws, ie., motion with reference to axesfixed in space.The severalassumptions which form thebasis for thisderivation will bepresented throughout the following discussion as they are needed to clarifythe various steps of thederivation. The first two of theseassumptions specifythe nature of the bodybeing stud ied and the atmosphere in which it is set.

Assumption I The airframe is assumed to be a rigid body; thus,the distance betweenany specifiedpoints in the body areinvarient.

Assumption II The earth is assumed to be fixed in space,and theearth's atmos- phereis assumed to be fixed with respect to the earth.

Table A-1, with the aid of Figure A-1, defines the direction of theaxes with respectto the airplane, as well as the nomenclature needed toapply Newton's laws.

Figure A-1. Directionof the axes with respect to theairplane.

31 2 ~~ " . .

L i near Angu tar Summat ion Summat ion Displace- Moments Velocity Velocity of Moments of Forces ments of Momen- Moment A I ong A I ong About A I ong About turn About of Axis Ax is Axis Axis Axis Axis Inertia

.~ .. .. __~. ~ ~~~ . .~ . - P U Rol I ing CL CFX Q Ihx Vel. IXX ~- I Q Y Pitchi ng CM 8 I CFY Vel. hy I IYY R I W Yaw i ng EN CFZ Y Vel.

Table A-1. Direction of theaxes with respect to the airplane and nomenclature needed to applyNewton's Laws.

Newton'ssecond law of motion states that the rate of change of momentum of a body isproportional to thenet force applied to the bodyand that the rate of change of the moment of momentum isproportional to thenet torque applied to the body.The mathematical statements of the law can be written d IFx = (mu)

and

dhz CN = - (A-2 1 dt

Assumption Ill The mass of theairplane is assumed to rema in constant for the duration for any partlcular dynamicana,lysis.

313 Assumption Ill permits the mass of the airplane to be written outsi de the d if- ferentiationsign in Equations (A-1).

The moments of momentum referred to in Equations (A-2) can be expanded by usingan element of mass of'the airplane dm which is rotating with the angu lar velocity ZJ(U = Pi + Q.j + Rk). Thiselement of mass is at the poin t (X,Y,Z) measured relative to-the c.g. ofthe airplane. The motion of theelement of mass ca'nbe approximated by sixlinear velocity components (Py, Pz, Qx, Qz, Rx, and Ry), as seen In Figure A-2.

# "". 0 0 0 0 "- -"

"_"" """

Figure A-2. Linearvelocity components of an elementof mass caused byan angular velocity E having components P, Q, and R.

The x, y, and z components of the moment of momentum are calculated by summing the moments of these velocity components about each axis and multiplying by mass dm. For examp le,

dhx = y(yP1dm + z(zP)dm - z(xRi)drn - y(xQ)drn.

Thus, if the moments aretaken about the x, y, and z axes, the followi ng setof equations is obta i ned :

dhx = (y2 + z2>P dm - zx R dm - yx Q dm

dhy = (z2 + x2)Q dm - xy P dm - yz R dm

dhz = (x2 + y2)R dm - zx P dm - zy Q dm (A-3)

For a finite mass, the components of the moment of momenturn are the integrals of Equations (A-3). Taking I,,= j(y2 + z2)dm, -Ixz = lxzdm, and Iyz= Izythe integralrelations become

hx = PIxx - QIxy -RTxz

314 The derivative dh/dt may befound by differentiating Equations (A-4) with respect to time. Thus, theequations of motionrelative to inertial axes become dU CF, = m dV CFy = m dW CF, = m dh EL =x=PIxx+ Pixx Q&y - kIxz Rtxz dt - 61xy - - dh CM = # = Gryy + Q?,, - - ~f~~ - PIxy- Pfxy

For ease in interpreting flight measurements,one desires to change the fixed axessystem to an Eulerianaxis system, ie., a right-handsystem of ortho- gona I coordinateaxes which has 'its origin at the center of gravity of the airplane and itsorientation fixed with respect to the airplane. Velocities of the airplane measured relative to these axes are absolute velocities, since, at any instant,the Eulerian axes are considered to be fixedin space. Also, moments and products of inertia in the Eulerian axis system areindependent of time, sincethese axes are fixed in the airplane; thus, dI/dt = 0. Sincethe Eulerian a2is system moves withrespect -Po inerital space, theabsolute acceleration (measured inthe Eulerian system) can be written

If U, V, and W are components of v-r and P, Q, and R are the components of w, then ax = IJ. + QW - RV ay = V. + RU - PW aZ = W + PV - QU (A-6)

In a similar manner, the change in the moment of momentum canbe written diiabs/dt = dK/dt + w x K ijk where w x K = P Q R hxhyhz 315 Thus,

"- dhz + hyP - hxQ (A-7 1 (%lbs dt

Assumption IV The xzplane is assumed to be a plane of symmetry.

UsingAssumption IV and theorientation convention of Figure A-I, the equations of motioncan be written . CF, = m(U + QW - RV)

Eu lerian angles are those angles throughwhich one axis system must be rotated to superimpose it uponanother having aninitial angular displacement from the first. Because theangles are not orthogonal, the order of rotation is important, if theindicated operations are to yieldcorrect results. Thesequence ofthese angularchanges are yaw, pitch, and roll. To carryout this superposition, one first yaws through a positiveangle + in accordancewith a right-handsystem so that - X, = X cos + + V sin @ - - - Y1 = Y cos $ -'X sin @

(see Figure A-3)

316 Figure A-3. Yaw through a positiveangle $ inaccordance with a r ig ht-hand system.

The nextrotation (Figure A-4) is a positivepitch through the angle 8, which g ives - - - x2 = X1 COS 8 - Z1 sin 8 - - Y* = Y1 - - z2 = Z, COS e + kl sin e (A-1 0)

Figure A-4. Positivepitch through the angle 8.

317 The finalrotation (Figure A-5) isthrough the roll angle 9, whichgives

(A-1 1)

Figure A-5. Rotationthrough the roll angle 4.

SubstitutingEquations (A-9) and (A-10) intoEquations (A-11) yieldsthe transformation needed to convert the initial axis system to the final system: - - - x3 = X cos e cos IC, + U cos e sin $ - z sin e - - Y3 = X(cos $ sin 8 sin $ - sin $ cos $1

+ Ytcos $ cos 4 + sin I,IJ sin e sin $1

The analysis of flight motions is concerned primarily with the vehicle's behaviorin response to disturbancesfrom initial conditions. It is convenient, therefore,to chose as the initial conditions flight behavior for which the velocities and accelerationsare well known and in whichthe aircraft spends mostof its f 1 ight time. Equi I ibrium (unaccelerated) f I ight is f I ight along a straight

31 8 I"

pathduring which the linear velocity vector measured relative to a fixed space isinvarient, and theangular velocity is zero."Steady flight" is flight dur- ingwhich the linear and angularvelocities in the Eulerian reference frame remain constant. Hence, equi I ibrium f I ight and f I ight with constant angular velocityare both forms of steady flight.

Alwaysacting on the aircraft even during periods of steady flight is gravity.Since it isunidirectional, it providesan orientation to themotion. The components of gravity acting along the steady f I ight aircraft axes relative to inertial spacecan be determined from Figure A-6 by direct resolution of the. gravityforce along the xo, yo, and zo (steadyflight) axes.

'YO

Figure A-6. Gravityacting on an airplane in steady f I ight with initial ang les 6, and @o with respect to the gravity vector.

Thus, X, = -W sin 6,

Yo = W cos 6, sin $o

2, = w cos 8, cos $o (A-1 3)

Thecomponents of gravity,acting along the disturbed Eulerian axes, are then

X3 = (-W sin 6,)cos 8 cos I) + (W cos 6, sin$o)cos 6 sin I)

- (W COS 8, COS $,)sin 8

319 Y3 = (-W sin BO)(cos J, sin 0 sin 4 - sin J, cos 4)

+ (W cos 8, sin $o)(cos $ cos $ + sin $ sin 8 sin $3

+ (W cos 8, cos @o) (cos 0 sin $1

Z3 = (-W sin B0>(cos I) sin 0 cos 4 + sin $ sin 4) + (W cos 8, sin $o)(sin Q sin 8 cos $ - cos Q sin $1

The right-handsides of Equations (A-8) expressthe aircraft acceleration interms of the linear and angularvelocities. The left-handsides of Equations (A-8) representthe unbalanced forces (thrust forces, aerodynamic forces, and gravityforces) which produce the airplane motion. The gravityforces have already beenexpanded and transformed tothe Eulerian axes (A-13); ideally,the same procedurecould be appliedto the thrust andaerodynamic forces. Because of thedifficulty in expressing the aerodynamic and thrust forces explicitly in termsof the linear and angularvelocities, it is customary torepresent these forces by Taylorseries expansions, taking a sufficient number ofterms to in- sureadequate accuracy for the maneuverbeing considered.

Because of thisspecial requirement, it is helpfulto separate the aerodynamic and thrust forces from thegravity forces, thus:

CF, = CF’, + X3

CFy = CF’), + Y3

CF, = IF’, + Z3 (A-1 5) where theprimed quantities are the summations of the aerodynamic and thrust forces, and X3,Y3, and Z3 arethe gravity componentsderived inEquations (A-14). It shouldbe noted that, if theinstant under consideration occurs during the steadyflight condition, then 8 = @ = $ = 0, and the components X3, Y3, and 23 reduce to Equations (A-13). The forcerelations from Equations (A-81, withthe gravityterms transposed to the right. side, are rewritten CF’, = m(U + QW - RV) - X3

CF’y = m(V + RU - PW) - Y3

CF’, = m(i + PV - QUI - z3 (A-1 6)

By substitutingEquations (A-14) inEquations (A-161, Equations (A-8) may be written

320 I'

CF', = mti + QW - RV) + (W sineo)cos e cos 9

-(W COS eo sin $O)COS 8 sin 1c) + (W cos 8, coscbo)sin 8 a CF'y = m(V + RU - PW) + (W sin B0>(cos $ sin 8 sin 9 - sin$ cos $1

-(W cos 8, sin $o)(cos 9 cos $ + sin $ sin 8 sin $1

-(w COS 8, COS $o)(~~s8 sin $1 CF', = m(W + PV - QUI + (W sin B0)(cos I) sin 8 cos $ + sin 9 sin $1 -(W cos 8, sin $,)(sin $ sin 8 cos $ - cos I) sin $1

-(w cos eo COS $o)(~~~e COS $1 CL = bTXx - ktxz + QRCI,, - I,,)- PQI,,

CM = GrYy+ PR(I,, - I,~) - R~I~~+ P~I,, IN = I%,,- kx, + po(rYy - T~,) + QRI,, (A-1 7)

Theseequations are then complete, except for the external forces and moments onthe left side, which include aerodynamic and thrustforces as we1 I as moments resulting from controlsurface deflections.

Definitionsof the Eulerian axis system and theEuleria? a;gles, discussed above, show thatthe rates of change ofthe Eulerian angles I), 8, and 6 arenot orthogonal. The fixedaxis angular velocities when writtenin terms of therates of change of the Eu lerian angles become P = 4 - $ sin.8. Q = 8 COS $I + I) sin $I COS e a . R = I) cos $I cos 8 - 8 sin 9 (A-1 8)

Equations (A-17) matchthe aerodynamic and thrust forces acting on an airplane to thegravity and resultinginertia forces. Theseequations are non-linear, since (1) theycan contain products of the dependent variables and (2) the dependentvariables appear as transcendental functions. The airframemotion can always beconsidered the result of disturbances to theairframe from some steady flight condition.Accordingly, each of thetotal instantaneous velocity components of theairframe can be written as the sum of a velocity componentduring the steady fl ight condition and a change in velocity caused by thedisturbance: u=uo+u

v=vo+v

W=Wo+w

32 1 P=Po+p

Q=Qo+q

R=Ro+r (A-1 9)

The zerosubscripts of Equations (A-19) indicatethe steady flight velocities, and thelower case letters represent the disturbance velocities. By substituting Equations (A-19) intoEquations (A-17) and notingthat derivatives with respect to time of thesteady state conditions are zero, Equations (A-17) become

IF”, = m[h + Q~W,+ woq + Q~W+ wq - RoVo - Rov - Vor - vr

+ (gsin Bo)cos 8 cos $ - (g cos 8, sin $o)cos 8 sin $

+ (9 COS e, COS $,)sin e]

CF’~= m[t + U,R, + Uor + R,U + ru - powo - pow - w0p

- wp + (g sin Bo)(cos $ sin 8 sin 4 - sin $ cos $1

- (g COS e, sin $I~)(COS cos 4 + sin 3, sin 8 sin $1

- (g cos e, COS $o)(cos e sin $11 CF’~ = m[i + PoVo + Pov + Vop + pv - QoUo - Qou - Uoq - qu + (g sineo)(cos $ sin 8 cos $ + sin $ sin $)

- (g COS e, sin $,)(sin 3, sin e cos 4 - cos $ sin $1

- g(cos e, COS $o)(~~~e COS $11

CL = ;Ixx - ;Ixz + (QoRo + Qor + Roq + qr) (IzZ - Iyy)

- (Po00 + Poq + QoP + Pq)Ixz CM = ;Iyy+ (PoRo + Por + Rop + pr)(Ixx - Izz)

- (Ro2 + 2R0r + r2)Ixz + (Po2 + 2Pop + p2 )Ixz

CN = FIZZ- bTxZ + (PoQo + Poq + QoP + Pq)(Tyy - IXX)

+ CQoRo + Qor+ Roq + qr)IxZ (A-20 1

322 Assumption V The disturbancesfrom the steady flight condition are assumed to besmall enough so thatthe pro- ductsand squares of thechanges in .velocities are negl igible in comparison to the changesihem- selves.Also, the disturbance anglesare assumed to besmall enough so that the sines of theseangles may be setequal to the angles and the cosines setequal to one.Products of these angles are a Is0 approxi- mate1 y zero and can be neg I ected . Since the disturbances are smal I, the change in air density encountered by theairplane during anydisturbance can be considered zero.

If Assumption V isapplied to Equations (A-201, they become

CF’, = rn[: + QoWo + Woq + Qow - RoVo - Rov - Vor

+ g sin eo - (gcos eo sin $o)$ + (gcos eo cos $o)eJ

CF~~= rn[G + U,R, + Uor + R,U - powo - P,W - w0p

- (g sin eo)$ - g cos eo sin +o - (gcos eo cos $0)+]

CF’, = m[G 3. POYO + Pov + Yop - QoUo - Qou - Uoq

+ (gsin eo)e + (gcos eo sin +o)+ - (g cos eo cos $0)]

CL = ;Ixx- ;Ixz + (QoRo + Qor+ Roq)(lZz - Iyy)

- ( PoQo + Poq + QoP)Ixz

CM = ;IrYy+ CP~R~+ Por + R~~)(I~~- 1~2)

- (Ro2 + 2Ror)I,, + (Po2 + 2P0p)Ixz CN = ;Izz - ;Ixz+ CPoQo + Poq + Qop) CTyy - Ixx)

+ (QoRo + Qo~(A-21+ Roq)Ixz 1

Equations(A-21) limit the applicability of the analysis to so-calledsmall perturbations.In the str ictlymathematical sense, Equations(A-23) are appli- cableonly to infinitesima I disturbances;however, .experience has shown that quite accurate results can be obtained by applyingthese equations to disturbances of f i ni te, non-zero magnitude. An additionalapplication of Assumption

323 V isthe reduction of Equations (A-18) to

P=&@

Q = 6 +$$

If theproducts of perturbations are neglected, the above equations are reduced to . P=$. Q=0. R=+ (A-23 1

Equations (A-231 show that,within the Iim its of sma I I perturbat iontheory, the instantaneousangular velocities P, Q, and R may be set equa I to the rates of change of the Eulerian angles.

