Comparison of Modal Analysis and Holography

E. Muller. W. Looser, P. E. Gygax, IWF-ETH Zurich - Submitted by Prof. E. Matthias. ETH Zurich (1)

A S H O R T D E S C R I P T I O N O F B O T H

M O D A L A N A L Y S I S

Yodal a n a l y s i s d e a l s w i t h t h e measu remen t , t h e eva- l u a t i o n and t h e r e p r e s e n t a t i o n o f s t r u c t u r a l mode s h a p e s . The p r i n c i p l e s o f modal a n a l y s i s h a v e b e e n known a l o n g time ago. With a n a l o g e q u i p m e n t s , mode s h a p e s were d e r i v e d from s i n u s o i d a l e x c i t a t i o n s a t t h e r e s o n a n c e f r e q u e n c y .

The s t r o n g deve lopmen t o f m i n i c o m p u t e r s and o f d i - g i t a l s i g n a l a n a l y s e r s g a v e new p o s s i b i l i t i e s wh ich a l - r e a d y r e a c h e d i n d u s t r i a l l a b o r a t o r i e s . E x p e r i m e n t a l modal a n a l y s i s h a s become a v e r y known t o o l i n t h e l a s t f i v e y e a r s and s e v e r a l m a n u f a c t u r e r s s u p p l y s u c h s y s - tems. Wi th t h e i n c r e a s e d m i n i a t u r i z a t i o n o f compute r h a r d w a r e and t h e deve lopmen t of more u s e r a d a p t e d s o f t - w a r e p a c k a g e s modal a n a l y s i s is becoming more and more p o p u l a r .

A v e r y i m p o r t a n t f e a t u r e o f a l l modal a n a l y s i s pack- a g e s is t h e v i s u a l i z a t i o n o f mode s h a p e s on a d i s p l a y . The undeformed s t r u c t u r e is v i b r a t e d s l o w l y w i t h t h e

a j Idealizsi s:rl*cture

d > Modal pararr .e iers !m = number o f Z . O ~ P S :

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Twenty y e a r s a g o t h e f i r s t l a s e r s t a r t e d t o work. Nowadays t h e l a s e r h a s many s p e c i f i c a p p l i c a t i o n s , o n e of them b e i n g h o l o g r a p h y . The b a s i c c h a r a c t e r o f t h i s m e a s u r i n g t e c h n i q u e is d e m o n s t r a t e d on a t y p i c a l and s i m p l e a r r a n g e m e n t ( f i g . 2 ) .

To g e t a ho log ram, a n o b j e c t h a s t o b e i l l u m i n a t e d by a c o h e r e n t l i g h t s o u r c e , t h e l a s e r ( f i q . 2 b ) . The ob- ject r e f l e c t s t h e l i g h t d i f f u s e l y . T h i s r e f l e c t e d l i g h t c o n t a i n s a l l i n f o r m a t i o n on t h e o b j e c t . The r e f e r e n c e wave r e f l e c t e d by t h e m i r r o r i n t e r f e r e s w i t h t h e l a s e r l i g h t cominq from t h e o b j e c t i n a r e g i o n o f s p a c e where a p h o t o s e n s i t i v e p l a t e is l o c a t e d .

To g e t t h e o b j e c t ' s image t h e d e v e l o p e d p h o t o s e n s i - t i v e p l a t e (now c a l l e d ho logram) is s h o n e t h r o u g h by a r e f e r e n c e wave ( f i g . 2 ~ ) .Using t h e same h o l o g r a p h i c set- up , a new wave f i e l d o r i g i n a t e s i n f r o n t o f t h e h o l o - gram by d i f f r a c t i o n . T h i s wave f i e l d is i d e n t i c a l w i t h

Annals of the CIRP Vol. 29/1/1980 397

measured a m p l i t u d e s and p h a s e s s u p e r p o s e d on. a l l res- p e c t i v e p o i n t s . An an ima ted d i s p l a y o f e a c h i d e n t i f i e d mode o f v i b r a t i o n r e s u l t s , wh ich c a n b e o b s e r v e d from any d e s i r e d d i r e c t i o n ( f i g . l e ) . Such a r e p r e s e n t a t i o n o f t h e d y n a m i c a l h e h a v i o u r is a v e r y p o w e r f u l t o o l f o r a d e s i g n e r l o o k i n g f o r s t r u c t u r a l weak p o i n t s o f an e x i s t i n g s t r u c t u r e . An e x p e r i e n c e d d e s i g n e r c a n a l r e a d y p r o p o s e v a l u a b l e improvements fr.om t h i s q u a l i t a t i v e e v a l u a t i o n o f mode s h a p e s .

