C H A P T E R 1
I N T R O D U C T I O N
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IIT Kharagpur
C H A P T E R 1
I N T R O D U C T I O N
1.1 I n t r o d u c t i o n to B o n d G r a p h s
T h e p r e s e n t w o r k is a i m e d at s t u d y i n g d y n a m i c s y s t e m s t h r o u g h s u b s t r u c t u r i n g a n d b o n d g r a p h t e c h n i q u e s . In t h i s w o r k , a t t e m p t h a s b e e n m a d e to e n h a n c e t he a r t of b o n d g r a p h s i m u l a t i o n s in t h e a n a l y s i s of c o m p l e x d y n a m i c s y s t e m s , t a k i n g a d v a n t a g e of the n a t u r e of s u b s y s t e m s a n d t h e i r i n t e r a c t i o n s . T h e i n s t a n c e s of a p p l i c a t i o n of b o n d g r a p h s in s y s t e m m o d e l
l i n g a r e c o n t i n u o u s l y i n c r e a s i n g . At t h e s a m e t i m e , the g e o g r a p h i c a l t e r r i t o r i e s , o v e r w h i c h b o n d g r a p h s a r e g a i n i n g p o p u l a r i t y , a r e a l s o e x p a n d i n g . H o w e v e r , t h e r e a r e s t i l l s o m e p l a c e s w h e r e b o n d g r a p h s h a v e j u s t a p p e a r e d on the h o r i z o n of s y s t e m a n a l y s i s . T h u s , to i m p r o v e r e a d a b i l i t y of t h e t h e s i s , b o n d g r a p h m n e m o n i c s a r e b r i e f l y e x p l a i n e d a l o n g w i t h s u i t a b l e
r e f e r e n c e s in A p p e n d i x A.
B o n d g r a p h s a r e u s e d to r e p r e s e n t p i c t o r i a l l y t h e e n e r g y e x c h a n g e , s t o r a g e a n d d i s s i p a t i o n a m o n g i n t e r a c t i n g p h y s i c a l s y s t e m s in a n e f f i c i e n t m a n n e r . T h e y a r e a l s o e x t e n d e d to no n- e n e r g e t i c s y s t e m s . U s e f u l n e s s of b o n d g r a p h s in s y s t e m m o d e l l i n g a n d s i m u l a t i o n l i e s in t he p r e c i s e w a y the a s s u m p t i o n s of s y s t e m m o d e l a n d t h e i r m o d i f i c a t i o n s a r e i n c o r p o r a t e d . T h e y a r e u s e d to m o d e l , w i t h t h e h e l p of a s m a l l n u m b e r of b a s i c e l e m e n t s , s y s t e m s o v e r d i f f e r e n t e n e r g y d o m a i n s , n a m e l y , e l e c
t r i c a l , m e c h a n i c a l , h y d r a u l i c , p n e u m a t i c , m a g n e t i c , f l u i d , t h e r m a l a n d o t h e r s in a u n i f i e d m a n n e r . T h t b o n d g r a p h t e c h n i q u e s g i v e r i s e to s e l e c t i o n of s t a t e v a r i a b l e s , d e r i v a t i o n of
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IIT Kharagpur
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s y s t e m e q u a t i o n s a n d t h e i r s o l u t i o n s y s t e m a t i c a l l y . T h e w h o l e p r o c e d u r e m a y be a u t o m a t e d if a d v a n t a g e is t a k e n of the e x i s t i n g c o m p u t e r p r o g r a m s s u c h as E N P O R T C 6 7 ,68 3. C o m p u t e r p r o g r a m s l i k e T U T S I M 0 5 1 , 8 8 3 a n d C A M P C 2 1 , 3 5 J , w h i c h a r e b a s e d on
c o n t i n u o u s s y s t e m s i m u l a t i o n l a n g u a g e s , m a y a l s o be u s e d for
•this p u r p o s e . T h e g r o w i n g a c c e p t a n c e of b o n d g r a p h t e c h n i q u e s , for t h e i r p o w e r in m o d e l l i n g , a n a l y s i s a n d s i m u l a t i o n of p h y s i c a l s y s t e m s , m a y be o b s e r v e d f r o m the u p d a t e d b i b l i o g r a p h y by B o s a n d B r e e d v e l d C4D.
1 . E S u b s t r u c t u r e S y n t h e s i s
In a n a l y s i n g a s y s t e m , v e r y o f t e n it is a d v a n t a g e o u s to r e g a r d it as a n a s s e m b l a g e of s u b s y s t e m s or s u b s t r u c t u r e s (in t h e p r e s e n t w o r k b o t h t e r m s w i l l be u s e d i n t e r c h a n g e a b 1 y ).
T h e s e s u b s t r u c t u r e s a r e a n a l y s e d s e p a r a t e l y a n d l a t e r c o u p l e d to g i v e t h e o v e r a l l s y s t e m m o d e l w h i c h m a y be s t u d i e d f o r t he r e q u i r e d d y n a m i c s of t h e s y s t e m . T h i s t e c h n i q u e of a n a l y s i n g a s y s t e m c o n s i d e r i n g it to be a c o l l e c t i o n of s u b s y s t e m s , e a c h r e p r e s e n t e d s u i t a b l y , is r e f e r r e d to as s u b s t r u c t u r e s y n t h e s i s C 5 4 D .
T h e u s e f u l n e s s of s u c h a n a p p r o a c h is i n c r e a s i n g w i t h t he a d v e n t of s m a l l c o m p u t e r s a n d t h a t c a p a b l e of p a r a l l e l p r o c e s s i n g . T h i s is b e c a u s e s u b s y s t e m s m a y be a n a l y s e d i n d e p e n d e n t l y a n d t h e i r r e d u c e d o r d e r m o d e l s m a y be u s e d in s t u d y i n g the o v e r a l l s y s t e m . T h i s is a l s o s u i t a b l e f or r e - a n a l y s i s i n c o
r p o r a t i n g c h a n g e s in s u b s y s t e m s , w h i c h is a v e r y s i g n i f i c a n t f a c e t of c o m p u t e r a i d e d d e s i g n of e n g i n e e r i n g s y s t e m s .
