Considerations in the determination of stability and control derivatives and dynamic characteristics from flight data

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REPORT 549—Part I

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Considerations in the determination of

stability and control derivatives and

dynamic characteristics from

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flight data.

by Chester H. Wolowicz

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(NASA CR OR TMX OR AD NUMHER) (CATEGORY) AGARD REPORT 549 - PART I r NORTH ATLANTIC TREATY ORGANIZATION ADVISORY GROUP FOR AEROSPACE RESEARCH AND DEVELOPMENT (ORGANISATION DU TRAITS DE L'ATLANTIQUE NORD)

CONSIDERATIONS IN THE DETERMINATION OF

STABILITY AND CONTROL DERIVATIVES AND

DYNAMIC CHARACTERISTICS FROM FLIGHT DATA

by Chester H.Wolowicz NASA Flight Research Center Edwards Air Force Base, California, USA This Report was prepared at the request of the Flight Mechanics Panel of AGARD 1',1 1 0T FILMED.

PRECEDING PAGE BLANK r CONTENTS Page LIST OF TABLES v LIST OF FIGURES v NOTATION x 1. INTRODUCTION 2. AXIS SYSTEMS AND COORDINATE TRANSFORMATIONS 2.1 Axis Systems 2 2.2 Coordinate Transformations 3. EqU ATIONS OF MOTION 3. 1 Inertial Quantities 3. 2 Gyroscopic Couples of Rotating Masses 3. 3 Gravitational Force 3.4 Aerodynamic Derivatives 3.5 Summary of the Equations of Motion 3.6 Determination of the Roots of the Determinant of the Lateral-Directional Small-Perturbation Equations 4. MASS CHARACTERISTICS 29 4.1 Weight and Center-of-Gravity Location 4.2 Moments of Inertia 4.3 Inclination of Principal Axis 5. INSTRUMENTATION 5.1 Mach Number, Altitude, and Dynamic Pressure 5.2 Control Position Transmitters 39 5.3 Angle-of-Attack anu Sideslip 39 5.4 Angular Velocities and Accelerations 5.5 Linear Accelerations 5.6 Phase T _•,g and Response 5.7 Ranges and Sensitivity 45 5.8 Pulse Code Modulation (PCM) Data-Acquisition Systems 6. FLIGHT TEST TECHNIQUES 6.1 Mach Number and Altitude 6.2 Angle-of-Attack and Load Factor 6.3 Aeroelasticity 6.4 Control Inputs 48 6.5 Maneuvers 48 6.6 General Comments 51 s a r Page 7. ANALYSIS OF FLIGHT DATA 7.1 Fundamentals of the Time-Vector Approach 51 7.2 Basic Flight Data 53 a From 7.3 Determination of and Free Oscillations in a the Absence of or Questionable and Data 54 7.4 Equations for Longitudinal Control and Stability Derivatives 56 7.5 Equations for Lateral-Directional Stability and Control Derivatives 62 7.6 The Graphical Time-Vector Technique 70 7.7 Other Analytical Techniques 72 7.8 Analog-Matching Techniques 74 8. APPLICATION OF FLIGHT DERIVATIVES Data and Theory 8.1 Verification of Wind-Tunnel 80 8.2 Effects of Aeroelasticity 81 8.3 Stability Criteria 82 8.4 Flight Guidance 87 9. CONCLUDING REMARKS 88 REFERENCES 89 TABLES 95-104 FIGURES 105 ' iv I LIST OF TABLES Page TABLE I Transformation of Derivatives from Stability to Body Axis 95 TABLE II Transformation of Derivatives from Body to Stability Axis TABLE III Transformation of Moments of Inertia from One Axis System to Another 97-98 TABLE General Equations of Motion IV 99 TABLE V Linearized Small-Perturbation Equations of Motion TABLE VI Laplace Transform Format of Small Perturbation Equations of Motion 101 VII Desirable Characteristics of Instruments for Free-Oscillation TABLE Maneuver TABLE VIII Format used by NASA Flight Research Center to Record Actual Conditions at Time of Maneuver 103 ,104 TABLE IX TABLE X TABLE 104 XI LIST OF FIGURES r Definition of b,dy, and wind axis systems, Fig.l stability, principal, control surface deflections, and force and moment coefficients 105 Fig.2 Relationship of body, stability, principal, wind, and spatial- reference orthogonal axes systems when body x-axis is rotated in sequence through and 106 — 6 , (^ q .

8 , Relationship of aerodynamic angles a , and axes angles

Fig.3 y

6 , , and E ; and Euler angles q , and 107

77 0

Several methods of considering Euler angle perturbations 108 Fig.4 Fig.5 Relation of p , q , and r about body axes and Euler angle rates ^ , 6 , and 109 ^ V \ Page Fig.6 Pertinent relationships of rotating mass for gyroscopic couple consideration. Rotating axis parallel to xz-plane of symmetry An example of the influence of ranges of disturbances such as Fig.7 (A/^ 1 and ('1'^ 2 on the value of a derivative Effect of time lag of modification of vortex flow about lifting Fig.8 surface on the change in CN following initial instant a change in 111 Fig.9 Direct propulsive effects of propeller 111 Fig.10 Direct propulsive effects of ,jet engine 112 Fig.11 Jet-exhaust inflow effect on horizontal tail 112 Fig.12 Determination of vertical position of center-of-gravity by tilting aircraft in roll Fig.13 Determination of vertical center-of-gravity and rolling moments of inertia by rolling oscillations 114 Determination of pitching moment of inertia Fig.14 115 i Fig.15 Determination of inclination of principal axis and yawing moment of inertia 115 Fig.16 Photograph showing a general arrangement for determining inclination of principal axis and yawing moment of inertia.

Springs attached to mounting brackets located below wings 116 Amplitude ratio 1A /I A rl Fig.17 as a function of spring restoring pI angle .

Fig.18 Details of total-pressure chamber and static-pressure orifices.

Reproduced from Reference 24 Fig.19 Photograph of a typical NASA installation of angle-of-attack and sideslip vanes on nose boom Effect of 'ratio of boom length to fuselage diameter on Mach number Fig.20 error. Reproduced from Reference 30 Reproduced from Fig.21 Variation of Mach number error with Mach number.

Reference 30 typical calibration curve for determination of true Mach number 120 Fig.22 A Determination of dynamic pressure from total pressure 121 Fig.23 vi Page Fig.24 Control position transmitter and recorder Fig.25 Theoretical effects on angle-of-attack mea:arements of upwash from the nose boom and fuselage at low speeds. Reproduced from Reference 30 Influence of flight-path curvature on vane indications of Fig.26 angle-of-attack Fig.27 Time-vector solution for correcting angle-of-attack records to the center-of-gravity of the aircraft Spherical Fig.28 flow-direction sensor. (Note location of pitch and yaw reaction-control nozzles of the airplane.)

Fig.29(a) Magnetically damped angular-velocity recorder Fig.29(b) Detail of linkage and damping system 128 Fig.30 Influence of interference angular velocity ("q" rate) about spin reference axis of a sensitive rate gyro 129 Fig.31 Equations for correcting; .rate gyro records for instrument misalinement 130 Fig.32 Equations for correcting records of linear accelerometers to the center-of-gravity of the aircraft Fig.33 Chart for correcting sensing-recording circuit of instrument for phase lag Fig.34 Chart for correcting sensing-recording circuit of instrument for dynamic amplification Fig.35 Functional schematics of pulse code moJulation (PCM) data- acquisition systems Fig.36 Results of analysis of flight data in region of rapid changes in characteristics aircraf t 134 Fig.37 Variation of the period of an F-100 series airplane as a function of Mach number, altitude, angle-of-attack, and load factor (from Reference 41) 135 Influence of angle-of-attack and load factor upon lateral.

Fig.38 characteristics of one aircraft A nomograph for planning flight test conditions in investigating Fig.39 aeroelastic effects on stability and control vii Page Fig.40 Comparison of lateral-directional responses to different types of rudder inputs Fig.41 Typical, time histories of the longitudinal response characteristics of the test airplane resulting; from abrupt stabilizer deflection Fig. 42 Typical time histories of the lateral and directional response characteristics of the test airplane resulting from abrupt yaw- damper deflection Ft g. 43 Comparison of wings-level and constant-heading sideslips Fig.44 Relation of small-perturbation rolling velocity and accelerat).on vectors to small-perturbation roll-displacement vector in a transient oscillation Fig. 45 Determination of period and phase angles from free-oscillation data 143 Fig.46 Determination of time-to-damp to one-half amplitude and amplitude ratios from free-oscillation data 144 Fig.47 Vector solution of Jn a I/jna using pitch rate as base for amplitude ratio when angl e-of-attack records are unavailable 145

Fig. 48 Vector solution of I.^1a t "oI

using yaw rate as base for

I/

amplitude ratios when sideslip records are unavailable Fig.49 Typical determination of flight quantities for the evaluation . of longitudinal control derivatives Fig. 50 A graphical time-vector solution for

and (C N, + CN a)

CNa Fig.51 Time histories of longitudinal pulses performed on the X-15 analog with the stability augmentation system engaged and disengaged (from Reference 42) Variation of lg lift coefficient and corresponding trim angle- Fig.52 of-attack with Mach number (from Reference 43) Longitudinal period and damping characteristics of the D-558-II Fig. 53 airplane as functions of Mach number and altitude Variation of static and dynamic longitudinal stability derivatives Fig.54 of the D-558-II airplane with Mach number (from Reference 43) Comparison of flight-determined horizontal-tail effectiveness with Fig.55 wind-tunnel results 153 Typical determination of flight quantities for the Evaluation of Fig.56 lateral control derivatives 154 viii I Page Fig. 57 Comparison of Cr t as determined by several different approximate methods with the time-vector method Fig.58 Comparison of results of determining C1, by time-vector method and steady-sideslip equations Fig. 59 A typical graphical time-vector solution of yawing and rolling stability derivatives 156-1.1.

Fig. 60 Results of graphical time-vector analysis of the effects of power on the Lateral-directional period, damping, and stability deriva- tives of the D-558-II research airplane (fre:^m Reference 43) 158-159 Fig.61 Grid plot used to trace source of incompatibility between flight and wind-tunnel data Fig. 62 Typical time history of maneuver to determine derivatives by least squaring the equations of motion (Reference 48) Fig.63 Lateral-directional derivatives determined by least squaring the equations of motion, as per Reference 48 162-163 Fig. 64 Typical analog-match of a "recovery -from-sideslip" maneuver of an experimental aircraft. M = 1.84 ; altitude = 49,400 ft Fig. 65 Influence of flexibility and air intake to engirt in the directional stability derivative, Cnp Fig.66 Comparison with flight data of results of analog simulation studies of 3600 rolls using flight-determined derivatives ix NOTATION The body system of axes, radian measure, and foot-pound-second unit- are used throughout the paper unless specifically stated or indicated otherwise. Basic sign conventions are shown in Figure 1. In Section 1, in which a number of axis systems are considered, the subscripts are used to denote quantities referred to the various systems except for the quantities referred to the body system of axes. The subscripts for U ese quantities are omitted for convenience except to identify coordinates, x, , and z b .

y b a perpendicular distance from spring to knife edge (Fig.13), s ft a,A polynomial coefficients (Section 3) A cross-sectional area of air-intake duct of jet engine at t entrance, ft2 A cross-sectional area of ,het-exhaust duct of het engine at j exit, ft2

a x , a t , a n longitudinal, l! accelerations of the air-

lateral, and norm l craft at the center of gravity relative 1.o the body system of axes; positive forward, and up, respec- to the right, tively, g units ax i , at i , an recorded values of a x , a t , and a n , respectively; corrected for phase lag and misalinement brk; not for loca- tion relative to the center of gravity, g units b wing span, ft b,B polynomial coefficients (Section 3) F mean aerodynamic chord, ft c,C polynomial coefficients (Section 3) spring couple (Section 4), ft lb C C coefficient of axial force along the body x-axis; positive c to the rear, –X/US c c Cc phugoid damping coefficient, V ac + — zc U au cos a cos '8 (C Cu )p contribution of power to phugoid damping coefficient, V aCT + 2C T - ) au cos a cos Q x ^Cc CCU ?a - C C ca =- ac ;a — 2V ^C = c cc q qc 2V ^C _ c Cr^e e drag coefficient; coefficient of axial force along the C stability x-axis, positive to the rear, -XS/qS C L lift coefficient; coefficient of lift force along the

stability z-hxis, positive up, -ZSrS

W C Lt = — qS

C La

lift-ci,rv- slope, aCL/aa

aCL _ CL& a ac 2V acL C L = v q ac — 2V aCL CL = 8e c'8 e ) w , ( C l ) C l , ( C l ) S , (C l o coefficient of rolling-moment about the body, stability, wind, and principal x-axis, respectively, (rolling moment) TSb aCl in.-roll derivative, damping - C1 a pb 2V ai l = C l r = . rb i a .2V xi aCl CI Q effective dihedral derivative, aC I CIS _ 2V

aCl

Clsa ab a aCI C l s r = ^b r Pitching-moment coefficient about the body, stability, wind, Cm0 = (C M ) s w' o o (C m ) (C m ) and principal y-axis, respectively, ( pitching moment)/aSc pitching - moment coefficient abcut the aerodynamic center Cm0 (Cm ) p contribution of power to pitching-moment coefficient aCm Cm a longitudinal-stability derivative, as a )p (Cm contribution of power to longitudinal stability ( Equations (35) and (44)) = aCM c ma a (Xc 2V Cm q = aCM a qc 2V _ aC 2C m _ m V Cmu au + cos a cos Q refer to Equations ( (Cm u )p 47), (48), and (49) Cm

_ a

Cmse as e — CN normal-force coefficient, coeffl A ent of force parallel to up, –Z/ZTS body z-axis; positive contribution of power to normal-force coefficient (CN ) p xi i aCN CN a - a (CN a variation of contribution of (C N ) P with angle of attack )p C ( variation of normal-force coefficient of horizontal tail Na)h.t.

with local angle of attack at the tail; coefficient based on horizontal-tail area and local dynamic pressure, -z h. t.

gh.t.Sh.t.

? 'h. t.

(CNa)v.t. variation of coefficient of force normal to vertical tail with vertical angle of attack of vertical tail; coefficient based on vertical-tail area and local dynamic pressure,

Y v. t. 1

q v. t. S v. t.)

^I-V.t.

aCN CN a.

.^ ac 2V

aC N

C N Q =

qc 2V longitudinal phugoid static-stability derivative, CNu 2C C)CN V -- + all cos OC cos _ aCN CN s e

a

e Yawing-moment coefficient about body, stability, wind, and C n o

I (Cn ) s # (C n ) w F (C d

principal z-axis, respectively, (yawing moment)/iSb aCn CnR static directional-stability derivative, a,8 acn Cn^ 7 -b 2 V aCn Cn r = rb 2V

xiii

aCn C np = pb 2V aCn __ Cns r a8 r ^Cn _ Cnb a ab a C T thrust coefficient, (thrJ,st) /Q-S T CTU u Xw (C x ) w coefficient of axial force along the wind x-axis, qS C y , (C y ) s , (C ) (C y ) o y w , side-force coefficient parallel to body, stability, wind, and principal y-axis, respectively, Cy = (Cy)S (Side force) AS

ac y

Cy ^ _ a'8 (Cy Q )p contribution of power to Cy,8 a Cy Cy Q --fib a /—^ 2V

acy

Cy r rb a- 2V

acy

Cy _ p a rb 2V Zw (C Z ) w z-axis, coefficient of force along the wind qS polynomial coefficients (Section 3) d,D e = 2.173 e,E polynomial coefficients (Section 3) xiv g acceleration of gravity, :t/sect - - sin ^, 9 1 V sec V F" sin ,/. , ?

cos V sec h altitude, ft H angular-momentum vector of a rotating mass, lb ft sec I rm Q , If fl z angular momentum of H about x , y z body axes, X . y , fi respectively, lb ft sec moment of inertia of rotating mass of engine about its Irm rotating axis, slug ft2

I T y , I r

moments of inertia of aircraft about x y z body axes, X, respectively, slug ft2

I Xa , Iyo, Izo x y

moments of inertia of aircraft about z principal

axes, respectively, slug ft2

Ix S ,Iy s moments of inertia of aircraft about x y z stability ,Iz s axes, respectively, slug ft2

Ix r , Iyr, moments of inertia of aircraft about x y z reference

Izr axes, respectively, slug ft2 moment of inertia of cradle supporting aircraft (Section 4), Ixc

slug ft 2

Ixz I _

X I x

Ixz i t _

z I z

I Xz product of inertia of aircraft referred to body x- and

z-axes, slug ft2

k stability-augmentation-system gain, sec K s linear spring constant, lb/ft K correlation constant (Section 5) ft K t torsional spring constant, A sa S , lb/rad ,. xv R- r 1 distance as defined locally at time of discussion, ft L,M,N rolling, pitching, and yawing moments about body x , y z axes, respectively, ft lb L i ,M i ,N i inertial rolling, pitching, and yawing moments about the respective body axes, ft lb pitching, and yawing moments due to gyroscopic rolling, L rm' M rm' Nrm action of rotating mass of engine, ft lb (L)S,(M)S,(N)S rolling, pitching, and yawing moments about the stability x , y axes, ft lb , z rolling acceleration about body x-axis, L (rolling moment)/Ix, Use c 2 gSh2 ^L = T p = Cl -- ft lb sec L P 2V Lp P = C 1 2VI sec x

Ixz N

+ L p P Ix 1 L^ = P IXz2 sec 1 — IxIz gSb2 ^L L r = ft lb sec ar = (:lr , 2V gSb 2 1 Cir 2VI L — x 0 sec Ixz + L N r r Ix , 1 _ r sec Ixz2 1 — IxIz aL ft lb L^ = = C IQ qSb , ,8 qSb 1 L^ = C lQ Ix sect = ft lb = Cls a qSb , Lsa bL a xvi qSh 1 Lga C 1 a sec I X Lg r C 1 qSh, ft lb r qSh 1 Lg r = 2 C 1 r I X sec m mass of airplane, W/g , slugs M mass rate of air intake of jat engine, slugs/sec a m i mass rate of jet exhaust, slug:,/sec M Mach number M i indicated Mach number M pitching acceleration about body y-axis, (pitching moment)/I Use C2 y ^M gSc2 M = = C m ft lb = e q q - q 2V gSc 2 1 Mq - Cmq 2VI sec y ^M qSc M u = au = Cm u V , lb sec _ qSc Mu - Cmu sect I a gSc , ft lb °,a as Cm qSc 1 M a - Cma sect I aM gSc2 Ma = Cma2V ft lb sec as = gSc 2 Ma Cma 2VI y sec aM = = Cmg e qSc , ft lb Mge

a8

e xvii qSc Mae =

I

c ms e t sec Y N yawing acceleration about body z-axis, (yawing moment)/Iz 1/sec aN qSb N p = )N Cn - ft lb sec; P 2V a P qSb N C n P 2VI z sec + Ix z L N P P Iz N^ sec P Ixz2 1 — Ixlz aN &Sb2 Nr ft lb sec c a r = — Cnr 2V _ gSb2 Nr C nr ' 2VI z sec ^— Xz Nr + Lr z _ 1 Nr sec r Ixz2 1 — IxIz aN Na = aN = CnagSb ft 1b qSb 1 NQ Cn,8 I — sect aN = = Cn 4Sb , ft lb Na r 8r a b r qSb NSr = Cnsr I z sect aN = = Cns a&Sb , ft lb Ns a aN a qSb Cnsa Nsa I z sect xviii p, q, r rolling, pitching, and yawing velocities, respectively, about body axes, rad/sec rolling, pitching, and yawing velocities, respectively, Pop q o O r about principal axes, rad/sec p g , q S , r s rolling, pitching, and yawing velocities, respectively, about stability axes, rad/sec p, q,r rolling, pitching, and yawing angular accelerations, respectively, about body axes, rad/sec' rolling, pitching, and yawing accell erations, respectively, p oi g o y ro about principal axes, rad/sect p static pressure, lb/ ft2 indicated static pressure, uncorrected recorded pressure, p i lb/ft 2 static pressures acting across inlet of air intake and p i ,pj exhaust, respectively, of jet engine, lb/ft2 stagnation pressure, lb/ft2 P + A q c T P period of short-period oscillation, sec z^V2 q dynarnic pressure, lb/ft2 impact pressure; dynamic pressure of compressible flow qc (Equations (98) and (99)), lb/ft2 indicated impact pressure, lb/ft2 qci dynamic pressures at the horizontal and vertical tail, gh.t.,gv.t.

respectively, lb/ft2 instantaneous radius of turn (Fig. 27), ft R reaction force (Section 4), lb R Laplacian operator, o- + iw , s sec s sensitivity S wing area, ft2 horizontal- aad vertical-tail areas, respectively, ft2 Sh.t.,Sv.t.

time, sec t xix T thrust due to power, lb time required for absolute value of transient short-period T1 /2 oscillation to damp to one-half amplitude, sec 1 1 roll-subsidence and spiral-divergence roots, respectively, TR Ts 1 of the lateral-directional characteristic equation, sec Tg,Ts roll-subsidence and spiral-divergence time constants, sec q(s) pitch-attitude time constant in numerator of To 5e(s) transfer function, sec u, v, w linear velocities relative to body x y z axes, respectively, ft/sec linear accelerations relative to body x y , z axes, respectively, ft/sect 4u 4u = — V Au Du = -- V V airspeed, ft/sec V i velocity of intake air at air intake of jet engine ft/sec V^ velocity of ,het exhaust, ft/sec W weight of aircraft, lb W of cradle (Section 4), lb

x, y, z distances from the center of gravity along body x , y , z

axes, respectively, ft forces along the body x , y , z axes, respectively; X,Y,Z to the right, and down, respectively, lb positive forward, components of gravitational force acting along the body x X g , Y g , Z g y z axes, respectively X qS = = – Cc u , lb sec/ft V X au xx gS _ - C C , x U mV sec ax X a S 1b = ^a = - C C( x _ 1 gS - X a a Cc mV ' sec ax Xg = _ - Cc S gS , lb gS 1 Xs - CC S ^ mV sec lb S v. t.

— (C N(x ) v, t. g v. t.

_ y v. t.

ay Y Q - ^^ = Cy QgS , lb FIS 1 CyQ YR mV sec (Y) p lateral force in plane of propeller disk due to propeller, lb ay Y .^ ^ = C Y OS lb gS 1 Ys CY6 sec mV az gS = = - CNu lb sec/ft z

aU V

gS z = - C N — , u " mV sec a z &SF Z q = = - CNq lb sec aU 2V WE z - C - N q 2m V2 az za = 70C = - CN aqS , 1 b

xx i

q'S _ Z(X CNa mV sec ;)Z WE Z _ -- - CN , lb sec Z& ^a a 2V qSc Za CNa 2m V2

(Z) p

contribution r.f propulsion . , ;ystem to Z force, lb

lb

h. t.

(^Na)h. t.^ih. t. S Z h. t.

a angle of attack of aircraft n a

C a

change in due to influence of flight-path curvature

a 0

angle of attack of aircraft for zero Z force

((X) p

angle of attack of thrust line relative to airstream velocity at propeller or air intake of jet engine; thrust line considered parallel to x axis

a p maximum positive or negative angle of attack obtained in a

roll maneuver ( ig.66) sideslip angle rate of change of /3 with time, rad/sec

y flight-path angle relative t.o horizontal

y adiabatic constant

8 a

aileron deflection; positive when left aileron is deflected down d8a aA d,Q

8 e elevator deflection; positive when trailing edge is

deflected down

8 r

rudder deflection; positive when trailing edge is deflected to the left

d8r

rQ dQ xxi i b sp angle included between reference x-axis and plane of spring couple (Fig.15) n increment E angle between body x-axis and principal x-axis; positive when reference is above principal axis at the nose upwash angle at the propeller due to such factors as E fuselage and wing r short-period ratio of actual damping to critical damping phugoid ratio of actual to critical damping Cph instrument damping ratio Cin angle of inclination of principal x-axis relative to stability x-axis; positive when principa l . x-axis is abuve stability x-axis at the nose pitch attitude of angular-velocity vector, of rotating Arm mass of zngine relative to body x-a.xis m μb relative aircraft density,

pSb

m relative aircraft density, ^ pSc μ o absolute viscosity, lb sec/in-2 mass density of air, slugs/ft3 P cr real part of Laplacian operator, s = o- + iw m r r time parameter, , sec

p VS

time constant for simplified stability augmentation system, sec yaw, pitch, and roll, Euler orientation angles, respectively.

(In general aircraft motions, they are normally the orienta- tion angles of the aircraft body axis syf em to a spatial (earth) reference system. In instrument alinement, they refer to the misalinement of the instrument reference axis system to the aircraft body axis system.)

xxiii rate of rotation of the Euler orientation ana • l.es, rad/sec

<

ngif,AP,Ar yaw, pitch, and rcll Ruler orientation angles of aircraft

body axis system durio g small perturbations relative to body axis system preceding the perturbations regardless of aircraft attitude preceding perturbations (Fig.4(b)),

f

AV)' fArdt , AP I -- Agdt A&' fApdt .,.amping angle fid 4) ij phase angle of vector quantity i relative a vector quantity j undamped and damped natural frequencies, respectively, of Wn'(`)nd the aircraft in short-period modes of oscillation, rad/sec undamped and damped natural frequencies, respectively, of ph1(41dph the aircraft in phugoid modes of oscillation, rad/sec undamped natural frequency of instrument, rad/sec 141in R ang,!lar rate of rotation of rotating mass of engine, rad/sec A

W absolute magnitude of a vector quantity j ; always positive

transformation matrix [L] inverse transformation matrix [L] -1 [L] s ,[L] b transform matrices to transform vector quantities from reference axis system to aircraft stability and body axis systems, respectively [a] s transformation matrix representing transformation from body to stability axis system relative to body, stability, wind, and principal axes ( ) b' ( ) s' ( ) w' ( )o systems, respectively ( ) p contribution due to power xxi v CONSIDERATIONS IN THE DETERMINATION OF STABILITY AND CONTROL DERIVATIVES AND DYNAMIC CHARACTERISTICS FROM FLIGHT DATA Chester H.Wolowicz 1. INTRODUCTION The determination of stability and control characteristics from flight data in the form of derivatives and other behavior parameters has become an important part of flight testing. As new concepts in airplanes are developed or the airplane flies in new Mach and altitude regimes, there is the need to verify theory and wind-tunnel data and the various influences on stability characteristics, to provide information not obtained in wind-tunnti studies, and to uncover the sources of discrepancies between prediction and actual flight behavior. Where wind-tunnel data are unavailable or where safety of flight into ut:tested regions is of concern, flight-determined derivatives have been extrapolated to predict airplane behavior prior to flight into these regions,.

Because of the exploratory nature of many of the investigations, the practical aspects of determining derivatives and other behavior parameters, such as oscillatory characteristics, from flight data are very important. Experience has shown that a maximum appreciation and understanding of the practical aspects is attained when back- ground knowledge includes an understanding of axis systems, transformations, the equations of motions and the limitations of the equations, techniques used to determine the mass characteristics of the airplane, the installation and behavior of flight test instrumentation, flight test techniques, and the theory and limitations of techniques used to determine the stability and control characteristics from flight data.

Although some of the factors mentioned above, such as axis systems and transformations as well as aspects of the equations of motion, may be found in textbooks, the treatment is generally not oriented toward flight testing. Some of the techniques used in deter- mining stability and control characteristics may be found in technical reports; however, limitations of the techniques occasionally may not be shown. This paper attempts to bring all the factors together to provide a rEady reference of pertinent information.

It is, in fact, a greatly expanded version of AGARD Report 224*.

It is the purpose of this paper to discuss the various factors that influence the determination of stability and control derivatives and other behavior characteristics from flight data. Included are illustrations of the application of flight derivatives to verification of predictions and to determination of aeroelastic effects, stability criteria, and flight guidance. This paper is intended not only for the practical engineer who is working with flight data but also for the scientist who is attempting to develop new, sophisticated analytical techniques.

• Stability-Derivative Determination From Flight Data by Chester H.Wolowicz and Euclid C.Holleman, October 1958.

Acknowledgement of investigators whose work has directly contributed to the present paper is made in each section. It is r^-%ognized that many noteworthy works of other investigators are not referenced.

2. AXIS SYSTEMS AND COORDINATE TRANSFORMATIONS 2.1 Axis Systems In the study of the dynamics of the airplane, as many as six orthogonal axis systems may be used simultaneously. An unders Landing of these systems or reference frames and their relation to the aircraft and its motions at various flight conditions is essential to the proper analysis of flight data. Although a comprehensive treatment of axis systems may be found in Reference 1, a brief treatment of the axis systems is presented in this section.

2.1.1 Body Systems The body axis system (x b , y b , zb ) is body-fixed with its origin at the center of gravity of the airplane. The x b axis is always parallel to the fuselage reference line and when the center of gravity is in the plane of symmetry, as it normally is, both the x b and z b axes are in the airplane's plane of symmetry, as shown in Figure 1. The axis is normal to the plane of symmetry; thus, the body system of y b axes is angularly invariant with respect to the aircraft structure.

Because of its angular invariance with respect to the aircraft, the body axis system is an excellent frame of reference for mounting flight test instruments. The orientation of the flight test instruments and their consequent output relative to the body axes — especially the linear accelerometer and angular rate and acceleration sensors — make it convenient to determine, from flight data, stability and control parameters with respect to this reference frame. Aside from convenience, this reference frame is the logical frame about which to orient rates, accelerations, and the stability and control para- meters in the study of handling-quality criteria, inasmuch as the orientation of the pilot is invariant relative to this frame.

2.1.2 Stability System

The stability axis system (x S , y S , z s ) is a special case of the body axis system.

Like the body system, the x s and zs P ies are in the plane of symmetry when the center of gravity is in this plane, and parallel to the plane of symmetry when the center of gravity is not in the plane. Unlike the body system, however, the x s and z s axes are angularly variant relative to the fuselage reference line. The z s axis is perpendicular to the resultant velocity vector and the x. axis is parallel to the component of the resultant velocity vector projected onto the plane of symmetry, as shown in Figure 1.

The important parametric relationship between the body and stability axes systems is the angle of attack, a , which is the angle between the x s and x b axes (Fig.1).

The stability axis system is commonly used in theoretical subsonic aerodynamics and subsonic, wind-tunnel force and moment investigations. It is also employed, on occasion, in place of body axes in flight test investigations of longitudinal stability and control characteristics.

2.1.3 Principal System The principal axis system ( x o , y o , z o ) defines the natural axes of rotation of the aircraft. They are the axes which result in maximum and minimum moments of inertia.

The orientation of this axis system in the aircraft is a function of the mass distri- bution of the aircraft and will remain fixed as long as the mass and mass distribution remain fixed. When the lateral distribution of mass is symmetrical relative to the plane of symmetry, which is generally the case, the y o axis will coincide with the y b axis, and the x o and z o axes will lie in the plane of symmetry, as shown in Figure 1.

The inclination of the x o axis (Fig. l) to the x axis of the reference axis system (generally body axes in flight test investigations) has a direct bearing on the inertial moments experienced about the reference axes as reflected in the product of inertia term in the equations of motion and, hence, on the lateral stability of IxZ the airplane.

When the principal axes are used as reference a pes, as they occasionally are in theoretical and simulator investigations, they are used to simplify the equations of motion by the elimination of the I xZ term.

2.1.4 Wind System The wind axis system is related to the resultant velocity vector and the plane of symmetry of the airplane. As shown in Figure 1, she x w axis is parallel to the resultant velocity vector and lies in the transverse plane of the stability axes (x sy s plane). Consequently, the z w axis is coincident with the z s axis. The xw and y w axe; coincide with their respective counterparts x s and y $ when the aircraft has zero sideslip.

