Considerations in the Determination of S&C Derivatives and Dynamic Characteristica From Flight Data

8/12/2019 Considerations in the Determination of S&C Derivatives and Dynamic Characteristica From Flight Data

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REPORT 549-Part

T"T



.. 1 I II .»■ II- —~ —- - w - — ----- -

A G A R D E P O R T 549 - A R T I

NOR TH A T L A N T I C TREATY O R G A N I Z A T I O N

A D V IS O R Y G R O U P F OR A E R O S P A CE RESEARCH N D D E V E L O P M E N T

( O R G A N I S A T I O N D U T R A I T E D E L' T L A N T I Q U E O R D )

C O N S I D E R A T I O N S IN TH E D E T E R M I N A T I O N OP

S T A B I L I T Y A N D C ON T R OL D E R I V A T I V E S A N D

D Y N A M I C C H A R A C T E R I S T I C S P R O M P L I G H T D A TA

by

Chester H.Wolowicz

N A S A Plight Research Center

Edwards Air Force ase, California, S A

This Report w as repared at he equest of he light Mechanics anel f G A R D



• . I . ' f>i

i *M »

CONTENTS

LIST OF TABLES

LIST OF FIGURES

NOTATION

1 . INTRODUCTION

2 . AXIS SYSTEMS AND COORDINATE TRANSFORMATIONS

2.1 xis Systems

2.2 oordinate Transformations

3 . EQUATIONS OF MOTION

3.1 nertial Quantities

3.2 yroscopic Couples of Rotating Masses

3.3 ravitational Force

3.4 erodynamic Derivatives

3.5 ummary of th e Equations of Motion

3.6 etermination of th e Roots of th e Determinant of th e

Lateral-Directional Small-Perturbation Equations

4 . MASS CHARACTERISTICS

4.1 eight and Center-of-Gravity Location

4. 2 oments of Inertia

4.3 nclination of Principal Axis

5 . INSTRUMENTATION

5. 1 ach Number, Altitude, and Dynamic Pressure

5.2 ontrol Position Transmitters

5.3 ngle-of-Attack and Sideslip 5.4 ngular Velocities and Accelerations

5.5 inear Accelerations

5. 6 hase La g and Response

5.7 anges and Sensitivity

5.8 ulse Code Modulation (PCM) Data-Acquisition Systems

6 . FLICKT TEST TECHNIQUES

6.1 ac h Number and Altitude

6.2 ngle-of-Attack and Load Factor

6.3 eroelasticity

6.4 ontrol Inputs

6.5 aneuvers

6.6 eneral Comments

Page

v

v

x

1

2 2 4

10 10 13 13 1 5 23

25

3 0 31

34 35 39

39 42

44 45 45

46 46 4 7 47 48 48

5 1

i i i



TZr-SXSSZ-TT- iW

1

£

7 . ANALYSIS OF PLIGHT DATA 7 .1 undamentals of th e Time-Vector Approach 7 .2 asic Flicht Data 7 .3 etermination of nd Fro* Free Oscillations

th e Absence of or Questionable nd Data 7 .4 quations for Longitudinal Control and Stability

Derivatives 7 .5 quations for Lateral-Directional Stability and

Control Derivatives 7 .6 he Graphical Tine-Vector Technique 7 .7 ther Analytical Techniques 7 .8 nalog-Matching Techniques

8 . APPLICATION OF FLIGHT DERIVATIVES 8.1 erification of Wind-Tunnel Data an d Theory 8 .2 ffects of Aeroelasticity 8 .3 tability Criteria 8 .4 light Guidance

9 . CONCLUDING REMARKS

REFERENCES

TABLES

FIGURES

in

Page

51 51 5 3

54

56

6 2 7 0 7 2 7 4

8 0 8 0 81 8 2 8 7

88

8 9

9 5 - 1 0 4

1 0 5

iv



LIST OF TABLES

T A B L E I

T A B L E I I

T A B L E I I I

T A B L E I V

T A B L E V

T A B L E V I

T A B L E V I I

T A B L E V I I I

T A B L E I X

T A B L E X

T A B L E X I

T r a n s f o r m a t i o n o f D e r i v a t i v e s f r o m S t a b i l i t y t o B o d y A x i s

T r a n s f o r m a t i o n o f D e r i v a t i v e s f r o m B o d y t o S t a b i l i t y A x i s

T r a n s f o r m a t i o n o f M o m e n t s o f I n e r t i a f r o m O n e A x i s S y s t e m t o A n o t h e r

G e n e r a l E q u a t i o n s o f M o t i o n

L i n e a r i z e d S m a l l - P e r t u r b a t i o n E q u a t i o n s o f M o t i o n

L a p l a c e T r a n s f o r m F o r m a t o f S m a l l P e r t u r b a t i o n E q u a t i o n s o f M o t i o n

D e s i r a b l e C h a r a c t e r i s t i c s o f I n s t r u m e n t s f o r F r e e - O s c i l l a t i o n

M a n e u v e r

F o r m a t u s e d b y N A S A F l i g h t R e s e a r c h C e n t e r t o R e c o r d A c t u a l C o n d i t i o n s a t T i m e o f M a n e u v e r

P a g e

9 5

9 6

9 7 - 9 8

9 9

1 0 0

1 0 1

1 0 2

1 0 3

1 0 4

1 0 4

1 0 4

LIST OF FIGURES

F i g . 1 D e f i n i t i o n o f b o d y , s t a b i l i t y , p r i n c i p a l , a n d w i n d a x i s s y s t e m s , c o n t r o l s u r f a c e d e f l e c t i o n s , a n d f o r c e a n d m o m e n t c o e f f i c i e n t s 1 0 5

F i g . 2 R e l a t i o n s h i p o f b o d y , s t a b i l i t y , p r i n c i p a l , w i n d , a n d s p a t i a l - r e f e r e n c e o r t h o g o n a l a x e s s y s t e m s w h e n b o d y x - a x i s i s r o t a t e d i n s e q u e n c e t h r o u g h -\p a n d > 0 6

F i g . 3 R e l a t i o n s h i p o f a e r o d y n a m i c a n g l e s a , ß a n d y a x e s a n g l e s T ) a n d a n d E u l e r a n g l e s p a n d t > 0 7

F i g . 4 S e v e r a l m e t h o d s o f c o n s i d e r i n g E u l e r a n g l e p e r t u r b a t i o n s 0 8

F i g . 5 R e l a t i o n o f p , q , a n d a b o u t b o d y a x e s a n d E u l e r a n g l e r a t e s / / , 9 a n d > 0 9



Page

Pig . 6 ertinent elationships of otating nass or yroscopic ouple consideration. Rotating xis parallel o z-plane of ymmetry o

Pig . 7 n xample f he nfluence f anges f isturbances uch s (A/SJj and (A/3)2 on he alue f derivative o

Pig.8 ffect f im e a g of m odification of vortex low about ifting surface n he hange n C N following nitial nstant

change n a 1 1

Pig.9 irect ropulsive ffects f propeller 11

Pig . 10 Direct ropulsive ffects of jet ngine 1 2

Pig . 1 1 Jet -exhaust nflow effect n orizontal ail 12

Pig . 12 Determ ination f ertical position f enter-of-gravity y tilting ircraft n oll 13

Pig.13 Determination f ertical enter-of-gravity nd olling moments of nertia y olling oscillations 14

F ig . 14 Determ ination f pitching m o m e n t of nertia 1 5

Pig . 1 5 Determination f nclination f principal xis nd yawing m o m e n t of nertia 15

Pig . 1 6 hotograph howing eneral rrangement or eterm ining inclination f principal xis nd yawing m o m e n t f nertia. Springs ttached o mounting rackets ocated elow ings 1 5

Pig . 1 7 mplitude atio |Ap|/|Ar| a s unction f pring estoring angle 1 6

P i g . 1 8 Details of total-pressure c h a m b e r and static-pressure o r i f i c e s .

Reproduced f r o m Reference 2 4 1 7

Pig.19 Photograph of a t y p i c a l NASA installation of angle-of-attack a n d sideslip vanes on nose b o o m 1 8

P i g . 2 0 Effect of r a t i o of b o o m l e n g t h t o f u s e l a g e diameter o n Mach number e r r o r . eproduced f r o m Reference 30 1 9

P i g . 2 1 Variation of Mach number error w i t h Mach n u m b e r . eproduced f r o m Reference 30 1 9

F i g . 2 2 A t y p i c a l calibration c u r v e f o r determination of true Mach number 120

F i g . 2 3 Determination of dynamic pressure f r o m t o t a l pressure

2 1

v i



P i g . 24 Control position transmitter and recorder

Pig.25 Theoretical e f f e c t s o n angle-of-attack measurements of u p w a s h f r o m t h e nose boom a n d f u s e l a g e a t l o w s p e e d s . eproduced f r o m Reference 30

Page

1 2 2

1 23

Pig. 26 nfluence of light-path urvature n ane ndications f angle-of-attack 24

Pig.27 ime-vector olution or orrecting ngle-of-attack ecords o the enter-of-gravity f he ircraft 2 5

Pig.28 pherical low-direction ensor. (Note ocation f pitch nd ya w eaction-control nozzles f he irplane.) 2 6

Pig.29(a) Magnetically amped ngular-velocity ecorder 27

Pig . 29(b) Detail f inkage nd amping ystem 28

Pig.30 nfluence f nterference ngu lar elocity "q" ate) bout pin reference xis f ensitive ate yr o 29

Pig.31 quations or orrecting ate yr o ecords or nstrum ent m i s a l i ne m e nt 3 0

Pig.32 Equations or orrecting ecords f inear ccelerometers o he center-of-gravity f he ircraft 3 1

Fig.33 hart or orrecting ensing-recording ircuit f nstrum ent for hase ag 3 2

Pig.34 hart or orrecting ensing-recording ircuit f nstrum ent or

dynamic mplification 3 2

Pig . 3 5 Functional chematics f pulse od e m odulat ion P C M ) ata-

acquisition y s t e m s 3 3

Pig.36 esults f nalysis f light a ta n egion f apid hanges n aircraft haracteristics 1 3 4

Fig.37 ariation f he period f n - 1 0 0 eries irplane s unction of Mach umber, altitude, angle-of-attack, and oa d actor from

Reference 1 ) 3 5

Pig.38 nfluence f ngle-of-attack nd oa d actor pon ateral characteristics f ne ircraft 3 6

Fig.39 omograph or planning light est onditions n nvestigating aeroelastic ffects n tability nd ontrol 1 3 7

n

I

«

vii

Pag e

Flg.40 Com parison of ateral-directional esponses o different ypes of udder nputs 3 8

Pig.41 Typical ime histories of he ongitudinal esponse haracteristics of he est irplane esulting rom brupt tabilizer deflection 3 9

Fig. 4 2 Typical im e histories of he ateral nd directional esponse

characteristics of he est irplane esulting rom brupt a w - damper deflection 4 0

Pig. 4 3 Comparison f ings-level nd onstant-heading ideslips 41

Fig. 4 4 Relation f mall-perturbation olling elocity nd cceleration vectors o mall-perturbation oll-displacement ector n transient scillation 4 2

Fig. 4 5 De term inat ion f period nd ha s e ngles rom ree-oscillation ata 1 4 3

Pig.46 Determ inat ion f im e - to-da m p o ne-half m plitude nd m p l itude ratios rom ree-oscillation ata 4 4

Pig. 4 7 Vector olution of Aa /|A<x| using pitch ate s a se or am pl itude atio hen ngle-of-attack ecords re unavailable 45

Pig.48 Vector solution f |Aat/|A/?| using aw ate s base or

amplitude atios when ideslip ecords are unavailable 46

Pig. 49 Typical etermination of flight uantities or he evaluation

of ongitudinal ontrol erivatives 47

Pig. 5 0 A graphical ime-vector solution or C N( X an d C N + N(3t) 48

Pig. 51 Time histories of ongitudinal ulses performed n he X-15

analog with he tability augmentation ystem engaged nd

disengaged from Reference 2) 49

Pig.52 Variation of g ift oefficient nd orresponding rim angle- of-attack with Mach um b e r from Reference 3 ) 50

Pig. 53 Longitudinal eriod nd amping characteristics of he D-558-II

airplane as unctions of Mach um b e r nd altitude 51

Pig.54 Variation of static nd ynamic ongitudinal tability erivatives

of he D-558-II irplane with Mach umber from Reference 3 ) 52

Pig. 55 Comparison f light-determined horizontal-tail ffectiveness with

wind-tunnel esults 53

Pig. 56 ypical etermination f light uantities or he evaluation of lateral ontrol erivatives 54

viii



Pig . 57 Comparison of Cn^ as etermined y evera l different pproximate methods ith he im e-vector method

F ig . 5 8 Comparison f esults f etermining Cu by im e-vector method and teady-sideslip quations

Fig .59 ypical raphical im e-vector olution f yawing nd olling stability derivatives

F ig . 60 esults f raphical im e-vector nalysis f he ffects of power on he ateral-directional eriod, amping, nd tability erivatives of th e -558-II esearch irplane from eference 3 )

Fig .61 rid plot se d o race ource f ncom patibil ity etween light and ind-tunnel ata

F ig . 62 ypical im e ictory f maneuver o etermine derivatives y least quaring he quations f motion Reference 8)

F i g . 6 3 Lateral-directional erivatives etermined y east quaring he equations f m otion, a s er Reference 8

F i g . 6 4 Typical nalog-match f recovery-from -sideslip" maneuver of a n xperimental ircraft. M .84 ltitude 49,400 t

Fig . 6 5 nfluence f lexibility nd ir ntake o ngine n he directional tability erivative , Cn«

Fig .66 omparison ith light ata f esults f nalog s im ulation studies f 6 0 ° olls sing l ight-determ ined derivatives

Page

155

155

156-157

158-159

1 60

161

1 6 2 - 1 6 3

164

165

166

A

ix

NOTATION

The body s y s t e m of a x e s , r a d i a n m e a s u r e , and foot-pound-second units a r e used throughout t h e paper unless specifically stated o r indicated o t h e r w i s e . asic s i g n conventions a r e s h o w n i n Figure 1 . a Section 1 , n which a number of a x i s systems a r e c o n s i d e r e d , t h e subscripts a r e used t o denote quantities referred t o t h e various s y s t e m s except f o r the quantities referred t o t h e b o d y system of a x e s . h e subscripts f o r these quantities a r e omitted f o r convenience e x c e p t t o i d e n t i f y c o o r d i n a t e s , b , y b , a n d b .

perpendicular distance f r o m s p r i n g t o k nife e d g e ( F i g . 1 3 ) , f t

a, A polynomial coefficients Section )

cross-sectional area of air-intake uct of et ngine at entrance, t2

cross-sectional area of et-exhaust uct of et ngine at exit, ft2

ax -afa

n longitudinal, lateral, nd normal accelerations of he aircraft at he center f gravity elative o he ody ystem

of xes; ositive orward, to he ight, and p, respectively, g units

axl,ati,Bni recorded values of ax at and aa respectively; corrected or hase ag nd misalinement ut no t or ocation elative o he center of gravity, g units

b

b . B

c

c , C

C

w i n g s p a n , t

polynomial coefficients ( S e c t i o n 3 )

mean aerodynamic c h o r d , t

p o l y n o m i a l coefficients ( S e c t i o n 3 )

s p r i n g c o u p l e ( S e c t i o n 4 ) , f t l b

coefficient of a x i a l f o r c e a l o n g t h e b o d y x - a x i s ; positive t o t h e r e a r , X/qS

phugold damping c o e f f i c i e n t , - + 2C„ 3u co s a co s ß

< C o> u'P contribution of ower o hugoid amping coefficient,

y31 + _2C1_

B u cosacos/3

x

Cc a

Cc. 3C „ ac

B — 2 V

2 k B*°

2 V

"CS. c

38.

C La

CLS,

c.,(c,)8.(c,)w ,(cI)0

ci„

drag oefficient; oefficient f xial orce long he stability -axis, ositive o he ear, -X s/q S

lift oefficient; oefficient f ift orce long he

stability -axis, ositive p, -Zs/q S

W qS

lift-curve lope, Bc L /B<x

B2V

B*2 V

BS_

coefficient o f rolling-moment a b o u t t h e b o d y , s t a b i l i t y , w i n d , a n d principal x - a x i s , r e s p e c t i v e l y , ( r o l l i n g moment)/qSb

damping-in-roll d e r i v a t i v e , B e ,

B 2V

ft

ch B e _l_

r b

2V

x i

clß

C lß

c

cm l(ca)s(cm )m .(cm)0

«Vp

<C.ap

effective dihedral d e r i v a t i v e ,

«VP

L«s ,

3 Cj

18

2 V

pitchir-g-moment oefficient bout he ody, tability, ind, and principal y-axis, espectively, (pitching moment)/^Sc

pitching-tnooent oefficient bout he erodynamic enter

contribution f power o itching-moment oefficient

3 CB

longitudinal-stability erivative , -~da

contribution of power to longitudinal stability (Equations ( 3 5 ) and ( 4 4 ) )

2C_

«V ,

ac 2 V

I3L 2 V

V —Ä + B u cosacos/3

refer o quations 47), (48), and 4 9)

norm al-force oefficient, coefficient f orce parallel o body -axis; ositive p , - Z / q T S

contribution f power o orm al-force oefficient

xii

4= r

C N c 3 c ,

3a

(CNJI variation of contribution of C N)p i t h a n g l e o f attack

( C Na h . t . variation of normal-force coefficient of horizontal t a i l with l o c a l angle of a t t a c k a t t h e t a i l ; coefficient based

on horiz ontal-tail area and l o c a l dynamic p r e s s u r e , -Zh.t. gh.t.Sh.t.

>%.t.

(C N (X)v.t. variation of coefficient of orce normal o vertical tail with vertical ngle of attack of vertical ail; oefficient based n vertical-tail area nd ocal ynamic pressure,

( Yv-t- )

Viy.tA.t./

'Vt.

C N & 25L 3

2V

CN,

3CN

3 H i 2V

CNU longitudinal hugoid static-stability derivative,

3 C „ 2C « 3 u co s a co s ß

3S„ C „.<C

n>s-<Cn>w<Cn>o yawing-moment coefficient bout ody, stability, wind, nd

principal -axis, respectively, (yawing moment)/qSb

- n / 3

3Cn

static directional-stability d e r i v a t i v e , ~öß

ri ß ö C , n

7 * 2 V

- n r 3 C „

r b

2 V

x i i i

3 C u«.p

Cn j

Cns.

<Cx>.

C y.(Cy)8(Cy)w.(Cy)0

cJß

(Cy^Jp

C*P

3<* 2V

3 8,

thrust oefficient, (thrust)/qS

3u

coefficient f xial orce long he ind -axis, qS

side-force oefficient arallel o ody, tability, ind, and principal -axis, respectively, C = C_ )8 (Side orce)/qS

3 /3 contribution f ower o C y

2V

3 C „

3* 2 V

3 2V

<Cz>w

d . D

e

e , E

coefficient of f o r c e along the wind z - a x i s , —- qS

polynom ial oefficients Section )

2.178

polynom ial oefficients Section )

xiv



K

acceleration f ravity, ft/sec2

B n 1 - sin£. V ec

g - os 6 sin 0, —V ec

h

H

H x.H y,H z

altitude, ft

angular-momentum ector f otating m a s s , Ir . b ft sec

angular omentum f H about x body xes, respectively, lb t ec

moment f nertia f otating m a s s f ngine bout ts rotating xis, slug t2

Wz

Ix o ,Iyo,Iz

o

Ix s 'Iy8

,Izs

Ix r I y r.Iz r

moments of inertia of aircraft a b o u t , y , z b o d y a x e s , r e s p e c t i v e l y , l u g f t

2

moments o f inertia of aircraft a b o u t , y , z principal a x e s , r e s p e c t i v e l y , s l u g f t

2

moments o f inertia of aircraft a b o u t , y , z stability a x e s , r e s p e c t i v e l y , s l u g f t

2

moments of inertia of aircraft about x , y , z reference a x e s , r e s p e c t i v e l y , s l u g f t

2

moment of i n e r t i a of c r a d l e supporting aircraft ( S e c t i o n 4 ) , s l u g f t

2

*; i xz

k

K

product f nertia f ircraft eferred o ody - nd z-axes, slug t2

stability-augmentation-system ain, sec

linear pring onstant, lb/ft

correlation onstant Section )

torsional pring onstant, 2 K ga2 ft b/rad

xv



—

I

L.M.N

distance as defined l o c a l l y a t time of d i s c u s s i o n , f t

r o l l i n g , p i t c h i n g , a n d yawing moments a b o u t b o d y , y , z a x e s , r e s p e c t i v e l y , f t l b

LJ,II|,NJ i n e r t i a l r o l l i n g , p i t c h i n g , a n d yawing moments a b o u t t h e respective body a x e s , t l b

Lr.» M rB'Nr. rolling, itching, and yawing moments ue o yroscopic action of otating m a s s f ngine, ft b

( L )8, ( I I )8, ( N )S r o l l i n g , p i t c h i n g , a n d yawing moments a b o u t t h e stability x , y , z a x e s , t l b

r o l l i n g acceleration a b o u t b o d y x - a x i s , r o l l i n g moment)/Ix , 1 / s e c

2

BL qSb2

= —= C ; . f t l b s e c Bp ' P V

= l, q S b2

2 V I .

1

s e c

CP

« ?

i,

>+1L„ + -*£ N „

1 I 7 '

_ xz

hh

BL q Sb2

B r = C'rl7

q Sb2 1 clr 2 VIX sec

Cr X

s e c

, f t l b s e c

x z

1

s e c

BL —= Cj^qSb. t l b

= C q S b 1

p Ix sec'

L« a

BL —- = C js qSb, t l b BS » a

x v i



L $r

L sr

m

ma

mi

M

M ,

M S ,

8 .

qSb

s e c

Cj j qSb. ft lb

qSb

x

mass of airplane, /g slugs

mass rate of air intake of jet engine, slugs/sec

mass rate of jet exhaust, slugs/sec

Mach number

ind jated Mach number

Ditching acceleration about body y-axis,

(pitching moment)/I 1/sec2

ro^2

= C„ q S c M

Bq " *" « V

qSc*

, t lb sec

" 1 2VIy ' sec

3M qSc

B u =

C f f l u T'

l b SCC

Cm q S c

u y sec

2

BM = —= C^qSc , t lb

B«

= Cm

q S c

—I y e e '

BM q S c2

B T=Cm (i1

f t l b s e c

q S c * 1 Cm- , a 2 V Iy s e c

BM

e f t l b

L

*

xvii



N

qSc l

yawing cceleration bout ody -axls, yawing m om ent) / I 1/sec2

BN Sb * = r-=C

3p CP 2V , ft b ec

qS5 z 1 aP V I, sec

2

*P 1 iz sec

BN Sb2

= nr ft b sec » on

= c ,

Br ' 2V

qS b2 1 nr

2V I2 sec

K *'*h1

'

1 - xz sec

BN = = n/3qS b ft b

'-n/J " i ' S b 1

Iz sec'

N $rBN ~= Cn8rqS b ft b

R « i - n s qSb 1

r z s e e r

Ns,

N sa

BN —=C n5aqSb . f t l b

q S b - n g

z e e2

^ 1

x v i i i

p.q.r

Wo

P8' V r8

p.q.r

P 0>c» o 'ro

p

p'i

Pi-Pj

PT

p

q

< > c

q ct

Vt.' V.t.

R

R

rolling, itching, nd awing velocities, respectively, about ody xes, ad/sec

rolling, itching, nd awing velocities, espectively, about principal xes, ad/sec

rolling, pitching, and awing elocities, respectively, about tability xes, ad/sec

rolling, itching, and awing ngu lar ccelerations, respectively, about ody xes , rad/sec2

rolling, pitching, and awing ccelerations, respectively, about rincipal xes, rad/sec2

static pressure, lb/ft2

Indicated tatic pressure, ncorrected ecorded pressure, lb/ft

2

static pressures cting cross nlet f ir ntake nd

exhaust, respectively, of et ngine, lb/ft2

stagnation pressure, qc lb/ft2

period of hort-period oscillation, sec

dynamic ressure, i£v2 lb/ft2

im pact ressure; dynamic pressure f ompressible lo w

(Equations 98) nd 99)), lb/ft2

indicated m p a c t ressure, lb/ft2

dynamic pressures t he orizontal nd ertical ail, respectively, lb/ft2

instantaneous adius f urn Pig. 27), ft

reaction orce Section ), lb

1 Laplacian perator, a cu

s e c

° i n

S

^.t.'^-.t.

t

instrument sensitivity

wing a r e a , t2

horizontal- a n d vertical-tail a r e a » , e s p e c t i v e l y , t:

t i m e , e c

x i x



1/2

1 1 T"~* T

TR 'T8

U.V.W

U.V.»

A V

Vi

VJ

w

" c

x . y . z

X.Y.Z

Xg. Yg.Zg

thrust due to p o w e r , lb

time required f o r absolute value of transient short-period oscillation t o damp to one-half a m p l i t u d e , s e c

roll-subsidence and spiral-divergence r o o t s , r e s p e c t i v e l y ,

of t h e lateral-directional characteristic e q u a t i o n , —s e c

roll-subsidence a n d spiral-divergence time c o n s t a n t s , sec

q(s) pitch-attitude im e onstant n numerator f

transfer unction, s e c Ms

> linear velocities relative to body i , y , z a x e s , r e s p e c t i v e l y , ft/sec

linear accelerations relative t o b o d y , y , z a x e s , r e s p e c t i v e l y , t / s e c

2

Au

V

AÜ

V

a i r s p e e d , ft/sec

velocity of i n t a k e a i r a t a i r i n t a k e o f j e t engine f t / s e c

velocity of j e t e x h a u s t , t / s e c

weight of a i r c r a f t , l b

w e i g h t of c r a d l e ( S e c t i o n 4 ) , l b

distances f r o m t h e center of gravity a l o n g b o d y , y , z a x e s , r e s p e c t i v e l y , t

forces along t h e body , y , z a x e s , r e s p e c t i v e l y ; positive f o r w a r d , o t h e r i g h t , and d o w n , e s p e c t i v e l y , l b

components of gravitational f o r c e acting along t h e body , y , z x e s , respectively

3 x _ qS —= -CCuT b s e c / f t

x x



-~~r

II

-Ccu

q S

m V

1

sec

3 x

3a = -Cca Ö S . lb

qS 1 - cc0

m V sec

3 X

38 -Ccsä * lb

lv.t.

lf

lß

n o .

3Y

= -c, qS cs

3 Y _

3 /3

m V sec

«Wv.t.Qv.t.Sv.t. lb

= Cy^S lb

qS

^ V sec

lateral orce n plane f propeller isk ue c propeller, lb

CySq S lb

• • ii

t

= C ys m V sec

3 Z qS -— = - N — lb ec/ft 3 u V - qS 1

" NU T — V sec

3z Sc — - = - N„

lb ec 3 q q 2 V

= " N0 rqSc

2mV 3 Z —= - NaqS lb

xxi

Ö S

( Z ) p

Vt. a

( a )p

4 r

r

Sa

s .

- CN<x— —

3 z

mV s e c

q S c • = - CN ;

q S c

2 V , l b s e c

-C«-

contribution o f propulsion s y s t e m t o Z o r c e , l b

- ( C Na h . t . < i h . t .s

h . t . . l b

angle of a t t a c k of a i r c r a f t

change i n d u e t o influence of flight-path curvature

a n g l e of a t t a c k of a i r c r a f t f o r zero Z o r c e

a n g l e of a t t a c k of thrust l i n e relative t o airstream velocity a t propeller or a i r i n t a k e of j e t e n g i n e ; thrust line considered parallel t o axis

maximum positive or negative angle of attack obtained i n a

r o l l maneuver ( F i g . 6 6 )

sideslip angle

r a t e of c h a n g e of ß w i t h t i m e , rad/sec

flight-path angle relative t o horizontal

adiabatic c o n s t a n t

a i l e r o n d e f l e c t i o n ; positive w h e n l e f t a i l e r o n i s deflected d o w n

iß elevator d e f l e c t i o n ; positive w h e n trailing edge i s deflected d o w n

rudder d e f l e c t i o n ; positive w h e n trailing e d g e i s deflected t o t h e l e f t

> rß

dS,

iß

x x i l

sp

r

£ph

V

angle ncluded etween eference x-axis nd plane of spring

couple Fig.15)

increment

a n g l e between body x - a x i s a n d principal x - a x i s ; positive w h e n reference i s a b o v e principal axis a t the nose

upwash a n g l e a t t h e propeller d u e t o s u c h f a c t o r s a s f u s e l a g e and wing

short-period r a t i o of a c t u a l d a m p i n g t o c r i t i c a l damping

phugoid r a t i o of a c t u a l t o c r i t i c a l damping

i n s t r u m e n t damping r a t i o

a n g l e of inclination of p r i n c i p a l x - a x i s r e l a t i v e t o stability x - a x i s ; positive w h e n principal x - a x i s i s a b o v e stability x-axis a t t h e nose

pitch attitude of angular-velocity v e c t o r , , of rotating

mass of engine relative t o b o d y x-axis

"b r e l a t i v e aircraft d e n s i t y , pS b

P a

relative aircraft d e n s i t y , pS c

absolute viscosity, lb sec/in"2

mass density öf ir, oiugs/ft3

real part of Laplacian operator, s a + w

t ime arameter , -— se c p\S



t ime constant or simplified stability ugmenta t ion ys tem, sec

yaw, pitch, and oll, Euler orientation ngles, respectively, (In eneral ircraft motions, they are normally he orientation ngles of he aircraft ody axis ystem o spatial (earth) eference ys tem. In nstrument l inement, they

refer o he misalinement f he nstrument eference axis system o he aircraft ody axis ys tem. )

xxiii



« A 0.0

$,

$. ij

"n'^d

"»in

n

IJ I

[L ]

[L] 1

[ L ]8[Ub

w. ( b( s( w.( 0

()„

rate of rotation of t h e Euler orientation a n g l e s , rad/sec

y a w , p i t c h , a n d r o l l Euler orientation angles of aircraft body axis system during small perturbations relative t o body axis system preceding t h e perturbations regardless of aircraft attitude preceding perturbations ( F i g . 4 ( b ) ) , A t / / ' 2 i /Ardt , Ö' ~ jAqdt . >' jApdt

damping a n g l e

phase a n g l e of vector quantity elative a vector quantity

undamped a n d damped natural f r e q u e n c i e s , r e s p e c t i v e l y , of the a i r c r a f t i n short-period modes of o s c i l l a t i o n , rad/sec

undamped a n d damped natural f r e q u e n c i e s , r e s p e c t i v e l y , of t h e a i r c r a f t i n phugoid modes of o s c i l l a t i o n , r a d / s e c

undamped n a t u r a l frequency of i n s t r u m e n t , r a d / s e c

angular r a t e of rotation of rotating mass of e n g i n e , rad/sec

absolute magnitude of a vector quantity ; a l w a y s positive

transformation matrix

inverse transformation matrix

transform matrices t o transform vector quantities f r o m reference axis s y s t e m t o a i r c r a f t stability a n d b o d y a x i s s y s t e m s , respectively

transformation matrix representing transformation f r o m body t o stability a x i s s y s t e m

relative t o b o d y , t a b i l i t y , w i n d , a n d principal a x e s s y s t e m s , respectively

contribution d u e t o p o w e r

xxiv



r-

CONSIDERATIONS IN TH E DETERMINATION OF STABILITY AND CONTROL DERIVATIVES AND DYNAMIC CHARACTERISTICS FROH FLIGHT DATA

Chester H.Wolowicz

1 . INTRODUCTION

T h e determination of stability and control characteristics f r o m f l i g h t data i n t h e f o r m of derivatives and other behavior parameters h a s become a n important part of f l i g h t t e s t i n g . s new concepts i n airplanes a r e developed or the airplane f l i e s i n new Mach a n d altitude r e g i m e s , t h e r e i s t h e need to verify theory a n d wind-tunnel data and t h e various influences on stability c h a r a c t e r i s t i c s , t o provide i n f o r m a t i o n not obtained i n wind-tunnel s t u d i e s , a n d t o uncover t h e s o u r c e s of discrepancies between prediction a n d a c t u a l f l i g h t b e h a v i o r . h e r e wind-tunnel data a r e unavailable o r where s a f e t y of f l i g h t i n t o untested regions i s of c o n c e r n , flight-determined derivatives have been extrapolated t o predict airplane behavior prior t o f l i g h t i n t o t h e s e r e g i o n s .

Because o f t h e exploratory nature of many of t h e i n v e s t i g a t i o n s , t h e practical aspects of determining derivatives and other behavior p a r a m e t e r s , u c h a s oscillatory c h a r a c t e r i s t i c s , r o m f l i g h t data a r e v e r y i m p o r t a n t . xperience h a s s h o w n t h a t a maximum appreciation and understanding of t h e practical a s p e c t s i s attained w h e n background knowledge i n c l u d e s a n understanding of axis s y s t e m s , t r a n s f o r m a t i o n s , t h e equations of motions and the limitations of t h e e q u a t i o n s , techniques used t o determine t h e mass characteristics of the a i r p l a n e , the installation a n d behavior of f l i g h t test I n s t r u m e n t a t i o n , l i g h t t e s t t e c h n i q u e s , and t h e t h e o r y a n d limitations of techniques used to determine the stability a n d c o n t r o l characteristics f r o m f l i g h t d a t a .

Although s o m e o f t h e f a c t o r s mentioned a b o v e , u c h a s a x i s s y s t e m s a n d transformations a s w e l l a s aspects of t h e equations of m o t i o n , may b e f o u n d i n t e x t b o o k s , t h e treatment i s generally n o t oriented t o w a r d f l i g h t t e s t i n g . o m e of t h e techniques used i n d e t e r - mining stability and c o n t r o l characteristics may b e found i n technical r e p o r t s ; h o w e v e r , limitations of t h e techniques occasionally may not b e s h o r n . his paper attempts t o bring a l l t h e f a c t o r s together t o provide a r e a d y reference of pertinent i n f o r m a t i o n . I t i s , i n f a c t , a greatly expanded version of AGARD Report 2 2 4 * .

I t i s t h e purpose of t h i s paper t o discuss t h e various f a c t o r s t h a t influence t h e determination of stability a n d c o n t r o l derivatives a n d other behavior characteristics f r o m f l i g h t d a t a . ncluded are illustrations of t h e application of f l i g h t derivatives t o verification o f predictions a n d t o determination of aeroelastic e f f e c t s , stability c r i t e r i a , and f l i g h t g u i d a n c e . his paper i s intended not o n l y f o r the practical engineer w h o i s w o / k i n g w i t h f l i g h t data b u t a l s o f o r the scientist w h o i s attempting t o develop n e w , sophisticated a n a l y t i c a l t e c h n i q u e s .

* StabilityDerivative Determination ro m Flight Data y Chester H.Wolowicz a n d Euclid C.Holleman,

O c t o b e r 1 9 5 8 .

Acknowledgement f nvestigators hose ork as directly ontributed o he resent paper s m a d e n ach ection. It s recognized that a n y noteworthy orks of ther Investigators re not eferenced.

2 . AXIS Y S T E M S N D O O R D I N A T E R A N S F O R M A T I O N S

2 .1 Axis ystems

In he tudy f th e ynamics f he irplane, as a n y s ix rthogonal xis ys tems m a y e se d im ultaneously. A n nderstanding of heae ys tems r eference rames nd their relation o he ircraft nd ts m otions t arious light onditions s ssential to th e roper nalysis f light ata. Although omprehensive reatment f xis sys tems m a y e ound n eference , brief reatment of he xis ys tems s resented in his ection.

2.1.1 Body Systems

The ody xis ystem xb, b. b) s ody-fixed ith ts rigin t he enter of gravity f he irplane. Th e xb axis s lways parallel o he u se l age eference line nd h e n he enter f ravity s n he lane of ymmetry, s t ormally s, both he xb and zb axes re n he irplane' s plane f symmetry, s h o w n n Figure . T h e yb axis s ormal o he lane f ymmetry; hus, he ody ystem f axes s ngularly nvariant ith e spect o he ircraft tructure.

Because f ts ngular nvariance ith e spect o he ircraft, the o d y xis ystem is an xcellent rame f eference or mounting light est nstruments. T h e rientation of he light est nstrum ents nd heir onsequent utput elative o he ody xe s - especially he inear ccelerometer nd ngular a te nd cceleration ensors - m a k e t convenient o etermine , f r o m light ata, stability nd ontrol arameters ith e spect to his eference raoe. Aside r o m onvenience, his eference rame s he ogical frame bout hich o rient a t e s , ccelerations, and he tability n d ontrol arameters n he tudy f handling-qual ity riteria, inasmuch s he rientation f th e pilot s nvariant elative o his rame.

s * ' s * z s) < s a s p e c i a l case of t h e body axis s y s t e m .

2.1.2 Stability System

The stability a x i s system ( x , Like t h e b o d y s y s t e m , t h e 8 n d g x e s a r e i n t h e plane of s y m m e t r y w h e n t h e center of g r a v i t y i s i n this p l a n e , a n d parallel t o t h e p l a n e o f s y m m e t r y w h e n the center of gravity i s not I n t h e p l a n e . nlike t h e body s y s t e m , h o w e v e r , t h e g nd zs x e s a r e a n g u l a r l y v a r i a n t relative t o t h e f u s e l a g e reference l i n e . he g x i s i s perpendicular t o t h e r e s u l t a n t velocity vector a n d t h e fl x i s 1s a r a l l e l t o t h e component of t h e resultant velocity vector projected onto t h e plane o t s y m m e t r y , s h o w n i n Figure 1 .

as

T h e mportant arametric elationship etween he ody n d tability xe s yst ems is he ngle f attack, a hich s he ngle etween he x and xb axes F i g . 1 ).