Modal a n a l y s i s is b a s e d on l i n e a r i t y , so t h a t t h e d y n a m i c a l b e h a v i o u r o f a s t r u c t u r e u n d e r t es t c a n b e d e s c r i b e d w i t h 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 s . I f a s e t o f s u c h f r e q u e n c y r e s p o n s e s is measured a l l o v e r t h e s t r u c t u r e , modal a n a l y s i s p r o c e d u r e s t h e n e % t r a c t t h e modal p a r a m e t e r s , i.s. t h e e i g e n f r e q u e n c i e s , t h e damp- i n g v a l u e s and t h e a m p l i t u d e s o f v i b r a t i o n ( e i g e n v e c - t o r s o r mode S h a p e s ) f o r a l l r e s o n a n c e s o f i n t e r e s t . T h e s e modal p a r a m e t e r s r e p r e s e n t t h e c o m p l e t e d y n a m i c a l b e h a v i o u r o f t h e s t r u c t u r e w i t h i n t h e p r e s c r i b e d f r e - quency r a n g e .

B e f o r e s t a r t i n g t h e a n a l y s i s , a c e r t a i n number o f r e p r e s e n t a t i v e p o i n t s a r e d e f i n e d on t h e s t r u c t u r e ( f i g . l a ) . The t r a n s l a t i o n s i n t h r e e d i r e c t i o n s o f e a c h p o i n t a r e measu red , w h e r e a s r o t a t i o n s g e n e r a l l y a r e o m i t t e d . The number o f d e g r e e s o f f reedom ( D O F s ) n i s t h e r e f o r e t h r e e times t h e number o f s e l e c t e d p o i n t s . About 1QQ t o 288 DOFS a r e u s u a l l y needed f o r m i d d l e s i z e d mach ine t o o l s t r u c t u r e s .

Impac t and random e x c i t a t i o n t e c h n i q u e s a r e m o s t l y used i n t h e measurement s t e p ( f i q . l b ) . Impac t t e s t i n g w i t h a hammer is e x t r e m e l y f a s t ( l ess t h a n o n e m i n u t e f o r o n e f r e q u e n c y r e s p o n s e ) . Random e x c i t a t i o n r e q u i r e s more a v e r a g i n g b u t g i v e s t h e b e s t l i n e a r a p p r o x i m a t i o n o f n o n l i n e a r e l e m e n t s i n a l e a s t s q u a r e s s e n s e . A l l i n - p u t - o u t p u t r e l a t i o n s be tween t h e DOFs t a k e t h e form o f a s q u a r e f r e q u e n c y r e s p o n s e m a t r i x o f s i z e n*n, which i s s y m m e t r i c a l ( M a x w e l l ' s r e c i p r o c i t y p r i n c i p l e ) . Modal t h e o r y s a y s t h a t o n l y o n e column o r row o f t h i s m a t r i x h a s t o b e measu red and i d e n t i f i e d , s i n c e t h e o t h e r co lumns and rows b e a r t h e same i n f o r m a t i o n . The p o i n t o f e x c i t a t i o n ( d r i v i n g p o i n t ) s h o u l d e x c i t e a l l i n t e r e s t i n g modes o f v i b r a t i o n a s good a s p o s s i b l e . Some p r e l i m i n a r y t es t s g r e a t l y h e l p t h e o p e r a t o r t o c h o o s e t h i s b e s t d r i v i n g p o i n t . The i n p u t f o r c e and t h e a c c e l e r a t i o n s (or d i s p l a c e m e n t s ) o f a l l r e q u i r e d DOFs a r e t h e n measu red . These measu remen t s and t h e c a l c u l a - t i o n o f t h e f r e q u e n c y r e s p o n s e c o m p l i a n c e s a r e p e r - formed v e r y e f f i c i e n t l y w i t h modern d i g i t a l s i g n a l ana- l y s e r s ; t h e r e s u l t s b e i n g s t o r e d on a mass s t o r a g e de - v i c e f o r f u r t h e r p r o c e s s i n g (mode i d e n t i f i c a t i o n s t e p ) . S p e c i a l t e c h n i q u e s improv ing t h e a c c u r a c y were deve - l o p e d r e c e n t l y ( e .g . m u l t i - p o i n t e x c i t a t i o n [ Z l , redun- d a n t measurement o f more t h a n o n e row or column [ 3 ] ) .

The se t o f measu red f r e q u e n c y r e s p o n s e s is p ro - c e s s e d by a c u r v e f i t t i n g a l g o r i t h m which i d e n t i f i e s t h e modal p a r a m e t e r s o f t h e s t r u c t u r e ( f i g . l c ; l d ) . Least s q u a r e s e r r o r p r o c e d u r e s a r e m a i n l y used t o s o l v e t h i s p rob lem [2 ;7 ;9 ;121 . The g e n e r a l a l g o r i t m which f i t s t h e c o e f f i c i e n t s of a sum o f s i n g l e DOF s y s t e m s t o t h e measu red 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 s is w e l l known. I t is n o t o n l y s l o w b u t a l s o d i v e r g e s e a s i l y i f t h e i n i t i a l e s t i m a t i o n is n o t good enough. An o t h e r v e r y s i m p l e p r o c e d u r e i s t h e c i r c l e f i t b a s e d on t h e s i m i l a r i t y o f f r e q u e n c y r e s p o n s e r e s o n a n c e s w i t h c i r c l e s . T h i s commonly used method is v e r y f a s t and r e - l i a b l e . Bo th p r o c e d u r e s u n c o u p l e t h e modes o f v i b r a - t i o n . Peak p i c k i n g me thods a r e somet imes used t o o f o r l i g h t l y damped s t r u c t u r e s where t h e modes a re a l r e a d y a l m o s t uncoup led . I n t h i s c a s e , t h e a m p l i t u d e s and p h a s e s a r e s i m p l y p i c k e d o u t a t r e s o n a n c e .