The important parameters associated with the wind system are the sideslip angle, and the angle of climb, y . By basic definition the angle of sideslip, [3 , is the angle between the x w axis and the plane of symmetry and thus lies in the trans- verse stability axes plane, as shown in Figure 1. It should be noted that not all ^_sensors necessarily measure this 8 ; this will be discussed in Section 5 on "Instrumentation". The angle of climb always lies in the vertical plane and is the angle included between the x w axis and the horizontal plane.

2.1.5 Spatial Reference System The preceding axis systems are tied in with the plane of symmetry of the airplane with their origins at the center of gravity; as shown in Figure 1. To complete the systems of axes used, at least one inertial, space-fixed, axis system is required.

In dealing with general motions of aircraft, this spatial system is generally earth- referenced to describe the motion of the air p lane with respect to _ me for short time intervals. Such a situation is indicated in Figure 2, which shows the relationships of the various axis systems previously described and the relationship of the body axis z r ).

system with respect to the spatial reference (x r , y r , Shown in the figure are

flight path y , angle of sideslip 8 , angle of attack a , as well as the Euler

qi , B ,

orientation angles, and of the airplane's body axes relative to the spatial

axis system. This is shown in a much simpler format in Figure 3. The sequence of rotations of the Euler angles, is important. Generally, the sequence of rotation is ^) , A , and this means that the airplane is initially yawed, then pitched, and finally rolled.

It should be noted that y

= P, — a only when the aircraft is unbanked 0).

2.1.6 Perturbation Reference Frames In using perturbation theory in stability analysis, Euler angle perturbations may be considered to be superimposed on the unperturbed angles, as shown in Figure 4(a), + Aq , P + AP ,

with the result that the perturbed angles are and + AO , or

qJ 0

they may be based on a secondary spatial reference frame which is the unperturbed

airplane axis system (xb o , y bo , zb o ), In Figure 4(b) the unperturbed body axes con-

stitute the secondary spatial reference frame and are oriented to the basic spatial reference frame through the angles P , and (t . However, the perturbed planes q ,

are oriented to the secondary spatial reference plane by Ay)' , AP' , and A' , which

generally are not the same as Aq , AA , and A(^ .

2.2 Coordinate Transformations Coordinate transformations are used so frequently in dynamic studies of aircraft that some consideration should be given to this subject. Literature on transformations is extensive and ranges from the classical mathematical treatments (Reference 2, for example) to engineering applications (References 3 and 4, for example). At this time, the most pertinent transformations are considered to serve as guidelines for other transformations that may be desired, 2. 2.1 Transformation from Earth Reference Axes to Airplane Axes r , Consider X Y r , and Z r as generalized vector quantities acting along the co-

ordinates x r , Yr , Zr , resp ectively. The transformed vector quantities X , Y , Z

acting along x b , y b , and z b axes, respectively, are obtwined by performing three 6 ,

successive rotations, and (^ , to define the airplane's orientation with

q ,

respect to the reference axes x r , and z r ,, through a transformation matrix [L]

Yr , as follows X Xr Xr

L I

= 101 161 [qJ Y r. ( 1 a)

Y = E Yr Z Zr [Zr

1 0 0 cos 6 sin 0 Xr

0 -sin B cos 0 cos sin 0 1 0 -sin kk cos V 0 Y r (1b) B 6 0 0 -sin cos sin 0 cos 0 1 1Zr q cos B cos tk cos 6 sin tk Xr -sin

B cos 6 B Yr

sin sin sin V sin sin sin (tcos

q'

-sin gcos 0

+cos Vcos (10

cos gcos 0sin sin gcos 0sin 6 cos 0cos B Zr

+sin q sin -cos q sin

i

2.2.2 Transformation from Airplane Axes to Earth Axes j Since pro ection from airplane axes to earth axes is an inverse process of the preceding transformation, premuitiplication of Equation ( la) by the inverse trans- formation matrix [L) -1 results in X r X Y Y r = LLJ -1 (2a) Z r Z However, since the orthogonal projections on the airplane axes are being transformed to orthogonal projections on the earth axes, the inverse of the transformation matrix [L] in Equation ( lc) is the same as its transpose; thus

X r cos cos 6

sin sin cos cos cos (h sin A X q q -sin cos 0 +sin sin

q q 0

%

6 A

Y r = cos sin sin gsin 6 sin 0cos 0sin

sin Y (2b) q +cos cos -cos sin

0 0 0

q

Z ri

sin A sin cos 6 cos cos Z

O 0

L J L

2.2.3 Relationship Between Airplane Rates p q

and r and Euler Rates 0 , E , and It should be recognized from Figure 5 that, although the airplane rate-vector quantities, p , q , and r are orthogonal, the Euler rate-vector quantities are not.

Thus, to obtain the relationships of p , q , and r as functions of ^ , B , and

it is necessary to transform ^ , B , and ^ to components along x r , Yr , and zr

axes and then epply Equation (lc). The first transformation is accomplished rapidly by applying Equation (2b) and considering each Euler quantity as a special case of

transformation of a body axis . quantity. 6 are

To wit: in Equation (2b) both and considered zero for ^ and B , and Hence, the re- is considered zero for sulting transformation to the reference axes will be X r B qi cos cos -sin 0 Y r = cos 6 sin tP cos 0 6 (3a) Z r -sin ^ 1 0 ^ ( Substituting Equation 3a) into Equation (1c) results in the following: X p 1 0 -sin 6 Y = q = 0 (3b) cos sin Ocos 6 6

Z r 0 cos 6 cos 0 ^

-sin

To obtain the inverse of Equation , (3b), it is necessary to solve for the inverse of the transformation matrix since 6 , and ^ are not orthogonal and hence do not 0 , permit the use of the transpose for the inverse. This is accomplished by solving for the inverse matrix [L) - ' in the relationship 1 0 0 0 1 0

[L) [L) -1 = (4)

L0 0 1 After solving for [L]-' , the inverse of Equation (3b) is determined to be

1 sin Stan 6 p

cos Otan

B = 0 cos (t -sin q (5) 0 sin Osec P cos Osec r 2.2.4 Transformation of Euler Angles from the Body to the Stability Axis System If two different rotation series give the saire starting and ending orientation, the matrices representing the rotation series are equal, element for element, in the two transformation matrices. Thus, the Euler angles, 68 ,

and 0. , of the stability

r S , axes can be derived from the Euler angles, B b , and ¢ b , of the body axes by the Ob , following transformation matrix relationship

[L] S =

[a) [L] b (6) s where [LJ S is the transformation matrix of Equation (lc; using stability axis orienta-

tion angles BS , and

in place of 6 , and and [L) b

is the same

0 , q , qS , 0.

t9 b , transformation matrix using body axis orientation angles and (^ in place qb b of qj , B and if the same successive rotation series is employed. The trans- 0 , formation Lxl s is the matrix representing the transformation from the body to the

stability axis system, cr

cos (X 0 sin a

1 01

8 = 0 1 0 (7) L-sin a 0 cos a Upon performing the matrix multiplicati.)n shown by Eq^^tion (6), and checking corresponding elements in the equated results to obtain the most feasible elements for the desired result, the following relationships are arrived at sin B S = cos a sin B b — sin a. cos B b cos Ob cos 6b b sin ^^ = sin ¢ cos 6S (8) + sin a ( sin cos sin Bb — cos iPb sin 0b) cos a cos B b sin qb Ob 'Pb = sin `PS BS cos

2.2.5 Transformation of Aerodynamic Coefficients to

Various Axis Systems The following transformations are accomplished readily by employing Equation (2b) and replacing tP , F , and (^ in the equation by -p , a , and 0 , respectively.

Thus, to transform from body to stability axes, set 8 = 0 thereby obtaining C D = CC Cos a + C N sin a (Cy ) s = Cy C L = — C C N sin a + C cos a (9) = C l cos a + C n sin a (CI)s (Cm ) s = Cm (Cn) s = —C l sin a + Cn cos (x Similarly, to transform from body to wind axes ,8 (C x ) w = —CC cos a cos,8 + Cy sin,8 — C N sin a cos C y ) w = C ( C cos a sin,8 + C y cos'3 + C N sin a sin,8 ( C Z ) w = — C L = C C sin a — C N cos a (10) ( C l ) w = C l cos acos ,3 m sin,8 + C n sin acos,8 + C = C ( C m ) w m cos /3 — C l cos a sin,8 — C n sin a sin Q ( C ) w = C n cos a — C l sin a .

n Also, for stability to wind axes, set a = 0 , obtaining = ( C x ) w — C D cr>s,3 + (C Y),sin,8 sin,3 + ( C y ) s cos,8 CD = ( ^'y ) w (CZ ) w = —CL (11) (C l ) w = (C l ), cos,8 + (C.) s sin,8 (Cm ) w = (Cm ) s cos,8 — (C I ), sin,8 ( Cn ) w = (Cds To transform from wind to Stability or body axes, or stability to body axes, use is made of Equation (lc).

2.2.6 Transformation of Derivatives The transformation of derivatives from one axis system to another goes beyond pure kinematic transformations. Longitudinal derivatives are relatively simple in their transformations; lateral directional derivatives are more complex in transformations.

The several examples will illustrate the procedure to obtain derivative transformation.

Influence of factors such as power is not considered at this time.

Transformation of longitudinal derivatives is accomplished by direct differentiation of the coefficient equations. This is possible because a and q are not modified by

the axis system used. For example, to obtain the derivative of C L

with respect to a in a transformation from body to stability a: , es, differentiate the equation for C L in Equation (9) obtaining, on a per radian basis,

C La = —C^ a sin a + CNac COS a — C . (12)

The transformation of the lateral-directional derivatives is more complicated, inas- much as the angular rate variables r and p are affected by the transformation. At this time, sideslip angle, P , is not considered to be affected by the transformations because of its definition; however, the type of O-sensor used in flight tests — whether it be a vane, floating cone, or ball nose — does have a bearing on the interpretation of the readout and the meaning of the derivatives with respect to the sensed ^3 .

This is discussed in Section 5.

Consi&r the transformation of lateral-directional derivatives from the stability to the body axis system. Transformation of the yawing and rolling moment equations is

accomplished by

N = sin a

(N) cos a + (L)

s s

(13)

L = (L)

cos a — (N) sin a

s s

where L and N represent rolling and yawing moments, respectively.

However, 'ES

( N ) s = ( Cnf3)s18 + (Cnr)s rsb + (Cn4)s ^b + (Cnp)s psb + ( Cng)s b

2V 2V 2V (14) j p s b + (C1 )s8 gSb .

(L) s =

b + (Clp)s 8

(Clr)s rsb + (Cn^)s ^ [ ( C l B)s'8 + 2V 2V 2V

It will be necessary to express Or s in Equation (14) as functions of

and

Aps

Ar and Ap using the tr9iisform

rs = r cos a— p sin a l

(15) p = p cos a + r sin a .

G Upon substituting Equation (15) into (14) and Equation (14) into (13), and regrouping terms, N

,3 )s sin a +

lCn,3)s COG a+ (C1

gSb rb F 2 2 + I (Cn r )s cou a + (Cl p )s sin a + (Cn p + 1r)s sin acos a 2V +

C

rj r

b (l6a)

+ (Cnp) s cosa + (C 14) s sin a+

2V L

np)s cos a - ( C a-(Cn acos a1pb+

t e

+ [ ( C 1r)s sin r - C: 1p)s sin

aV + [ (Cn 8 ) , 9 cos a + (C1 5 )s sin a b and L (C1^ );; cos a - (Cn )s sink + Q gSb pb r a + (Cn a - (Cn p + C1 r )s sin a cosa ZV

+ (C1 p )s cos t r ) sin 2

fi

(l6b)

^b

+PC 1R) s cos a- (Cnp) sin a + s > LV rb

t a - (Cn p )s sin a - ( Cnr - C lp)s sin acos a rb +

+ [(Clr)s cos

8 .

3)s Cosa - (Cns)s sin a

+ [M -

Summaries of transformations of aerodynamic derivatives from stability to body axis system, and vice versa, are given in Tables I and H.

2.2.7 Transformation of Moments of Inertia from One Axis System to Another Although this topic is covered in applied mechanics literature, an illustrative example is given as a refresher. Also included are tables of transformations for ready reference.

To obtain Ix s in terms of body axes quantities, use is made of the fundamental relation

(17)

I xs = f ( y s + zg) dm .

Substituting the following transform into Equation (17), x s = x ccs a + r, sin a (18) Z = z cos a — x sin a and expanding, t e a

1i + xh) a — b z bdn sin acos a

11(Yh + zb) dm] cos dm] sin

Ixs =

+ P 2Ifx

= I x cos t a + I Z sine (X — 2I xZ sin acos a I x ) — (

(IZ + I Z — I x ) cos a — sin 2 a

i IXZ 3. EQUATIONS OF {NOTION The equations of motion of an airplane as found in texts on aircraft dynamics (such as Reference 5) and as normally presented in the technical literature, i1though prosaic, in appearance, do contain complexities in the significance of the individual terms.

The following discussion is intended to acquaint the reader with the scope of the complexities which may be encountered and which should be recognized and managed in dealing with the equations of mo+i.on. An understanding of this matter is important in applying the equations to derivative determination from flight data.

3.1 Inertial Quantities In all considerations of the inertial portions of the equations of motion, the axis system used has a direct bearing on the expressions for inertial forces; the degree of asymmetry of the mass distribution of the aircraft and the magnitude and violence of the aircraft motions affect the format of the expressions for inertial moments.

It is assumed, for the purposes of this paper, that the aircraft behaves as a rigid body. Where aeroelasticity is a factor, it sS assumed that proper precautions will have been taken to provide assurance that the rigid-body concept will provide a good degree of approximation.

Inertial quantities arise from the inherent action of the aircraft whose various components act as a rigid-body assembly and from the rotating masses attached to the aircraft.

3.1.1 Inherent Aircraft General Inertial Force Expressions Inasmuch as our interest lies in the analysis of flight test data oriented to the body axis system, the inertial force expressions applicable to this axis system and for all attitudes of flight are X i = m(u + qw — rv) (19a) (19b) Y i = m (v + ru — pw) (19c) Z i = m(w — qu + pv) .

If the stability axis system were employed as the reference instead of the bodf axis system, qw - pw - w -- 0 , inasmuch as there is no linear velocity component al r,ng the z-stability axis.

3. 1. 2 General .Inert cal-Moment Expressions For the general case where the principal axes are asymmetric to the various planes of the reference axes, the inertial-moment expressions are _ 2 r L i ' )qr I,^p + I xy (rp - q) - I XZ (r + yz ( - q 2 ) + ( I z - y (20a) + I pq ) Mi _ t y q + I yz (rq - r) - I Xy (r + (20b) q r) + I xz (P 2 - r 2 ) + (IX - Id rp z (gr - p) - ? yz (q + rp) + I + ( ) pq (200 N i = I r + I Xz xy ( g 2 - p 2 ) I y - I X Fortunately, situations involving general asymmetry of the aircraft are rare. Normally, the vehicle will have a mass distribution symmetrical relative to the xz-body plane of symmetry, with the result that the principal y-axis coincides with the y-body axis.

Under such circumstances, I xy = I y „ - 0 and the general inertial-moment expressions reduce to the following normally employed form: L i = I x - I Xz (r + + ( I z - I y )qr ,21a) pq ) M i +(I X = 1Y + I XZ ( p 2 - r 2 ) - )rp( ,:,, I z N i = I z r - I Xz + '210 ( I y - I X ) q (P - qr ) P The inertial expressiuns in Equations (21a, b, c) are nonlinear and thus not suitable for use in the derivation of closed-form stability equations. However, they are re- quired in analog or digital computer study of the motion of the aircraft in general or violent maneuvers and in the analog matching of flight data from such maneuver in attempts to determine the effective values of the stability and control deriva',ives for the maneuver.

In violent maneuver., the terms involving pq and rp are particularly important.

These term3, as Krell as qr , are gyroscopic terms. Modern high-performance aircraft tend to have low values of I x compared to I sand I z , with the result that gyro- scopic action represented by ( I x - (I Y °• T,,)pq , in particu!.ar, has been I z )rp and responsible for the uncontrollable, catastrophic rc,ll-coupling behavior of at least one jet aircraft after a deliberate high roll rate ir,nut.

When the motions of the aircraft are small or gradual. the inertial-moment expressions may be simplified to Li = I x xz r (22a) p - I M i = I ycj (22b) N i (22c) = I z r - I xzp 3.1.3 Small-Perturbation Inertial Expressions The classical approach to the study of aircraft dynamic stability and control involves the use of small disturbances (perturbations). Restricting the motions to small de- viations from steady-state conditions allows the elimination of non-linear terms from the inertial expressions. Such motions are useful in defining the stability, control, and handling qualities of the aircraft, and the pilot effort or autopilot character- istics required to control the motion. 1U' has bee'. found that the use of small- perturbation theory gives good results and permits the development of analytical expressions.

To arrive at the small-perturbation inertial expressions, replace the individual acceleration and velocity terms in Equations (19a, b, c) and (22a, b, c) by accelera- tions and velocities made up of disturbances superimposed on equilibrium conditions so that u , etc., is replaced by u + A^ , etc., respectively; expand the product terms; neglect the second-order quantities (Arms, for example); and subtract the initial conditions from the final resulting conditions. The resulting small-perturbation inertial expressions are AX = m[46 + wAq + qOw — rAv — vArl (23a) AY = m [U wAp — pAw, (23b) + unr + rAu — 'n^Z i + 1;^v (23c) - m[Aw — unq — qAu + vAp] and IxZAi I x ^^ — (24a) ALi = AM i = IyOq (24b) AN i = IzOi — I xZA ( 24c) Equations (23a, b, c) show that lateral-directional-mode perturbations Uv , Ar and np appear in the longitudina .-triode equations AX i and AZ, , and that the longitudinal-mode perturbations Au and Lw appear in the lateral-directional-mode equation A Z i .

This coupling of the two; modes can normally be minimized to permit practical use of the uncoupled practical approxiwation of Equations (23a, b, c) shown in Equations (25a, b, c). This minimization is achieved in flight test maneuvers such as elevator pulses for perturbation of the longitudina l . mode and rudder or aileron pulses for perturbation of the lateral-directional mode i p.itiated during steady wings- level or steady turn flight.

AX = m[Ati + wOq] (25a) DY i - (25b) m[dv + uQr — wAPI AZ uAq — 25c) = m [A w — qAu) ( 3.2 Gyroscopic Couples of Rotating Masses Spinning masses mounted on the aircraft - such as propellers and rotating elements of engines - possess angular momentum relative to the reference (body) axes and pro- duce gyroscopic couples on the aircraft which could be significant, as was the case on the X-5 airplane 6 .

Normally, the gyroscopic couples are negligible; however, the advent of vertical-rising aircraft with tilting engines and the increase in size of propulsion units on high-performance aircraft make it inadvisable to arbitrarily ignore this coupling.

For a rotating mass having a rotating axis in or parallel to the xz-plane of symmetry but at an angle to the x-body axis (Fig.6), it can be shown from the moment of `Arm momentum relation, y H and H = I rm ^2 that the gyroscopic couple about each of the body axes is (26a) Lrm = qH Z - rh o , _ -I rm Qq sin Prm M rm = rHx - IrmQ (r cos Prm + p sin (26b) arm) pH Z - N rm I rm Qq rm .

= pH -- qH x = - cos P (260 These rotating mass contributions are added to the inertial moments expressed by Equations (20a, b, c), (21a, b, c), and (22a, b, c).

For small perturbations of the aircraft, the perturbations of the gyroscopic couples resulting from the rotating mass are expressed by rm rm (27a) QAq sin 6 ALrm = -I 6 rm ) (27b) Or cos 6rm + AP sin rm = IrmQ AM (27 c) I rm Mvq = - cos 0 r .

ANrm These perturbations are added to Equations (24a, b, c) when significant, in which case, Equations (24a, b, c) will become inter-dependent because of "tie coupling of the longitudinal-mode and lateral-directional-mode moment equations. It should be noted that, if 6 rm were variable, the above relations in Equations (27a, b, c) would have required further expansion and introduced an additional degree of freedom in the form of A6rm .

3.3 Gravitational Force The gravity force will not contribute to the moment equations as long as the origin of the axis system is at the centre of gravity.

3.3.1 Components of Gravity Force With the gravity force W acting along the z r axis, the expressions for the components of gravity force acting along the body axes are readily deduced from Figure 4(a) to be = —W sin 6 ( 28a) X P = W cos sin (^ (28b) Y Z = W cos p cos ( ( 28c) These components are subtracted from the inertial-force equations (19a, b, c).

3.3.2 Small Perturbations

Small perturbations of the components of the gravity force may be based on Euler angle perturbations superimposed on the unperturbed angles using the same basic reference frame, or on Euler angle perturbations relative to a secondary spatial reference frame made up of the unperturbed aircraft axis system as discussed in Section 2.1.6. and as shown in Figure 4(b). In this second approach, unperturbed body axes are used as the secondary spatial reference when ii , :terest is primarily in perturbations of body-oriented flight test data.

P 6 4 O6 Using the first approach, replace and ^ in Equations ( 28a, b, c) by

+ O

and (^ , respectively; expand the resulting trigonometric functions, consider cosO_^_- 1 , sinO _-- A _ , and A_0_^_^ 0 ; and subtract the initial conditions from the result. The resulting small-perturbation expressions are —WOO (29a)

cos 6

AX =

= WOO cos 6 cos 0 — OP sin 6 sin (^) (29b)

AY —W(OO

cos P sin + O6 sin 6 cos 0) . (29c)

AZ =

In the second approach, using the unperturbed body a:es as the reference and Oqi' , O6' , OO' and (Fig. 4(b) ) as the Euler ..files of the perturbations, the per- turbations of the components of gravity are obtained by using Equations (1c) and (28a, b, c). In Equation (lc) the generalized quantities X r , Y r , and Z r are replaced by the expressions for X g , Y , and Zg , respectively, as given in Equations ( 28a, b, c) ; and the Euler angles qi , 6 , and are replaced by Oq' , O6' , Off' , and respectively. The generalized quantities X b , Y b at:d Z b in

Equation (lc) are now equal to ( X g + AX +

( Y g OY 9 ), and ( Z g + OZg ) : respectively.

), By subtracting the initial conditions (equations (28a, b, c)) front U.i resulting O

perturbed equation after considering r.os _ , and O_O 0

1 . sin

the perturbation expressions for this second approach will have the f(.1lowing form: Ovi' — cos 6 cos (^ O6') (30a) = W (cos 6 sin

AX

= W (sin Oq' (30b)

+ cos 6 cos 0 -AO')

AY

(30c) AZ = —W (sin 6 O6' + cos 6 sin (^ -AO') .

The advantage in using Equations (30a, b, c) instead of Equations (29a, b, c) is that, for small perturbations during highly banked as well as wings-level flight, l

ntp

[Ar dt (31a)

AP , fnq dt

(31b)

A(k ti Jnp dt .

(310

^qi , nA ,

To apply such simple integrations to and n^6 in Equations (29a, b, c)

requires that (^ and be small.

Both Equations (29a, b, c) and (30a, b, c) show coupling of the longitudinal and

A

lateral modes. In both sets of equations, the longitudinal modes (AX Zg) and are uncoupled from the lateral-mode perturbations by performing a longitudinal pulse when initial conditions are steady-state. In performing a lateral-directional pulse from steady-state conditions, the lateral-mode expression (30b) is inherently un- coupled from longitudinal perturbations, whereas expression (29b) shows interaction LAB of the longitudinal perturbation which is excited by the lateral-directional pulse.

When banked and climbing flight are being considered, it may be surmised from the preceding that Equations (30a, b, c) are more amenable than Equations (29a, b, c) to theoretical stability analysis and for analysis of flight data when longitudinal or lateral pulses are applied from initial steady-state conditions.

3.4 Aerodynamic Derivatives In stability and control investigations based on flight data, the previously dis- cussed inertial, gyroscopic, and gravitational quantities are normally equated to aerodynamic parameters only. This is done primarily to facilitate the analysis of flight data. However, in doing this, the parameters are no longer pure aerodynamic parameters, inasmuch as they will have been modified by influences arising from power and aeroelasticity as well as possible other sources. Generally, these influences can be accounted for and the pure aer( jnamic parameter arrived at.

Inasmuch as the equations are set up under the principle of super-position of influences, situations may be encountered in which the accuracy of the results obtained from the equations will deteriorate. This is of particular concern where very rapid control inputs are encountered. Also, inasmuch as the aerodynamic parameters are in the form of derivatives, care must be exercised not to exceed the validity of the derivative.

Finally, there are some limitations in combining several of the derivatives, Cn r — CnQ , for example.

Consideration is given at this time to the above-mentioned factors which have significance in the utilization of aerodynamic derivatives in the equations of motion and in the determination of the derivatives from flight data. For convenience, the conventional derivatives are tabulated overleaf.

Longitudinal 1)erivat Ives aC 2C +r = V N N RC N RC N RCN CN a - ^ N CN A CNS CNu CNq a u Cos aCos cc ^b ^ ac ) qc ( 2V 2V 2Cc "Cc ^ C c c )Cc ^Cc C C CU Cc a Cca - = CvS = V ^ ^$

u + cos (X Cost

^a a qc a -- 2V 2V ^)C ^C m 2 C aCm ^C m m C ^Cm m C MU = V + Cm , _ mq CmS = Cma = as au cos acos ^1 ^ a a ^ a (qF) C 2V 2V Lateral-Directional Derivatives aC ac a C - y - a Cy y C Cy Q - ^^ Cyr = a b Cyp t;y - Cy s C" - -a

a

b

a

v

,v P 2

acl act acl act = a^

Clr

C1,8 C1 - pb CJS - ab

clA = a ,3b

a /rb a —

^2V) ^2 V ) 2V

ac

aCn a C _ ac

ac e ns CnQ Cnr Cn P a,8 as _ a pb ^b rb\ -a a — 1 a -- 2V 2V (2V) 3.41 .1 Significance of the Derivatives The aerodynamic derivative provides the slope of the curve of the aerodynamic force or moment coefficient, as the case may be, with respect to an independent variable - other independent variables being considered constant - at a particular value of the variable. In analog simulation studies, nonlinear curves are reduced to straight-line segments, each segment being valid only for an incremental range of the independent variable.

In the inverse problem of obtaining derivatives from flight data, the derivative is valid only for the incremental range of disturbance of the independent variable, at the steady-state condition, used in determining the derivative. An example involving a non l inear with 8 is shown in Figure 7. In this example, the variation of C n

origin, 0, represents steady state and (A,8), and (A,8 ) 2 represent two disturbance

ranges of the variable. It will be noticed that the derivative obtained may differ appreciably in magnitude because of the nonlinearity of the curve in the disturbance ranges (A,8) 1 and (A P) 2 .

Vnsteacl;v Flow l;ffect,; i.:' In dealing with the derivative concept in x counting for the influence of independent variables on an aerodynamic force or moment coefficient, for example rh ' h I)b Cn Cr' " 4 Cnr ^V + Cn : -^^, + Cnp ^V + Cn8r,^1r it is assumed that each derivative contributes to the total as though it acted alone and that the aerodynamic force and moment coefficients are functions of the instan- taneous, values of the disturbance displacements and velocities, control angles, and their dorivat.ives. Further, the derivatives are based on the variation of the co- efficients under near-steady-state conditions of the variable. Although the deriva- tive concept of treating aerodynamic force and moment perturbations has generally worked wall, the application to situations ol - r,Ltiid'.y changing independent variables (unsteady flow conditions), as in the case of a very rapid control displacement or a sharp-edged gust, does not necessarily give correct answers. This is due to apparent mass effects of the air, whose inertia will not produce instantaneous changes in circulation and consequently causes aerodynamic lag. This is illustrated in Figure 8, which shov es the variation of C N as a function of nondimensional time, F,/2V , as a result of a step gust. The derivative concept would show a constant slope curve, whereas the actual variation of C,(t) would show a lag at the initial instance of the step gust input.

When an aircraft is oscillating sinusoidally, the lift will follow the sinusoidal variation in angle of attack but will be of smaller magnitude and there will be a phase difference between the lift and angle of attack. This unsteady flow effect is a function of reduced oscillating frequency, ^ ,!^/2V , as well as Mach number. Although the magnitude of C Na is not normally affected appreciably for normal airplane oscillating frequency conditions, the phase lag may bring about a large change in CNa This may be of considerable importance in pitch damping of tailless aircraft (Ref.7).

In general, all the aerodynamic derivatives behave in a similar manner. Thus, it is seen that attempts to use the derivative concept in analog simulators involving very rapid changes of the independent variables can lead to errors; conversely, deter- mination of derivatives from flight data requires awareness of the maneuvering or unsteady flow factors mentioned which can influence the magnitude of the derivative.

3A.' Derivatives ik! ith Respect to u Aerodynamic derivatives with respect to u are of concern when phugoid modes are being investigated. Because this mode is often overlooked, these derivatives are generally unfamiliar. Thus, some consideration is given to them at this time for future reference as needed, Consider —Z = C Differentiation with respect to N gS u shows az ac aV

- a u = a u qs+r^p"Sau

(32) aC 2C 1 _

(v

a

u cos a cos ^) V

where ^V/ r 'u = 1/( cos aCOO)

from u = Y cos ^3 cos a . The 4Z due to change in u

is now expressed as qS - Z u = CNu V .

(33) Since 2CN V ' CN + d u cos acos Q1 is more than simply the variation of the normal coefficient with res p ect to velocity, u , it can fittingly be called the effective aerodynamic derivative of C with respect to u , or C N .

. Similarly, the effective aerodynamic derivatives of C m

and Cc

with respect to u are dC 2C dC 2t; m - V m + and ^' + c --- , ^) UCOs altos au ^ /1 Cos a oolig or and Cc u , respectively.

Cm 3./i.4 Derivatives with Respect to q and a , and r and It is customary in reporting flight-determined derivatives, wherein transient

oscillations are useL in the analysis, to pair the derivatives varying with respect

to q and & and those varying with respect to r and ^ . For example

Aqc Aqc

ac ti 2V (Cmq + CM&) 2V - + Cma - ' Cmq 2V

an d (34)

Inasmuch as the phenomenon involving a is different from that involving q , and the phenomenon involving r is different from that of Q , the pairing is valid only when small - perturbation transient oscillations of a maneuver are involved and satisfy the linearized equations of motion. In addition, although the pairing works well for the longitudinal equations whether or not stability or body axes are: employed, the validity of the paring for the lateral - directional equations is dependent on the use

of the stability system of axes; if body axes are used, the pairing of r and

derivatives is permissible at low angles of attack.

In performing a small-perturbation longitudinal transient oscilation, the center of gravity of the aircraft tends to move along the flight path as though it were not

disturbed; consequently, the amplitude ratio /IA&I is similar to 1.0 and the

Iogi A q

vector quantities A q and A& are approximately in phase. Thus can be sub-

stituted for A& . In the case of a lateral-directional (Dutch roll) transient

oscillation relative to the stability axis syst.3m, the aircraft, in tending to

maintain its center of gravity along the flight path as though it were not disturbed, will experience Ar , inasmuch as r and ' are now referred to the same axis system. Thus, the amplitude ratio lAr1; LV'I is similar to 1.0, but the phase relation is approximately 180 0 ; consequently, the sign of the A i^ derivative is changed to mines in pairing r and derivatives. In dealing with the body axis system, I•'1rl!I^ti^^ and fir; can differ appreciably from 1.0 and 180 0 , respectively, at high angles of attack.

