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ON IDENTIFYING FREQUENCIES AND D A M P I N G I N SUBCRITICAL FLUTTER TESTING John C. Houbolt A e r o n a u t i c a l Research A s s o c i a t e s of P r i n c e t o n , I n c .
SUMMARY A review i s given of v a r i o u s procedures t h a t might be used i n e v a l u a t i n g system response c h a r a c t e r i s t i c s as involved i n sub- c r i t i c a l f l i g h t and wind-tunnel f l u t t e r t e s t i n g o f a i r c r a f t .
Emphasis i s given t o the means f o r e l i m i n a t i n g o r minimizing the contamination e f f e c t s produced by an unknown n o i s e i n the i n p u t .
R e s u l t s o f a newly developed procedure f o r i d e n t i f y i n g modal fre- quency and damping v a l u e s , and a p o s s i b l e way f o r making a de- t a i l e d e v a l u a t i o n of system parameters, a r e a l s o given.
INTRODUCTION The purpose of t h i s r e p o r t i s t o g i v e a review of v a r i o u s procedures t h a t might be used i n e v a l u a t i n g system response c h a r a c t e r i s t i c s as involved i n 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 o f a i r c r a f t . The a i m i n such t e s t i n g i s g e n e r a l l y t o e v a l u a t e modal damping and f r e q u e n c i e s as a f u n c t i o n o f f l i g h t speed. I n some c a s e s , s t u d i e s a i m t o i d e n t i f y t h e s y s t e m para- meters i n g r e a t e r d e t a i l , such as 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 o f a modelled d i f f e r e n t i a l e q u a t i o n o f motion.
I n p r a c t i c a l 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 three main problems a r i s e : (1) t h e r e u s u a l l y i s an unknown n o i s e i n p u t , such as that due to t u r b u l e n c e , and t h i s contamination makes t h e s y s t e m response e v a l u a t i o n very d i f f i c u l t , u n c e r t a i n , o r impossibxe; ( 2 ) t i m e f o r a t e s t run must o f t e n be k e p t s h o r t , such as l e s s t h a n 1 0 seconds ( f o r example, t o a c h i e v e a given speed t h e a i r p l a n e may have t o b e put i n a s h a l l o w d i v e and t h e i n t e r v a l o f time over which t e s t c o n d i t i o n s are reasonably c o n s t a n t i s t h e r e f o r e l i m i t e d ) , s h o r t - n e s s o f r e c o r d s i n t u r n a g g r a v a t e s t h e n o i s e problem; and ( 3 ) an u n d e r l y i n g desire i s t o be able t o perform r a p i d a n a l y s e s o f t h e r e c o r d s s o t h a t t h e t e s t s may proceed almost immediately to the n e x t t e s t run. The procedures p r e s e n t e d h e r e i n r e p r e s e n t v a r i o u s a t t e m p t s t o cope w i t h these problems, w i t h emphasis b e i n g given t o means f o r minimizing o r o b v i a t i n g t h e n o i s e problem.
Much o f t h e material i n t h i s r e p o r t i s c o v e r e d i n r e f e r e n c e 1, which c o n t a i n s a number o f r e f e r e n c e s t o o t h e r work; no o t h e r r e f e r e n c e i s t h e r e f o r e c i t e d . Some new f i n d i n g s are i n c l u d e d , RELEVANT EQUATIONS L e t t h e g e n e r a l g o v e r n i n g d i f f e r e n t i a l e q u a t i o n f o r r e s p o n s e f o r t h e a i r p l a n e s u b c r i t i c a l f l u t t e r s y s t e m b e g i v e n by Dly = D2P where D1 and D2 are 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 t h e
r e s p o n s e t o t h e f o r c i n g f u n c t i o n P . The f o r c e P may be a
p r e s c r i b e d f o r c e , as o b t a i n e d from a s h a k e r , o r it may be some unknown q u a n t i t y , such as due t o a t m o s p h e r i c t u r b u l e n c e , and t h e s e f o r c e s may b e a c t i n g s i n g l y or i n combination.
I f t h e i n p u t f o r c e i s a D i r a c f u n c t i o n 6(0) a t t = 0 , e q u a t i o n (1) d e f i n e s t h e impulse r e s p o n s e f u n c t i o n h as f o l l o w s Dlh = D 2 6 ( 0 )
F o r a u n i t s i n u s o i d a l i n p u t , P = e iwt , and w i t h
i w t y = H e e q u a t i o n (1) y i e l d s t h e f r e q u e n c y r e s p o n s e f u n c t i o n a c c o r d i n g t o t h e e q u a t i o n
( A , + i A 2 ) ( A + i B ) = N 1 + i N 2
(4) where Al,N1 and A 2 , N 2 are t h e r e a l and i m a g i n a r y 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 com-
p o n e n t o f H i s symmetrical w i t h r e s p e c t t o t h e f r e q u e n c y w , t h e B component i s a n t i s y m m e t r i c a l .
The h a n d H f u n c t i o n s are r e l a t e d b y t h e F o u r i e r t r a n s - form p a i r Ro H = h e - i w t d t ( 5 ) h = - 1 HeiWt d w 2lT r p o s i t i o n theorem, t h e s o l u t i o n o f eq a t i o n (l), By t h e s u p
f o r any g e n e r a l f o r c i n g f u n c t i o n P , i s g i v e n by
Y = P ( T ) h ( t - T ) d T
( 7 ) -00 The F o u r i e r t r a n s f o r m o f t h i s e q u a t i o n i s from which H f o l l o w s as F H = X ( 9 1 FP E q u a t i o n ( 8 ) a l s o leads t o t h e well-known spectral result
I f P i s e q u a l t o P + Q , where P i s a known f o r c e , and
Q i s an unknown " n o i s e " f o r c e , e q u a t i o n ( 8 ) would a p p e a r
F = H(Fp + FQ)
Y
-
The m u l t i p l i c a t i o n t h r o u g h t h e complex c o n j u g a t e Fp l e a d s i n t u r n to t h e s p e c t r a l e q u a t i o n where
i s t h e c r o s s spectrum between P and y , @ p i s t h e
@PY
i s t h e c r o s s spectrum between P and spectrum o f P , and
@ P Q
= 0 , and t h u s Q . I f P and Q are u n c o r r e l a t e d , @ p Q
e q u a t i o n (11) y i e l d s t h e i m p o r t a n t c r o s s - s p e c t r u m e q u a t i o n I which appears as a c o m p l e t e l y n o i s e - f r e e r e s u l t .
R e f e r e n c e 1 g i v e s some s i g n i f i c a n t s p e c i a l s o l u t i o n s to e q u a t i o n (I), as f o l l o w s .
DIRh = D2h(-t) where Thus, 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 o f h i s t h e r e s p o n s e o f s y s t e m to a f o r c e i n p u t o f h ( - t ) 11: _I where Qn i s white n o i s e . For t h i s s i t u a t i o n , it can b e shown t h a t = Rh ; t h u s , t h e c o r r e l a t i o n f u n c t i o n o f t h e r e s p o n s e Ry n to 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 o f t h e i m p u l s e f u n c t i o n h .