T h e tability xis ystem s o m m o n l y se d n theoretical ubsonic erodynamics nd

subsonic ind-tunnel orce nd m o m e n t nvestigations. It s lso mployed, n ccasion, in lace f ody xe s n light est nvestigations f ongitudinal tability nd control haracteristics.



2.1.3 Principal System

T h e principal a x i s s y s t e m ( z r, y0 z 0) defines t h e natural axes of rotation of t h e a i r c r a f t . hey a r e t h e a x e s which r e s u l t i n maximum a n d minimum moments of i n e r t i a . The orientation of this axis s y s t e m i n the aircraft i s a function of t h e mass distribution o f the aircraft a n d w i l l remain f i x e d a s l o n g a s t h e mass and mass distribution r e m a i n f i x e d . hen the l a t e r a l distribution of mass i s symmetrical relative t o the plane of s y m m e t r y , which i s generally the c a s e , t h e y0 xis w i l l coincide w i t h the yb a x i s , and t h e 0 nd zQ axes w i l l l i e i n the plane o f s y m m e t r y , a s s h o w n i n Figure 1 .

T h e inclination of t h e Q xis ( P i g . 1 ) t o t h e x i s of t h e reference axis s y s t e m ( g e n e r a l l y body axes i n f l i g h t t e s t investigations) has a d i r e c t bearing on t h e i n e r t i a l moments experienced about t h e reference axes & & reflected i n the product of inertia t e r m J2 n t h e equations of motion a n d , h e n c e , on t h e l a t e r a l stability of t h e a i r p l a n e .

When t h e principal axes are used a s reference a x e s , as they occasionally a r e i n theoretical a n d simulator i n v e s t i g a t i o n s , they a r e used t o simplify t h e equations of motion b y the elimination of t h e xz t e r m .

2.1A Kind System T h e wind a x i s system i s related t o the resultant velocity vector and t h e plane of

s y m m e t r y of t h e a i r p l a n e . s s h o w n i n Figure 1 , t h e w xis i s parallel t o t h e resultant velocity vector and l i e s i n t h e transverse plane of t h e stability axes ( x sy8 p l a n e ) . o n s e q u e n t l y , t h e w xis i s coincident with t h e g a x i s . he w

a n d yw axes coincide w i t h their respective counterparts g nd yg h e n t h e a i r c r a f t h a s z e r o s i d e s l i p .

T h e important parameters associated w i t h t h e wind system a r e t h e sideslip a n g l e , ß and t h e angle of c l i m b , y basic definition t h e angle of s i d e s l i p , i s t h e a n g l e between t h e , x i s a n d t h e plane of s y m m e t r y and thus l i e s i n t h e transverse stability a x e s p l a n e , a s s h o w n i n Figure 1 . t should b e noted t h a t n o t a l l / 3 - s e n s o r s necessarily measure this ß this w i l l b e discussed i n Section 5 on " I n s t r u m e n t a t i o n " . h e a n g l e of c l i m b a l w a y s l i e s i n the v e r t i c a l p l a n e and i s t h e

a n g l e included between t h e w x i s a n d t h e horizontal p l a n e .

2.1.5 Spatial Reference System T h e preceding axis s y s t e m s a r e tied i n w i t h t h e plane of symmetry of t h e airplane

w i t h t h e i r origins a t t h e c e n t e r of g r a v i t y ; a s s h o w n i n Figure 1 . o complete t h e systems of a x e s u s e d , a t l e a s t one i n e r t i a l , p a c e - f i x e d , axis system i s r e q u i r e d . I n dealing with g e n e r a l motions o f a i r c r a f t , t h i s s p a t i a l s y s t e m i s generally earth- referenced t o describe t h e motion o f t h e airplane w i t h r e s p e c t t o t i m e f o r s h o r t time i n t e r v a l s . uch a situation i s indicated i n Figure 2 , w h i c h shows t h e relationships of t h e various a x i s systems previously described a n d t h e relationship o f t h e body a x i s s y s t e m w i t h r e s p e c t to t h e s p a t i a l r e f e r e n c e ( x r, y r, zr) . hown i n t h e f i g u r e a r e f l i g h t p a t h y a n g l e of sideslip ß a n g l e o f a t t a c k , a s w e l l a s t h e Euler

orientation a n g l e s , / ' , 6 and > of t h e a i r p l a n e ' s body axes relative t o the spatial a x i s s y s t e m . his i s s h o w n i n a m u c h s i m p l e r f o r m a t i n Figure 3 . h e sequence of rotations of t h e Euler a n g l e s i s i m p o r t a n t . e n e r a l l y , t h e sequence of rotation i s

y p and p this means t h a t t h e airplane i s initially y a w e d , then p i t c h e d , a n d finally r o l l e d .

I t should b e noted that y - 6 - a only w h e n t h e aircraft i s unbanked ( < t > = 0 ) .

2.1.6 Perturbation Reference rames

I n using perturbation theory i n stability a n a l y s i s , Euler angle perturbations may be considered t o be superimposed o n t h e unperturbed a n g l e s , a s s h o w n i n Figure 4 ( a ) , with the result that the perturbed angles are p + A i / / , 8 A#, and / > + A < / > , or t h e y may b e based o n a secondary s p a t i a l reference f r a m e w h i c h i s t h e unperturbed airplane axis system ( x t > 0 yb0 Zb0) . n Figure 4 ( b ) t h e unperturbed body axes constitute the secondary s p a t i a l reference f r a m e a n d a r e oriented t o t h e basic s p a t i a l reference f r a m e through t h e angles p and p o w e v e r , the perturbed planes a r e oriented to t h e secondary s p a t i a l reference plane b y t / ; ' , Ad' and A0 ' w h i c h generally a r e not t h e s a m e a s « / » , A0 , a n d A0 .

2.2 Coordinate Transformations

Coordinate transformations a r e used s o f r e q u e n t l y i n dynamic studies of aircraft that some consideration should b e g i v e n t o this s u b j e c t . iterature o n transformations is extensive and ranges f r o m t h e c l a s s i c a l mathematical treatments (Reference 2 , f o r e x a m p l e ) t o engineering applications (References 3 and 4 , o r e x a m p l e ) . t this t i m e , the most pertinent transformations a r e considered t o s e r v e a s guidelines f o r other transformations t h a t may b e d e s i r e d .

2. 2.1 Transformation ro m Earth Reference

Axes o Airplane Axes

Consider Xr , Yr , ordinates y z .

a n d Zr s generalized vector quantities acting along the c o - , r e s p e c t i v e l y . h e transformed vector quantities , Y , Z

acting along b , yb , a n d z b x e s , r e s p e c t i v e l y , a r e obtained b y performing t h r e e successive r o t a t i o n s , / / , 8 and > t o define t h e a i r p l a n e ' s orientation with r e s p e c t t o t h e reference a x e s r y r , and z r hrough a transformation matrix L ]

a s follows

Y \~

Y = [L ]

Y

r Z _

Zr =

& ]

(la)

1 0 cos 

0 -sin 4 > cos t>

L cos # cos l/ /

sin 4 > sin 8 cos \ p -sini/'cost/j

cos i f » cos 4 > sind . +sin</'sin<£

-sin 8

0

cos

" c o s 8 0

0 sin 0

cos 8 sin«/ '

sin i /' sind sin 

s ini />cosc />sin 8 -cosi />s in<£

cos /» in 0 -sin 0 os /*

0 -sin 8

sin-/>cosö

cos <£^os6>

(lb)

(lc)

2.2.2 Transformation rom Airplane Axes o Earth Axes Since projection rom airplane xe s o earth xe s s n nverse process of he

preceding ransformation, premultiplication of Equation la) y he nverse ransformation matrix [L]"

1 results n

" Xr x

Yr = [L]-1

Y

w Z

(2a)

H o w e v e r , i n c e t h e orthogonal projections o n t h e airplane axes are being transformed t o orthogonal projections on t h e e a r t h a x e s , t h e inverse of the transformation matrix [ L ] n Equation ( l c ) i s t h e s a m e a s i t s t r a n s p o s e ; t h u s

008 00081/)

cos#sini/>

-sin 8

sin t > sin Ocos^p -sini/icosc^

sin sin sin

+COS</>COS<£

sin sin 8 +sin</'sin<£

sin'/'cos^sinö

-cos i / > sine/)

co s 0co s 6 (2b)

2.2.3 Relationship Between Airplane Rates p q and r and Euler Rates \ p 8 and t>

It hould e ecognized rom Figure hat, although he airplane ate-vector quantities, p q and r are orthogonal, the Euler ate-vector quantities are not. Thus, to obtain he elationships of p q and r as unctions of \f i nd 4 > it s necessary o ransform \ p nd j> to omponents long xr r and zraxes nd he n pply Equation lc). The first ransformation s ccomplished apidly

by pplying Equation 2b ) nd considering ac h Euler quantity s special case of transformation f ody axis quantity. T o wit: in Equation 2b ) oth 4 > and 8 are

considered er o or \p and 8 and < £ is onsidered er o or 

-sin 8

- s i n i / '

cos^ ( 3 a )

Substituting Equation ( 3 a ) i n t o Equation ( l c ) results I n t h e f o l l o w i n g :

X P 1 0 -sin 8 '

Y = q = 0 COS0 sin 4 > c o s 8 e

Z r 0 -sin$ co s 8 co s 4 > t ( 3 b )

T o obtain t h e inverse of Equation , ( 3 b ) < f i t i s necessary t o s o l v e f o r t h e inverse of the transformation matrix since > 8 and ^ a r e n o t orthogonal and hence d o not

6

penult he use of th e ranspose or he nverse . This s ccom plished y olving or th e nverse matrix [L]"

1 n he elationship

10 0

[L ] & J"1 = 0 1 0 .

0 0 1

After olving or [L]"1 th e nverse f quation 3 b) s etermined o e

(4)

0 1 s in <£tan & cos0tani9 P

6 = 0 C O S 0 -sin seed cos<t>sec8 r

(5 )

2.2.4 Transformat ion of Euler Angles ro m he Body to he Stability Axis System

If wo different otation eries iv e he am e tarting nd nding orientation, he matr ices epresenting he otation eries re qual , e l ement or lement , In he wo transform ation m atrices . Thus, he uler ngles , i/ /g B nd 4 > s f he tability axes an e erived rom he Euler ngles , \ p b b nd 4^ f he ody xe s y he following ransform ation matrix elationship

w , .u . (6) where [L S is he ransform ation matrix of Equation lc) sing tability xis rientation ngles \p 8 & B and 4 > s in lace f 0 6 and 4 > and [L ] b is he a m e transform ation matrix sing ody xis rientation ngles \ pb & b and 4 \ > in lace of 4 > nd < p f he am e uccess ive otation eries s mployed. The ransform ation [a ]s is he matrix epresenting he ransform ation r o m he ody o he stability xis ystem , r

W . cos a 0 s in a

0 0 -sin a 0 co s a

(7)

U p o n erforming he matrix multiplication h o w n y quation 6), nd hecking corresponding l em ents n he quated esults o btain he ost easible l ements or th e esired esult, th e ollowing elationships re rrived t

sini9„

sin0s

sini//

co s a s in & b - in a co s 6b cos <^

s in < t \ y cos 6b

co s 6. cos acosöjjSinv^ in a sini/^, cos «^ sin & b - os</^ sin^j , )

C O S

(8)

2.2.5 Transformation of Aerodynamic Coefficients o

Var ious Axis Systems

The ollowing ransformations re ccom pl ished eadily y mploying quat ion 2 b) an d eplacing < / > and 4 > in he quation y -ß a . and 0 espectively. Thus, o ransform rom ody o tability xes, et ß hereby btaining

C„ = Cc cos a + N sin a

<CJ > s = < V

-Cc sin a N cos a

(C.) _ I = Cj os a n sin a

(9)

(10)

<C

B>s = C B

(Cn)s = -Cj sin a „ cos a

Similarly, o ransform rom od y o in d xes

(Cx)w Cc cosacos/3 y sin/3 - N sinacos/3

(C w c cosasinp y cosp N sinasin/5

(Czw CL = C c sin a - N cos a

(Cj)w j cosacos/S Bsn/3 n sin acos/0

(C B)W B cos/3 - j cosasin/3 - n sinasin/3

(Cn)w n cos a - j in a

Also, or tability o in d xes, et a obtaining

(Cx)w = -C„cos/3 Cy)8sn/Ö

(Cy)w = C Dsn/3 Cy)8cs/?

<Cz> w = -CL

(cl = (Cj)scos/3+ CB)8sn/S

«Vw = «V8 cos/3- Cj)8 in/3

T o transform f r o m wind t o stability o r b o d y a x e s , o r s t a b i l i t ) ' t o body a x e s , u s e i s

made of Equation ( l c ) .

( I D

2.2.6 Transformation of Derivatives The transformation of derivatives f r o m one axis system t o another goes beyond pure

kinematic transformations. ongitudinal derivatives a r e relatively simple i n their t r a n s f o r m a t i o n s ; lateral'directional derivatives are more complex i n t r a n s f o r m a t i o n s . The s e v e r a l examples w i l l illustrate t h e procedure to obtain derivative t r a n s f o r m a t i o n . Influence of factors such a s power i s not considered a t this t i m e .

Transformation of longitudinal derivatives i s accomplished by direct differentiation of the coefficient e q u a t i o n s . his is possible because and q are n o t modified by the axis system u s e d . or e x a m p l e , t o obtain the derivative of CL ith r e s p e c t t o

a n a transformation f r o m body t o stability a x e s , differentiate the equation f o r C , n Equation ( 9 ) o b t a i n i n g , on a per radian b a s i s ,

CLn -CCasin a + C N (xcos a - C„ ( 1 2 )

T h e transformation of the lateral-directional derivatives i s more c o m p l i c a t e d , i n a s - much a s t h e angular r a t e variables nd p r e affected b y t h e t r a n s f o r m a t i o n . t this t i m e , sideslip a n g l e , is n o t considered t o b e affected b y t h e transformations b e c a u s e of i t s d e f i n i t i o n ; h o w e v e r , t h e type of / 3 - s e n s o r used i n f l i g h t tests - whether i t b e a v a n e , floating c o n e , or b a l l nose - does have a bearing o n t h e interpretation of t h e ß r e a d o u t a n d the meaning of t h e derivatives with r e s p e c t t o t h e sensed ß T h i s i s discussed i n Section 5 .

CoiPider t h e transformation of lateral-directional derivatives f r o m t h e stability t o t h r b o d y a x i s s y s t e m . ransformation of t h e yawing a n d rolling moment equations i s a c c o n , ) 1 1 s h e d b y

N N )8 c o s a + ( L )ssina

L L )8cosa - ( N )s s i n a ,

where L and N represent olling nd yawing moments, espectively.

However,

( N )s

(D8 =

r«b

,n ., . ,n , P

8b A ,„ x s] < C aß)*ß + Cnr)8 -i- + Cn/3 )s—+ Cnp)8-A- Cn?)8S gSb

r 3b

(C iß)sß + Clr8 -i- Cnß)B — + C ip), A- + C /5)8S|gSb

(13)

(14)

It will e necessary o xpress Arg and Ap8 in Equation 14 ) s unctions of Ar and Ap using he ransform

r co s a - sin a

p co s a + sin a

(15)

: - ' M

9

Upon substituting Equation ( 1 5 ) i n t o ( 1 4 ) and Equation ( 1 4 ) i n t o ( 1 3 ) , a n d regrouping t e r m s ,

N

Üb

+ (C Er)8cs2a + Cjp)ssn2a+ CBn Cir3sn np wr "jrb

J V

ßb

pb

2V

and

KCn^sCOsa Ciß)asin a\ß +

+ KCn^scosa + C gslna —

+ (Cnp)8cs2a-C ir8sn2a- C„r - jp)8sin acos a

+ (Cnj)sCosa+ ClssSina

L o — = (Cl^s cos a - C n ^ g ) s sin a p

+ 'C /p)8cos2a + Cnr)8sn2oc - C„p jr8sinacos a —

T I 3 b + (Ci^s cos a - Cn/g)8sin a —

' ( 1 6 a )

[ < (C j Jscos'a- C„_)8sn<a- Cn - j 8 sin r -'p' -lrb

, a cos a — J V

[ ' (C ls8cosa- Cns8sn 2s.

( 16b)

Summari e s f ransformations f erodynam ic erivatives rom tability o ody xis sy s t e m , and ice ersa, a re iven n able s nd I.

2.2.7 Transformation of Mom ent s f Intrtia ro m

O ne Axis System o Anothe r

Although his opic s overed n pplied m echanic s iterature, an llustrative exa mp l e s iven s efresher. Also ncluded re ables of ransformations or ready eference.

T o btain IXg in e r m s f ody xes quantities, se s ade f he undamental

relation

i*s "<y.+ 8> * (17)

10

Substituting he ollowing ransform nto quation 1 7 ) ,

x g = x cos a + s in a

zs = z co s a - x sin a

and xpanding,

Ix B = [/(*£ Zjj) m ] co s2 a + /(y£ £) t a i l in2 a - r' xbzbd m | sin a co s a

= Ix cos2 a + I2 s in2 a - 2IJZ sin acos a

= i (Iz Ix) - i (Iz - Ix) co s a - IIZ 3 in 2 < x

( 1 8 )

3 . EQUATIONS OF MOTION

T h e equations of notion of a n airplane a s found i n t e x t s o n aircraft dynamics ( s u c h

as Reference 5 ) a n d a s normally presented i n t h e t e c h n i c a l l i t e r a t u r e , l t h o u g h prosaic

i n a p p e a r a n c e , d o c o n t a i n complexities i n the s i g n i f i c a n c e of the individual t e r m s .

T h e following discussion i s intended t o a c q u a i n t t h e reader w i t h the s c o p e of t h e

complexities which m a y be encountered and which should b e recognized and managed i n

dealing w i t h t h e equations of m o t i o n . n understanding o f t h i s matter i s important

i n applying t h e equations t o d e r i v a t i v e determination f r o m f l i g h t < < a t a .

3 . 1 n e r t i a l Quantities

I n a l l considerations of t h e i n e r t i a l portions of t h e e q u r n s of m o t i o n , t h e a x i s

system used has a d i r e c t bearing o n t h e expressions f o r i n e r t i a l f o r c e s ; t h e degree

of asymmetry of the mass distribution of t h e aircraft a n d the magnitude a n d violence

of t h e a i r c r a f t noti/ws a f f e c t t h e f o r m a t of the expressions f o r i n e r t i a l m o m e n t s .

I t i s a s s u m e d , f o r t h e purposes o f this p a p e r , t h a t t h e a i r c r a f t b e h a v e s a s a rigid

b o d y . here aeroelasticity i s a f a c t o r , i t i s assumed t h a t proper precautions w i l l

have been taken to provide assurance t h a t t h e rigid-body c o n c e p t w i l l provide a good

degree of a p p r o x i m a t i o n .

Inertial quantities a r i s e f r o m t h e I n h e r e n t a c t i o n of t h e a i r c r a f t w h o s e various

components a c t a s a r i g i d - b o d y a s s e m b l y and f r o m t h e rotating masses attached t o t h e

a i r c r a f t .

3.1.1 Inherent Aircraft General nertial orce Expressions Inasmuch a s our interest l i e s i n t h e analysis of f l i g h t t e s t data oriented t o t h e

b o d y axis s y s t e m , t h e i n e r t i a l f o r c e expressions a p p l i c a b l e t o this axis s y s t e m and

f o r a l l attitudes of f l i g h t a r e

X j m( ü + qw - r v )

Yj m( v + r u - p w )

Zi - m( w - q u + p v ) .

( 1 9 a )

( 1 9 b )

( 1 9 c )



1 1

If he tability xis ystem ere mployed s he eference nstead f he ody xis s y s t em , q w w inasmuch s here s o inear elocity omponent long he z-stability xis.

3.1.2 General Inertial-Moment Expressions

F or he eneral ase here he principal xes re sym m etr ic o he arious planes of he eference xes, th e nertial-moment xpressions re

Lj = Ixp (rp - ) X2(r q) yzr2 - 2) Iz - y)qr 20a)

Ui = Iyq yzrq - ) r r) xzp2 - 2) I, - zrp 2 0 b )

N j = Izf xz(qr - ) yzq p) + I q2 - 2) Iy - xpq 20c)

Fortunately, situations nvolving eneral symmetry f he ircraft re are. ormally , the ehicle will a ve m a s s distribution y m m et r i c a l elative o he z-body lane f symmetry , with he esult hat he principal -axis oincides ith he -body xis. Under uch ircumstances, ITV = and he eneral nertial-moment xpressions reduce o he ollowing orma lly mployed orm:

Li = TxP xz(i S) dz y)qr 2 1 a )

M j = Iyq xz(p2 -

2) Ix - zrp 2 1 b )

N j = Izr - xz(p - r) Iy x q 21c)

The nertial xpressions n quations 2 1 a , b, c) re nonlinear nd hus ot uitable for use n he erivation f losed-form tability quations. However, they re e- quired n nalog r igital omputer tudy f he m otion f he ircraft n eneral or iolent aneuvers nd n h e nalog matching f light a ta rom uch ma n euver s in ttempts o e t ermine he ffective alues of he tability nd ontrol erivatives for he aneuver.

In iolent maneuvers , the e r m s nvolving pq and rp a re particularly m portant . These e r m s , a s well s qr a re yroscopic e r m s . Modern igh-perform ance ircraft tend o ave ow alues f I% compared o I an d Iz with he esult hat yroscopic ction epresented y (Ix 2rp an d (Iy xpq in particular, ha s een responsible or he uncontrollable, catastrophic oll-coupling behavior f t east one et ircraft fter eliberate ig h oll ate nput.

W h e n he m otions f he ircraft re m a l l r radual , the nertial-moment xpressions m a y e implified o

( 2 2a )

( 22b)

(22c)

h = I«p- "W

Mi = ^

Ni

= v- w >



1 2

3.1.3 Small-Perturbation Inertial Expressions

T h e classical approach t o t h e s t u d y of a i r c r a f t dynamic stability and c o n t r o l involves the use of s m a l l disturbances ( p e r t u r b a t i o n » ) . estricting t h e motions t o s m a l l d e - viations f r o m steady-state conditions allows the elimination of non-linear t e r m s f r o m t h e i n e r t i a l e x p r e s s i o n s . uch motions a r e u s e f u l i n defining t h e s t a b i l i t y , c o n t r o l , and handling qualities of t h e a i r c r a f t , a n d t h e p i l o t effort o r autopilot character- i s t i c s required t o c o n t r o l t h e m o t i o n . t h a s b e e n found that t h e use of s m a l l - perturbation t h e o r y g i v e s good results and permits t h e development of a n a l y t i c a l e x p r e s s i o n s .

T o a r r i v e a t t h e small-perturbation i n e r t i a l e x p r e s s i o n s , r e p l a c e the i n d i v i d u a l acceleration and velocity terms i n Equations ( 1 9 a , b , c ) a n d ( 2 2 a , b , ) by accelerations a n d velocities made up of disturbances superimposed on equilibrium conditions so that ü , e t c . , i s replaced by ü + Aü , e t c . , r e s p e c t i v e l y ; expand t h e p r o d u c t t e r m s ; n e g l e c t t h e second-order quantities ( A r A u , f o r e x a m p l e ) ; and s u b t r a c t t h e i n i t i a l conditions f r o m t h e f i n a l resulting c o n d i t i o n s . n e resulting small-perturbation i n e r t i a l expressions a r e

AX j m[Aü + w Aq + qA w - rAv v A r ] ( 2 3 a )

A Y , i l A v + uA r + rAu - w Ap - pAw]

AZ j m[Aw - uAq - qA u + pAv + vAp]

( 2 3 b )

( 2 3 c )

a n d

A L ,

A M ,

I^P

IyAq

I xzAr

ANi A-I P

( 2 4 a )

( 2 4 b )

( 2 4 c )

Equations ( 2 3 a , b , c ) show t h a t lateral-directional-mode perturbations Av , Ar , and Ap p p e a r i n t h e longitudinal-mode equations Ax i n d Az i , a n d t h a t t h e longitudinal-mode perturbations Au n d Aw a p p e a r i n t h e lateral-directional-mode equation Azi . h i s coupling of t h e t w o modes c a n normally b e minimized t o p e r m i t

practical use of t h e uncoupled practical approximation of Equations ( 2 3 a , b , c ) s h o w n i n Equations ( 2 5 a , b , c ) . his m 'limization i s achieved i n f l i g h t t e s t maneuvers s u c h as elevator pulses f o r perturbation of t h e longitudinal mode a n d rudder or a i l e r o n pulses f o r perturbation of t h e lateral-directional mode initiated during s t e a d y w i n g s - l e v e l or steady turn f l i g h t .

Ax , m[Aü + wAq] ( 2 5 a )

AYA [A v A r Ap]

AzA - mfaw - A q - A u]

( 25b)

(25c)



L—' .

13

3.2 Gyroscopic Couples of Rotating M a s s e s

Spinning m a s s e s ounted n he ircraft uch s propellers nd otating l e m e n t s of ngines - possess ngu lar omentum elative o he eference body) xes nd produce yroscopic ouples n he ircraft hich ould e ignificant, s as he ase n the -5 irplane6. Normal ly , the yroscopic ouples re negligible; owever , he dvent of ertical-rising ircraft ith ilting ngines nd he ncrease n ize f propulsion units n igh-perform ance ircraft ak e t nadvisable o rbitrarily gnore his coupling.

F or otating m a s s aving otating xis n r parallel o he z-plane f ymmetry bu t t n ngle & rm to he -body xis Pig.6), it a n e hown rom he oment f momentum elation, ü > f and H rmQ hat he yroscopic ouple bout ach f he body xes s

Lr m = 1«z-ry -Irmn1snör m 26a

>

M rB = r H x-z = rBQ(reo8 0rB psinV 26b)

N rm = pHy-qHx = -IrBnqcos0rm . S6c)

These otating m a s s ontributions re dded o he nertial oments xpressed y Equations 2 0 a , , c), ( 2 1 a , b, c), and 2 2 a , b, c).

F or mall perturbations of he ircraft, he perturbations f br . yroscopic ouples resulting rom he otating m a s s re xpressed y

ALr» = -Irm QSin0rill 2 7 a )

AMr m = Irft(Arcos0riB+APsn0rill) 27b)

ANr m = -IrB qcos0rm . 27 c )

These erturbations re dded o quations 2 4 a , b, c) hen ignificant, in hich case, quations 2 4 a , , c) will ecome nter-dependent because f he oupling f he longitudinal-mode nd ateral-directional-mode oment quations. It hould e oted that, if 9ra were ariable, the bove elations n quations 2 7 a , b, c) ould ave required urther xpansion nd ntroduced n dditional egree f reedom n h e orm 0f Aör. • m

3 . 3 Gravitational F o r c e

T h e g r a v i t y f o r c e w i l l n o t contribute t o t h e moment equations a s I o n s a s t h e origin

of t h e a x i s s y s t e m i s a t t h e c e n t r e of g r a v i t y .

1 4

3.3.1 Components of Gravity F o r c e

With the gravity f o r c e W acting along the r x i s , the expressions f o r the

components of gravity f o r c e acting along the body axes a r e readily deduced f r o m Figure

4 ( a ) to be

Xg = -Wsind 2 8 a )

Yg Wcosdsin<£ 2 8 b )

Zg Wcosdcos$ 2 8 c )

T h e s e components a r e subtracted f r o m t h e inertial-force quations ( 1 9 a , b , c ) .

3.3.2 Small Perturbations

S n a i l perturbations of t h e components of t h e gravity f o r c e may b e based o n Euler

a n g l e perturbations superimposed on t h e unperturbed angles using the same basic

reference f r a m e , or on Euler a n g l e perturbations relative to a secondary s p a t i a l

reference f r a m e made up of t h e unperturbed aircraft axis s y s t e m a s discussed i n

Section 2 . 1 . 6 . and as s h o w n i n Figure 4 ( b ) . n this second a p p r o a c h , unperturbed

b o d y axes a r e used a s t h e secondary s p a t i a l reference w h e n i n t e r e s t i s primarily i n

perturbations of body-oriented f l i g h t t e s t d a t a .

Using the f i r s t a p p r o a c h , replace 9 and > i n Equations ( 2 8 a , b , c ) b y d + Ad

and > + A < / > , e s p e c t i v e l y ; expand t h e resulting trigonometric f u n c t i o n s , consider cosA_~ l sinA _~ A_, a n d A_A_~ 0 ; a n d subtract t h e i n i t i a l conditions f r o m

t h e r e s u l t . h e resulting small-perturbation expressions a r e

AX.

AZ „

-WAd cos e A yg = W(Ad cos 8 cos < t> - A d sin 6 sin < p )

-W(A<£ osdsin<£ + A d ind cos < £ )

( 2 9 a )

( 2 9 b )

( 2 9 c )

I n t h e s e c o n d a p p r o a c h , using t h e unperturbed b o d y axes a s t h e reference and & )' AÖ ' a n d A < £ ' F i g . 4 ( b ) ) a s t h e Euler angles of t h e p e r t u r b a t i o n s , t h e p e r -

turbations of t h e components of g r a v i t y a r e obtained b y using Equations ( l c ) a n d

( 2 8 a , b , c ) . n Equation ( l c ) t h e generalized quantities Xr , Yr , and Zr r e replaced b y t h e expressions f o r Xg g and Zg , r e s p e c t i v e l y , a s g i v e n i n

Equations ( 2 8 a , b , c ) ; a n d t h e Euler angles p 8 a n d 4 > a r e replaced by i / / ' ,

Ad' , and A < £ ' , r e s p e c t i v e l y . h e generalized quantities Xb , Yb , a n d Zb n

Equation ( l c ) a r e now e q u a l t o . ( Xg + Ax g) , Yg + Ayg) , a n d ( Z g + Azg) , r e s p e c t i v e l y .

B y subtracting t h e i n i t i a l conditions ( E q u a t i o n s ( 2 8 a , b , c ) ) f r o m t h e resulting

perturbed e q u a t i o n after considering osA_^ i sinA_2*A_ , and A_A _ 2 * o ,

t h e perturbation expressions f o r this s e c o n d approach w i l l have t h e following f o r m :

Axg = W(co*d sin0 A < / / ' - c o s dcos Ad')

AYg ( s i n A i / / ' + cosdcos^ A0')

A Z . -W (sind Ad'

osdsin<£A</>')

( 3 0 a )

( 30b )

(30c)

1 5

T h e advantage i n using Equations ( 3 0 a , b , c ) instead of Equations ( 2 9 a , b , c ) is t h a t , o r s m a l l perturbations during highly banked a s « e l l a s wings-level f l i g h t ,

A i / / ' ~ /Ar d t 3 1 a )

A ( 9 ' ~ /Aq d t 3 1 b )

A < £ ~ /Ap d t . 3 1 c )

T o a p p l y s u c h s i m p l e integrations t o A i / / , Ad , and A# n Equations ( 2 9 a , b , c )

requires t h a t > a n d 6 b e s m a l l .

Both Equations ( 2 9 a , b , c ) a n d ( 3 0 a , b , c ) s h o w coupling of t h e longitudinal and l a t o r a l m o d e s . n b o t h s e t s of e q u a t i o n s , t h e l o n g i t u d i n a l modes ( Ax and Az ) are uncoupled from t h e lateral-mode perturbations b y performing a longitudinal pulse when i n i t i a l conditions a r e s t e a d y - s t a t e . n performing a lateral-directional pulse f r o m steady-state c o n d i t i o n s , t h e lateral-mode expression ( 3 0 b ) i s inherently u n - coupled f r o m longitudinal p e r t u r b a t i o n s , whereas expression ( 2 9 b ) shows interaction of t h e l o n g i t u d i n a l perturbation AÖ which i s excited b y t h e lateral-directional p u l s e .

When banked a n d climbing f l i g h t a r e being c o n s i d e r e d , i t may b e surmised f r o m the preceding t h a t Equations ( 3 0 a , b , c ) a r e more amenable than Equations ( 2 9 a , b , c ) to

theoretical stability analysis a n d f o r analysis of f l i g h t data when l o n g i t u d i n a l or l a t e r a l pulses a r e applied f r o m i n i t i a l steady-state c o n d i t i o n s .

3 . 4 Aerodynamic Derivatives

I n stability and c o n t r o l investigations based on f l i g h t d a t a , the previously d i s - cussed i n e r t i a l , g y r o s c o p i c , a n d g r a v i t a t i o n a l quantities a r e normally equated t o aerodynamic parameters o n l y . b i s i s d o n e primarily t o facilitate t h e analysis of f l i g h t d a t a . o w e v e r , n doing t h i s , t h e parameters a r e no longer pure aerodynamic p a r a m e t e r s , inasmuch a s t h e y w i l l h a v e been modified b y influences arising f r o m power and aeroelasticity a s w e l l a s p o s s i b l e other s o u r c e s . e n e r a l l y , these i n f l u e n c e s c a n b e accounted f o r a n d t h e p u r e aerodynamic parameter arrived a t .

Inasmuch a s t h e equations a r e s e t up under t h e principle of super-position of i n f l u e n c e s , situations may b e encountered i n w h i c h t h e accuracy of t h e results obtained f r o m t h e equations w i l l d e t e r i o r a t e . his i s of particular c o n c e r n w h e r e v e r y r a p i d c o n t r o l inputs are e n c o u n t e r e d . l s o , n a s m u c h a s t h e aerodynamic parameters a r e i n t h e f o r m o f d e r i v a t i v e s , c a r e m u s t b e exercised n o t t o exceed t h e validity of t h e d e r i v a t i v e .

F i n a l l y , t h e r e a r e s o m e limitations i n combining s e v e r a l o f t h e d e r i v a t i v e s , C n r - C n / a , o r e x a m p l e .

Consideration i s g i v e n a t this t i m e t o t h e above-mentioned f a c t o r s which have significance i n t h e utilization o f aerodynamic derivatives i n t h e equations of motion a n d i n t h e determination o f t h e derivatives f r o m f l i g h t d a t a . o r c o n v e n i e n c e , t h e c o n v e n t i o n a l derivatives a r e tabulated o v e r l e a f .



1 6

Longitudinal Derivatives

Cl» u 2C„

3u cosacos/3

3CN CN

« 17 CN«

3CN _ 9CN

' ( f ) (f) CN S = 3C,

38

3r Cc. = V—£

2C„ 3C„ Bu cos acos/3 3a "ca

3C „

• ( S ) c = -** -C S BCg

3S

3cm 2 C „ , C.„ V-JS-+ 3cm P _ 3u cos acos ß 3a 3C_ 3C.

f) *"'© 3S

Lateral-Directional Derivatives

°Jß 3/3 uy/3

© 3c„

*

3 c.

(£) ^ ys

5L 3S

C|4

3Cj clfi

3C 3Ä

\2V,

C/, 3 c,

3 (V 2 V >

ciB = 3C 3

\2V>

C l> 3 C j 3?

3cn

* 3 3j/J

Cn ^ =

\2V

Cn, 3C„ 3

2 V

Cnr3C„ 3f'£

2V

-ns 3S

3.4.1 Significance f he Derivatives

Tlie erodynamic erivative rovides he lope f he urve f he erodynamic orce

or om ent oefficient, s he ase ay e, with espect o n ndependent ariable

other ndependent ariables eing onsidered onstant t particular alue f he variable. In nalog imulation tudies, onlinear urves re educed o traight-line

segments, ach egment eing alid nly or n ncremental ange f he ndependent

variable.

In he nverse roblem f btaining erivatives ro m light ata, he erivative s valid nly or he ncremental ange f isturbance i he ndependent ariable, at he

steady-state ondition, sed n etermining he erivative. An xample nvolving

nonlinear ariation f C n with ß is hown n igure . In his xample, he

origin, , epresents teady tate nd ( ß)x a nd (A/3)2 represent w o isturbance

ranges f he ariable. It will e noticed hat he erivative btained ay differ

appreciably n magnitude ecause f he onlinearity f he urve n he isturbance

ranges A/3), and A/3)2



17

.7.4. 2 Unsteady low Effects

3u

i?***'"

8

*s fy + 2C

N Algs, \ Bu co s acos /?/

(32)

1

.

I n dealing w i t h t h e derivative c o n c e p t i n accounting f o r t h e influence of Independent variables o n a n aerodynamic f o r c e or moment c o e f f i c i e n t , o r example

Arb A,?b Apb ACn n^A5 + C nr + Cnj +Cnp +C„SrASr

i t i s assumed t h a t each derivative contributes t o t h e t o t a l a s t h o u g h i t acted a l o n e

a n d t h a t t h e aerodynamic f o r c e a n d moment coefficients a r e functions of t h e i n s t a n - t a n e o u s values o f t h e disturbance displacements a n d v e l o c i t i e s , c o n t r o l a n g l e s , and t h e i r d e r i v a t i v e s . u r t h e r , t h e derivatives a r e based o n t h e variation of t h e c o - efficients under near-steady-state conditions of t h e v a r i a b l e . lthough t h e deriva- t i v e c o n c e p t of treating aerodynamic f o r c e a n d moment perturbations has generallyworked w e l l , t h e application t o situations of r a p i d l y changing i n d e p e n d e n t variables ( u n s t e a d y flow c o n d i t i o n s ) , s i n t h e c a s e o f a v e r y r a p i d c o n t r o l displacement or a sharp-edged g u s t , does n o t necessarily g i v e c o r r e c t a n s w e r s . his i s d u e t o apparent mass effects of t h e a i r , w h o s e i n e r t i a w i l l not produce instantaneous changes i n circulation and consequently c a u s e s aerodynamic l a g . h i s i s illustrated i n Figure 8 , w h i c h s h o w s t h e variation of C N s a function of nondimensional t i m e , /2 V , a s a r e s u l t of a step g u s t . h e derivative c o n c e p t would show a c o n s t a n t s l o p e c u r v e , whereas t h e a c t u a l variation of C N(t) o u l d show a l a g a t t h e i n i t i a l i n s t a n c e of

t h e s t e p gust i n p u t .