Modal p a r a m e t e r s o b v i o u s l y c a n a l s o b e used i n q u a n t i t a t i v e e v a l u a t i o n s , i . e . f o r t e s t i n g and improv- i n g f i n i t e e l e m e n t m o d e l s [ill, f o r p r o c e s s s i m u l a t i o n s o r f o r s t r u c t u r a l m o d e l l i n g p r o c e d u r e s [ l a ] .

t h e o b j e c t wave t h a t e x i s t e d d u r i n g t h e e x p o s u r e . The o b s e r v e r l o o k i n g t h r o u g h t h e hologram sees a v i r t u a l imaqe o f t h e o b j e c t where t h e o r i g i n a l was s i t u a t e d .

A p h o t o p l a t e c a n s t o r e and r e c o n s t r u c t more t h a n one i n t e r E e r e n c e f i e l d . I f d i f f e r e n t s t a t e s o f a n ob- j e c t , f o r example undeformed and de fo rmed , a r e exposed o n e a f t e r t h e o t h e r on t h e same p h o t o p l a t e , b o t h f o r m s a r e r e c o n s t r u c t e d . The c o r r e s p o n d i n g wave f i e l d s a g a i n i n t e r f e r e . When t h e d e f o r m a t i o n be tween t h e s e fo rms h a s t h e same o r d e r o f m a g n i t u d e a s t h e l a s e r - w a v e l e n g t h , t h e r e s u l t i n g i n t e r f e r e n c e f r i n g e s make t h e d e f o r m a t i o n p e r c e p t i b l e ( f i g . 2 d ) .

B e s i d e s c o n t i n u o u s l a s e r s ( e . g . He-we, Argon) and p u l s l a s e r s ( e . g . R u b i n ) , mixed s y s t e m s , ( e .g . d r i v e d c o n t i n u o u s l a s e r s o r p u l s e d l a s e r s w i t h h i g h r e p e t i t i o n r a t e s ) a r e a l s o a v a i l a b l e . The maximum a d m i s s i b l e move- ment o f t h e o b j e c t d u r i n g t h e e x p o s u r e s h o u l d n o t exceed some t e n t h s o f t h e l a s e r w a v e l e n g t h . O t h e r w i s e no image c o u l d b e r e c o n s t r u c t e d [ 1 3 ] .

The r e c o n s t r u c t i o n o f a ho logram u s i n g a l a s e r w i t h a n o t h e r w a v e l e n g t h d i s t o r t s t h e wave f i e l d g e n e r a t e d b e h i n d t h e ho logram. I n g e n e r a l , t h e r e c o n s t r u c t e d p o i n t s o f t h e o b j e c t f o c u s now i n a s m a l l a r e a and t h e image is n o t l o c a t e d were t h e o r i g i n a l o b j e c t was [141.

The most common ho logram t y p e i n measu r ing t e c h - n i q u e is t h e d o u b l e exposed ho logram ( f i g . 2 d ) . Rea l t ime- , t i m e a v e r a g e - ( f i g . 1 1 ) and sandwich - ho lograms a r e o f t e n used t o o . Holograms a r e u s u a l l y s t o r e d on p h o t o p l a t e s , p h o t o s e n s i t i v e f i l m s o r t h e r m o p l a s t i c f i l m s . T h e r m o p l a s t i c f i l m s a r e r e a d i l y d e v e l o p e d b u t h a v e less r e s o l u t i o n [ 1 3 ] .

The s e n s i t i v e d i r e c t i o n o f e a c h o b j e c t p o i n t is t h e b i s e c t o r a n g l e be tween t h e d i r e c t i o n s o f i l l u m i n a t i o n and o b s e r v a t i o n . Only t h e p r o j e c t i o n o f t h e d e f o r m a t i o n on t h i s s e n s i t i v i t y v e c t o r is r e g i s t e r e d . I f t h i s vec- t o r c o i n c i d e s w i t h t h e d i r e c t i o n o f o b s e r v a t i o n , t h e i n t e r p r e t a t i o n o f t h e i n t e r f e r e n c e f r i n g e s is v e r y s i m p l e . Each a d d i t i o n a l d e f o r m a t i o n o f h a l f t h e wave- l e n g t h i n t h e d i r e c t i o n o f s e n s i t i v i t y c r e a t e s an a d d i - t i o n a l f r i n g e . The f r i n g e s c a n b e i n t e r p r e t e d a s r e l i e f l i n e s 1161. I n p r a c t i c e , g r e a t d i s t a n c e s be tween o b j e c t and ho logram and s m a l l a n g l e s be tween i l l u m i n a t i o n and o b s e r v a t i o n l e a d t o above s i t u a t i o n ( f i g . 3 b , 3 c , 5 ) .