111: - D R = D2Rp I Py 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 o f an i n p u t P i s a p p l i e d to t h e s y s t e m as an i n p u t f o r c e , t h e r e s p o n s e i s t h e 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 P and t h e r e s p o n s e y due to P e C L A S S I F I C A T I O N OF THE SWEPT S I N E F U N C T I O N F o r c i n g i n p u t s are a c h i e v e d by s e v e r a l means, s u c h as i n - e r t i a l shakers or aerodynamic vane e x c i t e r s , e x p l o s i v e c h a r g e s , s t i c k raps, and t h e n a t u r a l t u r b u l e n c e o f t h e atmosphere. O f a l l these means, vane e x c i t e r s or shakers a r e most commonly u s e d . F o r the f o r c i n g f u n c t i o n , t h e swept s i n e wave has become a p o p u l a r c h o i c e , mainly because it c o v e r s a s i z a b l e f r e q u e n c y band i n a s h o r t p e r i o d o f t i m e and because t h e s p e c t r a l c o n t e n t o f t h i s f u n c t i o n resembles w h i t e n o i s e . The r a t e o f sweep and t o t a l d u r a t i o n are prime v a r i a b l e s ; w i t h some t e s t s t h e sweep r a t e i s fast, i n o t h e r s t h e r a t e i s q u i t e s l o w , For d i s c u s s i o n and t e s t i n g p u r p o s e s , it a p p e a r s d e s i r a b l e to make a c l a s s i f i c a t i o n o f t h e d u r a t i o n of sweep. The r a t e o f change o f f r e q u e n c y depends o f c o u r s e on t h e f r e q u e n c y r a n g e c o v e r e d and t h e d u r a t i o n r e q u i r e d to make t h e sweep, For t h e t e s t i n g o f most a i r c r a f t s y s t e m s , however, it a p p e a r s t h a t 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 s u g g e s t e d :
1) F a s t sweep - one made w i t h a d u r a t i o n o f a b o u t 5 t o 1 0
s e c o n d s
2 ) Moderate sweep - d u r a t i o n o f around 1 minute
3) Slow sweep - d u r a t i o n of a r o u n d 5 m i n u t e s
Each of these sweeps has c e r t a i n a d v a n t a g e s and c e r t a i n de- f i c i e n c i e s , depending on the a p p l i c a t i o n . The slow sweep i s t h e b e s t f o r m i n i m i z i n g n o i s e , b u t t h e drawback i s l o n g t e s t i n g and r e c o r d a n a l y s i s times. I n many i n s t a n c e s , t h o u g h , t e s t c o n d i t i o n s d i c t a t e t h e u s e o f f a s t sweeps.
D A M P I N G AND FREQUENCY EVALUATION FOR THE IDEAL CASE F i g u r e 1 i n d i c a t e s three b a s i c ways f o r e v a l u a t i n g the damping and f r e q u e n c y o f a mode. It i s assumed t h a t a t e s t has been made, s u c h as t h r o u g h a p p l i c a t i o n o f a swept s i n e wave f o r c i n g f u n c t i o n , and t h a t t h e r e s p o n s e has b e e n a n a l y z e d to ob- 2 2
t a i n H ( e q u a t i o n ( g ) ) , which y i e l d s B and A , C = IHI
= A + B2 , and h ( e q u a t i o n ( 6 ) ) . The s i t u a t i o n d e p i c t e d by t h i s f i g u r e i s ideal; t h a t i s , t h e r e i s no n o i s e p r e s e n t i n the i n p u t and o n l y a s i n g l e mode i s i n v o l v e d . The t o p s k e t c h d e p i c t s t h e
t r a n s f e r l o c i or a d m i t t a n c e p l o t i n v o l v i n g A and B . The
r e s o n a n t f r e q u e n c y f o i s i d e n t i f i e d a t t h e p o i n t where t h e r e i s the g r e a t e s t rate o f change o f a r c l e n g t h with r e s p e c t to a change
i n t h e f r e q u e n c y . The damping r a t i o - i s g i v e n by t h e
@ c r 2 2
e q u a t i o n shown. I n the s e c o n d scheme i n v o l v i n g C 2 = A + B
p l o t t e d a g a i n s t f , t h e modal f r e q u e n c y i s i d e n t i f i e d by t h e l o c a t i o n o f t h e peak, t h e damping by t h e width a t 1 / 2 power. I n t h e t h i r d scheme, i n v o l v i n g damped u n f o r c e d motion a f t e r some e x c i t a t i o n , f r e q u e n c y i s i d e n t i f i e d by t h e p e r i o d T , damping by t h e l o g decrement e q u a t i o n .
Note, t h e o f f h a n d appearance o f a peak ( s e c o n d s k e t c h o f f i g u r e 1) may a t first c a u s e a m i s i n t e r p r e t a t i o n of damping. I n f i g u r e 2 , f o r example, t h e peaks on t h e r i g h t v i s u a l l y seem to i n d i c a t e more damping t h a n t h e p e a k s on t h e l e f t ; a l l p e a k s on the same l i n e h a v e t h e same damping, however, as measured i n terms o f p e r c e n t o f c r i t i c a l damping. Likewise, t h e t h r e e peaks on t h e r i g h t o f t h e middle s k e t c h have t h e same damping, even though t h e s h o r t e s t peak seems to s u g g e s t a l a r g e r damping t h a n t h e t a l l e s t peak e O t h e r means f o r deducing frequency and damping i n v o l v e curve- f i t t i n g p r o c e d u r e s , such as f i t t i n g t h e e x p e r i m e n t a l l y d e r i v e d frequency r e s p o n s e f u n c t i o n H , or f i t t i n g t h e impulse r e s p o n s e f u n c t i o n h and t h e n deducing the roots from t h e f i t t e d c u r v e s .
When modes are c l o s e t o g e t h e r , or when 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 e t e c h n i q u e s o f f i g u r e 1 b r e a k down. It i s towards h a n d l i n g t h e s i t u a t i o n o f t h e p r e s e n c e o f a number of modes and the c o n t a m i n a t i o n due to a n unknown n o i s e s o u r c e t h a t t h e re- mainder o f t h i s r e p o r t i s devoted.
THE USE OF EXCITERS AND TRANSDUCERS I N COMBINATION It i s odd t h a t l i t t l e i n g e n e r a l h a s been done i n u s i n g t r a n s d u c e r s i n p a i r s as a way of h e l p i n g t o s o l v e t h e c l o s e l y s p a c e d mode s i t u a t i o n , p a r t i c u l a r l y i n s e p a r a t i n g t h e s y m m e t r i c a l and a n t i s y m m e t r i c a l modes which have f r e q u e n c i e s c l o s e t o g e t h e r .