W h e n a n aircraft i s oscillating s i n u s o i d a l l y , t h e l i f t w i l l follow t h e sinusoidal variation i n a n g l e of attack b u t w i l l b e of smaller magnitude a n d t h e r e w i l l b e a p h a s e difference between t h e l i f t a n d a n g l e of a t t a c k . his unsteady flow e f f e c t i s a f u n c t i o n of reduced oscillating f r e q u e n c y , > c / 2 V , a s w e l l a s M a c h n u m b e r . lthough t h e magnitude o f p |a i s not normally affected appreciably f o r n o r m a l airplane osci^iting frequency c o n d i t i o n s , t h e p h a s e l a g may bring a b o u t a l a r g e c h a n g e i n N . T h i s m a y b e of considerable i m p o r t a n c e i n p i t c h damping of tailless aircraft ( R e f .7) .

I n g e n e r a l , a l l t h e aerodynamic derivatives b e h a v e i n a similar m a n n e r . h i s , t i s s e e n t h a t attempts t o u s e t h e derivative c o n c e p t i n analog simulators involving v e r y rapid c h a n g e s of t h e independent variables c a n l e a d t o e r r o r s ; c o n v e r s e l y , d e t e r -

mination of derivatives f r o m f l i g h t d a t a r e q u i r e s awareness o f t h e maneuvering or unsteady f l o w f a c t o r s mentioned w h i c h c a n i n f l u e n c e t h e magnitude o f t h e d e r i v a t i v e .

i 3.4.3 Derivatives with Respect o u

Aerodynamic derivatives w i t h r e s p e c t t o r e of c o n c e r n w h e n phugoid modes a r e b e i n g i n v e s t i g a t e d . ecause t h i s mode i s o f t e n o v e r l o o k e d , h e s e derivatives a r e generally u n f a m i l i a r . h u s , s o m e consideration i s g i v e n t o t h e m a t this t i m e f o r f u t u r e reference a s n e e d e d . onsider -Z = C NqS . ifferentiation w i t h r e s p e c t t o u h o w s

BZ 3cN 3v

18

where Bv/Bu /(cos acos/9) fron u Vcos/Scosa Th e Az du e o hange n u is ow xpressed s

_ qS " i " CN

U "7 (33)

Since

2C„ cM 3u cosacos/Sy

is more ha n i m p l y he ariation of he ormal oefficient ith espect o elocity, u t an ittingly e alled he ffective erodynamic derivative f C „ with espect to u r C Nu

with espect o . Similarly, he ffective erodynamic derivatives of u are

3 C L 2 C _ 3 u cos a cos / 3>

or 5 i» u an d C c„ respectively.

and 2C„ \ f ^c

V + —= ] V du cosacospy

and C „

J.4.-5 Z)ert i>at ives with Respect o q and d, and r a/«f p

It s ustomary n eporting light-determined derivatives, herein ransient oscillations re se d n he nalysis, to pair he derivatives arying ith espect to q and a an d hose arying ith espect o r an d p F or xample

Aqc A dc A q c q - + ä-- - (Cm q <W —

and

Cn, A rb

2 V + , A/?b nß-

2 V Ä (Cnr - n/§ )

A rb

i v

(34)

Inasmuch s he henomenon nvolving d is ifferent rom hat nvolving q an d the henomenon nvolving r is different rom hat of p he pairing s alid nly when mall-perturbation ransient oscillations f maneuv er re nvolved nd atisfy the inearized quations f m otion. In ddition, l though he pairing orks well or the ongitudinal quations hether r not tability r ody xes re mployed, he validity f he paring or he ateral-directional quations s ependent n he use

of he tability ystem f xes ; if ody xe s re s ed , he pairing of r and p

derivatives s permissible t ow ngles of ttack.

In erform ing mall-perturbation ongitudinal ransient scillation, the enter of ravity of he ircraft ends o ove long he light a th s hough t ere not disturbed; consequently, he m plitude atio | A q | / | A & | is imilar o ,0 nd he

vector uantities A q an d A d a re pproxim ate ly n ha s e . Thus A q ca n e ub- stituted or A« In he ase of ateral-directional Dutch oll) ransient oscillation elative o he tability xis y s t e m , the ircraft, in ending o



1 9

m a i n t a i n i t s c e n t e r o f g r a v i t y a l o n g t h e f l i g h t p a t h a s t h o u g h i t w e r e n o t d i s t u r b e d , w i l l e x p e r i e n c e Ar ^-A/3 , i n a s m u c h a s a n d ß a r e n o w r e f e r r e d t o t h e s a m e a x i s s y s t e m . h u s , t h e a m p l i t u d e r a t i o A r l / l A / 3 l s s i m i l a r t o 1 . 0 , b u t t h e p h a s e r e l a t i o n i s a p p r o x i m a t e l y 1 8 0 ° ; c o n s e q u e n t l y , t h e s i g n o f t h e A / 3 e r i v a t i v e i s c h a n g e d t o m i n u s i n p a i r i . g r a n d ß d e r i v a t i v e s . n d e a l i n g w i t h t h e b o d y a x i s s y s t e m , A r | / | A / ? | a n d $tß c a n d i f f e r a p p r e c i a b l y f r o m 1 . 0 u n d l o O

0, r e s p e c t i v e l y , a t h i g h a n g l e s o f

a t t a c k .

I t i s r e i t e r a t e d t h a t p a i r i n g t h e d e r i v a t i v e s i s v a l i d o n l y f o r t h e s p e c i a l c o n - d i t i o n s m e n t i o n e d . n t h e o t h e r h a n d , i t h a s n o t b e e n p o s s i b l e t o s o l v e f o r t h e ä d e r i v a t i v e s i n d e p e n d e n t o f t h e q e r i v a t i v e s , a n d $ d e r i v a t i v e s i n d e p e n d e n t o f d e r i v a t i v e s , f r o m f l i g h t d a t a w i t h a n y d e g r e e o f c o n s i s t e n c y a n d c o n f i d e n c e .

3.U.5 Power Effects

T h e p r o p u l s i v e s y s t e m m a y h a v e a s i g n i f i c a n t i n f l u e n c e o n t h e s t a b i l i t y a s w e l l a s t h e t r i m o f t h e a i r p l a n e . t s f o r c e a n d m o m e n t c o n t r i b u t i o n s t o t h e e q u a t i o n s o f m o t i o n m a y b e p r e s e n t e d a s d e r i v a t i v e s i n t h e e q u a t i o n s . f t h e p o w e r c o n t r i b u t i o n s a r e n o t a c c o u n t e d f o r b y t h e i r o w n d e r i v a t i v e s , t h e y w i l l b e r e f l e c t e d i n t h e m a g n i t u d e s o f t h e a e r o d y n a m i c s t a b i l i t y a n d c o n t r o l d e r i v a t i v e s w h i c h w i l l t h e n b e c o m e , i n e s s e n c e , e f f e c t i v e d e r i v a t i v e s . c o m p r e h e n s i v e t r e a t m e n t o f p o w e r e f f e c t s i s c o m p l e x a n d b e y o n d t h e s c o p e o f t h i s p a p e r . n l y m a j o r e f f e c t s a r e c o n s i d e r e d , t o s h o w h o w p r o p u l s i o n s y s t e m d e r i v a t i v e s c o n t r i b u t e t o t h e e f f e c t i v e v a l u e s o f t h e a e r o d y n a m i c d e r i v a t i v e s .

I t i s e s s e n t i a l a t t h i s t i m e t o e m p h a s i z e a n i m p o r t a n t p o i n t r e g a r d i n g c o n s i d e r a t i o n o f t h e e f f e c t o f p o w e r o n s t a b i l i t y . r u e i n h e r e n t s t a b i l i t y o f t h e a i r c r a f t w i t h p o w e r o n c a n o n l y b e e v a l u a t e d b y k e e p i n g t h e s e t t i p g s o f t h e e n g i n e a n d p r o p e l l e r c o n t r o l s f i x e d d u r i n g t h e m a n e u v e r . n y m a n e u v e r t h a t e n t a i l s a l t e r a t i o n o f t h e p r o - p u l s i v e s y s t e m c o n t r o l s d u r i n g t h e m a n e u v e r w i l l n o t p r o v i d e a t r u e i n d e x o f t h e s t a b i l i t y f r o m a n a n a l y s i s o f t h e t i m e h i s t o r y o f t h e m a n e u v e r .

Influence of propellers: I n f l u e n c e s o f p r o p e l l e r s c o n s i s t o f d i r e c t p r o p e l l e r e f f e c t s a n d a l s o i n d i r e c t e f f e c t s d u e t o t h e p r o p e l l e r s l i p s t r e a m o n t h e w i n g - f u s e l a g e a n d t h e t a i l s u r f a c e s .

Direct propeller effects: D i r e c t p r o p e l l e r e f f e c t s , a s s h o w n i n F i g u r e 8 , c o n s i s t o f a d i r e c t t h r u s t a c t i n g a l o n g t h e t h r u s t a x i s , a n d a t r a n s v e r s e f o r v e ( Y )

a s w e l l a s a n o r m a l f o r c e ( - Z ) p e r p e n d i c u l a r t o t h e t h r u s t a x i s i n t h e p l a n e o f t h e p r o p e l l e r d i s k . h e t h r u s t T s a p r i m a r y f u n c t i o n o f a n d V . Q u a n t i t a t i v e d e t e r m i n a t i o n o f t h e n o r m a l a n d t r a n s v e r s e f o r c e s - Z )p a n d Y ) m a y b e a c c o m p l i s h e d b y s o l v i n g f o r C NäP n d ( C y / 3 )p , a s d i s c u s s e d i n R e f e r e n c e s 8 a n d 9 . c t u a l l y , t h e d e r i v a t i v e s a r e o f m o r e c o n c e r n f o r t h e p u r p o s e s o f t h i s p a p e r t h a n t h e a c t u a l m a g n i t u d e s o f t h e f o r c e s .

T h e c o n t r i b u t i o n s o f t h e d i r e c t p r o p e l l e r e f f e c t ( F i g . 9 ) o n l o n g i t u d i n a l a n d l a t e r a l s t a b i l i t y a r e r e f l e c t e d i n

( C ma p = CTa + ( CNa p X-£ 3 5 )

a n d

«Vp = ~(Cyß)v • 3 6 )

2 0

I t i s opportune t o note t h e influences of t h e d i r e c t propeller e f f e c t s o n « h e n performing a transformation f r o m t h e b o d y t o t h e stability a x i s s y s t e m .

C L( X

T he net

effective C L and C D of he aircraft, in he absence of angular ates nd or ixed

controls, ca n e xpressed s

Aero Po wer

cos a -sin a

sin a cos a

sina,,

-cos a„

cos a„

sin a.

CN

<Cn>p

(37)

where he direct hrust s ssumed o be vectored parallel o he od y x-axis nd

< x _ = a + e _ P Differentiating Equation ( 3 7 ) w i t h r e s p e c t t o f o r C ; -a

(38) L« = ( C Naosa - C (Xsina) D Tainap C Napcosa ,

where C D is he effective value as hown n Equation 37).

Sti'dy f Equation 38) hows hat ower ncrease? he ffective C Lä of he ircraft. O n ow-performance aircraft, the ower effect s enerally negligible.

Propeller lipstream: The propeller lipstream nfluences he istribution of he

aerodynamic orces n he aircraft tructure as a esult f i) he ncrease n ocal velocity over he tructure ue o nd n he propeller lipstream, an d ii) pwash nd

downwash ffects of he otating lipstream f he propeller. T he lipstream an e

stabilizing or estabilizing, depending po n he direction of otation of he propeller an d he position of he ail elative o he otating slipstream. Analytical echniques

to uantitatively ccount or he propeller slipstream ffects n he tability of he

aircraft have not ee n atisfactory. Generally, powered mode ls re used o provide

engineering data n ew designs.

Influence f jet npines: he jet ngine has he ounterpart of he ffects hat

were hown or he propeller. It rovides direct hrust, shows normal nd ransverse

force effects at he ntrance of he ntake duct, an d - depending n eometry - s capable of nfluencing he quilibrium nd tability f he ircraft y nflow of air

into he jet xhaust. Unlike he propeller, the nfluence of he jet ngine n he tail urfaces, an d hence he tability of he airplane, is menab le o analytical techniques o uantitatively ccount or hese ffects.

T he hrust roduced n he ircraft quipped with jet ngine s qual, as hown

in igure 0, to he ectorial hange n momentum of he ir nd uel assing hrough

the engine plus he esultant of he pressure orces acting across he nlet nd outlet

areas. Where he ntake nd exhaust re n ine with he hrust xis nd he -body

axis

T = CrS

= nijVj r^Vcoscxjj pjAj pjAj) ( 39 )

22 Th e variations of orces nd moments ue o he jet ngine are primarily unctions

of ß nd ssuming hat control ettings are constant. rom he preceding

it a y e eadily educed hat

_ m .V cos a_ 3a_ f v f 3(ah „ I(C„ap = -1— J-E + •« • * *• n.t.'p ——*-=r* + CNa)h.t ~

qS a S 3a (44a)

z„ m„Vcoso„x„3a„ A.t.'h.t ("h.t.>p (44b) _ m^v cos a_ _ o«

qSc

2«VP ma cos/3 (C N(x)v.t. 5ts

v.t. K.t.>{ qSb 3a (45a)

2<VP maVcos/3x tsv.t.xv.t. K.t.>i — -r - CiOv. t.

qS b Sb 3a (45b )

where xp is positive when he air ntake s orward of he center of gravity nd

xy t is negative wii;h ertical ail ft of he center of gravity, nd

/ C, 2C, (C cu)p = - V

on cos a cos pi

( C i nu)p 2C „ \C„

, 3u cos a cos ßj \ p

(46)

(47)

where

z„ m aVsin a„ „

c S (Cm}p ^ PAt.), qS c

(48)

an d

3 .J ,. 3cT zp a

sin p p

W n 3u c qS c (49)

3.4,6 Aeroelastic Effects Th e preceding discussions ssumed hat he ircraft as igid. This ssumption as

permissible n he past; owever , odern ircraft lying at high peed under ynamic-

pressure onditions re ubject o degrees of lexibility of omponent parts hich

cannot, t imes, b e gnored nd which affect he tability of he ircraft11"16. Th e

contribution of aeroelastic deformation o derivatives s dependent primarily n

aircraft eometry nd ynamic pressure as well s tructural igidity nd Mach umber .

Aeroelastic henomena m ay e onsidered n wo eparate parts: tatic nd ynamic

peroelastic effects.



2 3

When aerodynamic loading takes place a t a sufficiently slow r a t e i n comparison t o t h e n a t u r a l frequency of vibration of t h e pertinent p a r t of t h e structure t o permit t h e assumption of s t a t i c deformation cf t h e s t r u c t u r e , t h e influence of aeroelasticity c a n b e accounted f o r b y modifying t h e d e r i v a t i v e s . llustrations of steady-state distortions w h i c h have b e e n serious i n t h e p a s t a r e a i l e r o n r e v e r s a l a n d wing d i v e r g e n c e . T o d a y , u c h f a c t o r s a s t h i n n e r w i n g s a n d more f l e x i b l e fuselages have magnified t h e e f f e c t s of s t r u c t u r a l flexibility o n s t a b i l i t y a n d c o n t r o l of a i r c r a f t .

I f t h e aerodynamic l o a d i n g f r e q u e n c y w e r e t o approach t h e s t r u c t u r a l frequency of

t h e pertinent c o m p o n e n t , me s t r u c t u r a l deformation w o u l d produce perturbations i n t h e aerodynamic forces and moments which have t o b e accounted f o r by the introduction of a d d i t i o n a l appropriate derivatives i n t h e equations of motion a n d t h e introduction of a d d i t i o n a l e q u a t i o n s , w h i c h would b e elasticity e q u a t i o n s .

3.It. 7 Other Effects

T h e preceding discussion h a s included major f a c t o r s w h i c h influence analysis and a c c o u n t f o r discrepancies between wind-tunnel a n d f l i g h t d a t a ; h o w e v e r , i t d o e s not a c c o u n t f o r a l l f a c t o r s . ther f a c t o r s c o u l d i n c l u d e j e t p l u m i n g , l o w separations associated w i t h movements of s h o c k w a v e s , a n d f u e l s l o s h i n g . i n c e o n e never knows w h a t phenomena w i l l o c c u r , t i s i m p e r a t i v e t o have a n open min^ i n trying t o account f o r discrepancies i n comparisons of d a t a .

3 . 5 S u m m a r y o f t h e Equations of Motion

T h e v a r i o u s dynamic relations w h i c h h a v e b e e n discussed a r e pertinent * o a n understanding of t h e equations of motion a n d t h e conditioning of d a t a t o t h e e q u a t i o n s . h e i n f l u e n c e o f power a n d s t r u c t u r a l f l e x i b i l i t y o n t h e various aerodynamic parameters ( c o e f f i c i e n t a n d stability derivatives) was s t r e s s e d , and i t w a s pointed out that the n e t r e s u l t cf t h e s e i n f l u e n c e s , or m o d i f i e r s , w a s t h e emergence of a n effective a e r o - d y n a m i c p a r a m e t e r .

I t i s easily recognized t h a t t h e i n t r o d u c t i o n i n t o t h e equations of motion of e a c h i n d i v i d u a l modifier t o t h e aerodynamic parameters w o u l d r e s u l t i n a cumbersome s e t of e q u a t i o n s . t i s more practical t o l e t t h e normally accepted stability s y m b o l ( Cna , f o r e x a m p l e ) represent t h e effective value t h a n t o l i s t a l l m o d i f i e r s . n s o d o i n g , o n e should b e a w a r e of t h e v a r i o u s s o u r c e s w h i c h contribute t o t h e magnitude of t h e effective parameter i n order t o properly a c c o u n t f o r t h e s e contributions during a n analog i n v e s t i g a t i o n , or other s t u d y , i n w h i c h wind-tunnel and calculated data a r e u s e d . n t h e other h a n d , n t h e i n v e r s e p r o b l e m of determining coefficients and derivatives f r o m f l i g h t d a t a , a discrepancy i n t r e n d s a s w e l l a s magnitude b e t w e e n w i n d - t u n n e l a n d f l i g h t d a t a w i l l s u g g e s t possible influences f r o m s o u r c e s n o t accounted f o r b y t u n n e l d a t a .

3.5.1 General quations

T h e f o l l o w i n g assumptions a r e made w i t h r e g a r d t o t h e equations o f motion summarized i n Table I V :

( i ) T h e airplane behaves a s a r i g i d b o d y , n t h a t t h e moments of i n e r t i a , n c l i n a - t i o n of p r i n c i p a l a x e s , t c . , a r e n o t affected significantly



- f

2 4

( i i ) The airplane i s symmetrical about the xz-plane with regard t o geometry and mass d i s t r i b u t i o n .

( i i i ) The a x e s of rotating elements o n t h e aircraft a r e f i x e d i n a d i r e c t i o n relative t c the b o d y reference a x e s .

( i v ) The e a r t h i s f l a t . ircraft s p e e d s a r e assumed t o be insufficient t o i n c l u d e e a r t h curvature i n t h e e q u a t i o n s .

( v ) The forcing frequency of a disturbance i s sufficiently f a r removed f r o m t h e n a t u r a l frequency of t h e pertinent p a r t of t h e structural components t o p e r m i t

t h e disturbance t o b e considered a s a static load and t h e effect of deformation t o b e accounted f o r b y modification of t h e aerodynamic p a r a m e t e r s .

( v i ) E a c h aerodynamic parameter i s a n effective p a r a m e t e r , i n t h a t i t i n c l u d e s a l l sources contributing t o i t s net v a l u e .

Although listed i n Table IV f o r c o m p l e t e n e s s , experience has s h o w n

Cy/ j and C Cq and C c& t o b e normally n e g l i g i b l e . yp

uyr

3.5.2 Small-Perturbation Equations

T h e g e n e r a l equations of motion i n Table I V a r e suitable f o r analog a n d d i g i t a l programing w h i c h involves l a r g e disturbances a n d nonlinear t e r m s ; t h e y a r e not suitable f o r analytical p u r p o s e s . or s u c h p u r p o s e s , i t i s necessary t o linearize t h e equations

a t l e a s t t o a n engineering degree of a c c u r a c y . his i s accomplished b y restricting their applications t o s m a l l p e r t u r b a t i o n s , as has b e e n discussed p r e v i o u s l y . n a d d i t i o n , h e perturbations a r e referred t o a s e c o n d a r y s p a t i a l reference f r a m e , discussed i n Section 2 . 1 . 6 , w h i c h i s t h e unperturbed airplane axis s y s t e m s h o w n i n Figure 4 ( b ) . sing t h e secondary reference s y s t e m f o r s m a l l perturbations permits t h e u s e of Equations ( 3 1 a , , c ) , w h i c h simplifies analysis a n d extends t h e validity of t h e linearized perturbation equations t o maneuvers involving high pitch attitude a n d l a r g e b a n k a n g l e s .

T h e u n c o u p l e d , linearized perturbation equations a r e s h o w n i n Table V i n a f o r m a t w h i c h generally constitutes t h e b a s i s f o r application t o derivative d e t e r m i n a t i o n . T h e assumptions listed f o r t h e g e n e r a l equations of motion a r e a l s o valid f o r t h e equations i n t h i s t a b l e . n a d d i t i o n , i t i s assumed t h a t t h e maneuvers a r e s u c h a s t o minimize

t h e

errors

i n t h e e r m s

arising f r o m

t h e

approximation of

t h e

gravity t e r m s s h o w n i n Equations ( 4 0 a , b , c ) . l s o , i t i s assumed t h a t t h e gyroscopic couples of rotating elements a r e n o t s i g n i f i c a n t , w h i c h m a y n o t a l w a y s b e t h e c a s e . h e equations a r e complete within t h e limits of t h e a s s u m p t i o n s , and analysis would r e v e a l a l l modes of l o n g i t u d i n a l and l a t e r a l m o t i o n s .

Omission of the longitudinal f j r c e equation a n d t h e ' a i r terms i n t h e longitudinal equations ( 5 0 a , b , ) would r e m o v e t h e phugoid mcde f r o m t h e analysis o f t h e l o n g i t u d i n a l m o t i o n s , leaving o n l y t h e short-period m o d e . h i s short-period f o r m a t of the l o n g - i t u d i n a l equations i s t h e one u s - j a l l y e m p l o y e d . lthough t h e small-perturbation e q u a t i o n s , s h o w n i n T a b l e V a r e f r e q u e n t l y used i n t h e f o r m a t s h o w n t o develop r e l a t i o n s f o r derivative d e t e r m i n a t i o n , i t i s a l s o desirable t o l i s t t h e equations i n a n o p e r a - t i o n a l f o r m a t a s Laplacian transforms with Laplace operator s



2 5

Using aplace ransform s nables he ynamic properties f he irplane o e defined by eries f ransfer unctions elating he arious esponsive m otions of he irplane to isturbing nputs. H ie ransfer unctions re xtensively se d n tability nd control, andling ualities, nd utomatic light ontrol nvestigations o ssess he effects of onfiguration hanges , he ffects of particular tability erivatives, nd the ffects f hanges n utom atic ontrol ystems. They re lso helpful n btaining stability erivatives rom light a ta .

With ero nitial ondition? nd nputs ue nly o ontrol eflections, the aplace transforms f he mall-perturbation quations of m otion ake n he perational orms shown n able I s quations 6 2 a , b, c) nd 6 3 a , , c). The notations X ^ a

etc., shorn n he qut tions, a re onvenient eans of isting he a r a m e t ers .

3.6 De t erm ination f he oots f he eterm inant f he Lateral-Directional mall-Perturbation quations

Th e ollowing discussion egarding he etermination f he oots f he et erm inant of he ateral-directional mall-perturbation quations s ased n eference 7. Although he ain points re rought ut t his i m e , ecourse hould e ade o he reference or more etailed onsiderations.

3.6.1 Th e Determinant

Using he aplace ransform orma t f he ateral-directional quations Equations ( 6 3 a , , c) n able I) , the e t erm inant f hese quations ay e xpressed n ither of w o o r m a t s , a s ollows:

(i) h e n xpressed s

As" s3 s2 s = 0 ( 65 )

then

I'l'

B = (Ln IX) Nr 'L,) I Iiljtf p *x"p r z ' Vz'

C = ( N pLr - rLp) Lp I Yg - N r l£)Y >

- (1 ^ sin x ) N g - sin a - L^]

D = (N/p - pL^ ) N^p - p Lr)Y 0 Bl(fy ß>

-g{tfi $iß) N ^ C r - $j£ß) sin a

E = gl(L ß N p - pNg) 2( Cr Kfy) .

(66)

2 6

( i l ) W n e n expressed a s

su + b s

3 + c sz + d s + e 0 , ( 6 7 )

then

b EJ-M;-?.

c N ^ L ; N ; L ^ ) L^3 npß + sß-üß sm «

d (N N££> N;L;- *#)?, - glN ä +

gp - Hjfe - N ; L ^ ) in a

e = - gl(L - L ß) 2(N^I; - Üfiß)

where he rimed a lues re qual o

m huh. - I'I'

X 2

a n d l - I'I'

XX

( 6 8 )

( 6 9 )

T h e determination of t h e r o o t s of the determinant i s dependent upon t h e modes of

motion of t h e a i r c r a f t . h e modes m a y b e :

( a ) Lateral phugoid (coupling of s p i r a l and r o l l m o d e s ) and Dutch r o l l .

( b ) S p i r a l d i v e r g e n c e , r o l l s u b s i d e n c e , and oscillatory ( D u t c h r o l l ) .

3.6.2 Determination of he Roots hen Lateral Phugoid nd Dutch Roll Modes Exist

T h e determinant ( E q n . ( 6 7 ) ) c a n b e approximated b y t h e following biquadratic

d\ / bd s * + (b - - B + [ c - -

c/ s ' + - s +

c

C

J

( 7 0 )

i n w h i c h t h e f i r s t and second quadratics represent t h e D u t c h r o l l and l a t e r a l phugoid

m o d e s , respectively.

T w o s e t s of conditions must be satisfied i f Equation ( 7 0 ) i s t o b e applicable 7 .

( a ) T h e approximate nature of Equation ( 7 0 ) requires t h a t / c2 « 1 nd bd/c

2 « t o assure v a l i d i t y of t h e e q u a t i o n .

( b ) I t i s necessary t h a t 2 - 4 e d < 1 n order t h a t t h e l a t e r a l phugoid e x i s t .

Reference 1 7 points o u t , o n t h e b a s i s of l i m i t e d e x p e r i e n c e , t h a t , o r v a l u e s of / c2

of approximately 0 . 0 5 o r l e s s , d/c2 a n b e a s l a r g e a s 0 . 2 5 a n d

2/4ec s l o w a s

0 . 0 0 5 w i t h o u t compromising t h e engineering accuracy of Equation ( 7 0 ) . h u s , t h e

equation applies i f

2 7

e d -r « -j < 0.25 , cz z 0.005 —

4 e c ( 7 1 )

T h e second quadratic i n Equation ( 7 0 ) expresses t h e l a t e r a l phugoid v e r y s i m p l y ; t h u s ,

f r o m s ' + - s + - c

a n d » p h

2 £ „ h < 4 » h d

( 7 2 )

The irst uadratic n quation 7 0 ) s nwieldy. It s impler o e t ermine he characteristics of he utch oll ode y he ollowing actored orm f eterm inant

(a2 £a)ns +a#(s2 £phw „phs a^ph) = 0 .

Expansion f iiis e t erminant nd om parison ith quat ion 67) hows hat

(73 )

b = 2&n C p h^ nph

c = wn ph 2 ^ n > (2^ptf^ph>

d = (2£phwnphK & w n)^ph

e ^ptPn

( 7 4 )

S i n c e j | ph n d ( 2 £p , c ü npll) r e obtained f r o m Equation ( 7 2 ) , * n d 2 £ & )n) c a n now b e determined f r o m Equations ( 7 4 ) o r

2 ^V 2f taJnph

wn

=

c " ^ ph - (

2H(2^nph )

( 7 5 )

3.6.3 Determination of he Roots he n Spiral Divergence,

Roll Subsidence, an d Dutch Roll Mod es Exist

W h e n t h e s p i r a l d i v e r g e n c e , r o l l s u b s i d e n c e , and D u t c h r o l l modes constitute the lateral-directional characteristics of t h e a i r p l a n e , which i o normally t h e c a s e , t h e determinant a s represented b y Equation ( 6 7 ) may b e factored i n t h e following t e r m s

characterizing t h e s e m o d e s :

s + SK)( + 2 t o g + « f i ( 7 6 )

Th e oefficients b usd e (Equation 68)) n erms of he actors of

Equat ion 76) re

1 1

b = 2fyon — —D T T

c = < o2 2fiw,

VTB

+ TS/

TRTS

d - 2

) + 2H - n Va e = w . n

T R1s

(77)

W h e n he piral-mode oot, 1AS s much ess han he roll ubsidence oot,

1/TR s it usually s, he coefficients b c an d e may e approximated

to oo d degree of accuracy y

b ~ 2£aJ„ — TR

C 2 c o l + 2H ~1

d ~ » 1 — I. rp

(78)

Eliminating 1 , / T « in Equat ion 78)

c a ( 2 4 a> ) —+o,2

d ~ (2&u

or

d

(79)

Eliminating 2 £ o > n in Equations 79) rovides n ccurate solution of w2 within he

limitation hat

1 «±

or

(a >2)3 - (w 2)2 bd(a#

? = 0 (80)

2 9

Eliminating > * n Equations ( 7 9 ) t o s o l v e f o r 2£w* i t h i n t h e limitation t h a t 1A„ « 1/TR esults i n

(2 n)3 - 2b(2C«n)

2 + ( c + bz)(2 n) + ( d - c b ) . 8 1 ) T h e roll-subsidence r o o t , / T R , may n o w b e obtained f r o m coefficient o r

i n Equation ( 7 8 ) o r

1 d

TR <

1 — b - 2 £ w , TR

n

( 8 2 )

The spiral-divergence oot, 1/Tg a y o w e pproximated rom ny ne of he coefficient expressions n Equations 77 ) , such s

1 Ts «ja/ty

(83)

4 . MASS CHARACTERISTICS

T h e airplane mass characteristics - w e i g h t , location of t h e center of g r a v i t y , moments of i n e r t i a , a n d inclination of principal a x i s - significantly affect airplane m o t i o n s . rrors i n t h e knowledge o f - t h e s e quantities a r e reflected directly i n t h e flight-determined derivatives and may g o v e r n the validity of t h e derivatives i n c o m - parisons w i t h wind-tunnel d a t a . lthough possible inaccuracies i n t h e Knowledge of t h e inertia characteristics must b e g i v e n s e r i o u s consideration i n comparisons of flight-determined derivatives w i t h wind-tunnel d a t a , t h e s e derivatives have b e e n used effectively i n flight-guidance simulator s t u d i e s .

T h e w e i g h t a n d horizontal l o c a t i o n o f t h e c e n t e r of g r a v i t y a r e a l w a y s determined

e x p e r i m e n t a l l y . nasmuch a s t h e vertical l o c a t i o n of t h e center of g r a v i t y , moments of i n e r t i a , a n d l o c a t i o n of t h e principal a x i s a r e difficult t o determine e x p e r i m e n t a l l y , manufacturer1

s estimates a r e usually relied u p o n . hese estimates are considered t o b e cf sufficient a c c u r a c y f o r most w o r k involving f l i g h t t e s t s . f more precise data a r e r e q u i r e d , t h e y should b e determined b y using experimental t e c h n i q u e s .

I t would b e highly desirable t o determine a l l of t h e mass characteristics e x p e r i m e n t a l l y . his i s not a l w a y s f e a s i b l e because of t h e l a c k of proper f a c i l i t i e s . L a r g e , l e x i b l e a i r c r a f t , u c h u s t h e Boeing B - 5 2 , offer practical p r o b l e m s , n t h a t experimentally determined rolling moments of inertia w i t h w i n g s drooped would n o t b e representative of f l i g h t c o n d i t i o n s . he f o l l o w i n g discussion of the experimental determination of mass characteristics of a i r c r a f t i s intended t o s e r v e a s a guideline i n setting up suitable facilities f o r u s e w i t h most categories of a i r c r a f t .

3 0

ö

4 . 1 eight and Center-of-Gravity Location

T h e weight and longitudinal position of t h e center of gravity relative t o t h e

horizontal reference l i n e of t h e airplane f o r t h e e m p t y and g r o s s w e i g h t conditions

c a n be obtained easily by leveling t h e a i r p l a n e on suitable scales or electronic

weighing c e l l s . ith weighing c e l l s , t w o of t h e c e l l s ( R t and R2) are usually located

a t the wing jackpoints a n d t h e third c e l l ( R3) i s located a t some convenient d i s t a n c e ,

I forward or a f t of t h e wing j a c k p o i n t s . h e horizontal position of t h e c e n t e r of

g r a v i t y relative t o t h e jackpoints i s t h e n determined f r o m

M = hi IR ( 8 4 )

For aircraft operating on conventional f u e l s , the variation of the c e n t e r of g r a v i t y

with f u e l consumption c a n usually b e defined adequately b y weighing the airplane a t

s e v e r a l f u e l l e v e l s , providing t h e r e i s a predetermined sequence or mode of operation

i n obtaining t h e f u e l f r o m t h e various f u e l c e l l s . h e n t h e aircraft i s equipped w i t h

f u e l c e l l s f r o m which t h e f u e l c a n b e d r a w n s e l e c t i v e l y , t h e center of gravity position

becomes a f u n c t i o n of t h e sequence i n drawing off the f u e l f r o m t h e various c e l l s a s

w e l l a s t h e weight of t h e f u e l . n s o m e i n s t a n c e s , t has b e e n f o u n d necessary t o

account f o r fuel-tank s h a p e a n d airplane a t t i t u d e . here hazardous f u e l s a r e u s e d ,

the center of gravity i s determined experimentally f o r t h e no-fuel condition o n l y ; t h e

effect of f u e l on t h e center of gravity position i s c a l c u l a t e d . h e horizontal l o c a t i o n

of t h e center of g r a v i t y i s experimentally obtained a t l e a s t t o w i t h i n 0 . 0 1 mean a e r o -

dynamic c h o r d , h i c h i s considered adequate f o r derivative d e t e r m i n a t i o n .

During f l i g h t t e s t s , t h e c e n t e r of gravity i s obtained b y observing t h e t o t a l a m o u n t

of f u e l consumed and subtracting i t f r o m t h e takeoff w e i g h t . eference t o a c h a r t

showing t h e variation of w e i g h t w i t h center of g r a v i t y provides t h e desired a n s w e r .

A n accurate knowledge of t h e v e r t i c a l l o c a t i o n of t h e center of g r a v i t y i s pertinent

t o t h e experimental derivative s t u d i e s , insofar a s experimental determination of moments

of inertia a n d comparison w i t h wind-tunnel data a r e c o n c e r n e d . h e v e r t i c a l center of

g r a v i t y c a n b e obtained b y static or oscillatory t e c h n i q u e s . or t h e static t e s t

t e c h n i q u e s , h e airplane i s placed i n v a r i o u s p i t c h o r r o l l a t t i t u d e s . o r t h e r o l l

approach ( F i g . 1 2 ) , h e a i r p l a n e i s mounted i n a h o r i z o n t a l , w i n g s - l e v e l attitude on

knife e d g e s alined with r e s p e c t t o e a c h other i n t h e plane of s y m m e t r y of t h e a i r c r a f t .

By rolling t h e airplane to various attitude angles a n d measuring t h e reaction Rt , moment a r m yx and t h e r o l l a n g l e > using a c l i n o m e t e r , t h e v e r t i c a l position of t h e c e n t e r of gravity i s obtained f r o m t h e equation

RiYi -f

czcsn<

^

Wsin<£ ( 8 5 )

For r i g i d aircraft of t h e order of 1 5 , 0 0 0 l b , a n d u n d e r carefully controlled c o n d i t i o n s ,

t h e vertical position i s considered t o b e deterainable t o within 1 i n c h .

T o determine t h e v e r t i c a l position of t h e c e n t e r o f g r a v i t y f r o m f r e e - o s c i l l a t i - a

t e s t s , a n y o n e of s e v e r a l techniques may b e u s e d . h e s i m p l e s t technique consists of

changing t h e equivalent torsional spring constant f o r pitching or rolling moment of

inertia t e s t s . o r rolling-oscillation t e s t s w i t h t h e s e t u p s h o w n i n igure 1 3 a n d

3 1

w i t h s m a l l damping e f f e c t s - a necessary condition f o r successful t e s t s - t h e equations of motion f o r t h e t w o spring conditions a r e

( I , + IXc f mz2 + «„a*)«^ + ( Ktl - W z - W ) 0 8 6 a )

dx r xc z2 czc< J 6 > 2 K t2 - x - czc)02 = 0 86b)

Considering 4 >l Acosc^t and 4 > 2 Bcoso;2t it s ound pon olving quations (86a , b) or z the ertical istance rom he knife dg e o he enter f ravity, that

Kts-Kt /p,)2 W czc

wLi PX/P2)2

]

T h e equivalent t o r s i o n a l spring c o n s t a n t , t m a y b e changed f r o m t x o Kt2 y changing t h e l i n e a r s p r i n g s o r t h e d i s t a n c e w h i c h i s perpendicular t o t h e spring ( s e e Figure 1 3 ) . h e change i n linear springs i s probably t h e more desirable a p p r o a c h .

