Requirements Imposed on Polymeric Materials by Structural Damping

Chapter 17

Requirements Imposed on Polymeric Materials by Structural Damping Applications

Downloaded by MICHIGAN STATE UNIV on November 30, 2013 | http://pubs.acs.org Publication Date: May 1, 1990 | doi: 10.1021/bk-1990-0424.ch017

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 i s seen by the designer of damping treatments. Our objective i s to provide an overview of the ways i n which v i s o e l a s t i c damping materials are used, and of the resulting material-property requirements, and the corresponding polymer deformation types (extension, shear, or wavebearing). F i n a l l y , 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 i s to present an introduction to structural damping for those polymer chemists who are not already familiar with the f i e l d . We shall treat several subjects, including what structural damping can accomplish, the d e f i n i t i o n of damping i n terms of energy and loss factor, and the important classes of damping mechanisms. F i n a l l y , we look b r i e f l y at the range of problems i n 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 v i s c o e l a s t i c 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 f i e l d s of vibration isolation, shock mitigation, and underwater sound. 0097-6156/90/0424-0317$08.25/0 © 1990 American Chemical Society

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

Nevertheless, we h o p e t h a t t h i s presentation w i l l prove u s e f u l .


The

special-purpose

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 removal of energy from a v i b r a t i n g system. Damping, t h e r e f o r e , serves t o c o n t r o l " f r e e " responses, i . e . , those responses f o r which the e l a s t i c and i n e r t i a l forces balance, l e a v i n g the system response c o n t r o l l e d by the l o s s e s i nt h e system. In t h e absence o f l o s s e s , t h e responses would be f o r c e f r e e , a n d once e x c i t e d , would c o n t i n u e a r b i t r a r i l y f a r i n time o r space. 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 these t h a t a r e c o n t r o l l e d by system 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 steady-state, resonant response, • 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. Broader p o t e n t i a l r e s u l t s include t h ereduction 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, a n d o f s t r u c t u r a l fatigue. Damping, Energy,

and Loss

Factor

I f , i n a f r e e l y v i b r a t i n g system that contains a l o s s mechanism, t h e system energy i s seen t o decay 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 e~ . (This i s t h ecase f o r a l i n e a r system i nwhich 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 sindependent o f v i b r a t i o n amplitude.) T h e d e c a y c o e f f i c i e n t i s a = COTJ , w h e r e 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 n d T| i s d e f i n e d as t h e system l o s s f a c t o r . S i n c e t h eenergy decay 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. T h u s , we h a v e 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 in a v e r y b a s i c way i n e n e r g y a n d power t e r m s a s f o l l o w s : at

0

*n = n /cow dis

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 n u m b e r o f e l e m e n t s , e a c h o f w h i c h may h a v e i t s own l o s s f a c t o r T j i a n d i t s own c o n t r i b u t i o n W i t o t h e system v i b r a t o r y energy. These elements might be, f o r e x a 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 p o w e r 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 of the several energies. We, t h e r e f o r e , h a v e t h e f a m i l i a r and u s e f u l d e f i n i t i o n o f system l o s s f a c t o r as f o l l o w s ( 1 ) :


17. KERWIN AND UNGAR

Structural Damping Applications

319

Tl=


This i s an important 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 system element t o be e f f e c t i v e i ndamping t h e s y s t e m , t h a t element 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 energy o f t h e system. I t i s customary i nd e a l i n g w i t h l i n e a r (or l i n e a r i z e d ) b e h a v i o r o f dynamical systems 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 a s c o m p l e x q u a n t i t i e s (2). F o r e x a m p l e , we w r i t e Y o u n g ' s m o d u l u s E = E

1

- iE" = E

1

(1-iTj)

where_ E i s t h e complex modulus E i s t h e " s t o r a g e " modulus E" i s t h e " l o s s " m o d u l u s and f

T| = E " / E '

i s t h el o s s

factor

of the material

T h e l o s s f a c t o r T| i s s o m e t i m e s d e n o t e d a s t a n 8, w h e r e 8 i s t h e phase a n g l e between t h e s t o r a g e a n d l o s s components o f harmonic s t r e s s . Correspondingly,

we h a v e

f o r the shear

modulus

G = G' - i G " = G' ( l - i T | ) Other e l a s t i c moduli o r compliances, e.g., b u l k , d i l a t a t i o n , e t c . , may b e d e s c r i b e d a s a b o v e . Note 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 dependence e" . (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 arbitrary.) i c o t