F i g u r e 3 s e r v e s as a reminder o f what p r a c t i c e s s h o u l d b e f o l l o w e d i n g e n e r a l . With one shaker, s a y on t h e r i g h t , t h e u s e o f o n l y t h e s i g n a l from p o i n t 1 makes i t very d i f f i c u l t to d i s - t i n g u i s h t h e symmetric mode from t h e a n t i s y m m e t r i c mode. The a d d i t i o n o f t h e s i g n a l s from p o i n t 1 and p o i n t 2 , however, i d e n t i f i e s t h e symmetric mode and v i r t u a l l y e l i m i n a t e s t h e a n t i - symmetric mode. The s u b t r a c t i o n o f t h e s i g n a l s , on t h e o t h e r hand, i d e n t i f i e s t h e a n t i s y m m e t r i c mode t o t h e e x c l u s i o n o f t h e symmetric mode. T h i s s u b t r a c t i o n scheme a l s o p r o v i d e s f o r good r e j e c t i o n o f symmetric e x c i t a t i o n due to n o i s e .
F o r two shakers, one on t h e l e f t and one on t h e r i g h t , use o f y1 o r y1 i- y2 f o r in-phase e x c i t a t i o n g i v e s s y m m e t r i c mode i s o l a t i o n . or If t h e two shakers are 1 8 0 ' out o f p h a s e , g i v e s good a n t i s y m m e t r i c mode i s o l a t i o n . Agaig: i n t h i s
y 1 - y2
c a s e , a l s o g i v e s good r e j e c t i o n o f symmetric e x c i t a t i o n
y1 - y2
due to n o i s e .
The use o f t w o pick-ups i n a d i f f e r e n t chordwise p o s i t i o n , such as at p o i n t s 3 and 4, a l s o s h o u l d be c o n s i d e r e d a s a way of h e l p i n g t o i s o l a t e c l o s e l y spaced modes;. t h e idea h e r e i s t h a t e x c i t a t i o n o f d i f f e r e n t modes a p p e a r s i n a d i f f e r e n t r e l a t i v e s e n s e a c c o r d i n g to t h e c l o s e n e s s to t h e nodal l i n e s , F i g u r e 4 d e p i c t s r e s u l t s o b t a i n e d f o r a three-mode system, w i t h two symmetric modes o f 3 Hz and 10 Hz and one aiitisymmetric mode of 9 . 8 Hz; t h u s , t h e a n t i s y m m e t r i c mode had a frequency only 2 p e r c e n t d i f f e r e n t from one of t h e symmetric modes. With one shaker, a swept s i n e wave e x c i t a t i o n , and o n l y one pick-up, t h e
deduced r e s u l t s f o r A , B , C 2 , B vs A , and h , i n d i c a t e t h a t
o n l y two modes a r e p r e s e n t , one around 3 Hz and one around 1 0 Hz.
F i g u r e 5 a p p l i e s to one-shaker e x c i t a t i o n of t h e same s y s t e m , b u t t h e s i g n a l s from a r i g h t and a l e f t t r a n s d u c e r are s u b t r a c t e d .
The marked change i n t h e r e s u l t s i s a c l e a r i n d i c a t i o n t h a t two modes a r e p r e s e n t n e a r 1 0 Hz. F o r f i g u r e 6, t h e s i t u a t i o n i s t h e same as f o r f i g u r e 5 , e x c e p t t h a t a s t r o n g symmetric e x c i t a t i o n due to n o i s e i s a l s o p r e s e n t . The r e s u l t s , i n s p i t e o f t h e n o i s e , g i v e s a t i p - o f f t h a t t h e r e a r e two c l o s e l y spaced modes around 1 0 Hz. Thus, w i t h one s h a k e r o p e r a t i o n , t h e t e c h n i q u e o f a d d i n g t h e s i g n a l s from two o p p o s i t e t r a n s d u c e r s and o f s u b t r a c t i n g t h e s i g n a l s and comparing t h e deduced r e s u l t s a p p e a r s as a good way to e s t a b l i s h whether two modes with f r e q u e n c i e s c l o s e t o g e t h e r - one symmetric, one a n t i s y m m e t r i c - are p r e s e n t . Two s h a k e r s , first used s y m m e t r i c a l l y t h e n a n t i s y m m e t r i c a l l y , p r o v i d e a n even b e t t e r way t o i s o l a t e symmetric and a n t i s y m m e t r i c modes.
I N I T I A L SEQUENCE OF DATA A N A L Y S I S Some o f the f i r s t data a n a l y s i s checks t h a t s h o u l d b e made a r e o f t e n o v e r l o o k e d i n a t e s t i n g sequence. A review of c e r t a i n i n i t i a l s t e p s t h a t s h o u l d b e performed i s t h u s c o n s i d e r e d worth- w h i l e .
It i s assumed t h a t tests a r e b e i n g made w i t h a swept s i n e f o r c e i n p u t , The f i r s t a n a l y s i s t h a t should be made i s to make an a t t e m p t to i d e n t i f y modal f r e q u e n c i e s r o u g h l y , to c l a s s i f y t h e modes as to whether t h e y are symmetrical o r a n t i s y m m e t r i c a l , and to see i f t h e a p p a r e n t modes can be i d e n t i f i e d w i t h ground v i - b r a t i o n modes. Suggested f i r s t s t e p s a r e a s f o l l o w s : 1) Combine s i g n a l s as i n d i c a t e d i n t h e p r e v i o u s s e c t i o n .
2 ) Scan t h e combined t i m e h i s t o r y s i g n a l s and look f o r " b u r s t s " i n t h e r e s p o n s e ; t h e o b j e c t here i s t o o b t a i n a rough i d e a o f t h e modal f r e q u e n c i e s and to e s t a b l i s h whether t h e mode i s symmetric o r a n t i s y m m e t r i c and whether p r i m a r i l y bending or t o r s i o n .
3) From t h e s i g n a l s , e s t a b l i s h raw H v a l u e s ( e q u a t i o n ( 9 ) ) and i n t u r n h v a l u e s ( e q u a t i o n ( 6 ) ) . Clear h , a c c o r d i n g to t h e c l e a r e d h p r o c e d u r e d i s c u s s e d sub-
s e q u e n t l y , t r a n s f o r m back to first improved H , and form
C2 = 1 H I 2 = A2 + B2 . Examine t h e C 2 f u n c t i o n to ob-
t a i n a second check on the modal f r e q u e n c i e s ( v e r i f y t h o s e e s t a b l i s h e d by s c a n n i n g t h e t i m e h i s t o r y s i g n a l s , p i c k up o t h e r s t h a t may have been missed) and to o b t a i n a first e s t i m a t i o n of modal damping where p o s s i b l e .
4 ) From t h e appearance of t h e C2 f u n c t i o n s , an assessment o f t h e n o i s e problem can be made, and a judgment can b e r e n d e r e d as to what t y p e p r o c e d u r e s s h o u l d b e u s e d s u b s e q u e n t l y to minimize the n o i s e problem, E s s e n t i a l l y , t h e idea b e h i n d these s t e p s i s to do something q u i t e s i m p l e a t first s o as to o b t a i n a q u i c k i n s i g h t as to what t h e f r e q u e n c i e s might b e and to o b t a i n a q u i c k appraisal o f t h e s e v e r i t y and n a t u r e o f t h e n o i s e problem.