T h e r e a r e d i f f e r e n t t e c h n i q u e s of s u b s t r u c t u r e s y n t h e s i s ,
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IIT Kharagpur
d e p e n d i n g on the m o d e of r e p r e s e n t a t i o n of the s u b s y s t e m s . M e i r o v i t c h C 5 4 D d i s c u s s e d t h e s e p r i n c i p l e s . M e i r o v i t c h a n d H a l e £2 3, 24 , 5 5 1 ] p r o p o s e d t he u se of a d m i s s i b l e f u n c t i o n s of the s u b s y s t e m s for a n a l y s i n g c o m p l e x s y s t e m s . G r i e f a n d Wu
\
C 2 2 D s u m m a r i s e d d i f f e r e n t s u c h t e c h n i q u e s a n d t h e i r a p p l i c a t i o n s to a w i d e r a n g e of s y s t e m s .
1 . 3 B o n d G r a p h A d a p t e d S u b s r t u c t u r i n g
K r o n C 3 6 U i n t r o d u c e d the c o n c e p t s of s u b s t r u c t u r i n g for p i e c e w i s e a n a l y s i s of l a r g e s y s t e m s . P a y n t e r C 6 4 3 a n d K a r n o p p C 2 9 D d i s c u s s e d t h e r e l a t i o n s h i p b e t w e e n b o n d g r a p h m e t h o d s a n d K r o n ' s c o n c e p t s of d i a k o p t i c s .
K a r n o p p a n d R o s e n b e r g C 2 7 D a n d M a r g o l i s C 41□ p r e s e n t e d m e t h o d s to s t u d y t he i n t e r c o n n e c t e d d i s t r i b u t e d a n d l u p m e d -
s y s t e m s u s i n g n o r m a l m o d e s a n d b o n d g r a p h s . M a r g o l i s a n d Y o u n g C 4 2 D p r o p o s e d a t e c h n i q u e of r e d u c i n g t h e s y s t e m e q u a t i o n o b t a i n e d f r o m b o n d g r a p h s to the c o n v e n t i o n a l s e c o n d o r d e r f o r m s u i t a b l e f o r e i g e n a n a 1y s i s . In R e f . C 4 4 D M a r g o l i s d i s c u s s e d t he m e t h o d of a n a l y s i n g d i s t r i b u t e d s y s t e m s w i t h m i x e d c a u s a l i n p u t s (in b o n d g r a p h s e n s e ) . M a r g o l i s C 4 5 3 a l s o p r e s e n t e d a n a l g o r i t h m f o r s y n t h e s i s of s u b s y s t e m m o d e l s to o b t a i n t h e o v e r a l l s y s t e m r e p r e s e n t a t i o n e x p l o i t i n g t he c a u s a l i n f o r m a t i o n of t h e s u b s y s t e m b o n d g r a p h s . T h i s a p p r o a c h of b o n d g r a p h a d a p t e d s u b s t r u c t u r i n g w a s i l l u s t r a t e d in R e f . C 4 6 D by c o n s i d e r i n g a l a r g e s c a l e s t r u c t u r a l m o d e l . P a s t r n a k C61II p r e s e n t e d a p r o c e d u r e f o r s i m u l a t i n g d y n a m i c s of l a r g e d i s t r i b u t e d s t r u c t u r e t h r o u g h b o n d g r a p h m o d e l l i n g t e c h n i q u e s u s i n g e x p e r i m e n t a l m o d a l a n a l y s i s d a t a of t h e s u b s t r u c t u r e s . M a r g o l i s a n d K a r n o p p C43I3 p r e s e n t e d a b o n d g r a p h b a s e d t e c h n i
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q u e of a n a l y s i n g m u l t i b o d y d i s r t i b u t e d s y s t e m s w i t h g e o m e t r i c n o n l i n e a r i t i e s . M a r g o l i s C 4 8 D s u m m a r i s e d s u c h b o n d g r a p h b a s e d i n v e s t i g a t i o n s on the a n a l y s i s of s y s t e m s c o n s i s t i n g of b o t h d i s t r i b u t e d a n d l u m p e d s u b s y s t e m s .
1 . 4 C l a s s i f i c a t i o n of D y n a m i c S y s t e m s S t u d i e d in t h i s W o r k
In the p r e s e n t w o r k l i n e a r d y n a m i c s y s t e m s h a v e b e e n s t u d i e d t h r o u g h s u b s t r u c t u r i n g a n d b o n d g r a p h t e c h n i q u e s . In t h i s , a s y s t e m is r e g a r d e d as an a s s e m b l a g e of s u b s y s t e m s ( s u b s t r u c t u r e s ) e a c h of w h i c h c a n be m o d e l l e d s u i t a b l y . T h e s u b s y s t e m s a r e a s s u m e d to h a v e d e f i n i t e i n t e r a c t i o n s . T h e s y s t e m s a r e c l a s s i f i e d a c c o r d i n g to the t y p e of s u b s y s t e m s a n d t h e i r i n t e r a c t i o n s .
T h e t y p e s of s y s t e m s c o n s i d e r e d in t h i s w o r k m a y be b r o a d l y d i v i d e d i n t o t wo g r o u p s . T h e f i r s t g r o u p c o n s i s t s of s y s t e m s w i t h i d e n t i c a l s u b s y s t e m s , a n d f o r t h e s e c o n d , t h e s u b s y s t e m s m a y not be i d e n t i c a l . S y s t e m s w i t h i d e n t i c a l s u b s y s t e m s a r e r e f e r r e d to as " r e p e a t e d s y s t e m s " . R e p e a t e d s y s t e m s a r e f u r t h e r c l a s s i f i e d as " p a r a l l e l s y s t e m s " w h i c h c o n s i s t of i d e n t i c a l s u b s y s t e m s c o n n e c t e d in a t o p o l o g i c a l l y i d e n t i c a l m a n n e r w i t h l o n g - r a n g e a n d n o n - i d e n t i c a 1 i n t e r a c t i o n s . T h e y a r e d i f f e r e n t f r o m t h e " p e r i o d i c s y s t e m s " in a s e n s e t h a t t h e i n t e r c o n n e c t i o n s m a y n o t be i d e n t i c a l a n d t h e y n e e d n ot be r e s t r i c t e d to t h e i m m e d i a t e n e i g h b o u r h o o d .
