[acs symposium series] sound and vibration damping with polymers volume 424 || requirements imposed...

Chapter 17

Requirements Imposed on Polymeric Materials by Structural Damping Applications

Edward M. Kerwin, Jr., and Eric E. Ungar

Laboratories Division, BBN Systems and Technologies Corporation, 10 Moulton Street, Cambridge, MA 02138

We review the application of polymeric materials in structural damping, giving an introduction to the f i e l d as it is seen by the designer of damping treatments. Our objective is to provide an overview of the ways in which visoelastic damping materials are used, and of the resulting material-property requirements, and the corresponding polymer deformation types (extension, shear, or wavebearing). Finally, we show how the requirements of a given damping problem define the visoelastic properties required of candidate polymers, as influenced by the mechanical and environmental conditions that obtain.

The purpose of this paper is to present an introduction to structural damping for those polymer chemists who are not already familiar with the fi e l d . We shall treat several subjects, including what structural damping can accomplish, the definition of damping in terms of energy and loss factor, and the important classes of damping mechanisms. Finally, we look briefly at the range of problems in which damping may be applied, and at the resulting requirements imposed on the viscoelastic materials involved.

This approach presents a narrow view of the whole f i e l d of structural damping. It considers only those damping mechanisms that involve viscoelastic materials, and covers the wide variety of treatment schemes only generically with regard to the types of viscoelastic-material deformations of import. Thus, we are not able to cover the details of damping-treatment design and engineering, nor the material-property requirements from other areas of acoustics, such as the fields of vibration isolation, shock mitigation, and underwater sound.

0097-6156/90/0424-0317$08.25/0 © 1990 American Chemical Society

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In Sound and Vibration Damping with Polymers; Corsaro, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

318 SOUND AND VIBRATION DAMPING WITH POLYMERS

N e v e r t h e l e s s , we hope t h a t t h i s s p e c i a l - p u r p o s e p r e s e n t a t i o n w i l l p r o v e u s e f u l .

The F u n c t i o n s o f S t r u c t u r a l Damping

S t r u c t u r a l damping i s d e f i n e d as t h e i r r e v e r s i b l e r e m o v a l o f e n e r g y f r o m a v i b r a t i n g s y s t e m . Damping, t h e r e f o r e , s e r v e s t o c o n t r o l " f r e e " r e s p o n s e s , i . e . , t h o s e r e s p o n s e s f o r w h i c h t h e e l a s t i c a n d i n e r t i a l f o r c e s b a l a n c e , l e a v i n g t h e s y s t e m r e s p o n s e c o n t r o l l e d by t h e l o s s e s i n t h e s y s t e m . I n t h e a b s e n c e o f l o s s e s , t h e r e s p o n s e s w o u l d be f o r c e -f r e e , a nd once e x c i t e d , w o u l d c o n t i n u e a r b i t r a r i l y f a r i n t i m e o r s p a c e . The c a t e g o r y " f r e e " r e s p o n s e s t h u s i n c l u d e s b o t h r e s o n a n c e s a n d f r e e l y t r a v e l i n g waves. I t i s t h e s e t h a t a r e c o n t r o l l e d by s y s t e m damping.

Thus, damping c a n a c c o m p l i s h t h e f o l l o w i n g : • l i m i t s t e a d y - s t a t e , r e s o n a n t r e s p o n s e , • c a u s e t r a n s i e n t r e s p o n s e s t o d e c a y more r a p i d l y ,

and • a t t e n u a t e t r a v e l l i n g waves.

B r o a d e r p o t e n t i a l r e s u l t s i n c l u d e t h e r e d u c t i o n o f s t r u c t u r a l v i b r a t i o n , o f r a d i a t e d sound, and o f s t r u c t u r a l f a t i g u e .

Damping, E n e r g y , a n d L o s s F a c t o r

I f , i n a f r e e l y v i b r a t i n g s y s t e m t h a t c o n t a i n s a l o s s mechanism, t h e s y s t e m e n e r g y i s s e e n t o d e c a y a t a r a t e t h a t i s p r o p o r t i o n a l t o t h e i n s t a n t a n e o u s e n e r g y W, i . e . , -dW/dt=a*W, i t f o l l o w s t h a t t h e d e c a y i s e x p o n e n t i a l , W = W 0e~ a t. ( T h i s i s t h e c a s e f o r a l i n e a r s y s t e m i n w h i c h t h e l o s s c o e f f i c i e n t i t s e l f i s i n d e p e n d e n t o f v i b r a t i o n a m p l i t u d e . ) The d e c a y c o e f f i c i e n t i s a = COTJ ,where CO = 2ftf i s t h e a n g u l a r f r e q u e n c y o f v i b r a t i o n , a nd T| i s d e f i n e d a s t h e s y s t e m l o s s f a c t o r . S i n c e t h e e n e r g y d e c a y r a t e i s n d i s f t h e power d i s s i p a t e d , i t f o l l o w s d i r e c t l y t h a t n ^ i s = C0T|W. Thus, we have t h e s y s t e m l o s s f a c t o r T| d e f i n e d i n a v e r y b a s i c way i n e n e r g y and power t e r m s as f o l l o w s :

*n = n d i s/cow

I n many c a s e s o f i n t e r e s t , t h e v i b r a t i n g s y s t e m c o m p r i s e s a number o f e l e m e n t s , e a c h o f w h i c h may have i t s own l o s s f a c t o r Tji and i t s own c o n t r i b u t i o n Wi t o t h e s y s t e m v i b r a t o r y e n e r g y . These e l e m e n t s m i g h t be, f o r exa m p l e , t h e s e v e r a l l a y e r s o f a l a m i n a t e d beam o r p l a t e . I n s u c h a c a s e , t h e power d i s s i p a t i o n r a t e i s t h e sum o f t h e s e v e r a l c o n t r i b u t i o n s , a n d t h e s y s t e m e n e r g y i s t h e sum o f t h e s e v e r a l e n e r g i e s . We, t h e r e f o r e , have t h e f a m i l i a r a n d u s e f u l d e f i n i t i o n o f s y s t e m l o s s f a c t o r as f o l l o w s ( 1 ) :

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17. KERWIN AND UNGAR Structural Damping Applications 319

Tl=

T h i s i s an i m p o r t a n t r e s u l t . I t shows t h a t i n o r d e r f o r a g i v e n s y s t e m e l e m e n t t o be e f f e c t i v e i n damping t h e s y s t e m , t h a t e l e m e n t must n o t o n l y have a s i g n i f i c a n t l o s s f a c t o r , b u t i t must a l s o p a r t i c i p a t e s i g n i f i c a n t l y i n t h e t o t a l e n e r g y o f t h e s y s t e m .

I t i s c u s t o m a r y i n d e a l i n g w i t h l i n e a r ( o r l i n e a r i z e d ) b e h a v i o r o f d y n a m i c a l s y s t e m s t o d e s c r i b e t h e p r o p e r t i e s o f v i s c o e l a s t i c m a t e r i a l s as comp l e x q u a n t i t i e s (2). F o r exa m p l e , we w r i t e Young's modulus

E = E 1 - i E " = E 1 (1-iTj)

w h e r e _ E i s t h e co m p l e x modulus E f i s t h e " s t o r a g e " modulus E" i s t h e " l o s s " modulus

and T| =E"/E' i s t h e l o s s f a c t o r o f t h e m a t e r i a l

The l o s s f a c t o r T| i s sometimes d e n o t e d as t a n 8, where 8 i s t h e p h a s e a n g l e between t h e s t o r a g e and l o s s components o f h a r m o n i c s t r e s s .

C o r r e s p o n d i n g l y , we have f o r t h e s h e a r modulus

G = G' - i G " = G' ( l - i T | )

O t h e r e l a s t i c m o d u l i o r c o m p l i a n c e s , e.g., b u l k , d i l a t a t i o n , e t c . , may be d e s c r i b e d as abo v e . N o t e t h a t t h e use o f i = V-l i n t h e above e x p r e s s i o n s i m p l i e s a t i m e d e p e n d e n c e e " i c o t. (Some a u t h o r s p r e f e r + j ; t h e c h o i c e i s a r b i t r a r y . )

The F u n c t i o n o f Damping. The f u n c t i o n o f damping i s t o c o n t r o l t h e " f r e e " r e s p o n s e s o f a s y s t e m . Such r e s p o n s e s a r e t h o s e i n w h i c h t h e s y s t e m k i n e t i c and p o t e n t i a l e n e r g i e s a r e i n b a l a n c e , so t h a t d i s s i p a t i v e i n f l u e n c e s a r e c o n t r o l l i n g . F r e e r e s p o n s e s i n c l u d e (a) r e s o n a n c e i n f r e q u e n c y a n d (b) p r o p a g a t i o n o f f r e e waves i n s p a c e . I n t h e f a m i l i a r c a s e o f a r e s o n a n c e i n f r e q u e n c y , t h e r e s p o n s e t o a g i v e n o s c i l l a t i n g f o r c e i n t h e v i c i n i t y o f r e s o n a n c e o r n a t u r a l f r e q u e n c y i s p e a k e d , s h o w i n g a maximum a t t h e r e s o n a n c e f r e q u e n c y and a f i n i t e b a n d w i d t h o f h i g h r e s p o n s e . B o t h f e a t u r e s a r e c o n t r o l l e d b y t h e s y s t e m l o s s f a c t o r : See F i g u r e 1.

