Longitudinal Aerodynamic Derivatives of a Slender Delta-Wing Research Aircraft Extracted from Flight Data

VLIcGTUiCP Es2L£G Kluyv^rwcg 1 _D:£LFT

CRANFIELD

INSTITUTE OF TECHNOLOGY

1975

LONGITUDINAL AERODYNAMIC DERIVATIVES OF A SLENDER

DELTA-WING RESEARCH AIRCRAFT EXTRACTED FROM

FLIGHT DATA

by

CRANFIELD INSTITUTE OF TECHNOLOGY COLLEGE OF AERONAUTICS

Longitudinal Aerodynamic Derivatives of a Slender

Delta-Wing Research Aircraft Extracted from

F l i g h t Data

by

V. Klein

Contract NO. K/A72c/239

the l o n g i t u d i n a l aerodynamic d e r i v a t i v e s from f l i g h t data for the Handley Page HP 115 slender delta-wing research a i r c r a f t . The responses in the r a t e of p i t c h and v e r t i c a l a c c e l e r a t i o n were e x c i t e d from the h o r i z o n t a l s t e a d y - s t a t e f l i g h t s at d i f f e r e n t a i r s p e e d s by the elevon d e f l e c t i o n . In some c a s e s the measured time h i s t o r i e s were converted i n t o frequency response c u r v e s , from which the unknown parameters were e s t i m a t e d . For the low angle of attack measurements the l i n e a r model of the a i r c r a f t was adequate, whereas for the runs measured at high angle of a t t a c k s the n o n - l i n e a r model had t o be used. Some i d e n t i f i a b i l i t y problems were met because of the limited number of measured outputs and uncorrect d e s i g n of the experiment. The non-dimensional aerodynamic d e r i v a t i v e s e x t r a c t e d were compared with those from windtunnel t e s t s and s t e a d y -s t a t e mea-surement-s. A rea-sonable degree of c o n -s i -s t e n c y between the v a r i o u s s e t s of r e s u l t s was obtained.

CONTENTS

INTRODUCTION

TEST AIRCRAFT AND INSTRUMENTATION SYSTEM MEASURED DATA

PREVIOUS RESULTS

MATHEMATICAL MODEL OF THE AIRCRAFT PARAMETER ESTIMATION PROCEDURE DATA ANALYSIS

7.1 Low Angle of Attack Measurements 7.2 High Angle of Attack Measurements DISCUSSION OF RESULTS

CONCLUSION REFERENCES

APPENDIX A: Perturbation State and Output

Page

1

1

2

4

A

6

9

10 12 13 16 18 19 Equations TABLES FIGURES

LIST OF TABLES

T a b l e

1. G e o m e t r i c , Mass and I n e r t i a C h a r a c t e r i s t i c s of t h e A i r c r a f t . 2. C h a r a c t e r i s t i c s of t h e I n s t r u m e n t a t i o n System.

3 . Review of Runs Measured and c o r r e s p o n d i n g F l i g h t C o n d i t i o n s and C h a r a c t e r i s t i c s .

4 . Review of Cases Analysed.

5. Effect of V a r i o u s Model Forms on P a r a m e t e r s Determined from F l i g h t D a t a . F l i g h t 849, Run 6 .

6 . E s t i m a t e d P a r a m e t e r s and Normalized C o e f f i c i e n t s i n t h e S e n s i t i v i t y and I n f o r m a t i o n M a t r i c e s . F l i g h t 849, Run 6 . 7 . Comparison of P a r a m e t e r s Determined from F l i g h t Data i n Time

and Frequency Domain.

8 . E f f e c t of V a r i o u s Model Forms on P a r a m e t e r s Determined from F l i g h t D a t a . F l i g h t 8 3 1 , Run 1.

9. E f f e c t of V a r i o u s Model Forms on P a r a m e t e r s Determined from F l i g h t D a t a . High Angle of A t t a c k .

10. Estimated P a r a m e t e r s and Normalized C o e f f i c i e n t s in t h e S e n s i t i v i t y and I n f o r m a t i o n M a t r i c e s . F l i g h t 850, Run 5 , N o n - l i n e a r Model, Case N.

1 1 . E f f e c t of V a r i o u s Model Forms on P a r a m e t e r s Determined from Simulated D a t a .

LIST OF FIGURES F i g . 1 . H a n d l e y - P a g e HP 115 A i r c r a f t . 2 . G e n e r a l Arrangement of HP 115 A i r c r a f t . 3 . V a r i a t i o n o f A n g l e of A t t a c k w i t h True A i r s p e e d and E l e v o n A n g l e w i t h A n g l e o f A t t a c k Measured i n S t e a d y - s t a t e H o r i z o n t a l F1 i g h t s . 4 . V a r i a t i o n o f L i f t w i t h A n g l e of A t t a c k Measured i n Wind-t u n n e l (Ref . 4 ) .

5. V a r i a t i o n o f P i t c h i n g Moment w i t h L i f t Measured i n Wind-t u n n e l ( R e f . 4 ) .

6 . Comparison of Time H i s t o r i e s Measured i n F l i g h t w i t h t h o s e Computed (ot •= 7 . 2 d e g , C a s e B) .

7 . Time H i s t o r i e s and Sample A u t o c o v a r i a n c e F u n c t i o n s o f t h e R e s i d u a l s (a = 7 . 2 d e g , Case B) .

8 . Comparison o f F r e q u e n c y R e s p o n s e Curves Measured i n F l i g h t w i t h t h o s e Computed (a = 7 . 2 d e g ) .

9 . D i f f e r e n t Input Forms u s e d i n t h e E x c i t a t i o n o f t h e Motion a t a " 12 d e g .

e "

10. Harmonic Content of Different Input Forms.

11. Measured Frequency Response Curves for three Different Inputs (a " 12 deg).

e

1 2 . Comparison o f Time H i s t o r i e s Measured i n F l i g h t w i t h t h o s e Computed (a * 1 6 . 4 d e g , Case B ) .

1 3 . Comparison of Time H i s t o r i e s Measured i n F l i g h t w i t h t h o s e Computed (a • 1 6 . 4 d e g . Case M).

1 4 . Comparison o f Time H i s t o r i e s Measured i n F l i g h t w i t h t h o s e Computed (a - 2 0 . 3 d e g . C a s e B ) .

1 5 . Comparison o f Time H i s t o r i e s Measured i n F l i g h t w i t h t h o s e Computed (o » 2 0 . 3 d e g , C a s e N ) .

1 6 . Time H i s t o r i e s and Sample A u t o c o v a r i a n c e F u n c t i o n s o f t h e R e s i d u a l s (a • 2 0 . 3 d e g . Case B) .

1 7 . Time H i s t o r i e s and Sample A u t o c o v a r i a n c e F u n c t i o n s o f t h e R e s i d u a l s (o » 2 0 . 3 d e g . Case N) .

1 8 . Comparison o f F r e q u e n c y R e s p o n s e C u r v e s Measured i n F l i g h t w i t h t h o s e Computed (a « 2 0 . 3 d e g ) .

2 0 . V a r i a t i o n of P a r a m e t e r s , Determined from F l i g h t Data and W i n d - t u n n e l , with Angle of A t t a c k (Linear Model, Case B ) . 2 1 . D i f f e r e n t I n p u t Forms used in t h e E x c i t a t i o n of the Motion. 2 2 . Harmonic Content of D i f f e r e n t I n p u t Forms.

2 3 . Comparison of Time H i s t o r i e s Measured in F l i g h t w i t h t h o s e Computed.

24. Comparison of Time H i s t o r i e s Measured in F l i g h t w i t h t h o s e Computed (Sharp P u l s e ) .

2 5 . E f f e c t of Various Weighting M a t r i c e s on Parameter Z^ and V a r i a n c e E s t i m a t e s .

2 6 . Simulated L o n g i t u d i n a l Response of HP 115 A i r c r a f t .

27. V a r i a t i o n of Handling C r i t e r i a , Determined from F l i g h t Data, with Angle of Attack (Linear Model, Case B ) .

2 8 . V a r i a t i o n of Non-dimensional Aerodynamic D e r i v a t i v e s , Determined from F l i g h t Data and W i n d - t u n n e l , with Angle of A t t a c k (Linear Model for 7 deg<a <16 deg, N o n - l i n e a r Model for ot >16 deg) .

e "

2 9 . V a r i a t i o n of Non-dimensional C o n t r o l D e r i v a t i v e s , Determined from F l i g h t Data and W i n d - t u n n e l , w i t h Angle of A t t a c k

NOTATION s t a t e m a t r i x input m a t r i x t r a n s f o r m a t i o n m a t r i x l i f t c o e f f i c i e n t pitching-moment c o e f f i c i e n t mean aerodynamic chord

t r a n s f o r m a t i o n m a t r i x

frequency r e s p o n s e f u n c t i o n

2 a c c e l e r a t i o n due t o g r a v i t y , (m/s )

s e n s i t i v i t y m a t r i x

moment of i n e r t i a of a i r c r a f t about y body a x i s , (kgm )