P a r a l l e l s y s t e m s m a y be f u r t h e r c l a s s i f i e d on t h e b a s i s of i n t e r a c t i o n s a m o n g s u b s y s t e m s . W h e n t h e i n e r a c t i o n s a r e s u c h t h a t t h e y c a n be d e r i v e d f r o m a s e t of p o t e n t i a l f u n
c t i o n s , t h e y a r e r e f e r r e d to as " p o t e n t i a l i n t e r a c t i o n s " ,
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o t h e r w i s e t h e y a r e c a l l e d " n o n - p o t e n t i a l i n t e r a c t i o n s " . The p o t e n t i a l i n t e r a c t i o n s a r e d i v i d e d i n t o two g r o u p s . T h e y a r e
" c o m m u t a t i v e " , w h e n the i n t e r a c t i o n s a r e s u c h t h a t t he o p e r a t o r s or m a t r i c e s r e p r e s e n t i n g t h e m c o m m u t e . T h e i n t e r a c t i o n s a r e " n o n - c o m m u t a t i v e ", w h e n t he c o r r e s p o n d i n g m a t r i c e s do not c o m m u t e . T h e " c o m m u t a t i v e p o t e n t i a l " i n t e r a c t i o n s h a v e s o m e s p e c i a l r e l a t i o n s t h a t t h e y a r e a m e n a b l e to s i m u l t a n e o u s d e c o u p l i n g ( d i a g o n a l i s a t i o n ) by a s i n g l e t r a n s f o r m a t i o n . T h i s p r o p e r t y is f u l l y m a d e u se of in the p r e s e n t s t u d y . T h e " n o n - c o m m u t a t i v e p o t e n t i a l " i n t e r a c t i o n s h a v e b e e n s t u d i e d u s i n g t h e t e c h n i q u e of o p e r a t o r p e r t u r b a t i o n s . In the c a t e g o r y of n o n - p o t e n t i a l i n t e r a c t i o n s , o n l y the c o m m u t a t i v e c a s e h a s b e e n d i s c u s s e d .
T h e s y s t e m s w i t h n o n - i d e n t i c a l s u b s y s t e m s a n d a g e n e r a l k i n d of i n t e r a c t i o n s h a v e a l s o b e e n c o n s i d e r e d in t h i s w o r k . T h i s c a t e g o r y of s y s t e m s a r e s t u d i e d by a t e c h n i q u e k n o w n as d u a l f o r m u l a t i o n . T h e d u a l f o r m u l a t i o n is b a s e d on t he m o m e n t u m c o o r d i n a t e s u n l i k e t he u s u a l N e w t o n i a n f o r m u l a t i o n b a s e d on g e o m e t r i c c o o r d i n a t e s . In t h i s , s u b s y s t e m s a r e m o d e l l e d i n d i v i d u a l l y w i t h p r o p e r b o u n d a r y c o n d i t i o n s at t h e i n t e r f a c e s . T h e y a r e l a t e r c o u p l e d u t i l i s i n g t h e c o n s t r a i n t s of
c o n t i n u i t y a n d e q u i l i b r i u m at t h e i n t e r f a c e s . H e r e t h e s u b s y s t e m s m a y be f r o m d i f f e r e n t e n e r g y d o m a i n s . In t h e p r e s e n t
s t u d y t w o i m p o r t a n t r e p r e s e n t a t i v e c l a s s e s a r e c o n s i d e r e d . T h e y a r e s t r u c t u r a 1- s t r u c t u r a 1 a n d s t r u c t u r a 1 - a c o u s t i c s y s t e m s r e s p e c t i v e l y . T h e d i f f e r e n c e b e t w e e n t h e t w o c l a s s e s l i e s in t h e m e t h o d of d u a l i s a t i o n of t he s u b s y s t e m s . H o w e v e r , the p r o c e d u r e m a y be e x t e n d e d a n a l o g o u s l y to o t h e r s y s t e m s .
F i g u r e 1.1 s h o w s the c l a s s i f i c a t i o n of s y s t e m s s t u d i e d
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IIT Kharagpur
DYNAMICSYSTEMS
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in t h i s w o r k , a c c o r d i n g to th e s u b s y s t e m s a n d t h e i r i n t e r a c t i o n s . P e r i o d i c s y s t e m s a r e not c o n s i d e r e d in the p r e s e n t
i n v e s t i g a t i o n a n d a re s h o w n w i t h a d i f f e r e n t b o u n d a r y in F i g . 1.1. In the f o l l o w i n g s e c t i o n s a b r i e f r e v i e w of r e l e v a n t l i t e r a t u r e on a b o v e s y s t e m s is p r e s e n t e d . T h i s h e l p s in i d e n t i f y i n g t h e s c o p e of p r e s e n t w o r k .
1 . 5 S u r v e y of R e l e v a n t L i t e r a t u r e
In t h i s s e c t i o n a b r i e f s u r v e y of l i t e r a t u r e on r e l e v a n t t o p i c s is p r e s e n t e d . T h i s h e l p s in i d e n t i f y i n g the s c o p e of t h e e x i s t i n g m e t h o d s a n d , in t u r n , t h e n e e d for n e w t e c h n i q u e s . H o w e v e r , to i m p r o v e r e a d a b i l i t y of e a c h c h a p t e r , the
r e f e r e n c e s a r e c i t e d a g a i n at p r o p e r p l a c e s in the t e x t .
P e r i o d i c S y s t e m s
P e r i o d i c s y s t e m s h a v e b e e n s t u d i e d by m a n y i n v e s t i g a t o / s . B r i l l o u i n C 5 3 i n i t i a t e d the s t u d y of t he . p r o p a g a t i o n of h a r m o
n i c w a v e s t h r o u g h p e r i o d i c s t r u c t u r e s in d i s c r e t e c r y s t a l l a t t i c e s . M e a d £4911 i n t r o d u c e d a g e n e r a l t h e o r y of w a v e p r o p a g a t i o n in l i n e a r p e r i o d i c s y s t e m s w i t h m u l t i p l e c o u p l i n g . T h i s w a s s u b s e q u e n t l y e n h a n c e d in a t w o - p a r t w o r k C 5 0 3 to s t u d y t h e r e l a t i o n s h i p b e t w e e n the b o u n d i n g f r e q u e n c i e s of t he p r o p a g a t i o n z o n e s a n d n a t u r a l f r e q u e n c i e s of t h e i n d i v i d u a l s u b s t r u c t u r e s for b o t h " m o n o " - a n d " m u 1 1 i c o u p i e d s y s t e m s . D e n k e et al. C 141 p r e s e n t e d a m a t r i x d i f f e r e n c e t e c h n i q u e to o b t a i n h a r m o n i c r e s p o n s e of t he s y s t e m . M e i r o v i t c h a n d E n g l e s e x t e n d e d t h i s t e c h n i q u e , to s t u d y p e r i o d i c s y s t e m s , by u s i n g m e t h o d of Z - t r a n s f o r m a n d m o d a l a n a l y s i s £ 1 8 , 5 3 3 . S e n G u p t a £ 7 6 , 7 7 3 s u m m a r i s e d t h e o r e t i c a l a n d e x p e r i m e n t a l w o r k on t h i s s u b j e c t .