Assumption VI Duringthe steady f I ightcondition, the airplane is assumed to be fly- ingwith wings level and with all components ofvelocity zero except Uo and Wo. Thus, Vo = Po = Qo = Ro = $0 = l/Jo = 0.

Assumption VI reducesthe equations of motion to

CF’, = m [i + woq + g sin eo + ge cos eo]

CF’~= m [; + Uor - w0p - gl/J sin 0, - g$ cos eo]

CF’, = m[i - uoq + ge sin eo - g COS eo]

CL = &x - FIX, CM = qlyy. . CN = rIzz- pIxz (A-24 1

The aerodynamicforces and moments are then expressed in coefficient form as

L = C,-+~V*S = Lift

D = CDfpV2S = Drag

x = c,4pv s = AerodynamicForce A long x Axis

324 I 111

Y = CyfpV2S = AerodynamicForce A long y Ax is

Z = C,fpV2S = AerodynamicForce Along z Axis

L = C 1 &pV2Sb = Rol I i ng Moment

M = C,$pV2Sc = Pitching Moment

N = CnfpV2Sb = Yawing Moment (A-25 1

where

S = Wing Area

c = Mean AerodynamicChord = Thewing chord which has theaverage characteristics of all chords in the wing.

b = Wingspan

The lift anddrag are the forces acting normal and parallelrespectively to the f I ight path.

As notedpreviously, each of theforces andmoments canbe expressed as a function of thevariables by expanding the forces and moments in a Taylorseries. The series has the form

F = F, + (aF/aa),a + caF/awOB + (aF/a8),6 + ... (A-26)

where a, B, and 8 arevariables, and thesubscript zero indicates that the quantitiesare evaluated at the steady flight condition. In Equation (A-261, terms of the order la2F/aa2) (a2/2! 1 and a I I higher order terms are omitted in accordancewith Assumption V. Beforeexpanding each of theforces and moments in the aboveform, a simp1 ificationcan be made. Because thexz plane is a planeof symmetry, the rate of change of the X and Z forces and the moment M, withrespect to thedisturbance velocities p, r, andv, iszero. Thus, theforces and moments acting on a disturbedairplane can be expressed ax ax ax x=xo+-u+-ax ax il+-q+,~+-w+~w+-ax ax au a: aq aq aw asE 6~

325 N=N0+-v+7aN aN; +2J r +-aN r+-p++,p+-a aN aN aN 6, av av ar a;aP aP a6R (A-27 I

where

bE = Ang le of def lectionof elevator

6F = Ang leof def lectionof flaps

6~ = Ang le of def lection of ailerons

6, = Angleof deflection of rudder

The thrustforce previously mentioned in Equations (A-15) can be introduced intothe equations of motion in much the sameway as the gravity force was introduced.The thrustin considered to be a functionof the power plantrevolutions perminute and theforward speed of the airplane. With the power plant located in the plane of symmetry, the thrust contributes to the X and Z forces and to the moment M. Withthe aid of Figure A-7, it isevident that, by settingthe steady flight thrust equal to To, theequations for the steady flight condition become

X, X, = To cos 5

Zo = - To sin 5

Mo = To z (A-28 1 j where 5 = anglebetween x axis and thrustline = perpend icu lard i stance from C .g . to 'j thrust I ine

326 Y

Figure A-7. Thrustforce relationship to X and Z forces and moment M.

Sincethe Eulerian axes remain fixed with reference to the airplane during a disturbance,the thrust components relative to the disturbed axes become

Z = - Tj sin 5

M=T z (A-29 1 1j where T1 (thethrust during the disturbance) = To + AT

If a Taylorseries expansion is assumed, then

Thu sf

The individualcontributions to theequations of motion have now been examined in some detail,giving perhaps some insightinto the basis of thecomplete equationsof motion of the airframe. Before continuing, however, it iswell to

327 note that the equations for steady flight can be foundby substituting the steady flight values of the aerodynamic,weight and thrust forces and moments into Equa- tions (A-24) and settingthe disturbance terms equal to zero:

X, X, - \h; sin 8, + To cos 5 = 0 Yo + 0 + 0 =o

Z, Z, + W cos 0, - To sin 5 = 0 Lo + 0 + 0 =o

M, + 0 + To Zj =o No + 0 + 0 =o (A-31 1

The equationsof motion for the disturbed airplane are then found by substituting thedisturbed values of the forces and moments intoEquations (A-24):

- aT sin 5 - (s

328 (A-32)

The quantitiesin boxes disappear because of the steady flight conditions of Equations (A-31). Dividingthe force equations by the mass m and the moment equationsby the appropriate moments of inertia yields terms of the form

--1 ax u and --1 a' r. m au Ixx ar Replacing(l/m)(aX/au) by Xu and (l/Iyx)(aL/ar) by Lr simplifiesthe notation. These quantities are called either "dimensional stabi I ity derivatives" or simply "stabi I ity derivatives." By eliminating those terms whose sum, in accordance withEquations (A-321, is zerobecause of thesteady flight conditions, andby usingthe previous shorthand notation, Equations (A-32) are reduced to the form:

+ Uor - w0p - g$ sin 0, - g4cos eo = Yrr + Y~F+ Y~Vt

329 Assumption V I I The flow is assumed to be quasi -steady.

Because of Assumption VI I , allderivatives with resDect to the rates of change of velocities are omm itted,with the exception of thoseinvolving i, which are retained to account for the effect on the horizonta I ta:j I of the downwash from thewing. It shouldalso bepointed out that the change in angle of attackcan be approximated by Aa = w/U.

The only restrictions thus far imposedon the orientation of the Eulerian axeswith respect to the airplane are that the y axis be a principal axis and that the origin belocated atthe center of gravityof the airplane. When the x axisis oriented so that it is a principalaxis, the Eulerian axes are referred to as pr i nc i pa I axes, but when the x axis in the a i rpl~a-ne_i~?pa-ya_l;l_e;l to therelative wind during steaa flight, the Eulerian axe-s"are referredto as stab i I i ty axes. When the a i rf rGe7s-d isturbed from the steady f I ight .. condition,the Eulerian axes rotate with the airframe and do not change direc- tionwith respect to the airplane. Consequently, the disturbed x axis may or may not be parallelto the relative wind while the airplane is in the disturbedflight condition. It should benoted that for small angles of attackthe moments ofinertia about the stability axes are approximately equal tothose about the bodyaxes; however, at lowspeeds (highangles of attack)the moments ofinertia about the stability axes can differ significantly fromthose about the body axes and this fact should be considered in dynamic analyses. The use ofthe stability axeseliminates the terms containing Wo fromEquations (A-33) byeliminating the following quantities: (1) all terms containing Wo, whichdisappears because of the direction of the stability axes; (2) all aerodynamicpartial derivatives with respect to rates of change ofvelocities, except those with respect to W; and (3) all aerodynamic partial derivativeswith respect to rates of change ofcontrol surface deflections. Equations (A-33) thenreduce to Equations (A-34 and (A-35). To avoidconfusion with body axiscoordinate systems, where 8, is the inclination of +he x axis withrespect to the horizon, the angles between thehorizontal and the x stabilityaxis is called yo. It will be recognizedthat because of the way thestability axis system isdefined yo is indeedthe angle of theflight path with respect to the earth.

w - Uoq + g8 sin yo = - Tu(sinE)u - TG~~~~~~Msin 5

. z*m 4 = T,u + .Y?T6RpM6~p~ + Muu + Mqq + Mww IYY . IYY

330 An examinationof these equations shows thatEquations (A-34) are functions of thevariables u, 0, and w, whereasEquations (A-35) are functions of thevariables v, r, andp. As a resultof the assumptions made inthis analysis, the equationsof motion can be treatedas two independent sets of three equations withEquations (A-34) describing the longitudinal or x-z planemotions and Equations(A-35) the lateral motions.

33 1

APPENDIX B

333 DEFINITION OF STABILITY DERlVATlVES

In this appendixthe dimensional srability derivatives which appeared inthe equations of motion are defined. For each dimensionalstability derivativean equation is given which, for themost part, relates it to a non- dimensionalderivative thus simplifying the evaluation of thedimensional derivatives.

Longitudinal Stabi I ity Derivatives:

334 - 30 pUSc acT T6~~~m 30 c a (7~RPM)

LateralStability Derivatives:

Lg = U,LV

Y=pUSb acy P 4m

335 It shouldbe noted that derivatives with respect to angles or rates of angular changeare defined per radian.

336 APPENDIX C

337 DERlVATION OF THE TRANSFER FUNCTIONS

Equations-CA-34)and CA-35) fromAppendix A may beconverted by theuse of determinantsinto the transfer functions considered earlier in this study.

Theangle of attack and theangle of sideslip can be written i.n approximate form basedon the velocity componentsand the perturbation velocity components:

V ACY, N - and B - "0 "0

Theangle between the equilibrium flight path and thedisturbed flight path is definedas E. It shouldbe pointed out that only when equalszero is the sideslipangle B equal to thenegative of the yaw angle.

The long i tud i na I equat ionscan be written bymodifying Equations (A-341, as shown below:

- Xuu - (Tucos S)u - x q + gecos - xfii - xww 9 yo

-Zuu + (Tusin<)u - Uoq - Z q + gosin + \f, zfi4 - z,w 9 yo = Z 6 - (T6RpM~inS)BRpM + Zg 6 &E E FF

-M,u - -TUurn -t 4 - M,q - MgQ - MWw

(C-1)

The lateralequations of motion, Equations (A-351, arerearranged by substititing f3U for v and dividingby Uo. Thus, . Yr B - YvB - cos yo)$ + r - -r - (9sin yo)$ UO UO UO

338 The rightsides of Equations CC-I) and CC-2) arethe control forces and represent the means by which either the human pilot or an autopilot can control the motion ofthe airframe. The thrust and thecontrol surface inputs are the forcing functionswhich determine the resultant motion of the airframe. Since the air- frameequations of motion are linear equations, the principle of superposition may beused to obtain a solution.For instance, the response to simultaneous application of elevator and rudderdeflections can be determined by calculating theresponse to each of these deflections separately and then adding the results to complete the solution.

The transfer functions are obtained by applying the method of Laplace trans- forms. If Equations ((2-1) and CC-2) aretransformed into the Laplacian domain,

where z .m A’ = Tucos 5 F’ = ?gRm B’ = TsRpMcos 5 YY C’ = T,sin 5 D- = T6RPMsin 5 = TTu Ea B , =2 IYY 339 .. . It should be notedthat p, q, and r were replacedby 4, 8, and $ respectively. Thisis permitted becauseof the small perturbation approximation. Using Equations CC-3) and Cramer’s rulefor solving equations bydeterminants, the longitudinaltransfer function for U(S)/~E(S) with 6,=0 and 6~m=Ocan be

xgE -(sX; + X,) +xq - gcos yo)

[S(I z;) - z,] -[s(u, + zq) - gsin yo] z% -

I [S - (X, + A*)] -(sX\; + X,) -(sX - gcos yo) 9 -(Z, - C’) [dl - Z$ - z,] -[s(u, + z - gsi n yo] 4 -(Mu + E’) -(sM; + M,) (s2 - Mqs)I (C-4 I

The denom inator determinant is the determinant of the homogeneous equationsde- noted by Dl. The expansionof Dl gives

Dl = As4 + Bs3 + Cs2 + DS + E, (C-5)

where A = 1 - Zf

B = -(1 - Z6)[(Xu + A’) + Mq] - Z, - M6(Uo + Zq) - X;(Zu - C’)

C = (X, + A’)[M (1 - Z;) + Z, + Mi(Uo + Z,)] - (Mu + E’)[Xt(Uo + Zq) q + Xq( 1 - Zt)] + M Z, + (Zu - C’)[MqXg - X,) - XqMmI 9 + MRgsin yo - Mw(Uo + Zq)

D = gsin yo[ (Mu + E)X$ + M, - M~(XU+ A>)] + gcos yo[(Z, - C’IMfi

+ (Mu + E’) (1 - ZR)] + (Mu + E’)[-Xw(Uo + Zq) + ZwXqI

+ (Z, - C’IIXwMq - XqMwl + (X, + A’l[Mw(Uo + Zq) - MqZwl

E = gcos yo[Mw(Z, - C’) - Zw(M, + E’)]

+ gsi n yo[(Mu + E’IX, - (X, + A’IM,]

The numeratordeterminant is expanded in Equatron (C-6):

340 N,/bE = Aas3 + Bas’ + Cas + D, (C-7 1 where A, = Zg /U, E

+ (Mg E/Uo)[(Xu + A’lgsin yo - (2, - C’)gcos yo] 34 1 [S - (X, + A’)] -(sXfi + Xw) x8E -(Z, - C’) [dl - Z$ - zwl z6E

where A0 = ZgEM; + Mg (1 - Zt) E Be = X6E[(Zu - C’IM; + (1 - Z;)(MU + E’)]

+ ZgE[Mw - Mg(Xu + A’) + (MU + E’IXq]

+ Mg [-Z, - (1 - Zi) (Xu + A’) - Xm(Zu - C’)] E Ce = XgE[Mw(ZU - C’) - Zw(MU + E’)]

I- Mg [Zw(Xu + A’) - Xw(Zu - C’)] E + Z6E[-Mw(Xu + A) + Xw(Mu + E’)]

It should be notedfrom the mechanics of the above derivation that, had it been desirable to derive the transfer functions for anyone of the other controlinputs, it wouldhave been necessary only toreplace 6~ by theappropri- atederivative whenever 6~ appeared inthe transfer function. This knowledge canalso be applied to thelateral transfer functions about to be derived. To make thefollowing transfer functions applicable to aileron deflection 6, instead of rudder def I ect ion BR, it is necessary on I y to rep lace BR by 6~ whenever 6~ appears, and to replace Yg R, LgR, and NgR by YgA. LgA. and N6A respectively.

Fol lowingthis approach, the lateral transfer functions for rudder def lections, (8p,=O), became:

342 D2 = s(As4 + Bs3 + Cs2 + Ds + E) (C-9 1

N /BR = s(A s3 + B s2 + C s + DB) (C-10) B B B B

343 - LB (s2 - sLp) Theabove transfer functions completely describe the airframe within the limits of theassumptions made in Appendix A of this study.

By substituting ju for s thetransfer functions are transformed into the frequencydomain. If one thenwrites the numerator anddenominator each as a magnitudeand phase angle and plots the ratio of magnitudesand the difference in phaseangles against frequency, the results indicate the magnitude and phase relationshipof the aircraft response to a sinusoidalcontrol input of unitmagnitude at anygiven frequency. These "Bode" plotsare quite useful invisualizing frequency regions of excessive aircraft response (regions of poordamping), the frequency above which aperiodic motions decrease rapidly inamplitude, frequency regions which should be avoided by structural and controlsystem resonances because of the possibility of coupling, and, if measured inflight, the presence and effectof non-linearities. Control sys- temand autopilotdesigners find such plots particularly useful when the scalesare logarithmic since among otheradvantages the amplitude of theac- tualaircraft response is merely subtracted from the desired response to find the effective transfer function which the autopilot or controlsystem must SUPP I Y - The denominator of thetransfer function is the Laplace transform of the characteristicequation of the system. It canbe thought of as representing thegeneral solution in the mathematical sense, for theresponse of the system. The numeratorterms, combined with the time history of thecontrol surface motions,are responsible for the particular solution. To obtainthe time history of theresponse to a particularcontrol surface input it is necessary to obtain theinverse transform of thetransfer function. Fairly extensive tables of Laplacetransforms are available and the proper form can often be found there- in. It may, however,be necessary to perform a partialfraction expansion to reduce the transfer function to a sum of simplerfunctions whose inversesdo appearin a table.For details, the reader is referred to a standardtext on controlsystem design such as Savant (Ref.100).

