Subcritical flutter testing and system identification

w NASA CR-132480 A . R . A . P . REPORT NO. 219 SUBCRITICAL FLUTTER TESTING AND SYSTEM IDENTIFICATION (NASA-CR-132480) S U B C R I T I C A L FLUTTER N74-34471 TESTING AND SYSTEM IDENTIFICATION (Aeronautical Research Associates of Princeton) 113 p HC $4.50 C S C L OlC Unclas G3/02 51C93 John C . Houbolt' Prepared under Contract No.

of Princeton Inc.

Aeronautical R e search Associates New Jersey 08540 50 Washington Road, Princeton, for NATIONAL AERONAUTICS AND SPACE ADMINISTRATION August TABLE O F CONTENTS S . R Y 1 .....................................................

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

~ n e r a l 4 S i g n i f i c a n t d i f f e r e n t i a l equations i n terms of CHARACTERISTICS O F VARIOUS 1 N P . S ..........................

S p e c t r a l content ......................................... 1 1

C l a s s i f i c a t i o n of t h e swept s i n e f u n c t i o n ................ 1 2

SECOND-ORDER SYSTEM UNDER CONSTANT FREQUENCY E X C I T A T I O N . 0 0 . 16

D i f f e r e n t i a l Equation Formulation of Nonsteady s i n uot

VANE FORCE ... TOR..........................^ 29

A T

Analog Set-Up and Associated E x c i t a t i o n and Measuring l c t * i .qirc.rr(..y ( I W C : . I . I . e .........................

Correlat.iori (JL' iriput and o u t p u t ........................ 3'1

REFERENCES ................................................. 51 .

iii

By John C . Iloubolt

SUMMARY Treatmcmt i s given C J f system response evali a t i o n , e s p e c i a l l y i n a p p l i c a t i o n t o s u b c r i t i c a l f l i g h t and wind t u n n e l f l u t t e r t e s t i n g of a i r c r a f t . An e v a l u a t i o n i s made of v a r i o u s e x i s t i n g techniques, i n conjunction w i t h a companion survey r e p o r t .

T h e o r e t i c a l and analog experiments are made t c ) study the i d e n t i - f i c a t i o n of system response c h a r a c t e r i s t i c s . Various i n p u t e x c i t a t i o n s are considered. New techniques f o r a n a l y z i n g response a r e explored, p a r t i c u l a r l y i n r e f e r e n c e t o the p r e v a l e n t p r a c t i c a l case where unwanted i n p u t noise i s p r e s e n t , such as d u e t o g u s t s or wind t u n n e l turbulence. F u r t h e r developments are a l s o m a d e of system parameter i d e n t i f i c a t i o n techniques.

Theory on t h e s u b j e c t i s extended, and many a s p e c t s of i d e n t i f y i n g system response c h a r a c t e r i s t i c s are given i n hand- book summary f a s h i o n .

INTRODUCTION An important and v i t a l phase of the a e r o e l a s t i c study 0:' a i r c r a f t i s the s u b s t a n t i a t i o n of f l u t t e r 'by means of' sub- c r i t i c a l f l i g h t f l u t t e r or w i n d t u n n e l t e s t s . Becizuse of the commonality of t h k problem t o a i r c r a f t des.Lgris, a major Cor-ifeL - ence was h e l d on t h e s u b j e c t i n May 1358 i n Washington, D.C.

U n t i l r e c e n t l y , l i t t l e had been done toward maintaining a summary of t h e v a r i o u s techniques used, o r how t h e y compare; e s s e n t i a l l y the companies have independently pursued arid developed t h e i r own i n d i v i d u a l schemes.

(.: ommuni c a t i clns , smaller group meet i n g s , and c on f'erenc e

p a p e r s i n d i c a t e that t h e r e i s much mutual i n t e r e s t .in t h e s u b j e c t , t h a t a number. of d i f f e r e n t methods are being used, arld t h a t a surve.y and cr.i.ti.que of t h e s e methods would be v a l u a b l e .

Design c o n s i d e r a t i o n s of t h e space s h u t t l e system emphasize the n e e d f o r and t i m e l i n e s s of such a survey.

* T n t h e neroelastic: a n a l y s i s of t h e s h u t t l e c o n f i g u r a t i o n many q u e s t i o n s n a t u r a l l y a r i s e . What subcriticla1 f l i g h t f l u t t e r Lechniques are p r e s e n t l y being used, and has 'the e v e r ' i n c r e a s i n g modern computer developments l e d t o improvements i n t h e LechriiqiAes? A sfgnif:i.cant question i s whether t h e techniques u s e d on aircraft are s u i t a ' b l e f o r studying t h e f l u t t e r problem of tihe shll%t:nl+ ~ It i s r e a l i z e d that the f l i g h t parameters f o r 'the s h u t t l e w i l l be i n a c o n s t a n t l y changing s t a t e . Thus, s t e a d y - s t a t e - t y p e f l u t t e r t e s t i n g techniques may n o t be a p p l i - C a b l e , and those techniques b a s e d on transient e x c i t a t i o n may be I t h e only t y p e sii.ita'rJle. If p r e s e n t techniques do not appear s u i t a b l e i n a p p l l c - a t i o n t;cs t h e space ::Iiutt.le , then r e s e a r c h must be undertaken t o develop f l i g h t f l u t t e r p r e d i c t i o n methods which w i l l be a p p l i c a b l e .

These needs l e d t o the study e f f o r t t h a t i s covered i n t h i s r e p o r t , sponsored by Langley Research Center of NASA. P a r t of t h e e f f o r t w a s d i r e c t e d toward making a survey, r e f e r e n c e 1.

T h i s r e f e r e n c e should be regarded as a companion t o t h i s r e p o r t .

Treatment herein d e a l s w i t h the e v a l u a t i o n of v a r i o u s sub- c r i t i c a l f l u t t e r t e s t i n g techniques, w i t h the s e t up and conduct of numerical and a n a l o g experiments of various schemes, w i t h the development of improved procedures, e s p e c i a l l y f o r the case where i n p u t noise is p r e s e n t , such as due t o turbulence o r b u f f e t i n g , a n d w i t h t h e development of system i d e n t i f i c a t i o n techniques. The r e p o r t i s a l s o intended Lo be, i n p a r t , a hand- book, s i n c e many n o t i o n s used i n system response e v a l u a t i o n are summarized.

It i s of interest t o note t h a t t h e survey and work of t h i s r e p o r t brought o u t t h e f a c t t h a t n o t only I; t h e r e much i n t e r e s t i n f l i g h t f l u t t e r t e s t i n g i n t h e United S t a t e s , but a very deep rooted i n t e r e s t i n England and o t h e r European c o u n t r i e s as w e l l , and t h a t , i n f a c t , s e v e r a l o t h e r survey-type papers on t h e sub- have r e c e n t l y been w r i t t e n , r e f e r e n c e s 2, 3, and 4.

SYMBOZS a constant ; e l a s t i c a x i s p o s i t f o n f r o m l e a d i n g edge ; l l f t curve s l o p e r e a l p a r t of frequency response func:!; Lon a constant.. ; wing span i m a g i n a r y p a r t of frequency response f u n c t i o n wing chord amplitude of frequency response f u n c t i o n exponential function, posit.lon ot' c-g r e l a t i v e t o t h e e l a s t i c a x i s , p o s i t i v e a f t d i s t a n C e of f o r c e a p p l i c a t i o n from e l a s t i c a x i s , p o s l t i v e forward d i s t a m e of accelerometer from r.I.astic axis, p o s i t i v e forward frequency i r i c p s F o u r i e r transform of f u n c t i o n y ; g e n e r a l l y , the s u b s c r i p t denotes the f u n c t i o n h impulse response f u n c t i o n s i n uot response due t o F = hS c u t frequency response f u n c t i o n ,

H = A + i B

reduced frequency, k = 2v ccc k mass r a d i u s of g y r a t i o n m L l i f t m mass; a n i n t e g e r n an i n t e g e r o p e r a t o r ; a r o o t P

3€

9 dynamic p r e s s u r e , q = 1 2 pV r

nondimensional e l a s t i c a x i s p o s i t i o n , r = - e

C e nondimensional l o c a t i o n of accelerometer,

ro = - 0

ro C r nondimensional l o c a t i o n of i n p u t f o r c e , f e@ rL

rf = -

C c o r r e l a t i o n f u n c t i o n of y R y W 2Vt S

nondimensional time, s = -

C t t i m e

v

veloc i t y nondimensional d e f l e c t i o n , w = C g e n e r a l displacement response f u n c t i o n response due t o noise d e f l e c t i o n a t accelerometer l o c a t i o n damping c o e f f i c i e n t c r i t i c a l damping c o e f f i c i e n t Dirac delta function c1 mass parameter V measure of v e l o c i t y , v = - 2v C P air d e n s i t y ;?V

U reduced v e l o c i t y , (T -

C D C r T t i m e @ angular dinplacement @y (03) power sp+::,t~um of the f u n c t i o n y w c, i r c ular !‘r E. quency undamped frequency; a c u t - o f f frequency wO 0 re f e renc e f re quenc y r GENERAL THEORETICAL RELATIONS T h i s sec.tiorl p r e s e n t s a l i s t i n g of the p r i n c i p a l g e n e r a l r e l a t i o n s t h a t a r e a p p r o p r i a t e i n t h e t r e a t m e n t of t h e response of l i n e a r systems t o v a r i o u s f o r c i n g f u n c t i o n s , and e s p e c i a l l y For t h e most i n a p p l i c a t i o n t o m b c r i t i c a l f l u t t e r t e s t i n g .

p a r t , l i t t l e <xssociat?d d i s c u s s i o n i s given. Some d e r i v a t i o n i s given where it i s f ’ e l t a p p r o p r i a t e , e s p e c i a l l y where t h e r e l a t i o n s h i p s a r e r i c h g e n e r a l l y known o r used. The e q u a t i o n s a r e formulated i n terms of b a s i c concepts t h a t a r e involved i n d e s c r i b i n g the response c h a r a c t e r i s t i c s of a s t r u c t u r e , such as t h e impulse function, the frequency response f u n c t i o n , c o r r e - l a t i o n f u n c t i o n s , and F o u r i e r transform r e l a t i o n s . Some of t h e equat.ions presented may appear as new developments.

General. - Let t h e g e n e r a l governing d i f f e r e n t i a l e q u a t i o n

f o r response be given by Dly = D2F where D1 arid D ; , a r e d i f f e r e n t i a l o p e r a t o r s , and y i s the

response t o t h e forcing f u n c t i o n F . For a simple damped mass

o s c i l l a t o r e q u a t i o n (1) i s

my + f3$ + Ky = F

t = 0 , I f ‘ the input f ’ o r . ~ : r i s a Dir;;Lc impulse f u n c t i o n a t equatiorl (1) defines t h e i m p u l s e r’esponse f’unc t i o n h f o r displacerrlent as rollows : D l h = D,h(O) L For a u n i t s i n u s o i d a l i n p u t , F = e and w i t h ioJt y = He equation (1) y i e l d s t h e frequency response f u n c t i o n

H ( u ) = A(u) + i B ( w ) (3)

according t o the equation

where A 1 , N1 and A2 , N2 a r e the real and imaginary p a r t s

t h a t are a s s o c i a t e d w i t h t h e o p e r a t o r s D1 and D2 . The A

component of H i s symmetrical w i t h r e s p e c t t o the frequency co the B component i s antisymmetrical.

The h and H functions are r e l a t e d by the F o u r i e r t r a n s - form p a i r H = h,-i~t d t (5) -La

Since h i s zero f < J r t < 0 and because of the symmetry

p r o p e r t i e s of A and B , it may be shown that the f o l l o w i n g

r e l a t i o n s a l s o a p p l y h = z A cos cot dco 7T

h = 2 r B s i n cot dw

7T Useful l i m i t p r o p e r t i e s of the H and h f u n c t i o n s are Q) H ( 0 ) = J h d t ( 7 ) .. .

The impulce resi)ori:,c f iAric,+,ions and h , f o r v e l o c i t y

h , and are

and f o r a c c e l e r a t i o n , fo.L:ows as d e r i v a t i v e s of defined by t h e r e l a t i o n s -W =

d t = A1 + iB1

H1 -m -W OD -00 whe re

A1 - - C U B -

B1 = CUA

A 2 = - w2A

By the s u p e r p o s i t i o n theorem, the s o l u t i o n of e q u a t i o n (1) f o r any general f o r c i n g f u n c t i o n F i s given by CQ

y = [ F ( . r ) h ( t - T ) dz

(10) .OD The Fourier transform of t h i s e q u a t i o n i s where Fy and FF denote t,he F o u r i e r transforxns, . r e s p e c t i v e l y , of y and F , S i m i l a r expressions i n terms of h' Hi ,

and H2 a p p l y f o r the response v a r i a ' b l e s $ and y . If each

s i d e of t h i s equation i s m u l t i p l i e d by i t s complex c o n j u g a t e , result i s t h e F l ? ' = Hfi FFFF Y Y which leads t o the well known s p e c t r a l equation

r e l a t i n g the i n p u t spectrum $F t o the output spectrum aY

through the amplitude squared of the frequency response f u n c t i o n .

A s p e c i f i c response equation.- A s p e c i f i c form of equation (1) t h a t i s of prime concern i n later s e c t i o n s of t h i s r e p o r t i s ... I . .

yv + a4y1' + a 7 + a$ + ali + aly = b 3 F + b$ + bl$ + blF

(13) With iut

= (A + i B ) e

Y = Y,e and icut F = e t h i s e q u a t i o n y i e l d s where 4 2 A, = CD a4 - cu a2 + a .

I

S o l u t i o n f o r A and B yields t h e r e s u l t s I' Another solutlon of importance, but which has riot grown i n t o popular uoe, 1 : ; the s o l u t i o n for. impedance A -

A + iB) m ( i n s t e a d of f o r admit;tance It i s noted that

From e q u a t i o n s (15) and ( 9 ) , it follows t h a t

NIAl + N2A2 - & A 2

A - =3

7 - N : + N : %

These e q u a t i o n s are of' s p e c i a l s i g n i f i c a n c e i n the t r e a t m e n t of f l u t t e r .

A s p e c i a l t y p e response Function.- Let the Input f o r c e be s i n uAt t h i s i s a simple b u t very s p e c i a l i n p u t taken as F = ; UI t f o r c e which has riot r e c e i v e d ,the r e c o g n i t i o n i t deserves, and which g i v e s a response t h a t has many important and u s e f u l c h a r a c t e r i s t i c s . For t h i s f o r c e , e q u a t i o n (11) becomes where, symbolically, the n o t a t i o n i n d i c a t e s t h a t Fly i s simply H t r u n c a t e d b y t h e "box c a r " f u n c t i o n . The a c t u a l response,

hs , and t h e i n v e r s e transform of , i s h e r e i n denoted as

FY appears as ' r h u s if' the frequericy K ~ ~ s ~ J ( J T ~ u ~ furicticiri of a s y s t e m i s known,

and i s sharply c u t o f t ' t o z e r o b e y o n d a frequency ub , t h e n

I I t h e ''irripulse f u n c t i o n that i s a.tSsu(.iated with the t r u n c a t e d I'requericy response f ' u n c a Lion i s s i m p l y the response of' the s i n uot S stem t o an inp rt force equal t o ; as cuO i s made

Oot

approaches h . l a r g e r , the more a.nd more

hS terms of S i g n i f i c a n t d i f f e r e n t i a l e q u a t i o n s i n

c o r r e l a t i o n f u n c t i o n s . - Several e q u a t i o n s are developed here

which a r e n o t g e n e r a l l y known but which should be of g r e a t h e l p i n system i d e n t i f i c a t i o n s t u d i e s .