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IIT Kharagpur
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H o w e v e r , t h e r e e x i s t s a c l a s s of r e p e a t e d s y s t e m s in w h i c h the i n t e r c o n n e c t i o n s m a y not be i d e n t i c a l a n d t h e y n e e d not be r e s t r i c t e d to the i m m e d i a t e n e i g h b o u r h o o d . T h e i n t e r a c t i o n s m a y be e l a s t i c , e l a s t i c w i t h d a m p i n g or v i s c o - e l a s t i c . In s u c h s y s t e m s , t e r m e d h e r e " p a r a l l e l " s y s t e m s , t he p e r i o d i c s t r u c t u r e t h e o r y w i l l n ot be a p p l i c a b l e as s u c h . A s s u m p t i o n of e x i s t e n c e of h a r m o n i c s o l u t i o n is a l s o a r e s t r i c t i o n in t he p r e s e n t c a s e . It is d e s i r a b l e to a d o p t a s u i t a b l e m e t h o d of a n a l y s i s f o r t h e s e s y s t e m s t a k i n g a d v a n t a g e of the i d e n t i c a l s u b s y s t e m s a n d n a t u r e of t h e i r i n t e r a c t i o n s p r o p e r l y .
P a r a l l e l S y s t e m s w i t h P o t e n t i a l I n t e r a c t i o n s
In p a r a l l e l s y s t e m s w i t h m u l t i p l e i n t e r a c t i o n s a m o n g t h e s u b s y s t e m s , t h e m e t h o d of a n a l y s i s d e p e n d s o n t he n a t u r e of i n t e r a c t i o n s . P a r a l l e l s y s t e m s w i t h c o m m u t a t i v e , p o t e n t i a l i n t e r a c t i o n s b e t w e e n s u b s t r u c t u r e s h a v e b e e n s t u d i e d by S a m a n t a a n d M u k h e r j e e C 7 1 3 . T u r b i n e b l a d e s i n t e r c o n n e c t e d w i t h s h r o u d s a n d l a c i n g w i r e s a r e a m o n g t h e e x a m p l e s of p a r a l l e l s y s t e m s w i t h n o n - c o m m u t a t i v e , p o t e n t i a l i n t e r a c t i o n s . N a t u r a l f r e q u e n c i e s of b a n d e d g r o u p of t u r b i n e b l a d e s w e r e s t u d i e d by P r o h l C 6 6 3 , a n d W e a v e r a n d P r o hi C 8 9 3 c o n s i d e r i n g the d i s c r e - t i s e d b l a d e s c o u p l e d t h r o u g h rnassless e l a s t i c s h r o u d s a n d a d a p t i n g the p r o c e d u r e to d i g i t a l c o m p u t a t i o n . T h o m a s a n d B e l e k [1843 p r e s e n t e d a t e c h n i q u e f or o b t a i n i n g th e v i b r a t i o n c h a r a c t e r i s t i c s of a m u l t i b l a d e p a c k e t f r o m t h e f r e q u e n c y i n f e r e n c e d i a g r a m of a t w o - b l a d e p a c k e t . T h i s i n f e r e n c e d i a g r a m c a n be c o n s t r u c t e d f r o m t h e a n a l y s i s of i n d e p e n d e n t m o d e s of u n c o u p l e d c o m p o n e n t b l a d e s a n d s h r o u d s .
T u n c e l et al. L & 7 1 a n a l y s e d s u c h s y s t e m s in t w o s t e p s .
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F'irst, t h e u n c o u p l e d b l a d e s w e r e a n a l y s e d a n d t h e n t h e e f f e c t s of c o u p l i n g , d ue to the i n t e r c o n n e c t i o n s , w e r e i n c o r p o r a t e d as p e r t u r b a t i o n s . T h o u g h t h i s m e t h o d is a t t r a c t i v e f o r its n o v e l a p p r o a c h , it l a c k e d in b o t h q u a n t i t a t i v e a n d q u a l i t a t i v e r e s u l t s , o w i n g to the a p r i o r i a s s u m p t i o n of w e a k c o u p l i n g . C h e n a n d W a d a C 63 u s e d m a t r i x p e r t u r b a t i o n t e c h n i q u e s for s t u d y i n g the e f f e c t of s m a l l c h a n g e s on t h e d y n a m i c r e s p o n s e of s t r u c t u r a l s y s t e m s . S u - H u a n et al. C 7 8 3 d e v e l o p e d a p r o c e d u r e b a s e d on t h i s t e c h n i q u e to o b t a i n the f r e e - f r e e m o d e s a n d f r e q u e n c i e s of s t r u c t u r a l s y s t e m s f r o m the m e a s u r e d
d a t a of t he c o n s t r a i n e d s y s t e m s .
H o w e v e r , s u c h s y s t e m s c a n be s t u d i e d in a m o r e e f f e c t i v e m a n n e r t a k i n g a d v a n t a g e of b o t h b o n d g r a p h t e c h n i q u e s a n d a s u i t a b l e p r e - t r e a t m e n t f o l l o w e d by o p e r a t o r p e r t u r b a t i o n , w i t h o u t r e s o r t i n g to t h e a s s u m p t i o n of w e a k c o u p l i n g r i g h t f r o m t h e b e g i n n i n g .
P a r a l l e l S y s t e a s w i t h C o B s u t a t i v e , N o n - p o t e n t i a l I n t e r a c t i o n s P r o b l e m s of f l u i d e l a s t i c v i b r a t i o n s of t u b e a r r a y s in h e a t e x c h a n g e r s a n d t h a t of f u e l r o d s in n u c l e a r r e a c t o r s b e l o n g to t h i s c l a s s of p a r a l l e l s y s t e m s w i t h n o n - p o t e n t i a l i n t e r a c t i o n s . B l e v i n s C 1□ d i s c u s s e d d i f f e r e n t m e c h a n i s m s l i k e t u r b u l e n c e , v o r t e x s h e d d i n g , j e t s w i t c h i n g w h i c h c a n i n d u c e i n s t a b i l i t i e s of t u b e a r r a y s u n d e r t h e a c t i o n of c r o s s f l o w s . In t h i s , B l e v i n s p r e s e n t e d t he q u a s i - s t a t i c f l u i d f o r c e c o e f f i c i e n t s f o r c a l c u l a t i n g t h e c r i t i c a l f l o w v e l o c i t i e s .