/=r

non-dimensional moment of i n e r t i a of a i r c r a f t about y body a x i s

cost function

gain in the t r a n s f e r function r e l a t i n g q and n l i f t f o r c e , (N)

a) pitching moment, (Km) b) information matrix a) number of outputs b) aircraft mass, (kg) number of data points measurement noise vector

reading of the vertical accelerometer, (g units)

z

measurement noise covariance matrix autocovariance function

correlation lag number wing area, (m )

complex v a r i a b l e in the Laplace transform, ( l / s

standard error

variance estimate

time, (s)

a) longitudinal airspeed component b) input vector

augmented input vector true airspeed, (m/s) we ight ing matr ix vertical force, (N) state vector

output vector measurement vector angle of attack, (rad)

reading of the wind vane, (rad)

vector of unknown parameters in A, B, C and D damping ratio

elevon deflection, (rad) pitch angle, (rad)

relative density parameter residual

3

a i r d e n s i t y , (kg/m )

var iance

T time-lead c o n s t a n t , (s)

qn

(ö c i r c u l a r frequency, ( l / s ) <ü natural frequency, ( l / s ) Matr ix component s:

T i n d i c a t e s transpose matrix operation *-! i n d i c a t e s i n v e r s e matrix operation Supscr i p t s : E measured q u a n t i t y 0 i n i t i a l v a l u e e s t e a d y - s t a t e value Additional n o t a t i o n :

Im(*) imaginary part of a complex number Re(*) r e a l part of a complex number v a r ( * ) v a r i a n c e

Non-dimensional aerodynamic d e r i v a t i v e s : c • _La ^Lqa ' ^ L n -C mq C T^a ^ ^ L 3a ^2V 9C m ^2V 9^0 m 9n9a ^ L q -^Lna C ma 9n3a 3C m 9a 1 32c C 2 - i ~ | _{ma 2 3^2} 3C mn 9n ^1^2' ^ L n

^ m i

-C mqa C • -mn 1 ^ \ • 2 3„2 ^^L 3n dc m ^2V 3^0 m 3C m -nc 2V

1 . INTRODUCTION

The Handley-Page HP 115 a i r c r a f t h a s been used by t h e R.A.E. Bedford for t h e i n v e s t i g a t i o n of aerodynamic c h a r a c t e r i s t i c s and h a n d l i n g q u a l i t i e s of a s l e n d e r d e l t a - w i n g a i r c r a f t a t low a i r s p e e d s . S e v e r a l r e p o r t s on f l i g h t t e s t d a t a a n a l y s i s were p u b l i s h e d , d e a l i n g mainly with t h e comparison of t h e l a t e r a l aerodynamic d e r i v a t i v e s with t h o s e o b t a i n e d from w i n d - t u n n e l t e s t s .

I n 1972 t h e l o n g i t u d i n a l r e s p o n s e s of t h e a i r c r a f t were measured. A year l a t e r t h e C r a n f i e l d I n s t i t u t e of Technology ( C . I . T . ) was

provided with a s u b s t a n t i a l p a r t of t h e s e measurements f o r t h e comprehensive a n a l y s i s based on system i d e n t i f i c a t i o n . The main purpose of t h i s a n a l y s i s was t o d e m o n s t r a t e t h e t e c h n i q u e s d e s c r i b e d i n Ref. 1 u s i n g t h e d a t a on t h e a i r c r a f t of a s p e c i a l d e s i g n and a l s o t h e d a t a from t h e experiment which was not o r i g i n a l l y d e s i g n e d for system i d e n t i f i c a t i o n .

The second r e a s o n was t o e v a l u a t e the whole s e t of aerodynamic 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 d e s c r i b i n g t h e motion of t h e a i r c r a f t and t o compare t h e s e d e r i v a t i v e s with r e s u l t s from p r e v i o u s s t e a d y

-s t a t e f l i g h t -s and w i n d - t u n n e l mea-surement-s.

2. TEST AIRCRAFT AND INSTRUMENTATION SYSTEM

The Handley-Page HP 115 a i r c r a f t c o n s i s t s b a s i c a l l y of a slender d e l t a wing of 75 degree leading edge sweep with streamwise t i p s . The wing has a symmetrical bi-convex section based on c i r c u l a r a r c s . I t s r e l a t i v e t h i c k n e s s i s 6 per c e n t . The leading and t r a i l i n g edges of the wing are r a t h e r sharp with r a d i i about 2.5 m i l i m e t e r s .

The a i r c r a f t i s powered by a B r i s t o l - S i d d e l e y Viper 9 engine with s e a - l e v e l s t a t i c t h r u s t of approximately 8200 N. The undercarriage i s n o n - r e t r a c t i n g . All c o n t r o l s are manually operated.

The a i r c r a f t in f l i g h t i s shown i n F i g . 1, i t s general arrangement i s presented in F i g . 2 . The main geometric, mass and i n e r t i a c h a r a c t e r i s t i c s of the a i r c r a f t are summarised in Table 1.

The f l i g h t p a r a m e t e r s m e a s u r e d i n c o n n e c t i o n w i t h t h e l o n g i t u d i n a l d y n a m i c s a r e l i s t e d i n T a b l e 2 , t o g e t h e r w i t h t h e t r a n s d u c e r s u s e d and t h e i r c h a r a c t e r i s t i c s . The p a r a m e t e r s w e r e r e c o r d e d i n two c o n t i n u o u s - t r a c e g a l v a n o m e t e r r e c o r d e r s u s i n g a common t i m e b a s e . I n a d d i t i o n t o t h e m e a s u r e d q u a n t i t i e s t h e p i l o t n o t e d t h e a i r s p e e d , h e i g h t and f u e l l e f t b e f o r e e a c h t e s t r u n . More d e t a i l s a b o u t t h e a i r c r a f t and i n s t r u m e n t a t i o n s y s t e m c a n b e f o u n d i n R e f . 2 . 3 . MEASURED DATA The l o n g i t u d i n a l r e s p o n s e s of t h e a i r c r a f t w e r e e x c i t e d from t h e h o r i z o n t a l s t e a d y - s t a t e f l i g h t y a t d i f f e r e n t a i r s p e e d s by e l e v o n d e f l e c t i o n . T h r e e d i f f e r e n t i n p u t f o r m s were u s e d , however m o s t of t h e d a t a was o b t a i n e d by u s i n g a s t i c k - f o r w a r d p u l s e . I n some t e s t r u n s t h e r e c t a n g u l a r p u l s e ( p o s i t i v e d e f l e c t i o n ) and a s h a r p s t i c k - b a c k w a r d p u l s e w e r e a l s o a p p l i e d . B e c a u s e of t h e i n p u t f o r m u s e d and t h e a i r c r a f t c h a r a c t e r i s t i c s t h e a i r s p e e d c h a n g e s d u r i n g t h e t r a n s i e n t m o t i o n w e r e n e g l i g i b l e . F o r t h e i d e n t i f i c a t i o n t h e t i m e h i s t o r i e s of t h e p o r t and s t a r b o a r d e l e v o n a n g l e , r a t e of p i t c h , v e r t i c a l a c c e l e r a t i o n and a n g l e of a t t a c k w e r e a v a i l a b l e . The o r d i n a t e s of a l l v a r i a b l e s were e x p r e s s e d a s t h e l e n g t h s m e a s u r e d i n m i l i m e t e r s from t h e d a t u m l i r e on t h e t r a c e s . The s a m p l i n g i n t e r v a l was t a k e n a s 0 . 0 3 1 2 5 s e c o n d and t h e number of d a t a p o i n t s f o r e a c h v a r i a b l e was e q u a l t o 3 2 0 . The c o r r e c t e d d a t a of t h e i n p u t and o u t p u t v a r i a b l e s w e r e o b t a i n e d by u s i n g c a l i b r a t i o n c u r v e s a n d , where n e c e s s a r y , by i n t r o d u c i n g c o r r e c t i o n s f o r t h e p o s i t i o n and d y n a m i c c h a r a c t e r i s t i c s of a n i n s t r u m e n t . The e x p r e s s i o n s u s e d had t h e form

n = \ (npj, + n g ^ ) q ( t .) « q^ ( t . - A t )

" z ^ ' ^ i ^ - " z R ( ^ - A t ) + - q ( t . ) a ( t . ) " a^ -^ a^ a,^ ( t . ) * / q ( t . )

where the index R denotes the measured quantity in physical u n i t s ,

At i s the t i m e - s h i f t , x, and x are the distances of the accelerometer

' A V

and wind vane with respect to the e . g . of the a i r c r a f t r e s p e c t i v e l y

and a and a a r e the constants in the linear approximation of the

measured a (a) r e l a t i o n s h i p as presented in Ref. 3 .

To simplify the process of data conversion the value of i t in (3.1)

was taken equal to the sampling time i n t e r v a l , even if the measured

equivalent time constants of the r a t e gyro and accelerometer were

0.02 sec. and 0.03 sec. r e s p e c t i v e l y .

The values of the true airspeed were obtained from the t r a c e readings

after t h e i r c o r r e c t i o n s for the position error and assuming the

standard atmospheric c o n d i t i o n s . The determination of the

instantaneous mass of the a i r c r a f t was based on the fuel meter

r ead ing s.

The review of the t e s t runs a v a i l a b l e and the corresponding f l i g h t

conditions are given in Table 3.

Some of the data taken from the i n i t i a l s t e a d y - s t a t e part of the

t e s t runs was plotted in two r e l a t i o n s h i p s as presented in Fig. 3 .

In both cases the r e s u l t s obtained are c o n s i s t e n t .

The accuracy of the output v a r i a b l e s in the t r a n s i e n t motion was

checked by using the integrated form of the r e l a t i o n s h i p

a - q + 1 - (n + sinO 6) (3.2)

^ u z e '

e

In many cases the comparison of measured and computed angle of a t t a c k

time h i s t o r i e s r e s u l t e d in large d i s c r e p a n c i e s . Preliminary

c a l c u l a t i o n s and further i n v e s t i g a t i o n revealed that the wind vane

s t i c t i o n was the most probable cause of these d i f f e r e n c e s . For t h i s

reason the measured angle of a t t a c k was not included in the further

a n a l y s i s .