L e t the i n p u t force be h ( - t) , which 1 s the impulse

t response f u n c t i o n f o l d e d around t o f a l l a l o n g the n e g a t i v e a x i s . By equation (lo), the response would be

y := r h ( - .r)h(t - z ) d 7

.- 00 which, i n t u r n , may be w r i t t e n This equation is, however, t o w i t h i n some cons,,ant the d e f i n i t i o n of' t h e a u t o c o r r e l a t i o n f u n c t i o n of h ( t j . Thus, t h e a.utocorrelation function of h .is the response of t h e

system t o a f o r c e i n p u t of h ( - t ) , or

DIRh = D2h(- t ) A rela.ted e q u a t i o n i s a s s o c i a t e d w i t h the response of t h e

system t o pure white n o i s e . Thus, i f t h e i n p u t is white noise, -

e q u a t i o n (12) becomes s i n c e qF i s f:Lat. The inverse transform of t h i s equation i s R = r h ( i ) h ( t + T ) d7: Yn I, which i s t h e same as equation (18). Thus, the c o r r e l a t i o n f u n c t i o n of t h e response due t o white n o i s e i s seen t o be the same as the a u t o c o r r e l a t i o n f u n c t i o n of the impulse f u n c t i o n h .

An e q u a t i o n i n v o l v j n g t h e c r o s s - c o r r e l & t i o n between any i l l p u t and t h e a s s o c i a t e d response i s also of s i g n i f i c a n c e . The l"ourier transform of e q u a t i o n (1) i s

(Ai + I A , ) F '

-- (N1 + iN2)FF

2 Y t (Al f i.A,)F = (N1 4- iN )I? F Y F 2 F F which immediately leads t o the s p e c t r a l equation

where aFy i s t h e cross-spectrum between the f o r c e and t h e

response, lend @F i s t h e spectrum of t h e i n p u t f o r c e . The i n v e r s e transform of t h i s s p e c t r a l equation i s F i s a p p l i e d Thus, i f the a u t o c o r r e l a t i o n f u n c t i o n of am i n p u t t o the system as a n i n p u t force, the response i s the c r o s s - c o r r e l a t i o n f u n c t i o n between F and the response y due t o F .

Some s i g n i f i c a n t e q u a l i t i e s and transforms.- Some e q u a l i t i e s and transforms of impor+,nnc.e are l i s t e d here t o c l o s e o r d e r i v a t i o n .

t h i s s e c t i o n . They a r e giver, without proof 1 out Two b a s i c Fourier transforms: s i n u(;t Convolution: s i n w t

jW si;?T si11 (u

*(t - 2 )

d, = for

u*(t - T )

?Lt -a, .

s i n u2t I - for

'01 ' ' O 2

t a3

I ' s i n a2(t - 4

dz = s i n Yt f o r

J sin Y7 LD

d t -

-00

- s i n ult f o r

- 7 Z O for P r o p e r t i e s of a s p e c i f i c h f m c t i o n : - The impulse response f'unctioii h f o r a simple damped mass o s c i l l a t o r i s B - - a t I e QC, h = - sin udt ( 2 2 ) ~ T I ad where The a u t o c o r r e l a t i o n of this f u n c t l o n , see equation (18), i s as r e f e r e n c e 5 a l s o shows. The d e r i v a t i v e o f ' Rh i s

-

"ot

1 P C r ( > > l ! ) s i n (11 t

Rh == - E : d

B

(3 c r i s seen t o be: equal, w i t h i n a Lonstant, t u I n t e r e s t i n g l y , &, h ( e q u a t i o n 2 2 ) ) . It would be of i n t e r e s t , t o ;tudy what t y p e systems have ti Puric t i ons thdt obey this p r o p e r t g ,

CHARACTERISTICS OF VARIOUS Irwws

S p e c t r a l c o n t e n t . - F i g u r e 1 indic-a.tes i.n summary f a s h i o n

t h e s p e c t r a l c h a r a c t e r i s t i c s t h a t a r c a s s o c i a t e d w l t h v a r i o u s i n p u t s t h a t are of concern i n s u b c r i t i c a l f l u t t e r t e s t i n g .

Four d i s t i n c t l y d i f f e r e n t func Lions, tkte h f u n c t i o n , s i n cc, t w h i t e n o i s e , a swept s i n e , and -

, are seen t o l e a d t o

"0"

:til d s t e n s i b l y f l a t power spec'tra. Besl.rleS t h e 6 f u n c t i o n , which i s d i f f i c u l t t r - , achieve in pra.c'tice, t h e only f u n c t i o n which leads t o a t r u 1 . y f l a t p o w e r spectrum, and which extends s i n iu '1;

t o zero frequency i s ?;he --&r- (' !hric.Lion. The white n o i s e

spectrum i s u s u a l l y quite jagged. The spactriun f o r a swept s i n e has large lobes at t h e .Low a n d hlgh rrequency ends. 'Tne I s i n m o t c o n t r a s t between t h e swept s i n e and the i s i n t e r e s t i n g ; the swept s i n e f u n c t i o n has c o n s t a n t amplitude but varying f r e - s i n u t quency; the f u n c t i o n has a c o n s t a n t frequency b u t

'Uot s i n uot

varying amplitude. The f u n c t i o n i s a very a t t r a c t i v e

Wet

f u n c t i o n f o r use i n system i d e n t i f i c a t i o n s t u d i e s and has n o t

cu0 - Q) Because i n the l i m i t as

been e x p l o i t e d s u f f i c i e n t l y r s i n uot the f u n c t i o n approaches the 6 f u n c t i o n , it i s 0 s i n uot suggested t h a t be termed the impulse s i n e f u n c t i o n

"-'ot

f o r ready i d e n t i f i c a t i o n purposes C l a s s i f i c a t i o n of t h e swept s i n e f u n c t i o n . - The swept s i n e f u n c t i o n has become rather popular f o r use i n s u b c r i t i c a l f l u t t e r t e s t i n g , r e f e r e n c e 1. The rate of sweep o r t o t a l d u r a t i o n i s one of t h e prime v a r i a b l e s ; w i t h some tests the sweep rate i s fast, i n o t h e r s t h e rate i s q u i t e slow.. For d i s c u s s i o n and t e s t i n g purposes, it appears desirable t o make a c l a s s i f i c a t i o n of the rate o r d u r a t i o n of sweep. The rate of change of f r e - quency depends o f course on t h e frequency range covered and t h e d u r a t i o n required t o make t h e sweep. For the t e s t i n g of most a i r c r a f t systems, however, it appears that c l a s s i f i c a t i o n can be based mainly on d u r a t i o n a l o n e . The f o l l o w i n g c l a s s i f i c a t i o n i s suggested:

1) F a s t sweep - one made w i t h a d u r a t i o n of about

5 seconds.

2 ) Moderate sweep - d u r a t i o n of around 1 minute.

3) Slow sweep - d u r a t i o n of around 5 minutes.

Each of these s w e e p s has c e r t a i n advantages, and c e r t a i n d e f i c i e n c i e s , depending on t h e a p p l i c a t i o n , R e s u l t s l a t e r i n t h e r e p o r t w i l l t r y t o b r i n g o u t some of the r e l a t i v e merits.

Related d i f f e r e n t i a l e q u a t i o n s . - It i s perhaps of i n t e r e s t t o note t h a t d i f f e r e n t i a l e a u a t i o n s a s s o c i a t e d w i t h v a r i o u s swept s i n e wave laws, and w i t h the impulse s i n e f u n c t i o n , may be i d e n t i f i e d . T h i s s e c t i o n shows, i n t h e n a t u r e of a n aside, the c o n s t r u c t i o n of these d i f f e r e n t i a l e q u a t i o n s .

Consider t h e d i f f e r e n t i a l e q u a t i o n s o l u t i o n t o be of the f o l l o w i n g general form y = e f ( t ) s i n g ( t ) (25) any f u n c t i o n s of i n t e r e s t . If t h e where g ( t ) are f ( t ) and and second d e r i v a t i v e of e q u a t i o n (25) a r e formed, and i f f i r s t the:?,c d e r i v a t i v e s are c;omb.ined i n l i n e a r f a s h i o n w i t h the f u n c t i o n

y , then the follbw ir!g (1 i f I ' e r e n t i s l equation may be shown as a

re sul1; Equation (25) i s t h u s a s o l u t i o n of t h i s d i f f e r e n t i a l e q u a t i o n .

The d i f f e r e n t i a l equation for v a r i o u s swept s i n e laws and f o r the impulse sine f u n c t i o n follow d i r e c t l y from these e q u a t i o n s .

Linear sweep l a w : Consider t h a t t h e s o l u t i o n i s t h e swept s i n e wave o f t e n used as a f o r c i n g f u n c t i o n i n s u b c r i t i c a l f l u t t e r t e s t i n g , namely

y = s i n 8 = s i n (a. + b t ) t

(2'7) where i n s t a n t a n e o u s frequency is defined as or

C I , =1 a + 2bt

I n terms of t h e beginning frequency (bo and the erid frequency

u)1 , a . f t e r a sweep of T seconds, a and b are

a = c o 0 1 - b = 2 T s o that t

( 1 ) = LDo + ("I - Uo) T

Equation ( 2 5 ) reduces t o e q u a t i o n (27) f o r f ( t ) = 0 g ( t ) = (a i- b t ) t By e q u a t i o n ( 2 6 ) , t h e n , the d i f f e r e n t i a l e q u a t i o n y i e l d i n g equation ( 2 5 ) as a solution is

2b 9 S (a + 2 b t ) y = 0

- a +- 2bt

Linear sweep down: For thls case

y = sin (a - b t ) t

o r f ( t ) = O ' , and g ( t ) = (a - b t ) t . Equation (26) thus

i n d i c a t e s the a s s o c i a t e d d i f f e r e n t i a l e q u a t i o n t o be 2 2b

+ (a - 2 b t )

y = 0

+ a - 2bt

Exponential sweep: a t u = w e L e t s o that LD

e = g ( t ) =]udt = a o eat

Through equation ( 2 6 ) , w i t h f ( t ) = 0 , the swept s i n e

i s t h u s found t o be defined by the d i f f e r e n t i a l e q u a t i o n 2 2at

j ; - a9 + uoe

y = o (33) Linear p e r i o d sweep: For t h i s case o r - 1

0 = g ( t ) =I ccdt = - b l o g (a - b t )

and

y = s i n [- log (a - b t ) ]

(34)

By equation (26) w i t h f ( t ) = 0 , the d i f f e r e n t i a l e q u a t i o n i s

found t o be ..

? +

2 Y = O (35) ' - a - b t (a - b t ) .

The impulse s i n e f u n c t i o n :

With f ( t ) = - l o g cot

g ( t ) = u t equation (25) d e f i n e s t h e i m p u l s e s i n e f u n c t i o n s i n ut Y ' T By equation ( 2 6 ) , t h e a s s o c i a t e d d i f f e r e n t i a l equation is found t o be .

2 2 j; + c o y = 0 (37) T h i s equation i s a special case of Besselts d i f f e r e n t i a l equation, w i t h t h e s o l u t i o n

y - t T Z

I(- 4

- -

The f u n c t i o n cos ut y = at i s a l s o a s o l u t i o n .

0 ther second -0 r d e r time -varying sys terns : Equation (;i6), a n d t h e subcases given by equations (29) , (3l), ( 3 3 ) , (35), and ( 3 7 ) , a r e noted t o be a s s o c i a t e d w i t h l i n e a r systems w i t h time-varying parameters. A s a f u r t h e r aside, it may be noted t h a t equation (26) may be u s e f u l i n t h e s t u d y of v a r i o u s second-order systems having time-varying p r o p e r t i e s .

A common apl)r.oach i n dealing w i t h time-varyirlg systems i s t o model the system and then t o seek approximate s o l u t i o n s t o the mudeled system. Reference 6 is an e x c e l l e n t t r e a t i s e a l w g t h e s e l i n e s . Consideration of e q u a t i o n s (25) and (26) suggests an approach which i s j u s t t h e o p p o s i t e . Thus, it i s supposed t h a t t h e s o l u t i o n i s known; from t h e s o l u t i o n t h e d i f f e r e n t i a l e q u a t i o n is derived. This d i f f e r e n t i a l equation i s then examined t o see whether it r e p r e s e n t s the system being s t u d i e d , o r a t l e a s t i s a, c l o s e approximation t o the system, Consider, for example, f ( t ) = - p t and g ( t ) = ut ; e q u a t i o n (26 j tlien i n a i c a t e s t h e w e i i known damped o s c i i i a t o r e q u a t i o n

j ; + 283; + (a + p2)y = 0

l b

- 2b + ( a + 2 b t ) 2 -t W

.y' - ( 2 % + 4tt)i + Y = 0

'3

T h i s equation i s n o t e d t o a p p l y t o a system w i t h a l i n e a r change i n damping and a. q i i a d r a t i c cha.rlge i n frequency. I f , f u r t h e r , a

and b are srnsll r e l a t i v e t o cu , then the c o e f f i c i e n t of y is

The e q u a t j o n WomLd t h e n r e p r e s e n t a good roughly a, c.onstnnt.

approximation t o a system w i t h a l.lnear change i n damping. The n a t u r e of t h e homGgeneous resporlse behavior of such a system i s I n t u r n aukomatically given by equation ( 2 5 ) .

SECOND-ORDER SYSTEM UNDER CONSTANT F'REQlBNCY E X C I T A T I O N I n studying the response c h a r a c t e r i s t i c s of s t r u c t u r e s , one is to i d e n t i f y t h e f r e q u e n c i e s and d a m p i n g of t h e p r i m a r y g o a l s values of t h e v a r i o u s modes. Common or popular wa.ys of i d e n t i - f y i n g these q u a n t i t i e s are summarized I n t h i s s e c t i o n i n terms of 2 wc: ond-clrder s y s t s m .

F igure ,' (jcpicbts a popular t y p r COIlStruCt ion i n v o l v i n g 1,: frequency r e s p o n s e f i i n r t i o n , and s p e ( ' i f i c a 1 l y i n t h e form o f ; 1 .

p l o t o f B a g a i n s t A . This p r e s e n t a t i o n i s o f t e n r e f e r r e d t ( , as the Kennedy-Pancu method, r e f e r e n c e 7. Much d i s c u s s i o n U I I For a +,his type coristruction is a l s o given i n r e f e r e n c e 8.

second-order s y s t e m , the A ' s a n d B ' s a r e given by F Displacement, -2 m" A + 1 B =

1 - x2 + igx

V e l o c i t y , F

ix -

C h a r a c t e r i s t i c a l l y , t h e p l o t s f o r a l l three q u a n t i t i e s resemble c i r c l e s , and indeed t h e p l o t f o r v e l o c i t y Is a true c i r c l e .

The resonant frequency is i d e n t i f i e d at the p o s i t i o n on the " c i r c l e " where t h e r e is g r e a t e s t arc l e n g t h swept f o r equal frequency increments. Damping i s found i n two ways: the diameter of t h e c i r c l e i s l / g (assuming the response at z e r o frequency has been normalized t o u n i t y ) , or by the equation where u0 i s the resonant frequency.

The p l o t i n f i g u r e 2d i l l u s t r a t e s the results obtained if there i s a mixture of viscous damping and s t r u c t u r a l "g" type damping; s p e c i f i c a l l y , the frequency response f u n c t i o n is given by F A + i B = wO

1 - x2 + i ( g x + gs)

All r e s u l t s shown i n f i g u r e 2 are f o r g = 2 -@- = .1 and and a r e presented on the assumption %f; at the f a c t o r s gs = .1 - F F , and - F a r e u n i t y .

2 1 - m

mo-b

Other means f o r e v a l u a t i n g frequency and damping are shown The t o p s k e t c h i n f i g u r e 3 (again f o r a second-order system).

r e p r e s e n t s C2 , t h e square of the amplitude of the frequency

response f u n c t i o n . The resonant frequency i s a s s o c i a t e d w i t h Damping may be found as shown, e i t h e r the peak of the c u r v e , i s known) o r from the w i d t h at from the peak value ( i f m ' o the half-height p o s i t i o n . A s shown i n the bottom of the f i g u r e , the h f u n c t i o n , o r t h e response t h a t ensues after suddenly c u t t i n g o f f a resonant e x c i t a t i o n , is s t i l l a n o t h e r way t o e s t i m a t e frequency and damping, The frequency i s e v a l u a t e d from the p e r i o d T ; damping is estimated from t h e decay of t h e peaks. The curves presented are based on viscous damping and p r o v i d e a quick wak f o r e s t i m a t i n g damping from s u c c e s s i v e p e a k v a l u e s .