I f a q u a n t i t a t i v e e v a l u a t i o n is p r e f e r r e d , t h e ho- log ram h a s t o be e v a l u a t e d f o r e a c h p o i n t o f i n t e r e s t . By c o u n t i n g t h e f r i n g e s and t a k i n g i n t o a c c o u n t t h e d i - r e c t i o n s o f i l l u m i n a t i o n and o b s e r v a t i o n , o n e component o f d e f o r m a t i o n be tween b o t h e x p o s u r e s may h e c a l c u - l a t e d . A t h r e e - d i m e n s i o n a l d e f o r m a t i o n v e c t o r may b e d e f i n e d from t h e componen t s a l o n g t h r e e d i f f e r e n t s e n - s i t i v i t y v e c t o r s 1141.

Q u a n t i t a t i v e e v a l u a t i o n s c a n nowadays b e a u t o m a t e d i n some p a r t i c u l a r c a s e s [ 2 0 ; 2 1 ] . I n a c o u p l e o f y e a r s t h e o b j e c t and its e n l a r g e d d e f o r m a t i o n v e c t o r w i l l be d i s p l a y e d on a s c r e e n , g i v i n g a c o m p r e h e n s i v e r e p r e s e n - t a t i o n o f t h e r e a l d e f o r m a t i o n s . The main a d v a n t a g e o f h o l o g r a p h i c measu remen t t e c h n i q u e s i s t h e q u i c k and o v e r a l l measu r ing and r e c o r d i n g . T h i s a d v a n t a g e is how- e v e r o n l y o b t a i n e d f o r q u a l i t a t i v e and one -d imens iona l e v a l u a t i o n s pe r fo rmed by a n e x p e r i e n c e d o p e r a t o r .

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D i g i t a l t e c h n i q u e whose r e s u l t s (mode s h a p e s ) a r e n o r m a l l y e v a l u a t e d q u a l i t a t i v e l y .

P u r e l y d y n a m i c a l a n a l y s i s o n l y ( f i g . l e , 4 , 8, 1 0 ) .

Measurement o f s t r u c t u r a l v i b r a t i o n s w i t h a c c e - l e r o m e t e r s f i x e d on t h e s t r u c t u r e : l e a d s t o mass l o a d i n g d i s t o r t i o n s . Mounting o f an e x c i t e r a l s o g i v e s m e c h a n i c a l i n t e r a c t i o n s .

The measu red f r e q u e n c y r e s p o n s e s and t h e r e s u l t - i n g modal p a r a m e t e r s a r e s t o r e d o n a mass s t o - r a g e d e v i c e and a r e v i s u a l i z e d w i t h compute r pe- r i p h e r a l s .

The a n a l y s i s is o n l y d o n e a t d i s c r e t e p o i n t s d i s t r i b u t e d o v e r t h e who le s t r u c t u r e . A simp- l i f i e d mode s h a p e o f a n i d e a l i z e d s t r u c t u r e re- s u l t s . A g r e a t number o f p o i n t s is needed f o r h i g h f r e q u e n c y modes.

Modal a n a l y s i s g e n e r a l l y i n v o l v e s a b s o l u t e v i b - r a t i o n measu remen t s .

Use o f d i g i t a l equ ipmen t e n a b l e s s i g n a l c o n d i - t i o n i n g ( e . g . s i g n a l a v e r a g i n g and f i l t e r i n g ) : o n l y t h e most i n t e r e s t i n g p a r t o f t h e a c t u a l mo- t i o n is o b s e r v e d .

The s e n s i t i v i t y o f t h e measu remen t d e p e n d s on t h e t r a n s d u c e r s and a m p l i f i e r s which a r e c u r - r e n t l y a v a i l a b l e i n w i d e s e n s i t i v i t y r a n g e s .

The measu remen t r a n g e i s p r a c t i c a l l y u n l i m i t e d ( f rom low l e v e l a m p l i t u d e s t o h i g h shock l e v e l s )

I d e a l f o r medium o r b i g s i z e s t r u c t u r e s ( f i g . 4 ) . Weight and d i m e n s i o n s of t h e t r a n s d u c e r s may b e c r i t i c a l on small s t r u c t u r e s .

C a l i b r a t i o n and t h e measu remen t s a r e r e a d i l y done . The t i m e e f f o r t f o r t h e a n a l y s i s d e p e n d s on t h e number o f p o i n t s , t h e c o m p l e x i t y o f t h e f r e q u e n c y r e s p o n s e s and t h e number o f mode s h a p e s i d e n t i f i e d ( f i g . 6 ) . I t may be v e r y l o n g .