It is reiterated that pairing the derivatives is valid only for the specikl con- ditions mentioned. On the other hand, it has not been possible to solve for the a r derivatives independent of the q derivatives, and derivatives independent of r derivatives, from flight data with any degree of consistency and confidence, 14.5 Power Effects The propulsive system may have a significa ► .t influence on the stability as well as the trim of the airplane. Its force and moment contributions to the equations of motion may be presented as derivatives in the equations. If the power contributions w are not accounted for by their own derivatives, they will be reflected in the magnitudes of the aerodynamic stability and control derivatives which will then become, in essence, effective derivatives. A comprehensive treatment of power effects is complex and beyond the scope of this paper. Only i..,jor effects are considered, to show how propulsion system derivatives contribute to the effective values of the aerodynamic derivatives.

It is essential at this time to emphasize an important point regarding consideration of the effect of power on stability. True inherent stability of the aircraft with power on can only be evaluated by keeping the settings of the engine and propeller controls fixed during the maneuver. Any maneuver that entails alteration of the pro- • vstem pi,,lsi,1 , controls during the maneuver will not provide a true index of the stability i" gym an analysis of the time history of the maneuver.

Influence of propellers: Influences of propellers consist of direct propeller effects and also indirect effects due to the propeller slipstream on the wing-fuselage and the tail surfaces.

Direct propeller effects: Direct propeller effects, as shown in Figure 8, consist of a direct thrust T acting along the thrust axis, and a transverse forve (Y)p as well as a normal force (—Z) p , perpendicular to the thrust axis in the plane of the propeller disk. The thrust T is a primary function of a and 1, . Quantitative determination of the normal and transverse forces ( — Z) p and (Y) p may be accomplished by solving for (CN a)p and (Cy ,,)p , as discussed in References 8 and 9, Actually, the derivatives aie of more concern for the purposes of this paper than the actual magnitudes of the forces.

The contributions of the direct propeller effect (Fig.9) on lor,gitudinal and lateral stability are reflected in xp (Cm a ) p = C Ta Zp + (CN a ) p (35)

c c

an d XP (Cn,8)p = — (Cy Q . (36) 2U It is opportune to note the influences of the direct p ropeller effects on CL,

when performing a transformation from the body to the stability axis system. The net

effective C L and C D of the aircraft, in the absence of angular rates and for fixed controls, can be expressed as Aero Power C N cos a —sin x sin up cos a p Cc rCL (37) C sin a cos a - cos a p CT sin u D p (Cn)P where the direct thrust is assumed ^o be vectored parallel to the body x-axis and a p 6 .

= a + Differentiating Equation (37) with respect to a for CLa CL a (CNa cos a - Cc a a) - ) p Cos a , (38) sin CD + CT, sin up + (CN a where C is the effective value as shown in Equation (37).

Study of Equation ( 38) shows that power increases the effective of the aircraft.

CL, On low-performance aircraft, the power effect is generally negligible.

Propeller slipst°eam: The propeller slipstream influences the distribution of the aerodynamic forces on the aircraft structure as a result of (i) the increase in local velocity over the structure due to and in the propeller slipstream, and (ii) upwash ana downwash effects of the rotating slipstream of the propeller. The slipstream can be stabilizing or destabilizing, depending 110on the direction of rotation of the propeller and the position of the tail relative to the rotating slipstream. Analytical techniques to quantitatively account for the propeller slipstream effects on the stability of the aircraft have not been satisfactory. Generally, powered models are used to provide engineering data on new designs.

Influence of jet engines: The jet engine has the counterpart of the effects that were shown for the propeller. It provides a direct thrust, shows no rnia.l and transverse force effects at the entrance of the intake duct, and — depending on geometry — is ca,,)able of influencing the equilibrium and stability of the aircraft by inflow of air

ir.to the jet exhaust. Unlike the propeller, the influence of the jet engine on the

tail surfaces, and hence the stability of the airplane, is amenable to analytical techniques to quantitatively account for these effects.

The thrust produced on the aircraft equipped with a jet engine is equal, as shown in Figure 10, to the vectorial change in mcmentum of the air and fuel passing through the engine plus the resultant of the pressure forces acting across the inlet and outlet areas. Where the intake and exhaust are in line with the thrust axis and the x-body axis

T = CrgS

= m j V j — m aV cos up + ( p i A i -- p j A ) . (39)

J A change in direction of the momentum vectors at intake or exhaust relative to the x-body axis brings into being forces normal to the body x-axis. Where the jet exhaust Is parallel to the x-body axis, the component of the normal force in the xz-plane of symmetry is expressed by maV sin o^ (40)

/

m a V sin xE q S qS Similarly, a normal force in the transverse plane is in evidence during a sideslip, or y (Y) p = (C ) p QS —maV sin (41) MV sin _ — qS .

qS A jet-induced inflow toward the jet axis at the tail may affect the stability of the aircraft if design precautions have not been taken. As a result of the jet exhaust spreading out behind the engine, a turbulent mixing of the air outside of the jet stream with the jet exhaust takes place along the boundary of the jet stream (see Figure 11). The drawing in of the air from outside the stream is jet-induced inflow.

A horizontal tail located in this jet-induced inflow field will be subjected to jet- induced downwash angles. Thus, the angl.^ of attack of the tail would be modified and

pitching moments would be created that would affe c t the stability as well as the

equilibrium of the aircraft.

The quantitative effects of the jet-induced downwash at the tail can be calculated

by using the theory developed by Ribner lc . This theory allows for curvature of the

jet due to angle of attack of the aircraft. It is also applicable to determination of jet-induced sidewash of the vertical-tail surfaces at asymmetric power conditions or during sideslip.

In the absence of suitable design precautions, such as boattail.ing of the exhaust to shield the tail surfaces from the inflow effect, the change in pitching and yawing ) p moments resulting from the jet-induced downwash (A cc,. and sidewash (Aa v t )p t respectively, can be expressed by (42) ( C Na)h. t.

—

(AM h. t. ) p (A %. t. ) pg tSt xh. t.

(43)

(AN v. t. ) p — —(CN a,)v. t.(A N.t. ) 0tSt x v. t.

where xh t and x v t are negative values with tails aft of the center of gravity.

The variations of forces and moments due to the yet (ngine are primarily functions , and V , assuming that control settings are constant. From the preceding of x , it may be readily deduced that m a V cos x) '1;) g tsh. t. '1(aht.)

qS ,ir-Y. qS Ax + m a V cos -J.

+ 1) X I) "(J. 1) t S h. t. x h. t. c^h. t. )P

m ^) P Ct 7P g (44b)

(C (C^ x) ► ,. r..

c qS c qSc A a

a V cos

_ m gtsv. t. -(,J. V. t. ) p ) 45a) — – (

( )

CN D, v. t.

C Y, < P , qSb ri.

qS

m a V cos"' x" -

"

tsv. t. v. t. (,,I V. t. )P

g x

^ a (45b) - Na)v. t.

1Cn.3)P ( C qS b qSb where x is positive when the air intake is forward of the center of gravity and x V t is negative with vertical tail aft of the center of gravity, and

"C. 2C

( C Cl d P (46)

- ^V + T

i u os acos P

c i t

'- i c 2C

m ^' + (47) (Cm u V )p -

u cos acos

P where zP m a V sin a x , qA. t. x h. t, ^ . U (48) (Cm ) P = CT c + ( C Na) V. t.( V. t. ) P c + qSc QS and ^Cm _ ACT + m a sin ap Z XP (49) ^u ^)u c qS c P 3.4.6 Aeroelastic Effects The preceding discussions assumed that the aircraft was rigid. This assumption was permissible in the past; However, modern aircraft flying at high speed under dynamic- pressure conditions are subject to degrees of flexibility of component parts which li-16 the a^r craft The cannot, at times, be ignored and which affect the stability of contribution of aeroelastic deformation to derivatives is dependent primarily on aircraft geometry and dynamic pressure as well as structural rigidity and Mach number.

parts: static and dynamic Aeroelastic phenomena may be considered in t^vo separate aeroelastic effects.

When aerodynamic loading takes place at a sufficiently slow rate in coftlp: 4 rison to rc t^ ^,;it tho natural frequency of vibration of the pertinent part of th y, 5trnrt - pe- the assumption of static deformation of the structure, the influence of aeroelasti0 ty can be accounted for by modifying the derivatives. Illustrations of steady-state distortions which have been serious in the past are aileron reversal and wing divergence.

Today, such factors as thinner wings and more t'lexible fuselages have magnified the effects of structural flexibility on stability and control cf aircraft.

If the aerodynamic loading frequency were to approach the structural frequency of the pertinent component, the structural deformation would produce perturbations in the aerodynamic forces and moments which have to be accounted for by the introduct.io ► , of additional appropriate derivatives in the equations of motion and the introduction of additional equations, which would be ela9 t .icity equations.

3A.7 Other Effects The preceding discussion has included major fat'-irs which influence analysis and account for discrepancies between wind-tunrel and flight data; however, it do-3s not account for all factors. Other factors corld include jet pluming, flow separations associated with movements of shock waves, and fuel sloshing. Since one never knows what phenomena will occur., it is imperative to have an open mind in trying to ,account for discrepancies in comparisons of data.

3.5 Summary of the Equations of Motion The various dynamic relations which have been discussed are pertinent to an under- standing of the equations of motion and the conditioning of data to the e4dotions. The influence of power and structural flexibility on the various aerodynamic parameters (coefficient and stability derivatives) was stressed, and it was pointed out that the net result of these influences, or modifiers, was the emergence of an effective aero- dynamic parameter.

It is easily recognized that the introduction into the equations of motion of each individual modifier to the aerodynamic parameters would result in a cumbersome set of equations. It is more practical to let the normally accepted stability symbol (Cn,8, for example) represent the effe '.ive value than to list all modifiers. In so doing, one should be aware of the various sources which contribute to the magnitude of the effec t ive parameter in order to properly account for these contributions during an analog investigation, or oti:_r study, in which wind-tunnel and calculated data are used. On the other hand, In the inverse problem of de.ermining coeifi , ients and derivatives from flight data, a discrepancv in trends as well as magnitude between wind-tunnel and flight data will suggest possible influences from sources not accounted for by tunnel data.

3. 1 General Equat ions The following a' mptions are made with regard to the equations of motion su ► .amarized in Table IV: (i) The airplane behaves as a rigid body in that the moments of inertia, inclina- tion of principal arms, etc., are not affected significantly.

(ii) The airplane is symmetrical about the xz-plane with regard to geometry and mass distribution, (iii) The axes of rotating elements on the aircraft are fixed in a direction relative to the body reference axes.

(iv) The ea,*t.h is flat. Aircraft speeds are assumed to be insufficient to include earth curvature in the equations.

(v) The forcing frequency of a disturbance is sufficiently far removed from the natural frequency of the pertinent part of the structural components to permit the disturbance to be considered as a static load and the effect of deformation to be accounted for by modification of the aerodynamic parameters.

(vi) Each aerodynamic parameter is an effective parameter, in that it includes all sources contributing to its net value.

Although listed. in Table IV for completeness, experience has shown Cy p , Cyr Cy4 and and to be normally negligible.

Cccc Cc .Small-Perturbation. Equatrc:,s 3.5. 2 The general equations of motion in 'fable IV are suitable for analcr, and d gital programing which involves large disturbances and nonlinear terms, they are not suitable for analytical purposes. For such purposes, it is necessary to linearize the equations at le p >t to an engineering degree of accuracy. This is accom;. Lshed by rest:;r:'-ing their applications to small perturbations, as has been discussed previously. In addition, the Prturbations are referred to a secondary spatial reference frame, discuss,--' in Secti(,r 2.1.6, which is the unperturbed airplane axis system shown in the secondary reference system for small perturbations permits Figure 4(b). Using the use of Equations 01a, b, c), which simplifies analysis and extends the validity of the linearized perturb?ti:cr, equations to maneuvers involving high pitch attitude anO large bank angles.

, in Table V in a format The uncoupled, linear ized perturbation equations are sr _, which generally constitutes the basis for application to -.!vative determination.

The assumptions listed for the general equations of motion are also valid for the equations in this table. In addition, it is assumed that the maneuvers are such as to minimize the errors in the g terms ari:ir,g from the approximation of the gravity terms s"iown in Equation's (40a, b, c). Also, it is assumed that the gyroscopic couples of rotating elements are not significant, which may not always be the case. The equations are complete within the limits of the assumptions, and analysis would reveal all modes of longitudinal and lateral motions.

16 terms in the longitudinal Omission of the 1,-%glT.udinul force equation and the equations (50a, b, ---) would remove the phugoid mode from the analysis of the longitudinal motions, leaving only the short-period mode. This short-period format of the long- employed. Although the small-perturbation itudinal equations is the ono usual.ly equations shown in Tabie V are frequently used in the format shown to develop relations for derivative determination, it is also desirable to list the equations in an opera- tional format as Laplacian transforms with Laplace operator s .

Using; Laplace transforms enables the dynamic properties of the airplane to be defined by a series of transfer functions relating the various responsive motions of the airplane to disturbing inputs. The transfer functions are extensively used in stability and

control, handling qualities, and automatic flight: control investigations to aQ5 ess the

ur^tion changes, the effects of particular stability derivatives, and effects of config r the effects of chaa ges in automatic control systems. They are also helpful in obtaining stability derivatives from flight data.

With zero initial conditions and inputs clue only to control deflections, the Laplace transforms of the small-perturbation equations of motion take on the operatioaal _forms

shown in Table VI as Equations (62a, b, c) and (63a, b, c). The notations X u , Ma

etc., shown in the equations, are a convenient means of listing the parameters.

3.6 Determination of the Roots of the Determinant of the Lateral-Directional Small-Perturbation Equations The following discussion regarding the determination of the roots of the determinant of the lateral-directional small-perturbation equations is based on Reference 17.

Although the main points are brought out at this time, recourse should be made to the reference for more detailed considerations.

3.6.1 The Determinant Using the Laplace transform format of the lateral-directional equations (Equations (63a, b, c) in Table VI), the determinant of these equations may be expressed in either of two formats, as follows: (i) When expressed as

As u + Bs 3 + Cs' + Ds + E = 0 , (65)

then

A - I x 'I' - 1

B = (L +IN p ) +(Nr+IzL,,,)-(I-

IX"II ZI P + IZLdY Q - C = ( N p L r - N r L p ) - ( L p + I x 'N p )Yp - ( N r - [(1 - I' sin a)Np - (sin a - I I )F, I (66) D = (NRL p - N pLQ ) (N^ + I Q ) - +(N rL - ,^ 4L „) Ya + g i - a.

g (t,,; r i 'Y 3 } (N#!, N rL,3 ) sin a r E = g l (L/31? p - L pN B ) - g 2 (N^3L r - NrLQ) (ii) When expressed as s + bs 3 + cs + ds + e = 0 , 2 (67) then by -Yp

c= - (IYpLr - NrLp) + LpY Q + NrY 13 + NQ - GQ sin a

d = - (Na

8 P - NpLQ) - (NrL p '- N p 'L i ')Y Q 1 NQ +

- 9 (68)

g 2 LQ - (N ,8 Lr - N r V) sin a

e

- g 1 (LQNp - L pNQ + NrLQ)

2( N Q L

r -

wh —e the primed values are equal to It N + I I L L + I19 i

Nr = z i i x 1

Lr - (69) and IxIZ i 1 - Ixz The determination of the roots of the determinant is dependent upon the modes of motion of the aircraft. The modes may be: (a) Lateral phugoid ( coupling of spiral and roll modes) and Dutch roll.

(b) Spiral divergence, roll subsidence, and oscillatory (Dutch roll).

3.6.2 Determ;;,ation of the Roots when Lateral Phugoid and Dutch Roll Modes Exist The determinant ( Egn.(67)) can be approximated by the following biquadratic d1 a bd d e

s 2

+ (b - - ' s + c - - s2 + - s + - = 0

(70)

^ c/ ( c c c c

in which the first and second quadratics represent the Dutch roll and lateral phugoid modes, respectively.

Two sets of conditions must be satisfied if Equation ( 70) is to be applicable 17 : (a) The approximate nature of Equation (70) requires that e/c 2 « 1 and bd/c 2 << 1 to assure Q 11idity of the equation.

(b) It is necessary that d 2 - 4ed < 1 in order that tho lateral phugoid exist.

Reference 17 points out, on the basis of limited experience, that, for values of a/c2 of approximately 0.05 or less, bd/c 2 can be as large as 0.25 and d 2/4ec as low as 0.005 without compromising the engineering accuracy of Equation ( 70). Thus, the equation applies if

1 r

bd < 0.25 e << 0.005 < d < 1 (71) C c 4ec The second quadratic in Equation (70) expresses the lateral phugoid very simply; thus, d e from s2 +- s +- c c e

%ph = -

and c (72) d

2 ^ ph( h ph C

The first quadratic in Equation (70) is unwieldy. It is simpler to determine the characteristics of the Dutch roll modh by the following factored form of determinant

!L) 2)

(s 2

+ 2rcv n + cvn

s (s 2 + 2C ph ph

s + (J ) 2 = 0 (73)

Expansion of this determinant and comparison with Equation (67) shows that

b - 2rcwn +

2rphcvnph c

+ 4ph + (2^ w d (2^pjfl)nph)

n

(74) d = (2^phwnph)(4)n + (2^wn)4ph E = Gin J%h^n Since cvn ph and ( 2^ phcvnph ) are nt,

, ained from Equation (72), wn and (*od can

now be determined from Equations (74) or

2^con = b • - 2 ^ph nph

(75)

cvn = c - wn

(2 ^&Jn) (2 ^ph cv n ph ) .

ph - 3.6.3 Determination of the Roots when Spiral Divergence, Roll Subsidence, and Dutch Roll Modes Exist When the spiral divergence, roll subsidence, and Dutch roll modes constitute the

lateral-directional. characteristics of the airplane, which is normally the case, the

determinant as represented by Equation ( 67) may be factored in the following terms characterizing these modes:

2 +2^Wn +can = 0 (76)

s+— s+— s

T s

TR / The coefficients b , c , d and a (Equation (68)) in terms of the factors of Equation (76) are 1 1 b = 2 ^wn + — + — T R Ts 1 1 1 c = can n + 2 aw 1 + — + s TRTs (FR T (77) 1 l 1 d = CO2 — + — + 2cv n n TRTs T R Tu e = W2 n TRTs When the spiral-mode root, 1/T s , is much less than the roll subsidence root, 1/T R , as it usually is, the coefficients b , c , d , and e may be approximated to a good degree of accuracy by 2 ^w n b + — TR (78) c w + 2 ^c`'n — T d acv 2 1 r. T R R Eliminating 1/T in Equation (78) n ) d + ^ c = (2 ^cv

n

7n (79) d (2^w+ d n n Eliminating 2^w in Equations (79) provides an accurate solution of within the

^n

limitation that T s TR or c(u)2) 2 – d 2 = 0 (80) (ten) + bd(a) 3 – 2 Eliminating in Equations (79) to solve for 2^ron within the limitation that

'ten

1/T s << 1/T R results in

(2sycvn) 3 - 2b(2^^^^ 2 n )

+ (c + b2)(2r.^)n) + (d - cb) = Q (81) The roll-subsidence root, 1/T

R

, may now be obtained from coefficient b or d in Equation (78) or 1 d TR

^^^n

(82)

b - 2 ^a)n

T

The spiral-divergence root, 1/T s

, may now be approximated from a,iiy one of the coefficient expressions in Equations (77), such as A

— 2 (83)

T s

cvn(1/TR)

4. MASS CHARACTERISTICS The airplane mass characteristics - weight, location of the center of gravity, moments of inertia, and inclination of principal axis - significantly affect airplane motions. Errors in the knowledge of-these quantities are reflected directly in the flight-determined derivatives and may govern the validity of the derivatives in com- parisons with wind-tunnel data. Although possil;l-- inaccuracies in the knowledge of the inertia characteristics must be given serious consideration in comparisons of flight-determined derivatives with wind-tunnel data, these derivatives have been used effectively in flight-guidance Nimulator studies.

The weight and horizontal location of the center of gravity are always determined experimentally.

In p smuch as the vertical location of the center of gravity, moments of inertia, and loca-ion of the principal axis are difficult to determine experilrnntally, manufacturer's estimates are usually relied upon. These estimates are considered to be of sufficient accuracy for most work involving flight tests. If more precise data are required, they sh3uld be determined by using experimental, techniques.

It would be highly desirable to determine all of the mass characteristics experimentally. This is not always feasible because of the lack of proper facilities.

Large, flexible aircraft, such as the Boeing B-52, of;i`er practical problems, in that

experimentally determined rolling moments of inertia with wings drooped would not be representative of flight conditions. The following discussion of the experimental eletermina,tion of mass characteristics of aircraft is intended to serve as a guideline in setting up suitable facilities for use witli most categories of aircraft.

4.1 Weight and Center-of-Gravity Location The weight and longitudinal position of the center of gravity relative to the horizontal reference line of the airplane for the L...,;^;,y and gross weight conditions can be obtained easily by leveling the airplane on suitable scales or electronic weighing cells. With weighing cells, two of the cells (R 1 and R 2 ) are usually located at the wing ,jackpoints and the third cell (R 3 ) is located at some convenient distance, I , forward or aft of the wing jackpoints. The horizontal position of the center of gravity relative to the jackpoints is then determined from R3, n l = (84) Y R For aircraft operating on conventional fuels, the variation of the center of gravity with fuel consumption can usually be defined adequately by weighing the airplane at several fuel levels, providing there is a predetermined sequence or mode of operation in obtaining the fuel from the various fuel cells. When the aircraft is equipped with fuel cells from which the fuel can be drawn selectively, the center of g,'avity position becomes a function of the sequence in drawing off the fuel from the various cells as well as the weight of the fuel. In some instances, it has been found necessary to account for fuel-tank shape and airplane attitude. Where hazardous fuels are used, the center of gravity is determined experimentally for the no-fuel condition only; the effect of fuel on the center of gravity position is calculated. The horizontal locat1c;i of the center of gravity is experimentally obtained at least to within 0.01 mean aero- dynamic chord, which is considered adequate for derivative determination.

During flight tests, the center of gravity is obtained by observing the total amount of fuel consumed and subtracting it from the takeoff weight. Reference to a chart showing the variation of weight with center of gravity provides the desired answer.

An accurate knowledge of the verticel location of the center of gravity is pertinent to the experimental derivative studies, insofar as experimental determination of moments of inertia and comparison with wind-tunnel data are concerned. The vertical center of gravity can be obtained by static or oscillatory techniques. For the static test techniques, the airplane is placed in various pitch or roll attitudes. For the roll approach (Fig.12), the airplane is mounted in a horizontal, wings-level attitude on knife edges alined with respect to each other in the plane of symmetry of the aircraft.

By rolling thE. airplane to various attitude angles and measuring the reaction R 1 , moment arm , and the roil angle 0 , using a clinometer, the vertical position of y l the center of gravity is obtained from the equation _ R l y I — W ? c sink z r85) °^

W sin 0

For rigid aircraft of the order of 1:5,000 lb, and under carefully controlled conditions.

the vertical position is considered to be determinable to within 1 inch.

To determine the vertical position of the center of gravity from free-oscillation tests, any one of several techniques may be used. The simplest technique consists of changing the equivalent torsional spring constant for pitching or rolling moment of inertia tests. For rolling-oscillation tests with the setup shown in Figure A and with small damping effects - a necessary condition for succe-sful tests - the equations of motion for the two spring conditions are + mz 2 (Kt, - Wz -

+ mc z2 )d' 1 + W czc)O 1 = (86a)

(Ix + IXC

(I X + I XC

+ mz 2 + m Jr (K t2 - Wx - W 0 (86b)

C z c ) 2 C z C 2 =

)(^

moons; de ring 4) 1 = A cos 1 t

and fie = B cos (,) 2 t ,

it is found upon solving Equations

(86a, b) for z the vertical distance from the knife edge to the center of gravity,

that

Z

z _ K p l /p 2 ) 2 -

t2 - K tl( C W C (87) W[1 - W (p1/p2) 2] The equivalent torsional spring constant, K

t

, may be changed from Kt 1 to Kt 2 by

changing the linear springs or the distance a which is perpendicular to the spring (see Figure 13). The change in linear springs is probably the more desirable approach.

Inasmuch as the rolling-oscillation test setup discussed constitutes an inverted pendulum, it is imperative that the equivalent torsional spring constant, K

t , be

greater than Wz + W z c

C for stability of setup. Also, the accuracy of the results

depends upon avoiding secondary spring actions of tiebacks and structural flexibility, which could inadvertently result in a lower effective spring constant than expected because of an equivalent series action of the secondary unwanted spring action with thr• intended spring.

4.2 Moments of Inertia The moments of inertia of an airplane are usually calculated during the design phase and are based on estimated weights and cen'troid locations for various parts of the aircraft. These calculated moments of inertia are considered to be adequate for most analyses when the results are to be used in simulator studies. However, should experimental determina:p ion of the inertia be required, methods are available (see References 18 to 21). The methods are generally restricted to rigid aircraft and to aircraft whose weight, as well as the safety precautions of the experiment, will permit pivoting the aircraft on knife edges and, suspending it from overhead cables.

Schematic representation of typical methods for determining the rolling and pitching moments of inertia are illustrated in Figures 13 and 14, respectively. Equation (86) is applicable to the determination of rolling moments of inertia in accord with Figure 13, with consideration given to the proper interpretation of the lengths z and zc to the mountings shown. In Figure 14, cradle weight is zero. The yawing moment of inertia may be safely determined from a cable-suspension method used to determine the inclination of the principal a.xVs (Figures 15 and 16), which is discussed subsequently.

Unless precautions are taken in every dt.11 of an experimental setup, difficulties may be encountered because of flexibility of experimental components, which will alter

the ;ffective spring constant, i. t , or modify the free-oscillation pivotal point

relative to the center of gravity of the aircraft. In one instance cf determining the pitching moment of inertia when the aircraft was supported at the wing jackpoints and oscillated with the spring at the nose, the wing section which had been considered rigid was observed to flex as the aircraft oscillated. This flexing caused the axis of rotation to shift forward and downward from the line through the jackpoints.

A common fault is the use of flexible cables as tiebacks for the springs and con- nection from the spring to the aircraft. Under no condition should flexible connections be used, inasmuch as they constitute springs in series with the actual intended springs employed; thus, the system from tieback to aircraft represents a much softer spring than intended. It should also be noted that, on some aircraft, attaching the spring to the aft portion of the fuselage would be an error, since the aft portion of the fuselage would constitute a relatively flexible structure and alter the effective spring constant.

Serious errors can also result when knowledge of the center-of-gravity location is inaccurate and when the line of action of the spring from the attach point to the air- craft is not perpendicular to the plane formed by the axis of rotation and the roint of spring att%chment on the aircraft (Fig.13).

Gcnerally, the inertia characteristics are determined for no-fuel conditions because fue.i sloshing tends to bring in a beat action in the oscillatory motions. When dot^ mination is attempted with fuel onboard, the difference in oscillatory modes between the slashing fuel and the aircraft should be as large as is practical, with due regard to safety of the setup, to minimize the beat action and permit determination of the natural frequency of oscillation of the aircraft.

Measuring the inertias of very large aircraft is difficult and is compounded with flexible aircraft. Such measurements are not in the .calm of the methods discussed.

A unique facility designed to enhance the feasibility for determining the moments of inertia of large aircraft about all three axes is located at the US Air Force Flight Test Center, Edwards, California,, USA. Its capabilities cover a weight range from 30,000 to 300,000 lb and moments of inertia from 250,000 to 10 x 10 6 slug ft2.

The facility enables the determination of aircraft moments of inertia from measure- ments of changes in pendulum characteristics resulting from the addition of an aircraft to a freely oscillating platform. The ba,:;ic elements of the facility consist of the platform, a control console for activating various systems which ready the platform for oscillation, and an instrumentation console for regulating the amplitude and measuring the period of the oscillations. The platform is an integral cruciform structure 110 ft long and 80 ft wide, with its loading surface flash with the surround- ing floor space. The apparatus employs special hydrostatic bearings (identical to those used in the 200 in. Palomar telescope) to support the platform, which is lockable in two axes with oscillation about the axis of interest.

To contend with the problem of aircraft flexibility, stiffening jacks are used to support the aircraft structure. As a result of the stiffening operation, flexibility effects are considered to be less than 4% in ; poll and 2% in pitch.

The experimental error in the methods discussed is of the order of ±5% or less.

4.3 Inclination of Principal Axis The inclination of the principal axis of the a.;.rplare is one of the more difficult quantities to determine experimentally. An error of 1/4 0 in the value of the inclina- tion of the principal axis can significantly affect some of the derivatives. The method of Reference 22 is considered accurate to 1/60.

This method consists of finding the direction of the restoring-moment vector which produces no rolling moment relative to the body x-axis during the yawing oscillations of the airplane as a spring mass system while suspended by means of a cable attached to a hoisting sling. Figure 15 shows schematically, and Figure 16 shows photogra- phically, a general arrangement of the setup. The airplane is suspended at a horizontal pitch attitude, and yaw restraint is provided by two sets of springs whose lines of action lie in a common plane. The springs should provide a pure couple action. The restoring-moment vector acts normal to the plane of the springs. The springs may be attached to short, rigid mounting brackets located below the wings equidistant from the plane of symmetry or to brackets mounted below the fuselage ahead of and behind the center of gravity. In this respect, the wing mounting arrangement is most convenient and less time-consuming. It is essential that the springs provide a pure couple action.

As the airplane oscillates in yaw with various inclinations of the plane of the spring couple (angle Ssp in Figure 15), some coupling is present between yaw N an' roll L , which results in a certain amount of rolling oscillation. This is shown in the following equations where the subscript r denotes the reference attitude of the airplane: L (88) Ixrzr -° Ixrzrrr = (89) Izrrr — Ixrzrrr = N At some one value of Ssp , however, the rolling motion accompanying the yawing motion is zero (!pl/lrl = 0) . In this situation the preceding equations reduce to — I xrzr rr - L (90) = N .

I zr r r However, as shown in Figure 15, —L tan (91) S.P = N Hence tan F sp = Ixrzr (92) IZr Inasmuch as the inclination of the principal axis is given by the wellknown expression

2E - Ixrzr

tan (93) Iz r — Ixr substitution of Equation (92) fori XZr in Equation (93) gives 2T z tan bs P

r

2t = - tan (94) Iz r — Ixr The value of Iz r is determined as a byproduct of the test by using C cos sp I Zr = r, (95) However, Ix r must be determined from other tests.

Figure 17 shows a typical experimental plot of the variation of Jpj/irj with 8sp for determining the value of a sp at which 1pl /lrl is zero. In obtaining the tests points shown in the figure, the flight test roll- and yaw-rate gyros mounted in the airplane were used to obtain oscillograph records for determining jpj/jrj from the transient osc ? lations.

The measured values of moments of inert-'a relative to the reference axes and the determined inclination of the principal axes may be used to determine the principal moments of inertias, Ix o and I ZO , by using the following equations Ix o = Ix r – Ix r z r tan E (96) ? ZO = Iz r E .

+ Ixrzr tan (97) Although no mention was made of the effects of air mass on the experimental values of moments of inertia, the effects should be considered and corrections applied if necessary. Reference 23 provides formulas to correct for air-mass effects.

Formulas for transferring moments of inertia from one set of axes to another were presented in Section 3.1.

5. INSTRUMENTATION Basic to an analysis of flight data is the instrumentation. Considerable instrumen- tation research has been in progress and many flight test instruments have been deve- loped to improve the linearity of response, resolution, dynamic response characteristics, readability, ruv.— ,'.mess, and reliability of calibrations over varying operating con- ditions and oxt, -nded periods of time. In addition, the application of the instruments require, R ,Ijwledge of mounting accuracy, sources of error in the flight records, and methods of correcting the errors. Inadequate appreciation of the instrument character- istJ-? Lcf,, mounting accuracy, and possible influence of sources of error serves as a detriment to the successful application of new techniques of analysis as well as a detriment to the analysis by approximate methods.