TECHNIQUES FOR M I N I M I Z I N G OR ELIMINATING INPUT NOISE EFFECTS Use o f Both I n p u t and Output I n f o r m a t i o n S i x schemes f o r c o p i n g w i t h t h e problem o f h a v i n g n o i s e i n t h e i n p u t are p r e s e n t e d i n b r i e f f a s h i o n i n t h i s s e c t i o n . ( S e e r e f e r e n c e 1 f o r more d e t a i l . ) It i s assumed t h a t one or more s h a k e r s are u s e d to d r i v e t h e system, s u c h as by a swept s i n e wave, and t h a t a n unknown e x c l t a t i o n n o i s e f o r c e , such as due to b u f f e t i n g or a t m o s p h e r i c t u r b u l e n c e , i s a l s o p r e s e n t .
C l e a r i n g h .- F i g u r e 7 i s t y p i c a l o f t h e r e s u l t s t h a t are
o b t a i n e d for H and h , by means o f e q u a t i o n s (9) and ( 6 ) , when a l a r g e i n p u t n o i s e i s p r e s e n t a l o n g w i t h t h e swept s i n e wave e x c i t a t i o n . One way to e l i m i n a t e much o f the n o i s e c o n t a m i n a t i o n i s s i m p l y to c l e a r or erase the r e s u l t s f o r h beyond a p o i n t where u s e f u l i n f o r m a t i o n no l o n g e r seems to a p p e a r , s u c h as p o i n t a i n f i g u r e 7, and t h e n to t r a n s f o r m t h i s t r u n c a t e d h back to H ( e q e ( 5 ) ) . Example r e s u l t s are g i v e n i n f i g u r e 8.
The remarkable improvement t h a t i s o b t a i n e d for t h e A and B v a l u e s by d o i n g t h i s s i m p l e e x p e d i e n t is s e e n ,
Weighting h .- Another t e c h n i q u e i s shown i n f i g u r e 9. Here
t h e raw h i s weighted by an e x p o n e n t i a l f u n c t i o n ; t h e weighted r e s u l t i s t h e n t r a n s f o r m e d back to g i v e r e f i n e d A and B v a l u e s .
T h i s t e c h n i q u e , as w i t h f i g u r e 8 , r e d u c e s n o i s e e f f e c t s g r e a t l y .
With t h i s w e i g h t i n g t e c h n i q u e , a c o r r e c t i o n to t h e deduced v a l u e s o f damping must b e made to c o r r e c t f o r t h e a p p a r e n t damping t h a t i s added by t h e w e i g h t i n g f u n c t i o n u s e d .
Use o f cross c o r r e l a t i o n between i n p u t and o u t p u t . - F i g u r e lO(a) a p p l i e s to t h e raw r e s u l t s as o b t a i n e d b y u s e o f e q u a t i o n ( 9 ) . By c o n t r a s t , t h e r e s u l t s shown i n f i g u r e 1 0 ( b ) were o b t a i n e d by u s e of e q u a t i o n (121, which i n v o l v e s t h e c r o s s s p e c t r u m between t h e measured o u t p u t and t h e known shaker f o r c e i n p u t . T h i s c r o s s - c o r r e l a t i o n t e c h n i q u e i s s e e n to g i v e a marked improvement i n t h e deduced A and B values.. I n g e n e r a l , t h e l o n g e r t h e r e c o r d , the b e t t e r i s t h e n o i s e m i n i m i z a t i o n by t h i s t e c h n i q u e .
Peak s h i f t i n g . - F i g u r e 11 i s used to d e s c r i b e the peak s h i f t i n g t e c h n i q u e f o r e l i m i n a t i n g n o i s e e f f e c t s , The t o p s k e t c h d e p i c t s t h e swept s i n e wave i n p u t f o r c e , the bottom s k e t c h t h e noise-contaminated r e s p o n s e , F i r s t , s e l e c t a peak such as a Then s e l e c t peak b and s h i f t the e n t i r e r e c o r d s o as t o make peak b f a l l on peak a a Next, take peak c and s h i f t t h e D b t h i s f o r a number of r e c o r d to make peak c f a l l on a .
the r e s u l t s to form a peaks i n s u c c e s s i o n , and t h e n add a l l composlte i n p u t f o r c e d e s i g n a t e d by same way, b u t u s i n g t h e same The o u t p u t r e s p o n s e i s h a n d l e d i n t h e s h i f t s as used f o r t h e i n p u t ; t h e composite r e s p o n s e i s d e s i g n a t e d as
YT = E yn
Now deduce H from PT and yT , u s i n g e q u a t i o n ( 9 ) . The
concept i n t h i s t e c h n i q u e i s t h a t a s i n g l e s h o r t record may be used and t h a t t h e s h i f t i n g and a d d i n g o p e r a t i o n s cause t h e meaningful o r i n t e l l i g e n t p a r t o f t h e r e c o r d to b e enhanced, a m p l i f i e d , o r r e i n f o r c e d , w h i l e t h e n o i s e l e v e l remains t h e same ( o r t h e s i g n a l - t o - n o i s e r a t i o i n c r e a s e s ) e F i g u r e 1 2 g i v e s re- s u l t s o b t a i n e d i n a p a r t i c u l a r c a s e where o n l y 19 s h i f t s were made. I n t h e maln frequency range o f i n t e r e s t , around 1 0 H z , it i s s e e n that p r a c t i c a l l y n o i s e - f r e e r e s u l t s are o b t a i n e d . A f e a t u r e of t h e peak s h i f t i n g scheme i s t h a t it i s p o s s i b l e to c o n c e n t r a t e on v a r i o u s frequency r a n g e s even w i t h t h e u s e o f a s i n g l e r e c o r d . For example, i n f i g u r e 11, two " b u r s t s " i n the o u t p u t r e s p o n s e a r e n o t e d , s u g g e s t i n g two f r e q u e n c i e s of p o s s i b l e concern. To c o n c e n t r a t e on t h e lower f r e q u e n c y , peaks i n t h e v i c i n i t y o f peak a are s h i f t e d t o f a l l a t peak a ; t o concen- t r a t e on the h i g h e r f r e q u e n c y , peaks i n t h e v i c i n i t y o f peak p are s h i f t e d , . .