Inasmuch a s t h e rolling-oscillation t e s t s e t u p discussed constitutes a n inverted p e n d u l u m , i t i s i m p e r a t i v e t h a t t h e equivalent t o r s i o n a l s p r i n g c o n s t a n t ,

t , b e

greater t h a n z + Wcz„ o r stability o f s e t u p . l s o , t h e a c c u r a c y of t h e r e s u l t s d e p e n d s u p o n avoiding secondary spring actions of tiebacks a n d s t r u c t u r a l f l e x i b i l i t y , w h i c h c o u l d inadvertently r e s u l t i n a l o w e r effective spring constant t h a n expected because of a n e q u i v a l e n t series action of t h e s e c o n d a r y unwanted spring action w i t h t h e intended s p r i n g .

4 . 2 oments of Inertia

T h e moments o f inertia of a n airplane a r e usually calculated during t h e d e s i g n p h a s e a n d a r e b a s e d o n estimated w e i g h t s a n d centroid l o c a t i o n s f o r various parts of t h e a i r c r a f t . hese calculated moments of i n e r t i a a r e considered t o b e adequate f o r most analyses w h e n t h e re3jlts a r e t o b e used i n simulator s t u d i e s . o w e v e r , should e x p e r i m e n t a l determination of the i n e r t i a b e r e q u i r e d , methods a r e available ( s e e References 1 8 t o 2 1 ) . h e methods a r e generally restricted t o r i g i d a i r c r a f t a n d t o aircraft w h o s e w e i g h t , a s w e l l a s t h e s a f e t y precautions of t h e e x p e r i m e n t , w i l l permit pivoting t h e a i r c r a f t o n knife e d g e s a n d suspending i t f r o m overhead c a b l e s .

Schematic representation of t y p i c a l methods f o r determining t h e r o l l i n g a n d pitching moments of inertia a r e illustrated i n F i g u r e s i 3 a n d 1 4 , r e s p e c t i v e l y . quation ( 8 6 ) i s applicable t o t h e determination o f rolling moments of inertia i n accord w i t h Figure 1 3 , with consideration g i v e n t o t h e proper interpretation of t h e l e n g t h s and z c t o t h e mountings s h o w n . n Figure 1 4 , c r a d l e w e i g h t i s z e r o . h e yawing moment of inertia m a y b e s a f e l y determined f r o m a cable-suspension method used t o determine t h e inclination of t h e principal a x i s ( F i g u r e s 1 5 a n d 1 6 ) , w h i c h i s discussed s u b s e q u e n t l y .

unless precautions a r e t a k e n i n e v e r y d e t a i l of a n experimental s e t u p , difficulties m a y b e encountered b e c a u s e of f l e x i b i l i t y o f e x p e r i m e n t a l c o m p o n e n t s , w h i c h w i l l a l t e r t h e effective s p r i n g c o n s t a n t , t o r modify t h e free-oscillation p i v o t a l point r e l a t i v e t o t h e c e n t e r of g r a v i t y of t h e a i r c r a f t . n o n e i n s t a n c e of determining t h e pitching moment of i n e r t i a w h e n t h e a i r c r a f t w a s supported a t t h e w i n g jackpoints and



3 2

oscillated with the s p r i n g a t t h e n o s e , t h e wing section w h i c h haJ b e e n considered rigid was observed t o flex a s t h e aircraft o s c i l l a t e d . his f l e x i n g caused t h e axis of rotation t o s h i f t f o r w a r d and downward f r o m t h e l i n e through t h e j a c k p o i n t s .

A common f a u l t i s t h e u s e of f l e x i b l e cables as tlebacks f o r t h e springs and c o n - nection from t h e spring t o t h e a i r c r a f t . nder no condition should f l e x i b l e connections be u s e d , inasmuch a s they constitute springs i n s e r i e s with t h e a c t u a l intended springs e m p l o y e d ; t h u s , t h e s y s t e m f r o m tieback to aircraft represents a much softer spring than i n t e n d e d . t should also b e noted t h a t , o n s o m e a i r c r a f t , attaching t h e s p r i n g t o t h e a f t portion of t h e fuselage would b e a n e r r o r , s i n c e t h e a f t portion of t h e f u s e l a g e would constitute a relatively f l e x i b l e structure and alter t h e effective spring c o n s t a n t .

Serious errors c a n also r e s u l t w h e n knowledge of t h e center-of-gravity l o c a t i o n i s inaccurate and w h e n t h e l i n e of action of t h e spring f r o m t h e attach point t o t h e a i r - craft i s not perpendicular to the plane formed b y t h e axis of rotation a n d t h e point of spring attachment o n t h e aircraft ( F i g . 1 3 ) .

G e n e r a l l y , t h e inertia characteristics are determined f o r no-fuel conditions because f u e l sloshing t e n d s t o bring i n a b e a t action i n t h e oscillatory m o t i o n s . h e n determination i s attempted w i t h f u e l o n b o a r d , t h e d i f f e r e n c e i n oscillatory modes b e t w e e n t h e sloshing f u e l and t h e a i r c r a f t should b e a s l a r g e a s i s p r a c t i c a l , with d u e regard t o s a f e t y of t h e s e t u p , t o minimize t h e b e a t action a n d permit determination of t h e n a t u r a l f r e q u e n c y of oscillation of t h e a i r c r a f t .

Measuring t h e inertias of v e r y l a r g e aircraft i s difficult a n d i s compounded w i t h f l e x i b l e a i r c r a f t . u c h measurements a r e n o t i n t h e r e a l m of t h e methods d i s c u s s e d . A nique f a c i l i t y designed t o e n h a n c e t h e feasibility f o r determining t h e moments of inertia of l a r g e aircraft a b o u t a l l t h r e e axes i s located a t t h e US Air F o r c e Flight Test C e n t e r , E d w a r d s , C a l i f o r n i a , U S A . t s capabilities c o v e r a w e i g h t r a n g e f r o m 30,000 t o 3 0 0 , 0 0 0 a n d moments of i n e r t i a f r o m 2 5 0 , 0 0 0 t o 1 0 x 1 0

6 s l u g f t

2. The facility enables t h e determination o* aircraft moments of inertia f r o n measurements of changes i n pendulum characteristics resulting f r o m t h e addition of a n a i r c r a f t t o a f r e e l y oscillating p l a t f o r m . h e b a s i c elements of t h e f a c i l i t y c o n s i s t of t h e p l a t f o r m , a c o n t r o l c o n s o l e f o r activating various s y s t e m s w h i c h r e a d y t h e platform f o r o s c i l l a t i o n , and a n instrumentation c o n s o l e f o r regulating t h e amplitude and measuring the period of t h e o s c i l l a t i o n s . h e platform i s a n i n t e g r a l c r u c i f o r m structure 110 f t long a n d 80 f t w i d e , with i t s l o a d i n g surface f l u s h w i t h t h e s u r r o u n d -

i n g f l o o r s p a c e . h e apparatus e m p l o y s s p e c i a l hydrostatic bearings ( i d e n t i c a l t o those used i n t h e 2 0 0 i n . Palomar t e l e s c o p e ) t o s u p p o r t t h e p l a t f o r m , w h i c h i s l o c k a b l e i n t w o a x e s w i t h oscillation a b o u t t h e axis of i n t e r e s t .

T o contend w i t h the p r o b l e m of aircraft f l e x i b i l i t y , stiffening jacks a r e used t o support t h e aircraft s t r u c t u r e . s a r e s u l t of t h e stiffening o p e r a t i o n , flexibility effects a r e considered t o b e l e s s t h a n 4% i n r o l l a n d 2% i n p i t c h .

T h e experimental e r r o r i n t h e methods discussed i s of t h e order of ±5% or l e s s .

4.3 nclination of P r i n c i p a l Axis

T h e i n c l i n a t i o n of t h e principal a x i s of t h e a i r p l a n e i s o n e of t h e i n o r e d i f f i c u l t

quantities t o d e t e r m i n e e x p e r i m e n t a l l y . n e r r o r o f 1 / 4 ° i n t h e v a l u e of t h e i n c l i n a - t i o n of t h e principal a x i s c a n significantly a f f e c t s o m e of t h e d e r i v a t i v e s . h e method of Reference 2 2 i s c o n s i d e r e d accurate t o 1 / 6 ° .



3 3

This method consists of f i n d i n g t h e direction of t h e r e s t o r i n g - t n o m e n t vector w h i c h produces no rolling moment relative t o t h e b o d y x-axis during t h e yawing oscillations of t h e airplane a s a spring mass system while suspended b y means of a c a b l e attached t o a hoisting s l i n g . i g u r e 1 5 s h o w s s c h e m a t i c a l l y , and Figure 1 6 s h o w s photogra- p h i c a l l y , a g e n e r a l arrangement of the s e t u p . he airplane i s suspended a t a horizontal p i t c h a t t i t u d e , and yaw r e s t r a i n t i s provided b y t w o sets of springs whose l i n e s of a c t i o n l i e i n a common p l a n e . he springs should provide a pure couple a c t i o n . h e restoring-moment vector acts normal to t h e plane of t h e s p r i n g s . he springs may be attached t o s h o r t , rigid mounting brackets located below t h e w i n g s equidistant f r o m t h e plane of s y m m e t r y o r t o brackets mounted below t h e f u s e l a g e ahead of a n d behind t h e c e n t e r o f g r a v i t y . n t h i s r e s p e c t , t h e wing mounting arrangement i s most c o n v e n i e n t a n d l e s s t i K i e - c o n s u m i n g . t i s e s s e n t i a l t h a t t h e springs provide a pure couple a c t i o n .

As t h e airplane oscillates i n yaw w i t h v a r i o u s inclinations of t h e plane of t h e spring c o u p l e ( a n g l e 8 i n Figure 1 5 ) , s o m e coupling i s present between yaw N n d r o l l , w h i c h results i n a c e r t a i n a m o u n t of rolling o s c i l l a t i o n . his i s s h o w n i n t h e following equations where t h e subscript denotes t h e reference attitude of t h e a i r p l a n e :

I x rPr - IxrVr L 88>

^ r J - r " I x r z i - P r

N

• 8 9 ) A t s o m e o n e v a l u e of „ , h o w e v e r , h e rolling motion accompanying t h e yawing motion

Oft)

i s z e r o I p l / l r l = 0 ) . n this situation t h e preceding equations r e d u c e t o

( 9 0 ) X rZ rrr = L

Izjir = N

However a s hown n igure 5 ,

tan8sp = -I

N

Hence

tan8sp = Ixrzr

Jzr

( 9 1 )

( 9 2 )

Inasmuch a s t h e inclination of t h e principal a x i s i s g i v e n b y t h e wellknown expression

2 1 tan2e rZr

. 9 3 ) I z r - I xr

substitution of Equation ( 9 2 ) f o r XZr n Equation ( 9 3 ) g i v e s

t a n 2 £ = - a i

T*"J2 M) I zr - Ix-,



I

1

3 4

The value of Zr s determined as a byproduct of the t e s t by using

CeosS. lzr

_ 8 £ or ( 9 5 )

H o w e v e r , Xr must be determined f r o n other t e s t s .

Figure 1 7 shows a typical experimental plot of t h e variation of p | / | r j ith

f o r determining the v a l u e of Sgp a t which p | / | r | s z e r o . n obtaining t h e tests points shown in t h e f i g u r e , the f l i g h t t e s t r o l l - and yaw-rate gyros mounted i n t h e airplane were used t o obtain oscillograph records f o r determining p | / | r | r o m the transient o s c i l l a t i o n s .

The measured values of moments of inertia relative t o t h e reference axes and t h e determined inclination of the principal axes may be used t o determine t h e principal moments of i n e r t i a s , , 0 and Zo , by using the following equations

l20

I x r - I x rzr tan

e

I z r + J t x r Z r

tan e

( 9 8 )

( 9 7 )

Although no mention was made of the effects of air mass o n t h e experimental values of moments of i n e r t i a , t h e effects should be considered and corrections applied if n e c e s s a r y . eference 2 3 provides formulas t o correct f o r air-mass e f f e c t s .

Formulas f o r transferring moments of inertia f r o m o n e s e t of axes t o another w e r e presented i n Section 3 . 1 .

5 . INSTRUMENTATION

Basic t o a n analysis of f l i g h t data i s t h e i n s t r u m e n t a t i o n . onsiderable instrumen- t a t i o n research has been i n progress and many f l i g h t t e s t Instruments have b e e n d e v e - l o p e d t o improve the linearity of r e s p o n s e , r e s o l u t i o n , dynamic response c h a r a c t e r i s t i c s , r e a d a b i l i t y , r u g g e d n e s s , and reliability of calibrations over varying operating c o n - ditions a n d extended periods of t i m e . n a d d i t i o n , t h e application of the instruments requires knowledge of mounting a c c u r a c y , s o u r c e s of e r r o r i n t h e f l i g h t r e c o r d s , a n d methods of correcting t h e e r r o r s . nadequate appreciation of t h e i n s t r u m e n t c h a r a c t e r -

i s t i c s , mounting a c c u r a c y , and possible influence of s o u r c e s of e r r o r serves a s a detriment t o t h e successful application of new techniques of analysis as w e l l a s a d e t r i m e n t t o t h e analysis b y approximate m e t h o d s .

I n the following d i s c u s s i o n , sufficient guidelines a r e presented t o show t h e c a r e required i n the s e l e c t i o n , i n s t a l l a t i o n , and calibration of instruments t o minimize e r r o r s i n t h e analysis of f l i g h t d a t a . ndividual instruments may differ f r o m o n e organization t o another a n d t h e degree of sophistication i n instruments a n d r e c o r d e r s w i l l v a r y w i t h t h e i n d i v i d u a l I n v e s t i g a t i o n ; h o w e v e r , t h e principals of operation of t h e s e n s o r s a r e generally t h e s a m e .

"-:«»

1 4

;" ^ s » v

y M

I 135

5 . 1 S a c h N i n b e r , A l t i t u d e , a n d Dynamic Pressure

Accurate determination of Mach number i s of f u n d a m e n t a l importance i n f l i g h t testing

high-speed a i r c r a f t . i e principal m e t h o d s , discussed i n d e t a i l i n References 2 4 and

2 5 , a r e based u p o n t h e following relationship f : > r subsonic conditions ( M < 1 . 0 )

ß.ti-N""-0., t i ( 1 + 0 . 2 M )7/2 - 1 ( 9 8 )

F o r supersonic conditions ( M > 1 . 0 ) , t h e equation i s modified t o i n c l u d e t h e l o s s i n

t o t a l pressure behind t h e s h o c k wave

/ 7+i 2 (y-l)

% +1 ,1 5.76II2 \5/2- = 2 1 = 1.2M2 1. 99)

27 2 7-1 56M " -8/

\7 + +1

The m p a c t pressure Q C nd he tatic pressure p are m ea s u r ed y using pitot- static ea d nd pressure ecorders. The a x i m u m Mach umber s well s ynamic pressure hich a n e e t erm ined y using pitot-static eads s of he rder f 3.5. H i g h er peeds re primarily ependent pon nertial latforms nd adar. Dynamic

pressures t Mach umbers s he pproxim ate ange of .5 o .0 a n e e t erm ined through he s e f pherical low-direction ensor nd otal- stagnation) ressure technique.

5.1.1 Pitot-Static Head M .5)

M u c h esearch a s e en one n arious ypes nd onfigurations of otal-pressure heads o educe ngularity ffects26'27. The ype hown n Figure 8 s sed widely. Thi s e ad a s n xternal ylindrical hape , a ylindrical hamber , and 0 ° lant profile. It s nsensitive zero rror) o ngle-of-attack rom 5° o 0 ° nd p o 1 0 ° f ideslip. The rror s ess ha n % n he ngle-of-attack ange rom 10° o 2 5 ° nd 1 0 ° ideslip.

The rrangement f he tatic-pressure orifices n he ead a s een ound o e pertinent n ncreasing he ange f nsensitivity f he orifices o low ngularities. Th e rrangement s e d a s een e t erm ined rom ests of orifice onfigurations2"'29. Th e w o dentical ets r rrangements hown n igure 8 re a ch ircumferential, with ou r orifices n he op , six n he ottom , and ne n he ottom enterline behind he thers. Th e w o ets of tatic-pressure orifices re se d o provide or separate pressure ystems. One et f tatic orifices s se d or he pilot' s instruments, the ther et or light est ecording nstruments o m i n i m i z e he im e lag of esponse hat ould e ncountered ith o m m o n y s t e m . The rrangement f the orifices n ach et provides n ncreased ange of nsensitivity o ngle-of- attack; owever , It s not s nsensitive o ideslip. Large tatic-pressure rrors are ncountered t ideslip ngles reater han °, Since onstant ideslip ngles are eldom ncountered, he tatic-pressure a ta a n e eadily aired.



~ 1 36

Installation of he pitot-static ead: nstallation f he pitot-static ead equires consideration f he omplicated low ield of he irplane, hich s unction f th e airplane onfiguration s e ll s M a c h umber nd ttitude. Errors n pitot-static- head eadings esulting r o m his low ield re eferred o s petition rrors. T h e static-pressure orifices re particularly ffected y position rrors t ubsonic speeds; hus, recautions re aken o ount he pitot-static ead s a r head of he airplane s s practical.

Of he arious ypes f nstallations f he pitot-static ead - uch s os e o o m , wing o o m , nd use lage - he ose-boom nstallation s he most uitable or minimizing position rrors. In his nstallation, h o w n n igure 9, th e ead s mounted n b o o m xtending a s ar ahead of he nose of he irplane s s practical. A s eported in eference 9, he mount of error n M a c h number ue o position rror n he tatic-

pressure measurements an e elated o ertain hysical easurements n he irplane. This s h o w n n igures 0 nd 1 , which re eproduced r o m eference 0 . In igure 20 , he rror n M a c h u m b e r ue o tatic-pressure rror s plotted s atio f b o o m ength o he m a x i m u m ffective use l a g e i ameter or ubsonic, ransonic, nd supersonic peeds . In igure 1 , th e ariation n a c h umber rror ith M a c h umber is plotted or wo irplanes aving oom -length-to-fuse lage -diam eter atios of .6 0 and .95. Above M a c h umber of . 05 , th e position rror rops o ero. T h e a c h n u m b e r t hich he position rror rops o e ro s ependent pon he ope-boom geometry nd s he M a c h cumber t hich he hock ave head of th e irplane rosse s over he tatic-pressure orifices.

W i n g -b o o m nstallations f he pitot-static head re ubject o evera l i sadvantages , including possible usceptibility o he hock ave aused y he ing s e ll s he

shock ave aused y he use l a g e . This om plicates he alibration nd makes t more difficult or he pilot o ly t he esired M a c h u m b e r n he egions here he shoe* aves re n he icinity f he orifices. W i n g o o m s re sual ly o r e ensitive to ideslip nd ubject o ore a g n esponse ecause f he onger ubing equired.

Fuse lage nstallations f he ead re ubject o position rrurs, hich re difficult o s ti m at e .

Calibration: Calibration f he pitot-static ead, fortunately, involves nly he determ ination f he position rror or he tatic ressure - he otal ressure s not ffected y position rror. Various methods ha t ave ee n sed nclude he pacer method, th e ly-by tower-pass) ethod, and m odifications f he asic adar-phototheodolite ethod24. T h e acer method equires he se f acer irplane ith

cal ibrated ystem nd pecial lights or alibration urposes. T h e ly-by ethod requires g light t xtremely ow altitudes a st n nstrumented ourse. This latter method not nly equires pecial lights, bu t s azardous n d im i te d o M a c h umbers f bout .8 .

T h e adar-phototheodolite ethod ha s he dvantage f roviding alibration ata during outine esearch lights. T h e method makes se of adiosonde unit o measure static-pressure nd emperature ariations f he tmosphere ith ltitude. It ls o requires round quipment onsisting f adar nit, hototheodolite, a hronograph, and hree ameras. O n e f he ameras hotographs he adar cope n d ives he lant range; he arge t arcera ives he orrection o he l evation cales; nd he hird camera ives he levation cale. T h e irplane tself s quipped ith radar eacon

3 7

t o a s s i s t i n t r a c k i n g . he three c a m e r a s and the airplane's internal records a r e synchronized b y means of t h e c h r o n o g r a p h . he radar-phototheodolite unit determines t h e range and e l e v a t i o n a n g l e of the airplane f r o m w h i c h t h e true geometric altitude of the airplane i s determined . w i t h i n ±100 f t ) a s a f u n c t i o n of t i m e .

A cross plot of the radar-phototheodolite data (airplane altitude v s . t i m e ) with the radiosonde results ( f r e e - s t r e a m static pressure v s . a l t i t u d e ) provides a plot of t r u e free-stream static pressure a s a f u n c t i o n of t i m e . ince t h e time base of the a i r - p l a n e * s indicated static-pressure records i s synchronized with the r a d a r - p h o t o t h e o d o l i t e , a comparison of t h e a i r p l a n e ' s indicated static-pressure records with t h e cross p l o t

provides the position e r r o r , p of t h e static h e a d . h e corrected static pressure m a y now be obtained f r o m t h e relation p = p " j + Ap . he t r u e I m p a c t pressure i s now determined f r o m c = pT - p = q cl - Ap

True Mach umber: True Mach number i s determined f r o m tables of q c/p a s f u n c t i o n s o f Mach number based o n Equations ( 9 8 ) and ( 9 9 ) . he indicated Mach n u m b e r , i , a s determined f r o m 1 q cl and t h e t a b l e s , i s plotted against t h e corrected Mach n u m b e r , M , t c provide a calibration c u r v e , s u c h a s s h o w n i n Figure 2 2 , o r t h e pitot-static- head installation on the a i r p l a n e . e n e r a l l y , calibration data points f o r four or f i v e flights a r e used before t h e calibration c u r v e i s f i n a l i z e d . he s c a t t e r . AM , i n calibration points i s usually within 10.01 at subsonic and supersonic speeds and within ±0.02 a t t r a n s o n i c s p e e d s .

Pressare ltitude: ltitude i s generally expressed i n terms of "pressure a l t i t u d e " , w h i c h i s t h e a l t i t u d e i n t h e standard atmosphere tables corresponding t o t h e corrected s t a t i c p r e s s u r e . h e corrections f o r a g i v e n pitot-static pressure system a r e obtained i n t h e f o r m p^/p s . M . T h e c u r v e f o r this relationship i s derived f r o m the Mach number calibration of t h e system and t h e position e r r o r f o r t h e static pressure d e t e r - mined as a ratio of the true static pressure by the following equations f r o m Reference 2 5 :

When M < 1 . 0 ,

When M > 1 . 0 ,

Ap -1.4M2 AM\ = — 1 0 0 )

p 1 + 0 . 2 M2 \M /

Ap ( 4 . 0 \/A*i\ -2 - 1 0 1 )

w ...

I :

p \5.6M2 - 0 . 8 /\ M

I n routine t e s t s , t h r ? pressure r a t i o j / p s divided b y pi t o obtain p , w h i c h i s used to determine t h e pressure a l t i t u d e .

Dynamic pressure: The dynamic p r e s s u r e , , i s determined f r o m t h e s i m p l e r e l a t i o n

q 0 . 7 p M2 1 0 2 )

,



r

——#»—

5.1.2 U se of Spher ical Flow-Direc t ion Sensor o Obta in Dynamic Pressure

In he bsence of ru e M a c h umber, uc h s h e n light s eyond th e practical Unit of th e pitot-static ube I f ~ 3.5), technique has een volved o btain he dynamic ressure , n he higher upersonic nd hypersonic egions, irectly rom he total -pressure ort "f a pherics. ' lnw-direction ensor31. H ie low-direction ensor, described n ore detail n ection .3 , is movable phere mounted t he os e f th e irplane o orm "bal l ose". The otal-pressure or t ectors nto he esultant velocity t he ensor.

Inasmuch s q 0.7 M 2 and, from he Rayleigh pitot ormula,

— = f(M)

where

f(M) = (7 + )M '

7/(7+1) 2 > M Z - 7 - )

7+

1/(7+1)

( 1 03 )

th e ynamic pressure an e xpressed s

5 = 0 .7p [ f (M)]pT

or q P(M)

( 1 0 4 )

( 105 )

A plot of q/pT ersus M ( F i g . 2 3 ( a ) ) shows that this ratio varies only about 5% i n t h e Mach range above 2 . 5 . s c result of this s m a l l variation i n / p . n u m b e r s , i t w a s suggested t h a t a n as

' i n d i c a t e d " dynamic p r e s s u r e ,

a t t h e higher Mach , c o u l d b e expressed

» i K p T ( 1 0 6 )

Figure 3(b) hows he atio f ndicated o ru e ynamic ressure, qi/ for wo values f K Using K 0 . 5 26 is % high t M .1 and .5% low t M 7

5.1.3 Pressure-Recording Instruments

Selection f he ressure-recording nstrum ents n heir anges or iven nstallation epends n he ltitude nd M a c h umber ange - .er hich pecified ttainable M a c h umber ccuracy s es ired. W h e n ests re o e onducted t ne ltitude, it is o roblem o elect pressure-recording nstrument o rovide he equisite accuracy. F or ests onducted over ar ge ange of ltitudes, he equisite ccuracy m a y e ttained y sing om bination f im ited-range nstruments . Considerations f th e pressure im e a g equire ha t he nstrument olume emain s m a l l s possible, thus necessitating n valuation f nstrument ccuracy ith onsideration or he errors aused y he dded im e a g of m ult iple- instrum ent nstallation.



1

3 9

The ag n esponse at he ecorder ervo or pilot display, s he ase a y e, an

b e calculated rom he ollowing ormula from Reference 2) , hich akes nto ccount the ense ine, nstrument olume, nd pressure

(107)

where

K = 1 2 8 M 0I(V 0O

7rpD '

X a g n esponse, sec

M 0 iscosity of he luid, lb sec/in"2

I ength of ense ine, in

V ol nstrument olume, in"3

P ean pressure n ense ine, lb/in2

D iameter of ense ice, in .

Several anges of nstruments are available or oth he static-pressure nd otal- pressure ecorders. Fo r he static-pressure ecorders, the ower-range nstruments

require emperature calibration. Hysteresis nd riction errors, nd emperature errors, should e within 1/2% of ange or better.

5 . 2 o n t r o l Position Transmitters

C o n t r o l position t r a n s m i t t e r s , commonly referred t o a s CPT u n i t s , s e n s e the c o n t r o l -

s u r f a c e deflections a n d m u s t be a c c u r a t e and sensitive enough t o measure s m a l l

d e f l e c t i o n s . ransmitters of t h e sliding contactor t y p e c h a n g e t h e r a t i o of resistance

i n t w o a r m s of a Wheatstone b r i d g e c i r c u i t . ny variation i n t h e resistance of t h e

a r m s unbalances t h e c i r c u i t and c a u s e s c u r r e n t t o flow t o t h e r e c o r d i n g galvanometer

( s e e Figure 2 4 ) .

I n a properly installed s y s t e m , t h e p h a s e l a g b e t w e e n t h e transmitter a n d t h e r e -

corder should b e n e g l i g i b l e .

h e e r r o r s d u e t o h y s t e r e s i s , z e r o s h i f t , e m p e r a t u r e , a c c e l e r a t i o n s , or vibrations should a l s o b e n e g l i g i b l e .

The transmitters a r e f i r m l y mounted a t t h e c o n t r o l surfaces t o eliminate t h e e f f e c t

of control-system d e f o r m a t i o n s . h e s p a n w i s e l o c a t i o n of t h e transmitter g i v e s a n

approximate spanwise surface d e f l e c t i o n .

Zero checks a r e made before a n d after e a c h f l i g h t to d e t e c t a n y z e r o s h i f t i n t h e

galvanometer recording s y s t e m .

f * i

5 . 3 ngle-of A t t a c k a n d Sideslip

5.3.1 Vane-Type low-Direction Sensors O f he various ypes of low-direction devices or ensing ngle f attack nd

sideslip p o M a c h umber of pproximately 3.0, good ccuracy nd eliability s

r - - < . - . '

4 2

t h e n t h e spherical sensor i s i n t h e z e r o position ( a x i s alined w i ^ h t h e a i r p l a n e ) , t h e a-ports are 4 2 ° above and below t h e reference l i n e i n t h e v e r t i c a l plane o f symmetry of the aircraft and t h e / i - p o r t s a r e 42° on either s i d e of t h e reference l i n e i n the transverse p l a n e .

T h e sphere constitutes the outer gimbal of a two-gimbal pivot system i n w h i c h t h e outer g i m b a l i s pivoted t o t h e i n n e r g i m b a l w h o s e p i v o t a l axis i s f i x e d a n d i s normal t o t h e plane of symmetry of t h e a i r p l a n e . s t h e sensing sphere s e e k s n u l l readings i n e a c h of i t s t w o sets of static-pressure p o r t s , t h e gimbals rotate a b o u t t h e i r r e s - pective a x e s . h e inner g i m b a l , rotating a b o u t i t s f i x e d a x i s , w h i c h i s normal to t h e plane of s y m m e t r y , sweeps a n angle i n t h e plane of s y m m e t r y . h e outer g i m b a l , w h o s e pivotal a x i s i s mounted on t h e i n n e r g i m b a l and remains i n t h e plane of s y m m e t r y at a l l t i m e s , sweeps a n a\gle ß i n a plane w h i c h i s perpendicular t o t h e plane of s y m m e t r y ; this p l a n e i s t L * > transverse plane of t h e stability a x i s s y s t e m of t h e a i r c r a f t . h e and ß a n g l e s nicked off by synchros a r e t h e aerodynamic a n d ß ngles of t h e a i r p l a n e .

T h e inherent a c c u r a c y of t h e spherical sensor i s of the order o f ±0.5° or better f o r dynamic pressures i n excess of 2 0 l b / f t

2. t high angles of attack i n excess of

approximately 2 6 ° a t l o w dynamic pressures of a b o u t 4 0 l b / f t2 and l e s s , t h e i n d i c a -

t i o n s are s u b j e c t t o l a r g e e r r o r s , possibly d u e t o f l o w interference of t h e l i p on t h e collar of t h e h o u s i n g .

5. 4 Angular Velocities and Accelerations

T h e angular velocity and angular acceleration relative t o a n y o n e a x i s c a n b e sensed

b y i n d i v i d u a l s e n s o r s or sensed a n d recorded i n a convenient packaged u n i t . igures 3 9 ( a ) and 3 9 ( b ) show t h e details of t h e angular-velocity a s p e c t o f a NACA d e s i g n e d , packaged u n i t w h i c h includes t h e r e c o r d e r . h e angular acceleration sensing a n d r e - cording involves a relatively s m a l l extension of this u n i t . he operation of the u n i t depends upon t h e precessional f o r c e of a restrained g y r o motor w h e n t h e u n i t i s s u b - jected to a n angular rate a b o u t a n axis w h i c h i s perpendicular t o b o t h t h e a x i s of rotation of the g y r o motor and t h e axis of rotation of t h e g i m b a l r i n g s . he g y r o - s c o p i c e l e m e n t i s t h e rotor o f a synchronous m o t o r . h e sensitive element i s r e s - trained b y a precision helical s p r i n g . h e moving s y s t e m i s damped b y rotating a n aluminium d i s k i n t h e f i e l d of a strong permanent m a g n e t . h e angular-velocity measurement i s made b y optically recording on t h e f i l m t h e angular displacement of t h e g i m b a l . Sensitivity of t h e angular-velocity recorder c a n b e adjusted b y rotating t h e actuator a r m along t h e mirror s t a f f t a i l .

Angular acceleration i s obtained b y differentiating t h e g i m b a l m o t i o n . he differentiation i s accomplished by mounting a c o i l i n a magnetic f i e l d a r d driving i t f r o m t h e damping s h a f t s o t h a t i t r o t a t e s w i t h s p e e d proportional t o t h e angular velocity of t h e g i m b a l . h e output v o l t a g e , w h i c h i s proportional t o t h e angular a c c e l e r a t i o n , i s recorded o n t h e f i l m b y a self-contained reflecting g a l v a n o m e t e r .

5 . 4 . i nherent Accuracy

I n well-designed angular-velocity s y s t e m s , t h e reading a c c u r a c y i s of t h e order o f 0.5% of f u l l scale or b e t t e r ; t h e errors due t o friction a n d hysteresis a r e l e s s t h a n 1 f f u l l s c a l e , a n d the c h a n g e i n sensitivity f r o m l a r g e c h a n g e s i n temperature s h o u l d



\ >

.W ' I

4 3

b e a s s m a l l as p o s s i b l e . rrors d u e to linear accelerations of 5 g should be l e s s t h a n 1 % . he sensor should provide f l a t response characteristics within ±1% f o r all anticipated impressed f r e q u e n c i e s . he phase l a g ( t i n e l a g ) i s a f u n c t i o n of damping r a t i o and undamped natural f r e q u e n c y of t h e s e n s o r .

Recorded angular accelerations a r e s u b j e c t t o the errors f o u n d i n t h e a n g u l a r - velocity r e c o r d . n t h e NACA acceleration-velocity packaged u n i t , additional errors a r e introduced b y t h e acceleration recording g a l v a n o m e t e r ; inasmuch a s t h e a n g u l a r - acceleration pickup i s a differentiation d e v i c e , t h e response a n d phase l a g of t h e aicelerometer a n d velocity portions of t h e u n i t a r e s i m i l a r .

5.4.2 M ount ing nd Corrections

I t i s iuportant that the instrument mounting be r i g i d . lthough s m a l l - a m p l i t u d e , high-frequency vibrations may not b e apparent o n t h e velocity t r a c e , t h e vibrations c a n introduce considerable noise i n the acceleration t r a c e .

Angular-velocity g y r o s a r e subject t o coupling errors caused b y a n interference ( a i r p l a n e ) angular velocity about t h e spin axis of t h r g y r o r o t o r . are should be exercised i n orienting the instrument during mounting s o as t o s u b j e c t i t s spin axis t o t h e minimum interference angular v e l o c i t y . mathematical s t u d y of t h e coupling e r r o r i s presented i n Reference 3 9 . he interference angular velocity ( a l s o k n o w n a s the q r a t e ) affects t h e s e n s i t i v i t y of t h e i n s t r u m e n t , t h e undamped natural f r e - q u e n c y , a n d t h e damping r a t i o . h e e x t e n t of t h e e r r o r s i s a f u n c t i o n of t h e g i m b a l t i l t , w h i c h , i t s e l f , i s a f u n c t i o n of the g y r o sensitivity i n spring-restrained i n - struments and t h e magnitude of ' . l i e interference angular v e l o c i t y . his i s illustrated i n Figure 40 f o r a n a n g u l a r - ve city u n i t having a static sensitivity of 0 . 2 5 6 r a d i a n p e r r a d i a n per s e c o n d . > j c r ^ a s e i n sensitivity would r e d u c e t h e coupling e r r o r ; h o w e v e r , a decrease i s not always d e s i r a b l e . o minimize t h e coupling e r r o r f o r a n y o n e i n s t r u m e n t , t h e a x e s should b e oriented a s f o l l o w s :

Desired

Velocity

Input

Axis

Spin

Axis

Output

Axis

R o l l r a t e , X z y

Pitch r a t e , y z X

Yaw r a t e , z y X

i f

- n

4 i ' i 1

Alinement of t h e sensLig-recording units should be within ±0.2° of correct orientation with relation to t h e body a x e s , ndetected misalinement has b e e n known t o r e s u l t i n erroneous values of highly pertinent d e r i v a t i v e s , w h i c h resulted i n misleading r e s u l t s i n analog-simulated rolling c h a r a c t e r i s t i c s . n a n y correction f o r m i s a l i n e m e n t , t i s pertinent that t h e recorded values b e corrected f o r phase l a g of t h e instrument prior t o insertion i n t h e correction e q u a t i o n s . i m u l t a n e o u s l y , t h e response of t h e i n s t r u - ment should b e c h e c k e d , f t h e r e i s a n y appreciable deviation f r o m t h e damping r a t i o o f 0 . 6 5 , t o ascertain t h e percentage e r r o r i n magnitude of t h e indicated quantity d u e t o t h e dynamics of t h e i n s t r u m e n t . isalinements i n t h e mounting of t h e unit may b e accounted f o r b y using t h e equations s h o w n i n Figure 3 1 .



a ipg^ssrssrg^E ,..,,.,4 . . .■ ITT-» Tuff "* ' "

4

r 44

5.5 inear Accelerations

In eneral , flight esting s one ith he ean-type inear acceleroneters hich a re vailable a s ingle-component r hree-component units. Drag eterm ination s frequently a d e ith ingle-component units30. The eam-motion estraining orce s general ly upplied by a pair of opposed helical prings. The ensitivity nd u n d a m p e d natural requency re ependent pon he prings sed.

5.5.1 nherent Accuracy I n properly designed beam-type linear a c c e l e r o m e t e r s , sensitivity and zero changes

f r o m random causes a r e less t h a n 0.5% of f u l l s c a l e . he s e n s o r should have a damping r a t i o of 0.65 and a sufficiently high undamped natural f r e q u e n c y t o provide f l a t r e s -

ponse characteristics within ±1% f o r Impressed frequencies up t o 6 0% of the undamped

natural frequency of t h e s e n s o r . ach linear accelerometer i s affected by a n inter-

acting acceleration acting along the b e a m . he effect i s generally s m a l l b u t should

net be arbitrarily i g n o r e d .