In the following discussion, sufficient guidelines are presented to show the care required in the selection, installation, and calibration of instruments to minimize errors in the analysis of flight data. Individual instruments may differ from one organization to another and the degree of sophistication in instruments and recorders will vary with the individual investigation; however, the principals of operation of the sensors are generally the same.

.

5. 1 (Hach Number, Altitude, and Dyns m i c Pressure Accurate determination of Mach number is of fundamental importance in flight testing high-speed aircraft. The principal methods, discussed in detail in References 24 and 25, are haled upon the following relationship for subsonic conditions (M < 1.0) v- _ //( (98) Q - 1- Y 1 M 2 - 1 - (1 + 0. 2M) 1/ 2 - 1 p ^ 2 For supersonic conditions (M > 1.0), the equation is modified to include the loss in total pressure behind the shock wave 1/('Y -1)

y + 1

— M2 2 5/ 2 + 1 2

Q C _ y

-- 5.76 M ^ M2 - 1 = 1.2M2 1 (99) 2- 2 2-/ A -1 5.6M 0.8 M2

y+1

y +1

The impact pressure q and the static pressure p are measured by using a pitot- static head and pressure; recorders. The maximum Mach number as well as dynamic pressure which can be determined by using pitot-static heads is of the order of 3.5.

Higher speeds are primarily dependent upon inertial platforms and radar. Dynamic pressures at Mach numbers is the approximate range of 2.5 to 8.0 can be determined through the use of a spherical flow-direction sensor and a total- (stagnation) pressure technique.

5.1.1 Pi tot-Static Head (M < 3.5) Much research has been done on various types and configurations of total-pressure 26,27.

heads to reduce angularity effects The type shown in Figure 18 is used widely.

This head has an external cylindrical shape, a cylindrical chamber, and a lo o slant profile. It is insensitive (zero error) to angle-of-attack from -5 0 to 20 0 and up to 100 of sideslip. The error is less than 1% in the angle-of-attack range from -10 0 to 250 and ±100 sideslip.

The arrangement of the static-pressur, orifices on the head has been found to be pertinent in increasing the range of insensitivity of the orifices to flow angularities.

The arrangement used has been determined from tests of orifice configurations2e,29.

The two identical sets or arrangements shown in Figure 18 are each circumferential, with four orifices on the top, six on the bottom, and one on the bottom centerline behind the others, The two sets of static-pressure orifices are used to provide for separate pressure systems. One set of static orifices is used for the pilot's instruments, the other set for flight test recording instruments to minimize the time lag of response that would be encountered with a common system. The arrangement of the orifices in each set provides an increased range of insensitivity to angle-of- attack; however, it is not as insensitive to sideslip. Large static-pressure errors are encountered at sideslip angles greater than 3 0 . Since constant sideslip angles are seldom encountered, the static-pressure data can be readily faired.

Installation of the pitot-static head: Installation of the pitot-static head requires consideration of the complicated flow field of the airplane, which is a function of the airplane configuration as well as Mach number and attitude. Errors in pitot-static- head readings resulting from this flow field are referred to as position errors. The static-pressure orifices are particularly affected by position errors at subsonic speeds; thus, precautions are taken to mount the pitot-static head as far ahead of the airplane as is practical.

Of the various types of installations of the pitot-static head — such as nose boom, wing boom, and fuselage — the nose-boom installation is the most suitable for minimizing position errors. In this installation, shown in Figure 19, the head is mounted on a boom extending as far ahead of the nose of the airplane as is practical. As reported in Reference 29, the amount of error in Mach number due to position error in the static- pressur y measurements can be related to certain physical measurements on the airplane.

This is shown in Figures 20 and 21, which are reproduced from Reference 30. In Figure 20, the error in Mach number due to static-pressure error is plotted as a ratio of boom length to the maximum effective fuselage diameter for subsonic, transonic, and supersonic speeds. In Figure 21, the variation in Mach number error with Mach number is plotted for two airplanes having boom-length-to-fuselage-diameter ratios of 0.60 and 0.95. Above a Mach number of 1.05, the position error drops to zero. The Mach number at which the position error drops to zero is dependent upon the nose-boom geometry and is the Mach number at which the shock wave ahead o f the airplane crosses over the static-pressure orifices.

Wing-boom installations of the pitot-static head are subject to several disadvantages, including possible susceptibility to the shock wave caused by the wing as well as the shock wave caused by the fuselage. This complicates the calibration and makes it more difficult for the pilot to fly at the desired Mach number in the regions where the shock waves are in the vicinity of the orifices. Wing buoms are usually more sensitive to sideslip and subject to more lag in response because of the longer tubing required.

Fuselage installations of the head are subject to position errors, which are diffi- cult to estimate.

Calibration: Calibration of the pitot-static head, fortunately, involves only the determination of the position error for the static pressure — the total pressure is not affected by position error. Various methods that have been used include the pacer method, the fly-by (tower-pass) method, and modifications of the basic radar-photo- theodolite method 24 . The pacer method requires the use of a pacer airplane with a calibrated system and special flights for calibration purposes. The fly-by method requires lg flight at extremely low altitudes past an instrumented course. This latter method not only requires special flights, but is hazardous and limited to Mach numbers of about 0.8.

The radar-phototheodolite method has the advantage of providing calibration data of a radiosonde unit to measure during routine research flights. The method makes use static-pressure and temperature variations of the atmosphere with altitude. It also requires ground equipment consisting of a radar unit, a phototheodolite, a chronograph, and three cameras. One of the cameras photographs the radar scope and gives the slalt range; the target camera gives the correction to the elevation scales; and the third camera gives the elevation scale. The airplane itself is equipped with a radar beacon to assist in tracking. The three cameras and the airplane's internal records are synchronized by means of the chronograph. The radar-phototheodolite unit determines the range any elevation angle of the airplane from which the true geometric altitude of the airplane is determined (within ±100 ft) as a function of time.

A cross plot of the radar-phototheodolite data (airplane altitude vs. time) with the radiosonde results (free-stream static pressure vs, altitude) provides a plot of true free-stream static pressure as a functioi, of time. Since the time base of the air- plane's indicated static-pressure records is synchronized with the radar-phototheodolite, a comparison of the airplane's indicated static-pressure records with the cross plot provides the position error, '^p , of the static head. The corrected static pressure + AF may now be obtained from the relation p = The true impact pressure is now p i determined from q c - p T - p = ci - Lip q True Mach number: True Mach number is determined from tables of q c /p as functions of Mach number based on Equations (98) and (99). The indicated Mach number, M i , as determined from , q ci , and the tables, is plotted against the corrected Mach number, p i M , to provide a calibration curve, such as shown in Figure 22, for the pitot-static- head installation on the airplane. Generally, calibration data points for four or five flights are used before the calibration curve is finalized. The scatter, AM in calibration points is usually within ±0.01 at subsonic and supersonic speeds and within ±0.02 at transonic speeds.

Pressure altitude: Altitude is generally expressed in terms of "pressure altitude", which is the altitude in the standard atmosphere tables corresponding to the corrected static pressure. The corrections for a given pitot-static pressure system are obtained in the form vs. M . The curve for this relationship is derived from the Mach pig number calibration of the system and the position error for the static pressure deter- mined as a ratio of the true static pressure by the following equations from Reference 25: When M < 1.0 , - 1.4M2 ^M ^P - (100) M ]. + 0. 2M P When M > 1.0 , 4.0 2(101) p ,j 5.. 2 - 0.8 M In routine tests, the pressure ratio `p is divided by T i to obtain p , which p i / is used to determine the pressure altitude.

Dynamic pressure: The dynamic pressure, q , is determined from the simple relation = 0.7pM 2 (102) q

5.1.2 Use of Spherical Flow-Direction Sensor to

Obtain Dynamic Pressure In the absence of true Mach number, such as when flight is beyond the practical limit of the pitot-static tube (M _- 3.5), a technique has been evolved to obtain the dynamic pressure, in the higher supersonic and hypersonic regions, directly from the total-pressure port of a spherical flow-direction sensor 31 . The flow-direction sensor, described in more detail in Section 5.3, is a movable sphere mounted at the nose of the airplane to form a "ball nose". The total-pressure port vectors into the resultant velocity at the sensor.

Inasmuch as q = 0,7 pM 2 and, from the Rayleigh pitot formula, p f(M) Pr where f(M) _ (103) (y + 1)M 2 y the dynamic pressure can be expressed as q = 0.7p[f(M)]p T (104) q or = F(M) (105) PT A plot of q/p T versus M (Fig.23(a)) shows that this ratio varies only about 5% in the Mach range above 2.5. As a result of this small variation in q /p T at the higher Mach numbers, it was suggested that an "indicated" dynamic pressure, , could be expressed q i as qi = Kp T (106) Figure 23(b) shows the ratio of indicated to true dynamic pressure, 4 i /—q , for two values of K . Using K = 0.526 , q is 5% high at M = 2.1 and 2.5% low at M = 7 5.1.3 Pressure-Recording Instruments Selection of the pressure-recording instruments and their ranges for a given instal- lation depends on the altitude and Mach number range over which a specified attainable Mach number accuracy is desired. When tests are to be conducted at one altitude, it is no problem to select a pressure-recording instrument to provide the requisite accuracy. For tests conducted over a large range of altitudes, the requisite accuracy may be attained by using a combination of limited-range instruments. Considerations of the pressure time lag require that the instrument volume remain as small as possible, thus necessitating an evaluation of instrument accuracy with consideration for the errors caused by the added time lag of multiple-instrument installation.

The lag in response at the recorder servo or pilot display, as the case may be, can be calculated from the following formula (from Reference 32), which takes into account the sense line, instrument; volume, and pressure 128/j,01(V01) (107) TrpDU where X = lag in response, sec kt o = viscosity of the fluid, lb sec/in -2 l = length of sense line, in Vol = instrument volume, in -3 F = mean pressure in sense line, lb/in2 D = diameter of sense line, in .

Several ranges of instruments are available for both the static-pressure and total- pressure recorders. For the static-pressure recorders, the lower-range instruments require temperature calibration. Hysteresis and friction errors, and temperature errors, should be within t1/2% of range or :wetter.

5.2 Control Position Transmitters Control position transmitters, commonly referred to as CPT units, sense the coiitrol- surface deflections anu must be accurate and sensitive enough to measure small deflections. Transmitters of the sliding contactor type chant ;e the ratio of resistance in two arms of a Wheatstone bridge circuit. Any variation in the resistance of the arms unbalances the circuit and causes current to flow to the recording galvanometer (see Figure 24).

In a properly installed system, the phase lag between the transmitter and the re- corder should be negligible. The errors due to hysteresis, zero shift, temperature, accelerations, or vibrations should also be negligible.

The transmitters are firmly mounted at the control surfaces to eliminate the effect of control-system deformations. The spanwise location of the transmitter gives an approximate spanwise surface deflection.

Zero checks are made before and after each flight to detect any zero shift in the galvanometer recording system.

5.3 Angle-of-Attack and Sideslip 5.3.1 Vane-Type P l ow-Direct ion Sensors Of the various types of flow-direction devices for sensing angle of attack and sideslip up to a Mach number of approximately 3.0, good accuracy and reliability is '~0 obtained with a counterbalanced, freely turning vane mounted on a nose boom, which also serves as a mount for the pitot-static head (Fig.19). Each vane is directly connected to a synchro transmitter within the boom, which is electrically connected to a synchro receiver mounted in a recorder located within the airplane. It should be noted that, although the a-vane measures the aerodynamic a , the -vane measures

P

a P

referenced to the body axis system of the airplane.

Inherent accuracy: Hysteresis in the system is practically nil. Friction introduces an error of less than t0. l o .

In an optical recorder system, the unbalance of a balanced optical recorder element may cause a trace deflection equivalent to 0.05 0 per g of acceleration. Temperature has no direct effect on sensitivity. Natural frequency and damping of the system should be of the order to 10 c/s and 0.65, respectively, to pro- vide flat response to within tl% for sinusoidal inputs up to 6 c/s, Mounting: The angle-of-attack and angle -of -sideslip vanes are mounted on a nose boom extending forward as far as possible to minimize the effects of upwash and shock wave.

In this respect, vanes are mounted 1'/z maximum fuselage diameters ahead of the airplane when feasible. Figure 25, reproduced from Reference 30, shows the theoretical effects of upwash from the nose boom and fuselage at low speeds. Wing upwash was not considered.

The boom and its mount should be sufficiently stiff to minimize deflections due to inertia and air loads. Particular care must be taken to aline the longitudinal axis of the boom with the longitudinal body axis of the airplane and the vane struts to the boom so that the angle-of-attack and angle-of-sideslip vane struts are parallel to the body y and z axes, respectively. The rear vane is for sideslip and projects verti- cally downward. The recorder is mounted in any convenient location.

The final calibration of a transmitter-recorder combination is made in Field checks: place on the airplane with the aid of a calibration fixture that provides an eccurate alinement of the vane with the boom and the zero of the calibration quadrant. Calibrations should be made in increments of about 2 0 to detect nonlinearities. Calibration should be performed both before and after flight.

The vanes should be given periodic checks for alinement with their pivotal shafts and for friction. An extension of the chordlne of the vane should be within 0.005 in.

of alinement with the center of the shaft.

Correction of recorded data: The angle-of-attack and angle-of-sideslip vanes measure local flow direction. The effects of boom bending due to inertia and air loads, flow components resulting from angular velocities, flight-path curvature, and upwash due to the boom, fuselage, and wings introduce errors in the measured flow angles with respect to the true airplane angle-of-attack or sideslip. In addition, phase lag and dynamic amplification of the sensing-recording system introduce additional errors in the re- cording of the vane indications. The magnitude of each effect must be investigated and corrections made to the recorded data wherever pertinent to the analysis for deter- mination of derivatives.

Sending of Lae boom results in errors in vane indications, inasmuch as the vane is referenced to the axis of the boom. As pointed out in Reference 30, deflections due to aerodynamic loading have been negligible; however, where very long booms are used or extremely long, flexible fuselages are being dealt with, bending corrections may be dete.^,iined from calculated aerodynamic loadir,, 31 Boom-bending error resulting from inertia loads is acc, ted for through static deflection calibration of the boom.

Upwash error resulting from the boom, fuselage, and wing is generally considered negligible and within the accuracy of the methods of analysis employed. This may not necessarily be true. In a boom-vane installation on a large bomber where the vane was one fuselage diameter ehead of the nose, the upwash error in angle-of-attack at sub- sonic speeds was of the order of 4%. On other large aircraft having fuselages of larger cross section, the influence was much larger. Upwash error due to the boom itself may be measured by wind-tunnel calibration of the system 2e,34 .

The effect of upwash at subsonic speeds at the vane due to the fuselage can be calculated by the method of Reference 35. The effect of upwash at subsonic speeds due to the wing can be calculated by the equations in Reference 36 for unswept wings and the methods of Reference 37 for swept wings. At supersonic speeds, the wing and fuselage do not contribute any upwash effects to the vane.

The angle-of-attack sensor is also subject to pitch-rate effects of flight-path curvature. Corrections for flight-path curvature (Fig.26) may be significant at sub- sonic speeds; whereas, pitch-rate corrections may be significant from the subsonic through the low supersonic speed range. Corrections for flight-path curvature affect magnitude primarily; whereas, corrections for pitch rate affect phase angle primarily.

This is illustrated in Figure 27, which shows the graphical time-vector determination of the absolute amplitude of the corrected angle of attack as a ratio of the indicated amplitude for an aircraft performing small-perturbation, free-oscillation maneuvers at a Mach number of 0.8 at 40,000 ft. The presentation considers only corrections for flight-path curvature and pitch-rate affects and is based on the equation x g / x

a ai

^a n — cos P cos + V q (108)

V2

The solution shows the influence of the flight-path curvature to be of the order of 3 %.

Flight-path curvature in yaw has a negligible effect on the sideslip vane. Correction for yaw-rate and roll-rate effects should be considered. The approximate expression for correcting the indicated sideslip for angular-rate effects is x — V r + V p (109) 5.3.2 Spherical llypersonic Flow-Direction Sensor The spherical flow-direction sensor shown in Figure 28 was designed to replace the a and vane-type sensors at the higher supersonic Mach numbers and dynamic pressure where the combined temperature and aerodynamic loads exceed the limitations of the vane- type sensor 38 . The spherical sensor is a null-seeking, hydraulically operated, electro- nically controlled servo-mechanism. It has pressure measurements as its sole sensing inputs. It operates on the principle that when two static ports are located on the great circle of a sphere, a null reading will result when the bisector of the included angle of the two static ports is parallel to the fluid stream immediately In front of the sensor. The rotation of the bisecting line relative to a reference gives the inclination of the fluid stream relative to the reference.

4'2 When the spherical sensor is in the zero position (axis alined with the airplane), the a-ports are 42 0 above and below the reference line in the vertical plane of symmetry of the aircraft and the .`-ports are 42 0 on either side of the reference line in the transverse plane.

The sphere constitutes the outer gimbal of a two-gimbal pi'vot system in which the outer gimbal is pivoted to the inner gimbal whose pivotal axis is fixed ^ , nd is normal to the plane of symmetry of the airplane. As the sensing sphere seeks null readings in each of its two sets of static-pressure ports, the gimbals rotate about their res-, pective axes. The inner gimbal, rotating about its fixed axis, which is normal to the plane of symmetry, sweeps an angle a in the plane of symmetry. The outer gimbal, whose pivotal axis is mounted on the inner gimbal and remains in the plane of symmetry at all times, sweeps an angle /1 in a plane which is perpendicular to the plane of symmetry; this plane is the transverse plane of the stability axis system of the aircraft. The a and '^ angles picked off by synchros are the aerodynamic a and P angles of the airplane.

The inherent accuracy of the spherical sensor is of the order of t0.5 0 or better for dynamic pressures in excess of 20 lb/ ft 2 . At high angles of attack in excess of approximately 26 0 at low dynamic pressures of about 40 lb/ft 2 and less, the a indica- tions are subject to large errors, possibly due to flow interference of the lip on the collar of the housing.

5.4 Angular Velocities and Accelerations The angular velocity and angular - , cceleration relative to any one, axis can be sensed by individual sensors or sensed and recorded in a convenient packaged unit. Figures 39(a) and 39(b) show the details of the angular-velocity aspect of a NACA designed, packaged unit which includes the recorder. The angular acceleration sensing and re- cording involves a relatively small extension of this unit. The operation of the unit depends upon the precessional force of a restrained gyro motor when the unit is sub- jected to an angular rate about an axis which is ; p erpendicular to both the axis of rotation of the gyro motor and the axis of rotation of the gimbal rings. The gyro- scopic element is the rotor of a synchronous motor. The sensitii-, element is res- trained by a precision helical spring. The moving system is damped by rotating an aluminium disk in the field of a strong permanent magnet. The angular-velocity measure- ment is made by optically recording on the film the angular displacement of the gimbal.

Sensitivity of the angular-velocity recorder can be adjusted by rotating the actuator arm along the mirror staff tail.

Angular acceleration is obtained by differentiating the gimbal motion. The differ- entiation is accomplished by mounting a coil in a magnetic field and driving it from the damping shaft so that it rotates with speed proportional to the angular velocity of the gimbal. The output voltage, which is proportional to the angular acceleration, is recorded on the film by a self-contained reflecting galvanometer.

5.4.1 Inherent Accuracy In weal-designed angular-velocity systems, the reading accuracy is of the order of 0.5% of full scale or better; the errors due to friction and hysteresis are less than 1% of full scale, and the change in sensitivity from large changes in temperature should be as small as possible. Errors due to linear accelerations of 5g should be less than loo. The sensor should provide flat response characteristics within t1% for all anticipated impressed frequencies. The pha!ia lag (time lag) is a function of damping ratio and undamped natural frequency of the sensor.

Recorded angular accelerations are subject to the errors found in the angular- velocity record. In the NACA acceleration-velocity packaged unit, additional errors are introduced by the acceleration recording galvanometer; inasmuch as the angular- acceleration pickup is a differentiation device, the response and phase lag of the accelerometer and velocity portions of the unit are similar.

5. 11. 2 ,Mounting and Correct ions It is important that the instrument mounting be rigid. Although small-amplitude, high-frequency vibrations may not be apparent on the velo c ity trace, the vibrations can introduce considerable noise in the acceleration trace.

Angular-velocity gyros are sub,+,ect to coupling errors caused by an interference (airplane) angular velocity about the spin axis of the gyro rotor. Care should be exercised in orienting the instrument during mounting so as to subject its spin axis to the minimum interference angular velocity. A mathematical study of the coupling error is presented in Reference 39. The interference angular velocity (also known as the q rate) affects the sensitivity of the instrument, the undamped natural fre- quency, and the damping ratio. The extent of the errors is a function of the gimbal tilt, which, itself, is a function of the gyro sensitivity in spri,,g-restrained in- struments and the magnitude of the interference angular velocit y . This is illustrated in Figure 40 for an angular-velocity unit having a static sensitivity of 0.256 radian per radian per second. A decrease in sensitivity would reduca the coupling error; however, a decrease is not always desirable. To minimize the coupling error for any one instrument, the axes should be oriente^ as follows: Input Spin Output Desired Velocity Axis Axis Axis Roll rate, p y x z q y Pitch rate, z x Yaw rate, r z y x Alinement of the sensing-recording units should be within t0.2 0 of correct orientation with relation to the body axes. Undetected misalinement has been known to result in erroneous values of highly pertinent derivatives, which resulted in misleading results in analog-simulated rolling characteristics. In any correction for misalinement, it is pertinent that the recorded values be correc 4 7?d for phase lag of the instrument prior to insertion in the correction equations. Simuitaneously, the response of the instru- ment should be checked, if there is any appreciable deviation from the damping ratio of 0.65, to ascertain the percentage error in magnitude of the indicated quantity due to the dynamics of the instrument. Misalinements in the mounting of -the unit may be accounted for by using the equations shown in Figure 31.

5.5 Linear Accelerations In general, flight testing is done with the beam-type linear accelvromet-rs which are available as single-component or three-component units. Draj- determination is frequently made with single-component units 30 . The beam-motion restraining force is generally supplied by a pair of opposed helical springs. The sensitivity and undamped natural frequency are dependent upon the springs used„ 5.5. 1 Inherent Accuracy In properly designed beam-type linear accelerometers, sensitivity and zero changes from random causes are less than 0.5% of full scale. The sensor should have a damping ratio of 0.65 and a sufficiently high undamped natural frequency to provide flat res- ponse characteristics within tl% for impressed frequencies up to 60% of the undamped natural frequency of the sensor. Each linear accelerometer is affected by an inter- acting acceleration acting along the beam. The effect is generally small but should not be arbitrarily ignored.

and Corrections 5.5.2 Mounting The instrument should be mounted as close to the center of gravity of the airplane as possible. It should be rigidly fastened on a rigid mounting attached to the primary structure of the airplane to avoid or at least minimize extraneous vibratory accelera- tions. It should be alined to within t0.2" of correct orientation with relation to all three reference axes. When the instrument is not mounted at the center of gravity of the airplane, corrections of the indicated readings to the center of gravity must be made by using the expressions shown in Figure 32. The equations for normal accelera- tion, an , and transverse acceleration, a t , can be linearized and corrections thus simplified by mounting the acceleroniL.,ers in the plane of symmetry along the x-axis.

5.6 Phase Lag and Response Since several individually recorded quantities are utilized in the determination of various derivatives, it is important that the phase-lag (time-lag) charact?ristics of each recording instrument be taken into conFi,deration. For systems where all the quantities can be recorded on electrical galvanometers, it is generally possible to equalize the individual phase lags by proper choice of the frequency response of the recording system. Where this is not possible, as in the use of certain of the self- recording NASA inz^ truments, phase-lag corrections must be considered and applied to bring all pertinent quantities into correct time relationship.

Phase-lag corrections must be applied before making any corrections for misalinement.

Corrections for misalinement must be made before correcting the vane and linear-accelero- meter records to the center of gravity of the airplane.

Besause of the nature of the control inputs, phase-lag corrections can be applied simply by shifting the data time scale 40 , as in the determination of control derivatives, or by correcting phase-angle relationships, as in the time-vector method of analysis.

This is accomplished by determining the undamped natural frequency of the airplane from free-oscillation maneuvers and Figure 33. Amplitude corrections are not require;41, since the instruments have flat response characteristics.

When the instruments are not sufficiently damped to provide, flat response character- istics, corrections to the magnitudes of the recorded quantities may be determined from Figure 34.

5.7 Ranges and Sensitivity Instruments used for studies of general handling qualities have relatively low sensitivities in order to accommodate the normal flight range and are used for approxi- mate evaluation of derivatives in conjunction with these studies. For accurate evalua- tion of the derivatives, using small disturbance maneuvers, sensitive gyros and accelero- meters are installed to supplement or replace those used for the handling-qualities studies. The ranges and sensitivities of the instruments are usually selected after studying flight test records of small-perturbation maneuvers performed over a Mach number range during pilot familiarization flights when the airplane is equipped with general-purpose flight test instruments. The increase in sensitivity of any one in- strument must be accomplished with discretion, inasmuch q.s ar,. optimum sensitivity is attained beyond which any increase may result simply in a false sense of accuracy.

Table VII shows the characteristics of instruments which are desirable for derivative investigations for one high-performance airplane when the pulsed free-oscillation maneuver is employed. The listed instrument natural frequencies are more than adequate to maintain flat response characteristics during forced portions of the maneuver up to the anticipated maximum frequencies for all recorded quantities.

5.8 Pulse Code Modulation (PCM) Data-Acquisition Systems In the preceding considerations of instrumentation, emphasis was placed on factors that affect the accuracy of individual sensors. Self-contained sensor-recorded units are compact, reliable, and accurate. The use of sensors wired to remote recorders can introduce degradation in the accuracy of the ovirall sensor-recorder system; how- ever, such systems are used to keep the instrumentation volume to a minimum where space is a prime factor and a large number of parameters are involved. As the number of sensed and recorded parameters increases, the time lag in the recovery of the data for the user increases. In flight test investigations where the bulk of the instru- ment7tion is a serious problem or where the number of parameters recorded may constitute a serious time lag in the recovery of the data for the user, a sophisticates' data- acquisition system is available to alleviate these problems. This system. riginated to fulfill the needs of the space industry, in whieb transducers of superior quality are used, is capable of handling the data to reasonable accuracy (0.2% to 1%). The system, referred to as the PCM system, converts the analog signal from the sensor to digital format and records the digitized data on tape on a time-sharing basis.

Figure 35(a) show- a schematic drawing of an airbo rne PCM system. The analog signals from the sensors go to a PCM encoder to convert the signal to an identification coded, digitized format. The coded, digitized signals are then recorded in parallel on n onboard tape recorder on a time-sharing basis. To recover the data, the taped signals are processed through a PCM decommutation, which ide7ntifies (unscrambles) the individual sensor signals, to a format computer to provide rca3-tr.,ne data outputs in the form of strip charts or oscillograph readouts for an immediate look at the data. The real-time data are also transmitted to a general-purpose computer which tabulates, plots, or performs complex manipulation of the data in engineering units.

Where weight is a serious factor, Figure 35(b) shows a schematic drawing of the PCM system using telemetry. The main differences between the telemetered and airborne PCM systems involves the transmission of the coded, digitized signals in series to the decommutator (instead of parallel to the recorder), which provides time synchronization of the signals before the signals are taped. In processing the data, the format com- puter properly identifies the individual data channels for real-time data output.

As stated earlier, the PCM system is a sophisticated operation. One installation at the NASA Flight Research Center, Edwards, California, is designed to handle 15,400 data samples per second from 77 to a maximum of 800 data sources.

6. FLIGHT TEST TECHNIQUES Determination of the flight test techniques to be used in obtaining stability and control derivatives from flight data is governed by a number of factors, includi.ig the methods of analysis to be employed. Successful mathematical methods of analysis have '-peen limited to the linearized form of the equations of motion and thus restrict the maneuvers to small perturbations. Inasmuch as stability derivatives are functions of angle of attack and Mach number and, to some extent, aeroelasticity of the airframe, the controlled variables are Mach number, load factor, and pressure altitude. For safety of flight, the investigation of the stability and control characteristics is k usually initiated with a gradual buildup of maneuvers at high altitude where the natural frequency and damping of the airplane are lower than at low altitudes and thus permit better control. It is desirable, whon feasible, to have the maneuvers performed with the airplane weight within such limits over the derivative-determination phase of the flight test program that the effects of changes in center-of-gravity position and moments of inertia will be negligible.

The important factors to be considered in flight testing for stability and control derivatives are discussed in the following sections.

6.1 Mach Number and Altitude Flight test maneuvers are generally performed at lg initial conditions at constant Mach number and altitude. Normally, some variations in these quantities are accepted if the resultant change in dynamic pressure is not more than 5% over that portion of the maneuver encompassed in the analysis. In regions where large Mach number effects exist (Fig.36), tests should be conducted at close Mach number intervals with more rigid requirements at constant Mach number and altitude. Failure to trim the aircraft to the desired Mach number and to maintain that Mach number during the maneuver in regions of rapidly varying characteristics may produce a scatter of data and an erroneous analysis.

The very nature of flight testing requires, for expediency, plotting the results of analysis as a function of Mach number, with each curve representing a constant- altitude condition. Figure 37, taken from Reference 41, shows the influence of altitude on flight test data on one supersonic aircraft.

6.2 Angle of Attack and Load Factor The variation in airplane characteristics with angle of attack is determined by performing maneuvers at different altitudes with lg trim conditions prevailing prior to the perturbation, or at constant altitude with the maneuver performed during a stabilized constant-g pushover or turn. It should be noted that it is difficult to obtain good maneuvers during stabilized turns; exceptional piloting skill is required.

Figures 37 and 38 show the influence of load factor on stability characteristics. In instances where the aeroelasticity of the structure is nil (dynamic pressure effects are nil), a combination, of the two techniques will result in the determination of the variation of the derivatives over an extended range of angle of attack. Should aero- elasticity of the structure be a. factor to contend with, the results from the two techniques will differ for the same angle of attack, Mach number, and center of gravity..

6.3 Aeroelasticity Aeroelastic deformation of the structure assumes increasing significance as the aircraft increases in size and slenderness and operates at increasing dynamic pressures.

Supersonic transport designs are flexible in order to keep the structural weight down, the paylo«d high, and the range capability a maximum. To apply theoretical flexibility corrections to rigid wind-tunnel data for comparisons with flight data provides an intuitive basis in ascertaining flexibility effects. When such comparisons are em- Ployed and a definite disagreement is evident in the comparison in regard to level and trends of the stability and control parameters as a function of Mach number, it may become difficult to locate the source of the discrepancy — wind-tunnel data or predicted flexibility correcr,ions. Thus, a more positive approach is required to assess flexibi- lity effects.

The stability and control derivatives should be essentially invariant for a rigid airplane as long as Mach number, angle-of-attack, and the center of gravity are constant (assuming Reynolds number effects to be a minor factor). Thus, any direct approach to investigating flexibility effects based on flight data should show the variation of the stability parameters — obtained at the same Mach number, angle-of-attack, and center of gravity — as a function of dynamic pressure. Although Mach number and center-of- gravity control is straightforward, the angle-of-attack is a problem.

The location of the angle-of-attack sensor exposes the sensor to errors resulting from structural deformations, in addition to the other sources discussed in Section 5.3.

Hence, it is more judicious to use the life coefficient C L in lieu of angle-of-attack a .