PREVIOUS RESULTS

The f i r s t idea about t h e aerodynamic c h a r a c t e r i s t i c s of t h e a i r c r a f t was o b t a i n e d from t h e r e s u l t s of w i n d - t u n n e l measurements p u b l i s h e d

i n Ref. 4 . Two examples from t h e s e measurements a r e reproduced in F i g . 4 and F i g . 5 . They r e p r e s e n t t h e v a r i a t i o n s of t h e l i f t and p i t c h i n g moment c o e f f i c i e n t s w i t h t h e a n g l e of a t t a c k and l i f t c o e f f i c i e n t s r e s p e c t i v e l y . In both c a s e s t h e e l e v o n a n g l e i s t h e p a r a m e t e r . The w i n d - t u n n e l measurements i n d i c a t e t h a t t h e l i f t and p i t c h i n g moment a r e n o n - l i n e a r f u n c t i o n s of t h e a n g l e of a t t a c k and elevon a n g l e .

There a r e a l s o u n p u b l i s h e d r e s u l t s from wind-tunnel measurement on a s p e c i a l o s c i l l a t i n g r i g . The model for t h e t e s t s was o n l y

s i m i l a r i n shape to t h a t f o r t h e HP 1 1 5 . From t h i s measurement t h e d e r i v a t i v e s C, and C + C • can be e v a l u a t e d ,

Lq mq ma

During the f i r s t f l i g h t measurements described in Ref. 2 the

l o n g i t u d i n a l handling c h a r a c t e r i s t i c s were i n v e s t i g a t e d . Later the limited information on the l o n g i t u d i n a l s t a b i l i t y and c o n t r o l

d e r i v a t i v e s was obtained from the t e s t s using quasi-steady and dynamic techniques. The r e s u l t s a r e presented in Ref. 5 . They

include numerical values on the control d e r i v a t i v e s C. and C

Ln mn

and d e t a i l e d q u a l i t a t i v e a n a l y s i s on remaining d e r i v a t i v e s t o g e t h e r with t h e i n d i c a t i o n on t h e p o s s i b l e e x i s t e n c e of n o n - l i n e a r t e r m s . MATHEMATICAL MODEL OF THE AIRCRAFT

As t h e f i r s t s t e p i n t h e i d e n t i f i c a t i o n p r o c e d u r e t h e mathematical model of t h e a i r c r a f t was formulated in t h e form of t h e s t a t e and o u t p u t e q u a t i o n s

X = Ax + Bu

( 5 . 1 ) y = Cx + Du

where X and y i s t h e s t a t e and o u t p u t v e c t o r r e s p e c t i v e l y , and * * . . .

u " u ( x , u ) IS t h e augmented i n p u t v e c t o r which, in g e n e r a l , i n c l u d e s n o n - l i n e a r t e r m s i n x , ux and u , a s w e l l a s t h e i n p u t v a r i a b l e s u . The i n t r o d u c t i o n of t h e augmented v e c t o r u n a b l e s u s ,

n o n - l i n e a r m o d e l s w i t h one computed a l g o r i t h m , a s p o i n t e d i n R e f . 1

Taking i n t o a c c o u n t t h e w i n d - t u n n e l measurements and p r e l i m i n a r y f l i g h t t e s t r e s u l t s , t h e e q u a t i o n s of m o t i o n a r e c o n s i d e r e d i n t h e form 2 a » Z a + Z q + Z „ a + Z q a + Z na + a q^ (j2 qa na Z n + Z*n + Z n n o 2 q « M a + M q + M 2 0 + M q a + M na + ^ a q a q" fl» M n + M*n + M n n o (5.2) e « q

In these equations only one state, namely q, was measured directly. The second output variable n is related to the state variables by the equation * 1 2 n = Z a + Z q + Z„9 + Z o" + z a q^ 9 £,2 (5.3) 1 Z qa + Z na + Z n + z « n + z qa na n n o

The s t a t e and o u t p u t e q u a t i o n s a r e d e v e l o p e d i n Appendix A, where t h e p a r a m e t e r s c o n t a i n e d i n t h e s e e q u a t i o n s a r e a l s o d e f i n e d . The s i m p l i f i e d l i n e a r model was formed a s

a - Z o + Z q + Z n + Z a q n o d - M a + M q + M n + M ^ a q n o (5.4) 9 • q * 1 1 n - Z a + Z q + Z„9 + Z n + Z z a q^ 9 n o

From (5.4) the transfer function relating the pitch rate and

elevon angle can be expressed as

1 + T S

£(8) . K ,, '^^ , , (5.5)

^

qn 1 ^ 2c^ ^ _1 2

I») U ) *

o o

The t r a n s f e r f u n c t i o n c o e f f i c i e n t s i n ( 5 . 5 ) a r e v e r y o f t e n c o n s i d e r e d a s t h e l o n g i t u d i n a l h a n d l i n g q u a l i t i e s c r i t e r i a e x p r e s s i n g the e l e v o n e f f e c t i v e n e s s , a i r c r a f t l e a d - t i m e c o n s t a n t i n p i t c h , and damping r a t i o and n a t u r a l frequency of t h e l o n g i t u d i n a l s h o r t - p e r i o d m o t i o n .

6 . PARAMETER ESTIMATION PROCEDURE

For the e v a l u a t i o n of unknown p a r a m e t e r s i n t h e s t a t e and o u t p u t e q u a t i o n s t h e o u t p u t e r r o r method with t h e maximum l i k e l i h o o d (ML) e s t i m a t i o n t e c h n i q u e was u s e d . The method i s d e s c r i b e d in Ref. 1 t o g e t h e r w i t h t h e computing a l g o r i t h m .

The computing s t a r t s w i t h t h e a p p r o x i m a t e v a l u e s for t h e unknown p a r a m e t e r s in t h e s t a t e and o u t p u t e q u a t i o n s and t h e n i t e r a t e s t i l l t h e minimum of t h e c o s t f u n c t i o n

N

J - i E ( z . - y.)'^ K] ( Z - y^) + f Jtn|R I ( 6 . 1 ) i " l

i s found. I n ( 6 . 1 ) z i s t h e measurement v e c t o r z = y + n , where n i s t h e measurement n o i s e v e c t o r and R, i s t h e measurement n o i s e c o v a r i a n c e m a t r i x whose e l e m e n t s a r e a l s o unknown. N i s t h e number of d a t a p o i n t s .

After t h e kth i t e r a t i o n t h e new e s t i m a t e s of unknown p a r a m e t e r s g are found a s

where the vector A6 is computed from the expression N . - I N

A6 - {Z H ! RJ H.} Z H ! R : (z.-y.) (6.3)

i-1 i«l

In t h i s equation H i s the s e n s i t i v i t y matrix whose elements are

9y/96-. j • 1| 2, . . . , m, where m i s the number of measured o u t p u t s .

The matrix R i s updated a f t e r each i t e r a t i o n using the r e l a t i o n

R, - i ? v.v.T (6.4)

1 » !

in which v. • z . - y . a r e the r e s i d u a l s at the time t . .

1 1 ' i 1

Equation (6.3) can also be w r i t t e n a s

A6 - -M (^) (6.3a)

where M is the Fisher information matrix. Its inverse gives a lower bound on the error covariance matrix for the estimated parameters. For the ML estimation with the a priori weighting the cost function

(6.1) includes the additional term

t

AJ - Y (6-$^)'^W2(6-e^) (6.5)

which introduces a penalty for departure from a p r i o r i values 6 •

The confidence in the a p r i o r i values is expressed by the weighting

matrix W .

2

With the a priori weighting equation (6.4) is changed as N „ _, -1 A8 - {W2 + I H'. R { H^} X i"l (6.6) N {-W2(6-e„) + E HT Rj(z.-y.)}

equations iu F (i«) - AF (m) + B X X F - CF Urn) + D y X (6.7)

and the c o s t f u n c t i o n has the form

Iz^ [F^(ia,.)-Fy(i..)] 'R-J [F^(ia..)-Fy(i..)]

- f i^nlR^I

+

(6.8)

In ( 6 . 7 ) and ( 6 . 8 ) F and F are the frequency r e s p o n s e s r e l a t i n g X y

the state and output variables to the input respectively, and F z are the measured frequency response c u r v e s . As f o l l o w s from Ref. 6

2

the main diagonal elements of the matrix R are formed by 2o , where 2

20^ - var {Re F^^(iüi)} + var {Ln F^^(iaj)} r • 1, 2, . . . , m

The estimates A8 are found from the equation ,N _T - I r ^ ^ -'^ -1

( 6 . 9 )

- {E nf R | H.} E H R, {F ( i w . ) - F (iu.)}

( 6 . 1 0 ) The s e n s i t i v i t y matrix H has now the elements 3F /dB ,

_ T y r '

r • 1, 2 , . . . , m, and H i s the transpose complex conjugate matrix t o H.

DATA ANALYSIS

The main results of the ML estimation technique are the values of estimated parameters, their error covariance matrix (a lower bound on the standard errors and the correlation coefficients), the

measurement noise covariance matrix (the variance estimate of measured outputs) and the value of the cost function. For the given set of

input-output data these results will vary with the structure of the model used in the estimation procedure.