I n d u s t r i a l l e v e l o f t h e i n s t r u m e n t a t i o n is a l - r e a d y r e a c h e d .

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T e c h n i q u e w i t h m a i n l y q u a l i t a t i v e c h a r a c t e r i n most a p p l i c a t i o n s .

Almost i d e a l l y s u i t e d f o r s t a t i c measu remen t s ( f i q . 2 d ) b u t e a s i l y a p p l i c a b l e t o dynamica l phenomena ( f i g . 3 c , 5 , 9 , 11).

O p t i c a l c o n t a c t l e s s measu remen t w i t h o u t m e c h a n i c a l i n t e r a c t i o n s w i t h t h e o b j e c t : i t o n l y r e q u i r e s f r e e i l l u m i n a t i o n and o b s e r v a t i o n q f t h e o b j e c t .

The o b j e c t i n f o r m a t i o n is s t o r e d i n a q u a s i con t inuous form on t h e p h o t o p l a t e .

A ho log ram conta ins i n f o r m a t i o n on t h e d e f o r m a t i o n o f t h e who le v i s i b l e s u r f a c e of t h e o b j e c t . Object de - t a i l s and c o m p l i c a t e d h i q h f r e q u e n c y v i b r a t i o n f i e l d s a r e b o t h v i s i b l e .

Ho lography is t y p i c a l l y a r e l a t i v e measurement g i v i n g t h e d i f f e r e n c e be tween t w o s t a t e s o f d e f o r m a t i ? n ( f i g . 2 , 9 ) ( s t a t i c a l a s w e l l a s d y n a m i c a l ) .

I n t e r f e r e n c e f r i n g e s a r e t h e r e s u l t o f t h e a c t u a l mo- t i o n o f t h e o b j e c t , i n c l u d i n g a l l e x t e r n a l p e r t u r b a - t i o n s .

A h i g h s e n s i t i v i t y i n s p l i t s o f l a s e r - w a v e l e n g t h s is a v a i 1 a b l e . Holography h a s a l i m i t e d maximum r a n g e o f de fo rma- t i o n s g i v e n by t h e maximum a c c e p t a b l e f r i n q e d e n s i t y .

S u i t a b l e f o r measu remen t s o n smal l o r m i d d l e s i z e d o b j e c t s ( f i g . 5 , 9 ) .

I n s t a l l a t i o n and a d j u s t m e n t o f t h e s e t u p i s v e r y t i m e consuming. The measurement i s s i m p l y d o n e by o n e ex- p o s u r e .

A c t u a l e q u i p m e n t s a r e s t i l l d e l i c a t e i n s t r u m e n t s .

A r t i f i c i a l e x c i t a t i o n is g e n e r a l l y used ( i m p a c t o r random). The e x c i t a t i o n f o r c e i s a l w a y s mea- s u r e d f o r f r e q u e n c y r e s p o n s e c a l c u l a t i o n s . [ l ; 2 ; 5 ; 81

The v i s u a l i z a t i o n o f v i b r a t i o n s g e n e r a t e d by o p e r a t i n g m a c h i n e s r e q u i r e s a n u n c o n v e n t i o n a l a p p l i c a t i o n o f modal a n a l y s i s . (see ‘ s p e c i a l a p p l i c a t i o n s ” o f t h i s t a b l e )

The measu red 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 s h a v e t o be c u r v e f i t t e d t o - g e t t h e modal p a r a m e t e r s . The v i s u a l i z a t i o n o f mode s h a p e s i s d o n e v e r y e f f i c i e n t l y o n a compute r d i s p l a y . The v i e w i n g p o i n t , t h e s c a l i n g o f a m p l i t u d e s e tc . c a n b e changed w i t h o u t d i f f i c u l t i e s .

T h r e e - d i m e n s i o n a l measu remen t s a re u s u a l i n mod- a l s u r v e y s . A l l t h r e e DOFs o f a p o i n t c a n b e measu red s i m u l t a n e o u s l y I C a 4 -channe l s i g n a l a n a l y s e r i s used .

The c u r v e f i t t i n g n o t o n l y g i v e s mode s h a p e s b u t a l s o e i g e n f r e q u e n c i e s and damping v a l u e s .

Modal a n a l y s i s i n g e n e r a l f i t s complex modes. Modal i d e n t i E i c a t i o n o f h i g h l y damped and c o u p l e d r e s o n a n c e s is d i f f i c u l t i n mos t c a s e s .

S i n u s o i d a l e x c i t a t i o n a t r e s o n a n c e is u s u a l l y a p p l i e d ( f i g . 3 ~ ) . The e x c i t a t i o n f o r c e may b e measu red i n pa- r a l l e l w i t h c o n v e n t i o n a l t e c h n i q u e s .

Measurement o f o p e r a t i n g m a c h i n e s i s p o s s i b l e ( f i g . 9 b ) . The r e s u l t i n q i n t e r E e r e n c e f r i n g e s a r e r ep - r e s e n t a t i v e f o r t h e a c t u a l s t a t e o f v i b r a t i o n .