Thus, from a practical point of view, a direct investigation of aeroelastic effects should be based on a comparison of flight data for different dynamic-pressure conditions obtained at the same Mach number, lift coefficient, and center of gravity.

An effective, flexible, and simple flight-planning procedure to determine the flight test conditions as a function of weight and altitude to provide constant M , C L , and center of gravity can be achieved by using a nomograph such as that in Figure 39. In this nomograph W , M , C L , and q are variables, and center of gravity is constant.

The nomograph is based on the following two basic relations for lg flight: W = CLgS (110) (111) q = 0.7pM 2 .

and It assumes the weight distribution, which could influence structural deformation, to be essentially constant. Inasmuch as C is a constant for any one Mach number con- L dition being investigated, the following expression is readily derived from the above equations and constitutes the basis for the nomograph:

W 25

_ . ( 112) W 1 p1 The subscripts 1 and 2 denote the initial and compatible second condition. It will be noticed that, for any one initial weight W 1 at altitude h l (as typified by pressure F l ), the vehicle will have to be at a weight W 2 at altitude h 2 to maintain the same C at the selected constant Mach number.

L To illustrate the use of the nomograph, consider an aircraft to have a weight of 411 x 10 3 lb at the time a stability maneuver was performed at Mach 2.34 at 55 x 10 3 ft.

These initial conditions, which have been spotted on Figure 39, show the dynamic pressure to be 730 lb/ft 2 . If it is desired to perform the,next stability maneuver at , the intersection of = 450 lb/ft 2 (450) and the constant Mach line (2.34) q 2 determines the new altitude, h 2 , to be 65 x 10 3 ft. ThQ intersection of the constant-

altitude line with the constant M , S , center-of-gravit;r line extended from condition

1 determines the weight (W 2 x 10 lb) required to provide the same M and = 252 CL at condition 2 as was present at the time of the stability maneuver at condition 1 (center-of-gravity being constant).

The nomograph is invaluable in systematic flight planning for determination of aero- elastic effects. It permits on-the-spot changes in planned flight conditions. It also accentuates the large changes in weight required to obtain significant changes in dynamic pressure to assure aeroelastic flight data which will be outside the area of experimental error of uncertainty.

6.4 Control Inputs The method of analysis selected governs the control input. The magnitude and duration of the input influence the magnitude of the perturbation. In the case of an aerodynamic coefficient that is highly nonlinear with respect to an independent variable, different magnitudes of the perturbation may result in different magnitudes of the derivative of the coefficient in analyzing flight data. Thus, in comparing flight results with wind- tunnel data, it is essential that the wind-tunnel value of the derivative be based not only on the same trim condition but also on the same magnitude of perturbation as the flight data.

Where nonlinearity of the coefficients is not a factor and, in lieu of increase of instrument scale factor, larger perturbations of the Jn dependent variables are desired to provide more accurate readability of the records, larger control inputs or complex control inputs may be used. Figure 40 shows the increase in amplitudes of recorded quantities resulting from a change in control input.

6.5 Maneuvers Maneuvers performed for determination of stability and control derivatives from flight data should be compatible with the requirements of the method of analysis to be employed. Current practical methods of analysis, whether they involve approximate equations solving for individual derivatives or comprehensive techniques solving a number of derivatives, have limitations in their utility; as a result, different types of maneuvers are employed within the range of their .individual limitations to obtain the derivatives. As a generality, it might be said that typical handling-quality maneuvers are employed in the determination of derivatives wherein analytical techniques are used. Included are longitudinal elevator-pulse maneuvers, pullups and push-overs, pullups and releases, rudder-pulse and aileron-pulse maneuvers, constant-heading side- slips, recovery from sideslip, and rudder-fixed rolls.

When flight maneuvers applicable to analytical technique for derivative determination are not available or usable, the airplane response to random inputs is analyzed to give limited stability data. This is accomplished effectively with the aid of an analog computer, using a technique involving the matching of analog and flight time histories.

6.5.1 Puise Maneuvers The simple pulse maneuver, shown in Figure 41 for a longitudinal perturbation, is the current mainstay for derivative determination. Normally, for this maneuver the airplane is trimmed at the desired angle-of-attack, altitude, and Mach number, and a free oscillation is initiated by an abrupt pulse — an elevator pulse for longitudinal oscillation, a rudder or aileron pulse for lateral-directional oscillations. The resulting free-oscillation of the aircraft is allowed to damp out with the controls held fixed at the initial trim setting. With an irreversible control system, this is easily accomplished by releasing the controls. On tailless aircraft, even small in- advertent control inputs during the free oscillation can significantly affect the damping and, hence, the damping derivatives. Moderate inadvertent control inputs can affect the period of oscillation, as well as the damping, and then influence the static derivative results as well.

Free oscillations are also initiated by release of controls at the end of a side- slip maneuver and at the end of pullup and push-over maneuvers.

In investigating the effects of angle of attack and load factor when utilizing the pulse maneuver in an elevated g turn, the application of the pulse technique is limited by the difficulty of performing a good maneuver. Difficulty has been exper- ienced during the maneuver in holding the proper bank angle to maintain constant load factor and Mach number. With a conventional control system, exceptional piloting skill is required to maintain fixed control during the airplane oscillations at elevated g . The use of the airplane damper as a device for applying a known deflec- tion signal to excite 'he desired unaugmented oscillations (Fig.42) offers a means of improving the quality of the data for elevated g conditions as well as lg conditions.

In well-performed pulse maneuvers and lightly damped oscillations, it is possible to determire a 2-second period to within 0.02 second. Good accuracy in damping can be measured for damping ratios less than 0.2. The accuracy of period and damping measure- ments becomes rather poor for damping ratios greater than about 0.3.

6.5.2 Constant -Heading Sideslip Maneuvers In the absence of pertinent and applicable pulse-maneuver data or in an effort to complement such data, the constant-heading sideslip maneuver can be used to determine the weathercock and effective dihedral derivatives Cn 8 and ClQ , provided control- effectiveness derivatives are available from other maneuvers.

Because of frequent loose usage of terminology, the expression "steady sideslip" is us , :d when "constant-heading sideslip" is meant. Actually, a sideslip can be accom- plished, as sho m in Figure 43, as a wings-level sideslip in which .yaw rate and, hence, a changing heading is involved, as a constant-heading sideslip in which a constant linear flight path is maintained (r =r = 0), or as a combination of these two varia- tions of sideslipping maneuvers. The distinctions in the variation of the sideslip maneuver affect the parameters involved in the analysis of the flight data and the format of the equations employed.

It is difficult to perform the sideslip maneuver as a steadily increasing sideslip at a constant heading without experiencing angular rate and acceleration transients.

A more successful approach to the maneuver is to increase the sideslip in increments in order to damp out the angular rates at each increment before proceeding to the next increment. Although this manner of accomplishing the maneuver involves more time, it is justified by the refinement and resulting usable data.

6.5.3 Pullup and Push-Over Maneuver This maneuver, or any one of its variations, is intended primarily for handling- qualities investigations. However, the control-effectiveness parameter, Cms e , can be mathematically determined from the initial phases of the maneuver. The maneuver is useful also in determining the other longitudinal derivatives by analog-matching techniques.

6.5.4 Recovery-From-Sideslip Maneuver This maneuver has been valuable for determining lateral-directional derivatives by the analog-matching technique. Good conditioning is achieved by first reducing rudder input ^o half the value present at the end of a constant-heading sideslip and then releasing it. This maneuver is considered in more detail in Section 7.8.4.

6.5.5 Elevated-g Turn Maneuver The use of this maneuver in derivative determination was discussed in Section 6.5.1.

6.5.6 Roll Maneuver (Rudder-Fixed) Cls a This maneuver lends itself to the determination of C1and ever. though it is primarily a handling-qualities maneuver. In its execution, the roll is initiated by an abrupt aileron step input. The initial phase of the maneuver, up to maximum roll rate, is the useful portion for derivative analysis. The initial phase involves neg- ligible sideslip, an essential factor in its utility for derivative analysis.

6.6 General Comments The maneuvers discussed constitute those commonly used in mathematical analysis of flight data for derivative determination wherein approximate expressions for deter- mining individual derivatives or a more comprehensive technique, such as the graphical time-vector method, is employed. Many of the approximate expressions and the time- vector method are dependent upon control-fixed free-oscillation data which are not usable when damping is high, thus leaving a vacuum for mathematical analysis of suitable data. Least squaring of the equations of motion has not been too successful, inasmuch as proper conditioning of the motions is difficult to establish and the requisite accuracy of the recorded data appears to be lacking. In the absence of suitable mathe- matical techniques, recourse is made to analog matching of higher clamped oscillations and response to random inputs.

At times, it is desirable to perform maneuvers for power-off as well as power-on conditions to investigate the influence of inflow effects of jet exhausts and possibly other jet-exhaust effects. This may not be operationally feasible for jet engines.

Jet-exhaust effects of rocket-engine aircraft have been studied by performing free- oscillation maneuvers just prior to and immediately following power cutoff. Only limited ranges of the records were usable for the power-off oscillations because of the decelerations and changes in altitude.

The analysis of data of a complete flight program for the determination of stability derivatives can be tedious and exacting. The number of computations necessary for an effective analysis of the data makes it apparent that systematic procedures are helpful.

Tabulation forms, such as shown in Table VIII, that include many pertinent flight quantities have proved to be helpful.

7. ANALYSIS OF FLIGHT DATA Of the many methods proposed for determination of stability and control derivatives, only a few are practical for a relatively rapid determination of the derivatives using approximate equations. The limitations of these equations must be known in order to avoid improper applications. Of the more comprehensive techniques of analysis proposed, the graphical time-vector method appears to be the most practical and provides reliable results within the limits of its applications. When analytical techniques are not applicable, analog matching of flight data has proven to be a practical technique for determining derivatives from flight data. In the following sections, the preceding techniques are discussed at some length. Comments on other detailed methods are also included.

Inasmuch as flight-test instrument: are referenced to the body-fixed axes, the derivatives are considered with respect to these axes. Conversion of the derivatives from the body to the stability system of axes, if required, is accomplished by the equations listed in Section 2.2.

7.1 Fundamentals of the Time - Vector Approach Inasmuch as some of the approximate equations are based on time-vector considerations, it is opportune to briefly discuss time-vector properties. Time-vector methods of analysis make use of the time-invariance of the amplitude and phase relations between the degrees of freedom of an exponentially damped sinusoidal oscillating system (second- order linear system) and the differential and integrals of the degrees of freedom to determine the values of these amplitude and phase relations, or to determine the con- stants of the system of equations.

Consider the damped, transient, sinusoidal, small - perturbation oscillation of the rolling degree of freedom. This simple system is described by Al n p + 2 ^(,) n .+ (,,) 2 AO' = 0 .

(113) The solution to this equation is n^6' _ SAO' l e coscvndt - ^ w " t (114) bind = ca where n 3 (1 - ^ 2 ) . (115) Differentiating with respect to time t , -^^ Tr t I<<^ r

np = n " cos (ca nd + 7r) + 3 (1 — ^ cos c^n d t + 2

t Z ) (116) 7T -^^ t IW n e bon d '' cos t + — + (Pd where (P is the damping angle d (^d = tan-1 (117) 3 (1 - C2) Similarly ^p = ^0(f'1bv (bvn d t + 7r 2 e-tw"t cos + 2( (118) ^d ) - Equations ( 114), (116), and (118) show that the amplitudes of these equations shri;jk at the same rate and the phase relationship between the amplitudes is time invariant.

The amplitudes of the first and second derivatives of A0' are equal to the amplitude of CEO' multiplied by the undamped natural frequency, w n , and by respectively.

wn , , The phase of the derivatives is a function of the damping angle, (D which iq a d function of the damping rat4o, As shown in Figure 44, velocity vector ^p leads the displacement vector no' by (90 + (Pd ), and the acceleration vector p leads the displacement vector (180 + 2(^d).

Where more than one degree of freedom is involved in the damped, sinusoidal, tran- sient oscillation system, and the frequency is common to all the freedoms involved, the instantaneous absolute values of the rotating vectors may be considered as ratios (referred to as amplitude ratios) and the phase relations of the ratios established.

These ratios of the rotating vectors and their corresponding phase angles are time invariant. As a result, the instantaneous value of any one degree of freedom may be readily determined if the characteristics of any one of the motions are known and the amplitude ratio and phase angle relative to the characteristic motion are known.

For example, if the known characteristic motion is - !1r = ^Wnt cos ( r,

.fir e

(119) ^n d t) and, i f In,{1 / Inr I , IAp I / lAr I ` ' a r , and ( D tir are known, then I.^ I " - w t

Inrl a t COS ( + 'f r d

(120) nd (TA r l Ipl ^wnt cos ((.)nd t

p = Inrle- + (Ppr )

(121) I n ) The time invariance of the amplitude ratios and their phase angles permits the re- presentation of any one of the linearized equations of motion by vectors. For example, by substituting Equations (119), (120), and (121) and the differentials of Equations (119) and (121) into the linearized, small-perturbation, rolling-moment equation, the following format is obtained, using the A r vector as the reference for the amplitude ratios and phase angles IArl I^p I I^rl I x ' ^ pl L (D, - Ixz = 0 L^ L^ - -Cl b (Cl -Cl .) b L^ (122) pr r rr p pr Q qSb IArI qSb Inrl rr Inrl 2V I^rI 2V where I A P I Io p I Inr I Io r

"n

wn 1 and (Prr = 0 (123) - lAr l Io r I IArl IArl The vector properties described, plus the requirement that the vector polygon re- presenting any one equation must close, make possible the determination of two unknown derivatives in any one equation. The accuracy with which the unknown derivatives are determined is dependent not only on the accuracy of the amplitude ratios used but also on the accuracy of the phase angle and the sensitivity of the unknown derivative to small errors in the phase angles.

It should be noted that the introduction of cross-coupling terms into the equation& of motion would result in nonlinear equations and, hence, time-variant relations of the cross-coupling terms relative to the other terms.

7.2 Basic Flight Data Application of many of the simpler equations for determining derivatives requires an evaluation of the period and damping; whereas, application of the time-vector method requires, in addition, the determination of amplitude and phase relationships. These quantities are obtained from the free-oscillation portion of the pulse maneuver, as illustrated in Figure 45. The spacing of the peaks of the oscillatory motions deter- mines the damped natural. period, and a comparison of these peaks for the different oscillatory quantities determines their phase relationship. Determination of the I phase relationships by an averaging process, typified by the table in Figure 45, has provided more consistent data than obtained by single readings. The first line of the example table lists the time of occurrence of consecutive plur, and minus peaks of the roll rate Ap Similarly, the second line lists the plus and minus peaks of the yaw rate '1r . The third line lists the time difference of the first two lines in ea.h column. Since the yaw rate 1r is the reference in this instance, the signs in the third line indicate that the roll rate . gy p lags the yaw rate .fir The values in the third line are averaged and converted to degrees.

It will be noticed in Figure 45 that a yawing divergence is evident in the yaw-rate record. To isolate the oscillatory motions and determine the time to damp the oscilla- tions, exponential curves are drawn as shown. A semilog plot of the double amplitudes included between the exponential outlines of each motion versus time establishes the time to damp of the oscillations (Fig.46). A comparison of the plotted double tudes of the variables determines the amplitude ratios.

As stated earlier, accuracy of measuring period and damping becomes rather poor for damping ratios greater than about u.3. Generally, configurations tested at moderate and high altitudes and without damper augmentation have been rather lightly damped so that free-oscillation methods of analysis can be applied with good accuracy.

The damping ratio ^ , damping angle 1h d , and the undamped natural frequency «s n , are obtained, for both short-period and phugoid free-oscillations, from the following relations: 0.693P \ = sin Itan -1 (124) 27T1/21 ^ 0.69 3P = tan 1 ^ — (175) d \277 T1/2 ^7r 2 0.6 93 a 1 n = + cvn ^ 2 = -- + (126)

^^n d

P T1/2 7.3 Petermination of a and From Free Oscillations

in the Absence of or Questionable a and 8

Data 7.3.1 Longitudinal Free Oscillations Should the a records be unavailable or questionable in free-oscillation longi- tudinal data and the pitch-rate records available, lA al/lA qi and ' may be obtained ^aQ by using time-vector techniques. Once these quantities are determined, it is a simple matter to plot a as a furction of time or, of more immediate concern, to determine IDa n /IAod for use determining I in CNa .

1 , IL`.a) / IDq and IAa n I / IAai is shown The complete procedure for determining (Iaq , in Figure 47. The procedure involves the application of the following linearized , to the center of auxiliary equation to correct the sensed nonnal acceleration, a ni gravity of the aircraft, as shown, in Figure 47(a),

i

( I i x

IAg 1

n

an l = /^ '/( (127) q an an q - P A Ingl g I q1 g and the vector application of Equation (56b) (Table V) in Figure 47(b) in the format I / _ V I n l_q + V IA &I L%.

Iva n g I = o (128) anq aq qq I n ql g Ingl g I n q I to solve for and taq V I

n «i

I A ai g IAgI

(129) IA l - V q n - g which now permits the determination IAan1 Joa IQgI n) (130) IA :-- — I A ^ .

Iog

l

When the vector quantities 0 an i

and A a n are approximately in phase and A4 is

approximately 90 0

out of phase with Aa n , which is usually the case, the vector

Equation ( 127) may be solved by t,.- simple algebraic format LI x

I a n l

i I o an i _

n

(131) g w

+

Iogl Iog1

7.3.2 Lateral-Directional Free Oscillations Should the 13 records be unavailable or questionable in free-oscillation lateral- directional data, and yaw-rate records available, W/8I/IO rl and may be obtained IAr by using a vector solution of the following linearized auxiliary equation to correct the transverse accelerometer record to the center of gravity of the aircraft,

IDat i I x IAr i z IQP

I

I Da t

L^atr r

(132)

Iorl I^rl L ^ atir g' I^rl L ^' r

+ g lor^ L^p and the application of equation ( 59) in the format

IAp

I D r I = 001 r - Z(D - Z(b - 6

-2 , r I2 L(A r 2 r 2-ra I C L sin Ind / 1

(Dr IAr I IAr IAr 1 I

I

I zl ^ r - C^ Q t I L^ at r

— C L cos 6 cos 0 (^^ (133)

rl rl I where — I 4)rr in,r 'n # and 1 r I'^;,^ i n i 3 , r _ d ) (90 + 4d) rr - (90 + ' P -_ fP ;fir ,,n ;'1r I n Figure 48 shows the application of Equations ( 132) and ( 133) to the determination - of I n" ^I A v 'r I , l/!:'1;''1 , and 1Aa t upon solving for I VI ,/inr j and from t"Ir , 4)i3r the graphical solution of Equation ( 133), it is a simple matter to obtain parameters with ^^ as a base, for example AP

I'1 p i lA r l

and Apr - ' A r SPA - J 'As ^^I .

7.4 Equations for longitudinal Control and Stability Derivatives The nature of the input and the ensuing free oscillations of the longitudinal-pulse maneuver permit the use of relatively simple methods of analysis in determining longi- tudinal control and stability derivatives, These methods give results comparable to those from the more complicated methods investigated. Only the simple nethods are discussed at this time and only data from these methods are presented. U11less other- wise stated, it is to be assumed that stability augmentation systems are not operational during the maneuver and that the aircraft behaves similarly to a rigid structure, in that its behavior can be represented by the linearized small-perturbation equations.

7.4.1 Control-Effectiveness Derivative, Curse The control - effectiveness derivatives are determined from the initial portion, approximately 0.2 second, of a rapid pulse maneuver ( Fig.49). During this part of the maneuver, the airplane response is almost entirely pitch acceleration, with the re^ult that the pitch control-effectiveness derivative can be determined from

I 0g

Curse— (134) qSc 0 8e In similar fashion, the change in normal - force coefficient due to elevator deflection can be dete:-:iined from W Aan (135) CNse — qS ,Se .

With the preceding restriction in mind, it is desirable, for accuracy, to read the peak control input and acceleration response with a disregard of the phase lag between the two, as shown in Figure 49. It has been found that the time difference in peak values of contrul input and acceleration response is primarily the result of instrument phase lag and, to a lesser extent, air-mass inertia effects. Analysis by this method requires instruments with flat response characteristics extending to relatively high frequencies (8 c/s).

Pulses applied at slower rates, and thus extending over a longer time interval, may require inclusion of damping and angle-of-attack terms in the equation, especially a . This may necessitate the inclusion of instrument phase-lag correction.; for q and a .

7.4.2 .Slope of the Normal-Force-Coefficient Curve From the short-period free-oscillation data of the airplane with the controls fixed, the variation of the normal-force coefficient with angle of attack may be evaluated from I A a n I

_ InanI

-W (136) CNa CL ^S lAcxl CXI IA 'ibis expression neglects the pitching-velocity and angle-of-attack-rate terms of the short-period form of the normal-force equation (Equation (58), Table V). These terms have been found to be negligible, as will be noticed in the typical vector diagram (Fig.50) of the vector form of this equation wherein the pitch rate was used as the base of the amplitude ratios.

In instances where "free-oscillatior. c l tja" have inadvertent inputs of the elevator and the angle-of-attack data have been ascertained as reliable, may be deter- CNa mined by selecting those portions of the time history in which the elevator is at its steady-state position and plotting a n versus a for a number of data points which encompass the range of a n on the records. The slope of the plotted points is ID a n I /IA aI . This fundamental technique, which involves some labor, may still be the simplest technique where a -,ontrol-fixed free oscillation is heavily damped and thus precludes the determination of an I/IA Io aI by other means.

The derivative C Na may be converted to the effective lift-curve slope, CL a , which includes the contribution of power, by using Equation (38). The inclusion or exclusion of the power term depends upon the influence of power. For conventional low-performance aircraft, C La = C Na at small angles-of-attack.

7.4.3. The Derivative (CNq + CNa) As explained in Section 3.4, the phe:iomenon involving a is different from that involving q . The pairing of the derivatives as (CN q + CNa) is valid only for longi- tudinal small-perturbation, free-oscillation maneuvers. In this maneuver., A q and

a

A& vectors are approximately in phase and I46L j /jA q) 1 , thus permitting the pairing.

Determination of the individual derivatives CNq and has thus far defied solution.

CNoc The determination of (C Nq + CN&) itself is difficult. It may be readily deduced from the vector diagram (Fig.50) of the following vector form of the short-period mode of Equation (58),

Ina I to al

Iogl c L^

= ' C^ ° Le a q + C L

Na --- ^ aq (CN q

0 + + CNa) (137)

qq 2V

! Oq I n Ioq I Ioq !

that Da CNa ^q ^a n q 2V CNa r., ( C Nq ^Na) (138) ^' anq Inc - ^ I - I C q I ^

Ioq I 2V

The individual quantities in Equation (138) show that the degree of success in deter- Nq mining (C + C N a) is dependent upon the accuracy with which x is determined.

(Pan ( This phase angle is small, of the order of a few degrees, and, even with the best n records and instrumentation, the error in readability of (Pa q from the records could be of the order of the angle itself. Thus, it is very difficult to determine this derivative to a reasonable degree of accuracy.

7.4.4 Pitching-Moment Static Stability and Damping Derivatives, Cm a and (Cmq + Cma) The equations for the pitching-moment stability derivatives are based on the normal- force equation muAq — mAw CNagSO a (139) obtained from the short-period form of Equations (56b) and (58b) and on the short- period form of the pitching - moment equation ( Equation (58c)) c I y ^q = CM"A a + CmgAq c + CM6t 2V

I L)

Differentiating Equation (139) with respect to time and substituting for A4 and A q in Equation (58c) provides the following z m Cm g a &Sc A C N + Da + 1 CN e C + C ^ a— C a= 0 (140) ma ma mq 2'r a 2Iy ^ I Since ( 140) is a second - order linear differential equation of the form +a) 2nAOL = Aa + 2^cvnAa 0 (141) then _ Iy Ir z z ^n Cmg CNa n (142) Cma qSc ^ qSc 4μc an d Cma ) _ 47-^wn (Cm q + CNcx - me [ (143) -4,r CNCX me ( ^ T112 The approximate form of the Cm a equation ( Equation ( 142)), in which the term Cm q (C Na 4,ud is omitted, results in a small error of the order of 3% / or less.

Attempts to determine Cm and Cm as individual quantities required a precision & of flight data and analysis of these data that is difficult to achieve. The difficulty arises primarily from the acuteness of the phase angle, which is generally of ^Paq , the order of a few degrees; an error of 1 0 in this phase angle can result in large errors in the solution.

7.4.5 The Phugoid Static Stability and Damping Derivatives and CN Cc u u Unlike the short-period mode of oscillation in which the velocity is essentially non-variant and the angle of attack is variant, the long - period (phugoid) mode of oscillation involves velocity perturbations and essentially constant angle - of-attack.

This implies that any variations in aerodynlmnic forces during the phugoid are primarily the result of per'Curbations of the normal and axial forces due to the velocity per- turbations, that is to say that Cc u and CN u 58a) and (58b) are the in Equations ( only derivatives of concern.

Upon dividing Equations ( 56a) and ( 56b) by V and substituting these equations for Aa x and Da n in Equations ( 58a) and ( 53b), respectively, and neglecting second-order effects, the following approximate expressions are obtained for a phugoid initiated from steady - state horizontal flight: + C qS + g A9 o f = 0 (144) 'U u° ^II mV V - + Oq = 0 (145) and 21 6 Nu mV The characteristic equation of the phugoid described by these two equations is a second- order linear differential equation which takes the Laplace form S S s 2 (146) + C = 0 s+ g ( 6N cu mV V u mV It is readily recognized that S (147) V = 2^phcvn ph

Ccu

m S and (148) CNu = "'nph mV2 Transposing these two cluations results in the following approximate equations for Nu deterrAning C and Cc u from flight, data ^ 2m(`, ph (149) CN u — pgS and 2mV,"phc,bph 2m^p hl^)nph (150) Cc u —S P VS The flight values of nph and (1) are determined from the phugoid oscillations in ^ph accordance with Equations (124) and (126).

An interesting byproduct of this brief consideration of the phugoid parameters suggests itself. If N can be considered to be similar to 2C N , then Equation (148) U takes on the approximate form LCNgS = g V cv n g

V( (151)

ph V W V Thus, the phugoid frequency, c`'nph , is approximately a function of velocity, V , only.

7.4.6 Corrections for Effects of Stability Augmentation System in Determining Derivatives from Short-Period Oscillations In performing a pulse maneuver with the stability augmentation system engaged, the ensuing transient short-period oscillation of the aircraft will be characterized by a period of oscillation and a damping ratio which will be different from those obtained with the pitch stability augmentation system off (Fig.51). With the system on, the period will decrease with increasing damping provided by the system; whereas, normally, the period increases with increase in inherent unaugmented damping. This is due to the system gain and the time constant. Thus, the gain and time constant are factors to be considered in equations for determining the stability derivatives, as is brought out in Reference 42. The subsequent discussion is based on this reference.

The following procedure for determining and (Cm Q + Cma) from flight data which Cma includes stability augmentation effects has been useful but is of limited utility.

The principal value of the ensuing discussion is the insight gained into the complica- tions which may be encountered in data which include stability augmentation effects.

For rigid-aircraft perturbations about a mean flight path, the Laplace transformed short-period mode two-degree-of-freedom longitudinal equations of motion may be re- presented in approximate, but practical, form as + (—M —& (152) (s — M q )Aq s - M a )Aa = MB e e Ab (153) -.Aq + (s — Za)Da = ZB eMe In the absence of pilot input, the transfer function for a damper with a first-order time lag may be represente.l by (s) e i I r k - ; k (1 - s) (154) Aq(s) 1 + ,-'s Substituting Equation (154) into Equations (152) and (153) results in the following determinant C(1 + M8 ) q - Mg e kT' s + (-M e k] ( - Mas - Ma) _ = 0 (155) _ 1(Z8 e kr') s + (-1 - Z8 e_ k)] ( s - Z a) whose characteristic equation is (1 + M8 k-r' + MaG8 e l )s 2 + e kr a Mg + [21 - Mq - W61 Mg e k Z e k-r' - (Ma - RU)28 k]s + e ( x + ZaM + (-M Q + 26 . m8 e k - MaZg e k) = 0 (156) Considering only those terms in Equation (156) which thus far have been shown to be significant, the short-period longitudinal frequency and damping of the aircraft with a first-order time-lag pitch damper are -M

a

2 ^, (c<^') (157) n 1 + Mgek'rr ( x + M q + Ma + Mg e k) , 2 ^f W f ti -(Z (158) n 1 + MS k r' e k'blving these equations for and (Cm q + Cma) Cma

(cvn) 2 (159)

Cm a + Cmg e k 'r = qSc 2I (^^ 10. 693 m^ C mq m aj r ( + C = -- ^ 4 , 1 + M g kT^ - (160) kCmg C Na - e me Iy T112 From the above, it is seen that Cm a is readily determined for a first - order linear pitch - damper system. The determination of (Cm q + Cma), on the other hand, may offer a problem, inasmuch as Cm.

in Mq is not readily determined by itself.

If the pitch damper system is not a first - order linear system, which is the case - for many systems, analytical solutions for C ma and (Cm q Cma) are impractical. In + such instances, analog techniques are applied in attempts to extract these derivatives.

7.4.7 Representative Results Typical time histories, the flight-determined period and damping ratios, and the flight-determined longitudinal stability derivatives of the D-558-II research airplane have been reproduced in Figures 52, 53, and 54 from Reference 43. Most of the data were obtained from the all-rocket-powered version of the airplane; the remainder of the data is based on the jet- and rocket-powered version.

These data have been used to illustrate representative results because they show the need for a concentration of flight test data in the transonic zone to establish the extent of any abrupt changes of the derivatives and to show the influence of altitude on this particular aircraft. Because the results did not include control effectiveness, Figure 55 shows representative data from Reference 42 for Cms . All data shown were obtained from wings-level pulse maneuvers and are typical of those that can be obtained from good flight techniques — which include control of flight variables, pilot skill, and instrumentation — and careful application of the methods of analysis discussed.

The maximum deviation from the faired value in the stability derivatives shown in Figure 54 is of the order of 5% for CNa , 10% for Cmoc , and 20% for (Cm q + Cm&); deviation of this order of magnitude occ,-. , r in only a minor portion of the data analyzed.

The maximum deviation of Cms in Figure 55 is difficult to assess because the data shown were obtained over a lar g e range of altitudes and elevation trim settings; however, the maximum deviation from faired values would be of the order of 10%, which would be representative.

7.5 Equations for Lateral-Directional Stability and Control Derivatives The lateral-directional control and stability derivatives are not as readily and reliably determined by the use of approximate equations as are the longitudinal de- rivatives, because of the more complex behavior of the airplane and the larger number of derivatives involved. In the following discussion, unless otherwise stated, it is again assumed that stability augmentation systems are not operational during the maneuver and that the aircraft's perturbed behavior can be represented by the linearized perturbation equations.

7.5.1 Control-Effectiveness Derivatives The basic procedures for determining lateral and directional control effectiveness are similar to those previously discussed for longitudinal control effectiveness. How- ever, the expressions for lateral-directional control effectiveness are complicated by the need to account for the possible influence of the inclination of the principal axis as well as the aerodynamic terms. Tests with a conventional high-performance airplane utilizing a rapid control pulse or step input showed that the directional control de- rivative, Cns , could be determined to good accuracy by considering only the inertia term. For examp le, Cn s r O r — Ixz Ap — (Cn r — Cn4) V O r — Cn p 2 AP — Cn, A,8 = [qS_b q Sr (161) 100 = 98 — 0 0 + 2 — 0 — where the magnitudes of the individual terms are given as percentages of the answer.

This simplification in determining may not be applicable to other aircraft.