T a n a k a et al. C 6 0 , 8 1 - 8 3 3 m e a s u r e d f l u i d f o r c e s a n d u s e d t h e m to get c r i t i c a l v e l o c i t i e s f o r d i f f e r e n t a r r a n g e m e n t s of
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t u b e a r r a y s a n d f l u i d f l o w . C h e n 119,1011 d i s c u s s e d t wo m a i n m e c h a n i s m s of i n s t a b i l i t i e s b a s e d on d i s p l a c e m e n t a n d v e l o c i t y d e p e n d e n t f l u i d f o r c e s . C h e n C 8 , 11 D a l s o s u m m a r i s e d t h e a v a i l a b l e l i t e r a t u r e e m p h a s i s i n g on the a n a l y t i c a l t e c h n i q u e s , e x p e r i m e n t a l r e s u l t s a n d f u t u r e t r e n d s of r e s e a r c h in t h i s f i e l d .
B l e v i n s ' t h e o r y C 1 D is q u i t e h e l p f u l in u n d e r s t a n d i n g the p h e n o m e n a of f l o w i n d u c e d v i b r a t i o n s w i t h m o d e r a t e c o m p u t a t i o n . In t he a n a l y s i s of B l e v i n s , e f f e c t s of d i f f e r e n t c o n s t r a i n t s a m o n g a d j a c e n t t u b e s in the a r r a y a n d t he d y n a m i c d e t a i l s of i n d i v i d u a l t u b e s a r e n o t c o n s i d e r e d . T h e s e m i a n a - l y t i c a l t e c h n i q u e of T a n a k a et al. a n d C h e n r e q u i r e s a s i g n i f i c a n t a m o u n t of m e a s u r e m e n t a n d c o m p u t a t i o n to d e t e r m i n e the f l u i d f o r c e c o e f f i c i e n t s w h i c h a r e n e c e s s a r y f o r f u r t h e r a n a l y s i s .
H o w e v e r , B l e v i n s ' t h e o r y can be u t i l i s e d in a b e t t e r w a y if the n o n - p o t e n t i a l n a t u r e of the i n t e r a c t i o n a m o n g the t u b e s , du e to f l u i d f l o w , is i d e n t i f i e d a n d e x p l o i t e d p r o p e r l y . * T h i s h e l p s a l s o in r e d u c i n g t h e m e a s u r e m e n t a n d c o m p u t a t i o n c o s t s c o n s i d e r a b l y .
B o n d G r a p h B a s e d D u a l F o r m u l a t i o n
M o s t of t h e a n a l y s e s of d y n a m i c s y s t e m s a r e b a s e d on u s u a l g e o m e t r i c c o - o r d i n a t e s . H o w e v e r , t h e p o s s i b i l t y of an a l t e r n a t i v e ( d u a l ) f o r m u l a t i o n b a s e d on m o m e n t u m c o - o r d i n a t e s w a s p o i n t e d o u t by T r e n t C 8 6 3 , T o u p i n C 8 5 1 a n d C h r a n d a l l et a l . C 12 H . K a r n o p p C 2 6 3 e x t e n d e d t h e i d e a to s t u d y c o u p l e d v i b r a t o r y s y s t e m s . In R e f . C 3 2 3 , K a r n o p p s t u d i e d s y s t e m of a c o u s t i c f i l t e r s u s i n g d u a l f o r m u l a t i o n s a n d n o r m a l m o d e s .
1 0
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L e b r u n u t i l i s e d t h i s a p p r o a c h in a n a l y s i n g h y d r a u l i c l i n e s a n d f l u i d f l o w n e t w o r k s C 3 7 , 3 Q D .
In R e f . C3411, K a r n o pp p r e s e n t e d f o u r a l t e r n a t i v e f o r m u l a t i o n s b a s e d on m a s s d i s p l a c e m e n t s , m o m e n t a , s p r i n g i m p u l s e a n d s p r i n g d i s p l a c e m e n t c o o r d i n a t e s r e s p e c t i v e l y . T h e o r d e r of the e q u a t i o n v a r i e s w i t h the t y p e of f o r m u l a t i o n , as d i s c u s s e d in R e f . C 3 3 3 . K a r n o p p a l s o p r o p o s e d a n a l t e r n a t i v e p r o c e d u r e of m o d a l a n a l y s i s of the f i r s t o r d e r s t a t e e q u a t i o n s d e r i v e d f r o m t he s y s t e m b o n d g r a p h s , l e a d i n g to t h e r e p r e s e n t a t i o n of the s y s t e m in f o r m of a n o t h e r b o n d g r a p h . It r e t a i n s t h e f e a t u r e of s y m m e t r y of b o n d g r a p h s t a t e e q u a t i o n s in b o t h t y p e s of e n e r g y v a r i a b l e s ( d i s p l a c e m e n t s a n d m o m e n t a ) a n d b o t h t y p e s of e x c i t a t i o n s ( e f f o r t a n d m o t i o n ) . T h e b o n d g r a p h so o b t a i n e d is s o m e w h a t u n u s u a l in a s e n s e t h a t e a c h m o d a l o s c i l l a t o r (a c o m b i n a t i o n of I-C c o n n e c t e d to a n S or 1- j u n c t i o n ) g e t s d e p i c t e d by a n I a n d a C e l e m e n t c o n n e c t e d by a TF w i t h m o d u l u s e q u a l to the c o r r e s p o n d i n g n a t u r a l f r e q u e n c y . It g i v e s r i s e to a n i n c r e a s e in the n u m b e r of b o n d s in the f i n a l b o n d g r a p h .