5.5.2 ounting nd Corrections The nstrument hould e mounted s close o he center of gravity of he airplane

as possible. It hould e igidly astened on igid mounting attached o he rimary

structure of the airplane o void or at east minimize xtraneous vibratory accelerations. It hould e alined o within ±0.2° of correct orientation with elation o

all hree eference xes. W h e n he nstrument s no t mounted at he enLer of gravity of he airplane, corrections of the ndicated eadings o he center of gravity must be made e y using the expressions hown n Figure 2. The equations or normal acceleration, aQ -nd ransverse acceleration, at can e inearized nd corrections hus simplified y mounting he accelerometers n he plane of ymmetry long he x-axis.

5.6 hase La g and esponse

Since several ndividually ecorded quantities are utilized n he determination of various derivatives, t s mportant hat he phase-lag time-lag) haracteristics of each ecording nstrument e aken nto consideration. Fo r ys t ems where all he quantities an e ecorded n electrical alvanometers, it s generally possible o

equalize he ndividual hase ags y proper hoice of the requency esponse of he recording ys tem. Where his s no t possible, s n he se of certain cf he self-

recording N A S A nstruments, hase-lag corrections must e onsidered nd applied o bring all pertinent quantities nto correct im e elationship.

Phase-lag corrections must e applied efore making any corrections or mlsal inement . Corrections or mlsalinement must e made before correcting he ane nd inear-accelerometer ecords o he center of gravity of he airplane.

Because of the nature of he control nputs, hase-lag corrections an e applied

simply y shifting he data im e scale"0, s n he determination of control derivatives,

or y correcting phase-angle elationships, s n he ime-vector method of analysis. This s ccomplished y etermining he ndamped natural requency of he airplane

from ree-oscillation maneuvers nd Figure 3. Amplitude corrections are not equired, since he nstruments have flat esponse characteristics.



ill i iiiiiai IBI.I -.—>m.. n i ' ^ ' T i Li p

4 5

I

n

When t h e instruments are not sufficiently damped t o provide f l a t response character- i s t i c s , corrections t o t h e magnitudes of the recorded quantities m a y b e determined f r o m Figure 3 4 .

5 . 7 anges a n d Sensitivity

Instruments used f o r studies of g e n e r a l handling qualities have relatively l o w sensitivities i n order t o accommodate t h e n o r m a l f l i g h t r a n g e a n d a r e used f o r approximate evaluation of derivatives i n conjunction with t h e s e s t u d i e s . or accurate e v a l u a - t i o n

of t h e

d e r i v a t i v e s ,

using

s m a l l

disturbance m a n e u v e r s ,

sensitive

g y r o s

a n u a c c e l e r o -

meters a r e installed t o supplement or r e p l a c e those used f o r t h e handling-qualities s t u d i e s . he ranges and sensitivities of t h e instruments are usually selected a f t e r studying f l i g h t t e s t records of small-perturbation maneuvers performed over a Mach number r a n g e during p i l o t f a m i l i a r i z a t i o n f l i g h t s w h e n t h e airplane i s equipped with general-purpose f l i g h t t e s t i n s t r u m e n t s . h e i n c r e a s e i n sensitivity of a n y one i n - strument must b e accomplished with d i s c r e t i o n , inasmuch a s a n optimum sensitivity i s attained beyond which a n y i n c r e a s e m a y r e s u l t simply i n a f a l s e s e n s e of a c c u r a c y .

T a b l e V I I shows t h e characteristics of instruments which a r e desirable f o r derivative investigations f u r one high-performance airplane w h e n t h e pulsed free-oscillation maneuver i s e m p l o y e d . h e listed i n s t r u m e n t n a t u r a l frequencies a r e more t h a n adequate t o maintain f l a t response characteristics d u r i n g f o r c e d portions of t h e maneuver up to

t h e anticipated maximum frequencies f o r a l l recorded q u a n t i t i e s .

5 . 8 u l s e Code Modulation ( P C M ) Data-Acquisition Systems

I n the preceding considerations of i n s t r u m e n t a t i o n , emphasis was placed on f a c t o r s t h a t a f f e c t t h e accuracy o f i n d i v i d u a l s e n s o r s . elf-contained sensor-recorded units a r e c o m p a c t , r e l i a b l e , and a c c u r a t e . h e use o f sensors wired t o r e m o t e recorders c a n introduce degradation i n t h e a c c u r a c y of t h e o v e r a l l sensor-recorder s y s t e m ; how- e v e r , s u c h systems a r e used t o keep t h e instrumentation v o l u m e t o a minimum where s p a c e i s a prime f a c t o r a n d a l a r g e number o f parameters a r e i n v o l v e d . s t h e number of sensed a n d recorded parameters i n c r e a s e s , t h e t i m e l a g i n t h e r e c o v e r y of t h e data f o r t h e user i n c r e a s e s . n f l i g h t t e s t investigations w h e r e t h e bulk of t h e i n s t r u - mentation i s a serious problem or w h e r e t h e number of parameters recorded may constitute

a serious t i m e l a g i n t h e r e c o v e r y of t h e data f o r t h e u s e r , a sophisticated d a t a - acquisition system i s available t o alleviate t h e s e p r o b l e m s . his s y s t e m , originated t o f u l f i l l t h e needs of t h e s p a c e i n d u s t r y , n which transducers of superior quality a r e u s e d , s capable o f handling t h e d a t a t o reasonable accuracy ( 0 . 2 % t o 1 % ) . he s y s t e m , referred t o a s t h e P C M s y s t e m , c o n v e r t s t h e analog s i g n a l f r o m t h e s e n s o r t o d i g i t a l f o r m a t and records t h e digitized data o n t a p e o n a time-sharing b a s i s .

♦

Figure 3 5 ( a ) s h o w s a schematic d r a w i n g of a n airborne PCM s y s t e m . h e analog s i g n a l s f r o m t h e sensors g o t o a P C M encoder t o c o n v e r t t h e s i g n a l t o a n identification c o d e d , digitized f o r m a t . h e c o d e d , digitized s i g n a l s a r e t h e n reccrded i n p a r a l l e l o n a n onboard t a p e recorder o n a time-sharing b a s i s . o r e c o v e r t h e d a t a , t h e t a p e d signals a r e processed through a P C M d e c o m m u t a t i o n , w h i c h i d e n t i f i e s ( u n s c r a m b l e s ) t h e i n d i v i d u a l s e r . s o r s i g n a l s , t o a f o r m a t computer t o p r o v i d e real-time data outputs i n t h e f o r m of

strip charts or oscillograph readouts f o r a n immediate l o o k a t t h e d a t a . h e real-time d a t a a r e also transmitted t o a general-purpose computer w h i c h t a b u l a t e s , p l o t s , or performs complex manipulation of t h e data i n engineering u n i t s .



^ ' '. ■ -T

4 6

Where weight is a serious f a c t o r , Figure 3 5 ( b ) shows a schematic drawing of t h e P O M s y s t e m using t e l e m e t r y . h e main differences b e t w e e n t h e telemetered a n d airborne PCM systems involves the transmission of the c o d e d , digitized signals i n series t o t h e decommutator (instead of parallel t o t h e r e c o r d e r ) , which provides time synchronization of t h e signals before the signals a r e t a p e d . n processing t h e d a t a , t h e f o r m a t c o m - puter properly identifies t h e i n d i v i d u a l data c h a n n e l s f o r real-time data o u t ; n i t .

As stated e a r l i e r , the PCM system i s a sophisticated o p e r a t i o n . ne i n s t a l l a t i o n a t the NASA Flight Research C e n t e r , Edw d s , C a l i f o r n i a , i s designed to handle 1 5 , 4 0 0 data

samples per second from 7 7 t o a maximum of 800 data s o u r c e s

6 . FLIGHT TEST TECHNIQUES

Determination of the f l i g h t test techniques t o be used i n obtaining stability a n d control derivatives f r o m flight data i s governed by a number of f a c t o r s , including the methods of analysis t o b e e m p l o y e d . uccessful mathematical methods of analysis have been limited t o the linearized f o r m of the equations of motion a n d thus r e s t r i c t t h e maneuvers t o s m a l l p e r t u r b a t i o n s . nasmuch a s stability derivatives are f u n c t i o n s of angle of a t t a c k and Mach number a n d , t o some e x t e n t , aeroelasticity of t h e a i r f r a m e , the controlled variables a r e Mach n u m b e r , load f a c t o r , a n d pressure a l t i t u d e . o r safety of f l i g h t , t h e investigation of t h e stability and c o n t r o l characteristics i s

usually initiated with a g r a d u a l buildup of maneuvers a t high altitude w h e r e t h e natural frequency and damping of t h e airplane a r e l o w e r than a t l o w altitudes a n d thus p e r m i t better c o n t r o l . t i s d e s i r a b l e , w h e n f e a s i b l e , t o have t h e maneuvers performed with t h e airplane w e i g h t within s u c h l i m i t s over t h e derivative-determination phase of the f l i g h t test program t h a t t h e e f f e c t s of c h a n g e s i n centei-of-gravity position a n d moments of inertia w i l l b e n e g l i g i b l e .

The important factors t o b e considered i n f l i g h t testing f o r stability a n d c o n t r o l derivatives a r e discussed i n the following s e c t i o n s .

8 . 1 ach Number and Altitude

Flight t e s t maneuvers a r e generally performed a t l g i n i t i a l conditions a t constant Mach number a n d a l t i t u d e . o r m a l l y , s o m e variations i n t h e s e quantities a r e accepted i f t h e resultant change i n dynamic pressure i s n o t more t h a n 5% over t h a t p o r t i o n of t h e maneuver encompassed i n t h e a n a l y s i s . n r e g i o n s w h e r e l a r g e Mach number e f f e c t s e x i s t ( F i g . 3 6 ) , tests should b e conducted a t c l o s e Mach number intervals w i t h m o r e rigid requirements a t constant Mach n u m b e

r a n d a l t i t u d e . ailure t o trim t h e aircraft

t o t h e desired Mach number a n d t o mi:1 t h a t Mach number during t h e maneuver i n

regions of rapidly varying characteristics m a y produce a s c a t t e r of data a n d a n erroneous a n a l y s i s .

T h e v e r y nature of f l i g h t testing r e q u i r e s , o r e x p e d i e n c y , plotting t h e r e s u l t s of analysis a s a f u n c t i o n of Mach n u m b e r , w i t h e a c h c u r v e representing a c o n s t a n t - altitude c o n d i t i o n . igure 3 7 , a k e n f r o m Reference 4 1 , s h o w s t h e i n f l u e n c e of altitude o n f l i g h t t e s t data o n o n e supersonic a i r c r a f t .



4 7

6.2 ngle of Attack and Load Factor

i ' V

■

T h e variation i n airplane characteristics w i t h angle of attack i s determined b y performing maneuvers a t different altitudes with l g t r i m conditions prevailing prior t o t h e p e r t u r b a t i o n , or a t constant altitude with t h e maneuver performed during a stabilized constant-g pushover or t u r n . t should b e noted that i t i s difficult t o obtair g o o d maneuvers during stabilized t u r n s ; exceptional piloting s k i l l i s r e q u i r e d . Figures 3 7 and 3 8 show t h e i n f l u e n c e of load f a c t o r o n stability c h a r a c t e r i s t i c s . n instances where t h e aeroelasticity of the structure i s nil ( d y n a m i c pressure effects

a r e n i l ) , a combination of t h e t w o techniques w i l l r e s u l t i n the determination of t h e variation of t h e derivatives over a n extended r a n g e of angle of a t t a c k . hould a e r o - elasticity of t h e structure b e a factor to contend w i t h , t h e results f r o m t h e t w o techniques w i l l differ f o r t h e s u n e angle of a t t a c k . Mach n u m b e r , a n d c e n t e r of g r a v i t y .

6.3 Aeroelasticity

*. Aeroelast ic eform ation f he tructure s sumes ncreas ing ignificance s he

aircraft ncreases n ize nd l enderness nd perates t ncreasing ynamic ressures . Supersonic ransport es igns re lexible n rder o eep he tructural eight o w n , th e ayload igh, nd he ange apability m a x i m u m . T o pply heoretical lexibility corrections o igid ind-tunnel ata or omparisons ith light ata rovides n intuitive basis n scertaining lexibility ffects. W h e n uc h om parisons re m -

ployed n d definite i sagreement s vident n he omparison n egard o evel nd trends f he tability nd ontrol arameters s unction f M a c h umber, it a y become difficult o ocate he ource f he iscrepancy - wind-tunnel ata r redicted flexibility orrections. Thus, more positive pproach s equired o ssess lexibility ffects.

T h e tability nd ontrol derivatives hould e ssentially nvariant or igid airplane s ong s a c h umber, angle-of-attack, and he enter f ravity re onstant (assuming eynolds n u m b e r ffects o e minor actor). Thus, any direct pproach o investigating lexibility ffects ased n light ata hould how he ariation f th e tability arameters - obtained t he a m e M a c h umber, angle-of-attack, nd enter of ravity - s unction f ynamic ressure . Although a c h u m b e r nd enter-of- gravity ontrol s traightfrrward, th e ngle-of-attack s roblem.

T h e ocation f he ngle-of-attack ensor xposes he ensor o rrors esulting f r o m tructural eformations, in ddition o he ther ources iscussed n ection .3 . Hence, it s o r e udicious o se he ife oefficient C L in ieu f ngle-of-attack a Thus, f r o m practical oint f i ew, a direct nvestigation f eroelastic effects hould e ased n omparison f light ata or different ynamic-pressure conditions btained t he a m e a c h u m b e r , lift oefficient, and enter f ravity.

A n ffective, flexible, and imple light-planning rocedure o etermine he light test onditions s unction f eight nd ltitude o rovide onstant M L and center f ravity an e chieved y sing nomograph uch s ha t n igure 9. In this omograph C L and q are ariables , and enter f ravity s onstant.

T h e omograph s ased n he ollowing wo as ic elations or g light:

W = C LqS 1 1 0 )

and = 0 . 7 p M 2 Ill)



Lim - Ulli. . . . I • I.-IJ ..,..,,. V

4 9

e m p l o y e d . urrent practical methods of a n a l y s i s , whether they involve approximate equations solving f o r i n d i v i d u a l derivatives or comprehensive techniques solving a number of d e r i v a t i v e s , have limitations i n their u t i l i t y ; as a r e s u l t , different types of maneuvers a r e employed w i t h i n t h e r a n g e of their individual limitations t o obtain t h e d e r i v a t i v e s . s a g e n e r a l i t y , t might b e said that typical handling-quality maneuvers a r e employed i n t h e determination of derivatives wherein analytical techniques a r e u s e d . ncluded a r e longitudinal elevator-pulse m a n e u v e r s , pullups and p u s h - o v e r s , pullups a n d r e l e a s e s , rudder-pulse a n d aileron-pulse m a n e u v e r s , constant-heading s i d e - s l i p s , recovery f r o m s i d e s l i p , and rudder-fixed r o l l s .

When f l i g h t maneuvers applicable t t r m a l y t i c a l technique f o r derivative determination a r e n o t available or u s a b l e , t h e airplane response t o random inputs i s analyzed to give limited stability d a t a . his i s accomplished effectively with t h e aid of a n analog c o m p u t e r , using a technique involving t h e matching of analog and f l i g h t time h i s t o r i e s .

6.5.1 Pulse Maneuvers

T h e simple p u l s e m a n e u v e r , s h o w n i n Figure 4 1 f o r a longitudinal p e r t u r b a t i o n , s t h e current mainstay f o r derivative d e t e r m i n a t i o n . o r m a l l y , o r this maneuver t h e airplane i s trimmed a t t h e desired a n g l e - o f - a t t a c k , a l t i t u d e , and Mach n u m b e r , and a f r e e oscillation i s i n i t i a t e d b y a n abrupt pulse - a n elevator pulse f o r l o n g i t u d i n a l o s c i l l a t i o n , a rudder or a i l e r o n pulse f o r lateral-directional o s c i l l a t i o n s . h e

resulting free-oscillation of t h e aircraft i s allowed t o damp out with t h e controls held f i x e d a t t h e i n i t i a l t r i m s e t t i n g . ith a n irreversible c o n t r o l s y s t e m , t h i s i s easily accomplished b y releasing t h e c o n t r o l s . n tailless a i r c r a f t , e v e n s m a l l i n - advertent c o n t r o l i n p u t s during t h e f r e e collation c a n significantly affect t h e damping a n d , h e n c e , t h e damping d e r i v a t i v e s . oderate inadvertent c o n t r o l inputs c a n a f f e c t t h e period of o s c i l l a t i o n , a s w e l l a s t h e d a m p i n g , and t h e n influence t h e static derivative r e s u l t s a s w e l l .

Prse oscillations a r e a l s o initiated b y r e l e a s e o f controls a t the e n d o f a s i d e - s l i p maneuver e n d a t t h e e n d of pullup a n d push-over m a n e u v e r s .

I n investigating t h e effects of a n g l e o x attack and l o a d f a c t o r w h e n utilizing t h e pulse maneuver i n a n elevated g u r n , t h e application of the p u l s e technique i s

limited b y the difficulty of performing a good m a n e u v e r . ifficulty has b e e n e x p e r - ienced during t h e maneuver i n holding t h e proper b a n k a n g l e t o maintain constant l o a d f a c t o r and Mach n u m b e r . ith a c o n v e n t i o n a l c o n t r o l s y s t e m , exceptional piloting s k i l l i s required t o maintain f i x e d c o n t r o l during t h e airplane oscillations a t elevated . h e use of t h e airplane damper a s a device f o r applying a known deflec- t i o n s i g n a l to excite t h e desired unaugmented oscillations ( F i g . 4 2 ) offers a means of improving t h e q u a l i t y of t h e data f o r elevated conditions a s w e l l a s l g c o n d i t i o n s .

I n well-performed p w l ' - e maneuvers a n d l i g h t l y damped o s c i l l a t i o n s , t i s possible t o determine a 2 - s e c o n d terioi t o w i t h i n 0.02 s e c o n d . ood accuracy i n damping c a n b e measured f o r d«nping r a t i o s l e s s c i i a n 0 . 2 . h e accuracy o f period and damping measurements becomes r a t h e r p c > r f o r damping ratios greater t h a n about 0 . 3 .



j

T^ ~

50 6.5.2 Constant-Heading Sideslip Maneuvers

In he bsence of pertinent id pplicable ulse-maneuver a ta or n n ffort o complement uch ata , he onstant-heading ideslip maneuver an e sed o etermine th e eathercock nd ffective ihedral derivatives C n/3 and Cj » rovided ontrol - effectiveness derivatives re vailable rom ther maneuvers.

Because f requent oose sage of erminology, th e xpression steady ideslip" s used h e n constant-heading ideslip" s meant. Actual ly , ideslip an e ccomplished, s hown n igure 3 , s wings- leve l ideslip n hich aw rate nd, ence, a hanging heading s nvolved, s onstant-hea ng ideslip n hich onstant linear light ath s maintained r ). r a s om bination of thp.o w o ariations of ideslipping maneuvers. T h e distinctions n he ariation f he ideslip maneuver affect he arameters nvolved n he nalysis of he light ata and he format f he quations mployed.

It s difficult o er£-*m he ideslip maneuver s teadily ncreas ing ideslip at constant eading without xperiencing angular a te nd cceleration ransients. A more uccessful pproach o he maneuver s o ncrease he ideslip n ncrements in rder to a m p ou t he ngular ates t ach ncrement efore roceeding o he next ncrement. Although his anner f ccom plishing the aneuver nvolves o r e i m e , it s ustified y he efinement n d esulting sable ata.

6.5.3 ullup nd Push-Over Maneuver

This aneuver, r ny ne of ts ariations, is ntended rim arily or handling- qualities nvestigations. However, th e ontrol-effectiveness arameter , C n$ , can be mathemat ica l ly etermined rom he nitial hases f he maneuver. T h e maneuver is s e ful lso n eterm ining th e ther ongitudinal erivatives y nalog-m atching techniques.

6.5.4 e cov e r y - FromS i d e s I i p Maneuver

This aneuver as ee n aluable or eterm ining ateral-directional derivatives y th e nalog-m atching echnique. G o o d onditioning s chieved y irst educing udder input o half he alue resent t he nd f onstant-heading ideslip nd he n releasing t. This aneuver s onsidered n o r e detail n ection 7.8.4.

6.5.5 levated-g urn Maneuver

T h e use of this maneuver i n derivative determination w a s discussed i n Section 6 . 6 . 1 .

6 . 5 . 6 " Roll Maneuver Rudder-Fixed)

This maneuver l e n d s i t s e l f t o t h e determination o f C j n d Cl8 e v e n though i t i s primarily a handling-qualities m a n e u v e r . n i t s e x e c u t i o n , the r o l l i s initiated b y a n a b r u p t aileron step i n p u t . h e i n i t i a l phase of t h e m a n e u v e r , up t o maximum r o l l r a t e , i s t h e u s e f u l portion f o r derivative a n a l y s i s . h e i n i t i a l phase involves negligible s i d e s l i p , a n e s s e n t i a l factor i n i t s utility f o r derivative a n a l y s i s .

sg^uiw"jüj 'ii a e '1 n " * » - j - ' » . . SB l

" ' ---~»»> - ~W

.

52 th e egree s f reedoa f n xponentially a m p e d inusoidal scillating ystem second- order inear ys tem) nd he differential nd ntegrals of he egree s of reedoa o determine he alues of hes e mplitude nd phase elations, r o etermine he on - stants f he ystem f equations.

Consider th e amped, ransient, inusoidal, m all-perturbation scillation f he rolling degree of reedom. This imple ystem s escribed y

A p $w nA p + &>*A 0 ' = o

The solution to this equation i s

M > ' = A 4 > ' l e "{<Vcos^ n,t ,

«here " n d = V<» " £2 )

( 1 1 3 )

( 1 1 4 )

( 1 1 5 )

Differentiating w ith r e s p e c t t o time t

A p = lA^'l^e"1"»* £cos ( o > ndt T ) /(l - 2) cosf<undt 1

-{« t t - lA0'lo)ne n cosiest - $J .

•here $d is he amping angle

( 1 1 6 )

$H = tan"1 - (l - *)

Sim i lar ly

• t A p = | A < £ ' | < y *

n cos w nrtt T - 2 % )

( 1 1 7 )

( 1 1 8 )

Equations ( 1 1 4 ) , 1 1 6 ) , a n d ( 1 1 8 ) show t h a t t h e amplitudes of t h e s e equations s h r i n k a t t h e same rate and the phase relationship between t h e amplitudes I s t i m e i n v a r i a n t . The amplitudes of t h e f i r s t a n d second derivatives of A0 ' r e e q u a l t o the amplitude

of A < £ ' ultiplied b y t h e undamped natural f r e q u e n c y , an , and b y w * , r e s p e c t i v e l y . The phase of the derivatives i s a f u n c t i o n of t h e damping a n g l e , d , w h i c h I s a f u n c t i o n of t h e damping r a t ^ o , s s h o w n i n Figure 4 4 , e l o c i t y vector Ap l e a d s t h e displacement vector A0 ' y ( 9 0 + $ d) , and t h e acceleration vector Ap e a d s t h e displacement vector ( 1 8 0 + 2 $d) .

Where more than one degree of f r e e d o m i s i n v o l v e d i n t h e d a m p e d , s i n u s o i d a l , r a n - s i e n t oscillation s y s t e m , a n d t h e frequency i s c o m m o n t o a l l t h e freedoms i n v o l v e d , t h e instantaneous absolute v a l u e s of t h e rotating vectors may b e considered a s ratios (referred t o a s amplitude r a t i o s ) and t h e phase relations of t h e ratios e s t a b l i s h e d . These r a t i o s of t h e rotating v e c t o r s and their corresponding phase angles are time i n v a r i a n t . s a r e s u l t , t h e instantaneous v a l u e of a n y one d e g r e e of f r e e d o m may b e readily determined if t h e characteristics of a n y one of t h e motions a r e known and

%

, . . , . \

■ mm» m» i i, », _i II J inn .. mi, , i m m

- ~ * ~

1 I

5 3

t h e amplitude r a t i o and phase angle relative t o the characteristic motion are k n o w n . For e x a m p l e , f the known characteristic motion i s

-X a> t

Ar = Ar ' C O B ( < » ndt)

and, f |A/3|/|Arl lApi/IArl »_ nd $._ are nown, hen

ß- CW'Vco.K,t*V

(119)

(120)

a,

i

cos ( o )ndt $pr) (121)

Hie ime nvariance of he mplitude atios nd heir phase angles permits he e- presentation of ny ne of he inearized equations of motion y vectors. For xam ple , b y ubstituting Equations 119), 120), nd 121) nd id differentials f Equations (119) nd 121) nto he inearized, small-perturbation, rolling-moment quation, the

following ormat s obtained, sing.the Ar vector as he eference or he m pl itude ratios nd phase angles

I. IAp| lAfl _ Ap | b .. Ar l b .. m Ä 7 I

Z

-qÄ I Ä H

Z§

-%1^1 « -(C.r-CiP^ A„ - 0 ( 1 2 2 ) p Arl V

pr

lArl V

.4

where

jApl

I Ar I < H |Ap|

n Arl

|Af | I A r I = Ct)„

lArl

lArl = 1 an d *rr = 0. (123)

H ie vector properties described, lus he equirement hat he vector polygon e-

presenting ny ne quation must lose, ake possible he etermination of wo nknown

derivatives n ny ne quation. Ihe accuracy with which he nknown erivatives are

determined s dependent not only n he ccuracy of he mplitude atios used but lso

on he ccuracy of he phase ngle nd he ensitivity f he nknown erivative o

small rrors n he phase ngles.

It hould e noted hat he ntroduction of ross-coupling erms nto he quations

of mot ion ould esult n nonlinear equations nd , ence, ime-variant elations of

the ross-coupling erms elative o he other erms. i

7.2 Basic Flight Data

Application of many of t h e simpler equations f o r determining derivatives requires a n evaluation of t h e period a n d d a m p i n g ; w h e r e a s , application of t h e time-vector method r e q u i r e s , n a d d i t i o n , t h e determination o f amplitude and p h a s e r e l a t i o n s h i p s . hese quantities a r e obtained f r o m the free-oscillation portion of the p u l s e m a n e u v e r , as illustrated i n Figure 4 5 . h e spacing of t h e peaks of t h e oscillatory motions d e t e r -

mines t h e damped natural p e r i o d , a n d a c o m p a r i s o n of these peaks f o r t h e different oscillatory quantities determines t h e i r p h a s e r r l r t i o n s h i p . etermination of t h e

: W 5 ~ ~ ~ — . ; J | | I

54

phase relationships by an averaging p r o c e s s , typified by t h e t a b l e i n Figure 4 5 , has provided more consistent data than obtained b y single r e a d i n g s . h e f i r s t l i n e of t h e example t a b l e l i s t s t h e time of occurrence of consecutive plus and minus peaks of t h e r o l l rate Ap . i m i l a r l y , t h e second line l i s t s t h e plus a n d minus p e a k s of t h e yaw rate Ar . he third line l i s t s the time difference of t h e f i r s t t w o lines i n e a c h c o l u m n . ince t h e yaw rate Ar i s the reference i n this i n s t a n c e , t h e signs i n t h e t h i r d l i n e Indicate t h a t t h e r o l l r a t e Ap a g s t h e yaw r a t e Ar . he v a l u e s i n t h e third l i n e a r e averaged and converted t o d e g r e e s .

I t w i l l be noticed i n Figure 45 that a yawing divergence i s e v i d e n t i n t h e yaw-rate r e c o r d . o isolate t h e oscillatory motions and determine t h e time to damp the oscilla-

t i o n s , exponential curves are drawn as s h o w n . semilog p l o t of t h e double amplitudes included between t h e exponential outlines of e a c h motion v e r s u s t i m e establishes t h e t i m e t o d a m p of t h e oscillations ( F i g . 4 6 ) . comparison of the plotted d o u b l e a m p l i - tudes of t h e variables determines t h e amplitude r a t i o s .

As stated e a r l i e r , accuracy of measuring period a n d damping becomes rather poor f o r damping ratios g r e a t e r t h a n a b o u t 0 . 3 . e n e r a l l y , configurations tested a t moderate and high altitudes a n d without damper augmentation have b e e n rather l i g h t l y damped s o that free-oscillation methods of analysis c a n b e applied w i t h good a c c u r a c y .

The damping ratio , damping angle $d , and t h e undamped natural frequency wn are o b t a i n e d , r e l a t i o n s :

for both short-period nd phugoid ree-oscillations, from he ollowing

I = sin tan j '01693P

S

27TT (124)

$ , » = tan -l

' 0 .693PN

>277-T (125)

i/2>

2r2

< = ^d + <& (126)

7.3 Determination f a and ß From Free Oscillations

in he Absence of or Questionable a and ß Data

7.3.1 Longitudinal Free Oscillations Should he a records e unavailable or questionable n ree-oscillation ongi-

tudinal ata nd he pitch-rate ecords available, A«|/|Aq| an d $ may e obtained

b y using ime-vector echniques. Once hese quantities re etermined, it s simple

matter o plot a as a riction f ime or, f more mmed ia t e oncern, o etermine

lAaJ/IAal for use n determining CNa

T he omplete procedure or determining lAal/IAql , nd c x q I A a nl/IAal s s h o w n i n Figure 4 7 . h e procedure I n v o l v e s t h e application of t h e following linearized auxiliary e q u a t i o n t o c o r r e c t t h e sensed n o r m a l a c c e l e r a t i o n , nl t o t h e c e n t e r of

m gg t — » w e ys

;

gravity of t h e a i r c r a f t , a s s h o w n i n Figure 4 7 ( a ) ,

%*b. IA »nil

AqT *an Q

lÄ rZaDiQ g A q| M

55

( 1 2 7 )

an d he ector pplication f Equation 5 6b ) Table ) n Figure 47(b) n he onsat

|AaJ M A Q|

|Aq | nq g|Aq| *« Q g Aa «« ( 1 28 )

solve or $ and

|Ao|

|Aq|

V A a |

g|Aq|

V ( 1 29 )

&- which ow permits he etermination

|A a n|

|Aa|

lAaJ |Aq|

JAa|

|Aq |

( 1 3 0 )

When t h e v e c t o r quantities Aani n d A a j j r e approximately i n phase and Aq s approximately 90 ° out of p h a s e with ht^ , which i s usually t h e c a s e , t h e vector Equation ( 1 2 7 ) m a y b e solved b y t h e s i m p l e algebraic f o r m a t

I A

| A qD

|

|Aa01 ( 1 3 1 )

7.3.2 Lateral-Directional Free Oscillations Should t h e ß r e c o r d s b e unavailable or questionable i n free-oscillation l a t e r a l -

d i r e c t i o n a l d a t a , a n d yaw-rate records a v a i l a b l e , A / ? l / | A r l nd $^r may b e obtained b y using a vector s o l u t i o n of t h e f o l l o w i n g linearized auxiliary equation t o c o r r e c t t h e transverse accelerometer record t o t h e center of gravity of the a i r c r a f t ,

l A a \ - * a t r

l A r i ir-gArl +

g | A r l *

a n d t h e application of e q u a t i o n ( 5 9 ) i n t h e f o r m a t

-2r MZ*. 2 lArl ß

< - 2T 7T-: $., - To i r~ Z$„ - , in 6 — Zfy,r - | i

rr I Ar I pr |ArS

- Lts0cos<2> r £%,r - L --A.Z$atr

( 1 3 2 )

( 1 3 3 )

56

where

lAfl |Ar| = CJ. $ 0 rr

and

lAr

1 |Arl *

T Ä 7 T = 7TkT\ = r r = $rr < > + *d) = - ( 9 0 + *d) . o )n A r

Figure 8 hows he pplication f quations 1 3 2 ) nd 1 3 3 ) o he eterm ination

of Aßl / lArl £, nd A a J / | A / 3 | p o n olving or Ay8|/|Ar nd ßr r o m th e raphical olution f Equation 1 3 3 ) , t s imple matter o btain arameters with y d a s ase , for xample

A p

Aß J A p j Ar| |Arl A y S l a n d * vß V ßr

7 . 4 Equations f o r Longitudinal C o n t r o l and Stability Derivatives

T h e nature of t h e i n p u t a n d t h e ensuing f r e e oscillations of t h e longitudinal-pulse

maneuver p e r m i t t h e u s e of r e l a t i v e l y simple methods of analysis i n determining l o n g i -

t u d i n a l c o n t r o l and stability d e r i v a t i v e s . hese methods give r e s u l t s comparable t o

t h o s e f r o m t h e more complicated methods i n v e s t i g a t e d . nly t h e simple methods a r e discussed a t t h i s t i m e and only d a t a f r o m these methods a r e p r e s e n t e d . nless other-

wise s t a t e d , t i s t o b e assumed t h a t stability augmentation systems a r e n o t operational

during t h e maneuver a n d t h a t t h e aircraft b e h a v e s similarly t o a rigid s t r u c t u r e , n

t h a t i t s behavior c a n b e represented b y t h e linearized small-pertur ation e q u a t i o n s .

7.b.l Control-Effectiveness Derivative, C»SS

T h e control-effectiveness derivatives a r e determined f r o m t h e i n i t i a l p o r t i o n ,

approximately 0 . 2 s e c o n d , o i a rapid pulse maneuver ( F i g . 4 9 ) . uring t h i s p a r t of

t h e m a n e u v e r , t h e airplane response i s a l m o s t entirely pitch a c c e l e r a t i o n , w i t h t h e

r e s u l t that t h e p i t c h control-effectiveness derivative c a n b e determined f r o m

C J . _ Aq_

q S c ASe ( 1 3 4 )

I n similar f a s h i o n , t h e change i n normal-force coefficient dm t o elevator deflection

c a n b e determined f r o m

W A ;

' N S .

an

3SA8e * ( 1 3 5 )

With t h e preceding restriction i n m i n d , t i s d e s i r a b l e , o r a c c u r a c y , t o r e a d t h e

p e a k c o n t r o l i n p u t a n d acceleration response w i t h a disregard of t h e phase l a g between

t h e t w o , s s h o w n i n Figure 4 9 . t h a s been f o u n d t h a t t h e time difference i n peak

v a l u e s of c o n t r o l i n p u t a n d acceleration response i s primarily t h e r e s u l t of Instrument



h n

5 7 phase ag nd, o esser xtent, ir -mass nertia ffects. Analysis y his ethod requires nstrum ents ith lat esponse haracteristics xtending o elatively ig h frequencies 8 /s).

Pu l se s pplied t lower at e s , nd hus xtending over onger ine nterval, m a y equire nclusion f amping nd ngle-of-attack e r ms n he quation, specially a This a y necessitate he nclusion f nstrument hase- lag orrections or q and a

7.4.? Slope of he Nornal-Force-Coeff ic ient Curve

F r o m he hort-period ree-oscillation a ta f he irplane ith he ontrols ixed, th e ariation f he ormal -i'orce oefficient ith ngle f ttack a y e valuated f r o m

W Aa „ qS |Aal = °L

lAaJ lAal ( 1 3 6 )

This xpression neglects he pitching-velocity nd ngle-of-attack-rate erms f he short-perici orm ? he orm al-force quation Equation 58), Table ). These e r ms have een ound o e negligible, s will e oticed n he ypical ector iagram (P ig . 5 0 ) f he ector orm f his quation herein he itch a te as se d s he base f he m plitude atios.

In nstances here "free-oscillation ata" av e nadvertent nputs f he levator and he ngle-of-attack ata have een scertained s eliable, CNa m a y e determined y electing hose ortions f he im e history n hich he l evator s t ts steady-state position nd plotting an versus a for umoer f ata points hich encompass he ange f aQ on he ecords. T h e lope of he plotted oints s l A a J / I A a l This undamental echnique, hich nvolves o m e abor, a y still e he s imp l e s t echnique here ontrol-fixed re e scillation s eavi ly a m p e d nd hus precludes he eterm ination of |A ani/Aal by ther means.

6 f

T h e derivative a ay b e converted t o t h e effective lift-curve s l o p e , | ,a , w h i c h includes t h e contribution of p o w e r , b y using Equation ( 3 8 ) . h e inclusion or exclusion of t h e power tenn_depends upon t h e i n f l u e n c e of p o w e r . o r conventional low-performance a i r c r a f t , i ,a CNa t s m a l l a n g l e s - o f - a t t a c k .

7A.3. T h e Derivative ( CNq + CNi )

As explained i n Section 3 . 4 , t h e phenomenon i n v o l v i n g i s different f r o m that involving q . h e pairing of t h e derivatives a s ( C Nq + CN^) i s valid only f o r l o n g i - t u d i n a l s m a l l - p e r t u r b a t i o n , ree-oscillation m a n e u v e r s . n this m a n e u v e r , Aq and Ad v e c t o r s a r e approximately i n p h a s e a n d A ü | / | A q | ~1 thus permitting t h e p a i r i n g . Determination of t h e i n d i v i d u a l derivatives nq a n d CN& a s t h u s f a r defied s o l u t i o n .

T h e determination of ( C Nq + CN^ i t s e l f i s d i f f i c u l t . t may b e readily deduced f r o m t h e vector diagram ( F i g . 5 0 ) of the f o l l o w i n g vector f o r m of the short-period mode of Equation ( 5 8 ) ,

r_,«jj *.r.T_

58

CLl7?rZan«+CHa —«q + CH q C N & ) —-Z$aq = 0. 1 3 7 ) IAq|

that

(CKq C H& ) -| A q j £ |Aq| 2V

2 V C w g $, ana (138)

T h e ndividual quantities n quation 1 3 8 ) how that he egree f uccess n etermining C n q C N ^ ) S ependent pon the ccuracy ith hich $a„a is determined. This hase ngle s nail, f he rder of a ew egrees , nd, ve n ith he e st records n d nstrum entation, he rror n eadability f $anq from he ecords ould be of he rder f he ngle tself. Thus, t s ery difficult o etermine his derivative to easonable egree of accuracy.