APPENDIX D

347 SIMPLIFIED RESPONSE CHARACTERISTICS

The equationsdeveloped in Appendices A, B, and C, althoughalready linearized,are still difficult to solve byhand. (Forthose with digital computer facilities, the programspresented in the present report may beused to obtain timesolutions to thecomplete system.) For preliminary design purposes it is desirable to be able to make rapid, approximate calculations of the effect of geometric or loading changeson the dynamicsof the aircraft. It istherefore of interest to determine the extent to whichthe equations and their solutions canbe simplifiedbefore the characteristic behavior is lost.

Inthis connection, it is convenient to assume that

It has long been common knowledge thatfor the longitudinal casethe phugoid oscillation is accompaniedby little or nochange invertical velocity while the short period oscillation takes place at constant speed if themagnitude of the pitchangle change iskept small. This suggests that the general three-degree- of-freedomsystem can be approximated by two two-degree-of-freedom systems.

Equations A-34 may be written

lj+ge=xuu+xwW

I-uq=zu+zw+76&E0 U W E

q=Mu+Mq+Mww+MGP+MU 6 . 9 6E E It should be notedthat the pertubation velocities, i.e., - u,, w, and q,are zero when theunperturbed variable is constantduring a particular maneuver. Takingadvantage of this fact, one finds that the system (1) reduces to three equations in TWO unknowns, one equationof which is therefore redundant. For theshort period approximation, the equation describing linear acceleration alongthe x-axis is redundant. (The accelerationis zero and theremaining forcesreduce to an identity.)Similarly for the phugoid case, theequation describingpitch is redundant.

d-Uoq=Zw+~6W EE 4=Mq+Mw+MiJ+M 6E 9 W 6E for the short period mode and

ic=xu-ge U -ui = z u + (3) u $:E for the phugoid mode.

348 Equations (2) and (3) have been written to reflectthese considerations. Transformed to the frequency domain these become

short period and

phugo id .

Substitution of the first equation of (4) into the second yields

(MG+Mg IS + (MwZ6 -M6 Z e E EW *= E (6

The oscillatoryroots of thedenominator of (6) describethe short period damping and frequency.Since ingeneral the transfer function of an oscillatory mode canbe described by

I 2 2 s +2rw s+w nn it is readily seen thatthe frequency of theshort period mode is given by

whilethe damping ratio is

U0MG f Zw + M 5 =- SP 2wn

349 u= Z6EdUO (11) 6E s 2 - xus - -zug UO from which it isapparent that

(12)

(13)

The approximatelatera!-directional characteristicsare obtained in a simi- larfash ion.The reader will recall tha t thesolution of the secondequat I on of A-35

(14) for motionrestricted to that about the x-axis is given on page137 as

r 1

The first term is thesteady state portion of the so lution and thetime required forthe transient solution to attain 63% of its fina I valueis

TheDutch roll is assumed forpurposes of approximationto consist of a yawingmotion about the z-axis. Thus the motion lies entirely in the x-y plane, i.e.,the bank angle remains constant and thereis noLr, V, or Lv.With the additionalassumptions that V = UoB, and r = 4, $J = -6, Y~Rand Yr = 0. equations A-35 reduce to

which when transformed becomes

thus

350 Unfortunately,the spiral mode is notreadily approximated with accuracy by a one-or-two-degree-of-freedom system. However,by discardingsmall quantities from thegeneral fifth order characteristic equation of lateralmotion and groupingthe remaining terms to matchthe expansion of

Ref. 17 was able to show that

Notethat unless the yaw dampingand theside force due to sideslipare both verylarge numerica I ly,the first term in the numerator may be neglected. At cruise,the last term is generally about 10" timesthe second term so that one common I y sees

The stability of the spiral mode may bededuced by examining the sign of the netdenominator since the numerator will almostalways be negative. Cnr byde- finitionis negative while Cgr is positive. The signof CQ dependsupon the dihedralangle but the aircraft is usuallyconfigured so that C~Risnegative. CnB depends-upon the area distribution in' the xz-p I ane but is usCa I ly p Gsitive. Thus, if CgBCnr islarger than CnBCg,, theaircraft will be spiraliy st able. It shouldbe noted, however, that the geometric changes needed to improve spiral stability will usuallyresult in poorer Dutch roll performance so that some compromise is necessary inthe absence ofan artificial stability system. Usua I ly one will opt for a veryslightly unstable spiral mode inorder to obtai n satis- factoryDutch rol I performance.

35 1

APPENDIX E

353 USE OF THE NON-LINEAR FORM OF THE EQUATIONS OF MOTION

There are occasions in the analysis of flight motions when onewould wish to studylarge departures from equilibrium,circumstances where the aerodynamic forces and moments arehighly non-linear, and the rare situations where there isextensive directional-longitudinal cross-coupling.The general equations* of coursedescribe these situations as well as they do those involving small departures from equilibrium.There are, however, no extensively-developed tech- niquesanalogous to thetransfer-function, root-locus procedures for examining thecharacteristics of solutions of systems of non-linear,partial differential equations.With the advent of verylarge digital computers, converting the equations to differenceequations for solution or using a variety of forward integrationtechniques became feasib!e.In order for such techniques to retain sufficientaccuracy when computingslowly decaying oscillations, however, ex- tremeprecision must be maintained. Inherently, the computation requires con- siderabletime on a large machine.

Conceptually, a simp I er approach is to employan analog computer. Here, too, a largemachine with many functiongenerators is required. Also theavail- ability of skilled analog programmers seems to belimited while the number of "canned" d i g i ta I programs ismultiplying. Thus theuser who finds it necessary toanalyze non-linear mot ons isadvised to secure a suitableprogram from others who haveemployed t.

* See equations A-8 and A-27. 354 APPENDIX F

355 SOMENOTES ON THE CONSTRUCTIONAND INTERPRETATION OF BODE PLOTSAND ROOTLOCUS DIAGRAMS

The use of Bode plots to studyairframe response dates from the early 1950's. The Bode plot was by then a familiar tool to thecontrol system engineer and when he was givenresponsibility for developingadvanced autopilots, it was natura! for him to employ a representationof the airframe dynamics which would facilitatehis task. How it does thisis outlined below.

The aircraftalone can beconsidered as one block in a combined aircraft- automaticcontrol system feedback loop.

2 J Aero- Response , Serm dynamic Aircraft t b Control -

Surface a

Figure F-1. Sample blockdiagram.

Each blockis described by one or more differentialequations, according to NewtonIs Second Law ofMotion or itselectrical equivalent. To combinethe characteristics of eachblock so as to form the characteristics of the overall system is quite difficult because a signal is modifiedin both phase andampli- tudein going through each block. By applyingthe Laplace transform

tothc iescribing differential equations oneobtains an algebraic representa- tionfor eachblock in the system. Itis then convenient to arrange this re- presentationin the form of a transferfunction, i.e., as a ratioof block response to blockexcitation. These transferfunctions can then be multiplied together to yield systemresponse to systemexcitation. It is a relatively straightforward procedure because each transfer function is merely a ratio of polynomialsin the Laplace operator s.

A transferfunction can be made todisplay additional ph sicalsignificance byallowing jw toreplace s, where w is a frequencyand j = k.The re- sultingtransfer function is then a ratio of productsof vector quantities, such as for example (3+4j)(5+6j) , (2+j 1 (7+5j1 356 The rulesfor reducing this complex quotientto its simplest form are (1) write each factor as an amp I itude anda phase angle, (2) mu I tip ly the numerator amp I i tudes together, (3) multiply the denominator amp I itudes together, (4) add thenumerator phase angles, (5) add thedenominator phase angles, ('6) formthe ratio of numerator to denominatoramplitudes (7) formthe difference between numerator and denominatorphase angles. The totaltransfer function is then representedby a single amplitude anda slngle phase angle. The numerical values of amp I itude andphase angle are computed at eachfrequency of interest. (The am I itude and phaseangle of the first factor in the exampleare Amp = h = 5 andphase tan-1 4/3.) By choosinga log-log or db-logfrequency representation to plot the ampli- tuderatio and alinear-log frequency plot for phase angle, onecan simply add amp1 i tude ratios andphase angles of components graphical ly to obtain transfer functionsof the enti re system. One can also do this for the factors in a transfer function.

Certain shape amp1 i tude ratios-frequency plots canbe shown to be associatedwith certain time responses.For example, thetransfer function of a simpleseries resistor-capacitor circuit

Ei" Eout

F i gu re F-2. Samp I e res i stor-capac itor c i rcu i t .

can be written

The differentialequation describing the current flow in this situation is of first order; hence, thetransfer function shown is said to be that of a first ordersystem. I f Ei is astep, Eo wi I I riseinstantaneously to the va I ue of Ei and then decay with ti me, reaching 37% of Ei in RC seconds. Such ti me response is then characteristic of first order systems.

It wi I I be noted that when w>>l/RC, IE,/Ei 1~1, i .e., independentof frequency and when w<

db =2O log(Eo/Ei 1, 357 thisrepresents a slope of +6db peroctave or +20 db per decade. The slope for largeval ues of w is 0 db peroctave and the changebegins inthe neighborhood of w=l/RC. The phaseangle will change a total 90' ingolng from w=O to a==.

If E; is made a sine wave of frequency wl, the plot of ]E /Ei I indicates that for wt=1/2RC, Eo/Ei = 1/2; for w'=1/IO RC,Eo/E] = 1/10,and ?he phaseangle is about--90°. For w'>l/RC, Eo/Ei = 1 and the phaseangle is zero.Thus the circuit transmits sine wave withfrequencies greater than 1/RC essentially un- changed in phase or amplitudebut differentiates, i.e., changesphase angle by 90'and amp I i tudeby the factor w, sine waves withfrequencies less than 1/RC.

Interchangingthe resistor and capacitorresults in the transfer function for a low frequencyintegrating circuit 1

I 1 - + jw RC

Note thatthe denominator of the transfer function is the same for bothcircuits. Thus this circuit wi II also exhibit the characteristic of a first order system in response to a step i n Ei :Eo wi I I reach 63% or E in RC seconds. j Systemsdescribed by a secondorder differential equation, i .e., a mass- spring-dampersystem or a inductance-capacitance-resistance system, yield transferfunction denominators of theform

where w isthe natural frequency of the system, thatis, the frequency at whichtRe system wou Id osci I I ate or resonate indefinitely if there were no damping orresistance. 5 isthe damping ratio.It describes the envelope of a decayingsinusoid:

On the Bode plot, a secondorder denominator factor wi I1 begin as a horizon- talline at low frequencies. The amplitude wi I I slowlyincrease and peak inthe neighborhood of wn, theratio of horizontal asymptote to peakheight varying directlywith 5. Beyond wn theamplitude falls off rapidlywith a finalslope of - 12 db peroctave. The total phaseang le change is I 80° withthe 90' point occurring at w=wn.

It wi I I be appreciatedthat any orderpolynomial canalways be factored to appearas a product of first andsecond orderfactors. Thus given a transfer functionone canalways graph it readilyby factoring it, graphingthe factors, adding,and plottingthe sum.

To obtain a reasonably good indication of thecharacteristic motions of a new ai rp lane oneneed on ly eva I uate the constants in the transfer functions and plot. The graph of the [6/S,l transferfunction wi II probablyexhibit two peaks, a high,sharp peak at low frequenciesand a modestpeak at much larger frequencies.These of coursecorrespond to thephugoid and short period 358 ~~~~ ~

modes respect i ve I y . The f requenc ies at wh i ch they occur and the i r dampi ng are i mmed i ate I y evi denl- fromthe p lot. The zerofrequency value of 8/15, Ts the steady state pi tchi tlg ve locitv wh i ch can be produced by a un i t e I evator def lection at a givenforward speed, -e.g., location,altitude, weight, and angle of attack. It is thus a measure of elevatoreffectiveness.

By measuring 6 and 6, as functions of tlrne in flight andperforming a harmon ic analysis of the time histories it is possible to construct an experimen- taI ly-determined l6/s,l transferfunction. This canthen be compared withthe one calculated from theequations of motion.Note that the later wi I I have two peaks and on I y two peaks. F I ight test records are common ly unre I i ab le at f re- quenciesless than 1 rad/sec. so thatthe phugoid is seldom evident. Distinct peaks may appearon the f I ight test Bode p lot at frequencies other than the shortperiod frequency. Thesecan be due to

( I) structural resonancesexcited by the ai rframe motion (2) inertialcross coupling (from the dutch rol I) (3) nonlinearitiesin the motion producing harmonics of theshort period mode or intermodulation with other modes (4) absence ofsignificant harmonic content in the 6, trace atthe particular frequency (a spurious peak, therefore) (5) poorqua I i ty data or dataprocessing.

Carefulanalysis, however, will usually reveal the source of theextra peaks. Onemay then compare the measuredvalues of theshort period andphugoid frequenciesand damping ratios with the predicted values. Knowledge ofthe frequencyand damping ratio also permit one to extract flight values of Cmq and CD if Cm, is known.

Thisvery brief discussion of the construction and utility of Bode plots is sufficient to point out the fact that they are not very efficient means of studying the effect on the motion of aircraft of varying the amount of control systemfeedback since a new plot must be made for eachvalue of feedback gain. The same is true for the variation in basic airframe dynamiccharacteristics resulting from changes in geometry or mass distribution. The root locusdiagram was developed to overcome thisdifficulty. As the name implies i.i shows onone figure,the trajectory the frequency anddamping of characteristic modes follow assystem parameters are changed.

Consider the transfer function

0 K(s+a 1 (s2+bs+c) T= s(s+d)(s+e)(s2+fs+g) thedenominator of the transfer function represents the characteristic equation of the system,e.y., theequation describing the free motion of thesystem (the responseindependent of controlinput). It is responsible for thegeneral solution of thesystem of differentialequations. The particularsolution comes from thenumerator.

It will beobserved that all values of s which makes thedenominator zero are solutions of the characteristic equation and therefore contribute a term of thetype eAt to thetime response. Since for theseroots the transfer function isundefined, denominator roots are called poles. Numerator roots are 35 9 appropr i ate I y calledzeros. It is customary to plotthese PO les and zeroson a graph whose abscissa is the real part ofthe s and whose ordinate is imaginary part.Poles are commonly depictedas x’s and zerosas 0’s. A firstorder root, e.g., (s+d), will always lie on theabscissa (A second ordersystem has two roots. They may be real,in which case they lie on theabscissa) or they may becomplex, in whichcase they are placed equidistant above and below the abscissa.

Any polewhich lies in the right half s-plane represents an unstablemotion. Zeros in the right half plane are significant in terms of thetype of motion only if thesystem depicted is a feedbacksystem. In this casethe zeros representthe location of the poles when thefeedback gain is made infinite. For zeros inthe right half plane then, the system will then become unstable at some finitevalue of feedbackgain. Knowledge ofthe location of the basic aircraft zeros is needed by designers in order to combine thecontrol systemchar- acteristics with those of the aircraft so as toobtain the desired response withoutunexpected instabilities. Note also that a zeroplaced on topof a pole will eliminate the motion caused by that pole from the time history of the particularvariable associated with the numerator (8 in e/% for example)but f rom no other time hi story.