Ensemble a v e r a g i n g . - I n ensemble a v e r a g i n g , t h e concept i s to deduce, by r e p e a t r u n s , a number of raw estimates f o r t h e f u n c t i o n
h , and t h e n to add a l l t h e r a w f u n c t i o n s t o g e t h e r . The idea i s
t h a t t h i s a v e r a g i n g - t y p e o p e r a t i o n w i l l "average o u t " n o i s e e f f e c t s and l e a v e o n l y t h e meaningful s i g n a l . Example r e s u l t s , i n v o l v i n g an ensemble a v e r a g e o f 20 r a w f u n c t i o n s , are shown i n f i g u r e 1 3 . It i s s e e n that v i r t u a l l y n o i s e - f r e e r e s u l t s are ob- t a i n e d . T h i s i s one of t h e best schemes f o r e l i m i n a t i n g n o i s e , b u t t h e main drawback i s t h a t it r e q u i r e s making a number o f r e p e a t r u n s .
l i m i t e d frequency band.- F i g u r e 14 i s g i v e n as a Sweep o v e r h e l p to d e s c r i b e a l i m i t e d sweep approach. Suppose t h a t t e s t sweeps are made to c o v e r t h e range o f 3 Hz t o 25 Hz i n 1 0 s e c o n d s , and c o n s i d e r t h a t t h e a n a l y s i s of t h e r e s u l t s i n d i c a t e some modal i n f o r m a t i o n i n t h e r a n g e o f 1 0 Hz b u t t h a t t h e r e s u l t s are t o o n o i s y to b e i n t e r p r e t e d w i t h c o n f i d e n c e . A good way t o improve t h e s i t u a t i o n i s t o sweep o v e r o n l y t h e f r e q u e n c y r a n g e o f con- c e r n , s a y , i n t h i s c a s e , from 8 Hz to 1 2 Hz i n the 1 0 s e c o n d s o f sweep t i m e . Generally., a v a s t improvement i n t h e deduced re- s u l t s w i l l b e n o t e d . The d i s a d v a n t a g e , o f c o u r s e , i s t h e problem o f r e s e t t i n g t h e sweep r a n g e and o f h a v i n g to make a n o t h e r r u n .
Use o f Output I n f o r m a t i o n Only There are a t l e a s t two ways t o d e r i v e s y s t e m r e s p o n s e c h a r a c t e r i s t i c s by c o n s i d e r a t i o n o f t h e o u t p u t r e s p o n s e a l o n e .
The p r o c e d u r e s a p p l y i n g e n e r a l whether t h e r e s p o n s e i s due to a f o r c e d swept e x c i t a t i o n w i t h a n unknown n o i s e i n p u t o r w h e t h e r t h e r e s p o n s e i s due to n o i s e e x c i t a t i o n a l o n e .
One p r o c e d u r e i n v o l v e s t h e e s t a b l i s h m e n t o f 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 o f t h e o u t p u t r e s p o n s e . Each s i d e or Y h a l f of t h i s symmetric f u n c t i o n has c h a r a c t e r i s t i c s of t h e h f u n c t i o n . The F o u r i e r t r a n s f o r m o f R i s t h e s p e c t r u n of Y @ Y t h e r e s p o n s e . Examination o f t h i s s p e c t r u m g i v e s an i n d i c a t i o n o f t h e f r e q u e n c y and damping of t h e system modes. Ensemble a v e r a g i n g o f t h e R f u n c t i o n s i s found to b e a p o w e r f u l way to Y minimize n o i s e by t h i s a p p r o a c h , r e f e r e n c e 1. O t h e r ways t o use t h e Ry f u n c t i o n and minimize n o i s e w i l l b e i n d i c a t e d i n t h e sub- s e q u e n t s e c t i o n .
A second p r o c e d u r e f o r d e r i v i n g system r e s p o n s e c h a r a c t e r - i s t i c s u s i n g r e s p o n s e i n f o r m a t i o n a l o n e i s t h e f o r m a t i o n o f t h e "randomdec" s i g n a t u r e . The e s s e n t i a l s o f one t y p e o f c o n s t r u c t i o n f o r t h i s a p p r o a c h are shown i n f i g u r e 15. It can be r e a s o n e d t h a t t h e sum o f a l l t h e i n d i v i d u a l s i g n a l s s h o u l d form a p u r e s i g n a l which resembles or has c h a r a c t e r i s t i c s o f t h e h f u n c t i o n .
Damping and f r e q u e n c y f o l l o w from t h e r e s u l t i n g summed s i g n a l . A main d i f f i c u l t y of t h e approach i s that t h e summation must o f t e n i n v o l v e h u n d r e d s o f f u n c t i o n s b e f o r e converged v a l u e s o f t h e sum are a c h i e v e d . Another d i f f i c u l t y i s i n i d e n t i f y i n g c l o s e l y s p a c e d modes.
SUCCESSIVE, CORRELATIONS OF CORRELATION RESULTS -
A P R O M I S I N G SOLUTION TO THE NOISE PROBLEM Under a c o n t r a c t e f f o r t f o r AFFTC/AFSC, Edwards AFB, t h e a u t h o r has d e v e l o p e d a d d i t i o n a l t e c h n i q u e s f o r t r e a t i n g the n o i s e problem - t e c h n i q u e s which a p p e a r r e m a r k a b l e and i n a way u n b e l i e v a b l e . This s e c t i o n summarizes some o f t h e r e s u l t s o b t a i n e d . The p r o c e d u r e s i n v o l v e d are q u i t e v e r s a t i l e and r e p r e - sent subsequent m a n i p u l a t i o n s f o r improving the q u a l i t y o f t h e re- sults t h a t are o b t a i n e d by most all t h e p r o c e d u r e s d e s c r i b e d e a r l i e r i n t h i s r e p o r t . Two f i g u r e s are p r e s e n t e d first as a way to d e s c r i b e the p r o c e d u r e s i n v o l v e d . I n f i g u r e 1 6 , t h e t o p s k e t c h r e f e r s to a u t o c o r r e l a t i o n o f t h e raw h f u n c t i o n ( s e e eq. (14)) t h a t has been deduced b y any o f t h e p r o c e d u r e s d i s c u s s e d p r e v i o u s l y , o r it r e f e r s to t h e a u t o c o r r e l a t i o n R , o b t a i n e d by Y c o n s i d e r i n g o n l y t h e r e s p o n s e (due to n o i s e a l o n e , due to a swept s i n e wave a l o n e , o r due t o these fcrrcing f u n c t i o n s a c t i n g i n c o m b i n a t i o n ) . Note, t h e raw h s h o u l d a l w a y s be c l e a r e d as d i s - c u s s e d i n c o n n e c t i o n w i t h f i g u r e s 7 and 8. L i k e w i s e , i f 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 i s used, t h e " n o i s y " t a i l s ( t h e t a i l p o r t i o n s on e i t h e r s i d e which a p p e a r to be due to n o i s e o n l y ) s h o u l d be erased. Then the f o l l o w i n g s t e p s are performed: Make R1 one-sided; c a l l it rl
Form R2 , t h e a u t o c o r r e l a t i o n o f r1
Form , t h e F o u r i e r t r a n s f o r m o f R2 ; look a t t h i s
@ * f u n c t i o n f o r improvement ( r e d u c t i o n i n n o i s e c o n t e n t ) and f o r mode i d e n t i f i c a t i o n Go back to R2 Make R2 one-sided; c a l l it r Repeat t h e s e s t e p s as o f t e n as n e c e s s a r y u n t i l t h e s p e c t r u m a p p e a r s w i t h o u t d i s t o r t i o n due t o n o i s e .
@ n I n t h e a p p l i c a t i o n of these s t e p s , t h e f o l l o w i n g w i l l o c c u r : The modes which show up w i t h low power w i l l first d i s a p p e a r (means for r e c o v e r i n g these modes w i l l be d i s c u s s e d s u b s e q u e n t l y ) .