Cnsr For the roll-control derivative, C lS , consideration must be given to the aero- dynamic derivative terms. For example, using the same high-performance airplane and a rapid aileron control input, CIS = Ix Ap ^^ 1 IxzOr— Cl b —CIr b 0r—C l Q a [TSb qSb p 2V ^V 8a (162) 100 = C 73 — 4 + 31 — 0 — The cross-control derivatives, and C1 can be evQlua,ted by using Equations CnS (161) and (162), respectively. The cross y derivatives are usually of smaller -control magnitude and are therefore more difficult to determine. It appears that all aero- dynamic terms may require consideration, as shown in the following example of the analysis for Cnsa . The flight quantities were obtained from the records as shown , in Figure 56. The time difference in the Peaks of the control input and the accelera- tions is due to the phase lag of the instruments. The acceleration and angular-rate records have essentially the correct phase relationship with respect to each other in this instance. The magnitudes of the individual terms as percentages of the answer are Ar — qSb Ap — (Cnr—Cn4) 2 Dr — Cn p Op — Cn QA^ b CnS = a2V [ ^ _S b (163) 100 = 9 + 206 — 141 + 10 + 16 It will be noticed that the produce-of-inertia term is particularly significant in this example. An error in principal - axis inclination would significantly affect the answer. For instance, in this example an error of 1/4 0 in the inclination of the principal axis ( 3 0 ) would result in an error of 12% in Cn Sa .

7.5. 2 The Sifle-Force Derivative, Cyp

This derivative, which contributes to the Dutch roll mode of oscillation and is an index of the pilot's ability to sense transverse accelerations, can be determined from the equation w Ioatl ti

(164)

^yQ qS 1A,8I The ratio to a t l is obtained from the control-fixed transient oscillations /IA,81 record is suspect or missing, the ratio resulting from a pulse maneuver. If the t and r records as explained in Section 7.3.2 and may be determined from the a lAa t l/IA /3) is analogous to that Figure 48. This indirect technique for obtaining for obtaining «i and considers Cy p , Cy r , and CyQ as negligible.

lAan l /io 7.5.3 The Directional-Stability Derivative, Cnp The static directional-stability derivative is one of primary importance, and good accuracy is required in its measurement. Although a number of closed-form equations have been used, each possesses limitations which, if not recognized, can lead to erroneous answers. Several of the equations are based on various degrees of degradation of the following expression, the derivation of which was based on the solution of the determinant of the linearized lateral -directional small-perturbation equations (Equations (61a), (61b), and (610). The expression includes all but the most negligible quantities.

qS Iz 1 — Ixz sin a C.1Q = Iz can — (2^cv n ) 2 - 2^cvn Cy — Ixz — sin a Clp — I x qSb I ^Mv ) b I qSb b 2^cvn I (Cl p — sin a) + (Cn r — C n p) C1 ,3 + C lp( Cnr — Cnp) — ZV Z I 2V x x

b

g Iz

(165)

I cVs ClpCnr ^2 Cyp mV Cnr + CIp lV x n x This equation shows that when is small, that is, of the order of 0.08 per radian Cnp (0.0014 per deg) or less, the ordinarily insignificant damping terms become important.

In such instances, C1 is particularly significant.

When is of an order higher than 0.0014, Equation (165) can be reduced to the Cnp following workable equation Iz Iz C (166) cv 2 + a C Ixz C qSb n np 1p 1p .

Ix Ix This expression can also be obtained by differentiating an approximate form of Equation (61a) to provide 4S 0^3 = +

—0 r + a

Cy'a mV and also using Equations (61a) and (61c) with the assumption that C lp , C lr , C1 and Cn p are all equal to zero. Substitutions result in the following linear differ- ential equation, qS Cyp — qSb (Cn r — Cnp) A,8+ mV 2VIz ISb (167) 1,8 + I I C 1pgSb 0'8 _ 0 b Cnp — a C x Z x + q I F Z in which the frequency term is identical to Equation (166).

The fact; that is a function of C1p in Equation (166) may result in question- Cnp able values of Cnp if CIp is estimated from wind-tunnel data rather than flight data, especially when flexibility effects as well as other phenomena may appreciably j p .

alter the wind-tunnel values of C An approximation of Equation (166) provHes the following simple expression, which is of limited utility: C = I Z (A)2

(168)

np qSb n The expression has been used successfully on occasions when angle-of-attack and dihedral effects were small. At low indicated airspeeds, where these effects are not small, the discrepancy can be 50% or more.

Values of Cnp have also been obtained from constant-heading sideslip maneuvers using the expression a 8ap) .

— rp + Cns (169)

r Cnp = ( C ns 8 This simple expression is obtained from Equations (61b) and (61c) with the stipulation that angular rates and accelerations are zero during the sideslip maneuver. The successful use of this equation is dependent upon the accurate determination of the 8r apparent stability parameters ,3 and S a p as well as the control-effectiveness derivatives. The results obtained from Equation (169) have shown a relatively poor consistency in the supersonic speed range, primarily because of the difficulty of obtaining sufficient sideslip angle at supersonic conditions to make accurate deter- mination of the apparent stability parameters.

In instances where the influence of I xZ and is negligible, an accurate CnP equation for Cnp , without the necessity of relying on C1p , has been derived from the yawing-moment equation (Cn r A — , (170) — Cnp) b - L z - C np A 8 = 0 qSb 2V and the following expressions for a transient oscillatory sinusoidal motion:

WI nt -^^,nt COS

(cvndt W nd t 0r = IArle - ^^ cos + 2 + ^d wn lArIe +'rr = ) I^r I 2 ) -cc,nt

Ar = cos (cvn d t + Cwnt coscvndt (171)

IArle rr) = IArIe- e w t A,810 31 C r I e - n COS (cvn d t + 1pr) I Io

Ior

Substituting expressions (171) into Equation (170), expanding by trigonometric identi- ties, and regrouping results in A n eos't' d + C q sin `' r sin ^^ nd t — ; - qsb r

L

I I?

, , n sin'1' d cos ( " 1 (Cn r — cos "n d t 0 . (172) C n ?) b + Cn qSh 2V .r The first bracketed quantity is a summation of components perpendicular to the .1r vector; the second is a summation of components parallel to the A r vector. Hence ^ <I I I qSh „'n cos I" ) d + Cn,i I sin `h,3r -70 ( 173)

- , 1

and N b I''1^ IZ q ) qSh W n sin + (Cn r - Cn,i) 2V + 0 ( 174) C n j ^.^r cos `) d Ir - Considering only Equation (173) at this time, if the phase angle is of the order 1/3r of 9G° and the damping angle is small - which are the conditions normally encountered - then sin 4)a r and cos 'P will each be similar to 1 and Equation (173) can be trans- d posed to IZ In rl ^^ (185) n Q n qSb This equation provides accurate values of CnQ , provided it is used within the limita- tions imposed in its derivation.

Table IX lists the results of the application of Equations (165), (166), and (168) to flight data of the F-104 and YF-102. The values of Cn Q , as determined by Equation (165), are used as reference values. For the F-104, Equation (156) shows good corre- lation with the reference value because of the high value of Cn Q , whereas the simple frequency equation (Equation (168)) shows poor agreement. For the YF-102, which has a low value of C nQ for the flight condition shown, Equation (166) shows a significant discrepancy with reference CnQ and points up the influence of the damping terms when CnQ is small. For this same case, it will be observed that the simple frequency equation is unworkable.

A relative comparison of the results obtained for the F-100 airplane using Equations (166), (168), and (169) and the results obtained using the more comprehensive graphical time-vector method (to be discussed later) are shown in Figure 57. Considering the graphical time-vector results as most representative for the airplane, it will be observed that the simple frequency equation (Equation (168)) would show poorest corre- lation at low subsonic speeds due to angle-of-attack and dihedral effects not accounted for in the equation, whereas Equation (169) shows poorest results in the supersonic region because of the difficulty in obtaining accurate values of Sr,8 and S$,8 Table X compares the values of Cnp determined from analog matching of oscillatory maneuvers of the X-15 airplane with values of Cn Q determined from Equations (166) and (175). The values of and Cn p are essentially equal to zero on this vehicle.

Ixr The agreement between analog values of Cn Q and the equation] is good. In Equation (166), the agreement is due to the high value of Cn Q . Equation (175) would be the more desirable to use on this airplane because it does not depend on the use of Clp for a solution.

7.5.4 The Effective Dihedral Derivative, C1Q C1 Q Several simple equations for are available with limitations on their utility, as in the case with most simplified equations.

Values of C1 Q can be obtained from the constant-heading sideslip maneuver using the expression Srp + C i8a 8 ap) .

= —(Clg r (176) C1Q :'he derivation of this expression and circumstances limiting its accuracy are identical to that brought out for its counterpart Equation (168)).

C1 Q A comparison of determined by Equation (176) and the more comprehensive graphical time-vector method is shown in Figure 58 for the F-100 airplane. At low Mach numbers, the results from the sideslip equation (Equation (176)) compare favorably with the time-vector results; at high Mach numbers, P, large discrepancy exists between C 1Q the two methods. Even though is not one of the derivatives determined most accurately by the time-vector method, the vector method is the most practical analytical means available for evaluating this derivative.

In instances where the influence of pos.:ble to combine is negligible,, it is Ixz Equations (166) and (175) to obtain cv 2

l I I1 1 I 1

ti C 1 1 (177) Wn — (xgSb Iopl The use of this equation is subject to the additional restriction that it should not be used when Cnp is small, as was noted in the discussion of Equation (166). Also, the equation must be used with caution when the angle-of-attack is less than about ( /cv n )

3 0 or 4 0 . When the angle-of-attack is less, (JAr I /1A,81) 1 may approach 1.0 and

the error in reading /IAO from the flight records may result in an error in IArl (IAr1/10,81)(1/c)n ) that may exceed the net magnitude of the parenthesized quantity.

If the record is the major contributor to inaccuracy in the amplitude ratio, the technique discussed in Section 7.3.2 may be employed to determine the ratio without recourse to the actual ^8 record.

A final precaution regarding the use of Equation (177) is in order. At very low w.,gles-of-attack, the error in the flight-Determined values of a can produce large errors in the equation; also, as a appro a ches zero, the equation approaches an in- determinate form, inasmuch as the bracketed quantity itself approaches zero.

7.5.5 Me Damr,Lng-in-Roll nerivative, CIp Simple expressions for the determination of CI

p

are dependent upon a roll maneuver initiated from wings-level flight by a step input of the ailerons. The derivations of the expressions impose the restrictions that yaw due to aileron, CnS , sideslip due to

the effective dihedral, C 1 , 3 , rsud

product-of-inertia effect are negligible. If these highly restrictive conditions are satisfied, the following relation can be employed n8 _ a

CI p = - Cjoa _ ( 178 )

b

\pl 2V

In using this equation, CI S

can be determined from the initial part of the control

input as discussed in Sections 7.5.1 and A p l

is determined at some time point, tl on the roll-rate time history where np is zero - the region of steady-state roll.

If desired, the separate determination of CIS a can be avoided by solving for

CI S- - Ap l b

(179) G8al 2V C1 and substituting this ratio into the equation Cl = Ix nA (180) qSb

P b

Cl S ,

Ap2 2V + C^ a ASa

J p

resulting in the format

2 I x V Q, 2 1

C 1 (181) I p p gSb 08a2 np - Op i

A8 a l

In these last two equations, the subscript 2 indicates that 0 p and Ap were obtained at a time point 2 on the roll - rate time history, preferably at the point of maximum rolling acceleration.

Although the restrictions imposed at the beginning of this section seriously limit the application of these equations, the last equation ( Equation (181)) is interesting

in that it shows that CI p

can be obtained without requiring the solution of CISa

7.5.6 The Effective Damping -t;. -Yaw Derivative, (Cn r - CnQ)

It was pointed out in Section 3.4.4 that Cn r and may be combined as an

Cnp

equivalent derivative, (Cn r - Cn4), only for oscillatory maneuvers, providing the

stability r,xis system is being considered or that the angle-of-attack is small if the body axis system is used. When the body axis system is employed, this is tantamount

ti

IDw'1 /1o,81 1 , at a <. 3 0 or so, Cn r and

to saying that when the amplitude ratio

Cn4 may be combined as an equivalent derivative for yaw rates.

The combined derivatives are frequently shown in the results of analysis of oscilla- tory motions relative to body axc-3. even though this amplitude-ratio condition is exceeded. 'Shen this is done, it means that an effective value of Cn r has been obtained which includes the influence of Cnp and the results of the analysis based on the use of the actual IAO'1/IAPI have produced an answer which is equivalent to the net con- tribution of Ar and A^ to nC n in terms of !fir .

An approximate equation for (Cn r - Cnp) is obtained directly from the damping term of the second-order differential equation (Equation (167)). Inasmuch as gSb2 qS 2 ^^^ n = - (182) np) - 2VI Z (Car - C m VCy 13 a transposition results in

- 21 V C

z (Cn r - Cnp)

(183)

bz ^

+ mp

Q Considering the assumptions made Li deriving Equation (167), from which Equation (183)

was obtained, and the stipulations regar(' ng the combining of Cn r

and Cnp , it may

be stated that Equation (183) will prov'.de better accuracy when 1 and

Ioq'1 11nj31

as lopl/lA,81 decreases to satisfy the c,;n dition that C and C n p have a negligible

influence on the equation.

An approximate equation for (Cn r - Cnp) which bas been used successfully in the X-15 airplane flight test program was derived from Equation (174) I Z JD,Q1 - b n 0 .

sin ^d + ( - C nr -

I cos `DQ r

Cnlj' qSb ^ ZV + Cnp ^[^r This equation is a summation of yawing-moment components parallel to the Ar vector during a free-oscillation maneuver and is subject to the restrictions 'Lhat I xZ and Cn p have a negligiblc influence on the yawing moment.

Since for angles-of-attack less generally varies only a few degrees from 90 0 far than about 15 0 , and since the damping angle is small, the preceding equation can be reduced and transposed to I ,.

(Cnr - Cn4) Wn - ^ ` b qSb — 2V (184) 1.386VIZ

gSb2T1/2

Analog records of free-oscillation maneuvers of the X-15 airplane, on which Cnp and I xZ are essentially zero, were analyzed for (Cn r - Cnp) by using Equations (183) and (184). The results, presented in Table XI, show that the latter equation was better suited for determination of the effective damping-in-yaw derivative, for angles-of.- attack up to approximately 12 0 , than Equation (183) for this vehicle.

7.5.7 Correlation for Effects of Stability Augmentation System in Determining Lateral-Directional Derivatives from Dutch Roll Oscillations When lateral and directional stability augmentation systems having first-order time lags are operational during a Dutch roll (free-oscillation) maneuver, the effects of the augmentation system on the frequency and damping of the oscillations and on

1,7rl/I0,81 may be accounted fo y in the same manner as was done for the longitudinal

mode of oscillation in Section 7.4.6.

7.6 The Graphical Time-Vector Technique

The graphical time-vector method of analysis 41-17

, the principles of which were dis- cussed and applied in the initial part of this section, is the most common manual technique used for determining the lateral and directional derivatives. Successful

application is dependent upon availability , of control-fixed, Dutch roll oscillation

data wherein the damping ratio is less than approximately 0.3 to permit definition of the period of oscillations, the log decrement of the damping of oscillations, amplitude ratios, and phase angles.

7.6.1 Advantages One advantage of the method is that the procedure is manual, and the analyst is afforded a graphical presentation of various factors affecting the ,solution.

Another advantage is that it is possible to obtain solutions when the 8-vane records are available, suspect, or when it is desired to ovoid applying corrections to these records. Bypassing the /3 records was discussed i.i Section 7.3.2. It was shown that the vector polygon of the transverse-acceleration equation is essential in the solution of the amplitude ratio, /IDr) , and the phase angle, Io ,8I 11Qr . Both of these quanti- ties are used in the vector polygons of the rolling- and yawing-moment equations to determine C nQ and Cl Q when the vector is used as the base for the amplitude ratios in the equations, as in Figures 59(a) and 59(b).

The phase angle is used in the orientation of th

— , A,Q vector in relation to the A r vector and provides a more

,-3curate value of ^Athan can usually be obtained directly from flight records.

The amplitude ratio, I48I/IDr) , is used to extract Cn Q and Cl Q from the deter-

min , d values of Cn Q

I48I/IDrl and Cl Q in Figures 59(a) and 59(b).

A81 /IArl

7.6.2 Disadvantages One disadvantage is that the development of a definite technique is required on the part of the analyst to minimize what would otherwise constitute a rather time-consuming and tedious effort to obtain a consistent and reliable set of results.

Another disadvantage is that only two of the three derivatives in each of the rolling and directional moment equations may be determined by means of the vector diagram, thus necessitating an estimate or a wind-tunnel value of one of the derivatives in each of the equations. Since Cn p and C1. terms in the vector diagrams (Figures 59(a) and 59(b)) are the smallest vectors, it is customary to estimate these quantities. The errors in the estimated values of Cn p will affect (Cn r – CnQ) primarily; the errors in C lr will generally affect Cl p primarily, but to a much smaller extent. For low angles-of-attack, (Cn r – Cnp) may be estimated by using Equations (183) or (184) within the limits of their applicability.

7.6.3 Application of the Graphical Time-Vector Technique to the Determination of Cnp , (Cn r - Cnp), Clp , and C1 Figures 59(a) and 59(b) show the application of the graphical time-vector technique to the determination of Cnp , ( C nr - C n p), Clp , and Clp The amplitude ratio, IApI /lArl , and the phase angle, pr , were determined from a semilog plot such as d) /lArl that in Figure 46. The ratio IL^BI and phase angle A pr were obtained from a transverse-acceleration diagram as discussed in Section 7.3.2. The remaining required amplitude ratios and phase angles were determined as follows IAp I Inp I Apr = (90 + ^d) 'Jjn Inr Apr I Ivr I ' Ar I 1Prr = 90 + (D (185) - (te n d Arl Ivr 1 = 1 ; 4)rr = 0

(Or I

The derivatives Cn p and Cl r , which have relatively small influences in this instance, were obtained from wind-tunnel data. Assuming there is no question of the accuracy of the datr., the tunnel data should be based on oscillatory tests, inasmuch as the flight data are based on an oscillatory maneuver.

With the various known vector quantities properly oriented in the respective diagrams, the diagrams were closed and the unknown vectors determined by drawing the unknown vectors in their proper phase-angle directions, and Ap r . The newly determined (Ppr vectors, such as -Cn ,8 0/3I/IArb and (Cn r - Cnp) (b/2V) , were then reduced to obtain Cnp , (Cn r - Cnp), Clp , and Clp .

Figures 60(a) and 60(b), from Reference 43, show the results of the application of the graphical time-vector technique to the rocket-powered D-558-II research airplane.

An interesting aspect of the results is the influence of power on the stability characteristics of this airplane.

At times there may appear to be an incompatibility within wind-tunnel data when the data are compared to flight-determined derivatives. It then becomes imperative to resolve the discrepancy within the g unnel data and between the tunnel data and the flight data. This is illustrated in the following example wherein Cnp was relatively low.

Dynamic model tests of a relatively rigid high-performance aircraft at a set Mach IIp number and a = 6.6 0 showed that C = 0.01 and (Cn r - Cnp) = -0. 14 . Tunnel data also showed Cnp to be equal to 0.055 on the basis of static tests and equal to 0.0757 on the basis of oscillatory tests. Flight data obtained from time histories of con- vergent transient oscillations of the quality shown in Figure 45 indicated that, when the wind-tunnel value of Cn p = 0.01 was used in the time-vector solution, Cnp was equal to 0.071 and (Cn r - Cnp) was equal to 0.313. It was obvious that (Cnr - Cnp) = 0.313 was not representative of the true characteristics of the aircraft in the Dutch roll mode, since its positive value indicated an oscillatory divergence, whereas flight data showed oscillatory convergence.

( D

A check of the phase angle pr ( -

104 0 ) by several analysts showed agreement within a few degrees. It was decided that a reasonable spread of uncertainty for the quality of data - corrected for instrument phase lag - would permit pr

4)

to be 105 0 t5 0 ; at worst, the uncertainty would be t10 0 . Accordingly, solutions for and (Cn r - Cnp) CnA were obtained by using various values of C np

and ( D pr

(within the spread of un- certainty). The results shown in Figure 61 in the form of a grid plot indicate the sensitivity of the determined values of (Cn r - C np) to p

and Cn and I D pr and

Cn,3 the incompatibility of the wind-tunnel data. The tunnel data were incompatible even when allowances were made for unvertainties in inertia characteristics and readability of flight data.

Use was made of approximate Equation (183) with due consideration to the limitations of the equation for higher angle-of-attack conditions to aid in establishing the magni- tude of (Cn r - C n p).

For the test condition of an angle-of-attack of 6.6 0 , the -0.458 value of (Cn r - C np) obtained by Equation (183) could be in error to the extent of 10017, or so. Hence, it was estimated that the correct value of (Cn r - Cnp) was closer to --0.20 than -0.458. Also, considering that the state of the art in obtaining (Cn r - Cnp) from wind-tunnel tests was more reliable than in obtaining Cn p , the tunnel value (-0.14) of (Cn r - Cnp) was surmised to be representative of the true value of this derivative. Uncertainties in the inertia characteristics required that some deviation be allowed in this value in obtaining (Cn r - C np) from flight data. It was, therefore, concluded that the results of the analysis should lie within the shaded area shown in Figure 61. Within this area, the value of (0.054) compatible with Cn,8 4) = -104 0 and (Ctir - Cnp) = - 0.14 was considered to be a mean value and was used pr as an analytical result. The corresponding value of Cn p should Lave been approximately -0.04.

The best accuracy in determining Cnp (Cn r - Cnp) is

and obtained when IO p I/In rI

is small, at which time the influence of Cn p is relatively small. When the roll-to- yaw ratio is large, it may be advantageous to estimate (Cn r - Cn4) and attempt to solve for Cn p .

For low angles-of-attack, (Cn r - Cnp) may be estimated by using Equations (183) or (184) within the limitations of their applicability.

The best accuracy in determining C1 ,8 Cl p is obtained when the roll-to-yaw ratio is large. At this time, the influence of Cl r is relatively small. In either case, the static derivatives, and C1 ,8 are determined more accurately than the Cn,8 rotary derivatives, (Cn r - Cnp) and Cl p .

It was previously pointed out that the a.- ,;uracy of analysis becomes rather poor for damping ratios greater than 0.3. Although a good approximation of the damping ratio for heavily damped aircraft way be obtained by comparing flight records with records of heavily damped motions - the damping ratio of which is known - it becomes difficult to draw accurately the exponential envelopes of the oscillatory motions to obtain reliable values of amplitude ratios.

7.7 Other Analytical Techniques The preceding discussions regarding determination of derivatives from flight data have shown various limitations. The graphical time-vector technique, although the most successful, is not usable for damping ratios in excess of about 0.3, requires control-fixed transient oscillation data, and requires the assumption of some deriva- tives, which may, at times, cause difficulties in solutions. To overcome the limita- tions of the preceding techniques, a number of methods have been proposed for the comprehensive determination of derivatives (References 48-54, for example). Some have been successful in practice; others have not. In most instances, the degree of sophisti- cation involved in the proposals requires automatic data-reduction equipment and the time and effort does not warrant their use when analog equipment is available for application of analog-matching techniques. Several of the methods are considered in the following sections.

7. 7. 1 Least Squaring of the Equations of Motion A logical and straightforward method, on the sophisticated side, for determining derivatives from flight data is the application of the least-squaring technique to the linearized equations of motion. Flight quantities at discrete time points are sub- stituted into the equations of motion. Many more data points are selected than the number of unknowns, and a least-squares process is applied to evaluate the unknown derivat-.ves. As logical and simple as the approach may be, it has not been employed too successfully for several reasons, including: difficulty in properly conditioning the maneuver, instrumentation accuracy, phase lags between instruments, insufficient amplification of recorded data to provide precise readability, noise in data readout, and instrument alinement.

One of the more successful attempts to apply this technique was reported in Reference 48. To excite all the lateral-directional modes and give measurable control inputs without exceeding the limits of the linearized equations of motion, the following control input program was used: "From trimmed level flight, step the rudder causing the airplane to yaw and then roll due to dihedral effect. When the bank angle reaches approximately 20 degrees, apply a step aileron deflection such that the airplane will roll toward a level flight attitude. In order to obtain a sufficiently long record of the response to aileron, the airplane is allowed to roll to an opposite bank angle of 20 degrees before stopping the recording and initiating recovery".

A typical time history of this maneuver is shown in Figure 62. All instruments had similar response characteristics and high recording sensitivity which was compatible with calibration-sensitivity spread and calibration spread. Alinement of instruments was within t0.3 0 . Recorded data were clean. It was found that noise in the readout data significantly affected the results. Twenty discrete time points used for the least-squaring process were considered sufficient.

The results, reproduced in Figures 63(a) and 63(b), show the degree of consistency obtained after the greater-than-usual precautions were taken to provide conditions that would be compatible with the needs of the technique. The re_.iirements for this technique are undoubtedly similar to those necessary to make other promising techniques workable, such are the method of Reference 49. This method is also an equation-of- motion technique utilizing the Fourier transform, a method function to remove de- pendence on initial and end conditions, and a least-squaring procedure.

7.7. 2 Frequency-Response• Me ehod

Methods have been proposed (References 50-52 for example) to determine stability and control derivatives by using frequency-respurse data obtained from flight tests. The method of Reference 52 encompasses the solution of all derivatives through a complex procedure. Other methods, such as that of Reference 51, provide only limited results based on various degrees of approximation.

The method of Reference 52 replaces the time plane with the frequency plane. Amplitude ratios and phase relationships of airplane response to control input from frequency- 51,55 response anal;;sis of a pulse maneuver provide real and imaginary quantities. The complex quantities at discrete frequencies are substituted into least-squared equations solving for the derivatives desired. The method is simple in theory; however, con- siderable care, work, and time are involved in the application, and some experience is necessary in the selection of discrete frequencie,.;. These factors minimize interest in further studies of the method, especially where time is of the essence in obtaining a relatively quick look at the flight values. Automatic data-reduction equipment would greatly expedite the frequency-response analysis find would be useful for the other , computations required.

7.8 Analog-Matching Techniques When flight data are of such a nature as to preclude the successful use of the graphical time-vector technique or the approximate equations, and when time and expense will not permit the use of an experimentation with more sophisticated techniques, recourse is usually made to the analog to determine the derivatives that will provide the best match of the analog time history with the flight time history of a maneuver.

The use of the analog should be considered as a last resort, to be used only when other techniques cannot be applied. It is not a "cure-all", for it can produce erroneous answers under certain conditions and still provide a good match with the flight time history of a maneuver.

7.8.1 Conventional and High-Speed Repetitive Operation (REPOP) Analog Matching The mathematical model of the aircraft for the analog computer is provided by the

airplane equations of motion; when attitude records (such as and 0) are available

q and used in the matching process, transformation equations are included to transform aircraft angular rates about the body axes to angular rates about Euler axes.

Generally, the simplest mathematical model compatible with the needs of an investi- gation is used to reduce the number of analog components required and to expediate solutions. A five-degree-of-freedom mathematical model, involving the general equations of motion, is employed when longitudinal and lateral-directional cross-coupling effects are factors in the responses of the airplane during the maneuver. When such cross- coupling effects are not factors to be contended with, the longitudinal and lateral- directional motions can be treated independently and as two s parate analog progreuns using the linearized equations. Under such circumstances, the longitudinal program is treated as a two-degree-of-freedom case (with velocity a constant) unless phugoid is being considered, which is not often; and the lateral-directional program is t'r'eated as a three-degree-of-freedom case. Small-perturbation equations may be used to advantage in such Instances, particularly when datums of angular rates and Euler attitudes may be suspect and angular accelerations have excessive noise or are not available.

Initial estimates of stability and control derivatives to be used in the mathematical model are obtained from available theoretical and/or wind-tunnel values. If possible, flight-determined derivatives obtained through the use of the approximate equations are employed. In the absence of the preceding, the best estimates possible are made.

Initial estimates are required to establish reasonable scaling factors for the manually adjusted derivative potentiometers to save operational time.

Inasmuch as errors in initial conditions shift the amplitude or rotate the response time history, provisions are made on the analog to program initial conditions through manually controlled potentiometers.

Flight test inputs in the form of aileron and rudder deflections are reproduced on function generator components of the analog in as faithful a reproduction as possible within the limits of the function ^ nnerators, which have a finite number of breakpoints.

When these inputs are introduced into the mathematical model, the analog computes a response.

In conventional analog-matching, the response is recorded by a strip recorder. The recorded response is then compared with the actual flight time history, which is re- produced on clear plastic to overlay on the analog time history. A mismatch Lidicates the need to modify the values of the derivatives, possibly change signs of several of them, and possibly modify the initial conditions. These changes are made by using a ,judicious trial-and-error process until a match is obtained.

The conventional matching technique is laborious because of the need to manually match a strip record with the overlay every time a programed condition is modified in order to study the effect of the modification and assess the next condition to be modified. The conventional technique may require from several days to a week to obtain a match.

High-speed repetitive operation (REPOP) matching differs from the conventional in several basic aspects . The strip recorder is replaced by an oscilloscope and the response to inputs is projected onto the scope, which has an overlay fastened to it.

The projected response appears as a stationary time history as a result of an automatic high-speed recycling of the response computation. The maximum recycling speed for fidelity is governed by the time bpan of the time history to be matched and the frequency- response characteristics of the function generator. Where a cycling rate of 250 cycles per second may provide fidelity for a 3- or 4-second time history, it may cause serious distortions in projections onto the scope if a 10-second time history is projected.

High-speed repetitive operation matching relieves the operator of manual matching of the time history, permits him to make rapid modifications of derivatives and initial conditions, and allows him to observe effectively the influence of a modification on the response. When an optimum match is achieved on the scope, a strip record is made and matched with an overlay to check the fidelity of the scope match and to retain a record of the resulting match. A REPOP match can normally be achieved in 4 to H hours.

I

7.8.2 Advantages of the Analog-Matching Technique The analog-matching technique for derivative determination, in effect, accomplishes what sophisticated analytical techniques (see the preceding section) have attempted.

It enables the determination of derivatives under circumstances where approximate equations and the graphical time-vector technique fail. It does not rely upon detlnite restrictive maneuvers, although there are some maneuvers that cannot be solved for.

Test data showing inadvertent inputs and subsequent disturbances may be used.

7.8.3 Limitations of the Analog-Matching Technique The success of every technique discussed for determining derivatives was contingent upon the proper conditioning of the maneuvers involved. 'This is no less true of the analog-matching technique. A Dutch roll maneuver, induced by a control pulse, in which no spiral- or roll-subsidence modes are significantly evident, is generally impossible to match with a unique set of derivatives. It p ill be found that any number of com- binations of derivatives will provide a match. A maneuver involving continuous oscilla- tion of the control surfaces, as would be the case of lateral-directional oscillatory motions with the lateral-directional stability augmentation system on, will also be very difficult to match with a unique set of derivatives.

A properly conditioned lateral-directional maneuver for use on the analog to permit determination of a unique set of derivatives for a match should excite the roll and spiral modes as well as Dutch roll oscillations. The likelihood of obtaining a unique set of derivatives is increased when the maneuver is conditioned to include a rudder disturbance of a step-like nature, a transient oscillation, and an aileron disturbance — not necessarily in this order — as was mentioned in Section 7.7.1 and also illustrated in a recovery-from-sideslip maneuver which is considered in the next section.