F i g u r e 4 i l l u s t r a t e s the impedance method which is a n o t h e r good way f o r e s t i m a t i n g frequency 4nd damping, although t h e The p l o t s r e p r e s e n t schemes h a v e n t t been pursued g r e a t l y , 1 A

-

and - vs; CQ P s r z seccnd=crder system, these

v s . --

c c 2 C ' q u a n t i t i e s are defined by Damping and frequency are found as shown ( a g a i n presented on i s u n i t y ) .

t h e basis t h a t mE/F F i g u r e 5 i s provided as a convenient r e f e r e n c e f i g u r e t o i n d i c a t e the b a s i c c h a r a c t e r i s t i c s of the impulse response and v e l o c i t y , and frequency r e s p o n s e f u n c t i o n s f o r displacement, a c c e l e r a t i o n f o r a second-order system.

RESULTS FOR SIMPLE SYSTEMS WITH TIME-VARYING INPUTS The n a t u r e of t h e r e s u l t s t h a t are obtained through use of swept s i n e and impulsive s i n e e x c i t a t i o n s are brought out t h i s s e c t i o n . Most of the r e s u l t s given have been obtained i n through s t u d y of a simple damped mass o s c i l l a t o r - system, w i t h an undamped frequency f o = 9.95 cps , a n d a 1 = .05 .

Bcr swept s i n e run, F i g u r e 6 shows t h e r e s u l t s f o r a fast The t o p sweeping up f r o m 4.8 cps t o 24 cps i n 4 seconds.

f i g u r e is the input f o r c e , the second the r e s p o n s e . The f i g u r e a t lower l e f t r e p r e s e n t s the a u t o c o r r e l a t i o n of the i n p u t f o r c e ; t h e f u n c t i o n i s seen t o be composed of two sin t y p e f u n c t i o n s , t h u s implying a f l a t - t y p e spectrum

wt

The figure on bottom between an upper and a lower frequency.

r i g h t r e p r e s e n t s t h e a u t o c o r r e l a t i o n f u n c t i o n R of the response y . If the i n p u t f o r c e has a t r u l y whxte spectrum, then Ry can be shown t o be the same as the a u t o c o r r e l a t i o n funcLion of t h e h f u n c t i o n , see e q u a t i o n ( 2 3 ) and r e f e r e n c e 5.

F u r t h e r , it 1s a l s o Pound t h a t , at least f o r a s i n g l e degree of freedom system, frequency and damping i n d i c a t e d by the R h f u n c t i o n a r e t h e same as f o r t h e h f u n c t i o n . It i s of i n v o l v e s u s e of i n t e r e s t t o note t h a t p r o c e s s i n g i n t h i s c a s e t h e response only; t h e e s t a b l i s h m e n t of R p r o v i d e s a ready means f o r e s t i m a t i n g system frequency and h a p i n g .

Figure 7 shows r e s u l t s f o r h as o b t a i n e d from the y € u n c t i o n of f i g u r e 6 by a randomdec-type technique ( r e f . 9 ) .

I n c o n t r a s t t o t h e randomdec p r o c e s s described i n r e f e r e n c e 9 , the process advocated here i s developed i n terms of f u n c t i o n s khat r e s u l t f r o m zero-crossings c o n s i d e r a t i o n s ; f i g u r e 7 ( a ) %l:Liis't raI;t?s t r i v z r . r ' f ~ - ~ r c:::s.ir:p:; t a c h r ~ i q i l ~ . f'or c o n s t r u c t i n g the

r-a.nd~m(l(:c: :?-Ignati~rb(: . 'l'w.! .I,rw;; i i ; i s t;ha..t; the randomdec sigrlsl

:io oh't,xirled Is ,!;ti:.: trnpu.1 :;e ri3:;ponse fiinction n . The r e s u l t

showi-1 :in f i g u r e 7 ( ' b ) rwp.r'esent,s ,!;he sum of only 20 f u n c t i o n s , s~ta,rt;:irig at the p(j,i.r)?, marked a i n figi.lre 6. A . s seen, t h e 'beglrxing p c J r t i o n r e p r y s e n t s qui,& well the decaying s i n e wave c h a r s c t e r i s t 8 i c o f the h fimctiorl for a second-order sy.stem .

The r e s u l t s at larger. time values carmot be considered reliable because t h e randomdec summatjon involved only 20 terms. Here i s a case where 8 mode:ratx sweep rate would be b e t t e r f o r randcJmdec T f a moderate sweep h a d been u o e d , then the summation purposes.

could have involved many mor% terms, w i t h ,the consequence that h funct,ion derived would also be a c c u r a t e a t l a r g e r t i m e v a l u e s .

Figure 8 p r e s s n t s r e s u l t s obtained by sweeping down from ' 2 4 cps 'to 4 . 8 c p s in 4 seconds. The a u t o c o r r e l a t i o r l f u n c t i o n of +,he resporise y I s .fou.r!d t o 'be v i r t u a l l y the same as f o r t h e sweep-xp run of f i g u r e 6.

F i g w e 9 p r e s e n t s the swee&-up m s u l t s f o r the system w i t h zero damping, And i s gi.ven ,bo show the cauti.on t h a t must be used i n i n t e r p r e t i n g t h e aut;!:cc)r:rela.t,:ion f u n c t i o n . The f u n c t i o n shown :ir, f i g u r e 3 ind:ica.te,i +,hat t h e system appears to have some damping. The respclrisr y hw+':ver, shows a p e r s i s t e n c e i n o w l l l a t i o n s , or r..In{.:fng, a f t e r the: resonance freqi.Jency i s passed. T h i s i : . ;i ,tip -f.JPf , t h d . t the system h a ; : I . i . % t l c or no dampir:g. ' tiare aq;~.:h, 9t: a moder&te s i n e sweep I l n d 'iieeri u s e d , then m a n y m c x e c,yc.l t:z cli. persistonce w o u l d 'bc.: :i nciic;Zted, which i n .turn w c . u l d lead t,cj : . I . rioridof:syjng aiitoc,orr~:L~L.I,ion f u n c t i o n .

Figucl-: .LI :;h(jws t i i t : response that . r e s u l t s due t o u s e of a n i m p u l s i v e sint.: :i.nput,. Shown at the bottom of t h e f i g u r e is 'the randomdec s1gnat;ur.e f o r h t h a t is obtained from t h e response y (as o u t l i n e d i n f i g . 7 ) . The h obtained i s virt;urtlly a true , r e p l i c a o f ,the exac't h f o r t h e system.

T h i s t ' i g u ~ shows that t h e use CJ'f the impulsive sine function can be a. p(Iworfu1. too:l f ' o r iise i n evaluating system regponse c h a r a c , t e r i st i c s .

?'ne f ollowlng t a b l e mmmarizes the f requenciies and damping values t h a t are . i n d i c a t e d b y the various e v a l u a t i o n techniques f o r the second-oraer s y s i e m that K B S studied.

Lo 9.95 C P S 05 9.92 .051

Randomdec of y , sweep up 9.92 .050

By R y , sweep down 9.92 .051

From h , impulsive s h e 9.96 .048

THEORETICAL FLUTTER MODEL A s an a i d i n t h e s t u d y of various s u b c r i t i c a l f l u t t e r t e s t i n g techniques, a t h e o r e t i c a l f l u t t e r r e f e r e n c e model was developed. T h i s mod21 w a s used t o provide e x a c t answers; the model was a l s o set up on a n analog machine so as t o provide a means f o r s i m u l a t i n g s u b c r i t i c a l t e s t i n g . T h i s s e c t i o n d e s c r i b e s t h e t h e o r e t i c a l model used.

Dif P e r e r i t i a l Equation Formulation of Nons%eady Aerodynamic Forces A n o v e l a,ppronch i s given here f o r approximating the air f o r c e s t h a t d e v e l o p on an a i r f o i l having nonsteady motion. The development a u t o m a t i c a l l y accounts f o r lag i n lift e f f e c t s , but a.voids having t o give e x p l i c i t c o n s i d e r a t i o n t o t h e commonly used F and G f u n c t i o n s t h a t are due t o Theodorsen and G a r r ick f o r a n o n c i l l a t i n g a i r f o i l , A s t u d y of o s c i l l a t i n g a i r f o i l t h e o r y and r e s u l t s f o r two- dimensional incompressible f l o w i n d i c a t e s that t h e b a s i c lift f o r c e s on t h e a i r f o i l may be r e p r e s e n t e d as shown i n the f o l l o w i n g s k e t c h where a refers t u the p o s i t i o r : of the e l a s t i c a x i s of t h e air-

f o i l system, Besides 1 ; ~ arid L 2 , an a d d i t i o n a l force a n d a

moment a s s o c i a t e d w i t h the i n e r t l a of the a i r a c t on t h e a i r - foil; these i m r t i a f o r c e s will be neglected i n t h i s treatment and w i l l be assumed 30 be taken i n t o account simply as a d d i t i o n s t o the mass and r o t a r y i n e r t i a of the a i r f o i l . Expressions f o r L 1 and L2 may be w r i t t e n ' where ha ref'er:; t u .the l i f t that develops at the q u a r t e r chord posit,:ion due ,to an impulsive change in a n g l e of attack a t t h e 3/4-chord p o s i t i o n , and b i s a i r f o i l span.

The growth :in l i f t on an airfoil f o l l o w i n g a s t e p - f u n c t i o n change in ar1g.L:; of a t t a c k i s o f t e n given i n approximation by ari equn.tion of the f o r m

-bit

( lii 1 ',

'I - @(t) = 1

-

Sometimes t w c ; or rn(JL'e exponeritial ,terms arc: incl.uded, b u t f ' o ~ prest.:rLt ~ u K . I ) o ; ; ~ s A s.j.r;kle term i s considered a d e q u a t e . A goo i approx.irnatiwl t'or a wing of f i n i t e a s p e c t ratio is, i n f ' a z t ,

(set? ref' . 11)

L - ' P ( t ) = 1 - *&-'3S

L'Vt

where s .= - c *

The der-lvatlve of equation (41) y i e l d s ha t h u s

-bit

h a. (1 - al)b(0) + alble (42)

- - , ,. I . .

Tf equation 43 i s m u l t i p l i e d by b l , and the r e s u l t I s added

t o equation [bJ+], the following simple result, void of any i n t e g r a l s , i s found T h e development 01' t h e e q u a t i o n s f o r flutter ( o r any arbitrary motion) can now proceed on t h e basis of this equation, rather than through n formulation whlch involve8 the F and.' G f u n c t i o n s ; nonsteady aerodynamic e f f e c t s w i l l a u t o m a t i c a l l y be taken i n t o account. Note, a similar developmnt can be made w i t h two exponential terms i n e q u a t i o n (41) ; equation (45) would t h e n appear as a second-order d i f f e r e n t i a l equation for L1 .

Equations f o r F l u t t e t * Model Coii:;ider. f h e a e r o e l a s t i c system d e p i c t e d by the f o l l o w i f i g s k e t c h e s ?,>,/'!

n

v

Note: e , eo

e f p o s i t l v e as shown.

2 7

2 3 -1 W 1 -WP'

e=,,, F r f

U 0 where m T 2 p , = - Y ml = c pb m c u c

Y =+

r , $ - - a

r: c and where S . w i n g area - c b The response qu,Antitii.s o f i n t e r e s t klcreirl are f o r d i s - placement and accoL<-:ration at L l t t ? p i c k u p Location The st eo , q u a n t i t i e s are d e f ' i r i ~ - t i a : : t -!

z w + ;r- ll, (47) From equations (46) tihe zciliAt,ion fc,r z i s Found to take tkte form which is the 'basis f0.r eqiml,iclri ( " 3 ) presented earlier (equation (13) is %lie r e s u l t uf' norma,lizin@: by t h e c o e f f i c i e n t 95) Functions (1lC) and f'reqiiency response s o l u t i o n s (15) ant1 (16j are t h u s appiiclable tc, equation (49).

If the ri ht-hand s i d e i s set equal t o zero, a n d

z i s assumed to be e ght , t; 1 ;'ii

following chnracteris5l.c equation of the system is found A

+ a,. x + a

L k.

- .

" 35 ( 1 = .1 I n a l l ' b u t m e case, the imcoup:L?d t r l r s i o n frequency

q, was

taken as 1 0 c p s , the uncoupled 'bending frequency 9 w a s t a k e n

8 s 2 c.ps; 1.n t h e lone case t;o = w The v a r i a b l e parameters Y @ " were as follows 1 2v o = - z . - ; (ar .Is a reference freq,) kr r- C J

ro = -

C

System roots. - A I:ommorl way 'to fiva.Lus,te f'l.utter' speed arid

frequency i s t o scdve f o r t h e roots cu ot' as a f u n c t i o n of a i r s p e e d V ; t h e r o o t s of A 1 a n d A2 a,re r e f e r r e d t o hers as quasi-f~'lutter r o o t s , sinc.e they are f i c t i t i o u s values f o r a l l ;:peecis t ? x c e p t t h e f l i i t t e r speed. A f l u t t e r . c o n d i t i o n i s d e f ' i n e d w h e n the r o o t s o.f AI and A,) are e q u a l . Figure 1 : ' sliow:: . i l l u s t r a t f , v e hehztwi o r patterns f ~ w F i g u r e 13 i s f o r the Lone r:a,sc? whe.t-2 W;I =: UL ; this YJ s i t u a t i o n Leads t o ti very low i'1utIx.r speed, as has o f t e n been observed.

I l l u s t r a t i o n s of' t h e behavior p a t t e r n s of the tr?le root.:l, as obtained from the c h a r a c t e r i s t i c equation, are shown In f i g u r e 14. These r o o t s are of p h y s i c a l s i g n i f i c a n c e s i n c e t h e y i n d i c a t e t h e damping and frequency of t h e v a r i o u s modes t h a t a r e p r e s e n t i n any response e x c i t a t i o n . F l u t t e r occurs when one of t h e damping values (p) becomes zero ( c r o s s e s from p o s i t i v e t o negative damping). F i g u r e s 1 4 ( c ) through ( e ) a r e examples of a slow approach t o f l u t t e r , s i n c e t h e damping d e - grades t o zero i n a slow f a s h i o p as a1r;peed i s i n c r e a s e d . F o r such c a s e s a f l u t t e r speed p r e d i c t i o n can u s u a l l y be made 't)y e x t r a p o l a t i n g t h e damping r e s u i t s . F'.gures 1 4 ( a ) and ( b ) i l l u s t r a t e the behavior f o r an abrup't o r e x p l o s i v e type of f l u t t e r . The damping may appear w e l i behaved, b u t t h e n w i t t i 8.

very small speed i n c r e a s e can sudderly degrade t o a p o s i t i v e v a l u e . These cases a r e very d i f f i c u l t , o r impossible, t o p r e - d i c t i n p r a c t i c e , and are the c a s e s t h a t cause grave concern i n f l i g h t f l u t t e r and wind t u n n e l t e s t i n g , A comparison of f i g u r e s 1 4 ( a ) and (b) is i n t e r e s t i n g . I n one c a s e , the f r e - quencies of two modes c r o s s , while the corresponding damping values diverge; i n the o t h e r c a s e , the damping values c r o s s , while the frequencies approach one a n o t h e r but t h e n d i v e r g e .

For the o t h e r cases shown i n f i g u r e 14, the f r e q u e n c i e s t e n d t o come t o g e t h e r , but no c r o s s i n g i s noted.

F i g u r e 15 shows a comparison of the t r u e a.nd quasi-roots f o r f r e q u e n c i e s , Figure l 5 ( a ) shows t h a t f o r a m i l d approhck.

t o f l u t t e r ( f i g u r e 1 4 ( c ) , the t r u e r o o t s and q u a s i - r o o t s are markedly d i f f e r e n t (a t r u e and q u a s i - r o o t are of course t h e same a t f l u t t e r ) . F i g u r e s l 5 ( b ) and ( c ) , which a p p l y t o an explosive-type f l u t t e r , as s e e n i n f i g u r e s 1 4 ( a ) and ( b ) , s k Q v J t h a t one branch of t h e q u a s i - r o o t s i s c l o s e t o the t r u e r o c t s .

T h i s f a c t , and o t h e r c h a r a c t e r i s t i c s that a r e seen i n f i g u r e s 1 5 ( b ) and ( c ) , as c o n t r a s t e d t o f i g u r e 1 5 ( a ) , suggest 'Itip-offll a t s u b c r i t i c a l speeds as t h a t perhaps there may be a t o whether o r not the f l u t t e r may be e x p l o s i v e . A t l e a s t t , h r e C .

d i s t i n c t p a t t e r n s may be n o t e d .