The mode w i t h t h e n e x t l o w e s t power ( a c t u a l l y a combi- n a t i o n o f power and damping) w i l l t h e n d i s a p p e a r , and s o on, u n t i l f i n a l l y o n l y one mode r e m a i n s .
With e a c h i t e r a t i o n , t h e r e s u l t s become more a n d more n o i s e - f r e e .
Sometimes, depending on modal power and dampfng and on mode c l o s e n e s s , n o i s e - f r e e r e s u l t s w i l l o c c u r w i t h p e r h a p s two o r t h r e e modes s t i l l r e m a i n i n g .
The r e a d i n g o f t h e f r e q u e n c y and damping of these re- m a i n i n g modes, by t h e s e c o n d scheme o f f i g u r e 1, w i l l be an a c c u r a t e i n d i c a t i o n of t h e f r e q u e n c y and damping o f these modes, F i g u r e 1 7 i l l u s t r a t e s a companion t y p e m a n i p u l a t i o n , I n t h i s c a s e , t h e c o r r e l a t i o n f u n c t i o n s are k e p t i n t h e i r two-sided form; t h u s , a c o r r e l a t i o n f u n c t i o n o f a c o r r e l a t i o n f u n c t i o n i s found, i n s u c c e s s i o n , I n t h i s c a s e , t h e f o l l o w i n g s h o u l d b e ob s e r v e d 1) The modes with t h e l o w e s t power l o s e more and more power w i t h e a c h i t e r a t i o n and f i n a l l y d i s a p p e a r .
2 ) The p e a k s become more and more s p i k e d ; damping i s lost, b u t f r e q u e n c y i s more and more s h a r p l y p i n p o i n t e d .
Although t h e t h e o r y i s n o t g i v e n h e r e , i t s h o u l d b e n o t e d t h a t t h e consequences o f t h e two t y p e s o f m a n i p u l a t i o n d e s c r i b e d can b e e x p l a i n e d on a t h e o r e t i c a l b a s i s .
Means f o r r e c o v e r i n g any lost mode a r e as f o l l o w s . Go back to t h e o r i g i n a l s p e c t r u m t y p e f u n c t i o n $1 . I n f i g u r e 1 6 , peak a would p r o b a b l y have remained to t h e e n d . But, suppose i t was d e s i r e d to i d e n t i f y t h e mode i n i c a t e d by b more pre- c i s e l y . I n t h i s c a s e , s i m p l y e r a s e t h e 4 , f u n c t i o n above fre- quency w 2 and below w1 ( i n t h i s c a s e , e r a s i n g above w 2 i s a l l t h a t i s r e q u i r e d ) ; a p p l i c a t i o n o f t h e s t e p s d e s c r i b e d e a r l i e r w i l l t h e n b r i n g o u t mode b i n a p u r e form.
F i g u r e 1 8 shows r e s u l t s as o b t a i n e d b y t h e one-sided proce- d u r e , u s i n g h as e s t a b l i s h e d from a raw o r c o n t a m i n a t e d H The e x p e r i m e n t i n v o l v e d u s e o f an a n a l o g s i m u l a t i o n o f a s y s t e m ; e x c i t a t i o n was by means o f a l i n e a r swept s i n e wave, and an un- known random n o i s e . I n p a r t ( a ) , w e see f r e q u e n c i e s a r o u n d 3 Hz and 10 Hz, b u t t h e p r e c i s e l o c a t i o n and damping cannot be e s - t a b l i s h e d . I n p a r t ( b ) , which r e p r e s e n t s t h e first i t e r a t i o n , mode 1 has j u s t a b o u t d i s a p p e a r e d , and t h e r e s t o f t h e f u n c t i o n i s much more n o i s e - f r e e . By 5 i t e r a t i o n s , mode 2 has become very p u r e ; damping and f r e q u e n c y a r e n e a r l y p r e c i s e l y t h e v a l u e s set i n t h e a n a l o g s e t up ( i n t h i s c a s e , = 10 Hz , - ' - - 0 . 0 5 ) .
f o e c r F i g u r e 19 g i v e s r e s u l t s u s i n g t h e r e s p o n s e o n l y , and i t s a u t o c o r r e l a t i o n , f o r t h e same r u n o f f i g u r e 18. The raw s p e c t r u m i n d i c a t e s t h e two modes i n t h e v i c i n i t y o f 3 Hz and 10 Hz, By t h r e e i t e r a t i o n s , t h e 1 0 Hz mode i s i d e n t i f i e d p u r e l y .
I n f i g u r e 2 0 , end r e s u l t s are shown f o r convergence to t h e mode n e a r 3 Hz, I n t h i s c a s e , s t r a i n r e s p o n s e r a t h e r t h a n a c c e l e r a t i o n r e s p o n s e was used, and convergence went a u t o m a t i c a l l y to t h e l o w e s t mode ( n o s p e c t r u m e r a s i n g had to be p e r f o r m e d ) .
1 2 Note, d i s p l a c e m e n t o r s t r a i n emphasizes t h e l o w e r modes, w h i l e a c c e l e r a t i o n r e s p o n s e , due to t h e u2 w e i g h t i n g , emphasizes t h e h i g h e r modes e F i g u r e 2 1 s e r v e s to show the remarkable power o f t h e proce- d u r e to r e g e n e r a t e c o r r e c t f r e q u e n c y and damping i n f o r m a t i o n when s e v e r e t r u n c a t i o n s i n t h e f r e q u e n c y p l a n e are made, F i g u r e 21(a) i s the o r i g i n a l s p e c t r u m o f h o b t a i n e d f o r a one- mode s y s t e m and w i t h o u t n o i s e i n t h e i n p u t , The shaded areas were t h e n erased; arter s e v e r a l i t e r a t i o n s , s t a r t i n g w i t h t h i s t r u n - c a t e d spectrum, t h e s p e c t r u m as i n d i c a t e d by f i g u r e 2 1 ( b ) was found. Frequency and damping o f t h e mode i s s t i l l i n t a c t . The experiment was r e p e a t e d , t r u n c a t i n g f i g u r e 21(a) to t h e s e v e r e form shown by f i g u r e 2 1 ( c ) ; here t r u r , c a t i o n i s w i t h i n the Qalf- power l i m i t s . A f t e r s e v e r a l i t e r a t i o n s , t h e r e s u l t s shown i n f i g u r e 2 1 ( d ) were o b t z i n e d . Damping and f r e q u e n c y a r e s t i l l t h e same as t h e o r i g i n a l , even thoueh t h e o n l y i n f o r m a t i o n u s e d was t h a t g i v e n by f i g u r e 2 1 ( c ) , F i g u r e 22 shows r e s u l t s t h a t were o b t a i n e d w i t h a s y s t e m - ’ - - 0.05 .
h a v i n g f r e q u e n c i e s o f 9 and 1 0 Hz, b o t h w i t h ‘ Bcr F i g u r e 2 2 ( a ) r e p r e s e n t s t h e raw o r c o n t a m i n a t e d s p e c t r u m o f h a After s e v e r a l i t e r a t i o n s by t h e one-sided a p p r o a c h , t h e r e s u l t shown i n f i g u r e 2 2 ( b ) was o b t a i n e d ; t h e f r e q u e n c y and damping are i n e x c e l l e n t agreement w i t h t h e model v a l u e s , F i g u r e 2 2 ( c ) r e p r e - s e n t s t h e s p e c t r u m as o b t a i n e d by c o n s i d e r i n g t h e r e s p o n s e o n l y .