As follows from the previous chapter, the proposed model of the aircraft contains seventeen unknown parameters, three of them are the bias terms and the rest includes the stability and control derivatives. The parameter Z- is assumed to be known and there is a known relationship between the parameters Z and Z ^.

q q

The large number of unknown parameters contrasts sharply with the limited amount of measured data. This inconsistency can create difficulties if all unknown parameters are to be estimated. If the anticipated problem is to be avoided, a simplified model should be used for the first estimate then, in the following computing runs, the number of unknown parameters can gradually be increased.

Such an approach, however, requires decisions on the significance of the differences in results obtained from various models, and on the adequacy of these models. This is, in general, a task of great complexity which, to the best of the author's knowledge, has not yet been solved. Therefore only a rough idea can be obtained, as proposed in the following:

1. the significant difference in corresponding parameters from various models can be judged from their 2a confidence intervals, provided that the parameters have roughly the same variance and non-significant correlation (strong correlation is assumed for correlation coefficients greater than 0.85),

2. the adequacy of the model can be inferred from the time histories of residuals and their autocovariance functions. For the correct model the residuals should form a sequence of uncorrelated random variables with gaussian distribution and zero mean, see Ref. 1.

From the time histories of the residuals trends may be apparent and eventual deterministic components discovered. If the

residuals are to be uncorrelated, and in the normal case also independent, then the autocovariance functions R and R

qo nzo should be equal to zero for r + o, where r is the correlation lag number.

No rule is proposed for the comparison of the variance estimates 2 2

s (q) and s (n ) , and the cost functions for various forms of the mod el.

The r u n s for t h e f o l l o w i n g a n a l y s i s were s e l e c t e d from t h e

measurements a t low and high a n g l e of a t t a c k s . The r e v i e w of c a s e s a n a l y s e d , which d i f f e r i n t h e amount of unknown p a r a m e t e r s and

i n c l u d e a p r i o r i known v a l u e s , i s p r e s e n t e d i n Table 4 . 7 . 1 Low Angle of A t t a c k Measurements

The r e s u l t s of e x t e n s i v e a n a l y s i s of t h e f i r s t s e l e c t e d r u n a r e p r e s e n t e d i n T a b l e 5 . They i n c l u d e t h e e s t i m a t e d p a r a m e t e r s , t h e i r s t a n d a r d e r r o r s (lower b o u n d s ) , t h e v a r i a n c e e s t i m a t e s of the

measurement n o i s e and t h e v a l u e s of the c o s t f u n c t i o n s , r e p r e s e n t e d by t h e l o g a r i t h m of |R | . For comparison the p r e d i c t e d v a l u e s , based on w i n d - t u n n e l t e s t s (Z , Z , M , M ) and s t e a d y - s t a t e f l i g h t

a q a q ^ " measurements (Z , M ) , a r e a l s o i n c l u d e d .

n n

I n t h e f i r s t four c a s e s , t h e l i n e a r model was used with v a r i o u s numbers of unknown p a r a m e t e r s . I n c a s e C some of t h e p r e d i c t i o n s e n t e r e d t h e a n a l y s i s a s t h e a p r i o r i known v a l u e s . The b e s t f i t between t h e measured and computed o u t p u t s was o b t a i n e d from c a s e D but t h e v a l u e s of t h e e s t i m a t e d p a r a m e t e r s remained c l o s e t o t h o s e i n c a s e B. The e s t i m a t e s from c a s e A show t h e l a r g e s t e r r o r s and a r e , i n some c a s e s , far from t h e p r e d i c t e d v a l u e s . On the o t h e r hand, t h e v a l u e s i n c a s e C seem t o be b i a s e d t o w a r d s t h e p r e d i c t i o n s . To s i m p l i f y t h e computing, and t o avoid p o s s i b l e n o n - p h y s i c a l v a l u e s f o r C- and C , t h e model in c a s e B was assumed a s an a d e q u a t e one

Lq zn

In the remaining eight cases the non-linear model with a d i f f e r e n t

number of linear terms was applied. The addition of the

non-linear terms r e s u l t e d in strong c o r r e l a t i o n s for some parameters and

thus in u n r e l i a b l e estimates of Z , M , M , and M . After the

a ' a ' q' n

i n t r o d u c t i o n of a p r i o r i weighting based on r e s u l t s from case B

the strong c o r r e l a t i o n in a l l parameters was removed. In g e n e r a l ,

the estimation with the non-linear model and a p r i o r i weighting

2

improves the f i t in n , whereas the estimates s (q) and the values

z

of the parameters in the linear terms do not differ significantly from those in case B and D. All parameters connected with the non-linear terms show large errors, and therefore their estimates have little practical meaning.

Because the a priori weighting did not degrade the fit in the output variables, there was no attempt to investigate the influence of the weighting matrix for the a priori values on the estimates.

The calculated responses using the parameters from case B are presented in Fig. 6, where they are compared with measured data. This figure shows that some differences exist between the computed and measured outputs over the interval between 2.5 and 3.5 sec.

The corresponding residuals and their autocovariance functions are plotted in Fig. 7. Both quantities reveal the amount of

disagreement in a better way than does the previous comparison. Nevertheless, the model used seems to be adequate, whereas the quality of measured data is in doubt.

The results of the sensitivity analysis are given in Table 6 as the suras of the values of the sensitivity functions and the main

diagonal elements of the information matrix. The quantities mentioned were normalized with respect to the values of the parameters, therefore they are comparable among themselves.

The linear model in case B was also applied for the estimation of parameters in the frequency domain. The results from two runs together with the estimates from the time domain are given in

Table 7. No significant differences are apparent from the comparison which could again be the indication of the adequacy of the linear model

for these measurements. The computed and measured frequency response curves are presented in Fig. 8 for one of the two runs analysed.

The frequency domain was also used for the demonstration of the influence of various input forms on the measured data. The input forms investigated are shown in Fig. 9 and their harmonic content (Fourier transform) in Fig. 10. The resulting frequency response curves are plotted in Fig. 11. Comparison shows that the amplitude characteristics for sharp pulse input differ from those for the other two inputs. These differences could create some discrepancies

in the estimated parameters from the measured data which includes all three input forms.

7.2 High Angle of Attack Measurements

The results from the measured data at a - 16.4 deg are given in

e

Table 8. The main parameters in the l i n e a r and non-linear models

do not d i f f e r s i g n i f i c a n t l y , but the f i t in n was improved owing

to non-linear terms being included. This improvement i s also

apparent from the time h i s t o r i e s as plotted in Fig. 12 (case B)

and F i g . 13 (case M).

The next two runs analysed were measured at a = 20.5 deg. The

r e s u l t s are summarized in Table 9. As in the previous example the

main parameters are mostly not influenced by the form of the model

but the f i t in n is considerably b e t t e r if the non-linear model i s

z

used. It was also found that the parameters Z* and M» could be

n n

included only with the a d d i t i o n a l non-linear terms. In the l i n e a r

model (case D) the values of these estimates were completely

unexpected.

The comparison of the measured and computed outputs for one run i s

presented in F i g . 14 (case B) and F i g . 15 (case N) . The time h i s t o r i e s

of r e s i d u a l s and t h e i r autocovariance functions are shown in F i g s . 16

and 17. The d i s t o r t i o n of the autocovariance functions for the

r e s i d u a l s in q are probably due to uncorrected measurement e r r o r s

in t h i s v a r i a b l e .

The results of the sensitivity analysis are given in Table 10. They indicate the increasing influence of non-linear terms and the

parameter M* in the pitching moment equation.

The inadequacy of the linear model for the high angle of attack measurements is demonstrated in Figs. 18 and 19, where the measured

frequency response curves are compared with computed ones based on the estimates from the frequency domain and time domain (predicted curves in Fig. 19).

8. DISCUSSION OF RESULTS

To obtain the overall idea about the main parameters, all test runs were analysed using the linear model representation. The variation of these parameters with a is presented in Fig. 20 where they are also compared with wind-tunnel and steady-state flight results. The estimated parameters exhibit quite large scatter which does not correspond to the standard errors of tlie individual parameters, However, the trend in the variation of the estimated parameters agrees, in general, idth the results from other sources with the exception of Z in the region of higher angle of attacks.

The greater part of the large scattered values in Fig. 20 are from the responses to a sharp pulse input. The effect of this input form is further analysed by the comparison of results from two runs with identical flight conditions but different input forms. The inputs considered are shown in Fig. 21, the harmonic content of them in Fig. 22. The measured and computed responses are plotted in Figs. 23 and 24. The results from both sets of data differed mainly in the estimates for Z . The values of this parameter, and the coefficients in the sensitivity and information matrices related to Z are given in the following table:

Item a N ^(9q/3Z ) ^ i-1 «^

M * 2

^ 1 Flight 833 Run 4 -1.29 ± 0.02 0.0442 0.0387 4909 Flight 855 Run 5 -1.69 ± 0.06 0.0014 0.0028 450 Predicted value -1.3

-The comparison shows that the use of a sharp pulse resulted in an inaccurate value of Z due to small excitation of n and low

a z

s e n s i t i v i t i e s in the two output v a r i a b l e s with respect to Z .

To increase the contribution of the measured n in the run with a

z

sharp pulse input, weighting matrix in the cost function (6.1) was introduced and the element of this matrix corresponding to n

gradually increased by the multiplying factor k . Different values

W n Z n n

of k were tested. The resulting changes in Z , s (q) and s (n )

wnz a z are shown in Fig. 25.

The other reason for the large scatter in the estimated parameters, regardless of the input form, can be caused by the limited number of measured outputs. This is demonstrated in the example based on

simulated data, which also shows the necessity for the introduction of a priori weighting, if the increased number of unknown parameters

is to be estimated.