The peak a m p l i f i c a t i o n f a c t o r Q i s d e f i n e d as

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a n d d e s c r i b e s t h e peak r e s p o n s e a t r e s o n a n c e r e l a t i v e t o t h e r e s p o n s e t h a t w o u l d e x i s t a t t h e r e s o n a n c e f r e q u e n c y i f o n l y t h e s y s t e m s t i f f n e s s ( o r o n l y t h e mass) were c o n t r o l l i n g . ( C o n s i d e r t h e s i m p l e c a s e o f a mass on a s p r i n g . )

The r e s o n a n c e f r a c t i o n a l b a n d w i d t h i s A f / f n = T], f o r T| n o t t o o l a r g e , and r e p r e s e n t s t h e f r a c t i o n a l f r e q u e n c y d i f f e r e n c e b etween t h e " h a l f - p o w e r " o r "3-dB-down" p o i n t s i n t h e r e s p o n s e , i . e . , t h e p o i n t s where t h e s q u a r e d a m p l i t u d e d r o p s t o h a l f i t s peak v a l u e .

The above d i s c u s s i o n i m p l i e s s t e a d y - s t a t e r e s p o n s e i n t i m e . An e q u i v a l e n t r e c i p r o c a l v i e w o f s t e a d y - s t a t e r e s o n a n c e r e s p o n s e i s t h a t i n t h e v i c i n i t y o f r e s o n a n c e t h e r e i s a d i p i n t h e f o r c e r e q u i r e d t o m a i n t a i n a c o n s t a n t l e v e l o f r e s p o n s e . The f o r c e - r e d u c t i o n r a t i o i s Q, and t h e f r a c t i o n a l b a n d w i d t h o f t h e f o r c e r e d u c t i o n i s T|. I n c o n t r a s t , a t r u l y f o r c e - f r e e r e s p o n s e o f a r e s o n a n t s y s t e m (once e x c i t e d ) w o u l d i n v o l v e t h e e x p o n e n t i a l d e c a y o f v i b r a t i o n a m p l i t u d e w i t h t i m e . As we have m e n t i o n e d e a r l i e r , d e c a y i s a l s o c o n t r o l l e d b y t h e s y s t e m l o s s f a c t o r as f o l l o w s :

T e m p o r a l d e c a y r a t e : At = 27 . 3 T | f n (dB/sec) ,

where a v i b r a t i o n l e v e l L v w o u l d be e x p r e s s e d i n t h e c u s t o m a r y l o g a r i t h m i c d e c i b e l (dB) measure f o r a r e s p o n s e q u a n t i t y v, s u c h a s v e l o c i t y ,

L v = 10 l o g l—Z-Y ( d B r e V r e f ) . \ V r e f /

Thus a l e v e l change o f 10 dB c o r r e s p o n d s t o a f a c t o r 10 change i n an e n e r g y o r i n t e n s i t y - l i k e q u a n t i t y s u c h as v 2 .

The c o n c e p t o f t h e i n f l u e n c e o f damping on v i b r a t i o n d e c a y i s v e r y f a m i l i a r . However, t h e q u a n t i t a t i v e i m p l i c a t i o n o f a g i v e n v a l u e o f l o s s f a c t o r may be l e s s s o . To h e l p c o n v e y t h e c h a r a c t e r o f t h e t e m p o r a l d e c a y , T a b l e I shows t h e f o l l o w i n g f o r a r a n g e o f l o s s f a c t o r s : (a) t h e number o f c y c l e s o f v i b r a t i o n r e q u i r e d f o r t h e v i b r a t i o n a m p l i t u d e t o d e c a y t o o n e - t e n t h o f i t s i n i t i a l v a l u e , (b) a c o r r e s p o n d i n g r e p r e s e n t a t i v e sound, a n d (c) t h e c l a r i t y o r p i t c h o f t h e soun d . N o t e t h a t as t h e l o s s f a c t o r i n c r e a s e s , t h e d e c a y t i m e d e c r e a s e s , and t h e f r e q u e n c y b a n d w i d t h i n c r e a s e s . The r e s u l t i s t h a t a u n i q u e f r e q u e n c y o r p i t c h f o r t h e t r a n s i e n t becomes p r o g r e s s i v e l y l e s s w e l l d e f i n e d . T h i s i s i n k e e p i n g w i t h t h e " u n c e r t a i n t y p r i n c i p l e " r e l a t i n g t i m e and f r e q u e n c y Q ) .

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T a b l e I . Dependence o f t h e C h a r a c t e r o f a T r a n s i e n t Decay on L o s s F a c t o r

L o s s F a c t o r N: c y c l e s f o r d e c a y t o 0.1

a m p l i t u d e S o u n d — P i t c h

< 0.001 > 730 " C l a n g " — a p p r o a c h e s p u r e t o n e

0.01 73 " B o n g " — c l e a r p i t c h

0.1 7.3 " B u n k " — d i s c e r n a b l e p i t c h

0.5 1.5 " T h u d " — a l m o s t w i t h o u t p i t c h

I n a way t h a t i s p a r a l l e l t o t h e t e m p o r a l d e c a y o f v i b r a t i o n o f a r e s o n a n t s y s t e m , damping g o v e r n s t h e s p a t i a l a t t e n u a t i o n o f f r e e l y p r o p a g a t i n g waves. I n F i g u r e 2, we see a wave, assumed e x c i t e d somewhere on t h e l e f t , t r a v e l i n g t o t h e r i g h t w i t h o u t f u r t h e r e x c i t a t i o n . The wave t r a v e l s w i t h "phase s p e e d " c a t a f r e q u e n c y f , w i t h a w a v e l e n g t h X g i v e n as

X = c / f .

As we have n o t e d above, t h e s y s t e m l o s s f a c t o r d e f i n e s t h e t e m p o r a l d e c a y o f t h e s y s t e m e n e r g y . T h e r e f o r e , i n c o n s i d e r i n g t h e s p a t i a l d e c a y o f v i b r a t i o n , we f i n d t h e " e n e r g y " s p e e d ( o r "group v e l o c i t y " ) c g i n v o l v e d i n t h e r e s u l t :

S p a t i a l d e c a y r a t e : Ax = 27.3 -°- T] ( d B / w a v e l e n g t h ) c g

[The e n e r g y s p e e d i s d e f i n e d (4-5) as c g = d f / d ( l A ) . ] I n n o n - d i s p e r s i v e s y s t e m s , s u c h as a c o u s t i c waves i n a

f l u i d o r s i m p l e t e n s i o n waves on a s t r i n g , t h e wave s p e e d does n o t v a r y w i t h f r e q u e n c y . Thus t h e e n e r g y s p e e d c g i s t h e same as t h e p h a s e s p e e d c, so t h a t

A\ = 27.3 T| (6B/X) ( n o n - d i s p e r s i v e waves)

T h e r e a r e wave t y p e s , however, i n w h i c h t h e wave s p e e d has an i n h e r e n t dependence on f r e q u e n c y . A v e r y i m p o r t a n t p r a c t i c a l example i s t h e p r o p a g a t i o n o f b e n d i n g waves on beams o r p l a t e s (.5.) where (when t h e w a v e l e n g t h i s l a r g e w i t h r e s p e c t t o t h e t h i c k n e s s o f t h e p l a t e o r beam) t h e e n e r g y s p e e d i s t w i c e t h e p h a s e s p e e d : c g = 2c. F o r s u c h b e n d i n g waves, we have

A^= 13.6 T| (dB/X) ( b e n d i n g waves)

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10

o Q . C O < D

OC 5K "5 O

2 ro

OC

1.0

0.1

Af/fn=n J V- t \ Q-i

\ i / ? / /

// jy Controlling

^Stiffness Resis

\ \ N \

N X

\ \ Parameter:

i % w

tance Massx.

I X 0.1

F i g u r e 1,

1.0

Relative Frequency f/f n

10

R e s o n a n c e B a n d w i d t h and A m p l i f i c a t i o n D e t e r m i n e d by L o s s F a c t o r

AA = 27.3 77 dB/A c 9

= 27.3 77 dB/A for nondispersive waves

= 13.6 77 dB/A for bending waves

F i g u r e 2 . F r e e Wave S p a t i a l A t t e n u a t i o n Due t o Damping

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Damping C o n f i g u r a t i o n s t h a t U t i l i z e V i s c o e l a s t i c M a t e r i a l s

V i s c o e l a s t i c m a t e r i a l s have f o u n d a p p l i c a t i o n i n v i b r a t i o n d a mping i n a m y r i a d o f c o n f i g u r a t i o n s . Some o f t h e s e a r e v e r y much s p e c i f i c t o a p a r t i c u l a r a p p l i c a t i o n o r d e v i c e o r s t r u c t u r e , w h i l e o t h e r s a r e b r o a d l y u s e f u l . I n t h i s s e c t i o n , i t i s o u r p u r p o s e t o i n d i c a t e t h e g e n e r a l ways i n w h i c h v i s c o e l a s t i c m a t e r i a l s a r e a p p l i e d by d e s c r i b i n g two w i d e l y u s e d t y p e s o f damping t r e a t m e n t s , o u t l i n i n g t h e i r modes o f o p e r a t i o n , t h e dependence o f t h e i r p e r f o r m a n c e on m a t e r i a l p r o p e r t i e s and s y s t e m g e o m e t r y , s e v e r a l e l a b o r a t i o n s on t h e b a s i c c o n f i g u r a t i o n s , a nd a few ex a m p l e s o f a p p l i c a t i o n s . I n t h i s way, we hope t o c o n v e y some u n d e r s t a n d i n g o f why c e r t a i n p a r a m e t e r s a n d p r o p e r t i e s o f v i s c o e l a s t i c m a t e r i a l s a r e i m p o r t a n t i n damping a p p l i c a t i o n s .