A polelocated at s=-3, for example, means thatthere is a contribution to the t ime historygiven by e-3t. Thus, thefurther to the left the pole, the more rapid isthe subsidence.Conversely, a pole at s=3 means themotion has an unstab I e component described by e3’. Typ i ca I I y the sp i raI mode in aircraft lies slightly to the right. MIL F8785B requiresthat double amplitude in bank angle sha I I not be attained i n I ess thanI2 seconds. Since el .693=2 , TS must be more than l/0.141 or ~10.141.

Stableoscillatory modes, it will be recalled, have rootswhich can be expressed by

Figure F3 indicates how varyingeither frequency or damping ratio separately moves thepoles. It also shows thatthe product

The ordinateof the figure is wnJ1-S2 ca I ledthe damped natura I frequency. Thisis the frequency of oscillation which one would measure from flight records and is seen to depend on the damping ratio. Note that for a damping ratio of unity , the osci I I ation hasdecayed to a subsidence described by e-2wnt.

360 lmagi nry Axis +

Lines of constant time to half amplitude, run 1-Period increasing I Lines of Constant Damplag .Ratio \-?v-;/ >0 ;

%,/ 1 \ I I I.( '1 I Lines of constant damped period Lines of constant

"" undamped natural frequency, wn .-- -3 -Real Axis + II Time to half - amplitude increasing t+

Figure F-3. Variationsof frequency anddamping ratio.

36 1

APPENDIX G

363 .

LONGITUDINALSAMPLE CALCULATIONS

Presentedbelow is a step by stepprocedure for calculating the longitudinal stabilityderivatives for the Cessna 182 airplane. A tablecontaining the per- tinentgeometric dimensions of the airplane is given; geometric and aerodynamic datasuch as aspect ratio, downwash, and wing lift curveslope are estimated; and formulasfor the stability derivatives are delineated, with appropriate num- bersfor the Cessna 182. These formulas were takenfrom applicable sections in the text, and the derivatives were calculated only for the cruise condition.

S = 174 ft.2 St = 38.71 ft.2 SE = 16.61 ft.2 b = 35.88 ft. bt = 11.54 ft. X = 0.695 At = 0.65 c.g.located at 26.4% m.a.c. fuselagelength = 25 ft. max fuselagewidth = 4.17 ft. lengthfrom c.g. to tail quarter-chord = 14.6 ft. lengthfrom wing quarter-chord to tail quarter-chord = 14.6 ft. lengthfrom nose to wingquarter-chord = 6.84 f+ lengthfrom c.g. to winga.c. (chordwise) = 0.1163 ft. lengthfrom c.g. to winga.c. (vertical) = 1.67 ft. lengthfrom c.g. to thrust axis = 0.0 ft. -~ Table G-1. Pertinentlongitudinal dimensions for the Cessna 182. S I. mean aerodynamicchord c = = 174 - 4.86 ft. -b -35.83 c =" 38.71 - 3.35 ft. c =" 16*61- I .44 ft. t 11.54 E 11.54

35'83 - 7,378 I 1-54 - 3.44 2. aspect ratio AR = - = - ARt = - c -4.86 -I .44

3. incidenceangle thewing incidence angle was assumed to be 1.5O. the tail incidenceangle was assumed to be -3.0'.

4. angle of attack thewing angle of attack is assumed to be 1.5'.

5. wing CL the 2-D wing CL was obtainedfrom Ref. 7 from a plot shown inFigure I. A Reynolds Number of 5.7 x IO6 with an a=1.5O is used toobtain Cp,=O.39. The 3-D wing CL is now foundfrom - - Lp, - 0.39 - = 0.3068 cL - '1 + Z.O/AR 1+ 2.0/7.378 wing 0 25 (1/X~0.33.0~ 6. downwash angle E = 20.0 c ( 7) Lw (AR)0.725 t

364 7. horizontaltail a a = a - i + it - E: = 1.5-1.5-3.0-1.61 = -4.61O t W

8. tai I efficiency rlk = qt/q assumed.to be 0.85

9. 2-D lift curveslope The 2-D wing lift curveslope is taken from Ref. 7 in the linear region and is found to be 0.103 per degree. Similarly,the tail 2-D lift curveslope is found to be 0.1 perdegree (using 0009 section).

IO. efficiencyfactors It is now necessary to approximatethe induced-angle span efficiency factor el, for boththe wing and the tail as well as Oswald's efficiencyfactor, e, for thewing. For the wing, el = 1/(1+~1, where T is approximated(using a taper ratio of 0.695)from Figure 4 as 0.103; thus el = 1/(1+0.103) = 0.907. Forthe tail the same figureis again used (with a taperratio of 0.65) and ~=0.084,while el = 1/(1+0.084> = 0.92.Oswald's efficiencyfactor isalso taken from Figure 4, where e = 1/(1+6) and 6 is found to be 0.022, while e=0.98.

II. 3-D lift curveslope using steps 9 and IO, the 3-D lift curveslopes can now be calculated.

('La ('La 2-D - ('L ('L c1 'wing- ('L '2-D 57.3

- 0.103 = .085 per (0.103) 57.3 degree or

+ (3.1416) (.go71 (7.378) 4.61 per rad i an

= 0.0635 per 57.3) degree or 3.64 per rad ian

12. change in downwash with a

( l/O .695 1 20.0(0.085) (3(4-86) = 0.42 (7.378) 0*725 14.6

365 . , . "" ..._.,

13. 2-D wing CD The 2-Dwing CDa can be approximatedfrom Ref. 7, depending a on the angleof attack of the wing. If the angleof attack is relatively sma I I, it may be neglectedin many cases. For the cruise condition of the Cessna182 (CD 12-., = 0.0 a 14. elevator angle A procedure for approximating the elevator deflection required for equilibrium f I ight is given below. The taiI lift coefficient, based on the tail area, required for equilibrium flight can be approximatedby

C = CL Ltai I W = 0.0 13. The angle ofattack required to achieve this lift coefficient is

La+Jper degree The actual angle of attack of the tail from Ref. 7 is at=-4.6I0. Now the difference between areId and at is the effective angle of attack producedby 1eflecting the elevator. From Figure 13, dat sE "- 0.624, based on- = 0.43. Thus s, a -a ~ + = 7.710. d= req'd t 0.204 4.61 E dat 0.624

15. parasite drag, ('D 'airplane -" s , where S = wing area and f = CCD AT f Tr %IT is the drag coefficient of eacha i rp I ane component part and is multiplied by the area onwhich it is based and summed for all the components to obtainf.

= C S where CD = 0.0065 fw i ng Dw w' W 6 (taken from Ref.7 for RN=5.7 x 10 1; thus, 2 2 = 1 = 1.131 fwing (0.0065)(174 ft ft

max. fuselage height = 4.8 ft. height/length = 0.192 max. fuselage width = 4.2 ft. width/length = 0.168

366 I

Thus, f rom fuse I age data, CD~= 0.OQ63.Add i ng 20% for the canopy, CD, = 0.0756.Assuming a rectangular area, 2 AT = (4.8)(4.2) = 20.2 ft. , 2 2 = (0.0756)(20.2 ft. 1 = 1.525 ft. fuse I age ,

landing = (‘D An)mai n + (CDtn)nose geargear IT gear The value of f for themain gear is foundfrom Table 5 to be 0.74. Forthe nose gear,where thediameter = I foot and thewidth = 0.5 feet, CD, = 0.8 and AT = (0.5)(1.0) = 0.5 ft.2. Thus, 2 = 0.74 (0.8)(0.5) = 1.14 ft. landing + gear 2 = (0.007)(38.71) = 0.2715 ft. empennage = (‘D 71Aa)ernpennage 2 = 1.131 1.525 1.14 0.271 = 4.067 ft. tota I + + + Adding 10% for mutualinterference betweencomponent $arts and 5% for malI protuberances, fairplane = 4.677 ft. . Thus,

- = 0.0269.

(‘D ’ai rpI ane ~ (‘DIT)a i rp I ane f

St 16. airplane CL (‘L)ai rp lane = CL + CL vt=0.307 + 0.0129(-)(.85)38.7 I I74 tw = 0.309 CL2 17. airplane CD - +- (‘D)a i rp I ane (‘D ’airplane TeAR 71 - (0.307IL = 0.031 I - (3.1416)(.98)(7.378) Tz, T 18. airplane C and CT C = I , but T = 0. m- - Thus, Cm = 0.0 and CT = 0.0.

19. airplane CL CD CL = CD = = CT = 0.0 -, c rn T cm U U U U U U U U

- = 4.61 20. airplane C ‘L - (‘L )wi ng La a a dCd 2CL 0 21. airplane C -”+- (‘Da)ai rp I ane da TeAR ‘La Da 2(.309)(4.61) = 0.0 + (3.1416)(.98)(7.378) = 0.126 367 22. airplane C (‘rn ’airplane =cma - ‘m + ‘rn ma a a a wing tai I f us.

2cL (

(‘rn a ’wing = {[I *O + XR 57.3 a

57.3 a a

2(.309) (0.0) +

= {[1” + (3.1416)(.98)(7.378) 4.61 2( .309)

+ [(3.1416)(.98)(7.378)

= 0.048 ’wing (‘rna ds = CL (1- -) -’t ‘t (‘rn )tai I da Sw c ‘t a at = (3.64)- (.58)(*)(%)38 71 (.85) = I. I98

= 0.048 - 1.2 + 0.265 = -0.885 (‘m a )ai rp I ane ds ‘t ’ St 23. airplane C L& (‘L&)a i rp I ane

= (2.0)(3.64)(0.42)(%)(%)(.85) = I .74

24. airplane C CD = 0.0 D& & R R* ds t t ’t 25. airplane C m. (‘rn.)airplane = -2*ocL -da ---c c Sw ‘t a a a t = -2.0(3.64) I.42)(=)(&(~)(.85) 14.6 14 6 38.71

= -5.24

368 X* Rt 26. airplane CL = 2.0 - CL + 2.0 - c -St ('L ('L )a i rp I ane C C LtSw nt 4 4 a

+ 2.0(414 ,~6](3.64)(~)('.85) 6 38 61 = 3.9 74

27. airplane C = 0.0 D ('D )airp 1 ane 9 9 "Y

2.0 14 6 (3.64)(=)(.85) = -12.43 4.86 I74 - (--I-)4.86

canbe found from Figure 14 for cf/c = I .43 = .43 -3.35

= 0.06 perdegree or 3.41 perradian.

(a 1 cL can be foundfrom Figure 16 using a c /c ratioof .43. (a 1 f 6 ca. Thisratio gives a valueof 1 = -0.77. Usingthis (a6C value and the+ai I aspectratio, &3.44, (a 1 6 c,L = 1.04 fromthe lower part of Figure 16. Thus,

airplane = (3.41) (3.64)(l .04)(=)(.85) = 0.427 (5.73 ) I74 (5.73)

30. airplane CD To estimate CD~E, thetail surface in Figure 17 which is 6, most likethe one inquestion should be used. The num- L erica1value of canbe taken from plotsof CD versus a fordifferent elevator deflectidns. From Figure 17 fortail surface 5, CD perradian = 0.315. Thus,

369 - (‘Ds )airplane (‘Os )perradian nt E E 38 71 = (0.315>(-)(.85> = 0.0596 perradian I74

31. airplane Cm - ““CL C =-Im)14*‘ (0.427) = -1.28 6E (‘ms E )airp I ane 6E

370 LATERALSAMPLE CALCULATIONS

The followingis a detailedprocedure for calculatingthe lateral stability derivatives for the Cessna 182 airplane.In TableG-2the pertinent airplane characteristicsare given, from which certain geometric andaerodynamic data suchas effectivevertical tail aspect ratio, vertical tail lift curveslope, wing and horizontaltail aspect ratio, body side area, and fuselagevolume are calculated. Then theformulas for the stability derivatives, from the appro- priatesections in the text, are presented with the numbers corresponding to the Cessna 182 forthe cruise condition.

2 Sw = 174 ft. bw = 35.83 ft. CL = .307 r = I .730 zw = -1.835 ft. A = .7 Lb = 25 ft. x, = 7.0 ft. H1 = 4.8 ft. H2 = 1.8 ft. ba = 8.9 ft. Ca = 0.75 ft. SR = 6.95 ft.2 H = 4.85 ft. Sv = 18.57 ft.2 bv = 5.75 ft. R1 = 0.73 ft. zv = 2.82 ft. IIV IIV = .85 Rv = 14.8 ft. W = 4.02 ft. bh = 11.6 ft. sh = 38.71 ft.2 Ah = .66 Yi = 8.34 ft. CD~= ,0279 U = 219 ft/sec.

P = .00205 s I ugs/ft .3, density at 5,000 feet a I titude HNOSE = 2.7 ft., fuselageheight in nose region WNOSE = 2.8 ft., fuselagewidth in nose region HFCY = 3.5 ft., fuselage height at front of canopy WFCY = 3.6 ft., fuselagewidth at front of canopy LFCY = 3.12 ft., lengthalong body centerlinefrom nose to front of canopy LMH = 6.41 ft., lengthalong body centerlinefrom nose to pointof maximum fuselageheight HBCY = 2.9 ft., fuselageheight at back of canopy WBCY = 3.1 ft., fuselagewidth at back of canopy LBCY = 12.83 ft., lengthalong body centerlinefrom nose to back of canopy

Table G-2. Pertinentlateral dimensions for the Cessna 182.

1. effectiveaspect ratio and lift-curveslope of vertical tail b vz Ae = I .55 - - 2.76 sv From Figure 28, av = (C 1 = .0534/deg.= 3.06/rad.

2. aspectratio of winq and horizontaltail bh AR=" bwL - 7.378; (ARlh =(r)= 3.476 sW h

37 1 3. mean aerodynamicchord of wing

"-- sw - 4.86 ft. cw bw 4. estimatebodv side area

Thebody sidearea (S&) isestimated using four trape- zoi dsby the following formula:

- (HNOSE + HFCY)(LFCY) + (H + HFCY)(LMH - LFCY) 'B - 2.0 .o S 2 (HEY + 2R1)(Lb - LBCY) +. (H + HEY)(LBCY - LMH) +. 2.0 2.0

74.8 ft. 2

5. estimatefuselage volume

The fuselagevolume isestimated using four prismoids bythe following formulas:

Vl = LFCY[2.0(HNOSE*WNOSE+HFCY*WFCY) + HFCY-WNOSE + HNOSE WFCY]

V2 = (LMH-LFCY )C2.0(HFCY*WFCY+H-W) f H* WFCY + HFCY = W]

V3 = (LEY-LMH)[2.0(HBCY*WBCY+H*W) + H WBCY + W HBCYI

Now each ofthe stability derivafives will be calculatedusing the formula from the text which seems bestsuited for light aircraft.

6. C YB (C 1 = -.OOOl Irl = -.000173/deg = -.00991/rad

Y, winqI 'P Body ReferenceArea = -Ki (C 1 ( (cy Ifus La fus 1 B sW (CL~)~~~is assumed equal to O.l/rad. Ki from Figure 23 is 1.647. 2 Body ReferenceArea = (Fuselage V~lume)~/~= 38.19 ft. . Therefore, 38 19 (cy Ifus= -I .647(0. I)(-) = -.03616/rad. I74 B

372 K, fromFigure 24, is I .O S z a.O 9 v W (1 + -)- = .724 + I .53($) + .4 - + .009(.AR) = 0.802 9 W d where d is equal to H, the maximum fuselageheight.