Also, as was mentioned in Section 7.6.3, better accuracy will be achieved in the major directional derivatives when the !Opl/lorl ratio of the dynamic characteristics is low (minimizing influence of Cn p ), and in the major lateral derivatives when the 1ApI/lA rl rati-) is high (minimizing the influence of Cl r ). It may be concluded, then, that Cn p and Cl r are normally difficult to determine to any respectable degree of accuracy. The possibility of determining appears to improve with increasing Clr tendency of the aircraft to roll off during a maneuver.

7.8.4 Application of the Analog-Matching Technique Figure 64 shows the results of an analog match of a "recovery-from-sideslip" maneuver at a Mach number of 1.84 and an altitude of 49,400 ft. The match is typical for this aircraft, which had negative effective dihedral and adverse aileron yaw for the match shown. Rigid wind-tunnel data corrected for flexibility effects on the actual vehicle predicted practically zero effective dihedral and proverse yaw due to aileron. It was impossible to substantiate the predicted values on the analog, and only one combination of derivatives would provide the match.

The following procedure is typical of that employed in arriving at the analog match of the flight data which did not include rolling and yawing accelerations: ??

(i) The mathematical model was represented by three lateral - directional small- perturbation equations

s

l^p + mV (- A 8 +

0^ = V si n ( + (t) - ar + a o Cys Sr + Cys Sa) - V sin

( to A

0o

r

Q

I gfib b + Cis CIP + b ( I^P + CirAr) + cis

DP = I Z Ai + I C

I

xR R

4p

nr = IRZ + qsb Cn r + Cns a

+ b ( Or

C + Cnp L1 P) + Cns AS

nr AS

I Z IZ 2V

(ii) In addition, the following transformation was employed to determine the change

in Culer roll angle, which attained magnitudes of the order of 20 0 on occasions

in the maneuvers under consideration A^ = Ap -

( r o + Ar)6 ( 0 0 + AO) - r o 6 o cosOo

cos

(111) Finally, the outputs of the mathematical model were applied to the following

t

two equations to modify analog values of A/3 and Da to correspond to the indicated values of the flight data:

^Qi = A,i + V Ar - V

AP

v

Aat i = - sin + AO) + - (n 3 + Or - a n^p) -

( 0 0 + sin 0 d

g ti . 0+ ^P ^_

-r' Ar Ap

— y instr + X instr — Zinstr g

g g

t

(iv) Starting with the arbitrarily selected time Zero (as in Figure 64) for the time history to be matched: (a) Cnp was adjusted for approximate frequency match.

(b) The control derivatives were adjusted to provide an initial rough match in the magnitudes of r and 0 .

(c) Potentiometers forr o and were adjusted to roughly aline r and

Qo Q traces of the analog with flight data; similarly, potentiometers for and p traces. These ^o and po were adjusted to roughly aline actions involved the following analog integration + D,C dt ,

Ar = f(i•

o + ni•)dt , A,8 = f (NO

Ap)dt , AO

= f (^o + off) dt .

Ap = f(Po + (d) Since adjustment of ^ O and P modifies alinement of analog and flight O traces of r and ^3 , step (c) was reiterated as many times as necessary to obtain a rough alinement of analog and flight traces of r and p .

(e) Attention was then focused on the r trace to obtain a more refined match of this trace by more cautiously adjusting Ca r Ca Cns a d and Cnsr This operation necessitated adjustment of C1,8C18a and Cl r to Clp keep the p trace in line.

It should be noted that the preceding five stern (iv) (a)-(e), which constitute an initial phase of operation to obtain an approximate .: p atch, involve about 1 hour. The explanation of the procedure is, by necessity, brief. It will be readily appreciated that the steps are iterative to keep the frequency of disturbance, the magnitude of disturbances of the various traces, and alinement of analog and flight time histories compatible.

The second phase of the analog-matching process involved the following operations: (f) With the r trace roughly matched, attention was focused on the (^ and p traces by manipulating the C1,6 and derivatives and the Clr ClP , lateral-control derivatives as .necessary. During this operation, fine adjustments were required and made on the r and traces (as per step (iv) (e)).

(g) With ,^ , p , r , and traces matched as closely as possible, attention was focused on the a t C , trace. This involved Cyp , ano C ys ys r .

a The last phase of the analog-matching operation involved making fine adjustments to initial conditions (to compensate for probable errors) and fine adjustments to the derivatives, in essence, performing an iterative procedure of the preceding operations.

The second and last phase of the analog-matching process generally involved 3 to 4 hours and, at times, more.

7.8.5 Accuracy of Results in Analog-Matching of Flight Data As mentioned previously, the accuracy of the results in analog-matching of flight data is largely dependent upon the conditioning of the maneuver. For the longitudinal derivatives, results from a pullup-and-release maneuver of an advanced high-performance aircraft showed the following accuracies based on the amount the derivatives could be changed before a trend toward mismatch became evident: (1) For a strong maneuver: 10% 57o CN a Cm a Cmse CNse 20% to 30% 10% 20% to 30% (CNQ + Cry«) 200% or more (Cm q + Cma) (2) For a weak maneuver: 20% 10% CN a Cma 100% C MS 20% CNS e e C NQ ( + CNa) 200% or more 4017v (Cm Q + Cma) The accuracies of the lateral-directional derivatives obtained from analog-matching of well-conditioned, release-from-sideslip maneuvers of the same aircraft are shown in the following tabulation, along with the f p ttors which influence the accuracy: True for any rudder release involving Cn,3 — 5% more than one cycle of oscillation.

to C i e — 50 10 15% Depended upon oscillatory characteristics of q5 and magnitude of after 8 , rudder release.

Cy A — 57o to 20% Depended on the magnitude ^ f : , • at oscillations and the average. s,ope of ,Q from release to steady value.

Cnr 5% to 30% Depended upon the amount of transients — the aircraft was allowed to go through before controls were applied again..

30% Cn p — 57o to Depended upon the magnitude of the roll rate during oscillation. Hi3her roll rate showed better accuracy.

— 5% to 30% Depended upon the magnitude of the roll

C rate during oscillation. Higher roll rate showed better accuracy.

to 50% Depended upon magnitude arj oscillatory r — 5% C I characteristics of rolloff. Larger rolloff showed better accuracy.

Cns r — True for any rapid rudder input.

5% 5% to 15% CIS a — Cns a — 517o to 30% Depended upon the magnitude of the control CI Sr — 10% to 30% input.

Cy Sr — Mo to 50% or more Cys — 407o to 100% or more a These results m:;, be considered typical of ghat may be expected in analog-matching of flight data obtained from properly conditioned maneuvers. The accuracies may well be typical of those that may be expected when comprehensive analytical techniques are used.

8. APPLICATION OF BLIGHT aJERIVATIVES If wind-tunnel data and theory were infallible, it stands to reason that there would be no need for flight determination of derivatives. However, such is not the case.

As new concepts in aircraft were developed, either with regard to physical geometry or propulsion systems, and as aircraft fly in new Mach and altitude regimes, there is the need to verify aerodynamic theory and wind-tunnel data and various influences of aercelastic deformations of prototype structures on stability characteristics; to provide supplementary information not obtained in limited wind-tunnel studies; and to uncover the source of discrepancies between predictions and actual flight behavior.

The following discussions provide some insight into several of these areas.

8.1 Verification of Wind-Tunnel Data and Theory As the Mach capability of the airplane increases, the technology in wind-tunnel testing becomes more critical with regard to model constru^tion, support of the model, and interpretation of the tunnel data. Whereas theory depends upon wind-tunnel data for verification, or to fill in gaps where theory fails, the wind-tunnel may depend upon flight data, as new regimes of flight unfold, to verify testing techniques.

Flight data pointed out the need for a greater concentration of test points in the transonic region to accurately define the stability characteristics in this region (Fig.54). Flight data showed also that it was not sufficient to use a cold jet stream to simulate the exhaust of rocket en,ines. Figure 60 shows the effect of the jet exhaust of the D-558-II research airplane on the lateral-directional stability character- istics of the vehicle in the supersonic region. The destabilizing influence of power was the result of a pluming of the hot jet exhaust and consequent formation of a lambda shock wave at the juncture of the vertical tail and the fuselage. During D;Itch roll oscillations, the sho--k wave on the leeward side of the vertical tail moved forward, while on the windward side it remained attached to the jet exit. This pheno- menon it not common; it was the result of overexpansion of the jet exhaust and the proKimity of the trailing edge of the vertical stabilizer to the jet exhausts.

Another illustration of discrepancy between i>aht and predicted data involved elevator setting for lg flight. A comparison of the variation of predicted and flight- determined elevator settings with Mach number .showed increasing discrepancy with in- creasing Mach number for a constant center-of-gravity position. In this instance, involving aeroelastic effects, predictions showed reasonably close correlation of CNa with flight data; whereas, Cm and showed a difference in trend as well a CmS as level. Preliminary study of the problem showed a need to consider as well Cmo as Cm and Cmg e . Thus, the following pitching-moment equation for trimmed un- a accelerated level flight, based on Equation (53b) (Table IV), was used and constituted the major consideration in arriving at the most likely causes for the discrepancy between predicted and flight trim settings of the elevator Cm a + Cm Se a (a + acN =o) + Cms e = 0 . (186) The angle-of-attack (a + ac N =o) was replaced by its equivalent

pc + = 0 = CN (187)

aCN CNa to determine both predicted and flight values of by the followi.ng new format Cmo is the slope-intercept expression for solving C.n o , or of Equation (186), which Cma CN - Cm 8e 8 e C mo = (188) r Na Also, Equation (186) was transposed to solve for o - Cm a - Cm (a + aCN = o ) (189) 8e e Curse Curs e A comparison of predicted and flight values of the ratios in Equation (189) showed the values of the ratio Cm a /Cms e to be essentially the same; however, Cmo/Cmse 8 e differed in line with the discrepancy in . Calculation of the static margin using Cm a /C Na , which was employed in determining Cm o , also showed a discrepancy between prediction and flight. In the final analysis, it appeared that the major source of discrepan ^y between predicted and flight longitudinal trim elevator settings was due primarily to the aifferences in Cm o and Cmse An illustration of a discrepancy between wind-tunnel and flight data involving power effects and aeroelasticity is shown in Figure 65. This instance concerned the F-100 airplane (Fig.16, which is considered to be a relatively rigid aircraft and has its air-intake nozzle at the nose. As shown in Figure 65, the variation of the wind-tunnel value of Cn,8 with Mach number has roughly the same trend as the flight- determined value. However, there is an appreciable difference in level that is well beyond the difference to be expected due to the values of moments of inertias; values are known to within 5% at best. The results of an investigation to trace the sources of the discrepancy showed appreciable moment of momentum effects of air-intake flow and aeroelasticity effects of the vertical tail. When the basic rigid tunnel data were corrected for these two factors, fairly good correlation was achieved with the flight data (Fig. 65) .

A technique in tracking down inconsistencies in wind-tunnel data involving Cn,8 Cn p , and (Cn r - Cn4) was illustrated in Section 7.6.3.

8.2 Effects of Aeroelasticity

The effects of aeroelastic deformation of the structural components on the stability and control characteristics of the aircraft are of prime concern, particularly i.n large transport designs, as pointed out in Section 6.3. 'The illustration of aero- elastic effects shown in Figure 65 represents an intuitive approach in accounting for a discrepancy between wind-tunnel and flight data. This approach presumes the basic rigid tunnel data to be correct. It also presumes that aeroelasticity effects are simple enough to permit reasonably reliable calculation of corrections to the data.

As aircraft increases in size and slenderness, and operate at increasing dynamic pressures, aeroelastic deformations of the structure assume increasing significance.

The influence of aeroelastic deformations on the stability and control characteristics is difficult to predict on the basis of theory. The deformations of the various cot;onents of the structure affect the shock patterns of the airflow which, in turn, affect the stability and control characteristics in a much more complex manner than the aeroelastic deformation of one or two surfaces on a relatively rigid aircraft.

Rigid-model data may be questionable because of the uncertainties in the true rigidity of the model and model supports and interference effects. Thus, a more positive approach is required to assess flexibility effects to verify and improve theory and develop tunnel techniques.

A flight test technique for determining aeroelasticity effects on stability and control characteristics is outlined in Section 6.3. The technique, as presented, is somewhat simplified in that the lifting components of thrust is considered to be negligible. This approximation simplifies flight planning, monitoring, and making on-the-spot changes in flight conditions of W and h for maneuvers at constant M approximately constant C u due to aerodynamic lift alone, and constant center-of- gravity. An average of the postflight-determined values of B W - Tsin CL = (190) qS for the test points on the "constant M , C L , and center-of-gravity line" in Figure 39 - such as points 1 and 2 - will constitute the representative value of C L for these test points. The maximum deviation from actual CL is within the experimental error of the investigation. The stability and control derivatives of these points, when plotted against dynamic pressure, define a curve which shows the effect of aero- ele;sticity on the derivatives. The curve represents only one M CL due to aero- dynamic lift alone, and center-of-gravity condition.

8.3 Stability Criteria Considerations of the stability of an airplane include not only its inherent stability, which is its behavior without pilot inputs following an initial disturbance, but also its behavior in response to pilot inputs. In general, the stud y of the stability of an airplane involves the effect of derivatives on the increase or decrease of the stability. It is an objective study. When the stability of the airplane is considered in the light of the degree; of pilot's acceptance of the airplane, and pilot ratings are introduced, the study becomes subjective and is referred to as a handling- qualities study. As may be readily surmised, one study complements the other.

Any extensive discussion of handling qualities, which integrates the pilot as a human servosystem constituting a feedback loop in the control system, is beyond the scope of this paper. It would involve the stud y of human factors and is affected by the pilot's technical background as well as the depth of piloting background, the types of aircraft flown, orientation and types of displays in the cockpit, and general cockpit enrronment. The art and science of handling-qualities investigations is covered extensively in the literature (References 57-65, for example).

8.3.1 Longitudinal Short-Period Oscillation, cvn The response of the airplane to an elevator input or gust disturbance will normally include a longitudinal short-period oscillation. An oscillatory condition by itself indicates a static oscillatory stability, Positive, neutral, or negative dynamic oscillatory stability is dependent upon the presence of positive, zero, or negative damping characteristics, respectively. A study of the longitudinal characteristics involves both static oscillatory stability and damping.

The undamped natural frequency ( static oscillatory stability) is a measure of the longitudinal stiffness of the airplane - analogous to a spring - mass system. This longitudinal stiffness is represented by = - (M a + MgLa)

wn

(191) q Sc Cm a q + CmgCNa 7tμc I It will be noted that for any one mass distribution and configuration of the airplane, the longitudinal stiffness is a direct function of Cm O A primarily. Thus, the oscilla- tory frequency of the airplane will decrease with decreasing Cma and decreasing q .

It should be noted that when Cm a is zero, a degree of longitudinal stiffness (static oscillatory stability) will be present as evidenced by the Cm g C term in Na the equation, providing Cm is negative - a normal situation. The contribution of this term to longitudinal stiffness will increase with increase in CN a , decrease in mass - density parameter, μ c , and increase in dynamic pressure, q .

In maneuvering flight, the pilot feels the effect of longitudinal oscillatory stiffness in the stick force per unit normal acceleration.

8.3.2 Longitudinal Short-Period Damping The longitudinal short-period damping is expressed either as the actual damping coefficient or as a damping ratio. The damping coefficient ( ft lb sec/rad) is dependent upon the aerodynamic derivatives CN a and ( Cm q + Cma), as shown in the equation ^wn 2 rZ(X + (Mq + M& qSc aqS CN 2 - mV - (Cm q + Cma) 2VI (192) Y

_ S Sc 2

CNapV 2m P V (Cmq + Cma) 41 Y A decrease in CN or the negative value of ( Cm q + Cma) will decrease the damping co- a efficient. 22cv n . It :sill be noticed that the magnitude of the coefficient is also dependent upon the mass density of the air ^ and airspeed V , as well as upon the airplane ' s mass characteristics and configuration.

The damping ratio ^ as may readily be surmised from the preceding, is obtained from n + Ma) - 2 ^cv Z a, + ( M q 2cv n 2v/--AIa (193) Sc i SIY - (Cm + Cm,) C N c ^p 3 P a [jM _ q 2c °^' [ :32Iy j ^(-Cma), Thus, for any one mass characteristic and configuration of the airplane, the damping ratio Z is a function of 3 p , CN, , (Cm q + Cma), and 3 ( - Cm(x) 8.3.3 Longitudinal Short-Period lead Term, -ZrA The parameter - Z a , which is a function of CNa , is a longitudinal short-period lead term which affects the lead of the pitch rate q with respect to the control input 8e and angle-of - attack as shown by the transfer functions 1 Cms qSc C Na gS / s + -.-- -- - ^— I s + — q(s) - `^ TB Iy mV ^ TB (194a) s2 ^e(s) n + 2^cv s + Wn s2 + 2Cwns + cvn and 0((s) 'Se _ (194b) + 2 ^ wns + S2 ^ e (s)

Wn

As shown in Reference 64, the time for peak amplitude of q due to a step input de- creases with decreasing - Z a . If - Z a becomes sufficiently small in comparison to W , the response to a step input can be disconcerting. It may be characterized in a tracking task by an initial increase in pitch attitude of the airplane followed by dwell, possibly with the airplane aimed at the target; but, then, with no further control input, there will be a subsequent increase in the attitude. This type of behavior may give the pilot the feeling that the airplane is unstable.

A low value of A X may cause the pilot to experience a looseness in pitch, pitch- rate overshoot, lack of control precision, and higher control forces. On the other hand, a high value of - Z a may cause a tendency to overcontrol, exceed normal g and, in general, give the impression that the control is too sensitive.

8.3.4 The Dutch Roll Oscillation, wn The Dutch roll mode of oscillation, represent ' by the following equation, based on an approximation of the second equation in Equations ( 78), is a measure of direct- ional stiffness ^n = Np - L^' sin a+ (Nr + L 1)Y'3 OL CI L p n . ^

- Q + IXz Cl^ qSb . (195)

I Z I x IX z ( Insofar as derivatives are concerned, Cn,8 and Cl Q are normally the ( ,aly derivatives of any consequence in defining the frequency of this mode of oscillation,, Of these two derivatives, is dominant. It should be noticed that when the static direct- Cna ional stability is zero (Cn,8 = 0), there is still some degree of oscillatory stability, (Cl R = ) and the pror'ict of inertia is providing the effective dihedral is positive negative, or vice versa. Some aspects of the controllability of tb ,: , i airplane when Cn,8 is near zero and slightly negative are reported in Reference 60.

8.3.5 Dutch Roll Damping Coefficient, 2^wn The Dutch roll damping coefficient represented b, ► the following equation, based on t an approximation of the first equation in Equa ions (78), gives the meavure of the dynamic stability of the Dutch roll mode = -

N I - Y A - Lp

2^wn 2 2 (196) Cn4) + CM + 2VI Clp ^ 2VI (Cnr - z x This equation shows the interaction of the more dominant derivatives affecting the damping ratio. The equation is more accurate than that shown as Equation (182) in that it includes Clp .

8.3.6 Dutch roll damping Ratio, ^ On the basis of Equations ( 195) and (196), the damping ratio can be approximated to at least the first degree of approximation by

^Nr -YQ - Lp

2v/N _ 1' bC b i C lp (&Sb)2 -, - Cnp) + mVb + 2VI C 2VI ( z x ('197) 2 CnA + Ixz Cl 2

Q^

I z 1

x 1z

The Dutch roll damping ratio is strongly affected by Nr and NA An increase in the negative value of NT not only increases the damping ratio, but also improves the stability of the spiral mode. Increasing ^A not only increases direct- ional stiffness but also the Dutch roll .amping ratio, which may be desirable. Decreasing N increases the bank angle that is induced by a given amount of sideslip in the Dutch roll motion, a characteristic which could be detrimental to maneuvering control of the airplane. In addition, decreasing NR increases the amount of Dutch roll disturbance in the roll mode response to a step aileron input - as reflected in the parameter (wO/wn)2 to be discussed - and can disturb and mislead the pilot.

8.3.7 Stability Criteria for Aileron-Only Roll Control, w^/cvn

The roll parameter, wo/wn , is the roll numerator to Dutch roll frequency ratio of

/Se response function. It is represented by

i

^^a

- rv^ = 1 - w n Na ga C nS Ixz C1 Q I xz + - —^ + Cn C1 S Iz Iz

p

a

= (198)

1 -

C1,8 I xz 1 + Cap I x The parameter is a measure of the amount by which the Dutch roll motion is excited when aileron inputs (rudder fixed) are made by the pilot. It is particularly important in the roll tracking task in which the pilot-airplane combination can exhibit considerably different lateral-directional oscillatory tendencies than would be exhibited by the airplane alone. It provides a good index regarding the increase or decrease in stability of the airplane during the aileron-alone roll tracking task.

o /w n When w = 1 , there is no yaw due to aileron inputs and there is little or no Dutch roll motion iii response to aileron input. When wWw n the pilot-airplane < 1 , combination in an aileron-only tracking task will exhibit an effective damping ratio in roll tracking tasks greater than the Dutch roll damping ratio. When wWw n > 1 , the effective damping ratio will be less than the Dutch roll damping ratio and the roll that results from aileron input is augmented by the roll due to sideslip; this can cause stability problems in the roll tracking task, especially when the Dutch roll damping ratio is small and is large.

101/1,8 I Equation (198) shows significant interaction of stability, control, and inertia

parameters affecting wo /w n . The interplay of C n s Xz

a , C1 ,8 and I is important, inasmuch as these parameters may have either plus or minus values. Normally, Cnsa and Cie are the controlling parameters; thus, if the effective dihedral is positive 0/wn < 1 (Cie < 0), Cns will have to be adver-:e (Cns < 0) to assure and a stabilizing action during the roll tracking tas k.

8.3.8 Dutch Roll Stability Criteria, 10111,8 The amplitude ratio 10111!81 is a characteristic of the Dutch roll oscillations and is thus independent of any excitations of control inputs. Its mathematical relation- ship to derivatives is given by NrDr2 2 r z e i + E rg Lr

(199)

El 1,81 NR e P 1 + Nr e J The complex interaction of the derivative parameters makes it difficult to determine pilot sensitivity to However, if the airplane has high directional stiff- 10111,81 .

ness ( co > 1) , i ow 1 01 11 ,81 , reasonable ^ > 0.1 , and adverse yaw ' ie to aileron, n the pilot generally does not bother to coordinate turns by using rueler, inasmuch as 10111,81 the lateral-directional stiffness keeps sideslip small and the low value of keeps roll due to sideslip small (Ref.64).

8 1 is large (of the order of 4 or more), rudder coordination becomes If 10111, necessary in maneuvering to keep sideslip small in order to minimize the roll due to sideslip. If the airplane is characterized by favorable yaw due to aileron (Cns a > 0) the pilot uses a cross-coordination of rudder as well as high values of 8 I , 10111, and aileron controls (right aileron and left rudder) to prevent excessive rolls in maneuvers (Ref.64). It is not difficult to achieve coordination of controls, pro- viding the airplane is not excited by external disturbances. However, because this cross-coordination is unnatural, the pilot is more critical of favorable yaw due to aileron (Cns a > 0) than adverse yaw due to aileron (Cns a < 0).

8.3.9 Poll -Subs idence Root, 1/TR

The roll-subsidence root, 1/T. , is influenced most significantly by the parameters shown in the following equation, which is based on the third equation of Equations (78) 1 g L^ ti P P P Np V) R (200) gSb2 _ — Clp 2VI .

X As shown, the roll subsidence is dominated by the damping - in-roll derivative, C lp .

The roll-subsidence root has a direct influe! , c:, on :ne steady - state roll rate in response to a specific aileron deflection. When the root is large, the damping in roll is high rind the pilot controls the bank angle by commanding and adjusting roll rate.

When it is small, the pilot controls bank angle by commanding and adjusting rolling acceleration.

8.3.10 Spiral-Divergence Root, 1/Ts The spiral - divergence root, 1/T s , is affected primarily by the parameters shown in the following equation, which is based on Equation (83), g (LpNr — LrNA (201) T TR V N T s Q The spiral mode can be convergent, neutrally stable, or divergent. Thus, for the purpose of defining the spiral stability boundary, the equation can be shown as a spiral stability criterion LpN r — NpL r spirally convergent or, as an approximation, = 0 neutral spiral stability (202) ClpCn r — CnpCl r < 0 spirally divergent .

It will be noticed that spiral stability is dependent upon the interaction of four derivatives. Since is normally positive and Cn r and C1 ,8 normally negative, CnQ it is well to have Clr negative: Under any circumstance, ClsCn should be greater r than CnQCl r for spiral stability.

A divergent spiral mode will result in the airplane performing an increasing nose- down and tip^ ► tening turn accompanied by an increase in speed and loss in altitude.

8.4 Flight. Guidance Research vehicles that incorporate new concepts of aerodynamic configuration, or research vehicles designed for flight in previously unexplored regions of flight (Mach and altitude), usually have a considerable amount of wind- t unnel investigations per- formed on models to check their stability and control characteristics. Despite the comprehensiveness of the tunnel tests, there wall be gaps in the data. In addition, there is normally a certain amount of reserve in placing complete confidence in the data. As a result, the flight envelope is built up gradually, using stability and control maneuvers to obtain flight-determined stability and control derivatives to verify wind-tunnel data.

Agreement in the comparisons results in a more rapid buildup of the flight envelope; disagreement involves a slowdown until the flight data can be reduced and cautiously extrapolated. The most representative values of the stability and control. character- istics are used in stability criteria and are programed into a flight simulator, in which the pilot simulates the intended mission and emergency conditions to reduce the amount of risk that would otherwise be involved in actual flight. The simulator normally uses the general equations of motion for a mathematical model, When roll-coupling instability became a physical reality with the loss of several F-100 airplanes, considerable effort was expended at the NASA Flight Research Center 66,67 .

in flight and simulator studies of the problem Because ^i the complex nature of the motions, guidance of the flight program using analog computations was desirable.

In a roll investigation of this type, a small increase in aileron deflection can pro- duce large effects on airplane motions. It has bee,- graphically demonstrated on several occasions that flight guidance based on linear e y ^rapolatioi J flight data at small aileron deflections can be highly misleading and dangerous Figure 66 shows a repre- sentative comparison of the measured excursions in angle , af-attack and angle-of-sideslip obtained in 360 0 rolls with those predicted by using flight-determined derivatives.

The good agreement has been demonstrated in most instances in which flight-determined derivatives have formed the basis of calculations. Consequently, the use of such guidance in flight planning has proved invaluable. The use of wind-tunnel and theoreti- cal derivatives in analog studies has not been as successful.

9. CONCLUDING REMARKS This paper has attempted to bring together the various factors that should be known by the engineer who is concerned with the determination of stability ar:d control charac- teristics from flight data o. the use of these flight-determined characteristics in handling-qualities research.

The discussions have been tempered with practical considerations. The various factors discussed and the observations made are the result of experience in working with flight data, developing techniques, comparing the data with predictions, and investigating the causes of discrepancies, The theoretical background, approximations, and limitations of the mathematical relations employed have been given careful consideration. The problems encountered with several of the more sophisticated techniques have been presented with the hope that any new comprehensive technique that may be proposed will take into consideration maneuvers to some of the practical problems with instrumentation and development of properly condition the flight data for the technique.

The pulse maneuver, properly executed, his been Found to be generally adequate in exciting motions required for stability-derivative analysis as well as for determining the characteristics of the oscillatory modes if adequate instrumentation and alinement are provided.

For longitudinal-derivative analysis, simple equations utilizing period and damping of the oscillatory mode of the airplane were shown to be as satisfactory as more com- prehensive methods.

For lateral-directional derivative analysis, the graphical time-vector method was shown to be the most satisfactory manual method of analysis. Simple approximate methods are useful if applied with caution.

Control effectiveness can usually be obtained by relating the peak acceleration to rapid control inputs. Consideration must be given to aerodynamic contributions if reasonable accuracy is to be realized.

The analog-matching technique for determining derivatives from flight data was shown to be a valuable method of analysis for use in the absence of data suitable for analytical techniques. However, the analog-matchl.ag technique has limitations in that data must be properly conditioned in order to obtain unique answers. The accuracy of the results obtained from this technique and the effect of the type of maneuver on the accuracy may well provide the clue to what may be expected from sophisticated techniques that may be proposed.

The use of flight data to verify wind-tunnel results and theory was discussed and illustrated. The possible inadequacy of comparisons of flight data with predictions for determining aeroelastic effects was pointed out and a flight-planning technique explained to permit determination of aeroelastic effects from flight data alone.

Pr`sent instrumentation and methods of analysis are adequate for extracting deriva- tives from flight data for use in most flight-guidance simulator studies and detection of claNracteristics which have not been predicted in the wind-tunnel.

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,heory and Simulator Experiments. Tech. Doc. Report ASD-TDk-62-505, Wright-Patterson Air Force Base, US Air Force, November 1962.

An Analog Study of the Relative Importance of Various 66. We!.!, Joseph Cay, Richard E. Facifor•s Affecting Roll Coupling. NACA RM H56A06, 1956.

Application of' Analytical Techniques to Flight Evaluations 67. Weil, Joseph in Critical, Control Areas. AGARD Report 369, 19dl.

TABLE I Transformation of Derivatives from Stability to Body Axis

CL a

COs sin a + Cc

CNa = a + CDa

cos a - sin a - CN

CD a CL a

CCa =

C C ma) s

ma = (

CnA = (Cnp) s cos a + (C l Q) s sin a

a + (Cl a + ( Cn p sin acos a

Cn r = ( C t p )s sin e +

nr)s cos

C I d 8

np)s cos a + (Clp)s Sin x

= ( C

Cn4

a - ( C I ds sin a - (Cn r - Cl p )s sin acos a

Cn p = (Cn p )s cos t e

C n s = ( Cns)s cos a + ( C a

IS)s sin

lp)s cos a - ( Cnp)s sin a

C Ip = ( C

a - ( Cn p )s sin e a - ( C acos a

Clr = (CIds cos t

nr - C lp)s sin

a - ( Cnp)s sin a

C IA = ( C IQ)s cos

C np

p )s cos t a + ( Cn,,)s sin e a - ( a cos a

CI p = (CI

+ Clr)s sin

Cns)s sin a

a - (

C is = ( CIS)s cos

i TABLE II Transformation of Derivatives from Body to Stability Axis a - CD CLa = CNa Cos a - C C, sin CL CDa = C% Cos a + CNa sin a + (Cm a)s = Cma (CnQ ) s = CnQ Cos a - C lQ a sin a - ( C lr - Cn p ) aCOS a ( Cnr)s = Cn r Cos t a + Cl p sin e sin a lQ sin ( CnQ) s = CnQ Cos a - C (Cn p )s = Cn p Cos t a — Clr sin e a + (Cn r — Cl p ) sin aCOS a (Cns)s = Cns COs a — Cis sin a (C i Q) s = Cl Q Cos a + Cn Q sin a ( C lr)s = Ci r COS 2 a — Cn Q sin e a + (Cn r — Ci p ) sin aCOS a (C1Q)s = Clp Cos a + CnQ sin a (Cl p )s = Ci p Cos t a + Cn r sin e a + (Cl r + Cn p ) sin aCOS a Cos a + Cns sin a ( C ls)s = C

TABLE III

Transformation of Moments of Inertia from One Axis System to Another

body to Stab i l i ty

= '(Ix + I z ) — 2(Iz — Ix) cos 2a — Ix sin 2u

Ixs

= iy

IYs

= + Iz) + 2(Iz — Ix) cos 2a + Ixz sin 2a

'(Ix

Ins

= ( I x — Iz) sin 2a + Ixz cos 2a

Ixszs Stability to Body

+ Iz s ) — 2(Iz s — Ix 8 ) cos 2a + Ix sz s sin 2a

I x = 2(Ix e

Iy = IYs Iz =

2( I zs + Ix s ) + 2(Iz s — Ix s ) cos 2a — Ixszs sin 2a

Ixz = cos 2a —

—I zs ) sin 2a

Ixszs

2( lxs

Principal to Stability

xs = 2(Ix0 + Izo), — — Ix o ) cos 277

I 2( I z0

I Ys = IY o

= 2(Ixo ; Iz o ) + 2(Iz o — Ix o ) cos 277

I zs

tz o ) sin 277

=

Ixszs 2(Ixo — Stability to Principal

I zg ) — 2(I z s — Ix s ) cos 277 + sin 277

2(Ixs +

I xo =

Ixszs

IY o = IYs

Ix sz s sin 27)

I zo = z (Ix s + Iz s ) + 2(Iz s — Ix s ) cos 277 —

Ix o z o = 0 = Ixs zs cos 277 — 2(Ixs — I zs ) sin 277

(Continued) Principal to Body I x = 2(Ixo +Iz o ) — 2(I zo — I xo ) cos 2E Iy = Iy o Iz = o ) o ) cos 2F Z(Ixa + Iz + 2cIzo — Ix Ixz = — 2(Ix — I zo ) sin 2E Booty to Principal Ixo = 2U + Iz) — 2(Iz — Ix) cos 2E — Ixz sin 2E x Iy o = Iy = 2(Ix + I z ) + 2(I z — I x ) cos 2E + I xz sin 2E Iz o = 0 = Ixz cos 2E + 2( I x — Iz) sin 2E Ixozo ^ Ln LO In ^' Iv f^ w f 1 El U + W .-. r. ...

a N

V

N v B ^ + U > N '^ I Iv' p f ^'' W ' I Ca 1 > ^ oc, r 1 r.