7.4.4 Pitching-Moment Static Stability nd Damping Derivatives, C Ba and C B„ G »a

T h e quations or he itching-m onent tability derivatives re ased n he normal- force quation

m u A q -mAw ~ C NaSA<x (139)

obtained r o m he hort-period o r m f Equations 56b) nd 58b) nd n he hort- period o r m f he itching-oom ent quation Equation 58c))

I/q = (cBoAa + C n(IA q ±+c ä -j SSc .

Differentiating quation 139 ) ith e spect o ine n d ubstituting or Aq and Aq in quation 58c) rovides he ollowing

Aa + —2r

n c2

C N a _2T «* +

C n

« ) A&- C ma + C«qCN«

4M , qSc

Aa

Since 1 4 0 ) s econd-order inear differential quation f he o r m

Aä 2£c^A<x+o)*Aa = 0,

then

a Sc n 4 M C q a S c n

( 140 )

( 141 )

(142)



60

and c — nmV

2 = ph (148)

Transposing hese w o quations esults n he ollowing approximate equations or determining C M and Ce„ fram light data

CN„ ^v . 2»^ph

pg S (149)

and

ZmV^ph^ph ^Cph^ph / O V B (150)

T h e f l i g h t values of £ ph nd o nph r e determined f r o m t h e phugoid oscillations i n accordance with Equations ( 1 2 4 ) and ( 1 2 6 ) .

An interesting byproduct of this brief consideration of the phugoid parameters suggests I t s e l f . f CNu a n b e considered t o b e similar t o CN h e n Equation ( 1 4 8 ) t a k e s on the approximate f o r m

**?#£)•?* (151)

Thus, he hugoid requency, a ), i>ph is pproximately unction of velocity, V nly.

7.4.6 Corrections or Effects of Stability Augmentation System n Determining Derivatives rom Short-Period Oscillations

In performing pulse maneuver with he stability ugmenta t ion ystem ngaged, he

ensuing ransient short-period oscillation of he aircraft will e characterized y period of oscillation nd amping atio which will e different rom hose obtained

with he pitch stability ugmenta t ion ystem off Fig.51). With he ystem n, he period will ecrease with ncreasing amping provided y he ys tem; whereas, ormally, the period ncreases with ncrease n nherent naugmented amping. This s ue o he

system ain nd he im e constant. Thus, he ain nd im e constant are actors o e

considered n equations or etermining he stability derivatives, s s rought ut

in Reference 2. The ubsequent discussion s ased n his eference.

The ollowing rocedure or etermining Cma and C Bq n$) rom light data »hich

includes stability ugmentation effects ha s een useful ut s of imited utility. The principal alue of the nsuing discussion s the nsight ained nto he omplica- tions which a y e ncountered n data which nclude stability ugmentation effects. Fo r igid-aircraft perturbations bout mean light ath, he Laplace ransformed

short-period mode wo-degree-of-freedom ongitudinal equations of motion a y e e- presented n pproximate, b ut practical, form s

(s - M Aq -M^ - MaAa = M S(AS e

-Aq s - a)Aa ZSfiAS e e

(152)

(153)

61

4 f

In he bsence of pilot nput, he ransfer unction or a mp er ith irst-order tine ag a y e epresented y

ASe(s) Aq(s)

k 1 T'S

~ k(l - T'S) ( 1 54 )

Substituting Equation ( 1 5 4 ) i n t o Equations ( 1 5 2 ) a n d ( 1 5 3 ) results i n t h e following determinant

<-«* " «a>

(s a = 0

[(1 8ekr')s -llQ - $ekj ]

[ (Z8ekr')s -1 sek)]

whose haracteristic quation s

(1 sekr' ä Z 8ekT')s2

+ "Za - Bq - B & - M 5ek - Ö 5ekT ' S d - fia)Z8ek]s

+ -Ma aSq J ise - Sa ZSek) = 0 .

( 1 5 5 )

(156)

Considering nly hose e r m s n quation 1 5 6 ) hich hus ar a ve een hown c e

significant, he hort-period ongitudinal requency nd * * »ping of he ircraft ith a irst-order ime-lag itch a mp er re

(o )2 ~

2*>n

1 sekT '

-(Za+ Q fr + $ek)

1 M sekr'

Solving hese quations or C^ and Cj, + m ^),

,g #+KT')(^ > '

( 1 5 7 )

(158)

( 1 59 )

( C m , + W _ 2Il m c2

CN c 4T fO . 693>

T1 M„kT' - \ m V c

k C , «8,

1/2/

( 1 6 0 )

P r o m he bove, it s een hat C ma is eadily e t ermined or irst-order inear pi tch -dam per ys t e m . The etermination f C B + „a, n he ther and, ay ffer a roblem, inasmuch s C B in M q is ot eadily e t ermined y tself.

If he i t ch-damper ystem s not irst-order inear ys t e m , which s he ase for any ystems, analytical olutions or C „a and Cm + Da) re mpractical. In

such nstances, nalog echniques re pplied n ttempts o xtract hese derivatives.

<

62

7 . 4 . 7 Representative Results

Typical time h i s t o r i e s , the flight-determined period and damping r a t i o s , and the flight-determined longitudinal stability derivatives of the D-558-II research airplane have been reproduced i n Figures 5 2 , £ 3 . and 5 4 f r o m Reference 4 3 . ost of t h e data w e r e obtained f r o m t h e all-rocket-powered version of t h e a i r p l a n e ; t h e remainder of t h e data i s based o n t h e jet- a n d rocket-powered v e r s i o n .

These data have been used t o illustrate representative results because they show t h e need f o r a concentration of f l i g h t t e s t data i n t h e transonic zone t o establish t h e extent of a n y abrupt c h a n g e s of t h e derivatives and to show t h e influence of altitude on this particular a i r c r a f t . ecause t h e results did not include c o n t r o l e f f e c t i v e n e s s , Figure 5 5 shows representative data f r o m Reference 4 2 f o r C ^ g ll

data shown were obtained f r o m wings-level pulse maneuvers and are typical of t h o s e t h a t c a n b e obtained f r o m good f l i g h t techniques - which i n c l u d e c o n t r o l of f l i g h t v a r i a b l e s , pilot s k i l l , and instrumentation - and c a r e f u l application of t h e methods of analysis d i s c u s s e d .

T h e maximum deviation f r o m t h e f a i r e d v a l u e i n t h e stability derivatives shown i n Figure 5 4 i s of t h e order of 5% f o r CN„ , 1 0 % f o r , a . and 20% f o r ( CBq + n a ) > deviation of this order of magnitude occur i n only a minor portion of t h e data a n a l y z e d . T h e maximum deviation of CBg i n Figure 55 is difficult to assess because t h e data s h o w n were obtained over a l a r g e r a n g e of altitudes and elevation t r i m s e t t i n g s ; h o w e v e r , t h e maximum deviation f r o m faired values would b e of the order of 1 0 % , which would b e r e p r e s e n t a t i v e .

7 . 5 quations f o r Lateral-Directional Stability and C o n t r o l Derivatives

T h e lateral-directional c o n t r o l and stability derivatives a r e not as r e a d i l y and reliably determined b y t h e use of approximate equations a s a r e the longitudinal d e - r i v a t i v e s , e c a u s e of t h e more c o m p l e x behavior o f t h e airplane and the larger number of derivatives i n v o l v e d . n t h e following d i s c u s s i o n , unless otherwise s t a t e d , t i s a g a i n assumed that stability augmentation systems are not operational during t h e maneuver and t h a t t h e a i r c r a f t ' s perturbed behavior c a n b e represented b y t h e linearized perturbation e q u a t i o n s .

7.5.1 Control-Efftctiveness Derivatives

T h e basic procedures f o r determining l a t e r a l a n d directional c o n t r o l effectiveness a r e similar t o t h o s e previously discussed f o r longitudinal c o n t r o l e f f e c t i v e n e s s . ow - e v e r , t h e expressions f o r lateral-directional c o n t r o l effectiveness a r e complicated b y t h e need to account f o r t h e possible influence of the inclination of the principal axis as w e l l a s t h e aerodynamic t e r m s . ests with a conventional high-performance airplane utilizing a rapid c o n t r o l pulse or step i n p u t showed t h a t t h e directional c o n t r o l d e - r i v a t i v e , „ j c o u l d b e determined t o good accuracy b y considering only the inertia t e r m . or e x a m p l e ,

* • » = [ &A' - f e * - < * r - * * > ^

A' - * p ^ - V t f ] £

1 0 0 9 8 - 0 -

( 1 6 1 )

6 3

w h e r e t h e magnitudes of t h e Individual tens a r e given a s percentages of t h e a n s w e r . I b i s simplification i n determining Cn$r may not be applicable t o other a i r c r a f t .

For t h e roll-control d e r i v a t i v e , js consideration must be given to t h e aerodynamic derivative t e r m s . o r e x a m p l e , using t h e same high-performance airplane and a rapid aileron c o n t r o l i n p u t ,

Cli

1 00 =

Lb-Ap - 1 * 2 A r - Cu — A p - Cu — Af - CiAsl - | _ q S b l S b 2 V 2 V \

7 3 - 3 1

( 1 6 2 )

T h e cross-control d e r i v a t i v e s , ng and C is c a n b e evaluated h y using Equations ( 1 6 1 ) and ( 1 6 2 ) , r e s p e c t i v e l y . he cross-control derivatives are usually of smaller magnitude and a r e therefore more d i f f i c u l t t o d e t e r m i n e . t appears t h a t a l l a e r o - dynamic terms may r e q u i r e c o n s i d e r a t i o n , a s s h o w n i n the following example of t h e analysis f o r ns h e f l i g h t quantities were obtained f r o m t h e records as shown i n Figure 5 6 . h e t i m e difference i n t h e peaks of t h e c o n t r o l i n p u t a n d t h e accelera- t i o n s i s d u e t o t h e phase l a g of t h e i n s t r u m e n t s . h e acceleration and angular-rate r e c o r d s have essentially t h t c o r r e c t phase relationship w i t h respect t o e a c h other i n this i n s t a n c e . h e magnitudes of the i n d i v i d u a l terms a s percentages of the answer a r e

'ns .

100

= [ i t A * - f e A * -(C n r - £ A r -

c- p ^A p

- v*] i 206 - 1 4 1 1 0 1 6

( 1 63 )

It will e oticed that he roduce-of-inertia erm s particularly ignificant n this xample. A n rror n principal-axis nclination ould ignificantly ffect he answer. F or nstance, in his xample n rror f /4 ° n he nclination f he principal xis 3 °) ould esult n n rror f 2% in C „s .

7.5.2 T h e i de-Force Derivative, C y o

This derivative, which ontributes o he utch oll o d e f scillation nd s n index f he pilot' s bility o ense ransverse ccelerations, can e etermined r o m th e quation

c

( 164 ) W |A at

° * e - W

T h e atio |A at/A /S| is btained r o m he ontrol-fixed ransient scillations resulting r o m ulse maneuver. If ha ß record s uspect r miss ing , th e atio m a y e etermined r o m he at and r records s xplained n ection .3.2 nd Figure 8. This ndirect echnique or btaining IA atl/A/3 | is nalogous o ha t for btaining |A anl/A a | a n d onsiders C yp yr < u i d C y/3 a s negligible

--

f

64

7.5.3 The Directional-Stability Derivative, C ^ g

The tatic directional-stability derivative s ne f rimary mportance, and ood accuracy s equired n ts measurement . Although n u m b e r f losed-form quations have een sed, ach possesses imitations hich, f not ecognized, an e ad o erroneous nswers. Severa l of th e quations re ased on arious egrees of egradation of th e ollowing xpression, he erivation of which as ased cs he olution of he determinant of he inearized ateral-directional m all-perturbation quations (Equations ( 6 1 a ) , (61b) , nd 61c)). Th e xpression ncludes all ut he most negligible quantities.

• r \

lz Sb b Hk* Ch ~ ClßBina) C nr - C ) + Cip«^ C n/j) — —

qS / \ gcostf < h 0 C a r + % fj- j C l i f i nr

2 V

b

i v qSb

^4* ( 1 6 5 )

This equation shows that when n^ i s s m a l l , t h a t I s , of t h e order of 0 . 0 8 per radian ( 0 . 0 0 1 4 per d e g ) or l e s s , t h e ordinarily insignificant damping terms become i m p o r t a n t . I n such i n s t a n c e s , {p is particularly s i g n i f i c a n t .

When aß i s of an order higher than 0 . 0 0 1 4 , Equation ( 1 6 5 ) c a n be reduced t o t h e following workable equation

''iiß ~ . f.\2

q S b Z afClß- -Ciß ( 1 6 6 )

This xpress ion an lso e btained y differentiating n pproximate o r m f Equation 6 1 a ) o rovide

A /3 = -Af tA p + _cy/3A /3 m V

and lso sing Equations Cila)

nd 6 1c )

ith he ssumption ha t C ;p {r if i and C np arc ir ential quation, and C n arc all qua l o ero. Substitutions esult n he ollowing inear differ-

A/3- qS S b 2

m V *' " vT (C °r " WZ

A/3 +

q S b S b „ * X L ß — C l

ß+ tf c '^sb

X

A/ 3 0

in hich he requency erm s dentical o quation 1 6 6 ) .

(167)

»'% " —

66

Substituting xpressions

1 7 1 ) nto quation 1 7 0 ) , expanding y rigonom etric denti- ties, nd egrouping esults n

< w scos$d C nfl w 1

* Ä 7 8in^J sin w ndt -

^ „8in*d C nr-C ^ ) + n/3 £-*] os o t = 0 (172)

The irst racketed uantity s ummation f omponents erpendicular o he A r vector; he econd s ummation f omponents parallel o he A r vector. Hence

( 173 )

and

8in$d C Br-B4)~+C M

cos ßr (174)

Considering nly quation 1 7 3 ) t his i m e , if he hase ngle $g r is f he rder of 0 ° nd th e amping ngle s m a l l which re he onditions ormally ncountered - then sin $^r and cos $d will ach e im i lar o nd Equation 1 7 3 ) an e ransposed to

h |Arl

(185)

This equation provides accurate values of tions imposed i n i t s d e r i v a t i o n .

Cnß > provided i t i s used within t h e l i m i t a -

T a b l e I X l i s t s t h e results of t h e application of Equations ( 1 6 5 ) , 1 6 6 ) , and ( 1 6 8 ) to f l i g h t data of t h e f - 1 0 4 and Y F - 1 0 2 . h e values of n^ , a s determined by Equation ( l ' J 5 ) , a r e used a s reference v a l u e s . or t h e P - 1 0 4 , Equation ( 1 5 6 ) s h o w s good c o r r e - lation w i t h t h e reference value because of t h e h i g h v a l u e of C n/ 3 , whereas t h e s i m p l e frequency equation (Equation ( 1 6 8 ) ) shows poor a g r e e m e n t . o r t h e Y F - 1 0 2 , which has a low value o f n/3 o r t h e f l i g h t condition s h o w n , Equation ( 1 6 6 ) s h o w s a significant discrepancy with reference n/S a n d points up t h e influence of t h e doping t e r m s w h e n Cu£ i s s m a l l . or t h i s same c a s e , i t w i l l be observed t h a t t h e simple frequency equation 1B u n w o r k a b l e .

A relative comparison of the results obtained f o r the F-100 airplane U3ing Equations ( 1 6 6 ) , 1 6 8 ) , and ( 1 6 9 ) a n d t h e results obtained using t h e more comprehensive g r a p h i c a l time-vector method ( t o be discussed l a t e r ) a r e s h o w n i n F i g u r e 5 7 . onsidering t h e graphical time-vector results as most representative f o r t h e a i r p l a n e , t w i l l b e observed that t h e simple f r e q u e n c y equation ( E q u a t i o n ( 1 6 8 ) ) would show poorest c o r r e - lation a t l o w subsonic speeds d u e t o angle-of-attack a n d d i h e d r a l effects n o t accounted f o r i n t h e e q u a t i o n , whereas Equation ( 1 6 9 ) s h o w s poorest results i n t h e supersonic region because of t h e difficulty i n obtaining accurate values of r^ and a/ S .

S

'

A

6 7

T a b l e X compares t h e values of C^ determined f r o a analog matching of oscillatory maneuvers of t h e X-15 airplane with values of Cnß determined f r o m Equations ( 1 6 6 ) and ( 1 7 5 ) . h e values of xz and Cn a r e essentially e q u a l to zero on this v e h i c l e . The agreement between analog values of Cn^ and the equations i s g o o d . n aquation ( 1 6 6 ) , t h e agreement i s d u e t o t h e high v a l u e of C Uß quation ( 1 7 5 ) would be t h e more desirable to use on this airplane because i t does not depend on t h e use of Cj« f o r a s o l u t i o n .

7 . 5 . 4 Th e Effective Dihedral Derivative, Cjo

Several simple equations f o r Cj „ a r e available with limitations on their u t i l i t y , a s i n the c a s e with most simplified e q u a t i o n s .

Values of C\ß c a n be obtained f r o m t h e constant-heading i, Jdeslip maneuver using

t h e e x p r e s s i o n

C lß = -<Ci8r8r/ 3 + C iSa a/ 3 ) ( 1 7 6 )

The d e r i v a t i o n of t h i s e x p r e s s i o n a n d circumstances limiting i t s accuracy a r e i d e n t i c a l t o t h a t b r o u g h t c u t f o r i t s counterpart ( E q u a t i o n ( 1 6 9 ) ) .

A c o m p a r i s o n of Cj^ determined b y Equation ( 1 7 6 ) and the more comprehensive graphical tine-vector method i s s h o w n i n Figure 5 8 f o r t h e F-100 a i r p l a n e . t l o w Mach n u m b e r s , t h e r e s u l t s f r o m t h e sideslip equation (Equation ( 1 7 6 ) ) compare f a v o r a b l y with t h e time-vector r e s u l t s ; a t h i g h Mach n u m b e r s , a l a r g e discrepancy exists b e t w e e n t h e t w o m e t h o d s . v e n though C \a i s n o t o n e of t h e derivatives determined most accurately b y t h e time-vector m e t h o d , t h e vector method i s the most practical a n a l y t i c a l means available f o r evaluating this d e r i v a t i v e . f

I n instances where t h e influence of xz s n e g l i g i b l e , t i s possible t o c o m b i n e Equations ( 1 6 6 ) and ( 1 7 5 ) t o obtain

l A r l J _

M ; l ( 1 7 7 )

T h e u s e of t h i s equation i s s u b j e c t t o t h e a d d i t i o n a l restriction t h a t i t should n o t b e u s e d w h e n C D/ S i s s m a l l , a s w a s noted i n t h e discussion of Equation ( 1 6 6 ) . ' s o , t h e equation must b e used with c a u t i o n w h e n t h e angle-of-attack i s l e s s than about 3 ° o r 4 ° . h e n the angle-of-attack i s l e s s , | A r I / I A / S ) ( l / o > n) ay approach 1 . 0 a n d t h e e r r o r i n reading A r / I A ß l r o m t h e f l i g h t r e c o r d s may r e s u l t i n a n e r r o r i n ( | A r l / | A / 3 | ) ( i / « n) h a t may exceed t h e n e t magnitude of the parenthesized q u a n t i t y . I f t h e ß record i s the major contributor t o I n a c c u r a c y i n t h e amplitude r a t i o , t h e technique discussed i n Section 7,3.2 may b e employed to determine t h e ratio without recourse t o t h e a c t u a l r e c o r d .

A f i n a l precaution regarding t h e u s e of Equation ( 1 7 7 ) i s i n o r d e r . t v e r y l o w

a n g l e s - o f - a t t e c k , t h e e r r o r i n t h e flight-detentined values of c a n produce l a r g e errors i n t h e e q u a t i o n ; a l s o , a s approaches z e r o , t h e equation approaches a n i n - determinate f o r m , inasmuch as t h e bracketed quantity itself approaches z e r o .

< ~ 6 8

7.5.5 Th e Damping-in-Roll Derivative, C; _

Sim pl e xpressions or he determination f Cj are ependent pon oll aneuver Initiated rom win?s-level light y tep nput f he ilerons. The derivations f the xpressions mpose he estrictions hat aw ue o ileron, C n? ideslip ue o the ffective dihedral, Cj ^ nd product-of-inertia ffect re negligible. If hese highly estrictive onditions re atisfied, he ollowing elation an e mployed

C1D = -Cfc A S.

aA A p — 1 V

(178)

In using his quation, j$ an o e term ined rom he nitial part f he ontrol input s cussed n Section .5.1 nd pt s e term ined t om e im e point, x on he oll-rate im e history here A p is ero he egion f teady-state oll.

If desired, he eparate etermination f C jS ca n e voided y olving or

C js p. b 1 * = i 179)

C l, A Saj V

and substituting t h i s r a t i o i n t o t h e equation

C l , r^Ap. qSb

A p , b Cz s

2 V Cjr a2

(180 )

resulting n he orma t

Ch - = 2I.V

qSb< A pJ

1

Ap2 A 5a,

-Ap, az

A8a

(181)

In hese ast w o quations, the ubscript ndicates hat A p an d A p were btained at im e point n he oll-rate im e history, referably t he point f m a x i m u m

rolling cceleration.

Although he estrictions mp os ed t he beginning of his ection eriously imit the pplication of hese quations, he ast quation Equat ion 181)) s nteresting in hat t hows hat Cj ca n e btained without equiring he olution of Cj * .

7.5.6 Th e Effective Damping' in-Yaw Derivative, (C ar nö)

It as pointed ut n Section .4.4 hat C r> r and Cnö m ay e ombined s n equivalent erivative, Cnp - nß), nly or oscillatory aneuvers, roviding he stability xis ystem s eing onsidered r hat he ngle-of-attack s m a l l f he body xis ystem s sed. W h e n he ody xis ystem s mployed, this s antamount to aying hat he n he mpli tude atio |A</ / ' | /IA/?I at a < ° C

nö m a y e ombined s n quivalent erivative or aw ntes.

or o, C „ and



-y%F. —

I

6 9

m

The ombined derivatives re requently hown n he esults f nalysis of oscillatory m otions elative o ody xes, ven hough his mplitude-ratio ondition s exceeded . W h e n his s one, it eans hat n ffective alue f C „r ha s e en btained

which ncludes he nfluence f C nö a nd he esults of he nalysis ased n he se

of he ctual \&p'\/\&ß\,

tribution of r a nd A /3 have roduced n nswer hich s quivalent o he net on - to A c in e r m s f A r

A n pproxim ate quation or CDr - „^ ) s btained directly rom he amping erm

of he econd-order differential quation Equation 167)). Inasmuch s

24"„ = qS qSb2

m V C y ^ g 2VIZ

(Cnr - n/$ ) ( 1 3 2 )

a transposition results i n

(Cnr - „^ ) 2£0nV + Cy £

( 1 8 3 )

Considering he s sum pt ions ade n deriving quation 167), from hich quation 183) w as btained, nd he tipulations egarding he om bining f Cnf a nd C Bß it a y be tated hat quation 1 8 3 ) will rovide better ccuracy hen |Ai />' | / |A/3l a nd as lAp| / |A /3 | decreases o atisfy he ondition hat

influence n he quation.

Cip and C n have negligible

A n pproxim ate quation or Cnr - D /g) hich a s een se d uccessfully n he X-15 irplane light est rogram as erived rom quation 174)

qS b Ssn*d + Cnr C„/0 — ,

W nß i

2 V A i cos' ySr

This quation s umma t ion f awing-moment omponents parallel o he A r during ree-oscillation ma n euver nd s ubject o he estrictions hat I C n have negligible nfluence n he awing oment.

vector and xz

Since $gr generally aries nly

ew degrees rom 90° or ngles-of-attack ess than bout 5 °, nd ince he amping ngle s mall, the preceding quation an e reduced nd ransposed o

(Cnr - nÄ ) ~ - £ a > , I,

n c qSb —

2 V

1 . 386 V I z ( 184 )

5Sb*Tl/2

Analog ecords f ree-oscillation m aneuvers f he -1 5 irplane, on hich C „

and Ixz a re ssentially ero, were nalyzed or Cnr - n^ ) y sing quations 1 8 3 ) and 184). Th e esults, presented n able I, show hat he atter quation as better suited or etermination f he ffective amping-in-yaw derivative, for ngles-of- attack p o pproxim ate ly 2 ° , han quation 1 8 3 ) or his ehicle.



70 7.5.7 Correlation or Effects of Stability Augmentation

System n Determining Lateral-Directional Derivatives from Dutch Roll Oscillations

W h e n ateral nd directional tability ugmentation ys tems aving irst-order im e lags re perational uring Dutch oll free-oscillation) aneuver, th e ffects f th e ugmentation ystem on he requency nd amping of th e oscillations nd n |Ar| / |A/3 l m a y e ccounted or n he a m e anner s a s one or he ongitudinal m o d e f oscillation n ection 7.4.6.

7 . 6 he Graphical Time-Vector Technique

The graphical time-vector method of analysis44"u7, t h e principles of w h i c h w e r e d i s - cussed and applied i n t h e i n i t i a l part of t h i s s e c t i o n , s t h e most common m a n u a l technique used f a r determining t h e l a t e r a l and directional d e r i v a t i v e s . uccessful application i s dependent upon availability of c o n t r o l - f i x e d , D u t c h r o l l oscillation data wherein t h e damping r a t i o i s less t h a n approximately 0 . 3 t o permit definition of the period of o s c i l l a t i o n s , t h e l o g decrement of t h e damping of o s c i l l a t i o n s , amplitude r a t i o s , a n d phase a n g l e s .

7.6.1 Advantages

One advantage of the method i s that t h e procedure i s m a n u a l , and the analyst i s afforded a graphical presentation of various f a c t o r s affecting t h e s o l u t i o n .

Another advantage i s that i t i s possible t o obtain solutions w h e n t h e / 3 - v a n e r e c o r d s a r e a v a i l a b l e , u s p e c t , or when i t i s desired t o avoid applying corrections t o t h e s e r e c o r d s . ypassing t h e ß records w a s discussed i n Section 7 . 3 . 2 . t was s h o w n t h a t t h e v e c t o r polygon of the transverse-acceleration equation i s essential i n t h e solution of t h e amplitude r a t i o , A / 3 | / | A r l , a n d t h e phase a n g l e , gr . Both of t h e s e q u a n t i - ties a r e used i n t h e vector polygons of t h e r o l l i n g - a n d yawing-moment equations t o determine CLg and Cjo when t h e vector i s used a s t h e b a s e f o r the amplitude ratios i n t h e e q u a t i o n s , a s i n Figures 5 9 ( a ) a n d 5 9 ( b ) . h e p h a s e a n g l e i s used i n t h e orientation of t h e / 3 vector i n r e l a t i o n t o t h e Ar v e c t o r and provides a more accurate v a l u e o f $g than c a n usually be obtained d i r e c t l y f r o m flight r e c o r d s . T h e amplitude r a t i o , | A / 3 | / | A r l , s used t o e x t r a c t n3 a n d C iB f r o m t h e d e t e r - -3 mined alues of C n/ 3 A/31/IArl and C ^ A/3 | / |Ar | in igures 9(a) nd 9(b)

7 . 6 . 2 Disadvantages

One disadvantage i s t h a t t h e development o f a d e f i n i t e technique i s required o n t h e p a r t of t h e analyst t o minimize w h a t would otherwise c o n s t i t u t e a rather time-consuming and tedious e f f o r t t o obtain a consistent a n d reliable s e t of r e s u l t s .

Another disadvantage i s t h a t only t w o of t h e t h r e e derivatives i n e a c h of t h e r o l l i n g and directional moment equations may b e determined by means of t h e vector d i a g r a m , t h u s necessitating a n i n t i m a t e o r a wind-tunnel value of o n e o f t h e derivatives i n e a c h of t h e e q u a t i o n s . ince n and C jr erms i n t h e v e c t o r d i a g r a m s (Figures 5 9 ( a ) and 5 9 ( b ) ) a r e t h e smallest v e c t o r s , t i s customary t o e s t i m a t e t h e s e q u a n t i t i e s . h e errors i n the estimated values of ,

i n

jr w i l l generally affect

j

np i l l a f f e c t ( C „r - Cnß) p r i m a r i l y ; t h e errors

p r i m a r i l y , b u t t o a m u c h smaller e x t e n t .

or low a n g l e s - o f - a t t a c k , CDr - C n^ ) m a y b e estimated b y using Equations ( 1 8 3 ) or ( 1 8 4 ) w i t h i n t h e l i m i t s of t h e i r a p p l i c a b i l i t y .



v>

I . r

71

7.6.3 Application of he Graphical Time-Vector Technique o th e Df:termination of C Bo ( C n r - np. O lß an d C j

Figures 9(a) nd 9(b) how he pplication of he raphical im e-vector echnique to he eterm ination f C^ C,^ C^), Ci^ nd C {p T h e m plitude atio, lApl / lAr l nd he hase ngle , $pr ere etermined rom emi log plot uch s that n igure 6 . T h e atio A/3 | / |Ar | nd hase ngle $a r were btained r o m transverse-acce l erat ion iagram s iscussed n ection .3.2. T h e emaining equired amplitude atios nd phase ngles ere etermined s ol lows

lApl |AplArl = * A rl

lArl

l A r l = ffr = 90 + $d

l A r l

lArl ~ * $ =0 *rr "

C D p and C jr hich have elatively m a l l nfluen

(185)

The derivatives were obtained f r o m wind-tunnel d a t a . ssuming there i s no question of the accuracy of t h e d a t a , t h e t u n n e l data should be based o i . oscillatory t e s t s , i n a s m u c h as t h e f l i g h t

data a r e based on a n oscillatory m a n e u v e r .

With t h e various known vector quantities properly oriented i n the respective d i a g r a m s , t h e diagrams were closed and the unknown vectors determined b y drawing t h e unknown vectors i n t h e i r proper phase-angle d i r e c t i o n s , and $or . h e newly determined v e c t o r s , s u c h a s -Cn^ ( | A / S | / | A r l ) and Cnr - C^ ) ( b / 2 V ) , w e r e t h e n reduced t o obtain

C n / 3 • cnr - C n ^ ) . lß . and C ip

Figures 6 0 ( a ) and 6 0 ( b ) , f r o m Reference 4 3 , show t h e r e s u l t s of the application of t h e graphical time-vector technique t o t h e rocket-powered D-558-II research a i r p l a n e . A n interesting a s p e c t o f t h e results i s t h e i n f l u e n c e of power o n t h e stability characteristics of t h i s a i r p l a n e .

At t i m e s t h e r e m a y appear to be an incompatibility within wind-tunnel data when the data a r e compared t o flight-determined d e r i v a t i v e s . t t h e n becomes imperative to r e s o l v e t h e discrepancy within t h e t u n n e l d a t a and b e t w e e n t h e t u n n e l data a n d t h e f l i g h t d a t a . his i s illustrated i n t h e following e x a m p l e w h e r e i n na w a s relatively l o w .

Dynamic model tests of a relatively rigid high-performance aircraft a t a s e t Mach number and = 6.6° howed t h a t Dp = 0 . 0 1 nd C n r - Cnfi ) -0.14 . u n n e l data a l s o showed „ ^ t o b e e q u a l t o 0 . 0 5 5 o n t h e basis of s t a t i c t e s t s a n d e q u a l t o 0 . 0 7 5 7 o n t h e b a s i s of oscillatory t e s t s . light d a t a obtained f r o m t i m e histories of c o n - vergent transient oscillations of the q u a l i t y s h o w n i n Figure 45 Indicated t h a t , w h e n t h e wind-tunnel v a l u e of Cn 0.01 as used i n t h e time-vector s o l u t i o n , aß w a s e q u a l t o 0 . 0 7 1 and ( CDr - Cn^ ) as e q u a l t o 0 . 3 1 3 , t was obvious t h a t ( C n r - C n / j )

= 0 . 3 1 3 a s not representative of t h e true characteristics of t h e aircraft i n t h e D u t c h r o l l m o d e , i n c e i t s positive v a l u e i n d i c a t e d a n oscillatory d i v e r g e n c e , whereas f l i g h t data showed oscillatory c o n v e r g e n c e .



7 2

A c h e c k of the phase angle $ ( - 1 0 4 ° ) b y s e v e r a l analysts showed agreement w i t h i n a f e w d e g r e e s . t w a s decided t h a t a reasonable spread of uncertainty f o r t h e q u a l i t y of data - corrected f o r instrument phase l a g - would permit t o b e 1 0 5 ° ± 5 ° ; a t w o r s t , t h e uncertainty would b e ± 1 0 ° . c c o r d i n g l y , solutions f o r C„ » a n d C Dr - Cn/ § )

( w i t h i n t h e spread of u n - T h e results s h o w n i n Figure 61 i n t h e f o r m of a grid plot i n d i c a t e t h e

and the incompatibility of the wind-tunnel d a t a . he tunnel data w e r e incompatible e v e n when allowances were made f o r unvertainties i n i n e r t i a characteristics a n d readability of f l i g h t d a t a .

were obtained b y uring various values of Cn and

c e r t a i n t y ) . " sensitivity of t h e determined values of C üß and ( Cnr - C aö ) t o Cn„ a n d $

Use was made of approximate Equation ( 1 8 3 ) w i t h d u e consideration t o t h e limitations of t h e equation f o r higher angle-of-attack conditions to a i d i n establishing t h e magni-

tude of ( C „ r - Cn/ § ) . or t h e t e s t condition of a n angle-of-attack of 6 . 6 ° , t h e -0.458 value of ( C n r - ( ^ g ) obtained by Equation ( 1 8 3 ) c o u l d b e i n error t o t h e e x t e n t of 100% or s o . e n c e , i t was estimated t h a t t h e correct value of ( C Dr - C n, ; ) was closer to -0.20 than - 0 . 4 5 8 . l s o , considering t h a t t h e s t a t e of t h e a r t i n obtaining ( C ur

_ Cnß) f r o m wind-tunnel t e s t s w a s more reliable t h a n i n obtaining „p , t h e t u n n e l

value ( - 0 . 1 4 ) of ( C Dr - Cn^ ) w a s surmised to b e representative of the t r u e v a l u e of this d e r i v a t i v e . ncertainties i n t h e inertia characteristics required t h a t s o m e deviation be allow ed i n this v a l u e i n obtaining ( C nj . - C nn) from f l i g h t d a t a . t w a s , t h e r e f o r e , concluded t h a t t h e results o f t h e analysis should l i e w i t h i n t h e shaded area s h o w n i n Figure 6 1 . ithin this a r e a , h e v a l u e of Cnß ( 0 . 0 5 4 ) compatible w i t h

pr = -104° nd C nr - C ^ g ) = -0.14 w a s considered to b e a mean value and w*s used as a n analytical r e s u l t . h e corresponding value of C „ should have b e e n approximately - 0 . 0 4 .

T h e best accuracy i n determining Cn^ n d ( Cnr - C D/ g ) i s obtained wbn A p | / | A r l i s s m a l l , a t w h i c h time the influence of C np i s relatively s m a l l . h e n t h e roll-to- yaw r a t i o i s l a r g e , t may b e advantageous t o e s t i m a t e ( C „r - Cn^ ) and a t t e m p t t o s o l v e f o r Cn o r l o w a n g l e s - o f - a t t a c k , C „ r - C D / g ) m a y b e estimated by using Equations ( 1 8 3 ) or ( 1 8 4 ) within t h e limitations of their a p p l i c a b i l i t y .

T h e best accuracy i n determining C iß a n d C j _ r a t i o i s l a r g e . t this t i m e , t h e i n f l u e n c e of /r s relatively s m a l l

i s obtained w h e n t h e roll-to-yaw I n either

case , th e tatic derivatives, Cnß and C rotary derivatives, ( C nr - C D /g ) nd C jp

Iß a r e determined more accurately t h a n t h e

I t w a s previously pointed o u t t h a t t h e accuracy o f analysis becomes r a t h e r poor f o r

damping ratios greater than 0 . 3 . lthough a good approximation of e h e damping ratio f o r heavily damped aircraft m a y b e obtained b y comparing f l i g h t r e c o r d s w i t h records of heavily damped motions - t h e damping ratio of w h i c h i s known - i t becomes difficult t o draw accurately t h e exponential envelopes of t h e oscillatory motions t o o b t a i n reliable values of amplitude r a t i o s .

7 . 7 ther Analytical Techniques

T h e preceding discussions regarding determination of derivatives f r o m f l i g h t data have s h o w n various l i m i t a t i o n s . h e graphical time-vector t e c h n i q u e , although t h e most s u c c e s s f u l , s not usable f o r damping r a t i o s i n excess of a b o u t 0 . 3 , requires control-fixed t r a n s i e n t oscillation d a t a , a n d r e q u i r e s t h e assumption of s o m e d e r i v a -



h i— -

73 i

t i v e s , which m a y , a t t i m e s , c a u s e difficulties i n s o l u t i o n s . o overcome t h e l i m i t a - tions of t h e preceding t e c h n i q u e s , a number of methods have been proposed f o r t h e comprehensive determination of derivatives ( R e f e r e n c e s 4 8 - 5 4 , f o r e x a m p l e ) . ome have been successful i n p r a c t i c e ; others have n o t . n most i n s t a n c e s , the degree of sophistication involved i n t h e proposals requires automatic data-reduction equipment and t h e time and effort does not warrant their use when analog equipment i s available f o r application of analog-matching t e c h n i q u e s . everal of the methods a r e considered i n t h e following s e c t i o n s .