The two g e n e r a l t r e a t m e n t s t h a t we s h a l l e x p l o r e a r e t h e " f r e e " v i s c o e l a s t i c l a y e r , a nd t h e c o n s t r a i n e d v i s c o e l a s t i c l a y e r . (These a r e sometimes r e f e r r e d t o as " e x t e n s i o n a l " a nd " s h e a r " damping, r e s p e c t i v e l y . ) These a r e s k e t c h e d i n F i g u r e 3, w h i c h shows e a c h t r e a t m e n t as a p p l i e d t o a p l a t e ( o r beam), and a l s o shows t h e t y p e o f d e f o r m a t i o n t h a t t h e v i s c o e l a s t i c l a y e r u n d e r g o e s due t o b e n d i n g o f t h e b a s e p l a t e . B e n d i n g v i b r a t i o n s a r e v e r y i m p o r t a n t i n t h e f i e l d s o f n o i s e a n d v i b r a t i o n c o n t r o l , a n d much o f o u r d i s c u s s i o n w i l l c o n c e r n t h e damping o f b e n d i n g waves. I n a d d i t i o n , i n c o n s i d e r i n g t h e e f f e c t i v e n e s s o f damping t r e a t m e n t s , one f i n d s t h a t t h e r e a r e t y p i c a l l y one o r more g e o m e t r i c p a r a m e t e r s o f t h e s t r u c t u r e a n d t h e t r e a t m e n t , a n d one o r more e l a s t i c o r s t i f f n e s s p a r a m e t e r s t h a t government t h e s y s t e m p e r f o r m a n c e .

The F r e e V i s c o e l a s t i c L a y e r . As we see i n F i g u r e 3, t h e f r e e v i s c o e l a s t i c l a y e r i s b onded t o t h e p l a t e t o be damped. As t h e p l a t e v i b r a t e s i n b e n d i n g , t h e v i s c o e l a s t i c l a y e r i s d e f o r m e d p r i n c i p a l l y i n e x t e n s i o n and c o m p r e s s i o n i n p l a n e s p a r a l l e l t o t h e p l a t e s u r f a c e . Such damping l a y e r s have l o n g been known, and a t f i r s t were a p p l i e d m o r e - o r - l e s s e m p i r i c a l l y . I n t h e e a r l y 1950s O b e r s t (_£,Z) and L i e n a r d (£,j)) p u b l i s h e d a n a l y s e s d e s c r i b i n g q u a n t i t a t i v e a n a l y t i c a l m odels o f f r e e l a y e r b e h a v i o r .

The damping p e r f o r m a n c e o f a f r e e l a y e r t r e a t m e n t f o r p l a t e b e n d i n g waves i s shown i n F i g u r e 4 (JLQ., 11) • T h i s c h a r t , w h i c h i s O b e r s t 1 s r e s u l t , g i v e s t h e s y s t e m l o s s f a c t o r T| r e l a t i v e t o X\2r t h e l o s s f a c t o r o f t h e v i s c o e l a s t i c m a t e r i a l , a s a f u n c t i o n o f t h e t h i c k n e s s r a t i o H 2 / H 1 ( v i s c o e l a s t i c l a y e r t o p l a t e ) . E a c h o f t h e s e v e r a l c u r v e s c o r r e s p o n d s t o a p a r t i c u l a r v a l u e o f t h e r e l a t i v e Young's s t o r a g e modulus E 2 / E 1 ( v i s c o e l a s t i c l a y e r t o p l a t e ) .

T h i s d e s c r i p t i o n o f t h e damping e f f e c t i v e n e s s o f t h e f r e e l a y e r i s c l e a r and s i m p l e : I t s f e a t u r e s a r e t h a t s y s t e m l o s s f a c t o r i n c r e a s e s

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Visco-elastic Constraining

(a) Free layer (b) Constrained layer

F i g u r e 3 . F r e e and C o n s t r a i n e d V i s c o e l a s t i c - L a y e r Damping T r e a t m e n t s

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a) w i t h l a y e r t h i c k n e s s b) w i t h l a y e r m odulus, and, o f c o u r s e , c) w i t h t h e l o s s f a c t o r o f t h e v i s c o e l a s t i c l a y e r

( t h e r e s u l t a n t s y s t e m l o s s f a c t o r i s p r o p o r t i o n a l t o T|2, t h e l o s s f a c t o r o f t h e v i s c o e l a s t i c l a y e r . )

I n t h e c e n t r a l r e g i o n o f F i g u r e 4 a p a r t i c u l a r l y s i m p l e r e s u l t a p p l i e s : t h e l o s s f a c t o r r a t i o T|/T}2 i s p r o p o r t i o n a l t o ( H 2 / H 1 ) 2 . Thus, o v e r a u s e f u l r a n g e

* Et feF *•

Her e t h e i n f l u e n c e s o f a g e o m e t r i c p a r a m e t e r , H 2 / H 1 , a n d o f an e l a s t i c p a r a m e t e r , E 2 / E 1 , a r e c l e a r . T h i s a n a l y t i c a l m o d e l f o r f r e e - l a y e r damping makes c l e a r t h e d e s i r a b l e p r o p e r t i e s o f t h e v i s c o e l a s t i c m a t e r i a l . S p e c i f i c a l l y , t h e f r e e - l a y e r damping m a t e r i a l s h o u l d have b o t h l a r g e s t o r a g e m odulus E 2 and l a r g e l o s s f a c t o r T|2. Over much o f t h e r a n g e , i t i s f a i r t o s a y t h a t t h e s y s t e m l o s s f a c t o r i s m a x i m i z e d by m a x i m i z i n g t h e l o s s m o d u l u s , t h a t i s , t h e p r o d u c t E 2 T I 2 • However, a t t h e u p p e r end o f p e r f o r m a n c e shown i n F i g u r e 4, a h i g h e r v a l u e o f l o s s f a c t o r T|2 i s somewhat more e f f e c t i v e t h a n a h i g h e r modulus E 2 (M.L. D r a k e , p e r s o n a l c o m m u n i c a t i o n , 1 9 8 9 ) .

We n o t e h e r e t h a t f o r s i m p l i c i t y , we have i m p l i e d t h a t " b e s t p e r f o r m a n c e " means h i g h e s t s y s t e m l o s s f a c t o r . T h i s i s t h e c a s e f o r a number o f damping a p p l i c a t i o n s , b u t n o t f o r a l l c a s e s . D. J . Mead (12,12.) has n o t e d t h e q u a n t i t a t i v e i m p o r t a n c e o f o t h e r s y s t e m p a r a m e t e r s ( s t i f f n e s s a n d mass) i n o p t i m i z i n g a damping t r e a t m e n t i n c a s e s where maximum l o s s f a c t o r i s n o t t h e c r i t e r i o n o f b e s t p e r f o r m a n c e ( e . g . , m i n i m i z i n g s t r e s s , a c c e l e r a t i o n , e t c . ) . These c o n s i d e r a t i o n s a r e p a r t i c u l a r l y i m p o r t a n t i n c o n t r o l l i n g s t r u c t u r a l f a t i g u e and eq u i p m e n t m a l f u n c t i o n .

An i n t e r e s t i n g c h a r a c t e r i s t i c o f t h e f r e e v i s c o e l a s t i c l a y e r i s t h a t i t i s , t o f i r s t o r d e r , " l o c a l l y r e a c t i n g " . T h a t i s , t h e s t r a i n i n t h e l a y e r i s p r o p o r t i o n a l t o t h e l o c a l c u r v a t u r e o f p l a t e , so t h a t t h e e n e r g y d i s s i p a t e d i n t h e v i s c o e l a s t i c l a y e r i s p r o p o r t i o n a l l o c a l l y t o t h e e l a s t i c e n e r g y i n t h e p l a t e . T h e r e f o r e , f o r f u l l c o v e r a g e by t h e v i s c o e l a s t i c l a y e r , t h e damping p e r f o r m a n c e (as s y s t e m l o s s f a c t o r ) i s i n d e p e n d e n t o f t h e mode shape o f t h e v i b r a t i o n . T h i s f a c t s i m p l i f i e s t h e s p e c i f i c a t i o n o f a t r e a t m e n t f o r w i d e f r e q u e n c y c o v e r a g e (as t h e r e i s no i n h e r e n t f r e q u e n c y o r mode-shape d e p e n d e n c e ) . I t i s , h o wever, p o s s i b l e t o o p t i m i z e p a r t i a l c o v e r a g e f o r a p a r t i c u l a r mode o r a l i m i t e d c l a s s o f modes.

The C o n s t r a i n e d V i s c o e l a s t i c L a y e r . The s e c o n d o f o u r two g e n e r a l damping t r e a t m e n t s i s t h e c o n s t r a i n e d v i s c o e l a s t i c l a y e r shown i n F i g u r e 3 ( b ) . The c o m p l e t e c o n s t r a i n e d - l a y e r c o n f i g u r a t i o n i s a t h r e e - l a y e r l a m i n a t e c o m p r i s i n g b a s e l a y e r t o be damped, v i s c o e l a s t i c l a y e r , a n d c o n s t r a i n i n g

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l a y e r . As F i g u r e 3 s u g g e s t s , t h e p r i n c i p a l d e f o r m a t i o n o f t h e v i s c o e l a s t i c l a y e r o c c u r s as s h e a r as t h e c o m p o s i t e u n d e r g o e s t r a n s v e r s e ( b e n d i n g ) m o t i o n . (The b e h a v i o r o f t h e c o n s t r a i n e d - l a y e r s y s t e m i s d e s c r i b e d i n a number o f r e f e r e n c e s . See, f o r e x a m p l e , R e f e r e n c e s 2, 14-19.)