A ho log ram c a n be i n t e r p r e t e d immedia t e ly . ( f i g . 3 h ) . The p o s s i b i l i t i e s and l imits o f t h e e v a l u a t i o n de - p e n d s o n t h e s e t u p o n t h e s i z e of t h e measu red a r e a and on t h e q u a l i t y , t y p e and s i z e o f t h e ho logram.

U s u a l q u a l i t a t i v e e v a l u a t i o n s a r e o n l y one -d imens ion - a l . O t h e r d i r e c t i o n s c a n b e t a k e n i n t o a c c o u n t w i t h r e f l e c t o r s ( f i g . 5 b ) . A t h r e e - d i m e n s i o n a l d e f o r m a t i o n v e c t o r c a n b e c a l c u - l a t e d by a q u a n t i t a t i v e e v a l u a t i o n . T h i s e v a l u a t i o n is e i t h e r t i m e consuming o r r e q u i r e s a s o p h i s t i c a t e d equ ipmen t [ 1 4 ; 1 6 ; 19; 281

No f r e q u e n c y and damping v a l u e s a r e s t o r e d i n ho log - rams. Resonance f r e q u e n c i e s a r e found by t h e a d d i - t i o n a l v i b r a t i o n t r a n s d u c e r a l r e a d y used f o r t r i g g e r p u r p o s e s .

A p u r e r e a l mode o f v i b r a t i o n is o n l y o b t a i n e d f o r l i g h t l y damped s t r u c t u r e s . T h i s v i b r a t i o n p a t t e r n c a n b e d e f i n e d by o n e time a v e r a q e o r by a d o u b l e p u l s e d ho logram. Complex modes w i t h moving n o d a l l i n e s a r e a l m o s t i m - p o s s i b l e t o a n a l y s e . They c a n o n l y b e r e p r e s e n t e d by many s u c c e s s i v e ho lograms . T h e i r i n t e r p r e t a t i o n i s e x t r e m e l y d i f f i c u l t .

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* Modal a n a l y s i s p r o c e d u r e s d e c o u p l e t h e r e s o - n a n c e s . T h i s mode s e p a r a t i o n i s an i m p o r t a n t f e a t u r e f o r h i g h l y damped and o v e r l a p p i n g modes t h a t would be d i s t o r t e d o t h e r w i s e .

S i n c e t h e i n f o r m a t i o n , s t o r e d i n d i g i t a l fo rm, d e s c r i b e s c o m p l e t e l y t h e d y n a m i c a l b e h a v i o u r , t h e d e f o r m a t i o n o f e a c h p o i n t i n e a c h d i r e c t i o n w i t h r e f e r e n c e t o any g i v e n e x c i t a t i o n c a n b e c a l c u l a t e d .

No r e s t r i c t i n g r e q u i r e m e n t s o n t h e e n v i r o n m e n t : measu remen t s i n d i r t y o r n o i s y rooms and on a l - most any s u r f a c e q u a l i t y may b e d o n e w i t h p a r t i - c u l a r c a r e .

The f i x i n g and a c c u r a t e o r i e n t a t i o n o f a c c e l e r o - m e t e r s o n bended s u r f a c e s t a k e s s p e c i a l c a r e . Measurement p a i n t s s h o u l d p o s s i b l y be c h o o s e n on p l a n e s p o t s o f t h e s t r u c t u r e .

Compl i ca t ed s t r u c t u r e s ( i n c l u d i n g bended s u r - f a c e s ) a r e n o t e a s y t o i d e a l i z e w i t h few p o i n t s . The r e s o l u t i o n f o r h i g h e r modes o f v i b r a t i o n is a l s o l i m i t e d .

A r e p r e s e n t a t i o n o f o p e r a t i n g mach ines c a n be pe r fo rmed i f a r e f e r e n c e a c c e l e r a t i o n i s used i n s t e a d o f t h e r e f e r e n c e f o r c e ( f i q . 8 ) . No c u r v e f i t t i n g o f t h e measu red f r e q u e n c y r e s p o n s e func - t i o n s is made. The r e l a t i v e m a g n i t u d e s and p h a s e s o f t h e v i b r a t i o n a r e s i m p l y p i c k e d o u t a t t h e f r e q u e n c y o f i n t e r e s t and v i s u a l i z e d u s i n g t h e d i s p l a y f e a t u r e o f modal a n a l y s i s .

G e n e r a l improvements: A t r e n d t o compac t and low c o s t i n s t r u m e n t a t i o n is o b s e r v e d . T h i s makes modal a n a l y s i s a v a i l a b l e t o smal l l a b o r a t o r i e s and i n d u s t r i a l r e s e a r c h

g r o u p s . The s o f t w a r e is v e r y w e l l u s e r a d a p t e d . Compact e q u i p m e n t s o n t h e o t h e r s i d e l o o s e i n f l e x i b i l i t y and l e t o n l y few p o s s i b l i t i e s f o r d e v e l o p i n g own s p e c i a l measurement t e c h n i q u e s .