The simulated time histories for a given input are presented in Fig. 26. The measurement errors were simulated by adding a gaussian random

noise to the computed outputs. Three cases were considered. In the first one only q and n entered the analysis. In Case 2 a was added to the previous outputs, in Case 3 the set of "measured" outputs was completed by q. The results obtained from the ML estimation without a priori weighting are given in Table 11. Case 1 failed to converge because of a large number of unknown parameters. Case 2 resulted in

p a r a m e t e r s many of them b e i n g f a r a p a r t from t h e t h e o r e t i c a l v a l u e s and with l a r g e e r r o r s . Only Case 3 w i t h four o u t p u t s gave r e a s o n a b l y good r e s u l t s . However, s t r o n g c o r r e l a t i o n f o r some p a r a m e t e r s remains i n b o t h c a s e s . The o n l y way f o r i t s removal i s i n f i x i n g some c o r r e l a t e d

p a r a m e t e r s or i n u s i n g a p r i o r i w e i g h t i n g .

The f i n a l r e s u l t s of t h e complete a n a l y s i s i n c l u d e t h e e s t i m a t e s of

h a n d l i n g q u a l i t i e s c r i t e r i a and n o n - d i m e n s i o n a l aerodynamic d e r i v a t i v e s . The f i r s t s e t of r e s u l t s i s p r e s e n t e d i n F i g . 27 and shows a r e a s o n a b l e d e g r e e of c o n s i s t e n c y except for t h e v a l u e s of C. The " o u t l i e r s "

a r e m o s t l y from t h o s e r u n s where a s h a r p p u l s e input was u s e d . The n o n - d i m e n s i o n a l aerodynamic d e r i v a t i v e s were computed from t h e

e s t i m a t e d p a r a m e t e r s from t h e e x p r e s s i o n s p r e s e n t e d i n Appendix A. A l l pitching-moment d e r i v a t i v e s a r e s l i g h t l y d i s t o r t e d by terms

i n c l u d i n g C *, b e c a u s e t h e v a l u e of t h i s d e r i v a t i v e cannot be e s t i m a t e d ma

e x p l i c i t l y . The n o n - l i n e a r model was used for r u n s where a was g r e a t e r t h a n 16 d e g . However, t h e e s t i m a t e s of t h e c o n t r o l d e r i v a t i v e s C ,

Ln C . • , and C • , a r e based o n l y on t h e n o n - l i n e a r model r e p r e s e n t a t i o n .

Ln mn , The variation of the aerodynamic derivatives with a are presented

in Figs. 28 and 29, and compared with the wind-tunnel and previous flight r e s u l t s whenever possible. The reason for a disagreement of the

estimated derivatives C_ , C , and C + C • at high angle of attacks La* ma' mq ma * **

CONCLUSION

The a p p l i c a t i o n of i d e n t i f i c a t i o n on t h e measured d a t a of t h e HP 115 a i r c r a f t r e s u l t e d i n :

1. the establishment of the adequate form for the equations of the

short-period longitudinal motion of t h i s a i r c r a f t ,

2. the estimation of main longitudinal aerodynamic d e r i v a t i v e s ,

3 . the i n d i c a t i o n of some reasons for the inaccuracies in the

r e s u l t s obtained.

The Maximum Likelihood parameter-estimation algorithm was used for the

estimation of unknown parameters. From these parameters the handling

q u a l i t i e s c r i t e r i a and non-dimensional aerodynamic d e r i v a t i v e s were

then computed.

For the low angle of attack measurements the linear model of the

a i r c r a f t was adequate. The estimation with the non-linear model

improved the f i t in the v e r t i c a l a c c e l e r a t i o n only, whereas the r e s t

of the estimated q u a n t i t i e s did not differ s i g n i f i c a n t l y . Also the

estimates in the frequency domain were in a good agreement with those

from the time h i s t o r i e s .

For the data measured at high angle of attacks the non-linear model

had to be used. In a l l cases the addition of non-linear terms r e s u l t e d

in a strong c o r r e l a t i o n for some parameters. After the introduction

of a p r i o r i weighting the strong c o r r e l a t i o n in a l l parameters was

r emoved.

The large number of unknown parameters contrasted sharply with the

limited amount of measured d a t a . This inconsistency was one of the

reasons for the inaccuracy of r e s u l t s obtained. The other reason was

connected with the inproper design of the experiment. The s e n s i t i v i t y

a n a l y s i s and comparison of r e s u l t s showed that a sharp pulse input used

for the e x c i t a t i o n in some t e s t runs was not s u i t a b l e .

Despite these shortcomings in measured data, the extracted aerodynamic

d e r i v a t i v e s agreed reasonably well with those obtained from wind-tunnel

and s t e a d y - s t a t e f l i g h t measurements. Certain disagreements in some

derivatives exist at high angle of attacks but the reason for this has not been discovered. The degree of consistency in C, • and C •

' Ln mn throughout the angle of attack range seems to substantiate the existence of these control derivatives for slender delta-wing aircraft with

trailing edge controls.

The experience from this analysis shows that further research is needed in:

1. the development of a method for the design of an optimal input which would be acceptable for the pilot,

2. the development of methods for the judgement of an adequate model and for solving associated problems, e.g. a decision on significant differences in the values of estimated parameters, measurement noise variance estimates and the cost function from various models.

10. REFERENCES 1. KLEIN, V. 2. BISGOOD, P.L, O'LEARY, C O , 3. BISGOOD, P.L. POULTER, R.L. 4. ENGLER, P.B.E. MOSS, G.F. 5. BISGOOD, P.L. 6. KLEIN, V. TOSOVSKY, J,

Parameter Identification Applied to Aircraft.

Report CIT - F I - 73 - 0 1 8 , October 1973. I n t e r i m Report on Low-Speed F l i g h t T e s t s of a Slender-Wing Research A i r c r a f t (Handley Page HP 1 1 5 ) . C . P . No. 8 3 8 , 1966. F l i g h t Measurements of t h e L a t e r a l 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 of a Slender-Wing Research A i r c r a f t (HP 115) and a Comparison w i t h Wind-tunnel R e s u l t s . RAE TR 72186, September 1972.

Low-speed Wind-tunnel T e s t s on a ^th S c a l e Model of t h e Handley-Page HP 1 1 5 .

R & M. Report No. 3486, August 1965. S t a b i l i t y and C o n t r o l T e s t on a Slender Wing R e s e a r c h A i r c r a f t .

RAE TECH. MEMO, AERO 1311, A p r i l 1971. General Theory of Complex Random V a r i a b l e and i t s A p p l i c a t i o n t o the Curve F i t t i n g a Frequency Response (Summary R e p o r t ) .

APPENDIX A P e r t u r b a t i o n S t a t e and Output E q u a t i o n s

For t h e s t u d y , t h e f o l l o w i n g a s s u m p t i o n s a r e i n t r o d u c e d :

1. t h e a i r c r a f t i s c o n s i d e r e d t o be i n l e v e l s t e a d y - s t a t e f l i g h t p r i o r t o t h e time when the i n p u t in the e l e v o n d e f l e c t i o n i s a p p l i e d ,

2 . the e x c i t e d motion i s a s t r i c t l y l o n g i t u d i n a l manoeuvre with n e g l i g i b l e a i r s p e e d c h a n g e s ,

3 . t h e c o e f f i c i e n t s of l i f t f o r c e and p i t c h i n g moment have t h e form • C- = C- + c , a + c, 3£ + c_ n + c • - ^ + L Le La Lq 2V Zn Ln 2V e e CT ?« + C, ^ ^ + C^ an

La^ Lqa 2V Lna

C = C a + C ' ^ + C • 5 § + C n + ( A . l ) m mo ma 2V mq 2V ran

e ^ e

^n,; w * ^n« 2 ^ ^ + C ^ + C an mn <iv ma mqa zv mna

e e L M w h e r e C, - ^^ s— and C •TrPV S -sPV Sc 2 e 2 e U s i n g t h e a b o v e m e n t i o n e d a s s u m p t i o n s and i n t r o d u c i n g Z , M z = ~ir- and m mV 1 e y

the perturbation equations of motion referred to the wind axes can be written as

a - q - ' ' z a + z q + Z 2 a + 2 aq + z an a q a ^ qa na + z n + z«n n n q = m a + m*a + m q + m o a + m a q + ^ o a q^ a'' qa m an + m n + m*n n» n n

For the output variable n the following relation holds

(A. 2) u n = - ^ (a- q - -^ sin 0 9) z g ^ u e * e (A.3)

Because of the last equation the state vector x • {a,q, 6} will be considered. Then the equations of motion expressed in the form with the augmented input vector u = |a , aq, a n , n, n, are

z z

a q M M a q 0 1 "1 0 0 0 J X + (A.4) Z 2 Z Z Z Z- Z a^ qa na n n o M 2 M M M M' M a^ qa na n n o 0 0 0 0 0 0

where Z and M are the bias terms. The form of equations

o o ^ (A.4) is suitable for the computing algorithm developed in Ref. 1

The output equations are formulated as

q 9 X + 0 0 0 0 0 0 Z 2 Z Z Z Z« Z a*' qa na n n o (A.5)

The p a r a m e t e r s i n the s t a t e and o u t p u t e q u a t i o n s a r e d e f i n e d a s f o l l o w s : Z = -K, C, a 1 La Z 2 = -K. C, 2 a^ 1 La Z = -K, C, n 1 Ln z = l-K„ c , q 2 Lq Z = -K„ C^ qa 2 Lqa Z ; - -K, C . n 2 Ln z - -K- C, q 2 Lq Z = -K, C, n a I Lna _{"6 u - ' " - e} _ g s i n 0 M » K, C - K^ C, a 3 ma 5 lA M = K- C - K^ C, na 3 mna 5 Lna M » K, (C + C • ) - K, C, M = K, C - K. C, q 4 mq ma 6 Lq n 3 mn 5 Ln