I n b e n d i n g v i b r a t i o n , t h i s t h r e e - l a y e r s y s t e m e x h i b i t s an i n h e r e n t d i s p e r s i o n f r o m " c o u p l e d " b e n d i n g f o r l a r g e w a v e l e n g t h s (low f r e q u e n c i e s ) t o " u n c o u p l e d " b e n d i n g f o r s m a l l w a v e l e n g t h s ( h i g h f r e q u e n c i e s ) . I n t h e s e l o w - and h i g h - f r e q u e n c i e s r e g i m e s , t h e e l a s t i c e n e r g y o f d e f o r m a t i o n l i e s a l m o s t e n t i r e l y i n t h e b e n d i n g and e x t e n s i o n o f t h e b a s e a n d c o n s t r a i n i n g l a y e r s . Thus, i n t h e s e l i m i t s o f v e r y l o w and v e r y h i g h f r e q u e n c y , t h e v i s c o e l a s t i c l a y e r c a n n o t b r i n g a b o u t s i g n i f i c a n t damping o f t h e s y s t e m . (But see t h e l a t e r d i s c u s s i o n o f segmented c o n s t r a i n i n g l a y e r s . )

[The m e a n i n g o f t h e t e r m s i s as f o l l o w s : I n c o u p l e d b e n d i n g , t h e w a v e l e n g t h i s l o n g enough so t h a t t h e s h e a r s t i f f n e s s o f t h e v i s c o e l a s t i c l a y e r ( a c t i n g o v e r a l e n g t h o f h a l f o f t h e b e n d i n g w a v e l e n g t h b e t w e e n s t r e s s r e v e r s a l s ) i s a b l e t o c a u s e t h e b a s e and c o n s t r a i n i n g members t o s t r e t c h a nd compress r a t h e r t h a n a l l o w i n g s i g n i f i c a n t s h e a r s t r a i n i n t h e v i s c o e l a s t i c c e n t r a l l a y e r . A t s u f f i c i e n t l y l a r g e w a v e l e n g t h s , t h e c o m p o s i t e bends e s s e n t i a l l y a s t h o u g h i t were " p e r f e c t l y " b onded t o g e t h e r by a c e n t r a l l a y e r w i t h " i n f i n i t e " s h e a r s t i f f n e s s . ]

[ I n c o n t r a s t , f o r s h o r t w a v e l e n g t h s ( h i g h f r e q u e n c i e s ) t h e s h e a r s t i f f n e s s o f a h a l f w a v e l e n g t h o f t h e v i s c o e l a s t i c l a y e r i s s m a l l r e l a t i v e t o t h e e x t e n s i o n a l s t i f f n e s s o f t h e b a s e a n d c o n s t r a i n i n g l a y e r s . I n t h e l i m i t o f v e r y s h o r t w a v e l e n g t h s , t h e l a m i n a t e b e h a v e s as t h o u g h t h e two o u t e r l a y e r s were j o i n e d b y a c e n t r a l l a y e r o f v a n i s h i n g s h e a r s t i f f n e s s , b u t w h i c h m a i n t a i n s t h e s p a c i n g b etween l a y e r s . Thus t h e " u n c o u p l e d " b e n d i n g s t i f f n e s s o f t h e l a m i n a t e a p p r o a c h e s s i m p l y t h e sum o f t h e b e n d i n g s t i f f n e s s o f t h e o u t e r l a y e r s , i . e . , a l e s s e r s t i f f n e s s t h a n t h a t o f t h e l a y e r s t i g h t l y c o u p l e d a t l a r g e w a v e l e n g t h s . ]

However, i n t h e m i d - f r e q u e n c y r e g i o n , where t h e e l a s t i c e n e r g y i n t h e v i s c o e l a s t i c l a y e r c a n r e p r e s e n t a u s e f u l f r a c t i o n o f t h e t o t a l , t h e s y s t e m l o s s f a c t o r r i s e s a n d p a s s e s t h r o u g h a maximum. F i g u r e 5 shows t h i s b e h a v i o r w i t h t h e w a v e l e n g t h (X) dependence r e p r e s e n t e d by a " s h e a r p a r a m e t e r " y d e f i n e d as f o l l o w s :

where t h e r e c i p r o c a l o f t h e p a r a m e t e r a d e f i n e s a c h a r a c t e r i s t i c d i s t a n c e i n w h i c h a s h e a r p e r t u r b a t i o n i n t h e l a m i n a t e r e l a x e s t o 1/e o f i t s i n i t i a l v a l u e (1A, 17, ZD.) . (See F i g u r e 6.) T h e r e f o r e , t h e s h e a r p a r a m e t e r m e a s u r e s t h e s q u a r e o f t h e r a t i o o f t h e b e n d i n g w a v e l e n g t h t o t h i s c h a r a c t e r i s t i c l e n g t h . C l e a r l y a i n v o l v e s s i m p l y t h e p r o d u c t o f t h e s h e a r s t i f f n e s s G 2 / H 2 o f a u n i t l e n g t h

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yopt Shear Parameter y

F i g u r e 5 . C h a r a c t e r i s t i c Damping P e r f o r m a n c e o f a C o n t i n u o u s C o n s t r a i n e d V i s c o e l a s t i c L a y e r

F i g u r e 6 . S h e a r R e l a x a t i o n i n a C o n s t r a i n e d V i s c o e l a s t i c L a y e r T r e a t m e n t

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a n d w i d t h o f t h e v i s c o e l a s t i c l a y e r , and t h e summed e x t e n s i o n a l c o m p l i a n c e s 1 / K i , I/K3 o f u n i t l e n g t h and w i d t h o f t h e b a s e and c o n s t r a i n i n g l a y e r s .

S i n c e , f o r b e n d i n g o f a s i m p l e beam o r p l a t e , f r e q u e n c y and w a v e l e n g t h a r e r e l a t e d as f « 1/A,2, i t f o l l o w s t h a t t h e s h e a r p a r a m e t e r y i s i n v e r s e l y p r o p o r t i o n a l t o f r e q u e n c y .

The l o s s f a c t o r peak w h i c h o c c u r s a t y o p t c a n be l o c a t e d i n f r e q u e n c y by t h e a p p r o p r i a t e c h o i c e o f t h e p a r a m e t e r s t h a t d e f i n e y (see b e l o w ) . The maximum l o s s f a c t o r T| m a x a t t h e peak i s g o v e r n e d by T[2 t h e l o s s f a c t o r o f t h e v i s c o e l a s t i c l a y e r and by a " s t i f f n e s s " p a r a m e t e r Y s i m p l y d e f i n e d as (17)

v _ ^coupled ~ ^uncoupled X — ,

^uncoupled

where t h e c o u p l e d and u n c o u p l e d b e n d i n g s t i f f n e s s e s have b e e n d i s c u s s e d above. F i g u r e 7 (JLQ, ZX) shows T| m a x v e r s u s Y f o r a r a n g e o f v a l u e s o f T |2. We see t h a t T] i s p r o p o r t i o n a l t o Y f o r s m a l l e r v a l u e o f t h e s e p a r a m e t e r s , b u t i n c r e a s e s more s l o w l y as e i t h e r p a r a m e t e r becomes l a r g e r ( s a y Y > 0.3 o r T] > 0.5) .

The p a r a m e t e r Y depends o n l y on t h e p r o p e r t i e s o f t h e b a s e and c o n s t r a i n i n g l a y e r s . A l t h o u g h we have c a l l e d i t a s t i f f n e s s p a r a m e t e r , Y does become de p e n d e n t o n l y on ge o m e t r y when t h e two o u t s i d e l a y e r s have e q u a l e l a s t i c m o d u l i .

Above we r e f e r r e d t o t h e optimum s h e a r p a r a m e t e r Yopt, w h i c h d e f i n e s t h e l o c a t i o n o f t h e peak damping as a f u n c t i o n o f w a v e l e n g t h ( o r f r e q u e n c y ) . We have (11)

Yopt = . 1 — — V (1+Y) (X-Tll)

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

(1) The S t i f f n e s s / G e o m e t r i c p a r a m e t e r Y w h i c h d e f i n e s t h e " s e v e r i t y " o f t h e b e n d i n g - w a v e d i s p e r s i o n c h a r a c t e r i s t i c b e tween c o u p l e d and u n c o u p l e d c o n d i t i o n s . Y and t h e l o s s f a c t o r T|2 o f t h e v i s c o e l a s t i c l a y e r d e t e r m i n e t h e maximum s y s t e m damping f o r t h e t r e a t m e n t .

( 2 ) The S h e a r P a r a m e t e r y w h i c h d e f i n e s t h e l o c a t i o n w i t h i n t h e d i s p e r s i o n c h a r a c t e r i s t i c s o f a g i v e n f r e q u e n c y b y c o m p a r i n g t h e c o r r e s p o n d i n g b e n d i n g w a v e l e n g t h w i t h t h e c h a r a c t e r i s t i c s h e a r r e l a x a t i o n l e n g t h i n t h e l a m i n a t e . y i s a com b i n e d g e o m e t r i c and e l a s t i c p a r a m e t e r . The g o v e r n i n g p r o p e r t y o f t h e v i s c o e l a s t i c l a y e r i s i t s s h e a r s t i f f n e s s G 2 / H 2 .