The F u n c t i o n o f D a m p i n g . 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 " responses o f a system. Such r e s p o n s e s are those i n which t h e system k i n e t i c and p o t e n t i a l energies are i n balance, so that d i s s i p a t i v e influences a r e controlling. 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 w a v e s i n s p a c e . I n the f a m i l i a r case o f a resonance i n frequency, t h e response t o a given o s c i l l a t i n g force i n t h e v i c i n i t y o f resonance 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 resonance frequency and a f i n i t e bandwidth o f high response. Both features a r e c o n t r o l l e d by t h e system loss factor: S e e F i g u r e 1. The p e a k 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 a s


320


and d e s c r i b e s t h epeak response a t resonance r e l a t i v e t o the response t h a t would e x i s t a t t h e resonance frequency 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 controlling. ( 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 spring.) T h e 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 = T], f o r T| not t o ol a r g e , and represents t h e f r a c t i o n a l frequency d i f f e r e n c e b e t w e e n 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 drops t o h a l f i t s peak v a l u e . The a b o v e 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 time. 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 resonance response i s that i nt h ev i c i n i t y o f resonance t h e r e i sa d i p i nt 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 response. T h e f o r c e - r e d u c t i o n r a t i o i s Q, a n d 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|. In c o n t r a s t , a t r u l y f o r c e - f r e e response o f a resonant system (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 decay o f v i b r a t i o n amplitude with time. A s we h a v e m e n t i o n e d e a r l i e r , decay i s a l s o c o n t r o l l e d b y t h e system l o s s f a c t o r as f o l l o w s :


n

Temporal

decay

rate:

A = 27.3T|f t

n

(dB/sec) ,

where a v i b r a t i o n l e v e l L would 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) m e a s u r e f o r a r e s p o n s e q u a n t i t y v, such as v e l o c i t y , v

L

v

= 10 l o g l—Z-Y

(dBre V

r e f

).

V

\ ref /

T h u s a l e v e l c h a n g e 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 energy o r i n t e n s i t y - l i k e q u a n t i t y such as v . The c o n c e p t o f t h e i n f l u e n c e o f d a m p i n g o n v i b r a t i o n decay 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 b e 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 sound. Note 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 edecay time decreases, and t h e frequency bandwidth increases. 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 defined. This i si nkeeping with the "uncertainty p r i n c i p l e " r e l a t i n g time and frequency Q ) . 2


17. KERWIN AND UNGAR Table


Loss

I.

321

Structural Damping Applications

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 on L o s s F a c t o r

Factor

< 0.001

N: cycles for d e c a y t o 0.1 amplitude > 730

0.01

73

Decay

Sound—Pitch "Clang"—approaches tone "Bong"—clear

pure

pitch

0.1

7.3

"Bunk"—discernable

0.5

1.5

"Thud"—almost

pitch

without

pitch

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 resonant system, damping governs 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 s e e 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 the right without further 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 wavelength X given as X = c/f. A s we h a v e n o t e d a b o v e , 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 the temporal decay o f the system energy. Therefore,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 "energy" speed (or "group v e l o c i t y " ) c i n v o l v e d i n t h e result: g

Spatial

decay

rate:

A

x

= 27.3

-°- T] c

(dB/wavelength)

g

[The

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

g

A\ = 2 7 . 3 T| (6B/X)

(non-dispersive

waves)

T h e r e a r e wave t y p e s , h o w e v e r , i n w h i c h t h e wave s p e e d h a s 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 b e a m s o r p l a t e s (.5.) w h e r e (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 energy speed i s t w i c e the phase speed: c = 2c. F o r such b e n d i n g w a v e s , we h a v e g

A^=

1 3 . 6 T| (dB/X)

( b e n d i n g waves)


322


10

Af/f = n


o

n

J

Q. CO

Requirements Imposed on Polymeric Materials by Structural Damping

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