7.7.1 Least Squaring of h e Equations f M o t i o n

A l o g i c a l a n d straightforward m e t h o d , on t h e sophisticated s i d e , o r determining derivatives f r o m f l i g h t data i s t h e application of the least-squaring technique to the linearized equations of m o t i o n . l i g h t quantities a t discrete time points a r e s u b - stituted i n t o t h e equations of m o t i o n . any more data points are selected than the number of u n k n o w n s , a n d a least-squares process i s applied t o evaluate t h e unknown d e r i v a t i v e s . s l o g i c a l and simple as the approach may b e , i t has not been employed t o o successfully f o r s e v e r a l r e a s o n s , n c l u d i n g : difficulty i n properly conditioning t h e m a n e u v e r , instrumentation a c c u r a c y , phase l a g s between i n s t r u m e n t s , insufficient amplification of recorded data t o provide precise r e a d a b i l i t y , noise i n data r e a d o u t , and instrument a l i n e m e n t .

One of t h e more s u c c e s s f u l attempts t o a p p l y this technique was reported i n Reference 4 8 . o excite a l l t h e lateral-directional modes and give measurable c o n t r o l i n p u t s ; without exceeding t h e l i m i t s of t h e linearized equations of m o t i o n , t h e following control i n p u t program was u s e d :

r

" P r o m trimmed l e v e l f l i g h t , step t h e rudder causing the airplane t o yaw and t h e n r o l l d u e t o dihedral e f f e c t . hen t h e b a n k a n g l e reaches approximately 2 0 d e g r e e s , a p p l y a s t e p aileron deflection s u c h that t h e airplane will r o l l toward a l e v e l f l i g h t a t t i t u d e . n order to obtain a sufficiently l o n g record of t h e response t o a i l e r o n , h e airplane i s allowed t o r o l l t o a n opposite b a n k a n g l e of 20 degrees b e f o r e stopping t h e recording a n d i n i t i a t i n g r e c o v e r y " .

A typical time history of this maneuver i s s h o w n i n Figure 6 2 . l l instruments had similar r e s p o n s e characteristics and high recording sensitivity w h i c h w a s compatible

with calibration-sensitivity spread and calibration s p r e a d . linement of instruments w a s within ± 0 . 3 ° . ecorded d a t a were c l e a n . t w a s f o u n d t h a t noise i n t h e r e a d o u t data significantly affected t h e r e s u l t s . wenty discrete time points used f o r the least-squaring process were considered s u f f i c i e n t .

T h e r e s u l t s , reproduced i n Figures 6 3 ( a ) a n d 6 3 ( b ) , show t h e degree of consistency obtained a f t e r t h e greater-than-usual precautions w e r e t a k e n to provide conditions t h a t would b e c o m p a t i b l e with t h e needs of t h e t e c h n i q u e . he requirements f o r this technique a r e undoubtedly similar to t h o s e necessary to make other promising techniques w o r k a b l e , s u c h a s t h e method of Reference 4 9 . his method i s also a n equation-of- motion technique utilizing t h e Fourier t r a n s f o r m , a method f u n c t i o n t o r e m o v e d e - pendence on i n i t i a l and e n d c o n d i t i o n s , and a least-squaring p r o c e d u r e . k



74

7 . 7 . 2 Frequency-Response Me thod

Methods have been proposed (References 5 0 - 5 2 f o r e x a m p l e ) to determine stability and c o n t r o l derivatives by using frequency-response data obtained f r o m f l i g h t t e s t s . h e ethod of Reference 5 2 encompasses t h e solution of a l l derivatives through a complex

p r o c e d u r e . ther m e t h o d s , s u c h a s that of Reference 5 1 , provide only limited results based on various degrees of a p p r o x i m a t i o n .

The method of Reference 5 2 replaces the time plane w i t h the frequency p l a n e . Amplitude ratios and phase relationships of airplane response to c o n t r o l input f r o m f r e q u e n c y - response analysis of a pulse maneuver51*

ss provide r e a l and imaginary q u a n t i t i e s . he complex quantities at discrete frequencies a r e substituted i n t o least-squared equations solving f o r t h e derivatives d e s i r e d . he method i s simple i n t h e o r y ; h o w e v e r , c o n - siderable c a r e , w o r k , and t i m e a r e involved i n t h e a p p l i c a t i o n , and some experience i s necessary i n the selection of discrete f r e q u e n c i e s . hese factors minimize interest i n f u r t h e r studies of t h e m e t h o d , especially where t i m e i s of the essence i n obtaining a relatively quick l o o k a t t h e f l i g h t wlues. utomatic data-reduction e q u i p m e n t would g r e a t l y expedite t h e frequency-response uialysiä a n d v o u l d be useful f o r t h e other computations r e q u i r e d .

7 . 8 nalog-Matching Techniques

When f l i g h t data a r e of s u c h a nature as t o preclude the successful use of t h e graphical time-vector technique or the approximate e q u a t i o n s , and when time and expense will not permit t h e use of a n experimentation with more sophisticated t e c h n i q u e s ,

recourse i s usually made to the analog to determine t h e derivatives that w i l l provide t h e best match of the analog time history w i t h t h e f l i g h t time history of a m a n e u v e r .

The use of the analog should b e considered a s a l a s t r e s o r t , t o b e used o n l y w h e n other techniques c a n n o t b e a p p l i e d . t i s n o t a " c u r e - a l l " , o r i t c a n produce erroneous answers under c e r t a i n conditions a n d s t i l l provide a good match w i t h t h e f l i g h t time history of a m a n e u v e r .

7.8.1 Conventional nd High-Speed Repetitive Operation R E P O P ) Analog Matching

T h e mathematical model of t h e aircraft f o r t h e a n a l o g c o m p u t e r i s provided by t h e airplane equations of m o t i o n ; when attitude records ( s u c h a s \p and r f > ) a r e available

and used i n the matching p r o c e s s , transformation equations a r e included t o transform aircraft angular rates a b o u t the body axes t o angular r a t e s about Euler a x e s .

G e n e r a l l y , the simplest mathematical m o d e l compatible with t h e needs of a n i n v e s t i - gation i s used t o r e d u c e t h e number of analog components required and t o expediate s o l u t i o n s . five-dagree-of-freedom mathematical m o d e l , involving t h e g e n e r a l equations of m o t i o n , s employed w h e n longitudinal a n d lateral-directional cross-coupling effects a r e factors i n t h e responses o f t h e airplane during t h e m a n e u v e r . h e n s u c h c r o s s - coupling effects a r e n o t f a c t o r s t o b e contended w i t h , t h e longitudinal a n d l a t e r a l - directional motions c a n b e treated independently a n d a s t w o separate analog programs using t h e linearized e q u a t i o n s . nder s u c h c i r c u m s t a n c e s , t h e longitudinal p r o g r a m i s treated a s a two-degree-of-freedom c a s e ( w i t h v e l o c i t y a c o n s t a n t ) unless phugoid i s being c o n s i d e r e d , w h i c h i s not o f t e n ; a n d t h e lateral-directional program i s

treated a s a three-degree-of-freedom c a s e .

mall-perturbation equations may b e used



T h e conventional matching technique i s laborious because of t h e need t o manually match a strip record w i t h t h e overlay e v e r y t i m e a programed condition i s modified i n order t o study t h e e f f e c t of t h e modification and assess t h e next condition t o b e m o d i f i e d . he conventional technique may r e q u i r e f r o m s e v e r a l days t o a week t o obtain a m a t c h .

High-speed repetitive operation ( R E P O P ) matching differs f r o m t h e conventional i n s e v e r a l basic aspects56. h e s t r i p recorder i s replaced b y a n oscilloscope a n d t h e

response t o inputs is projected onto the s c o p e , which has an overlay fastened to i t . T h e projected response appears as a stationary time history a s a result of a n automatic high-speed recycling of t h e response c o m p u t a t i o n . h e maximum recycling speed f o r f i d e l i t y i s governed b y t h e t i m e s p a n of t h e t i m e history to b e matched a n d t h e f r e q u e n c y - response characteristics of t h e f u n c t i o n g e n e r a t o r . here a cycling rate of 2 5 0 cycles p e r second may provide f i d e l i t y f o r a 3 - o r 4-second t i m e h i s t o r y , t may c a u s e serious distortions i n projections onto t h e s c o p e i f a 10-second t i m e history i s p r o j e c t e d .

High-speed repetitive operation matching relieves t h e operator o f manual matching of t h e t i m e h i s t o r y , permits him t o make rapid modifications of derivatives and I n i t i a l c o n d i t i o n s , and allows h i m t o o b s e r v e effectively t h e influence of a modification o n t h e r e s p o n s e . hen a n optimum match i s achieved o n t h e s c o p e , a strip record i s made a n d matched w i t h a n overlay t o c h e c k t . i e f i d e l i t y of t h e s c o p e match a n d t o r e t a i n a

record of t h e resulting m a t c h . REPOP match c a n normally b e achieved i n 4 t o & h o u r s .

T*

7 5

t o advantage i n such i n s t a n c e s , particularly when datums of angular rates and Euler attitudes may be suspect and angular accelerations have excesnive noise or are not a v a i l a b l e .

Initial estimates of stability a n d c o n t r o l derivatives t o be used i n t h e mathematical model a r e obtained f r o m available theoretical and/or wind-tunnel v a l u e s . f p o s s i b l e , flight-determined derivatives obtained through t h e use of t h e approximate equations a r e e m p l o y e d . n the a b s e n c e of t h e p r e c e d i n g , t h e b e s t estimates possible a r e m a d e . I n i t i a l estimates are required t o establish reasonable scaling f a c t o r s f o r the manually adjusted derivative potentiometers to save operational t i m e .

Inasmuch as errors i n i n i t i a l conditions s h i f t t h e amplitude or r o t a t e the response time h i s t o r y , provisions a r e made o n the analog t o program i n i t i a l conditions through manually controlled p o t e n t i o m e t e r s .

Flight t e s t i n p u t s i n t h e f o r m of aileron and r u d d e r deflections a r e reproduced o n f u n c t i o n generator components of t h e analog i n a s f a i t h f u l a reproduction as possible within t h e limits of t h e f u n c t i o n g e n e r a t o r s , which have a f i n i t e number of b r e a k p o i n t s . When these inputs are introduced i n t o the mathematical m o d e l , the analog computes a r e s p o n s e .

I n conventional a n a l o g - m a t c h i n g , t h e response i s recorded by a strip r e c o r d e r . h e recorded response i s t h e n compared with t h e a c t u a l f l i g h t t i m e h i s t o r y , which i s r e -

produced o n clear plastic t o o v e r l a y on t h e analog t i m e h i s t o r y . mismatch indicates t h e need t o modify t h e values of t h e d e r i v a t i v e s , possibly c h a n g e signs of s e v e r a l of t h e m , and possibly modify t h e i n i t i a l c o n d i t i o n s . hese changes a r e made by using a judicious trial-and-error process u n t i l a match i s o b t a i n e d .

r



76

7.8.2 dvantages of h e Analog-Matching Technique

T h e nalog-Batching technique or derivative eterm ination, in ffect, ccomplishes «hat ophisticated analytical echniques see he preceding ection) ave ttempted. It nables he etermination of derivatives under ircumstances here pproxim ate equations nd he raphical im e-vector technique ail. It oe s no t ely p o n definite restrictive maneuvers, lthough here re om e maneuvers ha t annot e olved or . T es t a ta shoving nadvertent nputs nd ubsequent isturbances m a y e sed.

7.8.3 Limitations of h e Ar.alog-Matching Technique

The success of every technique discussed f o r determining derivatives «as contingent upon the proper conditioning of the maneuvers i n v o l v e d . his i s no less true of t h e analog-matching t e c h n i q u e . Dutch r o l l m a n e u v e r , induced b y a c o n t r o l p u l s e , I n «hich no spiral- or roll-subsidence modes are significantly e v i d e n t , i s generally Impossible

to match with a unique set of d e r i v a t i v e s . t «ill be found that any number of c o m - binations of derivatives « i l l provide a m a t c h . maneuver Involving continuous oscillation of the c o n t r o l s u r f a c e s , as would b e the case of lateral-directional oscillatory motiors with the lateral-directional stability augmentation system o n , « i l l also be very difficult t o match w i t h a unique s e t of d e r i v a t i v e s .

A properly conditioned lateral-directional maneuver for use on the analog t o permit determination of a unique set of derivatives f o r a match should excite t h e r o l l and s p i r a l modes as well as Dutch r o l l o s c i l l a t i o n s . h e likelihood of obtaining a unique s e t of derivatives i s increased « h e n t h e maneuver i s conditioned t o include a rudder disturbance of a step-like n a t u r e , a transient o s c i l l a t i o n , and an aileron disturbance not necessarily i n this order - a s « a s mentioned i n Section 7.7.1 and also illustrated I n a recover?-from-sideslip maneuver «hich i s considered i n t h e next s e c t i o n .

A l s o , as «as mentioned i n Section 7 . 6 . 3 , better accuracy «HI b e achieved I n t h e r a t i o of t h e dynamic characteristics ajor directional derivatives « h e n t h e A p | / | A r

ilmlzing Influence of C0p) , ratio is high (minimizing t h e influence of Cir

I s l o « (minimizing influence of Cn_ ) , and i n t h e major lateral derivatives « h e n t h e | A p | / | A r | i t «ay b e c o n c l u d e d , t h e n ,

C jr re normally difficult to determine to a n y respectable degree of accuracyr he possibility of determining C jr appears to improve w i t h increasing tendency of the aircraft t o r o l l off during a m a n e u v e r .

that Cn and

7,8A Application of h e Analog-Matching Technique

Figure 4 hows he esults f an nalog match f recovery-from -sidesl lp" maneuver a t M a c h u m b e r of .84 nd n ltitude f 9,400 t. T h e atch s ypical or his aircraft, hich ad negative ffective ihedral nd dverse i leron ya w or he match

shown. Rigid ind-tunnel ata orrected or lexibility ffects n he ctua l ehicle predicted practically e ro ffective ihedral nd roverse ya w due o ileron. It as im possible o ubstantiate he redicted a lue s n he nalog, nd nl y ne ombination of derivatives ould rovide he match.

T h e ollowing procedure s ypical f ha t mployed n rriving t he nalog match of th e light ata which id ne t nclude olling nd ya«lng ccelerations:

- . m

77

a s 1

•

(1 ) h e mathemat ica l odel as epresented y hree ateral-directional m a l l - perturbation quations

A/3 = ? in « f > 0 A 0 ) - A r „ A p (q S + Cj j A 5r C y? A «a) - sin0o

A p = £ A f + y L+ 1(CjjAp cJrAr) Ci$rA Sr C j ^ A S J

I S b TbA r = -£* Ap — C n ^ g A S + — (C^r CDpA p ) nSrA 8r e,,ASj

(ii) n ddition, he ollowing ransform ation as mployed o etermine he hange in uler oll ngle , hich ttained magnitudes f he rder f 0 ° n ccasions in he maneuvers nder onsideration

A 0 = Ap - r0 Ar)0 o co s (0O A ft - 06Q COS<£0

(iii) Finally he utputs f th e mathemat ica l model ere pplied o he ollowing tw o quations o modify nalog alues f / 3 nd at o orrespond o he indicated alues f he light ata:

A/3. = /3 + lAr- Ap 1 A a tt - - sin < £ „ + A < £ ) in0o - A/8 + A r - x , A p ) -

" yinstr

/ ~ o / •3rAr* + pAp\ A f Ap

- : - jj

*

+ xinstr g " zlnstr g

( i v ) Starting with t h e arbitrarily selected t i m e zero ( a s i n Figure 6 4 ) f o r t h e time history t o b e m a t c h e d :

( a ) Cüß was adjusted f o r approximate f r e q u e n c y n a t c h .

( b ) T h e control derivatives w e r e adjusted t o provide a n i n i t i a l r o u g h match i n t h e magnitudes of n d 0 .

( c ) Potentiometers f o r „ n d ß0 w e r e adjusted t o r o u g h l y aline n d ß races of the analog w i t h f l i g h t d a t a ; s i m i l a r l y , potentiometers f o r

< P 0 nd p0 ere adjusted t o r o u g h l y a l i n e 0 and p r a c e s . hese actions involved t h e following analog I n t e g r a t i o n

A r = / ( f 0+Ar ) d t . A£ = J ( 4 0+A/ 3 ) d t .

AP = J (P 0 +

AP >dt

A <

£ = M

A fedt



78

(d) ince djustaent of j > 0 and p0 m odifies l ineaent f nalog and light tracer: f r and ß , tep c) as eiterated a s a a n j r iaes s neces sary to obtain rough lineaent of analog nd light races of r f > and p

(e ) ttention a s hen ocused on he r trace o btain aore efined natch of this race y aore autiously djusting C nr V g This operation necessitated adjustaent of Cj ^ j j teep th e p trace n ine.

C n « and C « ,

C l « a n d C l r t o

I t should be noted t h a t the preceding f i v e steps (IT) ( a ) - ( e ) , which constitute a n i n i t i a l phase of operation to obtain a n approximate m a t c h , involve about 1 h o u r . he explanation of the procedure i s , by n e c e s s i t y , b r i e f . t « i l l be readily appreciated that the steps are iterative to keep the frequency of d i s t u r b a n c e , the aagnitude of disturbances of the various t r a c e s , and alineaent of analog and f l i g h t t i n e histories c o a p a t i b l e .

The second phase of the analog-aatching process involved the following operations:

( f ) With the trace roughly m a t c h e d , attention was focused on t h e > and p traces by aanipulating the Cj ^ , C /p , and C jr derivatives and the lateral-control derivatives a s n e c e s s a r y . uring this o p e r a t i o n , i n e adjustments were required and aade on the r n d ß races ( a s per step ( i v ) ( e ) ) .

( g ) With > p , r , and ß races matched a s closely as p o s s i b l e , attention

was focused o n t h e

t

r a c e .

his involved C y ß CJJ

and Prsr

The ast phase f th e nalog-aatching operation nvolved making ine djustments o initial onditions to ompensate or probable rrors) nd ine djustments o he derivatives, n s sence , erform ing n terative rocedure of th e preceding operations. T h e econd nd ast hase of th e nalog-aatching process eneral ly nvolved o tours nd, t iaes, ore.

7.8.5 Accuracy of Results n Analog-Matching of Flight Data

As entioned reviously, the ccuracy f he esults n nalog-matching f light data s argely ependent pon he onditioning of th e aneuver. F or th e ongitudinal derivatives, esults rom ullup-and-relaase maneuver of n dvanced Mgh-performance aircraft howed he ollowing ccuracies ased n he mount he derivatives ould e

changed efore trend oward niaoatch ecame vident:

(1 ) or strong maneuver:

C N( X 0 %

CN S 2 0 % o 0 %

( C Nq Na) 00% or more

Cn «

Cnse

(C ma m s . )

1 0%

2 0 % o 0 %



R

7 9

( 2 ) F o r a w e a k M a n e u v e r :

C n «

2 0 % C - a 1 0 %

1 0 0 % *•.

2 0 %

2 0 0 % o r m o r e «V, + c Bd) 4 0 % SNQ + CH6t )

T h e a c c u r a c i e s o f t h e l a t e r a l - d i r e c t i o n a l d e r i v a t i v e s o b t a i n e d f r o n a n a l o g - m a t c h i n g o f w e l l - c o n d i t i o n e d , r e l e a s e - f r o m - s i d e s l i p m a n e u v e r s o f t h e s a m e a i r c r a f t a r e s h o w n

i n t h e f o l l o w i n g t a b u l a t i o n , a l o n g w i t h t h e f a c t o r s w h i c h i n f l u e n c e t h e a c c u r a c y :

C n ß %

C i ^ % t o 1 5 %

C j / 3 % t o 2 0 %

C n r % t o 3 0 %

C np % t o 3 0 %

C ip - 5 % t o 3 0 %

C ir - 5 % t o 5 0 %

^ n s r

c* sB

C n sa

- 5 %

- 5 % t o 1 5 %

- 5 % t o 3 0 %

- 1 0 % t o 3 0 %

T r u e f o r a n y r u d d e r r e l e a s e i n v o l v i n g m o r e t h a n o n e c y c l e o f o s c i l l a t i o n .

D e p e n d e d u p o n o s c i l l a t o r y c h a r a c t e r i s t i c s o f > a n d m a g n i t u d e o f ß a f t e r r u d d e r r e l e a s e .

D e p e n d e d o n t h e m a g n i t u d e o f t h e t

o s c i l l a t i o n s a n d t h e a v e r a g e s l o p e o f ß f r o m r e l e a s e t o s t e a d y v a l u e .

D e p e n d e d u p o n t h e a m o u n t o f t r a n s i e n t s

t h e a i r c r a f t w a s a l l o w e d t o g o t h r o u g h b e f o r e c o n t r o l s w e r e a p p l i e d a g a i n .

D e p e n d e d u p o n t h e m a g n i t u d e o f t h e r o l l r a t e d u r i n g o s c i l l a t i o n . i g h e r r o l l r a t e s h o w e d b e t t e r a c c u r a c y .

D e p e n d e d u p o n t h e m a g n i t u d e o f t h e r o l l r a t e d u r i n g o s c i l l a t i o n . i g h e r r o l l r a t e s h o w e d b e t t e r a c c u r a c y .

D e p e n d e d u p o n m a g n i t u d e a n d o s c i l l a t o r y c h a r a c t e r i s t i c s o f r o l l o f f . a r g e r r o l l o f f s h o w e d b e t t e r a c c u r a c y .

T r u e f o r a n y r a p i d r u d d e r i n p u t .

D e p e n d e d u p o n t h e m a g n i t u d e o f t h e c o n t r o l i n p u t .

C y8 - 5 % t o 5 0 % o r m o r e

C ys - 4 0 % t o 1 0 0 % o r m o r e f t

These esults m a y e onsidered typical f what a y e xpected n nalog-matching cf light ata btained r o m roperly onditioned maneuvers. h e ccuracies m a y e ll be ypical f hose ha t ay e xpected h e n omprehensive nalytical echniques re used.

:- I



80

8 . APPLICATION OF PLIGHT DERIVATIVES

If wind-tunnel data and theory w e r e i n f a l l i b l e , i t stands t o reason that there would be no need f o r f l i g h t determination of d e r i v a t i v e s . o w e v e r , such is not the c a s e . As new concepts i n aircraft were d e v e l o p e d , either with regard t o physical geometry or propulsion s y s t e m s , and as aircraft f l y i n new Mach and altitude r e g i m e s , t h e r e i s the need t o verify aerodynamic t h e o r y and wind-tunnel data and various influences of aeroelastic deformations of prototype structures on stability c h a r a c t e r i s t i c s ; t o provide supplementary information not obtained i n limited winJ-tunnel s t u d i e s ; and t o uncover the source of discrepancies between predictions and a c t u a l f l i g h t b e h a v i o r . The following discussions provide some i n s i g h t i n t o s e v e r a l of these a r e a s .

8.1 Verification of Wind-Tunnel Data and Theory

As t h e Mach c a p a b i l i t y of t h e airplane i n c r e a s e s , t h e technology i n wind-tunnel testing becomes more c r i t i c a l w i t h regard t o m o d e l c o n s t r u c t i o n , s u p p o r t of t h e m o d e l , and interpretation of t h e t u n n e l d a t a . hereas t h e o r y depends upon wind-tunnel data f o r v e r i f i c a t i o n , or t o f i l l i n g a p s where t h e o r y f a i l s , t h e wind-tunnel may depend upon f l i g h t d a t a , a s new r e g i n . e s of f l i g h t u n f o l d , t o v e r i f y testing t e c h n i q u e s .

Plight data pointed out the need f o r a greater concentration of t e s t points i n t h e transonic region t o accurately define the stability characteristics i n t h i s r e g i o n ( P i g . 5 4 ) . light data showed a l s o t h a t i t w a s n o t sufficient t o use a cold j e t stream t o simulate the e x h a u s t of r o c k e t e n g i n e s . igure 60 s h o w s t h e effect of t h e j e t exhaust of the D-558-II r e s e a r c h airplane on t h e lateral-directional stability c h a r a c t e r - i s t i c s of t h e vehicle i n t h e supersonic r e g i o n . he destabilizing influence of power w a s t h e r e s u l t of a pluming of t h e h o t j e t e x h a u s t a n d consequent f o r m a t i o n of a lambda s h o c k w a v e a t t h e juncture of t h e v e r t i c a l t a i l a n d t h e f u s e l a g e . uring Dutch r o l l o s c i l l a t i o n s , the s h o c k wave on t h e leeward side of t h e v e r t i c a l t a i l moved f o r w a r d , while on t h e windward s i d e i t remained attached t o t h e Jet e x i t . his p h e n o - menon i t n o t c o m m o n ; t w a s t h e r e s u l t of overexpansion of t h e j e t e x h a u s t a n d t h e proximity of t h e tr^Uing e d g e of t h e vertical stabilizer t o t h e j e t e x h a u s t s .

Another illustration of discrepancy between f l i g h t and predicted data involved elevator setting f o r l g f l i g h t . comparison of t h e v a r i a t i o n of predicted a n d f l i g h t - determined elevator settings w i t h M a c h number showed increasing discrepancy with i n - creasing M a c h number f o r a c o n s t a n t center-of-gravity p o s i t i o n . n t h i s i n s t a n c e ,

involving aeroelastic e f f e c t s , predictions showed reasonably c l o s e correlation of C na w i t h f l i g h t d a t a ; w h e r e a s , ma n d ms showed a difference i n trend a s w e l l a s l e v e l . reliminary s t u d y of t h e problem snowed a need t o consider mn s w e l l as C m „ and C , m s - Thus, he ollowing itching-moroent quation or r immed n- accelerated evel light, based on Equation 53b) Tab le V ), as sed nd constituted

the major consideration n arriving at he most ikely auses or he iscrepancy

between predicted nd light rim settings of he elevator

The angle-of-attack a + cc N= o) as eplaced y ts equivalent

(186)

a + «CN=O CNP (187)

8 1

t o determine both predicted and f l i g h t values of CBo y the following new f o r m a t of Equation ( 1 8 6 ) , which i s t h e slope-intercept expression f o r solving C Bo , or

CN( X e

A l s o , Equation ( 1 8 6 ) was transposed t o s o l v e f o r

K - Sa.-<w"* ac*> 1 8 9 ) C-Se ° W 8e

A comparison of predicted and f l i g h t values of t h e ratios i n Equation ( 1 8 9 ) showed t h e values of the r a t i o g/C^ t o b e essentially t h e s a m e ; h o w e v e r , Bo/C„ s differed i n l i n e with t h e discrepancy i n e . alculation of t h e static margin using c«o/ cNa • " h i

00 was employed i n determining CBo , also showed a discrepancy between prediction and f l i g h t . n t h e f i n a l a n a l y s i s , i t appeared that the major source of discrepancy b e t w e e n predicted and f l i g h t longitudinal t r i m elevator settings w a s d u e primarily t o t h e differences i n ^ n d C „ s

An illustration of a discrepancy between wind-tunnel and f l i g h t data involving

power effects and aeroelasticlty i s s h o w n i n Figure 6 5 . his instance concerned t h e F-100 airplane ( F i g . 1 6 ) , w h i c h i s considered t o b e a relatively r i g i d aircraft and has i t s air-intake nozzle a t t h e n o s e . s s h o w n i n Figure 6 5 , t h e variation of t h e wind-tunnel value of CD/ S with Mach number has r o u g h l y t h e s a m e trend as t h e f l i g h t - determined v a l u e . o w e v e r , there i s a n appreciable difference i n l e v e l t h a t i s w e l l beyond t h e difference t o b e expected d u e t o t h e values of moments of i n e r t i a s ; values a r e known t o within 5% a t b e s t . he results of a n investigation t o t r a c e the sources of t h e discrepancy showed appreciable moment of momentum e f f e c t s of air-intake f l o w a n d aeroelasticlty effects of the vertical t a i l . hen t h e b a s i c r i g i d t u n n e l data w e r e corrected f o r these t w o f a c t o r s , a i r l y good correlation w a s achieved with t h e f l i g h t data ( F i g . 6 5 ) .

A technique i n tracking d o w n inconsistencies i n wind-tunnel data involving C n;3 ,

CDp , and ( Cnr - C C y £ ) was illustrated i n Section 7 . 6 . 3 .

8.2 Effects o f Aeroelasticlty

The effects of aeroelastic deformation of t h e s t r u c t u r a l components o n t h e stability a n d c o n t r o l characteristics of t h e aircraft a r e of p r i m e c o n c e r n , particularly i n l a r g e transport d e s i g n s , a s pointed out i n Section 6 . 3 . h e illustration o f a e r o - e l a s t i c effects s h o w n i n Figure 65 represents a n i n t u i t i v e approach i n accounting f o r a discrepancy b e t w e e n wind-tunnel a n d f l i g h t d a t a . his approach presumes t h e b a s i c rigid t u n n e l data t o be c o r r e c t . t also presumes t h a t aeroelasticlty effects are s i m p l e enough t o permit reasonably r e l i a b l e calculation of corrections t o t h e d a t a .

As aircraft Increases i n s i z e a n d s l e n d e r n e s s , and operate a t increasing dynamic

p r e s s u r e s , aeroelastic deformations of t h e structure a s s u m e increasing s i g n i f i c a n c e . T h e i n f l u e n c e o f aeroelastic deformations o n t h e stability a n d c o n t r o l characteristics i s difficult t o predict o n t h e b a s i s of t h e o r y . h e deformations of t h e v a r i o u s components of t h e structure a f f e c t t h e s h o c k patterns of t h e airflow w h i c h , n t u r n ,



82 affect he tability nd ontrol haracteristics n m u c h o r e omplex manner ha n th e eroelaatic eformation f ne r wo urfaces n elatively igid ircraft. Rigid-model a ta m a y e Questionable ecause f he uncertainties n he ru e igidity of he odel nd model upports nd nterference ffects. Thus, o r e positive approach s equired o ssess lexibility ffects o erify n d mprove heory nd develop tunnel echniques .

A light est echnique or etermining eroelasticity ffects n tability nd control haracteristics s utlined n ection 6.3. Th e echnique, s presented, s somewhat implified n ha t he lifting components of hrust s onsidered o e negligible. This pproxim ation implifies light lanning, onitoring, nd making

on-the-spot hanges n light onditions f W and h for maneuvers t onstant M approxim ately onstant C L due o erodynamic ift lone, and onstant enter-of- gravity. A n verage of th e ostfl lght-determ ined a lues f

W - T sin 6 (190)

for he est points n he constant I I L and enter-of-gravity ine" n igure 3 9 - such s oints nd - will onstitute he epresentative alue f C , for these est oints. T h e a x i m u m eviation r o m ctual C L is ithin he xperimental error of th e nvestigation. T h e tability nd ontrol derivatives f hese points, w h e n plotted ga inst ynamic ressure , define urve hich hows he ffect f eroelasticity n he derivatives,

dynamic lift lone, and enter The urve epresents nly ne

f-gravity ondition. d u e t o a e r o -

8.3 Stability Criteria

Considerations of t h e stability of a n airplane i n c l u d e not o n l y i t s inherent s t a b i l i t y , which i s i t s behavior without pilot i n p u t s f o l l o w i n g a n i n i t i a l d i s t u r b a n c e , b u t a l s o i t s behavior i n response t o p i l o t i n p u t s . n g e n e r a l , t h e s t u d y of t h e stability of a n airplane i n v o l v e s t h e effect of derivatives on t h e I n c r e a s e o r d e c r e a s e of the s t a b i l i t y . t i s a i objective s t u d y . h e n t h e stability of t h e airplane i s considered i n t h e l i g h t of t h e degree o f p i l o t ' s acceptance of t h e a i r p l a n e , and p i l o t ratings a r e i n t r o d u c e d , t h e s t u d y becomes subjective and i s referred to as a handling- qualities s t u d y . s may b e - e a d i l y s u r m i s e d , o n e s t u d y complements t h e o t h e r .

Any extensive discussion of handling q u a l i t i e s , which integrates t h e pilot a s a human s e r v o s y s t e r o constituting a f e e d b a c k l o o p i n t h e c o n t r o l s y s t e m , s beyond t h e s c o p e of this p a p e r . t would i n v o l v e t h e s t u d y of human f a c t o r s a n d i s affected b y t h e p i l o t ' s technical background a s w e l l a s t h e d e p t h of piloting b a c k g r o u n d , t h e t y p e s of aircraft f l o w n , orientation a n d t y p e s of displays i n t h e c o c k p i t , a n d g e n e r a l c o c k p i t e n v i r o n m e n t . he a r t and science of handling-qualities investigations i s covered extensively i n t h e literature ( R e f e r e n c e s 5 7 - 6 5 , o r e x a m p l e ) .

8.3.1 Longitudinal Short-Period Oscillation, co n

T h e r e s p o n s e of t h e airplane t o a n elevator i n p u t or g u s t disturbance w i l l normally i n c l u d e a longitudinal short-period o s c i l l a t i o n . n oscillatory condition b y itself indicates a s t e t i c oscillatory s t a b i l i t y . o s i t i v e , n e u t r a l , or negative dynamic

oscillatory stability i s dependent u p o n t h e presence of p o s i t i v e , z e r o , or negative damping c h a r a c t e r i s t i c s , respectively A s t u d y of t h e l o n g i t u d i n a l characteristics involves both static oscillatory stability and d a m p i n g .



84

Thus, or ny ne nass haracteristic nd onfiguration of he irplane, he amping ratio £ is unct'jn f \ p N (X (CBq ld, and v/(-cB(X ).

8.3.3 Longitudinal Short-Period ead Term, -Za

Th e a r a m e t e r Z ~a hich s unction f C N (X is ongitudinal hort-period lead erm hich affects he ead of he pitch ate q with espect o he ontrol input Be an d ngle-of-attack s hown y he ransfer unctions

q(s) -MSeZa 8

sz

2£a)ns o ü *

x ) m C NaqS\ / 1 s

m V

B2

4«ns co l

( 1 9 4 a )

and

o(s) M S .

S e(s) 2 £o; ns a > * (194b)

A s hown n eference 4 , the im e or eak m pli tude f q du e o tep nput e - creases ith decreasing -Z I f -Z „ . . ua becomes sufficiently s m a l l i n comparison t o

c t >n , t h e r e s p o n s e t o a s t e p i n p u t c a n b e d i s c o n c e r t i n g . t may b e characterized i n a tracking t a s k b y a n i n i t i a l i n c r e a s e i n p i t c h attitude of the airplane followed by d w e l l , possibly w i t h t h e airplane aimed a t t h e t a r g e t ; b u t , t h e n , w i t h no f u r t h e r c o n t r o l i n p u t , t h e r e w i l l b e a subsequent i n c r e a s e i n t h e a t t i t u d e . his t y p e of behavior m a y give t h e pilot t h e f e e l i n g t h a t t h e airplane i s u n s t a b l e .

A low v a l u e of -Za may c a u s e t h e p i l o t t o experience a looseness i n p i t c h , p i t c h - r a t e o v e r s h o o t , a c k of c o n t r o l p r e c i s i o n , a n d higher c o n t r o l f o r c e s . n t h e other h a n d , a high value of -Za may c a u s e a tendency to o v e r c o n t r o l , exceed normal , a n d , n g e n e r a l , g i v e t h e i m p r e s s i o n t h a t t h e c o n t r o l i s t o o s e n s i t i v e .

8.3.4 he Dutch Roll Oscillation, wn

T h e Dutch r o l l mode of o s c i l l a t i o n , represented b y t h e following e q u a t i o n , based on a n approximation of t h e second equation i n Equations ( 7 8 ) , i s a measure of d i r e c t - i o n a l stiffness

og = N£-L£sina+ N j L

^ /n /3 -JLc- - "iß

+ YYClßj 5Sb ( 195 )

x

Insofar as derivatives a r e c o n c e r n e d , n/3 and C, ^ a r e normally t h e only derivatives of a n y consequence i n defining t h e frequency of t h i s mode of o s c i l l a t i o n . f t h e s e t w o d e r i v a t i v e s , n/3 s d o m i n a n t . t should be noticed t h a t w h e n t h e static d i r e c t - i o n a l stability i s z e r o ( Cn^ = 0 ) , t h e r e i s s t i l l s o m e degree of oscillatory s t a b i l i t y , providing t h e effective d i h e d r a l i s positive (Ciß - and t h e p r o d u c t of inertia i s n e g a t i v e , or v i c e v e r s a . ome aspects of t h e controllability of t h e airplane w h e n jnß i s near z e r o a n d slightly negative a r e reported i n Reference 6 0 .