N e w a p p l i c a t i o n s : S i n c e modal p a r a m e t e r s f u l l y r e p r e s e n t t h e dyna- m i c b e h a v i o u r o f a s t r u c t u r e , t h e y c a n be used f o r m o d e l l i n g . T h e r e b y t h e e f f e c t o f p o t e n t i a l s t r u c t u r a l c h a n g e s c a n b e i n v e s t i g a t e d . Coup1 ing o f s u b s t r u c t u r e s is a l s o p r o v i d e d [ 1 0 1 .

Modal a n a l y s i s r e s u l t s may improve o r s u p p l y d a t a f o r f i n i t e e l e m e n t mode l s (111. Dynamical p a r a m e t e r s (damping and s t i f f n e s s v a l u e s ) o f mach ine e l e m e n t s s u c h a s s l i d e s , b o l t e d connec- t i o n s , found by e x p e r i m e n t a r e o b v i o u s l y more a c c u r a t e .

I n t e r p r e t a t i o n : G r a p h i c s o f t w a r e s o € modal s y s t e m s a r e becoming more and more c o m f o r t a b l e : - two mode s h a p e s o n t h e same d i s p l a y - i n c r e a s e d f l e x i b i l i t y o f d i s p l a y o p e r a t i o n s - s u b s t r u c t u r i n q o f mode s h a p e s on c o l o r d i s p - l a y s

I f modes o f v i b r a t i o n a r e c o u p l e d , t h e ho logram rep- r e sen t s t h e s u p e r p o s i t i o n o f d e f o r m a t i o n s r e s u l t i n q from s e v e r a l modes.

A d i g i t a l i s a t i o n o f h o l o g r a m s is p o s s i b l e [ 1 9 ; 2 0 1 , b u t o n l y p a y i n g f o r s p e c i a l a p p l i c a t i o n s o r f o r g r e a t s e r i e s of o b j e c t s u s i n g t h e same h o l o g r a p h i c s e t u p . I t a l l o w s t h e n f u r t h e r d a t a p r o c e s s i n g ( e .9 . s t r a i n and s t r e s s c a l c u l a t i o n s ) .

A new ho logram and a new s e t u p a r e n e c e s s a r y i f an o t h e r p o i n t o f v i ew i s d e s i r e d .

S p e c i a l r e f l e c t i v e p a i n t i n g o f t h e o b j e c t may b e ne- c e s s a r y . I t c a n a l s o b e used t o d r a s t i c a l l y r e d u c e t h e i l l u m i n a t i o n e n e r g y .

Room d a r k e n i n g d u r i n g e x p o s u r e is n e c e s s a r y i f no s p e c i a l f i l t e r s a r e used .

The maximum mesured a r e a o f t h e o b j e c t d e p e n d s pr ima- r i l y o n t h e i n t e n s i t y and on t h e c o h e r e n c e l e n g t h o f t h e l a s e r s o u r c e , t h e n on t h e o p t i c a l and g e o m e t r i c a l s e t u p , on t h e s u r f a c e and s h a p e o f t h e s t r u c t u r e and on t h e s e n s i t i v i t y o f t h e p h o t o p l a t e .

The measu remen t r a n g e is i n f l u e n c e d by t h e l a se r -wa- v e l e n q t h and t h e h o l o g r a p h i c s e t u p . I t a l s o c a n b e w i d e l y a d j u s t e d by c h o o s i n q a n a d e q u a t e t r i g g e r p o i n t and p u l s e d e l a y .

* Holograms o f r o t a t i n g o b j e c t s c a n b e t a k e n by a d e r o - t a t o r [ 2 2 1 .

* The sandwich t e c h n i q u e c a n b e used [ 1 5 ; 2 3 1 t o g e t t h e s i g n o f t h e d e f o r m a t i o n or t o remove a r i g i d body mo- t i o n from f r i n g e p a t t e r n s .

* G e n e r a l improvements: Development o f more u s e r a d a p t e d i n s t r u m e n t a t i o n ( f r o m l a b o r a t o r y t o i n d u s t r i a l e q u i p m e n t ) . T h i s means i n c r e a s e d d u r a b i l i t y , r e l i a b i l i t y and m a i n t e n a n c e and t h u s improved p e r f o r m a n c e s w i t h r e f e - r e n c e t o : - q u a l i t y , s t a b i l i t y and e n e r g y o f l a se r beam - s t a r t up t i m e r e d u c t i o n - deve lopmen t , s e n s i t i v i t y and r e s o l u t i o n of p h o t o

- h a n d l i n g o f e q u i p m e n t ( a d j u s t m e n t s ) - room d a r k e n i n g r e q u i r e m e n t s d u r i n g e x p o s u r e - t r i g g e r i n g o f l a se r p u l s e