1) A quasi-root branch f a l l s c l o s e t o the t r u e r o o t s .

The t o p p o r t i o n s of t h e q u a s i - r o o t branches tend t o 2 ) remain parallel over a large speed r a n g e .

3) The q u a s i - r o o t branches tend t o c o a l e s c e n e a r one another ( t h e c o a l e s c e n c e p o i n t s a r e n ' t separated g r e a t l y a l o n g t h e x - a x i s d i r e c t i o n ) .

Whether a l l t h e s e t h r e e c h a r a c t e r i s t i c s must appear s i m u l t a n e - o u s l y , o r whether any one i s s u f f i c i e n t t o i n d i c a t e t h e L i k e l i -

hood of a n explosive f l u t t e r , i s n o t known. F u r t h e r s t u d y t o -

i n v e s t i g a t e t h e s e t e n t a t i v e o b s e r v a t i o n s i s considered d e s i r a b l e .

There

C o e f f i c i e n t s of the governing d i f f e r e n t i a l e q u a t i o n . -

is a n o t h e r promising p o s s i b i l i t y f o r e x t r a p o l a t i n g r e s u l t s f o r - w a r d t o e s t i m a t e the f l u t t e r speed, which should a p p l y whether

I - -

t f - 1 ~ :Lpproacfi ~ ; C J f'1ui;tr:r 1:: m l l d o r of an exp1o:;ive t y p e . 'I'tie!

sckit-mc is based on the r i o t i o n of i d e n t i f y i n g nyctem paramcater :: i n a more complete way than i d e n t i f y i n g damping and frequency values o n l y .

S p e c i f i c a l l y , one concept i s t o i d e n t i f y the co- e f f i c i e n t s an i n t h e governing d i f f e r e n t i a l e q u a t i o n of motion, equation (49). The problem may be stated as f o l l o w s . Suppose t h e response z due t o a given e x c i t a t i o n f o r c e F i s e s t a b l i s h e d ; is it then p o s s i b l e t o use z and F t o e s t i m a t e the C o e f f i c i e n t s

an , t h u s e s t a b l i s h i n g the d i f f e r e n t i a l

e q u a t i o n .

If the c o e f f i c i e n t s are known, then the complete response c h a r a c t e r i s t i c s can be determined.

Involved a l s o i s the n o t i o n that perhaps there, i s a more o r d e r l y v a r i a t i o n of the c o e f f i c i e n t s w i t h air speed than found f o r the damping o r an frequency values.

Thus, t h e concept advanced i s that of i d e n t i f y i n g t h e c o e f f i c i e n t s a n a t s e v e r a l s u b c r i t i c a l speeds assuming a c e r t a i n o r d e r model a p p l i e s , and then e x t r a p o l a t i n g these. c o e f f i c i e n t s t o h i g h e r speeds, and i n t u r n t o use t h e e x t r a p o l a t e d values t o p r e d i c t the f l u t t e r speed. As a way of g a i n i n g some i n s i g h t t o t h i s concept, a s p e c i f i c e v a l u a t i o n of t h e c o e f f i c i e n t s i n equation (49) was made through use of e q u a t i o n s ( 4 6 ) .

A i r s p e e d and e l a s t i c a x i s l o c a t i o n were l e f t as variables, the o t h e r parameters were given the s p e c i f i c values i n d i c a t e d p r e v i o u s l y i n t h i s s e c t i o n . ' The r e s u l t s f o u n d are as follows: &4 4

- a = 5.72 - 30r2 - 8.6r

u r

a 0 : = 10.4 + (.66. - 9.3r)a

a2 2 2

- a = 4.3808 - (.I5 + 3 r ) a

u r &o 2

- = .1 - .012a

a 2v

where u = -

. The c o e f f i c i e n t s a r e noted t o be i n v a r i a n t

C r @r have a simple q u a d r a t i c v a r i a t i o n w i t h r e s p e c t t o the a i r - speed. Figure 16 shows t h e v a r i a t i o n of t h e c o e f f i c i e n t s , e

normalized t o make a5 unity, f o r r = - C = .1 and - C -

e -- .3 .

S i n c e t h e v a r i a t i o n w i t h s p e e d i s o r d e r l y , and s i n c e the t h e o r e t i c a l model i n d i c a t e s t h e type of v a r i a t i o n that each coef f'icient should e x h i b i t ( f l a t o r p a r a b o l i c ) , r e l i a b l e c ---+-n-nl~+~.d.. A CI I a p v-L- v d''n t o higher a i r s p e e d s ought t u be p o s s i b l e . The ' q u e s t i o n is: "How w e l l can the c o e f f i c i e n t s be evaluated Prom measured response data?" The c i r c l e d p o i n t s on f i g u r e 16(a) r e p r e s e n t e s t i m a t i o n s from response data, and w i l l be ?is- cussed f u r t h e r i n a subsequent s e c t i o n .

R e s u l t s of the f'orm presented i n f i g u r e s 1 6 ( a ) and l 6 ( b ) may have a special s i g n i f i c a n c e i n p r e d i c t i n g what type of f l u t t e r may be encountered, t h a t I s , whether f l u t t e r w i l l be of t h e m i l d type o r of t h e e x p l o s i v e type. P a r t (a) a p p l i e s t o a m i l d type f l u t t e r , see curve of f i g u r e 1 4 ( d ) . The t h e c o e f f i c i e n t w t h a i r s p e e d i s seen t o be

v a r i a t i o n of Bf

s l i g h t . By c o n t r a s t , p a r t ( b ) , which a p l i e s t o an explosive

f l u t t e r case (see ,B2 curve i n f i g u r e 1 E ( a ) ) , shows a much

g r e a t e r change of t h e c o e f f i c i e n t s w i t h a i r s p e e d , e s p e c i a l l y t h e a3 and a2 c o e f f i c i e n t s . Marked changes i n t h e c o e f f i c i e n t s t h e r e f o r e appear t o be a c l u e o r a " t i p - o f f " that explosive-type f l u t t e r can be expected.

Frequency response results - Figure 17 p r e s e n t s r e p r e -

s e n t a t i v e r e s u l t s f o r frequency response as obtained from t h e t h e o r e t i c a l model. Some r e s u l t s that were obtained from t h e analog computer a r e a l s o i n d i c a t e d , The f ollowlng t a b l e serves t o show t h e parameters that apply t o each f i g u r e : e 2v e f'

- -

v = - -

F i g .

C C C C

1 7 (4 60 0 1 -.I

Analog resultG

.1 1 1 7 ( b ) 100 e 1 - e 1

also 17 ( c ) 100 .1 .1 .1 100 *1 .I -.3 U ( d ) 1 7 ( 4 100 .3 .1 . 1 The g c n e r a l i n t e n t i s t o show the v a . r i a t i o n i n the freyuericy response f b n c t i o n as brough't ahout ' b y changes i n a i r speed, elastic a x i s p o s i t i o n , and I n the l o c a t i o n s of t h e a p p l i e d e x - c i t a t i o n f o r c e a n d the measuring t r a n s d u c e r .

F i g u r e s 1'7(a) a.rid ( b ) EilS(j i n c l u d e the r e s u l t s that wt:rc.

obta.incid from the a r i d l u g computer set-up of t h e system. The results i n d i c a t e t h a t the ana.log ?;ys;tarn d u p l i c a t e s t h e theo- r e t i c a l l y e x a c t e q u a t i m s oi' motion qu.Lte a c c u r a t e l y .

Figure 19(a) p r e s e n t s impedance r e s u l t s i n the form of

-

( s e e e q u a t i o n s ( 1 6 ) ) versus frequency. The d i p s o r v a l l e y s i n " the curve a r e of s p e c i a l signficance, s i n c e the h o r i z o n t a l p o s i t i o n i n d i c a t e s a mode resonant frequency, while the d i s t a n c e of the d i p from t h e h o r i z o n t a l a x i s i n d i c a t e s t h e damping of t h e mode. The right s i d e of the figure i n d i c a t e s t h e manner of using t h e s e v a l l e y s t o extrapolate t o the f l u t t e r s p e e d . I n t h i s case, an e x t r a p o l a t e d value p r e c i s e l y the same as the e x a c t value i s i n d i c a t e d . This f i g u r e i s considered a s i g n i f i c a n t type p l o t , s i n c e it gives a f a i r l y complete p i c t u r e of the development of a f l u t t e r mode, allows both damping and frequency t o be t r a c k e c r e a d i l y , and leads t o a f a i r l y d i r e c t e x t r a p o l a t i o n t o p r e d i c t f l u t t e r . Figure l g ( a ) a p p l i e s t o a " m i l d " f l u t t e r c a s e . The q u e s t i o n n a t u r a l l y arises as t o whether a similar type e x t r a p o - be v a l i d f o r an "explosive"-type f l u t t e r l a t i o n procedure would s i t u a t i o n . I n f i g u r e l g ( b ) r e s u l t s a r e given f o r an e x p l o s j v e case, s p e c i f i c a l l y , the case covered by f i g u r e lLt(a). The n i a r k e , !

c u r v a t u r e of the curve i s perhaps a c l u e that the f l u t t e r may be of the e x p l o s i v e t y p e . It i s s e e n that the v a r i a t i o n of t h e

1 curve i s not as gradual a s f o r the m i l d f l u t t e r case of

c2 f i g u r e 1 9 ( a ) ; t h e v a r i a t i o n is not n e a r l y as a b r u p t , however, as i s n o t i c e d f o r the /32 damping curve of f i g u r e 1 4 ( a ) . T h u s , t h i s e x t r a p o l a t i o n procedure shows promise of applying t o t h e e x p l o s i v e c a s e s as w e l l as the m i l d f l u t t e r c a s e s .

s i n 'Dot A VANE FORCE CENERATOR O O O s c i l l a t i n g vanes attached t o the wing s t r u c t u r e are used q u i t e o f t e n as a means f o r generating an e x c i t a t i o n f o r c e f o r s u b c r i t i c a l f l u t t e r t e s t i n g . The vanes are u s u a l l y d r i v e n i n a s w e p t s i n e f a s h i o n t o produce a swept s i n e f o r c e . By means of (45), it i s p o s s i b l e t o de- the simple form o f f e r e d by equation r i v e an e q u a t i o n f o r vane motion which allows t h e g e n e r a t i o n of v a r i o u s p r e s c r i b e d e x c i t a t i o n f o r c e s .

(39) and (40) i n a p p l i c a t i o n t o a vane Consider e q u a t i o n s e x e c u t i n g r o t a r y motion only about some a x i s ( s e e s k e t c h p r e - (39)) e The t o t a l l i f t on t h e vane i s s e t e q u a l c e d i n g equation

t o the d e s i r e d vane f o r c e F , t h u s

L = L 1 + L 2 = F From t h i s equation the following r e l a t i o n may be derived

i1 + blLl + i2 + blL2 = 6 + blF

Thrn12gh mPafiQ nf e c p a t i o n s /4n) E?.nrl (451, 2Ed t h e en,r?nt,inr! fcr Y f o l l o w i n g e q u a t i o n (43) (with t h e y motion suppressed), t h i s equation may be w r i t t e n S o l u t i o n of t h i s e q u a t i o n f o r @ f o r a s t i p u l a t e d F y i e l d s the vane r o t a t i o n a l motion that i s necessary t o produce F .

Equation (51) t h u s l e a d s t o a ready means f o r g e n e r a t i n g an impulsive s i n e e x c i t a t i o n . I n t h i s c a s e F i s set e q u a l t o s i n uot ; t h e a s s o c i a t e d s o l u t i o n d e s c r i b e s the vane motion t h a t i s r e q u i r e d 'to produce a vane e x c i t a t i o n f o r c e of s i n m o t

. It i s noted that t h i s development a p p l i e s f'or t h e

c u t case of a f l x e d wing; a p p l i c a t i o n t o the case of a f l e x i b l e o r mova'ble wing should be s a t i s f a c t o r y , however, as l o n g as the f o r c e being a p p l i e d t o the wing is measured.

DEDUCTION OF' SYSTEM RESPONSE C H A R A C T E R I S T I C S T h i s s e c t i o n o u t l i n e s t h e development o f v a r i o u s tecrl- n i q u c s f o r i d e n t i f y i n g the b a s i c response c h a r a c t e r i s t i c s ( i t a s u b c r i t i c a l f l u t t e r s y s t e m . A t t e n t i o n i s f'~x:u:;cd m a i r ~ l . ; ' r

t h e frequency response f u n c t i o n H , a n d the i r n p u l s r reSpC'!i-

furiction h . A primary ob j c c t i v e of f o r c e d excitatii'jn

t e s t i n g i s t o d e r i v e such Functions a c c u r a t e l y so t h a t a ) dampirig arid f r e q u e n c i e s can be i d e n t i f i e d r e l i a b l y , b ) o r that a rniich more d e t a i l e d system i d e n t i f ' i c a t i c i n may b e made, such as t h e r e l i a b l e e v a l u a t i o n of t h e c o e f f i c i e n t s of the governing e q u a t i o n of motion.

One of t h e b i g g e s t problems of f ' l u t t e r t e s t i n g i s t h a t (if' coping w i t h an unknown n o i s e " i n p u t , such as d u e t o atmospkkZ7:i: t u r b u l e n c e i n f l i g h t f l u t t e r t e s t i n g , o r d u e t o tunriel n o i s e i n w i n d t u n n e l t e s t i n g . Since noise e x c i t a t i o n r e p r e s e n t s such a s e r i o u s o b s t a c l e i n deducing a c c u r a t e and r e l i a b l e response c h a r a c t e r i s t i c s , c o n s i d e r a b l e a t t e n t i u n w a s d i r e c t e d towards developing means for. e l i m i n a t i n g o r masking noise e f f e c t s . For this e f f o r t , u s e w a s made of an analog computer i n c o n j u n c t i o n w i t h v a r i o u s input f o r c e g e n e r a t o r s and system response a n a l y z e r s . T h i s analog s i m u l a t i o n proved i n v a l u a b l e , s i n c e marly d i f f e r e n t schemes could be i n v e s t i g a t e d rather q u i c k l y and r e p e a t e d l y .

S u r p r i s i n g l y , a number of d i f f e r e n t ways f'or coping w i t h t h e riolse problem were found, each having d i f f e r e n t merits.

In the d e v e l o p m e n t of t h e procedures, the f o l l o w i n g i n t e r - r e l a t e d q u e s t i o n s were k e p t i n mind (it i s suggested that t h e s e p o i n t s also be kept i n mind as the r e s u l t s are read).

1. What type of' iriput is r e q u i r e d ?

What recor-d dura5ions are r e q u i r e d ?

2.

Can reliable r e s u l t s be obtained from a s i n g l e r e c o r d ?

3.

4. Should r e s u l t s be obtained i n the form of a s i n g l e record of r e l a t i v e l y l o n g d u r a t i o n , o r should t h e a n a l y s i s be based on numerous r e c o r d s of s h o r t d u r a t i o n ?

How much t i m e i s required t o analyze t h e r e s u l t s ?

5.

P o i n t 5 is, of course, of prime concern i n f l i g h t f l u t t e r and Shut-down t i m e , o r ground t i m e , between wind t u n n e l t e s t i n g .

test p o i n t s t o await data analyses i s n o t considered d e s i r a b l e The d e s i r e i s t o make a t e s t , analyze t h e r e s u l t s i n g e n e r a l .

i n a matter of seconds o r minutes s o that t e s t s can proceed almost immediately t o the next p o i n t .

the analog simulation The f o l l o w i n g s e c t i o n describes system that w a s used. I n the subsequent s e c t i o n s r e s d l t s the analog simulation study are described. The o b t a i n e d from i d e a l s i t u a t i o n of no noise i n t h e i n p u t i s described f i r s t e The f o l l o w i n g s e c t i o n s then describe 11 d i f f e r e n t schemes t h a t were developed t o e l i m i n a t e , o r at l e a s t minimize, t h e noise problem; 4 d e a l w i t h d i s c r e t e - f r e q u e n c y t e s t i n g , 5 with time- varying e x c i t a t i o n , and 2 deal w i t h t h e use of response i n f o r - w i s e ( i r o n i c a l l y , n o i s e response can be mation a l o n e due t o used t o e s t a b l i s h n o i s e - f r e e response c h a r a c t e r i s t i c s ) .