Thus (Cy ItaiI = -.2619/rad.Therefore, B = -.00991 - .03616 - .2619 = -.3086/rad. (‘y ’total B

+ (AC 1 % cE (C, )w = e)r, if the tai I shape is ignored. B cE - 2 r from Figure 26 i s - .000238/deg , so (C, Iw = -.0236/rad. B

Assuming that K = 1.25, thisgives (C 1 = - .043 I3/rad. R w,r=o B sv zv (C I = -a = -.02184/rad. R v vs-Rv TI, B w bw From Table 9, valuesof (ACgBI1 = -.0006and (ACgB)2 = .00016 aregiven. Therefore,

= -.0236 - .04313 - .02184 - .0006 -I- .00016 (‘E ’total B = -.089/rad.

BodyArea Side ‘b ,V 8. C ” (CnB)totaI = -K n sW bW (‘yB)tai I W “B E;- From Figure 30, Kn = .002539/deg = 0.1455/rad. From the C calculation, = -.2619/rad. YB (‘yB’tai I Therefore, 74 8 25 14 8 = -.l455(-)- - (-.2619 - ) (‘nB)tota I 174 35.83 35.83 = .06455/rad.

373 9. cII P Since zero wi -

so, f rom Substituting g i ves

7.378+4 .O (Ca Iw= (-.4794) (.0279) = -.4643/rad. P 7.378+4.0 ] - Sb (ARlh + 4.0 (C 1 = 0.5 h(L)2(C 1 ah P ’W b~ [ ‘p a~=z.16.-,(ao)h (AR) h+4 .O1 From Figure34,using the horizontal tail aspect and taperratios of 3.476and 0.66, respectively,gives

= -.29/rad. (5 )a =27r PO Aga’in from Ref. 7, with a 0009 airfoi I, (ao)h = 5.73/rad, which, when substitutedin the aboveequation, gives (Cap h =- - .00324/rad.

Thekef ore, (C, 1tots I = -.4708/rad. P C (AC Ir Y IO. c c = (&CL + ca yP yP L (‘a p P )r=o From Figure 32,

rl L (2)= -.0795/rad. L Figure 33 gives

(AC 1 yp = .02743.Therefore, (‘a )r=o P (c = (-.0795)(.307) + (.027431(-.4708) = -.0373/rad. yP

(‘n ’total = (Cn Iw+ (Cn )v P P P

374 Thewing contribution is given by the following formula:

r. 2L (‘n ’w = ‘L AR+4cosAAR”4 [,,6 ( I+ -)cosA AR -1 tan12 A (3)CL h=Oo P Figure 35, as a function of wingaspect and taper ratio,gives

n (@)A=o~ = -.0588/rad.Thus, L (Cn Iw= (-.0588)(.307) = -.0180/rad. P The verticaltail contribution is given by

(z sina+Rvcosa V P

2 - (Z,,COSCX-Q V bW

30, 1 From Figure 36, - - 0.24,where ht is assumed approxi- a% mately equa I to zv.

” [zv-(zvco;;-Q sinal V - 9.30 1’ e=0.0 8% for zeroangle of attack.Therefore, (Cn )v - - .01119/rad.and (Cn Itotal= -.0292/rad. P P

12. c (‘yr)tota I = (CYr + (Cy r ’+ai I Yr ’ (C 1 = .143CL - .05 = -.0061/rad Yr RV = .2 I64/rad. (‘yr’tai I = -2.0 - (Cy 1tai I bw s

(‘yr’tota I = .2103/rad.

13. CR -

(CRr ’tota I - ’wing + ’tai I r r r co -r From Figure 38, (- ) = 0.2568/rad. Thus, cL

375 % - (‘2 ’wing - (<)CL = 0.0788/rad. Also, r Rv zv =-2.o ” = 0.0170/rad.Thus, (Car’tai I (‘y ’tai I bw bw $

(Cer)tota I = 0.0958/rad.

14. Cn (‘n ’tota I = (Cn Iw+ (Cn ItaiI r r r r 1+3X AR-6 I-X 2 (Cn Iw= -.33(=)Cg -.02(1- -- r 0 13 mlcL = -.009852/rad.

= - .09924/rad. (‘n ’total r

15. C Thevalue of CysA is assumed zerofor conventional Y light aircraft. &A

16. Ck &A &A From Figureobtainedis by firstconsidering the outboard

a value of C k6A ”- 0.786, then using the inboard edge of the aileron T

Yi m=.466 and, againfrom Figure 42, getting a W correspondingvalue of

”- 0.231. These two values are then subtracted, T outboardstation minus inboard station, to give a

C26A ”- (.786-.231) = 0.555over the extent of unit T antisymmetricalangle of attack. Now from Figure 41, T = 0.319.Therefore, 376 ... "

= 0.177/rad.

17. C "A From Figures 43and 44, K isdetermined to be -0.1537. Theref ore, C = 2.0(-.1537)(.307)(.177) = -.0167/rad. 6A

These controlderivatives are given by thefollowing 18. Y6R ' c"R1 "&R formu I as:

C =a~-sV Y6R sw - zv sv

'"R - a "sw bw

C = -a --sv Rv 6R swbw nV From Figure 46as a function of rudderarea to vertical tai I arearatio,

sR "- 0.374, it isdetermined that T = 0.574.Thus, sV thesethree derivatives are evaluated as follows:

C = . 187/rad. Y6R = .0147/rad.

C "&R = -.0658/rad.

377

APPENDIX H

379 LONGITUDINALPROGRAM

129 C GAMMA. IUSUALLV ZERO FOR LEVEL FLIGHlI- 130 c 131 132 23 133 13* i35 136 137 138 139 c 1k0 Lkl c 1k2 c 1k3 c 1k* c 1*5 1k6 1k7 1k11 1k9 150 151 152 153 15k 155 156 157 158 159 160 MI 162 163 164 165 lbb 167 168 169 170 111 172 173 174 175 176 117 178 119 110 181 182 183 18k 185 186 181 1118 189 190 191 192

c 193 c CALCULATION UF DIIIENSIOIIAL STI8ILlTVOERlVAllYES 1% c 195 c 0 AN0 OC IRE JUS1 CONSTANTS USED TO CALCULATE 1HE OIMENSIONII 196 c SlABILIlT DERIVATIVES. Iqr c 198 D-RHO*U.SAHS 199 CC.R"O*".S.CH/II. 200 c 201 c OIHENSIONALSTA8lL11V DEIIVATIVES 202 c 203 20k 205 206 207 208 20P 210 211 212 213 21k 215 216 217 218 219 220 c 221 C COEFFICIENlS FOR TUNSFER FUNCTION 222 c 223 c AI. A21 A). A4v A5 ARE CONSTANTS USED 10SIllPLlFV THE CALCYUTION 224 c OF 1HE OENOMlNAlOR lND NUMERATOII COEFFICIENTS. c 225 226 221 228 229 230 c 231 c 232 c 233 c 236 235 236 237 238 239 ZW 2*1 c 2k2 L 213 C 2kk c 2k5 2k6 2k7 2*8 2k9 250 251 c 232 C CCOSGM AN0 GSIMCM ARETHE PRODUlS OF THE ICtELERAllOM OUE IO c 253 C GRAVITY IASSUIIE - 32.2 Fl/SEC..2FOR lH1SALTllWf RANGE1 ALD THE 2% C COSINE AN0 SINERESsECllVELV OF (HE IHlTlAL FLIGHT PAIHISLE. c c 155 2ab

38Q 251 2S8 259 260 2bl 262 263 2b4 2b5 2bb 267 2b8 397 269 398 270 399 211 400 272 401 273 402 274 403 275 404 276 405 r 277 Vlb 278 407 279 W8 280 409 281 282 410 411 283 412 284 285 C 413 28b C I1 AN0 IK ARE CCLNTERS USEO TO OETERMINE THE MAXIKUI VALUE OF KU* 41b 287 CEPENDIhGC ON THE NW8611 OF NATURAL FREPUEHCIES IN BOTH THE 415 41b 280 C NUIIERATOR AN0 THE OEhOMlNATOR OF 1 PARTIWLLR TRANSFER AIKTION- 417 289 290 418 291 519 292 420 293 421 294 422 C 423 295 C 424 C 296 297 425 C 42b CO 3 1-1.5 298 IFlOAOSlOSIII1~GT~UCI~I~l 299 427 3 COnTIwE 340 *28 lFlM.hE.OICO TO 4 301 429 C 302 430 C IF IIO - 0. THERE IS NO UMACTERISTIC EPUATIOH. THEREFORE TM 303 431 C PROGRAM IS IERMINAT(0. 304 432 C 305 433 CALL Ell1 30 b .." 434 C 307 IFIKYF~EQ.lblGO m 18 435 C THE 4 'IF.STbTEIILhTS BELOY TELL MHlW SEI OFNUMERATOR 308 KYF-11 43b C COEFFICIENTS TO EVUUITE OEPENO~NG ON THE VALUE OFNUMER. 309 C 437 C 310 C GETROT IS USEO TO FINO ROOTS (IF PARTICULbRA NUMERATOR UEPEWING 438 311 C CN THE VALUE OF UJNfR. 439 312 UO 313 18 IFINU)IER.EU.lIGO TO 19 -1 314 IFLhUIIER.EP.PIG0 TO 20 442 315 1FINUMER.EPAICO TO 21 443 31b CALL GElROTlNTHS~IIN~RPN~RINI 4H 317 445 318 Ub 319 U7 310 H8

321 322 C 323 325 325 32b 321 328 329 330 331 332 333 334 rn0.n.n 4b2 335 &3 336 M4 337 -5 338 &b 339 350 31 342 343 355 345 35b 347 348 349 350 351 352 353 351 355 351 357 358 359 3bO C 488 361 C MI AN0 IN1 AREUSEO TO PPEVLNT HAVING ZERO pJL)SCRIPlSWEN 48V 3b2 C ULCULATlhG THE WMERATOR UO OE~OllNAT~GAINS FOR THE OOOE PLOT 490 3b3 C SU8IOUTlhE. 491 C 3b4 c C GETROT A SUMWTINf USING CTHER SUMOLRINES. ULCUATES 492 IS WIW. 365 493 C MOTS, OAWINC RATIOSI AW NATURILFUPUENCIESI AN0 THESE ARE 3bb 4P4 C TRANSF.ERRED 70 7UE RAIIPINE 8I USE OF I 'CM)ION. STATERENT. 367 c 495 3b8 C 49b 3b9 C VALUE m KYF. 491 C 310 C C MR OAMPIIG VATIC6 GREATER TWH aYElA ION-OSCILLAT~Y MCOEI THE 498 371 499 C FOLLOMIIIG FOUR Cvl& PREVENT TAKIPE THE SPUME RMT OF A NEUTIVE 372 500 C YIKOER YHEN CALCULITIND THe OWE0 NAlUrUiL FREQUENCI. IF TM 373 C MMPIHS VATIDS IPE GREATER THAN ONETHEN IHE OIMPEO k).lURU 374 sa C FREPUENCICS REMAIN 0.0. 502 375 503 C 37b rn*D."." m4 317 505 378 506 379 507 310 25 508 381 509 382 510 383 511 384 112 38 1 513 514 515 51b 511 518 519 520 '0 28 521

I UP 2 110 3 131 6 132 5 133 6 134 7 135 8 C 13 b 9 C 131 10 C 111 I1 C 139 12 C 140 13 C 141 14 C 142 15 C 143 1h C 1- 11 C 145 18 14b I9 c 141 20 148 21 149 22 150 23 151 24 152 25 153 2h 1% 21 111 28 15b 29 151 30 151 31 159 32 33 1b0 34 ILL 35 1b2 3b 1b3 31 I3 1b4 38 IbY 39 1M 4a I b1 41 168 *Z I be 110 43 Ill 172 113 114 115 llb Ill 14 111 119 110 15 111 181 113 114 181 lob 181 111 Ib 189 190 191 let U--PGOlRI1IIWP 193 TO~P-IZ.P~~~IIIZPUIPI 194 TI2P-I.b9Jl4lIIIIP*IMPl 195 w TO 15 196 17 WSP.O.0 191 W.O.0 198 199 1sV.-RG01R111 C 1P.O.O 200 9 Tl2sr..b93L47IlsP 201 10 TO5S?.2.995711SP 202 11 203 12 2M 13 205 P.CI4lICI51 I* 206 P.CI31lClSI 15 207 R.CI2IICISI 16 200 S.ClIIICI51 17 209 18 210 19 211 23 212 21 213 22 214 23 21s 24 216 25 217 2b 218 21 219 18 220 29 2221 30 222 31 223 32 22% 33 US 34 Ub IS" 22) 36 228 229 31 230 I!1v 231 40 232 41 233 42 2% 43 23s 44 23 b 45 237 4b 238 239 47 240 48 241 49 242 50 243 51 211 52 245 53 24 b % 257 55 248 56 249 57 250 58 151 59 252 60 253 61 254 62 255 61 25b IO 64

TO5I?.2.995711LSPYSPI 257 1 co 10 35 258 2 21 IAI-LOOTIIII 259 3 I., 2t.l C 4 261 C THIS SUBRWTINE FACIGRS A THIRD WOER POLYHOMIAL d1 A CLOSE0 FORM 5 262 C PROCEDURE GIVEN IN'INTPIIOUCTIOY IO THE lHEUR1 OF EWATIONS* BY b 263 C CCNKURICHT bN0 #MIFIE0 81 THC PROCEOURC GIVEN 1H ~SIAYOAOMATH T 2b4 C TABLES. BY WEIIICAL RUBBER CMPAM. 8 2b5 266 IO 2b7 11 268 12 269 13 210 14 211 15 272 16 173 11 274 10 275 276 19 271 20 278 21 279 22 280 23 281 24 212 25 233 2b 284 27 285 28 286 29 281 30 288 31 289 32 295 33 291 34 292 35 293 3b a4 31 295 38 296 39 297 298 299 300 301 302

383 C C 8T VSlNG mt C C

io 41 42 43 4+ 45 4b 47 48 49 YO 51 52 53 54 55 Yb 5? 58 59 60 61

c 5 b 7 s Io

384 LATERALPROGRAM

c I i29 c 2 i10 C 3 131 c 4 1 I? C 5 131 6 13b C 1 135 c 1 13b C 9 131 c IO 138 C 11 1Y9 C 12 I so c 13 151 C I* 142 C I5 163 c I6 Ikk C I1 1b5 C I8 166 C IP 141 c 2* 148 C 21 I (9 c 72 150 c 73 151 c Zk 152 C 25 1 S3 c 26 154 c 11 1HfAIRFllIE IS 1SSUNfO 10 If A RIGID BOW. 21 155 21 I 'd c 29 151 C 10 158 31 159 c 32 IM C 31 IbI C 36 162 c 35 163 c 36 164 c ?.I 165 C 38 1bb c 39 I b1 C 40 168 c 41 169 c k2 I 10 C b3 111 c 41 172 L '5 C L 46 L C +I C C 48 .9 C 50 c 51 C 52 c 51 c L 5b r L 55 c 56 C C 51 c 58 c 59 C 50 bI 67 6. 6b

65 66 bl 68 OV 10 11 C 12 C C 13 C c 7* C C 15 C L lb c 11 18 19 10 dl 82 83 -.. 84 212 8S C 213 8b c 216 81 L 215 0 c 216 49 211 90 218 91 219 92 220 Q3 221 9' 9' 222 95 223 PG 224 PI C 225 98 C 226 99 L 221 100 218 101 229 I07 230 103 231 IO* 232 105 C 2a3 ICb C 23b 101 C 231 IOU C 23b I ov 231 I IO 138 111 111 t?l 113 116 I15 IIb 111 118 I I9 I20 I21 122 12s 126 125 12b 121 126

