DAMPING BY UNCOUPLED PARTICULATE COMPOSITES

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Submitted on 1 Jan 1987

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DAMPING BY UNCOUPLED PARTICULATE COMPOSITES

B. Rath, M. Imam, N. Louat

To cite this version:

B. Rath, M. Imam, N. Louat. DAMPING BY UNCOUPLED PARTICULATE COMPOSITES.

Journal de Physique Colloques, 1987, 48 (C8), pp.C8-365-C8-369. �10.1051/jphyscol:1987854�. �jpa-

00227158�

DAMPING BY UNCOUPLED PARTICULATE COMPOSITES

B.B. RATH, M.A. IMAM and N.P. LOUAT*

Naval Research Laboratory, Code 6300, Washington, DC 20375-5000, U.S.A.

"crystal Growth and Materials Testing Associates, Lanham, MD 20706, U.S.A.

ABSTRACT

Large damping can be achieved by s u i t a b l e design o f a composite m a t e r i a l w h i c h c o n t a i n s a s i g n i f i c a n t volume f r a c t i o n o f f r e e 1 y v i b r a t i n g p a r t i c l e s i n

i n t e r n a l c a v i t i e s o f i t s m a t r i x .

An approximate a n a l y s i s o f t h e damping b e h a v i o r o f such a m a t e r i a l i s

presented. It i s shown t h a t t h e damping v a r i e s l i n e a r l y w i t h amplitude, i n v e r s e l y w i t h t h e f i r s t power o f t h e d i s t a n c e o f m o t i o n and w i t h t h e second power o f (1-k), where k i s t h e c o e f f i c i e n t o f r e s t i t u t i o n .

ANALYSIS:

It i s assumed t h a t t h e number o f p a r t i c l e s i n v o l v e d i s so l a r g e t h a t s t a t i s t i c a l c o n s i d e r a t i o n s apply; t h a t t h e mass(m) o f a p a r t i c l e i s much s m a l l e r t h a n t h a t o f t h e m a t r i x ( M ) t o be a s s o c i a t e d w i t h i t ; t h a t p a r t i c l e - v o i d

combinations d i f f e r one from another i n s i z e and shape; t h a t t h e m a t r i x i s s u b j e c t e d t o a s i n u s o i d a l l y v a r y i n g displacement, a s i n w t , where w i s t h e angular v e l o c i t y , a i s t h e amplitude and t i s t h e t i m e and t h a t t h e r e s u l t i n g

a c c e l e r a t i o n s , -a w2 s i n w t , a r e s u f f i c i e n t l y l a r g e f o r t h e p a r t i c l e s t o f l y , bouncing from s u r f a c e t o surface, r a t h e r t h a n t o r o l l .

On t h e b a s i s t h a t M>>m and t h a t c o l l i s i o n s o f t h e p a r t i c l e s w i t h t h e v o i d s u r f a c e are e l a s t i c , i t i s e a s i l y shown from a r e c t i l i n e a r a n a l y s i s t h a t t h e change i n speed s u f f e r e d by t h e p a r t i c l e i s t h a t o f t h e m a t r i x ; i n c r e a s i n g i f t h e i n d i v i d u a l motions are opposed, decreasing i f t h e y are i n t h e same d i r e c t i o n . It i s t h e n suggested t h a t t h e times o f i n d i v i d u a l f l i g h t s a r e n o t c o r r e l a t e d w i t h t h e p e r i o d , l / w , and t h a t a p a r t i c l e encounters a moving s u r f a c e and s u f f e r s w i t h equal p r o b a b i l i t y , an i n c r e a s e o r a decrease i n speed.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987854

JOURNAL DE PHYSIQUE

I f t h e change i n speeds o f a p a r t i c l e have a c h a r a c t e r i s t i c mean value, i t l e a d s t o a c o n s i d e r a t i on o f t h e behavior o f a random w a l k e r i n speed space and t h u s t h a t o f a d i s t r i b u t i o n f ( v ) o f p a r t i c l e speeds. However t h i s s i t u a t i o n w i l l n o t occur because t h e system i s n o t p e r f e c t l y e l a s t i c and hence a l o s s o f energy, i .e. damping, i s i n v o l v e d a t each c o l l i s i o n .