Analog Set-Up arid Associated E x c i t a t i o n and Measuring Equipment T h i s s e c t i o n g i v e s a b r i e f d e s c r i p t i o n of the analog s e t - up t h a t was used to s i m u l a t e s u b c r i t i c a l f l u t t e r t e s t i n g of a n a i r p l a n e ; whether the t e s t s be i n f l i g h t o r i n a wind t u n n e l .

F i g u r e 20(a) i s a block diagram i n d i c a t i n g t h e various p i e c e s of equipment used. Four t y p e s of e x c i t a t i o n input were used, a s i n e wave generator, a swept s i n e g e n e r a t o r , a t a p e r e - c o r d e r which c o u l d supply any s p e c i f i e d i n p u t such as the sin cot I n g e n e r a l , t h e n o i s e f u n c t i o n , and a noise g e n e r a t o r .

ot

i n p u t was treatetx as an unknown (nonmeasurable) q u a n t i t y . The a n a l o g system w a s a r e p r e s e n t a t i o n of e q u a t i o n s (46). Low-pass f i l t e r s were used on both the i n p u t and output s i g n a l s t o ensure t h a t t h e s p e c t r a l c o n t e n t of the s i g n a l s d i d n o t go beyond a c e r t a i n frequency ( t o avoid a l i a s i n g ) . The F o u r i e r a n a l y z e r was used as a ready means f o r processing the s i g n a l s . The scope d i s p l a y and s t r i p c h a r t s allowed f o r a "quick look" data a n a l y s i s .

The p r i n t e r and p l o t t e r allowed for the r e c o r d i n g of data.

I Figure 20(b shows the analog schematic that was d e r i v e d from

e q u a t i o n s ( 26 ), and which w a s 1Ised for w i r i n g the a n a l o g

computer.

F i g u r e 21 i s a p i c t u r e of ‘the simulated f l u t t e r t e s t i n g system, While set up at the Langley Research Center of NASA.

equipment Erom a number of d i f f e r e n t sources would be s u i t a b l e f o r use, the following l i s t i n g i n d l c a t e s the s p e c i f i c hardware used.

TR-4-8 Analog/hyhrid computer EA1 PACE Hewlett-Packard 5451 F o u r i e r Analyzer 5Lc66~ Analog t o d i g i t a l c o n v e r t e r 547% Control u n i t 2100A Computer H01-37224 Noise g e n e r a t o r 546011 Display u n i t H 5 1 - 1 8 A Oscilloscope ASR35 Teletype Corp. p r i n t e r 7046A x-y r e c o r d e r 5 3 2 9 Automatic counter.

3403C True r m s voltmeter Sangamo Sabre I11 t a p e r e c o r d e r Tektronix R564B Storage o s c i l l o s c o p e S p e c t r a l Dynamics S D l l 2 VciLtme t t : r 1 og c Oliver t e r SDl27 MZ/TFA coritrol.

SD104A-5 Sweep o s c i l l a t o r SDlOgB Co/Quad a n a l y z e r SD105B Amplitude servo/monitor SD122 Tracking f i l t e r Rockland 1022F Dual Hi/Lo f i l t e r Datagraph 5-510 CEC S t r i p c h a r t r e c o r d e r 1-511 CEC D.C. preamp.

The following s e c t i o n s deal w i t h r e s u l t s o b t a i n e d from the a n a l o g simulation j u s t d e s c r i b e d . Most of the r e s u l t s g i v e n are for the following choice i n parameters

v = - - 2v - 100

C e

- = .1

C e f

- = .1

C Cases which depart from these v a l u e s are s o i n d i c a t e d . Because of the l i m i t e d s t o r a g e c a p a c i t y of the F o u r i e r a n a l y z e r used, most of the runs i n v o l v i n g the use of the F o u r i e r a n a l y z e r were made covering a d u r a t i o n of 5 seconds o n l y . Swept s i n e s t u d i e s were t h e r e f o r e r e s t r i c t e d t o fast sweeps o n l y .

REPRODUCLBIL1,ITY OF THE O)X~GI!P:AL PAGE IS POOR Four techniques are described here for e s t a b l i s h i n g H o r h when the i n p u t i s f r e e of noise.

Dwell e - Figure 22 i l l u s t r a t e s .thz frequency response r e -

sults- were obtained by a frequency d w e l l technique. I n t h i s case, t h e e x c i t a t i o n f o r c e i s s e t a t a c e r t a i n l e v e l and a t s p e c i f i e d f r e q u e n c i e s . The response and i n p u t f o r c e a r e analyzed j o i n t l y by means of a Co-Quad a n a l y z e r t o y i e l d A , t h e component of t h e response in phase w i t h t h e s i n u s o i d a l e x c i t a t i o n f o r c e , a n d B t h e component 90 degrees out of phase. This i s a good technique if t h e time of d w e l l a t each frequency i s not a l i m i t i n g f a c t o r .

Swept s i n e input w i t h Co-Quad ama.1 z e r . - F i g u r e 23 i s

t y p i c a l of t h e r e s u l t s t h a t HW o t t a i r L e 4k-r y USE: of a swept

s i n c e f o r c e i n p u t and t h e u s e of a Co-Quad a n a l y z e r t o es- t a b l i s h A and B as i n the dwell c a s e . The main troublt w i t h t h i s method i s that r e s u l t s depend on t h e sweep rate, and whether the sweep is up or down. Generally, a sweep up t e n d s t o d i s t o r t peaks $cowards the r i g h t of the c o r r e c t value, while a sweep down d i s t u y t s peaks t o the l e f t . Damping i n d i - c a t e d i s h i g h e r t h a n actiial. To avoid d i s t o r t i o n , a very slow sweep rate must be u s e d , and i n such c a s e s t h e d w e l l technique might j u s t as w e l l be i 1 s ~ ~ d .

F o u r i e r t ransf c j r m afJproach usilig a swept d i n e i n p u t . - 'The

basis f o r t h i s a p p r o a c r - i s equation (11). A swept s i n e i n p u t , which covers the frequency range of .irJT;erest, i s used f o r e x c i t a t i o n purposes. The frequency response f'imction i s then e v a l u a t e d from the F o u r i e r transforms of F and y according t o the e q u a t i o n .

(52) -

R e s u l t s g e n e r a l i y do n o t depend on the r a t e of sweep.

F i g u r e 24 illust,ates some analog r e s u l t s that were dgduced

by t h i s procedure. 8.Fh?. t o p ske'bch a p p l i e s to H2 , and h , as

m i g h t be obtained through a c c e l e r a t i o n measiirements The second

s k e t c h from t h e top a p p l i e s t o Li and h , as would be obtained

from s t r a i n gage measu.remc!nt,s. The s k e t c h n e a r t h e r c e n t e r i l l u s t r a t e s a novel way t o e v a l u a t e the f u n c t i o n

co2 . If the

f u n c t i o n s €I2 and €1 re pe.r:fectly formed, then t h e ratio L -,

- H2

should e v a l u a t e t o u'' , see equaticms ( 9 ) . The s k e t c h H shows t h e r a t h o as obtained i'rom ,the analog r e s u l t s ; t h e near- n e s s t o an uL v a r i a t i o n is n o t b a d , c o n s i d e r i n g t h a t no a t t e m p t w a s made t o establish € l L ~arid H as a c c u r a t e l y as possible. 'Ez,c?th H2 and H x ~ * e 01' fnteres.f; i n p r a c t i c e ; t h e lower frequency modes t ? n a t o be emphasl.zed by t h e H f u n c t i o n , while the higher f'requ.E:ni:y modes a r e emphasized. by H2 . The bottom s k e t c h o r l tt,c f ' i g u r e applies to H1 and h .

Impulsive :;iiIe . - _-I_ ex; ita.th1orl.- Figiwe 25 shows the natiire Of t h e response that; WIS o b t m l'rcm an impulsive s i n e e x c i t a t i o n .

f3;nction that was obtained f o r a Figure 26 shows tric ]rig s l i g h t l y impure impulsive s i n e f o r c e i n p u t ; also shown i s t h e The rather sharp c u t - a s s o c i a t e d frequentcy r'esponse functlor,.

o f f of t h e f u n c t i o n d u e t o the use of an impulsive sine i n p u t i s noted.

Peak s h i f t i n g . - I n f i g u r e 30, F denotes a d i s c r e t e f r e - quency input f o r c e ; y r e p r e s e n t s the measured response. If trie s i n u s o i d a l curves were not p r e s e n t on the f i g u r e , y w o u l i seem t o be a response t o noise only; it c o n t a i n s , however, a d e f i n i t e s i n u s o i d a l component. The concept of using peak s h i f t i n g o r peak enhancement can be stated w i t h r e f e r e n c e t o

F . Consider first the t r a c e as given, next consider a l i k e F

f u n c t i o n and s h i f t it so as t o a l i g n peak b w i t h a , c o n s i d e r

another l i k e F ' f u n c t i o n and s h i f t t o make peak c a l i g n w i t h

a , and s o on; then add a l l the r e s u l t s . The r e s u l t i s t h e

t r a c e labeled C 'F . Note, s h i f t i n g on the F o u r i e r a n a l y z e r

system i s such t o cause t h e information that i s s h i f t e d off t h e l e f t side t o s p i l l around i n b e l t f a s h i o n and appear on t h e right side. Summations i n t h e o v e r l a p r e g i o n a r e t h u s n o t v a l i d ; because of t h i s overlap problem, t h e ends of the I : t r a c e have been cleared. Next, do the same o p e r a t i o n s w i t h t h e y f u n c t i o n , u s i n g p r e c i s e l y the same shifts as f o r t h e F f'unction. The

r e s u l t i s C y . Frequency response Information can then be

e v a l u a t e d from the C F and C y f u n c t i o n s .

The concept i n t h i s technique i s t h a t t h e s h i f t i n g and adding o p e r a t i o n s causes the meaningful o r i n t e l l i g e n t p a r t of t h e r e c o r d t o be enhanced, a m p l i f i e d , o r r e i n f o r c e d , while t k L c n o i s e l e v e l remains t h e same. Note, t h e a d d i t i o n of a number c.)f u n c o r r e l a t e d n o i s e r e c o r d s should give a r e s u l t which i s similar t o any one r e c o r d . Thus, i f an average value of t r l e summation i s formed, then t h e peak l e v e l should be t h e same as the o r i g i n a l peak l e v e l , b u t the n o i s e c o n t e n t should be d e - c r e a s i n g as l , where n i s t h e number of samples involved.

The e s t a b l i s h m e n t of a f a i r l y c l e a n s i n e wave, as i n d i c a t e d

by t h e C . y , from the r a t h e r n o i s y f u n c t i o n y i n d i c a t e s tlie

i m e f u l n e s s of t h i s peak s h i f t i n g technique. The r e s u l t i s f o r only 1 2 a d d i t i o n s ; a l a r g e r number would lead t o an improved quaLity f o r the C . y .

Ensemble averagin4.- This technique i s based on t h e concept of adding t o g e t h e r a number of independently generated r e c v r d s , w i t h t h e a d d i t i o n s being made s o that the i n p u t r e c o r d s star+ i n the same way (phase maintained). A s i n t h e peak s h i f t i n g technique, the ensemble sum of t h e output should show a de- c r e a s i n g noise c o n t e n t as t h e number i n t h e ensemble i s i n - creased. Figure 31 i l l u s t r a t e s the procedure, The t o p s k e t c h d e n o t e s a s i n g l e i n p u t record, t h e second s k e t c h t h e measured response. The r e c o r d s at t h e bottom shows t h e ensemble sum f o r F and y as obtained from an ensemble s i z e of 30. Note how t h e n o i s e i n the output has been reduced. Frequency response i n f o r m a t i o n f o l l o w s from the C F and C y r e c o r d s .

Time-Varying E x c i t a t i o n w i t h Noise i n the Input The u s e of a swept s i n e i n p u t and t h e F o u r i e r transform r e l a t i o n given by equation (52) l e a d s t o results of t h e type shown i n f i g u r e 32 when a n unknown n o i s e i s p r e s e n t i n t h e i n p u t . T h i s f i g u r e should be compared w i t h t h e no-noise analoa resiilt, f i g u r e s 22 and 26, and w i t h t h e corresponding exact r e - s u l t , figure l 7 ( c ) . T h i s figure v i v i d l y i l l u s t r a t e s the prublem brought about by i n p u t n o i s e .

With r e s u l t s of the type shown, it i s v i r t u a l l y hopeless t o deduce meaningful response c h a r a c t e r - i s t i c s .

Techniques f o r o b v i a t i n g the n o i s e problem a r e t h e r e - f o r e of g r e a t i n t e r e s t . .

C l e a r i n g o r weighting of t h e h function.- It i s t o be noted t h a t throughout t h i s r e p o r t , d i s c u s s i o n s of H o r h a r e e s s e n t i a l l y synonymous, s i n c e , as e q u a t i o n s ( 5 ) and ( 6 ) show, knowledge of one f u n c t i o n a u t o m a t i c a l l y d e f i n e s t h e o t h e r . The

use of equation (52) l e a d s f i r s t t o H , but h t h e n follows

d i r e c t l y . C e r t a i n featurF.s i n f i g u r e 32 are worth n o t i n g . T h e p o s i t i o n l a b e l e d a on. h appears t o be the p o i n t where t h e c o r r e c t or n o i s e - f r e e h f u n c t i o n would have decayed t o near zero. Beyond t h i s p o i n t the information shown i s mostly due t o noise. I n turn, most of the jaggedness i n the A 2 and B2 f u n c t i o n s i s due t o the e r r a t i c behavior of h beyond a p o i n t

such as a . The simple technique of c l e a r i n g the h f u n c t i o n

beyond the point a i s t h u s suggested as a n easy means f o r v a s t l y improving the n o i s e problem, r e f e r e n c e 13. A r e c t a n g u l a r t r u n c a t i o n i s i m p l i e d , having u n i t y out t o a s e l e c t e d time, arid z e r o t h e r e a f t e r . Figure 33 i l l u s t r a t e s r e s u l t s of t h i s type The r e s u l t s on top i s a n o t h e r example of t h e c l e a r i n g p r o c e s s .

type of r e s u l t s shown i n f i g u r e 32, and a r e f o r the raw data.

C l e a r i n g the h f u n c t i o n beyond a p o i n t corresponding t o a c j i i f i g u r e 32, and t h e n r e d e r i v i n g H l e a d s t o t h e r e s u l t s shown i n t h e middle. A g r e a t improvement i s noted; n o i s e e f f e c t s a r e still p r e s e n t , b u t a t l e a s t some i n d i c a t i o n s of f r e q u e n c i e s and damping are p r e s e n t . The bottom s k e t c h a p p l i e s t o t h e no-noise case and i s i n c l u d e d f o r comparative purposes.

References 3 and 13 i n d i c a t e t h e use of a n e x p o n e n t i a l weighting f u n c t i o n on t h e r a w h f u n c t i o n , and then r e -

e v a l u a t i n g H , as a means f o r minimizing n o i s e . F i g u r e 34

shows r e s u l t s obtained by t h i s approach. T h i s technique appears t o be q u i t e e f f e c t i v e i n e l i m i n a t i n g n o i s e e f f e c t s . The i n t r o - duction of a weighting f u n c t i o n of course causes d i s t o r t i o n s i r i t h e derived H f u n c t i o n . C o r r e c t i o n s t h a t account f o r t h e weighting f u n c t i o n must subsequently be made t o t h e deduced damping values. F o r t u n a t e l y , t h e s e c o r r e c t i o n s a r e e a s y t o make when a n exponential f u n c t i o n i s used, s i n c e t h e correct-icm i s simply t o s u b t r a c t out the apparent damping t h a t has been added by the weighting f u n c t i o n . Other weighting f u n c t i o n s are not recommended, s i n c e the c o r r e c t i o n s a r e n o t known or cannot be made.