F i g u r e 2 2 ( d ) i s t h e r e s u l t o b t a i n e d by t h e o n e - s i d e d a p p r o a c h a f t e r i n f o r m a t i o n beyond f, was e r a s e d ; t h i s e r a s i n g was done to b r i n g out t h e lower mode. The damping and f r e q u e n c y i n d i c a t e d by f i g u r e 2 2 ( d ) f o r t h i s mode i s i n good agreement w i t h t h e c o r r e c t v a l u e s , even though t h e i n f o r m a t i o n c o n t a i n e d i n peak 1 was a l l t h a t was u s e d , F i g u r e 2 2 ( e ) i s t h e r e s u l t o b t a i n e d by a p p l y i n g t h e two-sided approach to t h e f u n c t i o n ; t h e tendency t o form Ry s h a r p s p i k e s i s shown by t h i s s k e t c h .
F i g u r e 2 3 a p p l i e s to a s y s t e m h a v i n g modes f a i r l y c l o s e t o g e t h e r as f o l l o w s : Mode f ,Hz B’Bcr 1 8 0 . 0 5 2 9 0 . 0 5 1 0 0 . 0 2 F i g u r e 2 3 ( a ) i s t h e raw spectrum o f h e If no e r a s i n g i s made, a p p l i c a t i o n o f t h e sequence o f s t e p s would r e s u l t i n t h e 1 0 Hz mode coming o u t i n p u r e form. C l e a r i n g beyond f a y i e l d e d t h e r e s u l t shown by f i g u r e 2 3 ( b ) by t h e one-sided approach; c l e a r i n g b e f o r e f a and beyond f b y i e l d e d t h e r e s u l t shown b y f i g u r e 2 3 ( c ) , Damping and f r e q u e n c i e s f o r b o t h modes are v e r y good, Thus, b o t h lower modes were e x t r a c t e d , i n s p i t e o f t h e c l o s e n e s s o f a n o t h e r mode h a v i n g a much lower v a l u e o f damping.
SYSTEM PARAMETER IDENTIFICATION -
POSSIBILITIES OF A NEW APPROACH A number o f d i f f e r e n t schemes have been s t u d i e d as means f o r o f s y s t e m p a r a m e t e r s .
o b t a i n i n g a more d e t a i l e d i d e n t i f i c a t i o n c a t e g o r i e s : These schemes g e n e r a l l y f a l l under t h r e e 1) Curve f i t t i n g o f t h e f r e q u e n c y r e s p o n s e f u n c t i o n 2 ) F i t t i n g o f t i m e p l a n e i n f o r m a t i o n , s u c h as t h e h f unc t i on 3) D i f f e r e n c e - e q u a t i o n a p p r o a c h e s i n which t h e c o e f f i c i e n t s of a d i f f e r e n c e - e q u a t i o n model a r e e v a l u a t e d , from which s y s t e m roots may i n t u r n be e x t r a c t e d C o l l o c a t i o n p r o c e d u r e s are sometimes u s e d f o r t h e c u r v e - f i t t i n g o p e r a t i o n s b u t , more g e n e r a l l y , t h e a p p r o a c h e s a r e based on t h e u s e o f l e a s t - s q u a r e s c o n c e p t s . Some o f t h e s y s t e m i d e n t i f i c a t i o n a p p r o a c h e s are reviewed and developed i n r e f e r e n c e 1 and t h e r e f e r e n c e s c o n t a i n e d t h e r e i n . Thus, t h e y w i l l n o t be d i s c u s s e d f u r t h e r h e r e i n . I n s t e a d , t h e n o t i o n s o f a p o s s i b l e new a p p r o a c h w i l l b e o u t l i n e d .
A commonly u s e d c o n c e p t 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 o f an a i r c r a f t i s to make a p l o t o f damping g v e r s u s V , f i g u r e 2 4 .
The b a s i c i d e a i s to e s t a b l i s h t h e t r e n d o f t h e damping c u r v e s and to e x t r a p o l a t e forward to e s t i m a t e t h e f l i g h t s p e e d a t which t h e damping v a n i s h e s (or r e d u c e s to some s t i p u l a t e d l o w e r l e v e l ) .
T h i s p r o c e d u r e i s r e a s o n a b l y s a t i s f a c t o r y f o r a m i l d a p p r o a c h t o t h e c r i t i c a l f l u t t e r s p e e d , curve a , b u t i s q u i t e t r e a c h e r o u s when a n e x p l o s i v e f l u t t e r s i t u a t i o n i s e n c o u n t e r e d , curve b , f o r i n t h i s s i t u a t i o n t h e damping can d e t e r i o r a t e v e r y q u i c k l y w i t h o n l y a small i n c r e a s e i n speed. A way to o b v i a t e t h i s problem i s s o u g h t . R e f e r e n c e 1 s u g g e s t s one p o s s i b l e p r o c e d u r e & The i d e a i s t o d e r i v e t h e c o e f f i c i e n t s o f t h e assumed g o v e r n i n g d i f f e r - e n t i a l e q u a t i o n model and to watch how these c o e f f i c i e n t s v a r y w i t h a i r s p e e d . F i g u r e 2 5 , t a k e n from r e f e r e n c e 1, d e p i c t s re- s u l t s f o r t h e s i t u a t i o n o f a m i l d a p p r o a c h to f l u t t e r . The n a t u r e o f the e x t r a p o l a t i o n i s known by a n a l y t i c a l c o n s i d e r a t i o n s ;
f o r example, the C o e f f i c i e n t s a3 , a2 , al, and a. a r e known
t o v a r y I n a q u a d r a t i c manner. E x t r a p o l a t i o n to h i g h e r speeds seems s t r a i g h t f o r w a r d , With t h e e x t r a p o l a t e d c o e f f i c i e n t s , system r o o t s f o r h i g h e r speeds may b e e v a l u a t e d , from which an estimate o f t h e c r i t i c a l f l u t t e r s p e e d may be made, F i g u r e 26 shows t h e b e h a v i o r of t h e c o e f f i c i e n t s f o r a s y s t e m which h a s e x p l o s i v e f l u t t e r c h a r a c t e r i s t i c s . I n f i g u r e 25 t h e v a r i a t i o n o f t h e coef- f i c i e n t s a p p e a r s g r a d u a l , w h i l e i n f i g u r e 26 two o f t h e coef-
f i c i e n t s , s p e c i f i c a l l y a2 and a? , are changing q u i t e markedly
T h i s r a p i d , b u t n o t a b r i p t , change i n t h e c o e f f i c i e n t s w i t h V a w i t h speed a p p e a r s as a t i p - o f f t h a t t h e s i t u a t i o n may b e o f t h e e x p l o s i v e f l u t t e r v a r i e t y .