H o w e v e r , t h e r e e x i s t s a p o s s i b i l i t y of a n o t h e r p r o c e d u r e w h i c h is s i m i l a r to the u s u a l f o r m u l a t i o n . It is a l s o n e c e s s a
ry, at t h e s u b s y s t e m l e v e l , to e l i m i n a t e t h e h i g h f r e q u e n c y m o d e s d ue to c o u p l i n g c o m p l i a n c e s i n c o r p o r a t e d a m o n g t he s u b s y s t e m s for e a s e of a n a l y s i s C 4 4 ,4 5 D , in c a s e of d i s t r i b u t e d s y s t e m s . T h i s w i l l be p a r t i c u l a r l y i m p o r t a n t in a n a l y s i n g r e s p o n s e of s u c h s y s t e m s . A n e c e s s i t y is f e l t to s t u d y the d u a l f o r m u l a t i o n t a k i n g f u l l a d v a n t a g e of th e b o n d g r a p h t e c h n i q u e s a n d d e v e l o p it as a u s e f u l t o o l f o r s y s t e m a n a l y s i s
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t h r o u g h s u b s t r u c t u r i n g .
M a r g o l i s a n d T a b r i z i £ 4 7 3 p r o p o s e d a p r o c e d u r e of r e p e a t e d m o d a l d e c o m p o s i t i o n of t h e s y s t e m e q u a t i o n to s t u d y t he d y n a m i c s of i n t e r a c t i n g l u m p e d a n d d i s t r i b u t e d s y s t e m s . It is f o u n d n e c e s s a r y to e x t e n d t h e s a m e c o n c e p t to the b o n d g r a p h a d a p t e d du al s p a c e f o r m u l a t i o n .
A c o u s t o e l a s t i c S y s t e m s
S y s t e m s c o n s i s t i n g of a c o u s t i c a n d s t r u c t u r a l c o m p o n e n - t e s, t e r m e d " a c o u s t o e 1 a s t i c " s y s t e m s £ 1 6 3 w e r e s t u d i e d by L y o n
£ 4 0 3 , D o w e l l a n d V o s s £ 1 5 3 , a n d M o r s e a n d I n g a r d £ 5 6 3 f or s i m p l e h a r m o n i c m o t i o n s of s t r u c t u r e s . D o w e l l et al. £ 1 6 3 d e v e l o p e d the e q u a t i o n s f o r a r b i t r a r y w a l l m o t i o n s u s i n g G r e e n ' s t h e o r e m f o r the a c o u s t i c r e g i o n . T h e c o u p l e d s y s t e m e q u a t i o n c o u l d be o b t a i n e d f r o m t h e in v a c u o s t r u c t u r a l m o d e s a n d r i g i d w a l l a c o u s t i c m o d e s . D o w e l l £ 1 7 3 s u r v e y e d t h e a v a i l a
b l e l i t e r a t u r e on v e h i c l e i n t e r i o r n o i s e p r e d i c t i o n . N e f s k e et al. £ 5 8 , 5 9 3 p r e s e n t e d r e v i e w s of c u r r e n t p r a c t i c e s f o r v e h i c l e i n t e r i o r n o i s e a n a l y s i s b a s e d on f i n i t e e l e m e n t m e t h o d . T a b a r r o k £ 8 0 3 p r e s e n t e d a d u a l f o r m u l a t i o n f or a c o u s t o e 1 a s t i c s y s t e m s . It is f e l t a d v a n t a g e o u s to e x t e n d c o n c e p t s of b o n d g r a p h a d a p t e d d u a l f o r m u l a t i o n to th e a n a l y s i s of s u c h
s y s t e m s .
1 . 6 S c o p e of t h e P r e s e n t W o r k
F r o m t h e a b o v e d i s c u s s i o n , it is f e l t t h a t t h e r e e x i s t s a s c o p e f o r a n o r g a n i s e d s t u d y of a w i d e r a n g e of d y n a m i c s y s t e m s , w i t h h e l p of b o n d g r a p h t e c h n i q u e s , t a k i n g f u l l a d v a n t a g e of t h e i r f l e x i b i l t y in m o d e l l i n g m u l t i - e n e r g y d y n a m i c
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s y s t e m s a n d u n i q u e n e s s in f o r m u l a t i o n of s y s t e m e q u a t i o n s . T h e u n i q u e n e s s l i e s in the s y m m e t r i c m a n n e r of t r e a t i n g m o m e n t u m a n d d i s p l a c e m e n t v a r i a b l e s a n d b o t h f l o w a n d e f f o r t t y p e of e x c i t a t i o n s . T h i s f e a t u r e h e l p s in e x t r a c t i n g u s e f u l i n f o r m a t i o n , f r o m t h e b o n d g r a p h of t he s y s t e m , for f u r t h e r p r o c e s s i n g in s i m u l a t i n g its d y n a m i c s .
T h e p r e s e n t w o r k is d i s c u s s e d in t h e n e x t e i g h t c h a p t e r s . C h a p t e r s 2 to 4 d e a l w i t h t h e a n a l y s i s of r e p e a t e d s y s t e m s . In C h a p t e r s 5 to 8 n o n - r e p e a t e d s y s t e m s h a v e b e e n s t u d i e d u s i n g d u a l s p a c e f o r m u l a t i o n . C o n t e n t s of e a c h c h a p t e r a r e s u m m a r i s e d in t h e f o l l o w i n g p a r a g r a p h s .
P a r a l l e l S y s t e m s w i t h C o m m u t a t i v e , P o t e n t i a l I n t e r a c t i o n s
In C h a p t e r 2 a p r o c e d u r e is d i s c u s s e d to s t u d y t h e d y n a m i c s of p a r a l l e l s y s t e m s w i t h c o m m u t a t i v e , p o t e n t i a l i n t e r a c t i o n s . S u b s y s t e m s a r e d e c o u p l e d by a s i m i l a r i t y t r a n s f o r m a t i o n
i n f l u e n c i n g o n l y the i n t e r a c t i o n s . T h e p r o c e s s of d e c o u p l i n g c o n s i s t s of t wo s t e p s . In t h e f i r s t s t e p , a p i l o t b o n d g r a p h is c o n s t r u c t e d r e p l a c i n g e a c h s u b s y s t e m by a u n i t m a s s at the p o i n t of i n t e r c o n n e c t i o n a n d r e t a i n i n g o n l y o n e i n t e r a c t i o n at a t i m e . T h e s e c o n d s t e p c o n s i s t s of c o n s t r u c t i o n of a d e c o u p l ed s u b s y s t e m b o n d g r a p h w i t h t he c o n s t r a i n i n g e l e m e n t s o b t a i n e d f r o m t h e e i g e n a n a 1 y s i s of t h e p i l o t b o n d g r a p h . In c a s e of s y s t e m s w i t h m o r e t h a n o n e c o m m u t a t i v e i n t e r a c t i o n s , t he p i l o t b o n d g r a p h w i t h o n e i n t e r a c t i o n n e e d be c o n s i d e r e d , a n d e f f e c t s of o t h e r i n t e r a c t i o n s m a y be s u i t a b l y i n c o r p o r a t e d in t h e d e c o u p l e d s u b s y s t e m m o d e l . T h e o v e r a l l s y s t e m d y n a m i c s c an be s t u d i e d w i t h t h e s a m e c o m p u t a t i o n a l e f f o r t n e c e s s a r y in a n a l y s i n g a s i n g l e s u b s y s t e m .