Cross-spectrum between F and y .- The t h e o r y f o r t h i s

technique i s as follows. Th e response due t o an a p p l i e d ex- the n o i s e environment may be w r i t t e n as c i t a t i o n and due t o

y = yF + y, =s (F + F n ) h ( t - z)d.r

(53) where Fn i s t h e unknown input n o i s e and Yn i s the a s s o c i a t e d noise contamination i n the response. The F o u r i e r transform of y i s F = F + F = ( F F + F F ) H Y YF Y n n If t h i s equation i s m u l t i p l i e d through by the complex conjugate

, t h e n the following spectrum equation is i n d i c a t e d

%

Because t h e r e i s no c o r r e l a t i o n between F and Fn , however, both the Cross s p e c t r a @FY, and @FF, should vanish. The e q u a t i o n t h e n y i e l d s @ H = - FYF (55) @F which appears as a completely n o i s e - f r e e r e s u l t . The technique i s t h u s t o f o r m the cross-spectrum @ between the a p p l i e d

i n p u t and t h e measured response, and % d i v i d e by t h e i n p u t

spectrum @F t o o b t a i n H .

Typical r e s u l t s are shown i n f i g u r e 35 f o r the c a s e of a swept s i n e e x c i t a t i o n . A s u b s t a n t i a l improvement i s noted.

The i l l u s t r a t i o n i s not a fair t e s t of the approach, however, because of t h e very s h o r t record l e n g t h s t h a t had t o be used.

The results shown i n f i g u r e 35 r e p r e s e n t only 2.5 seconds of data, because c o r r e l a t i o n w a s involved. I n s p i t e of t h i s l i m i t a t i o n , a marked improvement i n the n o i s e problem is noted.

It i s f e l t t h a t r e c o r d l e n g t h s of about 30 seconds (a moderate sweep) a r e needed f o r t h i s c o r r e l a t i o n technique, and that i f such l e n g t h s were involved, then almost p e r f e c t r e s u l t s would be o b t a i n e d . The a u t h o r considers t h i s t o be one of the best t e c h n i q u e s a v a i l a b l e f o r e l i m i n a t i n g noise e f f e c t s .

Peak s h i f t i n $ . - A peak s h i f t i n g technique similar t o t h a t d e s c r i b e d under d i s c r e t e frequency t e s t i n g w i t h n o i s e i n t h e i n p u t i s a l s o p o s s i b l e f o r a swept s i n e i n p u t e x c i t a t i o n . I n t h i s case, s h i f t s are based on the peaks of the swept s i n e i n - p u t f u n c t i o n ; t h e s h i f t s for the response a r e taken i d e n t i c a l t o the s h i f t s of the i n p u t . The sum of the i n p u t s h i f t s i s t r e a t e d then as a s i n g l e input f u n c t i o n , and the sum of the s h i f t e d output f u n c t i o n s is t r e a t e d as a corresponding s i n g l e r e s p o n s e f u n c t i o n . Note, t h e concept t h a t s o l u t i o n s f o r l i n e a r systems may be l i n e a r l y added i s involved. The summed r e s u l t s are t r e a t e d by equation (52) as though t h e y r e p r e s e n t e d a s i n g l e r e s p o n s e run. R e s u l t s obtained a r e shown i n f i g u r e 36. Rela- t i v e t o figure 32, a v a s t improvement i s found. Here a g a i n though, as w i t h t h e c o r r e l a t i o n example, t h e t e s t of the tech- nique is not f a i r . The short r e c o r d 1ens;th a v a i l a b l e allowed o n l y a small number of peaks t o be summed. Thus, enhancement "average of the meaningful s i g n a l p a r t w a s n o t s u f f i c i e n t t o o u t " the n o i s e . Record l e n g t h s of around 30 seconds (moderate sweeps) should allow c o n s t r u c t i o n w i t h many peak s h i f t s and should make t h i s a powerful tec.hniqiie.

Ensemble averaging.- The previous techniques are based on the u s e of a s i n g l e r e c o r d only., By c o n t r a s t , an ensemble a.veraging technique involves the use of many sweep runs, The idea i s s i m p l y t o e v a l u a t e H or h f o r each of t h e runs and then t o add the r e s u l t s t o form an ensemble average. The con- c e p t i s t h a t noise e f f e c t s w i l l "average o u t " t o zero. R e s u l t s f o r an ensemble of 20 sweep runs, each of 5 seconds d u r a t i o n , a r e shown i n f i g u r e 37. The f i g u r e gives t h e r e s u l t s f o r both ~ 2v I; and l i , and f o r values of v = - = 60, 80, 100, and 110 .

C 'The value of v f o r f l u t t e r i s 117 T h i s technique I s seen t o be very e f f e c t i v e in l e a d i n g t o reduced n o i s e e f f e c t s ; t h e p e n a l t y i s t h e problem of having t o make many r e p e a t e d r u n s .

If 20 t o 30 runs, each of 5 seconds d u r a t i o n a r e needed, t h e n a t o t a l t e s t time of 100 t o l5O seconds i s implied f o r each t e s t point (one s p e e d ) , n o t counting r e s e t times between r u n s . A t o t a l t i m e of 1 5 0 seconds approaches but i s s t i l l smaller t h a n the sweep d u r a t i o n r u n s of 4 minutes that are o f t e n used i n f l i g h t t e s t s . The question i s r a i s e d : "Would a s i n g l e run of only about 50 seconds, analyzed by t h e c o r r e - l a t i o n or peak s h i f t i n g technique, be b e t t e r ? " Unfortunately, t h i s q u e s t i o n can not be answered a t t h e moment.

Figure Y((a) also includes t h e r & r e s u l t , for I HzI , w h i c : l i

also r e p r e s e n t s t h e spectrum of h . T h i s f'unction i s ' s e e n

t o be q u i t e clean, and of all the f u n c t i o n s shown, allows f o r the e a s i e s t e v a l u a t i o n of system damping and frequency. The frequency i s i n d i c a t e d by the l o c a t i o n of' the peak, the damping by t h e w i d t h a t half-peak h e i g h t , see f i g u r e 3.

Combined ensemble averaging and h w e i g h t i n g . - Some 01' t h e methods described here can of course 'be used i n CombinatiGri.

Figure 38 shows the r e s u l t s obtained b y an ensemble averaging of only 5 r u n s , w i t h the subsequent use of t h e e x p o n e n t i a l weighting f u n c t i o n technique. The r e s u l t i n g curves a r e q u i t e smooth, b u t c o r r e c t i o n of the data must of course be kept i n m i r i d ; t h e r e s u l t s shown s h o u l d be compared w i t h the e x a c t r e - sults shown i n f i g u r e l 7 ( c ) .

Ensemble averaging u s i n g response t o noise o n l y . - The

treatment following e q u a t i o n (1 a u t o - c o r r e l a t i o n f u n c t i o n of t h e response t o white n o i s e i s t h e same as t h e a u t o c o r r e l a t i o n f u n c t i o n of t h e impulse f u n c t i o n

h . T h i s f a c t suggests t h a t a u s e f u l r e s u l t might be d e r i v a b l e

by working w i t h noise response r e c o r d s o n l y . A u t o c o r r e l a t i o n f u n c t i o n s of the response t o noise a l o n e were established f o r a number of i n d i v i d u a l r u n s . The r e s u l t of adding t o g e t h e r 20 such functions i s shown i n f i g u r e 39. The agreement w i t h the r e s u l t shown i n f i g u r e 37 i s remarkable. Damping and frequency a p p e a r t o be r e a d i l y i d e n t i f i a b l e . The F o u r i e r t r a n s f o r m of t h e a u t o c o r r e l a t i o n f u n c t i m is shown as the second f u n c t i o n from the t o p . The smoothness [Jf t h i s f u n c t i o n i n d i c a t e s that the r e s u l t s a r e e s s e r i t i a l l y nojse-free response r e s u l t s f o r t h e system. The F o u r i e r t r a n s f w m of' Lhe r i g h t half of' Ry i s o f t e n of i n t e r e s t . T h i s result i s shown a t the bottom of the f i g u r e .

Use of the randomdec techrilque . - The randomdec technique

i s another means f o r d e r i v i n g system response c h a r a c t e r i s t i c s from n o i s e response information o n l y . The technique described i n f i g u r e 7 was a p p l i e d t o ,the noise response of t h e f l u t t e r system. R e s u l t s a r e shown i n f i g u r e 40; t h e s e r e s u l t s are t o be compared w i t h the h results shown i n f i g u r e 24.

The randomdec technique may be a p p l i e d t o a c c e l e r a t i o n n o i s e response r e s u l t s , but the c o n s t r u c t i o n does n o t l e a d t o a system p h y s i c a l f u n c t i o n . The reasons i s t h a t t h e Dirac f u n c t i o n u s u a l l y found w i t h h f u n c t i o n s , see f i g u r e 5, &re not accounted f o r p r o p e r l y . The randomdec s i g n a t u r e f o u n d from a c c e l e r a t i o n should give, however, an i n d i c a t i o n of system frequency and damping.

SYSTEM PARAMETER IDENTIFICATION . Most 0 1 ' t h e keckirii.ques Uued f u r E ! V a l l i a t : i i i g i'requency a~icl damping of t h e various modes of' ir, multimode system are based on the behavior of a s i n g l e degree of freedom system. When the modal f r e q u e n c i e s are w e l l separated, reasonably good estimates of f r e q u e n c i e s and damping probably r e s u l t , but even i n such c a s e s , the values deduce.d a r e r e a l l y only "pseudo" e s t i m a t e s of the t r u e values. When frequencies are c l m e t o g e t h e r , i d e n t i - f i c a t i o n becomes' u n c e r t a i n or impossible, o r e s t i m a t e s may be i n large e r r o r . The i d e n t i f i c a t i o n ( J f the parameters of a

system i n g r e a t e r d e t a i l i.s t h e r e f o r e desl.rable . The- establish-

ment of t h e c o e f f i c i e n t s of the: governing d i . f f e r e n t i a 1 e q u a t i o n of motion (see e q u a t i o n (:~j)) from response measurements i s , f o r example, an extended s t e p 'to b e t t e r sys,Lem i d e n t i f i c a t i o n .

With these c o e f f i c i e n t s , a l l response c h a r a c t e r i s t i c s of t h e system can be e v a l u a t e d , whether f'reqiiencles are c l o s e t o g e t h e r o r n o t . The accuracy of' the determination of t h e c o e f f i c i e n t s i s of course a f a c t o r .

T h i s s e c t i o n d e s c r i b e s three means fur e v a l u a t i n g system d e t a i l from response measurements.

parameters i n g r e a t e r References 1 4 through 17 r e p r e s e n t noteworthy t r e a t m e n t s of the s u b j e c t .

C o l l o c a t i o n us.irig the t'requt:ncr:y response f u n c t i o n . - Assume

t h a t t h e system under ctorisidera.t:ion i s a 5th-order system, s o t h a t equation (1 3) applie:;. The frequency response e q u a t i o n f o r displacement response i n d i c a t e d by t h i s equaticm i s - .

w a4 - LU 2 a 2 + a. + i

3 + , , ) ] ( A + i B ) =

which when expanded l e a d s t o 4 2 = U B ( 5 7 ) w A a 4 - a2Aa2 + A a o + u%a 3 - &al + w b2 - b, 4 2 3 (58)

w B a 4 - w Ba2 + Bao - w A a -t dial -t a% - cobl = -w A

3 3

The m u l t i p l i c a t i o n hrou h of equation (56) by A - i B and

d i v i s i o n by C 2 = A ’ + B5 l e a d s t o the following two

forms a l t e r n a t i v e

w a 4 4 - w a 2 + a o 2 + u 3 $ b g + a 2 A B A b = O

The c o l l o c a t i o n s o l u t i o n proceeds by u s i n g t h e s e equati.ons s i n g l y o r j o i n t l y t o s o l v e f o r the a? and bn c o e f f i c i e n t s .

Consider equation (57) f o r example; f i v e an and two bn c o e f - f i c i e n t s appear in t h i s e q u a t i o n . Measured v a l u e s of A and B are s u b s t i t u t e d i n t h i s e q u a t i o n a t seven d i f f e r e n t values of

, l e a d i n g t o seven l i n e a r simultaneous e q u a t i o n s w i t h unknowns

an and bn . S o l u t i o n i s t h e n made f o r t h e s e c o e f f i c i e n t s .

(57) and f o u r O r , f i v e frequency values may be used i n e q u a t i o n i n e q u a t i o n (58), g i v i n g nine simultaneous e q u a t i o n s in terms of t h e t o t a l of nine unknown c o e f f i c i e n t s .

and A t e s t of t h e approach w a s made by u s i n g v a l u e s of A B as obtained from the e x a c t s o l u t i o n , e q u a t i o n (49). The i n good agreement w i t h the o r i g i n a l c o e f f i c i e n t s evaluated were c o e f f i c i e n t used t o o b t a i n the A and B v a l u e s ; t h i s compari- son i s shown i n f i g u r e 16, where the c i r c l e d p o i n t s r e f e r t o t h e c o e f f i c i e n t s as e v a l u a t e d by the c o l l o c a t i o n p r o c e d u r e , T h i s comparison i n d i c a t e s t h a t the scheme works, a t l e a s t i n p r i n c i p l e .

Analog values of A and B were a l s o used t o check the procedure. Some of t h e r e s u l t s obtained were good, some were F i g u r e 41 bad , depending on the f reqirency l o c a t i o n s chosen.

i n d i c a t e s a f e w of the r e s u l t s o b t a i n e d , and shows the q u a s i - r o o t s t h a t were obtained from the deduced c o e f f i c i e n t s i n comparison t o t h e e x a c t q u a s i - r o o t s . The r e s u l t s i n d i c a t e t h a t perhaps the b e s t procedure t o use i s t o e v a l u a t e the c o e f f i c i e n t s s e v e r a l times f o r d i f f e r e n t frequency choice6 ( t h e e v a l u a t i o n is very quick s i n c e o n l y a few simultarieous l i n e a r equations a r e involved) and then t o average the r e s u l t s .

To check f u r t h e r on t h e c o l l o c a t i o n procedure, a s e n s i - t i v i t y study was made t o establish how s e n s i t i v e the c o e f - f i c i e n t s an and bn were t o assumed changes i n A and B .

Figure 42 i n d i c a t e s the r e s u l t s of' t h i s study. The study

s t a r t e d w i t h the e x a c t values af A and B . These values

were given random v a r i a t i o n s through use of a random number g e n e r a t o r . With the v a r i e d values, s o l u t i o n was made f o r t h e an and bn c o e f f i c i e n t s . This,experiment was repeated 100 times. The UA value in f i g u r e 42 r e p r e s e n t s t h e standard d e v i a t i o n of a l l the v a r i a t i o n s of the A and B v a l u e s ; the u a value, the standard d e v i a t i o n of a l l t h e v a r i a t i o n s found f o r t h e an and b values. This f i g u r e shows t h a t the c o e f f i c i e n t s a r e q u f t e s e n s i t i v e t o the A and B values used (roughly, a magnification in e r r o r s of 200).

Least squares d i f f e r e n c e equation approach.- The d i f f e r e n c e equation e q u i v a l e n t of e q u a t i o n (13) may be w r i t t e n as where t h e y t s and F ' s r e p r e s e n t e q u a l l y spaced values w i L h

i n t e r v a l E . The an and bn c o e f f i c i e n t s used here a r e not

t h e same as the c o e f f i c i e n t s in e q u a t i o n ( l 3 ) , but rather are some combination .of t h e s e c o e f f i c i e n t s . Assume the Y n arid Fn v a l u e s are measured values as obtained from a t e s t , and re-- w r i t e e q u a t i o n (61) i n t h e form where En r e p r e s e n t s a p o s s i b l e error. because t h e Y n and Fn values a r e not e x a c t . The c o e f f i c i e n t s an mid bn arc now found u s i n g a least squares process involving t h e error The problem statement appears as 'n 'E =E e: = min.

Minimization y i e l d s k I t h u s l e a d i n g to ri!rle l i n e a r simultaneous equations i n nine uri- knowns. SolutloE y i e l d s t h e d e s i r e d values of a n .

W i t h the s o l u + , i o n Tor tihe a values, the r o o t s of the char: % e r i s t i c equation of t h e diFference equation may be found.

Assume t h e right-hand side of equation (61) i s zero and l e t A t y = e = e ; tkie r e s u l t i s 4 3 5

P + a4P + a3p + a,p + alp .t a = 0

0 C .

where p = ex' .