"a2

» S

C^2

- K3

C^^2

M; = K, C • - K, C, • n 4 mn 6 Ln M = K, C - K, C, qa 4 mqa 6 Lqo w h e r e 1 T 4 2 i T y ^2 ~ 27 ^3 . 2 1 T y ma T = 2m oSV u = ' 2 i . 2 y

c •

6 4 i PT y 2m aSc I

Span , (m)

Centre line chord, (m) Mean aerodynamic chord, (m)

Elevons: 2 Area, (m ) Half-span, (m)

Fin and Rudder:

Gross area (fin, rudder, external to Roof chord, (m)

Tip chord, (in) 2 Rudder area, (m )

Mass characteristics:

Aircraft mass at engine start, (kg) Fuel aiass, (kg)

Normal e.g. position at engine start

Inertia characteristics:

Rolling moment of inertia, (kgm^) Pitching moment of inertia, (kgm )

2 Yawing moment of inertia, (kgm )

fuselage), No fuel 1825 21860 23570

(mh

The above figures are the principal moments of inertia e.g. position of 0.548c . The inclinat

inertia axis to body datum axis is - 3

.ion of longitud] 95 deg. 6.1 12.2 9.27 4.32 2.67 2.9 2.22 1.19 0.74 2290 531 0.548 aft of LE of centre line chord Full fuel 2050 21870 23970 relevant to the inal principal

TABLE 1. GEOMETRIC, MASS AND INERTIA CHARACTERISTICS OF THE AIRCRAFT.

Pitching velocity Vertical acceleration Angle of attack

Port elevon angle SCarbord elevon angle Pitch angle Altitude Airspeed Rate gyroscope A.C. accelerometer Wind vane/ratiometer 1 Variable inductance J pick -^ off Pendulum inclinometer Static head

Pilot head & Venturi static Aero Flight Aero Flight IT 3-1-16 Aero Flight SFIM J.33 Mk 9 Mk 9 & Aero Flight ±10 deg/s 0-2g - 4 + 3 6 deg Or20 deg ±20 d«g

-10

12

35 ^>

-A

-0.7

0.7

0.1

-0.2

-x) at V - 75

m/s

830 831 832 j 833 6 104 100 100 88 89 90 92 83 110 120 118 108 A A B A A B B B A A A B 57.82 55.46 55.27 46.23 47.22 45.78 48.49 44.83 61.39 65.62 65.38 59.39 13.0 13.2 13.6 1 6 . 4 15.9 16.5 15.1 17.9 12.1 10.8 10.5 11.9 0.29 1 1.7013 1 0.31 0 . 3 0 0.45 0 . 4 3 0.45 0.39 0.48 0.25 0.22 0.22 0.26 1.7343 1.7124 2.1394 2.0865 2.1100 1.9303 2.1774 1.5488 1.4397 1.4375 1.5660 10.6116 10.3759 10.2097 10.6693 10.6283 10.4203 10.0971 10.5300 10.2568 10.1914 10.1358 10.0330 0.1203 0.1231

0.1251 1

0.1186 1

0.1201 1

0.1220 0.1246 0.1207 0.1264 0.1264 0.1276 0.1292 x) A - forward s t i c k p u l s e ; B - forward s t i c k square pulse

TABLES. REVIEW OF RUNS MEASURED AND CORRESPONDING FLIGHT CONDITIONS AND CHARACTERISTICS.

849 850 1 2 3 4 5 6 7 8 2 3 4 5 6 7 141 138 140 160 160 160 160 140 90 90 90 73 71 70 A A 75.81 75.54 79.00 88.31 84.95 8 6 . 3 1 8 6 . 3 1 74.66 50.06 50.01 50.01 39.28 38.92 39.12 8.8 8.9 8.5 7.4 7.6 7.2 7 . 1 8.4 1 5 . 3 1 5 . 3 15.2 2 0 . 3 2 0 . 7 2 0 . 5 0.16 0.17 0.15 0.12 0.12 0.12 0 . 1 1 0.15 0.37 0.37 0.37 0 . 5 8 0 . 5 8 0.57 1.2686 1.2726 1.1890 1.0423 1.0386 1.0168 1.0107 1.1720 1.9068 1.8957 1.8846 2.3306 2.3125 2.2733 10.3746 10.3703 10.1328 9.9298 9.5178 9.4671 9.4105 9.4393 10.2973 10.2270 10.1674 9.8755 9.7092 9.5937 0.1187 0.1193 0.1215 0.1232 0.1259 0.1269 0.1276 0.1296 0.1237 0.1246 0.1253 0.1287 0.1304 0.1314 x) A - forward stick pulse

TABLES. REVIEW OF RUNS MEASURED AND CORRESPONDING

851 855 861 I 90 5 j 90

7 1 88

8 1 80

9 2 3 4 5 9

1 10

3 80 110 110 120 118 139 158 150 C C

1 C

i 4 8 . 7 3 , 16.2 0 , 4 1 47.48 4 6 . 9 0 43.42 44.25 6 0 . 9 0 6 0 . 7 0 64.91 6 5 . 9 2 81.43 88.10

1 86.83

16.1 15.7 18.0 17.4 12.2 11.9 10.8 10.8 7.9 7.1

1 8.0

0 . 4 1 0 . 4 1 0 . 4 8 0.45 0.25 0.25 0 . 2 2 0 . 2 1 0 . 1 3 0 . 1 1 0.12 2.0559 1.9969 1.9774 2.1045 2.0502 1.5818 1.5497 1.4440 1.4188 1.0914 0.9974 1.0853 10.8073 10.2279 10.0041 9.8574 9.7866 10.3920 10.1473 10.1111 10.0891 ! 9 . 5 8 7 1 9.3840 10.1654

1

0.1199 j 0.1246 1 0.1247 1 0.1306 1 0.1316 0.1194 1 0.1215 ! 0.1229 0.1243 0.1308 0.1325 0.1252 1 x) C - backward stick pulse

TABLES. REVIEW OF RUNS MEASURED AND CORRESPONDING

B

C

D

F

H

J

K

L

M

M

P

LINEAR NONLINEAR

1

z„ z„ z „ . z ; - M* - 0 q, q» n» n n ^ ' % ' 0 Z Z 1 q» q 2q 1 Z„,Z;-M; -Z -Z ^ -M -O n' n Ti qa no qa no Z - Z„„ - M „ - M„^ - 0 qa n<i qa na

Za' -2na -«a' • ^ a - 0

Zna - *Via • 0

-Za^ • ^ a - »*a2 - 0

-Za. ^T,, ^a, M^,

-^a.

\ t \ ,

**n

-^.\,

V. «r,

2a Zq ^o.^ \ ^ \ . ' " Zfl

'°

\ \ Ma2 M qa ^ a \ % Mo

z 1

Zo^ 8 ^ ( q ) 8 2 ( n * )

1 ^n|i^ll

- 1 . 7 4 0 . 9 0 5 - 5 -— - 0 . 3 8 -0 - 6 . 8 5 - 1 . 6 3 -- 1 2 . 6 -0 - 0 . 0 9 5 0 -- 1 . 4 0 ± 0 . 0 6 0 . 9 3 ± 0 . 0 3 -— - 0 . 3 4 ± 0 . 0 2 -0 . -0 -0 -0 4 ± -0 . -0 -0 1 - 5 . 0 ± 0 . 2 - 2 . 0 6 ± 0 . 0 7 -- 1 3 . 9 ± 0 . 2 -0 . -0 1 -0 ± -0 . -0 -0 3 - 0 . 1 2 ± 0 . 0 1 - 0 . 0 0 1 0 ± 0 . 0 0 0 9 2 . 3 X 10*5 4 . 3 X 1 0 - 6 - 2 3 . 0 5 - 1 . 5 2 ± 0 . 0 3 0 . 9 0 5 -— - 0 . 3 8 -0 . -0 -0 -0 6 ± -0 . -0 -0 1 - 5 . 3 7 ± 0 . 0 9 - 1 . 9 9 ± 0 . 0 6 -- 1 4 . 1 ± 0 . 2 -0 . -0 1 3 ± -0 . -0 -0 3 - 0 . 0 9 5 - 0 . 0 0 0 3 ± 0 . 0 0 0 9 2 . 3 X 1 0 - 5 4 . 8 X 1 0 - 6 - 2 2 . 9 3 - 1 . 7 1 ± 0 . 0 3 0 . 8 5 ± 0 . 0 2 -— - 0 . 3 4 ± 0 . 0 2 -- 0 . 0 0 0 9 ± 0 . 0 0 1 - 6 . 1 ± 0 . 1 - 1 . 7 9 ± 0 . 0 6 -- 1 3 . 6 ± 0 . 2 -0 . -0 -0 9 ± -0 . -0 -0 4 - 0 . 0 9 ± 0 . 0 0 9 - O . 0 0 1 3 ± 0 . 0 O 0 9 2 . 3 X 1 0 - 5 5 . 0 X 1 0 - 6 - 2 2 . 8 8 - 1 . 5 0 ± 0 . 0 3 0 . 9 0 5 -— - 0 . 3 3 ± 0 . 0 1 3 - 0 . 0 0 9 ± 0 . 0 0 1 - 0 . 0 0 0 9 ± 0 . 0 0 0 9 - 5 . 3 0 ± 0 . 0 8 - 1 . 9 2 ± 0 . 0 6 -- 1 3 . 8 ± 0 . 2 - 0 . 0 4 ± 0 . 0 4 0 . 0 0 1 ± 0 . 0 0 3 - 0 . 0 9 5 - O . 0 0 0 7 ± 0 . 0 0 0 9 2 . 3 X 1 0 - 5 3 . 1 X 1 0 - 6 - 2 7 . 9 6

TABLE 5. EFFECT OF VARIOUS MODEL FORMS ON PARAMETERS DETERMINED FROM FLIGHT DATA. FLIGHT 849, RUN 6.