1 3
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P a r a l l e l S y s t e m s w i t h N o n - c o m m u t a t i v e , P o t e n t i a l I n t e r a c t i o n s C h a p t e r 3 d e a l s w i t h a t e c h n i q u e to s t u d y the d y n a m i c s of a c l a s s of r e p e a t e d s y s t e m s w i t h n o n - c o m m u t a t i v e i n t e
r a c t i o n s . T h e s y s t e m e q u a t i o n s a r e f o r m u l a t e d u s i n g t e n s o r p r o d u c t a l g e b r a . A p r e - a n a l y s i s of the s y s t e m e q u a t i o n s is c a r r i e d o u t to r e d u c e t h e i n t e r a c t i o n s to a s m a l l p e r t u r b i n g o p e r a t o r . T h e p r o c e d u r e of p r e - t r e a t m e n t is a d a p t e d to b o n d g r a p h t e c h n i q u e s fo r th e e a s e of r e p r e s e n t a t i o n of d y n a m i c s y s t e m s . It is s h o w n t h a t t h e m e t h o d of o p e r a t o r p e r t u r b a t i o n m a y a l s o be h a n d l e d v e r y c o n v e n i e n t l y by b o n d g r a p h a p p r o a c h . T h e o v e r a l l s y s t e m d y n a m i c s c a n be s t u d i e d f r o m t h e a n a l y s i s of t he s u b s y s t e m s c o u p l e d t h r o u g h t h e p e r t u r b i n g o p e r a t o r t h u s r e d u c i n g th e s i z e of the p r o b l e m to be c o n s i d e r e d at a t i m e . T h e p r o c e d u r e h e l p s in a s s e s s i n g t h e e f f e c t s of d i f f e r e n t i n t e r a c t i o n s in a s y s t e m a t i c m a n n e r .
P a r a l l e l S y s t e m s w i t h C o m m u t a t i v e , N o n - p o t e n t i a l I n t e r a c t i o n s C h a p t e r 4 is d e v o t e d to t h e d i s c u s s i o n of a p r o c e d u r e f o r s t u d y i n g t h e d y n a m i c s of p a r a l l e l r e p e a t e d s y s t e m s w i t h n o n
p o t e n t i a l i n t e r a c t i o n s w i t h i n the f r a m e w o r k of b o n d g r a p h t e c h n i q u e s . T h i s t y p e of i n t e r a c t i o n , w h e n d e c o u p l e d , g i v e s r i s e to c o m p l e x e l e m e n t s w h i c h c a n n o t be u s e d f o r further- p r o c e s s i n g in t h e c o n v e n t i o n a l b o n d g r a p h a n a l y s i n g p a c k a g e s l i k e E N P O R T . T h i s s h o r t c o m i n g is r e m o v e d by r e f o r m u 1 a t i n g t he d e c o u p l e d s u b s y s t e m e q u a t i o n s to a n e q u i v a l e n t f o r m w i t h a l l r e a l p a r a m e t e r s l e a d i n g to r e d u c t i o n in s i z e of t he p r o b l e m to be s t u d i e d at a t i m e . T h e p r o c e d u r e is i l l u s t r a t e d by t h e e x a m p l e of f l u i d e l a s t i c v i b r a t i o n of t u b e a r r a y s in h e a t e x c h a n g e r s . T h e t e c h n i q u e is s u i t a b l e f o r i n t e r a c t i v e d e s i g n
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IIT Kharagpur
p u r p o s e s b e c a u s e of its a b i l i t y in a s s e s s i n g the e f f e c t s of d i f f e r e n t s y s t e m p a r a m e t e r s on d e s i g n v a r i a b l e s w i t h m o d e r a t e
c o m p u t a t i o n a n d r e a s o n a b l e a c c u r a c y .
B o n d G r a p h A d a p t e d D u a l S p a c e F o r m u l a t i o n
In C h a p t e r 5 a t e c h n i q u e is p r e s e n t e d to s t u d y the o v e r a l l c h a r a c t e r i s t i c s of c o m p l e x d y n a m i c s y s t e m s f r o m t h o s e of t he s u b s y s t e m s w h i c h a r e e a s y to m o d e l i n d i v i d u a l l y . T h e c o m p l e x s y s t e m is d i v i d e d i n t o a n u m b e r of s i m p l e s u b s y s t e m s e a c h of w h i c h m a y be r e p r e s e n t e d by u s i n g e i t h e r p o s i t i o n - v e l o c i t y or i m p u l s e - f o r c e d e s c r i p t i o n s . T h e a p p r o p r i a t e b o u n d a r y c o n d i t i o n s a r e a s s i g n e d at the i n t e r f a c e s . T h e s u b s y s t e m s a r e t h e n i n t e r p r e t e d in f o r m of b o n d g r a p h s u s i n g a f i n i t e m o d e r e p r e s e n t a t i o n . T h e s e b o n d g r a p h s a r e c o u p l e d t h r o u g h p r o p e r e l e m e n t s s u c h t h a t t h e c o n d i t i o n s of e q u i l i b r i u m a n d c o n t i n u i t y at t h e i n t r e f a c e s a r e p r e s e r v e d . T h e o v e r a l l s y s t e m d y n a m i c s a r e s t u d i e d f r o m t he r e s u l t i n g b o n d g r a p h . T h e p r o c e d u r e is i l l u s t r a t e d t h r o u g h s i m p l e e x a m p l e s . T h e e n t i r e a n a l y s i s is b a s e d o n b o n d g r a p h t e c h n i q u e s a n d t h u s a l l o w s t h e e f f e c t i v e u s e of E N P O R T p a c k a g e s . T h e m a i n a d v a n t a g e of t h e p r o p o s e d t e c h n i q u e l i e s in u n a m b i g u o u s s u b s t r u c t u r i n g of m u l - t i - c o n n e c t e d d i s t r i b u t e d s y s t e m s a n d r e m o v a l of r e d u n d a n t s t a t e v a r i a b l e s . It, is a l s o s u i t a b l e f o r " s t i f f " s y s t e m s b e c a u s e of m o d a l r e d u c t i o n at s u b s y s t e m l e v e l .