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330 SOUND, AND VIBRATION DAMPING WITH POLYMERS

The d e s i g n o f a c o n s t r a i n e d - l a y e r t r e a t m e n t (IX, UL, Z) i n v o l v e s a number o f s t e p s a n d i t e r a t i o n s . T y p i c a l l y (but n o t a l w a y s ) o u r g o a l w o u l d be t o a c h i e v e a g i v e n maximum l o s s f a c t o r T j m a x

a n < d t o l o c a t e t h i s peak p e r f o r m a n c e a t a g i v e n f r e q u e n c y f o r t h e o p e r a t i n g t e m p e r a t u r e r a n g e e x p e c t e d . Out o f t h e d e s i g n p r o c e s s w i l l come d e s i r e d r a n g e s o f v a l u e s o f Y and T |2, hence a v a l u e o f Yopt/- a n c * e v e n t u a l l y a d e s i r e d r a n g e f o r t h e s h e a r s t i f f n e s s G 2 / H 2 o f t h e v i s c o e l a s t i c l a y e r . Thus, t h e m a t e r i a l - p r o p e r t y r e q u i r e m e n t s w i l l be

a) a u s e f u l l y l a r g e l o s s f a c t o r T |2, and b) a s e l e c t e d v a l u e o f G 2 / H 2 A w i t h a p r a c t i c a l l y

a c h i e v a b l e H 2 .

As a r e s u l t , one may be a b l e t o c h o o s e f r o m a s e t o f m a t e r i a l s , a d j u s t i n g t h e s h e a r s t i f f n e s s by a d j u s t i n g t h e l a y e r t h i c k n e s s H 2 . ( T h i s i s s o m e t h i n g o f an o v e r s i m p l i f i c a t i o n , b e c a u s e t h e v a l u e o f H 2 i n f l u e n c e s Y.) T h i s a b i l i t y t o a d a p t a v i s c o e l a s t i c m a t e r i a l t o a p a r t i c u l a r a p p l i c a t i o n c a n a l l o w t h e i n c l u s i o n o f a r a n g e o f c a n d i d a t e m a t e r i a l s h a v i n g d e s i r a b l e p r o p e r t i e s a s i d e f r o m t h e d y n a m i c e l a s t i c p r o p e r t i e s G 2 and T |2 .

V a r i a t i o n s on C o n s t r a i n e d V i s c o e l a s t i c L a y e r S y s t e m s . The c o n s t r a i n e d v i s c o e l a s t i c l a y e r t r e a t m e n t r as a p p l i e d t o a b a s e member, i s a t i t s s i m p l e s t , a c o n t i n u o u s 3 - l a y e r l a m i n a t e . Numerous v a r i a t i o n s and e l a b o r a t i o n s have been e x p l o r e d , a n d a number have f o u n d p r a c t i c a l a p p l i c a t i o n . I n t h i s s e c t i o n we r e v i e w a few o f t h e s e as f u r t h e r i l l u s t r a t i o n s o f t h e a p p l i c a t i o n o f v i s c o e l a s t i c m a t e r i a l s i n s h e a r - d e p e n d e n t c o n f i g u r a t i o n s . E x a m p l e s d i s c u s s e d a r e

• Segmented c o n s t r a i n i n g l a y e r • M u l t i p l e c o n s t r a i n e d - l a y e r t r e a t m e n t s • " F i s h - s c a l e " t r e a t m e n t : m u l t i p l e , o v e r l a p p i n g

segmented l a y e r s .

Segmented Constraining Layer. As we s a i d i n t h e p r e c e d i n g s e c t i o n , a c h a r a c t e r i s t i c o f t h e c o n t i n u o u s c o n s t r a i n e d v i s c o e l a s t i c l a y e r t r e a t m e n t i s i t s i n h e r e n t w a v e l e n g t h ( f r e q u e n c y ) dependence. The s y s t e m l o s s f a c t o r f a l l s o f f on e i t h e r s i d e o f a peak v a l u e (see F i g u r e 5). H a p p i l y , t h e damping p e r f o r m a n c e i n t h e l o w - f r e q u e n c y , l a r g e -w a v e l e n g t h r e g i o n b e l o w t h e damping peak c a n be i m p r o v e d s i g n i f i c a n t l y by s e g m e n t i n g ( c u t t i n g ) t h e c o n s t r a i n i n g l a y e r p e r i o d i c a l l y . (See F i g u r e 8.)

T h i s a p p r o a c h , s u g g e s t e d by D. J . Mead (22) and a n a l y z e d b y P a r f i t t et al. (2Z, 22.), and l a t e r b y P l u n k e t t and Lee (24.) and b y Z e i n e t d i n o v a e t al. (25), r e s u l t s i n l o c a l l y i n c r e a s e d s t r a i n s i n t h e v i s c o e l a s t i c l a y e r i n t h e r e g i o n s a r o u n d t h e c u t s , and t h e r e f o r e r e s u l t s i n a d d i t i o n a l d i s s i p a t i o n . P a r f i t t ' s e x p e r i m e n t a l r e s u l t s

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F i g u r e 8. Segmented C o n s t r a i n e d - L a y e r T r e a t m e n t

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a r e shown i n F i g u r e 9, where i n c r e a s i n g numbers o f c u t s i n t h e c o n s t r a i n i n g l a y e r o f a t r e a t m e n t a p p l i e d t o a beam c a u s e t h e l o w f r e q u e n c y damping t o r i s e , r e a c h a maximum, and t h e n f a l l . T h e r e i s , f o r a g i v e n t r e a t m e n t , an optimum segment l e n g t h , w h i c h i s shown i n b o t h t h e o r y a n d e x p e r i m e n t . As one w o u l d e x p e c t , t h e optimum l e n g t h b a l a n c e s segment l e n g t h a g a i n s t t h e s h e a r - r e l a x a t i o n l e n g t h f o r t h e l a m i n a t e . When t h e c u t s a r e t o o c l o s e t o e a c h o t h e r ( i n t e r m s o f t h e s h e a r r e l a x a t i o n l e n g t h ) , t h e i n d u c e d s t r a i n f i e l d s i n t e r f e r e w i t h e a c h o t h e r , r e d u c i n g t h e e f f e c t i v e n e s s o f s e g m e n t i n g . As F i g u r e 9 shows, t h e optimum d e s i g n r a i s e s t h e l o w - f r e q u e n c y b r a n c h o f t h e l o s s -f a c t o r c h a r a c t e r i s t i c p r a c t i c a l l y t o t h e l e v e l o f damping s e e n a t t h e peak.

I n c i d e n t a l l y , K e r w i n and S m i t h (2_Q.) have shown t h a t e s s e n t i a l l y t h e same s e g m e n t - l e n g t h o p t i m i z a t i o n y i e l d s b e s t damping o f l o n g i t u d i n a l o r e x t e n s i o n a l waves i n t h e b a s e p l a t e . The segmented c o n s t r a i n e d v i s c o e l a s t i c l a y e r i s one o f t h e few t r e a t m e n t s c a p a b l e o f p r o v i d i n g u s e f u l d amping o f s u c h waves, w h i c h a r e t r o u b l e s o m e i n c e r t a i n c a s e s .

M u l t i p l e , Continuous Constrained V i s c o e l a s t i c Layer Treatments. M u l t i p l e l a y e r s o f c o n s t r a i n e d - l a y e r t r e a t m e n t a r e an o b v i o u s p o s s i b i l i t y f o r c o n s i d e r a t i o n . Some p r a c t i c a l a d v a n t a g e s m i g h t a c c r u e , s u c h a s e a s i e r h a n d l i n g a n d i n s t a l l a t i o n on c u r v e d s u r f a c e s . I t m i g h t a l s o be p o s s i b l e t o m a n u f a c t u r e and s t o c k f e w e r " s i z e s " o f t r e a t m e n t , g a i n i n g a p p l i c a t i o n s f l e x i b i l i t y t h r o u g h t h e u s e o f m u l t i p l e l a y e r s . F u r t h e r , d e s i g n p r o b l e m s may be e a s e d i n f i n d i n g optimum c o n f i g u r a t i o n s w i t h o u t b e i n g f o r c e d t o use v e r y t h i n l a y e r s o f a good, b u t l o w - m o d u l u s , v i s c o e l a s t i c m a t e r i a l .

S e v e r a l s t u d i e s o f m u l t i p l e l a y e r t r e a t m e n t s have been made (2_£, 22., 22, 2J_. F o r s u c h t r e a t m e n t c o m p r i s i n g i d e n t i c a l l a y e r s , t h e r e does n o t a p p e a r t o be a s t r o n g p e r f o r m a n c e a d v a n t a g e o v e r a s i n g l e c o n s t r a i n e d - l a y e r t r e a t m e n t o f e q u a l w e i g h t (25., 22.) .

An i m p o r t a n t e x c e p t i o n i s t h e use o f m u l t i p l e l a y e r s o f d i f f e r e n t p r o p e r t i e s , e s p e c i a l l y where damping p e r f o r m a n c e must be a c h i e v e d o v e r a b r o a d e r t e m p e r a t u r e r a n g e , o r e q u i v a l e n t l y a t s e v e r a l w i d e l y s e p a r a t e d t e m p e r a t u r e s . I n s u c h an a p p l i c a t i o n , as H e n d e r s o n (27) p o i n t s o u t , an i n n e r c o n s t r a i n e d - l a y e r t r e a t m e n t c o n t a i n s a h i g h - t e m p e r a t u r e v i s c o e l a s t i c damping m a t e r i a l , w h i l e an o u t e r t r e a t m e n t u s e s a l o w e r - t e m p e r a t u r e damping m a t e r i a l . S u c h a c o n f i g u r a t i o n p r o v i d e s good s h e a r c o u p l i n g t o t h e o u t e r t r e a t m e n t i n t h e l o w e r t e m p e r a t u r e r a n g e where t h e i n n e r v i s c o e l a s t i c l a y e r i s s t i f f . A t h i g h e r t e m p e r a t u r e s , t h e i n n e r c o n s t r a i n e d - l a y e r t r e a t m e n t i s , o f c o u r s e , w e l l c o u p l e d t o t h e v i b r a t i n g s t r u c t u r e , a nd i s t h e r e f o r e a b l e t o p r o v i d e e f f e c t i v e damping.