S o l u t i o n o f t h i s equation y i e l d s t h e r o o t s pn = c -f. i d n n Roots which approximate the r.cjo,ts of t h e c h a r a c t e r i s t i c equation of e q u a t i o n ( 2 6 ) may be now esta'blished as f o l l o w s . Let X = f3 .f i w represent t h e s e approximate r o o t s ; then S o l u t i o n f o r pn and an y i e l d s

.. Application of t h i s l e a s t squares procedure was made t o t h e

h Y R E and R i f u n c t i o n s , as given i n f i g u r e 37. Since t h e s e f u n c t i o n s r e p r e s e n t homogeneous s o l u t i o n of e q u a t i o n (l3), no Fn terms had t o be considered; t h a t i s , e q u a t i o n s (63) were not involved. Note, t h e i n i t i a l values of the f u n c t i o n s were n o t used s o as t o avoid t h e problem of t h e i r Dirac f u n c t i o n type beginning. Results obtained f o r and p f o r t h e value of v = 110 a r e shown i n t h e following t a b l e . h e e s t i m a t e s f o r t h e f i r s t mode show a c o n s i d e r a b l e v a r i a t i o n ; second mode v a l u e s , however, are .in goed confirmatiun. It should be kept i n mind t h a t t h e s e l i m i t e d r e s u l t s dcr n o t r e p r e s e n t a f a i r t e s t of t h i s d i f f e r e n c e equation approach. F i r s t , the s p a c i n g E was r a t h e r l a r g e ; a value of c =- .02 see:. was used, and t h u s one c y c l e of t h e higher frequency mode i s represented by o n l y s i x p o i n t s .

Second, t h e maximum numbcr 0 1 p ( j i r l L : : u s e d w a s o n l y 6 1 1 . And, t h i r d l y , t h e r e 1 s v 9 r y l i t t , l e rc'sporise of the first; mode preserit i n t h e f u n c t i o n s .

c, U I OJ

k i cn

X co w m 0 L n = t c o co m v 3 v 3 M pc I \o *c cu co G M b e- I 0 0.1 t- L n M cn r- I b rl =t cn M v3 0 I I \o M 0 I-4 :c cu M 4 M C J rx M c, 3 CU '0 I I I I I !f \ 1 c- I \ I w o\ M v3 .-I U ?

..

E o a ...

v

y + aby" + a + a,,? 4- a l i $- a,y - kJ. F - 'b 2 l? - bli - boF = ~ ( t ' )

3 l , - 3 where E r e p r e s e n t s an e r r o r due t o the u s e of measured y and F v a l u e s . Assime that the response d a t a have been processed t o

l e a d t o y = hs , see equation ( ~ 7 ) . The f u n c t i o n hs c o r r e -

s i n CI, t

sponds t o t h e response due t o F = , and f a l l s out i n a

u . ' o

s t r a i g h t f o r w a r d n a t u r a l way when p r o c e s s i n g i s made on the basis of t h e F o u r i e r t r a n s f o r m r e l a t i o n , equal;ic,n ( 5 2 ) . Note, t h i s a n a l y s i s , a l t h o u g h presented i n terms of' and hS s i n u) t

, is n o t r e s t r i c t e d ' t c ? these f u n c t i o n s alone. F =

CD ot The c o e f f i c i e r l t s itr! arid b, a r e now e v a l u a t e d through use of a l e a s t squares staLement irivcJ1vi.ng E , t h u s 1 ; ,

bE 1 ' d"F d t

E - ;3b' = d t" I1 t , Equations (70) and (-(l) give : j i m i l l taneoiis l i n e a r e q u a t i o n s i n terms of t h e aII and bn coefficients, which t h e n allows t h e i r e v a l u a t i o n t o be made.

Equations ('(0) a.r~d ( ' [ I may be converted t o a mcjre amenubl-i.

f o r m as follows. W . i t , t i l i m i k n extending f r o m t = - oo t o t - w , t h e i n t e g r a l s w h i c h a p p e a r i n the simultaneous e q u a t i o n s may be written i n t h r e e g e n e r a l forms ( 7 2 ) , I,

- f f l

-Tu Equations (73) thus become simply

= o , m + n odd

3 , m + n even

cmn

, m + n odd

= o Through means of equations (72 through (74), it ie p o s s i b l e t o write equations (70) and (7 ) aa follows -F5 -E 4

'14

' 8 0 -D6 D4 ' 7 " 6

la ' 8

' 6 0 -D4 0 -E6 F5 "4 -F 3

J '2

ID6 0 D4 O -D2 -F5 4 4 "3 E2

D6

0 -D4 O D2 0 E4 -F3 2 F1

P O F 1 -E 0 D4 0 -D, O ' 0 '3 E2 I* 8

F7 -E6 -"5 "4 "3 a6 0 -G4

0 -Q2 *7

' 6 ' 5 -E4 -F3 E2 0 04

"6

' " 5 g4 F3 -E2 -F1 -44 0 O2

0 -02 0 -F5

-E4 -F3 "2 F1 d o

w

= 6 unC2 dw

where Dn Qo = cunA dw En QI) = c u % d w Fn m+n+l

- -

Gn m t n + l 0 S o l u t i o n of these e q u a t i o n s l e a d s t o the d i f f e r e n t i a l equ&t.1.cIt: c o e f f i c i e n t s an and bn . The symmetry of' e q u a t j o n s ('75),

and repeated appearance of t h e Dn , En and Fn c o e f f i c i e n t s

should be noted. I n a l l , only 1 4 such c o e f f i c i e n t s need e v a l u a t i o n (the f o u r Fn values are known One of the a p p e a l i n g merits of t h i s approach i s t h a t k e Dn'S and E r l l s * may be evaluated a u t o m a t i c a l l y by the F o u r i e r analyzer t h a t I s b e i n g used t o analyze the response. S o l u t i o n f o r t h e an v a l u e s should t h e r e f o r e be q u i t e easy and quick.

A minor t e s t of equations (75) was made u s i n g only approx'i- mate values f o r Dn , En and F , ; r e s u l t s f o r an were, however, i n reasonable agreement with t h e corresponding e x a c t v a l u e s . F u r t h e r study of the soundness of t h i s l e a s t squares technique i s worthwhile a CONCLUDING REMARKS Main emphasis i n t h i s r e p o r t has been on t h e developmenh of Improved s u b c r i t i c a l f l u t t e r t e s t i n g techniques, whether I r i f l i g h t o r i n t h e wind t u n n e l , and p a r t i c u l a r l y i n ref'erenct.

the s i t u a t i o n where i n p u t noise i s p r e s e n t . Discuosion was g i v e n t o a c e r t a i n e x t e n t as the m a t e r i a l was p r e s e n t e d , Sc)ain a d d i t i o n a l o b s e r v a t i o n s are made here i n the Form of conclucliriti remarks , Frequency and damping evaluation: A number of d i f f e r e n t schemes were o u t l i n e d f o r deducing frequency and d m p i n g from response measurements a Mode F r o - quency i s i d e n t i f i e d q u i t e w e l l by most a l l of the technlquee.

Damping determination, however, i s more nebuloue , The trrwnsf'clr, l o c u s scheme, which hopefully l e a d s t o i d e n t i f i a b l e c i r c l e s , .I5 pc=-,.;:arly ~ ; = $ e e b ~ t f,be author does not f a v o r t h i s cLpprowh The diameter of a c i r c l e obtained from acceleratii,ii too highly.

- ' -

measurements IS, f o r e x m p l e , (assuming of c,ourRe that g m

I

behavior i s t h a t of a s i n g l e degree of' freedom system).

Thus, e x p l l c l t g e v a l u a t i o n from the diameter i s precluded because F

t h e unknown - i s a l s o involved ( m , i n general., i s some

m g e n e r a l i z e d mass v a l u e ) . A c t u a l l y , i t would be b e t t e r t o m e e q u a t i o n (38) t o e v a l u a t e ( i f the c e n t e r of t h e apparent g c i r c l e can be f i x e d ) , and t h e n t o use the diameter t o e v a l u a t e F

- The methods p r e f e r r e d a r e those shown i n figure 3, and

m ' p a r t i c u l a r l y the scheme i n v o l v i n g use of the w i d t h of the spectrum peak a t half power. T h i s scheme i s simple and d i r e c t , arid s u b j e c t i v e i n t e r p r e t a t i o n i s a minimum. (Note, w i t h g e s t a b l i s h e d b y t h e w i d t h of t h e peak, t h e h e i g h t i n t u r n may be F used t o estimate The scheme of deducing d u p i n g from a .)

randomdec s i g n a t u r e i s a l s o considered good and r e l i a b l e . It, i s t o be n o t e d t h a t a l l of the schemes are subject t G a comm(Jri problem; s p e c i f i c a l l y , a l l the methods f o r deducing f r e q u e n r i c s arid damping a r e open t o q u e s t i o n f o r t h e s i t u a t i o r i where two (or more) frequencies of the system are c l o s e t o g e t h e r .

Means of b e i n g a b l e t o d e t e c t when f r e q u e n c i e s a r e c l o s e t o g e t h e r and, i n t u r n , of deducing the f r e q u e n c i e s and damping are i n need of f iir t, h e r de ve lopment .

A s i n d i c a t e d , a number of methods may be used f o r obviaL1r:g t h e input ncjise problem. The ensemble averaging t e c h n i q w i s a % t r a c t i v e b u t r e q u i r e s a s u b s t a n t i a l number of r e p e t i t i v e runs.

The use of exponential weighting of the h f u n c t i o n appears s a t i s f a c t o r y , but schemes which d o n o t l e a d to d i s t o r t i o n of i.tL.- d a t a , which then r e q u i r e c o r r e c t i o n , a r e judged p r e f e r a b l e . 'Cri.: ( ' r ' ~ ~ s - ( . ' ~ r ~ e l a t i o n approach (equation ('21) and e q u a t i o n ( 5 5 ) ) , and t h e peak s h i f t i n g technique seem t h e most a p p e a l i n g on thL overall, The technique of u s l n g t h e response o n l y , a s o b t a i f l r L d It f ' r u m a white" rioise environment, and forming t h e autocorreLatiorI t'unct.ion, w i t h ensemble averaging, I s q u i t e i n t r i g u i n g and s k i c J 1 ~ I d be used where p o s s i b l e ; i n t h i s c a s e , a f u n c t i o n r e p r e s e n t i n g thf a u t o c o r r e l a t i o n f u n c t i o n of h i s found. I n general, randcmwi sigriatbures may be i n t e r p r e t e d as h only for. displacement r e s p o n s e . Randomdec s i g n a t u r e s obtained f'rov velocAty o r a c c e l e r a t i o n response are not s t r i c t l y the h o r h functicJr1:.

because s t e p f u n c t i o n s o r . p i r a c f u n c t i o n s at the o r i g i n are rlcJL

reproduced (see h and h i n f i g u r e 5 . Dampirig arid frequent i(1.s

as e s t a b l i s h e d from the randomdec s i g n a k u e s f o r v e l o c i t y o r

a c c e l ? r a t i o n response should, however, be r e p r e s e n t a t i v e of' a c t u a l s y s t e m response c h a r a c t e r i s t i c s I System i d e n t i f i c a t i o n : A v a r i e t y of p o s s i b l e sysLem i d e n t i f i c a t i o n schemes wttrc- developed, but a t t e n t i o n was r e s t r i c t e d h e r e i n t o t h r e e q p r ' o a c h e s . The c o l l o c a t i o n scheme is q u i t e simple but suffer.; from t h e f a c t t h a t t h e frequency v a l u e s chosen f o r response REPRODUCIBILITY OF Tdti W G I N A L PAGE IS PG9R matching i s s o a r b i t r a r y . The question of how t o handle t h e i n combination (one a s s o c i a t e d w i t h t h e r e a l two e q u a t i o n s p a r t of the s o l u t i o n , one w i t h the imaginary p a r t ) i s some- what of a mystery. The d i f f e r e n c e equation approach i s con- s i d e r e d good and q u i t e a t t r a c t i v e . An analogous scheme, r e p o r t e d i n r e f e r e n c e s 2 and 13, appears t o be h i g h l y regarded.

The d i f f e r e n t i a l equation approach, which makes use of fre- quency plane information, needs more c o r r o b o r a t i v e study, but i s considered q u i t e promising.

A f i n a l word is given w i t h r e s p e c t t o s u b c r i t i c a l f l u t t e r t e s t i n g techniques which differ from the type discussed h e r e i n .

T a c i t i n t h e schemes mentioned i n t h i s r e p o r t i s t h e assumption t h a t the a i r d e n s i t y i s constant, and t h a t t e s t s proceed on an incremental i n c r e a s e i n air speed basis. Reference 17 d e s c r i b e s procedures f o r t e s t i n g on an i n c r e a s e of air d e n s i t y basis, h o l d i n g speed e s s e n t i a l l y c o n s t a n t . Figure 43, taken from this r e f e r e n c e , shows t h e e x c e l l e n t success t h a t w a s obtained i n e x t r a p o l a t i n g t o f l u t t e r through use of a d e n s i t y i n c r e a s e approach. T h i s technique is s t i l l considered t o be a good and u s e f u l approach and should be kept i n mind i n any f u r t h e r development s t u d i e s of s u b c r i t i c a l f l u t t e r t e s t i n g .

A p p l i c a t i o n t o t h e space s h u t t l e : F l i g h t of t h e space shuttle w i l l r e p r e s e n t a situati.cn of a time-varying system, s i n c e dynamic p r e s s u r e a n d Mach number i n p a r t i c u l a r w i l l change r a p i d l y w i t h time, see f i g u r e 7, of r e f e r e n c e 1. The. q u e s t i o n t h a t n a t u r a l l y arises i s whether t h e methods discussed i n t h i s r e p o r t , which apply mainly t o t i m e i n v a r i a n t systems, can be used f o r s u b c r i t i c a l f l u t t e r e v a l u a t i o n of t h e s h u t t l e system-, With r e s p e c t t o the various methods, the following recommendations a r e made. During t h e t r a n s o n i c region of f l i g h t (where t h e dynamic p r e s s u r e w i l l bc; qmax), the random f'orcl-ng i n p u t b s s o c i - nominally about 75% of ated w i t h t h e t r a n s o n i c flow w i l l probably be s t r o n g enough 'tu e x c i t e a s i z a b l e random response. It follows t h e n t h a t two of t h e methods discussed h e r e i n might be u s e f u l i n e v a l u a t i n g t h e s h u t t l e response c h a r a c t e r i s t i c . One i s t o b r e a k t h e record up i n t o 5-second segments (on the assumption that t h e syscern i s n e a r l y t i m e - i n v a r i a n t during such an i n t e r v a l ) and t h e n t o form the a u t o c o r r e l a t i o n f u n c t i o n of t h e response f o r each see- ment. The second approach i s t o form the randomdec s i g n a t u r e from the random response record.

A t maximum dynamic pressure, the Mach number w i l l nominally be around 1.5. It i s n o t known a t t h i s time whether the f l o w d u r i n g t h i s period w i l l be rough enough t o cause random e x c i t a t i o n . If t e s t a indicate that the f l o w should s t i l l be I t rough," then t h e same two means f o r analyzing t h e aata should be used.

To ensure t h a t system e x c i t a t i o n can be obtained over a l l regions of flight, it i s d e s i r a b l e t o have aerodynamic vanes o r i n e r t i a l shakers i n s t a l l e d . The e x c i t a t i o n recommended is t o go through a s i n e sweep, first up, and t h e n down, i n continuous succession. Sweep d u r a t i o n s of 5 seconds are suggested e Each f i v e seconds of response information could then be analyzed t o deduce H and h through means of the method involving t h e F o u r i e r transform of the output response t o the Fourier transform of the Input f o r c e . I n t h i s c a s e , the procedure i n v o l v i n g t h e peak s h i f t i n g technique would appear i d e a l l y s u i t e d . Note, the time-varying a s p e c t more or less precludes any ensemble averaging .approach. The auto- c o r r e l a t i o n of t h e response due t o the sweeps should a l s o be obtained as an a d d i t i o n a l means f o r e v a l u a t i o n of the response c h a r a c t e r i s t i c s .