Za Zq Za2 \ ^ Zn« Zn Zfl Zo \ " q Ma2 M qa M-na \

''h

Mo z 1 z 1 ^ 0 s 2 ( q ) s 2 ( n * ) ^ n Ri - 1 . 7 4 0 . 9 0 5 - 5 -— - 0 . 3 8 -0 - 6 . 8 5 - 1 . 6 3 -— - 1 2 . 6 -0 - 0 . 0 9 5 0 -- 1 . 4 0 ± 0 . 0 6 0 . 9 3 ± 0 . 0 3 -— - 0 . 3 4 ± 0 . 0 2 -0 . -0 -0 -0 4 ± -0 . -0 -0 1 - 5 . 0 ± 0 . 2 - 2 . 0 6 ± 0 . 0 7 -— - 1 3 . 9 ± 0 . 2 -0 . -0 1 -0 ± -0 . -0 -0 3 - 0 . 1 2 ± 0 . 0 1 -O.OOIO ± 0 . 0 0 0 9 2 . 3 X 1 0 " ^ 4 . 3 X l ü ~ 6 - 2 3 . 0 5 - 1 . 5 2 ± 0 . 0 3 0 . 9 0 5 -— - 0 . 3 8 -0 . -0 -0 -0 6 ± -0 . -0 -0 1 - 5 . 3 7 ± 0 . 0 9 - 1 . 9 9 ± 0 . 0 6 -— - 1 4 . 1 ± 0 . 2 -0 . -0 1 3 ± -0 . -0 -0 3 - 0 . 0 9 5 - 0 . 0 0 0 3 ± 0 . 0 0 0 9 2 . 3 X 1 0 - 5 4 . 8 X 1 0 " ^ - 2 2 . 9 3 - 1 . 7 1 ± 0 . 0 3 0 . 8 5 ± 0 . 0 2 -— - 0 . 3 4 ± 0 . 0 2 -- 0 . 0 0 0 9 ± 0 . 0 0 1 - 6 . 1 ± 0 . 1 - 1 . 7 9 ± 0 . 0 6 -— - 1 3 . 6 ± 0 . 2 -0 . -0 -0 9 ± -0 . -0 -0 4 - 0 . 0 9 ± 0 . 0 0 9 - O . Ü 0 1 3 ± 0 . 0 0 0 9 2 . 3 X 1 0 - 5 5 . 0 X 10"6 - 2 2 . 8 8 - 1 . 5 0 ± 0 . 0 3 0 . 9 0 5 -— - 0 . 3 3 ± 0 . 0 1 3 - 0 . 0 0 9 + 0 . 0 0 1 - 0 . 0 0 0 9 ± 0 . 0 0 0 9 - 5 . 3 0 ± 0 . 0 8 - 1 . 9 2 ± 0 . 0 6 -- 1 3 . 8 ± 0 . 2 - 0 . 0 4 ± 0 . 0 4 0 . 0 0 1 ± 0 . 0 0 3 - 0 . 0 9 5 - 0 . 0 0 0 7 ± 0 . 0 0 0 9 2 . 3 X 1 0 - 5 3 . 1 X 1 0 - 6 - 2 7 . 9 6

TABLES. EFFECT OF VARIOUS MODEL FORMS ON PARAMETERS DETERMINED FROM FLIGHT DATA. FLIGHT 849, RUN 6.

' '"^

z

q 2 a 2

Zq„ 1

z

na

1 ^

^^ Zo M a M q \ 2 M qa M na "n Ho \ \ '

'°'

8 2 ( q ) 2 / *. i n | R - | - 1 . 7 4 0 . 9 0 5 - 5 -— - 0 . 3 8 -0 - 6 . 8 5 - 1 . 6 3 -- 1 2 . 6

-°

- 0 . 0 9 5 0 -- 1 . 5 1 ± 0 . 0 2 0 . 9 0 5 -- 0 . 7 ± 0 . 3 — - 0 , 3 4 ± 0 . 0 1 - 0 . 0 0 9 ± 0 . 0 0 1 0 . 0 0 2 0 ± 0 . 0 0 0 9 - 5 . 4 4 ± 0 . 0 6 - 2 . 2 1 ± 0 . 0 5 -- 1 5 ± 3 - 1 4 . 0 7 ± 0 . 0 2 - 0 . 0 8 ± 0 . 0 3 0 . 0 1 9 ± 0 . 0 0 3 - 0 . 0 9 5 0 . 0 0 0 9 ± 0 . 0 0 0 7 2 . 1 X 1 0 - 5 2 . 7 X lO"^ - 2 3 . 5 9 - 1 . 5 4 ± 0 . 9 0 5 - 2 ± - 0 . 5 ± — - 0 . 3 5 ± - 0 . 0 0 9 ± 0 . 0 3 1 0 . 4 0 . 0 1 0 . 0 0 1 0 . 0 0 2 0 ± 0 . 0 0 0 9 - 5 . 4 2 ± - 2 . 2 1 ± 3 ± - 1 7 ± - 1 4 . 0 7 ± - 0 . 0 8 ± 0 . 0 1 8 ± - 0 . 0 9 5 0 . 0 0 0 9 ± 2 . 1 X 2 . 7 X - 2 3 . 6 0 0 . 0 9 0 . 0 6 5 3 0 . 0 2 0 . 0 3 0 . 0 0 3

o.oooy

1 0 - 5 1 0 - ^ - 1 . 5 3 ± 0 . 0 3 0 . 9 0 5 2 ± 1 - 0 . 6 ± 0 . 4 8 ± 3 - 0 . 3 0 ± 0 . 0 2 - 0 . 0 1 2 ± 0 . 0 0 1 0 . 0 0 2 1 ± 0 . 0 0 0 9 - 5 . 4 1 ± 0 . 0 9 - 2 . 2 1 ± 0 . 0 6 3 ± 7 - 1 7 ± 4 0 . 0 1 ± 32 - 1 4 . 0 7 ± 0 . 0 2 - 0 . 0 8 ± 0 . 0 3 0 . 0 1 8 ± 0 . 0 0 3 - 0 . 0 9 5 0 . 0 0 1 0 ± 0 . 0 0 0 7 2 . 1 X 10"5 2 . 6 X 1 0 - ^ - 2 3 . 6 4 - 1 . 5 1 ± 0 . 0 2 0 . 9 0 5 -- 0 . 7 ± 0 . 3 ^ - 0 . 3 4 ± 0 . 0 1 - 0 . 0 1 0 ± 0 . 0 0 1 0 . 0 0 2 0 ± 0 . 0 0 0 9 - 5 . 4 2 ± 0 . 0 8 - 2 . 2 0 ± 0 . 0 5 -- 1 6 ± 4 - 0 . 7 ± 23 - 1 4 . 0 7 ± 0 . 0 2 - 0 . 0 8 ± 0 . 0 3 0 . 0 1 8 ± 0 . 0 0 3 - 0 . 0 9 5 0 . 0 0 0 9 ± 0 . 0 0 0 7 2 . 1 X 1 0 ~ 5 2 . 7 X lO"^ - 2 3 . 5 9

TABLE 5. EFFECT OF VARIOUS MODEL FORMS ON PARAMETERS

Za Z q «a

M

Za2

V

^ a

Z

n

z«

n

^ 2 M qa M na M

n

M«

n

z 1

q -1.4 0.93

-5:o

-2.06 --0.34 -_ --13.9 --0.12 0.0266 0.0436 0.0561 0.0530 -0.0006 -0.2142 -0.0293 0.0241 0.0324 0.0205 -0.0028 -0.0889 -0 . -0 « J 2 8 8015 7542 10030 7110 -681 -30151 -63 5 -1.53 --5.41 -2.21 2 -0.6 8 -0.30 -0.012 3 -17 0.01 -14.07 -0.08 -0.0313 -0.0658 0.0770 0.0000 0.0000 0.0001 0.0005 0.0000 0.0000 0.0025 0.0000 0.2590 0.0001 -0.0278 -0.0334 0.0273 0.0000 0.0000 0.0001 0.0024 0.0005 0.0000 0.0009 0.0000 0.1019 0.0000 -12153 -15970 14179 3 1 43 942 191 l 466 0 51548 5

-TABLE 6. ESTIMATED PARAMETERS AND NORMALIZED COEFFICIENTS IN THE SENSITIVITY AND INFORMATION MATRICES. FLIGHT 849, RUN 6.

z

a

M

o

M

q

M

n

U^(q)

s^n*)

1

1 ^'ffnz»n>

-1.52 ± 0.03 -5.37 ± 0.09 -1.99 ± 0.06 -14.1 ± 0.2 2.3 X 10~5 4.8 X lo"^ -1.49 ± 0.06 -5.77 ± 0.2 -1.69 ± O.l -13.0 ± 0.4 8.0 X 10-^ 3.4 X lO"^ -1.54 ± 0.04 -6.0 ± 0 . 1 -1.74 ± 0.08 -13.3 ± 0.3 4.0 X 10~5 7.7 X lO"^ -1.53 ± 0.07 -6.1 ± 0.2 -1.6 ± 0.1 -12.7 ± 0.4 7.7 X lO"^ 5.0 X lO"^

TABLET. COMPARISION OF PARAMETERS DETERMINED FROM FLIGHT DATA IN TIME AND FREQUENCY DOMAIN.