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Multiple-Layer Segmented Treatments. A n o t h e r i n t e r e s t i n g clamping c o n f i g u r a t i o n i n v o l v e s t h e u s e o f m u l t i p l e - l a y e r s e gmented c o n s t r a i n e d l a y e r s . See F i g u r e 10. T h i s c o n f i g u r a t i o n was p a t e n t e d by P a i n t e r (2J1, 2_£) and has been a n a l y z e d by Warnaka, K e r w i n , and C a r b o n e l l (2Q) and b y P l u n k e t t a nd Lee (24.) .

The e l a s t i c segments ( p l a t e l e t s o r " f i s h s c a l e s " ) t a k e on t h e r o l e o f a m a c r o s c o p i c f i l l e r f o r t h e v i s c o e l a s t i c m a t e r i a l , r a i s i n g E 1 and t h e r e f o r e E" ( a t l e a s t a t f i l l e r c o n c e n t r a t i o n s t h a t a r e n o t s o h i g h as t o r e d u c e t h e e f f e c t i v e l o s s f a c t o r e x c e s s i v e l y ) . The r e s u l t i s an o r t h o t r o p i c c o m p o s i t e m a t e r i a l t h a t c a n e x h i b i t t h e p r o p e r t i e s o f a v e r y e f f e c t i v e " f r e e " l a y e r t r e a t m e n t . As one w o u l d e x p e c t , t h e d e s i g n p r o c e s s i n v o l v e s o p t i m i z i n g t h e i n c r e a s e d e x t e n s i o n a l l o s s modulus by b a l a n c i n g t h e e x t e n s i o n a l s t i f f n e s s o f t h e segments and t h e s h e a r s t i f f n e s s o f v i s c o e l a s t i c l a y e r . E s s e n t i a l l y t h e same p h y s i c a l c o n s i d e r a t i o n s a p p l y as i n o p t i m i z i n g t h e segment l e n g t h i n a s i n g l e s e g m e n t e d - c o n s t r a i n e d - l a y e r t r e a t m e n t .

T h i s p l a t e l e t - f i l l e d t r e a t m e n t , a l t h o u g h a t t r a c t i v e i n some r e s p e c t s , does n o t a p p e a r t o have f o u n d w i d e a p p l i c a t i o n , p e r h a p s b e c a u s e o f i t s c o m p l e x i t y .

S p a c e d Damping T r e a t m e n t s f o r B e n d i n g Waves. A homogeneous member t h a t v i b r a t e s i n b e n d i n g e x p e r i e n c e s o s c i l l a t i n g c u r v a t u r e o f i t s n e u t r a l p l a n e (and o f i t s s u r f a c e s ) . The r e s u l t i n g e x t e n s i o n a l s t r a i n s i n c r e a s e l i n e a r l y w i t h d i s t a n c e f r o m t h e n e u t r a l p l a n e . T h e r e i s , t h e r e f o r e , t h e p o s s i b i l i t y o f m a k i n g g e o m e t r i c m a n i p u l a t i o n s o f a damping t r e a t m e n t t o make i t more e f f e c t i v e . One s u c h c o n c e p t known as " s p a c e d damping" i s p a r t i c u l a r l y a p p l i c a b l e t o f r e e a n d c o n s t r a i n e d v i s c o e l a s t i c l a y e r s ( 3 1 - 3 6 r 2_i) , and i s i l l u s t r a t e d i n F i g u r e 11. T h e r e we see t h a t t h e d e f o r m a t i o n o f t h e v i s c o e l a s t i c l a y e r i n e x t e n s i o n o r s h e a r i s i n c r e a s e d t h r o u g h t h e u s e o f a s p a c e r t h a t p l a c e s t h e f r e e l a y e r o r t h e c o n s t r a i n i n g l a y e r f u r t h e r f r o m t h e n e u t r a l p l a n e . The r e s u l t s c a n be h i g h e r l o s s f a c t o r , r e d u c e d t r e a t m e n t w e i g h t , o r t h e n e e d f o r a s m a l l e r amount o f v i s c o e l a s t i c m a t e r i a l .

The p r o p e r t i e s d e s i r e d o f an i d e a l s p a c e r l a y e r a r e t h a t i t be s t i f f i n s h e a r , b u t t h a t t h e s p a c e r i t s e l f c o n t r i b u t e m i n i m a l l y t o t h e b e n d i n g s t i f f n e s s o f t h e b a s e s t r u c t u r e , s h i f t i n g t h e n e u t r a l p l a n e as l i t t l e as p o s s i b l e . We n o t e t h a t f o r t h e s p a c e d c o n s t r a i n e d l a y e r , t h e c o m b i n e d f u n c t i o n o f t h e v i s c o e l a s t i c l a y e r a nd s p a c e r i s t o p r o v i d e a t h i c k , d i s s i p a t i v e a nd a p p r o p r i a t e l y s t i f f ( i n s h e a r ) l a y e r b etween t h e c o n s t r a i n i n g a nd b a s e l a y e r s . T h e r e f o r e t h e o r d e r o f t h e v i s c o e l a s t i c a n d s p a c e r e l e m e n t s i s a r b i t r a r y a nd t h e y may be s u b d i v i d e d a s l o n g as t h e d e s i r e d p r o p e r t i e s a r e p r e s e r v e d . These p o s s i b i l i t i e s g i v e a d d i t i o n a l f r e e d o m i n a d a p t i n g v i s c o e l a s t i c m a t e r i a l s f o r e f f e c t i v e damping.

O t h e r s p a c e r - l i k e g e o m e t r i c m a n i p u l a t i o n s have been c o n s i d e r e d f o r c o n s t r a i n e d v i s c o e l a s t i c l a y e r s . These

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3 r of J 5eg snts 1

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A - Frequencies where cut spacing = A

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1/ 3 4 5 7 I00 3 4 5 7 IK 5K

Frequency (Hz)

F i g u r e 9. M e a s u r e d L o s s F a c t o r o f Segmented C o n s t r a i n e d L a y e r Damping f o r V a r i o u s Segment L e n g t h s on a 3 6 - i n . - L o n g B a r (The c u r v e f o r 3 segments may a c t u a l l y be f o r 2.) ( A d a p t e d from r e f . 22)

F i g u r e 10. M u l t i p l e Segmented C o n s t r a i n e d - L a y e r T r e a t m e n t

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Visco-Elastic. Layer V

Plate•

(a) Simple ' (b) Spaced

Spacer used to increase extensional strain in free layer.

Constraining y Layer ^

^M/isco-Elastic-r Layer

Plate

• Spacer

• Spacer

(a) Simple (b) Spaced

Spacer used to increase shear strain in constrained layer.

F i g u r e 11. S i m p l e and S p a c e d F r e e and C o n s t r a i n e d V i s c o e l a s t i c - L a y e r Damping T r e a t m e n t s

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i n c l u d e W h i t t i e r ' s (22., 2J)) a r c h e d t r u s s , shown i n F i g u r e 12(a) a n d P a i n t e r ' s ( M , 21) h a t - s e c t i o n o f F i g u r e 1 2 ( b ) .

R e s o n a n t Damping T r e a t m e n t s , where damping i s r e q u i r e d o v e r a m o r e - o r - l e s s l i m i t e d r a n g e o f f r e q u e n c y , r e s o n a n t damping t r e a t m e n t s c a n be o f use (2, 2.) . F i g u r e 13 shows e x a m p l e s o f r e s o n a n t o r " t u n e d " dampers t h a t u s e v i s c o e l a s t i c m a t e r i a l s a s t h e lumped e l a s t i c e l e m e n t i n c o m b i n a t i o n w i t h an a p p r o p r i a t e mass. The v i s c o e l a s t i c e l e m e n t c a n have i t s p r i n c i p a l d e f o r m a t i o n e i t h e r i n e x t e n s i o n o r i n s h e a r , d e p e n d i n g on t h e g e o m e t r y c h o s e n .

A r e s o n a n t damper f u n c t i o n s a s f o l l o w s : A t t h e r e s o n a n c e f r e q u e n c y (as d e t e r m i n e d by t h e mass and t h e s t i f f n e s s ) , t h e i n p u t impedance t o t h e b a s e o f t h e damper becomes r e s i s t i v e and r i s e s t o a r e l a t i v e l y h i g h v a l u e , w h i c h i s p r o p o r t i o n a l t o t h e moving mass and i n v e r s e l y p r o p o r t i o n a l t o t h e l o s s f a c t o r o f t h e v i s c o e l a s t i c m a t e r i a l . Power, t h e r e f o r e , f l o w s i n t o t h e damper a n d i s d i s s i p a t e d .