I n summary, three approaches appear u s e f u l for s u b c r i t i c a l f l u t t e r t e s t e v a l u a t i o n of t h e space s h u t t l e system, namely The a u t o c o r r e l a t i o n of system response, whether the 1) response i s due t o a n a t u r a l random e x c i t a t i o n o r due 60 a c o n t r o l l e d f o r c e e x c i t a t i o n .

2 ) The randomdec s i g n a t u r e approach.

The u s e of the peak s h i f t i n g approach.

3 ) REFERENCES 1. Rosenbaum, Robert, "Survey of A i r c r a f t S u b c r i t i c a l F l i g h t F l u t t e r T e s t i n g Methods," N A S A CR-132479, May 1974.

2. B a i r d , E.F. and Clark, W.B., "Recent Developments i n F l i g h t F l u t t e r T e s t i n g i n t h e United S t a t e s , " AGARD Report N o . 596.

3. Baldock, J . C . A . and Skingle, C.W., " F l u t t e r Technology i n t h e United Kingdom - A Survey," A I A A Paper No. 73-330.

4. D a t , Rolland, "The T h e o r e t i c a l and Experimental Methods Used i n France f o r F l u t t e r P r e d i c t i o n , " AIAA Paper No.

73-329 5. Kandianis, F., "Frequency Response of S t r u c t u r e s and t h e E f f e c t s of Noise on i t s Estimates from t h e T r a n s i e n t J. Sound Vib., l 5 ( 2 ) , March 1971, pp. 203-215.

Response," 6. Reed, W i l m e r H., 111, '%Effects of a Time-Varying T e s t Environment on the Evaluation of Dynamic S t a b i l i t y with J. of the A e r o h p a c e A p p l i c a t i o n t o F l u t t e r T e s t i n g , " Sciences, Vol. 2 5 , No. 7, J u l y 1958.

7' Kennedy, Charles C . and Pancu, C.D.P., "Use of Vectors i n V i b r a t i o n Measurement and A n a l y s i s , " J. of the Aero- n a u t i c a l Sciences, Vol. 14, No. 11, November 1947.

8. de V r i e s , Gerhard and B e a t r l x , C h r i s t i a n , I 1 General Measuring Processes of Vibratory C h a r a c t e r i s t i c s of L i g h t l y Damped L i n e a r Structures, " Progress i n Aero- n a u t i c a l Sciences, Vol. 9, Pergamvn P r e s s , Oxford and * N e w York, 1968.

9. Cole, Henry A . , J r . , "On-Line F a i l u r e Detection and Damping Measurement (jf Aerospace S t r u c t u r e s by Random Decrement S i g n a t u r e s , NASA CR-2205, March 1973, 10. White, R.G., "Evaluation of t h e Dynamic C h a r a c t e r i s t i c s of S t r u c t u r e s by T r a n s i e n t T e s t i n g , " J.

Sound Vib., l5('2), March 1971, p p . 14'7-161.

11. Houbolt, John C . , "A Recurrence Matrix Solutlon f o r t h e Dynamic Response of' Ai:rcraft i n Gusts," NACA T N 2060, March 1950.

12. R e e d , w. H., 111, H a l l , A.W., and Barker, L.E.J., "Analog

Techni ques for Measuring the Frequency Response of Linear P h y s i c a l Systems Excited by Frequency-Sweep I n p u t s , I' N A S A TN D-508, October 1960.

Newman, K.W., S k i n g l e , C . W . , and Gaukroger, D . R . , "The 1 3 .

Development of Rapid-Tes+,ing Techniques for F l u t t e r Experiments, It Royal A i r c r a f t Establishment, Tech. Report No. 73067, May 1973.

14. Waisanen, P.R. arid Perangelo, H.J., "Real Time F l i g h t 11 F l u t t e r T e s t i n g via Z-Transform Analysis Technique, A I A A Paper No. 72-784.

Gaukroger, D . H . , Skingle, C.W., and Heron, K . H . , Numerical 15.

Analysis of Vector Respunse Loc.I," J. of' Sound a n d Vib., 2 9 ( 3 ) , 1973, P P * 341-353.

16. Heron, K . H . , Gaukroger, D.R., and Skirigle, C . W . , "The D e r i v a t i o n of Equations of M r J t i c m from Respvnse Data and i t s Application i n F l u t t e r T e s t i n g , " Royal A i r c r a f t Establishment, T e c h . R e p o r t N o . '(?OLjl, J u n e 1973.

Taylor, Lawrence W . , Jr. and Ilif'f, Kenneth W., "Systems

a Modified Newton-Raphson Method -

I d e n t i f i c a t i o n Using A FORTRAN Program," NASA TN D-6734, May 1972.

18. Houbolt , Jcihn C . and Hainey, P. C l r a l d , "!In t h e P r e d i c t i o n

of C r i t i c a l F'lutcer Conditions r'rom S u b c r i t i c a l Response Data arid Sorntli Related Wind-Twinel Experience, presented at t h e Flight; Fliitter T e s t i n g Symposium, Wdsh., D . C . , May 15-16, 1958.

FORCE S PE C TR [ IM Dirar.

f unc t i o n a ) Sine wave:

sin y t

c ) W h i t e noise: 'VI, (t) d ) Swept sine: s i n ( a -t b t ) t a 2b e ) Impulse s i n e : sin cu, t A U ' t F i g . 1.- S p e c t r a for various input f o r c e s .

f ) Swept sine (same as d ) g) Swept triangular wave h) Swept square wave Fig. 1.- (Concluded) t: F i g . 2.- I l l u s t r a t i v e transfer l o c i (admittance) as o b t a i n e d from damped mass oscillator.

I C 2 w 1.0 .1 .04 .06 .08 .10 .12 0 .02

B

Bc r F i g . 3.- Frequency and damping as o b t a i n e d from t h e frequency response f u n c t i o n or from the decaying f r e e o s c i l l a t i o n .

57 .

cDO t w ~ I F i g . 4.- Frequency and damping by use of impedance curves.

I Response t o u n i t s t e p Response t o u n i t impulse

-

I rr;; CLI w -2

k w 2 cDO

b B

- - 'Dot

NOTE: h = - 1 e Bcr

s i n CLI t ; ( A ( t ) not t o be confused

H ( u ) = A ( u ) + FB(c0)

w i t h A ( w ) ) F i g . 5.- Impulsive and frequency response c h a r a c t e r i s t i c s of a simple mass o s c i l l a t o r .

F Y R RF Y F i g . 6.- R e s u l t s f o r 2nd-order system due t o swept sine wave (sweep u p ) .

S h i f t y2 t o a and i n v e r t S h i f t y3 t o a S h i f t y 4 t o a and i n v e r t "Randomdec" process t o d e r i v e h f u n c t i o n h b) Randomdec. s i g n a l f o r h obtained from y i n f i g u r e 6 Fig. 7. - Establishment of h by randomdeo% technique.

Y R *F Y F i g . 8.- R e s u l t s for 2nd-order system due t o swept s i n e wave (sweep down).

Y F i g . 9.- Results for 2nd-order system with zero damping due to swept sine wave (sweep up).

L Y R Y F i g . 10.- R e s u l t s f o r 4th-order system w i t h two f r e q u e n c i e s c l o s e t o g e t h e r due t o swept s i n e wave (sweep up).

i F Y h F i g . 11.- R e s u l t s for 2nd-order system due t o impulsive s i n e i n p u t .

(a) e / c = .35

( 2 V l w , c 1

(b) e/c = .2 F i g . 12.- Various behavior p a t t e r n s associated w i t h the roots

of Al and A2 ; Ut - - 59)

( d ) e/c = .05 1 I li R.l I i I I i i i R t I I I I

(2V/w, d2

( e ) e/c = - 0 . 1

Fig. 12.- (Concluded)

Toit sjrstzrii ..A W I L I I I + L . ..--.* V G L J

F i g . 13.- Roots of “1 ariii is2 low f l u t t e r speed; o = CI) .

@ Y .4 .2 L I J 3.0

- P

U ' r -.2 (Real r o o t only)

- .4

- .6

Note:

- .8 X t

e Y a -1.0 t e r 01 I 1 I I 1 m I 1 I I 1 I 1 1 1 1.0 2 . 0 3 . 0 2v - - - - cot kr r e

a) - C = . 3

Fig. 14.- R o o t s of Characteristic E q u a t i o n .

-

.4

-

.2

B

LD 3.0 r

-

.4

-

.6 .8 L I I 1 1 I I 1 I I I 1 I I I 0 1.0 2.0 3.0 1 2v

r = U ] r C r

Fig. 14.- (Cont.)

.4 I 2.0 3.0 w w 2 r I I 1 I I 1 1 I 1 I 1 0 2.0 1.0 3.0 1 2v

lr==

r r e

( c ) c = .2

Fig. 14.- (Cont.)

B

' " r '" ' " r 1 I I I I I I I , I I I I I I 1 1.0 2.0 3.0 e

( d ) c = .1

Fig. 14.- (Cant.)

-

.2 1.0- - u r Flutter

\

9 I

I I

Y

I

I 1 1 I I 1 1 I I 1 1 1 I 1 1 1.0 2 . 0 3.0

I Fig. 14.- (Concluded)

rl .

c II .rl ai0 n cd U d .c;, m d k a c, 0 3 k aj u N n G-4 L m cu Y Y . . . I , 1 ... . . . . . I . . . . . . . . . . . . . ' I " * . ,....

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

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

f+

. , .

. . . : I q : : i : ::;: :::: 1 : : : : : : i ;:: . . . . . . . . . . . . . . . . . . . . . . . . . . .

....... ( . . I . ,... ................

.I,, . * * I , . a t *..I 0 s .

9 . . e . .

;;:.,::;; . . . . . .... ::;: : : i ; : : : i i : : !

. . . . . . . . . . . . . . I . . . . . . . . .

::!I ................................

: . : , . , . . , ....

, , . I * .

; t i : :;:: : : : , I : : ' : : : : !

- ,

... ':l( See Fig. 14. a

.. , . . . . . .

. . . . . .

--.

. . . . . .

.......

. . , . I , , . . I.., . .

. I : : ; : ...-. .

. . .

. . . . . . . . . . .

'.I! : : I . . . . . . . . .

. . . ,.

. . , , ..-- : :I ::::r7- : , . . ..+ -: i l i ; ; : . . : : I . : . : . . . . .

.,.. .

: I : . : ' : : : : :: ......

. . . .

, ' .

. . . .

. I , , . . . , . . . . . , I . .

. ..- ....... .._.-..

. . . .

I , . , .

: ; : : : I : : ; : :: . . . , . . . . . . 1 1 : : : : . . . . .

. . . . . 1 . ; ; : ..

. , .

: I : : : ' : I . .

!..

. . . , . . . .

. ~ ..... -- .. -...- ..... -.

See Fig.

. .

. . I . J : : : I : : . .

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

. . . . . . . . .

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

. . . . . , , I . . .

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

.. .I... ....... I

. I I , . , . . . .

. . . . . . . . . . . . . . . I . . . . . . . . . . . . . . . . . .

. . I . . . . . I . . . .

,... ..... ..'..,. -.. -

e /c =. 3 I5

. I , I : : . .

I . I F i g . 15.- (Concluded) G cu rl h u c a , n d c 3 v k %I rl ,310 c n I1 a3 pc \D 1 1 I I I I I I I I I I ln rl r : d I n P W X = w/w, (a) Fig. 17.- I l l u s t r a t i v e frequency response p l o t s .

!

A

n

.....

.. .

" I rw X ' o/wr ( c ) F i g . 17.- (Cont.)

A A ( d ) Fig. 17.- (Concluded) A

(a) c =0 = 0 . 1

-

C 2 e 0 A

-

(b) - = -

0.3 C C 2 Fig. 18. - ' I l l u s t r a t i v e impedance loci p l o t s .

II Q) d a

s

d d Q) E u . n c d a rl .

Q)

! 2

I1 d n d p-c W c d I1 4J X d r l - k 4J Y w cn d ho d Erc I y = - 2 v C e explosive f l u t t e r (b) - - .3 ; F i g . 19.- (Concluded) Q) m C a m Q)

T

k n n d h W a

*

m

*

a Q) m cu

u q

U c

-

S Q

t

n W I .

cu

F i g . 22. - Ideal frequency response f u n c t i o n (no n o i s e ) obtained

by frequency d w e l l technique.

' 1 1 '

J ; j

' I. 1 I . . . ... I . !.-...I. : . 0 I F i g . 2 3 . - Frequency response f u n c t i o n s obtained by slow sweeps.

I A

- " 2

F F i g . 24.- Frequency response and h f u n c t i o n s o b t a i n e d from FF a c d m h Ca k k I I A 2 8 2 hS t = O F i g . 26.- Frequency response f u n c t i o n and h , as obtained from an impulse s i n e i n p u t .

.

h u c a , 0 ' a , k k i I f = 7.9 CPS ‘C = 1 sec. 5 10 f = 8.44 cps (resonance) T = 1 sec. 5 10 F i g . 28.- Dwell r e s u l t s obtained w i t h n o i s y i n p u t f o r v a r i o u s averaging times i n Co-Quad a n a l y z e r .

Fig. 29.- Noise elimination by c r o s s - c o r r e l a t i o n f o r frequency dwell technique.

F , . .

/ I .

, . . I . . .

, . . , . . .

, . .

. . .

C F Y Fig, 30.- Noise elimination by peak s h i f t i n g f o r frequency dwell technique.

F f = 5 cps . . - . . . . .., .- - . . . . , .. , .

, - . . , , , , .

b

f F

P F i g . 31.- Noise e l i m i n a t i o n by ensemble averaging f o r frequency dwell technique.

. . . . . . . . . . . . . . . . . . . . . . . , . . , --.& . . , . , . . . . . . . . . . , . . . . . . . . ! . . , : . .

I , I . I .

. . . . . . . . . . . . . . . . . . , . . i . ! . . : . . . . ( . . .

1 ' , I .

/ , 8 , , # . I . . . . I : . : , . ! . . : . . . . . A ; * " _ , . I ~.

' : i ' , , , I . . . . . . . . . . . . . . . . . . . . : . . I , . . . . . . . . . . .

. . . . . . . . . . . . . . . ! . . , . . . I . 1 . . . . . . . . . . . . . . . . . . . . . . . . .

i i

F i g . 32.- Frequency response and E f u n c t i o n s obtained by s i n g l e swept s i n e run w i t h n o i s e i n i n p u t .

I m m c z

z

a

I F i g . 34.- Improved frequency response f u n c t i o n by e x p o n e n t i a l weighting of h f u n c t i o n .

. I I I

i i j

Use of c r o s s - c o r r e l a t i o n between input a n d output t o F i g . 35.- e l i m i n a t e n o i s e .

..

h 36.- Use of peaking s h i f t i n g technique f o r swept s i n e F i g .

run t o e l i m i n a t e n o i s e .

w v = 110 ; average of 20 runs a ) F i g . 37.- Use of ensemble averaging of s i n e sweep runs t o e l i m i n a t e n o i s e .

I

b) v = 100 , average of 20 runs

F i g . 37.- (Cont.)

.

..

h v = 80 ; average of 20 runs c ) F i g . 37.' (Cont.)

t I I

h

F i g . 38.- Combination use of ensemble averaging and weighting of h f u n c t i o n t o e l i m i n a t e n o i s e , v = 100 .

. ..

-

I - - I , 4 .

i

. - - I I 1 . : , j ' , I I I I I . . . , . . I . . .

I I ! !

I

1 I I

-L- -.

I I - - !

i

,

--

I !

I

i- I

i .I_ L...

,_C.

F i g . 39.- Ensemble averaging of t h e c o r r e l a t i o n f u n c t i o n s for a c c e l e r a t i o n response due t o n o i s e I n p u t only, v = 100 .

Shift to a Shift t o a and invert h =

c Yn

Fig. 40.- Randomdec s i g n a t u r e obtained from response t o n o i s e only, v = 100 .

J $ uA

. 0.1

.005 .001 .5 10 F i g . 42.- S e n s i t i v i t y s t u d y of deduced d i f f e r e n t i a l equation c o e f f i c i e n t s .

c) C Q)

-i=t

‘0 Q) r n a !

P I I I I Q) cq T v, rs‘ _---- c, 0 c

- I - *

u Q) c, c

>I ,\A-

/

! ?

-1%