1 C 2 C 3 C k C 5 C b 7 8 C 9 C THE OUTPuI FaBGrH NUIERAla LYO OEhOMINATOR IS PRIMED 111 A FORM IO C UHICH REWIRES TYI OSCILL4T~YMMES. IF WE a 8OlH OF THE MOES 11 C ARE hON-OYILLIluLY THEN IHE FOLLOMINGPROCEWRE IS YSEDS 12 C 11- 1HE OARPING 11.110 IS CHOSEH TO BE 1HE SMALLER WHINDE OF 13 C THE UEAL RmlS. SlllCE 1HlS Q.WlMILL 0011lll~lEIH lHE TIME 11 C OIIUIN IA YGAlJVL OUPlWS UTI0 MWLD 1M)ICAlE AN 15 C Ib C 17 C la c C 19 C 20 C 21 C 22 C 23 21 25 2b C 21 C 155 C 1% C 26 C C 29 C 151 C 10 C 158 C 31 C 159 C >2 C 1b0 33 C Ibl 3k c 1b2 35 C 1b3 3b C lbb 31 1b5 38 1b6 39 1b7 C 40 1b8 C 11 1b9 C 42 170 C k3 171 C 44 172 kS 11 3 kb 174 41 175 41 17b k9 177 50 178 51 I4 179 52 180 53 54 55 5b 57 58 59 bo C b1 C I IS A CCWIIER WlW D€lEIuIIHES THE NuI(IEA OF 110011 WlU HAVE 62 C MlH A RE4 AH0 UI IIIAGIHAW PART. bl C bk

b5 193 bb 19k b7 195 b8 19b 61 191 10 198 11 199 72 200 73 201 74 202 75 20 3 7b 20k 77 205 78 ZOb 79 207 an 208 81 209 82 210 83 211 1k 212 8S 213 86 21k 87 215 88 21b 89 217 qn 91 211 92 219 91 220 96 221 95 222 96 213 91 221 91 225 99 226 100 227 101 228 102 229 103 230 104 211 105 232 106 213 107 23k LOB 215 109 236 110 217 111 218 112 239 113 2w It1 2k1 115 212 116 213 I17 2k4 I18 245 119 24b 120 211 121 248 122 249 123 230 1% 251 125 252 126 253 127 2% I28 255 25b 387 1 251 2 258 3 259 4 260 Zbl 2K 263 2b4 2b5 Zbb 2b1 268 2b9 210 211 212 213 " 21k 19 215 20 27b 21 211 22 278 23 219 2k 280 25 281 282 283 28* 285 2110 281 288 289 290 291 292 293 294 295 296 ..X1 298 29P 300 301 302

9 IO II 12 13 14 I5 lb I1 18 19 20 21 22 23 2k 25 Zb 21 28 29 30 31 32 33 2: .-2: 3b 37 38 39

50 51 52 53 5k 55 5b 57 58 59 60 bI 62 b3 bk

-I - " I. ,. SUBIIOUTINE CUAOlCOF~RE~RIMl C C lHIS SUBRCUIINEFAClORS A SECMO DROER POLYNOMIU BY USING THE WIS IS A SUBROUIIN UHIW CALCUI 3 C WAORAllC FORRILL CONSrlUCT A BWE PLOT UhCE THE NI 4 C POLYHCIIIALS HAVE BEEN FACTMED. 5 WE ILINLIHE BY USING A .CM*DX' b 7 s 1 9 9 10 11 I2 I3 14 15 I6 IT 10 19 10 22 23 24 25 2b

io 31 32 33 34 35 3b 37 31 19 10 41 42 43

5'1 SI 52 53

1 2 3 6 5 6 1 d 9 la

389 w \D 0 LONGITUDINALOUTPUT LATERALOUTPUT /

...... " ...... n.... ".." ...... ".."...... IllUY 114111111 OlllllIllll """"

C.0 .-0~108000 CLI - -0.OltOOO Cas . O.OLb100 CV? - -0,O)IIW LIP - -0.~10100 CR - -0~0I1100 . ClL. - 0.110000 TU . 0.015100 CHI - 4.011100

...... ".e...... U...... n...... u...... u...... "I...... "...... e ...... I ...... ~ILlI*lllI AlRPLlhl WIIICIIIIIIICI """_""""" """_""""" *1.11"1*1 AII.1UI WlICll"llIlC5 ma - o.nozoo IIM"11 :IIWOOOOO n.11 u.smw C.CUSICA*~AI - 11.aoo000 b IBQ O.0110.b llM0 bhlh .11~.000010 1111 II.l0101)0 CICOIIOIYII 1I.aOOeDO (011111 . 0.W8lDO Id . 0.0 U - lll.OJCO ClollD +.11LOQO 1.1" 11.#100 L*IIIILIUI - 0.0 u .ll*.WOOOO Cnom - 4.110000 1'11 - I~+L.OWO c.IIIIOAUII - 0.0 11Mx11 - 0.0.lllo 111. kl.0000 111 . 0.0 111 .IlbI.01DO ...... "......

...... U...... "...... *...... *.*I..I..... lIIPO%U 10 SLllUO. DlllSC11Ol8 LII?uIII 10 IUODL. DI?,UIIDI *...... CLIM"...... ".....#...... "...... - 0.411ICO CDII. 0.011M00 Call - -1.111000 L - 1 1CC. 0.00010001 ...... LIIN - 0.111100 CLIM. 0.01G100 CIIN. -0.011100 I .1 &U. 0.00010000 * ...... """""""."~"___.____._^____..OLI.(I"I.L CO1,l1C11"ll IO& 1111 OIYIIIIIO. ILD WIIIIM . 011N . 1.0000 "11.1 - 11.1611 OIlll . l..,Il. OIIIl . 1.5.l,11 OIlll - 1.1111 ...... *I,., . 0.081. 111s1 . 11.111. "1111 . I1..,. I1 *,I11 . -1.11.1 . mttw - -w.im NPII~II - -IIO.PIIS IIII~LI- -..*.I. UIIIII - -n.xwa .....1...... 1..."..1..1...... *...* ......

.rnI'l5 - -O.b*.,l ., -I..*,,* ...... IfDIIIl : -11..1,,. ., 0.0 a(LI11.1 -0,011.0 ., 1.0

..I ...."I ...... "I.."...... e"...... "" ...... ""l".."."".."...... C" .... """".."" TIME RESPONSE PROGRAM

The time response program ais program which wiI I give a tabulated output of a transfer function due anto impulse ora step input by taking the inverse Laplace transform of the transfer function. When using this program there are two important restrictions: 1) If an impulse is used the order of the numerator polynomial must be lower than the order of the denominator polynomial. 2) If a step is used the order of the numerator pol ynomiaI must be I ess than or equaI to the order of the denominator polynomial. Use of the program requires the input of the variables listed below: MN - order of the numerator polynomial MD - order of the denominator polynomial ITYPE - whichindicates the type of response desired = 0 is the signal for an impulse = 1 is the signal for a step response

GAIND - coefficient of the highest order termin the denominator polynomial FORCE - the magnitudeof the input (If FORCE is 3 .O then the responsew i I I correspond toa 3' control surface i nput) NS(1) - coefficients of the numerator polynomiaI beginning with the lowest ordered term

imagi nary parts of the roots to the denominator

The output variables are definedin the program, and a sample output has been included at the end of the program. It should be noted thatall the pertinent input informationneeded for this program hasbeen included as output information in both the Longitudinaland Lateral programs preceding th is program.

39 1- I 2 3 k S b C 7 C 8 C V 10 C I1 C 12 C 13 Ik 15 1b I? I8 19 20 21 22 21 2k 25 2b C 27 C 28 C 29 30 31 32 33 3k 35 3b 31 38 39 k0 51 *2 C k3 C u C +5 56 41 k8 kV M

1 2 C 3 C k C 5 b 7 8 V IO 11

1 :..l....t...U...... "...... "...... " ...... 5 C k C PIIIPOSEI 5 C b C 1 C 0 C V C 10 C I1 C 12 C 13 C C 14 C IS Ib C C I1 C 18 C IV C 7s C 21 C 22 C 23 1.m DECREES C 2k C 25 C 26 C 27 C 28 C 29 C 30 C 31 C 32 C 31 C 3k C 35 C 3b C 31 C 38 C 39 C H) C +I C k2 C k3 C U C k5 C kb ICOOE 0 C kl - C 48 C kV C 50 C SI C 52 TIME C 53 C n 0.0 C IS C 50 0.10 5 51 C 0.20 c 0.30

0.40

0.10 APPENDIX I

393 1. Knight, Montgomery; andWenzinger, Carl J.: "Rolling Moments Due to Rolling and Yaw for Four Wing Models in Rotation." NACA TR-379, 1931.

2. Freeman, HughB. : "The Effect of Sma I I Ang Ies of Yaw and Pitch on the Characteristicsof Airplane Propellers." NACA TR-389, 1931.

3. Knight, Montgomery; and Noyes, Richard W.: "Span-Load Distributionas a Factorin Stability in Roll." NACA TR-393, 1931.

4. Weick, Fred E.; andWenzinger, CarlJ.: "Wind-Tunnel ResearchComparing LateralControl Devices, Particularly at High Angles of Attack. I--OrdinaryAilerons on Rectangular Wings." NACA TR-419, 1932.

5. Weick, Fred E.; and Wenzinger, Carl J.: "Wind-Tunnel ResearchComparing LateralControl Devices, Particularly at High Angles of Attack. I I I--Ord inary Ai Ierons Rigged Up 1Oo When Neutra I . NACA TR-423 , 1932. 6. Weick, Fred E.; and Harris, Thomas A.: "Wind-Tunnel ResearchComparing LateralControl Devices, Particularly at High Angles of Attack. IV--FloatingTip Ailerons on Rectangular Wings." NACA TR-424, 1932.

7. Dryden, Hugh L.; andMonish, B.H.: "The Effect of Area and AspectRatlo on the Yawing Moments of Rudders at LargeAngles of Pitch onThree Fuselages." NACA TR-437, 1932.

8. Weick, Fred E.; and Shortal, Joseph A.: "Wind-Tunnel ResearchComparing Latera.1Control Devices, Particularly at High.Angles of Attack. V--Spoilers and Aileronson Rectangular Wings." NACA TR-439, 1932.

9. Soul6, Hartley A.; and Wheatley,John B.: "A ComparisonBetween the Theoretical andMeasured LongitudinalStability Characteristics of anAirplane." NACA TR-442, 1933.

10. Weick, Fred E.; and Soule', Hartley A.; and Gough, Melvin N.: "A FIight Investigationof the Lateral Control Characteristics of Short Wide Ailerons And Various Spoilers with Different Amounts of Wing Dihedral.'' NACA TR-494. 1934.

11. Weick, Fred E.; andWenzinger, Carl J.:"Wind-Tunnel ResearchComparing LateralControl Devices, Particularly at High Angles of Attack. XII--Upper-SurfaceAilerons on Wings with Split Flaps."NACA TR-499, 1934.

12. Weick, Fred E.; and Noyes, Richard W.:. "Wind-Tunnel ResearchComparing LateralControl Devices, Particularly at High Angles of Attack. Xlll--AuxilaryAirfoils Used asExternal Ailerons." NACA TR-510, 1935.

394 13. Soul&, H.A.; and McAvoy, W.H.: "FI ight Investigation of Lateral Control Devices for Use With Full-Span Flaps." NACA TR-517, 1935. 14. Shortal, JosephA.: "Effect of Tip Shape and Dihedral on Lateral-StabiI ity Character i st i cs. NACA TR-548, 1 936. 15. Jones, Robert T.: "A Simplified Application of the Method of Operatorsto. the Calculation of Disturbed Motions an of Airplane." NACA TR-560, 1936. 16. Weick, Fred E.; and Jones, Robert T.: "The Effect of Lateral Controls in Producing Motion ofan Airplane as Computed from Wind-TunnelData." NACA TR-570,1 936. 17. Soule', Hartley A.: "Flight Measurements of the Dynamic Lqngitudinal Stability of Several Airplanesand a Correlation of the Measurements with Pilots' Observations Handlingof Characteristics.11 NACA TR-578, 1937. 18. Jonas, Robert T.: "A Study of the Two-Control Operationof an Airplane." NACA TR-579, 1937. 19. Weick, Fred E.; and Jones, Robert T.: "R6sum6 and Analysis of N.A.C.A. Latera I Control Research." NACA TR-605, 1937. 20. Jones, Robert T.: "The Influence of Lateral Stability on Disturbed Motions of an Airplane with Special Reference to the Motions Produced by Gusts." NACA TR-638, 1938. 21. Sherman, Albert: "Interference of Tail Surfaces and Wing and Fuselage From Tests of17 Combinations in the N.A.C.A. Variable-Density Tunnel. NACA TR-678, 1939. 22. Katzoff, S.: llLo.ngitude Stabilityand Control with Special Reference to S I i pstream Effects. NACA TR-690, 7 940. 23. lmlay, Frederick H.: "A Theoretical Study of Lateral Stabi I ity with an AutomaticNACA TR-693,1940. 24. Soul6, H.A.: "PreI iminary Investigation of the Flying Qual ities of Airplanes." NACA TR-700, 1940. 25. House, Rufus 0.; and Arthur R. Wallace: "Wind-Tunnel lnvest,igation of Effect of Interference on Lateral-Stability Characteristics of Four NACA 23012 Wings,an Elliptical and a Circular Fuselage,and Vertical Fins.!' NACA TR-705, 1941 . 26.. Jones, RobertT.; and Cohen, Doris: "An Analysis of the Stabilityof an Airplane with Free Controls." NACA TR709,1941. 27. Gilruth, R.R.; and Turner, W.N.: "Lateral Control Required for Satisfactory Flying Qualities Based on Flight Tests of Numerous Airplane." NACA TR-715, 1941. 395 28. Greenberg, Harry; and Sternfield, Leonard: "A TheoreticalInvestigation of the Lateral Osci I lations of an Airplane with Free Rudder with SpecialReference to theEffect of Friction." NACA TR-762, 1943.

29. Cohen, Doris: "A TheoreticalInvestigation of theRolling Oscillations of an Airplanewith Ailerons Free." NACA TR-787,1944.

30. Jones, Robert T.; and Greenburg, Harry:"Effect of Hinge-Moment Para- meterson Elevator Stick Forces in Rapid Maneuvers." NACA TR-798, 1944.

31. Ribner,Herbert S.: "Formulas forPropellers in Yaw and Chartsof the Side-ForceDerivative." NACA TR-819, 1945.

32. Kayten, Gerald G.: "Analysisof Wind Tunnel Stability and Control Testsin Terms ofFlying Qualities of Full-Scale Airplanes." NACA TR-825, 1945.

33. Sawyer, Richard H.: "FI ight Measurements of theLateral Control Charac- teri stics ofNarrow-Chord Ai I erons on the Tra i I ing Edge of a Fu I I - Span SlottedFlap." NACA TR-883, 1947.

34. Weil, Joseph;and Sleeman, William C.,Jr.: "Predictionof the Effects of PropellerOperation on the Static Longitudinal Stability of Single- EngineTractor Monoplanes withFlaps Retracted." NACA TR-941, 1949.

35. Sternfield, Leonard; and Gates, Ordway B., Jr.: "A Simplified Method for the Determination and Analysis of the Neutral-Lateral-OsciIla- tory-Stability Eoundary." NACA TR-943, 1949.