W e now combine t h e t h o u g h t s a s s o c i a t e d w i t h f i g u r e s 25 and 26 w i t h t h e p r o c e d u r e s d i s c u s s e d i n t h e p r e v i o u s s e c t i o n . Suppose t h a t t h e p r o c e d u r e s o u t l i n e d i n t h e p r e v i o u s s e c t i o n s t a n d t h e t e s t o f more e x t e n s i v e s t u d y and t h a t t h e p r o c e d u r e s i n d e e d a r e r e l i a b l e i n e s t a b l i s h i n g t h e f r e q u e n c i e s and damping o f t h e v a r i o u s modes o f t h e s y s t e m under s t u d y . With t h e f r e q u e n c i e s and damping e s t a b l i s h e d , t h e governing d i f f e r e n t i a l e q u a t i o n can t h e n be formed. A s an example, c o n s i d e r t h a t t h r e e modes are i d e n t i f i e d ; r o o t s may t h e n b e w r i t t e n as where gn = 2 ( k ) , From t h e s e r o o t s , t h e g o v e r n i n g d i f f e r - C r n e n t i a 1 e q u a t i o n f o l l o w s as Expansion o f t h i s e q u a t i o n y i e l d s t h e c h a r a c t e r i s t i c e q u a t i o n P6 -4- a5P 4- "4P h -4- a3P 4- a2P 2 t alp t a. = 0 which i n t u r n d e f i n e s t h e c o e f f i c i e n t s an o f t h e g o v e r n i n g d i f f e r e n t i a l e q u a t i o n . I n accordance w i t h f i g u r e s 2 5 and 26, w e watch how t h e s e c o e f f i c i e n t s vary w i t h a i r s p e e d .
W e n o t e t h a t c u r v e f i t t i n g i n t h e f r e q u e n c y p l a n e o r t i m e p l a n e , o r any o t h e r e v a l u a t i o n o f c o e f f i c i e n t s t h r o u g h u s e o f l e a s t - s q u a r e s p r o c e d u r e s , i s p r e c l u d e d i n t h i s s u g g e s t e d approach.
The s u c c e s s depends simply on t h e r e l i a b l e e s t i m a t i o n o f t h e mode f r e q u e n c y and damping v a l u e s .
C O N C L U D I N G REMARKS Which one o f t h e p r o c e d u r e s o u t l i n e d h e r e i n f o r minimizing n o i s e e f f e c t s i s t h e b e s t ? No s p e c i f i c c h o i c e can r e a l l y be made.
A s y s t e m a t i c s t u d y i s needed to try e a c h p r o c e d u r e i n a number o f d i f f e r e n t a p p l i c a t i o n s a n d c i r c u m s t a n c e s . The c h o i c e of which i s b e s t w i l l undoubtedly depend on t h e s i t u a t i o n e n c o u n t e r e d , Never- t h e l e s s , some comment a b o u t c e r t a i n f e a t u r e s o r drawbacks o f t h e p r o c e d u r e s can be made.
The p r o c e d u r e o f c l e a r i n g t h e impulse r e s p o n s e f u n c t i o n h ( r e c t a n g u l a r t r u n c a t i o n ) s h o u l d a l w a y s be u s e d , no matter how h has been d e r i v e d . The e x p o n e n t i a l w e i g h t i n g o f t h e raw h i s n o t s u g g e s t e d i n g e n e r a l , s i n c e t h e c l e a r e d h p r o c e s s s e r v e s j u s t a b o u t as w e l l . The u s e of t h e cross-spectrum approach i s c o n s i d - ered one of t h e b e s t b u t g e n e r a l l y i s more a p p l i c a b l e f o r t h e l o n g e r sweep times. The peak s h i f t i n g t e c h n i q u e i s very a t t r a c t i v e b u t o f c o u r s e r e q u i r e s t h e i n t e r m e d i a t e s t e p o f s h i f t i n g and sum- ming t h e r e c o r d p o r t i o n s . Ensemble a v e r a g i n g i s perhaps t h e b e s t b u t i s p r o b a b l y p r e c l u d e d i n most i n s t a n c e s b e c a u s e of t h e n e c e s - s i t y f o r making a number of r e p e a t r u n s . Randomdec i s n o t advo- c a t z d u n l e s s a swept s i n e wave forming f u n c t i o n i s used ( w i t h a n o i s e i n p u t a l o n e , too many terms are r e q u i r e d i n t h e summation i n g e n e r a l ) . Where r e s p o n s e i n f o r m a t i o n o n l y i s a v a i l a b l e , t h e a u t o c o r r e l a t i o n approach (or e q u i v a l e n t l y , t h e spectrum o f t h e r e s p o n s e ) s h o u l d , o f c o u r s e , be used. I n t h i s approach, c a r e s h o u l d be t a k e n t o e r a s e t h e "noisy" t a i l s o f t h e c o r r e l a t i o n f u n c t i o n , as mentioned i n t h e body o f t h e r e p o r t . A l s o , i n t h i s a p p r o a c h it i s l i k e l y t h a t f a i r l y l o n g r e c o r d l e n g t h s a r e a v a i l - a b l e ; t h i s works t o t h e f a v o r of t h e approach b e c a u s e , on t h e whole, t h e l o n g e r t h e r e c o r d t h e b e t t e r t h e r e s u l t s (as i n t h e g e n e r a l r u l e f o r most a l l a p p r o a c h e s ) .
A s a g e n e r a l comment, w h i l e t h e r e i s a s c i e n c e t o t h e pro- c e d u r e s for minimizing t h e n o i s e problem, t h e r e i s a l s o an a r t i n t h e i r a p p l i c a t i o n s . Depending on t h e c i r c u m s t a n c e s and t h e t y p e 1 6 o f a n a l y s i s equipment a v a i l a b l e little " t r i c k s " can b e i n s e r t e d a t a p p r o p r i a t e p l a c e s t o g a i n an improvement 9n t h e end r e s u l t s , REFERENCE 1, Houbolt, John C . : sub critical^ F l u t t e r Testing an Z d e n t i f i c a t i o -132480, Aug 1 7 A f 0 Figure 1.- Three basic ways f o r estimating modal frequencies and damping.
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cd k a , d F i g u r e 2 0 . - P u r e r e s u l t s f o r t h e low-frequency mode o b t a i n e d by s u c c e s s i v e c o r r e l a t i o n s o f t h e one- s i d e d c o r r e l a t i o n f u n c t i o n .
Figure 21.- Regeneration of modal response characteristics after truncation in the frequency plane.
Figure 22.- Isolation of modes by frequency plane erasing, two-mode system.
Figure 2 3 . - Isolation of specific modes by frequency plane erasing, three-mode system.
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an . 5 Evaluated by collocation procedure I I I I I I it' 60 70 60 90 100 110 120 2 v v = - C Figure 25.- Variation of differential equation coefficients w i t h airspeed, mild flutter.
1 .o an 60 70 80 90 100 110 120 2v v = - C F i g u r e 26.- V a r i a t i o n o f d i f f e r e n t i a l e q u a t i o n 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 x p l o s i v e f l u t t e r .