L e t t h e change i n p a r t i c l e speed i n t h e absence o f damping have t h e mean v a l u e

and i n t h e presence o f damping when t h e speed o f a p a r t i c l e l i e s i n an i n t e r v a l 6v a t v,

where X = l - k and k i s t h e c o e f f i c i e n t o f r e s t i t u t i o n .

Again, t h e i n t e r v a l between impacts o f a p a r t i c l e w i t h t h e w a l l s o f t h e c a v i t y i s

where L and d r e p r e s e n t r e s p e c t i v e l y t h e l i n e a r dimensions o f t h e c a v i t y and p a r t i c l e .

C l e a r l y a f i r s t s t e p i n t h e c a l c u l a t i o n o f t h e energy l o s s e s s u f f e r e d by t h e assembly o f p a r t i c l e s i s t o determine o f t h e d i s t r i b u t i o n o f p a r t i c l e speeds, f ( v ) . A l l o w i n g t h a t t h e changes i n speed g i v e n i n ( 1 ) and ( 2 ) a r e o f equal p r o b a b i l i t y , t h e n e t f l u x o f p a r t i c l e s i n t o a range o f speeds v t o v&6v,

where 6 v / v < < l i s :

-

where C i s a constant.

Expanding f ( v ) i n a T a y l o r s e r i e s and assuming t h a t A / V and (1-k) a r e small 6 can be w r i t t e n as

F o r steady s t a t e

(for ~ i < l and i l , s i n c e IV i s always s m a l l e r t h a n A) A

Where A i s a constant.

I n t e g r a t i n g e q u a t i o n (3)

Where B i s a constant.

A p p l y i n g t h e boundary c o n d i t i o n s , f=O a t v=O and t h a t t h e r e a r e j u s t N p a r t i c l e s , f ( v ) i s g i v e n by

I n an i n t e r v a l o f speed 6v a t v a number, n = f ( v ) d v ,

o f p a r t i c l e s s u f f e r , i n u n i t time, a number o f c o l l i s i o n s :

I n t h i s process t h e r e i s a l o s s i n energy o f amount:

These events, corresponding r e s p e c t i v e l y t o c o l l i s i o n s g i v i n g increases and decreases o f energy, have been taken t o be o f equal p r o b a b i l i t y so t h a t t h e mean r a t e o f l o s s o f energy i s :

JOURNAL DE PHYSIQUE

The l o s s from t h e whole d i s t r i b u t i o n , obtained from equations ( 4 ) and (5) i s :

It may be shown by d i v i d i n g and m u l t i p l y i n g t h e integrand o f ( 6 ) by v t h a t ,

so t h a t ,

should represent a good approximation.

Loss per c y c l e can be expressed as:

and as a f r a c t i o n o f t h e t o t a l v i b r a t i o n a l energy: T = N M ~ ~ ~ ~ / Z ;

I n t h e range o f i n t e r e s t where

A-L,

S u b s t i t u t i n g t h e values:

1 - 1 A = T I T I + = T U

A c o n s i d e r a t i o n o f these expressions show t h a t damping i s : (1) 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 amp1 i t u d e o f v i b r a t i o n , ( 2 ) h i g h e s t f o r a decreasing m i s f i t between t h e c a v i t y and t h e p a r t i c l e , except i n t h e l i m i t i n g case where they are equal ; (3) i s frequency independent above a c r i t i c a l value such t h a t t h e p a r t i c l e s bounce w i t h i n t h e c a v i t y r a t h e r than r o l l which they w i l l do i f g r a v i t a t i o n f o r c e d o c t r i n a t e and (4) i s s e n s i t i v e t o t h e c o e f f i c i e n t o f r e s t i t u t i o n because o f t h e i n v e r s e q u a d r a t i c dependence on (1-k).