36. Johnson, Harold 1.: "FlightInvestigation of the Effect of Various Vertical-tailModifications on the Directional Stability and Control Characteristicsof a Propeller-DrivenFighter Airplane." NACA TR-973, .1950.

37. Curfam, Howard J.; and Gardiner,Robert A.: "Method forDetermining the Frequency-Response Characteristics of an Element or System from the System TransientOutput Response to a Known InputFunction." NACA TR-984, 1950.

38. McKinney, Marion O., Jr.:"Analysis of Means ofImproving the Uncon- trolledLateral Motions of Personal Airplanes." NACA TR-1035, 1951.

39. Donegan,James J.; andPearson, Henry A.: '!Matrix Methods ofDetermining the Longitudinal-Stability Coeffic-ients and Frequency Response of an Aircraftfrom Transient Flight Data." NACA TR-1070, 1952.

40. Stone,Ralph W., Jr.:"Estimation of the Maximum Angle ofSideslip for Determination of Vertical-Tail Loads inRolling Maneuvers." NACA TR-1136, 1953.

396 41. Schade, Robert0.; and Hassell, James L., Jr.: "The Effects on Dynamic Lateral Stability and Control of Large Artificial Variations in the Rotary Stability Derivatives." NACA TR-1151, 1953. 42. Martina, AlbertP.: "Method for Calculating the Rollingand Yawing Moments Due toRolling for Unswept Wings With or WithoutFlaps or Ailerons by Use of Nonlinear Section Lift Data." NACA TR-1167, 1954. 43. Donegan, James J.: "Matrix Methods for Determining the Longitudinal- Stability Derivatives ofan Airplane from Transient Flight Data." NACA TR-1169, 1954. 44. Campbel I, John P.; and McKinney, Marion O., Jr.: "A Study of the Problem of Designing Airplanes With Satisfactory Inherent Damping of the Dutch RolI Oscillation." NACA TR-1199, 1954. 45. Donegan, James J.; Robinson, SamuelW., Jr.; and Gates, Ordway B., Jr.: "Determination of Lateral-Stability Derivativesand Transfer-Function Coefficients from Frequency-Response Data for Lateral Motions." NACA TR-1225, 1955. 46. Riley, Donald R.: "Effect of Horizontal-Tail Span and Vertical Location on the Aerodynamic Characteristics an of Unswept Tail Assembly in Sides1 ip." NACA TR-1171, 1954. 47. Eggleston, John M.; and Mathews, CharlesW.: "Application of Several Methods for Determining Transfer Functionsand Frequency Response of Aircraft fromFI ight Data." NACA TR-1204, 1954. 48. Weick, Fred E.; and Wenzinger, Carl J.: "Effect of Length of Handley Page Tip Slots on the LateraI -Stabi I ity Factor Dampingin RoI I .I1 NACA TN-423, July, 1932. 49. Bamber, M.J.; and Zimmerman, C.H.: "Effect of Stabilizer Location Upon Pitching and Yawing Moments in Spins as Shownby Tests with the Spinning Balance." NACA TN-474, November, 1933. 50. Soul6, Hartley A.; and Wetmore, J.W.: "The Effect of Slotsand Flaps on-Latera I.Contro I of a LOW-W i ng M onop lane as Determinedin FI ight." NACA TN-478, November, 1933. 51. Bamber, Mi I lard J.: "Aerodynamic RoI ling and Yawing Moments Produced by Floating Wing-Tip Ailerons, AsMeasured by the Spinning Balance." NACA TN-493, March, 1934. 52. Shortal, J.A.: "Effect of Retractab e-Spoiler Location on Rollingand Yaw i ng-Moment Coeff i c i ents. I' NACA TN-499,J u I y , 1934. 53. Wenzinger, Carl J.; and Bamber, Mi I lard J.: "Wind-Tunnel Tests of Three Lateral Control Devices in Combination with a Full-Span Slotted Flap on an N.A.C.A. 23012 Airfoil." NACA TN-659, August, 1938. 397 54. Bamber, M.J.; and House, R.O.: "Kind-Tunnel Lnvestigation of Effect of Yay on Lateral-Stability Characteristics. !--Four NACA BO12 Wi,ngs of Various Plan Forms withand without Dihedral." NACA TN-703, Apri I, 1939. 55. Jones, RobertT. ; and Feh I ner, Leo F.: "Transient Effects of the Wing Wake on the HorizontalTail." NACA TN-771, August, 1940. 56. Pass, H.R.: "Analysis of Wind-Tunnel Data on Directional Stability and Control." NACA TN-775, September, 1940. 57. Barnber, Millard J.: "Effect of Some Present-Day Airplane Design Trends on Requirements for Lateral Stability." NACATN-814, June, 1941. 58. Recant, Csidore G.; and Wallace, Arthur R.: "Wind-Tunnel lnvest,Igation of Effect ofYaw of Lateral-Stability Characteristics. Ill--Sym- metrically TaperedWing at Various Positions on Circular FuseI,age with and without a Vertica I Tai I .I1 NACA TN-825, September,1941. 59. Donlan, C.J.; and Recant, I.G.: IIMethods of Analyzing Wind-Tunnel Data for Dynamic Flight Conditions.11 NACATN-828, October, 1941. 60. Jones, RobertT.: "Notes on the Stabilityand Control of Tailless Airplanes." NACA TN-837, December, 1941. 61. Ankenbruck, Herman 0.: "Effects of Tip Dihedral on Lateral Stability and Control Characteristics on Determinedby Tests of a Dynamic Model in the La.ngIey Free-Fl.ight Tunnel." NACA TN-1059,July, 1946. 62. Phillips, William H.: "An Investigation of Additional Requirements for Satisfactory Elevator Control Characteristics." NACA TN-1060, June, 1946. 63. Bishop, Robert C.; and Lomax, Harvard: "A Simplified Method for De- termining from Flight Data the Rateof Change of Yawing-Moment Coefficient with 'Sides1ip." NACA TN-1076,June, 1946. 64. Shortal, Joseph A.; and Maggin, Bernard: "Effect of Sweepback and Aspect Ratio on Longitudina I Stabi I ity Characteristics of Wings at Low Speeds." NACA TN-1093, July, 1946. 65. Drake, Hubert M.: "Experimental Determination of the Effects of Di- rectional Stability and Rotary Damping in Yaw on Lateral Stability and Control Characteristics." NACA TN-1104, July, 1946. 66. Spahr, J. .Richard: "Lateral-Control Characteristics of Various Spoiler Arrangements as Measuredin Flight." NACA TN-1123, January,1947. 67. Purser, Paul E.; and Spear, Margaret F.: llTests to Determine Effects of Slipstream Rotation on the Lateral Stability Characteristics of a Single-Engine Low-Wing Airplane Model." NACA TN-1146, September, 3 946.

398 68. Harper, Char1 es W.; and Jones, Arthur L. : "A Comparison of the LateraI Motion Calculated forTailless and Conventional Airplanes." NACA TN-1154, February, 1947. 69. Malvestuto, Frank S., Jr.: flFormulas for Additional-Mass Corrections to the Moments of Inertia of Airplanes." NACATN-1187, February, 1947. 70. Sawyer, Richard H.: "FI ight Measurements of the Lateral Control Char- acteristics of Narrow-Chord Ailerons on Trailing the Edge of a Full-Span Slotted Flap." NACA TN-1188, February, 1947. 71. Hunter, P.A.; and Vensel, J.R.: "A Flight Investigation to Increase the Safety of a Light Airplane." NACA TN-1203, Ma'rch, 1947. 72. Hanson, Carl M.; and Anderson, Seth B.: "Flight tests ofa Double- Hinged Horizontal Tail Surface with Reference to Longitudinal-Sta- bility and Control Characteristics." NACA TN-1224, May, 1947.

73. Wal lace, Arthur R.; Rossi, Peter F.;We1 Is, Eva lyn G. : ffWind-Tunnel I nvest igat ion of the Effects andof F PowerI aps on the Static i - Long tudinal Stability Characteristics aof Single-Engine Low-WingAir- plane Model." NACA TN-1239, April, 1947. 74. Rathert, George A.,Jr.: llFlight Investigation of the Effects on Airplane Static Longitudinal Stability aof Bungee and Engine- Tilt Modifications." NACA TN-1260, May, 1947. 75. Bates, William R.: "Collection and Analysis of Wind-Tunnel Data on the Characteristics of IsolatedTail Surfaces with and without End Plates." NACA TN-1291, May, 1947. 76. Murray, Harry E.; and Wells, Evalyn G.: "Wind-Tunnel Investigation of the Effect ofWing-Tip Fuel Tanks on Characteristics of Unswept Wings in SteadyNACA TN-1317, June, 1947. 77. Tamburello, Vito; and Weil, Joseph: "Wind-Tunnel Investigation of the Effect of Powerand Flaps on the Static Lateral Characteristics of a Single-Engine Low-Wing Airplane Model." NACA TN-1327, June, 1947. 78. Hagerman, John R.: llWind-Tunnel Investigation of the Effect of Power and Flaps on the Static Longitudinal Stabilityand Control Char- acteristics of a Single-Engine High-Wing Airplane Model." NACA TN-1339, July, 1947. 79. Maggin, Bernard: "Experimental Verification of the Rudder-Free Stability Theory for an Airplane Model Equipped witha Rudder Having Positive Floating Tendenciesand Various Amounts of Friction." NACA TN-1359, July, 1947.

399 80. Hagerman, John R.: Wind-Tunnel Investigation of the Effect of Power and Flaps on the Static Lateral StabiI ity and Control Characteristics of a Single-Engine High-Wing Airplane Model." NACA TN-1379, July, 1 947. 81. Fischel, Jack; and Ivey, Margaret F.: "Cot lection of Test Data for Lateral Control with Ful I-Span Flaps." NACA TN-1404, April, 1948. 82. Tosti, Louis P.: "Low-Speed Static Stability and Damping-in-Roll Characteristics of some Sweptand Unswept Low-Aspect-Ratio Wings." NACA TN-1468, October,1947. 83. Polhamus, Edward C.; and Moss, Robert J.: "Wind-Tunnel Investigation of the Stabilityand Control Characteristics ofa Complete Model Equipped with a Vee Tail." NACA TN-1478, November, 1947. 84. Lomax, Harvard: "Sides1 ip Angles and Vertical-Tai I Loads Developed by Periodic Control Deflections." NACA TN-1504, January, 1948.

85. Hunter, Paul H.: "Flight Measurementof the Flying Qualities of Five Light Airplanes." NACA TN-1573, May,1948. 86. WeiI, Joseph; and Sleeman, William C., Jr.: "Prediction of the Effects of Propeller Operation on the Static Longitudinal Stability of Single-Engine Tractor Monoplanes with Flaps Retracted." TN-NACA 1722, October, 1948. 87. Schneitez, Leslie E.; and Naeseth, Rodger L.: "Wind-Tunnel Investiga- tion at Low Speed of the Lateral Control Characteristics of Ailerons Having Three Spansand Three Trailing-Edge Angles aon Semispan Wing Model.11 NACA TN-1738, November,1948. 88. Johnson, Harold S.: "Wind-Tunnel Investigation of Effects of Tail Length on the Longitudinaland Lateral Stability Characteristics of a Single-Propeller Airplane Model." NACA TN-1766, December, 1948. 89. Kauffman, Smith, Liddell, and Copper: "Flight Tests ofan Apparatus for Varying Dihedral Effect in FI ight." NACA TN-1788, December, 1 948. 90. Goodman, Alex; and Fisher, Lewis R.: "Investigation at Low Speeds of the Effectof Aspect Ratio and Sweep on Rolling StabilityDeriva- tives of Untapered Wings." NACA TN-1835, March, 1949.

91. Curfman, Howard J.,Jr. ; and Gardiner, RobertA. : "Method for Deter- mini,ng the Frequency-Response Characteristics anof Element or System From the System Output Responsea Known to Input Function." NACA TN-1964, October, 1949.

400 92. Mazelsky, Bernard; and Diederich, Frank1 in W. : "Two Matrix Methods for Calculating Forcing Functions fromKnown Responses." NACA TN-1965, October, 1949. 93. McKinney, Marion O., Jr.: "Analysis of Meansof Improving the Uncon- trol I ed LateraI Mot ions of PersonaI Ai rp lanes." NACA TN-I 997, December, 1 949. 94. Mokrzycki, G.A.: "Application of the Laplace Transformation to the Solution of the Lateral and Longitudinal Stability Equations." NACA TN-2002, January, 1950. 95. Bird, John D.; and Jaquet, ByronM.: "A Study of the Use of Experi- mental Stability Derivativesin the Calculation of the Lateral Disturbed Motions ofa Swept-Wing Airplaneand Comparison with FI ightNACA TN-2013, January,1950. 96. Mazelsky, Bernard; and Diederich, Franklin W.: "A Method of Deter- mining the Effectof Airplane Stability on the GustLoad Factor." NACA TN-2035, February, 1950. 97. Schade, Robert 0.: "Free-Flight-Tunnel Investigation of Dynamic Longi- tudinal Stability as Influencedby the Static Stability Measured in Wind-Tunnel Force Tests Under Conditions of Constant Thrust and Constant Power." NACA TN-2075,Apri I , 1950. 98. Murray, Harry E.; and Grant, Frederick C.: "Method of Calculating the Lateral Motions of Aircraft Based on the Laplace Transform." NACA TN-2129, July, 1950. 99. Johnson, Harold S.; and Hagerman, John R.: "Wind-Tunnel Investigation at Low-Speed of the Lateral Control Characteristicsof an Unswept Unfapered SemispanWing of AspectRatio 3.13 Equipped with Various 25-percent-chord Plain Ailerons." NACA TN-2199, October, 1950. 100. Shinbrot, Marvin: "A Least Squares Curve Fii-ting Method withAppli- cations to the Calculationsof Stability Coefficients From Tran- sient-Response Data." NACA TN-2341, April, 1951. 101. Campbell, Hunter, Hewes,and Whitten: "Flight Investigation of the Effect of Control Centering Spri,ngs on 'the j raApparent I Stab i I i tySp of A Personal-Owner Airplane." NACA TN-2413, July, 1951. 102. Marino, Alfred A.; and Mastrocola, N.: "Wind-Tunnel Investigation of the Contribution ofa Vertical Tail To the Directional Stability of a Fighter Type Airplane." NACA TN-2488, January, 1952. 103. Goodman, Alex: "Effects of Wind Position and Horizontal-Tail Position on the Static StabiI ity Character istics of Models Unsweptand 45' Swept Back to Mutual Reference." NACA TN-2504, October, 1951. 104. Shrinbot, Marvin: "A Description and a Comparison of Certain Nonlinear Curve-Fitting Techniques with Applications to the AnalysisTran- of sient-Response Data." NACA TN-2622, February, 1952. 40 1 105.Donegan, James J.: "Matrix Methods for Determinrngthe Lqngitudinal- Stability Derivatives of an Airplane from Transient Flight Data." NACA TN-2902, March, 1953.

106. Briggs,Benjamin R.; andJones, Arthur L.: "Techniques forCalculating Parameters of Nonlinear DynamicSystems from Response Data." NACA TN-2977, July, 1953.

107. Eggleston,John M.; and Mathews, Charles. W.: "AppI icationof Several Methods forDetermining Transfer Functions and Frequency Response of Aircraftfrom Flight Data." NACA TN-2997, September, 1953.