Use o f p u l s e and c o n t i n o u s l asers w i t h i d e n t i c a l wa- v e l e n g t h and u s i n g t h e same l i g h t p a t h . T h i s d e v e l o p - ment t e n d s t o a c h i e v e t h e f o l l o w i n g p o i n t s : - no a b e r r a t i o n when r e c o n s t r u c t i n g ho lograms - v i e w i n g o f p u l s e d ho lograms i n r e a l time t e c h n i q u e

s e n s i t i v e p l a t e s

' N e w a p p l i c a t i o n s :

* I n t e r p r e t a t i o n : Q u a n t i t a t i v e a n a l y s i s o f h o l o g r a m s a r e now more and more au tomated . P r o t o t y p e e q u i p m e n t s a r e i n s a l e b u t t h e i r d a t a r e c o r d i n g and t h e p r e s e n t a t i o n o f r e s u l t s s h o u l d s t i l l b e improved [ 1 9 ] .

401

REFERENCES COMBINED APPLICATION OF BOTH TECNNIQUES

Both me thods p r e s e n t e d i n t h i s p a p e r h a v e t h e r e s p e c i f i c a d v a n t a g e s . Modal a n a l y s i s and h o l o q r a p h y may b e used i n p a r a l l e l f o r s o l v i n g complex v i b r a t i o n prob- lems i n order t o b e n e f i t from t h e i r r e s p e c t i v e advan- t a g e s . Holograms o f a n o p e r a t i n g mach ine c a n f o r ex- ample be used f i r s t t o f i n d o u t q u a l i t a t i v e l y , which s t r u c t u r a l e l e m e n t s a r e r e p r e s e n t a t i v e f o r t h e v i b r a - t i o n p rob lem. These e l e m e n t s a r e t h e n a n a l y s e d a c c u r a - t e l y w i t h modal a n a l y s i s w i t h o u t m e a s u r i n g n e c e s s a r i l y t h e who le mach ine . R e c i p r o c a l l y , h o l o g r a p h y c a n a l s o f o l l o w a rough modal a n a l y s i s i f l o c a l d e f o r m a t i o n s o f s m a l l c o m p l i c a t e d p a r t s mus t b e i n v e s t i g a t e d w i t h more d e t a i l s .

The f o l l o w i n g t y p i c a l f e a t u r e s c a n b e deduced from above c o m p a r i s o n o f b o t h t e c h n i q u e s :

Nodal a n a l y s i s

* q u a n t i t a t i v e measu remen t s * b i g and m i d d l e s i z e d s t r u c t u r e s * mode u n c o u p l i n g and v i s u a l i z a t i o n * advanced u s e o f modal d a t a

Ho lography

* q u a l i t a t i v e measu remen t s s t a t i c d e f o r m a t i o n p a t t e r n s c o m p l i c a t e d o b j e c t s o r d e t a i l s

* h i g h r e s o l u t i o n o v e r who le s u r f a c e * c o n t a c t l e s s measu remen t

Such a combined a p p l i c a t i o n is i l l u s t r a t e d w i t h o n e o f our f i r s t examples . F i g . l l a shows t h e m i l l i n g sup- po r t o f a t u r b i n e b l a d e . An i m p o r t a n t mode a t 270 Hz was o b t a i n e d by modal a n a l y s i s ( f i g . 1 0 ) . The d i s t r i b u - t i o n o f p o i n t s o n t h e s t r u c t u r e was t o o rough t o see d e t a i l s o f wha t h a p p e n s be tween s u p p o r t and b l a d e . On t h e o t h e r hand t h e r e is n o t enough s p a c e f o r moun t ing an a c c e l e r o m e t e r on t h e p r o l a o d p i n . The ho logram of f i g . l l b c l e a r l y shows t h e l o c a l d e f o r m a t i o n s o f t h i s weak e l e m e n t t h a t g r e a t l y i n f l u e n c e s t h e r e s o n a n c e o f t h e b l a d e .

More combined e x a m p l e s o f t h i s t y p e a r e however needed b e f o r e p r e c i s e r u l e s and s t r a t e g i e s f o r a n optimum com- mon a p p l i c a t i o n o f h o l o g r a p h y and modal a n a l y s i s c a n b e s e t up.

GYGAX P.E. - P o s s i b i l i t e s e t l imites d e l ' a n a l y s e modale . Mecanique No. 341/342, m a i / j u i n 1978.

Sys t ems . Shock and V i b r a t i o n I n f o r m a t i o n C e n t e r , Wash ing ton , May 1975.

RICHARDSON M . - Yodal A n a l y s i s u s i n g D i q i t a l T e s t

RICHARDSON M . , KNISKERN J. - I d e n t i f y i n g Modes o f L a r g e S t r u c t u r e s from M u l t i p l e I n p u t a n d - Response Measurements . S o c i e t y o f Au tomot ive E n g i n e e r s , SAE 768875, 1976. SNOEYS- R., ROESEMS D., VANDEURZEN U . , VANHONACKER P. - S u r v e y o f Modal A n a l y s i s A p p l i c a t i o n s . Anna l s o f t h e CIRP, Vo1.28/2, 1979.

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