R e p e a t e d H o d a l D e c o n p o s i t i o n in B o n d G r a p h A d a p t e d D u a l S p a c e .
T h e o v e r a l l s y s t e m d y n a m i c s c a n be s t u d i e d u s i n g d u a l f o r m u l a t i o n as d i s c u s s e d in t h e p r e v i o u s p a r a g r a p h . F r o m t h e
1 5
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c o u p l e d s u b s y s t e m b o n d g r a p h s in d u a l s p a c e the o v e r a l l s y s t e m m o d e l can be r e c o n s t r u c t e d in a n e q u i v a l e n t u n i f o r m c o
o r d i n a t e s p a c e . T h i s a g a i n c an be r e p r e s e n t e d in f o r m of a n o t h e r b o n d g r a p h t a k i n g a d v a n t a g e of a s e c o n d s t a g e m o d a l d e c o m p o s i t i o n . T h e e f f e c t s of d a m p i n g a n d e x t e r n a l e x c i t a t i o n s c a n be i n c o r p o r a t e d in t h i s m o d e l s u i t a b l y . T h e s y s t e m r e s p o n s e c a n be o b t a i n e d o v e r the r e q u i r e d f r e q u e n c y r a n g e r e t a i n i n g the a p p r o p r i a t e m o d e s . T h e p r o c e d u r e is p a r t i c u l a r l y s u i t a b l e f o r " s t i f f " s y s t e m s b e c a u s e of i t s s c o p e of m o d a l t r u n c a t i o n at b o t h the s t a g e s of a n a l y s i s . C h a p t e r 6 d e a l s w i t h t h i s a s p e c t of b o n d g r a p h a d a p t e d d u a l f o r m u l a t i o n .
R o l e of Z e r o F r e q u e n c y M o d e s in B o n d G r a p h A d a p t e d D u a l S p a c e F o r m u l a t i o n
T h e t e c h n i q u e of a n a l y s i n g d y n a m i c s y s t e m s m a k i n g u s e of d u a l f o r m u l a t i o n a n d b o n d g r a p h s has b e e n p r e s e n t e d in C h a p t e r s 5 a n d 6. T h e r e p r e s e n t a t i o n of t h e s u b s y s t e m s a n d t h e i r s u b s e q u e n t m o d a l d e c o m p o s i t i o n h a v e b e e n b a s e d on t h e i r d y n a m i c m o d e s , r e m o v i n g a l t o g e t h e r t h e n o n - d y n a m i c m o d e s c h a r a c t e r i z e d by z e r o f r e q u e n c i e s . H o w e v e r , s o m e t i m e s in a m o m e n t u m b a s e d f o r m u l a t i o n t h e r e a r e z e r o f r e q u e n c y m o d e s w h i c h p l a y a s i g n i f i c a n t r o l e in t h e o v e r a l l d y n a m i c s of t h e s y s t e m s a n d w h e n e l i m i n a t e d , l e a d to a n o m a l o u s r e s u l t s . In c h a p t e r 7, t h e r o l e of' s u c h n o n d y n a m i c m o d e s in b o n d g r a p h a d a p t e d d u a l f o r m u l a t i o n is d i s c u s s e d . A n a l g o r i t h m is p r e s e n t e d fo r r e t a i n i n g t h e s e m o d e s in t he o v e r a l l s y s t e m m o d e l . T h e p r o c e d u r e is e x t e n d e d to o b t a i n s y s t e m r e s p o n s e t a k i n g a d v a n t a g e of a s e c o n d s t a g e m o d a l d e c o m p o s i t i o n . T h e p r o c e d u r e is i l l u s
t r a t e d by s u i t a b l e e x a m p l e s . 1 6
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IIT Kharagpur
17
A c o u s t o e l a s t i c S y s t e m s
C h a p t e r 8 d i s c u s s e s a p r o c e d u r e f o r a n a l y s i n g the d y n a m i c s of a c o u s t o e l a s t i c s y s t e m s w i t h i n the f r a m e w o r k of b o n d g r a p h t e c h n i q u e s . T h e s u b s t r u c t u r e s , a c o u s t i c a n d s t r u c t u r a l , a r e m o d e l l e d i n d i v i d u a l l y in f o r m of b o n d g r a p h s w h i c h a r e c o u p l e d t h r o u g h s u i t a b l e e l e m e n t s s a t i s f y i n g the c o n d i t i o n s at t h e i n t e r f a c e s . F r o m t h i s b o n d g r a p h a s e c o n d s t a g e m o d a l d e c o m p o s i t i o n is p e r f o r m e d to r e p r e s e n t t he o v e r a l l s y s t e m in y e t a n o t h e r b o n d g r a p h w h i c h ca n be a n a l y s e d to o b t a i n t h e o v e r a l l s y s t e m d y n a m i c s . T h e p r o c e d u r e is i l l u s t r a t e d by the e x a m p l e s of c o u p l e d c a v i t i e s . T h e s y s t e m
r e s p o n s e s d u e to d i f f e r e n t e x t e r n a l e x c i t a t i o n s a r e a l s o o b t a i n e d . T h e t e c h n i q u e is s u i t a b l e f or a c o u s t o e 1 a s t i c s y s t e m s w h i c h a r e " s t i f f " in n a t u r e due to t h e w i d e r a n g e of t h e i r n a t u r a l f r e q u e n c i e s .
C o n c l u s i o n s
T h i s c h a p t e r is d e v o t e d to s u m m a r i s e t h e p r e s e n t w o r k as to, its a p l i c a b i l i t y in a n a l y s i n g d y n a m i c s y s t e m s of d i f f e r e n t c l a s s e s a l o n g w i t h its s a l i e n t a d v a n t a g e s . T h e f u t u r e s c o p e of t h e w o r k is a l s o d i s c u s s e d .
T h e w o r k c o n c l u d e s w i t h a l i s t of r e f e r e n c e s c i t e d in the
t e x t .