The p e r f o r m a n c e and m a t e r i a l r e q u i r e m e n t s f o r a r e s o n a n t damper a r e d i f f e r e n t f r o m t h o s e f o r , s a y , a f r e e o r c o n s t r a i n e d v i s c o e l a s t i c l a y e r . F o r a g i v e n b a s e v e l o c i t y a m p l i t u d e , t h e i n p u t r e s i s t a n c e , and, t h e r e f o r e t h e power d i s s i p a t e d a t t h e r e s o n a n c e f r e q u e n c y i s i n v e r s e l y p r o p o r t i o n a l t o t h e l o s s f a c t o r o f t h e v i s c o e l a s t i c m a t e r i a l . However, t h e b a n d w i d t h o v e r w h i c h t h e h i g h i n p u t r e s i s t a n c e i s s e e n i s d i r e c t l y p r o p o r t i o n a l t o t h e l o s s f a c t o r . Thus a l o w l o s s f a c t o r g i v e s a h i g h d amping peak t h a t c o v e r s o n l y a n a r r o w f r e q u e n c y r a n g e . C o n v e r s e l y ,a h i g h m a t e r i a l l o s s f a c t o r r e s u l t s i n a l o w e r maximum damping, b u t an i n c r e a s e d e f f e c t i v e b a n d w i d t h . O b v i o u s l y , t h e c h o i c e o f m a t e r i a l p r o p e r t i e s depends on t h e p r o b l e m a t hand: i . e . , on t h e b a n d w i d t h a n d m a g n i t u d e o f damping r e q u i r e d . A f u r t h e r c o n t r o l l i n g f a c t o r i s t h e e x p e c t e d t e m p e r a t u r e r a n g e b e c a u s e t h e t e m p e r a t u r e d ependence o f t h e d y n amic p r o p e r t i e s o f t h e v i s c o e l a s t i c m a t e r i a l w i l l g o v e r n t h e t e m p e r a t u r e v a r i a b i l i t y o f t h e damper p e r f o r m a n c e .

V i s c o e l a s t i c m a t e r i a l s c a n a l s o be u s e d a s w a v e b e a r i n g members i n r e s o n a n t dampers (21, 22.) . I n t h i s c a s e a v i s c o e l a s t i c l a y e r a p p l i e d t o a p l a t e o r beam becomes w a v e b e a r i n g t h r o u g h i t s t h i c k n e s s . Thus, w i t h i n c r e a s i n g f r e q u e n c y , t h e l a y e r c a n p r e s e n t a r e l a t i v e l y h i g h r e s i s t i v e impedance t o t h e p l a t e when t h e l a y e r t h i c k n e s s becomes an odd m u l t i p l e o f t h e q u a r t e r w a v e l e n g t h o f c o m p r e s s i o n a l waves. (The s y s t e m damping d i p s a t even m u l t i p l e s o f X/4.) The damping p e r f o r m a n c e i s d e p e n d e n t on t h e m a t e r i a l t h i c k n e s s , c h a r a c t e r i s t i c i m p edance, a n d l o s s f a c t o r ( e q u i v a l e n t l y on i t s d e n s i t y , wave s p e e d , and l o s s f a c t o r . )

An e xample o f t h e m e a s u r e d damping by s u c h a l a y e r i s shown i n F i g u r e 14 (22.) . R e l a t i v e l y h i g h damping i s shown

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Shear deformation for upward deflection of beam

F i g u r e 12(a) A r c h e d T r u s s Damping (Adapted from r e f . 29)

Tr e a t m e n t

1 -Constraining Layer

- Viscoelastic Layer

- Hat-Section Spacer

F i g u r e 1 2 ( b ) . " H a t - S e c t i o n " Damping T r e a t m e n t (Adapted from ref.29)

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[ 1 L (a) Extension (b) Shear

F i g u r e 13. R e s o n a n t Damper C o n f i g u r a t i o n s

0.2

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F i g u r e 14.

h 2 = 1.0 2.0

m t 1 3.0 4.0 5.0

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o / f i A ft / / - — —

1 / - — —

/

A 1 1

Test 15"x

No. 1 Is ' E Jar

1 J

o

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18"x15"x3/8" Plate Test No. 2 18"x15"x3/8" Plate

EtelT!P!nflMateriaJ.Mass = Q 2 9 Plate Mass

Temperature =74° F

-EtelT!P!nflMateriaJ.Mass = Q 2 9 Plate Mass

Temperature =74° F

-

1 1 II II Ml 1 I I 1000 10,000

Frequency(Hz)

4

50,000

M e a s u r e d Damping P e r f o r m a n c e o f T h i c k n e s s -R e s o n a n t V i s c o e l a s t i c L a y e r (X i s t h e w a v e l e n g t h o f c o m p r e s s i o n a l waves i n t h e v i s c o e l a s t i c l a y e r ) ( A d a p t e d w i t h p e r m i s s i o n f r o m R e f . 38«Copyright 1964 The A m e r i c a n I n s t i t u t e o f P h y s i c s . )

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o v e r a s i g n i f i c a n t f r e q u e n c y r a n g e f o r a t r e a t m e n t - t o - b a s e -member w e i g h t r a t i o o f 0.29.

Damping A p p l i c a t i o n s a n d T r e a t m e n t C h a r a c t e r i s t i c s

I n t h i s s e c t i o n we want t o i n d i c a t e t h e b r o a d r a n g e o f p r o b l e m s i n w h i c h damping c an be u s e f u l , a n d t o p r e s e n t a few s p e c i f i c e x a m p l e s . F u r t h e r , we l i s t some o f t h e p r i n c i p a l p r o p e r t i e s a nd c h a r a c t e r i s t i c s o f v i s c o e l a s t i c m a t e r i a l s t h a t must be c o n s i d e r e d i n s e l e c t i n g a p p r o p r i a t e c a n d i d a t e m a t e r i a l s f o r a p p l i c a t i o n s .

E x a m p l e s o f t h e A p p l i c a t i o n s o f Damping. As was p o i n t e d o u t e a r l i e r , s t r u c t u r a l damping c a n c o n t r o l " f r e e " r e s p o n s e s , i . e . , (a) r e s o n a n t r e s p o n s e a n d s p a t i a l d i s t r i b u t i o n i n s t e a d y - s t a t e r e s p o n s e , a n d (b) b o t h r i n g i n g ( t e m p o r a l d e c a y ) and s p a t i a l d e c a y i n t r a n s i e n t r e s p o n s e . U s e f u l a p p l i c a t i o n s o f damping have c o v e r e d a w i d e r a n g e o f v i b r a t i o n i s s u e s as i n d i c a t e d i n T a b l e I I . The l i s t o f p r o b l e m i s s u e s r a n g e f r o m v i b r a t i o n l e v e l s h i g h enough t o c a u s e d e s t r u c t i o n and f a i l u r e o f s t r u c t u r e a n d e q u i p m e n t , t h r o u g h i s s u e s o f p h y s i c a l harm and i n t e r f e r e n c e w i t h p e r s o n n e l p e r f o r m a n c e , t o a e s t h e t i c c o n s i d e r a t i o n s a n d t h e d i s r u p t i o n o f p r e c i s i o n p r o c e s s e s . The c o r r e s p o n d i n g r a n g e s o f v i b r a t i o n a m p l i t u d e s and f r e q u e n c i e s c o v e r many d e c a d e s .

T a b l e I I . G e n e r a l A r e a s o f A p p l i c a t i o n o f Damping i n C o n t r o l l i n g t h e E f f e c t s o f V i b r a t i o n

• S t r u c t u r a l F a t i g u e and F a i l u r e • Damage t o Equip m e n t • E q u i p m e n t M a l f u n c t i o n • V i b r a t i o n (and N o i s e ) C o n t r o l

f o r P e r s o n n e l S a f e t y a n d F u n c t i o n • C o m f o r t , A c c e p t a n c e , p r o d u c t " F e e l " • E f f e c t s on I n s t r u m e n t s , P r o c e s s e s , and

P r e c i s i o n E q u i p m e n t

I n T a b l e I I I a r e l i s t e d some i m p o r t a n t a r e a s i n w h i c h damping i s a p p l i e d . C l e a r l y , w i d e r a n g e s o f p a r a m e t e r s ( t e m p e r a t u r e , f r e q u e n c y , a m p l i t u d e , e n v i r o n m e n t , e t c . ) a r e e n c o u n t e r e d . These i s s u e s a r e d i s c u s s e d f u r t h e r i n t h e f o l l o w i n g s e c t i o n .

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T a b l e I I I . E x a m p l e s o f S t r u c t u r a l Damping A p p l i c a t i o n s

F i e l d

T r a n s p o r t a t i o n A u t o m o t i v e

A i r c r a f t

S h i p s

R a i l r o a d

S p a c e c r a f t

I n d u s t r y

Commerce

C o n s t r u c t i o n

Items t r e a t e d

body p a n e l s , v a l v e c o v e r , o i l p an

p a n e l s , a p pendages, t u r b i n e - e n g i n e components, f u s e l a g e s k i n p a n e l s

h u l l , d e c k s , m a c h i n e r y p i p i n g

c a r s t r u c t u r e , p a n e l s , w h e e l s

e q u i p m e n t p a n e l s s u p p o r t s , e l e c t r o n i c c i r c u i t b o a r d s , o p t i c a l e q u i p m e n t , a n t e n n a and s u p p o r t s t r u c t u r e s

m a c h i n e r y , e.g., v e h i c l e s , d r i l l i n g r o d s , g r i n d e r s c i r c u l a r - s a w b l a d e s

a p p l i a n c e s , o f f i c e m a c h i n e s , c o m p u t e r components, m e t a l c a b i n e t s , and f u r n i t u r e

b u i l d i n g p a r t i t i o n s , m u l t i l a y e r c o n c r e t e s l a b s , window p a n e s

M o t i v a t i o n

n o i s e c o n t r o l

a c o u s t i c f a t i g u e , n o i s e c o n t r o l

n o i s e c o n t r o l

n o i s e c o n t r o l , w h e e l s q u e a l

a c o u s t i c f a t i g u e , m a l f u n c t i o n , v i b r a t i o n c o n t r o l

f a t i g u e and n o i s e c o n t r o l

n o i s e a n d v i b r a t i o n c o n t r o l

n o i s e and v i b r a t i o n c o n t r o l

R e q u i r e m e n t s on M a t e r i a l s f o r Damping A p p l i c a t i o n .