8 5

8.3.5 Dutch Roll Damping Coefficient, 2 £ < y

T h e Dutch r o l l d a m p i n g coefficient represented b y t h e following e q u a t i o n , based o n a n approximation of t h e f i r s t equation i n Equations ( 7 8 ) , gives t h e measure of t h e dynamic stability o f t h e Dutch r o l l mode

2H «r - * / J - Lp

( Cnr - C „ f l ) + -?£ + C j _

2 V I Z °r

aß

mV V IX p q S ( 1 9 6 )

This equation s h o w s t h e interaction of t h e more dominant derivatives affecting t h e damping r a t i o . h e equation i s - s o r e accurate than t h a t s h o w n as Equation ( 1 8 2 ) i n t h a t i t i n c l u d e s C j

8.3.6 Dutch roll amping Ratio, £

On t h e basis of Equations ( 1 9 5 ) and ( 1 9 6 ) , t h e damping ratio c a n be approximated t o a t l e a s t t h e f i r s t d e g r e e of approximation b y

- % - Vc ;

-fe n

c „ 4 > mVb 2 V I . • ] lJ ( q S b ) >

f- 'Iß)

( 1 9 7 )

T h e Dutch r o l l damping r a t i o i s strongly affected b y N£ and No . n i n c r e a s e i n t h e negative v a l u e of N^ o t only increases t h e damping r a t i o , , b u t a l s o improves t h e stability of t h e s p i r a l m o d e . ncreasing T A not only i n c r e a s e s direct- i o n a l stiffness b u t a l s o t h e D u t c h r o l l damping r a t i o , which may b e d e s i r a b l e . Decreasing Ng n c r e a s e s t h e b a n k a n g l e t h a t i s i n d u c e d b y a g i v e n amount of sideslip i n t h e Dutch

r o l l m o t i o n , a characteristic w h i c h c o u l d b e detrimental t o maneuvering c o n t r o l of t h e a i r p l a n e . n a d d i t i o n , decreasing Ni ncreases t h e amount of Dutch r o l l disturbance i n t h e r o l l mode r e s p o n s e t o a s t e p a i l e r o n i n p u t - a s reflected i n the parameter ( o j ^ / a i g )2 o b e discussed - and c a n disturb and nislead t h e p i l o t .

8.3.7 Stability Criteria or Aileron-Only Roll Control, ^Joi

T h e r o l l p a r a m e t e r , t y / < u n , i s t h e r o l l numerator t o Dutch r o l l frequency r a t i o of < £ / 8e esponse f u n c t i o n . t i s represented by

% _ G - L

1 T.ITi K ßhL

i ß n/ 3

( 1 9 8 )

86

The parameter I s a measure of t h e a m o u n t by which t h e Dutch r o l l motion i s excited when aileron inputs ( r u d d e r f i x e d ) a r e made b y t h e p i l o t . t i s particularly Important i n the r o l l tracking task i n w h i c h the pilot-airplane combination c a n exhibit considerably different lateral-directional oscillatory tendencies t h a n would be exhibited by the airplane a l o n e . t provides a good i n d e x regarding the increase or decrease i n stability of the airplane during the aileron-alone r o l l tracking t a s k .

When f y / w „ = l , there i s no ya w due to aileron Inputs and there i s little or no Dutch r o l l motion i n response to aileron i n p u t . hen

UJ<\ t h e pilot-airplane

combination i n an aileron-only tracking task w i l l e x h i b i t a n effective damping r a t i o i n r o l l tracking tasks greater t h a n the Dutch r o l l damping r a t i o . hen

WV% > 1 .

the effective damping ratio w i l l be l e s s t h a n the Dutch r o l l damping r a t i o and t h e r o l l

that results from aileron i n p u t i s augmented by t h e r o l l d u e to s i d e s l i p ; t h i s c a n cause stability problems i n t h e r o l l tracking t a s k , especially when t h e Dutch r o l l damping ratio i s small and 4>\/\ß\ i s l a r g e .

Equation ( 1 9 8 ) s h o w s significant interaction of s t a b i l i t y , parameters affecting c^f/co B h e i n t e r p l a y of Cns Ciß

inasmuch a s these parameters may have either plus or minus v a l u e s . o r m a l l y ,

c o n t r o l ,

and x z

a n d i n e r t i a i s i m p o r t a n t ,

ns. and Ciß a r e t h e controlling p a r a m e t e r s ; t h u s , i f t h e effective d i h e d r a l i s positive ( C { d < 0 ) , C „5 w i l l have t o be adverse (CBi < ) t c assure t y A > n < 1 nd a stabilizing action during t h e r o l l tracking t a s k .

8.3.8 Dutch Roll Stability Criteria, \<f>\/\ß\

T h e amplitude r a t i o 4>\ / \ß\ i s a characteristic of t h e Dutch r o l l oscillations and i s t h u s i n d e p e n d e n t of a n y excitations o f c o n t r o l i n p u t s . t s mathematical r e l a t i o n - s h i p t o derivatives i s g i v e n b y

1 0 1

1 / 9 1 hi

i NT'

2

r/ 2

-12 1 +

H

( 1 9 9 )

T h e c o m p l e x Interaction of t h e derivative parameters makes i t difficult t o determine p i l o t sensitivity t o 0 I / I / 3 I . o w e v e r , i f t h e airplane h a s high directional s t i f f - ness ( f i > n > l ) , low 0 | / | / Ö | , reasonable > 0 . 1 a n d adverse yaw d u e t o a i l e r o n , t h e

pilot generally does n o t

bother t o

coordinate t u r n s

b y using

r u d d e r , n a s m u c h a s t h e lateral-directional stiffness keeps sideslip s m a l l a n d t h e low v a l u e of < £ | / / ? l

keeps r o l l d u e t o sideslip s m a l l ( R e f .64 ) .

If < £ l / l / 3 | s l a r g e ( o f t h e order of 4 or m o r e ) , rudder coordination becomes necessary i n maneuvering t o keep sideslip s m a l l i n order t o minimize the r o l l d u e t o s i d e s l i p . f t h e airplane i s characterized b y favorable yaw d u e t o aileron (CDi > 0 ) as w e l l a s high values of # | / | / 8 | , t h e pilot uses a cross-coordination of rudder a n d aileron controls ( r i g h t aileron and l e f t r u d d e r ) t o prevent excessive rolls i n maneuvers ( R e f .64 ) . t i s not difficult t o achieve coordination of c o n t r o l s , p r o - viding t h e airplane i s not excited by e x t e r n a l d i s t u r b a n c e s . o w e v e r , because t h i s cross-coordination i s u n n a t u r a l , t h e p i l o t i s more c r i t i c a l of favorable yaw d u e t o aileron ( C ns 0 ) t h a n adverse yaw d u e t o aileron ( C „ g 0 ) .

IM

8 7

8.3.9 Roll-Subsidence Root, 1 / T .

T h e roll-subsidence r o o t , / T R , i s Influenced most significantly b y t h e parameters s h o w n i n t h e following e q u a t i o n , which i s based on t h e third equation of Equations ( 7 8 )

—~ - L ' T „ p K

= -cj, qSb

2

2VT

( 2 0 0 )

A s s h o w n , the r o l l subsidence i s dominated by t h e damping-in-roll d e r i v a t i v e , j

T h e roll-subsidence r o o t has a d i r e c t influence on t h e steady-state r o l l r a t e i n response t o a specific aileron d e f l e c t i o n . hen t h e r o o t i s l a r g e , t h e damping i n r o l l i s high and t h e pilot controls t h e b a n k a n g l e by commanding and adjusting r o l l r a t e . When i t i s s m a l l , the pilot controls bank angle b y commanding and adjusting rolling a c c e l e r a t i o n .

8.3.10 Spiral-Divergence Root, 1 / T 8

The spiral-divergence r o o t , / T g , i s affected primarily by the parameters shown i n t h e following e q u a t i o n , w h i c h i s based on Equation ( 8 3 ) ,

~ , g ( i f f i ; - L;N^)

* h ( 2 0 1 )

T h e s p i r a l mode c a n b e c o n v e r g e n t , neutrally s t a b l e , or d i v e r g e n t . h u s , o r t h e purpose o f defining t h e s p i r a l stability b o u n d a r y , t h e equation c a n b e s h o w n a s a s p i r a l stability criterion

V*r N

/ 3Lr

o r , a s a n a p p r o x i m a t i o n ,

clßcnr C n ^ J r

' spirally onvergent

= neutral piral tability

< spirally ivergent

( 2 0 2 )

I t w i l l b e noticed t h a t s p i r a l stability i s dependent upon t h e interaction of f o u r d e r i v a t i v e s . ince Cn/ 3 s normally positive and Cnr nd Cj ^ a r e normally n e g a t i v e , i t i s w e l l t o have C »r e g a t i v e . nder a n y c i r c u m s t a n c e , j ^ C n j . hould b e greater t h a n „ £ C ir o r s p i r a l s t a b i l i t y .

A divergent s p i r a l mode w i l l r e s u l t i n t h e airplane performing a n increasing nose- d o w n and tightening t u r n accompanied b y a n i n c r e a s e i n s p e e d a n d l o s s i n a l t i t u d e .

f

8 . 4 light Guidance

Research vehicles t h a t i n c o r p o r a t e new concepts of aerodynamic c o n f i g u r a t i o n , or

r e s e a r c h vehicles designed f o r f l i g h t i n previously unexplored regions of f l i g h t ( M a c h a n d a l t i t u d e ) , usually have a considerable a m o u n t of wind-tunnel investigations performed o n models t o c h e c k t h e i r s t a b i l i t y and c o n t r o l c h a r a c t e r i s t i c s . espite t h e



83

comprehensiveness of t h e tunnel t e s t s , t b s r e w i l l b e gaps i n t h e d a t a . n a d d i t i o n , there i s normally a c e r t a i n , amount of reserve i n placing complete confidence i n t h e d a t a . s a r e s u l t , t h e f l i g h t envelope i s built up g r a d u a l l y , using stability and c o n t r o l maneuvers t o obtain flight-determined stability and c o n t r o l derivatives t o verify wind-tunnel d a t a .

Agreement i n the comparisons results i n a more rapid buildup of t h e f l i g h t e n v e l o p e ; disagreement involves a slowdown u n t i l t h e f l i g h t data c a n b e reduced and cautiously e x t r a p o l a t e d . he most representative values of t h e stability and c o n t r o l character- i s t i c s a r e used i n stability criteria a n d a r e programed i n t o a flight s i m u l a t o r , n

which the pilot simulates the intended mission and emergency conditions to reduce the amount of r i s k t h a t would otherwise be involved i n a c t u a l f l i g h t . h e simulator normally uses t h e general equations of motion f o r a mathematical m o d e l .

When roll-coupling instability became a physical reality with t h e l o s s of s e v e r a l F-100 a i r p l a n e s , considerable effort was expended a t t h e NASA Flight Research Center i n f l i g h t and simulator studies of t h e problem66 '

67. ecause of the complex nature

of t h e m o t i o n s , guidance of t h e f l i g h t program using analog computations was d e s i r a b l e . I n a r o l l investigation of this t y p e , a s m a l l increase i n aileron deflection c a n pro- d u c e l a r g e effects on airplane m o t i o n s . t has b e e n graphically demonstrated o n s e v e r a l occasions that f l i g h t guidance b a s e d on linear extrapolation of f l i g h t data a t s m a l l a i l e r o n deflections c a n be highly misleading a n d d a n g e r o u s . igure 6 6 shows a representative comparison of t h e measured excursions i n angle-of-attack and angle-of-sideslip

obtained i n 360° r o l l s with t h o s e predicted b y using flight-determined d e r i v a t i v e s . T h e good agreement has been demonstrated i n most instances i n w h i c h flight-determined derivatives have f o r m e d t h e b a s i s of c a l c u l a t i o n s . o n s e q u e n t l y , the use of s u c h guidance i n f l i g h t planning has proved i n v a l u a b l e . he use of wind-tunnel and theoreti- c a l derivatives i n analog studies has not b e e n a s s u c c e s s f u l .

9 . CONCLUDING REMARKS

This paper has attempted t o b r i n g together t h e various f a c t o r s t h a t should b e known b y t h e engineer w h o i s concerned with t h e determination of stability and c o n t r o l c h a r a c - teristics f r o m f l i g h t data o r t h e use of these flight-determined characteristics i n handling-qualities r e s e a r c h .

T h e discussions have b e e n tempered w i t h practical c o n s i d e r a t i o n s . h e various f a c t o r s discussed and t h e observations made a r e t h e r e s u l t of experience i n working with f l i g h t d a t a , developing t e c h n i q u e s , comparing the data with p r e d i c t i o n s , and investigating t h e causes of d i s c r e p a n c i e s ,

T h e theoretical b a c k g r o u n d , a p p r o x i m a t i o n s , and limitations of t h e mathematical relations employed have been g i v e n c a r e f u l c o n s i d e r a t i o n . h e problems encountered with s e v e r a l o i t h e more sophisticated techniques have b e e n presented w i t h t h e hope t h a t a n y new comprehensive technique that m a y be proposed w i l l take into consideration s o m e of t h e practical problems w i t h instrumentation a n d development of maneuvers t o properly condition t h e f l i g h t data f o r t h e t e c h n i q u e .

T h e p u l s e m a n e u v e r , properly e x e c u t e d , has b e e n f o u n d t o b e generally adequate i n exciting motions required f o r stability-derivative analysis a s w e l l a s f o r determining t h e characteristics of t h e oscillatory modes i f adequate instrumentation a n d alinement a r e p r o v i d e d .



8 9

F o r longitudinal-derivative a n a l y s i s , simple equations utilizing period and damping of t h e oscillatory mode of the airplang were s h o w n t o be a s satisfactory a s more c o m - prehensive m e t h o d s .

T ^ s

F or ateral-directional derivative nalysis, he raphical im e-vector method as shown o e he most atisfactory anual ethod of nalysis. Simple pproximate methods are useful f pplied ith aution.

*

Control effectiveness c a n usually b e obtained b y relating the peak acceleration t o

rapid c o n t r o l i n p u t s .

onsideration must be given to aerodynamic contributions i f reasonable accuracy i s t o be r e a l i z e d .

Hie analog-matching technique f o r determining derivatives f r o m f l i g h t data w a s s h o w n t o be a valuable method of analysis f o r use i n t h e absence of data suitable f o r analytical t e c h n i q u e s . o w e v e r , t h e analog-matching technique has limitations i n t h a t data must be properly conditioned i n order to obtain unique a n s w e r s . he accuracy of t h e results obtained f r o m this technique and t h e e f f e c t of the type of maneuver on t h e accuracy may w e l l provide the c l u e t o w h a t may be expected f r o m sophisticated techniques that may be p r o p o s e d .

The use of flight data to verify wind-tunnel results and theory was discussed and i l l u s t r a t e d . h e possible inadequacy of comparisons of f l i g h t data with predictions

f o r determining aeroelastic effects was pointed out and a flight-planning technique explained t o permit determination of aeroelastic effects f r o m f l i g h t data a l o n e .

Present instrumentation and methods of analysis a r e adequate f o r extracting d e r i v a - tives f r o m f l i g h t data f o r use i n most flight-guidance simulator studies and detection of characteristics w h i c h have not been predicted i n t h e w i n d - t u n n e l .

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Elementary M atr ices. Cambridge university ress , 1938.

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Stiffness Effects n Truncated M o d a l Analysis. ASD-TDR- 63-334 Contract o. F33(657)-8374), ero. y s t e m s Div., O S Air orce y s t e m s o m m a n d , uly 9 63 .

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Transfer Function Factors. A SD -TD R -62 -10 63 Contract o. AF33(616)-7906), Aero. y s t e m s Div., S ir orce ys t ems C o m m a n d , arch 9 63 .

1 4 . ilne, .D . Dynamics f h e Deforma b le Aeroplane.

A R C , e pt e mbe r 9 64 . R & 3 4 5 . British

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1 6 . ole, enry . r. et l.

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Th e Longitudinal Response f Flexible Slender Aircraft

to Rando* Turbulence. eport ero. 69 1 , oyal Aircraft Establishment, ugust 964 .

Experimental nd Predicted Longitudinal nd Lateral-

Directional Response Characteristics of Large Flexible

35° Swept-Wing Airplane t n Altitude f 35,000 Feet.

N A C A eport 3 3 0 , 1957 .

Approx imate Airframe ransfer Functions nd Application

to Single Sensor Control Systems. Tech. eport 58-82 (ASTIA o. D 5 1 9 2 5 ) , right ir eve lopm ent enter , U S ir orce, une 958.

tn

2 7 . racey, illiam

et l.

2 8. earson, lbin . B r o m , arold .

2 9. racey, illiam

Scheithauer, lwood .

Wind-Tunnel Investigation of Number of Total Pressure

Tubes t High Angles-of-Attack. Supersonic Speeds. N A C A N 2 6 1 , 19 5 1 .

Calibr itio n of C ombi ne d Pitot-Static ub e nd Vane-

Type F l o w Angularity Indicator at ransonic Speeds nd

at Large Angles-of-Attack or Yaw. N A C A M 52 P 2 4 , 1952 .

Flight Investigation at Large Angles-of-Attack of he

Static-Pressure Errors of Service Pitot-Static Tube

Hav ing a Modified Orifice Configuration. N A C A N 159 , 1 9 5 4 .

1 8. urner, oward .

1 9. otess, harles . W oodward, laude .

2 0 . oodward, laude .

2 1 . ole, enry . r. Bennion, rancis .

9 1

Measn>-e<iient of he M ome n ts of Inertia of an Airplane y

A Simplified M e t h o d . NACA TN 2 2 0 1 , 9 5 0 .

An Investigation of h e Experimental Determination of Aircraft Inertia Characteristics. e c h . R e p t . 5 3 - 2 0 7 , Wright Air Development D i v i s i o n , US Air F o r c e , July 1 9 5 3 .

Handbook of Instructions or Experimentally Determining

th e Moments of Inertia nd Product of Inertia of Aircraft

by h e Spring Oscillation Me thod . T e c h . R e p t . 55-415

( A S T I A N o . AD 9 7 1 0 4 ) , Wright Air development C e n t e r , US Air F o r c e , June 1 9 5 5 .

Measurement of h e Longitudinal Moment of Inertia of Flexible Airplane. NACA TN 3 8 7 0 , 9 5 1 . S u p e r s e d e s R M A 5 5 J 2 1 ) .

2 2 . B o u c h e r , Robert W . et a l .

2 3 . M a l v e s t u t o , Frank S . G a l e , Lawrence J .

2 4 . B r u n n , Cyril D . S t i l l w e l l , Wendell H .

2 5 . Z a l o v c i k , John A .

2 6 . G r a c e y , William e t a l .

A Me thod for M ea sur ing h e Product of Inertia nd he

Inclination of h e Principal Longitudinal Axis of Inertia

of n Airplane. NACA TN 3 0 8 4 , 9 5 4 .

J r . o rmulas or Additional Ma ss Corrections o h e M ome n ts

of Inertia of Airplanes. NACA T N 1 1 8 7 , 9 4 7 .

Mach Number Measurements nd Calibration During Flight

at Hi gh Speeds nd at High Altitudes Including Data or

th e D-558-II Research Airplane. NACA RM H 5 5 J 1 8 , 1 9 5 6 .

A Radar M e t h o d of Calibrating Airspeed Installations n

Airplane n M ane uve rs t High Altitudes nd at Transonic

and Supersonic Speeds. NACA TN 1 9 7 9 , 1 9 4 9 .

Wind-Tunnel Investigation of Number of Total Pressure

Tubes t High Angles-of-Attack. ubsonic Speeds. ACA TN 2 3 3 1 , 9 5 1 . Supersedes NACA RM L 5 0 G 1 9 . )

3 0 . eeler, e .

et al.

Flight echniques or Determining Airplane Drag at High

Mach Numbers . N A C A N 821 , 1956 .



92

3 1 . ischel, a ck Webb, annie .

Flight-Informational Sensors, Display, nd Space Control

of h e X-15 Airplane or Atmospheric m i Near-Space Flight

Missions. N A C A N - 24 07 , 1964 .

3 2 . lldhack, .A. Pressure Drop n Tubing n Aircraft Instrument Installa- tions . N A C A N 93 . 19 37 .

3 3 . llen, . ulian Perkins, dward .

A Study of Effects of Viscosity n F l o w O v e r Slender

inclined Bodies of Revolution. N A C A eport 048 , 19 5 1 . (Supersedes A C A N 044.)

3 4 . inclair, rchibald . Mace , illiam .

3 5 . a ggy . a u l .

3 6 . auert, .

Wind-Tunnel Calibration of Combined Pitot-Static ub e an d Vane-Type Flow-Angularity Indicator t Mach Numbers

of 1.61 nd 2.01. N A C A N 808, 1 9 5 6 .

A M e t h o d for Predicting he Upwash Angles Induced at he

Propeller Plane of C o m b i n a t i o n of Bodies with n Unswept

fl ing. N A C A N 528, 19 5 1 .

Th e Elements of Aerofoil nd Airscrew T h e o r y .

University Press, 1947 Reprinted 948). Cambridge

3 7 . R o g a l l o , Vernon L .

3 8 . W o l o w i c z , Chester H . G o s s e t t , Terrence D .

Effects of Wing Sweep n he Upwash at he Propeller

Planes f Multiengine Airplanes. NACA TN 2 7 9 5 , 1 9 5 2 .

Operational nd Per fo rma nce Characteristics of h e X - 1 5

Spherical, ypersonic Flow-Direction Sensor. NACA T N D - 3 0 7 0 , 9 6 5 .

3 9 . M u z z e y , C . L . K i d d , E . A .

Measurement nd Interpretation of Flight Test Data or

Dynamic Stability and Control. Report N o . C A L - 6 0 , Cornell Aeronautical L a b o r a t o r y , April 9 , 9 5 4 .

4 0 . S e c k e l , Edward Systematic Errors. V o l . I V , P a r t IV A , A G A R D Flight

Test Manual , North Atlantic Treaty O r g a n i z a t i o n , 9 6 3 ( r e v . ) .

4 1 . W o l o w i c z , Chester H .

4 2 . Y a n c e y , Roxanah B . e t a l .


4 4 . Doetsch, .H .

Time-Vector Determined Lateral-Derivatives of Swept-

Wing Fighter-Type

Airplane with

Three Different

ertical Tails t Mach Numbers Between 0.70 nd 1.48. N A C A M

H56C20, 19 5 6 .

Aerodynamic-Derivative Characteristics f h e X-15 Research

Airplane s Determined f rom Flight Tests or Mach Numbers

f rom 0.6 o 3.4. N A S A N - 1 0 6 0 . 19 62 .

Effects f Jet Exhausts n Flight-Determined Longitudinal

an d Lateral Dynamic Stability Characteristics of h e

Douglas D-558-II Research Airplane. N A S A M 57G09, 1957 .

Th e im e ector Me thod or Stability Investigations.

Report ero 495 , oyal Aircraft Establishment, ugust

1 9 5 3 .



-

9 3

I

4 5 . B r e u h a u s , Waldemar 0 .

4 6 . S t e r n f i e l d , L .


4 8. owin, orman .

4 9. hinbrot, arvin

5 0 . reenberg, arry

51 . riplett, illiam . et l.

5 2 . onegan, a m e s J. et l.

53 . l awans , ernard . W hite , a ck .

54 . chumacher , loyd .

55 . ggleston, oh n .

Mathews, Charle s .

56 . Rampy, oh n .

Berry, onald .

Resume of h e im e ector M e t h o d s Means or Analyzing

Aircraft Stability Problems. Tech. eport 2 - 29 9 Contract N o . A P 3 3 ( 0 3 8 ) - 2 0 65 9 , R D O o. 461-1-2), W right Air Develop- me nt Center , S ir orce, ovember 952.

A Vector M e t h o d Approach o he Analysis of h e Dynamic

Lateral Stability of Aircraft. Journa l of he ero- nautical Sciences, Vol.21. pril 954, p.251-256.

Time-Vector Determined Lateral Derivatives of Swept-

Wing Fighter-Type Airplane ith Three Different ertical

Tails t Mach Numbers Between 0.70 nd 1.48. N A C A M

K56C20, 19 56 .

Lateral Stability Investigation of n -100A Airplane.

Tech. Rept . 5 7 - 5 63 ASTIA o. D 3 1 0 5 2 ) , W right ir Deve lopm ent Center , S ir orce, ovember 957.

O n h e Analysis f Linear nd Nonlinear Dynamica l Systems

Fr om Transient-Response Data. N A C A N 288, 1954 .

A Survey of M e t h o d s or Determining Stability Parameters of n Airplane ro m Dynamic Flight M easurement s. N A C A

T N 3 4 0 , 1 9 5 1 .

Th e Dynamic-Response Characteristics f 35° wept-Wing Airplane s Determined Fr om Flight Meauurements . N A C A

Report 2 5 0 , 19 55 .

Determination of Lateral-Stability Derivatives nd Transfer-

Function Coefficients ro m Frequency -Response Data or

Lateral Motions. N A C A eport 2 2 5 , 1955 . (Supersedes

N A C A N 083.)

A Method Utilizing Data n

h e Spiral, oll-Subsidence, an d Dutch Roll Mode s or Determining Lateral Stability

Derivatives ro m Flight M easurem ents . N A C A N 0 6 6 , 19 57 .

A M e t h o d or Evaluating Aircraft Stability Parameters Fr om Flight Test Data. ech. ept. 2-71, right ir Deve lopm ent Center , S ir orce, une 952.

Application of Several Methods or Determining Transfer

Functions nd Frequency Response f Aircraft ro m Flight

Data. N A C A eport 2 0 4 , 19 54 .

Determination of Stability Derivatives ro m Flight est

Data

y Means

f High Speed Repetitive Operation Analog M atch ing . FTC-TDR-64-8, S ir orce Flight Test enter, M a y 9 64 .



94

5 7 . shkenas , Irving . McRuer, uane .

58. adoff. elvin

5 9. reer. rent . et l.

60. aylor, awrence . r. Day, ichard .

Approx imate Airfraue ransfer Functions nd Applications

to Single Sensor Control Systems. Tech. ept. 58-82 (ASTIA N o. D 5 1 0 2 5 ) , right ir eve lopment Center, O S Air orce, une 958.

Th e Effects of Longituiinal Control-System Dynamics n

Pilot Opinion nd Re speise Characteristics s Determined

f r om Flight Tests nd f rom Ground Simulator Studies.

N A S A E M O 0 -1 -58A . 1958.

A Pilot Opinion Study of Lateral Control Requirements or

Fighter-Type Aircraft. N A S A E M O -29-59A, 1959.

Flight Controllability Limits nd Related Human Transfer

Functions s Determined ro m Simulator nd Flight Tests.

N A S A N -746 , 19 61 .

6 1 . shkenas , rving . McRuer, uane .

Th e Determination of Lateral Handling Quality Requirements

f rom Airframe-Human Pilot System Studies. Tech. Rept. 5 9 - 1 3 5 , right ir eve lopment Center, S ir orce, une 1959.

6 2 . J e x , H e n r y . C r o m w e l l , Charles H .

Theoretical nd Experimental Investigation of Some New

I I I ongitudinal Handl ing Qualities Parameters. T e c h . R e p t . A S D - T R - 6 1 - 2 6 , Wright-Patterson Air Force B a s e , US Air

F o r c e , June 1 9 6 2 .

6 3 . A s h k e n a s , I . L . M c R u e r , D . T .

A Theor y of Handl ing Qualities Derived ro m Pilot-Vehicle

System Considerations. Aerospace E n g i n e e r i n g , Vol.21 . , p p . 6 0 , 6 1 , 8 3 - 1 0 2 , F e b r u a r y 1 9 6 2 .

6 4 . ewell, F .D . Criteria or Acceptable Representation of Airplane Dynamic

Responses n Simulators Used or Pilot raining. Tech. Rept. A V T R A D E V C E N 1 4 6 - 1 , S aval raining Device

Center Port W ashington, Y ) ctober 9 62 .

6 5 . D u r a n d , T . S . J e x , H . R .

Handl ing Qualities n Single-Loop Roll Tracking Tasks:

Theor y nd Simulator Experiments. T e c h . D o c . Report A S D - T D R - 6 2 - 5 0 5 , Wright-Patterson A i r F o r c e B a s e , US Air F o r c e , November 1 9 6 2 .

6 6 . W e i l , Joseph D a y , Richard E .

6 7 . W e i l , Joseph

An Ana log Study of he Relative mpor tance f Va r i ous

Factors Affecting Roll Coupling. N A C A M 56A06 , 19 5 6 .

Application of Analytical echniques o Flight Evaluations

in Critical Control Areas. A G A R D eport 6 9, 1 9 6 1 .



9 5

TABLE I

Transfonwtion of Derivatives froa Stability t o Body Axis

c«a = C Laos a + oasin a + Cc

C ca = C oaos a - Lasin a - N

C«a = (C.a8

cnß = ( C n y j ) a coa a + Cj^s sin a

Cnr = ( C n r)a co s2 a + Cip)B sin2 a + ( C n p Cir)8snacosa

CDß = (Cny§)sCosa+ C )8Bina

S = ( C n p)s cos2 a - Cjr)8 sin2 a - ( C nr - Cj 8snacosa

C»S = ( C n8)8cosa + Cij)8sna

C lß = (Ci^scosa- C n^ )Bsna

ch = (Cir8 cos2 a - C n p)8 s in2 a - ( C nr - Cip)8sinacosa

clß = (Ci^)8cosa- C n/8)ssna

Ch

= (Cip)8 co s2 a + C nr)8 sin

2 a - (° np C ir8snacosa

Cl| = (Cls)s co s a - C n s)8 sin a

9 6

TABLE II

Transformation of Derivatives f r o « Body t o Stability Axis

CL« = C ((acos a - Ccasin a - Cp

cD a = CCacc8 a + N( Xsin a + L

<*«>s = C-a

(cn s = C^„cosa - Cj„sin a

(C nrs = Cnrcos2 a + C jpsin

2 a - C jr- Ca sin a cos a

(Cn^s = Cnä cos a - C l ö sin a

«Vs =

Chp cos

2

a - C jrsin

2

a + CDr - C/p ) sin a cos a

(Cnj)s r C Dj cos a - zssin a

(Clß)B = C iß cos a + C nä sin a

«Vs = C jrcos2 a - Cnß sin

2 a CDr - C jp sinacos a

(Clß)s = Cj^cosa + n/gSin a

<% U = C /p cos2 a + nrsin

2 a + C [r+ n sin a cos a

(Cjj)s = Cjj cos a + n$ sin a

9 7

TABLE III

Transformation of Noaents of Inertia f r o « One Axis Systen to Another

B o d y o Stability

xx. = d z) idz - x) cos 2a - xz sin 2a

«t. = ly

Jz. = dx z) idz - t) cos 2 a + xz sin 2a

I»szs = t dx ~ z) sin 2a + xz cos 2a

Stability o B o d y

Ix = ±(Ix8 z8) - idz8 - l8 cos 2a + X8Zg sin 2a

ly = y.

Iz = dz8 x8) + idz8 - Xs cos 2a - Ixs2s sin 2 a

Ixz = Ix8z8COB2a - idx8 - I- sin 2a

S

Principal ^ o Sto6 :Zity

l*B = idxo + W-idzo-Ix0) cos 2 7 7

* y» lyo

Izs = idxo + z0) + idz0 Ix0) cos 2 7 7

IisZs = idx0 - Z0)sin2i7

Stability to Principal

^o = i<ix8 + w - idz8 - W cos 2 ? 7 + Ix8z8 sin 2T ;

ho = h3 lz0 = ittx8 + z8) + idz8 - X8>

CO S 27 ? - Ixaz8 sin 2 7 7

Ixozo = 0 = Ix8zscos 2 ) 7 - i(rX8-IZg) in 2 7 7

(Continued)



98

Principal o Body

I* = idxo W - idzo - Ixo) O S 2€

h = ly o

h = idio z0) + idzo - IXo) cos 2 e

Ix z = -idx0-i 0) Bin 2e

Body o Principal

xo = idx z) - idz - x) C O S 2 6 - xzsin 2 e

h o = h

Iz0 = idi z) idz - x) cos2e xz sin2e

JXoZo -0 = Ixz cos 2e (IX - Iz) sin 2 e

99

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CQJN % X>1>a; C M 5 + 8

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1 02

T A B L E II

Des irable Characteristics of nstrum ents or Free-Oscillation Maneuver

. I

Function Ra ng e

Sensitivity (per nch

deflection)

Undamped

natural

frequency c/s)

Damping

ratio

a deg ± 1 0 5 . 0 8 ov more 0.65

ß deg ± 1 0 4 . 0 8 or more 0.65

q , radian/sec ±0.2 0 . 2 8 or more 0.65

q , radian/sec2

±0.5 0 . 5 8 or more 0 . 6 5

r , radian/sec ±0.1 0 . 1 8 or more 0.65

f , radian/sec2 ±0.4 0 . 4 8 or more 0.65

p , radian/sec . ±0.2

±0.6

0 . 2 , rudder pulses

0 . 6 , aileron pulses

8 or more

8 or more

0.65

0.65

p , radian/sec2 • ±0.6

±6.0



8 o r more

8 or more

0.65

0.65

an , g units ± 1 1 . 0 8 o r more 0.65

a t , g units i

±0.3

±0.6



8 or more

8 or more

0.65

0.65

1 0 3

TABLE VIII

Format used b y NASA Flight Research Center t o Record A c t u a l Conditions a t Tine o f Maneuver

Trace Instrument Not, frea

I».SO

10,SO

Oom t-rafio St«i» gcfrri Fir*, o. ' 19 6 .MIO

10 3o

Wt V/9- vane ocation

• T * l i r i &o r accelerometer ocat ion

X »- •»y

l##flMt#fK ««.-M* ft

M fps

Dynamit

preuvro,

3 w» ;9ht,

ft

a C.6., «*,«

W P

/Ö Demit/ 1,

«*<• /7 44,330 0.743T 723 /«2 7?,8,?o 2g.7 a23-» .••*f7J

Trace Jnsfrur n«n< Scale factors A«, to-o D am o . ratio Fit*. 0 -* Fit's.

a /o.r .U ICC

n_ J2.0 .cs- €,n

--£" n.*t a * 7

Airplane

Configuration



r~ 1 0 4

TABLE X

Airplane I I Altitude,

ft a,

efeg

Reference Cng (Eq.(6.53))

Cg a er adian

Equation 6 ~54 ) Equation 6-56)

F-104

YP-102

0.94

0.74

41,000

40,000

4.9

6.6

0.46

0.054

0.46

0.043

0.57

0.106

TABLE

a/U/h 15/0.8/60 3,5/0.8/40 6.6/1.2/60 14/1.6/80 10/2.0/80 5/2.0/80

Analog value

of C iß

-0.084 -0.021 -0.032 -0.074 -0.0164 -0.0034

Analog value of Caß

Equation 166)

Equation 175)

0.259

0.286

0.278

0.641

0.661

0.679

0.640

0.6o3

0.674

0.367

0.360

0.383

0.445

0.434

0.451

0.508

0.498

0.504

TABLE I

a /M/h* 15/0.8/60 3.5/1.0/40 18/1.2/80 6.6/1.2/80 14/2.0/80 10/2.0/80 5/2.0/80

Analog value

of (Cn,.- ) -1.58 -1.734 -1.71 -1.92 -2.09 -2.55 -2.58

Equation

(184)

Equation

(183)

-2.50

-4.22

-1.43

-1.647

-2.96

-4.69

-1.81

-2.36

-4.77

-8.46

-2.76

-4.41

-2.66

-4.39

b ltitude/1000



"*—JL

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01

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r- T K OS OS a 1

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l » 4 § 43 p

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+ > +J in 1»

5 ? •& K l 5 Q>

s <H o

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M 1

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g-e. - a *s

ä H M l

a t iv 9 a

9 I O I8 VI o a - 9 §

r

I H



i-- -

1 0 8

( a ) Euler angle perturbation referred t o t h e z , . asic reference f r a m e

( b ) Euler a n g l e perturbation referred t o t,^,^^ xes serving as a secondary s p a t i a l reference

e ' i g . 4 Several methods of considering Euler a n g l e perturbations

- -~- * Mi

I

I

i 1

109

Pig. 5 Relation f p nd r about ody xes nd uler ngle ates /<

an d 4 >



1 1 0

Ctn ta r of gravi ty

W « .6 Pertinent elationships f otating .....

Pig . 7 A n xaaple f in nfluence f ranges f isturbances uc h s (Aß) l and

(äß)2 on he alue of a erivative

Ill

2

^

Fig .8 Effect f t ime a g of m odification f ortex low bout ifting urface n he change n C N following nitial nstant hange n a

<p upwaih u* o w i n g . tc .

P

/

Fig.9 Direct propulsive effects of propeller



1 1 2

~w ""

£p pwath t ir nra l t«

Pig. 10 Direct ropulsive ffects f jet ngine

F i g . 1 1 J e t - e x h a us t nflow ffect n orizontal ail



= - >Mt

1 1 3

at

o I u a )

o

g

o

§ iH

E

ft .

I

.



1 1 4

a) • -I

1 I

s °sS3

+» >> ol -P -H •H -( ► •* &a

o i

O i2 gB

i-H e s 5 •H +i h

c o

a > u

I-H

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Fig . 15 Determination f nclination f rincipal xis nd yawing o m e n t f nertia

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1 1 1 6

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mounting brackets located below wings

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f r o m Reference 2 4



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148

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Pig.54 Variation of static a n d dynamic l o n g i t u d i n a l stability derivatives of t h e D-558-II airplane w i t h Mach number ( f r o m Reference 4 3 )



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F i g .56 T o p i c a l determination of f l i g h t quantities f o r t h e evaluation of l a t e r a l c o n t r o l derivatives



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Fig.60 Results of graphical time-vector analysis of t h e effects of power o n t h e

lateral-directional p e r i o d , d a m p i n g , and stability derivatives of t h e

D-558-II r e s e a r c h a i r p l a n e ( f r o m Reference 4 3 ) ( c o n t i n u e d )

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Pig .60 Results f raphical ine-vector nalysis f th e ffects f ower n he lateral-directional eriod, amping, n d tability derivatives f he

D-558-II esearch irplane from eference 3 ) concluded)



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t h e equations of motion (Reference 4 8 )

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Fig.65 Influence of flexibility a n d a i r intake to engine on t h e directional stability d e r i v a t i v e , n/3



1 86

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Considerations in the Determination of S&C Derivatives and Dynamic Characteristica From Flight Data

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Transcript of Considerations in the Determination of S&C Derivatives and Dynamic Characteristica From Flight Data