A U O V1 (U B U Q O + U W `' ^'^ f^' v V1 > + > 1.4 r ^.t)^ p + + f/1 > V N •^•I u $ R. •^ A,I N II U R' O

W ^• U a

^

U

+

+ + + U I 'y + $ a N p + `-' IL3 .G i> > + Q I W O l y I z •^I N Q + EEa tkD y II O U 4 9- } I I +,, I v > + I I I I c as v' v o N 7 II v a •m '$ •^ ^

o ^ J

IN m ^- z a + ad U II v U it O Ln z ^ I m ti > + 1 U I p p I W l U 1 II +

^^' pl >

a V

d 1 ^ I I as ^o ^ v v v LO I r7 q

m

W Q $ $ •^ H .1 I d0 o + I a II '^ ^ p I N u I p v O f% m I(

CY

V •^ I ^: co p •^ U H H ca LO $ .3 + i ~ I ..

v' w v' W a a N m as $ a a r^ •^ + I I I

o 0 U o

N .a a o o + ► v .-e. •^ .^ v tl u N N + ^v N II II II W v U ^> I ^ I ^ p, a c.

o l > a tl ja o4 r4 O C!

O 1- AA H ~ O 1 I I N N tl I IU I ^..

v U ^ ^, a Q t?+ I ,^ ft^ ^ ^ v t^.

v a ' C ^', a0 o^ t03 H H .. .. ..

.. .. ...

o

cd '.

co ca co cc co II .1 I bb a ICi' Iv' f w — 'T f ^^ o

a a

a a LO pi0 GO

0 a CIO

I tin U W U O W W V - 4 h0 r _1 I + t + + + + cg

w

O m !Cg I o y .^ I N .a ( N N I N d d O U O •Q ..r U U ++ + W ^ j hD U e p # + + U + + + O lo, la ti a N N 4I 4I < 0 0

w

tp

.a

+ a v v .a

m

+ ao + + + d

a

V E p O d b 8 I > > R. N a I N I d Q I 4-)^8 8 CL

d

U ..4 U co rm Q U d z d e

V

b b d a Qz N .,.1 1w

4 4 4 I L^J ^^ UJ U

as a^ C II ^!

II II II 21 II 21 21

V

a

a

a4J-

os d

I I y O dai" 14 ► 4 N tl co p.

C W O •N N •N tl tl W ••I.a _ U tl ^

A

^

^ tl

tl a N N N M M ^ ^D cD tD cD tD co

U u

m GO a(10 IN ► IiC I^ ► ^l la IZ O ► a o0 CIO Q w II II O

QQ-

a

d

'^ ^ ^

d

IN ^ ^ ^ a I Z I I m + Q, Ia se + I I rn ^ I IN p I N .^ O w, r-^ U I I ^ w m Cb I y v I a a, I I c d1 U GL1 ^ m E bD h0

v v

F

W ^ _ II II O a I bb 4J la I + I + IN p ^ I ti I p Li.

+ ^ V1 ^ 4 y •y v U kr Vo1 N N d! ^ v F c4 Q I I N I,.

a I^ ^ I I I a

+ Iz

H d O O N ^J tl ' P1 P1 tV d ^4!

^ O TABLE VII Desirable Characteristics of Instruments for Free-Oscillation Maneuver Sensitivity Undamped Dram ing Function Range (per inch natural ratt io frequency (c/s) deflection) - " 1 0.65 t10 8 or more a , deg 5.0 deg t10 4.0 8 or more 0.65 q radian/sec t0.2 0.2 8 or more 0.65 ±0.5 0.5 8 or more 0.65 q radian/sec 2 or more 0.65 f0.1 0.1 8 r radian/sec or more 0.65 t0.4 0.4 8 r radian/sec t rudder pulses 8 or more 0.65 t0.2 0.2, p radian/sec t0.6 0.6, all p ron pulses 8 or more , 0.65 t0.6 rudder pulses 8 or more 0.65 0.6, p radian/sect 8 or more 0.65 t6.0 6.0, aileron pulses as ,g units 8 or more 0.65 ti 1.0 t0.3 rudder pulses 8 or more 0.65 0.3, a t g units t0.6 aileron pulses 8 or more 0.65 0.6, TABLE, V111 Format used by NASA Flight Research Center to Record Actual Conditions at Time of Maneuver —7n — strument scal 't

oi l

/d- vane /ocol%007 Trace It s.

Nat. fre Dorn ro io Flt's. IQ-*

e

X^ • --- Zr .7s 0_c s sal P 6 o.

_

— d . — 7F _6t

_ ^l?6 __ Lineor occe/erome/ sr loco in

.6-S' .4100 De 11, so _ S2 X a --- iy' ;Z- /0, to .6 f /0.30

7 10

I 2 4 5 8 9

3 6

Dens;ty, Dynom;e We;ghi, Cjt s ` pressu ►

Fli.- Ron V

00000nt h

M W

I /0

q

^^^ts

fps ^ ft ld 0.7-95 723 142 237 220-17 27, Soo 28.7 0. ,00or73 40,920

bc ale rac

TraceInstrUl e t 4h - p/one I^^ - ► Flt's.

Ak+ . r Dom .roho o r U. 6 orn ConfiguroV S, i 7 _G s

d 3 Z. o

/^ 7 8Q l7

1 3 15 16 /8 /9

Il 12 14 w^= Domp;nq Damping Undo^+pe ^,^ ► t 041-ol Per;od, Se _ angle ratio ^. not. /r09.

%nput s de • Y • p.

r im I t rim ^ A 8 j

^ sin ^" •

p

^d ?aaP 5

sec sec ^,,, 4/

g vn^ls dog

deg

116)

/^7-

?.6 f. 1.7Z 7.'lG 0./7 l

6•S •7.s S 2 .9z

.!3 A, TABLE IX per radian CnA , Altitude, a, _ Reference Cnp Airplane M ft deg (Eq.(6-53)) Equation (6-54) Equation (6-56) F-104 0.94 41 000 4.9 0.46 0.46 0.57

YF-102 0.7 0 , 40,000 6.6 0.054 0.043 0.106

TABLE X

oVM/h 15/0.8/60 3.5/0.8/40 6.6/1.2/60 14/1.6/80 10/2.0/ q 0 5/2.0/80

Analog value -0 . 084 032 -0.074 -0.0164 - 0.0034 -0.021 -0 .

of ClQ Analog value 0.259 0 . 0.640 0.367 0 . 445 0.508 of CnA Equation ( 166) 0 . 266 663 0 . 360 0 . 434 0.498 0 . 661 0 .

0 . 383 0 . 451 0.504 Equation (175) 0 . 278 0 . 679 0.674 TABLE XI

alM/h* 15/0.8/60 3.5/1.0/40 18/1.2/80 6.6/1.2/80 14/2.0/80 1o,^2.0/80 5/2.0/80

Analog value -

1.58 -1 . 734 -1 . 71 -1.92 -2 . 09 -2.55 -2.58

- Cnp)

Of ('Cn

r

I

- 2.50 -2.96 -1.81 -4.77 -2.76 -2.68

Equation -1.43

(184)

Equation - 4.22 647 -4 . 69 -2.36 -8.46 -4 . -4.39

-1 .

(183)

• h = altitude/1000

V r-1 O QH V c co N • r • 1 4J Cd U N b •U .0 N B W b O C U c^ 4J .

p O

a o

U S R bp it

a

a^ . U +^ O •ri W • ri "C7

A

cd W rA O O W .G U U 4-4 r•1 O W U A b O "-4 W R •r-1 W G3 A .-I 4D 1 ^ a ^

v

d c v CL a, C d v v c m a ar

>/

as _d V c (D V C1 C O '^ c c c x a v X as O d •`-' x x

a

o 3

x c o ` a^ x .a V ^ c > U c O c O CU N J d $4 Q) > U O d 4.4 V Q) a) -\ O +^ G c ^+ o > CL N v 'L O C a a ° cd •., N H to b c O dr 3 $ O • o ^ •- 0 0 r ai W +-) c°.

b o CL N ^ N b cd cd a ' 111 -; Z N S-4 .0°

9 -

>, N b o CO +.)

+-) +3 M U) F 4 N fq N O Q) .LT ^ 4.4 O cd a O r1 O bD q1 O O Ha CI C O e o CL) r Qd CL m a, c c cl o CL b.

a

-o

t

> c v e T " ^ d y T ^ ^ N r• b > V CI e •r c o O !- CL v > CL a N Q N W cd y O N O -^ K b W ral bD ti V] a) U ^r 1 'O e Ld i7 O l^r O W O

a

O .,., od x a^

a

M 4D ic Y. r (a) Euler angle perturbation referred to the x ry r z r basic reference frame Center of gravity

- xba

—Txr

I

I I

f

i— - — I

axes serving (b) Euler angle p^rturbation referred to xboybozbo as a secondary spatial reference F'ig.4 Several methods of considering Euler angle perturbations 4'ftxr Y ^.

b

Fig.5 Relation of p , q , and r about body axes and Euler angle rates

B ,

and ^

Center of gravity X i S2 B 'r Rot wring rm mass Fig.6 Pertinent relationships of rotating mass for gyroscopic couple consideration.

Rotating axis parallel to xz-plane of symmetry Fig.7 An example of the influence of ranges cf disturbances such as (0,3) 1 and on the value of a derivative (0,) 2 III

CMa

CNaa(1)

C

i

2V F'ig.8 Effect of time ldg of modification of vortex flow about lifting surface on t:ie change in C following initial instant change in a (-Z) p Ep = upwash du i to wing, etc.

ap = a + t T ap \ velac^tY p /atrea^n Free.

Plane of p ropeller disk 2b Fi g -9 Direct propulsive effects of propeller 1-ZIp Bp 2 upwash of air intake X X T Ce nfe j Of Pi Ai \tore \ e 0% v v\o^^tV

#IS#

Vi as F'ig.10 Direct propulsive effects of ,het engine F'ig.ii Jet-exhaust inflow effect on horizontal tail

w

^d c.

U I.^ cd M R . - 1 4J y cd .y A W G i•.

O U O Lr W O C O O a 4-4 ^v U 4J H N W O A O .,1 U

a

N fl0 G4 4J a.

y r_ w a a^ t^ e no a N ~ N b a N O Ca ;V r1 4.

Nd N O V 0 a

it rl

dl +) O a set d U p^ i .a Cd U L•, U W O a O •,.4 ca a •e ,4 t^ O d d

M

P4 •.4 w Fig. 14 Determination of pitching moment of inertia X x0 ^ ._. . ......a air.. .ba I z0 z (a) Test setup

Z

(b) Vector resolution Fig. 15 Determination of inclination cf principal axis and yawing moment of inertia

lj

^`vC ,^^ ...,^.

JA.

io f, F imp,, or Fig. 16 Photograph showing s general arrangement for determining Inclination of prinr^ipal axis and yawing moment of inertia. Springs attached to mounting brackets :ocated below wings Amplitude ratio, IAp J Ar j -.2 /" . • 4 L- .04 .06 .08 -.04 -.02 0 .02 Tangent of resinring spring angle, tan 5Sp Amplitude ratio IAr I a:- a functio n, of spring restoring angle

Fig. 17 Iopl/

` n 7.86

— &- -0.872

A A ---- --- --- e o B R 10° 0.048 0.872 0.096 F-0-0.640 • 0.043-inch - diameter orifice View A-A o 0.052-inch-diameter orifice 0.096 r-- 0.096 4.1.

R . - —

M096 0.095 0.640 View B-B Fig.18 Details of total-pressure chamber and static-pressure orifices. Reproduced from Reference 24 its

a

a c cri ctj w O i a ec r- ed

r

E 4-4 p O O +^ O cd

f

cd O ^M C U Q Q z cd U

a

cd w C6 co S.

O ,t 0

a

w .12 Transonic .08 AM Subsonic Supersonic -.04 L Fuselage diameter Fig.20 Effect of ratio of boom length to fuselage diameter on Mach number error.

Reproduced from Reference 30 .16 Boom length Fuselage diameter r-- 0.6 0 .12 AM .08 0.95 .04 01 1 1 1 1 . 1 1 1 1.2 .7 1.0 1.1 .5 .6 .8 .9

M

Fig-21 Variation of Mach number error with Mach number. Reproduced from Reference 30 1.4 1.3 1.2 1.1 M (true) 1.0 .9 .8 .6 .6 1.0 1.1 .7 .8 .9 1.2 1.3 1.4 M i (indicated) typical calibration curve for d6termination of true Mach number Fig.22 A A 6 .5 .56 .4 .52 B .3 .48 pT .2 .44 .1 .40 a 0 1 2 3 7 4 5 6 M (a) Variation of q/pT with Mach number A 4 1.3 q i = 3 1.2 KPt —_' 2 1.1 A K = 0.540 1 1.0 K = 0.526 0 9 0 1 2 3 4 5 6 M (b) Variation of qi/q with Mach number Determination of dynamic pressure from total pressure fig.23 E c.

a^ b a ^o a^ e c, a

a

c, a L9 ec x E O 11 ^ O A x a^ Q 11 M ` E w.

O U W U a^ x ^ ^ U cd ^-+ B^ C)

am

w

N w

4 b

4> C U U —x a i •1:1 e ti o

O

O p ao

^ cal can a

ri ^-

d la

4-) a cd ^ `0 o 0 0 a^ r, +) tin cd a^ bo ac a o ^ CL) W 02 w U Q) %0 w (1) r' > I .a ^ a c, N LO N OD .,.q Giw -1 p N1 tr

N

O C! I ^ a c Y > d ^, a O

a s

c v V S \ C P Y O d I v m a O a r m a a ' V N h C U a c a

I y 9

^ ► cr C V C i .^' C O O c N O ^ O W s ^ 01 O O `v d ^ o u °' CL r1 > a eo x ` t

t 12

Ot O ► w c a O v, O 1 O .A a 3 ` c 1 I O d C a O ► x d s c U a O II

r

V a > d ^ d O

a

r c o a `O r d CL d O r C a M Y 4) r• O r.

O O N H CL O L U.

Ot ca

a

4J 'o no w w d O a^ U b U r-1 W H ^ a O' tl x

A

C

d

w 6.

o :' c M a v I O ,^ it O b.

Ul aC ^ H

I

° o

Q a --

v^

xi

^N a a^

v

wU` W X , ^v c.

U d 1.^ .

x

4-) U i L, O cd d U

UO^

a cd w

C Q

ao ZA

N

X , ti ^o U it W ^^ U b O 9 W

v

U CD

^.I a

e

H

--

1 "6 O Cc is ca r, c^ .c: U C1 C1 O C C1 O L.

+•+ cd ca U y O C ^ a+ CL) w +- 0 %n a N N 4+ O O 1: C .•-^ 4+ O 4- do C: C".

O 0 • U y 'L7 is i O

w

I ai U C {•r a) N b0 G..

E

.. O

46.

Am 400

a

a..0

• c • c..

a, ^o c.

K U a^ a U O t, M .-r U

a

b

U 4J Up r0 Cd c^ G4 lY8 N e a^ En oc a

a

b b a^ oc cd a w Cc .a rn N CD G.

i Calibrated characteristics Output sensitivity S in = 0.265 radians/(radians/sec) Undamped natural frequency w n = 40 .8 radians/sec Damping ratio 6= 0.657 Sin Spin reference axis Output axis

q/p

nxi

I

Input reference axis

Percentage error -- fin.

In

^ r

In ' n in S in 6in -4 0 -3 -2 -1 1 3 2 4 q rate, ^P ra: ians/sec F'ig.30 Influence of interference angular veloc.,ty ("q" rate) about spin reference axis of a sensitive rate gyro 1.30

m _

V

` N

O C c '^ N N O c y C y O N N + Q 1 ry + O V' sp er ^.1 V t7 C C U N C N D O O _ N V

a

Q U2 + •N _fr y p^jq J 1 Vim+ n e C[71'

a

c m N

c

'N W N •N C 'N O C V O d _O .

r d M _C N N ^^ C wl N V O N V N N O v O K V y Q V Q V N V a a + + ` a + + c.

ci w O a cr Z b c.

U U i-/ O 4.1 U W cd S-4 bb U U it O Fr O U $4 O W C N d E A 0 3 G. Q .'4 +.)

N ad C cr W t"1 M b) .rq aw wIH `^' ^ N V x a. o^

+

N N Q 0) N x N + N + + .QI + .Q o, 4- N •x ,^I O c.

Q) a O o o` a^ U II II II U x c O O O O U2 L+ U U O

x

it U +^ >K r-1 4-4 D w D U U N •a ^ Fr r - 1 4a 4i O O b O cd U it a) bD t10 Li .r.4 U \ •O' U R.

t~ O U S-4 x 4.° O .H c .o +-) 0 N Qi a N M t1D Ol LL O C ► C m Y d

C 0

a

a

-16 `'in = .70 2 ^in in 4 = - tan- .60

,=n

-12 .50

- Indicated

- Corrected for phase lag Phase lag, 1 , deg -8 -4 .10 0 .04 .08 .20 .12 .16 .24 (Un.

to u Chart for correcting sensing-recording circuit of instrument for phase lag F'ig.33 1 08 -- — -- I "in - 0 1 - c i )l 1.06 1 Al - iL — .") + /2,' 1.04 Amplitude factor 1.02 1.00 I 1 .04 (1) (11n in Fig.34 Chart for correcting sensing-recording circuit of instrument for dynamic amplification d c c o c p O a b.

V a —° ' c d V CL a o E ---I o ` o c d

v ^ ~ ^ d d

v Ix +-) C7 O N ~ S O a o 0 ^+ 0 0 y - o E a E c o E s a o ' m I E'

_ O o

U o E cd o o c a > d E c L& .` W o d ..

c O U ° a- c .^ v cd N .^ V ^. D Cd N W ^

H O

+-) H T O cd IV L b .o a 1.

c.

O o ° U

O

^ Q I U F, A O ^ y O

O

Cd CL N d c ,c°o. o o o V a c a^ `c c O a, O w E Cd :.'

O a a^ a c ^ ^ c ^ b E v a O o -- --- ^ '»= U Er ^, o .c E h0 `o U d E

1O d

^ V

Ai d OC W

^ F ^ I

a

I

E

W w

CL IC o c ' ( ( pw a U } C d I I ( I

a

I

°' I

c I

,1J U a Io o e, d I a I o o Cd a

IE V O V V

o a I ( I V c a U C I 10 I LO

^a

la I I ^ _

C C N N bjO

I C O I p y N O Ca

c c )

I N 2 •N V I CL CL h - o v O p V O 4) G C -' p C O C a> I =. , I 'A a> I I 2.4 — .12 2.0 .10 CNQ, P, sec per deg .08 1.6 .06

1.2 I

4.0 32 3.2 Cm q + Cm&, per radian 2.4

I

1', 1.6 8 .8 .9 1.0 1.0 .8 .9 M M Fig.36 Results of analysis of flight data in region of rapid changes in aircraft characteristics

hP, ft

O ^2 000 Sri

3.6

3.4 3.21- 3.0 2.8 P sec 2.6 2.4 2.2 2.0 1.8 .7 1.0 .8 .8 1.1 1.2 1.3 1.4 1.5 M Fig.37 Variation of the period of an F-100 series airplane as a function of Mach number, altitude, angle-of-attack, and load factor (from Reference 41) 8 3.2

4 2.8

P, sec 2.4

0 2.0

.16 320 .12 I t I S deg I 1

I

a n , 9-units 340 08 I I I ^ I

1.0 1.9

.6

1.0 o n , g-units 1.9

.04 .14

-.1

Cn

C Zp' - .3 `1 .10

per rod

per rod

.06

-.5

- 02

a n , g-units

1 an, 9-units

.6

6 1.0 1.9 1.0

1.9

( C nr - CnO,

I

I

-.06 II

Cj ^ %

-.3

^ I

per rod I I per rod I i ' .__-J - .10 -.5 2 4 6 8 10 2 4 6 8 i0 a, deg a, deg Fig.38 Influence of angle-of-attack and load factor upon lateral characteristics of one aircraft

O

x O

N

O IV O p lw r x v .r, v O O a ei Y O `o

m U

C O N o^ M a 0) x ar O qw i- —.

00 ^p C O N N G C N r N N C 'i7 O C 'L7 ^ N C •^ .-r .a O a+ td O

N

O N in c +j o

O

o „

., 41 N w d w O pp t^ O C 0.^ O c O O a ao ^- IQ c.

w O Q O N O C].

m O O O IQ e N O C O d O V a) M D GO D d Gc.

D O ID

v

0 0 0 o O O O o

pp

00 .p a C4

Ln N m v

i

c c O O •N O v .0 ' N J ► ° •Q a a ^O a Fi En c -W c4 c 41 N

a

° •rl o aJ a ti a^ of `o a J W O Cll GL C CU it a c ti ` O

w

w O •r1 O b o °- c :-j

y O

P CL ^ N V a^ J C40. ^ °a a a V 0

a

^ a v fy ° V r-4 a Cd Q o O J Or J d U Cy 3.

b r--1 Cd k CU Cd W O O L7 C b.

Cd y O

a

` := c ro o

U c

CL '_ J 'N a O

v Qw

N • a a . V. .. N _ C2 _.

v on v r ♦ o ac a s a v` , J O Is c d u _E ^N O Im V r ^ v a I ^ ^ w C N N U R •^ O ., U c.

a^ a^ .-4

w

U U V O Lei it Cd O Cy rn rn U N a a N '^^ i^r II b cd o e +) o an w — cr a O on .;.^ W w o o c.

o ^ o a W •., ae r+ cd +-) ' O a ` a) t .,-1 o, .c +' V

. o a ,.i

w U II C14 d^ O O O 0D '" `; N p N Go d O N O N w v m a N of \ N V .Q ^, c o c c

Flight

-- — Analog simulation

n__ I

0 -

ayd , deg 0

M = 1.245 M 1.614

M = 0 948

-20

.2 -

r ' 2 0

vt

vvv

radian/sec

Y v

I I 1

1. viv

,21^---

r, ^

A AA i n ji A A

- A

radian/sec 0

j viv

v VI V IV

-.04 .8 P,

A f\ a a A A Ar

T-v

v

radian/sect

v I v i [ k l I: v IV

i

.4 - P, 1-

o ^^

radian/sec

vv

— Y L

-.4 1 ,--_1 1 1 I .10

A A A

at,g

V, v IV I

_j

-.10 U /, deg eJ ^/ -2

6 8

0 2 4 6 8 0 2 4 6 8 0 2 4

t, sec

t, sec

t, sec Fig.42 Typical time histories of the lateral and directional response characteristics of the test airplane resulting from abrupt yaw-damper deflection .

I ^ I +^--- Y lil `%ft so dop I I dop Y 1= -=k----^ I----C-A I I I

ICI

(a) Wings-level sideslip (b) Constant-heading sideslip (r = 0) F'ig.43 Comparison of wings-level and constant-heading sideslips i AA Relation of small-perturbation rolling velocity and acceleration vectors to Fig.44 small-perturbation roll-displacement vector in a transient oscillation ,o a Cd u u

w

E O O ^" w VI a^ G O^ Cd v a, u N Cd co Cd c N ^-1

a

w

c fz .2 +-1 Cd -n a^ a n)

n

LO ,^ rr V

a

a

a

Film scale factors 40 ! g ip 0.511 radians/sec/in.

A P Ar — 0.126 radians/sec/in.

t — 0.490 g /in.

Da

24.3 units ON 4P — 10.3 degrees/in.

X18.0 units ^.n Aat 8.8 units 0o 5 units do 1 '/z = 4.4 7 24. 0.511 jAp) 3 X = 5.48 18.0 0.126 J Ar ( JA I 5.5 10.3

x

x ---- =

^ 0.436

18.0 0.126 57.3

i

1 Ar)

- x 0490 - 1.91

18.0 0.126 Ar ( I 9 11 12 5 6 7 8 10 t, sec F'ig.46 Determination of I-ime-to-damp to one-half amplitude and amplitude ratios from free-oscillation data

O

no.

III

a, ^°^ ac

MII

M

^^ Q O

k 1 ^l

N ^

,Irzt,

N ` • ^ O^1 a

a v 3 .., I ^I Q

\ II

^^o, '^^' ^

rn III

^a

^a

d

I ^ \ Qr ItH v v

II_ \

II

Q Q \ ^ 01

Q Q

d

'a q Q 4 an

I

1 .^

g a a

^bD o+ a ^ v O ' Q'1 O Q Q ^9 I

tl

r

X 1111

11 O

^IQ7 _ W i,

I^

^ 111^1`11^t• O ?ter U

s'

\, Cd ^^ Oy .a a d

N

O

^) II

a^

Qc ^

' a 4

as --

I

^a

Q)

Ili

I

O ^N o,

N

II 1^ MII

^D Qa

I

II I.

vJ m O .,.q Q Q ^C Of _Q a 4J O1 i 0 Q' c.

o

M VJ a^ N v b I _II cla

a

I

Ia a

e --1 d VJ ti of

w

N N

I I I I a^ .tea .

N od d Q

^Ia

Q

N

I I 4.

CV . a Ia c^ cr `y b i cn o •^ a^i 41C En c1

N

e e 4 .d p6+ ,^Q I l — rn

I 1

c4-)

v

L

w X1 'CM cl4 O I t4 O •,-4 +i a O v ^4 O 4) ala xlo^ N .t.

1 N cu I.0,1 c .rj ^ ^• II w

v

_II i ^^^ ad ^L

4 4 9

N^fi

I lk

I10trw

v

^L

as

I

AS e

I

I

I

I

Iq-

Aq f

I

I

I

Aa

I

0 2 3 t, sec F'ig.49 Typical determination of flight quantities for the evaluation of longitudinal control derivatives 1 48

aI _ . ^ac G ' _ t c q I _

C

C-

^^ CN•^_"o L^ _Q

LT O

^An

hoc

®

! ^I a^2 vi

gs

Q

141 ^

A

,

-89,20

I

C 1°==is76

I

.2 `T 1 " 1

x a^

^

I

A3-7

aOC

_ 113- 7 1,57

C

=

3 4, 3S'

^o ..

.^.

I' d 50 ^•n6s3 _ ®.00 + CN.

^) _ `- -

X8.66

(C / V

ZV

F'ig.50 A gr aphical time-vector solution for and ( CN, + CNa) CNa -- , SAS off ---SAS on -6 -4 as h deg -2 10a

A4, dog /sac'

-50 -100 lQ

Aq, deg /sec 0

-5 -1 fr 4a, deg -I

7 8 10

1 5 6

0 2 3 4

N, sec Time histories of longitudinal pulses performed on the X-15 analog with the F'ig.51 stability augmentation system engaged and disengaged ( from Reference 42) d d'

00000

, + O O 000

I

0 0 000 OO in- V O •- 4J N M I q L/1 •O

V

O O O a.d

.^C / / d ^' w d / / I i

W

/ / e 1 1 Fr M QO0

^a

.,4 r-

bw

1 ^

e

N Q F^ P w A, co tr 0 0 ^ e I U , 4^-r Fr O i2 .a O y S • r-1 .rn 4-i U 4-r Cd U ze a O w •,.q ^ rn V-r 4-4 O

d

LN '^ a N O ^O N O N 01 ' Z ► - a

u

tl V 4.6 3.

-A 0 ,2

't, f, a, I ai -r ke • a,rpIEu,,, Kt' Ats

2. 8

^^ Nb^ tYlJ P Cy f 1 rt (I , 2.

7 and anJ 4 2.0 1.6 IP 1,2, and 4 \ J 2,3. and 4 and I . I 4 [ _ TI/2 .40 ".Q 2 13 b .2 ,I ,C -6 P 13 14 15 1-6 M

Fig.53

Longitudinal period and damping characteristics of the D-558-II airplane as

functions of Mach number and altitude .12 s r .3, JrJ...t iwd 3, .10 ani 1 ' 3 ^ and ^ i ' .08 CN&C 3-1 Z : ano 2 .06 Power off Power on hp, ft (:) 2,j, 000 —___- -.-^,- - 04 n 30,000 ^J Q ♦ 4b,000 Q 55,000 60,000 .02 Urtfla gge1 symbols, all-rocket airplane Flapgei symbols,rocket and het airplane -.0 -.0 CM -.0 -.0 2 -.3

4 -.2

6 • -.I C m^ ♦ Cmq`

5 — -. 0

1.2 1.3 1.4 1.5 1.6 17 .6 ID 1.1 .5 !

M Fig-54 Variation of static and dynamic longitudinal stability derivatives of the D-558-II airplane with Mach number (from Reference 43) NO c O M c

4A on

d c Q c N C U c N O V

M

c

3 q

N b A co v N O M E o 3 -c Ol N O a^ a O 0 a^ _E ^.

^O N '^ N Q w II U U d W W U

O

ri cd w "d cd w ^O O N w

O

.--4 O ^ U N N ^ U W N b 0 i W %O O

U

0 N 0 .H O d N w N O w O OO +- 0 O 4- }

O

o

O

O O II II c, 11 II cd a .c

a

-C C e U M N M O N

O O O

O

O ►

O O h

O i

i

i

i

C:.

Of W ^S 01 GO ^ E E a, V V CL

a

I

Asa I

1!

I

As a f^

i ^I i

%

II II I AP I 1 I A • I ^ r I I I I I^

II

AP

^I

I I I I i I

I

Ar

I

e^ I

i

I i Fig.56 Typical determination of flight quantities for the evaluation of lateral control derivatives .10 .14 .12 r^ \ .10 per radian C .08

\

.06 \ Method Time vector .04 Steady sideslip, equation (169) Equation ( 166 ) --^ Simple frequency, equation (1 68 ) .8 1.0 M 1.2 1.4 Fig.57 Comparison of Cnl8 as determined by several different approximate methods A with the time-vector method C1(3 - per radian -.04 -.08 1 ' ' ' .8 1.2 1.4 1.0 M Cl Q Comparison of results of determining by time-vector method and steady- Fig.58 sideslip equations rlz_1 C.

01 z ^

3I n b

l°I°I _ Iarl

L4_ 0 b

L

rr

pr —