O b v i o u s l y , a p o l y m e r o r o t h e r m a t e r i a l t o be u t i l i z e d i n a v i b r a t i o n damping a p p l i c a t i o n must p o s s e s s a p p r o p r i a t e p r o p e r t i e s o v e r t h e r a n g e s o f t e m p e r a t u r e , f r e q u e n c y , and s t r a i n a m p l i t u d e i n v o l v e d . I t i s a l s o a p p a r e n t t h a t

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c o n s i d e r a t i o n must be g i v e n c o s t , w e i g h t , d u r a b i l i t y , s t a b i l i t y , e t c .

Some o f what we see as t h e p r i n c i p a l p r o p e r t i e s a n d c h a r a c t e r i s t i c s o f i m p o r t i n t h e d e v e l o p m e n t a n d s e l e c t i o n o f v i s c o e l a s t i c m a t e r i a l s f o r s t r u c t u r a l damping a r e l i s t e d b e l o w . T a b l e I V n o t e s " p a s s i v e " p r o p e r t i e s a nd c h a r a c t e r i s t i c s , t h a t i s , t h o s e t h a t p e r t a i n t o t h e m a t e r i a l i t s e l f a n d t o i t s p e r f o r m a n c e i n t h e s p e c i f i e d o p e r a t i n g c o n d i t i o n s . Somewhat i n c o n t r a s t , T a b l e V l i s t s " A c t i v e o r I n t e r a c t i v e " p r o p e r t i e s a nd c h a r a c t e r i s t i c s t h a t c o n c e r n t h e i n t e r a c t i o n o f t h e v i s c o e l a s t i c m a t e r i a l w i t h o t h e r m a t e r i a l s a n d w i t h t h e e n v i r o n m e n t o f t h e t r e a t m e n t . These t a b u l a t i o n s a r e s u r e l y i n c o m p l e t e , b u t a r e i n t e n d e d t o s u g g e s t t h e b r e a d t h o f c o n s i d e r a t i o n s t h a t may be i n v o l v e d i n m a t e r i a l s e l e c t i o n .

T a b l e I V . V i s c o e l a s t i c M a t e r i a l s f o r Damping. P a s s i v e P r o p e r t i e s and C h a r a c t e r i s t i c s

• A c o u s t i c a l (See T a b l e V I )

• E n v i r o n m e n t - R e l a t e d

- t e m p e r a t u r e - w e a t h e r - m o i s t u r e - r a d i a t i o n - o u t g a s s i n g

• A p p l i c a t i o n - R e l a t e d

- a p p l i c a t i o n p r o c e d u r e s and m a t e r i a l s - c u t t i n g , s h a p i n g

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

T a b l e V. V i s c o e l a s t i c M a t e r i a l s f o r Damping: A c t i v e - I n t e r a c t i v e P r o p e r t i e s a n d C h a r a c t e r i s t i c s

• t o x i c i t y • o d o r • f l a m m a b i l i t y • o u t g a s s i n g • c o m p a t i b i l i t y • e l e c t r i c a l a n d t h e r m a l c o n d u c t i v i t y

- c h e m i c a l - b i o - d e g r a d a t i o n - i m p a c t - f a t i g u e - c h e m i c a l c o n t a m i n a t i o n

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I n t h e p a r t i c u l a r c a s e o f t h e a c o u s t i c a l r e q u i r e m e n t s m e n t i o n e d i n T a b l e IV, we show i n T a b l e V I t h e t e c h n i c a l a c o u s t i c a l r e q u i r e m e n t s s p e c i f i c a l l y f o r f r e e a n d c o n s t r a i n e d v i s c o e l a s t i c l a y e r t r e a t m e n t s . T h i s t a b l e w i t h i t s n o t e s s u m m a r i z e s t h e r e q u i r e m e n t s on m a t e r i a l d y n a m i c p r o p e r t i e s t h a t were p r e s e n t e d i n t h e e a r l i e r d i s c u s s i o n o f t h e s e b r o a d l y u s e f u l t r e a t m e n t t y p e s .

T a b l e V I . T e c h n i c a l A c o u s t i c R e q u i r e m e n t s on t h e Dynamic P r o p e r t i e s o f V i s c o e l a s t i c Damping M a t e r i a l s

F r e e VE L a y e r C o n s t r a i n e d VE L a y e r

P r i n c i p a l E x t e n s i o n S h e a r D e f o r m a t i o n :

R e q u i r e m e n t s f o r L a r g e l o s s modulus • L a r g e l o s s max T| (Note 1) : E" = E fT| (Notes 2, f a c t o r , T|

3) • S p e c i f i e d s h e a r s t i f f n e s s G'/H (Note 2)

1. Maximum l o s s f a c t o r i s n o t a l w a y s t h e d e s i g n g o a l . Combined mass, s t i f f n e s s , and damping c r i t e r i a o f t e n a p p l y .

2. Good p r o p e r t i e s a r e r e q u i r e d o v e r s p e c i f i e d r a n g e s o f f r e q u e n c y and t e m p e r a t u r e .

3. N o t e t h a t a t h i g h e r l o s s f a c t o r s i n c r e a s e d T| i s more e f f e c t i v e t h a n i n c r e a s e d E f ( P e r s o n a l c o m m u n i c a t i o n , M i c h a e l D r a k e , 1989)

Summary

M a t e r i a l s t h a t a r e u s e d i n s t r u c t u r a l damping e n c o u n t e r w i d e r a n g e s o f t e m p e r a t u r e , f r e q u e n c y , a n d s t r a i n . The p e r t i n e n t d e f o r m a t i o n s may be i n e x t e n s i o n - c o m p r e s s i o n , s h e a r , d i l a t a t i o n , o r c o m b i n a t i o n s o f t h e s e . F u r t h e r , t h e m a t e r i a l s c a n be s u b j e c t t o l a r g e s t a t i c o r t r a n s i e n t l o a d s , sometimes o f s i g n i f i c a n t m a g n i t u d e , e i t h e r i n f a b r i c a t i o n o r i n s e r v i c e . T o g e t h e r w i t h t h e f a c t t h a t d amping t r e a t m e n t s c a n e n c o u n t e r demanding e n v i r o n m e n t s , t h e above c o n d i t i o n s make i t c l e a r t h a t t h e s e l e c t i o n o r d e v e l o p m e n t o f m a t e r i a l s f o r damping i s a c o m p l e x t a s k .

The g o a l o f t h i s p a p e r has been t o e s t a b l i s h t h e d e f i n i t i o n a n d m e c h a n i c a l p r i n c i p l e s o f s t r u c t u r a l damping as w e l l as t h e f u n c t i o n a l t y p e s o f a p p l i c a t i o n s i n w h i c h damping c a n be u s e f u l .

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In a d d i t i o n , we have presented s e v e r a l of the important, p r a c t i c a l types of damping treatments and the q u a n t i t a t i v e r o l e of v i s c o e l a s t i c m a t e r i a l s i n such treatments. The reader should keep i n mind s e v e r a l general p r i n c i p l e s of s t r u c t u r a l damping as fo l l o w s :

1. Damping operates on free responses: • Resonances • Free waves

2. Damping does l i t t l e or nothing to reduce the amplitude of f o r c e d responses.

3. In an e f f e c t i v e damping treatment, the v i s c o e l a s t i c element(s) must p a r t i c i p a t e s i g n i f i c a n t l y i n the energy of v i b r a t i o n .

4. Damping may or may not reduce sound r a d i a t i o n .

As an example of p r i n c i p l e number 4 above, consider a l o c a l l y e x c i t e d large panel on which resonant bending waves account f o r most of the v i b r a t o r y response. However, assume that these resonant waves have a wavelength shorter than that of free waves i n the surrounding a i r . In such a case the resonant waves are poorly coupled to the a i r , and r a d i a t e very l i t t l e sound. What r a d i a t i o n there i s can be dominated by non-resonant forced motion around the d r i v e p o i n t (and at other d i s c o n t i n u i t i e s ) . As a r e s u l t , a p p l i e d damping can reduce the resonant response, but not the fo r c e d motion and the r a d i a t i o n of sound.

It i s our hope that what we have presented here w i l l be of help t o those i n the f i e l d of Polymer Science and Engineering who have an i n t e r e s t i n developing m a t e r i a l s f o r a p p l i c a t i o n i n s t r u c t u r a l damping.

Literature Cited 1. Ungar, E. E.; Kerwin, E. M., Jr. J. Acoust. Soc Am.

1962, 34, 954-57. 2. Nashif, A. D.; Jones, D. I. G.; Henderson, J. P.

Vibration Damping; John Wiley & Sons: New York, 1985. 3. Morse, P. M. Vibration and Sound (2nd ed.); McGraw

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4. Strutt, J. W. (Lord Rayleigh). The Theory of Sound (2nd ed.); Dover: New York, 1945; Vol. 1, p 302.

5. Cremer, L.; Heckl, M. A.; Ungar, E. E. Structure-Borne Sound (2nd ed.); Springer Verlag: Berlin, 1988.

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RECEIVED January 24, 1990

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[acs symposium series] sound and vibration damping with polymers volume 424 || requirements imposed...

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