108. Donegan, J .J .; andRobinson, S. W., Jr. ; andGates, Ordway B., Jr. : "Determinationof Lateral Stability Derivatives and Transfer-Func- tion Coefficients from Frequency Response Data for Lateral-Motions." NACA TN-3083, May, 1954.

109. Fisher,Lewis R.: "Some Effectsof AspectRatio and Tai I Le.ngth on theContribution of a Vertical Tail to UnsteadyLateral Dampi,ng and Directional Stability of aModel OscillatingContinuously in Yaw."NACA TN-3121, January, 1954.

110. Canning, ThomasN. : "A Simp le Mechan ica I Ana lpgue for Studyi.ng the Dynamic Stability of Aircraft Habing Nonlinear Moment Characteris- tics. NACA TN-3125, February, 1954.

111. Gates, Ordway B., Jr.; and Woodllng, C.H.: "A Mefhod forEstimatlng Variations in the Roots of the Lateral-Stability Quartic due to Changes in Mass and AerodynamicParameters of anAirplane." NACA TN-3134, January, 1954.

112. Fisher,Lewis R.; and Fletcher, Herman S.: "Effectof Lag Sidewash on the Vert ica I-Ta i I Contribution to Osc i I latory Dampi,ng in Yaw of Airplane Models." NACA TN-3356, January, 1955.

113. Bates, Wi I I iam R.: "Static Stabi I ity of Fuselages Havi,ng a--Relatlvely Flat CrossSection." NACA TN-3429, March, .1955.

134. Letko, Wi I liam;and Williams, JamesL.: lfExperimentalInvestigation at Low Speed of Effects of FuselageCross Section on Static Longi- tud ina I and Latera I Stab i I ity Characteristics of Mode Is Hav i,ng 00 and 450Sweptback Surfaces." NACA TN-3551,December, 1955.

115. Fisher,Lewis R.; and Lichtenstein,Jacob H.; and Williams,Katherine D.: "A Preliminary tnvestigation of the Effects of Frequency and Am- p I itude on the Ro I I ing Derivatives of an Unswept-Wing Mode I Oscl I- Iati.ngin RoI I .I1 NACA TN-3554, January, 1956.

116. Wolhart,Walter D.; and Thomas, David F., Jr.:flStatic Longitudinal and Lateral Stability Characteristics at Low Speed of Unswept-Mid- wingModels Having Wings with an Aspect Ratio of 2,4, or 6." NACA TN-3649, May, 1956. 402 117. Weick, Fred E.; and Abramson, H. Norman: [[lnvestlgation of Lateral Control Near the StatI . Ana I ysis For Required Lqng ina itud I Trim Characteristics and Discussion of Des.ign Variables.11 NACA TN-3677, June, 1956. 118. Klawans, Bernard B.: "A Simple Method for Calculating the Characteristics of the Dutch Roll Motion ofan Airplane." NACA TN-3754, October, 1956.' 119. Gault, Donald E.: "A Correlation of Low-Speed, Airfoil Section Stalling Characteristics with Reynolds Number and Airfoi I Geometry." NACA TN-3963, March, 1957. 120; Pratt, Kermit G.; and Bennett, Floyd V.: "Charts for Est imati,ng the Effects of Short-Period Stability Characteristcs on Airplane Ver- ti ca I -Acce I erat ionand Pi tch-Ang I e Response i n Continuous Atmospheric Turbulence.11 NACA TN-3992, June, 1957. 121. Eggleston, John M.; and Phillips, William H.: "A Method for the Cal- culation of the Lateral Responseof Airplanes o Random Turbulence." NACA TN-4196, February, 1958. 122. Gates, Ordway B., Jr.; and Woodling, C.H.: "A Theoretical Analysis of the Effect of EngineAngular Momentum on Longitudinaland Directional - Stab i I i ty in Steady RoII i ng Maneuvers .I1 NACA TN-4249,Apri I, 1958. 123. Crane, Robert M.: "Computation of Hi,nge-Moment Characteristics of Hor- izontal Tai Is from Section Data." NACA-WR A-11, Apri I, 1945. 124. Holtzclaw, R.N.; and Crane, R.M.: "Wind-Tunnel Investigation of Ailerons on a Low-Drag Airfoil. I I ]--The Effect of Tabs." NACAWR A-18, November; 1944. 125. Laitone, Edmund V.; and Summers, James L.: "An Additional Investigation of the High-speed Lateral-Control Characteristics of Spoilers." NACA WR A-21, July, 1945. 126. Spahr, J. Richard; and Christophersen, Don R.: "Measurements in Flight of the StabiI ity, LateraI -Control, and Sta I I ing Characteristics of an Airplane Equipped with Ful I-SpanZap Flaps and Spoiler-Type Ailerons." NACA WR A-28, December, 1943. 327. Spahr, J. Richard; and Christophersen, Don R.: "Measurements in F1 ight of the Lateral-Control Characteristics anof Airplane Equipped with Full-Span Zap Flaps and Simple Circular-Arc-Type Ailerons.11 NACA WR A-32,September 7 944. - 128. Crane, Robert M.; and HoItzcIaw, Ralph W.: llWind-Tunnel Investigation of Ailerons ona Low-Drag AirfoiI. t--The Effect o? Ai leron. Profi le." NACA WR A-55, January,1944.

403 129. Clousing, LawrenceA.; and McAvoy, William H.: "Fl,ight Measurements of the LateraI Control Characteristicsof an Airplane Equipped with a Combination Aileron-Spoiler Control System.1' NACAWR A-68, Sep- tember, 1942. 130. Turner, William N.; and Adams, Betty: "Flight Measurements of the Effect of Various Amounts Aiof leron Droop on the Low-Speed Latera I-Control Characteristics of an Observation Airplane." NACA WR A-79, August, 1 943. 131. Hollingsworth, ThomasA.: "Investigation of Effect of Sideslip on Lateral- Stability Characteristics. tl--RectangularMidwing on Circular Fuse- lage with Variations in Vertical-Tail Area and Fuselage Length with and without Horizontal Tail Surface." NACA WR L-8, April, 1945. 132. Fehlner, Leo F.; and MacLachlan, Robert: "Investigation of Effect of Sideslip on Lateral Stability Characteristics. I--Circular Fuselage with Variations in Vertical-Tail Area and Tail Length with and without Horizontal Tail Surface." NACA WR L-12, May, 1944. 133. Hollingsworth, ThomasA.: "Investigation of Effect of Sideslip on Lateral Stability Characteristics. Ill--Rectangular Low-Wing on Circular Fuselage with Variationsin Vertical-Tail Area and Fuselage Length with and without Horizontal Tai I Surface." NACA WR L-17, Apri I, 3 945. 134. Ribner, Herbert S.: "Notes on the Propellerand Slipstream in Relation to Stability." NACA WR L-25, October, 1944. 135. Sjoberg, S.A.: "Flight Tests of Two AirplanesHaving Moderately High Effective Dihedral and Different Directional Stabilityand Control Characteristics." NACA WR L-40, October, 1945. 136. Swanson, Robert S.; and Priddy, E. Laverne: llLifting-Surface-Theory Values of theDamping in Roll and of the Parameter usedin Estimating Aileron Stick Forces." NACA WR L-53, August, 1945. 137. McKinney, Marion O., Jr.: llExperimental Determinationof the Effect of Negative Dihedral on Lateral Stabilityand Control Characteristics at High Lift Coefficients." NACAWR L-54, January, 1946. 138. Campbell, John P.;and Paulson, John W.: "The Effects of StaticMargin and Rotational Damping in Pitch on the Longitudinal Stability Charac- teristics of an Airplane as Determinedby Tests of a Model in the NACA Free-Flight Tunnel." NACA WR L-55, January, 1944. 139. Drake, Huber+M. : "The Effect of Lateral Area on the I LateraStabi I ity and Control Characteristics anof Airplane as Determined by Tests of a Model in the Langley FreeFI ight' Tunnel.I1 NACA WR L-103, February, 1946.

404 140. Purser, Paul E.; and McKinney, Elizabeth G.: llComparison of Pitching Moments Produced by Plain Flaps and by Spoilers and Some Aerodynamic Characteristics of an NACA 23012 Airfoil with Various Types of Aileron." NACA WR L-124, April, 1945. 141. Vogeley, A.N.: "CI imb and High Speed Testsof a Curtiss No. 714-IC2-12 Four-Blade Propeller on the Republic P-47C Airplane." NACAWR L-177, December, 1944. 142. McKinney, Marion O., Jr.; and Maggin, Bernard: "Experimental Verifi- cation of the Rudder-Free StabilityTheory for an Airplane Model Equipped with RuddersHavi ng Negative Float i ng Tendency and Neg I i- gible Friction." NACA WR L-184, November, 1944. 143. Ribner, Herbert S.: "Formulas for Propellers in Yaw and Charts of the Side-Force Cerivative." NACA WR L-217, May, 1943. 144. Bailey, F.J., Jr.; and O'Sul livan, WiI liam J.: "A Theoretical Analysis of the Effect of Aileron Inertia and Hinge Momenton the Maximum Rolling Acceleration of Airplanes in Abrupt Aileron RoIIs.~' NACA WR L-302, February, 1942. 145. Donlan, Charles J.: llSome Theoretical Considerations ofLongitudinal Stabi I Ity in Pouer-on FI ight with Special Referenceto Nind-Tunnel Testing .I1 NACA WR L-309, November, 1922. 146. Harris, Thomas A.; and Purser, Paul E.: "Wind-Tunnel Investigation of Plain Ailerons for a Wing with a Full-Span Flap Consisting ofan Inboard Fowler and an Outboard Retractable SplitFlap." NACA WR L-317, March, 1941. 147. Imlay, Frederick H.; and Bird, J.D.: ltWind-Tunnel Tests of Hinge-Moment Characteristics of SpringTab Ailerons." NACA WR L-318, January, 1944. 148. Letko, W.; and Kemp, W.B.: !!Wind-Tunnel Tests of Ailerons at Various Speeds. Ill--Ailerons of a 0.2 Airfoil Chord and True Contour with .35-Aileron-Chord Frise Balance on the NACA23012 Airfoil." NACA WR L-325, September, 1943. 149. Fehlner, Leo F.: !'A Study of the Effects of VerticalTail Area and Dihedral on the Lateral Maneuverability of an Airplane." NACA WR L-347, October, 1941. 150. Cohen, Doris: "A Theoretical Investi@ion of the Rolling Oscillations of an Airplane with Ailerons Free." NACA WR L-361, January, 1944. 151. MacDougall, George F.: I'Tests of Inverted Spins in the NACA Free-Spinning Tunnels." NACA WR L-370, December, 1943.

405 152. Sears, R.I.; and Hoggard, H.P., JR.: "Characteristics of Plain and Balanced Elevators ona Typica I Pursuit FuseI.age at Attitudes Simu- lating Normal-Flight and Spin Conditions." NACA WR L-379, March, 1942. 153. Goranson, R. Fabian: "Calculated Effects of Full-SpanS otted and Fowler Flaps on Longitudinal Stabilityand Control Character stics fora Typical Fighter Type Airplane with VariousTail Modif cat ions.I' NACA WR L-392, July,1942. 154. Harmon, Sidney M.: "Determination of the Dampi.ng Momentin Yay i,ng for Tapered Wings with Partial-Span Flaps." NACA WR L-39 5, August, 1943. 155. Seidman, Oscar; and KITnar, J.W.: l'Elevator Stick Forcesin Spins as Computed from Wind-Tunnel Measurements." NACAWR L-422, October, 942. 156. Wallace, Arthur R.; and Turner, Thomas R.: '!Wind-Tunnel Investigation of Effect of Yaw on Lateral-Stability Characteristics. V--Symmetri- ca I I y Tapered Wi.ng with a Circular Fuse I.age Havi,nga Horizonta I and a Vertica I Ta i I .I1 NACA WR L-459, June,1943. 357. Campfiel I, John P.; and Seacord, Charles L.,Jr. : "Effect of Wing Loading and Altitude on Lateral Stabilityand Control Characteristics ofan Airplane as Determined by Tests ofa Model in the Free-FI.ight Tunnel." NACA UR L-522, April, 3943. 358. Nissen, J.M.; and Phillips, W.H.: I'Measurements of theFlyi,ng Qualities of a Hawker HurricaneAi rp lane. NACA WR L-565, Apri I, 1942. 359. ph~llips,W.H.; and Crane, H.L.: "Flight Tests of VariousTail Modifi- cations on the BrewsterXSBA-3 Airplane. Il--Measur~ments ofFlying Qual ities wi.thTail Configuration NumberTWO." NACA WR L-598, Decernher, 2, 1943. 160. Johnson, Harold 1.; "R6surnB of NACA Stabilityand Control Tests of the Bell P-63 Series Airplane." NACAWR L-601, October, 1944. 36-1. Johnson, H.1.; and Liddel I, C.J.; and Hoover, H.H.: "Veasurements of the Flyi,ng Qual itiesof a Bel I P-390-1 Airplane." NACA WR L-602, Sep- tember, J943. 362. Stahil ityand Control Section of Fl,ight Research Division at La,ngley Latjoratory; %harts Showi,ng Stabi I i ty and Control Characteristics of Airplanes in Flight.!' NACA NR L-706, December, 3944. 363. Lowry, John G.;and To1 I, Thomas A.: "Power-On Longitudinal-Stabi I ity and Control Tests of the 1/8-Scale Model of the BrewsterFZA Airplane Equ'ipped withFu I I-Span Slotted Flapsand A New Horizonta I Ta i I .I1 NACA WR L-709, March, 19'42.

406 164. Recant, lsidore G.; and Swanson, Robert S.: "Determination of theSta- bility and Control Characteristics of Airplanes from Tests of Powered Models.11 NACA WR L-710, July, 1942.

165. Johnson, Harold I.: "FlightInvestigation to Improve the Dynamic Longi- tudinal Stability and Control-FeelCharacteristics of the P-63A-1 Airp lane with Closely Ba I anced Exper imenta I E levators." NACA WR L-730, Ju I y, 1946.

166. Goett,Harry J.; and Pass, H.R.: "Effectof Propel ler Operation on the Pitchi,ng- Moments of Sing I e-Eng i ne Monop lanes. NACA WR L-761, May, 1941 . \ \ 167. Goett,Harry J.; and. Roy P.; and Belsley,Steven E.: "Wind- TunnelProcedure of Critical Stability andControl Characteristics WR W-5, April, 1944.

168. Conway, H.M.: llNotes on Maximum AirplaneAngular Velocities.11 NACA WR W-101,May, 1943.

169. Mathias,Gotthold: !'The Calculationof Lateral Stability with Free Controls.11 NACA TM-741, Apri I, 1934.

170. Hibner,Walter: "Additional Test Data on Static Longitudinal Stabi Iity." NACA T"752,August, 1934.

171. Schmidt,Rudolf: "The Effectof the Masses of theControls on the Longi- tudinalStability with Free Elevator." NACA TM-900, July, 1939.

172. Martinov, A.; and Kolosov, E.: "Some Dataon theStatic Longitudinal Stability and Control- of. Airplanes (Design and ControlSurfaces)." NACA TM-941,May, 1940.

173. Raikh, A.: "Calculation of theLateral-Dynamic bility of Aircraft." NACA TM-1264, February, 1952.

174. Kramer, M.; andZober, Th.; and Esche; C.G.: "LateralControl by Spoilers atthe DVL." NACA TM-1307, August, 1951.

175. Hoene, H.: "Influence of StaticLongitudinal Stability onthe Behavior of Airplanesin Gusts." NACA TM-1323, November, 1951.

407