K

h

z

qo

^na

z

n

1

z.

n 1 Z o M a "q *'a2 M qa ^ a \ M.

n

Mo \ ' Zo^ «^(q)

•^(n*:

in Ri -1.08 0.925 -0.27 0 -1.87 -1.12 -4.03 -0 -0.075 0 .

1

--1.05 ± 0.03 0.93 ± 0.02 --0.175 ± 0.008 -0.004 ± 0.001 -2.08 ± 0.06 -0.72 ± 0.03 -— — -4.15 ± 0.05 -0.023 ± 0.002 -0.112 ± 0.008 0.0008± 0.0008 4.2 X 10"5 4.1 X lo"* -22.50 -I.11 ± 0.02 0.925 , -— -0.27 -0.006 ± 0.002 -2.04 ± 0.03 -0.84 ± 0.03 -~ — -4.40 ± 0.06 -0.028 ± 0.002 -0.075 0.003 ± 0.001 3.2 X 10"5 13.1 X lO"^ -21.53 -1.12 ± 0.01 0.925 -— -0.189 ± 0.007 0.005 ± 0.001 0.006 ± 0.001 -2.13 ± 0.02 -0.57 ± 0.02 * -— — -3.91 ± 0.05 ^ -0.14 1 0.02 0.024 ± 0.001 -0.075 0.0021 ±0.0007 3.4 X 10-5 3.1 X lo'^ -22.98 -1.15 + 0.01 0.925 -2.3 ± 0.4 0.3 ± 0.2 — -0.207 ± 0.008 0.003 ± 0.008 0.007 ± 0.008 -2.15 ± 0.02 -0.76 ± 0.02 -4 ± 1 1 ± 1 ~ -4.33 ± 0.04 -0.11 ± 0.02 0.028 ± 0.002 -0.075 0.0034 ±0.0007 2.8 X 10-5 3.2 X 10-^ -23.12 X strong correlation

TABLE 8. EFFECT OF VARIOUS MODEL FORMS ON PARAMETERS DETERMINED FROM FLIGHT DATA. FLIGHT 831, RUN1.

Za \ \ 2 Z qa Z na z n z-n Z o M a M q M 2 a ^ M qo M n a M n M . n M o z 1 q z 1 o s ^ ( q ) 2 * s ^ ( n ^ ) I i n | R 1 - 1 . 0 2 0 . 9 2 5 - 0 . 2 6 -0 - 0 . 6 5 - 1 . 1 - 3 . 4 -0 - 0 . 0 7 5 0 -CASi: B - 1 . 4 3 H 0 . 0 6 0 . 9 2 5 -- 0 . 2 6 0 . 0 2 1 ± 0 . 0 0 2 - 1 . 0 + 0 . 1 ^ - 1 . 0 3 ± 0 . 0 7 ^ -- 3 . 2 2 ± 0 . 0 7 -0 . -0 3 -0 ± -0 . -0 -0 1 - 0 . 0 7 5 0 . 0 0 8 ' 0 . 0 0 2 2 . 3 X IU~^ 1 9 . 5 X l ü " ^ - 2 1 . 5 3 CASE N - 1 . 4 8 ± 0 . 0 4 0 . 9 2 5 - 7 ± 4 - 0 . 9 ± 2 - 0 . 6 ± 2 - 0 . 2 3 i 0 . 0 1 - 0 . 0 0 6 +Ü.Ü02 0 . 0 2 0 ± 0 . 0 0 1 - 0 . 9 7 ± Ü.09 - 0 . 8 1 ± 0 . 0 6 - 7 3 ± 11 50 ± 8 178 ± 37 - 3 . 3 2 ± 0 . 0 6 - 0 . 1 7 ± 0 . 0 2 0 . 0 3 4 ± 0 . 0 0 1 - 0 . 0 7 5 0.ÜU8 + 0 . 0 0 1 2 . 4 X l o " ^ 7 . 0 X 10~^ - 2 2 . 5 3 CASE B - 1 . 4 1 ± 0 . 0 4 0 . 9 2 5 -- 0 . 2 6 -0 . -0 -0 7 ± -0 . -0 -0 2 - 2 . 1 8 ± 0 . 0 8 ^ - 0 . 3 5 ± 0 . 0 5 ^ -~ - 2 . 8 7 ± 0 . 0 6 -0 . -0 2 2 ± -0 . -0 -0 2 - 0 . 0 7 5 0 . 0 0 2 + 0 . 0 0 1 1.9 X 10~^ 1 6 . 2 X 10~^ - 2 1 . 9 2 CASE M - 1 . 4 1 ± 0 . 0 3 0 . 9 2 5 4 ± 2 0 ± I -- 0 . 1 8 8 ± 0 . 0 0 7 - 0 . 0 0 7 ± 0 . 0 0 1 1 0 . 0 0 8 7 ± 0 . 0 0 0 8 - 2 . 3 6 ± 0-.06 - 0 . 3 2 ± 0 . 0 4 - 6 1 ± 9 19 ± 5 -- 2 . 9 9 ± 0 . 0 5 0 . 0 7 ± O.Ol 0 . 0 3 4 ± 0 . 0 0 1 - 0 . 0 2 5 0 . 0 0 3 2 ± 0 . 0 Ü Ü 7 1.7 X l o " ^ 4 . 8 X l o " ^ - 2 3 . 2 5

X strong c o r r e l a t i o n

TABLE 9. EFFECT OF VARIOUS MODEL FORMS ON PARAMETERS DETERMINED FROM FLIGHT DATA. HIGH ANGLE OF ATTACK.

Ma *^q ^ 2 Zqa

Zna

Zn

Zn

Ma2

V

Mna

Mn

M;; -0.97 -0.81 -7 -0.9 -0.6 -0.23 -0.006 -73 50 178 -3.23 -0.17 0.0175 0.0258 0.0004 0.0000 0.0000 0.0057 0.0002 0.0236 0.0173 0.0055 0.2782 0.0070 0.0146 0.0219 0.0004 0.0000 0.0000 0.0013 0.0005 0.0289 0.0222 0.0031 0.2671 0.0035 2 838 4 238 74 1 0 428 80 5 150 3 921 678 50 140 799

TABLE 10. EFFECT OF VARIOUS MODEL FORMS ON PARAMETERS DETERMINED FROM FLIGHT DATA. HIGH ANGLE OF ATTACK.

z

a Z q Z 2 a

z

qa

z

n

z^

M a M q M 2 M q a M n M-n Z 1 q s ^ a ) s ^ ( q ) 1 8 2 ( q ) C C C - 1 . 4 0 . 9 2 - 5 - 0 . 5 - 0 . 2 0 - 0 . 0 0 8 - 1 . 0 - 1 . 0 5 10 - 4 - 0 . 2 - 0 . 0 8 -ASE I : n z CO ASE 2 : n z ASE 3 : n z - 1 . 3 5 ± 0 . 0 3 -- 3 . 9 ± 0 . 7 - 0 . 6 i 0 . 2 - 0 . 1 3 2 ± 0 . 0 0 8 0 . 0 6 ± 0 . 0 0 2 x . x x - 0 . 8 4 + 0 . 0 ^ - 0 . 9 5 ± 0 . 0 3 11 ± 3 ^ ' ^ ' ^ ' ^ ^ ^ ^ x x . x x x - 3 . 6 6 ± 0 . 0 7 - 0 . 2 4 ± 0 . 0 2 -1 2 . 7 X l o " ^ 1.6 X 1 0 " 5 3 . 8 X l O " ^ -and q " m e a s u r e d ' n v e r g e , q and a " m e a s u i , q , a and q "mes - 1 . 3 9 - 0 . 0 3 -- 4 . 6 ± 0 . 6 ^\ - 0 . 5 - 0 . 2 - 0 . 2 0 9 ± 0 . 0 0 9 - 0 . 0 0 7 * 0 . 0 0 2 - 0 . 9 2 ± 0 . 0 5 - 1 . 0 2 ± 0 . 0 2 7 ± 1 ^ « ^ ^ 9 . 1 ± 0 . 6 ^ ' ' ' - 3 . 9 9 ± 0 . 0 2 - 0 . 2 0 0 ± 0 . 0 0 4 -9 . 1 X 1 0 - ^ 1.6 X 10"5 3 . 8 X l O " ^ 2 . 6 X 1 0 - 5 ' ; f a i l e d t o •ed" i s u r e d " X,XX,XXX s t r o n g c o r r e l a t i o n

TABLE 11. EFFECT OF VARIOUS MODEL FORMS ON PARAMETERS DETERMINED FROM SIMULATED DATA.

18 14 IO - J l Mo O ï p \ o

8>

"^o -^_ o 30 40 50 60 70 80 9 0

V^[nn/s]

12 16 2 0 24

[deg]

— 4 — 8 - 1 2 — f6

^ " ^ J

"^0 • no o w ^

^e

P«9]

FIG. 3 VARIATION OF ANGLE OF ATTACK WITH TRUE AIRSPEED AND ELEVON ANGLE WITH ANGLE OF ATTACK MEASURED IN STEADY-STATE HORIZONTAL FLIGHTS.